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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. 26 years ago…. 26 years ago….

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## Webs of soft gluons from QCD to N=4 super-Yang-Mills theory

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**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**26 years ago…**L. Dixon Webs of soft gluons KEK symposium**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: L. Dixon Webs of soft gluons KEK symposium**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 QCDas well Aybat, LD, Sterman, hep-ph/0606254, 0607309 Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248 Eden, Staudacher, hep-ph/0603157 Beisert, Eden, Staudacher, hep-th/0610251 Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/0404092 Benna, Benvenuti, Klebanov, Scardicchio, hep-th/0611135 L. Dixon Webs of soft gluons KEK symposium**IR Structure of QCD Amplitudes[Massless Gauge Theory**Amplitudes] • Expand multi-loop amplitudes ind=4-2e around d=4 (e=0) • Overlapping soft (1/e) + collinear (1/e) divergences at each • loop order imply leading poles are ~1/e2Lat Lloops • Pole terms are predictable,due to soft/collinear factorization and exponentiation, in terms of a • collection of constants (anomalous dimensions) • Same constants control resummation of large logarithms • near kinematic boundaries – as Jiro Kodaira understood so well 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 L. Dixon Webs of soft gluons KEK symposium**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 Soft/Collinear Factorization Magnea, Sterman (1990) Sterman, Tejeda-Yeomans, hep-ph/0210130 L. Dixon Webs of soft gluons KEK symposium**_**• 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 L. Dixon Webs of soft gluons KEK symposium**finite as e 0; contains all Q2dependence**Pure counterterm (series of 1/e poles); like b(e,as), single poles in e determine completely also obey differential equations (ren. group): cusp anomalous dimension Jet function Mueller (1979); Collins (1980); Sen (1981); Korchemsky, Radyushkin (1987); Korchemsky (1989); Magnea, Sterman (1990) • By analyzing structure of soft/collinear terms • in axial gauge, find differential equation • for jet function J[i] (~ Sudakov form factor): L. Dixon Webs of soft gluons KEK symposium**_**as = running coupling in D=4-2e Jet function solution Magnea, Sterman (1990) • Solution to differential equations can be extracted from fixed-order calculations of form factors or related objects E.g. at three loops Moch, Vermaseren, Vogt, hep-ph/0507039, hep-ph/0508055 L. Dixon Webs of soft gluons KEK symposium**Solution is a path-ordered exponential:**depends on massless 4-velocities ; momenta are Soft function • For generic processes, need soft functionS • Much less well-studied than J • Also obeys a (matrix) differential equation: Kidonakis, Oderda, Sterman, hep-ph/9803241 soft anomalous dimension matrix L. Dixon Webs of soft gluons KEK symposium**Equivalently, consider web functionW**or eikonal amplitude of n Wilson lines E.g. for n=4, 1 + 2 3 + 4: Computation of soft anomalous dimension matrix • Only soft gluons • couplings classical, spin-independent • Take hard external partons to be scalars • Expand vertices and propagators Remove jet function contributions by dividing by appropriate Sudakov factors L. Dixon Webs of soft gluons KEK symposium**Expansion of 1-loop amplitude**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! 1-loop soft anomalous dim. matrix Kidonakis, Oderda, Sterman, hep-ph/9803241 1/e poles in 1-loop graph yield: L. Dixon Webs of soft gluons KEK symposium**4E graphs factorize trivially into**• products of 1-loop graphs. • 1-loop counterterms cancel all 1/e poles, leave no contribution to Two 3E graphs – each looks as if it might give a complicated color structure depending on 3 legs! 2-loop soft anomalous dim. matrix • Classify web graphs according to number of eikonal lines (nE) L. Dixon Webs of soft gluons KEK symposium**But:**vanishes due to antisymmetry after changing to light-cone variables with respect to A, B = 0 and factorizes into 1-loop factors, allowing its divergences to be completely cancelled by 1-loop counterterms + L. Dixon Webs of soft gluons KEK symposium**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 The 2E graphs Korchemsky, Radyushkin (1987); Korchemskaya, Korchemsky, hep-ph/9409446 All were previously analyzed for the cusp anomalous dimension Same analysis can be used here (although color flow is generically different) L. Dixon Webs of soft gluons KEK symposium**Implications for resummation**• To resum a generic hadronic event shape • requires diagonalizing the exponentiated • soft anomalous dimension matrix in color space • Because of theproportionality relation, • same diagonalization at one loop (NLL) still works • at two loops (NNLL), and eigenvalue shift is trivial! • Result foreshadowed in the bremsstrahlung • (CMW) schemeCatani, 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 L. Dixon Webs of soft gluons KEK symposium**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 Ncfor SU(Nc) • An exactly scale-invariant (conformal) field theory: for any coupling g,b(g) = 0 • Connected to gravity and/or string theory by • AdS/CFT correspondence, a weak/strong duality • Remarkable “transcendentality” relations with QCD L. Dixon Webs of soft gluons KEK symposium**“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 theQCD result • transcendentality (weight): n for pn • n for zn Similar counting for HPLs and for related harmonic sums used to describe DGLAP kernels at finite j L. Dixon Webs of soft gluons KEK symposium**in QCD through 3 loops:**K from Kodaira, Trentadue (1981) Moch Vermaseren, Vogt (MVV), hep-ph/0403192, hep-ph/0404111 L. Dixon Webs of soft gluons KEK symposium**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 L. Dixon Webs of soft gluons KEK symposium**Perturbative expansion:**? An all-orders proposal Integrability, plus an all-orders asymptotic Bethe ansatz led to the following proposal for the cusp anomalous dimension in large Nc N=4 SYM: Eden, Staudacher, hep-ph/0603157 where is the solution to an integral equation with Bessel-function kernel L. Dixon Webs of soft gluons KEK symposium**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 be non-trivial at strong-coupling, the ES proposal needed checking via another perturbative method, particularly at 4 loops. L. Dixon Webs of soft gluons KEK symposium**Cusp anomalous dimension via AdS/CFT**Maldacena, hep-th/9711200; Gubser, Klebanov, Polyakov, hep-th/9802109 • 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 • – s-model classically integrable too • – world-sheet s-model coupling is • Cusp anomalous dimension should be given • semi-classically, by energy of a long string, • a soliton in the s-model, spinning in AdS5 • First two strong-coupling terms known Bena, Polchinski, Roiban, hep-th/0305116 Gubser, Klebanov, Polyakov, hep-th/0204051 Frolov, Tseytlin, hep-th/0204226 L. Dixon Webs of soft gluons KEK symposium**Four-loop planar N=4 SYM amplitude**BCDKS, hep-th/0610248 Very simple – only 8 loop integrals required! L. Dixon Webs of soft gluons KEK symposium**Soft function only defined up to a multiple of the identity**matrix in color space • Planar limit is color-trivial; can absorb S intoJi • If all n particles are identical, say gluons, then each “wedge” is the square root of the “gg 1” process (Sudakov form factor): Soft/collinear simplification in large Nc(planar) limit L. Dixon Webs of soft gluons KEK symposium**Sudakov form factor in planar N=4 SYM**b=0, so running coupling in D=4-2ehas only trivial (engineering) dependence on scale m , simplifying differential equations • Expand in terms of L. Dixon Webs of soft gluons KEK symposium**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, … General amplitude in planar N=4 SYM Insert result for form factor into n-point amplitude extract cusp anomalous dimension from coefficient of pole L. Dixon Webs of soft gluons KEK symposium**Independently…**Arutyunov, Frolov, Staudacher, hep-th/0406256; Hernandez, Lopez, hep-th/0603204; … 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 new kernel: with With the “2”, the result is to flip signs of odd-zeta terms in ES prediction, to all orders (actually, z2k+1i z2k+1) L. Dixon Webs of soft gluons KEK symposium**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 L. Dixon Webs of soft gluons KEK symposium**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 gK(as) 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?! L. Dixon Webs of soft gluons KEK symposium**Extra Slides**L. Dixon Webs of soft gluons KEK symposium**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 determined • by single ultraviolet poles in e of S: L. Dixon Webs of soft gluons KEK symposium**6E and 5E graphs**factorize trivially into products of lower-loop graphs; no contribution to thanks to 2-loop result 4E graphs use same (A,B) change of variables ??? also trivial Proportionality at 3 loops? Again classify web graphs according to number of eikonal lines (nE) and then there are more 4E graphs, and the 3E and 2E graphs… L. Dixon Webs of soft gluons KEK symposium**Consistency with explicit multi-parton 2-loop computations**• Results for • Organized according toCatani, hep-ph/9802439 • After making adjustments for different schemes, everything • is 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 match Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157 L. Dixon Webs of soft gluons KEK symposium

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