Next-to-next-to-leading order $N$-jettiness soft function for $tW$ production

We calculate the $N$-jettiness soft function for $tW$ production up to next-to-next-to-leading order in QCD, which is an important ingredient of the $N$-jettiness subtraction method for predicting the differential cross sections of massive coloured particle productions. The divergent parts of the results have been checked using the renormalization group equations controlled by the soft anomalous dimension.


Introduction
Precise calculation of cross sections for the processes at the Large Hadron Collider (LHC) or future high-energy hadron colliders is crucial for testing the Standard Model (SM) and for searching for new physics. In the last a few years, there is a burst of fully differential next-to-next-to-leading order (NNLO) results for a large number of processes in the SM; see a recent review in ref. [1]. One of the main difficulties in the higher-order QCD calculations is to develop a systematical method to deal with the infrared singularities caused by double real emissions. The N -jettiness subtraction has proven to be successful in computing the NNLO differential cross sections of processes with jets, for example, W/Z/H/γ + j [2][3][4][5]. This subtraction method is based on the soft-collinear effective theory (SCET) [6][7][8][9][10], which is an effective theory of QCD in the infrared regions. The N -jettiness T N is an observable, proposed in [11], to describe the event shape of jet processes or processes with initial-state hadrons, a generalization of thrust at lepton colliders and beam thrust at hadron colliders [12]. The application of this observable to the NNLO calculations started with the top quark decay [13], which involves a massive and massless quark, and became mature with the V + j production at hadron colliders, which has three massless partons. However, for more complicated processes, e.g., involving one massive and two massless partons, the results are still missing.
The N -jettiness event shape variable is defined by [11] where n i (i = a, b, 1, ..., N ) are light-like reference vectors representing the moving directions of massless external particles, and q k denotes the momentum of soft or collinear partons. In the infrared divergent regions, the observable T N → 0, and the cross section is approximated by [11,12] Here the hard function H encodes all the information about hard scattering. The beam functions B i , (i = 1, 2), describe the perturbative and non-perturbative contributions from initial state, and have been obtained up to NNLO [14][15][16][17]. The jet function J n describes the final-state jet with a fixed invariant mass and has been calculated at NNLO [18,19]. The soft function S contains soft interactions between all colored particles. It has been studied up to NNLO for massless processes [20][21][22][23][24][25].
The differential cross section for any observable O is given by where a small cut-off parameter ∆ on the right-hand side is imposed. For the NNLO calculations the first term on the right-hand side at the leading power can be obtained by expanding eq. (1.2) to the second order of the strong coupling α s . The second term, due to the phase-space constraint, can be dealt with the standard NLO subtraction method for the process with an extra parton in the final state. As mentioned before, this method has been utilized to calculate differential cross sections of processes at both hadron colliders [2][3][4][5][26][27][28] and electron-hadron colliders [29,30]. The extension of the N -jettiness subtraction to more complicated processes requires the calculation of the corresponding soft and hard functions. We have calculated the Njettiness soft function for one massive colored particle production up to NNLO in ref. [31], where we assume it is produced at rest. In this paper, we present the result for more general situations, i.e., the massive colored particle can carry any possible momentum. Our result can be used to construct the N -jettiness subtraction terms for tW production. This paper is organized as follows. In section 2, we briefly introduce the definition of the soft function in terms of soft Wilson lines. In section 3, we study the renormalization group (RG) equation of the soft function and thus derive the structure of the soft function. We provide the details of the techniques in our calculations in section 4. Then, in section 5, we present the numerical results of the NLO and NNLO soft functions and compare the divergent terms with the predictions from RG equation. We conclude in section 6.

Definition of the soft function
In this section we first discuss the kinematics and the factorization of the cross section for tW production. Then we present the definition of soft function.
We consider the process where P 1 and P 2 denote incoming hadrons, t/t and W ± represent the top/anti-top quark and the W -boson in the final state, respectively. And X includes any unobserved final state. The partonic process at leading order (LO) for tW − production is It is convenience to introduce two light-like vectors n µ = (1, 0, 0, 1),n µ = (1, 0, 0, −1) .
Any momentum can be decomposed as p µ = (p + , p − , p ⊥ ) with p + = p · n, p − = p ·n. The momenta given in eq. (2.2) can be written as where v 2 = 1. Specifically, we parameterize v by two variables, i.e., β t and θ t , which measure the magnitude and the angle of the velocity, The 0-jettiness event shape variable in this process is defined as (2.6) In the limit τ √ŝ , the final state contains no hard radiations, only soft and collinear radiations allowed. The cross section in this limit admits a factorized form which can be derived in the framework of SCET. According to the analysis in refs. [11,12], the cross section can be written as where dΦ 2 is the two-body phase space integral, dσ 0 is the LO partonic differential cross section, Y is the rapidity of the partonic colliding system in the laboratory frame, the momentum fractions x a = ŝ/se Y and x b = ŝ/se −Y with √ s the collider energy, and µ is the renormalization scale. In momentum space the soft function is defined as the vacuum matrix element where T(T) is the (anti-)time-ordering operator. And Y n , Yn and Y v are the soft Wilson lines defined explicitly as [9,32,33] where P andP are the path-ordering and the anti-path-ordering operators.P k in eq.(2.8) is the operator extracting the momentum of each soft emission. The purpose of this paper is to calculate the soft function defined above for tW production up to NNLO accuracy.

Renormalization
In SCET the bare soft function in eq.(2.8) contains ultra-violet divergences in perturbative calculations, which are cancelled by the conterterms defined in the standard renormalization procedure. The renormalized soft function is finite and can be used in the calculation of the cross section in eq. (2.7). The renormalization introduces the scale µ dependence in the soft function, as well as in the hard and beam function. Because of the fact that the physical cross section does not dependent on the intermediate scale, the RG equation of the soft function can be derived from the RG equations of the hard and beam function, which will be used to extract the anomalous dimensions of the soft function. Given anomalous dimension the divergences in the bare soft function, as well as the scale dependence of the renormalized soft function, can be predicted. In this section we briefly discuss the renormalization of the soft function and the expression of the soft anomalous dimension. We work in d = 4 − 2 dimensional space-time.
Based on dimensional analysis, the bare soft function, in perturbation theory, can be written as where we use renormalized strong coupling α s and its renormalization factor The soft function after the Laplace transformation can be written as Then the corresponding renormalized soft functions is defined as where the renormalization factor Z s satisfies the differential equation with γ s the anomalous dimension of the soft function. We will suppress the arguments of the renormalization factor, anomalous dimension and the soft function in the following text for convenience. Given the soft anomalous dimension γ s , following refs. [34,35], the closed expression for Z s is derived and can be written as (3.5) The expansion series and derivative of the soft anomalous dimension are given by From eq. (3.3), we obtain the renormalized NLO and NNLO soft functions in Laplace spacẽ Since the renormalized soft function is finite, the divergent terms in the bare soft functioñ S is related to the renormalization factor Z s and can be derived from the above equations. As discussed before, the soft anomalous dimensions γ s can be derived from the independence of the cross section on the renormalization scale µ, whereB i is the beam function in Laplace space, of which the NLO and NNLO results can be found in refs. [14][15][16][17]. And the RG equation of the beam function is exactly the same as the evolution equation of the jet function to all orders [14], where T i is the color generator associated with the i-th parton [36,37] and the anomalous dimension γ i B can be found in refs. [16,17]. The RG equation for the hard function can be obtained from refs. [38,39] where the two-loop divergences have been calculated for massive scattering amplitudes in non-abelian gauge theories. It is straightforward to organize the RG equation for the hard function as with s 12 = 2p 1 · p 2 + i0, s 13 = −2p 1 · p 3 + i0, s 23 = −2p 2 · p 3 + i0. The anomalous dimensions γ 1,2 and γ Q , associated with the initial-and final-state particles, can be found in refs. [38,39] and references therein. Inserting eqs. (3.9-3.10) to eq.(3.8), the anomalous dimension of the soft function is obtained of which each ingredient is available up to NNLO.

Techniques in calculation
In the calculation of the NLO and NNLO soft function, we have to deal with one and two soft radiations, respectively. The phase space integration is with The φ angle is measured in the frame with the top quark φ t = 0. More explicitly, we choose p 3⊥ = |p 3⊥ |(0; 0, 1), q 1⊥ = |q 1⊥ |(0; sin φ 1 , cos φ 1 ), q 2⊥ = |q 2⊥ |(sin φ 2 sin βn ; sin φ 2 cos β, cos φ 2 ). (4.4) For the integrand involving 1/(q 1 · q 2 ), the phase space integration is parameterize as with At NLO, the measurement function is defined as where q + = q · n and q − = q ·n. At NNLO, the measurement function is defined as One can see that for NNLO the whole phase space is partitioned to four pieces. We label them as Region-I, Region-II, Region-III and Region-IV, respectively. In the hemisphere with q + i = τ i , we parameterize q − i = τ i /t i with t i ∈ (0, 1) and And then all those singularities at NLO will appear as τ −1−2 i and t −1+ i . In the end, we define If the integrands do not involve 1/(q 1 · q 2 ), we perform the phase space integration straightforward after the parameterization. For the integrands involving 1/(q 1 · q 2 ), we have in the Region-I or Region-III, and q 1 · q 2 = 1 2 in the Region-II or Region-IV. Here φ 12 is the angle between q 1⊥ and q 2⊥ . In the Region-I and Region-III, the double-real corrections contain a new kind of singularities that appear when t 1 = t 2 and φ 12 = 0. Following the method in ref. [25], we change the integration variables from φ 2 , β to φ 12 , β 12 . All dependence on φ 2 (such as q 2⊥ · p 3⊥ ) can be expressed in terms of φ 12 and β 12 , cos φ 2 = cos φ 1 cos φ 12 − sin φ 1 sin φ 12 cos β 12 . (4.14) The β 12 angle integration can be transformed by defining cos β 12 = 1 − 2x, Define cos φ 12 = 1 − 2z, (4.16) By writing in this form, we have picked out the singular part as t 2 → t 1 . The parameter r is solved to be 17) and the Jacobian is The φ 12 angular integration is given by Combined with eq.(4.16), we see that the singular part of (q 1 · q 2 ) −1 ∼ |t 2 − t 1 | −1−2 and (q 1 ·q 2 ) −2 ∼ |t 2 −t 1 | −3−2 . However, we find that the coefficient of (q 1 ·q 2 ) −2 is proportional to (t 1 − t 2 ) 2 . Now we divide the integration region of t 1 , t 2 to two sectors, i.e., In each sector, |t 2 − t 1 | has a definite sign and thus is easy to deal with. In the Region-II and Region-IV, one can carry out the same procedure as above except the relation between r and z changes to With this paramertization above, all the divergences can be extracted through the expansion,

NLO soft function
The LO soft function is trivial and has been given explicitly in eq.(3.1). In this section, we present its NLO result. Expanding the soft Wilson lines in eq.(2.8) in a series of the strong coupling, we obtain the NLO soft function where e γ E is inserted because we use MS renormalization scheme. The factor J µ(0) a (q) is the LO one-gluon soft current, or the eikonal current, with a the color index. After performing the phase space integration, we obtain the NLO bare soft function where A i is a function of β t and cos θ t . Figure 1 shows the numerical results for the NLO soft function and the comparison of the divergent coefficients between the numerical calculations and RG predictions with fixed cos θ t or β t . The deviations are not larger than 0.2% except for the case of |A i | → 0 . The points at β t = 0 just reproduce our previous results in ref. [31], as expected. It can also be seen that when β t → 1, i.e., the top quark is highly boosted, the coefficients A i , i = 0, 1, 2, 3, become divergent. This is due to the logarithmic structures such as ln n (1 − β t ) in the limit of β t → 1. In principle, this kind of logarithms can be predicted from effective field theory for boosted top productions. Because the top quark mass is small compared with its energy in the limit, the scale hierarchy of the process is τ m t √ŝ , and thus a different factorization formula should be derived. We leave the detailed discussion to a future work. Notice that in eq. (5.3) and fig. 1 we also show A 2 and A 3 which do not contribute to the NLO result. However, they will contribute to the renormalized NNLO soft function.

NNLO soft function
The NNLO contribution consists of two parts, i.e., The first part is the virtual-real correction, i.e., the one-loop virtual corrections to LO soft gluon current J µ(1) a (q); the second part is the double-real correction, i.e., the corrections with a double-gluon soft current J µν(0) ab (q 1 , q 2 ) or a massless quark-pair emission. For the virtual-real contribution we use the soft limit of one-loop QCD amplitudes which has been studied in refs. [40][41][42] and ref. [43] for massless and massive external particles. As for the double-real contribution we make use of the results in refs. [44,45] where the infrared behaviour of tree-level QCD amplitudes at NNLO has been analyzed. The details of the virtual-real and double-real matrix element can be found in our previous paper [31].
With the techniques discussed in section 4, the double-real part is calculated numerically after sector decomposition. The bare soft function at NNLO, defined in eq. (3.1), can be written as Using eq. (3.8) and the anomalous dimensions in eq. (3.11) the divergent terms in the bare NNLO soft function can be predicted, which is an important cross check of our calculations. Table 1 shows the comparison of the divergent terms in different color structures with fixed β t = 0.3 and cos θ t = 0.5. We see that the maximum deviation is less than 0.2%. Figures 2 and 3 show the numerical calculations and the RG predictions with cos θ t in the  range of (−1, 1) but fixed β t and with β t in the range of (0, 1) but fixed cos θ t , respectively. We find that the numerical results are consistent with the RG predictions. For most of the cases the deviations are less than 0.2%, while the deviations can be about 1% only when the absolute values of the coefficient B i are close to zero. We have checked that the points at β t = 0 reproduce our previous results in ref. [31]. Similar to the NLO results, in the highly boosted region, the NNLO coefficients contains logarithmic structures such as ln n (1 − β t ). They are divergent when β t → 1. This fact explains the behaviour of the distributions of the points near the end point of β t in fig.3.

Conclusions
The N -jettiness subtraction method is one of the efficient methods to perform differential calculations of the NNLO cross sections. In this paper, we present the calculation of NNLO soft function for tW production which is one of the indispensable ingredients in N -jettiness subtraction method. Our calculation makes use of the one-loop soft current and infrared limit of the QCD matrix elements from refs. [42][43][44][45] to construct the integrand. The phase space integrals are performed with the sector decomposition method and the techniques are discussed in details. The divergent terms at NLO and NNLO soft functions from our calculations are in very good agreement with those from the RG predictions. Once the twoloop hard function is obtained, we can perform the NNLO calculation for the differential cross section of tW production at hadron colliders. Our method can also be applied to the calculation of the N -jettiness soft function for top quark pair production, which provides another way to study the NNLO differential cross section for this process. We leave this application in future study.