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Scattering Amplitudes in Quantum Field Theory

Scattering Amplitudes in Quantum Field Theory. Lance Dixon (CERN & SLAC) EPS HEP 2011 25 July 2011. The S matrix reloaded.

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Scattering Amplitudes in Quantum Field Theory

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  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. EPS HEP11 Grenoble 25 July

  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 EPS HEP11 Grenoble 25 July

  4. Poles • Branch cuts 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 perturbativeQCD as alternative to Feynman diagrams! Perturbative information now assists analyticity. Works for many other theories too. EPS HEP11 Grenoble 25 July

  5. Perturbativeunitarity bootstrap • S-matrix a unitary operator between in and out states • S†S = 1  unitarityrelations (cutting rules) for amplitudes • Reconstruct (multi-)loop amplitudes from cuts efficiently, due to simple structure of treeand lower-loophelicityamplitudes • Generalized unitarity reduces everything to (simpler) trees EPS HEP11 Grenoble 25 July

  6. Granularity vs. Plasticity EPS HEP11 Grenoble 25 July

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

  8. Black Holes • “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 EPS HEP11 Grenoble 25 July

  9. Scattering Amplitudes • 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 h h h h EPS HEP11 Grenoble 25 July

  10. g g Gauge Theory Amplitudes • Seem less universal at first sight: • Have to specify gauge group G, representation R for matter fields, etc. • However, full amplitudes An can be assembled from universal, G,R-independent “color-stripped” or “color-ordered” amplitudes An : g g color universal EPS HEP11 Grenoble 25 July

  11. Through the clouds of gas and dust • Obscure astrophysical black holes, but also make them detectable, their physics much richer EPS HEP11 Grenoble 25 July

  12. Through the clouds of soft interactions • Obscure hard QCD amplitudes at their core, but also make the physics much richer F. Krauss EPS HEP11 Grenoble 25 July

  13. Relativistic Gauge Amplitudes • In relativistic limit, particle masses mb,mW, mt , … become unimportant, making gauge amplitudes still more universal. • In QCD, if we could go to extremely high energies, then asymptotic freedom, as 0,  we would need only leading order cross sections, i.e. tree amplitudes = + … EPS HEP11 Grenoble 25 July

  14. Universal Tree Amplitudes • 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: = + … = N=4 SYM QCD EPS HEP11 Grenoble 25 July

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

  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: These formulas can immediately be used for QCD (with external quarks too)LD, Henn, Plefka, Schuster, 1010.3991 EPS HEP11 Grenoble 25 July

  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 Where the fun really starts – textbook methods quickly fail, even with very powerful computers EPS HEP11 Grenoble 25 July

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

  19. Gauge Hierarchyof Amplitude Simplicity EPS HEP11 Grenoble 25 July

  20. N=4 super-Yang-Mills theory all states in adjoint representation, all linked by N=4 supersymmetry • Interactions uniquely specified by gauge group, say SU(Nc), 1 coupling g • Exactly scale-invariant (conformal) field theory: b(g) = 0 for all g EPS 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 lfor large-Nc(planar) N=4 SYM should have hidden structure, because large l limit  weakly-coupled gravity/string theory on large-radius AdS5x S5 Maldacena EPS HEP11 Grenoble 25 July

  22. AdS/CFT in one picture EPS 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 Properties all related in some way to AdS/CFT. To be explored in more detail tomorrow in talk by N. Beisert Outstanding question: Can these structures be used to solve exactly for all planar N=4 SYM amplitudes? EPS HEP11 Grenoble 25 July

  24. x1 x4 x2 invariant under inversion: x5 x3 Dual Conformal Invariance Broadhurst (1993); Lipatov (1999);Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 Conformal symmetry acting in momentum space, on dual or sector variables xi First seen in N=4 SYM planar amplitudes in the loop integrals k EPS HEP11 Grenoble 25 July

  25. Dual conformal invariance at higher loops • Simple graphical rules: • 4 (net) lines into inner xi • 1 (net) line into outer xi • 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 EPS HEP11 Grenoble 25 July

  26. Constrained Amplitudes • After all loop integrations are performed, amplitude depends only on external momentum invariants, • Dual conformal invariance (xi 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: 6 1 5 2 + 2 cyclic perm’s 4 3 EPS 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: EPS HEP11 Grenoble 25 July

  28. Strategy works through at least 5 loops Bern, Carrasco, Johansson, Kosower, 0705.1864 EPS HEP11 Grenoble 25 July

  29. Beyond the planar approximation • Beyond the large-Nc 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. EPS HEP11 Grenoble 25 July

  30. Radiation Zeroes • Mikaelian, Samuel, Sahdev (1979) computed • Found “radiation zero” at • Held independent of (W,g) helicities • Implies a connection between • “color” (here electric charge Qd) • kinematics (cosq) g EPS 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 EPS HEP11 Grenoble 25 July

  32. Color-Kinematic Duality at loop level • Consider any 3 graphs connected by a Jacobi identity • Color factors obey • Cs = Ct – Cu • Duality requires • ns = nt– nu • Powerful constraint on structure of integrands • Can always check afterwards using generalized unitarity EPS HEP11 Grenoble 25 July

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

  34. UV Finiteness of N=8 Supergravity? • Quantum gravity is nonrenormalizable by power counting: the coupling, Newton’s constant, GN = 1/MPl2is 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 supergravityDeWit, 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 EPS HEP11 Grenoble 25 July

  35. 28 = 256 massless states, ~ expansion of (x+y)8 SUSY 24 = 16 states ~ expansion of (x+y)4 EPS 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, EPS 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, EPS HEP11 Grenoble 25 July

  38. AdS = CFT gravity = gauge theory weak strong KLT gravity = (gauge theory)2 weak weak AdS/CFT vs. KLT EPS 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 SUGRA N=4 SYM N=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! EPS 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 [ ]2 [ ]2 Linear in 4 loop amplitude has similar dual representation! BCDJR, to appear [ ]2 EPS HEP11 Grenoble 25 July

  41. N=8 SUGRA as good as N=4 SYM in UV • Amplitude representations have been found through 4 loops withsame UV behavior as N=4 SYM • same critical dimension Dc 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) EPS 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 isimpossible 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 supergravityhas amazingly good ultraviolet properties • More surprises are surely in store in the future! EPS HEP11 Grenoble 25 July

  43. From “science” to “technology” N=4 SYM QCD EPS HEP11 Grenoble 25 July

  44. Extra Slides EPS HEP11 Grenoble 25 July

  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 EPS HEP11 Grenoble 25 July

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

  48. Infrared Factorization Loop amplitudes afflicted byIR (soft & collinear) divergences jet function (spin-dependent) soft function (color-dependent) hard function (finite as e 0) In any theory, including QCD, S andJexponentiate. Surprise: for planar N=4 SYM, in some cases, full amplitude does too. EPS HEP11 Grenoble 25 July

  49. coefficient of • Planar limit color-trivial: absorb S intoJi • Each “wedge” is the square root of the well-studied process “gg  1” (the exponentiatingSudakovform factor): Large NcPlanar Simplification EPS HEP11 Grenoble 25 July

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