ar X iv :h ep -p h/ 04 04 04 1 v1 5 A pr 2 00 4 Collider Phenomenology of Light Strange-beauty Squarks Kingman Cheung1), Wei-Shu Hou2) 1 Department of Physics and NCTS, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. 2 Department of Physics, National Taiwan University, Taipei, Taiwan, R.O.C. (Dated: April 5, 2004) Abstract Strong mixing between right-handed strange and beauty squarks is a possible solution to the CP violation discrepancy in B → φKS decay as recently suggested by the Belle data. In this scenario, thanks to the strong mixing one of the strange-beauty squarks can be as light as 200 GeV, even though the generic supersymmetry scale is at TeV. In this work, we study the production of this light right-handed strange-beauty squark at hadronic colliders and discuss the detection in various decay scenarios. Detection prospect at the Tevatron Run II is good for the strange-beauty squark mass up to about 300 GeV. PACS numbers: 11.30.Pb,14.80.Ly,12.15.Ff,12.60.Jv 1 I. INTRODUCTION Supersymmetry (SUSY) is the leading candidate for physics beyond the Standard Model (SM), because it provides a weak scale solution to the gauge hierarchy problem as well as a dynamical mechanism for the electroweak symmetry breaking. Usual treatments of SUSY, however, do not address the problem of flavors. The flavor problem consists of the existence of fermion generations, their mass and mixing hierarchies, as well as the existence of CP violation in quark (and now also in neutrino) mixings, and probably has origins above the weak scale. In an interesting combination [1] of Abelian flavor symmetry (AFS) and SUSY, it was pointed out [2, 3] that a generic feature is the near-maximal s̃R–b̃R squark mixing. Such a near-maximal mixing allows for one state to be considerably lighter than the squark mass scale m̃. Such a state, called the strange-beauty squark s̃b1, carries both s and b flavors, and is bound to impact on b → s transitions. It is remarkable that we may have a hint for new physics in CP violation in B → φKS decay, which is a b → ss̄s transition. The SM predicts that the mixing-dependent CP violation in this mode, measured in analogous way as the well established CP violation in B → J/ψKS mode, should yield the same result. The Belle collaboration, however, has found an opposite sign in the B → φKS mode for two consecutive years [4, 5]. The current discrepancy with SM prediction stands at a 3.5σ level. The result from the BaBar collaboration in 2003 is at odds [6] with Belle, but the combined result is still in 2.7σ disagreement with SM expectation. While more data are needed to clarify the situation, it has been pointed out [7] that a light s̃b1 squark provides all the necessary ingredients to narrow this large discrepancy with SM prediction. It has (1) a large s–b flavor mixing, (2) a (unique) new CP violating phase, and (3) right-handed dynamics. The latter is needed for explaining why similar “wrong-sign” effects are not observed in the modes such as B → KSπ0 and η′KS. These modes yield consistent results as what was measured in B → J/ψKS. A detailed study of various B decays suggested [7] that m s̃b1 ∼ 200 GeV and mg̃ ∼ 500 GeV are needed, while the squark mass scale m̃ and other SUSY particles can be well above the TeV scale. It is clear that a squark as light as 200 GeV is of great interest since the Tevatron has a chance of seeing it. One should independently pursue the search for a relatively light s̃b1 squark, even if the B → φKS CP violation discrepancy evaporates in the next few years. 2 We note that a strange-beauty squark, carrying ∼ 50% in strange and beauty flavor, would lead to a weakening of bounds on beauty squark search based on b-tagging. In this work, we study direct strange-beauty squark-pair production, as well as the feed down from gluino- pair production and the associated production of s̃b1 with a gluino. It turns out that the dominant contribution comes from direct squark-pair production as long as the squark mass is below 300 GeV. However, for squark mass above 300 GeV, the feed down from gluino-pair production with mg̃ = 500 GeV becomes important. We also study various decay scenarios of the strange-beauty squarks at the Tevatron, which is of immediate interest. The most interesting decay mode is s̃b1 → b/s+χ̃01, which gives rise to a final state of multi-b jets plus large missing energies. The other scenarios considered are the s̃b1-LSP and the R-parity violating s̃b1 decay possibilities. The organization of the paper is as follows. In Sec. II we recapitulate the features of the model needed for our collider study. We discuss the production of the strange-beauty squark at hadronic machines in Sec. III, and its decay modes and detection in Sec. IV. Conclusion is given in Sec. V. II. INTERACTIONS We do not go into the details of the model, but mention that the d flavor is decoupled [3] to evade the most stringent low energy constraints. The generic class of AFS models [1, 2] imply a near-maximal sR–bR mixing, which is extended to the right-handed squark sector upon invoking SUSY. We focus only on the 2 × 2 right-handed strange and beauty squarks, which are strongly mixed. The mass matrix is given by L = −(s̃∗R b̃∗R)   m̃ 2 22 m̃ 2 23e −iσ m̃223e iσ m̃233     s̃R b̃R   . (1) Since the mass matrix is hermitian and the phase freedom has already been used for quarks, so there remains only one CP violating phase [2, 3, 7]. However, for collider studies it is not yet relevant. With the transformation   s̃R b̃R   = R   s̃b1 s̃b2   =   cos θm sin θm − sin θmeiσ cos θmeiσ     s̃b1 s̃b2   , (2) 3 the mass term is diagonalized as L = −(s̃b ∗ 1 s̃b ∗ 2)   m̃ 2 1 0 0 m̃22     s̃b1 s̃b2   . (3) The diagonalization matrix R enters the gluino-quark-squark and squark-squark-gluon interactions. Assuming the quarks are already in mass eigenbasis, the relevant gluino-quark- squark interaction in the mass eigenbasis is L = − √ 2gsT a kj [ −g̃aPRsjs̃b ∗ 1k cos θm + g̃aPRbjs̃b ∗ 1k sin θme −iσ −g̃aPRsjs̃b ∗ 2k sin θm − g̃aPRbjs̃b ∗ 2k cos θme −iσ + h.c. ] , (4) where PR = (1 + γ 5)/2, and a, j, k are the color indices for gluinos, quarks and squarks, respectively. The squark-squark-gluon interaction is L = −igsAaµT aij ( s̃b ∗ 1i ↔ ∂µs̃b1j + s̃b ∗ 2i ↔ ∂µs̃b2j ) +g2s (T aT b)ijA aµAbµ ( s̃b ∗ 1is̃b1j + s̃b ∗ 2is̃b2j ) , (5) where (T aT b)ij = 1 6 δabδij + 1 2 (dabc + ifabc)T c ij . The relevant Feynman rules are listed in Fig. 1. III. PRODUCTION AT HADRONIC MACHINES We have set the generic SUSY scale at TeV, except for the gluino and the light strange- beauty squark s̃b1, which could be as light as 500 and 200 GeV, respectively. These masses are still allowed by the squark-gluino search at the Tevatron [8]. In fact, these limits are more forgiving for the present case because s̃b1 does not decay into b quark 100% of the time. A. Processes and Formulas The production of the strange-beauty squark can proceed via the following processes. 1. qq̄ and gg fusion (Fig. 2(a) and (b)) qq̄, gg → s̃b1 s̃b ∗ 1 . (6) 4 (p − p ) * µ sb ~ * sb ~ 1i 1j a, µ a ij sb ~ 1k * sb ~ * 1i sb ~ 1j g µν −i g T s ~ s j g a (b )j a s i 2 g T cos )( θ a e −iσ γ1+ 2 5 −i 2 g T sin s b, a,µ ν i g s 2 ( δ + d abc ij Tab3 1 1ij ) c kj kj γ1+ θ 2 5 m m FIG. 1: The relevant Feynman rules used in this work. The momenta are going into the vertex. If the initial state is ss̄ or bb̄, there is an additional contribution from the t-channel gluino exchange diagram, shown in Fig. 2(c). Note that there are also sb̄, bs̄ → s̃b1 s̃b ∗ 1 contributions via the t-channel gluino exchange diagram only. 2. The ss,bb, s̄s̄, b̄b̄,sb, s̄b̄ initial state scattering via t- and u-channel gluino exchange diagrams ss, sb, bb → s̃b1 s̃b1, s̄s̄, s̄b̄, b̄b̄ → s̃b ∗ 1 s̃b ∗ 1 , (7) shown in Fig. 2(d). 3. Gluino pair production, followed by gluino decay, qq̄,gg → g̃g̃; g̃ → ss̃b ∗ 1, bs̃b ∗ 1, s̄s̃b1, b̄s̃b1 . (8) For ss̄, bb̄ in the initial states there are additional t- and u-channel diagrams. Note that sb̄, s̄b → g̃g̃ are also possible through the t- and u-channel diagrams. 5 sb ~ * 1 sb ~ 1 sb ~ * 1 sb ~ 1 sb ~ 1 sb ~ * 1s s g ~ sb ~ 1 s s sb ~ 1 g ~ (a) (b) g g q q (c) (d) FIG. 2: Contributing Feynman diagrams for (a) qq̄ → s̃b1s̃b ∗ 1, (b) gg → s̃b1s̃b ∗ 1, (c) ss̄ (bb̄) → s̃b1s̃b ∗ 1, and (d) ss (bb) → s̃b1s̃b1. 4. Associated production of s̃b1 with gluino sg, bg → s̃b1g̃ , (9) followed by gluino decay. Since the gluino has a mass of at least 500 GeV, we expect the t- or u-channel gluino- exchange diagrams to be much smaller than qq̄ annihilation diagrams. Moreover, the t- or u-channel gluino-exchange diagrams are only relevant for s or b in the initial state, so the contributions of which are further suppressed by their parton luminosities. Nevertheless, we include all those t-channel gluino diagrams when the initial state quarks are s or b. In gluino-pair production we also keep the t- and u-channel s̃b1-exchange diagrams for the initial state quarks s or b. 6 Direct production of s̃b1s̃b ∗ 1 Let us first introduce some short-hand notation. The ŝ, t̂, û are the usual Mandelstem variables. We define the following t̂g̃ = t̂ − m2g̃ , ûg̃ = û − m2g̃ , t̂sb = t̂ − m2s̃b1 , ûsb = û − m 2 s̃b1 , βsb = √ 1 − 4m2 s̃b1 ŝ , βg = √ 1 − 4m2g̃ ŝ , βsbg = √√√√ ( 1 − m2g̃ ŝ − m2 s̃b1 ŝ )2 − 4 m2g̃ ŝ m2 s̃b1 ŝ . The subprocess cross section for qq̄ → s̃b1s̃b ∗ 1 is given by dσ d cos θ∗ (qq̄ → s̃b1s̃b ∗ 1) = 2πα2s 9ŝ βsb [ 1 4 (1 − β2sb cos2 θ∗) − m2 s̃b1 ŝ ] , (10) where θ∗ is the central scattering angle in the parton rest frame. Integrating over the scattering angle θ∗, the cross section is given by σ(qq̄ → s̃b1s̃b ∗ 1) = 2πα2s 27ŝ β3sb . (11) The differential cross section for gg → s̃b1s̃b ∗ 1 is dσ d cos θ∗ (gg → s̃b1s̃b ∗ 1) = πα2s 256ŝ βsb ( 64 3 − 48ûsbt̂sb ŝ2 )( 1 − 2ŝm2 s̃b1 ûsbt̂sb + 2ŝ2m4 s̃b1 û2 sb t̂2 sb ) . (12) The integrated cross section is given by σ(gg → s̃b1s̃b ∗ 1) = πα2s ŝ [ βsb 5ŝ + 62m2 s̃b1 48ŝ + m2 s̃b1 6ŝ m2 s̃b1 + 4ŝ ŝ ln 1 − βsb 1 + βsb ] . (13) For completeness we also give the expressions for ss̄,bb̄ → s̃b1s̃b ∗ 1 cross sections, dσ d cos θ∗ (ss̄ → s̃b1s̃b ∗ 1) = 2πα2sβsb 9ŝ ( 1 4 (1 − β2sb cos2 θ∗) − m2 s̃b1 ŝ ) × [ 1 − 1 3 ŝ t̂g̃ cos2 θm + 1 2 ŝ2 t̂2g̃ cos4 θm ] . (14) Integrating over cos θ∗ gives σ(ss̄ → s̃b1s̃b ∗ 1) = 2πα2s 27ŝ3 { βsbŝ ( ŝβ2sb − 6ŝ cos4 θm + cos2 θm(ŝ + 2m2−) ) + cos2 θm ( 2m4− − 3ŝ cos2 θm(ŝ + 2m2−) + 2ŝm2g̃ ) log ( ŝ + 2m2− − βsbŝ ŝ + 2m2− + βsbŝ )} , (15) 7 where m2− = m 2 g̃ − m2s̃b1 . The cross section for bb̄ → s̃b1s̃b ∗ 1 can be obtained by replacing cos2 θm ↔ sin2 θm in Eqs. (14) and (15). On the other hand, the processes sb̄, bs̄ → s̃b1 s̃b ∗ 1 only have the t-channel gluino exchange diagram, and its differential cross section is given by dσ d cos θ∗ (sb̄ → s̃b1s̃b ∗ 1) = πα2sβsb 9 ŝ t̂2g̃ cos2 θm sin 2 θm ( 1 4 (1 − β2sb cos2 θ∗) − m2 s̃b1 ŝ ) . (16) Direct production of s̃b1s̃b1 Production of s̃b1s̃b1 (s̃b ∗ 1s̃b ∗ 1) pair requires ss, sb or bb (s̄s̄, s̄b̄, or b̄b̄) in the initial state. The process proceeds via t- and u-channel gluino-exchange diagrams, as shown in Fig. 2(d). The differential cross section is given by dσ d cos θ∗ (ss → s̃b1s̃b1) = πα2sβsb 18 cos4 θm m 2 g̃ [ 1 t̂2g̃ + 1 û2g̃ − 2 3 1 t̂g̃ 1 ûg̃ ] , (17) where we have explicitly put in the factor 1/2 and so cos θ∗ ranges from −1 to 1. Integrating over the angle the total cross section is σ(ss → s̃b1s̃b1) = πα2sβsb 18 cos4 θm m 2 g̃ [ 4 m4− + ŝm 2 g̃ + 8 3βsbŝ 1 ŝ + 2m2− log ( ŝ + 2m2− − βsbŝ ŝ + 2m2− + βsbŝ )] . (18) The cross section for bb → s̃b1s̃b1 can be obtained by replacing cos4 θm ↔ sin4 θm, while that for sb → s̃b1s̃b1 by replacing cos4 θm ↔ cos2 θm sin2 θm. Note that, for example, the amplitude of bb → s̃b1s̃b1 contains the phase factor e−2iσ. Obviously, when we calculate the cross section the phase factor drops out. Feed down from gluino-pair production We employ a tree-level calculation for gluino-pair production, though including the next- to-leading order (NLO) corrections [9] the cross section may increase by more than 50%. However, the overall gluino-pair production is small because we have chosen the gluino mass to be at least 500 GeV. Whether we include the NLO correction or not does not affect our conclusion. 8 Here we give the tree-level formulas for gluino-pair production, without the squark in the t- and u-channels, dσ d cos θ∗ (qq̄ → g̃g̃) = 2πα2s 3ŝ βg t̂2g̃ + û 2 g̃ + 2m 2 g̃ŝ ŝ2 , dσ d cos θ∗ (gg → g̃g̃) = 9πα2s 16ŝ βg ( 1 − t̂g̃ûg̃ ŝ2 ) ( ŝ2 t̂g̃ûg̃ − 2 + 4m2g̃ŝ t̂g̃ûg̃ − 4ŝ2m4g̃ t̂2g̃û 2 g̃ ) , (19) where we have put in the factor of 1/2 for identical particles in the final state, and cos θ∗ is from −1 to 1. The integrated cross sections are given by σ(qq̄ → g̃g̃) = 8πα2s 9ŝ βg ( 1 + 2m2g̃ ŝ ) , σ(gg → g̃g̃) = 3πα 2 s 4ŝ [ −βg ( 4 + 17 m2g̃ ŝ ) + 3 ( 4m4g̃ ŝ2 − 4m2g̃ ŝ − 1 ) log ( 1 − βg 1 + βg )] . (20) For completeness we also give the cross sections for ss̄ → g̃g̃, dσ d cos θ∗ (ss̄ → g̃g̃) = 2πα2s 3ŝ βg { t̂2g̃ + û 2 g̃ + 2m 2 g̃ŝ ŝ2 + 2 9 cos4 θm ( t̂2g̃ t̂2 sb + û2g̃ û2 sb ) + 1 2 cos2 θm 1 ŝ ( ŝm2g̃ + t̂ 2 g̃ t̂sb + ŝm2g̃ + û 2 g̃ ûsb ) + 1 18 cos4 θm ŝm2g̃ ûsbt̂sb } . (21) The formulas for bb̄ → g̃g̃ can be obtained by replacing cos θm by sin θm. Note that sb̄, s̄b → g̃g̃ only occur via the t- and u-channel diagrams, and the differential cross section is given by dσ d cos θ∗ (sb̄ → g̃g̃) = πα 2 s 27ŝ βg cos 2 θm sin 2 θm { 4 ( t̂2g̃ t̂2 sb + û2g̃ û2sb ) + ŝm2g̃ ûsbt̂sb } . (22) We have chosen the mass of gluino to be at least 500 GeV, in order not to upset lower energy constraints such as b → sγ rate, and not to violate the bound from direct search at the Tevatron [8]. The gluino so produced will decay into a strange or beauty quark plus the strange-beauty squark s̃b1. Therefore, gluino-pair production gives two more jets in the final state than direct production. Having more jet activities to tag on may help the detection, especially when b-tagging is employed. We shall discuss in more detail in the next section when we treat the decay of the s̃b1. Nevertheless, since the gluino mass is above 500 GeV, the production rate at the Tevatron is rather small. For a gluino mass of 500 GeV, the production cross section is 2.9 fb, which may increase to about 4 fb after taking into account NLO correction [9]. However, it helps only a little as far as the strange-beauty squark pair 9 production is concerned, unless the squark mass is above 300 GeV. We will take this into account in our analysis. Production of s̃b1g̃ There is another process s(b)g → g̃s̃b1 that can contribute to strange-beauty squark production, but it requires either s or b in the initial state. The differential cross section for the process is given by dσ d cos θ∗ (sg → s̃b1g̃) = πα2s 192ŝ βsbg cos 2 θm [ 24 ( 1 − 2ŝûsb t̂2g̃ ) − 8 3 ] × [ − t̂g̃ ŝ + 2(m2g̃ − m2s̃b1)t̂g̃ ŝûsb ( 1 + m2 s̃b1 ûsb + m2g̃ t̂g̃ )] . (23) For the bg initial state, the above formula is modified by changing cos2 θm ↔ sin2 θm. B. Production Cross Sections The cross sections for direct s̃b1s̃b ∗ 1 pair production at the Tevatron are shown in Fig. 3(a), where we give the individual gg and qq̄ contributions. As expected, the gluon fusion con- tribution is subdominant for m s̃b1 >∼ 100 GeV. We also show in Fig. 3(a) the same sign s̃b1s̃b1 + s̃b ∗ 1s̃b ∗ 1 production, and the associated s̃b1g̃ + s̃b ∗ 1g̃ production. These processes are three orders of magnitude smaller than s̃b1s̃b ∗ 1 pair production, and can be safely ignored at the Tevatron. Gluino-pair production cross sections at the Tevatron are given in Fig. 3(b). Similar to squark-pair production, gluino-pair production is dominated by qq̄ pair annihilation. For a gluino mass of 500 GeV the cross section is only a few fb, and thus this contribution becomes comparable to direct s̃b1 pair production only when ms̃b1 >∼ 300 GeV. Therefore, at low ms̃b1 the gluino contribution is very small, while at high m s̃b1 the gluino contribution can extend the sensitivity further. The situation is different at the LHC. We show the corresponding results in Figs. 4(a) and 4(b), respectively. We see that gluon fusion now dominates over qq̄ pair annihilation. Furthermore, gluino pair production and squark pair production cross sections are both above 10 pb for m s̃b1 = 200 GeV and mg̃ = 500 GeV, and both contributions have to be taken into account at the LHC. We also show the associated s̃b1g̃ + s̃b ∗ 1g̃ production cross 10 10-4 10-3 10-2 10-1 100 101 100 150 200 250 300 350 400 C ro ss S e ct io n (p b ) m~sb1 (GeV) ECM=1.96 TeV Tevatron ~sb1 ~sb*1 production sum gg q−q ~sb1 ~sb1 + ~sb*1 ~sb*1 ~sb1 ~g m~g = 500 GeV 10-4 10-3 10-2 10-1 100 101 200 250 300 350 400 450 500 C ro ss S e ct io n ( p b ) m~g (GeV) ECM = 1.96 TeV Tevatron m~sb1 = 200 GeV ~g ~g production ~sb1 ~g sum q-q gg FIG. 3: Total cross section for direct production of (a) the s̃b1s̃b ∗ 1 pair and (b) the g̃g̃ pair at the Tevatron. The individual gg fusion and qq̄ annihilation contributions are shown. In (a) we also show s̃b1s̃b1 + s̃b ∗ 1s̃b ∗ 1 production, and s̃b1g̃ + s̃b ∗ 1g̃ production, where we have fixed mg̃ = 500 GeV. In (b) we also show s̃b1g̃ + s̃b ∗ 1g̃ production, where we have fixed ms̃b1 = 200 GeV. 11 10-5 10-4 10-3 10-2 10-1 100 101 102 103 200 400 600 800 1000 1200 1400 1600 1800 2000 C ro ss S e ct io n (p b ) m~sb1 (GeV) ECM=14 TeV LHC ~sb1 ~sb*1 production sum gg q−q ~sb1 ~sb1 + ~sb*1 ~sb*1 ~sb1 ~g m~g = 500 GeV 10-3 10-2 10-1 100 101 102 103 200 400 600 800 1000 1200 1400 1600 1800 2000 C ro ss S e ct io n (p b ) m~g (GeV) ECM=14 TeV LHC m~sb1=200 GeV~g ~g sum ggq−q ~sb1 ~g production FIG. 4: Total cross section for direct production of (a) the s̃b1s̃b ∗ 1 pair and (b) the g̃g̃ pair at the LHC. The individual gg fusion and qq̄ annihilation contributions are shown. In (a) we also show s̃b1s̃b1 + s̃b ∗ 1s̃b ∗ 1 production, and s̃b1g̃ + s̃b ∗ 1g̃ production, where we have fixed mg̃ = 500 GeV. In (b) we also show s̃b1g̃ + s̃b ∗ 1g̃ production, where we have fixed ms̃b1 = 200 GeV. 12 section in Figs. 4(a) and 4(b). These curves are somewhat misleading, however, that their cross sections become larger than s̃b1-pair (or gluino-pair) production for large enough ms̃b1 (mg̃). This is simply because the mass of mg̃ is held fixed at 500 GeV in Fig. 4(a) while ms̃b1 is fixed at 200 GeV in Fig. 4(b). Therefore, for very large mass the s̃b1- or g̃-pair production become suppressed. Before we discuss detection, we need to understand how the s̃b1 squark decays, to which we now turn. IV. DECAY AND DETECTION OF THE STRANGE-BEAUTY SQUARK If the SUSY scale is set at TeV, all SUSY particles should be around this scale, unless one has cancellation mechanisms in the diagonalization of the neutralino, chargino, or sfermion mass matrices that allow some of them to become close to the electroweak scale. The lightness of the s̃b1 in our scenario is a particular example of this type. This in fact involves fine-tuning. However, the fine-tuning is comparable [3] to what is already seen in the quark mixing matrix. In any case, we do not discuss it further here. We concentrate on squark-pair production at the Tevatron. We put the LHC study aside as its discussion is more intricate, but of less immediate interest. It is clear from Fig. 3(a) that the dominant production channels are gg,qq̄ → s̃b1s̃b ∗ 1. Gluino-pair production with mg̃ = 500 GeV, followed by gluino decay, is only relevant for ms̃b1 >∼ 300 GeV. On the other hand, s̃b1s̃b1 and s̃b ∗ 1s̃b ∗ 1 pair production and the associated production can be safely ignored. In the following, we take on three situations for the decay of the strange-beauty squark: (i) When the s̃b1 is the lightest supersymmetric particle (LSP) and R-parity is conserved. This stable s̃b1 case also includes the case when the s̃b1 is stable within the detector but decays outside. (ii) The s̃b1 is the LSP but R-parity is violated such that it will decay into 2 jets or 1 lepton plus 1 jet. (iii) The s̃b1 is the next-to-lightest supersymmetric particle (NLSP), and either neutralino (in supergravity) or gravitino (gauge-mediated) is the LSP such that s̃b1 will decay into a strange or beauty quark plus the neutralino or gravitino. Among the three cases we particularly emphasize case (iii), which is the most popular. In the SUGRA models, one has s̃b1 → s/bχ̃01, while in gauge-mediated models s̃b1 → s/bG̃ 13 or s̃b1 → s/bχ̃01 → s/b γG̃. In any case, there will be b/s-quark jets plus a large missing energy in the final state. We simplify the picture by modelling the decay as s̃b1 → s/bχ̃01 and by varying the mass of the neutralino. A. Stable Strange-beauty Squark In this case, the s̃b1s̃b ∗ 1 pair so produced will hadronize into color-neutral hadrons by combining with some light quarks. Such objects are strongly-interacting massive particles, electrically either neutral or charged. If the hadron is electrically neutral, it will pass through the tracker with little trace. The interactions in the calorimetry would be rather intricate, since charge exchange (d̄ replaced by ū when passing by a nucleus) can readily occur. 1 However, the hadron could be electrically charged with equal probability. In this case, the hadron will undergo ionization energy loss in the central tracking system, hence behaves like a “heavy muon”. Let us discuss this possibility since it is more straightforward. The energy loss dE/dx due to ionization in the detector material is very standard [10]. Essentially, the penetrating particle loses energy by exciting the electrons of the material. Ionization energy loss dE/dx is a function of βγ ≡ p/M and the charge Q of the penetrating particle. The dependence on the mass M of the penetrating particle comes in through βγ for a large mass M and small γ [10]. In other words, dE/dx is the same for different masses if the βγ values of these particles are the same. For the range of βγ between 0.1 and 1 that we are interested in, dE/dx has almost no explicit dependence on the mass M of the penetrating particle. Therefore, when dE/dx is measured in an experiment, the βγ can be deduced, which then gives the mass of the particle if the momentum p is also measured. Hence, dE/dx is a good tool for particle identification for massive stable charged particles. In fact, the CDF Collaboration has made a few searches for massive stable charged particles [11]. The CDF analyses required that the particle produces a track in the central tracking chamber and/or the silicon vertex detector, and at the same time penetrates to the outer muon chamber. 2 1 An issue arises when the neutral hadron containing the s̃b1 may “bounce” into a charged hadron when the internal d̄ is knocked off and replaced by a ū, for example. The probability of such a scattering depends crucially on the mass spectrum of the hadrons formed by s̃b1. In reality, we know very little about the spectrum, so we simply assume a 50% chance that a s̃b1 will hadronize into a neutral or charged hadron. 2 In Run II, the requirement to reach the outer muon chamber may be dropped but it leads to a lower 14 The CDF detector has a silicon vertex detector and a central tracking chamber (which has a slightly better resolution in this regard), which can measure the energy loss (dE/dx) of a particle via ionization, especially at low βγ < 0.85 (β < 0.65) where dE/dx ∼ 1/β2. Once the dE/dx is measured, the mass M of the particle can be determined if the momentum p is measured simultaneously. Furthermore, the particle is required to penetrate through the detector material and make it to the outer muon chamber, provided that it has an initial β > 0.25 − 0.45 depending on the mass of the particle [11]. Therefore, the CDF requirement on β or βγ is (note βγ = β/ √ 1 − β2) 0.25 − 0.45 <∼ β < 0.65 ⇔ 0.26 − 0.50 <∼ βγ < 0.86 . The lower limit is to make sure that the penetrating particle can make it to the outer muon chamber, while the upper limit makes sure that the ionization loss in the tracking chamber is sufficient for detection. CDF has searched for such massive stable charged particles, but did not find any. The limits placed on the mass of these particles are model dependent [12]. Some theoretical studies on massive stable charged particles exist for gluino LSP models [13], colored Higgs bosons and Higgsinos [14], and scalar leptons [15]. We use a similar analysis for strange-beauty squark pair production with the squark remaining stable within the detector. We employ the following acceptance cuts on the squarks pT (s̃b1) > 20 GeV , |y(s̃b1)| < 2.0 , 0.25 < βγ < 0.85 . (24) In Fig. 5, we show the βγ distribution for direct s̃b1 pair production at the Tevatron. It is clear that more than half of the cross sections satisfy the βγ cut. This is easy to understand as the squark is massive such that they are produced close to threshold. In Table I we show the cross sections from direct s̃b1 pair production with all the acceptance cuts in Eq. (24), for detecting 1 massive stable charged particle (MCP), 2 MCPs, or at least 1 MCPs in the final state. The latter cross section is the simple sum of the former two. We have used a probability of 50% that the s̃b1 will hadronize into a charged hadron. In the table, we also give the feed down from gluino-pair production in the parentheses. It is obvious that the feed down is relatively small for m s̃b1 <∼ 300 GeV, but becomes significant for ms̃b1 >∼ 300 GeV. Requiring about 10 such events as suggestive evidence, the sensitivity can reach up to signal-to-background ratio. 15 0 1 2 3 βγ = p/m 0 0.5 1 1.5 2 1/ σ d σ /d (β γ) m ~ sb 1 = 400 GeV m ~ sb 1 = 300 GeV m ~ sb 1 = 200 GeV FIG. 5: The βγ ≡ p/m s̃b1 spectrum for squark-pair production at the Tevatron, where p is the squark momentum. TABLE I: Cross sections for direct strange-beauty squark pair production at the Tevatron, with the cuts of Eq. (24). Here σ1MCP, σ2MCP denote requiring the detection of 1, 2 massive stable charged particles (MCP) in the final state, respectively. Requiring at least one MCP in the final state corresponds to simply adding the two cross sections. In parentheses, we give the contribution fed down from direct gluino-pair production. m s̃b1 (GeV) σ1MCP (fb) σ2MCP (fb) σ≥1MCP (fb) 200 41 (0.46) 9.3 (0.02) 50 (0.48) 250 10.9 (0.96) 2.8 (0.14) 14 (1.1) 300 3.1 (1.2) 0.91 (0.3) 4.0 (1.5) 350 0.87 (1.3) 0.29 (0.43) 1.2 (1.8) 400 0.23 (1.4) 0.088 (0.48) 0.32 (1.8) 450 0.058 (1.4) 0.024 (0.51) 0.082 (1.9) almost m s̃b1 ≃ 300 GeV with an integrated luminosity of 2 fb−1. 16 B. s̃b1 as LSP but R-parity is violated In this case the s̃b1 pair so produced will decay via the R-parity violating terms λ ′LQDc or λ ′′ UcDcDc in the superpotential. In general, λ′ and λ′′ couplings are not considered simultaneously, otherwise it will lead to unwanted baryon decay. Since nonzero λ′′ couplings would give only multi-jets in the final state, which would likely be buried under QCD backgrounds, we only consider the λ′ coupling in the following. By the right-handed nature of the strange-beauty squark in our scenario, the third index in the λ′ coupling is either 2 or 3, and we only consider λ′ii3, λ ′ ii2 with i = 1, 2. The strange- beauty squark will decay into e−u or µ−c. Therefore, the strange-beauty squark behaves like scalar leptoquarks of the first or second generation, respectively. The decay mode of τ−t is not feasible at the Tevatron. The current published limits [16] from CDF are 213 GeV and 202 GeV for the first and second generation leptoquarks while DØ obtained limits of 225 and 200 GeV, respectively. The latest preliminary limits [16] from CDF are 230 and 240 GeV, respectively, while those from DØ are 231 and 186 GeV, respectively. In one of the preliminary plots, the combined limits from all CDF and DØ Run I and II data can push the first generation leptoquark limit to around 260 GeV, which is very impressive. 3 The sensitivity reach in Run II has been studied in TeV2000 report [17]. The reach on the first or second generation leptoquarks are 235 and 325 GeV with a luminosity of 1 and 10 fb−1, respectively. Apparently, the preliminary limits obtained by CDF and DØ with a luminosity of ∼ 200 pb−1 are already very close to or even surpass the sensitivity reach quoted in TeV2000 report. Therefore, we believe that the limit that can be reached at the end of Run II (2fb−1) is very likely above 300 GeV. With an order more luminosity, the limit may be able to reach 350 GeV: see the total cross section in Fig. 3(a) C. s̃b1 is the NLSP In this case the s̃b1 so produced will decay into a strange or beauty quark plus the neu- tralino in the supergravity framework or the gravitino (or via an intermediate neutralino into a photon and a gravitino) in the gauge-mediated framework. Experimentally, the signature 3 These limits are for the leptoquarks that decay entirely into charged leptons and quarks. If the leptoquark also decays into a neutrino and a quark, the corresponding limit is somewhat weaker. 17 is similar, except for the fact that the neutralino is of order 100 GeV while the gravitino is virtually massless compared to the collider energy. We simplify the picture by modelling the decay as s̃b1 → s/bχ̃01 and by varying the mass of the neutralino. In addition, we have to check if the strange-beauty squark will decay within the detector. In the SUGRA case, the decay rate is of electroweak strength hence the decay is prompt. However, in the gauge- mediated case, the decay rate scales as ∼ 1/F 2SUSY, where √ FSUSY is the dynamical SUSY breaking scale. Therefore, if √ FSUSY is so large, the strange-beauty squark behaves like a stable particle inside the detector. Reference [15] showed that, for √ FSUSY >∼ 107 GeV the scalar tau NLSP would behave like a stable particle inside a typical particle detector. This value applies to the strange-beauty squark as well, up to a color factor. If it is stable, one goes back to case (i). So here we focus on the prompt decay of the strange-beauty squark, which is considered to be the more popular case. There are 2 quark jets in the final state of s̃b1s̃b ∗ 1 pair production, each of them either strange or beauty flavored, and with large missing energy due to the neutralinos or gravitinos. We impose the following cuts on the jets and missing transverse momentum, and we choose the following b-tagging and mistag efficiencies 4 pT j > 15 GeV , |ηj| < 2.0 , 6pT > 40 GeV, ǫbtag = 0.6 , ǫmis = 0.05 . Note that the branching ratio of the strange-beauty squark into a b quark scales as sin2 θm. We have tested our parton-level Monte Carlo program as follows. Most events generated pass the jet (pT j and |ηj|) requirements, as long as the mass difference between the s̃b1 and χ̃01 is larger than 50 GeV. Taking B(s̃b1 → bχ̃01) = 1 (a standard b̃ squark), we verify our input b-tagging efficiency, i.e. the ratio of 0 : 1 : 2 b-tagged jets is 16 : 48 : 36. Choosing the B(s̃b1 → bχ̃01) = 0.5 value expected in our scenario, the ratio of 0 : 1 : 2 b-tagged jets becomes 49 : 42 : 9. The double-tag approach becomes far less effective, but if we only require at least one b-tagged jet in the final state, the overall efficiency is about 0.5. On one hand, this is a dilution compared to the standard b̃ squark pair production, which gives overall efficiency of 0.84. On the other hand, the prospect is still very good for Run II. In Table II, we show the cross sections in units of fb for direct squark-pair production, with 4 The mistag efficiency is the probability that a non-b jet is detected as a b-jet. 18 the squark decaying into either s/b plus a neutralino at the Tevatron with √ s = 1.96 TeV. We have set mχ̃0 1 = 100 GeV; other values of mχ̃0 1 do not affect the result in any significant way, so long as the mass difference between the squark and neutralino is larger than about 50 GeV. Note that the production cross section itself is almost independent of sin2 θm and mg̃. This is because the dominant production channel is the standard QCD s-channel qq̄ → s̃b1s̃b ∗ 1 process and we have imposed mg̃ >∼ 500 GeV. The gluino-pair production process, which is also independent of sin2 θm, is only 2.9 fb, and only becomes relevant for ms̃b1 >∼ 300 GeV. The branching ratio of the s̃b1 into a b quark, however, scales as sin 2 θm. We therefore give results for sin2 θm = 1, 0.75, 0.5, 0.25, and for 0, 1, and 2 b-tagged jet events. The case for sin2 θm = 1 is the same as a standard b̃ squark. We see that, for sin2 θm >∼ 0.5, if we only require at least one b-tagged jet rather than demanding double-tag, the cross section does not change drastically as sin2 θm decreases from 1 to 0.5. Requiring a minimum of 10 signal events as suggestive evidence for such a squark, with an integrated luminosity of 2 fb−1 the sensitivity is around 300 GeV, if sin2 θm >∼ 0.5. If the integrated luminosity can go up to 20 fb−1, then the sensitivity increases to 350 GeV. We emphasize that the double-tag vs single-tag ratio contains information on sin2 θm, while their sum, when compared with the standard b̃ squark pair production, provides additional consistency check on cross section vs mass. Such work would depend on more detailed knowledge of the detector, which we leave to the experimental groups. V. DISCUSSION AND CONCLUSION In this work, we have considered the SUSY scenario that the only light degrees of freedom are the right-handed strange-beauty squark (m s̃b1 >∼ 200 GeV) and gluino (mg̃ = 500 GeV). Such a light squark is a result of a near-maximal mixing in the 2-3 sector of the right-handed squarks, which is in turn a result of approximate Abelian flavor symmetry. We have performed calculations for direct strange-beauty squark-pair production, as well as the feed down from gluino-pair production and the associated production of s̃b1 with gluino. It turns out that the dominant contribution comes from direct squark-pair produc- tion as long as the squark mass is below 300 GeV. As one has to require mg̃ >∼ 500 GeV, which comes from low energy bounds, gluino pair production is in general subdominant. However, as the squark mass is above 300 GeV, the feed down from gluino-pair production 19 TABLE II: Cross sections in fb for direct squark-pair production at the Tevatron with √ s = 1.96 TeV, for 0, 1, 2 b-tagged events. The imposed cuts are pT j > 15 GeV, |ηj| < 2, and 6 pT > 40 GeV, b-tagging efficiency ǫbtag = 0.6, and a mistag probability of ǫmis = 0.05. In parentheses, we give the contribution fed down from direct gluino-pair production. m s̃b1 (GeV) 0 b-tag 1 b-tag 2 b-tag 0 b-tag 1 b-tag 2 b-tag sin2 θm = 1 sin 2 θm = 0.75 150 115(0.11) 288(0.54) 175(2.2) 190(0.29) 284(0.89) 104(1.6) 200 26(0.091) 70(0.49) 47(2.2) 44(0.27) 70(0.85) 28(1.7) 250 6.1(0.090) 17(0.49) 11(2.2) 11(0.27) 17(0.85) 6.8(1.7) 300 1.5(0.090) 4.2(0.49) 2.9(2.2) 2.6(0.27) 4.2(0.85) 1.7(1.7) 350 0.38(0.090) 1.1(0.49) 0.72(2.2) 0.66(0.27) 1.1(0.86) 0.43(1.7) 400 0.094(0.090) 0.26(0.49) 0.18(2.2) 0.16(0.27) 0.26(0.86) 0.11(1.7) 450 0.022(0.096) 0.06(0.51) 0.04(2.2) 0.038(0.28) 0.061(0.87) 0.025(1.7) sin2 θm = 0.5 sin 2 θm = 0.25 150 283(0.66) 243(1.2) 51(1.0) 395(1.3) 165(1.1) 17(0.40) 200 68(0.63) 61(1.1) 14(1.0) 96(1.3) 42(1.1) 4.6(0.42) 250 16(0.62) 15(1.1) 3.3(1.0) 23(1.3) 10(1.1) 1.1(0.42) 300 4.0(0.63) 3.7(1.1) 0.84(1.0) 5.8(1.3) 2.5(1.1) 0.28(0.42) 350 1.0(0.63) 0.93(1.1) 0.21(1.0) 1.4(1.3) 0.64(1.1) 0.071(0.43) 400 0.25(0.63) 0.23(1.2) 0.052(1.1) 0.35(1.3) 0.16(1.1) 0.017(0.43) 450 0.058(0.64) 0.053(1.2) 0.012(1.0) 0.083(1.3) 0.037(1.1) 0.004(0.42) with mg̃ = 500 GeV becomes sizable. Furthermore, many new avenues such as ss̄ (ss) → s̃b1s̃b (∗) 1 opens up. These are, however, very suppressed at Tevatron energies because of heavy gluino mass. We have studied three decay scenarios of the strange-beauty squarks that are relevant for the search at the Tevatron, which is of immediate interest because it can be readily done in the near future. The three decay modes that we have considered are (i) (quasi-)stable s̃b1 as in s̃b1-LSP SUSY or in gauge-mediated SUSY breaking with a very large √ F , (ii) R-parity violating decay of s̃b1 (hence s̃b1 behaves like a leptoquark), and (iii) the popular case of 20 s̃b1 → s/bχ̃01 decay, where χ̃01 is the LSP. In the first case, the s̃b1 once produced would hadronize into a massive stable charged particle like a “heavy muon”, which would ionize and form a track in the central tracking system and in the outer muon chamber. This is a very clean signature. The sensitivity for Run II with an integrated luminosity of 2 fb−1 is up to about 300 GeV, which may increase to about 350 GeV with an order more luminosity. In the second case, the s̃b1 decays like a leptoquark of the first or second generation. The best current limit is 260 GeV (preliminary [16]) for the first generation. This is already at the sensitivity level of the Tev2000 study [17] for 2 fb−1. With an order more luminosity, the limit should reach 350 GeV. In the last case, s̃b1 → s/bχ̃01 decay leads to multiple b-jets plus large missing energy in the final state. The number of b-tag events depends on the mixing angle sin θm, because the branching ratio of s̃b1 → bχ̃01 scales as sin2 θm. As long as sin2 θm >∼ 0.5, the sensitivity at the Run II with 2 fb−1 goes up to about 300 GeV. With improved b-tagging in Run II, one can also make use of the single versus double b-tag ratio as well as the b-tagged cross section to determine m s̃b1 and the mixing angle sin2 θm. At the LHC s̃b1s̃b ∗ 1 and g̃g̃ pair production cross sections are comparable, with gg fu- sion being the dominant mechanism. Unlike at the Tevatron, the associated production of sg → s̃b1g̃ becomes interesting at the LHC. Nevertheless, s̃b1s̃b1 or s̃b ∗ 1s̃b ∗ 1 pair production remains relatively unimportant. With s̃b1 as light as 200 GeV, s̃b1s̃b ∗ 1 pair production may be relatively forward. On the other hand, g̃g̃ events, followed by g̃ → s̃b1s̄/b̄, would have extra hard jets to provide more handles. The s̃b1g̃ final state, if it can be separated, can probe the mixing angle cos2 θm in the production cross section. Discovery of the strange- beauty squark at the LHC should be no problem at all, but the richness demands a more dedicated study, which we leave for future work. In conclusion, the recent possible CP violation discrepancy in B → φKS decay suggests the possibility of a light strange-beauty squark s̃b1 that carries both strange and beauty flavors. Such an unusual squark can be searched for at the Tevatron Run II, with the precaution that s̃b1 can decay into a beauty or strange quark, and the standard b̃ search should be broadened. Discovery up to 300 GeV is not a problem, and anomalous behavior in both production cross sections and the single versus double tag ratio may provide confirming evidence for the strange-beauty squark. 21 Acknowledgments This research was supported in part by the National Science Council of Taiwan R.O.C. under grant no. NSC 92-2112-M-007-053- and NSC-92-2112-M-002-024, and by the MOE CosPA project. [1] Y. Nir and N. Seiberg, Phys. Lett. B 309, 337 (1993); M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 420, 468 (1994). [2] C.K. Chua and W.S. Hou, Phys. Rev. Lett. 86, 2728 (2001). [3] A. Arhrib, C.K. Chua and W.S. Hou, Phys. Rev. D 65, 017701 (2002). [4] K. Abe et al. [Belle Collaboration], Phys. Rev. D 67, 031102 (2003). [5] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 91, 261602 (2003). 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Amidei et al., ”Future Electroweak Physics at the Fermilab Tevatron”, Report of the tev 2000 Study Group, Fermilab-Pub-96/082 (1996). 23 http://www-d0.fnal.gov/Run2Physics/np/ Introduction Interactions Production at Hadronic Machines Processes and Formulas Production Cross Sections Decay and detection of the strange-beauty squark Stable Strange-beauty Squark "0365sb1 as LSP but R-parity is violated "0365sb1 is the NLSP Discussion and Conclusion Acknowledgments References