The Uses of Conformal Symmetry in QCD
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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 xµdx dx 2 only change the scale of the metric g ¼ x¼ µ xµg xµ µ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts Dilatation (global scale transformation) x x ¼ x Inversion x x ¼ x x 2
5 Conformal Transformations 1 conserve the interval ds 2 g xµdx dx 2 only change the scale of the metric g ¼ x¼ µ xµg xµ µ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts Dilatation (global scale transformation) x x ¼ x Inversion x x ¼ x x 2 Special conformal transformation x x ¼ = inversion, translation x x a, inversion x a x 2 1 2a x a 2 x 2
6 Conformal Algebra The full conformal algebra includes 15 generators P M D K (4 translations) (6 Lorentz rotations) (dilatation) (4 special conformal transformations)
7 Collinear Subgroup P = P z Special conformal transformation x x ¼ x 1 2ax translations x x ¼ x c dilatations x x ¼ x form the so-called collinear subgroup SL 2 Rµ p p px p x 1 Ô 2 p 0 p zµ ½ 1 Ô 2 p 0 p zµ 0 ««¼ a «b ad bc 1 c «d «µ ¼ «µ c «dµ 2j a«b c«d where xµ x µ «n µ is the quantum field with scaling dimension and spin projection s living on the light-ray Conformal spin: j l sµ2
8 SL 2µ Algebra is generated by P, M, D and K L L 1 il 2 ip L L 1 il 2 i2µk L 0 i2µ D M µ E i2µ D M µ can be traded for the algebra of differential operators acting on the field coordinates L «µ L «µ L «µ L «µ L 0 «µ L 0 «µ L L L 0 d d««2 d d«2j ««d d«j They satisfy the SL 2µ commutation relations L 0 L L L L 2L 0
9 SL 2µ Algebra is generated by P, M, D and K L L 1 il 2 ip L L 1 il 2 i2µk L 0 i2µ D M µ E i2µ D M µ can be traded for the algebra of differential operators acting on the field coordinates L «µ L «µ L «µ L «µ L 0 «µ L 0 «µ L L L 0 d d««2 d d«2j ««d d«j They satisfy the SL 2µ commutation relations L 0 L L L L 2L 0 The remaining generator E counts the twist t s of the field E «µ 1 sµ «µ 2 collinear twist = dimension - spin projection on the plus-direction
10 Conformal Towers A local field operator «µ that has fixed spin projection on the light-cone is an eigenstate of the quadratic Casimir operator i012 with the operator L 2 defined as L i L i «µ j j 1µ «µ L 2 «µ L 2 L 2 0 L2 1 L2 2 L 2 L i 0 Moreover, the field at the origin 0µ is an eigenstate of L 0 with the eigenvalue j 0 j and is annihilated by L L 0µ 0 L 0 0µ j 0µ A complete operator basis can be obtained from 0µ by applying the raising operator L Ç k «µ k L L L 0µ µ «0 Ç 0 0µ From the commutation relations it follows that L 0 Ç k k jµç k non-compact spin: j 0 j k, k 0 1 2
11 Adjoint Representation Two different complete sets of states: «µ for arbitrary «Ç k for arbitrary k Relation = Taylor series: ½ «µ k «µ Ç k k k0 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 where «dd«ç k È k «µ «µ «0 È k uµ uµ 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 È uµ u u jµ È uµ L È uµ u È uµ L È uµ u u 2 2ju È uµ
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 sµ involves the spin projection on plus direction. This is not helicity! Example: j µ 1 2 j µ 12 2 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 2µ SO 2 1µ We are dealing with the infinite-dimensional representation of the collinear conformal group
13 Separation of variables In Quantum Mechanics: O(3) rotational Angular vs. radial µ symmetry dependence 2 2m V rµ E µ rµ R rµy lm µ Y lm µ are eigenfunctions of L 2 Y lm l l 1µY lm, À L 2 0. In Quantum Chromodynamics: SL(2,R) conformal symmetry µ 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: Ê 2 d d 2 u µ 1 0 dw À u wµ u µ µ u µ 6u 1 uµ 1 2 µc32 2 2u 1µ n µ n 0µ «s µ nb0 «s 0 µ Summation goes over total conformal spin of the qq pair: j j q j q n n 2 Gegenbauer Polynomials Cn 32 are spherical harmonics of the SL 2 Rµ group Exact analogy: Partial wave expansion in Quantum Mechanics For Baryons: Summation of three conformal spins ½ J 1 N u iµ 120u 1u 2u 3 J3 j2 Jj N µ Y 12µ3 Jj uµ 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: j 1 Å j 2 Å n0 j 1 j 2 n For two fields living on the light-cone O «1 «2 µ j1 «1 µ j2 «2 µ We define a conformal operator L 2 È j j j 1µÈ j L0 È j j 0 È j L È j 0 The solution reads j 0 j j 1 j 2 n and or, equivalently, local composite operators Ç j 0µ È j 1 2 µ j1 «1 µ j2 «2 µ «1«20 the two-particle SL(2) generators are L a L 1a L 2a a L 2 a012 L 1a L 2a 2 È j 1 j 2 j u 1 u 2 µ u 1 u 2 µ n P 2j 1 12j 2 1µ n u2 u 1 u 1 u 2 where P abµ n xµ are the Jacobi polynomials Call the corresponding operator Ç j 1 j 2 j
16 Spin Summation -continued Three-particle: local composite operators j 1 Å j 2 Å j 2 Å n0 j 1 j 2 j 3 N Impose following conditions: µ Ç 0µ È µ j1 «1 µ j2 «2 µ j3 «3 µ «i 0 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 È 12µ3 Jj u i µ 1 u 3 µ j j 1 j P 2j j 1µ J j j 3 1 2u 3 µ P 2j 1 12j 2 1µ u2 u 1 j j 1 j 2 1 u 3 if a different order of spin summation is chosen È 31µ2 Jj u i µ Å jj ¼ Jµ È 12µ3 Jj¼ u i µ j 1 j 2 j¼j j 3 the matrix Å defines Racah 6j-symbols of the SL 2 ʵ group Braun, Derkachov, Korchemsky, Manashov 99
17 Conformal Scalar Product Correlation function of two conformal operators with different spin must vanish: 0T Ç j 1 j 2 j xµç j 1 j 2 j¼ 0µ0 Æ jj ¼ Evaluating this in a free theory: 1 È nè m du u 2j u 2j È n u 1 u 2 µè m u 1 u 2 µ Æ mn 0 where du du 1 du 2 Æ u 1 u 2 1µ similar for three particles 1 È Jj È J ¼ j ¼ du u 2j u 2j u 2j È N ¼ q ¼ u 1 u 2 u 3 µè Nq u 1 u 2 u 3 µ Æ JJ ¼ Æ jj ¼ 0 Complete set of orthogonal polynomials w.r.t. the conformal scalar product
18 Pion Distribution Amplitude Definition: µ 0 d 0µ 5 u «µ pµ if p 0 d 0µ 5 i D nu 0µ pµ if p µ n du e iu«p u µ 0 du 2u 1µ n u µ 0 It follows 1 du P 11 n 2u 1µ u µ É11 n 0Én 11 0µ pµ if p n 1 É11 n 0 Since P 11 n xµ Cn 32 xµ form a complete set on the interval 1 x 1 u µ 6u 1 uµ n µ ERBL ½ n0 n µ C 32 n 2u 1µ 2 2n 3µ «s µ 3 n 1µ n 2µ É11 n µ n µ n 0 µ «s 0 µ 0µ n 0
19 Higher Twists Conformal symmetry is respected by the QCD equations of motion Example: Pion DAs of twist-three 0 d 0µi 5u «µ pµ R 1 0 du e iu«p p u µ R fm2 m u m d 0 d 0µ 5u «µ pµ ir«p du e iu«p u µ 0 0 d «2µ 5gG «3µu «1µ pµ 2ip 2 Conformal expansion: 3 u i µ 360u 1 u 2 u 2 3 R p uµ R C 12 2 µ 3 2 Ò R uµ 6u 1 uµ R 5 72 Braun, Filyanov 90 du i e ip «1 u 1 «2 u 2 «3 u 3 µ 3 u i µ u 3 3µ u 1 u 2 8u 3 8u 2 3 µ u 1 u 2 2u 3 3u 2 3 µ C 12 4 µ 1 92 C 32 2 µ C 32 4 µ Ó
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 «s 1 V 0 u vµ C F 2 2 d 1 d 2 u µ dv V u v «s µµ v µ 0 u 1 v 1 1 u u v vµ u 1 1 v uµ v 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 É «1 «2 µ «1 µ «2 µ gµ «sc F É «1 «2 µ g À 4 É «1 «2 µ À 2À 12µ v 2À 12µ e 1 Explicit calculation (a) (b) (c) À 12µ v É «1 «2 µ À 12µ e É «1 «2 µ Now verify du u 1 uµ É «u 12 «2µ É «1 «u 21 µ 2É «1 «2 µ du É «u 1 12 «u 2 21 µ «u 12 «1 1 uµ «2 u Balitsky, Braun 88 À L a É «1 «2 µ L a À É «1 «2 µ L aé «1 «2 µ L 1a L 2a µé «1 «2 µ À L À L À L 0 0
23 Conformally Covariant Representation if so, À must be a function of the two-particle Casimir operator L 2 12 «1 «2 «1 «2 µ 2 To find this function, compare action of À and L 2 12 on the set of functions «1 «2 µ n À 12µ v 2 J 12 µ 2µ L 2 12 J 12 J 12 1µ À 12µ e 1J 12 J 12 1µ 1L 2 12 À ERBL 4 J 12 µ 2µ 2J 12 J 12 1µ 1 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 0q z 1 µq z 2 µq z 3 µb p µ dx i e ip x 1z 1 x 2 z 2 x 3 z 3 µ ³B x i 2 µ q q q µ ³ 32 x i 2 µ q q q µ ³ 12 N x i 2 µ ³ 12 x i 2 µ
25 Spectrum of anomalous dimensions Operator product expansion µ mixing matrix: N k 1 k 2 k 3 E3/2(N) ³ x i µ ³ k i µ x i x k 1 1 xk 2 2 xk 3 3 ³ x i 2 µ q z 1 µq z 2 µq z 3 µ D k 1 qµ Dk 2 qµ Dk 3 qµ γ n=0,1,... N= N In the light-ray formalism Two-particle structure: gµ g À qqq À 12 À 23 À 13 Example: qqq 32 B À B B z 1 z 2 z 3 µ ³ q z 1 µq z 2 µq z 3 µ
26 Hamiltonian Approach write À as a function of Casimir operators L 2 ik z i z z i z k kµ 2 Jik Jik 1µ µ À v 12 2 J12µ 2µ À 32 qqq À v 12 À v 23 À v µ À eµ 12 1 L 2 1 J J 12 1µ À 12 qqq À 32 qqq À e 12 À e Have to solve À Nn Nn Nn A Schrödinger equation with Hamiltonian À
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 Æ xi 1µ x 2j x 2j x 2j x iµ 2 x i µ that is, conformal symmetry allows for the transformation of the mixing matrix: forward non-hermitian ßÞ Ð N 1 non-forward µ hermitian ßÞ Ð N N 1µ2 µ hermitian in conformal basis ßÞ Ð 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 Nn N ce Nn Nc 1 ÆE Nn Nn 0µ Nn N c 2 Æ Nn with the usual quantum-mechanical expressions etc. ÆE Nn 0µ Nn 2 0µ Nn À 1µ 0µ Nn
29 Complete Integrability Conformal symmetry implies existence of À L 2 À L 0 0 two conserved quantities: For qqq and GGG states with maximum helicity and for qgq states at N c ½ there exists an additional conserved charge qqq GGG qgq max µ N c½ µ Q L 2 12 L2 23 À Q 0 Q L 2 qg L2 Gq c 1L 2 qg c 2L 2 Gq À Q 0 BDM 98 Anomalous dimensions can be classified by values of the charge Q: À Q q µ qµ Lipatov 94 95, Faddeev, Korchemsky 95 Braun, Derkachov, Manashov 98 A new quantum number
30 Complete Integrability continued q/ 3 η E3/2(N) l = 0 l = 2 l = N Example: qqq N Double degeneracy: N qµ N qµ q N µ q N N µ BDKM 98
31 WKB expansion of the Baxter equation µ 1/N expansion equation Q q is much simpler as À non-integrable corrections are suppressed at large N can use some techniques of integrable models Korchemsky N µ 6 ln 3 ln E 3 2 1µ Ô N 3µ N 2µ An integer numerates the trajectories: (semiclassically quantized solitons: Korchemsky, Krichever, 97) Integrability imposes a nontrivial analytic structure
32 Breakdown of integrability: Mass gap and bound states Simplest case: Difference between 32 and N µ 12 À µ À 32 1 L L 2 23 Flow of energy levels (anomalous dimensions): À 1µ À 12 Scalar diquark BDKM 98
33 Advanced Example: Polarized deep-inelastic scattering Cross section: pqm E l E l ¼ d 2 8«2 E µ l¼ 2 dåde l ¼ Q 4 F 2 x Q 2 µ cos M F 1 x Q 2 µ sin 2 2 d 2 8«2 E l ¼ µ dåde l ¼ Q 4 x g 1 x Q 2 µ 1 El¼ 2Mx cos g E l 2 x Q 2 µ E l At tree level: g 1 xµ g T xµ ½ d ½ 2 eix p s zq 0µ n 5 q nµp s z 1 ½ d 2M ½ 2 eix p s q 0µ 5 q nµp s quark distributions in longitudinally and transversely polarized nucleon, respectively
34 Polarized deep inelastic scattering continued Beyond the tree level N p sµq z 1 µg z 2 µq z 3 µn p µ 1 dx 1 dx 2 dx 3 Æ x 1 x 2 x 3 µ e ip x 1z 1 x 2 z 2 x 3 z 3 µ D q x i 2 µ 1 q G q µ g p n 2 x B 2 µ q G q µ g p n G G G 2 x B 2 µ quark-gluon correlations in the nucleon
35 Renormalization of quark-gluon operators, flavor non-singlet where À qgq N cà 0µ 2 N c À 1µ À 0µ V 0µ qg J 12 µ U 0µ qg J 23 µ À 1µ V 1µ qg J 12 µ U 1µ qg J 23 µ U 1µ qq J 13 µ V 0µ qg Jµ J 32µ J 32µ 2 1µ 34 U 0µ qg Jµ J 12µ J 12µ 2 1µ 34 V 1µ qg Jµ U 1µ qg Jµ 1µ J 52 J 32µ J 12µ J 12µ 1µ J 52 2 J 12µ V 1µ qq Jµ Jµ 1µ 34 U 1µ qq Jµ 1 J 1µ J 1µ 1µ 34 2
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 q low N c 2j 1 j q high 19 N c 4 ln j 4 E 6 G low N c 4 ln j 4 E E 19 3j j 2 ln j E 3 N c 4 3 n f Ç 1j 3 µ ln j E µ Ç ln 2 jj 2 µ To the leading logarithmic accuracy, the structure function g 2 x Q 2 µ is expressed through the quark distribution 1 g 2 x Q 2 µ g WW 2 x Q 2 µ 1 2 q e 2 q x dy y q T y Q2 µ
38 Approximate evolution equation for the structure function g 2 x Q 2 µ Introduce flavor-singlet quark and gluon transverse spin distributions Q 2 d q S dq 2 T x Q 2 µ «s 1 4 x Q 2 d dq 2 g T x Q 2 µ «s 1 4 x with the splitting functions dy y P T qq xyµ q S T y Q 2 µ P T qg xyµ g T y Q 2 µ dy y PT gg xyµ g T y Q 2 µ P T 4CF qq xµ 1 x P T gg xµ 4Nc 1 x Æ 1 xµ C F 1 2 N c 2 1 Æ 1 xµ N c N c 2 N c ln 1 x 3 x P T qg xµ 4n f x 2 1 xµ 2 ln 1 xµ n f 6 2C 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 are conserved J D x J K«2x x «x 2 g «J D 0 J K«0 if and only if the improved energy-momentum tensor is divergenceless and traceless xµ xµ xµ 0 g xµ 0 leading to the conserved charges D d 3 x J D xµ 0 K«d 3 x J 0 K«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) µ ln M2 2 G a G a slow xµ Under the scale transformation x x, A xµ A xµ, xµ 32 xµ and with the fixed cutoff 1 ÆS ln d 4 x G a Ga xµ which implies Restoring the standard field definition J D xµ Ga G a xµ J D xµ g QCD EOM xµ gµ 2g Ga Ga xµ
42 Conformal Anomaly continued Hope is to find a representation, for a generic quantity É É É con gµ É where É power series in «s g 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 xµ ¼ xµ xµ Æ xµ Let N 0T x 1 µ x N µ0 Æ 1 x 1 µ x N µ e is µ then 0 0TÆ x 1 µ x N µ0 0T x 1 µ Æ x N µ0 0TiÆS x 1 µ x N µ0 Dilatation N i1 can x i i µ 0T x 1 µ x N µ0 i Special conformal transformation N 2x i can x i i µ 2 x i x 2 i i i1 where d 4 x0t D xµ x 1 µ x N µ0 N i d 4 x 2x 0T D xµ x 1 µ x N µ0 D xµ gµ 2g Ga G a xµ EOM
44 Conformal Ward Identities continued in particular, for conformal operators N i1 N i1 i Ç nl N à i Ç nl N i n m0 n m0 can l Æ nm nm Ç ml N a n lµæ nm c nm lµ Ç ml 1 N 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 lµ 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 lµ 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 µ n «sµ can n n «sµ
45 to summarize: Thank you for attention!
The Uses of Conformal Symmetry in QCD
The Uses of Conformal Symmetry in QCD V. M. Braun University of Regensburg DESY, 23 September 2006 based on a review V.M. Braun, G.P. Korchemsky, D. Müller, Prog. Part. Nucl. Phys. 51 (2003) 311 Outline
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