O. Kalasliev"t D. Semikozh, I. Tkachev"

" Institute for Nuclear Research, Russian Academy of Sciences 117312. Moscow, Russia

bLaboratory of AstroParticle and Cosmology (APC) 75205, Paris, France

Received October 27, 2014

Recently, the IceCube collaboration reported first evidence for the astrophysical neutrinos. Observation corresponds to the total astrophysical neutrino flux of the order of 3 • 10" 8 GeV •cm 2 • s 1 • sr 1 in a PeV energy range [1]. Active galactic nuclei (AGN) are natural candidate sources for such neutrinos. To model the neutrino creation in AGNs, we study photopion production processes on the radiation field of the Shakura-Sunyaev accretion discs in the black hole vicinity. We show that this model can explain the detected neutrino flux and at the same time avoids the existing constraints from the gamma-ray and cosmic-ray observations.

Contribution for the JETP special issue in honor of V. A. Rubakov60th birthday

DOI: 10.7868/S0044451015030222


Detection of astrophysical neutrinos by the IceCube collaboration fl] has opened a new era in the high-cncr-gy astrophysics. The reported excess of neutrinos at energies E > 30 TeV can be described by a power law 1 /Ea with a: = 2.3 ± 0.3, and corresponds to the flux 3 • 10-8 GeV • cm"2 • s-1 • sr-1 for the sum of three flavors, possibly with a cutoff at 3 PeV [1]. This observation lias a high significance of 5.7 a and calls for theoretical modeling and explanation.

There are three main production mechanisms of high-energy neutrinos. First, galactic cosmic rays produce neutrinos in the proton proton (proton nucleus) collisions in the interstellar gas in the disc of our Milky-Way Galaxy. Such neutrinos would have energies from sub-GeV to PeV, but can come only from directions close to the Galactic plane. Interestingly, the three-year IceCube data do show some excess in the direction of the Galactic plane with a 2 % chance probability fl], possibly exhibiting small-scale anisotropy near the Galactic center. Both signatures can be explained by the neutrino production in the galactic cosmic ray-interactions with the interstellar gas. It was shown in Ref. [2] that at most 0.1 of the observed neutrino events

* E-mail: kalashevflinr.ac.ru

in IceCube can be described by cosmic-ray interactions with matter inside the Milky Way assuming a local density of gas. However, the expected signal is dominated by the flux from spiral arms and/or the Galactic Bar, where supernova explosion rates, magnetic fields, and the density of the interstellar gas are all much higher than in the vicinity of the Sun [3]. Moreover, the neutrino flux detected by the IceCube is consistent [3] with the power-law extrapolation of the E > 100 GeV diffuse gamma-ray flux from the Galactic Ridge, as observed by the Fermi telescope, which suggests common origin. As result, the contribution of the Galaxy to the neutrino flux can be much higher than 10%.

Second, ultra-high energy cosmic rays (UHECR) interact with intergalactic radiation and produce secondary EeV neutrinos in pion decays. The latter are called cosmogcnic neutrinos and have been extensively-studied theoretically since 1969 [4] (see, e.g., [5, 6] and the references therein). The expected flux of cosmogcnic neutrinos is somewhat model dependent, but even optimistic estimates are at least two orders of magnitude below the IceCube signal at PeV energies. Hence, cosmogcnic neutrinos arc irrelevant in this energy range.

Finally, high-energy neutrinos in a wide range of energies, from TeV to 10 PeV, can be produced in a variety of astrophysical sources in decays of charged pions created in the proton photon or proton proton

collisions in situ. Various kinds of astrophysical sources of high-energy neutrinos were considered prior to the IceCube observation, including active galactic nuclei (AGN) [7 12], gamma-ray birsts [13], star burst galaxies [14].

After the IceCube observation, the interest in the problem has grown substantially. In a number of recent works [15 20], an attempt was made to explain the IceCube events by various astrophysical sources of high-energy neut rinos.

In this paper, we develop the model originally proposed in Ref. [8], where neutrinos arise in interactions of high-energy cosmic rays accelerated in AGNs with photons from the big blue bump. Compared to the previous papers developing this concept, we attempt to explain the IceCube observation using photopion production by cosmic rays on the anisotropic radiation field produced by the realistic Shakura Sunyaev model of accretion discs [21].

This paper is organized as follows. In Sec. 2, we present theoretical details of our calculation, also reviewing the observational knowledge about black hole accretion discs and their radiation fields. In Sec. 3, we confront our numerical calculations with the IceCube result and put constraints on the properties of such prospective neutrino sources.



AGN are long-sought potential sites for high-energy neutrino production. They can accelerate protons up to highest energies and are surrounded by high-intensity radiation fields where photo-nuclear reactions with subsequent neutrino emission can occur. At the heart of an AGN resides a super-massive black hole surrounded by the accretion disc. The accretion disc is hot and is emitting thermal radiation which gives a prominent feature in the observed AGN spectra usually referred to as the "Big Blue Bump". Accelerated particles move along two jets perpendicular to the accretion disc, crossing this radiation field.

In what follows, we use the following model for neutrino production. We assume that proton acceleration occurs directly near the black hole horizon (see, e.g., Refs. [22, 23]). High-energy neutrinos appear in charged pion decays created in jr/ nir+ and »7 -4—¥ jm~ reactions in collisions with "blue bump" photons. As a first step, we recall the observational phenomenology of accretion discs and estimate the optical depth for these photopion production reactions.

2.1. Accretion disc phenomenology

The effective temperature of the optically thick material 011 the scale of the gravitational radius is given

by [21]

T0 = 30 eV



\ 10M/.






where M is the IIlclSS of a black hole and is the efficiency of converting the gravitational potential energy into electromagnetic radiation, L = i/M. at a given accretion rate M. The eddington luminosity Lsdd is defined as

LEdd = 1.26-10


( M

\ I0M/.

erg • s

The temperature has a power-law profile with the radial coordinate 011 the disc, T oc r-^. In theory [21], 3 = 3/4. Observationally, the slope is consistent with the thin disc theory, 3 = 0.61 jlQ^i, but would also allow a shallower temperature profile that would reduce the differences between the microlensing and flux size estimates [24].

Within uncertainties and with an accuracy sufficient for our purposes, the observed disc sizes at the radiation frequency E1 = 5 oV can be fitted by the relation [25]

R = 1015 cm



\ 108 il/,..

which is about two orders of magnitude larger than the gravitational radius. This estimate is somewhat larger than the expectation from the thin disc theory. The photon density around the disc can be approximated by the relation

_ Ldisc

''h ~ 4itR'2Ec ' where Ec is a typical photon energy. On average, spectral energy distributions (SED) of AGNs are peaked at the energy Ec = 10 oV (see, e.g., Ref. [26] for a review).

The optical depth to photomeson production can be estimated as r = cn^R, where a k. 5 • 10 the cross section at the ¿1-resonance. This gives

~28 cm2 is

T ~ 103

Ldisc 10 eV



irrespective of the black hole rilciSS. There are tight correlations between monochromatic and bolometric luminosities of AGNs, e.g., ALa(5100 A) « O.lLhoi (see [27, 28]). An estimate for Ldigc is given by AL\. For a typical bolometric luminosity, we can assume

Lboi « 0.1 Lsdd (see, e.g., Rof. [29, 30]). Therefore, r ~ 10 would be a typical value for the optical depth to photomcson production after the traveling distance comparable to the accretion disc size.

2.2. Radiation fields and reaction rates

In the laboratory frame, the rate of reactions with the photon background is given by the standard expression

R = J <fpn( p)(l - coséO<r(iD), (2)

where n(p) is the photon density in the laboratory frame, a(ùi) is the cross section of the relevant reaction in the rest frame of the primary particle as a function of the energy of the incident photon u) = yp(l — cos8), and 7 is the gamma-factor of the primary particle in the laboratory frame.

For the black-body radiation with a temperature T, we have



(2tt)3 exp(p/T) - 1 '


We assume that the disc segment at a radius r emits black-body radiation with a local temperature T(r),

T(r) = T0F(r), where T0 is given by Eq. (1) and [21]





Here, ra = 'Ik.11 is Schwarzschild gravitational radius,

r - > in13 M ' ' ' 10« M,


and rin is the radius of the disc inner edge. The contribution of such a segment to the photon density at a point z along the disc axis is

no)/* dv

n(p) = 2 + m (p. r)


wliere n0 is tlie unit vector in tlie direction froni r to î. Its contribution to tlie reaction rate in Eq. (2) can

be expressed as

R(z,r,"f) =

1 — cos 0 4tt 3(r2 + c2

dpp2 (t(û) exp (p/T(r)) — 1 '


t/Ms 100




: r - 0 - CO "E?

- / To = 15

Î / T0 = 7 1

/ / To = 3 ]




10ly 10 E, eV

Fig. 1. Optical depth as a function of the proton energy for several values of T0 (in eV)

where cos 6 = z/\/r2 + z2. Finally,


R(z, 7) = 2tt J r drR(z, r, 7 ).


The disc inner edge rin is related to the radiation efficiency as



V =

r dr F (r).

I11 what follows, we use = 0.1, which is a usual assumption in the existing literature.

The optical depth with respect

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