Anomalies in (semi)-leptonic B decays B ± τ ± ν, B ± Dτ ± ν and B ± D*τ ± ν, and possible resolution with sterile neutrino

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1 Chinese Physics C PARTICLES AND FIELDS OPEN ACCESS Anomaies in (semi-eptonic B decays B ± τ ± ν, B ± Dτ ± ν and B ± D*τ ± ν, and possibe resoution with sterie neutrino To cite this artice: Gorazd Cveti et a 017 Chinese Phys. C Reated content - A faciity to search for hidden partices at the CERN SPS: the SHiP physics case Sergey Aekhin, Wofgang Atmannshofer, Takehiko Asaka et a. - Towards the identification of new physics through quark favour vioating processes Andrzej J Buras and Jennifer Girrbach - Testing epton favor universaity in terms of BESIII and charm-tau factory data Wang Bin, Zhao Ming-Gang, Sun Ke- Sheng et a. View the artice onine for updates and enhancements. This content was downoaded from IP address on 18/1/017 at 05:34

2 Chinese Physics C Vo. 41, No. 11 ( Anomaies in (semi-eptonic B decays B ± τ ± ν, B ± Dτ ± ν and B ± D τ ± ν, and possibe resoution with sterie neutrino * Gorazd Cvetič 1;1 Francis Hazen ; C. S. Kim 3;3 Sechu Oh 4;4 1 Department of Physics, Universidad Técnica Federico Santa María, Vaparaíso, Chie Wisconsin IceCube Partice Astrophysics Center and Department of Physics, University of Wisconsin, Madison, WI, 53706, USA 3 Department of Physics and IPAP, Yonsei University, Seou , Korea 4 University Coege, Yonsei University, Incheon , Korea Abstract: The universaity of the weak interactions can be tested in semieptonic b c transitions, and in particuar in the ratios R(D ( Γ (B D ( τν/γ (B D ( ν (where = µ or e. Due to the recent differences between the experimenta measurements of these observabes by BaBar, Bee and LHCb on the one hand and the Standard Mode predicted vaues on the other hand, we study the predicted ratios R(D ( =Γ (B D ( τ+ missing /Γ (B D ( ν in scenarios with an additiona sterie heavy neutrino of mass 1 GeV. Further, we evauate the newy defined ratio R(0 Γ (B τ+ missing /Γ (B µν in such scenarios, in view of the future possibiities of measuring the quantity at Bee-II. Keywords: sterie neutrino, epton universaity, semieptonic B decays PACS: St DOI: / /41/11/ Introduction The epton universaity of weak gauge theory can be tested in excusive semieptonic B decays, possiby through the existence of new charged currents, as we as the quark favor mixing structure [1] of the Standard Mode (SM. To achieve these goas, it is usuay necessary to cacuate accuratey the corresponding hadronic matrix eements. However, in ratios ike R(D ( Γ (B D( τν Γ (B D ( ν, (1 with =e or µ, most of the hadronic uncertainties cance, making such ratios particuary reevant for testing the universaity of weak interactions. In Tabe 1 we show the SM predictions which we wi use. The shown (hadronic uncertainties originate, respectivey, from attice cacuations [] and from estimated higher order corrections in HQET to the ratio A 0 /A 1 of form factors [3]. Recenty, new improved experimenta resuts from the Bee [4] and LHCb [5] Coaborations have appeared, in addition to the oder BaBar resuts [6]. As a resut, the word average vaues reported by the HFAG group[7, 8] are arger than the SM predicted vaues for R(D by 1.9 σ, and for R(D by 3.3 σ (cf. Tabe 1. Tabe 1. Experimenta resuts [4 6] and SM predictions [] of R(D and R(D. The SM prediction with the scaar form factor (SFF variations [9] is aso given. The first and second experimenta errors are statistica and systematic, respectivey. R(D R(D BaBar 0.440±0.058± ±0.04±0.018 Bee 0.375±0.064± ±0.030±0.011 LHCb 0.336±0.07±0.030 experimenta average [7, 8] 0.397±0.040± ±0.016±0.010 SM Prediction [] 0.300± ±0.003 SM Prediction (SFF Variation [9] Received 1 Juy 017 G.C. acknowedges the support by FONDECYT (Chie ( , The work of C.S.K. was supported in part by the NRF grant funded by the Korean government of the MEST (016R1D1A1A E-mai: gorazd.cvetic@usm.c E-mai: francis.hazen@icecube.wisc.edu 3 E-mai: cskim@yonsei.ac.kr, Corresponding author 4 E-mai: scohph@yonsei.ac.kr Content from this work may be used under the terms of the Creative Commons Attribution 3.0 icence. Any further distribution of this work must maintain attribution to the author(s and the tite of the work, journa citation and DOI. Artice funded by SCOAP 3 and pubished under icence by Chinese Physica Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pubishing Ltd

3 Chinese Physics C Vo. 41, No. 11 ( Many theoretica expanations have been proposed to expain these indications of possibe epton universaity vioation, incuding charged scaar exchanges [10], vector resonances [11] or a W boson [1 14], and eptoquarks (or, equivaenty, R-parity vioating supersymmetry [1, 14 16]. The effects of exchange of on-she sterie neutrinos have aso been evauated, cf. [17, 18]. Expanation of the anomaies within an effective fied theory approach with dimension-6 scaar, vector and tensor operators appeared in Refs. [16, 19]. It was shown in Ref. [0] that the perturbative QCD (pqcd combined with attice resuts reduces the difference from the experimenta vaues. The eectroweak effects in B-anomaies were investigated in Ref. [1]. In Ref. [9], the authors checked how robust the SM predictions are for the R(D ( ratios. In contrast to the predictions of the vector form factors for B Dν decays, which have been determined we in measurements of the branching ratios and q -distributions in ight epton channes, the scaar form factor (SFF is measurabe ony in decays with τ eptons. Even sma deviations of the SFF from the attice vaues can bring the SM prediction coser to the current measured vaues of R(D, as shown in Tabe 1. As cosey reated to the decays, B D ( τν, the branching fraction (BF of B + τ + ν decay 1 was measured by Bee and BaBar [], as shown in Tabe II [7, 8, 3]. The SM prediction for the decay BF is given by [4] B SM (B + τ + ν=( ( This impies that if the measurement is improved in future B-factory experiments such as Bee-II [5], the comparison can carify whether new physics scenarios are needed. Tabe. Branching fractions of B + e + ν, µ + ν, τ + ν in units of 10 6 [7, 8, 3]. B + e + ν µ + ν τ + ν BABAR <1.9 < ±48 Bee <0.98 <1.7 91±19±11 experimenta average <0.98 < ±19 As a probe of new physics beyond the SM, the eptonic decays B + + ν are very interesting. This is because these decay rates can be evauated very precisey, and even at the tree-eve new physics effects may appear, e.g., contributions of charged Higgs [6] in two- Higgs doubet modes [7]. In the SM, the eptonic decay rates of B + + ν are proportiona to the square of the charged epton mass, m. Thus, the decays of B ± to e ± ν and µ ± ν are strongy suppressed in comparison with the decays to τ ± ν. Here we define new ratio R(0 [6] as R(0 Γ (B τν Γ (B µν, (3 which is one of the most interesting to test the universaity of weak interactions, since a the hadronic uncertainties cance in the ratio, and the ratio is a function of M τ /M B (and Mµ/M B. Heavy sterie neutra partices (a.k.a. heavy neutrinos have suppressed mixing with SM neutrinos and appear in various new physics scenarios, among them the origina seesaw [8] with very heavy neutrinos, seesaw with neutrinos with mass TeV [9], or with mass 1 GeV [30]. For some studies of the production of very heavy neutrinos with mass 100 GeV at the LHC we refer to Ref. [31]. We wi incude in our considerations the reactions B ± τ ± N,B ± Dτ ± N,B ± D τ ± N, where N is any heavy sterie neutrino of the Dirac or Majorana type, and interpret the measured branching fractions in the new physics scenario. For exampe, even if N is invisibe in the detector, we can sti distinguish B ± ± N signas from B + + ν for = e or µ, because these are two-body decays and therefore the momentum of the charged epton in the B meson rest frame is fixed by the mass of N. However, in the case of decays B ± τ ± N, the produced τ ± partice decays fast and hence there is more than one neutrino in the fina state, and the decay signature of B ± τ ± N cannot be distinguished from the ordinary B ± τ ± ν. Therefore, the experimentay observed signa of B ± τ ± ν may incude contributions from B ± τ ± N, and this signa we wi denote as B ± τ ± + missing. Massive neutrinos N in genera mix with the standard favor neutrinos, e.g. as in a seesaw type new physics scenario. We denote as U N the mixing coefficient for the heavy mass eigenstate N with the standard favor neutrino ν ( = e,µ,τ. The standard sub-ev neutrino ν (=e,µ,τ can then be represented as ν = 3 U νk ν k +U N N, (4 k=1 where ν k (k=1,,3 are the ight mass eigenstates. The 3 3 matrix U νk is the usua PMNS matrix [34]. In the reations (4 we assume the existence of ony one additiona massive sterie neutrino N, however, it can be extended with any number of N. Then the extended (unitary PMNS matrix U woud be in this case a 4 4 matrix, impying the reations 3 U νk =1 U N. (5 k=1 1 Throughout this paper, formuas can aso be appied to charge-conjugate modes. Other notations for U N exist in the iterature, among others V 4 in Ref. [3] and B N in Ref. [33]

4 Chinese Physics C Vo. 41, No. 11 ( One of our scenarios wi be with this unitarity assumption. This wi modify the decay width, due to the existence of a massive neutrino N, by the amount Γ (B + τ + N Γ (B + τ + N M N =0 and Γ (B + D ( τ + N Γ (B + D ( τ + N MN =0, where the minus terms are due to the unitarity of U. In the other scenario, the 3 3 PMNS mixing matrix is unitary, and N wi be regarded as a neutra fermion which does not mix with the SM favor neutrinos ν, but coupes with charged eptons such as τ in the same form as in the first scenario, for exampe, L g τγ µ W µ N+h.c. δl= ( g U τn τw XµN+h.c., (6 via mediation of a new physics charged gauge boson W ±, where subscript X denotes either L or R (eft or righthanded projection. Here W ± has the ight SM gauge boson W ± component, and this may ead to the coupings in Eq. (6, where the suppression effects of such (or simiar scenarios are parameterized in the parameter U τn. These coupings have the same form as in the previous scenario, but now we have no condition of unitarity (5. Such a vioation of the unitarity indicates unknown new physics beyond the SM. For our anaysis, we want to keep in a most generic form both the scenarios which ead to Eqs. (4 (5 and those which ead to Eq. (6. Nonetheess, we wish to mention, as a representative exampe for the mechanism of Eq. (6, the LR-modes [35 37] with the gauge group SU( L SU( R U(1 B L. Such modes have in the scaar sector a LR-doubet φ and tripets L, R. The vacuum expectation vaue (VEV u R of R is much arger than the other VEVs, u R v, w u L (where v,w 10 GeV are VEVs in φ, eading to a hierarchica mixing of the charged favor bosons W ± L and W R. ± The favor boson W ± R has, as a resut, a sma component ( v /u R of the SM mass eigenstate boson W ±. More specificay, W ± R = e iλ sinξw ± +cosξw ±, (7 where W ± is the heavy mass eigenstate with mass M g u R /4 M W, λ is a rea phase, and the mixing ange ξ is ξ vw ( MW, (8 v + w M where ( v + w = 1/( G F. In such modes we can have a heavy neutrino N R which forms with τ R an SU( R -doubet (N R,τ R, where N R is a favor and mass eigenstate, i.e., it does not mix with other favor neutrinos such as (ν,l c ( = e,µ,τ and ν,r ( = e,µ. Then the gauge boson mixing (7 gives us in the terms which coupe W ± R with eptons the foowing contributions (we take g R =g L =g: g τγ µ ( 1+γ5 = g τγ µ ( 1+γ5 N W Rµ+h.c. N ( e iλ sinξw µ +cosξwµ +h.c. (9 = g ( U τn τγ µ 1+γ5 NW µ +h.c.+..., (10 where the eipsis stands for the coupings with the heavy W ± boson. Here we see that in such a case the heavyight mixing parameter is U τn =sin ξ ξ, which, according to Eq. (8 is then ( 4 U τn MW. (11 M According to the anaysis of genera LR-modes of Ref. [37], there is a ot of freedom in V R, the righthanded quark mixing matrix, resuting in a reativey generous bound M > 300 GeV coming predominanty from the K L -K S mass difference m K and by B d Bd mixing (and assuming g R = g L. This impies the upper bound U τn < in such LR-scenarios 1. We further point out that the right-handedness of the couping τ-w-n, Eq. (10, does not affect the formuas for the decay widths Γ (B (D ( τν that we use in the present paper: we checked that these formuas turn out to be the same as in the case of the eft-handedness of the τ-w-n couping; we reca that the coupings of quarks to W are, of course, aways eft-handed. In our numerica anaysis, we wi derive the resuts for two scenarios: either the (4 4 matrix U is unitary, or U without the unitarity assumption, i.e. anayses with and without the unitarity assumption. Our formuas, to be derived in the foowing sections, wi be appicabe aso to cases with more than one additiona massive neutra fermion N where the second fermion has a mass M N 10 GeV, such partices being too heavy to be produced on-she in the considered decays. At present, the upper bounds for the mixing parameters U τn are avaiabe from the measurements and anayses of the CHARM [40] and DELPHI [41] Coaborations. These were dedicated direct measurements and 1 The LR-modes are restricted to the minima (symmetric versions, where V R is cosey reated with V L (=V CKM, the resuting bound is more restrictive, M >.3 TeV [38]. Signas of W R were searched for at the LHC (CMS Coaboration [39], in specific minima LR-mode scenarios where W ± R woud decay in the =e,µ channes to N,R; mass excusion regions M >3.3 TeV were found [39] in such scenarios, but ony if M N >00 GeV

5 Chinese Physics C Vo. 41, No. 11 ( anayses for decays producing heavy neutrinos N 1. On the other hand, there are indirect indications that the heavy-ight mixing parameters U τn have more restrictive upper bounds, coming from the τ epton decays where the anayses were made under the assumption of the SM scenario of (practicay massess neutrinos and unitary 3 3 PMNS matrix U PMNS, and there the epton universaity of the eectroweak couping g was shown to a arge precision [8] (Sec. 9. there. However, in these atter anayses, unike in Refs. [40, 41], it was assumed that heavy neutrinos do not exist. In view of the ack of any new updated dedicated measurements and anayses of the τ decays with heavy neutrinos, we wi usuay present here the upper bounds on the mixing parameters U τn as those from Refs. [40, 41]. Nonetheess, in the next section we wi present an anaysis of the epton universaity resuts [8] in the scenario of one additiona heavy neutrino N, and in the subsequent anayses in this work we wi keep in mind the restrictions on the heavyight mixing from such an anaysis. In this paper, we wi consider the recent experimenta anomaies, R(D and R(D, with the theoretica assumption of one not very heavy neutrino N: M N 1 GeV. We wi aso predict the newy defined R(0, which can be measured at Bee-II, as a function of the unknown parameters, M N and U τn. Restrictions on U τ N from epton universaity tests in τ decays The Heavy Favor Averaging Group (HFAG [8] obtained restrictions coming from epton universaity tests of the SM. They anayzed, among other things, the measured widths Γ (τ e+missing and Γ (µ e+missing, where missing stands for ν τ ν e (γ and ν µ ν e (γ. They thus obtained ( gτ =1.0010±0.0015, (1 g µ where the above notation stands for ( ( 5 gτ Mµ Γ (τ ν τ e ν e (γ = g µ M τ Γ (µ ν µ e ν e (γ [ ] Rγ (µ R W (µ f(m e /Mµ R γ (τ R W (τ f(me /Mτ = 1+(± (13 The above ratio in their anaysis incudes the QED [ (γ ] [4] and other corrections [ (M τ /M W, (M e /M µ ]. However, the net contribution of a these effects to the above quantity, represented as the correction factor in the brackets in Eq. (13, turns out to be 10 4 and wi thus be ignored in the anaysis here. If, on the other hand, we have an additiona heavy neutrino N which coupes to τ (but not to e or µ, either in a scenario in which the 4 4 matrix U is nonunitary [e.g., scenarios as in Eq. (6], or in a scenario where the 4 4 mixing matrix U is unitary [cf. Eq. (5], the above anaysis changes significanty. Namey, in the scenario where the forma 4 4 matrix U is nonunitary (and the 3 3 PMNS mixing matrix is unitary, accounting for the nonzero mass M N changes the above anaysis in the foowing way. The quantity Γ (τ ν τ e ν e gets repaced by Γ (τ ν τ e ν e Γ (τ ν τ e ν e +Γ (τ Ne ν e (14 = Γ (τ ν τ e ν e + U τn Γ (τ Ne ν e, where Γ (τ N ν stands for the decay width to a (massive neutrino N and the couping parameter U τn =1 (1 zn (15 Γ (τ N ν = G F 96π M 5 3 τ dzλ 1/ (1,z N,z(z z z { ( (1+z N z 1 z z + [(1 z N 1z ][ z 1+ z ( z z ]}. z (16 Here, z N =(M N /M τ, z =(M /M τ (=e or =µ, and z = p W /M τ, where p W is square of the invariant mass of W =( ν. The first term on the right-hand side of Eq. (15 denotes the usua SM contribution Γ (τ ν τ e ν e = 3 U τνk Γ (τ ν k e ν e k=1 = Γ (τ νe ν e Mν=0, (17 where the above identities hod because of the (practica massessness of the SM neutrinos ν k (k =1,,3 and the unitarity of the 3 3 PMNS matrix. The foowing ratio is crucia in the present anaysis: G (M N = Γ (τ N ν Γ (τ ν τ ν = U τn G (M N, (18 where =e or =µ, and G (M N = Γ (τ N ν Γ (τ ν ν Mν=0 (19 is the corresponding canonica ratio. The vaues (13, together with Eq. (15 and the notations (18-(19, then ead in the mentioned scenario to 1 CHARM imits [40] were obtained for M N < 300 MeV, based on the absence of signas N ν τz 0 ν τe + e in a neutrino beam dump experiment, where N is produced from decays of D s mesons. DELPHI imits [41] were obtained for M N >50 MeV, based on the absence of signas e + e Z 0 ν N for ong-ived and short-ived N, and they appy to a three parameters U N (=e,µ,τ

6 Chinese Physics C Vo. 41, No. 11 ( the foowing predictions for U τn : G e (M N =(± U τn = (± (0 G e (M N We reca that N here is massive, on-she, coupes ony to τ (and not to µ or e, and the 3 3 PMNS matrix is as in the SM, i.e., unitary. The ratio function G (M N is presented in Fig. 1 for the cases U τn =1 and U τn =10 3. G G ( N ( U N MN(G e V U N 1 Fig. 1. The ratios G e and G µ, Eq. (18, as a function of M N, for two choices of the heavy-ight mixing: U τn =1 (then G =G, and U τn =10 3. If, on the other hand, the 4 4 heavy-ight mixing matrix U is unitary, i.e., Eq. (5, an anaogous anaysis eads to the concusion that U τn has a very restrictive upper bound. Namey, in such a case the ratio on the eft-hand side of Eq. (13 must be beow the vaue of unity, and the most generous upper bound for U τn is then obtained if the right-hand side of Eq. (13 is U τn < ( (G e (M N 1 = ( G e (M N 1. (1 The restrictions (0 and (1 are presented in Fig.. From Fig. (a we can see that in the scenario of the 3 3 unitary PMNS matrix (the 4 4 matrix U is then nonunitary, the bounds (13 impy for the heavyight mixing parameter U τn stringent upper bounds U τn if the mass of N is ow (M N < 0.4 GeV, and for higher masses the bounds are not stringent: U τn 10 1 for M N >1 GeV. On the other hand, if the 4 4 matrix U is considered unitary, Fig. (b impies that the upper bounds for U τn are quite stringent ( 10 3 for higher masses M N > 0.6 GeV, whie for ight masses ess stringent upper bounds appy. The resuts of Fig. (b are consistent with the resuts of the anaysis [43] of the experimenta constraints for the we-measured decay ratios B(τ νν where = e,µ, and ν and ν are ight neutrinos or the heavy neutrino N, in the scenario which corresponds here to the unitary 4 4 U matrix. The authors of Ref. [43] obtain the upper bound U τn < 0.01 for 0.3 GeV<M N <1 GeV. G e G We wi keep in mind these restrictions which come indirecty from the epton universaity tests Eq. (13, i.e., from the decays τ e+missing and µ e+missing. Nonetheess, in the graphs we wi account for the bounds on U τn coming from the dedicated direct measurements of the CHARM [40] and DELPHI [41] Coaborations mentioned in the Introduction. U N U N H F A G [8] S e c.9.: g g g g (b MN(G e V 3 x 3 P M NS u n itary (a MN(G e V H F A G [8] S e c.9.: g g x 4 P M NS u n ita ry Fig.. The upper bounds as obtained by epton universaity tests, Eq. (13: (a in the case when the 3 3 PMNS mixing matrix is unitary (and the 4 4 matrix U is nonunitary - the soid ine is the upper bound if the centra vaue ( is taken in the vaues of Eq. (13, the dashed ine when the argest vaue ( is taken there; (b in the case when the 4 4 mixing matix U is unitary, and in that case the owest vaue ( is taken in Eq. (13. 3 Theoretica detais of R(D, R(D and R(0 with Light Sterie Neutrino 3.1 R(D and decays of B DN In the SM the ampitude for hadronic transition B D is given in terms of vector and scaar form factors, F 1 (q and F 0 (q, defined as D(p D cγ µ b B (p B [ = (p D +q µ (M ] B M D q µ F q 1 (q + (M B M D q q µ F 0 (q, (

7 Chinese Physics C Vo. 41, No. 11 ( where q = p B p D is the momentum of the virtua W. For the process B DN, the decay width is given in Ref. [18] as Γ (B DN = U N Γ (B DN, Γ (B DN = 1 384π 3 G F V cb 1 { M B (MB M D (M N +M dq 1 (q λ1/ (1, q F 1 (q [(q q M N+M (M N q M 4 N M 4 +F 0 (q 3(M B M D [q M N+M (M N+q M 4 N M 4 ], M D λ (1, M 1/ MB MB q, M N q (3 ][(q MD MB(q +MD+M B] 4 }, (4 where the kinematicay aowed vaues of q are (M N + M q (M B M D. Notice that if N is repaced by ν (i.e., M N 0, Γ (B DN in Eq. (4 becomes the SM decay width of B Dν, i.e., Γ SM (B Dν. The form factor F 1 (q is we known [44] 1. It can be expressed in terms of the variabe w w = (M B+M D q, (5 M B M D w+1 z(w =, (6 w+1+ in the foowing approximate form [44]: F 1 (q = F 1 (w=1 ( 1 8ρ z(w+(51ρ 10z(w (5ρ 84z(w 3, (7 where the free parameters ρ and F 1 (w = 1 have been recenty determined with high precision by the Bee Coaboration, see Ref. [46]: ρ = 1.09±0.05, (8 V cb F 1 (w=1 = (48.14± (9 The vaue (9 was deduced from their vaue of η EW G(1 V cb =η EW F 1 (w=1 4r/(1+r=(4.9± , where r = M D /M B and η EW = [47]. In our numerica evauations, we wi use the centra vaues ρ =1.09 and V cb F 1 (w =1= A recent study [9] has shown that the mosty unknown scaar form factor F 0 (q, expressed as F 0 (q =(1+αq +βq 4 F 1 (q, (30 can enhance the vaue of R(D up to within the SM, as shown in Tabe 1. We wi use this scaing reation for F 0 (q with α=+0.16 GeV and β = GeV as used in Ref. [9]. Taking the contribution from B DτN decays into account, and assuming the unitarity (5 of the matrix U, the ratio of branching fractions R(D is R(D Γ (B Dτ+ missing { Γ (B Dτν+ UτN [ Γ (B DτN Γ (B Dτν ]} = Γ (B Dν Γ (B Dν (=e,µ, (31 where Γ (B Dτν is the expression (4 for zero neutrino mass M ν = 0 and M = M τ. If unitarity is not assumed, R(D becomes R(D= Γ (B Dτν+ U τn Γ (B DτN (=e,µ. (3 Γ (B Dν 3. R(D and decays of B D N The matrix eements for the B D transition are more compicated than those for the B D transition, because of the vector character of D, incuding four form factors: [ ε µναβ ɛ q ] H µ η = iη (M B +M D ɛ ν(p D α (p B β V (q (M B +M D ɛ µ A 1 (q (M B +M D (p B+p D µ A (q ɛ q +M D q qµ (A 3 (q A 0 (q, (33 1 For eary attempts to account for the favor symmetry breaking in form factors of heavy pseudoscaars, cf. Ref. [45]

8 Chinese Physics C Vo. 41, No. 11 ( where The four form factors are H µ (η= 1 D (p D c(1 γ 5 γ µ b B 0 (p B = D 0 (p D c(1 γ 5 γ µ b B + (p B (34 H µ (η=+1 D + (p D b(1 γ 5 γ µ c B 0 (p B = D 0 (p D b(1 γ 5 γ µ c B (p B, (35 A 1 (q = 1 R (w+1f (1[1 8ρ z(w+(53ρ 15z(w (31ρ 91z(w 3 ], (36 V (q = A 1 (q R (w+1 [R 1(1 0.1(w (w 1 ], (37 A (q = A 1 (q R (w+1 [R (1+0.11(w (w 1 ], (38 A 3 (q = (M B+M D M D A 1 (q (M B M D A (q. (39 M D Here, R = M B M D /(M B +M D, the variabes w and z(w are given by Eq. (5, and the vaues of the free parameters determined in Ref. [48] are ρ = 1.14(±0.035, 10 3 F (1 V cb =34.6(±1.0, (40 R 1 (1 = 1.401(±0.038, R (1=0.864(±0.05. (41 We wi use the centra vaues of these parameters. For the decays B D N, the width is given in Ref. [18] as Γ (B D N= U N Γ (B D N, (4 where the canonica decay width (i.e., without the heavyight neutrino mixing is Γ (B D N = 1 G F V cb (MB M D dq λ 1/ 64π 3 MB (M N +M + 8M B q (M B +M D V (q + M ( 4 B 4MD q [ ( M + N M q [ 1 (M B M D A (q (M B +M D A 1 (q q q { ( 1 (M N+M (M B +M D ] + (M [ N+M MB q q ] A 1 (q } M D q q ( 1 (q +M D M B M D (M B +M D (M D (M B +M D q 1 [ 3 λ (M B +M D A 1 (q 4 q ] A 1 (q (M B +M D A (q ], (43 ( where λ=λ 1, M N, M q 1 q and q = 1 M Bλ 1/ ( 1, q M B, M D. M B As in the case of B DN, Γ (B D N expresses the SM decay width Γ SM (B D ν, when N is repaced by ν (i.e., M N 0. Anaogous to the case of R(D, incuding the possibe contribution from B D τn decays and assuming unitarity (5 of the fu U matrix, the ratio of branching fractions R(D can be written as R(D Γ { (B D τ+ missing Γ (B D τν+ U τn [ Γ (B D τn Γ (B D τν ]} = Γ (B D ν Γ (B D ν (=e,µ, (44 where Γ (B D τν is the expression (43 for zero neutrino mass M ν =0 and M =M τ. When the matrix U is not assumed to be unitary, R(D becomes R(D = Γ (B D τν+ U τn Γ (B D τn Γ (B D ν (=e,µ. ( R(0 and decays of B N In the decay B + + ν ( = e,µ,τ, within the SM with M ν 0, the decay width is given by Γ SM (B + + ν = 1 8π G Ff B V ub M 3 By (1 y (46 where y = M /M B. Here, M B and f B are the B + meson mass and the decay constant, respectivey, V ub is the corresponding CKM matrix eement, and G F =

9 Chinese Physics C Vo. 41, No. 11 ( GeV is the Fermi couping constant. The decay width for the process B + + N (=e,µ,τ is, as shown in Ref. [18]: Γ (B + + N= U N Γ (B ± ± N, (47 where the canonica width Γ (i.e., without the heavyight mixing factor U N is Γ (B + + N = 1 8π G F f B V ub M 3 B λ 1/ (1,y N,y [(1 y N y N +y (1+y N y ], (48 where y N =M N/M B and the function λ 1/ is given by λ 1/ (x,y,z=(x +y +z xy yz zx 1/. (49 It is obvious that if N is repaced by ν (i.e., y N 0, Γ (B + + N in Eq. (48 becomes Γ SM (B + + ν in Eq. (46. The SM expectation for the newy defined R(0, which is competey independent of hadronic uncertainty, is given by R(0 SM y τ(1 y τ y µ (1 y µ =(.55± , (50 where the uncertainty comes amost entirey from the uncertainty in the τ epton mass [49]. For the decays B + τ + + missing momentum, taking the contribution from the decays B + τ + N into account, the widths can be obtained with the unitarity assumption (5 by Γ (B + τ + + missing momentum = Γ (B + τ + ν+ U τn [ Γ (B + τ + N Γ (B + τ + ν ], (51 where Γ (B + τ + ν is for zero neutrino mass M ν =0, i.e., it is equa to the SM expression (46 with =τ. Considering the contribution from B + τ + N, the ratio R(0 in the considered scenario with one heavy neutrino N and the unitarity (5 of the U matrix is R(0 Γ (B τ+ missing { Γ (B+ τ + ν+ U τn [ Γ (B + τ + N Γ (B + τ + ν ]} =. (5 Γ (B µν Γ (B + µ + ν Here, Γ (B + + N is given in Eq. (48, and Γ (B + + ν is the same expression with zero neutrino mass M ν =0, i.e., Eq. (46. On the other hand, without the unitarity assumption, we have instead of the reation in Eq. (51 the foowing reation: Γ (B + τ + + missing momentum = Γ (B + τ + ν+ U τn Γ (B + τ + N; (53 and the ratio R(0 becomes R(0= Γ (B+ τ + ν+ U τn Γ (B + τ + N. (54 Γ (B + µ + ν As we sha see ater, the observation of R(0 can give very usefu information on the vaues of U τn and the mass of a sterie neutrino M N. 4 Numerica anaysis and discussion 4.1 R(D, R(D and B DτN, D τn In this numerica anaysis, first we study the R(D and R(D anomaies in our scenarios: in one scenario the matrix U without the unitarity assumption; in the other scenario, the fu 4 4 matrix U is considered unitary. As mentioned earier, the atter scenario is reaized when seesaw-type mechanisms are used, and the former may appear when the heavy neutra fermion N originates from a different, unknown, mechanism in a high energy framework beyond the SM. Aso, we examine the possibiity of finding certain direct bounds on magnitudes of the matrix eements U N from this anaysis of R(D as we as R(D. 1 We first cacuate R(D incuding the effect of the process B DτN as given in Eqs. (3 and (31. By comparing the theoretica resut with the experimenta data shown in Tabe 1, the aowed parameter space for U τn is found in terms of the mass of the sterie neutrino, M N. The resuts are shown in Fig. 3 for the two scenarios. In Fig. 3(a, the resut is found for the scenario without the assumption of unitarity for the 4 4 matrix U. It shows the aowed parameter space for U τn and M N obtained from the experimenta average vaue of R(D. The red (ight grey region is aowed by the experimenta data at 1σ eve, i.e., when the deviation does not surpass 1σ. The bue (dark grey region is aowed by the data at σ eve, i.e., when the deviation does not surpass σ (and is above 1σ. The white region can be regarded as excuded, as the deviation there surpasses σ. For comparison, the known avaiabe upper bounds from the CHARM and DELPHI experiments [40, 41] for U τn are aso incuded, as back squares 1. We see that a certain range of vaues of U τn and M N can fit the experimenta data of R(D. At 1σ eve, there is a tendency that for a smaer vaue of M N, a smaer U τn can fit 1 These upper bounds were obtained from various physica processes searching for heavy sterie neutrino N and are given in Tabe 3, see footnote and Refs. [3, 40, 41]

10 Chinese Physics C Vo. 41, No. 11 ( a 0.8 b UΤN 0.4 UΤN M N GeV M N GeV Fig. 3. (coor onine The shaded regions represent the aowed parameter space obtained from the experimenta average vaue of R(D shown in Tabe 1. The red (ight grey region is aowed by experimenta data for R(D at 1σ eve, and the bue (dark grey region at σ eve. The white region is excuded (more than σ deviation: (a when U is nonunitary, cf. Eq. (3; (b when U is unitary, cf. Eqs. (5 and (31. In (a and (b, the known present upper bounds from the CHARM and DELPHI experiments [40, 41] for U τn are denoted by back squares. In the nonunitary case (a, if accounting for the epton universaity measurements, the three back squares at M N =0.3, 0.4 and 0.5 GeV decrease to the vaues 10 as given in Fig. (a. In the unitary case (b, when accounting for the restrictions coming from τ νν [43] and the epton universaity measurements Fig. (b, the back squares at M N 0.3 GeV decrease to the vaues U τn the data. For instance, for M N = 0.3 GeV, the smaest vaue of U τn aowed by the 1σ data is.8 10, whie for M N = 1.0 GeV, the smaest aowed vaue of U τn is In particuar, for M N = 0.3 GeV, the vaues of.8 10 U τn are aowed by the 1σ data, in comparison with the known [40, 41] upper bound U τn = Simiary, for M N =0.4 GeV, the 1σ data aows the vaues of U τn , compared to the DELPHI [41] upper bound of U τn = We note that for M N =0.3 GeV and 0.4 GeV, the smaest aowed vaue of U τn is smaer than its avaiabe [40, 41] upper bound. However, if M N 0.5 GeV, the vaues of U τn aowed by the 1σ data are arger than the known DELPHI upper bounds. However, if using the indirect upper bounds coming from the epton universaity measurements, Fig. (a, a the aowed points move out of the 1σ region in Fig. 3(a. In Fig. 3 (b, the resut corresponds to the case considering unitarity of the (4 4 matrix U, cf. Eq. (5. In this scenario, there is no aowed parameter space for U τn and M N by the experimenta data at 1σ eve. At σ eve, a certain region of the parameter space is aowed. For exampe, for M N 0.3 GeV, the vaues of 0 U τn 1 are aowed, whie for M N = 0.6 GeV and 1.0 GeV, 0 U τn 0.4 and 0 U τn 0. are aowed, respectivey. In the case of R(D, it is harder to fit the experimenta data by incuding the contributions from B D τn together with those from the SM processes. Simiary to the anaysis of R(D, we compute R(D incuding the effect of the decay B D τn as given in Eqs. (44 and (45, and again examine the two scenarios: (a one considering the 4 4 matrix U to be nonunitary, Eq. (45; (b the other considering U to be unitary, Eqs. (5 and (44. It turns out that in the atter scenario (b, there are no aowed vaues of U τn and M N. Namey, Tabe 1 shows that the SM vaue of R(D is beow the centra experimenta vaue by more than 3σ; furthermore, the theoretica vaue in scenario (b becomes even ower when U τn 0, cf. Eq. (44. In contrast, in the former scenario (a, certain vaues of U τn and M N are aowed at σ eve, but not at 1σ eve. The aowed parameter space is shown in Fig. 4 for the scenario (a. For instance, for M N =0.3 GeV and 0.4 GeV, the vaues of U τn and U τn , respectivey, woud be aowed at 1σ eve. However, as the back squares in Fig. 4 indicate, the present CHARM and DELPHI [40, 41] upper bounds on U τn give compatibiity of the experimenta R(D with the scenario (a to at best σ eve, and this ony if M N 0.3 GeV. Further, if we incude the indirect upper bounds coming from the epton universaity measurements, Fig. (a, the aowed points move out of the σ region in Fig. 4. We comment on a feature of the resuts for R(D ( when the 4 4 U matrix is unitary. To be specific, et us consider the canonica decay widths Γ for the decays

11 Chinese Physics C Vo. 41, No. 11 ( B D ( τn given in Eqs. (4 and (43. We find that Γ (B D ( τn < Γ (B D ( τν, i.e., Γ (B D ( τn is a monotonicay decreasing function of M N. Thus the U τn term in R(D (, i.e., in the numerator of Eqs. (31 and (44, becomes negative and the effect of this term is to reduce R(D ( with respect to its SM vaue. Ony in the scenario where U is nonunitary, cf. Eqs. (3 and (45, does the presence of a heavy neutra fermion N increase the ratio R(D. UΤN M N GeV Fig. 4. (coor onine The shaded regions represent the aowed parameter space obtained from the experimenta average vaue of R(D shown in Tabe 1, in the case of nonunitarity of the fu U matrix, cf. Eq. (45. The red (ight grey region is aowed by the experimenta data at 1σ eve, and the bue (dark grey region at σ eve. The known CHARM and DELPHI [40, 41] upper bounds for U τn are denoted by back squares. If accounting for the epton universaity measurements, the three back squares at M N = 0.3, 0.4 and 0.5 GeV decrease to the vaues 10 as given in Fig. (a. 4. R(0 and B τn We now anayze the decay process B τ + missing momentum by incuding the process B τn. We wi first consider the BF of B τ + missing momentum. Simiary to the previous cases of R(D and R(D, two scenarios are examined: (a one scenario considering the matrix U as nonunitary, and where Γ (B + τ + + missing momentum is consequenty given by Eq. (53; (b the other scenario where the 4 4 matrix U is considered to be unitary, Eq. (5, and where Γ (B + τ + + missing momentum is consequenty given by Eq. (51. The resuts of this anaysis are presented in Figs. 5 (a and (b. The aowed vaues of U τn and M N are in a wide range for both scenarios. For future experiments, we make predictions for R(0 given in Eqs. (5 and (54. They are summarized in Tabe 3. The known upper bounds for vaues of U τn, for various specific vaues of M N, are taken from Refs. [3, 40, 41]. Due to the smaness of these upper bounds for U τn, the predicted vaues of R(0 for various vaues of M N do not deviate much from the SM predicted vaue R(0 SM =.55 10, cf. Eq. (50. Ony for 0.3 GeV M N 0.8 GeV are sizabe deviations from R(0 SM expected in the scenario of nonunitarity of U. For exampe, for M N =0.3 GeV and 0.4 GeV, the predicted vaues are R 0 = and , respectivey. Tabe 3. Predicted vaues of R 0. The R 0 vaues are cacuated in two ways: (i considering U to be nonunitary; (ii considering U to be unitary. The resuts are given to four digits to faciitate comparison. The vaues of U τn for various vaues of M N are taken as equa to the known CHARM and DELPHI upper bounds [40, 41]. M N /GeV U τn R 0 [nonunitarity] R 0 [unitarity] [SM] [SM] More interesting predictions reevant to R(0 are depicted in Fig. 6(a and (b, corresponding to the above two scenarios, respectivey. The figures show the graphs of U τn versus M N for given vaues of R(0. Provided that the vaue of R(0 is determined in future experiments (e.g., by measuring the BFs of B µν and B τ+ missing precisey, one can obtain usefu information on U τn and M N from the figures. For exampe, if R(0 is determined to be.50 10, it corresponds to the case (ii (i.e., thick red ine in Fig. 6(a and (b, from which the vaue of U τn can be extracted for a given M N. However, to discriminate experimentay between cases (i (SM and (ii, the branching ratios for B τ + missing and B µν wi have to be measured with precision of 1% or better; it is possibe that such a precision cannot be achieved at Bee-II. Further, if U is nonunitary and if, simutaneousy, the τ-n-w couping comes from LR-mode scenarios, then we have the upper bound U τn <5 10 3, cf. discussion after Eq. (11. In such a case, we cannot expect to get vaues of R(0 over.4 10 in such scenarios. Furthermore, if we take into account the indirect upper bounds on U τn coming

12 Chinese Physics C Vo. 41, No. 11 ( from the epton universaity measurements, cf. Fig. (a, the three back squares at M N =0.3, 0.4 and 0.5 GeV in Fig. 6(a decrease to the vaues 10, and we cannot get vaues of R(0 over a 0.8 b UΤN 0.4 UΤN M N GeV M N GeV Fig. 5. (coor onine The shaded regions represent the aowed parameter space obtained from the experimenta average vaue of BF(B + τ + ν = (1.06± given in Tabe. The red (ight grey region is aowed by the experimenta data at 1σ eve, and the bue (dark grey region at σ eve. Pot (a is obtained by considering U to be nonunitary, Eq. (53. Pot (b is obtained by considering U to be unitary, Eq. (51. In (a and (b, the known CHARM and DELPHI [40, 41] upper bounds for U τn are denoted by back squares. If the nonunitary case is generated by a LR-mode scenario, we have an additiona upper bound U τn < If accounting for the epton universaity measurements, the three back squares at M N =0.3, 0.4 and 0.5 GeV decrease to the vaues 10 as suggested in Figs. (a,(b. UΤN a iv iii ii i M N GeV UΤN i ii iii iv b M N GeV Fig. 6. (coor onine The graphs of U τn versus M N when R 0 is given. Pot (a is obtained by considering U to be nonunitary, cf. Eq. (54. In pot (b the 4 4 matrix U is considered to be unitary, Eqs. (5 and (5. In (a and (b, the cases (i, (ii, (iii, and (iv correspond to the given vaue of R 0=.3 10,.5 10,.7 10, and.30 10, respectivey. In (a and (b, the known CHARM and DELPHI [40, 41] upper bounds for U τn are denoted by back squares. If the nonunitary case (a is generated by a LR-mode scenario, we have an additiona upper bound U τn < If accounting for the epton universaity measurements, according to Fig., the three back squares at M N =0.3, 0.4 and 0.5 GeV in (a decrease to the vaues 10, and in (b at M N 0.3 GeV the back squares decrease to vaues Concusions In this work we studied the experimenta anomaies of the ratios R(D and R(D reated to the semieptonic B decays B Dτν and B D τν, and the newy suggested observabe R(0 reated to the purey eptonic B decays B τν and B µν, considering possibe effects from the presence of a neutra fermion (sterie neutrino N with

13 Chinese Physics C Vo. 41, No. 11 ( mass 1 GeV. In theoretica estimations of R(D, R(D, and R(0, the possibe effects of the processes B DτN, B D τn, and B τn were incuded. For generaity we considered two possibe scenarios: with the assumption of unitarity of the fu 4 4 mixing matrix U (extended PMNS matrix, and without the assumption of unitarity. We anayzed each of R(D, R(D and R(0 separatey. Our findings are summarized as foows. 1 For the observabe R(D, assuming U to be nonunitary, the discrepancy between the experimenta data and the theoretica prediction can be resoved at 1σ eve [cf. Fig. 3(a] for M N < 0.5 GeV. The possibe vaues of U τn are found for various vaues of the mass M N. Especiay, we have found that the vaues of U τn aowed by the 1σ data are.8 10 U τn and U τn for M N =0.3 GeV and 0.4 GeV, which fa within these intervas, respectivey. These vaues can be compared with the known CHARM and DELPHI [40, 41] upper bounds U τn = and for M N = 0.3 GeV and 0.4 GeV, respectivey. However, if the nonunitary U has its origin in genera SU( L SU( R U(1 B L modes [37], these modes woud impy that U τn < Further, if accounting for the indirect upper bounds on U τn coming from the epton universaity measurements, Fig. (a, the otherwise generous CHARM and DELPHI upper bounds at M N 0.5 GeV get decreased to U τn 10. When the fu 4 4 matrix U is assumed to be unitary, in contrast to the above case, there is no resoution for the anomay of R(D within 1σ eve, as shown in Fig. 3(b. 3 We found it to be more difficut to resove the anomay of R(D, compared with R(D. The discrepancy in R(D can be resoved ony when we assume that U is nonunitary, and at best ony at σ eve and at M N 0.3 GeV, cf. Fig. 4. When taking into account the indirect upper bounds on U τn coming from the epton universaity measurements, then the discrepancy cannot be resoved even at σ eve. 4 We demonstrated that certain usefu information on the parameters U τn and M N can be extracted from the purey eptonic B meson decays B τν, B τn and B µν. If the observabe R(0, invoving the rates of these decays, is measured in the future experiments, such as at Bee-II, the vaue of U τn coud be determined without any hadronic uncertainties, depending on M N, cf. Fig. 6. We assumed that the sterie heavy partice N is stabe and invisibe, hence does not decay inside the detector, and thus manifests itsef as missing momentum in the measurements. However, depending on the vaues of U N, M N and the detector size, the produced sterie neutrino N can decay within or beyond the actua detector. When N is produced as B + τ + N or B + D ( τ + N, and if N aso decays within the detector, the main signature of N wi be N + π (if N is Majorana or N π + (if N is Dirac or Majorana, which wi appear experimentay as a resonance in M( ± π. Then the experimenta signatures woud be B D ( ± ± π and B ± ± π (if N is Majorana or B D ( ± π and B ± π (if N is Dirac or Majorana. The detais of such decays at Bee-II and LHCb have been discussed in Ref. [18], for sufficienty sma vaues of U N, and M N GeV. References 1 N. Cabibbo, Phys. Rev. Lett., 10: 531 (1963 doi: / PhysRevLett M. Kobayashi and T. Maskawa, Prog. Theor. Phys., 49: 65 (1973 doi: /ptp H. Na et a (HPQCD Coaboration, Phys. Rev. D, 9(5: (015; Phys. Rev. D, 93(11: (016 doi: /PhysRevD ; /PhysRevD [arxiv: [hep-at]] 3 S. Fajfer, J. F. Kamenik, and I. Nišandžić, Phys. Rev. D, 85: (01 doi: /physrevd [arxiv: [hep-ph]] 4 M. Husche et a (Bee Coaboration, Phys. Rev. D, 9(7: (015 doi: /physrevd [arxiv: [hep-ex]] A. Abdesseam et a (Bee Coaboration, [arxiv: [hep-ex]] 5 R. Aaij et a (LHCb Coaboration, Phys. Rev. Lett., 115(11: (015 Addendum: [Phys. Rev. Lett., 115(15: (015] doi: /physrevlett ; /Phys- RevLett [arxiv: [hep-ex]] 6 J. P. Lees et a (BaBar Coaboration, Phys. Rev. Lett., 109: (01 doi: /physrevlett [arxiv: [hep-ex]] 7 Heavy Favor Averaging Group (HFAG Coaboration, resuts avaiabe at 016/rad/OUTPUT/HTML/rad tabe4.htm 8 Y. Amhis et a, arxiv: [hep-ex] 9 C. S. Kim, G. Lopez-Castro, S. L. Tostado, and A. Vicente, Phys. Rev. D, 95: (017 doi: / PhysRevD [arxiv: [hep-ph]] 10 S. Fajfer, J. F. Kamenik, I. Nišandžić, and J. Zupan, Phys. Rev. Lett., 109: (01 doi: /physrevlett [arxiv: [hep-ph]] A. Crivein, C. Greub, and A. Kokuu, Phys. Rev. D, 86: (01 doi: /physrevd [arxiv: [hep-ph]] A. Ceis, M. Jung, X. Q. Li, and A. Pich, JHEP, 1301: 054 (013 doi: /jhep01( [arxiv: [hep-ph]] J. A. Baiey et a, Phys. Rev. Lett., 109: (01 doi: /physrevlett [arxiv: [hep-ph]] P. Ko, Y. Omura, and C. Yu, JHEP, 1303: 151 (013 doi: /jhep03( [arxiv: [hepph]] A. Crivein, J. Heeck, and P. Stoffer, Phys. Rev. Lett., 116(8: (016 doi: /physrevlett [arxiv: [hep-ph]] J. M. Cine, Phys. Rev. D, 93(7: (016 doi: /physrevd [arxiv: [hep-ph]] M. Wei and Y. Chong-Xing, arxiv: [hep-ph] A. Ceis, M. Jung, X. Q. Li, and A. Pich, arxiv: [hep-ph] L. Wang, J. M. Yang, and Y. Zhang, arxiv: [hep-ph] 11 D. Buttazzo, A. Grejo, G. Isidori, and D. Marzocca, JHEP, 1608: 035 (016 doi: /jhep08(

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