Vector leptoquark mass limits and branching ratios of K_{L}^{0},B^{0},B_{s}\to l^{+}_{i}l^{-}_{j} decays with account of fermion mixing in leptoquark currents

# Vector leptoquark mass limits and branching ratios of K0L,B0,Bs→l+il−j decays with account of fermion mixing in leptoquark currents

A. D. Smirnov
Division of Theoretical Physics, Department of Physics,
Yaroslavl State University, Sovietskaya 14,
150000 Yaroslavl, Russia.
E-mail: asmirnov@univ.uniyar.ac.ru
###### Abstract

The contributions of the vector leptoquarks of Pati-Salam type to the branching ratios of decays are calculated with account of the fermion mixing in the leptoguark currents of the general type. Using the general parametrizations of the mixing matrices the lower vector leptoquark mass limit is found from the current experimental data on these decays. The branching ratios of the decays predicted at are calculated. These branching ratios for the decays are close to the experimental data whereas those for the decays , and are by order of less than their current experimental limits. For the decays these branching ratios are of order and respectively. The predicted branching ratios will be usefull in the current and future experimental searches for these decays.

Keywords: Beyond the SM; four-color symmetry; Pati–Salam; leptoquarks; B physics; rare decays.

PACS number: 12.60.-i

The search for a new physics beyond the Standard Model (SM) is one of the aims of the current experiments at the LHC. There is a lot of new physics scenarios (such as supersymmetry, left-right symmetry, two Higgs model, extended dimension models, etc.) which are now under experimental searches at the LHC.

One of the possible variants of new physics can be induced by the well known Pati-Salam idea on the possible four color symmetry between quarks and leptons of the vectorlike type regarding leptons as the quarks of the fourth color in frame of the gauge group [1]. This group has three gauge coupling constants related to the strong electromagnetic and weak coupling constants and can be regarded as an intermediate stage in the symmetry breaking of the GUT group  [2, 3, 4] embedded in the more large group  [5, 6]. The four color symmetry of the vectorlike type immediately predicts new gauge particles - the vector leptoquarks which belong to the multiplet of the group and form the color triplet of the SM group .

In general case the leptoquarks as the vector or scalar particles carrying both the baryon and lepton numbers appear in many models and can led to varied new physics effects, the comprehensive review of physics effects generated by leptoquarks can be found in [7]. In the last years the vector leptoquarks are used to explain the known anomalies in the semileptonic meson decays with simultaneous satisfaying the other experimental data. Using the conventional Pati-Salam vector leptoquarks this goal is achieved by the appropriate choice of the vector leptoquark couplings to fermions [8], by account of the fermion mixing in leptoquark currents of the special form [9], in frame of new gauge leptoquark model extending the four color group by the additional factor  [10], by extending the fermion sector of the Pati-Salam model [11], in a three-site gauge model using the Pati-Salam gauge group for each fermion generation [12]. Another approach to the explanations of the -decay anomalies is attempted in the model with the composite leptoquarks which is based on the Pati-Salam SU(4) group in the context of a new strongly interacting sector [13, 14]. In this approach the vector leptoquarks couple primarily to the fermions of the third generation with their couplings to the the fermions of the first two generations beeng suppressed.

The lower mass limits for the vector leptoquarks from their direct searches are of about or less . The essentially more stringent lower mass limits are resulting from the rare decays of pseudoscalar mesons. The most stringent of them are resulting from the decay and with neglect of fermion mixing in leptoquark currents are of order of [15, 16, 17, 18, 19].

It should be noted however that the fermion mixing in leptoquark currents is quite natural. It is as natural as the fermion mixings in the week currents which are described by the well known matrices and in the quark and lepton sectors respectively. Indeed, the mass eigenstates of left- and right-handed quarks and leptons , can enter to interactions with gauge and scalar fields in general case through the superpositions

 Q′L,Rpaα=∑q(AL,RQa)pqQL,Rqaα,l′L,Ria=∑j(AL,Rla)ijlL,Rja, (1)

where and are unitary matrices describing the fermion mixing and diagonalizing the mass matrices of quarks and leptons, … are the quark and lepton generation idexes, and are the and indexes, , are the up and down quarks, are the mass eigenstates of neutrinos and are the charged leptons. In the weak interaction the matrices and form the CKM and PMNS matrices as . In analogous way in the interaction of quarks and leptons with leptoquarks the matrices and led to specific matrices

 KL,Ra=(AL,RQa)+AL,Rℓa (2)

of the fermion mixing in leptoquark currents. The fermion mixing in leptoquark currents can essenially lower the mass limits on leptoquark masses. The current experimental data on the decays

 K0L,B0,Bs→l+il−j (3)

give now the possibility to obtain new lower mass limits for the leptoquarks with account of the fermion mixing in leptoquark currents.

In this paper the contributions of the vector leptoquarks of Pati-Salam type to the decays (3) are calculated and analysed with account of the fermion mixing in leptoquark currents of the general form and the corresponding new lower mass limit for the vector leptoquarks is obtained from the current data on these decays. With account of this lower mass limit the expected branching ratios of these decays are calculated and discussed.

The interaction of the vector leptoquarks with down fermions which are responsible for the decays (3) can be written in general case as

 \emphLVdl = g4√2{(¯dpα[(KL2)piγμPL+(KR2)piγμPR]li)Vαμ+h.c.}, (4)

where is the gauge coupling constant related to the strong coupling constant at the mass scale of the symmetry breaking, are the left and right operators of fermions and are the mixing matrices (2) for down fermions. It should be noted that in the case of the chiral () mixing the interaction (4) of vector leptoquarks with quarks and leptons is not purely vectorlike.

Denoting the sums of the branching ratios of charge conjugated final states as

 BrV(P→e+μ−)+BrV(P→μ+e−)≡BrV(P→eμ), (5) BrV(P→e+τ−)+BrV(P→τ+e−)≡BrV(P→eτ), (6) BrV(P→μ+τ−)+BrV(P→τ+μ−)≡BrV(P→μτ) (7)

the branching ratios of the decays of pseudoscalar mesons into lepton-antilepton pairs induced by the vector leptoquarks in the case of neglecting the electron and muon masses

 me,mμ≪mτ,mK0,mB0,mBs (8)

can be written as

 BrV(P→ll′)=BPβ2P,ll′forll′=e+e−,μ+μ−,eμ, (9) BrV(P→lτ)=BP(1−m2τ/m2P)2β2P,lτforlτ=eτ,μτ, (10) BrV(P→τ+τ−)=BP√1−4m2τ/m2P[β2P,τ+τ−−(m2τ/m2P)|kLP,33−kRP,33|2], (11)

where

 BP=mPπα2st(Mc)f2P%$\Large¯m$2P(RVP)22m4VΓtotP (12)

are the typical branching ratios of these decays and , are the mixing factors depending on the mixing matrices , . The enterring into (12) form factors  parametrize the matrix elements of the axial and pseudoscalar quark currents in the standard way, the factors account the gluonic corrections to the pseudoscalar quark current, , , are the mass and total width of meson and are the masses of its valency quarks, is the mass of the vector leptoquark.

With account of the definitions (5)–(7) the mixing factors have the form

 β2P,e+e−=β2P,11,β2P,μ+μ−=β2P,22,β2P,τ+τ−=β2P,33, (13) (14)

where the mixing factors are related to the matrix elements of the mixing matrices , .

The mixing matrices , are the unitary matrices and in general case each of them can be parametrized by three angles and six phases as

 (15)

where

 (16) (17)

,   and are the arbitrary angles and phases of the matrices . Keeping in mind that the phases of the quark and lepton states are fixed by the standard choice of the matrices and for the matrices we use the general form (15)–(17).

With account of (15)–(17) the factors , can be expressed in terms of mixing angles and phases of the matrices . For the expressions have the form

 β2K0L,11=(|(sL12cL23eiεL+cL12sL23sL13ei(δL+εL))cR12cR13+ +(sR12cR23e−iεR+cR12sR23sR13e−i(δR+εR))cL12cL13|2+L↔R)/4, (18) β2K0L,22=(|(cL12cL23eiεL−sL12sL23sL13ei(δL+εL))sR12cR13+ +(cR12cR23e−iεR−sR12sR23sR13e−i(δR+εR))sL12cL13|2+L↔R)/4, (19) β2K0L,12=β2K0L,21=(|(sL12cL23eiεL+cL12sL23sL13ei(δL+εL))sR12cR13+ +(−cR12cR23e−iεR+sR12sR23sR13e−i(δR+εR))cL12cL13|2+L↔R)/4. (20)

The mixing factors , for are more complicated and can be presented in the form

 β2B0,11=(|sL12sL23−cL12cL23sL13eiδL|2(cR12cR13)2+L↔R)/2, (21) β2B0,22=(|cL12sL23+sL12cL23sL13eiδL|2(sR12cR13)2+L↔R)/2, (22) β2B0,12=(|sL12sL23−cL12cL23sL13eiδL|2(sR12cR13)2+L↔R)/2, (23) β2B0,21=(|cL12sL23+sL12cL23sL13eiδL|2(cR12cR13)2+L↔R)/2, (24) β2B0,31=(|cL23cL13−cR23cR13ei(χR1−εR−χL1+εL)mτ/(2\Large¯mB0RVB0)|2(cR12cR13)2+L↔R)/2, (25) β2B0,32=(|cL23cL13−cR23cR13ei(χR1−εR−χL1+εL)mτ/(2\Large¯mB0RVB0)|2(sR12cR13)2+L↔R)/2, (26) (27) (28) +L↔R)/2, (29) (30)

and

 β2Bs,11=(|sL12sL23−cL12cL23sL13eiδL|2|sR12cR23+cR12sR23sR13e−iδR|2+L↔R)/2, (31) β2Bs,22=(|cL12sL23+sL12cL23sL13eiδL|2|cR12cR23−sR12sR23sR13e−iδR|2+L↔R)/2, (32) β2Bs,12=(|sL12sL23−cL12cL23sL13eiδL|2|cR12cR23−sR12sR23sR13e−iδR|2+L↔R)/2, (33) β2Bs,21=(|cL12sL23+sL12cL23sL13eiδL|2|sR12cR23+cR12sR23sR13e−iδR|2+L↔R)/2, (34) β2Bs,31=(|cL23cL13−cR23cR13ei(χR1−εR−χL1+εL)mτ/(2\Large¯mBsRVBs)|2|sR12cR23+cR12sR23sR13e−iδR|2+ +L↔R)/2, (35) β2Bs,32=(|cL23cL13−cR23cR13ei(χR1−εR−χL1+εL)mτ/(2\Large¯mBsRVBs)|2|cR12cR23−sR12sR23sR13e−iδR|2+ +L↔R)/2, (36) +L↔R)/2, (37) +L↔R)/2, (38) β2Bs,33=(|cL23cL13sR23cR13−[cL23(cL13)2sL23ei(χR2+δR+εR−χL2−δL−εL)+ (39) |kLBs,33−kRBs,33|2=|cL23cL13sR23cR13−cR23cR13sL23cL13ei(χR1+χR2+δR−χL1−χL2−δL)|2, (40)

where

 χL,R1=φL,R0−φL,R1−φL,R2,χL,R2=φL,R0+φL,R3. (41)

The mixing factors (13), (14), (18)–(40) describe in the general form the effect of the fermion mixing in leptoquark currents in the case (8) on the branching ratios of the decays of pseudoscalar mesons into lepton-antilepton pairs. These mixing factors can be used for the analysis of the branching ratios (9)–(11) in dependence on the mixing angles and phases of the mixing matrices (15).

As seen from the (25)–(30), (35)–(40) the mixing factors ,   ,   ,   for depend on the phases , , , through their combinations (41). With fixed values of mixing angles and phases  these mixing factors can be minimized over phases (41) by the conditions

 χL1−εL−χR1+εR=0,χL2−χR2=0,δL+εL−δR−εR=0. (42)

The most stringent lower mass limits for vector leptoquark are resulting from the experimental data on the branching ratios . As a result for the more small masses the mixing factors must be very small (close to zero) and can be assumed in the further analysis to be equal to zero

 β2K0L,ll′=0 (43)

for , where are given by the relations (13), (14) and (18)–(20).

There are two solutions of the equations (43):

 solution A:θL23=θR23=π/2,θL13=θR13≡θ13,δL+εL+δR+εR=π (44)

and

 solution B:θL13=θR13=π/2. (45)

In both cases (44) and (45) the mixing factors and for are equal to zero, which gives that in these cases .

In the case (42), (44) we obtain from (21)–(40) the nonzero mixing factors (13), (14) in the form

 β2B0,e+e−=β2B0,μ+μ−=c213((sL12cR12)2+(sR12cL12)2)/2, (46) β2B0,eμ=c213((sL12sR12)2+(cL12cR12)2), (47) (48) (49)

and

 β2Bs,e+e−=β2Bs,μ+μ−=s213((sL12cR12)2+(sR12cL12)2)/2, (50) β2Bs,eμ=s213((sL12sR12)2+(cL12cR12)2), (51) (52) (53)

As seen from (46)–(53) the mixing factors , in the case (42), (44) depend in general case on three mixing angles .

In the case (42), (45) the mixing factors (13), (14) depend on two effective mixing angles with and can be obtained from (46)–(53) by the substitutions .

The branching ratios (9)–(11) with the factors depending on the mixing angles have been numerically analysed with account of experimental data on the decays (3). We vary the mixing angles in the case (42), (44) and the mixing angles in the case (42, (45) to find the minimal vector leptoquark mass satisfying these data.

In the numerical analysis we use the data on the masses of leptons and quarks, the data on the masses , life times () and the form factors

 fK0L=fK−=155.72MeV,fB0=190.9MeV,fBs=227.2MeV

of mesons from ref. [20]. The experimental data on the branching ratios are also taken from the ref. [20] except the branching ratios of the decays for which we use the current data

 Br(B0→μ+μ−)exp<3.4⋅10−10 % \@@cite[cite]{[\@@bibref{}{Aaij:2017vad}{}{}]}, (54) Br(B0→τ+τ−)exp<2.1⋅10−3 % \@@cite[cite]{[\@@bibref{}{Aaij:2017xqt}{}{}]}, (55) Br(Bs→μ+μ−)exp=(3.0±0.6+0.3−0.2)⋅10−9 \@@cite[cite]{[\@@bibref{}{Aaij:2017vad}{}{}]}, (56) Br(Bs→τ+τ−)exp<6.8⋅10−3 % \@@cite[cite]{[\@@bibref{}{Aaij:2017xqt}{}{}]} (57)

of refs.[21, 22] and the branching ratios of the decays for which we use the recent data

 Br(B0→eμ)exp<1.0⋅10−9 \@@cite[cite]{[% \@@bibref{}{Aaij:2017newdata}{}{}]}, (58) Br(Bs→eμ)exp<5.4⋅10−9 \@@cite[cite]{[% \@@bibref{}{Aaij:2017newdata}{}{}]} (59)

of the ref.[23]. For the mass scale of the symmetry breaking we choose the value .

Using (46)–(53) for the case (44) we have found the lower vector leptoquark mass limit

 mV>86TeV (60)

in the case of the chiral mixing and the mass limit in the case of the vectorkike () one. The case (45) with the chiral and vectorkike () mixings gives the mass limits and respectively. As seen, the mass limit (60) is the lowerest one.

The effect of the fermion mixing on the decays of type (3) has been also considered in [24] but for the particular choice of the mixing matrices correponding to the case (44) with the additional restrictions . Besides, in comparing the theoretical results for mesons with the experimental data only the first terms in the branching ratios (5)–(7) and in the mixing factors (14) have been taken into account, which contradicts the usual treatment of the experimental branching ratios as the sums over charge conjugated final states. By these two reasons the lower vector leptoquark mass limit of ref. [24] seems questionable (for example instead of this value the case (44) with gives in fact the mass limit ).

The mass limit (60) is resulting from the current experimental data on the branching ratios of the ref. [20] and (54)–(59) with account of the fermion mixing in leptoquark currents of the general form. As seen, it is essentially lower than that of order of obtained with neglect of the fermion mixing [15, 16, 17, 18, 19] and noticeably exceeds the mass limit of order of resulting from the current direct searches for the vector leptoquarks. It is worthy to note that the mass limit (60) can be further lowered by account of the possible destructive interference of the contributions from the vector leptoquarks with those from the scalar leptoquarks which are also predicted in the scalar sector of the models with the four color symmetry. The possibility of such interference in the minimal model with the four color symmetry based on the gauge group [25, 26, 27]) in the case of neglect of the fermion mixing () has been demonstrated in [18, 19].

We have calculated the branching ratios predicted by the vector leptoquarks with the lower allowed mass . This value of the vector leptoquark mass is ensured by the appropriate values of the mixing angles from the region

 θ13=1.183−1.187,θL12(θR12)=0.00(0.81)−0.763(π/2) (61)

with the branching ratios being invariant under exchange in accordance with (46)–(53). The analysis shows that the variations of the branching ratios under variations of the mixing angles within the region (61) for the decays do not exceed 3% and the branching ratios of the decays and are of order of and respectively. For definiteness in the second column of the Table 1 we present the branching ratios for at . In the third column of the Table 1 we present for comparision the current experimental data on the decays under consideration.

As seen from the Table 1 the branching ratios predicted by the vector leptoquarks with for the decays are close to (or compatible with) the corresponding experimental data. It means that these decays are now most suitable for search for the vector leptoquarks and for setting the new more stringent limits on their mass. The decays as the perspective ones for obtaining the new mass limit for the vector leptoquarks have been pointed out in ref. [28] at the data of ref. [20] and (54)–(57), this conclusion is now confirmed at the data (58)–(59) with improving the vector leptoquark mass limit from of ref. [28] to the current mass limit (60).

As concerns the decays , and