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The two-loop QCD correction to massive spin-2 resonance \( \rightarrow q \bar{q} g \)

  • Taushif Ahmed
  • Goutam DasEmail author
  • Prakash Mathews
  • Narayan Rana
  • V. Ravindran
Open Access
Regular Article - Theoretical Physics

Abstract

The two-loop QCD correction to massive spin-2 graviton decaying to \(q + \bar{q} + g\) is presented considering a generic universal spin-2 coupling to the SM through the conserved energy-momentum tensor. Such a massive spin-2 particle can arise in extra-dimensional models. The ultraviolet and infrared structure of the QCD amplitudes are studied. In dimensional regularization, the infrared pole structure is in agreement with Catani’s proposal, confirming the universal factorization property of QCD amplitudes, even with the spin-2 tensorial coupling.

1 Introduction

After the discovery of the Higgs boson one of the main motivation of large hadron collider (LHC) is to search for new physics beyond the standard model (BSM). It is well understood that the standard model (SM) is not complete on many accounts. One of the main motivation to look for BSM physics is the large scale hierarchy between the Electroweak scale and the Planck scale, which is commonly known as the hierarchy problem. There are a plethora of models to address this issue. The models with spin-2 particles in brane world scenarios are particularly interesting. Recently there has been a renewed interest in search of spin-2 particle in the context of the Higgs boson discovery. Of these models, theories with large extra dimension like Arkani-Hamed–Dimopoulos–Dvali (ADD) [1, 2, 3] and warped extra dimension like Randall–Sundrum (RS) [4] have gained a lot of attention as their signature can be tested at LHC energies. In fact, dedicated groups at the LHC are studying the search potentials of such exotic particles. In the simplest of these models gravity is allowed to explore the full space-time dimensions. The effect of gravity on the 3-brane where the SM resides is realized by a tower of spin-2 Kaluza–Klein (KK) excitations. When coupled to the SM, these KK modes may provide important signatures about these models to be ultimately tested by the LHC. The channels like \(\ell ^{+}\ell ^{-}, \gamma \gamma , ZZ, W^{+}W^{-}\), and di-jet are the main channels to look for any deviations from the SM as a result of the virtual spin-2 exchange. At hadron colliders, QCD corrections to these processes are necessary to make precise quantitative predictions and to have control over the theoretical uncertainties. To next-to-leading-order (NLO) accuracy, these processes have been studied in the context of ADD [5, 6, 7, 8, 9, 10] and RS [11, 12, 13, 14, 15]. The effect of parton shower (PS) has also been considered [16, 17, 18]. Moreover, the signature can also be found in tri-gauge boson productions [19, 20, 21, 22, 23]. Recently all the important processes are fully automatized [24] in MadGraph5aMC framework [25] with NLO+PS accuracy for a generic spin-2 particle. Nevertheless the NLO corrections are often very large, resulting in large K-factors and large scale uncertainties. To have precise predictions the next stage is to compute full next-to-next-to-leading-order (NNLO) correction. The higher-order correction involves pure virtual contributions, pure real corrections and appropriate interference terms at each order. For the spin-2 case, the quark and gluon form factors have been calculated at two-loop level [26] and at three-loop level [27] in peturbative QCD (pQCD). An attempt to consider the soft and collinear contribution at NNLO accuracy was made [28] in the context of spin-2. This reduces the unphysical scale uncertainties, thereby improving the predictions. Recently the full NNLO correction has been studied [29] for Drell–Yan process with spin-2 mediators in the context of ADD.

The processes with missing energy associated with a SM particle can also give significant information as regards new physics. Particularly, jet \(+\) missing energy is one important process studied at the LHC to look for BSM signature like dark matter in simplified models (see for example [30] and the references therein). A massive spin-2 particle which goes undetected could be an important dark matter candidate. The NLO correction for this process in large extra dimension has been considered in [31] and it is shown that the QCD corrections for this process could be as large as 50% and suffer from large scale uncertainties. Therefore a full NNLO correction is important for such process to have accurate prediction of cross-section and distributions and also to have the scale uncertainties under control. An attempt has been made in this direction in [32], where the virtual NNLO QCD correction has been studied for massive spin-2 \(\rightarrow g+g+g\). Although the gg initiated sub-process is the dominant contribution at the LHC, as the perturbative order increases the other sub-processes like \(q\bar{q}, q(\bar{q}) g \) begin to contribute significantly. In fact for full NNLO correction one needs to have the virtual contributions from all the sub-processes, the real-virtual piece and the pure real corrections. In this article we have considered the two-loop virtual QCD correction to the process massive spin-2 \(\rightarrow q + \bar{q} + g\). After appropriate analytical continuation of the kinematical variables to the respective regions [33], the result in this paper can be used for other scattering sub-processes viz. \(q + \bar{q} \rightarrow G + g\) and \(q(\bar{q})+g \rightarrow G +q (\bar{q})\), where G denotes the spin-2 field. This computation along with the \(G\rightarrow g+g+g\) [32] to two-loop now completes the genuine two-loop QCD corrections (i.e. two-loop diagrams at the amplitude level) to the production of spin-2\(+\)jet at a hadron collider.

Our main motivation for this work is two-fold; first to probe the structure of quantum field theory in the presence of a spin-2 field, to check the universality of infrared (IR) pole structure in QCD [34]. The correct IR pole structure has been already realized in the case of spin-2 \(\rightarrow g+g+g\) [32]. Here we demonstrate the same for spin-2 \(\rightarrow q+\bar{q}+g\). Second, we present one of the important ingredients for full two-loop QCD correction for real graviton production associated with a jet. We consider a minimal universal coupling of spin-2 to the SM through the energy-momentum tensor. Due to the conserved spin-2 current, no additional ultraviolet (UV) renormalization is needed other than the QCD one. We encounter more than 800 Feynman diagrams with higher tensorial integrals. The rank-2 nature of spin-2 increases the complexity of our calculation. Using integration-by-parts (IBP) identities [35, 36] and Lorentz invariance (LI) identities [37], we are able to reduce all the scalar integrals to a smaller set of Master Integrals (MIs). These MIs are available in [37, 38, 39, 40, 41, 42]. Finally we observe the universality of infrared factorization of QCD amplitudes as predicted by Catani [34] (see also [43]).

The paper is organized as follows: in Sect. 2, we discuss the theoretical framework for our work, particularly the spin-2 action of interaction with the SM, the notation used, the procedure of UV renormalization and infrared factorization. In Sect. 3, we discuss our approach to the computation. Section 4 and the appendix are devoted to a presentation of the results. We conclude in Sect. 5.

2 Theoretical framework

We consider a generic spin-2 particle minimally coupled to the SM fields through the conserved SM energy-momentum tensor. The effective action which describes a spin-2 particle (\(G^{\mu \nu }(x)\)) interacting with colored particles is given by [1, 2, 3, 4, 44, 45]
$$\begin{aligned} \mathcal {S}_\mathrm{int} = -\frac{\kappa }{2} ~ \int \mathrm{d}^{4}x ~ T_{\mu \nu }^\mathrm{QCD}(x) ~ G^{\mu \nu }(x), \end{aligned}$$
(2.1)
where \(\kappa \) is a dimensionful universal coupling which determines the strength of graviton coupling to the SM. \(T_{\mu \nu }^\mathrm{QCD}\) is given by
$$\begin{aligned} T_{\mu \nu }^\mathrm{QCD}&= -g_{\mu \nu } \mathcal {L}_\mathrm{QCD} - F_{\mu \rho }^{a} F^{a\rho }_{\nu } - \frac{1}{\xi }g_{\mu \nu }\partial ^{\rho }(A_{\rho }^{a}\partial ^{\sigma }A_{\sigma }^a)\nonumber \\&\quad + \frac{1}{\xi } (A_{\nu }^{a}\partial _{\mu }(\partial ^{\sigma } A_{\sigma }^{a}) + A_{\mu }^{a}\partial _{\nu }(\partial ^{\sigma } A_{\sigma }^{a})) \nonumber \\&\quad +\frac{i}{4} \big [\bar{\psi }\gamma _{\mu }(\partial _{\nu } -i g_{s} T^{a} A_{\nu }^{a})\psi -\bar{\psi }(\partial _{\nu } \nonumber \\&\quad + i g_{s} T^{a} A_{\nu }^{a})\gamma _{\mu }\psi + (\mu \leftrightarrow \nu )\big ] \nonumber \\&\quad + \big [\partial _{\mu }\bar{\omega }^{a}(\partial _{\nu }\omega ^{a} - g_{s} f^{abc} A_{\nu }^{c}\omega ^{b}) + (\mu \leftrightarrow \nu )\big ], \end{aligned}$$
(2.2)
where \(g_{s}\) is the strong coupling constant and \(\xi \) is gauge fixing parameter. Here \(\mathcal {L}_\mathrm{QCD}\) is the QCD Lagrangian, given byHere \(\omega ^{a}\) is the ghost field introduced in order to cancel the unphysical degrees of freedom associated with the gluon fields \((A_{\mu }^{a})\). \(T^{a}\) and \(f^{abc}\) represent the generator and structure constants of SU(N) gauge group, respectively. Throughout the computation, we consider SU(N) as our gauge group and the QCD corresponds to \(N=3\).
The decay process considered is
$$\begin{aligned} G(Q) \rightarrow q(p_{1}) + \bar{q}(p_{2}) + g(p_{3}). \end{aligned}$$
(2.4)
The corresponding Mandelstam variables for this process are defined as
$$\begin{aligned} s \equiv (p_{1}+p_{2})^2, \qquad t \equiv (p_{2}+p_{3})^2, \qquad u \equiv (p_{3}+p_{1})^2. \end{aligned}$$
(2.5)
They satisfy the following relation:
$$\begin{aligned} s + t + u = M_{G}^{2} \equiv Q^{2}. \end{aligned}$$
(2.6)
Here \(M_{G}\) is the mass of the spin-2 field. The following dimensionless invariants which appear in the argument of harmonic polylogarithms (HPL) [46] and two-dimensional HPLs [41] are also defined:
$$\begin{aligned} x \equiv s/Q^2, \qquad y \equiv u/Q^2, \qquad z \equiv t/Q^2. \end{aligned}$$
(2.7)
Accordingly, Eq. (2.6) becomes
$$\begin{aligned} x+y+z = 1. \end{aligned}$$
(2.8)

2.1 Ultraviolet renormalization

Beyond leading order in perturbation theory, the on-shell QCD amplitudes develop both ultraviolet and infrared divergences. The spin-2 coupling to the SM particles \(\kappa \) is free from such ultraviolet renormalization, which is due to the fact that spin-2 couples universally to the SM through conserved current. So the only UV renormalization required is for the strong coupling constant \(\hat{g}_{s}\). Before performing the renormalization, we need to regularize the theory in order to identify the true nature of the divergences. We regularize the theory under dimensional regularization where the space-time dimension is chosen to be \(d=4+\epsilon \). Expanding the scattering amplitude in powers of \(\hat{a}_{s}=\hat{g}_{s}^{2}/16\pi ^2\), the matrix element (ME) is given by
$$\begin{aligned} {\mid {\mathcal {M}} \rangle }= & {} \left( \frac{\hat{a}_{s}}{\mu _{0}^{\epsilon }} S_{\epsilon }\right) ^{\frac{1}{2}} \bigg ( {\mid {\hat{\mathcal {M}}^{(0)}} \rangle } + \left( \frac{\hat{a}_{s}}{\mu _{0}^{\epsilon }} S_{\epsilon }\right) {\mid {\hat{\mathcal {M}}^{(1)}} \rangle } \nonumber \\&+ \left( \frac{\hat{a}_{s}}{\mu _{0}^{\epsilon }} S_{\epsilon }\right) ^{2} {\mid {\hat{\mathcal {M}}^{(2)}} \rangle } + \mathcal {O}(\hat{a}_{s}^{3}) \bigg ) \end{aligned}$$
(2.9)
where \(\hat{g}_{s}\) is the unrenormalized strong coupling constant, \(S_{\epsilon } = \text {exp}[~\frac{\epsilon }{2} ( \gamma _{E} - \text {ln}~4\pi )]\) and \(\gamma _{E} = 0.5772 ~.~.~.\) is the Euler constant. \({\mid {\hat{\mathcal {M}}^{(i)}} \rangle }\) is the unrenormalized color-space vector representing the ith-loop amplitude. \(\mu _{0}\) is a mass scale introduced to make the strong coupling constant \((\hat{g}_{s})\) dimensionless in d dimensions. We work within the \(\overline{\text {MS}}\) scheme for performing the UV renormalization, in which the renormalized coupling constant \(a_{s}\equiv a_{s}(\mu _R^2)\) is defined at the renormalization scale \(\mu _R\) and is related to the unrenormalized \(\hat{a}_{s}\) by
$$\begin{aligned} \frac{\hat{a}_{s}}{\mu _{0}^{\epsilon }} ~ S_{\epsilon }&= \frac{a_{s}}{\mu _{R}^{\epsilon }} ~ Z(\mu _{R}^{2}),\nonumber \\&= \frac{a_{s}}{\mu _{R}^{\epsilon }} \bigg [ 1 + a_{s} \frac{2 \beta _{0}}{\epsilon } + a_{s}^{2} ~\bigg ( \frac{4 \beta _{0}^{2}}{\epsilon } + \frac{\beta _{1}}{\epsilon }\bigg ) + \mathcal {O}(a_{s}^{3}) \bigg ], \end{aligned}$$
(2.10)
where
$$\begin{aligned} \beta _0= & {} \left( \frac{11}{3} C_{A} - \frac{4}{3} T_{F}~ n_{f} \right) ,\nonumber \\ \beta _1= & {} \left( \frac{34}{3} C_{A}^{2} - \frac{20}{3} C_{A} T_{F}~ n_{f} - 4 C_{F} T_{F}~ n_{f} \right) . \end{aligned}$$
(2.11)
Here \(C_{A} = N\) and \(C_{F} = (N^{2} -1)/2N\) are the quadratic Casimir of the SU(N) group. \(T_{F} = 1/2 \) and \(n_{f}\) is the number of light active quark flavors.
The matrix element can also be expressed as a power series of renormalized strong coupling constant with UV finite matrix elements \({\mid {\mathcal {M}^{(i)}} \rangle }\),
$$\begin{aligned} {\mid {\mathcal {M}} \rangle } = (a_{s})^{\frac{1}{2}} ~ \big ( {\mid {\mathcal {M}^{(0)}} \rangle } + a_{s} {\mid {\mathcal {M}^{(1)}} \rangle } + a_{s}^2 {\mid {\mathcal {M}^{(2)}} \rangle } + \mathcal {O}(a_{s}^3) \big ) \end{aligned}$$
(2.12)
where
$$\begin{aligned} {\mid {\mathcal {M}^{(0)}} \rangle }&= \left( \frac{1}{\mu _{R}^{\epsilon }} \right) ^{\frac{1}{2}} ~ {\mid {\hat{\mathcal {M}}^{(0)}} \rangle }, \nonumber \\ {\mid {\mathcal {M}^{(1)}} \rangle }&= \left( \frac{1}{\mu _{R}^{\epsilon }} \right) ^{\frac{3}{2}} ~ \left[ {\mid {\hat{\mathcal {M}}^{(1)}} \rangle } + \mu _{R}^{\epsilon } \frac{r_{_{1}}}{2} ~ {\mid {\hat{\mathcal {M}}^{(0)}} \rangle } \right] , \nonumber \\ {\mid {\mathcal {M}^{(2)}} \rangle }&= \left( \frac{1}{\mu _{R}^{\epsilon }} \right) ^{\frac{5}{2}} ~ \bigg [ {\mid {\hat{\mathcal {M}}^{(2)}} \rangle } + \mu _{R}^{\epsilon } \frac{3 r_{_{1}}}{2} ~ {\mid {\hat{\mathcal {M}}^{(1)}} \rangle }\nonumber \\&\quad + \mu _{R}^{2\epsilon } ~ \bigg ( \frac{r_{_{2}}}{2} - \frac{r_{_{1}}^2}{8} \bigg ) ~ {\mid {\hat{\mathcal {M}}^{(0)}} \rangle } \bigg ] \end{aligned}$$
(2.13)
with
$$\begin{aligned} r_{_{1}} = \frac{2\beta _0}{\epsilon },\qquad r_{_{2}} = \left( \frac{4\beta _0^2}{\epsilon ^2} + \frac{\beta _1}{\epsilon } \right) \end{aligned}$$
(2.14)

2.2 Infrared factorization

In a higher-order calculation, the UV renormalized matrix elements contain singularities of infrared origin. Generally two kinds of singularities arise – the soft and collinear while working with massless QCD. According to the KLN theorem, these singularities get canceled against the similar contribution from real emission Feynman diagrams, resulting in infrared safe observables. The IR divergences have a universal structure in dimensional regularization, which was predicted in [34] to two loops, except for the two-loop single pole in \(\epsilon \). In [43, 47, 48] the infrared structure of scattering amplitudes are studied and connection of the single pole to soft anomalous dimension matrix is predicted. The factorization of the single pole in quark and gluon form factors in terms of soft and collinear anomalous dimensions was demonstrated to two-loop level [49] whose validity at three loops was later established in [50]. The proposal by Catani was generalized beyond two loops in [51, 52].

According to Catani’s prediction [34], the renormalized amplitude factorizes in dimensional regularization. The ME at a given order \({\mid {\mathcal {M}^{(i)}} \rangle }\) can be expressed as the sum of the lower-order MEs times appropriate insertion operators (\(\mathbf {I}_{q}^{(i)}(\epsilon )\)) and a finite piece \({\mid {\mathcal {M}^{(i) \mathrm{fin}}} \rangle }\). These insertion operators contain the infrared pole structures, which are universal. For the present case, we have two external massless quarks and a gluon, for which the one-loop and the two-loop ME can be written in the following form:
$$\begin{aligned} {\mid {\mathcal {M}^{(1)}} \rangle }&= 2 ~\mathbf {I}_{q}^{(1)}(\epsilon ) ~{\mid {\mathcal {M}^{(0)}} \rangle } + {\mid {\mathcal {M}^{(1) \mathrm{fin}}} \rangle }, \nonumber \\ {\mid {\mathcal {M}^{(2)}} \rangle }&= 2 ~\mathbf {I}_{q}^{(1)}(\epsilon ) ~{\mid {\mathcal {M}^{(1)}} \rangle } + 4 ~\mathbf {I}_{q}^{(2)}(\epsilon ) ~{\mid {\mathcal {M}^{(0)}} \rangle } + {\mid {\mathcal {M}^{(2) \mathrm{fin}}} \rangle }, \end{aligned}$$
(2.15)
where the one-loop and two-loop insertion operators are given by
$$\begin{aligned} \mathbf {I}_{q}^{(1)}(\epsilon )&= \frac{1}{2} ~ \frac{e^{-\frac{\epsilon }{2} \gamma _{E}}}{\Gamma (1+\frac{\epsilon }{2})} \bigg \{ \bigg ( \frac{4}{\epsilon ^2} - \frac{3}{\epsilon } \bigg )(C_A - 2 C_F) \bigg [ \bigg ( -\frac{s}{\mu _{R}^2} \bigg )^{\frac{\epsilon }{2}} \bigg ] \nonumber \\&\quad + \bigg (-\frac{4 C_A}{\epsilon ^2} + \frac{3 C_A}{2\epsilon } + \frac{\beta _0}{2\epsilon }\bigg ) \bigg [ \bigg ( -\frac{t}{\mu _{R}^2} \bigg )^{\frac{\epsilon }{2}} + \bigg ( -\frac{u}{\mu _{R}^2} \bigg )^{\frac{\epsilon }{2}} \bigg ] \bigg \},\nonumber \\ \mathbf {I}_{q}^{(2)}(\epsilon )&= \frac{1}{2} ~ \mathbf {I}_{q}^{(1)}(\epsilon ) ~ \bigg [ \mathbf {I}_{q}^{(1)}(\epsilon ) - \frac{2\beta _0}{\epsilon } \bigg ] \nonumber \\&\quad + \frac{e^{\frac{\epsilon }{2} \gamma _{E}} ~ \Gamma (1+\epsilon ) }{\Gamma (1+\frac{\epsilon }{2})} \bigg [ -\frac{\beta _0}{\epsilon } + K \bigg ]~ \mathbf {I}_{q}^{(1)}(2\epsilon )\nonumber \\&\quad + \bigg ( 2 \mathbf {H}_{q}^{(2)}(\epsilon ) + \mathbf {H}_{g}^{(2)}(\epsilon ) \bigg ), \end{aligned}$$
(2.16)
where
$$\begin{aligned} K = \bigg ( \frac{67}{18} - \frac{\pi ^2}{6} \bigg ) C_A - \frac{10}{9}T_{F}~n_{f}. \end{aligned}$$
(2.17)
The functions \(\mathbf {H}_{q}^{(2)}(\epsilon ), \mathbf {H}_{g}^{(2)}(\epsilon )\) are dependent on the renormalization scheme. In the \(\overline{\text {MS}}\) scheme these are given by
$$\begin{aligned} \mathbf {H}_{q}^{(2)}(\epsilon )&= \frac{1}{\epsilon } \bigg \{ C_{A}C_{F} \bigg (-\frac{245}{432} + \frac{23}{16}\zeta _{2} - \frac{13}{4}\zeta _{3} \bigg ) \nonumber \\&\quad + C_{F}^{2}\bigg ( \frac{3}{16} - \frac{3}{2}\zeta _{2} + 3\zeta _{3}\bigg )\nonumber \\&\quad + C_{F}~n_{f} \bigg ( \frac{25}{216} - \frac{1}{8}\zeta _{2}\bigg ) \bigg \},\nonumber \\ \mathbf {H}_{g}^{(2)}(\epsilon )&= \frac{1}{\epsilon } \bigg \{ C_{A}^{2} \bigg (-\frac{5}{24} - \frac{11}{48}\zeta _{2} - \frac{1}{4}\zeta _{3} \bigg ) \nonumber \\&\quad + C_{A}~n_{f}\bigg (\frac{29}{54} + \frac{1}{24}\zeta _{2} \bigg ) + \frac{1}{4} C_{F}~n_{f} - \frac{5}{54} n_{f}^{2} \bigg \}. \end{aligned}$$
(2.18)
Here \(\zeta _{i}\) is the Riemann Zeta function.

3 Calculation of amplitudes

In this section we discuss the calculational details of the amplitudes \({\mid {\hat{\mathcal {M}}^{(i)}} \rangle }\) for the process \(G \rightarrow q + \bar{q} + g\) to two-loop level in pQCD. Particularly we calculate the squared matrix elements \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)}} \rangle \) and \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)}} \rangle \). Due to the tensorial coupling of spin-2 with the SM, the computational procedure becomes tedious. Starting from the generation of Feynman amplitudes, we systematically automatize the calculational procedure using in-house codes based on FORM [53], Mathematica, Reduze 2 [54] and LiteRed [55, 56].

3.1 Generation of Feynman diagrams and simplification

QGRAF [57] is used to generate all the Feynman amplitudes in terms of symbolic expressions. For the process under consideration, we have 4 Feynman diagrams in the Born, 43 in the one-loop and 847 in the two-loop level; here we have excluded all the tadpoles and self-energy corrections to the external legs. The raw QGRAF output is then manipulated using in-house FORM routines to incorporate the Feynman rules [44, 45] and to take care of the color and Dirac matrix ordering. For the internal gluons the Feynman gauge is used and the ghost-graviton interactions [58] are introduced in the Lagrangian (see Eq. (2.2)) as is necessary for higher-order computations. For the external gluon the physical polarizations are summed using
$$\begin{aligned} \sum _{\lambda =\pm 1} \epsilon ^{\mu }(p_{3},\lambda )\epsilon ^{\nu *}(p_{3},\lambda ) = -g^{\mu \nu } ~+~ \frac{p_{3}^{\mu }n^{\nu } + n^{\mu }p_{3}^{\nu }}{p_{3}.n}.\qquad \end{aligned}$$
(3.1)
Here \(\lambda \) is the helicity and \(p_3\) is the momentum of the external gluon. n is an arbitrary light-like 4-vector. We choose \(n = p_1\), one of the external fermion momenta without loss of generality. The spin-2 polarization sum in d dimensions is given by [44, 58]:
$$\begin{aligned} B^{\mu \nu ;\rho \sigma }(q)= & {} \bigg ( g^{\mu \rho } - \frac{q^{\mu }q^{\sigma } }{q.q} \bigg ) \bigg ( g^{\nu \sigma } - \frac{q^{\nu }q^{\sigma } }{q.q} \bigg ) \nonumber \\&+ \bigg ( g^{\mu \sigma } - \frac{q^{\mu }q^{\sigma } }{q.q} \bigg ) \bigg ( g^{\nu \rho } - \frac{q^{\nu }q^{\rho } }{q.q} \bigg )\nonumber \\&-\frac{2}{d-1} \bigg ( g^{\mu \nu } - \frac{q^{\mu }q^{\nu } }{q.q} \bigg ) \bigg ( g^{\rho \sigma } - \frac{q^{\rho }q^{\sigma } }{q.q} \bigg ),\nonumber \\ \end{aligned}$$
(3.2)
where the metric \(g_{\mu \nu } = \mathrm{Diag}(1,-1,-1,-1)\). The squared matrix elements are further processed using in-house codes based on LiteRed and Mathematica.

3.2 Reduction of tensor integrals

The two-loop calculation involves a large number of higher rank Feynman integrals, particularly in the present case it contains tensorial integrals. The conventional approach is to convert the different Feynman integrals into scalar integrals. This generates thousands of scalar integrals which need to be properly classified. The idea is to connect all the scalar integrals to belong to a particular basis. The basis is chosen keeping in mind that any scalar products of loop momenta and external momenta can be expressed only in terms of linear combinations of the propagators. In the case of one loop there are four different scalar products which are written in terms of four propagators. The choice of basis for the one-loop and the two-loop cases is given in [32]; for completeness we present those here. It is straightforward to choose the following set as the basis for the one-loop case:
$$\begin{aligned} B_{11}&= \{ \mathcal {D}_1,\mathcal {D}_{1;1},\mathcal {D}_{1;12},\mathcal {D}_{1;123} \}, \nonumber \\ B_{12}&= \{ \mathcal {D}_1,\mathcal {D}_{1;2},\mathcal {D}_{1;23},\mathcal {D}_{1;123} \}, \nonumber \\ B_{13}&= \{ \mathcal {D}_1,\mathcal {D}_{1;3},\mathcal {D}_{1;31},\mathcal {D}_{1;123} \}, \end{aligned}$$
(3.3)
where
$$\begin{aligned}&\mathcal {D}_{1} = k_{1}^2,\quad \mathcal {D}_{1;i} = (k_1 - p_i)^2,\quad \mathcal {D}_{1;ij} = (k_1 - p_i- p_j)^2, \nonumber \\&\quad \mathcal {D}_{1;ijk} = (k_1 - p_i- p_j - p_k)^2 \end{aligned}$$
(3.4)
with \(i,j,k = 1,2,3\). At two loops one has nine independent scalar products of loop momenta and external momenta, viz. \(\{ (k_{\alpha } \cdot k_{\beta }), (k_{\alpha } \cdot p_{i}) \}\); \(\alpha ,\beta = 1,2\) and \(i=1,2,3\). The physical diagrams contain at most seven different propagators. Hence we need to increase the number of propagators to nine. With the help of Reduze 2, all the two-loop diagrams are classified into the six different auxiliary topologies presented below:
$$\begin{aligned} B_{21}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;1},~\mathcal {D}_{2;1},~\mathcal {D}_{1;12}, ~\mathcal {D}_{2;12},\nonumber \\&\mathcal {D}_{1;123},~\mathcal {D}_{2;123} \}, \nonumber \\ B_{22}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;2},~\mathcal {D}_{2;2},~\mathcal {D}_{1;23}, ~\mathcal {D}_{2;23}, \nonumber \\&\mathcal {D}_{1;123},~\mathcal {D}_{2;123} \}, \nonumber \\ B_{23}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;3},~\mathcal {D}_{2;3},~\mathcal {D}_{1;31}, ~\mathcal {D}_{2;31},\nonumber \\&\mathcal {D}_{1;123},~\mathcal {D}_{2;123} \}, \nonumber \\ B_{24}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;1},~\mathcal {D}_{2;1},~\mathcal {D}_{0;3}, ~\mathcal {D}_{1;12}, \nonumber \\&\mathcal {D}_{2;12},~\mathcal {D}_{1;123} \}, \nonumber \\ B_{25}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;2},~\mathcal {D}_{2;2},~\mathcal {D}_{0;1}, ~\mathcal {D}_{1;23},\nonumber \\&\mathcal {D}_{2;23},~\mathcal {D}_{1;123} \}, \nonumber \\ B_{26}= & {} \{ \mathcal {D}_{0},~\mathcal {D}_{1},~\mathcal {D}_{2}, ~\mathcal {D}_{1;3},~\mathcal {D}_{2;3},~\mathcal {D}_{0;2}, ~\mathcal {D}_{1;31},\nonumber \\&\mathcal {D}_{2;31},~\mathcal {D}_{1;123} \}, \end{aligned}$$
(3.5)
where
$$\begin{aligned}&\mathcal {D}_{0} = (k_{1} -k_{2})^2,\quad \mathcal {D}_{\alpha } = k_{\alpha }^2,\quad \mathcal {D}_{\alpha ;i} = (k_{\alpha } - p_{i} )^2,\nonumber \\&\mathcal {D}_{\alpha ;ij} = (k_{\alpha } - p_{i} - p_{j} )^2, \nonumber \\&\mathcal {D}_{0;i} = (k_{1} - k_{2} - p_{i} )^2,\quad \mathcal {D}_{\alpha ;ijk} = (k_{\alpha } - p_{i} - p_{j} - p_{k} )^2. \end{aligned}$$
(3.6)
Although properly classified, these many scalar integrals are not all independent. In fact they are related by the IBP identities [35, 36] and LI identities [37], which follow from the Poincaré invariance. At a fixed order, they result in a large linear system of equations for the integrals. The inclusion of LI identities accelerates the solution of the system of equations, although they are not independent from the IBP identities [59]. We generate the IBP relations and LI identities using Laporta algorithm [60] as implemented in LiteRed. Using LiteRed along with Mint [61, 62] we reduce all different scalar integrals to a smaller set of irreducible scalar integrals i.e. the MIs.

At one loop, two kinds of MIs appear, viz. the Bubble two-propagator MI and Box four-propagator MI. In the case of two loops we find a total of 24 topologies, out of which eight are non-planar topologies and 16 planar topologies. All the two-loop MIs in our calculation can be related to the MI computed in [41, 42]. At this point we would like to note that some of the MIs in our case do not appear as given in [41, 42]. The reason behind this is the different convention in the basis in LiteRed and in [41, 42]. Thus we found topologies containing higher power of propagators instead of the irreducible numerator in [41, 42]. Nevertheless those can be related by properly using the IBP and LI identities. In this way we reduce all the scalar integrals to the known set of MIs. We also found two extra topologies for the MI [32], viz. Kite and GlassS, which are basically products of two one-loop MIs. GlassS is found to be the product of two Bubbles. Kite is the product of one Bubble and one Box one-loop MI. Using all the MIs we finally find the unrenormalized one-loop \(\langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(1)}} \rangle \) and two-loop \(\langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(2)}} \rangle \) matrix elements, which are presented in the next section.

4 Results

We have checked that the gauge fixing term appearing at the g-g-G vertex cancels against the contributions coming from ghost-ghost-G and ghost-ghost-gluon-G vertices. This cancellation of gauge dependent terms involving spin-2 serves a crucial check on our computation. Following the renormalization prescription in Sect. 2.1, we compute the UV renormalized matrix elements \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)}} \rangle \) and \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)}} \rangle \) in terms of the unrenormalized ones,
$$\begin{aligned} \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)}} \rangle= & {} \bigg ( \frac{1}{\mu _{R}^{\epsilon }} \bigg )^{2} ~ \bigg [ \langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(1)}} \rangle +\mu _{R}^{\epsilon } \frac{r_{_{1}}}{2} \langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(0)}} \rangle \bigg ] \nonumber \\ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)}} \rangle= & {} \bigg ( \frac{1}{\mu _{R}^{\epsilon }} \bigg )^{3} ~\bigg [ \langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(2)}} \rangle +\mu _{R}^{\epsilon } \frac{3r_{_{1}}}{2} \langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(1)}} \rangle \nonumber \\&+\,\mu _{R}^{2\epsilon } \bigg ( \frac{r_{_{2}}}{2} - \frac{r_{_{1}}^2}{8} \bigg ) \langle {\hat{\mathcal {M}}^{(0)} | \hat{\mathcal {M}}^{(0)}} \rangle \bigg ]. \end{aligned}$$
(4.1)
Similarly the renormalized matrix elements according to Catani’s prescription can be written as
$$\begin{aligned} \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)}} \rangle= & {} 2~\mathbf {I}_{q}^{(1)}(\epsilon ) ~ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(0)}} \rangle + \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)\mathrm{fin}}} \rangle \nonumber \\ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)}} \rangle= & {} 2~\mathbf {I}_{q}^{(1)}(\epsilon ) ~ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)}} \rangle + 4~\mathbf {I}_{q}^{(2)}(\epsilon ) ~ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(0)}} \rangle \nonumber \\&+\, \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)\mathrm{fin}}} \rangle . \end{aligned}$$
(4.2)
From Eq. (4.1) we extract the coefficients of different poles viz. the \(1/\epsilon ^4, 1/\epsilon ^3, 1/\epsilon ^2, 1/\epsilon \) and we find that they exactly agree with the respective poles coming from Eq. (4.2). By comparing the \(\mathcal {O}(\epsilon ^0)\) terms from these two sets of equations (Eq. (4.1) and Eq. (4.2)), we obtain the unknown pieces \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)\mathrm{fin}}} \rangle \) and \(\langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)\mathrm{fin}}} \rangle \).
The final result is written in the following form:
$$\begin{aligned} \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(0)}} \rangle= & {} \mathcal {F}_{b}~ \mathcal {A}^{(0)}, \nonumber \\ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(1)\mathrm{fin}}} \rangle= & {} \mathcal {F}_{b} \Bigg \{\mathcal {A}_{0}^{(1)} ~ \text {ln}\bigg ( -\frac{Q^2}{\mu _{R}^2} \bigg ) + \Big (\mathcal {A}_{1}^{(1)} \zeta _{2} + \mathcal {A}_{2}^{(1)} \Big ) \Bigg \}, \nonumber \\ \langle {\mathcal {M}^{(0)} | \mathcal {M}^{(2)\mathrm{fin}}} \rangle= & {} \mathcal {F}_{b} \Bigg \{ ~ \mathcal {A}_{0}^{(2)} ~\text {ln}^{2}\bigg ( -\frac{Q^2}{\mu _{R}^2} \bigg ) \nonumber \\&+\, \Big (\mathcal {A}_{1}^{(2)} \zeta _{3} + \mathcal {A}_{2}^{(2)} \zeta _{2} + \mathcal {A}_{3}^{(2)} \Big ) ~\text {ln}\bigg ( -\frac{Q^2}{\mu _{R}^2} \bigg ) \nonumber \\&+\,\Big ( \mathcal {A}_{4}^{(2)} \zeta _{2}^{2} + \mathcal {A}_{5}^{(2)} \zeta _{3} + \mathcal {A}_{6}^{(2)} \zeta _{2} + \mathcal {A}_{7}^{(2)} \Big ) \Bigg \},\nonumber \\ \end{aligned}$$
(4.3)
where
$$\begin{aligned} \mathcal {F}_{b} = 16 \pi ^{2} \kappa ^{2} \big (N^{2} -1\big )~Q^{2}, \end{aligned}$$
(4.4)
$$\begin{aligned} \mathcal {A}^{(0)} = \frac{ \big ( 2 + y^2\big (3 - 9z\big ) - 4z + 3z^2 - z^3 + y^3\big (-1 + 4z\big ) + y\big (-4 + 12z - 9z^2 + 4z^3\big ) \big )}{4yz(1 - y - z)}, \end{aligned}$$
(4.5)
$$\begin{aligned} \mathcal {A}_{i}^{(1)}= & {} \mathcal {A}_{i;C_A}^{(1)} ~ C_A + \mathcal {A}_{i;C_F}^{(1)} ~ C_F + \mathcal {A}_{i;n_f}^{(1)} ~ n_f, \nonumber \\ \mathcal {A}_{i}^{(2)}= & {} \mathcal {A}_{i;C_{A}^{2}}^{(2)} C_{A}^{2} + \mathcal {A}_{i;C_{F}^{2}}^{(2)} C_{F}^{2} + \mathcal {A}_{i;n_{f}^{2}}^{(2)} n_{f}^{2} + \mathcal {A}_{i;C_{A}C_{F}}^{(2)} C_{A}C_{F}\nonumber \\&+\, \mathcal {A}_{i;C_{A}n_{f}}^{(2)} C_{A}n_{f} + \mathcal {A}_{i;C_{F}n_{f}}^{(2)} C_{F}n_{f}. \end{aligned}$$
(4.6)
We notice that, unlike the Higgs decay i.e. \(H \rightarrow b + \bar{b} + g\) [63], there is no \(C_{F}\) term in the \(\mathcal {A}_{0}^{(1)}\), which is due to the absence of a Yukawa-like term in the spin-2 case. All the one-loop and two-loop coefficients are presented in Appendices B and C, respectively, except for the \(A_{7}^{(2)}\) term, which is provided as the ancillary file in the arXiv.

5 Conclusions

In this article, we present the two-loop virtual QCD correction to massive \(G \rightarrow q +\bar{q}+ g\) considering the minimal and universal coupling between spin-2 and the SM particles. We confine ourselves within the framework of massless QCD where only the light quark degrees of freedom are taken into account. We employ the Feynman diagrammatic approach to achieve our goal. As expected, the computation becomes very tedious not only due to the presence of a large number of Feynman diagrams but also due to the involvement of a tensorial coupling. In-house codes and state-of-the-art techniques, in particular, IBP and LI identities, are employed extensively to execute the computation successfully. The bare matrix elements contain UV as well as IR divergences. The strong coupling constant renormalization is sufficient to make it UV finite. No extra UV renormalization is required for the spin-2 coupling as a consequence of the conserved SM energy-momentum tensor through which it couples universally to the SM fields. The UV finite matrix elements exhibit poles of infrared origin in dimensional regularization. The resulting infrared pole structures are in exact agreement with the Catani’s prescription which ensures the universal factorization property of QCD amplitudes even in the presence of spin-2 field. This serves a crucial check on the correctness of our computation. The result presented here is an important piece, which now completes the genuine two-loop calculation of real graviton production associated with a jet. The full NNLO computation to this process needs additional inputs that we reserve for future study.

Notes

Acknowledgements

We sincerely thank Thomas Gehrmann for providing the necessary MIs which are used in this calculation. GD would like to thank R. N. Lee for help with LiteRed. GD also thanks Kuntal Nayek for help with Mathematica. GD is supported by funding from Department of Atomic Energy, India.

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Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Theory DivisionSaha Institute of Nuclear PhysicsKolkataIndia
  3. 3.Homi Bhaba National InstituteTraining School ComplexMumbaiIndia

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