Electroweak renormalization based on gauge-invariant vacuum expectation values

We briefly review a recently proposed scheme for a gauge-invariant treatment of tadpole corrections in spontaneously broken gauge theories called Gauge-Invariant Vacuum expectation value Scheme (GIVS). The tadpole scheme matters in higher-order predictions of observables if not all free parameters are fixed by renormalization conditions based on S-matrix elements, such as in MSbar renormalization. In contrast to previously used tadpole schemes, the GIVS unifies the properties of gauge invariance and perturbative stability. The application of the GIVS to the Standard Model, for instance, leads to very moderate electroweak corrections in the conversion of on-shell-renormalized to MSbar-renormalized masses. Moreover, in models with extended Higgs sectors, the GIVS is less prone to perturbative instabilities in the MSbar renormalization of Higgs mixing angles than observed for the traditional gauge-independent tadpole treatment. We illustrate this by considering the next-to-leading-order (electroweak and QCD) corrections to the decay processes $h/H\to WW/ZZ\to4$fermions of the CP-even neutral Higgs bosons h and H in a singlet Higgs extension of the Standard Model and in the Two-Higgs-Doublet Model.


Introduction
Electroweak (EW) corrections in the Standard Model (SM) and its extensions are an important ingredient in precision calculations for present and future collider phenomenology. Renormalization, which is an important part of this task, in particular includes the definition of vacuum expectation value (vev) parameters, such as in the SM, in higher orders-an issue that is connected to the treatment of tadpole loop contributions. In case, all model parameters are fixed by renormalization conditions based on S-matrix elements, i.e. by so-called on-shell (OS) conditions, higher-order predictions of observables do not depend on the tadpole scheme. However, this is different, for instance, in MS renormalization schemes, which are frequently employed for mass parameters and mixing angles in extended Higgs sectors.
Explicit tadpole contributions are most conveniently removed via tadpole counterterms which can be generated in two different ways in the Lagrangian: in the course of parameter renormalization [1,2], or alternatively via Higgs field redefinitions [3]. The former, called Parameter Renormalized Tadpole Scheme (PRTS) in the following, typically leads to small corrections originating from tadpoles, but in general suffers from gauge dependences if MS renormalization conditions are used for mass parameters. The latter, usually called Fleischer-Jegerlehner Tadpole Scheme (FJTS) , is free from gauge dependences, but is prone to very large corrections in MS schemes, jeopardizing perturbative stability of MS predictions. More details on tadpole renormalization can, e.g., be found in Refs. [5][6][7][8][9][10].
In Refs. [5,6] we have proposed 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 , in nonlinear representations of Higgs multiplets. In this article we briefly summarize the salient features of the GIVS and its first applications within the SM [5], a Singlet Higgs Extension of the SM (SESM) [6], and to the Two-Higgs-Doublet Model (THDM) [6].

Non-linear representations of Higgs sectors
One of the basic ideas underlying the GIVS is the use of a field basis in which the Higgs fields developing non-vanishing vevs are gauge invariant. This implies that explicit tadpole contributions induced by loop diagrams are gauge independent. In such a field basis the would-be Goldstoneboson part of the scalar Lagrangian is necessarily parametrized in a non-linear fashion. In the following, we briefly sketch the structure of appropriate non-linear representations for the SM, the SESM, and the THDM. For the full details, we refer to Refs. [5,6].

Standard Model
Non-linear field representations are most conveniently formulated via matrix fields. Denoting the usual, two-component SM Higgs doublet Φ and its charge conjugate Φ c , we form the 2 × 2 matrix The FJTS [3] is equivalent to the scheme of Ref. [4]. Based on completely different considerations about symmetry breaking in gauges, a similar idea for a hybrid scheme has been mentioned in Sect. D of Ref. [10]. = (Φ c , Φ) and write it in some "polar representation" in which ℎ corresponds to the physical Higgs field, which is gauge invariant, ( = 1, 2, 3) are real would-be Goldstone-boson fields and denotes the Pauli matrices. Since the matrix ( ) is unitary, the gauge-invariant square tr † = ( +ℎ) 2 does not involve would-be Goldestone-boson fields . In the SM Higgs Lagrangian the first term is the kinetic part, which encodes the gauge interactions of the scalar fields in the covariant derivative . Owing to the non-linear representation of , this part is non-polynomial in and induces interaction vertices of arbitrarily many would-be Goldstone-boson fields, but this is only a minor complication, and the usual perturbative Feynman diagram calculus works as usual. Note that the SM Higgs potential with the free parameters 2 2 and 2 , is free from would-be Goldstone-boson fields. With the of help the Nielsen identities [11], it is easy to show that the one-point vertex function Γ ℎ nl of the gauge-invariant Higgs field ℎ is gauge independent; a simple one-loop calculation confirms this.

Singlet extension of the SM (SESM)
The SESM extends the SM by a real singlet scalar field , leading to a second CP-even Higgs boson H. The SESM Higgs Lagrangian is given by with a non-polynomial kinetic part similar as in the SM and the gauge-invariant Higgs potential which again does not involve would-be Goldstone-boson fields . The Higgs fields ℎ 1 and ℎ 2 corresponding to particle excitations in and , respectively, are identified by with vev parameters 1 and 2 , and ( ) denoting the same unitary would-be Goldstone-boson matrix as in the SM, where = 2 . The fields (ℎ 1 , ℎ 2 ) are rotated into a field basis of ℎ and corresponding to mass eigenstates, where is a real-valued mixing angle which is determined by the parameters of the Higgs potential.
In analogy to the SM case, the one-point functions Γ ℎ nl and Γ nl of the fields ℎ and are gauge independent in the non-linear representation.

Two-Higgs-Doublet Model (THDM)
The THDM comprises two complex Higgs doublets ( = 1, 2), which together contain eight real d.o.f.s. Denoting the vev parameters of by , we parametrize the 2×2 matrix fields according to with the unitary would-be Goldstone-boson matrix ( ) as in the SM with The gauge-invariant fields ℎ 1 and ℎ 2 are CP even (under appropriate assumptions on the Higgs potential) and are rotated into the fields ℎ and as in (7) corresponding to neutral CP-even mass eigenstates h and H. The three remaining fields describe a neutral CP-odd Higgs boson A 0 via the gauge-invariant field 3 and two charged Higgs bosons H ± via the fields ± = ( 2 ± i 1 )/ √ 2. The proper normalization of these fields fixes the constants as above. The two Higgs mixing angles and are frequently taken as basic input parameters of the THDM. The Higgs Lagrangian of the THDM is given by in which the kinetic term is again non-polynomial in the would-be Goldstone-boson fields but polynomial in the fields ℎ and corresponding to physical Higgs bosons. In the non-linear Higgs representation the gauge-invariant Higgs potential can be written as THDM with the two-dimensional projection operators Ω ± = 1 2 (1 ± 3 ), which select the original Higgs doublet Φ or its charge conjugate Φ c from the matrix field . Obviously, the unitary Goldstoneboson matrix ( ) again drops out in THDM . To avoid flavour-changing neutral currents at tree level, we assume the Z 2 symmetry Φ 1 → −Φ 1 and Φ 2 → Φ 2 that is only softly broken by the 2 12 term in THDM . Moreover, all couplings in THDM are assumed to be real in order to conserve CP. Owing to the gauge invariance of the fields ℎ and the one-point functions Γ ℎ nl and Γ nl are again gauge independent in the non-linear representation.

The GIVS in the SM
Before formulating the GIVS as introduced in Ref. [5] for the SM, we briefly sketch the FJTS and PRTS for treating tadpoles in the course of renormalizing the theory. The full renormalization procedure of the SM can be found in Ref. [9]. Renormalization starts with the transformation that replaces the original bare quantities in terms of renormalized ones and renormalization constants. We mark bare parameters by subscripts "0" and bare fields by subscripts "B".
In the following, we denote the physical Higgs-boson field of the usually adopted linear Higgs representation by + ( ). One-particle-irreducible Green function Γ ... , so-called vertex functions, involve tadpole contributions if the splitting 0 + B ( ) does not provide an expansion of the effective Higgs potential about its true minimum (see for instance App. C of Ref. [12]). Technically, it is desirable to eliminate such tadpole contributions by appropriate parameter and field definitions. Choosing 0 such that 2 0 = 4 2 2,0 / 2,0 at least to leading order (LO) avoids tadpole contributions at tree level. In higher orders, the explicit (unrenormalized) tadpole function Γ can be cancelled upon generating a tadpole counterterm in the counterterm Lagrangian L. This is achieved by a tadpole renormalization condition for the renormalized one-point function Γ R (in momentum space) of the physical Higgs field, Note that Γ is a gauge-dependent quantity in contrast to its counterpart Γ ℎ nl in the non-linear representation. The tadpole counterterm is generated in the Lagrangian by appropriately choosing 0 and, if needed, by a further redefinition of the bare Higgs field B . Inserting the field decomposition 0 + B ( ) into the bare SM Lagrangian L, produces a term 0 in L with at the one-loop level, where 0 can be viewed as bare tadpole constant. The tadpole schemes described below impose different conditions on 0 , partially accompanied by appropriate field redefinitions of B , in order to generate the desired tadpole counterterm ℎ in the counterterm Lagrangian L.
In the FJTS the bare tadpole constant is consistently set to zero, 0 = 0, so that no tadpole counterterm is introduced via parameter redefinitions. Instead, the tadpole counterterm is introduced by an additional field redefinition in the bare Lagrangian. The field shift (14) distributes tadpole renormalization constants to many counterterms in L (see, e.g., App. A of Ref. [9]). Since the field shift (14) is a mere reparametrization of the functional integral over the Higgs field, it does not alter predictions for observables. Omitting the field shift would mean that explicit tadpole diagrams had to be included, but the result would still be the same as in the FJTS. 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, 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 gauge invariance, i.e. the gauge independence of the parametrization of an observable in terms of 0 and the other bare parameters 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.
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 0 = + is renormalized in such a way that the renormalized parameter is fixed by the renormalized W-boson mass W and and the SU(2) gauge coupling 2 = / w according to = 2 W / 2 , where w = sin w is the sinus of the weak mixing angle w and the electric unit charge. The corresponding renormalization constant is directly fixed by the renormalization conditions on , W , and 2 w = 1 − 2 W / 2 Z . In order to guarantee the compensation of all tadpole contributions after renormalization, the bare tadpole constant 0 = PRTS + PRTS is split into a renormalized value PRTS and a corresponding renormalization constant PRTS and we demand PRTS = 0, so that where the second equality holds in one-loop approximation. Since the renormalized parameter , which is directly fixed by measurements, and the original bare parameters 2 2,0 and 2,0 are gauge independent, the gauge dependence of PRTS goes over to , where it shows up as gauge dependence in the mass renormalization constant 2 W . Trading the two bare parameters 2 2,0 and 2,0 of the Higgs sector for 0 and H,0 , the PRTS tadpole counterterms can be generated by the replacements in the bare Lagrangian with 0 = 0, i.e. several vertex counterterms receive contributions from PRTS (see, e.g., App. A of Ref. [9]). If MS-renormalized mass parameters are used as input, the gauge dependence of PRTS enters the parametrization of observables in the step where 2 2,0 and 2,0 are traded for 0 and H,0 . Note that this flaw of introducing gauge dependences does not invalidate the applicability of the PRTS. On the positive side, 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.
The GIVS aims 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, the vev of the Higgs field is tied to the true minimum of the effective Higgs potential. Gauge dependences are avoided by switching to the non-linear Higgs representation (1) where the condition 0 = + applies to the gauge-invariant component 0 + ℎ B ( ). In detail, in the non-linear Higgs representation the GIVS is identical to the PRTS described above, i.e. the tadpole renormalization constant is fixed according to which is gauge independent as pointed out in the previous section. However, actual next-to-leadingorder (NLO) calculations are typically carried out in the linear representation, and simply taking PRTS nl as tadpole renormalization constant there does not fully cancel explicit tadpole loops. In order to fix this, we calculate GIVS in the linear representations from GIVS nl plus an extra term to restore = −Γ as demanded in (12): between the MS mass ( ) and OS induced by NLO EW corrections using the FJTS, PRTS, or GIVS at the renormalization scale = OS (taken from Ref. [5]).
The gauge-independent part GIVS 1 occurs in PRTS-like tadpole contributions to counterterm which are generated as in (16) analogous to (14). In summary, knowing the tadpole counterterms of the FJTS and the PRTS, as e.g., given in the SM Feynman rules of App. A of Ref. [9], the generation of the one-loop GIVS tadpole counterterms is easily accomplished by the substitutions In Ref. [5] we have illustrated the perturbative stability of the GIVS by considering the conversion of OS-renormalized masses of SM particles to MS masses with the various tadpole schemes. Table 1  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. As emphasized in the literature [13][14][15] for the top quark before, the FJTS shift Δ MS−OS t,EW = 10.75 GeV in the conversion of fermion masses is much larger than the typical size of EW corrections of the percent level. For the lighter fermions b and , the relative corrections Δ MS−OS EW / OS are even larger than for the top quark in the FJTS, reaching up to ∼ 50%, while the shifts in the PRTS and GIVS remain all moderate. 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.

The GIVS in extended Higgs sectors
The generalization of the GIVS from the SM to extended Higgs sectors is fully straightforward. The tadpole renormalization constant ℎ = −Γ ℎ of any Higgs field + ℎ ( ) that can acquire a vev is generated in the GIVS from two parts as in (20). The gauge-independent PRTS-like part GIVS ℎ ,1 = −Γ ℎ nl is calculated in the non-linear Higgs representation, and the gauge-dependent FJTS-like part GIVS ℎ ,2 = Γ ℎ nl − Γ ℎ accounts for the remaining contributions. For the SESM and THDM, the application of the GIVS and its impact on the MS renormalization of the Higgs mixing angles are described in Ref. [6] in detail. For the MS renormalization of the THDM mixing angle we find that the PRTS in any gauge coincides with the GIVS, a fact that puts existing PRTS results on a gauge-independent basis when reinterpreted as GIVS results. We conjecture that this feature of the MS renormalization of also carries over to supersymmetric theories.
The renormalization of Higgs mixing angles as well as the weaknesses and strengths of various schemes was discussed for the SESM and THDM already in Refs. [7,8,12,16,17]. In particular, Ref. [12] analyzes existing and newly suggested renormalization schemes wrt. the following criteria: (i) gauge independence, (ii) symmetry wrt. mixing degrees of freedom, (iii) perturbative stability, and (iv) smoothness for degenerate masses or extreme mixing angles. While the suggested schemes based on field-theoretical symmetries or on OS conditions widely meet these requirements, MS renormalization with FJTS tadpole treatment turned out to be particularly prone to perturbative instabilities in specific parameter regions (large or degenerate Higgs masses, extreme mixing angles). These features were demonstrated in a comprehensive discussion of NLO predictions for various Higgs-boson production and decay processes in Ref. [12], extending the earlier discussions of Refs. [7,8,16,17].
The results of Ref. [6] for the Higgs-boson decays h/H → WW/ZZ → 4 in the SESM and THDM demonstrate that MS renormalization with GIVS tadpole treatment mitigates perturbative instabilities significantly and produces gauge-independent results very close to the gauge-dependent PRTS. Table 2   and A2 in different renormalization schemes, with the on-shell scheme OS12 as input scheme (and full conversion of the input parameters into the other schemes). The scale variation (given in percent) corresponds to the scales = 0 /2 and = 2 0 , where 0 is the average Higgs mass (for all details, see Ref. [6]).

Conclusions
Recently we have proposed a new scheme for a gauge-invariant treatment of tadpole corrections in spontaneously broken gauge theories called Gauge-Invariant Vacuum expectation value Scheme (GIVS). The GIVS is a hybrid scheme of the two most commonly used tadpole schemes, in which tadpole counterterms are introduced either via parameter or via field renormalization transformations. In the construction of tadpole counterterms, the GIVS makes use of non-linear representations of Higgs sectors, in which fields that can develop non-vanishing vacuum expectation values are gauge invariant, but actual loop calculations are performed in the GIVS as usual. In contrast to previously used tadpole schemes, the GIVS unifies the properties of gauge independence and perturbative stability. We have illustrated these virtues by discussing the conversion of on-shell-renormalized to MS-renormalized masses in the SM and the MS renormalization of Higgs mixing angles in two frequently considered models with extended Higgs sectors-a singlet Higgs extension of the SM and the Two-Higgs-Doublet Model. We expect that the generalization of the GIVS beyond the one-loop level works without problems.