Peng Guo Physics Department & NTC Indiana University ... · ψ(4040) ψ(4160) ψ(4415) FIG. 2: 1−− spectrum. Black lines are experimental states at various levels of conﬁrmation,

Peng Guo

Physics Department & NTCIndiana University - Bloomington, U.S.A.

Collaborators:A.P. Szczepaniak, H. Matevosyan, R. Mitchell and M. Shepherd

Rescattering effect in decay

JLab, Jan.25, 2010

Tuesday, January 26, 2010

How to improve isobar model? Unitarity + Analyticity

Preliminary result: rho-pi puzzle, effect of inelastic channels

Conclusion

Outline I. Charmonium Spectroscopy:

microscopic approach

Hybrid Spectrum in Coulomb Gauge QCD

Charmonium in Quark matter

II. Charmonium Spectroscopy data analysis


I. Charmonium Spectroscopy: microscopic approach


Charmonium Spectroscopy

!

JPC

!

P = ("1)L+1,C = ("1)

L+S

Missing states from naive quark model: Exotics

!

0"",0

+",1"+,2

+",! ?

Hall D at JLab


Charmonium Spectroscopy from Lattice QCD

Naive quark potential model

17

3000

3250

3500

3750

4000

4250

4500

4750

5000

5250

5500

A1 T1 T2 E A2

no ops

!5 sm, us a0(2)"#smb1"#sm$"Bsm, us

b1"#sm$"Bsm

%"Dsmb1"#sm$"Bsm, us

0, 4 ... 1, 3, 4 ... 2, 3, 4 ... 2, 4 ... 3 ...

!4!5 sm, us

b1"#sm, us$"Bsm, us

3000

3250

3500

3750

4000

4250

4500

4750

5000

5250

5500q.m. D-wavesq.m. S-waves

this workexpt. (PDG)

q.m. G-waves

0-+ 1-+ 2-+ 3-+ 4-+

exotic exotic

3000

3250

3500

3750

4000

4250

4500

4750

5000

5250



q.m. G-waves

0-+ 1-+ 2-+ 3-+ 4-+

exotic exotic

A1 3024(2), 3692(24), 4291(48), 4333(61)

T1 4327(36), 4696(103), 5017(103)

T2 3946(48), 4376(24), 4920(48)

E 3879(42), 4418(61)

FIG. 12: J!+. Top pane: extracted state masses by zero-momentum lattice irrep. Color coding indicates continuum spinassignment (black=0, red=1, green=2, blue=3, grey=undetermined). Lower panes: Comparison with experimental masses andquark potential model masses. Hatching indicates that the spin assignment is not based upon a Z-analysis. Lower left paneshows the 4!+ lighter hypothesis. Lower right pane shows the 1!+ lighter hypothesis. Tabulated masses in MeV

be clearly seen in the three graphs of figure 13 in [24]. Since the wavefunctions of higher and higher excited statessample larger distances we might expect to feel even larger e!ects in our study. We suspect that quenching may be amajor contributor to our systematically high masses.

VI. CONCLUSIONS

We have investigated the use of a large basis of interpolating fields on the extraction of excited charmonium mesonstates in many JPC channels. We propose an application of a variational method that systematically selects theparameter t0 to ensure the best possible description of the data at all times greater than t0. We use the eigenvalues(principal correlators) to determine the mass spectrum and the eigenvectors to determine the overlaps of our inter-polating fields on to the states extracted. These numbers are used herein to aid continuum spin assignment and arerequired quantities for study of three-point functions with the aim of extracting transition matrix elements.

T.Barnes, S. Godfrey, E.S. Swanson

J. Dudek, R.G. Edwards, N. Mathur, D.G. Richards

8

3000

3200

3400

3600

3800

4000

4200

4400

4600

ma

ss / M

eV

3106±2

3864±19

3746±18

4283±77

4400±60

13S!

13D!

23S!

33S!

23D!

43S!hyb

J/!

!(3686)

!(3770)

Y(4260)

13S!

23S!

13D!

hyb?

33S!

!(4040)

!(4160)

!(4415)

FIG. 2: 1!! spectrum. Black lines are experimental statesat various levels of confirmation, blue bars are lattice stateswith quark model assignments as described in the text. Alsoshown are quark model predictions and the hybrid predictionof the Coulomb gauge model.

is located at a mass near 4.4 GeV, and since the b1 ! B

operator has a quark spin-singlet component it seems rea-sonable to anticipate overlap with this state.

Finally we will consider applying our methods to chan-nels that in continuum house JPC exotic states notpresent in the simple cc quark model. The most straight-forward choice is A+!

1 which only houses even spins andhence is always exotic with this PC. The simplest op-erator transforming in this way is !"0! - we find thatunsmeared correlators containing this operator are con-sistent with zero, which matches with the fact that thisoperator contains no gluonic field to generate a hybridstate. By smearing this operator using a function of thegauge-covariant laplacian we introduce the gluonic fieldand have the possibility of overlap with a hybrid state.Using this operator and smeared and unsmeared a1 ! B

operators we extract a state at 4465(65) MeV which hasclear overlaps on to all three operators. The first excitedstate is much higher in mass, at 5570(270) MeV.

A less trivial channel is T!+1 which houses exotic

1!+, 3!+ but also non-exotic 4!+ which is realised in thequark model as 1G4. In [11] it proved to be not possible todecisively state whether the ground state in this channelwas indeed the exotic 1!+ state or a 4!+ nonexotic, herewe will consider this again using our model-dependentoverlap comparison. In Table VI we present the extractedoverlaps using an eight dimensional basis of operators.For the ground state we see rather large overlaps withthe quark-model forbidden operators # ! B, #2 ! B sug-gesting that it may well be the exotic hybrid 1!+ state.The first excited state has overlaps consistent with zero

for all operators except the smeared b1 !" and possiblythe a0(2) ! ". This may well be a signal for a latticeartifact 4!+ state, whose spin-singlet (1G4) nature couldappear through overlap with b1!" which has a dominantspin-singlet piece in its non-relativistic reduction.

2+! exotics appear in (T2, E)+!. The T2 receivescontributions also from non-exotic 3+! and indeed theground state in that channel is identified as such. Thefirst excited state in T +!

2 matches with the ground statein E+! and in both cases large overlaps with the a1 !B

operator are seen strongly suggesting that this is the 2+!

exotic at a mass of 4620(60).

V. FINITE MOMENTUM

The non-relativistic quark-model states as constructeddo not transform covariantly under boosts, but do trans-form properly under rotations in three-dimensions. Thiscan lead to overlap on to more states than are allowedby Lorentz symmetry. For example, consider the opera-tor !"µ$!! - insisting upon Poincare invariance one hasonly the following overlaps

#0|!"µ$!!|0++(%p, r)$ = Zgµ! + Z "pµp!

#0|!"µ$!!|1++(%p, r)$ = Z&µ!"#p" %#(%p, r)

#0|!"µ$!!|2++(%p, r)$ = Z %µ!(%p, r),

so that if, as we do in the lattice calculation, one considersonly the spatial derivatives, one has overlaps

#0|!"i$j!|0++(%p, r)$ = Z'ij + Z "pipj

#0|!"i$j!|1++(%p, r)$ = Z&ijk!

pk %0(%p, r) & E %k(%p, r)"

#0|!"i$j!|2++(%p, r)$ = Z %ij(%p, r).

But note that this is not the most general set allowed bythree-dimensional rotations, parity and charge conjuga-tion, giving up on boost invariance we also are allowedoverlaps

#0|!"i$j!|0!+(%p, r)$ = Z&ijkpk

#0|!"i$j!|2!+(%p, r)$ = Z&ikl %jk(%p, r)pl.

Within the quark model state construction, we can ex-plicitly compute these overlaps at finite momentum find-ing

Z(3P0) =#

2E$p'

(

$

q2dq(2%)3

1#2

q3m2

q)(q)

Z "(3P0) =#

2E$p'

(

$

q2dq(2%)3

83q

%

1 & q2

4m2q& p2

16m2q

&

)(q)

Z(3P1) =#

2E$p'

( ME2

!p

$

q2dq(2%)3

4#3q%

1 + q2

4m2q

+ p2

16m2q

&

)(q)

Z(3P2) =#

2E$p'

(

$

q2dq(2%)3

8#6q%

1 + q2

20m2q& p2

16m2q

&

)(q)

Z(1S0) =#

2E$p'

(

$

q2dq(2%)3

#2

3q2

m2q)(q)

Z(1D2) =#

2E$p'

(

$

q2dq(2%)3

2#15

q2

m2q)(q),

!

V (r) = "4

3

#s

r+ kr +V

LS+V

SS+V

T


!

JQQ

!

+!

Jg

Charmonium Spectroscopy in Coulomb Gauge QCD


17

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3250

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3750

4000

4250

4500

4750

5000

5250

5500

A1 T1 T2 E A2

no ops

!5 sm, us a0(2)"#smb1"#sm$"Bsm, us

b1"#sm$"Bsm

%"Dsmb1"#sm$"Bsm, us

0, 4 ... 1, 3, 4 ... 2, 3, 4 ... 2, 4 ... 3 ...

!4!5 sm, us

b1"#sm, us$"Bsm, us

3000

3250

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4500

4750

5000

5250



q.m. G-waves

0-+ 1-+ 2-+ 3-+ 4-+

exotic exotic

3000

3250

3500

3750

4000

4250

4500

4750

5000

5250



q.m. G-waves

0-+ 1-+ 2-+ 3-+ 4-+

exotic exotic

A1 3024(2), 3692(24), 4291(48), 4333(61)

T1 4327(36), 4696(103), 5017(103)

T2 3946(48), 4376(24), 4920(48)

E 3879(42), 4418(61)

FIG. 12: J!+. Top pane: extracted state masses by zero-momentum lattice irrep. Color coding indicates continuum spinassignment (black=0, red=1, green=2, blue=3, grey=undetermined). Lower panes: Comparison with experimental masses andquark potential model masses. Hatching indicates that the spin assignment is not based upon a Z-analysis. Lower left paneshows the 4!+ lighter hypothesis. Lower right pane shows the 1!+ lighter hypothesis. Tabulated masses in MeV

be clearly seen in the three graphs of figure 13 in [24]. Since the wavefunctions of higher and higher excited statessample larger distances we might expect to feel even larger e!ects in our study. We suspect that quenching may be amajor contributor to our systematically high masses.

VI. CONCLUSIONS

We have investigated the use of a large basis of interpolating fields on the extraction of excited charmonium mesonstates in many JPC channels. We propose an application of a variational method that systematically selects theparameter t0 to ensure the best possible description of the data at all times greater than t0. We use the eigenvalues(principal correlators) to determine the mass spectrum and the eigenvectors to determine the overlaps of our inter-polating fields on to the states extracted. These numbers are used herein to aid continuum spin assignment and arerequired quantities for study of three-point functions with the aim of extracting transition matrix elements.

!

J = Jg + LQQ + SQQFour degenerate states for spin-independent interaction

S-Wave:

!

SQQ = 0

!

SQQ =1

!

J"1# 4

!

JPC :1"" = 0"+ =1"+ = 2"+

!

JgPgCg :1+"

!

JPC :1+" = 0++ =1++ = 2++

!

JgPgCg :1""

P.Guo, A.P.Szczepaniak, G.Galata, A.Vassallo and E.SantopintoPhys. Rev. D 78, 056003 (2008), Phys. Rev. D 77, 056005 (2008)

J. Dudek, R.G. Edwards, N. Mathur, D.G. Richards


Charmonium at Finite density

!

"#$%!!

Melting Charmonium in quark matter


II. Charmonium data analysis(Current work)


Isobar Model: quasi two-body decays1

3

2


Isobar Model: quasi two-body decays1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)


Isobar Model: quasi two-body decays

How to extract coupling to resonancesHow one establishes new resonances

Answer?It’s important to construct amplitudes which contain all the known physics

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)





1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)





1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)





1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)


rho-pi puzzle

Experiment

vs

PQCD

Current analysis: J/Psi -> 3 pion

What do we learn from Dalitz plot ?



Subenergy Unitarity

++

1’

2’

3

2

11

2

3

1’

2’1’

2’

1

2

3

=

(1’2’)

(2’3)

(31’)

Disc(12)

T

sth

s

Dispersion Relation

(I) Solve double integral equations

(II) Parametrize cuts which are associated with rescattering


Single integral equation: suitable for data fitting

!" #$%!"& #

'() #

*

*

$%*"& #

!" # $%!"& #'

() *

+

+

R.Pasquier & J.Y.Pasquier Phys.Rev.170:1294,1968

(I) Solving double integral equations


0.5 1 1.5 2 2.5 3s1/2

ij (GeV)

-1

-0.5

0

0.5

1

Re[

T!(s

,s ij)]

Breit-WignerRescattering

J/ " #" $%%&' " #"(

0.5 1 1.5 2 2.5 3s1/2

ij (GeV)

0

0.5

1

1.5

2

Im[T

!(s,s ij)]


J/ " #" $%%&' " #"(

1 2 3s1/2

ij (GeV)

-0.08

-0.04

0

0.04

0.08

Re[

T!(s

,s ij)]


"#S)$ %$ "&&'( $ %$)

1 2 3s1/2

ij (GeV)

0

0.05

0.1

0.15

0.2

Im[T

!(s,s ij)]


"2S)# $# "%%&' # $#(



Short distance Long distance


Short distance Long distanceSummary

Long distance final states rescattering cannot be the cause of rho-pi puzzle.


Does rescattering effect help explaining the Dalitz plot?

1. Improve the single Breit-Wigner parametrization.

2. Not quite the solution.

!4" Contamination from J/#!($)e!e" is eliminated bythe requirement that the sum of the deposited energies of thetwo charged pions in the BSC is less than 2 GeV. Figure 2

shows the scatter plot of Esc! versus Esc

" , where Esc! and Esc

"

are the deposited energies of the %! and %" in the BSC,respectively. This criterion has almost no effect on %!%"%0

events.After the above requirements, 219691 %!%"%0 candi-

dates are selected. Remaining backgrounds are evaluated us-ing two different Monte Carlo !MC" simulations. In the first,specific background channels, shown in Table I, are gener-ated. The total background from these channels in the se-lected %!%"%0 events is determined to be less than 1%. Thesecond simulation uses 30 million inclusive J/# MC eventsgenerated with the LUND model &16'. After normalizing theselected background events to 58 million J/# events, 3799background events are obtained, yielding a contamination of1.7%. In this paper, the latter background estimate is used tocorrect the branching fraction, giving a correction factor of(98.3#1.7)%.The Dalitz plot of m%!%0 versus m%"%0 is shown in Fig.

3. Three bands are clearly visible in the plot, which corre-spond to J/#!(!%", J/#!(0%0, and J/#!("%!. Thecorresponding histograms of m%!%0, m%!%", and m%"%0 are

shown in Fig. 4. From the Dalitz plot !Fig. 3", we see thatJ/#!%!%"%0 is strongly dominated by (% . Therefore, thedetection efficiency is determined using the RHOPI &17' gen-erator with SIMBES and is found to be 17.83%. Monte Carlosimulations using other generators to represent the structurein the Dalitz plot provide very similar detection efficiencies.

B. Systematic error analysis

In this analysis, the systematic error on the branchingfraction comes mainly from the following sources:

FIG. 3. The Dalitz plot for J/#!%!%"%0.FIG. 2. Plot of Esc

! versus Esc" ; the solid line is for Esc

! !Esc"

$2 GeV.

TABLE I. Background contributions from different decay chan-

nels. Here Nbkg is the number of events generated, and Nbkgnorm is the

number of background events selected, normalized by the branch-

ing fractions quoted in Ref. &15'.

Decay channel Nbkg Nbkgnorm

J/#!K*!K"!c.c. (K!K"%0) 100 000 773

J/#!K*!K"!c.c. (KKS0%) 50 000 153

J/#!K*0K0!c.c. (KKS0%) 50 000 129

J/#!$)! ($$() 100 000 158 FIG. 4. The distributions of the invariant mass of two pions for

!a" m%!%0, !b" m%!%", and !c" m%"%0.

MEASUREMENT OF THE BRANCHING FRACTION OF J/#!%!%"%0 PHYSICAL REVIEW D 70, 012005 !2004"

012005-3

BES Collaboration Phys.Rev.D70:012005,2004

0.5 1 1.5 2 2.50

200

400

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1400

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1800

310!Hist


(II) Parametrize cuts which are associated with rescattering + Inelastic channel

0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

Sin(delta)

0.6 0.8 1 1.2 1.4 1.6 1.80

20

40

60

80

100

120

140

160

180

Phase shift

0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

Inelasticity

0 0.5 1 1.5 2 2.5 30

200

400

600

800

1000

1200

1400

1600

1800

310!

ampsq

0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

Sin(delta)

0.6 0.8 1 1.2 1.4 1.6 1.80

20

40

60

80

100

120

140

160

180

Phase shift

0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

Inelasticity

0 0.5 1 1.5 2 2.5 30

200

400

600

800

1000

1200

1400

1600

1800

310!

ampsq

Coupled channels:

=Disc

0.5 1 1.5 2 2.50

200

400

600

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1000

1200

1400

1600

1800

310!Hist


0.5 1 1.5 2 2.50

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1800

310!Hist

5. Summary of Cut from rescattering

First the discontinuity of Tj from rescattering

Tj(s, sij) ≡1

2(sij − 4m2

π)j2 |pk|L

Mππj (sij)

(sij − 4m2π)j

gj(s, sij), Im1

Mππj (sij)

= − 1

128π2

�

1− 4m2π

sij.

The discontinuity of Tj(s, sij) goes to

Discs12 [Tj12(s, s12)]−

�1− 4m2

πs12

128π2Mππ

j12(s12 − i0)Tj12(s, s12 + i0) (72)

=

�1− 4m2

πs12

128π2Mππ

j12(s12 − i0)�

j31=odd

� 1

−1

d cos θ3

�(2j12 + 1)(2j31 + 1)dj12

10 (θ3)dj3110 (θ2)Tj31(s, s31 + i0).

+

(12)1

2

3

(12)1 1

2

3(23)

(A)

(B)

(C)

(D)

2

3

For specific channel, for instance (12) pair channel, the Tj12(s, s12) has the following

analytic structure

(A) cut starting from (√

s−m3)2 to −∞.

(B) cut starting from (m1 −m2)2 to −∞.

(C) cut starting from 0 to −∞.

(D) cut starting from (m1 + m2)2 to ∞.

FIG. 12: Cuts diagram of triangle diagram crossing cuts of Tj12(s, s12)

s=P 2

s 12

m 3

m 2

s 12

s 12

m 1

s 12

s=P 2

(A)

m 3 m 2

m 1(B) (C) (D)

s 12s 12

s 12

m 1

m 2

s 23+

s 12

m 3

s 12(A)

(B)(C)

(D)

166

K

K

K

K

+


0.5 1 1.5 2 2.50

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310!Hist

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310!Hist

0.5 1 1.5 2 2.50

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310!Hist


Summary

Long distance final states rescattering cannot be the cause of rho-pi puzzle.

Inelastic KK channel contribution is important at 1.6 GeV region in J/Psi decay channel.


Documents

Peng Guo Physics Department & NTC Indiana University ... · ψ(4040) ψ(4160) ψ(4415) FIG. 2: 1−− spectrum. Black lines are experimental states at various levels of conﬁrmation,