# Resonant oscillations of GeV - TeV neutrinos in internal shocks from gamma-ray burst jets inside the stars

###### Abstract

High-energy neutrinos generated in collimated jets inside the progenitors of gamma-ray bursts (GRBs) have been related with the events detected by IceCube. These neutrinos, produced by hadronic interactions of Fermi-accelerated protons with thermal photons and hadrons in internal shocks, are the only signature when jet has not broken out or failed. Taking into account that the photon field is thermalized at keV energies and the standard assumption that the magnetic field maintains a steady value throughout the shock region (with a width of cm in the observed frame), we study the effect of thermal and magnetized plasma generated in internal shocks on the neutrino oscillations. We calculate the neutrino effective potential generated by this plasma, the effects of the envelope of the star, and the vacuum on the path to Earth. By considering these three effects, the two (solar, atmospheric and accelerator parameters) and three neutrino mixing, we show that although GeV - TeV neutrinos can oscillate resonantly from one flavor to another, a nonsignificant deviation of the standard flavor ratio (1:1:1) could be expected on Earth.

###### keywords:

Long Gamma-ray burst: High-energy Neutrinos: – Neutrino Oscillation## 1 Introduction

Long gamma-ray bursts (lGRBs) have been associated to core collapse of massive stars leading to supernovae (CCSNe) of type Ib,c and II. Type Ic supernovae are believed to be He stars with radius 10 cm, and type II and Ib are thought to have a radius of cm. Depending on the luminosities and durations, successful lGRBs have revealed a variety of GRB populations: low-luminosity (ll), ultra-long (ul) and high-luminosity (hl) GRBs (Mészáros &
Waxman, 2001; Liang
et al., 2007; Gendre
et al., 2013). While llGRBs and ulGRBs have a typical duration of ( 10 - 10 s), hlGRBs have a duration of tens to hundreds of seconds. Another important population associated with CCSNe, although unobservables in photons, are failed GRBs which could be much more frequent than successful ones, limited only by the ratio of type Ib/c and type II SNe to GRBs rates. This population has been characterized by having high-luminosities, mildly relativistic jets and durations from several to ten seconds (Huang
et al., 2002; Mészáros &
Waxman, 2001; Soderberg &
et al., 2006, 2010).

Neutrinos are useful for studying the insides of stars, especially where photons cannot be observed either because jet fails or has not broken out yet, so in this case, they could be the only signature that would display the dynamics of the star. High-energy (HE) neutrinos from this population of stars have been pointed out to contribute significantly to the extragalactic neutrino background (ENB) (Murase &
Ioka, 2013; Fraija, 2014a; Taboada, 2010; Murase
et al., 2014; Waxman, 2013; Razzaque, 2013; Murase et al., 2013) and to explain the recent detections of TeV- PeV neutrinos by IceCube (IceCube Collaboration et al., 2013; Aartsen et al., 2014).

Measurements of HE neutrino properties such as flavor content would be involved with new physics if a deviation of the standard flavor ratio were observed (Learned &
Pakvasa, 1995; Athar
et al., 2000; Kashti &
Waxman, 2005; Mena
et al., 2014). The neutrino flavor ratio is expected to be at the source, =1 : 2 : 0 and on Earth (due to neutrino oscillations between the source and Earth) =1 : 1 : 1 and =1 : 1.8 : 1.8 for neutrino energies lesser and greater than 300 TeV, respectively (Kashti &
Waxman, 2005). Also measurement of a non-zero mixing angle coming from astrophysical sources could be relevant to clarify the neutrino mass hierarchy as well as CP violation searches in neutrino oscillations (Nunokawa
et al., 2008; Bandyopadhyay &
et al., 2009; Forero
et al., 2012).

As known, neutrino properties are modified when they propagate in a thermal and magnetized medium. A massless neutrino acquires an effective mass and an effective potential. The resonant conversion of active neutrino from one flavor to another () due to thermal and magnetized medium has been explored in many astrophysical contexts and has had relevant consequences in the dynamics of them (Wolfenstein, 1978a; Ruffert &
Janka, 1999; Goodman
et al., 1987; Volkas &
Wong, 2000; Dasgupta et al., 2008; Erdas &
Isola, 2000; D’Olivo &
Nieves, 1996b, 1994, a; D’olivo
et al., 1992; D’Olivo
et al., 2003; Nötzold &
Raffelt, 1988; Enqvist et al., 1991). For instance, Fraija (2014b) has showed that the effect of magnetic field in the dynamics of the fireball evolution of GRB was to decrease the proton-to-neutron ratio aside from the number of multi-GeV neutrinos expected in a neutrino detector.

Neutrino oscillations in vacuum and by matter effects in the failed GRB framework (along the jet and envelope of the star) have been examined by many authors (Mena
et al., 2007; Razzaque &
Smirnov, 2010; Sahu &
Zhang, 2010; Osorio Oliveros et al., 2013; Fraija, 2014a) and although these authors have studied the oscillations on the surface of the star due to its envelope, the effect of thermal and magnetic field plasma generated on internal shocks has not been explored. In this paper we calculate the effect of the magnetized and thermal shocked plasma on neutrino oscillations and then we estimate the flavor ratio on Earth. The organization of the paper is as follows: In section 2, we show a brief description of internal shocks. In section 3, we derive firstly, the neutrino effective potential for as a function of the magnetic field, temperature, angle (between the neutrino propagation and magnetic field) and chemical potential and secondly, the neutrino effective potential produced by the envelope of the star. In section 4 we derive the resonance condition, the flip probability for two and three-neutrino mixing and the flavor ratio expected on Earth, and in section V we discuss our results. We hereafter use in c.g.s. units and k==c=1 in natural units.

## 2 Description of Internal shocks

One of the most prosperous theory to explain the prompt emission and the afterglow in successful GRBs is the fireball model (Zhang & Mészáros, 2004; Mészáros, 2006). A GRB is considered successful when the jet drills inside the progenitor and breaks through the stellar envelope, otherwise it is taken into account as a failed GRB. When the jet encounters the stellar envelope two shocks are involved: an outgoing, or forward, shock (Rees & Meszaros, 1994; Paczynski & Rhoads, 1993) and another one that propagates back decelerating the ejecta, the reverse shock (Meszaros & Rees, 1994; Rees & Meszaros, 1994). The jet dynamics is mainly dominated by the jet head, which is controlled by the ram pressure balance between the reverse and forward shock. If the luminosity () is low enough and/or the density of the stellar envelope is high enough, then the hydrodynamic jet is collimated and internal shocks might occur inside the progenitor (Mizuta & Ioka, 2013; Bromberg et al., 2011; Murase & Ioka, 2013). In this model, inhomogeneities in the jet lead to internal shell collisions, higher shells () catching slower shells (). The kinetic energy of ejecta is partially dissipated via these internal shocks which take place at a distance of , where is the variability time scale of the central object, is the bulk Lorentz factor of the propagating shock and is the radius of the progenitor’s stellar surface. The constraint gives rise to those shocks inside the star. The physical width of the internal shock is lower by a factor of , i.e. . These internal shocks are expected to be collisionless, so that particles may be accelerated. In internal shocks the total energy density is equipartitioned to generate and/or amplify the magnetic field (Piran, 2005) and to accelerate particles , where m is the proton mass. Then, the magnetic field generated at the shocks is written as

(1) |

It is important to say that the strength of the magnetic field falls out of the shocked region achieving some Gauss and although its direction might be random, it is mostly transverse to the jet direction (Razzaque & Smirnov, 2010). From the causality condition, the coherence length of such magnetic field is only the order of . On the other hand, electrons are accelerated up to ultra-relativistic energies and then are cooled down rapidly in the presence of the magnetic field, producing the prompt emission by synchrotron radiation. The opacity to Thomson scattering is and photons thermalize at a black body temperature with peak energy given by (Razzaque et al., 2004)

(2) |

where is the Thompson cross section.

Protons are also accelerated and cooled down in internal shocks via electromagnetic (synchrotron radiation and inverse Compton (IC) scattering) and hadronic (proton-photon and proton-proton interactions) channels. Proton-photon and proton-proton interactions take place when accelerated protons interact with thermal keV photons (eq. 2) and proton density at the shock, (Mészáros &
Waxman, 2001). In both interactions HE charged pions and kaons are produced; , and subsequently neutrinos and . In this approach, the neutrino created by these processes will lie in the TeV - PeV energy range (Murase &
Ioka, 2013; Fraija, 2014a; Razzaque &
Smirnov, 2010).

## 3 Neutrino Effective Potential

In this section we are going to compute the neutrino effective potential due to the magnetized and thermal shocked plasma, and the envelope of the star.

### 3.1 Magnetized and thermal plasma

Recently, Fraija (2014b) derived the neutrino self-energy and effective potential up to order at strong, moderate and weak magnetic field approximation as a function of temperature, chemical potential and neutrino energy for moving neutrinos along the magnetic field. In this subsection, we will calculate the neutrino effective potential at the moderate and weak magnetic field limit for any direction of neutrino propagation. Therefore, following Fraija (2014b) we will show the equations that are more relevant for deducing the neutrino effective potential.

The neutrino effective potential is calculated by means of the dispersion relation

(3) |

where is estimated through the neutrino field equation in a medium (Nötzold & Raffelt, 1988; Enqvist et al., 1991)

(4) |

and is obtained from the real part of its self-energy diagram. Here and are the momentum along and perpendicular to the magnetic field, respectively, stands for the 4-velocity of the center-of-mass of the medium given by , and are the projection operators and a, b, and c are the Lorentz scalars which are functions of neutrino energy, momentum and magnetic field. These scalars are calculated from the neutrino self-energy due to CC and NC interactions of neutrino with the background particles. The effect of the magnetic field is introduced through the 4-vector which is given by (Fraija, 2014b). Using the Dirac algebra and from the dispersion relation (eq. 3), the neutrino effective potential can be written as

(5) |

where is the angle between the neutrino momentum and the magnetic field vector. Otherwise, the effective potential that is applicable to the neutrino oscillations in matter is which depends only on electron density (Wolfenstein, 1978b; D’olivo et al., 1992). For that reason, although the one-loop neutrino self-energy comes from three parts; the -exchange, -exchange and tadpole (Babaev, 2004; Erdas et al., 1998; Sahu et al., 2009a, b), we will only consider the neutrino effective potential due to charged currents . We will use the finite temperature field theory formalism and the Schwinger’s propertime method to include the magnetic field (Schwinger, 1951). From the W-exchange diagram (see fig. 1), the self-energy can be explicitly written as

(6) |

where is the weak coupling constant, is the W-boson propagator that in unitary gauge can be written as (Erdas et al., 1998; Sahu et al., 2009b)

(7) |

here m is the W-boson mass, G is the Fermi coupling constant, is the metric tensor and is the electromagnetic field tensor. From eq. (6), S(p) is the charged lepton propagator which is split in two propagators; one in presence of an uniform background magnetic field () and the other in a magnetized medium (), then it can be written as

(8) |

We can express the charged lepton propagator in presence of an uniform background magnetic field as

(9) |

where the functions and are written as

(10) | |||||

(12) | |||||

where is the mass of the charged lepton, , are the projections of the momentum on the magnetic field direction and , being the magnitude of the electron charge. Additionally, the covariant vectors are given as follows, , and . The other term in eq. (8) (due to magnetized medium) is given by (D’Olivo & Nieves, 1996a)

(13) |

where contains the distribution functions of the particles in the medium which are given by:

(14) |

where and are the inverse of the medium temperature and the chemical potential of the charged lepton. By evaluating eq. (6) explicitly we obtain

(15) |

where the Lorentz scalars are given by (Fraija, 2014b)

(16) |

and

(18) |

Here the electron number density and electron distribution function are

(19) |

and

(20) |

respectively, with and with . Solving the integral terms in eqs. (16), (LABEL:conbw) and (18) and replacing them in eq. (5) we calculate the neutrino effective potential for two cases: the moderate and the weak magnetic field limit.

#### 3.1.1 Moderate Magnetic field limit

In the moderate field approximation (), the Landau levels are discrete and can be described by sums ( with n=1, 2, 3 ..). In this regime, the neutrino effective potential is written as

(21) |

where and the functions F, G, J, H are written in the appendix A. It is worth noting that as the magnetic field decreases the effective potential will depend less on the Landau levels.

#### 3.1.2 Weak Magnetic field limit

In the weak field approximation (), all levels are full and overlap each other. In this regimen, sums over the Landau levels can be described and approximated by an integral , then the effective potential does not depend on the Landau levels. The potential in this regimen can be written as

(22) |

where the functions F, G, J, H are shown in the appendix A.

### 3.2 Density profiles of envelopes

Models of density distributions in CCSNe have been widely explored (Bethe & Pizzochero, 1990; Chevalier & Soker, 1989; Woosley et al., 1993; Shigeyama & Nomoto, 1990). We will use two models with density profiles and . Explicitly, the first model corresponds to a polytropic hydrogen envelope

(23) |

and the second model is a power-law fit with an effective polytropic index as done for SN 1987A (Chevalier & Soker, 1989)

(24) | |||

In both cases, from the number density of electrons , the neutrino effective potential can be written as

(26) |

where N is the Avogadro’s number, 0.5 is the number of electrons associated per nucleon and is given by eqs. (23) and (3.2).

## 4 Neutrino resonant oscillations

When neutrino oscillations take place in matter, a resonance could occur that would dramatically enhance the flavor mixing and could lead to a maximal conversion from one neutrino flavor to another. This resonance depends on the effective potential and neutrino oscillation parameters. The equation that determines the neutrino evolution in matter in the two and three-flavor framework is (Fraija et al., 2014)

(27) |

where

(28) |

is the mass difference (Giunti & Chung, 2007), is the two- and three-neutrino mixing matrix (see appendix B, eq. 45), is the neutrino effective potentials calculated in section 3 (for =is and ss) and is the neutrino energy. We hereafter use the first and second line for two- and three-neutrino mixing, respectively, as written in eq. (28). From the conversion probabilities, we obtain that the oscillation lengths are

(29) |

with the resonance conditions

(30) |

In addition to the resonance condition, the dynamics of the transition from one flavor to another must be determined by adiabatic conversion through the adiabaticity parameter (Mohapatra & Pal, 2004)

(31) |

with 1 or the flip probability given by

(32) |

By considering that the flux ratio of is created in the internal shocks, neutrinos firstly oscillate in matter due to the magnetized and thermal plasma and secondly oscillate to the star envelope. In vacuum, after neutrinos have left the star, they start oscillating to the Earth. Hence, from these three effects: internal shocks, envelope of the star and vacuum, the flavor ratio expected on Earth will be

(33) |

where the probabilities are derived in appendix B.

The best fit values of the two neutrino mixing are: Solar Neutrinos: and (Aharmim &
et al., 2011), Atmospheric Neutrinos: and (Abe &
et al., 2011) and Accelerator Neutrinos: and (Zeitnitz, 1994; Athanassopoulos
& et al., 1996, 1998) .
Combining solar, atmospheric, reactor and accelerator parameters, the best fit values of the three neutrino mixing are, and and, for and , (Aharmim &
et al., 2011; Wendell &
et al., 2010).

## 5 Results and Conclusions

In this analysis we have considered HE neutrinos created in the energy range of 100 GeV 100 TeV (Murase &
Ioka, 2013; Fraija, 2014a; Razzaque &
Smirnov, 2010) and also we have assumed (in the CCSNe-GRB connection) progenitors such as Wolf-Rayet (WR) and blue supergiant (BSG) stars with radii cm and cm, respectively, with formation of jets leading to internal shocks inside of them.

In internal shocks, energy is equipartitioned to generate and/or amplify the magnetic field and to accelerate particles. Electrons and protons are expected to be accelerated in these shocks, and after to be cooled down by synchrotron radiation, inverse Compton and hadronic processes (p and p-hadron interactions). Photons produced by electron synchrotron radiation are thermalized at keV energies and serve as targets for production of HE neutrinos through K, and decay products in the proton- and proton-hadrons interactions. Therefore, this plasma is endowed with a magnetic field and made of protons, mesons, electrons, positrons, photons and neutrinos.

First of all, we consider those internal shocks that take place inside progenitors (r), as plotted in fig. 2. In this figure, we show the contour lines of bulk Lorentz factors and variability time scales for different internal shock radii. In these plots we observe that for a typical value of variability in the range of , the values of bulk Lorentz factors are for a WR (left-hand figure) and for a BSG (right-hand figure). Taking into account internal shocks at cm (left-hand figure) and cm (right-hand figure) we can see that the physical width of the internal shocks is restricted to cm and cm, respectively. Once obtained the values of and t for internal shocks inside the progenitor we compute the range of values associated to the magnetic field and temperature of the plasma, as shown in figs. 3 and 4, respectively. We plot the contour lines of the magnetic fields (fig. 3) and thermalized photons (fig. 4) for values of luminosity in the range . Colors in light- and dark-gray backgrounds represent the regions of a WR and a BSG, respectively.

In figs. 3 and 4 one can see that the values of magnetic field and thermalized photons lie in the ranges and , respectively. It is important to clarify that the magnetic field amplified in the internal shocks falls out of them to the magnetic field endowed by the progenitor (black hole (BH) or magnetar) (Razzaque &
Smirnov, 2010).

Following Fraija (2014b) and taking into account that the range of neutrino energy considered is larger than the W-boson mass (E), we have obtained the neutrino effective potential up to an order in the moderate below and weak regime as a function of the observable quantities in the internal shocks: thermalized photons, magnetic field, neutrino energy and angle between the direction of neutrino propagation and the magnetic field. We plot the neutrino effective potential in both limits (moderate and weak field limits) as shown in fig. 5.
The neutrino effective potential at moderate limit (figures above) and weak limit (figures below) are plotted for a magnetic field in the range of and , respectively. In both cases, we use the values of temperature T= (20, 24, 27 and 30) keV, angle = (, , and ) and the neutrino energy E=10 TeV. The neutrino effective potential at the weak limit is smaller than at the moderate limit. It is worth mentioning that in the range of the magnetic field considered, the contribution of Landau levels to the effective potential at the moderate-field limit is not significant due to for and . From these plots one can observe that the neutrino effective potential is positive, therefore due to its positivity ( 0) for k= m and w, neutrinos can oscillate resonantly. From the resonance condition (eq. 30) and the neutrino effective potential at moderate (eq. 3.1.1) and weak (eq. 3.1.2) limit, we plot the contour lines of temperature and chemical potential as a function of neutrino energy for which the resonance condition is satisfied, as shown in figs. 6 and 7, respectively. From these figures, one can see that temperature is a decreasing function of chemical potential and neutrino energy. As neutrino energy increases, temperature decreases steadily. Considering the values of neutrino energy ( =100 GeV, 500 GeV, 10 TeV and 100 TeV) and , we see that the temperature and chemical potential are in the range 10 keV to 100 keV and 60 eV to 50 keV, respectively. For instance, taking into account a neutrino energy of 10 TeV, from fig. 6 we can see that temperature lies in the range 22.2 to 15.3 keV for solar, 23.4 to 16.2 keV for atmospheric, 40 to 26.4 keV for accelerator and 28.3 to 15.4 keV for three-neutrino parameters, and as shown in fig. 7, temperature lies in the range 18.3 to 14.1 keV for solar, 19.8 to 15.1 keV for atmospheric, 32.1 to 21.9 keV for accelerator and 24.5 to 18.3 keV for three-neutrino parameters. In addition, we have obtained the resonance lengths which are shown in table 1. As shown in this table, the resonance lengths lie in the range (10 to ) cm, hence depending on the progenitor associated and the oscillation parameters, neutrinos would leave the internal shock region in different flavors of 1:2:0. For instance, taking into account the parameters of three-neutrino mixing, neutrinos with energy less than 0.5 (10) TeV will oscillate resonantly with a resonance length equal or less than the radius of the progenitor, either a WR or BSG. Considering parameters of accelerator experiments, neutrino energy around 100 TeV will oscillate in a BSG star before leaving it.

Energy | (cm) | |||
---|---|---|---|---|

(TeV) | Solar | Atmosph. | Accelerat. | Three flav. |

4.8 | 1.2 | 7.1 | 2.6 | |

0.5 | 2.4 | 5.9 | 3.6 | 1.3 |

10 | 4.8 | 1.2 | 7.1 | 2.6 |

4.8 | 1.2 | 7.1 | 2.6 |

As the dynamics of resonant transitions is not only determined by the resonance condition, but also by adiabatic conversion, we analyze the flip probability (eq. 32) to find the regions for which neutrinos can oscillate resonantly. First of all we derive the neutrino effective potential as the function of magnetic field , and assume that at internal shocks ( cm for WR and cm for BSGs), magnetic fields change a 10% of any variation around the radius shock scale. For instance, for a WR star, Gauss/cm. We plot the flip probability as a function of neutrino energy for two and three flavors (fig 8). We divide each plot of flip probability in three regions in order to analyze the whole range of probabilities: less than 0.2 (P 0.2, a pure adiabatic conversion), between 0.2 and 0.8 (0.2 P 0.8 represents the transition region) and greater than 0.8 (P 0.8 is a strong violation of adiabaticity)(Dighe &
Smirnov, 2000). In fig. 8, we use two flavors: solar (left-hand figure above), atmospheric (right-hand figure above), accelerator (left-hand figure below) and three flavor (right-hand figure below). When we use solar parameters, a pure adiabatic conversion occurs in a WR (BSG) star for neutrino energies of less than (10) eV and (10) eV which are endowed with and , respectively. Considering atmospheric parameters, only a pure adiabatic conversion takes place in a WR (BSG) star for neutrino energies less than (10) eV and ( 10) eV which are endowed with and , respectively. Taking into account accelerator parameters, a pure adiabatic conversion happens in a WR (BSG) star for neutrino energies of less than (10) eV and 10) eV with and , respectively and once again considering three neutrino mixing, a pure adiabatic conversion occurs in a WR (BSG) star for neutrino energy of less than (10) eV and (10) eV with and , respectively. Higher energies to those considered are found in regions of transition and/or those prohibited.

In addition, we have studied the HE neutrino oscillations from the neutrino effective potential generated in the star envelope (eq. 26), as shown in fig. 9. From the resonance condition (eq. 30), we obtain the contour plots of radius as a function of neutrino energy. One can see that for neutrino energy in the range 100 GeV 100 TeV the radius lies in the range cm cm. The flip probability for neutrino oscillations in the envelope of a star was studied by Fraija (2014a). The author has plotted this probability as a function of neutrino energy for density profiles [A] (eq. 23) and [B] (eq. 3.2) and neutrino oscillation parameters. From this analysis, Fraija (2014a) showed that neutrinos can oscillate depending on their energy and the parameters of neutrino experiments, obtaining that neutrino with energies above dozens of TeV can hardly oscillate.

Finally, considering a flux ratio , we estimate the neutrino flavor ratio coming from the surface of a WR and BSG to Earth, as shown in fig. 10. In this estimation, we take into account the contribution of thermal and magnetized plasma at moderate and weak limit generated by internal shocks; at cm (second panel), cm (upper panel), cm (bottom panel) and cm (third panel), the effective potential due to the envelope of star and oscillation neutrinos in vacuum, due to the path up to Earth. In this figure we take into account two values of mixing angle, 2 (left column) and 11 (right column). As shown, one can observe that a nonsignificant deviation of the standard ratio (:: ; 1:1:1) is expected, less than 10 % for and only 2 % for . In addition, we plot the neutrino flavor ratio expected on Earth as a function of neutrino energy when the magnetic field is oriented to different angles concerning neutrino direction, as shown in figs. 11 and 12. In fig. 11 we consider the neutrino effective potential at the moderate-field limit and internal shocks at cm with a physical width cm and in fig. 12, we consider the neutrino effective potential at the weak-field limit and internal shocks at cm with a physical width cm. From both plots, we can see that although the neutrino flavor ratio changes at different angles, distances of internal shocks, strength of magnetic field (moderate and weak limit) and neutrino energy in the range , this flavor ratio expected on Earth lies between 0.98 and 1.02, hence we can conclude that the directionality of magnetic fields does not affect our results. Although currently neutrino oscillations can hardly be detected, new techniques in the near future will allow us to perceive these oscillations and put limits on the neutrino mixing angles. Finally, it is worth noting that the estimated values of the bulk Lorentz factor, in particular those relying on variability time measurements, are only raw approximations, and variations by a factor of a few cannot be ruled out by existing data.

## Acknowledgements

We are thankful to the anonymous referee for a critical reading of the paper and valuable suggestions that helped improve the quality and clarity of this work. We also thank A. M. Sodelberg, J. Nieves, B. Zhang, K. Murase, W. H. Lee, F. de Colle, E. Moreno and A. Marinelli for useful discussions. NF gratefully acknowledges a Luc Binette-Fundación UNAM postdoctoral fellowship. This work was supported by the projects IG100414 and Conacyt 101958.

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## Appendix A Effective Potential

The functions of the neutrino effective potential at moderate magnetic field limit are

(34) |

and at weak magnetic field limit are

where , K is the modified Bessel function of integral order i, and .