Threshold resummation at N$^3$LL accuracy and soft-virtual cross sections at N$^3$LO

We consider QCD radiative corrections to the production of colourless high-mass systems in hadron collisions. We show that the recent computation of the soft-virtual corrections to Higgs boson production at N$^3$LO [1] together with the universality structure of soft-gluon emission can be exploited to extract the general expression of the hard-virtual coefficient that contributes to threshold resummation at N$^3$LL accuracy. The hard-virtual coefficient is directly related to the process-dependent virtual amplitude through a universal (process-independent) factorization formula that we explicitly evaluate up to three-loop order. As an application, we present the explicit expression of the soft-virtual N$^3$LO corrections for the production of an arbitrary colourless system. In the case of the Drell-Yan process, we confirm the recent result of Ref.[2].

The authors of Ref. [1] have recently presented the result of the calculation of the cross section for the threshold production of the Higgs boson at hadron colliders at the next-to-next-to-nextto-leading order (N 3 LO) in perturbative QCD. This result has prompted the observation [2] that the Higgs boson calculation contains information on soft-gluon radiation that can be implemented to explicitly determine the N 3 LO threshold cross section for the Drell-Yan (DY) process. In the present contribution, we exploit the universality (process-independent) structure [3] of softgluon contributions near partonic threshold and the specific calculation of Ref. [1]. We show how the results of Refs. [1] and [3] can be straightforwardly combined and used to extract the general expression of the hard-virtual coefficient that contributes to threshold resummation at next-to-next-to-next-to-leading-logarithmic (N 3 LL) accuracy for the cross section of a generic (and arbitrary) colourless high-mass system produced in hadron collisions. The threshold resummation formula for the production cross section can also be perturbatively expanded up to N 3 LO, and for the specific case of the DY process we recover the result of Ref. [2].
The N 3 LO Higgs boson results of Ref. [1] complete a cross section calculation that requires the evaluation of several independent ingredients related to collinear-counterterm factors [4,5] and to real- [6,7] and virtual-radiation [8][9][10] contributions. One of these ingredients is the three-loop virtual amplitude [9,10] gg → H for Higgs boson production through gluon fusion (the three-loop results of Refs. [9,10] use the large-m top approximation). As discussed in Ref. [3], all-order softgluon resummation [11][12][13] for the hadroproduction cross section of a generic colourless high-mass system can be expressed in a process-independent form, whose sole process-dependent information is encoded in the virtual amplitude of the specific process. Therefore, using the cross section of Ref. [1] and the virtual amplitude of Refs. [9,10] for the specific case of Higgs boson production, we can apply the formulation of Ref. [3] and we can explicitly determine the entire process-independent information that contributes to soft-gluon resummation for a generic production process up to the three-loop level. In the following we recall the formalism of soft-gluon resummation (by mainly following the notation of Sect. 5 in Ref. [3]) and we present and illustrate our three-loop results.
We consider the inclusive hard-scattering reaction where the collision of the two hadrons h 1 and h 2 with momenta p 1 and p 2 produces the triggered final state F , and X denotes the accompanying final-state radiation. The observed final state F is a generic system of one or more colourless particles (with momenta q i ), such as lepton pairs (produced by the DY mechanism), photon pairs, vector bosons, Higgs boson(s), and so forth. We focus on the total cross section † for the process in Eq. (1) at fixed value M of the invariant mass of the triggered final state F (i.e., we integrate the differential cross section over the momenta q i with the constraint ( i q i ) 2 = M 2 ). In the simplest case, the final-state system F consists of a single ('on-shell') particle of mass M (for example, F can be a vector boson or a Higgs boson). The total cross section σ F (p 1 , p 2 ; M 2 ) for the production of the system F is computable in QCD perturbation theory according to the following factorization formula: where s = (p 1 + p 2 ) 2 ≃ 2p 1 · p 2 ,σ F a 1 a 2 is the total partonic cross section for the inclusive partonic process a 1 a 2 → F +X and, for simplicity, the parton densities f a i /h i (z i , M 2 ) (i = 1, 2) are evaluated at the scale M 2 (the inclusion of an arbitrary factorization scale µ F in the parton densities and in the partonic cross sections can be implemented in a straightforward way by using the Altarelli-Parisi evolution equations of f a/h (z, µ 2 F )). The partonic cross sectionσ F a 1 a 2 (ŝ; M 2 ; α S (M 2 )) depends on the mass M of the system F , on the centre-of-mass energy √ŝ of the colliding partons a 1 and a 2 , and it is a renormalization-group invariant quantity that can be perturbatively computed as series expansion in powers of α S (M 2 ). Considering, for instance, the inclusive partonic channel cc → F + X, we can writê and σ (0) cc→F is the lowest-order cross section for the partonic process cc → F . Since the system F is colourless, the lowest-order cross section is determined by the partonic processes of quarkantiquark annihilation (c = q,q) and/or gluon fusion (c = g) (in the case of qq-annihilation the quark and antiquark can have different flavours, such as, for instance, if F = W ± ). Perturbative expressions that are analogous to Eq. (3) can be written for the partonic cross sectionsσ F a 1 a 2 of all the other partonic channels. Using the renormalization-group evolution of the QCD running coupling α S (q 2 ), we can equivalently expandσ F a 1 a 2 in powers of α S (µ 2 R ), with corresponding perturbative coefficients g F (n) a 1 a 2 that explicitly depend on M 2 /µ 2 R , where µ R is an arbitrary renormalization scale. Throughout the paper we use parton densities as defined in the MS factorization scheme, and α S (q 2 ) is the QCD running coupling in the MS renormalization scheme.
The kinematical variable z = M 2 /ŝ in Eq. (3) parametrizes the distance from the partonic threshold. The limit z → 1 specifies the kinematical region that is close to the partonic threshold. In this region the partonic cross sectionσ F a 1 a 2 receives large QCD radiative corrections that are proportional to the singular functions where the subscript '+' denotes the customary 'plus-distribution'. The all-order perturbative resummation of these logarithmic contributions (including all the singular contributions that are proportional to δ(1 − z))) can be systematically performed by working in Mellin (N-moment) space [11,12]. The Mellin transformσ N (M 2 ) of the partonic cross sectionσ(ŝ; M 2 ) is defined aŝ In Mellin space, the threshold region z → 1 corresponds to the limit N → ∞, and the plusdistributions of Eq. (5) become powers of ln N ( 1 1−z ln m (1 − z) + → ln m+1 N +'subleading logs ′ ). These logarithmic contributions are evaluated to all perturbative orders by using threshold resummation [11,12]. Neglecting terms that are relatively suppressed by powers of 1/N in the limit Note that we are considering only the partonic channel cc → F + X, with cc = qq and/or cc = gg, since the other partonic channels give contributions that are of O(1/N). In this paper, we use the Mellin-space formalism of threshold resummation [11,12] that we have just introduced. Related formulations of threshold resummation for hadron-hadron collisions can be found, for instance, in Ref. [17] (which is exploited to derive the results of Ref. [2]) and in Refs. [18][19][20].
The expressionσ cc, N in the right-hand side of Eq. (7) embodies all the perturbative terms that are logarithmically enhanced or constant in the limit N → ∞. The partonic cross section σ F (res) cc, N has a universal (process-independent) all-order structure that is given by the following threshold-resummation formula [11][12][13][21][22][23]: The factor σ (0) cc→F obviously depends on the produced final-state system F , and it is simply proportional to the square of the lowest-order scattering amplitude M cc→F (see Eq. (22)) of the partonic process cc → F . The factor C th cc→F also depends on the produced final-state system F and, therefore, it includes a process-dependent component. The factor ∆ c, N is process-independent: it does not depend on the final-state system F , and it only depends on the type (c = q or c = g) of colliding partons.
The factor ∆ c, N is entirely due to soft-parton radiation [11,12]. This radiative factor resums all the perturbative contributions α n S ln m N (including some constant terms, i.e. terms with m = 0), and it has the following all-order form: where A c (α S ) and D c (α S ) are perturbative series in α S , The function A c (α S ) is produced by radiation that is soft and collinear to the direction of the colliding partons c andc. The effect of soft non-collinear radiation is embodied in the function D c (α S ). The perturbative coefficients A where n F is the number of quark flavours, N c is the number of colours, and the colour factors are The colour coefficient C c depends on the type c of colliding partons, and we have Using Eq. (9), the coefficients A c [23] indicate that this coefficient can have a small quantitative effect in practical applications of threshold resummation. By direct inspection of Eqs. (12) and (13), we note that the dependence on c (the type of colliding parton) of the perturbative functions A c (α S ) and D c (α S ) is entirely specified up to O(α 3 S ) by the overall colour factor C c . To highlight this overall dependence, we introduce the notation so that γ c /C c (see Eq. (12)) are universal QCD coefficients (namely, they do not depend on the type c of colliding parton). This overall dependence on C c , which is customarily named as Casimir scaling relation, follows from the soft-parton origin of both A c (α S ) and D c (α S ), and it is eventually a consequence of non-abelian exponentiation [29] for soft-gluon radiation. The validity of the Casimir scaling relation (14) beyond O(α 3 S ) is a subject of current theoretical investigations (see Ref. [30] and references therein). More detailed comments on the structure of soft-gluon radiation are postponed below Eq. (42).
In this paper we focus on the threshold-resummation factor C th cc→F . The factor C th cc→F embodies all the remaining N-independent contributions (i.e., terms that are constant in the limit N → ∞) to the partonic cross section in Eq. (8). This factor is definitely process dependent, and it has the general perturbative expansion Despite its process dependence, in Ref. [3] we have discussed and shown that the all-order factor C th cc→F (α S ) involves a minimal amount of process-dependent information. This information is entirely due to the renormalized all-loop scattering amplitude M cc→F of the (elastic-production) partonic process cc → F . Having M cc→F , we can introduce the corresponding hard-virtual amplitude M th cc→F for threshold resummation by using a process-independent (universal) factorization formula that has the following all-order expression [3]: The subtraction operatorĨ th c (ǫ, M 2 ) in Eq. (16) is a renormalization-group invariant quantity that does not depend on the specific final-state system F : it only depends on the type (c = q or c = g) of colliding partons and on a scale that is set by the invariant mass M of the system F . The factor C th cc→F (α S ) is then directly related to M th cc→F . In the simple case where the system F consists of a single particle of mass M, the direct relation is [3] α 2k where the value k of the power of α S (M 2 ) and the lowest-order amplitude M cc→F are precisely defined in Eq. (22). The relation in Eq. (17) can be straightforwardly generalized to the more general case where the system F is formed by two or more particles with momenta q i (see Eq. (1)). The generalization simply follows from the fact that we are considering the cross section integrated over the final-state momenta q i and, therefore, we have  (16) is completely independent of the final-state momenta q i and, therefore, the q i -dependence of M th cc→F ({q i }) is entirely and directly given by the q i -dependence of the scattering amplitude M cc→F ({q i }). In Ref. [3] we obtained the explicit expression of the subtraction operatorĨ th c up to the second order in the QCD coupling α S . In this paper we extend those results and computeĨ th c to the third order in α S .
Before presenting our results, we give more details on the notation that is used in Eqs. (16)- (18). The all-loop scattering amplitude M cc→F of the partonic process cc → F contains ultraviolet (UV) and infrared (IR) singularities, which are regularized in d = 4 − 2ǫ space-time dimensions. To be definite we use the customary scheme of conventional dimensional regularization (CDR). Before performing renormalization, the multiloop QCD amplitude has a perturbative dependence on powers of α u S µ 2ǫ 0 , where α u S is the bare coupling and µ 0 is the dimensional-regularization scale. In the following we work with the renormalized on-shell scattering amplitude that is obtained from the corresponding unrenormalized amplitude by just expressing the bare coupling α u S in terms of the running coupling α S (µ 2 R ) according to the MS scheme relation where γ E is the Euler number, µ R is the renormalization scale and β 0 , β 1 and β 2 are the first three coefficients of the QCD β-function [8]: The renormalized all-loop amplitude M cc→F has the perturbative (loop) expansion where the value k of the overall power of α S depends on the specific process (for instance, k = 0 in the case of the vector boson production process qq → V , and k = 1 in the case of the Higgs boson production process gg → H through a heavy-quark loop). Note that the lowest-order term M cc→F is not necessarily a tree-level amplitude (for instance, it involves a quark loop in the cases gg → H and gg → γγ). If F is a multiparticle system, using the shorthand notation of Eq. (18), we can write the lowest-order cross section as which (implicitly) fixes the overall normalization of the phase space integration. The perturbative terms M (l) cc→F (l = 1, 2, 3, . . . ) are UV finite, but they still depend on ǫ: in particular, they contain ǫ-pole contributions and, therefore, they are IR divergent as ǫ → 0. The IR divergent contributions to the scattering amplitude M cc→F have a universal (process-independent) structure [31][32][33][34] that is explicitly known up to the three-loop (l = 3) level [35]. The subtraction operatorĨ th c (ǫ, M 2 ) in Eq. (16) has the perturbative expansioñ and the perturbative termsĨ th(n) c (ǫ) contain IR divergent contributions (ǫ-poles) and a definite amount of IR finite contributions. The IR divergent contributions toĨ th c (ǫ, M 2 ) are exactly those that are necessary to cancel the IR divergences of the renormalized all-loop amplitude M cc→F . Therefore, the hard-virtual amplitude M th cc→F in Eq. (16) is IR finite order-by-order in perturbation theory, and it can be evaluated in the limit ǫ → 0. The threshold resummation coefficient C th cc→F (α S (M 2 )) can be directly computed in the four-dimensional limit ǫ → 0 (though, this limit is not explicitly denoted in the right-hand side of Eqs. (17) and (18)). The perturbative expansion of M th cc→F is completely analogous to that of M cc→F (see Eq. (22)) with the replacement M (ǫ) at equal or lower orders, i.e. with l ≤ n (see, e.g., Eqs. (48) and (49) in Ref. [3]). For simplicity, the perturbative expansions on the right-hand side of Eqs. (22) and (24) are expressed in powers of α S (M 2 ). Note, however, that M cc→F andĨ th c (ǫ, M 2 ) are separately renormalization-group invariant quantities. Therefore, they can be equivalently expanded as powers series in α S (µ 2 R ), with corresponding perturbative terms that depend on M 2 /µ 2 R (see, e.g., Eqs. (50)-(57) in Ref. [3])). The equivalent expansions are simply obtained by using Eq. (19) to directly express α S (M 2 ) in terms of α S (µ 2 R ) and integer powers of (M 2 /µ 2 R ) −ǫ .
In Ref. [3] we derived the explicit expression of the first-order and second-order subtraction operatorsĨ th(1) c (ǫ) andĨ th(2) c (ǫ). To extend the results to the third order, we introduce a more compact (though completely equivalent) all-order representation. The operatorĨ th c (ǫ, M 2 ) can be written as where R c and Φ c are real functions. The function Φ c (ǫ, M 2 ) is the IR divergent Coulomb phase that originates from the virtual contributions to the all-loop amplitude M cc→F . Its explicit expression The function R c (ǫ, α S ) contains IR finite terms and all the remaining IR divergent terms (in the limit ǫ → 0) in the exponent of Eq. (25). This perturbative function can be decomposed as follows: where The two components R soft c and R coll c of Eq. (27) have a soft and collinear origin, respectively. The ǫ-dependent perturbative coefficients on the right-hand side of Eqs. (28) and (29) read The coefficients γ c , γ  (35) depend on the parton flavour c = q, g and they have a collinear origin. They are equal to the coefficients of the term proportional to δ(1 − z) (i.e., to the virtual contribution) in the leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) collinear splitting functions [4], and their explicit values ‡ are The coefficients d (1) and d (2) in Eqs. (32) and (34) have a soft origin, and their values read The coefficients R fin(1) and R fin(2) determine the IR finite part on the right-hand side of Eqs. (30) ‡ In Ref. [3] we used a slightly different notation, and the coefficient γ c(1) therein is related to γ The first-order and second-order results in Eqs. (30)- (33) were obtained in Ref. [3]. The three-loop expressions in Eqs. (34) and (35) We note that the phase factor e −iΦc in Eq. (25) is physically (and practically) harmless to the purpose of computing the threshold resummation coefficient C th cc→F in Eqs. (17) and (18). Indeed, e −iΦc produces a corresponding overall phase factor contribution to M th cc→F in Eq. (16) and, therefore, e −iΦc gives a vanishing contribution to | M th cc→F | 2 and, hence, to C th cc→F . We recall [3] that this phase factor has been introduced inĨ th c to the sole practical (aesthetical) purpose of cancelling the IR divergent Coulomb phase of the virtual amplitude M cc→F , so that M th cc→F itself (and not only | M th cc→F | 2 ) is IR finite in the limit ǫ → 0. We note that M th cc→F can also be redefined by including equally harmless contributions that are purely real (rather than phase factors). We can consider a multiplicative redefinition M th cc→F → F (α S , ǫ) M th cc→F , where F is an arbitrary perturbative function (i.e., F = 1 + O(α S )) such that it is equal to unity in the limit ǫ → 0 (i.e., F = 1 + O(ǫ m ) with m = 1, 2, . . . ). Since M th cc→F is IR finite, this multiplicative redefinition gives a vanishing contribution to M th cc→F in the four-dimensional limit ǫ → 0. Such harmless multiplicative redefinition corresponds to the replacement ( (ǫ) that are presented in Ref. [3] include contributions that are due to this type of harmless terms.
The derivation of the factorization formula (16), its origin and the general structure of the subtraction operatorĨ th c (ǫ, M 2 ) in Eq. (25) were discussed in Ref. [3]. Here we limit ourselves to § In Ref. [3], the IR finite part ofĨ is specified by using a different notation in terms of the coefficients δ th (1) and δ th (2) therein.
presenting the main conclusions of our reasoning [3] in a very concise form (we refer to Sects. 4.1 and 5 of Ref. [3] for an extended discussion). We have already recalled the origin of the phase factor e −iΦc in Eq. (25). We then recall [3] that the remaining contributions toĨ th c (i.e., the factor e Rc in Eq. (25)) have a soft and collinear origin, as specified by the decomposition in Eq. (27). The collinear contributions are embodied in the factor e R coll c , and they are entirely due to the virtual part of the collinear-counterterm factor that is introduced in the (bare) partonic cross sections to factorize the MS parton densities (see Eq. (2)). Since we are considering parton densities in the MS factorization scheme, this collinear-counterterm factor is completely and explicitly specified up to O(α 3 S ) [4] and, in particular, the perturbative function R coll c (ǫ, α S ) in Eq. (29) includes only ǫ-pole contributions (see Eqs. (31), (33) and (35)) with no additional IR finite terms. The soft contributions toĨ th c are embodied in the factor e R soft c . They are due to the soft part of the MS collinear counterterm [4] and to the inelastic processes cc → F + X, where the radiated finalstate system X includes only soft partons. The soft-parton contribution of the inelastic processes can be determined by using universal (process-independent) soft factorization formulae [36][37][38][39][40] of the corresponding scattering amplitudes. In Ref. [41], the soft-parton contribution to the total cross section was explicitly computed up to NNLO in a process-independent form by using soft factorization formulae up to O(α 2 S ) [37][38][39]. A corresponding process-independent calculation at N 3 LO can be performed by using soft factorization formulae at O(α 3 S ) [7,42]. As discussed in Ref. [42], soft-factorization results from Refs. [7,38,39,42] and the soft limit of the results in Refs. [6] can be combined and used to reproduce [42] the results of the N 3 LO cross sections for Higgs boson [1] and DY production [2]. However, as discussed and pointed out in Ref. [3], much information on the soft contribution toĨ th c can be obtained independently of detailed computations. Indeed, due to non-abelian eikonal exponentiation [29], the intensity of soft radiation from the parton c is simply proportional to the Casimir coefficient C c of that parton (this conclusion is certainly valid up to O(α 3 S ) [29]). Therefore, R soft c (ǫ, α S ) can be expressed by factorizing the overall coefficient C c as in Eq. (28). This Casimir scaling behaviour is completely analogous to that of the functions A c (α S ) (see Eq. (14)), D c (α S ) (see Eqs. (11) and (13)) and Φ c (ǫ, α S ) (see Eq. (26)), since all these functions are entirely due to soft-parton contributions [3]. The perturbative coefficients R soft(n) (ǫ), with n = 1, 2, 3, in Eq. (28) are completely process independent and they can be determined by considering a single specific process. In particular, R soft(n) (ǫ) contains IR divergent contributions (ǫ-pole terms) and IR finite contributions. These IR divergent terms of soft-parton origin are due to real emission contributions, but they are constrained (because of the real-virtual cancellation mechanism of IR divergences) to be exactly equal to the corresponding IR divergent terms due to virtual radiation. Therefore, the ǫ-pole terms in Eqs. (30), (32) and (34) are completely specified by the explicit calculation of either the quark or gluon form factors [35] (as recalled below, the process independence of these terms is consistent with the universality structure of the IR divergent contributions to the QCD scattering amplitudes [31,33,34]). It follows that the IR finite coefficients R fin(n) (n = 1, 2, 3) are the only terms that are not explicitly determined by using our general reasoning [3]. Owing to their universality, the explicit computation of a single process is sufficient to extract the values of these IR finite coefficients. As illustrated below, we use the N 3 LO Higgs boson results of Ref. [1] to obtain the value of R fin (3) in Eq. (42).
Before considering the evaluation of R fin(3) , we present some additional comments on the structure of Eqs. (25)- (39) and on the connection between real-and virtual-emission contributions. As we have discussed, the subtraction operator (1 −Ĩ th c ) in Eqs. (16) and (25) includes the Coulomb phase factor e −iΦc and an additional factor of soft and collinear origin. In Eq. (25) we express this additional factor by using the exponentiated form e Rc . The exponentiated form, which is completely equivalent to its direct expansion in powers of α S , is more compact in view of the factorization and exponentiation properties of both soft and collinear contributions. Owing to factorization we can write e Rc = e R coll c e R soft c , i.e. we can introduce the decomposition in Eq. (27). The collinear factor e R coll c is entirely due to the virtual part of the collinear counterterm of the MS parton densities, and its exponentiated structure is eventually a consequence of the customary solution of the Altarelli-Parisi evolution equations in terms of an exponentiated evolution operator. Indeed (as stated below Eq. (35)) the exponent R coll c is directly determined by the coefficients γ c , γ  (39). We also note that the real emission contribution to the partonic cross section of Eq. (8) is separated in two different factors: the N-independent factor e R soft c (which contributes to (1 −Ĩ th c ) and, hence, to C th cc→F ) and the ln N-dependent radiative factor ∆ c, N of Eq. (9). These two factors have a soft origin and they are not fully independent. In particular, the coefficients of the dominant IR poles of R soft c (ǫ, α S ) are directly related to the dominant ln N-dependence of ∆ c, N (as given by the perturbative function A c (α S )). The subdominant ln N-dependence of ∆ c, N is due to the soft-parton function D c (α S ), whose perturbative coefficients D (n) c are related to the soft-parton coefficients C c d (n−1) and C c R fin(n−1) of R soft c (ǫ, α S ): this relation between ln N terms, ǫ-poles and IR finite terms is discussed and worked out in Refs. [27,28]. We note that using the general analysis of Refs. [27,28] and our result for R fin (3) in Eq. (42), the fourth-order coefficient D (4) c of D c (α S ) can be determined in terms of the ǫ-poles at O(α 4 S ) (once they become available).
To evaluate the third-order coefficient R fin(3) , we consider the perturbative expansion of the resummation formula in Eq. (8), which contains all the terms which are not suppressed in the large-N limit, namely, the logarithmically-enhanced terms and the constant terms as N → ∞. We consider the N 3 LO contribution (see, e.g., the Appendix E in Ref. [22]) and we transform it back to z space to obtain the general expression of the N 3 LO term g where D m = D m (z) are the plus-distributions defined in Eq. (5), and the dots in the right-hand side of Eq. (43) denote additional terms that are less singular in the limit z → 1 (i.e., terms that are relatively suppressed by some powers of (1 − z)). The terms that are explicitly denoted in the right-hand side of Eq. (43) define the soft-virtual (SV) approximation of the N 3 LO contribution g (see Eqs. (12) and (13)) and on the process-dependent coefficients C th(n) cc→F with n ≤ 3.
In the case of Higgs boson production (gg → H) by gluon fusion, the SV N 3 LO expression in Eq. (43) exactly corresponds to the result of the explicit computation performed in Ref. [1]. The first-order and second-order coefficients C th(1) gg→F and C th (2) gg→F are known (they can be determined by our process-independent resummation formalism up to O(α 2 S ) or, equivalently, they can be extracted from the SV NNLO results of Refs. [26,43] To be precise, the coefficient C th (3) gg→H in Eq. (44) corresponds to the perturbative expansion that is defined by Eq. (3) after having rescaled the partonic cross section with the Wilson coefficient of the effective point-like coupling ggH [9] (this definition exactly corresponds to that used in Eq. (4) of Ref. [1]). Having the information in Eq. (44) and using Eqs. (16) and (17), we apply the operator (1 −Ĩ th c ) of Eq. (25) to the three-loop gluon form factor [10] and we can extract the coefficient R fin (3) in Eq. (34). We find the explicit value that is presented in Eq. (42).
The coefficient R fin(3) completely determines the explicit expression of the process-independent subtraction operatorĨ th c up to O(α 3 S ). Using this expression and Eqs. (16)-(18), the threshold resummation coefficient C th cc→F (α S ) for an arbitrary process cc → F is straightforwardly and explicitly computable up to the three-loop order once the corresponding three-loop scattering amplitude M cc→F for that process is known.
As an application of our general formalism and results, we can consider the production of a vector boson V (V = Z, W ± ) by the DY process qq → V . Using the subtraction operator (1 −Ĩ th c ) and the results for the quark form factor up to three-loop order [10], we can compute the coefficients C th(n) qq→V with n = 1, 2, 3. We find C th(1)