The Uses of Conformal Symmetry in QCD

Transcription

1 The Uses of Conformal Symmetry in QCD V. M. Braun University of Regensburg DESY, 23 September 2006

2 based on a review V.M. Braun, G.P. Korchemsky, D. Müller, Prog. Part. Nucl. Phys. 51 (2003) 311

3 Outline 1 Conformal Group and its Collinear Subgroup 2 Conformal Partial Wave Expansion 3 Hamiltonian Approach to QCD Evolution Equations 4 Beyond Leading Logarithms

4 Conformal Transformations 1 conserve the interval ds 2 g x dx dx 2 only change the scale of the metric g 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 x 2

5 Conformal Transformations 1 conserve the interval ds 2 g x dx dx 2 only change the scale of the metric g 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 x 2 Special conformal transformation x x x a x 2 1 2a x a 2 x 2 = inversion, translation x x a, inversion

6 Conformal Algebra The full conformal algebra includes 15 generators P (4 translations) M (6 Lorentz rotations) D (dilatation) K (4 special conformal transformations)

7 Collinear Subgroup P = P z x x x Special conformal transformation 1 2ax translations x x x c dilatations x x x form the so-called collinear subgroup SL 2 R 1 p p0 2 p z p 1 p0 2 p z 0 px p x a b c d ad bc 1 c d 2j a b c d where x x n is the quantum field with scaling dimension and spin projection s living on the light-ray Conformal spin: j l s 2

8 SL 2 Algebra is generated by P, M, D and K L L1 il 2 ip L L1 il 2 i2 K L 0 i2 D M E i2 D M can be traded for the algebra of differential operators acting on the field coordinates L L L L L0 L0 L d L L 0 d 2 d 2j d d d j They satisfy the SL 2 commutation relations L0 L L L L 2L0

9 SL 2 Algebra is generated by P, M, D and K L L1 il 2 ip L L1 il 2 i2 K L 0 i2 D M E i2 D M can be traded for the algebra of differential operators acting on the field coordinates L L L L L0 L0 L d L L 0 d 2 d 2j d d d j They satisfy the SL 2 commutation relations L0 L L L L 2L0 The remaining generator E counts the twist t s of the field E 1 s 2 collinear twist = dimension - spin projection on the plus-direction

10 Conformal Towers A local field operator that has fixed spin projection on the light-cone is an eigenstate of the quadratic Casimir operator i Li L i j j 1 L 2 with the operator L 2 defined as L 2 L 2 0 L 2 1 L 2 2 L 2 L i 0 Moreover, the field at the origin 0 is an eigenstate of L 0 with the eigenvalue j 0 annihilated by L L 0 0 L0 0 j 0 j and is A complete operator basis can be obtained from 0 by applying the raising operator L 0 0 k L L L 0 k 0 From the commutation relations it follows that L0 k k j k non-compact spin: j 0 j k, k 0 1 2

11 Adjoint Representation Two different complete sets of states: for arbitrary k for arbitrary k Relation = Taylor series: k 0 k k k In general: Two equivalent descriptions light-ray operators (nonlocal) local operators with derivatives (Wilson) Characteristic polynomial Any local composite operator can be specified by a polynomial that details the structure and the number of derivatives acting on the fields. For example k k 0 k where dd u u k The algebra of generators acting on light-ray operators can equivalently be rewritten as algebra of differential operators acting on the space of characteristic polynomials. One finds the following 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

12 to summarize: A one-particle local operator (quark or gluon) can be characterized by twist t, conformal spin j, and conformal spin projection j 0 j k Conformal spin j 12 s involves the spin projection on plus direction. This is not helicity! Example: j j The difference k j 0 j 0 is equal to the number of (total) derivatives Exact analogy to the usual spin, but conformal spin is noncompact, corresponds to hyperbolic rotations SL 2 SO 2 1 We are dealing with the infinite-dimensional representation of the collinear conformal group

13 Separation of variables In Quantum Mechanics: O(3) rotational symmetry Angular vs. radial dependence 2 2m V r E r R r Y lm Y lm are eigenfunctions of L 2 Y lm l l 1 Y lm, L 2 0. In Quantum Chromodynamics: SL(2,R) conformal symmetry Longitudinal vs. transverse dependence Migdal 77 Brodsky,Frishman,Lepage,Sachrajda 80 Makeenko 81 Ohrndorf 82

14 Separation of variables continued Momentum fraction distribution of quarks in a pion at small transverse separations: Dependence on transverse coordinates is traded for the scale dependence: 2 d d u dw u w u u 6u 1 u 1 2 C u 1 n n 0 s s 0 n b0 Summation goes over total conformal spin of the qq pair: j j q j q n n 2 are spherical harmonics of the SL 2 R group Gegenbauer Polynomials Cn 3 2 Exact analogy: Partial wave expansion in Quantum Mechanics For Baryons: Summation of three conformal spins N ui J 1 Jj 120u 1u 2u 3 N Y123 Jj u J3 j2 J 3 N N 0 1 total spin of three quarks j 2 n n 0 1 spin of the (12)-quark pair

15 Spin Summation Two-particle: j1 j 2 n 0 j1 j 2 n For two fields living on the light-cone O 1 2 j 1 1 j 2 2 or, equivalently, local composite operators j 0 j j 1 2 j the two-particle SL(2) generators are L a L1 a L 2 a a L 2 a L1 a L 2 a 2 We define a conformal operator L 2 The solution reads j 0 j1 j 2 j L0 j j j 1 j j j0 j L j 0 j j 1 j 2 n and u1 u 2 u1 u 2 n P 2j 1 1 2j 2 1 n u 2 u 1 where P a b n u 1 u 2 x are the Jacobi polynomials Call the corresponding operator j1 j 2 j

16 Spin Summation -continued Three-particle: j1 j 2 j 2 n 0 j1 j 2 j 3 N local composite operators j 1 1 j 2 2 j 3 3 i 0 Impose following conditions: fixed total spin J j 1 j 2 j 3 N annihilated by L fixed spin j j 1 j 2 n in the (12)-channel 12 3 Jj ui 1 u 3 j j1 j2 P J3 1 2j 2j 1 j j3 1 2u 3 P j1 2j 1 2j 2 1 u j1 2 j2 u 1 1 u 3 if a different order of spin summation is chosen 31 2 Jj ui jj j1 j2 j J J 12 3 ui Jj j3 the matrix defines Racah 6j-symbols of the SL 2 group Braun, Derkachov, Korchemsky, Manashov 99

17 Conformal Scalar Product Correlation function of two conformal operators with different spin must vanish: Evaluating this in a free theory: 0 0 T j1 j 2 j x j1 j 2 j 0 0 jj n m 1 du u 1 2j 1 1 u 2 2j 1 2 n u1 u 2 m u1 u 2 mn where du du 1 du 2 u 1 u 2 1 similar for three particles J j J j 1 0 du u 2j u 2j u 3 2j 1 3 Nq u1 u 2 u 3 N q u1 u 2 u 3 JJ jj Complete set of orthogonal polynomials w.r.t. the conformal scalar product

18 Pion Distribution Amplitude Definition: 0 d 0 5u p if p 0 d 0 5 It follows 1 du P 1 1 n 0 Since P 1 1 n i 1 D n u 0 p if p n 1 1 2u 1 u 1 1 n n 0 du e iu p u 0 du 2u 1 n u 0 p if p n 1 x Cn 3 2 x form a complete set on the interval 1 x 1 u 6u 1 u ERBL n 0 n 2 2n 3 3 n 1 n 2 n Cn 3 2 2u n 1 1 n n s n 0 s 0 0 n 0

19 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 du e iu p p u R f m 2 0 m u m d 0 d 0 5u 1 p ir 1 p du e iu p u d 2 5gG 3 u 1 p 2ip 2 du i e ip 1 u 1 2 u ui u Conformal expansion: 3 u i 360u1 u 2 u R p u R C u u 1 u 2 8u 3 8u R u 6u 1 u R C Braun, Filyanov 90 3u1 u 2 2u 3 3u C C 3 2 4

20 to summarize: Conformal symmetry offers a classification of different components in hadron wave functions The contributions with different conformal spin cannot mix under renormalization to one-loop accuracy Whether 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

21 Hamiltonian Approach to QCD Evolution Equations ERBL evolution equation d 2 d 2 u 1 dv V u v s v 0 V 0 u v C F s 2 1 u v u v u v u 1 1 v v u v u How to make maximum use of conformal symmetry? Bukhvostov, Frolov, Kuraev, Lipatov 85

22 Where is the symmetry? RG equation for the nonlocal operator g 1 g 2 sc F v 2 12 e 1 Explicit calculation 12 v e 1 2 Now verify 0 1 du du u 1 u L a 1 2 L a 1 2 L a 0 (a) (b) u u 21 2 u1 12u 2 21 u u 2u 1 2 L 1 a L 2 a (c) 1 2 Balitsky, Braun L L L 0 0

23 Conformally Covariant Representation if so, must be a function of the two-particle Casimir operator L To find this function, compare action of and L 2 12 on the set of functions 1 2 n 12 v 2 J 12 2 L 2 12 J 12 J e 1 J 12 J12 1 1L 2 12 ERBL 4 J J 12 J For local operators take L 2 12 in the adjoint representation L 2 12 Bukhvostov, Frolov, Kuraev, Lipatov 85

24 Three-particle distributions E.g. baryon distribution amplitudes B N 0 q z 1 q z 2 q z 3 B p dxi e ip x 1z 1 x2 z 2 x3 z 3 q q q 3 2 xi 2 xi 2 q q q N xi B xi 2

25 Spectrum of anomalous dimensions Operator product expansion E3/2(N) xi k i mixing matrix: N k1 k 2 k 3 x i x k 1 1 xk 2 2 xk 3 3 x i 2 q z 1 q z 2 q z 3 D k 1 q D k 2 q D k 3 q γ n=0,1,... N= N Example: qqq 3 2 In the light-ray formalism g g B B B z 1 z 2 z 3 q z 1 q z 2 q z 3 Two-particle structure: qqq

26 Hamiltonian Approach write as a function of Casimir operators L 2 ik zi z k zi z k 2 J ik Jik 1 v 12 2 J qqq v12 v23 v e L J 12J qqq 3 2 qqq e 12 e Have to solve N n N n N n A Schrödinger equation with Hamiltonian

27 Hamiltonian Approach continued Hilbert space? Polynomials in interquark separation related by a duality transformation BDKM 99 Polynomials in covariant derivatives (local operators) E.g. going over to local operators, in helicity basis (BFKL 85 ) obtain a HERMITIAN Hamiltonian w.r.t. the conformal scalar product dx i x i 1 x 1 2j 1 1 x 2 2j 1 2 x 3 2j xi 2 xi that is, conformal symmetry allows for the transformation of the mixing matrix: forward non-hermitian N 1 non-forward hermitian N N 1 2 hermitian in conformal basis N 1

28 Hamiltonian Approach continued (2) Premium? Eigenvalues (anomalous dimensions) are real numbers Conservation of probability possibility of meaningful approximations to exact evolution E.g. 1N c expansion N n NcE N n N 1 c E N n N n 0 N n N 2 c N n with the usual quantum-mechanical expressions etc. E N n 0 N n 2 0 N n N n 1 0

29 Complete Integrability L 2 L 0 0 Conformal symmetry implies existence of two conserved quantities: For qqq and GGG states with maximum helicity and for qgq states at N c an additional conserved charge there exists qqq GGG max Q L12 2 L 2 23 Q 0 qgq Q LqG 2 LGq 2 c 1 L 2 qg c 2 L 2 Gq N c Q 0 BDM 98 Anomalous dimensions can be classified by values of the charge Q: Q q q A new quantum number Lipatov 94 95, Faddeev, Korchemsky 95 Braun, Derkachov, Manashov 98

30 Complete Integrability continued q/ 3 η E3/2(N) l = 0 l = 2 l = N Example: qqq N Double degeneracy: Nq Nq q N q NN BDKM 98

31 WKB expansion of the Baxter equation 1/N expansion equation Q q is much simpler as non-integrable corrections are suppressed at large N can use some techniques of integrable models Korchemsky N 6 ln 3 ln E An integer numerates the trajectories: (semiclassically quantized solitons: Korchemsky, Krichever, 97) Integrability imposes a nontrivial analytic structure N 3 N 2

32 Breakdown of integrability: Mass gap and bound states Simplest case: Difference between 3 2 and N 1 2 Flow of energy levels (anomalous dimensions): L 2 12 L Scalar diquark BDKM 98

33 Advanced Example: Polarized deep-inelastic scattering Cross section: pqm E l E l At tree level: d 2 d de l 82 El 2 Q 4 F 2 x Q 2 cos M F 1 x Q 2 sin 2 d 2 d de l 8 2 E l Q 4 x g 1 x Q 2 1 E l cos E l 2Mx g 2 E l x Q 2 g 1 x g T x 1 2M d 2 ei x p s z q 0 n 5q n p s z d 2 p s eix q 0 5q n p s quark distributions in longitudinally and transversely polarized nucleon, respectively 2

34 Polarized deep inelastic scattering continued Beyond the tree level N p s q z 1 G z 2 q z 3 N p 1 1 dx 1 dx 2 dx 3 x 1 x 2 x 3 e ip x 1z 1 x2 z 2 x3 z 3 Dq xi 2 q G q g p n 2 xb 2 q G q G G G g p n 2 xb 2 quark-gluon correlations in the nucleon

35 Renormalization of quark-gluon operators, flavor non-singlet 1 V 1 qg qgq N c 0 2 N c 1 0 V 0 qg J12 U 1 qg J12 U 0 qg J23 J13 J23 U 1 qq where V 0 qg J J 32 J U 0 qg J J 12 J V 1 1 qg J J 5 2 J 32 J 12 J 12 U 1 qg J 1 J J 12 V 1 qq J J 1 34 U 1 1 qq J J 1 J

36 Spectrum of Anomalous Dimensions E N,α qgq levels: crosses Interacting open and closed spin chains N GGG levels: open circles The highest qgq level is separated from the rest by a finite gap and is almost degenerate with the lowest GGG state The lowest qgq level is known exactly in the large N c limit Ali, Braun, Hiller 91 Braun, Korchemsky, Manashov 01

37 Flavor singlet qgq and GGG operators: WKB expansion Energies: j N 3 low q high q low G N c N c N c 2 j E j 2 2 ln j E 3 2 N c ln j 4 E j 2 3 n f 1j 3 4 ln j 4 E j ln j E 2 3 ln 2 jj 2 To the leading logarithmic accuracy, the structure function g 2 x Q 2 is expressed through the quark distribution g 2 x Q 2 g WW 2 x Q q e 2 q 1 x dy q T y y Q 2

38 Approximate evolution equation for the structure function g 2 x Q 2 Introduce flavor-singlet quark and gluon transverse spin distributions Q 2 d dq 2 q S T Q 2 d x Q 2 s 4 1 dq 2 g T x Q 2 s with the splitting functions P T qq x P T gg x x 4 1 x dy y PT qq dy y PT gg 4C F 1 1 x x 4N c 1 x N c 2 1 x xy q S T y Q 2 P T qg xy g T y Q 2 xy g T y Q 2 C F 1 N c N c N c ln 1 x x Pqg T x 4n f x 2 1 x 2 ln 1 x n f C F

39 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 to N 4 SUSY and AdS/CFT correspondence

40 Conformal Symmetry Breaking Noether currents corresponding to dilatation and special conformal transformations J D x J K 2x x x 2 g are conserved J D 0 J K 0 if and only if the improved energy-momentum tensor x x leading to the conserved charges x 0 g D d 3 x J 0 D x K d 3 x J 0 K is divergenceless and traceless x 0 In a quantum field theory these conditions cannot be preserved simultaneously

41 Conformal Anomaly Consider YM theory with a hard cutoff M, integrate out the fields with frequencies above S eff 1 4 d 4 x 1 g (0) ln M2 Ga G a slow x Under the scale transformation x x, A x A x, with the fixed cutoff which implies Restoring the standard field definition S ln d 4 x G a G a x J D x Ga G a x J D x g QCD x g EOM 2g Ga G a x x 3 2 x and

42 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

43 Conformal Ward Identities The Ward identities follow from the invariance of the functional integral under the change of variables x x x x Let N 0 T x 1 x N 0 1 x1 x N e is then 0 0 T x 1 x N 0 0 T x 1 x N 0 0 Ti S x 1 x N 0 Dilatation N i 1 can x i i 0 T x 1 x N 0 i Special conformal transformation N i 1 where 2x i can x i i 2 x i x 2 i i N i g D x 2g Ga G a x EOM d 4 x 0 T D x x 1 x N 0 d 4 x 2x 0 T D x x 1 x N 0

44 Conformal Ward Identities continued in particular, for conformal operators N i 1 N i i nl N n nl N i n m 0 can l nm nm ml N a n l nm c nm l ml 1 N i 1 m 0 Dilation WI is nothing else as the Callan Symanzik equation and nm is the usual anomalous dimension matrix It is determined by the UV regions and diagonal in one loop c nm l is called special conformal anomaly matrix It is determined by both UV and IR regions and is not diagonal in one loop However, nondiagonal entries in c nm l can be removed by a finite renormalization After this is done, all conformal breaking terms can be absorbed through the redefinition of the scaling dimension and conformal spin of the fields can n n s can n n s

45 to summarize: Thank you for attention!