We analyze the possibility of testing local Lorentz invariance from the observation of tau decays. Future prospects of probing distances below the electroweak characteristic scale are discussed.Artículo enviado a Zeitschrift für Physik CFacultad de Ciencias Exacta

Anchordoqui, Luis Alfredo

Dova, María Teresa

Gómez Dumm, Daniel

English

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Possible test of local Lorentz invariance from τ decays
L. Anchordoqui, M. T. Dova, D. Go´mez Dumm, P. Lacentre
Laboratorio de F´ısica Teo´rica, Departamento de F´ısica,
U. N. L. P., c. c. 67, 1900 La Plata, Argentina.
We analyze the possibility of testing local Lorentz invariance from the observation of tau decays.
Future prospects of probing distances below the electroweak characteristic scale are discussed.
Journal Reference: Zeitschrift fu¨r Physik C 73 (1997) 465.
Introduction. The question of the existence of a universal length, below which our present notion of flat spacetime
geometry is not valid, is of great theoretical importance [1]. In this respect, it is important to point out that the current
success of the Standard Model (SM) in describing the electroweak processes serves as a constraint for any observable
breakdown of local Lorentz invariance (LLI), which should appear as a small deviation from the SM predictions. In
this way, any search for LLI nonconservation effects will require to test distances below the electroweak scale, i.e. of
order 10−16 cm or less.
In order to get some insights into this subject, one possible way is to consider the experimental information on the
decays of particles which move at high velocities in the lab system. The comparison between the measured lifetimes at
these energies and the corresponding values in the rest frame would provide a direct test of the time dilation formula.
In this direction, a proposal has been recently presented by Freeman et al. [2] to measure the lifetimes of 530 GeV
charged pions at Fermilab.
In this article, we concentrate on the possibilities of testing LLI from the observation of tau lepton decays. In
particular, we consider the strongly relativistic tau pairs produced at LEP-I, taking profit of the increasing statistic
and excellent experimental precision of the tau lifetime measurement. By comparing this lifetime value with the low
energy results, it is possible to determine an upper bound for the LLI-breaking characteristic length. In addition, we
consider the possibility of testing LLI from the measurement of particular τ decay branching ratios, instead of dealing
with the total decay width. The future prospects of probing distances below the electroweak characteristic scale are
also discussed.
Framework. The idea of testing LLI from particle decay rates is not new. Indeed, this possibility was suggested
many years ago by Re´dei [3], who studied possible LLI breakdown effects on the decay of charged pions and muons.
Now, considering the extraordinary evolution of the experimental knowledge, it is worth to recover this proposal
including present data on particle lifetimes. We will use this framework extended to the τ lepton decays.
Following the scheme of Re´dei, we begin by introducing into the theory a timelike unit vector n which characterizes
the chosen reference frame [4]. As a consequence, the transition amplitudes become (in general) not invariant under
a Lorentz transformation of the particle states if the reference system is kept unchanged. The theory does not exhibit
Lorentz invariance in the so-called “active” sense. However, one still has Lorentz invariance in the “passive” sense, i.e.,
under a simultaneous transformation of the particle variables and the vector n (this is necessary in order to guarantee
that the probabilities remain independent of the velocity of the observer).
We will study the main tau decay partial widths considering in each case a noncausal effective Hamiltonian of the
type proposed by Re´dei,∫
Hint d4x = GF√
2
∫ ∫
d4x d4y ν¯τ (x)γ
µ(1− γ5)τ(x)F (x − y)Jµ(y) , (1)
where the current Jµ depends on the decay channel, and the noncausality is introduced through the form factor F (x).
The latter is given by
F (x) =
3
4πα3
δ(n · x)Θ(α2 + x2 − (n · x)2 )
and satisfies F (x)→ δ(x) in the limit α→ 0. In the lab system (which can be considered at rest with respect to the
surrounding macroscopic bodies1), we may take n = ( 1 , ~0 ). Then the form factor reduces to
1Most of the low velocity experiments which test LLI consider a preferential reference frame comoving at 350 km/s with the
1
F (x) =
3
4πα3
δ(x0)Θ(α
2 − |~x|2) , (2)
where it becomes clear that α represents a characteristic length below which the noncausality effects will take place.
We are interested in computing the decay amplitudes in the lab system for strongly relativistic tau leptons. The τ
polarization will be also included, since it can in principle contribute non-negligibly to the LLI breakdown effects we
are looking for (in the case of LEP-I, the taus are produced with a polarization of approximately 14%)2.
Let us first consider the leptonic decays τ− → l ν¯l ντ (l = µ−, e−) for a τ lepton with polarization Pτ . The
corresponding decay amplitude is found to be
|M|2 = 64 GF 2 (qτ · qνl −mτ s · qνl) (qν · ql) [ g(n, qτ−qντ ) ]2 . (3)
where s stands for the tau polarization 4-vector (qτ · s = 0, s2 = −Pτ 2) and we have neglected the lepton mass ml.
The function g(n,Q) is the Fourier transform of F (x),
g(n,Q) = 1− 1
10
α2
[
(Q · n)2 −Q2]+O(α4) . (4)
In order to calculate the partial widths Γl(Eτ ), it is convenient to perform a transformation to the tau rest frame.
However, notice that one has to boost both the particle variables and the vector n to get the right result. The Fourier
transform of F (x) in the new reference system reads
g(n,Q)(rest) = 1− 1
10
α2
[
γ2v2 (Eντ −mτ )2 + E2ντ + 2 γ2v (Eντ −mτ )Eντ cos θ + γ2v2E2ντ cos2 θ
]
, (5)
where θ is the angle between the boost direction and the 3-momentum ~qντ and we have used γ
2 = (1− v2)−1, being
v the tau velocity in the lab system. After integration over the phase space, we find for the leptonic partial widths
Γl(Eτ ) =
G2F m
5
τ
192 π3
γ−1[1− α2δl(Eτ )] , l = e, µ , (6)
with
δl(Eτ ) =
1
5
E2τ
(
2
15
+
31
90
v2 +
1
18
Pτv
)
. (7)
In the limit α→ 0, we get the relation
Γl(Eτ ) = γ
−1Γl(mτ ) , (8)
i.e. we recover the usual time dilation formula.
We consider now the decays τ− → hντ , with h = π−, ρ−, a−1 . The contribution of these processes, together with
the leptonic channels treated above, add up to approx. 90% of the total tau decay width. In analogy with (6), the
corresponding partial widths for τ leptons with energy Eτ can be written as
Γh(Eτ ) = Γh(mτ )γ
−1[1− α2δh(Eτ )] , h = π−, ρ−, a−1 , (9)
where we have assumed δh(mτ ) ≪ δh(Eτ ). This is justified when we deal with ultrarelativistic taus. The deviation
factors δh can be easily calculated, resulting
δh(Eτ ) =
1
20
E2τ
[
(1− xh)2 + 1
3
(1 + 10 xh + x
2
h) v
2 +
2
3
Pτλh (1− x2h) v
]
, (10)
where xh ≡ m2h/m2τ , and λh is given by
center of mass of the Universe (as seen from the dipole asymmetry of the 3 K background radiation [5]). However, this cosmic
motion can be neglected for our purposes in view of the high LEP-I velocities.
2The spatial component of n is not zero in the rest frame of the decaying tau. Therefore, n introduces a spatial anisotropy
and the decay probability is in general a function of the tau polarization direction.
2
λh ≡
{
1 for h = π−
1−2xh
1+2 xh
for h = ρ−, a−1 .
(11)
In the same way, the corrected expression for the total τ decay rate will read
Γ(Eτ ) =
∑
i
Γi(Eτ ) = γ
−1Γ(mτ )
(
1− α2∆(Eτ )
)
(12)
with
∆(Eτ ) ≡
∑
i
Γi
Γ
δi(Eτ ) , (13)
where the sum extends in principle to all τ decay channels. We will neglect here the deviations corresponding to the
less significant branching ratios, restricting ourselves to the δi contributions given by (7) and (10). Notice that the
energy dependence of Γi and Γ in (13) has been omitted, since it would imply a correction of order α
4 to the total
decay rate (12).
It is important to remark that, within the scheme under consideration, the branching ratios are no longer fixed
numbers. They depend in general on the energy of the decaying particle. In the case of the τ leptons, the above
equations conduce to the (order α2) relations
Γi(Eτ )
Γ(Eτ )
=
Γi(mτ )
Γ(mτ )
[
1 + α2(∆(Eτ )− δi(Eτ ))
]
. (14)
In this way, the sole measurement of branching ratios provides another possible test of LLI.
Numerical analysis. As stated above, the experimental data on the τ lepton lifetime at both high and low
energies provide an upper bound for the characterstic LLI breakdown length α. In addition, it is possible to estimate
how much energy would be necessary to probe distances below the electroweak scale for a given accuracy in the τ
lifetime measurements.
Let us first consider the τ leptons which are produced at approximately 45 GeV in LEP-I and SLD. From the
corresponding experimental data, the weighted average value of the τ lifetime is found to be [6]
ττ = 291.4± 1.6 fs . (15)
For low energy τ ’s, the lifetime value is less accurate. The present measurements include a 2.5% error, which is indeed
the dominant one when determining the bound for α. By combination of the CLEO [7] and ARGUS [8] results, we
obtain
ττ = 292± 7 fs . (16)
As can be seen, the data in (15) and (16) are compatible with a null value of α. In order to obtain an upper bound,
we consider the modified time dilation formula in eq. (12), with the δi corrections given by (7) and (10). With a 95%
confidence level, we find
α ≤ 3.3× 10−16 cm , (17)
where the numerical evaluation of the total deviation factor ∆ has been performed taking the experimental values for
the particle masses and τ decay branching ratios from Ref. [9].
From the expression (12), one can also examine the future possibilities of probing even smaller distances. In figure
1, we plot the value of α that would be sensitive to this test for a given energy of the decaying tau. The accuracy
in the tau lifetime measurement (this means, for both high and low energies) is included as a parameter. If we take
into account τ -leptons of ∼ 50 GeV, it can be seen that this accuracy should be increased up to a 0.5% in order to
reach a sensitivity to α below the electroweak characteristic length of 10−16 cm. For energies of ∼ 100 GeV, however,
distances beyond this limit could be probed with a τ lifetime precision of about 2%.
Another possible test of the presence of LLI violation can be obtained by comparing the τ and µ leptonic partial
widths [10]. Let us consider the ratio
R(Eτ ) ≡ Γ(τ → ντeν¯e)
Γ(µ→ νµeν¯e) = Be(Eτ )
τµ(Eµ)
ττ (Eτ )
, (18)
3
where Be is the branching ratio for the τ → ντeν¯e decay and Eτ is the energy of the decaying τ in the lab system.
Within our framework, it is easy to show that
R(Eτ ) =
(
gτ
gµ
)2(
mτ
mµ
)5
(∆W∆γ)
−1(1 − α2δl(Eτ )) , (19)
where gl stands for the coupling constant corresponding to the weak charged current of a lepton l. The factors
∆γ ≃ 1+8.6× 10−5 and ∆W ≃ 1− 3.0× 10−4 come from the inclusion of radiative corrections to the tree level decay
amplitudes. We have not included the correction δl(Eµ), which has been assumed to be much smaller than δl(Eτ ).
From (18) and (19), we finally have
(
gτ
gµ
)2
(1 − α2δl(Eτ )) =
(
mµ
mτ
)5
τµ
γ−1ττ (Eτ )
Be(Eτ )∆W∆γ , (20)
i.e. the presence of the LLI-breakdown parameter α would be observed as a violation of the τ − µ universality.
In reference [10], the right hand side of (20) has been evaluated using the experimental values of the masses and
decay rates. Setting α = 0, the LEP measurements for τ decays conduce to
gτ
gµ
= 0.9954± 0.0043 . (21)
Within the LLI-violating scheme we are considering, the accurate value in the above expression, together with the
requirement of universality gτ = gµ, provide a new upper bound for the parameter α. In figure 2, we plot the value
of the ratio gτ/gµ that is obtained from eq. (20) for values of α from 10
−17 to 10−15 cm. The solid curve corresponds
to the average experimental values of the parameters entering the right hand side, while the dashed lines take into
account a 2σ deviation. Within this confidence level, the agreement with universality conduces to
α ≤ 2.3× 10−16 cm . (22)
Let us finally stress the fact that measurements of branching ratios at high and low energies can provide accurate
tests of LLI. In this sense, the τ leptons represent an ideal laboratory, in view of the various significant decay channels
which can be observed. The prospects are also encouraged by the present improvement in these measurements at LEP
[11] and the proposal of building a Tau-Charm factory [12], where the expected precision on the values of branching
ratios will be far below 1% in one-year data sample.
ACKNOWLEDGMENTS
We would like to thank L. Epele, H. Fanchiotti, C. Garc´ıa Canal and J. Swain for useful discussions and comments.
This work was partially supported by CONICET (Argentina).
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4
[7] C. White, Nucl. Phys. B (Proc. Suppl.) 40 (1995) 311.
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[11] ALEPH Collaboration, D. Buskulic et al., CERN preprint CERN-PPE/95-140 (1995); ibid.CERN preprint CERN-PPE/95-
127 (1995).
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5
Figures
0 50 100
1E-17
1E-16
1E-15
0.1 %
0.5 %
2.5 %
 
 α
 (c
m
)
Eτ (GeV)
FIG. 1. Distance that is probed by the LLI-test, as a function of the τ lepton energy. Different accuracies in the τ lifetime
measurements are considered.
1E-17 1E-16 1E-15
0.98
0.99
1.00
1.01
1.02
g τ
 / 
g µ
α (cm)
FIG. 2. Ratio gτ/gµ calculated from measured values of τ and µ masses and leptonic decays. Solid and dashed lines represent
average and 2σ values respectively.
6


Possible test of local Lorentz invariance from τ decays

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Possible test of local Lorentz invariance from τ decays

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