TIFR/TH/19-3

Tensors for tending to tensions in τ decays

Amol Dighe,1, ∗ Subhajit Ghosh,1, † Girish Kumar,1, ‡ and Tuhin S. Roy1, §

1Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India

We propose a manifestly gauge invariant effective tensor operator that can account for the CPasymmetry anomaly in τ decays, contrary to the claim made in literature. Additionally, this op-erator can also help in resolving the long-standing discrepancy in the value of Vus extracted frominclusive τ decays. By construction, the operator evades bounds from neutron electric dipole mo-ment, while keeping the extraction of Vus from exclusive τ decays unaffected. We explicitly providea renormalizable model of flavor symmetries that yields the desired effective tensor operator.

Decays of τ leptons provide a laboratory for pre-cision tests of the Standard Model (SM) of particlephysics [1, 2]. Leptonic τ decays are crucial for testinglepton flavor universality and searching for lepton flavorviolating decays [3, 4]. Hadronic τ decays offer us theopportunity to probe strong interactions in a variety ofways. Since the mass of τ is fairly above the QCD scale,we can use perturbative results derived in the effectivefield theory framework. These decays have been usedto extract fundamental parameters of SM, such as thestrong coupling constant αs [5–8]. As τ is massive enoughto decay to strange mesons, these decays have also beenused to determine the strange quark mass and the Vus el-ement of the Cabibbo–Kobayashi–Maskawa (CKM) ma-trix [9–11].

Consistency across various measurements of the aboveparameters vouches for the success of SM. However fur-ther precision measurements of τ decays can uncoverhints of new physics (NP) beyond SM, through devia-tions from SM predictions. In this work, we take a criti-cal look at two such long-standing anomalies in hadronicτ decays:

• The determination of Vus from inclusive τ decays tofinal states including the strange quark. The valueextracted from inclusive decays [3],

|Vus|τ,in = 0.2186± 0.0021 , (1)

shows a striking 3.1σ deviation from the CKM uni-tarity prediction,

|Vus|uni = 0.22582± 0.00089 . (2)

• The CP asymmetry in τ → Ksπντ is calculated inSM to be [12, 13]

Aτ, SMCP = (0.33± 0.01)% . (3)

However the asymmetry measured by the BaBarcollaboration [14],

Aτ, expCP = (−0.36± 0.23± 0.11)% , (4)

is 2.8σ below the SM prediction.

∗ amol@theory.tifr.res.in† subhajit@theory.tifr.res.in‡ girishk@theory.tifr.res.in§ tuhin@theory.tifr.res.in

Various attempts have been made in the literature toreconcile the Vus anomaly within SM. Proposals rangefrom inadequate theoretical understanding of τ spectralfunctions [15, 16] to possible mis-measurements of someexclusive τ decay channels [17]. (The measurement ofinclusive τ → s decays is through the summation over allexclusive decay modes with an odd numbers of kaons [3]in the final state.) It is, however, not possible to accountfor the discrepancy in CP asymmetry within SM [18–20].It was pointed out in Ref. [18] that scalar NP operatorscannot generate new CP asymmetry while tensors can.However Ref. [20] claimed that a demand of gauge in-variance brings the tensor operator in conflict with themeasurement of neutron electric dipole moment (EDM).This claim has recently been repeated in Ref. [21].

In this Letter, we address both the above τ anomalieswith a single source of NP in the effective field theory(EFT), and provide an explicit renormalizable model.The essential requirements we impose are (i) no effecton Vus extraction from exclusive τ decays, which is con-sistent with SM within ∼ 1σ, (ii) manifest electroweak(EW) gauge invariance.

To comply with the the first requirement, the new op-erator ONP should be such that

〈Kντ |ONP| τ〉 = 〈K|OhadNP |0〉 〈ντ |Olep

NP|τ〉 = 0 . (5)

This condition is satisfied automatically if OhadNP , the

hadronic part of the NP operator, has a tensorial struc-ture. This is because 〈K|q σµνγ5 q′|0〉 = 0 for any fla-vors q, q′, due to the antisymmetry of the Lorentz struc-ture of the operator. Note that the symmetry argumentwould not be valid if the outgoing state were to involvemore than one hadron. Therefore, by choosing the fla-vor structure of Ohad

NP appropriately, one can contributeto inclusive τ decays. Thus a suitable choice of ONP thataccounts for the Vus discrepancy would be the tensor op-erator (sσµνu) ( ντσ

µντ).Remarkably, the same operator can also contribute to

the CP asymmetry in τ → Ksπντ [18]. Therefore, it maybe possible to reconcile both these anomalies by a unifiedexplanation. On the other hand, according to Ref. [20],a demand of EW gauge invariance necessarily gives riseto an additional operator (uLσµνuR) (τLσ

µντR), which isstrongly constrained from neutron EDM measurements.Consequently, there appears to be a “no-go” for the ex-planation of the CP anomaly.

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In this Letter, we argue that the no-go theoremsketched in Ref. [20] only pertains to a specific way ofimposing gauge invariance. We demonstrate that thereexists at least one class of manifestly gauge invariant op-erators that can accommodate both anomalies in τ de-cays and yet is safe from neutron EDM constraints. Asa concrete example, we propose a renormalizable flavormodel (detailed later in this article) which appends to SMthe following gauge invariant effective tensor operator af-ter integrating out all beyond-SM degrees of freedom:

L ⊃ KΛ4

[(¯

3H†)σµντR

][(q2H)σµνuR] + h.c. , (6)

where Λ represents the scale of the operator, defined suchthat |K| = 1. In our model, after electroweak symmetrybreaking (EWSB), Eq. (6) yields a single operator whereall the fields are written in the mass basis:

L ⊃ −4GF√2VusCT [(sL σµνuR)(ντLσ

µντR)] . (7)

We have used

CT ≡ −K

2√

2Vus

v2

GFΛ4= − K

2Vus

( vΛ

)4, (8)

with v = 246 GeV is the EWSB scale, and GF is theFermi constant. (Note that the factor of Vus is used inthe normalization of CT , only to simplify further calcula-tions.) This corresponds to a 4-fermion interaction witheffective coupling strength Kv2/2Λ4. The NP couplingvia the Higgs doublet in Eq. (6) ensures that the danger-ous operator that would have contributed to the neutronEDM is not generated.

Before going to the details of the renormalizable model,we now quantitatively explore our solutions to the abovetwo τ anomalies, including constraints from other rel-evant measurements. In SM, τ decays to strange andnon-strange hadrons proceed via the standard 4-fermi in-teraction,

L ⊃ −4GF√2

[Vus(sLγµuL)(ντLγµτL)

+Vud(dLγµuL)(ντLγµτL)

]. (9)

Let us first explore the extraction of Vus from inclusivedecay channels. In this extraction, strange and non-strange spectral functions of τ are used along with veryprecisely determined value of Vud. The relevant quantityis [5]

δRτ ≡Rτ,ns|Vud|2

− Rτ,s|Vus|2

, (10)

where Rτ,ns and Rτ,s are the partial widths defined as

Rτ,ns(s) ≡Γ[τ → ντ +Xns(s)]

Γ[τ → eντ νe]. (11)

Here Xns(s) represents the final state with even (odd)strangeness. The quantity δRτ , which vanishes in theperturbative and massless quark limit, encodes the cor-rections due to finite quark masses and non-perturbativeeffects. Using QCD sum rules and operator product ex-pansion (OPE) [22–24], this factor is theoretically calcu-lated in SM to be [10, 25]

δRSMτ = 0.242± 0.032 . (12)

Using this theoretical input and experimentally deter-mined values of Rτ,s, Rτ,ns, and Vud [3], the CKM ele-ment Vus is extracted using Eq (10), and is found to benearly 3.1σ [3] below the CKM unitarity prediction.

Now we calculate the effect of our NP on this extrac-tion. We expect that the tensor operator will modifyRτ,s, thereby changing the theoretical value of δRτ . Theeffect of this operator on the inclusive mode can be cal-culated using QCD sum rules and OPE [24, 26]. Weparametrize the corrections due to NP as

δRNPτ (CT ) = δRτ − δRSM

τ , (13)

and find

δRNPτ (CT ) ≈ −288π2Re(CT )

〈0|qq|0〉m3τ

− 18|CT |2 , (14)

where qq ≡ (uu + ss)/2. In deriving the above, we haveignored higher order correction to OPE due to quarkmasses, gluon condensates, higher order corrections inαs, and duality violation [27–29] effects.

The first term in Eq. (14) arises from the interferenceof the SM operator in Eq. (9) with the tensor operatorthat we introduced in Eq. (7), and therefore is linear inCT . The second term, quadratic in CT , comes from thetensor-tensor correlator. We briefly sketch the derivationof Eq. (14), and specially the estimation of the tensor-tensor correlator, in Appendix A.

Let us turn towards the other anomaly, namely, theCP asymmetry in τ → Ksπντ decays:

AτCP =Γ(τ+ → Ksπ

+ντ )− Γ(τ− → Ksπ−ντ )

Γ(τ+ → Ksπ+ντ ) + Γ(τ− → Ksπ−ντ ). (15)

In SM, the source of this asymmetry is indirect CP vio-lation due to K −K mixing in the final state [12]. Afterthe initial inconclusive null results from CLEO [30] andBelle [31], the BaBar collaboration has measured thisCP asymmetry at the PEP-II asymmetric-energy e+-e−

collider at SLAC [14].The events in the signal channel τ → πK0

Sντ (C1)at BaBar also receive significant contamination fromthe two background channels, τ → KK0

Sντ (C2) andτ → πK0K0ντ (C3). Therefore, the actual asymmetrymeasured is [14]

A =f1A1 + f2A2 + f3A3

f1 + f2 + f3. (16)

3

Here A1, A2, A3 are CP asymmetries in the channels C1,C2, C3, respectively, while f1, f2, f3 are the correspond-ing sample fractions. Note that the above CP asymme-tries measured in the experiment are also affected by theKS → π+π− decay time dependence of the event selec-tion efficiency [13]. As argued by BaBar, this results ina multiplicative correction factor such that [14]

A1 = (1.08± 0.01)AτCP . (17)

In SM, A1 = −A2, and A3 = 0. Using these relationsand the measurement Aexp = (−0.27 ± 0.18 ± 0.08)%,BaBar extracted Aexp

1 = (−0.36 ± 0.23 ± 0.11)%. Thecorresponding SM prediction is ASM

1 = (+0.36± 0.01)%.This is the 2.8σ discrepancy in the CP asymmetry in τdecays.

While introducing NP, a very important point needsto be considered which has been overlooked in previousanalyses [18–20]. The extraction of A1 from A as doneby BaBar, while valid in SM, fails in the presence of NP.This is because in general, A1 6= −A2. Therefore thecomparison of predictions in the presence of NP needs tobe done with the measured quantity A, and not with A1.The NP operator as introduced by us in Eq. (7) affectsonly the channel C1. Therefore

A1 = ASM1 +ANP

1 , A2 = ASM2 , A3 = ASM

3 = 0 . (18)

While ASM1 is positive, the direct CP violation arising

from interference of the V –A operator in Eq. (9) and thetensor operator in Eq. (7) can lead to a negative ANP

1 .This would move the value of A1, and hence the value ofA, closer to the measurement.

In the presence of both, indirect CP violation from SM

(Aτ,SMCP ) and direct CP violation from NP (Aτ,NPCP ), the

net CP asymmetry may be written in the form [18]

AτCP =Aτ,SMCP +Aτ,NP

CP

1 +Aτ,SMCP Aτ,NPCP

. (19)

The value of Aτ,NPCP may be expressed in terms of the

vector and tensor form factors f+(s) and BT (s), definedthrough

〈K(pK)π(pπ)|sγµu|0〉 = (pK − pπ)µf+(s)

+ (pK + pπ)µf−(s) , (20)

〈K(pK)π(pπ)|sσµνu|0〉 = ipµKp

νπ − pνKpµπmK

BT (s) , (21)

where s ≡ (pπ + pK)2. The asymmetry, as may be ob-tained from the differential decay rate [18, 20] (see Ap-pendix B), is

Aτ,NPCP =

Im(CT )

ΓτBR(τ → Ksπντ )I , (22)

where

I =

∫ m2τ

sπK

κ(s)|f+(s)BT (s)| sin[δ+(s)− δT (s)] ds . (23)

Here the denominator ΓτBR(τ → Ksπντ ) is the averagedecay width of τ± → Ksπ

±ντ . In the integral I, wedenote sπK = (mπ +mK)2, and κ(s) is defined as

κ(s) =G2F |Vus|2SEW

256m2τmK

(m2τ − s)2s2

λ3/2(s,m2K ,m

2π) , (24)

where λ(a, b, c) = a2+b2+c2−2(ab+bc+ca), and SEW =1.0194 [32, 33] is the EW correction factor. Here δ+(s)and δT (s) are the phases of the form factors f+(s) andBT (s), respectively.

In the decay τ → Ksπντ , the relevant form factorsare dominated by the K∗(892) resonance in the elasticlimit and K∗(1410) in the inelastic region. Both theseresonances can have contributions from f+(s) as well asBT (s), with δ+(s) = δT (s) in the elastic limit as guar-anteed by Watson’s final-state theorem [34]. Thereforeonly the inelastic region contributes to I, and hence tothe CP asymmetry.

The value of I is highly uncertain, mainly due tolack of information on BT (s). Assuming only SM, thevalue of f+(s) is obtained by fitting to a superposition ofBreit-Wigner (BW) functions, [35] or by using the dis-persive parametrization [17, 36, 37]. In the context ofNP, Ref. [20] uses simplified “Omnes function” forms [38]for the magnitudes of f+(s) and BT (s), with the phasedifference δ+(s) − δT (s) proportional to the BW phase

of K∗(1410). The constraint |Aτ,NPCP | . 0.03 |Im CT |,obtained therein, is crucially dependent on the value ofδ+(s) used, and on the assumed relation δT (s) = −δ+(s)in the inelastic region, which has no strong theoreticalmotivation. Here we employ a more general treatment,with the same Omnes forms for |f+(s)| and |BT (s)|, butwith

δT (s)− δ+(s) = α×Arg[BW(K∗(1410))] , (25)

which is automatically consistent with Watson’s final-state theorem [34]. The free real parameter α acts asa proxy for our ignorance about I, and hence we canstudy the dependence of NP constraints for different val-ues of α (hence for different values of I). We have takenf+(0)|Vus| = 0.2166(5) [39], whereas BT (0)/f+(0) =0.676(27) [40] from lattice calculations.

We show our numerical results of the constraints onreal and imaginary parts of CT in Fig. 1. The 1σ al-lowed regions (1 d.o.f, ∆χ2 ≤ 1) explaining AτCP, for tworepresentative α values (α = 0.2, 0.6), are shown withhorizontal grey bands. Note that all our results are in-variant under {α → −α, Im(CT ) → −Im(CT )}. Thevalue α = 0.2 approximately corresponds to the phasechoices in Ref. [20].

The NP introduced will also affect the decay ratesof τ → Ksπν and τ → K−π0ν. Since the theoreti-cal predictions for these quantities are isospin-related,and hence completely correlated [17], we only use thebranching ratio (BR) measurement for τ → K0πν =0.8386 ± 0.0141 [3]. The SM value of the BR is takenfrom [17]. The 1σ constraints (2 d.o.f, ∆χ2 ≤ 2.3) in the

4

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.300.00

0.05

0.10

0.15

0.20

0.25

0.30

↑

●

●

FIG. 1. The 1σ allowed regions from AτCP (grey), BR ofτ → Ksπντ (light blue, light green) and, the Vus anomaly(pink). The combined 1σ and 90% C.L. regions for α = 0.2(unfilled ovals) and α = 0.6 (filled ovals) are also shown.

CT plane are shown in Fig. 1 for α = 0.2 and 0.6, withlight blue and light green bands, respectively.

Fig. 1 also shows the 1σ (2 d.o.f, ∆χ2 ≤ 2.3) allowed re-gions explaining the Vus anomaly from inclusive τ decays.The values of quark condensates are taken from Ref. [41]and of other relevant quantities from Refs. [3, 4]. Wehave included 10% theoretical errors in the coefficients ofthe two terms in Eq. (14).

We finally show the combined results from AτCP , theBR of τ → Ksπν, and the Vus anomaly, for α = 0.2, 0.6.Clearly the range of preferred values of NP parametersdepends of α. For example, for α = 0.6, the best-fitvalue of Im(CT ) is 0.067, which is substantially lowerthan the required value implied in Ref. [20]. Note thatthis best-fit point has |CT | ≈ 0.072, which correspondsto Λ ∼ 580 GeV. However, a reliable prediction for I isneeded to pin down the scale of NP.

We now provide a renormalizable model of flavor sym-metries that generates the effective operator in Eq. (6).This model preserves the subgroup

G =

(3∏i=1

U(1)qi × U(1)ui

)× U(3)d

× U(2)` × U(2)e × U(1)`3 × U(1)e3

(26)

of the U(3)5, which is the global flavor symmetry of SMin the absence of Yukawa couplings. Here we use left-handed notation for representing all fermionic fields. Thequark doublets qi have charges +1 under U(1)qi , whilethe quark singlets uci have charges −1 under U(1)ui . Theremaining quark singlets dci form a triplet of U(3)d. Thelepton doublets `i and singlets eci from the first two gener-

(SU(3)C , SU(2)W )Y U(1)q2 U(1)u1 U(1)`3 U(1)e3B (3, 1)−1/3 0 1 0 0

Bc (3, 1)1/3 -1 0 0 0

N (1, 1)0 0 0 -1 0Nc (1, 1)0 0 0 0 1Φc (3, 1)2/3 0 +1 0 -1

Φy (1, 1)1 0 0 0 0

TABLE I. Charge assignments of new particles

ations are doublets under U(2)` and U(2)e, respectively.The leptons in the third generation, `3 and ec3, carrycharges +1 and −1, under U(1)`3 and U(1)e3 , respec-tively.

The additional particles in our model are four Weylfermions B,Bc, N,N c, and two scalars Φc,Φy. Thegauge and flavor quantum numbers of these particles aregiven in Table I. The full set of marginal NP interactionsare given as

L ⊃ k1 H†q2B

c + k2 H`3N + k3 Φcuc1N

c

+ k4 Φ†cec3B + k5 Φyu

c1B + k6 Φ†ye

c3N

c

+ k7 |HΦc|2 + k8 |HΦy|2 + k9 |ΦcΦy|2 .(27)

Scalar masses preserve flavor symmetries. In order togive masses to NP fermions, we add two extra spurionsMB and MN , both charged (+1,−1), under symmetries(U(1)q2 , U(1)u1

) and (U(1)`3 , U(1)e3), respectively.

L ⊃ MBBBc + MNNN

c+ m2c |Φc|2+ m2

y |Φy|2 . (28)

The SM Yukawa couplings break G to U(1)B ×∏3i=1 U(1)Li , the familiar baryon number and flavor lep-

ton number symmetries. In our scheme, the down-typeYukawa coupling matrix Yd is such that it can be di-agonalized by using only the U(3)d symmetry. Theup-type Yukawa matrix Yu will then have the form

V †CKM×diag (yu, yc, yt). The charged-lepton Yukawa ma-trix Ye is diagonalizable only using the U(2)`×U(2)e sym-metry. This ensures that flavor rotations only affect op-erators involving left-handed up-type quarks, so that noextra operators are generated along with Eq. (7). Theseconsiderations would guide the mechanism that generatesthe flavor structure at a high scale.

q2 H† q2 H†

τ c ucτ cuc

ℓ3 H ℓ3 H

MB

Φc

MN

MB

Φy

MN

FIG. 2. Feynman diagrams that generate NP effective opera-tors responsible for τ decays.

5

Given the full set of our symmetries and symmetry-breaking spurions, there are no additional counter-terms.In Fig. 2 we show Feynman diagrams where the key effec-tive operators are generated. The left diagram gives riseto(q2H

†ec3)

(`3Huc1), which upon Fierz transformation

yields the tensor operator in Eq. (6) along with its scalarversion

(q2H

†uc1)

(`3Hec3). After matching we find

KΛ4

= − k1k2k3k42MBMNm2

c

. (29)

The diagram on the right of Fig. 2, on the other hand,gives rise to only the scalar operator

(q2H

†uc1)

(`3Hec3)

with a different effective coupling. Therefore, there existregions in the parameter space of the EFT where only thetensor operator dominates, thus realizing our scenario.

Note that our scenario also generates two additionaloperators of mass dimension 6, which would give rise toanomalous Zντντ couplings. These would constrain thecombination (k2/MN ) from invisible Z width measure-ments [4]. There will also be a series of dimension-8 op-erators, some breaking lepton flavor universality, whichmay have important phenomenological consequences rel-evant for collider experiments like LHC and Belle-II.However, the full analysis of this specific UV model isbeyond the scope of this paper and we leave it for futureendeavors.

To summarize, the renormalizable flavor model pro-posed by us can account for the anomalies in Vus andAτCP measurements, evading the no-go bound for thelatter. Further exploration of these anomalies and themodel could unearth hitherto uncharted ways of probingthe flavor generation mechanism at a high scale, via lowenergy flavor data.

ACKNOWLEDGEMENTS

TSR was supported in part by the Early CareerResearch Award by Science and Engineering ResearchBoard, Dept. of Science and Technology, Govt. of India(grant no. ECR/2015/000196). We also acknowledgethe workshop “Beyond the Standard Model: where dowe go from here?” hosted at the Galileo Galilei Insti-tute for Theoretical Physics where part of the work wascompleted.

Appendix A: Correction to δRτ due to NP

With the inclusion of the NP tensor operator, Rτ,s[see Eq. (11)] will receive additional contributions fromvector-tensor interference and tensor-tensor correlators.

The net correction to δRτ is

δRNPτ =

12π2

m2τ

∫ m2τ

0

ds

(1− s

m2τ

)[12 Re(CT )

ρTV (q2)

mτ

−16|CT |2(

1 +s

2m2τ

)(ρ

(Q)TT (q2) + ρ

(R)TT (q2)

], (A1)

with s = q2 being the invariant mass of the hadronicstates. To derive the above expression we have used thefollowing spectral functions:

ρµναTV (q) =

∫dΠn(2π)4δ3(q − pn)×∑

n

〈0|Tµν |n〉〈n|V α†|0〉 , (A2)

ρµν,αβTT (q) =

∫dΠn(2π)4δ3(q − pn)×∑

n

〈0|Tµν |n〉〈n|Tαβ†|0〉 , (A3)

with V α ≡ uγαs and Tµν ≡ uσµνs. The spectral func-tions can be Lorentz decomposed as [24]

ρµναTV (q) = i(gµαqν − gναqµ)ρTV (q2) , (A4)

ρµν,αβTT (q) = Qµν,αβ(q)ρ(Q)TT (q2) +Rµν,αβ(q)ρ

(R)TT (q2) ,

(A5)

where

Qµν,αβ(q) =(qµqβgνα + qνqαgµβ − qµqαgνβ − qνqβgµα) ,(A6)

Rµν,αβ(q) =Qµν,αβ(q) + q2(gµαgνβ − gµβgνα) . (A7)

Using the QCD sum rule framework, the integral in (A1)can be written as a contour integral on the circle s = m2

τ ,in terms of the associated correlators ΠTV and ΠTT .

δRNPτ =

6πi

m2τ

∮s=m2

τ

ds

(1− s

m2τ

)[12 Re(CT )

ΠTV (q2)

mτ

−16|CT |2(

1 +s

2m2τ

)(Π

(Q)TT (q2) + Π

(R)TT (q2)

]. (A8)

To compute the contour integral, we use the followingterms in OPE of the correlators [24] in the large-q2 limit:

ΠTV (q2) ≈q2→∞

2

q2〈0|1

2(uu+ ss)|0〉 , (A9)

Π(Q)TT (q2) = Π

(R)TT (q2) ≈

q2→∞− Nc

24π2ln(−q2) , (A10)

where Nc(= 3) is the color factor. Evaluating the integralin Eq. (A8), we obtain Eq. (14):

δRNPτ ≈ −288π2Re(CT )

〈0| 12 (uu+ ss)|0〉m3τ

− 18|CT |2 .(A11)

6

Appendix B: Decay rate of τ → Ksπντ

The differential decay rate of τ → Ksπντ in the pres-ence of NP is given by [20, 42]

dΓ

ds= G2

F |Vus|2SEW(m2

τ − s)2λ1/2(s)(m2K −m2

π)2

512mτs3×[

|f0(s)|2 + ζ(s)

(|f+(s)− T (s)|2 +

2(m2τ − s)2

9sm2τ

|T (s)|2)]

(B1)

where λ(s) ≡ λ(s,m2K ,m

2π), SEW, and the form factors

f+(s), f−(s) and BT (s) are defined in Eq. (20) and (21).

The remaining scalar form factor f0(s) can be related tothe above form factors as

f−(s) =m2K −m2

π

s

(f0(s)− f+(s)

). (B2)

The other quantities are defined as

ζ(s) ≡ (m2τ + 2s)λ(s)

3m2τ (m2

K −m2π)

, (B3)

T (s) ≡ 3smτ

mK(m2τ + 2s)

CTBT (s) . (B4)

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