Two-Higgs Doublet Model Matched to Nonlinear Eﬀective Theory

We use functional methods to match the Two-Higgs Doublet Model with heavy scalars in the nondecoupling regime to the appropriate nonlinear eﬀective ﬁeld theory, which takes the form of an electroweak chiral Lagrangian (HEFT). The eﬀective Lagrangian is derived to leading order in the chiral counting. This includes the loop induced h → γγ and h → Zγ local terms, which enter at the same chiral order as their counterparts in the Standard Model. An algorithm is presented that allows us to compute the coeﬃcient functions to all orders in h . Some of the all-orders results are given in closed form. The parameter regimes for decoupling, nondecoupling and alignment scenarios in the eﬀective ﬁeld theory context and some phenomenological implications are brieﬂy discussed.


Introduction
Indirect effects of New Physics (NP) at colliders can be consistently described with effective field theories (EFTs), where the new heavy particles are integrated out.Applying this approach to electroweak symmetry breaking and Higgs-boson properties, the nonlinear EFT in the form of an electroweak chiral Lagrangian (EWChL, also refered to as nonlinear Higgs-sector EFT, or HEFT) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] provides us with the most natural framework [20].It is economic and general and properly accounts for nondecoupling effects in the scalar sector.While the EFT is model independent, matching its parameters to a specific scenario connects the EFT coefficients to a given UV theory.Recently, there has been renewed interest in the Two-Higgs Doublet Model (2HDM) [21] and the treatment of its properties at the electroweak scale in an EFT approximation [22][23][24][25] (for earlier work see e.g.[26]).Our motivation for addressing this topic is essentially twofold.First, we would like to investigate the description of the 2HDM in the nondecoupling regime, which corresponds to interesting regions of parameter space.Second, our analysis exemplifies the structure of the Higgs-EWChL in the context of the 2HDM as a prototypical extension of the Higgs sector.In addition, we use functional methods throughout, which make the calculations rather efficient and transparent.Exploiting the advantages of the functional approach, we go beyond the existing literature in computing higher terms in the Higgs functions, including some all-orders results in powers of the Higgs field h, and an algorithmic prescription for their general derivation.The method used in the present study has been developed in detail in [27], where it was applied to the matching of a singlet extension of the SM to the nonlinear EFT.
The paper is organized as follows.In Section 2 we introduce polar coordinates for the scalar sector of the 2HDM, which are especially convenient for the matching to the nonlinear EFT.In Section 3 we perform the matching of the 2HDM in the nondecoupling regime to the leading-order (LO) chiral Lagrangian at tree level, integrating out the heavy scalars by functional methods.The matching calculation is extended to the one-loop induced h → γγ and h → Zγ local EFT operators in Section 4. Section 5 summarizes important aspects of the 2HDM parameter space with heavy-scalar masses (of order TeV), including the decoupling, nondecoupling and alignment regimes.Some phenomenological implications are discussed in Section 6, before we conclude in Section 7.An Appendix contains the solution H 0 (h) of the LO equations of motion (eom) for the heavy scalar field H 0 to all orders in h (Appendix A), the one-loop matching for h → γγ and h → Zγ to all orders in h (Appendix B), and explicit expressions for the parameters of the 2HDM scalar potential (Appendix C).

2HDM scalar sector in polar coordinates
The scalar sector of the 2HDM consists of two complex doublets H 1 , H 2 , both in the fundamental representation of the weak gauge group SU(2) and with weak hypercharge Y = 1/2.It is convenient to define the conjugate doublets Hn ≡ iσ 2 H * n , with n = 1, 2, and the matrix fields The Lagrangian of the scalar sector can then be expressed as where . . .denotes the trace, a sum over n is understood, and V is the potential to be discussed below.Following [28], the matrix fields S n can be written in polar coordinates as Here σ a ≡ 2T a , a = 1, 2, 3, are the Pauli matrices, and U ≡ exp(2iϕ a T a /v) is the matrix of the electroweak Goldstone bosons, where v = 246 GeV is the electroweak vacuum expectation value (vev).The vevs of the two Higgs doublet fields are v 1 and v 2 , respectively, with v 2 1 + v 2 2 = v 2 , and Using the decomposition in (3), the eight real degrees of freedom in the complex doublets H n are expressed through the eight real fields ϕ a , ρ a and h n .The electroweak quantum numbers of S n and U imply that the covariant derivative reads where W µ = W µ a T a and B µ are the gauge fields of SU(2) L and U(1) Y .It follows from (3) that Consequently, h 1,2 and ρ 3 are electroweak singlets, whereas ρ 1,2 are singlets of SU(2) L , but charged under U(1) Y .Hence, For a = 1, 2, this can also be written in terms of the eigenstates ρ ± of charge and hypercharge (with Q = Y = ±1) as Inserting (3) into (2), the kinetic term becomes The potential in (2) can be written as [28] V in terms of the matrix fields S n from (1).P ± = (1 ± σ 3 )/2 are projection operators.
Here we assume invariance under S 1 → −S 1 , S 2 → S 2 , softly broken by m 2 12 , and CP invariance, so that all parameters in (10) are real.
When S n is expressed as in (3), the Goldstone field U disappears from the potential V in (10), which becomes a function of h n and ρ a .The vevs v 1,2 are defined such that terms linear in h 1,2 vanish.The terms quadratic in the fields are diagonalised by ρ ± , ρ 3 , and by h and H, which are related to h 1,2 by Here and in the following, we define cos φ ≡ c φ and sin φ ≡ s φ for generic angles φ.The mass eigenstates of the scalar sector are then given by h, identified as the observed Higgs at m h = 125 GeV, and the additional scalars H ≡ H 0 , H ± ≡ ±iρ ± and A 0 ≡ −ρ 3 .The eight parameters of V in (10), m 2 11 , m 2 22 , m 2 12 , λ 1 , . . ., λ 5 , can be traded for the vevs, the particle masses, the Higgs mixing angle and the soft breaking term: Dropping an irrelevant additive constant, the potential finally takes the form The coefficients can be found in Appendix C. The scalar sector couples to fermions through Yukawa interactions.We assume a type-II Yukawa sector given by the Lagrangian [21] where q L = (u L , d L ) T and ℓ L = (ν L , e L ) T are the left-handed doublets, and u R , d R , e R the right-handed singlets.The latter may be collected into q R = (u R , d R ) T and ℓ R = (ν R , e R ) T .We suppress generation indices, which are understood for the fermion fields in (15).L Y can be written in terms of the matrix fields S n in (3) as 3 Tree-level matching in the nondecoupling regime The full Lagrangian of the 2HDM can be written as where the scalar sector is represented by L S,kin , V and L Y from ( 9), ( 14) and ( 16), and L 0 denotes the unbroken Standard Model (SM), In terms of the model parameters in (12), (13) the nondecoupling limit is defined by the hierarchy with t β and c β−α of order unity in general.To leading order, all terms in the effective Lagrangian that are unsuppressed by the heavy scale M S (of order (M S ) 0 ) have to be retained.
The procedure of integrating out the heavy scalars at tree level in the nondecoupling scenario has been described in detail in [27].It consists of the following steps: • The equation of motion (eom) is solved to obtain the heavy field H 0 (h) to LO in the heavy-mass limit, O(M 0 S ).This requires the LO terms in the full-theory Lagrangian of order M 2 S .A closed-form solution for H 0 (h) is derived in Appendix A. The O(M 2 S )-Lagrangian contains the heavy fields A 0 and H ± only at quadratic order or higher.Contributions with only internal lines from these fields, therefore, cannot arise at tree level.Integrating them out at tree level and to LO then implies A 0 = H ± = 0.
• The eom solutions H 0 = H 0 (h) and A 0 = H ± = 0 are inserted into the Lagrangian (17).The O(M 2 S )-terms cancel and an expression of O(M 0 S ) in the heavy-mass expansion is obtained.
is performed to achieve a canonically normalized kinetic term for the Higgs field h.
For notational convenience we will drop the tilde in the end, taking h → h.
Proceeding in this way, the effective theory takes the form of an electroweak chiral Lagrangian at chiral dimension two, where The mass matrices M q are related to the Yukawa matrices in (16) through The expressions for the charged leptons, M e , M e , are similar to those for the downquark case.
Our method reproduces the results of [25] and gives several new expressions, the cubic coefficient of F U , the coefficient of h 5 in V (h) and the fermionic couplings.More generally, the procedure summarized at the beginning of Section 3, together with the allorders expression for H 0 (h) in Appendix A, defines an algorithm to extend the tree-level matching to all orders in h.

Other Yukawa interactions
Besides the type-II Yukawa interactions discussed above, there are three other possibilities without tree-level flavour changing neutral currents.Conventionally, these are given by • Type-X (lepton-specific) • Type-Y (flipped) Using the results of the matching for the type-II 2HDM, it is straightforward to find the matching for the other Yukawa structures.For example, in the type-I model, all terms depend only on S 2 , so the matching will have the same form as the up-type terms of the type-II 2HDM.As a result, we find for the type-I 2HDM with M u,d,e = vY u,d,e s β / √ 2. It is straightforward to obtain similar expressions for the type-X and type-Y models.

Nondecoupling effects at one loop
The procedure of integrating out the heavy scalars can be extended to one loop using functional methods [29].The most important effects at this order are the local operators inducing h → γγ and h → Zγ transitions, because they are loop suppressed in the SM.The EFT corrections are then at the same loop order as the leading contributions.In the 2HDM, the contributions to h → γγ and h → Zγ due to the heavy sector come from charged scalars H ± within the loop, or equivalently from the real fields ρ 1,2 in (3).To obtain the one-loop contributions with internal ρ i from functional integration, the Lagrangian in (2) has to be expanded to quadratic order in these fields.The quadratic piece takes the form where with i, j ∈ {1, 2}, ε ij the two-dimensional Levi-Civita symbol, and Here e is the electromagnetic coupling, s W = sin θ W , c W = cos θ W with the Weinberg angle θ W , A the photon and Z the Z-boson field.Performing the Gaussian integration of exp(i d 4 x L (2) ρ 12 ) over ρ i gives the effective Lagrangian [29] In a weakly-coupled model of the heavy sector, a generic matrix Ŷ scales at most with the first power of M H .The series in (34) will then converge and only a finite number of terms will contribute to any given order in the 1/M H expansion.By contrast, in the present nondecoupling scenario we have Ŷ ∼ M 2 H , which is of the same order as the denominator p 2 − M 2 H . Therefore, an infinite number of terms in the sum over n contributes at a given order in the 1/M H expansion.However, higher powers of Ŷ come with higher powers of h, since Ŷ n = O(h n ) in the field expansion.As a consequence, the infinite series generates a Higgs-function F O (h) that accompanies an EFT operator O, as it is characteristic for the Higgs-electroweak chiral Lagrangian.At any given order in h n , the operator coefficient is well-defined and calculable.
Following this reasoning, we can extract the terms of interest here from (34).These contain two factors of the field strength Xµν , corresponding to four covariant derivatives D, along with powers of Ŷ .Neglecting contributions with three or more powers of h, we need to include terms of order Ŷ and Ŷ 2 .
The result is given by which simplifies to Using (33) and eliminating H in favour of h, we obtain with We note that the field redefinition of h, needed to make its kinetic term canonically normalized, plays no role for F X through order h 2 .The first term ∼ h in F X (h) agrees with the result of [25], the term ∼ h 2 is new.Employing the procedure described above, it is straightforward to extend the calculation of F X to higher orders in h.As discussed in Appendix B, in the alignment limit c β−α = 0 the function F X , to all orders in h, takes the simple form corresponding to the well-known low-energy theorems [30].

Custodial symmetry breaking
The scalar potential in (10) contains the custodial-symmetry violating term [21] ∆V When integrating out the heavy scalars, this term generates the two-derivative operator but only at the one-loop level.The coefficient is directly related to the parameter T of oblique electroweak corrections, β 1 = αT /2, with α the fine structure constant.One finds, up to a factor of order unity, that [21] In accordance with the phenomenological requirement of approximate custodial symmetry, L β 1 cannot be a leading-order effect.Therefore, the difference λ 4 − λ 5 must be a weak coupling of O(1) and carries chiral dimension 2. L β 1 is then counted as a next-to-leadingorder (NLO) term of chiral dimension 4, consistent with β 1 being small as a loop factor 1/16π2 [12].The general analysis of such NLO effects is beyond the scope of the present paper.
5 Parameter space and the decoupling limit For the construction of a low-energy EFT, we consider the phenomenologically viable scenario where the masses of the new scalar degrees of freedom in the 2HDM are taken to be much larger than the electroweak scale, i.e.
Depending on the numerical values of the parameters, we can discern two basic scenarios, corresponding to weak and strong coupling, respectively.They are given by • Nondecoupling regime1 (strong coupling, nonlinear EFT) While c β−α is a priori unconstrained in this regime, we will also consider the case c β−α ≪ 1, referred to as the nondecoupling regime with (quasi-)alignment.We also note that the model with m = 0, the Z 2 symmetric 2HDM without soft breaking, has no decoupling limit [31][32][33][34].
• Decoupling regime 2 (weak coupling, linear EFT) In the strong-coupling case, we require the λ i to be somewhat below the nominal strongcoupling limit M S ≈ 4πv corresponding to |λ i | ≈ 16π 2 .Otherwise a description of the heavy scalar dynamics in terms of resonances would no longer be valid.To be more precise, the magnitude of the couplings is constrained by perturbative unitarity [35][36][37][38][39][40][41].
For loop corrections to the constraints, see [42][43][44].Generally speaking, these give much stronger bounds, namely |λ i | 4π.Furthermore, the couplings are constrained such that the potential is bounded from below and that the symmetry breaking vacuum is the global minimum of the potential.For the 2HDM with (softly broken) Z 2 symmetry, the necessary and sufficient conditions on the couplings read [45][46][47][48][49] To satisfy these bounds, the absolute values of the couplings have to be taken large uniformly, which limits the possible mass splitting between the heavy scalars.Especially the perturbative unitarity constraints severely restrict the possible parameter space of the nondecoupling regime.Nevertheless, masses of M S 1 TeV are still possible for m ∼ v, which clearly fulfills the power counting of the nondecoupling scenario.
In the decoupling regime, all NP effects are suppressed by powers of the heavy mass scale M S as formalised by the Appelquist-Carazzone decoupling theorem [50].Several EFT matching calculations have been performed in the decoupling limit, see e.g.[24,51,52].A decoupling regime automatically implies the alignment limit c β−α = 0, where the h-couplings approach their SM values [31].An explicit calculation gives with λ 345 ≡ λ 3 +λ 4 +λ 5 .When m ≫ v, this indeed approaches zero.As mentioned above, there is no similar relation in the nondecoupling regime, and thus, c β−α is unconstrained a priori.
To illustrate the two regimes discussed above, we take the hH 2 0 -coupling d 3 , given in Appendix C, as an example.In the nondecoupling regime, M 0 ∼ M S ≫ m h , m, so S ), whereas in the decoupling regime, the masses and parameters of the model scale as leading to Evidently, all heavy mass dependence has cancelled.Similar calculations show that this cancellation works for all d i and z i .It is now easy to see that all nondecoupling effects vanish in the decoupling-regime.Obviously, all tree-level nondecoupling effects vanish in the decoupling limit, since they are all proportional to c β−α .Also the anomalous hγγ-and hZγ-couplings disappear as the ratios d i /M 2 S and z i /M 2 S go to zero in the limit M S → ∞.

Phenomenological considerations
The simplest way to confront the nondecoupling effects of the 2HDM with experiment is by using a global HEFT fit.Such a fit has been performed using LHC run 1 and 2 data [53], where the authors fit the couplings of the HEFT Lagrangian in the form Table 1: LO matching results for the 2HDM.c u is the same for all up-type quarks (u, c, t) and c d is the same for all down-type quarks (d, s, b) and charged leptons (e, µ, τ ).
with ψ ∈ {t, b, c, τ, µ}3 .Our matching results are given in Table 1.The strongest constraint is derived from the Higgs-vector boson coupling where the given error corresponds to the 68% probability interval.This motivates the (quasi-)alignment limit as it constrains c β−α ≪ 1.
Applying the above bound to the anomalous Higgs-photon coupling, we find This coupling is particularly important, as it is bounded from below in the alignment limit.We see here that this is a direct consequence of the bound on the Higgs-vector boson coupling.From the global HEFT fit, the bound on c γ is given by c γ = 0.05 ± 0.20 (51) which is consistent with the matching prediction.Nevertheless, with more data from the LHC, it is plausible that the limits on c γ could be sufficiently improved to exclude the nondecoupling regime experimentally.Local couplings with more than one Higgs in (38) and (39), such as h 2 A µν A µν , could in principle be probed at a photon collider [54] in a process like γγ → hh.Aside from using an EFT approach, there is a large amount of literature using global fits for the 2HDM directly.Depending on the structure of the Yukawa interactions, these fits can give much stronger bounds on s β−α than the global HEFT fit (see e.g.[55,56]).However, a detailed analysis lies beyond the scope of this work.

Conclusions
We presented a systematic derivation of the EFT at the electroweak scale for the 2HDM in the nondecoupling regime.In this regime, the EFT takes the form of an electroweak chiral Lagrangian (nonlinear EFT).Our discussion follows closely the detailed discussion given in [27] for the nondecoupling regime of the SM extension with a heavy scalar singlet.The scalar sector of the 2HDM is written in polar coordinates, with a nonlinear representation of the Goldstone fields, which facilitates the use of functional methods that we employ throughout.An algorithmic procedure is given, by which the LO EFT Lagrangian can be worked out to arbitrary order in the Higgs field h.We confirm previous results for the EFT Higgs couplings and extend the derivation to additional terms.The main results are displayed in ( 21) -( 25) and ( 38), (39).Some all-orders expressions are given in closed form (Appendix A and B).We derive the LO EFT Lagrangian, including the fermionic Yukawa interactions and the loop-induced local terms for h → γγ and h → Zγ.As already pointed out in [25,57], the latter terms have interesting nondecoupling effects that survive in the alignment limit.Those are still compatible with present data.They could be discovered or ruled out in future measurements of anomalous Higgs-boson couplings.

Note added
While this paper was being finalized, the article [58] appeared on arXiv.It also addresses the HEFT matching of models with extended scalar sectors and partially overlaps with our results on the 2HDM.

A Exact solution for H 0 (h)
The LO term for H is calculated from the equations of motion at O(M 2 S ).In particular, we can set H ± = A = 0 in this approximation.As a result, retaining only O(M 2 S ) terms the Lagrangian simplifies to where After expressing h 1 and h 2 through h and H, the two fields take the form the Lagrangian (A.1) can be rewritten as At this point we can solve the eom by analogy to the heavy singlet model studied in [27].
By direct comparison to the results in the Appendix of [27], we can identify which exactly reproduces the corresponding terms of the heavy singlet model.The solution to the eom is then given by This expression fulfills which, when inserted back into the Lagrangian (A.5), shows that the O(M 2 S )-terms cancel up to a constant.Furthermore, the solution starts at O(h 2 ) with coefficients that are functions of s α , c α , s β and c β .Note that the combination vanishes in the alignment limit.Then, the square root in (A.7) reduces to 1, which gives H 0 (h) = 0.As a result, all tree level nondecoupling effects vanish in the alignment limit.
B One-loop matching of h n X µν X µν to all orders in n When calculating the one-loop EFT contributions of the form h n X µν X µν , we noted that, since Ŷ = O(M 2 H ), the series does not converge.Therefore, in order to calculate the full Higgs function associated with the operator X µν X µν , we need all coefficients C n of the expression In writing the above, we made essential use of the fact that Ŷ ∝ 1 and thus commutes with Xµν .This is, however, a special case.In general, Ŷ does not commute, giving more possible operator structures for each n.
To derive an all-orders result for the C n , we start from the general expression for the one-loop effective Lagrangian given in (34).We now use a slightly adapted form of a trick explained in the Appendix of [59]: We evaluate expression (34) in the special configuration ∂ µ Xν = ∂ µ Ŷ = 0, allowing us to drop all derivatives.In this case where Ĝ is any matrix valued function of Ŷ and Xµ .In the final expressions, we can express everything through D µ and Xµν , regaining the general result.
In our special case, we are only interested in the terms of the form Ŷ n Xµν Xµν , which contribute to the h n γγ nondecoupling effects.Setting [ Xµ , Ŷ ] = 0 automatically removes all terms of the form D µ Ŷ .
In this way, it is easy to evaluate all terms from (34) with 4 derivatives (4 factors of X µ ) and n factors of Y , which reduce to the terms of interest due to the formal gauge invariance of the functional integral.Finally we obtain 3 The first two terms in the sum over n agree with those given in [29] and the third and fourth terms agree with those given in [60], in the special case that Ŷ and Xµν commute.
The second expression in (B.3) represents the full Higgs function after taking and expressing h in terms of the canonically normalized Higgs field h, h = h( h), through the inverse of the function h(h) defined in (20).Here d (0) 5 is the O(M 2 S )-part of d 5 , and similarly for the other coefficients, and H 0 (h) is the function derived in Appendix A.
In the alignment limit c β−α = 0, the Lagrangian in (B.3) can be given in closed form to all orders in h.In this case, H 0 (h) vanishes and Y in (33)  2 .Then (B.3) takes the form of (38) with the function F X (h) given by (40).

C Parameters of the 2HDM potential
The full scalar potential is given in (14).In this section we give all coefficients of the potential in terms of the input parameters The cubic couplings read and the quartic couplings are given by The absence of a decoupling limit for m = 0 is obvious from these formulas.