recovering qcd at large n c : two-point green-functions

Frascati-EU4CM, February 8th 2005

J.J. Sanz Cillero, IN2P3-Orsay

Recovering QCD at large NRecovering QCD at large NCC::

Two-point Green-functionsTwo-point Green-functions

J. J. Sanz Cillero, IN2P3 - Orsay



Outline:Outline:

DUALITYQCD at large NC (QCD∞) Resonance Theory

(RT∞)

•Analysis of QCD correlators:

a)- Physical QCD content

b)- Relation RT Sum-rules

c)- Truncated RT + “pQCD Continuum”

•Conclusions



Duality in QCDDuality in QCD



RRTT∞∞

…………

n ® ¥njpmjpm

RRTT(n)(n)

QCDQCD∞∞Infinite Infinite

resonanceresonance

RegularizationRegularization

CUT-OFFCUT-OFFRegularizationRegularization



LOCAL DUALITYLOCAL DUALITY

RRTT∞∞ QCDQCD∞∞

pQCDpQCDpQCDpQCD

Local Duality Local Duality NOT NOT

POSSIBLE POSSIBLE at LO in 1/Nat LO in 1/NC C !!

2q2q <0 ……



QCD Green-QCD Green-

functions functions NOT NOT

BEINGBEING order- order-

parametersparameters



æ ö÷ç ÷ç ÷ç ÷çè øO

22 C

LL s2 2cut

N -qΠ (q )=- ln +

12 Λp

-UV divergentUV divergent

-Scale dependentScale dependent

-Logarithmic Logarithmic behaviourbehaviour

A O2 CLL 2L2 s

2L

d- Π (

N(q )= +

12q )=

dlnq p

-Physical & Physical & finitefinite

independentindependent

-Logs within Logs within ss

•Short distance behaviour in pQCDpQCD for (V+A):

Left-left correlator inLeft-left correlator in QCDQCD∞∞

[ Shiftman et al. ’76 ]

& AA & AA correlatorcorrelator

& VV & VV

correlatorcorrelator



( )åA

2 2nj2

LL 22 2j=1 j

Q F(q ;n)=

M +Q

( )åA

22 2j

2 2j2

LL 22 2MM M j

Q F(q )= lim

M +Q¥¥

®¥£

UV finiteUV finite

when Mwhen M∞∞22∞∞

with qwith q22= -Q= -Q22 < < 00

å22 2j

2j2

LL 2 2MM M j

FΠ (q )= lim

M +Q¥¥

®¥£

UVUV

DIVERGENTDIVERGENT

when Mwhen M∞∞22∞∞

Resonance Effective Resonance Effective TheoryTheory

RRT at Large NT at Large NCC

å2nj2

LL 2 2j=1 j

FΠ (q ;n)=

M +Q

•Resonance Resonance description:description:



å2 2j

2j2

LL 2 2M M j

FΠ (q )=

M +Q¥£ ( )

å2 2j

2 2j2

LL 22 2M M j

Q F(q )=

M +Q¥£

A

{ }ˆ

ˆ

æ ö÷ç× ÷ç ÷çè øò O2

2 2

2M 22 2

LL 2 20

δM-

δM + Q1

dMΠ (q ) = H(M )

M +Q

¥

( ) { }ˆ

ˆ

æ ö÷ç× ÷ç ÷çè øò2

OA2

2 2

2M 2 22 2

LL 22 20

δM-

δM + Q1

Q dM(q ) = H(M )

M +Q

¥

From series to integral

2 2 2j j j-1δM M -Mº

{ }ˆ 2 2j j=1

δM max δM¥

ºand the interspacingwith the

function 0

j

2j2

V 2j

FH(M )

δMº ³

•Integral representation of discrete Integral representation of discrete series:series:

[ Peris et al. ´98 ]

[ Beane ’ 01 ]

[ Golterman & Peris ‘

01 ]

…



Relation with sum-Relation with sum-rules:rules:

ˆ

òA

22 2

M 2 2Q δM2 2

LL 2 2 20

Q dM(q ) H(M )

(M +Q )

¥?;

òAt 2

2LL LL2 2

0

Q dt 1(q ) = ImΠ (t)

(t+Q )

¥

p

Infinite series

from RRTT

•Short distance QCD : Information about higher Short distance QCD : Information about higher multipletsmultiplets ˆ

ˆ

ì æ öüï ïï ï÷ç× ÷ ¾¾í ýç ÷çï ïè¾

øï ïî þ¾®O

2

2 2

j

2Vj

j

M2 CV 2

2j 2

LL j2j

δM1

M + δM

F 1= ImΠ (M )

δMN

H(M )= 12

±pp

→∞

pQCDpQCD

QCD dispersion relations



High mass High mass resonances: resonances:

Origin of the Origin of the “Continuum”“Continuum”

( ) ( )å å

14444444244444443 14444444244444443

A2 2 2 2j c j c

2 2 2 2j j2

LL 2 22 2 2 2M M M >Mj j

non-perturbative perturbative

Q F Q F(q ) = +

M +Q M +Q≤

ˆ2 2ct M δM: ?Convergence of H(t)

around

( ) { }ˆ

ˆ

æ ö÷ç× ÷ç ÷è øò2

O

2

2 2 2

c

2

2c

δM-

δM + M + Q

M 2

22M

1Q dt

H(t) t+Q

¥

{ }ˆ

ˆ

æ ö÷ç ÷ç ÷æ ö÷ç×ç ÷ç è øè

÷÷ø ç

2

O

2

2 2 2

c

δM-

δM + M + Q

2C

2 2 2c

1N Q

12 M +Qp

TruncatedTruncatedRRTT

““pQCD pQCD ContinuuContinuu

m”m” )éë

2ct M ,Î ¥

; CLL 2

N1ImΠ (t)

12p p

ssMMcc

22

““CONTINUUM”CONTINUUM”



High and low energy High and low energy

HierarchiesHierarchies

å å144444424444443 144444424444443

2 2 2 2j c j c

2 2j j2

LL 2 2 2 2M M M >Mj j


F FΠ (q ) = +

M +Q M +Q≤

AnalyticalAnalytical

Non-Non-analyticalanalytical2 2

ρ ρ 22 4ρ ρ

F F- Q +...

M M:

2 2 2ρ ρ ρ

2 4

F F M- +...

Q Q:

ì üæ öï ïï ï÷ç ÷í ýç ÷ç ÷ï ïè øï ïî þ

22C c

2 2 2

N MMln + +...

12 Q Q¥

p:

ì üæ öï ï÷ï ïç ÷çí ý÷ç ÷çï ïè øï ïî þ

2 2C

2 2 2c c

N M Qln + +...

12 M M¥

p:

AnalyticalAnalytical

non-p. 2LLΠ (q ) pert. 2

LLΠ (q )

2 2cM Q=

2 2cQ M=

pQCDpQCD::

PT:PT:

DOMINANTDOMINANT

DOMINANTDOMINANT



Phenomenology:Phenomenology:V+A Adler-V+A Adler-

functionfunction

(without pion (without pion pole)pole)

ρM =770 MeV

1aF = 135 MeV

1aM =1250 MeV

ρF = 154 MeV


( )1

22 2 2 2ρ' α ρF = F +F - F = 55 MeV

ρ'M =1450 MeV

NO NO “CONTINUUM“CONTINUUM

””

Truncated RTruncated RTT

++

““CONTINUUM”CONTINUUM”

ˆ1

2 2 2 2a ρδM =M -M (1 GeV);

1

2cM

2

2cM

3

2cM

2j

2j

F

4M

2jM

( ){ }ˆ

ˆP O

2

2

δM1 ±

t + δMLL

1H(t)= Im (t)×

p



ConclusionsConclusions



- Duality: QCDQCD RRT T (local & (local & global)global)

- QCD at large NC :

• Infinite resonance summation

• Hierarchy

• Truncated RT

+ pQCD continuum

Logarithmic & Logarithmic & UV behaviourUV behaviour

PhenomenoloPhenomenologicalgical

successsuccess

Short distances: Short distances: Continuum dominance

Long distances: Long distances: Lightest resonances



OUTLOOK:OUTLOOK:

• Form-factorsForm-factors (Correlators at NLO in (Correlators at NLO in 1/N1/NCC))

• Other Other non-order-parameternon-order-parameter Green- Green-funtionsfuntions

• Scattering amplitudesScattering amplitudes

??

??

??



•In addition:In addition: LOCAL LOCAL DUALITYDUALITY

2q <0AVERAGED AVERAGED (GLOBAL)(GLOBAL) DUALITYDUALITY

2q >0

•In general:In general: é ùê úê úë û

n

(n) 2 2 n 22

1 d(q ) = (-q ) Π(q )

n! dqA

Example for the scalar for the scalar

correlatorcorrelator (2) 2SS(q ) A



High and low energy High and low energy

HierarchiesHierarchies

( ) ( )å å

14444444244444443 144444442444444432 2 2 2j c j c

2 2 2 2j j2

LL 2 22 2 2 2M M M >Mj j


Q F Q F(q ) = +

M +Q M +Q≤

A

2ρ 24ρ

FQ +...

M:

2 2 2ρ ρ ρ

2 4

F 2 F M- +...

Q Q:

ì üï ïï ïí ýï ïï ïî þ

2C c

2 2

N M1+ +...

12 Q:

p

ì üï ïï ïí ýï ïï ïî þ

2C

2 2c

N Q- +...

12 M:

p

non-p. 2LL (q )A pert. 2

LL (q )A

2 2cM Q=

2 2cQ M=



Phenomenology:Phenomenology:Vector Adler-Vector Adler-

functionfunction


ρM =770 MeV

1aF = 135 MeV

1aM =1250 MeV

ρF = 154 MeV


( )1

22 2 2 2ρ́ α ρF = F +F - F = 55 MeV

ρ́M =1450 MeV

NO NO CONTINUUMCONTINUUM

Truncated Truncated RRTT

++

CONTINUUMCONTINUUM

ˆ1

2 2 2 2a ρδM =M -M (1 GeV);

1

2cM

2

2cM

3

2cM

( ){ }ˆ

ˆP

2

2

δM1 ±

t + δMVV

1H(t)= Im (t)×

pO

recovering qcd at large n c : two-point green-functions

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