Presented by C.G. Papadopulos at the Cracow Epiphany Confer ence on LHC Physics, Cracow, Poland, 4–6 January 2008.The recently proposed method (OPP) to extract the coefficients of the scalar one-loop integrals to any multi-particle (sub)-amplitude is described. Within this method no analytical information on the structure of the amplitude is needed, allowing for a purely numerical, but still algebraic, implementation of the algorithm. The algorithm can be used to automatically perform one-loop calculation both in QCD and in the EW Theory. As an application, we give QCD one-loop results for the process pp → ZZZ at the LHC.Work supported by HEPTOOLS, MRTN-CT-2006-035505 and ALGOT OOLS, MTKD-CT-2004-014319

Ossola, Giovanni

Papadopoulos, Costas G.

Pittau, Roberto

Acta Physica Polonica B

English

Repositorio Institucional Universidad de Granada

Vol. 39 (2008) ACTA PHYSICA POLONICA B No 7REDUCTION OF ONE-LOOP AMPLITUDESAT THE INTEGRAND LEVEL —NLO QCD CALCULATIONS∗ ∗∗Giovanni Ossola, Costas G. PapadopoulosNCSR Demokritos, 15310, Athens, GreeceRoberto PittauDepartamento de Fisica Teorica y del Cosmos y CAFPEUniversidad de Granada, 18071 Granada, Spain(Received May 6, 2008)The recently proposed method (OPP) to extract the coefficients of thescalar one-loop integrals to any multi-particle (sub)-amplitude is described.Within this method no analytical information on the structure of the am-plitude is needed, allowing for a purely numerical, but still algebraic, im-plementation of the algorithm. The algorithm can be used to automaticallyperform one-loop calculation both in QCD and in the EW Theory. As anapplication, we give QCD one-loop results for the process pp → ZZZ atthe LHC.PACS numbers: 11.10.–z, 12.38.Bx, 12.15.Lk, 13.85.Hd1. IntroductionIn two recent papers [1], we proposed a reduction method (OPP) forarbitrary one-loop sub-amplitudes at the integrand level [2]. The methodis based on idea of expressing the integrand of the one-loop amplitude interms of the propagators that depends on the integration momentum. Thesolution of this equation can proceed in an hierarchical way, by exploitingnumerically the set of kinematical equations for the integration momentum,corresponding to the so-called quadruple, triple and double cuts used inthe unitarity-cut method [3–5]. The method requires a minimal information∗ Presented by C.G. Papadopulos at the Cracow Epiphany Conference on LHC Physics,Cracow, Poland, 4–6 January 2008.∗∗ Work supported by HEPTOOLS, MRTN-CT-2006-035505 and ALGOTOOLS,MTKD-CT-2004-014319.(1685)1686 G. Ossola, C.G. Papadopoulos, R. Pittauabout the form of the one-loop (sub-)amplitude and therefore it is well suitedfor a numerical implementation. The method works for any set of internaland/or external masses, so that one is able to study the full electroweakmodel, without being limited to massless theories.2. The OPP methodThe starting point of the OPP reduction method is the general expressionfor the integrand of a generic m-point one-loop (sub-)amplitudeA(q̄) =N(q)D̄0D̄1 · · · D̄m−1, D̄i = (q̄ + pi)2 − m2i , p0 6= 0 . (1)In the previous equation, we use a bar to denote objects living in n = 4 + ǫdimensions, and q̄2 = q2 + q̃2, where q̃2 is ǫ-dimensional and (q̃ · q) = 0.N(q) is the 4-dimensional part of the numerator function of the amplitude.If needed, the ǫ-dimensional part of the numerator should be treated sep-arately, as will be explained later. N(q) depends on the 4-dimensional de-nominators Di = (q + pi)2 − m2i as followsN(q) =m−1∑i0<i1<i2<i3[d(i0i1i2i3) + d̃(q; i0i1i2i3)]m−1∏i6=i0,i1,i2,i3Di+m−1∑i0<i1<i2[c(i0i1i2) + c̃(q; i0i1i2)]m−1∏i6=i0,i1,i2Di+m−1∑i0<i1[b(i0i1) + b̃(q; i0i1)]m−1∏i6=i0,i1Di+m−1∑i0[a(i0) + ã(q; i0)]m−1∏i6=i0Di+P̃ (q)m−1∏iDi . (2)Inserted back in Eq. (1), this expression simply states the multi-pole natureof any m-point one-loop amplitude, that, clearly, contains a pole for anypropagator in the loop, thus one has terms ranging from 1 to m poles. Thecoefficients of the poles can be further split into two pieces. A piece thatstill depends on q (the terms d̃, c̃, b̃, ã), that vanishes upon integration, anda piece that does not depend on q (the terms d, c, b, a). Such a separation isReduction of One-Loop Amplitudes at the Integrand Level — NLO . . . 1687always possible and the latter set of coefficients is immediately interpretableas the ensemble of the coefficients of all possible 4, 3, 2, 1-point one-loopfunctions contributing to the amplitude.Once Eq. (2) is established, the task of computing the one-loop amplitudeis then reduced to the algebraical problem of fitting the coefficients d, c, b, aby evaluating the function N(q) a sufficient number of times, at differentvalues of q, and then inverting the system. It can be achieved quite efficientlyby singling out particular choices of q such that, systematically, 4, 3, 2 or 1among all possible denominators Di vanishes. Then the system of equationsis solved iteratively1. First one determines all possible 4-point functions,then the 3-point functions and so on. For example, calling q±0 the twosolutions (in general complex) for whichD0 = D1 = D2 = D3 = 0 , (3)(there are 2 solutions because of the quadratic nature of the propagators)and since the functional form of d̃(q; 0123) is known, one directly finds thecoefficient of the box diagram containing the above 4 denominators throughthe two simple equationsN(q±0)=[d(0123) + d̃(q±0 ; 0123)]∏i6=0,1,2,3Di(q±0). (4)This algorithm also works in the case of complex denominators, namely withcomplex masses. Notice that the described procedure can be performed atthe amplitude level. One does not need to repeat the work for all Feyn-man diagrams, provided their sum is known: we just suppose to be able tocompute N(q) numerically.As is well known, even starting from a perfectly finite tensor integral, thetensor reduction may eventually lead to integrals that need to be regular-ized (we use dimensional regularization). Such tensors are finite, but tensorreduction iteratively leads to rank m m-point tensors with 1 ≤ m ≤ 5, thatare ultraviolet divergent when m ≤ 4. For this reason, we introduced, inEq. (1), the d-dimensional denominators D̄i, that differs by an amount q̃2from their 4-dimensional counterpartsD̄i = Di + q̃2 . (5)The result of this is a mismatch in the cancellation of the d-dimensionaldenominators of Eq. (1) with the 4-dimensional ones of Eq. (2). The rationalpart of the amplitude, called R1 [7], comes from such a lack of cancellation.1 An interesting method to optimize the solution of the system has been very recentlypresented in [6].1688 G. Ossola, C.G. Papadopoulos, R. PittauA different source of Rational Terms, called R2, can also be generated fromthe ǫ-dimensional part of N(q) (that is missing in Eq. (1). For the timebeing, it should be added by hand by looking at the analytical structure ofthe Feynman Diagrams or via a dedicated set of Feynman Rules. Exampleson how to compute R2 are reported in [7] and [8]. The Rational Terms R1are generated by the following extra integrals, introduced in [1]∫dnq̄q̃2D̄iD̄j= − iπ22[m2i + m2j −(pi − pj)23]+O(ǫ) ,∫dnq̄q̃2D̄iD̄jD̄k= − iπ22+O(ǫ) ,∫dnq̄q̃4D̄iD̄jD̄kD̄l= − iπ26+O(ǫ) .(6)The coefficients of the above integrals can be computed by looking at theimplicit mass dependence (namely reconstructing the q̃2 dependence) in thecoefficients d, c, b of the one-loop functions, once q̃2 is reintroduced throughthe mass shift m2i → m2i − q̃2. One getsb(ij; q̃2) = b(ij) + q̃2b(2)(ij) , c(ijk; q̃2) = c(ijk) + q̃2c(2)(ijk) . (7)Furthermore, by definingD(m)(q, q̃2) ≡m−1∑i0<i1<i2<i3[d(i0i1i2i3; q̃2)+d̃(q; i0i1i2i3; q̃2)]m−1∏i6=i0,i1,i2,i3D̄i , (8)the following expansion holdsD(m)(q, q̃2) =m∑j=2q̃(2j−4)d(2j−4)(q) , (9)where the last coefficient is independent on q, d(2m−4)(q) = d(2m−4). Inpractice, once the 4-dimensional coefficients have been determined, one canredo the fits for different values of q̃2, in order to determine b(2)(ij), c(2)(ijk)and d(2m−4). Such three quantities are the coefficients of the three extrascalar integrals listed in Eq. (6), respectively. Therefore, the OPP methodallows an easy and purely numerical computation of the Rational Terms oftype R1.3. pp → ZZZ at NLOThe calculation is composed of two parts: the evaluation of virtual cor-rections, namely one-loop contributions obtained by adding a virtual particleto the tree-order diagrams, and corrections from the real emission of one ad-ditional massless particle from initial and final states, which is necessary inReduction of One-Loop Amplitudes at the Integrand Level — NLO . . . 1689order to control and cancel infrared singularities. The virtual correctionsare computed using the OPP reduction [1]. In particular, we make use ofCutTools [9]. Concerning the contributions coming from real emission weused the dipole subtraction method [10] to isolate the soft and collinear di-vergences and checked the results using the phase space slicing method [11]with soft and collinear cutoffs, as outlined in [12].These results have been already obtained, following a very different ap-proach, by Lazopoulos et al. in Ref. [13]. We also presented some prelim-inary results in [14]. A more complete study, that also includes the caseof W+W−Z [15], W+ZZ, and W+W−W+ production, has been recentlypresented in [16].Let us begin with the evaluation of the virtual QCD corrections to theprocess qq̄ → ZZZ. We consider the processq(p1) + q̄(p2) −→ Z(p3) + Z(p4) + Z(p5) . (10)Fig. 1. Diagrams contributing to virtual QCD corrections to qq̄ → ZZZ.At the tree-level, there are six contributions to this process, obtained bypermuting the final legs in all possible ways. One-loop corrections are ob-tained by adding a virtual gluon to the tree-level structures. The calculationinvolves the reduction of 48 diagrams.We perform a reduction to scalar integrals using the OPP reduction meth-od [1]. The coefficients determined in this manner should be multipliedby the corresponding scalar integrals. Since, in the process that we arestudying, no q-dependent massive propagator appears, we will only needmassless scalar integrals. They are computed using the package OneLOopwritten by van Hameren [17].1690 G. Ossola, C.G. Papadopoulos, R. PittauWe turn now to the calculation of Rational Terms. As explained in Sec-tion 2, part of this contribution, that we call R1, is automatically includedby the reduction algorithm. The second term R2, coming from the explic-itly ǫ-dimensional part of the amplitude, has been added after having beencomputed separately; it turns out that only three- and two-point functionscontribute and the result is proportional to the tree-order amplitude.We checked that our results, agree with the results obtained by the au-thors of Ref. [13].In what concerns the real emission, we only have to deal with initialstate singularities, where we distinguish qq̄ and qg initial states. For the qginitial state, no soft singularity is present because the corresponding tree-level contribution vanishes. We recall that the structure of the NLO partoniccross sections is as follows:σNLOqq̄ =∫V V VdσBqq̄ + dσVqq̄ + dσCqq̄ +∫gdσAqq̄ +∫V V V g[dσRqq̄ − dσAqq̄],σNLOgq =∫V V V+dσCgq∫gdσAgq +∫V V V g[dσRgq − dσAgq], (11)where dσB , dσV , dσC , dσR, dσA are respectively the Born cross section, thevirtual, virtual counterterm, real and real-subtraction cross sections. Forthe qq̄ initial state two dipoles are needed as subtraction terms. If p6 is themomentum which can become soft or collinear, the dipole term for gluonemission off the quark is given byDq1g6,q̄2 = 8παsCF2x̃ p1 · p6(1 + x̃21 − x̃)|MBqq̄({p̃})|2 ,x̃ =p1 · p2 − p2 · p6 − p1 · p6p1 · p2, (12)where the {p̃} are redefined momenta, {p̃j} = {p̃16, p̃2, p̃3, p̃4, p̃5}, which areagain on-shell and go to {p1, . . . , p5} in the singular limit, e.g. p̃16 = x̃ p1.The regularised real emission part then readsdσRqq̄ − dσAqq̄ =161N12s12[CF∣∣MRqq̄({pj})∣∣2 −Dq1g6,q̄2 −Dq̄2g6,q1]dΦV V V g ,where the factor 1/6 accounts for the three identical bosons in the final state.More details can be found in [10, 16].The hadronic differential cross section with hadron momenta P1 andP2 is the sum over all partonic initial states convoluted with the partondistribution functionsReduction of One-Loop Amplitudes at the Integrand Level — NLO . . . 1691dσ(P1, P2) =∑ab∫dz1dz2fa(z1, µF)fb(z2, µF)dσab(z1P1, z2P2) , (13)where the sum runs over the partonic configurations qq̄, q̄q, gq, qg, gq̄, q̄g.As an explicit example we present the numerical results for the caseuū → ZZZ for √s = 14 TeV and using CTEQ6L1 [18]. Tree-order crosssection has been evaluated using the HELAC event generator [19]. The samegenerator has been appropriately developed and used for the calculation ofthe real corrections as outlined above. In the following table the results infb are presented for the tree-order cross section σ0, the ratio of the virtualto the tree-level cross section, and the real contribution, combining 5- and6-point contributions, as described above, for all channels, i.e., uū, ug, gū,for different values of the factorization(renomralization) scale (µ=µF =µR).Scale σ0 σV /σ0 σR σNLOµ = MZ 1.481(5) 0.536(1) 0.238(2) 2.512(2)µ = 2MZ 1.487(5) 0.481(1) 0.232(2) 2.434(2)µ = 3MZ 1.477(5) 0.452(1) 0.232(2) 2.376(2)µ = 4MZ 1.479(5) 0.436(1) 0.232(2) 2.355(2)µ = 5MZ 1.479(5) 0.424(1) 0.237(2) 2.343(2)Fig. 2. Scale dependence of the total cross section.1692 G. Ossola, C.G. Papadopoulos, R. PittauAs is evident from these results, the K−factor is quite sizeable (1.58–1.69),whereas the dependence on the scale µ (see also Fig. 2) is for both cases quiteweak, due mainly to the electroweak character of the process.4. ConclusionsIn conclusion we have presented, a reduction method at the integrand levelwhich is changing the way we are looking at the NLO calculations: a full numericalbut still algebraic method has been born.The efficiency of the OPP is quite good. The main future improvement is theefficiency with which the one-loop amplitude, at the integrand level, is computed.Finally, taking into account the speed, precision and easiness of the OPPmethod, a universal NLO calculator/event-generator seems feasible.REFERENCES[1] G. Ossola, C.G. Papadopoulos, R. Pittau, Nucl. Phys. B763, 147 (2007)[arXiv:hep-ph/0609007]; J. High Energy Phys. 0707, 085 (2007)[arXiv:0704.1271[hep-ph]].[2] F. del Aguila, R. Pittau, J. High Energy Phys. 0407, 017 (2004)[arXiv:hep-ph/0404120].[3] Z. 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Reduction of one-loop amplitudes at the integrand level - NLO QCD calculations

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Reduction of one-loop amplitudes at the integrand level - NLO QCD calculations

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