Prescription for finite oblique parameters $S$ and $U$ in extensions of the SM with $m_W \neq m_Z \cos{\theta_W}$

We consider extensions of the Standard Model with neutral scalars in multiplets of $SU(2)$ larger than doublets. When those scalars acquire vacuum expectation values, the resulting masses of the gauge bosons $W^\pm$ and $Z^0$ are not related by $m_W = m_Z \cos{\theta_W}$. In those extensions of the Standard Model the oblique parameters $S$ and $U$, when computed at the one-loop level, turn out to be either gauge-dependent or divergent. We show that one may eliminate this problem by modifying the Feynman rules of the Standard Model for some vertices containing the Higgs boson; the modifying factors are equal to $1$ in the limit $m_W = m_Z \cos{\theta_W}$. We give the result for $S$ in a model with arbitrary numbers of scalar $SU(2)$ triplets with weak hypercharges either $0$ or $1$.


Introduction
The present knowledge of particle physics is encapsulated in the Standard Model (SM) [1][2][3]. This is an SU(3) × SU(2) × U(1) gauge theory that describes all the fundamental particles observed until now and the way they interact with each other. The scalar sector of the SM contains just one doublet of SU (2); it is responsible for giving masses m W and m Z to the gauge bosons W ± and Z 0 , respectively, as well as to the fermions.
Despite being one of the most accurate theories in Science, there are phenomena that the SM cannot explain. Therefore particle physicists try to either extend or complete it. One of the ways to extend the SM is by enlarging its scalar sector [4]. The most studied extension of that sector is the addition of another SU(2) doublet, obtaining a two-Higgsdoublet model (2HDM) [5]. Extensions of the SM with scalar SU(2) singlets are also frequent in the literature (see for instance Refs. [6][7][8]). There are also some extensions of the SM with scalar SU(2) triplets in the literature (see for instance Refs. [9][10][11][12][13][14][15][16][17]).
A feature of the SM is custodial symmetry. This is a symmetry of the scalar potential of the SM that is broken both by the Yukawa interactions and by the gauge coupling of the U(1) group. Custodial symmetry leads at the tree level to the relation where c W is the cosine of the Weinberg angle. The relation (1) is in good agreement with observation: experimental results give [18] ρ ≡ m 2 W m 2 Z c 2 W = 1.00038 ± 0.00020.
Models in which only scalar SU(2) singlets and doublets have vacuum expectation values (VEVs) preserve the relation (1) at the tree level. That relation is broken at the loop level because the custodial symmetry is not exact, as noted above. Peskin and Takeuchi have identified three so-called 'oblique parameters' that they named S, T , and U [19,20]. 1 These three observables parameterize some effects of New Physics (NP), i.e. physics of extensions of the SM. At the one-loop level, vacuum polarization produces the tensors Π µν V V ′ (q), where V and V ′ are gauge bosons that may be either A and A (A stands for the photon), A and Z, Z and Z, or W and W , and q is the four-momentum of the gauge bosons. We write those tensors as We then define computed in a given extension of the SM and A V V ′ (q 2 )| SM is A V V ′ (q 2 ) computed in the SM. The oblique parameters are defined as , (4b) The oblique parameters have already been computed for several extensions of the SM, in particular for models with arbitrary numbers of scalar doublets and singlets [23,24], wherein they are finite. In models where larger scalar multiplets are added to the SM and have VEVs, m W is in general different from m Z c W . As we will show in this paper, this gives rise to complications when computing the oblique parameters. 2 First of all, the oblique parameter T is divergent at one-loop level for models that do not have custodial symmetry [29,30]. Furthermore, as noted above, when computing the oblique parameters one needs to subtract the functions A V V ′ (q 2 ) computed in the SM from the same functions computed in the extension of the SM. This subtraction is non-trivial, since in the SM the masses of the gauge bosons obey equation (1) while in the extension they do not obey it. This leads to divergent parameters S and U. We have found out that, in order to obtain finite (and gauge-independent) results for S and U at the one-loop level in models with m W = m Z c W , one needs to multiply the usual SM Feynman rules 3 for some vertices containing the SM Higgs boson by factors that become equal to 1 when m W = m Z c W but are different from 1 for m W = m Z c W . This paper is organized as follows. In section 2 we consider an extension of the SM with arbitrary numbers of arbitrarily large SU(2) × U(1) scalar multiplets, subject only to the constraint of the preservation of electric charge. In section 3 we illustrate the problem that arises in the computation of both S and U in that extension of the SM, and display our proposed solution to that problem. In section 4 we restrict the model of section 2 to the case where only scalar SU(2) triplets and singlets with U(1) charges 1 or 0 are added to the SM; we give the explicit expression for S in that model. Section 5 summarizes our findings. Several appendices contain technical material that may be avoided by a hurried reader.

General extension of the SM
We consider an SU(2) × U(1) electroweak model where the scalar sector consists of an arbitrary number of multiplets M JY labeled by their weak isospin J and weak hypercharge Y . We restrict ourselves to models where all the scalars have integer electric charges 4 ; therefore, if J is a (half-)integer then Y is a (half-)integer too. Furthermore, we consider only complex multiplets; if some multiplets are real, then our conclusions are still valid but there are some modifications in the intermediate steps. Finally, for the sake of simplicity we assume that there is only one M JY for each value of the pair (J, Y ); our conclusions are still valid otherwise, but the notation would become clumsier.
Each multiplet M JY has a component M Q JY with electric charge Q if and only if J − |Q − Y | ∈ N 0 . Some multiplets M JY may have, for some values of Q = 0, both a component M Q JY with electric charge Q and a component M −Q JY with electric charge −Q; in our notation, The covariant derivative for the SU(2) × U(1) electroweak model is, as usual, 5 where e = gs W is the unit of electric charge, the T i (i = 1, 2, 3) are the three components of weak isospin, and the T ± are given by T ± = (T 1 ± iT 2 ) √ 2 . We use the SU(2) representation for weak isospin J: where r stands for the row of the matrix and c stands for the column of the matrix, with 1 ≤ r, c ≤ 2J + 1. We write M JY as a column matrix with 2J + 1 rows upon which the matrices of Eqs. (6) act. Clearly, from Eq. (6a), M Q JY is in the row J + 1 − T 3 = J + Y + 1 − Q of that column matrix. Then, using Eqs. (5) and (6), We define A to be the set of multiplets that have a component with zero electric charge.
For every Q ∈ N, we define two sets: We emphasize that both R Q and S Q are defined only for Q > 0. For those Q, let n Q denote the total number of charge-Q scalars and let S Q a (a = 1, . . . , n Q ) denote the physical (mass eigenstate) charge-Q scalars. For definiteness, we fix S 1 1 ≡ G + to be the charged Goldstone boson. We define S −Q a ≡ S Q a * .
For every Q ∈ N and M JY ∈ R Q , we write where R Q JY is a 1 × n Q row matrix. For every Q ∈ N and M JY ∈ S Q , we write where S Q JY is a 1 × n Q row matrix. We form the n Q × n Q unitary (mixing) matrix U Q by piling up all the row matrices R Q JY and S Q JY (for fixed Q ∈ N) on top of each other. Since U Q is unitary, Each multiplet M JY that belongs to A has VEV v JY (which may in some cases be zero) in its component M 0 JY with electric charge zero. We write where A JY and B JY are real 1 × n 0 row matrices and the real fields S 0 b (b = 1, . . . , n 0 ) are the physical neutral scalars. Without lack of generality, we fix S 0 1 ≡ G 0 to be the neutral Goldstone boson. We form the n 0 × n 0 real orthogonal matrix V by piling up all the row matrices A JY and B JY on top of each other. Since V is orthogonal, Hence, The masses of the gauge bosons are given in terms of the VEVs of the scalar fields by In line (14b) we have used firstly line (7b) with Q = 1 and secondly line (7c) with Q = −1.
In passing from line (14b) to line (14c) we have used One sees in Eqs. (14) that m 2 Z c 2 W = m 2 W in general requires J 2 + J = 3Y 2 for all the nonzero v JY . 6 This holds for the standard case of doublets with J = Y = 1/2, and also for neutral singlets with J = Y = 0, but it does not hold for most other choices of J and Y .

Prescription for S and U
We consider the vacuum-polarization one-loop diagram with two scalars as internal particles, see Fig. 1. Let the Feynman rules for its vertices be Figure 1: One-loop vacuum-polarization diagram with two scalars as internal particles.
Then the contribution of that diagram to is where "div" is a divergent quantity defined in Eq. (68) of appendix A. That quantity is independent both of q 2 and of the masses of the scalars in the diagram of Fig. 1. We want to compute the divergent contributions to the quantitites (16) for V V ′ = AA, AZ, ZZ, and W W . Those contributions originate solely in diagrams like the one in Fig. 1. (Other types of diagrams give either vanishing or finite contributions.) We therefore need the factors K and K ′ for all the diagrams, i.e. we need the Lagrangian terms that describe the interactions between one gauge boson and two scalars. Those terms are derived from Eq. (7) and may be found in appendix B.
Firstly consider A AA (q 2 ). This is generated by loops with charged scalars S Q a and S −Q a . According to Eq. (69) of appendix B, they generate Secondly consider A AZ (q 2 ). This is once again generated by loops with charged scalars S Q a and S −Q a . According to Eqs. (69) and (70) of appendix B, they generate where we have used Eqs. (10b) and (10c). Thirdly consider A ZZ (q 2 ). This is generated either by loops with charged scalars S Q a and S −Q a ′ , where a ′ may be different from a, or by loops with neutral scalars S 0 b and S 0 b ′ , where b ′ may differ from b. The relevant equations are Eqs. (70) and (71) of appendix B, respectively. They generate This leads to where we have used Eqs. (10b)-(10d) and (13a). Finally consider A W W (q 2 ). This is generated either by loops with one charged scalar S Q a and one neutral scalar S 0 b , or by loops with two charged scalars S ±Q a and S . The relevant equations are Eqs. (72) and (73) of appendix B, respectively. They generate where we have used Eqs. (10b)-(10d) and (13b), and also It is now clear that the combinations relevant for the parameters S and U in Eqs. (4b) and (4c), respectively, namely are given by It is proven in appendix C that ψ and θ are both zero, no matter what the set of M JY (and no matter what the VEVs of their neutral components are) present in any particular SU(2) × U(1) electroweak model, i.e. either in the SM or in any extension thereof.
Since the quantities ψ and θ vanish for the New Physics model and, identically, also for the Standard Model, one might think that S and U would turn out finite. However, if we subtract the SM result for ψ and θ, computed by using the usual SM Feynman rules for the Goldstone-boson vertices, from the New Physics result, which uses a different set of Feynman rules for those vertices, then we get a gauge-dependent result for both S and U. This happens because the gauge-dependent parts of the NP and SM results are different when Eq. (1) does not hold in the NP model. Indeed, as is shown in appendix D, some Goldstone-boson vertices depend on ρ, hence they are different in the SM and in the NP model. In Eqs. (26) we display the relevant Goldstone-boson vertices and their ρ-dependent value, as derived in appendix D; we also display, for each vertex and crossed over, its value in the SM, i.e. when ρ = 1, as given in Ref. [31].
We thus have a problem. We may obtain ψ = θ = 0 in both the SM and its extension by using, respectively, the Feynman rules adequate for each of them, with different values of ρ (viz. ρ = 1 in the SM and ρ = 1 in the extension); but, since the Feynman rules for the Goldstone-boson vertices differ in the two cases, we will then end up with a gaugedependent result. Or else we may use in both the SM and its extension the same Feynman rules, as they are written in Eqs. (26), i.e. with the same value of ρ, and then we will get a gauge-independent result-but ψ and θ will be non-zero in the SM and, therefore, S and U, respectively, will diverge.
We propose to make S and U finite by multiplying other SM Feynman rules-not the ones in Eqs. (26)-by factors that are equal to 1 when ρ = 1. This is possible because in the SM there is the Higgs particle H, with mass m h , that participates in the computation of A ZZ (q 2 )| SM through a loop with G 0 and H, and in the computation of A W W (q 2 )| SM through a loop with G + and H. We thus propose to use, in the SM, the following Feynman rules for the vertices ZG 0 H, ZZH, W ± G ∓ H, and W + W − H, respectively: The vertices ZG 0 H and ZZH are multiplied by the same factor √ ℵ due to gauge invariance. The same happens with the vertices W ± G ∓ H and W + W − H, which are both multiplied by √ . Obviously, in the true SM, i.e. when relation (1) holds, both ℵ and are 1.
In the computation of the SM contribution to S and U, we must consider six diagrams like the one in Fig. 1: 1. A diagram with inner scalars G + and H contributing to A W W (q 2 ).

A diagram with inner scalars
3. A diagram with inner scalars G 0 and H contributing to A ZZ (q 2 ). 4. A diagram with inner scalars G + and G − contributing to A ZZ (q 2 ).

5.
A diagram with inner scalars G + and G − contributing to A AA (q 2 ). 6. A diagram with inner scalars G + and G − contributing to A AZ (q 2 ).
A factor ℵ multiplies the diagram 3 and a factor multiplies the diagram 1.
Using Eqs. (24) and the results for the quantities (16) computed in the SM, we get for the parameter S α where the second to fifth terms inside the square brackets originate in diagrams 3-6, respectively. We have used Eq. (26b) in the third and fifth terms inside the square brackets. Since ψ NP = 0, we obtain a finite result for S if Thus, we must choose Notice that ℵ = 1 when Eq. (1) holds, viz. when the extension of the SM only has scalar doublets and/or singlets.
There is then a total of n 2 = n t 1 + n s 2 complex scalar fields with electric charge 2, n 1 = n d + n t 1 + n t 0 + n s 1 complex scalar fields with electric charge 1, n 0 = 2n d + 2n t 1 + n t 0 + n s 0 real scalar fields with electric charge 0.
The neutral fields are allowed to have non-zero VEVs where the v k and the w p are in general complex and the x q and the u l are real. The masses of the gauge bosons W ± and Z 0 are given in terms of the VEVs of the scalar fields as where we have defined Comparing the second Eq. (41) with Eq. (14d), note that there is an extra factor 1/2 multiplying x. This is because the field λ 0 q is real, hence it has a factor 1/2 in its gauge-kinetic term.
We expand the neutral fields around their VEVs as We define the mixing matrices of the scalar fields as where the dimensions of the matrices are The matrices V 3 and V 4 are real while the other ones are complex. The matrix is n 2 × n 2 unitary; it diagonalizes the Hermitian mass matrix of the scalars with charge 2. The matrixŨ is n 1 × n 1 unitary; it diagonalizes the Hermitian mass matrix of the scalars with charge 1. The matrixṼ is n 0 × n 0 real orthogonal; it diagonalizes the symmetric mass matrix of the real components of the neutral scalar fields.
One may obtain the form of the Goldstone bosons by applying to the vacuum state the generators of the gauge group that are spontaneously broken. Using this method, we obtain while the first columns of U 4 , V 3 , and V 4 are identically zero.

The formula for S
There are two kinds of diagrams that produce the parameter S: the ones like in Fig. 1 and those like in Fig. 2. Diagrams of the type of Fig. 1  Let M c denote the mass of the scalar S ++ c , m a denote the mass of the scalar S + a , and µ b denote the mass of S 0 b . Using the Feynman diagrams from appendix E and the prescription described in section 2, we have obtained for the oblique parameter S the following result: (53) One may use the unitarity of the matricesT andŨ and the orthogonality ofṼ to write the following relations: We use Eqs. (54)-(58) to simplify Eq. (53), obtaining The coefficients of the logarithms in lines (59g)-(59k) add up tp zero as they should: where in the last step we have used Eqs. (51). Equation (59) for S generalizes the result given in ref. [24] for the case without any scalar SU(2) triplets and without charge-2 singlets. In that case the matrices T 2 , T 4 , U 2 , U 3 , V 2 , and V 3 and zero. Ref. [24] then gave It must be stressed that ref. [24] used a different definition of S [33], wherein in Eq. (4b) was substituted by This caused the appearance of the functions G (I, J, m 2 Z ) andĜ (J, m 2 Z ) in Eq. (61). In order to make the connection with the present work, one must use

Conclusions
In this paper we have discovered a problem that arises in the computation of the oblique parameters S and U whenever the New-Physics model at hand does not obey relation (1), because of scalars in multiplets of SU(2) larger than doublets having VEVs. Namely, the Feynman rules for some Goldstone-boson vertices depend on ρ-see Eqs. (26)-and this causes a mismatch between the New-Physics model, wherein ρ = 1, and the Standard Model, wherein ρ = 1. Depending on how one (mis)handles this mismatch, S and U may turn out either gauge-dependent or divergent. We have proceeded by suggesting a solution for this problem through a redefinition of the Standard-Model Feynman rules for some vertices containing the Higgs boson H. Namely, we have inserted by hand into those Feynman rules factors √ ℵ and √ -see Eqs. (27)-that are equal to 1 when relation (1) holds. One easily finds out that, if ℵ and take the values in Eqs. (30) and (33), respectively, then the problem is solved.
We have utilized our insight to explicitly compute the value of S in a New-Physics model containing arbitrary numbers of triplets of SU(2) (and also extra doublets and singlets) with weak hypercharges either 0 or 1.
In the future, it may be interesting to study whether the changes to the Feynman rules that we have suggested may somehow be justified, whether they cure the problems in the computation of S and U also at the two-loop level, and whether they may cure other possible problems elsewhere. We leave these questions for consideration by other authors.

A Feynman integrals
We define the function We have used the following results: and div = −1 48π 2 In Eqs. (66) and (68), µ is an arbitrary mass scale. In Eq. (68), γ is Euler's constant. The quantity named "div" diverges in the limit d → 4.

B Interaction terms of one gauge boson with two scalars
In this appendix we derive some consequences of Eq. (7). The interaction terms of the photon with the charge-Q scalars (Q > 0) are given by where we have used Eq. (10a). The interaction terms of the Z with two charged scalars are given by The interaction terms of the Z with two neutral scalars are given by The interaction terms of the W with one singly-charged scalar and one neutral scalar are given by where we have used Eqs. (15). The interaction terms of the W with two charged scalars are given by We show in this appendix that the quantities ψ and θ in Eqs. (25) are equal to 0. We shall use the sums 1 + 2 + 3 + · · · + n = n (n + 1) 2 , 1 + 4 + 9 + · · · + n 2 = n (n + 1) (2n + 1) 6 , which are valid for any positive integer n. We shall separately consider the following types of multiplets M JY : 1. Y < −J. A multiplet of this type belongs neither to A nor to R Q for any Q > 0. It belongs to S Q for Q = −Y − J, . . . , −Y + J.
2. Y = −J. A multiplet of this type belongs to A. It also belongs to S Q for Q = 1, 2, . . . , −2Y . It does not belong to R Q for any value of Q > 0.
3. −J < Y < J. A multiplet of this type belongs to A, to R Q for Q = 1, . . . , J + Y , and to S Q for Q = 1, . . . , J − Y .
4. Y = J. A multiplet of this type belongs to A and also to R Q for Q = 1, 2, . . . , 2Y . It does not belong to S Q for any value of Q > 0.

Y > J.
A multiplet of this type belongs neither to A nor to S Q for any Q > 0. It belongs to R Q for Q = Y − J, . . . , Y + J.
This proves that the Feynman rule in Eq. (26c) is correct for a model with an arbitrary scalar content. From Eq. (7), Therefore, the ZW ± G ∓ interaction terms are This proves the Feynman rule in Eq. (26a) for a model with an arbitrary scalar content.

E The Feynman diagrams that produce the parameter S in a model with triplets
In order to compute the oblique parameter S we need to compute the vacuum-polarization tensors Π µν V V ′ , where V and V ′ may be either A and A (A is the photon), A and Z, or Z and Z.
The diagrams that contribute to ∂ δA ZZ (q 2 ) ∂q 2 q 2 =0 at the one-loop level are 7 The diagrams that contribute to ∂ δA AA (q 2 ) at one-loop level are The diagrams that contribute to ∂ δA AZ (q 2 ) at one-loop level are