Complete two-loop QCD amplitudes for tW production at hadron colliders

Published for SISSA by Springer Received: May 27, 2023 Accepted: July 3, 2023 Published: July 11, 2023 Complete two-loop QCD amplitudes for tW production at hadron colliders a School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China b School of Physics, Shandong University, Jinan, Shandong 250100, China c Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China d School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China e Center for High Energy Physics, Peking University, Beijing 100871, China E-mail: , , , , , Abstract: We have calculated the complete two-loop QCD amplitudes for hadronic tW production by combining analytical and numerical techniques. The amplitudes have been first reduced to master integrals of eight planar and seven non-planar families, which can contain at most four massive propagators. Then a rational transformation of the master integrals is found to obtain a good basis so that the dimensional parameter decouples from the kinematic variables in the denominators of reduction coefficients. The master integrals are computed by solving their differential equations numerically. We find that the finite part of the two-loop squared amplitude is stable in the bulk of the phase space. After phase space integration and convolution with the parton distributions, it increases the LO cross section at the 13 TeV LHC by about 3%. Keywords: Higher-Order Perturbative Calculations, Top Quark ArXiv ePrint: 2212.07190 1 Corresponding author. c The Authors. Open Access, ⃝ Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2023)089 JHEP07(2023)089 Long-Bin Chen,a Liang Dong,b Hai Tao Li,b,1 Zhao Li,c,d,e Jian Wangb,1 and Yefan Wangb,1 Contents 1 2 Calculation method 3 3 Numerical results 5 4 Conclusion 9 1 Introduction The top quark, which was first discovered at the Fermilab Tevatron [1, 2], is the heaviest elementary particle in the Standard Model (SM), and is believed to play a fundamental role in the electroweak symmetry breaking [3]. The single top-quark production processes deserve detailed studies because they can be used to measure the W tb coupling structure, which may be modified by new physics [4]. Additionally, these processes are often considered as important backgrounds for many new physics searches. In this work, we focus on the tW associated production, which has the second-largest rate after the t-channel at the large hadron collider (LHC). The inclusive and differential cross sections for this process have been measured with an accuracy of 10% [5–8]. In order to provide precise theoretical predictions for the rate and kinematical distributions, higher-order quantum corrections are indispensable. The next-to-leading order (NLO) quantum chromodynamics (QCD) correction has been obtained for stable tW production [9–12]. The correction to the process including decays was calculated in [13]. The NLO QCD result was interfaced with parton shower within both the MC@NLO and POWHEG formalisms [14–16]. The NLO electroweak correction was computed in [17]. There have been some efforts devoted to the calculation of the corrections beyond the NLO in QCD. The approximate next-to-next-to-next-to-leading order total cross section was obtained by expanding the threshold resummation formula [18–21]. The corrections induced by soft gluons have been resummed to all orders in the strong coupling [22]. All these studies show that higher-order corrections are so significant that they should be taken into account in comparison to data at the LHC. However, the next-to-next-to-leading order (NNLO) QCD corrections are still missing for tW associated production, although the NNLO cross sections of the other two singletop production processes have been obtained in the last decade [23–28]. The N-jettiness soft function, one of the ingredients in a full NNLO QCD correction using the N-jettiness subtraction, has been calculated by two of the authors [29, 30]. Recently, we reported the analytical results of the leading color and light quark-loop contributions to the NNLO virtual corrections [31] using the two-loop master integrals obtained in [32–34]. Nevertheless, –1– JHEP07(2023)089 1 Introduction P1 P2 P3 P4 P5 P6 P7 P8 it is ideal to have the full two-loop virtual correction to provide precise predictions. To the best of our knowledge, tW associated production is the last 2 → 2 process in the SM that has not been computed at the two-loop level in QCD. We will address this gap in this work. This is made possible due to the impressive progress in both analytical and numerical methods for computing Feynman integrals [35, 36]. We calculate the two-loop corrections to the process g(k1 ) + b(k2 ) → W (k3 ) + t(k4 ) under the kinematical conditions k12 = k22 = 0, k32 = m2W and k42 = (k1 + k2 − k3 )2 = m2t . The Lorentz invariant Mandelstam variables are defined by s = (k1 + k2 )2 , t = (k1 − k3 )2 , u = (k2 − k3 )2 , (1.1) so that s + t + u = m2W + m2t . The amplitude is expanded in a series of αs as follows: M=  ∞  X αs i 4π i=0 M(i) , (1.2) where αs is the strong coupling. In this work, we do not keep the polarization information and focus solely on amplitude squared. We have presented the analytic result of the oneloop squared amplitude in [37]. In this work, we will calculate the interference between the two-loop and tree-level amplitudes. According to the color structure and fermion-loop contribution, it is decomposed as X   M(0)∗ M(2) + M(0) M(2)∗ = NC4 A + NC2 B + C + spin,color  +nl NC3 El + NC Fl + 1 D NC2 1 1 Gl + nh NC3 Eh + NC Fh + Gh NC NC    , (1.3) where NC is the number of quark colors, nl (nh ) is the total number of massless (massive) quark flavors. The coefficients of various color structures are denoted by A, B, C, D, E, F and G. –2– JHEP07(2023)089 Figure 1. Planar integral topologies for gb → tW . The black and red thick lines represent the top quark and the W boson, respectively. The black thin lines denote massless particles. NP1 NP2 NP5 NP4 NP3 NP6 NP7 2 Calculation method We generate the two-loop Feynman diagrams using FeynArts [38] and perform the Dirac algebra with the assistance of FeynCalc [39]. In our calculation, the anticommuting γ5 scheme proposed in [40] is implemented. The traces with two γ5 ’s are easy to evaluate, while the traces with one γ5 vanish because there are only three independent momenta in the tW production process. Furhter discussion on the implementation can be found in [37]. After the spin and polarization summation, all Lorentz indices are contracted, and the squared amplitude is expressed as a combination of a large number of scalar integrals. These integrals are reduced to a set of basis integrals, called master integrals, making use of the integration-by-part (IBP) identities [41, 42], which are automatically generated in FIRE [43]. It turns out that all (...truncated)