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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
,α
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
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
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
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
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
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
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
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
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
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