In the context of constructing one-loop amplitudes using a unitarity
bootstrap approach we discuss a general systematic procedure for obtaining the
coefficients of the scalar bubble and triangle integral functions of one-loop
amplitudes. Coefficients are extracted after examining the behaviour of the cut
integrand as the unconstrained parameters of a specifically chosen
parametersiation of the cut loop momentum approach infinity.Comment: 5 Pages. To appear in the proceedings of the 8th International
  Symposium on Radiative Corrections (RADCOR 2007): Application of Quantum
  Field Theory to Phenomenology, Florence, Italy, 1-6 Oct 200

Forde, Darren

English

arXiv

In the context of constructing one-loop amplitudes using a unitarity bootstrap approach we discuss a general systematic procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted after examining the behavior of the cut integrand as the unconstrained parameters of a specifically chosen parameterization of the cut loop momentum approach infinity. Measurements of new physics at the forthcoming experimental program at CERN's Large Hadron Collider (LHC) will require a precise understanding of processes at next-to-leading order (NLO). This places increased demands for the computation of new one-loop amplitudes. This in turn has spurred recent developments towards improved calculational techniques. Direct calculations using Feynman diagrams are in general inefficient. Developments of more efficient techniques have usually centered around unitarity techniques [1], where tree amplitudes are effectively 'glued' together to form loops. The most straightforward application of this method, in which the cut loop momentum is in D = 4, allows for the computation of 'cut-constructible' terms only, i.e. (poly)logarithmic containing terms and any related constants. QCD amplitudes contain, in addition to such terms, rational pieces which cannot be derived using such cuts. These 'missing' rational parts can be extracted using cut loop momenta in D = 4-2 {var_epsilon}. The greater difficulty of such calculations has restricted the application of this approach, although recent developments [3, 4] have provided new promise for this technique. Recently the application of on-shell recursion relations [5] to obtaining the 'missing' rational parts of one-loop processes [6] has provided an alternative very promising solution to this problem. In combination with unitarity methods an 'on-shell bootstrap' approach provides an efficient technique for computing complete one-loop QCD amplitudes [7]. Additionally other new methods have also proved fruitful for calculating rational terms [8]. Such developments have again refocused attention on the optimization of the derivation of the cut-constructable pieces of the amplitude. Deriving cut-constructible terms for any one-loop amplitude reduces to the computation of coefficients of a set of scalar bubble, scalar triangle and scalar box integral functions. Box coefficients may be found with very little work, directly from the quadruple cut of the relevant box function [9]. A unique box coefficient contributes to each distinct quadruple cut. Unfortunately triangle and bubble coefficients cannot be derived in quite so direct a manner. Multiple scalar integral coefficients appear inside a two-particle cut or triple cut. It is therefore necessary to disentangle the relevant bubble or triangle coefficients from any other coefficients sharing the same cut [1, 4, 10, 11]. The large number of NLO processes of interest for the LHC suggests that a completely automated computational procedure is highly desired. To this end we discuss, in this proceeding, a recently proposed method [12, 13] for the direct, efficient and systematic extraction of bubble and triangle coefficients which is well suited to automation

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SLAC-PUB-13124UCLA/08/TEP/4Constructing QCD one-loop amplitudesDarren Forde∗†Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA,&Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095–1547, USA.E-mail: dforde@slac.stanford.eduIn the context of constructing one-loop amplitudes using a unitarity bootstrap approach we discussa general systematic procedure for obtaining the coefficients of the scalar bubble and triangle in-tegral functions of one-loop amplitudes. Coefficients are extracted after examining the behaviourof the cut integrand as the unconstrained parameters of a specifically chosen parametersiation ofthe cut loop momentum approach infinity.8th International Symposium on Radiative CorrectionsOctober 1-5, 2007Florence, Italy∗Speaker.†Research supported in part by the US Department of Energy under contracts DE–FG03–91ER40662 and DE–AC02–76SF00515.February 2008Measurements of new physics at the forthcoming experimental program at CERN’s LargeHadron Collider (LHC) will require a precise understanding of processes at next-to-leading order(NLO). This places increased demands for the computation of new one-loop amplitudes. This inturn has spurred recent developments towards improved calculational techniques.Direct calculations using Feynman diagrams are in general inefficient. Developments of moreefficient techniques have usually centred around unitarity techniques [1], where tree amplitudes areeffectively “glued” together to form loops. The most straightforward application of this method, inwhich the cut loop momentum is in D = 4, allows for the computation of “cut-constructible” termsonly, i.e. (poly)logarithmic containing terms and any related constants. QCD amplitudes contain,in addition to such terms, rational pieces which cannot be derived using such cuts. These “missing”rational parts can be extracted using cut loop momenta in D = 4−2ε [2]. The greater difficulty ofsuch calculations has restricted the application of this approach, although recent developments [3,4] have provided new promise for this technique.Recently the application of on-shell recursion relations [5] to obtaining the “missing” rationalparts of one-loop processes [6] has provided an alternative very promising solution to this prob-lem. In combination with unitarity methods an “on-shell bootstrap” approach provides an efficienttechnique for computing complete one-loop QCD amplitudes [7]. Additionally other new methodshave also proved fruitful for calculating rational terms [8].Such developments have again refocused attention on the optimisation of the derivation ofthe cut-constructable pieces of the amplitude. Deriving cut-constructible terms for any one-loopamplitude reduces to the computation of coefficients of a set of scalar bubble, scalar triangle andscalar box integral functions. Box coefficients may be found with very little work, directly fromthe quadruple cut of the relevant box function [9]. A unique box coefficient contributes to eachdistinct quadruple cut. Unfortunately triangle and bubble coefficients cannot be derived in quite sodirect a manner. Multiple scalar integral coefficients appear inside a two-particle cut or triple cut.It is therefore necessary to disentangle the relevant bubble or triangle coefficients from any othercoefficients sharing the same cut [1, 4, 10, 11]. The large number of NLO processes of interest forthe LHC suggests that a completely automated computational procedure is highly desired. To thisend we discuss, in this proceeding, a recently proposed method [12, 13] for the direct, efficient andsystematic extraction of bubble and triangle coefficients which is well suited to automation.1. Triangle coefficientsFollowing in the spirit of the box coefficient [9] we would like to apply a triple cut to extracta triangle coefficient. Such a triple cut isolates a unique triangle coefficient but also containscontributions from scalar box integrals which share three of their four propagators with the triangle.The separation of the coefficient of a particular scalar triangle integral from any box coefficientscan be effected byc0 =− [InftA1A2A3] (t)∣∣∣t=0. (1.1)Equation 1.1 instructs us to start by taking the triple cut of the desired triangle coefficient,Atree1 Atree2 Atree3 (t) = Atreec3−c1+2(−l,c1, . . . ,(c3−1), l1)Atreec2−c3+2(−l1,c3, . . . ,(c2−1), l2)×Atreen−c2+c1+2(−l2,c2, . . . ,(c1−1), l), (1.2)2shown in figure 1(a), with l1 = l−K1 and l2 = l +K2, where K1 and K2 are sums of external mo-a)K1ll1 = l −K1l2 = l −K2−K2K3c1c3 − 1c3c2 − 1c2c1 − 1b) K1ll1 = l −K1c1 + 1c2 c2 + 1c1Figure 1: a) The triple cut used to compute a scalar triangle coefficient. b) The two-particle cut used tocalculate a scalar bubble coefficient.menta. The cut momentum l of the triple cut depends on a single parameter t, and is parameterisedin the specific form [11, 12],〈l−|= t〈K,−1 |+S1 (γ−S2)(γ2−S1S2)〈K,−2 |, 〈l+|=S2 (γ−S1)(γ2−S1S2) t 〈K,+1 |+ 〈K,+2 |. (1.3)Here we have expressed the cut momentum in terms of a convenient basis of null vectors K1 andK2K,μ1 =Kμ1 − (S1/γ)Kμ21− (S1S2/γ2) , K,μ2 =Kμ2 − (S2/γ)Kμ11− (S1S2/γ2) , (1.4)with two solutions for γ = (K1 ·K2)±√Δ with Δ = (K1 ·K2)2−K21K22 .The Inft is instructing us to series expand this cut integrand around t = ∞. The t0 componentof this series expansion gives the desired triangle coefficient. For the three-mass case describedabove we must also average over the two solutions to γ . Analytic continuation of l to complexmomenta allows one- and two-mass triangles, containing three-point vertices, to be computed in asimilar manner after setting the relevant masses in eq. (1.3) and eq. (1.4) to zero. In these casesonly one solution to γ survives.This procedure succeeds because of the specific momentum parametrisation we have chosen.The series expansion of the Inft would in general give us rational coefficients ai multiplying inte-grals over powers of t. Seen schematically this is−1∑i=−∞ai∫dt ti +a0∫dt +a1∫dt t + . . .+amax∫dt tmax, (1.5)and we would expect contributions to the scalar triangle coefficient from every term. It is easy toshow though that all integrals over t will vanish, eliminating any such contributions. For example,∫dtt ∼∫d4l 〈K,−2 |/l|K,−1 〉l2l21 l22∼ 〈K,−2 | /K1|K,−1 〉C1 + 〈K,−2 | /K2|K,−1 〉C2 = 0, (1.6)with a similar result for other non-zero powers of t. The Ci are Passarino-Veltman reduction coef-ficients.32. Bubble coefficientsA similar procedure applies to the extraction of bubble coefficients. To compute the coefficientof a particular bubble we use a two-particle cut and must disentangle our desired coefficient fromthe scalar boxes and triangles which this will also contain. We start from the generic two-particlecut Atree1 Atree2 (t,y) = Atreec2−c1+2(−l,(c1 +1), . . . ,c2, l1) Atreec1−c2+2(−l1,(c2 +1), . . . ,c1, l) shown in fig-ure 1(b), with l1 = l−K1. Having isolated a single bubble coefficient we parameterise the cut loopmomentum l, which now depends upon two free parameters t and y, using〈l−|= t〈K,−1 |+S1γ (1− y) 〈χ−|, 〈l+|= yt〈K,+1 |+ 〈χ+|. (2.1)This is expressed in terms of a basis of massless on-shell momenta K1 and χ . χ is an arbitraryfree vector, which the final result is independent of, used to define K,μ1 = Kμ1 − (S1/γ)χμ withγ = 〈χ±| /K1|χ±〉 ≡ 〈χ±| /K1|χ±〉.The equivalent expression to eq. (1.1) is then given byb0=−i [Inft [InfyA1A2] (y)] (t)∣∣∣t→0, ym→ 1m+1− 12 ∑Ctri [InftA1A2A3] (t)∣∣∣tm→T (m). (2.2)The first term is the natural extension of the single Inft of eq. (1.1) to the case of two free parame-ters. In this double series expansion we expand around y = ∞ and then t = ∞ and again drop termsproportional to t, because the corresponding integrals disappear. Integrals over y are non-vanishingthough and are related to the scalar bubble integral, B0(K21 ), via∫dyym = B0(K21 )/(m+1).The naively unexpected second term of eq. (2.2) involves a sum over all the triangles Ctri thatcontain the original two-particle cut. Writing the two-particle cut integrand schematically in theforma0(t)+a1(t)y+ . . .+amax(t)ymax +∑CtriAL(yi(t), t)AR(yi(t), t)ξi (y− yi(t)) , (2.3)allows us to understand why contributions from triangles arise. Solving (l(y, t)−K2)2 = 0, theadditional propagator present in the triangles in the last term above, gives us yi(t). Inserting thissolution into the momentum parameterisation of l given by eq. (2.1) leaves us with the momentumparametrisation of the triangles that we wish to separate from the bubble. This parametrisationdiffers importantly from eq. (1.3) in that the integrals over t do not vanish, as can be seen forexample with∫dtt ∼∫d4l〈χ−|/l|K,−1 〉l2l21 l22∼ 〈χ−| /K1|K,−1 〉C1 + 〈χ−| /K2|K,−1 〉C2 	= 0. (2.4)The remaining contributions to the bubble coefficient are then found by relating these non-vanishingintegrals over t to scalar bubbles usingT (m) =(S1γ)m 〈χ−| /K2|K,−1 〉m(K1 ·K2)m−1Δm(m∑l=1CmlSl−12(K1 ·K2)l−1)Bcut0 (K21 ), (2.5)and T (0) = 0. The coefficients Cml are given byC11 =12, C21 =−38 , C22 =−38 , C31 =−112Δ(K1 ·K2)2 +516 , C32 =58 , C33 =516 .4I would like to thank Zvi Bern, Lance Dixon, David Kosower and Carola Berger for manyhelpful discussions.References[1] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B425:217 (1994)Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B435:59 (1995)[hep-ph/9409265].Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B 513, 3 (1998) [arXiv:hep-ph/9708239].[2] Z. Bern and A. G. Morgan, Nucl. Phys. B 467 (1996) 479 [arXiv:hep-ph/9511336].[3] C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Phys. Lett. B 645, 213 (2007)C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. 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Constructing QCD one-loop amplitudes

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