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Hidden Structures in Field Theory Amplitudes and their Applications Niels Bohr Institute August 12, 2009 Zvi Bern, UCLA 1 Hidden Structures in Amplitudes Why have a conference on hidden structures? Why should we care? 1. Some of the most important advances stem from identifying and exploiting hidden structures found in explicit calculations. 2. Hidden structures help us calculate! 3. Leads to deeper understanding of basic properties of quantum field theory! Will go through various examples. 2 How to Hide Structures Vertices and propagators involve gauge-dependent off-shell states. An important origin of the complexity. To get at root cause of the trouble we must rewrite perturbative quantum field theory so all pieces are ZB, Dixon, Dunbar and Kosower gauge invariant. 3 How to Expose Structures Use gauge invariant on-shell formalisms on-shell amplitude BCFW recursion for trees Britto, Cachazo, Feng and Witten BCFW recursion uses only on-shell quantities on-shell Modern unitarity method for loops Application of generalized unitarity ZB, Dixon, and Kosower ZB, Dixon, Dunbar and Kosower (BDDK) complex momenta to solve cuts Britto, Cachazo and Feng Buchbinder and Cachazo “method of maximal cuts” used in 4 loop calculations. ZB, Carasco, Johansson, Kosower 4 Famous Example: Twistor Space Curves Witten conjectured that in twistor space gauge theory amplitudes have delta-function support on curves of degree: Connected picture Disconnected picture Structures imply an amazing simplicity in the scattering amplitudes. Nair; Witten Roiban, Spradlin and Volovich Cachazo, Svrcek and Witten Gukov, Motl and Neitzke Bena Bern and Kosower 5 MHV Vertices Cachazo, Svrcek and Witten MHV amplitude non-MHV amplitude twistor space momentum space The MHV amplitudes are vertices for building new amplitudes. The MHV rules follow directly from twistor space structure 6 Twistor Space Structures At one-loop the coefficients of all box integral functions have beautiful twistor space structure. Box integral Twistor space support Three negative helicities Four negative helicities Twistor structures are mapped for all N = 4 gauge theory amplitudes. Related to box coefficients. Bern, Dixon and Kosower Britto, Cachazo and Feng Zigzag structure as number of negative helicities increases. Korchemsky and Sokatchev Arkani-Hamed, Cachazo, Cheung, Kaplan 7 Talks from Spradlin, Hodges, Korchemsky, etc A Remarkable Twistor String Formula The following “connected” formula encapsulates the tree-level S-matrix of N = 4 sYM. Integral over the Moduli and curves Witten Roiban, Spradlin and Volovich Degree d polynomial in the moduli ak Strange formula from Feynman diagram viewpoint. But it’s true: careful checks by Roiban, Spradlin and Volovich This is an example of a remarkable structure, yet no applications. It’s a very intriguing formula! Formula for gravity? 8 Structure in Gravity Amplitudes Consider the gravity Lagrangian flat metric graviton field metric +… Infinite number of complicated interactions Compare to Yang-Mills Lagrangian on which QCD is based Only three and four point interactions Gravity seems so much more complicated than gauge theory. Off Shell structure is very well hidden! 9 Gravity as the Square of Gauge Theory Instead work on shell: gauge theory: gravity: “square” of Yang-Mills vertex. On-shell squaring structure is immediately exposed 10 KLT Relations • At tree level Kawai, Lewellen and Tye derived a relationship between closed and open string amplitudes. • In field theory limit, relationship is between gravity and gauge theory. Gravity amplitude where we have stripped all coupling constants Color stripped gauge theory amplitude Holds for any external string states. See review: gr-qc/0206071 11 Gravity as the Square of Gauge Theory ZB, Carrasco, Johansson Gauge theory: color factor kinematic numerator factor sum over diagrams with only 3 vertices Feynman propagators For every color factor Jacobi identity there exists a corresponding numerator equation. Used to derive a set of identities between See Vanhove’s talk amplitudes Gauge theory: Einstein Gravity: Used at four loops See Carrasco’s and Johansson’s talks Cries out for a unified description of the sort given by string theory! Not as well understood as we would like. 12 A Higher-loop Structure Consider the four gluon all-positive helicity amplitude in QCD. If you expand it in polylogs it is some moderate mess. Instead let’s write it in a special basis of integralsZB, Dixon, Kosower planar double box non-planar double box hep-ph/0001001 Why do planar and non-planar double box numerators look the same? last contribution happens to vanish for all plus helicity ZB, Carrasco, Johansson The numerator Jacobi-like identity explains it. We find a general structure for all helicities! 13 Loop Iteration of N = 4 Amplitudes The planar four-point two-loop amplitude undergoes fantastic simplification. ZB, Rozowsky, Yan Anastasiou, ZB, Dixon, Kosower is universal function related to IR singularities This gives two-loop four-point planar amplitude as iteration of one-loop amplitude. Three loop satisfies similar iteration relation. Rather nontrivial. ZB, Dixon, Smirnov 14 All-Loop Generalization Identification of the interative structure directly leads to “BDS ansatz”. Recognize exponential and resum. constant independent of kinematics. all-loop resummed amplitude IR divergences cusp anomalous dimension finite part of one-loop amplitude Anastasiou, ZB, Dixon, Kosower ZB, Dixon and Smirnov Gives a definite prediction for all values of coupling given BES integral equation for the cusp Beisert, Eden, Staudacher anomalous dimension. 15 Alday and Maldacena Strong Coupling For four point amplitude: constant independent of kinematics. all-loop resummed amplitude IR divergences cusp anomalous dimension finite part of one-loop amplitude Wilson loop In a beautiful paper Alday and Maldacena confirmed the conjecture for 4 gluons at strong coupling from an AdS string theory computation. Minimal surface calculation. Very suggestive link to Wilson loops even at weak coupling. Drummond, Korchemsky, Henn, Sokatchev ; Brandhuber, Heslop and Travaglini ZB, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich; Anastasiou, Branhuber, Heslop, Khoze, Spence, Travagli, • Identification of new symmetry: “dual conformal symmetry” Drummond, Henn, Korchemsky, Sokatchev; Beisert, Ricci, Tseytlin, Wolf; Brandhuber, Heslop, Travaglini; 16 Berkovits and Maldacena All-loop Trouble at Higher Points For various technical reasons it is difficult to solve for minimal surface for larger numbers of gluons. Alday and Maldacena realized certain terms can be calculated at strong coupling for an infinite number of gluons. Disagrees with BDS conjecture L T Explicit numerical computation at 2-loop six points. Need to modify conjecture! ZB, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich Drummond, Henn, Korchemsky, Sokatchev Dual conformal invariance and equivalence to Wilson loops persists Can the BDS conjecture be repaired for six and higher points? 17 Symmetry and Ward Identities A key to understanding the structure of amplitudes is symmetry. Planar N = 4 YM has three distinct interlocked symmetries: Relatively simple at tree level for generic kinematics susy: QA t ree = 0 conformal: dual variables dual conformal: Together these forms a Yangian structure Drummond, Henn Plefka; Bargheer, Beisert, Galleas, Loebbert, McLoughlin • Clearly connected to integrability. But can this be made precise and useful? • What about loop level? Conformal invariance not so simple. Infrared singularities and “holomorphic anomaly” confusion.18 In Search of the Holy Grail When in doubt calculate! log of the amplitude discrepancy Can we figure out the discrepancy? Important new information from regular polygons should serve as a guide. Explicit solution at eight points Alday and Maldacena (2009) Solution valid only for strong coupling and special kinematics, but it is explicit! Need to exploit this. 19 N = 8 Supergravity No-Triangle Property ZB, Dixon, Perelstein, Rozowsky; ZB, Bjerrum-Bohr and Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager; Proofs by Bjerrum-Bohr and Vanhove; Arkani-Hamed, Cachazo and Kaplan. One-loop D = 4 theorem: Any one loop amplitude is a linear combination of scalar box, triangle and bubble integrals with Brown, Feynman; Passarino and Veltman, etc rational coefficients: • In N = 4 Yang-Mills only box integrals appear. No triangle integrals and no bubble integrals. • The “no-triangle property” is the statement that same holds in N = 8 supergravity. Non-trivial constraint on analytic form of amplitudes. 20 N = 8 Supergravity No-Triangle Property No triangle property looks like a esoteric technical point about one loop structure. However, it gives us the most potent means for making solid non-trivial statements about UV properties of N = 8 supergravity to all loop orders. 21 N = 8 L-Loop UV Cancellations 2 .. 1 From 2 particle cut: ZB, Dixon, Roiban 3 numerator factor 4 1 in N = 4 YM L-particle cut numerator factor • Numerator violates one-loop “no-triangle” property. • Too many powers of loop momentum in one-loop subamplitude. • After cancellations behavior must be same as N = 4 Yang-Mills! • UV cancellation exist to all loop orders. (not a proof of finiteness) • Even pure gravity displays nontrivial cancellations! ZB, Carrasco, Forde, Ita, Johansson • These all-loop cancellations not potentially explained by susy or by Berkovits’ string theory nonrenormalization theorem. Green, Russo; Vanhove; Bossart, Howe, Stelle 22 Some Interesting Open Problems • More to do at tree level. — further unraveling of twistor-space structure. — understanding relation between different (BCFW) recursions and dual formulation. — gravity as the square of YM. Not as well understood as we would like. Crucial for understanding gravity. • Interface of string theory and field theory– certain features clearer in string theory, especially at tree level. KLT classic example. • Can we carry over Berkovits string theory pure spinor formalism to field theory? Should help expose full susy. • Higher-dimensional methods: helicity, on-shell 23 superspace. Some Interesting Open Problems • Repairing BDS ansatz at six and higher points. Explicit • • • • analytic information from Alday and Maldacena. Proof of dual conformal symmetry to all loop orders. Derivation of Ward Identities. Role of conformal invariance and Yangian structure at loop level? IR divergences and “holomorphic anomaly”? Connection of amplitudes to Wilson loops. N = 4 and QCD. Non-MHV amplitudes? What do the no-triangle property of N = 8 supergravity mean for the effective action? We will hear a lot about some of these points at this conference. 24 Summary • Remarkable progress in a broad range of topics: AdS/CFT, quantum gravity and LHC physics. • Identification of hidden structures lead to ever more potent means of computation which in turn leads to new understanding and further identification of new hidden structures. • Many exciting new developments in scattering amplitudes at this conference. • Expect a lot more progress in coming years! 25 Fortune Cookie 26 Extra Transparancies 27 Gravity Echoes of Structure Gravity has same twistor structure as gauge theory except derivative of delta function support. Witten; ZB, Bjerrum-Bohr, Dunbar A recent relation: Gauge: Elvang, Freedman Drummond, Spradlin, Volovich, Wen Dual conformal invariants Gravity: Symmetries and structures of gauge theory have echo in gravity! 28 Schematic Illustration of Status loops Same power count as N=4 super-Yang-Mills UV behavior unknown from feeding 2, 3 and 4 loop All-loop UV finiteness. calculations into iterated cuts. No potential susy or string non-renormalization Explanations. finiteness unproven Berkovits string theory non-renormalization theorem points to good L = 5, 6 behavior. Needs to be redone in field theory! No triangle property explicit 2, 3, 4 loop computations terms 29 State-of-the-Art One-Loop Calculations In 1948 Schwinger computed anomalous magnetic moment of the electron. 60 years later typical example we can calculate: Only two more legs than Schwinger! Key processes for the LHC: four or more final state objects • Never been done via Feynman diagrams, though many failed attempts. • Widespead applications to LHC physics pp ! W + 3 jets 30 Impact on LHC Physics Tree level: Pretty much solved 20 years ago by off-shell (Berends-Giele) numerical recursion. One loop: Lots of important unsolved problems! 31 Application: State of the Art QCD for the LHC Berger, ZB, Dixon, Febres Cordero, Forde, Gleisberg, Ita, Kosower, Maitre (BlackHat collaboration) Apply on-shell methods Data from Fermilab Never been done via Feynman diagrams, though many failed attempts. Excellent agreement between NLO theory and experiment. Triumph of on-shell methods! 32 New W + 3-Jet Predictions for LHC Triumph for on-shell methods See David Kosower’s talk 33