Electroweak renormalization based on gauge-invariant vacuum expectation values of non-linear Higgs representations: 1. Standard Model

The renormalization of vacuum expectation value parameters, such as $v$ in the Standard Model (SM), is an important ingredient in electroweak renormalization, where this issue is connected to the treatment of tadpoles. Tadpole counterterms can be generated in two different ways in the Lagrangian: in the course of parameter renormalization, or alternatively via Higgs field redefinitions. The former typically leads to small corrections originating from tadpoles, but in general suffers from gauge dependences if MSbar renormalization conditions are used for mass parameters. The latter is free from gauge dependences, but is prone to very large corrections in MSbar schemes, jeopardizing perturbative stability in predictions. In this paper we propose a new scheme for tadpole renormalization, dubbed Gauge-Invariant Vacuum expectation value Scheme (GIVS), which is a hybrid scheme of the two mentioned types, with the benefits of being gauge independent and perturbatively stable. The GIVS is based on the gauge-invariance property of Higgs fields, and the corresponding parameters like $v$, in non-linear representations of Higgs multiplets. We demonstrate the perturbative stability of the GIVS in the SM by discussing the conversion between on-shell and MSbar renormalized masses.


Introduction
After roughly one decade of data taking, very many analyses carried out by the LHC collaborations ATLAS and CMS in fact produce precision measurements of cross sections and model parameters at a level where electroweak (EW) corrections play an important role. In the calculation of EW corrections, the procedure of renormalization is of crucial importance, involving issues like the choice of renormalization scheme and input parameters. Electroweak renormalization was worked out in the 1970-90s in various variants [1][2][3][4][5][6][7] and meanwhile is a standard procedure in modern next-to-leading order (NLO) calculations (see, e.g., the review [8] for details and further original references).
In some EW renormalization schemes, details of the parametrizations of vacuum expectation values (vevs) of Higgs fields play a role in higher-order calculations, in others they do not. Technically, these details concern the treatment of tadpole diagrams or subdiagrams, which have only one external leg and can be interpreted as contributions to vevs of the corresponding external field. In on-shell (OS) renormalization schemes, for instance, all renormalized model parameters are directly related to physical observables, so that in predictions for other observables all tadpole contributions cancel between loop diagrams and counterterms. In renormalization schemes, however, in which not all mass parameters are tied to physical observables, such as in schemes with MS-renormalized mass parameters, the treatment of tadpole contributions has an impact on the parametrization of predicted observables in terms of renormalized parameters and on the running behaviour of the MS masses. An analogous statement also applies to models with extended Higgs sectors in which Higgs mixing angles may be renormalized with MS conditions. In the literature two completely different types of tadpole treatments are in use. Both types fully absorb explicit tadpole diagrams upon introducing a tadpole counterterm δL δt = δt h in the Lagrangian for each Higgs field component φ(x) = v + h(x) that might acquire a non-vanishing vev v. This means δt is adjusted in such a way that the vev of h vanishes, 0|h(x)|0 = 0. The explicit tadpole terms absorbed by δt are redistributed to counterterms of other couplings as implicit tadpole contributions. In the two variants for the tadpole treatment, however, the tadpole counterterms δL δt are generated in a very different way. One possibility is to include the renormalization constant δt in the parameter renormalization transformation which expresses bare parameters in terms of renormalized parameters, as, e.g., done in Refs. [4,6]. We call this scheme Parameter Renormalized Tadpole Scheme (PRTS) in the following. The renormalized parameter v quantifying the Higgs vev then corresponds to the value of φ in the minimum of the renormalized (corrected) Higgs potential. This scheme, however, has the unpleasant feature that δt, which is gauge dependent in general, enters the relations between bare parameters of the Higgs potential, and this results in a potentially gauge-dependent parametrization of predicted observables if MS masses or Higgs mixing angles are used (see discussions in Refs. [9][10][11]).
The second possibility to introduce δL δt is the Fleischer-Jegerlehner Tadpole Scheme (FJTS) as introduced in Ref. [12], 1 where the field h(x) is redefined by a transformation h(x) → h(x) + ∆v with a constant ∆v in the bare Lagrangian. This transformation is only a change in parametrization of the functional integral of the generating functional of Green functions and does not change any physical prediction. Choosing the constant ∆v = −δt/M 2 h , with M h denoting the mass of the corresponding Higgs boson, implies 0|h(x)|0 = 0 as demanded. The FJTS redistributes the tadpole corrections to other counterterms without changing the parametrization of observables at all, i.e. the scheme produces the same result as if just including all explicit tadpole diagrams wherever they appear. In other words, in this scheme the perturbative expansion of φ(x) proceeds about the bare value of v, which is related to the minimum of the bare Higgs potential, not the corrected one. On the upside, the FJTS does not introduce gauge dependences in the parametrization of predictions, but by experience issues with perturbative stability potentially exist if OS conditions for masses are not used entirely. Comparing corrections to observables calculated within renormalization schemes with MS mass parameters that differ only by using the PRTS or FJTS, the differences reflect the additional correction ∆v required in the FJTS to shift the expansion point of φ to the renormalized value of v. These additional corrections appearing in the FJTS can be very large and jeopardize the perturbative stability of predictions. In the SM, for instance, the differences between OS-and MS-renormalized masses are unnaturally large [14]. For the top-quark mass this difference is of the order of 10 GeV and thus of similar size as the QCD correction (see Refs. [14][15][16] and references therein). In models with MS-renormalized mixing angles in extended Higgs sectors such as the Two-Higgs Doublet Model, the FJTS is very prone to huge corrections in extreme parameter scenarios, as discussed in Refs. [9-11, 17, 18].
In this paper, we propose a new tadpole scheme, dubbed Gauge-Invariant Vacuum expectation value Scheme (GIVS), which aims at unifying the good features of the PRTS and FJTS. It is designed to expand Higgs fields about the corrected minimum of the effective Higgs potential, so that no additional corrections arise from correcting the expansion point. We, thus, expect that the GIVS produces only moderate corrections in predictions, very similar to the PRTS and in contrast to the FJTS. This expectation is confirmed in the explicit phenomenological example discussed below. To avoid any gauge dependences in tadpole contributions, the construction of the GIVS makes use of non-linear parametrizations of the Higgs fields [19,20]. In these non-linear representations CP-even neutral components of Higgs multiplets-and thus also their potential constant contributions-are gauge invariant, and the Higgs potentials are completely free of would-be Goldstone-boson fields, so that tadpole renormalization constants become gauge independent. Simply using these tadpole counterterms in actual calculations based on linear Higgs representations, would not lead to a full compensation of explicit tadpole diagrams. This mismatch can be resolved upon generating a second type of tadpole counterterm by field shifts as in the FJTS, so that all explicit tadpole diagrams are cancelled by the sum of the two types of tadpole counterterms, which renders the GIVS a hybrid scheme. We formulate the GIVS for the SM and demonstrate its perturbative stability by evaluating the differences between OS-and MS-renormalized masses of SM particles. In a forthcoming publication we will apply the GIVS to non-standard Higgs sectors, e.g., with additional singlet or doublet scalar fields.
The article is organized as follows: In Section 2 we describe the non-linear Higgs representation of the SM in detail, including the calculation of the tadpole constants in the linear and non-linear representations. Moreover, we generally show the gauge independence of tadpoles in the non-linear representation there. The formulation of the GIVS as well as the application to the conversion of masses between OS and MS definitions are presented in Section 3. Our conclusions are given in Section 4, and the appendix briefly describes the application of the GIVS within the background-field method.

Linear and non-linear Higgs representations of the Standard Model
In this section, we introduce the linear and non-linear Higgs representations for the SM. All parameters and fields are considered as "bare" in this section, i.e. the renormalization transformation for introducing renormalized quantities and renormalization constants, including the choice of the tadpole scheme, will be the next step after this section. In the formulation of the SM in the linear Higgs representation and the definition of field-theoretical quantities we consistently follow the notation and conventions of Ref. [8]. The transition to the non-linear Higgs representation uses Refs. [19][20][21][22] as guideline.

Kinetic Higgs Lagrangian
Most commonly, the SM Higgs doublet is introduced as a complex two-component field Φ with charge conjugate Φ c = iσ 2 Φ * , where σ j (j = 1, 2, 3) are denoting the Pauli matrices in the following. In the conventions of Ref. [8], Φ and Φ c are parametrized according to with the complex would-be Goldstone-boson fields φ + , φ − = (φ + ) * , the real would-be Goldstone-boson field χ, the physical Higgs field η (called H in Ref. [8]), and the constant v parametrizing the vev of the Higgs doublet. For the transition from the linear to the nonlinear Higgs representation it is convenient to first switch to the (2 × 2) matrix notation for the Higgs doublet with the 2×2 unit matrix 1 and φ j (j = 1, 2, 3) are the three real would-be Goldstone-boson degrees of freedom in a more generic notation. Note that we use summation convention over the Goldstone index j, which is sometimes replaced by a vector-like notation φ = (φ 1 , φ 2 , φ 3 ) T , σ = (σ 1 , σ 2 , σ 3 ) T , etc., and boldface characters like φ to indicate matrix structures. The new field components φ j can be identified according to 3) The complex square Φ † Φ of the Higgs doublet Φ, which is the field combination entering the Higgs potential, translates into the trace of Φ † Φ, The matrix field Φ can be parametrized in the non-linear form in which h corresponds to the physical Higgs field and ζ = (ζ 1 , ζ 2 , ζ 3 ) T to real would-be Goldstone-boson components. Since the matrix U (ζ) is unitary, the square of the Higgs field does not involve would-be Goldestone-boson fields ζ j , so that the Higgs potential does not involve ζ j either. Making use of the shorthands the relations between the component fields of the two representations are explicitly given by [20] where the expansions in the second equalities neglect terms that are of higher order in the Goldstone fields ζ j . Our conventions are such that (η, φ ) and (h, ζ ) agree up to higher powers in the Goldstone fields. The Higgs doublet and its charge conjugate carry weak hypercharges Y w,Φ = +1 and Y w,Φ c = −1, respectively, so that the matrix field Φ transforms under SU(2) w × U(1) Y gauge transformations as [20] Φ with the transformation matrices where g 2 and g 1 are the SU(2) w and the U(1) Y gauge coupling constants, respectively, and θ = (θ 1 , θ 2 , θ 3 ) T and θ Y the corresponding gauge parameters. For the field h and the matrix Hence, the Higgs field h is invariant under gauge transformations while the fields ζ j change under gauge transformations in a non-trivial way. Note also that the parameter v, which quantifies the Higgs vev, is directly associated to the gauge-invariant component h, in contrast to the linear parametrization, where the constant v is attributed to the field component η, which is not gauge invariant.
In the non-linear representation the Lagrangian for the kinetic terms of the Higgs and Goldstone fields is given by where D µ is the covariant derivative acting in matrix notation as with the matrix-valued SU(2) w gauge field W µ and the U(1) Y gauge field B µ . For later purposes, it is very convenient to define the following combination of gauge fields, with Z µ denoting the Z-boson field and c w = cos θ w the cosine of the weak mixing angle θ w . Inserting the non-linear representation (2.5) of Φ into L H,kin , terms with arbitrary powers of Goldstone fields ζ j emerge. This complicated structure becomes rather transparent after introducing the matrix fields which absorb the complete ζ dependence of L H,kin , µ is identical with the gauge field W µ in the unitary gauge after performing the corresponding field transformation in the full Lagrangian [19,20]. The full dependence of L H,kin on the Goldstone fields can be easily derived from the components of the matrix fields W (u) µ and C (u) µ , which are given by where × denotes the usual cross product of 3-dimensional vectors. Accordingly, an expression for C (u) µ · C (u),µ in Eq. (2.16) can be derived as To facilitate the derivation of Feynman rules, it is convenient to insert the expanded version of C (u) µ into L H,kin up to the desired order in ζ j fields. Up to quadratic order in ζ j , the Higgs kinetic Lagrangian reads The gauge-fixing Lagrangian for the non-linear representation is given by 2 corresponding to the W ± bosons, and A µ being the photon field. The parameters ξ a (a = A, Z, W ) are the usual arbitrary gauge parameters, and M V (V=Z,W) the Z-and W-boson masses.

Higgs potential and tadpoles
The complex Higgs doublet field Φ undergoes self-interactions as described by the Higgs potential with µ 2 2 and λ 2 being real, positive free parameters. 2 In the linear Higgs representation, the potential V involves would-be Goldstone-boson fields which is not the case in the non-linear representation, where it reads In lowest order, the vev parameter v is chosen such that the field configurations η ≡ 0 and h ≡ 0 correspond to the classical minimum of V , so that no terms linear in η or h remain in V . Beyond lowest order, however, loop corrections induce non-vanishing contributions T η and T h nl to the one-point vertex functions Γ η and Γ h nl , known as tadpole constants. To prevent any confusion w.r.t. to the two types of Higgs representations, we mark vertex functions Γ ... and tadpole contributions T ... in the non-linear representation by a suffix "nl" throughout. For later purposes, we compute and compare these one-loop tadpole contributions T η and T h nl of the physical Higgs fields η and h of the linear and non-linear representations, respectively. In the linear representation Γ η is given by In order to avoid a clash of notation with the reference mass scale µ of dimensional regularization, we denote the parameters in the potential µ 2 2 and λ2, with the "2" hinting on the Higgs doublet.
with M H denoting the Higgs-boson mass and m f the mass of the fermion f , which has colour multiplicity N c f . The field label u generically stands for Faddeev-Popov ghost fields. Here we made use of the scalar one-point one-loop integral in D = 4 − 2 dimensions, with the arbitrary reference mass µ and the standard UV divergence in which γ E denotes the Euler-Mascheroni constant. In the non-linear representation Γ h nl is given by Note that no ghost loops contribute to the tadpole T h nl in the non-linear representation, because the field h does not couple to ghost fields, since Goldstone fields ζ j and the gaugeinvariant Higgs field h do not mix under gauge transformations. As already anticipated in the introduction, the gauge invariance of h in the non-linear representation has the consequence that the corresponding tadpole contribution T h nl is gauge independent, while T η is not. For later convenience, we introduce the parameter quantifying the gauge-dependent difference between the tadpole parameters in the linear and non-linear Higgs representations. It is interesting to note that ∆v ξ vanishes in the Landau gauge, where all ξ a = 0. This is due to the fact that the Goldstone-boson modes are massless in Landau gauge, so that the corresponding one-loop tadpole diagrams lead to vanishing scaleless integrals in dimensional regularization. We do not necessarily expect that an analogous statement holds beyond the one-loop level. We conclude this section by considering the gauge dependences of the tadpole functions Γ η and Γ h nl from a more general point of view, in particular in order to get an idea about a possible generalization beyond the one-loop level. The gauge-parameter dependences of any irreducible Green function are controlled by the so-called Nielsen identities [23][24][25], expressing the invariance under extended BRST variations, which take into account variations of gauge parameters in addition to BRS variations of the fields. In more detail, the derivative ∂ ξa Γ ϕ 1 ϕ 2 ... of the vertex function Γ ϕ 1 ϕ 2 ... of some fields ϕ j w.r.t. a gauge parameter ξ a is expressed in terms of local operator insertions involving the corresponding (Grassmann-valued) BRST sources γ ϕ j and χ a for the fields ϕ j and the gauge parameter ξ a , respectively. The general formulation of the Nielsen identities within the SM, using conventions very close to ours, can be found in Ref. [25]. In particular, the gauge-parameter dependence of the tadpole vertex function Γ H (with H generically denoting the Higgs field) is derived in Sect. 3 there, with the result where the arguments 0 on the vertex functions with more than one external leg express the fact that external momenta are zero. Note that this result does not depend on the Higgs representation (linear or non-linear), which has to be specified when calculating the occuring vertex functions.
where we have used the lowest-order relations Γ η 0 = 0 and Γ ηη 0 (0) = −M 2 H and the fact that Γ χaγ H H and Γ χaγ H are one-loop induced. A very simple one-loop calculation explicitly yields with the scalar one-loop two-point function . (2.32) Making use of for all gauge parameters ξ a , because the BRST variation of the gauge-invariant fields h vanishes. The proven gauge independence of the tadpole function Γ h nl is a crucial requirement for a possible generalization of the GIVS beyond one loop.

Schemes for tadpole and vacuum expectation value renormalization
Before formulating our new proposal for handling tadpole contributions, the GIVS, we first recapitulate the FJTS and PRTS variants for treating tadpoles in the linear Higgs representation in the SM. The following description of the FJTS and the PRTS is fully equivalent to the one given in Sect. 3.1.6 of Ref. [8], although we have switched to a notation for the renormalization in the Higgs sector that is closer to the treatment of extended Higgs sectors described in Refs. [17,18], in order to prepare the generalization of the GIVS beyond the SM. All aspects of the renormalization procedure not spelled out explicitly below, are exactly as described in Ref. [8].
We start out by considering the Higgs potential V , as defined in (2.21), and denote bare parameters by subscripts "0" and bare fields by subscripts "B" in the following. The classical ground-state configuration Φ 0 minimizes V , so that We separate the ground-state configuration Φ 0 = (0, v 0 / √ 2) T from the bare Higgs doublet Φ B by introducing a bare vev parameter v 0 , the precise definition of which is specific to the chosen tadpole scheme as described below, 3 (3.2) Higher-order corrections contain tadpole diagrams, i.e. Feynman diagrams containing subdiagrams of the form given in Eq. (2.23). The vertex functions, defined via a Legendre transformation from the connected Green functions, involve such tadpole contributions if the splitting v 0 + η B (x) of the physical Higgs field does not provide an expansion of the effective Higgs potential about its true minimum (see for instance App. C of Ref. [11]). Technically, it is desirable to organize the perturbative bookkeeping by appropriate parameter and field definitions and renormalization in such a way that the occurrence of tadpole contributions is widely suppressed. Choosing v 0 such that v 2 0 = 4µ 2 2,0 /λ 2,0 at least to leading order avoids tadpole contributions at tree level. We will always assume this in the following. In higher orders, the explicit (unrenormalized) tadpole contribution T η of (2.23) can be cancelled upon generating a tadpole counterterm δt η in the counterterm Lagrangian δL. This is achieved by a tadpole renormalization condition for the renormalized one-point function Γ η R (in momentum space) of the physical Higgs field, The tadpole counterterm is generated by appropriately choosing v 0 and, if needed, by a further redefinition of the bare Higgs field η B . Inserting the field decomposition (3.2) into the bare Lagrangian L, produces a term t 0 η in L with at the one-loop level, where t 0 can be viewed as bare tadpole constant. The tadpoles described below impose different conditions on t 0 , partially accompanied by appropriate field redefinitions of η B , in order to generate the desired tadpole counterterm δth in the counterterm Lagrangian δL.

Fleischer-Jegerlehner tadpole scheme (FJTS) [12]:
In the FJTS the bare tadpole constant is consistently set to zero, so that no tadpole counterterm is introduced via parameter redefinitions, and the bare Higgs-boson mass is fixed by . The field shift (3.7) distributes tadpole renormalization constants to many counterterms in δL. Each term of L containing a Higgs field η B produces such a counterterm upon replacing the η leg in the Feynman rule by a factor ∆v FJTS (see, e.g., App. A of Ref. [8]).
Since the field shift (3.7) is a mere reparametrization of the functional integral over the Higgs field, as long as all parameter renormalization constants are kept fixed, this shift does not influence any physical observable, but only redistributes terms in the calculation of observables. Setting ∆v FJTS = 0 would be possible without changing any prediction; the only difference in this variant is that explicit tadpole diagrams are not cancelled by counterterms and have to be included in the calculation of corrections to observables. This consideration, in particular, makes clear that in the FJTS tadpole contributions correct for the fact that the effective Higgs potential is not expanded about the location of its minimum, but about the minimum of the potential in lowest order, which in the course of renormalization receives further corrections. For this reason, renormalization constants to mass parameters receive tadpole corrections in the FJTS, which are rather large by experience. In OS renormalization schemes these corrections cancel in predictions, because these tadpole corrections systematically cancel between self-energies and mass counterterms, but in other renormalization schemes such as MS schemes this cancellation is only partial, and large corrections typically remain.
On the positive side, the FJTS respects the gauge-invariance requirement mentioned above. To see this, recall that physical observables are always parametrized in a gaugeindependent way in terms of the original bare parameters of the theory, i.e. in terms of µ 2 2,0 and λ 2,0 in the Higgs sector. This gauge independence is neither disturbed by the gaugeindependent reparametrization in terms of the parameters v 0 and M H,0 from Eqs. (3.5)and (3.6), nor by any pure field shift such as the one provided by (3.7) even though ∆v FJTS is gauge dependent. Finally, the gauge independence of the parametrization of an observable in terms of v 0 and M H,0 carries over to the renormalized version of these parameters if the corresponding renormalization constants do not introduce gauge dependences, which is for instance the case in OS and MS schemes in the FJTS.

Parameter-renormalized tadpole scheme (PRTS) [6]:
The idea behind the PRTS is to achieve an expansion of the Higgs field about the true minimum of the renormalized effective Higgs potential (as obtained from the effective action after renormalization) by appropriate relations among the parameters of the theory. To this end, the bare parameter is renormalized in such a way that the renormalized parameter v is fixed by the renormalized parameters M W and g 2 = e/s w , which are directly related to measured values, where s w = sin θ w is the sinus of the weak mixing angle θ w and e the electric unit charge. The corresponding renormalization constant is, thus, directly fixed by the renormalization conditions on e, M W , and s 2 w = 1 − M 2 W /M 2 Z . In order to guarantee the compensation of all tadpole contributions after renormalization, the bare tadpole constant t 0 given in Eq. (3.4) is split into a renormalized value t PRTS and a corresponding renormalization constant δt PRTS , where the second equality holds in one-loop approximation. Since the renormalized parameter v, which is directly fixed by measurements, and the original bare parameters µ 2 2,0 and λ 2,0 are gauge independent, the gauge dependence of δt PRTS goes over to δv, where it shows up as gauge dependence in the mass renormalization constant δM 2 W . In the renormalization procedure, the two bare parameters µ 2 2,0 and λ 2,0 of the Higgs sector are tied to two renormalized parameters, for which we take v as specified above and the Higgs-boson mass M H . The link to M H is provided by the squared bare Higgs mass where again Eq. (3.9) was used in the last equality. From Eqs. (3.13) and (3.14), we see that the PRTS tadpole renormalization constant δt PRTS can also be introduced by the replacements [8,10] in the bare Lagrangian with t 0 = 0. As a result of the described procedure, some vertex counterterms receive contributions from δt PRTS ; the corresponding counterterm Feynman rules can, e.g., be found in App. A of Ref. [8].
As mentioned before, these gauge dependences of the PRTS fully drop out in predictions based on OS-renormalized parameters. If MS-renormalized mass parameters are used as input, the gauge dependence of δt PRTS enters the parametrization of observables in the step where µ 2 2,0 and λ 2,0 are traded for v 0 and M H,0 via Eqs. (3.13) and (3.14). However, these gauge dependences do not invalidate the applicability of the PRTS. In spite of the gauge dependences, consistent predictions can either be produced upon fixing a gauge once and for all, or by translating measured input parameters between different gauge choices. By experience, the PRTS has the practical advantage over the FJTS that contributions to mass renormalization constants are much smaller, which, in particular, implies that conversions of renormalized mass parameters between OS and MS renormalization schemes are typically much smaller in the PRTS as compared to the FJTS (see also Section 3.2).

Gauge-Invariant Vacuum expectation value Scheme (GIVS):
The aim in the new proposal of this paper is to unify the benefits of the FJTS and the PRTS: the gauge-invariance property of the former and the perturbative stability of the latter. To avoid potentially large corrections induced by tadpole loops as inherent in the FJTS, we tie the vev of the Higgs field to the "true" minimum of the effective Higgs potential, i.e. to the Higgs potential expressed in terms of renormalized parameters, as done in the PRTS. The gauge dependences in the PRTS result from the fact that the location of the minimum of the renormalized effective Higgs potential, quantified by the parameter v, is translated into a condition v 0 = v + δv for the non-gauge-invariant component v 0 + η B (x) of the Higgs doublet Φ (2.1) in the linear Higgs representation. This problem is avoided by switching to the non-linear Higgs representation (2.5) where the condition v 0 = v + δv applies to the gauge-invariant component v 0 + h B (x), a fact that gives the GIVS its name. In detail, we fix the tadpole counterterm by where the tadpole contribution T h nl results from the one-point function of the h field in the non-linear Higgs representation, Γ h nl = T h nl . Generating now tadpole counterterms from the bare Lagrangian according to Eq. (3.15) with δt PRTS nl instead of δt PRTS , this procedure is just the application of the PRTS in the non-linear representation. Note, however, that δt PRTS nl is a gauge-independent constant, so that the PRTS in the non-linear Higgs representation does not suffer from gauge dependences. This procedure already fully defines the GIVS in the non-linear representation, but almost all explicit calculations of EW corrections are carried out in the linear Higgs representation.
The GIVS is defined in the linear Higgs representation in such a way that the effect of tadpole renormalization is exactly the same as in the non-linear representation. This means that we set which is the (gauge-independent) part of the tadpole renormalization that goes into relation (3.13) between bare parameters. The tadpole counterterms proportional to δt GIVS 1 are exactly the ones as generated in the PRTS according to Eq. (3.15) with δt PRTS replaced by δt GIVS 1 . Since, however, T h nl = T η , these tadpole counterterms are not sufficient to cancel all explicit tadpole diagrams, which go with T η in the linear representation. We achieve the complete cancellation of explicit tadpole diagrams upon generating additional tadpole counterterms as in the FJTS by a field shift η B → η B + ∆v GIVS in the bare Lagrangian with δt GIVS 2 = −M 2 H ∆v GIVS and demand with ∆v ξ representing the gauge-dependent quantity defined in (2.27). The constant δt GIVS 2 is gauge dependent, but does not have any effect on physical observables, analogously to its role in the FJTS.
To summarize, the GIVS is a hybrid version of the PRTS and the FJTS with two types of tadpole counterterms: the ones connected to δt GIVS 1 = v 0 (µ 2 2,0 − λ 2,0 v 2 0 /4) as δt PRTS in the PRTS and the ones connected to ∆v GIVS in the same way as ∆v FJTS in the FJTS. The GIVS tadpole counterterms are generated from the bare Lagrangian with t 0 = 0 by the substitutions which combines the substitutions (3.7) and (3.15) of the FJTS and PRTS, respectively. Alternatively, with the tadpole counterterms of the FJTS and the PRTS already generated, the generation of the one-loop GIVS tadpole counterterms is easily accomplished by the substitutions (3.20) These substitutions can, e.g., be directly applied to the SM Feynman rules given in App. A of Ref. [8]. If both δt PRTS and δt FJTS contribute to a counterterm vertex, in which case simply δt is written in those Feynman rules, the full GIVS tadpole constant δt GIVS has to be taken, This is, in particular, the case for the counterterm in the Higgs one-point function Γ η R , which receives the counterterm δt = −T η so that Γ η R = 0 as demanded.

Relation between on-shell and MS renormalized masses in the SM
In order to compare the different tadpole renormalization schemes, we consider the relation between MS and OS renormalized masses, M and M OS , respectively. The link between M and M OS is provided by the bare mass parameter M 0 , which is split into a renormalized mass and a corresponding mass renormalization constant δM or δM OS in the two schemes, Taking into account that the MS renormalization constant δM only consists of the UVdivergent contributions proportional to the standard UV divergence ∆ defined in Eq. (2.25), the mass difference ∆M MS−OS is given by where the suffix "finite" means that ∆ is set to zero. The mass difference, thus, can be calculated from the OS mass renormalization constant δM OS upon setting the UV-divergent constant ∆ to zero and specifying a value for the scale µ which now plays the role of a renormalization scale. For expressing the OS constants δM OS in terms of self-energy functions Σ(p 2 ) at on-shell points p 2 = (M OS ) 2 , we follow the notation and conventions of Ref. [8] where self-energies Σ(p 2 ) do not only include the contribution Σ 1PI (p 2 ) from one-particle irreducible (1PI) diagrams, but also all explicit tadpole loops and tadpole counterterms (see Eq. (141) in Ref. [8]). Omitting the superscript "OS" for on-shell masses throughout, we obtain at the one-loop level Explicit expression for self-energy functions can, e.g., be found in Ref. [6]. 5 The sum of all tadpole contributions (explicit loop diagrams and renormalization constants) is contained in the ∆v term, which is chosen according to the applied tadpole scheme,  [14] and find agreement. The values obtained in the PRTS and the GIVS are of comparable size while in general the FJTS leads to larger differences between the OS and the MS masses. An exception is the conversion of the Z-boson mass, for which all three tadpole schemes produce mass shifts of the moderate size that is naively expected from EW corrections. As emphasized in the literature [14][15][16] Despite these large corrections, the FJTS often is favoured in the literature in this context, since it leads to a gauge-independent result in contrast to the PRTS. Note, however, that these large EW one-loop corrections entail an enhancement of the theoretical uncertainties due to missing higher-order corrections. The GIVS, on the other hand, provides gauge-independent mass shifts that are moderate and, thus, leads to smaller EW theory uncertainties, when those uncertainties are estimated by the propagation of the known corrections to higher order as typically done. 6

Conclusions
Extensive discussions in the literature have shown that the two mostly used prescriptions for tadpole contributions in EW renormalization lead to unsatisfactory results in predictions based on MS renormalization conditions. The tadpole prescription (called PRTS in this paper) in which relations between parameters are exploited to generate tadpole counterterms show decent perturbative stability, but suffer from gauge dependences; on the other hand, generating tadpole counterterms from Higgs field redefinitions (called FJTS) avoid gauge dependences, but potentially suffers from perturbative instabilities. The difference between the two tadpole schemes can be interpreted as different choices of vevs for the Higgs field at higher orders, i.e. the separation of the physical Higgs field into a constant contribution and field excitation is different in the two schemes. The PRTS expands about the "true" (corrected) vev, while the FJTS leads to potentially large corrections in the renormalization of mass parameters originating from the perturbative shift in the Higgs vev. In the SM, this issue concerns MS-renormalized mass parameters, in models with extended Higgs sector this concerns MS-renormalized Higgs mixing angles in addition. 6 There is a large cancellation in the mass shift ∆m MS−OS t,EW between the one-loop QCD and EW corrections in the FJTS scheme, as pointed out in Ref. [14]. Since this cancellation is, however, accidental, it does not lead to a reduction of theoretical uncertainties from missing higher orders.
Motivated by this unsatisfactory situation, we have proposed a hybrid scheme of the PRTS and FJTS variants unifying the strengths and avoiding the weaknesses of the PRTS and FJTS schemes by generating the gauge-dependent part of the tadpole counterterm al la FJTS, where it does not enter predictions for observables, and the potentially large gauge-independent part a la PRTS, where it is absorbed into parameter relations which in turn protects observables from large corrections. The new scheme is called Gauge-Invariant Vacuum expectation value Scheme (GIVS), because it exploits the fact that parameters v i determining Higgs vevs like the famous parameter v in the SM, appear as parts of truly gauge-invariant field components of Higgs multiplets Φ i if these multiplets are represented in an appropriate non-linear fashion. These non-linear Higgs representations factorize the would-be Goldstone-boson parts from the remaining Higgs field components in such a way that gauge-invariant combinations of the fields Φ i , such as Φ † i Φ j , do not involve Goldstone fields. Thus, Goldstone fields do not appear in the Higgs potential at all. The condition that determines the v i by minimizing the effective Higgs potential does not involve Goldstone fields, resulting in gauge-independent tadpole corrections that can be absorbed into parameter relations as in the PRTS. The GIVS, thus, expands Higgs fields about the "true" minimum of the effective Higgs potential, like the PRTS, but in a representation in which the vevs acting as expansion points are gauge invariant. The hybrid character of the GIVS comes into play by fixing tadpole renormalization constants δt i in the non-linear representation of the Higgs sector and making use of these δt i in the linear representation where these δt i are supplemented by FJTS-like contributions to fully cancel all explicit tadpole diagrams. We stress that actual loop calculations in the GIVS can be entirely carried out in the linear Higgs representations like for the PRTS and FJTS, once the simple tadpole constants are known, i.e. calculations in the GIVS are not more complicated than usual.
We have described the GIVS for the SM in such detail that further applications of this scheme at the one-loop level should be simple. Owing to the gauge-invariance property of the Higgs field and its one-point function in the non-linear representation, which follows from a Nielsen identity, we expect that the GIVS can be generalized to higher loop levels without major obstacles. To illustrate the perturbative stability of one-loop results based on MS renormalization with the GIVS tadpole treatment, we have discussed the mass parameter conversion between OS and MS-renormalized masses in the SM. As expected, the GIVS leads to small shifts between MS-and OS-renormalized masses, in contrast to the FJTS. In a forthcoming publication, we will apply the GIVS to a scalar singlet extension of the SM and to the Two-Higgs-Doublet Model and investigate the perturbative stability of MS renormalization of the Higgs mixing angles. We expect that the GIVS outperforms the FJTS in view of stability, very similar to the PRTS, but without the downside of the PRTS of leading to gauge dependences.
with the covariant derivative in the adjoint representation defined aŝ where X stands for any matrix-valued field transforming in the adjoint representation. To be in line with the BFM formulation in the linear Higgs representation [7,8], we take a common gauge parameter ξ Q for both the SU(2) w and U(1) Y gauge fields, although it would be possible to introduce different gauge parameters for the two group factors. The derivation of the corresponding Faddeev-Popov Lagrangian proceeds as usual, and the result involves neither h norĥ in the non-linear Higgs representation.
In the BFM, each conventional Feynman rule splits into different versions with different numbers of quantum and background fields; for the linear Higgs representation the Feynman rules are explicitly given in Refs. [7,8]. For the calculation of 1PI Green functions (vertex functions), only Feynman rules with exactly two quantum fields are needed, corresponding to the fact that exactly two loop lines are attached to each vertex. For the formulation of the GIVS, we only need the Higgs one-point function Γĥ nl = Tĥ nl in the non-linear Higgs representation, the calculation of which only requires allĥΨ † Ψ terms for the quantum fields Ψ of the Lagrangian at one loop. The intermediate steps of this calculation are straightforward and simple, so that we only quote the result that the tadpole constants in the BFM completely agree with the corresponding tadpoles of the conventional formalism after setting all gauge parameters ξ a to ξ Q , although the break-up into bosonic diagramatic contributions of gauge bosons, would-be Goldstone bosons, and ghost fields is somewhat different, Γĥ nl = Tĥ nl = T h nl ξa=ξ Q . (A.7) The implementation of the GIVS in the BFM renormalization procedure works exactly as described in Section 3 for the conventional formalism. The tadpole renormalization constant δt GIVS consists of the same two parts δt GIVS 1 and δt GIVS 2 as defined in Eq. (3.18). Nominally the tadpole constants T h (nl) have to be replaced by Tĥ (nl) , but according to Eq. (A.7) those quantities do not change in the transition to the BFM for ξ a = ξ Q . The generation of the tadpole counterterms follows the same strategy as in the conventional formalism as well. Making use of the BFM Feynman rules given in App. A of Ref. [8], the GIVS tadpole counterterms are obtained by the substitutions δt PRTS → δt GIVS 1 , δt FJTS → δt GIVS 2 , and δt → δt GIVS 1 + δt GIVS 2 .