bootstraps old and new
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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 . - PowerPoint PPT PresentationTRANSCRIPT
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Bootstraps Old and New
L. Dixon, J. Drummond, M. von Hippeland J. Pennington
1305.nnnnAmplitudes 2013
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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
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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 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
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Amplitudes 2013, May 2 4L. Dixon Bootstraps Old and New
Duality
=
Veneziano (1968)
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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
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Amplitudes 2013, May 2 6L. Dixon Bootstraps Old and New
Crossing symmetry condition
=
f
f
ii
Unitarity:
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Amplitudes 2013, May 2 7
Ising Model in D=3
L. Dixon Bootstraps Old and New
El-Showk et al., 1203.6064Anomalous dimension bounds from unitarity + crossing + knowledge of conformal blocks+ scan over intermediate states + linear programming techniques
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Amplitudes 2013, May 2 8
Conformal Bootstrap for N=4 SYM
L. Dixon Bootstraps Old and New
Beem, Rastelli, van Rees, 1304.1803
Would be very interesting to make contact with perturbativeapproaches e.g. as in talk by Duhr
planar limit
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Amplitudes 2013, May 2 9
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]
L. Dixon Bootstraps Old and New
=
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Amplitudes 2013, May 2 10L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 11L. Dixon Bootstraps Old and New
• 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
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L. Dixon Bootstraps Old and New
Amplitudes 2013, May 2 12
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]
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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
L. Dixon Bootstraps Old and New
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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
L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 15
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
L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 16L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 17
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:
L. Dixon Bootstraps Old and New
+
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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]
L. Dixon Bootstraps Old and New
A(z) R(z)rational part
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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
L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 20L. Dixon Bootstraps Old and New
• 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
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Amplitudes 2013, May 2 21L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 22L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 23L. Dixon Bootstraps Old and New
Professor of symbology at Harvard University, has usedthese techniques to make a series of important advances:
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Amplitudes 2013, May 2 24L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 25L. Dixon Bootstraps Old and New
S[ R6(2)(u,v,w) ] in these variables
GSVV, 1006.5703
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Amplitudes 2013, May 2 26L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 27L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 28L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 29L. Dixon Bootstraps Old and New
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]
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Amplitudes 2013, May 2 30L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 31L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 32L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 33L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 34L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 35L. Dixon Bootstraps Old and New
How many functions? First entry ; non-product
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Amplitudes 2013, May 2 36
R6(3)(u,v,w)
L. Dixon Bootstraps Old and New
Many relations among coproduct coefficients for Rep:
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Only 2 indep. Rep coproduct coefficients
L. Dixon Bootstraps Old and New
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 limit2. Multi-regge limit
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Amplitudes 2013, May 2 39L. Dixon Bootstraps Old and New
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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 = 0,1fixes last 2 constants (a2 agrees with Caron-Huot+He and BSV)
L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 41L. Dixon Bootstraps Old and New
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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Amplitudes 2013, May 2 42L. Dixon Bootstraps Old and New
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
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Amplitudes 2013, May 2 43L. Dixon Bootstraps Old and New
NNLLA impact factor now fixedResult from DDP, 1207.0186 still had 3 beyond-the-symbol ambiguities
Now all 3 are fixed:
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Amplitudes 2013, May 2 44
Simple slice: (u,u,1) (1,v,v)
L. Dixon Bootstraps Old and New
Includes base point (1,1,1) :
Collapses to 1d HPLs:
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Amplitudes 2013, May 2 45L. Dixon Bootstraps Old and New
Plot R6(3)(u,v,w)
on some slices
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Amplitudes 2013, May 2 46L. Dixon Bootstraps Old and New
Indistinguishable(!) up to rescaling by:
cf. cusp ratio:
ratio ~
(u,u,u)
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Amplitudes 2013, May 2 47
Proportionality ceases at large u
L. Dixon Bootstraps Old and New
0911.4708
ratio ~ -1.23
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Amplitudes 2013, May 2 48L. Dixon Bootstraps Old and New
(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)
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Amplitudes 2013, May 2 49
Ratio for (u,u,1) (1,v,v) (w,1,w)
L. Dixon Bootstraps Old and New
ratio ~ - 9.09
ratio ~ ln(u)/2
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Amplitudes 2013, May 2 50
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, …
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Amplitudes 2013, May 2 51L. Dixon Bootstraps Old and New
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?
L. Dixon Bootstraps Old and New
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Extra Slides
L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 54L. Dixon Bootstraps Old and New
Multi-Regge kinematics1 2
345
6
Very nice change of variables[LP, 1011.2673] is to :
2 symmetries: conjugationand inversion
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Numerical integration contours
L. Dixon Bootstraps Old and New
base point (u,v,w) = (1,1,1)
base point (u,v,w) = (0,0,1)
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Amplitudes 2013, May 2 56L. Dixon Bootstraps Old and New
• 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
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Amplitudes 2013, May 2 57L. Dixon Bootstraps Old and New
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Amplitudes 2013, May 2 58L. Dixon Bootstraps Old and New
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