Determining Strong Phase of $a_1$ Meson Decay Amplitude using $W \to \nu \tau(\to \nu a_1(\to\pi^\mp \pi^\mp \pi^\pm))$ Process

To measure the helicity of a spin-1 meson from the triple vector product of the three-momenta of its decay products, one needs information about the strong phase of the decay amplitude. In this paper, taking $a_1(1260)$ meson as an example, we present a method to extract information about the strong phase from the triple vector product of the pion momenta in $W \to \nu \tau(\to \nu a_1(\to\pi^\mp \pi^\mp \pi^\pm))$ process, where the $a_1$ helicity is known a priori from electroweak theory. This process is advantageous in that highly-boosted $a_1^-$ mesons from $\tau_L^-$ decays have nearly maximal helicity asymmetry and thus most reflect the strong phase. We revisit the theoretical calculation of the $a_1$ meson helicity in $W \to \nu \tau(\to \nu a_1)$ process. We formulate the differential decay rate of polarized $a_1$ mesons in a manner convenient for the study of the $a_1$ meson helicity asymmetry. Finally, we present the method for extracting information about the strong phase, and assess its feasibility at the LHC.


Introduction
The helicity of spin-1 mesons can be a probe for physics beyond the Standard Model (SM). For example, in B − → K − π − π + γ process induced by b → sγ, the SM predicts that the K − π − π + system is mostly left-handed because W boson loop gives an amplitude with a left-handed photon, while various extensions of the SM contain an extra amplitude with a right-handed photon. The helicity of the K − π − π + system can be determined from the triple vector product of the three-momenta of K − , π − , π + , and indeed a non-zero polarization of the system has been confirmed experimentally [1]. Nevertheless, the helicity has not been measured. The difficulty lies in the fact that the triple vector product of three-momenta is a naïve T-odd quantity [2] (odd under the reversal of all three-momenta and spins), and in CP-conserving theories like QCD its expectation value is non-zero only with the strong phase of the decay amplitude of K − π − π + resonances, which is poorly understood. Since K 1 (1270) and K 1 (1400) resonances (the latter is much suppressed) contribute to B − → K − π − π + γ process [3], efforts have been made to theoretically or phenomenologically determine the strong phase of K 1 (1270) and K 1 (1400) decay amplitudes [4,5,6,7,8,9]. Notably, Ref. [8] has pursued a purely phenomenological approach where one extracts, from experimental data on B − → K − π − π + J/ψ process, information about the strong phase necessary for the K 1 (1270) helicity measurement.
In this paper, we study experimental determination of the strong phase of a spin-1 meson's decay amplitude which utilizes a hadronic decay of τ lepton from a W boson decay. Since the helicity of a spin-1 meson in the decay of a polarized τ is known a priori from electroweak theory, we can use W → ντ (→ νA) events (A denotes a spin-1 meson) to determine the strong phase. Moreover, W → ντ (→ νA) process is advantageous in that highly-boosted spin-1 mesons in W → ντ (→ νA) events have nearly maximal helicity asymmetry (i.e. almost purely left-handed or right-handed) and hence the impact of the strong phase is maximized.
Once the strong phase of the a − 1 → π − π − π + decay amplitude is determined, one can use it to search for new physics through the a 1 polarization. Moreover, we expect that the strong phase of the K − 1 → K − π − π + decay amplitude is determined in basically the same manner, which is then utilized for the most interesting case, the photon polarization measurement in This paper is organized as follows: In Section 2, we revisit the theoretical calculation of the helicity of a − 1 meson in W − → ν τ τ − (→ ν τ a − 1 ) process. The a − 1 helicity is calculated as a function of the energy fraction of a − 1 in τ − decay in the laboratory frame, z = E a 1 /E τ . We will confirm that a − 1 with z 0.8 (i.e. highly-boosted a − 1 ) is almost purely left-handed. In Section 3, we express the differential decay rate of polarized a − 1 mesons using the following parametrization: Let p 1 , p 2 , p 3 respectively denote the four-momenta of π − , π − , π + , with Q·p 1 > Q · p 2 (Q = p 1 + p 2 + p 3 ). In an a − 1 rest frame, we write the angle between p 3 × p 1 and a − 1 's boost direction in the laboratory frame as Ψ, and write the angle between p 3 and the projection of a − 1 's boost direction onto the a − 1 decay plane as φ. The angles Ψ, φ and the Dalitz variables s 13 = (p 1 + p 3 ) 2 , s 23 = (p 2 + p 3 ) 2 completely parametrize the differential decay rate. A benefit of this parametrization is that that part of the differential decay rate which reflects the a − 1 helicity asymmetry is simply linear in cos Ψ and is independent of φ. 2 In Section 4, we present a method to determine the strong phase using W → ντ (→ νa 1 (→ π ∓ π ∓ π ± )) events, based on the theoretical calculation of the a − 1 helicity in Section 2 and the parameterization of the differential decay rate in Section 3. Statistical uncertainty in the above determination at the 14 TeV LHC with 300 fb −1 of data is further estimated. Section 5 summarizes the paper.
In Appendix, we give a simple derivation of the Wigner rotation, which is used in the calculation of the a − 1 helicity in boosted τ − decays in Section 2.
Integrating out the azimuthal angle of a − 1 momentum around the τ − helicity axis, we remove interference among amplitudes with different a − 1 helicities. The differential decay rate is then factorized into the one for τ − → ν τ a − 1 process and the one for a − 1 → π − π − π + process, and is expressed as whereθ denotes the angle between the a − 1 momentum and the τ − helicity axis in a τ − rest frame. For convenience, we tradeθ for the energy fraction of a − 1 in τ − decay in the laboratory frame, z, where β denotes the speed of τ − in the laboratory frame. Eq. (2) is then rewritten as with dΓ(τ − h → ν τ a − 1,λ )/dz corresponds to the differential decay rate of τ − with helicity h decaying into a − 1 with helicity λ, for a specific value of z. dΓ(a − 1,λ → π − π − π + ) corresponds to the differential decay rate of a − 1 with helicity λ.
In the rest of the section, we evaluate dΓ(τ − h → ν τ a − 1,λ )/dz. The helicity amplitude M(τ − h → ν τ a − 1,λ ) is given by where we retain Q 2 dependence of the polarization vector ǫ µ , since a 1 is a broad resonance.
The helicity amplitude M(τ − h → ν τ a − 1,λ ) is specified in terms of the a − 1 helicity along the a − 1 boost direction in a τ − rest frame, λ τ , and the angle between the a − 1 momentum and the τ − helicity axis in a τ − rest frameθ; we find Experimentally, what we measure is the a − 1 helicity along the a − 1 boost direction in the laboratory frame, λ lab , not the helicity along the a − 1 boost direction in a τ − rest frame λ τ . Hence, we want to rewrite Eq. (8) in terms of λ lab . For this purpose, we expand the a − 1 polarization vectors in the laboratory frame in terms of those in a τ rest frame, as where d J=1 λ ′ λ is a d-function, andθ is the angle between the a − 1 boost direction in a τ − rest frame and that in the laboratory frame, measured in an a − 1 rest frame.θ is expressed in terms of the angle between the a − 1 momentum and the τ − helicity axis in a τ − rest frameθ, and the speed and boost factor of τ − in the laboratory frame β and γ = 1/ 1 − β 2 , as (see Appendix for the derivation) or equivalently, in terms of the energy fraction of a − 1 in τ − decay in the laboratory frame z, as The helicity amplitude is rewritten in terms of the a − 1 helicity in the laboratory frame λ lab , as withθ given in Eq. (12) or Eq. (13). Assembling Eqs. (5),(8), (14),(13), we numerically calculate dΓ(τ − h → ν τ a − 1,λ )/dz and present it in the form of the normalized differential decay rate (15) in Fig. 2, for each τ − helicity h = −1/2 (τ − L ) and h = 1/2 (τ − R ). The boost factor of τ − is fixed as γ = 23, corresponding to the boost factor of τ − from the decay of a W boson at rest. However, the plots are almost independent of γ when γ 3. Three different values of the a − 1 invariant mass, Q 2 = 1.13 GeV, 1.23 GeV, 1.33 GeV, are considered.
The left panel of Fig. 2 presents the a − 1 helicity in W − →ν τ τ − (→ ν τ a − 1 ) process. In the left panel, we observe that a − 1 meson with z 0.8 is almost purely left-handed (λ lab = −1). We note that Fig. 2 is in agreement with the preceding study Ref. [13].

Differential Decay Rate of Polarized a − 1 Mesons
We formulate the differential decay rate of a − 1 → π − π − π + process for each a − 1 helicity, Eq. (6), in a manner convenient for the study of the helicity asymmetry.
The helicity amplitude is written as where ǫ µ (Q, λ) is the polarization vector of a − 1 with helicity λ, and J µ is the hadronic current, The most general parametrization for the hadronic current J µ that respects (i) Lorentz covariance, (ii) the current conservation Q µ J µ = 0, and (iii) Bose symmetry of two π − 's, is given as follows: Let p 1 , p 2 , p 3 respectively denote the momenta of π − , π − , π + (then Q µ = (p 1 +p 2 +p 3 ) µ ), where the two π − 's are distinguished by Q · p 1 > Q · p 2 . The most general parametrization is where MeV is the pion decay constant, and F (Q 2 , s 13 , s 23 ) is a general function of three Lorentz scalars. F (Q 2 , s 13 , s 23 ) is normalized in such a way that if ππ resonances were absent, we would have F (Q 2 , s 13 , s 23 ) = 1 by chiral perturbation theory [14].
We explicitly write the a − 1 decay helicity amplitude Eq. (16) in the a − 1 rest frame whose z-axis is along the a − 1 boost direction in the laboratory frame (thus λ lab is along this z-axis). In this frame, the momenta of the three pions and their sum can be parametrized as where x i andx i are defined in terms of Q 2 , s 13 , s 23 as θ i (i = 1, 2) denotes the angle between p i and p 3 , which is given in terms ofx i as Ψ is the angle between p 3 × p 1 vector and the z-axis, and φ is the angle between p 3 vector and the projection of the z-axis onto the a − 1 decay plane, which satisfy The polarization vectors are given by From Eqs. (16), (18), (27), the helicity amplitudes are expressed as where A and B are structure functions with mass dimension +2 defined as 3 (remind thatx i is related to Q 2 , s 13 , s 23 through Eq. (24)) Finally, we plug Eqs. (28),(29) into the formula for the polarized a − 1 differential decay rate  3 In the m π → 0 limit, they asymptote as Consider a general a 1 production process, not limited to the W → ντ (→ a 1 ν) process. In terms of the transverse and asymmetric helicity fractions in the laboratory frame defined by , the a − 1 differential decay rate in a general a 1 production process satisfies (general process) where N nor = 2 3 ds 12 ds 13 (|A| 2 + |B| 2 ).
From Eq. (37), we find that the cos Ψ asymmetry is proportional to both the helicity asymmetry P A and the term Im(A · B * )/N nor , the latter of which is non-zero only with the strong phase.
cos Ψ is a naïve T-odd quantity, and its expectation value is non-zero only with the strong phase, in accordance with what is stated in Section 1.
Once the function Im(A · B * )/N nor is known, one can measure P A using asymmetry of the number of events with cos Ψ > 0 and cos Ψ < 0. Conversely, if the helicity asymmetry of a − 1 is known a priori, one can determine Im(A · B * )/N nor by measuring asymmetry of the number of events with cos Ψ > 0 and cos Ψ < 0. This is indeed feasible in τ − L → ν τ a − 1 (→ π − π − π + ) process, for which the a − 1 helicity is theoretically calculable as done in Section 2. To be specific, we write the differential decay rate of the τ − L → ν τ a − 1 (→ π − π − π + ) process in terms of 1 Γ dΓ λ lab dz Eq. (15) for τ − L as Since 1 Γ dΓ λ lab dz can be computed theoretically, it is possible to determine Im(A · B * )/N nor from the cos Ψ asymmetry of τ − L → ν τ a − 1 (→ π − π − π + ) events. A problem is that when we use pp → W ∓ → ν τ ∓ (→ ν π ∓ π ∓ π ± ) events to collect τ − L , it is difficult to reconstruct z, since two neutrinos contribute to the missing transverse momentum. In this paper, we evade the reconstruction of z by exploiting a positive correlation between z and a − 1 's transverse mass M T . We impose a tight selection cut on M T and thereby select events with large z. Im(A · B * )/N nor is determined from the cos Ψ asymmetry of events, divided by the convolution of theoretically calculated 1 Γ dΓ + dz − 1 Γ dΓ − dz and the reweighting function of z under given selection cuts (the reweighting function is obtainable from a Monte Carlo simulation).
An advantage of the above method is that, for τ − L and for large z, 1 is maximized (see the left panel of Fig. 2). Hence, the cos Ψ asymmetry is maximized and the statistical uncertainty in the determination of Im(A · B * )/N nor is reduced.
We comment that, for τ − L and for z → 1, 1 dz quickly approaches to its value at z = 1 and is insensitive to the precise value of z. Hence, once we collect large-z events, the convolution of 1 Γ dΓ + dz − 1 Γ dΓ − dz and the reweighting function of z is not affected by details of the reweighting function, which reduces the systematic uncertainty associated with the estimation of the reweighting function. However, confirming this reduction of the systematic uncertainty is beyond the scope of the present paper.
4 Method to determine Im(A · B * )/N nor
Remind that N(Q 2 ) and Im(A · B * )/N nor do not depend on z.
In real experiments, we cannot measure the right hand side of Eq. (42), since we do not reconstruct z. Instead, we propose to measure the following quantity: where f cut (z, Q 2 , s 13 , s 23 ) denotes the fraction of events with specific values of z, Q 2 , s 13 , s 23 that pass the selection cuts, which is common for cos Ψ > 0 and cos Ψ < 0. From Eqs. (42),(45) and z-independence of N(Q 2 ) and Im(A · B * )/N nor , we have The right hand side is measured in experiments. As for the left hand side, N(Q 2 ) is known from the pp → W ∓ total cross section and branching fractions of has been calculated theoretically in Section 2. f cut (z, Q 2 , s 13 , s 23 ) can be evaluated with a Monte Carlo simulation by exploiting generator-level information on z. Therefore, it is possible to determine Im(A · B * )/N nor .
(48) w(z, Q 2 , s 13 , s 23 ) is interpreted as the reweighting of events with specific values of z, s 13 , s 23 , Q 2 due to the selection cuts, which is again common for cos Ψ > 0 and cos Ψ < 0. In terms of w(z, Q 2 , s 13 , s 23 ), Eq. (46) is recast in the form, In Section 4.2, we generate detector-level Monte Carlo events for the 14 TeV LHC and impose selection cuts on them. Using these events, in Section 4.3, we evaluate w(z, Q 2 , s 13 , s 23 ).
In Section 4.4, we estimate statistical uncertainty in a measurement of the right hand side of Eq. (49) with 300 fb −1 of data.

Monte Carlo Event Generation and Selection Cuts
Using MadGraph5 aMC@NLO [15] with the TauDecay package [16], we generate parton-level events for the process (charged-conjugated process is also considered), for √ s = 14 TeV pp collisions. Then, we use PYTHIA8 [17] to simulate parton showering. The groups of events with 0, 1, 2 parton(s) are matched with MLM-matching [18] algorithm.
For the events generated, we use the Delphes3 program [19] to simulate the CMS detector effects, considering |η τ -jet | < 2.5, p τ -jetT > 1 GeV and 60% tagging efficiency for the identification of a three-prong τ -jet. We reconstruct an a − 1 meson from a three-prong τ -jet by requiring that the three charged tracks have charges summed to that of a − 1 and that the invariant mass (calculated by assuming that each charged track is a pion) be less than 2 GeV. The variables cos Ψ, s 13 , s 23 are calculated as described in Section 3.
We impose the following selection cuts on the above samples: • Event must contain exactly one reconstructed a − 1 meson.
Additionally, we impose • Transverse mass for the a − 1 , where p a 1 T denotes the transverse momentum of the a − 1 , and φ a 1 p / T is the azimuthal angle between the a − 1 and the missing transverse momentum, must satisfy either M T > 50 GeV, 60 GeV or 70 GeV.
The number of the sum of W + and W − events with 300 fb −1 of data at each stage of event selection and for each M T cut is tabulated in Table 1. Table 1: Number of the sum of pp → W ∓ → ν τ ∓ (→ ν π ∓ π ∓ π ± ) events at the 14 TeV LHC with 300 fb −1 of data, after each selection cut.
Selection cut Number of W + and W − events One reconstructed a − 1 and p / T > 25 GeV 40.  We have confirmed that large-z events are efficiently collected with a tight M T cut such as M T > 70 GeV.

Statistical Uncertainty
We estimate statistical uncertainty in a measurement of the right hand side of Eq. (49) with 300 fb −1 of data.
The statistical uncertainty is given as follows: Let δN + (δN − ) denote the number of events after the selection cuts of Section 4.2, in a bin of Q 2 , s 13 , s 23 with cos Ψ > 0 (cos Ψ < 0). If the bins are sufficiently narrow, the right hand side of Eq. (49) is approximated by for which the ratio of the statistical uncertainty over its value is given by This corresponds the relative statistical uncertainty in the determination of Im(A · B * )/N nor , and hence we obtain In Fig. 4, we present the relative statistical uncertainty Eq. (54) in each bin of (s 13 , s 23 ) with 1.26 GeV> Q 2 >1.20 GeV, for various M T cuts, at the 14 TeV LHC with 300 fb −1 of data. We observe that the relative statistical uncertainty is below 2% in multiple bins of (s 13 , s 23 ), and so the determination of Im(A · B * )/N nor ) is feasible at the 14 TeV LHC with 300 fb −1 of data, at least in light of statistics.
A tighter M T cut diminishes overall statistics, but it enhances the relative cos Ψ asymmetry and the latter is largest for z ∼ 1. Nevertheless, we do not find improvement in relative statistical uncertainty with tigher M T cuts. This is because in the present simulation, the loss of overall statistics is more significant than the enhancement of the relative cos Ψ asymmetry.

Summary
We have presented a method to extract information about the strong phase of the a − 1 → π − π − π + decay amplitude necessary for the a 1 helicity measurement. Our method utilizes W → ντ (→ νπ ∓ π ∓ π ± ) events, for which the a 1 helicity is theoretically calculable. The method has an advantage that a − 1 mesons from τ − L decays with large boost (i.e. with z = E a 1 /E τ ∼ 1 in the laboratory frame) have nearly maximal helicity asymmetry and thus most reflect the strong phase. We have revisited the theoretical calculation of the a − 1 helicity in the laboratory frame in W − →ν τ τ − (→ ν τ a − 1 ) process. We have formulated the differential decay rate of polarized a 1 mesons, where the information about the strong phase necessary for the helicity measurement is encapsulated by the term Im(A · B * )/N nor . Finally, we have proposed a method to determine Im(A · B * )/N nor from pp → W → ντ (→ νπ ∓ π ∓ π ± ) events, and by estimating the statistical uncertainty at the 14 TeV LHC with 300 fb −1 of data, we have revealed that this method is feasible at least in light of statistics. In Fig. 5, we show all the relevant Lorentz frames: The Lorentz frame 1 and 4 is a vector boson rest frame, where angular-momentum-quantization axis z 1 is chosen along the vector boson's three-momentum in the τ rest frame 2 , while axis z 4 is chosen along the vector boson's three-momentum in the laboratory frame 3 . The τ rest frame 2 is obtained from the laboratory frame 3 by a boost along the τ momentum direction z in the laboratory frame.
Since the above successive transformations are all on the (z, x) plane, the quantization axis z 1 in the vector boson rest frame is recovered by a rotation byθ about the common y-axis, namely, we have where R y (θ) = e −iJ 2 θ denotes rotation about the common y-axis and B z (y) = e −iK 3 y denotes boost along the corresponding z-direction depicted in Fig. 5. Only two generators of the Lorentz transformations appear in Eq. (A.1), whose non-zero components are (J 2 ) jk = −iǫ 2jk , (K 3 ) 0k = (K 3 ) k0 = iδ k3 . In Eq. (A.1),θ denotes the angle between z-axis and the vector boson three-momentum in the τ rest frame, θ denotes the angle between z-axis and the vector boson's three-momentum in the laboratory frame, andθ is the Wigner rotation angle we want to derive. The rapidity along each direction satisfies tanh y τ = p τ /m τ = γβ , (A.2b) tanh y v = p v /m v = γ 2 (βω +k cosθ) 2 + (k sinθ) 2 , (A.2c) wherep v denotes the vector boson's three-momentum in the τ rest frame, p v denotes the vector boson's three-momentum in the laboratory frame, and p τ denotes the τ lepton's threemomentum in the laboratory frame. Here, we have parametrized the τ lepton's four-momentum in the laboratory frame 3 as p µ τ = (E τ , 0, 0, p τ ) = m τ (γ, 0, 0, γβ) .

β→1
(1 + a 2 ) cosθ + 1 − a 2 (1 − a 2 ) cosθ + 1 + a 2 , (A. 13) which is a good approximation for a vector meson in the decay of a τ lepton coming from the decay of W, Z. Finally, it is worth noting that the expression Eq. (A.9) gives the helicity conservation cosθ −−→ a→0 1 , (A.14) in the massless limit of the vector meson with a = m v /m τ → 0. There is no rotation (θ = 0) in the massless limit, because the helicity of a massless particle is an invariant of Lorentz transformations.