New Physics Effects in Higgs Decay to Tau Leptons

We study the possible effects of TeV scale new physics (NP) on the rate for Higgs boson decays to charged leptons, focusing on the tau tau channel which can be readily studied at the Large Hadron collider. Using an SU(3)_C X SU(2)_L X U(1)_Y invariant effective theory valid below a NP scale Lambda, we determine all effective operators up to dimension six that could generate appreciable contributions to the decay rate and compute the dependence of the rate on the corresponding operator coefficients. We bound the size of these operator coefficients based on the scale of the tau mass, naturalness considerations, and experimental constraints on the tau anomalous magnetic moment. These considerations imply that contributions to the decay rate from a NP scale Lambda ~ TeV could be comparable to the prediction based on the SM Yukawa interaction. A reliable test of the Higgs mechanism for fermion mass generation via the h->tau tau channel is possible only after such NP effects are understood and brought under theoretical control.


I. INTRODUCTION
The search for the Higgs boson and the study of its properties is a primary task of the Large Hadron Collider (LHC). Currently, global fits to precision electroweak data find the Higgs mass to be 84 +33 −24 GeV with an upper bound given by m h < 150 GeV at 95% CL [1]. LEP has also placed a lower bound of m h > 114. 4 GeV [2]. Assuming the Standard Model (SM) of electroweak interactions, the Higgs is expected to be found in early physics runs at LHC or in the near future at the Tevatron. If a new scalar particle is found at LHC or the Tevatron, studying its self coupling and its couplings to fermions and gauge bosons will be important steps in determining whether or not it is the SM Higgs boson. One promising channel for testing the coupling to fermions is h → τ + τ − [3]. One can look at Higgs production via Weak Boson Fusion (WBF) which has distinctive signals (see [3] for a general review, see [4] for NP effects on WBF) allowing one to eliminate most of the QCD background. In addition, in WBF the Higgs is typically produced with p T ∼ 100 GeV ≫ m τ which facilitates a relatively precise invariant mass reconstruction of the τ + τ − pair. Realistically, at the LHC a measurement of the hτ τ coupling for m h < 140 GeV is expected to be made with about 100 fb −1 of data to ∼ 10% accuracy [5,6].
In this paper, we examine how new physics (NP) at or above the TeV scale could effect the decay rate Γ(h → τ + τ − ). If such effects are large, they could complicate a test of the SM Higgs mechanism for fermion mass generation using this decay channel. Of course identical statements can be made for h → e + e − , µ + µ − although these channels are too suppressed by small Yukawa couplings to be experimentally interesting (we will briefly comment on these particular channels). In order to analyze possible NP effects on Γ(h → τ + τ − ) in a model-independent manner, we employ an effective field theory approach where NP effects are encoded in SU (3)×SU (2)×U (1) invariant higher dimension operators built out of SM fields: where µ is renormalization scale, n ≥ 4 denotes the operator dimension, and j is the index running over all independent operators.
In what follows, we will take the NP scale Λ to be at or above the TeV scale. New physics at such a scale is expected for at least two reasons: triviality asserts that the Higgs mass vanishes in the absence of a cut off [7,8], and the radiative instability in the Higgs sector in the absence of additional NP leads to the hierarchy problem. Such an EFT approach, with Λ ∼ TeV, has been applied to precision electroweak observables [9,10,11,12,13,14] and has recently been the subject of further investigations for LHC and ILC phenomenology [15,16,17,18,19,20,21,22,23] as well as neutrino properties and interactions [24,25,26,27].
We use naturalness and/or experimental constraints to bound the Wilson coefficients of the relevant higher dimension operators. We find for h → ℓ + ℓ − , where is the Higgs vacuum expectation value, y ℓ is the charged lepton Yukawa coupling, and C denotes a combination of Wilson coefficients of the higher dimension operators.
Given that 1/y ℓ ≫ 1, one might naively expect that very large deviations from the SM rate could be observed. As we show, naturalness considerations generally imply that C ∼ y ℓ , thereby counteracting the 1/y ℓ enhancement.
Nevertheless, when Λ is not too large compared to v, we find that ∆Γ/Γ can be of order unity. In this case, a reliable test of the Higgs mechanism for lepton mass generation would require additional studies to disentangle the effects of NP in the h → ℓ + ℓ − channel.

II. HIGHER DIMENSION OPERATORS
The lowest dimension operator that contributes to h → ℓ + ℓ − is the n = 4 SM Yukawa interaction where L and e are lepton SU(2) L doublet and right handed charged lepton singlet fields respectively and φ is the Higgs doublet 1 . The effects of new physics first 1 We work in a basis where the charged lepton Yukawa matrices are diagonal.
appear at n = 6. Since Λ ∼ TeV≫ v, contributions from n > 6 operators can be safely omitted. Using the basis of Buchmuller and Wyler [12], the operators relevant to the h ℓ + ℓ − coupling at tree level are It is also useful to consider the symmetric and antisymmetric combinations of the last two operators in Eq. (3) It is straightforward to show that the symmetric com- where the currentL(D µ φ)e is gauge invariant and nonanomalous and the scalar potential V H (φ) is given by The total derivative on the LHS of Eq. III. n = 6 CONTRIBUTIONS TO h → ℓ + ℓ − DECAY Contributions to Γ(h → ℓ + ℓ − ) can be obtained by expanding the Higgs field as usual around its vacuum expectation value v, so that Here U(x) = e i ξ a (x) σa/v and ξ a (x) are the Goldstone bo- where ℓ is the charged lepton field, P R is the right handed projection operator and where, in the obtaining the last line of Eq. (8) and replacing y ℓ byȳ ℓ +3 C eH v 2 /2Λ 2 + C + m 2 h /Λ 2 , wherē y ℓ denotes the coefficient of O eY in the presence of NP. In general, the appearance of a higher-dimension operator that contributes to the lepton mass will change the relationship between the Yukawa coupling and m ℓ , implying thatȳ ℓ = y ℓ . In the SM, this relationship is For the n = 6 operators considered here, O eH generates a tree-level contribution to m ℓ . In this case, Eq. (10) no longer gives the relationship between the lepton mass that appears in the Lagrangian density after EWSB and the coefficient of O eY , and we must replace it by 3 where y ℓ is given by its SM value as in Eq. (10) and δy ℓ gives the shift due to the presence of a non-vanishing C eH . In contrast, the O + contributes to m ℓ only at the 3 We thank Mark Wise for pointing out the need to include this correction.
one-loop level through its mixing with O eH and the effects of matching onto O eY at the scale Λ (see below).
In this case, the mixing of O + with O eH implies a nonvanishing δy.
The resulting expression for the relative change in the decay rate is where where we have taken all operator coefficients to be real for purposes of this analysis.
To the extent that the effects of O eH and O + on m ℓ and ∆Γ are suppressed by v 2 /Λ 2 , we may write Eq. (12) as and we have used C + = (C De + CD e )/2. We have crosschecked the result in Eqs. (13,14) by using the operators O De and OD e directly without employing the equations of motion while noting that C + = (C De + CD e )/2.
We observe that the contribution from O eH depends on indicating the possibility of significant NP effects for Λ ∼ TeV and reasonable choices for the Wilson coefficient.
We will explore bounds on the size of the Wilson coefficients in later sections and show that naturalness considerations imply that they are generally proportional to The expressions in Eq. (12,14) allow us to estimate the size of possible new physics contributions to the h → ℓ + ℓ − rate. As discussed in Ref. [28], the operators O eH , O De , and OD e could be generated by tree level effects of new physics above the scale Λ. As a result, the corresponding operator coefficients could in principle be O(1) rather than O(1/16π 2 ) as in a naive application of naive dimensional analysis (NDA) [29]. Setting v 2 /(Λ 2 y ℓ ) = 1 one finds that NP can have an O(1) effect In deriving the resulting naturalness expectations for the n = 6 operator coefficients, we will follow the approach used recently in Refs. [24,25,26,27]  The constraints that follow from the latter low energy effective theory are too weak to be interesting.

A. Naturalness constraints on OeH
The analysis of naturalness considerations for O eH is particularly straightforward, as it generates a tree-level Here, we have omitted corrections associated with the running of C eH (µ) from the scale Λ to v as they are loop and coupling suppressed and do not substantially affect the corresponding naturalness expectation 4 . In the absence of large cancellations between this contribution and SM Yukawa contribution, we have |δm τ | < ∼ m τ or 4 Recall that we are taking C eH to be real in this analysis.
The resulting shift in the Yukawa coupling is δy τ = −y τ (y τ ) for positive (negative) C eH . Using Eq. (12) we obtain the naturalness bounds ∆Γ/Γ = 8 (∆Γ/Γ = 0). It is interesting to note that these bounds are lepton speciesindependent since the RHS of Eq. (17) is proportional to the Yukawa factor, thereby canceling the factor of y 2 τ in the denominator of Eq. (12).
When Λ ≫ v, the upper bound on |C eH |v 2 /Λ 2 Eq. (17) can allow the magnitude of the operator coefficient to be much larger than unity. In addition to appearing phys-  These contributions will appear when the full theory above the scale Λ is used to compute renormalization of O eY and, thus, would generate a matching correction to the effective theory below the scale Λ. Without knowing the full theory, we cannot compute this matching contribution precisely. Nevertheless, it is possible to estimate its magnitude using NDA. Doing so yields leading to The resulting expectation for the possible size of ∆Γ/Γ becomes more stringent than the tree-level bound when Λ > ∼ 4πv since the corresponding contribution to the RHS of Eq. (12) decreases as v 2 /Λ 2 .
It is possible that details of a specific model for NP above the scale Λ will preclude any contributions from O eH to O eY , in which case the naturalness expectation in Eq. (19) would not apply. In the absence of such a specific scenario, however, Eq. (19) gives a reasonable estimate of the magnitude of C eH (Λ).
We illustrate the expectations for ∆Γ/Γ obtained from Eqs. (17) and (19)  . 5 The effects of the O eH operator on higgs Yukawa couplings has been recently studied within the context of multi-scalar doublet models in [30]. Large regions of parameter space were found where order one effects are realized in the higgs decay rate in agreement with the large effects found to be possible in our naturalness bounds.

B. Naturalness constraints on O+
We first observe that O + does not contribute to m τ at tree-level since it contains a covariant derivative acting on φ. Alternately, we can express Eq. (5) as is sufficient. We again use NDA and obtain leading to The mixing with O eH is given to lowest order in the lepton Yukawa coupling by the diagrams in Fig 4. In principle, one can obtain this mixing by comput- and the dots above denote contributions from self renormalization and the mixing of other operators into O eH .
We obtainγ from the one loop computation of the diagrams in Fig.(4), we find Requiring that the resulting contribution to the τ mass be of the same order of magnitude as, or smaller than, m τ leads to the constraint Substituting this inequality into Eq. (12) leads to an upper bound on the contribution from O + to ∆Γ/Γ that decreases logarithmically as Λ increases but grows quadratically with m h . This bound is generally weaker than the expectation based on one-loop matching, but it will apply even in specific models that give a negligible renormal- Before looking at the implications of the above naturalness constraints on the bounds for ∆Γ(h → τ + τ − )/Γ, in the next section we explore possible constraints arising from the measurement of the τ anomalous magnetic moment.

C. Anomalous magnetic moment constraints on O+
Since the coefficients C ± of O ± depend on linear com- and O He and O Hℓ by using the identity and suitable integrations by parts, leading to Hℓ + h.c..
After EWSB, one has where F µν and Z µν are the field strength tensors for the Z 0 and photon respectively. Since g 1 cos θ W = g 2 sin θ W = e, we have Using this result, together with Eq. (30) and the definition of the anomalous magnetic moment a ℓ as the coefficient of the operator e 4m ℓl σ µν ℓ F µν (34) we obtain The τ anomalous magnetic moment has never been directly measured. The best bound is given by DELPHI [31] which finds the 95% CL − 0.052 < a τ < 0.013.
The current standard model calculation [32] of a τ is This leads to a conservative estimate of the deviation from the SM, considering the lack of data, given by Using this bound and Eq. (35) leads to If NP at high scales leads to Although h → µ + µ − is experimentally not a promising channel it is interesting to note that ∆Γ(h → µ + µ − )/Γ could be as large as 20% in spite of the extremely stringent constraint coming from the experimental bound on the muon anomalous magnetic moment. The muon anomalous magnetic moment has been measured very precisely [1]: The corresponding constraint in this case will be from which we conclude a possible 20% effect in ∆Γ(h → µ + µ − )/Γ. This is due to an enhancement coming from two Yukawa factors. The first Yukawa factor appears in the standard way as shown in Eq. (12). An additional Yukawa factor appears as seen in Eq. (iv) C eH (Λ) = 0 = C + (Λ), leading to no deviation.
We observe that, in all but scenario (iv), an order one shift in the h → τ + τ − rate is possible when the scale of NP is ∼ TeV.
The LHC can look for a deviation from the SM rate of up to 10% or more in Γ(h → τ + τ − ). Any such deviation would not invalidate the Yukawa mechanism but would be consistent with NP at TeV scales in addition to the SM Higgs. Thus, it is necessary to disentangle TeV scale effects before drawing any conclusions on the Higgs mechanism for fermion mass generation. The naturalness considerations discussed above imply that a 10% or larger deviation for the decay rate from the SM prediction would be associated with a NP mass scale < ∼ 10 TeV.
It is also interesting to examine these conclusions in the minimal lepton flavor violation (MLFV) [33,34,35,36,37,38,39,40]  The off diagonal flavor changing effects due to O eH and O + contribute at one loop to the flavor-changing decays τ − → ℓ − j γ, where ℓ j = µ, e. From a straightforward dimensional analysis, we find that the naturalness expectations discussed above imply contributions to the decay branching ratios B τ →ℓjγ = Γ(τ → ℓ j γ)/Γ(τ → ℓ jν ν) that are well below the present experimental limits. For example, the contribution from O eH to B τ →eγ is roughly 10 −8 (v/m H ) 4 (C eH v 2 /Λ 2 ) 2 , so that for C eH v 2 /Λ 2 ∼ y τ as implied by the tree-level naturalness considerations and for m H ∼ v, one obtains a contribution to B τ →eγ of order 10 −12 , a result that is seven orders of magni-tude smaller the experimental limit. Thus, naturalness considerations lead to considerably more stringent expectations for the dimension six operators than one would infer from these flavor-changing decays.