Resummation for $2\rightarrow n$ processes in single-particle-inclusive kinematics

We present a formalism and detailed analytical results for soft-gluon resummation for $2\rightarrow n$ processes in single-particle-inclusive (1PI) kinematics. This generalizes previous work on resummation for $2 \rightarrow 2$ processes in 1PI kinematics. We also present soft anomalous dimensions at one and two loops for certain $2 \rightarrow 3$ processes involving top quarks and Higgs or $Z$ bosons, and we provide some brief numerical results.


Introduction
In theoretical calculations of hard-scattering cross sections of relevance to hadron colliders, the state-of-the-art has been moving steadily towards higher orders, more loops, and resummations at higher logarithmic accuracy; it has also been gradually expanded to processes with larger numbers of final-state particles. In particular, soft-gluon resummations have become a very useful tool in making predictions for additional corrections beyond complete fixed-order results. In many cases these soft-gluon corrections are large, and in fact they numerically dominate the complete corrections and can be thought of as very good approximations to complete results.
Soft-gluon resummation follows from factorization properties of the cross section [1][2][3][4][5] and it has been applied to a large number of processes in hadron collisions. Most of the applications for total cross sections and differential distributions have been done for 2 → 2 processes in single-particle-inclusive (1PI) as well as pair-invariant-mass (PIM) kinematics, most notably for top-quark production (see [6] for a review) but also many other processes. Applications to 2 → 3 processes using extensions of the PIM formalism, e.g. three-particle-invariant-mass kinematics, have also been made [7][8][9][10][11][12][13][14]. In this paper, we generalize resummation to processes with n particles in the final state in 1PI kinematics. We also give more details for 2 → 3 processes with top quarks and Higgs or Z bosons.
We begin in Section 2 with the development of the formalism, starting with elementary considerations and kinematics for 2 → 2 processes, and then for 2 → 3 processes, before moving on to the generalization to 2 → n processes and the derivation of the resummed cross section in the general case. We also provide results for the expansion of the cross section to fixed order, in particular next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). In Section 3, we provide details about the cross section calculation at the partonic and hadronic levels. Section 4 has details about the soft anomalous dimensions through two loops for 2 → 3 processes involving a top quark and a Higgs or Z boson, and a brief numerical application to t-channel tqH and tqZ production which shows the power of the formalism. We conclude in Section 5 and include an appendix with an alternate kinematical calculation of the cross section.
2 Resummation for 2 → n processes In this section we develop the formalism for resummation in 1PI kinematics with multi-particle final states. We begin with some simple considerations and definitions for 2 → 2 processes in the next subsection, and extend them to 2 → 3 processes in subsection 2.2 and to 2 → n processes in subsection 2.3. The complete resummation formalism for 2 → n processes is given in subsection 2.4. Fixed-order expansions of the resummed cross section are provided in subsection 2.5.

Kinematics and threshold for 2 → 2 processes
We first consider processes that are 2 → 2 at lowest order, p a + p b → p 1 + p 2 (e.g. qq → tt). We define the usual kinematical variables where, depending on the process, the masses m 1 and m 2 can be zero or finite. As we approach partonic threshold, s th → 0 and there is vanishing energy for any additional radiation. If we have an additional gluon with momentum p g being emitted in the final state, then by using momentum conservation, p a + p b = p 1 + p 2 + p g , it is straightforward to show that the above definition of s th is equivalent to s th = (p 2 + p g ) 2 − p 2 2 . It is clear that s th goes to 0 as p g goes to 0 (soft gluon). The physical meaning is also more clear from this way of writing s th : it is the invariant mass squared of the "particle 2 + gluon" system minus the invariant mass squared of particle 2, i.e. it describes the extra energy in the soft emission. Note that particle 1 is the observed particle in this single-particle-inclusive kinematics.
If the incoming partons a and b come from hadrons A and B, then we also define the hadronlevel variables S = (p A + p B ) 2 , T = (p A − p 1 ) 2 , U = (p B − p 1 ) 2 , and S th = S + T + U − p 2 1 − p 2 2 . Assuming that p a = x a p A and p b = x b p B , where x a and x b denote the fraction of the momentum carried by partons a and b in hadrons A and B, respectively, then we have the relations Then, using the above relations and after some algebra, we find that The last term, involving (1 − x a )(1 − x b ), is higher order and can be ignored near threshold, as x a → 1 and x b → 1.

Kinematics and threshold for 2 → 3 processes
We next consider processes that are 2 → 3 at lowest order, p a + p b → p 1 + p 2 + p 3 (e.g. bq → tq ′ H). We define the parton-level variables s, t, u, and the hadron-level variables S, T , U, as before. If we have an additional gluon with momentum p g in the final state, then momentum conservation is p a + p b = p 1 + p 2 + p 3 + p g . We can define the threshold variable as s th = (p 2 + p 3 + p g ) 2 − (p 2 + p 3 ) 2 . This clearly gives the same physical meaning as extra energy from gluon emission and clearly vanishes as p g → 0.
One can also show after some work that this is equivalent to s th = s + t + u − p 2 1 − (p 2 + p 3 ) 2 . We also define S th = S + T + U − p 2 1 − (p 2 + p 3 ) 2 , and find, after some algebra, the relation s . (2. 2) The last term, involving (1 − x a )(1 − x b ), can be ignored in the threshold limit, as x a → 1 and x b → 1. We see that our results here are a natural extension of the relations for 2 → 2 kinematics.

Kinematics and threshold for 2 → n processes
These relations can be extended to an arbitrary number of particles: we consider processes that are 2 → n at lowest order, p a + p b → p 1 + p 2 + · · · + p n . Again, we define the parton-level variables s, t, u, and the hadron-level variables S, T , U, as before. With an additional gluon with momentum p g in the final state, momentum conservation is p a +p b = p 1 +p 2 +· · ·+p n +p g .
Then the threshold variable is s th = (p 2 +· · ·+p n +p g ) 2 −(p 2 +· · ·+p n ) 2 with the same physical meaning as before, and vanishing as p g → 0. Using the abbreviation p 2···n = p 2 + · · · + p n , we can rewrite the threshold variable as s th = (p 2···n + p g ) 2 − p 2 2···n . We can also show that this variable can also be written as s th = s + t + u − p 2 1 − p 2 2···n . We also define S th = S + T + U − p 2 1 − p 2 2···n , and find that Again, the last term, involving (1 − x a )(1 − x b ), can be ignored as x a → 1 and x b → 1. Finally, we note that one can appropriately redefine the above relations if, instead of particle 1, the observed particle is n or any of the other particles.

Resummation
The factorized form of the double-differential cross section in proton-proton collisions in 1PI kinematics is where E 1 is the energy of the observed particle 1, φ a/A (φ b/B ) are parton distribution functions (pdf) for parton a (b) in hadron A (B), andσ ab→1···n is the hard-scattering partonic cross section. For simplicity we do not explicitly show in the above equation the depenendence on µ F and µ R , the factorization and renormalization scales.
The resummation of soft-gluon corrections follows from the factorization of the cross section in integral transform space [1,3]. We define Laplace transforms (indicated by a tilde) of the partonic cross section asσ(N) = s 0 (ds th /s) e −N s th /sσ (s th ), where N is the transform variable, and note that logarithms of s th transform into logarithms of N, with the latter exponentiating. We also define transforms of the pdf asφ(N) We also consider the parton-parton cross section E 1 dσ ab→1···n /d 3 p 1 , of the same form as Eq. (2.4) but with the incoming hadrons replaced by partons [1][2][3][4][5] and define its transform (again indicated by a tilde) as Taking a transform of Eq. (2.5), as defined in Eq. (2.6) above, and using Eq. (2.3) (ignoring the higher-order terms), we have where N a = N(p 2 2···n − u)/s and N b = N(p 2 2···n − t)/s. Next, we proceed with a refactorization of the cross section in terms of a new set of functions [1][2][3][4][5]. We first rewrite Eq. (2.3) as where the w's denote dimensionless weights. Note that w a = 1 − x a and w b = 1 − x b since they refer to different functions. Then, a refactorized form of this cross section [1,3,5] is The infrared-safe hard function H ab→1···n does not depend on N, and it describes contributions from the amplitude and from the complex conjugate of the amplitude. The soft function S ab→1···n describes the emission of noncollinear soft gluons in the 2 → n process. Both the hard and the soft functions are process-dependent matrices in color space in the partonic scattering, and the trace of their product is explicit in the above result. The functions ψ are distributions for incoming partons at fixed value of momentum, that describe the dynamics of collinear emission from those partons. The J i denote functions that describe collinear emission from final-state colored particles. Taking a transform of Eq. (2.9), of the form defined in Eq. (2.6), and using Eq. (2.8), we then have (2.10) Comparing Eqs. (2.7) and (2.10), we get the following expression for the transform-space hard-scattering partonic cross section, (2.11) The N-dependence of the soft matrixS ab→1···n is resummed via renormalization group evolution [1]. We haveS is the unrenormalized quantity and Z ab→1···n S is a matrix of renormalization constants. Thus,S ab→1···n obeys the renormalization group equation where g 2 s = 4πα s and β is the QCD beta function, (2.14) The lowest-order term in the above series for the beta function is given by β 0 = (11C A − 2n f )/3 where C A = N c , with N c the number of colors, and n f is the number of light quark flavors. The evolution of the soft function is controlled by the soft anomalous dimension matrix, Γ ab→1···n S , which is calculated from the coefficients of the ultraviolet poles of eikonal diagrams [1,4,15,16].
The transform-space resummed cross section is derived from the renormalization-group evolution of the soft function and the other N-dependent functions in Eq. (2.11), and it is given by [1,3,6] ) . (2.15) The first exponential resums universal soft and collinear contributions from the incoming partons [17,18].
for a quark or antiquark and C i = C A for a gluon, while A (2)  The expression for the final-state exponential, involving E ′ i , depends on whether we have massless or massive particles or jets. For massive particles or for colorless particles it is 1. For massless quarks or gluons we have

Fixed-order expansions
We can expand the formula for the resummed cross section, Eq. (2.15), to any fixed order [6,19] and invert it back to momentum space. Below we provide explicit results for the soft-gluon corrections at NLO and NNLO.
The NLO soft-gluon corrections are

19)
F LO = α d s tr{H (0) S (0) } denotes the leading-order (LO) coefficient, and c 2 is given by denoting the terms involving logarithms of the scale, and denoting the scale-independent terms. Also, With regard to the δ(s 4 ) terms, we split them into a term c 1 , that is proportional to the Born cross section, and a term T c 1 that is not. We write c 1 = c µ 1 + T 1 , with (2.24) denoting the terms involving logarithms of the scale. We note that T 1 and T c 1 cannot be calculated from the resummation formalism but they can be determined from a comparison to a complete NLO calculation.
The NNLO soft-gluon corrections are 27) We note that at next-to-next-to-leading-logarithm (NNLL) resummation accuracy for a given process, all soft-gluon terms in the expansion through NNLO can be fully calculated.

Cross section and kinematics
In this section we provide some formulas that are needed for the calculation of partonic and hadronic cross sections with multi-particle final states.

Frame-invariant integration variables
It has been shown by Byckling and Kajantie [20,21] that one can write the expression for the phase space integration of a 2 → n scattering process while integrating over only invariant variables. For processes with massless initial states, we have the phase space integral R n (s) = 1 4s dp 2 1...n−1 dt n−1 dφ dp 2 1...n−2 dt n−2 ds n−1 Θ(−∆ 4 (n − 1)) with s = (p a + p b ) 2 and p 1···n = (p 1 + · · · + p n ) 2 . We define the generalised kinematic invariants is the four-dimensional Gram determinant which can be written as The limits of integration are given by where λ(x, y, z) = (x − y − z) 2 − 4yz, and G(i) and V (i) are given by and The angle φ describes a rotation of the process around the beam axis and is trivial for our purposes. Integrating it out, including the flux factor and the matrix element |M|, and using the identity p 2 1...n−1 = s + t n−1 + u n−1 − m 2 n , we obtain the differential partonic cross section

Hadronic cross section
The LO hadronic cross section is obtained by convoluting the differential partonic cross section with the appropriate parton distribution functions: where S, T n−1 , and U n−1 are the hadronic analogues of the partonic invariants. We extend 2 → 3 particle kinematic definitions [22] to 2 → n particle kinematics, giving the conditions which yield the integration bounds for x a and x b : For an arbitrary 2 → n process, there are 1 2 (n − 2)(n − 3) relations between all possible kinematic invariants that are not fixed by momentum conservation. These must instead be fixed by the condition that any five or more vectors are always linearly dependent in four-dimensional space and their symmetric Gram determinant vanishes: ∆ l+1 (p 1 , p 2 , · · · , p l , −p b ) = 0 , 4 ≤ l ≤ n . (3.10) The Gram determinant condition ∆ l+1 = 0 can be equivalently written as a Cayley determinant condition [20] as ∆ l+1 (p 1 , p 2 , · · · , p l , −p b ) = 4 Soft-gluon corrections for 2 → 3 processes with a top quark and a Higgs or Z boson In this section we consider several processes involving a three-particle final state with a top quark and a Higgs boson, or a top quark and a Z boson. We present the soft anomalous dimension matrices for these processes at one and two loops. We also give some brief numerical results for tqH and tqZ production to illustrate the use of the formalism.
We begin with the s-channel processes q(p a ) +q ′ (p b ) → t(p 1 ) +b(p 2 ) + H(p 3 ) and q(p a ) + q ′ (p b ) → t(p 1 ) +b(p 2 ) + Z(p 3 ). We define s, t, and u as in Section 2, and further define s ′ = (p 1 + p 2 ) 2 , t ′ = (p b − p 2 ) 2 , and u ′ = (p a − p 2 ) 2 . We choose the color basis c 1 = δ ab δ 12 and c 2 = T c ba T c 12 . Then, at one loop, the four elements of the s-channel soft anomalous dimension matrix are given by Γ s (1) where m t is the top-quark mass.
We continue with the t-channel processes b(p a ) + q(p b ) → t(p 1 ) + q ′ (p 2 ) + H(p 3 ) and b(p a ) + q(p b ) → t(p 1 ) + q ′ (p 2 ) + Z(p 3 ). We define the kinematical variables as before and choose the color basis c 1 = δ a1 δ b2 and c 2 = T c 1a T c 2b . The four elements of the t-channel soft anomalous dimension matrix at one loop for these processes are given by Γ t (1) .

(4.2)
At two loops, the soft anomalous dimension matrices for each process can be written compactly in terms of the one-loop results. We have We also note that soft anomalous dimension matrices at one loop for processes with three colored particles in the final state have appeared in Refs. [23,24].
To illustrate the usefulness of our formalism, we now briefly apply our methods to the cross section for the t-channel processes b(p a ) + q(p b ) → t(p 1 ) + q ′ (p 2 ) + H(p 3 ) and b(p a ) + q(p b ) → t(p 1 ) + q ′ (p 2 ) + Z(p 3 ). NLO calculations for these processes have appeared in Refs. [25,26]. We use a renormalisation and factorization scale of µ = m t = 173.0 GeV, and MMHT2014 pdf [27]. Our higher-order soft-gluon corrections are computed from resummation at next-toleading-logarithm (NLL) accuracy. In our discussion below, we denote the sum of the LO cross section and the NLO soft-gluon corrections as approximate NLO (aNLO); and we denote the sum of the aNLO cross section and the NNLO soft-gluon corrections as approximate NNLO (aNNLO).
For Z associated production, we find aNLO enhancements of the total top + antitop LO cross section of 13.4% at 8 TeV, 30.9% at 13 TeV, and 34.1% at 14 TeV. At aNNLO, we find enhancements over the aNLO cross section of 5.3% at 8 TeV, 5.6% at 13 TeV, and 5.7% at 14 TeV. Using MadGraph5 aMC@NLO [28], we find a total NLO enhancement at 8 TeV of 13.4%, showing that our results at this energy approximate very well the total NLO cross section. This is also consistent with the results in Ref. [25]. However, at higher energies, our enhancements become larger than the full NLO corrections.
For Higgs associated production, we find aNLO enhancements of the total top + antitop LO cross section of 5.2% at 8 TeV, 14.9% at 13 TeV, and 16.4% at 14 TeV. At aNNLO, we find enhancements over aNLO of 5.0% at 8 TeV, 4.4% at 13 TeV, and 4.5% at 14 TeV. The NLO enhancements from MadGraph5 aMC@NLO and from Refs. [25,26] are higher than ours, but still the soft-gluon corrections are a significant and dominant portion of the full corrections.
A detailed phenomenological study of these processes is beyond the scope of this work. We plan to further study these and other processes in future work.