The cosmic-ray air-shower signal in Askaryan radio detectors Krijn D. de Vriesa, Stijn Buitinka, Nick van Eijndhovena, Thomas Meuresb, Aongus O´ Murchadhab, Olaf Scholtena,c aVrije Universiteit Brussel, Dienst ELEM, B-1050 Brussels, Belgium bUniversit´eLibre de Bruxelles, Department of Physics, B-1050 Brussels, Belgium cUniversity Groningen, KVI Center for Advanced Radiation Technology,Groningen, The Netherlands Abstract We discuss the radio emission from high-energy cosmic-ray induced air showers hitting Earth's surface before the cascade has died out in the atmosphere. The induced emission gives rise to a radio signal which should be detectable in the currently operating Askaryan radio detectors built to search for the GZK neutrino flux in ice. The in-air emission, the in-ice emission, as well as a new component, the coherent transition radiation when the particle bunch crosses the air-ice boundary, are included in the calculations. Key words: Cosmic rays, Neutrinos, Radio detection, Coherent Transition Radiation, Askaryan radiation 1. Introduction needed. Due to its long attenuation length, the induced radio signal is an excellent means to detect these GZK We calculate the radio emission from cosmic-ray-induced neutrinos. This has led to the development of several air showers as a possible (background) signal for the Askaryan radio-detection experiments [1-6,26-30]. Nevertheless, the radio-detection experiments currently operating at Antarc- highest-energy neutrinos detected so-far are those observed tica [1, 2, 3]. A high-energy neutrino interacting in a recently by the IceCube collaboration [31] and have ener- medium like (moon)-rock, ice, or air will induce a high- gies up to several PeV, just below the energies expected energy particle cascade. In 1962 Askaryan predicted that from the GZK neutrino flux. during the development of such a cascade a net negative In this article we calculate the radio emission from charge excess arises mainly due to Compton scattering [4]. cosmic-ray-induced air showers as a possible (background) This net excess charge by itself will induce a radio signal signal for the Askaryan radio-detection experiments cur- that can be used to measure the original neutrino. This rently operating at Antarctica [1, 2, 3]. Besides the emis- Askaryan radio emission [4, 5, 6] has been confirmed ex- sion during the cascade development also transition radi- perimentally at SLAC [7], and more recently the Askaryan ation should be expected when the cosmic ray air shower effect was also confirmed in nature by the radio emission hits Earth's surface [32, 33]. It follows that the induced from cosmic-ray induced air showers [8, 9, 10]. emission is very hard to distinguish from the direct Askaryan For high-energy cosmic-ray air showers, along with the emission from a high-energy neutrino induced cascade in Askaryan emission, there is another emission mechanism a dense medium such as ice. due to a net transverse current that is induced in the shower front by Earth's magnetic field [11-14]. Recently the radio emission from cosmic-ray air showers has been 2. Radio emission from a particle cascade measured in great detail by the LOFAR collaboration [10, 15, 16], confirming the predictions from several indepen- We start from the Li´enard-Wiechert potentials for a dent radio emission models [17-20]. point-like four current from classical electrodynamics and Most Askaryan radio detectors [1-3,21-23] search for closely follow the macroscopic MGMR [34] and EVA [20] models. Both models were developed to describe the ra- arXiv:1503.02808v2 [astro-ph.HE] 25 Nov 2015 so-called GZK neutrinos that are expected from the in- dio emission from cosmic-ray-induced air showers. The teraction of ultra-high-energy cosmic-ray protons with the µ cosmic microwave background [24, 25]. The expected GZK Li´enard-Wiechert potentials for a point charge, APL(t; ~x), neutrinos are extremely energetic with energies in the EeV as seen by an observer positioned at ~x at an observer time t range, while the flux at these energies is expected to fall be- are obtained directly from Maxwell's equations after fixing low one neutrino interaction per cubic kilometer of ice per the Lorenz gauge [35], year. Therefore, to detect these neutrinos an extremely µ µ 1 J large detection volume, even larger than the cubic kilo- A (t; ~x) = : (1) PL 4π jDj meter currently covered by the IceCube experiment, is 0 ret The point-like current is defined by J µ = eV µ, where e µ Email address: [email protected] (Krijn D. de Vries) is the charge, and V is the four-velocity for a particle at Preprint submitted to Elsevier April 9, 2018 ~ ξ(tr) where the retarded emission time is denoted by tr. where the distance di, the distance covered by the emission The denominator of the vector potential, D, is the retarded in layer i, is obtained by using a ray-tracing procedure four-distance. For an extended current with longitudinal based on Snell's law. Following [36], the retarded distance dimension h and lateral dimensions ~r, the vector potential for a signal traveling through different media is given by, has to be convolved with the charge distribution given by dt the weight function w(~r; h), D = L : (5) dtr Z µ µ 1 2 J w(~r; h) A (t; ~x) = dh d r ; (2) In this work the index of refraction is assumed to be in- 4π jDj 0 ret dependent of frequency within the radio frequency range where the vector potential has to be evaluated at the re- starting from a few MHz, up to several GHz. In the simpli- tarded emission time tr. The corresponding geometry is fied situation where the signal travels through a medium with constant index of refraction n, the retarded distance can be written in the more common form, x/ y− plane Cascade direction z D = nR(1 − nβ cos(θ)) ; (6) Observer position: where θ denotes the opening angle between the line of sight (x=0, y=0, z=0) (x , y ,z=0) from the emission point to the observer and the direction of movement of the emitting charge. d= (x−r )2 +( y−r )2 Charge cloud: √ x y ⃗ξ=( =− ) 0,0, z ct r 2.1. Cherenkov effects for a single electron V μ=(1,0,0,−β) 2 2 For a single electron moving at a highly relativistic R=√(−ct r +h) +d velocity β~ = ~v=c ≈ 1 along the z-axis (by definition), n=n h 2 the current is given by J µ = e (1; 0; 0; −β). The electric z=z b n=n1 r Charge position: field is now obtained directly from the Li´enard-Wiechert (x−rx , y−r y , z=−ct r +h) potentials through, dA0 dAi Ei(t; ~x) = − − ; (7) dxi dct Figure 1: The geometry used to calculate the radiation where i = x; y gives the polarization of the field in the i emitted from a charge cloud crossing a boundary at z = transverse direction, and x denotes the observer position in the transverse plane (x1 = x; x2 = y). For the moment zb. The observer is positioned at an impact parameter p 2 2 we will ignore the electric field in the longitudinal direction d = (x − rx) + (y − ry) . and, since Ai / J i = 0 for i = 1; 2 (there is no transverse current), we only have to consider the spatial derivative of denoted in Fig. 1. We consider an observer positioned at the scalar potential. The electric field in the longitudinal an impact parameter d = p(x − r )2 + (y − r )2 perpen- x y direction will in general be small and can easily be calcu- dicular to the charge track, where r , and r denote the x y lated following the gauge condition ~k · ~ = 0, where ~k is lateral position of the considered charge within the charge the momentum vector of the photon and ~ the polarization. cloud. Defining the element in the plane of the observer Hence the photon cannot be polarized along its direction perpendicular to the charge-track as z = 0, we can define of motion. Starting at the zeroth component of the vector the time at which the front of the charge cloud crosses this potential, the spatial derivative can be evaluated by, plane to be t = 0. Using these definitions the position of the charge along the track is now given by z = −ctr + h. dA0 @ = A0; (8) Fixing the geometry, the vector potential can now be dxi @xi evaluated. The retarded emission time is obtained from the light-cone condition with respect to the optical path which corresponds to the radiation from a net charge mov- length L, ing through the medium. For a relativistic electron (β ≈ 1) moving in a medium with a refractive index n > 1 this c(t − tr) = L; (3) term becomes, from which the relation between the observer time and the i @ 0 Est(t; ~x) = − A emission time, tr(t), can be obtained. It should be noted @xi that tr is a negative quantity. For a medium consisting −e (1 − n2)xi out of m layers with different index of refraction n , the = 3 : (9) i 4π0 jDj optical path length can be defined by m Where the label 'st', denotes that the field is due to a X highly relativistic non time-varying steady charge. The L = nidi; (4) i=1 emission shows a radial polarization direction and vanishes 2 linearly with the distance of the observer to the shower evaluated as, core. This component of the electric field is suppressed by @t @ the factor 1 − n2, which vanishes in vacuum.