The tau lepton is the only lepton for which hadronic decay modes are kinematically
allowed. Tau lepton is the heaviest member of the third generation of elementary
particles and it has own lepton number and one associated neutrino does not
interact through strong interactions. The σ lepton provides an ideal tools
to study the interaction between the weak charged current and hadrons which
is not available by the other leptons. Hadrons from σ decays are produced
by the weak charged current from the QCD vacuum. The hadronic physics factors
and the characteristic of each decay channels are related by spectral functions
as far as the total decay rate. Spectral functions parameterize the transition
probability of creating hadrons out of the QCD vacuum (out of the charged weak
current) as a function of hadronic mass (Gentile and Pohl,
1996). In heavy lepton decay to hadronic final state, only final states
with J = 1 or J = 0 are allowed which these final states can have either positive
or negative parity. Scalar and axial-vector mesons have positive parity but
vector and pseudo scalar mesons have negative parity. In these decays, fundamental
quantities, called spectral functions, describe properties of the hadronic systems.
Specific relationships and predictions for these spectral functions can be obtained
using CVC, PCAC and certain assumptions about the symmetries, as is usual in
the phenomenology of weak hadronic decays (Berger et
|| Properties of the weak current
|The vector current can produce scalar (0+) and
vector (1¯) hadronic final states, The axial-vector current can produce
pseudoscalar (0¯) and axial-vector (1+) hadronic final states
Properties of the weak charged current: It is easy to shown that the vector and axial-vector currents are even and odd, respectively under G-parity. Recall that G-parity operation is a combination of a rotation in isospin space and charge conjugation. Table 1 shows the properties of the weak current.
If vector current is conserved which is true for non-strange final states, it is impossible to have the scalar (0+) in the final states. The decay products of vector current interactions are always vectors (1¯). Strange decays are not obeyed from CVC, so the final states can be scalar (0+) or vector (1¯).
In the non-strange tau decays which G-parity is conserved, the separation of
vector and axial-vector components in hadronic final states, only have been
observed by pions. Even number of pions (with G-parity = 1) is related to vector
states and odd number of pions (with G-parity = -1) is related to axial-vector
states (Lyon, 2004). The hypotheses Conserved Vector
Current (CVC) that used to describe the weak current and its conjugate,
together with the electromagnetic current, form an isospin triplet of conserved
currents. For even G-parity, only the vector current contributes in final states
and therefore CVC connects the decay rate to the cross-section for e+
HADRONIC TAU DECAYS
The tau lepton is the only lepton which is sufficiently heavy to be able to decay into hadrons. The theoretical framework for these decays is based on the well tested assumptions that the structure of weak hadronic current is (V-A). The Feynman diagram for hadronic tau decay is shown in Fig. 1.
where, v1, a1, v0 and a0. Tsai
has been derived the general form of hadron production in heavy-lepton decays
through the (V-A) interactions by (Tsai, 1971):
where, v1, a1 v0 and a0 are the spectral functions corresponding to the non-strange vector, axial-vector, scalar and pseudo-scalar final states, respectively and are functions of q2, the invariant mass of hadronic final state and θc is the Cabibbo angle. VS1, aS1, vS0 and aS0 are the corresponding spectral functions for the strange final states. Each spectral function refers to final states having unique spin-parity and strangeness assignments. The spectral functions v1, a1 vS1 and aS1 is related to final states with J = 1, whereas, a0, vS0 and aS0 comes from the final states with J = 0. v0 is due to CVC.
|| Feynman diagram for hadronic tau decay (τ¯→(hadrons)¯Vτ)
CALCULATE THE BRANCHING FRACTION FOR ONE PIONIC TAU DECAY
The most general form of the hadronic decay width has been given in Eq.
1. For any single particle final state this equation is reduced to the term
containing the corresponding spectral function only and the spectral function
is then described by the matrix element of the corresponding weak current between
the vacuum state and given particle final state (Tsai, 1971).
The corresponding spectral function for the decay τ¯→πvτ
is given by:
Equation 1 and 2 gives:
The muonic tau decay width for pion is given by:
Equation 3 and 4 reduces to:
Therefore, the branching fraction of the pionic tau decay is determined by the pion and tau life times (τπ and τ):
|| Branching fraction for decay mode τ¯→π¯vτ
|Br: Branching ratio
By using the particle data group (Lyon, 2004) we obtain:
This is in excellent agreement with experimental measurement shown in Table 2.
CALCULATION OF THE BRANCHING FRACTION FOR ONE KAONIC TAU DECAY
The corresponding spectral function for kaonic tau decay is:
The strength of the axial-vector current coupling to the kaon, is fk sin θc and is measured in kaon decay via k¯→¯¯+vτ. As the same as pionic tau decay, we can derive kaonic tau decay branching fraction as:
From particle data group (Amsler et al., 2008),
we have obtained the branching fraction of kaonic tau decay as:
which is in excellent agreement with experimental data shown in Table 3. In another way we have:
where, h¯ means π¯ or k¯. By using particle data group
(Amsler et al., 2008) and Eq.
11 the other value for the branching fraction of kaonic tau decay is obtained
|| Branching fraction for hadronic tau decay
|Br: Branching ratio
TAU DECAYS INTO VECTOR MESONS
A precise formulation of the weak hadronic decays depending only on the unknown spectral functions is represented by Eq. 1, so that each of which isolates particular hadronic channels. Eq. 1 represents a precise formulation of the weak hadronic decays depending only on the unknown spectral functions, each of which isolates particular hadronic channels. Specific relationships and predictions for these spectral functions can be obtained using CVC, PCAC and certain assumptions about the symmetries, as is usual in the phenomenology of weak hadronic decays. The CVC theorem relates the vector part of the strangeness conserving charged weak current to the isovector part of the total cross section for e¯ e+ annihilations into hadrons:
is the point cross section for the interaction e+ e¯→μ+ μ¯. Therefore, CVC yields a definite prediction for the rate of tau decay into vector mesons.
THE BRANCHING FRACTION OF TWO PIONIC TAU DECAY
The decay width of tau into vector mesons τ¯→ρ¯vτ→π¯π°vτ
(with G-parity equals 1), is obtained by considering the vector spectral function
(Bisello et al., 1989):
where, g2ργ = 1/8π, gργ is the vector current coupling the ρ and the ρ decay constant, fρ, can be determined from the experimental data. From Eq. 1 and the vector spectral function we have:
|| Branching fraction for decay τ¯→π¯π0vτ
|Br: Branching ratio
Eq. 14 reduces to:
By considering the experimental data mρ = 775.5 MeV and mτ = 1776.84 MeV from Eq. 15 we have:
Which is almost in agreement with experimental data shown in Table 4.
THE BRANCHING FRACTION OF 4 PIONIC TAU DECAY
The phenomenology of the 4 pionic final state tau decay is entirely analogous
to that for the tau decay into ρvτ. Since the decay proceeds
through the vector current, one can again relate the decay rate to an integral
over the cross section for the e+ e¯ annihilation into 4 pions.
However, there are two possibility for the final states of e+ e¯
annihilation, namely π+π¯π0π0
and π+π+π¯π¯. As the same
in τ decay, there are two channels namely, π¯π0π0π0vτ
The isospin constrains implies the following linear relations between the corresponding
rates (Gilman and Rhie, 1985):
Therefore, the spectral functions for 4 pionic final states tau decay are:
From Eq. 1 and 19 we have:
|| Branching fraction for tau decays into 4 pions
|Br: Branching ratio
By considering the experimental data of FRASCATI-ADONE, MEA (Esposito
et al., 1980) and SLAC-PEP2-BABAR (Aubert et
al., 2005) in center of mass energy
And by using the data group FRASCATI-ADONE, GAMMA-GAMMA 2 (Bacci
et al., 1981):
From Eq. 20 and 21 we obtain:
Substituting electronic tau decay width in Eq. 22-23
Which is approximately in agreement with experimental data shown in Table 5.
THE ESTIMATION OF THE BRANCHING RATIO FOR TAU DECAY INTO STRANGE PARTICLES
In the framework of standard model we have (Kuhn and Mirkes,
There are four possibilities for the decay of the τ, two of these involves hadronic final states and two involve leptonic final states.
Since, quarks can mix in the standard model, a factor of |Vud|2
≅ cos2 θc ≅ 0.05 is introduced to accompany
the hadronic current in the ud state and similarly a factor |Vud|2
≅ sin2 θc ≅ 0.05 is present for the us
case, where, |Vud| and |Vus| are absolute
CKM elements and θc is the Cabibbo angle. For the leptonic current
there is no mixing and so the equiva lent factor is 1. A consequence of Eq.
1 and 25 is:
The factor 3 in front of the |Vij|2, is correspond to the 3 possible quark colors.
The CVC (Conserved Vector Current) theorem relates the vector part of the strangeness conserving charged weak current to the isovector part of the total cross section for e+ e¯ annihilations into hadrons.
Conservation of angular momentum requires that the ensemble of final state hadrons produced in tau decays can only have total spin J = 0 or J = 1. In general, these final states should have either positive or negative parity. Therefore, the number of possible combinations of spin-parity and strangeness in the final state is restricted to 8 which in turn restricts the number of spectral functions we have needed to consider to obtain a complete description of the partial width of the tau decay to hadrons.
The relation between the cross sections and rates for producing the 4 pion
combinations (π¯, 3π0vτ, 2π¯π+π0πτ,
in tau decays and e+ e¯ annihilation gives important hints on
the validity of isospin symmetry and the order of isospin breaking terms. The
dependency of rates and cross sections on ,
the invariant mass of the four-pion system and the differential mass distributions
of 2 and 3 pions gives informations about the structures of amplitude resonance.