Multi-Loop Amplitudes with Maximal Supersymmetry Lance Dixon (SLAC) work with Z. Bern, J.J Carrasco, H. Johansson, R. Roiban, to appear International workshop on gauge and string amplitudes  IPPP Durham, 30 March 2009

Why maximal supersymmetry? Two maximally supersymmetric theories in D=4: N=4 super-Yang-Mills theory N=8 supergravity Each of interest in its own right: N=4 conformal, related to gravity by AdS/CFT N=8 amazingly well-behaved in ultraviolet, for a point-particle theory of gravity The two theories are very closely linked: Tree-level KLT (and more recent) relations KLT leads to relations between multi-loop amplitudes L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY vs. spectra 28 = 256 massless states, ~ expansion of (x+y)8 SUSY 24 = 16 states ~ expansion of (x+y)4 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Why multi-loop amplitudes? N=4 super-Yang-Mills theory: AdS/CFT connection – especially strong in planar (large Nc) limit  classical strings/gravity Integrability (connection to amplitudes could be strengthened) Infrared singularities resemble QCD, give insight into IR structure there Interesting patterns emerge pseudoconformality of (planar) loop integrals dual superconformal invariance (in planar limit) homogeneous maximal transcendentality in D = 4 – 2e; still not well understood; also extends to nonplanar terms color structure of UV poles L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Why multi-loop amplitudes? N=8 supergravity: Possibility of ultraviolet finiteness Study infrared singularities in gravity Interpret what “ gravity ~ (gauge theory)2 ” means L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Gravity ~ (gauge theory)2 at 1 and 2 Loops Green, Schwarz, Brink; Grisaru, Siegel (1981) “color dresses kinematics” Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998) 2 loops: N=8 supergravity: just remove color, square prefactors! L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Kawai-Lewellen-Tye relations KLT, 1986 Derive from relation between open & closed string amplitudes. Low-energy limit gives N=8 supergravity amplitudes as quadratic combinations of N=4 SYM amplitudes , consistent with product structure of Fock space, L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

More recent tree relations Britto, Cachazo, Feng, hep-th/0412308; BCFW, hep-th/0501052 Elvang, Freedman, 0710.1270 Drummond, Spradlin, Volovich, Wen, 0901.2363 Derived from BCFW recursion relations. Also gives N=8 supergravity amplitudes as quadratic combinations of (pieces of) N=4 SYM amplitudes, consistent with Dual superconformal invariants – e.g. Drummond, Henn, 0808.2475 Dual superconformal invariants – Drummond, Henn, 0808.2475 where N=4 trees are Dual superconformal invariants – Drummond, Henn, 0808.2475 Dual superconformal invariants – Drummond, Henn, 0808.2475 Dual superconformal invariants – Drummond, Henn, 0808.2475 Dual superconformal invariants – Drummond, Henn, 0808.2475 Dual superconformal invariants – Drummond, Henn, 0808.2475 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Comparison of KLT and DSVW Both give N=8 supergravity amplitudes as quadratic combinations of (pieces of) N=4 SYM amplitudes. Both respect (very useful for exploiting at higher loops) KLT has different permutations appearing, [EF ] DSVW always has same permutation in a given term KLT has “gravity factors” built out of Lorentz dot products DSVW gravity factors are built from spinor strings, such as L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Multi-loop generalized unitarity Bern, LD, Kosower, hep-ph/0001001; Bern, Czakon, LD, Kosower, Smirnov hep-th/0610248; Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/0702112; BCJK, 0705.1864; Cachazo, Skinner, 0801.4574; Cachazo, 0803.1988; Cachazo, Spradlin, Volovich, 0805.4832 Ordinary cuts of multi-loop amplitudes contain loop amplitudes. But it is very convenient to work with tree amplitudes only. For example, at 3 loops, one encounters the product of a 5-point tree and a 5-point one-loop amplitude: Cut 5-point loop amplitude further, into (4-point tree) x (5-point tree), in all 3 inequivalent ways: cut conditions satisfied by real momenta L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Chop further Allowing for complex cut momenta, one can chop an amplitude entirely into 3-point trees  maximal cuts or ~ leading singularities Advantage is that these cuts are maximally simple, yet give an excellent starting point for constructing the full answer. For example, in planar (leading in Nc) N=4 SYM they find all terms in the complete answer for 1, 2 and 3 loops L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Finding missing terms Maximal cut method: Allowing one or two propagators to collapse from each maximal cut, one obtains near-maximal cuts These near-maximal cuts are very useful for analyzing N=4 SYM (including nonplanar) and N=8 SUGRA at 3 loops BCDJKR, BCJK (2007); Bern, Carrasco, LD, Johansson, Roiban, 0808.4112 Leading singularity method: Uses consistent behavior with respect to “hidden singularities” Cachazo, Skinner; Cachazo (2008) Recent supersum advances for more complicated cuts Drummond, Henn, Korchemsky, Sokatchev, 0808.0491; Arkani-Hamed, Cachazo, Kaplan, 0808.1446; Elvang, Freedman, Kiermaier, 0808.1720; Bern, Carrasco, Ita, Johansson, Roiban, to appear L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Multi-loop “KLT copying” Bern, LD, Dunbar, Perelstein, Rozowsky (1998) Suppose we know an N=4 SYM amplitude at some loop order – both planar and nonplanar terms. Then we have “simple” forms for all of its generalized cuts, i.e. products of N=4 SYM trees, already summed over all internal states The KLT relations let us write the N=8 SUGRA cuts, which are products of N=8 SUGRA trees, summed over all internal states, very simply in terms of sums of products of two copies of the N=4 SYM cuts. L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Example of KLT copying at 3 loops Using it is easy to see that N=8 SUGRA N=4 SYM N=4 SYM rational function of Lorentz products of external and cut momenta; all state sums already performed L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY The rung rule Many higher-loop contributions to gg  gg scattering deduced from a simple property of the 2-particle cuts at one loop Bern, Rozowsky, Yan (1997) Leads to “rung rule” for easily computing all contributions which can be built by iterating 2-particle cuts L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Rung rule for gravity “KLT copying” for 4-point amplitudes is very simple, leads to simple rung rule for N=8 supergravity as well. BDDPR (1998) N=4 SYM rung rule N=8 SUGRA rung rule 2 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY 3 loop amplitude Nine basic integral topologies Seven (a-g) were already known (2-particle cuts  rung rule) BDDPR (1998) Two new ones (h,i) have no 2-particle cuts BCDJKR (2007); BCDJR (2008) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 numerators at 3 loops Omit overall manifestly quadratic in loop momentum L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=8 numerators at 3 loops Omit overall also manifestly quadratic in loop momentum BCDJR (2008) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Applications of the (full color) three-loop amplitudes L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=8 SUGRA in the UV N=8 supergravity in higher dimension D still has same critical dimension for UV divergence as N=4 SYM: Dc = 8 L = 1 Dc = 4 + 6/L L = 2,3 If this persists to all loop orders then N=8 supergravity would be perturbatively finite. We can also evaluate the precise 3-loop divergence in Dc = 6 – 2e (mainly to verify that it’s nonzero): Note homogeneous transcendentality (z3, no 1) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 SYM in the UV Determine the full color dependence of the UV divergences in N=4 SYM in the critical dimension. Bern, Carrasco, LD, Johansson, Roiban, to appear Dc = 8 (L = 1) Dc = 4 + 6/L (L = 2,3) Of interest in light of recent investigations of potential counterterms Bossard, Howe, Stelle, 0901.4661 We have results for general gauge group G. Here we just give the case G = SU(Nc), in terms of the color structures: L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 SYM in UV at one loop Box integral in Dc = 8 - 2e with color factor  where Corresponds to counterterms such as and (no extra derivatives) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 SYM in UV at two loops Planar and nonplanar double box integrals in Dc = 7 - 2e easy [BDDPR 1998] with color factors  “20” terms linked to each other by group theory Corresponds to counterterms such as and (two extra derivatives) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=4 SYM in UV at three loops UV poles of integrals (e), (f), (g), (i) in Dc = 6 - 2e [BCDJR 2008] with color factors  Corresponds to type counterterms. Absence of double-trace terms of form at L = 3 calls out for explanation. Curiously, full divergence does not have uniform transcendentality – but each coefficient of Nck does! L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=4 SYM in IR at three loops Full color dependence of IR singularities in gauge amplitudes in D = 4 - 2e is dictated by two constants in collinear jet functions, plus the soft anomalous dimension matrix Akhoury (1979); Mueller (1979); Collins (1980); Sen (1981); Sterman (1987); Botts, Sterman (1989); Catani, Trentadue (1989); Korchemsky (1989); Magnea, Sterman (1990); Korchemsky, Marchesini (1992); Catani (1998); Sterman, Tejeda-Yeomans (2002) See talk by Lorenzo Magnea L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=4 SYM in IR at three loops (cont.) Constants in jet functions already known in N=4 SYM at three loops, where they are purely planar Bern, LD, Smirnov, hep-th/0505205 Also equal to the maximal transcendentality part of the QCD constants! Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/0404092 Moch, Vermaseren, Vogt, hep-ph/0403192, hep-ph/0404111, hep-ph/0508055 transcendentality (weight): n for pn n for zn What’s not completely known at 3 loops (although there are many constraints & conjectures) is the soft anomalous dimension matrix – see talk by Magnea L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=4 SYM in IR at three loops (cont.) What is known at three loops is the matter-dependent part of LD, 0901.3414 The difference between QCD and N=4 SYM is “just matter” So a 3-loop quantity needed for QCD resummation can be extracted by evaluating, through 1/e, 9 integrals from an N=4 SYM amplitude! The 2 planar integrals are known, but unfortunately the 7 non-planar integrals are still intractable… Recent progress though with one fewer leg: Heinrich, Huber, Kosower, Smirnov, 0902.3512; Baikov, Chetyrkin, Smirnov2, Steinhauser, 0902.3519 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=8 SUGRA in IR at three loops Similarly, once the (related) integrals are evaluated (even numerically), one could test the exponentiation of IR divergences in gravity, as observed so far at two loops, Naculich, Nastase, Schnitzer, 0805.2347; Brandhuber, Heslop, Nasti, Spence, Travaglini, 0805.2763 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

On to four loops (full color N=4 SYM) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

The 4 loop 4 point amplitude in N=4 SYM Scope of the problem illustrated by the number of cubic 4-point graphs with nonvanishing coefficients and various topological properties L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Vacuum cubic graphs at 4 loops To decorate with 4 external legs cannot generate a nonvanishing (no-triangle) cubic 4-point graph only generate rung rule topologies the most complex L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Planar terms all known Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248 Planar result has pseudoconformal invariance, acting on dual or sector variables xi No such guide for the nonplanar terms Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Box cut Bern, Carrasco, Johansson, Kosower, 0705.1864 If the diagram contains a box subdiagram, can use the simplicity of the 1-loop 4-point amplitude to compute the numerator very simply Planar example: Only five 4-loop cubic topologies do not have box subdiagrams. But there are also “contact terms” to determine. L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Twist identity If the diagram contains a four-point tree subdiagram, can use a Jacobi-like identity to relate it to other diagrams. Bern, Carrasco, Johansson, 0805.3993 Relate non-planar topologies to planar, etc. For example, at 3 loops, (i) = (e) – (e)T [ + contact terms ] 2 3 1 4 - = L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Checks on the final result Lots of different products of MHV tree amplitudes. NMHV7 * anti-NMHV7 and MHV5 * NMHV6 * anti-MHV5 – evaluated by Elvang, Freedman, Kiermaier, 0808.1720 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Simple (rung rule) diagrams L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

(mostly) box cut diagrams L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Most complex diagrams L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 in UV at 4 loops Need UV poles of 4-loop vacuum graphs (doubled propagators represented by blue dots). By injecting external momentum in right place, can rewrite as 4-loop propagator integrals that factorize into product of -1-loop propagator integral with UV pole - finite 3-loop propagator integral Do this in multiple ways  Either “gluing relations” or cross-check. Only 3 vacuum integrals required Dc = 4 + 6/4 = 11/2 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY N=4 in UV at 4 loops (cont.) Reduce 3-loop propagator integrals to master integrals using integration by parts (IBP), a la MINCER. Chetyrkin, Tkachov (1981) Most nontrivial integral is nonplanar master integral, for which we only have numerical results (obtained using Gegenbauer polynomial x-space technique Chetyrkin, Tkachov (1981); Bekavac, hep-ph/0505174 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

N=4 SYM in UV at four loops Combining UV poles of integrals with color factors  Again corresponds to type counterterms. Absence of double-trace terms at L = 4 again calls out for explanation. Related to better UV behavior of abelian theories? Transcendentality = 3 ?? L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Conclusions Full color 4 point 4 loop amplitude has been computed in N=4 super-Yang-Mills theory, using a combination of rung rule, box cut, twist identity, and ansatze. UV divergences have been extracted, and compared with results for L = 1,2,3. Double-trace terms drop out after L = 2. Why? Next task (in progress) is to compute the 4-point 4-loop amplitude in N=8 supergravity, and inspect its UV behavior. Will it continue to be as well-behaved as N=4 super-Yang-Mills theory? Wealth of IR information in gauge theory & gravity is also available … once technology is developed for doing non-planar 4-point integrals (even numerically) in D = 4 – 2e at L = 3,4. L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Extra Slides L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

L. Dixon Multi-loop Amp's with Max. SUSY Vacuum integrals at four loops (before cancellations between different topologies) L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

Integrals for planar amplitude at 5 loops Bern, Carrasco, Johansson, Kosower, 0705.1864 L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009

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)  includes cusp anomalous dimension 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 pn n for zn Similar counting for HPLs and for related harmonic sums used to describe DGLAP kernels at finite j L. Dixon Multi-loop Amp's with Max. SUSY Amp's Durham 30 March 2009