The status of the theoretical predictions for the top-anti top production in hadronic collisions is shortly reviewed, paying a articular attention to the analytic calculation of the two-loop QCD corrections to the parton-level matrix elements

Bonciani, R

Ferroglia, A

Gehrmann, T

von Manteuffel, A

ZORA

PoS(ICHEP 2010)098Two-Loop Corrections to Top-Antitop Production atHadron CollidersR. Bonciani∗Laboratoire de Physique Subatomique et de Cosmologie, Université JosephFourier/CNRS-IN2P3/INPG, F-38026 Grenoble, FranceE-mail: bonciani@lpsc.in2p3.frA. FerrogliaNew York City College of Technology 300 Jay Street, NY 11201 Brooklyn, USAEmail: AFerroglia@citytech.cuny.eduT. GehrmannInstitut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, SwitzerlandEmail: Thomas.Gehrmann@uzh.chA. von ManteuffelInstitut für Theoretische Physik, Universität Zürich, CH-8057 Zurich, SwitzerlandEmail: manteuffel@physik.uzh.chC. StuderusFakultät für Physik Universität Bielefeld, D-33501 Bielefeld, GermanyEmail: cedric@physik.uni-bielefeld.deThe status of the theoretical predictions for the top-anti top production in hadronic collisions isshortly reviewed, paying a particular attention to the analytic calculation of the two-loop QCDcorrections to the parton-level matrix elements.35th International Conference of High Energy Physics - ICHEP2010,July 22-28, 2010Paris France∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(ICHEP 2010)098Two-Loop Corrections to Top-Antitop Production at Hadron Colliders R. BoncianiSince its discovery in 1995, the top quark properties were extensively studied at the FermilabTevatron. In the last 15 years, many observables concerning top-quark physics were measuredwith remarkable accuracy. Among others, the t ¯t total cross section, σt¯t , was measured with anaccuracy of ∆σt¯t/σt¯t ∼ 9%. Data coming from the LHC are expected to improve significantly themeasurement of several observables related to the top quark. Already within a couple of yearsof data taking in the low-luminosity low-energy phase (L ∼ 100pb−1/year at 7TeV of center ofmass energy), tens of thousands t ¯t events before selection will be available. Consequently, alreadyin this first phase, the accuracy of the cross section measurement is supposed to match the onereached to date at the Tevatron. In the high-luminosity phase ( L ∼ 100fb−1/year at 14TeV ofcenter of mass energy) it will be possible to reach an accuracy ∆σt¯t/σt¯t ∼ 5% [1]. This accuracyin the experimental measurements motivates theorists to refine the existing predictions, both forthe total top-quark pair production cross section and for the related differential distributions. Inthe following, we will briefly outline the research program aiming to the full calculation of thenext-to-next-to-leading order (NNLO) corrections to top-pair production cross section.The full NLO QCD corrections to the total cross section were calculated in [2] in the case of“stable” on-shell top-quarks. In [3] several differential distributions were calculated at the sameaccuracy level. In [4] the NLO corrections to the top-pair production were evaluated by takinginto account the top-quark decay in narrow-width approximation. The resummation of soft-gluonenhanced terms near the t ¯t production threshold is implemented at the leading [5], next-to-leading[6], and next-to-next-to-leading [7] logarithmic accuracy. Approximate NNLO formulas for thetotal cross section were recently obtained by several groups [8].The calculation of the corrections to the partonic cross section beyond leading order can besplit in the calculation of the real corrections, in which the final state includes extra partons inaddition to the top-quark pair, and in the calculation of virtual (loop) corrections to the partonicprocesses already present at the tree level. A complete calculation of the NNLO corrections re-quires the knowledge of the two-loop matrix elements for the processes qq¯→ t ¯t (quark-annihilationchannel) and gg→ t ¯t (gluon-fusion channel), as well as the 2→ 3 matrix elements at the one-looplevel and the 2→ 4 matrix elements at the tree-level [9]. Moreover, in order to be able to deal withIR singularities, a NNLO subtraction method has to be implemented [10].From the technical point of view, the calculation of the two-loop (virtual) corrections is par-ticularly challenging. The squared matrix element for the processes q(p1)+q(p2)→ t(p3)+ t(p4)and g(p1)+g(p2)→ t(p3)+ t(p4), summed over spin and color, can be expanded in powers of thestrong coupling constant αS as follows:|M |2(s, t,m,ε) = 16pi2α2S[A0 +(αspi)A1 +(αspi)2A2 +O(α3s)]. (1)where p2i = 0 for i = 1,2 and p2j = −m2t for i = 3,4. The Mandelstam variables are defined inthe usual way: s = −(p1 + p2)2, t =−(p1− p3)2, u = −(p1− p4)2. Conservation of momentumimplies that s+ t + u = 2m2t .The tree-level term A0 in the r. h. s. of Eq. (1) is well known in both production channels. TheO(αS) term A1 arises from the interference of one-loop diagrams with the tree-level amplitude.The O(α2S ) term A2 consists of two parts: the interference of two-loop diagrams with the Bornamplitude and the interference of one-loop diagrams among themselves, A2 = A (2×0)2 +A(1×1)2 .2PoS(ICHEP 2010)098Two-Loop Corrections to Top-Antitop Production at Hadron Colliders R. BoncianiThe latter term, A (1×1)2 , was calculated for both channels in [11]. The first term, A(2×0)2 , originat-ing from the two-loop diagrams, can be decomposed according to the color and flavor structures asfollows:A(2×0)qq¯2 = (N2c−1)[N2c Aqq¯+Bqq¯+Cqq¯N2c+NcNlDqq¯l +NlNcEqq¯l +NcNhDqq¯h +NhNcEqq¯h +N2l Fqq¯l+NlNhFqq¯lh +N2hFqq¯h],A(2×0)gg2 = (N2c−1)[N3c Agg + NcBgg +1NcCgg + 1N3cDgg + N2c NlEggl + N2c NhEggh + NlFggl + NhFggh+NlN2cGggl +NhN2cGggh +Nc(N2l Hggl +N2h Hggh +NlNhHgglh)+N2lNcIggl +N2hNcIggh +NlNhNcIgglh], (2)where Nh and Nl are the number of heavy-quark flavors (in our case, only the top quark is consid-ered heavy) and light-quark flavors, respectively. The coefficients A,B, . . . , Ilh in both channels arefunctions of s, t, and mt , as well as of the dimensional regulator ε . These quantities were calcu-lated in [12] in the approximation s, |t|, |u| ≫ m2t . However, the results in [12] are not sufficient toobtain accurate NNLO predictions, since a large fraction of the events are characterised by valuesof the partonic center of mass energy which do not satisfy the ultra-relativistic limit. The com-plete top mass dependence of the color coefficients A,B, . . . is required. A numerical calculation ofA(2×0)qq¯2 , exact in s, t, and mt , was presented in [13]. Analytic expressions for all of the IR polesin A (2×0)qq¯2 and A(2×0)gg2 are also available [15]; they were calculated by employing the expres-sion of the IR divergences in a generic two-loop QCD amplitude with both massive and masslessparticles, derived in [14].The analytic calculation of the coefficients Aqq¯, Dqq¯l , ...,Fqq¯h which appear in the quark anni-hilation channel, as well as of the coefficient Agg for the gluon fusion channel, was carried out in[16]. It must be observed that the coefficients with a l or h subscript receive contributions onlyfrom diagrams involving a closed light or heavy fermion loop, respectively. The leading color co-efficients in the two production channels, Aqq¯ and Agg, involve planar diagrams only. The resultsreported in [16] were obtained by employing the Laporta Algorithm [17], implemented in the C++code REDUZE 2 [18, 19], for the reduction to the Master Integrals [20, 16]. Subsequently, themaster integrals were evaluated by means of the differential equation method [21]. The analyticexpression of the master integrals can be written in terms of one- and two-dimensional harmonicpolylogarithms [22]. The analytic results were evaluated numerically by employing codes whichmake use of a GiNaC package for the evaluation of generalized polylogarithms [23]. In [16] theanalytic results were also expanded in the s ≫ m2t limit, in order to reproduce the results alreadyobtained in [12]. Starting from the exact results of [16], it was also possible to obtain new analyticformulas which are valid in the production threshold limit s→ 4m2t .A complete numerical result for the two-loop corrections in the gluon fusion channel is stillmissing, and to date only the coefficient Agg was evaluated. Among the remaining color coefficientsin Eq. (2), Eggl , Fggl , Gggl , Hggl , Hggh , Hgglh , Iggl , Iggh , Igglh can be calculated using the same techniquealready employed in [16]; their evaluation is in progress [24]. The remaining color coefficients bothin the quark-antiquark channel (Bqq¯ and Cqq¯) and in the gluon fusion channel (Bgg, Cgg, Dgg, Eggh ,Fggh , and Gggh ) contain either crossed box topologies, or complicated massive sub-topologies. In the3PoS(ICHEP 2010)098Two-Loop Corrections to Top-Antitop Production at Hadron Colliders R. Boncianifirst case, the calculation of the color coefficients in a closed functional form using the differentialequation method is very difficult, because of the large number of master integrals that occurs forsome specific topologies. In the second case, the problems arise from the fact that it is known thatalready the sunrise diagram with three virtual propagators of mass m and an external momentum psuch that p2 6= m2 can be expressed analytically only in terms of elliptic integrals [25]. This three-denominator diagram appears as a sub-topology in the color coefficients with subscript h. A viablesolution for both issues is the semi-numerical approach adopted, for instance, for the two-loopequal-mass sunrise diagram [26].To conclude, the calculation of the two-loop corrections to the top-quark pair production is anessential step needed for the full evaluation of the NNLO corrections to the top-quark productioncross section and differential distributions. In the last few years several results were obtained forthe quark annihilation channel. The calculation of the two-loop corrections in the gluon fusionchannel is technically more complicated because of the larger number of diagrams involved andbecause the functional basis of the harmonic polylogarithms is known to be insufficient to obtaina full analytic result. However, the calculation of a large number of the color coefficients can becarried out with available methods and is in progress. The calculation of the color coefficientsfor which standard analytic techniques are insufficient can in principle be carried out with semi-analytic methods already applied to other related problems.Acknowledgments. The work of R. B. is supported by the Theory-LHC-France initiative ofCNRS/IN2P3, T. G. and A. v.M. are supported by the Schweizer Nationalfonds (grants 200020-126691 and 200020-124773). The work of C. S. was supported by the Deutsche Forschungsge-meinschaft (DFG SCHR 993/2-1).References[1] W. Bernreuther, J. Phys. G 35 (2008) 083001; arXiv:1008.3819; R. Frederix, arXiv:1009.6199.[2] P. Nason et al. Nucl. Phys. B 303 (1988) 607; W. Beenakker et al. Phys. Rev. D 40 (1989) 54; Nucl.Phys. B 351 (1991) 507; M. Czakon and A. Mitov, Nucl. Phys. B 824 (2010) 111.[3] P. Nason et al. Nucl. Phys. B 327 (1989) 49 [Erratum-ibid. B 335 (1990) 260]; M. L. Mangano et al.Nucl. Phys. B 373 (1992) 295; S. Frixione et al. Phys. Lett. B351 (1995) 555.[4] W. Bernreuther et al. Nucl. Phys. B 690 (2004) 81; K. Melnikov and M. Schulze, JHEP 0908 (2009)049; W. Bernreuther and Z. G. Si, Nucl. Phys. B 837 (2010) 90.[5] E. Laenen et al. Nucl. Phys. B 369 (1992) 543; Phys. Lett. B 321 (1994) 254; E. L. Berger andH. Contopanagos, Phys. Lett. B 361 (1995) 115; Phys. Rev. D 54 (1996) 3085; Phys. Rev. D 57(1998) 253; S. Catani et al. Phys. Lett. B 378 (1996) 329; Nucl. Phys. B 478 (1996) 273.[6] N. Kidonakis and G. Sterman, Phys. Lett. B 387 (1996) 867; Nucl. Phys. B 505 (1997) 321;R. Bonciani et al. Nucl. Phys. B 529 (1998) 424 [Erratum-ibid. B 803 (2008) 234]; Phys. Lett. B 575(2003) 268.[7] M. Beneke et al. Nucl. Phys. B 828 (2010) 69; M. Czakon et al. Phys. Rev. D 80 (2009) 074017;V. Ahrens et al. JHEP 1009 (2010) 097; arXiv:1006.4682. N. Kidonakis, arXiv:1009.4935.[8] U. Langenfeld et al. Phys. Rev. D 80 (2009) 054009; M. Cacciari et al. JHEP 0809 (2008) 127;N. Kidonakis and R. Vogt, Phys. Rev. D 78 (2008) 074005.4PoS(ICHEP 2010)098Two-Loop Corrections to Top-Antitop Production at Hadron Colliders R. Bonciani[9] S. Dittmaier et al. Phys. Rev. Lett. 98 (2007) 262002; Eur. Phys. J. C 59 (2009) 625; G. Bevilacqua etal. Phys. Rev. Lett. 104 (2010) 162002; K. Melnikov, M. Schulze, Nucl. Phys. B840 (2010) 129.[10] D.A. Kosower, Phys. Rev. D 67 (2003) 116003. A. Daleo, T. Gehrmann and D. Maître, JHEP 0704(2007) 016; A. Gehrmann-De Ridder and M. Ritzmann, JHEP 0907 (2009) 041; A. Gehrmann-DeRidder et al. JHEP 0509 (2005) 056; E. W. N. Glover and J. Pires, JHEP 1006 (2010) 096;R. Boughezal et al. PoS RADCOR2009 (2010) 052; S. Catani and M. Grazzini, Phys. Rev. Lett. 98(2007) 222002; M. Czakon, Phys. Lett. B693 (2010) 259; C. Anastasiou, F. Herzog andA. Lazopoulos, arXiv:1011.4867 [hep-ph].[11] J. G. Korner et al. Phys. Rev. D 77 (2008) 094011; C. Anastasiou and S. M. Aybat, Phys. Rev. D 78(2008) 114006; B. Kniehl et al. Phys. Rev. D 78 (2008) 094013.[12] M. Czakon et al. Phys. Lett. B 651 (2007) 147; Nucl. Phys. B 798 (2008) 210.[13] M. Czakon, Phys. Lett. B 664 (2008) 307.[14] T. Becher and M. Neubert, Phys. Rev. Lett. 102 (2009) 162001; JHEP 0906 (2009) 081; Phys. Rev. D79 (2009) 125004; A. Mitov et al. Phys. Rev. D 79 (2009) 094015; A. Ferroglia et al. Phys. Rev. Lett.103 (2009) 201601; E. Gardi and L. Magnea, Nuovo Cim. C 32N5-6 (2009) 137;[15] A. Ferroglia et al. JHEP 0911 (2009) 062.[16] R. Bonciani et al. JHEP 0807 (2008) 129; R. Bonciani et al. JHEP 0908 (2009) 067; R. Bonciani etal. arXiv:1011.6661 [hep-ph].[17] S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087; F.V. Tkachov, Phys. Lett. B 100 (1981) 65;K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B 192 (1981) 159.[18] A. von Manteuffel and C. Studerus, Reduze 2, to be published.[19] C. Studerus, Comput. Phys. Commun. 181 (2010) 1293.[20] M. Argeri et al. Nucl. Phys. B 631 (2002) 388; R. Bonciani et al. Nucl. Phys. B 661 (2003) 289[Erratum-ibid. B 702 (2004) 359]; Nucl. Phys. B 690 (2004) 138; Nucl. Phys. B 676 (2004) 399;J. Fleischer et al. Nucl. Phys. B 547 (1999) 343; U. Aglietti and R. Bonciani, Nucl. Phys. B 668(2003) 3; Nucl. Phys. B 698 (2004) 277; A.I. Davydychev and M.Y. Kalmykov, Nucl. Phys. B 699(2004) 3. M. Czakon et al. Phys. Rev. D 71 (2005) 073009; G. Bell, arXiv:0705.3133; R. Boncianiand A. Ferroglia, JHEP 0811 (2008) 065.[21] A.V. Kotikov, Phys. Lett. B 254 (1991) 158; Phys. Lett. B 259 (1991) 314; Phys. Lett. B 267 (1991)123; E. Remiddi, Nuovo Cim. A 110 (1997) 1435.[22] A B. Goncharov, Math. Res. Lett. 5 (1998), 497; E. Remiddi and J.A.M. Vermaseren, Int. J. Mod.Phys. A 15 (2000) 725; T. Gehrmann and E. Remiddi, Nucl. Phys. B 601 (2001) 248; Nucl. Phys. B601 (2001) 287; Comput. Phys. Commun. 141 (2001) 296; Comput. Phys. Commun. 144 (2002) 200;D. Maître, Comput. Phys. Commun. 174 (2006) 222; hep-ph/0703052.[23] J. Vollinga and S. Weinzierl, Comput. Phys. Commun. 167 (2005) 177.[24] R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel, and C. Studerus, in preparation.[25] S. Laporta, E. Remiddi, Nucl. Phys. B704, 349-386 (2005).[26] S. Pozzorini and E. Remiddi, Comput. Phys. Commun. 175 (2006) 381; U. Aglietti et al. Nucl. Phys.B 789 (2008) 45.5

Two-loop corrections to top-antitop production at hadron colliders

http://www.zora.uzh.ch/41371/1/ICHEP_2010_098.pdf

Two-loop corrections to top-antitop production at hadron colliders

Abstract

Similar works

Full text

Available Versions

ZORA