One-loop effects of MSSM particles in e^-e^+ \to Zh and e^-e^+ \to \nu\bar\nu h at the ILC

The 1-loop effects of the MSSM at the ILC are investigated through numerical analysis. We studied the higgs production processes $e^-e^+\rightarrow Zh$ and $e^-e^+\rightarrow \nu\bar{\nu}h$ at the ILC. It is found that the magnitude of the MSSM contribution through the 1-loop effects is sizable enough to be detected. In the study, three sets of the MSSM parameters are proposed, which are consistent with the observed higgs mass, the muon $g$-$2$, the dark matter abundance and the decay branching ratios of $B$ mesons. In the $e^-e^+\rightarrow Zh$ process, the 1-loop effects of the MSSM are visible and the distinction of the parameter sets is partially possible. For the study of $e^-e^+\rightarrow \nu\bar{\nu}h$, we used the equivalent $\it W$-boson approximation in the evaluation of the 1-loop cross section. While the 1-loop effect of the MSSM is visible, the distinction of the parameter sets might not be possible in this process under the value of realistic luminosity at the ILC.


Introduction
The standard model (SM) is completed by the discovery of the last piece, the higgs particle.However, it is argued that it is not the final theory of the fundamental particles.For example, the SM includes a lot of free parameters.When one calculates the mass of higgs, the values of these parameters are intentionally selected to cancel the quantum correction.This cancellation is as precise as up to around 17 order.Some researchers regard it as "unnatural".The supersymmetric (SUSY) model [1] is considered as one of the promising candidates for the theory beyond the standard model.In this theory, each particle in the SM has its supersymmetric partner, or, a sparticle.The quadratic divergence in the calculation of quantum correction to the higgs mass is canceled by other contributions from the sparticles, so that the fine tuning problem disappears.The search of sparticles is the important subject of the present and future collider experiments to prove the SUSY.In spite of hard efforts in the large hadron collider (LHC) experiments, slightest signature of their existence has not been obtained.For example, the scalar top particle (stop) seems not to exist under O(1)TeV mass region [2,3].Though sparticles are so heavy that it is difficult to be produced directly, the indirect signature would be expected at the international linear collider (ILC).
Since high luminosity is expected at the ILC experiments [4], the quite small experimental error is expected.Correspondingly, therefore, the quite accurate calculations of the physical observables are required.As an explicit model, the Minimal Supersymmetric Standard Model (MSSM) is studied in this paper.We have calculated cross sections and decay branching ratios at the 1-loop level using the GRACE/SUSY-loop system [5,6,7,8,9].In this paper we report numerical results on the cross section of e − e + → Zh [10].Our results are consistent with those given in the previous work [11] under the same setting of MSSM parameters.We have also calculated the cross section of e − e + → ν νh, with the equivalent W -boson approximation (EWA) [12].The complete SM 1-loop correction of σ(e − e + → Zh) and σ(e − e + → ν νh) have already been calculated using the GRACE [13] and we reproduced the results for comparison.We calculate the MSSM 1-loop correction to perform the indirect search of sparticles as they contribute to the cross sections through 1-loop diagrams.The parameter sets of the MSSM are chosen so as to meet the various experimental constraints.They include the anomalous magnetic moment of muon g-2 [14,15], the dark matter (DM) thermal relic density [16,17,18], B meson rare decay branching ratio Br(b → sγ) [19] and the higgs mass [20,21].By the constraints the mass spectra of MSSM are quite limited.For the selection of MSSM mass spectra we utilized the following program packages.MicrOMEGAs [22] was used in the estimation of the DM thermal relic density and SuSpect2 [23] was used for g-2 , Br(b → sγ) and m h .

The selection of the MSSM parameter sets
In Table 1 we show experimental constraints we have considered, and in Table 2 we show three MSSM parameter sets we have selected.Detailed methods for the selection of sets have been explained in the previous work [6].
The first common settings in the three sets are (A) M 1 = 350GeV, M 2 = 450GeV, µ = 1TeV, tan β =50 and m l < 500GeV (ℓ = e, µ).They are necessary for sets to satisfy the constrains (1), ( 2) and (3)  They are necessary for sets to be satisfied the constraint (4), because the co-annihilation occurring between τ1 and the LSP is required for meeting this constraint in the Bino LSP case.The difference among the three sets exists in the settings of masses of the strongly interacting sparticles (m g (= M 3 ) and m q ), gluino and squarks.Considering the LHC bounds (5), we take (m g and m q) ≃ (2TeV, 1.5TeV), (5TeV, 5TeV) and (10TeV, 10TeV) for set 1, set 2 and set 3, respectively.For each set, the left-right mixing parameter θ t and masses m t1,2 in the stop t sector are tuned to satisfy the higgs mass constraint (6).
Table 1: The experimental constraints for MSSM parameters.

The GRACE system
There are more than twice as many different types of particles in the MSSM as those in the SM ; therefore, there are various possible sparticle production processes in the collider experiments.A large number of Feynman diagrams appearing in each production process requires tedious and lengthy calculations in evaluating the cross sections.Accurate theoretical prediction requires an automated system to manage such large scale computations.The GRACE system for the MSSM calculations [28,9] has been developed by the KEK group (the Minami-tateya group) to meet the requirement.The GRACE system uses a renormalization prescription that imposes mass shell conditions on as many particles as possible, while maintaining the gauge symmetry by setting the renormalization conditions appropriately [9].In the GRACE system for the SM, the usual 't Hooft-Feynman linear gauge condition is generalized to a more general non-linear gauge (NLG) that involves five extra parameters [29,30].We extend it to the MSSM formalism by adding the SUSY interactions with seven NLG parameters [9,31].We can check the consistency of the gauge symmetry by verifying the independence of the physical results from the NLG parameters.We ascertain that the results of the automatic calculation are reliable by carrying out the following checks: • Electroweak (ELWK) non-linear gauge invariance check (NLG check) • Cancellation check of ultraviolet divergence (UV check) • Cancellation check of infrared divergence (IR check) • Check of soft photon cut-off energy independence (k c check) Actually, the 1-loop differential cross sections (distributions) are separated into two parts, where the suffix M indicates SM or MSSM, and each part is computed separately.The loop and the counter term contribution dσ M virtual should be gauge invariant and the UV finite but IR divergent.We regularize the IR divergence by the fictitious photon mass λ, so both dσ M virtual and the soft photon contribution dσ soft are λ dependent.The λ dependence is canceled in dσ M L&S .Finally, the k c independent 1-loop physical cross sections can be obtained by where, k and Ω are the energy and the solid angle of the photon respectively.We selected the values of the MSSM parameters related to the higgs sector so that the tree level cross section are numerically identical between the MSSM and the SM.Therefore, the suffix M can be neglected in dσ tree and dσ hard .
For the confirmation of the verifiability of 1-loop correction, we defined following correction ratios, For the estimation of effects of the MSSM virtual particles, we defined following ratio [32], For the process e − e + → Zh, we calculated differential cross sections where θ Z is the scattering angle of the Z-boson.

Equivalent W-boson Approximation (EWA)
For the process e − e + → ν νh, we should calculate the cross section, However, the estimation of the MSSM full 1-loop correction for ( 6) is difficult even using the GRACE system because the number of Feynman diagrams of the process e − e + → ν νh becomes about 10 4 .In this paper, therefore, we use the EWA formulae [12], where ŝ ≡ xs and σn (ŝ) denotes the center of mass energy squared and cross section of the sub-process e − W + n → ν e h.Then we set x min = (m e + m W ) 2 /s.f n (x) are the energy distribution functions for W n -boson with helicity n = −1, 0, +1, where For the electron and W -boson coupling, g L = e/( √ 2 sin θ W ) and g R = 0.Moreover, the EWA cross section ( 7) is numerically indistinguishable from in the energy range considered here, because the contribution of W ± is almost negligible as shown in Figures.
In Figure 2, we show the tree level energy distribution of the higgs dσ/dE h from (6) and dσ ′ 0 /dE h in (15) at √ s=500 GeV.The statistical errors were calculated assuming L=500fb −1 .The difference is within the error range in the region E h =150∼230 GeV.We could figure out that the EWA reproduce the e − e + → ν νh cross section in this energy region.The peak from Zh production around E h =250 GeV cannot be reproduced by the EWA. 4 The numerical results

The SM contribution
Table 3 shows the SM parameters which we used in the calculations.All the following numerical results (include the MSSM cross section) are computed in these parameters.Figure 3 shows the δ SM NLO in (3) for e − e + → Zh at √ s=250 GeV (left) and e − e + → ν νh used EWA at √ s=500GeV (right).The plotted errors were calculated assuming L=250 fb −1 and 500 fb −1 for e − e + → Zh and e − e + → ν νh respectively.In advance, we should confirm that the 1-loop correction does not buried in this error.In the left figure, the δ SM NLO is estimated to be −(6 ∼ 9)% in entire region and it is larger than the error.Similarly, in the right figure, δ SM NLO is estimated to be −4.5 ∼ +1.5% in entire region and this value is larger than the error.
From the results we figure out that the 1-loop correction is necessary for reliable theoretical predictions in both processes.[%]

The MSSM contribution
Figure 4 shows the 1-loop corrected angular distribution of Z , dσ/dcosθ Z (left) and the ratio δ susy in (4) (right) for the process e − e + → Zh.From the left figure, we find that the both the SM and the MSSM 1-loop contributions are negative and they are satisfied the relations, In the right figure, the δ susy are estimated to be 1.17∼1.25%at cosθ Z = 0.In the entire region these ratios are larger than the statistical error assuming planned luminosities at the ILC.It means that the 1-loop contribution of the MSSM could be measured at √ s =250GeV.Also, the set 1 will be possibly distinguished from set 2 and set 3. Figure 5 shows the 1-loop corrected energy distribution of higgs (left) and δ susy (right) in e − e + → ν νh.The left (right) figure shows differential cross section dσ/dE h (δ susy ).In the left figure the SM and MSSM 1-loop contribution are indistinguishable at a glance.However, δ susy are estimated to be 1.5∼1.7% and they are larger than the statistical error assuming 500 fb −1 in the entire region.It means that the 1-loop contribution of the MSSM could be measured at √ s =500 GeV.Three proposed sets can not be distinguished each other at least assuming the planned luminosities at the ILC.The precision measurements of the higgs production processes at the ILC will bring us important information on the heavy sparticles through the virtual 1-loop effects.

Summary and conclusions
We investigated the indirect effects of the MSSM at the ILC.We focused on the center of mass energies of 250 GeV and 500 GeV in the processes e − e + → Zh and e − e + → ν νh, respectively.We selected the mass spectra of the MSSM which are consistent with the observed mass of higgs, the thermal relic density of the dark matter, the low energy experiments and the LHC bounds of sparticles.The parameter sets proposed in our calculations have the squarks and the gluino with masses of 1.5, 5.0, 10.0 TeV and the sleptons and gauginos with masses less than 0.5 TeV.With using our developed GRACE system, we calculated the 1-loop corrected cross sections of the processes e − e + → Zh and e − e + → ν νh at the ILC.For the analysis of latter process, we adopted the equivalent W -boson approximation.We confirmed that both the SM and the MSSM 1-loop correction is necessary for the accurate theoretical predictions at the ILC.We found that the 1-loop effects of MSSM are verifiable both at 250 GeV and 500 GeV.Moreover the difference between set 1 and set 2 and 3 will be possibly observed with e − e + → Zh at √ s=250GeV.

Figure 1 :
Figure 1: √ s dependence of tree level total cross sections of e − e + → ν νh.The solid line and the dashed line and the dotted points correspond to σ, σ ′ and σ ′ 0 , respectively.

Table 2 :
Masses and MSSM parameters for three sets (masses in unit of GeV)

Table 3 :
The SM parameters used in this report.