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Vacuum Expectation Value Renormalization in the Standard Model and Beyond

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Vacuum Expectation Value Renormalization in the Standard Model and Beyond 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 “origi- nal”) 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 renorm- alization constants.
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