Connecting muon anomalous magnetic moment and multi-lepton anomalies at LHC

In a previous paper by several of the authors a number of predictions were made in a study pertaining to the anomalous production of multiple leptons at the Large Hadron Collider (LHC). Discrepancies in multi-lepton final states have become statistically compelling with the available Run 2 data. These could be connected with a heavy boson, H, which predominantly decays into a standard model Higgs boson, h, and a singlet scalar, S, where GeV and GeV. These can then be embedded into a scenario where a two-Higgs-doublet is considered with an additional singlet scalar, 2HDM+S. The long-standing discrepancy in the muon anomalous magnetic moment, , is interpreted in the context of the 2HDM+S type-II and type-X, along with additional fermionic degrees of freedom. The 2HDM+S model alone, with constraints from the LHC data, does not seem to explain the anomaly. However, adding fermions with mass of order GeV can explain the discrepancy for sufficiently low values of fermion-scalar couplings.


Introduction
The anomalous magnetic moment of the muon, , is one of the most important precision observables to test the standard model (SM) and provide a complementary, non-collider constraint of beyond the standard model (BSM) physics. Currently, the discrepancy between the experimental measurement and the SM prediction is [1][2][3][4][5][6][7][8], where: Exp µ − a SM µ = 2.87(80) × 10 −9 . (1) This opens a window of opportunity for quantum corrections driven by BSM particles [9][10][11][12][13]. In a model independent scenario, a detailed study [10] showed the contribution to for BSM particles of masses of a few 100 GeV. A complete two-loop contribution to in the two-Higgs-doublet model (2HDM) is performed in [14,15] to  In this study, we draw on our previous work, in which we studied the effects of a new scalar, H, heavier than the SM Higgs, in relation to Run 1 results from the LHC [16,17]. From an effective Lagrangian approach, the best fit mass of H was determined as GeV, where these past studies drew on (but were not limited to) the production of multiple leptons (in association with bquarks), as had been studied in the search for SM Higgs. These studies were associated with the top quark. As a result of our earlier studies, the introduction of a scalar mediator, S, was necessary, such that our effective vertices were constructed using HSh, HSS, and Hhh interactions. Furthermore, the S could decay (in a Higgs-like manner) to SM particles [16].
We made a number of predictions at high energy proton-proton collisions, related to the production of multiple leptons in [16,17]. Assuming that the singlet scalar behaves like a SM Higgs-like boson, the data can be described with GeV and GeV. These discrepancies have become statistically compelling with the available Run 2 data [18], where the mass points and parameters were fixed from our earlier studies, and as such were not altered in our model to better explain the data. The final states were selected as per the predictions in [16,17], which predate the Run 2 data. These include the anomalous production of opposite-sign, same-sign, and three leptons in the presence and absence of bquarks.
The discrepancies that arise in final states and regions of the phase space, where different processes dominate the SM description, do not point to a likely mismodeling of a specific SM process. Rather, the anomalies and their kinematic characteristics are reasonably well described by a simple ansatz, where is produced via gluon-gluon fusion and in association with top quarks in high-energy proton-proton collisions. It is therefore appropriate to understand the possible connection of the above-mentioned spectroscopy with the anomaly via loop corrections. ∆a µ The above-mentioned H and S can be embedded into a 2HDM scenario with an additional scalar, where S is a singlet under the SM gauge groups [16,19,20]. This was done explicitly in [19], where a study of this embedding's 2HDM+S parameter space was made that can accommodate the discrepancies between the SM and the data reported in [21]. Here, we investigate whether the 2HDM+S model with the parameter space identified in [21] can account for the anomaly or whether new degrees of freedom are necessary.
While the multi-lepton anomalies reported in [18,21] seem to be fairly well described with the simple ansatz mentioned above, in [18] a more complex picture was indicated in the data than provided by this 2HDM+S model. The specific processes that indicated this greater complexity were in the dilepton system, including the invariant mass, transverse mass, and the missing transverse energy. These tended to be wider than what is predicted by the decay chain. New leptonic degrees of freedom could significantly alter the decays of S, thus modifying the differential distribution predicted by the model [21]. In this light, we explore what one can learn from the anomaly with regard to these potential new degrees of freedom. ∆a µ In this short article, we connect with the constrained parameter space of the 2HDM+S at the LHC. We briefly explain the model considered for this study in Section 2, along with the constraints on the parameter space from previous studies. The one-and two-loop formulae are discussed in Section 3, and the results of the study are presented in Section 4. The implications of this study to other processes are discussed in Section 5. Finally, a summary and conclusion of this study is presented in Section 6.

Model
∆a µ As mentioned in Section 1, we consider the 2HDM+S model as a possible explanation for . Following [19,20,22] this model is, in summary, based on the wellknown 2HDM with the addition of a real singlet scalar S. The potential is given by the following: where the fields and are the Higgs doublets, while is the singlet scalar field. The first three lines correspond to the terms in the real 2HDM potential. The final four terms relate to the singlet S field. Models that have more than one Higgs doublet can have tree-level flavor changing neutral currents (FCNCs). To avoid these tree-level currents, the usual approach is to couple all quarks with the same charge to a single double.
Because of the addition of a singlet scalar, this model has three physical CP-even scalars h, S, and H, with one CP-odd scalar A and charged scalar . Other parameters of this model are the mixing angles , and , vacuum expectation values (vev) , and the masses . As discussed in Section 1, the masses of a large number of these parameters are fixed a priori from previous studies [16,19,20], where the as yet constrained mass , and to a lesser extent , will be addressed in this study.
The relevant Yukawa couplings between the SM fermions and 2HDM+S scalar mass eigenstates are given as follows: For details on the couplings and other information, we refer the readers to [19,20]. Furthermore, for our studies, we only consider type-II and lepton-specific (type-X) models within the parameter space considered in [19].
Chinese Physics C Vol. 44, No. 6 (2020) 063103 Because these types of models are highly constrained by many studies, we also considered a scenario by adding BSM leptons, which are singly charged. Specifically, we consider light leptons with a mass of GeV, which are not directly produced at colliders. This indicates that these leptons are to be treated as mediators, and would contribute via loop corrections to the anomaly. For the sake of simplicity, and without any loss of generality, we consider singly charged SM singlet vector-like leptonic fermions with chiral components, which transform as follows [10]:  4), can be easily cast in terms of the scalar mass eigenstates, as in Eq. (3). By adding these interactions to the 2HDM+S, we expanded our model to what we label as a 2HDM+S+f model. However, these fermions are also constrained by collider searches in terms of masses and model-dependent couplings, which we further explain in Section 5.
The overall coupling should be constrained as , though it should be noted that all the couplings that appear in the interactions are functions of the mixing angles and used to diagonalize the mass matrix appropriately in the model. Without loss of generality, we can assume , scanning over values in the coming sections, along with the mass of the new vector-like fermion, f.  [14,15], where these calculations also apply for the 2HDM+S with appropriate coupling arrangements. The one-and twoloop diagrams contributing to are shown in Fig. 1. The type-II and type-X (lepton specific) 2HDMs are suitable to explain the discrepancy with positive contributions to the . In these models, the lepton couplings to the new bosons are enlarged, while the top Yukawa coupling are kept favorably small.
The one loop contribution from neutral and charged scalars is given by the expression: where , , and The two loop contribution from neutral scalars is given by the expression: where , , , and , , and are the mass, charge, and number of color degrees of freedom of the fermion in the loop. The functions are: where and . In this study, we go one step further and use the , as given by [10]: , and . The function is defined as: ∆a µ m f m A We now employ these formulae, inputting the numerical values of parameters from previous studies [16,19,20], to generate the , scanning across the parameters , , and the mixing angles in the following section.

Results
For numerical calculations, we considered the masses of the scalars to be GeV, = 140 -150 GeV, = 250 -270 GeV, and = 400 -600 GeV and . In a few cases, we have also scanned over light masses of A and discussed this where appropriate. The parameter space chosen here and in [20] is consistent with the following: a) theoretical constraints, such as vacuum stability from the global boundedness and minimum of the potential, as well as tree-level perturbative unitarity, etc. [23][24][25][26] and from [23,27]. c) compatibility with S, T, and U, i.e., the oblique parameters.
As a first test, we start with 2HDMs, where the positive one-loop contributions stem from the terms involving H ± tan β m A h and H, whereas the terms with A and yield negative contributions. Conversely, h and H provide the negative two-loop contributions, whereas A provides a positive contribution. In the region of large and small , the 2HDM two-loop contributions become larger than the one-loop contributions, allowing for an explanation of the discrepancy. The required parameter space of the type-II and type-X 2HDMs is shown in Fig. 2. For a complete analysis of the 2HDM contributions, see [15].
Although the addition of S is necessary for the explanation of multi-lepton anomalies, it does not have a significant effect on the parameter space of the 2HDM required to account for , since the contributions from the CP-even scalars are suppressed with respect to that of the CP-odd scalar. This can be seen in a comparison of the plots in Fig. 2 and those in Fig. 3. In contrast to the existing constraints on the 2HDM+S, Fig. 3 shows that it requires a high value of and a relatively small mass of A. We also showed the constraints on the parameter space in the -plane excluded by the direct searches at the LEP and the measurement of at 95% C.L. In all cases considered here, -decays provide the most stringent constraints for both models and types. To calculate the leptonic decay, we consider the leading one-loop diagrams and use the formula given in [28].
Following the model used in [19], where GeV and , the full two-loop contribution from 2HDM+S is shown in Fig. 4. From the plots in Fig. 4, it is clear that the existing constraints on and the particle masses in the 2HDM+S make the model not suitable to account for . Addition of BSM fermions: Introducing additional fermionic degrees of freedom that interact with the 2HDM+S particles allows the one-loop contributions to become larger than the two-loop contributions. Fermions type-X 2HDM. One and two sigma regions are shown in dark blue and light blue, respectively. This parameter space can be further constrained by experimental data. Regions I and II (above dashed red and blue lines respectively) are excluded by direct search at LEP and decay ( ), respectively, at 95% C.L. [28].
Chinese Physics C Vol. 44, No. 6 (2020) 063103 063103-4 ∆a µ ∆a µ in the mass range 100 GeV to 1000 GeV enlarge the oneloop contributions sufficiently to account for . The one-loop diagram with the new fermions is explained in Fig. 1. In Fig. 5, we depict the contribution to the using Eq. (11) for different choices of couplings. Because the contributions from the 2HDM+S without the BSM fermions and the two-loop diagrams containing the SM fermions are much smaller, only the one-loop contribution is considered here. In Fig. 6, we depict some of the couplings required by different fermion masses.

Constraints and implications ∆a µ
Thus far, we constrained the parameter space of 2HDM+S and 2HDM+S+f models by calculating the contributions to . However, there are other sources from which these model parameters can also be constrained. We discuss some of them in this section.
Low energy bounds: The observed data of highly constrains the parameter space of 2HDMs, and hence they also constrain the extended models considered here. The branching fraction of has an enhancement of in type II 2HDMs, arising from the mediation of the extra Higgses possible in the box and penguin type diagrams, as both charged leptons and down-type quarks have a coupling related to for the extra bosons. However, the same couplings in a type X 2HDM are enhanced by for leptons and for quarks, which implies no such relationship in the branching fraction in the type X case. In fact, for type X, the leading contribution is largely independent for large . A caveat to this (when comparing type X to type II 2HDM) is related to the contributions from  type-X 2HDM+S. One and two sigma regions are shown in dark blue and light blue, respectively. Note the similarity to the parameter space required by the 2HDM without S in Fig. 2. Regions I and II (above dashed red and blue lines respectively) are excluded by direct search at LEP and decay ( ), respectively, at 95% C.L [28].
A, the light CP-odd Higgs. A can provide large contributions to the branching fraction of , needed to explain the anomaly. From the formula in [29], constraints on the parameter space are obtained in [28]. A similar observation is allowed for the models and parameters considered in this work. , where the first uncertainty combines the experimental statistical and systematic contributions, and the second is due to the uncertainty in the ratio of the and the fragmentation functions [30]. This is in aggreement with our analysis.
f ′ Muon mass corrections: Because of the addition of singly charged SM singlet vector-like leptonic fermions given in Eq. (4), the mixing between muon and vectorlike singlet leptonic fermion can provide corrections to the muon mass. The quantum corrections from the vector- like leptons to the SM lepton masses are avoided by minimal flavor protections [10]. Here, the Yukawa coupling between the muon and the corresponding vector-like lepton will contribute to the muon mass proportional to the Yukawa coupling, suppressed by the usual loop factor of . Perturbativity and the accepted mass correction impose constraints on the relevant Yukawa couplings in the muon sector. In this work, our choice of the couplings are limited by these requirements, as shown in Fig. 5.