Eur. Phys. J. C (2017) 77:701DOI 10.1140/epjc/s10052-017-5272-0

Regular Article - Theoretical Physics

Lepton polarization asymmetries in rare semi-tauonic b → sexclusive decays at FCC-ee

J. F. Kamenik1,2, S. Monteil3, A. Semkiv3,4, L. Vale Silva1,a

1 Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia2 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia3 Université Clermont Auvergne, CNRS/IN2P3, LPC, 63000 Clermont-Ferrand, France4 Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine

Received: 8 August 2017 / Accepted: 1 October 2017 / Published online: 24 October 2017© The Author(s) 2017. This article is an open access publication

Abstract We consider measurements of exclusive raresemi-tauonic b-hadron decays, mediated by the b → sτ+τ−transition, at a future high-energy circular electron–positroncollider (FCC-ee). We argue that the high boosts of b-hadronsoriginating from on-shell Z boson decays allow for a fullreconstruction of the decay kinematics in hadronic τ decaymodes (up to discrete ambiguities). This, together with thepotentially large statistics of Z → bb, opens the door forthe experimental determination of τ polarizations in theserare b-hadron decays. In the light of the current experimen-tal situation on lepton flavor universality in rare semilep-tonic B decays, we discuss the complementary short-distancephysics information carried by the τ polarizations and sug-gest suitable theoretically clean observables in the form ofsingle- and double-τ polarization asymmetries.

1 Introduction

Rare (semi)leptonic b-hadron decays allow for some of themost sensitive tests of the standard model (SM) descriptionof flavor. Consequently they constitute powerful probes ofpossible flavor dynamics beyond the SM. In recent yearsthese processes have attracted a lot of attention, in partdue to several experimental measurements defying theo-retical expectations within the SM. Deviations are foundin B → K (∗)μ+μ− [1] (cf. [2]) and Bs → φμ+μ−[3] branching ratios, as well as in the angular analysis ofB → K ∗μ+μ− decays at large K ∗ recoil energies [4–7],see also [8]. While not conclusive at present, these resultsmay constitute first hints of new physics (NP). Interestinglyenough, an anomaly is also found in the measurements ofthe ratios R(∗)

K = B(B → K (∗)μ+μ−)/B(B → K (∗)e+e−)

a e-mail: luiz.vale@th.u-psud.fr

[9,10], thus suggesting violations of lepton flavor universal-ity (LFU). Such effects can only arise from physics beyondthe SM. Unexpected phenomena are currently also observedin charged current mediated semileptonic B meson decaysinvolving τ leptons in the final state: the measurements ofRD(∗) = B(B → D(∗)τ−ν)/B(B → D(∗)�−ν) ratios,where � = e, μ [11,12] (cf. [13]), exhibit tensions with thecorresponding very precise SM predictions [14–19], there-fore again pointing towards possible violations of LFU andthus physics beyond the SM.

On the other hand, the rare semi-tauonic decay processb → sτ+τ− has not been observed so far. The presentupper bound on the branching ratio B(B+ → K+τ+τ−) <

O(10−3) [20] is expected to be improved by one or two ordersof magnitude in the coming decade by the Belle II experi-ment. Unfortunately, this is still far above the SM-predictedrates of O(10−7) [21] (and Sect. 3.4 below). Consequently,possible NP effects in rare semi-tauonic B meson decaysare poorly constrained at present, cf. Refs. [22–24], and thesituation is not expected to improve much in the near future.

We want to argue, however, that a new generation of high-energy particle collider experiments could allow to constrainthe relevant (sb) (τ τ ) operators much better. The NP case forthe next generation collider facilities is usually built aroundthe SM electroweak hierarchy problem, particle dark matterand heavy neutrinos (see e.g. [25–33]). However, the highinvolved costs and risks motivate an exhaustive set of appli-cations. Indeed the worst-case scenario, where no new res-onances are seen at the highest available collider energies,could well materialize. In that case, the capacity of flavorprocesses in general, and flavor changing neutral currents inparticular, to unveil short-distance physics at very high scales,potentially much above direct collider reach, would becomeinvaluable. In the present analysis we focus on the potential-ities of a Future Circular Collider, and more particularly its

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electron–positron collider phase (FCC-ee) [28], previouslyknown as TLEP [34]. The high luminosity of the FCC-eemachine complemented by an excellent vertexing system ofthe FCC-ee detectors under consideration would allow forunique b-hadron rare decay studies, for instance those involv-ing final-state tau leptons. Moreover, decays with tau lep-tons in the final state open up novel NP search opportunities.Contrary to light leptons, polarizations of final-state taus canin principle be reconstructed through their hadronic decaykinematics. This in turn allows one to construct a new classof observables, with complementary sensitivities to short-distance physics, as it will become clear later in the text.

The τ lepton polarization observables have already beendiscussed in the past [35–43] (see also [44–48] for otherfinal states). In the context of semileptonic charged currents,τ polarization observables have been considered in detailsin, e.g., [49], and [50] discusses experimental prospects forBelle II. Here we present a proof of principle study of theviability of the complete B → K (∗)τ+τ− reconstruction atthe FCC-ee (in Sect. 2). We then focus on the precision withwhich polarization asymmetries can be predicted within theSM as well as their impact on NP directions singled out bythe current experimental anomalies in rare semi-muonic Bdecays (in Sect. 3). Our main conclusions are summarized inSect. 4. Some of the lengthier and more technical derivationsand expressions are relegated to the appendices.

2 The B0 → K∗0(892)τ+τ− experimentalreconstruction and sensitivity at high-luminosityZ-factory

A possible long-term strategy for high-energy physics at col-liders, after the exploitation of the LHC and its High Lumi-nosity upgrade, considers a tunnel in the Geneva area ofabout 100 km circumference, which takes advantage of thepresent CERN accelerator complex. The future circular col-lider (FCC) concept builds upon the successful experienceand outcomes of the LEP-LHC machines. Therefore, a pos-sible first step of the project is to fit in the tunnel a high-luminosity e+e− collider aimed at studying comprehensivelythe electroweak scale with center-of-mass energies rangingfrom the Z pole up to beyond the t t production threshold [34].A 100 TeV proton–proton collider is then considered as theultimate goal of the project. Let us mention that an electronproton collider is also considered as an option of this project.

The goal of the high-luminosity e+e− collider is to pro-vide collisions in the beam energy range of 40–175 GeV. Thiswould allow to study with unprecedented precision the fourelectroweak energy thresholds: 91 GeV (Z -pole), 160 GeV(W -pair production), 240 GeV (Higgs production in associ-ation with a Z -boson) and 350 GeV (t t-pair production). Inparticular, the circulation of about 10,000 bunches for oper-

[GeV]s210 310

]-1 s

-2 c

m34

Lum

inos

ity [1

0

1

10

210

FCC-ee (Baseline, 2 IPs)

ILC (Baseline)

ILC (Lumi/Energy Upgrade)

CLIC (Baseline)

-1s-2 cm36 10×Z (91.2 GeV) : 2.8

-1s-2 cm35 10× (161 GeV): 3.8 -W+W

-1s-2 cm35 10×HZ (240 GeV) : 1.0

-1s-2 cm34 10× (350 GeV) : 2.6 tt

-1s-2 cm34 10×HZ : 1.5

-1s-2 cm34 10× : 1.0 tt

-1s-2 cm34 10×500 GeV : 1.8

Fig. 1 Comparison of e+e− collider luminosities

ation at the Z -pole allows one to envision the productionof O(1012−13) Z decays. Figure 1 gathers the luminosityprofiles of several e+e− collider projects and supports theabove-mentioned event yields at the Z -pole.

The decay B0 → K ∗0(892)τ+τ− is characterised by atleast two neutrinos in the final state. Their presence makes thecorresponding experimental search very challenging. How-ever, an excellent knowledge of the decay vertices, which canbe obtained thanks to the multibody hadronic τ decays, mayhelp to fully solve the kinematics of these decays. Namely,the reconstruction of the primary vertex and the decay vertexof the B0 are defining the direction of the B0 meson, fix-ing two degrees of freedom. The reconstruction of the two τ

leptons’ decay vertices are providing four further constraints(not all independent though). Eventually, the knowledge ofthe mass of the τ lepton closes the system, up to a quadraticambiguity. FCC-ee experiments are defining unique featuresto perform this kinematical fit: the clean leptonic machineenvironment allowing to place the vertex detector as close as2 cm from the interaction point and the boost experienced bythe B0-meson at the Z -pole.

We have studied the B0 → K ∗0(892)τ+τ− decay in thecontext of the FCC-ee machine by means of Monte Carlosimulated events (signal and background) generated with afast simulation featuring a parametric detector. The detec-tor performance considered in the study is inspired by thatobtained for the ILD vertex detector [51]. For the sake ofsimplicity we have chosen to consider the τ decays into threecharged pions (mostly proceeding through the two-body pro-cess τ− → a−

1 ντ and its charge conjugate).Figure 2 displays the reconstructed invariant mass distri-

bution of signal and background events, simulated accord-ing to the branching fractions predicted in the SM, andcorresponding to 1013 Z -boson decays. About a thousandof events, cleanly reconstructed, can be expected, openingthe way to measurements of the angular properties of thedecay. To our knowledge, these FCC-ee performances areunequaled at any current or foreseeable experiment. The

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2c, GeV/0dB

m4.5 5 5.5 6 6.5

)2

cE

vent

s / (

0.02

GeV

/

0

20

40

60

80

100

120eeFCC-

6.565.554.55<4<3<2<1<012345

Fig. 2 Invariant mass reconstruction of B0 → K ∗0(892)τ+τ− candi-dates. The τ particles are decaying into three prongs τ− → π−π+π−ντ

allowing the τ decay vertex to be reconstructed. The primary vertex (Zvertex) is reconstructed from primary tracks and the secondary ver-tex (B0 vertex) is reconstructed thanks to the K ∗(892) daughter par-ticles (K ∗(892) → K+π−). Two dominant sources of backgroundsare included in the analyzed sample, namely Bs → D+

s D−s K ∗0(892)

and B0 → D+s K ∗0(892)τ−ντ . They are modeled by the red and pink

probability density functions (p.d.f.), respectively. The signal p.d.f. isdisplayed in green

reconstruction of the τ decays into three charged pions andan additional π0 can be also used in the partial reconstruc-tion of the decay, doubling the expected signal yields. Thefully charged 5-body decays are relevant as well for the par-tial reconstruction technique, providing an additional 20%statistics to the baseline study.

3 Tau polarization observables in rare semi-tauonicB(s) decays: SM expectations and NP sensitivity

3.1 Effective Hamiltonian

In the SM, the effective weak Hamiltonian relevant for thequark-level b → sτ+τ− transitions at scales μ � μW ∼O(MW ,mt ) is

HeffSM = −4

GF√2VtbV

∗ts

(C1(μ) Oc

1(μ) + C2(μ) Oc2(μ)

+8∑

i=3

Ci (μ) Oi (μ) + Cττ9 (μ) O9(μ)

+ Cττ10 (μ) O10(μ)

)+ h.c. . (1)

In order to simplify the discussion in Sect. 3.4 we neglectdoubly Cabibbo-suppressed contributions proportional toVubV

∗us ∼ O(λ2)VtbV

∗ts . We use the standard operator basis

Oc1 = (sLγμT

acL)(cLγ μT abL),

Oc2 = (sLγμcL)(cLγ μbL),

O3 = (sLγμbL)∑

(qγ μq),

O4 = (sLγμTabL)

∑(qγ μT aq),

O5 = (sLγμ1γμ2γμ3bL)∑

(qγ μ1γ μ2γ μ3q),

O6 = (sLγμ1γμ2γμ3TabL)

∑(qγ μ1γ μ2γ μ3T aq),

O7 = e

16π2 mb(sLσμνbR)Fμν,

O8 = gs16π2 mb(sLσμνT abR)Ga

μν,

O9 = e2

16π2 (sLγ μbL)τ γμτ,

O10 = e2

16π2 (sLγ μbL)τ γμγ5τ, (2)

where the sums run over all light quark flavors q, and mb isthe MS bottom quark mass. In the following, we will dropthe superscript “ττ” of the Wilson coefficients.

The matching of the full SM theory at the EW scale (μW )onto the effective Hamiltonian of Eq. (1) is discussed inRef. [52]. For completeness, the numerical values of the Wil-son coefficients calculated up to the NNLO order in αs(μW )

and RGE evolved to the b-hadron mass scale μb = 4.8 GeVare listed in Table 1. We use mb(mb) = 4.2 GeV, andαs(MZ ) = 0.1184 (and αs(μb) = 0.216).

Perturbative matrix element corrections from the four-quark operators O1 − O6 and the magnetic dipole O8 at thescale μb are absorbed into the effective Wilson coefficientsCeff

7 ,Ceff8 ,Ceff

9 given below (cf. e.g. [59])

Ceff7 ≡ C7 − 1

3

[C3 + 4

3C4 + 20C5 + 80

3C6

]

+ αs

4π

[(C1 − 6C2)A(q2) − C8F

(7)8 (q2)

], (3)

Ceff8 ≡ C8 + C3 − 1

6C4 + 20C5 − 10

3C6, (4)

Ceff9 ≡ C9 + Y (q2) + 8

m2c

q2

[4

9C1 + 1

3C2 + 2C3 + 20C5

]

+ αs

4π

[C1

(B(q2)+4C(q2)

)

−3C2

(2B(q2)−C(q2)

)− C8F

(9)8 (q2)

], (5)

together with Ceff10 ≡ C10, where Ceff

8 is given for complete-ness and will not be relevant in our discussion. The explicitdependence of the (effective) Wilson coefficients and αs onthe scale μb have been suppressed, and only the first power

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701 Page 4 of 19 Eur. Phys. J. C (2017) 77 :701

Table 1 SM Wilson coefficients in the MS (NDR) scheme at the scale μb = 4.8 GeV, Refs. [53–58]. Below C7 ≡ C7− 13 (C3+ 4

3C4+20C5+ 803 C6)

C1 C2 C3 C4 C5 C6 C7 Ceff8 C9 C10

−0.2632 1.0111 −0.0055 −0.0806 0.0004 0.0009 −0.2923 −0.1663 4.0749 −4.3085

in m2c/q

2 is kept (where mc is the MS charm-quark mass,mc(mc) = 1.27 GeV), consistently with [60], where thecharm is treated as massless. In Eq. (5), Y (q2) reads

Y (q2) = h(q2)

[4

3C1+C2+11

2C3−2

3C4+52C5−32

3C6

]

− 1

2h(mb, q

2)

[7C3 + 4

3C4 + 76C5 + 64

3C6

]

+ 4

3

[C3 + 16

3C5 + 16

9C6

], (6)

and the functions A, B,C (or equivalently F (7)1,u , F

(7)2,u , F

(9)1,u ,

F (9)2,u ) are found in [60], while F (7)

8 , F (9)8 are found in [61].

Finally, the charm-loop function entering the expression ofY (q2) is

h(q2) = 8

27+ 4

9

(log

μ2

q2 + iπ

)(7)

and

h(mb, q2) = 4

9

(log

μ2

m2b

+ 2

3+ z

)

−4

9(2 + z)

√z − 1 arctan

1√z − 1

, z = 4m2b

q2 . (8)

In the presence of NP as hinted to by the present b →sμ+μ− measurements, the effective Wilson coefficients canreceive corrections of the form

Ceffi → Ceff

i + δCi , i = 9, 10. (9)

In addition, new operators can also be induced, contributingto the processes at hand at the tree level. Here we are includingthe chirally flipped operators

O ′9=

e2

16π2 (sRγ μbR)τ γμτ , O ′10=

e2

16π2 (sRγ μbR)τ γμγ5τ,

(10)

while additional scalar and tensorial operators, not favored bythe anomalies in muon data, will not be considered. We alsodo not consider a shift in δC7 nor the operator O ′

7, which areLFU-conserving (see, e.g., [62] for constraints on δC7,C ′

7).Note that in the SM the operator O ′

7 is present with a sup-pression factor ms/mb, giving a (higher-order) contribution

that we neglect. Of course, factors of MK (∗)/MB or Mφ/MBsare kept throughout our analysis. In summary, we considerthe following effective Hamiltonian at the scale μ:

Heff = HeffSM − 4

GF√2VtbV

∗ts

×[δC9(μ) O9(μ) + δC10(μ) O10(μ)

+C ′9(μ) O ′

9(μ) + C ′10(μ) O ′

10(μ)

]+ h.c. (11)

In the present analysis, we do not discuss possible UV com-pletions of Heff but instead refer the interested reader to theexisting literature on the subject [24,63–65].

3.2 τ polarization observables

Next we introduce the (pseudo-)observables characterizingthe polarizations of on-shell-produced τ± leptons [35,36].The decay kinematics of B(s) → Mτ+τ−, M = K (∗), φ, isdepicted in Fig. 3. Note that in the following we consider sta-ble final-state leptons and the M meson, which is equivalentto working in the narrow width approximation.

The longitudinal, transverse and normal polarizations ofthe τ± can be probed using (single) polarization asymmetriesP±

A , for A = L , T, N . They can be defined relative to a basisof space-like directions onto which we project the spinors ofthe τ±. For definiteness, we focus first on the τ− where wechoose

s−L =

(0,

−→p −|−→p −|

), s−

N =(

0,

−→p × −→p −|−→p × −→p −|

),

s−T =

(0,

−→p × −→p −|−→p × −→p −| ×

−→p −|−→p −|

), (12)

with s−L , s−

T , s−N given in the rest-frame of the τ− while−→p −,

−→p are the three-momenta of, respectively, the τ− lep-ton and the final-state M meson, defined in the rest-frame ofthe τ−τ+ pair. The s+

A directions (in the τ+ rest-frame) canbe defined analogously with the replacement −→p − → −→p +:

s+L =

(0,

−→p +|−→p +|

), s+

N =(

0,

−→p × −→p +|−→p × −→p +|

),

s+T =

(0,

−→p × −→p +|−→p × −→p +| ×

−→p +|−→p +|

). (13)

Orthonormality of sA can now be used to define polarized τ±spinors as u(±e−

A ) = u · P−A (±) and v(±e+

A ) = P+A (±) · v,

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Eur. Phys. J. C (2017) 77 :701 Page 5 of 19 701

R

B(s) K(∗ ), φ

τ−

zτ

xτ

τ+

zτ

xτ

yτ

Rχ

θτ

Fig. 3 Relevant reference frames. The four-momentum of the B(s)meson is k, while the one of the final-state meson is p. The four-momenta of the τ− and τ+ are p− and p+, respectively, and q2 =(p− + p+)2 is the dilepton invariant mass. The Rττ ,R′

ττ referenceframes correspond to a boost (indicated by “

∫ ∫”) from the B(s) meson

rest-frame towards the reference frame of the τ+τ− pair, and the refer-ence frame Rττ is determined from the rotation of R′

ττ (with x ′τ , y

′τ , z

′τ

constituting a right-handed frame). The longitudinal polarization of theτ− lepton is defined along the z′τ axis, while the transverse (or normal)polarization is parallel (respectively, orthogonal) to the plan x ′

τ Oz′τ . Asimilar construction applies to the τ+ lepton. The angle θτ describesthe orientation of the negatively charged lepton. For M = K ∗, φ, theirsubsequent decay products define a plane, also indicated, with relativeorientation given by χ

where PaA(±) ≡ (14 ± γ5/saA)/2 such that Pa

A(±) · PaA(±) =

PaA(±) and Pa

A(±) · PaA(∓) = 0 [66]. The τ± polarization

asymmetries (or simply polarizations) are then defined as

P∓A (q2) =

[d�dq2 (e∓

A , e±A ) + d�

dq2 (e∓A ,−e±

A )]

−[d�dq2 (−e∓

A , e±A ) + d�

dq2 (−e∓A ,−e±

A )]

d�dq2 (e∓

A , e±A ) + d�

dq2 (e∓A ,−e±

A ) + d�dq2 (−e∓

A , e±A ) + d�

dq2 (−e∓A ,−e±

A ), (14)

for A = L , T, N , after integration over the angles θτ and χτ .In this definition, the indices ±e−

A (±e+A ) indicate the states

that are left invariant by the projector P−A (±) (P+

A (±)). Inthe massless limit mτ → 0+ in the SM the longitudinalasymmetry reduces to the left–right asymmetry, while thetransverse and normal asymmetries vanish. Also note thatP±

Aonly require the knowledge of one of the two polarizations.One can also consider polarization asymmetries defined forboth τ± polarization states, as follows:

PAB(q2) =[d�dq2 (e−

A , e+B ) − d�

dq2 (−e−A , e+

B )]

−[d�dq2 (e−

A ,−e+B ) − d�

dq2 (−e−A ,−e+

B )]

d�dq2 (e−

A , e+B ) + d�

dq2 (e−A ,−e+

B ) + d�dq2 (−e−

A , e+B ) + d�

dq2 (−e−A ,−e+

B ), (15)

for A, B = L , T, N . The complete analytic expressions forthe polarization observables are given in Appendix B.

The first application of the tau polarization asymmetriesdefined above concerns the ratios of the longitudinal andtransverse asymmetries. It turns out that all dependence onthe Wilson coefficients cancels out in these quantities. Forthe decay B → K τ+τ− we have

P±L (K )(q2)

P±T (K )(q2)

= f+(q2)

f0(q2)

4√q2 − 4m2

τ

√λ(M2

B, M2K , q2)

3π mτ (M2B − M2

K ),

(16)

where λ(a, b, c) ≡ a2 + b2 + c2 − 2 (a b + b c + a c). Notethat this expression does not depend on the Wilson coeffi-cients, but only on the ratio of hadronic form factors (param-eterizing local matrix elements of operators entering Heff

as defined in Appendix A) f+(q2)/ f0(q2), therefore being aclean (differential) probe of the relevant form factor determi-nations. Importantly, this remains true even in the presenceof possible NP effects of the form δC9,10,C ′

9,10 (and alsoδC7,C ′

7, but not necessarily in the presence of scalar and

tensor semileptonic four fermion operator effects). To obtainsimilar observables in B → K ∗τ+τ− or Bs → φτ+τ−decays, we need to project to the longitudinal polarization ofthe K ∗ or φ. Then we can define

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701 Page 6 of 19 Eur. Phys. J. C (2017) 77 :701

P±L (VL)(q2)

P±T (VL)(q2)

=[(MB(s) + MV )2 (M2

B(s)− M2

V − q2)A1(q2)

A0(q2)

− λ(M2B(s)

, M2V , q2)

A2(q2)

A0(q2)

]

× 2√q2 − 4m2

τ

3π MV mτ (MB(s) + MV )√

λ(M2B(s)

, M2V , q2)

,

V = K ∗, φ, (17)

which is again independent of the relevant Wilson coeffi-cients and thus defines a clean unbinned experimental con-straint on the relevant form factors (or possibly indicates thepresence of scalar or tensor semileptonic four fermion oper-ators).

3.3 Sources of uncertainty

In order to derive quantitative predictions for the observablesin rare exclusive semi-tauonic B(s) decays, we need to evalu-ate the corresponding exclusive hadronic matrix elements ofoperators in Heff . Generically, the relevant decay amplitudescan be written as a sum over products of the so-called helic-ity amplitudes Hλ

F (u �F v)(λ), F = V, A, P, (S, T, T 5),

times a combination of Wilson coefficients and kinematicalfactors. Here, λ is the helicity of the K (∗), φ, or the lep-tonic pair, and �V , �A, �P denote, respectively, the Lorentzstructures γ μ, γ μγ5, γ5 (or 14, σμν, σμν γ5 for �S, �T , �T 5,respectively, not considered here), where the bilinear u γ5 v

is derived from qμ · u γμγ5 v.An important source of theoretical uncertainty in rare

semileptonic b-decays is the so-called charm-loop effectsappearing when the (s�1b)(c�2c) operators in Heff are con-tracted with the EM current jμe.m.. Since the physical phasespace of B(s) → Mτ+τ−, M = K (∗), φ, decays is restrictedto the small hadronic recoil region, one can use the corre-sponding hard scale

√q2 EM to control the size of such

effects in a perturbative expansion in powers of �QCD/mb

(∼ ms/mb) and αs (a power of the latter being integratedin Eqs. (3) and (5)) [67,68]. Of course, such a pertur-bative approach cannot capture the long-distance hadronicdynamics at the origin of the broad cc resonances such asψ(3770), ψ(4040), ψ(4160), ψ(4415), standing above thesharp ψ(2S) peak. To take into account �QCD/mb powercorrections, and corrections stemming from intermediate ccrescattering effects, we adopt a simplified treatment and con-sider an uncertainty of 10 % × Ceff

9 at the level of the HλV

helicity amplitude introduced above. We also allow for arbi-trary relative strong phase differences of this correction wheninterfering with the local matrix elements of operators inHeff

(see also Refs. [59,69,70]). In the following we refer to thisset of corrections as “Charm” (though more generally alsoweak annihilation topologies, for instance, are addressed inthe OPE). Regarding long-distance hadronic dynamics, goodresults are expected for inclusive observables, i.e., integratedover the full physical q2 range [68]. We furthermore cross-check our simplified treatment of “Charm” long-distanceeffects in q2 binned observables by employing a model forthe charmonium and open-charm resonances [36] that is dif-ferential in q2. Though the model assumes the factorizationof resonant effects, known to be inaccurate [71], the result-ing uncertainty in the two approaches is similar in size. Inour numerical analysis we highlight the cases where the sim-plified treatment of “Charm” leads to uncertainty estimatessubstantially different from the resonance model.

Another important source of uncertainties is the knowl-edge of the relevant hadronic form factors. The completeexpressions for their parameterizations can be found inAppendix A. For the B → K (∗) form factor extractionused here [72] statistical lattice ensemble uncertainties dom-inate, while for Bs → φ [73] they are an important compo-nent. Obviously, correlations between individual parametersentering form factor parameterizations have to be taken intoaccount (see Appendix A for details).

In the next section we give numerical values for the the-oretical uncertainties on asymmetries and branching ratioscoming from the form factors, “Charm” uncertainties, and a2% uncertainty on the Wilson coefficients in Table 1, meantto include parametric uncertainties such as the one from αs , aswell as perturbative higher-order effects, similarly to [74].1

Finally, we note that S-wave pollution is not expected to bea problem at large dilepton momenta [76].

3.4 SM predictions

In this section we give the SM predictions for the consideredobservables and discuss their theoretical uncertainties. Someof the numerical inputs used in the calculation are given by[77,78]2

|Vtb| = 0.999119 (+24, −12), |Vts | = 0.04108 (+30, −57),

MB = 5.28 GeV, MBs = 5.366 GeV, mτ = 1.777 GeV,

τB = 1.520 × 10−12 s, τBs = 1.510 × 10−12 s,

MK = 0.50 GeV, MK ∗ = 0.892 GeV, Mφ = 1.020 GeV, (18)

1 QED corrections to inclusive b → s�+�− decays have been evaluatedin Ref. [57]. We note, however, that logarithmically enhanced QEDcorrections [75] proportional to log(m2

b/m2�) are not expected to have

the same impact in the case of taus as they have for light leptons.2 Though the extraction of the CKM matrix elements is made under thehypothesis of LFU, the observables involved in such extraction do notinvolve the LFUV considered here.

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Eur. Phys. J. C (2017) 77 :701 Page 7 of 19 701

Table 2 SM predictions for B− → K−τ+τ− observables. The firstuncertainties correspond to the quadratic sum of the effects of thevariations of the form factor uncertainties over their quoted intervals(after diagonalization to take into account the quoted correlations). The“Charm” uncertainties correspond to corrections of 10 %×Ceff

9 times an

arbitraryCP-even phase ei θ , which is allowed to vary freely. Finally, thecolumn labeled “WC” corresponds to the effect of variating the Wilsoncoefficients given in Table 1 by 2 %, and include for the branching ratiothe effect of the uncertainty on the value of |Vts |. The asymmetries AFBand P±

N vanish in this case. All uncertainty intervals are symmetrized

Asyms., BR CV FF “Charm” WC

〈P−L (K )〉 − 0.246 ± 0.004 ± 0.006 ± 0.002

〈P+L (K )〉 + 0.246 ± 0.004 ± 0.006 ± 0.002

〈P−T (K )〉 − 0.75 < 0.001 ± 0.02 ± 0.006

〈P+T (K )〉 + 0.75 < 0.001 ± 0.02 ± 0.006

〈P−L (K )〉/〈P−

T (K )〉 + 0.33 0.005 ± 0.01 < 0.001

〈PLL (K )〉 + 0.30 ± 0.01 ± 0.06 ± 0.02

〈PT T (K )〉 − 0.68 ± 0.005 ± 0.02 ± 0.007

〈PNN (K )〉 − 0.20 ± 0.01 ± 0.09 ± 0.04

〈PLT (K )〉 − 0.33 < 0.001 ± 0.03 ± 0.01

〈PT L (K )〉 − 0.33 < 0.001 ± 0.03 ± 0.01

〈PLN (K )〉 + 0.11 < 0.001 ± 0.07 ± 0.002

〈PNL (K )〉 + 0.11 < 0.001 ± 0.07 ± 0.002

〈PT N (K )〉 − 0.03 < 0.001 ± 0.03 < 0.001

〈PNT (K )〉 − 0.03 < 0.001 ± 0.03 < 0.001

〈BR(K )〉 × 107 1.61 ± 0.07 ± 0.12 ± 0.06 ± 0.03|Vts |

while the value of the hyperfine constant is αEM (μb) =1/133. The difference in the masses of the charged and theneutral B and K mesons can be neglected, and then we donot generally differentiate their charge or flavor.

We define the binned total rate, polarization observables,Eqs. (14)–(15), and the FB asymmetry, Eq. (20) below, viaq2 integration of the corresponding differential partial rates.For instance,

〈P−A 〉 = 1

〈�〉∫ q2

max

q2min

dq2{ [

d�

dq2 (e−A , e+

A ) + d�

dq2 (e−A ,−e+

A )

]

−[d�

dq2 (−e−A , e+

A ) + d�

dq2 (−e−A ,−e+

A )

]},

A = L , T, N , (19)

where 〈�〉 ≡ ∫ q2max

q2min

dq2d�/dq2. Similarly, 〈BR〉 = 〈�〉τB(s) .

Here we consider a single bin within q2min = 14.18 GeV2,

in order to avoid the sharp ψ(2S) resonance at√q2 =

3.686 GeV, and the zero hadronic recoil kinematical end-point q2

max = (MB−MK (∗) )2 or (MBs −Mφ)2. Therefore, therange [q2

min, q2max] includes the broad resonances ψ(3770),

ψ(4040), ψ(4160), and possibly ψ(4415). As discussed inRef. [68], the integration over q2 damps resonant effects upto some extent so that they are captured by the effective“Charm” contribution introduced in the previous Section.A further reason to discuss only the binned observables isthat, in any case we expect at first to measure the integrated

polarization asymmetries, prior to the measurement of theirdifferential distributions.

Our numerical SM predictions are given in Tables 2, 3and 4. We note that the central value of the B → K τ+τ−branching ratio is in broad agreement (12% difference) withRef. [21]. The difference may be at least partly traced backto different sets of form factors and is roughly consistentwith stated form factor uncertainties. We note that the SMvalues of polarization observables exhibit a distinct hierarchywith 〈P±

T (K )〉, 〈PT T (K )〉 (or 〈P±L (V )〉, 〈P−

T (V )〉) being thelargest asymmetries in the case of B → K τ+τ− (respec-tively, B → K ∗τ+τ− and Bs → φτ+τ−) and also the clean-est ones. For all decay channels, the uncertainties are domi-nated by “Charm” effects. Moreover, the form factor uncer-tainties are highly suppressed in the case of B → K τ+τ−.We also point out that there are a few numerical coincidencesin these tables that are purely due to the choice of q2

min,for instance |〈P±

L (V )〉| |〈P−T (V )〉|, or |〈P±

L (K )〉| 1/4while |〈P±

T (K )〉| 3/4.A few more comments are in order concerning the numer-

ical values in Tables 2, 3, and 4. First, the expressions ofsome quantities are identical up to a sign, e.g., |P−

T (K )| =|P+

T (K )|, |PLT (K )| = |PT L(K )|, etc., which can be under-stood in terms of discrete C, P and T symmetry transfor-mations. Note that the signs of the different asymmetrieson the other hand depend on the convention we use for theprojection vectors Eqs. (12)–(13). For the K ∗ and φ, theirtransverse polarizations are at the origin of the differences|P−

T (V )| �= |P+T (V )| and |PLT (V )| �= |PT L(V )|. For the

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Table 3 SM predictions for theB → K ∗τ+τ− observables.The “Charm” uncertaintiescorrespond to independentcorrections of 10% × Ceff

9 timesarbitrary phases ei θλ , λ = 0,±1,which are allowed to vary freelyand independently. In this table,K ∗

long. denotes the longitudinalpolarization of the K ∗. SeeTable 2 for more comments

Asyms., BR CV FF “Charm” WC

〈P−L (K ∗)〉 − 0.56 ± 0.007 ± 0.03 ± 0.01

〈P+L (K ∗)〉 + 0.56 ± 0.007 ± 0.03 ± 0.01

〈P−T (K ∗)〉 − 0.53 ± 0.02 ± 0.04 ± 0.002

〈P+T (K ∗)〉 + 0.02 ± 0.04 ± 0.1 ± 0.01

〈P−N (K ∗)〉 − 0.02 ± 0.001 ± 0.01 < 0.001

〈P+N (K ∗)〉 − 0.02 ± 0.001 ± 0.01 < 0.001

〈P−L 〉/〈P−

T 〉 (K ∗long.) + 0.68 ± 0.03 ± 0.01 < 0.001

[〈P−L 〉2 + 〈P−

T 〉2]1/2 (K ∗) + 0.77 ± 0.01 ± 0.02 ± 0.008

〈PLL (K ∗)〉 − 0.35 ± 0.02 ± 0.02 ± 0.007

〈PT T (K ∗)〉 + 0.05 ± 0.03 ± 0.09 ± 0.01

〈PNN (K ∗)〉 + 0.09 ± 0.02 ± 0.08 ± 0.01

〈PLT (K ∗)〉 − 0.001 ± 0.02 ± 0.03 ± 0.007

〈PT L (K ∗)〉 − 0.28 ± 0.009 ± 0.03 ± 0.01

〈PLN (K ∗)〉 + 0.05 ± 0.002 ± 0.02 ± 0.001

〈PNL (K ∗)〉 + 0.05 ± 0.002 ± 0.02 ± 0.001

〈PT N (K ∗)〉 − 0.002 ± 0.002 ± 0.03 < 0.001

〈PNT (K ∗)〉 − 0.002 ± 0.002 ± 0.03 < 0.001

〈AFB(K ∗)〉 + 0.20 ± 0.01 ± 0.03 ± 0.004

〈BR(K ∗)〉 × 107 1.30 ± 0.09 ± 0.22 ± 0.07 ± 0.03|Vts |

latter, the numerical values of 〈P+T (V )〉 and 〈PLT (V )〉 are

suppressed due in part to the approximate pure left-handedlepton currents we have for Heff

SM, i.e., Ceff9 −Ceff

10 . On theother hand, the single normal asymmetry P±

N , and the cor-related asymmetries PLN ,PNL ,PT N ,PNT , depend on theimaginary phases of Ceff

7 and Ceff9 and are therefore greatly

suppressed in the SM. Instead, PNN does not vanish in theabsence of complex phases.

As advocated in Sect. 3.2, the ratio of the unbinned longi-tudinal and transverse asymmetries is insensitive to the effec-tive Wilson coefficients, which absorb the effects of charmloops. In our implementation of the approach of Ref. [68],with a correction absorbed into Ceff

9 independent on q2, thisproperty is retained to a very good extent even when con-sidering the experimentally accessible ratio of asymmetriesintegrated over finite q2 bins, leading to a “Charm” uncer-tainty smaller than ±0.001 in all cases. However, since inthe binned ratio the cancellation of the (effective) Wilsoncoefficients is due to their small variation over the q2 bin,the cancellation breaks down when employing a model ofcharmonium and open-charm resonances that is differentialin q2 [36]. Employing it here, and only here, we give anestimate of the corresponding uncertainty in the case of theratio of longitudinal and transverse polarizations, shown inTables 2, 3 and 4.

For B → K ∗ and Bs → φ decays we also give the valuesfor the polarization vector modulus, i.e., [(P−

L )2+(P−T )2]1/2.

In this observable there is a somewhat better control of the

“Charm” related uncertainty due to a partial cancellation ofthe term sensitive to the CP-even phase exp{i θ−}, which isallowed to vary freely. Still, the resulting uncertainty is of thesame order as the quoted “Charm” uncertainties for the indi-vidual longitudinal and transverse asymmetries and consis-tent with the estimate within the charm resonance model [36].

Finally, touching upon experimental aspects, note thatthe processes B± → K±τ+τ−, and B0 → K ∗0τ+τ−or B0 → K ∗0τ+τ− are self-tagging, while the finalstate with a φ meson requires tagging to determine theflavor of the parent meson (bottom or anti-bottom). Inthis context we note that the expressions of the asymme-tries P±

L ,P±T ,PLN ,PNL ,PT N ,PNT are P-odd, while the

expressions of the asymmetries P±N ,PLT ,PT L ,PLL ,PT T ,

PNN are P-even. In absence of CP violation then, which isthe case in the SM when terms proportional to VubV

∗us are

neglected, the former set is C-odd, while the latter is C-even.Therefore, the former set vanishes in the untagged sample,while the latter does not.

3.5 Beyond SM effects

In general, NP affecting τ polarization asymmetries can alsobe probed using more traditional observables, such as thetotal rate or the various angular asymmetries. A global sen-sitivity comparison is beyond the scope of our work. For thesake of illustration, we compare the sensitivities of the sim-plest and most accessible observables, namely the total rate

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Table 4 SM predictions forBs → φτ+τ− observables. SeeTables 2 and 3 for morecomments

Asyms., BR CV FF “Charm” WC

〈P−L (φ)〉 − 0.55 ± 0.006 ± 0.03 ± 0.008

〈P+L (φ)〉 + 0.55 ± 0.006 ± 0.03 ± 0.008

〈P−T (φ)〉 − 0.51 ± 0.02 ± 0.04 < 0.001

〈P+T (φ)〉 + 0.07 ± 0.04 ± 0.1 ± 0.009

〈P−N (φ)〉 − 0.02 ± 0.001 ± 0.01 < 0.001

〈P+N (φ)〉 − 0.02 ± 0.001 ± 0.01 < 0.001

〈P−L 〉/〈P−

T 〉 (φlong.) + 0.68 ± 0.01 ± 0.01 < 0.001

[〈P−L 〉2 + 〈P−

T 〉2]1/2 (φ) + 0.76 ± 0.006 ± 0.03 ± 0.009

〈PLL (φ)〉 − 0.34 ± 0.01 ± 0.02 ± 0.005

〈PT T (φ)〉 + 0.04 ± 0.03 ± 0.09 ± 0.009

〈PNN (φ)〉 + 0.11 ± 0.02 ± 0.08 ± 0.01

〈PLT (φ)〉 − 0.02 ± 0.02 ± 0.03 ± 0.005

〈PT L (φ)〉 − 0.27 ± 0.01 ± 0.03 ± 0.008

〈PLN (φ)〉 + 0.05 ± 0.002 ± 0.02 ± 0.001

〈PNL (φ)〉 + 0.05 ± 0.002 ± 0.02 ± 0.001

〈PT N (φ)〉 − 0.004 ± 0.002 ± 0.03 < 0.001

〈PNT (φ)〉 − 0.004 ± 0.002 ± 0.03 < 0.001

〈AFB(φ)〉 + 0.18 ± 0.01 ± 0.03 ± 0.002

〈BR(φ)〉 × 107 1.24 ± 0.09 ± 0.21 ± 0.05 ± 0.03|Vts |

Fig. 4 New physics effects in the form of a real δC9 ≡ δNP, whosevalues are given in the horizontal axes. The vertical axes give the val-ues of the branching ratio, the FB asymmetry and the longitudinal andtransverse polarizations of the τ−, in the SM (solid, filled blue, inde-

pendent on the value of δNP) and in the NP under consideration (dashedorange). The blue band corresponds to the errors seen in Table 3, whichare recalculated for NP. See Fig. 7 for plots concerning B → K τ+τ−

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Fig. 5 New physics effects in the form of a real δC9 = −C ′9 ≡ δNP, whose values are given in the horizontal axes. See Fig. 4 for more comments

on the reading of the graphics

Fig. 6 New physics effects in the form of a real δC9 = −C ′9 = −δC10 = −C ′

10 ≡ δNP, whose values are given in the horizontal axes. See Fig. 4for more comments on the reading of the graphics. See Fig. 8 for plots concerning B → K τ+τ−

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Fig. 7 New physics effects in the form of a real δC9 ≡ δNP, whosevalues are given in the horizontal axes. The vertical axes give the valuesof the branching ratio, the transverse polarization of the τ−, and thecorrelated longitudinal–longitudinal and normal–normal polarizations,

in the SM (solid, filled blue, independent on the value of δNP) and in theNP under consideration (dashed orange). The blue band corresponds tothe errors seen in Table 2, which are recalculated for NP

and the forward–backward (FB) asymmetry. The FB asym-metry is defined as the branching ratio for θτ ∈ [0, π

2 ] minusthe one for θτ ∈ [π

2 , π ] (see Fig. 3),

AFB(q2) = d�(zτ > 0)/dq2 − d�(zτ < 0)/dq2

d�(zτ > 0)/dq2 + d�(zτ < 0)/dq2 . (20)

A broader class of FB asymmetries for the inclusive processB → Xsτ

+τ− have been previously discussed in Ref. [40].The operators O9, O10, O ′

9, O′10, O7, O ′

7 contribute at thetree level, and therefore deserve special attention when study-ing the impact of NP effects in B(s) → M�+�− decays. Con-tributions from O7 or the chirality flipped O ′

7 are LFU andtheir non-standard effects are already severely constrainedby existing measurements of rare B radiative and semi-electronic (semi-muonic) decay modes. We thus focus onthe following illustrative NP scenarios:

NP1 : δC9 real, (21)

NP2 : δC9 = −C ′9 real, (22)

NP3 : δC9 = −C ′9 = −δC10 = −C ′

10 real. (23)

Scenarios NP1 and NP2 for δCμμ9 ≈ −1 and δCμμ

9 =−C

′μμ9 ≈ −0.9, respectively, are actually favored by cur-

rent anomalies in muon data [79,80]. Scenario NP3 forCμμ

9 = −C′μμ9 = −δCμμ

10 = −C′μμ10 ≈ −0.7 is also

favored in some analyses [79]. Below we will also commenton other motivated scenarios such as with real δC9 = −δC10

or with complex C (′)9,10. In the following, we focus on the

B → K ∗τ+τ− mode, which was also our subject in Sect. 2.Since we are interested in the cases that the polarization

asymmetries exhibit better NP sensitivity than the branch-ing ratio, we choose the ranges of variation for the NPcontributions such that the branching ratio changes by atmost a factor ≈ 2. Note that existing direct bounds onB+ → K+τ+τ− [23] and Bs → τ+τ− [81] constrain NPeffects to |CNP

mn | � 103, where CNPmn is the NP Wilson coeffi-

cient of the operator (sγ μPmb) (τ γμPnτ), m, n = L , R (seealso [22,82] for indirect bounds). In addition, for heavy NPrespecting SM SU (2)L gauge invariance, existing bounds onB → K (∗)νν constrain |CNP

mL | � O(10), m = L , R [24]. Inthe NP ranges considered here, none of these existing boundsis violated.

In Figs. 4, 5, 6, 7, and 8 we present the NP induced vari-ations in the branching ratio together with the asymmetries

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Fig. 8 New physics effects in the form of a real δC9 = −C ′9 = −δC10 = −C ′

10 ≡ δNP, whose values are given in the horizontal axes. See Fig. 7for more comments on the reading of the graphics

for which we have the most notable effects in NP scenar-ios NP1, NP2, NP3 (for NP2 the values for the processB → K are unchanged with respect to the SM, and there-fore not shown). Similar plots are obtained for the decayBs → φ and are not displayed here. In the figures, thehorizontal blue bands represent the SM predictions, with“Charm”, FF and WC uncertainty contributions combinedlinearly. In presence of the NP manifestations consideredhere, |P−

L | = |P+L |, |P−

T (K )| = |P+T (K )|, |P−

N | = |P+N |,

|PLT (K )| = |PT L(K )|, |PLN | = |PNL |, |PT N | = |PNT |and AFB(K ) = 0.

We highlight 〈P±L (V )〉 whose SM-like values would

exclude large regions of the NP parameter space. Moreover,we note that some SM-like branching ratio solutions actuallyflip the sign of 〈P±

L (V )〉. To illustrate the use of the longitudi-nal asymmetry to discriminate models, note from Figs. 4 and5 that though the longitudinal asymmetries for NP1 and NP2have similar shapes, they evolve differently with the value ofδNP, which parameterizes the size of NP effects: for instance,around δNP ≈ −2 the value of 〈P−

L (V )〉 for the case NP2 isvery different compared to the SM, while that is not true forNP1.

Apart from the longitudinal polarization, the FB asym-metry and the transverse polarization may also show signflips, for both B → K ∗τ+τ− and B → K τ+τ−. Thoughthe FB asymmetry is not enhanced, it is useful for distin-

guishing cases NP2 and NP3, since as seen from Figs. 5 and6 the branching ratios and 〈P±

L (V )〉, 〈P−T (V )〉 have similar

shapes. Moreover, NP2 and NP3 can also be distinguishedby looking at B → K τ+τ−, cf. Fig. 8, since for NP2 all theobservables have SM-like values for this channel.

Considering other NP scenarios, note from Fig. 9 that thecase of a real δC9 = −δC10 (also favored by muon data forCμμ

9 = −δCμμ10 ≈ −0.7 [79]) shows a large suppression

of the branching ratio 〈BR(K ∗)〉. For obvious reasons then,if this case is realized for a large interval of the NP shiftsto the Wilson coefficients, B → K ∗τ+τ− asymmetries willprobably be out of reach experimentally.

Imaginary phases have also been considered in the con-text of muon data anomalies in, for instance, Refs. [80,83]and the analyses indicate possible large imaginary compo-nents. Such an example is illustrated in Fig. 10 for imaginaryδC9. We observe that no large effects in single tau polar-ization asymmetries are induced in this case (similar com-ments for 〈P−

N (V )〉 would also apply for an imaginary δC10,and, moreover, imaginary values for C ′

9,C′10).3 A significant

enhancement is only observed in 〈PLN (K )〉; see Fig. 11. Inany case, observables other than the tau polarizations may bemore sensitive to complex phases; see e.g. Ref. [84].

3 Note, however, that the tensorial operator (sσαβb) εαβμν (τσμντ ) notconsidered here can significantly enhance the normal polarization.

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Fig. 9 New physics effects in the form of a real δC9 = −δC10 ≡ δNP, whose values are given in the horizontal axes. See Figs. 4, 7 for morecomments on the reading of the graphics

Fig. 10 New physics effects in the form of an imaginary δC9 ≡ i δNP. The values for the real δNP are given in the horizontal axes. See Fig. 4 formore comments on the reading of the graphics

Fig. 11 New physics effects coming as δC9 ≡ i δNP (left), or δC10 ≡ i δNP (right). See Fig. 7 for more comments on the reading of the graphics

To conclude, in Table 5 we gather the asymmetries thathave the largest values in the presence of the NP scenariosconsidered here. For instance, scenarios NP1 and NP2 canbe distinguished by an enhanced value of 〈P+

T (V )〉, whiledifferent values of δC9 in scenario NP1 can be distinguishedby measuring an anomalously enhanced value of 〈P+

T (V )〉,〈PLT (V )〉, 〈PT T (V )〉, 〈PT L(V )〉, or a SM-like 〈P±

L (V )〉.Of course, a combined correlated measurement of the varioustau polarization asymmetries considered here can in principle

provide even more powerful tests and discriminants betweenthe SM and various NP scenarios.

4 Conclusions

In the light of the recent intriguing experimental results indi-cating possible LFU violations in B → K (∗)�+�− decays, itis important to investigate possible NP effects in rare semi-

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Table 5 Non-exhaustive table making the correspondence between theobservation of an asymmetry at or beyond the level ±0.4, and the phys-ical scenarios among NP1, NP2, NP3 this observation implies. BelowV = K ∗, φ. Of course, the non-observation of 〈P±

L (V )〉, 〈P−T (V )〉,

〈P±T (K )〉, 〈PT T (K )〉 at a sizable level also implies the existence of

physics beyond the SM, which is not discussed in this table

〈P−L (V )〉 ≈ −0.5,

〈P+L (V )〉 ≈ +0.5, SM-like

〈P−T (V )〉 ≈ −0.5

δC9 � −5,

〈P−L (V )〉 ≈ +0.5, δC9 = −C ′

9 � −3,

〈P+L (V )〉 ≈ −0.5 δC9 = −C ′

9 =−δC10 = −C ′

10 � −3

δC9 = −C ′9 � −3,

〈P−T (V )〉 ≈ +0.5 δC9 = −C ′

9 =−δC10 = −C ′

10 � −3

〈P+T (V )〉 ≈ −0.5 δC9 � −6

δC9 ≈ −3,

〈PT L (V )〉 ≈ −0.5 δC9 = −C ′9 ≈ −2,

δC9 = −C ′9 =

−δC10 = −C ′10 ≈ −2

〈PLT (V )〉 ≈ −0.5 δC9 ≈ −5

〈PT T (V )〉 ≈ −0.4 δC9 ≈ −4

δC9 ≈ −4,

〈PNN (V )〉 ≈ −0.4 δC9 = −C ′9 ≈ −2,

δC9 = −C ′9 =

−δC10 = −C ′10 ≈ −2

〈P−T (K )〉 ≈ −0.8,

〈P+T (K )〉 ≈ +0.8, SM-like

〈PT T (K )〉 ≈ −0.7

〈P−T (K )〉 ≈ +0.8 δC9 � −8,

〈P+T (K )〉 ≈ −0.8 δC9 = −C ′

9 =−δC10 = −C ′

10 � −3

〈PT L (K )〉 ≈ −0.5, δC9 ≈ −5

〈PLT (K )〉 ≈ −0.5

〈PT T (K )〉 ≈ −0.8 δC9 ≈ −5

〈PT T (K )〉 ≈ −0.4 δC9 = −C ′9 =

−δC10 = −C ′10 � −2

〈PNN (K )〉 ≈ −1 δC9 ≈ −5

〈PNN (K )〉 ≈ +1 δC9 = −C ′9 =

−δC10 = −C ′10 � −2

〈PLL (K )〉 ≈ +0.8 δC9 ≈ −5

〈PLL (K )〉 ≈ −0.4 δC9 = −C ′9 =

−δC10 = −C ′10 � −2

tauonic b-hadron decays. Currently these modes are onlypoorly constrained from the experimental side, allowing forpotentially large NP effects. Given the difficulty to recon-struct the B(s) → Mτ+τ− decays, where M is a pseu-doscalar or vector meson, we have argued that a future high-

energy e+e− collider could play a crucial role in gainingexperimental access to these rare processes.

In particular, working with the current FCC-ee colliderand detector design parameters we have presented a case forthe viability of a full reconstruction of the B0 → K ∗0τ+τ−decays in e+e− collisions at the Z -resonance mass. Ourresults indicate that of the order of a few thousands fullyreconstructed events can be expected at the baseline colliderluminosity. Based on these encouraging results we have con-sidered observables other than the branching ratio. In particu-lar, we have investigated the lepton polarization asymmetries,which are inaccessible in light lepton modes and thus provideuniquely new probes of NP in rare semileptonic b-hadrondecays. Investigating closely the different sources of theo-retical uncertainty we have provided precise SM predictionsfor these observables in B → K (∗)τ+τ− decays and inves-tigated their sensitivity to some motivated and representativeNP scenarios. In particular, the single longitudinal and trans-verse asymmetries of the taus in the B → K ∗τ+τ− decaysare sensitive to the NP scenarios considered here, whilefor the B → K τ+τ− decay the correlated longitudinal–longitudinal and normal–normal asymmetries may show siz-able enhancements. Table 5 summarizes our findings on thepossible enhancement or modulation of a variety of tau polar-ization asymmetries. For completeness we have computedthese observables in the SM also for the Bs/Bs → φτ+τ−decays which are not self-tagging and thus seem less suitablefor experimental tau polarization studies.

Further dedicated experimental studies will certainly berequired in order to firmly establish the NP sensitivity of taupolarization observables in rare semi-tauonic B decays in thecontext of the FCC-ee. More generally, however, we hope ourpreliminary results will help to strengthen the (flavor) physicscase for the FCC-ee.

Acknowledgements JFK and LVS thank Wolfgang Altmannshoferand David Straub for enlightening discussions and acknowledge thefinancial support from the Slovenian Research Agency (research corefunding No. P1-0035 and J1-8137).

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

A Form factors

We employ the parameterization of the hadronic matrix ele-ments in terms of the form factors as found in Ref. [72,85].First, for B → K transitions we have

123

Eur. Phys. J. C (2017) 77 :701 Page 15 of 19 701

〈K (p)|sγ μb|B(k)〉 = f+(q2)

(kμ + pμ − M2

B − M2K

q2 qμ

)

+ f0(q2)M2

B − M2K

q2 qμ, (24)

〈K (p)|i sσμνb|B(k)〉 = 2 fT (q2)

MB + MK(kμ pν + kν pμ), (25)

〈K (p)|sb|B(k)〉 = M2B − M2

K

mb − msf0(q

2). (26)

Similarly for B(s) → V , where V = K ∗, φ, we employ

〈V (p, ε)|sγ μb|B(s)(k)〉 = 2iV (q2)

MB(s) + MVεμνρσ ε∗

ν pρkσ , (27)

〈V (p, ε)|sγ μγ 5b|B(s)(k)〉 = 2MV A0(q2)

ε∗ · qq2 qμ

+ (MB(s) + MV )A1(q2)

(ε∗μ − ε∗ · q

q2 qμ

)

− A2(q2)

ε∗ · qMB(s) + MV

×[(k + p)μ −

M2B(s)

− M2V

q2 qμ

], (28)

qν〈V (p, ε)|sσμνb|B(s)(k)〉 = 2T1(q2)εμρτσ ε∗ρkτ pσ , (29)

qν〈V (p, ε)|sσμνγ5b|B(s)(k)〉

= iT2(q2)[(ε∗ · q)(k + p)μ − ε∗

μ(M2B(s)

− M2V )]

+ iT3(q2)(ε∗ · q)

[q2

M2B(s)

− M2V

(k + p)μ − qμ

],

(30)

while 〈V (p, ε)|qγ 5b|B(s)(k)〉 can be determined fromqμ〈V (p, ε)|qγ μγ 5b|B(s)(k)〉. In both cases we neglect themass of the strange quark over the mass of the bottom quark.

For the B → M�+�− form factors we employ determi-nations using lattice QCD. Fortunately, the B → Mτ+τ−phase space is restricted to the high-q2 (small M recoil)region, where the current lattice QCD form factor calcula-tions are directly applicable. For the B → K�+�− process,we use the form factors given in Ref. [85]. They are parame-terized in terms ofB = {b+

0,1,2, b00,1,2, b

T0,1,2} in the following

way:

f+,T (q2) = 1

P+,T (q2)

K−1∑m=0

b+,Tm

×[zm − (−1)m−K m

KzK

],

f0(q2) = 1

P0(q2)

K−1∑m=0

b0mz

m, K = 3, (31)

where

z(q2, t0) =√t+ − q2 − √

t+ − t0√t+ − q2 + √

t+ − t0,

P+,0,T (q2) = 1 − q2/M2+,0,T , (32)

with t0 = (MB+MK )(√MB−√

MK )2, t+ = (MB+MK )2,M+,T = MB∗

s= 5.4154 GeV, M0 = MB∗

s0= 5.711 GeV,

MB = 5.27958 GeV and MK = 0.497614 GeV.Lattice extractions of the form factors are also available

for B → K ∗�+�− and Bs → φ�+�−, and we use the resultsof Refs. [72,73] (the differences with respect to [21], such asa smaller lattice spacing, are commented therein). We notethat in these calculations the K ∗ or φ have been treated asstable particles (see e.g. [86] for a discussion related to thisapproximation). Here form factors are parameterized in termsof A = {aF

0,1}, F = V, A0,1,12, T1,2,23, as follows:

F(q2) = aF0 + aF

1 z(q2, t0)

1 − q2/(MB(s) + �MF )2 ,

F = V, A0,1,12, T1,2,23, (33)

with t0 = 12 GeV, t+ = (MB(s) +MV )2, V = K ∗, φ, MBs =5.366 GeV, MK ∗ = 0.892 GeV and Mφ = 1.020 GeV, whilethe mass differences �MF are given in Ref. [72].

B Full expressions

We now list the different expressions for the asymmetries andbranching ratios discussed in the main text. In what follows,U = B(s) and V = K ∗, φ. First we have, for B → K ττ ,

�(K ) = �

3q2 4π( | f A|2 (

δλ + 24m2τ M2

K q2)

+ 6q2(2 Re[ f ∗A fP ]mτ ξ + | fP |2q2)

+ δλ | fV |2) , (34)

P−L (K ) × �(K ) = �

3√q2

8π Re[ f ∗A fV ]√�λ, (35)

P−L (K ) + P+

L (K ) = 0, (36)

P−T (K ) × �(K ) = �√

q28π2

√λ( Re[ f ∗

A fV ]mτ ξ

+ Re[ f ∗P fV ]q2), (37)

P−T (K ) + P+

T (K ) = 0, (38)

P−N (K ) × �(K ) = −�8π2 Im[ f ∗

A fP ]√�√

λ, (39)(P−

N (K ) − P+N (K )

)× �(K ) = −�16π2 Im[ f ∗

A fP ]√�√

λ, (40)

PLL (K ) × �(K ) = �

3q2 4π(

| f A|2 (10λm2

τ + 24m2τ M2

K q2 − λq2)

+ q2(12 Re[ f ∗A fP ]mτ ξ + 6 | fP |2q2 − λ | fV |2) + 2λm2

τ | fV |2),

(41)

PT T (K ) × �(K ) = − �

3q2

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701 Page 16 of 19 Eur. Phys. J. C (2017) 77 :701

16π(

| f A|2 (10λm2

τ + 24m2τ M2

K q2 − λq2)

+ q2(12 Re[ f ∗A fP ]mτ ξ + 6 | fP |2q2 + λ | fV |2)

− 2λm2τ | fV |2

), (42)

PNN (K ) × �(K ) = − �

3q2 16π(

| f A|2 (δλ + 24m2

τ M2K q

2)

+ 6q2(2 Re[ f ∗A fP ]mτ ξ + | fP |2q2)

+ δλ(− | fV |2)), (43)

PLT (K ) × �(K ) = − �

q2 4π2√

�√

λ( | f A|2 mτ ξ

+ Re[ f ∗A fP ]q2), (44)

PLT (K ) − PT L (K ) = 0, (45)

PLN (K ) × �(K ) = − �√q2

4π2√

λ( Im[ f ∗A fV ]mτ ξ

+ Im[ f ∗P fV ]q2), (46)

PLN (K ) − PNL (K ) = 0, (47)

PT N (K ) × �(K ) = �

3√q2

32π Im[ f ∗A fV ]√�λ, (48)

PT N (K ) − PNT (K ) = 0, (49)AFB(K ) = 0. (50)

Now, for B → V ττ ,

�(V ) = �

3 M2V q

24π

(λ(

2 |A|2δ M2V q

2 + 2ξ(δ( Re[B∗ C]

+ Re[F∗ G]) + 6 Re[G∗ H ]m2τq

2)

+ |C |2δλ+2� |E |2 M2

V q2 + 12 Re[F∗ H ]m2

τq2 + δ |G|2λ

+ 24 |G|2 m2τ M2

V q2 + 6 |H |2 m2

τ (q2)2

)

+|B|2δ(λ + 12 M2

V q2)

+ |F |2(δλ + 12� M2

V q2))

, (51)

P−L (V ) × �(V ) = �

3 M2V

√q2

8π√

�(λ

(2 Re[A∗ E] M2

V q2

+ ξ( Re[B∗ G] + Re[C∗ F])+ Re[C∗ G]λ) + Re[B∗ F]

(λ + 12 M2

V q2))

, (52)

P−L (V ) + P+

L (V ) = 0, (53)

P−T (V ) × �(V ) = �

M2V

√q2

8π2√

λmτ

(4 Re[A∗ B] M2

V q2

+ ξ( Re[B∗ F] + Re[B∗ H ]q2)

+ Re[B∗ G](λ + 4 M2

V q2)

+ λ( Re[C∗ F] + Re[C∗ G]ξ + Re[C∗ H ]q2))

, (54)(P−T (V ) + P+

T (V ))

× �(V )

= �64π2 Re[A∗ B]√λmτ

√q2, (55)

P−N (V ) × �(V ) = − �

M2V

8π2√

�√

λmτ

×(−2 M2

V ( Im[A∗ F] + Im[B∗ E] − 2 Im[F∗ G])+ Im[F∗ H ]ξ + Im[G∗ H ]λ)

, (56)(P−

N (V ) − P+N (V )

)× �(V )

= − �

M2V

16π2√

�√

λmτ

(4 Im[F∗ G] M2

V

+ Im[F∗ H ]ξ + Im[G∗ H ]λ), (57)

PLL (V ) × �(V ) = �

3 M2V q

24π

(λ(

2 |A|2 M2V q

2(

2m2τ − q2

)

+ 2ξ(

2m2τ ( Re[B∗ C] + 5 Re[F∗ G])

− q2(

Re[B∗ C] + Re[F∗ G]− 6 Re[G∗ H ]m2

τ

))+ 2 |C |2λm2

τ − |C |2λq2

− 2� |E |2 M2V q

2 + 12 Re[F∗ H ]m2τq

2

+ 10 |G|2λm2τ + 24 |G|2 m2

τ M2V q

2

− |G|2λq2 + 6 |H |2 m2τ (q

2)2)

+ |B|2(

2m2τ − q2

)(λ + 12 M2

V q2)

+ |F |2λ(

10m2τ − q2

)− 12� |F |2 M2

V q2), (58)

PT T (V ) × �(V )

= �

3 M2V q

216π

(|B|2

(2λm2

τ + 24m2τ M2

V q2 − λq2

)

− λ(

− 4 |A|2 m2τ M2

V q2 − |A|2 M2

V (q2)2

+ 2ξ(q2

(Re[B∗ C] − Re[F∗ G]

+ 6 Re[G∗ H ]m2τ

)− 2m2

τ ( Re[B∗ C]−5 Re[F∗ G])

)− 2 |C |2λm2

τ + |C |2λq2

+ � |E |2 M2V q

2 + 10 |F |2 m2τ

− |F |2q2 + 12 Re[F∗ H ]m2τq

2

+10 |G|2λm2τ + 24 |G|2 m2

τ M2V q

2

− |G|2λq2 + 6 |H |2 m2τ (q

2)2))

, (59)

PNN (V ) × �(V )

= − �

3 M2V q

216π

(λ(q2

(� M2

V ( |A|2 − |E |2)

+ 6 |H |2 m2τq

2)

+ 2ξ(

− Re[B∗ C]δ+ δ Re[F∗ G]+6 Re[G∗ H ]m2

τq2)+δ( |F |2 − |C |2λ)

+ 12 Re[F∗ H ]m2τq

2 + |G|2(δλ + 24m2

τ M2V q

2))

− |B|2(δλ + 24m2

τ M2V q

2))

, (60)

PLT (V ) × �(V ) = − �

M2V q

24π2

√�

√λmτ

×(

− 2 Re[A∗ F] M2V q

2 − 2 Re[B∗ E] M2V q

2

+ |F |2ξ + 2 Re[F∗ G]λ + 4 Re[F∗ G] M2V q

2

+ Re[F∗ H ]ξq2 + |G|2λ M2U − |G|2λ M2

V

− |G|2λq2 + Re[G∗ H ]λq2), (61)

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Eur. Phys. J. C (2017) 77 :701 Page 17 of 19 701

(PLT (V ) − PT L(V )

)

×�(V ) = �16π2√

�√

λmτ ( Re[A∗ F]+ Re[B∗ E]), (62)

PLN (V ) = �

M2V

√q2

4π2√

λmτ

(ξ( Im[B∗ F]

+ Im[B∗ H ]q2) + Im[B∗ G](λ + 4 M2

V q2)

+ λ(Im[C∗ F] + Im[C∗ G]ξ + Im[C∗ H ]q2))

, (63)

PLN (V ) − PNL(V ) = 0, (64)

PT N (V ) × �(V )

= − �

3 M2V

√q2

32π√

�λ(− Im[A∗ E] M2

V q2 + Im[B∗ F]

+ ξ( Im[B∗ G] + Im[C∗ F]) + Im[C∗ G]λ), (65)

PT N (V ) − PNT (V ) = 0, (66)

AFB(V ) × �(V )

= �8π√

�√

λ

√q2( Re[A∗ F] + Re[B∗ E]).

(67)

Above we employ the following notation:

� =√

�/q2√

λ

64 M3U π

, (68)

f A = 4 f+(q2)(C10 + C ′10 + δC10) , (69)

fP = 1

q2 4mτ (C10 + C ′10 + δC10)(( f0(q

2) − f+(q2))

×( MU − MK )( MU + MK ) + f+(q2)q2) , (70)

fV = 1

MU + MK4( f+(q2)( MU + MK )

(Ceff9 + C ′

9 + δC9)

−2 fT (q2)mb(Ceff7 + δC7 + C ′

7)) , (71)

A = 4V (q2)(Ceff9 + C ′

9 + δC9)

MU + MV

+8mb T1(q2)(Ceff7 + δC7 + C ′

7)

q2 , (72)

B = − 1

q2 2( MU + MV )(q2 A1(q2)(Ceff

9 − C ′9 + δC9)

+2mb( MU − MV ) T2(q2)(Ceff

7 + δC7 − C ′7)),

(73)

C = 2 A2(q2)(Ceff9 − C ′

9 + δC9)

MU + MV

+4mb(Ceff7 + δC7 − C ′

7)

×(

T3(q2)

M2U − M2

V

+ T2(q2)

q2

), (74)

E = 4V (q2)(C10 + C ′10 + δC10)

MU + MV, (75)

F = 2 A1(q2)( MU + MV )(−C10 + C ′

10 − δC10) , (76)

G = 2 A2(q2)(C10 − C ′10 + δC10)

MU + MV, (77)

H = 1

q2( MU + MV )2(−C10

+C ′10 − δC10)(2 MV ( MU + MV )

×( A0(q2) − A3(q

2)) − q2 A2(q2)) , (78)

λ = M4U + M4

V + (q2)2 − 2 (M2U M2

V

+M2V q2 + q2 M2

U ) , (79)

� = −4m2τ + q2 , (80)

ξ = M2U − M2

V − q2, (81)

δ = 2m2τ + q2. (82)

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