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Scalar leptoquarks and the rare B meson decays Suchismita Sahoo, Rukmani Mohanta School of Physics, University of Hyderabad, Hyderabad - 500 046, India 5 1 Abstract 0 2 We study some rare decays of B meson involving the quark level transition b ql+l− (q = d,s) → r p in the scalar leptoquark model. We constrain the leptoquark parameter space using the recently A measured branching ratios of B µ+µ− processes. Using such parameters, we obtain the 8 s,d → 2 branching ratios, direct CP violation parameters and isospin asymmetries in B Kµ+µ− and → ] h B πµ+µ− processes. We also obtain the branching ratios for some lepton flavour violating p → p- decays B l+l−. We find that the various anomalies associated with the isospin asymmetries of → i j e h B Kµ+µ− process can be explained in the scalar leptoquark model. [ → 3 v PACS numbers: 13.20.He, 14.80.Sv 3 9 1 5 0 . 1 0 5 1 : v i X r a 1 I. INTRODUCTION TheraredecaysofB mesonsinvolving flavorchangingneutralcurrent (FCNC)transitions b s/d, provide an excellent testing ground to look for new physics. In the standard model → (SM), these transitions occur at one-loop level and hence, they are very sensitive to any new physics contributions. Although, so far we have not seen any clear indication of new physics in the b sector, but there appears to be some kind of tension with the SM predictions in some b s penguin induced transitions. It should be noted that the recent measurement → by LHCb collaboration [1] shows several significant deviations on angular observables in the rare decay B K∗0µ+µ− from their corresponding SM expectations. In particular, the → most significant discrepancy of 3.7σ, arises in the variable P′ [2] (the analogue of S in [3]) 5 5 provides high sensitivity to new physics (NP) effects in b sγ, sl+l− transitions. Further → results from LHCb experiment in combination with the critical assessment of the theoretical uncertainties will be necessary to clarify whether the observed deviations are a real sign of NP or simply the statistical fluctuations [4, 5]. Anotherindicationofnewphysicsisrelatedtotherecentmeasurement ofisospinasymme- try in B Kµ+µ− process by LHCb experiment [6], which gives a negative deviation from → zero at the level of 4σ taking into account the entire q2-spectrum. The isospin-asymmetry in B Kll is expected to be vanishingly small in the SM and hence, the measured asymmetry → provides another smoking-gun signal for new physics. More recently another discrepancy occurs in the measurement of the ratio of branching fractions of B Kl+l− decays into dimuons over dielectrons by the LHCb collaboration → [7], BR(B Kµ+µ−) R = → , (1) K BR(B Ke+e−) → and the obtained value in the dilepton invariant mass squared bin (1 . q2 < 6) GeV2 is RLHCb = 0.745+0.090 0.036 . (2) K −0.074 ± Combining the statistical and systematic uncertainties in quadrature, this observation corresponds to a 2.6σ deviation from its SM prediction R = 1.0003 0.0001 [8], where cor- K ± rections of order α and (1/m ) are included. In contrast to the anomaly in the rare decay s b B K∗µ+µ−, which is affected by unknown power corrections, the ratio R is theoretically K → clean and this might be a sign of lepton flavour non-universal physics. 2 Although it is conceivable that these anomalies mostly associated with b sl+l− tran- → sitions are due to statistical fluctuations or under-estimated theory uncertainties, but the possible interplay of new physics could not be ruled out. These LHCb results have attracted many theoretical attentions in recent times [2, 4, 5, 9] both in the context of some new physics model or in model independent way. In this paper we would like to investigate some of the rare decay modes of B meson involving the FCNC transitions b (s,d)l+l−, → e.g., B Kl+l−, B πl+l− and B l+l− using the scalar leptoquark (LQ) model. In → → → i j particular, we would like to see whether the leptoquark model can accommodate some of the anomalies discussed above, in particular the ones associated with B Kll processes. → It is well-known that leptoquarks are color-triplet bosonic particles that can couple to a quark and a lepton at the same time and can occur in various extensions of the standard model [10]. They can have spin-1 (vector leptoquarks), which exist in grand unified theories based on SU(5), SO(10) etc., or spin-0 (scalar leptoquarks). Scalar leptoquarks can exist at TeV scale in extended technicolor models [11] as well as in quark and lepton composite models [12]. The phenomenology of scalar leptoquarks have been studied extensively in the literature [13–15]. It is generally assumed that the vector leptoquarks tend to couple directly to neutrinos, and hence it is expected that their couplings are tightly constrained from the neutrino mass and mixing data. Therefore, in this paper we consider the model where leptoquarks can couple only to a pair of quarks and leptons and hence may be inert with respect to proton decay. Hence, the bounds from proton decay may not be applicable for such cases and leptoquarks may produce signatures in other low-energy phenomena [15]. Thepaperisorganizedasfollows. InsectionIIwebrieflydiscusstheeffectiveHamiltonian describing the process b sl+l− and the new contributions arising due to the exchange of → scalar leptoquarks. The constraints on the leptoquark parameter space are obtained using the recently measured branching ratios of the decay modes B µ+µ− and B X e+e− s,d s → → processinsectionsIII.Thebranchingratiosandvariousasymmetries oftheraredecaymodes B Kl+l− and B πl+l− are discussed in sections IV and V respectively. In Section → → VI we present the lepton flavour violating decays B l+l− and Section VII contains the → i j conclusion. 3 II. EFFECTIVE HAMILTONIAN FOR b sl+l− PROCESS → In the standard model effective Hamiltonian describing the quark level transition b sll → is given as [16] 6 4G e = FV V∗ C (µ)O +C s¯σ (m P +m P )b Fµν Heff − √2 tb ts" i i 716π2 µν s L b R Xi=1 (cid:16) (cid:17) α α +Ceff (s¯γµP b)¯lγ l+C (s¯γµP b)¯lγ γ l , (3) 9 4π L µ 104π L µ 5 # where GF is the Fermi constant and Vqq′ are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, α is the fine-structure constant, P = (1 γ )/2 and C ’s are the Wilson L,R 5 i ∓ coefficients. The values of the Wilson coefficients are calculated at the next-to-next-leading order (NLL) by matching the full theory to the effective theory at the electroweak scale and subsequently solving the renormalization group equation (RGE) to run them down to the b-quark mass scale [3], and the values used in this analysis are listed in Table-1. C C C C C C Ceff Ceff C C 1 2 3 4 5 6 7 8 9 10 0.3001 1.008 0.0047 0.0827 0.0003 0.0009 0.2969 0.1642 4.2607 4.2453 − − − − − − TABLE I: The SM Wilson coefficients evaluated at the scale µ = 4.6 GeV [17]. A. New Physics Contributions due to Scalar Leptoquark exchange The effective Hamiltonian (3) will be modified in the leptoquark model due to the additional contributions arising from the exchange of scalar leptoquarks. Here, we will consider the minimal renormalizable scalar leptoquark models [15], containing one single additional representation of SU(3) SU(2) U(1) and which do not allow proton decay at the tree × × level. It has also been shown that this requirement can only be satisfied by two models and in these models, the leptoquarks can have the representation as X = (3,2,7/6) and X = (3,2,1/6) under the gauge group SU(3) SU(2) U(1). Our objective here is to × × consider these scalar leptoquarks which potentially contribute to the b (s,d)µ+µ− transi- → tions and constrain the underlying couplings from experimental data on B µ+µ−. The s,d → 4 details of these new contributions are explicitly discussed in Ref. [18], and here we simply outline the main points. The interaction Lagrangian for the coupling of scalar leptoquark X = (3,2,7/6) to the fermion bilinears is given as = λij u¯i (V ej Y νj) λij e¯i V†uj +Y†dj +h.c. . (4) L − u αR α L − α L − e R L αL α αL (cid:16) (cid:17) Using the Fierz transformation, one can obtain from Eq. (4), the contribution to the interaction Hamiltonian for the b sµ+µ− process as → λ32λ22∗ λ32λ22∗ = µ µ [s¯γµ(1 γ )b][µ¯γ (1+γ )µ] µ µ O +O . (5) HLQ 8M2 − 5 µ 5 ≡ 4M2 9 10 Y Y (cid:16) (cid:17) One canthus write the leptoquark effective Hamiltonian (5) analogousto its SM counterpart (3) as G α = F V V∗(CNPO +CNPO ) , (6) HLQ −√2π tb ts 9 9 10 10 with the new Wilson coefficients π λ32λ22∗ CNP = CNP = µ µ . (7) 9 10 −2√2G αV V∗ M2 F tb ts Y Similarly the interaction Lagrangian for the coupling of X = (3,2,1/6) leptoquark to the fermion bilinear can be expressed as = λij d¯i (V ej Y νj)+h.c. , (8) L − d αR α L − α L and after performing the Fierz transformation, the interaction Hamiltonian becomes λ22λ32∗ λ22λ32∗ = s b [s¯γµ(1+γ )b][µ¯γ (1 γ )µ] = s b O′NP O′NP , (9) HLQ 8M2 5 µ − 5 4M2 9 − 10 V V (cid:16) (cid:17) where O′ and O′ are the four-fermion current-current operators obtained from O by 9 10 9,10 making the replacement P P . Thus, due to the exchange of the leptoquark X = L R ↔ (3,2,1/6), one can obtain the new Wilson coefficients C′NP and C′NP associated with the 9 10 operators O′ and O′ as 9 10 π λ22λ32∗ C′NP = C′NP = s b . (10) 9 − 10 2√2 G αV V∗ M2 F tb ts V The analogous new physics contributions for b dµ+µ− transitions can be obtained from → b sµ+µ− process by replacing the leptoquark couplings λ32λ22∗ by λ32λ12∗ and the CKM → elements V V∗ by V V∗ in Eqs. (6-10). After having the idea of new physics contributions tb ts tb td to the process b (s,d)µ+µ−, we now proceed to constrain the new physics parameter → space using the recent measurement of B µ+µ−. s,d → 5 III. B µ+µ− DECAY PROCESS s,d → The rare leptonic decay processes B µ+µ−, mediated by the FCNC transition b s,d → → s,d are strongly suppressed in the standard model as they occur at one-loop level as well as suffer from helicity suppression. These decay processes are very clean and the only nonperturbative quantity involved is the B meson decay constant, which can be reliably calculated using the non-perturbative methods such as QCD sum rules, lattice gauge theory etc. Therefore, they are considered as one of the most powerful tools to provide important constraints on models of new physics. These processes have been very well studied in the literature and in recent times also they have attracted a lot of attention [19–25]. Therefore, here we will point out the main points. The constraint on the leptoquark couplings from B µ+µ− are recently extracted by one of us in Ref. [18]. s → The most general effective Hamiltonian describing these processes is given as G α = F V V∗ CeffO +C′ O′ , (11) Heff √2π tb tq" 10 10 10 10# where q = d or s, Ceff = CSM +CNP and C′ = C′NP. The corresponding branching ratio 10 10 10 10 10 is given as G2 2 4m2 BR(B µ+µ−) = F τ α2f2 M m2 V V∗ 2 Ceff C′ 1 µ. (12) q → 16π3 Bq Bq Bq µ| tb tq| 10 − 10 − M2 (cid:12) (cid:12) s Bq (cid:12) (cid:12) (cid:12) (cid:12) However, as discussed in Ref . [19], the average time-integrated branching ratios BR(B q → µ+µ−) depend on the details of B B¯ mixing, which in the SM, related to the decay q q − widths Γ(B µ+µ−) by a very simple relation as BR(B µ+µ−) = Γ(B µ+µ−)/Γq , q → q → q → H where Γq is the total width of the heavier mass eigen state. H Including the corrections of (α ) and (α2), the updated branching ratios in the O em O s standard model are calculated in [25] as BR(B µ+µ−) = (3.65 0.23) 10−9, s SM → | ± × BR(B µ+µ−) = (1.06 0.09) 10−10 . (13) d SM → | ± × These processes are recently measured by the CMS [26] and LHCb [27] experiments and the current experimental world average [28] is BR(B µ+µ−) = (2.9 0.7) 10−9, BR(B µ+µ−) = 3.6+1.6 10−10 , (14) s → ± × d → −1.4 × (cid:0) (cid:1) 6 which are more or less consistent with the latest SM prediction (13), but certainly they do not rule out the possibility of new physics as the experimental errors are still quite large. We will now consider the additional contributions arising due to the effect of scalar leptoquarks in this mode. Including the contributions arising from scalar leptoquark exchange, one can write the transition amplitude for this process from Eq. (11) as G (B0 µ+µ−) = µ+µ− B0 = F V V∗αf M m CSMP, (15) M q → h |Heff| qi −√2 π tb tq Bq Bq µ 10 where C C′ CNP C′NP P 10 − 10 = 1+ 10 − 10 = 1+reiφNP , (16) ≡ CSM CSM 10 10 with reiφNP = (CNP C′NP)/CSM , (17) 10 − 10 10 r denotes the magnitude of the ratio of NP to SM contributions and φNP is the relative phase between them. As discussed in section II, the exchange of the leptoquarks X(3,2,7/6) and X(3,2,1/6) give new additional contributions to the Wilson coefficients C and C′ 10 10 respectively. Thus, the branching ratio in both the cases will be BR(B µ+µ−) = BR(B µ+µ−) (1+r2 2rcosφNP) . (18) q q SM → → | − Using the theoretical and experimental branching ratios from (13) and (14), the constraints on the combination of LQ couplings can be obtained by requiring that each individual leptoquark contribution to the branching ratio does not exceed the experimental result. The allowed region in r φNP plane which are compatible with the 1 σ range of the − − experimental data are shown in Fig.-1 for B µ+µ− (left panel) and for B µ+µ− d s → → (right panel). From the figure one can see that for B µ+µ− the allowed range of r and d → φNP as 0.5 r 1.3 , for 0 φNP π/2 or 3π/2 φNP 2π , (19) ≤ ≤ ≤ ≤ ≤ ≤ (cid:0) (cid:1) (cid:0) (cid:1) which can be translated to obtain the bounds for the leptoquark couplings using Eqs. (7), (10) and (17) as λ32λ12∗ 1.5 10−9 GeV−2 | | 3.9 10−9 GeV−2 , (20) × ≤ M2 ≤ × S 7 2 0.7 0.6 1.5 0.5 0.4 r 1 r 0.3 0.2 0.5 0.1 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 φNP φNP FIG. 1: The allowed region in the r φNP parameters space obtained from the BR(B µ+µ−) d − → (left panel) and BR(B µ+µ−) (right panel). s → with M as the leptoquark mass. For B µ+µ− process for 0 r 0.1 the entire range S s → ≤ ≤ for φNP is allowed, i.e., 0 r 0.1 , for 0 φNP 2π . (21) ≤ ≤ ≤ ≤ However, in our analysis we will use relatively mild constraint as 0 r 0.35 , with π/2 φNP 3π/2 . (22) ≤ ≤ ≤ ≤ This gives the constraint on leptoquark couplings as λ32λ22∗ 0 | | 5 10−9 GeV−2 for π/2 φNP 3π/2 . (23) ≤ M2 ≤ × ≤ ≤ S One can also obtain the constrain on the leptoquark couplings λ32λ22∗ /M2 from the inclu- | | S sive measurements BR(B¯0 X µ+µ−). However, as shown in Ref. [18], these constraints d → s are more relaxed than those obtained from B µ+µ−. So in our analysis we will use the s → constraints obtained from B µ+µ−. s → For other leptonic decay channels i.e., B e+e−, τ+τ− only the experimental upper s,d → limits exists [29]. Now using the theoretical predictions for these branching ratios from [25], we obtain the constrain on the upper limits of the various combinations of leptoquark couplings as presented in Table-II. However, the constraints obtained from such processes are rather weak. 8 Decay Process Couplings involved Upper bound of the couplings in GeV−2 B e±e∓ |λ31λ11∗| < 1.73(cid:0) 10−5 (cid:1) d → MS2 × B τ±τ∓ |λ33λ13∗| < 1.28 10−6 d → MS2 × B e±e∓ |λ31λ21∗| < 2.54 10−5 s → MS2 × B τ±τ∓ |λ33λ23∗| < 1.2 10−8 s → MS2 × TABLE II: Constraints obtained from the leptoquark couplings from various leptonic B l+l− s,d → decays. For the analysis of B Ke+e− process, we need to know the values of the leptoquark → couplings λ31λ21∗/M2, which can be extracted from the inclusive decay rates B X e+e−. S → s To obtain such constraints, we closely follow the procedure adopted in Ref. [18]. Using the SM predictions and the corresponding experimental measurements from [30] for both low-q2 (1 6) GeV2 and high-q2 (& 14.2 GeV2) as − BR(B → Xsee)|q2∈[1,6] GeV2 = (1.73±0.12)×10−6 (SM prediction) = (1.93 0.55) 10−6 (Expt.) ± × BR(B → Xsee)|q2>14.2 GeV2 = (0.2±0.06)×10−6 (SM prediction) = (0.56 0.19) 10−6 (Expt.) (24) ± × In Fig-2, we show the allowed region in CNP φNP parameter space due to exchange of 10 − the leptoquark X(3,2,7/6) in the left panel. The right panel depicts the allowed region in the C′NP C′NP space due to exchange of the leptoquark X(3,2,1/6), where green (red) 9 − 10 region corresponds to high-q2 (low-q2) limits in both the panels. Thus, it can be noticed that the bounds coming from the high-q2 measurements are rather weak. Considering the exchange of the X(3,2,7/6) leptoquark as an example, we obtain the bound on CNP as 10 2.0 CNP 3.0 for the entire range of φNP, which gives the bound on r as − ≤ 10 ≤ 0 r 0.7 . (25) ≤ ≤ After obtaining the bounds on various leptoquark couplings, we now proceed to study the rare decays B K/πl+l− and B l+l−. → → i j 9 5 5 NPC9 0 ’NPC10 0 -5 -5 0 50 100 150 200 250 300 350 -5 0 5 Φ C’NP 9 FIG. 2: The allowed region in CNP φNP parameter space (left panel) and C′NP C′NP (right 10 − 9 − 10 panel) obtained from BR(B X e+e−), where the green (red) region corresponds to high-q2 s → (low-q2) limits. IV. B Kl+l− PROCESS → We now consider the semileptonic decay process B Kl+l−, which is mediated by the → quark level transition b sl+l− and hence, it constitutes a quite suitable tool of looking → for new physics. The isospin asymmetries of B Kµ+µ− and the partial branching ratios → of the decays B0 K0µ+µ− and B+ K+µ+µ− are recently measured as functions of → → the dimuon mass squared (q2) by the LHCb collaboration [31]. In this paper we will study the process in the large recoil region i.e., 1 q2 6 GeV2, in order to be well below the ≤ ≤ radiative tail of the charmonium resonances, using the QCD factorization approach [32–34]. LHCb has measured the branching ratio in this region and the updated result is [31] BR(B+ → K+µ+µ−)|q2∈[1,6] GeV2 = (1.19±0.03±0.06)×10−7 . (26) This mode has also been analyzed by various authors [35–37] and the SM predictions is given as BR(B+ K+µ+µ−) SM = 1.75+0.60 10−7 . (27) → |q2∈[1,6] GeV2 −0.29 × (cid:0) (cid:1) Although, there is no significant discrepancy between these two results, the SM predictions is slightly higher than the experimental measurement. To calculate the branching ratio, one use the effective Hamiltonian presented in Eq. (3) and obtain the transition amplitude for this process. The matrix elements of the various 10