Preprint typeset in JHEP style - HYPER VERSION

hep-ph/yymmnnn, TTP05–07, SFB/CPP-05-17 — July 2005

Pseudo-scalar Higgs production at next-to-leading order SUSY-QCD

Robert V. Harlander and Franziska Hofmann

Institut für Theoretische Teilchenphysik, Universität Karlsruhe

D-76128 Karlsruhe, Germany

  E-mail: robert.harlander@cern.ch, fhofmann@particle.uni-karlsruhe.de

Abstract: The production rate of the CP-odd Higgs boson in the Minimal Supersymmetric Standard Model is evaluated through next-to-leading order in the strong coupling constant. The divergent integrals are regulated using Dimensional Reduction, with a straightforward implementation of γ_{5}. The well-known Standard Model result is recovered in this scheme through a cancellation between the quark and squark contributions as the masses of the supersymmetric particles tend to infinity.

Keywords: Higgs production, hadron collider, supersymmetry, next-to-leading order calculations.

1. Introduction

The MSSM predicts a fundamental CP-odd scalar particle A, commonly referred to as the pseudo-scalar Higgs boson. One of the most important properties that distinguishes it from its CP-even analogues h and H is the absence of tree-level couplings to the electro-weak gauge bosons W and Z. Decay and production processes through these particles, which have been shown to be extremely helpful for CP-even Higgs searches and studies, are thus very much suppressed. This leaves associated tt̄A and bb̄A production as well as the loopinduced gluon fusion process as the most important production modes of a pseudo-scalar Higgs boson at the LHC (for a recent review, see Ref. [?]). In this paper, we present the evaluation of the inclusive gluon fusion cross section through NLO in the strong coupling constant.

Despite the fact that the tree-level ggϕ coupling vanishes (ϕ ∈ {h, H, A}), gluon fusion in general has a comparatively large cross section due to the high gluon luminosity at the LHC, and the large top-Yukawa coupling. In the limit where squarks and gluinos are decoupled, QCD corrections have been evaluated through NNLO [?, ?, ?, ?, ?]; they have been shown to be numerically significant but perturbatively well-behaved.

A precise determination of the gluon fusion cross section at the LHC could yield sensitivity to as yet undiscovered particles that may mediate the ggϕ coupling apart from the top and bottom quarks. Within the MSSM, for example, top squarks can play a significant role if they are lighter than around 400 GeV (recall that the Yukawa coupling for squarks is typically proportional to m_{q}^{2} rather than m2 ; q̃). In the case of CP-even Higgs bosons, such effects occur already at LO (i.e., 1-loop). The NLO result was obtained in Ref. [?] within an effective theory for top, stop, and gluino masses much larger than the Higgs mass.

For the CP-odd Higgs boson, however, squarks do not affect the ggA vertex at 1-loop level due to the structure of the Aq̃q̃ coupling, as will be shown below. The two-loop effects are thus expected to have a larger influence than for the CP-even Higgs production. What adds to this is that, in the limit of large m_{q}, the quark mediated contribution to the ggA vertex does not receive any QCD corrections due to the Adler-Bardeen theorem. As we will show, the only 2-loop QCD effects to this coupling are due to mixed gluino-quark-squark diagrams in this limit, leading to a potentially increased sensitivity to the gluino mass. Of course, the heavy quark limit is not applicable for the bottom mediated gluon-Higgs coupling which does receive QCD corrections [?, ?].

The calculation involves a technical issue that deserves special mention, namely the implementation of γ_{5} within DRED. Its mathematically consistent and practically feasible formulation has been a subject of interest for many years now [?] (for recent developments concerning DRED, see Ref. [?, ?]). Here we adopt an approach close to the prescription of Refs. [?, ?]. An important consistency check is obtained from the SM limit: even for an infinitely heavy SUSY spectrum, the SUSY diagrams give a non-vanishing contribution which exactly cancels the 2-loop contributions arising from the SM diagrams in this scheme. The well-known result of vanishing higher order corrections to the quark-mediated ggA coupling, as required by the Adler-Bardeen theorem, is thus recovered in a non-trivial way. Combining this observation with the considerations of Ref. [?], it seems likely that in a supersymmetric theory the formal difficulties of DRED do not pose serious technical problems in practical calculations [?].

The outline of this paper is as follows: In Sect. ?? we introduce our notation and quote the LO result for the partonic process gg → A. In Sect. ??, the effective Lagrangian underlying our calculation is introduced and the treatment of γ_{5} is discussed. Sect. ?? outlines the method of the calculation and discusses the general structure of the result. It also provides the analytic formulae in some limiting cases. The numerical influence of the NLO terms is discussed in Sect. ??. In Sect. ?? our findings are summarized and an outlook on possible extensions of this work is given.

2. Notation and Leading Order Result

2.1 Lagrangian

We write the underlying Lagrangian in the following form:

L = L_{QCD} + L_{qA} + L_{SQCD} + L_{q̃A} ,  (2.1)

where

L_{qA} = ∑_{q} \frac{m_{q}}{v} g_{q}^{A} q̄γ_{5}qA ,  L_{q̃A} = ∑_{q} ∑_{i=1}^{2} \frac{m_{q}^{2}}{v} g̃_{q,ij}^{A} q̃_{i}q̃_{j} A .  (2.2)

The sums ∑_{q} run over all quark flavors. m_{q} is the mass of quark q and v ≈ 246 GeV the Higgs vacuum expectation value; the coupling constants g_{q}^{A} and g̃_{q,ij}^{A} will be defined below. L_{QCD} denotes the full QCD Lagrangian with six quark flavors, while L_{QCD} + L_{SQCD} is the supersymmetric extension of L_{QCD} within the MSSM, i.e., L_{SQCD} incorporates kinetic, mass, mixing, and interaction terms of all the squarks and gluinos. Since we will be concerned with higher orders in the strong coupling α_{s} only, A does not appear as a dynamical field and does not require a kinetic term.

q̃_{1}, q̃_{2} denote the squark mass eigenstates, related to the chiral eigenstates q̃_{L}, q̃_{R} (the superpartners of the left- and the right-handed quark q) through

( q̃_{1} ; q̃_{2} ) = ( \begin{matrix}cos θ_{q} & sin θ_{q} \\ − sin θ_{q} & cos θ_{q} \end{matrix} ) ( q̃_{L} ; q̃_{R} ) .  (2.3)

The coupling constants relevant for the following discussion are^{1}

g_{b}^{A} = tan β ,  g_{t}^{A} = cot β ,  g̃_{t,11}^{A} = g̃_{t,22}^{A} = 0 ,
g̃_{t,12}^{A} = −g̃_{t,21} = \frac{m2 ; t̃_{1} − m2 ; t̃_{2}}{2m_{t}^{2}} sin θ_{t} cot β + \frac{μ_{SUSY}}{m_{t}} (cot^{2} β + 1) .

(2.4)

1For a more complete collection of Feynman rules, see Ref. [?], for example.

The effect of quarks other than bottom and top can be neglected due to their small Yukawa couplings; squark effects, on the other hand, are typically suppressed by at least one power of m_{q}/mq̃, so that only top squarks will be considered in this paper.

2.2 Hadronic cross section

The cross section for the hadronic process pp → A + X at a center-of-mass energy s is determined by the formula^{2}

σ(z) = ∑_{i,j∈\{q,q̄,g\}} ∫_{z}^{1} dx^{1} ∫_{z/x_{1}}^{1} dx^{2} φ^{i}(x^{1})φ^{j}(x^{2}) σ̂_{ij} (\frac{z}{x_{1}x_{2}} ) ,  z := \frac{M_{A}^{2}}{s} ,  (2.5)

where φ_{i}(x) is the density of parton i inside the proton. σ̂_{ij} is the cross section for the process ij → A + X, where, as indicated in Eq. (??), i and j are parton labels. For our numerical analysis, we will use the MRST parton density sets throughout this paper [?, ?].

Sample diagrams for i=j=g and i=g, j=q that contribute to the inclusive Higgs production rate at LO and NLO are shown in Fig. ??.

(a)  (b)

(c)  (d)  (e)

(f)  (g)  (h)  (i)

Figure 1: Diagrams contributing to the process gg → A + X in the MSSM at (a) LO and (b)–(i) NLO. According to the definition of the main text, (a)–(e) are referred to as “SM diagrams”, (f)–(i) as “SUSY diagrams”. The straight-curly line connecting quarks and squarks represents a gluino (denoted by g̃ in the text).

2The modifications for pp̄ collisions are obvious and shall not be pointed out explicitely in this paper.

2.3 Standard Model limit and leading order result

In the limit where all SUSY masses tend to infinity (denoted in the following by M_{SUSY} → ∞), the Lagrangian of Eq. (??) reduces to

L^{SM} = L_{QCD} + L_{qA} .  (2.6)

This will be called the “Standard Model” limit in what follows, despite the fact that the SM does not contain a CP-odd Higgs boson. Consequently, diagrams without any squark and gluino lines will be called “SM diagrams” (e.g. Fig. ?? (a)–(e)), while “SUSY diagrams” contain at least one propagator of a SUSY particle^{3} (Fig. ?? (f)–(i)).

Note that due to the antisymmetric structure of the g̃_{q,ij}^{A}, Eq. (??), there are no SUSY diagrams at the one-loop level. The LO result for the partonic process gg → A is thus determined solely by diagrams of the type shown in Fig. ?? (a). It reads explicitely:

\begin{aligned}σ̂_{gg}(x) & = σ_{0}^{A} δ(1 − x) + O(α_{s}^{3}) , \\ σ_{0}^{A} & = \frac{π}{256v^{2}} (\frac{α_{s}}{π} )^{2} | ∑_{q∈\{t,b\}} A_{q}(τ_{q})|^{2} , \\ A_{q}(τ_{q}) & = g_{q}^{A} τ_{q}f(τ_{q}) ,  τ_{q} = \frac{4m_{q}^{2}}{M_{A}^{2}} , \end{aligned}  (2.7)

where

f(τ) = \{ \begin{matrix}arcsin^{2} \frac{1}{\sqrt{τ}} , & τ ≥ 1 , \\ −\frac{1}{4} [ln \frac{1 + \sqrt{1 − τ}}{1 − \sqrt{1 − τ}} − iπ]^{2} , & τ < 1 . \end{matrix}  (2.8)

Throughout this paper, α_{s} denotes the strong coupling constant, renormalized in the \overline{MS} scheme for QCD with five massless quark flavors.

3. Higher orders

3.1 Standard Model result – Effective Lagrangian

The SM contributions, based on L_{SM} of Eq. (??), are known through NLO in terms of 1-dimensional integral representations [?], implemented in the program HIGLU [?]. They include both top and bottom quark loops as shown in Fig. ?? (a)–(e) (q ∈ {b, t}).

It has been shown in Ref. [?, ?] that the NLO top quark contributions are well approximated by the formula

σ_{t}^{NLO} = K_{∞} σ_{t}^{LO} , where  K_{∞} = \frac{σ_{∞}^{NLO}}{σ_{∞}^{LO}} .  (3.1)

3In the following we will strictly distinguish between SM/SUSY diagrams and SM/SUSY results: As it turns out, the SM diagrams do not give the SM result in our scheme.