Oblique Corrections in the MSSM at One-Loop—Part I: Scalars

Abstract

This paper is devoted to reconsider the one-loop oblique corrections arising from the scalar superpartners in the MSSM, that is, the squarks, the sleptons, and the scalars in Higgs sector. We explicitly present the complete one-loop forms of self-energy corrections to the gauge bosons of SM electroweak gauge groups, as well as their descendants the 𝑆, 𝑇, and 𝑈 parameters, which can be directly applied to constrain the parameter space of the MSSM. Our results about one-loop self energies are found to agree with Drees et al. (1992) and Pierce et al. (1997). Nevertheless, the 𝑆, 𝑇, and 𝑈 parameters are not in agreement with Dobado et al. (1999).

1. Introduction

With LHC keeping running on searches of the standard model (SM) Higgs with mass around 125 GeV [1, 2], we are at the dawn when studies of Higgs physics become a field of the precise test, and we find out whether low-scale supersymmetry is going to show up.

Among frameworks in which the implication of the recent LHC results in new physics are explored, the electroweak precise test always stands as an important way for studying phenomenology of Higgs physics. The logic is that SM (naturally including the Higgs particle ℎ) and its extensions such as supersymmetric SM can be analyzed through considering the quantum corrections, more concretely oblique corrections [3, 4], in these theories to the electroweak precise test observables. In SM, the oblique corrections mainly depend on the Higgs particle mass 𝑚_ℎ and top quark mass, it follows that electroweak precise test provides useful constraint on 𝑚_ℎ [5]. For example, it is very successful in analyzing technicolor models [4].

This method can be similarly applied in SUSY models, from which the masses of supersymmetric particles can be constrained (see, e.g., [6]). But this method is not very efficient due to some reasons. One is that there are so many mass parameters in SUSY models. Another is that the bounds on these masses are not yet sufficient before the running of the LHC collider [7–9]. So even in the minimal supersymmetric standard model (MSSM), it is impossible to make firm claims. In order to extract useful information, as discussed in the literature, various limits such as highly degenerate [5, 10] or super heavy masses [11–13] of supersymmetric particles are considered in SUSY models.

However, these approximations are too simple to be suitable when the present LHC data is incorporated into the MSSM. Also, it can be verified that the analytic results related to our discussion usually are not consistent in the literature. Due to these observations, we should ascertain the oblique corrections in the MSSM first. In this paper, we will address this issue by using the two-component formalism of Lagrangian for supersymmetric particles under the electroweak group. The main outcomes are two fold. At first, the results we obtain can be applied to the most general cases without taking any assumptions about the quark- and lepton scalar masses. Moreover, our results cannot be reduced to those of [11] when we use the same limit as that the authors took. The difference is probably due to the missing or the over counting of Feynman loops involving heavy scalars as the internal lines.

We want to emphasize that while this work is being prepared, the LHC results [1, 2] indicate that its main task has been transformed from discovery to precise tests of the SM-like Higgs and discovery of supersymmetry. The oblique corrections to electroweak observables can still serve as a window to explore the parameter space of superpartners' masses by combining the data collected by the LHC and other colliders. The implications to MSSM and NMSSM along this line will be addressed elsewhere [14].

The paper is organized as follows. In Section 2, we derive the bosonic oblique corrections in the MSSM. We divide the task into three parts, that is, the squark sector, the slepton sector, and the Higgs sector. We will consider the fermionic contributions in the paper [15]. In Section 3, we then make preliminary checks on the results presented in Section 2 in terms of the fact that the radiative corrections to electroweak mixing angle is finite. In Section 4, we present the bosonic results of 𝑆, 𝑇, and 𝑈 parameters in the MSSM. The property that these parameters are also finite is more stringent examination on the results than the one in Section 3. In Section 5, we conclude and discuss our main results. We find that our results about one-loop self energies are found to agree with [6, 16]. Nevertheless, the 𝑆, 𝑇, and 𝑈 parameters do not math with those of [11]. In the appendix explicit expressions for the functionals related to one-loop graphs are presented. We would like to emphasize that the on-shell renormalization scheme is used throughout this paper.

2. One-Loop Bosonic Contributions

Since we deal with the analytic calculation of one-loop self energy of SM electroweak gauge bosons, it is convenient to use the two-component formalism both for the supersymmetric scalars and fermions.

2.1. Lagrangian for Electroweak Scalar Doublets

To derive the Feynman rules, we refer to the Lagrangian for scalar doublets Φ^𝑇=(𝜙₁,𝜙₂) under the representation of electroweak gauge group, which can be written as ℒ_𝐻∼𝐷_𝜈Φ^†𝐷^𝜈Φ=Φ^†𝐷_𝜈𝐷^𝜈Φ (1) with the definitions𝐷_𝜈𝜕Φ=^𝜈𝑒−𝑖−2𝑠𝑐𝑒𝑠𝑐𝑄₁𝑍_𝜈−𝑖𝑒𝑄₁𝐴_𝜈𝑒−𝑖√𝑊2𝑠^+𝜈𝑒−𝑖√𝑊2𝑠^−𝜈𝜕_𝜈𝑒+𝑖+2𝑠𝑐𝑒𝑠𝑐𝑄₂𝑍_𝜈−𝑖𝑒𝑄₂𝐴_𝜈Φ. (2)Here 𝑄₁ and 𝑄₂ represent the 𝑌-charges of up-type 𝜙₁, and down-type 𝜙₂, respectively. AndΦ^†𝐷_𝜈=Φ^†𝜕_𝜈𝑒+𝑖−2𝑠𝑐𝑒𝑠𝑐𝑄₁𝑍_𝜈+𝑖𝑒𝑄₁𝐴_𝜈𝑖𝑒√𝑊2𝑠^+𝜈𝑖𝑒√𝑊2𝑠^−𝜈𝜕_𝜈𝑒−𝑖+2𝑠𝑐𝑒𝑠𝑐𝑄₂𝑍_𝜈+𝑖𝑒𝑄₂𝐴_𝜈, (3)where parameters 𝑠=sin𝜃_𝑊 and 𝑐=cos𝜃_𝑊.

For EW singlet scalars 𝑆 such as right-hand squarks and sleptons, the Lagrangian is given by ℒ_𝐻∼𝐷_𝜈𝑆^†𝐷^𝜈𝑆=𝑆^∗𝜕_𝜈𝑠−𝑖𝑐𝑒𝑄𝑍_𝜈+𝑖𝑒𝑄𝐴_𝜈×𝜕^𝜈𝑠+𝑖𝑐𝑒𝑄𝑍^𝜈−𝑖𝑒𝑄𝐴^𝜈𝑆. (4) Here, 𝑄 denotes the 𝑌-charge of 𝑆 field.

From (1) to (4), we can derive the Feynman rules for MSSM scalars (see, e.g., [17]). In what follows, we take the ’t Hooft-Feynman gauge for nonabelian gauge fields involved. It turns out that there are three types of graphs as shown in Figure 1 needed to be considered.

(a)

(b)

(c)

Figure 1 

Graphs that contribute to the self-energy of SM gauge bosons due to scalar superpartners.

2.2. Squark Sector

The squark contributions are composed of those coming from three-generation left-hand squarks 𝑞_𝐿𝑖=(𝑢_𝑖,𝑑_𝑖) and their right-hand 𝑢_𝑅𝑖, 𝑑_𝑅𝑖. According to the Lagrangian (1), we find that there are two types of Feynman diagrams that contribute to the one-loop self-energy of vector bosons, as shown in Figures 1(a) and 1(b). (In the next subsection there will be an extra Feynman diagram that needs to be counted due to the Higgs VEVs.) We find (We have neglected 1/16 factor in each Π^𝑉𝑉′ in this section, which will be restored in Section 4.)Π_𝐿^𝛾𝛾𝑞²=𝑁_𝑐𝑒²9𝜋²_𝑖=1,2,3−16𝐴𝑞²;𝑚𝑢²_𝑖,𝑚𝑢²_𝑖−4𝐴𝑞²;𝑚²𝑑_𝑖,𝑚²𝑑_𝑖+8𝑎𝑚𝑢²_𝑖+2𝑎𝑚²𝑑_𝑖,Π_𝐿^𝑍𝑍𝑝²=𝑁_𝑐1𝜋²_𝑖=1,2,3𝑒−4−2𝑠𝑐2𝑒𝑠3𝑐²×𝐴𝑞²;𝑚𝑢²_𝑖,𝑚𝑢²_𝑖𝑒−4−2𝑠𝑐𝑒𝑠3𝑐²×𝐴𝑞²;𝑚²𝑑_𝑖,𝑚²𝑑_𝑖𝑒+2−2𝑠𝑐2𝑒𝑠3𝑐²𝑎𝑚𝑢²_𝑖𝑒+2−2𝑠𝑐𝑒𝑠3𝑐²𝑎𝑚²𝑑_𝑖,Π_𝐿^𝛾𝑍𝑞²=𝑁_𝑐𝑒3𝜋²_𝑖=1,2.3𝑒−8−2𝑠𝑐2𝑒𝑠𝑞3𝑐×𝐴²;𝑚𝑢²_𝑖,𝑚𝑢²_𝑖𝑒−4−2𝑠𝑐𝑒𝑠𝑞3𝑐×𝐴²;𝑚²𝑑_𝑖,𝑚²𝑑_𝑖𝑒+4−2𝑠𝑐2𝑒𝑠𝑎𝑚3𝑐𝑢²_𝑖𝑒+4−2𝑠𝑐𝑒𝑠𝑎𝑚3𝑐𝑑²_𝑖,Π_𝐿^𝑊𝑊𝑞²=𝑁_𝑐12𝜋²𝑒²𝑠²_𝑖=1,2.3𝑎𝑚𝑢²_𝑖+𝑎𝑚²𝑑_𝑖−4𝐴𝑞²;𝑚𝑢²_𝑖,𝑚²𝑑_𝑖 (5)(6) for left-hand squark doublets, and Π_𝑅^𝛾𝛾𝑞²=𝑁_𝑐𝑒²9𝜋²_𝑖=1,2,3−16𝐴𝑞²;𝑚𝑢²_𝑅𝑖,𝑚𝑢²_𝑅𝑖−4𝐴𝑞²;𝑚²𝑑_𝑅𝑖,𝑚²𝑑_𝑅𝑖+8𝑎𝑚𝑢²_𝑅𝑖+2𝑎𝑚²𝑑_𝑅𝑖,Π_𝑅^𝑍𝑍𝑝²=𝑁_𝑐𝑒²9𝜋²𝑠²𝑐²_𝑖=1,2,3−16𝐴𝑞²;𝑚𝑢²_𝑅𝑖,𝑚𝑢²_𝑅𝑖−4𝐴𝑞²;𝑚²𝑑_𝑅𝑖,𝑚²𝑑_𝑅𝑖+8𝑎𝑚𝑢²_𝑅𝑖+2𝑎𝑚²𝑑_𝑅𝑖 (7) together with Π_𝑅^𝛾𝑍𝑞²=𝑁_𝑐𝑒²9𝜋²𝑠𝑐_𝑖=1,2.316𝐴𝑞²;𝑚𝑢²_𝑅𝑖,𝑚𝑢²_𝑅𝑖+4𝐴𝑞²;𝑚²𝑑_𝑅𝑖,𝑚²𝑑_𝑅𝑖−8𝑎𝑚𝑢²_𝑅𝑖−2𝑎𝑚²𝑑_𝑅𝑖,Π_𝑅^𝑊𝑊𝑞²=0 (8) for right-hand squarks. Here, 𝐴(𝑝²;𝑥,𝑦) and 𝑎(𝑥) are one-loop integral functions, which are defined in the appendix. Note that the right-hand sfermions 𝑢_𝑅𝑖, 𝑑_𝑅𝑖 and 𝑒̃_𝑅𝑖 have no contributions to the self-energy of 𝑊 boson. Also, we want to mention that there are four-point vertexes for 𝑊 bosons from the Lagrangian (1), however they contribute at two-loop.

The effects on Π due to FCNC can be taken into account through the reexpression of squark and slepton sectors with their corresponding mass eigenstates. For example, the mixing in mass matrix of 𝑢_𝐿 and 𝑢_𝑅 can be diagonalized via a unitary matrix 𝑈(𝛼₁): 𝑢_𝐿𝑢_𝑅=cos𝛼₁−sin𝛼₁sin𝛼₁cos𝛼₁𝑢₁𝑢₂ (9) such that the vertexes for sfermions coupled to vector bosons can be read from the mass eigenstates 𝑢_1,2.

2.3. Slepton Sector

In this sector, the contributions to self-energy of SM vector bosons stem from the three-generation left-hand sleptons 𝑙̃_𝐿𝑖=(𝜈_𝑖,𝑒_𝑖̃) and right-hand 𝑒̃_𝑅𝑖. In terms of the general expression in (1), we derive that Π^𝛾𝛾𝑞²=𝑒²𝜋²_𝑖=1,2,3−4𝐴𝑞²;𝑚𝑒²_𝑖̃,𝑚𝑒²_𝑖̃−4𝐴𝑞²;𝑚𝑒²̃_𝑅𝑖,𝑚𝑒²̃_𝑅𝑖+2𝑎𝑚𝑒²_𝑖̃+2𝑎𝑚𝑒²_𝑅̃,Π^𝑍𝑍𝑝²=𝑒²𝜋²_𝑖=1,2,31−2𝑠²²𝑠²𝑐²×−𝐴𝑞²;𝑚𝑒²_𝑖̃,𝑚𝑒²_𝑖̃+12𝑎𝑚𝑒²_𝑖̃+12𝑠²𝑐²𝑎𝑚𝜈²_𝑖−𝑠²𝑐²×4𝐴𝑞²;𝑚𝑒²̃_𝑅𝑖,𝑚𝑒²̃_𝑅𝑖−2𝑎𝑚𝑒²̃_𝑅𝑖−𝑠²𝑐²4𝐴𝑞²;𝑚²𝜈_𝑖,𝑚𝜈²_𝑖,Π^𝛾𝑍𝑞²=𝑒²𝜋²_𝑖=1,2,31−2𝑠²×𝑞𝑠𝑐−2𝐴²;𝑚𝑒²_𝑖̃,𝑚𝑒²_𝑖̃+𝑎𝑚𝑒²_𝑖̃+𝑠𝑐4𝐴𝑞²;𝑚𝑒²̃_𝑅𝑖,𝑚𝑒²̃_𝑅𝑖−2𝑎𝑚𝑒²̃_𝑅𝑖,Π^𝑊𝑊𝑞²=12𝜋²𝑒²𝑠²_𝑖=1,2.3𝑎𝑚𝜈²_𝑖+𝑎𝑚𝑒²_𝑖̃−4𝐴𝑞²;𝑚𝜈²_𝑖,𝑚𝑒²_𝑖̃. (10) As we have mentioned in the previous subsection, there is no contribution to Π^𝑊𝑊 from the right-hand 𝑒̃_𝑅𝑖. The discussion about taking the masses mixings among left- and right-hand sclars is similar to that about (9) via introducing a unitary matrix 𝑈(𝛼₂): 𝑒_𝐿̃𝑒_𝑅̃=cos𝛼₂−sin𝛼₂sin𝛼₂cos𝛼₂𝑒₁̃𝑒₂̃. (11) So far, we have dealt with evaluating the first two graphs in Figure 1, now we proceed to discuss the Higgs sector in which the last graph has to be included.

2.4. Higgs Sector

The calculation of contributions coming from the neutral 𝐴⁰, 𝐻⁰ as well as the charged real scalar 𝐻^± is similar to those of sfermion sector but more involved. One reason is that there is an extra Feynman diagram Figure 1(c) that needs to be considered due to the new couplings with the Higgs VEVs 𝑣_𝜇 and 𝑣_𝑑. The other reason is that the expressions when we transform from Higgs gauge eigenstates to their mass eigenstates are complicated.

Consider the following: 𝐻_𝜇=sin𝛽𝐺⁺+cos𝛽𝐻⁺1√2𝑣_𝜇+1√2cos𝛼ℎ⁰+1√2sin𝛼𝐻⁰+𝑖√2sin𝛽𝐺⁰+𝑖√2cos𝛽𝐴⁰𝐻_𝑑,=1√2𝑣_𝑑−1√2sin𝛼ℎ⁰+1√2cos𝛼𝐻⁰−𝑖√2cos𝛽𝐺⁰+𝑖√2sin𝛽𝐴⁰−cos𝛽𝐺⁻+sin𝛽𝐻⁻. (12)(13)Here 𝐺⁰ and 𝐺^± are Nambu-Goldstone modes under the ’t Hooft-Feynman gauge. tan𝛽=𝑣_𝜇/𝑣_𝑑 and angle 𝛼 is introduced in the unitary matrix so as to diagonalize the mass matrix of scalars. The mass spectra for scalars in the Higgs sector can be explicitly found in [18].

Evaluate the Feynman diagrams gives rise to the following: Π^𝛾𝛾𝑞²=𝑒²𝜋²−4𝐴𝑞²;𝑚²_𝐺⁺,𝑚²_𝐺⁺−4𝐴𝑞²;𝑚²_𝐻⁺,𝑚²_𝐻⁺+2𝑎𝑚²_𝐺⁺+2𝑎𝑚²_𝐻⁺+𝜋²𝑒²2𝑠²sin𝛽𝑣_𝜇+cos𝛽𝑣_𝑑²×𝑏₀𝑞²;𝑚²_𝑊,𝑚²_𝐺⁺,Π^𝑍𝑍𝑞²=𝑒²4𝜋²𝑠²𝑐²𝑎𝑚²_ℎ0+𝑎𝑚²_𝐻0+𝑎𝑚²_𝐺0+𝑎𝑚²_𝐴0+21−2𝑠²²×𝑎𝑚²_𝐺⁺+𝑎𝑚²_𝐻⁺+𝑒⁴4𝜋²𝑠⁴𝑐⁴cos𝛼𝑣_𝜇−sin𝛼𝑣_𝑑²×𝑏₀𝑞²;𝑚²_𝑍,𝑚²_ℎ0+sin𝛼𝑣_𝜇+cos𝛼𝑣_𝑑²×𝑏₀𝑞²;𝑚²_𝑍,𝑚²_𝐻0+2𝑠⁴𝑐²sin𝛽𝑣_𝜇+cos𝛽𝑣_𝑑²×𝑏₀𝑞²;𝑚²_𝑊,𝑚²_𝐺⁺−𝑒²𝜋²𝑠²𝑐²1−2𝑠²²𝐴𝑞²;𝑚²_𝐺⁺,𝑚²_𝐺⁺+𝐴𝑞²;𝑚²_𝐻⁺,𝑚²_𝐻⁺+cos²𝛼−𝛽𝐴𝑞²;𝑚²_ℎ0,𝑚²_𝐴0+𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐺0+sin²×𝐴𝑞𝛼−𝛽₂;𝑚²_ℎ0,𝑚²_𝐺0+𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐴0,Π^𝛾𝑍𝑞²𝑒=−²1−2𝑠²2𝜋²𝑚𝑠𝑐2𝑎²_𝐺⁺+2𝑎𝑚²_𝐻⁺+4𝐴𝑞²;𝑚²_𝐺⁺,𝑚²_𝐺⁺−4𝐴𝑞²;𝑚²_𝐻⁺,𝑚²_𝐻⁺−𝑒⁴2𝜋²𝑠𝑐sin𝛽𝑣_𝜇+cos𝛽𝑣_𝑑²𝑏₀𝑞²;𝑚²_𝑊,𝑚²_𝐺⁺,Π^𝑊𝑊𝑞²=𝑒²4𝜋²𝑠²2𝑎𝑚²_𝐻⁺+2𝑎𝑚²_𝐺⁺+𝑎𝑚²_𝐻0+𝑎𝑚²_𝐴0+𝑎𝑚²_𝐺0+𝑎𝑚²_ℎ0+𝑒⁴4𝜋²𝑠²𝑐²sin𝛽𝑣_𝜇+cos𝛽𝑣_𝑑²×𝑐²𝑏₀𝑞²;0,𝑚²_𝐺⁺+𝑠²𝑏₀𝑞²;𝑚²_𝑍,𝑚²_𝐺⁺+𝑒⁴4𝜋²𝑠⁴cos𝛼𝑣_𝜇−sin𝛼𝑣_𝑑²𝑏₀𝑞²;𝑚²_𝑊,𝑚²_ℎ0+sin𝛼𝑣_𝜇+cos𝛼𝑣_𝑑²×𝑏₀𝑞²;𝑚²_𝑊,𝑚²_𝐻0−𝑒²𝜋²𝑠²sin²×𝐴𝑞𝛽−𝛼₂;𝑚²_ℎ0,𝑚²_𝐺⁺+𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐻⁺+cos²×𝐴𝑞𝛼+𝛽₂;𝑚²_ℎ0,𝑚²_𝐻⁺+𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐺⁺+𝐴𝑞²;𝑚²_𝐴0,𝑚²_𝐻⁺+𝐴𝑞²;𝑚²_𝐺0,𝑚²_𝐺⁺. (14) In terms of the corrections arising from the SM, the superpartners’ contributions in MSSM can be separated from (14) as Π^𝛾𝛾𝑞²=𝑒²𝜋²−4𝐴𝑞²;𝑚²_𝐻⁺,𝑚²_𝐻⁺+2𝑎𝑚²_𝐻⁺,Π^𝑍𝑍𝑞²=𝑒²4𝜋²𝑠²𝑐²×𝑎𝑚²_𝐻0+2𝑎𝑚²_𝐴0+21−2𝑠²²𝑎𝑚²_𝐻⁺+𝑒²𝜋²𝑠²𝑐⁴𝑚²_𝑊cos²×𝑏𝛼−𝛽₀𝑞²;𝑚²_𝑍,𝑚²_𝐻0−𝑏₀𝑞²;𝑚²_𝑍,𝑚²_𝐻0−𝑒²𝜋²𝑠²𝑐²×1−2𝑠²²𝐴𝑞²;𝑚²_𝐻⁺,𝑚²_𝐻⁺+cos²×𝐴𝑞𝛼−𝛽₂;𝑚²_ℎ0,𝑚²_𝐴0+𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐺0−cos²𝛼−𝛽𝐴𝑞²;𝑚²_ℎ0,𝑚²_𝐺0+sin²𝛼−𝛽𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐴0,Π^𝛾𝑍𝑞²𝑒=−²1−2𝑠²𝜋²𝑞𝑠𝑐−2𝐴²;𝑚²_𝐻⁺,𝑚²_𝐻⁺+𝑎𝑚²_𝐻⁺,Π^𝑊𝑊𝑞²=𝑒²4𝜋²𝑠²2𝑎𝑚²_𝐻⁺+𝑎𝑚²_𝐻0+𝑎𝑚²_𝐴0+𝑒²𝜋²𝑠²cos²𝛼−𝛽𝑚²_𝑊×𝑏₀𝑞²;𝑚²_𝑊,𝑚²_𝐻0−𝑏₀𝑞²;𝑚²_𝑊,𝑚²_ℎ0+𝑒²𝜋²𝑠²cos²𝛼−𝛽𝐴𝑞²;𝑚²_ℎ0,𝑚²_𝐺⁺−𝐴𝑞²;𝑚²_ℎ0,𝑚²_𝐻⁺−𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐺⁺−𝑒²𝜋²𝑠²sin²𝛼−𝛽𝐴𝑞²;𝑚²_𝐻0,𝑚²_𝐻⁺+𝐴𝑞²;𝑚²_𝐴0,𝑚²_𝐻⁺. (15)

3. Preliminary Checks on the Results

In electroweak theory, there are various observables that can be precisely measured, however, they depend only on three parameters which are composed of the gauge coupling constants parameters 𝑔, 𝑔^′, and VEV 𝑣 associated with the scale of spontaneous symmetry breaking. For example, one can define the weak mixing angle as 𝑠²_𝑊𝑚≡1−^2𝑊𝑚²_𝑍. (16) Another definition uses 𝑠²_∗≡𝑔^′2𝑔²+𝑔^′2. (17) Alternatively, we do this via the accurately known weak-interaction observables 𝛼, 𝐺_𝐹, and 𝑚_𝑍: sin2𝜃₀≡4𝜋𝛼(𝑚²_𝑍)√2𝐺_𝐹𝑚²_𝑍^1/2=0.2307±0.0005 (18) which gives us an accurate standard of reference. The corrections to 𝛼, 𝐺_𝐹, and 𝑚_𝑍 due to quantum effects of new particle states beyond SM will lead to deviation from the reference point. So the measured value can be used to constrain the content of these new particles and estimate the bounds of their masses.

Now we use the weak mixing angle as an example to check our results presented in the previous section. The rational is as follows. The quantum corrections to observables such as 𝑚_𝑍, 𝑚_𝑊, 𝛼, and 𝐺_𝐹 in low-energy electroweak theory include corrections to the vector boson propagator, vertex, and box corrections. The last two types are usually proportional to the ratio of masses of external (light) fermions over masses of heavy particle states. With the limit that these light masses of SM fermions are set to zero, only the first type is important practically. This type of correction is known as oblique correction, as they enter the low-energy weak interactions only indirectly. The different definitions in (16) to (19) all agree at zero order but receive different radiative corrections. However, the differences, for instance [19], 𝑠²_𝑊−𝑠²_∗𝑐≡−²𝑚²_𝑊Π^𝑊𝑊𝑚²_𝑊−𝑐²Π^𝑍𝑍𝑚²_𝑍+𝑠𝑐³𝑚²_𝑊Π^𝛾𝑍𝑚²_𝑍 (19) due to the radiative corrections must be finite and are free of ultraviolet divergence.

For the part of slepton sector, substituting (10) into (19) gives 𝑠²_𝑊−𝑠²_∗=𝑐²𝑒²𝜋²𝑠²𝑚²_𝑊−1−2𝑠²𝐴𝑚²_𝑍;𝑚𝑒²_𝑖̃,𝑚𝑒²_𝑖̃−𝐴𝑚²_𝑍;𝑚𝜈²_𝑖,𝑚𝜈²_𝑖+2𝐴𝑚²_𝑊;𝑚𝑒²_𝑖̃,𝑚𝜈²_𝑖−𝑠²𝑎𝑚𝑒²_𝑖̃ (20) for each generation of left-hand sleptons, which then gives us the part of divergence that is proportional to −1−2𝑠²12𝑚𝑒²_𝑖̃−1𝑚12^2𝑍−12𝑚𝜈²_𝑖−1𝑚12^2𝑍1+24𝑚𝑒²_𝑖̃+14𝑚²𝜈_𝑖−1𝑚12^2𝑊−𝑠²𝑚𝑒²_𝑖̃=0. (21) For each generation of right-hand slepton, we find that the contribution to the deference in (19) vanishes.

Now we proceed to examine the results in the squark sector. Substituting (5) into (19) yields 𝑠²_𝑊−𝑠²_∗=𝑒²𝑐²𝜋²𝑠²𝑚²_𝑊−31−4𝑠²𝐴𝑚²_𝑍;𝑚𝑢²_𝑖,𝑚𝑢²_𝑖−23𝑠²𝑎𝑚𝑢²_𝑖−13𝑠²𝑎𝑚²𝑑_𝑖−21−3𝑠²𝐴𝑚²_𝑍;𝑚²𝑑_𝑖,𝑚²𝑑_𝑖+2𝐴𝑚²_𝑊;𝑚𝑢²_𝑖,𝑚²𝑑_𝑖 (22) for the left-hand squarks, from which the part of divergence is proportional to −31−4𝑠²12𝑚𝑢²_𝑖−1𝑚12^2𝑍−23𝑠²𝑚𝑢²_𝑖−13𝑠²𝑚²𝑑_𝑖−21−3𝑠²12𝑚²𝑑_𝑖−1𝑚12^2𝑍1+24𝑚𝑢²_𝑖+14𝑚²𝑑_𝑖−1𝑚12^2𝑍=0. (23)

For the part of Higgs sector, substituting (14) into (19) gives that the divergent parts in (19) are composed of those arising from 𝑎(𝑥), 𝐴(𝑥), and 𝑏₀ functions. It turns out that the contribution from 𝑎(𝑥) function is proportional to −(𝑒²𝑐²/𝜋²𝑚²_𝑊)(𝑚²_𝐺⁺+𝑚²_𝐻⁺), which exactly cancels the part from 𝐴 function. And the divergent parts in 𝑏₀ function cancel themselves.

4. 𝑆, 𝑇, and 𝑈 in the MSSM

In this section, we derive a set of parameters which estimate the oblique corrections to precise electroweak observables. These parameters are known as 𝑆, 𝑇, and 𝑈 in the literature [3, 4]. As a byproduct we also use the property that these parameters are finite values to examine the results shown in Section 2, independently of what we have done in Section 3. The dependence of 𝑆, 𝑇, and 𝑈 parameters on Π^𝐼𝐽(𝑝²) can be perturbatively expanded in powers of the external momentum squared. It turns out that these corrections are quite simple [3, 4]: 𝑆≡−16𝜋𝑒²𝑠𝑐𝑠𝑐Π^𝛾𝛾′(0)−𝑠𝑐Π^𝑍𝑍′(0)+𝑐²−𝑠²Π^𝛾𝑍′(0)𝑇≡4𝜋𝑒²Π^𝑊𝑊(0)𝑚²_𝑊−Π^𝑍𝑍(0)𝑚²_𝑍−2𝑠𝑐Π^𝛾𝑍(0)𝑚²_𝑍,𝑈≡16𝜋𝑠²𝑒²×Π^𝑊𝑊′(0)−𝑐²Π^𝑍𝑍′(0)−𝑠²Π^𝛾𝛾′(0)−2𝑐𝑠Π^𝛾𝑍′(0), (24) where Π^𝐼𝐽′(0)=𝑑²Π^𝐼𝐽𝑑𝑝²_𝑝2=0, (25)Π^𝐼𝐽 is the part with metric as the coefficient in Π^𝐼𝐽_𝜇𝜈=𝑔_𝜇𝜈Π^𝐼𝐽+⋯.

In this paper, we consider the calculations of the oblique corrections in MSSM, which are composed of four parts given a specific vacuum polarization: Π_SUSY𝑆𝑙̃=Π+Π+Π_𝐻+Π_𝑁𝐶. (26) We derive the first three parts in (26) in this paper, and will explore the last part in the future [15].

Firstly, the slepton sector gives 𝑆̃𝑙=112𝜋³_𝑖=1𝑚ln²𝜈_𝑖𝑚𝑒²_𝑖̃,𝑇̃𝑙=116𝜋𝑠²𝑚²_𝑊³_𝑖=1𝑚²𝜈_𝑖+𝑚𝑒²_𝑖̃−2𝑚𝑒²_𝑖̃𝑚𝜈²_𝑖ln𝑚𝜈²_𝑖/𝑚𝑒²_𝑖̃𝑚²𝜈_𝑖−𝑚𝑒²_𝑖̃,𝑈̃𝑙=1𝜋³_𝑖=1𝑚𝑒²_𝑖̃3𝑏^′₀0,𝑚𝑒²_𝑖̃,𝑚𝑒²_𝑖̃−𝑚𝑒²_𝑖̃+𝑚𝜈²_𝑖6𝑏^′₀0,𝑚𝑒²_𝑖̃,𝑚𝜈²_𝑖+𝑚𝑒²_𝑖̃−𝑚²𝜈_𝑖²𝑏24^′′00,𝑚𝑒²_𝑖̃,𝑚²𝜈_𝑖+1𝑓𝑚12𝑒²_𝑖̃,𝑚𝜈²_𝑖−1𝑓𝑚12𝑒²_𝑖̃,𝑚𝑒²_𝑖̃. (27) For the squark sector, we have 𝑆𝑆=112𝜋³_𝑖=1𝑚ln²𝑑_𝑖𝑚𝑢²_𝑖,𝑇𝑆=316𝜋𝑠²𝑚²_𝑊׳_𝑖=1𝑚𝑢²_𝑖+𝑚²𝑑_𝑖−2𝑚𝑢²_𝑖𝑚²𝑑_𝑖ln𝑚𝑢²_𝑖/𝑚²𝑑_𝑖𝑚𝑢²_𝑖−𝑚²𝑑_𝑖,𝑈𝑆=³_𝑖=11𝜋𝑚𝑢²_𝑖𝑏^′₀0,𝑚𝑢²_𝑖,𝑚𝑢²_𝑖+𝑚²𝑑_𝑖𝑏^′₀0,𝑚²𝑑_𝑖,𝑚²𝑑_𝑖−𝑚𝑢²_𝑖+𝑚²𝑑_𝑖𝑏^′₀0,𝑚𝑢²_𝑖,𝑚²𝑑_𝑖+12𝑓𝑚𝑢²_𝑖,𝑚²𝑑_𝑖−14𝑓𝑚𝑢²_𝑖,𝑚𝑢²_𝑖−14𝑓𝑚²𝑑_𝑖,𝑚²𝑑_𝑖 (28) with the finite function 𝑓(𝑥,𝑦) is defined as 𝑓=1𝑥,𝑦.𝑥−𝑦𝑥ln𝑥−𝑦ln𝑦 (29) Note that 𝑏^′₀ and 𝑏^′′₀ in (27) to (28) are given in appendix, which are also finite.

Finally, the Higgs sector yields 𝜋𝑆_𝐻=ℎ𝑚²_𝐻⁺,𝑚²_𝐻⁺−sin²𝛼−𝛽ℎ𝑚²_𝐴0,𝑚²_𝐻0+cos²×ℎ𝑚𝛼−𝛽_2ℎ0,𝑚²_𝐺0−ℎ𝑚²_𝐻0,𝑚²_𝐺0−ℎ𝑚²_ℎ0,𝑚²_𝐴0−𝑚²_𝑊𝑐²cos²×𝑏𝛼−𝛽_′00;𝑚²_𝑍,𝑚²_ℎ0−𝑏^′₀0;𝑚²_𝑍,𝑚²_𝐻0,𝜋𝑇_𝐻=14𝑠²𝑚²_𝑊×𝑚²_𝐻⁺ln𝑚²_𝐻⁺+cos²×𝑚𝛼−𝛽_2𝑊𝑓𝑚²_𝑊,𝑚²_𝐻0−𝑓𝑚²_𝑊,𝑚²_ℎ0+𝑚²_𝑍𝑓𝑚²_𝑍,𝑚²_ℎ0−𝑓𝑚²_𝑍,𝑚²_𝐻0+cos²×𝑔𝑚𝛼−𝛽_2ℎ0,𝑚²_𝐺⁺+𝑔𝑚²_ℎ0,𝑚²_𝐴0+𝑔𝑚²_𝐻0,𝑚²_𝐺0−𝑔𝑚²_ℎ0,𝑚²_𝐻⁺−𝑔𝑚²_𝐻0,𝑚²_𝐺⁺−𝑔𝑚²_ℎ0,𝑚²_𝐺0−sin²𝛼−𝛽𝑔𝑚²_𝐻0,𝑚²_𝐻⁺+sin²𝛼−𝛽𝑔𝑚²_𝐴0,𝑚²_𝐻0−𝑔𝑚²_𝐴0,𝑚²_𝐻0, (30) together with 𝜋𝑈_𝐻=ℎ𝑚²_𝐻+,𝑚²_𝐻⁺−ℎ𝑚²_𝐴0,𝑚²_𝐻0−cos²𝛼−𝛽𝑚²_𝑊×𝑏^′₀0;𝑚²_𝑊,𝑚²_𝐻0−𝑏^′₀0;𝑚²_𝑊,𝑚²_ℎ0+cos²𝛼−𝛽𝑚²_𝑍×𝑏^′₀0;𝑚²_𝑍,𝑚²_ℎ0−𝑏^′₀0;𝑚²_𝑍,𝑚²_𝐻0+cos²×ℎ𝑚𝛼−𝛽_2ℎ0,𝑚²_𝐺⁺−ℎ𝑚²_ℎ0,𝑚²_𝐻⁺−ℎ𝑚²_𝐻0,𝑚²_𝐺⁺+ℎ𝑚²_ℎ0,𝑚²_𝐴0−ℎ𝑚²_ℎ0,𝑚²_𝐺0+ℎ𝑚²_𝐺0,𝑚²_𝐻0−sin²𝛼−𝛽ℎ𝑚²_𝐻0,𝑚²_𝐻⁺−ℎ𝑚²_𝐴0,𝑚²_𝐻0 (31) with the finite functions 𝑔(𝑥,𝑦) and ℎ(𝑥,𝑦) defined as 𝑔=1𝑥,𝑦+12𝑥ln𝑥+𝑦ln𝑦𝑥+𝑦6+1−1+𝑓𝑥,𝑦24𝑥+𝑦−2𝑥𝑦𝑥𝑥−𝑦ln𝑦,ℎ=1𝑥,𝑦6𝑥+𝑦𝑏^′₀−10;𝑥,𝑦24𝑥−𝑦²𝑏^′′₀−10;𝑥,𝑦𝑓.12𝑥,𝑦 (32) Note that the combination cos(𝛼−𝛽) is given by cos²=𝑚𝛼−𝛽^2ℎ𝑚²_𝐴𝑚²_𝑍−𝑚²_ℎ𝑚²_𝑍+𝑚²_𝐴−2𝑚²_ℎ. (33)

Here, a few comments are in order for the results (27) to (31). First, it is obvious that each part in MSSM contributes to the finite 𝑆, 𝑇, and 𝑈 values as required. There are some constants terms in the expression of 𝑆, 𝑇, and 𝑈 parameters, which naively violate the property that large superpartner masses (compared with 𝑚_𝑍) leading to the conclusion that the theory decouples from the SM. But expanding the relevant functions we have defined implies that these constants are cancelled and the conclusion is restored. Second, from (30) to (31) it seems that the SM contributions has not been separated from the MSSM contributions due to some terms involved with SM fields. But it is crucial to notice that these terms are multiplied by a factor cos²(𝛼−𝛽), as given by (33). Under the limit that the superpartners decouple from SM, which results in 𝛼=𝛽, the results in (30) to (31), are reduced to pure MSSM ones. Finally, unlike in the case of SM, where the sensitivity of these parameters to the SM Higgs mass 𝑚_ℎ is logarithmic, the results in (30) to (32) demonstrate that this is not strictly true in the MSSM anymore. According to the simulations about 𝑆, 𝑇, and 𝑈 shown in [14], the sensitivity of these parameters to the mass of Higgs boson can be either stronger or weaker, which depends on choices of parameters left in the Higgs sector. Overall, the discrepancy is not so significant as one expects roughly.

Under the assumption that the breaking of 𝑆𝑈(2) symmetry is weak, that is, Δ_𝑞𝑖=𝑚𝑒²_𝑖̃−𝑚𝜈²_𝑖≪𝑚𝜈²_𝑖,𝑚𝑒²_𝑖̃ and Δ_𝑆𝑖=𝑚²𝑑_𝑖−𝑚𝑢²_𝑖≪𝑚𝑢²_𝑖,𝑚²𝑑_𝑖, we find 𝑆𝑞⟶112𝜋³_𝑖=1Δ_𝑆𝑖𝑚²𝑑_𝑖,𝑇𝑞3⟶−16𝜋𝑠²³_𝑖=1Δ_𝑆𝑖𝑚²_𝑊,𝑈𝑞1⟶−4𝜋³_𝑖=1Δ_𝑆𝑖𝑚²𝑑_𝑖,𝑆̃𝑙1⟶−12𝜋³_𝑖=1Δ_𝑞𝑖𝑚𝑒²_𝑖̃,𝑇̃𝑙1⟶−16𝜋𝑠²³_𝑖=1Δ_𝑞𝑖𝑚²_𝑊,𝑈̃𝑙1⟶−24𝜋³_𝑖=1Δ_𝑞𝑖𝑚𝑒²_𝑖̃. (34) From (34), one sees that both in the squark and slepton sector the relative ratio |𝑆/𝑈| is approximatively around the unity, while their values relative to 𝑇 depend on the ratios of 𝑚²/𝑚²𝑑_𝑖 and 𝑚²/𝑚𝑒²_𝑖̃.

5. Discussions and Conclusions

In this paper, we have revisited the oblique corrections in the context of the MSSM. The motivation for exploring these contributions is quite clear since they are useful to interpret the latest LHC data about the Higgs mass and bounds on superpartner masses. (Note that there is no need to consider the gauge invariance of these contributions once again, as this problem in the SM contributions has been properly treated [20–22].) Thus, we reconsider the theoretic calculations under ’t Hooft-Feynman gauge, with the bosonic part, as the first step towards the complete answer.

The results presented in this paper are checked by two examinations. In one examination, we directly confront our results to the finite radiative correction to the deference 𝑠²_𝑊−𝑠²_∗ in each sector. It shows that the individual contribution deriving from each sector indeed respects this property. The other examination is by using the property that the 𝑆, 𝑇, and 𝑈 parameters are finite. We also verify the expectation.

In summary, our results about one-loop self energies are found to exactly agree with [6, 16]. (The formulae presented in [16] take the SM part into account, however, they do not include the contribution arising from the pure gauge part. As we missed this point, we made the claims about disagreement with [16] in the previous version of this paper.) Nevertheless, the 𝑆, 𝑇, and 𝑈 parameters do not match with those of [11], which cannot be explained by the possible difference of renormalization scheme performed. In particular, our results do not take the scalar mass mixing among left and right hand into account, and therefore correspond to the simple case with vanishing mass mixing. Taking this difference into account and comparing our results with [11], one finds the following.(i)The 𝑆 and 𝑈 parameters arising from squark and slepton sector do not agree. (ii)The 𝑇 parameter in (27) and (28) agrees with [11].

We leave the study of oblique corrections arising from the fermions, that is, the neutralinos and charginos, elsewhere [15].

Appendix

One-Loop Integrals

Two-point functions 𝐴(𝑝²;𝑥,𝑦), 𝑎(𝑥), and 𝑏₀(𝑝²;𝑥,𝑦) are defined as the integrals [23–25] 𝜇^𝜖𝑑^𝑛𝑘(2𝜋)^𝑛𝑖𝑘²−𝑚²=1(4𝜋)²𝑎𝑚²,𝜇^𝜖𝑑^𝑛𝑘(2𝜋)^𝑛𝑖𝑘_𝜇𝑘_𝜈𝑘²−𝑚²₁𝑘+𝑝²−𝑚²₂=1(4𝜋)²𝑔_𝜇𝜈𝐴𝑝²;𝑚²₁,𝑚²₂,𝜇+⋯_𝜖𝑑^𝑛𝑘(2𝜋)^𝑛𝑖𝑘²−𝑚²₁𝑘+𝑝²−𝑚²₂=1(4𝜋)²𝑏₀𝑝²;𝑚²₁,𝑚²₂, (A.1) respectively, where 𝑝² is the external line vector boson momentum squared. Here, we have ignored terms that are irrelevant for discussions in the definition of 𝐴(𝑝²;𝑥,𝑦). Explicitly, 𝑎(𝑥)=−𝜂𝑥+𝑎_𝐹(𝑥),𝑏₀𝑝²;𝑥,𝑦=−𝜂+𝑏_𝐹𝑝²;𝑥,𝑦, (A.2) where the divergent factor in these functions is carried by 𝜂 as follows: 1𝜂=𝜖+ln(4𝜋)−𝛾_𝐸. (A.3) Here, 𝑑=4−2𝜖 and 𝜇 is the RG scale. The finite parts 𝑎_𝐹 and 𝑏_𝐹 in (A.2) are given by 𝑎_𝐹(𝑥)𝑥=𝑥−1+ln𝜇²,𝑏_𝐹𝑝²;𝑥,𝑦=¹₀()()𝑝𝑑𝑡ln𝑡𝑥+1−𝑡𝑦−𝑡1−𝑡²𝜇², (A.4) respectively. And 𝐴𝑝²=𝑝;𝑥,𝑦²−12𝑥+𝑦4𝜂+1𝑎12^𝐹(𝑥)+𝑎_𝐹𝑦+16𝑝𝑥+𝑦−²2𝑏_𝐹𝑝²+;𝑥,𝑦𝑥−𝑦12𝑝²×𝑎_𝐹(𝑥)−𝑎_𝐹𝑦−𝑥−𝑦𝑏_𝐹𝑝²−1;𝑥,𝑦6𝑝𝑥+𝑦−²3. (A.5) Note that the 𝐴(𝑝²;𝑥,𝑦) and 𝑏₀(𝑝²;𝑥,𝑦) functions are symmetric in 𝑥 and 𝑦.

It is also useful to notice some descents of 𝑏₀ functions: 𝑏^′₀10;𝑥,𝑦=−2𝑥+𝑦𝑥−𝑦²+𝑥𝑦𝑥−𝑦³𝑥ln𝑦,𝑏^′′₀=𝑥0;𝑥,𝑦³+9𝑥²𝑦−9𝑥𝑦²−6𝑥𝑦𝑥+𝑦ln𝑥/𝑦−𝑦³3𝑥−𝑦⁵. (A.6)

Conflict of Interests

The authors do not have a direct financial relation with any commercial identity mentioned in the paper that might lead to a conflict of interests for any of the authors.

Acknowledgment

Sibo Zheng would like to thank Professor Ken-ichi Hikasa for his correspondence. The work is supported in part by National Natural Science Foundation of China with Grant no. 11247031.