Two Higgs doublets to explain the excesses pp → γγ (750 GeV ) and h → τ ± µ ∓

: The two Higgs doublet model emerges as a minimal scenario in which to address, at the same time, the (cid:13)(cid:13) excess at 750 GeV and the lepton (cid:13)avour violating decay into (cid:28) (cid:6) (cid:22) (cid:7) of the 125 GeV Higgs boson. The price to pay is additional matter to enhance the (cid:13)(cid:13) rate, and a peculiar pattern for the lepton Yukawa couplings. We add TeV scale vector-like fermions and (cid:12)nd parameter space consistent with both excesses, as well as with Higgs and electroweak precision observables.


Introduction
The recently presented indications for a diphoton excess at ATLAS and CMS at an invariant mass of 750 GeV [1,2] have caused much excitement in the high-energy phenomenology community . At the same time, some hints of anomalies persist in the LHC run-I data. Notably there is a 2.4σ excess at CMS in the h → τ ± µ ∓ decay of the 125 GeV Standard Model-like Higgs boson h [98], corresponding to a best-fit branching ratio BR(h → τ ± µ ∓ ) = 0.84 +0. 39 −0.37 %. This is compatible with the ATLAS analysis which finds BR(h → τ ± µ ∓ ) = 0.77 ± 0.62 % [99].
We consider a CP-conserving 2HDM of type I in the decoupling limit [121], where the second doublet has a mass ∼ 750 GeV. We work in the "Higgs basis", where H 1 = [0, (v + h 1 )/ √ 2] denotes the doublet which gets a vacuum expectation value (vev) v 246 GeV, and which has Standard Model Yukawa couplings. The second doublet H 2 = [H + , (h 2 +iA)/ √ 2] does not couple to Standard Model fermions, except for a LFV Yukawa to τ ± µ ∓ . The physical Higgs bosons are the CP-even h and H, the pseudoscalar A and the charged

JHEP03(2016)073
Higgses H ± . In the decoupling limit, the light h is almost aligned on the vev, making it the Standard-Model-like Higgs of 125 GeV. In section 2 we show how to enhance the H and A couplings to gluons and photons, by introducing new vector-like charged fermions, while respecting the bounds from electroweak precision tests and h signal strengths. We neglect the charged Higgs H + because it contributes little to H, A → γγ. A small mixing with h 2 allows the LFV decay h → τ ± µ ∓ , as discussed in section 3. In section 4, we demonstrate that one can accommodate the 750 GeV excess from the decays of H and A, in agreement with the LFV excess.

Two Higgs doublets coupling to extra matter
In this section we neglect the misalignment between the CP-even mass basis, and the "Higgs" basis, and focus on the Higgs couplings to new fermions. That is, we consider the limit where the Standard Model Higgs boson h is identified with h 1 , and the second Higgs doublet H 2 does not couple to the Standard Model, except for its gauge interactions. Therefore, H = h 2 and A cannot decay to Standard Model particles at tree-level. We include the misalignment in the following section, in order to obtain h → τ ± µ ∓ .
In order for H and/or A to play the role of the 750 GeV resonance, we need to introduce a large effective coupling to γγ, as well as to gg, in the hypothesis that the resonance is produced via gluon fusion. If the production is dominated by quarks, that have a smaller parton density function, one needs an even larger coupling to γγ. We will discuss quantitatively these two possibilities in section 4.
To provide an explicit realization for such effective couplings, we introduce two vectorlike fermions, that transform under SU (3) The state of electric charge Q+1 has mass M D and no Yukawa couplings. The two states of charge Q couple to the Higgs doublets, and their mass matrix is non-diagonal because of the vev of H 1 . We will denote the mass eigenvalues by M 1 ≤ M 2 . Note that, in order to induce the couplings Hγγ and Aγγ (and analogously for gluons), one needs either λ S 1 λ D 2 = 0 or λ D 1 λ S 2 = 0. This is illustrated diagrammatically in figure 1, and it amounts to generate the effective operator H † 2 H 1 F µν F µν via a fermion loop. The couplings λ D,S 1 are constrained as they contribute to the h-decays into γγ and gg, as well as by the precision electroweak parameters S and T . Indeed, vector-like charged fermions were employed in the past to explain the transient excess in the h → γγ channel, see e.g. [123,124]. A detailed analysis of the allowed parameter space is provided in ref. [125]. Here we describe two illustrative cases: stronger bound comes from the Higgs signal strengths. For R c = 8 and |Q| ≤ 3, one needs (λ S 1 v)/( √ 2M 1 ) 0.12. In this case the bound comes from the hgg coupling.
(2) One vanishing Yukawa coupling, e.g. λ D 1 = 0. This pattern strongly suppresses the correction to the couplings hγγ and hgg, because, in the limit of heavy fermions, they are proportional to λ D 1 λ S 1 . However, an upper bound on λ S 1 still exists, coming from the T parameter, ( (8) and M 1 = 1 TeV. Note that T does not depend on the hypercharge, therefore it turns out that one can take it very large, say Q ∼ 10, without violating the constraints.
Let us now turn to the heavy Higgs doublet H 2 . Its couplings to the fermion mass eigenstates are easily derived [125] in terms of the parameters in the Lagrangian of eq. (2.1). Then, one can compute the decay width into two photons for the scalar H and the pseudoscalar A. The result is particularly compact in the limit M H 2M 1,2 , since in this case the loop form factor is the same for both fermions in very good approximation, A 1/2 (0) = 4/3. Similarly, for A we use the loop form factorÃ 1/2 (0) = 2. Then, one obtains In the same approximation, the widths into two gluons read where C(R c ) is the index of the color representation. Note that the ratio of H-rates over A-rates is given by a factor (2|λ

JHEP03(2016)073
For definiteness, consider the case (2) described above, λ D 1 = 0, and take M H M A 750 GeV. Then, one obtains 35, that is the largest value allowed by the T parameter for R c = 3. In the case of a colour octet, N c = 8 and C(R c ) = 3, there is a slightly stronger upper bound, (λ S 1 v)/( √ 2M 1 ) 0.25: therefore, one gains a factor ∼ 3 in γγ and a factor ∼ 20 in gg.
Note that one can reproduce the same rates with smaller Yukawa couplings: taking N pairs of vector-like fermions, all with equal charges and coupling λ D 2 , the rates scale as (N λ D 2 ) 2 . From a theoretical point of view, it may be more justified to introduce several vector-like fermions, but with charges related to the Standard Model ones, such as one or more vector-like families, composed of t, b and τ partners. Adding over their contributions one could obtain a qualitative similar effect.
One should also remark that the heavy fermion loops also induce decays of H and A to Zγ, ZZ and W W , with width of the same order as (or slightly smaller than) for γγ. However, the upper bounds from the 8 TeV LHC are weaker than the one on γγ, as discussed e.g. in ref. [55]. Therefore, they are presently unconstraining. At run 2, the better perspective appears to be the observation of the Zγ channel.
3 The τ ± µ ∓ decay of the 125 GeV Higgs boson Flavour-changing Higgs couplings are generic in the 2HDM, but their effects are not seen in low energy precision experiments searching for lepton or quark flavour change. So a discrete symmetry, which forbids flavour-changing Yukawa couplings, is usually imposed on the 2HDM. To allow for LFV h decays, without generating undesirable flavour-changing processes, we suppose that our 2HDM almost has a discrete symmetry: all the Standard Model fermions have the usual Yukawa couplings to H 1 ("type I" model), and the only two couplings of H 2 to Standard Model fermions are the µτ LFV ones, [105,122] for a more formal analysis). Recall that the diagonalisation of the fermion mass matrices diagonalises the Yukawa couplings of H 1 , which carries the vev. Therefore, the LFV couplings are attributed, by definition, to the doublet H 2 with zero vev. Note that eq. (3.1) amounts to assume that µ and τ numbers are not conserved, while electron number remains a good symmetry at the renormalizable level. Such symmetry has to be slightly broken to allow for viable neutrino masses. In general, this breaking will propagate radiatively to the H 2 Yukawa couplings, however the size of this effect can be sufficiently small, as it strongly depends on the specific neutrino mass model. In section 4

JHEP03(2016)073
we will also consider a scenario where H 2 is produced from an additional Yukawa coupling to b quarks, that can be added without phenomenological problems. The CP-even mass eigenstates h and H are misaligned with respect to the vev by an angle that is commonly parametrized as β − α: (3.2) In the decoupling limit [121,122], sin(β − α) 1 and where the Higgs potential contains a term Λ 6 H † 2 H 1 H † 1 H 1 + h.c., in the basis where H 1 has no vev. The coupling of h to τ ± µ ∓ is therefore proportional to cos(β −α)ρ, and one obtains The CMS best-fit is BR(h → τ ± µ ∓ ) = 0.0084 [98], which gives where the width was taken at its Standard Model value, Γ h 4.1 MeV.
In the 2HDM, the CMS excess in h → τ ± µ ∓ is consistent with the current upper bound BR(τ → µγ) ≤ 2.6 × 10 −7 BR(τ → µνν) [118,119]. However, the extra fermions which enhance H, A → γγ as in eqs. (2.2)-(2.3), will also enhance the rate for τ → µγ [104]: if a neutral Higgs is exchanged between its γγ andτ µ vertices, and one of the photons connects to the lepton line, a diagram for τ → µγ is obtained. Such diagrams with a top loop were calculated in the 2HDM in [120]. From their results, the combined contribution of H and A can be estimated, for M 1 M 2 and λ D 1 = 0, as where the experimental bound is 384π 2 (A 2 L + A 2 R ) ≤ 2.6 × 10 −7 . With the definition of Yukawa couplings given in eq. (2.1), it turns out that choosing a large λ D 2 (λ S 2 ) leads to a destructive (constructive) interference among the H and A amplitudes. This was taken into account in eq. (3.6), where the difference in loop integral functions was chosen 1/2, as given in [120] for M 2 1 /M 2 H 2. A similar estimate can be made for A R . We neglect the h contribution to τ → µγ, because its coupling to γγ is not enhanced, see scenario (2) in section 2. So the Babar-Belle bound on τ → µγ could be satisfied for which sets a lower bound on cos(β − α) when combined with eq. (3.5): If the masses and couplings were purposefully tuned, it might be possible to suppress the τ → µγ amplitude even further, so we will consider eq. (3.8) to be a preference but not an exclusion.

JHEP03(2016)073 4 Reproducing the 750 GeV excess
Let us discuss the decay widths of H and A as a function of the Higgs mixing cos(β − α) and of the LFV couplings ρ µτ,τ µ . The mixing does not affect the couplings of the pseudoscalar A, for which the discussion of section 2 applies. On the other hand, the misalignment parametrised in eq. Since several Higgs signal strengths have been tested at LHC-8 TeV with 10% precision, the Higgs mixing is bounded from above cos 2 (β − α) 0.1 .

(4.2)
This is consistent with eq. (3.8). As discussed in section 2, the corrections to h → γγ and h → gg may lead to a slightly stronger upper bound on cos(β − α), if the couplings λ D,S 2 are very large. However, such bound drops for λ S 1 · λ D 1 → 0, see case (2) in section 2. Finally, the contributions to S and T from scalar loops are small in the 2HDM close to the decoupling limit [105,127], as we explicitly checked for our choice of the parameters.
The mixing has an important effect on the total width of H, since the latter can decay to Standard Model particles n's, with coupling g Hnn = cos(β − α)g SM hnn . The dominant contributions read, at the tree level, where, for the latter numerical estimate, we used the accurate values of the widths for M H 750 GeV, as given in ref. [128]. Here we neglected the channel H → hh, because the corresponding trilinear scalar coupling may be suppressed, by conveniently choosing the scalar potential parameters. Recall that the cross-section for pp → H → γγ is proportional to Γ(H → gg)/Γ tot H , where the numerator corresponds to the assumed dominant H production mode, and the denominator is the total width of H. Therefore, the contribution of H to the excess degrades as soon as Γ(H → gg)/M H 0.33 cos 2 (β − α).
The LFV couplings ρ µτ,τ µ also open an additional decay channel for both H and A, with a width , (4.4) where the last equality comes from eq. (3.5).
One should also mention that the presently preferred width of the excess, Γ ∼ 45 GeV, could be mimicked by two narrow resonances close in mass. Indeed, the mass split between H and A is given, in the decoupling limit, by M 2 .c. appears in the Higgs potential. This is naturally of the correct order of magnitude for Λ 5 1. Note, however, that the H-mediated cross-section tends to be suppressed relatively to the A-mediated one by two factors: the additional Higgs width in eq. (4.3), and the factor 4/9 from the loop form factors, see eqs. (2.6)-(2.7).
Let us put all the constraints together to identify the possible windows of parameters that allow to reproduce the 750 GeV excess in agreement with the preferred h → τ ± µ ∓ rate. The resonant LHC total cross-section, in the crude zero-width approximation, reads where s = (13 TeV) 2 , M H(A) 750 GeV, and the P i coefficients are the integrals for convoluting over parton densities, that define the parton luminosities for each species i: Consistency with the absence of resonances at 8 TeV favours i to be either gluons or bs, for which the luminosity is Pb b 14 and P gg 2000 (we used for eq. (4.6) the latest pdfs from ref. [126]). We focus first on gluon-gluon fusion as the dominant production mechanism. This channel enjoys the largest parton density functions, so it is sufficient to have Γ(H, A → γγ)/M H,A 10 −6 [3], as long as Γ H,A tot Γ(H, A → gg). However, the latter is loopsuppressed as shown in eq. (2.7). The total cross-section for some choices of the parameters is shown in figure 2 as a function of cos(β −α). Note that for completeness and cross-check, we have also compared with the more elaborated invariant mass distribution dσ/dM (gg → H(A) → γγ) where M ≡ √ŝ is the γγ invariant mass, that we have calculated taking into account the exact width dependence, and integrating this expression over an appropriate large range for M around the resonance. The numerical differences with the narrow-width approximation expression in eq. (4.5) is at most 2-3 % for all the relevant parameter choices discussed below, as could be intuitively expected since the total width of either A or H remains in all cases sufficiently moderate with respect to the resonance mass, such that the narrow width approximation is justified a posteriori. We can envisage two scenarios: Let us compare with the alternative possibility that the production of H and A is not dominated by gluon fusion, rather by bb → H, A. The parton density functions give a suppression of order 100 with respect to gluons, so that the excess requires Γ(H, A → γγ)/M H,A 2 · 10 −4 [3]. The advantage is that a Yukawa coupling (ρ b / √ 2)b(h 2 + iγ 5 A)b can easily overcome the other tree-level widths in eq. (4.3) and eq. (4.4), Indeed, one can reproduce the preferred value Γ 45 GeV for ρ b 1. Moreover, there is no constraint from dijet searches at 8 TeV, as the b-quark parton density function is very small. Therefore, one identifies the following scenario: (C) When Γ H,A tot Γ(H, A → bb), both H and A contribute to the excess, as long as Γ(H, A → γγ) 2 · 10 −4 . Confronting with eq. (2.6), one needs a pair of vector-like fermions with R c = 3 and Q = 7, or R c = 8 and Q = 5. Note that is difficult to avoid such large exotic charges by augmenting the number of multiplets in the loop, as the signal scales with Q 4 . As discussed in section 2, such large Q can be compatible with Higgs decays and the S and T parameters, however the bound of eq. (3.8) from τ → µγ is exceeded by a factor of few.
The total cross-sections, combining both the gluon fusion and bb production channels, are shown in figure 3 as a function of cos(β − α), for Q = 5 and other parameters as in figure 2. Here the cross-sections are calculated with the exact width dependence and integrating dσ/dM (gg, bb → H(A) → γγ). In fact due to the dominant contribution of the bb decay to the total width Γ tot in this case, the bb production channel largely dominates (for instance the gluon fusion process contributes to the total cross-section by about ∼ 10% only for R c = 8, and much less for R c = 3). Note that in this case the discrepancy with the cross-sections in the narrow width approximation of eq. (4.5) amounts to 7-8 %, for the parameter choices discussed above, that is roughly of order Γ H,A tot /M H,A .

Final comments
We entertained the possibilities that both the γγ excess at 750 GeV and the h → τ ± µ ∓ excess are due to new physics. A minimal way to introduce (renormalisable) flavour viola-

JHEP03(2016)073
tion and extra bosons to the Standard Model is to add a second Higgs doublet. Its τ ↔ µ coupling may be connected to large 2 − 3 mixing in the neutrino sector, in scenarios where the Yukawa couplings of charged leptons and neutrinos are related. The neutral scalars H and A can play the role of the 750 GeV resonance, even though the strength of the excess in the early 13 TeV data is significantly larger than the one expected in the 2HDM alone. We take this as a hint that additional states close to the TeV are present in the underlying theory, with large Yukawa couplings to the second Higgs doublet. We have shown that a pair of vector-like fermions is sufficient to reproduce the right cross-section, and respect all other constraints. However, such fermions must have gauge charges larger than the Standard Model fermions: indicatively, for a Yukawa 3 and R c ≤ 8, one needs |Q| ≥ 2 in scenarios (A) and (B), and |Q| ≥ 5 in scenario (C), see section 4. Alternatively, several pairs of fermions have to be introduced. These are important indications to constrain those well-motivated extensions of the Standard Model that predict vector-like fermions, such as top partners.
Were the heavy Higgses to have no couplings to Standard Model fermions, then gg → H, A → γγ is a natural discovery channel. However, to explain h → τ ± µ ∓ , the heavy Higgses must interact with τ ± µ ∓ , and mixing is required between h and H. Both requirements gives Standard Model decay channels to H and A, which reduces BR(H, A → gg, γγ); nonetheless we find three scenarios that fit both excesses. In addition, the mixing must respect both a lower bound to reproduce the LFV excess, and an upper bound to protect the 125 GeV Higgs couplings: 10 −3 cos(β − α) 0.3.
The decay τ → µγ is a particular challenge for this model, because the heavy Higgses couple to τ ± µ ∓ and have an enhanced coupling to γγ. In combination, these interactions give a "Barr-Zee" contribution to τ → µγ which is dangerously large. By choosing the Yukawas to obtain destructive interference between A and H, we find that at least two of the scenarios are compatible with the current experimental limit on τ → µγ.