aa r X i v : . [ h e p - ph ] F e b Prepared for submission to JHEP
The invariant space of multi-Higgs doublet models
M. P. Bento
CFTP, Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa,Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
E-mail: [email protected]
Abstract:
In a model with more than one scalar doublet, the parameter space enclosesboth physical and unphysical information. Invariant theory provides a detailed descriptionof the counting and characterization of the physical parameter space. The Hilbert seriesfor the 3HDM is computed for the first time using partition analysis, in particular Omegacalculus, giving rise to the possibility of a full description of its physical parameters. Arigorous counting of the physical parameters is given for the full class of models with N scalars as well as a decomposition of the Lagrangian into irreducible representations ofSU( N ). For the first time we derive a basis-invariant technique for counting parameters ina Lagrangian with both basis-invariant redundancies and global symmetries. ontents A.1 The 2HDM 26A.2 The 3HDM 26A.3 The NHDM 27
B Hilbert series in
SU(3) B.1 One ’s 28B.3 Three ’s 29B.4 One and one
29– 1 –
Introduction
In high-energy physics symmetries play a fundamental role in model building of both theoryand phenomenology multi-scalar theories. Nevertheless, we are often interested on theinvariants of these symmetries as they may relate to gauge invariance, physical parametersor the construction of the Lagrangian.The scalar potential has been thoroughly studied in the physics literature, where var-ious basis-invariant methods [1–4] and more group-theoretical methods such as billinears[5–14] were used. Recent work has developed a new perspective on the group structure ofthe parameter space by using invariant theory. The characterization of invariants has alsobeen instrumental to study physical parameters and CP violation. The Hilbert series ofthe 2HDM was first obtained in [15] and later studied in the context of CP violation in[16]. With this technique, the complete roadmap to the basis-invariant description of the2HDM built on invariant theory was achieved in [17].Invariant theory, a field of algebraic geometry, concerns the study of precisely theseinvariants and was developed by many prominent mathematicians such as David Hilbert,Emmy Noether and physicist Hermann Weyl. It has been used in the context of stringtheory [18]. More recently, an excellent review of these methods was given in [19] along withstrategies for handling couplings with derivatives in EFTs. As a theory, it also provides aframework for a full group-theoretical perspective of the parameter space.As complete as it may be, invariant theory relies heavily on the computation of a for-mal quantity, known as the Hilbert series. In it lies the full count and characterization ofany physical parameter in a theory. A shortcoming of this strategy is the sometimes insur-mountable calculation of a very large number of residues of multivariate integrals. Here,we introduce a technique developed by P. MacMahon [20], later called Omega calculus.With it, many challenging complex integrals become attainable, as we will show with thevery complicated case of the 3HDM.By using several results obtained throughout the years in the mathematical literature,we extract properties for the class of NHDM (N Higgs Doublet models). In particular, weshow that the Hilbert series is not needed to compute the number of physical parametersin multi-Higgs doublet models.With the knowledge on how the vector space of a Lagrangian decomposes in irreduciblerepresentations of any group, we derive a technique that counts the number of parameters ina Lagrangian with both basis-invariant redundancies and global symmetries. This methoddoes not require knowledge of invariant theory, only of the group structure of the symmetrygroup G . There are essentially two perspectives regarding the group structure of the scalar sectorof a NHDM: the fields and their representations [9, 12], and the parameter space and itsrepresentation [15, 16], both under a basis transformation group. We will follow the latterby decomposing the parameter space into irreducible representations of SU( N ). Assigning– 2 –his structure will allow us to build the invariant analogue of the parameter space underbasis transformations. In other words, build the physical parameter space of any multi-Higgs doublet model.In the most general NHDM, the Lagrangian potential can be written as V H = µ ij (Φ † i Φ j ) + z ij,kl (Φ † i Φ j )(Φ † k Φ l ) , (2.1)where the matrices follow hermiticity and symmetry properties µ ij = µ ∗ ji and z ij,kl = z kl,ij = z ∗ ji,lk . It is well known that eq. (2.1) is not unique and that we can always perform abasis transformation SU( N ) such that we generate the same physical theory. Furthermore,the fields transform under the fundamental representation r f . Thus, µ ij and z ij,kl transformas µ ij → ¯ r f ⊗ r f ,z ij,kl → Sym ( r f ⊗ r f ⊗ ¯ r f ⊗ ¯ r f ) , (2.2)where Sym denotes essentially the symmetry property of z ij,kl . With this decompositionwe findSym ( r f ⊗ r f ⊗ ¯ r f ⊗ ¯ r f ) = Sym (cid:0)(cid:2) Sym ( r f ) ⊕ Alt ( r f ) (cid:3) ⊗ (cid:2) Sym (¯ r f ) ⊕ Alt (¯ r f ) (cid:3)(cid:1) = (cid:2) Sym ( r f ) ⊗ Sym (¯ r f ) (cid:3) ⊕ (cid:2) Alt ( r f ) ⊗ Alt (¯ r f ) (cid:3) , (2.3)where we follow the well known group theory result r ⊗ r = Sym ( r ) ⊕ Alt ( r ), where Symand Alt are respectively the symmetric and antisymmetric parts of the tensor product. Insection 7 we will revisit eq. (2.3).The results above point to a full decomposition of the parameter space of the scalarpotential in terms of irreducible representations of SU( N ). Thus, we define the vectorspace of the parameters as V , defined as the space that transforms with V = µ ⊕ z → ¯ r f ⊗ r f ⊕ (cid:2) Sym ( r f ) ⊗ Sym (¯ r f ) (cid:3) ⊕ (cid:2) Alt ( r f ) ⊗ Alt (¯ r f ) (cid:3) . (2.4)The dimension dim V is the number of parameters in the potential. This can be readilycomputed from eq. (2.4). With dim r f = N , the number of doublets, anddim [Sym( r f )] = N ( N + 1)2 , dim [Alt( r f )] = N ( N − , (2.5)we finally getdim V = N + (cid:18) N ( N + 1)2 (cid:19) + (cid:18) N ( N − (cid:19) = N ( N + 3)2 , (2.6)as the number of parameters in the NHDM.– 3 – xample. In the 2HDM we may decompose µ ij → ⊗ = ⊕ , (2.7)and similarly we decompose z ij,kl → Sym ( ⊗ ⊗ ⊗ ) = [ ⊗ ] ⊕ [ ⊗ ]= 2( ) ⊕ ⊕ . (2.8)Then the vector space of parameters V transforms as V → ) ⊕ ) ⊕ , (2.9)and dim V = 14, the number of parameters in the 2HDM.The decomposition of V is instrumental to the analysis of the physical parametersof the NHDM. Throughout this paper we won’t concern ourselves with the structure ofthe representation themselves, as it is not needed for any of our results. Nevertheless, asystematic approach to this calculation is given in ref. [16] with the use of projectors. The decomposition of V into irreducible representations provides a framework for how agroup G acts on V . Nevertheless, no physical parameter is given in experiment in matrixform. The computation of physical parameters is often built on contracted tensors forwhich the answer is a number, a polynomial in the Lagrangian parameters [1]. Thus, it isimportant to introduce the notion of polynomial rings and their properties.We will be rather formal with our notation in order to compare with the mathematicalliterature. Consequently we will provide examples to map the formalism to the analysis ofthe NHDM matrices µ ij and z ij,kl . Let us consider a vector space V as the space with dimension dim V spanned by the basis x , . . . , x n . The polynomial ring K [ V ] = K [ x , . . . , x n ] is then composed by polynomialfunctions in x i and span every algebraic combination of such parameters in the field K .We will consider the field K to be the complexes C . Furthermore, we consider the actionof a group G on V , for which each element g ∈ G has a representation ρ ( g ) acting on V .We will abuse the notation by stating g instead of ρ ( g ). Then we may define the ring ofinvariants K [ V ] G to be K [ V ] G := { x ∈ K [ V ] | g.x = x } , (3.1)which comprises all algebraic combinations of the parameters in V which are invariantunder the action of G . The ring of invariants K [ V ] G = K [ f , . . . , f r ] is then generated by f , . . . , f r , the primary invariants.There are several noteworthy explanations so far. We begin with a space V to whichwe apply the group G . Then, we collect the invariants of the action of G such that the– 4 –emaining space is generated by elements called the primary invariants. We already seethat K [ V ] G ⊆ K [ V ], i.e. the invariant ring is contained in the original one. The dimensionof a ring is called the Krull dimension and it is the minimum number of generators of thering. The dimension of the initial ring K [ V ] is given bydim( K [ V ]) = dim V = dim( K [ x , . . . , x n ]) = n , (3.2)while for the ring of invariants we define the Krull dimension asdim( K [ V ] G ) = dim( K [ f , . . . , f r ] G ) = r . (3.3)Thus, r ≤ n . The Krull dimension of K [ V ] G also has a crucial interpretation, it is the num-ber of physical parameters of the theory and it will be a meaningful quantity throughoutthis paper. Example.
Let us consider a scalar potential with only µ ij . Then, V = A , the space of 2 × µ ∈ A . The dimension of V is then dim V = 4, the parameters µ , µ , µ ∗ , µ . Hence, the polynomial ring in the complexes is C [ V ] = C [ µ , µ , µ ∗ , µ ] = { µ , µ + 2 µ , . . . } (3.4)consisting on all of the polynomials in the parameters. If we choose to study the invariantsunder basis transformations G = SU(2) we may define the invariant ring as C [ V ] G = C [Tr µ, det µ ] SU(2) = { Tr µ, det µ + (Tr µ ) , . . . } , (3.5)which consists on all polynomials which are SU(2)-invariant. The Krull dimension of C [ V ] G is given by dim C [ V ] G = 2. We chose the two corresponding parameters to be the traceand the determinant of µ ij .The ring of invariants K [ V ] G is solely responsible for all the physical information ina theory. Knowing its generators is equivalent to knowing all of the physical parameters,their CP properties, and in principle, even their impact on physical processes. To thisend, we need a tool to compute the generators of the invariant space, the Hilbert series.For simplicity, while keeping some of the common mathematical notation, we willinterchange K [ V ] with R and K [ V ] G with R G . The Hilbert series is a very powerful tool for the characterization of K [ V ] G , the ring ofinvariants. The series itself is given by H ( K [ V ] G , t ) = ∞ X k =0 dim( R Gk ) t k , (3.6)where dim( R Gk ) are the number of invariants of degree k which are invariant under thegroup G and t is a token variable describing the degree of the invariants. Contrary to the The degrees of the generators may be used in principle to track the order of a process, e.g. an invariantof degree six should not appear in low order Feynman diagrams. – 5 –rimary invariants (generators), these need not be independent, as they total the numberof invariants.The Hilbert series can also be written in a closed form, generally as H ( K [ V ] G , t ) = P ( t ) Q ( t ) = P ( t )(1 − t ) d (1 − t ) d . . . (1 − t m ) d m , (3.7)where P ( t ) is a polynomial. The denominator of the Hilbert series describes the degreeand number of invariants under the action of a group G . In eq. (3.7) we count d invariantsof degree one, d invariants of degree two, etc.From eq. (3.7) we can read several properties of the invariant ring K [ V ] G . In thecontext of the Hilbert series, the Krull dimension r is such that the limitlim t → (1 − t ) r H ( K [ V ] G , t ) = γ (3.8)is neither infinite nor zero. Alternatively, this also means that in eq. (3.7) we can read thedimension as dim( K [ V ] G ) = r = d + d + · · · + d m . (3.9)In other words, the minimum number of invariants needed to generate the invariant spaceis given by the Krull dimension, r . As such, we can always expand the Hilbert series as H ( K [ V ] G , t ) = γ (1 − t ) r + τ (1 − t ) r − + . . . , (3.10)where the significance of γ and τ will be more clear later on. Example.
In the 2HDM the Hilbert series has been fully computed [16], both the ungraded(all token variables equal t ) and the multi-graded (a token variable for each representation).Excluding the three singlets of eq. (2.9), the ungraded series is given by H ( K [ V ] G , t ) = 1 + 4 t + 4 t + 15 t + 18 t + 53 t + O (cid:0) t (cid:1) , (3.11)or in closed form H ( K [ V ] G , t ) = 1 + t + 4 t + 2 t + 4 t + t + t (1 − t ) (1 − t ) (1 − t ) , (3.12)where we can read that K [ V ] G is generated by 4 degree two, 3 degree three and 1 degreefour generators. Along with the three singlets, this yields a total of 11 physical parameters.We can also expand it around t = 1 such that H ( K [ V ] G , t ) = 7 / − t ) + 7 / − t ) + . . . . (3.13) Until now we have just stated general properties of invariant rings and Hilbert series butmade no comment on its computation. For this, we separate two cases.– 6 –et G be a finite group and ρ ( g ) a representation in GL n . Then we may compute theMolien formula as H ( K [ V ] G , t ) = 1 | G | X g ∈ G − tρ ( g )) . (3.14)We note that although we sum over the elements of the group, we need only to sum overone element for each conjugacy class times the number of elements in it.Similarly, one can compute the Hilbert series for an infinite group. Let G be a reductivegroup, e.g. SU( N ), SO( N ), SL( N ). Then we define the Weyl integration formula to be H ( K [ V ] G , t ) = Z G dµ G − tρ ( g )) , (3.15)where dµ G stand for the Haar measure. A number of them can be found in [21] for Liegroups, where it is defined as Z G dµ G = 1(2 πi ) m I | z | =1 · · · I | z m | =1 dz z . . . dz m z m Y α + − m Y l =1 z α + l l ! , (3.16)where α + are the positive roots of the group.Finally, we add the notion of plethystic exponential and plethystic logarithm, whichas far as we know was first introduced in [18]. The plethystic exponential is defined asPE[ z j , t, r ] := exp X k ≥ t k χ r ( z kj ) k , (3.17)where χ r ( z kj ) is the character of the representation r = ρ . It can be interpreted with sometrivial steps to be 1det( − tρ ( g )) = exp X k ≥ t k Tr (cid:8) ρ ( g ) k (cid:9) k . (3.18)The plethystic logarithm is defined asPL (cid:2) H ( K [ V ] G , t ) (cid:3) := X k ≥ µ ( k ) k ln h H ( K [ V ] G , t k ) i , (3.19)where µ ( k ) is the M¨obius function. The significance of eq. (3.19) as a series is in thecounting of possible primary invariants in the positive terms and the determination of thesyzygies in negative terms, and has been extensively discussed in [18].In the context of Lie groups we will always take the integration to be over the maximaltorus T of the group G . This will be the Abelian group which intersects all conjugacyclasses of G and will greatly simplify our analysis. Until now we have discussed several known results in invariant theory as it applies tophysics problems. Here we present a collection of formal results in invariant theory. These– 7 –ill be instrumental to describe the class of models with more than one scalar. As with themore formal sections of this text, we will give an example at the end to guide the readerthrough the properties of the Hilbert series.We have stated before that we will consider to be working on K = C , unless statedotherwise. We find that the characteristic of the field C is char( C ) = 0. In characteristiczero fields it suffices to say that for K [ V ] G to be finitely generated, G must be reductive.All semi-simple groups, finite groups and tori are examples, as described in [22]. Examplesof semi-simple groups are SL( N ), SU( N ) and O( N ). Theorem 1. If G is semi-simple and connected, then K [ V ] G is Gorenstein [23] . If K [ V ] G is Gorenstein then H ( K [ V ] G , t − ) = ( − r t q H ( K [ V ] G , t ) , (3.20) where r is the Krull dimension and q ∈ Z as shown in [24] . In other words, eq. (3.20) also implies that the numerator of the Hilbert series shouldbe palindromic.
Theorem 2.
A theorem in [25] states that for almost all representations of a connected,semi-simple group G we have q = dim V , (3.21) where q is defined in eq. (3.20) and dim V is the dimension of our initial space. We will always assume that this is true. The computation of the Hilbert series will confirmit at the end.
Remark.
The degree of the Hilbert series is defined by [25]deg (cid:0) H ( K [ V ] G , t ) (cid:1) = deg (cid:18) P ( t ) Q ( t ) (cid:19) = deg ( P ( t )) − deg ( Q ( t )) = − q , (3.22) with q defined in eq. (3.20) . Thus, it follows that for almost all representations of G wehave that deg (cid:0) H ( K [ V ] G , t ) (cid:1) = − q = − dim V . (3.23)This is an important result which will enable us to know how to find the correct form ofthe Hilbert series at the end of the computation.
Theorem 3.
A theorem of Knop and Littelmann [26] confirms that for all representationsof G we have r ≤ − deg (cid:0) H ( K [ V ] G , t ) (cid:1) ≤ dim V . (3.24)In [25] another important corollary follows.
Corollary 1. If G is a semi-simple, connected group G , then for almost all representationswe have dim G = 2 τγ , (3.25) where γ and τ are defined in eq. (3.10) . – 8 –he most important result we present here is given in [27]. Remark. If G is semi-simple and connected, the Krull dimension r is given by r = q − τγ −→ dim V − dim G , (3.26) where the arrow means “for almost all representations of G ”, in accordance with eq. (3.21) . Finally we can state a non-uniqueness property of the Hilbert series. Let H ( K [ V ] G , t )be a Hilbert series respecting all of the above properties. Then there may exist H ( K [ V ] G , t )with the same properties. This result is discussed in [28] along with an algorithm to searchfor an optimal solution which is often, but not always, the minimal solution. We will alwayssearch for the minimal solution, i.e. the one where the degrees of the Hilbert series areminimal. In [22] the author exemplifies H ( K [ V ] G , t ) = 1 + t + t + 2 t + 2 t + 2 t + t + t + t (1 − t )(1 − t ) , (3.27)and H ( K [ V ] G , t ) = 1 + t + t + t + t + t + t + t + t + t + t + t (1 − t )(1 − t ) . (3.28)While in another context, eqs. (3.27)-(3.28) serve as an example of the Hilbert series non-uniqueness. Example.
Let H ( K [ V ] G , t ) be the Hilbert series of the 2HDM, written in eq. (3.12). Theo-rem 1 states that its numerator is palindromic, which is true. While all of Popov’s resultsare true for all but a finite number of representations, the 2HDM is one of them, i.e. therepresentation 2( ) ⊕ is one of the “almost all representations”. Then, q = 11 = dim V , (3.29)deg (cid:0) H ( K [ V ] G , t ) (cid:1) = −
11 = − dim V , (3.30)dim G = 2 τγ = 2 × , (3.31) r = dim K [ V ] G = dim V − dim G = 11 − . (3.32)– 9 – Computing the Hilbert series
While the computation of the Hilbert series by the Molien formula for finite groups enjoysa large amount of software and information, the computation for infinite groups is muchless straightforward.Calculating eq. (3.15) is usually achieved by the use of the plethystic exponential asthe integrand using the characters of the representations of G . In [19] there is a collectionof such character functions in the appendix. Ungraded and multi-graded Hilbert series arethen computed with the residue theorem and the fact thatPE[ z j , t, r ⊕ r ] = PE[ z j , t, r ] × PE[ z j , t, r ] , (4.1)as we know that Tr ( a ⊕ b ) = Tr( a ) + Tr( b ). Nevertheless, the use of the residue theoremhas a stark impact on the complexity of the computation. For large representations ormultivariate integrations the closed form of the Hilbert series may take too long to compute,too much memory or even be impossible with current technology. This problem has sinceprevented the use of invariant theory in physics for more complicated problems. Combinatorics has been a constant intersection with invariant theory. In more recent years,the same field has been essential for the computation of invariants.In the work of Percy MacMahon [20] the author illustrates partition analysis by solvingcombinatorics problems. Suppose we want to find all non-negative integer solutions to3 a − b + c = 0. Then, the generating function will be an Elliot-rational function, arational function which can be written as products in the denominator of the type A − B ,where A and B are monomials in the variables. Then it is characterized by X a − b + c =0 a,b,c ≥ t a t b t c . (4.2)Next, we introduce a new variable λ and use an operator Ω = to force the constant term ofthe series, such that X a − b + c =0 a,b,c ≥ t a t b t c = Ω = X a,b,c ≥ λ a − b + c t a t b t c . (4.3)It can be shown that both the function before applying the operator and the one after areElliot-rational functions [20]. In the 3 a − b + c = 0 example it is written in closed form as1(1 − λ t )(1 − λ − t )(1 − λ t ) . (4.4)Alternatively, an operator Ω ≥ can be shown to solve problems with inequalities.In general one defines the operator Ω = as in [20]Ω = ∞ X j = −∞ · · · ∞ X j m = −∞ a j ,...,j m λ . . . λ m := a ,..., , (4.5)– 10 –here the variables λ i are restricted to the neighbourhood of | λ i | = 1. The computation ofsuch operation has been extensively covered by [29] which culminated with the developmentof the Omega package for Mathematica and later, by Guoce Xin, for Maple [30, 31]. Besidesthe difference in platform, Guoce Xin’s software uses a faster algorithm based on a differentapproach detailed in his paper.The fast algorithm in [30] is a very powerful tool for computing Hilbert series with theWeyl formula. It contrasts with the residue theorem as a faster and less resource hungrymethod and it is based on the following. Let G be a semi-simple group with a maximaltorus T with an action on V given by diag[ m ( z ) , . . . , m n ( z )] where m ( z ) are Laurentmonomials [22] in z , . . . , z m . Then the character is given by χ r = n X i =1 m i ( z ) , (4.6)with n = dim V . Let us define with eq. (3.15) and eq. (3.16) the Hilbert series H ( K [ V ] G , t ) = 1(2 πi ) m I | z | =1 · · · I | z m | =1 dz z . . . dz m z m Q α + (cid:18) − Q ml =1 z α + l l (cid:19) (1 − m ( z ) t ) . . . (1 − m n ( z ) t ) . (4.7)It then follows [22] that the Hilbert series H ( K [ V ] G , t ) is the coefficient of 1 as series in z , . . . , z n of Q α + (cid:18) − Q ml =1 z α + l l (cid:19) (1 − m ( z ) t ) . . . (1 − m n ( z ) t ) . (4.8)Thus, using Omega calculus, one can use eq. (4.5) to write the important equality H ( K [ V ] G , t ) = Ω = Q α + (cid:18) − Q ml =1 z α + l l (cid:19) (1 − m ( z ) t ) . . . (1 − m n ( z ) t ) (4.9)where instead of λ i we have z i . We note that the assumption that | z i | = 1 is a straight-forward assumption in the Weyl formula. It is noteworthy that this algorithm will alwayswork in a function of the type F ( z , . . . , z m ) ∈ K [ z , . . . , z m , z − , . . . , z − m ], i.e. it is de-scribed by powers of its variables and reciprocals alone, which is a common property inmany physical applications.The speed of Xin’s algorithm is owed to partial fraction decomposition and with it weare able to remove entire rational functions which do not contribute to the final answer.This idea is first attributed to Richard P. Stanley in [32]. The use of Xin’s algorithm is fairly straightforward. It was used in [33] with Ell.mpl shortlyafter its introduction and in [34] where a version of eq. (4.9) is introduced and the packageEll2.mpl is used. The package is rather straightforward with the main function of ourinterest to be the command
E OeqW(f, v, ve) , where f is the integrand, v and ve are allthe variables and the variables to integrate, respectively.– 11 – xample. We will compute the Hilbert series for the 2HDM in [16] starting with eq. (38).Thus, H ( K [ V ] G , q, y, t ) = 12 πi I | z | =1 dzz (1 − z )PE[ z, q, ]PE[ z, y, ]PE[ z, t, ] , (4.10)wherePE[ z, q, ] PE[ z, y, ] PE[ z, t, ] == 1(1 − t )(1 − tz )(1 − tz )(1 − y )(1 − yz )(1 − yz )(1 − q )(1 − qz )(1 − qz )(1 − qz )(1 − qz ) , (4.11)and (1 − z ) = Q α + (cid:16) − z α + (cid:17) . Then, using eq. (4.9) we have H ( K [ V ] G , t ) = Ω = (cid:2) (1 − z )PE[ z, q, ] PE[ z, y, ] PE[ z, t, ] (cid:3) , (4.12)where z is the variable to eliminate and q , y and t are the remaining variables. In Maplewe write: restart :read ("/ path / to / Ell2 . mpl ")integrand :=( q ,y ,t ,z) -> write_our_eq .(52) * (1 - z ^2)f := E_OeqW ( integrand (q ,y ,t ,z ), [q ,y ,t ,z], [z ]):g := normal (f) By simply multiplying the numerator and denominator by (1 − q y )(1 − q t ) / [(1 − qy )(1 − qt )] it becomes clear that we have successfully reproduced eq. (39) of [16]. Example.
The ungraded Hilbert series in eq. (3.12). While it is clear that we may just do q = y = t in our first example, we want to demonstrate the method when there are powersin the denominator. Hence, we start withPE[ z, t, ] PE[ z, t, ] PE[ z, t, ] = 1(1 − t ) (1 − tz ) (1 − tz ) (1 − tz )(1 − tz ) , (4.13)and thus H ( K [ V ] G , t ) = 12 πi I | z | =1 dzz (1 − z )(1 − t ) (1 − tz ) (1 − tz ) (1 − tz )(1 − tz )= Ω = " (1 − z )(1 − t ) (1 − tz ) (1 − tz ) (1 − tz )(1 − tz ) . (4.14)Then we use Maple and write: – 12 – estart :read ("/ path / to / Ell2 . mpl ")integrand :=( t ,z ) -> (1 - z ^2)/((1 - t )^3(1 - t/z ^2)^3(1 - t*z ^2)^3(1 - t/z ^4)(1 - t*z ^4))f := E_OeqW ( integrand (t ,z), [t ,z], [z ]):g := normal (f) The output will be H ( K [ V ] G , t ) = 1 − t + t + 5 t + t − t + t (1 − t ) (1 − t ) (1 − t ) (1 + t ) (1 + t + t ) , (4.15)which after some algebra we can write as H ( K [ V ] G , t ) = 1 + t + 4 t + 2 t + 4 t + t + t (1 − t ) (1 − t ) (1 − t ) , (4.16)in agreement with eq. (3.12). The case of the full characterization and counting of invariants in the 3HDM is still anopen problem. It is clear that the computation of their properties mirrors the one of the2HDM, albeit the fact that it is much more complicated. We will present here for the firsttime the full computation of the Hilbert series of the 3HDM, in both expanded and closedform.
Following our previous results in decomposing V we may quickly arrive at the relevantdecomposition of the 3HDM. The decomposition of z ij,kl is given by z ij,kl → (cid:2) Sym ( ) ⊗ Sym (¯ ) (cid:3) ⊕ (cid:2) Alt ( ) ⊗ Alt (¯ ) (cid:3) = [ ⊗ ¯ ] ⊕ [¯ ⊗ ]= [ ⊕ ⊕ ] ⊕ [ ⊕ ]= ⊕ ⊕ ⊕ ⊕ . (5.1)The decomposition of µ ij is straightforward and hence, µ ij → ⊕ ,z ij,kl → ⊕ ⊕ ⊕ ⊕ , (5.2)– 13 – result that we compute differently in appendix A. From here, we can already definethe Hilbert series as in eq. (3.15) with the plethystic exponentials. Thus, the multigradedHilbert series we are interested in is defined as H ( K [ V ] G , s, t, u, q ) = 1(2 πi ) I | z | =1 dz z I | z | =1 dz z (1 − z z ) (cid:18) − z z (cid:19) (cid:18) − z z (cid:19) × PE[ z , z , s, ] PE[ z , z , t, ] PE[ z , z , u, ] PE[ z , z , q, ] , (5.3)where the token variables are s , t , u for the three adjoint representations and q for the .From eq. (3.17) we can compute the plethystic exponentials of the ’s and the . Theplethystic exponential depends only on the character polynomials that we construct withthe weight system of the irreducible representations. With LieART [35] it is straightforwardto compute χ ( z , z ) = z z + z z + z z + 2 + z z + z z + 1 z z , (5.4)and χ ( z , z ) = z z + z z + z z + z z + z + 1 z + z z + z z + z z + 1 z z + 2 z z + 2 z z + 2 z z + 2 z z + 2 z z + 2 z z + z + 1 z + 3 . (5.5)Then, through eq. (3.17) we havePE[ z , z , s, ] = (cid:20) (1 − s ) (cid:18) − s z z (cid:19) (cid:18) − s z z (cid:19) (cid:18) − s z z (cid:19) × (1 − sz z ) (cid:18) − s z z (cid:19) (cid:18) − s z z (cid:19)(cid:21) − , (5.6)andPE[ z , z , q, ] = " (1 − q ) (cid:18) − q z (cid:19) (cid:0) − qz (cid:1) (cid:18) − q z z (cid:19) (1 − qz z ) (cid:18) − q z z (cid:19) × (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19) (cid:0) − qz z (cid:1) × (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19) (cid:18) − q z (cid:19) (cid:0) − qz (cid:1) (cid:18) − q z z (cid:19) × (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19) (cid:18) − q z z (cid:19)(cid:21) − , (5.7) That the 3HDM decomposed in 3( ) ⊕ ) ⊕ was first mentioned by Andreas Trautner. – 14 –hich we already recognize as Elliot-rational functions. It may be inferred by examiningeqs. (5.6) and (5.7) that the computation of this particular integral is difficult due to theexistence of higher order poles, cubic and quartic polynomials as well as the plain factthat we are dealing with multivariate residues. These are known to be specially difficult tohandle. An alternative to the direct computation of the Hilbert series is an expansion as a formalseries in the token variables. As it turns out this expansion is well-behaved and easy tocompute. With it we may apply the residue theorem for z , z → H ( K [ V ] G , s, t, u, q ) =1 + q + u + tu + t + su + st + s + 2 q + q u + qu + u + q t + qtu + tu + qt + t u + t + q s + qsu + su + qst + 2 stu + st + qs + s u + s t + s + 4 q + 2 q u + 4 q u + qu + u + 2 q t + 5 q tu + 3 qtu + tu + 4 q t + 3 qt u + 3 t u + qt + t u + t + 2 q s + 5 q su + 3 qsu + su + 5 q st + 6 qstu + 4 stu + 3 qst + 4 st u + st + 4 q s + 3 qs u + 3 s u + 3 qs t + 4 s tu + 3 s t + qs + s u + s t + s + 6 q + 8 q u + 11 q u + 5 q u + 2 qu + u + 8 q t + 17 q tu + 14 q tu + 6 qtu + 2 tu + 11 q t + 14 q t u + 10 qt u + 3 t u + 5 q t + 6 qt u + 3 t u + 2 qt + 2 t u + t + 8 q s + 17 q su + 14 q su + 6 qsu + 2 su + 17 q st + 27 q stu + 17 qstu + 6 stu + 14 q st + 17 qst u + 8 st u + 6 qst + 6 st u + 2 st + 11 q s + 14 q s u + 10 qs u + 3 s u + 14 q s t + 17 qs tu + 8 s tu + 10 qs t + 8 s t u + 3 s t + 5 q s + 6 qs u + 3 s u + 6 qs t + 6 s tu + 3 s t + 2 qs + 2 s u + 2 s t + s + O (cid:16) [ stuq ] (cid:17) . (5.8)The ungraded Hilbert series is then given by equaling t = s = u = q , H ( K [ V ] G , t ) =1 + 7 t + 22 t + 94 t + 438 t + 1971 t + 8376 t + 34973 t + 138426 t + 525486 t + 1912602 t + 6685563 t + 22488737 t + 72974065 t + 228829031 t + 694812413 t + 2046440237 t + 5856320772 t + 16308266932 t + 44255437022 t + O (cid:0) t (cid:1) , (5.9)where we see a direct interpretation with eq. (3.6). It is important to note that theseinvariants are not necessarily algebraically independent, as this distinction will be computedonly with the closed form of the Hilbert series.The plethystic logarithm will be given by eq. (3.19) and it will allow us to know thetype of invariants relevant for our study. The expansion is very long and we present thedegree two and three invariants along with the first syzygy, i.e. the first negative term.– 15 –he expansion isPL[ H ( K [ V ] G , s, t, u, q )] = q + u + tu + t + su + st + s + 2 q + q u + qu + u + q t + qtu + tu + qt + t u + t + q s + qsu + su + qst + 2 stu + st + qs + s u + s t + s + . . . − s t u + . . . . (5.10)From here, we are only missing the information from the closed form of Hilbert series, howmany invariants and of what degree. The Hilbert series of the 3HDM would naively be computed with the use of the residuetheorem. By doing so, one quickly finds the computation to be very complex as variousproblems come into play. First, by solving higher degree polynomials in the denominatorand integrating the first time we arrive at a second integration plagued with square, cubicand quartic roots of the integration variable. Secondly, it is not trivial to deal with thecomplex roots nor to use substitution of variables in the integrand. Omega calculus, usedhere for the first time for NHDM physical applications, offers the solution for all of theseshortcomings.In the 3HDM we have Y α + − m Y l =1 z α + l l ! = (1 − z z ) (cid:18) − z z (cid:19) (cid:18) − z z (cid:19) (5.11)and then by eq. (4.9) and eq. (5.2) we write the ungraded Hilbert series as H ( K [ V ] G , t ) = Ω = (cid:20) (1 − z z ) (cid:18) − z z (cid:19) (cid:18) − z z (cid:19) PE[ z, t, ] PE[ z, t, ] (cid:21) . (5.12)The code in Maple is straightforward and described in subsection 4.2. It ran for 56 minutesusing 5 Gb of memory in a personal laptop equipped with an Intel Core i7-8750H. Thesolution, while not immediately in the form most useful to us, consists on the rationalfunction H ( K [ V ] G , t ) = P ( t )(1 − t ) (1 + t ) (1 + t ) (1 + t + t ) (1 + t + t + t + t ) × t + t )(1 + t + t + t + t + t + t ) , (5.13)where we refrained from writing the full palindromic polynomial of degree 146 in thenumerator.Before going forward we note several interesting properties of eq. (5.13). First, theKrull dimension is 43, or equivalently the number of physical parameters minus the three– 16 –inglets. This comes directly from eq. (3.26) as (27 + 3 × − t = 1 we get H ( K [ V ] G , t ) = λ (1 − t ) + τ (1 − t ) + O ((1 − t ) − ) , (5.14)in agreement with eq. (3.10), and λ = 193687318524832031333716902552613683200000000 ,τ = 19368731852483203138429225638153420800000000 . (5.15)From eq. (3.25) we also add validity to our earlier assumptions that eq. (3.26) is valid inthe 3HDM. Thus, 2 τγ = 8 = dim G , (5.16)achieving the expected result.From eq. (5.13) it is not trivial to find a minimal Hilbert series that satisfies all therequired properties. Simple algebraic manipulations lead us to thousands of solutions. Afteran intensive search using the Numpy package in Python we get to a seemingly minimalsolution. Thus, the Hilbert series describing the most general 3HDM is given by H ( K [ V ] G , t ) = P ( t )(1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t ) (5.17)where the palindromic polynomial P ( t ) is too large to write here but we will leave it asan attachment to this paper. We already see that eq. (5.17) also agrees with eq. (3.22) andeq. (3.23) in that deg (cid:0) H ( K [ V ] G , t ) (cid:1) = −
51 = − dim V = − q . (5.18)Furthermore, we note the large degree invariants in eq. (5.17) contrasting with the caseof the 2HDM. The question remains if this Hilbert series is minimal. Although we areconfident with the result, only a subsequent study on the invariants themselves can pointto whether this is an optimal solution. This will be the topic of a future paper [36]. In this section we will work out a number of interesting properties of the NHDM, which wecan learn from the tools used so far. Our first result concerns the counting of the physicalparameters of the NDHM.
Theorem 4.
Let the model be the most general NHDM. Then, the number of physicalparameters is given by N physical = N + N + 22 , (6.1) In the spirit of appendix B this is the SU(3) Hilbert series of three ’s and one . – 17 – here N are the number of doublets. Proof.
Let the group G = SU( N ) be the family transformations of the NHDM and letus define a physical parameter as a family invariant parameter. Then we may define theinvariant ring K [ V ] G as having Krull dimension N physical and field K = C . The dimensionof the initial space dim V is then given bydim V = X i dim r i = N ( N + 3)2 , (6.2)where r i are the representations of the decomposition of the matrices µ and z , and the lastequality is given by the counting of total parameters [37, 38]. Then, we use a theorem in[39], asserting that the Krull dimension of K [ V ] G is given bydim K [ V ] G = dim V − dim G + dim G v , (6.3)where G v is the stabilizer of G . Because we already know that dim G = N −
1, we onlyneed to compute the dimension of the stabilizer for our case. Specifically, whether is zero( G acts freely on V ) or not. In ref. [40] the authors establish that if the action is reducible,and it is in our case, then G acts freely if at least one irreducible action acts freely. Inparticular, for the irreducible representation of a simple group, which is the case of SU( N ),dim G v = 0 if and only if dim r i > dim G . Thus, we only have to show that in the NHDM,there is always an irreducible representation with dimension greater than N −
1. This istrivial because the decomposition of the NHDM always implies the computation of r a ⊗ r a for r a being the adjoint representation. This will always result in at least a representationof higher dimension than dim G which will always be needed for the decomposition. Hence,in the NHDM dim K [ V ] G = dim V − dim G + dim G v = N ( N + 3)2 − ( N −
1) + 0= N + N + 22 . (6.4)Our proof sheds light on the conditions of eq. (3.26) and shows its validity for a numberof cases. This result does not hold in general for cases where symmetries are enforced inthe Lagrangian. It does however bound the number of physical parameters in any NHDM.We summarize Theorem 4 in table 1.Another result that we provide is on the form of a conjecture for multi-Higgs doubletmodels. This result was first conjectured and stated by J. P. Silva in a private discussion built on table 1 of[37]. Here, we show a formal proof of it. In fact there is a distinction here between the notion of transcendence degree and Krull dimension.Nevertheless, we don’t need to worry about it as they are the same in finitely generated algebras. – 18 – Number of parameters (dim V ) N physical = dim K [ V ] G N N ( N +3)2 N + N +22 Table 1 . Physical parameters of the NHDM with a group of family transformations SU( N ). Conjecture 1.
Let the model be the NHDM with
N > . Then the vector space of param-eters V is decomposed as V = 3( ) ⊕ (cid:0) N − (cid:1) ⊕ (cid:18) N ( N − − ) (cid:19) ⊕ (cid:18) N ( N + − ) (cid:19) (6.5) with dim V = 3 + 3 (cid:0) N − (cid:1) + (cid:18) N ( N − N − (cid:19) + (cid:18) N ( N + 2 N − (cid:19) = N ( N + 3)2 . (6.6) The decomposition of µ ij and z ij,kl is given by µ ij = ⊕ ( N − ) ,z ij,kl = 2( ) ⊕ (cid:0) N − (cid:1) ⊕ (cid:18) N ( N − − ) (cid:19) ⊕ (cid:18) N ( N + − ) (cid:19) . (6.7)It is noteworthy that we have proved this result explicitly for all representations N <
15, in agreement with our conjecture. Furthermore, is is straightforward to show thatConjecture 1 is always a combinatorical solution and also that it is unique. The conditionsfor the conjecture to hold are discussed in detail in appendix A where the decompositionof SU( N ) adjoint representations is assumed to exist as illustrated in eq. (A.14).We summarize our results in table 2, where we used the LieART package as well asOmega package for Mathematica [29] to confirm results. The functions b N and d N aredefined as b N = N ( N − N − d N = N ( N + 2 N − µ ij z ij,kl Number of parameters (dim V )2 ⊕ ) ⊕ ⊕ ⊕ ) ⊕ ) ⊕ ⊕ ) ⊕ ) ⊕ ⊕ ⊕ ) ⊕ ) ⊕ ⊕ ⊕ ) ⊕ ) ⊕ ⊕ ⊕ ) ⊕ ) ⊕ ⊕ N ⊕ ( N − ) 2( ) ⊕ (cid:0) N − (cid:1) ⊕ b N ⊕ d N N ( N +3)2 Table 2 . Representation decomposition of the NHDM where b N and d N are given in (6.8). and the last line is representative of Conjecture 1. So far we have discussed both the decomposition of the matrices of the Lagrangian, and theinvariants of the most general multi-Higgs scalar models. However, we have not presentedany result towards the use of symmetries in the Lagrangian. In this section we show howto count all of the remaining parameters after imposing a symmetry.
Theorem 5.
Let us consider a symmetry by the action of a group G . We choose a rep-resentation ρ ( g ) = r = L r i for the fields. Then the number of parameters is given by thenumber of singlets in µ ij → ¯ r ⊗ r ,z ij,kl → (cid:2) Sym ( r ) ⊗ Sym (¯ r ) (cid:3) ⊕ (cid:2) Alt ( r ) ⊗ Alt (¯ r ) (cid:3) . (7.1) Proof.
By imposing a symmetry in the Lagrangian we are decomposing the vector space V = µ ⊕ z in irreducible representations of G . In contrast with the strategy for basisinvariants, we know that only the degree one invariants can remain. These correspond toelements (1 − t ) in the denominator of the Hilbert series. A remarkable property of theseterms is that they can be factored out from the summation as they do not depend on the– 20 –epresentation. This can be understood by decomposing r T = (¯ r ⊗ r ) ⊕ (cid:2) Sym ( r ) ⊗ Sym (¯ r ) (cid:3) ⊕ (cid:2) Alt ( r ) ⊗ Alt (¯ r ) (cid:3) = n ( ) ⊕ M r j = r j , (7.2)where r T is the representation of the full decomposition. Then, H ( K [ V ] G , t ) = XZ − t r T ) = 1(1 − t ) n XZ (cid:16) − t L r j = r j (cid:17) , (7.3)and thus, H ( K [ V ] G , t ) = (1 − t ) − n H ( K [ V ′ ] G , t ) . (7.4)The remaining irreducible representations r j are not invariant by themselves and requirehigher degrees to form an invariant. Consequently, we need not compute the Hilbert seriesto know how many invariants of degree one exist. This quantity is given by n in eq. (7.2),the number of singlets. Example.
We consider the 2HDM with a Z symmetry. We choose the representation r = ⊕ ′ corresponding to the transformation diag(1 , − ′ ⊗ ′ = , (7.5)we get µ ij → ( ⊕ ′ ) ⊗ = 2( ) ⊕ ′ ) . (7.6)For z ij,kl we need first to know what corresponds to Sym and Alt. This can easily beachieved with the character table and with χ Sym ( g ) = 12 (cid:0) χ ( g ) + χ ( g ) (cid:1) ,χ Alt ( g ) = 12 (cid:0) χ ( g ) − χ ( g ) (cid:1) , (7.7)where χ ( g ) is the character of g . This system always has a solution. Here, by choosing g = a , with a = e , we have χ Sym ( a ) = 12 (0 + 2) = 1 ,χ Alt ( a ) = 12 (0 −
2) = − . (7.8)The only possible solution is that( ⊕ ′ ) ⊗ = Sym ( ⊕ ′ ) ⊕ Alt ( ⊕ ′ ) = ( ⊕ ⊕ ′ ) ⊕ ( ′ ) . (7.9) In fact we have already silently agreed to this when we left out the three singlets of the 3HDM fromthe computation of the Hilbert series. – 21 –onsequently, the matrix z ij,kl decomposes as z ij,kl → (cid:2) ( ⊕ ⊕ ′ ) ⊗ ( ⊕ ⊕ ′ ) (cid:3) ⊕ (cid:2) ′ ⊗ ′ (cid:3) = 6( ) ⊕ ′ ) . (7.10)Hence, we conclude that the 2HDM has 8 parameters left after imposing Z , 2 from µ ij and6 from z ij,kl . We note four important facts. First, these 8 parameters are not physical, but7 will be. Indeed, λ can be made real after rephasing. Second, this result is completelybasis-invariant. We might have chosen an equivalent matrix other than diag(1 , − Z is imposed. Third, wemay even do better than just count the number of parameters in z ij,kl . By knowing howmany singlets come from the Alt part, in this case just one, we can use the decompositionin SU(2) to assign it to a particular group of parameters. Similarly, we can do the samefor the Sym part. Example.
We consider the 2HDM with a Z symmetry. We define ω = exp(2 iπ/
3) andchoose r = ′ ⊕ ′′ corresponding to the action of diag( ω, ω ). The product rules are givenby ′ ⊗ ′ = ′′ , ′′ ⊗ ′′ = ′ , ′ ⊗ ′′ = . (7.11)Then, µ ij → ( ′ ⊕ ′′ ) ⊗ = 2( ) ⊕ ′ ⊕ ′′ . (7.12)Choosing the element g = a with a = e , the identity element, we find χ Sym ( a ) = 12 (1 + ( − ,χ Alt ( a ) = 12 (1 − ( − , (7.13)where we used that χ ( a ) = Tr (cid:2) diag( ω, ω ) (cid:3) = − χ ( a ). Then, the only possibility is( ′ ⊕ ′′ ) ⊗ = Sym ( ′ ⊕ ′′ ) ⊕ Alt ( ′ ⊕ ′′ ) = ( ⊕ ′ ⊕ ′′ ) ⊕ ( ) . (7.14)Therefore, we decompose z ij,kl as z ij,kl → (cid:2) ( ⊕ ′ ⊕ ′′ ) ⊗ ( ⊕ ′ ⊕ ′′ ) (cid:3) ⊕ [ ⊗ ]= 4( ) ⊕ ′ ) ⊕ ′′ ) . (7.15)Hence, we conclude that the 2HDM with Z symmetry has 6 parameters, 2 from µ ij and 4 from z ij,kl . Here, the number of parameters coincides with the number of physicalparameters. – 22 – xample. We consider the 2HDM with a U (1) symmetry. We use the transformationdiag( e iξ , e − iξ ), which corresponds to r = ¯ ′ ⊕ ′ . The product rules are given by ′ ⊗ ′ = ′′ , ¯ ′ ⊗ ¯ ′ = ¯ ′′ , ¯ ′ ⊗ ′ = , (7.16)where contrarily to the other examples, we have another representation appearing in theproduct rules, a consequence of G infinite. With it, µ ij decomposes as µ ij → (¯ ′ ⊕ ′ ) ⊗ = 2( ) ⊕ ′′ ⊕ ¯ ′′ . (7.17)The characters, if we choose a to be the element with representation diag( e iξ , e − iξ ), aregiven by χ Sym ( a ) = 12 (cid:0) ( ξ ) + 2 cos(2 ξ ) (cid:1) = 1 + 2 cos(2 ξ ) ,χ Alt ( a ) = 12 (cid:0) ( ξ ) − ξ ) (cid:1) = 1 . (7.18)Thus, the only solution is given by(¯ ′ ⊕ ′ ) ⊗ = Sym (¯ ′ ⊕ ′ ) ⊕ Alt (¯ ′ ⊕ ′ ) = ( ⊕ ′′ ⊕ ¯ ′′ ) ⊕ ( ) . (7.19)Therefore, we decompose z ij,kl as z ij,kl → (cid:2) ( ⊕ ′′ ⊕ ¯ ′′ ) ⊗ ( ⊕ ′′ ⊕ ¯ ′′ ) (cid:3) ⊕ [ ⊗ ]= 4( ) ⊕ ′′′′ ) ⊕ ′′′′ ) . (7.20)This result is remarkably similar to the case of Z . This is not coincidental as in fact theylead to the same symmetry constraint in the 2HDM [37]. This can be seen from the factthat it only differs in the representations that are primed, those that we will not keep. Example.
We consider the 2HDM with a S symmetry. We choose r = corresponding tothe action of doublet representation in the fields. The product rules are given by ′ ⊗ ′ = , ′ ⊗ = , ⊗ = ⊕ ′ ⊕ . (7.21)Then µ ij decomposes as µ ij → ⊗ = ⊕ ′ ⊕ . (7.22)Choosing g = (1 ,
2) and consulting the character table we get χ Sym ( g ) = 12 (0 + 2) = 1 ,χ Alt ( g ) = 12 (0 −
2) = − . (7.23)Then, the only possibility is ⊗ = Sym ( ) ⊕ Alt ( ) = ( ⊕ ) ⊕ ( ′ ) . (7.24)– 23 –hus, z ij,kl decomposes as z ij,kl → [( ⊕ ) ⊗ ( ⊕ )] ⊕ (cid:2) ′ ⊗ ′ (cid:3) = 3( ) ⊕ ′ ⊕ ) . (7.25)Consequently, this model has 4 parameters, 1 from µ ij and 3 from z ij,kl . This can bechecked against [41]. Example.
We consider the 3HDM with a A symmetry. We choose r = corresponding tothe action of doublet representation in the fields. The product rules are given by ′ ⊗ ′ = ′′ , ′′ ⊗ ′′ = ′ , ′ ⊗ ′′ = , ⊗ = ⊕ ′ ⊕ ′′ ⊕ ) . (7.26)Then µ ij decomposes as µ ij → ⊗ = ⊕ ′ ⊕ ′′ ⊕ ) . (7.27)Choosing g = (1 , ,
4) and consulting the character table we get χ Sym ( g ) = 12 (cid:0) ( − + 3 (cid:1) = 2 ,χ Alt ( g ) = 12 (cid:0) ( − − (cid:1) = − . (7.28)where we used g = e , the identity element. The only possibility is ⊗ = Sym ( ) ⊕ Alt ( ) = ( ⊕ ′ ⊕ ′′ ⊕ ) ⊕ ( ) . (7.29)Thus, z ij,kl decomposes as z ij,kl → (cid:2) ( ⊕ ′ ⊕ ′′ ⊕ ) ⊗ ( ⊕ ′ ⊕ ′′ ⊕ ) (cid:3) ⊕ [ ⊗ ]= 5( ) ⊕ ′ ) ⊕ ′′ ) ⊕ ) . (7.30)With this we conclude that this model has 6 parameters, 1 from µ ij and 5 from z ij,kl . Thiscan be checked against [42]. Example.
Finally we consider the 3HDM with a S symmetry. We will not write everytensor product rule in this case they are many. Choosing r = we get µ ij → ⊗ = ⊕ ⊕ ⊕ ′ . (7.31)Choosing g = (1 ,
2) we get the characters χ Sym ( g ) = 12 (cid:0) + 3 (cid:1) = 2 ,χ Alt ( g ) = 12 (cid:0) − (cid:1) = − , (7.32)– 24 –nd then z ij,kl → [( ⊕ ⊕ ) ⊗ ( ⊕ ⊕ )] ⊕ (cid:2) ′ ⊗ ′ (cid:3) = 4( ) ⊕ ′ ⊕ ) ⊕ ) ⊕ ′ ) . (7.33)Thus, the 3HDM with S symmetry has 5 parameters, 1 from µ ij and 4 from z ij,kl . Thiscan be checked against [42].There are many interesting analysis that one can make from these examples. It is easyto check that choosing the 2HDM with a symmetry diag( i, − i ) yields a similar result tothe one we obtained with Z . However, choosing diag(1 , i ) yields a similar result to theone of Z and U(1). Both are cases in which a group effectively acts as another.One result we can infer from this and previous sections is that due to the decompositionof any NHDM into representations of SU( N ), we are always guaranteed to have threesinglets. This result follows from the fact that any G that we choose will be a subgroup ofPSU( N ). If SU( N ) guarantees three singlets, so will any symmetry groups. Furthermore,these three will be physical parameters. We studied in detail the group structure of the matrices in the scalar potential of multi-Higgs doublet models. We show its decomposition under irreducible representations ofSU( N ) with a simple formula using the symmetric and antisymmetric part of the tensorproduct. With this decomposition, the study of the physical parameters of the theorybecomes attainable.We have used a tool from partition theory, Omega calculus, to compute complicatedHilbert series without using the residue theorem. Its use in high-energy physics is a first, asmost computations depend on the residue theorem. In particular, we compute for the firsttime the closed form of the Hilbert series of the 3HDM, a result previously very difficult toobtain by standard methods. From this function, we will be able to completely characterizethe physical parameters of the 3HDM.Using a number of formal results in invariant theory we proved that the most generalNHDM has ( N + N + 2) / N <
15. We leave the formula as aconjecture for all N .For the first time we derived a basis-invariant method for counting parameters in aLagrangian with both basis-invariant redundancies and global symmetries. We show thatthe knowledge of tensor product decomposition and character theory is enough for attainingthis purpose. Furthermore, this technique does not require analysis of the Lagrangian itself.With invariant theory, we hope that a clear path to a full basis-invariant overview tothe physical parameters of NHDM will soon be possible. There are still many unansweredquestions on the CP properties and the physical parameters in theories with symmetries,both of which we have not addressed in this paper.– 25 – cknowledgments M. P. B. is very grateful to J. P. Silva for all the useful discussions on scalar models andendless advice. M. P. B. is also grateful to A. Trautner for the time spent explaininginvariant theory in scalar models. This work is supported in part by the PortugueseFunda¸c˜ao para a Ciˆencia e Tecnologia (FCT) under contract SFRH/BD/146718/2019.This work is also supported in part by FCT under contracts CERN/FIS-PAR/0008/2019,PTDC/FIS-PAR/29436, UIDB/00777/2020, and UIDP/00777/2020.
A An algorithm for the decomposition of the NHDM
In section 2 we decomposed the matrices µ and z of the scalar potential V H = µ ij (Φ † i Φ j ) + z ij,kl (Φ † i Φ j )(Φ † k Φ l ) (A.1)such that its bare elements transform under a direct sum of irreducible representationsof SU( N ). In order to do that we have followed Trautner [16] in the use of projectionoperators to enforce hermiticity and symmetrization in indices. In this appendix we willconcern ourselves with the combinatorics of such decomposition. We show an alternativemethod and reason in favor of Conjecture 1. A.1 The 2HDM
In the 2HDM the family group SU(2) acts on the matrices µ and z under the fundamentalrepresentations, i.e. the two doublets form a family space doublet. Then the decompositionis given by ⊗ = ⊕ , (A.2)and ⊗ ⊗ ⊗ = 2( ) ⊕ ) ⊕ , (A.3)The projection operators then reduce the representations to µ ij = ⊕ , (A.4)and z ij,kl = 2( ) ⊕ ⊕ , (A.5)which was done in section 2. A.2 The 3HDM
Given a multi-Higgs doublet model we may ask if it is possible to guess the representationsneeded to compute the Hilbert series. A simple solution would be to find how many ofthe initial irreducible representations sum to the number of parameters the model has. Itturns out that we can follow a simple algorithm with partition analysis, the same tool weuse to compute H ( K [ V ] G , t ). In the 3HDM we know that the number of parameters is 54– 26 –cf. table 1 of [37]). The product decomposition before the projection operators is givenby ¯3 ⊗ = ⊕ , (A.6)and ¯3 ⊗ ⊗ ¯3 ⊗ = 2( ) ⊕ ) ⊕ ¯10 ⊕ ⊕ . (A.7)Without the use of the projection operators we already know that the non-singlet rep-resentations must add to 51 and that the octet in eq. (A.6) must be included. Then, itis a combinatorical task to find the non-singlet representations of eq. (A.7) that sum to51 − X a +10 a +27 a − a,b,c ≥ t a t a t a = Ω = (cid:20) λ − (1 − λ t )(1 − λ t )(1 − λ t ) (cid:21) = t t = ⇒ ) ⊕ . (A.8)Then, in the 3HDM we have µ ij = ⊕ , (A.9)and z ij,kl = 2( ) ⊕ ) ⊕ , (A.10)for a total sum of 54 parameters. It is noteworthy that we haven’t mentioned projectionoperators at all in the 3HDM and it agrees with previous computations. A.3 The NHDM
For a general multi-Higgs doublet model we can easily generalize the procedure we usedfor the 3HDM. The algorithm goes as follows. Let the model be the NHDM with familytransformation group SU( N ) with r i , r f , and r a a generic, a fundamental, and adjointrepresentation respectively. We may then decompose,¯ r f ⊗ r f = ⊕ r a , (A.11)¯ r f ⊗ r f ⊗ ¯ r f ⊗ r f = 2( ) ⊕ M i m i r i , (A.12)where m i are the multiplicities. Then, the remaining representations of z ij,kl are given byΩ = " λ − (dim V − r a − Q i (1 − λ r i t i ) , (A.13)where r i are the unique dimensions of the decomposition and dim V = N ( N + 3) / < N <
15 it becomes clear that eq. (A.13) yields an uniqueand equal answer among them. With this in mind we are able to conjecture for cases
N >
3. If the decomposition of the adjoint representation in SU( N ) is given by r a ⊗ r a = ⊕ a N ) ⊕ b N ⊕ c N ⊕ c N ⊕ d N , (A.14)– 27 –here a N = N − , b N = N ( N − N − ,c N = (cid:18) N − (cid:19)(cid:18) N + 22 (cid:19) , d N = N ( N + 2 N − . (A.15)Then, it is straightforward to prove that z ij,kl → ) ⊕ a N ) ⊕ ( b N ) ⊕ ( d N ) , (A.16)for N > Proof.
If eq. (A.14) holds true then the expansion of eq. (A.13) will always have to cancelthe λ terms to be a solution. Thus, all solutions are solutions of x a N + x b N + x c N + x d N = N ( N + 3)2 − ( N − − , (A.17)which is a simple linear equations problem where we must match the polynomial on theLHS with the one of the RHS. The solution is unique2 a N + b N + d N = N ( N + 3)2 − ( N − − . (A.18) B Hilbert series in
SU(3)
For completeness and such that we may provide results for the reader in case there is needfor Hilbert series in SU(3) we list the ungraded Hilbert series that we computed beforegoing to the case of the 3HDM.
B.1 One 8
For the case of one , dim V = 8 and the Hilbert series is given by H ( K [ V ] G , t ) = 1(1 − t )(1 − t ) , (B.1)where the Krull dimension dim K [ V ] G = 2. B.2 Two 8’s
For the case of two ’s, dim V = 16 and the Hilbert series is given by H ( K [ V ] G , t ) = 1 + t (1 − t ) (1 − t ) (1 − t ) , (B.2)where the Krull dimension dim K [ V ] G = 8.– 28 – .3 Three 8’s For the case of three ’s, dim V = 24 and the Hilbert series is given by H ( K [ V ] G , t ) = 1 + 3 s + 7 s + 9 s + 16 s + 18 s + 25 s + 30 s + 34 s + · · · + s (1 − t ) (1 − t ) (1 − t ) , (B.3)where we omitted terms in the numerator but since it is palindromic, they are easy tocompute. The Krull dimension dim K [ V ] G = 16. B.4 One 27
For the case of one , dim V = 27 and the Hilbert series is given by H ( K [ V ] G , t ) = P ( t )(1 − t )(1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t ) (1 − t )(1 − t ) , (B.4)where the numerator is too large to show in this form. It is given by a palindromicpolynomial of degree 74 for which we list the first 37 coefficientsCoefficients = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . . . } . (B.5)The Krull dimension dim K [ V ] G = 19. B.5 One 27 and one 8
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