NNLO hard functions in massless QCD

Journal of High Energy Physics, Dec 2014

We derive the hard functions for all 2 → 2 processes in massless QCD up to next-to-next-to-leading order (NNLO) in the strong coupling constant. By employing the known one- and two-loop helicity amplitudes for these processes, we obtain analytic expressions for the ultraviolet and infrared finite, minimally subtracted hard functions, which are matrices in color space. These hard functions will be useful in carrying out higher-order resummations in processes such as dijet and highly energetic top-quark pair production by means of soft-collinear effective theory methods.

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NNLO hard functions in massless QCD

Alessandro Broggio 1 4 Andrea Ferroglia 1 2 3 Ben D. Pecjak 0 1 Zhibai Zhang 1 2 3 CH- 1 Villigen PSI 1 Switzerland 1 DH 1 LE Durham 1 U.K. 1 Open Access 1 c The Authors. 1 0 Institute for Particle Physics Phenomenology, University of Durham 1 The City University of New York 2 Physics Department, New York City College of Technology 3 The Graduate School and University Center, The City University of New York 4 Paul Scherrer Institut We derive the hard functions for all 2 2 processes in massless QCD up to next-to-next-to-leading order (NNLO) in the strong coupling constant. By employing the known one- and two-loop helicity amplitudes for these processes, we obtain analytic expressions for the ultraviolet and infrared finite, minimally subtracted hard functions, which are matrices in color space. These hard functions will be useful in carrying out higher-order resummations in processes such as dijet and highly energetic top-quark pair production by means of soft-collinear effective theory methods. 1 Introduction Hard functions to NNLO: calculational procedure Hard functions to NNLO: results Four-quark scattering Two-quark two-gluon scattering q(p1) + q(p2) Q(p3) + Q(p4) q(p1) + Q(p2) q(p3) + Q(p4) q(p1) + Q(p2) q(p3) + Q(p4) q(p1) + Q(p2) Q(p3) + q(p4) q(p1) + q(p2) q(p3) + q(p4) q(p1) + q(p2) q(p3) + q(p4) g(p1) + g(p2) q(p3) + q(p4) q(p1) + g(p2) q(p3) + g(p4) q(p1) + g(p2) g(p3) + q(p4) Four-gluon scattering Introduction Some of the most fundamental processes at hadron colliders such as the LHC are mediated at leading order (LO) in perturbation theory by 2 2 scattering processes of colored particles two prime examples within the Standard Model are dijet and top-quark to all orders in perturbation theory. The factorization formulas underlying such resummations depend on the way in which the observable is sensitive to soft and collinear emissions, and are thus in general different for inclusive jet production [13], highly boosted top-quark pair production [4, 5], and by means of Soft Collinear Effective Theory (SCET) methods [8, 9]. However, a common ingredient to all resummations for 2 2 processes are so-called hard functions, which hard function for each possible 2 2 partonic process involving massless quarks and gluons, although all can be derived from those for qq QQ, qg qg, and gg gg scattering, where q and Q are distinct quarks. The hard functions are related to the interference of dimension for the different partonic scattering processes 2 2 for four-quark processes, 3 3 for qg qg processes, and 9 9 for the four-gluon process. The next-to-leading order logarithmic (NNLL) accuracy. The goal of the current work is to build on previous results by presenting the complete set of next-to-next-to-leading order (NNLO) hard functions. The main building blocks for 2 2 massless QCD processes calculated in [1115]. We turn these computations into results for the hard functions by performing an IR renormalization procedure on the with the arXiv submission of this work. To facilitate use by other groups, we also provide a Mathematica interface to the results. The hard functions we calculate in the present work are a necessary ingredient for pushing any resummed calculation of a dijet observable at hadron colliders to next-tofor the renormalization-group evolution equations to that order. However, especially in cases where NNLO results are known, it is a frequent practice to include these boundary terms on top of NNLL resummations to achieve NNLL+NNLO accuracy,1 even in the for practitioners of higher-order resummation in the near and distant future. The organization of the paper is as follows: we describe our calculational procedure in section 2, give results in section 3, and conclude in section 4. Hard functions to NNLO: calculational procedure The goal of this paper is to obtain the NNLO hard functions for all scattering processes with two incoming and two outgoing partons in massless QCD. These processes can be 1Here we use the nomenclature of, e.g., [16]. Including all possible crossings, the four-quark processes are where q and Q indicate quarks of different flavors. For scattering processes involving two quarks and two gluons, we focus on the following three possibilities: q(p1) + q(p2) Q(p3) + Q(p4) , q(p1) + Q(p2) q(p3) + Q(p4) , q(p1) + Q(p2) q(p3) + Q(p4) , q(p1) + Q(p2) Q(p3) + q(p4) , q(p1) + q(p2) q(p3) + q(p4) , q(p1) + q(p2) q(p3) + q(p4) , g(p1) + g(p2) q(p3) + q(p4) , q(p1) + g(p2) q(p3) + g(p4) , q(p1) + g(p2) g(p3) + q(p4) . There is also a qq gg process, but with our definitions its hard function is the same as (2.7) up to an overall factor (N 2 which accounts for the color average over the incoming quarks rather than the incoming involving quarks. We therefore omit these from the discussion. Finally, we consider the four-gluon scattering process g(p1) + g(p2) g(p3) + g(p4) . Here and in the following, we associate to the particle carrying momentum pi a helicity representation of SU(3) if the particle is a quark, and in the adjoint representation if the particle is a gluon. For use later on, we introduce the invariants s = (p1 + p2)2 , t = (p1 p3)2 , u = (p1 p4)2 , and r = t/s . which we will use to write our end results as functions of s, the t Hooft scale , and the dimensionless ratio r. A unique hard function is associated to each of the processes listed above. These can all be extracted using the two-loop helicity amplitudes calculated in [1115]. To describe color-space formalism of [17], which allows us to treat the different cases with a uniform in color space, whose perturbative expansions we define as |Mh(, r, s)i = 4s |M(h0)i + a set of coefficients |Mh i 2L|Mh i the helicity amplitudes. Moreover, we have suppressed the arguments of the expansion coefficients on the right-hand side.2 Finally, in order to follow SCET conventions for the The amplitudes can be further decomposed in a particular color-space basis as |M(hL)i = X where |CI i are basis vectors. The basis includes two vectors in processes involving four quarks and three vectors in processes involving two quarks and two gluons. For the four The helicity amplitudes contain IR poles in the dimensional regulator . 19]. We thus define renormalized amplitudes according to The exact form of the renormalization factor Z was determined up to two-loops by means of For reasons that will become apparent later on, we define the perturbative expansion of the renormalization factor as (with a slight abuse of notation inherited from [20]) Z1 (, r, s, ) = 1 + and (2.14). With this notation it is now a simple matter to write expressions for the hard functions to NNLO. We first define expansion coefficients through H(r, s, ) = states with two quarks, NR = N (N 2 1)2 for initial states with two gluons. In terms of the color-decomposed, IR and UV matrix elements read HI(0J) = HI(1J) = HI(2J) = 1 X M (h0I) M (h0J) , M (h0I) M (h1J) + M (h1I) M (h0J)i , M (h1I) M (h1J) + M (h0I) M (h2J) + M (h2I) M (h0J)i . The factor 1/4 in (2.18) is related to the average over the spin of the two incoming partons. The normalization of the expansion coefficients above (but not that of the hard function itself) then coincides with the mt 0 limit of the corresponding results for the production of massive top pairs [21]. The hard functions are 2 2 matrices for scattering processes involving four quarks and 3 3 matrices for processes involving two quarks and two gluon. The hard functions for the process involving four gluons are 9 9 matrices with our choice of color basis. All of the matrices are Hermitian. With our definitions, the hard function is related to the square of the renormalized amplitude as The matrix s(0) is a tree-level soft function, whose elements are defined as s(I0J) = hCI |CJ i . The color bases for the various processes we consider are specified in the next section, in s and r by NR With this conceptual framework in place, we now address the more practical issue of In all cases, we have used the helicity amplitudes evaluated in the t Hooft-Veltman (HV) scheme. The most straightforward way to use the information in those papers would be indeed used this straightforward (and tedious) method in obtaining our results. A slightly more streamlined method, detailed recently in [20], uses that [1115] do in the structure of IR poles written down in [22]. Therefore, constructing the MS subtracted schemes. To understand how to perform this switch, we first consider the typical split of As a concrete example, the one-loop helicity amplitudes calculated in [12] are written as (see (4.10) of that work) |M(h1)i = I(1)()|M(h0)i + |Mh while the two-loop helicity amplitudes are written as (see (4.11) of that work) is defined as I(, r, s, ) = 1 + malization factor Z in (2.15). The difference is that Z contains only pole terms while I contains both pole terms and some finite terms. Moreover, the 1/ pole term and finite parts of the two-loop coefficient I(2) were not fully specified in [22], but are instead parameterized in a function HR(2.)S. defined in equation (19) of that work. The authors of [1115] provide explicit expressions for this function in their calculations, but in such a way that the finite parts of I(2) are not the same in each paper. For these reasons, the finite remainders quoted in [1115] differ from the MS renormalized amplitudes (2.15). Instead, they can be viewed as renormalized amplitudes in a scheme defined by equations (2.22) of HR(2.)S.. To convert these finite-remainders to the MS scheme, one can insert (2.22), (2.23) into (2.15) to find i =|Mh (1),fini + I(1)() + Z(1)() |Mh i i =|Mh (2),fini + I(1)() + Z(1)() |Mh + hI(2)() + I(1)() + Z(1)() I(1)() + Z(2)()i |Mh i One can then recast this equation into an explicitly IR finite form. As explained above, the result depends on the choice of the single pole term in I(2). In the case where this term is identical to that in Z, i.e. adds no extra finite parts to I(2), one can write the result in as in [20] I(1)()+Z(1)() =C0 , 8 Ti Tj cusp ln C0 = X C1 = X sij sij sij 4 0 ln sij sij 96 0 The sums run over the unordered tuples (i, j) of distinct partons, Ti is the color generator many places, for example in appendix A of [20]; an explicit expression for the commutator appendix. We emphasize that in [1115] the single pole term in HR(2.)S. is multiplied by to the right-hand side of the second line of (2.26) upon expansion in . We do not list these MS-renormalized helicity amplitudes from those references using (2.25) and (2.26). We have calculated the MS-renormalized helicity amplitudes using both methods described above, and checked that they agree. We then used these amplitudes to construct we have performed on our channel and basis-dependent results, which we give in the next section. First, for the qq QQ and gg qq channels, we verified that the trace of funcin the latter reference were tested against the squared NLO and NNLO matrix elements for the processes in (2.1), (2.7), which can be found in [2426]. For the gg gg channel, on the other hand, we have checked (2.19) against the UV-renormalized squared matrix elements given in [27, 28]. In order to carry out this last comparison, it was necessary to renormalized away the IR poles from the squared amplitudes in [27, 28], this was done by employing once more the IR renormalization method of [18, 19]. Second, the hard functions for the channels in (2.2), (2.3), (2.4) were assembled not for the processes in (2.8), (2.9) were also obtained a second time from the amplitudes for the process in (2.7) by applying crossing symmetries. Third, we have checked that the hard functions satisfy the renormalization-group equations implied by (2.15). These take the form amplitudes is exactly known up to two loops [18, 19]. In those papers it is also conjectured and basis dependent results for the anomalous dimension in the next section. As a byproduct of our calculation we also evaluated the NLO hard functions, which were previously calculated in [10]. We find agreement with the results in that work, after we account for differences in notation.3 Hard functions to NNLO: results We now present our results for the hard functions. We split the discussion into three in turn are subdivided according to momentum crossings. In each case we define the channel-dependent color basis in which the hard function is calculated, and give analytic be either in the fundamental or adjoint representation, depending on the process). These vectors satisfy the relation h{a}|{b}i = a1b1 a2b2 a3b3 a4b4 . The action of the color operators Ti on the vectors |{a}i, which is needed to construct the basis-dependent expressions, is discussed in many references, see for example section 3.2 The main results of this work are the NNLO hard functions obtained through the last line of (2.18). The analytic results for these functions would fill about 100 pages, were they printed out explicitly. As by now customary in such situations, we instead include the results in electronic format with the arXiv submission of this work. All of the hard functions are stored in Mathematica input files which can be loaded in the accompanying Mathematica notebook. In the latter file, a simple function allows the user to obtain numerical values for the hard functions for the processes listed in (2.1)(2.10) once the desired perturbative order (LO, NLO, or NNLO) and the values of r, s, and Nl (the number of fermions) are specified. As a reference for other groups which might desire to carry out this calculation, we give explicit numerical results for the NLO and NNLO hard functions at a specific benchmark point in the subsections that follow. In all cases we use N = 3, Nl = 5, r = 5 1 , s = 2. 3Equation (55) in [10] defines a real hard function in the channels with two quarks and two gluons, while when applying the crossing relations listed in table 2 of [10] in cases where fermions are switched between the initial and final state, as this necessitates extra minus signs. Finally, there is a minor typo in table 5 of [10], which is related to the four-gluon channel: the color factors listed in the next to the last column of that table apply to the helicities labeled 7 and 8, while the ones listed in the last column of the table apply to the helicities labeled by the numbers 916. we use singlet-octet type color bases defined below, for which the tree-level soft function is4 s(0) = where CF = (N 2 lous dimensions are gathered in the subsections below. q(p1) + q(p2) Q(p3) + Q(p4) The color basis which we employ to describe the four-quark process in (2.1) is C1 h{a}|C1i = a1a2 a3a4 , C2 h{a}|C2i = tca2a1 ta3a4 . c with this choice, the tree-level hard function is H(0) = 1 2r + 2r2 point (3.2): H(1) = H(2) = 0.139210 + i 0.192224 22.1589 + i 70.2433 0.139210 i 0.192224 ! 22.1589 i 70.2433 ! 1 r sources, for example appendix A in [20]. 4The soft function in (3.3) differs by an overall factor N from the one some of us employed in previous work involving four quark partonic processes (see for example [21]). This is due to the definition of s(0) to the soft function for the channels involving two gluons, which can be found in (3.20). H(0) = 2 2r + r2 (N21)2 2NN221 N N N21 ! The NLO and NNLO matrices at the benchmark point (3.2) are H(1) = H(2) = 47.5473 + i 13.8859 1575.43 + i 608.692 47.5473 i 13.8859 ! 1575.43 i 608.692 ! q(p1) + Q(p2) q(p3) + Q(p4) The color basis which we employ to describe the four-quark process in (2.3) is The color basis which we employ to describe the four-quark process in (2.2) is the same one introduced in (3.4). The tree-level hard matrix is C1 h{a}|C1i = a3a2 a4a1 , C2 h{a}|C2i = tca3a2 ta4a1 . c In this channel, the tree-level hard matrix is identical to the one in (3.9). The NLO and NNLO matrices at the benchmark point (3.2) are H(1) = H(2) = 50.2365 i 13.8859 1350.52 i 209.622 50.2365 + i 13.8859 ! 1350.52 + i 209.622 ! N23 H(0) = 1 1 r (1 r)2 1 r 2CF 0 ! . (3.13) q(p1) + Q(p2) Q(p3) + q(p4) The color basis employed for the process in (2.4) is the one we wrote in (3.11). The LO hard function in this channel is H(1) = H(2) = 14.8225 + i 15.5573 538.071 i 2019.16 14.8225 i 15.5573 ! 538.071 + i 2019.16 ! q(p1) + q(p2) q(p3) + q(p4) We consider here the scattering process in (2.5), where we employ the color basis in (3.4). The tree-level hard matrix is in this case given by H(0) = (N 2 1) r N (r 1)2 + r 2 2 r N (r 1)2 + r 2 2 N{2r[N2r(2(r1)r+1)+2N(r1)2+r2]+4} N21 H(1) = H(2) = 126.555 i 2.12346 5787.25 i 2270.79 126.555 + i 2.12346 ! 5787.25 + i 2270.79 ! The NLO and NNLO matrices at the benchmark point (3.2) are H(1) = H(2) = 54.4697 + i 14.8627 1745.69 + i 590.364 54.4697 i 14.8627 ! 1745.69 i 590.364 ! q(p1) + q(p2) q(p3) + q(p4) The color basis which we employ in the process in (2.6) is the one in (3.11). The tree-level hard matrix is H(0) = (N 2 1) (N21)(1r)2 The NLO and NNLO matrices at the benchmark point (3.2) are is the same for each, and reads s(0) = V 0 N2 0 0 N24 With this basis, the tree-level hard matrix becomes H(0) = N1 21r 2(1r) + 2r 1 1 21r + 2(1r) + 4r 4r2 3 21r 2(1r) + 2r 1 . 1 1 The NLO and NNLO matrices at the benchmark point (3.2) are functions and anomalous dimensions are gathered in the subsections below. g(p1) + g(p2) q(p3) + q(p4) The quark-antiquark pair production in the gluon fusion channel, (2.7), is studied by employing the color basis C1 h{a}|C1i = a1a2a3a4 , C3 h{a}|C3i = da1a2ctca3a4 C2 h{a}|C2i = if a1a2ctc H(1) = 1.90813 i 0.304853 8.37732 + i 2.53415 H(2) = 25533..1240808+ii316434.2.08583 106.493 i 180.767 52.2319 106.493 + i 180.767 , (3.23) 5.83592 i 0.880136 5.83592 + i 0.880136 , 1 r 2 2 2 N N22N4 . (3.24) The coefficients of the expansion up to NNLO can be found for example in appendix A The tree-level hard function is N 2 r 2(1r) 2 2 r 2(1r) 2 21 1 r + 1 1r while the NLO and NNLO matrices at the benchmark point (3.2) are H(1) = 21.674.4093952+ii62.939.7971682 89.5442 + i 24.3766 278.010 H(2) = 717601.944.769ii33278..4690399 3080.95 + i 2025.95 9379.39 0.960743 89.5442 i 24.3766 , 3080.95 i 2025.95 , (3.27) 1 r 2 2 2 1 2 N N22N4 . 2 4N2 N 2N The color basis that we adopt in order to describe the process in (2.8) is C1 h{a}|C1i = a4a2a3a1 , C2 h{a}|C2i = if a4a2ctc C3 h{a}|C3i = da4a2ctca3a1 The tree-level hard function is H(0) = N1 1 21r 1 2r + 2r 2 1 4r + 21r + (1r)2 + 2r 1 21r 1 2r + 2r . (3.30) 4 1 2 N1 1 21r 1 2r + 2r 1 21r 1r + 2r The color basis employed in order to describe the scattering process in (2.9) is C1 h{a}|C1i = a3a2a4a1 , C2 h{a}|C2i = if a3a2ctc C3 h{a}|C3i = da2a3ctca4a1 H(1) = 17.7615 i 29.2185 H(2) = 3153.14 + i 6008.72 2.78334 2.76521 + i 6.64060 124.257 + i 3.26543 786.213 i 1640.19 4658.03 + i 1019.52 124.257 i 3.26543 , 4658.03 i 1019.52 . NLO and NNLO matrices at the benchmark point (3.2) are Four-gluon scattering For the four-gluon scattering case (2.10) we adopt the color basis used in [15], namely C1 h{a}|C1i = 4Tr [ta1 ta2 ta3 ta4 ] , C2 h{a}|C2i = 4Tr [ta1 ta2 ta4 ta3 ] , C3 h{a}|C3i = 4Tr [ta1 ta4 ta2 ta3 ] , C4 h{a}|C4i = 4Tr [ta1 ta3 ta2 ta4 ] , C5 h{a}|C5i = 4Tr [ta1 ta3 ta4 ta2 ] , C6 h{a}|C6i = 4Tr [ta1 ta4 ta3 ta2 ] , C7 h{a}|C7i = 4Tr [ta1 ta2 ] Tr [ta3 ta4 ] , C8 h{a}|C8i = 4Tr [ta1 ta3 ] Tr [ta2 ta4 ] , C9 h{a}|C9i = 4Tr [ta1 ta4 ] Tr [ta2 ta3 ] . H(0) = c e f The color basis in (3.32) is over-complete. The factor of 4 in the r.h.s. of (3.32) arises from the fact that the authors of [15] define their color basis by employing color matrices The tree-level hard function for the process in (2.10) is where the elements a, ,f are b = + r 1 r d = (1 r)2 1 r + r 1, The NLO hard function for the four-gluon scattering process of (2.10) depends on nine independent functions and has the following structure: where the non-zero elements at the benchmark point (3.2) are a1 = 68.8613, d1 = 179.607, b1 = 111.212 + i18.1565, e1 = 256.410 i42.2061, The NNLO matrix has the structure c1 = 158.807 i55.4626, f1 = 359.541, i1 = 62.8973 + i58.5533. c1 e 1 e 1 h a1 b1 c1 c1 b1 a1 g1 g1 g1 b1 d1 e1 e1 d1 b1 h1 h1 h1 1 f1 f1 e1 c1 i1 i 1 f1 f1 e1 c1 i1 i H(1) = b 1 d1 e1 e1 d1 b1 h1 h1 h1 , a1 b1 c1 c1 b1 a1 g1 g1 g1 1 i 1 i 1 i 1 i 1 h 1 h 1 g 1 g 1 0 0 0 1 0 0 0 1 h1 i1 i1 h 1 g1 0 0 0 a2 b2 c2 c2 b2 a2 g2 j2 m2 b2 d2 e2 e2 d2 b 2 h2 k2 n2 2 e 2 e 2 f2 f2 e2 c 2 f2 f2 e2 c H(2) = b 2 d2 e2 e2 d2 b 2 h 2 i 2 i 2 h 2 g 2 l 2 l 2 k 2 j m2 n2 o2 o2 n2 m2 p2 p2 p2 2 i2 l2 o2 2 h2 k2 n2 , 2 p2 p2 p2 and the value of the 16 independent elements at the benchmark point (3.2) is a2 = 2106.67 , d2 = 5188.15 , g2 = 1930.67i 5041.68 , j2 = 9.86728+i 3871.76 , m2 = 148.769i 202.044 , b2 = 3196.18+i 3422.95 , e2 = 8129.75+i 692.435 , h2 = 2747.40i 8529.46 , k2 = 392.448+i 5892.80 , n2 = 135.769i 698.760 , p2 = 1041.49 . c2 = 4797.70i 4902.02 , f2 = 13732.3 , i2 = 3470.79+i 13852.3 , l2 = 1609.60i 9483.43 , o2 = 1194.28+i 1181.94 , = 2N cusp (s) ln 2 i + 4g (s) 1 where the matrices M1 and M2 are M1 = 0 0 M2 = 0 0 N N 1 1 0 0 N 0 0 0 0 1 0 0 0 1 0 0 0 1 0 , 0 N 0 0 1 0 1 1 0 0 0 0 1 N 0 1 1 r 1 1 Finally, the tree-level soft function is s(0) = C2 N C2 N N V N V N N N V N V N 2V N N V N V N V N V N N V N N V N V N N V N N N N , with C1 = N 4 We have given results for the spin-averaged hard functions for all 2 2 scattering processes in massless QCD up to NNLO in the strong coupling constant. These hard functions are a necessary ingredient for resummations in processes mediated by 2 2 scatterings at Born level, typical examples being dijet and boosted top production. We extracted our results from NNLO calculations of UV-renormalized helicity ampliidea is to interpret the IR poles in the color-decomposed helicity amplitudes as the UV which we specified in section 3. In all cases we performed several non-trivial cross-checks 2 2 matrices for four-quark processes, 3 3 matrices for two-quark two-gluon processes, and 9 9 matrices for the four-gluon process. We have listed their explicit numerical values at a benchmark point in section 3, which will facilitate future cross-checks. Moreover, we paper to ensure their easy accessibility. Our results will thus be useful for practitioners of higher-order resummations in the near and distant future. Acknowledgments No. 66590-00-44 and by the National Science Foundation Grant No. PHY-1417354. 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Alessandro Broggio, Andrea Ferroglia, Ben D. Pecjak. NNLO hard functions in massless QCD, Journal of High Energy Physics, 2014, 5, DOI: 10.1007/JHEP12(2014)005