Eur. Phys. J. C (2017) 77:344 DOI 10.1140/epjc/s10052-017-4888-4

Regular Article - Theoretical Physics

New physics effects in charm meson decays involving → + −( ∓ ± ) c ul l li l j transitions

Suchismita Sahoo, Rukmani Mohantaa School of Physics, University of Hyderabad, Hyderabad 500046, India

Received: 18 January 2017 / Accepted: 5 May 2017 © The Author(s) 2017. This article is an open access publication

Abstract We study the effect of the scalar leptoquark and these branching fractions, the so-called R(D(∗)), defined as  (∗) ¯ (∗) ¯ (∗) Z boson on the rare decays of the D mesons involving R(D ) = BR(B → D τντ )/BR(B → D lνl ), where → + −( ∓ ± ) = ,μ . σ flavour changing transitions c ul l li l j . We con- l e , exceed the SM prediction by 3 5 [6], thus opening strain the new physics parameter space using the branching an excellent window to search for new physics (NP) in the ratio of the rare decay mode D0 → μ+μ− and the D0 − D¯ 0 up quark sector. oscillation data. We compute the branching ratios, forwardÐ Mixing between a neutral meson and its anti-meson with a backward asymmetry parameters and flat terms in D+(0) → specific flavour provides an useful tool to deal with problems π +(0)μ+μ− processes using the constrained parameters. The in flavour sector. For example, in the past the K 0 − K¯ 0 and branching ratios of the lepton flavour violating D meson B0 − B¯ 0 oscillations, involving mesons made of up- and decays, such as D0 → μe,τe and D+(0) → π +(0)μ−e+ down-type quarks, have provided information as regards the are also investigated. charm and top quark mass scale, much before the discovery of these particles in the collider. On the other hand, the D0 − 1 Introduction D¯ 0 system involves mesons with up-type quarks and in the SM the mixing rate is sufficiently small, so that the new The rare B and D meson decay processes driven by a flavour physics component might play an important role in this case. changing neutral current (FCNC) transitions constitute a sub- The mixing parameters required to describe the D0 − D¯ 0 ject of great interest in the area of electroweak interactions mixing are defined by x = M/ and y = /2, where and provide an excellent testing ground to look for new M () is the mass (width) difference between the mass physics beyond the standard model (SM). The FCNC decays eigenstates. are highly suppressed in the SM and occur only at one-loop In this paper, we focus on the analysis of rare charm meson + − ∓ ± level. Of particular interest among the FCNC decays are the decays induced by c → uμ μ and c → uμ e FCNC rare semileptonic B meson decays involving the transitions transitions. We calculate the branching ratios, forwardÐ b → sl+l−, where several anomalies at the level of few backward asymmetry parameters and the flat terms in +( ) +( ) + − sigma have been observed recently in the LHCb experiment D 0 → π 0 μ μ processes both in the scalar lep-  [1Ð4]. To complement these results, efforts should also be toquark (LQ) and the generic Z model. These processes made towards the search for new physics signal in the up suffer from resonance background through c → uM → + − () quark sector, mainly in the rare charm meson decays involv- ul l , where M denotes η (pseudoscalar), ρ,φ,ω(vector) ing c → ul+l− quark level transitions. Recently the LHCb mesons. However, to reduce the background coming from experiment has searched for the branching ratio of the lep- these resonances, we work in the low and high q2 regimes, ton flavour violating (LFV) D0 → μ∓e ± decays and put i.e., q2 ∈[0.0625, 0.275] GeV2 and q2 ∈[1.56, 4.00] GeV2, the limit as BR(D0 → μ∓e ± )<1.3 × 10−8 [5] at 90% which lie outside the mass square range of the resonant confidence level (CL). On the other hand, both the Belle and mesons. We also compute the branching ratios of lepton +( ) +( ) − + the BaBar experiments have reported significant deviations flavour violating D0 → μe,τe and D 0 → π 0 μ e ¯ (∗) on the measured branching fractions of B → D τντ pro- processes. These LFV processes have negligible contribu- cesses from the corresponding SM predictions. The ratio of tions from the SM, as they proceed through the box diagrams with tiny neutrino masses in the loop. However, they can  a e-mail: [email protected] occur at tree level in the LQ and Z models and are expected 123 344 Page 2 of 16 Eur. Phys. J. C (2017) 77:344 to have significantly large branching ratios. Leptoquarks transitions can be investigated well theoretically. The change are hypothetical colour triplet bosonic particles, which cou- in charm quantum number for rare FCNC charm meson ple to quarks and leptons simultaneously and contain both decays is either of two units or one unit, and hence, they baryon and lepton quantum numbers. It is interesting to study involve either C = 2orC = 1 transitions. The D0 − D¯ 0 flavour physics with leptoquarks as they allow quarkÐlepton mixing takes place via a C = 2 transition and the decay transitions at tree level, thus explaining several observed processes with C = 1 transitions are c → ul+l− and anomalies, e.g., the lepton non-universality (LNU) parameter c → uγ . | = ( → μ+μ−)/ ( → + −) RK q2∈[1,6] GeV2 BR B K BR B Ke e If we integrate out the heavy degrees of freedom associated in rare B decays. The existence of the scalar leptoquark is with the new interactions at a scale M, an effective Hamilto- predicted in the extended SM theories, such as grand unified nian in the form of a series of operators of increasing dimen- theory [7Ð11], the PatiÐSalam model, the extended techni- sions can be obtained. However, the operators of dimension colour model [12,13] and the composite model [14]. In this d = 6 have important contributions to charm meson decays work, we consider the model which conserves baryon and or mixing. In general, one can write the complete basis of lepton numbers and does not allow proton decay. Here we these effective operators in terms of chiral quark fields for would like to see how this model affects the leptonic and both D0 − D¯ 0 mixing and the process D0 → l+l− as [52,53] semieptonic decays of the D0 meson induced by c → ul+l−   |H| = (μ)  | | (μ), transitions. The phenomenology of scalar leptoquarks and f i G Ci f Qi i (1) their implications to the B and D sector has been extensively i=1 studied in the literature [15Ð51]. where G has inverse-mass squared dimensions, Ci are the The Z  boson is a colour singlet vector gauge boson and it Wilson coefficients.1 is electrically neutral in nature. By adding an additional U(1) The effective operators for D0 − D¯ 0 mixing at the heavy gauge symmetry, the new Z  gauge boson could be naturally mass scale M are given by [52,53] derived from the extension of electroweak symmetry of the = ( γ )( γ μ ), = ( σ )( σ μν ), SM, such as superstring theories, grand unified theories and Q1 uL μcL uL cL Q5 u R μνcL u R cL μ μ theories with large extra dimensions. The processes mediated Q2 = (uL γμcL )(u Rγ cR), Q6 = (u RγμcR)(u Rγ cR), via c → u FCNC transitions could be induced by the generic Q3 = (uL cR)(u RcL ), Q7 = (uL cR)(uL cR), Z  model at tree level. The theoretical framework of the heavy = ( )( ), = ( σ )( σ μν ), new Z  gauge boson has been studied in the literature [52, Q4 u RcL u RcL Q8 uL μνcR uL cR 53,56,57]. In this paper, we investigate the Z  contribution (2) to the rare D0 meson decay processes within the parameter + − q ( ) = L(R)q space constrained by D0 − D¯ 0 mixing and D0 → μ μ where L R are the chiral quark fields with L(R) = ( ∓γ )/ processes. 1 5 2 as the projection operators. The paper is organized as follows. In Sect. 2, we discuss In the standard model, the effective weak Hamiltonian for c → u μ = m the effective Hamiltonian describing C = 1 transitions the transitions at the scale c can be written as i.e., c → ul+l−, and C = 2 transition, which is respon- the sum of three contributions [49,54,55], 0 − ¯ 0 d s peng sible for D D mixing. The new physics contribution to Heff = λd H + λsH + λbH , (3) c → u transitions and the constraint on leptoquark couplings ∗ 0 ¯ 0 0 + − λ = from the D − D oscillation and the process D → μ μ where q VuqVcq is the product of CabibboÐKobayashiÐ are discussed in Sect. 3. We calculate the constraint on Z  Maskawa (CKM) matrix elements. The explicit form of peng + − couplings from D0 − D¯ 0 mixing and leptonic D0 → μ+μ− H , which basically is responsible for the c → ul l decays in Sect. 4. In Sect. 5, we compute the branching ratios, transition is given by ⎛ ⎞ forwardÐbackward asymmetry parameters and the flat terms   +(0) → π +(0)μ+μ− 4G F   of the process D in both these mod- Hpeng =− √ ⎝ C O + C O ⎠ , +(0) → π +(0)μ− + i i i i els. The lepton flavour violating D e and 2 i=3,...10,S,P i=7,...10,S,P D0 → μe,τe processes are discussed in Sects. 6 and 7. (4) Finally we summarize our findings in Sect. 8. where G F is the Fermi constant, the Ci ’s are the Wilson coef- ficients evaluated at the charm quark mass scale (μ = mc) at 2 Effective Hamiltonian for C = 1 and C = 2 Next-Next-to-Leading-Order (NNLO) [58]. We use the two eff ( ) transitions loop result of Ref. [59]fortheC7 mc Wilson coefficients,

1 Though the rare charm decays are affected by large non- We denote the Wilson coefficients for C = 2 operators as ci and perturbative effects, the short distance structure of FCNC those for C = 1 operators as Ci throughout this work 123 Eur. Phys. J. C (2017) 77:344 Page 3 of 16 344

∗ eff = ∗ ( . + . )( ± . ) L = ¯ L † +¯ R˜ † + . ., VcbVubC7 Vcs Vus 0 007 0 020i 1 0 2 and the cor- lRY Q u RY L h c (6) responding effective operators for the c → ul+l− transitions ˜ ∗ are given by [49] where  = iτ2 represents the conjugate state. The transi- () e μν tion of weak basis to mass basis divides the Yukawa couplings O = mc(u¯σμν R(L)c)F , 7 16π 2 to two part of couplings pertinent for the upper and lower dou- 2 blet components. The left-handed quark and lepton doublets () e ¯ μ O = (u¯γμ L(R)c)(γ ) , are represented by Q and L and u (l ) is the right-handed 9 16π 2 R R 2 quark (charged-lepton) singlet. We use the basis where CKM () e ¯ μ O = (u¯γμ L(R)c)(γ γ ), and PMNS rotations are assigned to down-type quarks and 10 π 2 5 16 → ν → ν 2 neutrinos, i.e., dL VCKMdL and L VPMNS L . () e L R O = (uR¯ (L)c)()¯ , Here Y and Y are the leptoquark couplings in the mass S π 2 16 basis of the up-type quarks and charged leptons. Now writ- 2 () e ¯ ing the leptoquark doublets in terms of its components as O = (uR¯ (L)c)(γ5), P 16π 2  = ((5/3),(2/3))T , where the superscripts denote the 2 e ¯ μν electric charge of the LQ components and expanding the OT = (u¯σμνc)(σ ) , 16π 2 terms in Eq. (6), one can obtain the interaction Lagrangian 2 e ¯ μν for different components of LQs given as [50] OT = (u¯σμνc)(σ γ5). (5) 5 16π 2   ∗ L(2/3) = ¯ [ L ] (2/3) The contributions from the primed operators as well as the lR Y VCKM dL   scalar, pseudoscalar and tensor operators are absent in the SM R (2/3) + u¯ R[Y VPMNS]νL  + h.c., and arise only in scenarios beyond the standard model. The     O ( / ) ( / )∗ ( / ) renormalization group running does not affect the 10 oper- L 5 3 = l¯ Y L u  5 3 − u¯ Y Rl  5 3 + h.c.. 2 2 R L R L ator, i.e., C10(mc) = C10(MW ) ∝ (m , /m ) and, hence, d s W (7) the Wilson coefficient C10 is negligible in the SM.

Thus, one can see from (7) that only (5/3) component medi- 3 New physics contribution due to the exchange of ates the interaction between up-type quarks and charged scalar leptoquarks lepton. Now applying the Fierz transformation, we obtain additional contributions to the SM Wilson coefficients for → μ+μ− The presence of leptoquarks can modify the SM effective c u transition as [49] Hamiltonian of c → u transitions, giving appreciable devi- π Y L Y L∗ ation from the SM values. These colour triplet bosons can LQ = LQ =− √ μc μu , C9 C10 2 be either scalars or vectors. There exist three scalar and four 2 2G F αemλb m vector relevant leptoquark states which potentially contribute π Y R∗Y R → + − LQ =− LQ =− √ cμ uμ , to the c ul l transitions and are invariant under the SM C9 C10 2 2 2G F αemλb m gauge group SU(3)C × SU(2)L × U(1)Y , where the hyper- π Y L∗Y R∗ charge Y is related to the electric charge and weak isospin LQ = LQ =− √ μu cμ , = − CS CP 2 (I ) through Y Q I3. Out of three possible scalar lepto- 2 2G F αemλb m quarks with the quantum numbers (3, 3, −1/3), (3, 1, −1/3) L R   π Yμ Y μ and (3, 2, 7/6) [49,50], only the leptoquark with multiplet C LQ =−C LQ =− √ c u , S P α λ 2 (3, 2, 7/6) conserves both baryon and lepton numbers and, 2 2G F em b m ∗ ∗ thus, avoids rapid proton decay at the electroweak scale. π Y R Y L + Y R Y L LQ =− √ uμ μc cμ μu , Similarly out of the vector multiplets (3, 3, 2/3), (3, 1, 5/3), CT 2 8 2G F αemλb m (3, 2, 1/6) and (3, 2, −5/6), only the first two leptoquark R L R ∗ L ∗ π −Y μYμ + Y μ Yμ states do not allow baryon number violation and can be CLQ =− √ u c c u , (8) T5 2 considered to study the observed anomalies in flavour sec- 8 2G F αemλb m tor. In this work we consider the baryon number conserving X = (3, 2, 7/6) scalar leptoquark which induces the inter- where αem is the fine structure constant. After having an action between the up-type quarks and charged leptons and, idea about the new Wilson coefficients, we now proceed to thus, contributes to the semileptonic decay amplitudes. constrain the combination of LQ couplings using the experi- The interaction Lagrangian of the X = (3, 2, 7/6) scalar mental data on D0 − D¯ 0 mixing and the process D0 → l+l−, leptoquark with the SM bilinears is given by [49,50] where l = μ, e. 123 344 Page 4 of 16 Eur. Phys. J. C (2017) 77:344

3.1 Constraint on leptoquark couplings from D0 − D¯ 0 and its coupling to electron or tau is negligible. The SM con- mixing tribution to the mass difference is very small and hence can be neglected. The corresponding experimental value is given In the standard model, D0 − D¯ 0 mixing proceeds through by [60] the box diagrams with an internal down-type quarks and + . − M = 0.0095 0 0041 ps 1. (14) W-boson exchange and the weak interaction boxes are sup- D −0.0044 pressed due to the GIM mechanism because of the smallness Now comparing the mass difference with the 1σ range of of the down quark mass in comparison to the weak scale. In experimental data, the bound on leptoquark coupling for a ¯ the LQ model, there will be a contribution to the D0 − D0 TeV scale LQ is given by mass difference from the box diagrams with the leptoquark   ∗ −3 M R R and leptons flowing in the loop. Since the SM contribution 7.73 × 10 ≤|Yμ Yμ | 1TeV c u to the mass difference is very small, we consider its value   − M to be saturated by new physics contributions. Furthermore, ≤ 1.26 × 10 2 , (15) the couplings to the left-handed quarks are considered to 1TeV be zero in order to avoid strict constraints in the down-type which can be translated with Eq. (8) to give the constraint on quark sector. Thus, considering only right-handed couplings, the new Wilson coefficients one can write the effective Hamiltonian due to the lepto-     M LQ LQ M quark X(3, 2, 7/6) and charged lepton/neutrinos in the loop 0.1 ≤ λbC =−λbC ≤ 0.17 . 1TeV 9 10 1TeV as [25,26] (16)  ( R R∗)2 2 Ylc Ylu 1 ml 1 Heff = I + 0 → μ+μ−( + −) 128π 2 M2 M2 M2 3.2 Constraint from D e e process l    × (¯γ μ )(¯γ ), c PRu c μ PRu (9) The rare leptonic D0 → μ+μ−(e+e−) processes, mediated → + − where the first term is due to the charged lepton and second by the FCNC transitions c ul l at the quark level, are term is due to neutrinos in the loop (ignoring the effect of highly suppressed in the SM due to a negligible C10 Wilson neutrino mixing). The loop function I (x) is given as coefficient and also suffer from CKM suppression. These processes occur only at one-loop level and are considered 1 − x2 + 2x log x I (x) = , (10) as some of the most powerful channels to constrain the new (1 − x)2 physics parameter space in the charm sector. Analogous to the which is very close to 1, i.e., I (0) = 1. Using the relation leptonic B meson decay processes, the only non-perturbative quantity involved is the decay constant of the D meson, which  ¯ 0|(¯γ μ )(¯γ )| 0=2 2 2 , can be reliably calculated using non-perturbative methods D c PRu c μ PRu D BD fD MD (11) 3 such as QCD sum rules, lattice gauge theory and so on. The + − we obtain the contribution due to leptoquark exchange as branching ratio of the process D0 → l l isgivenby[48,49]  ( R R∗)2   2 2 5 2 2 2 LQ 1 Y Y + − G α M f |λb| 4m M = D¯ 0|H |D0= l lc lu B f 2 M . BR D0 → l l = τ F em D D 1 − l 12 eff 2 2 D D D D π 3 2 2MD 192π M 64 MD ⎡     2 (12) 4m2  CLQ − C LQ  × ⎣ 1 − l  S S  M2  m  Since MD = 2|M12|, we get D c     ⎤ ∗ 2  LQ LQ  2  R R   C − C 2m    Y Y  +  P P + l CLQ − C LQ  ⎦ . LQ 2 2 l lc lu  2 10 10  M = 2|M |= M f B , (13) mc M D 12 D D D 2 2 D 3 64π M (17) where l denotes the charged-lepton flavours. In our analysis, + − ∗ ∗ the mass of the D0 meson is taken from [60], the value of The process D0 → μ μ has a dominant intermediate γ γ +2.3 the decay constant fD = 222.6 ± 16.7− . MeV [62] and state in the SM, which is electromagnetically converted to a 2 4 μ+μ− γ ∗γ ∗ BD(3GeV) = 0.757(27)(4) [63]. To obtain the bound on pair. After including the contribution of the the leptoquark coupling, we assume that the individual lepto- intermediate state, the predicted branching ratio of this pro- + − − quark contribution to the mass difference does not exceed the cessisBR(D0 → μ μ ) 2.7 × 10 5 × BR(D0 → γγ) − 1σ range of the experimental value. Since we are interested [64]. Using the upper bound BR(D0 → γγ) < 2.2 × 10 6 R R∗ in obtaining the bounds on the YμcYμu couplings, here we at 90% CL reported in [65], the estimated limit on the branch- SM assume that the leptoquark has dominant coupling to muons ing ratio is BR D0 → μ+μ−  10−10 [49]. The present 123 Eur. Phys. J. C (2017) 77:344 Page 5 of 16 344

0.1 0.3

0.2 0.05 0.1 S S ′ ′ ~ 0 ~ 0 + C + C S S ~ ~ C C -0.1 -0.05 -0.2

-0.1 -0.3 -0.1 -0.05 0 0.05 0.1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ~ ~′ ~ ~′ CS - C S CS - C S

˜ ± ˜  0 → μ+μ− 0 → + − Fig. 1 The allowed region for the CS CS Wilson coefficient obtained from D (left panel)andD e e processes (right panel) experimental limits on the branching ratios of the dileptonic Table 1 The allowed values of the Wilson coefficients obtained from + − + − decays of the D meson are [60] the upper bound of the process D0 → μ μ (e e ). The constraint on ˜ LQ ˜ LQ   the Ci coefficients can also be applicable to Ci Wilson coefficients 0 → μ+μ− < . × −9, BR D 6 2 10 0 + − 0 + −   Wilson coefficient D → μ μ D → e e 0 + − −8 BR D → e e < 7.9 × 10 . (18) ˜ LQ . C10 0 8 600 ˜ LQ . . Using the above experimental bounds, the constraint on the CS 0 053 0 186 ˜ LQ . . leptoquark coupling can be obtained by imposing the condi- CP 0 053 0 186 tion that the individual leptoquark contribution to the branch- ing ratio does not exceed the experimental limit. In this anal- ysis, we neglect the new physics contribution to the Wilson the experimental bound on the branching fraction of the pro- coefficient C , as the scalar and pseudoscalar Wilson coeffi-  10 cess D0 → μ+μ−(e+e−). The bounds on C˜ LQ Wilson cients will be dominating due to the additional multiplication i coefficients will be the same as those for C˜ LQ. factor MD/ml as noted from Eq. (17). Now, redefining the i Wilson coefficients by If we impose chirality on the scalar leptoquarks i.e., they couple to either left-handed or right-handed quarks, but not () ()  C˜ LQ = λ C LQ, (19) ( ) i b i to both, then the CS,P Wilson coefficients will vanish and () LQ LQ LQ ˜ − ˜ we get only the additional contribution of the C , Wilson we show in Fig. 1, the allowed region in the CS CS , 9 10 ˜ LQ + ˜ LQ 0 → μ+μ− coefficients to the SM. Now comparing the theoretical and CS CS plane, obtained from the D (left ()LQ + − C˜ panel) and D0 → e e processes (right panel). Here we experimental branching ratios, the allowed range of 10   Wilson coefficients are given in Table 1. have used the relations C LQ = C LQ and C LQ =−C LQ S P S P In order to evade the strict bounds in the down-type quark from Eq. (8). From the figure, we find the allowed range sector, we consider the leptoquark couplings to the left- for the above combinations of Wilson coefficients from the L 0 + − handed quarks (Y ) as zero. Therefore, the only contribu- process D → μ μ to be ˜  ˜      tion to the rare charm decays comes from the C =−C     9 10  ˜ LQ − ˜ LQ  . ,  ˜ LQ + ˜ LQ  . , Wilson coefficients, which are related to the right-handed CS CS 0 06 CS CS 0 06 (20) quark couplings. Now, to include the (pseudo)scalar and whereas the bounds obtained from the process D0 → e+e− (pseudo)tensor Wilson coefficients and to extract the respec- are rather weak, i.e., tive upper bound complying with the constraints from B and      ˜ LQ ˜ LQ  ˜ LQ ˜ LQ K physics, we consider a numerically tuned example as dis- C − C   0.2, C + C   0.2. (21) R S S S S cussedin[49]. We assume√ that the Y coupling is perturba- | R| < π It is obvious that the bounds obtained in Eqs. (20) and (21) tive, i.e., Y 4 . In particular, we consider a large R R = . could not give us proper information about the bounds on value for Ycμ coupling, e.g., Ycμ 3 5. We compute the  ˜ LQ ˜ LQ bound on Y R coupling by using the constraint on the C˜ LQ individual CS and CS coefficients. Therefore, we con- uμ 10 sider only one Wilson coefficient at a time to extract the Wilson coefficients from D0 → μ+μ− process, which is R < . × −3 upper bound on individual coefficients. In Table 1, we report found to be comparatively small, Yuμ 8 76 10 .Now ˜ LQ the constraint on the CS,P Wilson coefficients obtained from we instigate a non-zero coupling to the left-handed quark 123 344 Page 6 of 16 Eur. Phys. J. C (2017) 77:344  L R  2 Yuμ, which along with the large Ycμ coupling provides non- (Z ) fD MD BD 2 M = (c1(mc) + c6(mc)) zero values for the C , and C , coefficients. However, the D 2 S P T T5 2MZ  3 0 → μ+μ−      process D imposes a strong bound on the coeffi- 1 η 1 η cient C , which, together with the large Y R coupling, limits − + c2(mc) + + c3(mc) . S cμ 2 3 12 2 Y L < . × −3 the left-handed coupling to μu 1 14 10 . Thus, from (25) the above discussion we observe that At the charm mass scale, the Wilson coefficients in terms of ˜ LQ =−˜ LQ = . , ˜ LQ = ˜ LQ  C9 C10 0 8 CS CP the Z couplings are expressed as = ˜ LQ = ˜ LQ =− . . 2 4CT 4CT 0 053 (22) c1(mc) = r(mc, MZ  ) g  , 5  Z 1  4 1/2 −4 Our predicted bounds on the leptoquark coupling are in c3(mc) = r(mc, MZ  ) − r(mc, MZ  ) gZ 1gZ 2, 3 agreement with the constraints obtained in Refs. [48,49] and / c (m ) = 2 r(m , M  )1 2g  g  , also with the constraints obtained from B, K physics [66]. 2 c c Z Z 1 Z 2 ( ) = ( ,  ) 2 , c6 mc r mc MZ gZ 2 (26)

where r(mc, MZ  ) is the RG factor at the heavy mass scale   4 New physics contribution in Z model and r(mc, MZ  ) = 0.72 (0.71) for Z mass, MZ  = 1(2) TeV [52]. The new heavy Z  gauge boson can exist in many extended Now we consider two possible cases to constrain the cou- SM scenarios and can mediate the FCNC transitions among plings gZ 1 and gZ 2. One comes with only left-handed cou- the fermions in the up quark sector at tree level. The most pling present, i.e., (gZ 2 = 0) and the second with both general Hamiltonian for c → u transition in the Z  model is left-handed and right-handed couplings present with equal given as [53] strengths (gZ 1 = gZ 2 = gZ  ). Here, we make the simple assumption that the NP part dominates over the SM contri- q μ μ HFCNC = H =  γ +  γ . 0 − ¯ 0 Z  Z  gZ 1uL μcL Z gZ 2u R μcR Z (23) bution in D D mixing. Thus, for the first case, on substi- tution of gZ 2 = 0inEqs.(25) and (26), the mass difference Analogously, one can write the Hamiltonian for the leptonic becomes l sector H  as 2 2 Z  f M B r(m , M  ) g   (Z ) = D D D c Z Z 1 . MD 2 (27)  μ  μ 3  HL =  γ  +  γ  . MZ Z  gZ 1 L μ L Z gZ 2 R μ R Z (24) Now varying the mass difference MD within its 1σ allowed    Here gZ i and gZ i are the couplings of the Z boson with range [60], we obtain the quarks and leptons, respectively, where i = 1or2for  gZ 1 −7 −1 the Z μ vector boson coupled to left-handed or right-handed = (4.4 − 7.2) × 10 GeV , (28) MZ  currents.    = After having obtained the possible Z couplings with and for a representative Z mass MZ 1 TeV, the value of quarks and leptons, we proceed to constrain the new param- the coupling is found to be eter space using the results from charm sector, e.g., the −4 gZ 1 = (4.4 − 7.2) × 10 . (29) experimental data on D0 − D¯ 0 mixing and the branching + − ratios of D0 → l l processes. The constraint on the cou- Analogously for the second case, i.e., gZ 1 = gZ 2 = gZ  ,  the constraint pling of Z with the leptonic part is obtained from the upper   limit on the branching ratio of the lepton flavour violating − MZ  − − + − g  = (1.2 − 2.0) × 10 4 (30) τ(μ) → e e e processes. Z 1TeV is obtained. 4.1 Constraint from D0 − D¯ 0 mixing 4.2 Constraint from D0 → μ+μ− process In this subsection, we calculate the constraint on the Z  couplings from the mass difference of the charm meson The effective Hamiltonian for the process D0 → l+l− in the 0 ¯ 0 mass eigenstates, which characterizes the D − D mix- Z  model is given by [53] 0 − ¯ 0 ing phenomenon. The D D oscillation arises from the  Z  1     |C = 2| transition that generates off-diagonal terms in the H + − =   +   c→u  2 gZ 1gZ 1 Q1 gZ 1gZ 2 Q7 0 ¯ 0 M  mass matrix for D and D mesons. The mass difference of Z    0 − ¯ 0 μ = +   +   , D D mixing at the scale mc is given by [53] gZ 1gZ 2 Q2 gZ 2gZ 2 Q6 (31) 123 Eur. Phys. J. C (2017) 77:344 Page 7 of 16 344

  τ −(μ−) → − + − where the operators Q1,2 are 4.3 Constraints on gZ 1 from e e e process     ¯ μ Q = l γμl u¯ γ c ,  1  L L  L L  Considering only the left-handed coupling to the Z boson,  ¯ μ − − + −  Q2 = lL γμlL u¯ Rγ cR , (32) the branching ratio of the process μ → e e e in the Z model is given by [67,68]   and Q6,7 can be obtained from Q1,2 by the substitutions of 5 L L 2   τμmμ |g μ g | q → q and q → q . μ− → e−e+e− = e ee , L R R L BR 3 4 (38) 768π M  Comparing Eq. (31) with the SM effective Hamiltonian (4) Z yields the additional contributions to the Wilson coefficients where we have explicitly shown the indices in the couplings. ()Z  C9,10 as The experimental upper limit on the branching ratio of this mode is BR(μ− → e−e+e−)<10−12 [60]. For the analysis,     π g  (g  ± g  ) Z ( Z ) =−√ Z 1 Z 2 Z 1 , we use the mass and lifetime of the muon from [60] and C9 C10 L  α λ 2 consider the coupling g = g  as SM-like with its value 2G F em b MZ  ee Z 1    as presented in Eq. (35). Thus, using the experimental upper  Z π g  (g  ± g  )  Z (  ) =−√ Z 2 Z 2 Z 1 . L C9 C10 2 (33) limit, we get the bound on the g μe coupling as 2G α λ M    F em b Z  | L | < . × −5 MZ . g μe 5 69 10 (39) The branching ratio for the process D0 → μ+μ− in the Z  1TeV model is given as [53] Analogously using the branching ratio of τ − → e−e+e−  − − + − − process, BR(τ → e e e )<2.7 × 10 8 [60], the con- 2 2 2  0 + − fDm MD 4m L BR(D →   )|  = τ 1 − straint on the lepton flavour violating g τe coupling is found Z D π 4 2 32 MZ  MD to be      2   2 L MZ × (g  − g  ) g  − g  . |g τ | < 0.02 . (40) Z 1 Z 2 Z 1 Z 2 e 1TeV (34)

 = For simplicity, we consider here gZ 2 0. Now considering +( ) +( ) + −  5 D 0 → π 0 μ μ process the couplings of the Z boson to the final leptons as the same form as the SM-like diagonal couplings of Z boson to leptons as discussed in [53], i.e., In this section, we study the rare semileptonic decay process + → π +μ+μ−   D , which is mediated by the quark level tran- 2 θ sition c → uμ+μ− and constitutes a suitable tool to search  = g −1 + 2 θ ,  = sin W , gZ 1 sin W gZ 2 g cos θW 2 cos θW for new physics. The dominant resonance contributions come φ ρ ω η() (35) from the , and vector mesons and the effects of mesons are comparatively negligible. These decay modes where g is the gauge coupling of Z boson and θW is the have recently been studied in Refs. [48,49,61] in various Weinberg mixing angle. Now using the experimental upper new physics scenarios and it is found that the model with limit on the branching ratio BR(D0 → μ+μ−)<6.2 × scalar/vector leptoquarks and the minimal supersymmetric 10−9 [60], we obtain model with R-parity violation can give significant contri- g  butions. The matrix elements of various hadronic currents Z 1 < . × −8 −2. π 2 7 67 10 GeV (36) between the initial D meson and the final meson can be MZ  parametrized in terms of three form factors f0, f+ and fT For MZ  = 1 TeV, the constraint on the gZ 1 coupling is [49]: π( )|¯γ μ( ± γ ) | ( ) k u 1 5 c D pD  < . , gZ 1 0 077 (37)   2 − 2 2 μ MD Mπ μ = f+ q (pD + k) − q which is rather weak compared with the constraint obtained q2 0 ¯ 0 from D − D mixing.   2 − 2  2 MD Mπ μ It should be noted that the constraint on Z couplings from + f0 q q , (41) q2 the D0 → μ+μ− decay process and the D0 − D¯ 0 mixing data have been computed in [53]. Similarly, the constraint on μν π(k)|¯uσ (1 ± γ )c|D(p ) the couplings from D0 − D¯ 0 oscillation are obtained in Ref. 5 D ( 2) [52]. We found that our constraints are consistent with the fT q μ ν = i [(pD + k) q above predictions, if we use the updated values of various MD + Mπ ν μ μναβ input parameters. − (pD + k) q ± i (pD + k)α qβ ], (42) 123 344 Page 8 of 16 Eur. Phys. J. C (2017) 77:344

2 where p and k are the four momenta of the D and π mesons, d l D = a (q2) + b (q2) θ + c (q2) 2 θ, 2 l l cos l cos (46) respectively, and q = pD − k is the momentum transfer.√ The dq dcos θ 0 0 form factors for D → π are scaled as fi → fi / 2by isospin symmetry. For the q2-dependence of the form fac- where tors, we use the parameterization from Refs. [69,70], as √   λ 2 2 2 2 2 2 2 al (q ) = 0 λβl 2q β |S| +|P| + (|A| +|V | ) ( ) ( ) l 2 2 f+ 0 2 f+ 0 f+(q ) = , f0(q ) = , ( − )( − ) ( − ( / )) + ( 2 − 2 + 2) ( ∗) + 2 2 | |2 , 1 x 1 ax 1 x b 4ml MD Mπ q Re AP 8ml MD A ( ) ( 2) = fT 0 , √ ! fT q (43) ( 2) =  λβ 2β2 ( ∗) + 2 ( ∗) (1 − xT )(1 − aT xT ) bl q 4 0 l q l Re ST q Re PT5 √ " + ( 2 − 2 + 2) ( ∗) + λβ ( ∗) , = 2/ 2 = . ( ) = ml MD Mπ q Re AT5 l ml Re VS where x q m pole with m pole 1 90 8 GeV, a # . ( ) = . ( ) √ λβ 2 0 28 14 and b 1 27 17 are the shape parameters [49] 2 l 2 2 cl (q ) = 0 λβl − (|V | +|A| ) measured from D → πlν decay process and f+(0) = 2

0.67(3) [71]. The parameters in the fT form factor are 2 2 2 2 1/2 ∗ = 2/ 2 ( ) = . ( ) = . ( ) + 2q (β |T | +|T5| ) + 4ml βl λ Re(VT ) , (47) xT q MD∗ , fT 0 0 46 4 and aT 0 18 16 [70]. l Thus, one can write the transition amplitude for the process D+ → π +μ+μ− as [49,72] with   M( + → π + + −) λ = 4 + 4 + 4 − 2 2 + 2 2 + 2 2 , D l l MD Mπ q 2 MD Mπ MDq Mπ q λ α   G F b em μ ¯ μ ¯ = i √ Vp [lγμl]+Ap [lγμγ l] π D D 5 m2 2  β = 1 − 4 l , (48) ¯ ¯ l q2 + (S + T cos θ) [ll]+(P + T5 cos θ) [lγ5l] , (44) where θ is the angle between the D meson and the negatively and charged lepton in the rest frame of the dilepton. The functions 2 2 2 G α |λb| V , A, S and P are defined in terms of the Wilson coefficients  = F em . (49) 0 ( π)5 3 as 4 MD

( 2) 2mc fT q 2 NP NP Thus, the branching ratio is given by V = C + f+(q )(C + C + C ), + 7 9 9 9   MD Mπ  dBR = τ ( 2) + 1 ( 2) . = ( 2) + NP + NP , 2 D al q cl q (50) A f+ q C10 C10 C10 dq2 3 M2 − M2 = D π ( 2)( NP + NP), The forwardÐbackward asymmetry (AFB) is another useful S f0 q CS CS 2mc observable to look for new physics; it is defined by [49] 2 2  M − Mπ $ P = D f (q2)(C NP + CNP) 1 d2 0 P P A (q2) = θ 2mc FB d cos 2 0 dq d cos θ 2 − 2   $ % 2 MD Mπ 2 2 0 2 − m f+(q ) − f (q ) − f+(q ) d  d l 2 0 − d cos θ q 2 θ 2   −1 dq d cos dq NP NP 2 × C10 + C + C , b (q ) 10 10 = l . (51) / ( 2) + 1 ( 2) 2 f (q2)β λ1 2 al q 3 cl q T = T l C NP, + T MD Mπ Since the coefficient bl depends only on the scalar and pseu- 2 f (q2)β λ1/2 doscalar Wilson coefficients the forwardÐbackward asym- T = T l C NP. (45) 5 T5 MD + Mπ metry is zero in SM. However, the additional new physics contribution can give a non-zero contribution to the forwardÐ () () Here C NP, C NP and C NP are the new Wilson coef- backward asymmetry parameter. Another interesting observ- 9,10 S,P T,T5 ficients arising from either the scalar leptoquark model or able is the flat term, defined as [72] the generic Z  model. Using Eq. (44), the double differential $ 2 &$ 2   2 θ qmax qmax decay distribution with respect to q and , for the lepton l = 2 ( + ) 2 + 1 , FH dq al cl dq al cl (52) flavour l is given by [49,72] 2 2 3 qmin qmin 123 Eur. Phys. J. C (2017) 77:344 Page 9 of 16 344 where the uncertainties get reduced due to the cancelation between the numerator and denominator. For numerical evaluation, we take the particle masses and the lifetime of the D meson from [60]. For the CKM matrix elements, we use the Wolfenstein parametrization = . +0.023 λ = . ± . with values A 0 814−0.024, 0 22537 0 00061, ρ¯ = 0.117 ± 0.021 and η¯ = 0.353 ± 0.013 [60]. With these input parameters, we compute the resonant/non-resonant branching ratios of the process D+ → π +μ+μ− by integrat- ing the decay distribution with respect to q2. We parametrize the contributions from the resonances with the BreitÐWigner → res ρ,ω,φ → res shapes for C9 C9 ,for (vector) and CP CP for η() (pseudoscalar) mesons [48,49]: Cres 9 Fig. 2 The resonant contributions to the branching ratio of D+ → iδρ 1 1 1 = aρe − π +μ+μ− in the SM. The band arises due to the uncertainties in BreitÐ q2 − m2 + imρρ 3 q2 − m2 + imωω ρ ω Winger parameters and the variation of relative phases. The horizontal δ aφei φ black line represents the experimental upper bound from [60] + , 2 2 q − mφ + imφφ δ aηei η aη With all the input parameters from [60] along with the Cres = + . (53) P 2 − 2 +  2 2 SM Wilson coefficients [58,59], we present in Table 2 the q mη imη η q − mη + imη η predicted values of the branching ratios for the D+(0) → +( ) + − Here m M (M ) denotes the mass (total decay width) of π 0 μ μ processes by integrating the decay distribu- () the resonant state M, where M corresponds to η ,ρ,ω,φ tion in low and high q2 bins. Here we have used the q2 + + mesons. With the approximation of BR(D → π M(→ regimes as q2 ∈[0.0625, 0.275] GeV2 and q2 ≥ 1.56 GeV2 + − + + + − μ μ )) BR(D → π M)BR(M → μ μ ) and con- to reduce the background coming from the dominant reso- sidering the experimental upper bound from [60], the mag- nances. The theoretical uncertainties in the SM are associated nitudes of the BreitÐWigner parameters are given by [48] with the lifetime of the D meson, the CKM matrix elements +0.05 2 2 and the hadronic form factors. In Fig. 3, the variation of aφ = 0.24 GeV , aρ = 0.17 ± 0.02 GeV , −0.06 the SM branching ratios of the process D+ → π +μ+μ− aω = aρ/3, in the very low and high q2 regimes are shown in red +0.00004 2  2 dashed lines and the green bands represent the SM theoretical aη = 0.00060− . GeV , aη ∼ 0.0007 GeV . (54) 0 00005 uncertainties. The detailed procedure of SM resonant contributions to Now using the constraint on the leptoquark parameter the D+ → π +μ+μ− process can be found in [48,49,70]. space obtained in Sect. 3, we show in Fig. 3,theq2 vari- In Fig. 2, we show the q2 variation of the branching ratio of ation of the branching ratio of the process D+ → π +μ+μ− D+ → π +μ+μ− process including the resonant contribu- in low q2 (left panel) and high q2 (right panel) regimes both in tion in the SM. The band in the figure is due to the uncer- the scalar leptoquark and Z  models. Here the orange (blue) tainties associated with the aM parameters as given in (54) band represents the contributions from the scalar leptoquark and the random variation of relative phases within −π and (Z ) model. The 90% CL experimental upper bounds on the π. For simplicity we have assumed the same phase for all the branching ratios from [73], resonances. From the figure, one can observe that in the low + + + − − 2 ( → π μ μ )| < . × 8, and high q regions the long distance resonant contributions BR D low q2 2 0 10 are approximately one order of magnitude below the current ( + → π +μ+μ−)| < . × −8, BR D high q2 2 6 10 (55) experimental sensitivity, and hence these regions are suitable to look for new physics beyond the SM. Thus, both in the SM are shown in thick black lines. In Table 2, we present and in the leptoquark and Z  models, we study the process the integrated branching ratios of the process D+(0) → D+ → π +μ+μ− only at the very low and high q2 regimes. π +(0)μ+μ−es in both the low and the high q2 regions in the However, it should be emphasized that the uncorrelated vari- leptoquark and Z  models. We find that the predicted branch- ation of the unknown resonant phases affects the branching ing ratios in the leptoquark model have significant devia- ratio in the low q2 region significantly, which makes it quite tions from the corresponding SM values due to the effect difficult to infer the possible role of new physics. of the scalar leptoquark and are well below the experimen- 123 344 Page 10 of 16 Eur. Phys. J. C (2017) 77:344

Table 2 The predicted Decay process D+ → π +μ+μ− D0 → π 0μ+μ− branching ratios for D+(0) → π +(0)μ+μ− Low q2 processes in both the low q2 and −13 −13 the high q2 region in the scalar Non-resonant SM (3.02 ± 0.483) × 10 (1.19 ± 0.19) × 10  X (3, 2, 7/6) LQ and Z model. Resonant SM (1.36 − 2.4) × 10−10 (5.66 − 9.89) × 10−11 This also contains the resonant − − LQ model (2.6 − 8.68) × 10 10 (1.02 − 3.4) × 10 10 and non-resonant SM branching  −12 −13 ratios Z model (0.65 − 1.18) × 10 (2.55 − 4.62) × 10 Expt. limit (90% CL) 2 × 10−8 [73] ··· High q2 Non-resonant SM (5.14 ± 0.82) × 10−13 (2 ± 0.32) × 10−13 Resonant SM (1.25 − 3.29) × 10−10 (0.456 − 1.24) × 10−10 LQ model (1.32 − 3.36) × 10−9 (0.513 − 1.3) × 10−9 Z  model (1.4 − 2.78) × 10−12 (0.545 − 1.08) × 10−12 Expt. limit (90% CL) 2.6 × 10−8 [73] ···

Fig. 3 The variation of branching ratio of D+ → π +μ+μ− with non-resonant SM and the cyan bands are for the resonant SM. The green respect to low q2 (left panel) and high q2 (right panel). The orange bands stand for the theoretical uncertainties from the input parameters bands represent the contributions from the scalar leptoquark, the blue in the SM. The solid black line denotes the 90% CL experimental upper bands are due to the Z  contributions, the red dashed lines are for the limit [73]

 2 tal upper limits. However, the effect of the Z boson to the AFB=−0.083 → 0.042 in low q , +( ) +( ) + − branching ratios of D 0 → π 0 μ μ processes is very 2 AFB=−0.087 → 0.062 in high q , marginal. A =−0.095 → 0.06 in full q2. (56) In the leptoquark model, the variation of the forwardÐ FB + → π +μ+μ− backward asymmetry for the process D The Z  model provides additional contributions only to the in low q2 (left panel) and high q2 (right panel) is pre- C9,10 Wilson coefficients, and there are no new contributions sented in Fig. 4. The forwardÐbackward asymmetry depends () to scalar or tensor terms. Thus, the forwardÐbackward asym- ,   on the combinations of the CS and CT T5 Wilson coeffi- metry vanishes in the Z model. In both the LQ and the Z cients, thus they have zero value in the SM. However, the + + + −  model, the plot for flat term of D → π μ μ process additional contributions of CS,P Wilson coefficients due to with respect to low q2 (left panel) and high q2 (right panel) scalar leptoquark exchange give non-zero contribution to the is given in Fig. 5. The predicted values in low q2 range are forwardÐbackward asymmetry, though it is not so significant. The integrated forwardÐbackward asymmetry for the process FH |SM = 0.4 ± 0.064, FH |LQ = 0.336 → 0.46, D+ → π +μ+μ− is given as FH |Z = 0.4 → 0.41, (57) 123 Eur. Phys. J. C (2017) 77:344 Page 11 of 16 344

Fig. 4 The variation of the forwardÐbackward asymmetry of D+ → π +μ+μ− with respect to low q2 (left panel) and high q2 (right panel)inthe scalar leptoquark model

Fig. 5 The variation of the flat term of D+ → π +μ+μ− with respect to low q2 (left panel) and high q2 (right panel) in the scalar leptoquark and Z  models and in the region of high q2 6 D+(0) → π +(0)μ−e+

Since the individual lepton flavour number is conserved in FH |SM = 0.03 ± 0.005, FH |LQ = 0.34 → 0.5, the standard model, the observation of lepton flavour viola- F |  = 0.08 → 0.095. (58) H Z tion in the near future will provide unambiguous signal of new physics beyond the SM. The observation of neutrino  In addition to the leptoquark and Z models, the rare charm oscillation implies the violation of lepton flavour in neutral meson decays mediated by the c → u transitions have also sector and it is expected that there could be FCNC transi- been investigated in various new physics models such as the tions in the charged lepton sector as well, such as li → l j γ , → ¯ → ∓ ± → (∗) ∓ ± minimal supersymmetric standard model [61,64,74,75], the li l j lklk, B li l j and B K li l j etc. The LFV two Higgs doublet model [74], a warped extra dimensions decay modes proceed through box diagrams with tiny neu- model [75] and the up vector like quark singlet model [76,77]. trino masses in the loop, thus become very rare in the SM. + − In Refs. [48,49], the process D → πμ μ is studied in the However, these modes can occur at tree level in the leptoquark context of both scalar and vector leptoquark models. Our and Z  models, thus can provide observable signature in the results on predictions are found to be consistent with the high luminosity experiments. In this section, we would like to literature. 123 344 Page 12 of 16 Eur. Phys. J. C (2017) 77:344 study the lepton flavour violating semileptonic decay process √  + + − + λ λ → π μ 2 1 2 2 2 ∗ D e . Due to the absence of intermediate states, B (q ) = 2 f+(q ) (mμ + m )Re(F F ) l 0 q2 e S V these LFV processes have no long distance QCD contribu-  tions and dominant φ, ω resonance backgrounds. The general + ( − ) ( ∗) , expression for the transition amplitude of D+ → π +μ−e+ me mμ Re FP FA (63) process in a generalized new physics model, is given by   α λ  (λ λ )3/2 G F em b 2 2 2 2 1 2 2 2 M =− √ f+(q ) F (μ¯ e) + F (μγ¯ e) Cl (q ) =−20 f+(q ) |FA| +|FV | , π S P 5 4q6 2     + μ μγ¯ + μ μγ¯ γ , (64) FV pD μe FA pD μ 5e (59) where the functions Fi , i = V, A, S, P are defined as and

 F = K NP + K NP , λ = λ( 2 , 2 , 2), λ = λ( 2, 2 , 2). V 9 9 1 MD Mπ q 2 q mμ me (65) = NP + NP , FA K10 K10 M2 − M2 ( 2) For numerical estimation in the leptoquark model, we use the = 1( NP + NP) D π f0 q FS KS KS constrained leptoquark couplings obtained from the process 2 m f+(q2) + −   c D0 → μ μ and assume that the coupling between differ- + 1 NP + NP ( − ) K9 K9 me mμ ent generation of quarks and leptons follow the simple scaling 2 L(R) L(R) 1/2   laws, i.e. Y /Y = (mi /m j ) with j > i.Asdis- 2 − 2 2 ij ii M Mπ f0(q ) cussed in [47,78,79], such pattern of ansatz can explain the × D − 1 − 1 , q2 f+(q2) decay widths of radiative LFV decay μ → eγ .Nowusing such ansatz, the variation of branching ratio with respect 1 M2 − M2 f (q2) = ( NP + NP) D π 0 2 +(0) → π +(0)μ− + FP K P K P to q for D e process in the leptoquark 2 m f+(q2)   c model is shown in left panel of Fig. 6 and the correspond- 1 + NP + NP ( + ) ing integrated value is given in Table 3. In this mode, the K10 K10 mμ me  2 ( )NP   forwardÐbackward asymmetry depends on K , Wilson 2 2 2 9 10 M − Mπ f (q ) × D 0 − 1 − 1 . (60) coefficients which give non-zero contribution. The left panel q2 f+(q2) of Fig. 7 shows the q2 variation of the forwardÐbackward asymmetry and the corresponding integrated value is found ( NP) ( . → . ) Here the Wilson coefficients Ki involve the combination to be 0 039 0 047 . The variation of the flat term with L L/R∗ L L/R∗ respect to q2 is presented in the left panel of Fig. 8 and the of LQ couplings as YμcYeu instead of YμcYμu in Eq. (8). Now using Eq. (59), the differential decay distribution for the integrated value is (0.137 → 0.33).  D+ → π +μ−e+ process with respect to q2 and cos θ (θ is For the Z model, we consider the constraint on the cou-  ¯ the angle between the D and μ− in the μ − e rest frame) is pling of Z boson to the quarks, obtained from the D0 − D0 + − given as mixing and the process D0 → μ μ as given in Eqs. (29) and (37). For the lepton flavour violating coupling, the con- μ− → − + − 2 straint is taken from the process e e e , as discussed d 2 2 2 2 = Al (q ) + Bl (q ) cos θ + Cl (q ) cos θ, (61) in Sect. 4. Thus, using Eqs. (29), (37) and (39), the predicted dq2d cos θ +( ) +( ) − +  branching ratio of D 0 → π 0 μ e process in the Z model is given in Table 3 and the q2 variation of the pro- where cess D+ → π +μ−e+ is shown in Fig. 6 (right panel). The √  λ λ λ   forwardÐbackward asymmetry variation is shown in right 2 1 2 2 2 1 2 2 −3 Al (q ) = 20 f+(q ) |FV | +|FA| panel of Fig. 7 and the predicted value is −1.15 × 10 , q2 4   which is very small. In Fig. 8 (right panel), we show the plot 2 2 2 2 +|FS| q − (mμ + me) for q variation of the flat term and the integrated value is   0.158. +| |2 2 − ( − )2 FP q mμ me From Table 3, one may note that the predicted branch- 2 2 2 2 2 2 ing ratios are well below the present experimental limit +|FA| M (mμ + me) +|FV | M (mμ − me)  D  D + → π +μ− + ∗ for the D e process. Although there is no + 2 − 2 + 2 ( + ) ( ) − + MD Mπ q mμ me Re FP FA experimental bound on the process D0 → π 0μ e so  ∗ far, the experimental upper limit on the branching ratios + (m − mμ)Re(F F ) , (62) e S V of the D0 → π 0μ∓e ± process is well known, which is 123 Eur. Phys. J. C (2017) 77:344 Page 13 of 16 344

Fig. 6 The variation of branching ratio of the LFV process D+ → π +μ−e+ in the leptoquark model (left panel) and generic Z  model (right panel) with respect to q2.Thesolid black lines represent the 90% CL experimental upper bound [60]

− + − + Table 3 The predicted branching ratios for D+(0) → π +(0)μ−e+ 7 D0 → μ e (τ e ) LFV decay process lepton flavour violating processes in the scalar X (3, 2, 7/6) LQ and Z  model. The present upper limit on the branching ratio BR(D0 → ∓ ± − + + − − Recently LHCb put the upper limit on branching ratio of the π 0μ e ) = BR(D0 → π 0μ e + π 0μ e )<8.6 × 10 5 [60] D0 → μ∓e ± lepton flavour violating decay mode [5]: Decay process D+ → π +μ−e+ D0 → π 0μ−e+ ( 0 → μ∓ ± ) ( 0 → μ− + + μ+ −) LQ model (1.67 − 3.72) × 10−11 (0.56 − 1.4) × 10−11 BR D e BR D e e −8 Z  model (2.95 − 7.8) × 10−12 (1.15 − 3.04) × 10−12 < 1.3 × 10 . (66) Experimental limit <2.9 × 10−6 [60] ··· Neglecting the mass of electron, the branching ratio of the process D0 → μ−e+ is given by [48]

0 0 ∓ ± 0 0 − + − + given as BR(D → π μ e ) = BR(D → π μ e + BR(D0 → μ e ) π 0μ+ −)< . × −5 0 → e 8 6 10 . Our results for the process D 2 0 − +  2 α2 5 2 |λ |2 2 π μ e in both the leptoquark and the Z models are found G e M f b mμ = τ F D D 1 − D π 3 2 to be within the above experimental bound. 64 MD

Fig. 7 The variation of forwardÐbackward asymmetry of the LFV process D+ → π +μ−e+ in the leptoquark model (left panel) and generic Z  model (right panel) with respect to q2

123 344 Page 14 of 16 Eur. Phys. J. C (2017) 77:344

Fig. 8 The variation of flat term of the LFV process D+ → π +μ−e+ in the leptoquark model (left panel) and generic Z  model (right panel) with respect to q2

    NP NP  2  K − K mμ   So far there is no experimental evidence on the LFV decay ×  S S + K NP − K NP  0 → τ − + 0 →  m M2 9 9  process D e . Our results for the process D c D τ −(μ−) +    e is comparable with [48,51,53].  NP NP  2  K − K mμ   +  P P + K NP − K NP  . (67)  2 10 10  mc MD 8 Conclusion We use the scaling ansatz as discussed in the previous sec- 0 In this paper we have studied the rare decays of the D tion to compute the required leptoquark coupling for D →  meson in both scalar leptoquark and generic Z models. We μ−e+ process and the predicted branching ratio is found to have considered the simple renormalizable scalar leptoquark be model with the requirement that proton decay would not − + − BR(D0 → μ e ) = (3.18 − 4.8) × 10 11. (68) be induced in perturbation theory. The leptoquark param- eter space is constrained using the present upper limit on 0 + − 0 ¯ 0 Now using Eqs. (29), (37) and (39), the predicted branching branching ratio of D → μ μ process and the D − D   ratio of this LFV process in the Z  model is oscillation data. For the Z model, the constraints on the Z couplings are obtained from the mass difference of D0 − D¯ 0 − + − + − BR(D0 → μ e ) 6.1 × 10 17. (69) mixing, the process D0 → μ μ and the lepton flavour violating τ −(μ−) → e−e+e− processes. Using the con- The predicted branching ratio is although small, but can be strained parameter space, we estimated the branching ratios, searched at LHCb experiment. The exploration/observation forwardÐbackward asymmetry parameters and the flat terms of this decay mode would definitely shed some light in the in the D+(0) → π +(0)μ+μ− processes. The branching ratios leptoquark scenarios. in the LQ model are found to be ∼ O(10−10), which are Similarly using the new Wilson coefficient generated via larger than the corresponding SM predictions in the very low leptoquark exchange, the branching ratio for the process and very high q2 regimes. If these branching ratios will be − + D0 → τ e is found to be observed in near future they would provide indirect hints of leptoquark signal. Furthermore, we estimated the branching ( 0 → τ − +) = ( . − . ) × −14. BR D e 2 84 9 75 10 (70) ratios of lepton flavour violating D+(0) → π +(0)μ−e+ and D0 → μ(τ)−e+ processes, which are found to be rather However, there is no experimental observation of the lepton small. We also estimated the forwardÐbackward asymmetry flavour violating process D0 → τ −e+. The constraint on parameter and the flat term for the LFV decays. Z  coupling to tau and electron is obtained from the process τ − → − + − e e e .UsingEq.(40), the branching ratio for the Acknowledgements − +  We would like to thank the Science and Engineer- process D0 → τ e in Z model is given as ing Research Board (SERB), Government of India for financial support through Grant No. SB/S2/HEP-017/2013. − + − BR(D0 → τ e ) = (0.73 − 1.94) × 10 15. (71) 123 Eur. Phys. J. C (2017) 77:344 Page 15 of 16 344

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