Stop Decay with LSP Gravitino in the final state: $\tilde{t}_1\to\widetilde{G}\,W\,b$

In MSSM scenarios where the gravitino is the lightest supersymmetric particle (LSP), and therefore a viable dark matter candidate, the stop $\tilde{t}_1$ could be the next-to-lightest superpartner (NLSP). For a mass spectrum satisfying: $m_{\widetilde{G}}+m_t>m_{\tilde{t}_1}>m_{\widetilde{G}}+m_b+m_W$, the stop decay is dominated by the 3-body mode $\tilde{t}_1\rightarrow b\,W\,\tilde{G}$. We calculate the stop life-time, including the full contributions from top, sbottom and chargino as intermediate states. We also evaluate the stop lifetime for the case when the gravitino can be approximated by the goldstino state. Our analytical results are conveniently expressed using an expansion in terms of the intermediate state mass, which helps to identify the massless limit. In the region of low gravitino mass ($m_{\widetilde{G}}\ll m_{\tilde{t}_1}$) the results obtained using the gravitino and goldstino cases turns out to be similar, as expected. However for higher gravitino masses $m_{\widetilde{G}} \lesssim m_{\tilde{t}_1}$ the results for the lifetime could show a difference of O(100)\%.


Introduction 2 The Stop Lifetime within the MSSM
We start by giving some relevant formulae for the input parameters that appear in the Feynman rules of the gravitino within the MSSM. The (2x2) stop mass matrix can be written as: where the entries take the form: The corresponding mass eigenvalues are given by: and where The mixing angle θt appears in the mixing matrix that relate the weak basis (t L ,t R ) and the mass eigenstates (t 1 ,t 2 ), and it is given . From these expressions it is clear that in order to obtain a very light stop one needs to have a very large value for the trilinear soft supersymmetry-breaking parameter [25,30]. It turns out that such scenario helps to obtain a Higgs mass value in agreement with the mass measured at LHC (125-126 GeV) in a consistent way within the MSSM. Following Ref. [31], we derived the expressions for all the relevant interactions vertices that appear in the amplitudes for the decay width (t 1 → G W b), whose Feynman graphs are shown in Figures [1][2][3]. We shall need the following vertices: − m W γ ν γ µ (V i2 sin βP R + U i2 cos βP L ) , wheret 1 denotes the lightest stop, while t is the top quark and G denotes the gravitino. With b we denote the bottom quark, while W is the gauge boson, χ + i denotes the chargino andb i is de sbottom. With P R and P L corresponding to the left and right projectors, a i b i , S i , P i are defined in Appendices A, B, as well the mixing matrices V 1i , U 1i that diagonalize the chargino factor.
For the case when the gravitino approximates to the goldstino state, the interaction vertices that will appear in the amplitudes for the decay width (t 1 → G W b) are the following: whereas the vertices V 2 (t b W ), V 3 (t 1 Wb i ) and V 5 (t 1 b χ + i ) remain the same as in the gravitino case.

The Amplitude fort
The decay lifetime of the stop was calculated in Ref. [26], where the chargino contribution was approximated by including only the dominant term. Here we shall calculate the full amplitude and determine the importance of the neglected term for the numerical calculation of the stop lifetime. In what follows we need to consider the Feynman diagrams shown in Figures [1,2,3], which contribute to the decay amplitude fort 1 (p) → G(p 1 ) W (k) b(p 2 ), with the momenta assignment shown in parenthesis.

Figure 3: Chargino mediated diagram
The total amplitude is given by: where M t , Mb i , M C χ + i denotes the amplitudes for top, sbottom and chargino mediate diagrams, respectively. In the calculation of Ref. [26], the chargino-mediated diagram included only part of the vertex V 6 (χ + i W G). Here, in order to keep control of the vertex V 6 and therefore M c where ψ a = (t,b j , χ + k ). The functions P ψa (q a ) correspond to the propagators factors, thus for the chargino ψ a = χ + i , we have Similar expressions hold for the sbottom and the top contributions, Pb(q 2 ) and P t (q 1 ) respectively. The terms W ψaψa include the traces involved in each squared amplitudes W 0 For simplicity, we have written the completeness relations for the gravitino field and the vector polarization sum of the boson W as follows: The functions W ψaψa depend on the scalar products of the momenta p, p 1 , p 2 , k, q 1 , q 2 and q 3 . After carefully analyzing the resulting traces (handed with FeynCalc 1 [37,38]) we find that these functions can be written as powers of the intermediate state masses, as follows: Full expressions for each function w iψaψa ∀ i = 1, 2, 3 are included in Appendix A. Furthermore, we also find that the interference terms can be written in a similar form, namely: Again, as in the previous case, the function W ψaψ b include the traces appearing in the interferences, specifically we have It turns out that the functions W ψaψ b can be expressed also in powers of the intermediate masses: The w jψaψ b ∀ j = 1, 2, 3, 4 are as the w iψaψa 4-momentum's scalar products functions completely determined by the kinematics of our decay. We consider that [30] and [38] are an useful way to present our results as well an easy manner to compute complicated and messy traces. Then the decay width can be obtained after integration of the 3-body phase-space The variables x and y are defined as x = 2 Numerical results for the lifetime τ = 1 Γ will be presented and discussed in Section 3.

The Amplitudest 1 → G W b with the goldstino approximation
In this section we shall present the calculation of the stop decay using the gravitino-goldstino high energy equivalence theorem [28]. In the high energy limit (m G ≪ mt 1 ) we could consider the gravitino field (spin 3 2 particle) as the derivative of the goldstino field (spin 1 2 particle). We shall consider in this section the same Feynman diagrams Figures [1,2,3] that we used in Section 2.1, but with the proviso that the gravitino field shall be described by the goldstino fields. Making the replacement Ψ G → i 2 3 1 m G ∂ µ Ψ in the gravitino interaction lagrangian, one obtain the effective interaction lagrangian for the goldstino as is show in [31]. The averaged squared amplitude for the Goldstino is then written as As in the previous Section 2.1, we can build the amplitudes from the inclusion of all the vertices into the expressions from each graph, namely: Where the superindex "G" that appears in the amplitudes [41-43] refers to the goldstino amplitudes. The constants appearing in front of each amplitudes are: . We obtain similar expressions to [??] for the squared amplitudes of the goldstino case, namely: where the function W G ψaψa includes traces corresponding to the goldstino squared amplitudes, which are given as follows: the functions W G ψaψa depend on the scalar products of the momenta p, p 1 , p 2 , k, q 1 , q 2 and q 3 , these functions will also be written as powers of the intermediate state masses, namely: All the full expressions for each function w G iψaψa ∀ i = 1, 2, 3 can be foud in Appendix B. Again, the interferences terms for the goldstino are also written in the form: The functions W ψaψ b correspond to the traces involved in the interference terms, i.e.
The W G ψaψ b functions also expressed as powers of the intermediate masses: The full expressions for w G jψaψ b ∀ j = 1, 2, 3, 4 can be found in the Appendix B.

Numerical Results
The decay width is obtained by integrating the differential decay width over the dimensionless variables x, y which have limits given by 2µ After integrating numerically the expressions for the differential decay width, we obtain the values for the decay width, for a given set of parameters. We consider two values for the stop mass, mt 1 = 200 GeV and mt 1 = 350 GeV , we also fix the chargino mass to be m χ + i = 200, 500 GeV , while the sbottom mass is fixed to be mb i = 300, 500 GeV . In Figures [4,5] we show the lifetime of the stop, as function of the gravitino mass, within the ranges 200-250 GeV for the case with mt 1 = 350 GeV , and 50-100 GeV for mt 1 = 100 GeV . We show the results for the case when one uses the full expression for chargino-gravitino-W vertex (circles), as well as the case when the partial inclusion of such vertex, as it was done in [26] (triangles) and in the limit of the goldstino approximation (squares). We noticed that for low gravitino masses (m G → 0) the full gravitino result becomes almost indistinguishable from the goldstino case, while the partial gravitino result has also similar behavior. For larges gravitino masses (m G ∼ = mt 1 ) the results for the stop lifetime using the full gravitinio and goldstino approximation could be very different, up to O(50%) different.
On the other hand, the values for the stop life-time using the full gravitino and partial gravitino limit are very similar for low gravitino masses, while for the largest allowed masses the difference in results is at most of order O(50%). The value of the lifetime obtained in all theses cases turns out to be of order 10 7 − 10 12 sec, which results in an scenario with large stop lifetime that has very special signatures both at colliders and has also important implications for cosmology, as it was discussed in ref. [26]. For instance, regarding the effect on BBN, the Stopt 1 have to form quasi stable sbaryons (t 1 qq) and mesinos (t 1q ), whose late decays could have affected the light element abundance obtained in BBN, while negatively charged stop sbaryons and mesinos could contribute to lower the Coulomb barrier for nuclear fusion process occurring in the BBN epoch. However, as argued in [26] the great majority of stop antisbaryons would have annihilated with ordinary baryons to make stop antimesinos and most stop mesinos and antimesinos would have annihilated. The only remnant would have been neutral mesinos which would be relatively innocuous, despite their long lifetime because they would not have important bound state effects. Further discussion of BBN issues of Ref. [29] divide the stop lifetime into regions that could have an effect, but the larges ones (which represent our results) do not pose problems for the success of BBN. Then, regarding the effect of late stop decay on the Cosmic Microwave Background (CMB), we have included some comments in the text, to estimate the main effects. The arguments which read as follows: Very long lifetimes (τ > 10 12 s) would have been excluded if one uses the approximate results of Ref. [39], which present bounds on the lifetime τ (for the case when stau is the NLSP) using the constrain in the chemical potential µ < 9 × 10 −5 . However, it was discussed in Ref. [40], that a more precise calculation reduces the excluded region for lifetimes, ending at about τ ∼ 10 11 s − 10 12 s. Thus, the region with very large stop lifetimes could also survive. Specific details that change from the stop decay (3-body) as compared with stau decays (2-body), such as the energy release or stop hadronization, will affect the calculation, but the numerical evaluation of such effect is beyond the scope of our paper.

Conclusions
In this paper we have calculated the stopt 1 lifetime in MSSM scenarios where the massive gravitino is the lightest supersymmetric particle (LSP), and therefore is a viable dark matter candidate. The lightest stopt 1 corresponds to the next-to-lightest supersymmetric particle (NLSP). We have focused on the kinematical domain m G +m t > mt 1 > m G +m b +m W , where the stop decay width is dominated by the modet 1 → b WG.
The amplitiude for the full calculation of the stop 3-body decay width includes contributions from top, sbottom and chargino as intermediate states. We have considered the full chargino-gravitino vertex, which improves the calculation presented in ref. [26]. Besides performing the full calculation with massive gravitino, we have also evaluated the stop decay lifetime for the limit when the gravitino can be approximated by the goldstino state. Our analytical results are conveniently expressed, in both cases, using an expansion in terms of the intermediate state mass, which helps in order to identify the massless limit.
We find that the results obtained with the full chargino vertex are not very different from the approximation used in ref. [26], in fact they only differ approximately in a 50%. The comparison of the full numerical results with the ones obtained for the goldstino approximation, show that in the limit of low gravitino mass (m G ≪ mt 1 ) there is not a significant difference in values of the stop lifetime obatined from each method. However, for m G mt 1 the difference in lifetime could be as high as 50%. Numerical results for the stop lifetime give value of order 10 7 − 10 12 sec, which makes the stop to behave like a quasi-stable state, which leaves special imprints for LHC search. Our calculation shows that the inclusion of the neglected term somehow gives a decrease in the lifetime of the stop. However, it should be pointed out that the region of parameter space correspond to the NUHM model.

A Analytical Expressions for Amplitudes with Gravitino in the final state
In this appendix we present explicitly the full results for the 10 w ψaψa functions that arose from a convenient way to express the large traces that appear in the squared amplitudes [21], as well as the 18 w ψaψ b functions in the interferences [31] of the 3-body stopt 1 decay with gravitino in the final state. First, we shall present the contributions for the squared amplitudes, then we shall present the interferences.

A.1 Top Contribution
For the averaged squared amplitude of the top quark contribution, the functions w 1tt , w 2tt and w 3tt are: The functions f 1 , f 2 and f 3 are functions of the variables x and y that were defined previously in Section 3, they are f 1 = We have also used in [56-58] the following substitutions t and a 3 = (At + Bt) 2 , with At = cos θt + sin θt and Bt = cos θt − sin θt.

A.2 Sbottom Contribution
For the averaged squared amplitude of the squark sbottom contribution, the function w 1b ibi is: With We have done in the amplitude [??] the following substitution a and with a i = (sin θb, cos θb), b i = (cos θb, − sin θb) and κ i = (cos θt cos θb, − cos θt cos θb).

A.3 Partial Chargino Contribution
For the averaged squared amplitude of the chargino contribution, the functions w 0 are as follows w 0 with h 4 = 2m 2 W m 2G + f 2 3 and h 5 = m 2 G + 2f 3 + m 2 W , we have also used the following substitutions For the low-to-moderate range of tan β we have: where cos φ L , ± sin φ L are elements of the matrix V that diagonalizes the chargino mass matrix, expressions for S 2 and P 2 may be obtained by replacing cos φ L → − sin φ L and sin φ L → cos φ L in where we have defined With h 6 = 3m 2 W m 2G + 2f 2 3 , we have used the substitution V i1 P R − U i1 P L = T i + Q i γ 5 in the first term of the interaction vertex V 6 (χ + i WG), we have also done the following substitutions in the functions [65-67]:

A.5 Interference Terms
functions ∀ k = 1, 2, 3, 4, are: In order to have control in the calculations with huge expressions, we have done the following substitutions in the functions [71-73]: For the interference term M 0 † χ + i Mb i , the functions w jχ + ib i ∀ j = 1, 2 are: In the functions [78,79], we have done the following substitutions:

Interference
For the interference term M † t M 0 χ + i the functions w jtχ + i ∀ j = 1, 2, 3, 4 are: With h 8 = 2f 3 − m 2 W and h 9 = 3m 2 W m 2G + f 2 3 . We have done the following substitutions in the functions [83][84][85][86]: t Mb i , the functions w jtb i ∀ j = 1, 2 are: For the interference term M † χ + i Mb i , the functions w jχ + ib i ∀ j = 1, 2 are: We have done the following substitutions in the functions [89,90], We have done the following substitutions in the functions [92

B Analytical expressions for the amplitudes for the Goldstino approximation
In this appendix we present explicitly the full results for the 7 w G ψaψa functions that arose from the squared amplitudes [44], as well as the 8 w G ψaψ b functions that appear in the interference terms [49] of the 3-body stopt 1 decay with goldstino in the final state. First, we shall present the contribution for the squared amplitudes, then we shall present the interferences. We shall shown that the w G ψaψa and w G ψaψ b functions are very compacts expressions, opposed to the resulting functions in the gravitino case that we have presented in Appendix A. The approximation of the gravitino field by the derivative of the goldstino field is good in the high energy limit (m G ≪ mt 1 ), in the sense that in this limit they behave similar and also in the simplification of the computations.

B.1 Top Contribution
For the averaged squared amplitude of the top quark contribution, the resulting functions w jtt ∀ j = 1, 2, 3 are: With a 1 , a 2 and a 3 defined previously in Appendix A.

B.2 Sbottom Contribution
For the averaged squared amplitude of the sbottom squark contribution, with the w 1b ibi function as: with D ij1 defined previously in Appendix A.

B.3 Chargino Contribution
For the averaged squared amplitude of the chargino contribution, the resulting functions w jχ + i χ + i ∀ j = 1, 2, 3 are: where P ij1 and P ij2 are defined above in Appendix A.

B.4 Interference Terms
, the functions w jtb i ∀ j = 1, 2 are: Where ∆ i1 and ∆ i2 are defined above in Appendix A.
with C ij1 and C ij2 defined above in Appendix A.

Interference
For the interference term M G † t M G χ + i , the functions w G jtχ + i ∀ j = 1, 2, 3, 4 are: with R i1 , R i2 , R i3 and R i4 defined above in Appendix A.