bootstraps old and new l. dixon, j. drummond, m. von hippel and j. pennington 1305.nnnn amplitudes...

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]



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)

• Analytic

results for

rational

one-loop

amplitudes:


+


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 limit

2. 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 fixed

Result 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

3

456

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 l. dixon, j. drummond, m. von hippel and j. pennington 1305.nnnn amplitudes...

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