Upload
philip-floyd
View
217
Download
0
Tags:
Embed Size (px)
Citation preview
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…
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 3
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:
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 4
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 5
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 6
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 7
• 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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 8
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 9
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 10
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 11
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 12
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 13
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!
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 14
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
+
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 15
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 16
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 17
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 18
“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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 19
Moch Vermaseren, Vogt (MVV), hep-ph/0403192, hep-ph/0404111
in QCD through 3 loops:
K from Kodaira, Trentadue (1981)
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 20
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 21
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 22
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.
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 23
• 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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 24
Four-loop planar N=4 SYM amplitudeBCDKS, hep-th/0610248Very simple – only 8 loop integrals required!
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 25
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):
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 26
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 27
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, …
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 28
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; …
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 29
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 30
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?!
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 31
Extra Slides
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 32
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:
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 33
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
Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 34
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