Hidden Structures in Field Theory Amplitudes and their Applications 1 Niels Bohr Institute August 12, 2009 Zvi Bern, UCLA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAA

2 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.

3 How to Hide Structures ZB, Dixon, Dunbar and Kosower Vertices and propagators involve gauge-dependent off-shell states. An important origin of the complexity. 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 gauge invariant.

4 How to Expose Structures Modern unitarity method for loops ZB, Dixon, Dunbar and Kosower (BDDK) ZB, Carasco, Johansson, Kosower on-shell BCFW recursion uses only on-shell quantities Britto, Cachazo, Feng and Witten on-shell amplitude Use gauge invariant on-shell formalisms BCFW recursion for trees Application of generalized unitarity “method of maximal cuts” used in 4 loop calculations. Britto, Cachazo and Feng Buchbinder and Cachazo ZB, Dixon, and Kosower complex momenta to solve cuts

5 ves 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

6 MHV Vertices The MHV amplitudes are vertices for building new amplitudes. Cachazo, Svrcek and Witten momentum spacetwistor space MHV amplitude The MHV rules follow directly from twistor space structure non-MHV amplitude

7 Twistor Space Structures At one-loop the coefficients of all box integral functions have beautiful twistor space structure. Twistor space supportBox integral Three negative helicities Four negative helicities Zigzag structure as number of negative helicities increases. Twistor structures are mapped for all N = 4 gauge theory amplitudes. Related to box coefficients. Bern, Dixon and Kosower Britto, Cachazo and Feng Korchemsky and Sokatchev Arkani-Hamed, Cachazo, Cheung, Kaplan Talks from Spradlin, Hodges, Korchemsky, etc

8 A Remarkable Twistor String Formula The following “connected” formula encapsulates the tree-level S-matrix of N = 4 sYM. Strange formula from Feynman diagram viewpoint. Degree d polynomial in the moduli a k Integral over the Moduli and curves But it’s true: careful checks by Roiban, Spradlin and Volovich Witten Roiban, Spradlin and Volovich This is an example of a remarkable structure, yet no applications. It’s a very intriguing formula! Formula for gravity?

9 Structure in Gravity Amplitudes Gravity seems so much more complicated than gauge theory. Infinite number of complicated interactions Consider the gravity Lagrangian Compare to Yang-Mills Lagrangian on which QCD is based + … Only three and four point interactions Off Shell structure is very well hidden! flat metric metric graviton field

10 Gravity as the Square of Gauge Theory gauge theory: gravity: “square” of Yang-Mills vertex. Instead work on shell: On-shell squaring structure is immediately exposed

11 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. where we have stripped all coupling constants Color stripped gauge theory amplitude Gravity amplitude Holds for any external string states. See review: gr-qc/

12 Gravity as the Square of Gauge Theory kinematic numerator factor Feynman propagators color factor  sum over diagrams with only 3 vertices For every color factor Jacobi identity there exists a corresponding numerator equation. Used to derive a set of identities between amplitudes Gauge theory: Einstein Gravity: ZB, Carrasco, Johansson See Vanhove’s talk Gauge theory: Cries out for a unified description of the sort given by string theory! Not as well understood as we would like. Used at four loops See Carrasco’s and Johansson’s talks

13 Consider the four gluon all-positive helicity amplitude in QCD. Why do planar and non-planar double box numerators look the same? If you expand it in polylogs it is some moderate mess. Instead let’s write it in a special basis of integrals planar double box non-planar double box ZB, Dixon, Kosower hep-ph/ A Higher-loop Structure The numerator Jacobi-like identity explains it. We find a general structure for all helicities! last contribution happens to vanish for all plus helicity ZB, Carrasco, Johansson

14 Loop Iteration of N = 4 Amplitudes The planar four-point two-loop amplitude undergoes fantastic simplification. 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 ZB, Rozowsky, Yan

15 All-Loop Generalization Identification of the interative structure directly leads to “BDS ansatz”. Recognize exponential and resum. Anastasiou, ZB, Dixon, Kosower ZB, Dixon and Smirnov all-loop resummed amplitude IR divergences cusp anomalous dimension finite part of one-loop amplitude constant independent of kinematics. Gives a definite prediction for all values of coupling given BES integral equation for the cusp anomalous dimension. Beisert, Eden, Staudacher

16 Alday and Maldacena Strong Coupling 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. all-loop resummed amplitude IR divergences cusp anomalous dimension finite part of one-loop amplitude constant independent of kinematics. For four point amplitude: Wilson loop 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; Berkovits and Maldacena

17 For various technical reasons it is difficult to solve for minimal surface for larger numbers of gluons. Can the BDS conjecture be repaired for six and higher points? Alday and Maldacena realized certain terms can be calculated at strong coupling for an infinite number of gluons. T L Explicit numerical computation at 2-loop six points. Need to modify conjecture! Disagrees with BDS conjecture ZB, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich Drummond, Henn, Korchemsky, Sokatchev Dual conformal invariance and equivalence to Wilson loops persists All-loop Trouble at Higher Points

18 Symmetry and Ward Identities A key to understanding the structure of amplitudes is symmetry. Planar N = 4 YM has three distinct interlocked symmetries: Together these forms a Yangian structure 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. Relatively simple at tree level for generic kinematics dual variables Drummond, Henn Plefka; Bargheer, Beisert, Galleas, Loebbert, McLoughlin dual conformal: conformal: susy:

19 Important new information from regular polygons should serve as a guide. Solution valid only for strong coupling and special kinematics, but it is explicit! Need to exploit this. Alday and Maldacena (2009) Can we figure out the discrepancy? Explicit solution at eight points In Search of the Holy Grail log of the amplitude discrepancy When in doubt calculate!

20 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. 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. One-loop D = 4 theorem: Any one loop amplitude is a linear combination of scalar box, triangle and bubble integrals with rational coefficients: Brown, Feynman; Passarino and Veltman, etc N = 8 Supergravity No-Triangle Property

21 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. N = 8 Supergravity No-Triangle Property

22 N = 8 L-Loop UV Cancellations From 2 particle cut: L-particle cut UV cancellation exist to all loop orders. (not a proof of finiteness) Even pure gravity displays nontrivial cancellations! These all-loop cancellations not potentially explained by susy or by Berkovits’ string theory nonrenormalization theorem. 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! numerator factor in N = 4 YM ZB, Dixon, Roiban Green, Russo; Vanhove; Bossart, Howe, Stelle ZB, Carrasco, Forde, Ita, Johansson

23 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 superspace.

24 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.

25 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. Summary Many exciting new developments in scattering amplitudes at this conference. Expect a lot more progress in coming years!

26 If structure you pay attention reward shall you receive! Fortune Cookie

27 Extra Transparancies

28 Symmetries and structures of gauge theory have echo in gravity! Gravity Echoes of Structure A recent relation: Elvang, Freedman Drummond, Spradlin, Volovich, Wen Dual conformal invariants Gauge: Gravity: Gravity has same twistor structure as gauge theory except derivative of delta function support. Witten; ZB, Bjerrum-Bohr, Dunbar

29 Schematic Illustration of Status finiteness unproven loops No triangle property explicit 2, 3, 4 loop computations Same power count as N=4 super-Yang-Mills UV behavior unknown terms from feeding 2, 3 and 4 loop calculations into iterated cuts. Berkovits string theory non-renormalization theorem points to good L = 5, 6 behavior. Needs to be redone in field theory! All-loop UV finiteness. No potential susy or string non-renormalization Explanations.

30 State-of-the-Art One-Loop Calculations 60 years later typical example we can calculate: In 1948 Schwinger computed anomalous magnetic moment of the electron. 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 Only two more legs than Schwinger!

31 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!

32 Berger, ZB, Dixon, Febres Cordero, Forde, Gleisberg, Ita, Kosower, Maitre (BlackHat collaboration) Excellent agreement between NLO theory and experiment. Triumph of on-shell methods! Data from Fermilab Never been done via Feynman diagrams, though many failed attempts. Apply on-shell methods Application: State of the Art QCD for the LHC

33 New W + 3-Jet Predictions for LHC See David Kosower’s talk Triumph for on-shell methods