An explicit calculation of pseudo-goldstino mass at the leading three-loop level

Pseudo-goldstinos appear in the scenario of multi-sector SUSY breaking. Unlike the true goldstino which is massless and absorbed by the gravitino, pseudo-goldstinos could obtain mass from radiative effects. In this note, working in the scenario of two-sector SUSY breaking with gauge mediation, we explicitly calculate the pseudo-goldstino mass at the leading three-loop level and provide the analytical results. In our calculation we consider the general case of messenger masses (not necessarily equal) and include the higher order terms of SUSY breaking scales. Our results can reproduce the numerical value estimated previously at the leading order of SUSY breaking scales with the assumption of equal messenger masses. It turns out that the results are very sensitive to the ratio of messenger masses, while the higher order terms of SUSY breaking scales are rather small in magnitude. Depending on the ratio of messenger masses, the pseudo-goldstino mass can be as low as ${{\cal O} (0.1)}$ GeV.


I. INTRODUCTION
Albeit lack of direct evidence at the LHC, supersymmetry (SUSY) as a generalization of space-time symmetry in quantum field theory remains one of the most popular extensions of the Standard Model of particle physics. The SUSY breaking and mediation mechanism plays a key role in the phenomenology of SUSY. If the spontaneous SUSY breaking is generated independently in different hidden sectors, then each sector provides a massless goldstino η i with SUSY breaking scale F i . One linear combination of η i is identified as the true massless goldstino that is absorbed by gravitino, other orthogonal combinations named pseudo-goldstinos could acquire masses from radiative corrections. However, unconstrained by the supercurrent, the interactions between pseudo-goldstinos and ordinary superparticles could be sizable enough to induce unconventional signatures in collider physics and cosmology [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. For example, these physical pseudo-goldstinos with loop induced masses could be lighter than the lightest neutralino so that the neutralino could decay to a pseudogoldstino plus a gauge boson or Higgs boson, leading to intriguing phenomenology at the LHC and future lepton colliders [15][16][17]. On the other hand, since the thermally produced bino-like neutralinos may decay readily to pseudo-goldstinos (the pseudo-goldstinos finally decay to gravitino dark matter), the stringent constraints on SUSY from dark matter detection and relic density can be relaxed (note that most GUT-constrained SUSY models like CMSSM/mSUGRA suffer from such a tension between dark matter constraints and muon g-2 explanation if the dark matter is the lightest bino-like neutralino [19]).
Obviously, the mass of pseudo-goldstino is a crucial parameter for all the phenomenological analysis in the multi-sector SUSY breaking scenario. Due to the intrinsic property of supergravity, there is a universal mass which is twice the gravitino mass m 3/2 at tree level. Another tree-level contribution coming from electroweak symmetry breaking in the low energy part is negligible. The possible loop corrections to the mass of pseudo-goldstino have been discussed in Ref. [1]. If SUSY breaking sectors only communicate via gauge interactions which are common from the viewpoint of model building, it is argued that the leading order contribution to the pseudo-goldstino mass arises at the three-loop level and it could be described through the two-point correlation functions defined in General Gauge Mediation [2]. Also a numerical value of the pseudo-goldstino mass was estimated from the evaluation of these functions assuming the SUSY breaking scales M i in the two sectors are equal [1]. Note that the analysis in Ref. [1] is based on arguments and approximation, while an independent cross-check or an explicit calculation of the three-loop contributions with different messenger masses is still missing to date. Thus in this note we perform an explicit three-loop calculation for the mass corrections to the pseudo-goldstino, which is also applicable to the minimal gauge mediation. Although the realistic SUSY breaking and gauge mediation models might not be as simple as the minimal one mentioned above, the physics behind them for the mass corrections at the leading three-loop level is almost the same.
So our result could be easily generalized to other cases or used to make model-dependent estimations.
This work is organized as follows. In Section II we will make a brief review on the framework with pseudo-goldstinos and mention some technical details used for our calculation.
Section III contains the analytical results and related discussions. Finally, we conclude in Section IV.

II. TECHNICAL DETAILS
The superpotential with multi-sector SUSY breaking in the minimal gauge mediation is given by with goldstino η i contained in the θ component of chiral spurion superfield X i , whose auxil- whose two eigenvalues are |M i | 2 ± F i . In order to avoid tachyonic scalar masses and gauge symmetry breaking, usually it is assumed that F i ≪ |M i | 2 .
Next, let us turn to the goldstino part in Eq. 1. In the two hidden sector scenario we define F = F 2 1 + F 2 2 and tan θ = F 2 /F 1 , then the combination G = η 1 cos θ + η 2 sin θ is eaten by the super-Higgs mechanism, while one pseudo-goldstino G ′ = −η 1 sin θ + η 2 cos θ is left. Due to the fact that G has to be massless before the gravitational effects are taken into account, the radiatively generated mass matrix for goldstinos η 1,2 has to be of the following where M 12 denotes the loop induced mass mixing between the two goldstinos η 1,2 , i.e., − 1 2 η 1 M 12 η 2 . Now it is easy to get the Majorana mass of the pseudo-goldstino The typical non-vanishing Feynman diagrams contributing to M 12 which start at three loops are shown in Fig. 1 [20][21][22][23][24][25][26][27][28][29]. In the case of hierarchical momenta and masses, asymptotic expansions (see, e.g., Refs. [30,31]) could be applied to evaluate Feynman integrals. As mentioned above, the scalar mass of the messengers will be split to |M i | 2 ± F i after SUSY breaking. So there are four different mass scales in the first four diagrams and six scales in the last one of Fig. 1. Although these multi-scale integrals can not be evaluated directly, we can perform Taylor expansion on the scalar propagators to reduce the number of scales appearing in the integral This trick has been used by one of us in calculating the next-to-next-to-leading order hadronic correction to the muon anomalous magnetic momentum [32]. We also checked that in this way the same expanded results for the one-loop gaugino mass corrections would be reproduced as in Ref. [33]. Thus, at the cost of increasing the number of integrals, we are left with two-scale vacuum integrals which could be solved analytically.
In our calculation the algebraic part is done with the help of FORM [34]. FIRE [35] which implements the Laporta algorithm [36] is used to reduce all the scalar integrals into a basis of master integrals (MI). The analytical results for these MIs which require the introduction of harmonic polylogarithms (HPLs) H(a 1 , ..., a k ; x) are already known [23]. For simplicity, we show the simplest HPLs and more details on their properties could be found in Refs. [37][38][39]:

III. RESULTS AND DISCUSSIONS
From Eq. 4 we learn that the mass of pseudo-goldstino could be easily derived from the radiative corrections to the matrix element M 12 . In the following we choose to present M 12 by grouping terms according to the power of two independent SUSY breaking scales F 1,2 : where C N = (N 2 −1)/4 is the color factor for SU(N) gauge group and g is the corresponding From above equations, it is easy to see that there are no contributions with even powers of F i . The reason is rather simple: perturbative expansion of F i in the Feynman diagrams plays the role of converting the scalar components of the superfields from one to another, and even times of these transformations lead to SUSY-preserving interactions which should vanish due to the symmetry constraint. Same behaviors have also been found for the oneloop gaugino mass corrections in the model of minimal gauge mediation [33], where even powers of F i also do not contribute.   Next, another important check besides UV-finiteness 1 on the correctness of our result will be explained. When x is fixed to be 1, the result for the related functions expressed in terms of complicated HPLs could be greatly simplified. In this case, one will get the expressions listed in Table I  order to get an estimation of the numerical effects of the higher order terms expanded in F i , two plots are shown in Fig. 2 Note that comparing to the usual low energy soft SUSY breaking parameter m sof t , the pseudo-goldstino mass is suppressed by a factor g 2 /(16π 2 ) 2 when both the SUSY breaking and messenger scales are comparable. Considering the summation effects over different gauge sectors, the authors in Ref. [1] concluded that the mass of pseudo-goldstino is around the GeV scale in this case. Thus, it is rather simple to see from Table II that  Before the end of this section, let us make a direct comparison with the result obtained in [1]. The authors considered the scenario of equal messenger scales and then performed numerical evaluation. The approximated value 4.21 in Appendix B of Ref. [1] agrees well with our analytical result at O(F i ) which corresponding to 7 2 ζ 3 , when the same convention as theirs is used.

IV. CONCLUSION
We calculated the three-loop contribution to the mass of pseudo-goldstino explicitly.
Since the exact evaluation of a basis of MIs for general vacuum integrals at the three-loop level is still unavailable, the implicit conditions F i ≪ M 2 i are used to reduce the number of scales we have to deal with. After performing expansions in F i , we provided analytical results which are in good agreement with the numerical value in the literature. It has been proved that the mass correction is dominated by the leading order contribution at O(F i ). We also found that the mass correction could be greatly reduced if there is a hierarchy between different messenger scales. Note that although the whole calculation is based on the twosector SUSY-breaking scenario, our results could also be generalized to multi-sector cases as long as these sectors communicate with each other via gauge interactions. So, our result presented in this note could be useful for phenomenological analysis or model-buildings on pseudo-goldstino in the future.