Abstract
We considered the Inverse Compton Scattering (ICS) of charged particles onto photons whose distribution is a Black Body Radiation (BBR) deriving the exact energy and angular differential distribution in the general case and in its most useful expansions. These results can be successfully applied in high energy accelerators experiments to evaluate the ICS contribution from the thermal photons in the cavity as well as in astrophysics where the ICS of cosmic rays plays a relevant role in a variety of phenomena. In particular we show how our formulae reproduce the ICS energy spectrum recently measured at LEP, how it could be considered a key tool in explaining the Gamma Ray Bursts (GRB) , SGRs energy spectrum. Finally we predicted the presence of a low gamma flux, nearly detectable at hundred of TeV from SNRs SN1006 as well as, at lower energy (tens TeV, due to gamma ray cascading in cosmic BBR), from relic extragalactic highest cosmic rays sources born by jets in AGN,as blazars 3C279,Mrk421,Mrk 501.
ROME prep. INFN n.1134
INVERSE COMPTON SCATTERING ONTO BBR
IN HIGH ENERGY PHYSICS AND GAMMA
(MeVTeV) ASTROPHYSICS
Daniele Fargion and Andrea Salis
Dipartimento di Fisica, Universita’ di Roma “La Sapienza”,
INFNSezione di Roma I
P.le A. Moro 2, 00185 Rome, Italy.
ROME prep. INFN n.1134
23 February 1996
Introduction
The Inverse Compton Scattering (ICS) plays a relevant role in highest energy astrophysics (cosmic rays and gamma astronomy) [16] and high energy physics (LEP I, LEP II, accelerators) [79]. Indeed from one hand the ICS of high relativistic cosmic rays (either electron at GeV or proton and nuclei at much higher energies) onto electromagnetic fields (either cosmological Black Body Radiation (BBR) at , interstellar lights, radio waves or even stationary magnetic fields) is the source of high energy photons (X, gamma rays) which we do observe in the Universe as diffuse or point source, on the other hand the ICS is often the main process responsible for the slowing down (i.e. for the energy losses) of energetic charged particles; indeed ICS is often the main cause of cosmic rays lifetime behaviour as well as of their energy spectrum depletion at harder regions (i.e. of their detailed spectrum shape and evolution) [56]. Moreover, ICS of relativistic jets by compact objects onto thermal photons by a star companion in a binary system or accretion disk might be, as recently proposed [10], the key process able to produce ”gamma jets” responsible (by their rotation and blazing into different directions) of the puzzling Gamma Ray Bursts (GRB). So even if this subject seems apparently settled [34] we revisit the ICS process in order to obtain an analytic and compact formula able to describe the differential energy and angular ICS scattered photon number spectrum of relativistic charges onto BBR photon spectrum. One of the main feature of our final expressions is that we can easily cover the entire range of energy of the ICS energy spectrum. These results improve the Montecarlo simulation because the latter must consider only a thin portion of the energy range in order to inspect in detail the ICS spectrum [9]. In the following we describe our approach and we show some remarkable results.
The ICS onto BBR spectrum
We follow a standard procedure to get the general ICS spectrum of a relativistic charge (electron, proton, nuclei in cosmic rays or in accelerators bunches) hitting photons whose distribution is a BBR spectrum. We consider first the photon target distribution in the Laboratory Frame (LF) where the BBR is isotropic and homogeneous, then we transform it to the Electron Frame (EF) where the BBR is still homogeneous but highly anisotropic, therefore we evaluate in the EF the usual Compton scattering and finally we transform back the diffused differential photon number to the LF. The starting photon target distribution, in the LF, is the wellknown isotropic and homogeneous BBR whose number density per unit energy and solid angle is given by the Planck formula
(1) 
where is the Boltzmann constant and T the BBR temperature. We transform this distribution to the EF by standard Lorentz boosts choosing as z axis the direction coincident with the initial electron momentum and reminding that is a relativistic invariant [2]. In the following we label by a the quantities related to the electron frame EF, by a subscript 0 if they are considered before the scattering, by a subscript 1 if after; so we have
where is the adimensional electron velocity and the corresponding Lorentz factor. The transformed BBR number density distribution exhibits, in this relativistic limit and in the EF, a clear dipole anisotropy and it is becomes more and more peaked around as increases. An analogous dipole signature, at non relativistic regime, is the cosmological one found at millikelvin level in the BBR due to the Earth motion. The associated energy spectrum is mostly ”green shifted” showing a maximum around with respect to the original ”red” BBR spectrum with a maximum around . The next step is to derive the total number of diffused photons in the EF; this number can be obtained as follows:
(2) 
where is the Compton differential cross section
(3) 
is the speed of light and the scattering angle must be expressed as a function of the other angles involved, i.e. the incoming and the outcoming angles
(4) 
The last step is to derive the final exact ICS differential number distribution in the Laboratory Frame (where the scattered photons will be observed) by using the inverse Lorentz transformations; the result is:
(5) 
where from eq.4 must be expressed as a function of . This ”blue shifted” distribution is different from the previous ones because now the expected peak around a given energy has been spread into a wide plateau from up to energies. The most general ICS differential distribution can be obtained by means of a numerical integration of eq.5 over but as we are more interested in high energy phenomena we also show how to simplificate the above formula in this cases. First of all we remind that in particle accelerators we are dealing with ultrarelativistic particles, i.e. ; moreover for most astrophysical problems the photon energy in the EF is much smaller than the electron rest mass, i.e. ; so it is often possible to approximate the Compton differential cross section by the Thomson one; as a third further approximation we may consider the ultrarelativisticThomson limit where the two previous conditions and are both satisfied. Let us discuss these three different expansions. In the first case () the BBR photons, in the EF, are pratically all incident headon so the incident angle can be approximately written as . Consequently the scattering angle is related to the angle by the simple formula and within . Moreover the kinematics of the ICS shows that the scattered radiation, in the LF, is strongly concentrated in a narrow cone along the direction of motion of the ultrarelativistic particle. The ICS differential distribution in eq.5, in the ultrarelativistic expansion, reduces to an analytical expression
(6) 
where and . The second possible approximation is the Thomson limit . In this case we can neglect all terms of order with respect to 1 in eq.5 and the resulting differential distribution becomes
(7) 
Indeed, from the previous eq.67, we can get the last expansion, the Thomsonultrarelativistic formula. It stems from assuming and at the same time. This means that the ICS differential distribution can be obtained by setting in eq.6 and by setting in eq.7. In the first derivation we obtain:
(8) 
In the second derivation the integral contained in eq.7 can be simplified remembering that so and the integrand is significatively different from zero only inside a thin cone whose aperture is just . Thus the result is a formula differing from eq.8 only by a factor of order and in our assumptions this term is completely negligible. So the still analytical eq.8 is our ultrarelativisticThomson distribution. In fig.1 we show the surprising metamorphosis of a Planckian Black Body into a taller (by a suppression factor ) serial of smooth truncated hills at larger relativistic regimes. This process is mainly due to the overlapping of a series of blueshifted spectra obtained at different angular directions; the softer photons are the more isotropic ones while the harder photons are strongly anisotropic and located in the inner cone of the beam. The ”Rayleigh” regions of the differential ”BBR” spectra overlap each other even if calculated at different angles () while their peaks are higher and more and more blueshifted for angles approaching zero. Consequently the exponential decay of the ICS spectrum on the right side reflects the ”Wien” regions behaviour of the angular ”BBR” spectra at . We give here a qualitative summary of the ICS energy spectrum behaviour in the whole energy range. For the spectrum exhibits a linear growth (like the Rayleigh region of a BBR one); for energies the linear behaviour is modified by a logaritmic correction and the growth is proportional to ; for the spectrum is spread into a very slowly linearly decreasing plateau up to ; for the spectrum decays as (we remember the reader that the Wien region of a BBR distribution decays as ). We remind that the commonly and widely applied analytical formulae on ICS are the ones derived by F.Jones [3] and based on ICS onto monochromatic and isotropic radiation; this ficticious and artificial ”BBR” leads to a final spectrum in disagreement with the experimentally observed ones. In the next two sections we show that we can evaluate in the right way ICS spectra in high energy physics and astrophysics by means of our formulae.
The ICS spectrum in LEP experiments
We could directly verify the validity of present results by a (successful) fit of experimental ICS spectra obtained at LEP by A.Melissinos group [7] and by G.DiambriniPalazzi group [8] (at a higher degree of precision). Indeed the LEP vacuum pipe can be considered as a black body cavity at room temperature (); hence the electromagnetic radiation in thermal equilibrium is scattered and beamed ahead by the electron (and positron) bunches whose energy is . At this energy corresponds a Lorentz factor so we can apply our ultrarelativistic formulae for ICS to evaluate the interesting region of the spectrum. We compare our results with the Montecarlo simulations performed at LEP and able to fit the ICS effect in the experimental data. We compute either the Thomson and the Compton spectra in order to show the differences between the two curves and with respect to the Montecarlo simulation. The spectra can be obtained by a numerical integration over the angle in expressions 68 respectively for Compton or Thomson limit:
(9) 
where is the number of particles in a bunch and is the flight time in the LEP straight section [9]. The three ICS spectra, derived from our equations and from the Montecarlo simulation are shown in fig.2 and labelled respectively as:
1) M: Montecarlo simulation (ref.[9] fig.6)
2) T: Thomson approximation, from eq.8
3) C: Compton approximation, from eq.6
Note that our Compton spectrum shows a good agreement with the Montecarlo simulation while the Thomson one, at high energies, is a factor 3 higher. This overestimate of the Thomson spectrum is clearly related to the independence of the Thomson cross section with respect to the photon energy. We notice that our ICS spectrum, in the Compton limit, is nearly coincident with the Montecarlo approximation at low energy but for higher energies () it is a 26 above. This discrepancy seems not to be related to our approximations because from our data we obtain, for the total event number, a value pratically coincident (within ) with found by Di Domenico [9] by Montecarlo simulations. The difference might be due to the Montecarlo method whose statistical procedure implies a smaller number of events at higher energies. However this discrepancy does not affect the beam lifetime because in its evaluation only the number rate is involved. We remind that the ICS may also be used to study the bunch internal structure and the result of a coherent emission by the charges, in this case the optimal experimental set up is reached when the relativistic bunches are hit by collinear back photon emitted by a laser [11].
The ICS in Astrophysics
The astronomy traces mostly the presence of relativistic electrons (or, at lower level, of relativistic nuclei) by their synchrotron radiation or their ICS onto infrared, interstellar or cosmic radiation. Moreover the same ICS may become the main slowing down process for relativistic charges once magnetic field energy densities are below the corresponding cosmic photon energy densities . This situation generally occurs in extragalactic spaces where . In this framework cosmic rays electrons around SN1006 have been recently discovered indirectly by their non thermal Xray emission due to synchrotron radiation from ultrarelativistic electrons () [12]. It is therefore of great interest and actuality to provide not just an order of magnitude for the corresponding ICS ray flux but also for its detailed spectrum for such relativistic and ultrarelativistic cosmic rays electrons. We show in fig.3 these ICS spectra for . It is important to note the existence of a high energy transition in the ICS from the Thomson to the Compton behaviour. This change occurs dramatically at the highest energy range of the spectra. The Lorentz factor for Compton behaviour occurs for electrons () and for protons (, …) in ICS with BBR respectively at huge energy values
(10) 
The ICS behaviour in the Compton limit may be qualitatively predicted keeping in mind that the ICS energy spectrum falls off at photon energies near and the energy conservation calls for a cutoff of the extension of the ICS spectrum plateau from to instead of reaching the usual extreme value . Let us better understand this ICS behaviour change from Thomson to Compton regime as follows: as long as the incident photon, in the EF, has a characteristic energy the diffused photon, in the EF, mantains most of its original energy and it is spread around nearly isotropically. Therefore, once the spectrum is reviewed in the LF after the Lorentz boost, the previous different angular distribution of the photons in the EF becomes, in the LF, a different energy distribution of the same photons. The angular integral of these differential ”BBR” spectra , whose ”Rayleigh” regions overlap, produces in the energy range the wider plateau shown in fig.4. However, at Compton regime, when the differential cross section leads, in the EF, to a diffused and anisotropic photon distribution which becomes more and more beamed in a ”Compton cone” at angles related to the boost transformations). All the photons diffused outside the cone can reach in the LF final energies of order . On the other side, around and inside the Compton cone but far from the edge of the inner Lorentz cone (for example we may consider (not to be confused with the thinner ”Lorentz” angles
The main scenarios for ICS applications
For a synthetic but complete picture of all the above described ICS behaviours
we suggest to define the following characteristic regimes each labelled by the
relevant parameters involved.
1) (, ) The
nonrelativistic Compton scattering limit onto coldwarm BBR (fig.6).
Here the simplest ICS spectrum becomes a self similar Planck spectrum; its
reflectivity efficiency depends on the usual quantities ,
, … One of the mostly celebrated application is the
SunyaevZeldovich effect in cosmology.
2) (, ) The ICS at
relativistic Thomson limit onto coldwarm BBR (fig.12). This is
the case with most application in high energy physics and astrophysics (LEP I,
LEP II, GRBs, cosmic rays at energies and ,…). Now the ICS spectrum deviates from a pure Planckian spectrum
leading to a ”cuthill” spectrum with its smooth edge covering the energies
from up to photon energies
. This ”simple” ICS spectrum
may be ruling the spectra of ICS by charge (electron) jets () onto nearby stellar companion photons () leading
to successful gamma jets able to explain the integral spectra of GRBs [13].
This arguments have widely been developed by the authors
and are still under consideration [14]. Because of the ”smooth” behaviour of
the edge of the spectrum there is not a one to one relation between the cosmic
rays electron or proton spectral index and the final ICS rays
spectral index. Another scenario where ICS leads to rays takes
place in extragalactic blazars which eject beamed cosmic rays (protons,
nuclei,…) for huge distances. Their ICS onto 2.73 K BBR may also lead to
collinear jets; their energy is dominated by the electron presence
(over the nuclei one) in the cosmic rays jets. Such gamma jet is the large
scale version of the minijet model plus ICS considered by the authors for
GRB production in galactic binary systems.
3) (, )
The ICS at relativistic Compton limit onto coldwarm BBR (fig.7). These
energy windows are relevant for the highest cosmic rays interactions
onto BBR (, ). The ICS spectrum
exhibits a pile up of photons at the edge of the highest energies
leading to a peaked maximum at and a sharp cut
off at energies just above it. The presence of such a peak has important
consequences in the cosmic rays rays links. The primordial incident
cosmic rays spectrum at highest energies (power law, …) leaves its original
inprint into a similar ”photocopy” ray spectrum. Moreover the new
born high energies rays () may also successfully
interact with the BBR photons (electron pairs production) leading to
electronproton cascades or electromagnetic showers at lower and lower
energies. The showers will arrest their growth as soon as the last degraded
photon energies will reach the threshold . It is important to remind that the process at ultrarelativistic limit has a very similar cross section (
BreitWheeler, 1934) as the annihilation (Dirac, 1930) or the Klein
Nishina (1929) cross section. One of the consequences is the expected
cosmic background presence of
relic TeV rays components (due to the direct cosmological ICS or
due to its secondary showers at energies ). This
argument will be discussed in detail elsewhere.
4) (, ) The non relativistic
Compton scattering onto ”hot” BBR (fig.8). The resulting ICS spectrum
deviates from the
original one and exhibits a small peak at energies . The effect as above is related to the energy dependence of the
KleinNishina cross section.
5) (, ) The
relativistic ICS onto ”hot” BBR. The spectrum shows a new ”plateau”
extending from
energies up to
energies with a marked peak at energies and a
sharp cut off above, this behaviour is similar of point 3).
6) (, ) The equilibrium
regime of ICS onto ”hot” BBR (fig.9). The spectrum shows a marked peak at
energies with a sharp cutoff above. The known
astrophysical scenarios where such ICS may play a role could be the earliest
hottest (thermal equilibrium) cosmological epochs (, ) and
in the hot thermal cores of supernovae explosions (). It
is interesting to notice the nature of the non equilibrium ICS spectrum and
it may be
worthfull to derive the exact kinetic equations (due to such ICS) for the
multicomponent thermal bath of the early universe as well as the SN
explosive
processes. Finally if the fireball model is a real event as the one needed to
explain the GRB puzzle then such ICS (onto the last layers of the fireball
explosion) would be smeared out by multiscattering during last stages of
the fireball into a
final nearby thermal GRB spectrum. We do not recognize such a presence of
thermal inprint in GRB [13].
7) () The ultrarelativistic
”ultrahot” ICS (fig.10). This ICS spectrum exhibits a peaked spectrum as
usual at energies but also a significative
decaying component at higher energies. We can see a ”shoulder” which may
become, in the
extreme cases , a slowly decaying ”plateau”. Such a
peculiar process (at the present) seems very hypothetical but we have shown
it in order to cover the entire range of possible ICS behaviours. Possible
exotic
scenarios where such extreme conditions (5,6,7) may take place efficiently
occur also near miniblackhole evaporations in the early Universe where the
cosmological BBR has a temperature lower or comparable or even greater than
the corresponding earliest miniblackhole temperature (). This hybrid thermal bath, depending on miniblackhole
primordial masses and distribution, may be dominant in non standard early
cosmology bariogenesis and even in later cosmological nucleosynthesis.
Conclusions and Applications
The developed ICS formulae have a wide range of applications; in particular in fitting LEP I, LEP II experimental data and in understanding recent GRB puzzling spectra (in this last case we modified eq.8 in order to take into account the diluted and anisotropic BBR spectrum seen from the jet and we assumed a ringlike photon source ([14] eq.20). Moreover the extreme spectra of ICS at relativistic Compton limit onto coldwarm BBR, its peak at highest energies, may be probed at LEP I, LEP II using a diffused thermal optical light (a flash) in the beam pipe during the bunch crossing. Finally the recent evidence for cosmic rays electron at energies above hundreds TeV by their observed synchrotron radiation at soft X spectra implies the coexistence of a low but nearly detectable component of high cosmic rays at energies . Because of arguments in 3) the cosmic rays electrons spectrum () will be reflected also in the ”photocopy” spectrum. Their total energy flux will be of the same order of magnitude of the X rays one:
(11) 
The corresponding ray flux number will be extremely suppressed because of the energy ratio
(12) 
and it follows that the needed area for such a low flux is possibly below the present air shower arrays sensibility. However an accurate directionflux correlation might be able to observe in a near future a tiny 100 TeV flux. We suggest to seriously consider a detailed observational program in order to detect above the tiny 100 TeV flux from SN1006 and possibly the other suggested candidates (Cas A, IC443, Tycho SN, …) from SNRs. Their presence at the above fluxes is a necessary and compelling consequence of fundamental QED and cosmological BBR theories combined in present ICS models. Finally we interpret the two X lobes around SN1006 as generated by an electron jet scattering onto the relic giant shell contrary to the more popular idea of a Fermi shock acceleration mechanism. In these beaming models one should also expect a rare and strongly time dependent TeV rays ”burst” which could be observed in a short period of time (hours) once the observer is inside the thin jet cone direction, with an amplified integral intensity by a factor Lorentz larger then a diffused spherical source. Well known candidates are blazars or quasars as 3C279 , AGNs, as NGC 3079, or most recent TeV Mrk sources. Similar arguments leaded us to expect a low variable relic background of gamma (ten TeV) noise due to the pile up of cosmic integral ICS gamma rays and their electromagnetic cascades just below the electron pairs creation threshold on cosmic BBR.
Acknowledgements
We wish to thank Prof. G.DiambriniPalazzi and Dott. A.Di Domenico for useful discussions and support.
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Figure Captions

Fig.1: The ICS spectrum from eq.8 for T=291 K and non relativistic (BBR) (dot), and relativistic (dot dash), (dash), (continuous) Lorentz factor

Fig.2: The ICS spectra at LEP: Montecarlo simulation M (ref.9 fig.6), Thomson approximation T (eq.8), Compton approximation C (eq.6)

Fig.3: The ICS spectrum for T=2.73 K and (dot dash), (dot), (continuous) Lorentz factor

Fig.4: The ICS ”Thomson” spectrum for T=2.73 K and Lorentz factor

Fig.5: The ICS spectrum evolution from ”hilllike” Thomson spectra (dash), (lower continuous), 1 (dot dash) to Compton peaked spectra (dot), (higher continuous) for Lorentz factor

Fig.6: The ICS Thomson spectra for T=291 K and (BBR) (dot), (dash) Lorentz factor

Fig.7: The ICS Compton spectra for and Lorentz factor

Fig.18: The ICS Compton spectrum for and Lorentz factor

Fig.9: The ICS Compton spectrum for and Lorentz factor ()

Fig.10: The Compton ICS spectrum for (continuous), (dot) and Lorentz factor