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Home , Gaugino, Higgsino

I. AND

The and the electroweak 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 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 , or unless R-parity is not conserved, because it is the only MSSM that can make a good dark 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 -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 - 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 and the . (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 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.

The is a color octet , 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 exchange and -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. (23)] due to its color octet nature, and because it couples to all the quark-squark pairs. The leading two-loop corrections to the gluino pole mass typically increase the prediction by another 1 2%. −

III. SQUARKS AND SLEPTONS

In principle, any scalars with the same electric charge, R-parity, and color quantum num- bers can mix with each other. This means that with completely arbitrary soft terms, the mass eigenstates of the squarks and sleptons of the MSSM should be obtained by diago- nalizing the three 6 6 mass-squared matrices for up-type squarks (˜u , c˜ , t˜ , u˜ , c˜ , t˜ ), × L L L R R R ˜ ˜ ˜ ˜ down-type squarks (dL, s˜L, bL, dR, s˜R, bR), and charged sleptons (˜eL, µ˜L, τ˜L, e˜R, µ˜R, τ˜R), and one 3 3 mass-squared matrix for sneutrinos (˜ν , ν˜ , ν˜ ). The generl form of the 6 6 matrix × e µ τ × is 2 2 2 † ∗ m +(T3f Qf s )c2βm + Mf M af vf µ Yf vf6 2 f˜L W Z f M ˜ = − − . (25) f † † 2 2 2 † af vf µYf vf6 m ˜ Qf sW c2βmZ + Mf Mf ! − fR − The situation is considerably simplified if two special conditions are imposed on the flavor structure of the soft breaking terms:

1. Squark and slepton mass-squared matrices are universal:

2 2 1 2 2 1 2 21 2 2 1 2 21 MQ˜ = mQ˜ , Mu˜c = mu˜ , Md˜c = md˜ , ML˜ = mL˜ , Me˜c = me˜ . (26)

2. The A-matrices of trilinear scalar couplings are proportional to the corresponding Yukawa matrices:

au = AuYu, ad = AdYd, ae = AeYe. (27)

We will later come back to a detailed discussion whether this structure is theoretically motivated and phenomenologically unavoidable. For now, we analyze the spectrum that

5 would follow if (26) and (27) hold. Then, most of the mixing angles are very small. The third-generation squarks and sleptons can have masses very different from those first- and second-generation counterparts, because of the effects of large Yukawa (yt,yb,yτ ) and soft

(at, ab, aτ ) couplings in the RG equations: d 32 2 1 16π2 m2 = X + X g2 M 2 6g2 M 2 g2 M 2 + g2S, (28) dt Q˜3 t b − 3 3| 3| − 2| 2| − 15 1| 1| 5 1 2 d 2 32 2 2 32 2 2 4 2 16π m c = 2X g M g M g S, (29) dt u˜3 t − 3 3| 3| − 15 1| 1| − 5 1 2 d 2 32 2 2 8 2 2 2 2 16π m ˜c = 2Xb g M3 g M1 + g S, (30) dt d3 − 3 3| | − 15 1| | 5 1 d 6 3 16π2 m2 = X + X 6g2 M 2 g2 M 2 g2S, (31) dt L˜3 τ b − 2| 2| − 5 1| 1| − 5 1 2 d 2 24 2 2 6 2 16π m c = 2Xτ g M1 + g S, (32) dt e˜3 − 5 1| | 5 1 where

2 2 2 2 2 2 2 2 S Tr[Y m ] = m m + Tr[M M 2M c + M + m c ], ≡ j φj Hu − Hd Q˜ − L˜ − u˜ d˜c e˜ 2 2 2 2 2 X = 2 y (m + m + m c )+2 a , t | t| Hu Q˜3 u˜3 | t| 2 2 2 2 2 Xb = 2 yb (m + m ˜ + m ˜c )+2 ab , | | Hd Q3 d3 | | 2 2 2 2 2 X = 2 y (m + m + m c )+2 a . (33) τ | τ | Hd L˜3 e˜3 | τ | Furthermore, the third generation squarks and charged sleptons can have substantial mixing in pairs (t˜L, t˜R), (˜bL, ˜bR), (˜τL, τ˜R). In contrast, the first and second generation squarks and sleptons have negligible Yukawa couplings (and, consequently, under our assumption (27), also negligible a-couplings), so they end up in seven very nearly degenerate, unmixed pairs

(˜eR, µ˜R), (˜νe, ν˜µ),(˜eL, µ˜L), (˜uR, c˜R), (d˜R, s˜R), (˜uL, c˜L), (d˜L, s˜L). Let us first consider the spectrum of the first- and second-generation squarks and slep- tons. In models with universal boundary conditions, the running masses-squared can be conveniently summarized as follows: 1 m2 = m2 = m2 + K + K + K , Q˜1 Q˜2 0 3 2 36 1 4 m2 = m2 = m2 + K + K , u˜1 u˜2 0 3 9 1 1 m2 = m2 = m2 + K + K , d˜1 d˜2 0 3 9 1 1 m2 = m2 = m2 + K + K , L˜1 L˜2 0 2 4 1 2 2 2 me˜1 = me˜2 = m0 + K1. (34)

6 A key point is that the same K3, K2 and K1 appear in these equations, since all the chiral supermultiplets couple to the same gauginos with the same gauge couplings. The different couplings in front of K1 just correspond to the various values of the . The

Ki’s arise from the RG running proportional to the gaugino masses. Explicitly,

3/5 ln Q0   1 2 2 Ka(Q)=  3/4  dtga(t) Ma(t) (a =1, 2, 3). (35)   × 2π3 ln Q | |   Z 4/3     Here Q0 is the input RG scale at which the universal boundary conditions apply, and Q should be taken to be evaluated near the squark and slepton mass under consideration, presumably less than 1 TeV. The running parameters ga(Q) and Ma(Q) obey d 1 β g = b g3, (b , b , b ) = (33/5, 1, 3), ga ≡ dt a 16π2 a a 1 2 3 − 2 2 2 2 M1/g1 = M2/g2 = M3/g3 = m1/2/gU . (36)

If the input scale is taken to ne the scale of coupling unification, M 2 1016 GeV, one U ≈ × finds that numerically

K 0.15m2 , K 0.5m2 , K (4.5 6.5)m2 (37) 1 ≈ 1/2 2 ≈ 1/2 3 ≈ − 1/2 for Q m . Note that K K K ; this is a consequence of the relative sizes of the ∼ Z 3 ≫ 2 ≫ 1 gauge couplings. The large uncertainty in K3 is due, in part, to the experimental uncertainty in the QCD coupling constant, and in part due to the uncertainty in where to choose Q, since K3 runs rather quickly below TeV. 2 Quite generally (for example, in gauge mediation, where there is a different m0 for each type of gauge representation), one expects that the squarks are heavier than the sleptons. Regardless of the type of model, there is also a “hyperfine” splitting in the squark and slepton mass spectrum produced by electroweak symmetry breaking. Each squark and slepton φ gets a contribution ∆φ to its mass-squared, coming from the SU(2)L and U(1)Y D-term quartic interactions of the form (squark)2(Higgs)2 and (slepton)2(Higgs)2, where the 0 0 neutral Higgs scalars Hu and Hd get VEVs. They are model independent for a given value of tan β: ∆ =(T g2 Y g′2)(v2 v2)=(T Q sin2 θ )cos2β m2 , (38) φ 3φ − φ d − u 3φ − φ W Z where T3φ, Yφ and Qφ are the third component of , the weak hypercharge, and the electric charge of the left-handed chiral supermultiplet to which φ belongs. For

7 1 2 2 2 1 1 2 2 example, ∆u˜ = ( sin θW )cos2β m , ∆ ˜ = ( + sin θW )cos2β m , and ∆u˜ = L 2 − 3 Z dL − 2 3 Z R 2 2 2 2 3 sin θW cos2β mZ . These D-term contributions are typically smaller than the m0 and

K1,2,3 contributions, but should not be neglected. They split apart the components of

SU(2)L doublets. The mass splittings between the members of the doublet within each of the first two generations is governed by the model independent sum rules

2 2 2 2 2 2 2 2 m m = m ˜ m = g (v v )/2= cos2βm . (39) e˜L − ν˜e dL − u˜L u − d − W

2 In the allowed range tan β > 1. It follows that me˜L > mν˜e and md˜L > mu˜L , with the magnitude of the splitting constrained by electroweak symmetry breaking. Next we consider the masses of the top squarks, for which there are several non-negligible ˜† ˜ ˜† ˜ contributions. First, there are the mass-squared terms for tLtL and tRtR, that are just equal ro m2 + ∆ and m2 + ∆ , similar as for the first two generations. Second, there are Q˜3 u˜L u˜3 u˜R 2 ˜† ˜ ˜† ˜ contributions equal to mt for each of tLtL and tRtR. These come from the F -terms in thd 2 0∗ 0˜ † ˜ ˜† ˜ scalar potential of the form yt Hu Hu(tLtL + tRtR), with the Higgs fields replaced by their VEVs. (Of course, there are similar contributions to all other squarks and sleptons, but, for most purposes, they are negligibly small.) Third, there are contributions to the scalar potential from F -terms of the form µ∗y t˜† t˜ H0∗ + h.c.. These contribute to the stop mass − t R L d termsd as µ∗vy cos βt˜† t˜ +h.c.. Finally, there are contributions to the scalar potential from − t R L 3 ˜† ˜ the soft (scalar) couplings attRQ3Hu + h.c.. Putting all of these contributions together, we have a mass-squared 2 2 matrix for the top squarks, which in the interaction basis (t˜ , t˜ ) × L R is given by t˜ ˜† ˜† 2 L mt˜ = (tL tR)Mt˜ , (40) L − t˜R ! where m2 + m2 + ∆ v(a∗ sin β µy cos β) 2 Q˜3 t u˜L t − t Mt˜ = . (41) v(a sin β µ∗y cos β) m2 + m2 + ∆ ! t − t u˜3 t u˜R This hermitian mass matrix can be diagonalized by a unitary matrix to give the mass eigenstates: ∗ t˜1 ct˜ s˜ t˜L = − t , (42) t˜2 ! st˜ ct˜ ! t˜R ! 2 2 2 2 of masses m < m . Here c˜ + s˜ = 1. If the off-diagonal terms of Eq. (41) are real, t˜1 t˜2 | t| | t| then ct˜ and st˜ are the cosine and sine of a stop mixing angle θt˜, which can be chosen in the range 0 θ˜ < π. Because of the large RG effects proportional to X in Eqs. (28) and (29), ≤ t t at the electroweak scale one often finds that m2 < m2 , and both of these quantities are u˜3 Q˜3

8 usually significantly smaller than the squark masses-squared for the first two generations. The diagonal terms in (41) tend to somewhat mitigate this effect. The off-diagonal terms typically induce, however, significant mixing, which always reduces the lighter top-squark mass-squared eigenvalue. Therefore, models often predict that t˜1 is the lightest squark of all, and that it is predominantly t˜R. A very similar analysis can be performed for the bottom squarks and for the charged sleptons, which in the interaction bases (˜bL, ˜bR) and (˜τL, τ˜R) have the following mass-squared matrices: 2 2 ∗ m + m + ∆ ˜ v(a cos β µy sin β) 2 Q˜3 t dL b b M˜ = − , (43) b ∗ 2 2 v(ab cos β µ yb sin β) m + m + ∆ ˜ ! − d˜3 t dR m2 + m2 + ∆ v(a∗ cos β µy sin β) 2 L˜3 t e˜L τ τ Mτ˜ = − . (44) v(a cos β µ∗y sin β) m2 + m2 + ∆ ! τ − τ e˜3 t e˜R ˜ ˜ These can be diagonalized to give the mass eigenstates b1, b2 andτ ˜1, τ˜2, in exact analogy to Eq. (42). The magnitude and importance of mixing in the sbottom and stau sectors depen on how big tan β is. If tan β is not very large (usually this means tan β < 10), the sbottoms and ∼ staus are not significantly affected from the mixing terms and the RG effects from Xb and X , because y ,y y . In that case, the mass eigenstates are very nearly the interaction τ b τ ≪ t eigenstates, ˜bL, ˜bR, τ˜L andτ ˜R. The latter three, andν ˜τ , are then nearly degenerate with their first- and second-generation counterparts. However, even in the case of small tan β, ˜bL feels the effects of the large top Yukawa coupling. In particular, from Eq. (28) we see that X acts to decrease m2 as it is RG-evolved from the input scale to the electroweak scale. t Q˜3

Consequently, the mass of ˜bL can be significantly lighter than those of d˜L ands ˜L, even if the boundary conditions are universal. For larger values of tan β, the mixings in Eqs. (43) and (44) can be quite significant, because yb,yτ and ab, aτ are non-negligible. Just as in the case of top squarks, the lighter ˜ sbottom (b1) and stau (˜τ1) can be significantly lighter than the first and second generation counterparts, even for universal boundary conditions. Similarly,ν ˜τ can be significantly lighter thanν ˜µ, ν˜e. The requirement that the third generation squarks and sleptons have positive masses- squared implies limits on the magnitudes of a∗ sin β µy cos β, a∗ cos β µy sin β, and t − t b − b a∗ cos β µy sin β. If they are too large, then the smaller eigenvalue of Eqs. (41), (43) or τ − τ

9 TABLE I: Interaction and mass eigenstates of the MSSM.

Name Spin R-parity Interaction e.s. Mass e.s.

0 0 0 ± Higgs 0 +1 Hu,Hd h ,H , A ,H Squarks 0 1 Q˜ , u˜c, c˜c, t˜c, d˜c, s˜c, ˜bc u˜ , d˜ − 1,2,3 1,2,3,4,5,6 1,2,3,4,5,6 Sleptons 0 1 L˜ , e˜c, µ˜c, τ˜c ℓ˜ , ν˜ − 1,2,3 1,2,3,4,5,6 1,2,3 Neutralinos 1 1 B˜0, W˜ 0, H˜ 0, H˜ 0 N˜ 2 − u d 1,2,3,4 Charginos 1 1 W˜ ±, H˜ +, H˜ − C˜± 2 − u d 1,2 Gluinos 1 1g ˜ g˜ 2 − Gravitino 1 , 3 1 G˜ G˜ 2 2 −

(44) is driven negative, implying that a squark or a charged slepton gets a VEV, breaking

SU(3)C and/or U(1)EM.

IV. SUMMARY

The MSSM interaction and mass states are presented in Table I.

V. THE SUPERSYMMETRIC FLAVOR PROBLEM

A. The new physics flavor puzzle

We now derive bounds on the scale of new physics and/or its flavor structure. We do so in a model-independent language, and will later apply it to supersymmetry. Consider, for example, the following dimension-six, four-fermion, flavor changing operators:

zsd 2 zcu 2 zbd 2 zbs 2 ∆F =2 = 2 (dLγµsL) + 2 (cLγµuL) + 2 (dLγµbL) + 2 (sLγµbL) . (45) L ΛNP ΛNP ΛNP ΛNP Each of these terms contributes to the mass splitting between the corresponding two neutral . For example, the term (d γ b )2 contributes to ∆m , the mass difference L∆B=2 ∝ L µ L B B 1 0 0 between the two neutral B-mesons. We use M = B ∆F =2 B and 12 2mB h |L | i 1 B0 (d γµb )(d γ b ) B0 = m2 f 2 B . (46) h | La La Lb µ Lb | i −3 B B B

10 (Analogous expressions hold for the other neutral mesons.) This leads to ∆mB/mB = 2 M B /m (z /3)(f /Λ )2. Experiments give: | 12| B ∼ bd B NP

ǫ 2.3 10−3, K ∼ × ∆m /m 7.0 10−15, K K ∼ × −14 ∆mD/mD < 2 10 , ∼ × ∆m /m 6.3 10−14, B B ∼ × ∆m /m 2.1 10−12. (47) Bs Bs ∼ ×

These measurements give then the following constraints (the bound on m(z ) is stronger I sd by a factor of (2√2ǫ )−1 than the bound on z ): K | sd| m(z ) 2 104 T eV ǫ I sd × K  q 3 √zsd 1 10 T eV ∆mK   × Λ >  2 (48) NP  √zcu 9 10 T eV ∆mD ∼  ×  2 √zbd 4 10 T eV ∆mB  ×  1  √zbs 7 10 T eV ∆mBs  ×  If the new physics has a generic flavor structure, that is z = (1), then its scale must ij O be above 103 104 TeV (or, if the leading contributions involve electroweak loops, above − 102 103 TeV). If indeed Λ T eV , it means that we have misinterpreted the hints from − NP ≫ the fine-tuning problem and the puzzle. There is, however, another way to look at these constraints:

−9 2 m(zsd) < 6 10 (ΛNP/TeV ) , I ∼ × −7 2 zsd < 8 10 (ΛNP/TeV ) , ∼ × −6 2 zcu < 1 10 (ΛNP/TeV ) , ∼ × −6 2 zbd < 6 10 (ΛNP/TeV ) , ∼ × −4 2 zbs < 2 10 (ΛNP/TeV ) . (49) ∼ × It could be that the scale of new physics is of order TeV, but its flavor structure is far from generic. One can use that language of effective operators also for the SM, integrating out all significantly heavier than the neutral mesons (that is, the top, the Higgs and the

11 weak gauge bosons). Thus, the scale is Λ m . Since the leading contributions to neutral SM ∼ W 2 mixings come from box diagrams, the zij coefficients are suppressed by α2. To identify the relevant flavor suppression factor, one can employ the spurion formalism. For example, the flavor transition that is relevant to B0 B0 mixing involves d b which transforms as − L L u u† 2 ∗ (8, 1, 1) 3 . The leading contribution must then be proportional to (Y Y ) y V V . SU(3)q 13 ∝ t tb td Indeed, an explicit calculation (using VIA for the matrix element and neglecting QCD corrections) gives B 2 2 2M12 α2 fB ∗ 2 2 S0(xt)(VtbVtd) , (50) mB ≈ −12 mW 2 2 where xi = mi /mW and x 11x x2 3x2 ln x S0(x)= 1 + . (51) (1 x)2 " − 4 4 − 2(1 x)# − − Similar spurion analyses, or explicit calculations, allow us to extract the weak and flavor suppression factors that apply in the SM:

m(zSM) α2y2 V V 2 1 10−10, I sd ∼ 2 t | td ts| ∼ × zSM α2y2 V V 2 5 10−9, sd ∼ 2 c | cd cs| ∼ × zSM α2y2 V V 2 7 10−8, bd ∼ 2 t | td tb| ∼ × zSM α2y2 V V 2 2 10−6. (52) bs ∼ 2 t | ts tb| ∼ × SM (We did not include zcu in the list because it requires a more detailed consideration. The naively leading short distance contribution is α2(y4/y2) V V 2 5 10−13. However, ∝ 2 s c | cs us| ∼ × 2 2 higher dimension terms can replace a ys factor with (Λ/mD) . Moreover, long distance contributions are expected to dominate. In particular, peculiar phase space effects have been identified which are expected to enhance ∆mD to within an order of magnitude of the present upper bound.) It is clear than that contributions from new physics at Λ 1 T eV should be suppressed NP ∼ by factors that are comparable or smaller than the SM ones. Why does that happen? This is the new physics flavor puzzle. The fact that the flavor structure of new physics at the TeV scale must be non-generic means that flavor measurements are a good probe of the new physics. Perhaps the best- studied example is that of supersymmetry. Here, the spectrum of the and the structure of their couplings to the SM fermions will allow us to probe the mechanism of dynamical supersymmetry breaking.

12 B. The supersymmetric flavor puzzle

We consider the contributions from the box diagrams involving the squark doublets of 0 0 the first two generations, Q˜ , to the D0 D and K0 K mixing amplitudes. The L1,2 − − u u∗ u u∗ contributions that are relevant to the neutral D system are proportional to K2iK1i K2jK1j , where Ku is the mixing matrix of the gluino couplings to a left-handed and their supersymmetric squark partners. (In the language of the mass insertion approximation, we calculate here the contribution that is [(δu ) ]2.) The contributions that are relevant to ∝ LL 12 d∗ d d∗ d d the neutral K system are proportional to K2i K1iK2j K1j, where K is the mixing matrix of the gluino couplings to a left-handed and their supersymmetric squark partners ( [(δd ) ]2 in the mass insertion approximation). We work in the mass basis for both ∝ LL 12 and squarks. A detailed derivation can be found in [2]. It gives:

2 2 2 2 D αsmDfDBDηQCD ˜ (∆mu˜) u u∗ 2 M12 = 2 [11f6(xu)+4xuf6(xu)] 4 (K21K11 ) , (53) 108mu˜ mu˜ 2 2 (∆˜m2)2 K αsmK fKBK ηQCD ˜ d˜ d∗ d 2 M12 = 2 [11f6(xd)+4xdf6(xd)] 4 (K21 K11) . (54) 108md˜ m˜ d

2 Here mu,˜ d˜ is the average mass of the corresponding two squark generations, ∆mu,˜ d˜ is the 2 2 mass-squared difference, and xu,d = mg˜/mu,˜ d˜. One can immediately identify three generic ways in which supersymmetric contributions to neutral meson mixing can be suppressed:

1. Heaviness: m 1 T eV ; q˜ ≫ 2. Degeneracy: ∆m2 m2; q˜ ≪ q˜ 3. Alignment: Kd,u 1. 21 ≪ When heaviness is the only suppression mechanism, as in [3], the squarks are very heavy and supersymmetry no longer solves the fine tuning problem.1 If we want to maintain supersymmetry as a solution to the fine tuning problem, either degeneracy or alignment or a combination of both is needed. This means that the flavor structure of supersymmetry is not generic, as argued in the previous section.

1 When the first two squark generations are mildly heavy and the third generation is light, as in effective supersymmetry [4], the fine tuning problem is still solved, but additional suppression mechanisms are needed.

13 C. Non-degenerate squarks at the LHC?

The 2 2 mass-squared matrices for the relevant squarks have the following form: × ˜ 2 2 1 2 2 2 † MUL =m ˜ QL + sW mZ cos2β + MuMu, 2 − 3  ˜ 2 2 1 1 2 2 † MDL =m ˜ QL sW mZ cos2β + MdMd . (55) − 2 − 3  We note the following features of the various terms:

m˜ 2 is a 2 2 hermitian matrix of soft supersymmetry breaking terms. It does not • QL × ˜ 2 ˜ 2 break SU(2)L and consequently it is common to MUL and MDL . On the other hand,

it breaks in general the SU(2)Q flavor symmetry.

The terms proportional to m2 are the D-terms. They break supersymmetry (since • Z they involve D = 0 and for D = 0) and SU(2) but conserve SU(2) . T3 6 Y 6 L Q The terms proportional to M 2 come from the F - and F -terms. They break the • q UR DR

gauge SU(2)L and the global SU(2)Q but, since FUR = FDR = 0, conserve supersym- metry.

Given that we are interested in squark masses close to the TeV scale (and the experimental 2 lower bounds are of order 300 GeV), the scale of the eigenvaluesofm ˜ QL is much higher than 2 2 † mZ which, in turn, is much higher than mc , the largest eigenvalue in MqMq . We can draw the following conclusions:

1. m2 = m2 m2 up to effects of order m2 , namely to an accuracy of (10−2). u˜ d˜ ≡ q˜ Z O 2. ∆m2 = ∆m2 ∆m2 up to effects of order m2, namely to an accuracy of (10−5). u˜ d˜ ≡ q˜ c O 3. Since K V V˜ † and K V V˜ † (the matrices V diagonalize the quark mass- u ≃ uL L d ≃ dL L qL † ˜ 2 u d squared matrices MqMq while VL diagonalizesm ˜ QL ), the mixing matrices K and K are different from each other, but the following relation to the CKM matrix holds to an accuracy of (10−5): O KuKd† = V. (56)

Eqs. (53) and (54) can be translated into our generic language:

ΛNP = mq˜, (57)

14 2 zcu = z12 sin θu,

2 zsd = z12 sin θd, ˜ 2 2 11f6(x)+4xf6(x) 2 ∆˜mq˜ z12 = αs 2 , (58) 18 mq˜ ! with Eq. (56) giving sin θ sin θ sin θ =0.23. (59) u − d ≈ c We now ask the following question: Is it possible that the first two generation squarks, ˜ 2 2 QL1,2, are accessible to the LHC (mq˜ < 1 T eV ), and are not degenerate (∆mq˜/mq˜ = (1))? ∼ O To answer this question, we use Eqs. (49). For Λ < 1 T eV , we have z < 1 10−6 NP cu × −8 ∼ ∼ and, for a phase that is 0.1, zsd < 6 10 . On the other hand, for non-degenerate 6≪ ∼ × squarks, and, for example, 11f˜ (1)+4f (1) = 1/6, we have z =8 10−5. Then we need, 6 6 12 × simultaneously, sin θu < 0.11 and sin θd < 0.03, but this is inconsistent with Eq. (59). ∼ ∼ There are three ways out of this situation:

1. The first two generation squarks are quasi-degenerate. The minimal level of degeneracy

is (˜m2 m˜ 1)/(˜m2 +m ˜ 1) < 0.12. It could be the result of RGE [5]. − ∼ 2. The first two generation squarks are heavy. Putting sin θ =0.23 and sin θ 0, as in u d ≈ models of alignment [6, 7], Eq. (48) leads to

mq˜ > 2 T eV. (60) ∼

2 2 3. The ratio x =m ˜ g/m˜ q is in a fine-tuned region of parameter space where there are acci- ˜ dental cancellations in 11f6(x)+4xf6(x). For example, for x =2.33, this combination is 0.003 and the bound (60) is relaxed by a factor of 7. ∼ Barring such accidental cancellations, the model independent conclusion is that, if the first two generations of squark doublets are within the reach of the LHC, they must be quasi- degenerate [8, 9]. Exercise: Does Kd V suffice to satisfy the ∆m constraint with neither degeneracy 31 ∼| ub| B nor heaviness? (Use the two generation approximation and ignore the second generation.) Is there a natural way to make the squarks degenerate? Examining Eqs. (55) we learn that degeneracy requiresm ˜ 2 m˜ 21. We have mentioned already that flavor universality QL ≃ q˜

15 is a generic feature of gauge interactions. Thus, the requirement of degeneracy is perhaps a hint that supersymmetry breaking is gauge mediated to the MSSM fields.

[1] S. P. Martin, arXiv:hep-ph/9709356. [2] G. Raz, Phys. Rev. D 66, 037701 (2002) [arXiv:hep-ph/0205310]. [3] N. Arkani-Hamed and S. Dimopoulos, JHEP 0506, 073 (2005) [arXiv:hep-th/0405159]. [4] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B 388, 588 (1996) [arXiv:hep- ph/9607394]. [5] Y. Nir and G. Raz, Phys. Rev. D 66, 035007 (2002) [arXiv:hep-ph/0206064]. [6] Y. Nir and N. Seiberg, Phys. Lett. B 309, 337 (1993) [arXiv:hep-ph/9304307]. [7] M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 420, 468 (1994) [arXiv:hep-ph/9310320]. [8] M. Ciuchini, E. Franco, D. Guadagnoli, V. Lubicz, M. Pierini, V. Porretti and L. Silvestrini, arXiv:hep-ph/0703204. [9] Y. Nir, JHEP 0705, 102 (2007) [arXiv:hep-ph/0703235].

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