Neutral Higgs boson production at \(e^+e^\) colliders in the complex MSSM: a full oneloop analysis
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Abstract
For the search for additional Higgs bosons in the Minimal Supersymmetric Standard Model (MSSM) as well as for future precision analyses in the Higgs sector precise knowledge of their production properties is mandatory. We evaluate the cross sections for the neutral Higgs boson production at \(e^+e^\) colliders in the MSSM with complex parameters (cMSSM). The evaluation is based on a full oneloop calculation of the production mechanism \(e^+e^ \rightarrow h_i Z,\, h_i \gamma ,\, h_i h_j\) \((i,j = 1,2,3)\), including soft and hard QED radiation. The dependence of the Higgs boson production cross sections on the relevant cMSSM parameters is analyzed numerically. We find sizable contributions to many cross sections. They are, depending on the production channel, roughly of 10–20 % of the treelevel results, but can go up to 50 % or higher. The full oneloop contributions are important for a future linear \(e^+e^\) collider such as the ILC or CLIC. There are plans to implement the evaluation of the Higgs boson production cross sections into the code FeynHiggs.
Keywords
Higgs Boson Large Hadron Collider Minimal Supersymmetric Standard Model Production Cross Section Loop Correction1 Introduction
The discovery of a new particle with a mass of about \(125 \,\, \mathrm {GeV}\) in the Higgs searches at the Large Hadron Collider (LHC), which has been announced by ATLAS [1] and CMS [2], marks the culmination of an effort that has been ongoing for almost half a century and opens a new era of particle physics. Within the experimental and theoretical uncertainties the properties of the newly discovered particle measured so far are in agreement with a Higgs boson as predicted in the Standard Model (SM) [3].
The identification of the underlying physics of the discovered new particle and the exploration of the mechanism of electroweak symmetry breaking will clearly be a top priority in the future program of particle physics. The most frequently studied realizations are the Higgs mechanism within the SM and within the Minimal Supersymmetric Standard Model (MSSM) [4, 5, 6, 7]. Contrary to the case of the SM, in the MSSM two Higgs doublets are required. This results in five physical Higgs bosons instead of the single Higgs boson in the SM. In lowest order these are the light and heavy \(\mathcal{CP}\)even Higgs bosons, h and H, the \(\mathcal{CP}\)odd Higgs boson, A, and two charged Higgs bosons, \(H^\pm \). Within the MSSM with complex parameters (cMSSM), taking higherorder corrections into account, the three neutral Higgs bosons mix and result in the states \(h_i\) (\(i = 1,2,3\)) [8, 9, 10, 11, 12]. The Higgs sector of the cMSSM is described at the tree level by two parameters: the mass of the charged Higgs boson, \(M_{H^\pm }\), and the ratio of the two vacuum expectation values, \(\tan \beta \equiv t_\beta = v_2/v_1\). Often the lightest Higgs boson, \(h_1\) is identified [13] with the particle discovered at the LHC [1, 2] with a mass around \(\sim 125\,\, \mathrm {GeV}\) [14]. If the mass of the charged Higgs boson is assumed to be larger than \(\sim 200\,\, \mathrm {GeV}\) the four additional Higgs bosons are roughly mass degenerate, \(M_{H^\pm }\approx m_{h_{2}} \approx m_{h_{3}}\) and referred to as the “heavy Higgs bosons”. Discovering one or more of the additional Higgs bosons would be an unambiguous sign of physics beyond the SM and could yield important information as regards their possible supersymmetric origin.
Results for the cross sections (1)–(3) have been obtained over the last two decades. Treelevel calculations for \(e^+e^ \rightarrow AH,HZ,Ah\) in the rMSSM have been presented in Ref. [57]. First loop corrections to hZ, hA and AZ production in the rMSSM were published in Refs. [58, 59, 60], respectively. A first (nearly) full calculation of the production channels (1) and (2) in the rMSSM was presented in Ref. [61] (leaving out only a detailed evaluation of the initial state radiation).^{1} A complete oneloop calculation in the rMSSM of channel (1) with \(h_i = h, H\) and \(h_j = A\) was presented in Ref. [62]. Unfortunately, only treelevel results were given. A treelevel evaluation of the channels (1) and (2) in the cMSSM was presented in Ref. [63], where higherorder corrections were included via effective couplings. The loopinduced processes (3) with \(h_i = h,H,A\) in the rMSSM were published in Ref. [64] and the (also loopinduced) channels (2) and (3) with \(h_i = A\) in the cMSSM were published in Ref. [65]. Higherorder corrections to the channels (1) and (2) in the cMSSM were given in Ref. [29], where the thirdgeneration (s)fermion contributions to the production vertex as well as Higgs boson propagator corrections were taken into account. Another full oneloop calculation of \(e^+e^ \rightarrow h Z\) was given in Ref. [66], but again restricted to the rMSSM. In Refs. [67, 68] “supersimple” expressions have been derived for the processes \(e^+e^ \rightarrow hZ, h\gamma \) in the rMSSM. The production of two equal Higgs bosons in \(e^+e^\) collisions in the rMSSM, where only box diagrams contribute, were presented in Ref. [69] and further discussed in Ref. [70]. Finally, the process \(e^+e^ \rightarrow hZ\) with leading corrections in the rMSSM has been computed in Ref. [71]. A numerical comparison with the literature will be given in Sect. 4.
In this paper we present for the first time a full and consistent oneloop calculation for neutral cMSSM Higgs boson production at \(e^+e^\) colliders in association with a SM gauge boson or another cMSSM Higgs boson. We take into account soft and hard QED radiation, collinear divergences and the \(\hat{Z}\) factor contributions. In this way we go substantially beyond the existing calculations (see above). In Sect. 2 we very briefly review the renormalization of the relevant sectors of the cMSSM. Details as regards the calculation can be found in Sect. 3. In Sect. 4 various comparisons with results from other groups are given. The numerical results for all production channels (1)–(3) are presented in Sect. 5. The conclusions can be found in Sect. 6. There are plans to implement the evaluation of the production cross sections into the Fortran code FeynHiggs [45, 46, 47, 48, 49, 50].
1.1 Prolegomena

FeynTools \(\equiv \) FeynArts + FormCalc + LoopTools + FeynHiggs.

\(s_\mathrm {w}\equiv \sin \theta _W\), \(c_\mathrm {w}\equiv \cos \theta _W\).

\(s_{\beta  \alpha }\equiv \sin (\beta \alpha )\), \(c_{\beta  \alpha }\equiv \cos (\beta \alpha )\), \(t_\beta \equiv \tan \beta \).
2 The complex MSSM
The cross sections (1)–(3) are calculated at the oneloop level, including soft and hard QED radiation; see the next section. This requires the simultaneous renormalization of the Higgs and gaugeboson sector as well as the fermion sector of the cMSSM. We give a few relevant details as regards these sectors and their renormalization. More information can be found in Refs. [30, 31, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81].
The renormalization of the Higgs and gaugeboson sector follows strictly Ref. [72] and references therein (see especially Ref. [45]). This defines in particular the counterterm \(\delta t_\beta \), as well as the counterterms for the Z boson mass, \(\delta M_Z^2\), and for the sine of the weak mixing angle, \(\delta s_\mathrm {w}\) (with \(s_\mathrm {w}= \sqrt{1  c_\mathrm {w}^2} = \sqrt{1  M_W^2/M_Z^2}\), where \(M_W\) and \(M_Z\) denote the W and Z boson masses, respectively).
3 Calculation of diagrams
In this section we give some details as regards the calculation of the treelevel and higherorder corrections to the production of Higgs bosons in \(e^+e^\) collisions. The diagrams and corresponding amplitudes have been obtained with FeynArts (version 3.9) [82, 83, 84], using the MSSM model file (including the MSSM counterterms) of Ref. [72]. The further evaluation has been performed with FormCalc (version 8.4) and LoopTools (version 2.12) [85]. The Higgs sector quantities (masses, mixings, \(\hat{Z}\) factors, etc.) have been evaluated using FeynHiggs [45, 46, 47, 48, 49, 50] (version 2.11.0).
3.1 Contributing diagrams
Furthermore, in general, in Figs. 1, 2, and 3 we have omitted diagrams with selfenergy type corrections of external (onshell) particles. While the contributions from the real parts of the loop functions are taken into account via the renormalization constants defined by OS renormalization conditions, the contributions coming from the imaginary part of the loop functions can result in an additional (real) correction if multiplied by complex parameters. In the analytical and numerical evaluation, these diagrams have been taken into account via the prescription described in Ref. [72].
Within our oneloop calculation we neglect finite width effects that can help to cure threshold singularities. Consequently, in the close vicinity of those thresholds our calculation does not give a reliable result. Switching to a complex mass scheme [86] would be another possibility to cure this problem, but its application is beyond the scope of our paper.
3.2 Ultraviolet divergences
As regularization scheme for the UV divergences we have used constrained differential renormalization [87], which has been shown to be equivalent to dimensional reduction [88, 89] at the oneloop level [85]. Thus the employed regularization scheme preserves SUSY [90, 91] and guarantees that the SUSY relations are kept intact, e.g. that the gauge couplings of the SM vertices and the Yukawa couplings of the corresponding SUSY vertices also coincide to oneloop order in the SUSY limit. Therefore no additional shifts, which might occur when using a different regularization scheme, arise. All UV divergences cancel in the final result.^{3}
3.3 Infrared divergences
Soft photon emission implies numerical problems in the phase space integration of radiative processes. The phase space integral diverges in the soft energy region where the photon momentum becomes very small, leading to infrared (IR) singularities. Therefore the IR divergences from diagrams with an internal photon have to cancel with the ones from the corresponding real soft radiation. We have included the soft photon contribution via the code already implemented in FormCalc following the description given in Ref. [92]. The IR divergences arising from the diagrams involving a photon are regularized by introducing a photon mass parameter, \(\lambda \). All IR divergences, i.e. all divergences in the limit \(\lambda \rightarrow 0\), cancel once virtual and real diagrams for one process are added. We have (numerically) checked that our results do not depend on \(\lambda \).
We have also numerically checked that our results do not depend on \(\Delta E = \delta _s E = \delta _s \sqrt{s}/2\) defining the energy cut that separates the soft from the hard radiation. As one can see from the example in the upper plot of Fig. 4 this holds for several orders of magnitude. Our numerical results below have been obtained for fixed \(\delta _s = 10^{3}\).
3.4 Collinear divergences
Numerical problems in the phase space integration of the radiative process arise also through collinear photon emission. Mass singularities emerge as a consequence of the collinear photon emission off massless particles. But already very light particles (such as e.g. electrons) can produce numerical instabilities.
Comparison of the oneloop corrected Higgs production cross sections (in fb) with FeynHiggsXS at \(\sqrt{s} = 1000\,\, \mathrm {GeV}\) and \(M_{H^\pm }= 310.86\) and higgsmix = 3 as input in FeynHiggs. FeynTools: \(m_h = 123.17\,\, \mathrm {GeV}\), \(m_H = 300.00\,\, \mathrm {GeV}\), \(m_{A} = 301.70\,\, \mathrm {GeV}\). FeynHiggsXS: \(m_h = 118.68\,\, \mathrm {GeV}\), \(m_H = 301.84\,\, \mathrm {GeV}\), \(m_{A} = 300.00\,\, \mathrm {GeV}\)
Process  FeynHiggsXS  FeynTools  

Full  Self \(+\) vert  Self  Full  Self \(+\) vert  Self  
\(e^+e^ \rightarrow h_1 Z\, ({\approx } h Z)\)  15.2845  14.1038  14.7896  12.0972  14.6641  12.3536 
\(e^+e^ \rightarrow h_3 Z\, ({\approx } H Z)\)  0.0221  0.0174  0.0245  0.0251  0.0275  0.0181 
\(e^+e^ \rightarrow h_1 h_2\, ({\approx } h A)\)  0.0262  0.0242  0.0165  0.0220  0.0253  0.0292 
\(e^+e^ \rightarrow h_2 h_3\, ({\approx } A H)\)  6.1456  7.0250  6.7694  5.8913  6.8347  6.0994 
In the PSS method, the phase space is divided into regions where the integrand is finite (numerically stable) and regions where it is divergent (or numerically unstable). In the stable regions the integration is performed numerically, whereas in the unstable regions it is carried out (semi) analytically using approximations for the collinear photon emission.
The collinear part is constrained by the angular cutoff parameter \(\Delta \theta \), imposed on the angle between the photon and the (in our case initial state) electron/positron.
MSSM default parameters for the numerical investigation; all parameters (except of \(t_\beta \)) are in GeV (calculated masses are rounded to 1 MeV). The values for the trilinear sfermion Higgs couplings, \(A_{t,b,\tau }\) are chosen such that charge and/or colorbreaking minima are avoided [100, 101, 102, 103, 104, 105, 106], and \(A_{b,\tau }\) are chosen to be real. It should be noted that for the first and second generation of sfermions we chose instead \(A_f = 0\), \(M_{\tilde{Q}, \tilde{U}, \tilde{D}} = 1500\,\, \mathrm {GeV}\) and \(M_{\tilde{L}, \tilde{E}} = 500\,\, \mathrm {GeV}\)
Scen.  \(\sqrt{s}\)  \(t_\beta \)  \(\mu \)  \(M_{H^\pm }\)  \(M_{\tilde{Q}, \tilde{U}, \tilde{D}}\)  \(M_{\tilde{L}, \tilde{E}}\)  \(A_{t,b,\tau }\)  \(M_1\)  \(M_2\)  \(M_3\) 

\(\mathcal S\)  1000  7  200  300  1000  500  \(1500 + \mu /t_\beta \)  100  200  1500 
\(m_{h_{1}}\)  \(m_{h_{2}}\)  \(m_{h_{3}}\)  

123.404  288.762  290.588 
4 Comparisons

In Ref. [57] the processes \(e^+e^ \rightarrow AH,HZ,Ah\) have been calculated in the rMSSM at tree level. As input parameters we used their parameters as far as possible. For the comparison with Ref. [57] we successfully reproduced their upper Fig. 2.

A numerical comparison with the program FeynHiggsXS [61] can be found in Table 1. We have neglected the initial state radiation and diagrams with photon exchange, as done in Ref. [61]. In Table 1 “self”, “self+vert” and “full” denote the inclusion of only selfenergy corrections, selfenergy plus vertex corrections or the full calculation including box diagrams. The comparison for the production of the light Higgs boson is rather difficult, due to the different FeynHiggs versions. As input parameters we used our scenario \(\mathcal S\); see Table 2 below. (We had to change only \(A_{t,b,\tau }\) to \(A_{t,b} = 1500\) and \(A_{\tau } = 0\) to be in accordance with the input options of FeynHiggsXS.) It can be observed that the level of agreement for the “self+vert” calculation is mostly at the level of 5 % or better. However, the box contributions appear to go in the opposite direction for the first three cross sections in the two calculations. This hints toward a problem in the box contributions in Ref. [61], where the box contributions were obtained independently from the rest of the loop corrections (see also the comparison with Ref. [58] below), whereas using FeynTools all corrections are evaluated together in an automated way. It should be noted that a selfconsistent check with the program FeynHiggsXS gave good agreement with Ref. [61] as expected (with tiny differences due to slightly different SM input parameters).

In Ref. [62] the processes \(e^+e^ \rightarrow HA, hA\) (and \(e^+e^ \rightarrow H^+ H^\)) have been calculated in the rMSSM. Unfortunately, in Ref. [62] the numerical evaluation (shown in their Fig. 2) are only treelevel results, although the paper deals with the respective oneloop corrections. For the comparison with Ref. [62] we successfully reproduced their lower Fig. 2.

In Ref. [63] a treelevel evaluation of the channels (1) and (2) in the cMSSM was presented, where higherorder corrections were included via (\(\mathcal{CP}\) violating) effective couplings. Unfortunately, no numbers are given in Ref. [63], but only twodimensional parameter scan plots, which we could not reasonably compare to our results. Consequently we omitted a comparison with Ref. [63].

We performed a comparison with Ref. [29] for \(e^+ e^ \rightarrow h_i Z, h_i h_j\) (\(i,j = 1,2,3\)) at \(\mathcal{O}(\alpha )\) in the cMSSM. In Ref. [29] only selfenergy and vertex corrections involving \(t, {\tilde{t}}, b, \tilde{b}\) were included, and the numerical evaluation was performed in the CPX scenario [98] (with \(M_{H^\pm }\) chosen to yield \(m_{h_{1}} = 40\,\, \mathrm {GeV}\)) which is extremely sensitive to the chosen input parameters. Nevertheless, using their input parameters as far as possible, we found qualitative agreement for \(t_\beta < 15\) with their Fig. 20. For larger \(t_\beta \) values the CPX scenario appears to be too sensitive to small deviations in the input parameters, and the agreement worsened.

\(e^+e^ \rightarrow hZ\) at the full oneloop level (including hard and soft photon bremsstrahlung, as well as \({\hat{\mathbf {Z}}}\) matrix contributions) has been analyzed in Ref. [66]. While complex parameters in this work are mentioned, all formulas and numerical examples only concern the rMSSM. They also used FeynArts, FormCalc and LoopTools to generate and simplify their code. Unfortunately no numbers are given in Ref. [66], but only twodimensional parameter scan plots, which we could not reasonably compare to our results. Consequently, we omitted a comparison with Ref. [66].

In Refs. [67, 68] “supersimple” expressions have been derived for the processes \(e^+e^ \rightarrow hZ, h\gamma \) in the rMSSM. We successfully reproduced Fig. 4 (right panels) of Ref. [67] in the upper plots of our Fig. 5 and Fig. 5 (right panels) of Ref. [68] in the lower plots of our Fig. 5. As input parameters we used their (SUSY) parameter set S1. The small differences in the differential cross sections are caused by the SM input parameters (where we have used our parameters; see Sect. 5.1 below) and the slightly different renormalization schemes and treatment of the Higgs boson masses.

The Higgsstrahlung process \(e^+e^ \rightarrow hZ\) with the expected leading corrections in “Natural SUSY” models [i.e. a oneloop calculation with thirdgeneration (s)quarks] has been computed in Ref. [71] for real parameters. This work also used FeynArts, FormCalc, and LoopTools to generate and simplify their code. Unfortunately, again no numbers are given in Ref. [71], but mostly twodimensional parameter scan plots, which we could not reasonably compare to our results. Only in the left plot of their Fig. 4 they show (fractional) corrections to the Higgsstrahlung cross section. However, the MSSM input parameters are not given in detail, rendering a comparison again impossible.
 In Ref. [58] the box contributions to the processes \(e^+e^ \rightarrow hZ,hA\) were computed. We used their input parameters as far as possible and reproduced Figs. 5 and 8 (solid lines, “box”) of Ref. [58] in our Fig. 6. The small differences are due to slightly different SM input parameters. However, we disagree in the sign of the box contributions in \(e^+e^ \rightarrow hA\), except for the sneutrino loops. Consequently, the sign difference of the full box contributions w.r.t. our result depends on the choice of the MSSM parameters. It should be noted that the code of Ref. [58] is also part of the code from Ref. [59] (see the next item) and Ref. [61].

In Ref. [59] the processes \(e^+e^ \rightarrow hZ, hA\) are computed within a complete oneloop calculation. Only the QED (including photon bremsstrahlung) has been neglected. We used their input parameters as far as possible and (more or less successfully) reproduced Figs. 5 and 6 (upper rows, solid lines) of Ref. [59] qualitatively in our Fig. 7. Smaller differences are mainly due to different Higgs boson masses and the use of Higgs boson wave function corrections in Ref. [59], while we used an effective mixing angle \(\alpha _{\text {eff}}\). In order to facilitate the comparison we used the same simple formulas for our Higgs boson masses and \(\alpha _{\text {eff}}\) as in their Eqs. (4)–(7). Therefore our \(\sigma _{\text {tree}}\) correspond rather to their \(\sigma ^{\epsilon }\) and our \(\sigma _{\text {full}}\) rather to their \(\sigma ^{\text {FDC}}\). The larger differences in the loop corrections of \(e^+e^ \rightarrow hA\) (right plots of our Fig. 7) are due to the different sign of the leading box contributions of Ref. [59]; see also the latter item. It should be noted that the code of Ref. [59] is also part of the code from Ref. [61]. Using FeynHiggsXS with the input parameters of Ref. [59] (as far as possible) gave also only qualitative agreement with the Figs. of Ref. [59].

In Ref. [64] the loopinduced processes \(e^+e^ \rightarrow h\gamma , H\gamma , A\gamma \) have been computed. We used the same simple formulas for our Higgs boson masses and \(\alpha _{\text {eff}}\) as in their Eqs. (3.48)–(3.50). We also used their input parameters as far as possible, but unfortunately they forgot to specify the trilinear parameters \(A_f\). Therefore we chose arbitrarily \(A_f = 0\) for our comparison. In view of this problem the comparison is acceptable; see our Fig. 8 vs. Figs. 4, 5 and 7 of Ref. [64]. It should be noted that the code of Ref. [64] is also part of the code from Refs. [59, 61].

In Ref. [60] the loopinduced process \(e^+e^ \rightarrow A Z\) has been computed in the rMSSM using FeynArts, FormCalc and LoopTools. We used their input parameters (as far as possible) and are in good agreement with their Figs. 2 and 4; see our Fig. 9. We only significantly differ quantitatively for \(t_\beta = 4\) in combination with their case L (which denotes light SUSY particles). However, we could not find why in this particular case the comparison failed.

In Ref. [65] the loopinduced processes \(e^+e^ \rightarrow A\gamma , A Z\) have been computed in the cMSSM using also FeynArts, FormCalc and LoopTools. We used their input parameters and are in good qualitative agreement with their Figs. 3 and 5. But again we differ quantitative significantly by roughly a factor 1.3 (1.7) in \(e^+e^ \rightarrow A\gamma \) (\(e^+e^ \rightarrow A Z\)). As in Ref. [60] (see above) this is due to the low \(t_\beta \) and SUSY masses used in Ref. [65]. Unfortunately the code of Ref. [65] is no longer available, making further investigations impossible. But we repeated successfully (9 digits agreement) our calculations with the older versions of FeynArts (i.e. MSSM model files) and FormCalc as they were used in Ref. [65], i.e. the different versions of FeynArts and FormCalc can be excluded as a reason for this discrepancy.
A final comment is in order. We argue that the problems in the comparison with Ref. [61] (i.e. FeynHiggsXS), Refs. [59, 64] are due to the fact that all three papers are based (effectively) on the same calculation/source, where we discussed the differences in particular in the sign of some box contributions. Consequently, these three papers should be considered as one rather than three independent comparisons, and thus do not disprove the reliability of our calculation. It should also be kept in mind that our calculational method/code has already been successfully tested and compared with quite a few other programs; see Refs. [30, 31, 72, 73, 74, 75, 76, 77, 78, 79, 80].
5 Numerical analysis
In this section we present our numerical analysis of neutral Higgs boson production at \(e^+e^\) colliders in the cMSSM. In the various figures below we show the cross sections at the tree level (“tree”) and at the full oneloop level (“full”). In case of extremely small treelevel cross sections we also show results including the corresponding purely loopinduced contributions (“loop”). These leading twoloop contributions are \(\propto \mathcal{M}_{\text {1loop}}^2\), where \(\mathcal{M}_{\text {1loop}}\) denotes the oneloop matrix element of the appropriate process.
5.1 Parameter settings
 Fermion masses (onshell masses, if not indicated differently):According to Ref. [99], \(m_s\) is an estimate of a socalled “current quark mass” in the \(\overline{\mathrm {MS}}\) scheme at the scale \(\mu \approx 2\,\, \mathrm {GeV}\). \(m_c \equiv m_c(m_c)\) and \(m_b \equiv m_b(m_b)\) are the “running” masses in the \(\overline{\mathrm {MS}}\) scheme. \(m_u\) and \(m_d\) are effective parameters, calculated through the hadronic contributions to$$\begin{aligned} \begin{array}{l@{\quad }l} m_e = 0.510998928\,\, \mathrm {MeV}, &{} m_{\nu _e} = 0, \\ m_\mu = 105.65837515\,\, \mathrm {MeV}, &{} m_{\nu _{\mu }} = 0, \\ m_\tau = 1776.82\,\, \mathrm {MeV}, &{} m_{\nu _{\tau }} = 0, \\ m_u = 68.7\,\, \mathrm {MeV}, &{} m_d = 68.7\,\, \mathrm {MeV}, \\ m_c = 1.275\,\, \mathrm {GeV}, &{} m_s = 95.0\,\, \mathrm {MeV}, \\ m_t = 173.21\,\, \mathrm {GeV}, &{} m_b = 4.18\,\, \mathrm {GeV}. \end{array} \end{aligned}$$(8)$$\begin{aligned} \Delta \alpha _{\text {had}}^{(5)}(M_Z)&= \frac{\alpha }{\pi }\sum _{f = u,c,d,s,b} Q_f^2 \left( \ln \frac{M_Z^2}{m_f^2}  \frac{5}{3}\right) \nonumber \\&\approx 0.027723. \end{aligned}$$(9)
 Gauge boson masses:$$\begin{aligned} M_Z = 91.1876\,\, \mathrm {GeV}, \quad M_W = 80.385\,\, \mathrm {GeV}. \end{aligned}$$(10)
 Coupling constant:$$\begin{aligned} \alpha (0) = 1/137.0359895. \end{aligned}$$(11)
The SUSY parameters are chosen according to the scenario \(\mathcal S\), shown in Table 2, unless otherwise noted. This scenario constitutes a viable scenario for the various cMSSM Higgs production modes, i.e. not picking specific parameters for each cross section. The only variation will be the choice of \(\sqrt{s} = 500\,\, \mathrm {GeV}\) for production cross sections involving the light Higgs boson.^{4} This will be clearly indicated below. We do not strictly demand that the lightest Higgs boson has a mass around \(\sim 125\,\, \mathrm {GeV}\), although for most of the parameter space this is given. We will show the variation with \(\sqrt{s}\), \(M_{H^\pm }\), \(t_\beta \) and \(\varphi _{A_t}\), the phase of \(A_t\).
The numerical results shown in the next subsections are of course dependent on the choice of the SUSY parameters. Nevertheless, they give an idea of the relevance of the full oneloop corrections.
5.2 Full oneloop results for varying \(\sqrt{s}\), \(M_{H^\pm }\), \(t_\beta \), and \(\varphi _{A_t}\)
The results shown in this and the following subsections consist of “tree”, which denotes the treelevel value and of “full”, which is the cross section including all oneloop corrections as described in Sect. 3.
We begin the numerical analysis with the cross sections of \(e^+e^ \rightarrow h_i h_j\) (\(i,j = 1,2,3\)) evaluated as a function of \(\sqrt{s}\) (up to \(3\,\, \mathrm {TeV}\), shown in the upper left plot of the respective figures), \(M_{H^\pm }\) (starting at \(M_{H^\pm }= 160\,\, \mathrm {GeV}\) up to \(M_{H^\pm }= 500\,\, \mathrm {GeV}\), shown in the upper right plots), \(t_\beta \) (from 4 to 50, lower left plots) and \(\varphi _{A_t}\) (between \(0^{\circ }\) and \(360^{\circ }\), lower right plots). Then we turn to the processes \(e^+e^ \rightarrow h_i Z\) and \(e^+e^ \rightarrow h_i \gamma \) (\(i = 1,2,3\)). All these processes are of particular interest for ILC and CLIC analyses [18, 19, 20, 21, 22, 24, 25] (as emphasized in Sect. 1).
5.2.1 The process \(e^+e^ \rightarrow h_i h_j\)
We start our analysis with the production modes \(e^+e^ \rightarrow h_i h_j\) (\(i,j = 1,2,3\)). Results are shown in the Figs. 10, 11, 12, and 13. It should be noted that there are no hHZ couplings in the rMSSM (see Ref. [124]). For real parameters this leads to vanishing treelevel cross sections if \(h_i \sim h\) and \(h_j \sim H\) (or vice versa). Furthermore there are no hhZ, HHZ, and AAZ couplings in the rMSSM, but also in the complex case the tree couplings \(h_i h_i Z\) (\(i = 1,2,3\)) are exactly zero (see Ref. [124]). In the following analysis \(e^+ e^ \rightarrow h_i h_i\) (\(i = 1,2,3\)) are loop induced via (only) box diagrams and therefore \(\propto \mathcal{M}_{\text {1loop}}^2\).
We begin with the process \(e^+e^ \rightarrow h_1 h_2\) as shown in Fig. 10. As a general comment it should be noted that in \(\mathcal S\) one finds that \(h_1 \sim h\), \(h_2 \sim A\) and \(h_3 \sim H\). The hAZ coupling is \(\propto c_{\beta  \alpha }\), which goes to zero in the decoupling limit [125, 126, 127, 128], and consequently relatively small cross sections are found. In the analysis of the production cross section as a function of \(\sqrt{s}\) (upper left plot) we find the expected behavior: a strong rise close to the production threshold, followed by a decrease with increasing \(\sqrt{s}\). We find a relative correction of \(\sim 15\,\%\) around the production threshold. Away from the production threshold, loop corrections of \(\sim +27\,\%\) at \(\sqrt{s} = 1000\,\, \mathrm {GeV}\) are found in \(\mathcal S\) (see Table 2). The relative size of loop corrections increase with increasing \(\sqrt{s}\) and reach \(\sim +61\,\%\) at \(\sqrt{s} = 3000\,\, \mathrm {GeV}\) where the tree level becomes very small. With increasing \(M_{H^\pm }\) in \(\mathcal S\) (upper right plot) we find a strong decrease of the production cross section, as can be expected from kinematics, but in particular from the decoupling limit discussed above. The loop corrections reach \(\sim +27\,\%\) at \(M_{H^\pm }= 300\,\, \mathrm {GeV}\) and \(\sim +62\,\%\) at \(M_{H^\pm }= 500\,\, \mathrm {GeV}\). These large loop corrections are again due to the (relative) smallness of the treelevel results. It should be noted that at \(M_{H^\pm }\approx 350\,\, \mathrm {GeV}\) the limit of 0.01 fb is reached. This limit corresponds to 10 events at an integrated luminosity of \(\mathcal{L}= 1\, \text{ ab }^{1}\), which can be seen as a guideline for the observability of a process at a linear collider. The cross sections decrease with increasing \(t_\beta \) (lower left plot), and the loop corrections reach the maximum of \(\sim +38\,\%\) at \(t_\beta = 36\) while the minimum of \(\sim +26\,\%\) is at \(t_\beta = 5\). The phase dependence \(\varphi _{A_t}\) of the cross section in \(\mathcal S\) (lower right plot) is at the 10 % level at tree level. The loop corrections are nearly constant, \(\sim +28\,\%\) for all \(\varphi _{A_t}\) values and do not change the overall dependence of the cross section on the complex phase.
Not shown is the process \(e^+e^ \rightarrow h_1 h_3\). In this case, for our parameter set \(\mathcal S\) (see Table 2), one finds \(h_3 \sim H\). Due to the absence of the hHZ coupling in the MSSM (see Ref. [124]) this leads to vanishing treelevel cross sections in the case of real parameters. For complex parameters (i.e. \(\varphi _{A_t}\)) the treelevel results stay below \(10^{5}\) fb. Also the loopinduced cross sections \(\propto \mathcal{M}_{\text {1loop}}^2\) (where only the vertex and box diagrams contribute in the case of real parameters) stay below \(10^{5}\) fb for our parameter set \(\mathcal S\). Consequently, in this case we omit showing plots to the process \(e^+e^ \rightarrow h_1 h_3\).
In Fig. 11 we present the cross section \(e^+e^ \rightarrow h_2 h_3\) with \(h_2 \sim A\) and \(h_3 \sim H\) in \(\mathcal S\). The HAZ coupling is \(\propto s_{\beta  \alpha }\), which goes to one in the decoupling limit, and consequently relatively large cross sections are found. In the analysis as a function of \(\sqrt{s}\) (upper left plot) we find relative corrections of \(\sim 37\,\%\) around the production threshold, \(\sim 5\,\%\) at \(\sqrt{s} = 1000\,\, \mathrm {GeV}\) (i.e. \(\mathcal S\)), and \(\sim +6\,\%\) at \(\sqrt{s} = 3000\,\, \mathrm {GeV}\). The dependence on \(M_{H^\pm }\) (upper right plot) is nearly linear above \(M_{H^\pm }\gtrsim 250\,\, \mathrm {GeV}\), and mostly due to kinematics. The loop corrections are \(\sim 8\,\%\) at \(M_{H^\pm }= 160\,\, \mathrm {GeV}\), \(\sim 5\,\%\) at \(M_{H^\pm }= 300\,\, \mathrm {GeV}\) (i.e. \(\mathcal S\)), and \(\sim 52\,\%\) at \(M_{H^\pm }= 500\,\, \mathrm {GeV}\) where the tree level goes to zero. As a function of \(t_\beta \) (lower left plot) the treelevel cross section is rather flat, apart from a dip at \(t_\beta \approx 10\), corresponding to the threshold \(m_{\tilde{\chi }_{1}^\pm } + m_{\tilde{\chi }_{1}^\pm } = m_{h_2}\). This threshold enter into the treelevel and the loop corrections only via the \({\hat{\mathbf {Z}}}\) matrix contribution (calculated by FeynHiggs). The relative corrections increase from \(\sim 5\,\%\) at \(t_\beta = 7\) to \(\sim +7\,\%\) at \(t_\beta = 50\). The dependence on \(\varphi _{A_t}\) (lower right plot) is very small, below the percent level. The loop corrections are found to be nearly independent of \(\varphi _{A_t}\) at the level of \(\sim 4.6\,\%\).
We now turn to the processes with equal indices. The tree couplings \(h_i h_i Z\) (\(i = 1,2,3\)) are exactly zero; see Ref. [124]. Therefore, in this case we show the pure loopinduced cross sections \(\propto \mathcal{M}_{\text {1loop}}^2\) (labeled as “loop”) where only the box diagrams contribute. These box diagrams are UV and IR finite.
In Fig. 12 we show the results for \(e^+e^ \rightarrow h_1 h_1\). This process might have some special interest, since it is the lowest energy process in which triple Higgs boson couplings play a role, which could be relevant at a highluminosity collider operating above the two Higgs boson production threshold. In our numerical analysis, as a function of \(\sqrt{s}\) we find a maximum of \(\sim 0.014\) fb, at \(\sqrt{s} = 500\,\, \mathrm {GeV}\), decreasing to \(\sim 0.002\) fb at \(\sqrt{s} = 3\,\, \mathrm {TeV}\). The dependence on \(M_{H^\pm }\) is rather small, as is the dependence on \(t_\beta \) and \(\varphi _{A_t}\) in \(\mathcal S\). However, with cross sections found at the level of up to 0.015 fb this process could potentially be observable at the ILC running at \(\sqrt{s} = 500\,\, \mathrm {GeV}\) or below (depending on the integrated luminosity).
Overall, for the neutral Higgs boson pair production we observed an increasing cross section \(\propto 1/s\) for \(s \rightarrow \infty \); see Eq. (4). The full oneloop corrections reach a level of 10–\(20\,\%\) or higher for cross sections of 0.01–10 fb. The variation with \(\varphi _{A_t}\) is found to be rather small, except for \(e^+e^ \rightarrow h_1 h_2\), where it is at the level of \(10\,\%\). The results for \(h_i h_i\) production turn out to be small (but not necessarily hopelessly so) for \(i = 1\), and negligible for \(i = 2,3\) for Higgs boson masses above \(\sim 200\,\, \mathrm {GeV}\).
5.2.2 The process \(e^+e^ \rightarrow h_i Z\)
In Figs. 14 and 15 we show the results for the processes \(e^+e^ \rightarrow h_i Z\), as before as a function of \(\sqrt{s}\), \(M_{H^\pm }\), \(t_\beta \) and \(\varphi _{A_t}\). It should be noted that there are no AZZ couplings in the MSSM (see [124]). In the case of real parameters this leads to vanishing treelevel cross sections if \(h_i \sim A\).
Not shown is the process \(e^+e^ \rightarrow h_2 Z\). In this case, for our parameter set \(\mathcal S\) (see Table 2), one finds \(h_2 \sim A\). Because there are no AZZ couplings in the MSSM (see [124]) this leads to vanishing treelevel cross sections in the case of real parameters. For complex parameters (i.e. \(\varphi _{A_t}\)) the treelevel results stay below \(10^{5}\) fb. Also the loopinduced cross sections \(\propto \mathcal{M}_{\text {1loop}}^2\) (where only the vertex and box diagrams contribute in the case of real parameters) are below \(10^{3}\) fb for our parameter set \(\mathcal S\). Consequently, in this case we omit showing plots to the process \(e^+e^ \rightarrow h_2 Z\).
Overall, for the Z Higgs boson production we observed an increasing cross section \(\propto 1/s\) for \(s \rightarrow \infty \); see Eq. (5). The full oneloop corrections reach a level of \(20\,\%\) (\(50\,\%\)) for cross sections of 60 fb (0.03 fb). The variation with \(\varphi _{A_t}\) is found to be small, reaching up to \(10\,\%\) for \(e^+e^ \rightarrow h_3 Z\), after including the loop corrections.
5.2.3 The process \(e^+e^ \rightarrow h_i \gamma \)
In Figs. 16 and 17 we show the results for the processes \(e^+e^ \rightarrow h_i \gamma \) as before as a function of \(\sqrt{s}\), \(M_{H^\pm }\), \(t_\beta \) and \(\varphi _{A_t}\). It should be noted that there are no \(h_i Z \gamma \) or \(h_i \gamma \gamma \) (\(i = 1,2,3\)) couplings in the MSSM; see Ref. [124]. In the following analysis \(e^+ e^ \rightarrow h_i \gamma \) (\(i = 1,2,3\)) are purely loopinduced processes (via vertex and box diagrams) and therefore \(\propto \mathcal{M}_{\text {1loop}}^2\).
We start with the process \(e^+e^ \rightarrow h_1 \gamma \) shown in Fig. 16. The largest contributions are expected from loops involving top quarks and SM gauge bosons. The cross section is rather small for the parameter set chosen; see Table 2. As a function of \(\sqrt{s}\) (upper left plot) a maximum of \(\sim 0.1\) fb is reached around \(\sqrt{s} \sim 250\,\, \mathrm {GeV}\), where several thresholds and dip effects overlap. The first peak is found at \(\sqrt{s} \approx 283\,\, \mathrm {GeV}\), due to the threshold \(m_{\tilde{\chi }_{1}^\pm } + m_{\tilde{\chi }_{1}^\pm } = \sqrt{s}\). A dip can be found at \(m_t+ m_t= \sqrt{s} \approx 346\,\, \mathrm {GeV}\). The next dip at \(\sqrt{s} \approx 540\,\, \mathrm {GeV}\) is the threshold \(m_{\tilde{\chi }_{2}^\pm } + m_{\tilde{\chi }_{2}^\pm } = \sqrt{s}\). The loop corrections for \(\sqrt{s}\) vary between 0.1 fb at \(\sqrt{s} \approx 250\,\, \mathrm {GeV}\), 0.03 fb at \(\sqrt{s} \approx 500\,\, \mathrm {GeV}\) and 0.003 fb at \(\sqrt{s} \approx 3000\,\, \mathrm {GeV}\). Consequently, this process could be observable for larger ranges of \(\sqrt{s}\). In particular in the initial phase with \(\sqrt{s} = 500\,\, \mathrm {GeV}\) [107] 30 events could be produced with an integrated luminosity of \(\mathcal{L}= 1\, \text{ ab }^{1}\). As a function of \(M_{H^\pm }\) (upper right plot) we find an increase in \(\mathcal S\) (but with \(\sqrt{s} = 500\,\, \mathrm {GeV}\)), increasing the production cross sections from 0.023 fb at \(M_{H^\pm }\approx 160\,\, \mathrm {GeV}\) to about 0.03 fb in the decoupling regime. This dependence shows the relevance of the SM gaugeboson loops in the production cross section, indicating that the top quark loops dominate this production cross section. The variation with \(t_\beta \) and \(\varphi _{A_t}\) (lower row) is rather small, and values of 0.03 fb are found in \(\mathcal S\).
We finish the \(e^+e^ \rightarrow h_i \gamma \) analysis in Fig. 17 in which the results for \(e^+e^ \rightarrow h_i \gamma \) (\(i = 2,3\)) are displayed, where \(e^+e^ \rightarrow h_2 \gamma \ (h_3 \gamma )\) is shown as solid red (dashed blue) line. In \(\mathcal S\), as discussed above, one finds \(h_2 \sim A\) and \(h_3 \sim H\). While both Higgs bosons have reduced (enhanced) couplings to top (bottom) quarks, only the H can have a nonnegligible coupling to SM gauge bosons. As function of \(\sqrt{s}\) (upper left plot) we find that for the \(h_2\gamma \) (\(h_1\gamma \)) production maximum values of about \(0.006\ (0.001)\) fb are found. However, due to a similar decay pattern and similar masses (for not too small \(M_{H^\pm }\), \(300\,\, \mathrm {GeV}\) here) it will be difficult to disentangle those to production cross sections, and the effective cross section is given roughly by the sum of the two. This renders these loopinduced processes at the border of observability. The peaks observed are found at \(\sqrt{s} \approx 540\,\, \mathrm {GeV}\) due to the threshold \(m_{\tilde{\chi }_{2}^\pm } + m_{\tilde{\chi }_{2}^\pm } = \sqrt{s}\) for both production cross sections. They drop to the unobservable level for \(\sqrt{s} \gtrsim 1 \,\, \mathrm {TeV}\). As a function of \(M_{H^\pm }\) (upper right plot) one can observe the decoupling of \(h_3 \sim H\) of the SM gauge bosons with increasing \(M_{H^\pm }\), lowering the cross section for larger values. The “knee” at \(M_{H^\pm }\approx 294\,\, \mathrm {GeV}\) is the threshold \(m_{\tilde{\chi }_{1}^\pm } + m_{\tilde{\chi }_{1}^\pm } = m_{h_2}\). This threshold enter into the loop corrections only via the \({\hat{\mathbf {Z}}}\) matrix contribution (calculated by FeynHiggs). The loop corrections vary between 0.008 fb at \(M_{H^\pm }\approx 160\,\, \mathrm {GeV}\) and far below 0.001 fb at \(M_{H^\pm }\approx 500\,\, \mathrm {GeV}\). The dependence on \(t_\beta \) (lower left plot) is rather strong for the \(h_2\gamma \) production going from 0.007 fb at \(t_\beta = 4\) down to 0.0035 fb at \(t_\beta = 50\). The dip at \(t_\beta \approx 10\) is the threshold \(m_{\tilde{\chi }_{1}^\pm } + m_{\tilde{\chi }_{1}^\pm } = m_{h_2}\). This threshold enter into the loop corrections again only via the \({\hat{\mathbf {Z}}}\) matrix contribution (calculated by FeynHiggs). For the \(h_3\gamma \) production the cross section stays at the very low level of 0.001 fb for all \(t_\beta \) values. The dependence on the phase \(\varphi _{A_t}\) of the cross sections (lower right plot) is very small in \(\mathcal S\), with no visible variation in the plot.
Overall, for the \(\gamma \) Higgs boson production the leading order corrections can reach a level of 0.1 fb, depending on the SUSY parameters. This renders these loopinduced processes in principle observable at an \(e^+e^\) collider. The variation with \(\varphi _{A_t}\) is found to be extremely small.
6 Conclusions
We evaluated all neutral MSSM Higgs boson production modes at \(e^+e^\) colliders with a twoparticle final state, i.e. \(e^+e^ \rightarrow h_i h_j, h_i Z, h_i \gamma \) (\(i,j = 1,2,3\)), allowing for complex parameters. In the case of a discovery of additional Higgs bosons a subsequent precision measurement of their properties will be crucial to determine their nature and the underlying (SUSY) parameters. In order to yield a sufficient accuracy, oneloop corrections to the various Higgs boson production modes have to be considered. This is particularly the case for the high anticipated accuracy of the Higgs boson property determination at \(e^+e^\) colliders [23].
The evaluation of the processes (1)–(3) is based on a full oneloop calculation, also including hard and soft QED radiation. The renormalization is chosen to be identical as for the various Higgs boson decay calculations; see, e.g., Refs. [30, 31].
We first very briefly reviewed the relevant sectors including some details of the oneloop renormalization procedure of the cMSSM, which are relevant for our calculation. In most cases we follow Ref. [72].
We have discussed the calculation of the oneloop diagrams, the treatment of UV, IR, and collinear divergences that are canceled by the inclusion of (hard, soft, and collinear) QED radiation. We have checked our result against the literature as far as possible, and in most cases we found acceptable or qualitative agreement, where parts of the differences can be attributed to problems with input parameters (conversions) and/or special scenarios. Once our setup was changed successfully to the one used in the existing analyses we found good agreement.
For the analysis we have chosen a parameter set that allows simultaneously a maximum number of production processes. In this scenario (see Table 2) we have \(h_1 \sim h\), \(h_2 \sim A\), and \(h_3 \sim H\). In the analysis we investigated the variation of the various production cross sections with the centerofmass energy \(\sqrt{s}\), the charged Higgs boson mass \(M_{H^\pm }\), the ratio of the vacuum expectation values \(t_\beta \) and the phase of the trilinear Higgs–top squark coupling \(\varphi _{A_t}\). For light (heavy) Higgs production cross sections we have chosen \(\sqrt{s} = 500\, (1000)\,\, \mathrm {GeV}\).
In our numerical scenarios we compared the treelevel production cross sections with the full oneloop corrected cross sections. In certain cases the treelevel cross sections are identical zero (due to the symmetries of the model), and in those cases we have evaluated the oneloop squared amplitude, \(\sigma _{\text {loop}} \propto \mathcal{M}_{\text {1loop}}^2\).
We found sizable corrections of \(\sim \) 10–20 % in the \(h_i h_j\) production cross sections. Substantially larger corrections are found in cases where the treelevel result is (accidentally) small and thus the production mode likely is not observable. The purely loopinduced processes of \(e^+e^ \rightarrow h_ih_i\) could be observable, in particular in the case of \(h_1 h_1\) production. For the \(h_i Z\) modes corrections around 10–20 %, but increasing to \(\sim 50\,\%\), are found. The purely loopinduced processes of \(h_i\gamma \) production appear to be observable for \(h_1\gamma \), but they are very challenging for \(h_{2,3}\gamma \).
Only in very few cases a relevant dependence on \(\varphi _{A_t}\) was found. Examples are \(e^+e^ \rightarrow h_1 h_2\) and \(e^+e^ \rightarrow h_3 Z\), where a variation, after the inclusion of the loop corrections, of up to \(10\,\%\) with \(\varphi _{A_t}\) was found. In those cases neglecting the phase dependence could lead to a wrong impression of the relative size of the various cross sections.
The numerical results we have shown are, of course, dependent on the choice of the SUSY parameters. Nevertheless, they give an idea of the relevance of the full oneloop corrections.
Following our analysis it is evident that the full oneloop corrections are mandatory for a precise prediction of the various cMSSM Higgs boson production processes. The full oneloop corrections must be taken into account in any precise determination of (SUSY) parameters from the production of cMSSM Higgs bosons at \(e^+e^\) linear colliders. There are plans to implement the evaluation of the Higgs boson production into the public code FeynHiggs.
Footnotes
 1.
A corresponding computer code is available at http://www.feynhiggs.de.
 2.
We found that using loop corrected Higgs boson masses in the loops leads to a UV divergent result.
 3.
It should be noted that some processes are UV divergent if the electron mass is neglected (see Sect. 3.1). The full processes including the terms proportional to the electron mass are, of course, UV finite. Dropping the divergence, the numerical difference between the two calculations was found to be negligible. Therefore we used the (faster) simplified code with neglected electron mass for our numerical analyses below.
 4.
In a recent reevaluation of ILC running strategies the first stage was advocated to be at \(\sqrt{s} = 500\,\, \mathrm {GeV}\) [107].
 5.
It should be noted that a calculation very close to the production threshold requires the inclusion of additional (nonrelativistic) contributions, which is beyond the scope of this paper. Consequently, very close to the production threshold our calculation (at the tree and loop level) does not provide a very accurate description of the cross section.
 6.
This limit corresponds to ten events at an integrated luminosity of \(\mathcal{L}= 1\, \text{ ab }^{1}\), which can be seen as a guideline for the observability of a process at a linear collider.
Notes
Acknowledgments
We thank A. Arhrib, G. Gounaris, T. Hahn, F. Renard, J. Rosiek and K. Williams for helpful discussions. The work of S.H. is supported in part by CICYT (Grant FPA 201340715P) and by the Spanish MICINN’s ConsoliderIngenio 2010 Program under Grant MultiDark CSD200900064.
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