$CP$ violating mode of the stoponium decay into $Zh$

We show that a novel decay mode $Zh$ of the bound state of stop-anti-stop pair in the ground state $^1S_0(\widetilde t_1\widetilde t_1^*)$ may have a significant branching ratio if the $CP$ violating mixing appears in the stop sector, even after we apply the stringent constraint from the measurement of the electric dipole moment (EDM) of the electron. We show that the branching ratio can be as large as 10% in some parameter space that it may be detectable at the LHC.


I. INTRODUCTION
So far the Higgs boson discovered in 2012 is the only fundamental particle of scalar in nature [1]. On the other hand, colored scalar bosons are definitely signs of physics beyond the standard model (SM), which often appears in many new physics models. One outstanding example is the scalar-top (stop) quark -superpartner of the top quark -in the Minimal Supersymmetric Standard Model (MSSM). Their strong interaction allows them to be produced abundantly at hadron colliders if kinematically allowed. The current search for the stop at LHC has pushed its mass above about 500 GeV [2]. To escape its detection, the mass of the lightest stop state t 1 is compressed just above the lightest neutralino mass so that there is not much missing momentum for tagging the event at the LHC. In such a scenario, the stop state is rather long lived in comparison to the time scale of QCD hadronization.
Therefore, the stop-anti-stop pair can form the bound state, called the stoponium [3], which is produced through the gluon-gluon fusion [3][4][5] as in the squarkonium [6] production. The ground state η ≡ 1 S 0 ( t 1 t * 1 ) of the stoponium can then be identified by its distinctive decay modes, such as hh, W W , ZZ, γγ, etc. Among them, the channel hh stands out [3] for its significant decay rate with clean detection signature. Recent studies of the stoponium at LHC can be found in [7][8][9][10][11][12]. There are also efforts in studying the QCD corrections [13,14], the lattice calculation [15], the mixing between the Higgs boson and the stoponium [16] , and the role of the stoponium [17] in the dark matter co-annihilation.
Surprisingly in all studies about the stoponium decay, the channel Zh is not given. In fact, the process is forbidden by the underlying assumption of the CP conservation, which implies the cancellation of amplitudesá la the Furry theorem. However, there is no strong argument against CP violation in the stop sector. We are going to show in this article that η → hZ can have a significant branching ratio when CP violating parameters are chosen yet within the experimental constraint due to the electron electric dipole moment (eEDM) measurement.
If the mass of the stoponium is close to the mass m A of the pseudoscalar Higgs boson, substantial enhancement of the Zh decay mode happens due to the resonance effect. Neverthless, for a stoponium mass around 1.2 TeV ∼ m A the eEDM places a very stringent constraint on the choice of the CP -violating parameter such that the B( η → Zh) ∼ 10 −3 .
On the other hand, if the mass of the second stop is not too far from the lightest stop, substantial cancellation between the stop contributions to the eEDM can happen, such that the CP -violating parameter can be chosen to be much larger and the branching ratio B( η → Zh) ∼ 10 −1 . In the extreme case that the m A → ∞ when the eEDM is not effective, the branching ratio can reach a large value, B( η → Zh) ∼ O(0.5). This is the major result of our work. Furthermore, due to the heavy stoponium decay the Z and h bosons are very boosted, in which both bosons can be identified as boosted objects with advanced boost techniques to suppress backgrounds. Such rather straightforward detection of the Z and h bosons makes the mode Zh a wonderful place to look for the new particle as well as CP violation.
The organization is as follows. In the next section, we give details about the mixing in the stop sector, as well as the CP -violating couplings to the Higgs boson and Z boson. In Sec. III, we analyze the decay mode Zh together with the eEDM constraint. In Sec. IV, we estimate the observability of the Zh mode at the LHC. We summarize in Sec. V.

II. CP -VIOLATION IN THE STOP SECTOR
Let us start with the Z boson couplings to the stops t i (i = 1, 2). The convective current among stop states is Our convention for the Feynman vertex amplitude is for the incoming p j and the outgoing p i . Under the charge conjugation C, t i The negative sign in the transformation of J comes from that in ↔ ∂ .
Consequently, we need to make the C-odd transformation for the Z gauge boson, Z µ C ←→ −Z µ . The hermiticity of the unitary interaction L ⊃ ij g Z ij J µ ij Z µ requires g Z ij = g Z * ji . If the charge conjugation is a good symmetry, we have g Z ij = g Z ji . From this, we know that a complex g Z ij (for i = j) if its phase is not removable implies C-parity violation In general, if the states t L,R mix with each other by the complex 2×2 matrix into the mass eigenstates t 1,2 , we expect the complex off-diagonal g Z 12 coupling to the Z boson. However, we can set g Z 12 real by redefining the relative phase between the two stop fields t 1 , t 2 . Indeed in the next section, we adopt such a choice in our convention. To have a genuine C-parity non-conservation, we need additional complex coupling coefficient y, which appears in the Higgs vertex of yh( t * 2 t 1 ). Then there is no more freedom to remove its phase.
For the renormalizable interaction of the pure bosonic sector, operators of dim 4 or less do not involve the P -odd Levi-Civita -symbol. Therefore, the P -parity is conserved in the Z vertex. Consequently, the C-parity violation is the CP -violation. Our example is the decay of the ground state of the stoponium in 1 S 0 ( t 1 t * 1 ) into Zh. The exchange of t 2 can appear in the t-channel and in the u-channel, as shown in the first two diagrams in Fig. 1. The phase of g Z ij is tied with another vertex yh t * 1 t 2 , and thus overall unremovable. The two amplitudes of the u and t channels cancel if the coupling factor is real, but add up if imaginary. The production of Zh from such a decay is a sign of CP -violation. Furthermore, there exists the direct coupling of the pseudoscalar A 0 to the stops, , which is CP -violating. The ground state of the stoponium η can annihilate into the virtual A 0 in the s-channel as in the third diagram in Fig. 1, and then become Zh via the ZA 0 h gauge vertex. If the mass of the stoponium is close to the mass m A of the pseudoscalar Higgs boson, substantial enhancement of the Zh decay mode happens, indeed the Zh mode is significant in such a scenario. Nevertheless, it is restricted by the eEDM especially when the mass eignestates of the stop sector is widely separated and m A is moderate. When m A is chosen to be very heavy, then the constraint of eEDM disappears and the CP parameter can be chosen very large and the branching ratio into Zh can be as large as O(0.5).
Input parameters in the calculation of the η → Zh decay mode include masses m t 1 , m t 2 , mixing parameters θ t , δ u , Re[µ * e −iδu ], Im[µ * e −iδu ] , and tan β is the ratio of the VEV of the two Higgs doublet.
The relative phase between the µ parameter and the trilinear A t parameter can be established in the following t L t R * term in the Lagrangian: where where c β , s β are shorthand notation for cos β and sin β, c α , s α are for cos α and sin α, respectively, tan β ≡ v u /v d is the ratio of the VEV of the two Higgs doublet, and α is the mixing angle between the two neutral components of the Higgs doublets.
The stop mass matrix can be expressed as We can define a phase δ u by then the mass matrix can be diagonalized by an orthogonal transformation with an angle θ t into mass eigenstates t 1 and t 2 : The stop mass matrix can be re-expressed in terms of m t 1 , m t 2 , θ t , and δ u as By comparing the off-diagonal elements of the above two stop mass matrix, we can express

B. Relevant Couplings for Zh decay mode
The interaction between h and t L,R is where and For the interaction between the heavy Higgs H and stops t 1,2 , we need to change the above h, t 1,2 interactions by substitutions On the other hand, the interaction between A 0 and t L,R is The interaction between Z boson and t L,R is where the two-way derivative i ↔ ∂ µ applies only to the stop fields, and picks up (p − p ) µ of the stop momenta p, p flowing into the vertex in the Feynman diagram.
The process t 1 t * 1 → hZ involve the s-channel diagram going by the A 0 exchange, as well as the t-channel and the conjugated u-channel by the t 2 exchange, as shown in Fig. 1.
In the non-relativistic approximation, the overall amplitude is where g Z Ah = g 2 √ 1−x W cos(β − α). The overall transition rate requires the polarization sum, Here we use 2P · p Z = s + m 2 Z − m 2 h and s = m 2 η 4m 2 t 1 . The kinematic function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc). Note that all amplitudes are suppressed by the non-alignment factor cos(β − α), which appears in both g Z Ah and Im[y h The partial decay width in the non-relativistic approximation is where the bound state wave function at the origin is estimated by the Coulomb type expression, In comparison, we show the partial decay width of the gluon-gluon mode .
In MSSM, the relevant contribution to the eEDM based on the CP violating parameters in the stop sector t 1,2 arises via the two-loop Barr-Zee diagrams [19].
where α em = e 2 /(4π), v 246 GeV, and F (z) is a two-loop function given by In fact the eEDM contribution vanishes in two different limits, first when A 0 becomes heavy and decoupled, and second when m t 1 m t 2 so that their effects cancel each other.
Our numerical results show that even in general cases, an ample parameter space satisfies the eEDM constraint, but still gives significant branching ratio mode of hZ.
Although the one-loop contributions to the eEDM 1-loop also exist in the neutralinoselectron diagram, and the chargino-sneutrino diagram, they involve totally different CP violating parameters and can be tuned to give tiny eEDM [20]. Therefore, we ignore their one-loop effect in eEDM. In another approach [21,22], one can allow the sole contribution of one type of diagrams to exceed the current experimental limit, where one can expect that there might be other types of diagrams that would cancel one another.

III. ANALYSIS
The input parameters that are relevant for the stoponium decay into Zh are: m t 1 , m t 2 , Re[µ * e −iδu ], Im[µ * e −iδu ], θ t , tan β, and m A . In the computation of the branching ratios of the stoponium, it also involves the gluino mass mg and cos(β − α).
Since we expect the pseudoscalar resonance can enhance the decay rate when m η is around the heavy pseudoscalar A 0 mass, we study the following cases, 1. Near and below the pole, m η < m A by setting 2m t 1 = 1200 GeV and m A = 1.5 TeV.
2. Well below the pole, m η m A by setting 2m t 1 = 1200 GeV and m A = 2.5 TeV.
3. Far from the pole for an extremely heavy m A . We set 2m t 1 = 1200 GeV m A . In this case, we simply remove the s-pole contribution. Note that in this limiting case, the two-loop contribution of the pseudoscalar boson A 0 to the eEDM vanishes as well.
Note that we do not choose m A very close to m η in case (1), because for such a low m A the contribution to the eEDM would be large. In Fig. 2  In the extreme case of case (3), the mass of the pseudoscalar A 0 is set to be very heavy.
Practically, we ignore the term involving the A 0 exchange. We show in Fig. 4 the branching ratios for the stoponium with m t 1 = 600 GeV and m t 2 = 1000 GeV, except for the lowerright panel where m t 2 = 650 GeV. Since there are no more A 0 contribution to the eEDM, we can set the parameter Im[µ * e −iδu ] large enough to achieve a dominant branching ratio for the Zh mode. We have chosen Im[µ * e −iδu ] = 100, 200, 5000, and 5000 GeV, respectively.
Note that increasing Im[µ * e −iδu ] will also increase the hh mode, because the partial width Γ( η → hh) ∝ |y h t 1 t 2 | 2 , and Γ(stoponium → hZ) ∝ Im[y h t 1 t 2 ]. In the most favorable case, the branching ratio into Zh can be of order O(0.5), as indicated in the lower-right panel. The other input parameters are the same as Fig. 2.

IV. OBSERVABILITY AT THE LHC
The leading order(LO) production process for η at LHC is through the gluon-gluon fusion, The cross section section can be expressed in term of its gluonic decay width as [7] where g(x, Q) is the gluon parton distribution function, and τ ≡ m 2 η /s with the center of mass energy of pp collision √ s. For the parton distribution function, we used CTEQ6 [23] with the factorization scale Q = m η . The K-factor, which is the ratio between the next leading order (NLO) and the LO cross sections, we take a reasonable value about 1.4. For more detailed NLO calculation, we refer to Ref. [14]. At NLO, we obtain the production cross section for m η 1.2 TeV at the LHC of √ s = 13 TeV.
The Zh decay mode of the stoponium can be searched for via h → bb and Z → + − or Z → jj. At the LHC, such searches have been performed [24][25][26][27], in which hadronic or leptonic modes of the Z boson and bb mode of the Higgs boson have been used. It is clear that the leptonic mode of the Z boson is clean but suffers from a small branching ratio.
The hadronic mode of Z boson was believed to be suffered from large QCD background.
Nevertheless, with the advance of various boosted-jet techniques the hadronic decays of the Z boson and h can be performed with reasonable success. Since the stoponium is rather heavy ∼ 1.2 − 1.5 TeV here, the Z boson and the Higgs boson are very boosted with p T ∼ 0.6 − 0.75 TeV. The opening angle between the decay products of the Z or the Higgs boson is ∼ 2M/p T ∼ 0.3 − 0.5. This is in the right ballpark for excellent detectability of boosted jets in contrast to the conventional QCD background.
The recent search for pp → X → Zh → jjbb performed by ATLAS [24] at the LHC gave an upper limit on σ(pp → X → Zh) × B(h → bb + cc) < 20 − 30 fb around the resonance mass 1.2 − 1.5 TeV. On the other hand, the search pp → X → Zh → + − bb was also performed [26]. The upper limit on σ(pp → X → Zh) × B(h → bb + cc) < 10 fb. Note that these searches was designated for vector resonances. In the same paper, they also gave σ(pp → A → Zh) × B(h → bb) < 10 fb for m A ≈ 1.2 TeV. Therefore, the production cross section of the stoponium times the branching ratio into Zh is well below the current limits at the LHC.
With a project luminosity of 300 fb −1 at the end of Run II, we can expect about 15 events for η → Zh → (jj, ) + bb for an optimistic branching ratio B( η → Zh) ∼ 10%. We emphasize again that in CP -conserving case the stoponium would not decay into Zh, yet a small branching ratio into Zh would signal a violation of CP symmetry.

V. CONCLUSIONS
We have demonstrated that the decay mode of the ground state of the stoponium, η → Zh, can have a dominant or significant branching ratio if we choose suitable CP violating mixing in the stop sector, which is still allowed by the eEDM measurement. Observation of such a decay mode of the stoponium is clean signal of CP violation. The detailed phenomenology will be investigated in a separate analysis.
Our framework for the decay mode Zh from the scalar pair in the ground state can be extended to other models that have fundamental colored scalar bosons, such as the technipion [28] or the colored octet Higgs [29].
We offer a few comments before closing.
1. Both the partial width of η → Zh and eEDM increase with increase in the parameter Im[µ * e −iδu ]. Therefore, we cannot make it arbitrarily large. When m A = 1.5 TeV and m t 1 = 600 GeV, Im[µ * e −iδu ] can only be 100 − 200 GeV.