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Higgs-photon associated production at hadron colliders

We present cross sections for the reactions p pbar -->H gamma and pp -->H\gamma arising from the subprocess q qbar -->H gamma. The calculation includes the complete one-loop contribution from all light quarks as well as the tree contributions from light quarks and c and b quarks. These are the main sources of Higgs-photon associated production at hadron colliders. At Tevatron energies, the cross section is typically between 0.1fb and 1 fb for 80 Gev<m_H<160 GeV, and at LHC energies it exceeds about 2 fb over the same range of m_H. While substantial, these cross sections are not large enough to produce a signal which is discernable above the backgrounds from q qbar-->b bbar gamma and gg-->b bbar gamma.


I. INTRODUCTION
The production of intermediate mass Higgs bosons at hadron colliders arises primarily from the gluon fusion process gg → H, which is dominated by the top quark loop [1]. Depending on the collider energy, there can also be a substantial contribution from the gauge boson fusion processes W + W − , ZZ → H [2]. For Higgs boson masses m H less than 2m W , detection of the Higgs via its dominant decay mode H → bb must contend with a large background from gg → bb. The production of Higgs bosons in association with a gauge boson, specifically Z, W ± [3,4] or a photon [5,6], is potentially helpful in dealing with these backgrounds, although at a significant cost in the rate.
In this report, we examine the possibility of using Higgs-photon associated production in hadron collisions as a means of studying properties of the Higgs boson. The main source of the Hγ final state is quark-antiquark annihilation. Two-gluon annihilation, which accounts for single H production, is forbidden by Furry's theorem because the gluons are in a color singlet state. For light quarks, the direct annihilation into Hγ is suppressed by the ratio m q /m W and is negligible. This means that one-loop electroweak corrections involving W 's, Z's and top quarks dominate the light quark contribution to qq → Hγ. The diagrams involved in the one-loop quark-antiquark annihilation calculation are similar to those encountered in the calculation of eē → Hγ, which has been performed for the Standard Model [5,6] and its supersymmetric generalization [7]. As in the case of µμ → Hγ [8,9], there are also tree-level contributions from cc and bb annihilation, which are enhanced by their larger couplings but suppressed by parton distribution function effects.
In the next section, we outline the extension of our previous tree [9] and one-loop results [6] to qq → Hγ. This is followed by a discussion.

A. Tree-level amplitudes
The tree-level diagrams are illustrated in Fig. 1 and the resulting amplitude is where p 1 is the momentum of the quark, p 2 the momentum of the antiquark, k the momentum of the γ and ǫ its polarization and e q is the quark charge in units of the proton charge. For massless quarks, only the helicity non-flip contributions from the factorsv(p 2 )u(p 1 ) and v(p 2 )σ µν u(p 1 ) are non-zero. Thus, we expect the helicity flip tree amplitudes to contain a factor of order m q /E relative to the non-flip amplitudes. Explicitly, we find, where E is the quark energy in the center of mass, |p| = E 2 − m 2 q , ω is the photon energy, θ is the photon scattering angle and λ γ = ±1 is the photon helicity. It can be seen that the helicity flip amplitudes have an additional factor of m q .

B. One-loop amplitudes
The one-loop amplitudes for qq → Hγ receive contributions from pole diagrams involving virtual photon and Z exchange and from various box diagrams containing quarks, gauge bosons and/or Goldstone bosons. There are also double pole diagrams whose contribution vanishes. This is illustrated in Fig. 1. The main difference between eē annihilation and qq annihilation occurs in the crossed box diagram in the last row of Fig. 1. Since both members of a quark doublet are charged, there are additional crossed box diagrams containing W 's. This contribution can be obtained from our Z box results by merely changing the coupling and replacing m Z by m W .
In the non-linear gauges we chose [6], the full amplitude consists of four separately gauge invariant terms: a photon pole, a Z pole, Z boxes and W boxes. These amplitudes can be written as In terms of the scalar functions defined in the appendices of our previous paper [6], we have [10] with I 3 denoting the third component of the external quark weak isospin, m t being the top quark mass and the prime denoting the replacement m Z → m W .ẽ q is the charge of the internal quark in units of the proton charge. In this case, it is the helicity flip contributions from the factorsv(p 2 )γ µ u(p 1 ) andv(p 2 )γ µ γ 5 u(p 1 ) which survive in the m q → 0 limit. This can be seen by noting that, in the center of mass, we havē then M loop is predominantly helicity flip.

C. Differential cross section
The differential cross section dσ(qq → Hγ)/dΩ γ is where the invariant amplitude M = M tree + M loop . Because of the helicity structure discussed above, the interference terms turn out to be negligible and one can simply add the tree and one-loop cross sections.
To obtain the collider cross sections, we convolute Eq. (13) with the appropriate quark and antiquark distribution functions using dσ dm 2 Here, τ = m 2 Hγ /s , z = cos θ γ and z 0 and z 1 are determined by the choice of the rapidity and transverse momentum cuts. We used CTEQ3-1M parton distribution functions [11] and imposed a rapidity cut of 2.5 and a transverse momentum cut of 10 GeV on both the Higgs boson and the photon.

III. DISCUSSION
The total cross section for pp(pp) → Hγ is illustrated in Fig. 2 for an upgraded Tevatron energy of 2.0 TeV (left panel) and an LHC energy of 14 TeV (right panel). In each case, we show the tree contributions from light quarks (u, d, s), light quarks plus the c quark and light quarks plus c and b quarks. In addition, the loop contribution from the light quarks is shown. The loop contributions from the c and b quarks are negligible. As asserted in the Introduction, Fig. 2 confirms that the light quark tree contribution can always be neglected. Interestingly, the largest tree contribution is due to the c quark for both the Tevatron and the LHC, and the b quark contribution is always important. At the Tevatron energy, the loop diagrams contribute about as much as the tree diagrams, while, for the LHC, the tree diagrams dominate.
In our calculation of the heavy quark tree contributions, we simply convoluted the cross section for QQ → Hγ with the complete heavy parton distribution functions [11], instead of subtracting from them the lowest order logarithmic correction to avoid potential double counting of higher order gluonic contributions [12,13]. Based on an earlier study of Higgs boson production at the SSC [14], this procedure is likely to somewhat overestimate the heavy quark fusion contribution. The effects of QCD corrections have not been considered, although some of these, in particular the K factor, usually increase the heavy quark cross sections [15]. We have not included top quark fusion in our results since the top quark is almost certainly too massive to be treated as a parton at LHC energies.
Although there are non-negligible numbers of events for a Tevatron with an upgraded luminosity of 20 -30fb −1 /year [16] or the LHC with a design luminosity of 100fb −1 /year, the background to the signal pp(pp) → Hγ → bbγ must be considered. To do this, we have calculated the direct production of the bbγ final state arising from the channels pp(pp) → qq → bbγ [17] and pp(pp) → gg → bbγ. In Table I, the background contributions are shown for cuts on the bb invariant mass m bb in the vicinity of m H = 100 GeV and 200 GeV and rapidity cuts y max = 1.0 and 2.5 on the b,b and γ. In addition to these cuts, we require the separation ∆R between the γ and the b and the γ and theb to be greater than 0.4, the transverse momenta of the b,b and γ to be greater than 15 GeV, and the bbγ invariant mass to be greater than 170 GeV. The latter two cuts reduce the background in the m H = 100 GeV region by about a factor of 5. They reduce the loop signal by about 40% at √ s = 1.8 TeV and 14 TeV, but the corresponding reduction in the tree contribution is a factor of 15 at 1.8 TeV and a factor of 6.8 at 14 TeV. Thus the tree contribution, though substantial, is difficult to isolate. The background is compared to the signal in Table II for Higgs boson masses of 100 GeV and 200 GeV. Despite the application of numerous cuts, it appears that the background is prohibitively large for the observation of a Higgs boson with Standard Model couplings produced in association with a photon.
Finally, to determine the sensitivity of associated production to changes in the ttH coupling, we computed the cross section including a factor λ multiplying the Standard Model coupling [18]. For a 2 TeV Tevatron upgrade, this effect is illustrated in Fig. 3 as a function of λ. The most obvious effect is the uniform decrease in the cross section as λ varies from λ = 0 to the Standard Model value at λ = 1.0. With increasing λ, the cross sections eventually exceed the Standard Model result. The enhancement in the cross section for larger values of λ is still not sufficient to produce favorable S/ √ B value. For example, for λ = 10 and m H = 100 GeV, the cuts used for Table II give S/ √ B ∼ .22. It is clear that any anomalous Hγ coupling must result in a cross section of several femtobarns and a fairly isotropic angular distribution in order to produce a signal that will survive the cuts required to reduce the bbγ background.

ACKNOWLEDGMENTS
We would like to thank G. Kane