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Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithm The Uses of Conformal Symmetry in QCD V. M. Braun University of Regensburg DESY, 23 September 2006 V. M. Braun The Uses of Conformal Symmetry in QCD

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Page 1: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

The Uses of Conformal Symmetry in QCD

V. M. Braun

University of Regensburg

DESY, 23 September 2006

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 2: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

based on a review

V.M. Braun, G.P. Korchemsky, D. Müller, Prog. Part. Nucl. Phys. 51 (2003) 311

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 3: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Outline

1 Conformal Group and its Collinear Subgroup

2 Conformal Partial Wave Expansion

3 Hamiltonian Approach to QCD Evolution Equations

4 Beyond Leading Logarithms

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 4: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Transformations

1 conserve the interval ds2 � g�� �x�dx� dx�2 only change the scale of the metricg��� �x� � � � �x�g�� �x�

�preserve the angles and leave the light-cone invariant

Translations

Rotations and Lorentz boosts

Dilatation (global scale transformation) x� x� � �x�Inversion x� x� � x� �x2

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 5: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Transformations

1 conserve the interval ds2 � g�� �x�dx� dx�2 only change the scale of the metricg��� �x� � � � �x�g�� �x�

�preserve the angles and leave the light-cone invariant

Translations

Rotations and Lorentz boosts

Dilatation (global scale transformation) x� x� � �x�Inversion x� x� � x� �x2

Special conformal transformation

x� � x�� � x� � a�x2

1 � 2a � x � a2x2

= inversion, translationx� � x� � a� , inversion

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 6: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Algebra

The full conformal algebra includes 15 generators

P� (4 translations)

M �� (6 Lorentz rotations)

D (dilatation)

K � (4 special conformal transformations)

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 7: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Collinear Subgroup

P = Pz

p� � 1�2

�p0 � pz� � �

p� � 1�2

�p0 pz� � 0

px � p� x�

Special conformal transformation

x � x� � x1 � 2ax

translationsx � x� � x � c

dilatationsx � x� � �xform the so-calledcollinear subgroup SL

�2�R�

� � � a � b

c � d� ad � bc � 1� � � � � � � � � �

c � d�2j� �a � b

c � d �where

� �x� � � �

x � � � � n � is the quantumfield with scaling dimension� and spin projections“living” on the light-ray

Conformal spin:

j � �l � s��2

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 8: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

SL �2� Algebra

is generated byP� , M � , D andK L� � L1

� iL2� �iP�

L � L1 � iL2� �

i�2�K L0

� �i�2��D � M � �

E � �i�2��D � M � �

can be traded for the algebra of differentialoperators acting on the field coordinates

�L� � � � �� � L�� � ��L � � � �� � L� � ��L0 � � � �� � L0

� � �

L� � � d

d L � � 2 d

d � 2j �L0

� � d

d � j�They satisfy theSL

�2� commutation relations

�L0 �L� � � �L��

L �L�� � 2L0

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 9: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

SL �2� Algebra

is generated byP� , M � , D andK L� � L1

� iL2� �iP�

L � L1 � iL2� �

i�2�K L0

� �i�2��D � M � �

E � �i�2��D � M � �

can be traded for the algebra of differentialoperators acting on the field coordinates

�L� � � � �� � L�� � ��L � � � �� � L� � ��L0 � � � �� � L0

� � �

L� � � d

d L � � 2 d

d � 2j �L0

� � d

d � j�They satisfy theSL

�2� commutation relations

�L0 �L� � � �L��

L �L�� � 2L0

The remaining generatorE counts thetwist t � � � s of the field�

�E � � � �� � 1

2

�� � s�� � �collinear twist = dimension - spin projection on the plus-direction

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 10: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Towers

A local field operator� � � that has fixed spin projection on the light-cone is an eigenstate of

the quadratic Casimir operator

�i�0�1�2

�L i � �L i � � � ��� � j

�j � 1�� � � � L2� � �

with the operatorL2 defined as

L2 � L20� L2

1� L2

2�L2 �Li � � 0 �

Moreover, the field at the origin� �

0� is an eigenstate ofL0 with the eigenvaluej0� j and is

annihilated byL �L � � �0�� � 0 � �

L0 � � �0�� � j� �

0� �A complete operator basis can be obtained from

� �0� by applying the “raising” operatorL�

�k� �

L� � � � � � �L� � �L� � � �0���� � ����� �k� � �� ������0� �

0 � � �0� �

From the commutation relations it follows that

�L0 ��k � � �

k � j��k

non-compact spin: j0� j � k, k � 0�1�2 � � �

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 11: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Adjoint Representation� Two different complete sets of states:� � � for arbitrary �k for arbitraryk

Relation = Taylor series:

� � � � ��k�0

�� �kk� �

k �� In general: Two equivalent descriptions

light-ray operators (nonlocal)

local operators with derivatives (Wilson)

Characteristic polynomial

Any local composite operator can be specifiedby a polynomial that details the structure andthe number of derivatives acting on the fields.For example

�k� ��

k��� �� � �� ������0�

k�u� � ��u�k

where�� � d�d The algebra of generators acting on light-ray operators canequivalently be rewritten as algebraof differential operators acting on the space of characteristic polynomials. One finds thefollowing ‘adjoint’ representation of the generators acting on this space:

�L0� �

u� � �u�u

� j� � �u� ��

L� �u� � �u

� �u� ��

L�� �u� � �u�2

u� 2j�u� � �

u� �V. M. Braun The Uses of Conformal Symmetry in QCD

Page 12: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

to summarize:

A one-particle local operator (quark or gluon) can be characterized by twistt , conformalspin j, and conformal spin projectionj0

� j � k

Conformal spinj � 1�2�� � s� involves the spin projection on “plus” direction. This isnot helicity!Example: �

� � j � 1

2

�3�2 � 1�2� � 1�

� j � 1

2

�3�2 � 1�2� � 1�2

The differencek � j0 � j � 0 is equal to the number of (total) derivatives

Exact analogy to the usual spin, but conformal spin is noncompact, corresponds tohyperbolic rotationsSL

�2� � SO

�2�1�

We are dealing with the infinite-dimensional representation of the collinearconformal group

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 13: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Separation of variables

In Quantum Mechanics:O(3) rotational

symmetry

� Angular vs. radialdependence

� �2

2m� � V� �

r��� � � E� � � ��r� � R

�r�Ylm

� ��Ylm

� �� are eigenfunctions ofL2Ylm � l�l � 1�Ylm, �

L2� � 0.

In Quantum Chromodynamics:SL(2,R) conformal

symmetry

� Longitudinal vs. transversedependence

Migdal ‘77Brodsky,Frishman,Lepage,Sachrajda ‘80Makeenko ‘81Ohrndorf ‘82

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 14: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Separation of variables —continued

Momentum fraction distribution of quarks in a pion at small t ransverse separations:

� Dependence on transverse coordinates is traded for the scale dependence:

�2 dd�2

�� �u � � � � 1

0dw �

uw� �� �u � � � �� �

u � � � 6u

�1u� �1 � �2� �� �C3�2

2

�2u1� � � � ���n� �� � � �n� ��

0� � s � �s �0 � � n�b0

� Summation goes over total conformal spin of the�qq pair: j � jq � j�q � n � n � 2� Gegenbauer PolynomialsC3�2n are ‘spherical harmonics’ of the SL

�2R� group� Exact analogy: Partial wave expansion in Quantum Mechanics

For Baryons: Summation of three conformal spins

�N�ui � � 120u1u2u3 ��

J�3

J�1�j�2

�J �jN

�� � Y 12�3J �j �

u�J � 3 � N

N � 0

1 � � � total spin of three quarks

j � 2 � nn � 0

1 � � � spin of the (12)-quark pair

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 15: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Spin Summation

Two-particle:

�j1� � �

j2� � �n�0

�j1� j2

� n�

For two fields “living” on the light-cone

O�

1 � 2� � �j1

� 1� � j2

� 2�

or, equivalently, local composite operators

�j�0� � �

j��1 � �2�� j1

� 1� � j2

� 2� �����1��2�0

the two-particle SL(2) generators are

La� L1�a � L2�a � a � 0�1�2

L2 � �a�0�1�2

�L1�a � L2�a�2

We define aconformal operator�L2 �

j� j

�j � 1�� j�

L0�

j� j0

�j�

L� j� 0

The solution readsj0 � j � j1� j2

� n and

� j1�j2j

�u1 �u2� �

� �u1

� u2�nP�2j11�2j21�n

�u2 � u1

u1� u2 �

whereP �a �b�n�x� are the Jacobi polynomials

Call the corresponding operator� j1 �j2j

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 16: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Spin Summation -continued

Three-particle:�j1� � �

j2� � �j2� � �

n�0

�j1� j2

� j3� N�

local composite operators

� �0� � � ��1 � �2 � �2�� j1

� 1� � j2

� 2� � j3

� 3� ����� i ��0� Impose following conditions:

fixed total spinJ � j1� j2

� j3� N

annihilated by�L

fixed spinj � j1� j2

� n in the (12)-channel�� �12�3

Jj

�ui � � �

1 � u3�jj1j2 P �2j31�2j1�Jjj3

�1 � 2u3� P �2j11�2j21�

jj1j2

�u2 � u1

1 � u3 �if a different order of spin summation is chosen

� �31�2Jj

�ui � � �

j1�j2�j� �Jj3

�jj� �J� � �12�3

Jj��ui �

the matrix�

defines Racah 6j-symbols of the SL�2�� � group

Braun, Derkachov, Korchemsky, Manashov ‘99

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 17: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Scalar Product

Correlation function of two conformal operators with different spin must vanish:

�0�T �� j1 �j2

j

�x�� j1 �j2

j��0�� �0� � �jj�

Evaluating this in a free theory:

��n ��m� �

� 1

0

�du� u2j11

1 u2j212

�n�u1 �u2��m

�u1 �u2� � �mn �

where�du� � du1du2 � �u1

� u2 � 1�similar for three particles

��J �j �� J� �j� � �

� 1

0

�du� u2j11

1 u2j212 u2j31

3�

N �q� �u1 �u2 �u3��N �q�u1 �u2 �u3� � �JJ � �jj�

Complete set of orthogonal polynomials w.r.t. the conformal scalar product

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 18: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Pion Distribution Amplitude

Definition: �0��d �0��� �5u

� � ��� �p�� � if� p�� 1

0du e

iu�p� �� �u � ���

�0��d �0��� �5

�i�D� �nu

�0� ��� �p�� � if�

�p� �n�1

� 1

0du

�2u � 1�n�� �u � � �

It follows� 1

0du P1�1

n�2u � 1��� �u � �� � ���1�1

n �� � �0��1�1

n�0� ��� �p�� � if� pn�1� ���1�1

n ��

SinceP1�1n

�x� � C3�2

n�x� form a complete set on the interval�1 � x � 1

���u � �� � 6u

�1 � u�

��n�0

�n�� � C3�2

n�2u � 1� �

�n�� � � 2

�2n � 3�

3�n � 1� �n � 2�

���1�1n

�� ��� � �

n�� � � �

n��0� �

s�� �

s��0� �

0�n �0

ERBL ‘79 – ‘80

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 19: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Higher Twists�Conformal symmetry is respected by the QCD equations of motion

Example: Pion DAs of twist-three�0��d �0�i�5u

�� � ��� �p�� � R� 1

0du e

�iup� �p�u � � R � f

�m2�

mu � md�0��d �0�����5u

�� � ��� �p�� � 1

6iR�

p� � 1

0du e

�iup� �� �u � ��0��d��2�����5gG�� ��3�u��1� ��� �p�� � 2ip2� � �dui

� e�ip� 1u1�2u2�3u3� �3� �

ui � �

Conformal expansion:

�3��ui � � 360u1u2u2

3 � 7�2 � � 9�2 1

2

�7u3 � 3� � � 11�2

1

�2 � 4u1u2 � 8u3

� 8u23�

� � 11�22

�3u1u2 � 2u3

� 3u23� � � � �

R �p�u� � R � 30� 7�2C1�2

2

�� � � 3

2 4� 11�21 � � 11�2

2 � 2� 9�2 C1�24

�� �R �� �u� � 6u

�1 � u� R � 5� 7�2 � 1

2� 9�2C3�2

2

�� � � 1

10 4�11�21 � � 11�2

2 C3�24

�� ��Braun, Filyanov ‘90

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 20: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Partial Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

to summarize:

Conformal symmetry offers a classification of different components in hadron wavefunctions

The contributions with different conformal spin cannot mixunder renormalization toone-loop accuracyWhether this is enough to achieve multiplicative renormalization depends on the problem

QCD equations of motion are satisfied

Allows for model building: LO, NLO in conformal spin

The validity of the conformal expansion is not spolied by quark masses

Convergence of the conformal expansion is not addressed

Close analogy: Partial wave expansion in nonrelativistic quantum mechanics

V. M. Braun The Uses of Conformal Symmetry in QCD

Page 21: The Uses of Conformal Symmetry in QCD - DESYtheosec/sino-german-workshop06/talks/sat/Braun.pdf · that has fixed spin projection on the light-cone is an eigenstate of the quadratic

Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Hamiltonian Approach to QCD Evolution Equations

ERBL evolution equation

�2 d

d�2��

�u � �� � � 1

0dv V

�u � v � s

�� �� ��

�v � � �

V0�u � v� � CF

s

2�

�1 � u

1 � v

�1 �

1

u � v� � �u � v� � u

v

�1 �

1

v � u� � �v � u���How to make maximum use of conformal symmetry?

Bukhvostov, Frolov, Kuraev, Lipatov ‘85

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Where is the symmetry?

RG equation for the nonlocal operator � � 1 � 2� � �

��

1�����

2��� ���

� � �g� ��g� � � 1 � 2� � � sCF

4���

� �� � 1 � 2�

� � 2� �12�v � 2� �12�

e� 1

(c)(a) (b)Explicit calculation

�� �12�v � �� � 1 � 2� � � � 1

0

du

u

�1 � u� �� � u

12� 2� � � � 1 � u21� � 2� � 1 � 2�� �

�� �12�e � �� � 1 � 2� � � 1

0

�du� � � u1

12 � u221� u

12 � 1�1 � u� �

2u

Balitsky, Braun ‘88Now verify

��� La �� � 1 � 2� � La

��� �� � 1 � 2� � La�

� 1 � 2� � �

L1�a � L2�a �� � 1 � 2�

�� � L�� � �� � L � � �� � L0� � 0

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformally Covariant Representation

if so,�

must be a function of the two-particle Casimir operator

L212� ���1 ��2

� 1 �

2�2 �To find this function, compare action of

�andL2

12 on the set of functions�

1 � 2�n

� �12�v

� 2�� �

J12� � � �2�� � L2

12� J12

�J12 � 1�

� �12�e

� 1� �J12�J12 � 1�� � 1�L2

12

�ERBL

� 4�� �

J12� � � �2�� � 2� �J12

�J12 � 1�� � 1

For local operators takeL212 in the adjoint representation

�L2

12

Bukhvostov, Frolov, Kuraev, Lipatov ‘85

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Three-particle distributions

E.g. baryon distribution amplitudes B � N �� � � � ��0�q�z1�q�z2�q�z3� �B�p � � �� � � � �

� �dxi� eip �x1z1�x2z2�x3z3��B

�xi ��2�

q�

q�

q� � ���3�2� �

xi ��2� �q�

q�

q� � � ���1�2

N

�xi ��2����1�2� �xi ��2�

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Spectrum of anomalous dimensions

Operator product expansion�

mixing matrix: N � k1� k2

� k3

� �xi � � � �ki � � � �xi xk1

1 xk22 xk3

3� �xi � �2�

q�z1�q�z2�q�z3� � �

Dk1�q� �Dk2�q� �Dk3�q�

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

E3/

2(N

)

N

γ n=0,1,...N=14

Example:qqq� � 3�2

In the light-ray formalism ��

���

� � �g� ��g �B � �

� B � B�z1 � z2 � z3� � q

�z1�q�z2�q�z3�

Two-particle structure:

�qqq

� �12� �

23� �

13

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Hamiltonian Approach

write � as a function of Casimir operators L2ik � �� zi

��zk

�zi zk �2 � �Jik

��Jik 1�

1 2 3

�� � v12 � 2�� �J12� � �2�� � ��3�2

qqq� � v

12� � v

23� � v

13

1 2 3

�� � e�1�2 � 1

L212

� 1J12J12�1� � ��1�2

qqq� � ��3�2

qqq � � e12 � � e

23

Have to solve� �

N �n � �N �n�N �n

— A Schrödinger equation with Hamiltonian�

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Hamiltonian Approach —continued

Hilbert space?� Polynomials in interquark separation� Polynomials in covariant derivatives (local operators)— related by a duality transformationBDKM ’99

E.g. going over to local operators, in helicity basis (BFKL 85’) obtain a HERMITIANHamiltonian w.r.t. the conformal scalar product

��1 ��2� �

� 1

0dxi � �� xi � 1� x2j11

1 x2j212 x2j31

3

� �

1�xi ��2

�xi �

that is, conformal symmetry allows for the transformation of the mixing matrix:

� forwardnon-hermitian �

� �� �

N � 1

� � non-forwardhermitian �

� �� �

N�N � 1��2

� � hermitianin conformal basis�

� �� �

N � 1

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Hamiltonian Approach —continued (2)

Premium?�

Eigenvalues (anomalous dimensions) are real numbers�

“Conservation of probability” – possibility of meaningfulapproximations to exact evolution�

E.g. 1�Nc expansion

�N �n � NcEN �n � N

1c �EN �n � � � �

N �n � � �0�N �n � N2c ��N �n � � � �

with the usual quantum-mechanical expressions

�EN �n � ��� �0�N �n ��2 �� �0�N �n �� �1� �� �0�N �n�etc.

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Complete Integrability

� Conformal symmetry implies existence oftwo conserved quantities:

�� �L2� � �� �L0� � 0

� For qqq andGGG states with maximum helicity and forqGq states atNc� � there exists

an additional conserved charge

qqqGGG

��max� Q � �L2

12�L223��� �Q� � 0

qGqNc��� Q � �L2

qG �L2Gq� � c1L2

qG� c2L2

Gq�� �Q� � 0

BDM ’98

� Anomalous dimensions can be classifiedby values of the chargeQ:

�� � � �

Q� � q

� �� � � � �q�

�A new quantum number

Lipatov ‘94–‘95, Faddeev, Korchemsky ‘95Braun, Derkachov, Manashov ‘98

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Complete Integrability —continued

Double degeneracy:

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30

q/3

N

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

E3/

2(N

)

η

N

l = 0

l = 2

l = 7

Example:qqq� � 3�2

� �N �q� � � �N � �q�q�N � � � � �q

�N �N � � �

BDKM ’98

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

WKB expansion of the Baxter equation� 1/N expansion

Korchemsky ’95-’97�

equationQ� � q

�is much simpler as

� � � ��

�non-integrable corrections are suppressed at largeN�can use some techniques of integrable models

� �N � � � � 6 ln � � 3 ln 3 � 6 � 6�E� 3

��2� � 1�

� 1

�2�5�2 � 5� � 7�6�

� 1

72�3�464�3 � 696�2 � 802� � 517�

� � � �� � � �

N�3��N�2�An integer� numerates the trajectories:(semiclassically quantized solitons: Korchemsky, Krichever, ’97)�

Integrability imposes a nontrivial analytic structure

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Breakdown of integrability: Mass gap and bound states

Simplest case:

Difference between���3�2 andN�� ���1�2

� ��� � � 3�2 � � � 1

L212

� 1

L223�

� �� � 1� � �1�2

Flow of energy levels(anomalous dimensions):

Scalar diquark BDKM ’98

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Advanced Example: Polarized deep-inelastic scattering

Cross section: � � pq�M � El � El�d2�

d�

dEl�������� � 8 2E2

l�Q4� F2

�x �Q2� cos2 �

2� 2�

MF1�x �Q2� sin2 �

2 �

d2�d�

dEl������� � 8 2El�

Q4x g1

�x �Q2� �1� El�

Elcos�� � 2Mx

Elg2�x �Q2�

At tree level:

g1�x� � � �� d�

2�ei�x �p � sz ��q�0� �n�5q

��n� �p � sz �

gT�x� � 1

2M

� �� d�2�

ei�x �p � s� ��q�0��� �5q��n� �p � s� �

— quark distributions in longitudinally and transversely polarized nucleon, respectively

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Polarized deep inelastic scattering —continued

Beyond the tree level

�N�p � s� ��q�z1�G�

z2�q�z3� �N �p � � �� �

� � � �� 11

dx1dx2dx3 � �x1� x2

� x3� eip�x1z1�x2z2�x3z3�Dq�xi � �2�

�q G q�

gpn2

�xB � �2�

�q G qG G G � �

gp�n2

�xB � �2�

quark-gluon correlations in the nucleon

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Renormalization of quark-gluon operators, flavor non-singlet

� qGq� Nc� �0� � 2

Nc� �1� �

� �0� � V �0�qg�J12� � U �0�qg

�J23� �

� �1� � V �1�qg�J12� � U �1�qg

�J23� � U �1�qq

�J13� �

where

V �0�qg�J� �

��J � 3�2� �

��J � 3�2� � 2

��1� � 3�4 �

U �0�qg�J� �

��J � 1�2� �

��J � 1�2� � 2

��1� � 3�4 �

V �1�qg�J� � ��1�J5�2�

J � 3�2� �J � 1�2� �J � 1�2� �U �1�qg

�J� � � ��1�J5�2

2�J � 1�2� �

V �1�qq�J� �

��J� �

��1� � 3�4 �

U �1�qq�J� � 1

2

���J � 1� �

��J � 1�� �

��1� � 3�4 �

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Spectrum of Anomalous Dimensions

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20

EN

N

qGq levels: crosses GGG levels:open circles

The highest qGq level is separated fromthe rest by a finite gap and is almost de-generate with the lowest GGG state

The lowest qGq level is known exactlyin the largeNc limit

Ali, Braun, Hiller ‘91

� Interacting open and closed spin chains

Braun, Korchemsky, Manashov ’01

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Flavor singletqGq andGGG operators: WKB expansion

Energies: j � N � 3

� lowq

� Nc

�2

��j� � 1

j� 1

2� 2�E� � 2

Nc

�ln j � �

E� 3

4� �2

6� � � ��

� highq

� Nc

�4 ln j � 4�E � 19

6� 19

3j2� � 2

3nf� � �

1�j3�� low

G� Nc

�4 ln j � 4�E

� 1

3� �2

3� 1

j

�2�2�18

3

�ln j � �

E � � �2

3 �� � � �ln2 j�j2�

To the leading logarithmic accuracy, the structure function g2�x �Q2� is expressed through the

quark distribution

g2�x �Q2� � gWW

2�x �Q2� � 1

2

�q

e2q

� 1

x

dy

y�q

�T

�y �Q2�

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Approximate evolution equation for the structure functiong2 �x � Q2�

Introduce flavor-singlet quark and gluon transverse spin distributions

Q2 d

dQ2�q

� �ST

�x �Q2� �

s

4�� 1

x

dy

y PTqq�x�y��q

� �ST

�y �Q2� � PT

qg�x�y��gT

�y �Q2�

Q2 d

dQ2�gT

�x �Q2� �

s

4�� 1

x

dy

yPT

gg�x�y��gT

�y �Q2�

with the splitting functions

PTqq�x� � �

4CF

1 � x�� � � �1 � x�

�CF

� 1

Nc

�2 � �2

3 �� � 2CF �PT

gg�x� � �

4Nc

1 � x�� � � �1 � x�

�Nc

��2

3� 1

3� � 2

3nf �

� Nc��2

3� 2� � Nc ln

1 � x

x

�2�2

3� 6� �

PTqg�x� � �4nf x � 2

�1 � x�2 ln

�1 � x�

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

to summarize:

It is possible to write evolution equations in the SL(2)–covariant form

obtain quantum mechanics with hermitian Hamiltonian

uncover hidden symmetry (integrability)

powerful tool in applications to many-parton systems

Similar technique is used for studies of the interactions between reggeized gluons

Interesting connections toN � 4 SUSY and AdS/CFT correspondence

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Symmetry Breaking

Noether currents corresponding to dilatation and special conformal transformations

J�D� x���� � J

�K �� � �2x� x� � x2g�� � ��� �

are conserved

��J�D� 0 � ��J

�K �� � 0

if and only if the improved energy-momentum tensor��� is divergenceless and traceless

��� �x� � � �� �x� � � ���� �x� � 0 � g����� �x� � 0 �leading to the conserved charges

D � �d3x J0

D�x� � K� � �

d3x J0K �� �

In a quantum field theory these conditions cannot be preserved simultaneously

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Anomaly

Consider YM theory with a hard cutoffM, integrate out the fields with frequencies above�Seff

� � 1

4

�d4x

� 1�g(0)�2 � �

0

16�2ln M2��2� �

Ga��Ga�� �slow

�x� � � � �

Under the scale transformationx� � �x� , A� �x� � �A� ��x�,��x� � �3�2

���x� and

� � ��� with the fixed cutoff

�S � � 1

32�2�

0 ln � �d4x Ga��Ga�� �x�

which implies

��J�D

�x� � � �

0

32�2Ga��Ga�� �x�

Restoring the standard field definition

�� J�D

�x� � g���QCD�� �x� EOM� � �g�

2gGa��Ga�� �x�

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Anomaly —continued

Hope is to find a representation, for a generic quantity�

� � �con �� �g�

g� � � where � � � power series in s �

Main result (D.Müller): possible in a special renormalization scheme

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Ward Identities

The Ward identities follow from the invariance of the functional integral under the change ofvariables

� �x� � � � �x� � � �

x� � �� �x�Let

��N � � �

0�T� �x1� � � �� �xN � �0� � � 1� � � � �

x1� � � �� �xN � eiS�� �

then

0 � �0�T�� �x1� � � �� �xN � �0� � � � � � �

0�T� �x1� � � � �� �xN � �0� � �0�Ti�S

� �x1� � � �� �xN � �0�

Dilatation

N�i�1

��can� � xi � �i � �0�T� �x1� � � �� �xN � �0� � �i

�d4x

�0�T�D

�x� � �x1� � � �� �xN � �0� �

Special conformal transformation

N�i�1

�2x�i

��can� � xi � �i � � 2��� x�i � x2i � �i � ��

N � � �i�

d4x 2x� �0�T�D

�x� � �x1� � � �� �xN � �0� �

where

�D�x�� � �g�

2gGa��Ga�� �x� � EOM

V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

Conformal Ward Identities —continued

in particular, for conformal operators

N�i�1

�i� ��nl ��N � � � n�

m�0

��canl �nm

� ��nm � � ��ml ��N � �

N�i�1

� i

� ��nl ��N � � in�

m�0

�a�n � l��nm

� �� cnm�l�� � ��ml1 ��N � �

Dilation WI is nothing else as the Callan–Symanzik equationand ��nm is the usualanomalous dimension matrix

It is determined by the UV regions and diagonal in one loop

�� cnm�l� is called special conformal anomaly matrix

It is determined by both UV and IR regions and isnot diagonal in one loop

However, nondiagonal entries in�� cnm�l� can be removed by a finite renormalization

After this is done, all conformal breaking terms can be absorbed through the redefinition of thescaling dimension and conformal spin of the fields�can

n� �n

� s � � �can

n� �

n�

s � �V. M. Braun The Uses of Conformal Symmetry in QCD

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Conformal Group and its Collinear Subgroup Conformal Parti al Wave Expansion Hamiltonian Approach to QCD Evolution Equations Beyond Leading Logarithms

to summarize:

Thank you for attention!

V. M. Braun The Uses of Conformal Symmetry in QCD