Two-loop Beta Function for Complex Scalar Electroweak Multiplets

We present the general form of the renormalizable four-point interactions of a complex scalar field furnishing an irreducible representation of SU(2), and derive a set of algebraic identities that facilitates the calculation of higher-order radiative corrections. As an application, we calculate the two-loop beta function for the SM extended by a scalar multiplet, and provide the result explicitly in terms of the group invariants. Our results include the evolution of the Higgs-portal couplings, as well as scalar"minimal dark matter". We present numerical results for the two-loop evolution of the various couplings.


Introduction
Complex scalar elds furnishing a general representation of the electroweak gauge group SU (2) × U (1) of the standard model (SM) received increased interest in recent years. For instance, they can provide a viable dark ma er candidate in so-called minimal dark-ma er models [1]. e renormalization group (RG) evolution of coupling constants is an invaluable tool in phenomenological analyses [2]. It plays a particularly important role when interpreting and comparing the results of experiments performed at widely di erent energy scales, such as dark ma er direct detection and production of dark ma er at particle colliders. A framework for consistent RG analysis for fermionic dark ma er in the context of e ective eld theories has been presented in Ref. [3]. e rst consistent and complete basis of e ective operators for scalar dark ma er up to mass dimension six has been wri en down in Ref. [4]; however, the RG evolution has not yet been calculated.
For scalar dark ma er it is possible to write down self interactions, as well as interactions with the SM, at the renormalizable level -the so-called Higgs-portal dark ma er [5][6][7]. To our knowledge, the rst classi cation of the self interactions of scalar elds with electroweak charges has been given in Ref. [8]. In this work, we rederive the scalar potential in a slightly di erent form that is well suited for the calculation of radiative corrections.
As an application, we calculate the beta functions for all scalar couplings, as well as the new scalar contributions to all SM couplings, at the two-loop level. To this end, we prove a set of algebraic relations that allows to express all two-loop matrix elements in terms of tree-level matrix elements of the basis operators in the scalar potential. While these algebraic relations simply rely on the algebra of Clebsch-Gordan coe cients as well as SU (2) gauge symmetry, many of them turn out to be quite non-trivial, and have not been derived before, to the best of our knowledge. Among other results, we show how to express a product of two SU (2) generators, contracted over their adjoint indices, in terms of Clebsch-Gordan coe cients. e resulting relations can be used to manipulate general representations of the SU (2) algebra in an algorithmic way.
Our results are valid for a scalar eld furnishing an arbitrary irreducible representation of SU (2) and for arbitrary hypercharge. While these results are known in principle [9,10], we present them in closed form and explicitly in terms of group invariants for the rst time. We believe that this form of the beta functions makes them more suitable for practical applications. Auxiliary les with our analytic results in computer-readable form are available via a gitlab repository (see Sec. 6).
As a cross check of our results, we also calculated the two-loop beta function for most of the SM Higgs, gauge and Yukawa couplings. We nd a result consistent with the SM beta function extracted from Ref. [9], see Ref. [11], if we take into account the corrections pointed out in Refs. [12,13]. See also Refs. [14][15][16][17][18] for recent results at the three-and four-loop level.
Depending on the representation, the impact of the one-and two-loop contributions to the running of the scalar as well as the SM couplings can be sizeable. We discuss a few examples, focusing on a scalar septuplet ("minimal dark ma er") and the running of the SM quartic Higgs and SU (2) gauge coupling.
is paper is organized as follows. In Sec. 2 we de ne our setup and construct the scalar potential. In Sec. 3 we present our results for the beta functions. e required algebraic relations are collected and proven in Sec. 4. Sec. 5 contains numerical illustrations of our results. We conclude in Sec. 6. Supplementary material is presented in two appendices. In App. A we describe the various analytic checks that we performed on our calculation, and derive explicit formal expressions for the beta functions. In App. B we provide all eld and mass renormalization constants that are necessary in intermediate steps of the calculation. For completeness, we also include all quadratic poles of the coupling renormalization constants.

Construction of the operator basis
We consider a complex scalar eld ϕ with mass M ϕ which furnishes a (2j ϕ + 1)-dimensional irreducible representation of the Standard Model SU (2) × U (1) gauge group, where j ϕ = 0, 1/2, 1, . . . is any integer or half integer. e Lagrangian for this model is given by e summation convention over Lorentz and adjoint gauge indices is in use here and in the following. e covariant derivative acting on the scalar eld is given by with the corresponding eld strength tensors Here, B µ and W a µ (with a = 1, 2, 3) are the U (1) and SU (2) gauge elds, respectively. eτ a kl are SU (2) generators in the (2j ϕ + 1)-dimensional representation, de ned by τ 1 ± iτ 2 kl = δ k,l±1 (j ϕ ∓ l)(j ϕ ± l + 1) , with k, l running over the values −j ϕ , −j ϕ + 1, . . . , j ϕ − 1, j ϕ , while Y ϕ is the scalar hypercharge. We now derive the general form of the scalar potential V ϕ . Any Hermitian, renormalizable fourscalar operator has the general form e form of the real coe cients v jϕ irks must be determined such that the operator O ϕ is invariant under the SU (2) gauge group (the U (1) invariance is immediately apparent). Ignoring all quantum numbers that do not transform under SU (2), the operator coe cients can be wri en as v jϕ irks ≡ j ϕ , r; j ϕ , s| V |j ϕ , i; j ϕ , k (6) where V are the reduced matrix elements. Inserting two complete sets of states, we have where C jj (JM ; mm ) are Clebsch-Gordan coe cients (we use the notation of Ref. [19]). De ning the composite eld operator Writing a general SU (2) transformation as D (J) = exp iθ aτ (J),a , whereτ (J),a are here the SU (2) generators in the 2J + 1-dimensional representation, gauge invariance requires Using the unitarity of the D matrices, this can be wri en as the condition By Schur's Lemma, v is either zero or has the form where λ (J) ϕ is a constant. We de ne a set of "Sigma matrices" as (note that we regard the isospin j ϕ of the scalar multiplet to be xed in this work). We then write the general potential as 1 e symmetry properties of the Clebsch-Gordan coe cients imply the corresponding properties of the Sigma matrices, Σ is restricts the number of independent operators in the basis. Obviously, the coe cients v jϕ irks in Eq. (5) can be chosen symmetric under exchange of i ↔ k and r ↔ s. Hence, the only non-zero operators in our basis are those involving Sigma matrices that are symmetric in their lower indices, is immediately tells us that there are N ϕ ≡ oor(j ϕ + 1) operators in our basis. As a related consequence, the sum over J in Eq. (14) e ectively runs only over even values for integer j ϕ , while for half-integer j ϕ only terms with odd J contribute.
We illustrate this construction by the example of an electroweak doublet. e Sigma matrices for j ϕ = 1/2 are e potential operator for J = 0 vanishes identically: and only the operator for J = 1 remains: is is equivalent to the fact that we can, employing the more standard de nition of operators, express (ϕ † σ a ϕ) 2 in terms of (ϕ † ϕ) 2 , using the Fierz relation σ a ij σ a kl = 2δ il δ kj −δ ij δ kl . Here, σ a ij are the usual Pauli matrices.

Beta function for a scalar multiplet
In this section, we present the beta function of the full SM extended by a scalar ϕ furnishing a rep- Figure 1: Feynman diagrams corresponding to the contributions from the scalar eld ϕ to the one-loop standard model beta function. Fig. (a) shows the contribution to the gauge boson eld counterterms which must be subtracted when gauge bosons appear in external states in Green's functions. Fig. (b) shows the ϕ loop contributing to the one-loop Higgs quartic coupling beta function.
consider is given by where L ϕ is given in Eq. (1), is the gluonic QCD Lagrangian, and are the kinetic terms for the SM fermions, where Q L and L L denote the le -handed quark and lepton doublets, and u R , d R , and R the right-handed up-quark, down-quark, and lepton elds. e sums run over the three fermion generations, k = 1, 2, 3. Furthermore, is the Higgs doublet Lagrangian, and the Yukawa Lagrangian is given by where H c = iσ 2 H * is the charge-conjugated Higgs eld. In this work, we neglect the Yukawa couplings of all light fermions, keeping only the top, bo om, charm, and τ Yukawas y t , y b , y c , and y τ non-zero. is implies that we can assume the Yukawa matrices to be diagonal and neglect CKM mixing. Finally, the Higgs-portal Lagrangian is given by Here, τ a ≡ σ a /2 in terms of the usual Pauli matrices. Note that the second term in Eq. (25) is absent in the case j ϕ = 0. e Lagrangian (20) is renormalized in the usual way by introducing eld and coupling renormalization constants. For instance, we express the unrenormalized scalar couplings (denoted by the superscript "0") in terms of renormalized couplings as and similarly for all other couplings and elds. e superscripts (1) and (2) denote the one-and two-loop contributions, respectively. e ellipsis stands for higher-order terms. We extract the beta function in the MS scheme from the 1/ poles of the coupling counterterms, as explained in App. A. We employ dimensional regularization in d = 4 − 2 space-time dimensions, and we can treat all particles as massless in our calculation.
We determine all renormalization constants by calculating the divergent parts of Green's functions with suitably chosen external states (sample Feynman diagrams are shown in Figs. [1][2][3][4]. In the calculation of the coupling counterterms, it is necessary to subtract eld counterterms corresponding to the external elds. For this reason, all eld renormalization constants are calculated in addition to the coupling renormalization constants (the results are collected in App. B).
In order to isolate the ultraviolet poles, we employ the infrared (IR) rearrangement described in Ref. [20], to which we refer for more details. In short, the method amounts to an exact decomposition of all propagators in terms of propagators with a common IR regulator mass, which we call M IRA . E ectively, we introduce a common mass M IRA for the scalar, the gauge-boson, and the ghost elds, ese masses get renormalized at higher orders, and we introduce corresponding mass counterterms Z M IRA ,i , i = W, B, ϕ, in the usual way (M 2 bare = Z M 2 M 2 ). e explicit results needed for our work are collected in App. B. We explicitly veri ed that all our results are independent of the regulator mass M IRA , as it should be.
All O(10 000) Feynman diagrams were calculated using self-wri en FORM [21] routines, encoding the algorithm presented in Ref. [22]. e Feynman diagrams were generated using qgraf [23]. e SU (2) group algebra and renormalization was performed independently by the two authors; the results are in complete agreement. We describe further analytic checks of our calculation in App. A. e beta functions are de ned as the logarithmic derivatives of the couplings with respect to the renormalization scale, µ d dµ ey are given in terms of the coupling counterterms by for all couplings, denoted here collectively by g i = g 1 , g 2 , g s , λ (J) ϕ , λ ϕH , λ ϕH , λ H , y t , y b , y c , y τ . Here, Z g i ,1 is the residue of the 1/ pole of the counterterm and a k = 1 when g k is a gauge or Yukawa coupling while a k = 2 when g k is a quartic scalar coupling. Expanding the beta function by loop order as β g i = β (1) g i + β (2) g i + . . ., we nd for the one-loop contributions Here and in the following, a prime on the summation sign indicates a restricted sum over indices, de ned by e sum e ectively runs over even or odd values of J only, if the weak isospin j ϕ of the scalar multiplet is integer or half-integer, respectively. (For instance, for a SU (2) septuplet with j ϕ = 3 we have J = 0, 2, 4, 6.) e group-theory functions are de ned as J (j ϕ ) ≡ j ϕ (j ϕ +1), D(j ϕ ) ≡ 2j ϕ +1, and in terms of the Wigner 9j symbol [24] -see Sec. 4 for more details. Moreover, n g = 3 denotes the number of SM fermion generations. Our one-loop results for the pure SM contributions agree with those in Ref. [11]. e scalar contribution to β (1) g 2 agrees with the expression given in Ref. [8]. e remaining results are new.
We note here that, at one-loop, the only beta functions which receive contributions from the complex scalar are the gauge and quartic scalar couplings. e contributions to the gauge coupling beta functions arise in our calculation from the gauge boson eld counterterms (Fig. 1a). In addition to SM terms, the Higgs quartic coupling beta function gains two terms from diagrams with scalar loops, shown in Fig. 1b. e beta functions for the Higgs-portal couplings and quartic scalar couplings are subdivided into three classes: scalar only terms, mixed scalar-gauge terms, and gauge-only terms. Sample diagrams of each of these classes are shown in Fig. 2.
e Higgs-portal coupling beta functions also receive contributions from Yukawa couplings, coming from the eld counterms for the external Higgs elds in the four-point Green's functions.
In order to express these contributions in terms of the operators in the scalar potential (14), we rewrite all SU (2) generators appearing in the W -boson vertices in terms of the Sigma matrices dened in Eq. (13), and use completeness relations for the Clebsch-Gordan coe cients to simplify the terms. A similar strategy is applied for the "mixed" contributions involving both gauge and scalar interactions. e detailed relations that we use are discussed in Sec. 4. In several cases, particular care has to be taken, as the sum over indices in the completeness relations runs over all possible values of the J spin quantum number, while the local scalar interactions can only involve the restricted sums over odd or even values. Gauge invariance ensures that the nal result can be expressed in terms of restricted sums only.
A comment on our treatment of γ 5 is in order. Diagrams containing fermion triangles can contribute terms with an odd number of γ 5 matrices to the gauge-boson eld counterterm and gauge coupling counterterms. We took the corresponding contributions to the gauge coupling beta functions from the literature [9,11], and calculated only the additional scalar contributions at one-and two-loop (sample Feynman diagrams showing these contributions are given in Fig. 3a). For all other (scalar and Yukawa) beta functions, we performed the two-loop calculation including also the full set of SM particles. We veri ed explicitly that, in our calculation, only traces with an even number of γ 5 matrices in closed fermion loops appeared. According to common lore [25], we evaluated these traces using naive anticommuting γ 5 . We nd the following two-loop results: where the coe cients are given, for k = λ For the Higgs-portal couplings we nd Our results for the quartic Higgs self coupling are Our pure SM results agree with those in Refs. [9,11], apart from three terms which are consistent with the corrections made in Ref. [13]. All other analytic results are presented here in closed form for the rst time. 2 For the two-loop calculation, we use the same strategy to express all SU (2) generators in terms of Sigma matrices and to simplify the expressions using the relations given in Sec. 4. It is again possible to express all results in terms of the operators in the scalar potential (14), as required by gauge invariance.
At two-loops, all beta functions receive contributions from scalar elds. In the Yukawa beta functions, the only additional terms from scalar elds arise from the external Higgs eld counterterm (Fig.  3b). e Feynman diagrams required to extract the quartic scalar coupling beta functions again split into di erent classes: those including zero, one, two, or three internal gauge bosons. In Fig. 4, we give sample diagrams from each class which give contributions to the quartic scalar coupling beta functions.

Group Theory Relations
To express all results in terms of matrix elements of our basis operators, and to check the gaugeparameter independence and locality of our two-loop counterterms explicitly, we had to use a number of algebraic relations. ese relations arise from the gauge invariance of the underlying theory as well as the properties of the Clebsch-Gordan coe cients, and are collected and proven below. For clarity, the summation convention is suspended in this section. All summations are indicated explicitly.
To begin, we collect some orthogonality properties of the Sigma matrices that follow directly from the corresponding standard properties of the Clebsch-Gordan coe cients: e exchange of W gauge bosons introduces explicit SU (2) generators that need to be rewri en in terms of Sigma matrices. Since the Clebsch-Gordan coe cients describe a transformation between two complete sets of orthonormal state vectors, they are used to rewrite the product of two SU (2) generators: Diagrams with multiple scalar couplings likewise need to be expressed in terms of the basis operators. is is facilitated by the following "sum rule" for Sigma matrices: In the following, we give explicit expressions for C(J) and K(J 1 , J 2 , J 3 ). We then derive further relations between these quantities that can be used to simplify the results of our calculation. Our general strategy is to express all results in terms of our operator basis and the group theory invariants J (j ϕ ) ≡ j ϕ (j ϕ + 1), the eigenvalue of the SU (2) Casimir operator, lτ a ilτ a and D(j ϕ ) ≡ 2j ϕ + 1, the dimension of the SU (2) multiplet representation with isospin j ϕ .
We begin by showing in terms of the Wigner 9j symbol [24]. Starting with Eq. (98), we multiply both sides by Σ (J),M rl and sum over r, l, to obtain e Sigma matrices can be wri en in terms of the Wigner 3j symbols as [24] Σ (J),M In this way, Eq. (101) becomes Since M 1 , M 2 ∈ Z, the factor of −1 disappears. We can also freely change −M 1 , −M 2 → M 1 , M 2 since these indices are summed over. We also take M → −M on both sides. Now, we use the symmetry properties of the 3j symbols to rewrite Comparison of the last two equation yields Eq. (100). Note that, as expected, K(J 1 , J 2 , J 3 ) is symmetric in its rst two indices. Next, we show × j ϕ m|τ (jϕ),a j ϕ m j ϕ n|τ (jϕ),a j ϕ n .
Noting that, by de nition,τ (jϕ),a are the spin-j ϕ generators, we introduce the notationτ a mn ≡ j ϕ m|τ (jϕ),a |j ϕ n , i.e. we label the generators with j ϕ as well as a = 1, 2, 3, and the states by j ϕ and their "magnetic" quantum numbers m, n, m, n = −j ϕ , . . . , j ϕ . Using the symmetry relation of the Clebsch-Gordan coe cients where we use m = M − n and m = M − n , as well as the fact that M is always integer, to rewrite the phase factor. We also take −M → M using the symmetry of the sum over M . Now, we arti cially regard each spin-j ϕ state as belonging to the spin-j ϕ subspace of the Clebsch-Gordan decomposition of the tensor product of a spin-j ϕ and a spin-J state. For instance, j ϕ m| = nN C jϕJ (j ϕ m; nN ) j ϕ n; JN | and analogous relations lead to j ϕ m|τ (jϕ),a j ϕ m = nN n N C jϕJ (j ϕ m; nN )C jϕJ (j ϕ m ; n N ) j ϕ n, JN | τ (jϕ),a ⊗ 1 J + 1 jϕ ⊗τ (J),a j ϕ n , JN .
Hence, we nd the explicit tensor decomposition for the generators we see that where we use the fact that the generators are traceless. In practice, the interchange of indices in the Sigma matrices gives rise to additional phase factors, hence we also need the relation To prove this relation, we again sum Eq. (108) over J, now taking into account the symmetry properties of the Clebsch-Gordan coe cients We note in passing that this relation can be used to calculate the "restricted" sum over J, as e proof proceeds similar to the above, except we must use the (anti-)symmetry of the Clebsch-Gordan coe cients: We then use the SU (2) algebra to re-write ab imknτ which gives the relation (129). Again, we use this to calculate the restricted sum over J, as Another pair of rules cubic in C(J) is necessary to reduce the algebra in diagrams involving the SM Higgs. e rst is given by Rewriting the le -hand side and using the appopriate orthogonality relations for the Clebsch-Gordan coe cients gives where we make use of the de nition of the totally symmetric tensor, which vanishes for SU (2). Next, we simply use the group algebra to nd Squaring this expression and using abc abc abc = 6 (138) gives the result in Eq. (133). e second relation cubic in C(J) is As before, we re-write the le -hand side of this expression and use orthogonality relations which, when the trace is performed over the identity, gives Eq. (139). In order to derive the necessary algebraic relations involving the factor K, we rst prove the following useful relation: To derive this, we note that the only object in our basis with a single adjoint representation index and two isospin-j ϕ representation indices is the generatorτ a ir (any other objects with only these free indices can be reduced to this generator). erefore, we make the ansatz Multiplying both sides byτ a ri , summing over a, i, r, and using Eq. (108) gives the relation (144). Using this result, we now prove To this end, we consider the product Equating these expressions and using the orthogonality relations, the result (146) follows.
A further important relation incorporates the condition of gauge invariance: All interaction vertices must be gauge-invariant. A scalar SU (2) spin-j ϕ eld multiplet transforms as ϕ i → ϕ i = ϕ i + δϕ i under an in nitesimal gauge transformation, where while Hence, we have the relation and summing over i, k, r, n yields Eq. (149). We now derive a few relations involving K and two powers of C. e rst is For a proof, consider the product of generators but it is also expressed as Equating these two expressions, multiplying both sides by M Σ To prove it, we perform the sum over J 2 using Eq. (146) and the fact that K is symmetric in its rst two indices to nd where in the last equality we used Eq. (129). e relation without phase factors is shown similar to the above, performing the sum over J 2 and using Eq. (126). Finally, we prove the following symmetry relation for a contraction of two K factors: First, consider the sum is simpli es to   Table 1: Numerical input used to determine the initial conditions of the coupling constants. All values are taken from Ref. [27].
Equating these two expressions gives the nal result (162).

Numerics
In this section, we present numerical results for the running of the scalar and gauge couplings. All the numerical inputs are taken from Ref. [27], see Tab. 1. We employ the expressions given in Ref. [28] to determine the initial conditions for the strong coupling g s (M Z ) = 1.1626, the top Yukawa coupling y t (M Z ) = 0.9320, and the quartic Higgs coupling λ H (M Z ) = 0.5040. We determine g 1 (M Z ) and g 2 (M Z ) directly via the relation  Figure 6: Two-loop running of scalar quartic couplings for j ϕ = 3 with Y ϕ = 0. e notation is the same as in Fig. 5.
Note that G F is RG invariant, and we neglect the QED running of m τ . We obtain y c (M Z ) = 0.0036 and y b (M Z ) = 0.0164 in the six-avor theory by four-loop QCD running and decoupling of the corresponding quark masses and subsequent conversion using an expression analogous to Eq. (167). As we are only interested in the qualitative behaviour of our results, we neglect uncertainties throughout. We solve the coupled system of RG equations numerically, using the python package pywigxjpf [29] and the Mathematica code found in Ref. [30] for the numerical evaluation of the Wigner 9j symbols.
In Fig. 5 we show the one-loop running of all scalar couplings for j ϕ = 3, with scalar hypercharge Y ϕ = 0.
is case corresponds to the "minimal scalar dark ma er" (MSDM) scenario in Ref. [31], amended by the two Higgs-portal couplings λ ϕH and λ ϕH . In the le panel, we assumed an initial condition of λ i (M Z ) = 0.5 for all four scalar couplings and the two Higgs-portal couplings. e highenergy behaviour is largely independent of these assumptions; in fact, even if the couplings are all zero at the weak scale, large values get generated via weak gauge-boson exchange (with the exception of λ ϕH ). e couplings quickly enter a non-perturbative regime and run into a Landau pole around 10 5 GeV.
Next, we study the impact of the two-loop corrections to the RG evolution of the scalar couplings in the same scenario, see Fig. 6. Again, we display the results for the two sets of initial conditions. Note that the Landau pole around µ = 10 5 GeV is shi ed to the higher scale µ = 10 7 GeV, with a plateau-like behaviour in between. However, these features appear at Fig. 6. Again, we display the results for the two sets of initial conditions. Note that the Landau pole around µ = 10 5 GeV is shi ed to the higher scale µ = 10 7 GeV, with a plateau-like behaviour in between. However, these features appear at non-perturbative values for the coupling constants and should therefore not be taken too literally. e only signi cant change is that the "triplet" Higgs-portal coupling λ ϕH turns out to be asymptotically free.
Finally, we examine the impact of the new scalar degrees of freedom on the running of the SM couplings. We keep assuming vanishing hypercharge for the new scalars, Y ϕ = 0, and focus on the evolution on the gauge coupling g 2 rst. e running of g 2 is displayed in Fig. 7. In the le panel, we show the one-loop evolution. We see that, at one-loop, the SU (2) gauge coupling exhibits a Landau pole at around 10 15 GeV for j ϕ = 3 (MSDM), while for higher representations the Landau pole appears close to or below the TeV scale. is behaviour has been qualitatively described in, for instance, Ref. [8]. Looking at the two-loop results in the right panel in Fig. 7, we see that the Landau pole for j ϕ = 3 is signi cantly shi ed down to 10 7 GeV, while all other poles lie below the TeV scale. Apparently, the SM extended by MSDM cannot be perturbative up to the Planck scale.
As our last example, we show the evolution of the quartic Higgs coupling in Fig. 8. Again we display the one-loop results in the le panel, and the two-loop results in the right panel. While the SM evolution of λ H (black dashed line) is only marginally a ected by the presence of an additional scalar multiplet with j ϕ = 1, higher representations lead to a drastic departure from this picture. For j ϕ = 3 (MSDM), the Higgs quartic runs into a Landau pole around 10 5 GeV, while the pole lies at the TeV scale for j ϕ = 5. Interestingly, the two-loop results show that this pole is in fact negative.
We relegate a more detailed discussion of the phenomenological implications of these results to future work.

Conclusions
In this work, we constructed the form of the potential involving four-point interaction of a complex scalar eld furnishing a general irreducible representation of the electroweak gauge group SU (2) × U (1), in terms of Clebsch-Gordan coe cients. We presented the beta functions determining the RG evolution of the scalar as well as the SM couplings explicitly in terms of SU (2) group invariants, up to the two-loop level. As an important ingredient of our calculation we proved a set of algebraic relations that we used to express the results for the one-and two-loop Green's functions in terms of our basis operators. For convenience, auxiliary les containing the analytic results of the beta functions in the form of a python module, as well as a mathematica package, are available at https://gitlab.com/complex-beta-function . Our results are completely general and might have applications in many elds. As one example, we studied the RG ow of the self interactions of scalar dark ma er in minimal dark ma er models [1], and the impact of the scalar elds on the RG evolution of the SM couplings. Moreover, the beta functions will be a necessary ingredient in the RG analysis of scalar dark ma er interacting via higher dimension operators [3,4].
A generalization of our results in this direction would be to consider the self interactions of fermionic dark ma er. is case is more complicated since the interactions start at mass dimension six, and additional Fierz relations associated with the Dirac-matrix structure restrict the form of all possible operators. is investigation is relegated to future work.

A Analytic checks of our calculation
As a check of our results we used a generalized R ξ gauge for the W , B and G elds and veri ed that all beta functions are gauge-parameter independent. For completeness, we provide here the gauge-xing and ghost terms in our Lagrangian: As a second consistency check of our calculation, we veri ed that all two-loop counterterms are local, i.e. they do not contain any explicit logarithms of the renormalization scale µ. As a third check of our calculation, we derive the explicit expressions of the beta function in terms of the coupling counterterms (see below). e niteness of the beta function as → 0 yields consistency relations that allow to calculate the quadratic pole of the two-loop coupling renormalization constants in terms of the one-loop results. ese quadratic poles are in full agreement with the results of our calculcation. For completeness, we provide the expressions for the quadratic poles in App. B.
In the remainder of this section, we derive the relation between the beta function and the residua of the coupling renormalization constants, as well as the relation between the linear one-loop poles and the quadratic two-loop poles. As above, we denote the all couplings generically by a coupling vector g i . e bare couplings g i,0 are expressed in terms of the renormalized couplings g i as g i,0 = µ a i Z g i g i , where a i = 1 if g i is a gauge or Yukawa coupling, and a i = 2 if g i is a scalar coupling (the coe cients a i are chosen such that all couplings remain dimensionless in d space-time dimensions). Here, µ is the renormalization scale, and the Z g i are the coupling renormalization constants. We expand the Z g i by order of pole as and use standard methods [32] to express the beta function in terms of the derivatives of the linear poles of the coupling counterterms: e fact that the 1/ contributions to the beta function have to cancel leads to the following consistency condition on the counterterms: Further conditions can be derived by requiring the cancelation of the higher poles; however, they do not lead to additional constraints on the two-loop counterterms. e relation (171) is made more explicit by expanding the counterterms by loop-order, Keeping only terms at two-loop order, and using the fact that the counterterms are polynomials in the couplings, we arrive at k a k g k ∂(δZ We then rewrite Eq. (171) as We checked explicitly that this relation is satis ed for all our coupling counterterms.

B Renormalization constants
In this appendix we collect all renormalization constants that were needed in intermediate steps of the calculation, namely, all eld and arti cial-mass counterterms. e 1/ pole parts of the coupling counterterms give rise to the beta functions, as explained in App. A, and are not repeated here. For completeness, however, we show the 1/ 2 pole parts. e MS scheme is used throughout. For the one-loop eld renormalization constants we nd δZ (1) δZ (1)