Higgs mediated CLFV processes $\mu N(eN)\rightarrow\tau X$ via gluon operators

We revisit charged lepton flavor violating (CLFV) scattering processes $\ell_{i} N \to \tau X \, (\ell_{i} \ni e, \mu)$ mediated by Higgs. We point out that a new subprocess $\ell_{i} g \to \tau g$ via the effective interactions of Higgs and gluon gives the dominant contribution to $\ell_{i} N \to \tau X$ for an incident beam energy of $E_{\ell} \lesssim 1\,\text{TeV}$ in fixed target experiments. Furthermore, in the light of quark number conservation, we consider quark pair-production processes $\ell_{i} g \to \tau q \bar{q}$ ($q$ denotes quarks) instead of $\ell_{i} q \to \tau q$. This corrects the threshold energy of each subprocess contributing to $\sigma(\ell_{i} N \to \tau X)$. Reevaluation of $\sigma(\ell_{i} N \to \tau X)$ including all of relevant subprocesses shows that the search for $\ell_{i} N \to \tau X$ could serve a complementary opportunity with other relevant processes to shed light on the Higgs CLFV.


I. INTRODUCTION
One of the most puzzling issues in particle physics is the flavor sector; why there are three generations, what the origin of mass hierarchy is, and so on. Many types of solutions for this puzzle have been proposed in UV completions of the standard model (SM), and in general predict a misaligned Yukawa couplings in the mass basis which give rise to flavor violating interactions of Higgs and fermions. The Higgs field therefore could be a promising probe to the puzzle.
Charged lepton flavor violation (CLFV) is an important process to search for the Higgs induced flavor violation due to its high sensitivity and a variety of observables [1,2]. We focus on the CLFV involving taus as they are relatively less constrained and a sizable effect could be naturally expected. Besides that an interesting deviation in h → τ µ [3] also motivates us to focus on the tau CLFV. Once one of the tau CLFV processes is discovered, e.g., τ → ℓ i γ, τ → ℓ i ℓ j ℓ k , h → τ ℓ i , etc., where i, j, k denote charged lepton flavor indices, other CLFV processes and their correlation should be studied for shedding light on the UV structure responsible for the CLFV interactions. Among the complementary reactions the tau production ℓ i N → τ X is relatively less attention paid to. Here N is a nucleus and this process is expected to happen in fixed target experiments at a sizable rate ∝ ρ, where ρ being target density [4,5]. The beam intensity is planned to reach at 10 22 electrons per year in ILC [6], and at 10 20 muons per year in neutrino factories and in the muon collider experiment [7]. Furthermore, LHeC and its optional plans facilitate the electron-proton collision at the center of mass energy of √ s 1 TeV with the luminosity L 10 33 cm −2 s −1 [8].
In the light of these upcoming experiments, ℓ i N → τ X would be a promising probe for the Higgs induced CLFV.
In this letter, we point out that the gluon initiated partonic subprocess, having not considered in the literatures, provides a dominant contribution to ℓ i N → τ X in fixed target experiments for incident beam energy of E l 1 TeV. Furthermore, we stress the importance of quark number conservation. Since sea quarks are generated through gluon splitting, we have to include the final state with a quark pair, instead of sea quark single-production ℓ i q → τ q. Here q = s, c, b, t. The related subprocess ℓ i q → τ q has been studied in Ref. [9] in the context of a supersymmetric extension of SM, where they consider only the effective 4-Fermi operators involving the sea quarks and evaluate the cross sections assuming massless patrons. The difference of the available phase space between ℓ i g → τ qq and ℓ i q → τ q is not negligible at the fixed target experiments with the relatively low collision energy √ s ≃ √ 2M E l ≃ 13.7 GeV E l /100 GeV, where M denotes the nucleon mass. For example, production of τ bb is kinematically allowed only when the beam energy exceeds E ℓ 55 GeV, while it would be E ℓ 19 GeV when simply the τ b threshold is considered. In this letter, we reformulate the Higgs mediated tau production ℓ i N → τ X by taking the new effects: (i) the effective interaction of Higgs and gluons, and (ii) the quark number conservation, into account. Due to the altered formulations, the cross section of ℓ i N → τ X drastically changes from the previous estimation for the beam energy E ℓ 1 TeV. It would largely affect the search for this process in the next generation experiments.
The relevant Lagrangian terms for the Higgs mediated CLFV scattering ℓ i N → ℓ j X are shown in Eq. (3).
where G a µν is gluon field strength, and g hgg is an effective coupling. In the literatures, the scattering is described only by sea quark contributions ℓ i q → τ q through the quark Yukawa interactions. We investigate the effects of the effective operator of Higgs and gluons given by the second term and the effects of the quark number conservations. CLFV interactions are parametrized by the couplings ρ ij , where i and j are flavor indices of charged leptons, and i = j. Current bounds on the couplings come from the measurements of CLFV Higgs decays at the LHC [10][11][12], and are |ρ eτ | 2 + |ρ τ e | 2 < 2.4 × 10 −3 and |ρ µτ | 2 + |ρ τ µ | 2 < 3.16 × 10 −3 , where an interesting excess was reported [3].

A. effective gluon Higgs operator
The effective coupling g hgg is generated by the triangle diagrams where quarks are running (see Fig. 1 (a)), and is derived as a function of the momentum transfer of the Higgs q h (q 2 h = −Q 2 < 0) as follows [13,14], where v = 246 GeV is the vacuum expectation value of the Higgs field, α s = g 2 s /4π, and m q represents the mass of the running quark. The function f (r) is given as, Note that the function f (x) has no imaginary part since q 2 h < 0 for the t-channel scattering ℓ i g → τ g. This is different from the on-shell Higgs production at the LHC, where the scale is fixed at q 2 h = m 2 h , and therefore, the lighter quark contributions induce the imaginary part.
We count only charm, bottom, and top loop contributions for the effective coupling. Each loop contribution becomes larger and approach to a constant value for smaller Q 2 since the loop function has the following asymptotic form, Therefore, in addition to the top quark loop, the relatively heavy quarks (b, c) also contribute while the lighter quarks (u, d, s) contributions are still suppressed in the DIS regime Q 2 1 (GeV) 2 . This is different from the case at the LHC, where the dominant contribution is essentially only via the top quark loop as q 2 h = m 2 h . Figure 2 shows the Q 2 dependence of g hgg and each quark contribution. Due to the constructive contributions there exists a sizable enhancement of the cross sections relative to the case with top contribution only. Note that the gluon operator for the CLFV hadronic tau decays is derived by integrating out the heaviest three quarks since energy scales of those reactions are at O(100) MeV [15]. The gluon operator derived under the same condition makes the correlation of the decays and the scattering ℓ i g → τ g to be clear, and leads to comprehensive understanding of the CLFV interactions.

B. cross sections
The Lagrangian (3) describes the two types of subprocesses, ℓ i g → τ g ( Fig. 1 (a)) and ℓ i g → τ qq ( Fig. 1 (b)). The total cross section is formulated as where x is the Bjorken variable and y is the measure of inelasticity; Here, P and p i denote momenta of the initial nucleon and the initial lepton. Note that the momentum transfer q h = p i −p τ is defined only with the initial lepton and the final tau momenta but not with the momentum related withX. The ranges of x and y are obtained as [16,17], Here M and m τ denote the masses of nucleon and tau lepton, respectively. The gluon parton distribution function (PDF) is denoted as f g (ξ, Q 2 ) and ξ is the fourmomentum fraction of the nucleon carried by the parton, p g = ξP . The range of ξ depends on the subprocess. The parton level differential cross section of ℓ i g → τ g in a massless limit of incident lepton is given by The invariant mass of the system is denoted byŝ = (p i + p g ) 2 . We have the relation ξ = x as the outgoing gluon is massless. In the same limit, the parton level differential cross section of ℓ i g → τ qq is given by where K ≡ 1 − 4m 2 q /w 2 , and w 2 = (p g + q h ) 2 = (p q + pq) 2 = Q 2 (ξ/x − 1) is the invariant mass of the final quark and anti-quark system. For ℓ i g → τ qq, to correct the finite mass effect of the outgoing quarks m q , ξ = x(Q 2 + w 2 )/Q 2 is taken [18].

C. general form of CLFV gluon operator
We briefly discuss the CLFV scatterings ℓ i N → τ X mediated by other heavy particles which (in)directly couples with the gluon field, for example, heavy Higgses H and A in two Higgs doublet models. As long as those particles are heavy enough we can describe the subprocess ℓ i g → τ g using the following effective operators, The constraints on C ij come from the searches for CLFV tau decays, BR(τ → µπ + π − ) < 2.1 × 10 −8 and BR(τ → eπ + π − ) < 2.3 × 10 −8 [19], which are corresponding to |C µτ | 2 + |C τ µ | 2 < 7.40 × 10 −18 GeV −6 and |C eτ | 2 + |C τ e | 2 < 8.10 × 10 −18 GeV −6 [20]. Note that these constraints are much weaker than those assuming only via the SM Higgs, where the stringent constraints on ρ ij is applicable through the relation C ij = ρ ij g hgg /m 2 h . The differential cross section of the subprocess ℓ i g → τ g is easily calculated by ignoring Q 2 term in the denominator in Eq. (10),

III. NUMERICAL ANALYSIS
We perform a numerical analysis. In our analysis, the maximally allowed CLFV Yukawa coupling for electron, |ρ eτ | 2 + |ρ τ e | 2 = 2.4 × 10 −3 [10] is taken. Other input parameters are m h = 125.7 GeV, m τ = 1.777 GeV, m t = 173.2 GeV, m b = 4.180 GeV, m c = 1.275 GeV, and M = 0.9383 GeV [21]. We have used CTEQ6L1 for the nucleon PDF [22]. We restrict the phase space integration with W 2 > (1.5 GeV) 2 and Q 2 > (1 GeV) 2 to ensure that the parton model picture is valid, where W 2 = (P + q h ) 2 is the hadronic invariant mass. We consider only the DIS regime and ignore other resonant effects which are known to be sub-dominant [16].

fixed target experiments
First we consider the prospects at fixed target experiments. Figure 3 shows the cross section of the Higgs mediated scattering ℓ i N → τ X as a function of the incident lepton beam energy. We show the contributions from each subprocess separately, and also the sum in a thick solid line. Due to the large gluon PDF and no phase space suppression, the new subprocess ℓ i g → τ g leads to the large enhancement in σ(ℓ i N → τ X). The ratio between σ(ℓ i N → τ X) with and without the new subprocess is 7.8 (1.8) for E ℓ = 50 GeV (500 GeV). The subprocess ℓ i g → τ cc (ℓ i g → τ bb) only starts to be relevant at E ℓ ∼ 100 GeV (500 GeV) due to the phase space suppression by the production of a pair of quark and anti-quark. Therefore, for E ℓ 1 TeV, the dominant contribution for the CLFV scattering in fixed target experiments comes from the new subprocess. Inclusion of the g hgg coupling enhancement shown in Fig. 2 is also important. Typically, it provides a factor of 3 ∼ 7 enhancement relative to the case with top contribution only.
Event rate of ℓ i N → τ X process at the fixed target experiments is estimated as [23], where N ℓi is the intensity of ℓ i per year, and T m is the target mass in unit of g cm −2 . According to the formula, O(10) (O(10 3 )) events of ℓ i N → τ X are expected per year for the electron beam energy of an upgrade option in ILC (PWFA), E e = 500 GeV (5 TeV), T m = 100 g cm −2 , and the electron intensity N e = 10 22 /year.
The scattering cross section assuming the contact operator with the maximally allowed value is shown in a dot-dashed line. For muon options, we require the neutrino factories, which would reach at N µ = 10 20 /year [7] with a beam energy of O(100) GeV to provide O(10) events/year. The currently available intensity is not enough, for example, N µ = 10 15 /year in COMPASS II experiment [24] and N µ = 10 19 /year with a lower beam energy in COMET [25] in future.

collider experiments
Next, we turn to the prospects at collider experiments. The total cross section of the Higgs mediated scattering as a function of the collision energy is shown in Fig. 4. It is essentially the same quantity shown in Fig. 3 but in different regime since the collision energy at fixed target experiments is related with √ s ≃ √ 2M E ℓ ∼ 1.4 GeV E ℓ /GeV. As the collision energy √ s increases the cross section grows rapidly. When it reaches at ∼ 2m t , a subprocess ℓ i g → τ tt starts to contribute and becomes dominant for √ s 1 TeV.
A future electron-proton beam collision experiment TLHeC (VHE-TLHeC) plans to achieve a √ s Although such a high energy provides an opportunity to discover ℓ i N → τ X, it is challenging to identify the SM Higgs as the CLFV mediator due to the high dominance of ℓ i g → τ tt. By measuring each cross section of the subprocess both at the fixed target and at the beam collision experiments, we can comprehensively understand the CLFV interactions with the search results for CLFV tau decays including the gluon operator. It could also judge whether the CLFV scattering is mediated by the SM Higgs or the other particles.
Experimental feasibility of measuring each subprocess should be also studied further and we can study e.g., the angular distributions of the tau and the hadronic system X, the compositions of the final states, and so on. We leave these issues for our future works.

IV. SUMMARY AND OUTLOOK
In summary, the Higgs mediated CLFV scattering ℓ i N → τ X has been reconsidered by taking (i) new subprocess ℓ i g → τ g, and (ii) sea quark number conserving subprocess ℓ i g → τ qq (q = c, b, t), into account. At the fixed target experiments with E ℓ 1 TeV, the total cross section σ(ℓ i N → τ X) is enhanced more than about twice by the subprocess ℓ i g → τ g. It has been pointed out that the associated quarks are only produced in pairs in ℓ i N → τ X, and hence σ(ℓ i N → τ qq) starts to be relevant on higher beam energy than that estimated in the previous works wherein the phase space suppression by a quark pair production is not considered. O(10) − O(10 3 ) events of ℓ i N → τ X could be expected in both the fixed target experiments and the beam collision experiments in future, and it would provide opportunities to shed light on the nature of Higgs mediated CLFV complementarily with other experiments, such as CLFV tau decay searches. Our results hold for other CLFV mediators which couple with gluon and/or heavy quarks.