Flavors and Phases in Unparticle Physics
Inspired by the recent Georgi’s unparticle proposal, we study the flavor structures of the standard model (SM) particles when they couple to unparticles. At a very high energy scale, we introduce charges for the SM particles, which are universal for each generation and allow fields to distinguish flavor generations. At the scale, operators and charges are matched onto unparticle operators and charges, respectively. In this scenario, we find that tree flavor changing neutral currents (FCNCs) can be induced by the rediagonalizations of the SM fermions. As an illustration, we employ the Fritzsch ansatz to the SM fermion mass matrices and we find that the FCNC effects could be simplified to be associated with the mass ratios denoted by , where is the mass of the heaviest particle in each type of fermion generations and are the flavor indices. In addition, we show that there is no new CP violating phase for FCNCs in down type quarks beside the unique one in the CKM matrix. We use as examples to display the new FCNC effects. In particular, we demonstrate that the direct CP asymmetries in the decays can be due to the peculiar CP conserving phase in the unparticle propagator.
In the standard model (SM), it is known that flavor changing processes at tree level can only be generated for charged currents mediated the W gauge boson in the quark sector. These charged currents will induce flavor changing neutral currents (FCNCs) via quantum loops. Consequently, the most impressive features of flavor physics are the Glashow-Iliopoulos-Maiani (GIM) mechanism GIM () and the large top quark mass PDG06 (). For instance, the former makes the ( and ) mixings and rare decays naturally small while the latter leads to large mixings () as well as the time-dependence CP asymmetry for the decay of . Among these effects, the most important measured quantities are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements CKM (), coming from the unitary matrices which diagonalize the left-handed up and down-type quark matrices. Although there are no disagreements between the SM and experiments, which might give us some clue as to what may lie beyond the SM, it is important to keep searching for any discrepancies. In particular, the next generation of flavor factories such as SuperKEKB SuperKEKB () and LHCb LHCb () with design luminosities of and , respectively, may provide some hints for new flavor effects. Thus, theoretically it should be interesting to explore the possible new phenomena related to flavor physics newphys1 (); newphys2 ().
Recently, Georgi has proposed that an invisible sector dictated by the scale invariant may weakly couple to the particles of the SM Georgi1 (); Georgi2 (). Since the scale invariant stuff cannot have a definite mass unless it is zero, it should be made of unparticles Georgi1 () as the SM particles have definite masses. In terms of the two-point function with the scale invariance, it is found that the unparticle with the scaling dimension behaves like a non-integral number of invisible particles Georgi1 (). Consequently, the unparticle physics phenomenology has been extensively explored in Refs. Georgi1 (); Georgi2 (); CKY (); LZ (); CH12 (); Ding_Yan (); Y_Liao (); Aliev (); Catterall (); Li (); Lu (); Steph (); Fox (); Greiner (); Davou (); CGM (); XG (); MR (); Zhou (); FFRS (); Bander (); Rizzo (); GN (); Zwicky (); Kikuchi (); M_Giri (); Huang (); Krasnikov (); Lenz (); C_Ghosh (); Zhang (); Nakayama (); Deshpande (); DEQ (); Neubert (); Hannestad (); Das (); Bhattacharyya (); Majumdar (); Alan-Pak (). Some illustrative examples such as and have been given to display the unparticle properties. It is also suggested that the unparticle production in high energy colliders might be detected by searching for the missing energy and momentum distributions Georgi1 (); Georgi2 (). Nevertheless, flavor factories with high luminosities mentioned above should also provide good environments to search for unparticles via their virtual effects.
Besides the Lorentz structure, so far there is no rule to govern the interactions between the SM particles and unparticles. The flavor physics associated with unparticles is quite arbitrary, i.e., the couplings could be flavor conserving or changing. Moreover, there is no any correlation in the transitions among three generations for flavor changing processes. In this note, we will study the possible flavor structures for the SM particles when they couple to unparticles. Since the gauge structure of unparticles involves more theoretical uncertainties, we only pay attention to the interactions with the charged fermion sectors. We will not discuss the neutrino sector because the nature of neutrino flavors is still unclear.
We start from the scheme proposed in Ref. Georgi1 (). For the system with the scale invariance Georgi1 () there exist so-called Banks-Zaks () fields that have a nontrivial infrared fixed point at a very high energy scale BZ (). Above the electroweak scale, since all SM particles are massless, we cannot tell the differences between down-type quarks or up-type quarks. In the SM, we have flavor symmetries MM (), where and denote the singlet states for down and up-type quarks, respectively, while stands for the quark doublet. Therefore, if fields are flavor blind, plausibly new flavor mixing effects cannot be generated for vector and axial-vector currents after the electroweak symmetry breaking (EWSB). It is worthy to mention that scalar-type couplings, illustrated by and in weak eigenstates, could basically produce FCNCs after the spontaneous symmetry breaking. For example, would be induced for the coupling of after the EWSB, where are unitary matrices to diagonalize the down-type quark Yukawa matrix. Note that since () have to be singlets, the -quark has to be either left-handed or right-handed before the EWSB as it should be. For the convention of , is just the CKM matrix. Immediately, we suffer a serious problem from the mixing due to the coupling for being associated with () where is the Wolfenstein parameter Wolfenstein (). To avoid the large FCNC problem, one can set the Yukawa matrix be hermitian so that . As a result, the FCNCs at tree level via scalar-type interactions are removed. In any event, despite the property of Yukawa matrices, to get natural small FCNCs at tree level for scalar and vector-type interactions, we need some internal degrees of freedom for fermions that could differentiate flavors by the scale invariant stuff.
In order to reveal the new flavor mixing effects due to the involvement of unparticles, we assume that the SM particles carry some kind of charges so that fields could distinguish flavor species. In terms of the prescription in Ref. Georgi1 (), the interactions between and SM fields are given by
where is the high energy mass scale of the messenger, is a free parameter, denote the 3-generation of fermions in the SM, are the corresponding charges, is the possible Dirac matrix and is the operator composed of fields. We note that although are different for each generation, the interactions are still flavor conserved. To simplify our discussion, we regard that all fermions in each generation have the same charge at the high energy scale and we assume that the interactions with the fields are invariant under parity. Subsequently, with the dimensional transmutation at the scale, the operators in Eq. (1) will match onto unparticle operators. The effective interactions are obtained to be
where is a Wilson-like coefficient function and is the scaling dimension of the (unparticle) operator. Here the unparticle operators have been set to be hermitian Georgi2 (). In principle, could be related to by a complicated matching procedure. However, at the current stage, it is impossible to give any explicit calculations for the matching. With the property of the diagonal matrix, we know that should be also a diagonal one, parametrized by . Hence, below the scale, could be regarded as unparticle charges carried by the SM fermions to distinguish the flavors by the unparticle stuff.
When the energy scale goes down below the EWSB scale, described by the vacuum expectation value (VEV) of the Higgs field , the flavor symmetry will be broken by the Yukawa terms and the charged fermions become massive. Afterward, the weak eigenstates of the fermions appearing in Eq. (2) need to be transformed to the physical eigenstates by proper unitary transformations. Hence, Eq. (2) is found to be
where we have redefined the coefficient functions to be dimensionless free parameters denoted by , and , respectively. In Eq. (3), the power of is taken to fit the dimension of the effective Lagrangian in four-dimension spacetime and the explicit terms represent the main FCNC effects. Note that we have separated the interactions in terms of the fermion chirality. In addition, are the unitary matrices to diagonalize the Yukawa matrix of F-type fermions, where F could be up and down-type quarks and charged leptons. According to Eq. (3), in general we have two types of sources for new FCNCs, i.e., and . As known, the determination of flavor mixing matrices is governed by the detailed patterns of the mass matrices. For convenience, we just focus on the quark sector. It has been known that the CKM matrix, defined by , is approximately an unity matrix. This indicates that the quark mass matrices are very likely aligned and have the relationship of with qm1 (); qm2 (); qm3 (). In Ref. qm3 (), it showed that the Fritzsch quark mass matrices, given by Fritzsch (); qm2 ()
where and are diagonal phase matrices, could lead to reasonable structures for the mixing angles and CP violating phase in the CKM matrix just in terms of the quark masses. From the hierarchy , it is found that the interesting equalities qm3 ()
are satisfied. Although the extensions of the Fritzsch ansatz could have more degrees of freedom to fit the experimental data MN (), however, since our goal of this study is to explore the flavor structure affected by unparticles, we will take the simplest version of the Fritzsch ansatz in Eq. (4) as our working base. In addition, we have checked that due to the character of mass hierarchy, the extensions of Eq. (4) do not change our following results.
Since the SM has been extended to include new flavor interactions, we have to be careful to use the phase convention because the rotated phases will flow to Eq. (3). To avoid the phase ambiguity, we should start from the flavor basis in Eqs. (2) and (4). At first, we rotate away and from by redefining the phases of the up-type quarks. In order to make the weak charged currents to be invariant under this transformation, left-handed down quarks should make the transformation simultaneously. Then, the interactions in Eq. (2) for up and down quarks to the scalar unparticle become
Due to , and being all diagonal matrices, the phase redefinition will not influence the vector current interactions. Since is a real and symmetric matrix, it can be diagonalized by a orthogonal matrix such that . Accordingly, we obtain , , and . Then, the flavor structures could be expressed by
We note that the phases in and appear only in the scalar-type interactions. With , and and the convention of where denote and , we find that , and . As a result, the orthogonal matrix could be obtained as qm3 ()
Since the CKM matrix expressed by in general has six phases, we can redefine the phases in up and down-type quarks again so that satisfies one single CKM phase convention PDG06 (). With the new phases in , and , Eq. (6) becomes
where , and the vector-type interactions are parity conserved. We note that the flavor structures in Eq. (8) have some restrictions on . To see the problem clearly, we decompose the flavor changing effects to be
with . Since all phase matrices are in diagonal forms, an analysis on will not lose the generality. Using the elements in Eq. (7), the possible flavor changing effects are explicitly given by
Phenomenologically, and are dictated by , , and mixings, respectively. From Eq. (9), one can easily see that if , a strict constraint on is inevitable due to . If and are in the same order of magnitude, it will make the FCNC effects involving the third generation be less interesting. Motivated by the successful SM results for pseudoscalar meson oscillations in the down-type quark systems, where the related CKM matrix elements for , and have the ratios , we find that in should be satisfied, i.e., . In addition, the sign and the specific magnitude should be chosen to somewhat cancel out the second term of the first line in Eq. (9). With this scheme, we then have
which is needed for the phenomenological reason.
With the experimental data and Fritzsch ansatz, we have obtained the FCNC effects from the couplings of quarks and unparticles. According to the results in Eq. (8), we highlight some interesting characters as follows:
If the phase matrices and are independent, from Eq. (8) we find that only scalar-type FCNCs could have different couplings for different chiralities. Even there are some new CP violating phases in , due to , we see that the scalar-type interactions are in fact associated with . Thus, there are no new physical CP violating effects unless the processes involve . It is also true for cases with the vector current couplings.
If , from Eq. (4) we can easily find that the corresponding mass matrices are hermitian. The FCNC effects are all related to which is also hermitian. As a result, in terms of the quark currents, the couplings of fermions and unparticles are parity-even and no new CP phase is induced for down type quarks in this case. It should be worthy to mention that the hermitian mass matrices could be naturally realized in gauge models such as left-right symmetric models hermitian_y (). The hermiticity could help us to solve the CP problem in models with supersymmetry (SUSY) ABKL () and it has an important implication on CP violation in Hyperon decays Chen_PLB521 ().
To illustrate the peculiar phenomena in Eq. (8) associated with unparticles, we take as examples. For simplicity, we adopt scalar-type interactions as the representative. The effective interactions are
the decay amplitudes for by due to unparticles are expressed by
where the angle is from with
Due to , the function of can be simplified to BBL (). Here, we have used the measured decay to remove the uncertainties from and . Besides the BRs, from Eq. (11) we can also study the direct CP asymmetries (CPAs) in the two-body exclusive B decays, defined by
It is known that in a process the direct CPA needs CP conserving and unrotated CP violating phases simultaneously. Since the unparticle stuff provides a CP-conserved phase, if the process carries a physical CP violating phase, a nonvanishing CPA is expected. In , the new free parameters are , and (), which can be constrained by () of the mixings (lepton anomalous magnetic dipole moments). Explicitly, we find that
To estimate the numerical values, we take , with , , GeV, GeV and . For the mixing parameters of and , measured to be GeV and GeV CDF (), respectively, we will use their central values as the inputs to constrain . For , we will concentrate on the muon one with . The difference between the experimental value and the SM prediction on the muon is given by PDG06 (). We will take the upper limit to bound the free parameter . For simplicity, we set TeV, , and . To see the effects of the scalar unparticle on the muon and mixings, we first show the results in Fig. 1, where the solid, dotted, dashed and dash-dotted lines stand for , , and , respectively. We note that is treated as a free parameter due to . From the figures, we find that the muon and mixings are very sensitive to the scale dimension . The smaller it is, the stronger constraint on we get.
Furthermore, with the inputs and the allowed values of and , the BRs for [solid] and [dashed] and CPA for as functions of are presented in Fig. 2. Due to the current upper limit on , should be less than . The flat curves in Fig. 2(a) correspond to the SM predictions. Amazingly, from Fig. 2(b) we see that an unusual direct CPA of is generated in . Besides the necessary weak CP violating phase existed in the SM, the result mainly depends on the CP conserving phase carried by the unparticle in its propagator. This is a unique phenomenon since it is supposed to be vanishing small even some new CP violating phases are introduced. In other words, if a signal of the CPA in is observed, it must be the unparticle effect. Similar results are also expected in the dielectron and ditau modes. However, there is no direct CPA for decays due to in the SM.
In summary, we have studied the flavor structures of the SM fermions
when they couple to the scale invariant stuff. In order to get
naturally suppressed FCNC effects at tree level and
transitions among three generations, we have introduced the charges that
are universal in each generation
but generation un-blind.
The charges could be regarded as the internal
of freedom carried by the fermions for which
the -fields can distinguish
the flavor generations.
By the dimensional transmutation, the
charges are matched onto the unparticle charges when
onto the unparticle operators. After the EWSB, the FCNCs are induced
by the rediagonalizations of the fermion mass matrices.
To demonstrate the FCNC
effects, we have adopted the simplest
Fritzsch ansatz for quarks.
Consequently, we have found that the FCNC
effects are associated with the square roots of the mass ratios,
In addition, although the couplings of the FCNCs could be
there is no more new CP violating phase available
because the matrices responsible to
the FCNC effects are symmetric. Moreover, we have used decays to illustrate the influence of
unparticles. In particular, with the peculiar CP conserving phases
carried by unparticles, a unique phenomenon is generated in the
direct CPAs of .
If any CP violating signal is found in these decays, it must indicate the
existence of unparticles.
We would like to thank Prof. Ling-Fong Li for useful discussions. This work is supported in part by the National Science Council of R.O.C. under Grant #s: NSC-95-2112-M-006-013-MY2 and NSC-95-2112-M-007-059-MY3.
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