Integrability in QCD and beyond Vladimir M. Braun University of Regensburg thanks to Sergey Derkachov, Gregory Korchemsky and Alexander Manashov KITP, QCD and String Theory Integrability in QCD and beyond - p. 1/35
Regensburg KITP, QCD and String Theory Integrability in QCD and beyond - p. 2/35
Outline In this talk: Renormalization in conformally-covariant form Integrability on the example of three-quark operators in QCD Violation of integrability: Mass gaps SUSY extensions: From Æ ¼ to Æ Further QCD example: Polarized deep-inelastic scattering not included: High energy scattering in QCD Full Æ SUSY and gauge/string correspondence KITP, QCD and String Theory Integrability in QCD and beyond - p. 3/35
Heisenberg spin chain 1926: Heisenberg Complete integrability À Ä Ò Ë Ò Ë Ò 1981-83: Generalizations for arbitrary spin Ñ Number of degrees of freedom = number of conservation laws À À  ÒÒ µ Ä Ò Â ÒÒ Â ÒÒ µ Ë Ò Ë Ò µ Conceptually interesting Powerfull machinery: Algebraic Bethe Ansatz (ABA) Method of separation of Variables (SoV) À ܵ µ Ü µ KITP, QCD and String Theory Integrability in QCD and beyond - p. 4/35
ÃÄ Ò µ Æ at the same time, in QCD... 1973-74: Anomalous dimensions of leading twist operators Æ µ Æ µ Æ µ µ Æ 1976-78: High-energy (Regge) asymptotics of scattering amplitides Ò 1991: Anomalous dimensions of quark-antiquark-gluon operators, Æ 40 35 γ N 30 25 Hidden symmetry? 20 15 10 Exact analytic solution 5 10 15 20 25 30 N KITP, QCD and String Theory Integrability in QCD and beyond - p. 5/35
Õ Õ Õ µ ³ quark fields live on a light-ray Þ ¼ Example: Twist-three operators Leading-twist three-particle parton distributions E.g. baryon distribution amplitudes Æ ¼Õ Þ µõ Þ µõ Þ µ Ô µ Ü Ü Ü Æ È Ü µ Ô Ü Þ Ü Þ Ü Þ µ ³ Ü µ ¼ Ü µ Æ Ü µ ³ Õ Õ Õ µ Ü µ ³ KITP, QCD and String Theory Integrability in QCD and beyond - p. 6/35
Problem: Prolifiteration of degrees of freedom Moments of distribution amplitudes local operators: ³ Ü µ ³ µ Ü Ü Ü ³ Ü µ Ü µõ Þ µõ Þ µ Õµ Õµ Õµ Õ Þ Mixing matrix: Æ E3/2(N) 16 14 12 10 8 6 4 2 0 γ n=0,1,... N=14 Example: qqq 0 5 10 15 20 25 30 N Rich spectrum of anomalous dimensions reflects complexity of genuine degrees of freedom KITP, QCD and String Theory Integrability in QCD and beyond - p. 7/35
¼ Þ Þ Þµ ¼ Þµ Þ Þ Þ Þ second-order differential operators on the space of Þ Þ Þ µ Þµ Þ Þ Þµ Þ Þ µ Conformal Symmetry on the Light-Cone: ËÄ Êµ Õ, is conformal spin of the field Ä Þµ Generators obey the ËÄ µ algebra Ä Þµ Þ Þ Þµ Þ Ä ¼ Þµ Casimir operators Ä Ä ¼ Ä ¼ Ä Ä Ä Þµ µ Þµ Summation of spins Ä Ä ««µ Ä È Ä «Ä Ä «Ä «µ «¼ «¼ È KITP, QCD and String Theory Integrability in QCD and beyond - p. 8/35
À ÕÕÕ À À À RG equations in ËÄ µ-covariant form Light-ray operators µ Two-particle structure: À Þ Þ Þ µ ³ Õ Þ µõ Þ µõ Þ µ Renormalization = Displacement: Balitsky, V.B. 88 À Þ Þ Þ µ «««µ «Þ «Þ «Þ «Þ Þ µ where Ê «Ê «««Æ «««µ, ««KITP, QCD and String Theory Integrability in QCD and beyond - p. 9/35
µ ³ ܵ Æ Üµ µ ³ ܵ RG equations in ËÄ µ-covariant form continued Conformal symmetry Ä ¼ À ¼ «µ «««««««³ ««Explicit calculation 1 2 3 1 2 3 KITP, QCD and String Theory Integrability in QCD and beyond - p. 10/35
Þ À Ú Â µ µ À µ À Ú ÀÚ ÀÚ ÕÕÕ À µ µ Ä RG equations in ËÄ µ-covariant form Hamiltonian approach À write as a function of Casimir Ä operators B FK L 85 Þ Þ µ Â Â µ Þ 1 2 3 Â µ Â ÕÕÕ À À ÕÕÕ À À 1 2 3 Have to solve À ÆÒ ÆÒ ÆÒ A Schrödinger equation with Hamiltonian À KITP, QCD and String Theory Integrability in QCD and beyond - p. 11/35
Hamiltonian approach continued going over to local operators: In helicity basis (BFKL 85 ) obtain a HERMITIAN Hamiltonian with respect to the conformal scalar product Æ Ü µ Ü Ü Ü Ü µ Ü µ Ü ¼ that is, conformal symmetry allows for the transformation of the mixing matrix: ¼ forward non-hermitian µ ¼ non-forward hermitian µ ¼ ÖÑØÒ Ò ÓÒÓÖÑÐ ßÞ Ð Æ ßÞ Ð Æ ßÞ Ð Æ µ Æ KITP, QCD and String Theory Integrability in QCD and beyond - p. 12/35
ÆÒ Æ ÆÒ Æ Æ ÆÒ ÆÒ ¼µ Hamiltonian approach continued (2) Eigenvalues (anomalous dimensions) are real numbers Conservation of probability possibility of meaningful approximations to exact evolution Systematic Æ expansion Æ ÆÒ Æ ÆÒ with the usual quantum-mechanical expressions ÆÒ ¼µ ÆÒ Æ ¼µ À µ ¼µ ÆÒ ÆÒ etc. KITP, QCD and String Theory Integrability in QCD and beyond - p. 13/35
Ä Ä É É ¼ À É Ä Õ Ä Õ Ä Õ Ä Õ À Õ É µ Õµ Complete Integrability Conformal symmetry implies existence of two conserved quantities: À Ä À Ä ¼ ¼ For ÕÕÕ and states with maximum helicity and for ÕÕ states at Æ there exists an additional conserved charge ÑÜ µ ÕÕÕ BDM 98 Æ µ ÕÕ À É ¼ Anomalous dimensions can be classified by values of the charge É: A new quantum number KITP, QCD and String Theory Integrability in QCD and beyond - p. 14/35
WKB expansion of the Baxter equation µ 1/N expansion G. Korchemsky 95-97 equation É Õ is much simpler as À non-integrable corrections are suppressed at large Æ can use some techniques of integrable models Æ µ ÐÒ ÐÒ µ ¼ An integer numerates the trajectories: (semiclassically quantized solitons: Korchemsky, Krichever, 97) Integrability imposes a nontrivial analytic structure Ô µ Æ µ Æ KITP, QCD and String Theory Integrability in QCD and beyond - p. 15/35
Ô Æ µ Æ µ WKB expansion of the Baxter equation µ 1/N expansion continued The dispersion curve Õµ 15.0 Ν = 30 14.0 Õµ ÐÒ E(q) 13.0 Ê Æ µ Ç µ 12.0 11.0 Example: qqq exact WKB Æ are defined as roots of the cubic equation: 10.0-0.20-0.10 0.00 0.10 0.20 q/η 3 Æ Æ Õ ¼ KITP, QCD and String Theory Integrability in QCD and beyond - p. 16/35
WKB expansion of the Baxter equation µ 1/N expansion continued (2) q/ 3 η 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2 l = 0 l = 2 l = 7 Double degeneracy: 0 5 10 15 20 25 30 N Æ Õµ Æ Õµ 16 Õ Æ µ Õ Æ Æ µ 14 12 10 E3/2(N) 8 6 4 2 Example: qqq 0 0 5 10 15 20 25 30 N BDKM 98 KITP, QCD and String Theory Integrability in QCD and beyond - p. 17/35
The ground state solution (Õ ¼) for the wave function of For even Æ Õ ¼ is the solution with lowest energy Ü Ü ÆÕ¼ Ü µ Ü Ü µ Æ Ü µ Ü Ü Ü µ Æ Ü µ Ü Ü µ Æ Ü µ The lowest anomalous dimension Æ Õ ¼µ Æ µ Here Ü Ü Ü are fractions of the momentum carried by the three quarks KITP, QCD and String Theory Integrability in QCD and beyond - p. 18/35
Ä À µ À Breakdown of integrability: Mass gaps and bound states experiment Simplest case: Difference between and Æ µ Flow of energy levels (anomalous dimensions): À µ À Ä KITP, QCD and String Theory Integrability in QCD and beyond - p. 19/35
Ä Breakdown of integrability: Mass gaps and bound states theory In lower part of the ÕÕÕ spectrum Perturbation Õ ¼ Ä Ä Õ Æ ÐÒ Level splitting Æ ÐÒ Of order ÐÒ Æ lower levels have to be rediagonalized µ θ q Õ ¼ Õ ÐÒ Æ Ó Õ Õ ¼µ Ä Phases of the cyclic permutations: µ È µ Matrix (ÐÒ Æ ÐÒ Æ ): ÈÕ Õ Õ ¼ ¼ ÐÒ Æ ¼ ¼ ¼ ¼ ÐÒ Æ ¼ ¼........... ¼ ¼ ¼ ¼... ÐÒ Æ ÐÒ Æ! A mass gap! KITP, QCD and String Theory Integrability in QCD and beyond - p. 20/35
Ƽ «µ ¼ ÆÒ¼ ³ «µ ¼ ÆÒ¼ ³ Ü µ Æ Ü µ Ü ÆÒ¼ È µ Æ Ü µ È µ Æ Ü µ Breakdown of integrability: Mass gaps and bound states interpretation wave function Nucleon and wave functions ÆÒ¼ Ü Üµ Æ Ü µ Ü Üµ Æ Ü µ ³ ܵ «¼µ ³ ܵ ÜÜÜ Æ¼ «¼µ free motion 1 2 3 [ 1 2 3 ] -+ ( 1 3) scalar diquarks? KITP, QCD and String Theory Integrability in QCD and beyond - p. 21/35
Supersymmetry: from Æ ¼ to Æ A. Belitsky, S. Derkachov G. Korchemsky and A. Manashov [hep-th/0403085] One expects that SUSY enhances integrability as one goes from Æ ¼ Æ QCD-like integrability is extended and new features may appear Need a universal approach to treat simultaneously the operator mixing in various extended SYM Æ The light-cone superspace formalism Quantize SYM in the light-cone gauge ܵ ¼ Decompose fermions and gauge fields into good µ and bad µ components Integrate out nonpropagating ( bad ) fields and rewrite the action in terms of propagating ( good ) fields Ë Æ Ë Æ µ Combine good fields into light-cone superfields Ü µ Kogut, Soper 70 KITP, QCD and String Theory Integrability in QCD and beyond - p. 22/35
Ë Æ µ Æ light-cone SYM Light-cone superfield formulation of Æ SYM Mandelstam, Brink et al. 83 Ü µ Complex scalar Æ chiral superfield Chiral superfield spans all helicities of the particles in the supermultiplet The anti-chiral superfield is not independent All single-trace light-cone operators are built from a single chiral superfield Ç Ä µ ØÖ Þ µ Þ Ä Ä µ KITP, QCD and String Theory Integrability in QCD and beyond - p. 23/35
µ ܵ ܵ Ü µ ܵ ܵ Ü Üµ ܵ... and back from Æ to Æ ¼ Method of truncation : ËÆ ËÆ ËÆ ËÆ ¼ Brink et al. 83 µ Æ light-cone chiral superfield µ Æ light-cone chiral superfield ܵ µ Æ ¼ light-cone chiral (super)field chiral superfield and its conjugate are independent Single-trace light-cone operators can be built from two superfields Integrable Ç sector Ä µ ÌÖ µ Ä µ ÖÐ Non-integrable Ç sector Ä µ ÌÖ µ Ä µ ÑÜ KITP, QCD and String Theory Integrability in QCD and beyond - p. 24/35
Ç µ Ê ¼ Î Projection operator, µ Þ Þ µ Ê ¼ À ܵ ܵ A two particle kernel One-loop dilatation operator in chiral sector À Î µ acts as a displacement in the light-cone superspace Þ µ ( Æ ) ««Ç µ Ç ««µ µ Ç ««µ µ «µ«ç µ µ Ç µ µ Ç µ Compare this with QCD expression for Þ Þ Þ µ Õ Þ µõ Þ µõ Þ µ: ««Þ Þ µ «Þ «µþ Þ µ Þ «Þ «µþ µ «[For quarks Õ, for chiral superfield, hence different power of «] KITP, QCD and String Theory Integrability in QCD and beyond - p. 25/35
Dilatation operator on the light-cone superspace Þ µ The one-loop dilatation operator has the same, universal form for Æ ¼, Æ, and Æ SYM Æ Dilatation operator displaces superfields in the Þ superspace µ The number of supercharges Æ defines the dimension of the superspace Natural generalization of the QCD formulae Þ ¼µ Two reductions of the Æ dilatation operator Expand in even light-cone coordinates, project on ¼ Maximal helicity gluon operators À ËÄ Êµ magnet [Braun,Derkachov,Manashov; Belitsky; G.K 98] Expand in odd -coordinates, project on Þ ¼ À ËÇ µ magnet [Minahan, Zarembo 02] BMN-like operators: Ç Ð ÌÖ ¼µ Ä Ä ¼µ In general: collinear subgroup of the superconformal symmetry group: Heisenberg (super) spin magnet with ËÄ Æ µ the symmetry KITP, QCD and String Theory Integrability in QCD and beyond - p. 26/35
Æ beyond the light-cone The case Æ of SYM is still special: All light cone operators belong to ËÄ Æ µ the sector since is not dynamicaly independent µ Integrability extends to all operators (including those with bad components) and the collinear subgroup ËÄ Æ µ extends to the full superconformal symmetry group È ËÍ Æ µ. [Beisert, Staudacher, 03] Why? Crucial: Superconformal symmetry relates operators of different twist Superconformal transformations applied to light-cone operators Æ yield combinations of operators of different twist but having the same anomalous dimension This is not the case in QCD where operators of different twists are believed to be dynamically independent Probably persists to all loops KITP, QCD and String Theory Integrability in QCD and beyond - p. 27/35
ÓÒÓÖÑÐ µ Outlook One-loop dilatation operator (evolution equations) in QCD is/are integrable in some sectors Classical bremsstrahlung: Cusp anomalous dimension: Æ for Æ Ù Ô «µ ÐÓ Æ Integrability appears as a consequence of the existence of massless vector particles Offers powerful machinery Open problems in QCD context: Analytic structure of the spectrum Parton interpretation in higher twists From Æ to Æ ¼: Rethinking of the role of non-quasipartonic operators; properties of anomalous dimensions in all twists Beyond one loop: Formal conformal limit Integrability? Particular models? Breaking of integrability vs. breaking of conformal symmetry: physics issues? KITP, QCD and String Theory Integrability in QCD and beyond - p. 28/35
Cross section: ÔÕÅ Ð Ð ¼ Ð ¼ Å Ð ¼ Å µ «Ð ¼ ܵ Ì Üµ Ü Ô Þ Õ ¼µÒ Õ ÒµÔ Þ ÅÜ Ü É µ Ð Ü Ô Õ ¼µ Õ ÒµÔ Addendum: Polarized deep-inelastic scattering Ü É µ Ó Å Ü É µ Ò É «Ð ¼ É Ü É µ Ð ¼ Ü Ð Ó µ At tree level: Å quark distributions in longitudinally and transversely polarized nucleon, respectively KITP, QCD and String Theory Integrability in QCD and beyond - p. 29/35
Ü Ü Ü Æ Ü Ü Ü µ Ô Ü Þ Ü Þ Ü Þ µ Õ Ü µ Õ Õ µ Ô Ò Ô Ò Ü µ µ Polarized deep-inelastic scattering continued Beyond the tree level Æ Ô µõ Þ µ Þ µõ Þ µæ Ô µ Ü µ µ Õ Õ quark-gluon correlations in the nucleon KITP, QCD and String Theory Integrability in QCD and beyond - p. 30/35
À ÕÕ Æ À ¼µ À µ Æ Î ¼µ Õ Â À ¼µ µ Í ¼µ Õ Â µ Î µ Õ Â À µ µ Í µ Õ Â µ Í µ ÕÕ Â µ ¼µ Õ Âµ  µ  µ µ Î ¼µ Õ Âµ  µ  µ µ Í µ Õ Âµ Í µ  µ µ ÕÕ Âµ µ µ Î µ ÕÕ Âµ Í Renormalization of quark-gluon operators, flavor non-singlet where µ Õ Âµ Î µ  µ  µ  µ    µ  µ µ KITP, QCD and String Theory Integrability in QCD and beyond - p. 31/35
Open inhomogeneous spin chain 40 35 γ N 30 25 20 The lowest level is special and is separated from the rest of the spectrum by a mass gap 15 10 Exact analytic solution 5 10 15 20 25 30 N This special level was found by ABH 91 and it determines the evolution of Ü É µ to the leading logarithmic accuracy It corresponds to the particular combination of quark-antiquark-gluon operators that can be reduced to the quark-antiquark twist-three operator using equations of motion Detailed study: Belitsky; Derkachov, Korchemsky, Manashov 00 KITP, QCD and String Theory Integrability in QCD and beyond - p. 32/35
Flavor singlet ÕÕ and operators BDM 01 E N,α 45 40 35 30 25 20 15 10 The highest qgq level is separated from the rest by a finite gap and is almost degenerate with the lowest GGG state 0 2 4 6 8 10 12 14 16 18 20 N levels: crosses levels: open circles ÕÕ Interacting open and closed spin chains KITP, QCD and String Theory Integrability in QCD and beyond - p. 33/35
Ü É µ Ï Ï ÐÒ Æ Õ Flavor singlet ÕÕ and operators: WKB expansion Energies: Æ ÐÓÛ Æ Õ Õ Æ ÐÒ Ò Ç µ ÐÒ µ Ç ÐÒ µ ÐÓÛ Æ ÐÒ To the leading logarithmic accuracy, the structure function Ü É µ is expressed through the quark distribution É µ Ü Ý Õ Ì Ý É µ Ý Ü Õ KITP, QCD and String Theory Integrability in QCD and beyond - p. 34/35
É É Ì Ü É µ «É È Ì ÕÕ Üµ È Ì Üµ Æ Æ Ý Ý Ý È Ì Üݵ Ì Ý É µ Ý Æ Æ Approximate evolution equation for the structure function Ü É µ Introduce flavor-singlet quark and gluon transverse spin distributions Õ Ë Ì Ü É µ «É ÕÕ Üݵ Õ Ë Ì Ý É µ È Ì Õ Üݵ Ì Ý É µ Ì È Ü Ü with the splitting functions Æ Üµ Ü Ò Æ Üµ Ü Æ ÐÒ Ü Ü È Ì Õ Üµ Ò Ü Üµ ÐÒ Üµ KITP, QCD and String Theory Integrability in QCD and beyond - p. 35/35