Supersymmetry with a Chargino NLSP and Gravitino LSP
PPreprint typeset in JHEP style - PAPER VERSION
Supersymmetry with a Chargino NLSP and Gravitino LSP
Graham D. Kribs a , Adam Martin b and Tuhin S. Roy a a Department of Physics and Institute of Theoretical Science,University of Oregon, Eugene, OR 97403 b Department of Physics, Sloane Laboratory, Yale University, New Haven, CT 06520
Abstract:
We demonstrate that the lightest chargino can be lighter than the lightest neu-tralino in supersymmetric models with Dirac gaugino masses as well as within a curiousparameter region of the MSSM. Given also a light gravitino, such as from low scale super-symmetry breaking, this mass hierarchy leads to an unusual signal where every superpartnercascades down to a chargino that decays into an on-shell W and a gravitino, possibly witha macroscopic chargino track. We clearly identify the region of parameters where this signalcan occur. We find it is generic in the context of the R -symmetric supersymmetric standardmodel, whereas it essentially only occurs in the MSSM when sign( M ) (cid:54) = sign( M ) = sign( µ )and tan β is small. We briefly comment on the search strategies for this signal at the LHC. Keywords:
BTSM. a r X i v : . [ h e p - ph ] J u l . Introduction If the fundamental scale of supersymmetry breaking is low, which can happen with gaugemediation [1], the lightest supersymmetric particle (LSP) is the gravitino. Sparticles pro-duced at colliders will rapidly decay down to the next-to-lightest sparticle (NLSP) whichthen slowly decays into a particle and gravitino. This decay chain is assured assuming thereis no excessively small kinematic suppression for heavier sparticles to decay into the NLSP(see e.g. [2]). The identity of the NLSP becomes paramount to determine the collider signal;many possibilities for the NLSP have been considered [3–9] including the lightest neutralino,the stau, and the gluino.In this paper, we demonstrate that the NLSP could be a chargino, leading to a dramati-cally distinct signal of supersymmetry. Every superpartner cascades down to a chargino thatdecays into an on-shell W and a gravitino, possibly with a macroscopic chargino track. Thefinal decay ˜ χ ± → W ± ˜ G is 2-body, at least for m ˜ G <
21 GeV, due to the LEPII bound on themass of charginos, m ˜ χ ± >
101 GeV [10].Common lore asserts that the lightest neutralino is always lighter than the lightestchargino in the minimal supersymmetric standard model (MSSM). This is certainly true inthe bino limit M (cid:28) M , µ, M Z , and has been studied and confirmed in the wino limit [11–14]and Higgsino limit [15, 16], at least without excessively large radiative corrections. Generi-cally, radiative corrections to the mass difference between the chargino and neutralino aresmall, less than a GeV [17]. The exception is if the lightest gauginos are Higgsino-like withsignificant contributions from top and bottom squarks, where Ref. [15] found it could be aslarge as a few GeV. It is not clear if this parameter region remains viable, in light of presentdirect search constraints and electroweak precision corrections. Nevertheless, as we will see,there are qualitatively distinct regions of tree-level gaugino parameters resulting in a charginoNLSP regardless of radiative corrections. This is the focus of the paper.We find two qualitatively distinct scenarios where the chargino can be the NLSP. Thefirst, and by far the most significant, is the minimal R -symmetric supersymmetric model(MRSSM). Generically, this model can have the lightest chargino lighter than the lightestneutralino due to the fundamentally different neutralino mass matrix that results from theDirac gaugino masses. The second scenario is, remarkably, a curious and relatively unexploredregion of the MSSM parameter space, where sign( M ) (cid:54) = sign( M ) = sign( µ ) and tan β issmall [18]. The mass difference between the lightest neutralino and the lightest chargino inthe R -symmetric scenario can be tens of GeV or more, whereas it can be up to about 5 GeV(at tree-level) in this curious region of the MSSM.The organization of this paper is as follows. Sec. 2 reviews Dirac gauginos, the MRSSMmodel, and demonstrates that a chargino is NLSP in a wide region of parameter space. Sec. 3is devoted to identifying the curious region of the MSSM parameter space where a charginocan be the NLSP. In Sec. 4 the decay width of the chargino NLSP into the LSP is calculated.In Sec. 5 the collider phenomenology of a chargino NLSP is discussed. Finally, in Sec. 6 weconclude. – 1 –any analytical results are presented to concretely demonstrate that the chargino canbe the NLSP in the wino and Higgsino limits of the MRSSM and the MSSM. This discussionis somewhat technical; readers interested in just knowing the parameter space that results ina chargino NLSP may go directly to Sec. 2.4 for the MRSSM (especially Figs. 3 and 4), andto the latter half of Sec. 3 for the MSSM (especially Figs. 10 and 11). Readers unfamiliarwith Dirac gauginos are encouraged to read up to the end of Sec. 2.1. Readers interested injust the new signals can skip directly to Sec. 5.
2. Neutralinos and Charginos with Dirac Gaugino Masses
Dirac gaugino masses result when the gaugino is married with a fermion in the adjoint rep-resentation through the operator (cid:90) d θ W (cid:48) α M W αi Φ i , (2.1)after the spurion W (cid:48) α = Dθ α acquires a D -term. Here Φ i are chiral superfields in the adjointrepresentation of the SM groups. M represents the messenger scale where these operators aregenerated. This possibility has been contemplated for weak scale supersymmetry some timeago [19–21] and more recently [22–29].Gauginos which acquire Dirac masses from Eq. (2.1) are not necessarily Dirac fermionsonce electroweak symmetry is broken and the gauginos mix with Higgsinos. Charginos areobviously Dirac fermions, since charginos carry a conserved U (1) charge, i.e., electric charge.Neutralinos are Dirac fermions only if a global U (1) is preserved by all neutralino interac-tions. In the minimal R -symmetric supersymmetric standard model (MRSSM) [29], a U (1) R symmetry is preserved, and thus the neutralinos are Dirac fermions. By contrast, in theFox-Nelson-Weiner (FNW) model [22], while gauginos acquire Dirac masses, the Higgsinosacquire mass through an ordinary µ -term. The Higgsino mass violates the U (1) R -symmetry,and thus leads to neutralinos that are (pseudo-Dirac) Majorana fermions. For our purposes,the most illuminating scenario with Dirac gaugino masses is the MRSSM.The remarkable feature of the MRSSM is that it drastically ameliorates the supersym-metric flavor problem, with no excessive contributions to electric dipole moments, but withorder one squark and slepton mass mixings among nearly all flavors [29, 30]. This is possiblefor several reasons: left-right squark and slepton mixing is absent; the gaugino masses M canbe naturally 4 π/g heavier than the scalar masses; and several flavor-violating operators aremore suppressed than in the MSSM due to the absence of R -violating operators.A low energy model with U (1) R symmetry can arise, for example, if supersymmetrybreaking hidden sector preserves U (1) R , which happens in a wide class of supersymmetrybreaking models (see e.g., [31]). Nevertheless, cancellation of the cosmological constant withan R -violating constant in the superpotential [32] is generally expected to cause R -violationto be communicated from the hidden sector to the MRSSM via anomaly mediation. A naturalway to minimize the size of the R -symmetry violation is to take the gravitino mass small, such– 2 –ields SU (3) C SU (2) W U (1) Y U (1) R Q U ¯3 1 - D ¯3 1 L E ˜ B ˜ W ˜ g H u H d R u R d Table 1:
Gauge and R -charges of all chiral supermultiplets in the MRSSM. as in a low scale supersymmetry breaking scenario. It is thus very natural to imagine an R -symmetric model with a light gravitino, making the resulting experimental signals importantto study.There are several other models that have Dirac gaugino masses where we find that thechargino NLSP phenomenon can also occur. In Appendix A we briefly comment on the FNWmodel [22], showing that there are specific limits where the neutralinos become Dirac fermionsand the mass matrices reduce to the ones found in the MRSSM. Hence, our results apply tothis model as well. In the MRSSM, gaugino masses arise from Eq. (2.1) which generate Dirac masses that pairthe gauginos (cid:16) ˜ g, ˜ W and ˜ B (cid:17) with their fermionic partners (cid:0) ψ ˜ g , ψ ˜ W and ψ ˜ B (cid:1) . Higgsino massesarise from pairing Higgs superfields H u and H d with partner fields R u and R d through a pairof mass terms (cid:90) d θ (cid:104) µ u H u R u + µ d H d R d (cid:105) . (2.2) R d,u transform identically to H u,d under the electroweak group, except that the R -charges are2 rather than 0. This R -charge assignment forbids Yukawa-like couplings of the R -fields to thematter fields. Hence, only Higgses acquire electroweak symmetry breaking expectation values.Upon electroweak symmetry breaking, the electroweak gauginos mix with the Higgsinos (justlike in the MSSM). For completeness, all the multiplets in the MRSSM described in Ref. [29]are listed in Table 1 along with their matter and R -charges.Let us first investigate the neutralino mass matrix in the MRSSM. The R -charges de-termine which neutralinos mix with each other and provide a guiding principle to determinethe gauge-eigenstate basis. The vector N + ≡ (cid:16) ˜ W , ˜ B, ˜ R u , ˜ R d (cid:17) carries R -charge +1, while– 3 –he vector N − ≡ (cid:16) ψ W , ψ ˜ B , ˜ H u , ˜ H d (cid:17) carries a R -charge −
1. Field rotations do not mix upfields with different R -charges. Hence, just like with charginos in the MSSM, two independentrotation matrices are required to diagonalize the mass matrix.The Lagrangian for neutralino masses can be written in the gauge-eigenstate basis as L neutralino mass = N T+ M ˜ N N − + c.c. , (2.3)where the neutralino mass matrix is given as M ˜ N = M − gv u / √ gv d / √ M g (cid:48) v u / √ − g (cid:48) v d / √ − λ u v u / √ λ (cid:48) u v u / √ µ u λ d v d / √ − λ (cid:48) d v d / √ µ d . (2.4)Here (cid:104) H u (cid:105) ≡ v u and (cid:104) H d (cid:105) ≡ v d with v u + v d = v / (cid:39) (174 GeV) . Notice the apparentlyunusual location of the µ -terms in Eq. (2.4) is a direct result of the Dirac nature of theneutralino mass matrix. The physical mass-squareds are given by eigenvalues of M ˜ N M T˜ N .There are four new parameters λ u , λ d , λ (cid:48) u and λ (cid:48) d that arise from the superpotential terms (cid:90) d θ (cid:104) H u (cid:0) λ u Φ ˜ W + λ (cid:48) u Φ ˜ B (cid:1) R u + H d (cid:0) λ d Φ ˜ W + λ (cid:48) d Φ ˜ B (cid:1) R d (cid:105) . (2.5)These couplings are unnecessary to the structure of the MRSSM but are nevertheless allowedunder all of the charge assignments. Various checks have been performed on Eq. (2.4) toverify that every entry in this mass matrix is correct, see Appendix B.For charginos, the mass matrix is even simpler because of the conservation of electro-magnetic charge as well as R -charge. In the MRSSM there are eight two-component fermionsfrom the winos, Higgsinos, and R -fields. Based on the R -charges and the electromagneticcharges, the eight fermions can be grouped into following four different classes which do notmix among themselves:charges Q = +1 Q = − R = +1 χ ++ ≡ (cid:16) ˜ W + , ˜ R + d (cid:17) χ + − ≡ (cid:16) ˜ W − , ˜ R − u (cid:17) R = − χ − + ≡ (cid:16) ψ +˜ W , ˜ H + u (cid:17) χ −− ≡ (cid:16) ψ − ˜ W , ˜ H − d (cid:17) (2.6)These charges imply that the charginos pair up as L chargino mass = χ T++ M χ χ −− + χ T − + M χ χ + − + c.c. , (2.7)where the chargino mass matrices in our basis are M χ = (cid:34) M λ d v d gv d µ d (cid:35) and M χ = (cid:34) M λ u v u gv u µ u (cid:35) . (2.8)– 4 –ince each of these matrices are diagonalized by bi-unitary transformations, four independentrotations are needed for the four vectors listed in Eq. (2.6) in order to diagonalize the massmatrices.We are now ready to calculate the mass eigenstates under various assumptions aboutthe parameters of the MRSSM. One qualitative difference from the MSSM is that the Diracnature of the gauginos allows us to take the gaugino masses M , M and Higgsino masses µ u , µ d real and positive (rotating phases into the holomorphic masses for the scalar adjointsand the λ parameters) [29]. Examining the gaugino mass matrices, we explore several limitswhere we can analytically demonstrate that the chargino is the NLSP and obtain a goodestimate of the mass difference. Our first treatment is to take λ u,d = λ (cid:48) u,d = 0. This ismotivated in part to simplify our analysis, but also to emphasize that nonzero values of thesecouplings are not necessary to obtain a chargino lighter than the lightest neutralino. Furthersimplifications can result by taking the large tan β limit and the large M limit. In Sec. 2.3we generalize and expand the discussion, while still working at tree-level. As we will see, thelightest chargino can be significantly lighter than the lightest neutralino when the λ couplingsare present with O ( g ) values. M with λ = λ (cid:48) = 0The gaugino mass matrices, Eqs. (2.4) and (2.8), simplify in the limit λ u,d = λ (cid:48) u,d = 0. Herewe will also take µ u = µ d = µ , which will prove extremely convenient in our analytic analysisin this section. Equal Higgsino masses will also allow us to illustrate the contrast betweenthe MSSM and the MRSSM. We are obviously not interested in the case where the bino isthe lightest gaugino, hence we take large M , consistent with the motivations of Ref. [29].Integrating out the bino and taking tan β (cid:29)
1, the neutralino and chargino mass matricesare given by M ˜ N = M − M W µ
00 0 µ and M χ = (cid:34) M √ M W µ (cid:35) . (2.9)For charginos, in the case tan β >
1, we need only consider M χ in order to find the lightestchargino mass.The block diagonal form of M ˜ N in Eq. (2.9) shows that a pair of neutral Higgsinos acquirea (Dirac) mass µ and do not mix with the other neutralinos. This allows us to further simplifythe neutralino mass matrix down to just the upper 2 × × M ˜ N is identical to M χ , except that the off-diagonalelement is smaller for the neutralino mass matrix. Simple 2 × both the Higgsino limit( µ < M , M ) and the wino limit ( M < µ, M ),– 5 –iggsino limit: ∆ + = − µM W M + O (cid:18) M (cid:19) , wino limit: ∆ + = − M M W µ + O (cid:18) µ (cid:19) . (2.10)where ∆ + ≡ m ˜ χ ± − m ˜ χ .It is well known that the one-loop electromagnetic radiative correction increases thechargino mass [11–14]. But, since the size of the mass difference shown above is not paramet-rically small compared to the loop contribution, the tree-level splitting can easily dominateso long as one of the diagonal elements does not far exceed the other.The analytical results are strikingly confirmed by examining a larger region of the MRSSMparameter space numerically. In Figs. 1 and 2 we plot the contours of ∆ + , the difference ofthe lightest chargino mass to the lightest neutralino mass. Fig. 1 explores the mostly-Higgsinolimit and was generated by holding µ = 150 GeV and tan β = 10. Similarly, Fig. 2 exploresthe mostly-wino limit and was generated holding M = 150 GeV and tan β = 10. The regionsunder the dashed lines in these Figures result in m ˜ χ ± <
101 GeV at tree level and thus willbe ignored from further consideration.In these Figures, it is clear that a chargino is the lightest gaugino in the regions with∆ + <
0. This occurs throughout the tree-level parameter space of the wino limit, shownin Fig. 2. Note that we have not included radiative corrections in these numerical resultsbecause a full calculation requires knowing the full spectrum of the model. Nevertheless, itis clear that in a wide range of parameter space, the tree-level mass difference is much largerthan are expected from radiative corrections, demonstrating that the chargino can indeed bethe NLSP. λ couplings The full parameter space of the MRSSM is much larger than the MSSM. To keep things man-ageable we consider the following simplifications: λ u = λ d ≡ λ and λ (cid:48) u = λ (cid:48) d ≡ λ (cid:48) . Moreover,it will be useful to define the parameters M λ = λv/ θ λ = λ (cid:48) /λ , analogous to theelectroweak parameters M W = gv/ θ W = g (cid:48) /g . We use perturbation techniques onthis mass-squared matrix to find approximate analytical expressions for neutralino masses. µ u = µ d = µ In this case, the neutralino mass matrix simplifies in a very interesting way: M ˜ N = M − sin β M W cos β M W M sin β tan θ W M W − cos β tan θ W M W − sin β M λ sin β tan θ λ M λ µ β M λ − cos β tan θ λ M λ µ . (2.11)– 6 – igure 1: Contours of ∆ + (GeV) in the sim-plified MRSSM at tan β = 10 and µ =150 GeV. Figure 2:
Contours of ∆ + (GeV) in the sim-plified MRSSM at tan β = 10 and M =150 GeV. The lower 2 × × β . Therefore a rotation of the last two rows andcolumns by an angle β will make the third element of the first two rows and columns vanishsimultaneously while leaving the lower 2 × µ from the other eigenstates.The chargino mass-squared are similarly found after diagonalizing M χ M T χ and M χ M T χ .These rank two matrices are straightforward to evaluate. For the purpose of comparing withthe neutralino masses, however, we expand the eigenvalues in various limits. When tan β > M χ . For the mass difference, we find • Higgsino limit: M , M > µ ∆ + = − M λ M W (cid:18) β − M − tan θ W tan θ λ M (cid:19) + O (cid:18) M , M (cid:19) , (2.12) • wino limit: M , µ > M ∆ + = − (cid:0) β − (cid:1) M λ M W µ + O (cid:18) µ , M (cid:19) . (2.13)To leading order in this expansion, a chargino is clearly the lightest gaugino in thewino limit. In the Higgsino limit, the ratio of M and M is important. When M /M > tan θ W tan θ λ / (2 sin β − .3.2 One Higgsino Heavy: µ d (cid:29) M , M and µ u = µ Another limit which can be analyzed analytically is when one Higgsino is much heavier thanthe other mass parameters. Taking µ d (cid:29) M , M , we can immediately integrate out down-type Higgsinos resulting in a 3 × M χ . Again using perturbation techniques, we find • Higgsino limit: M , M > µ ∆ + = − M λ M W sin β (cid:18) M − tan θ W tan θ λ M (cid:19) + O (cid:18) M , M (cid:19) , (2.14) • wino limit: M , µ > M ∆ + = − sin β M λ M W µ + O (cid:18) µ , M (cid:19) . (2.15)The mass differences calculated in this case are quite similar to the case of equal Higgsinomasses, except for the dependency on sin β . In fact, in the large tan β limit, Eqs. (2.14) and(2.15) reduce to Eqs. (2.12) and (2.13) respectively. A careful gaze reveals that, at sizeabletan β , the relevant portion of the gaugino mass matrices are identical whether one considersthe equal Higgsino case or one Higgsino heavy case. Having demonstrated analytically that the chargino can be the NLSP in several limits, we nowturn to analyzing a larger region of the MRSSM parameter space numerically. The neutralinoand chargino masses are determined by nine parameters: M , M , µ u , µ d , tan β , λ u , λ d , λ (cid:48) u ,and λ (cid:48) d . As it is too cumbersome to do a complete scan, we restrict to the simplificationsintroduced in the previous subsection: λ u = λ d = λ and λ (cid:48) u = λ (cid:48) d = λ (cid:48) . As before, we alsotrade the parameters λ and λ (cid:48) for the mass parameters M λ and the angle θ λ .In the first part of this subsection we keep M λ and tan θ λ fixed and scan the rest of theparameter space in order to understand the variation of ∆ + . Later we will choose a particularpoint in the M , M , µ u , µ d and tan β space, and see the dependence of ∆ + on our choice of M λ and tan θ λ .Note that given renormalization group evolution of the superpotential parameters λ and λ (cid:48) , evidently a natural choice is tan θ λ = tan θ W . We also use M λ = M W to begin ourdiscussion.First, consider the case of equal Higgsino masses with tan β = 10. (The one heavyHiggsino scenario, as we described before, is similar to the equal Higgsino scenario at sizeabletan β ). In Fig. 3 we hold µ constant and vary M and M . The shape of the contours arethe same as in Fig. 1, where we neglected the “ λ ” couplings. The striking difference is thatnow the mass difference (∆ + ) can be as large as −
30 GeV. In fact, this is why we chose µ = 200 GeV, so that the lightest chargino remains above the LEPII limit. In Fig. 4, the– 8 – igure 3: Contours of ∆ + (GeV) in theMRSSM at tan β = 10 and µ = 200 GeV inthe equal Higgsino mass limit. Figure 4:
Contours of ∆ + (GeV) in theMRSSM at tan β = 10 and M = 200 GeVin the equal Higgsino mass limit. same analysis is repeated in the wino limit, with M = 200 GeV, where again we see thatmass difference ∆ + is negative and can be up to −
30 GeV.It is interesting to investigate the dependence of ∆ + on tan β in the wino and Higgsinolimits. In Fig. 5, we take M = 500 GeV, M = 200 GeV, and vary µ and tan β . In Fig. 6,we take M = 500 GeV, µ = 200 GeV, and vary M and tan β . Both the Figures showsimilar features. For a given set of mass parameters, the contours are largely insensitive tothe value of tan β as long as tan β is sizeable. Close to tan β = 1 the contours change rapidly.At tan β = 1 we find ∆ + ≥ β < β > β → / tan β .We can also investigate how M λ and tan θ λ affects the chargino/neutralino mass hierarchy.This is most easily done by considering a specific point in the M , M , µ and tan β spacewhere ∆ + is large. From Figs. 3 and 4, we find large (negative) ∆ + when M ∼ µ . Inaddition, ∆ + is almost independent of M for large enough M . Finally, from Figs. 5 and 6we see that large ∆ + occurs at large tan β .In Fig. 7, the variation of ∆ + in the M λ − tan θ λ plane is shown. We take M = 600 GeVand tan β = 10 with µ = M = 250 GeV. We do not vary M λ beyond ±
200 GeV, since evenat this value λ already exceeds 1. The Figure clearly shows that we get the largest (negative)splitting when M λ is positive and tan θ λ is negative, up to ∆ + ∼ −O (50 GeV). The shadedarea above the dashed line is excluded since it results in a chargino with mass m ˜ χ ± < µ = M is reduced, ∆ + becomes larger (negative). On the other hand, the upperbound on M λ from the bound on the chargino mass also decreases rapidly. For larger values of µ = M , the upper bound on M λ increases and effectively one can also find a larger splitting.– 9 – igure 5: Contours of ∆ + (GeV) in theMRSSM at M = 500 GeV and M =200 GeV in the equal Higgsino limit. Figure 6:
Contours of ∆ + (GeV) in theMRSSM at M = 500 GeV and µ = 200 GeVin the equal Higgsino limit. Figure 7:
Contours of ∆ + (GeV) in the tan θ λ − M λ plane for M = 600 GeV , tan β = 10 and µ = M = 250 GeV. From various analytical results and numerical figures it is clear that a chargino can be sig-nificantly lighter than the lightest neutralino. The mass difference can far exceed the size ofradiative corrections to the gaugino masses.– 10 –
The expressions for the mass differences between the lightest neutralino and the lightestchargino in Eq. (2.12) and in Eq. (2.13) were obtained for tan β >
1. The case tan β < β ↔ cos β . • The expressions for ∆ + are given in Eq. (2.14) and in Eq. (2.15) when µ d is large.An alternate limit where µ u is greater than all other relevant masses may be found bysubstituting sin β ↔ cos β in Eq. (2.14) and in Eq. (2.15). • Given the results from the various limits, combined with Figs. 5 and 6, we find that attan β = 1, a neutralino is always the lightest gaugino. • Finally, in the absence of λ couplings, the mass difference ∆ + can be obtained in theHiggsino limit without decoupling M . Starting with Eqs. (2.12) and (2.14), hold tan θ λ finite, while independently taking M λ →
0. We obtain µ u = µ d = µ → ∆ + = − µ M W (cid:18) β − M − tan θ W M (cid:19) , Large µ d → ∆ + = − sin β µ M W (cid:18) M − tan θ W M (cid:19) . (2.16)These expressions demonstrate the chargino can be lighter than the neutralino in theMRSSM without λ couplings.
3. Neutralino and Chargino Masses in the MSSM
We now turn to studying the neutralino and chargino masses in the MSSM. The neutralinomass matrix in the MSSM is rank four, and although exact analytical expressions for theeigenvalues exist [33], they are not particularly transparent. We instead consider severalwell-known limits where the mass difference between the chargino and neutralino can beeasily calculated analytically. Later in this section we generalize our results using numericalcalculations. We allow the mass parameters M , M , µ to have arbitrary sign, though withoutloss of generality we can take M >
0. We do not consider arbitrary phases, since they areseverely constrained in the MSSM from the absence of electric dipole moments [34]. Theneutralino gauge eigenstates include a bino with mass M , hence in the small M limit amostly-bino neutralino will always be the lightest gaugino. The nontrivial cases of interest tous will occur when M is not the smallest parameter in the mass matrix.The two interesting limits that could have a chargino NLSP are the Higgsino limit andthe wino limit. In the Higgsino limit M , M > µ, M W , calculations of the masses of lightgauginos including radiative corrections can be found in [15,16]. The mass difference betweenthe lightest chargino and lightest neutralino is∆ + = (cid:34) (cid:18) tan θ W M M + 1 (cid:19) + (cid:18) tan θ W M M − (cid:19) µ | µ | sin 2 β (cid:35) M W M + O (cid:18) M (cid:19) , (3.1)– 11 – igure 8: Contours of ∆ + (GeV) in theMSSM at tan β = 2 and µ = 150 GeV. Figure 9:
Contours of ∆ + (GeV) in theMSSM at tan β = 2 and M = 150 GeV. The neutralino-chargino mass splitting grows as M is reduced.In the wino limit M < M , µ , the lightest neutralino and the lightest chargino are highlydegenerate. The mass difference was calculated including one-loop effects in [11–14]. Thetree-level splitting can be obtained by expanding in powers of 1 /µ , with the leading ordersplitting occurring at O (cid:0) /µ (cid:1) :∆ + = M W µ M W M − M tan θ W sin β + O (cid:18) µ (cid:19) . (3.2)One loop corrections to Eq. (3.2) are positive and typically small, of order 0.1 GeV [14].In the well-known case sign( M ) = sign( M ), it is evident from the leading terms inEqs. (3.1) and (3.2) that m ˜ χ ± > m ˜ χ for any sign of µ in either the Higgsino limit or winolimit. The lightest neutralino persists as the lightest gaugino throughout the MSSM parameterspace with sign( M ) = sign( M ). For completeness and comparison to the MRSSM, wedemonstrate this explicitly in Appendix C.More interestingly, when M < M >
0, the tree-level mass difference betweenthe lightest chargino and the lightest neutralino is no longer positive definite. This is clearalready at leading order in both the Higgsino limit Eq. (3.1) and the wino limit Eq. (3.2). Themass difference ∆ + grows as tan β →
1, and it is also generally larger in the Higgsino limit.To calculate size of the splitting, especially in regions of the parameter space not covered bythe limits given above, we turn to numerical evaluation of the masses.The following figures summarize the numerical results. First we fix tan β = 2. This isabout the smallest value of tan β that could possibly yield a lightest Higgs boson mass abovethe LEP II bound [35]. In Figs. 8 and 9 we take µ = 150 GeV and M = 150 GeV respectively.– 12 – igure 10: Contours of ∆ + (GeV) in theMSSM at M = 600 GeV and µ = 150 GeV. Figure 11:
Contours of ∆ + (GeV) in theMSSM at M = 150 GeV and µ = 250 GeV. The regions under the dotted lines in these Figures result in m ˜ χ ± <
101 GeV at tree level andwe ignore them from further consideration. When µ in Fig. 8 and M in Fig. 9 are reducedfurther, the dashed line for M ˜ χ ± = 101 GeV rises.As before, we show contours of ∆ + . In both of these Figures there are considerableregions with negative ∆ + , demonstrating that the chargino can be NLSP. Notice that thesplitting is largest in the Higgsino limit, up to about 5 GeV. We deliberately show a smallerrange of values of M and µ in Fig. 9 so that the gradient of ∆ + may be clearly seen. Here M is held fixed and we start with large µ . ∆ + increases as µ is decreased. The rate of increaseof splitting peaks as µ becomes comparable to M .We show the tan β dependence of the size of the splitting in Fig. 10. We set M = 600 GeVand µ = 150 GeV, which is a point in Fig. 8 where the splitting is the largest. We find thatthe splitting between lightest neutralino and the lightest chargino is maximized at tan β = 1for a fixed value of M and for M (cid:39) − µ for a given tan β . For the sake of completeness wealso evaluated the mass splitting in the wino limit, for the parameters M = 150 GeV and µ = 250 GeV, in Fig. 11. Not surprisingly, the splitting is smaller than that in the Higgsinolimit. However, the purpose of the Figure is to demonstrate that a wino-like chargino NLSPalso occurs at small tan β .Finally, we comment on the gaugino mass hierarchy within a common extension of theMSSM that incorporates a gauge singlet S . In the next-to-minimal supersymmetric standardmodel (NMSSM), the superpotential includes W ⊃ λ S SH u H d + 16 kS , (3.3)where the Higgsino mass µ = λ S (cid:104) S (cid:105) is generated after S acquires an scalar expectation value.– 13 –he chargino mass matrix is unchanged while the neutralino mass matrix is enlarged to a 5 × m S = kµ/λ S and the singlino-Higgsino mixing λ S . Of these parameters, λ S can always be chosen to be positive, while thesign of m S is arbitrary (as is the sign of µ ). Like the MSSM, we find the neutralino is heavierthan the chargino whenever all mass parameters ( M , M and m S ) are positive. However, if m S < M , M >
0. This result can be understoodby observing that bino mixing is entirely analogous to singlino mixing with other neutralinostates, replacing the bino mass with the singlino mass and the g (cid:48) v u,d Higgsino mixing terms by λ S v u,d . Therefore, for the purposes of determining eigenvalues, the singlino acts just like thebino. As a result, the chargino can be lightest when either M or m S is negative. However,just like the MSSM, the chargino-neutralino splitting remains small, (cid:46)
4. Chargino Decay
Having demonstrated that the chargino can be the NLSP, we now turn to considering thedecay of the chargino into the gravitino. Given that the mass splitting between the charginoNLSP and a neutralino next-to-next-to-lightest supersymmetric particle (NNLSP) can besmall, it may be possible for the neutralino NNLSP to decay directly to a gravitino. Hence,in the discussion below, we consider both chargino and neutralino 2-body decays to thegravitino. In the next section we will compare the rates for 2-body decay of the NNLSPdirectly into a gravitino against the 3-body decay of the NNLSP into the NLSP.Charginos could decay to W ± plus gravitino, or to an electrically charged ( ± R -chargeneutral) scalar plus gravitino. Similarly, neutralinos decay into a photon, Z or a neutralscalar plus gravitino. In the MSSM, these scalars are contained in H u , H d , while in theMRSSM, there are extra scalars in Φ ˜ W , Φ ˜ B . In addition R u , R d contain scalars, but theycarry R -charge 2 and cannot be involved in the decay. The form of the decay width of achargino (or neutralino) ˜ χ into a gravitino and a spectator particle ( X ) is [5]:Γ( ˜ χ → ˜ G + X ) ∼ κ m χ πM pl ˜ m / (cid:16) − m X m χ (cid:17) (4.1)where κ is an order one mixing angle. The decay width (4.1) is sensitive to the mass ofthe spectator particle ( m X ), and decays to heavier final states are kinematically suppressed.The kinematic factor makes the decays to the lightest particle possible, W ± in the case of achargino and γ for a neutralino, the preferred mode. Additionally, the charged scalar massmatrices contains at least one extra parameter, namely b (or m A ), compared to the gauginomass matrices. By adjusting the additional parameter(s) we are always free to focus on thesimpler scenario where decays to charged scalars are kinematically forbidden. Converting theabove width to a decay length for a chargino at rest, L = 3 . κ − (cid:16)
100 GeV m ˜ χ (cid:17) (cid:16) ˜ m /
10 eV (cid:17) (cid:16) − . (cid:16) m ˜ χ
100 GeV (cid:17) − (cid:17) − mm (4.2)– 14 –he characteristic decay length is proportional to m / . In weakly-coupled messengersectors, the gravitino mass can be related to the sparticle masses. If the mediation scaleis low, the gravitino is typically too light to produce a visible charged track. With stronginteractions present in the hidden sector, however, the gravitino mass can be significantlyenhanced with respect to the rest of the spectrum [36–38], opening up the possibility of achargino track. In the following, the gravitino mass is taken to be a free parameter, and thusour analysis applies regardless of the scale of messenger interactions.Two important applications of formula (4.1) for us are: the decay of the lightest charginothrough ˜ χ ± → W ± ˜ G , and the decay of the lightest neutralino into γ + gravitino, Z + grav-itino. Within the MRSSM, working in the Higgsino limit to leading order in 1 /M : κ W ˜ G = sin β, κ γ ˜ G = sin θ W sin β M W M , κ Z ˜ G = sin β . (4.3)Alternatively, in the wino limit we have: κ W ˜ G = 1 , κ γ ˜ G = sin θ W , κ Z ˜ G = cos θ W . (4.4)to leading order in O (1 /µ ). To lowest order in either limit, the κ couplings are independentof λ, λ (cid:48) .When M is small, the lightest neutralino is mostly the neutral wino, and thus the decaysto γ, Z are simply proportional to the photino or zino fraction of ˜ W . On the other hand,when µ is small and the lightest neutralino is primarily a Higgsino, the coupling to the Z remains order one, while the coupling to the photon is suppressed.Shifting to the MSSM, in the limit µ ∼ − M , and | µ | , | M | (cid:28) | M | the lightest neutralinois a maximal mixture of Higgsino and bino, while the chargino is purely Higgsino. This contentis reflected in the κ mixing angles: κ W ˜ G = 1 + sin β M W M , κ γ ˜ G = cos θ W (cid:34) √ θ W (1 − sin 2 β ) M W M (cid:35) ,κ Z ˜ G = 12 cos β + sin θ W (cid:34) √ θ W (cid:16) − sin 2 β cos β + sin β (cid:17) M W M (cid:35) − √ θ W (cid:16) β − β + 3 sin 3 β β (cid:17) M W M . (4.5)
5. Collider Phenomenology
In low-scale supersymmetry models, the NLSP is typically either a neutralino or a chargedslepton. As all particles in a low-energy supersymmetry-breaking model eventually decaydown to the NLSP, its properties such as spin, mass, charge, and decay width form thefoundation upon which all collider studies are built. By demonstrating that a chargino canbe the NLSPs, we are opening the door to an entirely new class of sparticle signatures, witha plethora of exciting phenomenological consequences.– 15 –
00 400 500 600 700 800 900 10001.00.55.00.110.050.0100.0 m q ! ! GeV " Σ S U S Y pp W $ W % $ X $ ! pb " Figure 12:
Lowest order cross section σ SUSY ( pp → W + W − + X ) at the LHC (14 TeV). We haveapproximated the inclusive sparticle cross section by the QCD production cross section of all 12squarks. For simplicity the squarks were taken to be degenerate, and sleptons were assumed to beheavier than the squarks. Rather than study a particular exclusive process, we focus here on the inclusive crosssections for sparticle production with chargino NLSP. A detailed study of the optimal cutsto pick out a given sparticle spectrum over the background is beyond the scope of this paper;instead, our aim is to identify search channels and the most important SM backgrounds. Thiseffort is in the same spirit as Ref. [5], which studied the inclusive signal with a neutralinoNLSP, namely γγ + / E T . We also comment on the possibilities for distinguishing the charginoNLSP from other potential NLSPs. We model the inclusive sparticle signal by the total production cross section for all twelvespecies of squarks. We do not consider the leptons or Higgses, and assume all squarks tobe degenerate in mass. The inclusive squark production cross section as a function of thecommon squark mass is shown in Fig. 12 below. In this scenario, the squarks first decayinto a quark plus chargino or neutralino. The subsequent decays of the heavy (non-NLSP)charginos/neutralinos depend on the details of the gaugino spectrum: if kinematically allowed,heavy gauginos decay to a light gaugino plus a gauge bosons, otherwise they will decay intothree-body decay final state containing a light gauginos plus two fermions. Finally, thechargino NLSPs each decay into W + gravitino. Thus, every supersymmetric event containsat least two on-shell W s plus missing energy. Using this model for sparticle production, wewant to discuss the sparticle discovery potential via excesses in W + W − + / E T + X . Somepossible decay chains are shown in Fig. 13. where ˜ χ +2 , ˜ χ are the second-lightest chargino– 16 – igure 13: Possible decay chains resulting from squark production. Exactly what path is typicaldepends on details of the sparticle spectrum. and lightest neutralino respectively. As we can see, in addition to on-shell W pairs, inclusivesparticle events involve cascades containing additional (hard) quarks and leptons. Theseextra leptons/jets are especially important in scenarios where the chargino lives longer than ∼
25 ns: by the time the long-lived charginos decay to objects which can be triggered on, thenext bunch-crossing will have occurred in the detector, causing the charginos to be associatedwith the wrong event. However, if there are extra leptons/jets present in the sparticle decays,one can trigger on those objects instead of on the chargino decay products. Then, by refiningthe analysis offline to search for charged tracks, the presence of charginos could be revealed.Model Limit ξ L + ξ R MRSSM µ (cid:28) M , M MRSSM M (cid:28) µ, M µ ∼ − M , M (cid:28) M (cid:104) − √ tan θ W (cid:16) − sin 2 β sin β +cos β (cid:17) M W M (cid:105) Table 2:
The parameter values in different limits and models that determine the ratio R Γ . Another complication in this scenario is the possibility that the second-heaviest charginoand/or the lightest neutralino decay directly into gravitino + X rather than decay via three-body decays. If the heavier chargino decays directly, the signal will still contain two W bosonsplus a pair a gravitinos, while if a neutralino decays directly one of the W s is replaced by ahard photon or Z . The effect of the direct gravitino decays depends on the relative rates: R Γ = Γ( ˜ χ H → ˜ χ L f f (cid:48) )Γ( ˜ χ H → X + ˜ G ) = N f g M pl ˜ m / π M W (cid:16) ξ L + ξ R κ (cid:17)(cid:16) ∆ m ˜ χ (cid:17) = 0 . N f (cid:16) ξ L + ξ R κ (cid:17) (cid:16) ˜ m /
10 eV (cid:17) (cid:16)
100 GeV m ˜ χ (cid:17) (cid:16) ∆2 GeV (cid:17) , (5.1)where ∆ is the mass difference between the heavy chargino/neutralino ( ˜ χ H ) and the NLSP( ˜ χ L ): ∆ = m ˜ χ H − m ˜ χ L . The prefactor N f is the number of fermionic degrees of freedom ˜ χ L Neutral Higgses are also a possibility, which we ignore for simplicity here. – 17 – .011. 100.0.011 1000.011 1000.001 0.01 0.1 1. 10.246810 m (cid:144) (cid:64) eV (cid:68) (cid:68) (cid:64) G e V (cid:68) Figure 14:
Decay ratio R Γ as a function of the gravitino mass and the mass splitting ∆. R Γ < β = 10 , M = 500 GeV) is the dotted line (green),the wino limit of the MRSSM (100 GeV NLSP, tan β = 10) is the dashed line (blue), and the MSSMcase (100 GeV NLSP, M = −
200 GeV = − µ, tan β = 2) is shown as the solid line. In all three casesthe contours indicate values, from left to right, of R Γ = 0 . , R Γ = 1 , and R Γ = 100. is kinematically allowed to decay into, and ξ L , ξ R , ( κ ) are mixing angles from the chargino-neutralino- X ( ˜ χ H -gravitino- X ) interactions; values in the limits of interest are given in Ta-ble 2. We plot R Γ for the neutralino as a function of the mass splitting ∆ and gravitinomass in Fig. 14 below. The region where direct decays are important is restricted to lightgravitinos and small splittings. For gravitinos heavier than a few tenths of an eV, the threebody decays are always dominant in either of the MRSSM limits we consider. For the MSSM,the large bino component of the lightest neutralino leads to a large κ γ ˜ G and suppressed ξ L,R ,thus direct decays can be important for gravitinos as heavy as 1 eV. Neglecting direct decaysof heavy gauginos to gravitinos, we now explore various search strategies and backgrounds to W + W − + / E T . The experimental search strategy will depend greatly on the lifetime of the chargino. If thechargino is long-lived but decays within the detector, it will leave a charged track, possiblyleading to a displaced vertex. As mentioned above, triggering on the long-lived charginos istroublesome. However, provided they can be found, displaced vertices are great discriminantof new physics from background . Lifetimes in the vicinity of 5 ns would actually be ideal: 5 ns is small enough that triggering/resolution is – 18 –f the charginos decay promptly, conventional variables such as H T = (cid:80) i = (cid:96),j p T and M eff = H T + / E T [39] will likely be the first to indicate discovery. Additionally, these variableswill be effective handles for separating signal from background. The optimum value for cutsin these variables depends on the superpartner mass scale.In the case of prompt charginos, the dominant SM backgrounds in W + W − / E T + X are:¯ tt +jets , single top + jets, W + W − /W ± Z +jets, and W ± /Z +jets . Exactly which backgroundis most important will depend on the strategy for W pair detection.Irreducible SM backgrounds that result from W + W − → (cid:96) + ν + jj , contain only one truesource of missing energy - the neutrino from the leptonic W decay. In addition to providingless / E T to SM events (compared to two neutrinos), with a single source of missing energy onecan reconstruct the transverse mass of the W ( m T,W ) from the p T of the lepton and the / E T .For the SM events the transverse mass distribution will exhibit a peak near M W , followed bysteep dropoff. Meanwhile, the m T distribution for the signal will fall much slower (after M W )due to the excess / E T carried by the gravitinos. Therefore, by cutting on m T,W (cid:38)
100 GeVwe can remove a large fraction of the background while maintaining the signal. The signalmay still be polluted by events where two leptons are produced but one of them is missed bythe detector. An important background which falls into this category is ¯ tt → (cid:96)ν(cid:96) (cid:48) ν (cid:48) ¯ bb .A second channel to search for W pairs, (cid:96)ν(cid:96) (cid:48) ν (cid:48) , has the advantage of an additional lepton,which greatly reduces the W ± /Z + jets background. Unfortunately, when both W ’s decayleptonically, the SM backgrounds have higher missing energy, so the efficiency of / E T cuts willbe reduced. The p T and η of the two leptons remain useful variables for discriminating signalfrom background.The third possibility is for both W s to decay hadronically. In that case, the signal is4+ jets + / E T , with four of the jets breaking up into two pairs, each pair reconstructing to a W .Although reconstructing W s can be difficult with realistic jet resolution, the background canbe greatly suppressed by imposing hard cuts (100 GeV or more) on the p T of the reconstructed W s and the missing energy. However, unless there is an additional lepton in these eventsfrom an earlier cascade decay, multijet QCD becomes an important and difficult background.Further detailed study, in all of the channels mentioned above as well as for a more diversespectrum are needed. If the chargino NLSP is long-lived, a charged track will be visible in the detector. Dependingon the exact lifetime of the chargino, the charged track will either exit the detector or end ina displaced vertex. Either way, a visible track allows us to rule out the majority of neutralino not an issue, yet long enough that timing information in the calorimeter could be used to differentiate signalfrom background. We thank Dirk Zerwas for bringing this point to our attention. By “+jets” we are including heavy quark flavors ( b, c ) as well as light. The ¯ tt and single-top backgrounds can be reduced further by rejecting any events with a b -tag, at theexpense of compromising the stop/sbottom contribution to (12), while W/Z + jets can be reduced by requiringtwo jets to reconstruct a W . – 19 –LSP scenarios. However, distinguishing a chargino NLSP from a charged slepton NLSPrequires more careful study.As sleptons are much heavier their decay products, the lepton coming from the displacedvertex will tend to be collinear with the slepton track. On the other hand, because W sare so massive they will emerge from chargino decays without significant boosting; thereforechargino tracks will demonstrate a distinct kink feature. Additionally, W bosons decay to jetsand (democratically) into all three leptons - therefore chargino decays will lead to e, µ, τ eachin equal amounts, up to reconstruction and detector efficiency effects. Selectron and smuonNLSPs also decay to lepton plus / E T , but their final state is a particular lepton flavor (forat least approximately flavor-diagonal slepton masses). By simply counting the number andtype of leptons in a data sample, one should easily be able to distinguish selectrons/smuonsfrom charginos. Stau NLSPs are somewhat trickier because taus decay democratically toelectrons and muons; however, provided adequate tau-tagging capability at the LHC, stausshould also be easily distinguishable from charginos.Short-lived charginos are somewhat trickier. In principle, the identity of the particleproduced along with the gravitino can tell us something about the NLSP; for neutralinosthis spectator particle is likely a photon, for a chargino it will be a W , and for sleptons thespectator is a lepton. Additionally, the maximum energy achieved by the spectator dependson the LSP mass, allowing one to distinguish WIMP LSP scenarios from gravitino LSPscenarios [4]. In practice, the success of spectator-identification will depend on the details ofthe sparticle spectrum, and it is easy to dream up tricky scenarios.To distinguish the chargino NLSP limit of the (N)MSSM from the MRSSM additionalobservables are needed. One handle is that the hallmark same-sign-lepton signal will be ab-sent [24] whenever the gauginos have a Dirac mass. A second, though less robust discriminantis the mass difference between the lightest neutralino and the lightest chargino. In the MSSMthe NLSP chargino occurs when several gauginos are nearly degenerate, thus it is difficult toarrange mass differences (even with large radiative corrections) greater than ∼
10 GeV. Inthe MRSSM large neutralino-chargino mass differences are easier to accommodate, especiallyif λ, λ (cid:48) (cid:54) = 0.
6. Conclusions
In this paper we have studied a new signal of supersymmetry that results when a chargino isthe NLSP and the gravitino is LSP. A necessary condition for this scenario to occur is thata chargino must be the lightest gaugino. We have found • The chargino can be the lightest gaugino in a wide range of parameter space whenneutralinos are Dirac fermions, such as in the MRSSM. • In the MSSM a chargino can be the NLSP essentially only in the case sign( M ) (cid:54) =sign( M ) = sign( µ ). – 20 – There is qualitative difference between the generated gaugino mass hierarchies depend-ing on whether the neutralinos are Dirac fermions or Majorana fermions. In the MSSM,the splitting is large when tan β is small, as opposed to the case of Dirac gauginos whenthe splitting is maximized for large tan β . • In addition we also observe quantitative difference between the two cases. In the MSSMthe splitting is small ( (cid:46) O (5 GeV)) and in the MRSSM the splitting can be as big as O (30 GeV) at tree level.Given a light gravitino, a chargino NLSP will decay into a gravitino and an on-shell W .Summarizing the phenomenology: • If the gravitino mass is far larger than O (100 eV), a chargino produced at a colliderwill escape the detector leaving a charged track. • If the gravitino mass is of order O (1 −
100 eV), a chargino produced at a collider canhave a displaced vertex and/or track, resulting in a decay well away from the interactionvertex. • If the gravitino mass is far smaller than O (1 eV), a chargino produced at a collider willdecay promptly into a W and a gravitino.In the first two cases, the charged track provides a great way to discriminate signal frombackground. Distinguishing a chargino NLSP from a slepton NLSP requires exploiting thesmaller boost and flavor-democratic decay of the W . When the chargino decays promptly,the pair of W s in each event provide a striking signature of sparticle production. Standardmethods may suffice to reduce background and extract the signal, however additional benefitcould be achieved by taking advantage of the characteristic W W + / E T + X final state. Dedi-cated studies of exclusive sparticle production may provide more promising opportunities todiscover supersymmetry with a chargino NLSP. Acknowledgments
We thank S. Thomas, N. Weiner and D. Zerwas for helpful conversations. GDK and AMthank the Aspen Center for Physics for hospitality where part of this work was completed.GDK acknowledges the support of the KITP Santa Barbara, where some of this work wasperformed, and the support in part by the National Science Foundation under Grant No.PHY05-51164. This work was supported in part by the DOE under contracts DE-FG02-96ER40969 (GDK, TSR) and DE-FG02-92ER40704 (AM).
A. The Fox-Nelson-Weiner model with Dirac gaugino masses
The FNW model [22] contains the same content as the MRSSM but without the R u , R d fields.Gaugino masses arise exclusively from the “supersoft” operator, Eq. (2.1). The Higgs sector– 21 –n this model, however, has an ordinary µ -term in place of Eq. (2.2). Just like the MSSM,the µ -term marries the Higgsinos ˜ H u and ˜ H d with each other, and after expanding aroundthe nonzero vevs of the two Higgses, one finds mass terms between the gauginos and theHiggsinos. The resulting neutralino mass terms are given by L ⊃ N T0 M n N , (A.1)where N = ˜ W ˜ B ˜ H d ψ W ψ ˜ B ˜ H u and M n = gv d / √ M − gv u / √
20 0 − g (cid:48) v d / √ M g (cid:48) v u / √ gv d / √ − g (cid:48) v d / √ µM M − gv u / √ g (cid:48) v u / √ µ . (A.2)The fields ψ i are the fermions in the supermultiplets Φ i . In particular, ψ ˜ W is a SU (2) W triplet containing a neutral ψ W and as well as charged components ψ ± ˜ W ≡ √ (cid:16) ψ (1)˜ W ± iψ (2)˜ W (cid:17) .The chargino mass terms in this model are given by: L ⊃ (cid:104) ˜ W + ψ +˜ W ˜ H + u (cid:105) M gv d M gv u µ ψ − ˜ W ˜ W − ˜ H − d . (A.3)Similar to the discussion of the MRSSM, in the limit of large tan β (where v d →
0) theneutralino mass matrix simplifies drastically. A linear combination of U (1) R under which thegauginos are charged and U (1) Y under which the Higgsinos are charged is preserved and thusneutralinos become Dirac fermions with the following mass matrix: L ⊃ (cid:104) ˜ W ˜ B ˜ H d (cid:105) M − gv u / √ M g (cid:48) v u / √
20 0 µ ψ W ψ ˜ B ˜ H u . (A.4)The chargino mass matrix also simplifies: L ⊃ (cid:104) ˜ W + ψ +˜ W ˜ H + u (cid:105) M M gv u µ ψ − ˜ W ˜ W − ˜ H − d . (A.5)Despite U (1) R broken by the Higgs sector in the model, it is easy to see that a charginocan be the NLSP. In the large tan β limit, the mass matrices in Eqs. (A.4) and (A.5) areidentical to what was obtained in the MRSSM in the case when the two Higgsinos have equalmasses (i.e. µ u = µ d = µ ). Similarly, when µ d (cid:29) M , M , µ u = µ and tan β is large, therelevant part of the mass matrices in the MRSSM in Eqs. (2.4) and (2.8) become identical toEqs. (A.4) and (A.5) respectively. – 22 – igure 15: Contours of ∆ + (GeV) in theMSSM at tan β = 2 and µ = 150 GeV. Figure 16:
Contours of ∆ + (GeV) in theMSSM at tan β = 2 and M = 150 GeV. B. Verifying the Form of the MRSSM Gaugino Mass Matrices
One important verification results by setting all supersymmetry breaking parameters as wellas all λ parameters to zero. In this limit, the masses of the neutralinos and charginos arisedue to the vev of the Higgses and their kinetic terms. The kinetic terms of the Higgses areobviously independent of R -symmetry. This implies that the masses of the heavy charginosand neutralinos should be identical whether in the MSSM or in the MRSSM. This requirementverifies that the elements in the mass matrices proportional to gauge couplings have the correctform.In order to verify the rest of the couplings we expanded the corresponding terms in theLagrangian in terms of electromagnetic eigenstates in the place of weak eigenstates. Thenumerical factors between the couplings of the charge neutral states and the charged statesnow determine various elements in the mass matrices. H u R u → H u R u + H + u R − u H u Φ ˜ W R u ⊂ √ H u (cid:18) √ W R u + Φ +˜ W R − u (cid:19) W Φ ˜ W → W + Φ − ˜ W + W − Φ +˜ W + W Φ W , (B.1)where the charge eigenstates are defined the usual way. C. Chargino and Neutralino Masses in the MSSM with sign( M ) = sign( M ) It is illuminating to numerically explore a broader range of the MSSM parameter space. InFigs. 15 and 16 we plot the contours of ∆ + , the difference of the lightest chargino mass to– 23 –he lightest neutralino mass. Fig. 15 explores the mostly-Higgsino limit and was generatedby holding µ = 150 GeV and tan β = 2. Similarly, Fig. 16 explores the mostly-wino limit andwas generated holding M = 150 GeV and tan β = 2. In each case, the Figures clearly show∆ + > m ˜ χ ± <
101 GeV at tree level. When µ in Fig. 15 and M inFig. 16 are reduced further, the dashed line for M ˜ χ ± = 101 GeV rises. For example setting µ = 120 GeV excludes the entire region below M = 450 GeV.The numerical results carry additional insight into the chargino/neutralino system. Acommon feature of Fig. 15 and Fig. 16 is that the contours decrease as one moves to largervalues of both the x − and y − coordinates (to the top-right corner). In Fig. 15 this can beunderstood by noting that if both M , M are heavy, the gauginos associated with them(the bino and winos) can be integrated out; the low energy effective theory is left with justHiggsinos, which contain a pair of degenerate neutralinos and charginos, i.e. ∆ + → µ and M ), shown in Fig. 16 where the decoupling is much more rapid. This provides additionalverification of our analytical results in Eq. (3.1) where we saw that the splitting was moresuppressed in the wino case. References [1] G. F. Giudice and R. Rattazzi,
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