Hadronic tau decays as New Physics probes in the LHC era

We analyze the sensitivity of hadronic tau decays to non-standard interactions within the model-independent framework of the Standard Model Effective Field Theory (SMEFT). Both exclusive and inclusive decays are studied, using the latest lattice data and QCD dispersion relations. We show that there are enough theoretically clean channels to disentangle all the effective couplings contributing to these decays, with the $\tau \to \pi\pi\nu_\tau$ channel representing an unexpected powerful New Physics probe. We find that the ratios of non-standard couplings to the Fermi constant are bound at the sub-percent level. These bounds are complementary to the ones from electroweak precision observables and $p p \to \tau \nu_\tau$ measurements at the LHC. The combination of tau decay and LHC data puts tighter constraints on lepton universality violation in the gauge boson-lepton vertex corrections.

Hadronic tau decays have been extensively used in the last decades to learn about fundamental physics [1,2]. The inclusive decays are used to accurately extract fundamental Standard Model (SM) parameters such as the strong coupling constant [3][4][5], the strange quark mass or the V us entry of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [6,7]. They represent also a valuable QCD laboratory, where chiral parameters or properties of the QCD vacuum can be extracted with high precision in a model-independent fashion through dispersion relations [8,9]. On the other hand, exclusive hadronic tau decays are much harder to predict within QCD with high accuracy and they are thus useful to learn about hadronic physics. The only exceptions are the two-body decays τ → πν τ , Kν τ , thanks to the precise lattice calculations of the pion and kaon decay constants [10].
The agreement between the above-mentioned determinations of SM and QCD parameters with determinations using other processes represent a non-trivial achievement, which is only possible thanks to the impressive effort carried out in several fronts: experimental, lattice and analytical QCD methods. Needless to say, this agreement is easily spoiled if non-standard effects are present. However, the use of hadronic tau decays as New Physics (NP) probes has been marginal so far (see e.g. Refs. [11,12]), with the exception once again of the simple τ → πν τ , Kν τ channels. The goal of this letter is to amend this situation presenting an unprecedented comprehensive analysis of the NP reach of hadronic tau decays.
We focus for sake of definiteness on the non-strange decays, which are governed by the following low-energy effective Lagrangian [13] 1 where we use σ µν = i [γ µ , γ ν ]/2 and G F is the Fermi constant. The only assumptions are Lorentz and U (1) em × SU (3) C invariance, and the absence of light nonstandard particles. In practice we also assume that the subleading derivative terms in the EFT expansion (suppressed by m τ /m W ) are indeed negligible. The Wilson coefficients i parametrize non-standard contributions, and they vanish in the SM leaving the V − A structure generated by the exchange of a W boson. The nonstandard coefficients i can be complex, but the sensitivity of the observables considered in this work to the imaginary parts of the coefficients is very small. Thus, the results hereafter implicitly refer to the real parts of i . Through a combination of inclusive and exclusive τ decays, we are able to constrain all the Wilson coefficients in Eq. (1) -this is the main result of this paper. In the MS scheme at scale µ = 2 GeV we find the following central values and 1σ uncertainties: 1 We have not included right-handed (and wrong-flavor) neutrino fields [14], which in any case do not interfere with the SM amplitude and thus contribute at O( 2 i ) to the observables. where e L,R parametrize electron couplings to the first generation quarks and are defined in analogy to their tau counterparts. They affect the G F V ud value obtained in nuclear β decays [15], which is needed in the analysis of hadronic tau decays. The correlation matrix associated to Eq. (2) is Below we summarize how Eqs. (2)-(3) were derived.

I. EXCLUSIVE DECAYS
The τ → πν τ channel [16] gives the following 68% CL constraint: where B 0 = m 2 π /(m u + m d ). We included the SM radiative corrections [17][18][19] and the latest lattice average for the pion decay constant, f π ± = 130.2 (8) MeV (N f = 2 + 1) [20], from Refs. [21][22][23]. We stress that the lattice determinations of f π ± are a crucial input to search for NP in this channel, and despite its impressive precision, it represents the dominant source of error in Eq. (4), followed by the experimental error (2.4 times smaller), and the radiative corrections uncertainty. Because of this, significant improvement in the bound above can be expected in the near future. Alternatively, as often seen in the literature [1], one can obtain tighter constraints on the effective theory parameters by considering "theoretically clean" ratios of observables where the f π dependence cancels out. For example, from the ratio Γ(τ → πν)/Γ(π → µν) one can deduce This and similar constraints are not included in Eq. (2), which only summarizes the input from hadronic tau decays without using any meson decay observables. Instead, we later combine Eq. (2) with the results of Ref. [24], which derived a likelihood for the effective theory parameters based on a global analysis of pion and kaon decays. The combination effectively includes constraints from Γ(τ → πν)/Γ(π → ν), once correlations due to the common f π uncertainty are properly taken into account. The τ → ππν τ channel, which is sensitive to vector and tensor interactions, is much more complicated to predict within QCD in a model-independent way. However, a stringent constraint can be obtained through the comparison of the spectral functions extracted from τ → ππν τ and its isospin-rotated process e + e − → π + π − , after the proper inclusion of isospin-symmetry-breaking corrections. The crucial point here is that heavy NP effects (associated with the scale Λ) can be entirely neglected in e + e − → π + π − at energy √ s Λ due to the electromagnetic nature of this process. We can benefit from past studies that exploited this isospin relation to extract from data the ππ component of the lowest-order hadronic vacuum polarization contribution to the muon g − 2, usually denoted by a had,LO µ [ππ], through a dispersion integral. Such approach assumes implicitly the absence of NP effects, which however may contribute to the extraction from tau data. In this way we find a sub-percent level sensitivity to NP effects: where a τ µ = (516.2±3.6)×10 −10 [25] and a ee µ = (507.14± 2.58) × 10 −10 [26] are the values of a had,LO µ [ππ] extracted from τ and e + e − data. The ∼ 2σ tension with the SM reflects the well-known disagreement between both datasets [26]. 2 . In order to estimate the factor multiplying T in Eq. (6), we have (i) assumed that the proportionality of the tensor and vector form factors, which is exact in the elastic region [28,29], holds in the dominant ρ resonance region (as is the case within the resonance chiral theory framework [30]); (ii) used the lattice QCD result of Ref. [31] for the ππ tensor form factor at zero momentum transfer (see also Refs. [32,33]).
The constraint above can be strengthened by directly looking at the s-dependence of the spectral functions (instead of the a µ integral), which would also allow us to disentangle the vector and tensor interactions. 3 Moreover, the a τ,ee µ uncertainties include a scaling factor due to internal inconsistencies of the various datasets [26], which hopefully will decrease in the future. In fact, new analyses of the ππ channel are expected from CMD3, BABAR, and possibly Belle-2 [26,27]. All in all, we can expect a significant improvement in precision with respect to the result in Eq. (6) in the near future.
As recently pointed out in Ref. [12], a third exclusive channel that can provide useful information is τ → ηπν τ , since the non-standard scalar contribution is enhanced with respect to the (very suppressed) SM one. Because of this, one can obtain a nontrivial constraint on τ S even though both SM and NP contributions are hard to predict with high accuracy. Using the latest experimental results for the branching ratio [16,34] and a very conservative estimate for the theory errors [12,35,36] we find 2 Recently Ref. [27] found a ee µ = 503.74 ± 1.96 using similar data but a different averaging method than Ref. [25]. This increases to 3σ the tension between a τ µ and a ee µ 3 Useful angular and kinematic distributions including NP effects were recently derived in Ref. [29].
which will significantly improve if theory or experimental uncertainties can be reduced. The latter will certainly happen with the arrival of Belle-II, which is actually expected to provide the first measurement of the SM contribution to this channel [37] (see also Ref. [38] for Belle results). This is the only probe in this work with a significant sensitivity (via O( 2 S ) effects) to the imaginary part of i coefficients. Including the latter does not affect the bound in Eq. (7) though.

II. INCLUSIVE DECAYS
Summing over certain sets of decay channels one obtains the so-called inclusive vector (axial) spectral functions ρ V (A) , which are nothing but the sum of the hadronic invariant mass distributions up to some constants and kinematic factors [1,2]. In the SM they are proportional to the imaginary parts of the associated V V (AA) two-point correlation functions, Π V V (AA) (s), but these relations are modified by NP effects [39,40]. Thus, one could directly use the latest measurements of these spectral functions to constrain such effects if we had a precise theoretical knowledge of their QCD prediction. However, perturbative QCD is known not to be valid at √ s < 1 GeV, especially in the Minkowskian axis, where the spectral function lies. Nevertheless, one can make precise theoretical predictions for integrated quantities exploiting the well-known analiticity properties of QCD correlators [3]. Here we extend the traditional approach to include also NP effects, finding [39,40] s0 where ω(x) is a generic analytic function and ρ exp V ±A (s) is the sum/difference of the vector and axial spectral functions, extracted experimentally under SM assumptions [2,25]. We also introduced the couplings V /A ≡ τ L±R − e L+R . Last, the contributions X V V /AA and X V T can be calculated via the Operator Product Expansion, as discussed in Appendix A. Eq. (8) shows how the agreement between precise SM predictions (RHS) and experimental results (LHS) for inclusive decays can be translated into strong NP constraints.
In the V +A channel, we find two clean NP constraints using ω(x) = (1 − x) 2 (1 + 2x), which gives the total nonstrange BR, and with ω(x) = 1. They provide respectively In the V − A channel, where the perturbative contribution is absent, two strong constraints can be obtained using ω(x) = 1 − x and ω(x) = (1 − x) 2 : τ L+R − e L+R + 1.9 τ R + 8.0 τ T = (10 ± 10) · 10 −3 . (12) DV dominate uncertainties for the first constraint, while experimental and f π uncertainties dominates the latter one. This constraint could be improved with more precise data and f π calculations, but at some point DV, much more difficult to control, would become the leading uncertainty. The non-neglibible correlations between the various NP constraints derived above (due to f π and experimental correlations) have been taken into account in Eq. (2).

III. ELECTROWEAK PRECISION DATA
If NP is coming from dynamics at Λ m Z and electroweak symmetry breaking is linearly realized, then the relevant effective theory at E m Z is the socalled SMEFT, which has the same local symmetry and field content as the SM, however the Lagrangian contains higher-dimensional operators encoding NP effects [13,41]. The SMEFT framework allows one to combine in a model-independent way constraints from lowenergy measurements with those from Electroweak Precision Observables (EWPO) and LHC searches. Moreover, once the SMEFT is matched to concrete UV models at the scale Λ, one can efficiently constrain masses and couplings of NP particles. The dictionary between low-energy parameters in Eq. (1) and Wilson coefficients in the Higgs basis [42,43] where we approximated V CKM ≈ 1 in these O(Λ −2 ) terms. δg W f L/R are corrections to the SM W f f vertex and c i are 4-fermion interactions with different helicity structures; see Appendix B for their precise definitions. Note that R is lepton-universal in the SMEFT, up to dim-8 corrections [13,44]. We perform this matching at µ = M Z , after taking into account the QED and QCD running of the low-energy coefficients i up to the electroweak (EW) scale [45]. Electroweak and QCD running to/from 1 TeV is also carried out in the comparison with LHC bounds below.
Our results are particularly relevant for constraining lepton universality (LU) violation. To illustrate this point, in Fig. 1 we compare the different LEP and lowenergy observables that are directly sensitive to a different coupling of the W boson with leptons of the first and third generation, which we assume here to be the dominant NP effect. In fact, W → ν decays in LEP-2 display a ∼ 2σ preference for LU violation [16,46], corresponding to the best fit δg W τ L − δg W e L = 0.022 (12). Combining the information from the hadronic τ decay observables discussed in this letter we find δg W τ L − δg W e L = 0.0062 (32). This is ∼3.5 times more sensitive than the LEP-2 constraint, and competitive with other sensitive LU probes. Much like W decays in LEP-2, the hadronic τ decays also display a ∼ 2σ preference for δg W τ L − δg W e L > 0, mainly due to the tension in Eq. (6).
One can do a more sophisticated analysis in the case when all dimension-6 SMEFT operators are present simultaneously. As a matter of fact, Ref. [43] carried out a flavor-general SMEFT fit to a long list of precision observables, which did not include however any observable sensitive to qqτ τ interactions. As a result, no bound was obtained on the four-fermion Wilson coefficients, [c i ] τ τ 11 . From Eq. (13), given that [c (3) q ] ee11 and the vertex corrections δg are independently constrained, hadronic tau decays imply novel limits on these coefficients. We find (14) after marginalizing over the remaining SMEFT parameters. These are not only very strong but also unique low-energy bounds. On the other hand Ref. [43] did access the right-handed vertex correction: δg W q1 R = −(1.3± 1.7) × 10 −2 , from neutron beta decay [24,47]. Including hadronic tau decays in the global fit improves this significantly: δg W q1  (2), whereas the fourth one includes also clean LU ratios such as Γ(τ → πν)/Γ(π → µν).

IV. LHC BOUNDS
It is instructive to compare the NP sensitivity of hadronic tau decays to that of the LHC. While the experimental precision is typically inferior for the LHC, it probes much higher energies and may offer a better reach for the Wilson coefficients whose contribution to observables is enhanced by E 2 /v 2 . We focus on the highenergy tail of the τ ν production. This process is sensitive to the 4-fermion coefficients [c (3) q , c equ , c edq , c (3) equ ] τ τ 11 , which also affect tau decays. Other Wilson coefficients in Eq. (13) do not introduce energy-enhanced corrections to the τ ν production, and can be safely neglected in this analysis. 5 In Table I we show our results based on a recast of the transverse mass m T distribution of τ ν events in √ s = 13 TeV LHC collisions recently measured by AT-LAS [50]. We estimated the impact of the Wilson coefficients on the dσ(pp → τ ν)/dm T cross section using the Madgraph[51]/Pythia 8 [52]/Delphes [53] simulation chain. We assign 30% systematic uncertainty to that estimate, which roughly corresponds to the size of 4 A 50% stronger (weaker) bound on δg W q 1 R is obtained using the recent lattice determination of the axial charge in Ref. [48] (Ref. [49]). 5 Other energy-enhanced operators do not interfere with the SM.
Thus, their inclusion would not change our analysis.
the NLO QCD corrections to the NP terms (not taken into account in our simulations) [54]. The SM predictions are taken from [50], and their quoted uncertainties in each bin are treated as independent nuisance parameters. We find that for the chirality-violating operators the LHC bounds are comparable to those from hadronic tau decays. On the other hand, for the chiralityconserving coefficient [c (3) q ] τ τ 11 the LHC bounds are an order of magnitude stronger thanks to the fact that the corresponding operator interferes with the SM qq → τ ν amplitude. Let us stress that SMEFT analyses of highp T data require additional assumptions though, such as heavier NP scales and suppressed dimension-8 operator contributions. Last, we observe an O(2)σ preference for a non-zero value of [c (3) q ] τ τ 11 due to a small excess over the SM prediction observed by ATLAS in several bins of the m T distribution. (green) after using the LHC input to constrain [c (3) q ]ττ11 and [c (3) q ]ee11. We also show separate 95% CL contours from hadronic τ decays (red) and from previous EWPO [43] (blue), which are correlated (see discussion below Eq. (5).
The LHC and τ decay inputs together allow us to sharpen the constraints on LU of gauge interactions. From Table I , complementing the information from previous low-energy EWPO [43]. The interplay between the two is shown in Fig. 2. The input from hadronic tau decays leads to the model independent constraint on LU of W boson interactions: δg W τ L −δg W e L = 0.0134 (74). The error is more than factor of two larger than in the previously considered scenario where δg W τ L − δg W e L was the only deformation of the SM, however the present bound holds for generic heavy NP scenarios where all dimension-6 SMEFT operators can be generated with arbitrary coefficients.

V. CONCLUSIONS
We have shown in this work that hadronic τ decays represent competitive NP probes, thanks to the very precise measurements and SM calculations. This is a change of paradigm with respect to the traditional approach, which considers these decays as a QCD laboratory where one can learn about hadronic physics or extract fundamental parameters such as the strong coupling constant. From this new perspective, the agreement between such determinations [3][4][5] and that of Ref. [10] in the lattice is recast as a stringent NP bound. Our results are summarized in Eq. (2) and can be easily applied to constrain a large class of NP models with the new particles heavier than m τ . Hadronic τ decays probe new particles with up to O(10) TeV masses (assuming order one coupling to the SM) or even O(100) TeV masses, for strongly coupled scenarios. They can be readily combined with other EWPO within the SMEFT framework to constrain NP heavier than m Z . Including this new input in the global fit leads to four novel constraints in Eq. (14), which are the first model-independent bounds on the corresponding τ τ qq operators. Moreover, it leads to tighter bounds on the W boson coupling to right-handed quarks. Hadronic τ decays represent a novel sensitive probe of LU violation (τ vs. e), which competes with and greatly complements EWPO and LHC data. This is illustrated in Fig. 2 and Table I for vertex corrections and contact interactions respectively. Our constraints can thus be useful in relation with the current hints of LU violation in certain B mesons decays, or the old tension in W decays. For instance, our model-independent O(1)% constraints in Eq. (2) imply that the hints for O(10)% LU violation observed in B → D * τ ν decays [56][57][58] cannot be explained by NP effects in the hadronic decay of the τ lepton, but must necessarily involve (as is the case in most models) non-standard LU-violating interactions involving the bottom quark.
The discovery potential of these processes in the future is very promising since the constraints derived in this work are expected to improve with the arrival of new data (e.g. from Belle-II) and new lattice calculations. The τ → ππν τ channel represents a particularly interesting example through the direct comparison of its spectrum and e + e − → π + π − data. Last, the extension of our analysis to strange decays of the tau lepton represents another interesting research line for the future.
where = τ, e. In the Higgs basis the δg's are treated as independent parameters spanning the space of dimension-6 operators; in the Warsaw basis they can be expressed as linear combinations of several Wilson coefficients [42,43]. The vertex correction δg W q1 L (parametrizing the W coupling to left-handed up and down quarks) does not appear in Eq. (13) because its effect cancels in the τ L − e L difference. The four-fermion coefficients c i are defined by L ⊃ [c (3) lq ] τ τ 11 (lγ µ σ i P L l)(qγ µ σ i P L q) +[c lequ ] τ τ 11 (lP R e)(qP R u) +[c ledq ] τ τ 11 (lP R e)(dP L q) +[c (3) lequ ] τ τ 11 (lσ µν P R e)(qσ µν P R u) ) where l = (ν τ , τ ) T , q = (u, d) T , and v ≈ 246 GeV is the vacuum expectation value of the Higgs field. [c (3) lq ] ee11 is defined analogously with l = (ν e , e) T . Both in the Higgs and the Warsaw basis the c i coefficients are independent parameters.