Bootstraps Old and New L. Dixon, J. Drummond, M. von Hippel and 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. 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. L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 2

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 Heisenberg two 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 L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 3

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 4 Duality = Veneziano (1968)

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  c m, h p,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., ] L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 5

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 6 Crossing symmetry condition =   i i Unitarity:

Ising Model in D=3 L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 7 El-Showk et al., Anomalous dimension bounds from unitarity + crossing + knowledge of conformal blocks + scan over intermediate states + linear programming techniques

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

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] L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 9 =

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 10 Integrability and planar N=4 SYM For N=4 SYM, Hamiltonian is integrable: – infinitely many conserved charges – scattering of quasi-particles (magnons) via 2  2 S matrix obeying YBE Also: integrability of AdS 5 x S 5  -model Bena, Polchinski, Roiban (2003) Lipatov (1993); Minahan, Zarembo (2002); Beisert, Kristjansen, Staudacher (2003) ; … Single-trace operators  1-d spin systems Anomalous dimensions from diagonalizing dilatation operator = spin-chain Hamiltonian. In planar limit, Hamiltonian is local, though range increases with number of loops

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 11 Solve system for any coupling by Bethe ansatz: – multi-magnon states 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 Integrability  anomalous dim’s Staudacher, hep-th/ ; Beisert, Staudacher, hep-th/ ; Beisert, hep-th/ , hep-th/ ; Eden, Staudacher, hep-th/ ; Beisert, Eden, Staudacher, hep-th/ ; talk by Schomerus

L. Dixon Bootstraps Old and New Amplitudes 2013, May 2 12 to all orders Full strong-coupling expansion Basso,Korchemsky, Kotański, [th] Benna, Benvenuti, Klebanov, Scardicchio [hep-th/ ] Beisert, Eden, Staudacher [hep-th/ ] Agrees with weak-coupling data through 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/ ; Cachazo, Spradlin, Volovich, hep-th/ Agrees with first 3 terms of strong-coupling expansion Gubser Klebanov, Polyakov, th/ ; Frolov, Tseytlin, th/ ; Roiban, Tseytlin, [th]

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 line Basso, And scattering of these excitations S(u,v) And the related pentagon transition P(u|v) for Wilson loops …, Basso, Sever, Vieira [BSV], ; talk by A. Sever L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 13

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 L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 14

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 L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 15

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

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/ ; hep-ph/ ) Analytic results for rational one-loop amplitudes: L. Dixon Bootstraps Old and NewAmplitudes 2013, May

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

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 L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 19

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 20 n = 6 first place BDS Ansatz must be modified, due to dual conformal cross ratios Six-point remainder function MHV

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 21 Formula for R 6 (2) (u 1,u 2,u 3 ) First worked out analytically from Wilson loop integrals Del Duca, Duhr, Smirnov, , pages of Goncharov polylogarithms. Simplified to just a few classical polylogarithms using symbology Goncharov, Spradlin, Vergu, Volovich,

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 22 Wilson loop OPEs 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, Here, promote symbol to unique function R 6 (3) (u 1,u 2,u 3 ) Alday, Gaiotto, Maldacena, Sever, Vieira, ; GMSV, ,

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 23 Professor of symbology at Harvard University, has used these techniques to make a series of important advances:

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 24 What entries should symbol have? We assume entries can all be drawn from set: with + perms Goncharov, ; GSVV, ; talks by Duhr, Gangl, Volovich

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 25 S[ R 6 (2) (u,v,w) ] in these variables GSVV,

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 26 First entry Always drawn from GMSV, Because first entry controls branch-cut location Only massless particles  all cuts start at origin in  Branch cuts all start from 0 or ∞ in

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 27 Final entry Always drawn from Seen in structure of various Feynman integrals [e.g. Arkani-Hamed et al., ] related to amplitudes Drummond, Henn, Trnka ; LD, Drummond, Henn, , V. Del Duca et al., ,… Same condition also from Wilson super-loop approach Caron-Huot,

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 28 Generic Constraints Integrability (must be symbol of some function) S 3 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

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 29 OPE Constraints R 6 (L) (u,v,w) vanishes in the collinear limit, v = 1/cosh 2   0   ∞ In near-collinear limit, described by an Operator Product Expansion, with generic form   ∞ Alday, Gaiotto, Maldacena, Sever, Vieira, ; GMSV, ; ’ Basso, Sever, Vieira [BSV], ; talk by A. Sever   [BSV parametrization different]

L. Dixon Bootstraps Old and NewAmplitudes 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.  conjugate to twist p, spin m of conformal primary fields (flux tube excitations) Expand anomalous dimensions in coupling g 2 : Leading discontinuity  L-1  of R 6 (L) needs only one-loop anomalous dimension

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 31 OPE Constraints (cont.) As   ∞, v = 1/cosh 2    L-1 ~ [ln v]  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 for L > 3 New results of Basso, Sever, Vieira give v 1/2 e ±i  [ln v]  k, k = 0,1,2,…L-1 and even v 1 e ±2i  [ln v]  k, k = 0,1,2,…L-1

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 32 Constrained Symbol Leading discontinuity constraints reduced symbol ansatz to just 2 parameters: DDH, f 1,2 have no double- v discontinuity, so  1,2 couldn’t be determined this way. Determined soon after using Wilson super-loop integro-differential equation Caron-Huot, He,  1 = - 3/8  2 = 7/32 Also follow from Basso, Sever, Vieira

L. Dixon Bootstraps Old and NewAmplitudes 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-product  n-1,1 (F) Duhr, Gangl, Rhodes, which specifies all first derivatives of F :

L. Dixon Bootstraps Old and NewAmplitudes 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  (k). Most ambiguities resolved by equating 2 nd order mixed partial derivatives. Remaining ones represent freedom to add globally well-defined weight n-k functions multiplied by  (k).

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 35 How many functions? First entry ; non-product

R 6 (3) (u,v,w) L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 36 Many relations among coproduct coefficients for R ep :

Only 2 indep. R ep coproduct coefficients L. Dixon Bootstraps Old and NewAmplitudes 2013, May pages of 1-d HPLs Similar (but shorter) expressions for lower degree functions

Integrating the coproducts Can express in terms of multiple polylog’s G(w;1), with w i drawn from { 0, 1/y i, 1/(y i y j ), 1/(y 1 y 2 y 3 ) } 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: 1.Near-collinear limit 2.Multi-regge limit L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 38

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 39

Fixing all the constants 11 bts constants (plus  1,2 ) before analyzing limits Vanishing of collinear limit v  0 fixes everything, except  2 and 1 bts constant Near-collinear limit, v 1/2 e ±i  [ln v]  k, k = 0,1 fixes last 2 constants (  2 agrees with Caron-Huot+He and BSV ) L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 40

L. Dixon Bootstraps Old and NewAmplitudes 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, , ; Lipatov, ; Lipatov, Prygarin, , ; Bartels, Lipatov, Prygarin, , ; LD, Drummond, Henn, ; Fadin, Lipatov, ; LD, Duhr, Pennington, ; talk by Schomerus Factorization and exponentiation in this limit provides additional source of “boundary data” for bootstrapping!

L. Dixon Bootstraps Old and NewAmplitudes 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, ; LD, Duhr, Pennington, ; Pennington, Put LLA, NLLA results into bootstrap; extract N k LLA, k > 1

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 43 NNLLA impact factor now fixed Result from DDP, still had 3 beyond-the-symbol ambiguities Now all 3 are fixed:

Simple slice: (u,u,1)  (1,v,v) L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 44 Includes base point (1,1,1) : Collapses to 1d HPLs:

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 45 Plot R 6 (3) (u,v,w) on some slices

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 46 Indistinguishable(!) up to rescaling by: cf. cusp ratio: ratio ~ (u,u,u)(u,u,u)

Proportionality ceases at large u L. Dixon Bootstraps Old and NewAmplitudes 2013, May ratio ~ -1.23

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 48 (1,1,1) (1, v,v )( u,1, u ) on plane u + v – w = 1 collinear limit w  0, u + v  1 R 6 (3) (u,v,u+v-1) R 6 (2) (u,v,u+v-1)

Ratio for (u,u,1)  (1,v,v)   (w,1,w) L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 49 ratio ~ ratio ~ ln(u)/2

On to 4 loops In the course of , 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 v 1/2 e ±i   4 constants Adding BSV’s v 1 e ±2i   0 constants!! [Thanks to BSV for supplying this info!] Next step: Fix bts constants, after defining functions (globally? or just on a subspace?) L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 50 LD, Duhr, Pennington, …

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 51 Conclusions Bootstraps are wonderful things Applied successfully to D=2 integrable models To CFTs in D=2 and now D > 2 To perturbative amplitudes & integrands To anomalous dimensions in planar N=4 SYM Now, nonperturbatively to whole D=2 scattering problem on OPE/near-collinear boundary of phase-space for scattering amplitudes With knowledge of function space and this boundary data, can determine perturbative N=4 amplitudes over full phase space, without need to know any integrands at all

What about (quantum n=8 super)gravity? L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 52

Extra Slides L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 53

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 54 Multi-Regge kinematics Very nice change of variables [LP, ] is to : 2 symmetries: conjugation and inversion

Numerical integration contours L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 55 base point (u,v,w) = (1,1,1) base point (u,v,w) = (0,0,1)

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 56 A pure function f ( k ) of transcendental degree k is a linear combination of k -fold iterated integrals, with constant (rational) coefficients. We can also add terms like Derivatives of f ( k ) can be written as for a finite set of algebraic functions  r Define symbol S [Goncharov, ] recursively in k: Iterated differentiation

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 57

L. Dixon Bootstraps Old and NewAmplitudes 2013, May 2 58 Wilson loops at weak coupling Computed for same “soap bubble” boundary conditions as scattering amplitude: One loop, n=4 Drummond, Korchemsky, Sokatchev, One loop, any n Brandhuber, Heslop, Travaglini, Two loops, n=4,5,6 Wilson-loop VEV always matches [MHV] scattering amplitude! Drummond, Henn, Korchemsky, Sokatchev, , , ; Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, Weak-coupling properties linked to superconformal invariance for strings in AdS 5 x S 5 under combined bosonic and fermionic T duality symmetry Berkovits, Maldacena, ; Beisert, Ricci, Tseytlin, Wolf,