Vacuum Cherenkov Radiation In Quantum Electrodynamics With High-Energy Lorentz Violation

Abstract

We study phenomena predicted by a renormalizable, CPT invariant extension of
the Standard Model that contains higher-dimensional operators and violates
Lorentz symmetry explicitly at energies greater than some scale Lambda_{L}. In
particular, we consider the Cherenkov radiation in vacuo. In a rather general
class of dispersion relations, there exists an energy threshold above which
radiation is emitted. The threshold is enhanced in composite particles by a
sort of kinematic screening mechanism. We study the energy loss and compare the
predictions of our model with known experimental bounds on Lorentz violating
parameters and observations of ultrahigh-energy cosmic rays. We argue that the
scale of Lorentz violation Lambda_{L} (with preserved CPT invariance) can be
smaller than the Planck scale, actually as small as 10^{14}-10^{15} GeV. Our
model also predicts the Cherenkov radiation of neutral particles.

Full text

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arXiv:1101.2019v2 [hep-ph] 12 Mar 2011
IFUP-TH 2010/46
Vacuum Cherenkov Radiation
In Quantum Electrodynamics
With High-Energy Lorentz Violation
Damiano Anselmia,band Martina Taiutib,c
aInstitute of High Energy Physics, Chinese Academy of Sciences,
19 (B) Yuquanlu, Shijingshanqu, Beijing 10049, China,
bDipartimento di Fisica “Enrico Fermi”, Università di Pisa,
Largo B. Pontecorvo 3, I-56127 Pisa, Italy,
cINFN, Sezione di Pisa,
Largo B. Pontecorvo 3, I-56127 Pisa, Italy
damiano.anselmi@df.unipi.it, martina.taiuti@df.unipi.it
Abstract
We study phenomena predicted by a renormalizable, CPT invariant extension of the Standard Model
that contains higher-dimensional operators and violates Lorentz symmetry explicitly at energies greater
than some scale ΛL. In particular, we consider the Cherenkov radiation in vacuo. In a rather general class
of dispersion relations, there exists an energy threshold above which radiation is emitted. The threshold is
enhanced in composite particles by a sort of kinematic screening mechanism. We study the energy loss and
compare the predictions of our model with known experimental bounds on Lorentz violating parameters
and observations of ultrahigh-energy cosmic rays. We argue that the scale of Lorentz violation ΛL(with
preserved CPT invariance) can be smaller than the Planck scale, actually as small as 1014-1015GeV. Our
model also predicts the Cherenkov radiation of neutral particles.
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1 Introduction
Lorentz symmetry is one of the most precise symmetries in nature [1]. Nevertheless, the possibility
that it might be violated at high energies or large distances is still open and has been widely
investigated. If we assume that Lorentz symmetry is not exact, several phenomena that are
otherwise forbidden can occur. Examples are the Cherenkov radiation in vacuo and the photon
decay into an electron-positron pair. Studying phenomena of this type and comparing predictions
with experimental data, we can look for signs of Lorentz violation and put bounds on the values
of Lorentz violating parameters.
From the theoretical point of view, it is interesting to know that if Lorentz symmetry is
(explicitly) violated at high energies, vertices that are non-renormalizable by power counting can
become renormalizable by a modified power-counting criterion, which assigns different weights
to space and time [2]. Consistent models, where the dispersion relations are modified by higher
powers of momenta, can contain operators of higher dimensions, such as two-scalar–two-fermion
vertices and four-fermion vertices; they are multiplied by inverse powers of some energy ΛL, which
can be interpreted as the scale of Lorentz violation. Lorentz violating gauge theories [3, 4] can
be formulated, as well as Lorentz violating extensions of the Standard Model [5, 6], which we
call, for brevity, LVSM. In the common perturbative framework, these theories are unitary, local,
polynomial and causal.
Various phenomena that are forbidden in Lorentz invariant theories, but allowed in Lorentz
violating ones, have been studied in the literature, mainly using the modified dispersion relations
of low-energy effective models. Here we plan to study some of those phenomena in the realm of the
LVSM, where the dispersion relations are crucial for renormalizability, therefore more constrained
and valid, in principle, at arbitrarily high energies (when gravity is switched off). Our purpose is
to derive bounds on the magnitude of ΛL. We believe that the scale of Lorentz violation may be
smaller than the Planck scale. If this were true, our understanding of physics around the Planck
scale, in particular quantum gravity, would have to be reconsidered from scratch.
We assume that CPT is preserved (or that it is violated at energies much larger than ΛL). The
value of ΛLoriginally suggested in ref. [5] from considerations about neutrino masses and bounds
on proton decay was ΛL∼ 1014-1015GeV. (In the appendix we briefly review those arguments
and the minimal LVSM.) In this paper we show that such values are indeed compatible with
experimental data on Lorentz violating phenomena.
Experimental bounds on the parameters that multiply higher-dimensional operators can be
read from the tables of Kostelecky and Russell [1]. At present, the best results belong to the
photon sector, and concern the quadratic terms
Fkλ∂α1···∂αnFµν.
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In particular, from astrophysical birefringence and astrophysical dispersion it is found that the
coefficients of the terms of dimensions 6 and 8 are bounded by
? 10−29GeV−2and ? 10−25GeV−4,
respectively. Interpreting such coefficients as ∼ 1/Λ2
data are consistent with our claim that ΛLcould be as small as 1014-1015GeV.
Under some assumptions, ultrahigh-energy cosmic rays have been claimed to raise the bound
on ΛLwell above the Planck scale [7]. However, the nature of ultrahigh-energy cosmic rays has
not been firmly established, yet, so it is not obvious how to use them to put unambiguous bounds
on the scale of Lorentz violation. In this paper we give several scenarios that are consistent with
a value of ΛLwell below the Planck scale, assuming that ultrahigh-energy cosmic rays are protons
or heavy nuclei. For our purposes, it will be sufficient to restrict to the minimal QED subsector
of the LVSM, which we call LVQED.
We focus on the Cherenkov radiation in vacuo. For a very general class of dispersion relations
we prove that there exists an energy threshold above which radiation is emitted and below which
it is not emitted. Quite interestingly, the threshold is enhanced in composite particles by a sort of
kinematic screening mechanism. We study the energy loss as a function of time and prove that in
all cases of our interest it is so rapid that the radiation is practically governed by pure kinematics.
Our models also predict the Cherenkov radiation of neutral particles.
The paper is organized as follows. In section 2 we present the LVQED model we are going
to study and some basic formulas. In section 3 we study the Cherenkov radiation in the low-
energy expansion. From section 4 onwards we investigate situations where the standard low-
energy expansion does not apply. Some results can be derived using exact dispersion relations.
For other purposes a different kind of expansion can be used. In section 4 we study kinematic
constraints and derive the energy threshold for Cherenkov radiation. In section 5 we compare
two typical scenarios with experimental data. In section 6 we study composite particles and show
that compositeness favors larger thresholds. In section 7 we discuss the Cherenkov radiation of
neutrons and neutrinos, while section 8 contains our conclusions.
Land ∼ 1/Λ4
Lwe see that these experimental
2 Preliminaries
In this section we write the models we are going to study and derive a general formula for the
energy loss per unit time.
The LVQED model we consider is the minimal QED subsector of the LVSM recalled in the
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appendix, to which we refer for the notation. Its Lagrangian reads
L=LF+¯ψ
?
?
iγ0D0+ib0
Λ2
L
¯ D /3+ ib1¯D / − m −
b′
ΛL
¯ D /2
?
?
ψ
(2.1)
+e
ΛL
¯ψb′′σijFij+b′
0
ΛLγi∂jFij
?
ψ + ieb′′
0
Λ2
L
Fij
¯ψγi
← →
¯Dj
2
ψ
?
,
where the covariant derivative is Dµ= ∂µ+ ieAµ, σµν= −i[γµ,γν]/2. Moreover,
LF=1
2F2
0i−1
4Fij
?
τ2− τ1
¯∂2
Λ2
L
+ τ0(−¯∂2)2
Λ4
L
?
Fij
is the Lagrangian of free photons. The quantization of this theory has been studied in ref. [8].
More details can be found there, together with an analysis of its renormalization. The dispersion
relations of fermions and photons are
E(¯ p2) =
?
¯ p2
?
b1+b0
Λ2
L
¯ p2
?2
+
?b′
ΛL¯ p2+ m
?2
,ω(¯k2) =
?
τ2¯k2+ τ1(¯k2)2
Λ2
L
+ τ0(¯k2)3
Λ4
L
, (2.2)
respectively.
The low-energy limit of (2.1) is
Llow=1
2F2
0i−τ2
4F2
ij+¯ψ?iγ0D0+ ib1¯D / − m?ψ,
(2.3)
which formally coincides with the Lagrangian of QED in a medium. The parameters τ2and b1
are related to the dielectric constant ε and the magnetic permeability µ by the formulas
τ2=ε
µ,
b1= ε.
(2.4)
Moreover,
n =√εµ =
b1
√τ2
is the refractive index. Performing the replacements (2.4) and the rescalings
xi→ εxi,Ai→Ai
ε,
ψ →
ψ
ε3/2,
in the action of (2.3), we obtain the more common Lagrangian
Lmedium=ε
2F2
0i−
1
4µF2
ij+¯ψ(iD / − m)ψ.
(2.5)
We use for (2.5) the gauge-fixing term of Lorenz type
LGF= −1
2µ(εµ∂0A0− ∂iAi)2.
(2.6)
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We assume that a particle above threshold continuously loses its energy through the process
e → eγ. Then the emitted radiation is made of a large number of low-frequency photons. The
particle remains above threshold, but its energy asymptotically tends to the threshold value. This
conservative assumption is sufficient for our purposes. Indeed, we are going to show that in all
cases we are interested in the energy loss calculated in this way is so rapid that we can assume that
the particle (practically) reaches the threshold instantaneously. Other types of emission occur.
For example, the model (2.1) also contains vertices with two fermions and two or more photons,
which allow elementary processes such as e → eγγ and e → eγγγ. It may also happen [9] that the
particle loses most of its energy emitting a single sufficiently energetic photon, or a finite number
of photons. Then the deceleration is not continuous. These effects can increase the rate of energy
loss, but do not affect the conclusions of this paper.
It is convenient to derive a general formula for the energy loss per unit time without making
assumptions on the dispersion relations. It will be applied to both (2.1) and (2.3). Consider a
charged fermion of energy E and momentum p emitting a photon of frequency ω. Call E′and p′
the energy and momentum of the fermion after emission. The expression of the differential width
is
1
2E|M|2(2π)δ(E − ω − E′)(2π)3δ3(p − k − p′)
dΓ =
d3k
2ω(2π)3
d3p′
2E′(2π)3,
(2.7)
where |M|2is the squared modulus of the transition amplitude, summed over the final states and
averaged over the initial states.
As usual, the integral over p′is done eliminating the delta function associated with momentum
conservation. The surviving integral is reduced to an integral over ω and u = cosθ, θ being the
angle between the momentum of the incoming fermion and the momentum of the emitted photon.
Next, the delta function of energy conservation can be used to perform the u-integral. It gives
u as a function of p and k. Finally, the condition |u(p,k)| ? 1 determines the range of the final
k-integration. We find
dΓ
dω=
16πEpωdω
|M|2
kdk
?
u∗
1
???E′
p′dE′
dp′
???u=u∗
,
(2.8)
where the sum is over the solutions u∗(p,k) to the condition of energy conservation. In the case
of (2.5), the solution is unique. Instead, the dispersion relations of our Lorentz violating models
admit multiple solutions, in general. Yet, the solution remains unique under quite reasonable
assumptions (see section 4). In this case the k-range is of the standard form 0 ? k ? kmax, for
some kmax.
The differential width can be used to calculate the energy loss per unit time, using the formula
?ωmax
where ωmax= ω(k2
max).
dE
dt
= −
0
ωdΓ
dωdω,
(2.9)
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