EURO EUROFLA LAVOU OUR 20 R 2008, 22 08, 22-26 -26 Sep ep, DU DURHAM RHAM A duality relation between loops and trees Germán Rodrigo in collaboration with Stefano Catani, T anju Gleisberg, Frank Krauss, and Jan C. Winter JHEP 09 (2008) 065 [arXiv:0804.3170 [hep-ph]] germán rodrigo a duality relation between loops and trees euroflavour08 1
multiparticle final states at next-to-leading order (NLO) ● LO @ LHC: 100% uncertainty typically ● NLO @ LHC necessary for 2→3 (many recent results) and 2→4 (not yet a cross section) ● Radiative Return @ NLO: at least 2→3 germán rodrigo a duality relation between loops and trees euroflavour08 2
multiparticle final states at next-to-leading order (NLO) ● LO @ LHC: 100% uncertainty typically ● NLO @ LHC necessary for 2→3 (many recent results) and 2→4 (not yet a cross section) ● Radiative Return @ NLO: at least 2→3 new feature wrt LO: NLO = ∫ m 1 d R ∫ m d V combine m with m+1 virtual contribution real radiation germán rodrigo a duality relation between loops and trees euroflavour08 3
kinematics: momentum real radiation conservation + observable dependent function ∫ m 1 d R = ∫ d m 1 { p i } M m 1 { p i } F m 1 { p i } split phase-space integrand in two parts: several well (...) fin + (...) div known/tested working methods (subtraction, IR finite: computable IR singular: analytically dipole, slicing, mixed, ...) numerically as LO computable up to O(ε) virtual contribution ∫ m d V = ∫ d m { p i } ∫ d d q M m { p i } F m { p i } loop integral: in multiparton processes (m≥5) regarded as main practical bottleneck many new developments in recent years (OPP, generalized Unitarity, ...) germán rodrigo a duality relation between loops and trees euroflavour08 4
general goal I transform loop integral into customary phase space integral for real radiation (loop ⇔ phase-space duality) ∫ loop d d q M m { p i } ,q = ∫ d q M m q { p i } ,q d d q q 2 ∫ m q ... II then treat ∫ m 1 ... similarly to the real emission contribution III Monte Carlo integration [see also Soper, Nagy, Kramer, Kleinschmidt, Moretti, Piccinini, Polosa] germán rodrigo a duality relation between loops and trees euroflavour08 5
Outline ● The Feynman Tree Theorem ● A duality theorem between one-loop integrals and single-cut phase-space integrals ● Relating the FTT and the duality relation ● Massive integrals, unstable particles, and gauge poles ● Duality at the amplitude level ● Final remarks germán rodrigo a duality relation between loops and trees euroflavour08 6
Notation T o simplify the presentation: massless internal lines only (more on massive particles later) Scalar one-loop integral L N p 1 , ... , p N = − i ∫ d d q N 1 2 d ∏ 2 i0 q i i = 1 q μ is the loop momentum (anti-clockwise) internal lines i N q i = q ∑ ∑ p i = 0 , p N i = p i . p k , k = 1 i = 1 shorthand notation: d q d − 1 q − i ∫ d dq 0 ∫ d ∞ ⋅ ≡ ∫ dq 0 ∫ q ⋅ ⋅ ≡ ∫ q ⋅ − i ∫ d ⋅ ⋅ ⋅ ⋅ , d − 1 ⋅ ⋅ ⋅ ⋅ −∞ 2 2 2 q ≡ 2 i q germán rodrigo a duality relation between loops and trees euroflavour08 7
Feynman and Advanced propagators 1 Feynman propagator +i0: positive frequencies are G q ≡ propagated forward in time, 2 i0 q and negative frequencies backward Advanced propagator both poles displaced above 1 G A q ≡ the real axis (independently 2 − i0 q 0 q of the sign of the energy) and are related by G A q = G q q x ± i0 = PV 1 1 x ∓ i x germán rodrigo a duality relation between loops and trees euroflavour08 8
Feynman Tree Theorem RP Feynman, Acta Phys. Polon. 24 (1963) 697 Advanced one-loop integral: Feynman propagators replaced by advanced propagators N N p 1 , ... , p N = ∫ q ∏ G A q i L A i = 1 Cauchy residue theorem N p 1 , ... , p N = 0 L A N N L 2 − cut N ... L N − cut = ∫ q ∏ [ G q i N L 1 − cut N q i ] = L i = 1 then N p 1 , ... , p N ... L N − cut N p 1 , ... , p N = − [ L 1 − cut p 1 , ... , p N ] N L in four-dimensions, 4-cut at most (4 delta functions) germán rodrigo a duality relation between loops and trees euroflavour08 9
Duality Theorem S.Catani et al. JHEP09(2008)065 Cauchy residue theorem close the contour at ∞ on the lower half plane ⇨ select residues with positive definite energy N p 1 , ... , p N = L − 2 i ∫ q ∑ Res Im q 0 0 [ ∏ G q i ] N i = 1 G q j ] = [ Res ith − pole G q i ] [ ∏ Res ith − pole [ ∏ G q j ] ith − pole N N j = 1 j ≠ i 1 = ∫ dq 0 q i 2 Res ith − pole 2 i0 q i ● equivalent to cut that line and set it on-shell ● one-loop integral represented as a linear combination of N single-cut phase-space integrals ● shift q i → q in each term ⇔ single phase-space integral over N terms germán rodrigo a duality relation between loops and trees euroflavour08 10
Duality Theorem S.Catani et al. JHEP09(2008)065 Cauchy residue theorem N p 1 , ... , p N = L − 2 i ∫ q ∑ Res Im q 0 0 [ ∏ G q i ] N i = 1 G q j ] = [ Res ith − pole G q i ] [ ∏ Res ith − pole [ ∏ G q j ] ith − pole N N j = 1 j ≠ i 1 = ∫ dq 0 q i 2 Res ith − pole 2 i0 q i [ ∏ q j i0 ] ith − pole N N 1 1 = ∏ 2 − i0 q j − q i q j j ≠ i j ≠ i ● the customary +i 0 prescription is modified ● Lorentz covariant dual prescription ● η is a future-like vector: η 0 > 0 , η 2 ≥ 0 ● analytic continuation: s ij → s ij - i 0 wrong germán rodrigo a duality relation between loops and trees euroflavour08 11
The calculation is elementary, but involves some subtle points [ q k j 2 i0 ] q 2 =− i0 ,q 0 = q 0 1 1 = k j0 − 2 q ⋅ k j k j 2 2 q 0 = q 2 − i0 q 0 where germán rodrigo a duality relation between loops and trees euroflavour08 12
The calculation is elementary, but involves some subtle points [ q k j 2 i0 ] q 2 =− i0 ,q 0 = q 0 1 1 = k j0 − 2 q ⋅ k j k j 2 2 q 0 2 − i0 ≃ ∣ q ∣− i0 = q 2 q 0 2 ∣ q ∣ O i0 where germán rodrigo a duality relation between loops and trees euroflavour08 13
The calculation is elementary, but involves some subtle points [ q k j 2 i0 ] q 2 =− i0 ,q 0 = q 0 1 1 = k j0 − 2 q ⋅ k j k j 2 2 q 0 1 = 2 − i0 k j0 /∣ q ∣ 2 qk j k j 2 − i0 ≃ ∣ q ∣− i0 = q q 0 2 ∣ q ∣ germán rodrigo a duality relation between loops and trees euroflavour08 14
The calculation is elementary, but involves some subtle points [ q k j 2 i0 ] q 2 =− i0 ,q 0 = q 0 1 1 = k j0 − 2 q ⋅ k j k j 2 2 q 0 1 = 2 − i0 k j0 /∣ q ∣ 2 qk j k j 2 − i0 ≃ ∣ q ∣− i0 = q q 0 2 ∣ q ∣ ● only the sign matters: - i 0 k j 0 /| q | → - i 0 k j 0 → - i 0 ηk j where η μ = ( η 0 ,0) with η 0 >0 ● different choices of the future-like vector η are equivalent to different choices of the coordinate system germán rodrigo a duality relation between loops and trees euroflavour08 15
Duality theorem Duality relation between one-loop integrals and single-cut phase-space integrals N p 1 , ... , p N = − N p 1 , ... , p N L L = − [ I N − 1 p 1 , p 12 , ... , p 1, N − 1 cyclic perms. ] where n 1 n k 1 , ... ,k n = ∫ q q ∏ I 2 − i0 k j 2 q k j k j j = 1 N one-particle phase-space integrals ⇔ one phase-space integral over N tree quantities germán rodrigo a duality relation between loops and trees euroflavour08 16
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