LHC $\tau$-rich Tests of Lepton-specific 2HDM for $(g-2)_\mu$

The lepton-sepcific (or type X) 2HDM (L2HDM) is an attractive new physics candidate explaining the muon $g-2$ anomaly requiring a light CP-odd boson $A$ and large $\tan\beta$. This scenario leads to $\tau$-rich signatures, such as $3\tau$, $4\tau$ and $4\tau+W/Z$, which can be readily accessible at the LHC. We first study the whole L2HDM parameter space to identify allowed regions of extra Higgs boson masses as well as two couplings $\lambda_{hAA}$ and $\xi_h^l$ which determine the 125 GeV Higgs boson decays $h\to \tau^+\tau^-$ and $h\to AA/AA^*(\tau^+\tau^-)$, respectively. This motivates us to set up two regions of interest: (A) $m_A \ll m_{H} \sim m_{H^\pm}$, and (B) $m_A \sim m_{H^\pm} \sim {\cal O}(100) \mbox{GeV} \ll m_H$, for which derive the current constraints by adopting the chargino-neutralino search at the LHC8, and then analyze the LHC14 prospects by implementing $\tau$-tagging algorithm. A correlated study of the upcoming precision determination of the 125 GeV Higgs boson decay properties as well as the observation of multi-tau events at the next runs of LHC will be able to shed light on the L2HDM option for the muon $g-2$.


I. INTRODUCTION
The muon g − 2 anomaly has been a long standing puzzle since the announcement by the E821 experiment in 2001 [1]. During the past 15 years, development in both experimental and theoretical sides has been made to reduce the uncertainties by a factor of two or so, and thus establish a consistent 3σ discrepancy. Although not significant enough, it could be a sign of new physics beyond the Standard Model (SM). Since the first announcement of the muon g − 2 anomaly, quite a few studies have been made in the context of two-Higgs-doublets models (2HDMs) [2][3][4][5]. Recently, it was realized that the "lepton-specific" (or "type X") 2HDM (L2HDM) * with a light CP-odd Higgs boson A and large tan β is a promising candidate accommodating a large muon g − 2 while escaping all the existing theoretical and experimental constraints [7]. Some of the following studies showed that the allowed parameter space is further resrticted, in particular, by the consideration of the 125 GeV Higgs boson decay to light CP-odd Higgs bosons h → AA [8], and the tau decay τ → µνν combined with the lepton universality conditions [9].
In this paper, we attempt to make a thorough study of the whole L2HDM parameter space in favor of the muon g − 2 explanation, and analyze the LHC tests of the favoured parameter space leading to τ -rich signatures like 3τ , 4τ and 4τ + W/Z. First, we show how the SM Higgs exotic decays h → AA as well as h → AA * (τ + τ − ) constrain the parameter space. It is connected to the determination of the allowed ranges of the normalized tau (lepton) Yukawa coupling in the right-or wrong-sign domain, and thus more precise measurement of the 125 GeV Higgs boson properties will put stronger bounds on the L2HDM parameter space. As we will see, the hAA coupling can be made arbitrarily small by a cancellation for m H m A only in the wrong-sign limit of the tau Yukawa coupling [10], and it opens up the region of m A < m h /2 [8]. In the region of m A > m h /2, the three-body decay h → Aτ + τ − should be suppressed and the SM (right-sign) limit of the tau Yukawa coupling is allowed for m A 70 GeV. The allowed parameter space is further restircited by the lepton universality tests of HFAG which measures the leptonic decay processes at the level of 0.1% [11]. For this, we improve the analysis of [9] to single out proper constraints on the tree and loop contributions to the tau decay. useful formulas, and explain why a large (g−2) µ is easily accommodated with a light CP-odd Higgs boson A and large tan β. In Section III, we summarize all the relevant theoretical and experimental constraints, and quote some of the latest results which are not included in our analysis. By using the profile likelihood method, we identify the allowed L2HDM parameter regions under these constraints and show them at 68% and 95% confidence level. In Section IV, we discuss τ -rich signatures at the LHC expected in the identified parameter regions. We analyze the 3τ events to identify the parameter regions excluded already by the current LHC 8 TeV data. In addition, we show the prospect for the future LHC14 run with a dedicated simulation. We conclude in Section V.

II. 2HDM WITH A LEPTON-SPECIFIC DOUBLET (L2HDM)
Let us first introduce the L2HDM to present useful formulas for our analysis heavily relying on the paper by Gunion and Haber [12]. Among various types of 2HDMs classified by the Yukawa coupling patterns of the two Higgs doublets Φ 1,2 with the same SM quantum numbers, the L2HDM allows the following Yukawa couplings: where family indices have been omitted and Φ 2 = iσ 2 Φ * 2 . This pattern may be a result of a discrete Z 2 [13]: Φ 2 → Φ 2 and Φ 1 → −Φ 1 combined with e R → −e R while the other fermions are invariant under the Z 2 transformation. The most general form of the 2HDM scalar potential is given by The Z 2 symmetry enforces λ 6 = λ 7 = 0. However, the m 2 12 term that softly breaks Z 2 should be allowed. All couplings are assumed to be real. In the desired vacuum, both doublets acquire VEVs, denoted as v 1 and v 2 for Φ 1 and Φ 2 , respectively. Large VEV hierarchy, i.e., tan β ≡ v 2 /v 1 1, is of our interest for the explanation of the muon g − 2.
By decomposing the doublets as 2) T , we see the model has three mass squared matrices of A i , H ± i and h i , which can be diagonalized by two angles α and β. The physical Higgs particles in mass eigenstates are given by where s α and s β are abbreviations for sin α and sin β, etc. In this paper, we adopt the convention 0 < β < π/2 and −π/2 ≤ β − α ≤ π/2. Then the SM-like Higgs boson is h ≈ c α h 2 with either positive or negative sign for c α . In the very large tan β limit, two Higgs doublets are almost decoupled. But some degree of non-decoupling effects, encoded in 0 ≤ c β−α 1, will play very important roles in our study.
The mass spectrum can be calculated analytically in terms of the coupling constants in the Higgs potential, but practically it is more convenient to take masses as inputs and inversely express coupling constants with them: One can see that we require an intolerably large λ 1 ≈ tan 2 βm 2 H /v 2 O(10 4 ) in the large tan β region if m 2 12 = 0. Thus, the soft Z 2 breaking term m 12 needs to be non-vanishing, and it is determined to be m 2 12 ≈ m 2 H / tan β. The mass splittings among the extra Higgs bosons are controlled by two parameters λ 4,5 : Immediately, we need λ 5 ≈ −λ 4 ∼ O(1) to get the favored mass pattern m A m H m H ± by Electroweak precision test constraints. In addition, from Eq. (4) we know that in the large tan β limit we determine λ 2 ≈ m 2 h /v 2 ≈ 0.26, just as in SM. In general, the Yukawa couplings of the five physical Higgs bosons, h, H, A and H ± in the 2HDM are given by where f runs over all of the quarks and charged leptons, and furthermore u, d, and l refer to the up-type quarks (u, c, t), down-type quarks (d, s, b), and charged leptons (e, µ, τ ), respectively. Specified to the L2HDM, we have In any type of the 2HDM, the Higgs-to-gauge boson couplings read: where V refers to Z and W ± gauge bosons. For very large value of tan β, we have |ξ u,d H | |ξ u,d A | = cot β and |ξ l H | |ξ l A | = tan β, in short, the quark Yukawa couplings of H and A are highly suppressed while the lepton Yukawa couplings of H and A are highly enhanced. This feature helps to shed a light on the muon g − 2 problem while evading various experimental constraints.

III. CONSTRAINTS ON L2HDM PARAMETERS
In this section we first describe all the relevant theoretical and experimental constraints on the L2HDM parameter space. Based on these constraints we present our results in 2dimensional profile likelihood maps. The 68% (95%) contours will be presented in dark (light) green in all the likelihood maps.
A. Enhanced (g − 2) µ with large tan β and light A Recent progress in determining the muon anomalous magnetic moment a µ = (g −2) µ /2 establishes a 3σ discrepancy: which is in a good agreement with various group's determinations [7]. Such an excess can obviously be attributed to a new physics contribution. In the framework of 2HDMs, the Barr-Zee 2-loop correction with a light A and τ running in the loop [14] can generate a large positive ∆a µ due to an enhancement factor of |ξ l A | 2 (m τ /m µ ) 2 in the large tan β limit. Let us note that the Barr-Zee diagram with H running in the loop gives a negative contribution to ∆a µ and thus a heavier H is preferred to enhance ∆a µ . For more details, we refer the readers to Ref. [7].

B. Theoretical constraints
There are several theoretical constraints; the perturbativity, vacuum stability and unitarity bounds to be considered. All of them are implemented at the weak scale. In particular, the first imposes the highest mass scale for the Higgs states. † Alternative option is the public Mathematica code [16].
An immediate consequence of this bound can be obtained from Eq. (5): saturated for λ 5 −λ 4 = 4π. Assuming a small contribution from m A , it gives the upper bound m H + ∼ m H 900 GeV. Note that with the large tan β approximation, λ 1 becomes an independent parameter and its magnitude is in principle allowed to run within 4π by perturbativity.
• Vacuum stability demands The last condition can be rewritten as One of the key features in our discussion is that the couplings and thus the upper limits on the heavy Higgs masses show quite different behaviors in the right-sign (SM) and wrong-sign limit of the normalized Yukawa coupling ξ l h [10]. Using a trigonometric identity, ξ l h can be expressed by As found at the LHC, the 125 GeV Higgs boson h is very much SM-like requiring, in particular, |s β−α | 1 and |ξ τ h | ≈ 1. Notice that this can be reached in the SM limit t β c β−α ≈ 0 (leading to the right-sign lepton coupling ξ l h ≈ +1), or in the large tan β limit with t β c β−α ≈ 2 (leading to the wrong-sign couplig ξ l h ≈ −1). Using the relation (11), one finds in the large tan β limit. Now, in the right-sign limit (ξ l h s β−α → +1), we have which puts a bound m H < 250 GeV for m A = 0, which is consistent with [7]. On the other hand, in the wrong-sign limit (ξ l h s β−α → −1), m H can be arbitrarily large allowing a fine-tunnig s 2 β−α + ξ l h s β−α ≈ 0. These properties will be clearly shown in our Figs. 2 and 3.
• Tree-level unitarity for the scattering of Higgs bosons and the longitudinal parts of the EW gauge bosons.
The numerical evaluation of the necessary and sufficient conditions for the treelevel unitarity in the general 2HDM has been encoded by the open-source program 2HDMC [15]. We deal with this constraints relying on it. Here, we point out that this constraint is rather loose in the following reason. In the limit of large tan β, the parameter λ 1 decouples from the other parameters λ 2,3,4,5 , and is allowed to run between 0 and 4π independently. Therefore, one can always track down a value of λ 1 to meet the requirement of the tree-level unitarity without affecting any other physical observables significantly.

C. Electroweak precision test
Electroweak precision test (EWPT), commonly referred to as the ρ parameter bound, is taken into account by calculating the oblique parameters, S, T and U in the 2HDMC code. As we are interested in a splitting spectrum of A and H, H ± , the custodial symmetry is potentially violated significantly. However, as analyzed in detail in Ref. [7], taking the SM limit s β−α → 1, the custodial symmetry can be restored if m H ± ≈ m H (m A ) for arbitrary value of m A (m H ) [17]. In our scan study, we reproduce the previous results as clearly demonstrated in Fig. 2. Let us remark that we have updated the central values, error bars, and correction matrix adopted in Ref. [7], using the new PDG data [18].

D.
Light A and Higgs exotic decay As we are interested in the case of a light CP-odd scalar A, the SM Higgs boson can have an exotic decay of (i) h → AA for At the moment, the current LHC data on the SM Higgs boson put a strong constraint on the hAA coupling λ hAA and m A . On the other hand, it will be an interesting channel to test the hypothesis of the L2HDM explaining the muon g − 2 at the next runs of the LHC. The partial decay widths of these processes are where The function G(x) is a very fast monotonically decreasing function with respect to x. For instance, we have G(0.3) ≈ 0.28 to be compared with G(0.5) ≈ 0.0048.
Generically, λ hAA is expected to be around the weak scale hence leading to a large decay width at the GeV scale, which is readily excluded. To avoid this situation, one may require ‡ In type-I and type-II 2HDM, Ref. [19] studied the possibility of two-body decay mode h → AA while the three-body decay mode was ignored. m A > m h /2 or arrange a mild cancelation to get sufficiently small λ hAA . Interestingly, one can find where λ 3 + λ 4 − λ 5 is given in Eq. (12). This relation says that there could be a cancellation among three contributions from m A , m h and m H . In particular, for m H m h,A of our interest, the cancellation is obtained only in the wrong-sign limit with ξ l h −1. This can be explicitly seen by taking λ hAA as a free parameter (traded with λ 1 ) and expressing the normalized tau (lepton) coupling as In the limit of m H m A and λ hAA → 0, it can be further approximated h ) −1, and thus we have ξ l h −1. § We demonstrate this behavior in the right panel of Fig. 3.
The presence of a light A may leave hints at Higgs exotic decay through the channel h → AA(A * ) →4τ . The upper bound of the exotic branching ratio of the Higgs decay is known to be 60 %, however, a mildly more stringent bound on the h → AA mode using multilepton searches by CMS [20] can be set: Br(h → AA → 4τ ) 20% almost independent on m A [21]. In this paper we impose a conservative cut Br(h → AA(A * )) 40%.

E. Collider and other constraints
• Collider searches on the SM and exotic Higgs bosons For various Higgs constraints from LEP, Tevatron and LHC, we rely on the package HiggsBounds-4.2.0 [22] incorporating the most updated data on BR(h → τ τ ). We notice that the DELPHI search [23] on the process is sensitive to our model. The Fig. 15 in the Ref. [23] shows the region m A + m H 185 GeV is excluded at 95% confidence level.
Specific to our study, the 125 GeV Higgs decay h → τ + τ − is of particular concern as it can deviate significantly from ±1 as indicated in Eq. (17). We use the new data from CMS [24] and ATLAS [25], weighted by their statistic error bars: The case with s β−α ≈ −1 (or equivalently cos α ≈ −1), i.e., the reversed couplings of other SM couplings, is completely excluded from our numerical results. So, we have s β−α ≈ +1 in this paper.
In our analysis, we do not include this constraint as it is irrelevant for m A > 15 GeV. More details can be found in Refs. [8,9].
• τ decays and lepton universality In the limit of light H ± and large tan β, the charged Higgs boson can generate significant corrections to τ decays at tree and 1-loop level [26]. Recent study [9] attempted to put a stringent bound on the charged Higgs contributions from the lepton universality bounds obtained by the HFAG collaboration [11]. Given the precision at the level of 0.1 %, the HFAG data turned out to provide most stringent bound on the L2HDM parameter space in favor of the muon g − 2. Thus, it needs to be considered more seriously. For this, we improve the previous analysis treating the HFAG data in a proper way.
From the measurements of the pure leptonic processes, τ → µνν, τ → eνν and µ → eνν, HFAG obtained the constraints on the three coupling ratios, (g τ /g µ ) = Γ(τ → eνν)/Γ(µ → eνν), etc. Defining δ ll ≡ (g l /g l ) − 1, let us rewrite the data: In addition, combing the semi-hadronic processes π/K → µν, HFAG also provided the averaged constraint on (g τ /g µ ) which is translated into We will impose the above lepton universality constraints in our parmeter space. Now, it is important to notice that only two ratios out of three leptonic measurements are independent and thus they are strongly correlated as represented by the correlation coefficients [11]. Therefore, one combination of the three data has to be projected out.
One can easily check that the direction δ l τ µ − δ l τ e + δ l µe has the zero best-fit value and the zero eigenvalue of the covariance matrix, and thus corresponds to the unphysical direction. Furthermore, two orthogonal directions δ l τ µ + δ l τ e and −δ l τ µ + δ l τ e + 2δ l µe are found to be uncorrelated in a good approximation. In the L2HDM, the deviations from the SM prediction δ ll are calculated to be Here δ tree and δ loop are given by [26]: (20,21) and (22), one obtains the following three independent bounds: Based on the constraints Eq. (24) on the two fundamental free parameters δ tree and δ loop , we can draw the the lepton universality likelihood contours, where we found the minimum value χ 2 min = 0.229. In Fig. 1, we present profile likelihood contours on the m H ± -tan β plane the red, blue, and black lines are corresponding to 99%, 95%, and 90% confidence level, respectively. Note that the δ loop is always negative in the region of our interest listed in Table I. On the other hand, δ tree depends only on the parameter tan β/m H ± and negative in most of the region but can be also positive. In a fine-tuned region located tan β/m H ± ∼ 1 GeV −1 as we can see in the large tan β and small m H ± corner in Fig. 1, where the positive δ tree and the negative δ loop cancel.
We also found that lepton universality likelihood is practically not sensitive to the heavy neutral Higgs mass m H and cos(β − α) in our region of interest. Hence, we show the lepton universality contours only on the m H ± -tan β plane (Fig. 1) and on the m A -tan β plane (Fig. 4 left panel).
Let us finally remark that we use Gaussian distribution or hard cut for the likelihood functions to impose the experimental constraints. When the central values, experimental errors and/or theoretical errors are available, Gaussian likelihood is used. Otherwise the hard cut is adopted such as the Higgs limits implemented in HiggsBounds.

F. Results
Our input parameters and the scan ranges of them are summarized in Table I. Some comments are in order. (i) We focus on the case that the SM-like Higgs boson h is the lighter CP-even Higgs boson with mass 125 GeV [27]. ¶ (ii) We require cos(α − β) ≤ 0.1, which guarantees that h couples to quarks and vector bosons without appreciable deviation from the SM predictions. The updated LHC results can be found in Ref. [28]. (iii) The upper bound on m H,H ± < 400 GeV is put by hand since we are interested in the relatively light region testable at the LHC near future. In principle, they can be as heavy as about 900 GeV according to the perturbativity constraints. (iv) We restrict ourselves to tan β ≤ 150.
We show the scan results in several 2 dimensional profile likelihood maps from Fig. 2 to Fig. 4. The inner green (outer light green) contours are 68% (95%) confidence region. The points are summarized in the following: • The left panel of Fig. 2  of the right panel of Fig. 2, one finds a mild degeneracy between A and H ± with m A ≈ 100 − 180 GeV and m H ± 160 GeV. For m A > 100 GeV, tan β needs to be larger than about 70, see Fig. 4. We call the former region as Region A and the latter as Region B. Note that the fragmentation of the plots, particularly in the region B of the left panel of Fig. 2, is due to a coarse-tuning likelihood. As we will see in the next section, Region B is already excluded by the current LHC 8TeV data.
• The left panel of Fig. 3 shows the relation between λ hAA and m A . We see only |λ hAA | ∼ 0 is allowed for m A 60 GeV, while larger |λ hAA | is allowed for m A 60 GeV. The right panel of Fig. 3 shows the relation between ξ τ h vs. m A . In the region m A 70 GeV, only the wrong-sign region (ξ l h < 0) is allowed. It is consistent with suppressed λ hAA seen in the left panel as discussed in Eq. (17). For heavier A, there appears the right-sign region.
• Remarkably, the m A 60 GeV region tends to show an enhancement in Br(h → τ τ ), up to a factor |ξ l h | 2 ∼ 4. While above it both (mild) enhancement and suppression are possible. Further precise measurement of Br(h → τ τ ) helps to shrink the allowed parameter regions.
• In the left panel of Fig. 4, The contours of lepton universality likelihood are also presented in 99% (red), 95% (blue), and 90% (black) confidence limit. The region with tan β < 140 with small m A allowed by other constraints are very constrained by lepton universality. However, the region located at the large tan β > 140 are always allowed by the fine-tuning cancellation between δ tree and δ loop by selecting an  appropriate m H ± . The lower tan β region allowed at 95% appears to be a consistent combination of the same 95% contour lines with different values of m H ± in [9].
• A light A with m A ∼ 20 − 63 GeV is of our particular interest. * * In this region, the wrong-sign limit (ξ l h ∼ −1) has to be realized and thus the lower bound on tan β is correlated with the upper bound on cos(α − β), which can be seen from the right panel of Fig. 4. We can also see that the two discrete regions correspond to the right-sign limit (tan β cos(β − α) 0) and wrong-sign limit (tan β cos(β − α) 2) as described around Eq. (11).
• The exotic Higgs decay h → AA or h → Aτ τ is a promising channel to probe the L2HDM explanation of the muon g − 2 as its branching ratio can be quite sizable unless there is a particular reason to suppress λ hAA as shown in Fig. 5.

IV. τ −RICH SIGNATURE AT LHC
In the previous section, we identified two favored regions of the L2HDM parameter space. In this section, we discuss how the current LHC search results can constrain this model further. Since the relationship between m A and tan β is constrained by the (g − 2) µ , * * Remark again this region is further reduced by considering the tau decay and lepton universality data [9].  as shown in the left panel of Fig. 4, we can simply parametrize tan β as a function of m A : which will be assumed in this section. We left with three Higgs mass parameters m A , m H , and m H ± , which determine phenomenologies at the LHC.
The bulk parameter space with m A m H ∼ m H ± is a clear prediction of the leptonspecific 2HDM considered in this paper. Since the extra Higgs bosons are mainly from the "leptonic" Higgs doublet with a large tan β, all the three members are expected to dominantly decay into the τ −flavor, leading to τ −rich signatures at LHC [29][30][31] via the following production and ensuing cascade decay chains: As seen in Fig. 2, we can also find a small island at the right-lower corner of the plot where m H ± ∼ m A ∼ 100 GeV, which we call Region B while the above bulk region we call Region A. In the following we fix m H ± in the two regions based on the best fit point: where λ(1, x, y) = (1 − x − y) 2 − 4xy. It can be compared with the partial decay width of From Eqs. (30) and (31) one can see that the W A channel turns out to dominate over the τ ν channel when m H + > √ 2m τ tan β as shown in the left panel of Fig. 6, where we plotted the branching ratio of H ± → AW ± . We can get the decay width Γ(H → ZA) by replacing m H + and M W with m H and M Z , respectively, in the above expression, and its branching ratio is also shown in the right panel.
Even if H/H ± undergoes the decay involving Z/W ± , the associated A will eventually decay into τ τ and thus multiple τ signature up to 4τ + W or/and Z would be one of the peculiar signatures of the model at the LHC.

A. Current Constraints
Current LHC 8 TeV data already set the constraints in the parameter space we are interested in. In both Region A and Region B we take model point grid with m A ∈ [20,200] GeV and m H ∈ [140, 320] GeV both with 20 GeV steps, that is, 100 model points for each region. We generate the 50,000 signal events with MadGraph [32] for each parameter point and interfaced to CheckMATE 1.2.0-beta [33] for checking the current bound with 20 fb −1 data at 8 TeV LHC. The analyses implemented in the CheckMATE are listed in the Table IV A. We checked all the analyses and considered that a model point is excluded when at least one analysis excludes it at 95% C.L. Fig. 7 shows the estimated 95% C.L. exclusion contours. For most of the parameter space, the strongest constraint comes from the chargino-neutralino search in ATLAS [34]. Especially, it is from the signal region "SR2τ a" therein, which requires two τ leptons and an additional isolated lepton, with m max T 2 > 100 GeV, E / T > 50 GeV and b-veto. Heavier m H > 200 GeV (Region A) or m H > 280 GeV (Region B), and light m A < 50 GeV are still allowed and we will show later that the next run of LHC can explore some of the regions. For the heavier m H regions the sensitivities are weaker just because of the smaller cross sections, while for light m A region it is because τ s from lighter A decays become softer and thus the acceptance quickly decreases. Moreover, the H/H ± → AZ/W ± decay modes also start open to decrease the number of hard τ s from direct H/H ± decays. Note that the exclusion of the lighter m A parameter space is of interest only for Region A, since for Region B the interesting m A in our scenario to explain (g − 2) µ is confined to be lie above 100 GeV as you can see in Fig. 2.

B. 14 TeV prospects
In this section we estimate the reach of the LHC 14 TeV in Region A and B based on the model point grids defined previously for the LHC 8 TeV study. The signal cross sections depend on heavy Higgs masses, and in Fig. 8 we show the contour plots of total cross section on the m A − m H plane for Region A (Region B) in the left (center) panel. Actually, for relatively small m A the dominant contribution comes from the H ± A production while the HA production contributes secondarily; HH ± and H + H − contributions are subdominant.  As this model predicts τ -rich signatures the signal is sensitive to τ -tagging and we im-plement τ -tagging algorithm using track and calorimeter information from Delphes 3.0 [38] as described in Ref. [39], which basically is a simplified version of the ATLAS τ -tagging algorithm [40,41]. We use two variables: where p j is the jet center direction and the distance of the furthest track among p i (with p T > 1 GeV) to p j is denoted as R max ; E calo T is the E T deposited in each calorimeter tower and the summations run over the calorimeter towers within the cones centered around p j with cone size R < 0.1 and 0.2 for the numerator and the denominator, respectively. Both R max and f core measure essentially how narrow the jet is; τ -jet is expected to be narrow and gives a smaller R max and f core ∼ 1. We found these two variables are most relevant for the discrimination.
We show R max and f core distribution in Fig. 9. We also show the ROC curve obtained by changing the cut value R cut max for R max < R cut max with fixing f cut core = 0.95 for f core > f cut core . Compared with the plot shown in Ref. [41], our simulation is reasonably conservative up to the signal efficiency ∼ 60%. We select the working point with R cut max = 0.1, which gives τ = 59% with the background jet rejection 1/ BG = 97. We apply the following event selection cuts to the signal and BG events. First, we require events with at least 3 τ -tagged jets, based on the algorithm explained above. At this stage the dominant background becomes tt, W +jets and Z+jets. Next, we require enough missing momentum E / T > 100 GeV to efficiently reduce the W +jets and Z+jets contributions. The normalized E / T distributions are shown in the right panel of Fig. 8. Finally, to reduces the tt background, we veto events with any b-tagged jet with p T > 25 GeV nor any jet with p T > 50 GeV. This cut efficiently reduces the remaining backgrounds. Table III summarizes the number of events after the successive selection cuts in unit of fb for the various BG processes and for the signal benchmark model point C. We compute the signal to background ratio S/B and significance based on statistical uncertainty S/ √ B. The significance quoted here is based on the integrated luminosity of 25 fb −1 . We can use the µµ modes as suggested in Ref. [30] to improve the sensitivity and to reconstruct the events but we mainly focus on τ -rich signatures, which require a relatively low statistics to set limit and expected sensitive at the early stage of LHC run 2.
We show the results for several selected benchmark points A to F in detail.   Based on the significance values we show the expected discovery reaches at LHC 14 TeV in Fig. 10. The left panel corresponds to Region A and the right panel does to Region B. Both panels show the expected 2σ, 3σ and 5σ discovery reach contours with assumed integrated luminosity of 25 fb −1 . It is seen that most of the interesting parameter regions can be covered. Only limitation is for the region with light m A and heavy m H where the sensitivity becomes weak even though the intrinsic signal cross sections are not so small. The reasons are again because of the smaller acceptance for the softer τ and longer decay chains involving Z/W as explained in the previous section on 8 TeV analysis. Moreover, in such a region, a light A from heavy H + /H decay will be boosted, resulting in a collimated τ −pair which becomes difficult to be tagged as two separated τ -jets. It is one of the reasons to have less acceptance for this parameter region. We can estimate the separation R τ τ of the τ leptons from A decay: For example, R τ τ ∼ 0.4 for m H = 300 GeV and m A = 30 GeV, and R τ τ ∼ 0.3 for m H = 400 GeV and m A = 30 GeV. Since the jets are usually defined with R = 0.5, the τ −pair starts overlapping. We indicated the region with the overlapping τ problem in red lines in the left panel of Fig. 10. In that region, we have to think of how to capture the kinematic features of the boosted A → τ + τ − . We may be able to take the overlapping τ problem as an advantage by utilizing jet substructure study, which is already proven useful [42][43][44].
For example, using di-tau tagging as proposed in Ref. [45] might be beneficial, although we leave this for future work.
[GeV]  Table IV are indicated with filled boxes. Red lines indicate the region with expected smaller τ separation of R τ τ ∼ 0.5 and 1.

V. CONCLUSIONS
The lepton-sepcific (or type X) 2HDM is an interesting option for the explanation of the muon g − 2 anomaly which requires a light CP-odd Higg boson A and large tan β. In this paper, we made a scan of the L2HDM parameter space to identify the allowed ranges of the extra Higgs boson masses as well as the related two couplings ξ l h and λ hAA of the 125 GeV Higgs boson which govern its standard and exotic decays h → τ + τ − and h → AA/AA * (τ + τ − ), respectively. The tau Yukawa coupling is found to be either in the wrong-or right-sign limit depending on the mass of A. More precise determination of the standard tau Yukawa coupling and a possible observation of one of the above exotic modes would provide a hint for the current scenario.
There appear two separate mass regions in favor of the muon g − 2: (A) m A m H ∼ m H ± and (B) m A ∼ m H ± ∼ 100GeV m H , which lead us to set up two regions of interest for the LHC study: (A) m H ± = m H + 15GeV, and (B) m H ± = max(90GeV, 0.8m A + 10GeV) with tan β parametrized by tan β = 1.25(m A /GeV) + 25. In these parameter spaces, one expects to have τ -rich signatures readily accessible at the LHC through the extra Higgs productions pp → AH ± /AH/H ± H ± /HH followed by H → AZ/τ + τ − H ± → AW ± /τ + ν and A → τ + τ − . Indeed, the current LHC8 data start to exclude (yet mild) some of the above two regions: m H up to about (A) 200 GeV and (B) 280 GeV for m A > 50 GeV from the consideration of the ATLAS neutralino-chargino search results. However, the region of m A 30 GeV (with tan β 40) which also satisfies the tau decay and lepton universality data [9] is hardly tested by the τ -rich signatures in near future even though HL-LHC should be able to over the region. Thus, further study, for example, on the boosted A → τ τ will be required in the next runs of LHC to cover all of the L2HDM parameter space explaining the muon g − 2 anomaly.