Impact of Righthanded Strangebeauty Squark on Transitions
Abstract
As the hint for violating new physics in has weakened, we reconsider the possibility of near maximal mixing between  squarks. Such a righthanded strangebeauty squark can be realized by combining supersymmetry with an approximate Abelian flavor symmetry, and comes with a unique new violating phase from righthanded quark mixing. Naturally heavy strangebeauty squark and gluino, of order 0.5 to 1 TeV, are easily accommodated by recent timedependent violation measurements in and . Because of near maximal mixing, even with such heavy masses, the and can still strongly impact on mass difference and generate violation in the mixing, which can still be probed at Tevatron Run II. But if the scenario is realized, the LHC will provide definitive information on the new phase, and possibly discover the squark. Timedependent violation in can be probed at the future factory upgrades. Other decays influenced by large righthanded dynamics are also discussed.
pacs:
11.30.Er, 11.30.Hv, 12.60.Jv, 13.25.HwI Introduction
The existence of flavor, and the observed patterns associated with it, are not understood. violation (CPV) also seems to be closely linked to the flavor problem. In the lefthanded quark sector, which can be probed by the weak interaction, information is summarized in the CabibboKobayashiMaskawa (CKM) quark mixing matrix, which contains a unique CPV phase (usually taken as ) if there are just 3 generations. The CKM matrix elements exhibit a hierarchical pattern , , , which echo the hierarchy in quark masses. However, since the weak interaction is purely lefthanded, we have no information on righthanded flavor physics.
It was observed some time ago that Nir ; CHPRL01 , if there is an approximate Abelian flavor symmetry (AFS) in nature, then the order of magnitude of the elements of and mass matrices, in powers of , can be inferred from our current knowledge of quark masses and mixings. It turns out that the largest mixing effect, of order 1, would be between the righthanded and quarks. Such flavor mixing, though hidden in the Standard Model (SM), would be brought forward if supersymmetry (SUSY) is also realized. Near maximal mixing between the and squarks would give rise to two flavormixed “strangebeauty” squarks , with a single associate new CPV phase defined as CHPRL01 ; ArhribPRD01 . The strong interaction would now contain flavorchanging  couplings, and should affect transitions.
Sure enough, some “anomalies” have been uncovered recently in CPV in transitions, notably in , which has illustrated the frontier nature of such studies in the past few years. As the B factories mature, it is exciting that the LHC would turn on in 2007, making CPV studies involving the system accessible. Furthermore, one can directly search for the flavormixed squarks and probe a broad range of parameter space.
Timedependent CPV (TCPV) in decay () was established in 2001. Because the CKM factor for the dominantly tree level transition is expected to be almost real, is identical to , the phase in mixing (i.e. ) to very good approximation. The current world average is HFAG ( for PDG2005 PDG ). TCPV in decay, , is of interest because, in SM one expects to the percent level. This is because the loopinduced transition that underlies is controlled by the CKM factor , which again is very close to being real in SM. This makes an excellent probe of violating new physics (NP).
Interestingly, for the two consecutive years of 2002 and 2003, the combined result of BaBar and Belle strongly contradicted the SM prediction of . Even the sign was inconsistent. This socalled sign anomaly stimulated many theoretical studies which showed that, to account for the sign anomaly, one in general would need large  flavor mixing, new CPV phases, and possibly righthanded dynamics (see, for example, rhSUSYKagan ; rhSUSYKhalil ; rhSUSYKane ; rhSUSYHarnik ; CHNPRL04 ). This seemed to be an ideal situation for the strangebeauty squark. Indeed, our previous work CHNPRL04 was stimulated by the 2003 result of HFAG . We showed that a rather light and not too heavy gluino , such as GeV, together with a large new phase , could give . However, for SUSY scale at TeV, GeV would require finetuning to in the squark mass matrix ArhribPRD01 .
Since 2004, however, the experimental discrepancy has weakened considerably, and we need not adhere to our previous conclusion. Fine tuning of the mass is no longer necessary, and one could reconsider the model in a more natural setting. The Belle updated result of Bellebsss , based on 386 million events, is in agreement with BaBar result of based on 227 million events BabarphiK . The combined result of HFAG is 2.5 away from the sign anomaly of . But there is still a hint of deviation between and , which could be due to .
On the other hand, several other modes also show some discrepancy. The decay involves both penguin and tree contributions. The latter is, however, suppressed by , so is also expected in the SM. From same number of events with , BaBar finds BabarpiK , and Belle finds Bellebsss . The combined result of HFAG is different from with more than 1.4 significance, and is in the same direction as .
Similarly, is also predicted in the SM. Yet this is again in some conflict with the combined result of HFAG . However, the results of BaBar BabaretaPRK and Belle Bellebsss are at some odds with each other, so the error of 0.09 probably should be rescaled to 0.13. Note that, thanks to the large decay rate, the measurement of has better accuracy compared with . However, the large rate of is not well understood, and is likely generated not by new physics. In contrast to , the mode is simpler and more transparent. Hence we focus on the two modes of and in this work.
The main purpose of this work is to update the picture of righthanded strangebeauty squarks . We identify new preferred regions for and from recent data, and revisit the implications for mixing and the associated violating phase. We believe this update would be useful for LHC experiments, which would start in 2007. We follow the observations outlined in Ref. HouICHEP04 , which was written after 2004 data revealed drastic softening of results. With smaller deviations, it is now more customary to use the difference
(1) 
which measures the deviation from SM expectations. We put up useful benchmarks to pin down our model parameters. Since the error in the data is still large, let us take the central value of data as a criterion to be more conservative. We find three scenarios:

Scenario 1: ,

Scenario 2: ,

Scenario 3: ,
which could provide hint for NP. In addition, there is a fourth possibility,

Scenario 4: ,
which, evidently, does not discriminate between NP and the SM.
In this work, we pay attention to the issue of naturalness, that is, that and are comparable to SUSY scale. We shall see that GeV and GeV can be accounted for by the recent data. These mass regions are well within the discovery ranges at LHC CMSBrochure . Measurements of oscillations and the associated phase will provide further information on , and . In fact, these measurements may discover NP, even if further B factory results confirm Scenario 4.
This paper is organized as follows: Sec. II concentrates on the formalism for violation observables. In Sec. III we give the SM expectations. In Sec. IV we briefly recapitulate our model of near maximal mixing. Sec. V and VI gives our NP results. Discussion and conclusion are given in Sec. VII. We refer the hadronic parameters accompanied by the chromodipole effects to Appendix A.
Ii Timedependent Violation
In this section we present the formalism for TCPV. For neutral B meson decays into eigenstate , the CPV are studied by means of the time evolution asymmetry timeCPV ,
(2) 
The coefficients , and are described in terms of decay amplitudes and as
(3) 
with , where is the phase in mixing. The state satisfies . For the system, due to , one finds the simpler form which is given by
(4) 
The phase corresponds to .
With Wolfenstein parameterization, can be expressed in terms of violating parameters and as Buras98
(5) 
where . As we stated, the value of is basically determined by . Although there is a small discrepancy between unitarity fit of and direct experimental result of , the approximation remains reasonable. Therefore, in order to evade uncertainties brought from decay amplitude, we use Eq. (5) for calculation of . In what follows, we take and . We then find for , and for .
Impact of violating NP on TCPV in transitions can be understood as follows. Let us take arising from purely as an example. We write its amplitude as . The first term denotes the SM contribution which contains the strong phase but approximately no weak phase because of . The second term, instead, corresponds to the NP contribution with strong phase , accompanied by a new phase . One finds the simple expression for up to SfGronau ; rhSUSYKhalil
(6) 
where . basically follows around zero. We see from Eq. (6) that, unless , vanishes for any additional contributions that carry only strong phases. Therefore, either SM or NP without CPV mechanism, i.e. or , would give . However, we note that Eq. (6) could be diluted by the relative strong phase . For as a typical case, any NP effect is washed away in (one would then in general get large , which is not the case).
To calculate decay amplitudes we need to evaluate hadronic matrix elements. It is known that the naive factorization (NF) framework NF , which involves small strong phase, has difficulties in explaining current experimental measurements on decay rates and on direct CP violation. Recent developments of factorization frameworks, QCDF QCDF and PQCD (see, for example, piKPQCD ; pipiPQCD ), have shown the importance of annihilation processes which can generate sizable strong phases (but not at level). Such annihilation effects could manifest differently in SM and NP, and in different modes. Our interest is the genuine effects on from NP. Because of the small strong phases, the analysis within NF framework would be rather transparent. In the following we use NF in our calculation and assume absence of final state interactions. We will illustrate the possible dilution from presence of additional strong phases by introducing a heuristic term.
Iii Standard Model Expectations
This section is devoted to SM calculations. Besides following Ref. NF , we will take into account the socalled chromodipole effects from ChromoDipole . For hadronic decays from transition, the effective Hamiltonian is given by
One has NF currentcurrent, strong penguin and electroweak penguin operators, ,  and . In addition, the chromodipole operator is defined as
(8) 
where () is a generator of SU(3) with color indices , and denotes the momentum carried by the virtual gluon. The operators arise from righthanded dynamics, which are represented by exchanging everywhere. Since the weak interaction probes only lefthanded dynamics, the shortdistance coefficient in SM is suppressed by a factor of . In what follows, we will neglect the SM contributions in the primed coefficients.
Let us start from . The decay does not occur at tree level. This decay had been studied within QCDF phiKQCDF ; QCDF and PQCD framework phiKPQCD , giving decay rate in agreement with the experimental result of HFAG .
With NF, the amplitude for is given by,
(9)  
where , and is the scalar product of the meson polarization vector with meson momentum. The definition of the parameters can be found in Ref. NF . Table 1 enumerates the numerical values of which we use for calculation. We take that would be derived from . The hadronic parameter is taken for , and from the evaluation in NF, which can be found in the Appendix.
Taking MeV and , we find . Without the chromodipole effects, i.e. , one has . Due to , the chromodipole contribution is substantial, and gives large reduction for the rate. But even for , the decay rate obtained by NF calculation is far below the experimental data.
For the the decay rate, NF seems deficient. However, the SM result for is independent of the factorization framework. As we noted, any additional effects do not change Eq. (6) unless an extra CPV phase enters. Taking account of , we find , for , which is in good agreement with Refs. BenekePLB05 ; FSIPRD05 .
We turn to . For the decay rate, the QCDF result QCDF and the PQCD result piKPQCD are comparable to the current data PDG . In the NF framework, the amplitude is
(10)  
where for the transition, while for the transition. The chromodipole contribution appears only in the latter, and we shall use . Taking MeV and MeV, Eq. (10) leads to . Just as , reduces the rate substantially. But even for remains problematic.
For , we are not hampered by using NF. Eq. (10) is more complicated than Eq. (9). Furthermore, would be smeared with the tree contributions carrying although is highly suppressed. However, Eq. (6) persists. The amplitude in Eq. (10) gives , for , again in agreement with Refs. BenekePLB05 ; FSIPRD05 .
In similar way, we evaluate TCPV observables in , , modes. The amplitudes of these modes are rather complicated. But the trend of our results are consistent with the results in Refs. BenekePLB05 ; FSIPRD05 . For , we find , , and , respectively. We also calculate TCPV observables in and . We assume these final states to be the CP even state, via . As anticipated, we find and , respectively, with minus sign coming from . The relevant chromodipole hadronic parameters , and can be found in Appendix A.
Iv Strangebeauty squarks
In this section, we briefly summarize our model without going into details. The lefthanded flavor mixing is well understood in terms of the CKM matrix elements. The pattern of flavor mixing in left and righthanded dynamics could be different. But within the SM we are unable to probe righthanded quark mixing.
Our model is one of the possibilities to address violation in the righthanded sector. If there is an underlying approximate Abelian flavor symmetry (AFS), the righthanded and quarks can have near maximal mixing Nir ; CHPRL01 . The effective AFS implies downtype quark mass matrix to be of the form ArhribPRD01 ,
(11) 
where we quote 23 sector only. The ratio is approximately . The near maximal righthanded mixing may be the largest offdiagonal element. However, its effect is hidden from our view in the SM for absence of righthanded dynamics. The combination of AFS and SUSY brings forth righthanded dynamics involving squarks, as well as realizing a near maximal  squark mixing CHPRL01 ; ArhribPRD01 .
As pointed out in CHPRL01 ; ArhribPRD01 , applying four texture zeros is needed to be safe from kaon low energy constraints. Decoupling flavor is implied by lack of NP indication in mixing. From this backdrop, we focus on 23 generation subsystem. With decoupling of flavor, the down squark mass matrix is reduced from to , which is split up into
(12) 
where each submatrix would in principle be complex, and the Hermitian nature implies . With squark mass scale , one finds and , while near democratic structure in Chua01 ; ArhribPRD01 .
We now parameterize as
(13) 
with a unique new phase , which cannot be rotated away since phase freedom is already used in quark sector. Transforming
(14)  
one obtains the mass eigenstates , with the corresponding eigenvalues . In general, with , strangebeauty squark could be made lighter as a consequence of level splitting.
We assume that SUSY is at TeV scale, and the AFS scale is not too far above the SUSY scale. To make as light as 200 GeV for TeV scale SUSY, one needs finetuning of to level ArhribPRD01 . In fact, the scenario of our previous work CHNPRL04 used and for TeV. The sign anomaly in 2003 seemed to demand such finetuning for light . Furthermore, we also demanded to be not heavier than 600 GeV. With the weakening of the current data, the previous somewhat extreme model parameters can be relaxed. In the following, we continue to pursue near maximal  mixing for convenience, i.e. , but make the tuning less strict.
Let us not give the explicit expressions Chua01 of gluinoquarksquark and squarksquarkgluino interactions in mass eigenbasis. Instead, we present the righthanded chromodipole contributions as an example. The relevant coefficient is described as
where
with , , and , and is obtained by the replacement and . The expressions of , , and in Eq. (LABEL:eq:C12pr) are given by
(16) 
The coefficients at scale can be calculated by means of the leading order renormalization equations Chua01 ,
where we assume .
Righthanded strong penguins are also taken into account in our calculation, where and mainly enter and squark mixings, respectively. The size of is smaller than the size of by at least two orders of magnitude.
Before moving to the numerical study in the next session, one remark should be made. In general, the new effect on lefthanded dynamics arise from and squark mixing. Here we assume that lefthanded squark is at TeV scale. In addition, one finds ArhribPRD01 that  mixing and  mixing are suppressed by a factor . Therefore, we expect the dominance of the SM contribution in lefthanded dynamics, and neglect the lefthanded SUSY effects. It is important to note that, in this specific framework, a light can well survive the constraint of decay rate ArhribPRD01 . As shown in Refs. ArhribPRD01 ; CHNPRL04 , for GeV, neither the mass nor the phase are constrained. Furthermore, the impact of on the direct CPV in bsphotonACP , , is rather small. This is based on the fact that there is no righthanded tree level weak interaction, and no operator mixing between right and lefthanded dynamics. We found that, even though had persisted, the modulation from the SM prediction for (around +0.6%) is as small as .
V Impact of on
We now introduce the SUSY effects into and . As stated in previous sections, for simplicity we neglect the righthanded SM contributions to transition, as well as the lefthanded SUSY contributions. Consequently, in our calculation, the lefthanded coefficients are identical to the SM calculation, while the righthanded coefficients arise purely from NP.
The decay amplitude of in Eq. (9) is modified as
(18) 
where are the terms shown in Eq. (9), modified by . Because of and , the chromodipole SUSY effect is the dominant NP effect in Eq. (18).
Fig. 1(a) illustrates our result for vs new CP phase . We fix to be 1.5 TeV for illustration. Since NF contains small strong phases, our results reflect genuine NP contributions. Here we set as a target for extracting the preferred regions for and . We illustrate with four examples spanning some range for and , but with values remaining natural. We see that, for , the range – GeV with – GeV can be easily accommodated by the current data. These mass regions are well within the discovery range at LHC CMSBrochure , which would be commissioned in 2007. On the other hand, one has for , and at .
Note that , or as well as with small phase CHNPRL04 can still be accommodated and need not be thrown away. However, these ranges are not of interests in this work as we emphasize naturalness for on the SUSY scale. In the following, we take GeV as our standard value with at 1.5 TeV.