a_1 properties in hadronic tau decays
aa r X i v : . [ h e p - ph ] F e b a properties in hadronic tau decays Ina Lorenz ∗ Department of Physics, Indiana University, Bloomington, IN 47405, USACenter for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USAE-mail: [email protected]
Emilie Passemar
Department of Physics, Indiana University, Bloomington, IN 47405, USACenter for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USATheory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USAE-mail: [email protected]
Hadronic tau decays belong to the processes that show a resonance-like structure in the axialvector current in the 1 − a meson, seemsto show different properties in different processes. The process τ → πν τ allows for a cleanseparation of weak and strong effects and a clear production mechanism. We examine how thisstructure can be related to interactions between the three pions that emerge in the final state. Inparticular we start from the interactions between all two body combinations. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ properties in hadronic tau decays Ina Lorenz
1. Interest in τ → πν τ The axial vector current in the few GeV range plays a role in many processes, for example decaysof B and D mesons, Higgs decays and neutrino scattering. The models employed to describe thiscurrent partly boil down to summing up Breit Wigner lineshapes and fitting to the invariant massof the final state system, e.g. in τ → πν τ . The dominant resonance-like structure is denoted as a meson. However, the properties of this meson, as listed in the PDG, depend on the productionmechanism, and contrast significantly between π P → π P and the tau decay. This situation callsfor a conceptual improvement of the models. On one hand, of course the Dalitz plot distributionsshould be considered in any resonance model. On the other hand, more information on the spinstructure is available from structure functions, observables that are directy related to helicity ampli-tudes. In order to use this additional information we refrain from any Breit Wigner parametrizationand start from the two pion interactions which are known from ππ scattering.
2. Definitions
We consider the semileptonic decay, see Fig. 2, τ ( l ) → ν τ ( l ) + π ( p ) + π ( p ) + π ( p ) , (2.1)and its description following Refs. [1, 2]. The general amplitude is M = cos θ C G F √ L µ H µ . (2.2)where θ C is the Cabibbo angle. The leptonic part can be written in the standard model as L µ = ¯ u ( l ) γ µ ( − γ ) u ( l ) . For the hadronic part of the tau decay, we can write the generic matrix element H i jkl µ = h π i ( p ) π j ( p ) π k ( p ) | V l µ ( ) − A l µ ( ) | i , V l µ =
12 ¯ q τ l γ µ q , A l µ =
12 ¯ q τ l γ µ γ q , (2.3)where i jkl are isospin indices and τ l the Pauli matrices in isospin space. For an odd number of final τ ν τ W π
Figure 1:
Schematic rescattering of pions from the tau decay. state pions, the vector contribution vanishes and we consider only the axial part in the following.One possible decomposition is into form factors, as used e.g. in Ref. [2]. However, we use instead adecomposition of the matrix element into helicity amplitudes, since these have a simple expansioninto partial waves. The partial waves yield a simple form of the unitarity relations that we areinterested in.The helicity amplitudes can be expressed in the Mandelstam plane where s , t , u correspond to the1 properties in hadronic tau decays Ina Lorenz two-by-two scattering element h π ( p ) π ( p ) | π ( − p ) A µ ( Q µ ) i . For three body decays often denotedas Dalitz plot invariants s , s and s , here we use s = ( p + p ) , t = ( p + p ) , u = ( p + p ) , Q = s + t + u − M π . (2.4)The center-of-mass scattering angle in each channel, θ s , θ t and θ u , respectively, are related to theKacser function K ( s ) = t − u cos θ s = q λ ( s , M π , M π ) q λ ( s , Q , M π ) . (2.5)The Källén function λ ( a , b , c ) = a + b + c − ( ab + bc + ca ) can be written λ ab ( s ) = λ ( s , M a , M b ) = [ s − ( M a − M b ) ][ s − ( M a + M b ) ] . (2.6)With the definitions L µν = L µ L † ν and H µν = H µ H † ν of the leptonic and hadronic tensor the differ-ential decay rate is given by d Γ ( τ → ν τ π ) = m τ | M | d Φ = G F m τ cos θ C L µν H µν d Φ , L µν H µν = ∑ X L X W X , (2.7)where d Φ is the phase space element. L µν and H µν , can be combined to form 16 symmetric andantisymmetric structure functions W X . One useful basis for the hadronic structure functions W X isdefined via the polarization of the final state system. Consider the polarization vectors ε µ ( λ ) of thethree pions in their c.m. frame or the W boson in its rest frame, respectively. We can now definethe helicity amplitudes A i jkl λ : = h π i ( p ) π j ( p ) π k ( p ) | A l µ ( ) ε µ ( λ ) | i , (2.8)where the subscript denotes the helicity. The outgoing pions have the two possible physical states | π π π ± i and | π + π − π ± i , that can be related by their isospin structure and crossing symmetry. Inthe following we will consider A π π π ± λ ( s , t , u ) ˆ = A λ ( s , t , u ) and neglect isospin breaking.
3. Method and parametrization
We approximate the transverse amplitude similar to Refs. [3, 4], A + ( s , t , u ) ∝ l max ∑ l = ∑ I ( l + ) (cid:20) d l ( θ s ) (cid:18) K ( s ) s (cid:19) l − P I a + , Il ( s ) + d l ( θ t ) (cid:18) K ( t ) t (cid:19) l − P I a + , Il ( t )+ d l ( θ u ) (cid:18) K ( u ) u (cid:19) l − P I a + , Il ( u ) (cid:21) , (3.1)where P i jmnI is the isospin projection operator. The relevant Wigner d-matrix is given by d l ( θ ) = − sin θ / p l ( l + ) P ′ l ( cos θ ) , where the prime denotes a derivative of the Legendre polynomial. Theabove expansion results in partial waves a Il that contain no kinematical but only dynamical cuts.This allows us to relate the parts of the partial waves that contain the left- and right-hand cuts a right / le ftIl ( s ) in an iterative procedure suggested by Khuri and Treiman [5].2 properties in hadronic tau decays Ina Lorenz
In the following we always consider a Il = a + , Il . For each channel, we can write the discontinuityas a sum of the unitarity cut in this channel and those from the crossed channel as a le ftIl Disc a Il ( s ) = ρ ( s ) t ∗ l ( s ) (cid:16) a rightIl ( s ) + a le ftIl ( s ) (cid:17) , (3.2)where ρ ( s ) = p − M π / s and t l ( s ) is the partial wave of the two-pion system, well-known from ππ scattering. This discontinuity enters the standard dispersion relation, e.g. unsubtracted, a rightIl ( s ) = π Z ∞ s ds ′ Disc a rightIl ( s ′ ) s ′ − s , s = M π . (3.3)Expanding A + ( s , t , u ) in the s -channel physical region, comparing to Eq. (3.1), multiplying bothsides with P ′ l ( z s ) and integrating over z s = cos θ s we can write a le ftIl ( s ) ∝ ∑ I ′ , l ′ ( l ′ + ) Z + − dz s ( − z s ) P ′ l ( z s ) (cid:16) P ′ l ′ ( z t ) C II ′ st a I ′ l ′ ( t ( s , z s )) + P ′ l ′ ( z u ) C II ′ su a I ′ l ′ ( u ( s , z s )) (cid:17) , (3.4)where C st / su are the standard crossing matrices, see e.g. Ref. [4]. To find a solution of this set ofequations, we parametrize the transverse partial wave amplitudes similiar to Ref. [6], as a Il ( s ) = Ω Il ( s ) n − ∑ i c i s i + s n π Z ∞ s ds ′ s ′ n ρ ( s ′ ) t ∗ l ( s ′ ) Ω ∗ Il ( s ′ ) a le ftIl ( s ′ )( s ′ − s ) ! , Ω Il ( s ) = exp (cid:18) s π Z ∞ s ds ′ s ′ δ Il ( s ′ ) s ′ − s (cid:19) , (3.5)where the Omnès functions Ω Il ( s ) contain the unitary cut in s , and we use their parametrizationfrom Ref. [7]. The term in brackets in Eq. (3.5) contains the cuts from the crossed channels andcorresponds to an n -times subtracted dispersion relation with the subtraction constants c i . In afirst step the left-hand cuts can be set to zero. However, three main restrictions of this approachare relevant in our case. First, the framework relies on the assumption that two body interactionsdominate. This assumption is only justified at low energy, Q ≪ . Second, the truncationof Eq. ( . ) induces an uncertainty that has to be tested in practice. Third, a precise knowledge ofthe individual waves decreases with increasing energy.
4. Preliminary results
Our calculation for the helicity amplitudes can directly be compared to the experimentally deter-mined structure functions. All structure functions that are not compatible with zero according toCLEO [8] can be related to W A ( s , t , u ) ∝ | A + ( s , t , u ) | + | A − ( s , t , u ) | [1]. In Fig. 2 we showthe structure function W A given by the CLEO collaboration in the corresponding bins and our fitresult. Here we ignore the left hand cuts which corresponds to the first iteration step in a KhuriTreiman approach. For a complete analysis, see Ref. [9]. The dotted lines show the binning in Q , the solid line bars correspond to bins in s and t and the red dashed line to our preliminary fit.Changing the variables by Eq. (2.4) and integrating W A ( Q , s , t ) over s and t yields the integratedstructure functions w A , int ( Q ) shown in Fig. 3. Here, a three body resonance-like structure occurs3 properties in hadronic tau decays Ina Lorenz
Bin number W A ( G e V ) CLEO fit
Figure 2:
Structure functions from CLEO [8]. and can be reproduced qualitatively by our parametrization based on two body interactions. Bothfigures on structures functions show a better agreement with the data for lower bin numbers or lower Q values, respectively. For this kinematical region the Omnès functions are known with higherprecision. For close to vanishing Q values the Khuri Treiman approach would be justified, asthe dominating two body interaction corresponds to first order contributions in chiral perturbationtheory. The CLEO measurement [8] found the contributions from an off-shell W to be compatible Q (GeV ) w A , i n t ( G e V ) CLEOfit to W A (Q , s, t) Figure 3:
Integrated structure functions from CLEO [8]. with zero. We thus approximate the decay rate by the transverse component [1] dNNdQ (cid:12)(cid:12)(cid:12)(cid:12) λ = ∝ (cid:18) ( M τ − Q ) Q (cid:19) (cid:0) + Q / M τ (cid:1) w A , int ( Q ) . (4.1)The comparison to the decay rates from CLEO and ALEPH is given in Fig. 4. Due to the verycoarse grained bins in s and t , we show the binning in Q . Again, the fit does not contain a specificparametrization of the three body resonance like a Breit Wigner, but merely two body interactions.This might hint towards an interesting origin of the a meson, and/or towards the necessity for4 properties in hadronic tau decays Ina Lorenz s(GeV ) −0.050.000.050.100.150.200.250.30 ( N ) d N / d s ALEPH_2005: τ→π2π CLEO, from W A (Q , s, t)fit to CLEO-W A (Q , s, t) Figure 4:
Decay rate from a fit to CLEO structure functions, also compared to ALEPH [10]. more iterations in the partial wave procedure or to include also three body unitarity. As a feasibilitystudy, this work shows a good first description of the structure function and the tau decay rate.Therefore for a future detailed analysis it would be desirable to obtain the Dalitz plot distributionsfor a direct analysis, in particular from more precise measurements by Belle and BABAR. The fullDalitz plot information will help to separate the different uncertainties, namely the knowledge ofthe Omnès functions, the range of applicability of the approach and the truncation error.
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