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QCD Corrections to Production of Higgs Pseudoscalars

Models of electroweak symmetry breaking with more than a single doublet of Higgs scalars contain a neutral pseudoscalar boson. The production of such a pseudoscalar in hadron collisions proceeds primarily via gluon fusion through a top-quark loop (except for those models in which the pseudoscalar coupling to bottom quarks is strongly enhanced). We compute the QCD corrections to this process in the heavy-quark limit, using an effective Lagrangian derived from the axial anomaly.

The primary mission of the next generation of colliders is the study of the physics of electroweak symmetry breaking. Since the Higgs mechanism provides the most promising scenario for the symmetry breaking, the search for the particles of the Higgs sector is of primary importance. The minimal model of electroweak symmetry breaking contains one complex scalar doublet, three components of which become the longitudinal degrees of freedom of the W ± and Z. The remaining component of the doublet is the so-called Higgs boson. However, the minimal model with one doublet has no a priori justification and there are several motivations for considering models with enlarged Higgs sectors, either containing more doublets, singlets or more exotic representations. For example, supersymmetric models require at least two doublets. Likewise, at least two doublets are required to produce CP violation in the Higgs sector [1].
Models with enlarged Higgs sectors have a richer particle content than the minimal model; in general, neutral pseudoscalars (with respect to their fermion couplings) and charged scalars as well as extra neutral scalars are present. In this letter we will study the QCD corrections to the production of a Higgs pseudoscalar (A 0 ) via gluon fusion. This process proceeds primarily through a top-quark loop (unless the coupling to bottom quarks is greatly enhanced). We will focus on the case of a light pseudoscalar and work in the heavy-top-quark limit : m t ≫ M A .
To fix our normalization we take the coupling of the pseudoscalar to top quarks to be m t γ 5 /v where v = 246 GeV[2]. The lowest-order amplitude for gg → A 0 is well known and takes the form [3] with τ = 4m 2 t /M 2 A , g A = α s /2πv, and . We see that in the large m t limit the amplitude for gg → A 0 is independent of the top quark mass, just as in the scalar case: τ f (τ ) → 1 as τ → ∞. The heavy-top approximation of the amplitude is accurate to within 5% for m 2 t > 2M 2 A and to within 10% for m 2 t > M 2 A .
The amplitude in Eq. (1) can be computed by evaluating the triangle diagram with a top quark in the loop and taking the limit m t → ∞ or instead by noticing that this amplitude is related to the axial anomaly [4]. Let j 5 =ψγ 5 ψ be the axial current and j µ 5 =ψiγ µ γ 5 ψ be the axial vector current. The anomaly equation reads: where G a µν is the field-strength tensor for SU(3) andG a µν is its dual,G a µν = ǫ µνρσ G a ρσ . In the heavy quark limit the left side of Eq. (2) vanishes; it is proportional to more powers of external momenta than the other terms [5]. Therefore, in the heavy quark limit the matrix element of j 5 between gluon states is given Equivalently, one may treat the interactions of gluons with the A 0 in the heavyquark limit as arising from the effective Lagrangian[6] The power of Eq. (4) for our problem comes from the Adler-Bardeen theorem [7] which states that Eq.
(2) is true to all orders in perturbation theory. Therefore, Eq.
(4) provides the correct effective Lagrangian with which to compute radiative corrections. The advantage is that amplitudes which correspond to two-loop diagrams in the original theory are one-loop diagrams in the effective theory. A similar Lagrangian can be written for the interaction of photons with the pseudoscalar [8,9]; in that case one sees immediately that the O(α s ) corrections to γγ → A 0 vanish in the m t → ∞ limit.
The effective Lagrangian, Eq. (4), leads to two-and three-gluon vertices with the A 0 (as well as a four-gluon vertex which is irrelevant to the current study): where the k i are the gluon momenta directed inward.
We use dimensional regularization in computing the radiative corrections. However, the ǫ-tensors in Eq. (5) are intrinsically four-dimensional objects and must be treated as such. The product of two ǫ-tensors can be written in terms ofḡ µν the metric tensors in the four-dimensional sub-space: whereḡ µνḡ µν = 4. For simplicity, we take the incoming momenta to be in four dimensions (this is simply a choice of frame). The lowest order cross section is then, averaged over colors and polarizations in n = 4 − 2ǫ dimensions, where z = M 2 A /s and s = (k 1 + k 2 ) 2 .
The virtual radiative corrections are given by the diagrams in Figure 1, where the amplitude involving the four-gluon vertex vanishes due to the antisymmetry of the ggA 0 -vertex. The other diagram yields the result for the virtual corrections where The real corrections are given by the diagrams in Figure 2. The amplitudes squared for the various processes are, averaged over colors and spins, where s, t, and u are the familiar Mandelstam variables and k 3 is the momentum of the final state quark or gluon, with the hat denoting the (n − 4)-dimensional components. Since the initial state particles are taken to be in four dimensionŝ k 3 ·k 3 is the only quantity which depends on the (n − 4)-dimensional components.
Under integration over the angular variables transverse to the incoming particles we find [10] where k T is the transverse momentum of the outgoing gluon. So the terms witĥ k 3 ·k 3 will contribute terms of O(1), but only when multiplied by double poles, t −2 or u −2 (this was used in simplifying Eq. (9)).
Integration over n-dimensional phase space gives the cross sections where It is interesting to note that the real corrections, when written in terms of the lowest-order cross section are identical in form to those for the scalar case [6].
We see that the terms of O(1/ǫ 2 ) cancel between the real and virtual diagrams.
The terms of O(1/ǫ) may be absorbed into redefinitions of the parton distribution functions in the usual factorization procedure. We use the MS prescription which fixes the subtraction terms as where the splitting functions are defined where β 0 = 11 6 N c − 1 3 N f with N f being the number of light quark flavors. The charge renormalization subtraction term is When all the contributions are included the final results arê Since the real corrections in our result are the same as for the scalar case, the only difference between the scalar and pseudoscalar results is the coefficient of the