LOG#046. The Cherenkov effect.

The Cherenkov effect/Cherenkov radiation, sometimes also called Vavilov-Cherenkov radiation, is our topic here in this post.

In 1934, P.A. Cherenkov was a post graduate student of S.I.Vavilov. He was investigating the luminescence of uranyl salts under the incidence of gamma rays from radium and he discovered a new type of luminiscence which could not be explained by the ordinary theory of fluorescence. It is well known that fluorescence arises as the result of transitions between excited states of atoms or molecules. The average duration of fluorescent emissions is about $\tau>10^{-9}s$ and the transition probability is altered by the addition of “quenching agents” or by some purification process of the material, some change in the ambient temperature, etc. It shows that none of these methods is able to quench the fluorescent emission totally, specifically the new radiation discovered by Cherenkov. A subsequent investigation of the new radiation ( named Cherenkov radiation by other scientists after the Cherenkov discovery of such a radiation) revealed some interesting features of its characteristics:

1st. The polarization of luminiscence changes sharply when we apply a magnetic field. Cherenkov radiation luminescence is then causes by charged particles rather than by photons, the $\gamma$-ray quanta! Cherenkov’s experiment showed that these particles could be electrons produced by the interaction of $\gamma$-photons with the medium due to the photoelectric effect or the Compton effect itself.

2nd. The intensity of the Cherenkov’s radiation is independent of the charge Z of the medium. Therefore, it can not be of radiative origin.

3rd. The radiation is observed at certain angle (specifically forming a cone) to the direction of motion of charged particles.

The Cherenkov radiation was explained in 1937 by Frank and Tamm based on the foundations of classical electrodynamics. For the discovery and explanation of Cherenkov effect, Cherenkov, Frank and Tamm were awarded the Nobel Prize in 1958. We will discuss the Frank-Tamm formula later, but let me first explain how the classical electrodynamics handle the Vavilov-Cherenkov radiation.

The main conclusion that Frank and Tamm obtained comes from the following observation. They observed that the statement of classical electrodynamics concerning the impossibility of energy loss by radiation for a charged particle moving uniformly and following a straight line in vacuum is no longer valid if we go over from the vacuum to a medium with certain refractive index $n>1$. They went further with the aid of an easy argument based on the laws of conservation of momentum and energy, a principle that rests in the core of Physics as everybody knows. Imagine a charged partice moving uniformly in a straight line, and suppose it can loose energy and momentum through radiation. In that case, the next equation holds:

$\left(\dfrac{dE}{dp}\right)_{particle}=\left(\dfrac{dE}{dp}\right)_{radiation}$

This equation can not be satisfied for the vacuum but it MAY be valid for a medium with a refractive index gretear than one $n>1$. We will simplify our discussion if we consider that the refractive index is constant (but similar conclusions would be obtained if the refractive index is some function of the frequency).

By the other hand, the total energy E of a particle having a non-null mass $m\neq 0$ and moving freely in vacuum with some momentum p and velocity v will be:

$E=\sqrt{p^2c^2+m^2c^4}$

and then

$\left(\dfrac{dE}{dp}\right)_{particle}=\dfrac{pc^2}{E}=\beta c=v$

Moreover, the electromagnetic radiation in vaccum is given by the relativistic relationship

$E_{rad}=pc$

From this equation, we easily get that

$\left(\dfrac{dE}{dp}\right)_{radiation}=c$

Since the particle velocity is $v, we obtain that

$\left(\dfrac{dE}{dp}\right)_{particle}<\left(\dfrac{dE}{dp}\right)_{radiation}$

In conclusion: the laws of conservation of energy and momentum prevent that a charged particle moving with a rectilinear and uniform motion in vacuum from giving away its energy and momentum in the form of electromagnetic radiation! The electromagnetic radiation can not accept the entire momentum given away by the charged particle.

Anyway, we realize that this restriction and constraint is removed and given up when the aprticle moves in a medium with a refractive index $n>1$. In this case, the velocity of light in the medium would be

$c'=c/n

and the velocity v of the particle may not only become equal to the velocity of light $c'$ in the medium, but even exceed it when the following phenomenological condition is satisfied:

$\boxed{v\geq c'=c/n}$

It is obvious that, when $v=c'$ the condition

$\left(\dfrac{dE}{dp}\right)_{particle}=\left(\dfrac{dE}{dp}\right)_{radiation}$

will be satisfied for electromagnetic radiation emitted strictly in the direction of motion of the particle, i.e., in the direction of the angle $\theta=0\textdegree$. If $v>c'$, this equation is verified for some direction $\theta$ along with $v=c'$, where

$v'=v\cos\theta$

is the projection of the particle velocity v on the observation direction. Then, in a medium with $n>1$, the conservation laws of energy and momentum say that it is allowed that a charged particle with rectilinear and uniform motion, $v\geq c'=c/n$ can loose fractions of energy and momentum $dE$ and $dp$, whenever those lost energy and momentum is carried away by an electromagnetic radiation propagating in the medium at an angle/cone given by:

$\boxed{\theta=arccos\left(\dfrac{1}{n\beta}\right)=\cos^{-1}\left(\dfrac{1}{n\beta}\right)}$

with respect to the observation direction of the particle motion.

These arguments, based on the conservation laws of momenergy, do not provide any ide about the real mechanism of the energy and momentum which are lost during the Cherenkov radiation. However, this mechanism must be associated with processes happening in the medium since the losses can not occur ( apparently) in vacuum under normal circumstances ( we will also discuss later the vacuum Cherenkov effect, and what it means in terms of Physics and symmetry breaking).

We have learned that Cherenkov radiation is of the same nature as certain other processes we do know and observer, for instance, in various media when bodies move in these media at a velocity exceeding that of the wave propagation. This is a remarkable result! Have you ever seen a V-shaped wave in the wake of a ship? Have you ever seen a conical wave caused by a supersonic boom of a plane or missile? In these examples, the wave field of the superfast object if found to be strongly perturbed in comparison with the field of a “slow” object ( in terms of the “velocity of sound” of the medium). It begins to decelerate the object!

Question: What is then the mechanism behind the superfast  motion of a charged particle in a medium wiht a refractive index $n>1$ producing the Cherenkov effect/radiation?

Answer:  The mechanism under the Cherenkov effect/radiation is the coherent emission by the dipoles formed due to the polarization of the medium atoms by the charged moving particle!

The idea is as follows. Dipoles are formed under the action of the electric field of the particle, which displaces the electrons of the sorrounding atoms relative to their nuclei. The return of the dipoles to the normal state (after the particle has left the given region) is accompanied by the emission of an electromagnetic signal or beam. If a particle moves slowly, the resulting polarization will be distribute symmetrically with respect to the particle position, since the electric field of the particle manages to polarize all the atoms in the near neighbourhood, including those lying ahead in its path. In that case, the resultant field of all dipoles away from the particle are equal to zero and their radiations neutralize one to one.

Then, if the particle move in a medium with a velocity exceeding the velocity or propagation of the electromagnetic field in that medium, i.e., whenever $v>c'=c/n$, a delayed polarization of the medium is observed, and consequently the resulting dipoles will be preferably oriented along the direction of motion of the particle. See the next figure:

It is evident that, if it occurs, there must be a direction along which a coherent radiation form dipoles emerges, since the waves emitted by the dipoles at different points along the path of the particle may turn our to be in the same phase. This direction can be easiy found experimentally and it can be easily obtained theoretically too. Let us imagine that a charged particle move from the left to the right with some velocity $v$ in a medium with a $n>1$ refractive index, with $c'=c/n$. We can apply the Huygens principle to build the wave front for the emitted particle. If, at instant $t$, the aprticle is at the point $x=vt$, the surface enveloping the spherical waves emitted by the same particle on its own path from the origin at $x=0$ to the arbitrary point $x$. The radius of the wave at the point $x=0$ at such an instant t is equal to $R_0=c't$. At the same moment, the wave radius at th epint x is equal to $R_x=c'(t-(x/v))=0$. At any intermediate point x’, the wave radius at instant t will be $R_{x'}=c'(t-(x'/v))$. Then, the radius decreases linearly with increasing $x'$. Thus, the enveloping surface is a cone with angle $2\varphi$, where the angle satisfies in addition

$\sin\varphi=\dfrac{R_0}{x}=\dfrac{c't}{vt}=\dfrac{c'}{v}=\dfrac{c}{vn}=\dfrac{1}{\beta n}$

The normal to the enveloping surface fixes the direction of propagation of the Cherenkov radiation. The angle $\theta$ between the normal and the $x$-axis is equal to $\pi/2-\varphi$, and it is defined by the condition

$\boxed{\cos\theta=\dfrac{1}{\beta n}}$

or equivalently

$\boxed{\tan\theta=\sqrt{\beta^2n^2-1}}$

This is the result we anticipated before. Indeed, it is completely general and Quantum Mechanics instroudces only a light and subtle correction to this classical result. From this last equation, we observer that the Cherenkov radiation propagates along the generators of a cone whose axis coincides with the direction of motion of the particle an the cone angle is equal to $2\theta$. This radiation can be registered on a colour film place perpendicularly to the direction of motion of the particle. Radiation flowing from a radiator of this type leaves a blue ring on the photographic film. These blue rings are the archetypical fingerprints of Vavilov-Cherenkov radiation!

The sharp directivity of the Cherenkov radiation makes it possible to determine the particle velocity $\beta$ from the value of the Cherenkov’s angle $\theta$. From the Cherenkov’s formula above, it follows that the range of measurement of $\beta$ is equal to

$1/n\leq\beta<1$

For $\beta=1/n$, the radiation is observed at an angle $\theta=0\textdegree$, while for the extreme with $\beta=1$, the angle $\theta$ reaches a maximum value

$\theta_{max}=\cos^{-1}\left(\dfrac{1}{n}\right)=arccos \left(\dfrac{1}{n}\right)$

For instance, in the case of water, $n=1.33$ and $\beta_{min}=1/1.33=0.75$. Therefore, the Cherenkov radiation is observed in water whenever $\beta\geq 0.75$. For electrons being the charged particles passing through the water, this condition is satisfied if

$T_e=m_ec^2\left(\dfrac{1}{\sqrt{1-\beta^2}}-1\right)=0.5\left( \dfrac{1}{\sqrt{1-0.75^2}}-1\right)=0.26MeV$

As a consequence of this, the Cherenkov effect should be observed in water even for low-energy electrons ( for isntance, in the case of electrons produced by beta decay, or Compton electrons, or photoelectroncs resulting from the interaction between water and gamma rays from radioactive products, the above energy can be easily obtained and surpassed!). The maximum angle at which the Cherenkov effec can be observed in water can be calculated from the condition previously seen:

$\cos\theta_{max}=1/n=0.75$

This angle (for water) shows to be equal to about $\theta\approx 41.5\textdegree=41\textdegree 30'$. In agreement with the so-called Frank-Tamm formula ( please, see below what that formula is and means), the number of photons in the frequency interval $\nu$ and $\nu+d\nu$ emitted by some particle with charge Z moving with a velocity $\beta$ in a medium with a refractive indez n is provided by the next equation:

$\boxed{N(\nu) d\nu=4\pi^2\dfrac{(Zq)^2}{hc^2}\left(1-\dfrac{1}{n^2\beta^2}\right) d\nu}$

This formula has some striking features:

1st. The spectrum is identical for particles with $Z=constant$, i.e., the spectrum is exactly the same, irespectively the nature of the particle. For instance, it could be produced both by protons, electrons, pions, muons or their antiparticles!

2nd. As Z increases, the number of emitted photons increases as $Z^2$.

3rd. $N(\nu)$ increases with $\beta$, the particle velocity, from zero ( with $\beta=1/n$) to

$N=4\pi^2\left(\dfrac{q^2Z^2}{hc^2}\right)\left(1-\dfrac{1}{n^2}\right)$

with $\beta\approx 1$.

4th. $N(\nu)$ is approximately independent of $\nu$. We observe that $dN(\nu)\propto d\nu$.

5th. As the spectrum is uniform in frequency, and $E=h\nu$, this means that the main energy of radiation is concentrated in the extreme short-wave region of the spectrum, i.e.,

$\boxed{dE_{Cherenkov}\propto \nu d\nu}$

And then, this feature explains the bluish-violet-like colour of the Cherenkov radiation!

Indeed, this feature also indicates the necessity of choosing materials for practical applications that are “transparent” up to the highest frequencies ( even the ultraviolet region). As a rule, it is known that $n<1$ in the X-ray region and hence the Cherenkov condition can not be satisfied! However, it was also shown by clever experimentalists that in some narrow regions of the X-ray spectrum the refractive index is $n>1$ ( the refractive index depends on the frequency in any reasonable materials. Practical Cherenkov materials are, thus, dispersive! ) and the Cherenkov radiation is effectively observed in apparently forbidden regions.

The Cherenkov effect is currently widely used in diverse applications. For instance, it is useful to determine the velocity of fast charged particles ( e.g, neutrino detectors can not obviously detect neutrinos but they can detect muons and other secondaries particles produced in the interaction with some polarizable medium, even when they are produced by (electro)weak intereactions like those happening in the presence of chargeless neutrinos). The selection of the medium fo generating the Cherenkov radiation depends on the range of velocities $\beta$ over which measurements have to be produced with the aid of such a “Cherenkov counter”. Cherenkov detectors/counters are filled with liquids and gases and they are found, e.g., in Kamiokande, Superkamiokande and many other neutrino detectors and “telescopes”. It is worth mentioning that velocities of ultrarelativistic particles are measured with Cherenkov detectors whenever they are filled with some special gasesous medium with a refractive indes just slightly higher than the unity. This value of the refractive index can be changed by realating the gas pressure in the counter! So, Cherenkov detectors and counters are very flexible tools for particle physicists!

Remark: As I mentioned before, it is important to remember that (the most of) the practical Cherenkov radiators/materials ARE dispersive. It means that if $\omega$ is the photon frequency, and $k=2\pi/\lambda$ is the wavenumber, then the photons propagate with some group velocity $v_g=d\omega/dk$, i.e.,

$\boxed{v_g=\dfrac{d\omega}{dk}=\dfrac{c}{\left[n(\omega)+\omega \frac{dn}{d\omega}\right]}}$

Note that if the medium is non-dispersive, this formula simplifies to the well known formula $v_g=c/n$. As it should be for vacuum.

Accodingly, following the PDG, Tamm showed in a classical paper that for dispersive media the Cherenkov radiation is concentrated in a thin  conical shell region whose vertex is at the moving charge and whose opening half-angle $\eta$ is given by the expression

$\boxed{cotan \theta_c=\left[\dfrac{d}{d\omega}\left(\omega\tan\theta_c\right)\right]_{\omega_0}=\left(\tan\theta_c+\beta^2\omega n(\omega) \dfrac{dn}{d\omega} cotan (\theta_c)\right)\bigg|_{\omega_0}}$

where $\theta_c$ is the critical Cherenkov angle seen before, $\omega_0$ is the central value of the small frequency range under consideration under the Cherenkov condition. This cone has an opening half-angle $\eta$ (please, compare with the previous convention with $\varphi$ for consistency), and unless the medium is non-dispersive (i.e. $dn/d\omega=0$, $n=constant$), we get $\theta_c+\eta\neq 90\textdegree$. Typical Cherenkov radiation imaging produces blue rings.

THE CHERENKOV EFFECT: QUANTUM FORMULAE

When we considered the Cherenkov effect in the framework of QM, in particular the quantum theory of radiation, we can deduce the following formula for the Cherenkov effect that includes the quantum corrections due to the backreaction of the particle to the radiation:

$\boxed{\cos\theta=\dfrac{1}{\beta n}+\dfrac{\Lambda}{2\lambda}\left(1-\dfrac{1}{n^2}\right)}$

where, like before, $\beta=v/c$, n is the refraction index, $\Lambda=\dfrac{h}{p}=\dfrac{h}{mv}$ is the De Broglie wavelength of the moving particle and $\lambda$ is the wavelength of the emitted radiation.

Cherenkov radiation is observed whenever $\beta_n>1$ (i.e. if $v>c/n$), and the limit of the emission is on the short wave bands (explaining the typical blue radiation of this effect). Moreover, $\lambda_{min}$ corresponds to $\cos\theta\approx 1$.

By the other hand, the radiated energy per particle per unit of time is equal to:

$\boxed{-\dfrac{dE}{dt}=\dfrac{e^2V}{c^2}\int_0^{\omega_{max}}\omega\left[1-\dfrac{1}{n^2\beta^2}-\dfrac{\Lambda}{n\beta\lambda}\left(1-\dfrac{1}{n^2}\right)-\dfrac{\Lambda^2}{4\lambda^2}\left(1-\dfrac{1}{n^2}\right)\right]d\omega}$

where $\omega=2\pi c/n\lambda$ is the angular frequency of the radiation, with a maximum value of $\omega_{max}=2\pi c/n\lambda_{min}$.
Remark: In the non-relativistic case, $v<, and the condition $\beta n>1$ implies that $n>>1$. Therefore, neglecting the quantum corrections (the charged particle self-interaction/backreaction to radiation), we can insert the limit $\Lambda/\lambda\rightarrow 0$ and the above previous equations will simplify into:

$\boxed{\cos\theta=\dfrac{1}{n\beta}-\dfrac{c}{nv}}$

$\boxed{-\dfrac{dE}{dt}=\dfrac{e^2 v}{c^2}\int_0^{\omega_{max}}\omega\left(1-\dfrac{c^2}{n^2v^2}\right)d\omega}$

Remember: $\omega_{max}$ is determined with the condition $\beta n(\omega_{max})=1$, where $n(\omega_{max})$ represents the dispersive effect of the material/medium through the refraction index.

THE FRANK-TAMM FORMULA

The number of photons produced per unit path length and per unit of energy of a charged particle (charge equals to $Zq$) is given by the celebrated Frank-Tamm formula:

$\boxed{\dfrac{d^2N}{dEdx}=\dfrac{\alpha Z^2}{\hbar c}\sin^2\theta_c=\dfrac{\alpha^2 Z^2}{r_em_ec^2}\left(1-\dfrac{1}{\beta^2n^2(E)}\right)}$

In terms of common values of fundamental constants, it takes the value:

$\boxed{\dfrac{d^2N}{dEdx}\approx 370Z^2\sin^2\theta_c(E)eV^{-1}\cdot cm^{-1}}$

or equivalently it can be written as follows

$\boxed{\dfrac{d^2N}{dEdx}=\dfrac{2\pi \alpha Z^2}{\lambda^2}\left(1-\dfrac{1}{\beta^2n^2(\lambda)}\right)}$

The refraction index is a function of photon energy $E=\hbar \omega$, and it is also the sensitivity of the transducer used to detect the light with the Cherenkov effect! Therefore, for practical uses, the Frank-Tamm formula must be multiplied by the transducer response function and integrated over the region for which we have $\beta n(\omega)>1$.

Remark: When two particles are close toghether ( to be close here means to be separated a distance $d<1$ wavelength), the electromagnetic fields form the particles may add coherently and affect the Cherenkov radiation. The Cherenkov radiation for a electron-positron pair at close separation is suppressed compared to two independent leptons!

Remark (II): Coherent radio Cherenkov radiation from electromagnetic showers is significant and it has been applied to the study of cosmic ray air showers. In addition to this, it has been used to search for electron neutrinos induced showers by cosmic rays.

CHERENKOV DETECTOR: MAIN FORMULA AND USES

The applications of Cherenkov detectors for particle identification (generally labelled as PID Cherenkov detectors) are well beyond the own range of high-energy Physics. Its uses includes: A) Fast particle counters. B) Hadronic particle indentifications. C) Tracking detectors performing complete event reconstruction. The PDG gives some examples of each category: a) Polarization detector of SLD, b) the hadronic PID detectors at B factories like BABAR or the aerogel threshold Cherenkov in Belle, c) large water Cherenkov counters liket those in Superkamiokande and other neutrino detector facilities.

Cherenkov detectors contain two main elements: 1) A radiator/material through which the particle passes, and 2) a photodetector. As Cherenkov radiation is a weak source of photons, light collection and detection must be as efficient as possible. The presence of a refractive material specifically designed to detect some special particles is almost vindicated in general.

The number of photoelectrons detected in a given Cherenkov radiation detector device is provided by the following formula (derived from the Tamm-Frank formula simply taking into account the efficiency in a straightforward manner):

$\boxed{N=L\dfrac{\alpha^2 Z^2}{r_em_ec^2}\int \epsilon (E)\sin^2\theta_c(E)dE}$

where $L$ is the path length of the particle in the radiator/material, $\epsilon (E)$ is the efficiency for the collector of Cherenkov light and transducing it in photoelectrons, and

$\boxed{\dfrac{\alpha^2}{r_em_ec^2}=370eV^{-1}cm^{-1}}$

Remark: The efficiencies and the Cherenkov critical angle are functions of the photon energy, generally speaking. However, since the typical energy dependen variation of the refraction index is modest, a quantity sometimes called Cherenkov detector quality fact $N_0$ can be defined as follows

$\boxed{N_0=\dfrac{\alpha^2Z^2}{r_em_ec^2}\int \epsilon dE}$

In this case, we can write

$\boxed{N\approx LN_0<\sin^2\theta_c>}$

Remark(II): Cherenkov detectors are classified into imaging or threshold types, depending on its ability to make use of Cherenkov angle information. Imaging counters may be used to track particles as well as identify particles.

Other main uses/applications of the Vavilov-Cherenkov effect are:

1st. Detection of labeled biomolecules. Cherenkov radiation is widely used to facilitate the detection of small amounts and low concentrations of biomolecules. For instance, radioactive atoms such as phosphorus-32 are readily introduced into biomolecules by enzymatic and synthetic means and subsequently may be easily detected in small quantities for the purpose of elucidating biological pathways and in characterizing the interaction of biological molecules such as affinity constants and dissociation rates.

2nd. Nuclear reactors. Cherenkov radiation is used to detect high-energy charged particles. In pool-type nuclear reactors, the intensity of Cherenkov radiation is related to the frequency of the fission events that produce high-energy electrons, and hence is a measure of the intensity of the reaction. Similarly, Cherenkov radiation is used to characterize the remaining radioactivityof spent fuel rods.

3rd. Astrophysical experiments. The Cherenkov radiation from these charged particles is used to determine the source and intensity of the cosmic ray,s which is used for example in the different classes of cosmic ray detection experiments. For instance, Ice-Cube, Pierre-Auger, VERITAS, HESS, MAGIC, SNO, and many others. Cherenkov radiation can also be used to determine properties of high-energy astronomical objects that emit gamma rays, such as supernova remnants and blazars. In this last class of experiments we place STACEE, in new Mexico.

4th. High-energy experiments. We have quoted already this, and there many examples in the actual LHC, for instance, in the ALICE experiment.

Vacuum Cherenkov radiation (VCR) is the alledged and  conjectured phenomenon which refers to the Cherenkov radiation/effect of a charged particle propagating in the physical vacuum. You can ask: why should it be possible? It is quite straightforward to understand the answer.

The classical (non-quantum) theory of relativity (both special and general)  clearly forbids any superluminal phenomena/propagating degrees of freedom for material particles, including this one (the vacuum case) because a particle with non-zero rest mass can reach speed of light only at infinite energy (besides, the nontrivial vacuum itself would create a preferred frame of reference, in violation of one of the relativistic postulates).

However, according to modern views coming from the quantum theory, specially our knowledge of Quantum Field Theory, physical vacuum IS a nontrivial medium which affects the particles propagating through, and the magnitude of the effect increases with the energies of the particles!

Then, a natural consequence follows: an actual speed of a photon becomes energy-dependent and thus can be less than the fundamental constant $c=299792458m/s$ of  speed of light, such that sufficiently fast particles can overcome it and start emitting Cherenkov radiation. In summary, any charged particle surpassing the speed of light in the physical vacuum should emit (Vacuum) Cherenkov radiation. Note that it is an inevitable consequence of the non-trivial nature of the physical vacuum in Quantum Field Theory. Indeed, some crazy people saying that superluminal particles arise in jets from supernovae, or in colliders like the LHC fail to explain why those particles don’t emit Cherenkov radiation. It is not true that real particles become superluminal in space or collider rings. It is also wrong in the case of neutrino propagation because in spite of being chargeless, neutrinos should experiment an analogue effect to the Cherenkov radiation called the Askaryan effect. Other (alternative) possibility or scenario arises in some Lorentz-violating theories ( or even CPT violating theories that can be equivalent or not to such Lorentz violations) when a speed of a propagating particle becomes higher than c which turns this particle into the tachyon.  The tachyon with an electric charge would lose energy as Cherenkov radiation just as ordinary charged particles do when they exceed the local speed of light in a medium. A charged tachyon traveling in a vacuum therefore undergoes a constant proper-time acceleration and, by necessity, its worldline would form an hyperbola in space-time. These last type of vacuum Cherenkov effect can arise in theories like the Standard Model Extension, where Lorentz-violating terms do appear.

One of the simplest kinematic frameworks for Lorentz Violating theories is to postulate some modified dispersion relations (MODRE) for particles , while keeping the usual energy-momentum conservation laws. In this way, we can provide and work out an effective field theory for breaking the Lorentz invariance. There are several alternative definitions of MODRE, since there is no general guide yet to discriminate from the different theoretical models. Thus, we could consider a general expansion  in integer powers of the momentum, in the next manner (we set units in which $c=1$):

$\boxed{E^2=f(p,m,c_n)=p^2+m^2+\sum_{n=-\infty}^{\infty}c_n p^n}$

However, it is generally used a more soft expansion depending only on positive powers of the momentum in the MODRE. In such a case,

$\boxed{E^2=f(p,m,a_n)=p^2+m^2+\sum_{n=1}^{\infty}a_n p^n}$

and where $p=\vert \mathbf{p}\vert$. If Lorentz violations are associated to the yet undiscovered quantum theory of gravity, we would get that ordinary deviations of the dispersion relations in the special theory of relativity should appear at the natural scale of the quantum gravity, say the Planck mass/energy. In units where $c=1$ we obtain that Planck mass/energy is:

$\boxed{M_P=\sqrt{\hbar^5/G_N}=1.22\cdot 10^{19}GeV=1.22\cdot 10^{16}TeV}$

Lets write and parametrize the Lorentz violations induced by the fundamental scale of quantum gravity (naively this Planck mass scale) by:

$\boxed{a_n=\dfrac{\Xi_n}{M_P^{n-2}}}$

Here, $\Xi_n$ is a dimensionless quantity that can differ from one particle (type) to another (type). Considering, for instance $n=3,4$, since the $n<3$ seems to be ruled out by previous terrestrial experiments, at higer energies the lowest non-null term will dominate the expansion with $n\geq 3$. The MODRE reads:

$E^2=p^2+m^2+\dfrac{\Xi_a p^n}{M_P^{n-2}}$

and where the label $a$ in the term $\Xi_a$ is specific of the particle type. Such corrections might only become important at the Planck scale, but there are two exclusions:

1st. Particles that propagate over cosmological distances can show differences in their propagation speed.
2nd. Energy thresholds for particle reactions can be shifted or even forbidden processes can be allowed. If the $p^n$-term is comparable to the $m^2$-term in the MODRE. Thus, threshold reactions can be significantly altered or shifted, because they are determined by the particle masses. So a threshold shift should appear at scales where:

$\boxed{p_{dev}\approx\left(\dfrac{m^2M_P^{n-2}}{\Xi}\right)^{1/n}}$

Imposing/postulating that $\Xi\approx 1$, the typical scales for the thresholds for some diffent kind of particles can be calculated. Their values for some species are given in the next table:

We can even study some different sources of modified dispersion relationships:

1. Measurements of time of flight.

2. Thresholds creation for: A) Vacuum Cherenkov effect, B) Photon decay in vacuum.

3. Shift in the so-called GZK cut-off.

4. Modified dispersion relationships induced by non-commutative theories of spacetime. Specially, there are time shifts/delays of photon signals induced by non-commutative spacetime theories.

We will analyse this four cases separately, in a very short and clear fashion. I wish!

Case 1. Time of flight. This is similar to the recently controversial OPERA experiment results. The OPERA experiment, and other similar set-ups, measure the neutrino time of flight. I dedicated a post to it early in this blog

https://thespectrumofriemannium.wordpress.com/2012/06/08/

In fact, we can measure the time of flight of any particle, even photons. A modified dispersion relation, like the one we introduced here above, would lead to an energy dependent speed of light. The idea of the time of flight (TOF) approach is to detect a shift in the arrival time of photons (or any other massless/ultra-relativistic particle like neutrinos) with different energies, produced simultaneous in a distant object, where the distance gains the usually Planck suppressed effect. In the following we use the dispersion relation for $n=3$ only, as modifications in higher orders are far below the sensitivity of current or planned experiments. The modified group velocity becomes:

$v=\dfrac{\partial E}{\partial p}$

and then, for photons,

$v\approx 1-\Xi_\gamma\dfrac{p}{M}$

The time difference in the photon shift detection time will be:

$\Delta t=\Xi_\gamma \dfrac{p}{M}D$

where D is the distance multiplied (if it were the case) by the redshift $(1+z)$ to correct the energy with the redshift. In recent years, several measurements on different objects in various energy bands leading to constraints up to the order of 100 for $\Xi$. They can be summarized in the next table ( note that the best constraint comes from a short flare of the Active Galactic Nucleus (AGN) Mrk 421, detected in the TeV band by the Whipple Imaging Air Cherenkov telescope):

There is still room for improvements with current or planned experiments, although the distance for TeV-observations is limited by absorption of TeV photons in low energy metagalactic radiation fields. Depending on the energy density of the target photon field one gets an energy dependent mean free path length, leading to an energy and redshift dependent cut off energy (the cut off energy is defined as the energy where the optical depth is one).

2. Thresholds creation for: A) Vacuum Cherenkov effect, B) Photon decay in vacuum. By the other hand, the interaction vertex in quantum electrodynamics (QED) couples one photon with two leptons. When we assume for photons and leptons the following dispersion relations (for simplicity we adopt all units with M=1). Then:

$\omega_k^2=k^2+\xi k^n$                $E^2_p=p^2+m^2+\Xi p^n$

Let us write the photon tetramomentum like $\mathbb{P}=(\omega_k,\mathbf{k})$ and the lepton tetramomentum $\mathbb{P}=(E_p,\mathbf{p})$ and $\mathbb{Q}=(E_q,\mathbf{q})$. It can be shown that the transferred tetramomentum will be

$\xi k^n+\Xi p^n-\Xi q^n=2(E_p\omega_k-\mathbf{p}\cdot\mathbf{k})$

where the r.h.s. is always positive. In the Lorentz invariant case the parameters $\xi, \Xi$  are zero, so that this equation can’t be solved and all processes of the single vertex are forbidden. If these parameters are non-zero, there can exist a solution and so these processes can be allowed. We now consider two of these interactions to derive constraints on the parameters $\Xi, \xi$. The vacuum
Cherenkov effect $e^-\rightarrow \gamma e^-$ and the spontaneous photon-decay $\gamma\rightarrow e^+e^-$.

A) As we have studied here, the vacuum Cherenkov effect is a spontaneous emission of a photon by a charged particle $0.  These effect occurs if the particle moves faster than the slowest possible radiated photon in vacuum!
In the case of $\Xi>0$, the maximal attainable speed for the particle $c_{max}$ is faster than c. This means, that the particle can always be faster than a zero energy photon with

$\displaystyle{c_{\gamma_0}=c\lim_{k\rightarrow 0}\dfrac{\partial \omega}{\partial k}=c\lim_{k\rightarrow 0}\dfrac{2k+n\xi k^{n-1}}{2\sqrt{k^2+\xi k^n}}=c}$

and it is independent of $\xi$. In the case of $\Xi<0$, i.e., $c_{par}$ decreases with energy, you need a photon with $c_\gamma. This is only possible if $\xi<\Xi$.

Therefore, due to the radiation of photons such an electron loose energy. The observation of high energetic electrons allows to derive constraints on $\Xi$ and $\xi$.  In the case of $\Xi<0$, in the case with n=3, we have the bound

$\Xi<\dfrac{m^2}{2p^3_{max}}$

Moreover, from the observation of 50 TeV photons in the Crab Nebula (and its pulsar) one can conclude the existens of 50 TeV electrons due to the inverse Compton scattering of these electrons with those photons. This leads to a constraint on $\Xi$ of about

$\Xi<1.2\times 10^{-2}$

where we have used $\Xi>0$ in this case.

B) The decay of photons into positrons and electrons $\gamma\rightarrow e^+e^-$ should be a very rapid spontaneous decay process. Due to the observation of Gamma rays from the Crab Nebula on earth with an energy up to $E\sim 50TeV$. Thus, we can reason that these rapid decay doesn’t occur on energies below 50 TeV. For the constraints on $\Xi$ and $\xi$ these condition means (again we impose n=3):

$\xi<\dfrac{\Xi}{2}+0.08, \mbox{for}\; \xi\geq 0$

$\xi<\Xi+\sqrt{-0.16\Xi}, \mbox{for}\;\Xi<\xi<0$.

3. Shift in the GZK cut-off. As the energy of a proton increases,the pion production reaction can happen with low energy photons of the Cosmic Microwave Background (CMB).

This leads to an energy dependent mean free path length of the particles, resulting in a cutoff at energies around $E_{GZK}\approx 10^{20}eV$. This is the the celebrated Greisen-Kuzmin-Zatsepin (GZK) cut off. The resonance for the GZK pion photoproduction with the CMB backgroud can be read from the next condition (I will derive this condition in a future post):

$\boxed{E_{GZK}\approx\dfrac{m_p m_\pi}{2E_\gamma}=3\times 10^{20}eV\left(\dfrac{2.7K}{E_\gamma}\right)}$

Thus in Lorentz invariant world, the mean free path length of a particle of energy 5.1019 eV is 50 Mpc i.e. particle over this energy are readily absorbed due to pion photoproduction reaction. But most of the sources of particle of ultra high energy are outside 50 Mpc. So, one expects no trace of particles of energy above $10^{20}eV$ on Earth. From the experimental point of view AGASA has found
a few particles having energy higher than the constraint given by GZK cutoff limit and claimed to be disproving the presence of GZK cutoff or at least for different threshold for GZK cutoff, whereas HiRes is consistent with the GZK effect. So, there are two main questions, not yet completely unsolved:

i) How one can get definite proof of non-existence GZK cut off?
ii) If GZK cutoff doesn’t exist, then find out the reason?

The first question could by answered by observation of a large sample of events at these energies, which is necessary for a final conclusion, since the GZK cutoff is a statistical phenomena. The current AUGER experiment, still under construction, may clarify if the GZK cutoff exists or not. The existence of the GZK cutoff would also yield new limits on Lorentz or CPT violation. For the second question, one explanation can be derived from Lorentz violation. If we do the calculation for GZK cutoff in Lorentz violated world we would get the modified proton dispersion relation as described in our previous equations with MODRE.

4. Modified dispersion relationships induced by non-commutative theories of spacetime. As we said above, there are time shifts/delays of photon signals induced by non-commutative spacetime theories. Noncommutative spacetime theories introduce a new source of MODRE: the fuzzy nature of the discreteness of the fundamental quantum spacetime. Then, the general ansatz of these type of theories comes from:

$\boxed{\left[\hat{x}^\mu,\hat{x}^\nu\right]=i\dfrac{\theta^{\mu\nu}}{\Lambda_{NC}^2}}$

where $\theta^{\mu\nu}$ are the components of an antisymmetric Lorentz-like tensor which components are the order one. The fundamental scale of non-commutativity $\Lambda^2_{NC}$ is supposed to be of the Planck length. However, there are models with large extra dimensions that induce non-commutative spacetime models with scale near the TeV scale! This is interesting from the phenomenological aside as well, not only from the theoretical viewpoint. Indeed, we can investigate in the following whether astrophysical observations are able to constrain certain class of models with noncommutative spacetimes which are broken at the TeV scale or higher. However, there due to the antisymmetric character of the noncommutative tensor, we need a magnetic and electric background field in order to study these kind of models (generally speaking, we need some kind of field inducing/producing antisymmetric field backgrounds), and then the dispersion relation for photons remains the same as in a commutative spacetime. Furthermore, there is no photon energy dependence of the dispersion relation. Consequently, the time-of-flight experiments are inappopriate because of their energy-dependent dispersion. Therefore, we suggest the next alternative scenario: suppose, there exists a strong magnetic field  (for instance, from a star or a cluster of stars) on the path photons emitted at a light source (e.g. gamma-ray bursts). Then, analogous to gravitational lensing, the photons experience deflection and/or change in time-of-arrival, compared to the same path without a magnetic background field. We can make some estimations for several known objects/examples are shown in this final table:

In summary:

1st. Vacuum Cherenkov and related effects modifying the dispersion relations of special relativity are natural in many scenarios beyond the Standard Relativity (BSR) and beyond the Standard Model (BSM).

2nd. Any theory allowing for superluminal propagation has to explain the null-results from the observation of the vacuum Cherenkov effect. Otherwise, they are doomed.

3rd. There are strong bounds coming from astrophysical processes and even neutrino oscillation experiments that severely imposes and kill many models. However, it is true that current MODRE bound are far from being the most general bounds. We expect to improve these bounds with the next generation of experiments.

4th. Theories that can not pass these tests (SR obviously does) have to be banned.

5th. Superluminality has observable consequences, both in classical and quantum physics, both in standard theories and theories beyond standard theories. So, it you buid a theory allowing superluminal stuff, you must be very careful with what kind of predictions can and can not do. Otherwise, your theory is complentely nonsense.

As a final closing, let me include some nice Cherenkov rings from Superkamiokande and MiniBoone experiments. True experimental physics in action. And a final challenge…

FINAL CHALLENGE: Are you able to identify the kind of particles producing those beautiful figures? Let me know your guesses ( I do know the answer, of course).

Figure 1. Typical SuperKamiokande Ring.  I dedicate this picture to my admired Japanase scientists there. I really, really admire that country and their people, specially after disasters like the 2011 Earthquake and the Fukushima accident. If you are a japanase reader/follower, you must know we support your from abroad. You were not, you are not and you shall not be alone.

Figure 2. Typical MiniBooNe ring. History: I used this nice picture in my Master Thesis first page, as the cover/title page main picture!