# Scalar pair production in a magnetic field in de Sitter universe

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## Abstract

The production of scalar particles by the dipole magnetic field in de Sitter expanding universe is analyzed. The amplitude and probability of transition are computed using perturbative methods. A graphical study of the transition probability is performed obtaining that the rate of pair production is important in the early universe. Our results prove that in the process of pair production by the external magnetic field the momentum conservation law is broken. We also found that the probabilities are maximum when the particles are emitted perpendicular to the direction of magnetic dipole momentum. The total probability is computed and is analysed in terms of the angle between particles momenta.

## 1 Introduction

The problem of particle generation in magnetic field in Minkowski space-time was first studied by Heisenberg and Euler [1] and this paper represents the basis for the today well known Schwinger effect. This paper proves for the first time how to solve the Dirac equation when a magnetic field is present, and these results represent the basis for the more recent studies that imply particle production in the presence of external fields [2, 3, 4, 5, 6, 7, 8, 9] in curved backgrounds. An important result was obtained recently in [9], where the minimally coupled Klein–Gordon equation was solved in four dimensions in a de Sitter geometry. Further in [9] the rate of pair production was computed with the help of Bogoliubov method and the Minkowski limit was addressed. Another result which uses the nonperturbative treatment of scalar pair production in magnetic field on Robertson–Walker universe was discussed in [10], where the density number was computed using the Bogoliubov coefficients. In [10] the de Sitter case is not contained since the scale factor *a*(*t*) was chosen to have a linear time dependence. The result of [10] has proved how to work out the density number when a magnetic field is present in Robertson–Walker spacetime. While the vast majority of nonperturbative computations are done in two dimensions, in [10] the computations were made in four dimensions. These results need to be completed with a perturbative treatment in the case of scalar field since the perturbative method allows a complete study of the case when the gravitational fields are strong in comparison with the magnetic fields.

In this paper we will study the problem of scalar pair production in dipolar magnetic field on de Sitter geometry. We will use the perturbative methods presented in [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], for computing the amplitude/probability of scalar pair production in magnetic field on de Sitter expanding universe. The problem of particle production when a magnetic field and a gravitational field are present seems to receive little attention in literature because of the technical difficulties in solving the field equations in the presence of magnetic fields. The perturbative approach to the problem of fermions production in magnetic field on de Sitter geometry was used in [13], and the main results prove that the probability of pair production is nonvanishing only in strong gravitational fields. However for a complete picture of the problem of pair production in gravitational fields one needs to combine the nonperturbative results [2, 3, 4, 5, 6, 7, 8], with the perturbative treatment of this problem. This will imply as we mentioned above that the nonperturbative computations to be extended in four dimensions, since only in these cases we will have a complete picture on how the field equations with external field will look like and how these equations can be solved. However some time ago is was argued [12], that the production due to the field interactions should also be taken into consideration. Despite of this observation, the perturbative approach only recently received attention and the results are based on the calculations of QED transition amplitudes that generates particle production in a curved background [13, 14, 20]. The problem of space expansion that generate particle production was first discussed in [22], and important results were also obtained in [4, 5]. Since the de Sitter space-time could describe our universe, we believe that it is important to study the problem of scalar particle production in magnetic fields in this geometry by using both perturbative and nonperturbative methods. The first step in our study will be the computation of the first order QED transition amplitude that generates scalar particles in the field of a magnetic dipole. Then we will present the main steps required for computing the total probability for the process of pair production in magnetic field.

The origin of magnetic fields in the Universe is a subject that has been approached by many authors [23, 24, 25, 26, 27, 38], mostly motivated by the astronomical observations which show that, the galaxies and galaxy clusters have a proper magnetic field of weak intensity. The vast majority of the papers propose technical solutions of accurate measurements of the large scales magnetic fields, but an interesting idea that is analysed in the literature is that, these magnetic fields own their origin in the early stages of Universe evolution.

The paper begins with an introduction in the problem of pair production in magnetic field. In section two we compute the amplitude/probability of pair production in magnetic field. Section three is dedicated to the graphical study of our analytical results and limit cases, while in section four we compute the total probability. Our conclusions are presented in section five.

## 2 Probability calculation

We must specify that we work in Coulomb gauge i.e. \(\nabla _{i}(\sqrt{-g}A^{i})=0\), and our amplitude is gauge invariant since the transformations \(A^{i}\rightarrow A^{i}+\partial ^{i}\Lambda \), leave the amplitude unchanged [36].

*K*functions [34, 35]:

## 3 Graphical results and limit cases

*k*and we considered for modulus of magnetic moment the value \(\mathcal {M}=1\).

From Figs. 1, 2, 3 and 4 we observe that the probability drops to zero for large values of parameter *k*. The probability is nonvanishing only for \(k\in (1.51,2)\), where the gravity is still strong and we can conclude that this process is possible only in strong gravitational fields. Another observation is that the probability of pair production is sensibly larger when the angle between momenta vectors approaches \(\pi \) and the momenta ratio \(p/p'\) are close to unity as seen from Figs. 1 and 4.

*k*and \(p/p'\) are fixed (Figs. 5, 6).

Finally we conclude that there are higher probabilities for pair production processes that have as result particles with the momenta ratio close to unity and with the momenta on the same direction but opposite as orientation.

Let us study now the Minkowski limit for amplitude and probability. First we observe from our graphs that for \(k\rightarrow \infty \) our probability is vanishing. This result is in fact the Minkowski limit which is obtained when \(k=\infty \). Indeed in the Minkowski scalar QED [37] the first order processes of pair production in external fields are not allowed by the simultaneous energy–momentum conservation. In de Sitter case the translational invariance with respect to time is lost and as a consequence the amplitudes of particle production in the first order of perturbation theory are nonvanishing.

*k*, the parameter \(\mu \simeq k\) . Then by using the Eqs. (27) and (28) from Appendix the function \(f_{k}\) that defines the amplitude can be brought for large

*k*to the form:

*k*and will vanish for \(k=\infty \). Here we remark that the probability is proportional with the factor \(e^{-2\pi k}\), which is the same factor found in [6, 7], where the problem of pair production in external fields on de Sitter geometry was studied by using a nonperturbative approach [6, 7]. We must mention that a nonperturbative calculation of pair production in magnetic field on de Sitter geometry has been done in [9].

## 4 The total probability

In the present paper we restrict to compute the amplitude of transition and probability for the process of pair production in the field of a magnetic dipole, using the QED formalism developed in [14, 39]. Our calculations are valid only in early universe when the gravitational fields were strong in comparative with the magnetic fields and present the first derivation of the total probability for the process of pair production in magnetic field using a perturbative method. The result obtained in [9] study the problem of scalar pair production in strong magnetic fields using a nonperturbative method. The interesting idea is now to combine the two methods using the formalism developed in [3] for obtaining the contribution of both cosmological particle production and perturbative particle production. That will require some work to be done to translate the calculations in terms of density number of particles, as was shown in [12].

## 5 Concluding remarks

We present in this paper the first perturbative approach to the problem of scalar pair production in a magnetic field in a de Sitter geometry. The main result of our paper is related to the fact that the first order transition amplitude and probability are nonvanishing only in strong gravitational fields that corresponds in our case to the early universe conditions. Our study proves that the particles will be most probably emitted perpendicular to the magnetic field direction. This conclusion shows that there are differences comparatively with the problem of pair production in electric field where the particles are emitted parallel with the electric field direction [6, 7, 15]. In the Minkowski limit the amplitude and probability are vanishing since in Minkowski scalar QED [37] this process is forbidden by the energy–momentum conservation.

The problem of particle production when a gravitational field and a magnetic field are present seems to receive little attention in literature. This study is important since it is well known that black holes and neutron starts have also strong magnetic fields and it is fundamental to understand the problem of fields interactions in these geometries. For further study it will be important to combine the nonperturbative and perturbative results related to the problem of pair production in magnetic field on de Sitter spacetime for obtaining a complete picture of this phenomenon. This can be done by using the formalism proposed in [3], and we hope that in a future study to present these results.

## Notes

### Acknowledgements

This work was supported by a grant of the Ministry of National Education and Scientific Research, RDI Programme for Space Technology and Advanced Research - STAR, project number 181/20.07.2017.

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