Di-Higgs Decay of Stoponium at
Future Photon-Photon Collider
Hayato Ito, Takeo Moroi and Yoshitaro Takaesu
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
We study the detectability of the stoponium in the di-Higgs decay mode at the photon-photon collider option of the International Linear Collider (ILC), whose center-of-mass energy is planned to reach TeV. We find that detection of the di-Higgs decay mode is possible with the integrated electron-beam luminosity of if the signal cross section, , of fb is realized for the stoponium mass smaller than 800 GeV at 1 TeV ILC. Such a value of the cross section can be realized in the minimal supersymmetric standard model (MSSM) with relatively large trilinear stop-stop-Higgs coupling constant. Implication of the stoponium cross section measurement for the MSSM stop sector is also discussed.
Low energy supersymmetry (SUSY) is an attractive candidate of the physics beyond the standard model even though the recent LHC experiment is imposing stringent constraints on the mass scale of superparticles. Importantly, there is still a possibility that there exist superparticles with their masses below TeV scale. In particular, a scalar top-quark (stop) with the mass of is still allowed if there exists a neutralino whose mass is just below that of the stop mass; in such a case, even if the stop is produced at the LHC experiments, its decay products are too soft to be observed so that it can evade the detection at the LHC.
If there exists stop with its mass of , it will become an important target of future collider experiments. If the signal of the stop is discovered at the LHC, the LHC and the International Linear Collider (ILC) may play important role to study its basic properties (like the mass and left-right mixing angle). It is, however, also important to study the strength of the stop-stop-Higgs coupling because the Higgs mass in the supersymmetric model is sensitive to it; measurement of the stop-stop-Higgs coupling is crucial to understand the origin of the Higgs mass in supersymmetric model. It motivates the study of the stop-stop bound state (so-called stoponium which is denoted as in this paper) because decay rate of the stoponium is crucially depends on such a coupling. If we observe the process of , we can acquire information about the stop-stop-Higgs coupling.
Photon-photon colliders may be useful to perform such a study [1, 2].#1#1#1For the stoponium studies at other colliders, see . A photon-photon collider is one of the options of the ILC and can be realized by converting high-energy electron (or positron) beam of the ILC to the backscattered high-energy photon. It has been intensively discussed that the photon-photon collider can be used to study the properties of Higgs and other scalar particles . One of the advantages of photon-photon colliders is that the single production of some scalar particles (including the Higgs boson) is possible so that the kinematical reach is close to the total center-of-mass (c.m.) energy; this is a big contrast to other colliders with and collision. The single production of the stoponium bound state is also possible with the photon-photon collision, and hence it is interesting to consider the stoponium study at the photon-photon collider.
In this paper, we investigate how and how well we can study the property of the stoponium at the photon-photon collider, paying particular attention to the process of . We calculate the cross section of the stoponium production process at the photon-photon collider. We also estimate backgrounds, and discuss the possibility of observing the stoponium production process at the photon-photon collider. Implication of the cross section measurement of the stoponium di-Higgs decay mode for the MSSM stop sector is also discussed.
The organization of this paper is as follows. In Section 2, we discuss the theoretical framework of the stoponium and its production cross section and decay widths at the photon-photon collider. The detectability of the stoponium in the di-Higgs decay mode is investigated in Section 3. In Section 4 we discuss the implication of the cross section measurement of the stoponium production and di-Higgs decay for the MSSM stop sector. Then we provide our summary in Section 5.
2 Stoponium: basic properties
Let us first summarize the framework of our analysis. We assume the minimal supersymmetric standard model (MSSM) as the underlying theory. The relevant part of the superpotential for our study is given by
where is the top Yukawa coupling, , , and denote up- and down-type Higgses, right-handed top quark, and third-generation quark-doublet, respectively, and “hat” is used for the corresponding superfields. In addition, and are indices, while the color indices are omitted for simplicity. The relevant part of the soft SUSY breaking terms is
where “tilde” is used for superpartners.
Neglecting the effects of flavor mixing, the stop mass terms are expressed as
where is the top-quark mass, ,
with being the Weinberg angle, the -boson mass, and . and parameters are taken to be real. The mass eigenstates are given by the linear combination of the left- and right-handed stops; we define the mixing angle as
where and are lighter and heavier mass eigenstates with the masses of and , respectively.
Before closing this subsection, we comment on the lightest MSSM Higgs boson mass. The lightest Higgs mass is sensitive to the masses of stops as well as to the parameter through radiative corrections. With the stop masses being fixed, the lightest Higgs mass becomes equal to the observed Higgs mass (which is taken to be throughout our study) for four different values of ; two of them are positive and others are negative. We call these as positive-large, positive-small, negative-large, and negative-small solutions of , where large and small solutions correspond to those with large and small values of .
2.2 Stoponium production at a photon-photon collider and its decay
Due to the strong interaction, a stop and an anti-stop can form a bound state, called stoponium. In this analysis, we concentrate on the case where the decay rate of a stop is negligibly small. Because we are interested in the collider study of the stoponium, we concentrate on the bound state of the lighter stop. The lowest bound state, , has the quantum number of , and hence its resonance production does not occur at colliders. At photon-photon colliders, on the contrary, the process may occur, where denotes final-state particles of the stoponium decay. High energy photon-photon collisions can be achieved by photons originating from backscattered lasers off electron beams, and this possibility was discussed in detail [5, 6]. For a concrete discussion, we assume a photon-photon collider utilizing an upgraded International Linear Collider (ILC) whose energy is planned to reach .
With the c.m. energy of colliding photons, , being fixed,#2#2#2In this article, we denote the center-of-mass energy of colliding photons by and that of colliding electrons and positrons by . the cross section for the process can be written as 
Here and are the Stokes parameters of the initial-state photons where corresponds to the photons with helicity . In this study we only consider axially symmetric electron beams, and then other components of the Stokes parameters () are negligible . (= ) is the cross section for photon collisions with circular polarization and given with the Breight-Wigner approximation by
where and are the total and partial decay widths of the stoponium, respectively, and are polarizations of initial-state photons, and is the mass of the stoponium, which is roughly estimated as
throughout this paper. Detailed calculations in Ref.  shows that the error of this estimation is and negligible for our discussion.
Since backscattered photons off electron beams are not monochromatic, the cross section at the photon-photon collider is given by#3#3#3In this article, we define cross sections for the photon-photon collider as the number of events normalized by the luminosity of electron beams, .
using the luminosity function of backscattered photons [5, 6, 9] denoted by (with and being the photon energies normalized by the energy of the electron beam in the laboratory frame).#4#4#4For more details about the luminosity function, see Appendix A Here, with being the averaged energy of the laser photons in a laboratory frame, and and are the luminosity and c.m. energy of the electron beams, respectively. We take to maximize without spoiling the photon luminosity . Since we consider the case where , we use the narrow-width approximation and obtain
with and being the mean helicities of initial electrons and laser photons, respectively, and . In our numerical calculation, we take and .
The decay rates of the stoponium are related to the matrix elements for the pair-annihilation processes of the stop and anti-stop. For the case of two-body final states, i.e., , the decay width is related to the matrix element of the scattering process as 
where . The matrix elements for the stop pair-annihilation processes relevant to our study are summarized in Appendix B. As we have mentioned, we consider the case where the decay width of the stop is much smaller than the total decay width of the stoponium. This is the case when the mass difference between the lighter stop and the lightest supersymmetric particle (LSP), which is assumed to be the lightest neutralino in our analysis, is small enough.
In Fig. 1, we plot the stoponium production cross section
taking , which maximizes the cross section. The stoponium production cross section can be as large as fb for .
In Fig. 2, we show the branching ratios of the stoponium as functions of the lightest stop mass, taking , , , and negative-large or negative-small solutions of the parameter. The positive solutions give similar branching ratios. The decay mode dominates the stoponium decay, and this may be an useful mode for stoponium searches at photon-photon colliders. However, the branching ratio is completely determined by the strong coupling constant, and measuring this mode does not give much information on SUSY interactions. Therefore, we do not investigate this mode in this study. Although and decay modes have non-negligible branching ratios of , they suffer from large SM backgrounds [1, 2, 12]. We therefor investigate decay mode in the following sections as a probe to the SUSY interactions, especially to the stop sector. We will see in the next section that the signal-to-background ratio of the process may be large enough to be observed at the ILC-based photon-photon collider.
3 search at a photon-photon collider
In this section, we discuss a search strategy for the decay mode of the stoponium at the photon-photon collider. We assume an upgraded ILC, whose energy is planned to reach  with the integrated luminosity of .
For our numerical calculation, we adopt four sample SUSY model points with , , , and as summarized in Table 1.
|Point 1||Point 2||Point 3||Point 4|
We assume that the lighter stop and the lightest neutralino, , will be discovered before the photon-photon collider experiment is carried out. In addition, if the stop is within the kinematical reach of the photon-photon collider, detailed studies of the stop will have been already performed with the collisions at the ILC, and hence we also assume that the basic properties of the lighter stop such as the mass and the chirality will be measured at the ILC before the start of the photon-photon collider. We consider the cases where the bino is the lightest SUSY particle (LSP) with , and GeV for each , respectively.#5#5#5We don’t consider the relic abundance of the LSP in this study. Other SUSY particles are assumed to be sufficiently heavy ( TeV) and irrelevant to our photon collider study. and parameter are taken to be 10 and 2 TeV, respectively, for all the sample points, and the Higgs mass of 125.7 GeV is realized by adjusting the parameter and the heavier stop mass, . (For our numerical calculation of the Higgs mass, we use FeynHiggs v2.11.3 .) In order to maximize the stoponium production cross section, we adjust the c.m. energy of the electron beams as for each sample point shown in Table 1. The cross sections of the process for those sample points are also shown in the Table.
3.1 Signal event selection
From Table 1 we see that the cross sections for the sample points are of (0.1) fb, and then the process would give only (100) events at . Therefore, we use the main decay mode of the Higgs boson for the signal process, i.e., .
In order to simulate the signal process, we generate events where a scalar particle (which corresponds to the stoponium) is produced at the photon-photon collider and decayed to a Higgs pair with their subsequent decays to , using MadGraph5_aMC@NLO v2 ; the luminosity function of the colliding photons [6, 9] are implemented by modifying the electron PDF routines in MadGraph5. The cross section of the events is normalized to that of the signal process according to Eq. (12). The generated events are then showered with PYTHIA v6.4  and passed to DELPHES v3  for fast detector simulations. In the detector simulation, we assume energy resolutions of 2%/% and 50%/% for an electromagnetic calorimeter and hadron calorimeter, respectively, based on ILC TDR . FastJet v3  is employed for jet clustering using the anti- algorithm  with the distance parameter of 0.5.
From the generated events, we first select events containing more than four jets and satisfying GeV and for all of the four highest jets (Preselection), where and are the transverse momentum and pseudo-rapidity, respectively. We then impose the following cuts successively:
Here is the invariant mass of the four highest jets. is the number of b-tagged jets in each event, where we assume 80% b-tag efficiency, and 10% and 0.1% mis-tag rates for jets and jets, respectively. In S3 and S4, and are defined as follows. We first divide the leading four jets into two jet pairs. Among three possible pairings, we choose the one which minimizes , where and are the invariant masses of the jet pairs such that
is defined as
where and are the differences of pseudo-rapidities and azimuthal angles between the paired jets with the invariant mass of , respectively.
After imposing the selection cuts, the relevant background processes are the non-resonant , , , , (where ), , , and production processes. The event numbers after all the selection cuts are imposed are estimated for the above background processes as in the signal process case, except the non-resonant and backgrounds, which are loop induced processes.
As for and backgrounds, we use approximate estimations; the event numbers of the non-resonant background after each cuts are estimated with 15% uncertainty, and the event numbers of the background are estimated as the upper bounds after all cuts are applied. We will see in Sec. 3.3 that even these rough estimations are enough for our study and leave more detailed estimations for future works.
In the following, we describe our procedure to estimate the non-resonant and backgrounds. The production cross sections of background processes at the photon-photon collider can be expressed in the similar way as for the signal process discussed in Sec. 2.2:
with . In the second line, contributions from and are negligible since we consider axial symmetric electron beams; the sign in front of the stokes parameters are taken to be positive (negative) for .
First, let us discuss the non-resonant Higgs pair production process. The dominant background contributions are from Higgs pairs decaying to bottom quarks. Based on Eq. (22), the cross section after all selection cuts are imposed, , is given by
where is the total efficiency of all the selection cuts for events with a c.m. energy , a total energy measured in the laboratory frame , and Higgs scattering angle in the c.m. frame of the collision.
We approximate this expression by neglecting the angular dependence of the Higgs production cross section. In Fig. 3, we plot the luminosity-weighted differential cross section
for and with GeV (Point 1).
In evaluating the differential cross section , we use the one-loop expressions given by Ref. .
From the figure, we see that the luminosity-weighted differential cross sections for the photon helicity are larger than those for the photon helicity. Those cross sections do not change significantly over the whole range of for both choices. We have also checked that Point 4 shows a similar behavior with GeV. Therefore, we approximate that Higgs pairs are produced almost isotropically in the c.m. frame of the photon collision. Then, Eq. (24) is written as
Note that we approximate the differential cross section by its averaged value over . The total cut efficiency is estimated by generating event samples of isotropically produced Higgs pairs with the luminosity function, setting the c.m. energy of the Higgs pairs to and imposing all the selection cuts on the generated events.
Finally, we comment on our approximation that the Higgs pairs are produced isotropically. Using the maximum value of the differential cross section over the range instead of the averaged one in Eq. (26), we obtain the upper bound of . We check that the differences between the upper bounds and our approximated cross section, Eq. (26), are less than 15 %. This can be regarded as the uncertainty of our approximation, which is sufficient for our study as we will see in the next subsection.
Next, we discuss the background. The dominant background contributions are from and decay modes. Instead of directly estimating the background cross section with all the selection cuts being imposed (), we set an upper bound on the cross section by removing the Preselection and S4 cuts since estimation of the efficiencies of those cuts needs more detailed simulation. The upper bound is written as
where and are the efficiencies of the S2 cut for the decay mode and the S3 cut, respectively. In evaluating , we use the one-loop expressions given in Ref. . In Eq. (29), we approximately take into account the efficiencies of the S1, S2 and S3 cuts as follows. The effect of the S1 cut is approximated by limiting the integration interval, setting the upper and lower limits to those of the S1 cut , i.e., GeV and GeV for each sample model points. The efficiency of the S2 cut corresponds to the probability that three or four jets are -tagged from the () final state and is obtained as . The S3 cut efficiency is estimated from simulated event samples of boson pairs produced in the collision, setting the c.m. energy of the boson pair at since the peak region of the photon-photon luminosity is tuned at around this energy. This upper bound on the background will be used in estimating the upper bound on the total background in the next subsection.
We present expected signal and background event numbers with all the selection cuts imposed for the sample model points in Table 2.
|Point 1||Point 2||Point 3||Point 4|
|[ GeV ]|
Here, we assume the integrated electron-beam luminosity of . More than ten signal events are expected for all the sample points, while background events are effectively reduced to less than four events. We estimate the expected significance of detecting the decay mode using an approximated formula based on the Poisson distribution :
with being the expected signal (total background) event number.#6#6#6This significance approaches to when . The significance for each sample point is also presented in Table 2. Because we only estimate the upper bounds on the background, the expected significances are regarded as lower bounds. We see that in order for the detection, the signal cross sections, , of fb are required for the stoponium masses of 500 – 800 GeV, respectively. In the rest of this section, we discuss how background events are reduced by imposing the selection cuts.
After imposing all the selection cuts, the major background source is the non-resonant Higgs pair () production process, and the contributions from other background sources except , and are negligibly small. We present the cut-flow information along with the cut efficiencies in parentheses for the sample Point 1 in Table 3 and for other points in Table 4, where only the non-negligible background processes are presented.
|Point 1||Preselection S1||S2||S3||S4|
|Point 2||Preselection S1||S2||S3||S4|
|Point 3||Preselection S1||S2||S3||S4|
|Point 4||Preselection S1||S2||S3||S4|
The selection cut S2, which requires three or four jets are b-tagged, then plays an important role in reducing the large portion of the background events which needs some non- jets to be mistagged to pass the cut.
The selection cut S3, relevant to the di-jet invariant masses, also reduce most of the background events efficiently, except for non-resonant , by imposing Higgs mass constraints on two pairs of jets. At this stage, only the non-resonant , , and backgrounds remains sizable.
The selection cut S4, which is based on distributions, further reduces the remained and backgrounds. In Figs. 4, we present the