International Conference on QCD and Hadronic Physics, Beijing, China , 16-20 June 2005

MATCHING MESON MATCHING MESON RESONANCES TO OPE IN QCDRESONANCES TO OPE IN QCD

A.A. Andrianov*#, V.A. Andrianov*, S.S. Afonin** and D. Espriu**

# INFN, Sezione di Bologna * St. Petersburg State University** Universitat de Barcelona

Based on S.A., A.A., V.A., D.E., JHEP 0404, 039 (2004)

Triggered by M. A. Shifman, hep-ph/0009131 S. Beane, Phys. Rev. D64 (2001) 116010 M. Golterman and S. Peris, Phys. Rev. D67 (2003) 096001

Linear mass spectrum withuniversal slope

Large-Nc QCD

IntroductionIntroduction

narrow resonances

In two-point correlators of quark currents:

Sum of narrow resonances? Operator Product

Expansion (OPE)

Constraints on meson mass spectrum?

Hadron string Nonlinear corrections to mass spectrum?

2 4

20 12 2

( ) exp ( ) (0)

( )

( )

cJ

J JJ

Jn

NQ d x iQx q q x q q

Z nD D Q

Q m n

;,,, AVPSJ ;,,, 55 γγγγi μμΓ const.JJ DD 10 ,

Two-point correlators in Euclidean space (q denotes u- or d-quarks):

)(2)( 2 nFnZ VAVA — residues —are related to some observables from

,Vee ... τντ),()(2)( 22 nmnGnZ SPSPSP

)(

)(2)(2

nm

nGmnF

P

PqP weak decay constants—

In the vector and axial-vector cases the decay constants are normalized as follows:

em 3

2

em 3

2

10 | (0) | ( , ) ,

(2 )

10 | (0) | ( , ) .

(2 )

V V

A A

j V e k eF m e

A A e k eF m e

Relations with observables:

1 1 1

1

1 1

1

1

2 2

2 3 2 2 2

2 2

32 2

2 2

4,

3

21 1 ,

16

1 .24

VV e e

V

F a a aa

a aa

a

F

m

G m F m m

m m

F m m

f m

2222 cut

2 2 2 4 6

22 1( ) ln

8 8 3c s

S s

G qqNQ

Q Q Q QO

2222 cut

2 2 4 6 8

44 1( ) ln

12 12 9c s

A s

G qqNQ

Q Q Q QO

Operator Product ExpansionOperator Product Expansion (chiral limit, LO of PT, large-Nc )

2222 cut

2 2 4 6 8

28 1( ) ln

12 12 9c s

V s

G qqNQ

Q Q Q QO

2222 cut

2 2 2 4 6

11 1( ) ln

8 8 3c s

P s

G qqNQ

Q Q Q QO

gluon condensate four-quark condensate

After summing over resonances and comparing with the OPE (at eachpower of ) one arrives at the so called asymptotic sum rules.

21 Q

2

2 26

( ) ( ) 8V A s

qqQ Q

Q

from the pert. theory

In order to sum over resonances we use Euler-Maclaurin formula:

1

0 0

(2 1) (2 1)12

1( ) ( ) (0) ( ) '( ) '(0)

2 2!

'''( ) '''(0) ... ( 1) ( ) (0) ...4! (2 2)!

NN

n

k k kk

Bf n f x dx f f N f N f

BBf N f f N f

k

where B1=1/6, B2=1/30, ... (Bernoulli numbers)

Improving the linear ansatz …Improving the linear ansatz … Phenomenologically it is plain that Regge trajectories are not linear

for small “n”. However arbitrary ansätze for m2(n) and F2(n) result in appearance of terms which are absent in the standart OPE: 1)1) fractional or odd power of Q; 2)2) Q-2kln(Λ2/Q2).

We want to construct the parametrization that does not lead to the unwanted terms and reproduces the parton-model logarithm.

To reproduce the leading log2 2 2

2 2 2 2

( ) ( ) 1ln

( )

F x dx Q m xC O

Q m x Q

Condition 2)2) is satisfied only if 2

2 ( )( ) ( ) , ( ) ( ),

dm xF x t x t x C t x

dx

where Δt(x) is a rapidly decreasing function to be determined.

If we do not consider the running coupling constant and anomalous dimensions, the direct expansion of the integral

2 2

2 2

( ) ( )212 2

2 2 20(0) (0)

( ) ( ( )) 1( ) ( ) ( ( ))

( )

km N m N

k

km m

t x d m xt x m x d m x

Q m x Q

must be defined at any order [many proposals do not meet this criterium: Beane, Simonov, ...]

Apart from the constant solution (linear Regge ansatz) Δt(x) may drop as an exponential of some power of x (perhaps modulated by some powers of x)

,

,0

( ) i FB xi F

i

t x C A e

This is the simplest ansatz compatible with the OPE.

Let us consider the generalization of the Weinberg sum rules:

2 2 2 2 ( )

0

( ) ( ) ( ) ( )N

i i iV V A A

n

F n m n F n m n C

Here the C(i) (i=0,1, ...) represent the corresponding condensate. For the absolute convergence of the series at a given i one needs

2 21

2 2

1( ) ( ) , ,

1( ) ( ) , .

V A

V A

F n F n in

m n m n in

Consequently, for the convergence at any i one has to have mV = mA and it is natural to expect that δ(n) decrease exponentially too.

Let us discuss corrections to the linear mass spectrum 2 2

0( ) ( ).m n m an n

Vector and axial-vector mesonsVector and axial-vector mesons

Consider:2 2

2

( ) ( ),

( ) ( )

mJ J J J

FJ J

m n M a n n

F n const n

,J V A

String picture: universality (agrees with phenomenology – Pancheri, Anisovich)

2V A S Pa a a a a string tension

Conditions: agreement with the analytical structure of the OPE & convergence of sum rules for ПV(Q2) – ПA(Q2)

1). ( ) 0n linear trajectories V AM M degeneratespectrum

2). ( ) 0n

2

2 2

22 1

8

( ) ,

( )( )

JF

J BnJ m

B nJJJ F

m n M an A e

dm nF n A e

dn

( , 0)JFB B

corrections to linear spectrum

(n is the principalquantum number)

Scalar and pseudoscalar mesonsScalar and pseudoscalar mesons

Following the same arguments (J=S,P):

2

2 2

22 3

16

( ) ,

( )( )

JF

J BnJ m

B nJJJ F

m n M an A e

dm nG n A e

dn

( , 0)JFB B

Important: sum rules over chiral partners (cutoff!) – there are two variants

I. Linear σ-model: (0) 0Pm m (π-in)

II. Non-linear σ-model: π-meson is out of the trajectory,

(1300)(0) 1300 100 MeVPm m (π-out)

It is possible to use this analysis for some predictions of phenomenological interest. For instance:

2

2

2 2 210 2 0

2 2 2 22

0

22 2 2

8 2 2 0

1( ) ( ) ,

8

3( ) ( ) ( ) ,

16

( ) ( ) .32

V A

Q

V A

S P

Q

dL Q Q Q

dQ

m d Q Q Q Qm f

f dL Q Q Q

dQqq

An example of input masses (in MeV) for the mass spectra of our work and resulting constants. The corresponding experimental values (if any) are displayed in brackets.

Mass spectrum (in MeV) and residues (in MeV) for the inputs from the previous table (for the first 4 states).

π-in

π-out

Remark 1: D-wave vector mesonsRemark 1: D-wave vector mesons

D S

V.V. Anisovich at al.

2 ( ) DB nD DF n A e

( 0)DB

D-wave vector states decouple!

Remark 2: dimension-two gluon condensate Remark 2: dimension-two gluon condensate λλ22

In the OPE:2

23 2

22 2 cut

3 2

( ) ,4

3( ) ln

2

sVA

sSP

QQ

QQ

VA-channels - no problem

regular due to λ refers to pion only

On the other hand:

2

2(0) 2P

qqZ

f

(from current algebra)

If and π-meson belongs to the trajectory:0Z 2 22 GeV

Phenomenological bounds (B.L. Ioffe et al.):2 2(0.05 0.08) GeV

22 2 2

,03

3 ( )( ) ( ) 2 ( ) ( )

2s SP

SP SP SP SP n

dm nZ n Z n G n m n Z

dn

SP-channels:

Remark 3: perturbation theoryRemark 3: perturbation theory

Consider the vector correlator:2

22

0

1 Im ( )( ) Subt.Const.

sQ ds

s Q

Resonance saturation:

where

2

1Im ( ) 1 ( )

12c

s

Ns s

2 2

0

Im ( ) 2 ( ) ( ( ))N

n

s F n s m n

Smoothness: Euler-Maclauren summation

2 2 2 2

0 0 non-pertub.

( ) ( ( )) ( ) ( ( )) 1NN

n

F n s m n F n s m n O s

Result: 20

2

0

2 ( )Im ( )

( )

F ns

dm ndn n n

0 0 ( )n n s

Check for the first two states:

1

1 1

22

2 2 2 2 2' '

11 (1GeV)

24a c

sa a

FF N

m m m m

Numerically (without the factor 10-2):

1.5 0.2 1.3 0.6 1.4

One-loop: no anomalous dimensions, running αs(Q2).

QCD

2 22

0 2

24

1ln

dQ QpartondQQ

In OPE:

In order to reproduce this behaviour we should accept the following ansatz for the residues:

QCD

22

22

0 2

( )

1 ( ) 4( ) 1

8ln

m n

dm nF n

dn

The influence on the spectrum is negligible.