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Research ArticleΥ(119899119878) rarr 119861
lowast
119888120587 119861
lowast
119888119870 Decays with Perturbative QCD Approach
Junfeng Sun1 Yueling Yang1 Qin Chang1 Gongru Lu1 and Jinshu Huang2
1 Institute of Particle and Nuclear Physics Henan Normal University Xinxiang 453007 China2College of Physics and Electronic Engineering Nanyang Normal University Nanyang 473061 China
Correspondence should be addressed to Yueling Yang yangyuelinghtueducn
Received 23 March 2016 Accepted 17 May 2016
Academic Editor Chao-Qiang Geng
Copyright copy 2016 Junfeng Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
Besides the traditional strong and electromagnetic decay modes Υ(119899119878)meson can also decay through the weak interactions withinthe standard model of elementary particle With anticipation of copious Υ(119899119878) data samples at the running LHC and comingSuperKEKB experiments the two-body nonleptonic bottom-changing Υ(119899119878) rarr 119861
lowast
119888120587 119861
lowast
119888119870 decays (119899 = 1 2 3) are investigated
with perturbative QCD approach firstly The absolute branching ratios for Υ(119899119878) rarr 119861lowast
119888120587 and 119861lowast
119888119870 decays are estimated to reach up
to about 10minus10 and 10minus11 respectively which might possibly be measured by the future experiments
1 Introduction
The upsilon Υ(119899119878) meson is the spin-triplet 119878-wave state ofbottomonium (bound state consisting of bottom quark 119887 andantibottom quark 119887) with well-established quantum numberof 119868119866119869119875119862 = 0minus1minusminus [1] The characteristic narrow decay widthsof Υ(119899119878) mesons for 119899 = 1 2 and 3 provide insight intothe study of strong interactions (see Table 1 and note thatfor simplicity Υ(119899119878) will denote Υ(1119878) Υ(2119878) and Υ(3119878)mesons in the following content if not specified definitely)The mass of Υ(119899119878) meson is below 119861 meson pair thresholdΥ(119899119878) meson decays into bottomed hadrons through strongand electromagnetic interactions are forbidden by the law ofconservation of flavor number The bottom-changing Υ(119899119878)decays can occur only via the weak interactions within thestandard model although with tiny incidence probabilityBoth constituent quarks of upsilons can decay individuallywhich provide an alternative system for investigating theweak decay of heavy-flavored hadrons In this paper wewill study the nonleptonic Υ(119899119878) rarr 119861
lowast
119888119875 (119875 = 120587 and
119870) weak decays with perturbative QCD (pQCD) approach[2ndash4]
Experimentally (1) over 108 Υ(119899119878) data samples havebeen accumulated at Belle and BaBar experiments [5] Moreand more upsilon data samples will be collected at the
running hadron collider LHC and the forthcoming 119890+119890minus col-lider SuperKEKB (the SuperKEKB has started commission-ing test run (httpwwwkekjpenNewsRoomRelease))There seems to exist a realistic possibility to explore Υ(119899119878)weak decay at future experiments (2) Signals of Υ(119899119878) rarr119861lowast
119888120587 119861
lowast
119888119870 decays should be easily distinguished with ldquocharge
tagrdquo technique due to the facts that the back-to-back finalstates with different electric charges have definitemomentumand energy in the rest frame of Υ(119899119878) meson (3) 119861lowast
119888meson
has not been observed experimentally by now 119861lowast119888meson
production via the strong interaction is suppressed due tothe simultaneous presence of two heavy quarks with differentflavors and higher order in QCD coupling constant 120572
119904
Υ(119899119878) rarr 119861lowast
119888120587 119861lowast
119888119870 decays provide a novel pattern to
study 119861lowast119888meson production The identification of a single
explicitly flavored 119861lowast119888meson could be used as an effective
selection criterion to detect upsilon weak decays Moreoverthe radiative decay of 119861lowast
119888meson provides a useful extra
signal and a powerful constraint (the investigation on theradiative decay of 119861lowast
119888meson can be found in eg [6] with
QCD sum rules) Of course any discernible evidences of ananomalous production rate of single bottomed meson fromupsilon decays might be a hint of new physics
Theoretically many attractive QCD-inspired methodshave been developed recently to describe the exclusive
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 4893649 9 pageshttpdxdoiorg10115520164893649
2 Advances in High Energy Physics
Table 1 Summary of mass decay width on(off)-peak luminosity and numbers of Υ(119899119878)
Meson Properties [1] Luminosity (fbminus1) [5] Numbers (106) [5]Mass (MeV) Width (keV) Belle BaBar Belle BaBar
Υ(1119878) 946030 plusmn 026 5402 plusmn 125 57 (18) sdot sdot sdot 102 plusmn 2 sdot sdot sdot
Υ(2119878) 1002326 plusmn 031 3198 plusmn 263 249 (17) 136 (14) 158 plusmn 4 983 plusmn 09
Υ(3119878) 103552 plusmn 05 2032 plusmn 185 29 (02) 280 (26) 11 plusmn 03 1213 plusmn 12
nonleptonic decay of heavy-flavored mesons such as thepQCD approach [2ndash4] the QCD factorization approach[7ndash9] and soft and collinear effective theory [10ndash13] andhave been applied widely to vindicate measurements on 119861meson decaysThe upsilon weak decay permits one to furtherconstrain parameters obtained from 119861 meson decay andcross comparisons provide an opportunity to test variousphenomenologicalmodelsThe upsilonweak decay possessesa unique structure due to the Cabibbo-Kobayashi-Maskawa(CKM) matrix properties which predicts that the channelswith one 119861(lowast)
119888meson are dominant Υ(119899119878) rarr 119861
lowast
119888119875 decay
belongs to the favorable 119887 rarr 119888 transition which should inprinciple have relatively large branching ratio among upsilonweak decays However there is still no theoretical studydevoted to Υ(119899119878) rarr 119861
lowast
119888119875 decay for the moment In this
paper we will present a phenomenological investigation onΥ(119899119878) rarr 119861
lowast
119888119875weak decay with the pQCD approach to supply
a ready reference for the future experimentsThis paper is organized as follows Section 2 focuses on
theoretical framework and decay amplitudes for Υ(119899119878) rarr119861lowast
119888120587 119861
lowast
119888119870 weak decays Section 3 is devoted to numerical
results and discussion The last section is a summary
2 Theoretical Framework
21 The Effective Hamiltonian Theoretically Υ(119899119878) rarr 119861lowast
119888120587
119861lowast
119888119870 weak decays are described by an effective bottom-
changing Hamiltonian based on operator product expansion[19]
Heff =119866119865
radic2
sum
119902=119889119904
119881119888119887119881
lowast
119906119902119862
1(120583)119874
1(120583) + 119862
2(120583)119874
2(120583)
+ hc
(1)
where 119866119865≃ 1166 times 10
minus5 GeVminus2 [1] is the Fermi couplingconstant the CKM factors 119881
119888119887119881
lowast
119906119889and 119881
119888119887119881
lowast
119906119904correspond
to Υ(119899119878) rarr 119861lowast
119888120587 and 119861lowast
119888119870 decays respectively with the
Wolfenstein parameterization theCKM factors are expandedas a power series in a smallWolfenstein parameter 120582 sim 02 [1]
119881119888119887119881
lowast
119906119889= 119860120582
2
minus
1
2
1198601205824
minus
1
8
1198601205826
+ O (1205827
)
119881119888119887119881
lowast
119906119904= 119860120582
3
+ O (1205827
)
(2)
The local tree operators 11987612
are defined as
1198741= [119888
120572120574120583(1 minus 120574
5) 119887
120572] [119902
120573120574120583
(1 minus 1205745) 119906
120573]
1198742= [119888
120572120574120583(1 minus 120574
5) 119887
120573] [119902
120573120574120583
(1 minus 1205745) 119906
120572]
(3)
where 120572 and 120573 are color indices and the sum over repeatedindices is understood
The scale 120583 factorizes physics contributions into short-and long-distance dynamics The Wilson coefficients 119862
119894(120583)
summarize the physics contributions at scale higher than 120583and are calculable with the renormalization group improvedperturbation theory The hadronic matrix elements (HME)where the local operators are inserted between initial andfinal hadron states embrace the physics contributions belowscale of 120583 To obtain decay amplitudes the remaining workis to calculate HME properly by separating from perturbativeand nonperturbative contributions
22 Hadronic Matrix Elements Based on Lepage-Brodskyapproach for exclusive processes [20] HME is commonlyexpressed as a convolution integral of hard scattering subam-plitudes containing perturbative contributions with univer-sal wave functions reflecting nonperturbative contributionsIn order to effectively regulate endpoint singularities andprovide a naturally dynamical cutoff on nonperturbativecontributions transverse momentum of valence quarks isretained and the Sudakov factor is introduced within thepQCD framework [2ndash4] Phenomenologically pQCDrsquos decayamplitude could be divided into three parts theWilson coef-ficients119862
119894incorporating the hard contributions above typical
scale of 119905 process-dependent rescattering subamplitudes 119879accounting for the heavy quark decay and wave functions Φof all participating hadrons which is expressed as
int119889119896119862119894(119905) 119879 (119905 119896)Φ (119896) 119890
minus119878
(4)
where 119896 is the momentum of valence quarks and 119890minus119878 is theSudakov factor
23 Kinematic Variables The light cone kinematic variablesin Υ(119899119878) rest frame are defined as follows
119901Υ= 119901
1=
1198981
radic2
(1 1 0)
119901119861lowast
119888
= 1199012= (119901
+
2 119901
minus
2 0)
1199013= (119901
minus
3 119901
+
3 0)
119901plusmn
119894=
(119864119894plusmn 119901)
radic2
119896119894= 119909
119894119901119894+ (0 0
119896119894perp)
120598
1=
1199011
1198981
minus
1198981
1199011sdot 119899
+
119899+
Advances in High Energy Physics 3
120598
2=
1199012
1198982
minus
1198982
1199012sdot 119899
minus
119899minus
120598perp
12= (0 0 1)
119899+= (1 0 0)
119899minus= (0 1 0)
119904 = 21199012sdot 119901
3
119905 = 21199011sdot 119901
2= 2119898
11198642
119906 = 21199011sdot 119901
3= 2119898
11198643
119901 =
radic[1198982
1minus (119898
2+ 119898
3)
2
] [1198982
1minus (119898
2minus 119898
3)
2
]
21198981
(5)
where119909119894and
119896119894perpare the longitudinalmomentum fraction and
transverse momentum of valence quarks respectively 120598119894and
120598perp
119894are the longitudinal and transverse polarization vectors
respectively and satisfy relations 1205982119894= minus1 and 120598
119894sdot 119901
119894= 0
the subscript 119894 on variables 119901119894 119864
119894 119898
119894 and 120598
119894corresponds
to participating hadrons namely 119894 = 1 for Υ(119899119878) meson119894 = 2 for the recoiled 119861lowast
119888meson and 119894 = 3 for the emitted
pseudoscalar meson 119899+and 119899
minusare positive and negative null
vectors respectively 119904 119905 and 119906 are the Lorentz-invariantvariables 119901 is the common momentum of final states Thenotation of momentum is displayed in Figure 2(a)
24 Wave Functions With the notation in [16 21] wavefunctions are defined as
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598
1[119898
1Φ
VΥ(119896
1) minus 1199011Φ
119905
Υ(119896
1)]
119895119894
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
perp
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598perp
1[119898
1Φ
119881
Υ(119896
1) minus 1199011Φ
119879
Υ(119896
1)]
119895119894
⟨119861lowast
119888(119901
2 120598
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989631198901198941198962sdot119911
120598
2[119898
2Φ
V119861lowast
119888
(1198962) + 1199012Φ
119905
119861lowast
119888
(1198962)]
119895119894
⟨119861lowast
119888(119901
2 120598
perp
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989621198901198941198962sdot119911
120598perp
2[119898
2Φ
119881
119861lowast
119888
(1198962) + 1199012Φ
119879
119861lowast
119888
(1198962)]
119895119894
⟨119875 (1199013)
10038161003816100381610038161003816119906119894(0) 119902
119895(119911)
100381610038161003816100381610038160⟩ =
119894119891119875
4
sdot int 11988911989631198901198941198963sdot119911
1205745[1199013Φ
119886
119875(119896
3) + 120583
119875Φ
119901
119875(119896
3)
+ 120583119875(119899minus119899+ minus 1)Φ
119905
119875(119896
3)]
119895119894
(6)
where 119891Υ 119891
119861lowast
119888
and 119891119875are decay constants of Υ(119899119878) 119861lowast
119888 and
119875mesons respectivelyConsidering mass relations of 119898
Υ(119899119878)≃ 2119898
119887and 119898
119861lowast
119888
≃
119898119887+ 119898
119888 it might assume that the motion of heavy valence
quarks in Υ(119899119878) and 119861lowast
119888mesons is nearly nonrelativis-
tic The wave functions of Υ(119899119878) and 119861lowast
119888mesons could
be approximately described with nonrelativistic quantumchromodynamics (NRQCD) [22ndash24] and time-independentSchrodinger equation For an isotropic harmonic oscillatorpotential the eigenfunctions of stationary statewith quantumnumbers 119899119871 are written as [17]
1206011119878(119896) sim 119890
minus119896
2
21205732
1206012119878(119896) sim 119890
minus119896
2
21205732
(2119896
2
minus 31205732
)
1206013119878(119896) sim 119890
minus119896
2
21205732
(4119896
4
minus 20119896
2
1205732
+ 151205734
)
(7)
where parameter 120573 determines the average transversemomentum that is ⟨119899119878|1198962
perp|119899119878⟩ sim 120573
2 Employing the substi-tution ansatz [25]
119896
2
997888rarr
1
4
sum
119894
119896
2
119894perp+ 119898
2
119902119894
119909119894
(8)
where 119909119894and 119898
119902119894
are the longitudinal momentum fractionand mass of valence quark respectively then integrating out119896perpand combining with their asymptotic forms the distribu-
tion amplitudes (DAs) for Υ(119899119878) and 119861lowast119888mesons can be
written as [17]
120601V119879Υ(1119878)
(119909) = 119860119909119909 expminus119898
2
119887
81205732
1119909119909
120601119905
Υ(1119878)(119909) = 119861119905
2 expminus119898
2
119887
81205732
1119909119909
120601119881
Υ(1119878)(119909) = 119862 (1 + 119905
2
) expminus119898
2
119887
81205732
1119909119909
120601V119905119881119879Υ(2119878)
(119909) = 119863120601V119905119881119879Υ(1119878)
(119909) 1 +
1198982
119887
21205732
1119909119909
120601V119905119881119879Υ(3119878)
(119909) = 119864120601V119905119881119879Υ(1119878)
(119909)(1 minus
1198982
119887
21205732
1119909119909
)
2
+ 6
120601V119879119861lowast
119888
(119909) = 119865119909119909 expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
4 Advances in High Energy Physics
120601119905
119861lowast
119888
(119909) = 1198661199052 expminus
1199091198982
119888+ 119909119898
2
119887
81205732
2119909119909
120601119881
119861lowast
119888
(119909) = 119867(1 minus 1199052
) expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
(9)
where 119909 = 1 minus 119909 119905 = 119909 minus 119909 According to NRQCD powercounting rules [22] 120573
119894≃ 120585
119894120572119904(120585
119894) with 120585
119894= 119898
1198942 and QCD
coupling constant 120572119904 The exponential function represents 119896
perp
distribution Parameters of 119860 119861 119862 119863 119864 119865 119866 and 119867 arenormalization coefficients satisfying the conditions
int
1
0
119889119909120601119894
Υ(119899119878)(119909) = int
1
0
119889119909120601119894
119861lowast
119888
(119909) = 1
for 119894 = V 119905 119881 119879(10)
The shape lines of normalized DAs for Υ(119899119878) and 119861lowast119888
mesons are shown in Figure 1 It is clearly seen that (1) DAsforΥ(119899119878) and119861lowast
119888mesons fall quickly down to zero at endpoint
119909 119909 rarr 0 due to suppression from exponential functions (2)DAs for Υ(119899119878) meson are symmetric under the interchangeof momentum fractions 119909 harr 119909 and DAs for 119861lowast
119888meson are
basically consistent with the feature that valence quarks sharemomentum fractions according to their masses
Our study shows that only the leading twist (twist-2) DAsof the emitted light pseudoscalar meson 119875 are involved indecay amplitudes (see Appendix) The twist-2 DAs have theexpansion [16]
120601119886
119875(119909) = 6119909119909sum
119894=0
11988611989411986232
119894(119905) (11)
and are normalized as
int
1
0
120601119886
119875(119909) 119889119909 = 1 (12)
where 11986232
119894(119905) are Gegenbauer polynomials
11986232
0(119905) = 1
11986232
1(119905) = 3119905
11986232
2(119905) =
3
2
(51199052
minus 1)
(13)
and each term corresponds to a nonperturbative Gegenbauermoment 119886
119894 note that 119886
0= 1 due to the normalization
condition equation (12) 119866-parity invariance of the pion DAsrequires Gegenbauer moment 119886
119894= 0 for 119894 = 1 3 5
25 Decay Amplitudes The Feynman diagrams for Υ(119899119878) rarr119861lowast
119888120587 weak decay are shown in Figure 2 There are two types
One is factorizable emission topology where gluon attachesto quarks in the samemeson and the other is nonfactorizable
emission topology where gluon connects to quarks betweendifferent mesons
With the pQCD master formula equation (4) the ampli-tude for Υ(119899119878) rarr 119861
lowast
119888119875 decay can be expressed as [26]
A (Υ (119899119878) 997888rarr 119861lowast
119888119875) = A
119871(120598
1 120598
2) +A
119873(120598
perp
1 120598
perp
2)
+ 119894A119879120598120583V120572120573120598
120583
1120598V2119901120572
1119901
120573
2
(14)
which is conventionally written as the helicity amplitudes[26]
A0= minus119862Asum
119895
A119895
119871(120598
1 120598
2)
A= radic2119862Asum
119895
A119895
119873(120598
perp
1 120598
perp
2)
Aperp= radic2119862A1198981
119901sum
119895
A119895
119879
119862A = 119894119881119888119887119881lowast
119906119902
119866119865
radic2
119862119865
119873119888
120587119891Υ119891119861lowast
119888
119891119875
(15)
where 119862119865= 43 and the color number119873
119888= 3 the subscript 119894
on 119860119895
119894corresponds to three different helicity amplitudes that
is 119894 = 119871119873 119879 the superscript 119895 on 119860119895
119894denotes indices of
Figure 2 The explicit expressions of building blocks A119895
119894are
collected in Appendix
3 Numerical Results and Discussion
In the center-of-mass ofΥ(119899119878)meson branching ratioB119903 forΥ(119899119878) rarr 119861
lowast
119888119875 decay is defined as
B119903 =1
12120587
119901
1198982
ΥΓΥ
1003816100381610038161003816A
0
1003816100381610038161003816
2
+1003816100381610038161003816A
1003816100381610038161003816
2
+1003816100381610038161003816A
perp
1003816100381610038161003816
2
(16)
The input parameters are listed in Tables 1 and 2 If notspecified explicitly we will take their central values as thedefault inputs Our numerical results are collected in Table 3where the first uncertainty comes from scale (1 plusmn 01)119905
119894and
the expression of 119905119894is given in (A4) and (A5) the second
uncertainty is from mass of119898119887and119898
119888 the third uncertainty
is from hadronic parameters including decay constants andGegenbauer moments the fourth uncertainty is from CKMparameters The followings are some comments
(1) Branching ratio for Υ(119899119878) rarr 119861lowast
119888120587 decay is about
O(10minus10) with pQCD approach which is well withinthe measurement potential of LHC and SuperKEKBFor example experimental studies have showed thatproduction cross sections forΥ(119899119878)meson in p-p andp-Pb collisions are a few 120583119887 at the LHCb [27 28] andALICE [29 30] detectors Consequently there willbe more than 1012 Υ(119899119878) data samples per 119886119887minus1 datacollected by the LHCb and ALICE corresponding toa few hundreds of Υ(119899119878) rarr 119861
lowast
119888120587 events Branching
ratio for Υ(119899119878) rarr 119861lowast
119888119870 decay O(10minus11) is generally
Advances in High Energy Physics 5
060402 08 10000
1
2
3
4
120601Blowast119888(x)
120601Υ(3S)(x)120601Υ(2S)(x)
120601Υ(1S)(x)
(a)
00 02 04 06 08 100
1
2
3
4
5
6
120601tBlowast119888(x)
120601tΥ(3S)(x)120601tΥ(2S)(x)
120601tΥ(1S)(x)
(b)
000
1
2
3
4
04 06 08 1002
120601VBlowast119888(x)
120601VΥ(3S)(x)120601VΥ(2S)(x)
120601VΥ(1S)(x)
(c)
Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast
119888mesons
u k3
bb
p3
p2p1
120587
d(k3)
G Blowastc
b(k1) c(k2)
Υ
)(
(a)
bb
u
120587
G Blowastc
b c
d
Υ
(b)
bb
u
G Blowastc
b c
d
120587
Υ
(c)
G
b c
120587
bb
u
Blowastc
d
Υ
(d)
Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast
119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and
nonfactorizable emission diagrams (c d)
less than that for Υ(119899119878) rarr 119861lowast
119888120587 decay by one order of
magnitude due to the CKM suppression |119881lowast
119906119904119881
lowast
119906119889|2
sim
1205822
(2) As it is well known due to the large mass of119861lowast
119888 the momentum transition in Υ(119899119878) rarr 119861
lowast
119888119875
decay may be not large enough One might naturally
wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast
119888120587 decay from different 120572
119904120587 region are shown
in Figure 3 It can be clearly seen that more than
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
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2 Advances in High Energy Physics
Table 1 Summary of mass decay width on(off)-peak luminosity and numbers of Υ(119899119878)
Meson Properties [1] Luminosity (fbminus1) [5] Numbers (106) [5]Mass (MeV) Width (keV) Belle BaBar Belle BaBar
Υ(1119878) 946030 plusmn 026 5402 plusmn 125 57 (18) sdot sdot sdot 102 plusmn 2 sdot sdot sdot
Υ(2119878) 1002326 plusmn 031 3198 plusmn 263 249 (17) 136 (14) 158 plusmn 4 983 plusmn 09
Υ(3119878) 103552 plusmn 05 2032 plusmn 185 29 (02) 280 (26) 11 plusmn 03 1213 plusmn 12
nonleptonic decay of heavy-flavored mesons such as thepQCD approach [2ndash4] the QCD factorization approach[7ndash9] and soft and collinear effective theory [10ndash13] andhave been applied widely to vindicate measurements on 119861meson decaysThe upsilon weak decay permits one to furtherconstrain parameters obtained from 119861 meson decay andcross comparisons provide an opportunity to test variousphenomenologicalmodelsThe upsilonweak decay possessesa unique structure due to the Cabibbo-Kobayashi-Maskawa(CKM) matrix properties which predicts that the channelswith one 119861(lowast)
119888meson are dominant Υ(119899119878) rarr 119861
lowast
119888119875 decay
belongs to the favorable 119887 rarr 119888 transition which should inprinciple have relatively large branching ratio among upsilonweak decays However there is still no theoretical studydevoted to Υ(119899119878) rarr 119861
lowast
119888119875 decay for the moment In this
paper we will present a phenomenological investigation onΥ(119899119878) rarr 119861
lowast
119888119875weak decay with the pQCD approach to supply
a ready reference for the future experimentsThis paper is organized as follows Section 2 focuses on
theoretical framework and decay amplitudes for Υ(119899119878) rarr119861lowast
119888120587 119861
lowast
119888119870 weak decays Section 3 is devoted to numerical
results and discussion The last section is a summary
2 Theoretical Framework
21 The Effective Hamiltonian Theoretically Υ(119899119878) rarr 119861lowast
119888120587
119861lowast
119888119870 weak decays are described by an effective bottom-
changing Hamiltonian based on operator product expansion[19]
Heff =119866119865
radic2
sum
119902=119889119904
119881119888119887119881
lowast
119906119902119862
1(120583)119874
1(120583) + 119862
2(120583)119874
2(120583)
+ hc
(1)
where 119866119865≃ 1166 times 10
minus5 GeVminus2 [1] is the Fermi couplingconstant the CKM factors 119881
119888119887119881
lowast
119906119889and 119881
119888119887119881
lowast
119906119904correspond
to Υ(119899119878) rarr 119861lowast
119888120587 and 119861lowast
119888119870 decays respectively with the
Wolfenstein parameterization theCKM factors are expandedas a power series in a smallWolfenstein parameter 120582 sim 02 [1]
119881119888119887119881
lowast
119906119889= 119860120582
2
minus
1
2
1198601205824
minus
1
8
1198601205826
+ O (1205827
)
119881119888119887119881
lowast
119906119904= 119860120582
3
+ O (1205827
)
(2)
The local tree operators 11987612
are defined as
1198741= [119888
120572120574120583(1 minus 120574
5) 119887
120572] [119902
120573120574120583
(1 minus 1205745) 119906
120573]
1198742= [119888
120572120574120583(1 minus 120574
5) 119887
120573] [119902
120573120574120583
(1 minus 1205745) 119906
120572]
(3)
where 120572 and 120573 are color indices and the sum over repeatedindices is understood
The scale 120583 factorizes physics contributions into short-and long-distance dynamics The Wilson coefficients 119862
119894(120583)
summarize the physics contributions at scale higher than 120583and are calculable with the renormalization group improvedperturbation theory The hadronic matrix elements (HME)where the local operators are inserted between initial andfinal hadron states embrace the physics contributions belowscale of 120583 To obtain decay amplitudes the remaining workis to calculate HME properly by separating from perturbativeand nonperturbative contributions
22 Hadronic Matrix Elements Based on Lepage-Brodskyapproach for exclusive processes [20] HME is commonlyexpressed as a convolution integral of hard scattering subam-plitudes containing perturbative contributions with univer-sal wave functions reflecting nonperturbative contributionsIn order to effectively regulate endpoint singularities andprovide a naturally dynamical cutoff on nonperturbativecontributions transverse momentum of valence quarks isretained and the Sudakov factor is introduced within thepQCD framework [2ndash4] Phenomenologically pQCDrsquos decayamplitude could be divided into three parts theWilson coef-ficients119862
119894incorporating the hard contributions above typical
scale of 119905 process-dependent rescattering subamplitudes 119879accounting for the heavy quark decay and wave functions Φof all participating hadrons which is expressed as
int119889119896119862119894(119905) 119879 (119905 119896)Φ (119896) 119890
minus119878
(4)
where 119896 is the momentum of valence quarks and 119890minus119878 is theSudakov factor
23 Kinematic Variables The light cone kinematic variablesin Υ(119899119878) rest frame are defined as follows
119901Υ= 119901
1=
1198981
radic2
(1 1 0)
119901119861lowast
119888
= 1199012= (119901
+
2 119901
minus
2 0)
1199013= (119901
minus
3 119901
+
3 0)
119901plusmn
119894=
(119864119894plusmn 119901)
radic2
119896119894= 119909
119894119901119894+ (0 0
119896119894perp)
120598
1=
1199011
1198981
minus
1198981
1199011sdot 119899
+
119899+
Advances in High Energy Physics 3
120598
2=
1199012
1198982
minus
1198982
1199012sdot 119899
minus
119899minus
120598perp
12= (0 0 1)
119899+= (1 0 0)
119899minus= (0 1 0)
119904 = 21199012sdot 119901
3
119905 = 21199011sdot 119901
2= 2119898
11198642
119906 = 21199011sdot 119901
3= 2119898
11198643
119901 =
radic[1198982
1minus (119898
2+ 119898
3)
2
] [1198982
1minus (119898
2minus 119898
3)
2
]
21198981
(5)
where119909119894and
119896119894perpare the longitudinalmomentum fraction and
transverse momentum of valence quarks respectively 120598119894and
120598perp
119894are the longitudinal and transverse polarization vectors
respectively and satisfy relations 1205982119894= minus1 and 120598
119894sdot 119901
119894= 0
the subscript 119894 on variables 119901119894 119864
119894 119898
119894 and 120598
119894corresponds
to participating hadrons namely 119894 = 1 for Υ(119899119878) meson119894 = 2 for the recoiled 119861lowast
119888meson and 119894 = 3 for the emitted
pseudoscalar meson 119899+and 119899
minusare positive and negative null
vectors respectively 119904 119905 and 119906 are the Lorentz-invariantvariables 119901 is the common momentum of final states Thenotation of momentum is displayed in Figure 2(a)
24 Wave Functions With the notation in [16 21] wavefunctions are defined as
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598
1[119898
1Φ
VΥ(119896
1) minus 1199011Φ
119905
Υ(119896
1)]
119895119894
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
perp
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598perp
1[119898
1Φ
119881
Υ(119896
1) minus 1199011Φ
119879
Υ(119896
1)]
119895119894
⟨119861lowast
119888(119901
2 120598
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989631198901198941198962sdot119911
120598
2[119898
2Φ
V119861lowast
119888
(1198962) + 1199012Φ
119905
119861lowast
119888
(1198962)]
119895119894
⟨119861lowast
119888(119901
2 120598
perp
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989621198901198941198962sdot119911
120598perp
2[119898
2Φ
119881
119861lowast
119888
(1198962) + 1199012Φ
119879
119861lowast
119888
(1198962)]
119895119894
⟨119875 (1199013)
10038161003816100381610038161003816119906119894(0) 119902
119895(119911)
100381610038161003816100381610038160⟩ =
119894119891119875
4
sdot int 11988911989631198901198941198963sdot119911
1205745[1199013Φ
119886
119875(119896
3) + 120583
119875Φ
119901
119875(119896
3)
+ 120583119875(119899minus119899+ minus 1)Φ
119905
119875(119896
3)]
119895119894
(6)
where 119891Υ 119891
119861lowast
119888
and 119891119875are decay constants of Υ(119899119878) 119861lowast
119888 and
119875mesons respectivelyConsidering mass relations of 119898
Υ(119899119878)≃ 2119898
119887and 119898
119861lowast
119888
≃
119898119887+ 119898
119888 it might assume that the motion of heavy valence
quarks in Υ(119899119878) and 119861lowast
119888mesons is nearly nonrelativis-
tic The wave functions of Υ(119899119878) and 119861lowast
119888mesons could
be approximately described with nonrelativistic quantumchromodynamics (NRQCD) [22ndash24] and time-independentSchrodinger equation For an isotropic harmonic oscillatorpotential the eigenfunctions of stationary statewith quantumnumbers 119899119871 are written as [17]
1206011119878(119896) sim 119890
minus119896
2
21205732
1206012119878(119896) sim 119890
minus119896
2
21205732
(2119896
2
minus 31205732
)
1206013119878(119896) sim 119890
minus119896
2
21205732
(4119896
4
minus 20119896
2
1205732
+ 151205734
)
(7)
where parameter 120573 determines the average transversemomentum that is ⟨119899119878|1198962
perp|119899119878⟩ sim 120573
2 Employing the substi-tution ansatz [25]
119896
2
997888rarr
1
4
sum
119894
119896
2
119894perp+ 119898
2
119902119894
119909119894
(8)
where 119909119894and 119898
119902119894
are the longitudinal momentum fractionand mass of valence quark respectively then integrating out119896perpand combining with their asymptotic forms the distribu-
tion amplitudes (DAs) for Υ(119899119878) and 119861lowast119888mesons can be
written as [17]
120601V119879Υ(1119878)
(119909) = 119860119909119909 expminus119898
2
119887
81205732
1119909119909
120601119905
Υ(1119878)(119909) = 119861119905
2 expminus119898
2
119887
81205732
1119909119909
120601119881
Υ(1119878)(119909) = 119862 (1 + 119905
2
) expminus119898
2
119887
81205732
1119909119909
120601V119905119881119879Υ(2119878)
(119909) = 119863120601V119905119881119879Υ(1119878)
(119909) 1 +
1198982
119887
21205732
1119909119909
120601V119905119881119879Υ(3119878)
(119909) = 119864120601V119905119881119879Υ(1119878)
(119909)(1 minus
1198982
119887
21205732
1119909119909
)
2
+ 6
120601V119879119861lowast
119888
(119909) = 119865119909119909 expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
4 Advances in High Energy Physics
120601119905
119861lowast
119888
(119909) = 1198661199052 expminus
1199091198982
119888+ 119909119898
2
119887
81205732
2119909119909
120601119881
119861lowast
119888
(119909) = 119867(1 minus 1199052
) expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
(9)
where 119909 = 1 minus 119909 119905 = 119909 minus 119909 According to NRQCD powercounting rules [22] 120573
119894≃ 120585
119894120572119904(120585
119894) with 120585
119894= 119898
1198942 and QCD
coupling constant 120572119904 The exponential function represents 119896
perp
distribution Parameters of 119860 119861 119862 119863 119864 119865 119866 and 119867 arenormalization coefficients satisfying the conditions
int
1
0
119889119909120601119894
Υ(119899119878)(119909) = int
1
0
119889119909120601119894
119861lowast
119888
(119909) = 1
for 119894 = V 119905 119881 119879(10)
The shape lines of normalized DAs for Υ(119899119878) and 119861lowast119888
mesons are shown in Figure 1 It is clearly seen that (1) DAsforΥ(119899119878) and119861lowast
119888mesons fall quickly down to zero at endpoint
119909 119909 rarr 0 due to suppression from exponential functions (2)DAs for Υ(119899119878) meson are symmetric under the interchangeof momentum fractions 119909 harr 119909 and DAs for 119861lowast
119888meson are
basically consistent with the feature that valence quarks sharemomentum fractions according to their masses
Our study shows that only the leading twist (twist-2) DAsof the emitted light pseudoscalar meson 119875 are involved indecay amplitudes (see Appendix) The twist-2 DAs have theexpansion [16]
120601119886
119875(119909) = 6119909119909sum
119894=0
11988611989411986232
119894(119905) (11)
and are normalized as
int
1
0
120601119886
119875(119909) 119889119909 = 1 (12)
where 11986232
119894(119905) are Gegenbauer polynomials
11986232
0(119905) = 1
11986232
1(119905) = 3119905
11986232
2(119905) =
3
2
(51199052
minus 1)
(13)
and each term corresponds to a nonperturbative Gegenbauermoment 119886
119894 note that 119886
0= 1 due to the normalization
condition equation (12) 119866-parity invariance of the pion DAsrequires Gegenbauer moment 119886
119894= 0 for 119894 = 1 3 5
25 Decay Amplitudes The Feynman diagrams for Υ(119899119878) rarr119861lowast
119888120587 weak decay are shown in Figure 2 There are two types
One is factorizable emission topology where gluon attachesto quarks in the samemeson and the other is nonfactorizable
emission topology where gluon connects to quarks betweendifferent mesons
With the pQCD master formula equation (4) the ampli-tude for Υ(119899119878) rarr 119861
lowast
119888119875 decay can be expressed as [26]
A (Υ (119899119878) 997888rarr 119861lowast
119888119875) = A
119871(120598
1 120598
2) +A
119873(120598
perp
1 120598
perp
2)
+ 119894A119879120598120583V120572120573120598
120583
1120598V2119901120572
1119901
120573
2
(14)
which is conventionally written as the helicity amplitudes[26]
A0= minus119862Asum
119895
A119895
119871(120598
1 120598
2)
A= radic2119862Asum
119895
A119895
119873(120598
perp
1 120598
perp
2)
Aperp= radic2119862A1198981
119901sum
119895
A119895
119879
119862A = 119894119881119888119887119881lowast
119906119902
119866119865
radic2
119862119865
119873119888
120587119891Υ119891119861lowast
119888
119891119875
(15)
where 119862119865= 43 and the color number119873
119888= 3 the subscript 119894
on 119860119895
119894corresponds to three different helicity amplitudes that
is 119894 = 119871119873 119879 the superscript 119895 on 119860119895
119894denotes indices of
Figure 2 The explicit expressions of building blocks A119895
119894are
collected in Appendix
3 Numerical Results and Discussion
In the center-of-mass ofΥ(119899119878)meson branching ratioB119903 forΥ(119899119878) rarr 119861
lowast
119888119875 decay is defined as
B119903 =1
12120587
119901
1198982
ΥΓΥ
1003816100381610038161003816A
0
1003816100381610038161003816
2
+1003816100381610038161003816A
1003816100381610038161003816
2
+1003816100381610038161003816A
perp
1003816100381610038161003816
2
(16)
The input parameters are listed in Tables 1 and 2 If notspecified explicitly we will take their central values as thedefault inputs Our numerical results are collected in Table 3where the first uncertainty comes from scale (1 plusmn 01)119905
119894and
the expression of 119905119894is given in (A4) and (A5) the second
uncertainty is from mass of119898119887and119898
119888 the third uncertainty
is from hadronic parameters including decay constants andGegenbauer moments the fourth uncertainty is from CKMparameters The followings are some comments
(1) Branching ratio for Υ(119899119878) rarr 119861lowast
119888120587 decay is about
O(10minus10) with pQCD approach which is well withinthe measurement potential of LHC and SuperKEKBFor example experimental studies have showed thatproduction cross sections forΥ(119899119878)meson in p-p andp-Pb collisions are a few 120583119887 at the LHCb [27 28] andALICE [29 30] detectors Consequently there willbe more than 1012 Υ(119899119878) data samples per 119886119887minus1 datacollected by the LHCb and ALICE corresponding toa few hundreds of Υ(119899119878) rarr 119861
lowast
119888120587 events Branching
ratio for Υ(119899119878) rarr 119861lowast
119888119870 decay O(10minus11) is generally
Advances in High Energy Physics 5
060402 08 10000
1
2
3
4
120601Blowast119888(x)
120601Υ(3S)(x)120601Υ(2S)(x)
120601Υ(1S)(x)
(a)
00 02 04 06 08 100
1
2
3
4
5
6
120601tBlowast119888(x)
120601tΥ(3S)(x)120601tΥ(2S)(x)
120601tΥ(1S)(x)
(b)
000
1
2
3
4
04 06 08 1002
120601VBlowast119888(x)
120601VΥ(3S)(x)120601VΥ(2S)(x)
120601VΥ(1S)(x)
(c)
Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast
119888mesons
u k3
bb
p3
p2p1
120587
d(k3)
G Blowastc
b(k1) c(k2)
Υ
)(
(a)
bb
u
120587
G Blowastc
b c
d
Υ
(b)
bb
u
G Blowastc
b c
d
120587
Υ
(c)
G
b c
120587
bb
u
Blowastc
d
Υ
(d)
Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast
119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and
nonfactorizable emission diagrams (c d)
less than that for Υ(119899119878) rarr 119861lowast
119888120587 decay by one order of
magnitude due to the CKM suppression |119881lowast
119906119904119881
lowast
119906119889|2
sim
1205822
(2) As it is well known due to the large mass of119861lowast
119888 the momentum transition in Υ(119899119878) rarr 119861
lowast
119888119875
decay may be not large enough One might naturally
wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast
119888120587 decay from different 120572
119904120587 region are shown
in Figure 3 It can be clearly seen that more than
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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Superconductivity
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ThermodynamicsJournal of
Advances in High Energy Physics 3
120598
2=
1199012
1198982
minus
1198982
1199012sdot 119899
minus
119899minus
120598perp
12= (0 0 1)
119899+= (1 0 0)
119899minus= (0 1 0)
119904 = 21199012sdot 119901
3
119905 = 21199011sdot 119901
2= 2119898
11198642
119906 = 21199011sdot 119901
3= 2119898
11198643
119901 =
radic[1198982
1minus (119898
2+ 119898
3)
2
] [1198982
1minus (119898
2minus 119898
3)
2
]
21198981
(5)
where119909119894and
119896119894perpare the longitudinalmomentum fraction and
transverse momentum of valence quarks respectively 120598119894and
120598perp
119894are the longitudinal and transverse polarization vectors
respectively and satisfy relations 1205982119894= minus1 and 120598
119894sdot 119901
119894= 0
the subscript 119894 on variables 119901119894 119864
119894 119898
119894 and 120598
119894corresponds
to participating hadrons namely 119894 = 1 for Υ(119899119878) meson119894 = 2 for the recoiled 119861lowast
119888meson and 119894 = 3 for the emitted
pseudoscalar meson 119899+and 119899
minusare positive and negative null
vectors respectively 119904 119905 and 119906 are the Lorentz-invariantvariables 119901 is the common momentum of final states Thenotation of momentum is displayed in Figure 2(a)
24 Wave Functions With the notation in [16 21] wavefunctions are defined as
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598
1[119898
1Φ
VΥ(119896
1) minus 1199011Φ
119905
Υ(119896
1)]
119895119894
⟨0
10038161003816100381610038161003816119887119894(119911) 119887
119895(0)
10038161003816100381610038161003816Υ (119901
1 120598
perp
1)⟩ =
119891Υ
4
sdot int 1198891198961119890minus1198941198961sdot119911
120598perp
1[119898
1Φ
119881
Υ(119896
1) minus 1199011Φ
119879
Υ(119896
1)]
119895119894
⟨119861lowast
119888(119901
2 120598
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989631198901198941198962sdot119911
120598
2[119898
2Φ
V119861lowast
119888
(1198962) + 1199012Φ
119905
119861lowast
119888
(1198962)]
119895119894
⟨119861lowast
119888(119901
2 120598
perp
2)
10038161003816100381610038161003816119888119894(119911) 119887
119895(0)
100381610038161003816100381610038160⟩ =
119891119861lowast
119888
4
sdot int
1
0
11988911989621198901198941198962sdot119911
120598perp
2[119898
2Φ
119881
119861lowast
119888
(1198962) + 1199012Φ
119879
119861lowast
119888
(1198962)]
119895119894
⟨119875 (1199013)
10038161003816100381610038161003816119906119894(0) 119902
119895(119911)
100381610038161003816100381610038160⟩ =
119894119891119875
4
sdot int 11988911989631198901198941198963sdot119911
1205745[1199013Φ
119886
119875(119896
3) + 120583
119875Φ
119901
119875(119896
3)
+ 120583119875(119899minus119899+ minus 1)Φ
119905
119875(119896
3)]
119895119894
(6)
where 119891Υ 119891
119861lowast
119888
and 119891119875are decay constants of Υ(119899119878) 119861lowast
119888 and
119875mesons respectivelyConsidering mass relations of 119898
Υ(119899119878)≃ 2119898
119887and 119898
119861lowast
119888
≃
119898119887+ 119898
119888 it might assume that the motion of heavy valence
quarks in Υ(119899119878) and 119861lowast
119888mesons is nearly nonrelativis-
tic The wave functions of Υ(119899119878) and 119861lowast
119888mesons could
be approximately described with nonrelativistic quantumchromodynamics (NRQCD) [22ndash24] and time-independentSchrodinger equation For an isotropic harmonic oscillatorpotential the eigenfunctions of stationary statewith quantumnumbers 119899119871 are written as [17]
1206011119878(119896) sim 119890
minus119896
2
21205732
1206012119878(119896) sim 119890
minus119896
2
21205732
(2119896
2
minus 31205732
)
1206013119878(119896) sim 119890
minus119896
2
21205732
(4119896
4
minus 20119896
2
1205732
+ 151205734
)
(7)
where parameter 120573 determines the average transversemomentum that is ⟨119899119878|1198962
perp|119899119878⟩ sim 120573
2 Employing the substi-tution ansatz [25]
119896
2
997888rarr
1
4
sum
119894
119896
2
119894perp+ 119898
2
119902119894
119909119894
(8)
where 119909119894and 119898
119902119894
are the longitudinal momentum fractionand mass of valence quark respectively then integrating out119896perpand combining with their asymptotic forms the distribu-
tion amplitudes (DAs) for Υ(119899119878) and 119861lowast119888mesons can be
written as [17]
120601V119879Υ(1119878)
(119909) = 119860119909119909 expminus119898
2
119887
81205732
1119909119909
120601119905
Υ(1119878)(119909) = 119861119905
2 expminus119898
2
119887
81205732
1119909119909
120601119881
Υ(1119878)(119909) = 119862 (1 + 119905
2
) expminus119898
2
119887
81205732
1119909119909
120601V119905119881119879Υ(2119878)
(119909) = 119863120601V119905119881119879Υ(1119878)
(119909) 1 +
1198982
119887
21205732
1119909119909
120601V119905119881119879Υ(3119878)
(119909) = 119864120601V119905119881119879Υ(1119878)
(119909)(1 minus
1198982
119887
21205732
1119909119909
)
2
+ 6
120601V119879119861lowast
119888
(119909) = 119865119909119909 expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
4 Advances in High Energy Physics
120601119905
119861lowast
119888
(119909) = 1198661199052 expminus
1199091198982
119888+ 119909119898
2
119887
81205732
2119909119909
120601119881
119861lowast
119888
(119909) = 119867(1 minus 1199052
) expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
(9)
where 119909 = 1 minus 119909 119905 = 119909 minus 119909 According to NRQCD powercounting rules [22] 120573
119894≃ 120585
119894120572119904(120585
119894) with 120585
119894= 119898
1198942 and QCD
coupling constant 120572119904 The exponential function represents 119896
perp
distribution Parameters of 119860 119861 119862 119863 119864 119865 119866 and 119867 arenormalization coefficients satisfying the conditions
int
1
0
119889119909120601119894
Υ(119899119878)(119909) = int
1
0
119889119909120601119894
119861lowast
119888
(119909) = 1
for 119894 = V 119905 119881 119879(10)
The shape lines of normalized DAs for Υ(119899119878) and 119861lowast119888
mesons are shown in Figure 1 It is clearly seen that (1) DAsforΥ(119899119878) and119861lowast
119888mesons fall quickly down to zero at endpoint
119909 119909 rarr 0 due to suppression from exponential functions (2)DAs for Υ(119899119878) meson are symmetric under the interchangeof momentum fractions 119909 harr 119909 and DAs for 119861lowast
119888meson are
basically consistent with the feature that valence quarks sharemomentum fractions according to their masses
Our study shows that only the leading twist (twist-2) DAsof the emitted light pseudoscalar meson 119875 are involved indecay amplitudes (see Appendix) The twist-2 DAs have theexpansion [16]
120601119886
119875(119909) = 6119909119909sum
119894=0
11988611989411986232
119894(119905) (11)
and are normalized as
int
1
0
120601119886
119875(119909) 119889119909 = 1 (12)
where 11986232
119894(119905) are Gegenbauer polynomials
11986232
0(119905) = 1
11986232
1(119905) = 3119905
11986232
2(119905) =
3
2
(51199052
minus 1)
(13)
and each term corresponds to a nonperturbative Gegenbauermoment 119886
119894 note that 119886
0= 1 due to the normalization
condition equation (12) 119866-parity invariance of the pion DAsrequires Gegenbauer moment 119886
119894= 0 for 119894 = 1 3 5
25 Decay Amplitudes The Feynman diagrams for Υ(119899119878) rarr119861lowast
119888120587 weak decay are shown in Figure 2 There are two types
One is factorizable emission topology where gluon attachesto quarks in the samemeson and the other is nonfactorizable
emission topology where gluon connects to quarks betweendifferent mesons
With the pQCD master formula equation (4) the ampli-tude for Υ(119899119878) rarr 119861
lowast
119888119875 decay can be expressed as [26]
A (Υ (119899119878) 997888rarr 119861lowast
119888119875) = A
119871(120598
1 120598
2) +A
119873(120598
perp
1 120598
perp
2)
+ 119894A119879120598120583V120572120573120598
120583
1120598V2119901120572
1119901
120573
2
(14)
which is conventionally written as the helicity amplitudes[26]
A0= minus119862Asum
119895
A119895
119871(120598
1 120598
2)
A= radic2119862Asum
119895
A119895
119873(120598
perp
1 120598
perp
2)
Aperp= radic2119862A1198981
119901sum
119895
A119895
119879
119862A = 119894119881119888119887119881lowast
119906119902
119866119865
radic2
119862119865
119873119888
120587119891Υ119891119861lowast
119888
119891119875
(15)
where 119862119865= 43 and the color number119873
119888= 3 the subscript 119894
on 119860119895
119894corresponds to three different helicity amplitudes that
is 119894 = 119871119873 119879 the superscript 119895 on 119860119895
119894denotes indices of
Figure 2 The explicit expressions of building blocks A119895
119894are
collected in Appendix
3 Numerical Results and Discussion
In the center-of-mass ofΥ(119899119878)meson branching ratioB119903 forΥ(119899119878) rarr 119861
lowast
119888119875 decay is defined as
B119903 =1
12120587
119901
1198982
ΥΓΥ
1003816100381610038161003816A
0
1003816100381610038161003816
2
+1003816100381610038161003816A
1003816100381610038161003816
2
+1003816100381610038161003816A
perp
1003816100381610038161003816
2
(16)
The input parameters are listed in Tables 1 and 2 If notspecified explicitly we will take their central values as thedefault inputs Our numerical results are collected in Table 3where the first uncertainty comes from scale (1 plusmn 01)119905
119894and
the expression of 119905119894is given in (A4) and (A5) the second
uncertainty is from mass of119898119887and119898
119888 the third uncertainty
is from hadronic parameters including decay constants andGegenbauer moments the fourth uncertainty is from CKMparameters The followings are some comments
(1) Branching ratio for Υ(119899119878) rarr 119861lowast
119888120587 decay is about
O(10minus10) with pQCD approach which is well withinthe measurement potential of LHC and SuperKEKBFor example experimental studies have showed thatproduction cross sections forΥ(119899119878)meson in p-p andp-Pb collisions are a few 120583119887 at the LHCb [27 28] andALICE [29 30] detectors Consequently there willbe more than 1012 Υ(119899119878) data samples per 119886119887minus1 datacollected by the LHCb and ALICE corresponding toa few hundreds of Υ(119899119878) rarr 119861
lowast
119888120587 events Branching
ratio for Υ(119899119878) rarr 119861lowast
119888119870 decay O(10minus11) is generally
Advances in High Energy Physics 5
060402 08 10000
1
2
3
4
120601Blowast119888(x)
120601Υ(3S)(x)120601Υ(2S)(x)
120601Υ(1S)(x)
(a)
00 02 04 06 08 100
1
2
3
4
5
6
120601tBlowast119888(x)
120601tΥ(3S)(x)120601tΥ(2S)(x)
120601tΥ(1S)(x)
(b)
000
1
2
3
4
04 06 08 1002
120601VBlowast119888(x)
120601VΥ(3S)(x)120601VΥ(2S)(x)
120601VΥ(1S)(x)
(c)
Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast
119888mesons
u k3
bb
p3
p2p1
120587
d(k3)
G Blowastc
b(k1) c(k2)
Υ
)(
(a)
bb
u
120587
G Blowastc
b c
d
Υ
(b)
bb
u
G Blowastc
b c
d
120587
Υ
(c)
G
b c
120587
bb
u
Blowastc
d
Υ
(d)
Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast
119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and
nonfactorizable emission diagrams (c d)
less than that for Υ(119899119878) rarr 119861lowast
119888120587 decay by one order of
magnitude due to the CKM suppression |119881lowast
119906119904119881
lowast
119906119889|2
sim
1205822
(2) As it is well known due to the large mass of119861lowast
119888 the momentum transition in Υ(119899119878) rarr 119861
lowast
119888119875
decay may be not large enough One might naturally
wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast
119888120587 decay from different 120572
119904120587 region are shown
in Figure 3 It can be clearly seen that more than
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
120601119905
119861lowast
119888
(119909) = 1198661199052 expminus
1199091198982
119888+ 119909119898
2
119887
81205732
2119909119909
120601119881
119861lowast
119888
(119909) = 119867(1 minus 1199052
) expminus119909119898
2
119888+ 119909119898
2
119887
81205732
2119909119909
(9)
where 119909 = 1 minus 119909 119905 = 119909 minus 119909 According to NRQCD powercounting rules [22] 120573
119894≃ 120585
119894120572119904(120585
119894) with 120585
119894= 119898
1198942 and QCD
coupling constant 120572119904 The exponential function represents 119896
perp
distribution Parameters of 119860 119861 119862 119863 119864 119865 119866 and 119867 arenormalization coefficients satisfying the conditions
int
1
0
119889119909120601119894
Υ(119899119878)(119909) = int
1
0
119889119909120601119894
119861lowast
119888
(119909) = 1
for 119894 = V 119905 119881 119879(10)
The shape lines of normalized DAs for Υ(119899119878) and 119861lowast119888
mesons are shown in Figure 1 It is clearly seen that (1) DAsforΥ(119899119878) and119861lowast
119888mesons fall quickly down to zero at endpoint
119909 119909 rarr 0 due to suppression from exponential functions (2)DAs for Υ(119899119878) meson are symmetric under the interchangeof momentum fractions 119909 harr 119909 and DAs for 119861lowast
119888meson are
basically consistent with the feature that valence quarks sharemomentum fractions according to their masses
Our study shows that only the leading twist (twist-2) DAsof the emitted light pseudoscalar meson 119875 are involved indecay amplitudes (see Appendix) The twist-2 DAs have theexpansion [16]
120601119886
119875(119909) = 6119909119909sum
119894=0
11988611989411986232
119894(119905) (11)
and are normalized as
int
1
0
120601119886
119875(119909) 119889119909 = 1 (12)
where 11986232
119894(119905) are Gegenbauer polynomials
11986232
0(119905) = 1
11986232
1(119905) = 3119905
11986232
2(119905) =
3
2
(51199052
minus 1)
(13)
and each term corresponds to a nonperturbative Gegenbauermoment 119886
119894 note that 119886
0= 1 due to the normalization
condition equation (12) 119866-parity invariance of the pion DAsrequires Gegenbauer moment 119886
119894= 0 for 119894 = 1 3 5
25 Decay Amplitudes The Feynman diagrams for Υ(119899119878) rarr119861lowast
119888120587 weak decay are shown in Figure 2 There are two types
One is factorizable emission topology where gluon attachesto quarks in the samemeson and the other is nonfactorizable
emission topology where gluon connects to quarks betweendifferent mesons
With the pQCD master formula equation (4) the ampli-tude for Υ(119899119878) rarr 119861
lowast
119888119875 decay can be expressed as [26]
A (Υ (119899119878) 997888rarr 119861lowast
119888119875) = A
119871(120598
1 120598
2) +A
119873(120598
perp
1 120598
perp
2)
+ 119894A119879120598120583V120572120573120598
120583
1120598V2119901120572
1119901
120573
2
(14)
which is conventionally written as the helicity amplitudes[26]
A0= minus119862Asum
119895
A119895
119871(120598
1 120598
2)
A= radic2119862Asum
119895
A119895
119873(120598
perp
1 120598
perp
2)
Aperp= radic2119862A1198981
119901sum
119895
A119895
119879
119862A = 119894119881119888119887119881lowast
119906119902
119866119865
radic2
119862119865
119873119888
120587119891Υ119891119861lowast
119888
119891119875
(15)
where 119862119865= 43 and the color number119873
119888= 3 the subscript 119894
on 119860119895
119894corresponds to three different helicity amplitudes that
is 119894 = 119871119873 119879 the superscript 119895 on 119860119895
119894denotes indices of
Figure 2 The explicit expressions of building blocks A119895
119894are
collected in Appendix
3 Numerical Results and Discussion
In the center-of-mass ofΥ(119899119878)meson branching ratioB119903 forΥ(119899119878) rarr 119861
lowast
119888119875 decay is defined as
B119903 =1
12120587
119901
1198982
ΥΓΥ
1003816100381610038161003816A
0
1003816100381610038161003816
2
+1003816100381610038161003816A
1003816100381610038161003816
2
+1003816100381610038161003816A
perp
1003816100381610038161003816
2
(16)
The input parameters are listed in Tables 1 and 2 If notspecified explicitly we will take their central values as thedefault inputs Our numerical results are collected in Table 3where the first uncertainty comes from scale (1 plusmn 01)119905
119894and
the expression of 119905119894is given in (A4) and (A5) the second
uncertainty is from mass of119898119887and119898
119888 the third uncertainty
is from hadronic parameters including decay constants andGegenbauer moments the fourth uncertainty is from CKMparameters The followings are some comments
(1) Branching ratio for Υ(119899119878) rarr 119861lowast
119888120587 decay is about
O(10minus10) with pQCD approach which is well withinthe measurement potential of LHC and SuperKEKBFor example experimental studies have showed thatproduction cross sections forΥ(119899119878)meson in p-p andp-Pb collisions are a few 120583119887 at the LHCb [27 28] andALICE [29 30] detectors Consequently there willbe more than 1012 Υ(119899119878) data samples per 119886119887minus1 datacollected by the LHCb and ALICE corresponding toa few hundreds of Υ(119899119878) rarr 119861
lowast
119888120587 events Branching
ratio for Υ(119899119878) rarr 119861lowast
119888119870 decay O(10minus11) is generally
Advances in High Energy Physics 5
060402 08 10000
1
2
3
4
120601Blowast119888(x)
120601Υ(3S)(x)120601Υ(2S)(x)
120601Υ(1S)(x)
(a)
00 02 04 06 08 100
1
2
3
4
5
6
120601tBlowast119888(x)
120601tΥ(3S)(x)120601tΥ(2S)(x)
120601tΥ(1S)(x)
(b)
000
1
2
3
4
04 06 08 1002
120601VBlowast119888(x)
120601VΥ(3S)(x)120601VΥ(2S)(x)
120601VΥ(1S)(x)
(c)
Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast
119888mesons
u k3
bb
p3
p2p1
120587
d(k3)
G Blowastc
b(k1) c(k2)
Υ
)(
(a)
bb
u
120587
G Blowastc
b c
d
Υ
(b)
bb
u
G Blowastc
b c
d
120587
Υ
(c)
G
b c
120587
bb
u
Blowastc
d
Υ
(d)
Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast
119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and
nonfactorizable emission diagrams (c d)
less than that for Υ(119899119878) rarr 119861lowast
119888120587 decay by one order of
magnitude due to the CKM suppression |119881lowast
119906119904119881
lowast
119906119889|2
sim
1205822
(2) As it is well known due to the large mass of119861lowast
119888 the momentum transition in Υ(119899119878) rarr 119861
lowast
119888119875
decay may be not large enough One might naturally
wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast
119888120587 decay from different 120572
119904120587 region are shown
in Figure 3 It can be clearly seen that more than
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
060402 08 10000
1
2
3
4
120601Blowast119888(x)
120601Υ(3S)(x)120601Υ(2S)(x)
120601Υ(1S)(x)
(a)
00 02 04 06 08 100
1
2
3
4
5
6
120601tBlowast119888(x)
120601tΥ(3S)(x)120601tΥ(2S)(x)
120601tΥ(1S)(x)
(b)
000
1
2
3
4
04 06 08 1002
120601VBlowast119888(x)
120601VΥ(3S)(x)120601VΥ(2S)(x)
120601VΥ(1S)(x)
(c)
Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast
119888mesons
u k3
bb
p3
p2p1
120587
d(k3)
G Blowastc
b(k1) c(k2)
Υ
)(
(a)
bb
u
120587
G Blowastc
b c
d
Υ
(b)
bb
u
G Blowastc
b c
d
120587
Υ
(c)
G
b c
120587
bb
u
Blowastc
d
Υ
(d)
Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast
119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and
nonfactorizable emission diagrams (c d)
less than that for Υ(119899119878) rarr 119861lowast
119888120587 decay by one order of
magnitude due to the CKM suppression |119881lowast
119906119904119881
lowast
119906119889|2
sim
1205822
(2) As it is well known due to the large mass of119861lowast
119888 the momentum transition in Υ(119899119878) rarr 119861
lowast
119888119875
decay may be not large enough One might naturally
wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast
119888120587 decay from different 120572
119904120587 region are shown
in Figure 3 It can be clearly seen that more than
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
6906
3811 014 017
2477
049 038 00806050402 0701 0803
120572s120587
0
20
40
60
80(
)
(1S)rarrBlowastc 120587Υ
(a)
550
021 013 01606050402 0701 0803
120572s120587
3988
328 088 039 0070
20
40
60
80
()
(2S)rarrBlowastc 120587Υ
(b)
011 013
(3S)rarrBlowastc 120587
06050402 0701 0803120572s120587
0
20
40
60
80
()
50674531
247 074 033 019 005
Υ
(c)
Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast
119888120587 decay (a) Υ(2119878) rarr 119861
lowast
119888120587 decay (b) and Υ(3119878) rarr 119861
lowast
119888120587 decay (c) from
different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions
Table 2 The numerical values of input parameters
TheWolfenstein parameters119860 = 0814
+0023
minus0024[1] 120582 = 022537 plusmn 000061 [1]
Mass decay constant and Gegenbauer moments119898
119887= 478 plusmn 006 GeV [1] 119891
120587= 13041 plusmn 020MeV [1]
119898119888= 167 plusmn 007 GeV [1] 119891
119870= 1562 plusmn 07MeV [1]
119898119861lowast
119888
= 6332 plusmn 9MeV [14] 119891119861lowast
119888
= 422 plusmn 13MeV [15]c119886119870
1(1 GeV) = minus006 plusmn 003 [16] 119891
Υ(1119878)= 6764 plusmn 107MeV [17]
119886119870
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(2119878)= 4730 plusmn 237MeV [17]
119886120587
2(1 GeV) = 025 plusmn 015 [16] 119891
Υ(3119878)= 4095 plusmn 294MeV [17]
cThe decay constant 119891119861lowast
119888
cannot be extracted from the experimental data because of no measurement on 119861lowast
119888weak decay at the present time Theoretically the
value of 119891119861lowast
119888
has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast
119888
is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861
lowast
119888119875 decays
Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast
119888119875 decays
Modes Υ(1119878) rarr 119861lowast
119888120587 Υ(2119878) rarr 119861
lowast
119888120587 Υ(3119878) rarr 119861
lowast
119888120587
1010
timesB119903 435+029+019+044+017
minus024minus041minus031minus030228
+013+026+040+009
minus003minus035minus016minus015214
+012+009+048+007
minus012minus041minus015minus015
Modes Υ(1119878) rarr 119861lowast
119888119870 Υ(2119878) rarr 119861
lowast
119888119870 Υ(3119878) rarr 119861
lowast
119888119870
1011
timesB119903 345+023+013+038+013
minus021minus035minus027minus025191
+011+007+036+007
minus009minus031minus015minus014165
+009+008+040+005
minus021minus033minus013minus012
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 7
93 (97) contributions come from 120572119904120587 le 02
(03) region implying that Υ(119899119878) rarr 119861lowast
119888120587 decay
is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation
(3) Because of the relations among masses 119898Υ(3119878)
gt
119898Υ(2119878)
gt 119898Υ(1119878)
resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ
Υ(3119878)lt
ΓΥ(2119878)
lt ΓΥ(1119878)
in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861
lowast
119888119875) gt
B119903(Υ(2119878) rarr 119861lowast
119888119875) gt B119903(Υ(1119878) rarr 119861
lowast
119888119875) for
the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982
Υ(119899119878)in (16) has
almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912
Υ(119899119878)Γ
Υ(119899119878)with the maximal
value 1198912
Υ(1119878)Γ
Υ(1119878)for 119899 le 3 Besides contributions
from 120572119904120587 isin [02 03] regions decrease with 119899 (see
Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other
factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation
4 Summary
Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast
119888120587 and119861lowast
119888119870 final states is investigated with
the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572
119904120587 le 03
which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861
lowast
119888120587 (119861lowast
119888119870) decay with branching ratio about
10minus10 (10minus11) at the future experiments
Appendix
Building Blocks for Υrarr 119861lowast
119888119875 Decays
The building blocksA119895
119894 where the superscript 119895 corresponds
to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows
A119886
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
VΥ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)
sdot 120601V119861lowast
119888
(1199092) [119898
2
1119904 minus (4119898
2
11199012
+ 1198982
2119906) 119909
2]
+ 120601119905
119861lowast
119888
(1199092)119898
2119898
119887119906
A119886
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886)119898
1120601
119881
119861lowast
119888
(1199092)119898
2(119906 minus 119904119909
2)
+ 120601119879
119861lowast
119888
(1199092)119898
119887119904
A119886
119879= minus2119898
1int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119886 119887
1 119887
2) 119864
119886119887(119905
119886) 120601
119881
Υ(119909
1)
sdot 120572119904(119905
119886) 119886
1(119905
119886) 120601
119879
119861lowast
119888
(1199092)119898
119887+ 120601
119881
119861lowast
119888
(1199092)119898
21199092
A119887
119871= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
V119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
VΥ(119909
1) [119898
2
2119906 minus 119898
2
1(119904 minus 4119901
2
) 1199091]
+ 120601119905
Υ(119909
1)119898
1119898
119888119904
A119887
119873= int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887)119898
2120601
119881
Υ(119909
1)119898
1(119904 minus 119906119909
1)
+ 120601119879
Υ(119909
1)119898
119888119906
A119887
119879= minus2119898
2int
1
0
1198891199091int
1
0
1198891199092int
infin
0
1198871119889119887
1
sdot int
infin
0
1198872119889119887
2119867
119886119887(120572
119890 120573
119887 119887
2 119887
1) 119864
119886119887(119905
119887) 120601
119881
119861lowast
119888
(1199092)
sdot 120572119904(119905
119887) 119886
1(119905
119887) 120601
119881
Υ(119909
1)119898
11199091+ 120601
119879
Υ(119909
1)119898
119888
A119888
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119888) 119862
2(119905
119888) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199091minus 119909
3)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119906119909
1minus 119904119909
2minus 2119898
2
31199093)
sdot 120575 (1198871minus 119887
2)
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
8 Advances in High Energy Physics
A119888
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
1minus 119909
3) + 119898
2
2119906 (119909
3minus 119909
2)
A119888
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119888 119887
2 119887
3) 119864
119888119889(119905
119888) 119862
2(119905
119888) 120572
119904(119905
119888)
sdot 120575 (1198871minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3) 119898
2
1(119909
3minus 119909
1)
+ 1198982
2(119909
2minus 119909
3)
A119889
119871=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 120601
119886
119875(119909
3)
sdot 120572119904(119905
119889) 119862
2(119905
119889) 120601
VΥ(119909
1) 120601
V119861lowast
119888
(1199092) 4119898
2
11199012
(1199093minus 119909
2)
+ 120601119905
Υ(119909
1) 120601
119905
119861lowast
119888
(1199092)119898
1119898
2(119904119909
2+ 2119898
2
31199093minus 119906119909
1)
sdot 120575 (1198871minus 119887
2)
A119889
119873=
1
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1119904 (119909
3minus 119909
1) + 119898
2
2119906 (119909
2minus 119909
3)
A119889
119879=
2
119873119888
int
1
0
1198891199091int
1
0
1198891199092int
1
0
1198891199093int
infin
0
1198891198871int
infin
0
1198872119889119887
2
sdot int
infin
0
1198873119889119887
3119867
119888119889(120572
119890 120573
119889 119887
2 119887
3) 119864
119888119889(119905
119889) 119862
2(119905
119889)
sdot 120572119904(119905
119889) 120575 (119887
1minus 119887
2) 120601
119879
Υ(119909
1) 120601
119879
119861lowast
119888
(1199092) 120601
119886
119875(119909
3)
sdot 1198982
1(119909
1minus 119909
3) minus 119898
2
2(119909
2minus 119909
3)
(A1)
where 119909119894= 1 minus 119909
119894 variable 119909
119894is the longitudinal momentum
fraction of the valence quark 119887119894is the conjugate variable of the
transverse momentum 119896119894perp and 120572
119904(119905) is the QCD coupling at
the scale of 119905 1198861= 119862
1+ 119862
2119873
119888
The function119867119894is defined as follows [17]
119867119886119887(120572
119890 120573 119887
119894 119887
119895) = 119870
0(radicminus120572119887
119894)
sdot 120579 (119887119894minus 119887
119895)119870
0(radicminus120573119887
119894) 119868
0(radicminus120573119887
119895)
+ (119887119894larrrarr 119887
119895)
119867119888119889(120572
119890 120573 119887
2 119887
3) = 120579 (minus120573)119870
0(radicminus120573119887
3)
+
120587
2
120579 (120573) [1198941198690(radic120573119887
3) minus 119884
0(radic120573119887
3)]
sdot 120579 (1198872minus 119887
3)119870
0(radicminus120572119887
2) 119868
0(radicminus120572119887
3)
+ (1198872larrrarr 119887
3)
(A2)
where 1198690and119884
0(119868
0and119870
0) are the (modified) Bessel function
of the first and second kind respectively 120572119890(120572
119886) is the
gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573
119894corresponds
to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]
120572 = 1199092
1119898
2
1+ 119909
2
2119898
2
2minus 119909
11199092119905
120573119886= 119898
2
1minus 119898
2
119887+ 119909
2
2119898
2
2minus 119909
2119905
120573119887= 119898
2
2minus 119898
2
119888+ 119909
2
1119898
2
1minus 119909
1119905
120573119888= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
120573119889= 119909
2
1119898
2
1+ 119909
2
2119898
2
2+ 119909
2
3119898
2
3minus 119909
11199092119905 minus 119909
11199093119906
+ 11990921199093119904
(A3)
The typical scale 119905119894and the Sudakov factor 119864
119894are defined
as follows where the subscript 119894 corresponds to the indices ofFigure 2
119905119886(119887)
= max(radicminus120572radicminus120573119886(119887)
1
1198871
1
1198872
) (A4)
119905119888(119889)
= max(radicminus120572radic1003816100381610038161003816120573119888(119889)
1003816100381610038161003816
1
1198872
1
1198873
) (A5)
119864119886119887(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) (A6)
119864119888119889(119905) = exp minus119878
Υ(119905) minus 119878
119861lowast
119888
(119905) minus 119878119875(119905) (A7)
119878Υ(119905) = 119904 (119909
1 119901
+
1
1
1198871
) + 2int
119905
11198871
119889120583
120583
120574119902 (A8)
119878119861lowast
119888
(119905) = 119904 (1199092 119901
+
2
1
1198872
) + 2int
119905
11198872
119889120583
120583
120574119902 (A9)
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 9
119878120587119870(119905) = 119904 (119909
3 119901
+
3
1
1198873
) + 119904 (1199093 119901
+
3
1
1198873
)
+ 2int
119905
11198873
119889120583
120583
120574119902
(A10)
where 120574119902= minus120572
119904120587 is the quark anomalous dimension
the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]
Competing Interests
The authors declare that there are no competing interestsrelated to this paper
Acknowledgments
Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)
References
[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014
[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995
[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997
[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997
[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014
[6] Z Wang ldquoThe radiative decays 119861119888
plusmn
rarr 119861plusmn
119888120574 with QCD sum
rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013
[7] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999
[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000
[9] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001
[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001
[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002
[12] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002
[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002
[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012
[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015
[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006
[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861
119888119875 decays with perturbative QCD approachrdquo Physics
Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons
with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013
[19] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996
[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980
[21] P Ball and G Jones ldquoTwist-3 distribution amplitudes ofKlowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007
[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992
[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995
[24] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005
[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002
[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870
decaysrdquo Physical Review D vol 66 Article ID 0540132002
[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014
[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904
119873119873= 5TeVrdquo
Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon
production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013
[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of