Eﬀective ﬁeld theory with two Higgs doublets

: In this article we extend the eﬀective ﬁeld theory framework describing new physics eﬀects to the case where the underlying low-energy theory is a Two-Higgs-Doublet model. We derive a complete set of independent operators up to dimension six assum-ing a Z 2 -invariant CP-conserving Higgs potential. The eﬀects on Higgs and gauge boson masses, mixing angles in the Higgs sector as well as couplings to fermions and gauge bosons are computed. At variance with the case of a single Higgs doublet, we ﬁnd that pair production of SM-like Higgses, arising through dimension-six operators, is not ﬁxed by fermion-fermion-Higgs couplings and can therefore be sizable.


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example of such a theory, but also the L µ − L τ model of refs. [16,17] reduces to a 2HDM if the Z ′ and the L µ − L τ -breaking singlet are heavy.
In general, any theory of new physics (NP) super-seeding the 2HDM at higher energies must satisfy the following requirements (in close analogy to the SM case): (i ) Its gauge group contains the SM gauge group SU(3) C ×SU(2) L ×U(1) Y as a subgroup.
(ii ) It contains two Higgs doublets as dynamical degrees of freedom, either as fundamental or composite fields.
(iii ) At low energies it reproduces the 2HDM, barring the existence of weakly coupled light particles, like axions or sterile neutrinos (in which case the EFT should include in addition these particles as dynamical degrees of freedom).
In this approach, heavier NP particles are integrated out and their effects are parameterized in terms of Wilson coefficients of higher-dimension operators suppressed by inverse powers of Λ. In our 2HDM case we have (1.1) Here L (4) 2HDM is the standard renormalizable 2HDM Lagrangian (to be specified in the next section) which contains only dimension-two and dimension-four operators. Q (5) k generalize the Weinberg operator [18] giving rise to neutrino masses and Q (6) k denote the dimensionsix operators. C (5) k and C (6) k are their dimensionless Wilson coefficients. In this paper we neglect the effects of operators of dimension seven and higher, which are suppressed by at least three powers of Λ.
The purpose of this paper is to extend the SM-EFT approach to the case of two Higgs doublets. 2HDMs are considered here as low-energy theories and using our framework the effects of heavier NP particles can be clarified and studied in a systematic way. For example, effective operators in 2HDMs have been recently considered to account for a diphoton excess in LHC data [68,69] with the outcome that an explanation requires to extend the field content of pure 2HDMs [70][71][72][73][74][75][76][77][78][79] 1 2 Table 1. The matter content of the 2HDM. l (e) is the lepton doublet (singlet), u and d the righthanded up and down quark singlets and q the quark doublet. Here, j = 1, 2, α = 1, 2, 3, and p = 1, 2, 3 stand for isospin, color and generation indices, respectively.
This article is organized as follows. In section 2 we will introduce our notation and conventions. The complete list of operators up to dimension six before spontaneous EW symmetry breaking will be given in section 3. We will then discuss EW symmetry breaking and the definition of the physical basis with diagonal mass matrices in section 4. In section 5 we will conclude and point out an interesting phenomenological application of our formalism.

Notation and conventions
In this section we establish our notation and conventions following ref. [23] and ref. [3]. The renormalizable 2HDM Lagrangian before spontaneous EW symmetry breaking, reads with T A = 1 2 λ A and S I = 1 2 τ I denoting the SU(3) and SU(2) generators, and λ A (τ I ) the Gell-Mann (Pauli) matrices. It is useful to define the following Hermitian derivative terms: The gauge field strength tensors are given by while X µν = 1 2 ǫ µνρσ X ρσ , with ǫ 0123 = +1, denotes the dual tensor (X = {G A , W I , B}).

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We consider a CP-conserving scalar potential [3] V (ϕ 1 , with a Z 2 symmetry 3 (softly broken by a dimension-two term) preventing the existence of terms with odd powers of ϕ 1 and ϕ 2 . Here all parameters are assumed to be real. 4 In order to study fluctuations around the VEVs that minimize the potential, 5 the two complex scalar fields are parameterized as The Lagrangian for the mass terms of the CP-odd (η a ), CP-even (ρ a ) and charged (φ + a ) Higgses is where we eliminated m 2 11 and m 2 22 by the minimization conditions. The charged and CPodd mass matrices have both one vanishing eigenvalue, which corresponds to the Goldstone bosons giving masses to the W and the Z, and a non-zero eigenvalue named m 2 H ± and m 2 A , respectively. Both matrices are diagonalized by the same angle β defined as Another independent rotation angle, called α, enters the definition of the CP-even mass eigenstates h and H, with eigenvalues m h and m H respectively: 3 Note that assigning Peccei-Quinn charges to Higgs doublets and fermions has the same effect on the potential and the Yukawa couplings as imposing a Z2 symmetry. 4 For an analysis of the conditions for a CP-conserving Higgs sector we refer to [81]. 5 Since no CP violation is involved, both v1 and v2 can be taken to be real. Table 2. Couplings of right-handed fermion singlets to Higgs doublets in 2HDMs with natural flavor conservation. These couplings can be enforced by an appropriate assignment of Z 2 (or Peccei-Quinn) charges to the scalar doublets and right-handed fermions.

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where usually the lighter one is identified with the SM Higgs, with mass m h ≈ 125 GeV. Let us finally turn to the Yukawa part of the Lagrangian in eq. (2.1): Here we suppressed fermion flavor indices and defined ϕ j = ǫ jk (ϕ k ) ⋆ , using the totally antisymmetric ǫ jk with ǫ 12 = +1. The Yukawa couplings Y f 1,2 are understood to be 3 × 3 matrices in flavor space. If Y f 1 and Y f 2 are simultaneously non-zero, in general flavor changing neutral currents arise [82][83][84]. However, there are four 2HDMs with natural flavor conservation (see table 2) where only one of these couplings is present. As for CPconservation in the potential, this can be achieved by an appropriate Z 2 charge assignment to right-handed fermions. When we discuss the extension to dimension six, we will assume that the terms are made Z 2 -invariant in the same way.

Gauge invariant operators
In this section we list the independent gauge invariant operators up to dimension six in the 2HDM-EFT. They are defined before the EW symmetry breaking takes place, meaning that they are given in the interaction basis, as the mass basis is not yet defined. After EW symmetry breaking, the fermions acquire masses and also the Higgs mass matrices receive additional contributions compared to the 2HDM with dimension-four operators only.
At dimension five the generalization of the Weinberg operator reads where C denotes the charge conjugation matrix. The procedure we follow to obtain a complete set of independent operators at dimension six is the same as the one applied and thoroughly described in ref. [23] for SM-EFT. Obviously, the operators involving no Higgs doublets do not change compared to SM-EFT, and for them we refer the reader to ref. [23]. Using classical equations of motion, neglecting total derivatives, and imposing the constraint of vanishing total hypercharge, we derived a set of independent operators, which we classify like in the case of the SM-EFT as follows: • ϕ 6 : operators with Higgs doublets only (table 3), which modify the Higgs potential. We assumed that these operators respect the Z 2 symmetry present at dimension four.
• ϕ 4 D 2 : operators with four Higgs doublets and two derivatives (table 4), which modify the kinetic terms of the Higgs fields, the Higgs-gauge boson interactions and the W and Z masses.
• Ψ 2 ϕX: operators with two fermion fields, one field strength tensor and one Higgs doublet (table 5), which give rise to dipole interactions after EW symmetry breaking.
• Ψ 2 ϕ 2 D: operators with two fermions fields, two Higgs doublets and one covariant derivative (table 7), which contribute to the fermion-Z and fermion-W couplings after EW symmetry breaking.
As in the case of the dimension-four Lagrangian, we assume that the discrete Z 2 symmetry for operators involving Higgs and fermion fields is restored by an appropriate charge assignment to right-handed fermions. Concerning table 4, we stress that, as in the case of one Higgs doublet, operators with derivatives acting on two conjugated or two unconjugated fields are not independent.

Physical basis
In this section we discuss the modifications of the Higgs potential and the relation between the Yukawa couplings and the fermion masses induced by the dimension-six contributions.    Table 6. Operators with two scalar fields and two field strength tensors. X denotes G A , W I or B.

Kinetic terms, Higgs and gauge boson masses
The effect of the operators in table 4 on the kinetic terms of the Higgs fields amounts to The kinetic terms are made canonical by the shifts

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The operators with gauge and Higgs fields in table 6 lead to the shifts where c w and s w denote cosine and sine of the weak mixing angle, respectively. These relations, together with the ϕD operators, affect the W and Z masses as where v 2 ≡ v 2 1 + v 2 2 . The shift of the Higgs fields applied to the dimension-four potential and the ϕ 6 operators modify the Higgs mass terms in the Lagrangian. Choosing again to eliminate m 2 11 and m 2 22 via the minimization conditions, we obtained with ∆m 2 η = ∆m 2 ϕDη + ∆m 2 ϕ 6 η ∆m 2 ρ = ∆m 2 ϕDρ + ∆m 2 ϕ 6 ρ ∆m 2 φ ± = ∆m 2 ϕDφ ± + ∆m 2

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and as well as and ∆m 2 Interestingly, the mass matrices for the CP-odd and the charged Higgs bosons are not diagonalized anymore by the angle β as in the case of the dimension-four potential. For this purpose, one needs two angles β η and β φ ± , which do not satisfy tan β η,φ ± = v 1 /v 2 .

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This relation holds at dimension four but is broken by the ϕD operators. However, in the presence of and ϕ 6 operators only, the CP-odd and the charged Higgs mass matrices would still be diagonalized by β defined as tan β = v 1 /v 2 . Also the angle α diagonalizing the CP-even mass matrix gets modified compared to the case of 2HDM without dimensionsix operators. However, this effect can be accounted for by an appropriate redefinition of α, which is anyway a free parameter in the 2HDM. 6 The eigenvalues of the mass matrices also change, except those corresponding to the pseudo-Goldstone bosons, which are still zero at O(1/Λ 2 ), as we checked. Therefore, the effects on the eigenvalues can be absorbed into the definitions of m h , m H , m H ± and m A .

Yukawa sector
After EW symmetry breaking, the fermion mass matrices in the presence of the dimensionsix operators of table 8 are given by with f = e, d, u. Here m f , Y f 1,2 as well as the coefficients of the dimension-six operators are in general arbitrary 3 × 3 matrices in flavor space. However, since the eigenvalues of m f are the physical fermion masses, by working in the basis where m f is diagonal, the rotations that map onto this basis get implicitly absorbed in the definitions of Y f 1,2 and C f ϕ . Depending on which version of the four 2HDMs with natural flavor conservation we are interested in (or which one is assumed to be the limiting case of the 2HDM of type III 7 ), one can choose to eliminate either Y f 1 or Y f 2 from the Yukawa Lagrangian. Afterwards, it is straightforward to calculate the couplings of φ + 1,2 , η 1,2 and ρ 1,2 to fermions. Taking as an example the lepton Yukawa couplings and eliminating Y e 1 , we find with Nevertheless, one should keep in mind that in the limit of heavy Higgses A 0 , H 0 , H ± , α has to vanish as tan β → ∞ modulo corrections of order v 2 /m 2 H . When dimension-six operators are included, this is still the case but additional (even smaller) corrections of order v 2 /Λ 2 are present. 7 Due to stringent flavor constraints (see ref. [85] for a recent analysis), a 2HDM with generic Yukawa couplings (i.e. of type III) should only slightly differ from one of the four 2HDMs with natural flavor conservation.

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and P R denoting the right-handed projector. The outcome in the case that Y e 2 is eliminated is obtained by simply interchanging 1 ↔ 2. In order to express these couplings in terms of the physical Higgs fields, the rotations by (redefined) α, β η and β φ ± have to be applied. The analogous of eq. (4.14) for down-quarks has exactly the same structure. For up-type quarks all terms involving a charged or CP-odd Higgs switch sign and φ + 1,2 get replaced by φ − 1,2 . We can now point out an interesting effect arising in the 2HDM-EFT related to Higgspair production [86][87][88][89]. In the SM-EFT there is only one operator giving rise to fermionfermion-Higgs-Higgs interactions and this is directly correlated to fermion-fermion-Higgs couplings [60,90]. Therefore, its effect in the four-particle interaction is limited due to the constraints on the fermion-fermion-Higgs couplings. In our 2HDM-EFT three terms affect the fermion-fermion-Higgs couplings while the part of the Lagrangian relevant for Higgs pair production is (4.16) By comparing with eq. (4.14), it is clear that a cancellation between C 222 f ϕ and C 122 f ϕ could have quite different effects in fermion vertices with one or two Higgses. Therefore a suppressed modification to fermion-fermion-Higgs couplings accompanied by a sizable effect in pair production of SM-like Higgs bosons is possible in 2HDM-EFT.

Conclusions and outlook
In this article we extended the SM-EFT approach to NP to the case in which two Higgs doublets are dynamical degrees of freedom (2HDM-EFT). In this framework the effects of additional heavy particles (e.g. heavy SUSY partners in the MSSM) are parameterized in terms of higher-dimension operators and their Wilson coefficients. Our analysis enables systematic studies of the role played by additional degrees of freedom beyond the 2HDM field content. We derived a complete set of independent gauge-invariant dimension-six operators with two Higgs doublets of the same hypercharge under the assumption of a Z 2 symmetry involving the Higgs field and the right-handed fermions. These operators modify the Higgs potential and the relation between the fermion masses and Higgs-fermion-fermion couplings. We performed the transition to the physical basis by re-diagonalizing the Higgs, gauge boson and fermion kinetic terms and mass matrices. We showed that the CP-odd and charged Higgs mass matrices are in general not diagonalized by one angle β (defined by tan β = v 1 /v 2 ) but rather by two different angles. Finally, we derived the expressions for the Higgs-fermion-fermion couplings in the presence of dimension-six operators.
Even though a study of phenomenological applications of our framework is beyond the scope of this article, we pointed out one interesting example illustrating the differences between 2HDM-EFT and SM-EFT. In the latter, there is only one dimension-six operator giving rise to fermion-fermion-Higgs-Higgs interactions and its effect on pair production of SM Higgses is limited by constraints on Higgs-fermion-fermion couplings. In the 2HDM-EFT instead, there are three operators entering Higgs-fermion-fermion couplings, whose contributions can cancel each other. The same operators, as well as additional ones, also con-

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tribute to the couplings of Higgs pairs to fermions but with different prefactors. Higgs pair production is decoupled from single-Higgs-fermion interactions and could thus be sizable.