Webs of soft gluons from QCD to N=4 super-Yang-Mills theory Lance Dixon (SLAC) KEK Symposium “Towards precision QCD physics” in memory of Jiro Kodaira

Webs of soft gluons from QCD to

N=4 super-Yang-Mills theory

Lance Dixon (SLAC)

KEK Symposium“Towards precision QCD physics”

in memory of Jiro Kodaira March 10, 2007

Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 2

26 years ago…


26 years ago…• I was also at SLAC in the summer of 1981 – but as an undergraduate working on the Mark III experiment at SPEAR• I had no idea what “Summing Soft Emission” meant – although I did sneak into some of the SLAC Summer Institute lectures on The Strong Interactions• So I could not yet appreciate the beauty of this formula:


Outline

• The two-loop soft anomalous dimension matrix in QCD – it’s all about K

• Multi-loop analogs of K (cusp anomalous dim.)– we now (probably) know them to all loop orders in large Nc N=4 super-Yang-Mills theory, and tantalizing pieces of them in QCD as well

Aybat, LD, Sterman, hep-ph/0606254, 0607309

Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248Eden, Staudacher, hep-ph/0603157Beisert, Eden, Staudacher, hep-th/0610251Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/0404092Benna, Benvenuti, Klebanov, Scardicchio, hep-th/0611135


IR Structure of QCD Amplitudes [Massless Gauge Theory Amplitudes]• Expand multi-loop amplitudes in d=4-2around d=4 (=0) • Overlapping soft (1/) + collinear (1/) divergences at each loop order imply leading poles are ~ 1/2L at L loops

• Pole terms are predictable, due to soft/collinear factorization and exponentiation, in terms of acollection of constants (anomalous dimensions)

Mueller (1979); Akhoury (1979); Collins (1980), hep-ph/0312336; Kodaira, Trentadue (1981);Sen (1981, 1983); Sterman (1987); Botts, Sterman (1989); Catani, Trentadue (1989); Korchemsky (1989); Magnea, Sterman (1990); Korchemsky, Marchesini, hep-ph/9210281; Giele, Glover (1992); Kunszt, Signer, Trócsányi, hep-ph/9401294; Kidonakis, Oderda, Sterman, hep-ph/9801268, 9803241; Catani, hep-ph/9802439; Dasgupta, Salam, hep-ph/0104277; Sterman, Tejeda-Yeomans, hep-ph/0210130; Bonciani, Catani, Mangano, Nason, hep-ph/0307035; Banfi, Salam, Zanderighi, hep-ph/0407287; Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157

• Same constants control resummation of large logarithms near kinematic boundaries – as Jiro Kodaira understood so well


Soft/Collinear FactorizationMagnea, Sterman (1990) Sterman, Tejeda-Yeomans, hep-ph/0210130

• S = soft function (only depends on color of ith particle; matrix in “color space”)• J = jet function (color-diagonal; depends on ith spin) • H = hard remainder function (finite as ; vector in color space) color: Catani, Seymour, hep-ph/9605323; Catani, hep-ph/9802439


• For the case n=2, gg 1 or qq 1,the color structure is trivial, so the soft function S = 1

• Thus the jet function is the square-root of the Sudakov form factor (up to finite terms):

_

The Sudakov form factor


Jet function

• By analyzing structure of soft/collinear terms in axial gauge, find differential equation for jet function J[i] (~ Sudakov form factor):

finite as contains all Q2dependence

Mueller (1979); Collins (1980);Sen (1981); Korchemsky, Radyushkin (1987); Korchemsky (1989); Magnea, Sterman (1990)

Pure counterterm (series of 1/ poles);like (,s), single poles in determine completely

also obey differential equations (ren. group): cusp anomalous dimension


can be extracted from fixed-order calculations of form factors or related objects

• Solution to differential equations

s = running coupling in D=4-2_

E.g. at three loops Moch, Vermaseren, Vogt, hep-ph/0507039, hep-ph/0508055

Magnea, Sterman (1990)

Jet function solution


Soft function

• For generic processes, need soft function S• Much less well-studied than J• Also obeys a (matrix) differential equation:

Kidonakis, Oderda, Sterman, hep-ph/9803241

soft anomalous dimension matrix

Solution is a path-ordered exponential:

depends on massless 4-velocities ; momenta are


Equivalently, consider web function W or eikonal amplitude of n Wilson lines

E.g. for n=4, 1 + 2 3 + 4:

Computation of soft anomalous dimension matrix

Remove jet function contributions by dividing by appropriate Sudakov factors

Only soft gluons couplings classical, spin-independent • Take hard external partons to be scalars• Expand vertices and propagators


1-loop soft anomalous dim. matrix1/ poles in 1-loop graphyield:

Kidonakis, Oderda, Sterman, hep-ph/9803241

Agrees with known divergences of generic one loop amplitudes:

Giele, Glover (1992); Kunszt, Signer, Trócsányi, hep-ph/9401294; Catani, hep-ph/9802439

Finite, hard parts scheme-dependent!

Expansion of 1-loop amplitude


2-loop soft anomalous dim. matrix

• Classify web graphs according to number of eikonal lines (nE)

• 4E graphs factorize trivially into products of 1-loop graphs. • 1-loop counterterms cancel all 1/ poles, leave no contribution to

Two 3E graphs – each looks as if it might give a complicated color structure depending on 3 legs!


But:

vanishes due to antisymmetry after changing to light-cone variables with respect to A, B= 0

factorizes into 1-loop factors, allowing its divergences to becompletely cancelled by 1-loop counterterms

and

+


The 2E graphsAll were previously analyzedfor the cusp anomalous dimension

Korchemsky, Radyushkin (1987); Korchemskaya, Korchemsky, hep-ph/9409446

Same analysis can be used here (although color flow is generically different)

All color factors become proportional to the one-loop ones,

Proportionality constant dictated by cusp anomalous dimension

2-loop soft anomalous dimension – it’s all about K


Implications for resummation

• To resum a generic hadronic event shape

requires diagonalizing the exponentiated soft anomalous dimension matrix in color space• Because of the proportionality relation, same diagonalization at one loop (NLL) still works at two loops (NNLL), and eigenvalue shift is trivial!• Result foreshadowed in the bremsstrahlung (CMW) scheme Catani, Marchesini, Webber (1991)

for redefining the strength of parton showering using

Kidonakis, Oderda, Sterman, hep-ph/9801268, 9803241; Dasgupta, Salam, hep-ph/0104277; Bonciani, Catani, Mangano, Nason, hep-ph/0307035; Banfi, Salam, Zanderighi, hep-ph/0407287


Why N=4 super-Yang-Mills theory?

• Most supersymmetric theory possible without gravity• Uniquely specified by local internal symmetry group

– e.g., number of colors Nc for SU(Nc)• An exactly scale-invariant (conformal) field theory:

for any coupling g, (g) = 0• Connected to gravity and/or string theory by

– AdS/CFT correspondence, a weak/strong duality

• Remarkable “transcendentality” relations with QCD


“Leading transcendentality” relation between QCD and N=4 SYM

• KLOV (Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/0404092)

noticed (at 2 loops) a remarkable relation between kernels for • BFKL evolution (strong rapidity ordering)• DGLAP evolution (pdf evolution = strong collinear ordering)

in QCD and N=4 SYM:• Set fermionic color factor CF = CA in the QCD result and keep only the “leading transcendentality” terms. They coincide with the full N=4 SYM result (even though theories differ by scalars) • Conversely, N=4 SYM results predict pieces of the QCD result

• transcendentality (weight): n for n

n for n

Similar counting for HPLs and for related harmonic sums used to describe DGLAP kernelsat finite j


Moch Vermaseren, Vogt (MVV), hep-ph/0403192, hep-ph/0404111

in QCD through 3 loops:

K from Kodaira, Trentadue (1981)


in N=4 SYM through 3 loops:

KLOV prediction

• Finite j predictions confirmed (with assumption of integrability) Staudacher, hep-th/0412188

• Confirmed at infinite j using on-shell amplitudes, unitarity Bern, LD, Smirnov, hep-th/0505205

• and with all-orders asymptotic Bethe ansatz Beisert, Staudacher, hep-th/0504190• leading to an integral equation Eden, Staudacher, hep-th/0603157


An all-orders proposalIntegrability, plus an all-orders asymptotic Bethe ansatz led to the following proposal for the cusp anomalous dimensionin large Nc N=4 SYM:

where

is the solution to an integral equation with Bessel-function kernel

Perturbative expansion:

?

Eden, Staudacher, hep-ph/0603157


ES proposal (cont.)

Eden, Staudacher, hep-ph/0603157

Because of various assumptions made, particularly an overall dressing factor, which could affect the entire“world-sheet S-matrix”, and which was known to benon-trivial at strong-coupling, the ES proposal needed checking via another perturbative method, particularly at 4 loops.


• AdS/CFT duality suggests that weak-coupling perturbation series for planar N=4 SYM should have very special properties: strong-coupling limit is equivalent to weakly-coupled strings in large-radius AdS5 x S5 background

– -model classically integrable too – world-sheet -model coupling is

• Cusp anomalous dimension should be given semi-classically, by energy of a long string, a soliton in the -model, spinning in AdS5

• First two strong-coupling terms known

Cusp anomalous dimension via AdS/CFT

Maldacena, hep-th/9711200;Gubser, Klebanov, Polyakov, hep-th/9802109

Gubser, Klebanov, Polyakov, hep-th/0204051

Bena, Polchinski, Roiban,hep-th/0305116

Frolov, Tseytlin, hep-th/0204226


Four-loop planar N=4 SYM amplitudeBCDKS, hep-th/0610248Very simple – only 8 loop integrals required!


Soft/collinear simplification in large Nc (planar) limit

• Soft function only defined up to a multiple of the identity matrix in color space• Planar limit is color-trivial; can absorb S into Ji

• If all n particles are identical, say gluons, then each “wedge” is the square root of the “gg 1” process (Sudakov form factor):


Sudakov form factor in planar N=4 SYM

• Expand in terms of

=0, so running coupling in D=4-2has only trivial (engineering) dependence on scale , simplifying differential equations


General amplitude in planar N=4 SYM

Insert result for form factor into n-point amplitude

extract cusp anomalous dimension from coefficient of pole

We found a numerical result consistent with:

compared with ES prediction, a single sign flip at four loops!

We also argued that at order the signs of terms containing should be flipped as well, …


Independently…

At the same time, investigating the strong-coupling properties of the dressing factor led Beisert, Eden and Staudacher [hep-th/0610251] to propose an integral equation with a newkernel: with

With the “2”, the result is to flip signs of odd-zeta terms in ES prediction, to all orders (actually, 2k+1 i 2k+1 )

Arutyunov, Frolov, Staudacher, hep-th/0406256;Hernandez, Lopez, hep-th/0603204; …


Soon thereafter …

Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135]solved BES integral equation numerically, by expanding in basis of Bessel functions.

Solution agrees stunningly well with “KLV approximate formula,” which incorporates the known strong-coupling behavior


Conclusions for part 2

• Combining a number of approaches, an exact solution for the cusp anomalous dimension in planar N=4 SYM appears to be in hand.• Result provides a very interesting test of the AdS/CFT correspondence. •Through KLOV conjecture, the exact solution provides all-loop information about certain “most transcendental” terms in K(s) in perturbative QCD.• The multi-loop analogs of K are related to the energy of a spinning string in anti-de Sitter space!• What would Jiro Kodaira make of all this?!


Extra Slides


Soft computation (cont.)• Regularize collinear divergences by removing Sudakov-type factors (in eikonal approximation), from web function, defining soft function S by:

• Soft anomalous dimension matrix determinedby single ultraviolet poles in of S:


Proportionality at 3 loops?

Again classify web graphs according to number of eikonal lines (nE)

6E and 5E graphs factorize trivially into products of lower-loop graphs; no contribution to thanks to 2-loop result

4E graphs

also trivial???

and then there are more 4E graphs, and the 3E and 2E graphs…

use same (A,B) change of variables


Consistency with explicit multi-parton 2-loop computations

• Results for

• Organized according to Catani, hep-ph/9802439• After making adjustments for different schemes, everythingis consistent

Anastasiou, Glover, Oleari, Tejeda-Yeomans (2001); Bern, De Freitas, LD (2001-2); Garland et al. (2002); Glover (2004); De Freitas, Bern (2004); Bern, LD, Kosower, hep-ph/0404293

And electroweak Sudakov logs for 2 2 also matchJantzen, Kühn, Penin, Smirnov, hep-ph/0509157

Documents

Webs of soft gluons from QCD to N=4 super-Yang-Mills theory Lance Dixon (SLAC) KEK Symposium “Towards precision QCD physics” in memory of Jiro Kodaira