The Higgsinos and the Electroweak Gauginos Mix with Each Other Because of the Effects ˜ 0 ˜ 0 of Electroweak Symmetry Breaking
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I. NEUTRALINOS AND CHARGINOS The higgsinos and the electroweak gauginos mix with each other because of the effects ˜ 0 ˜ 0 of electroweak symmetry breaking. The neutral higgsinos (Hu and Hd ) and the neutral gauginos (B˜ and W˜ 0) combine to form the mass eigenstates called neutralinos. The charged ˜ + ˜ − ˜ + ˜ − higgsinos (Hu and Hd ) and winos (W and W ) mix to form the mass eigenstates with charge 1 called charginos. We denote the neutralinos mass eigenstates by N˜ with i = ± i ˜± 1, 2, 3, 4. We denote the chargino mass eigenstates by Ci with i = 1, 2. (Another common 0 ± notation isχ ˜i andχ ˜i .) By convention, these mass eigenstates are labeled in ascending mass order, so that mN˜1 < mN˜2 < mN˜3 < mN˜4 and mC˜1 < mC˜2 . The lightest neutralino, N˜1, is often assumed to be the LSP, unless there is a lighter gravitino, or unless R-parity is not conserved, because it is the only MSSM particle that can make a good dark matter candidate. In this section, we describe the mass spectrum and mixing of neutralinos and charginos in the MSSM. 0 ˜ ˜ 0 ˜ 0 ˜ 0 In the interaction eigenstate basis, ψ = (B, W , Hd , Hu), the neutralino-mass part of the Lagrangian reads 1 0 T 0 M ˜ = (ψ ) M ˜ ψ + h.c., (1) L N −2 N where M 0 g′v /√2 g′v /√2 1 − d u 0 M2 gvd/√2 gvu/√2 M ˜ = − . (2) N ′ g vd/√2 gvd/√2 0 µ − − ′ √ √ g vu/ 2 gvu/ 2 µ 0 − − The entries M1 and M2 come directly from the MSSM soft Lagrangian, while µ comes from the superpotential. Electroweak symmetry breaking generates the gaugino-higgsino mixing, when the Higgs scalar is replaced by its VEV in the Higgs-higgsino-gaugino coupling. Eq. (2) can be rewritten as M 0 c s m s s m 1 − β W Z β W Z 0 M2 cβcW mZ sβcW mZ M = − , (3) N˜ cβsW mZ cβcW mZ 0 µ − − sβsW mZ sβcW mZ µ 0 − − where sβ = sin β, cβ = cos β, sW = sin θW . and cW = cos θW . The mass matrix can be diagonalized by a unitary matrix VN to obtain mass eigenstates: ˜ 0 Ni =(VN )ijψj , (4) 1 so that ∗ −1 VN MN˜ VN = diag(mN˜1 , mN˜2 , mN˜3 , mN˜4 ) (5) has real positive entries on the diagonal. In general, the parameters M1, M2 and µ can have arbitrary complex phases. A redef- inition of the phases of B˜ and W˜ always allows us to choose a convention in which M1 and M2 are both real and positive. The phase of µ within that convention is then really a physical parameter and cannot be rotated away. (We have already used the freedom to redefine the phases of the Higgs fields to make b, H0 and H0 real and positive.) How- h ui h d i ever, if the phase of µ is different from zero and, more precisely, of order one, there can be potentially disastrous predictions for CP violation, particularly unacceptably large EDMs for the electron and the neutron. (This situation can, in principle, be avoided by cancela- tions of contributions involves a-terms against those involving gaugino masses.) Therefore, it is often assumed that µ is real in the same set of conventions where M , M , b, H0 , H0 1 2 h ui h d i are real. The sign of µ is still undetermined by this constraint. We would like to stress that imposing the simultaneous vanishing (or smallness) of these CP violating phases is an ad-hoc requirement, meant the render the MSSM phenomenologically viable, not a generic prediction. In models that satisfy M1 M2 M3 m1/2 2 = 2 = 2 = 2 , (6) g1 g2 g3 gU one has the prediction 5 M tan2 θ M 0.5M (7) 1 ≈ 3 W 2 ≈ 2 at the electroweak scale. If so, then the neutralino masses and mixing angles depend on only three unknown parameters. This assumption has been made in most phenomenological studies; nevertheless, it is an assumption, that can (and should) be experimentally tested. There is a not-unlikely limit in which the electroweak symmetry breaking effects can be viewed as a small perturbation on the neutralino mass matrix. If m µ + M , µ + M , (8) Z ≪| 1| | 2| then the neutralino mass eigenstates are very nearly a “bino-like” N˜ B˜; a “wino-like” 1 ≈ N˜ W˜ 0; and a “higgsino-like” N˜ , N˜ (H˜ 0 H˜ 0)/√2, with mass eigenvalues 2 ≈ 3 4 ≈ u ± d 2 2 mZ sW (M1 + µ sin 2β) m ˜ = M + , (9) N1 1 − µ2 M 2 ··· − 1 2 2 mW (M2 + µ sin 2β) mN˜2 = M2 2 2 + , (10) − µ M2 ··· 2 − 2 2 mZ (I sin 2β)(µ M1cW M2sW ) m ˜ = µ + ∓ ± ± + , (11) N3,4 | | 2(µ M )(µ M ) ··· ± 1 ± 2 where we take M , M real and positive, and µ real and with sign I = 1. The subscripts of 1 2 ± the mass eigenstates may need to be rearranged depending on the numerical values of the parameters. In particular, the above ordering assumes M < M µ . 1 2 ≪| | The chargino spectrum can be analyzed in a similar way. In the interaction eigenstate ± ˜ + ˜ + ˜ − ˜ − basis ψ =(W .Hu , W , Hd ), the chargino mass terms in the Lagrangian are 1 ± T ± M ˜ = (ψ ) M ˜ψ + h.c., (12) L C −2 C where, in a 2 2 block form, × T 0 XC MC˜ = , (13) XC 0 ! with M2 gvu M2 √2sβmW XC = = . (14) gvd µ ! √2cβmW µ ! The mass eigenstates are related to the interaction eigenstates by two unitary 2 2 matrices × V+ and V− according to ˜+ ˜ + ˜− ˜ − C1 W C1 W = V+ , = V− . (15) ˜+ ˜ + ˜− ˜ − C2 ! Hu ! C2 ! Hd ! Note that the mixing matrix for the positively charged left-handed fermions is different from that for the negatively charged left-handed fermions. They are chosen so that ∗ −1 V−XC V+ = diag(mC˜1 , mC˜2 ), (16) with positive real entries m ˜ . Because these are 2 2 matrices, it is not difficult to solve Ci × for the masses explicitly: 2 1 2 2 2 2 2 2 2 2 2 m ˜ = M2 + µ +2m ( M2 + µ +2m ) 4 µM2 m sin 2β . (17) C1,2 2 | | | | W ∓ | | | | W − | − W | q † These are the (doubly degenerate) eigenvalues of the 4 4 matrix M M ˜ , or, equivalently, × C˜ C † the eigenvalues of XC XC . In the limit of Eq. (8), with real M2 and µ, the chargino mass ˜± ˜± eigenstates consist of wino-like C1 and a higgsino-like C2 , with masses 2 mW (M2 + µ sin 2β) mC˜1 = M2 2 2 + , (18) − µ M2 ··· 2 − mW I(µ + M2 sin 2β) m ˜ = µ + . (19) C2 | | − µ2 M 2 ··· − 2 3 The labeling here, again, corresponds to M < µ , and I =sign(µ). Note that, in the limit 2 | | that electroweak symmetry breaking effects are somewhat smaller than the electroweak conserving ones, the charged C˜1 and the neutral N˜2 winos are nearly degenerate, with mass of order M2. Similarly, the higgsinos C˜,N˜3 and N˜4 are nearly degenerate, with mass of order µ . | | The Feynman rules involving neutralinos and charginos may be inferred in terms of VN ,V+,V− from the MSSM Lagrangian. II. GLUINOS The gluino is a color octet fermion, so it cannot mix with any other particle in the SSM, even if R-parity is violated. In this regard, it is unique among all the SSM sparticles. In models with gaugino mass unification, or in models of gauge mediation, the gluino mass parameter M3 is related to the bino and wino mass parameters M1 and M2 by Eq. (6), so α 3 α M = s sin2 θ M = s cos2 θ M (20) 3 α W 2 5 α W 1 at any RG scale, up to small two-loop corrections. This implies a rough prediction, M : M : M 6:2:1 (21) 3 2 1 ≈ near the TeV scale. It is therefore reasonable to suspect that the gluino is considerably heavier than the lighter neutralinos and charginos (even in many models where the gaugino mass unification condition is not imposed). For more precise estimates, one must take into account the fact that M3 is really a running mass parameter with an implicit dependence on the RG scale Q. Because the gluino is a strongly interacting particle, M3 runs rather quickly with Q, d 1 β M = b g2M , b = 33/5. (22) M3 ≡ dt 3 8π2 3 3 3 3 A more useful quantity physically is the RG scale independent mass mg˜ at which the renor- malized gluino propagator has a pole. Including one-loop corrections to the gluino propa- gator due to gluon exchange and quark-squark loops, one finds that the pole mass is given in terms of the running mass in the DR scheme by α m = M (Q) 1+ s [15 + 6 ln(Q/M )+ A ] , (23) g˜ 3 4π 3 q˜ X 4 where 1 2 2 2 2 Aq˜ = dx x ln[xmq˜/M3 + (1 x)mq/M3 x(1 x) iǫ]. (24) Z0 − − − − The sum in Eq. (23) is over all 12 quark-squark multiplets, and we neglect small effects due to squark mixing. [As a check, requiring mg˜ to be independent of Q in Eq. (23) reproduces the one-loop RG equation in Eq. (22).] The correction terms proportional to αs in Eq. (23) can be quite significant, because the gluino is strongly interacting, with a large group theory factor [the 15 in Eq.