• Introduction & motivation

• Baryons in the 1/Nc expansion of QCD

• Photoproduction of excited baryons

• Summary & conclusions

N.N. ScoccolaDepartment of Physics - Comisión Nacional de Energía Atómica - Buenos Aires – Argentina

Photoproduction of Excited Baryons in Large Nc QCD

Based on work with J. Goity and N. MatagneGoity, NNS, Phys. Rev. Lett. 99 (07) 062002; Goity, Matagne, NNS, Phys. Lett. B663 (08) 222

INTRODUCTION & MOTIVATION

Baryon photocouplings have been the subject of many studies over a period of 40 years, and are key elements in the understanding of baryon structure and dynamics.

The most commonly used tool of analysis of these quantities has been constituent quark model, which relation with QCD is, at least, unclear. In this sense, it is important to have a model independent approach to the physics of excited baryon photoproduction.

One possible systematic approach of this type is the 1/Nc expansion of QCD.

SU(6) irrep

SU(3)f

irrep

JP

S = 0 S= - 1

I = 0 I = 1

S = - 2

S = - 3 56+(l=0) 28 1/2+ N(939) (1116) (1193) (1318)

410 3/2+ (1232) (1385) (1533) (1672) 70-(l=1) 28 3/2- N(1520) (1690) (1580)** (1820)

1/2- N(1535) (1670) (1620)** 48 1/2- N(1650) (1800) (1750) 5/2- N(1675) (1830) (1775) 3/2- N(1700) ? (1670) 210 1/2- (1620) 3/2- (1700) 21 1/2- (1405) 3/2- (1520)

56+(l=0’) 28 1/2+ N(1440) (1600) (1660) 410 3/2+ (1600) (1690)**

56+(l=2) 28 3/2+ N(1720) (1890) ? 5/2+ N(1680) (1820) (1915)

410 1/2+ (1910) 3/2+ (1920) (2080)** 5/2+ (1905) 7/2+ (1950) (2030)

70+(l=0’) 28 3/2- N(1710) (1800) (1880)** 48 1/2- N(1540)* 210 1/2- (1550)* 21 1/2- N(2100)*

SU(6) Classification

BARYONS IN THE 1/Nc EXPANSION OF QCD

QCD has no obvious expansion parameter. However, t’Hooft (’74) realized that if one extends the QCD color group from SU(3) to SU(Nc), where Nc is an arbitrary (odd) large number, then 1/ Nc may treated as the relevant expansion parameter of QCD. To have consistent theory , QCD coupling constant g2 ~ 1/Nc

Witten (’79) observed that baryons are formed by Nc “valence” quarks with Mb~

O(Nc1) and rB ~ O(Nc

0) . In the Large Nc limit a Hartree picture of baryons emerges: each quark moves in a self-consistent effective potential generated by the rest of the (Nc-1) quarks.

cN

GSBaryon

It is also possible to show that BB’ coupling is Nc1/2). Thus, since f ~ (Nc

1/2), it can expressed as

22

20

' | [ , ] | ( )jb iacc

π

N iB X X B O N

f k

a iacA i

π

Ng π X

f

In order to preserve unitarity for Large Nc

Xia spin-flavor operator of (Nc0)

Correspondingly for pion-nucleon scattering we have

Gervais-SakitaDashen-Manohar consistency relation

, ,i a ia ia

c

GS T X

N form contracted SU(2 Nf ) algebra

In this limit, GS baryons can be chosen to fill SU(2 Nf) completely symmetric irrep.

Therefore, in the Large Nc limit

n-body operator obtained as the product of n generators of SU(2 Nf) , i.e. Si, Ta, Gia

• n-body operators need at least n quarks exchanging (n-1) gluons according to usual rules it carries a suppression factor (1/Nc)n-1

e.g a 3-body operator g4~Nc

-2

• Some operators may act as coherent matrix elements of order Nc

e.g. Gia~ Nc

Any color singlet QCD operator can be represented at the level of effective theory by a series of composite operators ordered in powers of 1/Nc

Unknown coefficients to be fitted

Rules for Nc counting

As example we construct the three flavor GS baryon masses. Up to 1/Nc

contributions and leading order in the SU(3) flavor breaking we get

It looks like there are many contributions up to this order. However, since for GS baryons operators are acting on fully symmetric irrep of SU(6), several reduction formulae appear (Dashen, Jenkins, Manohar, PRD49(94)4713; D 51(95), 3697).

Using these relations M simplifies to

This type of analysis has been applied to analyze axial couplings and magnetic moments, etc. (Dai,Dashen,Jenkins,Manohar, PRD53(96)273, Carone,Georgi, Osofsky, PLB 322(94)227, Luty,March-Russell, NPB426(94)71, etc). For reviews, see A.Manohar, hep-ph/9802419; R.Lebed, nucl-th/9810080.)

One gets parameter free testable relations

Carone et al, PRD50(94)5793, Goity, PLB414(97)140; Pirjol and Yan, PRD57(98)1449; PRD 57(98)5434 proposed to extend these ideas to analyze low lying excited baryons properties.

Take as convenient basis of states: multiplets of O(3) x SU(2 Nf) (approximation since they might contain several irreps of SU(2 Nf)c (Schat, Pirjol,

PRD67(03)096009, Cohen, Lebed, PLB619(05)115))

Core (Nc-1 quarks in S irrep) +

Excited quarkwith l

Coupled to MS or S irrep of spin-flavor

Excited baryon composed by

1cN

Low lyingExcited Baryon

For lowest states relevant multiplets are [0+, 56] , [1+, 70] , [2+, 56] …

In the operator analysis, effective n-body operators are now

( ) ( )n nO R (n)where is an (3) operator and an (2 ) tensorfR O SU N

a( )

a1c c

1

1 = , ,... where = , ,

i ian

i iac cnc cc n

λ t s gλT S GN

1cN

This approach has been applied to analyze the excited baryon massesCarlson, Carone, Goity and Lebed, PLB438 (98) 327; PRD59 (99) 114008. Schat, Goity and NNS, PRL88(02)102002, PRD66 (02) 114014, PLB564 (03) 83 .

Remarks on [1-,70]-plet masses

Although spin-flavor symmetry is broken at (Nc0), the corresponding

operators (e.g. spin-orbit) have small coefficients.

Hyperfine operator ScSc/Nc of (Nc-1) give chief spin-flavor breaking.

determined only by spin-orbit operator. For the rest of spin-orbit partners other operators appear. Important is li ta Gc

ia/Nc

Remarks on [2+,56]-plet masses

Spin-flavor symmetry only broken at (1/Nc). Hyperfine operator ScSc/Nc gives the most important contribution.

Testable model independent relation well satisfied.

Strong decay widths have been also analyzed Goity, Schat, NNS, PRD71(05)034016; Goity, NNS, PRD71(05)034016

PHOTOPRODUCTION OF EXCITED BARYONS IN LARGE Nc QCD

The helicity amplitudes of interest are defined as• =1/2, 3/2 is helicity along -momentum (z-axis)

• ê+1 is -polarization vector

(B*) sign factor which depends on sign strong amplitude N → B*

The e.m. current can be represented as a linear combination of effective multipole baryonic operators with isospin I =0,1. Their most general form is

Tensor to be expressed in terms of spherical harmonics of photon momentum

[ , ][ , ] ( ) [ ', ] L IL I l l I

Acts orbital wf of excited q

Acts of spin-flavor wfwhere

[ ]* [ , ] *, 3 3[10]

,

[ ]* [ , ] *, 1 3 3[10]

,

3( ) ( ) ; , | | ; 1,

4

3( 1)( ) ( ) ; , | | ; 1,

4(2 1)

L IML L Icn L n

a I

L IEL L Icn L n

a I

NA B g B I N I

L NA B g B I N I

L

Then, the electric and magnetic components of the helicity amplitudes can be expressed in terms of the matrix elements of the baryonic operators as

For the -dependence of g’s we use the standard penetration factor expression

'[ , ] [ , ], ' , '( ) /

LL I L In L n Lg g m

* 1/ 2* *( ) 1 sign || ||

J

QCDB l N H B

Strong signs are determined as follows

l partial wave for pion strong decay

determined up to an overall sign for each partial wave. When B* can decay through 2 partial waves (e.g. S or D; P or F ) undetermined sign (e.g. =S/D , ’=P/F)

Helicity amplitudes for [1-,70]-plet non-strange resonances

= -1 clearly favored by fits. Only this case is shown.

Helicity amplitudes for [2+,56]-plet non-strange resonances

’ = -1 clearly favored by fits. Only this case is shown.

Helicity amplitudes for [0+,56’]-plet non-strange resonances

CONCLUSIONS

•The 1/Nc expansion of QCD provides a useful and systematic method to analyze the excited baryon properties.

•Hierarchies implied by the 1/Nc power counting well respected in the analysis of the e.m. helicity amplitudes.

•Only a reduced number of the operators in the NLO basis turns out to be relevant. Some of these operators can be identified with those used in QM calculations. However, there are also 2B operators (not included in QM calculations) which are needed for an accurate description of the empirical helicity amplitudes.