Hunting for B + → K + τ + τ − imprints on the B + → K + μ + μ − dimuon spectrum
Claudia Cornella, Gino Isidori, Matthias König, Sascha Liechti, Patrick Owen, Nicola Serra
ZZH-TH 01/20January 13, 2020
Hunting for B + → K + τ + τ − imprintson the B + → K + µ + µ − dimuon spectrum C. Cornella, G. Isidori, M. K¨onig, S. Liechti, P. Owen, N. Serra
Physik-Institut, Universit¨at Z¨urich, CH-8057, Switzerland
Abstract
We investigate the possibility of indirectly constraining the B + → K + τ + τ − decayrate using precise data on the B + → K + µ + µ − dimuon spectrum. To this end, weestimate the distortion of the spectrum induced by the B + → K + τ + τ − → K + µ + µ − re-scattering process, and propose a method to simultaneously constrain this (non-standard)contribution and the long-distance effects associated to hadronic intermediate states. Thelatter are constrained using the analytic properties of the amplitude combined with dataand perturbative calculations. Finally, we estimate the sensitivity expected at the LHCbexperiment with present and future datasets. We find that constraints on the branchingfraction of O (10 − ), competitive with current direct bounds, can be achieved with thecurrent dataset, while bounds of O (10 − ) could be obtained with the LHCb upgrade-IIluminosity. a r X i v : . [ h e p - ph ] J a n Introduction
In recent years, discrepancies between the observed values and the Standard Model (SM)predictions of the lepton-flavour universality (LFU) ratios R D ( ∗ ) [1–5] and R K ( ∗ ) [6–9], char-acterizing the semileptonic transitions b → clν and b → sll , have sparked great interest. Thepattern of anomalies seems to point to intriguing new-physics (NP) scenarios, with possibleconnections to the SM flavour puzzle. A large class of NP models proposed to explain thesehints of physics beyond the SM, and in particular those aiming for a combined explanation ofthe R K ( ∗ ) and R D ( ∗ ) anomalies, imply dominant couplings to third-generation fermions, whichshould also enter other semileptonic b -quark decays.A general expectation, confirmed by many explicit NP constructions, is that of a large en-hancement of b → sτ + τ − transitions (see e.g. [10–17]). While flavour-changing neutral-current(FCNC) decays with muon and electron pairs have been observed both at the exclusive andat the inclusive level, probing rare decays with a τ + τ − pair in the final state is experimentallyvery challenging. The current experimental limits for all processes mediated by the b → sτ + τ − amplitude are still very far from the corresponding SM predictions [18, 19], leaving the NPexpectation of possible large enhancements unchallenged.In this work we investigate the possibility of indirectly constraining the b → sτ + τ − am-plitude via its imprint on the B + → K + µ + µ − dimuon spectrum. In presence of a largeNP enhancement, the b → sτ + τ − amplitude would induce a distinctive distortion of the B + → K + µ + µ − spectrum via the (QED-induced) re-scattering process B + → K + τ + τ − → K + µ + µ − [10]. The latter has a discontinuity at q = 4 m τ ( q ≡ m µµ ), namely at the thresholdwhere the tau leptons can be produced on-shell. This gives rise to a “cusp” in the dimuon-invariant mass spectrum, which could in principle be detected with sufficient experimentalprecision. More generally, the lightness of the τ -leptons implies a well-defined deformationof the B + → K + µ + µ − spectrum, which is determined only by the analytic properties of there-scattering amplitude.It should be stressed that the phenomenon we are considering here is different from theQED mixing between dimension-six FCNC operators with different lepton species analysed inRef. [20]. If NP is heavy and the b → sτ + τ − amplitude is strongly enhanced, the operatormixing can give rise to sizable modifications of the Wilson coefficients of the dimension-sixeffective Hamiltonian relevant to b → sl + l − decays ( l = e, µ ). However, this phenomenoncannot be distinguished in a model-independent way from other NP effects of short-distanceorigin (at least using low-energy data only). On the contrary, the non-local effect we areinterested in can be unambiguously attributed to the re-scattering of light intermediate statescharacterised by the tau mass, hence it can be translated into a model-independent constrainton the B + → K + τ + τ − amplitude.The main difficulty in extracting such bound is obtaining a reliable description of the B + → K + µ + µ − dimuon spectrum within the SM, or better in the limit where the τ + τ − → µ + µ − re-scattering is negligible. This is non trivial, given that the B + → K + l + l − spectrum is plaguedby theoretical uncertainties originating from B → K form factors and hadronic long-distancecontributions. While the former are smooth functions in the q region of interest and canbe well described using lattice QCD [21, 22] and/or light-cone sum rules [23], long-distanceeffects induced by intermediate hadronic states, such as the charmonium resonances, are more1roblematic. They are genuine non-perturbative effects and introduce physical discontinuitiesbelow and above the q = 4 m τ threshold. Far from the resonance region, these effects canbe estimated using perturbative constraints derived at q <
0, with | q | (cid:29) Λ , combinedwith a Λ /q or Λ /m c expansion to incorporate the leading non-perturbative correc-tions [24, 25]. However, this approach is not suitable for our purpose, which requires a reliabledescription of the whole spectrum, and in particular of the resonance region. To achieve thisgoal, we adopt a data-driven approach which takes full advantage of the known analytic prop-erties of the amplitude: knowing the precise location of all one- and two-particle hadronicthresholds, we use subtracted dispersion relations to describe the q –dependence of the wholespectrum in terms of a series of ( q –independent) hadronic parameters, which are fitted fromdata. This method can be considered a generalisation of the approaches proposed in Ref. [26]and, to some extent, in Refs. [27–29], with a few key differences, the most notable ones beingthe use of subtracted dispersion relations and the explicit inclusion of two-particle thresholds.To reduce the number of independent free parameters, perturbative constraints derived fromthe low- q region are also implemented. Proceeding this way we obtain a description of thespectrum that is flexible enough to extract the non-perturbative parameters characterisingthe various hadronic thresholds from data, but retains a significant predictive power in thesmooth region within and below the two narrow charmonium states, allowing us to set usefulconstraints on the B + → K + τ + τ − → Kµ + µ − re-scattering.The method we propose is particularly well suited for the LHCb experiment, which hasalready collected a large sample of B + → K + µ + µ − events and has an excellent resolution inthe dimuon spectrum [30]. In order to estimate the sensitivity of LHCb in view of the fullrun II dataset, we generate pseudo-experiments based on the yields and ampltiudes obtainedin Ref. [30], and calculate the expected limit under the background-only hypothesis using theCLs method [31].The paper is organised as follows: in Section 2 we introduce the theoretical frameworknecessary to describe the B + → K + µ + µ − dimuon spectrum within and beyond the SM, sepa-rating short-distance contributions (Section 2.1), long-distance contributions due to interme-diate hadronic states (Section 2.3), and long-distance contributions due to the τ + τ − → µ + µ − re-scattering (Section 2.4). The analysis of the LHCb sensitivity is presented in Section 3.The results are summarised in the Conclusions. The dimension-six effective Langrangian describing b → sll transitions, renormalized at lowenergies [ µ = O ( m b )], can be decomposed as L eff = 4 G F √ V tb V ∗ ts (cid:88) i C i ( µ ) O i , (1)2here the leading FCNC effective operators are defined as O = e π m b (¯ sσ µν P R b ) F µν , O l = e π (¯ sγ µ P L b )(¯ lγ µ l ) , O l = e π (¯ sγ µ P L b )(¯ lγ µ γ l ) , (2)and the most relevant four-quark operators ( q = u, c ) as O q = (¯ sγ µ P L q )(¯ qγ µ P L b ) , O q = (¯ s α γ µ P L q β )(¯ q β γ µ P L b α ) . (3)Within the class of models we are considering, all relevant NP effects are encoded in the valuesof the Wilson coefficients C l , , . Given the normalisation in Eq. (1), C l , , and C c , are realand O (1) within the SM, whereas C u , = ( V ub V ∗ us /V tb V ts ) × O (1) (see Ref. [26] for the precisevalues of the Wilson coefficients and the complete basis of operators).The matrix elements (cid:104) K + µ + µ − |O i | B + (cid:105) are non-vanishing at the tree level only in the caseof the FCNC operators (with l = µ ). Considering only these contributions, the B + → K + µ + µ − decay rate can be written as: d Γ dq = α G F | V tb V ∗ ts | π κ ( q ) β ( q ) (cid:26) κ ( q ) β ( q ) (cid:12)(cid:12) C µ f + ( q ) (cid:12)(cid:12) + m µ ( m B − m K ) q m B (cid:12)(cid:12) C µ f ( q ) (cid:12)(cid:12) + κ ( q ) (cid:20) − β ( q ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) C µ f + ( q ) + 2 C m b + m s m B + m K f T ( q ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:41) , (4)where κ ( q ) = λ / ( m B , m K , q ) / m B is the kaon momentum in the B -meson rest frame, β ( q ) = (cid:113) − m µ /q , and f i ( q ) with i = + , , T are the vector, scalar and tensor B → K form factors. The non-local contributions generated by the non-leptonic operators in L eff and by the operator O τ can be encoded in Eq. (4) by replacing C µ with a q -dependent function: C µ → C µ, eff9 ( q ) = C µ + Y c ¯ c ( q ) + Y light ( q ) + Y τ ¯ τ ( q ) , (5)where Y I ( q ) denotes the non-local contributions corresponding to the intermediate state I ,which can annihilate into a dimuon pair via a single-photon exchange.The functions Y c ¯ c ( q ) and Y light ( q ) encode non-perturbative hadronic contributions, whichcannot be estimated reliably in perturbation theory, at least in a large fraction of the accessible q spectrum. Adopting a notation similar to that of Ref. [26], we can express Y c ¯ c ( q ) as Y c ¯ c ( q ) = 16 π f + ( q ) H ( BK ) c ¯ c ( q ) , (6)3 + K + l − l + V B + K + M ′ l − l + M Figure 1:
Diagrammatic representations of the long-distance contributions to C eff9 . The left-handside depicts the exchange of a single vector resonance. The graph on the right-hand side shows thecontribution from two-particle intermediate states. where H ( BK ) c ¯ c ( q ) is defined by the gauge-invariant decomposition of the following non-localhadronic matrix element i (cid:90) d xe iq · x (cid:104) K ( p ) | T (cid:40) j em µ ( x ) , (cid:88) i =1 , C ci O ci (cid:41) | B ( p + q ) (cid:105) = (cid:2) ( p · q ) q µ − q p µ (cid:3) H ( BK ) c ¯ c ( q ) , (7)with j em µ = (cid:80) q Q q ¯ qγ µ q . The function Y light ( q ), containing the contribution of the subleadingnon-leptonic operators in L eff , is defined in a similar way via the replacement (cid:88) i =1 , C ci O ci → (cid:88) i =3 − , C i O i + (cid:88) i =1 , C ui O ui . (8)Our main strategy is to write the non-perturbative functions Y c ¯ c ( q ) and Y light ( q ) usinghadronic dispersion relations. More precisely, for the leading charm contribution we considerone- (1P) and two-particle (2P) intermediate states (see Fig. 1), using dispersion relationssubtracted at q = 0, while for the subleading Y light ( q ) function we consider only one-particleintermediate states and use unsubtracted dispersion relations.Under these conditions, C µ, eff9 ( q ) in Eq. (5) is finally decomposed according to C µ, eff9 ( q ) = C µ + Y (0) c ¯ c + ∆ Y c ¯ c ( q ) + ∆ Y c ¯ c ( q ) + Y ( q ) + Y τ ¯ τ ( q ) , (9)with ∆ Y c ¯ c (0) = ∆ Y c ¯ c (0) = 0.In the next section we analyse the structure of ∆ Y c ¯ c ( q ), ∆ Y c ¯ c ( q ), and Y ( q ) in detail.The expression of Y τ ¯ τ ( q ), which is the only term in Eq. (9) that can be fully evaluated inperturbation theory, is given in Sect. 2.4. The general structure of the subtracted dispersion relation used to determine ∆ Y c ¯ c ( q ) is∆ Y c ¯ c ( q ) = 16 πq f + ( s ) (cid:90) ∞ m J/ Ψ ds s ( s − q ) 12 i Disc (cid:104) H ( BK ) c ¯ c ( s ) (cid:105) ≡ q π (cid:90) ∞ m J/ Ψ ds ρ c ¯ c ( s ) s ( s − q ) . (10)The function ρ c ¯ c ( s ) is the spectral-density function describing the hadronic states I c ¯ c , charac-terized by valence charm-quarks and invariant mass s , contributing as real intermediate states4n the re-scattering B → K I c ¯ c → Kµ + µ − . As noted before, we decompose ρ c ¯ c ( s ) into one-and two-particle intermediates states, ρ c ¯ c ( s ) = ρ c ¯ c ( s ) + ρ c ¯ c ( s ), ρ c ¯ c ( s ) ∝ (cid:88) j A ( B → KV j ) δ ( s − m j ) , (11) ρ c ¯ c ( s ) ∝ (cid:88) j (cid:90) dp j δ ( s − p j ) (cid:90) d (cid:126)p j d (cid:126)p j π E j E j A ( B → KM + j M − j ) δ (4) ( p j − p j − p j ) , (12)neglecting the phase-space suppressed contribution with three or more particles. For the sake of simplicity, in Eq. (11) we have treated the single-particle states as infinitelynarrow resonances. The effect of finite widths can be incorporated via Breit-Wigner functions,yielding∆ Y c ¯ c ( q ) = (cid:88) j =Ψ(1 S ) ,..., Ψ(4415) η j e iδ j q m j A res j ( q ) , A res j ( s ) = m j Γ j ( m j − s ) − im j Γ j , (13)where the sum runs over all the charmonium vector resonances in the accessible kinematicalrange. Here η j and δ j are real parameters which must be determined from data, similarly towhat has been performed by the LHCb collaboration in [30]. The fitted η j ’s can be put inone-to-one correspondence with the product of the B + → K + V j and V j → µ + µ − branchingfractions via B ( B + → K + V j ) × B ( V j → µ + µ − ) = τ B + G F α | V tb V ∗ ts | π m B − m K ) (cid:90) m µ dq κ ( q ) ×× (cid:20) β ( q ) − β ( q ) (cid:21) (cid:12)(cid:12) f + ( q ) (cid:12)(cid:12) | η j | (cid:12)(cid:12)(cid:12)(cid:12) q m j A res j ( q ) (cid:12)(cid:12)(cid:12)(cid:12) . (14)The expression (13) differs from the decomposition adopted in Ref. [30] by the q /m j term,which arises from the subtraction procedure in the dispersion relation. On the one hand, theuse of subtracted dispersion relations for the charm contribution is necessary to ensure theconvergence of the integral in the two-particle intermediate states (see sect. 2.3.2). On theother hand, choosing the subtraction point at q = 0 allows us to decouple the determinationof the resonance parameters of the spectrum from the overall normalisation of the rate, andhence from the determination of C µ from data. The price to pay is the appearance of theundetermined constant term Y (0) c ¯ c = Y c ¯ c (0) in Eq. (9). This term plays no role in the descriptionof the dimuon spectrum, but is relevant for the extraction of the value of C µ . To this purpose,we note that the estimate presented in Ref. [26], which is based on a Λ /m c expansion andalso takes next-to-leading O ( α s ) corrections on the pure partonic result into account (seesect. 2.3.4), yields Y (0) c ¯ c ≈ − . ± . , (15)which is about − (2 ± C µ, SM9 ≈ .
23. 5 igure 2:
Real (solid) and imaginary (dashed) parts of the normalised hadronic two-particle con-tributions to Y c ¯ c ( q ), as defined in Eq. (16). Proceeding in a similar way, we can decompose the two-particle contributions as∆ Y c ¯ c ( q ) = (cid:88) j η j e iδ j A j ( q ) , A j ( q ) = q π (cid:90) ∞ s j dss ˆ ρ j ( s )( s − q ) , (16)where ˆ ρ j ( s ) are normalised spectral densities for the two-body intermediate states charac-terised by the threshold s j = ( m j + m j ) .While we do not have a precise estimate of these spectral densities at generic kinematicalpoints, an excellent description of their behaviour around the respective thresholds is obtainedby approximating them with powers of the K¨all´en function, with an exponent determined bythe lowest partial wave allowed in the B + → K + M M → K + µ + µ − re-scattering. This isbecause higher-order partial waves, characterised by higher powers of the K¨all´en function,are both phase-space suppressed and, most importantly, give rise to a less singular behaviourat the threshold. From angular momentum conservation we can then determine the leadingpartial wave and obtain the following estimates for the normalised spectral densities of thetwo-particle intermediate states of lowest mass:ˆ ρ DD ( s ) = (cid:18) − m D s (cid:19) / , ˆ ρ D ∗ D ∗ ( s ) = (cid:18) − m D ∗ s (cid:19) / , ˆ ρ DD ∗ ( s ) = (cid:18) − m D s (cid:19) / . (17)In the case of the DD ∗ intermediate state we have replaced the complete expression dependingon both masses with a simplified one depending only on m ¯ D = ( m D + m D ∗ ) /
2, which provides6n excellent approximation. With these estimates in place we find:∆ Y c ¯ c ( q ) = η ¯ D e iδ ¯ D h S (cid:0) m ¯ D , q (cid:1) + (cid:88) j = D,D ∗ η j e iδ j h P (cid:0) m j , q (cid:1) , (18)with h P (cid:0) m, q (cid:1) = 23 + (cid:18) − m q (cid:19) h S (cid:0) m, q (cid:1) , h S (cid:0) m, q (cid:1) = 2 − G (cid:18) − m q (cid:19) , (19)and G ( y ) = (cid:112) | y | (cid:26) Θ( y ) (cid:20) ln (cid:18) √ y − √ y (cid:19) − iπ (cid:21) + 2 Θ( − y ) arctan (cid:18) √− y (cid:19)(cid:27) . (20)It is worth noting that, while the lowest threshold is at q = 4 m D , the contribution from the DD ∗ intermediate state is the only one which can occur in the S -wave, corresponding to asingular (square-root) behaviour at the threshold (see Fig. 2). The remaining hadronic contribution we need to estimate is Y light ( q ), defined by Eqs. (6)–(7) via the replacement (8). The Wilson coefficients of the effective operators appearing in H ( BK )light ( q ) are either loop- or CKM-suppressed. As a result, we can limit ourselves to includeonly one-particle hadronic intermediate states. In principle, such operators describe transitionsalso to states with valence charm quarks; however, since we fit the hadronic coefficients η j fromdata, these terms are naturally absorbed in the η j appearing in ∆ Y c ¯ c ( q ). We are thus left onlywith vector resonances containing light valence quarks. Among them, we can further restrictthe attention to the ρ , ω , and φ resonances, since the leptonic decay rates of the heavier statesare very small.There is no clear advantage in using subtracted vs. unsubtracted dispersion relations indescribing the contributions of the light vector resonances. The convergence of the dispersiveintegrals does not pose a problem, and the subtraction at q = 0 is not particularly usefulsince the light-quark contributions are in a non-perturbative regime at q = 0. However, whenfitting data, the subtraction at q = 0 retains the advantage of decoupling the determinationof the spectrum from that of the Wilson coefficient. As default option, we adopt unsubtracteddispersion relations. As discussed in Sect. 2.3.5, checking the stability of the result usingsubtracted vs. unsubtracted dispersion relations for the light vector resonances provides anestimate of the “model error” of the proposed approach.Given these considerations, we decompose Y ( q ) as Y ( q ) = (cid:88) j = ρ,ω,φ η j e iδ j A res j ( q ) , (21)in perfect analogy with the decomposition adopted in Ref. [30] for these light states.7 .3.4 Theoretical constraints on the hadronic parameters The hadronic decompositions in Eqs. (13), (18) and (21) contain 12 free complex parameters: 6in ∆ Y c ¯ c ( q ), 3 in ∆ Y c ¯ c ( q ), and 3 in Y ( q ). In principle, since they correspond to differentfunctional forms, they could all be fitted from data. In practice however, an unconstrained fitwould leave significant degeneracies in the parameter space. It is therefore useful to restrictthe variability of such parameters using theoretical constraints. In the following we discussthree conservative conditions which can be imposed using perturbative arguments. I. Constraint on the slope of ∆ Y c ¯ c ( q ) at q = 0 . The lowest-order perturbative estimate of ∆ Y c ¯ c ( q ) is obtained by factorising the matrix ele-ment (cid:104) K ( p ) | ¯ sγ µ b | B ( p + q ) (cid:105) in Eq. (7) and computing the charm-loop at O ( α s ):∆ Y pert c ¯ c ( q ) = 2 (cid:18) C + 13 C (cid:19) × Q c × q (cid:90) ∞ m c ds (cid:113) − m c s (cid:16) m c s (cid:17) s ( s − q )= 2 (cid:18) C + 13 C (cid:19) (cid:20) h S ( m c , q ) − h P ( m c , q ) (cid:21) . (22)This expression is certainly not a good approximation of ∆ Y c ¯ c ( q ) close to the resonance region;however, it is expected to provide a reasonable approximation at q ≈
0, up to O (Λ QCD /m c )corrections. We can thus use it to set bounds on the slope of ∆ Y c ¯ c ( q ) in the vicinity of q = 0.The perturbative result implies ddq ∆ Y pert c ¯ c ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = 415 (cid:18) C + 13 C (cid:19) m c ≈ (1 . ± . × − GeV − , (23)where the numerical value has been obtained setting m b / < µ < m b and m c = 1 . O (Λ QCD /m c , α s ) corrections (which involvenew hadronic matrix elements) modifies the above prediction to − (0 . ± . × − GeV − .Given these considerations, in the numerical analysis we employ the following constraintsRe (cid:88) j =Ψ(1 S ) ,... η j e iδ j Γ j m j + η ¯ D e iδ j m D + (cid:88) j = D,D ∗ η j e iδ j m j = (1 . ± . × − GeV − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j =Ψ(1 S ) ,... η j e iδ j Γ j m j + η ¯ D e iδ j m D + (cid:88) j = D,D ∗ η j e iδ j m j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ × − GeV − , (24)where we slightly enlarged the error from (23), such that the 1 σ range covers the differencebetween the central value in (23) and the one including O (Λ QCD /m c , α s ) corrections estimatedin Ref. [26]. 8 I. Upper bound on the | η j | in ∆ Y c ¯ c ( q ) . The comparison of the perturbative result with ∆ Y c ¯ c ( q ) also allows us to define the naturalrange for the η ¯ D,D,D ∗ parameters, which are poorly constrained by data. Focusing the attentionon the leading S -wave contribution, it turns out that the perturbative quark loop can besaturated by the DD ∗ meson loop, in the limit m c → m ¯ D , setting η ¯ D = 2( C + C / ≈ (0 . ± . . On general grounds, each of the exclusive meson contributions should be significantlysmaller than the inclusive quark contribution. As a result, in the following we set the upperlimit (cid:12)(cid:12) η ¯ D,D,D ∗ (cid:12)(cid:12) ≤ . . (25) III.
Upper bound on | Y ( q = 0) | . Using an unsubtracted dispersion relation and taking into account only one-particle intermedi-ate states for the light-quark contributions implies Y ( q ) → q , while Y (0) (cid:54) =0. More precisely, one finds a power-like suppression of the type Y ( q ) ∼ Λ QCD /q atlarge q , whereas Y (0) is not parametrically suppressed by any scale ratio. However, sincethe Wilson coefficients entering Y are strongly suppressed, either by loop factors or bysubleading CKM factors, | Y (0) | cannot be too large. Parametrically we expect | Y (0) | < O (1) × max {|C ... | , |C u , |} . (26)Taking the size of the C i into account, we set the conservative bound (cid:12)(cid:12) Y (0) (cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j = ρ,ω,φ η j Γ j m j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . , (27)which should be interpreted as a constraint on the relative phases of the light resonances. Despite not being entirely dictated by first principles, the parameterisation of long-distancehadronic contributions discussed so far contains all the relevant one- and two-particle disconti-nuities of the amplitude, with free coefficients to be fixed by data. It should therefore providea sufficiently general (and unbiased) description of the impact of hadronic contributions onthe B + → K + µ + µ − spectrum. Still, it may be worthwhile to assess whether the proposedparameterisation influences the extraction of information on Y τ ¯ τ ( q ) and, correspondingly, theextraction of a bound on B ( B + → K + τ + τ − ). An estimate of this “model error” can be ob-tained by examining the stability of the obtained bound on B ( B + → K + τ + τ − ) under smallvariations of the model assumptions. The latter include: i) the use of subtracted vs. unsub-tracted dispersion relations for the light resonances; ii) the use of q -dependent widths for bothcharmonium and/or light resonances; iii) strengthening or relaxing the theoretical constraintsin Eqs. (24), (25), and (27). The largest perturbative contribution is the one induced by strange-quark loops, yielding ∆ Y pert s ¯ s ( q ) = C s [ h S ( m s , q ) − h P ( m s , q )], with |C s | = (2 / | C + 4 C + 3 C + C | ≈ . ± . .4 Tau-lepton contribution The contribution from the intermediate τ -leptons can be computed in perturbation theory,yielding Y τ ¯ τ ( q ) = − α em π C τ (cid:20) h S ( m τ , q ) − h P ( m τ , q ) (cid:21) , (28)with the functions h L ( m, s ) defined in Eq. (19). The functional form is identical to the one ofthe perturbative charm contribution and, to a large extent, to the one of the DD ∗ contribution,illustrated in Fig. (2). However, the cusp is located at q = 4 m τ , sufficiently well separatedfrom the various hadronic thresholds.In principle, the short-distance b → sτ + τ − amplitude does not need to be controlled bythe CKM matrix in a generic NP model. However, in most realistic scenarios the weak phasesof all b → sl + l − amplitudes are aligned to the SM one, implying Im( C τ ) = Im( C µ ) = 0. In thefollowing, we adopt this (motivated) simplifying assumption.An estimate of the maximal allowed size of |C τ | can be derived from the experimentalupper bound on B ( B + → K + τ + τ − ) < . × − at 90% CL by Babar [19], which is morethan four orders of magnitude larger than B ( B + → K + τ + τ − ) SM ≈ . × − [34]. Neglectingthe contributions from operators other than O τ and O τ , we find B ( B + → K + τ + τ − ) ≈ (cid:40) . × − × |C τ | C τ = C τ , . × − × |C τ | C τ = 0 . (29)In the case C τ = C τ ( C τ = 0) the Babar result then implies |C τ | ≤ . × (9 . × ), to becompared to C τ, SM9 ≈ .
2. As we discuss below, saturating this bound leads to a pronouncedditau cusp in the spectrum (see Figure 3), opening the possibility of extracting a more stringentbound on B ( B + → K + τ + τ − ) from a precise measurement of the B + → K + µ + µ − dimuonspectrum. In order to assess the sensitivity to the branching ratio B ( B + → K + τ + τ − ) at the LHCbexperiment, we generate pseudo-experiments corresponding to the signal yields obtained inRef. [30] and scaled to the full run II dataset, taking into account the collected luminosityand b -hadron cross-section increase at 13 TeV [32]. This leads to around 40,000 non-resonant B + → K + µ + µ − candidates. As the efficiency is reasonably flat as a function of dimuon massand the background level is very low, we neglect these effects. Fig. 3 shows the fit modelwith a dataset generated at the expected yield. This illustrates the visible sensitivity to ahypothetical signal component generated according to the current experimental limit [19].The size and phase of the one-particle resonant contributions are determined from thebranching fractions reported in Ref. [32], which are used to determine the initial values of η j and δ j for the data to be generated. Due to the complicated experimental resolutioneffects near the J/ψ and ψ (2 S ) resonances, the regions 9 . < q < . /c and 13 . < igure 3: Example pseudodata expected from the full run II dataset collected by the LHCb exper-iment assuming the SM. The distribution expected if the B + → K + τ + τ − branching fraction werepresent at the current experimental limit of 2 . × − is overlaid. q < .
95 GeV /c are excluded from the fit and the phase differences associated with theseresonances are constrained to the uncertainties in Ref. [30]. Outside of this region, finite-resolution effects in q are ignored as all the components are broad. In order to mimic thesensitivity one would have when fitting the data, Gaussian constraints are applied to the J/ψ and ψ (2 S ) resonant parameters according to the uncertainties reported in Ref. [32].The component which most closely resembles the signal is the contribution from two-particle hadronic intermediate states. We allow the phases and magnitudes of these statesto vary in the fit. As the shape of the ˆ ρ DD and ˆ ρ D ∗ D ∗ spectral densities are very similar, wecombine them with an equal contribution to avoid large correlations in the fit. The correlationcoefficient between the two-particle contribution and the signal is around 0 .
6, reflecting theirsimilar shapes.The form factor uncertainties are taken from Ref. [23] and are implemented in the fit as amultivariate Gaussian constraint. The data slightly helps constrain the form factor parameters,but this affects the sensitivity on C τ only in a mild way.As discussed above, a possible B + → K + τ + τ − → K + µ + µ − re-scattering leads to twofeatures in the B + → K + µ + µ − dimuon spectrum. One is the cusp in-between the J/ψ and ψ (2 S ) resonances, the other is a distortion in the shape of the non-resonant B + → K + µ + µ − component. The cusp is not the most sensitive signature of the ditau signal, due to itsrelatively small contribution. Instead, it is the shape of the non-resonant part which generatesthe largest sensitivity. Consequently, neglecting the resolution is justified and the shape of thecharmonium contribution is important in constraining a possible B + → K + τ + τ − signal.11 cenario C τ (90% CL) B ( C τ = − C τ ) B ( C τ = 0)Run I-II dataset 533 2 . × − . × − Run I-V dataset 139 1 . × − . × − Run I-II dataset, improved form factors 533 2 . × − . × − Run I-V dataset, improved form factors 127 1 . × − . × − Table 1:
Sensitivity to C ττ according to various LHCb scenarios. The expected sensitivity on the C τ contribution is determined using the CL s method [31].The sensitivity with the current dataset is reported in Table 1, along with two other potentialfuture scenarios corresponding to the LHCb upgrade-II luminosity and a hypothetical improve-ment of the form factor uncertainties by a factor of three. The estimated sensitivity utilisingthe run I-II datset corresponds to a limit on the B + → K + τ + τ − branching ratio which isslightly more stringent than the current constraints placed by the BaBar collaboration and isexpected to compete with the projected sensitivity of the Belle-II experiment when more datais collected. If the branching ratio B ( B + → K + τ + τ − ) were significantly enhanced over its SM value, itwould induce a peculiar distortion of the B + → K + µ + µ − spectrum, characterised by a cuspat q = 4 m τ and by a distortion of the dimuon distribution. In this work we have proposed amethod that uses this effect as a tool to extract a bound on B ( B + → K + τ + τ − ) from futureprecise measurements of dΓ( B + → K + µ + µ − ) / d q .A necessary ingredient to achieve this goal is a reliable description of the B + → K + µ + µ − dimuon spectrum, within the SM, in the full kinematical range, especially in the region of thenarrow charmonium states. As we have shown, this can be obtained by means of a data-drivenapproach which takes full advantage of the known analytic properties of the decay amplitude,supplemented by robust theoretical constraints. Our approach differs from previous attemptsof including non-local hadronic contributions to the B + → K + µ + µ − decay amplitude bythree main points: i) the use of dispersion relations subtracted at q = 0 for the charmoniumstates; ii) the inclusion of two-particle thresholds; iii) the use of short-distance constraintsat low q to reduce the number of free parameters. In this way one separates the problemof the normalisation of the B + → K + µ + µ − rate, and the corresponding extraction of short-distance Wilson coefficients, from the problem of obtaining a reliable description of the dimuonspectrum. While within our approach there is no significant progress on the first problem, thereis a tangible advantage on the second one. The parameterisation of the amplitude we proposeis flexible enough to allow the extraction of all the relevant parameters characterising hadronicthresholds in the dimuon spectrum from data, while retaining significant predictive power inthe smooth region within and below the two narrow charmonium resonances. This fact is thekey property which allows us to set useful constraints on the B + → K + τ + τ − → K + µ + µ − B + → K + µ + µ − ) / d q .The method we have proposed is particularly well suited for the LHCb experiment, whichhas already collected a large sample of B + → K + µ + µ − events and has an excellent resolutionin the dimuon spectrum [30]. As we have shown, the data already collected in run II shouldallow to set a bound on B ( B → Kτ + τ − ) of O (10 − ), competitive with current direct bounds(see Table 1). Bounds of O (10 − ) could be obtained with the LHCb upgrade-II luminosity. Acknowledgments
This project has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme under grant agreement833280 (FLAY), and by the Swiss National Science Foundation (SNF) under contract 200021-159720.
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