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Refining MHV Diagrams

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Refining MHV Diagrams Refining MHV Diagrams by Andreas Brandhuber Queen Mary, University of London Munich, June 19, 2007 hep-th/0612007 AB-Spence-Travaglini arxiv:0704.0245 [hep-th] AB-Spence-Travaglini-Zoubos Outline • Motivation • MHV Diagrams • Tree MHV Diagrams (Cachazo-Svrcek-Witten) • Loop MHV Diagrams (AB-Spence-Travaglini) • One-loop Amplitudes in Pure Yang-Mills from MHV diagram • The all-minus helicity amplitudes • MHV diagrams from LC Lagrangian (Gorsky-Rosly, Mansfield, Morris-Ettle) • Completion of MHV diagram method • all-plus helicity amplitude from Lorentz violating counterterm • non-canonical, “holomophic” field transformation • Conclusions Goal • The goal of this talk is to address two questions: I. Do MHV Diagrams (a la CSW) provide a new, complete, perturbative expansion of Yang-Mills? II. And if not can we find a suitable completion of the MHV diagram method? Motivations • Simplicity of scattering amplitudes unexplained by textbook Feynman methods: e.g. Park-Taylor formula for MHV tree amplitude <--+++...+> • New ideas from twistor string theory (initiated by Witten 2003) help to “explain” this simplicity • Novel, practical tools to calculate physically relevant amplitudes ➡ LHC is coming • New reorganisations of conventional perturbation theory Spinor helicity Amplitudes formalism ColourSpinor decomposition helicity formalism • Main idea:A =disentangleA( λi, colourλi,;hi ) Scattering p µ null vector Amplitudes { } • At tree lep µ v el n, Yullang-Mills vector interactions are planar •Color Ordering:• µ µ DefineColour p a-or a ˙ = der p µ edσ a a ˙par whertial amplitudesµe σ = ( 1 , !σ µ ) • • Define p a a ˙ = p µ σ a a ˙ where σ = ( 1 , !σ ) tree • aσ aσn! A (2 pi,εi ) = ∑Tr(T 1 T ) A(σ(p1,ε1),...,σ(pn,εn)) If p { = 0 } thenσ 2 d··e·t p = 0 • ! momenta• If and p polarisation = 0 then v ectors Colourd eexprt-orpderessed=ed 0par intial termsamplitude of spinors and helicities ˜ ˜ ˜ ˜ HenceSpinor Helicity: p aa˙Hence= λaλ a˙p aa˙ = λ·aλ(a˙λ) positiv· eλ (negativ(λ) positive) helicitye (negativ e) helicity Next• example:- Include• only diagrams with fixed cyclicspinors orderingspinors of gluons ˙ ! colour indices stripped off a12b := ε λaλb [12a˙] :=b˙ ˜ a˙˜ b - Inner Maximall Analproductsytic- Inner ystructur H elicity pr 12e oducts is : simpler= V ε iolatingab λ λ , amplitude ab [12 1 ] 2 : =, ε ˙λ˜, λ˜ εa˙b˙λ1λ2 - ! " !1 2" a˙b 1 2 or MHV amplitude2(p p ) = 12 [12] 2(p p ) += +121 · [122 +] ! " MHVFirst Tree examples: Amplitudes: 1 ·A2(1 ,2! ,..".n ) = 0 , •(ParkAte-Ta ylor)loop level: multi-trace contributionsat tree level ! helicities• are a permutation of !!+++....++ Simple & A(1−,2 ,...n ) = 0 - subleading in 1/N Holomorphic! 4 + + i j A (1 ...i− ... j− ...n ) = " # Parke-Taylor formula MHV 12 23 n1 " #" #···" # ! ! Amplitudes in Twistor Space • Witten (2003): Simplicity can be “explained” • MHV Amplitudes holomorphic ऄline in twistor space • moreover: L-loop amplitudes with Q negative helicity gluons localise in twistor space on curves of degree=Q-1+L and genus<=L Q=4 Q=3 MHV Diagrams • MHV amplitude (holomorphic) = line in twistor space = pointlike interaction in Minkowski space (Witten, Cachazo-Svrcek- Witten) • CSW Rules (Cachazo-Svrcek-Witten) • MHV amplitudes promoted to local vertices off-shell continuation L = l + z! , " L !˜ a˙ • µ µ µ a ∼ aa˙ • Connect MHV vertices with scalar propagators • Note: η is related to LC gauge fixing MHV Diagrams cont’d typical MHV diagram contributing to 1 1 an NNMHV amplitude 2 2 L L! • • Covariance (= η-independence) is achieved after summing all MHV diagrams (CSW) • Equivalence with results of Feynman diagram calculations • On-Shell Recursion Relations (Britto-Cachazo-Feng-Witten) • MHV diagrams = special recursion relation (Risager) (shifts invisible since MHV amplitudes are holom.) From Trees to Loops (AB-Spence-Travaglini) • Original prognosis from twistor string theory was negative (Berkovits-Witten), Conformal SUGRA modes spoil duality • Try anyway: • Connect MHV vertices, using the same off-shell continuation as for trees • sum over all MHV diagrams and internal particle species • Find measure, perform loop integration • different from unitarity-based approach dM ! Z m1,m2,h The loop measure 4 4 d L1 d L2 (4) dM = 2 2 " (L2 L1 + PL) L1 + i!L2 + i! − • Use L = l + z η and L (l, z) → d4L dz = d4l"(+)(l2) L2 + i! z × dipersive phase space measure x measure • Loop integral becomes (use dimensional regularisation) (Dispersion integral) x (2-particle LIPS integral) • Sum of all MHV diagrams is covariant • MHV 1-loop amplitudes in N=4/N=1 SYM (agree with Bern- Dixon-Dunbar-Kosower) Evidence for MHV method • Tree Level Amplitudes ऄ complete proof (CSW, Britto-Cachazo- Feng-Witten, Risager) • One-Loop Amplitudes in Supersymmetric Yang-Mills • MHV method at one-loop ऄExplicit calculation of complete MHV one-loop Amplitudes in N=4 SYM (AB-Spence-Travaglini) & N=1 SYM (Bedford-AB-Spence-Travaglini, Quigley-Rozali) • “Proof” at one-loop for generic amplitudes in SYM • Feynman Tree-Theorem ऄ covariance • unitarity cuts • soft & collinear limits Without Supersymmetry • Cut-constructible part of MHV 1-loop amplitudes in pure Yang-Mills from MHV diagrams (Bedford-AB-Spence-Travaglini) • First result for QCD from MHV diagrams • Amplitudes in pure Yang-Mills not 4D cut-constructible • rational terms are missed by MHV diagrams • Look for alternative techniques, like on-shell recursions... (Bern, Dixon, Kosower, Berger, Forde...) • ... or try to refine the MHV diagram method Two related problems •RelatedMissing Pr all-plusoblems amplitude in pure Yang-Mills 1 γ5 i Tr( − lˆ lˆ lˆ lˆ ) 1 loop( +,..., +) = 2 1 2 3 4 An− 1 n − 2 ∑ 48π 1 l <l <l <l n 12 23 n1 ≤ 1 2 3 4≤ # $# $···# $ (Bern-Chalmer-Dixon-Kosower, Mahlon) ! all-plus equivalently computed in Self-Dual Yang-Mills • All-plus helicity one-loop amplitude in pure YM is missing !•vanishesfinite in supersymmetric theories • vanishes in supersymmetric theories • MHV method reproduces cut-constructible par• cant of be amplitudes computed in Self-Dual-Y in pure Yang-Millsang-Mills (Bedford, Brandhuber, Spence, GT) • we claimed that MHV diagrams miss rational terms with one exception... ! rational terms missing...with one exception: The all-minus amplitude (AB, Spence, Travaglini) • Can be drawn with MHV vertices (unlike the all-plus) ➡ MHV diagrams should be able to produce it! • For n-point <--- ....-> amplitude we need n 3-point MHV vertices: • Recall: D=Q-•1Thr+L ee-point scalar-scalar-negative helicity • Crucial: 3-point MHVgluon ver ticesvertex: = LC vertices ➡ result a priori correct (3) η L1 k] v¯ (L1,k,L2) = ! | | φAzφ ηk ! " • MHV three-point vertices are the same as in lightcone Yang-Mills ! expected on the basis of Mansfield’s field redefinition Explicit calculation Explicit Calculation Supersymmetric• Use supersymmetric decomposition decomposition:of 1-loop amplitudes A = (A + 4A + 3A ) 4(A + A ) + A g g f s − f s s • N=4 and N=1 SUSY contributions are zero • N=4 and N=1 contributions vanish • Only need to consider a complex scalar running in the loop • •Gluontechnicall ➡ y simpler scalar running in the loop • Use dimensional regularisation (DR): 3pt MHV vertices 4- -dim’lsimpler (spinor to brackcalculateets) while propagators are D-dim’l 2− 3− 2− MHV diagrams+ for the <----> 2− 3− − 2− + + + 3− 2− − − + − 2− + 33−− 22−− 3− − + + pms − ++ − −− − ++ + − + + pms 3− + ++ − + 3− 4− −− − + − − −− ++ + pms + + + 4−−− + pms − − + − + − 1− + 1−+ 4− + − + + 44−− −− 1− 1− 4− ++ −− −− 4− 11−− + perms 11−− 4− + 2− − 3− + + + 2− − − 3− − + pm + + bubbles give − − − ++ ++ pm 22−− 1 − − + perms33−− 4− + − + vanishing contribution 1− ++ + − −− 4−− ++ pm pm + 1−− + 1 −− 44−− 12 34 D=8 2ε 1 ResultAnswer ~ = ! "! " K4 K4 = ε(1 ε)I4 − 1•2 34 1 ε 0 [12] [34] D=8 2ε − − −→→ −6 Result ~ ! "! " K4 K4 = ε(1 ε)I4 − − − −ε →0 −6 [12] [34] 1122 3344 → D=8 2ε 11 K = ε(1 ε)I D=8− 2ε ResultResult ~~ !! ""!! ""KK44 K44 = ε(1 ε)4I4 − (Originally deriv[ ed] b[y Bern,] Kosower, and Bern and− Morgan)− −ε ε→00−6 [1122] [3344] − − −→→→ −6 (Originally derived by Bern, Kosower, and Bern and Morgan) (Originall(Originallyy deriv deriveded b byy Bern, Bern, K Kosoosowwerer, ,and and Bern Bern and and Morgan) Morgan) FinitenessFinitenessFiniteness of of the theof theall-min all-min all-minusus amplitude usamplitude amplitude Finiteness of the all-minus amplitude 2 2 2 2 2 InDefine (DR): L D = L 4 + L 2 ε with 2 LD 2= L 2+ L :2= L 2 µ •Define• L D = L 4 + L 2 ε −with LD = L4 + L4 2ε :=2εL4 µ4 • − − − − − 2 2 2 2 2 Define L D = L 4 + L 2 ε with LD = L + L := L µ • •− 4 2ε 4 A finiteA finite, non-zer, non-zero resulto− result arises− arises from from • •finite, non-zero result due to incomplete cancellations of • incomplete cancellations of propagators • A finite, non-zeroincomplete r(inesultverse) arises propagators cancellations from of propagators incomplete cancellations of propagators2 2 2 2 2 L2 L4L2 Lµ24 +µµ2+ µ µ2 µ • 4 = 4= − = += 1+ 2 L2 − 2 L2 1 2 L2 2 2 2 2 LD2 D LD D LD D L4 L4 µ + µ µ 2 = −• 2 = 1+ 2 LD LD! MHV verLticesD are 4-dimensional !• MHVFiniteness vertices comes are as4-dimensional an ε/ε-effect (smells like an anomaly) ! MHV vertices are 4-dimensional !• D-dimensionalAll-min! D-dimensionalus amplitude propagators pr obtainedopagators fr om MHV diagrams, but all-plus amplitude remains mysterious ! D-dimensional propagators Where is the all-plus amplitude? • Go back to the path integral! • Mansfield’s approach: • LC quantisation of Yang-Mills: A-=0 • integrate out A+ (no derivatives w.r.t. LC time x-) • Schematically the action becomes (Scherk-Schwarz) -+ --+ -++ --++ S = S
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