application/xml[formula omitted]–[formula omitted] mixing in [formula omitted] models with flavor-changing neutral currentsVernon BargerCheng-Wei ChiangJing JiangPaul LangackerPhysics Letters B 596 (2004) 229-239. doi:10.1016/j.physletb.2004.06.105journalPhysics Letters BCopyright © 2004 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26935963-426 August 20042004-08-26229-23922923910.1016/j.physletb.2004.06.105http://dx.doi.org/10.1016/j.physletb.2004.06.105doi:10.1016/j.physletb.2004.06.105http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.3

PLB21181S0370-2693(04)00967-010.1016/j.physletb.2004.06.105Elsevier B.V.Phenomenology

Fig. 1

Three-dimensional plot of xs (a) and sin2ϕs (b) versus ρL and ϕL with a Z′-mediated FCNC for left-handed b and s quarks. The color shadings in both plots have no specific physical meaning.

Fig. 2

Contour plot of xs and sin2ϕs with a Z′-mediated FCNC for left-handed b and s quarks. The shaded region is for xs<20.6, which is excluded at 95% CL by experiments. The hatched region corresponds to 1σ ranges around the SM value xs=26.3±5.5 (black curve). The solid curves open to the left are contours for xs=50 (red in the web version), 70 (green in the web version) and 90 (blue in the web version) from left to right. The curves open to the right are contours for sin2ϕs=0.5 (dotted), −0.5 (solid) and the SM value −0.07±0.01 (dashed).

Fig. 3

xs (a) and sin2ϕs (b) as functions of ϕL and ϕR for ρL=ρR=0.001. The color shadings have no specific physical meaning.

Bs–B¯s mixing in Z′ models with flavor-changing neutral currents

Vernon

Barger

barger@oriole.physics.wisc.edu

Cheng-Wei

Chiang

chengwei@physics.wisc.edu

Jing

Jiang

jiangj@hep.anl.gov

Paul

Langacker

pgl@electroweak.hep.upenn.edu

aDepartment of Physics, University of Wisconsin, Madison, WI 53706, USAbHEP Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USAcDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA

Editor: M. Cvetič

Abstract

In models with an extra U(1)′ gauge boson family non-universal couplings to the weak eigenstates of the standard model fermions generally induce flavor-changing neutral currents. This phenomenon leads to interesting results in various B meson decays, for which recent data indicate hints of new physics involving significant contributions from b→s transitions. We analyze the Bs system, emphasizing the effects of a Z′ on the mass difference and CP asymmetries.

Introduction

The study of B physics and the associated CP-violating observables has been suggested as a good means to extract information on new physics at low energy scales [1–7]. Since B–B¯ mixing is a loop-mediated process within the standard model (SM), it offers an opportunity to see the footprints of physics beyond the SM. The currently observed ΔMd=0.489±0.008 ps−1[8] and its mixing phase sin2β=0.736±0.049 extracted from the Bd→J/ψKS mode [9] agree well with constraints obtained from other experiments [10]. However, no such information other than a lower bound ΔMs>14.5 ps−1[11] is available for the Bs meson yet.Based upon SM predictions, ΔMBs is expected to be about 18 ps−1 and its mixing phase ϕs only a couple of degrees. In contrast to the Bd system, the more than 25 times larger oscillation frequency and a factor of four lower hadronization rate from b quarks pose the primary challenges in the study of Bs oscillation and CP asymmetries. Since the Bs→J/ψϕ decay is dominated by a Cabibbo–Kobayashi–Maskawa (CKM) favored tree-level process, b→cc¯s, that does not involve a different weak phase in the SM, its asymmetry provides the most reliable information about the mixing phase ϕs.Although new physics contributions may not compete with the SM processes in most of the b→cc¯s decays, they can play an important role in Bs–B¯s mixing because of its loop nature in the SM [12]. In particular, the mixing can be significantly modified in models in which a tree-level b→s transition is present. Thus, measurement of the properties of Bs meson mixing is of high interest in future B physics studies as a means to reveal new physics [13,14]. Since the current B factories do not run at the ϒ(5S) resonance to produce Bs mesons, it is one of the primary objectives of hadronic colliders to study Bs oscillation and decay in the coming years [15,16].Flavor changing neutral currents (FCNC) coupled to an extra U(1)′ gauge boson arise when the Z′ couplings to physical fermion eigenstates are non-diagonal. One way for this to happen is by the introduction of exotic fermions with different U(1)′ charges that mix with the SM fermions [17–21] as occurs in E6 models. In the E6 case, mixing of the right-handed ordinary and exotic quarks, all SU(2)L singlets, induces FCNC mediated by a heavy Z′ or by (small) Z–Z′ mixing, so the quark mixing can be large. Mixing between ordinary (doublet) and exotic (singlet) left-handed quarks induces FCNC mediated by the SM Z boson [21]. We will also allow for this possibility, but in this case the quark mixing must be very small.Another possibility involves family non-universal couplings. It is well-known that string models naturally give extra U(1)′ groups, at least one of which has family non-universal couplings to the SM fermions [22–25]. Generically, the physical and gauge eigenstates do not coincide. Here, unlike the above-mentioned E6 case, off-diagonal couplings of fermions to the Z′ boson can be obtained without mixing with additional fermion states. In these types of models, both left-handed and right-handed fermions can have family non-diagonal couplings with the Z′, while couplings to the Z are family diagonal (up to small effects from Z–Z′ mixing).The Z′ contributions to Bs–B¯s mixing are related to those for hadronic, semileptonic, and leptonic B decays in specific models in which the diagonal Z′ couplings to qq¯, ℓ+ℓ−, etc., are known, but are independent in general.11

Bs–B¯s mixing in leptophobic E6 models was considered in Ref. [21]. A model with non-universal right-handed couplings was discussed in [26]. Mechanisms for flavor change in dynamical symmetry breaking models are described in [27].

We have found that in specific models, Bs–B¯s mixing effects can be significant while being consistent with the other constraints; these results will be presented elsewhere. In the present Letter, we will treat the mixing in a model-independent way.Recently, we have studied the implications of a sizeable off-diagonal Z′ coupling between the bottom and strange quark in the indirect CP asymmetry of B→ϕKS decay [28], which appears to show a significant deviation from the SM prediction [5,6,29,30]. Here we extend our analysis to Bs–B¯s mixing where the Z′ contributions also enter at the tree level. Applications to the B→πK anomaly are under investigation [31].The Letter is organized as follows. In Section 2, we review the basic formalism of Bs–B¯s mixing. In Section 3, we evaluate ΔMs in the SM. In Section 4, we include the Z′ contributions, allowing both left-handed and right-handed couplings in the mixing, and study their effects on observables. Our main results are summarized in Section 5.2

Bs–B¯s mixing

In the conventional decomposition of the heavy and light eigenstates (1)|Bs〉L=p|Bs0〉+q|B¯s0〉,|Bs〉H=p|Bs0〉−q|B¯s0〉, the mixing factor (2)(qp)SM≃M12SM*M12SM, has a phase (3)ϕsSM=2arg(VtbVts*)=−2λ2η¯≃−2°,sin2ϕsSM≃−0.07, where the theoretical expectation Γ12SM≪M12SM is used. The approximate formula Eq. (2) receives a small correction once Γ12SM is included. Model independently, this only shifts ϕs is at the few percent level. With errors on λ and η¯ included, we have the SM expectation that sin2ϕsSM≃−0.07±0.01.The off-diagonal element of the decay matrix, Γ12SM, is evaluated by considering decay channels that are common to both Bs and B¯s mesons, and M12 is the off-diagonal element of the mass matrix. Due to the CKM enhancement, Γ12SM is dominated by the charm-quark contributions over the up-quark contribution in a box diagram. Unlike the Kaon system, Γ12SM is much smaller than M12SM for B mesons because the former is related to the B meson decays and set by the scale of its mass, whereas the latter is proportional to mt2. We can safely assume that Γ12 is not significantly modified by new physics because Γ12 receives major contributions from CKM favored b→cc¯s decays in the SM, and the SM result Γ12≪M12 is unlikely to change.The mass difference of the two physical states is (4)ΔMs≡MH−ML≃2|M12| . The width difference is (5)ΔΓ≡ΓH−ΓL=2Re(M12*Γ12)|M12|=2|Γ12|cosθ, where the relative phase is θ=arg(M12/Γ12). Since Γ12 is dominated by the contributions from CKM favored b→cc¯s decays, we have θ=arg(−(VtbVts*)2/(VcbVcs*)2)≃π[32], and thus ΔΓ≃−2|Γ12| is negative in the SM. Although Γ12 is unlikely to be affected by new physics, the width difference always increases as long as the weak phase of M12 gets modified [33].The observability of Bs–B¯s oscillations is often indicated by the parameter (6)xs≡ΔMsΓs, where Γs=(4.51±0.18)×10−13 GeV, converted from the world average lifetime τs=1.461±0.057 ps[8]. The expected large value of xs is a challenge for experimental searches. Currently, the result from all ALEPH [34], CDF [35], DELPHI [36], OPAL [37], and SLD [38] studies of ΔMs with a combined 95% confidence level (CL) sensitivity on ΔMs of 18.3 ps−1 gives [11](7)ΔMs>14.5 ps−1andxs>20.8. It is also measured that mBs=5369.6±2.4 MeV[8] and ΔΓs/Γs=−0.16−0.16+0.15(|ΔΓs|/Γs<0.54) (with the 95% CL upper bound given in parentheses [11]) consistent with recent next-to-leading-order (NLO) QCD estimates [39]. In comparison, the Bd system has mBd=5279.4±0.5 MeV, ΔMd=(0.489±0.008) ps−1, xd=0.755±0.015, and τBd=1.542±0.076 ps[8].3

ΔMs in the SM

The |ΔB|=2 and |ΔS|=2 operators relevant for our discussions are: (8)OLL=[s¯γμ(1−γ5)b][s¯γμ(1−γ5)b],O1LR=[s¯γμ(1−γ5)b][s¯γμ(1+γ5)b],O2LR=[s¯(1−γ5)b][s¯(1+γ5)b],ORR=[s¯γμ(1+γ5)b][s¯γμ(1+γ5)b]. Because of the V–A structure, only the operator OLL contributes to Bs–B¯s mixing in the SM. The other three operators appear in the Z′ models because of the right-handed couplings and operator mixing through renormalization, as considered in the next section.In the SM the contributions to (9)M12SM≃12mBs〈Bs0|HeffSM|B¯s0〉 are dominated by the top quark loop. The result, accurate to NLO in QCD, is given by [40](10)M12SM=GF212π2MW2mBsfBs2(VtbVts*)2η2BS0(xt)[αs(mb)]−6/23[1+αs(mb)4πJ5]BLL(mb), where xt=(mt(mt)/MW)2 and (11)S0(x)=4x−11x2+x34(1−x)2−3x3lnx2(1−x)3. Using mt(mt)=170±5 GeV, we find S0(xt)=2.463. The NLO short-distance QCD corrections are encoded in the parameters η2B≃0.551 and J5≃1.627[40]. The bag parameter BLL(μ) is defined through the relation (12)〈B¯s|OLL|Bs〉≡83mBs2fBs2BLL(μ). In the following numerical analysis, we will use GF=1.16639×10−5 GeV−2 and MW=80.423±0.039 GeV[8], and write the SM part of ΔMs as (13)ΔMsSM=1.19|VtbVts*0.04|2(fBs23 MeV)2(BLL(mb)0.872)×10−11 GeV.Current lattice calculations still show quite large errors on the hadronic parameters fBs=230±30 MeV and BLL(mb)=0.872±0.005[41–43]. However, the ratio (14)ξ≡fBsBˆBsfBdBˆBd can be determined with a much smaller theoretical error, where BˆBq is the renormalization-independent bag parameter for the Bq meson (q=d,s). Therefore, the error on ΔMs within the SM can be evaluated by comparing with ΔMd, i.e., (15)ΔMsSM=ΔMdSMξmBsmBd(1−λ2)2λ2[(1−ρ¯)2+η¯2]. Using the measured values of the Wolfenstein parameters [44]λ=0.2265±0.0024, A=0.801±0.025, ρ¯=0.189±0.079, and η¯=0.358±0.044[10], ξ=1.24±0.07[45], and the mass parameters quoted above, we obtain the SM predictions (16)ΔMsSM=(1.19±0.24)×10−11 GeV=18.0±3.7 ps−1,xsSM=26.3±5.5. As noted above, the central value of xs is slightly larger than the current sensitivity based upon the world average. Recent LHC studies show that with one year of data, ΔMs can be explored up to 30 ps−1 (ATLAS), 26 ps−1 (CMS), and 48 ps−1 (LHCb) (corresponding to xs up to 46, 42, and 75); the LHCb result is based on exclusive hadronic decay modes [16]. The sensitivity of both CDF and BTeV on xs can also reach up to 75 using the same modes [15], for a luminosity of 2 fb−1. The sensitivity on sin2ϕs is correlated with the value of xs, and it becomes worse as xs increases. A statistical error of a few times 10−2 can be reached at CMS and LHCb for moderate xs≃40[16].4

Z′ contributions

For simplicity, we assume that there is no mixing between the SM Z and the Z′ (small mixing effects can be easily incorporated [17]). A purely left-handed off-diagonal Z′ coupling to b and s quarks results in an effective |ΔB|=2, |ΔS|=2 Hamiltonian at the MW scale of (17)HeffZ′=GF2(g2MZg1MZ′BsbL)2OLL(mb)≡GF2ρL2e2iϕLOLL(mb), where g2 is the U(1)′ gauge coupling, g1=e/(sinθWcosθW), MZ′ is the mass of the Z′, and BsbL is the FCNC Z′ coupling to the bottom and strange quarks. The parameters ρL and the weak phase ϕL in the Z′ model are defined by the second equality. Generically, we expect that g2/g1∼1 if both U(1) groups have the same origin from some grand unified theory, and MZ/MZ′∼0.1 for a TeV-scale Z′. If |BsbL|∼|VtbVts*|, then an order-of-magnitude estimate gives us ρL∼O(10−3), which is in the ballpark of giving significant contributions to the Bs–B¯s mixing. The Z′ does not contribute to Γ12 at tree level because the intermediate Z′ cannot be on shell. After evolving from the MW scale to mb, the effective Hamiltonian becomes [40](18)HeffZ′=GF2[1+αs(mb)−αs(MW)4πJ5]R6/23ρL2e2iϕLOLL(mb), where R=αs(MW)/αs(mb). Although the above effective Hamiltonian is largely suppressed by the ratio (g2MZ)/(g1MZ′), it contains only one power of GF in comparison with the corresponding quadratic dependence in the SM because the Z′-mediated process occurs at tree level.The full description of the running of the Wilson coefficient from the MW scale to mb can be found in [40]. We only repeat the directly relevant steps here. The renormalization group equation for the Wilson coefficients C→, (19)ddlnμC→=γT(g)C→(μ), can be solved with the help of the U matrix (20)C→(μ)=U(μ,MW)C→(MW), in which γT(g) is the transpose of the anomalous dimension matrix γ(g). With the help of dg/dlnμ=β(g), U obeys the same equation as C→(μ). We expand γ(g) to the first two terms in the perturbative expansion, (21)γ(αs)=γ(0)αs4π+γ(1)(αs4π)2. To this order the evolution matrix U(μ,m) is given by (22)U(μ,m)=(1+αs(μ)4πJ)U(0)(μ,m)(1−αs(m)4πJ), where U(0) is the evolution matrix in leading logarithmic approximation and the matrix J expresses the next-to-leading corrections. We have (23)U(0)(μ,m)=V([αs(m)αs(μ)]γ→(0)/2β0)DV−1, where V diagonalizes γ(0)T, i.e., γD(0)=V−1γ(0)TV, and γ→(0) is the vector containing the diagonal elements of the diagonal matrix γD(0). In terms of G=V−1γ(1)TV and a matrix H whose elements are (24)Hij=δijγi(0)β12β02−Gij2β0+γi(0)−γj(0), the matrix J is given by J=VHV−1.The operators OLL and ORR do not mix with others under renormalization. Their Wilson coefficients follow exactly the same RGEs, where the above-mentioned matrices are all simple numbers. The factor (25)[1+αs(mb)−αs(MW)4πJ5]R6/23 in Eq. (18) reflects the RGE running. On the other hand, O1LR and O2LR form a sector that is mixed under RG running. Although the Z′ boson only induces the operator O1LR at high energy scales, O2LR is generated after evolution down to low energy scales and, in particular, its Wilson coefficient C2LR is strongly enhanced by the RG effects [46].With contributions from both the SM and the Z′ boson with only left-handed FCNC couplings included, the Bs mass difference is (26)ΔMs=ΔMsSM(1+ΔMsZ′ΔMsSM)=18.0|1+3.858×105ρL2e2iϕL| ps−1. The corresponding result for the oscillation parameter is (27)xs=26.3|1+3.858×105ρL2e2iϕL|. With couplings of only one chirality, the physical observables ΔMs, xs, and sin2ϕs are periodic functions of the new weak phase ϕL with a period of 180°.

Fig. 1

(a) shows the effects of including a Z′ with left-handed coupling. We see that if ρL is small, xs is dominated by the SM contribution and has a value ∼26. For ϕL around 90° and ρL between 0.001 and 0.002, the Z′ contribution tends to cancel that of the SM and reduces xs to be smaller than the SM value of 26.3. In Eq. (27) and Fig. 1(a), we see that the Z′ has a comparable contribution to the SM if ρL≳0.002, independent of the actual value of ϕL. The planned resolution of Fermilab run II and LHCb are both about xs≲75[15,16]. Thus, a ρL greater than about 0.003 will result in an xs beyond the planned sensitivity. If xs is measured to fall within a range, one can read from the plot what the allowed region is for the chiral Z′-model parameters. The same discussion can easily be applied to a Z′ model with only right-handed couplings. Fig. 1(b) shows sin2ϕs as a function of ρL and ϕL. As ρL increases, sin2ϕs goes through more oscillations when ϕL varies from 0 to π.

In Fig. 2

, we show the overlayed plot of the contours of fixed xs and those of fixed sin2ϕs. The shaded region in the center shows the experimentally excluded points in the ϕL–ρL plane that induce xs values smaller than 20.6. The hatched area corresponds to the parameter space points that produce xs values falling within the 1σ range of the SM value of 26.3. Contours for higher values of xs are also shown. The SM predicted sin2ϕs≃−0.07±0.01 would appear as narrow bands around the sin2ϕs=−0.07 curves. Note that even if the xs measurement turns out to be consistent with the SM expectation, it is still possible that the new physics contributions, such as the Z′ model considered here, can alter the sin2ϕs value significantly. It is therefore important to have a clean determination of both quantities simultaneously. Once xs and sin2ϕs are extracted from Bs decays, one can determine ρL up to a two-fold ambiguity and ϕL up to a four-fold ambiguity, except for the special case with sin2ϕs≃0.

ΔΓs can be determined with a high sensitivity by measuring the lifetime difference between decays into CP-specific states and into flavor-specific states. Using the J/ψϕ mode, simulations determine [16] that the LHC can measure the ratio ΔΓs/Γs with a relative error ≲10% for an actual value around −0.15. Tevatron simulations show that ΔΓs/Γs can be measured with a statistical error of ∼0.02. For a sufficiently large ρL, the cosθ factor in Eq. (5) increases from −1 at ϕL=0° (mod 180°) to the maximum 1 at ϕL=90° (mod 180°). We are left with the phase ranges 0°≲ϕL≲30°, 60°≲ϕL≲120°, and 150°≲ϕL≲180° (mod 180°) where a 3σ determination of ΔΓs can be made.Once the right-handed Z′ couplings are introduced, we also have to include the new |ΔB|=2 operators O1LR, O2LR, and ORR defined in Eq. (8) into the effective Hamiltonian that contributes to the Bs–B¯s mixing. The matrix element of ORR is the same as that of OLL, while those of O1LR and O2LR are (28)〈B¯s|O1LR|Bs〉=−43(mBsmb(mb)+ms(mb))2mBs2fBs2B1LR(mb),(29)〈B¯s|O2LR|Bs〉=2(mBsmb(mb)+ms(mb))2mBs2fBs2B2LR(mb).For the Z′ coupling to right-handed currents, we define new parameters ρR and the associated weak phase ϕR: (30)ρReiϕR≡g2MZg1MZ′BsbR. At the MW scale, we have additional contributions to the effective Hamiltonian due to the right-handed currents, similar to Eq. (17). The terms due to the left–right mixing are (31)HeffZ′⊃GF2ρLρRei(ϕL+ϕR)(O1LR+O1RL,O2LR+O2RL)(10). In the RGE running, the Wilson coefficient of O1LR mixes with that of O2LR; the relevant anomalous dimension matrices are [46](32)γ(0)=(6Nc120−6Nc+6Nc),(33)γ(1)=(1376+152Nc2−223Ncf2003Nc−6Nc−443f714+9Nc−2f−2036Nc2+4796+152Nc2+103Ncf−223Ncf), where Nc is the number of colors and f is the number of active quarks. At the scale of the B meson masses, the value of f is 5.We take mb(mb)=4.4 GeV, ms(mb)=0.2 GeV, and ΛMS¯(5)=225 MeV. Following Eqs. (23) and (24), we find the effective Hamiltonian terms for the operators O1,2LR at mb to be (34)HeffZ′⊃GF2ρRρLei(ϕL+ϕR)(O1LR+O1RL,O2LR+O2RL)(0.930−0.711). Note that there is no contribution of the operator O2LR at the MW scale. It is induced through the operator mixing in RGE running and actually has an important effect at the mb scale, as one can see from its Wilson coefficient in Eq. (34).In the numerical analysis, we use the central values of B1LR(mb)=1.753±0.021 and B2LR(mb)=1.162±0.007 given in Ref. [42] with the decay constant fBs the same as before. The predicted mass difference with all the Z′ contributions included is then (35)ΔMs=18.0|1+3.858×105(ρL2e2iϕL+ρR2e2iϕR)−2.003×106ρLρRei(ϕL+ϕR)| ps−1. The overall contribution to xs from the SM and Z′ is (36)xs=26.3|1+3.858×105(ρL2e2iϕL+ρR2e2iϕR)−2.003×106ρLρRei(ϕL+ϕR)|. To illustrate the interference among different contributions, we set ρL=ρR=0.001 and plot xs and sin2ϕs versus the weak phases ϕL and ϕR in Fig. 3

(a) and (b), respectively.

First, we note that after the RGE running the operators O1LR and O2LR interfere constructively. This can be seen from the relative minus sign between the Wilson coefficients in Eq. (34) and a corresponding relative minus sign in the hadronic matrix elements given in Eqs. (28) and (29). Because of the constructive interference and the fact that the bag parameters B1LR and B2LR are twice as large as BLL, the LR and RL operators together become the dominant contributions. The interference of all the terms makes xs reach its maximum when one of the weak phases is 180° and the other is 0° (mod 360°). If ρL and ρR are both much smaller than 10−3, the variation in xs in the ϕL–ϕR space will be indistinguishable from the predicted SM range. Compared to Fig. 1(a) for Z′ with only LL couplings, Fig. 3(a) shows that even for large values of ρL and ρR, xs can be smaller than 20.6 due to the interference among all the terms in Eq. (36). The current xs⩾20.6 bound excludes the regions with ϕL+ϕR≃0° (mod 360°). Because of the assumed equal values of ρL and ρR, the parameter space points with the same sin2ϕ output lie along directions that are roughly parallel to the ϕL+ϕR=360° line. For the more general cases of different ρL and ρR values, the crests and troughs in Fig. 3(b) are no longer parallel to the ϕL+ϕR=360° line.5

Conclusions

In this Letter we discuss the effects of a Z′ gauge boson with FCNC couplings to quarks on the Bs–B¯s mixing. We show how the mass difference and CP asymmetry are modified by the left-handed and right-handed b–s–Z′ couplings that may involve new weak phases ϕL and ϕR. In the particular case of a left-chiral (right-chiral) Z′ model, one can combine the measurements of ΔMs (or xs) and sin2ϕs to determine the coupling strength ρL (ρR) and the weak phase ϕL (ϕR) up to discrete ambiguities. Once comparable left- and right-chiral couplings are considered at the same time, we find the left–right interference terms dominate over the purely left- or right-handed terms, partly because of the renormalization running effects and partly because of the larger bag parameters.

Acknowledgments

C.-W.C. would like to thank the hospitality of the high energy theory group at University of Pennsylvania. This work was supported in part by the United States Department of Energy, High Energy Physics Division, through Grant Contract Nos. DE-FG02-95ER40896, W-31-109-ENG-38, and EY-76-02-3071.

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