NNLO corrections to the total cross section for Higgs boson production in hadron–hadron collisions

Nuclear Physics B 665 (2003) 325–366 www.elsevier.com/locate/npe

NNLO corrections to the total cross section for Higgs boson production in hadron–hadron collisions V. Ravindran a , J. Smith b,1 , W.L. van Neerven c a Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India b C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook,

Stony Brook, NY 11794-3840, USA c Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

Received 18 February 2003; received in revised form 16 May 2003; accepted 21 May 2003

Abstract We present the next-to-next-to-leading order (NNLO) corrections to the total cross section for (pseudo-) scalar Higgs boson production using an alternative method than those used in previous calculations. All QCD partonic subprocesses have been included and the computation is carried out in the effective Lagrangian approach which emerges from the standard model by taking the limit mt → ∞ where mt denotes the mass of the top quark. Our results agree with those published earlier in the literature. We estimate the theoretical uncertainties by comparing the K-factors and the variation with respect to the mass factorization/renormalization scales with the results obtained by lower order calculations. We also investigate the dependence of the cross section on several parton density sets provided by different groups.  2003 Elsevier B.V. All rights reserved. PACS: 12.38.-t; 12.38.Bx; 13.85.-t; 14.80.Gt

1. Introduction The Higgs boson, which is the corner stone of the standard model, is the only particle which has not been discovered yet. Its discovery or its absence will shed light on the mechanism how particles acquire mass as well as answer questions about supersymmetric extensions of the standard model or about compositeness of the existing particles and the E-mail address: [email protected] (W.L. van Neerven). 1 Partially supported by the National Science Foundation grant PHY-0098527.

0550-3213/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0550-3213(03)00457-7

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Higgs boson. The LEP experiments [1] give a lower mass limit of about mH ∼ 114 GeV/c2 and fits to the data using precision calculations in the electro-weak sector of the standard model indicate an upper limit mH < 200 GeV/c2 with 95% confidence level. After the end of the LEP program the search for the Higgs will be continued at hadron colliders, in particular, at the TEVATRON and the LHC. In this paper we concentrate on Higgs production where the lowest order reaction proceeds via the gluon–gluon fusion mechanism. In the standard model the Higgs boson couples to the gluons via heavy quark loops. Since the coupling of the scalar Higgs boson H to a fermion loop is proportional to the mass of the fermion (for a review see [2]), the topquark loop is the most important. The latter contribution is also dominant for the pseudoscalar Higgs boson A provided tan β is small where β denotes the mixing angle in the two-Higgs-doublet model (2HDM). In lowest order (LO) the gluon–gluon fusion process g + g → B with B = H, A, represented by the top-quark triangle graph, was computed in [3]. The two-to-two body tree graphs, given by gluon bremsstrahlung g + g → g + B, g + q(q) ¯ → q(q) ¯ + B and q + q¯ → g + B were computed for B = H in [4] and for B = A in [5]. From these reactions one can derive the transverse momentum (pT ) and rapidity (y) distributions of the (pseudo-) scalar Higgs boson. The total integrated cross section, which also involves the computation of the QCD corrections to the top-quark loop, has been calculated in [5]. This calculation is rather cumbersome since it involves the computation of two-loop triangular graphs with massive quarks. Furthermore also the two-to-three parton reactions have been computed in [6] using helicity methods. From the experience gained from the next-to-leading (NLO) corrections presented in [5] it is clear that it will be very difficult to obtain the exact NLO corrections to one-particle inclusive distributions as well as the NNLO corrections to the total cross section. Fortunately one can simplify the calculations if one takes the large top-quark mass limit mt → ∞. In this case the Feynman graphs are obtained from an effective Lagrangian describing the direct coupling of the (pseudo-) scalar Higgs boson to the gluons. The LO and NLO contributions to the total cross section in this approximation were computed in [7]. A thorough analysis [5,8] reveals that the error introduced by taking the mt → ∞ limit is less than about 5% provided mH
2mt . The two-to-three body processes were computed with the effective Lagrangian approach for the scalar and pseudo-scalar Higgs bosons in [9] and [10] respectively using helicity methods. The one-loop corrections to the two-to-two body reactions above were computed for the scalar Higgs boson in [11] and the pseudo-scalar Higgs boson in [12]. These matrix elements were used to compute the transverse momentum and rapidity distributions of the scalar Higgs boson up to NLO in [13–15] and the pseudo-scalar Higgs boson in [12]. The effective Lagrangian method was also applied to obtain the NNLO total cross section for scalar Higgs production by the calculation of the two-loop corrections to the Higgs-gluon–gluon vertex in [16], the soft-plus-virtual gluon corrections in [17,18] and the computation of the two-to-three body processes in [19,20]. These calculations were repeated for pseudo-scalar Higgs production in [21,22]. In this paper we recalculate the NNLO corrections to the total cross section for (pseudo-) scalar Higgs production except for the two-loop virtual contributions leading to the δ(1 − x) part of the coefficient functions of which the result is taken over from [16].

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The complete coefficient functions have been computed recently in [19–22] and we found full agreement with their analytic results when the number of colours is set N = 3. However, our method differs from the one presented in [19,21] and the approach followed in [20,22]. The authors in the latter references compute the total cross section using the Cutkosky [23] technique where one- and two-loop Feynman integrals are cut in certain ways. These Feynman integrals can be computed using various techniques (for more details see the references in [20]). Furthermore this method been also applied to compute rapidity distributions in NLO which is presented in [24]. The authors in [19] choose a more conventional method which was already used in [25–27] to compute the coefficient functions for the Drell–Yan process. However, instead of an exact computation of the 2 → 2 and 2 → 3 body phase space integrals they expand them around x = 1. Here √ x = m2 /s where m and s denote the (pseudo-) scalar Higgs mass and the partonic centre-of-mass energy, respectively. Since the coefficient functions can be expressed into a finite number of polylogarithms of the types Lin (x), Sn,p (x) (see, e.g., [28]) and logarithms of the types lni x lnj (1 − x), which are all multiplied by polynomials in x, one can expand them in the limit x → 1 and match them with the expressions coming from the phase space integrals. In this way the coefficients of the above functions are determined. However, as is shown in [27] this procedure is not necessary since one can obtain the Polylogarithms in a direct way by expanding the hypergeometric functions [29] in ε = n − 4 where n is the number of dimensions characteristic of n-dimensional regularization. The point is that there is no essential difference between Higgs production in the effective Lagrangian approach and the Drell–Yan production mechanisms except that the matrix elements in the former case contain higher powers of the invariants in the numerators. In [27] a program was made using the algebraic manipulation program FORM [30] to evaluate the phase space integrals analytically. For this calculation we have extended the program so that it can accommodate integrals having higher powers of invariants in the numerator. Our approach can be also used for differential distributions, in particular, for jet production (see, e.g., [31]). Here one cannot avoid multi-particle phasespace integrals. Of course two-to-four body processes are even more cumbersome to deal with but one can at least try to compute the pole terms (1/ε)k , which arise from infrared and collinear divergences, analytically or numerically (for numerical methods see [32]). Another difference with the previous results is that we present that part of the coefficient functions which is not proportional to δ(1 − x) for general colour factors of the local gauge group SU(N ). The latter is important if one wants to resum the dominant contributions (see [8,33]). Our paper will be organized as follows. In Section 2 we give definitions and discuss the regularization method to compute the partonic cross sections in the effective Lagrangian approach. In Section 3 we present the calculation of the coefficient functions. In particular, we discuss the various frames in which the three-particle phase space integrals are computed. In Section 4 we present the total cross sections for scalar Higgs production at the LHC and the TEVATRON. The long expressions for the coefficient functions expressed in the various colour factors are presented in Appendix A (scalar Higgs) and Appendix B (pseudo-scalar Higgs).

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2. Application of the effective Lagrangian approach to Higgs production In the large top-quark mass limit the Feynman rules (see, e.g., [9]) for scalar Higgs production (H) can be derived from the following effective Lagrangian density 1 a H a,µν (x), LH eff = GH Φ (x)O(x) with O(x) = − Gµν (x)G 4 whereas pseudo-scalar Higgs (A) production is obtained from
 A  LA with eff = Φ (x) GA O1 (x) + GA O2 (x) 1 λσ O1 (x) = − µνλσ Gµν a Ga (x), 8

f 1  O2 (x) = − ∂ µ q¯i (x)γµ γ5 qi (x), 2

(2.1)

n

(2.2)

i=1

where Φ H (x) and Φ A (x) represent the scalar and pseudo-scalar fields, respectively, and nf µν denotes the number of light flavours. Furthermore the gluon field strength is given by Ga and the quark field is denoted by qi . The factors multiplying the operators are chosen in such a way that the vertices are normalised to the effective coupling constants GH , GA and A . The latter are determined by the top-quark triangular loop graph, including all QCD G corrections, taken in the limit mt → ∞ which describes the decay process B → g + g with B = H, A, namely,     1/2   µ2 GB = −25/4as µ2r GF τ FB (τ )CB as µ2r , r2 , mt  

2   A = − as µ2r CF 3 − 3 ln µr + · · · GA , G (2.3) 2 m2t and as (µ2r ) is defined by   αs (µ2r ) , as µ2r = 4π

(2.4)

where αs (µ2r ) is the running coupling constant and µr denotes the renormalization scale. Further GF represents the Fermi constant and the functions FB are given by FH (τ ) = 1 + (1 − τ )f (τ ),

FA (τ ) = f (τ ) cot β,

1 f (τ ) = arcsin2 √ , for τ  1, τ √ 2  1 1− 1−τ f (τ ) = − ln √ + πi , 4 1+ 1−τ

for τ < 1,

τ=

4m2t , m2

(2.5)

where cot β denotes the mixing angle in the two-Higgs-doublet model. Further m and mt denote the masses of the (pseudo-) scalar Higgs boson and the top quark, respectively. In the large mt -limit we have lim FH (τ ) =

τ →∞

2 , 3τ

lim FA (τ ) =

τ →∞

1 cot β. τ

(2.6)

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The coefficient functions CB originate from the corrections to the top-quark triangular graph provided one takes the limit mt → ∞. We have presented the Born level couplings GB in Eq. (2.3) for general mt for on-shell gluons only in order to keep some part of the top-quark mass dependence. This is an approximation because the gluons which couple to the H and A bosons via the top-quark loop in partonic subprocesses are very often virtual. The virtual-gluon momentum dependence is neither described by FB (τ ) nor by CB . However, for total cross sections the main contribution comes from the region where the gluons are almost on-shell so that this approximation is better than it is for differential cross sections with large transverse momentum. The coefficient functions are computed up to order αs2 in [8,34] for the H and in [35] for the A. They read as follows    2  µ2r C H a s µr , 2 mt    (5) 2 2 27 2 100 1063 2 (5) 2 = 1 + as µr [5CA − 3CF ] + as µr CF − CA CF + C 2 3 36 A   µ2 4 5 − CF Tf − CA Tf + 7CA2 − 11CA CF ln r2 3 6 mt  

2 47 µ + nf Tf −5CF − CA + 8CF ln r2 , (2.7) 9 mt    2  µ2r CA as µr , 2 = 1, mt

(2.8)

(5)

where as is presented in the five-flavour number scheme. Notice that the coefficient function in Eq. (2.7) is derived for general colour factors of the group SU(N ) from Eq. (6) in [34]. These factors are given by N2 − 1 1 , Tf = . (2.9) 2N 2 Notice that Tf is also incorporated into GB in Eq. (2.3) where it is set to the value Tf = 1/2. Using the effective Lagrangian approach we will calculate the total cross section of the reaction CA = N,

CF =

H1 (P1 ) + H2 (P2 ) → B(−p5 ) + X ,

(2.10)

where H1 and H2 denote the incoming hadrons and X represents an inclusive hadronic state. The total cross section is given by πG2B σtot = 8(N 2 − 1) B = H, A,



1 dx1

a,b=q,q,g ¯ x

with x =

1

m2 , S



   dx2 fa x1 , µ fb x2 , µ2 ∆ab,B 2

x/x1

S = (P1 + P2 )2 ,

p52 = m2 ,



 x m2 , , x 1 x 2 µ2 (2.11)

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where the factor 1/(N 2 − 1) originates from the colour average in the case of the local gauge group SU(N ). Further we have assumed that the (pseudo-) scalar Higgs boson is mainly produced on-shell, i.e., p52 ∼ m2 . The parton densities denoted by fa (y, µ2 ) (a, b = q, q, ¯ g) depend on the mass factorization/renormalization scale µ. The same scales also enter the coefficient functions ∆ab,B which are derived from the partonic cross sections    n  m2 π d p5 +  2 δ p5 − m2 Tab,B (p5 , p1 , p2 ), σab,B z, 2 = (2.12) n s (2π) µ where the incoming parton momenta, denoted by p1 and p2 , are related to the hadron momenta by p1 = x1 P1 ,

p2 = x2 P2 ,

s = (p1 + p2 )2

⇒

s = x1 x2 S,

z=

m2 . s

The amplitude Tab,B can be written as  Tab,H (p5 , p1 , p2 ) = G2H d 4 y eip5 ·y a, b|O(y)O(0)|a, b,    A O2 (y) Tab,A (p5 , p1 , p2 ) = d 4 y eip5 ·y a, b| GA O1 (y) + G   A O2 (0) |a, b. × GA O1 (0) + G

(2.13)

(2.14)

(2.15)

The expressions above for the amplitude Tab are similar to those given for the Drell–Yan process except that the conserved electroweak currents are replaced by the operators O and O1 , O2 . The latter are not conserved so that they acquire additional renormalization constants defined by O(y) = ZO O(y),

j (y), Oi (y) = Zij O

(2.16)

where the hat indicates that the quantities under consideration are unrenormalized. Insertion of the above equations into Eqs. (2.14), (2.15) leads to the renormalized expressions  2 O(0)|a, b, d 4 y eip5 ·y a, b|O(y) Tab,H (p5 , p1 , p2 ) = G2H ZO (2.17) Tab,A (p5 , p1 , p2 ) 
 2 2 1(y)O 2A Z21 1 (0)|a, b A Z11 Z21 a, b|O +G + 2GA G = d 4 y eip5 ·y G2A Z11 

2 2 2 2(y)O A Z12 Z22 a, b|O 2A Z22 2(0)|a, b + 2GA G + GA Z12 + G 

2 2A Z22Z21 + GA G 1 (y)O 2(0) A (Z11Z22 + Z12 Z21 ) a, b|O + GA Z11 Z12 + G  2 (y)O 1 (0)|a, b . +O (2.18) The operator renormalization constants depend on the regularization scheme, in particular, on the prescription for the γ5 -matrix and the Levi-Civita tensor in Eq. (2.2). The computation of Tab,H will be carried out by choosing n-dimensional regularization where in the case of Tab,A we adopt the HVBM prescription [36,37] for the γ5 -matrix. For this

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choice the contraction of the Levi-Civita tensors proceeds as  µ2 ν λ σ   δµ1 δµ21 δµ21 δµ21    µ2  δν1 δνν12 δνλ12 δνσ12  µ2 ν2 λ2 σ2  =  µ2 µ1 ν1 λ1 σ1  (2.19) ν2 λ2 σ2  ,  δλ 1 δλ 1 δλ 1 δλ 1   µ2 ν λ σ  δσ1 δσ21 δσ12 δσ12 where all Lorentz indices are taken to be n-dimensional. To facilitate the calculation one can replace γµ γ5 in Eq. (2.2) by (see [38]) i γµ γ5 = µρσ τ γ ρ γ σ γ τ , (2.20) 6 which is equivalent to the HVBM scheme. Choosing the MS subtraction scheme the renormalization constant corresponding to the operator O becomes [39]

  2   2 4 ZO = 1 + as µ2r Sε β0 + as2 µ2r Sε2 2 β02 + β1 + · · · , (2.21) ε ε ε where Sε denotes the spherical factor characteristic of n-dimensional regularization. It is defined by   ε Sε = exp [γE − ln 4π] . (2.22) 2 The lowest order coefficients β0 and β1 originate from the beta-function given by     β(αs ) = −as2 µ2r β0 − as3 µ2r β1 + · · · , 11 4 34 20 β1 = CA2 − 4nf Tf CF − nf Tf CA . β0 = CA − nf Tf , (2.23) 3 3 3 3 The operator renormalization constants corresponding to O1 and O2 are computed in [40] and they read

 2 2   1 2 2 2 4 2 Z11 = Zαs = 1 + as µr Sε β0 + as µr Sε 2 β0 + β1 + · · · , (2.24) ε ε ε where Zαs denotes the coupling constant renormalization factor defined by   aˆ s = Zαs as µ2r . (2.25) The remaining constants are Z21 = 0,



 2

and

Z5s

(2.26)

1 Z12 = as µr Sε [−6CF ], ε s Z22 = ZMS Z5s , where by

s ZMS

(2.27) (2.28)

are the constants characteristic of the HVBM scheme. They are given



  1 20 44 s ZMS = 1 + as2 µ2r Sε − CA CF − nf Tf CF , ε 3 3

    107 31 CA CF + nf Tf CF . Z5s = 1 + as µ2r [−4CF ] + as2 µ2r 22CF2 − 9 9

(2.29)

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The latter renormalization constant is determined in such a way that the Adler–Bell–Jackiw anomaly [41]   O2 (y) = 4as µ2r nf Tf O1 (y), (2.30) is preserved in all orders in perturbation theory according to the Adler–Bardeen theorem [42].

3. Computation of the partonic cross section up to NNLO In this section we will give a short outline of the computation of the partonic cross sections σab,B in Eq. (2.12) up to next-to-next-leading order (NNLO). Insertion of a complete set of intermediate states and using translational invariance in Eq. (2.17) we obtain for scalar Higgs production   ∞ m    πG2H 2 d n pi +  2  n + 2 2 ZO d p5 δ p5 − m δ pi σab,H = Kab s (2π)n−1 m=3 i=3,i=5  m   × δ (n) pj |Mab→X H |2 , j =1

Mab→X H = p1 , p2 |O(0)|X, p5  with |X, p5  = |p3 , p4 , p6 · · · pm , p5 ,

(3.1)

where Kab represents the spin and colour average over the initial states. Further one can write down a similar expression for the pseudo-scalar Higgs (see Eq. (2.18)). Up to NNLO we have to compute the following subprocesses. On the Born level we have the reaction g + g → B.

(3.2)

In NLO we have in addition to the one-loop virtual corrections to the above reaction the following two-to-two body processes g + g → B + g,

g + q(q) ¯ → B + q(q), ¯

q + q¯ → B + g.

(3.3)

In NNLO we receive contributions from the two-loop virtual corrections to the Born process in Eq. (3.2) and the one-loop corrections to the reactions in Eq. (3.3). To these contribution one has to add the results obtained from the following two-to-three body reactions g + g → B + g + g,

g + g → B + qi + q¯i ,

g + q(q) ¯ → B + q(q) ¯ + g, q + q¯ → B + g + g, q1 + q2 → B + q1 + q2 , q + q → B + q + q.

(3.4) (3.5)

q + q¯ → B + qi + q¯i , q1 + q¯2 → B + q1 + q¯2 ,

(3.6) (3.7) (3.8)

In the case of pseudo-scalar Higgs production one also has to add the contributions due to 2 in Eq. (2.18). Up to order αs2 we 1 and O interference terms coming from the operators O

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have to compute the following expression for the reactions in (3.4) σgg,A π = Kab s 



  d n p5 δ + p52 − m2

 m   2  d n pi +  2  (n)    × δ pi δ pj p1 , p2 |O1 |X, p5  (2π)n−1 m=3 i=3 j =1

 1 |p5 p5 |O 2 |p1 , p2  A δ (n) (p1 + p2 + p5 ) p1 , p2 |O + GA G   2 |p5 p5 |O 1 |p1 , p2  . (3.9) + p1 , p2 |O 2 G2A Z11

4  m  

1 |p5  and p1 , p2 |O 2 |p5  can be found in Figs. 1(a) The Feynman graphs for p1 , p2 |O and 2(a) of [21], respectively. From the above equation one infers that the interference term A is proportional to contributes as a delta-function δ(1 − z) to σgg,A . Furthermore since G GA (see Eq. (2.3)) one can extract the latter constant as an overall factor from the above equation. For the reactions in Eq. (3.5) we have to compute    πG2A σgq,A = Kab d n p5 δ + p52 − m2 s   m   4  m   2  d n pi +  2  (n)  2   × Z11 δ pi δ pi p1 , p2 |O1 |X, p5  (2π)n−1 m=3 i=3 j =1   d n p3 +  2  (n) + Z12 δ p3 δ (p1 + p2 + p3 + p5 ) (2π)n−1  1 |p5 , p3 p5 , p3 |O 2 |p1 , p2  × p1 , p2 |O   (3.10) . + p1 , p2 |O2 |p5 , p3 p5 , p3 |O1 |p1 , p2  1 |p3 , p5  and p1 , p2 |O 2 |p3 , p5  can be found in The Feynman graphs for p1 , p2 |O Figs. 1(b) and 2(b) of [21], respectively. In the HVBM-scheme the latter matrix element is proportional to ε = n − 4. However, this contribution does not vanish in the limit ε → 0 because of the single pole term present in Z12 in Eq. (2.27). The same expression as in originating from the first reaction in Eq. (3.6). In this Eq. (3.10) also exists for σq q,A ¯ 1 |p3 , p5  and p1 , p2 |O 2 |p3 , p5  are case the Feynman graphs corresponding to p1 , p2 |O shown in Figs. 1(d) and 2(d) of [21], respectively. Notice that these contributions also survive in the calculation of differential distributions, see [12]. The computation of the phase-space integrals proceeds as follows. First we take over the expressions from the one- and two-loop corrections to the Born reaction in Eq. (3.2) which are computed in [7] (one-loop) and [16] (two-loop). Unfortunately the two-loop result is not presented for arbitrary colour factors so that we cannot distinguish between nf CA Tf and nf CF Tf . Hence we have shuffled all contributions proportional to nf into the term proportional to nf CA Tf . To compute the two-to-two body processes including the

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virtual corrections we have chosen the centre-of-mass frame of the incoming partons given by 1 1 p1 = P12 (1, 0, . . . , 0, 1), p2 = P12 (1, 0, . . . , 0, −1), 2 2 P12 − m2 −p3 = √ (1, 0, . . . , − sin θ, − cos θ ), 2 P12      1  P12 + m2 , 0, . . . , P12 − m2 sin θ, P12 − m2 cos θ , −p5 = √ (3.11) 2 P12 where we have introduced the invariants Pij = (pi + pj )2 ,

P12 = s.

(3.12)

In this frame the two-to-two body phase space integral becomes   2 3−n P12 1−n/2 1 2 πGH 2 σab→c H = Kab ZO P12 (4π)n/2 Γ (n/2 − 1) µ2   π P12 − m2 n−3 × dθ sinn−3 θ |Mab→c B |2 . µ2

(3.13)

0

Here µ indicates the dimension of the strong coupling constant g → gµ(4−n)/2 which is characteristic of n-dimensional regularization. In order to make all ratios dimensionless we have already included the factor µ(4−n)/2 in the phase space integrals. Notice that, in principle, the scale µ has nothing to do with the factorization or renormalization scale. However, for convenience one puts the latter scales equal to µ. The evaluation of the integral in Eq. (3.13) is rather easy even when |Mab→cB | contains virtual contributions. Since the integrals can be evaluated analytically we have made a routine using the algebraic manipulation program FORM which provides us with the results. A similar routine is made for the two-to-three body phase space integrals but here the computation is not so easy unless one chooses a suitable frame. Since we integrate over the total phase space the integrals are Lorentz invariant and, therefore, frame independent. The matrix elements of the partonic reactions can be partial fractioned in such a way that maximally two factors Pij in Eq. (3.12) depend on the polar angle θ1 and the azimuthal angle θ2 . Furthermore one factor only depends on θ1 whereas the other one depends both on θ1 and θ2 . Therefore, the following combinations show up in the matrix elements l Pijk Pmn ,

l Pijk Pm5 ,

k l Pi5 Pm5 ,

2 pi2 = pj2 = pm = pn2 = 0,

4  k  −2, 4  l  −2, p52 = m2 .

(3.14)

For the first combination it is easy to perform the angular integration since all momenta represent massless particles and one obtains a hypergeometric function (see, e.g., Eq. (4.19) in [14]). The angular integral of the second combination is more difficult to compute because one particle is massive and the result is an one-dimensional integral over a hypergeometric function which, however, can be expanded around ε. Examples of these types of integrals can be found in Appendix C of [43]. The last combination is very

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difficult to compute in n dimensions because both factors contain the massive particle indicated by p5 . Therefore, one has to avoid this combination at any cost. This is possible if one chooses the following three frames. The first one is the centre-of-mass frame of the incoming partons. The momenta are 1 1 P12 (1, 0, . . . , 0, 0, 1), p2 = P12 (1, 0, . . . , 0, 0, −1), 2 2 P12 − P45 −p3 = √ (1, 0, . . . , 0, sin θ1 , cos θ1 ), 2 P12 P12 − P35 −p4 = √ (1, 0, . . . , sin ψ sin θ2 , cos ψ sin θ1 + sin ψ cos θ2 cos θ1 , 2 P12 cos ψ cos θ1 − sin ψ cos θ2 sin θ1 ),

p1 =

sin2

P12 P34 ψ = , 2 (P12 − P35 )(P12 − P45 )

2 σab→cd H = Kab ZO

(3.15)

  P12 G2H 1 1 P12 1−n/2 dP35 2P12 (4π)n Γ (n − 3) µ2 m2



2 P12 +m  −P35

×

dP45

P35 P45 − P12 m2 µ4

n/2−2

P12 m2 /P35



P12 + m2 − P35 − P45 × µ2

n/2−2 π dθ1 sinn−3 θ1 0

π ×

dθ2 sinn−4 θ2 |Mab→cd B |2 .

(3.16)

0

For the next frame we choose the centre-of-mass frame of the two outgoing particles indicated by the momenta p3 and p4 . p2 = (ω2 , 0, . . . , 0, |p5 | sin ψ, |p5 | cos ψ − ω1 ), p1 = ω1 (1, 0, . . . , 0, 0, 1), 1 −p3 = P34 (1, 0, . . . , sin θ1 sin θ2 , sin θ1 cos θ2 , cos θ1 ), 2 1 −p4 = P34 (1, 0, . . . , − sin θ1 sin θ2 , − sin θ1 cos θ2 , − cos θ1 ), 2 −p5 = (ω5 , 0, . . . , 0, |p5 | sin ψ, |p5 | cos ψ), ω1 =

P12 + P15 − m2 , √ 2 P34

cos ψ =

ω2 =

P12 + P25 − m2 , √ 2 P34

(P34 − m2 )(P15 − m2 ) − P12 (P25 + m2 ) ,  (P12 + P15 − m2 ) (P15 + P25 )2 − 4m2 P34

ω5 = −

P15 + P25 , √ 2 P34 (3.17)

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2 σab→cd H = Kab ZO

  G2H P12 1−n/2 1 1 2P12 (4π)n Γ (n − 3) µ2

0 dP25

m2 −P12

m2 +P

12

m2 /(P



25

−m2 )

×



dP15 m2 −P

(P15 − m2 )(P25 − m2 ) − P12 m2 µ4

n/2−2

12 −P25



P12 + P15 + P25 − m2 × µ2

n/2−2 π dθ1 sinn−3 θ1 0

π ×

dθ2 sinn−4 θ2 |Mab→cd B |2 .

(3.18)

0

For the last frame we choose the centre-of-mass frame of one of the outgoing partons and the (pseudo-)scalar Higgs boson indicated by the momenta p4 and p5 , respectively, p1 = ω1 (1, 0, . . . , 0, 0, 1),

p2 = (ω2 , 0, . . . , 0, ω3 sin ψ, ω3 cos ψ − ω1 ),

−p3 = ω3 (1, 0, . . . , 0, sin ψ, cos ψ), −p4 = ω4 (1, 0, . . . , sin θ1 sin θ2 , sin θ1 cos θ2 , cos θ1 ), −p5 = (ω5 , 0, . . . , −ω4 sin θ1 sin θ2 , −ω4 sin θ1 cos θ2 , −ω4 cos θ1 ), P12 + P13 P12 + P23 P13 + P23 , ω2 = √ , ω3 = − √ , ω1 = √ 2 P45 2 P45 2 P45 P45 − m2 P45 + m2 , ω5 = √ , √ 2 P45 2 P45 P12 P23 − P45 P13 , cos ψ = (P12 + P13 )(P13 + P23 ) ω4 =

2 σab→cd H = Kab ZO

(3.19)

  G2H P12 1−n/2 1 1 2P12 (4π)n Γ (n − 3) µ2

0 dP23

m2 −P



0 ×

dP13

P13 P23 µ4

n/2−2 

12

P12 + P13 + P23 − m2 µ2

n−3

m2 −P12 −P23



P12 + P13 + P23 × µ2

1−n/2 π dθ1 sin 0

× |Mab→cd B |2 .

π n−3

dθ2 sinn−4 θ2

θ1 0

(3.20)

The integration over the angular independent invariants Pij can be performed by rescaling them with respect to P12 (see Appendix E in [26]). The integrals are such that they can be

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performed in an algebraic way by a program based on FORM [30] which has been also used to compute the NNLO coefficient functions of the Drell–Yan process in [25–27] (for more details see [44]). The partonic cross sections Eq. (2.12) can be written as follows     πG2H m2 m2 1 2 Z W σab,H z, 2 = z, , ab,H µ 8(N 2 − 1) 1 + ε/2 O µ2     πG2A 1+ε 2 m2 m2 Z Wab,A z, 2 , σab,A z, 2 = (3.21) µ 8(N 2 − 1) 1 + ε/2 11 µ gg,B z−1 W





 m2 ε/2 2  (0) (1) (1) P − 2β0 + wgg,B + εw¯ gg,B = δ(1 − z) + aˆ s Sε µ2 ε gg  2 ε  m 1  (0) (0) (0) (0) (0) 2Pgg ⊗ Pgg + aˆ s2 Sε2 + Pgq ⊗ Pqg − 10β0Pgg + 12β02 2 2 µ ε  1  (1) (1) (1) (1) (0) (0) + 2Pgg ⊗ wgg,B + 2wqg,B ⊗ Pqg + zB + Pgg − 6β0 wgg,B ε

(1) (1) (1) (2) (0) (0) − 6β0 w¯ gg,B + 2Pgg ⊗ w¯ gg,B + 2w¯ qg,B ⊗ Pqg + wgg,B

with zH = −4β1 ,

zA = −2β1 .

(3.22)

The coefficients zB , multiplying the single pole terms in the equations above, originate from the operators O in Eq. (2.1) and O1 in Eq. (2.2) and they are removed via the operator renormalization constants ZO in Eq. (2.21) and Z11 in Eq. (2.24).  2 ε/2

m 2 (0) (1) (1) −1 P + wgq,B + εw¯ gq,B z Wgq,B = aˆ s Sε µ2 ε gq  2 ε   1 3 (0) 1 (0) 2 2 m (0) (0) (0) + aˆ s Sε + Pgg ⊗ Pgq + Pqq ⊗ Pgq − 5β0 Pgq 2 2 µ2 ε2  1 1 1 (1) (1) (1) (1) (0) (0) P − 6β0 wgq,B + Pqg ⊗ wq q,B + ¯ + Pgq ⊗ wgg,B ε 2 gq 2   (0)  1 (0) (1) (1) (0) + Pgg + Pqq ⊗ wgq,B − 6β0 w¯ gq,B + Pqg ⊗ w¯ q(1) q,B ¯ 2

  (0) (1) (1) (2) (0) (0) ⊗ w¯ gq,B + wgq,B , + Pgq ⊗ w¯ gg,B + Pgg + Pqq (3.23)  m2 ε/2
(1) (1)  wq q,B ¯ q q,B ¯ + εw ¯ 2 µ  2 ε m 1 (0) 1 (1) (0) (0) + aˆ s2 Sε2 Pgq ⊗ Pgq + −6β0 wq(1) q¯ + 2Pgq ⊗ wgq,B 2 2 µ ε ε  (1) (0) (0) − 6β0 w¯ q(1) + 2Pqq ⊗ wq(1) ¯ gq,B q,B ¯ q,B ¯ + 2Pgq ⊗ w

(2) (0) , + 2Pqq ⊗ w¯ q(1) + w (3.24) q,B ¯ q q,B ¯

q q,B ˆ s Sε z−1 W ¯ =a



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q1 q2 ,B = z−1 W q1 q¯2 ,B , z−1 W  2 ε 2 (0) m 1 (0) (1) (0) = aˆ s2 Sε2 Pgq ⊗ Pgq + Pgq ⊗ wgq,B 2 2 µ ε ε

(1) (2) (0) + 2Pgq ⊗ w¯ gq,B + wq1 q2 ,B ,

(3.25)

where ⊗ denotes the convolution symbol defined by 1 f ⊗ g(z) =

1 dz2 δ(z − z1 z2 )f (z1 )g(z2 ).

dz1 0

(3.26)

0

For identical quark–quark scattering we have the same formula as in Eq. (3.25) except (2) (2) that wqq,B = wq1 q2 ,B . The expressions above follow from the renormalization group equations. They are constructed in such a way that become finite after coupling constant renormalization, operator renormalization and mass factorization are carried out. The splitting functions Pab (z) and the coefficients wab (z) with z = m2 /s also occur in the (1) coefficient functions given below except for the NLO terms w¯ ab (z), which are proportional to ε. They are given by    8 3 (1) 2 2 − 16 + 8z − 8z w¯ gg,H = CA 8D2 (z) − 6ζ (2)D0 (z) + ln (1 − z) − ζ(2) z 4    8 8 1 + + − 16 + 8z − 8z2 ln2 z − ln z ln(1 − z) 1−z z 4   1 55 67 55 67 22 (1 − z)3 ln(1 − z) − ln z − + + z − z2 − 3 z 2 3 9z 3 9

+ 2δ(1 − z) ,    1 3 4 (1) − 4 + 2z ln2 (1 − z) − ln z ln(1 − z) + ln2 z − ζ(2) w¯ gq,H = CF z 4 4   

3 1 7 3 + − + 6 − z ln(1 − z) − ln z + −5+ z , z 2 2z 2  

3 1 1 (1) 2 8 (1 − z) w¯ q q,H (3.27) ln(1 − z) − ln z + , ¯ = CF 3 z 2 6 where Di denotes the distribution  i  ln (1 − z) . Di = 1−z +

(3.28)

The difference between scalar and pseudo-scalar Higgs production is given by
 (1) w¯ gg,A−H = CA −8(1 − z) − 14δ(1 − z) , (1) = 0, w¯ gq,A−H

w¯ q(1) = 0. q,A−H ¯

(3.29)

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To render the partonic cross sections finite one has first to perform coupling constant renormalization. This is done by replacing the bare coupling constant by the renormalized one see Eq. (2.25). Then one has to carry out operator renormalization which is achieved by multiplying the expressions in Eqs. (3.22)–(3.25) with the operator renormalization constants in Eqs. (2.21), (2.24). The remaining divergences, which are of collinear origin, are removed by mass factorization 2 z−1 ZO Wab,B



1 m2 , ε µ2





=

Γca

c,d=q,q,g ¯

     2 1 1 m ⊗ Γdb ⊗ z−1 ∆cd,B , ε ε µ2 (3.30)

where Γca (z) denote the kernels containing the splitting functions which multiply the collinear divergences represented by the pole terms 1/εk . Note that in the case of pseudoscalar Higgs production the ZO (see Eq. (2.21)) in the above equation has to be replaced by Z11 in Eq. (2.24). For the four different subprocesses the mass factorization relations in Eq. (3.30) become equal to 2 z−1 ZO Wgg,B = Γgg ⊗ Γgg ⊗ z−1 ∆gg,B + 4Γgg ⊗ Γqg ⊗ z−1 ∆gq,B ,

z

−1

2 Wgq,B ZO

= Γqq ⊗ Γgg ⊗ z

−1

∆gq,B + Γgq ⊗ Γgg ⊗ z

+ Γqq ⊗ Γqg ⊗ z z

−1

2 ZO Wq q,B ¯

= Γgq ⊗ Γgq ⊗ z

−1

−1

−1

∆gg,B

∆q q,B ¯ ,

∆gg,B + Γqq ⊗ Γqq ⊗ z

(3.32) −1

∆q q,B ¯

+ 2Γgq ⊗ Γqq ⊗ z−1 ∆gq,B , z

−1

2 ZO Wq1 q2 ,B

= Γgq ⊗ Γgq ⊗ z

−1

∆gg,B + 2Γgq ⊗ Γqq ⊗ z

+ Γqq ⊗ Γqq ⊗ z

−1

∆q1 q2 ,B ,

(3.31)

(3.33) −1

∆gq,B (3.34)

where we have identified Γqg = Γqg ¯ ,

Γgq = Γg q¯ ,

gq,B = W g q,B qg,B = W qg,B W , ¯ =W ¯ ∆gq,B = ∆g q,B . ¯ = ∆qg,B = ∆qg,B ¯

(3.35)

Since we need the finite expressions up to order αs2 it is sufficient to expand the kernels Γca up to the following order in the renormalized coupling constant

  1 (0) Γqq = δ(1 − z) + as µ2 Sε Pqq , ε

 2 1 (0) Γqg = as µ Sε P , 2ε qg

  1 (1),PS 1 (0) PS (0) Γqq = as2 µ2 Sε2 P ⊗ P + P , gq 4ε2 qg 4ε2 qq

(3.36) (3.37) (3.38)

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  1 (0) Γgq = as µ2 Sε Pgq ε  

  1 1 (0) 1 (1) 1 (0) (0) (0) (0) Pgq ⊗ Pgg + as2 µ2 Sε2 2 + Pgq , + Pqq ⊗ Pgq + β0 Pgq 2 2ε ε 2 (3.39)

 2 1 (0) Γgg = δ(1 − z) + as µ Sε Pgg ε  

  1 (0) 1 (1) 2 2 2 1 1 (0) (0) (0) (0) + as µ Sε 2 P ⊗ Pgg + Pgq ⊗ Pqg + β0 Pgg + Pgg , 2 2ε ε 2 gg (3.40) (PS denotes pure-singlet). After renormalization and mass factorization the coefficient functions have the following algebraic form

 2  (0) m2 (1) −1 z ∆gg,B = δ(1 − z) + as µ Pgg ln 2 + wgg,B µ     1 (0) 1 (0) 5 m2 2 2 (0) (0) (0) 2 Pgg ⊗ Pgg + Pgq ⊗ Pqg − β0 Pgg + 3β0 ln2 2 + as µ 2 4 2 µ 2 

(1) m (1) (1) (1) (0) (0) + Pgg − 3β0 wgg,B + Pgg ⊗ wgg,B + wgq,B ⊗ Pqg + zB ln 2 µ

(2) , + wgg,B (3.41)

  1 (0) m2 (1) Pgq ln 2 + wgq,B z−1 ∆gq,B = as µ2 2 µ     3 (0) m2 1 (0) 5 2 2 (0) (0) (0) Pgg ⊗ Pgq + Pqq ⊗ Pgq − β0 Pgq ln2 2 + as µ 8 8 4 µ  1 (1) 1 (0) 1 (0) (1) (1) (1) P − 3β0 wgq,B + Pqg ⊗ wq q,B + ¯ + Pgq ⊗ wgg,B 2 gq 4 2 

2  m 1  (0) (1) (2) (0) (3.42) ln 2 + wgq,B , + Pgg + Pqq ⊗ wgq,B 2 µ 2  2 
(1)    2 2 1 (0) (0) 2 m z−1 ∆q q,B Pgq ⊗ Pgq µ w + a µ = a ln 2 ¯ s s q q,B ¯ 4 µ

m2 (1) (1) (1)  (2) (0) (0) ln 2 + wq q,B + −3β0 wq q,B , ¯ + Pgq ⊗ wgq,B + Pqq ⊗ wq q,B ¯ ¯ µ (3.43) z−1 ∆q1 q2 ,B = z−1 ∆q1 q¯2 ,B

2   m2 (1) (2) 2 2 1 (0) (0) 2 m (0) P ⊗ Pgq ln 2 + Pgq ⊗ wgq,B ln 2 + wq1 q2 ,B . = as µ 4 gq µ µ (3.44) In Appendices A and B we give the explicit expressions for the coefficient functions so (k) (k) and wab,B . Our results which are expressed in that one can determine the coefficients Pab

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the colour factors CA , CF and Tf agree with those published in [20–22] for N = 3. In the representation of the coefficient functions above we have put the renormalization scale µr equal to the mass factorization scale µ. If one wants to distinguish between both scales one can make the simple substitution

    αs (µ2r ) µ2 β0 ln r2 . αs µ2 = αs µ2r 1 + (3.45) 4π µ 4. Total cross sections for the process p + p → H + X In this section we will present total cross sections (see Eq. (2.11)) for Higgs-boson production in proton–proton collisions at the LHC and in proton–antiproton collisions at the TEVATRON. The cross section can be written in two ways (for the definitions see [17, 45]) πG2B σtot = 8(N 2 − 1)



1

a,b=q,q,g ¯ x

    x m2 2  dy Φab y, µ ∆ab , , y µ2

(4.1)

ab is the momentum fraction luminosity defined by where x = m2 /S and Φ   ab y, µ2 = Φ

1

   y 2 du  fa u, µ2 fb ,µ . u u

(4.2)

y

However, Eq. (4.1) can be also cast in the form    1    y πG2B x m2 2 ∆ , , y, µ σtot = x dy Φ ab ab x y µ2 8(N 2 − 1) a,b=q,q,g ¯ x     2 −1  with Φab y, µ = y Φab y, µ2 ,

(4.3)

where Φab denotes the parton luminosity. For the exact cross section the expressions in Eqs. (4.1) and (4.3) lead to the same results but if one wants to study the various approximations in the literature like the soft-plus-virtual (SV) gluon approximation then it makes a difference if ∆ab (z) or z−1 ∆ab (z) will be expanded around z = 1. It turns out that for the expression in Eq. (4.3) the soft plus virtual gluon terms represented by Di (z) in Eq. (3.28) and the leading ln(1 − z) terms, which are of collinear origin (see [17,45]), dominate the cross section. For their resummation see [46]. A determination of the leading ln(1 − z) terms in the case of deep inelastic structure functions is given in [47]. In the subsequent part of this section we study the dependence of the cross section on the input parameters like the QCD scale Λ, the renormalization/factorization scale µ, the mass m of the Higgs boson and the input parton densities. We are also interested in the theoretical uncertainty of the NNLO cross section. One uncertainty is due to the corrections which show up beyond NNLO. They can be guessed from the rate of convergence represented by the K-factor and the variation with respect to the scale µ. Another uncertainty is

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Table 1 Various parton density sets with their values for the QCD scale Λ and αs (MZ ) MRST02 (LO, lo2002.dat) MRST01 (NLO, alf119.dat) MRST02 (NNLO, vnvalf1155.dat) CTEQ6 (LO, cteq6l.tbl) CTEQ6 (NLO, cteq6m.tbl) GRV98 (LO, grv98lo.grid) GRV98 (NLO, grvnlm.grid)

ΛLO 5 = 167 MeV = 239 MeV ΛNLO 5 NNLO Λ5 = 176 MeV = 226 MeV ΛNLO 5 = 226 MeV ΛNLO 5 LO Λ5 = 132 MeV ΛNLO = 174 MeV 5

αsLO (MZ ) = 0.130 NLO αs (MZ ) = 0.119 NNLO αs (MZ ) = 0.115 αsNLO (MZ ) = 0.118 αsNLO (MZ ) = 0.118 αsLO (MZ ) = 0.125 NLO αs (MZ ) = 0.114

caused by the input parton densities. Parton density sets differ from each other in several aspects. They are based on fitting data from different experiments. Furthermore the shapes of the densities at the specific input scale chosen for µ differ from each other. Finally these sets are based on different theoretical assumptions, for instance, in the treatment of heavy flavours. In some sets the heavy flavour is treated as a massless particle which is described by a parton density similar to the treatment of the light quarks. In other sets the mass of the heavy flavour is considered to be on the same order of magnitude as the other large scales. Then the heavy flavour production is described by exact perturbation theory at a certain order so that threshold effects are fully taken into account. Although the coefficient functions in the effective Lagrangian approach are computed exactly in NNLO this is not the case for the parton densities because the exact three-loop splitting functions (anomalous dimensions) are not known yet. Until now only a finite number of moments are available (see [48]) which are used in [49] together with other constraints to approximate the splitting functions. These approximations are very reliable as long as y > 10−4 in Eqs. (4.1), (4.3). The approximated splitting functions were used in [50] to obtain NNLO parton density sets. One of them called MRST02 (see Table 1) will be used in our paper. For the approximations in LO and NLO we use the sets in [50] and [51], respectively. For the LO, NLO and NNLO plots we employ the one-, two-, and three-loop asymptotic forms of the running coupling constant as given in Eq. (3) of [52]. In order to make a comparison with other parametrizations for the parton densities we also choose the sets made by CTEQ [53] and GRV [54]. However, these sets do not contain NNLO versions so that we can present the NLO results only. All sets are listed in Table 1 2 together with the corresponding QCD scale Λnf determined for nf = 5. The same number of flavours is also chosen for the coefficient functions. Notice that the GRV sets do not contain densities for charm and bottom quarks. For simplicity the factorization scale is set equal to the renormalization scale µr . For our plots we take µ = m and use the MRST sets for the LO, NLO and NNLO computations (see Table 1) unless mentioned otherwise. Here we want to emphasize that the magnitude of the σtot is extremely sensitive to the renormalization scale because the effective coupling constant GB ∼ αs (µr ), which implies that σ LO ∼ αs2 . The sensitivity to the factorization scale is much smaller. 2 Notice that according to the prescription of the CTEQ group [53] also the LO corrected quantities have to be computed with the NLO running coupling constant.

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√ Fig. 1. The total cross section σtot plotted as a function of the Higgs mass at S = 14 TeV with µ = m. The NLO plots are presented for the total (solid line) and the subprocesses gg (long-dashed line), 10 × abs(gq + g q) ¯ (dot-dashed line) and 100 × (q q) ¯ (short-dashed line). Also shown is the LO result (dotted line).

For the computation of the effective coupling constant GB in Eq. (2.3) we choose the top quark mass mt = 173.4 GeV/c2 and the Fermi constant GF = 1.16639 GeV−2 = 4541.68 pb. In this paper we will only study scalar Higgs production and omit pseudoscalar Higgs production. The cross sections of the latter are about 9/4 cot2 β larger than those for the standard model Higgs so that all our conclusions also apply to pseudo-scalar Higgs production. We give results for both proton–proton√collisions at the centre-of-mass √ energy S = 14 TeV and proton–antiproton collisions at S = 2 TeV. In Fig. 1 we have plotted the contributions coming from the various subprocesses up to NLO. From this figure we infer that σtot is completely dominated by the gg reaction whereas the other contributions are down by two (gq + g q-subprocess) ¯ or even by three (q q-subprocess) ¯ orders of magnitude. This picture is different from the behaviour of the ¯ transverse momentum pT - and rapidity y-distributions in [13–15] where the (gq + g q)reaction amounted to about 1/3 of that for the gg-reaction when pT > 30 GeV/c. This is because σtot receives all its contribution from the small pT -region (pT < 30 GeV/c) where the gg subprocess overwhelms all other reactions. This picture is not changed when we study the cross section in NNLO (see Fig. 2) where new subprocesses contribute given by the qq, q q¯ and q¯ q¯ reactions. Notice that the incoming (anti-)quarks can be identical and non-identical. The relative order of magnitude is the same as in NLO and the new reactions are down by three orders of magnitude. In Fig. 3 we have plotted σtot in LO, NLO and NNLO as a function of m. The cross section decreases when m increases due to a reduction in the available phase space. In NNLO the curve shows a steeper decrease than in NLO and LO. Furthermore the growth of the cross section slows down while going to

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√ Fig. 2. The total cross section σtot plotted as a function of the Higgs mass at S = 14 TeV with µ = m. The NNLO plots are presented for the total (solid line) and the subprocesses gg (long-dashed line), abs(gq + g q) ¯ (dot-dashed line), 100 × (q q) ¯ (dotted line) and 100 × (qq + q¯ q) ¯ (short-dashed line).

higher orders. To show more clearly how the cross sections change with respect to m we have put them√in Table 2. We have also shown them for the TEVATRON (proton–antiproton collisions at S = 2 TeV) in Table 3 where they are appreciably smaller than in the case of the LHC. The NNLO cross sections in Tables 2, 3 have some theoretical errors which can be estimated in various ways. First one can study their variation with respect to the scale µ. This can be achieved by computing the ratio 

µ N µ0

 =

σtot (µ) , σtot (µ0 )

(4.4)

where µ0 = m and µ is varied in the range 0.1 < µ/µ0 < 10. In Fig. 4 a plot of N is shown for m = 100 GeV/c2 . Here one observes a clear improvement while going from LO to NNLO. In particular, the curve for NNLO is flatter than that for NLO. The improvement becomes even better when we choose a larger Higgs mass see, e.g., Fig. 5 (m = 200 GeV/c2 ). However, for still larger masses there is hardly any improvement anymore (see Fig. 6 where m = 300 GeV/c2 ). This is in contrast to the NNLO corrected cross section for vector boson production in [27] which is insensitive to the choice of scale chosen in the range above. A second way to study the reliability of our prediction is to study the rate of convergence of the perturbation series, which is represented by the K-factor. We choose the following

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Fig. √ 3. The total cross section σtot with all channels included plotted as a function of the Higgs mass at S = 14 TeV with µ = m; NNLO (solid line), NLO (dashed line) and LO (dotted line).

Table 2 Total cross sections in pb for Higgs masses between 100 and 300 GeV at the LHC. The LO, NLO and NNLO results are generated with the MRST parton densities listed in Table 1 Mass

LO

NLO

NNLO

100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

30.35 25.53 21.77 18.77 16.36 14.38 12.74 11.37 10.23 9.254 8.425 7.714 7.102 6.573 6.115 5.718 5.375 5.080 4.827 4.615 4.441

52.75 44.75 38.43 33.37 29.27 25.88 23.06 20.69 18.69 17.00 15.53 14.28 13.20 12.26 11.45 10.74 10.13 9.604 9.152 8.772 8.468

60.84 51.80 44.62 38.85 34.17 30.29 27.07 24.36 22.09 20.10 18.43 16.97 15.71 14.62 13.69 12.86 12.16 11.55 11.03 10.59 10.24

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Table 3 Total cross sections in pb for Higgs masses between 100 and 300 GeV at the TEVATRON. The LO, NLO and NNLO results are generated with the MRST parton densities listed in Table 1 Mass

LO

NLO

NNLO

100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

0.6116 0.4608 0.3530 0.2745 0.2161 0.1721 0.1385 0.1124 0.0920 0.0758 0.0629 0.0526 0.0442 0.0374 0.0318 0.0270 0.0234 0.0202 0.0176 0.0154 0.0136

1.351 1.025 0.791 0.619 0.491 0.393 0.318 0.259 0.213 0.177 0.147 0.124 0.105 0.089 0.076 0.065 0.056 0.049 0.043 0.038 0.033

1.781 1.363 1.060 0.835 0.666 0.538 0.438 0.359 0.293 0.248 0.207 0.176 0.149 0.127 0.109 0.095 0.082 0.072 0.063 0.055 0.049

√ Fig. 4. The quantity N (µ/µ0 ) (see Eq. (4.3) at S = 14 TeV plotted in the range 0.1 < µ/µ0 < 10 with µ0 = m and m = 100 GeV/c2 . The results are shown for LO (dotted line), NLO (dashed line) and NNLO (solid line).

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

Fig. 5. Same as Fig. 4 but now for m = 200 GeV/c2 .

Fig. 6. Same as Fig. 4 but now for m = 300 GeV/c2 .

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√ Fig. 7. The K-factors in NLO and NNLO at S = 14 TeV as a function of the Higgs mass using the MRST-sets; NNLO NLO (solid line), K (dot-dashed line). K

definitions K NLO =

NLO σtot LO σtot

,

K NNLO =

NNLO σtot LO σtot

.

(4.5)

In Fig. 7 one observes that both K-factors slowly increase when m gets larger. Moreover, there is a considerable improvement in the rate of convergence if one goes from NLO to NNLO. At m = 100 GeV/c2 we have K NLO ∼ 1.74 whereas K NNLO ∼ 2.00. Still the corrections for Higgs boson production are larger than those obtained from Z-boson production at the LHC where one gets K NLO ∼ 1.22 and K NNLO ∼ 1.16 (see [27]). In Fig. 8 we compare the K-factor in NLO obtained from the MRST with the results computed by using the parton density sets given by the CTEQ [53] and GRV [54] collaborations (see Table 1). From this figure we infer that the K-factor computed using GRV98 and CTEQ6 is larger than the one obtained from MRST01(NLO) at m = 100 GeV/c2 . However, at large m there is a cross over and we observe K CTEQ > K MRST > K GRV98 . The large result shown by the CTEQ6-set is due to the behaviour of their gluon density in LO. In NLO this density behaves in the same way as those in the MRST and GRV sets. We also investigated older sets provided by the CTEQ-Collaboration, namely, CTEQ5 and CTEQ6. It turns out that all these LO CTEQ sets lead to the same large value for K NLO as shown by Fig. 8. A study of K NLO , using the most recent parton densities, was also made in [45] but only for the TEVATRON. After the K-factor we now make a comparison between σtot computed from the various sets which leads to the third uncertainty in the theoretical prediction. For that purpose we

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349

√ Fig. 8. The K-factors in NLO at S = 14 TeV as a function of the Higgs mass using the following parton density sets MRST01 (solid line), GRV98 (dashed line) and CTEQ6 (dotted line).

plot the ratios R CTEQ =

CTEQ σtot

, MRST

σtot

R GRV =

GRV σtot MRST σtot

,

(4.6)

which are shown in NLO in Fig. 9. In this figure we see that the CTEQ6 set leads to the same cross section as the one obtained by MRST01. The cross section given by the GRV98 is 20% above the MRST01 result and the ratio decreases with increasing m to 5% above at m = 300 GeV/c2 . The reason that the CTEQ6 result is closer to the MRST sets can be attributed to the fact that their parton density sets are more recent and are constructed to fit the same experiments. Notice that a study of the ratios in Eq. (4.6) was also made for the TEVATRON in [45] using the most recent parton densities. The factorization scale dependence in Figs. 4–6 and the K-factor in Fig. 7 of the NNLO cross section can be used to give an error estimate of the latter for the LHC. Since (σ NNLO − σ NLO )/σ NLO = K NNLO/K NLO − 1 the error on σ NNLO is equal to 15%, 19% and 21% for m = 100 GeV/c2 , m = 200 GeV/c2 and m = 300 GeV/c2 , respectively (see Fig. 7). This is corroborated by the scale variation if we choose the range 0.25 < µ/µ0 < 4.0. For µ/µ0 = 0.25 the errors above become 18%, 15% and 14%, respectively. In the case µ/µ0 = 4.0 they are given by 16% for all m given above. Notice that the error due the K-factor increases when m gets larger whereas the opposite happens if the error estimate is derived from the mass factorization/renormalisation scale. On top of this we have to add the error due to the chosen parton density. For this we choose the NLO plots in Fig. 9 where the cross section predicted by CTEQ6 hardly differs from the

350

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MRST (Eq. (4.5)) in NLO at Fig. 9. The ratios R = σtot /σtot GRV CTEQ (solid line), R (dotted line). mass; R

√

S = 14 TeV and µ = m as a function of the Higgs

one given by MRST01 but the GRV98 plot deviates from the latter by 20% for small m to 5% for large m. An important feature of the Higgs cross section, which is very often emphasised in the literature, is that the integral in Eq. (4.1) is dominated by y ∼ x = m2 /S. Since x is small the gg part of σtot dominates the contributions coming from the other partonic subprocesses because of the steep rise of the gluon–gluon luminosity in Eq. (4.2) at small y. The importance of the small y-region is revealed when we impose an upper cut on the integral in Eq. (4.1). Hence we compute    πG2B gg x, µ2 Φ σtot (xmax ) = 2 8(N − 1) xmax   ∞    2   (i) x m2 i 2 ab y, µ ∆ + (4.7) , , as µ dy Φ ab y µ2 a,b=q,g i=1

x

and plot the ratio σtot (xmax ) EXACT (4.8) , σtot (1) = σtot , σtot (1) which is shown for NLO and NNLO in Fig. 10(a) for the choice xmax = 5x. The figure reveals that more than 95% of the cross section comes from the integration region x
y
5x where x < 5.1 × 10−5 . When m gets larger R(xmax ) will increase because the available phase space for Higgs boson production decreases. The latter also happens √ when the centreof-mass energy decreases like in the case for the TEVATRON where S = 2 TeV (here R(xmax ) =

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351

(a)

(b) Fig. 10. (a) The ratio R(xmax ) (see Eq. (4.7)) for√ proton–proton collisions (LHC) where xmax = 5x with (solid line), x = m2 /S. The CM energy and scale are given by S = 14 TeV and µ = m, respectively; NNLO √ NLO (dashed line). (b) Same as in (a) but for proton–antiproton collisions (TEVATRON) at S = 2 TeV and µ = m.

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x < 2.5 × 10−3) see Fig. 10(b). (Note Fig. 10(a) and (b) have different scales.) In the latter case more than 99% of the cross section receives its contribution from x
y
5x which becomes 100% when m = 300 GeV/c2 . Summarising our results we have recalculated that part of the NNLO corrections to the coefficient functions for (pseudo-) scalar Higgs production which is not proportional to δ(1 − z). Here we used a different method than those presented in [21] and [22]. In particular, our approach differs from the one given in [21] since we directly evaluated the multi-particle phase space integrals without fitting the coefficient functions to an expansion in the terms (1 − z)k lnl (1 − z). Further we presented that part of the coefficient functions which is not proportional to δ(1 − z) for general colour factors and checked that for N = 3 our answers are in agreement with the literature. It turns out that the total cross section is almost completely determined by the gg-subprocess which is in contrast to the differential cross section where also the gq + g q-channel ¯ gives a considerable contribution. We studied the K-factors and the dependence of the cross sections on the chosen mass factorization/renormalization scale and the adopted parton density set from which one can infer an error estimate of the cross section. If we exclude the GRV98 parton density the error can be mainly attributed to the missing higher order contributions to the coefficient functions which we estimate to lie between 14% and 21%.

Acknowledgements V. Ravindran would like to thank C. Anastasiou and K. Melnikov for providing him with the unpublished results. In particular, he could check that we had the same partonic cross section for g + g → g + g + H(A) prior to renormalization and mass factorization were carried out.

Appendix A In this appendix we present the coefficient functions for scalar Higgs boson production. In the case of the gg subprocess one can split the corresponding coefficient function into       m2 m2 m2 H ∆gg,B z, 2 = ∆S+V z, + ∆ z, , gg,B gg,B µ µ2 µ2

(A.1)

where the superscripts S + V and H denote the soft-plus-virtual and hard gluon parts, respectively. The former contains the distributions δ(1 − z) and Di (z) where the latter is defined in Eq. (3.28). We find (1),S+V ∆gg,H

 2

m = CA 8D0 (z) ln + 16D1(z) + 8ζ(2)δ(1 − z) , µ2

(A.2)

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

∆(1) gg,H = CA

(1) ∆gq,H

 2

 m −16z + 8z2 − 8z3 ln µ2     1 + −32z + 16z2 − 16z3 ln(1 − z) − ln z 2

8 22 − ln z − (1 − z)3 , 1−z 3

353

(A.3)

  2 

   m 1 2 2 = CF 4 − 4z + 2z ln + 8 − 8z + 4z ln(1 − z) − ln z µ2 2

− 3 + 6z − z2 , (A.4)

(1)

2 ∆q q,H ¯ = CF

∆(2),S+V gg,H

8 (1 − z)3 , 3

(A.5)

   2 m 44 64D1 (z) − D0 (z) − 32ζ(2)δ(1 − z) ln2 3 µ2    176 536 D1 (z) + − 80ζ(2) D0 (z) + 192D2(z) − 3 9    2  88 m + δ(1 − z) −24 − ζ(2) + 152ζ(3) ln + 128D3(z) 3 µ2   176 1072 − D2 (z) + − 160ζ(2) D1 (z) 3 9   1616 176 + ζ(2) + 312ζ(3) D0 (z) + − 27 3  

536 220 4 2 + δ(1 − z) 93 + ζ(2) − ζ(3) − ζ (2) 9 3 5    2  16 m 64 160 D0 (z) ln2 D1 (z) − D0 (z) + + nf Tf CA 2 3 3 9 µ   2   m 320 32 64 D1 (z) + δ(1 − z) 16 + ζ(2) ln + D2 (z) − 3 µ2 3 9   

 448 64 80 2024 160 + − ζ (2) D0 (z) + δ(1 − z) − − ζ(2) + ζ(3) 27 3 27 9 9  2

m + 4δ(1 − z) . + nf Tf CF 8δ(1 − z) ln (A.6) µ2

= CA2

Notice that the colour decomposition of the δ(1 − z) term into nf Tf CA and nf Tf CF is arbitrary due to our ignorance of the same parts appearing in the two-loop virtual corrections computed in [16]. They are only correct when N = 3. (2),C 2

(2),CA Tf 2 F Tf A ∆(2) + nf Tf CF ∆(2),C , gg,H gg,H = CA ∆gg,H + nf Tf CA ∆gg,H

(A.7)

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V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366 (2),C 2

∆gg,H A =

   32 −128z + 64z2 − 64z3 ln(1 − z) + −96z2 + 32z3 − ln z 1−z   2 332 2 m 352 376 3 + z− z + 132z ln2 − 3 3 3 µ2    32  Li2 (−z) + ln z ln(1 + z) + 64z + 32z2 + 32z3 + 1+z     2 − 256z + 256z Li2 (1 − z) + −384z + 192z2 − 192z3 ln2 (1 − z)   224 + 192z − 480z2 + 224z3 − ln z ln(1 − z) 1−z   8 40 2 3 + 128z − 48z + − ln2 z 1−z 1+z   1856 2 1760 3 2032 z− z + z ln(1 − z) + −528 + 3 3 3   1328 2 1496 3 88 1 904 ln z z+ z − z + + 176 − 3 3 3 3 1−z   1262 3562 3026 2 16 + 192z − 64z2 + 96z3 + ζ(2) + − z+ z 1+z 3 9 9   2 4322 3 m z ln − 9 µ2   64 2 3 + 128z + 64z + 64z + ln z ln(1 − z) ln(1 + z) 1+z     1−z 1−z + ln(1 − z) Li2 (−z) + Li3 − − Li3 1+z 1+z    16 2S1,2(−z) + ζ(2) ln(1 + z) + −24 + 8z − 32z2 − 32z3 + 1+z  + 2 ln(1 + z) Li2 (−z) + ln z ln2 (1 + z)   16 3 + −8 + 24z − 64z + Li3 (−z) 1+z   64 144 − + −16 − 208z − 624z2 + 112z3 − S1,2 (1 − z) 1−z 1+z   64 8 + + −16 + 672z + 536z2 + 88z3 − Li3 (1 − z) 1−z 1+z   64 2 3 + 24 − 104z − 16z + 16z − ln z Li2 (−z) 1+z   48 56 2 3 − + 8 + 8z − 88z + 8z − ln z Li2 (1 − z) 1−z 1+z − 512z(1 + z) ln(1 − z) Li2 (1 − z)

 

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355

  248 + 240z − 504z2 + 248z3 − ln z ln2 (1 − z) 1−z   16 128 − ln2 z ln(1 − z) + −96z + 304z2 − 144z3 + 1−z 1+z   56 + 20 − 92z − 16z2 − 16z3 − ln2 z ln(1 + z) 1+z   + −256z + 128z2 − 128z3 ln3 (1 − z)   136 2 16 1 56 1 8 16 3 + z− z + 24z − + ln3 z + 3 3 3 3 1−z 3 1+z   32 2 3 + 384z − 128z + 192z + ζ(2) ln(1 − z) 1+z   24 144 2 3 − ζ(2) ln z + 8 − 72z + 392z − 168z + 1−z 1+z     112 88 − 32z + 56z2 + z3 Li2 (−z) + ln z ln(1 + z) + − 3 3   548 2 572 3 44 1 1112 464 Li2 (1 − z) + z+ z − z − + − 3 3 3 3 3 1−z   2848 2 2992 3 176 1 1232 2144 − z+ z − z + + ln z ln(1 − z) 3 3 3 3 3 1−z   1856 2 1760 3 2032 z− z + z ln2 (1 − z) + −528 + 3 3 3   286 748 3 22 1 ln2 z + 164z − 302z2 + z − + − 3 3 3 1−z   7576 2 8108 8636 − z+ z − 1020z3 ln(1 − z) + 9 9 9   1582 2 8108 3 536 1 3602 2188 ln z + z− z + z − + − 9 3 3 9 9 1−z   8 + −608z + 320z2 − 352z3 + ζ(3) 1+z   1016 2 1408 3 16705 18523 1220 1156 − z+ z − z ζ(2) − + z + 3 3 3 3 27 27

19751 2 7055 3 z + z , − (A.8) 27 9 (2),C T

∆gg,H A f =

   2 32 16 16 m − z + z2 − z3 ln2 3 3 3 µ2   64 128 64 z + z2 − z3 ln(1 − z) + − 3 3 3

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 64 568 2 32 32 32 1 296 728 z − z2 + z3 − + z− z ln z − 3 3 3 3 1−z 9 9 9   2    152 3 m z ln + + −16 + 16z − 8z2 S1,2 (1 − z) + Li3 (1 − z) 3 µ2   8 2 32 3 16 1 16 160 − z+ z − z + Li2 (1 − z) + 3 3 3 3 3 1−z     64 128 64 z + z2 − z3 ln2 (1 − z) − ζ(2) + − 3 3 3   128 2 64 3 64 1 64 z− z + z − + ln z ln(1 − z) 3 3 3 3 1−z   16 8 1 8 − 8z + 20z2 − z3 + ln2 z + 3 3 3 1−z   1160 2 304 3 592 1456 + z− z + z ln(1 − z) + − 9 9 9 3   268 2 3142 6188 160 1 296 208 3 ln z + − z+ z − 64z + − z + 9 3 3 9 1−z 27 27

5218 2 3068 3 + (A.9) z − z , 27 27 +

 (2),C T

∆gg,H F f =

  2 32 32 m + 8z − 8z2 − z3 ln2 16z(1 + z) ln z + 3 3 µ2    + 32z(1 + z) 2 Li2 (1 − z) + 2 ln z ln(1 − z) − ln2 z   128 128 3 + + 32z − 32z2 − z ln(1 − z) 3 3   352 2 128 3 256 448 64 z ln z − − z+ z + − − 64z − 48z2 + 3 3 9 3 3   2   m 544 3 z ln + 32 + 32z + 80z2 S1,2 (1 − z) + 9 µ2   + 32 − 160z − 112z2 Li3 (1 − z)   64 3 64 2 − 48z − 160z + z Li2 (1 − z) + 3 3  + z(1 + z) 128 ln(1 − z) Li2 (1 − z) − 32 ln z Li2 (1 − z) − 64ζ(2) ln z  40 3 ln z + 64 ln z ln2 (1 − z) − 64 ln2 z ln(1 − z) + 3     128 128 + + 32z − 32z2 − z3 ln2 (1 − z) − ζ(2) 3 3

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

  128 256 3 − 128z − 96z2 + z ln z ln(1 − z) + − 3 3   32 + 52z + 28z2 − 32z3 ln2 z + 3   704 2 1088 3 512 896 − z+ z + z ln(1 − z) + − 9 3 3 9   256 1088 3 + 192z − 128z2 − z ln z + 9 9

3280 2 1984 3 608 4144 + z− z − z , − 27 9 9 27 (2),C 2

(2)

(2),C CF

∆gq,H = CF2 ∆gq,H F + CA CF ∆gq,H A (2),C 2

∆gq,H F =

(2),C T

+ nf Tf CF ∆gq,H F f ,

357

(A.10)

(A.11)

 2    

 m 8 − 8z + 4z2 ln(1 − z) + 4z − 2z2 ln z + 4z − z2 ln2 µ2     + 16 Li2 (1 − z) + 8 − 8z + 4z2 3 ln2 (1 − z) − 4ζ(2)



    + −16 + 32z − 16z2 ln z ln(1 − z) + −8z + 4z2 ln2 z     16 + −36 + 64z − 28z2 ln(1 − z) + −8z + 38z2 + z3 ln z 3   2 106 m 124 3 − 12z − 4z2 − z ln + 9 9 µ2    2 + −32 + 48z − 24z S1,2 (1 − z) − ζ(2) ln z    + −16 − 16z + 8z2 Li3 (1 − z) − ln(1 − z) Li2 (1 − z)    13 3  2 ln (1 − z) − 16ζ(2) ln(1 − z) + 8ζ(3) + 4 − 4z + 2z 3     + −16 + 8z − 4z2 ln z Li2 (1 − z) + 8 − 24z + 12z2 ln2 z ln(1 − z)     10 5 + −24 + 40z − 20z2 ln z ln2 (1 − z) + z − z2 ln3 z 3 3     32 32 + 16z + 16z2 + z3 Li2 (−z) + ln z ln(1 + z) + 3 3   92 16 3 2 + − + 56z + 42z + z Li2 (1 − z) 3 3  2  2 + −60 + 94z − 45z ln (1 − z)   32 + 36 − 56z + 88z2 + z3 ln z ln(1 − z) 3

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    63 40 232 16 − 104z + 60z2 + z3 ζ(2) + −14z − z2 − z3 ln2 z + 2 3 3 3   248 878 − 144z + 48z2 − z3 ln(1 − z) + 9 9   106 214 2 112 3 + − + 69z − z + z ln z 9 3 9

17 2 1304 3 1393 130 − z+ z + z , − (A.12) 54 9 18 27  (2),C C ∆gq,H A F

=

    24 − 24z + 12z2 ln(1 − z) + −24 − 24z − 24z2 ln z

  2 4 m 230 188 + z − z2 + 8z3 ln2 3 3 3 µ2    + −48 − 144z − 72z2 Li2 (1 − z)    + 16 + 16z + 8z2 Li2 (−z) + ln z ln(1 + z)     + 72 − 72z + 36z2 ln2 (1 − z) + −128 − 64z − 112z2 ln z ln(1 − z)   + 24 + 32z + 28z2 ln2 z + 16zζ (2)   40 2 868 736 3 + z + z + 32z ln(1 − z) + − 3 3 3   8 2 448 3 z + z − 32z ln z + 124 − 3 3   2 2422 1724 362 2 32 3 m + − z− z − z ln 9 9 9 9 µ2   + −136 − 248z − 148z2 S1,2 (1 − z)        1−z + 104 + 344z + 148z2 Li3 (1 − z) + 8 + 8z + 4z2 4 Li3 − 1+z   1−z + 2 Li3 (−z) − 4 ln z Li2 (−z) + 4 ln(1 − z) Li2 (−z) − 4 Li3 1+z  + 4 ln z ln(1 − z) ln(1 + z) − 3 ln2 z ln(1 + z)   + −80 − 304z − 136z2 ln(1 − z) Li2 (1 − z)   + −48 − 32z − 32z2 ln z Li2 (1 − z)     70 2 140 140 − z + z ln3 (1 − z) + −8 − 12z − 10z2 ln3 z + 3 3 3   + 64 + 48z + 64z2 ln2 z ln(1 − z)   + −132 − 60z − 114z2 ln z ln2 (1 − z) + 32zζ (2) ln(1 − z) −

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

    + 64 + 112z + 80z2 ζ(2) ln z + 136 − 112z + 68z2 ζ(3)   538 40 + − + 40z + 34z2 − z3 Li2 (1 − z) 3 3     88 16 + − − 40z − 12z2 − z3 Li2 (−z) + ln z ln(1 + z) 3 3   97 784 634 + z + z2 + 32z3 ln2 (1 − z) + − 3 3 3   692 832 40 2 3 − z − z − 64z ln z ln(1 − z) + 3 3 3   10 2 68 3 344 z − z + z ln2 z + −62 + 3 3 3   574 104 − 128z − 58z2 − z3 ζ(2) + 3 3   64 4342 904 − z − 116z2 − z3 ln(1 − z) + 9 3 9   1079 2 224 3 2402 2660 + z+ z + z ln z + − 9 9 9 9

1171 2 238 3 21539 9962 + z+ z − z , − 54 27 54 27

359

(A.13)



(2),C T ∆gq,H F f

  2 8 2 16 16 m − z + z ln2 = 3 3 3 µ2     232 304 16 32 32 − z + z2 ln(1 − z) − ln z) − + z + 3 3 3 9 9   2 152 2 m − z ln 9 µ2    4 2  2 8 8 − z+ z ln (1 − z) + 2 ln2 z − 4 ln z ln(1 − z) + 3 3 3     104 128 152 2 232 304 2 + − + z − 24z ln(1 − z) + − z+ z ln z 3 3 9 9 9

716 2 1060 1672 − z+ z , + (A.14) 27 27 27 (2),C 2

(2) 2 F ∆(2) q1 q2 ,H = ∆q1 q¯2 ,H = CF ∆q1 q2 ,H , (2),C 2

F ∆q1 q2 ,H =



  2   m −16 − 16z − 4z2 ln z − 24 + 16z + 8z2 ln2 µ2

(A.15)

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+

 

  16 + 16z + 4z2 −4 Li2 (1 − z) − 4 ln z ln(1 − z) + ln2 z

  + −96 + 64z + 32z2 ln(1 − z)

  2 m + 48 − 32z − 20z ln z + 102 − 72z − 30z ln µ2    1 + 16 + 16z + 4z2 −8S1,2 (1 − z) + 8 Li3 (1 − z) − ln3 z 3 

2



2

− 8 ln(1 − z) Li2 (1 − z) − 2 ln z Li2 (1 − z) − 4 ln z ln2 (1 − z)    2 + 2 ln z ln(1 − z) + 4ζ(2) ln z + −48 + 32z + 8z2 Li2 (1 − z)    + −96 + 64z + 32z2 ln2 (1 − z) − ζ(2)   + 96 − 64z − 40z2 ln z ln(1 − z)     + −24 + 32z + 8z2 ln2 z + 204 − 144z − 60z2 ln(1 − z)

  2 2 + −118 + 88z + 58z ln z − 210 + 188z + 22z , (2),C CF2

A 2 ∆(2) qq,H = CA CF ∆qq,H

(2),C CF2

=

∆qq,H A

(2),C 3

∆qq,H F =

(2),C 2

(2),C 3

(2),C 2

+ CF3 ∆qq,H F + CF2 ∆qq,H F ,

(A.17)


    −16 + 16z − 8z2 S1,2 (1 − z) + 8z + 12z2 ln2 z    + −16z − 36z2 ln z − 10 − 32z + 42z2 ,

(A.18)

  
   32 − 32z + 16z2 S1,2 (1 − z) + −16z − 24z2 ln2 z + 32z + 72z2 ln z  + 20 + 64z − 84z2 , (A.19) (2),C 2

F ∆qq,H F = ∆q1 q2 ,H ,

(A.20)

(2),C CF2

(2)

A 2 ∆q q,H ¯ ¯ = CA CF ∆q q,H

(2),C 3

(2),C 2

F F + CF3 ∆q q,H + CF2 ∆q q,H ¯ ¯

(2),C 2 T

F f + nf Tf CF2 ∆q q,H , ¯

(2),C CF2

A ∆q q,H ¯

(A.16)

(A.21)

  2 m 88 88 3 2 + 88z − 88z z ln = − + 3 3 µ2   2 + 8 + 8z + 4z −4S1,2 (−z) − 6 Li3 (−z) − 4 ln(1 + z) Li2 (−z) 

+ 6 ln z Li2 (−z) − 2 ln z ln2 (1 + z) + 3 ln2 z ln(1 + z)  − 2ζ (2) ln(1 + z) − 4ζ(3)

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361



  16 16 − 16z + 16z2 − z3 5 Li2 (1 − z) + 2 ln2 (1 − z) 3 3  − 8 ln(1 − z)    32 3  2 + 32z + 16z + z Li2 (−z) + ln z ln(1 + z) 3     176 32 3 2 3 + − + 192z − 168z + 64z ζ(2) + −40z − z ln2 z 3 3   500 2 128 3 890 2 88 296 − z+ z − z ln z + 130 − 316z + z + 3 3 3 3 3

332 3 z , − (A.22) 3

+

(2),CF3 ∆q q,H ¯



   64 64 3 64 3 2 2 − 64z + 64z − z ln(1 − z) + 32z − 32z + z ln z = 3 3 3   2   64 m 64 + 16 + 16z + 8z2 4S1,2 (−z) + 6 Li3 (−z) + z − z2 ln 2 3 3 µ + 4ζ (3) + 4 ln(1 + z) Li2 (−z) − 6 ln z Li2 (−z) + 2 ln z ln2 (1 + z)  − 3 ln2 z ln(1 + z) + 2ζ(2) ln(1 + z)   160 3 z Li2 (1 − z) + −32 + 128z − 128z2 + 3    2 + −64z − 32z Li2 (−z) + ln z ln(1 + z)   32 32 3 2 − 32z + 32z − z ln2 (1 − z) + 3 3   128 3 64 2 z ln z ln(1 − z) + − + 96z − 96z + 3 3     16 3 224 224 3 2 2 2 − 256z + 208z − z ζ(2) + 40z + 48z − z ln z + 3 3 3   352 2 352 z− z + 32z3 ln(1 − z) + −32 + 3 3   200 2 260 2 164 256 3 z− z − 32z ln z − − 64z + z + − 3 3 3 3

+ 32z3 , (A.23)

(2),C 2

(2),C 2

F F ∆q q,H = ∆q1 q2 ,H , ¯

(2),C 2 T

F f ∆q q,H = ¯



  2 32 32 m − 32z + 32z2 − z3 ln 3 3 µ2

(A.24)

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 64 64 64 64 − z + z2 − z3 ln(1 − z) 9 3 3 9   128 2 128 3 688 2 32 160 368 592 z− z + z ln z − + z− z + − + 3 3 3 9 27 9 9

656 3 + (A.25) z . 27

+

The expressions above which survive in the limit z → 1 are given by

 2 m (1),CA − 32 ln(1 − z) + 8 , lim ∆gg,H = −16 ln µ2 z→1 (2),C 2

lim ∆gg,H A =

z→1

lim z→1

(2),C T ∆gg,H A f

   2 184 m −128 ln(1 − z) + ln2 3 µ2    2 1336 1024 m 2 ln(1 − z) + 160ζ(2) − + −384 ln (1 − z) + ln 3 9 µ2   1096 2 2660 ln (1 − z) + 320ζ(2) − − 256 ln3 (1 − z) + ln(1 − z) 3 9

4228 784 ζ(2) + − 624ζ (3) − (A.27) , 3 27

     2 416 32 2 m2 128 m ln(1 − z) + = − ln + − ln 3 µ2 3 9 µ2

128 1232 808 128 2 , ln (1 − z) + ln(1 − z) + ζ(2) − − 3 9 3 27

 2 m F lim ∆(1),C + 4 ln(1 − z) + 2 , = 2 ln gq,H µ2 z→1 (2),C 2

lim ∆gq,H F =

z→1



 (2),C CF

z→1

(A.28)

(A.29)

 2

 m 4 ln(1 − z) + 3 ln2 µ2

 2 

m 26 3 ln (1 − z) + 12 ln2 (1 − z) − 16ζ(2) − 18 ln + 2 µ 3   − 11 ln2 (1 − z) + −32ζ(2) − 26 ln(1 − z) + 16ζ(3)

+ 12ζ (2) + 9 ,

lim ∆gq,H A

(A.26)

=

12 ln(1 − z) −

  2 22 m ln2 3 µ2

(A.30)

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

363

  2  304 m 4 ln + 36 ln2 (1 − z) + ln(1 − z) − 4ζ(2) + 3 9 µ2   43 2 70 3 ln (1 − z) + ln (1 − z) + −8ζ(2) + 58 ln(1 − z) + 3 3

460 + 62ζ (3) + 14ζ(2) − (A.31) , 27

lim z→1

(2),C T ∆gq,H F f

     2 80 8 2 m2 16 4 m ln ln(1 − z) − = + + ln2 (1 − z) ln 2 2 3 µ 3 9 µ 3

104 − 16 ln(1 − z) + (A.32) . 27

Appendix B In this appendix we present the coefficient functions for pseudo-scalar Higgs boson production. To shorten the expressions we give the difference between the coefficient functions originating from pseudo-scalar and scalar production only, namely,
 ∆(1),S+V gg,A−H = CA 8δ(1 − z) , (1)

(B.1)

∆gg,A−H = 0,

(B.2)

∆(1) gq,A−H = 0,

(B.3)

(1)

= 0. ∆q q,A−H ¯

(B.4)

In the NNLO corrections below we indicate by O12 the contribution coming from the interferences between the operators O1 and O2 in Eqs. (3.9), (3.10). In the coefficient functions one has to put O12 = 1, ∆(2),S+V gg,A−H

  2  m 20 64D0 (z) − δ(1 − z) ln 3 µ2 

 215 + 128D1 (z) + δ(1 − z) + 64ζ(2) 3   2  m 8 + CA Tf nf − δ(1 − z) ln 3 µ2  2 

  µr 32 16 196 + O12 ln − + δ(1 − z) − 9 3 3 m2t  2

 m + CF Tf nf −8δ(1 − z) ln − 4δ(1 − z) . µ2

= CA2

(B.5)

364

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366

For the colour decomposition of the nf -part of the expression above see the remark below Eq. (A.6). Also  2 2

 m (2),CA 2 3 − 16z ln2 z ∆gg,A−H = −128z + 64z − 64z ln µ2   + −256z + 128z2 − 128z3 ln(1 − z)   44 232 452 2 440 64 z − 64z2 + 64z3 − ln z + + z− z + 32 + 3 1−z 3 3 3

176 3 + (B.6) z , 3

(2),CA Tf ∆gg,A−H

16 16 32 2 32 z ln z + + z− z , = 3 3 3 3

(B.7)


 2 2 F Tf ∆(2),C gg,A−H = 16z ln z − 16 + 32z − 16z ,

(B.8)

 
(2),CF2 = 8z ln2 z − 24z ln z − 4 + 16z − 12z2 + O12 −12 + 12z2 , ∆gq,A−H

(B.9)

(2),C C

A F ∆gq,A−H =

 2  m 32 − 32z + 16z2 ln − 16z ln2 z µ2     + 64 − 64z + 32z2 ln(1 − z) + 80z − 16z2 ln z + 60 − 48z

2 + 4z , (B.10)



F Tf ∆(2),C gq,A−H = 0,

(B.11)


(2),CF2 = −16z ln2 z + (32 + 48z) ln z + 88 − 96z + 8z2 , ∆q1 q2 ,A−H

(B.12)


 (2),CA CF2 ∆qq,A−H = −16z ln2 z + 32z ln z + 32 − 32z ,

(B.13)


 (2),CF3 = 32z ln2 z − 64z ln z − 64 + 64z , ∆qq,A−H

(B.14)

(2),C 2

(2),C 2

F F = ∆q1 q2 ,A−H , ∆qq,A−H

(2),C 2

(2),C 2

F F ∆q1 q¯2 ,A−H = ∆q1 q2 ,A−H ,

(2),CA CF2 ∆q q,A−H ¯



8 56 2 64 3 80 2 = 16z ln z + z ln z + + z − z , 3 3 3 3

(B.15) (B.16) (B.17)

V. Ravindran et al. / Nuclear Physics B 665 (2003) 325–366


(2),CF3 ∆q q,A−H = −16z ln2 z − 32z ln z + 64 − 160z + 96z2 ¯   + O12 −48 + 96z − 48z2 , (2),C 2

(2),C 2

F F ∆q q,A−H = ∆q1 q2 ,A−H , ¯

(2),CF2 Tf ∆q q,A−H ¯

32 32 2 64 + z . = − z ln z − 3 3 3

The expressions above which survive in the limit z → 1 are given by  2

2 m (2),CA lim ∆gg,A−H = −128 ln 2 − 256 ln(1 − z) + 64 , µ z→1 lim z→1

(2),CA CF ∆gg,A−H

 2

m = 16 ln + 32 ln(1 − z) + 16 . µ2

365

(B.18) (B.19) (B.20)

(B.21)

(B.22)

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