Transcript Document

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.
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Main Physics Entry
• Geoffrey Chew and others … sought to derive as
much information as possible about the strong
interaction from plausible assumptions about the Smatrix, … 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
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Duality
=
Veneziano (1968)
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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]
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Crossing symmetry condition
f
f
=
i
i
Unitarity:
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Ising Model in D=3
El-Showk et al., 1203.6064
Anomalous dimension bounds from
unitarity + crossing + knowledge of conformal blocks
+ scan over intermediate states + linear programming techniques
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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
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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]
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Integrability and planar N=4 SYM
• 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
• For N=4 SYM, Hamiltonian is integrable: Lipatov (1993);
Minahan, Zarembo (2002);
– infinitely many conserved charges
– scattering of quasi-particles (magnons) Beisert, Kristjansen,
Staudacher (2003); …
via 2  2 S matrix obeying YBE
• Also: integrability of AdS5 x S5 s-model Bena, Polchinski, Roiban (2003)
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Integrability  anomalous dim’s
• 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
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
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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
Beisert, Eden, Staudacher
[hep-th/0610251]
Agrees with first 3 terms
of strong-coupling expansion
Gubser Klebanov, Polyakov,
th/0204051;
Frolov, Tseytlin, th/0204226;
Roiban, Tseytlin, 0709.0681 [th]
Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135]
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New
Bootstraps Old and
Full strong-coupling
expansion
Basso,Korchemsky,
Kotański,
0708.3933 [th]
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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
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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
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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
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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 
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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:
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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)
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rational part
R(z)
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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
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Six-point remainder function
• n = 6 first place BDS Ansatz must be modified, due to
dual conformal cross ratios
MHV
6
1
5
2
4
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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 symbology
Goncharov, Spradlin, Vergu, Volovich, 1006.5703
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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 function R6(3)(u1,u2,u3)
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Professor of symbology at Harvard University, has used
these techniques to make a series of important advances:
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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
+
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perms
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S[ R6(2)(u,v,w) ] in these variables
GSVV, 1006.5703
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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
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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
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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
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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]
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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
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OPE Constraints (cont.)
• As t  ∞ , v = 1/cosh2t 
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-1
and even
v1 e±2if [ln v] k , k = 0,1,2,…L-1
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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
a1 = - 3/8
Caron-Huot, He, 1112.1060
a2 = 7/32
• Also follow from Basso, Sever, Vieira
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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:
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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).
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How many functions?
First entry
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; non-product
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R6(3)(u,v,w)
Many relations among coproduct coefficients for Rep:
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Only 2 indep. Rep coproduct coefficients
2 pages of 1-d HPLs
Similar (but shorter) expressions for lower degree functions
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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
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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 = 0,1
fixes last 2 constants
(a2 agrees with Caron-Huot+He and BSV)
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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!
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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
Put LLA, NLLA results into bootstrap;
extract NkLLA, k > 1
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Fadin, Lipatov,
1111.0782;
LD, Duhr, Pennington,
1207.0186;
Pennington, 1209.5357
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NNLLA impact factor now fixed
Result from DDP, 1207.0186 still had
3 beyond-the-symbol ambiguities
Now all 3 are fixed:
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Simple slice: (u,u,1)  (1,v,v)
Collapses to 1d HPLs:
Includes base point (1,1,1) :
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Plot R6(3)(u,v,w)
on some slices
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(u,u,u)
Indistinguishable(!) up to rescaling by:
ratio ~
cf. cusp ratio:
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Proportionality ceases at large u
ratio ~ -1.23
0911.4708
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(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
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Ratio for (u,u,1)  (1,v,v)   (w,1,w)
ratio ~ ln(u)/2
ratio ~ - 9.09
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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  0 constants!! 
[Thanks to BSV for supplying this info!]
• Next step: Fix bts constants, after defining
functions (globally? or just on a subspace?)
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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
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What about (quantum n=8 super)gravity?
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Extra Slides
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Multi-Regge kinematics
2
1
6
5
3
4
Very nice change of variables
[LP, 1011.2673] is to
:
2 symmetries: conjugation
and inversion
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Numerical integration contours
base point (u,v,w) = (1,1,1)
base point (u,v,w) = (0,0,1)
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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:
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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
• Two loops, n=4,5,6
Brandhuber, Heslop, Travaglini, 0707.1153
Drummond, Henn, Korchemsky, Sokatchev,
0709.2368, 0712.1223, 0803.1466;
Bern, LD, Kosower, Roiban, Spradlin,
Vergu, Volovich, 0803.1465
Wilson-loop VEV always matches [MHV] scattering amplitude!
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
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