Classification of effective operators for interactions between the Standard Model and dark matter

We construct a basis for effective operators responsible for interactions between the Standard Model and a dark sector composed of particles with spin less or equal to 1. Redundant operators are eliminated using dim-4 equations of motion. We consider simple scenarios where the dark matter components are stabilized against decay by Z2 symmetries. We determine operators which are loop-generated within an underlying theory and those that are potentially tree-level generated.


Introduction
Understanding the nature of dark matter (DM) is one of the most pressing current issues in astroparticle physics. Of the many hypotheses proposed, one of the most fruitful and promising is based on the assumption that DM is composed of one or more new elementary particles. This possibility has been extensively studied in a variety of specific models, most prominently in realistic supersymmetric models that are characterized by predicting that DM is the lightest supersymmetric particle whose mass should be in the O(100 GeV) range.
In this paper we will be interested in constructing a model-independent description of the interactions between the dark and SM sectors using an effective Lagrangian approach. We will assume the standard sector with one doublet scalar field, but augmented by a number of right-handed neutrinos ν R to allow for the possibility of Dirac neutrino masses. We denote this enlarged model as the νSM.
Concerning the dark sector we allow the dark matter to be multi-component, containing fermions, scalars and massive Abelian vectors. Each component is protected against decay by a symmetry we denote G dark , and which we need not to specify at this point, but which can include discrete or continuous (local or global) subgroups; it also contains the local symmetry related to mass generation of the dark vector. It is important to note that even though the dark sector may contain several stable particles, not all have to contribute significantly to the DM relic density inferred from the WMAP and PLANCK data [1].
Within the scenarios we consider, the dark and νSM sectors interact through the exchange of heavy particles whose mass is much larger than the typical momentum transfer in all processes being considered and their effects decouple at low-energies. In addition, we will assume that the underlying theory is weakly coupled and renormalizable. Under these circumstances the DM-νSM interactions can be described by a series of effective operators where O DM and O νSM are composed of fields belonging to the respective sectors, and it is assumed that they are invariant under corresponding symmetries. The effective Lagrangian consists of a linear combination of terms of this type. The coefficients are suppressed by appropriate powers of the heavy-physics scale Λ (the power is determined by the dimension of O DM −νSM ), and contain unknown dimensionless couplings that parameterize at low energies all interactions between the standard and dark sectors. In addition to the hierarchy generated by operator dimensionality it is also useful to note that some of the operators are necessarily generated by heavy-particle loops, so that their coefficients are correspondingly suppressed, the remaining operators can be generated at tree level, but whether this is the case depends on the details of the underlying theory. In this paper we will construct all operators O DM −νSM of dimension ≤ 6 and determine whether they can be generated at the tree level.
In constructing the effective Lagrangian, we will eliminate operators that vanish when dim-4 equations of motion are imposed, since they give no contribution to on-shell matrix elements, both in perturbation theory (to all orders) and beyond [2]; we call such operators redundant. A given type of heavy physics may generate a basis of operators different from the one listed below; such a basis may be transformed in the one we use by applying equations of motion.
Dim-6 effective operators for interactions between the Standard Model and DM have been already present in various contexts in the literature [3]. However the goal of this paper is to construct a basis of operators [4] that then could be consistently adopted to describe different aspects of DM physics.
The paper is organized as follows. In section 2 we define the model and list νSM operators up to dimension 4. In section 3 we present dark operators needed to construct dim-6 effective operators. Section 4 contains our main results, i.e. the basis of operators up to dimension 6. In section 5 we summarize of our findings. Appendix A specifies our conventions, while appendix B reviews mechanisms for dark vector boson mass generation.

νSM operators
As mentioned above we will consider the Standard Model of electroweak interactions supplemented by a number of right-handed neutrinos ν R (νSM ); this model contains the matter fields collected in table 1. We assume 3 quark families, 3 lepton SU(2) doublets and charged right-handed lepton singlets, and n ν right-handed neutrinos. We use these fields to construct the gauge-invariant operators O νSM appearing in (1.1), which we classify according to their canonical dimension (up to dim ≤ 4) Table 1. νSM matter field content and their hypercharge quantum numbers. Weak isospin, colour and generation indices are denoted by j = 1, 2, α = 1, 2, 3 and p = 1, 2, 3 respectively. We assume the presence of n ν right-handed neutrinos, so k = 1, · · · , n ν .
operators are collected in table 2 (Hermitian conjugation of operators containing fermions are not listed separately but should be included when constructing the effective Lagrangian in order to ensure it is Hermitian). It should also be noted that, in the presence of fermionic fields there exists operators where Dirac matrices might appear between the two factors in (1.1). We also use the fact that four-fermion operators can always be rearranged into the form (1.1) by using Fierz transformations. All νSM fields are assumed to be singlets under symmetries stabilizing dark fields. At this stage, we retain terms that are total derivatives and also we do not apply equations of motion at this point (that will be done when constructing the effective Lagrangian). dim scalars vectors tensors Table 2. νSM operators that are singlets of SU ( representations. X µν stands for B µν , W I µν or G A µν , ψ ∈ {l, ν R , e, q, u, d}. This list includes operators that are total derivatives, equations of motion were not adopted at this stage.

Dark operators
In this section we construct the list of operators 1 up to dim 4, that consist of DM fields: a real scalar Φ, left and right chiral fermions Ψ L , Ψ R and an Abelian vector field V µ . For simplicity, we assume that the symmetry stabilizing the dark fields is of the form where the dark scalars Φ are odd with respect to the first factor and even with respect to the others, the Ψ R(L) are odd with respect to the second (third) factor and even with respects to the others, and similarly for the V µ . We also introduce a set of right-handed fermions N R l (l = 1, · · · , n N ) that transform in the same way as Φ under G dark . As we will show shortly, their presence allows for Yukawa interactions involving DM and ν R , which might be relevant for DM phenomenology. As a consequence of these assumptions the lightest particle in each of these dark sectors (Φ and N R , Ψ L , Ψ R , V µ ) is stable separately and the effective Lagrangian will not contain terms having odd number of fields from any sector.
In appendix B we review two procedures for generating the mass of Abelian vector bosons V µ : the Stuckelberg and Higgs mechanisms. The dark sector operators for both mechanisms are collected in table 3. Within the Stuckelberg approach, the U (1) gauge invariance requires that the V µ appears only through the operator V µ or the field strength V µν ; for the Higgs approach V µ appears only within the covariant derivatives of the complex scalar field φ (see appendix B). It should be stressed that φ is not a dark field (as it is explained in the appendix |φ| belongs to the heavy sector), nevertheless we retain φ in the table to ensure manifest gauge invariance. Note that all operators contained in table 3 are neutral under G dark . Since all the dark fields are assumed to be singlets under νSM gauge symmetries, so are the operators contained in the table.

The effective Lagrangian
In the scenario being considered, the full theory contains not only the dark and standard sectors, but also a heavy sector responsible for generating effective operators, the theory is assumed to be weakly coupled, and the scale of heavy physics Λ is assumed to be substantially larger than the electroweak scale v 246 GeV. At energies significantly below Λ the dynamical content of such a theory is well described by an effective Lagrangian obtained by "integrating out" the heavy degrees of freedom, and which takes the form k is of the form (1.1) and is multiplied by an unknown dimensionless (Wilson) coefficient C (n) k . It should be stressed that the right-handed neutrinos ν R belong dim no space-time indices Table 3. DM operators built of Φ, Ψ ∈ {Ψ L , Ψ R , N R } and V µ symmetric under (3.1); note the presence of operators containing N R and Φ, allowed by the assumption of their both being odd under (Z 2 ) Φ . For the sake of dark-sector gauge invariance the table contains also the vector field V µ defined in (4.12) (highlighted in red) and covariant derivatives of the scalar field φ (blue). They are relevant when the vector boson mass is generated through the Stuckelberg and Higgs mechanisms, respectively. Vector operators and symmetric tensors of dim-4 are not listed, because we are interested in the νSM × DM operators up to dim-6 and the νSM does not contain vector operators nor symmetric tensors with dimension less or equal 2.
to the light sector, so the scale of their mass must be Λ; in the following we will assume that it is at most of the order of v.
The effective Lagrangian is a useful tool for parametrizing the low-energy effects of a theory in a consistent and controlled way. In the case where the low-energy theory is described by the SM the full set of such operators up to dim-6 was constructed in [5] and then refined and systematized in [6].
As we have already mentioned, we allow in the dark sector for the presence of N R , right-handed fermions which transform the same way as Φ under G dark . As we will shortly demonstrate, this choice has important consequences since then Yukawa-type interactions between the right-handed neutrinos ν R , the N R and Φ (the last two are odd under (Z 2 ) Φ ) are allowed [7]. This represents a new renormalizable portal coupling between the νSM and the dark sector.
Note that since the N R are odd under G dark , they cannot mix with the ν L to generate a Dirac mass term; the ν R , however, can.
The terms of dimension ≤ 4 in the effective Lagrangian, L (4) , consists of 3 parts The first part is the Standard Model Lagrangian with right-handed neutrinos where Γ e,u,d are 3 × 3 matrices, Γ ν is a 3 × n ν matrix and Majorana mass m ν is n ν × n ν matrix. Note that we can always choose a field basis such that m ν is diagonal. The second term in (4.2) contains only dark fields 2 L (4) where m 2 Φ > 0 in order to preserve (Z 2 ) Φ . m N is a n N ×n N matrix that, as in the case of the ν R , can be assumed to be diagonal; possible mechanisms for vector-boson-mass generation are reviewed in appendix B.
The last term in (4.2), responsible for dim-4 interactions between νSM and DM, reads where Y Φ is a n ν × n N matrix. Except of the standard Higgs portal, ϕ † ϕΦ 2 , following [7], we have added above a possible Yukawa interactions between ν R , N R and the dark scalar Φ. Here we have also assumed that the dark fields can carry only their own Z 2 quantum numbers, so a given dark field can not transform non-trivially under Z 2 that stabilizes a different dark sector component, this eliminates some operators that otherwise would be present, e.g. ν T R CΨ R Φ. Note that the kinetic mixing between the U (1) Y and the additional U (1) corresponding to the dark vector boson V µ is forbidden by the stabilizing symmetry G dark . The Lagrangian L (4) νSM ×DM contains all possible renormalizable interactions between the νSM and DM that are allowed within the assumptions specified above.
We wish to make a comment concerning stability of fermions that appear in our scenario and which are neutral under SM gauge symmetries. We assumed that there are n ν righthanded neutrinos ν R . In general ν R 's decay by standard Yukawa interactions, however the lightest of them is stable. Besides ν R 's there are n N of N R 's, Ψ R and Ψ L . Among then only Ψ R and Ψ L are guarantied to be stable (by the virtue of (Z 2 ) Ψ R × (Z 2 ) Ψ L ). Since both N R 's and Φ are odd under G dark , therefore the lightest of them is stable, in other words N R 's might be unstable.
Equations of motion for νSM fields derived from the L (4) for νSM fields are the same as in the SM with two exceptions. Equations for the Higgs doublet (ϕ) and right-handed neutrinos (ν R ) contain terms, that originate from interactions present in L (4) νSM ×DM . The complete list of equations of motion for DM and SM ν R fields is: (4.8) Many operators of the form νSM × DM are redundant through the application of the equations of motion [2], and should be omitted from the basis. Below we provide an illustration of this process of elimination; the notation we use is the following. If an operator includes LHS of one of the above equations, then it can be written as a sum of the operators that consists of the RHS of that equation and an operator, denoted by EOM , which vanishes due to that equation of motion. The purpose is to express a given operator as a linear combination of other operators, total derivatives T D and EOM . Such operators are redundant in effective Lagrangian. Operators vanishing due to the Bianchi identity are denoted by BI . Using tab. 2 and 3 and these rules one can construct an irreducible basis of νSM × DM operators up to dim 6. We provide two examples of how the equations of motion can be used to eliminate some operators. First we show thatψγ µ ψ∂ µ (Φ 2 ) (ψ ∈ {l, ν R , e, q, u, d}) is redundant. After integrating by parts and applying equation of motion (4.7) we obtain the followinḡ νSM × Φ 2 denotes operators made as a product of some operator belonging to L (4) νSM and Φ 2 . If all operators O (4) νSM × Φ 2 are included in our list then there is no need to haveψγ µ ψ∂ µ (Φ 2 ) as well.
A bit more involved algebra is needed to show redundancy of B µνΨ γ µ ∂ ν Ψ where we used Again, if operators iϕ † ← → D µ ϕΨγ µ Ψ andψ p γ µ ψ qΨ γ µ Ψ were present then B µνΨ γ µ ∂ ν Ψ should be omitted from the operator basis. Similar arguments apply if B µν is replaced byB µν .
Before proceeding to the final table collecting all the effective operators we discuss the mechanisms of dark vector boson mass generation, as they are relevant for the final output.

The Stuckelberg mechanism
In this scenario the Lagrangian (B.2) is invariant under Z 2 symmetry, with both σ and V µ being odd. The equations of motion read (4.12) The Stuckelberg Lagrangian and equations of motion reduce to a part of the Lagrangian (4.4) and equations (4.8) when the σ = 0 (unitary) gauge is adopted. It is worth, at this point, to discuss in some detail the operator composed of two dark vector fields and two Higgs boson doublets: V µ V µ ϕ † ϕ, the effects of which have been investigated in the literature e.g. in [8]. Because of gauge invariance this operator can only be generated by V µ V µ ϕ † ϕ of mass dimension 6. It should be noticed that in the σ = 0 (4.13) therefore there appears an unavoidable suppression factor m 2 V /Λ 2 even though formally V µ V µ ϕ † ϕ is dim-4 operator. Note that higher dimensional operators of that sort would be suppressed by higher powers of Λ.

The Higgs mechanism
Another method to generate vector mass is the Higgs mechanism (B.3). In this case (Z 2 ) V corresponds to charge conjugation: writing φ = ρ exp(iθ), V µ and θ are odd while ρ is even. Since we require V µ to be stable, the Lagrangian must be invariant under (Z 2 ) V and this symmetry must remain unbroken. This is indeed what happens since only ρ develops a vacuum expectation value, for details see appendix B. The vector-boson mass m V is of order of the Higgs field vacuum expectation value v provided the gauge coupling constant g ∼ v/Λ and ρ ∼ Λ. The remaining physical scalar has a mass ∼ Λ and decouples from the low-energy theory.
Let us again focus on the V µ V µ ϕ † ϕ operator. It is easy to see that it can only be generated by |D µ φ| 2 (ϕ † ϕ). Using now the above values for g and ρ we find (4.14) so we observe that this coupling is also suppressed in the Higgs scenario. Few other comments are here in order. First, note that the operator φ * D µ D µ φ is not invariant under (Z 2 ) V . In order to make it invariant we have to add its conjugate: Note that this operator is symmetric in its Lorentz indices and vanishes after contraction with B µν .

νSM × DM operators
The resulting effective operators obtained via (1.1) using tables 2 and 3 are contained in table 4. There are several comments here in order. Note that operators denoted by "TREE" in table 4 are those for which there exists a new physics model where they are generated at tree level, but it is not yet possible to determine whether this is the case for the situation realized in Nature; a specific model may generate those potentially tree generated (PTG) operators at one or higher loops, or may not generate it at all because of the details of its particle content and symmetries.
All operators present in table 4 are of the form of (1.1) with O νSM and O DM being separately invariant under symmetries of νSM and G dark , respectively. Such operators can be generated at tree-level by the exchange of heavy particles that may or may not be neutral under the νSM and dark symmetries (both options always exist, though existing data may constrain the properties of non-neutral particles more severely). For example ϕ † ϕΨ T L CΨ L can be generated by the exchange of a neutral heavy scalar S with couplings S|ϕ| 2 and Ψ T L CΨ L S; or by a heavy Dirac fermion F with the same SM gauge transformation properties as ϕ, odd under (Z 2 ) Ψ L , and with couplingsF ϕΨ L andF ϕ(Ψ L ) C (C denotes the usual charge conjugation operation). For an illustration, in table 5, we draw generic diagrams (within an underlying theory) that could be responsible for operators contained in table 4.
It should be noticed that the stabilizing symmetries imply that neither Ψ L nor Ψ R can appear separately in any interaction. Therefore it is sufficient to restrict ourself just to one chirality of Ψ, consequently one could drop e.g. Ψ R , such that Ψ in table 4 would just correspond to Ψ L or N R .
Loop: Table 4. List of all νSM × DM operators up to dim 6, that are suppressed by at most Λ −2 . Dark matter sector consists of a real scalar Φ, chiral fermions Ψ ∈ {Ψ L , Ψ R , N R } and vector field V µ . Tree and loop-generated operators are collected in the upper and lower part of the table, respectively. Operator ϕ † ϕV µ V µ appears in both categories, because within the Higgs mechanism, it can be generated at the tree-level approximation, while within the Stuckelberg model it requires a loop. Note that one entry in the table may refer to various operators, because (∼) over X µν denotes X µν orX µν , X µν stands for B µν , W I µν or G A µν and ψ ∈ {l, ν R , e, q, u, d}. The bosonic operators are all Hermitian. In case of the operators containing fermions, iϕ † ← → D µ ϕΨγ µ Ψ is Hermitian and conjugation ofψ p γ µ ψ qΨ γ µ Ψ is equivalent to transposition of the generation indices. For the remaining operators Hermitian conjugations are not listed explicitly.

Summary
In this paper we have constructed a basis of operators of dim ≤ 6 which describe interactions between Dark Matter composed of an Abelian vector, chiral fermions and a real scalar with the Standard Model. Our assumptions were the following: • Each component of the dark sector is stable by the virtue of an independent Z 2 symmetry, • Each component of the dark sector transforms non-trivially only under the symmetry which is responsible for its own stability, • The Standard Model fields are neutral under any symmetries of the dark sector, • The dark sector contains neutral fermions, N R , which are odd under (Z 2 ) Φ symmetry responsible for stability of Φ, the N R has no other quantum numbers, • The dark sector fields are neutral under any symmetry of the Standard Model, mediator operator neutral charged
Sign convention for covariant derivative is exemplified by (3) generators with Gell-Mann matrices λ A and S I = 1 2 τ I are SU(2) generators with Pauli matrices τ I . It is useful to define Hermitian derivative term Gauge field strength tensors and their covariant derivatives are

B Mass generation for Abelian vector bosons
In this appendix we review possible mechanisms of Abelian vector-boson mass generation. A massive vector field can be described by the Proca Lagrangian (B.1), since the mass term spoils the gauge invariance therefore renormalizabilty of this theory is not apparent.

B.1 The Stuckelberg mechanism
The Stuckelberg mechanism is a way to restore the gauge symmetry of (B.1) by introducing a real scalar field σ (see e.g. [9], [10]) with appropriate transformation rules: The field σ can be eliminated from the model by choosing χ = −σ/m V ; in this gauge the Stuckelberg Lagrangian becomes the same as in the Proca theory (B.1). It should be emphasized that when the mass of the vector field is generated by the Stuckelberg mechanism (B.2) the gauge invariance requires that the vector field appears only as V µν , V µν or V µ = ∂ µ σ − m V V µ , both of which have mass dimension 2.

B.2 The Higgs mechanism
Another way to make an Abelian vector field massive is the Higgs mechanism. It uses complex scalar field φ that acquires a vacuum expectation value φ = f / √ 2 which spontaneously breaks a U (1) local symmetry and thus generates a mass m V = gf for the associated gauge vector field V µ :

(B.3)
In contrast to the Stuckelberg mechanism the scalar field cannot be completely eliminated by the gauge transformation. We can write the complex scalar field as φ = 2 −1/2 (f +h)e iθ/f in terms of which the Lagrangian becomes The field h has a mass m h = 2 √ λf 3 . The Stuckelberg mechanism can be seen as a limit of the Higgs mechanism when f → ∞, g → 0 with m V fixed. The scalar field h decouples from θ and V µ as its mass tends to infinity m h → ∞ as a result of an arbitrarily large dimensional parameter (f ). Then L H becomes identical as the Stuckelberg one. (B.5) where L h is Lagrangian of the decoupled field h. It is worth to see the decoupling also in the effective field theory framework. If we assume that V µ is a light particle with mass of the order v, which is the SM Higgs vev, we can assign h into the heavy sector of the mass scale Λ. Since the scalar mass is m h = 2 √ λf ∼ √ λv/g, therefore for λ ∼ 1 it is enough to set coupling g ∼ v/Λ. Gauge invariant quantities containing a vector field in the model with the Higgs mechanism (B.3) are built from V µν ,Ṽ µν and covariant derivatives of complex scalar field D µ φ. It is assumed that νSM fields are singlets under the Higgs U(1) symmetry. Operators built of V µν andṼ µν only are the same as in Stuckelberg case. Operators with φ appear at dimension 3 (or higher), because they must contain φ * to ensure gauge invariance and at least one covariant derivative that contains the vector field. 3 It is worth noticing that the field θ could be removed from the Lagrangian choosing unitary gauge φ → φe −iθ/f , Vµ → Vµ + ∂µθ/f . In this gauge the interactions of h and Vµ are: 2g 2 f hVµV µ , g 2 h 2 VµV µ , −2λh 3 f, −λ/2h 4