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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

Bootstraps Old and New

L. Dixon, J. Drummond, M. von Hippel

and J. Pennington

1305.nnnn

Amplitudes 2013

from wikipedia
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

main physics entry
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

d uality
Duality

=

Veneziano (1968)

Amplitudes 2013, May 2

c onformal bootstrap
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

crossing symmetry condition
Crossing symmetry condition

f

  • f
  • i

=

  • i

Unitarity:

Amplitudes 2013, May 2

ising model in d 3
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

conformal bootstrap for n 4 sym
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

scattering in d 2
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

integrability and planar n 4 sym
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

integrability anomalous dim s
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

to all orders

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

many o ther integrability applications to n 4 sym anom dim s
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

what about amplitudes in d 4
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

early partial integrand bootstrap
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

slide16

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

one loop integrated bootstrap
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

one loop amplitudes with cuts
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

beyond integrands one loop
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

six point remainder function
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

formula for r 6 2 u 1 u 2 u 3
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

wilson loop opes
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

slide23

Professor of symbologyat Harvard University, has used

these techniques to make a series of important advances:

Amplitudes 2013, May 2

what entries should symbol have
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

s r 6 2 u v w in these variables
S[ R6(2)(u,v,w) ]in these variables
  • GSVV, 1006.5703

Amplitudes 2013, May 2

first entry
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

final entry
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

generic constraints
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

ope constraints
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

ope constraints cont
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

ope constraints cont1
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

constrained symbol
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

reconstructing the function
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

reconstructing functions cont
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

how many functions
How many functions?

First entry ; non-product

Amplitudes 2013, May 2

r 6 3 u v w
R6(3)(u,v,w)

Many relations among coproduct coefficients for Rep:

Amplitudes 2013, May 2

only 2 indep r ep coproduct coefficients
Only 2 indep. Rep coproduct coefficients

2 pages of 1-d HPLs

Similar (but shorter) expressions for lower degree functions

Amplitudes 2013, May 2

integrating the coproducts
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

fixing all the constants
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

m ulti regge limit
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

physical 2 4 multi regge limit
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

nnlla impact factor now fixed
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

s imple slice u u 1 1 v v
Simple slice: (u,u,1)  (1,v,v)

Collapses to 1d HPLs:

Includes base point (1,1,1) :

Amplitudes 2013, May 2

slide45

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

on some slices

Amplitudes 2013, May 2

slide46

(u,u,u)

Indistinguishable(!) up to rescaling by:

ratio ~

cf. cusp ratio:

Amplitudes 2013, May 2

proportionality ceases at large u
Proportionality ceases at large u

ratio ~ -1.23

0911.4708

Amplitudes 2013, May 2

slide48

(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

r atio for u u 1 1 v v w 1 w
Ratio for (u,u,1)  (1,v,v)   (w,1,w)

ratio ~ - 9.09

ratio ~ ln(u)/2

Amplitudes 2013, May 2

on to 4 loops
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

slide51

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

Amplitudes 2013, May 2

extra slides
Extra Slides

Amplitudes 2013, May 2

multi regge kinematics
Multi-Regge kinematics

2

1

6

3

5

4

Very nice change of variables

[LP, 1011.2673] is to :

2 symmetries: conjugation

and inversion

Amplitudes 2013, May 2

numerical integration contours
Numerical integration contours

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

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

Amplitudes 2013, May 2

iterated differentiation
Iterated differentiation
  • 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:

Amplitudes 2013, May 2

wilson loops at weak coupling
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

Drummond, Henn, Korchemsky, Sokatchev, 0709.2368, 0712.1223, 0803.1466;

Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, 0803.1465

  • Two loops, n=4,5,6

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

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

Amplitudes 2013, May 2