Renormalization of vacuum expectation values in spontaneously broken gauge theories: Two-loop results

We complete the two-loop calculation of beta-functions for vacuum expectation values (VEVs) in gauge theories by the missing O(g^4)-terms. The full two-loop results are presented for generic and supersymmetric theories up to two-loop level in arbitrary R_xi-gauge. The results are obtained by means of a scalar background field, identical to our previous analysis. As a by-product, the two-loop scalar anomalous dimension for generic supersymmetric theories is presented. As an application we compute the beta-functions for VEVs and tan(beta) in the MSSM, NMSSM, and E6SSM.


Introduction
The renormalization of vacuum expectation values (VEVs) in general gauge theories with R ξgauge has been studied in our earlier work [1]. We showed that in R ξ -gauge the VEVs renormalize differently from the respective scalar fields and explained the origin and behaviour of this difference. We computed VEV-counterterms and β-functions at one-loop and leading two-loop level. The purpose of this subsequent paper is to complete the two-loop renormalization of VEVs in general gauge theories and generic supersymmetric theories.
The renormalization of a VEV v can generically be written in the two equivalent forms with √ Z being the field renormalization constant of the corresponding scalar field. The main insight of Ref. [1] has been that δv can be interpreted by the field renormalization Ẑ of a suitable chosen scalar background field. Thus, a simple computation becomes possible in terms of a single two-point function.
In the present paper we address the following points: 1. The missing two-loop terms of the order g 4 in Ẑ are computed and the complete two-loop VEV β-function for general gauge theories with R ξ gauge fixing can be provided.
2. Gauge kinetic mixing in case of several U (1) gauge factors is taken into account in the computation of the g 4 terms.
3. The complete results are specialised to general supersymmetric theories in the DR scheme. 4. As a by-product the anomalous dimension γ (2) for generic N = 1 supersymmetric theories is derived in DR for arbitrary values of ξ.

5.
As application, the concrete results for anomalous dimensions and β-functions of VEVs and tan β are provided in the well-known supersymmetric models MSSM, NMSSM, and E 6 SSM. These results can be readily applied in practical applications. Moreover, they highlight various characteristic features of the general results.
This paper is organized as follows: Sec. 2 provides a brief summary of the formalism and notation. Sec. 3 is centred on the computation of the full two-loop results for general gauge theories and supersymmetric theories. The application to the MSSM, NMSSM, and E 6 SSM is carried out in Sec. 4. Generally this paper provides a complete picture up to two-loop level and summarizes all relevant expressions, but the one-loop and Yukawa-enhanced two-loop results have already been published in [1].

General Gauge Theory and Scalar Background Fields
The renormalization of vacuum expectations can be cast in an elegant scheme by employing a scalar background field. As elaborated in our previous publication [1], we use the general setting of real scalar fields ϕ a , Weyl 2-spinors ψ pα , and real (non-abelian) gauge fields V A µ in the notation of [2][3][4][5]. The Lagrangian is given as The VEVs v a are replaced in this formalism by scalar background fields (φ a +v a ). These auxiliary fields allow to formulate a rigid (global) gauge invariant gauge fixing; analogous to Ref. [6] the gauge-fixing functional reads By settingφ a to zero, one recovers the gauge theory in standard R ξ -gauge. But the inclusion of ϕ a and the rigid (global) gauge invariant gauge fixing imply that the following renormalization transformations are sufficient An additional VEV counterterm is then prohibited. In the standard approach, without background fields, the most generic renormalization transformation of the scalar fields with shifts reads The two formalisms are equivalent, with the following identifications As a result, the β function of the VEV can be obtained as with the anomalous dimensions γ andγ corresponding to the field renormalizations √ Z and Ẑ , respectively. One of the main results of Ref. [1] was that the computation of δẐ can be reduced to the very simple, unphysical two-point function Here K ϕ b are the sources of the BRS transformation of the scalar field, andq a is the BRS transformation ofφ a . Both of these unphysical fields appear in a very simple and well prescribed way in the Lagrangian. Our formalism is independent of the actual value assigned tov a . We can therefore choosev a as the minimum of the full loop-corrected scalar potential. Hence, our β-functions describe the running of the full VEV, which is required, for example, in many supersymmetry applications such as spectrum generators [7,8]. Note that this running VEV has to be distinguished from other definitions used for example in the Standard Model [9,10], which corresponds to the VEV defined explicitly in terms of the running tree-level potential parameters Ref. [9] contains a diagram exposing the difference in the running between the different definitions.

General Gauge Theory
The one-loop results for the anomalous dimensions γ ab (S),γ ab (S) and β-functions β(v a ) in a general gauge theory have been presented in [1] and read At the two-loop level, the terms of O(g 2 Y Y † ) ofγ (2) [1] and the full γ (2) [2,5] have already been published. Therefore, the computation of O(g 4 )-terms inγ (2) remains at two-loop. Fig. 1 contains the four relevant graphs that generate the divergencies in the loop corrections of Γq a,Kϕ b , wherein we implicitly understand one-loop subdivergencies to be subtracted. As before, all calculations are carried out in MS or equivalently MS scheme. In analogy to the presentation of Machacek & Vaughn [2][3][4], we provide the contributions of each diagram of Fig. 1 in Tab. 1 with the notation The completed two-loop results in the MS scheme read as follows

Kinetic Mixing
The results of Sec. 3.1 hold for simple gauge groups. The generalization to product groups is obvious, except for gauge kinetic mixing of U (1) field strength tensors. In the recent literature, the impact of gauge kinetic mixing on RGEs has been studied quite extensively up to two-loop level [11][12][13]. Following the approach of Refs. [12,13], we need to provide substitution rules for γ to take kinetic mixing into account. A generic gauge group G can be decomposed into with the simple groups G k and the two (finite) sets I, J ⊂ N. The part of the Lagrangian describing kinetic mixing reads Analogously to Refs. [12,13], we definê with the root defined by √ Ξ √ Ξ = Ξ. The inspection of the graphs in Fig. 1 implies that there do not exist any gauge kinetic mixing contributions toγ (1) and the O(g 2 Y Y † )-part ofγ (2) , because BRS-ghost and -antighost are not affected by kinetic mixing. Graphs 1(b) and 1(c) are not affected either, as U (1)-gauge fields do not interact with the corresponding Faddeev-Popov-ghosts. Hence, the only change for kinetic mixing stems from graph 1(a), in particular from the one-loop insertion of the scalar self-energy. The relevant substitution rule is given by Here g k denote the non-abelian gauge couplings and g d the abelian ones, with the corresponding quantum numbers Q d . Further, X denotes the field under consideration, e.g. up-or down-type Higgs. The substitution rules for γ can be found in [12,13].

Supersymmetric Gauge Theory
The treatment of supersymmetric theories requires to take three subtleties into account: (i) supersymmetric theories are formulated in terms of complex scalar fields, (ii) the coupling structure is severely restricted by supersymmetry, and (iii) the use of the supersymmetry-preserving renormalization scheme DR. The first two points are merely computational issues, in the sense that one needs to take care of the changed coupling structure and the scalar field representation. Hence, these aspects will not be spelled out in detail and we directly present the results for complex scalar fields in a notation based on Ref. [14]. We will, however, give some details on the conversion to DR, which requires transition counterterms for parameters [15] and fields [16]. The existence of such transition counterterms is due to the equivalence of dimensional reduction and dimensional regularisation as shown in Ref. [17].
At one-loop level the results have been provided earlier [1] and read The first two-loop renormalization studies of softly broken N = 1 SUSY theories in DR have been performed in [18][19][20], though not always in component fields as used here. To our knowledge, the full result for γ (2) in a general supersymmetric theory is not available in the literature, except for Landau gauge (ξ = 0) [14]. In order to obtain the result for arbitrary ξ we proceed in the following steps. We first reevaluate the Feynman graphs in Ref. [2] with a generic N = 1 supersymmetric Lagrangian. 1 Then we apply transition counterterms for the conversion from MS to DR. This step differs from the case of the DR β-functions computed in Ref. [18]. Since the β-functions in that reference are gauge invariant, physical quantities, only transition counterterms for physical parameters were required, and those were provided in Ref. [15]. In the present case of γ-functions, also transition counterterms for field renormalization and gauge parameters are necessary. These were presented in Ref. [16]. Fortunately, however, the needed additional transition counterterms for the scalar field renormalization and for the gauge parameter are zero, The transition forγ to supersymmetry and DR could be carried out in an analogous way, by employing transition counterterms. However, it is also possible and simpler to use the fact that there is no difference between MS and DR for any diagram contributing to δẐ at the twoloop level. Hence,γ is equal in the MS and DR schemes. From this knowledge, one can then derive additional transition counterterms as a by-product: δẐ (1),trans = 0, and owing to the non-renormalization of the gauge fixing, where δZ (1),trans V denotes the transition counterterm for the gauge field, as obtained in Ref. [16]. With these ingredients, the full gauge-dependent two-loop results for the anomalous dimensions γ andγ as well as for the VEV β-function can be obtained. In DR they read

Application to Concrete Supersymmetric Models
This section provides the explicit two-loop results for the renormalization of all VEVs in the MSSM, NMSSM, and E 6 SSM, using the notation of Ref. [1]. For completeness and convenience, we provide the full results including previously known ones. (22b)

MSSM
(23b) The β-function of tan β follows then as two-loop The application of the general two-loop results yields for the MSSM The explicit calculations confirm our earlier statement [1] that the same R MSSM terms inγ (2) appear for up-and down-Higgs. Thus, we obtain the two-loop β-function for tan β as Tr y e y e † + 3N c Tr y u y u † y u y u † − 3N c Tr y d y d † y d y d † − 3 Tr y e y e † y e y e † + 1 (4π) 2 ξξ The gauge-dependence of tan β at two-loop stems solely from theγ terms.

NMSSM
one-loop The one-loop anomalous dimensions for the Higgs doublets H u,d in the NMSSM resemble the corresponding MSSM results: The NMSSM Higgs singlet S has the following RGE coefficients: N c Tr y d y d † + Tr y e y e † + |λ| 2 (33b) with R NMSSM = R MSSM . Again, the R NMSSM terms inγ (2) are equal for up-and down-Higgs. Next, we can provide the results for the two-loop gauge singlet: (4π) 4 γ

E 6 SSM
The E 6 SSM introduces a new feature: The U (1) N -extension of the SM-gauge group leads inevitably to gauge kinetic mixing. The notations for kinetic mixing of Sec. 3.2 can be specialized to the E 6 SSM asĝ = g 1 g 11 g 1 1 g 1 and Q(X) := Note that Eq. (36) contains the GUT-normalized U (1) Y -and U (1) N -charges for any field X.
The quantum-numbers Q Y (X) and Q N (X) are those of Ref. [21].
one-loop In comparison to our earlier results [1] the one-loop anomalous dimensions γ andγ are now extended for the general case of gauge kinetic mixing already present at tree-level. For the Higgs-doublets H u/d,3 and the SM-singlet S 3 our computations yield (38b) Thus, the one-loop β-function for tan β is given by Eq. (40) illustrates once more the gauge dependence of tan β at one-loop level due to the different U (1) N -quantum numbers of the Higgs doublets, see [1].
The connection with the more conventional treatment [21,22] of the kinetic mixing in the E 6 SSM is established by the coupling matrix (c.f. Eq. (36)) g = g 1 −g 1 tan χ 0 g 1 cos χ . (45b)

Conclusions
We completed the calculation of the two-loop VEV β-functions for general gauge theories and generic supersymmetric theories. The result complements the well-known set of RGE coefficients of Refs. [2][3][4][5] for general gauge theories as well as the supersymmetric gauge theories of Refs. [14,18]. In particular, we achieved the following • Completion ofγ (2) by the missing O(g 4 )-contributions of our earlier results [1].
• Extension of γ (2) DR SUSY to arbitrary values of the gauge fixing parameter ξ. As a consequence, we were able to provide the full VEV β-function for general and supersymmetric gauge theories in the MS and DR scheme up to the two-loop level. The result was applied to the MSSM, NMSSM, and E 6 SSM and we proved the statements made in [1] on the O(g 4 )-terms: 1. R u − R d = 0 in the MSSM and NMSSM, 2. R u − R d = 0 for the E 6 SSM.