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PHYSICAL REVIEW D 101, 035010 (2020)

Resolving the ðg − 2Þμ and B anomalies with and a dark Higgs

† ‡ Alakabha Datta,1,2,* Jonathan L. Feng ,2, Saeed Kamali ,1,2, and Jacky Kumar 3,§ 1Department of Physics and Astronomy, University of Mississippi, 108 Lewis Hall, Oxford, Mississippi 38677, USA 2Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA 3Physique des Particules, Universite de Montreal, C.P. 6128, Succursale Centre-Ville, Montreal, Qu´ebec H3C 3J7, Canada

(Received 8 September 2019; accepted 16 January 2020; published 7 February 2020)

At present, there are outstanding discrepancies between predictions and measurements of the ’s g − 2 and several B- properties. We resolve these anomalies by considering a two- Higgs-doublet model extended to include leptoquarks and a dark S. The leptoquarks modify B-meson decays and also induce an Sγγ coupling, which contributes to the muon’s g − 2 through a Barr- Zee diagram. We show that, for TeV-scale leptoquarks and dark Higgs boson masses mS ∼ 10–200 MeV, a consistent resolution to all of the anomalies exists. The model predicts interesting new decays, such as B → KðÞeþe−, B → KðÞγγ, K → πγγ, and h → γγγγ, with branching fractions not far below current bounds.

DOI: 10.1103/PhysRevD.101.035010

I. INTRODUCTION The most longstanding anomaly we consider is in the At present, there are a number of anomalies in low- anomalous magnetic moment of the muon. A recent energy measurements. Among these are the anomalous evaluation of the standard model (SM) prediction [1] finds a 3.7σ discrepancy with the experimental measurement [2]: magnetic moment of the muon, ðg − 2Þμ, and several in the decays of B . Although none of these currently rises ð −2Þexp −ð −2ÞSM ¼ 27 4ð2 7Þð2 6Þð6 3Þ 10−10 ð Þ to the level of a 5σ anomaly on its own, they are significant g μ g μ . . . . × : 1 deviations, and it is interesting to investigate them, par- ticularly if there are parsimonious explanations and if these The first two uncertainties are theoretical, and the last is explanations motivate new analyses of current and near- experimental. The experimental uncertainty is currently the future data. largest, but it is expected to be reduced by a factor of 4 by In this work, we explain all of these anomalies in a the Muon g − 2 Experiment [3], which is currently collect- concrete model: a two-Higgs-doublet model (2HDM) ing data at . extended to include TeV-scale leptoquarks and a light In the B sector, there are a large number of anomalies scalar S with mass mS ∼ 10–200 MeV. We find solutions with various levels of significance; for a review, see that depend on only a small number of parameters and Ref. [4]. These anomalies may be divided into charged − show that these explanations motivate interesting new current (CC) processes, such as b → cτ ν¯τ, and neutral searches, particularly for rare meson decays to diphoton current (NC) processes, such as b → slþl−. The CC ðÞ final states and Higgs boson decays to four . decays B → D τντ have been measured by the BABAR [5,6], Belle [7–9], and LHCb [10] Collaborations. These results may be expressed in terms of the ratios RðDðÞÞ ≡ ¯ ðÞ − ¯ ðÞ − BRðB → D τ ν¯τÞ=BRðB → D l ν¯lÞ, where l ¼ e, μ, *[email protected] † in which many theoretical and systematic uncertainties [email protected] ‡ [email protected] cancel. By averaging the most recent measurements, the §[email protected] HFLAV Collaboration has found [11]

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2470-0010=2020=101(3)=035010(14) 035010-1 Published by the American Physical Society DATTA, FENG, KAMALI, and KUMAR PHYS. REV. D 101, 035010 (2020) where, here and in the following, the first uncertainty is RðKðÞÞ anomalies, at least for the central q2 data. Weak- statistical and the second is systematic. These measure- scale states do not fully resolve the low-q2 discrepancy, ments exceed the SM predictions RðDÞSM ¼ 0.299 since a larger effect is required to modify the larger SM 0.003 [12] and RðDÞSM ¼ 0.258 0.005 [13] by 2.3σ widths near the pole, but the U does and 3.4σ, respectively. A combined analysis of RðDÞ also reduce the discrepancy for the low-q2 data to and RðD Þ, including measurement correlations, finds a roughly 1.7σ [41]. 4 1σ deviation of . from the SM prediction [11]. A new The U leptoquark does not, however, resolve the ðg − 2Þμ measurement [14] by the Belle Collaboration, using semi- anomaly; it contributes at one loop, but this contribution is leptonic tagging, gives too small. We must therefore introduce additional if we are also to explain the ðg − 2Þμ discrepancy. Explanations ð Þexp ¼ 0 307 0 037 0 016 ð Þ R D . . . ; 4 in terms of additional weak-scale states, such as sleptons and [63], remain viable, but the implications of ð Þexp ¼ 0 283 0 018 0 014 ð Þ R D . . . ; 5 these explanations for experiments are very well known. Alternatively, the ðg − 2Þμ anomaly could be resolved by which reduces the deviation of the combined measurements light and very weakly coupled particles. Dark photons with from the SM predictions to about 3.1σ. þ þ þ − mass ∼10 MeV–1 GeV were previously proposed as pos- In the NC sector, the ratio RK ≡ BRðB → K μ μ Þ= þ þ þ − sible solutions [64,65], but these solutions are now excluded BRðB → K e e Þ [15,16] has been precisely measured [66]. However, other light- solutions remain viable. by LHCb, most recently in Ref. [17], which finds For example, a light leptophilic scalar can contribute exp þ0.060þ0.016 2 2 significantly to ðg − 2Þμ for large tan β ∼ 200, while its R ¼ 0.846−0 054−0 014 ; 1 ≤ q ≤ 6.0 GeV ; ð6Þ K . . relatively weak hadronic couplings allow it to avoid stringent 2 2 bounds [67]. where q ¼ m þ − . This is lower than the SM prediction l l In this work, we consider a different and novel light, SM ¼ 1 00 0 01 2 5σ ≡ RK . . [18] by . . The related ratio RK weakly coupled particle solution to the ðg − 2Þμ problem: BRðB0 → K0μþμ−Þ=BRðB0 → K0eþe−Þ has been mea- 1 a light scalar S with mass m ∼ 10–200 MeV that is an sured by LHCb to be [19] S extension of the standard Type II 2HDM model. The scalar 0 66þ0.11 0 03 0 045 ≤ 2 ≤ 1 1 2ð 2Þ S, which we will often refer to as the dark Higgs boson, exp . −0.07 . ; . q . GeV lowq R ¼ : couples to both and , but with couplings that K 0 69þ0.11 0 05 1 1 ≤ 2 ≤ 6 0 2ð 2Þ . −0.07 . ; . q . GeV centralq are suppressed both by Yukawa couplings and by a small θ ð7Þ mixing parameter sin . At the one-loop level, its contri- bution to ðg − 2Þμ is too small to resolve the anomaly. SM ¼ However, motivated by the leptoquark solution to the B These are also lower than the SM predictions [18] RK 2 SM 2 anomalies, we note that leptoquarks (as well as other 0.906 0.028 (low q ) and R ¼ 1.000.01 (central q ) K γγ by 2.3σ and 2.5σ, respectively. Taken together, the general TeV-scale particles) will generically induce an S cou- ð − 2Þ consensus is that these B-decay branching ratios differ pling, and this can resolve the g μ anomaly through a significantly from SM predictions, and theoretical hadronic two-loop Barr-Zee diagram. In this way, the solutions to the uncertainties [22–24] alone may not explain the data. ðg − 2Þμ and B anomalies proposed here are connected. An interesting question, then, is whether the B anomalies (As an aside, we note that, for values of mS just below 2mμ, have a common explanation in terms of new physics. our explanation can also completely remove the discrep- 2 Early work on the simultaneous explanation of the CC ancy in the low-q RK measurement, following a pos- and NC anomalies [25–28] has been followed by many sibility noted previously in Ref. [68].) model calculations; an incomplete list can be found in In addition to resolving longstanding anomalies, the Refs. [29–61]. Remarkably, there appears to be a rather proposed explanation predicts new signals. In particular, simple explanation for both the CC and NC anomalies in given the light state S and its couplings to and terms of a single vector leptoquark U with SM quantum photons, the model predicts new meson decays, such as 2 þ − numbers ð3; 1; 3Þ that couples dominantly to left-handed B → KS and K → πS, followed by S → e e ; γγ, leading quarks and leptons. A clear guide to the combined to dilepton and diphoton signals that could be discovered in explanation of the anomalies may be found in Ref. [62]. current and near-future experiments. The model also predicts → → γγγγ For a mass mU ∼ 1 TeV and Oð1Þ couplings to the third exotic Higgs boson decays h SS , which may generation, the U leptoquark can explain the RðDðÞÞ and appear in detectors as a contribution to the h → γγ signal. In Sec. II, we present the model, including the new fields 1 we introduce and the relevant model parameters. In Sec. III, The Belle II Collaboration has also measured RKðÞ [20,21] ð − 2Þ recently, but these measurements currently have relatively larger we determine the parameter values that resolve the g μ uncertainties, and so have little effect on our analysis. anomaly. In Sec. IV, we then discuss constraints on the

035010-2 RESOLVING THE ðG − 2Þμ AND B ANOMALIES … PHYS. REV. D 101, 035010 (2020) model from hadronic physics and show that a resolution in terms of the physical Higgs boson masses. As shown in to the ðg − 2Þμ and B constraints exists in a viable region Appendix A, the results are of parameter space. The interesting implications for exotic B, K, and Higgs boson decays are discussed in Sec. V. vA 2vA m2 sin θ ≈− ; sin θ0 ≈− 1 − h : ð10Þ We summarize our conclusions in Sec. VI. Appendixes A m2 m2 2m2 and B contain details of the 2HDM model and the effective H h H Sγγ coupling, respectively. The last term of Eq. (9) is an Sγγ coupling governed by the parameter κ, which has dimensions of inverse mass. This II. THE MODEL coupling is generically induced by heavy states, such as Our model is an extension of the Type II 2HDM. The leptoquarks, as will be discussed in Sec. III. Finally, as discussed in Sec. I, we add a vector leptoquark Type II 2HDM contains two Higgs doublets Hu and Hd, U with SM quantum numbers ð3; 1; 2Þ and Lagrangian which get vacuum expectation values (VEVs) vu and vd 3 and give mass to the up-type and down-type , 2 ϕ 1 respectively. We extend this by adding a singlet scalar , U Uμν 2 μ U ¯ μ L ¼ − FμνF − m UμU − ½h ðQ γ L ÞUμ þ H:c: which couples to the Higgs doublets through the portal U 4 U ij iL jL interactions μ − gmUSUμU : ð11Þ † † † † V ¼ AðH H þ H H Þϕ þ½λ H H þ λ H H portal u d d u u u u d d d The U leptoquark’s couplings to left-handed quarks and þ λ ð † þ † Þϕϕ ð Þ ’ ud HuHd HdHu ; 8 leptons resolve the B-meson anomalies. The leptoquark s couplings to right-handed quarks and leptons are con- where CP conservation is assumed. In this extension, we strained to be small [69]. We have also included the consider parameters such that Hu and Hd get VEVs, but ϕ leptoquark’s couplings to S. This interaction is allowed does not. After electroweak symmetry breaking, then, the by all symmetries, but it will not play an important role in trilinear scalar couplings mix the new scalar with the Higgs any of the phenomenology discussed below. As we will of the 2HDM, and the quartic scalar couplings discuss later, we consider the U leptoquark coupling to contribute to new Higgs boson decays h → ϕϕ and to the photons to be the same as the one between the W boson mass of the ϕ. and photons. Since the leptoquark is colored, it couples to More precisely, to determine the physical states of the also [70]. This coupling leads to their pair produc- theory, we minimize the full Higgs potential and diago- tion at high energies, but it does not affect our phenom- nalize the mass matrices; for details, see Appendix A. In the enology here. end, the physical states include the SM-like Higgs boson h In summary, the model we consider consists of a 2HDM and the heavy Higgs bosons H, A, and H of the 2HDM, model extended to include a light dark Higgs boson S and ’ U but also a new real scalar, the dark Higgs boson S, with a leptoquark U. The leptoquark s couplings hij are chosen Lagrangian to resolve the B anomalies [53]. In addition to these, the parameters of the theory that are most relevant for the 1 1 X m L ¼ ð∂ Þ2 − 2 2 − θ β f ¯ phenomenology we discuss below are S 2 μS 2 mSS sin tan ffS f¼d;l v X m 1 m ; tan β; sin θ;m ; κ; ð12Þ − θ0 β f ¯ − κ μν ð Þ S H sin cot ffS 4 SFμνF ; 9 ¼ v f u 0 where tan β, sin θ, and mH fully determine sin θ and the S couplings to fermions, and κ determines the S where v ≃ 246 GeV and tan β ¼ vu=vd. The couplings to fermions are inherited from the mixing of the dark Higgs couplings to photons. We will be primarily interested in ∼ 10–200 boson with the 2HDM Higgs bosons: they are suppressed the parameter ranges mS MeV, moderate to by Yukawa couplings, and the down-type couplings are large tan β ∼ 10–60, small mixing angles sin θ ∼ 0.005, ∼ 1 κ ∼ ð1 Þ−1 enhanced by tan β, while the up-type couplings are sup- mH TeV, and TeV . pressed by cot β. In addition, they are modified by the θ θ0 mixing angles sin and sin . For weak portal interactions III. RESOLVING THE MUON MAGNETIC ≪ β A mh and large tan , these mixing angles can be written MOMENT ANOMALY Given a 2HDM extended to include a dark Higgs boson 2Although we will not be considering or supersymmetric states in this work, we note that the Type II S and a vector leptoquark U through the Lagrangian terms 2HDM is the Higgs sector of the minimal supersymmetric of Eqs. (9) and (11), respectively, we can now calculate the standard model. beyond-the-SM contributions to ðg − 2Þμ.

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FIG. 1. Contribution of the effective Sγγ coupling to ðg − 2Þμ.

A. Dark Higgs boson contribution from effective Sγγ coupling Let us first consider the dark Higgs boson contribution from the Sγγ effective coupling shown in Fig. 1. This contribution is dominated by the log-enhanced term [71] 2 ð β κÞ γγ 1 mμ Λ FIG. 2. The region of the tan ; plane where an effective S Δðg − 2ÞSγγ ≈ θ β κ ; ð Þ ð − 2Þ μ 4π2 sin tan ln 13 coupling induces a Barr-Zee contribution to g μ that enhan- v mS ces the theoretical prediction to be within 1σ of the measured Λ value. The subdominant one-loop contribution from a virtual S where is the cutoff scale, which we may take to be of the has also been included. We fix sin θ ¼ 0.005, Λ ¼ 2 TeV, and order of the mass of the particles that induce the effective show results for mS ¼ 100 MeV and 200 MeV, as indicated. Sγγ coupling. Parameters required to resolve the ðg − 2Þμ anomaly are presented in Fig. 2. For dark Higgs mixing ð − 2Þ angle sin θ ∼ 0.005 and tan β ∼ 10–60, we see that the diagram. The Barr-Zee contribution to g μ with a W effective coupling required is κ ∼ ð1 TeVÞ−1. In our cal- boson in the loop has been calculated in Ref. [73] in the culations we also include the contribution to the context of 2HDMs. As leptoquarks are not gauge bosons, anomalous magnetic moment at the one-loop level, which there might be ambiguities in the leptoquark two-loop ð1Þ has been calculated to be [72] contribution. For an O estimate of this contribution, we model the effect of this leptoquark loop by the W loop. Z 2 1 ð1 þ Þð1 − Þ2 We find that the leptoquark contributions to ðg − 2Þμ are δ ð1-loopÞ ¼ gl z z ð Þ al 2 dz 2 −2 ; 14 always positive—that is, in the right direction—and they 8π 0 ð1 − zÞ þ r z induce an effective Sγγ coupling parameter where r ¼ ml=mS and, in our case, gl ¼ sin θ tan βðml=vÞ. α XNLQ NcQ2g κ ¼ EM Vi F ð4m2 =m2Þ; ð16Þ B. Dark Higgs boson contribution from Sγγ W Vi S 4π mV coupling induced by V leptoquarks i¼1 i

How could such values of κ be induced? As an example, 5 where α ≃ 1=137; Nc ¼ 3 and Q ¼ are the number of motivated by the effectiveness of leptoquarks for explain- EM 3 ing the B anomalies, we consider adding N vector colors and of the leptoquarks Vi, respec- LQ tively; g parametrizes the SV V coupling in Eq. (11); and ¼ 1 … Vi i i leptoquarks Vi, i ; ;NLQ, with Lagrangians FW is a loop function defined in Ref. [74]. For large leptoquark masses m ≫ m , the loop func- 1 μ Vi S Vi Viμν 2 V ¯ μ LV ¼ − FμνF − m ViμV − ½h ðQjRγ LkRÞViμ ≃ 7 i 4 Vi i jk tion is FW . In the simple case where we have NLQ ¼ μ copies of degenerate leptoquarks with mass mV mLQ þ H:c: − g m SV μV ; ð15Þ i Vi Vi i i ¼ and coupling gVi gV, Eq. (16) reduces to where for simplicity we add only leptoquarks with SM 5 quantum numbers ð3; 1; Þ and assume that their couplings NLQgV 3 κ ≃ 0.034 : ð17Þ to right-handed quarks and leptons are identical. mLQ V Assuming small couplings hjk, the leading way in which ð − 2Þ −1 these Vi leptoquarks contribute to g μ is by inducing Setting gV ¼ 3 and requiring κ ≈ TeV , the mass and an Sγγ coupling, which then contributes through a Barr-Zee number of leptoquarks required to resolve the ðg − 2Þμ

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ð Þ ð3 1 5Þ FIG. 3. The region of the mLQ;NLQ plane where NLQ vector leptoquarks Vi with mass mLQ and SM quantum numbers ; ; 3 induce an effective Sγγ coupling that resolves the ðg − 2Þμ anomaly. In all panels, we set mS ¼ 100 MeV. In the upper and lower panels, we fix ðsin θ; tan βÞ¼ð0.01; 60Þ and (0.005,40), respectively. For the left panels, we set gV ¼ 3 and show the bands where the ðg − 2Þμ discrepancy is reduced to 1σ. For the right panels, we consider the several values of gV indicated and plot the lines on which the theoretical prediction for ðg − 2Þμ exactly matches its experimentally measured value. [In the upper- and lower-right panels, the induced couplings are κ ≃ ð3.2 TeVÞ−1 and ð0.9 TeVÞ−1, respectively.]

≈ ð100 Þ ð − 2Þ 1σ anomaly are related by mLQ NLQ GeV . The required viable, one can reduce the discrepancy in g μ to . ¼ parameters are shown graphically in Fig. 3. Alternatively, one can achieve the same result with NLQ γγ 10 ¼ 4 We see that it is not difficult to induce an effective S leptoquarks with mass mLQ TeV, which is likely coupling large enough to resolve the ðg − 2Þμ anomaly. challenging even for searches at the High Luminosity LHC. β ¼ 60 ¼ 5 For the tan case shown, with even just NLQ For the tan β ¼ 40 case shown, one requires roughly twice ¼ 2 leptoquarks with mass mLQ TeV, which is currently as many leptoquarks, but the number is still not very large.

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FIG. 4. U leptoquark contributions to ðg − 2Þμ. Left: Two-loop Barr-Zee diagram involving also the dark Higgs boson S. Center and right: One-loop diagrams that are independent of the dark Higgs boson.

Of course, the assumed new physics that is necessarily light summary, then, the U leptoquark contributions to ðg − 2Þμ is the dark Higgs boson S. This will have interesting are negligible in our model and do not modify our observable consequences, as we discuss in Sec. IV. discussion about the V leptoquark requirements to resolve the ðg − 2Þμ anomaly. C. U Leptoquark contribution ð − 2Þ In addition to the contributions to g μ mediated by IV. RESOLVING THE B ANOMALIES AND the dark Higgs boson and independent of the U leptoquark, HADRONIC CONSTRAINTS there are also the contributions that depend on the U leptoquark shown in Fig. 4. These include the two-loop A. The U leptoquark and B anomalies Barr-Zee contribution from a Sγγ coupling mediated by the The couplings of the U leptoquark in Eq. (11) can resolve U leptoquark, similar to those discussed above for V all the B anomalies. Let us start with the b → sμþμ− leptoquarks in Sec. III B, and also two one-loop contribu- anomalies, which include the RK and RK measurements. tions independent of the dark Higgs boson. The procedure to fit for new physics is the following. The two-loop Barr-Zee diagram’s contribution is as The b → sμþμ− transitions are defined via an effective discussed above. The contribution of a single U leptoquark Hamiltonian with vector and axial vector operators: with mass ∼TeV is not sufficient to raise the theoretical ð − 2Þ αG X prediction for g μ to the experimental value. H ¼ − pffiffiffiF V V ðC O þ C0 O0 Þ; eff 2π tb ts a a a a In addition, however, there are the one-loop contribu- a¼9;10 tions from the coupling of U to the muon and down-type ¼½¯γ ½μγ¯ μðγ Þμ ð Þ U ¯ γνμ ¼ O9ð10Þ s μPLb 5 ; 20 quarks, hiμdiL LUν, where i d, s, b. These contribu- tions to ðg − 2Þμ are [75] where the Vij are elements of the Cabibbo-Kobayashi- X c U 2 2 2 Maskawa (CKM) matrix, and the primed operators are N ðh μÞ 4mμ 5mμ Δðg − 2ÞU ¼ − i Q − Q ; obtained by replacing L with R. The Wilson coefficients μ 16π2 3 2 i 3 2 U ¼ i¼d;s;b mU mU include both SM and new physics contributions: Ca þ ð Þ Ca;SM Ca;NP. One now fits to the data to extract Ca;NP. 18 There are several scenarios that give a good fit to the data, 1 and the results of recent fits can be found in Refs. [69,70, where Nc ¼ 3 is the number of colors, and Q ¼ − and μμ μμ i 3 76–79]. One of the popular solutions is C ¼ −C , ¼ − 2 9;NP 10;NP QU 3 are the electric charges of the down-type quarks which can arise from the tree-level exchange of the U and the U leptoquark. Substituting these charges and the leptoquark in Eq. (11). Following the results of Ref. [69], value for the muon mass, we find fitting to the b → sμþμ− data constrains the central values of X 2 the U couplings to satisfy U −10 U 2 TeV Δðg − 2Þμ ¼ − 1.4 × 10 ðh Þ : ð19Þ iμ −4 mU U U ¼ 8 10 ð Þ i¼d;s;b hbμhsμ × : 21

This contribution is of the wrong sign to explain the The framework to explain all the B anomalies, including ð − 2Þ U g μ anomaly and depends on the couplings hiμ.In both the CC and the NC anomalies, involves the U U U → leptoquark coupling to the third-generation quarks and particular, the couplings hbμ and hsμ contribute to b þ − þ − ð1Þ U ∼ 1 sμ μ and are used to explain the RðK Þ and b → sμ μ leptons in the gauge basis with O coupling, hbτ anomalies [53,55]. As we show in the next section, [53]. As one moves from the gauge to the mass basis, for the U U however, the couplings hiμ have small enough values that quarks and leptons, the couplings hbμ and hsμ are generated. ð − 2Þ U ∼ 1 U U U we can ignore the one-loop contribution to g μ.In Hence, one has the hierarchy hbτ >hbμ >hsμ >hdμ.

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U ∼ 0 1–0 6 Using the allowed values of hbμ . . [53] and Eq. (21), U we see that the one-loop U contribution to ðg − 2Þμ in Eq. (19) cannot resolve the ðg − 2Þμ discrepancy. The ðg − 2Þμ anomaly therefore requires additional new physics, such as the S boson discussed in Sec. III.

B. Hadronic constraints In this model, the S boson inherits its couplings from the Higgs boson, and so it necessarily couples to both leptons and . The lepton couplings, specifically the muon coupling, are desired to resolve the ðg − 2Þμ anomaly. Here we begin to examine the implications of the hadronic couplings, which may either constrain the model or lead to predictions of interesting new signals. Particularly stringent are constraints on FCNC processes, since couplings like bsS are induced through a penguin loop. Integrating out the W-top loop induces the effective bsS vertex [80] FIG. 5. Contours of constant flight distance (excluding the boost factor) (d0 ¼ cτ0) of the light scalar S in the ðmS; κÞ plane. pffiffiffi We fix sin θ ¼ 0.005 and tan β ¼ 40. In the pink shaded region, sin θ0 3 2G m2V V ð − 2Þ 1σ L ¼ F t ts tb m sP¯ bS þ H:c:; ð22Þ the g μ anomaly is reduced to . bs v tan β 16π2 b R 2 2 2 3 ð Þj⃗ j τ 0 g A0 mS pK B sin θ BRðB → K SÞ¼ bs ; ð24Þ where the factor β comes from the top coupling 2 v tan 8πðmb þ msÞ to S. By the same loop process, but replacing b and s quarks with s and d quarks, respectively, the sdS vertex is also where mb and ms are the bottom and generated. Note that the FCNC amplitude depends on the masses, respectively; f0 and A0 are form factors, which θ0 2 mixing angle sin in Eq. (10), which is suppressed by mh, are taken from Refs. [83,84]; and gbs is the flavor-changing ð − 2Þ while the g μ in Eq. (13) is controlled by the mixing b → s coupling with the normalization Lbs ¼ gbssP¯ RbS. θ 2 angle sin in Eq. (10), which is suppressed by mH.Ifa higher value of mH is compensated by a larger value of the mixing parameter A to keep the same sin θ, then sin θ0 can become too large and be inconsistent with FCNC data. The FCNC interactions will induce two-body decays B → KðÞS and K → πS. To determine the signature of these processes, it is important to determine how the S decays. For mS ∼ 10–200 MeV, the possible decays are S → eþe−; γγ. In Figs. 5 and 6, we show the S lifetime and branching fraction to eþe−, respectively. We see that for most of the parameters of interest, the S flight distance (excluding the boost factor) is cτ0 ≲ 1 mm, and so the S decay is effectively prompt. We also see that the dominant decay is to diphotons, with BRðS → eþe−Þ ∼ 10−5 − 10−3 in the parameter region of interest. We now determine the rates for the two-body decays B → KðÞS and K → πS. For the two-body decays B → KðÞS we have [81,82]

2 2 2 2 2 2 ð → þ −Þ g f0ðm Þðm − m Þ jp⃗ jτ FIG. 6. Contours of constant branching fraction BR S e e BRðB → KSÞ¼ bs S B K K B ð23Þ ð κÞ θ ¼ 0 005 β ¼ 40 32π 2 ð − Þ2 in the mS; plane. We fix sin . and tan .Inthe mB mb ms pink shaded region, the ðg − 2Þμ anomaly is reduced to 1σ,andin the purple shaded region, BRðB → Keþe−Þ is within 1σ of its and measured value.

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Given the prompt S decays to eþe− and γγ,wehave uncertainty. The expression for the mass difference BRðB → KðÞeþe−Þ dominantly coming from BRðB → due to the new scalar is [82,89] KðÞSÞBRðS → eþe−Þ and BRðB → KðÞγγÞ dominated ðÞ 5 g2 by BRðB → K SÞBRðS → γγÞ. One can extend this to Δ NS ¼ − bs 2 ð Þ M 2 2 f mB : 26 K decays also. Bs 24 m − m B s Bs S We now discuss constraints from B and K decays on this model. In this subsection, we will consider a variety of We use a similar equation for the K − K¯ mixing nonleading constraints and show that they are far from mass difference and use the experimental value excluding the favored parameter space of this model. These Δ exp ¼ð52 93 0 09Þ 108 −1 MK . . × s [87]. observables are listed in Table I and are the following: (iv) K decay: The rare decay Kþ → μþνeþe− has been (i) B total decay width: In the first two rows of Table I, measured by the NA48=2 Collaboration to be ð → ðÞ Þ þ þ þ − −8 we require that BR B K S not exceed the BRðK → μ νe e Þ¼ð7.81 0.23Þ × 10 [90], uncertainty in the SM prediction of the width of where the measurement is restricted to the kinematic the , which we take to be around 10% [85]. region with m þ − ≥ 140 MeV. To study this decay → μþμ− e e (ii) Bs decay: The process Bs is mediated by an mode, we calculate the branching ratio of the decay s-channel dark Higgs boson S, where the matrix K → μνμS, where the scalar particle S is radiated off gbsgμ element is M →μþμ− ¼ 2 2 ðsP¯ bÞðμμ¯ Þ. We use Bs m −m R the muon leg [91]. The total branching ratio is then Bs S flavio [86] to calculate the contribution of the determined through light scalar S to this decay mode. The branching þ þ þ − ratio of this decay is measured to be ð3.0 0.4Þ × BRðK → μ νμe e Þ −9 10 [87]. The process B → γγ is also mediated by þ þ þ − s ¼ BRðK → μ νμSÞBRðS → e e Þ: ð27Þ an s-channel S. The SM prediction for BRðBs → γγÞ is around 5 × 10−7 [88], and there exists an exper- 3 1 10−6 imental upper bound of . × [87] for this The K → πeþe− mode also has been measured observable. The branching ratio of the decay in by the NA48=2 Collaboration to be BRðK → terms of the effective Sγγ coupling κ is πeþe−Þ¼ð3.110.12Þ×10−7 [92]. For this proc- 2 2 2 7 ess, we find the two-body decay rate K → π S, jg j jκj f m BRðB → γγÞ¼ bs Bs Bs τ : ð25Þ and the branching ratio of the desired process is s 64π m2ðm2 − m2Þ2 Bs b Bs S determined by

(iii) B and K mixing: In the SM, the B mass difference þ − s s BRðK → π e e Þ is ΔMSM ¼ð17.4 2.6Þ ps−1 [53]. We require that Bs ¼ ð → π Þ ð → þ −Þ ð Þ the new scalar contribution not exceed the SM BR K S BR S e e : 28

(v) KS;L decays: The decays KS;L → γγ are mediated TABLE I. Values of the contribution of the new scalar S to through s-channel dark Higgs bosons S, just as in the various meson observables. We fix the dark scalar mass to case Bs → γγ discussed above. The new contribu- mS ¼ 100 MeV. References for the experimental constraints are given in the text. tions to these decay modes and their Particle Data Group values [87] are presented in Table I. New scalar (vi) Last, although not a hadronic constraint, we also list ð − 2Þ contribution the model prediction for g e. Just as there is a θ ¼ 0 005 sin . , Existing constraints/ Barr-Zee contribution to ðg − 2Þμ, there is an analo- Observable tan β ¼ 40 measurements ð − 2Þ gous Barr-Zee contribution to g e. In contrast BRðB → KSÞ 1.7 × 10−4 < 10% to the muon case, the measured value for ðg − 2Þ −4 e BRðB → K SÞ 1.7 × 10 < 10% is smaller than the SM prediction, and so our ð → μþμ−Þ 4 2 10−14 ð3 0 0 4Þ 10−9 BR Bs . × . . × model’s contribution to ðg − 2Þ is in the wrong ð → γγÞ 7 4 10−11 3 1 10−6 e BR Bs . × < . × direction. However, as can be seen in Table I, the ΔMNS −2.5 × 10−17 GeV < 1.7 × 10−12 GeV ð − 2Þ Bs contribution to g e is very small, and does not Δ NS −6 3 10−24 5 9 10−18 MK . × GeV < . × GeV significantly worsen the agreement between theory þ þ þ − −14 −8 BRðK → μ νe e Þ 3.3 × 10 ð7.81 0.23Þ × 10 and experiment. þ − −11 −7 BRðK → π e e Þ 8.7 × 10 ð3.11 0.12Þ × 10 We see that none of the constraints listed in Table I is a ð → γγÞ 3 3 10−16 ð2 63 0 17Þ 10−6 BR KS . × . . × significant constraint on the model. In the next section, we ð → γγÞ 3 2 10−14 ð5 47 0 04Þ 10−4 BR KL . × . . × will consider the leading constraints, which do constrain δðg − 2Þ 6.3 × 10−14 ð−87 36Þ × 10−14 e parts of the model parameter space, but also provide

035010-8 RESOLVING THE ðG − 2Þμ AND B ANOMALIES … PHYS. REV. D 101, 035010 (2020) interesting predictions for signals that could be seen in the almost identical, and they range from roughly 1 × 10−4 to near future. 3 × 10−4 for tan β ¼ 40. Because the γγ comes from a light S, for a sufficiently γ γ V. NEW SIGNALS OF THE MODEL low mS, the two may be collinear and look like a single . One of the γ may also be soft, in which case again the 2γ ðÞ − A. B → K e + e will look like a single γ. Hence, experimentally one should ðÞ As noted above, the model contributes to the decay B → check the B → K γ signal carefully to look for signs of a KðÞeþe− with branching fraction BRðB → KðÞeþe−Þ¼ diphoton resonance. We should also point out that our ðÞ þ − predictions for the B → Kγγ rates should be considered as BRðB → K SÞBRðS → e e Þ. The region of the ðmS; κÞ parameter space that is consistent with the measured value ballpark estimates, as one can choose a more general of BRðB → KðÞeþe−Þ¼ð3.1þ0.9þ0.2 0.2Þ × 10−7 [93] 2HDM model to relax the branching ratio predictions. If −0.8−0.3 π0 is shown in Fig. 6, along with the region in which the the mass of the S is close to the mass, the final states for → ðÞπ0 → ðÞ π0 ðg − 2Þμ anomaly is resolved. We see that the existing B K and B K S, with both and S decaying γγ constraint on BRðB → KðÞeþe−Þ excludes the very lowest to , are the same, and one will have to consider carefully adding the two contributions. As nonleptonic decays are values of mS ∼ 10 MeV, but most of the parameter space ðÞ þ − very difficult to calculate, it will be difficult to detect the is allowed. Future measurements of BRðB → K e e Þ presence of the S particle in this case or obtain constraints with increased sensitivity may therefore see a deviation → ðÞπ0 predicted by this model. There is also a measurement of on the model from the B K measurement. In the → γγ → þ − 0 1 2 SM, the nonresonant decay B Xs has a branching ratio the inclusive B Xse e decay [94] for .

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D. h → γγγγ and implications for h → γγ coupling, or both, and if there is a large coupling, it could The model discussed here may also modify Higgs boson blow up just above the TeV scale, requiring a number of decays through the process h → SS, followed by S → γγ.3 additional states in any UV-complete theory. An UV Since the SM Higgs boson is much heavier than the scalar completion of our model is beyond the scope of this work, S, the two photons from S decay are boosted and highly but we believe that in any UV framework, the essential collimated. Therefore, the decay h → Sð→ γγÞSð→ γγÞ features of our model will remain valid. ∼ 100 contributes to the h → γγ signal [96]. We can calculate For dark Higgs mass mS MeV and dark Higgs θ ∼ 0 005 β ∼ 40 ∼ 10 1 mixing angle sin . , tan , and NLQ V the couplings appearing in the 2 ghSShSS interaction in terms of the parameters of the potential and mixing leptoquarks with masses at the TeV scale, the correction resolves the ðg − 2Þ anomaly. The introduction of a new parameters. The resulting branching ratio is μ light scalar S has many possible effects on SM meson sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi phenomenology. We have checked that all current bounds g2 4m2 BRðh → SSÞ¼ hSS 1 − S: ð29Þ on K and B properties, as well as the current constraint on 32πm Γ m2 ð − 2Þ h h h g e, are respected for the parameters that solve the ðg − 2Þμ and B meson anomalies; see Table I. The signal strengths measured by CMS and ATLAS In the near future, however, there are measurements that μγγ ¼ 1 18þ0.17 μγγ ¼ 1 06þ0.14 are . −0.14 [97] and . −0.12 [98], respec- could uncover beyond-the-SM effects and provide evidence tively. By a naive combination of these two measurements, for this model. In particular, the dark Higgs boson is light γγ we find μ ¼ 1.11 0.10. (We averaged the CMS enough to be produced in meson decays, and it then decays γγ and ATLAS measurements to μ ¼ 1.18 0.16 and through S → eþe−; γγ. The S boson has cτ ∼ 0.01–1 mm, γγ μ ¼ 1.06 0.13, respectively.) and so for most model parameters the decay is indistin- In the parameter region of our interest in the model, guishable from prompt, which yields interesting new ðÞ þ − we can find values for parameters of the potential such that dielectron events from B → K e e with m þ − ¼ m → → γγγγ e e S the addition of the process h SS to the SM rate and diphoton signals from B → KðÞγγ and K → πγγ with of h → γγ does not exceed the measured signal strength. mγγ ¼ m . The branching ratios for some of these modes θ ¼ 0 005 β ¼ 40 S As an example, for sin . and tan , and are shown in Figs. 6 and 7. In all cases, the predicted taking mdu ¼ 200 GeV, λ1 ¼ 0.6, λ2 ¼ 0.3, λ345 ¼ 2.8, λ ¼ −0 3 λ ¼ 0 0005 λ ¼ 0 005 branching ratios are not far from current sensitivities, d . , u . , and ud . , the signal although current measurements typically explore ranges of strength becomes μγγ ≈ 1.08. Of course, this also implies γγ meþe− and mγγ outside the considered range of mS.As that as the experimental constraints on μ become more ðÞ −4 4 examples, the model predicts values BRðB → K γγÞ∼10 precise, a deviation from the SM expectation may appear. þ þ 0 −6 and BRðK → π γγÞ;BRðKL → π γγÞ∼10 . Provided the S is not too degenerate with the neutral π0, these signals VI. CONCLUSIONS could be observed above background in the near future—for In this study, we have proposed a concrete model that example, at Belle II, providing a motivation to look for these resolves both the ðg − 2Þμ and B-meson anomalies, which exotic diphoton modes and an avenue for testing this model. are currently among the leading discrepancies between SM More generally, these decay modes test many models where predictions and experimental data. The model is a Type II the ðg − 2Þμ anomaly is resolved by a two-loop Barr-Zee 2HDM model, such as the Higgs sector of the minimal contribution generated by a light S with an Sγγ coupling. supersymmetric model, extended to include a light dark In addition, there are potentially observable contribu- Higgs boson S, a leptoquark U, and additional leptoquarks tions to exotic Higgs decays h → SS → γγγγ, which, given V. The U leptoquark resolves the B anomalies, and the V that the S is very light, typically lead to signals indistin- leptoquarks generate a Sγγ coupling. This coupling induces guishable from h → γγ. For the desired model parameters, a two-loop Barr-Zee contribution to ðg − 2Þμ, which is the contribution to h → γγ is within current constraints, shown in Fig. 1. The model makes interesting predictions but improved measurements could uncover a deviation for exotic signals that can be looked for in current and from SM predictions. Of course, electromagnetic calorim- upcoming data. Our proposed resolution to the ðg − 2Þμ eters with extremely fine spatial resolution that could θ ∼ problem requires either a large number of LQs or a large differentiate photons separated by opening angles of mS=mh ∼ mrad would be able to distinguish the γγγγ signal from the γγ signal, which would provide a smoking gun 3The model also predicts heavy Higgs boson decays H → SS, −6 signal of new physics. but the branching ratio for this is very small, of the order of 10 . 4As noted below Eq. (8), after electroweak symmetry breaking, the quartic interactions contribute to the ϕ mass. For the quartic ACKNOWLEDGMENTS coupling values given here, we require some fine-tuning between this contribution and the bare mass for the mass of the physical We thank W. Altmannshofer, T. Browder, L. Cremaldi, scalar S to be in the desired range mS ∼ 10–200 MeV. R. Harnik, and T. M. P. Tait for discussions. A. D. and S. K.

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2 are supported in part by NSF Grant No. PHY-1414345. M13 ¼ vA sin β; ðA9Þ A. D. thanks the hospitality of the T. D. Lee Institute, where part of the work was completed. J. L. F. is supported in part M2 ¼ vA β; ð Þ by NSF Grants No. PHY-1620638 and No. PHY-1915005 23 cos A10 and in part by Simons Investigator Grant No. 376204. The work of J. K. is financially supported by NSERC of 2 2 2 2 2 2 2 M33 ¼ m0 þ v λd cos β þ v λu sin β þ 2v λud cos β sin β; Canada. ðA11Þ APPENDIX A: CALCULATION OF S COUPLINGS λ ¼ λ þ λ þ λ IN TERMS OF 2HDM MODEL PARAMETERS where 345 3 4 5, and vd and vu are the VEVs of the two doublets Hd and Hu, with tan β ¼ vu=vd and We now explicitly calculate the parameters in the 2 þ 2 ¼ 2 ¼ð246 Þ2 vd vu v GeV . Lagrangian in Eq. (9), following the analysis of We assume A ≪ v; mdu, so we can consider the portal Ref. [67]. We start with the Type II 2HDM with the terms as small perturbations. In this case, we diagonalize Yukawa couplings the mass matrix perturbatively, where the nonperturbed mass matrix is the usual 2HDM mass matrix. We define the −L ¼ ¯0 0 0 þ ¯ 0 0 0 þ ¯ 0 0 ˜ 0 þ Y L YeHdeR Q YdHddR Q YuHuUR H:c: mixing matrix that diagonalizes the mass matrix as ðA1Þ 0 1 0 10 1 ρd − sin α cos αδ13 h B C B CB C Here the superscript denotes the quantities that are in @ ρu A ≈ @ cos α sin αδ23 A@ H A; ðA12Þ flavor space. ϕ δ31 δ32 1 S We write the scalar potential as δ ’ ð ϕÞ¼ ð Þþ ðϕÞ where ij s are small mixing angles that mix the light scalar V Hd;Hu; V2HDM Hd;Hu Vϕ with the other two scalars of the 2HDM. When we þ ð ϕÞ ð Þ Vportal Hd;Hu; ; A2 diagonalize the mass matrix of the 2HDM, the parameter α satisfies the usual equation where 2M2 ¼ 2 † þ 2 † − 2 ð † þ † Þ 2α ¼ 12 ð Þ V2HDM mddHdHd muuHuHu mdu HdHu HuHd tan 2 2 ; A13 M11 − M22 λ1 † 2 λ2 † 2 † † þ ðH H Þ þ ðH H Þ þ λ3ðH H ÞðH H Þ 2 d d 2 u u d d u u and the masses of the two CP-even Higgs bosons are λ † † 5 † 2 † 2 given by þ λ4ðH H ÞðH H Þþ ½ðH H Þ þðH H Þ ; d u u d 2 d u u d h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ðA3Þ 2 1 2 2 2 2 2 2 2 m ¼ M þ M ∓ ðM − M Þ þ 4ðM Þ : h;H 2 11 22 11 22 12

1 Aϕ λϕ ðA14Þ ¼ ϕ þ 2ϕ2 þ ϕ3 þ ϕ4 ð Þ Vϕ B 2 m0 2 4 ; A4 To determine expressions for the δij’s, we write the mass ¼ ð † þ † Þϕ þ½λ † þ λ † Vportal A HuHd HdHu uHuHu dHdHd matrix as þ λ ð † þ † Þϕϕ ð Þ 0 1 ud HuHd HdHu : A5 M2 M2 0 B 11 12 C 2 B 2 2 C After each doublet obtains a VEV, we write the neutral M ¼ @ M12 M22 0 A ¼ þ ρ real components of the doublets as Hi vi i, where 00 2 ¼ M33 i d, u. After expanding the potential, the elements of 0 1 ðρ ρ ϕÞ the mass matrix of the CP-even scalars in the d; u; 00vA β B sin C basis are B C þ @ 00vA cos β A; ðA15Þ 2 ¼ 2 β þ λ 2 2 β ð Þ M11 mdu tan 1v cos ; A6 vA sin β vA cos β 0 2 ¼ 2 β þ λ 2 2 β ð Þ M22 mdu cot 2v sin ; A7 where the second matrix is considered as a small pertur- ¼ β 2 2 2 bation. Below, we use the shorthand notation sβ sin M12 ¼ −m þ λ345v cos β sin β; ðA8Þ du and cβ ¼ cos β.

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1 We require the lighter Higgs h to have SM-like couplings 2 † μ 2 μ −L ¼ ξ ð2m WμW þ m ZμZ ÞS; ðA21Þ to gauge bosons and fermions, so that we have VVS V v W Z β − α ¼ π=2. Assuming M33 ≪ mh;mH, and writing α ¼ β − π=2, we find that the small mixing parameters are where the coupling is the same for both W and Z:   3 3 2 2 −2vAsβcβ 2vAsβ m m ξ ¼ δ þ δ ¼ ð1 þ 2βÞ ð Þ δ ¼ − h þ 2β 1 − h W;Z cβ 13 sβ 23 2 cot : A22 13 2 2 cot 2 ; m 2m 2m mh h  H H 2 2 β β ≈ β β → 1 δ ¼ − vA 2 1 − mh ð1 − 2βÞ In the large tan limit, we write cos cot and sin 23 2 sβcβ 2 2 cot ; β mh mH so that we can write this coupling in terms of cot only: vAs δ ¼ 2β −2vA β 31 2 ; ξ ¼ cot ð1 þ 2 βÞ ð Þ mh W;Z 2 cot : A23 mh vAc2β δ ¼ − ð Þ 32 2 : A16 mH In summary, we have the following couplings in terms of tan β: In the Yukawa sector, after rotating to the mass basis and   2 2 2 defining the mass matrices of fermions, the interaction vA mh 2 mh ξl ¼ − β þ β 1 − ; ð Þ terms between the physical light scalar S and the fermions ;d 2 tan 2 2 cot 2 2 A24 mh mH mH become   2vA m2 δ δ δ ξ ¼ − β 1 − h ð1 − 2βÞ ð Þ 13 13 ¯ 23 u 2 cot 2 2 cot ; A25 −LffS ¼ eM¯ eeþ dMddþ uM¯ uu S; ðA17Þ mh mH vcβ vcβ vsβ 2vA ’ ξ ¼ − cot βð1 þ cot2 βÞ: ðA26Þ where the Mf s are the diagonal mass matrices of the W;Z m2 fermions. To better compare with SM Higgs couplings, h we write these couplings as X m −L ¼ ξ f ffS:¯ ðA18Þ APPENDIX B: COUPLING TO TWO PHOTONS ffS f v f¼l;d;u To calculate the scalar coupling to two photons, we use expressions from Ref. [99], where the decay width for Then, using the expressions for the mixing parameters in Higgs to two photons is given in terms of generic -1, Eq. (A16), we find that the couplings of the scalar S to 1 spin- , and spin-0 particles in the loop. Although the fermions are 2 contribution to S → γγ is dominated by the effective   κ 2vAs2 2 2 coupling in the parameter region we are interested in, ξ ¼ − β β mh þ 2β 1 − mh ð Þ l;d 2 tan 2 cot 2 ; A19 we include all other possible particles in the loop for m 2m 2m 1 h H H completeness. In our case, there are only spin-1 and spin-2   2 2 particles in the loop, so the rate can be written as 2vAsβ m ξ ¼ − cot β 1 − h ð1 − cot2βÞ ; ðA20Þ u 2 2 2 α2 3 4π mh mH EMmS gSVV 2 ΓðS → γγÞ¼ κ þ N Q A1ðr Þ 1024π3 α m2 c;V V V where the couplings to down-type quarks and leptons are EM V 2g ¯ 2 enhanced by tan β and the couplings to up-type quarks are þ Sff 2 ð Þ ð Þ Nc;fQfA1=2 rf ; B1 suppressed by cot β. In the limit of large tan β, we may take mf β → π=2 and α → 0 so that sβ → 1 in the equations above, ¼ 4 2 2 1 and we can write the couplings purely in terms of tan β. where ri mi =mS. V and f represent spin-1 and spin-2 We can find the couplings of S to the weak gauge bosons particles, respectively; Q and Nc are the particle’s electric by expanding the kinetic terms of the two scalar doublets. charge and number of colors; and the expressions for A1 We find and A1=2 are given in Ref. [99].

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