Implications of a vector-like lepton doublet and scalar Leptoquark on R( D (∗) )
aa r X i v : . [ h e p - ph ] J un HRI-RECAPP-2018-003
Implications of a vector-like lepton doublet and scalarLeptoquark on R ( D ( ∗ ) ) . Lobsang Dhargyal ∗ and Santosh Kumar Rai † Regional Centre for Accelerator-based Particle Physics,Harish-Chandra Research Institute, HBNI, Jhusi, Allahabad - 211019, India (Dated: June 5, 2018)
Abstract
We study the phenomenological constraints and consequences in the flavor sector, of introducinga new fourth generation Z odd vector-like lepton doublet along with a Standard Model (SM)singlet scalar and an SU (2) L singlet scalar leptoquark carrying electromagnetic charge of +2 / ,both odd under a Z . We show that with little fine tuning among the various Yukawa couplings inthe new physics (NP) Lagrangian along with the CKM parameters, the model is able to push thetheoretical value of R ( D ∗ ) th from . ± . to . ± . and R ( D ) th from . ± . to . ± . compared to the SM value. Especially the NP contributions are able to reduce thediscrepancy between experiment and theory of R ( D ∗ ) substantially compared to SM. This is quiteimpressive given that the model satisfy all other very stringent constrains coming from neutralmeson oscillations and precision Z -pole data. PACS numbers: ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION. The Standard Model (SM) of particle physics has been very successful in accountingfor particle interactions and gives an admissible explanation for the electroweak symmetrybreaking (EWSB) mechanism that agrees with all observed data. It has also been tested toa very high degree of precision and the discovery of the Higgs boson at the Large HadronCollider (LHC) completed the last missing piece of the framework. However there are alsoseveral experimental observations such as that of neutrino-oscillation suggesting neutinos tohave mass, existence of dark-matter (DM) and dark-energy in the Universe which definitelypoint to the incompleteness of our understanding and to physics beyond the SM (BSM).With very little hint of any new physics (NP) discovery by direct searches at LHC atthe moment there is huge interest in discrepencies observed in the flavor sector of particlephysics. Recently many experiments have reported observed deviations from SM predictionsin few observables such as R ( D ∗ ) , R K ( ∗ ) , muon ( g − , etc. with statistical significance inthe range of ∼ (2 − σ , which could be, if not due to statistical fluctuations, strong hintsof NP. The focus will therefore be on the upcoming precision machines such as Belle-II and
LHCb .In this work we carry out a phenomenological study by extending the SM with newparticles and show how this extension can explain the deviations in R ( D ( ∗ ) ) without violatingany experimental constraints. To do this we add an additional color-singlet matter multipletin the form of a vector-like lepton doublet under SU (2) L . We also add a neutral scalar aswell as an SU (3) c triplet scalar leptoquark (LQ), both singlets under the SU (2) L gaugegroup. The only additional requirement on all the additional particles is that they areodd under a discrete Z symmetry. We then proceed to calculate the contributions fromsuch a model to the R ( D ( ∗ ) ) and compare it to the observed deviations. We take intoaccount all stringent constraints coming from flavor precision data on the model parametersincluding those coming from K − ¯ K and B i − ¯ B i ( i = d, s ) oscillations, Br ( Z → f ¯ f ) ( f = u, d, s, b, e, µ, τ ) and from Peskin-Takeuchi S, T and U parameters. We show that thenew particles can contribute substantially to R ( D ∗ ) provided the parameters are tuned in2 phenomenological way.Our paper is organised as follows. In section II we give the new particle content andtheir interaction Lagrangian. In section III we discuss the relevant constraints that wouldrestrict out fit on the parameters of the model with respect to limits coming from b → cτ ν τ ,neutral meson oscillation data and Z -pole data. Finally in section IV we summarize ourresults and conclude. II. MODEL DETAILS.
In this work we study a model which extends the SM particle content with a vector-likelepton L = ( F F − ) T , doublet under the SU (2) L and odd under a discrete Z symmetry,an SU (3) c triplet scalar leptoquark ( φ LQ ) odd under the Z and singlet under the SU (2) L gauge group carrying + unit of electic charge and a neutral complex scalar ( S ) singlet underthe SM gauge group and odd under the Z . The quantum numbers under the SM gaugesymmetry and the new discrete Z for the new particles are shown in Table I. Note thatall the SM particles are even under the Z . We write the most general Yukawa interactionLagrangian involving the new set of particles that is consistent with all the symmetries ofthe model as L NP = X i =1 h i ¯ Q iL L R φ LQ + X j =1 h j ¯ L j L L R S + m F ¯ L L L R + h.c. (1)where i = 1 , , represent the SM quark generations and the couplings can be put as( h u , h c , h t ) or equivalently ( h d , h s , h b ). Similarly j = 1 , , represent the SM lepton gener-ations and the couplings can be put as ( h e , h µ , h τ ) for the leptons while m F is the mass ofthe new vector-like leptons. With the additional scalar LQ φ LQ and the complex scalar S ,the most general scalar potential that is invariant under the full symmetry of the model can3 articles SU (3) c SU (2) L U (1) Y Z L φ LQ Z . be written as [1] V ( H, φ LQ , S ) = m H † H + m φ LQ φ † LQ φ LQ + m S S † S + λ H † H ) + λ HLQ ( H † H )( φ † LQ φ LQ )+ λ φS ( φ † LQ φ LQ )( S † S ) + λ HS ( H † H )( S † S ) + λ φ LQ φ † LQ φ LQ ) + λ S S † S ) + (cid:18) m S S + λ S S + λ S | S | S + λ ′ HS | H | S + h.c. (cid:19) + m φ LQ φ LQ + λ φ LQ φ LQ + λ φ LQ | φ LQ | φ LQ + λ ′ Hφ LQ | H | φ LQ + h.c. ! (2)where H represents the SM Higgs doublet. The new scalar fields do not get any vacuumexpectation value (VEV) and can be expressed as φ LQ = φ R + iφ I √ , S = S R + iS I √ . (3)Then we have a mass relation for the real and imaginary components of the scalars given by m S R − m S I = m S + λ ′ HS v where v is the electroweak VEV for the SM Higgs. Note thatfor m S R − m S I > the S R becomes the lightest component of the neutral singlet scalar S .As the Z remains unbroken, with m F larger than m S R this will be stable and can be a DMcandidate. However we find that to fit our results for R ( D ∗ ) we require that its Yukawacouplings have to be large with the fermions as suggested by b → cτ ν τ data. This wouldlead to large annihilation cross section and therefore its contribution to the present relicdensity is expected to be small [1] which is acceptable and not ruled out. In this analysis,for simplicity we take m S = λ ′ HS ≈ and m φ LQ = λ ′ Hφ LQ ≈ which means that for boththe new scalars their real part and complex part are symmetric in all respect.4 II. CONSTRAINTS FROM NEUTRAL MESON OSCILLATION DATA, Z -POLEAND b → cτ ν τ . We know that the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix with three realand one imaginary physical parameters can be made manifest by choosing an explicitparametrization. With the standard parameterization [2] in terms if θ ′ ij s and the Kobayashi-Maskawa phase δ we find it interesting and worth pointing out that the requirement of apositive contributions from NP to R ( D ∗ ) and constraints from neutral meson oscillationsare favored when π ≤ θ ≤ π and π ≤ θ , θ ≤ π . This implies that the sign of the firsttwo rows of the CKM matrix elements are negative relative to the third row compared toinstead the usual convention where all angles have been fixed in the first quadrant. Whenexpressed in terms of the mass eigenstates of the down quarks ( d ′ , s ′ , b ′ ), we have h ′ i = j = t X j = u h j V ji (4)for the down quark Yukawa couplings, where i = d, s, b . Thus we note that the effectivecoupling of the down-type quarks with the new vector-like leptons and the scalar LQ aremodified in the mass basis via the CKM mixing matrix while the up-type quark couplingsremain the same. Now we would like to point out that if we impose the condition h ′ d = − h d V ud − h s V cd + h b V td = 0 , (5)then the NP has no contribution to the K − ¯ K and B − ¯ B oscillations. Since theseobservables are very precisely measured and no deviations from the SM prediction have beenreported, the above condition seems a quite natural experimental imposition. Note that inEq. (5) the sign change of the first two rows of the CKM matrix elements is explicitly shown.In addition to this it is favorable to have significantly large Yuakawa coupling strength for The notation we use on the right side of Eq. (4) is by representing h j as h u , h c , h t to write it in a compactway. However these are the same as h d , h s , h b respectively, as pointed out below Eq. (1) and as writtenexplicitly in Eq. (5). h b = 3 . < √ π which favors the b → cτ ν τ data and parameterize h s ≈ h b a . Now ifwe demand a to be real then to satisfy Eq. (5), h d has to be complex. Additional constrainton the parameters also come from the respective mass bounds on the new charged andneutral leptons as well as bounds from B s − ¯ B s oscillation on Re (∆ M NPB s ) and Im (∆ M NPB s ) .This is discussed in more detail in section III B. A. Contribution to b → cτ ν τ . In recent works in [3, 4], it was shown that observed deviations from SM in the muon ( g − ), generation of small neutrino masses, Baryogenesis as well as the observed anomalies in R K ( ∗ ) could be explained with new exotic scalars and leptons. Therefore it is very interestingto see whether exotic scalars and leptons can also explain the R ( D ∗ ) , where R ( D ∗ ) = Br ( B → D ( ∗ ) τν τ ) Br ( B → D ( ∗ ) ℓν ℓ ) with ℓ = e, µ . A deviation from SM predictions in R ( D ( ∗ ) ) was first reportedby Babar [5] followed by
Belle [6–8] and
LHCb [9, 10], with the latest HFAG average of theexperimental result amounting to [11] R ( D ) Exp = 0 . ± . ± .
024 ; R ( D ∗ ) Exp = 0 . ± . ± . . (6)These when compared to the SM predictions as given in [12, 13] respectively: R ( D ) SM = 0 . ± .
008 ; R ( D ∗ ) SM = 0 . ± . , (7)and taking the correlation between the two observables into account, the combined deviationfrom SM is around 4.1 σ in these observables. Although the present HFAG world averagesare well above the SM predicted values, the Belle results agrees with both the SM value aswell as the HFAG world averages [14], where HFAG world averages in these observables arestill dominated by the
Babar ’s data due to it having the least error of all the measurementstill date.In this model, there is no contribution to b → cτ ν τ transition at tree level, but at thebox loop level there is contribution from NP to the quark level transition due to the Yukawa6nteractions shown in Eq. (1). The box loop diagram shown in Figure 1 from NP addcoherently to the SM contribution and so we can express the effective Hamiltonian as [14] H eff = 4 G F √ V cb (1 + C NP )[( c, b )( τ, ν τ )] (8)where ( c, b )( τ, ν τ ) is the usual SM left handed vector four current operator and C NP in ourmodel can be expressed as C NP = N ( − V ub h d − V cb h s + V tb h b ) ∗ | h s || h τ | π m F S ( x i , x j ) , (9)where S ( x i , x j ) = 1(1 − x i )(1 − x j ) + x i ln( x i )(1 − x i ) ( x i − x j ) − x j ln( x j )(1 − x j ) ( x i − x j ) are the Inami-Lim functions [15, 16] with N = 4 G F | V cb |√ , x i = m φ LQ m F − and x j = m S m F . bc LQ Φ F F S τν FIG. 1: Contributions to the b → cτ ν τ from the new particles at box loop level. We have cross checked our calculations and find it to agree with a similar result evaluatedin context of b → sµ + µ − given in [16] and we taken λ = 0 . ± . , A = 0 . ± . , ¯ η = 0 . ± . and ¯ ρ = 0 . +0 . − . , ignoring corrections of O ( λ ) and above, where λ , A , ¯ η and ¯ ρ being the parameters in the Wolfenstein parametrization of the CKM matrixelements [2]. 7e choose h e , h µ << h τ = 3 . along with fixing the benchmark value of the masseswell above the present respective experimental bounds [2]. For our calculation we assume m F ± = m F = 200 GeV, m φ LQ = 900 GeV and m S = 150 GeV. Further we choose a ( a ≈ h b h s )to be real along with the constraints on Yukawa couplings from Eq. (5) as well as requiring Re [( h ′ s h ′ b ) ] ≤ . × − from ∆ M errorSM and Im [( h ′ s h ′ b ) ] ≤ . × − from CP violationdata in B s − ¯ B s oscillation (see section III B for details). We get for the best fit values of theparameters as a = − . , Re ( h d ) = − . × − and Im ( h d ) = − . which gives R ( D ∗ ) NP = 0 . ± .
051 and R ( D ) NP = 0 . ± . . (10)Compared to the experimental values there is substantial contribution especially to the R ( D ∗ ) from the NP, where the errors quoted here are the experimental errors scaled by p χ . The contribution from NP has reduced the deviation in R ( D ∗ ) from . σ to . σ and FIG. 2: Plot of | C NP | vs m F for m LQ = 900 GeV and m S = 150 GeV. deviation in R ( D ) from 2.3 σ to 0.6 σ . In Figure 2 we plot | C NP | as a function of m F while fixing m LQ = 900 GeV and m S = 150 GeV. The NP contribution goes down as m F increases as shown in Figure 2 and falls to 1% of the SM value as m F ∼ TeV. where χ = [ ( R ( D ) Exp − R ( D ) NP ) σ Exp ( D ) + ( R ( D ∗ ) Exp − R ( D ∗ ) NP ) σ Exp ( D ∗ ) ] . ( c, b )( τ, ν τ ) current can also contribute to B c → τ ν τ with same | C NP | = 1 . ,however this is much smaller than the present allowed limit given as | C NP | allowed = 1 . [17]. Similarly NP contribution to D s → τ ν τ is negligible compared to SM as well. Notethat with the additional particle content of the model and their Yukawa interaction termswe also get NP contributions to hadronic decay of τ . The decay τ → ( Kπ ) ν τ is proportionalto the product of couplings h ′ s h d | h τ | and gives C ′ NP = O (10 − ) . Thus the NP contributionis quite small and negligible compared to SM contributions in this mode. Even though h d is complex, it still cannot contribute to the CP violation in τ → ( Kπ ) ν τ or τ → ρπν τ etc. This is because the NP contributions in this model to the vector and the axial-vectoreffective four current come with same magnitude and phase (see Ref. [18, 19] and referencesthere in for more details). NP contributions to C in B → K ( ∗ ) µ + µ − via photon penguinis about | C NP | = 3 . × − which is again too small to have any effect on the reportedanomaly in C [16] and NP contributions to b → sγ is | C NP + 0 . C NP | ≈ − which isabout two orders smaller than the 2 σ present experimental bound [16]. In this model wecan also get contributions to B → K ( ∗ ) τ + τ − , B s → τ + τ − and D → ( π ) ν τ ¯ ν τ which arenot properly measured yet but NP contributions to these modes are less than a percent-level, at the order of | C NP | = 1 . or smaller and so negligible compared to the SMcontribution. We also note that NP contribution to the anomalous magnetic moment of τ is ∆ a NPτ ≈ − . × − compared to the experimental bound − . < ∆ a Exp.τ < . [2]which is again negligible. B. Neutral meson oscillation.
Similar to the SM, the new particles in our model also contribute to neutral mesonoscillations via the box loop. From the condition that we imposed in Eq.(5) our model givesno contribution to the K − ¯ K and B − ¯ B . However for B s − ¯ B s oscillations we do havenon vanishing contributions which can be put as L NPeff = 2 C NPB s ¯ s α γ µ P L b α ¯ b β γ µ P L s β (11)9here C NPB s = ( h ′ s h ′ b ) π m F S ( x, x ) where S ( x, x ) are again the Inami-Lim functions with x = m LQ m F and the factor 2 to account for the contributions from the two diagrams in Figure 3. Then bs bs sb sbFF F F Φ Φ ΦΦ
LQ LQ LQLQ
FIG. 3: Contributions to the B s − ¯ B s mixing from the new particles. The F in the loop is thecharged component of the vector-like lepton doublet. introducing a factor of to take into account the Wick contraction and color structureover counting , we have h B s | ¯ s α γ µ P L b α ¯ b β γ µ P L s β | ¯ B s i = 14 × × M B s f B s B ( µ ) 12 M B s (12)and so with ∆ M NPB s = 2 Re ( h B s |L NPeff | ¯ B s i ) we have ∆ M NPB s = 14 × × M B s f B s B ( µ ) × C NPB s , (13)where f B s is the B s decay form factor and B ( µ ) is a QCD scale correction factor, theirvalues are taken from [22, 23].Then with the values of the Yukawa couplings and masses given in the previous section weget Re (∆ M NPB s ) = 1 . ps − . This when compared to the error in experimental measure-ment of the same observable given as ∆ M ExpB s = (17 . ± . ps − , the NP contribution When expanded in terms of creation and annihilation operators, there will be four terms, two from eachdiagrams, only two will contribute to the real process but when summed all four are summed so a factor to compensate it; and also when summed over the colors, we sum over the two different possible colorsinglet arrangements but only one actually contribute, so a factor to compensate that, actually theseover counted factor of 4 are in factor in the Eqs.(13), see [20, 21] for details.
10s well above the error in the experimental measurement taken from PDG [2]. But given thatthere are still large errors in the SM calculations with the latest estimate of SM calculationpredicting a 1.8 σ deviation above the experimental average as ∆ M SMB s = (20 . ± . ps − [23], the above NP contribution is well within the error in the latest SM calculation. Wewould like to point out that the previous SM calculations in Refs. [22, 24] agrees with theexperimental value but their errors are larger than the latest SM prediction. Since NP con-tribution is allowed to be as large as the SM and experimental errors added in quadrature,if we take the previous SM predictions, NP contribution is allowed to be little larger thanthe above value. Note that in all the above calculations we have taken the hadronizationparameters from Refs. [22, 23] and the experimental values from PDG [2].Due to h d being complex, we also have a non-zero imaginary component of ∆ M NPB s given as Im (∆ M NPB s ) = − . × Γ ExpB s . This can contribute to the CP violation in the B s − ¯ B s mixing which is parametrized in terms of Re ( ǫ B s )1+ | ǫ B s | , where ǫ B s = − Im (∆ M B s ) ∆Γ B s − i ∆ M B s . Inour case with ∆Γ B s << ∆ M B s the CP violating parameter due to NP can be approximatelyexpressed as Re ( ǫ NPB s )1+ | ǫ NPB s | ≈ − Im (∆ M NPB s ) × ∆Γ ExpB s M ExpB s ) ≈ +1 . × − compared to Re ( ǫ Exp )1+ | ǫ Exp | ≈ ( − . ± × − [2]. Note that the NP contribution is an order of magnitude smaller than thepresent experimental limit. There is no contribution to the ∆Γ B s from NP since none ofthe intermediate particles in Figure 3 can go on shell. For the D − ¯ D oscillation with 2 σ bound from [16] given as | C ExpD | < . × − T eV − we compare | C NPD | ≈ . × − T eV − and find the NP contribution to be around two orders of magnitude smaller thanthe present experimental bound at 2 σ . 11 . Z pole constrains. For theoretical calculations of contribution from new fermions to the Z decay into twofermions via higher order loops, we have used [1] Br ( Z → f i f i ) = G F √ π m Z (16 π ) Γ tot.Z ( T i − Q i sin ( θ W )) | h ′ i | | [ F ( m F , m φ ) + F ( m F , m φ )] | (14)where F ( a, b ) = Z dx (1 − x ) ln [(1 − x ) a + xb ] (15)and F ( a, b ) = Z dx Z − x dy ( xy − m Z + ( a − b )(1 − x − y ) − ∆ ln ∆∆ (16) ∆ = − xym Z + ( x + y )( a − b ) + b (17)with Γ totZ = 2 . .Now with the numerical values of the Yukawa couplings given before and with m F ± = m F = 200 GeV, m φ LQ = 900 GeV and m S = 150 GeV, we get Br ( Z → ¯ dd ) NP = 0 due toEq. (5) while Br ( Z → ¯ uu ) NP , Br ( Z → ¯ ss ) NP << Br ( Z → ¯ cc ) NP ≈ O (10 − ) well withinthe experimental errors given by Br ( Z → ¯ uu ) Experror ≈ . , Br ( Z → ¯ ss ) Experror ≈ . and Br ( Z → ¯ cc ) Experror ≈ . . Even for the decay mode where the large Yukawa choices can besignificant we find Br ( Z → ¯ bb ) NP = 4 . × − as compared to Br ( Z → ¯ bb ) Experror ≈ × − putting the NP contribution an order of magnitude smaller than the experimental error.The contributions from NP to Br ( Z → ¯ ee ) and Br ( Z → ¯ µµ ) are negligible compared tothe experimental errors since we assume that h e , h µ << (which is required to explainthe R ( D ( ∗ ) ) anomalies). For the third generation lepton where we have h τ large we get Br ( Z → ¯ τ τ ) NP ≈ . × − and Br ( Z → ¯ νν ( invisible )) NP ≈ . × − compare to Br ( Z → ¯ τ τ ) Experror ≈ × − < Br ( Z → invisible ) Experror . Here again the NP contributionsare negligible. All the experimental values are taken from the latest PDG averages [2].12egarding the contributions of the new states [25], to the Peskin-Tekeuchi S, T and Uparameters, we find that with the above given masses of the new fermions we have S ≈ . , T ≈ and U ≈ in our model, which are well within the present experimentalbounds on these parameters [26]. IV. CONCLUSIONS.
In this work we have introduced a vector like fourth generation lepton doublet ( F , F − )along with an SU (3) c triplet scalar leptoquark φ LQ and a neutral scalar ( S ) both singletunder the SU (2) L gauge group. All the newly added particles are odd under a discretesymmetry Z . With these new particles we have done a comprehensive analysis of thephenomenological consequences of the model taking all the very stringent constraints from K − ¯ K and B i − ¯ B i ( i = d, s ) oscillations as well as Br ( Z → f ¯ f ) and Peskin-Tekuchiparameters into account. We find that such a model can give a substantial contribution to R ( D ( ∗ ) ) , and is able to reduce the tension between theoretical prediction and experimentalmeasured value of R ( D ∗ ) from . σ to . σ and deviation in R ( D ) from . σ to . σ .Especially the NP contribution is able to reduce the discrepancy between experiment andtheory in R ( D ∗ ) substantially. In addition the mass of the newly introduced states requiredto give a large contribution to the R ( D ∗ ) lie in a range which will be directly probedat the LHC with higher luminosity. Thus the model presents robust phenomenologicalconsequences accessible at both the high energy collider experiment such as the LHC aswell as leaving imprints in the flavor sector.We find that while accommodating the large contributions to R ( D ( ∗ ) ) the model does notviolate any other observations and is found to satisfy all other stringent constraints comingfrom neutral meson oscillations and precision Z-pole data. Acknowledgments
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