Stop Decay with LSP Gravitino in the final state: t ~ 1 → G ˜ Wb
aa r X i v : . [ h e p - ph ] F e b Stop Decay with LSP Gravitino in the final state: ˜ t → e G W b
J.Lorenzo D´ıaz-Cruz ∗ and Bryan O. Larios † Facultad de Ciencias F´ısico - Matem´aticas, BUAPApdo. Postal 1364, C.P. 72000, Puebla, Pue. M´exico
September 18, 2018
Abstract
In MSSM scenarios where the gravitino is the lightest supersymmetric particle (LSP), andtherefore a viable dark matter candidate, the stop ˜ t could be the next-to-lightest superpartner(NLSP). For a mass spectrum satisfying: m e G + m t > m ˜ t > m e G + m b + m W , the stop decay isdominated by the 3-body mode ˜ t → b W ˜ G . We calculate the stop life-time, including the fullcontributions from top, sbottom and chargino as intermediate states. We also evaluate the stoplifetime for the case when the gravitino can be approximated by the goldstino state. Our analyticalresults are conveniently expressed using an expansion in terms of the intermediate state mass, whichhelps to identify the massless limit. In the region of low gravitino mass ( m e G ≪ m ˜ t ) the resultsobtained using the gravitino and goldstino cases turns out to be similar, as expected. However forhigher gravitino masses m e G . m ˜ t the results for the lifetime could show a difference of O(100)%. ∗ [email protected] † [email protected] Introduction
The properties of Supersymmetric theories, both in the ultraviolet or the infrared domain have hada great impact in distinct domains of particle physics, including model building, phenomenology,cosmology and formal quantum field theory [1]. In particular, Supersymmetric extensions of theStandard Model can include a discrete symmetry, R parity, that guarantees the stability of the lightestsupersymmetric particle (LSP) [2], which allows the LSP to be a good candidate for dark matter (DM).Candidates for the LSP in the minimal supersymmetric extension of the Standard Model (MSSM)include sneutrinos, the lightest neutralino χ and the gravitino e G . Most studies has focused on theneutralino LSP [3], while scenarios with the sneutrino LSP seem more constrained [4].Scenarios with gravitino LSP as DM candidate have also been considered [5, 7, 6]. In such scenarios,the nature of the next-to-lightest supersymmetric particle (NLSP) determines its phenomenology [8, 9].Possible candidates for NLSP include the lightest neutralino [10, 11], the chargino [12], the lightestcharged slepton [13], or the sneutrino [14, 15, 16, 17]. The NLSP could have a long lifetime, due tothe weakness of the gravitational interactions, and this leads to scenarios with a metastable chargedsparticle that could have dramatic signatures at colliders [18, 19] and it could also affect the Big Bangnucleosynthesis (BBN) [20, 21, 22].Squark species could also be the NLSP, and in such case natural candidates for NLSP could bethe sbottom [23, 24, 25] or the lightest stop ˜ t . There are several experimental and cosmologicalconstraints for the scenarios with a gravitino LSP and a stop NLSP that were discussed in [26]. Itturns out that the lifetime of the stop ˜ t could be (very) long, in which case the relevant colliderlimits are those on (apparently) stable charged particles. For instance the limits available from theTevatron collider imply that m ˜ t >
220 GeV [27] . Thus, knowing in a precise way the stop lifetimeis one of the most important issues in this scenario, and this is precisely the goal of our work. Inthis paper we present a detailed calculation of the stop lifetime, for the kinematical region where the3-body mode ˜ t → e G W b dominates . Besides calculating the amplitude using the full wave functionfor the gravitino, we have also calculated the 3-body decay width (and lifetime) using the gravitino-goldstino equivalence theorem [28]. It should be mentioned that this scenario is not viable withinthe Constrained Minimal Supersymmetric Standard Model (CMSSM). However there are regions ofparameter space within the Non-Universal Higgs Masses model (NUHM) that pass all collider andcosmological constraints (relic density, nucleosynthesis, CMB mainly) [29].The organization of our paper goes as follows, we begin Section 2 by giving some formulae for thestop mass. In Section 2.1 we compute the squared amplitudes for the stop decay with gravitino inthe final state (˜ t → e G W b ) including the chargino, sbotom and top mediated states. After carefullyanalyzing the results for the squared amplitude, we have identified a convenient expansion in termsof powers of the intermediate particle mass, which only needs terms of order O ( m i ) , O ( m i m j ). It isour hope that such expansion could help in order to relate the calculation of the massive and masslesscases. In future work we plan to reevaluate this decay using the helicity formalism suited for thespin- case. In Section 2.2 we compute the squared amplitudes for the stop decay considering thegravitino-goldstino high energy equivalence theorem that allow us approximate the gravitino as thederivative of the goldstino. We present in Section 3 our numerical results, showing some plots wherewe reproduce the stop lifetime for the approximate amplitude considered in [26], and compare it withour complete calculation, we also compare these results with goldstino approximation. Conclusionsare included in Section 4, finally all the analytic full results for the squared amplitudes are left inAppendices A,B. The LHC will probably be sensitive to a metastable ˜ t that is an order of magnitude heavier. Our calculation of stop lifetime improves the one presented in [26] where an approximation was used for the chargino-mediated contribution that neglected a subdominant term in the expression for the vertex χ + i e G W . The Stop Lifetime within the MSSM
We start by giving some relevant formulae for the input parameters that appear in the Feynman rulesof the gravitino within the MSSM. The (2x2) stop mass matrix can be written as: f M t = (cid:18) M LL M LR M † LR M RR (cid:19) , (1)where the entries take the form: M LL = M L + m t + 16 cos 2 β (4 m W − m Z ) ,M RR = M R + m t + 23 cos 2 β sin θ W m Z , (2) M LR = − m t ( A t + µ cot β ) ≡ − m t X t . The corresponding mass eigenvalues are given by: m t = m t + 12 ( M L + M R ) + 14 m Z cos 2 β − ∆2 , (3)and m t = m t + 12 ( M L + M R ) + 14 m Z cos 2 β + ∆2 , (4)where ∆ = (cid:0) M L − M R + cos 2 β (8 m W − m Z ) (cid:1) + 4 m t | A t + µ cot β | . The mixing angle θ ˜ t appearsin the mixing matrix that relate the weak basis (˜ t L , ˜ t R ) and the mass eigenstates (˜ t , ˜ t ), and it is givenby tan θ ˜ t = ( m t − M LL ) | M LR | . From these expressions it is clear that in order to obtain a very light stop oneneeds to have a very large value for the trilinear soft supersymmetry-breaking parameter [25, 30]. Itturns out that such scenario helps to obtain a Higgs mass value in agreement with the mass measuredat 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 appearin the amplitudes for the decay width (˜ t → e G W b ), whose Feynman graphs are shown in Figures [1-3].We shall need the following vertices: V (˜ t t e G ) = − √ M ( γ ν γ µ p ν )(cos θ ˜ t P R + sin θ ˜ t P L ) , (5) V ( t b W ) = ig √ γ ρ P L , (6) V (˜ t W ˜ b i ) = − ig κ i √ p + q ) µ , (7) V (˜ b i b e G ) = − √ M ( γ ν γ µ q ν )( a i P R + b i P L ) , (8) V (˜ t b χ + i ) = − i ( S i + P i γ ) , (9) V ( χ + i W e G ) = − √ M (cid:16) − /pγ ρ γ µ ( V i P R − U i P L ) (10) − m W γ ν γ µ ( V i sin βP R + U i cos βP L ) (cid:17) , where ˜ t denotes the lightest stop, while t is the top quark and e G denotes the gravitino. With b wedenote the bottom quark, while W is the gauge boson, χ + i denotes the chargino and ˜ b 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 AppendicesA, B, as well the mixing matrices V i , U i that diagonalize the chargino factor.3or the case when the gravitino approximates to the goldstino state, the interaction vertices thatwill appear in the amplitudes for the decay width (˜ t → G W b ) are the following: e V (˜ t t G ) = m t − m t √ M m e G ! (cos θ ˜ t P R + sin θ ˜ t P L ) , (11) e V (˜ b i b G ) = m b − m b i √ M m e G ! ( a i P R + b i P L ) , (12) e V ( χ + i W G ) = − m χ + i √ M m e G [ /pγ ρ ( V i P R − U i P L )] , (13)whereas the vertices V ( t b W ) , V (˜ t W ˜ b i ) and V (˜ t b χ + i ) remain the same as in the gravitino case. ˜ t → e G W b
The decay lifetime of the stop was calculated in Ref.[26], where the chargino contribution was approx-imated by including only the dominant term. Here we shall calculate the full amplitude and determinethe importance of the neglected term for the numerical calculation of the stop lifetime. In what followswe need to consider the Feynman diagrams shown in Figures [1,2,3], which contribute to the decayamplitude for ˜ t ( p ) → e G ( p ) W ( k ) b ( p ), with the momenta assignment shown in parenthesis.˜ t t b W Ψ µ V V Figure 1: Top mediated dia-gram ˜ t ˜ b i V Ψ µ bWV Figure 2: Sbottom mediateddiagram˜ t χ + i V Ψ µ WbV Figure 3: Chargino mediated diagramThe total amplitude is given by: M = M t + M ˜ b i + M Cχ + i , (14)where M t , M ˜ b i , M Cχ + i denotes the amplitudes for top, sbottom and chargino mediate diagrams, re-spectively. In the calculation of Ref. [26], the chargino-mediated diagram included only part of thevertex V ( χ + i W e G ). Here, in order to keep control of the vertex V and therefore M cχ + i , we shall split4 cχ + i into two terms as follows M cχ + i = M χ + i + f M χ + i , (15)where M χ + i denotes the amplitude considered in Ref. [26], which only includes the second term of [10](with two gamma matrices), while f M χ + i includes the first term (with 3 gamma matrices). Then, theaveraged squared amplitude [14] becomes | M | = | M t | + | M ˜ b i | + | M χ + i | + | f M χ + i | +2 Re (cid:16) M † χ + i f M χ + i + M † t M ˜ b i + M † t M χ + i + M † t f M χ + i + M † ˜ b i M χ + i + M † ˜ b i f M χ + i (cid:17) . (16)From the inclusion of the vertices V i from each graph, we can build each amplitudes, as follows: M t = C t P t ( q ) Ψ µ p µ ( A t + B t γ )( /q + m t ) γ ρ ǫ ρ ( k ) P L u ( p ) , (17) M ˜ b i = C ˜ b i P ˜ b i ( q ) Ψ µ q µ ( a i P l + b i P R ) p ρ ǫ ρ ( k ) P L u ( p ) , (18) M χ + i = C χ + i P χ + i ( q ) Ψ µ γ ρ ǫ ρ ( k ) γ µ ( V i + Λ i γ )( /q + m χ )( S i + P i γ ) u ( p ) , (19) f M χ + i = C χ + i P χ + i Ψ µ /pγ ρ γ µ ( T i + Q i γ ) ǫ ρ ( k )( /q + m χ )( S i + P i γ ) u ( p ) . (20)Where C t = g M , C ˜ b i = g κ i M , C χ + i = m W M and C χ + i = M . We have defined q ≡ p − p , q ≡ p − k and q ≡ p − p , and ǫ ρ ( k ) denotes the W polarization vector. Expressions for A ˜ t , B ˜ t , a i , b i , κ i , V i , A i , S i and P i are presented in the Appendices A,B. Then, after performing the evaluation of each expression,we find convenient to express each squared amplitude, as follows: | M ψ a | = C ψ a | P ψ a ( q a ) | W ψ a ψ a , (21)where ψ a = ( t, ˜ b j , χ + k ). The functions P ψ a ( q a ) correspond to the propagators factors, thus for thechargino ψ a = χ + i , we have P χ + i ( q ) = 1 q − m χ + i + iǫ . (22)Similar expressions hold for the sbottom and the top contributions, P ˜ b ( q ) and P t ( q ) respectively.The terms W ψ a ψ a include the traces involved in each squared amplitudes W tt = Tr h M ρσ D µν p µ p ν ( A ˜ t + B ˜ t γ )( /q + m t ) γ ρ P L /p P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ ) i , (23) W ˜ b i ˜ b i = Tr h p ρ p σ M ρσ D µν q µ q ν ( R i + Z i γ ) /p ( R j − Z j γ ) i , (24) W χ + i χ + i = Tr h M ρσ D ρσ ( V i + Λ i γ )( /q + m χ )( S i + P i γ ) /p (25)( S j − P j γ )( /q + m χ )( V j − Λ j γ ) i ,W χ + i χ + i = Tr h M ρσ D µν /pγ ρ γ µ ( T i + Q i γ )( /q + m χ )( S i + P i γ ) /p ( S j − P j γ ) (26)( /q + m χ )( T j − Q j γ ) γ ν γ σ /p i . X λ =1 ǫ ρ ( ~k, λ ) ǫ ∗ σ ( ~k, λ ) = − g ρσ + k ρ k σ m W = M ρσ (27) X ˜ λ =1 Ψ µ ( ~p , ˜ λ )Ψ ν ( ~p , ˜ λ ) = − ( /p + m ˜ G ) × ( g µν − p µ p ν m G ! (28) − g µσ − p µ p σ m G ! g νλ − p ν p λ m G ! γ σ γ λ ) = D µν . (29)The functions W ψ a ψ a depend on the scalar products of the momenta p, p , p , k, q , q and q . Aftercarefully analyzing the resulting traces (handed with FeynCalc [37, 38]) we find that these functionscan be written as powers of the intermediate state masses, as follows: W ψ a ψ a = w ψ a ψ a + m ψ a w ψ a ψ a + m ψ a w ψ a ψ a . (30)Full expressions for each function w iψ a ψ a ∀ i = 1 , , M † ψ a M ψ b = C ψ a C ψ b P ∗ ψ a ( q a ) P ψ b ( q b ) W ψ a ψ b . (31)Again, as in the previous case, the function W ψ a ψ b include the traces appearing in the interferences,specifically we have f W χ i + χ i + = Tr h M ρσ D µν /pγ ρ γ µ ( T i + Q i γ )( /q + m χ ) /p ( S i − P i γ )( S j − P j γ ) (32)( /q + m χ )( V j − Λ j γ ) γ ν g σ i ,W t ˜ b i = Tr h M ρσ p ρ /p P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ ) p µ D µν q ν ( R i + Z i γ ) i , (33) W tχ + i = Tr h M ρσ /p P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ ) p µ D µρ (Λ i + V i γ )( /q + m χ ) (34)( S i + P i γ ) i , f W tχ + i = Tr h M ρσ D µν /pγ ρ γ µ p ν ( T i + Q i γ )( /q + m χ )( S i + P i γ ) /p P R γ σ (35)( /q + m t )( A ˜ t − B ˜ t γ ) i ,W χ + i ˜ b i = Tr h M ρσ p ρ ( p ν − k ν ) /p ( S i − P i γ )( /q + m χ )(Λ i − V i γ ) (36) D νσ ( R j + Z j γ ) i , f W χ + i ˜ b i = Tr h M ρσ D µν ( p ν − k ν )( R i + Z i γ ) /p ( S i − P i γ )( /q + m χ ) (37)( T i − Q i γ ) γ µ γ ρ /pp σ i . It turns out that the functions W ψ a ψ b can be expressed also in powers of the intermediate masses: W ψ a ψ b = w ψ a ψ b + m ψ a (w ψ a ψ b + m ψ b w ψ a ψ b ) + m ψ b w ψ a ψ b . (38) Progress in automatic calculation of MSSM processes with gravitino have appeared recently [32], some of our resultshave been checked by the authors of Ref. [33] and they found agreement in the results (private communications). jψ a ψ b ∀ j = 1 , , , iψ a ψ a d Γ dx dy = m t π | M | . (39)The variables x and y are defined as x = 2 E e G m ˜ t and y = 2 E W m ˜ t . Numerical results for the lifetime τ = will be presented and discussed in Section 3. ˜ t → G W b with the goldstino approximation
In this section we shall present the calculation of the stop decay using the gravitino-goldstino highenergy equivalence theorem [28]. In the high energy limit ( m e G ≪ m ˜ t ) we could consider the gravitinofield (spin particle) as the derivative of the goldstino field (spin (cid:0) (cid:1) particle). We shall consider in thissection the same Feynman diagrams Figures [1,2,3] that we used in Section 2.1, but with the provisothat the gravitino field shall be described by the goldstino fields. Making the replacement Ψ e G → i q
23 1 m e G ∂ µ Ψ in the gravitino interaction lagrangian, one obtain the effective interaction lagrangian forthe goldstino as is show in [31]. The averaged squared amplitude for the Goldstino is then written as | M G | = | M Gt | + | M G ˜ b i | + | M Gχ + i | (40)+ 2 Re ( M G † t M G ˜ b i + M G † t M Gχ + i + M G † ˜ b i M Gχ + i ) . As in the previous Section 2.1, we can build the amplitudes from the inclusion of all the vertices intothe expressions from each graph, namely: M Gt = e C t P t ( q )Ψ( A ˜ t + B ˜ t γ )( /q + m t ) γ ρ P L ǫ ρ ( k ) u ( p ) , (41) M G ˜ b i = e C ˜ b i P ˜ b i ( q )Ψ( R i + Z i γ ) u ( p ) p σ ǫ σ ( k ) , (42) M Gχ + i = e C χ + i P χ + i ( q ) /pγ ρ ( T i + Q i γ )Ψ ǫ ρ ( k )( /q + m χ )( S i + P i γ ) u ( p ) . (43)Where the superindex “G” that appears in the amplitudes [41-43] refers to the goldstino ampli-tudes. The constants appearing in front of each amplitudes are: e C t = − g (cid:18) m t − m t √ Mm e G (cid:19) , e C ˜ b i = g κ i (cid:18) m b − m bi √ Mm e G (cid:19) and e C χ + i = − m χ + i √ Mm e G . We obtain similar expressions to [ ?? ] for the squared am-plitudes of the goldstino case, namely: | M Gψ a | = e C ψ a | P ψ a ( q a ) | W Gψ a ψ a , (44)where the function W Gψ a ψ a includes traces corresponding to the goldstino squared amplitudes, whichare given as follows: W Gtt = Tr h ( /p + m ˜ G )( A ˜ t + B ˜ t γ )( /q + m t ) γ ρ P L M ρσ /p (45) P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ ) i ,W G ˜ b i ˜ b i = Tr h p ρ p σ M ρσ ( /p + m ˜ G )( B i + Z i γ ) /p ( B j − Z j γ ) i , (46) W Gχ + i χ + i = Tr h M ρσ ( /p + m ˜ G ) /pγ ρ ( T i + Q i γ )( /q + m χ )( S i + P i γ ) /p (47)( S j − P j γ )( /q + m χ )( T j − Q j γ ) γ σ /p i , W Gψ a ψ a depend on the scalar products of the momenta p, p , p , k, q , q and q , thesefunctions will also be written as powers of the intermediate state masses, namely: W Gψ a ψ a = w G ψ a ψ a + m ψ a w G ψ a ψ a + m ψ a w G ψ a ψ a . (48)All the full expressions for each function w Giψ a ψ a ∀ i = 1 , , M G † ψ a M Gψ b = e C ψ a e C ψ b P ∗ ψ a ( q a ) P ψ b ( q b ) W Gψ a ψ b . (49)The functions W ψ a ψ b correspond to the traces involved in the interference terms, i.e. W Gt ˜ b i = Tr h M ρσ /p P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ )( /p + m ˜ G )( B i + Z i γ ) p ρ i , (50) W Gtχ + i = Tr h M ρσ ( /p + m ˜ G ) /pγ ρ ( T i + Q i γ )( /q + m χ )( S i + P I γ ) (51) /p P R γ σ ( /q + m t )( A ˜ t − B ˜ t γ ) i ,W Gχ + i ˜ b i = Tr h M ρσ /p ( S i − P i γ )( /q + m χ )( T i − Q i γ ) γ ρ /p ( /p + m ˜ G )( R j + Z j γ ) p σ i . (52)The W Gψ a ψ b functions also expressed as powers of the intermediate masses: W ψ a ψ b = w G ψ a ψ b + m ψ a (w G ψ a ψ b + m ψ b w G ψ a ψ b ) + m ψ b w G ψ a ψ b . (53)The full expressions for w Gjψ a ψ b ∀ j = 1 , , , The decay width is obtained by integrating the differential decay width over the dimensionless variables x, y which have limits given by 2 µ G < x < µ ˜ G − µ W with µ i = m i m t and y ± = (2 − x ) (cid:0) µ ˜ G + µ W − x + 1 (cid:1) ± p x − µ ˜ G (cid:0) µ ˜ G − µ W − x + 1 (cid:1) (cid:0) µ ˜ G − x + 1 (cid:1) , (54)Γ = Z µ G − µ W µ G Z y + y − m t π | M | dy dx. (55)After integrating numerically the expressions for the differential decay width, we obtain the valuesfor the decay width, for a given set of parameters. We consider two values for the stop mass, m ˜ t =200 GeV and m ˜ t = 350 GeV , we also fix the chargino mass to be m χ + i = 200 , GeV , while thesbottom mass is fixed to be m ˜ b i = 300 , GeV .In Figures [4,5] we show the lifetime of the stop, as function of the gravitino mass, within theranges 200-250 GeV for the case with m ˜ t = 350 GeV , and 50-100 GeV for m ˜ t = 100 GeV . We showthe results for the case when one uses the full expression for chargino-gravitino-W vertex (circles), aswell as the case when the partial inclusion of such vertex, as it was done in [26] (triangles) and in thelimit of the goldstino approximation (squares). We noticed that for low gravitino masses ( m e G → m e G ∼ = m ˜ t ) the results forthe stop lifetime using the full gravitinio and goldstino approximation could be very different, up to O (50%) different. 8n the other hand, the values for the stop life-time using the full gravitino and partial gravitinolimit are very similar for low gravitino masses, while for the largest allowed masses the difference inresults is at most of order O (50%). The value of the lifetime obtained in all theses cases turns out tobe of order 10 − sec, which results in an scenario with large stop lifetime that has very specialsignatures both at colliders and has also important implications for cosmology, as it was discussed inref. [26]. Figure 4: Stop lifetime 1Figure 5: Stop lifetime 2For instance, regarding the effect on BBN, the Stop ˜ t have to form quasi stable sbaryons (˜ t qq )and mesinos (˜ t ¯ q ), whose late decays could have affected the light element abundance obtained inBBN, while negatively charged stop sbaryons and mesinos could contribute to lower the Coulomb9arrier for nuclear fusion process occurring in the BBN epoch. However, as argued in [26] the greatmajority of stop antisbaryons would have annihilated with ordinary baryons to make stop antimesinosand most stop mesinos and antimesinos would have annihilated. The only remnant would have beenneutral mesinos which would be relatively innocuous, despite their long lifetime because they wouldnot have important bound state effects. Further discussion of BBN issues of Ref. [29] divide thestop 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 theCosmic Microwave Background (CMB), we have included some comments in the text, to estimate themain effects. The arguments which read as follows: Very long lifetimes ( τ > s ) would have beenexcluded if one uses the approximate results of Ref. [39], which present bounds on the lifetime τ (forthe case when stau is the NLSP) using the constrain in the chemical potential µ < × − . However,it was discussed in Ref. [40], that a more precise calculation reduces the excluded region for lifetimes,ending at about τ ∼ s − 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), suchas the energy release or stop hadronization, will affect the calculation, but the numerical evaluationof such effect is beyond the scope of our paper. In this paper we have calculated the stop ˜ t lifetime in MSSM scenarios where the massive gravitinois the lightest supersymmetric particle (LSP), and therefore is a viable dark matter candidate. Thelightest stop ˜ t corresponds to the next-to-lightest supersymmetric particle (NLSP). We have focusedon the kinematical domain m e G + m t > m ˜ t > m e G + m b + m W , where the stop decay width is dominatedby the mode ˜ t → b W ˜ G .The amplitiude for the full calculation of the stop 3-body decay width includes contributions fromtop, sbottom and chargino as intermediate states. We have considered the full chargino-gravitinovertex, which improves the calculation presented in ref. [26]. Besides performing the full calculationwith massive gravitino, we have also evaluated the stop decay lifetime for the limit when the gravitinocan be approximated by the goldstino state. Our analytical results are conveniently expressed, in bothcases, using an expansion in terms of the intermediate state mass, which helps in order to identify themassless limit.We find that the results obtained with the full chargino vertex are not very different from theapproximation used in ref. [26], in fact they only differ approximately in a 50%. The comparison ofthe full numerical results with the ones obtained for the goldstino approximation, show that in thelimit of low gravitino mass ( m e G ≪ m ˜ t ) there is not a significant difference in values of the stop life-time obatined from each method. However, for m e G . m ˜ t the difference in lifetime could be as highas 50%. Numerical results for the stop lifetime give value of order 10 − sec, which makes thestop to behave like a quasi-stable state, which leaves special imprints for LHC search. Our calculationshows 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 finalstate
In this appendix we present explicitly the full results for the 10 w ψ a ψ a functions that arose from aconvenient way to express the large traces that appear in the squared amplitudes [21], as well as the18 w ψ a ψ b functions in the interferences [31] of the 3-body stop ˜ t decay with gravitino in the finalstate. First, we shall present the contributions for the squared amplitudes, then we shall present the10nterferences. A.1 Top Contribution
For the averaged squared amplitude of the top quark contribution, the functions w tt , w tt and w tt are: w tt = 4 a h m W m G (cid:16) f ( m W (6 m G + 2 h m t − q ) + 6 f m W + 4 f ) − f (cid:0) m G (cid:0) − (cid:0) f + 3 m W (cid:1)(cid:1) + f (4 f + 3 m W ) + f q (cid:1) + ( q − m G )( m W m G + 3 f m W + 2 f )+ 4 f ( f − m G ) − m W m G h m t − f m W (cid:17) , (56)w tt = 8 a h m W m ˜ G ( m W ( m G + m t − f ) − f (4 f + 3 m W ) + 3 f m W + 2 f + 2 f ) , (57)w tt = 4 a h m W m G ( m W ( f − m G ) − f m W − f + 2 f f ) . (58)The functions f , f and f are functions of the variables x and y that were defined previously inSection 3, they are f = m t y, f = m t x, f = m t ( − − µ e G − µ W + x + y ), with µ e G = m e G m t and µ W = m W m t . We have also used in [56-58] the following substitutions h = ( f − m G m t ), a = ( A ˜ t − B ˜ t ) , a = A t − B t and a = ( A ˜ t + B ˜ t ) , with A ˜ t = cos θ ˜ t + sin θ ˜ t and B ˜ t = cos θ ˜ t − sin θ ˜ t . A.2 Sbottom Contribution
For the averaged squared amplitude of the squark sbottom contribution, the function w b i ˜ b i is:w b i ˜ b i = 8 D ij h h (( f − f ) − q m G )3 m W m G . (59)With h = f − f − m G and h = f − m W m t . We have done in the amplitude [ ?? ] the followingsubstitution a i P R + b i P L = ( R i + Z i γ ) such that D ij = R i R j + Z i Z j , with R i = a i + b i , Z i = a i − b i , R j = a j + b j and Z j = a j − b j , and with a i = (sin θ ˜ b , cos θ ˜ b ) , b i = (cos θ ˜ b , − sin θ ˜ b ) and κ i = (cos θ ˜ t cos θ ˜ b , − cos θ ˜ t cos θ ˜ b ). A.3 Partial Chargino Contribution ( M χ + i ) For the averaged squared amplitude of the chargino contribution, the functions w kχ + i χ + i , ∀ k = 1 , , χ + i χ + i = − ij1 h m W m G (cid:16) ( m G + f )(2( m G + m W ) + 4 f − q )+ f ( − m G − f + q ) − f ( m G + f ) (cid:17) , (60)w χ + i χ + i = − h (Σ ij1 + Σ ij2 )( h − f − f )3 m W m ˜ G , (61)w χ + i χ + i = 8Σ ij3 h h m W m G , (62)11ith h = 2 m W m G + f and h = m G + 2 f + m W , we have also used the following substitutionsΣ ij = ( S i S j + P i P j )( V i V j − Λ i Λ j ) − ( S i P j + P i S j )(Λ i V j − V i Λ j ), Σ ij = ( S i S j + P i P j )( V i V j − Λ i Λ j ) +( S i P j + P i S j )(Λ i V j − V i Λ j ), Σ ij = ( S i S j + P i P j )( V i V j + Λ i Λ j ) + ( S i P j + P i S j )(Λ i V j + V i Λ j ), with V i = V i sin β + U i cos β and Λ i = V i sin β − U i cos β . For the low-to-moderate range of tan β wehave: S = 12 (cid:18) − g cos φ L + g m t sin φ L sin θ ˜ t √ m W sin β (cid:19) , (63) P = 12 (cid:18) − g cos φ L − g m t sin φ L sin θ ˜ t √ m W sin β (cid:19) , (64)where cos φ L , ± sin φ L are elements of the matrix V that diagonalizes the chargino mass matrix, ex-pressions for S and P may be obtained by replacing cos φ L → − sin φ L and sin φ L → cos φ L in [63]and [64]. A.4 Full Chargino Contribution ( f M χ + i ) For the averaged squared amplitude f M χ + i of the chargino contribution, the functions w kχ + i χ + i ∀ k =1 , ,
3, are: w χ + i χ + i = q P ij1 h , (65)w χ + i χ + i = 16 m ˜ G ( P ij1 + P ij2 )3 m W ( h − f − f )(2 f − m W m t ) , (66)w χ + i χ + i = P ij2 h , (67)where we have defined h = 163 m W m G (cid:16) f ( f (2( f − f ) f − m G (2 f + m W )) − f m G m t )+ h (2 f m W − m t h ) + f (4 f m G − m G ) (cid:17) . (68)With h = 3 m W m G + 2 f , we have used the substitution V i P R − U i P L = T i + Q i γ in the first termof the interaction vertex V ( χ + i W ˜ G ), we have also done the following substitutions in the functions[65-67]: P ij = ( S i S j + P i P j )( T i T j + Q i Q j ) − ( S i P j + P i S j )( T i Q j + Q i T j ) , (69) P ij = ( S i S j + P i P j )( T i T j + Q i Q j ) + ( S i P j + P i S j )( T i Q j + Q i T j ) . (70)12 .5 Interference Terms M † χ + i f M χ + i Interference
For the interference term M † χ + i f M χ + i , the e w kχ + i χ + i functions ∀ k = 1 , , ,
4, are: e w χ i + χ i + = 16 S ij1 m W m ˜ G (cid:16) f ( m G (8 f + 2 m W − q ) + 4 f ) + f (cid:0) f f (2 m G + f − q ) − ( m G (4 f + m W ) + 2 f )(2( m G + m W ) + 4 f − q ) (cid:1) + f m W ( − m G (2 f − f − m W + q ) + 2 m G + f q ) + f q m t (cid:17) , (71) e w χ i + χ i + = 16( S ij2 + S ij3 )3 m W m G (cid:16) m t ( − f m G ( f − m W ) + 2 m W m G − f )+ 2 f (cid:0) f m G h + 2 f ( m W m G + f ) (cid:1) − f m W (5 m G h + f (2 f − m G )) − f (4 f m G + m G ) (cid:17) , (72) e w χ i + χ i + = − S ij4 m W m ˜ G (cid:16) f m W ( f − m G ) + f ( m G (4 f + m W ) + 2 f ( f − f )) − f m G + f m t (cid:17) , (73)In order to have control in the calculations with huge expressions, we have done the following substi-tutions in the functions [71-73]: S ij = ( S i S j + P i P j )( T i V j + Q i Λ j ) − ( S i P j + P i S j )( Q i V j + T i Λ j ) , (74) S ij = ( S i S j + P i P j )( T i Λ j + Q i V j ) − ( S i P j + P i S j )( Q i Λ j + T i V j ) , (75) S ij = ( S i S j + P i P j )( T i Λ j + Q i V j ) + ( S i P j + P i S j )( Q i V j + T i Λ j ) , (76) S ij = ( S i S j + P i P j )( T i V j + Q i Λ j ) + ( S i P j + P i S j )( Q i V j + T i Λ j ) . (77) M † χ + i M ˜ b i Interference
For the interference term M † χ + i M ˜ b i , the functions w jχ + i ˜ b i ∀ j = 1 , χ + i ˜ b i = − η ij1 m W m ˜ G (cid:16) − f (cid:0) m t ( m W m G + f ) − f f ( m G + 3 f − m W )+ 2 f h + f (2 f − m W ) (cid:1) + m W (cid:0) m t bigl ( m G ( − f + 4 f + m W )+ 2 m G + f (cid:1) + f ( − f (2 m G + 6 f + m W ) + 2 f h + 2 f ) (cid:1) + f ( − m G ( − f + 4 f + m W ) − m G + 2 f ) + f m G (cid:17) , (78)w χ + i ˜ b i = 8 η ij2 h m W m G ( m W ( m G h m t + f ( f − f )) − f m G + f ( f − f ) f ) . (79)In the functions [78,79], we have done the following substitutions: η ij = R j (Λ i S i − V i P i ) + Z j (Λ i P i − V i S i ) , (80) η ij = R j (Λ i S i + V i P i ) + Z j (Λ i P i + V i S i ) . (81)(82)13 † t M χ + i Interference
For the interference term M † t M χ + i the functions w jtχ + i ∀ j = 1 , , , tχ + i = 2Ω i1 m W m ˜ G (cid:16) f (cid:0) m t ( m G ( f + 2 m W ) − f ) − f ( m G h + 2 f (3 f + m W )) − f m G h + f (2 f − m W ) (cid:1) + m t (cid:0) − m W m G ( − f + 6 f + m W ) − m W m G + f ((2 f + f ) m W + 2 f ( f − f )) (cid:1) + f ( m G ( − f + 10 f + 3 m W ) + 4 m G + 8 f f ) + f (cid:0) f ( m G + m W )+ f m W (4 m G − f + m W ) + 4 f f m W + 4 f (cid:1) − f m G (cid:17) , (83)w tχ + i = 4Ω i2 m W m G (cid:16) − m t ( f ( f − m W m G ) + 2 m W m G ) + f (cid:0) f m G m t + m G ( m W − f ) − f m G (2 f + m W ) + 2 f f (cid:1) + f ( m G ((2 f − f ) m W + f ) + f ( f − f ) m W ) + 2 f m G ( m G − f ) (cid:17) , (84)w tχ + i = 2Ω i3 m W m ˜ G (cid:16) m t h − f m G ( f − f + 2 m W ) − f m W + 2 f f (cid:0) m W − f (cid:1)(cid:17) , (85)w tχ + i = − i4 m W m G (cid:16) − m G m t (4 f m W + h ) − f (cid:0) f m G h + f (2 f − m W m G ) (cid:1) + 2 f f h + f m W (3 m G + 2 f ) + f (4 f m G + m G ) (cid:17) . (86)With h = 2 f − m W and h = 3 m W m G + f . We have done the following substitutions in thefunctions [83-86]: Ω i = ( A ˜ t − B ˜ t )( S i − P i )(Λ i + V i ), Ω i = ( A ˜ t − B ˜ t )( S i − P i )(Λ i − V i ), Ω i =( A ˜ t + B ˜ t )( S i − P i )(Λ i − V i ) and Ω i = ( A ˜ t + B ˜ t )( S i − P i )(Λ i + V i ). M † t M ˜ b i Interference
For the interference term M † t M ˜ b i , the functions w jt ˜ b i ∀ j = 1 , t ˜ b i = 2 (∆ i1 + ∆ i2 )3 m W m G (cid:16) f (cid:0) f m G ( − m t + h ) − m G m t ( f − m G ) + m G h − f ( m G + f ) + 4 f (cid:1) + 2 f (cid:0) m t (cid:0) − m G ( f − m W ) + f m W m G + f f ( f − f ) (cid:1) + f m G m t + f (cid:0) f m G ( f − f + m W ) − m W m G + f ( f − f ) m W (cid:1)(cid:1) + m W (cid:0) m t (cid:0) m G (2 f + m W ) + (4 f − f f − f ) m G − f ( f − f ) (cid:1) − m G m t (2 m G − f + f ) − f m G (2 f − h ) (cid:1) + 4 f m G ( f − m G ) (cid:17) , (87)w t ˜ b i = 2 (∆ i1 − ∆ i2 )3 m W m ˜ G (cid:16) f (2 f − m G (2 m t + h )) + f (cid:0) m t ( m G h − f )+ 2 f ( m W m G − f m W + f ) − f (2 f + m W ) (cid:1) + m W (cid:0) m t ( − m G (cid:0) f + m W ) − f + 2 f f + f ) + 2 m G m t + f (2 f − h ) (cid:1) + f m G (cid:17) . (88)with ∆ i = ( R i − Z i ) A ˜ t and ∆ i = ( Z i − R i ) B ˜ t . 14 M † χ + i M ˜ b i Interference
For the interference term f M † χ + i M ˜ b i , the functions e w jχ + i ˜ b i ∀ j = 1 , e w χ + i ˜ b i = 8 f C ij1 m W m G (cid:16) f (cid:0) f m G ( m t + h ) + f (2 m W m G + m G − f )+ f (2 f − m G ) (cid:1) − m t (cid:0) m G (2 f + m W ) + m G (cid:0) f (3 f + 4 m W ) − f ( f + m W ) (cid:1) + 2( f − f ) f (cid:1) − f m W (cid:0) m G ( − f + 4 f + 2 m W )+ 2 m G + 2 f ( f − f ) (cid:1) + f ( m G − f m G ) (cid:17) , (89) e w χ + i ˜ b i = − f C ij2 h ( − f h m t − f m W + f ( f + f ))3 m W m ˜ G . (90)We have done the following substitutions in the functions [89,90], C ij = T i ( R j S i + Z j P i ) − Q i ( R j P i + Z j S i ) ,C ij = T i ( R j S i + Z j P i ) + Q i ( R j P i + Z j S i ) . (91) M † t f M χ + i Interference
For the interference term M † t f M χ + i , the functions e w jtχ + i ∀ j = 1 , , , e w tχ + i = 8 R i1 m W m G (cid:16) − m G m t (4 f m W + h ) + m t (cid:0) f m W (3 m G + 4 f ) + m G (cid:0) m W m G − f (cid:1) h + 2 f h ( m W m G + f ) (cid:1) + f (cid:0) m G m t ( m G + 4 f ) − m G h + f (4 m W m G + 6 m G ) + f (8 f − m G ) (cid:1) + 2 f (cid:0) m t (cid:0) − (2 m G ( f + m W )+ ( f − f ) m G ( f + m W ) + 2 f f ) (cid:1) − f ( f − m G )( f h + f m W ) (cid:1) − f m W ( m G + 2 f ) h + f (6 m G − f m G ) (cid:17) , (92) e w tχ + i = 8 R i2 m W m ˜ G (cid:16) f ( m t ( m G ( f + 2 m W ) − f ) − f ( f h + f m W ))+ h ( m t ( f − m W m G ) + f m W ) + f (cid:0) m G ( − f + 6 f + 3 m W )+ 3 m G + 8 f f (cid:1) − f m G (cid:17) , (93) e w tχ + i = 8 R i3 m W m G (cid:16) f ( − f m G h m t + 2 f m G ( f − m W ) − f f )+ m t (2 f ( m W m G + f ) − f m G + m W m G ) − f m W ( m G + 2 f − f )+ f (4 f m G − m G ) (cid:17) , (94) e w tχ + i = 8 R i4 m W m ˜ G (cid:16) m t h − m t (cid:0) − m G ( f − f m W ) + m W m G + f ((3 f − f ) m W + 2 f ) (cid:1) + 2 f (cid:0) f (2 m G ( m W − f ) + f (2 f − m W )) − m t ( m G ( m W − f )+ 2 f f ) (cid:1) + f ( − f m G − m G m t + 3 m G + 4 f ) + f m W ( m G + 2 f − f ) (cid:17) . (95)We have done the following substitutions in the functions [92 -95] R i1 = ( A ˜ t − B ˜ t )( S i + P i )( T i − Q i ), R i2 = ( A ˜ t + B ˜ t )( S i − P i )( T i + Q i ), R i3 = ( A ˜ t − B ˜ t )( S i + P i )( T i + Q i ) and R i4 = ( A ˜ t + B ˜ t )( S i − P i )( T i − Q i ).15 Analytical expressions for the amplitudes for the Goldstino ap-proximation
In this appendix we present explicitly the full results for the 7 w Gψ a ψ a functions that arose from thesquared amplitudes [44], as well as the 8 w Gψ a ψ b functions that appear in the interference terms [49]of the 3-body stop ˜ t decay with goldstino in the final state. First, we shall present the contributionfor the squared amplitudes, then we shall present the interferences. We shall shown that the w Gψ a ψ a and w Gψ a ψ b functions are very compacts expressions, opposed to the resulting functions in the gravitinocase that we have presented in Appendix A. The approximation of the gravitino field by the derivativeof the goldstino field is good in the high energy limit ( m e G ≪ m ˜ t ), in the sense that in this limit theybehave 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 e w jtt ∀ j =1 , , e w tt = 4 2 a m W (cid:16) f ( m W (6 m G + 2 h m t − q ) + 6 f m W + 4 f ) − f ( − m G (4 f + 3 m W )+ f (4 f + 3 m W ) + f q ) + ( q − m G )( m W m G + 3 f m W + 2 f )+ 4 f ( f − m G ) − m W m G h m t − f m W (cid:17) , (96) e w tt = 4 a m ˜ G m W (cid:0) m W ( m G + m t − f ) − f (4 f + 3 m W ) + 3 f m W + 2 f + 2 f (cid:1) , (97) e w tt = 2 a ( m W ( f − m G ) − f m W − f + 2 f f ) m W . (98)With a , a and a defined previously in Appendix A. B.2 Sbottom Contribution
For the averaged squared amplitude of the sbottom squark contribution, with the e w b i ˜ b i function as: e w b i ˜ b i = 4 D ij1 h h m W , (99)with D ij defined previously in Appendix A. B.3 Chargino Contribution
For the averaged squared amplitude of the chargino contribution, the resulting functions e w jχ + i χ + i ∀ j = 1 , , e w χ + i χ + i = 4 q P ij1 m W (cid:16) m t ( m W ( m G + f ) + 3 f m W + 2 f − f f )+ 2 f (2 f ( f − f ) − (3 f + f ) m W ) (cid:17) , (100) e w χ + i χ + i = 12 m ˜ G h m t ( P ij1 + P ij2 ) ( h − f − f ) , (101) e w χ + i χ + i = 4 P ij2 m W (cid:16) m t ( m W ( m G + f ) + 3 f m W + 2 f − f f )+ 2 f (2 f ( f − f ) − (3 f + f ) m W ) (cid:17) , (102)16here P ij and P ij are defined above in Appendix A. B.4 Interference Terms M G † t M G e b i Interference
For the interference term M G † t M G e b i , the functions w jt ˜ b i ∀ j = 1 , t ˜ b i = 2(∆ i1 + ∆ i2 ) m W (cid:16) − f ( f ( m t − m G ) + f m W ) + m W (cid:0) m t (2 m G + f ) − f ( m G + m t ) (cid:1) + 2 f ( f − m G ) (cid:17) , (103)w t ˜ b i = 2 m ˜ G (∆ i1 − ∆ i2 )( m W ( f − m t ) + f − f f ) m W . (104)Where ∆ i1 and ∆ i2 are defined above in Appendix A. M G † χ + i M G e b i Interference
For the interference term M G † χ + i M G e b i , the functions w Gjχ + i ˜ b i ∀ j = 1 , G χ + i ˜ b i = 4 C ij1 m ˜ G m W (cid:0) m t ( m W ( h − f ) + f f ) − f h (cid:1) , (105)w G χ + i ˜ b i = 4 C ij2 m W (cid:0) − m W h m t h + f (2 f − m G ) − f ( f h m t + f m W ) (cid:1) , (106)with C ij1 and C ij2 defined above in Appendix A. M G † t M Gχ + i Interference
For the interference term M G † t M Gχ + i , the functions w Gjtχ + i ∀ j = 1 , , , G tχ + i = 4 R i1 m ˜ G m W (cid:16) m t ( m W (4 m G − f + 3 m W ) + 5 f m W − f )+ f m W (2 f − h ) − f m G + f (4 f ( f + m W ) − m W h m t ) (cid:17) , w G tχ + i = 4 R i2 m W (cid:16) m t ( f (2 f + m W ) − m W m G ) + f m W (3 h − f )+ 2 f m G − f f ( f + m W ) (cid:17) , (107)w G tχ + i = 4 R i3 m ˜ G m W (cid:0) m W ( m t − f ) − f (2 f + 3 m W ) + 2 f (cid:1) , (108)w G tχ + i = R i4 m W (cid:16) f ( f − f ) − (3 f + f ) m W )(2 f − m G ) (109)+ 4 h m t (( f + 3 f ) m W + 2 f ( f − f )) (cid:17) , with R i1 , R i2 , R i3 and R i4 defined above in Appendix A.17 cknowledgments We would like to acknowledge the support of CONACYT and SNI. We also acnknowledge to Ab-del Perez for his valuable comments. B.O. Larios is supported by a CONACYT graduate studentfellowship.
References [1] For a review see: S.P. Martin, A Supersymmetry primer, Adv.Ser.Direct.High Energy Phys. (2010) 1-153 [arXiv:hep-ph/9709356].[2] J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Nucl. Phys. B (1984)453; H. Goldberg, Phys. Rev. Lett. (1983) 1419.[3] J. R. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, Phys. Rev. D , 055005 (2004)[arXiv:hep-ph/0405110].[4] T. Falk, K. A. Olive and M. Srednicki, Phys. Lett. B (1994) 248 [arXiv:hep-ph/9409270].[5] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. (2003) 011302[arXiv:hep-ph/0302215]; Phys. Rev. D (2003) 063504 [arXiv:hep-ph/0306024].[6] J. L. Feng, S. Su and F. Takayama, Phys. Rev. D (2004) 075019 [arXiv:hep-ph/0404231].[7] J. R. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, Phys. Lett. B (2004) 7[arXiv:hep-ph/0312262].[8] F. D. Steffen, JCAP , 001 (2006) [arXiv:hep-ph/0605306].[9] A. De Roeck, J. R. Ellis, F. Gianotti, F. Moortgat, K. A. Olive and L. Pape,[arXiv:hep-ph/0508198].[10] F. D. Steffen, [arXiv:hep-ph/0711.1240].[11] M. Johansen, J. Edsj, S. Hellman, J. Milstead , JHEP (2009) 023, [arXiv:hep-ph/0807.4936].[13] J. Heisig, J. Heising, JCAP 04 (2014) 023 [arXiv:1310.6352].[14] J. L. Feng, S. F. Su and F. Takayama, Phys. Rev. D (2004) 063514 [arXiv:hep-ph/0404198].[15] T. Kanzaki, M. Kawasaki, K. Kohri and T. Moroi, [arXiv:hep-ph/0609246].[16] L. Covi and K. Sabine JHEP 0708 (2007) 015, [arXiv:hep-ph/0703130v3].[17] K. Kadota, K. A. Olive, L. Velasco, Phys. Rev. D (2009) 055018, [arXivhep-ph/0902.2510v3].[18] J. R. Ellis, A. R. Raklev and O. K. Oye, JHEP , 061 (2006) [arXiv:hep-ph/0607261].[19] K. Hamaguchi, Y. Kuno, T. Nakaya and M. M. Nojiri, Phys. Rev. D (2004) 115007[arXiv:hep-ph/0409248].[20] R. H. Cyburt, J. R. Ellis, B. D. Fields and K. A. Olive, Phys. Rev. D (2003) 103521[arXiv:astro-ph/0211258]; J. R. Ellis, K. A. Olive and E. Vangioni, Phys. Lett. B (2005)30 [arXiv:astro-ph/0503023]. 1821] M. Kawasaki, K. Kohri and T. Moroi, Phys. Lett. B (2005) 7 [arXiv:astro-ph/0402490]; Phys.Rev. D (2005) 083502 [arXiv:astro-ph/0408426].[22] K. Kohri and Y. Santoso, Phys. Rev. D , 043514 (2009) [arXiv:0811.1119 [hep-ph]].[23] J. R. Ellis and S. Rudaz, Phys. Lett. B (1983) 248.[24] C. Boehm, A. Djouadi and M. Drees, Phys. Rev. D (2000) 035012 [arXiv:hep-ph/9911496].[25] J. R. Ellis, K. A. Olive and Y. Santoso, Astropart. Phys. (2003) 395 [arXiv:hep-ph/0112113].[26] J. L. Diaz-Cruz, John Ellis, Keith A. Olive, Yudi Santoso, JHEP 0705 (2007) 003[arXiv:hep-ph/0701229].[27] T. Phillips, talk at DPF 2006, Honolulu, Hawaii, October 2006, .[28] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio, R. Gatto, Phys. Lett. B (1988) 313.[29] L. Covi and F. Dradi,
JCAP 1410 (2014) 10, 039, [arXivhep-ph/1403.4923].[30] J.E. Molina, et al, Phys. Lett.
B737 (2014) 156-161, [arXiv:hep-ph/1405.7376].[31] T. Moroi, [arXiv:hep-ph/9503210].[32] H. Eberl and V. C. Spanos, [arXiv:1509.09159 [hep-ph]].[33] H. Eberl and V. C. Spanos, [arXiv:1510.03182 [hep-ph]].[34] E. Brubaker et al. [Tevatron Electroweak Working Group], [arXiv:hep-ex/0608032].[35] W. M. Yao et al. [Particle Data Group], J. Phys. G (2006) 1.[36] CDF Collaboration, .[37] V. Shtabovenko, R. Mertig and F. Orellana, arXiv:1601.01167.[38] R. Mertig, M. Bhm, and A. Denner, Comput. Phys. Commun., , 345-359, (1991).[39] J. l. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D68