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Bootstraps Old and New

Bootstraps Old and New. L. Dixon, J. Drummond, M. von Hippel a nd J. Pennington 1305.nnnn Amplitudes 2013. From Wikipedia. Bootstrapping : a group of metaphors which refer to a self-sustaining process that proceeds without external help .

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Bootstraps Old and New

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  1. Bootstraps Old and New L. Dixon, J. Drummond, M. von Hippel and J. Pennington 1305.nnnn Amplitudes 2013

  2. From Wikipedia • Bootstrapping:a group of metaphors which refer to a self-sustaining process that proceeds without external help. • The phrase appears to have originated in the early 19th century United States (particularly in the sense "pull oneself over a fence by one's bootstraps"), to mean an absurdly impossible action. Amplitudes 2013, May 2

  3. Main Physics Entry • Geoffrey Chew and others … sought to derive as much information as possible about the strong interaction from plausible assumptions about the S-matrix, … an approach advocated by Werner Heisenbergtwo decades earlier. • Without the narrow resonance approximation, the bootstrap program did not have a clear expansion parameter, and the consistency equations were often complicated and unwieldy, so that the method had limited success. • With narrow resonance approx, led to string theory Amplitudes 2013, May 2

  4. Duality = Veneziano (1968) Amplitudes 2013, May 2

  5. Conformal bootstrap Polyakov (1974): Use conformal invariance, crossing symmetry, unitarity to determine anomalous dimensions and correlation functions. • Powerful realization for D=2, c < 1 [Belavin, Polyakov, Zamolodchikov, 1984]: Null states  cm, hp,q differential equations. • More recently: Applications to D>2 [Rattazzi, Rychkov, Tonni, Vichi(2008)] • Unitarity  anom. dim. inequalities, saturated by e.g. D=3 Ising model [El-Showk et al., 1203.6064] Amplitudes 2013, May 2

  6. Crossing symmetry condition f • f • i = • i Unitarity: Amplitudes 2013, May 2

  7. Ising Model in D=3 Anomalous dimension bounds from unitarity + crossing + knowledge of conformal blocks + scan over intermediate states + linear programming techniques El-Showket al., 1203.6064 Amplitudes 2013, May 2

  8. Conformal Bootstrap for N=4 SYM Beem, Rastelli, van Rees, 1304.1803 planar limit Would be very interesting to make contact with perturbative approaches e.g. as in talk by Duhr Amplitudes 2013, May 2

  9. Scattering in D=2 • Integrability:infinitely many conserved charges Factorizable S-matrices. 22 S matrix must satisfy Yang-Baxter equations • Many-body S matrix a simple product of 22 S matrices. • Consistency conditions often powerful enough to write down exact solution! • First case solved: Heisenberg antiferromagnetic spin chain [Bethe Ansatz, 1931] = Amplitudes 2013, May 2

  10. Integrability and planar N=4 SYM • Single-trace operators  1-d spin systems • Anomalous dimensions from diagonalizingdilatation operator =spin-chain Hamiltonian. • In planar limit, Hamiltonian is local, • though range increases with number of loops • For N=4 SYM, Hamiltonian is integrable: • – infinitely many conserved charges • – scattering of quasi-particles (magnons) • via 2  2 S matrix obeying YBE • Also: integrabilityof AdS5 x S5s-model Bena, Polchinski, Roiban (2003) Lipatov (1993); Minahan, Zarembo (2002); Beisert, Kristjansen, Staudacher (2003); … Amplitudes 2013, May 2

  11. Integrability anomalous dim’s • Solve system for any coupling by Bethe ansatz: • – multi-magnonstates with only phase-shifts • induced by repeated 2  2 scattering • – periodicity of wave function  Bethe Condition • depending on length of chain L • – As L ∞, BC becomes integral equation • – 2  2 S matrix almost fixed by symmetries; • overall phase, dressing factor, not so easily deduced. • – Assume wrapping corrections vanish for large spin operators Staudacher, hep-th/0412188; Beisert, Staudacher, hep-th/0504190; Beisert, hep-th/0511013, hep-th/0511082; Eden, Staudacher, hep-th/0603157; Beisert, Eden, Staudacher, hep-th/0610251 ; talk by Schomerus Amplitudes 2013, May 2

  12. Beisert, Eden, Staudacher [hep-th/0610251] to all orders Agrees with weak-coupling data through 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248; Cachazo, Spradlin, Volovich, hep-th/0612309 Agrees with first 3 terms of strong-coupling expansion GubserKlebanov, Polyakov, th/0204051; Frolov, Tseytlin, th/0204226; Roiban, Tseytlin, 0709.0681 [th] Full strong-coupling expansion Basso,Korchemsky, Kotański, 0708.3933 [th] Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135] Amplitudes 2013, May 2

  13. Many other integrability applications to N=4 SYM anom dim’s • In particular excitations of the GKP string, defined by which also corresponds to excitations of a light-like Wilson lineBasso, 1010.5237 • And scattering of these excitations S(u,v) • And the related pentagon transition P(u|v) for Wilson loops …, Basso, Sever, Vieira [BSV], 1303.1396; talk by A. Sever Amplitudes 2013, May 2

  14. What about Amplitudes in D=4? • Many (perturbative) bootstraps for integrands: • BCFW (2004,2005) for trees (bootstrap in n) • Trees can be fed into loops via unitarity Amplitudes 2013, May 2

  15. Early (partial) integrand bootstrap • Iterated two-particle unitarity cuts for 4-point amplitude in planar N=4 SYM solved by “rung rule”: • Assisted by other cuts (maximal cut method), obtain complete (fully regulated) amplitudes, especially at 4-points talk by Carrasco • Now being systematized for generic (QCD) applications, especially at 2 loops talks by Badger, Feng, Mirabella, Kosower Amplitudes 2013, May 2

  16. All planar N=4 SYM integrands Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka, 1008.2958, 1012.6032 • All-loop BCFW recursion relation for integrand  • Or new approach Arkani-Hamed et al. 1212.5605, talk by Trnka • Manifest Yangian invariance  • Multi-loop integrands in terms of “momentum-twistors”  • Still have to do integrals over the loop momentum  Amplitudes 2013, May 2

  17. One-loop integrated bootstrap • Collinear/recursive-based bootstraps in n for special integrated one-loop n-point amplitudes in QCD (Bern et al., hep-ph/931233; hep-ph/0501240; hep-ph/0505055) • Analytic results for rational one-loop amplitudes: + Amplitudes 2013, May 2

  18. One-Loop Amplitudes with Cuts? • Can still run a unitarity-collinear bootstrap • 1-particle factorization information assisted by cuts Bern et al., hep-ph/0507005, …, BlackHat [0803.4180] rational part A(z) R(z) Amplitudes 2013, May 2

  19. Beyond integrands & one loop • Can we set up a bootstrap [albeit with“external help”] directly for integrated multi-loop amplitudes? • Planar N=4 SYM clearly first place to start • dual conformal invariance • Wilson loop correspondence • First amplitude to start with is n = 6 MHV. talk by Volovich Amplitudes 2013, May 2

  20. Six-point remainder function • n = 6 first place BDS Ansatzmust be modified, due to dual conformal cross ratios MHV 6 1 5 2 4 3 Amplitudes 2013, May 2

  21. Formula forR6(2)(u1,u2,u3) • First worked out analytically from Wilson loop integrals Del Duca, Duhr, Smirnov, 0911.5332, 1003.1702 • 17 pages of Goncharovpolylogarithms. • Simplified to just a few classical polylogarithms using symbology • Goncharov, Spradlin, Vergu, Volovich, 1006.5703 Amplitudes 2013, May 2

  22. Wilson loop OPEs Alday, Gaiotto, Maldacena, Sever, Vieira, 1006.2788; GMSV, 1010.5009, 1102.0062 • Remarkably, can be recovered directly • from analytic properties, using “near collinear limits” • Wilson-loop equivalence  this limit is controlled by an operator product expansion (OPE) • Possible to go to 3 loops, by combining OPE expansion with symbol LD, Drummond, Henn, 1108.4461 • Here, promote symbol to unique functionR6(3)(u1,u2,u3) Amplitudes 2013, May 2

  23. Professor of symbologyat Harvard University, has used these techniques to make a series of important advances: Amplitudes 2013, May 2

  24. What entries should symbol have? Goncharov, 0908.2238; GSVV, 1006.5703; talks by Duhr, Gangl, Volovich • We assume entries can all be drawn from set: • with • + perms Amplitudes 2013, May 2

  25. S[ R6(2)(u,v,w) ]in these variables • GSVV, 1006.5703 Amplitudes 2013, May 2

  26. First entry • Always drawn from GMSV, 1102.0062 • Because first entry controls branch-cut location • Only massless particles •  all cuts start at origin in •  Branch cuts all start from 0 or ∞ in Amplitudes 2013, May 2

  27. Final entry • Always drawn from • Seen in structure of various Feynman integrals [e.g. Arkani-Hamed et al., 1108.2958] related to amplitudes Drummond, Henn, Trnka 1010.3679; LD, Drummond, Henn, 1104.2787, V. Del Duca et al., 1105.2011,… • Same condition also from Wilson super-loop approach Caron-Huot, 1105.5606 Amplitudes 2013, May 2

  28. Generic Constraints • Integrability (must be symbol of some function) • S3 permutation symmetry in • Even under “parity”: • every term must have an even • number of – 0, 2 or 4 • Vanishing in collinear limit • These 4 constraints leave 35 free parameters Amplitudes 2013, May 2

  29. OPE Constraints Alday, Gaiotto, Maldacena, Sever, Vieira, 1006.2788; GMSV, 1010.5009; 1102.0062’ Basso, Sever, Vieira [BSV], 1303.1396; talk by A. Sever • R6(L)(u,v,w)vanishes in the collinear limit, • v = 1/cosh2t 0 t  ∞ • In near-collinear limit, described by an Operator Product Expansion, with generic form s f t  ∞ [BSV parametrization different] Amplitudes 2013, May 2

  30. OPE Constraints (cont.) • Using conformal invariance, send one long line to ∞, put other one along x- • Dilatations, boosts, azimuthal rotations preserve configuration. • s, f conjugate to twist p, spin m of conformal primary fields (flux tube excitations) • Expand anomalous dimensions in coupling g2: • Leading discontinuity t L-1of R6(L) needs only one-loop anomalous dimension Amplitudes 2013, May 2

  31. OPE Constraints (cont.) • As t  ∞ , v = 1/cosh2t  tL-1 ~ [lnv]L-1 • Extract this piece from symbol by only keeping terms with L-1 leading v entries • Powerful constraint: fixes 3 loop symbol up to 2 parameters. But not powerful enough forL > 3 • New results of Basso, Sever, Vieira give • v1/2 e±if [lnv]k , k = 0,1,2,…L-1 • and even • v1 e±2if[lnv]k , k = 0,1,2,…L-1 Amplitudes 2013, May 2

  32. Constrained Symbol • Leading discontinuity constraints reduced symbol ansatz to just 2 parameters: DDH, 1108.4461 • f1,2 have no double-v discontinuity, so a1,2 • couldn’t be determined this way. • Determined soon after using Wilson super-loop integro-differential equation • Caron-Huot, He, 1112.1060 • a1 = - 3/8 a2= 7/32 • Also follow from Basso, Sever, Vieira Amplitudes 2013, May 2

  33. Reconstructing the function • One can build up a complete description of the pure functions F(u,v,w)with correct branch cuts iteratively in the weight n, using the (n-1,1) element of the co-productDn-1,1(F) Duhr, Gangl, Rhodes, 1110.0458 • which specifies all first derivatives of F: Amplitudes 2013, May 2

  34. Reconstructing functions (cont.) • Coefficients are weight n-1 functions that can be identified (iteratively) from the symbol of F • “Beyond-the-symbol” [bts] ambiguities in reconstructing them, proportional to z(k). • Most ambiguities resolved by equating 2nd order mixed partial derivatives. • Remaining ones represent freedom to add globally well-defined weight n-k functions multiplied by z(k). Amplitudes 2013, May 2

  35. How many functions? First entry ; non-product Amplitudes 2013, May 2

  36. R6(3)(u,v,w) Many relations among coproduct coefficients for Rep: Amplitudes 2013, May 2

  37. Only 2 indep. Rep coproduct coefficients 2 pages of 1-d HPLs Similar (but shorter) expressions for lower degree functions Amplitudes 2013, May 2

  38. Integrating the coproducts • Can express in terms of multiple polylog’sG(w;1), with widrawn from {0,1/yi ,1/(yiyj),1/(y1y2y3)} • Alternatively: • Coproducts define coupled set of first-order PDEs • Integrate them numerically from base point (1,1,1) • Or solve PDEs analytically in special limits, especially: • Near-collinear limit • Multi-regge limit Amplitudes 2013, May 2

  39. Amplitudes 2013, May 2

  40. Fixing all the constants • 11 bts constants (plus a1,2) before analyzing limits • Vanishing of collinear limit v  0 fixes everything, except a2 and 1 bts constant • Near-collinear limit, v1/2e±if [lnv]k, k = 0,1 fixes last 2 constants (a2 agrees with Caron-Huot+He and BSV) Amplitudes 2013, May 2

  41. Multi-Regge limit • Minkowski kinematics, large rapidity separations between the 4 final-state gluons: • Properties of planar N=4 SYM amplitude in this limit studied extensively at weak coupling: • Bartels, Lipatov, Sabio Vera, 0802.2065, 0807.0894; Lipatov, 1008.1015; Lipatov, Prygarin, 1008.1016, 1011.2673; Bartels, Lipatov, Prygarin, 1012.3178, 1104.4709; LD, Drummond, Henn, 1108.4461; Fadin, Lipatov, 1111.0782; LD, Duhr, Pennington, 1207.0186; talk by Schomerus • Factorization and exponentiation in this limit provides additional source of “boundary data” for bootstrapping! Amplitudes 2013, May 2

  42. Physical 24 multi-Regge limit • Euclidean MRK limit vanishes • To get nonzero result for physical region, first let • , then u 1, v, w  0 Fadin, Lipatov, 1111.0782; LD, Duhr, Pennington, 1207.0186; Pennington, 1209.5357 Put LLA, NLLA results into bootstrap; extractNkLLA, k > 1 Amplitudes 2013, May 2

  43. NNLLA impact factor now fixed • Result from DDP, 1207.0186still had • 3 beyond-the-symbol ambiguities Now all 3 are fixed: Amplitudes 2013, May 2

  44. Simple slice: (u,u,1)  (1,v,v) Collapses to 1d HPLs: Includes base point (1,1,1) : Amplitudes 2013, May 2

  45. Plot R6(3)(u,v,w) on some slices Amplitudes 2013, May 2

  46. (u,u,u) Indistinguishable(!) up to rescaling by: ratio ~ cf. cusp ratio: Amplitudes 2013, May 2

  47. Proportionality ceases at large u ratio ~ -1.23 0911.4708 Amplitudes 2013, May 2

  48. (1,1,1) R6(3)(u,v,u+v-1) R6(2)(u,v,u+v-1) (1,v,v) (u,1,u) collinear limit w 0, u + v1 on plane u + v – w = 1 Amplitudes 2013, May 2

  49. Ratio for (u,u,1)  (1,v,v)   (w,1,w) ratio ~ - 9.09 ratio ~ ln(u)/2 Amplitudes 2013, May 2

  50. On to 4 loops LD, Duhr, Pennington, … • In the course of 1207.0186, we “determined” the 4 loop remainder-function symbol. • However, still 113 undetermined constants • Consistency with LLA and NLLA multi-Regge limits  81 constants  • Consistency with BSV’s v1/2 e±if 4 constants  • Adding BSV’s v1 e±2if 0constants!!  [Thanks to BSV for supplying this info!] • Next step: Fix btsconstants, after defining functions (globally? or just on a subspace?) Amplitudes 2013, May 2

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