Twistor Spinoffs for Collider Physics

Twistor Spinoffs for Collider PhysicsLance Dixon, SLACFermilab ColloquiumJune 7, 2006

L. Dixon Twistor Spinoffs for Collider Physics

Physics at very short distances Unification: particle interactions simpler at short distances


Physics at very short distances Supersymmetry predicts a host of new massive particles including a dark matter candidate Typical masses ~ 100 GeV/c2 1 TeV/c2 (mproton = 1 GeV/c2) Einstein (E = mc2): heavy particles require high energies Heisenberg (Dx Dp > h): short distances require high energies (and large momentum transfers) Many other theories of electroweak scale mW,Z = 100 GeV/c2 make similar predictions: new dimensions of space-time new forces etc.How to sort them all out?


The Tevatron The present energy frontier right here! Proton-antiproton collisions at 2 TeV center-of-mass energy


The Large Hadron Collider Proton-proton collisions at 14 TeV center-of-mass energy, 7 times greater than Tevatron Luminosity (collision rate) 10100 times greater New window into physics at the shortest distances opening 2007


LHC DetectorsATLASCMS


What will the LHC see?


What might the LHC see?


A better way to compute?Backgrounds (and many signals) require detailed understanding of scattering amplitudes for many ultra-relativistic (massless) particles especially quarks and gluons of QCD


The loop expansionAmplitudes can be expanded in a small parameter, as = g2/4p At each successive order in g2, draw Feynman diagrams with one more loop the number grows very rapidly! For example, gluon-gluon scattering


Why do we need to do better?Leading-order (LO), tree-level predictions are only qualitative, due to poor convergence of expansion in strong coupling as(m) ~ 0.1NLO corrections can be 30% - 80% of LOstate of the art:


Tevatron Run II exampleAzimuthal decorrelation of di-jets at D0 due to additional radiationZ. Nagy (2003)


LHC Example: SUSY SearchEarly ATLAS TDR studies using PYTHIA overly optimistic ALPGEN based on LO amplitudes, much better than PYTHIA at modeling hard jets What will disagreement between ALPGEN and data mean? Hard to tell because of potentially large NLO correctionsGianotti & Mangano, hep-ph/0504221Mangano et al. (2002) Search for missing energy + jets. SM background from Z + jets.


Dialogue between theorists & experimenters


The dialogue continuesExperimenters to theorists:OK, wed really like these at NLO, by the time LHC startsLes Houches 2005


How do we know theres a better way?AnAnfrom only 10 diagrams! Because Feynman diagrams for QCD are too complicated


How do we know theres a better way?Because many answers are much simpler than expected!For example, special helicity amplitudes vanish or are very short


Mathematical Tools for Physics


Simplicity in Fourier spaceExample of atomic spectroscopyt


The right variablesScattering amplitudes for massless plane waves of definite 4-momentum: Lorentz vectors kim ki2=0


Adding spinsFrom two non-identical non-relativistic spin objects, make spin 1


Spinor productsAntisymmetric product of two spin is spin 0 (rotationally invariant)


Spinor Magic Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes


Twistor SpaceStart in spinor space:


Twistor Transform in QCDWitten (2003)


More Twistor MagicBerends, Giele;Mangano, Parke, Xu (1988) =A6


Even More Twistor Magic Now it is clear how to generalize


MHV rulesLed to MHV rules:Cachazo, Svrcek, Witten (2004) Twistor space picture:More efficientalternative to Feynman rulesfor QCD trees


MHV rules for treesRules quite efficient, extended to many collider applications massless quarksGeorgiou, Khoze, hep-th/0404072;Wu, Zhu, hep-th/0406146;Georgiou, Glover, Khoze, hep-th/0407027 Higgs bosons (Hgg coupling)LD, Glover, Khoze, hep-th/0411092;Badger, Glover, Khoze, hep-th/0412275 vector bosons (W,Z,g*)Bern, Forde, Kosower, Mastrolia, hep-th/0412167


Twistor structure of loops Simplest for coefficients of box integrals in a toy model, N=4 supersymmetric Yang-Mills theoryCachazo, Svrcek, Witten;Brandhuber, Spence,Travaligni (2004)Bern, Del Duca, LD, Kosower;Britto, Cachazo, Feng (2004)Again support is on lines,but joined into rings, to match topology of theloop amplitudes


Whats a (topological) twistor string?Whats a normal string?Abstracting the lessons often the best! E.g., Bern, Kosower (1991)


Even better than MHV rulesBritto, Cachazo, Feng, hep-th/0412308On-shell recursion relationsTrees are recycled into trees!Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs,evaluated with momenta shifted by a complex amountAnAk+1An-k+1[Off-shell antecedent: Berends, Giele (1988)]


A 6-gluon example220 Feynman diagrams for gggggg Helicity + color + MHV results + symmetries


Simple final form


Relative simplicity grows with n


Proof of on-shell recursion relationsBritto, Cachazo, Feng, Witten, hep-th/0501052


Speed is of the Essence For collider phenomenology, in the end all one needs are numbers But one needs them many times to do integrals over phase space For LHC, n ~ 6 9, they do pretty well


On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005 New features arise compared with tree case Same techniques work for one-loop QCD amplitudes much harder to obtain by other methods than are trees.


Rational functions in loop amplitudesAfter computing cuts using unitarity, there remains an additive rational-function ambiguity Determined using - tree-like recursive diagrams, plus - simple overlap diagramsNo loop integrals required in this step Bootstrap rational functions from cuts and lower-point amplitudesMethod tested on 5-point amplitudes, used to get new QCD results:

Now working to generalize method to all helicity configurations, and to processes on the realistic NLO wishlist.

Forde, Kosower, hep-ph/0509358 Berger, Bern, LD, Forde, Kosower, hep-ph/0604195, hep-ph/0606nnn,


Example of new diagramsrecursive:overlap:Compared with 1034 1-loop Feynman diagrams (color-ordered)7 in all


Revenge of the Analytic S-matrixReconstruct scattering amplitudes directly from analytic propertiesChew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne;Veneziano; Virasoro, Shapiro; (1960s)Analyticity fell somewhat out of favor in 1970s with rise of QCD;to resurrect it for computing perturbative QCD amplitudesseems deliciously ironic!


ConclusionsExciting new computational approaches to gauge theories due (directly or indirectly) to development of twistor string theorySo far, practical spinoffs mostly for trees, and loops in supersymmetric theoriesBut now, new loop amplitudes in full, non-supersymmetric QCD needed for collider applications are beginning to fall to twistor-inspired recursive approachesExpect the rapid progress to continue


Extra slides


Initial data


Supersymmetric decomposition for QCD loop amplitudesgluon loopN=4 SYMN=1 multipletscalar loop--- no SUSY,but also nospin tanglesN=4 SYM and N=1 multiplets are simplest pieces to compute because they are cut-constructible determined by their unitarity cuts, evaluated using D=4 intermediate helicities


Loop amplitudes with cutsGeneric analytic properties of shifted 1-loop amplitude, Cuts and poles in z-plane:


Direct proof of MHV rules via OSRRMHV rules:K. Risager, hep-th/0508206There is a different complex momentum shift for which the on-shell recursion relations (OSRR) for NMHV are identical, graph by graph, to the MHV rules. Proof is inductive in


Why does it all work?In mathematics you don't understand things. You just get used to them.


Twistor Spinoffs for Collider Physics

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