6
1 þþ nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate Kwei-Chou Yang * Department of Physics, Chung Yuan Christian University, Chung-Li 320, Taiwan (Received 30 June 2011; published 22 August 2011) Two strategies are taken into account to determine the f 1 ð1420Þ-f 1 ð1285Þ mixing angle . (i) First, using the Gell-Mann-Okubo mass formula together with the K 1 ð1270Þ-K 1 ð1400Þ mixing angle K 1 ¼ ð34 13Þ extracted from the data for BðB ! K 1 ð1270ÞÞ, BðB ! K 1 ð1400ÞÞ, Bð( ! K 1 ð1270Þ# ( Þ, and Bð( ! K 1 ð1420Þ# ( Þ, gave ¼ð23 þ17 23 Þ . (ii) Second, from the study of the ratio for f 1 ð1285Þ! 0 and f 1 ð1285Þ! & 0 branching fractions, we have a twofold solution ¼ð19:4 þ4:5 4:6 Þ or ð51:1 þ4:5 4:6 Þ . Combining these two analyses, we thus obtain ¼ð19:4 þ4:5 4:6 Þ . We further compute the strange quark mass and strange quark condensate from the analysis of the f 1 ð1420Þ-f 1 ð1285Þ mass difference QCD sum rule, where the operator-product-expansion series is up to dimension six and to Oð 3 s ;m 2 s 2 s Þ accuracy. Using the average of the recent lattice results and the value that we have obtained as inputs, we get h ssi=h uu0:41 0:09. DOI: 10.1103/PhysRevD.84.034035 PACS numbers: 12.38.Lg, 12.38.Aw, 14.40.Be I. INTRODUCTION The f 1 ð1285Þ and f 1 ð1420Þ mesons with quantum num- ber J PC ¼ 1 þþ are the members of the 1 3 P 1 states in the quark model language and are mixtures of the pure octet f 8 and singlet f 1 , where the mixing is characterized by the mixing angle . The BABAR results for the upper bounds of B ! f 1 ð1285ÞK , f 1 ð1420ÞK were available recently [1]. The relative ratio of these two modes is highly sensitive to [2]. On the other hand, in the two-body B decay involving the K meson in the final state, the amplitude receives large corrections from the chiral enhancement a 6 term which is inversely proportional to the strange quark mass. The quark mass term mixes left- and right-handed quarks in the QCD Lagrangian. The spontaneous breaking of chiral symmetry from SUð3Þ L SUð3Þ R to SUð3Þ V is further broken by the quark masses m u;d;s when the baryon number is added to the three commuting conserved quanti- ties Q u , Q d , and Q s , respectively, the numbers of q q quarks for q ¼ u, d, and s. The nonzero quark condensate which signals dynamical symmetry breaking is the impor- tant parameter in QCD sum rules [3], while the magnitude of the strange quark mass can result in the flavor symmetry breaking in the quark condensate. In an earlier study h ssi=h uui 0:8 < 1 was usually taken. However, very re- cently the Jamin-Lange approach [4] together with the lattice result for f B s =f B [5] and also the Schwinger-Dyson equation approach [6] can give a central value larger than 1. In this paper, we shall embark on the study of the f 1 ð1420Þ and f 1 ð1285Þ mesons to determine the mixing angle , strange quark mass, and strange quark condensate. In Sec. II, we shall present detailed discussions on the determination of the mixing angle . Substituting the K 1 ð1270Þ-K 1 ð1400Þ mixing angle, which was extracted from the B ! K 1 and ( ! K 1 # ( data, to the Gell- Mann-Okubo mass formula, we can derive the value of . Alternatively, from the analysis of the decay ratio for f 1 ð1285Þ! 0 and f 1 ð1285Þ! & 0 , we have a more accurate estimation for . In Sec. III we shall obtain the mass difference QCD sum rules for the f 1 ð1420Þ and f 1 ð1285Þ to determine the magnitude of the strange quark mass. From the sum rule analysis, we obtain the constraint ranges for m s and as well as for h ssi. Many attempts have been made to compute m s using QCD sum rules and finite energy sum rules [713]. The running strange quark mass in the MS scheme at a scale of " 2 GeV is m s ¼ 101 þ29 21 MeV given in the particle data group (PDG) aver- age [14]. More precise lattice estimates have been recently obtained as m s ð2 GeVÞ¼ 92:2ð1:3Þ MeV in [15], m s ð2 GeVÞ¼ 96:2ð2:7Þ MeV in [16], and m s ð2 GeVÞ¼ 95:1ð1:1Þð1:5Þ MeV in [17]. These lattice results agree with strange scalar/pseudoscalar sum rule results which are m s 95ð15Þ MeV. In the present study, we study the m s from a new frame, the f 1 ð1420Þ-f 1 ð1285Þ mass difference sum rule, which may result in larger uncertainties due to the input parameters. Nevertheless, it can be a cross-check compared with the previous studies. Further using the very recent lattice result for m s ð2 GeVÞ¼ 93:6 1:0 MeV as the input, we obtain an estimate for the strange quark condensate. II. SINGLET-OCTET MIXING ANGLE OF THE 1 þþ NONET A. Definition In the quark model, a 1 ð1260Þ, f 1 ð1285Þ, f 1 ð1420Þ, and K 1A are classified in 1 þþ multiplets, which, in terms of spectroscopic notation n 2Sþ1 L J , are 1 3 P 1 p-wave mesons. Analogous to and 0 , because of SUð3Þ breaking effects, f 1 ð1285Þ and f 1 ð1420Þ are the mixing states of the pure octet f 8 and singlet f 1 , * [email protected] PHYSICAL REVIEW D 84, 034035 (2011) 1550-7998= 2011=84(3)=034035(6) 034035-1 Ó 2011 American Physical Society

nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

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Page 1: nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

1þþ nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

Kwei-Chou Yang*

Department of Physics, Chung Yuan Christian University, Chung-Li 320, Taiwan(Received 30 June 2011; published 22 August 2011)

Two strategies are taken into account to determine the f1ð1420Þ-f1ð1285Þ mixing angle �. (i) First,

using the Gell-Mann-Okubo mass formula together with the K1ð1270Þ-K1ð1400Þ mixing angle �K1¼

ð�34� 13Þ� extracted from the data for BðB ! K1ð1270Þ�Þ, BðB ! K1ð1400Þ�Þ, Bð� ! K1ð1270Þ��Þ,and Bð� ! K1ð1420Þ��Þ, gave � ¼ ð23þ17�23Þ�. (ii) Second, from the study of the ratio for f1ð1285Þ ! ��

and f1ð1285Þ ! �0� branching fractions, we have a twofold solution � ¼ ð19:4þ4:5�4:6Þ� or ð51:1þ4:5

�4:6Þ�.Combining these two analyses, we thus obtain � ¼ ð19:4þ4:5

�4:6Þ�. We further compute the strange quark

mass and strange quark condensate from the analysis of the f1ð1420Þ-f1ð1285Þ mass difference QCD sum

rule, where the operator-product-expansion series is up to dimension six and to Oð�3s ; m

2s�

2sÞ accuracy.

Using the average of the recent lattice results and the � value that we have obtained as inputs, we get

h�ssi=h �uui ¼ 0:41� 0:09.

DOI: 10.1103/PhysRevD.84.034035 PACS numbers: 12.38.Lg, 12.38.Aw, 14.40.Be

I. INTRODUCTION

The f1ð1285Þ and f1ð1420Þ mesons with quantum num-ber JPC ¼ 1þþ are the members of the 13P1 states in thequark model language and are mixtures of the pure octet f8and singlet f1, where the mixing is characterized by themixing angle �. The BABAR results for the upper bounds ofB� ! f1ð1285ÞK�, f1ð1420ÞK� were available recently[1]. The relative ratio of these twomodes is highly sensitiveto � [2]. On the other hand, in the two-body B decayinvolving the K meson in the final state, the amplitudereceives large corrections from the chiral enhancement a6term which is inversely proportional to the strange quarkmass. The quark mass term mixes left- and right-handedquarks in the QCD Lagrangian. The spontaneous breakingof chiral symmetry from SUð3ÞL � SUð3ÞR to SUð3ÞV isfurther broken by the quark masses mu;d;s when the baryon

number is added to the three commuting conserved quanti-ties Qu, Qd, and Qs, respectively, the numbers of q� �qquarks for q ¼ u, d, and s. The nonzero quark condensatewhich signals dynamical symmetry breaking is the impor-tant parameter in QCD sum rules [3], while the magnitudeof the strange quark mass can result in the flavor symmetrybreaking in the quark condensate. In an earlier studyh �ssi=h �uui � 0:8< 1 was usually taken. However, very re-cently the Jamin-Lange approach [4] together with thelattice result for fBs

=fB [5] and also the Schwinger-Dyson

equation approach [6] can give a central value larger than 1.In this paper, we shall embark on the study of the

f1ð1420Þ and f1ð1285Þ mesons to determine the mixingangle �, strange quark mass, and strange quark condensate.In Sec. II, we shall present detailed discussions on thedetermination of the mixing angle �. Substituting theK1ð1270Þ-K1ð1400Þ mixing angle, which was extractedfrom the B ! K1� and � ! K1�� data, to the Gell-

Mann-Okubo mass formula, we can derive the value of�. Alternatively, from the analysis of the decay ratio forf1ð1285Þ ! �� and f1ð1285Þ ! �0�, we have a moreaccurate estimation for �. In Sec. III we shall obtain themass difference QCD sum rules for the f1ð1420Þ andf1ð1285Þ to determine the magnitude of the strange quarkmass. From the sum rule analysis, we obtain the constraintranges forms and � as well as for h �ssi. Many attempts havebeen made to compute ms using QCD sum rules and finiteenergy sum rules [7–13]. The running strange quark mass

in the MS scheme at a scale of � � 2 GeV is ms ¼101þ29

�21 MeV given in the particle data group (PDG) aver-

age [14]. More precise lattice estimates have been recentlyobtained as msð2 GeVÞ ¼ 92:2ð1:3Þ MeV in [15],msð2 GeVÞ ¼ 96:2ð2:7Þ MeV in [16], and msð2 GeVÞ ¼95:1ð1:1Þð1:5Þ MeV in [17]. These lattice results agree withstrange scalar/pseudoscalar sum rule results which arems ’ 95ð15Þ MeV. In the present study, we study the ms

from a new frame, the f1ð1420Þ-f1ð1285Þ mass differencesum rule, which may result in larger uncertainties due tothe input parameters. Nevertheless, it can be a cross-checkcompared with the previous studies. Further using the veryrecent lattice result for msð2 GeVÞ ¼ 93:6� 1:0 MeV asthe input, we obtain an estimate for the strange quarkcondensate.

II. SINGLET-OCTET MIXING ANGLE �OF THE 1þþ NONET

A. Definition

In the quark model, a1ð1260Þ, f1ð1285Þ, f1ð1420Þ, andK1A are classified in 1þþ multiplets, which, in terms ofspectroscopic notation n2Sþ1LJ, are 1

3P1 p-wave mesons.

Analogous to and 0, because of SUð3Þ breaking effects,f1ð1285Þ and f1ð1420Þ are the mixing states of the pureoctet f8 and singlet f1,*[email protected]

PHYSICAL REVIEW D 84, 034035 (2011)

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jf1ð1285Þi ¼ jf1i cos�þ jf8i sin�;jf1ð1420Þi ¼ �jf1i sin�þ jf8i cos�:

(1)

In the present paper, we adopt

f1 ¼ 1ffiffiffi3

p ð �uuþ �ddþ �ssÞ; (2)

f8 ¼ 1ffiffiffi6

p ð �uuþ �dd� 2�ssÞ; (3)

where there is a relative sign difference between the �sscontents of f1 and f8 in our convention. From the Gell-Mann-Okubo mass formula, the mixing angle � satisfies

cos 2� ¼ 3m2f1ð1285Þ � ð4m2

K1A�m2

a1Þ3ðm2

f1ð1285Þ �m2f1ð1420ÞÞ

; (4)

where

m2K1A

¼ hK1AjH jK1Ai¼ m2

K1ð1400Þcos2�K1

þm2K1ð1270Þsin

2�K1; (5)

with H being the Hamiltonian. Here �K1is the

K1ð1400Þ-K1ð1270Þ mixing angle. The sign of the mixingangle � can be determined from the mass relation [14]

tan� ¼ 4m2K1A

�m2a1 � 3m2

f1ð1420Þ3m2

18

; (6)

where m218 ¼ hf1jH jf8i ’ ðm2

a1 �m2K1A

Þ2 ffiffiffi2

p=3< 0, we

find � > 0. Because of the strange and nonstrange lightquark mass differences, K1A is not the mass eigenstate andit can mix with K1B, which is one of the members in the11P1 multiplets. From the convention in [18] (see alsodiscussions in [19,20]), we write the two physical statesK1ð1270Þ and K1ð1400Þ in the following relations:

jK1ð1270Þi ¼ jK1Ai sin�K þ jK1Bi cos�K;jK1ð1400Þi ¼ jK1Ai cos�K � jK1Bi sin�K:

(7)

The mixing angle was found to be j�K1j � 33�, 57� in [18]

and� �37�,�58� in [21]. A similar range 35� & j�K1j &

55� was obtained in [22]. The sign ambiguity for �K1is due

to the fact that one can add arbitrary phases to j �K1Ai andj �K1Bi. This sign ambiguity can be removed by fixing thesigns of decay constants fK1A

and f?K1B, which are defined

by

h0j �c���5sj �K1AðP; Þi ¼ �ifK1AmK1A

�ðÞ� ; (8)

h0j �c���sj �K1BðP; Þi ¼ if?K1B���� �

�ðÞP

; (9)

where �0123 ¼ �1 and c � u or d. Following the con-vention in [20], we adopt fK1A

> 0, f?K1B> 0, so that �K1

should be negative to account for the observable BðB !K1ð1270Þ�Þ � BðB ! K1ð1400Þ�Þ [23,24]. Furthermore,from the data of � ! K1ð1270Þ�� and K1ð1400Þ�� decaystogether with the sum rule results for the K1A and K1B

decay constants, the mixing angle �K1¼ ð�34� 13Þ� was

obtained in [24]. Substituting this value into (4), we thenobtain �quad ¼ ð23þ17�23Þ� [25], i.e., �quad ¼ 0� � 40�.1

B. The determination of �

Experimentally, since K �K and K �K� are the dominantmodes of f1ð1420Þ, whereas f0ð1285Þ decays mainly to the4� states, this suggests that the quark content is primarily

s�s for f1ð1420Þ and n �n ¼ ðu �uþ d �dÞ= ffiffiffi2

pfor f1ð1285Þ.

Therefore, the mixing relations can be rewritten to exhibitthe n �n and s�s components which decouple for the ideal

mixing angle �i ¼ tan�1ð1= ffiffiffi2

p Þ ’ 35:3�. Let �� ¼ �i � �,we rewrite these two states in the flavor basis,2

f1ð1285Þ ¼ 1ffiffiffi2

p ð �uuþ �ddÞ cos ��þ �ss sin ��;

f1ð1420Þ ¼ 1ffiffiffi2

p ð �uuþ �ddÞ sin ��� �ss cos ��:

(10)

Since the f1ð1285Þ can decay into ��, we know thatf1ð1285Þ has the s�s content and � deviates from its idealmixing value. To have a more precise estimate for �, westudy the ratio of f1ð1285Þ ! �� and f1ð1285Þ ! �0�branching fractions. Because the electromagnetic (EM)interaction Lagrangian is given by

LI ¼ �A�EMðeu �u��uþ ed �d��dþ es �s��sÞ

¼ �A�EM

�ðeu þ edÞ

�u��uþ �d��d

2

þ ðeu � edÞ�u��u� �d��d

2þ es �s��s

�; (11)

with eu ¼ 2=3e, ed ¼ �1=3e, and es ¼ �1=3e being theelectric charges of u, d, and s quarks, respectively, weobtain

1Replacing the meson mass squared m2 by m throughout (4),we obtain �lin ¼ ð23þ17�23Þ�. The difference is negligible. Ourresult can be compared with that using �K1

¼ �57� into (4);one has �quad ¼ 52�.

2In PDG [14], the mixing angle is defined as � ¼ �� �i þ�=2. Comparing it with our definition, we have � ¼ �=2� ��.

KWEI-CHOU YANG PHYSICAL REVIEW D 84, 034035 (2011)

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Page 3: nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

Bðf1ð1285Þ ! ��ÞBðf1ð1285Þ ! �0�Þ ¼

� h�jes �s��sjf1ð1285Þih�jðeu � edÞð �u��u� �d��dÞ=2jf1ð1285Þi

�2 ¼

�m2f1�m2

m2f1�m2

�3

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}phase factor

¼� �e=3

2e=3þ e=3

�2

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}EM factor

� h�j�s��sjf1ð1285Þih�jð �u��u� �d��dÞ=2jf1ð1285Þi

�2�m2

f1�m2

m2f1�m2

�3

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}phase factor

� 4

9

�m�f�m�f�

�2tan2 ��

�m2f1�m2

m2f1�m2

�3; (12)

where f1 � f1ð1285Þ, and f� and f� are the decay con-stants of � and �, respectively. Here we have taken thesingle-pole approximation3:

h�j�s��sjf1ð1285Þih�jð �u��u� �d��dÞ=2jf1ð1285Þi

� m�f�gf1��

m�f�gf1��=ffiffiffi2

p sin ��

cos ��=ffiffiffi2

p �m�f�m�f�

�2tan ��: (13)

Using f� ¼ 209� 1 MeV, f� ¼ 221� 3 MeV [27], andthe current dataBðf1ð1285Þ ! ��Þ ¼ ð7:4� 2:6Þ � 10�4

and Bðf1ð1285Þ ! �0�Þ ¼ ð5:5� 1:3Þ% [14] as inputs,we obtain �� ¼ �ð15:8þ4:5

�4:6Þ�, i.e., the twofold solution � ¼ð19:4þ4:5

�4:6Þ� or ð51:1þ4:5�4:6Þ�. Combining with the analysis

� ¼ ð0–40Þ� given in Sec. II A, we thus find that � ¼ð19:4þ4:5

�4:6Þ� is much preferred and can explain experimentalobservables well.

III. MASS OF THE STRANGE QUARK

We proceed to evaluate the strange quark mass from themass difference sum rules of the f1ð1285Þ and f1ð1420Þmesons. We consider the following two-point correlationfunctions:

���ðq2Þ ¼ iZ

d4xeiqxh0jTðj�ðxÞjy�ð0ÞÞj0i¼ ��1ðq2Þg�� þ�2ðq2Þq�q�; (14)

�0��ðq2Þ ¼ i

Zd4xeiqxh0jTðj0�ðxÞj0y� ð0ÞÞj0i

¼ ��01ðq2Þg�� þ�0

2ðq2Þq�q�: (15)

The interpolating currents satisfying the relations,

h0jjð0Þ� ð0Þjfð0Þ1 ðP; Þi ¼ �iffð0Þ1m

fð0Þ1�ðÞ� ; (16)

are

j� ¼ cos�jð1Þ� þ sin�jð8Þ� ; (17)

j0� ¼ � sin�jð1Þ� þ cos�jð8Þ� ; (18)

where

jð1Þ� ¼ 1ffiffiffi3

p ð �u���5uþ �d���5dþ �s���5sÞ; (19)

jð8Þ� ¼ 1ffiffiffi6

p ð �u���5uþ �d���5d� 2�s���5sÞ; (20)

and we have used the shorthand notations for f1 �f1ð1285Þ and f01 � f1ð1420Þ. In the massless quark limit,we have �1 ¼ q2�2 and �0

1 ¼ q2�02 if one neglects the

axial-vector anomaly.4 Here we focus on �ð0Þ1 since it

receives contributions only from axial-vector (3P1) mesons,

whereas �ð0Þ2 contains effects from pseudoscalar mesons.

The lowest-lying fð0Þ1 meson contribution can be approxi-

mated via the dispersion relation as

m2

fð0Þ1

f2fð0Þ1

m2

fð0Þ1

� q2¼ 1

Z sfð0Þ

0

0ds

Im�ð0ÞOPE1 ðsÞ

s� q2; (21)

where �ð0ÞOPE1 is the QCD operator-product-expansion

(OPE) result of �ð0Þ1 at the quark-gluon level [20], and s

fð0Þ1

0

is the threshold of the higher resonant states. Note that thesubtraction terms on the right-hand side of (21), which arepolynomials in q2, are neglected since they have no con-tributions after performing the Borel transformation. Thefour-quark condensates are expressed as

h0j �q�iaq �q�i

aqj0i ¼ �a21

16N2c

Trð�i�iÞ

� TrðaaÞh �qqi2; (22)

3The following approximation was used in [26]:

h�j�s��sjf1ð1285Þih�jð �u��u� �d��dÞ=2jf1ð1285Þi � 2 tan ��:

4Considering the anomaly, the singlet axial-vector current issatisfied with

@�jð1Þ� ¼ 1ffiffiffi3

p ðmu �uuþmd�ddþms �ssÞ þ 3�s

4�G ~G:

1þþ NONET SINGLET-OCTET MIXING ANGLE, . . . PHYSICAL REVIEW D 84, 034035 (2011)

034035-3

Page 4: nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

where a2 ¼ 1 corresponds to the vacuum saturation ap-proximation. In the present work, we have � ¼ �� and

���5, for which we allow the variation a2 ¼ �2:9–3:1

[9,28,29]. For �ð0ÞOPE1 , we take into account the terms

with dimension 6, where the term with dimension ¼ 0(D ¼ 0) is up toOð�3

sÞ, withD ¼ 2 (which is proportionalto m2

s) up to Oð�2sÞ and with D ¼ 4 up to Oð�2

sÞ. Notethat such radiative corrections for terms can read from[30–32]. We do not include the radiative correction to theD ¼ 6 terms since all the uncertainties can be lumped intoa2. Note that such radiative corrections for terms with

dimensions ¼ 0 and 4 are the same as the vector mesoncase and can read from [30,31].Further applying the Borel (inverse-Laplace) transfor-

mation,

B ½fðq2Þ� ¼ limn!1

�q2!1�q2=n2¼M2fixed

1

n!ð�q2Þnþ1

�d

dq2

�nfðq2Þ; (23)

to both sides of (21) to improve the convergence of theOPE series and further suppress the contributions fromhigher resonances, the sum rules thus read

f2f1m2f1e�m2

f1=M2 ¼

Z sf10

0

sdse�s=M2

4�2

�1þ �sð

ffiffiffis

p Þ�

þ F3

�2sð

ffiffiffis

p Þ�2

þ ðF4 þ F04cos

2�Þ�3sð

ffiffiffis

p Þ�3

�� ðcos�� ffiffiffi

2p

sin�Þ2½ �msð��Þ�2

�Z s

f10

0ds

1

2�2e�s=M2

�1þ

�H1 ln

s

�2�þH2

��sð��Þ

�þ

�H3aln

2 s

�2�þH3b ln

s

�2�þH3c �H3a�

2

3

���sð��Þ

�2�� 1

12

�1� 11

18

�sðMÞ�

���s

�G2

��

�4

27

�sðMÞ�

þ�� 257

486þ 4

3�ð3Þ � 2

27 1�E

��2sðMÞ�2

� Xqi�u;d;s

h �mi �qiqii þ 1

3ð ffiffiffi

2p

cos�þ sin�Þ2�2a1 �mqh �qqi � 352��s

81M2a2h �qqi2

�þ 1

3ðcos�� ffiffiffi

2p

sin�Þ2

��2a1 �msh�ssi � 352��s

81M2a2h �ssi2

�; (24)

f2f01m2

f01e�m2

f01

=M2

¼Z s

f01

0

0

sdse�s=M2

4�2

�1þ �sð

ffiffiffis

p Þ�

þ F3

�2sð

ffiffiffis

p Þ�2

þ ðF4 þ F04sin

2�Þ�3sð

ffiffiffis

p Þ�3

�þ ðsin�þ ffiffiffi

2p

cos�Þ2½ �msð��Þ�2

�Z s

f01

0

0ds

1

2�2e�s=M2

�1þ

�H1 ln

s

�2�þH2

��sð��Þ

�þ

�H3aln

2 s

�2�þH3b ln

s

�2�þH3c �H3a�

2

3

���sð��Þ

�2�� 1

12

�1� 11

18

�sðMÞ�

���s

�G2

��

�4

27

�sðMÞ�

þ�� 257

486þ 4

3�ð3Þ � 2

27 1�E

��2sðMÞ�2

� Xqi�u;d;s

h �mi �qiqii þ 1

3ð ffiffiffi

2p

sin�� cos�Þ2�2a1 �mqh �qqi � 352��s

81M2a2h �qqi2

�þ 1

3ðsin�þ ffiffiffi

2p

cos�Þ2

��2a1 �msh�ssi � 352��s

81M2a2h �ssi2

�; (25)

where

F3¼1:9857�0:1153nf ’1:6398 fornf¼3; F4¼�6:6368�1:2001nf�0:0052n2f ’�10:2839 for nf¼3;

F04¼�1:2395�; H1¼� 8

81 2

1¼�2; H2¼2

9 2þ4 2

��1

1

��2

2

��8

9 2

1�4 1’3:6667;

H3a¼4:2499; H3b¼�23:1667; H3c¼29:7624;

�mqh �qqi�1

2ð �muh �uuiþ �mdh �ddiÞ; h �qqi2�1

2ðh �uui2þh �ddi2Þ; a1¼1þ7

3

�sðMÞ�

þ�85

6�7

6 1�E

��2sðMÞ�2

; (26)

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Page 5: nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

with 1 ¼ ð2nf � 33Þ=6, 2 ¼ ð19nf � 153Þ=12, �1 ¼ 2,�2 ¼ 101=12� 5nf=18, and nf ¼ 3 being the number offlavors and � ¼ 1, and 0 for f1 (singlet) and f8 (octet),respectively [32]. In the calculation the coupling constant�sð

ffiffiffis

p Þ in Eqs. (24) and (25) can be expanded in powers of�sðMÞ:�sð

ffiffiffis

p Þ�

¼ �sðMÞ�

þ 1

2 1 ln

s

M2

��sðMÞ�

�2

þ�1

2 2 ln

s

M2þ 1

4 2

1ln2 s

M2

���sðMÞ�

�3

þ� 3

2ln

s

M2þ 5

8 1 2ln

2 s

M2þ 1

8 3

1ln3 s

M2

���sðMÞ�

�4 þ � � � ; (27)

where 3 ’ �20:1198. Using the renormalization-groupresult for the m2

s term given in [31], we have expanded thecontribution to the order Oð�2

sm2sÞ at the subtraction scale

�2� ¼ 2 GeV2 for which the series has better convergencethan at the scale 1 GeV2; however, the convergence of theseries has no obvious change if using a higher referencescale. As in the case of flavor-breaking � decay, the D ¼ 2series converges slowly; nevertheless, we have checkedthat this term, which intends to make the output ms to besmaller in the fit, is suppressed due to the fact that the masssum rules for f1ð1285Þ and f1ð1420Þ are obtained byapplying the differential operator M4@ ln=@M2 to bothsides of (24) and (25), respectively. Nevertheless, the dif-ferential operator will instead make the D ¼ 4 term con-taining msh �ssi become much more important than the m2

s

term in determining the f1ð1285Þ-f1ð1420Þ mass differ-ence although they are the same order in magnitude.

In the numerical analysis, we use�ð3ÞNLOQCD ¼ 0:360 GeV,

corresponding to �sð1 GeVÞ ¼ 0:495, �ð4ÞNLOQCD ¼

0:313 GeV, and the following values (at the scale � ¼1 GeV) [9,28,29,33]:

��s

�Ga

��Ga��

�¼ ð0:009� 0:007Þ GeV4;

h �mq �qqi ¼ �f2�þm2

�þ=4;

h �qqi2 ’ ð�0:247Þ6 GeV6;

h�ssi ¼ ð0:30–1:3Þh �qqi;a2 ¼ �2:9–3:1;

(28)

where the value of h �qqi2 corresponds to ðmu þmdÞ�ð1 GeVÞ ’ 11 MeV, and we have cast the uncertainty ofh �qqi2 to a2 in the D ¼ 6 term. We do not consider theisospin breaking effect between h �uui and h �ddi sinceh �ddi=h �uui � 1 � �0:007 [34] is negligible in the present

analysis. The threshold is allowed by sf10 ¼ 2:70�0:15 GeV2 and determined by the maximum stability ofthe mass sum rule. For an estimate on the threshold differ-

ence, we parametrize in the form ðffiffiffiffiffiffisf01

0

q�

ffiffiffiffiffiffisf10

qÞ=

ffiffiffiffiffiffisf10

�� ðmf01�mf1Þ=mf1 , with � ¼ 1:0� 0:3. In other words,

we assign a 30% uncertainty to the default value. Wesearch for the allowed solutions for strange quark massand the singlet-octet mixing angle � under the followingconstraints: (i) Comparing with the observables, the errorsfor the mass sum rule results of the f1ð1285Þ and f1ð1420Þin the Borel window 0:9 GeV2 M2 1:3 GeV2 are con-strained to be less than 3% on average. In this Borelwindow, the contribution originating from higher reso-nances (and the continuum), modeled by

1

Z 1

sfð0Þ

0

dse�s=M2Im�ð0ÞOPE

1 ðsÞ; (29)

is about less than 40% and the highest OPE term (withdimension six) at the quark level is no more than 10%.(ii) The deviation between the f1ð1420Þ-f1ð1285Þ massdifference sum rule result and the central value of thedata [14] is within 1� error: jðmf0

1�mf1Þsum rule �

144:6 MeVj 1:5 MeV. The detailed results are shownin Table I. We also check that if by further enlarging the

uncertainties of sf10 and �, e.g. 25%, the changes of results

can be negligible. We obtain the strange quark mass withlarge uncertainty: msð1 GeVÞ ¼ 106:3� 35:1 MeV[i.e. msð2 GeVÞ ¼ 89:5� 29:5 MeV] and h �ssi=h �uui ¼0:56� 0:25 corresponding to � ¼ ð19:4þ4:5

�4:6Þ�, where the

values and ms and h�ssi are strongly correlated.Further accounting for the average of the recent lattice

results [15–17]: msð2 GeVÞ ¼ 93:6� 1:0 MeV and usingthe � value that we have obtained as the inputs, we geth�ssi=h �uui ¼ 0:41� 0:09 which is less than 1 and incontrast to the Schwinger-Dyson equation approach in[6] where the ratio was obtained as ð1:0� 0:2Þ3. Ourprediction is consistent with the QCD sum rule result ofstudying the scalar/pseudoscalar two-point function in [35]where the authors obtained h �ssi=h �uui ¼ 0:4–0:7, depend-ing on the value of the strange quark mass.

TABLE I. The fitting results in the f1ð1284Þ-f1ð1420Þ mass difference sum rules. In fit II, wehave taken the average of the recent lattice results forms, which is rescaled to 1 GeVas the input.

msð1 GeVÞ h�ssi=h �uui hð�s=�ÞG2i a2

Fit I 106:3� 35:1 0:56� 0:25 0:0106� 0:0042 0:89� 0:62Fit II [124:7� 1:3] 0:41� 0:09 0:0108� 0:0037 0:95� 0:45

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Page 6: nonet singlet-octet mixing angle, strange quark mass, and strange quark condensate

IV. SUMMARY

We have adopted two different strategies for determin-ing the mixing angle �: (i) Using the Gell-Mann-Okubomass formula and the K1ð1270Þ-K1ð1400Þ mixingangle �K1

¼ ð�34� 13Þ� which was extracted from the

data for BðB ! K1ð1270Þ�Þ, BðB ! K1ð1400Þ�Þ, Bð� !K1ð1270Þ��Þ, and Bð� ! K1ð1420Þ��Þ, the result is � ¼ð23þ17�23Þ�. (ii) On the other hand, from the analysis of the

ratio of Bðf1ð1285Þ ! ��Þ and Bðf1ð1285Þ ! �0�Þ, wehave �� ¼ �i � � ¼ �ð15:8þ4:5

�4:6Þ�, i.e., � ¼ ð19:4þ4:5�4:6Þ� or

ð51:1þ4:5�4:6Þ�. Combining these two analyses, we deduce the

mixing angle � ¼ ð19:4þ4:5�4:6Þ�.

We have estimated the strange quark mass and strangequark condensate from the analysis of the f1ð1420Þ-

f1ð1285Þ mass difference QCD sum rule. We have ex-panded the OPE series up to dimension six, where theterm with dimension zero is up to Oð�3

sÞ, withdimension ¼ 2 up to Oðm2

s�2sÞ and with dimension ¼ 4

terms up to Oð�2sÞ. Further using the average of the recent

lattice results and the � value that we have obtained as theinputs, we get h �ssi=h �uui ¼ 0:41� 0:09.

ACKNOWLEDGMENTS

This research was supported in part by the NationalCenter for Theoretical Sciences and the National ScienceCouncil of R.O.C. under Grant No. NSC99-2112-M-003-005-MY3.

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