Edinburgh Research Explorer

Transcription

1 Edinburgh Research Explorer Higgs production via gluon-gluon fusion with finite top mass beyond next-to-leading order Citation for published version: Marzani, S, Ball, RD, Del Duca, V, Forte, S & Vicini, A 8, 'Higgs production via gluon-gluon fusion with finite top mass beyond next-to-leading order' Nuclear Physics B, vol 8, no. 1-, pp DOI: 1.116/j.nuclphysb Digital Object Identifier DOI: 1.116/j.nuclphysb Link: Link to publication record in Edinburgh Research Explorer Document Version: Accepted author manuscript Published In: Nuclear Physics B Publisher Rights Statement: Author's Post-print: author can archive post-print ie final draft post-refereeing General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the authors and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately and investigate your claim. Download date:. Feb. 18

2 IFUM-911-FT Edinburgh 8/1 CERN-PH-TH/8-9 Higgs production via gluon-gluon fusion with finite top mass beyond next-to-leading order arxiv:81.544v1 [hep-ph] 16 Jan 8 Simone Marzani, a, Richard D. Ball, a,b Vittorio Del Duca, c1 Stefano Forte d and Alessandro Vicini d a School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK b CERN, Physics Department, Theory Division, CH-111 Genève 3, Switzerland c INFN, Laboratori Nazionali di Frascati, Via E. Fermi 4, I-44 Frascati, Italy d Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-133 Milano, Italy Abstract We present a computation of the cross section for inclusive Higgs production in gluon gluon fusion for finite values of the top mass in perturbative QCD to all orders in the limit of high partonic center of mass energy. We show that at NLO the high energy contribution accounts for most of the difference between the result found with finite top mass and that obtained in the limit m t. We use our result to improve the known NNLO order result obtained at m t. We estimate the effect of the high energy NNLO m t dependence on the K factor to be of the order of a few per cent. CERN-PH-TH/8-9 January 8 1 On leave from INFN, Sezione di Torino, Italy

3 1 The cross section in the soft limit and in the hard limit The determination of higher order corrections to collider processes [1], and specifically Higgs production [] in perturbative QCD is becoming increasingly important in view of forthcoming phenomenology at the LHC. The dominant Higgs production mechanism in the standard model is inclusive gluon gluon fusion gg H + X through a top loop. The next to leading order corrections to this process were computed several years ago [3, 4] and turn out to be very large of order 1%. The bulk of this large correction comes from the radiation of soft and collinear gluons [5], which give the leading contribution in the soft limit in which the partonic center-ofmass energy ŝ tends to the Higgs mass m H, and which at LHC energies turns out to dominate the hadronic cross section after convolution with the parton distributions. This dominant contribution does not resolve the effective gluon-gluon-higgs ggh coupling induced by the top loop. As a consequence, the NLO correction can be calculated [6,7] quite accurately in the limit m t, where it simplifies considerably because the ggh coupling becomes pointlike and the corresponding Feynman diagrams have one less loop. Recently, the NNLO corrections to this process have been computed in the m t limit [8]. The NNLO result appears to be perturbatively quite stable, and this stability is confirmed upon inclusion [9] of terms in the next few orders which are logarithmically enhanced as ŝ m H, which can be determined [1] using soft resummation methods. This suggests that also at NNLO the large m t approximation should provide a good approximation to the yet unknown exact result. However, this is only true for the total inclusive cross section: for example, if one looks at the production of Higgs plus jets, if the transverse momentum is large the infinite m t approximation fails [11]. Indeed, even though the m t independent contribution from soft and collinear radiation turns out to dominate the cross section at the hadronic level, it does not necessarily provide a good approximation to the partonic cross section in a fixed kinematical region. In particular, the infinite m t approximation, which becomes exact in the soft limit, fails in the opposite hard limit of large center of mass energy. This is due to the fact that the ggh vertex is pointlike in the infinite m t limit, whereas for finite m t the quark loop provides a form factor as we shall see explicitly below. Clearly, a point like interaction has a completely different high energy behaviour than a resolved interaction which is softened by a form factor: in fact one can show [1] that a point like interaction at n th perturbative order has double energy logs while a resolved interaction has only single logs. This means that as ŝ the gg H + X partonic cross section ˆσ behaves as ˆσ ŝ k=1 αk s ln k 1 k=1 αk s ln k 1 ŝ m H ŝ m H pointlike: m t resolved: finite m t 1 Hence, as the center-of-mass energy grows, eventually m t ceases to be a good approximation to the exact result. It is clear from eq. 1 that this high energy deviation between the exact and approximate behaviour is stronger at higher orders, so one might expect the relative accuracy of the infinite m t approximation to the k th order perturbative contribution to the cross section to become worse as the perturbative order increases. Conversely, this suggests that 1

4 it might be worth determining the high energy behaviour of the exact cross section and use the result to improve the infinite m t result, which is much less difficult to determine. Eventually, a full resummation of these contributions might also become relevant. The leading high energy contributions to this process in the infinite m t limit have in fact been computed some time ago in Ref. [13]: this amounts to a determination of the coefficient of the double logs eq. 1, in the pointlike case. In this paper, we compute the coefficients of the single logs eq. 1 in the resolved exact case. Our result takes the form of a double integral, whose numerical evaluation order by order in a Taylor expansion gives the coefficient of the logs eq. 1 at the lowest perturbative order the integral can be computed in closed form. After checking our result against the known full NLO result of Ref. [3, 4], we will discuss the way knowledge of the exact high energy behaviour of the cross section at a given order can be used to improve the infinite m t result, using the NLO case, where everything is known, as a testing ground. We will show that in fact, at NLO the different high energy behaviour eq. 1 accounts for most of the difference between the exact and infinite m t cross sections. We will then repeat this analysis in the NNLO case, where only the infinite m t result is currently known. We will show that in fact at this order the contribution of the logarithmically enhanced terms which dominate the partonic cross section at high energy is substantial even for moderate values of the partonic center-of-mass energy, such as ŝ m H. The calculation of the leading high energy logs is presented in section, while in section 3 we discuss its use to improve the NLO and NNLO results. The appendix collects the explicit expressions of the form factors which parametrize the amplitude for the process gg H with two off shell gluons, which is required for the calculation of sect.. Determination of the leading high energy logarithms.1 Definitions, kinematics and computational procedure We compute the total inclusive partonic cross section ˆσgg H + X in an expansion in power of α s, as a function of the partonic center-of-mass energy ŝ: ˆσgg H + X = ˆσ gg αs ; τ; y t, m H, where the dimensionless variables τ and y t parametrize respectively the partonic center-of-mass energy and the dependence on the top mass: τ m H ŝ 3 y t m t. 4 m H

5 The corresponding contribution to the hadronic cross section σ can be obtained by convolution with the gluon-gluon parton luminosity L: σ gg τ h ; y t, m H = 1 Lw τ h dw ˆσ gg 1 w dx x g h1 α s ; τ h w ; y t, m H w x, m H Lw 5 g h x, m H, 6 where g hi x i, Q is the gluon distribution in the i-th incoming hadron and in eq. 5 the dimensionless variables τ h parametrizes the hadronic center-of-mass energy s τ h m H s. 7 Note that τ h τ 1, and that if y t < 1 4 then the intermediate t t pair produced by the gluon-gluon fusion can go on shell. It is convenient to define a dimensionless hard coefficient function Cα s m H ; τ, y t ˆσ gg αs ; τ; y t, mh = σ y t Cα s m H, τ, y t 8 Cα s m H, τ, y t = δ1 τ + α sm H C 1 αs m H τ, y t + C τ, y t, 9 π π where σ δ1 τ is the leading order cross section, determined long ago in ref. [14]: σ y t = α s G F 56π 4y t y ts y t, 1 where s y t = We also define the Mellin transform ln 1 1 4yt y t + π i if y t < 1 4 i tan 1 1 = i sin 1 1 if y t Cα s m H, N, y t = 4y t 1 1 4y t dτ τ N 1 Cα s m H, τ, y t, 1 denoted with the same symbol by slight abuse of notation. We are interested in the determination of the leading high energy contributions to the partonic cross section ˆσgg H + X, namely, the leading contributions to Cα s m H, τ, y t as τ to all orders in α s m H. Order by order in α sm H, these correspond to the highest rightmost pole in N in the expansion in powers of α s m H of Cα sm H, N, y t. The leading singular contributions to the partonic cross section ˆσgg H + X to all orders can be extracted [1] from the computation of the cross section for a slightly different process, namely, the cross section σ off gg H computed at leading order, but with incoming off-shell gluons, a suitable choice of kinematics and a suitable prescription for the sum over polarizations. The procedure used for this determination is based on the so-called high energy or k t factorization [1], and consists of the following steps. 3

6 One computes the matrix element M µν ab k 1, k for the leading-order process gg H at leading order with two incoming off-shell gluons with polarization indices µ, ν and color indices a, b. The momenta k 1, k of the gluons in the center-of-mass frame of the hadronic collision admit the Sudakov decomposition at high energy k i = z i p i + k i, 13 where p i are lightlike vectors such that p 1 p, and k i are transverse vectors, k i p j = for all i, j. The gluons have virtualities k i = k i = k i. 14 The cross section σ off gg H is computed averaging over incoming and summing over outgoing spin and color: σ off = 1 1 J 56 Mµν ν ab M µ ba ε λ1 µ k 1ε λ 1 µ k 1 ε λ ν k ε λ ν k dp, 15 λ 1 λ where the flux factor J = k 1 k k 1 k 16 is determined on the surface orthogonal to p 1, p eq. 13, and the phase space is dp = π 1 δ m H z 1 k 1 + k. 17 m H Note that the kinematics for a 1 process is fixed, so eq. 15 gives the total cross section and no phase space integration is needed. The sums over gluon polarizations are given by ε λi µ k iε λ i ν k i = kµ i kν i ; i = 1,. 18 λ i Here, the virtualities will be parametrized through the dimensionless variables k i ξ i k i. 19 m H The reduced cross section σ, obtained extracting an overall factor m H, m H σ offgg H σy t ; ξ 1, ξ, ϕ, z, is then a dimensionless function σy t ; ξ 1, ξ, ϕ, z of the parameter y t eq. 4 and of the kinematic variables ξ 1, ξ, the relative angle ϕ of the two transverse momenta ϕ = cos 1 k1 k, 1 k 1 k and z m H z 1 z p 1 p = m H k 1 k k 1 k. Note that, in the collinear limit k 1, k, z eq. reduces to τ eq. 3. 4

7 The reduced cross section is averaged over ϕ, and its dependence on z eq. is Mellintransformed: 1 π σn, ξ 1, ξ = dz z N 1 dϕ π σy t; ξ 1, ξ, ϕ, z. 3 The dependence on ξ i is also Mellin-transformed, and the coefficient of the collinear pole in M 1, M is extracted: hn, M 1, M = M 1 M dξ 1 dξ ξ M ξ M 1 σn, ξ 1, ξ. 4 Note that the integral in eq. 4 has a simple pole in both M 1 = and M =. The residue of this pole is the usual hard coefficient function as determined in collinear factorization, which is thus CN = hn,,. The leading singularities of the hard coefficient function eq. 1 are obtained by expanding in powers of α s at fixed α s /N the function obtained when M 1 and M in eq. 4 are identified with the leading singularities of the largest eigenvalue of the singlet anomalous dimension matrix, namely with [15] m Hσ y t Cα s m H, N, y t = h N, γ s αs N αs, γ s [1 + O α s ]. 5 N Here, γ s is the leading order term in the expansion of the large eigenvalue γ + of the singlet anomalous dimension matrix in powers of α s at fixed α s /N: γ + αs αs α s, N = γ s + γ ss +..., 6 N N αs γ s = N n=1 n CA α s c n ; c n = 1,,, ζ3,..., 7 πn where C A = 3. So far, this procedure has been used to determine the leading nontrivial singularities to the hard coefficients for a small number of processes: heavy quark photo and electro production [1], deep inelastic scattering [16], heavy quark hadroproduction [17,18], and Higgs production in the infinite m t limit [13].. Cross section for Higgs production from two off-shell gluons The leading order amplitude for the production of a Higgs in the fusion of two off shell gluons with momenta k 1 and k and color a,b is given by the single triangle diagram, and it is equal to [ M µν ab = 4i δ ab g sm t k µ k1 ν A v m 1 ξ 1, ξ ; y t g µν A ξ 1, ξ ; y t H k1 k k1 k k + A m 1 ξ 1, ξ ; y t A ξ 1, ξ ; y t 1k µ ν k1k k µ ν kk µ ] 1k1 ν, 8 H k1k 5

8 where the strong coupling is α s = g s and the top Yukawa coupling is given by h 4π t = mt in v terms of the Higgs vacuum-expectation value v, related to the Fermi coupling by G F = 1 v. The dimensionless form factors A 1 ξ 1, ξ ; y t and A ξ 1, ξ ; y t have been computed in ref. [11]; their explicit expression is given in the appendix. They were subsequently rederived in Ref. [19], where an expression for the Higgs production cross section from the fusion of two off-shell gluons was also determined, but was not used to obtain the high energy corrections to perturbative coefficient functions. The spin- and colour-averaged reduced cross section eq. is then found using eq. 15, with the phase space eq. 17. We get σy t ; ξ 1, ξ, ϕ, z = 8 π 3 αsg F m yt 1 H ξ 1 ξ z A 1 A 1 δ z 1 ξ 1 ξ ξ 1 ξ cosϕ. 9 Because of the momentum conserving delta, the Mellin transform with respect to z is trivial, and the reduced cross section eq. 3 is given by π σn, ξ 1, ξ = 8 π 3 αs G Fm dϕ 1 1 H y t π 1 + ξ 1 + ξ N α cosϕ [ N A1 cos ϕ + ξ 1 ξ A ] [ A1 1 + ξ 1 + ξ A 1 ξ1 ξ A + A 1 A ] cos ϕ, where we have defined the dimensionless variable α 4ξ 1 ξ 1 + ξ 1 + ξ. 31 The three form factors A i are independent of ϕ, so all the angular integrals can be performed in terms of hypergeometric functions, with the result σn, ξ 1, ξ = 8 { π 3 αs G Fm 1 A 1 H y t F 1 + ξ 1 + ξ N 1 N, N + 1,, α + α 4 NN + 1 F 1 N +, N + 3, 3, α + ξ 1 ξ A 3 F 1 N, N + 1, 1, α N [ } A ξ 1 + ξ A 1 A + A 1 A ] 1 F 1 N + 1, N +,, α ξ 1 + ξ In the limit m t, using the behaviour of the form factors eq. 6 the term in square brackets in eq. 3 as well as the term proportional to A 3 are seen to vanish. The remaining terms, proportional to A 1, give the result in the pointlike limit. The reduced cross section in this limit was already derived in ref. [13] see eq. 9 of that reference: our result differs from that of ref. [13], though the disagreement is by terms of relative ON, hence it is immaterial for the subsequent determination of the leading singularities of the hard coefficient function. 6

9 .3 High energy behaviour The leading singularities of the coefficient function can now be determined from the Mellin transform hn, M 1, M eq. 4 of the reduced cross section eq. 3, letting M 1 = M = γ s α s /N according to eq. 5, and expanding in powers of α s i.e. effectively in powers of M 1, M and then in powers of N about N =. In the pointlike case m t the Mellin integral eq. 4 diverges for all M 1, M when N =, and it only has a region of convergence when N >. As a consequence, the function hn, M 1, M eq. 4 has singularities in the M plane whose location depends on the value of N, namely, simple poles of the form 1 N M 1 M : the expansion in powers of M i has finite radius of convergence M i < N, leading to an expansion in powers of M i N and thus double poles when M i = γ s. In the resolved case finite m t we expect the Mellin integral to converge when N = at least for < M i < M, for some real positive M. We can then set N =, and obtain the leading singularities of the coefficient function from the expansion in powers of M of h, M, M, letting M = γ s. This turns out to be indeed the case: when N =, σn, ξ 1, ξ only depends on ξ 1, ξ through the form factors, and the combination of form factors which appear in σ eq. 3 is regular when ξ 1, ξ see eq. 63, while it vanishes when ξ 1, ξ see eq. 64. Hence, we can let N = in σ, and get h, M 1, M = 8 π 3 αs G Fm H y t + M 1 M dξ 1 ξ M [ ] dξ ξ M 1 1 A 1 + ξ 1 ξ A Because the term in square brackets in eq. 33 tends to a constant as ξ 1, ξ, the integrals in eq. 33 have an isolated simple pole in M 1 and M, and thus the Taylor expansion of hn, M 1, M has a finite radius of convergence. We can then determine the Taylor coefficients by expanding the integrand of eq. 33 and integrating term by term. It follows from eqs. 5-7 that knowledge of the coefficients up to k-th order in both M 1 and M is necessary and sufficient to determine the leading singularity of the coefficient function up to order α k s. Let us now determine the leading singularities of first three coefficients of the expansion of the coefficient function eq. 8. The constant term determines the leading order result σ eq. 8: m H σ y t = h,,. 34 Using the on-shell limit of the form factors see eq. 63 of the appendix in eq. 33 we reproduce the well-known result eq. 1. The next-to-leading order term C 1 N, y t is determined by noting that h, M, = 4 π 3 α sg F m Hy t M + dξ ξ M 1 A 1 ξ, = h,, 8 π 3 α s G Fm H y t M + dξ ln ξ d A 1ξ, dξ + OM. 35 7

10 m H C 1 y t C y t Table 1: Values of the coefficients eq. 36 and eq. 38 of the Oα s /N and Oα s /N of the leading singularities of the coefficient function Cα s m H ; N, y t eq. 1. Equations 5-7 then immediately imply that C 1 N, y t = C 1 y t C A [1 + ON], N C 1 8π = y 4 ts y t + dξ ln ξ d A 1ξ,. 36 dξ The value of the coefficient C 1, determined from a numerical evaluation of the integral in eq. 36, is tabulated in table 1 as a function of the Higgs mass. Upon inverse Mellin transformation, one finds that lim τ C1 τ, y t = C A C 1 y t. 37 The values given in table 1 are indeed found to be in perfect agreement with a numerical evaluation of the small τ limit of the full NLO coefficient function C 1 τ, y t [4], for which we have used the form given in ref. []. Turning finally to the determination of the hitherto unknown NNLO leading singularity, we 8

11 evaluate the OM terms in the expansion eq. 35: by using again eqs. 5-7 we find C N, y t = C y t C A [1 + ON] N C 8π y t = y ts y t dξ 1 + { + dξ ln ξ d A 1ξ, dξ dξ [ ln ξ 1 ln ξ A 1 ξ 1, ξ ξ 1 ξ + A 3 ξ 1, ξ ]}. 38 The value of the NNLO coefficient C y t obtained from numerical evaluation of the integrals in eq. 38 is also tabulated in table 1. This is the main result of the present paper. 3 Improvement of the NLO and NNLO cross sections Knowledge of the leading small τ behaviour of the exact coefficient function Cα s m H ; τ, y t eq. 8 can be used to improve its determination. Indeed, as discussed in section 1, we expect the pointlike m t approximation to be quite accurate at large τ, whereas we know that it must break down as τ. Specifically, the small τ behaviour of the coefficient function is dominated by the highest rightmost singularity in Cα s m H ; N, y t eq. 1, which for the exact result is a k-th order pole but becomes a k-th order pole in the pointlike approximation. Hence the pointlike approximation displays a spurious stronger growth eq. 1 at small enough τ. Having determined the exact small τ behaviour up to NNLO, we can improve the approximate pointlike determination of the coefficient function by subtracting its spurious small τ growth and replacing it with the exact behaviour. We discuss first the NLO case, where the full exact result is known, and then turn to the NNLO where only the m t result is available. 3.1 NLO results At NLO the small τ behaviour of the coefficient function in the pointlike approximation is dominated by a double pole, whereas it is given by the simple pole eq. 36 in the exact case. This corresponds to an exact NLO contribution C 1 τ, y t which tends to a constant at small τ, whereas the pointlike approximation to it grows as lnτ: C 1 τ, = d 1 point τ + O τ ; d1 point τ = c1 ln τ + c C 1 τ, y t = d 1 ex τ, y t + O τ ; d 1 ex τ, y t = 3C 1 y t, 4 where C 1 y t is tabulated in table 1, while from Refs. [4,6,7] we get c 1 = 6; c 1 1 = The NLO term C 1 τ, y t eq. 8 is plotted as a function of τ in fig. 1, both in the pointlike m t approximation, and in its exact form computed with increasing values of the Higgs 9

12 Figure 1: The hard coefficient C 1 τ, y t eq. 9 parton level coefficient function normalized to the Born result plotted as a function of τ. The curves from top to bottom on the left correspond to m t = black, and to m t = 17.9 GeV red, with m H = 13, 18, 3, 8 GeV. mass, i.e. decreasing values of y t. It is apparent that the pointlike approximation is very accurate, up to the point where the spurious logarithmic growth eq. 39 sets in. We can construct an approximation to C 1 τ, y t by combining the pointlike approximation with the exact small τ behaviour: [ ] C 1,app. τ, y t C 1 τ, + d 1 ex τ, y t d 1 point τ Tτ 4 where d 1 ex τ, y t and d 1 point τ are defined as in eq. 4 and eq. 39 respectively, while Tτ is an interpolating function, which we may introduce in order to tune the point where the small τ behaviour given by d 1 ex τ, y t sets in. Clearly, as τ the approximation eq. 4 reproduces the exact small τ behaviour of the exact coefficient function eq. 4, provided only the interpolating function lim τ Tτ = 1. Furthermore, as discussed in section 1, the behaviour of the coefficient function C 1 τ, y t as τ 1 is to all orders controlled by soft gluon radiation, which leads to contributions to C 1 τ, y t which do not depend on y t and diverge as τ 1. Hence, the pointlike approximation is exact as τ 1. Because the functions d 1 ex τ, y t and d 1 point τ are regular as τ 1, this exact behaviour is also reproduced by the approximation eq. 4, provided only lim τ 1 Tτ is finite. Hence, C 1,app. τ, y t reproduces the exact C 1 τ, y t as τ up to terms 1

13 Figure : The hard coefficient C 1 τ, y t eq. 9 with m H = 13 GeV left and m H = 8 GeV right. The solid curves correspond to m t = black and m t = 17.9 GeV red, same as fig. 1. The three blue curves correspond to the approximation eq. 4,43, with k = dot-dashed, k = 5 dotted, k = dashed. that vanish as τ and as τ 1 up to terms that are nonsingular as τ 1, even when Tτ = 1. Nevertheless, we may also choose Tτ in such a way that T1 = while T = 1 always, so that C 1 τ, y t agrees with the pointlike approximation C 1 τ, in some neighborhood of τ = 1. For instance, we can let Tτ = 1 τ k, 43 with k real and positive, so that the first k orders of the Taylor expansion about τ = 1 of C 1,app. τ, y t and the pointlike approximation coincide. By varying the value of k, we can choose the matching point τ, such that C 1,app. τ, y t only differs significantly from the pointlike approximation if τ < τ : a larger value of k leads to a smaller value of τ. In fig. we compare the approximate NLO term eq. 4 to the exact and pointlike results, for two different values of y t, with Tτ given by eq. 43 and a choice of k which leads to different values of the matching between approximate and pointlike curves. It appears that an optimal matching is obtained by choosing k in such a way that the approximation eq. 4 matches the pointlike result close to the point where the logarithmic growth of the latter intersects the asymptotic constant value of the exact result. Note that this optimal matching could be determined without knowledge of the exact result. With this choice, the approximation eq. 4 differs from the exact result for the NLO contribution to the partonic cross section by less than 5% for all values of τ. 11

14 Figure 3: The hard coefficient C τ, y t eq. 9 parton level coefficient function normalized to the Born result plotted as a function of τ. The curves from top to bottom on the left correspond to m t = black, and to the approximation eq. 47 with Tτ eq. 43 and k = 5, and m t = 17.9 GeV red, with m H = 13, 18, 3, 8 GeV. 3. NNLO and beyond At NNLO, the pointlike approximation to the coefficient function has a quadruple pole at N =, corresponding to a ln 3 τ rise, while the exact result only has a double pole, and thus it rises linearly with ln τ: C τ, = d point τ + O τ ; d point τ = c 4 ln 3 τ + c 3 ln τ + c ln τ 44 C τ, y t = d ex τ, y t + O τ ; d ex τ, y t = 9 C y t lnτ, 45 where C y t is tabulated in table 1, while from Ref. [8] we get c 4 = 6; c 3 = n 17 f 18 ; c = π n f 8 18, 46 where n f the number of flavors. At this order, the exact form of C τ, y t is not known. However, analogously to the NLO case, we construct an approximation to it based on its determination [8] in the pointlike limit, 1

15 Figure 4: The hard coefficient C τ, y t eq. 9 with m H = 13 GeV, plotted versus τ on a logarithmic left or linear right scale. The solid black curve corresponds to m t = black, same as fig. 3, and the three blue curves are the approximation eq. 47 with m t = 17.9 GeV and Tτ eq. 43 with and k = dot-dashed, k = 5 dotted, same as fig. 3, k = dashed. combined with the exact small τ behaviour eq. 38: [ ] C,app. τ, y t C τ, + d ex τ, y t d point τ Tτ 47 with d ex τ, y t and d point τ defined in eq. 45 and eq. 44 respectively, and Tτ an interpolating function as discussed in section 3.1. Note that as τ the approximation eq. 47 only reproduces the exact result up to a constant, whereas at NLO the approximation eq. 4 reproduces the exact result up to terms which vanish at least as Oτ. The approximation to the exact result C,app. τ, y t, computed using C from table 1 with four different values of the Higgs mass, and taking Tτ eq. 43 with k = 5 is compared in fig. 3 to the pointlike approximation C,app. τ, y t of ref. [8] with n f = 5. In figures 4-5 we further compare the results obtained with different choices of the matching function Tτ eq. 43, and the same two values of the Higgs mass used to produce figs. -3 at NLO. At this order, the contribution from the leading small τ logs to the pointlike C,app. τ, is sizable even for large τ. Indeed, figs. 4-5 show that the behaviour of C around its local maximum at τ.65 receives a sizable contribution from the lnτ rise and ln τ drop eq. 44. If these are removed by using eqs. 47,43 with k =, the shape of C around the maximum is affected significantly, but if the matching is moved to smaller τ by choosing k > 5 the maximum is reproduced. Hence, whereas we can still obtain a rather smooth matching at any desired value of τ the choice of the optimal value of τ is not obvious. In particular, matching at a value of τ 13

16 Figure 5: Same as fig. 4, but now with m H = 8 GeV. where the contribution of the asymptotically spurious ln τ becomes significant leads to rather large values of the matching point τ >.6. Anyway, it is clear that the pointlike approximation breaks down for τ <.1. Contributions beyond NNLO in the expansion of hn, γ s, γ s eq. 5 in powers of αs can be N determined by pursuing the expansion of h, M, M eq. 33 in powers of M, and determining numerically the ensuing integrals, which have the form of eqs. 36,38, but with higher order powers of ln ξ 1, ln ξ. The series of contributions to the coefficient function eq. 8 thus obtained has a finite radius of convergence in N Mellin space, dictated by the location of the rightmost singularity in γ s, and thus in τ space it converges for all < τ 1 []. Therefore, its resummation can be accomplished to arbitrary accuracy by computation of a finite number of terms. This resummation, however, induces spurious singularities in the N space coefficient function, which can be removed by the inclusion of a suitable class of formally subleading running-coupling corrections, as recently shown in Ref. [3]. 3.3 K factors The accuracy of the various approximations at the level of hadronic observables clearly depends on the individual process. For the total inclusive cross section eq. 5, as is well known, the pointlike approximation is actually very good, and thus the impact of the improvement eq. 4 14

17 κ NLO κ NNLO m H = 13 GeV pointlike exact n.a. appr., k = appr., k = m H = 8 GeV pointlike exact n.a. appr., k = appr., k = Table : The NLO and NNLO contributions to the K factor eq. 48, computed with center-ofmass energy s = 14 TeV, and m t, denoted with pointlike, or m t = 17.9 GeV, denoted with exact or approximate. The approximate result uses eqs. 4,47, with Tτ eq. 43 and the value of k given in the table. The MRST [1] gluon distribution has been used. is moderate. To give a quantitative assessment, we define a K factor by letting: σ gg τ h ; y t, m H = σ gg τ h; y t, m H Kτ h; y t, m H Kτ h ; y t, m H = 1 + α sm H κ NLO τ h ; y t, m π H + αs m H π κ NNLO τ h ; y t, m H + O α 3 s m H, 48 where σgg is the leading order form of the contribution eq. 5 of the gluon gluon channel to total hadronic cross section. The value of the NLO and NNLO contributions to the K factors, determined using the MRST [1] gluon distribution in eq. 5 are given in table at LHC energies for two values of the Higgs mass, both in the pointlike, exact and approximate eq. 4 and eq. 47 cases. At NLO with m H = 13 GeV light, the pointlike approximation to κ NLO deviates by.3% from the exact result, and even with m H = 8 GeV heavy it only deviates by 1.6%. It should be kept in mind, however, that κ NLO itself is quite large: for α s.1, it amounts to a 1% contribution to the K factor eq. 48. Hence, the error made using the pointlike approximation is between the per mille and the per cent level, and thus not entirely negligible in a precision analysis. Using the approximation eqs with the values k = for light Higgs and k = 5 for heavy Higgs, which are seen from fig. to give good matching, the deviation can be reduced to.% and.7% respectively, and even more accurate results could be obtained by an optimization of the matching. However, a poor choice of the matching such as k = 5 for light Higgs or k = for heavy Higgs can lead to a result at the hadronic level which is actually closer to the pointlike approximation, or even worse than it. It is clear that at the partonic level the small τ behaviour eq. 39 accounts for most of the discrepancy between the exact and pointlike results, and even 15

18 the determination of a hadronic observable which depends very little on the parton-level small τ behaviour can be improved very substantially for values of τ H relevant for LHC by using the approximation eq. 4. The NNLO contribution κ NNLO is not known. Its values computed in the pointlike approximation, or with the approximation eqs. 47,43 and different choices of k are shown in table. Even at the inclusive hadronic level, now the size of the NNLO contribution can change up to about 5 1% if the matching is performed at large τ. Furthermore, κ NNLO is also quite large: with α s.1, it amounts to a 5% correction to the leading order, and thus to a further 5% correction to the K factor. Therefore, the impact of the pointlike approximation at NNLO is up to several per cent of the total K factor, rather larger that the impact of the pointlike approximation at NLO, and comparable to uncertainties which are currently discussed in precision studies at NNLO. 4 Outlook In this paper we have determined the leading high energy i.e. small τ = m Hŝ singularities of the cross section for Higgs production in gluon gluon fusion to all orders in the strong coupling, by providing an expression eq. 33 whence the coefficients of these singularities can be obtained by Taylor expanding and computing a double integral. We have given explicit numerical expressions for these coefficients up to NNLO. The high energy behaviour of this cross section is different according to whether it is determined with finite m t or with m t pointlike approximation. It turns out that at NLO this different high energy behaviour is responsible for most of the discrepancy between the pointlike approximation and the exact result. As a consequence, an accurate approximation to the exact result can be constructed by combining the pointlike approximation at large τ with the exact small τ behaviour. Some care must be taken in matching, but very accurate results can be obtained by simply choosing the matching point as that where the spurious small τ behaviour of the pointlike behaviour sets in. At NNLO, where the exact result is not known, the impact of the high energy behaviour turns out to be large even for moderate values of τ.5. Hence, an approximation constructed analogously to that which is successful at NLO, namely matching the pointlike limit to the asymptotic exact behaviour at the point where the asymptotically spurious terms become significant, leads to an approximate result which differs significantly from the pointlike approximation for most values of the partonic center-of-mass energy. The effect of these high energy terms on the total inclusive hadronic cross section remains quite small, because the latter is dominated by the region of low partonic center of mass energy, partly due to shape of the gluon parton distributions, which are peaked in the region where the gluons carry a small fraction of the incoming nucleon s energy, and partly because the partonic cross section is peaked in the threshold τ 1 region. Even so, the pointlike determination of the NNLO contribution to the total hadronic cross section can be off by almost 5-1% due to this spurious high energy behaviour, especially for relatively large values of m H > GeV. Because the NLO and NNLO corrections to the cross section are quite large, the overall effect of 16

19 these terms on the cross section is at the per cent level, and in particular their effect at NNLO is rather larger than at NLO. A study of the phenomenological implications of these results is thus relevant for a precision determination of the Higgs production cross section. Acknowledgements: We thank Thomas Binoth and Fabio Maltoni for discussions, and Guido Altarelli for a critical reading of the manuscript. This work was partly supported by the Marie Curie Research and Training network HEPTOOLS under contract MRTN-CT and by a PRIN6 grant Italy. The work of S. Marzani and R.D. Ball was done with the support of the Scottish Universities Physics Alliance. A Form factors The form factors in eq. 8 are given by [ 4yt A 1 ξ 1, ξ, y t = C ξ 1, ξ, y t 1 + ξ 1 + ξ 1 4ξ 1ξ + 1 ξ ] 1ξ 1 + ξ ξ 3 [ [B ξ B 1] ξ + 1 ξ ] 1ξ 1 + ξ 3 1 ξ 3 [ [B ξ 1 B 1] ξ ξ ] 1ξ 1 ξ 1 + ξ 3 with π 1 + ξ 1 + ξ, 49 [ A ξ 1, ξ, y t = C ξ 1, ξ, y t y t ξ 1 + ξ + ξ ] 1ξ [ ] 3 ξ + [B ξ B 1] + [B ξ 1 B 1] 1 ξ 1 + ξ [ 3 ] ξ1 1 + ξ 1 ξ π, 5 3 = 1 + ξ 1 + ξ ξ 1 ξ + ξ 1 + ξ = 1 + ξ 1 + ξ 4ξ 1 ξ. 51 It is also convenient to define the form factor A 3 ξ 1, ξ, y t 1 [ ] 1 + ξ1 + ξ A 1 A. 5 ξ 1 ξ The scalar integrals B and C are B ρ = 1 4yt ρ ρ tan 1 8π ρ 4y t ρ, if < ρ < 4y t ; B ρ = 1 ρ 4yt 1 + ρ ρ 4y t ln, if ρ < or ρ > 4y 16π t ; 53 ρ 1 ρ ρ 4y t 17

20 C ξ 1, ξ 1 1 { 1 y δ + 1 ln1 y 16π ln 3 1 y δ1 1 x δ + 1 z δ ln1 x ln 1 x δ + ln1 z ln 1 z δ3 +Li y + δ Li y δ 1 + Li y + δ1 Li y δ1 +Li x + δ + + Li x δ + Li x + δ Li x δ +Li z + δ Li z δ + 3 Li z + δ 3 Li z δ 3 }, 54 where δ 1 ξ 1 + ξ 1 3, δ ξ 1 ξ 1 3, δ 3 ξ 1 + ξ + 1 3, 55 and δ ± i 1 ± δ i x ± ξ 1 ± y t y ± ξ 1 y t 1 ±, y t, ξ 1 + 4y t, ξ 1 z ± 1 y t 1 ± i 4y t In the infinite top mass limit the scalar integrals become lim B 1 ρ = + ρ 1 + O, 58 y t 16π 6y t yt lim C ξ 1, ξ = ξ 1 ξ 1 + O, 59 y t 3π y t 1y t yt 3 so that the form factors reduce to lim m m t ta 1 = m 1 H 48π ; These limits also imply that In the on-shell limit the scalar integrals are lim B ξ i = 1 ξ i 8π, lim C ξ 1, ξ, y t = ξ 1 lim C ξ 1, ξ, y t = ξ 1,ξ α s lim m 4m t ta = m 1 + ξ 1 + ξ H. 6 48π lim m m t A 3 =. 61 t 1 1 3π 1 3π 1 + ξ ln z z + 18 ln z, z + ln x, 6 x +

21 so that A 1, = 1 8π + 1 3π A, = 1 16π + 1 3π ln z 4y t 1 z + ln z z + y t 1 63 The high energy limit of the form factors is trivially determined when ξ 1, ξ with ξ 1 ξ : lim A 1ξ 1, ξ, y t = ; ξ 1,ξ lim A 3ξ 1, ξ, y t = ; ξ 1, ξ If ξ 1, ξ with ξ 1 = ξ the limit is more subtle. In this case we get lim A 1ξ, ξ, y t = ξ C ξ, ξ, y t lim ξ 4 1 +O ξ lim A ξ 1, ξ, y t = 1 ξ 1,ξ 4π. 64 [ 1 1 ξ 16π ln y t ξ 1 + 4y t 1tan 1 1 ] 4y t 1, 65 where we have defined C ξ 1, ξ, y t C ξ 1, ξ, y t However, it turns out that lim ξ C ξ, ξ, y t = [ 1 16π ln y t ξ ξ y t 1tan 1 hence we conclude that eq. 64 holds also when ξ 1 = ξ. ] 1 + O 4y t 1 1, 67 ξ References [1] G. P. Salam, Int. J. Mod. Phys. A [] R. Harlander, Acta Phys. Polon. B [3] D. Graudenz, M. Spira and P. M. Zerwas, Phys. Rev. Lett [4] M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, Nucl. Phys. B [5] M. Kramer, E. Laenen and M. Spira, Nucl. Phys. B [6] A. Djouadi, M. Spira and P. M. Zerwas, Phys. Lett. B [7] S. Dawson, Nucl. Phys. B

22 [8] C. Anastasiou and K. Melnikov, Nucl. Phys. B 646 ; R. V. Harlander and W. B. Kilgore, Phys. Rev. Lett ; V. Ravindran, J. Smith and W. L. van Neerven, Nucl. Phys. B [9] S. Moch and A. Vogt, Phys. Lett. B [1] S. Catani, D. de Florian, M. Grazzini and P. Nason, JHEP [11] V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld, Nucl. Phys. B [1] S. Catani, M. Ciafaloni and F. Hautmann, Phys. Lett. B ; S. Catani, M. Ciafaloni and F. Hautmann, Nucl. Phys. B [13] F. Hautmann, Phys. Lett. B [14] J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B ; M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl. Phys [Yad. Fiz ]. [15] T. Jaroszewicz, Phys. Lett. B [16] S. Catani and F. Hautmann, Nucl. Phys. B [17] R. D. Ball and R. K. Ellis, JHEP [18] G. Camici and M. Ciafaloni, Nucl. Phys. B [Erratum-ibid. B ]. [19] R. S. Pasechnik, O. V. Teryaev and A. Szczurek, Eur. Phys. J. C [] R. Bonciani, G. Degrassi and A. Vicini, JHEP [1] A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J. C [] R. D. Ball and S. Forte, Phys. Lett. B [3] R. D. Ball, arxiv: [hep-ph].