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 $\BZ$ charges for the SM particles, which are universal for each generation and allow $\BZ$ fields to distinguish flavor generations. At the $\Lambda_{\UP}$ scale, $\BZ$ 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 $\sqrt{m_{i}m_{j}/m^2_{3}}$, where $m_3$ is the mass of the heaviest particle in each type of fermion generations and $i, j$ are the flavor indices. In addition, we show that there is no new CP violating phase beside the unique one in the CKM matrix. We use $\bar B_{q}\to \ell^{+} \ell^{-}$ as examples to display the new FCNC effects. In particular, we demonstrate that the direct CP asymmetries in the decays can be $O(10%)$ 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 [1] and the large top quark mass [3]. For instance, the former makes the P 0 −P 0 (P = K and D) mixings and rare P decays naturally small while the latter leads to large B q −B q mixings (q = d, s) as well as the time-dependence CP asymmetry for the decay of B d → J/ΨK S . Among these effects, the most important measured quantities are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [2], 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 [4] and LHCb [5] with design luminosities of 5 × 10 35 and 5 × 10 32 cm −2 s −1 , 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 [6,7].
Some illustrative examples such as t → u + U and e + e − → µ + µ − 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 [8,9]. 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. [8]. For the system with the scale invariance [8] there exist so-called Banks-Zaks (BZ) fields that have a nontrivial infrared fixed point at a very high energy scale [49]. 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 [50], where D and U denote the singlet states for down and up-type quarks, respectively, while Q stands for the quark doublet. Therefore, if BZ 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 byd dO andd γ 5 dO in weak eigenstates, could basically produce FCNCs after the spontaneous symmetry break- would be induced for the coupling ofd P L dO after the EWSB, where V D R,L are unitary matrices to diagonalize the down-type quark Yukawa matrix. Note that sincedΓd (Γ = 1, γ 5 ) have to be SU(2) L singlets, the d-quark has to be either left-handed or right-handed before the EWSB as it should be. For the convention of V U L = 1, V D L is just the CKM matrix. Immediately, we suffer a serious problem from the K 0 −K 0 mixing due to the coupling fordsO being associated with (V D R12 − λ) where λ is the Wolfenstein parameter [51]. To avoid the large FCNC problem, one can set the Yukawa matrix be hermitian so that V D R = V D L . 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 BZ charges so that BZ fields could distinguish flavor species. In terms of the prescription in Ref. [8], the interactions between BZ and SM fields are given by where M U is the high energy mass scale of the messenger, g BZ is a free parameter, are the corresponding BZ charges, Γ is the possible Dirac matrix and O BZ is the operator composed of BZ fields. We note that although Q BZ i 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 BZ charge at the high energy scale and we assume that the interactions with the BZ fields are invariant under parity. Subsequently, with the dimensional transmutation at the Λ U scale, the BZ operators in Eq. (1) will match onto unparticle operators. The effective interactions are obtained to be where C F U is a Wilson-like coefficient function and d BZ(U ) is the scaling dimension of the BZ (unparticle) operator. Here the unparticle operators have been set to be hermitian [9]. In principle, Q U could be related to Q BZ 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 Q BZ matrix, we know that Q U should be also a diagonal one, . Hence, below the Λ U scale, Q U 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 H = v/ √ 2, 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 . As known, the determination of flavor mixing matrices V F L,R 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 V U L V D † L , is approximately an unity matrix. This indicates that the quark mass matrices are very likely aligned and have [53,54,55]. In Ref. [55], it showed that the Fritzsch quark mass matrices, given by [52,54] where R F and H F 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 m u(d) ≪ m c(s) ≪ m t(b) , it is found that the interesting equalities [55] are satisfied. Although the extensions of the Fritzsch ansatz could have more degrees of freedom to fit the experimental data [56], 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.
Due to Q U , R F and H F being all diagonal matrices, the phase redefinition will not influence the vector current interactions. SinceM F is a real and symmetric matrix, it can be diag- Then, the flavor structures could be expressed bȳ We note that the phases in R F and H F appear only in the scalar-type interactions. With trM F , trM 2 F and detM F and the convention of diaM dia As a result, the orthogonal matrix could be obtained as [55] O Since the CKM matrix expressed by V CKM = O U H U H † D O T D in general has six phases, we can redefine the phases in up and down-type quarks again so that V CKM = XO U H U H † D O T D Y † satisfies one single CKM phase convention [3]. With the new phases in diaX = e i(α−β) (−i, 1, 1), diaY = e iα (−1, 1, 1) and diaH U H † D = e iβ (−i, 1, 1), Eq. (6) becomes where diaZ U = (−i, 1, 1), diaZ D = (−1, 1, 1) and the vector-type interactions are parity conserved. We note that the flavor structures in Eq. (8) have some restrictions on Q U . To see the problem clearly, we decompose the flavor changing effects to be Since all phase matrices are in diagonal forms, an analysis on O F Q U O T F will not lose the generality. Using the elements in Eq. (7), the possible flavor changing effects are explicitly given by and ∆m Bs have the ratios λ 3 : λ : 1, we find that |r 21 . 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 R F and H F are independent, from Eq. (8)   It is also true for cases with the vector current couplings.
• If R F = H F , from Eq. (4) we can easily find that the corresponding mass matrices are hermitian. The FCNC effects are all related to 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 [57]. The hermiticity could help us to solve the CP problem in models with supersymmetry (SUSY) [58] and it has an important implication on CP violation in Hyperon decays [59].
• From Eq. (9), we find that ∆m Bs /∆m B d ≈ m d /m s ∼ λ 2 , which is consistent with the experimental data [3].
• Since the masses of the charged leptons also have the mass hierarchy m e ≪ m µ ≪ m τ , if we take the same phase convention, Eqs. (8) and (9) should be straightforwardly extended to the charged lepton sector.
To illustrate the peculiar phenomena in Eq. (8) associated with unparticles, we takeB q → ℓ + ℓ − as examples. For simplicity, we adopt scalar-type interactions as the representative.
The effective interactions are Since the coefficient functions are always associated with U-charges, we defineQ U With the propagator of the scalar unparticle operator proposed in Refs. [8,9], given by the decay amplitudes forB q → ℓ + ℓ − by due to unparticles are expressed by We note that φ U is a CP conserving phase [9,10,12]. Combining with the SM contributions, the corresponding branching ratios (BRs) are where the angle β q is from V tq = |V tq |e −iβq with Here, we have used the measured B − → τν τ decay to remove the uncertainties from f B and |V tq |. 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 B q → ℓ + ℓ − , the new free parameters are d U , Λ U and N qb (N ℓℓ ), which can be constrained by ∆m Bq (∆a ℓ ) of the B q −B q mixings (lepton anomalous magnetic dipole moments). Explicitly, we find that GeV and (11.69 ± 0.08) × 10 −12 GeV [61], respectively, we will use their central values as the inputs to constrain N d(s)b . For ∆a ℓ , we will concentrate on the muon one with ℓ = µ.
The difference between the experimental value and the SM prediction on the muon g − 2 is given by ∆a µ = a exp µ − a SM µ = (22 ± 10) × 10 −10 [3]. We will take the upper limit to bound the free parameter N ℓℓ . For simplicity, we set Λ U = 1 TeV, 1 < d U < 2, ReN qb ∼ ImN qb and ReN ℓℓ ∼ ImN ℓℓ . To see the effects of the scalar unparticle on the muon g − 2 and B q −B q mixings, we first show the results in Fig. 1, where the solid, dotted, dashed and dash-dotted lines stand for d U = 1.2, 1.4, 1.6 and 1.8, respectively. We note that ImN db is treated as a free parameter due to N sb = m d /m s N db . From the figures, we find that the muon g − 2 and B q −B q mixings are very sensitive to the scale dimension d U . The smaller d U it is, the stronger constraint on ImN µµ(db) we get. Furthermore, with the inputs and the  and CPA for B d → µ + µ − as functions of d U are presented in Fig. 2. Due to the current upper limit on B(B s → µ + µ − ), d U should be less than 1.66. The flat curves in Fig. 2(a) correspond to the SM predictions. Amazingly, from Fig. 2  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, i.e., m i m j /m 2 3 . In addition, although the couplings of the FCNCs could be complex, there is no more new CP violating phase available because the matrices O F Q U R F H † F O T F responsible to the FCNC effects are symmetric. Moreover, we have usedB q → ℓ + ℓ − 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 B d → ℓ + ℓ − . If any CP violating signal is found in these decays, it must indicate the existence of unparticles.