NLO Supersymmetric QCD Corrections to $t \bar t h^0$ Associated Production at Hadron Colliders

We calculate NLO QCD corrections to the lightest neutral Higgs boson production associated with top quark pair at hadron colliders in the minimal supersymmetric standard model(MSSM). Our calculation shows that the total QCD correction significantly reduces its dependence on the renormalization/factorization scale. The relative correction from the SUSY QCD part approaches to be a constant, if either $M_S$ or $m_{\tilde{g}}$ is heavy enough. The corrections are generally moderate(in the range of few percent to 20%) and under control in most of the SUSY parameter space. The relative correction is obviously related to $m_{\tilde{g}}$, $A_t$ and $\mu$, but not very sensitive to $\tan\beta$, $M_S$ at both the Tevatron and the LHC with our specified parameters.


I. Introduction
One of the major objectives of future high-energy experiments is to search for scalar Higgs particles and investigate the symmetry breaking mechanism of the electroweak interactions.In the standard model (SM) [1], one doublet of complex scalar fields is introduced to spontaneously break the symmetry, leading to a single neutral Higgs boson h 0 .But there exists the problem of the quadratically divergent contributions to the corrections to the Higgs boson mass.This is the so-called naturalness problem.One of the hopeful methods, which can solve this problem, is the supersymmetric (SUSY) extension to the SM.In these extension models, the quadratic divergences of the Higgs boson mass can be cancelled by loop diagrams involving the supersymmetric partners of the SM particles exactly.The most attractive and simplest supersymmetric extension of the SM is the minimal supersymmetric standard model (MSSM) [2,3].In this model, there are two Higgs doublets H 1 and H 2 to give masses to up-and down-type fermions.The Higgs sector consists of three neutral Higgs bosons, one CP -odd particle (A 0 ), two CP -even particles (h 0 and H 0 ), and a pair of charged Higgs bosons (H ± ).
However, these Higgs bosons haven't been directly explored experimentally until now.The published experimental lower mass bounds for the Higgs bosons presented by LEP experiments are: M h 0 > 114.4 GeV (at 95% CL) for the SM Higgs boson, and for the MSSM bosons M h 0 > 91.0 GeV and M A 0 > 91.9 GeV (at 95% CL, 0.5 < tan β < 2.4 excluded).The SM fits to precision electroweak data [4] indirectly set a limitation of the light Higgs boson, M h 0 < 200 GeV, while there should has a scalar Higgs boson lighter than about 130 GeV in MSSM.[5].This lightest Higgs boson with mediate mass is certainly in the exploring mass range of the present and future colliders, such as the Tevatron Run II, LHC and LC.At a LC the cross section for e + e − → t th is small, about 1 fb for √ s = 500 GeV and m h = 100 GeV [6,7].But it has a distinctive experimental signature and can potentially be used to measure the top quark Yukawa coupling in the intermediate Higgs mass region at a LC with very high luminosity.S. Dawson and L. Reina calculated the NLO QCD corrections to e + e − → t th 0 process at LC's in Ref. [8].And in references [9,10,11] the SM electroweak corrections to the process e + e − → t th 0 are calculated.H. Chen et al., have studied the QCD and electroweak corrections to the process γγ → t th 0 in the SM at LC's [12].All these works show that the evaluation of radiative corrections is a crucial task for all accurate measurements of t th 0 production process.There are various channels which can be exploited to search for the Higgs boson h 0 with intermediate mass at TeV energy scale hadron colliders, such as gluon-gluon fusion Higgs boson production(gg → h 0 ), the associated production with a weak intermediate boson (qq ′ → W h 0 , Zh 0 ).Recently, the production channels pp/pp → t th 0 + X attracted the physicist's attentions, because these channels offer a spectacular signature (W + W − b bb b) [13] and provides a possibility in probing the Yukawa coupling [14,15].The total cross section for pp/pp → t th 0 + X at tree level and NLO QCD corrections in the SM have been studied in Refs.[14,15,16,17,18].
The supersymmetric (SUSY) electroweak corrections to the e + e − → t th 0 process can be over ten percent for favorable parameter values [19].In Ref. [20], it was found that the SUSY QCD interactions by exchanging gluinos and squarks can impact on the Yukawa coupling vertex in the process e + e − → t th 0 at LC.At pp/pp hadron colliders with a center-of-mass energy of TeV scale, the dominated contributions to t th 0 production are from subprocesses q q, gg → t th 0 .To these high energy t th 0 production processes, the SUSY radiative corrections, especially the SUSY QCD corrections, may be remarkable.
In this paper, we calculated the cross section for the associated production of the Higgs boson with top quark pair in the MSSM at hadron colliders including the NLO QCD corrections.In section 2, we present the calculations of the leading order cross sections to pp/pp → t th 0 + X in the MSSM.In section 3, we present the calculations of the O(α 3 s ) QCD corrections to pp/pp → t th 0 + X in the MSSM.The numerical results and discussions are presented in section 4. Finally, a short summary is given.

II. The leading order cross sections
The Feynman diagrams at leading order(LO) for the subprocess in the MSSM are plotted in Fig. 1.They are s-channel, gluon exchange diagrams with Higgs boson radiation off top-quark and anti-top-quark, respectively.The process at the tree level in the MSSM are described by the Feynman diagrams of Fig. 2. The LO Feynman diagrams for both subprocesses in the MSSM are the same with their corresponding ones in the SM.
In above two channels we use p 1,2 and k 1,2,3 to represent the four-momenta of the incoming partons and the outgoing particles, respectively.Because of the small mass of u-and d-quark, we neglect the diagrams which involve h 0 − u − ū and h 0 − d − d Yukawa vertexes.
Fig. 1.The tree-level Feynman diagrams for the q q → t th subprocess.
The explicit expression for the amplitudes of subprocess q q → t th 0 at tree level can be written as: But in the MSSM, Y is given as where α is the mixing angle which leads to the physical Higgs boson eigenstates h 0 and H 0 .The angle β is defined as tan β = v 2 /v 1 , where v 1 and v 2 are the vacuum expectation values.
According to the different topologies of Feynman diagrams, the explicit expression for the amplitudes of subprocess gg → t th 0 in the MSSM at tree level can be divided into three parts.
where M gg1 tree , M gg2 tree and M gg3 tree correspond to the amplitudes of Fig. 2(a-b), Fig. 2(c-f) and Fig. 2(g-h), respectively.For the amplitude parts M ggi tree (i = 1, 2, 3), we have the expressions as: where ǫ µ 1 and ǫ ν 2 are the polarization four-vectors of the incoming gluons.The SU(3) structure constants are given by f abc .Then the lowest order cross sections for the subprocesses q q, gg → t th 0 in the MSSM are obtained by using the following formula: where dΦ 3 is the three-body phase space element.The summation is taken over the spins and colors of initial and final states, and the bar over the summation recalls averaging over the spins and colors of initial partons.The LO total cross section of pp/pp → t th 0 + X can be expressed as: where σij LO (ij = q q, gg) is the LO parton-level total cross section for incoming i and j partons, G p/p i 's are the LO parton distribution functions (PDF) with parton i in a proton/antiproton.
From the above deduction, we can see that the ratio between the tree level cross sections of subprocess q q(gg) → t th 0 in the SUSY model and the SM, is written as

III. NLO QCD Corrections in the MSSM
In the calculation of the NLO QCD corrections in the MSSM, we adopt the dimensional regularization in D = 4 − 2ǫ dimensions to isolate the ultraviolet(UV), infrared(IR) and collinear singularities.Renormalization and factorization are performed in the modified minimal substraction(M S) scheme, and the wave functions of the external fields, and top quark's mass in propagators and in the Yukawa couplings are renormalized in the on-shell(OS) scheme.We divide the O(α 3 s ) QCD correction to the subprocess q q(gg) → t th 0 in the MSSM into two parts.One is the so-called SM-like QCD correction part, another is SUSY-QCD correction part arising from virtual gluino/squark exchange contributions.Then the total NLO QCD corrections and relative corrections in the MSSM can be expressed as ∆σ (q q,gg) where we define the relative correction as δ = ∆σ NLO σLO .The NLO SM-like QCD correction part(relative correction part) in the MSSM has following relation with the NLO SM QCD one In our calculation we introduce the following counterterms.
where g s denotes the strong coupling constant, t, u, d and G µ denote the fields of top-, up-, downquark and gluon.The definitions and the explicit expressions of these renormalization constants can be found in Ref. [21].For the renomalization of the QCD coupling constant g s , we use the M S scheme except that the divergences associated with the colored SUSY particle loops are subtracted at zero momentum [22].Since we have δg = δg (SM −like) + δg (SQCD) , the terms should be obtained as δg where N = 3 ,n f = 5 and 1/ǭ = 1/ǫ U V − γ E + ln(4π).The summation is taken over the indexes of squark and generation.The M S scheme violates supersymmetry explicitly, and the q qg Yukawa coupling ĝs , which should be the same with the qqg gauge coupling g s in the supersymmetry, takes a finite shift at one-loop order as shown in Eq.(3.7) [23].
with N = 3 and C F = 4/3.In our numerical calculation we take this shift between ĝs and g s into account.
Actually, the calculation of the NLO SM-like QCD corrections in the MSSM for the subprocesses q q, gg → t th 0 is the same as that of the NLO SM QCD corrections in Refs.[14,15], except their numerical results satisfied the relations shown in Eq.(3.2).
The NLO SUSY-QCD contribution part to the q q(gg) → t th 0 subprocess comes from the oneloop diagrams involving virtual gluino/squark exchange.For demonstration, we show the pentagon diagrams which contribute to the NLO SUSY-QCD correction part for the subprocesses q q → t th 0 and gg → t th 0 in Fig. 3, where the upper indexes s, t, u = 1, 2. Because there is no massless particle in the loop, all these diagrams with gluino/squark loop are IR finite.The pentagon and box diagrams in SUSY-QCD part are UV finite, but the self-energy and vertex diagrams in this part contain UV divergences.That is renormalized by the proper related counterterms defined in Eq.(3.3).   3. The pentagon diagrams for the q q → t th 0 and gg → t th 0 subprocess.
The O(α 3 s ) supersymmetric QCD correction part of the cross section in the MSSM to the subprocesses q q, gg → t th 0 can be expressed as ∆σ (q q,gg) where M (q q,gg) tree are the Born amplitudes for q q, gg → t th 0 subprocesses, and M (q q,gg) SQCD are the renormalized amplitudes of all the one-loop Feynman diagrams involving virtual gluino/squark.
In the calculations of loop diagrams we adopt the definitions of one-loop integral functions of Ref. [24].The Feynman diagrams and the relevant amplitudes are generated by FeynArts 3 [25], and the Feynman amplitudes are subsequently reduced by FormCalc32.The phase space integration is implemented by using Monte Carlo technique.The numerical calculations of integral functions are implemented by using developed LoopTools.
We write the NLO QCD corrected parton-level total cross section σij N LO (x 1 , x 2 , µ) as: σ(qq,gg) The NLO QCD corrected total cross section of pp/pp → t th 0 + X in the MSSM can be expressed as: where σ(ij) NLO (ij = q q, gg) is the NLO QCD corrected parton-level total cross section for incoming i and j partons, and G p/p i are the NLO parton distribution functions (PDF) for parton i in a proton/antiproton.The equation include two channels: q q, gg → t th 0 .In our calculations, we choose the factorization scale equals the renormalization scale, i.e., µ f = µ r = Q.The partonic center-of-mass energy squared, ŝ, is given in terms of the total hadronic center-of-mass energy squared ŝ = x 1 x 2 s.

IV. Numerical Results and Discussion
In our numerical calculation, we adopt the MRST NLO parton distribution function [28] and the 2-loop evolution of α s (µ 2 ) to evaluate the hadronic NLO QCD corrected cross sections, while for the hadronic LO cross sections we use the MRST LO parton distribution function and the one-loop evolution of α s (µ 2 ).We take the SM parameters as α ew (m 2 Z ) −1 = 127.918,m W = 80.423 GeV , m Z = 91.188GeV , m t = 174.3GeV , m u = m d = 66 M eV [29].There we use the effective values of the light quark masses (m u and m d ) which can reproduce the hadron contribution to the shift in the fine structure constant α ew (m 2 Z ) [30].The other relevant parameters, such as mixing angle of the Higgs fields α and masses of the lightest Higgs boson, gluino, stop-quarks, are obtained by adopting the FormCalc package, except otherwise stated.The input parameters for the FromCalc program are M S , M 2 , A t , m A 0 , µ and tan β.The related parameters for the MSSM Higgs sector are obtained from the CP-odd mass m A 0 and tan β with the constraint tan β ≥ 2.5.In the program the grand unification theory(GUT) relation M 1 = (5/3) tan 2 θ W M 2 is adapted for simplification and the gluino mass m g is evaluated by For the sfermion sector, the relevant input parameters are M S , A f and µ, and there we take the assumptions of and the soft trilinear couplings for sfermions q and l being equal, i.e., A q = A l = A f .We present the dependence of the cross section on the renormalization/factorization scale Q/Q 0 in Fig. 4(a-b) for the Tevatron and the LHC separately, where we denote Q 0 = m t + m h 0 /2 and the input parameters are taken as A t = 800 GeV , M S = 400 GeV , M 2 = 110 GeV , m A 0 = 270 GeV , µ = −200 GeV and tan β = 6.With these input parameters, we get all the other supersymmetric parameters, among them cos α = 0.954, m h 0 = 120 GeV , m t1 = 207.75GeV and m t2 = 577.63GeV , but the value of m g is a function of the energy scale Q(m g (Q 0 ) = 317.9GeV ).In order to show the cross section dependence on the renormalization/factorization scale, we fix m g = 300 GeV in Fig. 4(a-b).There we plot the curves for cross sections σ LO , σ N LO and σ SM −like N LO of the processes pp/pp → t th 0 + X.The notations σ N LO and σ SM −like N LO represent the cross sections involving complete QCD and SM-like QCD corrections.Fig. 4(a) shows that the NLO QCD contributions to the process pp → t th 0 + X in the MSSM at the Tevatron, in which the dominant subprocess is q q → t th 0 , has a negative NLO QCD corrections near the position of Q = Q 0 .While Fig. 4(b) shows that the NLO QCD contributions to the process pp → t th 0 + X in the MSSM at the LHC, in which the dominant subprocess is gg → t th 0 , will give positive corrections near the position of Q = Q 0 .Here we should note that if Q goes down to a very low value, i.e., Q << Q 0 , large logarithmic corrections spoil the convergence of perturbation theory in the proton-antiproton colliding energy of the Tevatron.That can be seen from our numerical results for the Tevatron.It shows that the total NLO QCD corrected cross section σ N LO in the MSSM tends to have a negative value when Q → 0 [15].From Fig. 4(a-b), we can conclude that the dependence of the NLO QCD corrected cross section σ N LO on the scale Q is significantly reduced comparing with σ LO , and is slightly weakened comparing with σ SM −like N LO .In the following calculation, we fixed the value of the renormalization/factorization scale being Q 0 .In Fig. 5(a) and (b) we show the LO and total NLO QCD cross sections σ LO and σ N LO in the MSSM as the functions of tan β(m h 0 ) at the Tevatron and the LHC respectively, taking A t = 800 GeV , M S = 400 GeV , M 2 = 110 GeV , m A 0 = 270 GeV and µ = −200 GeV .The corresponding relative corrections δ of both cross sections versus tan β(m h 0 ), where the relative correction is defined as , are plotted in Fig. 5(c).From these figures, we can see that the cross sections σ N LO and σ LO decrease rapidly as tan β varies in the range from 2 to 10, and then goes down very slowly when tan β changes from 10 to 40.We can read from Fig. 5(a-b) that when tan β increases from 2 to 40, the total NLO QCD corrected cross section σ N LO in the MSSM decreases roughly from 9.1 f b and 1078 f b to 5.2 f b and 641 f b for the Tevatron and the LHC, respectively.The two curves of relative corrections δ for the Tevatron and the LHC in Fig. 5(c) look like rather stable when tan β runs from 2 to 50.We can read from this figure that the NLO QCD relative correction values in the MSSM at the Tevatron and the LHC are generally about −17% and 26% in these varying range of tan β, respectively.
In Fig. 6, we show the relative NLO QCD correction δ in the MSSM as a function of M S , taking A t = 800 GeV , M 2 = 110 GeV , m A 0 = 270 GeV , µ = −200 GeV and tan β = 6.The figure demonstrates that the relative NLO QCD corrections in the MSSM at the Tevatron and the LHC, are stable when M S changes from 400 GeV to 2 T eV .Their values are about 25% at the LHC and −18% at the Tevatron.We find from our calculation that when M S is taken as a large value, the correction from the NLO SUSY QCD correction part decreases to a constant due to the decouple of heavy stop quarks.Fig. 7 shows the total QCD relative correction δ in the MSSM as a function of m g with the input parameters same as in Fig. 4. We can see from this figure that the δ have a concave structure in the vicinity of m g ∼ 140 − 150 GeV , where the masses satisfy the relation m g + m t ≈ m t1 + m h 0 and the re-scattering enhancement t * 1 → g + t → t1 + h 0 takes place.When m g goes from 400 GeV to 2000 GeV , the relative corrections are very stable, they are about 24% for the LHC and −18% for the Tevatron.Similar with the case in Fig. 6, due to the decouple effect the correction of the SUSY QCD correction part decreases to a constant when g is getting heavy.
Fig. 8 presents the total NLO QCD relative correction δ in the MSSM as a function of the SUSY parameter A t , assuming M S = 400 GeV , M 2 = 110 GeV , m A 0 = 270 GeV , µ = −200 GeV and tan β = 6.We can see from the figure that the total NLO QCD relative corrections in the MSSM are very sensitive to A t in the region near the position of A t = 1000 GeV (where m t1 = 108.6GeV ).Actually, the reason for that is because of the large contribution from the light stop quark t1 loops.When the chosen parameters A t and µ make a large mass splitting between the two scalar top-quarks, then the t1 becomes light.We can read from the figure the total NLO QCD relative correction δ in the MSSM can reach −24% at the Tevatron and 7% at the LHC when A t is near 1000 GeV .
In Fig. 9, we show the total NLO QCD relative correction δ in the MSSM as a function of the SUSY parameter µ, assuming A t = 800 GeV , M S = 400 GeV , M 2 = 110 GeV , m A 0 = 270 GeV and tan β = 6.We can see that the total NLO QCD relative corrections in the MSSM increase slowly when µ goes up from −1000 GeV to 1000 GeV , this is because the absolute values of the negative corrections from the SUSY QCD part are becoming smaller as µ increases.The value of δ at the Tevatron can be beyond −22% when µ is about −1000 GeV .
In this paper we calculated the NLO QCD corrections to the processes pp/pp → t th 0 + X in the MSSM at the Tevatron and the LHC.We analyzed the dependence of the corrected cross sections or relative corrections on the renormalization/factorization scale Q, SUSY parameters tan β, M S , m g, A t and µ, respectively.It shows that the dependence of the total NLO QCD corrected cross section in the MSSM on the scale Q is significantly reduced comparing with the σ LO .With our chosen parameters, the numerical results demonstrate that the relative correction is obviously related to m g, A t and µ in some parameter regions, but not very sensitive to tan β, M S at both the Tevatron and the LHC for our specified parameters.We conclude that the total NLO QCD corrections are generally moderate, which have the values in the range of few percent to about 20% in most of the SUSY parameter space.We find that the relative correction from the NLO SUSY QCD correction part becomes to be constant when either M S or m g has large value.We find also the relative correction of the SUSY QCD part will be largely enhanced when the mass splitting between stop-quarks is large, the total NLO QCD relative correction in the MSSM δ can reach −24% at the Tevatron and 7% at the LHC.
Fig.3.The pentagon diagrams for the q q → t th 0 and gg → t th 0 subprocess.

Fig. 4 .
Fig.4.The cross sections σ LO at the leading order and σ N LO involving the NLO QCD corrections in the MSSM as the functions of the renormalization/factorization scale Q with m g = 300 GeV and the other parameters are from FormCalc by using the input SUSY parameters: M S = 400 GeV , M 2 = 110 GeV , A t = 800 GeV , m A 0 = 270 GeV , µ = −200 GeV and tan β = 6.Fig.4(a) is for the process pp → t th 0 + X at the Tevatron and Fig.4(b) for the process pp → t th 0 + X at the LHC.

Fig. 5 .Fig. 6 .Fig. 7 .
Fig. 5.The cross sections σ LO at the leading order and σ N LO involving the NLO QCD corrections in the MSSM as the functions of tan β with the input parameters A t = 800 GeV , M 2 = 110 GeV , m A 0 = 270 GeV , M S = 400 GeV , µ = −200 GeV .Fig.5(a) is for the process pp → t th 0 + X at the Tevatron and Fig.5(b) for the process pp → t th 0 + X at the LHC.Fig.5(c) shows the relative NLO QCD correction as the function of tan β in both the Tevatron and LHC.

Fig. 8 .Fig. 9 .
Fig.8.The total NLO QCD relative corrections(δ) of the processes pp/pp → t th 0 + X at the Tevatron and the LHC, as the functions of A t .
The tree-level Feynman diagrams for the gg → t th subprocess.h 0− t − t Yukawa coupling in the SM Y