Non-electromagnetic emissions from high energy particles in extreme environments has been studied in the literature by using several variations of the semi-classical formalism. The detailed mechanisms behind such emissions are of great astrophysical interest since they can alter appreciably the associated energy loss rates. Here, we review the role played by the source proper acceleration $a$ in the particle production process. The acceleration $a$ determines the typical scale characterizing the particle production and, moreover, if the massive particle production is inertially forbidden, it will be strongly suppressed for $a$ below a certain threshold. In particular, we show that, for the case of accelerated protons in typical pulsar magnetospheres, the corresponding accelerations $a$ are far below the pion production threshold.
Processes involving particle production in the presence of classical electromagnetic fields F ab can be considered in the context of semi-classical approximation, in which the source particle is described by a classical current with a prescribed trajectory and the emitted particles are considered as fully quantized fields. For the case of accelerated protons, the inertially forbidden decays via the strong interaction channel like, for instance, p + a → p + π 0 and p a → nπ + , and via the weak channel, as p + a → ne + ν, have been studied in detail in the semi-classical approximation. For the case of motion under magnetic fields, the validity of the semi-classical approximation is warranted if [1][2][3],
where B ⋆ = γB is the magnitude of the magnetic field measured in the instantaneous reference frame of the proton, γ is the usual Lorentz factor, and B cr = m 2 p /e ≈ 1.5 × 10 20 G is a critical value for the magnetic field, denoting the region where a full quantized analysis is mandatory. Provided that condition (1) is obeyed, the semi-classical calculations are accurate and the associated emitted power depends only on the proper acceleration of the source [4]. For the case of a magnetic field B, for instance, the proper acceleration of a (relativistic) proton is a ≈ γeB/m p and, consequently, Eq. ( 1) can be rewritten as
The parameter χ can be defined in a frame independent way as χ = (eF ab p a ) 2 /m 3 p , where p a stands for the proton momentum. For the case of inertially forbidden particle production, as those ones explicitly listed above, one naturally expects a strong suppression for quasi-inertiall (χ ≈ 0) protons.
We review here the role played by the proper acceleration a of the source in the massive particle production, with emphasis on the ranges favoring the massive particle emission channels. We also discuss the possibility of observing signals of curvature pion radiation from accelerated protons in the magnetosphere of strongly magnetized pulsars. Unless otherwise stated, natural units where c = h = 1 and the spacetime signature (+, -, -, -) are adopted through this work.
We consider here the following kind of decay process
where p 1 and p 2 are the source particles, described by a classical current corresponding to the states of a two level system following a prescribed classical trajectory with proper acceleration a, and q i are the emitted particles, described by quantized fields. (For further details, see [5,6], for instance.) We assume that the respective masses m qi of the emitted particles q i obeys m qi < m p1,2 , where m p1 and m p2 stands for, respectively, the masses of the source particles p 1 and p 2 . If m p1 < m p2 + i m qi , the process is known to be forbidden for inertial trajectories (a = 0). It is intuitive to expect that, if the source is supposed to follow a prescribed trajectory, the momentum k i of the emitted particle q i , measured in the rest frame of the source, must be constrained to [5,6]. Moreover, the mean energy ωi of the emitted particles (also measured in the proton’s reference frame) is of the order of the source proper acceleration [5],
i.e., ωi ∼ a.
Then, from the condition m qi ≪ m 1,2 and ω2
q1 one has a ≪ m 1,2 . In order to clarify these points, let us consider the explicit example of the emission of neutral pions by uniformly accelerated protons,
Without loss of generality, let us assume a uniformly accelerated trajectory along the z direction with worldline given by
x µ = a -1 (sinh(aτ ), 0, 0, cosh(aτ )) .
Following the same procedure employed in the Refs. [5,6], one obtains the differential decay rate as function of the energy of the emitted pion
where K 0 is the modified Bessel function of order 0, m π ≈ 140 MeV is the π 0 mass, and G eff is the effective coupling constant. The total emitted power is
where G pq mn stands for the Meijer G-function [7]. The corresponding normalized energy distribution,
is plotted in Fig. (1), from which it is clear that mean energy of the emitted particles are ω ≈ a. From (4) and Fig.
(1), one can argue that, if the acceleration is such that
corresponding process (3) becomes energetically favored. In fact, more precise estimates reveal that the threshold is a ≥ ∆m + i m q1 , where ∆m = m 2 -m 1 . Hence, the neutral pion production ( 5) is expected do be favored for a ≥ m π ≈ 140 MeV. For the physical relevant case of protons in circular motion, one has also exactly the same threshold (10). For such case, the formulas for the neutral pion production (5) can be obtained from the previously one calculated in [6] for scalar emissions
We notice that, in order to obtain that the total power emission (11) from Eq. (4.7) of [6], one must perform the following modifications on the particle states: |π + → |π 0 and |n → |p . These substitutions ensure the charge conservation and that the initial and final charged nucleon states are coupled in the same way to the magnetic field. In ( 11), G
eff is related
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