1. Scattering Amplitudes in Quantum Field Theory Lance Dixon (CERN & SLAC) EPS HEP 2011 25 July 2011

2. The S matrix reloaded Almost everything we know experimentally about gauge theory is based on scattering processes with asymptotic, on-shell states, evaluated in perturbation theory. Nonperturbative, off-shell information very useful, but often more qualitative (except for lattice gauge theory). All perturbative scattering amplitudes can be computed with Feynman diagrams – but that is not necessarily the best way, especially if there is hidden simplicity. On-shell methods can be much more efficient, and provide new insights. Use analytic properties of the S matrix directly, not Feynman diagrams. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 2

3. Three Applications of On-Shell Methods QCD: a very practical application, needed for quantifying LHC backgrounds to new physics talk by G. Zanderighi N=4 super-Yang-Mills theory: lots of simplicity, both manifest and hidden. A particularly beautiful application of on-shell and related methods – see also talk by N. Beisert N=8 supergravity: amazingly good UV behavior, beyond expectations, unveiled by on-shell methods L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 3

4. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 4 The Analytic S-Matrix Bootstrap program for strong interactions: Reconstruct scattering amplitudes directly from analytic properties: “on-shell” information Landau; Cutkosky; Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne; Veneziano; Virasoro, Shapiro; … (1960s) Analyticity fell out of favor in 1970s with the rise of QCD & Feynman rules Now resurrected for computing amplitudes in perturbative QCD as alternative to Feynman diagrams! Perturbative information now assists analyticity. Works for many other theories too. Poles Branch cuts

5. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 5 Perturbative unitarity bootstrap Reconstruct (multi-)loop amplitudes from cuts efficiently, due to simple structure of tree and lower-loop helicity amplitudes S-matrix a unitary operator between in and out states S † S = 1  unitarity relations (cutting rules) for amplitudes Generalized unitarity reduces everything to (simpler) trees

6. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 6 Granularity vs. Plasticity

7. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 7 Generalized Unitarity (One-loop Plasticity) Ordinary unitarity: put 2 particles on shell Generalized unitarity: put 3 or 4 particles on shell Trees recycled into loops!

8. “The most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time” S. Chandrasekhar Black Holes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 8

9. The most perfect microscopic objects there are in the universe? When gravitons scatter, the only elements in the construction are again our concepts of space and time Scattering Amplitudes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 9 h h h h

10. Seem less universal at first sight: Have to specify gauge group G, representation R for matter fields, etc. However, full amplitudes A n can be assembled from universal, G,R-independent “color-stripped” or “color-ordered” amplitudes A n : Gauge Theory Amplitudes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 10 g gg g coloruniversal

11. Through the clouds of gas and dust Obscure astrophysical black holes, but also make them detectable, their physics much richer L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 11

12. Through the clouds of soft interactions Obscure hard QCD amplitudes at their core, but also make the physics much richer L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 12 F. Krauss

13. In relativistic limit, particle masses m b,m W, m t, … become unimportant, making gauge amplitudes still more universal. In QCD, if we could go to extremely high energies, then asymptotic freedom,  s  0,  we would need only leading order cross sections, i.e. tree amplitudes Relativistic Gauge Amplitudes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 13 =+ …

14. For pure-glue trees, matter particles (fermions, scalars) cannot enter, because they are always pair-produced Pure-glue trees are exactly the same in QCD as in maximally supersymmetric gauge theory, N=4 super-Yang-Mills theory: Universal Tree Amplitudes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 14 =+ …= QCD N=4 SYM

15. Parke-Taylor formula (1986) When very simple QCD tree amplitudes were found, first in the 1980’s … the simplicity was secretly due to N=4 SYM Tree-Level Simplicity L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 15

16. All N=4 SYM trees in closed form Drummond, Henn, 0808.2475 For example, for 3 (-) gluon helicities and [n-3] (+) gluon helicities, extract from: 5 (-) gluon helicities and [n-5] (+) gluon helicities: L. Dixon Scattering Amplitudes in QFT 16 EPS HEP11 Grenoble 25 July These formulas can immediately be used for QCD (with external quarks too) LD, Henn, Plefka, Schuster, 1010.3991

17. Beyond trees For precise Standard Model predictions at colliders [talk by Zanderighi] For investigating of formal properties of gauge theories and gravity  need loop amplitudes L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 17 Where the fun really starts – textbook methods quickly fail, even with very powerful computers

18. Different theories differ at loop level QCD at one loop: L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 18 rational part well-known scalar one-loop integrals, master integrals, same for all amplitudes coefficients are all rational functions – determine algebraically from products of trees using (generalized) unitarity N=4 SYM

19. Gauge Hierarchy of Amplitude Simplicity L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 19

20. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 20 N=4 super-Yang-Mills theory Interactions uniquely specified by gauge group, say SU(N c ), 1 coupling g Exactly scale-invariant (conformal) field theory:  (g) = 0 for all g all states in adjoint representation, all linked by N=4 supersymmetry

21. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 21 Planar N=4 SYM and AdS/CFT In the ’t Hooft limit, fixed, planar diagrams dominate AdS/CFT duality suggests that weak-coupling perturbation series in for large- N c (planar) N=4 SYM should have hidden structure, because large limit  weakly-coupled gravity/string theory on large-radius AdS 5 x S 5 Maldacena

22. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 22 AdS/CFT in one picture

23. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 23 Four Remarkable, Related Structures Unveiled Recently in Planar N=4 SYM Scattering Exact exponentiation of 4 & 5 gluon amplitudes Dual (super)conformal invariance Strong coupling “soap bubbles” Equivalence between (MHV) amplitudes & Wilson loops Outstanding question: Can these structures be used to solve exactly for all planar N=4 SYM amplitudes? Properties all related in some way to AdS/CFT. To be explored in more detail tomorrow in talk by N. Beisert

24. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 24 Dual Conformal Invariance Conformal symmetry acting in momentum space, on dual or sector variables x i First seen in N=4 SYM planar amplitudes in the loop integrals Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 x5x5 x1x1 x2x2 x3x3 x4x4 k invariant under inversion:

25. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 25 Dual conformal invariance at higher loops Simple graphical rules: 4 (net) lines into inner x i 1 (net) line into outer x i Dotted lines are for numerator factors All integrals entering planar 4-point amplitude at 2, 3, 4, and 5 loops are of this form! Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248 Bern, Carrasco, Johansson, Kosower, 0705.1864

26. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 26 Constrained Amplitudes After all loop integrations are performed, amplitude depends only on external momentum invariants, Dual conformal invariance ( x i inversion) fixes form of amplitude, up to functions of invariant cross ratios: Since, no such variables for n=4,5  4,5 gluon amplitudes totally fixed, to exact-exponentiated form (BDS ansatz) For n=6, precisely 3 ratios: + 2 cyclic perm’s 1 2 34 5 6

27. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 27 Constrained Integrands Before loop integrations performed, even 4-gluon amplitude has invariant integrands. But only a limited number of possibilities. Use generalized unitarity to determine correct linear combination, by matching to a general ansatz for the integrand. Convenient to chop loop amplitudes all the way down to trees. 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 inequivalent ways:

28. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 28 Strategy works through at least 5 loops Bern, Carrasco, Johansson, Kosower, 0705.1864

29. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 29 Beyond the planar approximation Beyond the large- N c limit, or in theories other than N=4 SYM, dual conformal invariance doesn’t hold. However, another recently discovered set of relations for integrands can be used to get many contributions from a handful of planar ones: “Color-kinematics duality”, or BCJ relations. Old history (at 4-point tree level) dating back to discovery of radiation zeroes.

30. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 30 Radiation Zeroes Mikaelian, Samuel, Sahdev (1979) computed Found “radiation zero” at Held independent of (W,  ) helicities Implies a connection between –“color” (here electric charge Q d ) – kinematics (cos  ) 

31. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 31 Color-Kinematic Duality Extend to other 4-point non-Abelian gauge amplitudes Zhu (1980), Goebel, Halzen, Leveille (1981) Massless all-adjoint gauge theory: Group theory  3 terms not independent (color Jacobi identity): In suitable “gauge”, find “kinematic Jacobi identity”: Structure extends also to arbitrary number of legs Bern, Carrasco, Johansson, 0805.3993

32. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 32 Color-Kinematic Duality at loop level Consider any 3 graphs connected by a Jacobi identity Color factors obey C s = C t – C u Duality requires n s = n t – n u Powerful constraint on structure of integrands Can always check afterwards using generalized unitarity

33. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 33 Simple 3 loop example Using we can relate non-planar topologies to planar ones = - 2 3 1 4 In fact all N=4 SYM 3 loop topologies related to (e) Carrasco, Johansson, 1103.3298

34. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 34 UV Finiteness of N=8 Supergravity? Quantum gravity is nonrenormalizable by power counting: the coupling, Newton’s constant, G N = 1/M Pl 2 is dimensionful String theory cures divergences of quantum gravity by introducing a new length scale at which particles are no longer pointlike. Is this necessary? Or could enough (super)symmetry, allow a point particle theory of quantum gravity to be perturbatively ultraviolet finite? A positive answer would have profound implications, even if it is just in a “toy model”. Investigate by computing multi-loop amplitudes in N=8 supergravity DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978-9) and then examining their ultraviolet behavior. Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/0702112; BCDJR, 0808.4112, 0905.2326, 1008.3327, 1108.nnnn

35. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 35 2 8 = 256 massless states, ~ expansion of (x+y) 8 SUSY 2 4 = 16 states ~ expansion of (x+y) 4

36. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 36 Gravity from gauge theory (tree level) Kawai, Lewellen, Tye (1986) derived relations between open & closed string amplitudes Low-energy limit gives N=8 supergravity amplitudes as quadratic combinations of N=4 SYM amplitudes, respecting factorization of Fock space,

37. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 37 Gravity from color-kinematics duality Given a color-kinematics satisfying gauge amplitude, e.g. with, then gravity amplitude is simply given by erasing color factors and squaring numerators, e.g. Bern, Carrasco, Johansson, 0805.3993, 1004.0476; Bern, Dennen, Huang, Kiermaier, 1004.0693 Like KLT relations, gravity = (gauge theory) 2, amplitudes respect factorization of Fock space,

38. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 38 AdS/CFT vs. KLT AdS = CFT gravity = gauge theory weak strong KLT gravity = (gauge theory) 2 weak

39. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 39 KLT Copying N=4 SYM amplitude (full color, not just planar), plus KLT relations & generalized unitarity, gives all information needed to construct N=8 amplitude at same loop order. For example, at 3 loops: N=8 SUGRAN=4 SYM rational function of Lorentz products of external and cut momenta; all state sums already performed With a color-kinematics-duality respecting gauge amplitude, passing to the gravity amplitude is trivial!

40. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 40 “Color-kinematics duality” at 3 loops Bern, Carrasco, Johansson, 0805.3993, 1004.0476 N=4 SYM N=8 SUGRA [ ] 2 [ ] 2 Linear in 4 loop amplitude has similar dual representation! BCDJR, to appear

41. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 41 N=8 SUGRA as good as N=4 SYM in UV Amplitude representations have been found through 4 loops with same UV behavior as N=4 SYM  same critical dimension D c for UV divergences: But N=4 SYM is known to be finite for all L. Therefore, either this pattern has to break at some point ( L=5??? ) or else N=8 supergravity would be a perturbatively finite point-like theory of quantum gravity, against all conventional wisdom! L=5 results critical (several bottles of wine at stake)

42. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 42 Conclusions Scattering amplitudes have a very rich structure, not only in gauge theories of Standard Model, but especially in highly supersymmetric gauge theories and supergravity. Much of this structure is impossible to see using Feynman diagrams, but has been unveiled with the help of unitarity-based methods Among other applications, these methods have had practical payoffs in the “NLO QCD revolution” provided clues that planar N=4 SYM amplitudes might be solvable in closed form shown that N=8 supergravity has amazingly good ultraviolet properties More surprises are surely in store in the future!

43. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 43 From “science” to “technology” N=4 SYM QCDQCD

44. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 44 Extra Slides

45. Many more QCD trees from N=4 SYM Gluinos are adjoint, quarks are fundamental Color not a problem because it’s easily manipulated – common to work with “color stripped” amplitudes anyway No unwanted scalars can enter amplitude with only one fermion line: Impossible to destroy them once created, until you reach two fermion lines: Can use flavor of gluinos to pick out desired QCD amplitudes, through (at least) three fermion lines (all color/flavor orderings), and including V+jets trees LD, Henn, Plefka, Schuster, 1010.3991 L. Dixon Scattering Amplitudes in QFT 45 EPS HEP11 Grenoble 25 July

46. Infinities “The infinite can be appreciated only by the finite.” - J. Brodsky, Watermark L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 46 “The finite can be appreciated only after removing the infinite.” - Quantum Field Theory Two types of infinities: Ultraviolet  renormalization Infrared  factorization

47. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 47 Dimensional Regulation in the IR One-loop IR divergences are of two types: Soft Collinear (with respect to massless emitting line) Overlapping soft + collinear divergences imply leading pole is at 1 loop at L loops

48. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 48 Infrared Factorization In any theory, including QCD, S and J exponentiate. Surprise: for planar N=4 SYM, in some cases, full amplitude does too. Loop amplitudes afflicted by IR (soft & collinear) divergences soft function (color-dependent) jet function (spin-dependent) hard function (finite as   0)

49. L. Dixon Scattering Amplitudes in QFTEPS HEP11 Grenoble 25 July 49 Large N c Planar Simplification Planar limit color-trivial: absorb S into J i Each “wedge” is the square root of the well-studied process “gg  1” (the exponentiating Sudakov form factor): coefficient of