bootstraps old and new

Bootstraps Old and New

L. Dixon, J. Drummond, M. von Hippeland J. Pennington

1305.nnnnAmplitudes 2013

Amplitudes 2013, May 2 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.

L. Dixon Bootstraps Old and New


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 theoryL. Dixon Bootstraps Old and New

Amplitudes 2013, May 2 4L. Dixon Bootstraps Old and New

Duality

=

Veneziano (1968)


Conformal bootstrapPolyakov (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]L. Dixon Bootstraps Old and New


Crossing symmetry condition

=

f

f

ii

Unitarity:


Ising Model in D=3


El-Showk et al., 1203.6064Anomalous dimension bounds from unitarity + crossing + knowledge of conformal blocks+ scan over intermediate states + linear programming techniques


Conformal Bootstrap for N=4 SYM


Beem, Rastelli, van Rees, 1304.1803

Would be very interesting to make contact with perturbativeapproaches e.g. as in talk by Duhr

planar limit


Scattering in D=2• Integrability: infinitely many conserved charges Factorizable S-matrices.

22 S matrix must satisfy Yang-Baxter equations

• Many-body S matrix a simple product of 22 S matrices. • Consistency conditions often powerful enough to write

down exact solution!• First case solved: Heisenberg antiferromagnetic spin chain [Bethe Ansatz, 1931]


=


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 AdS5 x S5 s-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


• 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/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



to all orders

Full strong-couplingexpansionBasso,Korchemsky,Kotański, 0708.3933 [th]

Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135]

Beisert, Eden, Staudacher [hep-th/0610251]

Agrees with weak-couplingdata through 4 loopsBern, Czakon, LD, Kosower, Smirnov, hep-th/0610248;Cachazo, Spradlin, Volovich,hep-th/0612309

Agrees with first 3 termsof strong-coupling expansionGubser Klebanov, Polyakov,th/0204051; Frolov, Tseytlin, th/0204226; Roiban, Tseytlin, 0709.0681 [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, 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



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



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



All planar N=4 SYM integrands

• 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

Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka, 1008.2958, 1012.6032


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)

• Analyticresults forrationalone-loopamplitudes:


+


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]


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



• n = 6 first place BDS Ansatz must be modified, due to dual conformal cross ratios

Six-point remainder function

MHV

1

2

34

5

6


Formula for R6(2)(u1,u2,u3)

• First worked out analytically from Wilson loop integrals Del Duca, Duhr, Smirnov, 0911.5332, 1003.1702

17 pages of Goncharov polylogarithms.

• Simplified to just a few classical polylogarithms using symbologyGoncharov, Spradlin, Vergu, Volovich, 1006.5703


Wilson loop OPEs

• Remarkably, can be recovered directlyfrom 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 function R6(3)(u1,u2,u3)

Alday, Gaiotto, Maldacena, Sever, Vieira, 1006.2788; GMSV, 1010.5009, 1102.0062


Professor of symbology at Harvard University, has usedthese techniques to make a series of important advances:


What entries should symbol have?

• We assume entries can all be drawn from set:

with + perms

Goncharov, 0908.2238; GSVV, 1006.5703; talks by Duhr, Gangl, Volovich


S[ R6(2)(u,v,w) ] in these variables

GSVV, 1006.5703


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


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


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


OPE Constraints

• 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

t ∞

Alday, Gaiotto, Maldacena, Sever, Vieira, 1006.2788; GMSV, 1010.5009; 1102.0062’Basso, Sever, Vieira [BSV], 1303.1396; talk by A. Sever

f

s

[BSV parametrization different]


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-1 of R6(L) needs only

one-loop anomalous dimension


OPE Constraints (cont.) • As t ∞ , v = 1/cosh2 t t 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 v1/2 e±if [ln v] k , k = 0,1,2,…L-1and even v1 e±2if [ln v] k , k = 0,1,2,…L-1


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


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 Dn-1,1(F) Duhr, Gangl, Rhodes, 1110.0458

which specifies all first derivatives of F:


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).


How many functions? First entry ; non-product


R6(3)(u,v,w)


Many relations among coproduct coefficients for Rep:


Only 2 indep. Rep coproduct coefficients


2 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 wi drawn from {0, 1/yi , 1/(yi yj), 1/(y1 y2 y3) }• 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 limit2. Multi-regge limit



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, v1/2 e±if [ln v] k , k = 0,1fixes last 2 constants (a2 agrees with Caron-Huot+He and BSV)



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!


Physical 24 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, 1209.5357Put LLA, NLLA results into bootstrap;

extract NkLLA, k > 1


NNLLA impact factor now fixedResult from DDP, 1207.0186 still had 3 beyond-the-symbol ambiguities

Now all 3 are fixed:


Simple slice: (u,u,1) (1,v,v)


Includes base point (1,1,1) :

Collapses to 1d HPLs:


Plot R6(3)(u,v,w)

on some slices


Indistinguishable(!) up to rescaling by:

cf. cusp ratio:

ratio ~

(u,u,u)


Proportionality ceases at large u


0911.4708

ratio ~ -1.23


(1,1,1)

(1,v,v) (u,1,u)

on planeu + v – w = 1

collinear limit w 0, u + v 1

R6(3)(u,v,u+v-1)

R6(2)(u,v,u+v-1)


Ratio for (u,u,1) (1,v,v) (w,1,w)


ratio ~ - 9.09

ratio ~ ln(u)/2


On to 4 loops• 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 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 New

LD, Duhr, Pennington, …


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?



Extra Slides



Multi-Regge kinematics1 2

345

6

Very nice change of variables[LP, 1011.2673] is to :

2 symmetries: conjugationand inversion


Numerical integration contours


base point (u,v,w) = (1,1,1)

base point (u,v,w) = (0,0,1)


• 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 fr

• Define symbol S [Goncharov, 0908.2238] recursively in k:

Iterated differentiation


Wilson loops at weak coupling

Computed for same “soap bubble” boundary conditions as scattering amplitude:

• One loop, n=4

Drummond, Korchemsky, Sokatchev, 0707.0243

• One loop, any n

Brandhuber, Heslop, Travaglini, 0707.1153

• Two loops, n=4,5,6

Wilson-loop VEV always matches [MHV] scattering amplitude!

Drummond, Henn, Korchemsky, Sokatchev, 0709.2368, 0712.1223, 0803.1466;Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, 0803.1465

Weak-coupling properties linked to superconformal invariance for strings in AdS5 x S5 under combined bosonic and fermionic T duality symmetry Berkovits, Maldacena, 0807.3196; Beisert, Ricci, Tseytlin, Wolf, 0807.3228

bootstraps old and new

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