Higgs masses of the general 2HDM in the Minkowski-space formalism
aa r X i v : . [ h e p - ph ] O c t Higgs masses of the general 2HDM in theMinkowski-space formalism
A. Deg´ee and I. P. Ivanov , IFPA, Universit´e de Li`ege, All´ee du 6 Aoˆut 17, bˆatiment B5a, 4000 Li`ege, Belgium Sobolev Institute of Mathematics, Koptyug avenue 4, 630090, Novosibirsk, Russia
November 4, 2018
Abstract
We study the masses of the Higgs bosons in the most general two-Higgs-doubletmodel in a basis-independent approach. We adapt the recently developed Minkowski-space formalism to this problem and calculate traces of any power of the mass-matrix ina compact and reparametrization-invariant form. Our results can be used to gain insightinto the dynamics of the scalar sector of the general 2HDM.
The two-Higgs-doublet model (2HDM) is one of the simplest extensions of the Higgs mech-anism of the electroweak symmetry breaking beyond the Standard-Model, [1, 2, 3]. In thismodel one introduces two doublets of Higgs fields, φ and φ , which interact with the matterfields and also self-interact via an appropriate Higgs potential.Higgs potential of 2HDM contains many free parameters, which are not constrained byexperiment. Extensive studies conducted over past decades have shown that playing evenwith a small subset of these free parameters one can get a rich spectrum of models withdifferent phenomenologies (see examples in [3, 4]).Recently, it has become clear that not all of these free-parameters are equally important.One has a certain freedom in choosing the basis in the Higgs field space when writing thelagrangian. This basis change shifts the values of the parameters of the lagrangian, but byconstruction it has no effect on the physical observables. Thus, it is only the basis-invariantfeatures of the theory, and not the entire set of free parameters, that really shapes the phe-nomenology of the model.When using reparametrization transformations, one is immediately led to the most general2HDM, whose Higgs potential contains all possible electroweak-invariant quadratic and quarticcombinations of the two doublets. Several groups have recently focused on the properties ofthe general 2HDM and developed a set of basis-invariant tools adequate for this task. Themotivation behind this interest is not to provide the most accurate description of the real world,but rather to understand the whole spectrum of possibilities offered by the second doublet.1n this way, the most general 2HDM with no a priori restriction on its free parameters shouldbe viewed as a useful tool for building specific models with predefined properties, and it isdefinitely worth studying in as much detail as possible. The main problem with the general 2HDM is that it cannot be worked out with straightforwardalgebra. The obstacle arises at the very first step: when minimizing the Higgs potential, onearrives at algebraic equations of high order, which cannot be solved in the general case. Inthis situation, any method that would give any non-trivial insight into the model is welcome.Following the early suggestion of [5], a very elaborate basis-independent treatment of gen-eral 2HDM was presented in [6] and further developed in [7, 8]. In this approach one writesthe Higgs potential as V = Y ab ( φ † a φ b ) + 12 Z abcd ( φ † a φ b )( φ † c φ d ) , and manipulates with Y ab and Z abcd as tensors rather than just a collection of parameters.Instead of finding explicitly the vector of vacuum expectation values, v a , one adds it to the setof objects to manipulate with, keeping in mind, however, that it satisfies the extremum condi-tion. Along these lines, one can find several algebraically independent invariants constructedas full contractions of the available tensors, and some of the properties of the model could beseen through the prism of these invariants.Unfortunately, this powerful technique lacks intuition, as the results arise from lengthy(and often computer-assisted) algebra of invariant polynomials. A more appealing approachto the general case was suggested and developed in [9, 10, 11, 12, 13, 14]. In this approachone works not in the space of Higgs fields φ a , but in the real four-dimensional space of gauge-invariant bilinears ( φ † a φ b ) (the orbit space), which has the Minkowski-space signature. Manyof properties of the Higgs potential can be derived in a very intuitive way based on simplegeometric considerations.This approach was developed further in [15, 16], where the reparametrization group wasextended also to non-unitary transformations of the fields. In the 1 + 3-dimensional orbitspace it leads to the full Lorentz group of transformations, and this freedom provides evenmore insight into the properties of the 2HDM potential. In particular, many of the statementsabout the general 2HDM are much more naturally formulated in terms of the four eigenvaluesof the Minkowski tensor Λ µν (see the next Section) rather than its space-like part. Anotherkey point of [15, 16] was that all essential results were formulated exclusively in terms ofthe parameters of the potential, without using the yet-unknown vacuum expectation values.For example, within this approach one could formulate conditions for the spontaneous CP -violation and draw the full phase diagram of the model exclusively in term of the parametersof the potential, without using the unknown vacuum expectation values.Thanks to all these approaches, we have now a fairly detailed understanding of the prop-erties of the Higgs potential and of the vacuum in the general 2HDM. The next step in the study of the general 2HDM is to understand its dynamics. This includesthe mass spectrum of the physical Higgs bosons, the pattern of their interactions, as well as2heir couplings to the fermions. All this must be done within a basis-independent approach.Let us stress once again that if one chooses a restricted Higgs potential, for example, anexplicitly CP -symmetric one, the entire calculation is drastically simplified. One can explicitlyfind the minimum of the potential and calculate the masses and the interaction of the Higgsbosons. This straightforward approach fails for the most general 2HDM, which calls uponmore involved techniques for the analysis of its properties.The mass spectrum of the general 2HDM was studied in a number of recent papers. Forexample, in [11, 14, 15] the mass matrix was explicitly calculated in a specific basis and notin a reparametrization-invariant form. An interesting study was also presented in [17], wherecertain bounds and relations between the masses and the parameters of the potential wereobserved, however that work relied only on numerical analysis. Finally, very recently a verydetailed account of the dynamics of the general 2HDM was presented in [8]. Among otherresults, explicit expressions of the mass matrix were derived in U (2)-invariant way in terms ofvarious full contractions of tensors Y ab and Z abcd as well as vacuum expectation values.In the present paper we show how to analyze the masses of the physical Higgs bosons inthe Minkowski-space formalism. We obtain compact SO (1 , In this work we focus on the scalar sector of 2HDM. The Higgs potential of the most generalrenormalizable 2HDM, V H = V + V , is conventionally parametrized as V = − h m ( φ † φ ) + m ( φ † φ ) + m ( φ † φ ) + m ∗ ( φ † φ ) i ; V = λ φ † φ ) + λ φ † φ ) + λ ( φ † φ )( φ † φ ) + λ ( φ † φ )( φ † φ ) (1)+ 12 h λ ( φ † φ ) + λ ∗ ( φ † φ ) i + nh λ ( φ † φ ) + λ ( φ † φ ) i ( φ † φ ) + h . c . o . It contains 14 free parameters, four in the quadratic and 10 in the quartic terms, whichmakes the phenomenology of 2HDM very rich even at tree level. However, not all pointsin this 14-dimensional space of parameters lead to distinct physics: if two sets of parame-ters can be mapped into each other by a certain linear transformation between the doublets(reparametrization transformation, or basis change), they will lead to the same physics, [6, 4].Usually one insists that the kinetic term be invariant, so one considers only global unitarytransformations between the two doublets, U (2). However, as shown in [15, 16], one can extendthis reparametrization group to the general linear group GL (2 , C ). The Higgs kinetic terms is3ot invariant under non-unitary transformations, but it can be treated in a reparametrization-covariant way, so that all the physical observables still remain invariant under this extendedreparametrization group. This approach has provided several new insights, which would bevery difficult to see using the more traditional unitary reparametrization group.Technically, the extended reparametrization group can be implemented as follows. Weswitch from the fields to bilinears and introduce the four-vector r µ = ( r , r i ) = (Φ † Φ , Φ † σ i Φ),where Φ = ( φ , φ ) T is a 2-dimensional vector of Higgs doublets and σ i are the Pauli matrices.This four-vector is gauge invariant and parametrizes the gauge orbits in the space of the Higgsfields. The general reparametrization group GL (2 , C ) can be written as C ∗ ⊗ SL (2 , C ), where C ∗ is the group of simultaneous multiplication of both φ i with the same complex number,while SL (2 , C ) is the special linear transformation group. It is the latter group that leads tonon-trivial transformations of the Higgs potential, which we now focus on.Transformations of Φ under SL (2 , C ) correspond to the SO (1 ,
3) transformations of r µ ,equipping the gauge orbit space with the Minkowski-space structure. It follows from thedefinition of r µ that r = ( φ † φ ) + ( φ † φ ) ≥ , r µ r µ = 4 h ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) i ≥ , (2)so that the physically realizable vectors r µ populate not the entire 1+3-dimensional Minkowskispace, but the future lightcone ( LC + ). The Higgs potential in the r µ -space can be written ina very compact form: V = − M µ r µ + 12 Λ µν r µ r ν . (3)Here the four-vector M µ is built from parameters m ij in (1), while the symmetric four-tensorΛ µν is constructed from the quartic coefficients λ i . Their explicit expressions as well as someproperties can be found in [13, 15, 16]. Here we just note the most important property of Λ µν for potentials stable in a strong sense : Λ µν can always be diagonalized by a certain SO (1 , r µ -space, and after diagonalization it takes formΛ µν = diag(Λ , − Λ , − Λ , − Λ ) with Λ > , Λ > Λ i , i = 1 , , , (4)where the inequalities among the eigenvalues result from the positivity constraint on thepotential. The minus signs in front of the “space-like” eigenvalues arise from the pseudo-euclidean metric in the orbits space.It is known that the potential (3) can have three types of minima: (i) the electroweak(EW) conserving, (ii) the EW-breaking but charge conserving (i.e. neutral), and (iii) the EW-and charge-breaking ones. One can use the v.e.v.s of the two doublets h φ i i to construct h r µ i .Then, the three type of minima correspond to: (i) h r µ i = 0 (the apex of the forward lightcone LC + ), (ii) h r µ i 6 = 0 but h r µ ih r µ i = 0 (the surface of LC + ), (iii) h r µ i 6 = 0 and h r µ ih r µ i > LC + ). The position of the charge-breaking extremum h r ν i ch is given by thefollowing equations: Λ µν h r ν i ch = M µ , (5) We use here the terminology of [11]: the potential is stable in a strong sense, if its quartic part increasesalong all rays starting from the origin in the Higgs field space. The potential is called stable in a weeak sense,if the quartic part has flat directions, but the quadratic potential increases along them. For the Minkowski-space analysis of potentials stable in a weak sense, see [18], where a similar condensed-matter problem wasconsidered.
4f Λ µν is not singular, a solution of this system always exists and is unique: h r µ i ch = m µ ≡ (Λ − ) µν M ν . However, the requirement that h r ν i ch lies inside the forward lightcone placesbounds on M µ that could yield physically realizable solutions. In addition, the charge-breakingextremum is minimum only if all Λ i < i = 1 , ,
3, i.e. if the tensor Λ µν is positive-definite inthe entire space of non-zero vectors r µ . When searching for the neutral extrema, we use theLagrange multiplier technique. The positions of all neutral extrema h r µ i are the solutions ofthe following simultaneous equations:Λ µν h r ν i − ζ µ = M µ , ζ µ ≡ ζ h r µ i , (6)where ζ is a Lagrange multiplier. This system can have up to six solutions, [10, 11, 15], amongwhich there are at most two minima, while the other are saddle points, [16].Finally, following [15], we write the Higgs kinetic term covariantly as K = K µ ρ µ , ρ µ = ( ∂ α Φ) † σ µ ( ∂ α Φ) , (7)where α denotes the usual space-time coordinates, while µ , as before, refers to the orbit space.The reparametrization transformation properties of ρ µ are the same as for r µ . In the “default”frame, K µ = (1 , , , SO (1 ,
3) transformation, K µ acquires non-zero “space-like”coordinates, however the condition K µ K µ = 1 is always satisfied. The four-vector K µ is notinvolved in the search for the minimum of the potential, however it affects the mass matrix atthis minimum. This generalized kinetic term effectively incorporates the non-diagonal kineticterm, which, as was argued in [19], must be introduced in the initial lagrangian to restorerenormalizability of the model. In the previous studies, [15, 16] the Minkowski-space formalism was used to understand variousproperties of the 2HDM lagrangian and of the vacuum state. The next logical step is to studythe dynamics of the model in an reparametrization-covariant way. In this paper we makea step towards fulfilling this program. We obtain expressions for the mass matrices of thephysical Higgs bosons in the most general 2HDM and study some of their properties.When doing so, we stick to the Minkowski space formalism, but we adapt it to our prob-lem. Although the masses are physical observables and are reparametrization-invariant, themass-matrix is, obviously, basis-dependent. So, for intermediate calculations we switch backfrom the bilinears to the Higgs fields themselves, derive the mass-matrix in a specific basis,then calculate the traces of the powers of this matrix, and return to the Minkowski-spaceformalism. Although the resulting equations do not yield the masses in a simple closed form,they nevertheless can be useful for the analysis of the general 2HDM.In the subsection devoted to the neutral vacuum below, we also comment on relation ofour results with some of the previous studies of the mass spectrum.
Let us denote the complex fields as φ i,α , where i = 1 , α = ↑ , ↓ indicates the upper and lower components in a given doublet. Let us then introduce the5-component real vector of scalar fields ϕ a , a = 1 , ...,
8, with the following components: ϕ a = (Re φ , ↑ , Im φ , ↑ , Re φ , ↑ , Im φ , ↑ , Re φ , ↓ , Im φ , ↓ , Re φ , ↓ , Im φ , ↓ ) . (8)The four-vector r µ can be rewritten in terms of ϕ a as r µ = ϕ a Σ µab ϕ b . (9)Here, Σ µ are four real symmetric 8-by-8 matrices; Σ is just the unit matrix, while explicitform of Σ i can be immediately reconstructed from the definitions (see Appendix). Since theupper and lower components of the doublets are not mixed by the Higgs potential, matricesΣ µ have a block-diagonal form, composed of identical 4-by-4 matrices. Below, we will oftendeal with these 4-by-4 matrices, denoting them by the same letter Σ µ . Which set of matricesis being used, 4-by-4 or 8-by-8, should be clear from the context.In contrast to σ µ , the matrices Σ µ do not form a closed algebra, but they belong to a largeralgebra (Σ µ , Π µ ), described in the Appendix. They also share with σ µ an important property: (cid:8) Σ i , Σ j (cid:9) = 2 δ ij · I , (10)where brackets denote the anticommutator. It follows then that if a regular real symmetric8-by-8 matrix A is written as a µ Σ µ , then its inverse is A − = a µ ¯Σ µ a µ a µ , ¯Σ µ ≡ (Σ , − Σ i ) . (11)Below we will encounter products of matrices Σ’s and ¯Σ’s. When simplifying these products,the following results prove useful:12 (Σ µ ¯Σ ν + ¯Σ ν Σ µ ) = g µν · I , (12)12 (cid:0) Σ µ ¯Σ ρ Σ ν + Σ ν ¯Σ ρ Σ µ (cid:1) = g µρ Σ ν + g νρ Σ µ − g µν Σ ρ . With this notation, we can give a compact expression for the mass matrix in a specific basis.Let us write the expansion of the scalar lagrangian near an extremum as L ≈ ( K ρ Σ ρab )( ∂ α ϕ a )( ∂ α ϕ b ) − H ab ( ϕ a − h ϕ a i )( ϕ b − h ϕ b i ) , H ab ≡ ∂ V∂ϕ a ∂ϕ b , where the hessian H ab is calculated at the extremum. The 8-by-8 mass matrix can then beexpressed as M ac = ( K ρ Σ ρ ) − ab H bc = K ρ ¯Σ ρab H bc . (13)In the rest of this Section we calculate this mass matrix and analyze its eigenvalues for thethree possible types of vacua: electroweak-symmetric, charge-breaking and neutral. The masses of the Higgs bosons in the electroweak-symmetric vacuum are determined onlyby the quadratic term of the potential and can be easily calculated in a straightforward way.The eight masses are grouped into two quartets with values (1) are m , = 14 (cid:18) ( − m ) + ( − m ) ± q ( m − m ) + 4 | m | (cid:19) . (14)6hese masses squared are positive, if m < m < m m > | m | . However, wefind it useful to work out this simple case in the reparametrization-covariant formalism justto illustrate how it works.The hessian H ab comes only from the M µ r µ term of the potential and is equal to − M µ Σ µab .The mass matrix is then M ab = K ρ ( − M µ )( ¯Σ ρ Σ µ ) ab . (15)Matrices Σ’s have a block-diagonal form, and therefore so does the mass matrix (15). It isbuilt of two identical 4-by-4 blocks ( M ) ab , with a, b = 1 , , ,
4, whose form is still given bythe same expression but now with 4-by-4 matrices Σ µ . In order to find its eigenvalues, let uscalculate the trace of its successive powers:Tr[ M ] = K ρ ( − M µ )Tr[ ¯Σ ρ Σ µ ] = − KM ) , Tr[( M ) ] = K ρ ( − M µ ) K ρ ′ ( − M µ ′ )Tr[ ¯Σ ρ Σ µ ¯Σ ρ ′ Σ µ ′ ]= 2( KM ) K ρ M µ Tr[ ¯Σ ρ Σ µ ] − K ρ K ρ ′ M Tr[ ¯Σ ρ Σ ρ ′ ] = 8( KM ) − K M , Tr[( M ) n ] = − KM )Tr[( M ) n − ] − K M Tr[( M ) n − ] . (16)These relations among the traces prove the mass matrix has only two independent eigenvalues.A simple analysis shows that there are two pairs of different eigenvalues, which are equal to m , = − ( KM ) ± p ( KM ) − M , (17)where we used K = 1. This expression is reparametrization-invariant and can be calculatedin any frame. In particular, in the original frame, where K µ = (1 , , , m , = − M ± | ~M | . (18)Using the definition of M µ , one can immediately recover (14). Eq. (18) also shows that inorder for the EW-symmetric extremum to be minimum, the four-vector M µ must lies insidethe backward lightcone. Let us now find the mass matrix of the general 2HDM in the case of a charge-breaking vacuum.The hessian has the following form: H bc = 2Λ µν Σ µbb ′ Σ νcc ′ ϕ b ′ ϕ c ′ . (19)All fields here must be understood as v.e.v.’s h ϕ a i , but to keep the notation simple, we willsuppress the brackets. Thus, the mass matrix can be written as M = 2 K ρ Λ µν ¯Σ ρ Σ µ ( ϕ ⊗ ϕ )Σ ν . (20)By construction, this is a 8-by-8 matrix. However, we know that it must have four flatdirections corresponding to the Goldstone modes. We shall now get rid of these four flatdirections by showing that there exists a 4-by-4 matrix M such that trace of any power of M is equal to the trace of the same power of M .7ndeed, consider the trace of M . Thanks to the properties of Σ’s, we haveTr [ M ] = 2 K ρ Λ µν ϕ Σ ν ¯Σ ρ Σ µ ϕ = 2 K ρ Λ µν ( g µρ m ν + g νρ m µ − g µν m ρ )= 2 K ρ m µ (2Λ ρµ − TrΛ g ρµ ) ≡ S · Λ] . Here, the matrix S · Λ is a symbolic form of the tensor S µα Λ αν ≡ S µα Λ αν , where S νµ ≡ K ν m µ + K µ m ν − ( Km ) g νµ . (21)Note that the matrix S · Λ is defined in the euclidean space, and although it contains the tensors S µα and Λ αν , they are contracted according to the usual rules of matrix multiplication.Consider now the trace of the square of M :Tr (cid:2) ( M ) (cid:3) = 4 K ρ Λ µν K ρ ′ Λ µ ′ ν ′ · ϕ Σ ν ¯Σ ρ ′ Σ µ ′ ϕ · ϕ Σ ν ′ ¯Σ ρ Σ µ ϕ . (22)Note that this expression does not factorize because Λ µν and Λ µ ′ ν ′ couple the first and thesecond threads of Σ’s. Consider one of these threads, e.g. ϕ a (Σ ν ¯Σ ρ ′ Σ µ ′ ) ab ϕ b . This is a quadratic form in ϕ a ; therefore, only the ab -symmetric part of the product of Σ’ssurvives. This effectively leads to the ν ↔ µ ′ symmetrization, and one can again apply (12)to obtain K ρ ′ ϕ a (cid:16) Σ ν ¯Σ ρ ′ Σ µ ′ (cid:17) ab ϕ b = S νµ ′ . The trace of the square of the mass matrix is thenTr (cid:2) ( M ) (cid:3) = 4Λ µν S νµ ′ Λ µ ′ ν ′ S ν ′ µ = Tr (cid:2) (2 S · Λ) (cid:3) . This calculation is easily generalizes to any power of the mass matrix:Tr [( M ) n ] = Tr [(2 S · Λ) n ] . (23)The fact that the trace of any power of M is equal to the trace of the same power of the 4-by-4 matrix 2 S · Λ, means that there are four zero-modes in M and that all the four non-zeroeigenvalues of M coincide with the eigenvalues of 2 S · Λ. Thus, the four eigenvalues of thematrix 2 S · Λ gives the masses squared of the physical Higgs bosons in the charge-breakingvacuum.There is no simple way to calculate the masses themselves. However, the product of allfour masses squared can be easily inferred from the above expression: Y i m i = det(2 S · Λ) = 16 det S · detΛ . (24)Both tensors here are written in the euclidean space. Determinant of euclidean Λ αβ is theproduct of the eigenvalues of Minkowski Λ µν : detΛ = Λ Λ Λ Λ . In order to calculate theother determinant, let us take a closer look at S µν . The way it is defined, Eq. (21), allows Note a subtlety here: in a generic basis, the eigenvalues of the euclidean matrix Λ αβ , which is not evensymmetric, are different from the eigenvalues of the Minkowski tensor Λ µν , i.e. Λ and Λ i . However, theproduct of all the eigenvalues of these two matrices are equal.
8s to immediately find its eigenvalues. Indeed, consider first a reduced version of this tensor, K µ m ν + K ν m µ . In general, K µ and m µ are non-parallel four vectors, both lying strictlyinside the forward lightcone. Within the subspace spanned by them, one can identify twoeigenvectors of this reduced tensor, e µ ± = K µ √ K ± m µ √ m , e µ + e − µ = 0 , whose eigenvalues are ( Km ) ±√ K m . Note that e µ + lies inside the forward lightcone, while e µ − lies outside it. In addition, there are two eigenvectors in the subspace orthogonal to K µ and m µ ,with zero eigenvalues. Since adding a term proportional to g µν does not change the eigenvectorsbut just shifts all the eigenvalues by a common constant, we get the following result: S µν isdiagonalizable by an appropriate SO (1 ,
3) transformation, and after diagonalization it takeform: S µν = diag( S , − S , − S , − S ) , S = √ m , S = −√ m , S = S = − ( Km ) . (25)Therefore, we obtain: Y i m i = 16Λ ( − Λ )( − Λ )( − Λ ) · m ( Km ) . (26)As said above, a charge-breaking extremum exists, if m µ lies inside the future lightcone, i.e.if m > Km ) >
0. It is also known that the charge-breaking extremum is a minimumif the tensor Λ µν is positive-definite in the entire Minkowski space, i.e. if all its spacelikeeigenvalues Λ , , are negative . Thus, all factors in (26) are positive.Another observation concerns cases when the potential has an explicit symmetry. Consider,for example, the lowest possible explicit symmetry, a Z -symmetry , which consists in reflectionof, say, second axis. This explicit symmetry means that K = 0, M = 0, and that Λ µ = 0for µ = 2. It is known that the position of the charge-breaking minimum preserves all thediscrete symmetries, so that m is also zero. In this case one can immediately calculate themass squared of the excitation that violates this symmetry: m = 2( − Λ )( Km ) . (27) Let us now consider the neutral vacuum. The four-vector r µ corresponding to a neutralvacuum must lie on the surface of the forward lightcone (again, we always refer to the v.e.v.’s,so that the brackets h· · ·i are implicitly assumed). Therefore, the minimization procedureinvolves a Lagrange multiplier ζ , which brings up a new lightcone four-vector, ζ µ , defined as ζ µ = Λ µν r ν − M µ = ζ · r µ . This new four-vector gives rise to an additional term in the massmatrix: M = 2 K ρ Λ µν ¯Σ ρ Σ µ ( ϕ ⊗ ϕ )Σ ν + K ρ ζ µ ¯Σ ρ Σ µ . (28) We checked that these conditions can be also inferred from the positive-definiteness of the mass matrixjust derived. This symmetry is known in the literature as a generalized CP -symmetry. The “conventional Z ” corre-sponds, strictly speaking, to a ( Z ) -symmetry of the potential, see details in [16]. ϕ a ) and neutral(the last four components of ϕ a ) modes, which do not mix.Before we proceed, let us note that essentially this expression for the mass matrix of themost general 2HDM, but with a trivial kinetic part, was obtained in other works, [11, 15, 8].All these papers followed then the standard procedure: one switches to the basis where onlythe first doublet has non-zero v.e.v. (the Higgs basis), and then the entries of the mass matrixcan then be written in a simple way via the parameters of the potential in this specific basis aswell as v . We show in this subsection that the basis-invariant features of the mass matrix canbe written in an SO (1 , µν -diagonalbasis. We checked that in the canonical basis, our result reproduce those of [15, 8].Consider first the charged excitations. Their masses arise solely from the last term in (28): M ch. = K ρ ζ µ ¯Σ ρ Σ µ , (29)where Σ’s are now 4-by-4 matrices. By explicit calculations and using the fact that ζ = 0,we get: Tr M ch. = 4( Kζ ) , Tr[( M ch. ) ] = 8( Kζ ) , Tr[( M ch. ) n ] = 2[2( Kζ )] n . (30)It means that this matrix has only two non-zero eigenvalues, which are identical and equal to m H ± = 2( Kζ ) . (31)This implies, in particular, that in order for the extremum to be minimum, ζ must lie on thesurface of the forward, not backward lightcone.For the neutral modes one has the same expression as in (28), but with 4-by-4 matricesΣ µ : M n. = 2 K ρ Λ µν ¯Σ ρ Σ µ ( ϕ ⊗ ϕ )Σ ν + K ρ ζ µ ¯Σ ρ Σ µ . (32)Let us calculate the trace of the mass matrix of the neutral Higgs bosons:Tr M n. = 2 K ρ Λ µν ϕ Σ ν ¯Σ ρ Σ µ ϕ + 4( Kζ ) = 4Λ µν K µ r ν − Kr ) + 4( Kζ )= 2(4Λ µν − TrΛ g µν ) K µ r ν − KM ) . (33)We expect that among the four neutral modes there will be one goldstone, which makes thedeterminant of M n. zero. To check it explicitly, we first factor out the matrix K ρ ¯Σ ρ andcheck by a direct calculation that its determinant is equal to ( K µ K µ ) = 1. The remainingdeterminant det [2Λ µν Σ µ ( ϕ ⊗ ϕ )Σ ν + ζ µ Σ µ ]is equal to zero, which can be best seen in the Higgs basis, where the second row and thesecond column have only zeros. In the generic basis, the goldstone mode is w i = (Π ) ij φ j ,where the matrix Π is the generator of the SO (2) rotations between the real and imaginaryparts, see Appendix. 10 .5 The extra symmetry of the neutral modes The appearance of the tensor 4Λ µν − TrΛ g µν in (33) is not accidental, but reflects an extrasymmetry of the neutral mass matrix. If we consider the neutral vacuum and if we analyzeonly neutral excitations, we always stay on the surface of the lightcone: we consider only r µ such that g µν r µ r ν = 0. This means that if we shift the tensor Λ µν in the potential asΛ µν → Λ µν + Cg µν (34)with an arbitrary C , the purely neutral contribution to the potential does not change, andneither does the neutral mass matrix. The tensor 4Λ µν − TrΛ g µν is precisely the combinationthat is invariant under such a shift. In terms of the original parametrization of the quarticpotential (1), this symmetry means that the neutral Higgs boson masses do not depend onthe value of TrΛ = λ − λ .One can make use of this extra symmetry to simplify the neutral Higgs boson mass matrix.First, note that the neutral mass matrix (32) is invariant under the transformation (34) thanksto the following relation: 2 g µν Σ µ ( ϕ ⊗ ϕ )Σ ν + r µ Σ µ = 0 . (35)Let us recall now that ζ µ is proportional to r µ : ζ µ = ζ · r µ , where ζ is the Lagrange multiplierof the minimization problem. Then, we can group the two terms in (32) together: M n. = 2 K ρ ˜Λ µν ¯Σ ρ Σ µ ( ϕ ⊗ ϕ )Σ ν where ˜Λ µν ≡ Λ µν − ζ g µν . (36)It is remarkable that the new tensor ˜Λ µν is itself invariant under (34) as this shift is accom-panied by ζ → ζ + C : ζ r µ ≡ ζ µ = Λ µν r ν − M µ → (Λ µν + Cg µν ) r ν − M µ = ζ µ + Cr µ = ( ζ + C ) r µ . (37)With this expression in hand, we can again use the trick from the analysis of the charge-breaking vacuum and state that all the neutral boson masses are given by the eigenvalues ofthe following matrix written in a manifestly covariant form:˜ M n. = 2 ˜ S · ˜Λ , where ˜ S µν = K µ r ν + K ν r µ − ( Kr ) δ µν . (38)Therefore, one can immediately write the trace of any power of the mass matrix:Tr[( M n. ) k ] = 2 k ˜ S µ ν ˜Λ ν µ · · · ˜ S µ k ν k ˜Λ ν k µ , (39)and calculate the determinant of ˜ S µν using (25):det ˜ S = − r ( Kr ) = 0 , (40)which proves the existence of a goldstone mode in a basis-invariant fashion. The principal result of this paper is a demonstration that the mass spectrum of the general2HDM can be studied in a reparametrization-invariant way within the Minkowski-space for-malism of [15, 16]. This means that the scalar propagators can be now written explicitly andcan be used, for example, to improve the thermal one-loop calculations of [20].11nother interesting issue that one can now address is to understand to what extent theperturbativity/tree-level unitarity bounds on the Higgs potential restrict the values of theHiggs boson masses. In the Standard Model, there is a strong correlation between the valueof the quartic coupling constant λ and the Higgs boson mass. Therefore, an upper limiton λ implies a corresponding upper limit on M H . In the 2HDM, due to a large number offree parameters, the situation is more complicated, see [22, 4, 21, 23, 24]. It was noted thatin certain cases masses of some of the Higgs bosons can be very high without violating thetree-level unitarity conditions. With an explicit expression for the trace of the mass matrix,one could now attack this problem in the most general case within the Minkowski-spacetechnique. The only piece still missing is a reparametrization-covariant expression for thetree-level unitarity constraints.In conclusion, we showed that the Minkowski-space approach to the most general 2HDMcan also be used to analyze the mass spectrum of the physical Higgs bosons. We calculatedthe traces of the powers of the mass matrix and its determinant for all types of vacuum thatcan exist in 2HDM. These results can now be used to get even more insight into the propertiesof the general 2HDM.We are thankful to J.-R. Cudell for helpful discussions. This work was supported by theBelgian Fund F.R.S.-FNRS via the contracts of Charg´e de recherches (I.P.I.) and of Aspirant(A.D.). The work of I.P.I. was in part supported by grants RFBR 08-02-00334-a and NSh-1027.2008.2 A Algebra of matrices Σ µ and Π µ The four-vector of matrices Σ µ is introduced via Eq. (9). The full 8-by-8 matrices Σ µ haveblock-diagonal form and are built from two identical 4-by-4 matrices, which we also denote bythe same letter Σ’s and whose properties we describe here.Σ is just the unit matrix, while the explicit expressions of Σ i are:Σ = , Σ = − − , Σ = − − . (41)These matrices satisfy the Clifford algebra condition: { Σ i , Σ j } = 2 δ ij I . (42)The set of Σ’s is not closed under taking commutators. Instead, they can be expressed viareal antisymmetric matrices Π i :Π i ≡ Π Σ i , where Π = − − . (43)The matrix Π is the generator of the simultaneous SO (2) rotations between the real andimaginary parts the two doublets; it commutes with all Σ i and its square is equal to −
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