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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

- 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

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

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

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

- 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]

=

Amplitudes 2013, May 2

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

- 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

[hep-th/0610251]

to all ordersAgrees 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 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?

- 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

- 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

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

- 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?

- 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

- 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

- 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 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

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

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?

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

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

- 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

- 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

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.)

- 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 (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

- 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

- 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.)

- 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

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

- 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

- 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

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 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, 1209.5357

Put LLA, NLLA results into bootstrap;

extractNkLLA, k > 1

Amplitudes 2013, May 2

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

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

Collapses to 1d HPLs:

Includes base point (1,1,1) :

Amplitudes 2013, May 2

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 + v1

on plane

u + v – w = 1

Amplitudes 2013, May 2

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

- 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

What about (quantum n=8 super)gravity?

Amplitudes 2013, May 2

Extra Slides

Amplitudes 2013, May 2

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

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

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

Amplitudes 2013, May 2

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

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

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