Collider Prospects for Muon $g-2$ in General Two Higgs Doublet Model

Recent progress on muon $g-2$ measurement prompts one to take it even more seriously. In the general two Higgs doublet model that allows extra Yukawa couplings, we take a simplified approach of single enhanced coupling. We fix the charged lepton flavor violating coupling, $\rho_{\tau\mu} = \rho_{\mu\tau}$, via the one-loop mechanism, for illustrative masses of the heavy scalar $H$ and pseudoscalar $A$, where we assume $m_A = m_{H^+}$. Since extra top Yukawa couplings are plausibly the largest, we turn on $\rho_{tt}$ and find that LHC search for $gg \to H,\,A \to \tau\mu$ gives more stringent bound than from $\tau\to \mu\gamma$ with two-loop mechanism. Turning on a second extra top Yukawa coupling, $\rho_{tc}$, can loosen the bound on $\rho_{tt}$, but LHC constraints can again be more stringent than from $B \to D\mu\nu$ vs $De\nu$ universality. This means that evidence for $H,\,A \to \tau\mu$ may yet emerge with full LHC Run 2 data, while direct search for $\tau^\pm\mu^\mp bW^+$ or $t\bar cbW^+$ (plus conjugate) may also bear fruit.

The well-known 2HDM Model I and II invoke a Z 2 symmetry to implement the Natural Flavor Conservation (NFC) condition of Glashow and Weinberg [14], i.e. just one Yukawa matrix per quark charge (and for charged leptons as well). But this is "special", if not ad hoc, so in the general 2HDM (g2HDM) one drops the Z 2 symmetry and let Nature reveal her flavor design. First called Model III [15], and following the footsteps of the Cheng-Sher ansatz [16], indeed the emergent fermion mass mixing hierarchies can be exploited to ease the worries [14] of flavor changing neutral couplings (FCNC): extra Yukawa matrices should trickle off when going off-diagonal. The recent emergent alignment phenomenon, that the observed h boson at 125 GeV resembles very closely [17] the SM Higgs boson, brought in a flavor-independent surprise: alignment suppresses [18] FCNC involving the h boson. Nature's designs for flavor seem intricate.
The alignment control of FCNC is illustrated by h → τ µ search. The CMS experiment found initially [19] an intriguing 2σ hint, which subsequently disappeared [20]. The full Run 2 data at 13 TeV gives [21], But since the FCNC ρ τ µ arises from the heavy exotic doublet Φ ( Φ = 0) rather than the mass-giving doublet Φ (sole source of vacuum expectation value), the hτ µ couples as ρ τ µ c γ , where c γ ≡ cos γ is the h-H mixing angle between the two CP -even scalar bosons. Thus, alignment, that c γ or h-H mixing is small, can account for Eq. (3) without requiring ρ τ µ to be small, which is analogous [22] to another FCNC process, t → ch (with coupling ρ tc c γ ). This is the starting point for a oneloop mechanism (see Fig. 1(left)) to account for the muon g − 2 anomaly [9,10], originally stimulated by the CMS hint [19] for h → τ µ. Refs. [6,11]  Invoking the one-loop mechanism of Fig. 1(left) to account for muon g − 2 for m A, H at the weak scale implies rather large ρ τ µ [6][7][8][9][10][11], at several tens times λ τ ∼ = 0.01, the tau Yukawa coupling in SM. Eq. (3) then demands |c γ | 1, i.e. near the alignment limit. On one hand this calls for a symmetry, which we do not get into. On the other hand, one should turn to H, A → τ µ (and H + → τ + ν (µ) , µ + ν (τ ) ) search [23], as it is not hampered by small c γ but at full strength of s γ (≡ sin γ) → −1. This needs finite ρ tt for gluon-gluon fusion production, so let us articulate our approach.
With one Higgs doublet of the SM already fully affirmed, adding a second doublet is the most conservative and simple extension. But, while simple, without a Z 2 symmetry to enforce NFC, the g2HDM possesses many new parameters. We therefore take a simplified approach of one large extra Yukawa coupling at a time, especially if it is greatly enhanced compared with analogous SM couplings. By analogy with the known top Yukawa coupling, λ t ∼ = 1, however, it seems plausible that the extra top Yukawa coupling ρ tt is the strongest. Taking ρ τ τ O(λ τ ) to keep the one-loop effect small, the large ρ τ µ needed for muon g − 2 can induce τ → µγ with finite ρ tt [8] through the two-loop diagram of Fig. 1(right), which places a bound on ρ tt ρ τ µ , where the Belle experiment has recently updated [24] with full data. This constrains ρ tt to be considerably smaller than the ρ τ µ needed for muon g − 2. We will show that a recent search [25] for gg → H → τ µ by CMS with 36 fb −1 data at 13 TeV, when interpreted in g2HDM, would place bounds on ρ tt that are more stringent than from Belle.
To enlarge the allowed range for ρ tt , we turn on a second extra top Yukawa coupling, ρ tc , which can dilute [6] the branching ratio B(H, A → τ µ), thereby extend the allowed range for ρ tt . However, the product of ρ tc ρ τ µ can induce B → Dµν τ through H + exchange and affect the measured B → Dµν rate (as the ν τ flavor cannot be detected), thereby break universality [6] with B → Deν. Depending on the H + mass, ρ tc comparable in strength to ρ τ µ can be allowed. We will argue that searching for τ ± µ ∓ bW + or tcbW + at the LHC should be of interest.
The purpose of this paper is therefore threefold. First, g2HDM can [6-11] account for muon g−2 anomaly, which is not new. Second, possible astounding signatures may emerge soon, not just at Belle II, but also at the LHC, if large ρ τ µ is behind the muon g − 2 anomaly. Our two LHC contact points above imply that gg → H, A → τ µ may suddenly emerge, perhaps even with full Run 2 data, or detailed work may discover novel signatures such as τ ± µ ∓ bW + or tcbW + at the LHC. Third, this illustrates how limited our knowledge of g2HDM really is.
In Sec. II we discuss the one-loop mechanism and find the bound on ρ τ µ , then discuss flavor constraints on ρ tt and ρ tc that follow, as well as various flavor concerns; in Sec. III we compare τ → µγ constraint on ρ tt with direct search for gg → H, A → τ µ, and B → D ν universality ( = e, µ) constraint on ρ tc with τ ± µ ∓ bW + or tcbW + search at the LHC; after some discussion in Sec. IV, we offer our summary.
The Yukawa couplings in g2HDM are [26,27] − where i, j are summed over generations, L, R = (1∓γ 5 )/2 are projection operators, V is the CKM matrix, with lepton matrix taken as unity due to vanishing neutrino masses. One can therefore read off the ρ µτ , ρ τ µ and ρ tt couplings indicated in Fig. 1. We do not write down the Higgs potential V (Φ, Φ ) (except assuming it is CP conserving), as it can be found in many papers traced to Ref. [26]. We shall take m H = 300 GeV as benchmark to keep ρ τ µ "reasonable", and illustrate with m A = m H + = 340, 420, 500 GeV. That is, we assume custodial symmetry (namely η 4 = η 5 , in the notation of Ref. [18]) to reduce tension with oblique parameter constraints. We follow, for example, Ref. [28] and perform a parameter scan utilizing 2HDMC [29] to demonstrate that there is parameter space that satisfy perturbativity, unitarity and positivity as well as precision electroweak constraints, as shown in Fig As we will soon see, to provide 1σ solution to the muon g − 2 anomaly for benchmark m H , m A values, ρ τ µ = ρ µτ (which we take to simplify) need to be ∼ 20 times the strength of λ τ 0.01. Eq. (3) then implies So ρ τ µ ∼ 0.2 gives c γ 0.005, which is close to the alignment limit. In this paper we take the alignment limit of c γ → 0 and s γ → −1 to simplify. Setting c γ = 0 and using the well-known one-loop formula [6][7][8][9][10][11], we plot ∆a µ vs ρ τ µ = ρ µτ for m H = 300 GeV and ∆m = m A − m H = 10, 40, 120, 200 GeV in Fig. 3, together with the 1σ and 2σ ranges from Eq. (2). We note that A and H exactly cancel one another in ∆a µ when degenerate, which is illustrated by ∆m = 10 GeV. Such near degenerate masses would require extremely large ρ τ µ values. With m H = 300 GeV as benchmark, the three values of m A = 340, 420, 500 GeV are taken for illustration. One sees that, as m A increases toward decoupling, H starts to dominate and the ρ τ µ strength needed for 1σ solution decreases, namely ρ τ µ 0.289, 0.192, 0.167, (6) which are ∼ 20 times larger than λ τ 0.01. If one replaces the muon in the left of Fig. 1(left) by a tau, a finite ρ τ τ can induce τ → µγ. We have checked that for ρ τ τ = O(λ τ ) or smaller [30], the one-loop chiral suppression gives a rate that is quite below existing bounds [20]. It is known [8] that a two-loop mechanism can in fact dominate for appropriate ρ tt values in the fermion loop, as seen from Fig. 1(right). The mechanism is similar to µ → eγ [31], from which the formulas are adapted. Using the known formulas and 1σ solution values of Eq. (6), and assuming ρ τ τ ∼ 0, one finds |ρ tt ρ τ µ | < 0.014, 0.016, 0.017 for m H = 300 GeV and m A = 340, 420, 500 GeV, or |ρ tt | 0.05, 0.08, 0.1, which are rather small compared with λ t ∼ = 1. With ρ tt considerably smaller than 1, it is generally easier to accommodate flavor constraints, especially in the quark sector [32,33], and we do not explore it further. As we will show in the next section, search for gg → H, A → τ µ at the LHC turns out to provide more stringent bound on ρ tt than Eq. (7). One thought [6] then is to dilute B(H, A → τ µ) with a second extra top Yukawa coupling, namely ρ tc . However, B → Dµν vs B → Deν universality provides a bound [6] on ρ tc ρ τ µ through H + exchange, where the H +c b and H + µν τ couplings can be read off Eq. (4). The not so intuitive point is that the former is V tb ρ tc [34] and not CKM suppressed! Thus, ρ tc ρ τ µ would generate B → Dµν τ , which would add incoherently [11] to B → Dµν that is measured by experiment rather precisely, but cannot discern the neutrino flavor. From Eq. (6) and using Eq. (84) of Ref. [11], we compare with the Belle measurement [35], Adding errors in quadrature and allowing 2σ range, we find |ρ tc ρ τ µ | 0.024, 0.037, 0.053 for m H = 300 GeV and m H + = m A = 340, 420, 500 GeV, giving |ρ tc | 0.08, 0.19, 0.32.
The decreasing value of ρ τ µ with increasing m A , viz. Eq. (6), implies the opposite for ρ tc , which for m A 400 GeV can surpass the ρ τ µ strength needed for 1σ solution of muon g − 2.
In fact, a good part of the dilution effect for B(H, A → τ µ) is driven by the opening of A → HZ decay for m A 400 GeV, which is one reason why we increased ∆m = m A − m H > m Z . With m A = m H + , this means H + → HW + also opens up. As we shall see in the next section, a sizable ρ tc can give rise to cg → bH + production [34] (again, thecbH + coupling is not CKM-suppressed), as well as [36] cg → tH, tA, with cg → tA suppressed for higher m A . H + → HW + decay with H → tc, τ µ would lead to additional signatures at the LHC.
The coupling ρ µτ (ρ τ µ ) enters the H +ν µ τ (H +ν τ µ) coupling directly, and can generate τ → µν µντ , where the neutrino flavors are swapped compared with W boson exchange, which the experiment cannot distinguish. Compared with τ (µ) → eνν, this constitutes another test of lepton universality violation at the per mille level, which has been recorded by HFLAV [40]. We have checked that ρ τ µ = ρ µτ 0.9 is still allowed for our benchmark masses, which is more accommodating than muon g − 2.
Finally, the ρ τ µ = ρ µτ coupling affects Z → τ τ, µµ decays, which have been precisely measured [20], again at the per mille level. We find the bound to be weaker than the bounds from τ decays discussed here. All other ρ f ij s are set to zero. Bounds from τ → µγ [24] and CMS search [25] for gg → H → τ µ (reinterpreted in case of A) are indicated.

III. FLAVOR PROBES VS LHC SEARCH
The connection between muon g − 2 (or τ → µγ) with H, A → τ µ search [6][7][8][9][10][11] at hadron colliders is well known. In part because of the τ → µγ constraint on ρ tt , Ref. [6] did not explore gg → H, A → τ µ but emphasized the electroweak pair production of exotic scalar boson pairs instead. However, we are interested in keeping ρ tt as CMS has recently performed a search for heavy H → τ µ (and τ e) channel. Using ∼ 36 fb −1 data at 13 TeV and assuming narrow width, the bound [25] on σ(gg → H)B(H → τ µ) ranges from 51.9 fb to 1.6 fb for m H ranging from 200 GeV to 900 GeV. The H notation is intentional: CMS followed Ref. [41] and studied the scalar H case, but not for pseudoscalar A.
It is of interest to check the current LHC bound on ρ tt within the large ρ τ µ = ρ µτ interpretation of muon g − 2. We evaluate the pp → H, A → τ µ + X cross section for LHC at 13 TeV, assuming gg → H, A as the dominant subprocess and convolute with parton distribution functions. Taking ρ tt as real for our benchmark m H and m A values, we fold in the decay branching ratios B(H, A → τ µ) and give our results in Fig. 4 as blue (red) solid curves for H (A), with the corresponding CMS bound given as horizontal lines for our three benchmark sets. The bounds from τ → µγ, Eq. (7) are given as (dark pink) vertical lines. We have checked that Γ H ∼ 1 GeV, while Γ A < 2 GeV for m A < 400 GeV, and remains below 10 GeV for m A ∼ 500 GeV, hence satisfy the narrow width approximation. We do not combine the two separate states.
We see that, with only a subset of Run 2 data, the CMS bound is more stringent than the bound from τ → µγ, as one should always take the more stringent bound out of H vs A. For the cases of m A = 340, 420 GeV, the larger production cross sections for A imply more stringent limit than from H, but for m A = 500 GeV, parton densities have dropped too low, and the bound from H is more stringent than A, but still more stringent than from τ → µγ. We remark that the case of m H , m A = 300, 340 GeV, i.e. Fig. 4(left), does provide a 1σ solution to muon g − 2. But if all other extra Yukawa couplings, ρ f ij s (f = u, d, ), are not stronger than λ max(i, j) , the Yukawa strength in SM, then there are little other consequences, except in the gg → A → τ µ probe of ρ tt itself. The reinterpreted CMS bound of ρ tt 0.02 from Fig. 4(left) is below ρ tt 0.05 allowed by τ → µγ, which is really small compared with λ t ∼ = 1. But there is still discovery potential with full Run 2 data. The allowed range for ρ tt from CMS gets partially restored for heavier m A , where gg → H, A → τ µ production becomes predominantly through H in Fig. 4(right). The easing of the bound on ρ tt is due to the opening of A → tt, HZ, as can be seen from the branching ratio plot, Fig. 5(left).
In addition to ρ tt needed for τ → µγ and gg → H, A production, we give the effect of turning on ρ tc , the second extra top Yukawa coupling, in Fig. 5(left), as it can dilute the gg → H, A → τ µ cross section. However, as discussed in the previous section, the product of ρ tc ρ τ µ can make B → Dµν deviate from B → Deµ and break e-µ universality. From the 1σ solution to muon g − 2, Eq. (6), for our benchmark m H = 300 GeV, m A = 340, 420, 500 GeV, Belle data [35] therefore leads to the bound on ρ tc , giving Eq. (8). Taking ρ tc 0.08, 0.19, 0.32, and the ρ tc -induced dilution for the three corresponding m A values in Fig. 5(left), we draw the dashed curves in Fig. 4. Not much is changed in Fig. 4(left), but for Fig. 4(center) and Fig. 4(right), the gg → H, A → τ µ cross sec- tion weakens progressively. It is less significant for A production at m A = 500 GeV in Fig. 4(right), because A → HZ is much stronger. But the corresponding drop in gg → H → τ µ is rather significant. This is because the ρ tc 0.32 value is about twice as large as the ρ τ µ value of 0.167. The large ρ tc brings about interest, thereby possible constraints, from production processes at the LHC.
Before discussing ρ tc -induced processes, we remark that Fig. 5, where we also give H + decay branching fractions, is not just for our three benchmark values of m A = 340, 420, 500 GeV. The plot is made by the same approach: for m H = 300 GeV and a given m A , we find the 1σ muon g − 2 solution value for ρ τ µ analogous to Eq. (6), then find the allowed upper bounds for ρ tt and ρ tc , analogous to Eqs. (7) and (9), respectively. The plot therefore scans through m A , with m H fixed at 300 GeV.
Having ρ tc alone with ρ tt small, it can generate [36] cg → tH, tA → ttc or same-sign top plus jet final state, which can [42] feed the ttW control region (CRW) of 4t search by CMS (by different selection cuts, AT-LAS is less stringent), where there is now a full Run 2 data study [43]. More significant is a study [28] of cg → bH + → bAW + production followed by A → tc and both the top and the W + decay (semi-)leptonically, which can also feed the CRW of CMS 4t search. In this study, H + → AW + decay was treated with m H + = m H , i.e. swapping H ↔ A from our present case. The bH + cross section is sizable due to a light b quark and no CKM suppression for thecbH + coupling [34]. It was found that ρ tc ∼ 0.15 is barely allowed for m H + ∼ 500 GeV. This may seem to make the large values of ρ tc for our dilution effect untenable.
It is the m A = m H + = 500 GeV benchmark that CRW of CMS 4t may put a limit on ρ tc : H + → cb is slightly below 30%, while τ ν (µ) + µν (τ ) is no longer a dilution factor. This is due to the dominance of H + → HW + , now at 70%. With H → tc at ∼ 35%, it would feed 4t CRW abundantly, as pointed out in Ref. [28] for the analogous twisted custodial case for bAW + production. If we take the number there as a guesstimate, then ρ tc ∼ 0.15 would be borderline. This just means that ρ tc should be considerably less than 0.32 from Eq. (8), which means that one cannot reach ρ tt ∼ 0.1 as allowed by τ → µγ.
We note that, from Fig. 2 and for m H ∼ 300 GeV, m A = m H + at 500 GeV is already at the "border" of the scan space. Thus, we conclude that for large splitting of m A = m H + from m H , the solution to muon g − 2 would not allow a ρ tc value larger than ρ τ µ for the oneloop solution. As one moves m H higher, the discussion should be similar for higher m A = m H + , requiring larger ρ τ µ values to solve muon g − 2.

IV. DISCUSSION AND CONCLUSION
Referring to the 4.2σ "white space of disagreement" between theory and experiment, the question "What monsters may be lurking there?" from the April 7 announcement presentation by the Muon g−2 experiment, became the quote of the day. We stress that ρ τ µ /λ τ ∼ 20, while large, is not "monstrous", and is up to Nature to choose (and the counterbalancing ρ tt /λ t 1/10 as well). This can be compared, for example, with tan β = O(10 3 ) in the muon-specific 2HDM [44], tailor-made for the muon g − 2 by some Z 4 symmetry. In a different Z 4 arrangement, the µ-τ -philic extra scalar doublet [45] was introduced with only the ρ τ µ = ρ µτ coupling, with further variations of [46] m A − m H < 50 GeV by considering the ∼ 2σ deviation from SM in τ decays, or pushing m H to be very light [47]. We hope we have illustrated that such elaborations are not necessary, that the g2HDM is versatile enough with its arsenal of extra Yukawa couplings, to provide solution to muon g − 2 while hiding from view so far, but with promising LHC implications.
With enhanced ρ τ µ implied by muon g − 2, it has been suggested [48,49] that, by the natural complexity of extra Yukawa couplings, they could possibly drive electroweak baryogenesis (EWBG), i.e. be the CP violating source of the baryon asymmetry of the Universe. In fact, part of our motivation for considering ρ tt and ρ tc are their potential as drivers [50] for EWBG. Our findings show that ρ tt may still suffice, but ρ tc may not be strong enough. But whether one deals with large ρ τ µ or small but finite ρ tt as the EWBG driver, one may have to face the issue of the alignment limit, or extremely small h-H mixing. We note that, by the arguments we have given, the simultaneously large ρ τ µ , ρ τ τ and ρ tt as given in Ref. [48] cannot survive collider bounds such as from CMS search for H, A → τ µ [25], and ATLAS [51] and CMS [52] searches for H, A → τ τ , which we have checked explicitly.
We note in passing that g2HDM has been invoked in weaker (considered as a perturbation hence keeping tan β) form [53], where having rather small ρ τ µ called for relatively light m A . In a similar vein, a reinterpretation [54] of the CMS gg → H → τ µ search [25] also urged for probing below m H = 200 GeV. These suggestions are not invalid, but we chose m H within the search range of CMS at a typical weak scale to illustrate how g2HDM may resolve muon g − 2.
The flavor conserving coupling ρ µµ can in principle induce new contributions to muon g − 2 via one-and twoloop diagrams. The one-loop effect [55] involves ρ µµ only, but because of chiral suppression from muon in the loop, exorbitantly large ρ µµ 0.85 (∼ 1400λ µ : monstrous!) is needed for 1σ solution to muon g − 2 for the same benchmark masses as before. The two-loop effect [56,57], by replacing the τ on the left of Fig. 1(right) with µ, can be enhanced by extra top Yukawa coupling ρ tt . But it again calls for the product of ρ tt ρ µµ to be too large. Taking, for example, ρ tt ∼ −0.1 for m H = 300 GeV, m A = m H + = 340 GeV, one can achieve 1σ solution to muon g−2 for ρ µµ 0.42, which is again rather large [58]. Indeed, this runs into conflict with the CMS search [39] (the ATLAS bound [59] at similar data size is weaker) for gg → H, A → µ + µ − with ∼ 36 fb −1 , which restricts ρ tt 10 −2 for such a strong ρ µµ value. This shows, by power of the CMS bound [39] on gg → H, A → µ + µ − even with just a fraction of LHC Run 2 data, that the two-loop ρ µµ mechanism is not viable.
Finally, let us comment on the electroweak production processes suggested in Ref. [6], namely pp → HA, H ± H, H ± A, H + H − through vector bosons. For large ρ τ µ values needed for muon g − 2, dileptonic decays of H, A, H ± may prevail, which allow exquisite reconstruction [6] of exotic extra Higgs boson masses, hence rather attractive. However, the electroweak production cross section, at the fb level, is rather small. Take the case of m H = 300 GeV, m A = m H + = 340 GeV for example. If we allow the lowest ρ tc value of 0.08 in Eq. (9), nominally from B → D ν universality constraint, we find that the cg → bH + cross section, being a strong production process involving a gluon (with not too significant a suppression from charm parton distributions) and only one heavy particle in final state, is almost two orders of magnitude higher than the electroweak production processes. This effect persists for the other two benchmarks where A and H + are heavier, which allow for larger ρ tc values. Furthermore, as we have illustrated, multiple decay modes of the heavier A and H + (see Fig. 5), as well as H → τ ± µ ∓ , tc (tc) would dilute one another, which works also for the decays of the extra scalar bosons in electroweak pair production. Thus, neither the search for the electroweak production of extra Higgs boson pairs, nor the reconstruction of the extra boson masses, may be as rosy as argued in Ref. [6].
In summary, we employ large CLFV Yukawa coupling ρ τ µ = ρ µτ ∼ 0.2 in the general 2HDM to account for muon g − 2 anomaly through the known one-loop mechanism. The extra Higgs bosons have mass at the weak scale, but one is close to the alignment limit of very small h-H mixing to evade h → τ µ. This motivates the check with gg → H, A → τ µ search, where a subset of LHC Run 2 data already puts more stringent bound on the extra top Yukawa coupling ρ tt than from τ → µγ through the two-loop mechanism. The stringent constraint on ρ tt can be eased by allowing a second extra top Yukawa coupling, ρ tc , which motivates the search for cg → bH + → τ ± µ ∓ bW + , tcbW + (same-sign dilepton plus two b-jets and additional jet, with missing p T ) at the LHC, on top of possible same-sign top plus jet signatures from cg → tH, tA → ttc. Whether these extra Yukawa couplings can drive electroweak baryogenesis should be further studied.