Vacuum Expectation Value Renormalization in the Standard Model and Beyond

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PHYSICAL REVIEW D 103, 076010 (2021)

Vacuum expectation value renormalization in the standard model and beyond

Vytautas Dūdenas˙ * Institute of Theoretical Physics and Astronomy, Faculty of Physics, Vilnius University, 9 Sauletekio,˙ LT-10222 Vilnius, Lithuania † Maximilian Löschner Institute for Theoretical Physics, Karlsruhe Institute for Technology, Wolfgang-Gaede-Straße 1, 76131 Karlsruhe, Germany

(Received 29 October 2020; accepted 8 March 2021; published 13 April 2021)

We show how the renormalization constant of the Higgs vacuum expectation value fixed by a tadpole condition is responsible for gauge dependences in various definitions of parameters in the Rξ-gauge. Then we show the relationship of this renormalization constant to the Fleischer-Jegerlehner (FJ) scheme, which is used to avoid these gauge dependences. In this way, we also present a viewpoint on the FJ scheme complementary to the ones already existing in the literature. Additionally, we compare and discuss different approaches to the renormalization of tadpoles by identifying the similarities and relations between them. The relationship to the Higgs background-field renormalization is also discussed.

DOI: 10.1103/PhysRevD.103.076010

I. INTRODUCTION methods that are commonly used in precision calculations for the Standard Model (SM), as, e.g., in [1] and more In modern particle physics, high precision calculations formal discussions of VEV renormalization in general are of increasing importance for finding signs of new gauge theories as, e.g., in [9,10]. physics in the comparisons of theory predictions to We want to emphasize the known fact that an indepen- experimental data. An integral part of these calculations dent VEV (or tadpole) renormalization constant is neces- is the subject of renormalization. Even though the main sary in addition to the renormalization of the parameters principles of renormalization are well understood (see, and fields of the unbroken theory in order to render all e.g., [1] for a recent review of electroweak radiative n-point Green’s functions finite in the Rξ-gauge (this was corrections) and represent a standard textbook subject, some subtleties are still actively discussed to the present already noted in, e.g., [11,12]). In the broken phase, the day. One of these is the subject of vacuum expectation usage of the VEV in gauge-fixing functions affects the value (VEV) renormalization in conjunction with so-called global symmetry properties of the theory [9], which leads to the need of this additional degree of freedom.1 Hence, in tadpole schemes. For existing examples of discussions in the literature, see, e.g., [2–8] or [1] for a list of tadpole spontaneously broken gauge theories such as the SM, this schemes. However, we find that a unified exposition of can be understood as an artifact of the gauge-fixing procedure rather than a direct consequence of the mecha- the relationships between such schemes is still missing in the literature. Hence, in this paper, we want to elucidate the nism of spontaneous symmetry breaking itself. This intro- relation between the renormalization of vacuum expect- duction affects the definitions of the parameters, leading to gauge dependences in some of them. In principle, these ation values, tadpole schemes, gauge dependences, and the special role of Goldstone boson tadpoles in this respect. gauge dependences will always cancel in physical observ- More specifically, we show the connections between ables. Moreover, the S-matrix can even be made finite without renormalizing tadpoles at all, hence, leaving the one-point Green’s functions infinite [11]. Nevertheless, it *[email protected] can be favorable to demand gauge-independent physical † [email protected] parameter definitions in perturbative calculations, i.e., for

Published by the American Physical Society under the terms of 1 the Creative Commons Attribution 4.0 International license. Note that in some older literature, when the Rξ-gauge was not Further distribution of this work must maintain attribution to as commonly used as it is nowadays, this fact might not be the author(s) and the published article’s title, journal citation, mentioned. As an example, in [13–15], this fact is not discussed and DOI. Funded by SCOAP3. due to the use of R-gauge fixing.

2470-0010=2021=103(7)=076010(15) 076010-1 Published by the American Physical Society VYTAUTAS DŪDENAS˙ and MAXIMILIAN LÖSCHNER PHYS. REV. D 103, 076010 (2021) intermediate expressions such as mass counterterms. The constant reparametrization. This section also relates the Fleischer-Jegerlehner tadpole scheme (FJ scheme) [16] was FJ scheme to the findings presented in [9], and moreover, proposed to avoid these spurious gauge dependences. This we illustrate the differences between the tadpole schemes becomes even more important if one goes beyond the by comparing VEV-renormalization constants numerically Standard Model (BSM), where the usual on-shell (OS) and comment on the outcomes. We conclude our presen- renormalization is not possible for all parameters (see [17] tation in Sec. V. Some details of our calculations can be for a discussion of the problems arising here) and explains found in the appendixes. These include a short note on the part of the renewed attention to the subject in the context of construction of the Rξ-gauge in the background-field the two-Higgs-doublet models [5–8,18]. formalism in Appendix A, the calculation of the purely The FJ scheme is closely related to the aforementioned gauge-dependent divergences using the background fields additional independent VEV-renormalization constant. As in the SM (an adaptation from [9]) in Appendix B, we will see, the FJ scheme makes sure that this degree of consequences of different renormalization conditions of freedom will not enter the parameter and counterterm this approach in Appendix C, explicit divergences of definitions, and hence, allows for gauge-independent renormalization constants in Appendix D, and numerical definitions. However, neither in the original paper, nor in input values that we used to calculate VEV-renormalization the more recent ones on the scheme, is this relationship constants in Appendix E. explicitly exposed. Instead, the notions of the “proper VEV” [16] or the “correct one-loop minimum” [1,5,8,18] are used II. AN ADDITIONAL COUNTERTERM IN Rξ to motivate the gauge cancellations in the FJ scheme. We try The necessity of an independent VEV-renormalization to fill the gap by exposing this feature and also suggest constant for renormalizing one-point Green’s functions has looking at the scheme as simply being a convenient set of been noted before (e.g., see [11,12,33]). However, one might counterterm redefinitions. Using this viewpoint, we also come to the conclusion that this is done purely for conven- show translations between different tadpoles schemes. ience. The reasons for this impression are given by the fact With a pedagogical purpose in mind, we carry out our that S-matrix elements are finite even without renormalizing study for the SM at the one-loop level, but also comment on tadpoles at all [11] or the fact that a gauge fixing other than the implications for BSM. Moreover, the paper is set up – such that our results can be easily reproduced using, e.g., the Rξ-gauge is used, such as in [13 15], where indeed all ’ the native FeynArts [19] SM file together with FeynCalc [20– n-point Green s functions can be made finite via multipli- 22] or FormCalc [23,24] (both with and without background cative renormalization of the parameters of the unbroken fields). theory. Here, we want to clarify that the latter statement is not Note that in order to extend our discussion of gauge true in the Rξ-gauge. As discussed rather recently in [9],the dependences to higher-loop orders, one has to adopt the explanation comes from the fact that this gauge fixing complex mass scheme (CMS) [25–31] in addition to using explicitly breaks a global SUð2Þ × Uð1Þ symmetry. Then, the FJ scheme. The reason being that in the presence of a VEV counterterm cannot be forbidden on the grounds of unstable particles, propagator poles can acquire imaginary symmetry arguments, or in other words, its divergence parts from two loops onward, and the usual OS scheme structure is not fixed by the field strength renormalization leads to gauge-dependent mass definitions in that case. One of a physical scalar. In order to fix its divergence structure, the can therefore only prove the gauge independence of the authors of [9] restore the original global symmetries of the complex propagator poles [27] and needs to include this in theory by the introduction of background fields2 and express the discussion via the CMS. the VEV renormalization in terms of the background-field We start our presentation in Sec. II by explaining why it renormalization. Additionally, via the use of Becchi-Rouet- is necessary to have an additional renormalization constant Stora-Tyutin (BRST) sources, the difference between the in the spontaneously broken phase of the SM as compared VEV’s divergence structure and the physical scalar field’s to the unbroken phase and introduce a tadpole condition to renormalization is isolated. We will now clarify these points fix the former. In Sec. III, we present the translations explicitly for the SM. between the renormalization constants of different param- We consider the usual Higgs potential of the SM: eter sets that are used as independent in the renormalization procedure. In particular, we show the relations between VðϕÞ¼μ2ϕ†ϕ þ λðϕ†ϕÞ2: ð1Þ renormalization constants of symmetry-based (or “original”) parameters of the theory to the ones used in the usual The neutral component of the Higgs doublet ϕ acquires a OS scheme in [32]. These translations illuminate the gauge VEV v as in dependences in the definitions of the usual mass renormalization constants. In Sec. IV, we define the FJ scheme as 2This setup is also related to more formal studies of algebraic known from the literature, show how it provides gauge- renormalization [34–39]. In these studies, the term rigid sym- independent counterterm definitions, and present a new metry is used instead of global symmetry, but the meaning is viewpoint on the scheme in terms of renormalization the same.

076010-2 VACUUM EXPECTATION VALUE RENORMALIZATION IN THE … PHYS. REV. D 103, 076010 (2021) ! 0 1 Gþ qffiffiffiffiffiffi qffiffiffiffiffiffi W 1 ϕ ¼ 1 ; ð2Þ ˆ ¯ B ˆ ˆC pffiffi ðv þ h þ iG Þ ϕ þ ϕ → Zϕ@qffiffiffiffiffiffi ϕ þ ZϕϕA ð8Þ 2 Z ˆ Zϕ þ where GW and GZ are the Goldstone boson fields and h is the physical Higgs field. The tree-level minimum condition and identify the renormalization of the components of the for Eq. (1) gives physical scalar and VEV by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ μ2 pffiffiffiffiffiffi V 2 ¯ ˆ ¼ 0 ⇒ v ¼ − ; ð3Þ Zh ¼ Zϕ=Zϕ; ð9aÞ ∂h λ h¼GW=Z¼0 pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ¯ ˆ which leads to the Higgs mass being Zv ¼ Zhˆ ¼ ZϕZϕ: ð9bÞ 2 2λ 2 −2μ2 mh ¼ v ¼ : ð4Þ It is now comfortable to introduce, schematically,

Following [9], we introduce background fields denoted δ ¼ Z − 1; ð10Þ by “hats” via ! ! ˆ þ for every renormalization constant. Then from Eqs. (9a) ˆ GW GW and (9b) at one loop we have5 ϕ → ϕ þ ϕ ¼ 1 þ : pffiffi 1ffiffi ˆ ˆ 2 ðh þ iGZÞ p ðv þ h þ iG Þ 2 Z 1 δ δ¯ − δˆ δ δ¯ δˆ ð5Þ h ¼ ϕ ϕ; v ¼ 2 ð ϕ þ ϕÞ: ð12Þ

The Rξ-gauge-fixing function is modified in such a way It is thus clear that the VEV renormalization can coincide ˆ that the gauge fixing and ghost part of the Lagrangian are with the Higgs field renormalization only if δϕ vanishes. invariant under the global gauge transformation, where the ˆ We show how one can get the divergent part of δϕ with the gauge parameters are restricted to3 help of BRST sources in Appendix. B. The result for the UV divergent part is ξ ¼ ξW ¼ ξZ ¼ ξA: ð6Þ 2 ξ An explicit construction of the gauge-fixing functions using δˆ 2 2 2 ϕjUV ¼ 4 − 16π2 2 ð mW þ mZÞ; ð13Þ background fields in the SM is explained in Appendix A.In D v this case, one finds that all n-point Green’s functions can be where D is the number of space-time dimensions, and rendered finite using only multiplicative renormalization m and m are masses of W and Z bosons, respectively. constants, W Z Hence, it clearly vanishes for ξ → 0, and can only then let pffiffiffiffiffiffi qffiffiffiffiffiffi the VEV renormalization coincide with the Higgs field → → ˆ → ˆ p Zpp; f Zff; f Zfˆf ð7Þ renormalization. ˆ To further see the role of δϕ, we consider the Higgs one- ˆ for all parameters p, fields f, and background fields f of point function. By inserting renormalization constants into the theory. However, in order to isolate purely gauge- Eq. (1), collecting all the terms linear in h, and using the 4 ˆ dependent divergences in a single constant Zϕ, we write tree-level minimum condition Eq. (3), we get a counterterm the field and background-field renormalization as for the one-point function of h, i.e.,

3 δ −λ δ − δ 2 δ¯ δˆ 3An explicit calculation shows that at one loop, it is enough to th ¼ v ð λ μ þ ϕ þ ϕÞ; ð14Þ use a single additional counterterm even without the equality of the gauge parameters of Eq. (6). However, it is unclear whether 5 this holds at higher-loop orders. One can construct a gauge-fixing Note that since we started with the multiplicative constants, function which preserves the global invariance even when all parameter renormalization constants including the one of the Eq. (6) does not hold instead. Such a gauge-fixing function VEV are defined dimensionless, leading to simpler relations was introduced in [39], yet we want to focus our discussion on between the constants. The translation to the dimensionful the widely used Rξ-gauge. constants can be easily done by replacing 4 ¯ Note that Zϕ can nevertheless have gauge-dependent finite parts. This effectively comes about due to the Passarino-Veltmann δp 2 2 2 δp → ð11Þ function B0ðp ;m ξ;m ξÞ carrying gauge-independent UV di- p ¯ vergent but gauge-dependent finite terms. Hence, Zϕ can just be used as a tool to disentangle divergence structures. in all our expressions.

076010-3 VYTAUTAS DŪDENAS˙ and MAXIMILIAN LÖSCHNER PHYS. REV. D 103, 076010 (2021) which is fixed by the tadpole condition, i.e., dependence. As we will see in the next section, this is usually the case for mass counterterms.

δth þ Th ¼ 0; ð15Þ A. Remarks where Th are the one-loop tadpole contributions. Employing the tadpole condition of Eq. (15) is of special Throughout the rest of this paper, we keep this as a fixed interest when working with the generating functional for condition in all different tadpole schemes. The gauge- the one particle irreducible (1PI) Green’s functions, which dependent part of the tadpole function Th is is defined as the Legendre transformation of the generating functional for the connected Green’s functions. This transformation is only well defined for vanishing one-point Green’s functions, making Eq. (15) essential [27,41,42]. ð16Þ The 1PI generating functional in turn is used for deriving functional identities such as the Slavnov-Taylor [43] or Nielsen identities [44]. Questions about gauge dependence in connection with VEV renormalization can also be very relevant in studies of where (ξ) denotes that we take only the gauge-dependent effective loop potentials. It has already been noted in [45], tadpoles, and A0 is a one-point Passarino-Veltman function in the context of an Abelian-Higgs model, that Goldstone [40]. Checking the UV divergences in Eq. (16) and using boson tadpoles violate the so-called Higgs-low-energy Eq. (14) in Eq. (15), we see that the gauge-dependent theorem (HLET). This relates to an older finding that divergences cancel as one can not get Goldstone boson tadpoles from any potential by taking a derivative with respect to the Higgs ðξÞ − λ 3δˆ 0; VEV [3]. The violation of the HLET is in correspondence Th jUV v ϕjUV ¼ ð17Þ ˆ to the necessity of δϕ in the Rξ-gauge. In [45], the Rξ-gauge ˆ is traded for the so-called Rξ σ-gauge, which reinstates a hence, δϕ alone absorbs all the gauge-dependent divergen- ; ces in the tadpole condition. As shown in [7] for the multi- global symmetry of the Lagrangian and in this way avoids Higgs-doublet SM, one can come to the same conclusions the necessity of an independent VEV counterterm though. by simply demanding the finiteness of all scalar n-point Similarly, in studies of finite temperature phase transi- ˆ ˆ tions (see, e.g., [46,47]), one finds gauge-dependent posi- functions via δ 2 ; δλ; δϕ; δ , where δϕ is introduced ad hoc 7 f μ hg tions of the minima of effective loop potentials. Here, this as an additional VEV-renormalization constant and without 6 is a result of using an Rξ-gauge and defining the effective any reference to the background-field formalism. For a action as the sum of 1PI graphs, i.e., without tadpole and derivation of this kind in the SM, we show the divergences other external leg contributions. Moreover, it is interesting of the relevant n-point functions in Appendix D. to find the diagram in Fig. 1 of [47], which determines the When using the Rξ-gauge fixing in the SM, there are no Nielsen coefficient of a one-loop effective potential. Our gauge-dependent divergences in other renormalization ˆ definition of δϕ via BRST sources shown in Fig. 2 of constants (see Appendix D for explicit expressions); hence, Appendix B is the equivalent of this in terms of its we have divergence structure. As a last note in this section, we would like to stress ∂ ∂ ∂ ¯ δ δ 2 δ 0 the differences in the use of background fields in [9,48]. ∂ξ λjUV ¼ ∂ξ μ jUV ¼ ∂ξ ϕjUV ¼ : ð18Þ It might appear as if the authors state exact opposites, namely, that a nonzero VEV counterterm is strictly δˆ ξ → 0 Since ϕjUV vanishes when , while the gauge-inde- necessary versus the statement that no VEV renormali- pendent part is obviously untouched by this limit, we can zation in addition to the Higgs field renormalization is ˆ 8 conclude that δϕ renormalizes purely the spurious diver- needed. However, both statements are not contradictory, gences that are caused the gauge-fixing procedure in the as the respective contexts differ. In [48], the authors do Rξ-gauge. It is therefore only necessary as an independent not renormalize quantum fields at all, as they are renormalization constant when ξ ≠ 0. In this sense, it is the interested only in the Green’s functions of the back- minimal addition to the set of renormalization parameters ground fields. Then, the statement that no genuine VEV of the unbroken theory in order to render all n-point counterterm is needed translates to the fact that no ˆ Green’s functions finite. Moreover, any inclusion of δϕ 7 in counterterm definitions will carry over its gauge Nevertheless, the values of the potentials at these points are found to be gauge independent. 8Both references use the notation δv for different quantities 6 ˆ In [7], δvk is the equivalent of vδϕ. leading to potential confusion.

076010-4 VACUUM EXPECTATION VALUE RENORMALIZATION IN THE … PHYS. REV. D 103, 076010 (2021) renormalization in addition to Eq. (9b) is necessary. In In the SM, the relevant set of tree-level relations is [9], however, the focus lies on the renormalization of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi quantum fields, while the background fields are still used 2 2 2 v v 2 2 m ¼ μ þ 3λv ;m¼ g2;m¼ g1 þ g2; to preserve the symmetry structure of the theory. Then, h W 2 Z 2 ˆ g1g2 Zϕ in Eq. (9b) is interpreted as an additional counterterm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − μ2 2λ e ¼ 2 2 ;th ¼ vð þ v Þ: ð21Þ to the one of Eq. (9a), due to a mismatch between g1 þ g2 quantum field renormalization and the renormalization of its VEV. In their notation, δv ≠ 0 as they parametrized it Using Eq. (20), we get the relations between the renorm- in a relationship with the quantum instead of the back- alization constants of the parameter renormalized tadpole ground-field renormalization as compared to [48]. scheme (PRTS) [32], usually used with an OS scheme, to the ones of the original parameters, namely,

III. TRANSLATION OF RENORMALIZATION 2 ft ;m ;m ;m ;eg ↔ fv; μ ; λ;g1;g2g; ð22Þ CONSTANTS h h W Z 1 2 Before a renormalization procedure is carried out, where g1 and g2 are the Uð Þ- and SUð Þ-couplings, one has to choose a set of independent renormalization respectively. The results are shown in the first column of δ constants. In the SM, one usually chooses experimentally Table I, where v is expressed in terms of the field well-accessible physical parameters as independent renormalization constants of Eq. (12). This is to show where the gauge-fixing-induced ξ-dependences appear via renormalization constants, while in BSM studies, it can ˆ be convenient to use the set of original theory parameters δϕ. Inspecting the first column of Table I, one clearly sees ˆ and the VEVs, especially when the use of an MS scheme that δϕ enters the definition of the usual mass counterterms. cannot be avoided. For the comparability of different This means that the latter are necessarily gauge dependent choices, it is instructive to have a translation between if one defines them as in the PRTS. In the next section, the sets of renormalization constants. To get these relations, we will present the FJ scheme and show its role in consider that we have a parameter set fpg related at tree the cancellation of gauge dependences by virtue of the level to a parameter set fp0g by some function f: renormalization constant redefinitions shown in the second column of Table I. 0 p ¼ fiðfpgÞ: ð19Þ i IV. RELATIONS AMONG DIFFERENT Introducing renormalization constants as in Eq. (B9) and VEV SCHEMES expanding to one-loop order induces the relations A. FJ scheme 1 ∂ The FJ scheme is a procedure of reinstating tadpole δ δ p0 ¼ · ð p pjÞ fðfpgÞ: ð20Þ contributions in perturbative calculations so that parameter i f ðfpgÞ j ∂p i j definitions can be defined in a gauge-independent way.

TABLE I. The mass, VEV, and electric charge renormalization constants are expressed in terms of the renormalization constants of gauge couplings g1, g2 and potential parameters λ, μ together with the (background) field renormalization constants in the two tadpole schemes. This is to emphasize the relations of divergence structures between the different renormalization constants. Δ is the “FJ term” used to relate counterterms from the usual tadpole scheme to the FJ scheme (see Sec. IVA).

Usual tadpole scheme [32] FJ scheme [16]

Δ 0 Th 1 ¯ ˆ ¼ Δ ¼ 2 ¼ 2 ðδλ − δμ2 þ δϕ þ δϕÞ vmh 1 1 ¯ ˆ δ δ 2 − δ δv ¼ 2 ðδϕ þ δϕÞ vjFJ ¼ 2 ð μ λÞ 3 ¯ ˆ δ 0 δth ¼ −λv ðδλ − δμ2 þ δϕ þ δϕÞ thjFJ ¼ 3 ¯ ˆ 1 δ 2 δ 2 δ 2 ¼ 2 ðδλ þ δϕ þ δϕÞ − 2 δμ2 M jFJ ¼ μ Mh H ¯ ˆ δ 2 2δ δ 2 − δ δ 2 ¼ 2δ þ δϕ þ δϕ M jFJ ¼ g2 þ μ λ MW g2 W 2δ 2δ 2δ 2δ g2 g2 þg1 g1 g2 g2 þg1 g1 δ 2 2 δ¯ δˆ δ 2 2 δ 2 − δ M ¼ 2 2 þ ϕ þ ϕ M j ¼ 2 2 þ μ λ Z g1þg2 Z FJ g1þg2 1 ¯ ˆ 1 δ δ δ δ δ j ¼ δ þ ðδμ2 − δλÞ mf ¼ y þ 2 ð ϕ þ ϕÞ mf FJ y 2 δ δ 1 2δ 2δ e ¼ ej ¼ 2 2 ðg1 g2 þ g2 g1 Þ FJ g1þg2

076010-5 VYTAUTAS DŪDENAS˙ and MAXIMILIAN LÖSCHNER PHYS. REV. D 103, 076010 (2021)

ˆ In the current literature, this procedure is often paraphrased of [9], or equivalently, δϕ: The FJ scheme cancels the as a shift of the VEV to the correct minimum of the gauge-dependent divergences which come about due to the loop-corrected scalar potential [1,5,8,18]. This means breaking of the global gauge symmetry via the Rξ-gauge that one first shifts the bare VEV by the full tadpole fixing. We will discuss the gauge dependence of finite parts contributions, i.e., at the end of the section.

Δ Δ Th vbare ¼ vbarejFJ þ v; v ¼ 2 ; ð23Þ B. FJ reparametrization mh Equation (23) together with the tadpole condition where we indicated the shifted VEV by “FJ.” Only then, Eq. (15) might seem as two renormalization steps. one adopts multiplicative renormalization constants and However, Eq. (23) represents a mere redefinition of a bare inserts Δv at each appearance of the VEV in the Lagrangian parameter. In the following, we make this redefinition before using the parameter relations of Eq. (21). Since more explicit and thereby give an equivalent way of Eq. (23) constitutes a redefinition of the bare VEV, it no formulating the FJ scheme. This will also simplify the longer can be interpreted in terms of the background-field comparison of the different tadpole schemes. renormalization constant, and we introduce an independent We first generate counterterms via multiplicative δ 2 vjFJ for its renormalization. Then, the initial bare VEV is renormalization of the parameters fv; μ ; λ;g1;g2g and related to the renormalized VEV v as then do a simple zero insertion of what we call a FJ term Δ in the bare VEV, i.e., δ Δ vbare ¼ v þ v vjFJ þ v: ð24Þ v ¼ vð1 þ δ − Δ þ ΔÞ; ð27Þ The full tadpole counterterm becomes bare v

3 2 δ −λ δ − δ 2 2δ − Δ where Δ is the dimensionless equivalent of the VEV shifts th ¼ v ð λ μ þ vjFJÞ vmh: ð25Þ used in Eq. (23), namely, Using the definition of the FJ-VEV shift, Eq. (23), we see that the first term of Eq. (25) must vanish in order to fulfill T Δ ¼ h : ð28Þ the tadpole condition of Eq. (15), which leads to the vm2 identification h 1 We then redefine the VEV counterterm by δ δ 2 − δ vjFJ ¼ 2 ð μ λÞ: ð26Þ δ δ − Δ vjFJ ¼ v ; ð29Þ This shows that in the FJ scheme, one recovers the tree- level relation between the VEV and the parameters of the scalar potential of Eq. (3), so that the VEV renormalization which gets us back Eq. (24), and thereby make the relation can be fully expressed in terms of shifts of the potential between the two tadpole schemes explicit. Similarly, we parameters μ2 and λ. In this sense, the shift of Eq. (23) can can identify be paraphrased as shifting the VEV to the correct one-loop δ δ − Δ minimum. Or, as presented in [16], one chooses the mf jFJ ¼ m ; ð30aÞ proper VEV. Introducing the shift of Eq. (23) into the Lagrangian everywhere is effectively equivalent to including regular δ 2 j ¼ δ 2 − 2Δ; ð30bÞ mV FJ mV tadpole contributions in the Green’s and vertex functions in addition to the 1PI contributions, i.e., effectively setting δt ¼ 0 as well as Δ ¼ 0. Nevertheless, the FJ scheme has δ 2 j ¼ δ 2 − 3Δ; ð30cÞ h mh FJ mh the appeal that renormalized one-point functions vanish exactly with the tadpole condition Eq. (15) being fulfilled, and thereby parametrize the change in going from the while they would remain formally divergent in the latter renormalization constants of the PRTS to the ones of the FJ δˆ case. Since Eq. (26) does not include ϕ in its definition, it scheme. In deriving the last line of Eq. (30c), care needs to has no gauge-dependent UV divergences in contrast to the be taken by imposing the tadpole condition only after the PRTS. This is shown in Table I where we also see that none shift Eq. (29) was inserted. ˆ of the other parameters carries a δϕ-dependence in the FJ Diagrammatically, Eq. (27) can be understood as a scheme. Here, we see the relationship between the FJ simple reassignment of the two instances of Δ. In the case scheme and the additional VEV-renormalization constant of one-loop corrections to a fermion mass, this means

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contributions for mpole to be gauge independent (see, e.g., [8]).

C. Numerical comparison In this section, we want to show an explicit deter- ð31Þ mination of the VEV-renormalization constants in the various schemes and give a numerical example for their where one can identify the FJ mass renormalization comparison. constant of Eq. (30a) in the first diagram, which has no We can use the relations Eq. (21) to define the VEV ˆ counterterms via the renormalization constants δg2 and gauge-dependent divergences due to the cancellation of δϕ δ 2 in its definition, while the third diagram represents implicit mW as Δ 0 tadpole contributions. Instead, if we choose ¼ , the first δ 2 δ diagram represents a gauge-dependent mass counterterm δ δ δ δ δ 2 mW − g2 vðf pgÞ ¼ vð g2; mWÞ¼v 2 ; ð34Þ 2m g2 δm as defined in the PRTS. In any case, the usual tadpole W condition of Eq. (15) provokes the cancellation of the last 2 two diagrams. which holds for both tadpole schemes. The SUð Þ-coupling Generically, we can write g2 can in turn be expressed as δ δ 2 − 2 δ 2 δ e e þ mW cw mZ ð32Þ g2 ¼ 2 2 − 2 : ð35Þ sw mZ mW

This step is usually done such that physical quantities in order to emphasize the inclusion of (implicit) tadpole known to high accuracy can be used as input parameters. contributions in the FJ scheme. Here, the superscript The charge renormalization constant δe can be defined in “Pr” stands for renormalization prescriptions such as terms of the γ-γ self energy and the γ-Z mixing [1]. It does using the MS or OS scheme, which needs to be specified not depend on the choice of a tadpole scheme (because the for discussing the gauge dependence of finite parts. Higgs field h does not couple to the photon, and there is no Concerning the latter, one finds that the FJ scheme also γ-Z-h vertex) and neither does δg2 as defined in Eq. (35). leads to gauge-independent one-loop mass parameter Instead, the difference between the VEV-renormalization definitions in the OS scheme. δ 2 constants comes about via the definition of mW.As There is a simple explanation of why the reintroduction explained in Sec. IVA, the FJ scheme is equivalent to of tadpoles in the FJ scheme gives a gauge-independent including tadpoles in the counterterm definitions either result. As an example, we consider a fermion one-loop pole implicitly as in Eq. (32), or explicitly as if they were not mass in a bare perturbation theory: renormalized at all. This means we can define

ð33Þ ð36Þ

The whole expression Eq. (33) is gauge independent, as it is for the on-shell W-boson mass counterterm. This definition a one-loop pole mass [27], while mbare is gauge indepen- is gauge independent. In contrast to that, no tadpole dent by principle. Thus, the mass shift, i.e., the term in contributions enter Eq. (34) in the PRTS, leading to the parentheses in Eq. (33), is gauge independent as well. This definition means that the gauge dependence of the tadpole contributions induced by the Goldstone boson tadpoles of Eq. (16) ð37Þ is canceled when added to the 1PI contributions. With the FJ scheme implicitly including the tadpole contributions, one realizes that the mass renormalization constants of which is a gauge-dependent quantity. Hence, using Eq. (36) Eq. (30) therefore are gauge independent in terms of their or Eq. (37) in Eq. (34) defines the VEV counterterm in the finite parts as well when defined in an OS scheme. Note FJ or the PRTS scheme, respectively. A third version of the that in the PRTS, where Eq. (15) enforces tadpoles to VEV counterterm is the definition via the Higgs back- δ vanish in Eq. (33) and no VEV shift is introduced, mbare ground-field renormalization hˆ using Eq. (B9).One needs to compensate for the gauge dependence of the 1PI possible renormalization condition for the latter is

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∂ renormalization scale μR such that the top-quark tadpole ΣR 2 2 0 2 ˆ ˆjp ¼m ¼ ; ð38Þ 9 ∂p h h h vanishes, this would not be of help in theories with particles at much higher mass scales. This is because even R where Σ ˆ ˆ is the one-loop renormalized self-energy of the if μR is chosen to minimize the effect of the heaviest particle h h μ2 2 Higgs background field. Then, we can define tadpoles [i.e., via minimizing the effect of lnð R=mheavyÞ in the resulting A0 functions], the lighter ones would in turn v δvj ¼ δ ˆj : ð39Þ yield sizable shifts due to the large-scale difference BG 2 h OS μ between R and mlight. These aspects are in accordance with the findings of [17] A definition of the VEV counterterm in terms of field where it was shown that tadpole contributions can be a renormalization constants can be expected to be gauge source of numerical instabilities. Another example is the dependent, and this is indeed the case. multi-Higgs-doublet Standard Model discussed by one of Now, using Eq. (34) together with Eq. (36) for the FJ scheme and Eq. (37) the PRTS, one can verify by an the authors in [7], where a MS scheme is used and heavy Δ explicit one-loop calculation that Majorana fermions give large contributions to v, ulti- mately leading to very large corrections for one-loop ∞ neutrino masses. δvj ¼ δvj þ Δv ¼ δvj : ð40Þ PRTS FJ BG In this sense, the FJ scheme can present a trade-off between gauge-independent quantities and numerical sta- This serves as a consistency check of Eq. (29) together with bility in a perturbative calculation. Eq. (12). Note that the second equal sign only holds for the UV parts of the counterterms, which is to be expected, because there is no simple relation between Eq. (38) and the D. Remarks other renormalization conditions in terms of finite parts. (i) The considerations in Sec. IVA are helpful for Nevertheless, we think it is illuminating to see a direct adjusting parameter input values in one-loop calcu- comparison of three rather different approaches to tadpole lations from the FJ scheme to ones without tadpole and VEV renormalization. contributions. As an example, we mention the MS Using the numerical input parameters of Appendix E, the scheme, where from Eq. (29) one finds expressions above yield MS MS 1 − Δ δ 10 155 v jFJ ¼ v ð finiteÞ; ð42Þ vfinjPRTS ¼ . GeV; ð41aÞ and similar relations for the masses via Eq. (30).In δ −138 457 vfinjFJ ¼ . GeV; ð41bÞ addition, since the lhs of Eq. (42) is gauge independent, one can fully account for the gauge depend- Δ 148 612 vfin ¼ . GeV; ð41cÞ ences on the rhs via the Goldstone contributions of Δ. δ 15 2504 vfinjBG ¼ . GeV: ð41dÞ (ii) It is rather straightforward to generalize the procedure of Sec. IVA to BSM models with altered With these quantities being renormalization scale de- scalar sectors, and we refer to the existing presenta- pendent and all of them except for Eq. (41b) being gauge tions of [1,8]. The two-Higgs-doublet model is one dependent, one cannot put too much of an interpretation on example. Here, a simple way to generalize the FJ these values. Nevertheless, it allows us to discuss two procedure is to choose the Higgs basis [49–51]. interesting aspects. Then, one can straightforwardly use Eq. (28) with Δ First, we see that Eq. (41b) being defined OS and being identified by Eq. (28). The only difference is Eq. (41c) show a large cancellation when combined to that Th and δth now represent the tadpole contribu- give Eq. (41a), which we tested for a wide range of tions and the tadpole counterterms of the extended renormalization scale values. In theories where not all scalar sector in the Higgs basis. Another example is model parameters can be defined via process-independent the multi-Higgs-doublet model as discussed in [7], Δ physical observables in an OS scheme, such cancellations where one can attribute a FJ term k to each doublet ϕ can be absent, potentially leading to sizable shifts when Δ k and define these via the respective tadpole is included in parameter definitions (for the sake of gauge contributions Tk. independence). (iii) One can think about solving the issues of gauge δ dependence, numerical stability, and moreover, the Second, both the VEV shift as well as vfinjFJ receive their largest contribution from the heaviest SM particle, 9 δ μ2 ≃ the top quark. While in the SM, one could minimize To be precise, vfinjFJ vanishes at a scale of R Δ δ 182 2 1 the numerical effect of v or vfinjFJ by choosing a ð GeVÞ = expð Þ.

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ˆ compatibility of the background-field approach to δϕ is equivalent to using Eq. (39). The numerical VEV renormalization by investigating alternative comparison in Eq. (41) shows no coincidence with definitions of Δ. One alternative is to define the other VEV renormalizations and therefore in- dicates that the renormalization conditions used in ξ Tð Þ the PRTS are incompatible with Eq. (45). Δ ¼ h ; ð43Þ vm2 In Appendix C, we show that when promoting h Eq. (17) to a condition on finite terms as well, so that ˆ ξ δϕ absorbs the full Goldstone tadpoles, it leads to a where Tð Þ are only the gauge-dependent parts of the h gauge-dependent charge renormalization condition. tadpole contributions, i.e., the Goldstone boson δˆ This setting is practically equivalent to using ϕjOS, tadpoles of Eq. (16). This could be justified by 2 the claim that these contributions come about as an apart from subtraction point being p ¼ 0 instead of 2 2 artifact of the global symmetry breaking effect of the p ¼ mh. Hence, it shows the incompatibility of δˆ Rξ-gauge. In this approach, gauge independence as ϕjOS with the usual charge renormalization con- in the original FJ scheme would still be guaranteed, dition as discussed in Sec. IV C. because the difference in the choices of Δ would In general, there seems to be no obvious way to ˆ only lie in gauge-independent contributions. Then, define the finite parts of δϕ in order to mimic the large contributions to Δ, e.g., from the top quark or effect of the FJ procedure, and it therefore remains even heavier particles in BSM models, would be purely as a tool for studying divergence structures. absent and lead to numerically smaller corrections. Δ Nevertheless, one could argue that this choice of V. CONCLUSIONS seems somewhat arbitrary in the sense that not all tadpole contributions are treated on the same footing The symmetry breaking effect of the Rξ-gauge fixing and that moreover, the extraction of the gauge- leads to the necessity of a renormalization constant in dependent tadpole contributions might become less addition to the ones for parameters and fields for the obvious at higher-loop orders. option to render all n-point Green’s functions finite. By ˆ (iv) As discussed in Sec. II, δϕ absorbs the gauge- adapting the findings of [9] to the SM, we showed dependent divergences coming from the introduc- explicitly how this independent renormalization constant tion of the Rξ-gauge. One could try to mimic the is related to the Higgs background-field renormalization effect of a FJ-VEV shift via an appropriate choice of and to Goldstone boson tadpoles. Effectively, this degree of ˆ the finite parts of δϕ. An obvious attempt would freedom was used in the tadpole condition already in [40], be to use an OS condition for the renormalized yet we wanted to emphasize its origin lying in the gauge two-point function of the physical Higgs field fixing as opposed to being a direct consequence of and the Higgs background field,10 i.e., spontaneous symmetry breaking. We showed how this degree of freedom leads to gauge dependences in all the ∂ ∂ counterterms it enters, such as the mass counterterms in Σ 2 2 0 Σ 2 2 0 2 hhjp ¼m ¼ ; 2 hˆ hˆjp ¼m ¼ : ð44Þ ∂p h ∂p h Table I. The FJ scheme [16] manages to avoid these gauge This yields dependences in parameter and counterterm definitions. The scheme was originally presented with arguments about 1 using the proper VEV, while in the recent literature, the δˆ δ − δ ϕjOS ¼ 2 ð hjOS hˆjOSÞ notion of true one-loop minimum [1] was employed. We, however, showed that one can look at this scheme as being 1 2 1 2 2 2 ¼ ξg2 2B0ðm ; ξm ; ξm Þ simply a set of convenient counterterm redefinitions and in 4 ð4πÞ2 h W W this way provide some independence from the interpreta- 1 2 2 2 tion of these notions. þ B0ðm ; ξm ; ξm Þ : ð45Þ cos2θ h Z Z The global symmetry argument, which lets us claim that W ˆ we need only one additional renormalization constant δϕ, Remarkably, it gives the same functional expression breaks down whenever ξW ≠ ξZ. On the other hand, the FJ as the unphysical Green’s function used to check the scheme generalizes to any loop order straightforwardly also divergences in Eq. (B11), except that the subtraction when ξW ≠ ξZ, even though it implicitly uses the degree of 2 2 2 0 point is p ¼ mh instead of p ¼ . This choice of freedom of the Higgs background-field renormalization. This hints at the possibility that a single renormalization 10 δˆ This is in contrast to the approach of [9], where only infinite constant ϕ is enough also in this case, yet it is unclear contributions are taken into account. whether there exists a rigorous symmetry argument for that.

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Hence, there is a subtle interplay between the VEV is invariant. The mass eigenstate gauge-fixing functions are renormalization interpreted as the renormalization of the recovered by Higgs background field, and the FJ scheme. 3 We used the SM as a playground to test various FA ¼ FWsW þ FBcW; ðA4Þ aspects of the renormalization in the Rξ-gauge with a 3 − special emphasis on tadpole conditions and the connection FZ ¼ FWcW FBsW; ðA5Þ between different approaches to the subject. This becomes especially relevant in BSM models with extended scalar 1 1 2 F ¼ pffiffiffi ðF ∓ iF Þ; ðA6Þ sectors, where, e.g., numerical effects of tadpole contribu- W 2 tions and the discussion of gauge dependences in mixing angles remain actively discussed [17,52]. We advocate the where sW and cW are the sine and cosine of the Weinberg use of the FJ prescription for keeping track of gauge angle, respectively. Taking the limit dependences in intermediate expressions and as a useful 0 tool for consistency checks in perturbative calculations. ˆ ϕ → 1 ðA7Þ Nevertheless, we also draw attention to settings where the pffiffi 2 v FJ scheme can yield large corrections in renormalized quantities, potentially leading to numerical instabilities in in Eqs. (A1) and(A2) and inserting them into Eqs. (A4)– non-OS schemes and discuss a possible alternative. (A6), we recover the usual Rξ-gauge-fixing functions:

ACKNOWLEDGMENTS μ FA ¼ ∂ Aμ; μ The authors are grateful to Thomas Gajdosik for dis- FZ ¼ ∂ Zμ − ξmZGZ; cussions and a careful reading of the manuscript. We would ∂μ ∓ ξ also like thank João P. Silva and Dominik Stöckinger for FW ¼ Wμ i mWGW: ðA8Þ encouraging us to carry out this project. M. L. also thanks Marcel Krause and Stefan Liebler for many helpful discussions. V. D. thanks the Lithuanian Academy of Sciences APPENDIX B: CALCULATION USING for their support via Project No. DaFi2019. M. L. is BRST SOURCES supported partially by the DFG Collaborative Research ˆ Center TRR 257 “Particle Physics Phenomenology after Following [9], we find the divergence structure of δϕ the Higgs Discovery.” directly from the unphysical Green’s functions that include BRST sources. The Higgs doublet of the SM is decom- APPENDIX A: GAUGE-FIXING FUNCTIONS posed into quantum and background field:

In the background-field formalism, the gauge-fixing Φ ¼ ϕ þ ϕˆ: ðB1Þ functions for the Uð1Þ and SUð2Þ gauge-fixing parts, respectively, are given by The fields Φ and ϕˆ correspond to ϕeff and ϕˆ þ vˆ of [9], respectively. The BRST transformation of the background Y Y ∂μ − ξ ϕˆ† ϕ − ϕ† ϕˆ field is postulated to be in a contractible pair with another FB ¼ Bμ i g1 2 2 ; ðA1Þ background field qˆ, i i μ † σ † σ ˆ i ∂ i − ξ ϕˆ ϕ − ϕ ϕˆ sϕ ¼ qˆϕ;sqˆϕ ¼ 0; ðB2Þ FW ¼ Wμ i g2 2 2 ; ðA2Þ so neither ϕˆ nor qˆ contribute to the BRST cohomology where ϕ is the Higgs doublet field, and ϕˆ is its background ˆ [53]. In other words, they do not contribute to the physical field. Both fields ϕ and ϕ transform in the same way under spectrum of the theory. The BRST transformation for the the global gauge transformation; hence, it is easy to show field Φ is that Eq. (A1) is invariant under the gauge transformation, while Fi of Eq. (A2) transforms as a vector in the adjoint σk W Φ ϕ ϕˆ i ϕ ϕˆ representation of SUð2Þ. The antighost c¯i also transforms s ¼ g2i 2 ð þ Þck þ 2 g1ð þ ÞcB: ðB3Þ as a vector in the adjoint representation; hence, the gauge- fixing term From Eqs. (B2) and (B3), we get ξ σk i L ¯i i i ϕ ϕ ϕˆ ϕ ϕˆ − ˆ GF ¼ s c F þ 2 B ðA3Þ s ¼ g2i 2 ð þ Þck þ 2 g1ð þ ÞcB qϕ ; ðB4Þ

076010-10 VACUUM EXPECTATION VALUE RENORMALIZATION IN THE … PHYS. REV. D 103, 076010 (2021) where σ are Pauli matrices, and ck and cB are the ghost fields of the SUð2Þ and Uð1Þ gauge group, respectively. Finally, we include the BRST source Kϕ in the Lagrangian

† † LK ¼ Kϕsϕ þ sϕ Kϕ: ðB5Þ

The renormalization transformations of the field and background field from Eq. (8) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ¯ ˆ ˆ ¯ ˆ ˆ ϕ → Zϕ=Zϕϕ; ϕ → ZϕZϕϕ; ðB6Þ thus, the introduced “technical” background field qˆ transform as qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ˆ ¯ ˆ ˆ ¯ ˆ ¯ ˆ sϕ → s ZϕZϕϕ ¼ ZϕZϕqˆ ⇒ qˆ → ZϕZϕq:ˆ ðB7Þ

ˆ FIG. 1. Feynman rules for calculating δϕ. BRST sources transform as the inverse renormalization transformation of the corresponding field. Then, the relation

δΓ ¼hsϕi; ðB8Þ δKϕ

ˆ where Γ is the effective vertex functional and is unchanged FIG. 2. One-loop diagram for calculating δϕ. after the renormalization transformation. From Eq. (B6), 1 1 we get that the transformation for the BRST source of the Γˆ½1 − δˆ ξ 2 2 0 ξ 2 ξ 2 i ˆ ¼ i ϕ þ i g2 2 B0ð ; m ; m Þ Higgs doublet quantum field is qhKh 4 ð4πÞ W W qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 B0 0; ξm ; ξm ; ˆ ¯ þ 2θ ð Z ZÞ ðB11Þ Kϕ → Zϕ=ZϕKϕ: ðB9Þ cos W

2 2 2 2 g2v or, using A0ðm Þ¼m ð1 þ B0ð0;m ;m ÞÞ, mW ¼ 2 , and Including all these renormalization transformations into mW ¼ cos θWmZ: Eq. (B5), we get 1 Γˆ½1 −δˆ 2 2 ξ − ξ 2 ˆ ¼ ϕ þ 2 2 f ½A0ðm Þ m k qhKh 4π W W † σ ˆ ˆ i ˆ ˆ ð Þ v L ¼ K g2i ðϕ þ ZϕϕÞc þ g1ðϕ þ ZϕϕÞc K ϕ 2 k 2 B 2 ξ − ξ 2 þ½A0ðmZ Þ mZg: ðB12Þ ˆ † − ZϕKϕqˆϕ þ H:c: ðB10Þ The finiteness of this two-point function fixes the diver- ˆ gences of δϕ. Moreover, Eq. (B12) immediately shows that The last term gives the counterterm for an unphysical ˆ its divergences coincide with the ones of the Goldstone Green’s function that includes only Zϕ. We will look at the ˆ boson tadpoles Eq. (16), which means that δϕ indeed ’ Γ ˆ Green s function qhKh , where h is the Higgs field compo- ’ absorbs all the gauge-dependent divergences in the tadpole nent of the doublet. The counterterm of this Green s condition Eq. (15). function is shown in the last diagram of Fig. 1. To calculate this Green’s function at one loop, one only needs the APPENDIX C: GAUGE DEPENDENCE IN THE qˆ Kϕ c Feynman rules for interactions between , , and , which BACKGROUND-FIELD-MODIFIED OS SCHEME can be read out from Eqs. (B10) and (A3), and are shown in Fig. 1. The loop diagram that we will need to calculate is In this section, we present the consequence of promoting shown in Fig. 2. The result of the sum of Fig. 2 and the last Eq. (17) to a renormalization condition on finite parts, diagram of Fig. 1 is namely,

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ξ 1 ð Þ − λ 3δˆ 0 2 2 Th v ϕ ¼ : ðC1Þ δ jξ ¼ − f½A0ðm ξÞþ2A0ðm ξÞ h ð4πvÞ2 Z W In principle, this is a valid renormalization condition, which − 2 2 2 ξ 2 ξ 2 2 2 ξ 2 ξ mh½B0ðp ;mZ ;mZ Þþ B0ðp ;mW ;mW Þg: allows us to absorb all the tadpole finite gauge dependences ˆ ðC8Þ into δϕ. One can verify that the FJ term Δ is ˆ 1 By using the expression for Δ from Eq. (C2) and δϕ, Δ δˆ ¼ 2 ϕ þ gauge independent ðC2Þ fixed by Eq. (C1), we see that the FJ-OS mass counterterm, as defined in Eq. (30c), is truly gauge independent: when Eq. (C1) holds. This allows for a more direct 3 interpretation of a FJ term in the sense of a background- δ 2 δ 2 − 3Δ δ 2 − δˆ 0 ð m jFJÞjξ ¼ m jξ ξ ¼ m jξ ϕ ¼ : ðC9Þ field renormalization also in finite parts, leading to a h h h 2 gauge-independent OS mass renormalization constant. δ¯ Also, the form of Eq. (C2) gives a possibility to modify From Eq. (12), we can get the field renormalization part ϕ that does not have gauge-dependent divergences. However, the FJ procedure to include only the gauge-dependent ¯ ˆ it turns out that in the OS, the finite part of δϕ is gauge term, namely, δϕ. However, we will show that this choice, together with the OS conditions on the two-point functions, dependent: leads to a gauge-dependent charge renormalization con- δ¯ δ δˆ stant and cannot be used with the usual charge renormal- ϕjξ ¼ hjξ þ ϕ ization condition presented in, e.g., [1,32,33]. 1 2 2 2 2 ¼ m ½B0ðp ;m ξ;m ξÞ To show this, we first need to get the gauge-dependent ð4πvÞ2 h Z Z ¯ part of δϕ in this scheme. For that, we look at the gauge- 2 2 2 þ 2B0ðp ;m ξ;m ξÞ: ðC10Þ dependent part of the Higgs self-energy, which can be W W written as Note that the divergences in this term are gauge independent as they should be. 1 2 2 Σξ p − m f 2f Now using Eq. (C1) together with Eq. (28), we see that ¼ 4π 2 ð hÞð Z þ WÞ ð vÞ we must have 1 3 2 2 ξ 2 2 ξ þ mh 2 ðA0ðmZ ZÞþ A0ðmW WÞÞ; ðC3Þ ¯ ð4πvÞ 2 δλjξ þ δϕjξ − δμ2 jξ ¼ 0: ðC11Þ

2 1 2 2 2 2 2 From the fact that the FJ mass renormalization constant of f ¼ A0ðm ξÞ − ðp þ m ÞB0ðp ;m ξ;m ξÞ: ðC4Þ V V 2 h V V the Higgs boson coincides with δμ2 (see Table I) and is gauge independent, the gauge dependence of δλ is The gauge-dependent part of the renormalized one-loop ¯ self energy function is δλjξ ¼ −δϕjξ: ðC12Þ

R 2 2 Σξ ¼ δ jξp − ðδ 2 þ δ Þjξm þ Σξ; ðC5Þ From Table I, we see that δλ enters the definition of the FJ h mh h h mass renormalization constants, which are gauge independent. Hence, from the gauge independence of δ 2 j , where we write jξ to denote the gauge-dependent terms of MW FJ the renormalization constants. The OS conditions give δ 2 j and δ j , we get MZ FJ mf FJ ∂ 0 2δ − δ 2δ δ¯ ΣR 2 2 0 ΣR 2 2 0 ¼ g1 2 jξ λjξ ¼ g1 2 jξ þ ϕjξ; ðC13Þ 2 jp ¼m ¼ ; jp ¼m ¼ : ðC6Þ ; ; ∂p H H 1 1 0 δ − δ δ δ¯ Inserting Eqs. (C3)–(C5) into OS conditions Eq. (C6) ¼ yjξ 2 λjξ ¼ yjξ þ 2 ϕjξ; ðC14Þ to check the gauge-dependent parts, we get the gauge dependences of the mass and field renormalization con- which leads to stants of the OS scheme (i.e., in the tadpole scheme of 1 Ref. [32]): 2δ 2δ δ¯ δ δ¯ 0 2 2 ðg1 g2 jξ þ g2 g1 jξÞþ ϕjξ ¼ ejξ þ ϕjξ ¼ : g1 þ g2 1 3 2 2 δ 2 ξ 2 ξ m jξ ¼ 2 ½A0ðmZ ZÞþ A0ðmW WÞ; ðC7Þ ðC15Þ h ð4πvÞ 2

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From Eq. (C15), we see that the charge renormalization fermions. Note that neither A nor B are gauge dependent, constant δ is gauge dependent, because it needs to cancel since ghost and vector boson gauge dependences cancel e ¯ the gauge dependence of δϕjξ given in Eq. (C10). Note that exactly. The divergences abbreviated as S come from dia- this is solely because we enforced Eq. (C1). Thus, we have grams with the Higgs boson loop contribution. Finally, the two tadpole conditions, pole and residue conditions for the only gauge-dependent UV divergent term G corresponds to W, Z, and Higgs two-point function, which are in total divergences of the Goldstone boson loop and vanishes in the ξ → 0 eight conditions; hence, they fully determine g1;2, δλ, δμ2 , case of . Note that we wrote all these functions in terms ¯ ˆ δϕ, δϕ, and the field renormalization constants of the Z and offour abbreviated constants; thus, we need four independent W bosons. This means that there is no freedom left to conditions and 4 degrees of freedom to uniquely fix them. In the case of ξ → 0, the number is reduced to only 3. The impose a charge renormalization condition as in, e.g., ¯ ˆ [1,32,33], which would give a gauge-independent charge 4 degrees of freedom are μ, λ, δϕ,andδϕ. They appear in the renormalization constant otherwise. Nevertheless, it is counterterms of one-, two-, three-, and four-point functions interesting to see that the gauge-independent definition of the Higgs boson, respectively, of the charge renormalization constant is possible also in 1 ¯ 2 ¯ ˆ this “scheme” by absorbing δϕ into their definitions, as δΓ − m v δ − δ 2 δ δ ; h ¼ 2 h ð λ μ þ ϕ þ ϕÞ ðD8Þ suggested by Eq. (C15). Hence, in principle, it is possible to 1 use Eq. (C1) instead of the usual charge renormalization 2 ¯ ˆ 2 ¯ ˆ δΓhh ¼ − m ð3δλ − δμ2 þ 5δϕ þ δϕÞþp ðδϕ − δϕÞ; condition and even define a gauge-independent charge 2 h renormalization constant. Yet to understand all the conse- ðD9Þ quences of this unconventional choice, a more thorough 2 mh ¯ ˆ study is needed which is beyond the scope of this work. δΓ ¼ −3 ðδλ þ 2δϕ − δϕÞ; ðD10Þ hhh v 2 APPENDIX D: EXPLICIT DIVERGENCES mh ¯ ˆ δΓ ¼ −3 ðδλ þ 2δϕ − 2δϕÞ: ðD11Þ hhhh v2 We used the SM file from FeynArts together with FormCalc to get the explicit expressions for the divergences. The To make sure that the renormalized n-point functions are divergences of the one-, two-, three-, and four-point finite, we solve four equations: function of the Higgs boson in the SM, respectively, are δΓUV ΓUV 0 h þ h ¼ ; ðD12Þ ΓUV 3 h ¼ v ðA þ G þ SÞ; ðD1Þ δΓUV ΓUV 0 hh þ hh ¼ ; ðD13Þ 1 2 ΓUV ¼ p2 B þ G þ v2ð3A þ 5S þ GÞ; ðD2Þ ∂ hh v2 m2 ðδΓUV þ ΓUVÞ¼0; ðD14Þ h ∂p2 hh hh ΓUV 6 2 − hhh ¼ vðA þ S GÞ; ðD3Þ δΓUV ΓUV 0 hhh þ hhh ¼ : ðD15Þ ΓUV 6 2 − 2 hhhh ¼ ðA þ S GÞ; ðD4Þ They give us where we abbreviated 1 1 1 δμ2 ¼ B þ S; δλ ¼ 2 B þ S þ A; λ λ λ 1 X X − 4 4 3 4 1 m2 A ¼ 2 2 m þ mq δˆ δ¯ − λ h 4πv f ϕ ¼ G; ϕ ¼ B; ¼ 2 : ðD16Þ ð Þ f¼e;μ;τ q¼u;d;s;c;t;b λ 2v 4 4 Inserting these expressions into Eq. (D11), we automatically − 3ð2m þ m Þ ; ðD5Þ W Z get X X 1 2 2 δΓUV ΓUV 0 B 2 m 3 m hhhh þ hhhh ¼ ; ðD17Þ ¼ 4π 2 f þ q ð vÞ μ τ f¼e; ; q¼u;d;s;c;t;b ΓUV where hhhh is given in Eq. (D4). − 3ð2m2 þ m2 Þ ; ðD6Þ The VEV counterterms presented in Sec. IV C have W Z divergences, which can be read out from Table I, inserting 3 m4 1 m2 the expressions for the renormalization constants shown in S ¼ h ;G¼ h ð2m2 ξ þ m2 ξÞ; ðD7Þ Eq. (D16): 2 ð4πv2Þ2 2 ð4πv2Þ2 W Z 2 1 ˆ ¯ 1 1 and omitted a global factor of 4−D. The divergences in A and δv ¼ vðδϕ þ δϕÞ¼ v G − B ; ðD18Þ B come from loop diagrams with ghosts, vectors, and 2 2 λ

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1 1 1 1 LoopTools [54] at its standard renormalization scale of δ δ 2 − δ − μ 1 vjFJ ¼ 2 vð μ λÞ¼ 2 v B þ λ S þ λ A ; ðD19Þ R ¼ GeV,

ξ ¼ 1;mW ¼ 80.398 GeV;mZ ¼ 91.1876 GeV; δvj ¼ δv þ Δv FJ m ¼ 125.09 GeV;v¼ 246.221 GeV; 1 1 1 1 1 h ⇒ Δv ¼ − v S þ A þ G ¼ − ΓUV: 0 000510999 0 105658 2 λ λ λ 2 h me ¼ . GeV;mμ ¼ . GeV; mh 1 77684 ðD20Þ mτ ¼ . GeV; mu ¼ 0.19 GeV;mc ¼ 1.4 GeV;mt ¼ 172.500 GeV; APPENDIX E: INPUT VALUES md ¼ 0.19 GeV;ms ¼ 0.19 GeV;mb ¼ 4.75 GeV; Here we present the numerical input values used in e ¼ 0.308147: Eqs. (41a)–(41d) together with the software package

[1] A. Denner and S. Dittmaier, Phys. Rep. 864, 1 (2020). [22] V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. [2] P. A. Grassi, B. A. Kniehl, and A. Sirlin, Phys. Rev. D 65, Commun. 256, 107478 (2020). 085001 (2002). [23] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. [3] S. Weinberg, Phys. Rev. D 7, 2887 (1973). 118, 153 (1999). [4] S. Actis, A. Ferroglia, M. Passera, and G. Passarino, Nucl. [24] T. Hahn, S. Paßehr, and C. Schappacher, Proc. Sci., LL2016 Phys. B777, 1 (2007). (2016) 068 [arXiv:1604.04611]. [5] M. Krause, R. Lorenz, M. Muhlleitner, R. Santos, and H. [25] A. Sirlin, Phys. Rev. Lett. 67, 2127 (1991). Ziesche, J. High Energy Phys. 09 (2016) 143. [26] A. Denner, S. Dittmaier, M. Roth, and D. Wackeroth, Nucl. [6] V. Dūdenas˙ and T. Gajdosik, Phys. Rev. D 98, 035034 Phys. B560, 33 (1999). (2018). [27] P. Gambino and P. A. Grassi, Phys. Rev. D 62, 076002 [7] W. Grimus and M. Löschner, J. High Energy Phys. 11 (2000). (2018) 087. [28] A. Denner, S. Dittmaier, M. Roth, and L. Wieders, Nucl. [8] A. Denner, L. Jenniches, J.-N. Lang, and C. Sturm, J. High Phys. B724, 247 (2005). Energy Phys. 09 (2016) 115. [29] A. Denner and S. Dittmaier, Nucl. Phys. B, Proc. Suppl. [9] M. Sperling, D. Stöckinger, and A. Voigt, J. High Energy 160, 22 (2006). Phys. 07 (2013) 132. [30] B. A. Kniehl and A. Sirlin, Phys. Lett. B 530, 129 (2002). [10] M. Sperling, D. Stöckinger, and A. Voigt, J. High Energy [31] D. Espriu, J. Manzano, and P. Talavera, Phys. Rev. D 66, Phys. 01 (2014) 068. 076002 (2002). [11] T. Appelquist, J. Carazzone, T. Goldman, and H. R. Quinn, [32] A. Denner, Fortschr. Phys. 41, 307 (1993). Phys. Rev. D 8, 1747 (1973). [33] M. Böhm, A. Denner, and H. Joos, in Gauge Theories [12] P. H. Chankowski, S. Pokorski, and J. Rosiek, Phys. Lett. B of Strong and Electroweak Interactions, 3rd ed. (B. G. 274, 191 (1992). Teubner, Stuttgart, 2001). [13] B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3121 (1972). [34] O. Piguet and S. Sorella, Algebraic Renormalization: [14] B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3137 (1972); Perturbative Renormalization, Symmetries and Anomalies 8, 4654(E) (1973). (Springer, Berlin, 1995), Vol. 28. [15] B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3155 (1972). [35] R. Haussling and E. Kraus, Z. Phys. C 75, 739 (1997), [16] J. Fleischer and F. Jegerlehner, Phys. Rev. D 23, 2001 arXiv:hep-th/9608160. (1981). [36] E. Kraus, Ann. Phys. (N.Y.) 262, 155 (1998). [17] A. Freitas and D. Stöckinger, in Proceedings of the 10th [37] E. Kraus and K. Sibold, Z. Phys. C 68, 331 (1995). International Conference on Supersymmetry and Unification [38] E. Kraus and S. Groot Nibbelink, in Proceedings of the 4th of Fundamental Interactions (SUSY02) (DESY, Hamburg, National Summer School for German-Speaking Graduate Germany, 2002), pp. 657–661 [arXiv:hep-ph/0210372]. Students of Theoretical Physics (1998) [arXiv:hep-th/ [18] M. Krause, Master’s thesis, Karlsruhe Institute of Technol- 9809069]. ogy, Institute for Theoretical Physics, (2016). [39] E. Kraus, Acta Phys. Pol. B 29, 2647 (1998), https://www [19] T. Hahn, Comput. Phys. Commun. 140, 418 (2001). .actaphys.uj.edu.pl/fulltext?series=Reg&vol=29&page=2647. [20] R. Mertig, M. Böhm, and A. Denner, Comput. Phys. [40] G. Passarino and M. J. G. Veltman, Nucl. Phys. B160, 151 Commun. 64, 345 (1991). (1979). [21] V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. [41] C. Becchi, in Proceedings of the 2nd Triangle Graduate Commun. 207, 432 (2016). School in Particle Physics (1996) [arXiv:hep-ph/9705211].

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[42] C. Itzykson and J. Zuber, Quantum Field Theory, [49] S. Davidson and H. E. Haber, Phys. Rev. D 72, 035004 International Series in Pure and Applied Physics (2005); 72, 099902(E) (2005). (McGraw-Hill, New York, 1980). [50] H. E. Haber and D. O’Neil, Phys. Rev. D 74, 015018 [43] A. A. Slavnov, Theor. Math. Phys. 10, 99 (1972). (2006); 74, 059905(E) (2006). [44] N. K. Nielsen, Nucl. Phys. B101, 173 (1975). [51] H. E. Haber and D. O’Neil, Phys. Rev. D 83, 055017 [45] L. P. Alexander and A. Pilaftsis, J. Phys. G 36, 045006 (2011). (2009). [52] A. Denner, S. Dittmaier, and J.-N. Lang, J. High Energy [46] H. H. Patel and M. J. Ramsey-Musolf, J. High Energy Phys. Phys. 11 (2018) 104. 07 (2011) 029. [53] G. Barnich, F. Brandt, and M. Henneaux, Phys. Rep. 338, [47] M. Garny and T. Konstandin, J. High Energy Phys. 07 439 (2000). (2012) 189. [54] T. Hahn and M. P´erez-Victoria, Comput. Phys. Commun. [48] A. Denner, G. Weiglein, and S. Dittmaier, Nucl. Phys. B440, 118, 153 (1999). 95 (1995).

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