Loop Amplitudes from MHV Diagrams - Part 1

Loop Amplitudes from MHV Diagrams - part 1 -

Gabriele Travaglini Queen Mary, University of London

Brandhuber, Spence, GT hep-th/0407214, hep-th/0510253 Bedford, Brandhuber, Spence, GT hep-th/0410280, hep-th/0412108 Brandhuber, McNamara, Spence, GT in progress 2 HP Workshop, Zürich, September 2006 Outline

• Motivations and basic formalism • MHV diagrams (Cachazo, Svrcek,ˇ Witten) - Loop MHV diagrams (Brandhuber, Spence, GT) - Applications: One-loop MHV amplitudes in N=4, N=1 & non-supersymmetric Yang-Mills • Proof at one-loop: Andreas’s talk on Friday Motivations

• Simplicity of scattering amplitudes unexplained by textbook Feynman diagrams - Parke-Taylor formula for Maximally Helicity Violating amplitude of gluons (helicities are a permutation of −−++ ....+) • New methods account for this simplicity, and allow for much more efﬁcient calculations • LHC is coming ! Towards simplicity

• Colour decomposition (Berends, Giele; Mangano, Parke, Xu; Mangano; Bern, Kosower)

• Spinor helicity formalism (Berends, Kleiss, De Causmaecker, Gastmans, Wu; De Causmaecker, Gastmans, Troost, Wu; Kleiss, Stirling; Xu, Zhang, Chang; Gunion, Kunszt) Colour decomposition • Main idea: disentangle colour • At tree level, Yang-Mills interactions are planar

tree a a A ( p ,ε ) = Tr(T σ1 T σn) A(σ(p ,ε ),...,σ(p ,ε )) { i i} ∑ ··· 1 1 n n σ Colour-ordered partial amplitude

- Include only diagrams with ﬁxed cyclic ordering of gluons - Analytic structure is simpler

• At loop level: multi-trace contributions - subleading in 1/N Spinor helicity formalism

• Consider a null vector p µ µ µ = ( ,! ) • Deﬁne p a a ˙ = p µ σ a a ˙ where σ 1 σ • If p 2 = 0 then det p = 0 ˜ ˜ Hence p aa˙ = λaλa˙ · λ (λ) positive (negative) helicity • spinors a b a˙ b˙ Inner products 12 : = ε ab λ λ , [12] := ε ˙λ˜ λ˜ • ! " 1 2 a˙b 1 2 Parke-Taylor formula

4 + + i j A (1 ...i− ... j− ...n ) = " # MHV 12 23 n1 " #" #···" #

☞

☞ • Colour decomposition and spinor helicity formalism make the simplicity manifest... • ...but we still have to explain it !

Twistor space • Simple geometrical structure of the amplitudes in twistor space (Witten)

MHV Diagrams (Cachazo, Svrcek,ˇ Witten)

• String theory on twistor space (Witten) Recursion relations for YM tree amplitudes • (Britto, Cachazo, Feng +Witten) Twistor space (in one slide) • Half Fourier transform on spinor variables ∂ (λ ,λ˜ ) (λ ,µ˜ ) λ˜ i a a˙ −→ a a˙ −→ ∂µ ∂ a,a˙ = 1,2 µ i −→ − ∂λ˜ • Twistor space is a copy of C 4

1 (λ,λ˜ ) (tλ,t− λ˜ ) (λ,µ) (tλ,tµ) −→ ➡ −→

3 4 ☜ # of supersymmetries Projective twistor space, CP | • Fourier transform can be explicitly carried out for MHV amplitudes

‣ Holomorphicity in the spinor variables • Result ≠ 0 if coordinates of gluons satisfy a˙ aa˙ µ + x λa = 0 x = point in Minkowski space • Line in twistor space (Penrose’s incidence relation)

‣ Maps points in Minkowski space to twistors What about generic amplitudes ?

• Scattering amplitudes are supported on curves in twistor space (Witten, 2003)

• d = q − 1 + l q = # negative helicity gluons, l = # loops • g ≤ l

‣ Tree MHV: q=2, l=0 ➡ d=1, g=0 (complex line) Amplitude Twistor space structure MHV diagrams

! "

! " M MHV ! !

! " " ! " M M nMHV ! !

! !

" ! " " ! ! ! nnMHV " M M M " " Why MHV diagrams ?

• MHV amplitudes localise on lines in twistor space (Witten)

• A line in twistor space corresponds to a point in Minkowski space (Penrose) ➡ • An MHV amplitude can be thought of as a local interaction in space-time ! (Cachazo, Svrcek,ˇ Witten) MHV Rules (tree level) (Cachazo, Svrcek,ˇ Witten)

• MHV amplitude ➡ MHV vertex • Off-shell continuation for internal (possibly loop) momenta needed: · Internal momentum is off-shell M M · Need to deﬁne spinor λ for an off-shell vector!

• Need propagators to connect the MHV vertices Off-shell continuation

If L 2 = 0 , we can write • ! Laa˙ = laa˙ + zηaa˙

‣ η a a˙ := η a η˜ a ˙ is a null reference vector ‣ z = L 2 / 2 ( L η ) is a real number · ‣ l a a˙ : = l a l˜ a˙ is the off-shell continuation, ‣ l L η˜ a˙ (equivalent to CSW’s) a ⇒ aa˙ Internal propagators

i Just scalar propagators • P2 + iε • At loop level, the i ε prescription is crucial in correctly determining the integration range Proofs of MHV diagrams Tree level

• Covariance (η-independence); singularities (Cachazo, Svrcek,ˇ Witten; Britto, Cachazo, Feng, Witten) • MHV diagrams are a special BCFW recursion relation (Risager) • Lagrangian derivation (Mansﬁeld; Gorsky, Rosly) Loop MHV diagrams (Brandhuber, Spence, GT)

• Initial prognosis terribly poor... ‣ Twistor string theory dual to conformal supergravity (not Yang-Mills, not even ordinary supergravity) at the quantum level • ...but we tried anyway ! • Simplest amplitude: 1-loop MHV amplitude in N=4 super Yang-Mills • Computed in 1994 by Bern, Dixon, Dunbar, From TreesKoso tow erLoops, cont’d

1 loop tree A − = A Atree MHV∑× the all order in ! 2-mass easy box function:

" " 2me 2 2 c! s − Two-mass easyt bo− x function F (s,t,P ,Q ) = − 2F1 (1, ",1 ",as) + − 2F1 (1, ",1 ",at) −"2 µ2 − − µ2 − − !" # " # P2 " Q2 " − − F 1, ",1 ",aP2 − − F 1, ",1 ",aQ2 − µ2 2 1 − − − µ2 2 1 − − & " # $ % " # $ % 2(pq) with a := P2Q2 st − MHV rules for Loops • Sew d MHV vertices q = # negative helicity gluons, d = q − 1 + l • l = # loops • As at tree level, we use a. CSW off-shell continuation

b. Scalar propagators • MHV, 1-loop: d = 2 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 • Chose measure, perform loop integration MHV 1-loop amplitudes in N=4/N=1 SYM (agrees with BDDK) • Calculation from MHV diagrams

d M∑ d M Z ! Z m1,m2,h • The sum is over ‣ all possible MHV diagrams

‣ internal particle species (g, f, s) and helicities

‣ different from unitarity-based approach of BDDK • We have to ﬁnd the measure... 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 • Chose measure, perform loop integration • MHVThe 1-loop amplitudes integration in N=4/N=1 SYM measur (agrees with BDDK)e

· P L is the momentumdM on the left ! Z m1,m2,h

4 4 d L1 d L2 (4) dM := 2 2 δ (L2 L1 + PL) L1 + iε L2 + iε − Use L = l + z η , and L ( l , z ): • → d4L dz d3l ➡ 2 = L + iε z + isgn (l0η0)ε 2l0 dispersive measure X phase-space measure (Nair measure) Loop integral becomes:

(Dispersion integral) X (2-particle LIPS integral) • LIPS integral: ‣ computes the cut of the amplitude

‣ regularise IR divergences: 4−2ε dimensions • Dispersion integral reconstructs the amplitude from cuts The return of the analytic S-matrix !

(1966) Comments

• Final result is covariant and agrees with BDDK

‣ Non-trivial cancellation of η-dependence • MHV diagrams work for loops ! • Result incorporates a large number of Feynman diagrams

‣ 1 MHV diagram vs 227,585 Feynman diagrams (7 points) More Comments

• Dispersion integrals are simple - no subtractions needed (van Neerven) • No ghosts i Scalar propagators • P2 + iε ‣ MHV diagrams related to lightcone gauge quantisation (more on this later) New form of the 2-mass easy box function

2 2 1 ε ε 2 ε 2 2 2 F(s,t,P ,Q ) = ( s)− + ( t)− ( P )− ( Q )− + B(s,t,P ,Q ) −ε2 − − − − − − ! "

B(s,t,P2,Q2) = Li (1 aP2) + Li (1 aQ2) Li (1 as) Li (1 at) 2 − 2 − − 2 − − 2 − From Trees to Loops, cont’d P2 + Q2 s t a = − − P2Q2 st − tree AMHV s = (P + p)2 t = (P + q)2 × the all order in ! 2-mass easy box function:

" " 2me 2 2 - Morc!e compacts − than usual expressiont − F (s,t,P ,Q ) = − 2F1 (1, ",1 ",as) + − 2F1 (1, ",1 ",at) −"2 µ2 − − µ2 − − - Simpler!" anal# ytic continuation " # P2 " Q2 " − − F 1, ",1 ",aP2 − − F 1, ",1 ",aQ2 − µ2 2 1 − − − µ2 2 1 − − & " # $ % " # $ % 2(pq) with a := P2Q2 st − Further applications

• One-loop MHV amplitudes in N=1 super Yang-Mills (Bedford, Brandhuber, Spence, GT; Quigley, Rozali) ‣ Result expressed in terms of ﬁnite boxes, and triangles ‣ Agreement with BDDK ‣ No twistor string theory for N=1 Super Yang-Mills... ‣ ...nevertheless MHV diagram method works ! • Cut-constructible part of 1-loop MHV amplitudes in non-supersymmetric Yang-Mills (Bedford, Brandhuber, Spence, GT) ‣ Extends 5-pt and adjacent negative helicity cases of BDK and BDDK ‣ First new result at 1-loop in pure YM

‣ Non-supersymmetric amplitudes are not cut-constructible in 4 dimensions

‣ rational terms are missed by MHV diagrams Next goal:

Prove MHV diagrams at one loop

• Covariance (η-independence) • Correct singularity structure

Andreas Brandhuber’s talk: Friday at 11:25 Summary

• MHV diagrams provide a new diagrammatic method to calculate scattering amplitudes at tree and one-loop level in Yang-Mills • Proof for generic one-loop amplitudes: ‣ Feynman Tree Theorem ✂ ‣ Singularity structure