Effect of scalar leptoquarks on the rare decays of B_s meson
aa r X i v : . [ h e p - ph ] J a n Effect of scalar leptoquarks on the rare decays of B s meson Rukmani Mohanta
School of Physics, University of Hyderabad, Hyderabad - 500 046, India
Abstract
We study the effect of scalar leptoquarks on some rare decays of B s mesons involving the quarklevel transition b → s l + l − . In particular we consider the decays B s → µ + µ − , ¯ B d → X s µ + µ − and B s → φ µ + µ − . The leptoquark parameter space is constrained using the recently measuredbranching ratio of the B s → µ + µ − process at LHCb and CMS experiments. Using such parameterswe obtain the branching ratio, forward backward asymmetry and the CP asymmetry parametersin the angular distribution of B s → φ µ + µ − process. PACS numbers: 13.20.He, 14.80.Sv . INTRODUCTION The standard model (SM) of electroweak interaction is very successful in explaining theobserved data so far and is further supported by the recent discovery of a Higgs-like bosonin the mass range of 126 GeV. But still there are many reasons to believe that it is not theultimate theory of nature, rather some low energy limit of some more fundamental theorywhose true nature is not yet well understood. It is therefore an ideal time to test thepredictions of the standard model more carefully and try to identify the nature of physicsbeyond it. If there would be new physics (NP) at the TeV scale associated with the hierarchyproblem, it is natural to expect that it would first show up in the flavor sector and in thiscontext the rare decays of B mesons induced by flavor changing neutral current (FCNC)transitions play a very crucial role. The FCNC transitions are one-loop suppressed in the SMand thus provide an excellent testing ground to look for possible existence of new physics.In this paper we would like to investigate some rare decay modes of B s meson using thescalar leptoquark (LQ) model. The study of B s meson has attracted a lot of attention inrecent times as large number of B s mesons are produced in the LHCb experiment and thiswould open up the possibility to study the rare decays of B s meson with high statisticalprecision. The most important and sought after rare decay mode is the B s → µ + µ − processmediated by the FCNC transition b → s , has been recently observed by the LHCb [1] andCMS [2] collaborations. This mode is very interesting as it is theoretically very clean andhighly suppressed in the standard model and hence well suited for constraining the newphysics parameter space. Another important rare decay channel mediated by the quarklevel transition b → sµ + µ − is the inclusive decay process ¯ B d → X s µ + µ − . The integratedbranching ratio for this process has been measured by both Belle [3] and BaBar [4] collab-orations. It is expected that in the low q region (1 GeV ≤ q ≤ ) as well as inthe high q region ( q ≥ ) the theoretical predictions are dominated by perturba-tive contributions and hence a theoretical precision of order 10% is in principle possible [5].We will use the measured branching ratios of these processes to constrain the leptoquarkparameters and subsequently apply these parameters to study the semileptonic rare decaymode B s → φµ + µ − .Leptoquarks are color-triplet bosons that can couple to a quark and a lepton at the sametime and can occur in various extensions of the SM [6]. Scalar leptoquarks are expected2o exist at the TeV scale in extended technicolor models [7] as well as in models of quarkand lepton compositeness [8]. The general classification of leptoquark models are discussedin [9] and the phenomenology of scalar leptoquarks have been studied extensively in theliterature [10–12]. Here, we will consider the model where leptoquarks can couple only toa pair of quarks and leptons and thus may be inert with respect to proton decay. In suchcases, proton decay bounds would not apply and leptoquarks may produce signatures inother low-energy phenomena [11].The paper is organized as follows. In section II we briefly discuss the effective Hamiltoniandescribing the process b → sl + l − . The new contributions arising due to the exchange of scalarleptoquark are presented in section III. We present the rare decay modes B s → µ + µ − and¯ B d → X s µ + µ − in sections IV and V respectively and obtain the constraints on leptoquarkparameters. The decay mode B s → φµ + µ − is discussed in Section VI and section VIIcontains the Conclusion. II. EFFECTIVE HAMILTONIAN FOR b → sl + l − PROCESS IN THE STANDARDMODEL
Within the standard model the effective Hamiltonian describing the quark level transitionis given as [13] H eff = − G F √ V tb V ∗ ts " X i =1 C i ( µ ) O i + C e π (cid:16) ¯ sσ µν ( m s P L + m b P R ) b (cid:17) F µν + C eff α π (¯ sγ µ P L b )¯ lγ µ l + C α π (¯ sγ µ P L b )¯ lγ µ γ l , (1)where G F is the Fermi constant and V qq ′ are the Cabibbo-Kobayashi-Maskawa (CKM) matrixelements, α is the fine structure constant, P L,R = (1 ∓ γ ) / C i ’s are the Wilsoncoefficients. The values of the Wilson coefficients are calculated at the next-to-next-leadingorder (NNLO) by matching the full theory to the effective theory at the electroweak scale andsubsequently solving the renormalization group equation to run them down to the b-quarkmass scale i.e., µ b = 4 . C eff contains a perturbative part and a resonance part which comes fromthe long distance effects due to the conversion of the real c ¯ c into the lepton pair l + l − . Thus,3 eff can be written as C eff = C + Y ( s ) + C res , (2)where s = q and the function Y ( s ) denotes the perturbative part coming from one loopmatrix elements of the four quark operators and is given in Ref. [13]. The long distanceresonance effect is given as [15] C res = 3 πα (3 C + C + 3 C + C + 3 C + C ) X V i = J/ψ,ψ ′ κ m V i Γ( V i → l + l − ) m V i − s − im V i Γ V i , (3)where the phenomenological parameter κ is taken as 1.7 and 2.4 for the two lowest two ¯ cc resonances J/ψ and ψ ′ [14]. III. NEW PHYSICS CONTRIBUTIONS DUE TO SCALAR LEPTOQUARK EX-CHANGE
In the leptoquark model the effective Hamiltonian describing the process b → sl + l − willbe modified due to the additional contributions arising from the exchange of leptoquarks.Here, we will consider the minimal renormalizable scalar leptoquark models [11], wherethe standard model is augmented only by one additional scalar representation of SU (3) × SU (2) × U (1) and which do not allow proton decay at the tree level. It has been shown in[11] that there are only two models which can satisfy this requirement. In these models theleptoquarks have the representation as X = (3 , , /
6) and X = (3 , , /
6) under the SU (3) × SU (2) × U (1) gauge group. Our aim here is to consider these scalar leptoquarks whichpotentially contribute to the b → sµ + µ − transitions and constrain the underlying couplingsfrom experimental data on B s → µ + µ − and ¯ B d → X s µ + µ − . Although the decay modes¯ B d → ¯ K µ + µ − and ¯ B d → K ∗ µ + µ − are also mediated by the same quark level transition b → sµ + µ − , we do not consider the measured branching ratios of such processes to constrainthe NP parameter space as these measurements involve additional uncertainties due to theform factors. However, we will comment on the recent observation of several anomalies onangular observables in the rare decay B → K ∗ µ + µ − by the LHCb collaboration [16].Now we consider all possible renormalizable interactions of such leptoquarks with SMmatter fields consistent with the SM gauge symmetry in the following subsections.4 . Model I: X = (3 , , / In this model the interaction Lagrangian for the coupling of scalar leptoquark X =(3 , , /
6) to the fermion bilinears is given as [11] L = − λ iju ¯ u iR X T ǫL jL − λ ije ¯ e iR X † Q jL + h.c. , (4)where i, j are the generation indices, Q L and L L are the left handed quark and leptondoublets, u R and e R are the right handed up-type quark and charged lepton singlets and ǫ = iσ is a 2 × X = V α Y α , L L = ν L e L , and ǫ = − . (5)After expanding the SU (2) indices the interaction Lagrangian becomes L = − λ iju ¯ u iαR ( V α e jL − Y α ν jL ) − λ ije ¯ e iR (cid:16) V † L u jαL + Y † α d jαL (cid:17) + h.c. . (6)Thus, from Eq. (6), one can obtain the contribution to the interaction Hamiltonian for the b → sµ + µ − process after Fierz rearrangement as H LQ = λ µ λ ∗ µ M Y [¯ sγ µ (1 − γ ) b ][¯ µγ µ (1 + γ ) µ ] ≡ λ µ λ ∗ µ M Y (cid:16) O + O (cid:17) , (7)which can be written analogous to the SM effective Hamiltonian (1) as H LQ = − G F α √ π V tb V ∗ ts ( C NP O + C NP O ) (8)with the new Wilson coefficients C NP = C NP = − π √ G F αV tb V ∗ ts λ µ λ ∗ µ M Y . (9) B. Model II: X= (3,2,1/6)
Analogous to the previous subsection the interaction Lagrangian for the coupling of X =(3 , , /
6) leptoquark to the fermion bilinear can be given as L = − λ ijd ¯ d iR X T ǫL jL + h.c. , (10)5here the notations used are same as the previous case. Expanding the SU (2) indices onecan obtain the interaction Lagrangian as L = − λ ijd ¯ d αR ( V α e jL − Y α ν jL ) + h.c. . (11)After performing the Fierz transformation the interaction Hamiltonian describing the process b → sµ + µ − is given as H LQ = λ s λ ∗ b M V [¯ sγ µ P R b ][¯ µγ µ (1 − γ ) µ ] = λ s λ ∗ b M V (cid:16) O ′ NP − O ′ NP (cid:17) , (12)where O ′ and O ′ are the four-fermion current-current operators obtained from O , bymaking the replacement P L ↔ P R . Thus, the exchange of the leptoquark X = (3 , , / C ′ NP = − C ′ NP = π √ G F αV tb V ∗ ts λ s λ ∗ b M V . (13)After obtaining the new physics contributions to the process b → sµ + µ − , we will proceedthe constrain the new physics parameter space using the recent measurement of B s → µ + µ − . IV. B s → µ + µ − DECAY PROCESS
The rare decay process B s → µ + µ − , mediated by the FCNC transition b → s is stronglyhelicity suppressed in the standard model. Furthermore, it is very clean and the onlynonperturbative quantity involved is the decay constant of B s meson which can be reliablycalculated by the well known non-perturbative methods such as QCD sum rules, latticegauge theory etc. Therefore, it is believed to be one of the most powerful tools to look fornew physics beyond the standard model. This process has been very well studied in theliterature and in recent times also it has attracted a lot of attention [17–22]. Therefore, herewe will quote the important results.The most general effective Hamiltonian describing this process H eff = G F α √ π V tb V ∗ ts " C eff O + C ′ O ′ , (14)where C eff = C SM + C NP and C ′ = C ′ NP . The branching ratio for this process is given asBR( B s → µ + µ − ) = G F π τ B s α f B s m B s m µ | V tb V ∗ ts | (cid:12)(cid:12)(cid:12) C eff − C ′ (cid:12)(cid:12)(cid:12) s − m µ m B s . (15)6ow using α = 1 / | V tb V ∗ ts | = 0 . ± . f B s = 227 ± C SM = − . B s meson from [23] we obtain the SM branchingratio for this process as BR( B s → µ + µ − ) = (3 . ± . × − , (16)which is consistent with the latest SM prediction Br( B s → µ + µ − ) = (3 . ± . × − [19].The branching ratio for this mode has recently been measured by both LHCb [1] and CMS[2] collaborations. Analyzing the data corresponding to an integrated luminosity of 1 fb − at √ − at √ B s → µ + µ − ) = (2 . +1 . − . ) × − . (17)The CMS collaboration [2] also obtained analogous resultBR( B s → µ + µ − ) = (3 . +1 . − . ) × − , (18)where they have used the data samples corresponding to integrated luminosities of 5 and20 fb − at √ s = 7 and 8 TeV. The weighted average of these two measurements yieldsBR( B s → µ + µ − ) = (2 . ± . × − , (19)which is consistent with the latest SM prediction (16), but certainly it does not rule out thepossibility of new physics in this mode. While new physics can still affect this decay mode,but certainly its contribution is not the dominant one.However, as discussed in Ref . [17], in the experiment the time integrated untagged decayrate is measured, whereas in the above theoretical calculation the effect of meson oscillationis not taken into account. Therefore, while comparing the SM prediction for B s → µ + µ − decay rate with the experimental result one should take into account the sizable widthdifference ∆Γ s between B s mass eigenstates. i.e., y s ≡ Γ ( s ) L − Γ ( s ) H Γ ( s ) L + Γ ( s ) H = ∆Γ s s = 0 . ± . , (20)where Γ s = τ − B s denotes the average B s decay width. Hence, the experimental result isrelated to the theoretical prediction asBR th ( B s → µ + µ − ) = (cid:20) − y s A ∆Γ y s (cid:21) BR( B s → µ + µ − ) exp , (21)7here the observable A ∆Γ equals +1 in the SM. Thus, using the experimental value of y s we obtain the branching ratio in the standard modelBR( B s → µ + µ − ) th | SM = (3 . ± . × − . (22)We will now consider the effect of scalar leptoquarks in this mode. One can write thetransition amplitude for this process from Eq. (14) as A ( B s → µ + µ − ) = h µ + µ − |H eff | B s i = − G F √ π V tb V ∗ ts αf B s m B s m µ C SM P, (23)where P ≡ C − C ′ C SM = 1 + C NP − C ′ NP C SM = 1 + re iφ NP , (24)with re iφ NP = ( C NP − C ′ NP ) /C SM , (25)denotes the new physics contribution and φ NP is the relative phase between SM and the NPcouplings. In general P ≡ | P | e φ P carries the CP violating phase φ P . The phases φ P and φ NP are related to each other by the relationtan φ P = r sin φ NP rφ NP . (26)As discussed in section III, the exchange of the leptoquark X (3 , , /
6) gives new contributionto C and X (3 , , /
6) gives additional contribution C ′ the branching ratio in both thecases will be BR( B s → µ + µ − ) th = (cid:20) A ∆Γ − y s (cid:21) BR SM (1 + r − r cos φ NP ) . (27)In the leptoquark model the observable A ∆Γ becomes [17] A ∆Γ = cos 2 φ P . (28)In order to find the constrain on the combination of LQ couplings we require that eachindividual leptoquark contribution to the branching ratio does not exceed the experimentalresult. Now using the SM value from (16), we show in Fig. 1 the allowed region in r − φ NP plane which is compatible with the 2 σ range of the experimental data. From the figure onecan see that for 0 ≤ r ≤ . φ NP is allowed, i.e.,0 ≤ r ≤ . , for 0 ≤ φ NP ≤ π . (29)8 r φ NP FIG. 1: The allowed region in the r − φ NP parameters space obtained from the BR( B s → µ + µ − ). V. ANALYSIS OF ¯ B d → X s µ + µ − MODE
Now we would like to constrain the NP couplings from the measured branching ratioof the inclusive decay ¯ B d → X s µ + µ − . The integrated branching ratio for this process hasbeen measured by both Belle [3] and BaBar [4] collaborations and the average value of thesemeasurements in the two regions are [5]BR( B d → X s µ + µ − ) = (1 . ± . × − low q = (0 . ± . × − high q , (30)where the low- q and high- q regions correspond to 1 GeV ≤ q ≤ and q ≥ . , respectively. The decay mode has been very well studied in the literature and herewe are presenting only the main results. The differential branching ratio for this process inthe standard model is given as [24] d BR ds (cid:12)(cid:12)(cid:12)(cid:12) SM = B
83 (1 − s ) s − t s × (cid:20) (2 s + 1) (cid:18) t s + 1 (cid:19) | C eff | + (cid:18) − s ) t s + (2 s + 1) (cid:19) | C | + 4 (cid:18) s + 1 (cid:19) (cid:18) t s + 1 (cid:19) | C | + 12 (cid:18) t s + 1 (cid:19) Re( C C eff ∗ ) (cid:21) , (31)9here t = m µ /m poleb and s = q / ( m poleb ) . The normalization constant B is related toBR( ¯ B → X c e ¯ ν e ) through B = 3 α BR( ¯ B → X c e ¯ ν e )32 π f ( ˆ m c ) κ ( ˆ m c ) | V tb V ∗ ts | | V cb | , (32)where ˆ m c = m polec /m poleb . f ( ˆ m c ) is the lowest order phase space factor for the ¯ B → X c e ¯ ν process, i.e., f ( ˆ m c ) = 1 − m c + 8 ˆ m c − ˆ m c −
24 ˆ m c ln ˆ m c , (33)and the function κ ( ˆ m c ) is the power correction to BR( ¯ B → X c e ¯ ν ), which includes both the O ( α s ) QCD corrections and the leading order (1 /m b ) power corrections κ ( ˆ m c ) = 1 − α s ( m b )3 π g ( ˆ m c ) + h ( ˆ m c )2 m b . (34)Here the two functions are given as g ( ˆ m c ) = (cid:18) π − (cid:19) (1 − ˆ m c ) + 32 ,h ( ˆ m c ) = λ + λ f ( ˆ m c ) (cid:2) − m c −
72 ˆ m c + 72 ˆ m c −
15 ˆ m c −
72 ˆ m c −
72 ˆ m c ln ˆ m c (cid:3) , (35)where λ and λ are the kinetic energy and magnetic moment operators.In the leptoquark model there will be additional contribution arising due to the exchangeof leptoquarks which will introduce the new couplings C NP , C NP , C ′ NP and C ′ NP as dis-cussed in section III. Including these NP contributions and neglecting the sub-leading termswhich are suppressed by m µ /m b and m s /m b , the branching ratio can be given as (cid:18) d BR ds (cid:19) Total = (cid:18) d BR ds (cid:19) SM + B h
163 (1 − s ) (1 + 2 s )[Re( C eff C NP ∗ + Re( C C NP ∗ ]+ 83 (1 − s ) (1 + 2 s ) h | C NP | + | C NP | + | C ′ NP | + | C ′ NP | i + 32(1 − s ) Re( C C NP ∗ ) i . (36)For numerical evaluation we use the input parameters as ˆ m c = 0 . ± .
02 [25], BR( ¯ B → X c e ¯ ν ) = (10 . ± . | V tb V ∗ ts | / | V cb | = 0 . ± .
009 [26] and the parameters λ and λ as λ = − (0 . ± .
05) GeV and λ = 0 .
12 GeV [27]. With these parameters the branchingratio in the SM is found to beBR( ¯ B → X s µ + µ − ) = (1 . ± . × − low q = (0 . ± . × − high q . (37)10hese predicted branching ratios are in agreement with the corresponding experimental val-ues within their 1- σ range. To constrain the new physics couplings coming from the exchangeof scalar leptoquarks X (3 , , /
6) and X (3 , , / X (3 , , /
6) only the NP couplings C NP and C NP will arise whereas for X (3 , , /
6) thecouplings C ′ NP and C NP ′ will contribute. Furthermore, as shown in Eqs. (9) and (13) themagnitudes of these couplings in each case will be same. With the additional assumptionthat these two couplings will have the same phase φ NP and neglecting the small phase dif-ference between C eff and C NP we obtain the constraint equations for these NP couplingsfrom Eqs. (30), (36) and (37) as C NP h (0 .
58 + 0 . C + 0 . C ) cos φ NP + 0 .
02 sin φ NP i + 0 . | C NP | = − . ± . q ) C NP h (0 .
11 + 0 . C + 0 . C ) cos φ NP + 0 .
009 sin φ NP i + 0 . | C NP | = 0 . ± . q ) (38)The corresponding 1- σ allowed region in the | C NP | - φ NP plane is shown in the Figure-2where the green region corresponds to the constraint coming from high- q bound and themagenta region coming from the low- q limit. From the figure one can see that the boundscoming from the high- q measurement is rather weak. From the low- q constraint one caninfer that for the value − ≤ C NP ≤ φ NP is allowed. These boundscan be translated to the bounds on r and φ NP as done for B s → µ + µ − process as0 ≤ r ≤ . , for 0 ≤ φ NP ≤ π . (39)Thus, from Eqns. (29) and (39) one can see that the bounds on NP couplings coming fromBR( ¯ B d → X s µ + µ − ) is slightly weak in comparison to BR( B s → µ + µ − ).Next we will consider the contributions coming from the X (3 , , /
6) exchange. In thiscase the new couplings C ′ NP and C ′ NP will come into picture. Proceeding in a similarfashion as done for X (3 , , /
6) leptoquark case, we obtain the constraint equations forthese parameters as0 . h | C ′ NP | + | C ′ NP | i = ( − . ± .
51) (low q )0 . h | C ′ NP | + C ′ NP | i = (0 . ± .
12) (high q ) . (40)11
50 100 150 200 250 300 350 - - Φ NP C N P FIG. 2: The allowed region in the C NP − φ NP parameters space obtained from the BR( ¯ B d → X s µ + µ − ), where the green (magenta) region corresponds to high- q (low- q ) limits. The corresponding allowed region in C ′ NP - C ′ NP plane is shown in Figure-3, where the greenregion corresponds to the bounds coming from high- q limit and magenta region correspondsto the low- q bound. Thus, from the low- q bounds one can obtain the limits on C ′ NP and C ′ NP as − . ≤ | C ′ NP | , | C ′ NP | ≤ .
5. Again translating the above bounds into the boundon r one can obtain 0 ≤ r ≤ . , (41)which is again much weaker than the bounds coming from B s → µ + µ − measurements.However, in our analysis we will use relatively mild constraint, consistent with bothBR( B s → µ + µ − ) and BR( ¯ B d → X s µ + µ − ) measurements as0 ≤ r ≤ . , with 60 ◦ ≤ φ NP ≤ ◦ . (42)This limit on r can be translated to give us bound on leptoquark coupling using Eqs. (9),(13) and (25) as (cid:12)(cid:12)(cid:12)(cid:12) λ µ λ ∗ µ M Y (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ s λ ∗ b M V (cid:12)(cid:12)(cid:12)(cid:12) ≤ . × − GeV − . (43)If we use the values of the couplings as | λ d,e | ≈ .
1, allowing the perturbation theory to bevalid, we get the lower bound on the scalar leptoquark mass as M X ≥ . . (44)12 - - - C
9' NP C ' N P FIG. 3: The allowed region in the C ′ NP − C ′ NP parameters space obtained from the BR( ¯ B d → X s µ + µ − ), where the green (magenta) region corresponds to high- q (low- q ) limits. It should be noted that the recent measurement by LHCb collaboration [16] shows severalsignificant deviations on angular observables in the rare decay B → K ∗ µ + µ − from their cor-responding SM expectations. In particular an anomalously low value of S at high q at 2 . σ level and an opposite sign of S at low q region at 2 . σ level. Although it is conceivable thatthese anomalies are due to statistical fluctuations or under estimated theory uncertainties[29], but the possible indication of new physics could not be ruled out. It has been shownin Ref. [30] that a consistent explanation of most of the anomalies associated with b → s rare decays can be obtained by NP contributing simultaneously to the semileptonic operator O and its chirally flipped counterpart O ′ with C NP ≃ − (1 . ± .
3) and C ′ NP ≃ . ± . C NP and C NP contribute simultaneously it may notbe possible to explain these anomalies.After obtaining the allowed range for the leptoquark coupling we will now proceed tostudy the semileptonic decay process B s → φµ + µ − . VI. B s → φ l + l − PROCESS
Here we will consider the decay mode B s → φµ + µ − . At the quark level, this decay modeproceeds through the FCNC transition b → sl + l − , which occurs only through loops in the13M, and therefore, it constitutes a quite suitable tool of looking for new physics. Moreover,the dileptons present in this process allow us to formulate many observables which can serveas a testing ground to decipher the presence of new physics [28].Recently the branching ratio of this decay mode has been measured by the LHCb collab-oration [31] using the data corresponding to an integrated luminosity of 1 . − collectedat √ s =7 TeV as BR( B s → φµ + µ − ) = (cid:0) . +0 . − . ± . ± . (cid:1) × − . (45)They have also performed the angular analysis and determine the angular observables F L , S , A and A , which are consistent with the standard model expectations. This process hasbeen very well studied in the literature, both in the SM and in various extensions of it [32].The branching ratio predicted in the standard model is in the range (14 . − . × − which is significantly higher than the present experimental value (45). This deviation maybe considered as a smoking gun signal of new physics in this mode or more generally in theprocesses involving b → s transitions.Using the effective Hamiltonian presented in Eq. (1) one can obtain the transition am-plitude for this process. The matrix elements of the various hadronic currents between theinitial B s meson and the final vector meson φ can be parameterized in terms of various formfactors as [33] h φ ( k, ε ) | ( V − A ) µ | B s ( P ) i = ǫ µναβ ε ∗ ν P α k β V ( q ) m B + m φ − iε ∗ µ ( m B + m φ ) A ( q )+ i ( P + k ) µ ( ε ∗ q ) A ( q ) m B + m φ + iq µ ( ε ∗ q ) 2 m φ q h A ( q ) − A ( q ) i , h φ ( k, ε ) | ¯ sσ µν q ν (1 + γ ) b | B s ( P ) i = iǫ µναβ ε ∗ ν P α k β T ( q ) + h ε ∗ µ ( m B − m φ ) − ( ε ∗ q )( P + k ) µ i T ( q ) + ( ε ∗ q ) h q µ − q m B − m φ ( P + k ) µ i T ( q ) , (46)where V and A denote the vector and axial vector currents, A , A , A , A , V, T , T and T are the relevant form factors and q is the momentum transfer.14hus, with eqs. (1) and (46) the transition amplitude for B s → φl + l − is given as M ( B s → φ l + l − ) = G F α √ π V tb V ∗ ts (cid:26) ¯ lγ µ l h − Aǫ µναβ ε ∗ ν k α q β − iBε ∗ µ + iC ( P + k ) µ ( ε ∗ · q ) + iD ( ε ∗ · q ) q µ i + ¯ lγ µ γ l h − Eǫ µναβ ε ∗ ν k α q β − iF ε ∗ µ + iG ( ε ∗ · q )( P + k ) µ + iH ( ε ∗ · q ) q µ i(cid:27) , (47)where the parameters A, B, · · · H are given as [34] A = 2 (cid:16) C eff SM + C NP + C ′ NP (cid:17) V ( q ) m B + m φ + 4 m b q C T ( q ) ,B = ( m B + m φ ) (cid:18) C eff SM + C NP − C ′ NP ) A ( q ) + 4 m b q ( m B − m φ ) C T ( q ) (cid:19) ,C = 2( C eff SM + C NP − C ′ NP ) A ( q ) m B + m φ + 4 m b q C T ( q ) + q m B − m φ T ( q ) ! ,D = 4( C eff SM + C NP − C ′ NP ) m φ q (cid:16) A ( q ) − A ( q ) (cid:17) − C m b q T ( q ) ,E = ( C SM + C NP + C ′ NP ) 2 V ( q ) m B + m φ ,F = 2( C SM + C NP − C ′ NP )( m B + m φ ) A ( q ) ,G = ( C SM + C NP − C ′ NP ) 2 A ( q ) m B + m φ ,H = 4( C SM + C NP − C ′ NP ) m φ q (cid:16) A ( q ) − A ( q ) (cid:17) . (48)The differential decay rate is given as d Γ ds = G F α π | V tb V ∗ ts | m B τ B λ / (1 , r φ , ˆ s ) v l ∆ , (49)where ˆ s = q /m B , r φ = m φ /m B , v l = p − m l /s , λ ≡ λ (1 , r φ , ˆ s ), is the triangle functionand ∆ = 13 r φ h λm B ˆ s (cid:16) (3 − v l ) | A | + (12 r φ ˆ s + λ )(3 − v l ) | B | + λ m B (3 − v l ) | C | + 16 v l m B r φ ˆ sλ | E | + (24 r φ ˆ sv l + λ (3 − v )) | F | + m B λ (cid:0) s (1 + r φ )(1 − v ) − s (1 − v l ) + λ (3 − v ) (cid:1) | G | + 3 λm B ˆ s (1 − v l ) | H | + 2 Re [ F G ∗ ] m B λ (cid:0) r φ (3 − v ) + v l (1 + 2ˆ s ) − (cid:1) − Re [ F H ∗ ] m B ˆ s (1 − v l ) λ + 6 Re [ GH ∗ ] m B ˆ sλ (1 − r φ )(1 − v )+ 2 Re [ BC ∗ ] m B λ (3 − v )( r φ + ˆ s − i . (50)15nother observable is the lepton forward backward asymmetry ( A F B ), which is also a verypowerful tool for looking into new physics signature. In particular the position of the zerovalue of A F B is very sensitive to the presence of new physics. The normalized forward-backward asymmetry is defined as A F B ( s ) = Z d Γ d ˆ sd cos θ d cos θ − Z − d Γ d ˆ sd cos θ d cos θ Z d Γ d ˆ sd cos θ d cos θ + Z − d Γ d ˆ sd cos θ d cos θ , (51)where θ is the angle between the directions of l + and B s in the rest frame of the lepton pair.The forward-backward asymmetry can also be written in the form [34] A F B ( q ) = −
1∆ 8 m B √ λ v l ˆ s Re [ A ∗ F + B ∗ E ] (52)As seen from [31], the actual decay being observed is not B s → φµ + µ − but B s → φ ( → K + K − ) µ + µ − . Thus, the angular analysis of the four body final state offers a large numberof observables in the differential decay distribution [35]. The angular distribution of thedecay process ¯ B s → φ ( → K + K − ) µ + µ − can be defined by the decay angles θ K , θ l and Φ,where θ K ( θ l ) denotes the angle of K − ( µ − ) with respect to the direction of flight of the ¯ B s meson in the K + K − ( µ + µ − ) center-of-mass frame and Φ denotes relative angle of the µ + µ − and the K + K − decay planes in the ¯ B s meson center of mass frame and is given as [14] as d Γ dq d cos θ l d cos θ K d Φ = 932 π I s sin θ K + I c cos θ K + ( I s sin θ K + I c cos θ K ) cos 2 θ l + I sin θ K sin θ l cos 2Φ + I sin 2 θ K sin 2 θ l cos Φ+ I sin 2 θ K sin θ l cos Φ + ( I s sin θ K + I c cos θ K ) cos θ l + I sin 2 θ K sin θ l sin Φ + I sin 2 θ K sin 2 θ l sin φ + I sin θ K sin θ l sin 2Φ . (53)The corresponding expression for CP conjugate process B s → φ ( → K + K − ) µ + µ − ( d ¯Γ) canbe obtained from (53) by the replacement of I i ’s by ¯ I i ’s where these observables are relatedto each other through I ( a )1 , , , , −→ ¯ I ( a )1 , , , , , I ( a )5 , , , −→ − ¯ I ( a )5 , , , , (54)with all weak phases conjugated. The angular coefficients I ( a ) i , usually expressed in terms16f the transversity amplitudes which are given as [14] A ⊥ L,R = N p λ h (cid:16) ( C eff + C NP + C ′ NP ) ∓ ( C + C NP + C ′ NP ) (cid:17) V ( q ) m B + m φ + 2 m b sC T ( q ) i ,A k L,R = − N √ m B − m φ ) h (cid:16) ( C eff + C NP − C ′ NP ) ∓ ( C + C NP − C ′ NP ) (cid:17) A ( q ) m B − m φ + 2 m b s C T ( q ) i ,A L,R = − N m φ √ s h (cid:16) C eff + C NP − C ′ NP ) ∓ ( C + C NP − C ′ NP ) (cid:17) × (cid:18) ( m B − m φ − s )( m B + m φ ) A ( q ) − λ A ( q ) m B + m φ (cid:19) +2 m B C ( m B + 3 m φ − s ) T ( q ) − λ m B − m φ ! i , (55) A t = N λ s h C + C NP − C ′ NP ) i A ( q ) (56)where N = V tb V ∗ ts (cid:20) G F α · π m B sv p λ (cid:21) / , (57)with λ = ( m B + m φ + s ) − m B m φ . With these transversity amplitudes the angularcoefficients are given as I s = 2 + v l h | A L ⊥ | + | A L k | + ( L → R ) i + 4 m µ s Re (cid:0) A L ⊥ A R ∗⊥ + A L k A R ∗k (cid:1) I c = | A L | + | A R | + 4 m µ s (cid:16) | A t | + 2 Re ( A L A R ∗ ) (cid:17) I s = v l (cid:16) | A L ⊥ | + | A L k | + ( L → R ) (cid:17) ,I c = − v l (cid:16) | A L | + ( L → R ) (cid:17) ,I = v l (cid:16) Re ( A L A L ∗k + ( L → R ) (cid:17) ,I = v l (cid:16) Re ( A L A L ∗k − ( L → R ) (cid:17) ,I = √ v l (cid:16) Re ( A L k A L ∗⊥ − ( L → R ) (cid:17) ,I s = 2 v (cid:16) Re ( A L k A L ∗⊥ − ( L → R ) (cid:17) ,I = √ v l (cid:16) Im ( A L A L ∗k − ( L → R ) (cid:17) ,I = v l √ (cid:16) Im ( A L A L ∗⊥ ) + ( L → R ) (cid:17) ,I = v l (cid:16) Im ( A L ∗k A L ⊥ ) + ( L → R ) (cid:17) . (58)17rom these angular coefficients one can construct twelve CP averaged angular coefficients S ( a ) i and twelve CP asymmetries A ( a ) i as S ( a ) i = ( I ( a ) i + ¯ I ( a ) i ) . d (Γ + ¯Γ) dq , A ( a ) i = ( I ( a ) i − ¯ I ( a ) i ) . d (Γ + ¯Γ) dq . (59)All the physical observables can be expressed in terms of S i and A i . For example the CPasymmetry in the dilepton mass distribution can be expressed as A CP = d (Γ − ¯Γ) dq . d (Γ + ¯Γ) dq = 34 (2 A s + A c ) −
14 (2 A s + A c ) . (60)The q average of these observables are defined as follows: h S ( a ) i i = Z dq ( I ( a ) i + ¯ I ( a ) i ) , Z dq d (Γ + ¯Γ) dq h A ( a ) i i = Z dq ( I ( a ) i − ¯ I ( a ) i ) , Z dq d (Γ + ¯Γ) dq . (61)After getting familiar with the different observables associated with B s → φµ + µ − decayprocess we now proceed for numerical estimation. For this purpose we use the form factorscalculated in the light-cone sum rule (LCSR) approach [33], where the q dependence ofvarious form factors are given by simple fits as f ( q ) = r − q /m fit , (for A , T ) f ( q ) = r − q /m R + r − q /m fit , (for V, A , T ) f ( q ) = r − q /m fit + r (1 − q /m fit ) , (for A , ˜ T ) . (62)The values of the parameters r , r , m R and m fit are taken from [33]. The form factors A and T are given as A ( q ) = m B + m V m φ A ( q ) − m B − m φ m φ A ( q ) ,T ( q ) = m B − m φ q (cid:16) ˜ T ( q ) − T ( q ) (cid:17) . (63)The particle masses and the lifetime of B s meson are taken from [23]. The quark masses(in GeV) used are m b =4.8, m c =1.5, the fine structure coupling constant α = 1 /
128 andthe CKM matrix elements as V tb V ∗ ts = 0 . d B r (cid:144) d s - - A F B FIG. 4: Variation of the differential branching ratio (in units of 10 − ) (left panel) and the forward-backward asymmetry with respect to the momentum transfer s (right panel) for the B s → φµ + µ − process. variation of differential decay rate (left panel) and the forward backward asymmetry (rightpanel) in the standard model with respect to the di-muon invariant mass.In the leptoquark model, this process will receive additional contribution arising fromthe leptoquark exchange. Hence, in the leptoquark model the Wilson coefficients C , willreceive additional contributions C NP , as well as new Wilson C ′ , associated with the chirallyflipped operators O ′ , will also be present as already discussed in Section III. The boundson these new coefficients can be obtained from the constraint on r (42) extracted from theexperimental results on BR( B s → µ + µ − ) and BR( ¯ B d → X s µ + µ − ). For the leptoquarks X = (3 , , /
6) and X = (3 , , / r ≤ .
35 for φ in the range(60 − ◦ . This constraint can be translated with eqns (9), (13) and (42) which gives thevalue of the new Wilson coefficients as | C LQ | = | C LQ | ≤ | r C SM | (for X = (3 , , / | C ′ LQ | = | C ′ LQ | ≤ | r C SM | (for X = (3 , , / . (64)Using these values we show the variation of differential decay rate and forward-backwardasymmetry for X = (3 , , /
6) in Figure-5 and for X = (3 , , /
6) in Figure-6. From thesefigures it can be seen that the branching ratio could have significant deviation from its SMvalue both in the upward as well as downward direction. However, the zero position of theforward-backward asymmetry does not have any significant deviation.We now proceed to calculate the total decay rate for B s → φ µ + µ − . It should be notedthat the long distance contributions arise from the real ¯ cc resonances with the dominant19 d B r / d s s in GeV -0.4-0.2 0 0.2 0.4 0 5 10 15 20 A F B s in GeV FIG. 5: Same as Figure-4, where the red curves represent the SM values and the grey regionsrepresent the results due to X = (3 , , /
6) leptoquark contributions. d B r / d s s in GeV -0.4-0.2 0 0.2 0.4 0 5 10 15 20 A F B s in GeV FIG. 6: Same as Figure-4, where the red curves represent the SM values and the grey regionsrepresent the results due to X = (3 , , /
6) leptoquark contributions. contributions coming from the low lying resonances
J/ψ and ψ ′ (2 S ). In order to minimizethe hadronic uncertainties it is necessary to eliminate the backgrounds coming from theresonance regions. The resonant decays B s → J/ψφ and B s → ψ ′ (2 S ) φ with ψ/ψ ′ (2 S ) → µ + µ − are rejected by applying the vetos on the dimuon mass regions around the charmoniumresonances, i.e., (2946 < m ( µ + µ − ) < / c and (3592 < m ( µ + µ − ) < / c [31]. Using these veto windows we obtain the branching ratio for the B s → φµ + µ − decay20 ABLE I: Allowed range of the CP violating observables (in units of 10 − ), in the leptoquarkmodel. Observables Allowed range Observables Allowed range(in units of 10 − ) (in units of 10 − ) h A s i (0 . → . h A i − (60 → h A c i (8 → h A s i (7 . → . h A s i (0 . → . h A i (3 . → . h A c i − (7 . → . h A i − (46 → h A i − (1 . → . h A i − (0 . → . h A i (2 . → . h A CP i (8 . → . mode asBR( B s → φµ + µ − ) = 13 . × − (in SM) , = (5 . − . × − (in LQ Model − I (X = 3 , , / , = (8 . − . × − (in LQ Model − II (X = 3 , , / . (65)Thus, one can see that the observed branching ratio (45) can be accommodated in the scalarleptoquark model.Our next objective is to study the effect of leptoquark in the CP asymmetry parameters A ( a ) i . The q variation of these observables in the low q regime is shown in Figure-7. Herewe have varied new weak phase between (60-90) degree and fixed the r value at 0.35. Thetime integrated value in the low q region is shown in Table-1. Some of these asymmetriesare measured by the LHCb collaboration, which are almost in agreement with the standardmodel predictions but with large error bars. Future measurement with large data samplescould possibly minimize these errors and help to infer the presence of new physics, if thereis any from these observables. VII. CONCLUSION
In this paper we have studied the effect of the scalar leptoquarks in the rare decays of B s meson. The large production of B s mesons at the LHC experiment opens up the possibility21 A s A c A s - - - - - - A c - - - - - A A - - A - - - A s - - - - - - A A - - - - - - A A C P FIG. 7: Variation of the CP violating observables with di-muon invariant mass q . to study the rare decays of B s meson with high statistical precision. We have consideredthe simple renormalizable leptoquark models which do not allow proton decay at the treelevel. Using the recent results on BR( B s → µ + µ − ) and the value of BR( ¯ B d → X s µ + µ − ),the leptoquark parameter space has been constrained. Using such parameters we obtainedthe bounds on the product of leptoquark couplings. We then estimated the branching ratioand the forward backward asymmetry for the rare decay process B s → φµ + µ − . The SMprediction for BR( B s → φµ + µ − ) is found to be higher than the corresponding experimentalobserved value. We found that the branching ratio has been deviated significantly fromthe corresponding SM value and the observed branching ratio can be accommodated in this22odel. However, the zero-position of the forward-backward rate asymmetry does not havesignificant deviation in the leptoquark model but there is a slight shifting towards right.We have also shown the variation of different CP asymmetry parameters A ( a ) i in the low- q region. The time-integrated values of some of the asymmetry parameters are found tobe significantly large, the observation of which in the LHCb experiment would provide thepossible existence of leptoquarks. Acknowledgments
We would like to thank Council of Scientific and Industrial Research, Government ofIndia for financial support through grant No. 03(1190)/11/EMR-II. [1] R. Aaij et al. , [LHCb Collaboration], Phys. Rev. Lett. , 101805 (2013), arXiv:1307.5024[hep-ex].[2] S. Chatrchyan et al. , [CMS Collaboration], arXiv:1307.5025 [hep-ex].[3] M. Iwasaki et al. , [Belle Collaboration], Phys. Rev. D , 092005 (2005),arXiv:hep-ex/0503044.[4] B. Aubert et al. , [BaBar Collaboration], Phys. Rev. Lett. , 081802 (2004),arXiv:hep-ex/0404006.[5] T. Huber, T. Hurth and E. Lunghi, Nucl. Phys. B , 40 (2008), arXiv:0712.3009 [hep-ph].[6] H. Georgi and S. L. Glashow, Phys. Rev. Lett. , 438 (1974); J. C. Pati and A. Salam, Phys.Rev. D , 275 (1974).[7] B. Schrempp and F. Shrempp, Phys. Lett. B 153 , 101 (1985); B. Gripaios, JHEP , 045(2010) (arXiv:0910.1789 [hep-ph]).[8] D. B. Kaplan, Nucl. Phys.
B 365 , 259 (1991).[9] W. Buchmuller, R. Ruckl and D. Wyler, Phys. Lett.
B 191 , 442 (1987) [Erratum- ibid.
B 448 ,320 (1999)]; A. J. Davies and X. G. He, Phys. Rev.
D 43 , 225 (1991).[10] S. Davidson, D. C. Bailey and B. A. Campbell, Z. Phys.
C 61 , 613 (1994), hep-ph/9309310; I.Dorsner, S. Fajfer, J. F. Kamenik, N. Kosnik, Phys. Lett. B , 67 (2009); arXiv:0906.5585[hep-ph]; S. Fajfer, N. Kosnik, Phys. Rev. D , 017502 (2009), arXiv:0810.4858 [hep-ph];R. Benbrik, M. Chabab, G. Faisel, arXiv:1009.3886 [hep-ph]; A. V. Povarov, A. D. Smirnov,arXiv:1010.5707 [hep-ph]; J. P Saha, B. Misra and A. Kundu, Phys. Rev. D 81 , 095011 (2010), rXiv:1003.1384 [hep-ph]; I. Dorsner, J. Drobnak, S. Fajfer, J. F. Kamenik, N. Kosnik, JHEP , 002 (2011), arXiv: 1107.5393 [hep-ph].[11] J. M. Arnold, B. Fornal and M. B. Wise, Phys. Rev. D , 035009 (2013), arXiv:1304.6119[hep-ph].[12] N. Kosnik, Phys. Rev. D , 055004 (2012), arXiv:1206.2970 [hep-ph].[13] A. J. Buras and M. M¨unz, Phys. Rev. D 52 , 186 (1995); M. Misiak, Nucl. Phys. B , 23(1993); ibid. , 461 (E) (1995).[14] W. Altmannshofer, P. Ball, A. Bharucha, A. J. Buras, D. M. Straub, M. Wick, JHEP ,001 (2009), arXiv:0811.1214 [hep-ph].[15] C. S. Lim, T. Morozumi and A. I. Sanda, Phys. Lett. B. , 343 (1989); N. G. Deshpande,J. Trampetic and K. Panose, Phys. Rev. D , 1461 (1989); P. J. O’Donnell and H. K. K.Tung, Phys. Rev. D , R2067 (1991); P. J. O’Donnell, M. Sutherland and H. K. K. Tung,Phys. Rev. D , 4091 (1992) F. Kr¨uger and L. M. Sehgal, Phys. Lett. B , 199 (1996).[16] R. Aaij et al. , [LHCb Collaboration], arXiv:1308.1707 [hep-ex].[17] K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, A. Pellegrino, N. Tuning,Phys. Rev. Lett. , 041801 (2012), arXiv:1204.1737 [hep-ph].[18] A. J. Buras, R. Fleischer, J. Girrbach, R. Knegjens, JHEP, , 77 (2013); arXiv:1303.3820[hep-ph].[19] A. J. Buras, J. Girrbach, D. Guadagnali and G. Isidori, Eur. Phys. J. C 72 , 2172 (2012),arXiv:1208.0934 [hep-ph].[20] G. Yeghiyan, arXiv:1305.0852 [hep-ph].[21] H. K. Dreiner, K. Nickel, F. Staub, arXiv:1309.1735 [hep-ph].[22] Damir Becirevic, Nejc Kosnik, Federico Mescia, Elia Schneider, arXiv:1205.5811 [hep-ph].[23] J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).[24] A. K. Alok, A. Datta, A. Dighe, M. Duraiswamy, D. Ghosh and D. London, JHEP , 121(2011).[25] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D , 034002 (2002).[26] J. Charles et al. , CKMFitter Group, Eur. Phys. Journ. C 41 , 1 (2005).[27] A. Ali, G. Hiller, L. T. Handoko and T. Morozumi, Phys. Rev. D , 4105 (1997).[28] A. K. Alok, A. Datta, A. Dighe, M. Duraisamy, D. Ghosh, D. London, arXiv:1008.2367 [hep-ph]; arXiv:1103.5344 [hep-ph].
29] A. Khodjamirian, T. Mannel, A. Pivovarov and Y. M. Wang, JHEP , 089 (2010); M.Beylich, G. Buchalla and T. Feldmann, Eur. Phys. J.
C 71 , 1635 (2011); D. Becirevic and A.Tayduganov, Nucl. Phys.
B 868 , 368 (2013); J. Matias, Phys. Rev.
D 86 , 094024 (2012).[30] W. Altmannshofer and D. M. Straub, arXiv:1308.1501 [hep-ph].[31] R. Aaij et al. , [LHCb Collboration], arXiv:1305.2168 [hep-ex].[32] C. Bobeth, G. Hiller and G. Piranishvili, JHEP , 106 (2008), arXiv:0805.2525 [hep-ph];C. Bobeth, G. Hiller and D. van Dyk, arXiv:1105.2659 [hep-ph]; G. Erkol and G. Turan,Eur. Phys. J. C 25 , 575 (2002), arXiv:hep-ph/0203038; R. Mohanta and A. K. Giri, Phys.Rev.
D 75 , 035008 (2007), arXiv:hep-ph/0611068; U. Yilmaz, Eur. Phys. J.
C 58 , 555(2008), arXiv:0806.0269 [hep-ph]; Q. Chang and Y. -H. Gao, Nucl. Phys.
B 845 , 179 (2011),arXiv:1101.1272 [hep-ph]; Y. -G. Xu, L. -H. Zhou, B. Z. Li, R. M. Wang, arXiv:1305.5010[hep-ph]; I. Ahmed, M. Jamil Aslam and M. Ali Paracha, arXiv:1307.5359 [hep-ph].[33] P. Ball and R. Zwicky, Phys. Rev. D , 014029 (2005).[34] A. K. Alok, A. Dighe, D. Ghosh, D. London, J. Matias, M. Nagashima and A. Szynkman,JHEP , 053 (2010), arXiv:0912.1382 [hep-ph].[35] F. Kr¨uger, L. M. Sehgal, N. Sinha and R. Sinha, Phys. Rev. D 61 , 114028 (2000); [Erratum ibid
D 63019901 (2000)].