On Neutrino Masses in the MSSM with BRpV

One loop corrections to the neutrino mass matrix within the MSSM with Bilinear R Parity Violation are calculated, paying attention to the approach in which an effective $3\times 3$ neutrino mass matrix is used. The full mass matrix is block diagonalized, it is found that second and third order terms can be numerically important, and this is analytically understood. Top-stop loops do not contribute to the effective $3\times 3$ at first order, nevertheless they contribute at third. An improved $3\times 3$ approach that include these effects is proposed.

If neutrinos are massive Majorana particles, lepton number violating terms must be present. In the Minimal Supersymmetric Standard Model (MSSM) [16] with Bilinear R-Parity Violation (BRpV) [17], R-parity is broken via lepton number violation, introducing a bilinear term at the superpotential level [18][19][20][21]. Therefore, neutrino masses and mixing angles are generated via a low-energy see-saw mechanism, mixing neutrino flavor-eigenstates and neutralinos. Although this solution is appealing to explain neutrino masses and mixing angles, signals for supersymmetry at the LHC have not been seen [22]. Since the majority of the searches are based on supersymmetry with bilinear R-parity conserved, there is an open window for it.
In the MSSM with R-Parity violation, one neutrino mass is generated at tree-level, while the other two neutrinos remain massless. To reconcile theoretical predictions with the experimental data requires going beyond the tree-level approximation [23]. Several authors have shown the dependence of the neutrino masses in terms of the parameter which bilinearly violate R-parity, and also how to determine these from collider physics [24]. Improvements in the precision measurement of the neutrino parameters [25], as it will be discussed, suggest to go beyond one loop order in the calculation of the neutrino masses.
The most convenient way to numerically introduce one loop corrections to neutrino masses in this model is through the 7 × 7 mass matrix, which includes 4 neutralinos and 3 neutrinos. If this mass matrix is block diagonalized, an effective 3 × 3 neutrino mass matrix is generated, and it is very convenient when an algebraical understanding is sought. Nevertheless, the 3 × 3 approach can miss important numerical effects. This motivates a more careful treatment of the block diagonalization, leading to an improved 3 × 3 approach.
The paper is organized as follow: In section 2, introductory remarks about neutrino mass generation in BRpV are provided. Section 3 shows how loop corrections are treated in this article. Section 4 develops algebraic approximations that explain the numerical effects. Finally, conclusions about the findings are provided.

Neutrino Masses in Bilinear R-Parity Violation
Models with BRpV include a bilinear term in the superpotential that violates simultaneously R-Parity and lepton number. The superpotential has the following form, where in W Y uk one has the usual R-Parity conserving Yukawa terms. Here the explicit bilinear terms are shown, with µ the higgsino mass and ǫ i the BRpV mass parameters. In this work trilinear R-Parity violating terms are not considered, motivated by models that generate BRpV and not TRpV [26]. The terms shown in eq. (2.1) induce a mixing between neutralinos and neutrinos, forming a set of seven neutral fermions F 0 i . The corresponding tree level mass terms can be written by a 7 × 7 mass matrix as follows, where F 0 k are the mentioned neutral fermions and S 0 ℓ are scalars formed by the mixing between Higgs bosons and sneutrinos [20]. These contributions can be calculated approximately in the block-diagonalized basis, obtaining a generalization to the neutrino mass matrix in eq. (2.6), which is customary to write as, where the parameter A receives tree-level contributions given in eq. (2.6), while the parameters B and C are loop generated. It is also worth mentioning that the parameter C is scale invariant, while B is not. As mentioned, the neutrino/neutralino tree-level mass matrix is completely diagonalized. This is done by applying an extra rotation to the one shown in eq. (2.4). This is, The matrix N ν diagonalizes the effective tree-level neutrino mass matrix given in eq. (2.6) [21], and the N matrix diagonalizes the 4 × 4 neutralino mass matrix. The net effect is to have, It is at this point that quantum corrections are included, where δM χ are one-loop corrections within the neutralino 4×4 sub-matrix, δM ν the one-loop corrections to the 3 × 3 neutrino sub-matrix, and δm refers to the one-loop corrections to the neutralino/neutrino mixing sector. The above matrix can be block-diagonalized again, obtaining the following result, where there have been defined, Notice that the last two terms in equation (2.12) are of second and third order in our blockdiagonalization expansion, and thus they are susceptible to be neglected. Nevertheless, since the neutrino masses are several orders of magnitude smaller than the neutralino masses, the two terms are numerically important.

High Order Effects on Neutrino Masses
In order to show these effects, one-loop corrected neutrino masses in a specific supersymmetric scenario are calculated. A few of the parameters that define this benchmark are shown in Table 1, where the given scalar masses correspond to the third generation. In addition, in Table 2 are shown the masses of a few relevant particles. This scenario was generated using the code SUSPECT [28] for the R-Parity conserving part. In particular, the Higgs boson mass is 126 GeV, as measured by experiments [29]. In addition, SUSPECT allows the calculation for: (i) the deviation from unity of the ρ parameter ∆ρ = 7.7 × 10 −6 [30,31], (ii) the anomalous magnetic moment of the muon ∆a µ = 5.7 × 10 −11 [30,32], and (iii) the branching ratio for the radiative decay of a bottom quark B(b → sγ) = 3.3 × 10 −4 [33]. The BRpV part is handled by our own code. Since BRpV parameters are much smaller Parameter Value Units  than the supersymmetric scale represented by the Higgsino mass parameter µ, the extra contributions to the above loop quantities from BRpV are negligible. The selected BRpV parameters are given in Table 3. Note that the values for ǫ i are large enough to make the radiative corrections to neutrino masses very important. The experimental values for the neutrino parameters are given in Table 4.
First of all, a study on how important are the different loops in the determination of the neutrino parameters has been performed. In Fig. 1 one works in the plane formed by the atmospheric ∆m 2 23 and the solar ∆m 2 12 neutrino mass parameters. In vertical and horizontal dashed lines the 3σ experimental limits for these parameters are shown. At approximately the center of this allowed region one has the predictions from our scenario using the full 7 × 7 mass matrix, represented by a dark (black) diamond. Flowing from this point one has several arrows ending in circles (red), one for each loop. What it is done here is to omit in every entry of the 7 × 7 mass matrix the contribution from the corresponding loop, and show the prediction for the mass differences in these conditions.
The contributions from the bottom-sbottom, neutralino-neutral scalar, and charginocharged scalar loops are large as expected ( Fig. 1-top). The not-so-known effect is the importance of the top-stop loops, which are large enough to move the prediction outside the 3-σ region when it is not included (Fig. 1-bottom). The reason for the unexpectedness of this result is that these loops do not contribute to the neutrino masses in the 3×3 approach, which is very popular. The contribution by these loops appears through the last term in eq. (2.12), (c)  which is of third order. As explain in the next section, this contribution is proportional to the top quark Yukawa coupling and needs the presence of the bottom-sbottom loops as well. One may also see that in this particular scenario, the 3 × 3 approximation does not work since it gives a prediction for the solar and atmospheric mass squared parameters which are off by several orders of magnitude, represented by a cross (blue).  Second of all, in Fig. 2 a similar process is performed. This time a specific loop in a given entry in the 7×7 mass matrix is omitted. For the arrows ending in a square (magenta), one is omitting all the loops at each (3,3), (3,4) and (4, 4) matrix elements. For the arrows ending in a circle (red), one is omitting the up-sup loops for the same matrix elements. Finally, for the arrows ending in a triangle (blue), one is omitting all the loops except up-sup, also for the same matrix elements. The lesson draw from the figure is that the importance of the top-stop loops lies in the higgsino section of the mass matrix. This is clear since the corrections in that sector are proportional to the top quark Yukawa coupling.
When it is convenient to work with a 3 × 3 neutrino mass matrix, the second and third order terms in eq. (2.12) should be included, because they are numerically important. Once that is done, the precision obtained with the 7 × 7 approach is recovered. In Table 5 the prediction for the neutrino observables in the same scenario introduced before is shown. In the second and third column the usual 7 × 7 and 3 × 3 approaches are shown. In the last column the extra terms in eq. (2.12), calling the approach as 3 × 3 f ull , is included. It is clear the recovery in precision.
The second order is given by the third term in eq. (2.12). In the chosen scenario, this term is also very important. That can be understood from Fig. 1-top and Fig. 2. In Fig. 1top the effect of the first order is seen by the cross (blue). The fact that this prediction is so small is an indication that this first order effect is also small. On the other hand, the effect  of the third order seen in Fig. 2, although large when compared to experimental errors, is small compared to full expansion (first plus second plus third order), therefore, the second order is very important.

Algebraic Approximations
Here, approximated algebraic expressions for second and third order terms from the topstop contribution to the solar mass are found, in order to better understand the numeric results shown in the previous section. These numerical calculations show that top-stop loops contribute importantly. The contribution from top-stop loops to the second order term in eq. (2.12) is studied. In the higgsino sector the relevant matrix elements of the inverse neutralino mass matrix, following the Appendix B is, (4.14) Therefore, and it does not contribute to the solar mass, since it is proportional to Λ i Λ j . In fact, since the top-stop coupling to neutrinos does not include ǫ terms, none of the quantities δm tt ij will produce a contribution to the solar mass. Thus, third order term is studied next.
The third order term in eq. (2.12), given by is written in the basis where the tree-level neutralino mass matrix has already been diagonalized. If work is to be done in the original basis instead, the term to analyze is, where δm (and δMχ) in eq. (4.16) is written in the diagonal basis, while δm (and δMχ) in eq. (4.17) is written in the original basis. The same notation is used for both out of simplicity.
In order to algebraically understand the issues mentioned in the previous section a few approximations are performed. First, notice that down-type quarks contribute to δm with a term proportional to ǫ i , while up-type quarks do not, as can be seen from the Appendix A. Thus, in this approximation, Second, notice that the (4, 4) matrix element in the neutralino sector makes a strong numerical effect on the neutrino parameters, and up-type quarks contribute to it. To isolate this effect it is assumed, (δM χ ) ij = δM χ,44 δ i4 δ j4 . With this, the contribution from top-stop loops to the third order term in eq. (2.12) is, Approximating further the ǫǫ term is, which gives the same order of magnitude of the solar mass squared difference, thus it should not be neglected.

Conclusions
It was shown that the 3×3 approach in the calculation of neutrino masses in the MSSM with BRpV, in the light of the present accuracy of the experimental results, sometimes does not give an acceptable answer. This was understood by studying the 3×3 second and third order terms in the block diagonalization of the 7 × 7 mass matrix. When it is convenient to work with 3 × 3 matrices, it was shown also that keeping these terms gives a very similar result compared to the ones extracted from the 7 × 7 neutrino mass matrix. In addition, in the 3 × 3 approach, the top-stop loops do not contribute, nevertheless, they can be numerically important. These loops contribute through the already mentioned third order term, and it was shown that the contribution is dependent on the bottom as well as the top quark Yukawa couplings. The second order term in eq. (2.12) can also be very important. In fact, a scenario was chosen where it is crucial. All these issues motivate a two-loop calculation of neutrino masses in this model.

A.1 Top-stop loops in δM χ
It is numerically observed that among the 16 matrix elements of δM χ , the (4, 4) is the one that gives the largest contribution. In addition, the top-stop loops have an important effect on this matrix element. In order to algebraically understand the phenomena, this contribution is calculated. The coupling between neutral fermions and top-stop quarks is, 22) and where h t is the top quark Yukawa coupling, Rt is the (assumed real) 2×2 rotation matrix that diagonalizes the stop quark mass matrix, N is the (assumed real) 7 × 7 rotation matrix that diagonalizes the neutralino sector, and η j is the sign of the corresponding fermion j.
Notice that the complex conjugated N * is kept only for reference, since one assumes it is real. If this coupling is specialized to the case when the neutral fermion is a neutralino one finds, In this case, N is the (real) 4 × 4 rotation matrix that diagonalizes the neutralino mass sub-matrix, and η j is the sign of the j-th neutralino mass. The relevant loop is formed with those couplings,t k t χ 0 (1.24) Here the three dots mean that only the terms proportional to h 2 t are shown. Also, the fact that the matrix N is real was already used.
When evaluating δM ij χ it should be understood that in the basis where the neutralinos are diagonal, one wants to evaluate the neutralino mass at p 2 , and symmetrize over i and j. Thus, (1.25) The contribution to the (4, 4) neutrino/neutralino mass matrix element is therefore, which is an approximation for the top-stop loop contribution to δM 44 χ .

A.2 Bottom-sbottom loops in δM χ
Bottom-sbottom loops contribute importantly to δM χ , and through it, also contribute importantly to the third term in eq. (2.12). Bottom-sbottom loops contribute importantly to δM ν too, but they are not the focus of this study. The neutral fermion coupling to bottom-sbottom quarks is, 27) and where h b is the bottom quark Yukawa coupling, Rb is the (assumed real) 2 × 2 rotation matrix that diagonalizes the sbottom quark mass matrix, N is the already defined (and real) 7 × 7 rotation matrix that diagonalizes the neutralino sector, and η j is the already defined sign of the corresponding fermion j. Specializing this coupling to the case when the neutral fermion is a neutralino, one finds, 28) and where N is the already defined (real) 4 × 4 rotation matrix that diagonalizes the neutralino mass sub-matrix. The bottom-sbottom loops are, 29) and again, only the terms proportional to h 2 b are shown. The contribution to δM ij χ is therefore, (1.30) The contribution to the (4, 4) neutrino/neutralino mass matrix element is therefore, which is an approximation for the bottom-sbottom loop contribution to δM 44 χ .

A.3 Top-stop loops in δm
In δm one has mixing between neutralinos and neutrinos. Therefore, to find the quantum corrections in this region of the mass matrix the neutralino-top-stop coupling in eq. (1.23) is needed. Also, to specialize the general coupling in eq. (1.22) to the neutrino-top-stop coupling is needed. One finds, In the last equalities, the O couplings are defined as, 33) and the ξ ij and ξ i parameters are defined in Appendix C. The loops contributing to δm are, (1.34) where only terms proportional to the Yukawa coupling squared are kept. The above leads to the following contribution to δm, From this result one learns that the second order term in eq. (2.12) will never generate a solar neutrino mass from top-stop loops. This last conclusion arises because there is no term proportional to ǫ i in eq. (1.35).

A.4 Bottom-sbottom loops in δm
As it was mentioned before, in δm one has mixing between neutralinos and neutrinos. The contribution from bottom-sbottom loops to this quantity starts with the neutral fermion coupling to bottom-sbottom quarks, which is given in eq. (1.27). Specializing that coupling to the case when the neutral fermion is a neutrino one finds, In the last equalities, the O couplings are defined as, In addition, the neutralino coupling to bottom-sbottom, given in eq. (1.28), is needed. The bottom-sbottom loops contributing to δm are therefore, (1.38) where again only terms proportional to the Yukawa coupling squared are kept. The above leads to the following contribution to δm, From this result one learns that the second order term in eq. (2.12) can generate a solar neutrino mass from bottom-sbottom loops, because of the term proportional to ǫ i in eq. (1.39). But that fact is known. More importantly, one learns that the top-stop loops can contribute to the solar mass through the third order term in eq. (2.12), in combination with the bottom-sbottom loops.

B Inverse Neutralino Mass Matrix
For the reader's benefit, the inverse of the tree-level neutralino mass matrix is given. Its matrix elements are equal to, with the following expressions for each sub-matrix, and I hg = (I gh ) T .