a review of uhe neutrino detection using the askaryan effect

Transcription

1 a review of uhe neutrino detection using the askaryan effect J. C. Hanson (CCAPP, The Ohio State University) March 1, 216 The Kavli Institute for Cosmological Physics

2 outline I. The Continuing Story of GZK neutrinos and Radio II. Regathering our knowledge of the Askaryan effect A. ZHS (1991), RB (21), ARVZ (2-11) B. The basic effect, some definitions and the LPM Effect III. Towards a complete analytic formalism A. Our implementation of RB, agreement with ZHS B. Accounting for the LPM effect IV. Numerical Work A. Constructing the shower, given numerical constraints B. Deriving the general form factor provided by Geant4 V. Results in the Fourier domain, LPM and form factor together VI. Newest work: analytic equations in the time-domain A. Graphs of asymmetric vector potential 2

3 regathering of knowledge

4 regathering our knowledge of the askaryan effect The basic effect is coherent radiation from a negative charge excess: Fraunhofer regime: E(ω) has spherical symmetry ( 1/R) Fresnel regime: E(ω) cylindrical symmetry ( 1/ ρ) η = (a/ z coh ) 2 (1) Feynman s formula: E rad sgn(1 nû β) θ 4

5 feynman s formula - imagine charge passing right to left. 5

6 regathering our knowledge of the askaryan effect Some definitions and remarks. Longitudinal: refers to the z coordinate, or shower axis. Lateral: refers to the ρ coordinate (z 2 + ρ 2 = R 2, ρ/r = sin θ). θ, the viewing angle k, wavevector in the medium Shower width: a (m) 3/2ln(E) Excess charged particles: n max : E/ ln(e) Energy-scaling: Product of n max a E (area under Gaussian to first order) 6

7 regathering our knowledge of the askaryan effect Papers and other references. ZHS - Zas, Halzen, Stanev (1991). Calculates radiation from tracks of all particles and sums them. Produces coherence argument naturally. Expensive, but proves the concept. RB - Ralston and Buniy (21). Completely analytic, in Fresnel and Fraunhofer regime. Factors the E-field into form factor and charge evolution. ARZ - Alvarez-Muniz, Romero-Wolf, Zas (2-11). Semi-analytic approach that requires simulation of charge evolution, but provides analytic formula for E-field. Greisen parameterization (Prog. in Cosmic Ray Physics, 1956, ch. 1) E&M shower model. Leads to Rossi B appoximation etc. Solution for lateral charge evolution (see below). 7

8 regathering our knowledge of the askaryan effect The main result from RB: J = vn(z )f(z ct, ρ ) (2) R E(ω) = a n max ν m GHz F( q)ψ E(θ, η) (3) ψ = i exp(ikr) sin θ (4) E(θ = θ C, η) = e θ (1 iη) 1/2 (5) q = (ω/c, k ρ/r) (6) Rossi showed that the Greissen solution for n(z ) with depth a can be approximated as a gaussian with width a The linear ω dependence comes from acceleration factor in Lienard-Wiechert fields. 8

9 coherence zones Coherence zones, and other useful approximations. The Fraunhofer approximation leads to an insight: R = x x ρ (7) R = x x λ (8) i k x x ikr ik ρ(τ) (9) Beginning with the Lienard-Wiechert retarded potentials for decelerating charge, and focusing only on the radiation term, one can show (y = πνδt(1 nβ cos θ)): E(ω, x) iω eikr R sin y y () Similar to single-slit diffraction with length a. 9

10 coherence zones A more subtle approximation, keeping another order... x x = (z z ) 2 + (ρ ρ ) 2 (11) x x R(z ) ρ ( ) ρ ρ 2 R + (12) R R(z ) = (z z ) 2 + ρ 2 (13) Scale of the instantaneous charge excess is small compared to the longitudinal shower development. Keep the first two terms, drop the third. Integrals decouple into the form factor, and the Fresnel-Fraunhofer integrals.

11 definition of the form factor The 3D Fourier transform of the charge distribution, f, the normalized charge excess distribution. (Dropping bold font for vectors). d 3 x f(x ) = 1 (14) F(q) = d 3 x exp ( iq x )f(x ) (15) ( ω q = c, k ) R ρ (16) The structure of the Askaryan electric field is derived in RB, parameterized in ZHS, and fit in the time domain by ARVZ. In addition to the LPM effect, the main thrust of this work is to analytically derive F(q), and match to Monte Carlo simulations from Geant4. 11

12 landau-pomeranchuk-migdal effect Simple incorporation: draw the a-parameter from the EM curve below, rather than Greisen. 1.e+4 EM Electron (Gerhardt+Klein) EM Neutrino (Greisen) Had. Electron (Gerhardt+Klein) Had Neutrino (Gaisser-Hillas) Shower Length [m] 1.e+3 1.e+2 1.e+1 1.e+ 1.e+14 1.e+16 1.e+18 1.e+2 1.e+22 1.e+24 1.e+26 Neutrino/Electron Energy [ev] 12

13 askaryan fields

14 analytic forms of askaryan fields 14

15 analytic forms of askaryan fields 15

16 analytic forms of askaryan fields 16

17 analytic forms of askaryan fields 6 E ν = 17 ev, ρ = m -1, m 1 Viewing Angle (deg) Time (ns) -1 17

18 accounting for the lpm effect - scaling 18

19 accounting for the lpm effect - scaling 19

20 accounting for the lpm effect - scaling 2

21 accounting for the lpm effect - scaling 21

22 accounting for the lpm effect - scaling 22

23 accounting for the lpm effect - viewing angle σ θ (deg) EM Showers, E ν Freq -1 (GHz -1 ) ev, RB Model LPM On LPM Off 23

24 accounting for the lpm effect - viewing angle σ θ (deg) EM Showers, E ν Freq -1 (GHz -1 ) ev, RB Model LPM On LPM Off 24

25 accounting for the lpm effect - viewing angle σ θ (deg) EM Showers, E ν Freq -1 (GHz -1 ) ev, RB Model LPM On LPM Off 25

26 numerical work

27 geant4 simulations Limitations: Few hundred jobs at once. Charged RUs from finite account, 1 RU = CPU-hours Memory use < 8 GB, (MC thresholds) Strategy: Implement pre-shower sub-shower strategy, with Geant4. Utilizes back-fill (each sub-shower is cpu-minutes). Obeys memory constraints Cost: few hundred RUs, courtesy of Dr. Amy OSU Goal: F(ω, θ) using Geant4 pre-showers and sub-showers. 27

28 geant4 simulations Number of Particles Greisen,.1 EeV Geant4, e+ and e-,.1 EeV Geant4, e+, e-, gammas,.1 EeV Shower Longitudinal Axis (mm) 28

29 geant4 simulations Fractional negative charge excess, q. (MC threshold-dependent) q Shower Axis, z (mm) 29

30 geant4 simulations - z -form factor dependence? All tracks in ps 5 ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 3

31 geant4 simulations - z -form factor dependence? All tracks in ps ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 31

32 geant4 simulations - z -form factor dependence? All tracks in ps 15 ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 32

33 geant4 simulations - z -form factor dependence? All tracks in ps 2 ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 33

34 geant4 simulations - z -form factor dependence? All tracks in ps 25 ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 34

35 geant4 simulations - z -form factor dependence? All tracks in ps 3 ns after primary interaction: 4 3 e- e+ Electron Tracks in z Shower Axis, z (mm) 35

36 geant4 simulations - conclusions The instantaneous form factor in z is so small, it doesn t limit the Askaryan radiation... Unless the time-scale that matters is actually the Nyquist frequency of the RF detectors (1 GHz or 1 ns). If that were true, then the z-shape could matter (long tail, flat top is limited by time-window). One can show that the phase shift due to any z-dependence in form factors goes like { ν t ϕ/ θ 2πn θ R } λ sin θ C (17) 36

37 geant4 simulations - ρ -form factor dependence ) -2 Charged Particle Density (cm OSC Results Greisen, 5 rad. lengths Greisen, rad. lengths Greisen, 2 rad. lengths ρ (Moliere units =.4, g cm -2 ) 37

38 zhs form factor Necessary to explain why deccelerating charge doesn t radiate up to optical frequencies: E(k) k. 1 k 2 F ZHS (k) = ( ) 2 = k k (18) k2 k What does the corresponding charge distribution (inverse Fourier transform) resemble? Must treat the poles carefully. f(ρ ) = k2 2π f(ρ )/k 2 = 1 2πi e ikρ dk, k C (19) (k + ik )(k ik ) ie ikρ (k + ik ) 1 (2) (k ik ) 38

39 zhs form factor f(ρ )/k 2 = 1 ie ikρ (k + ik ) 1 (21) 2πi (k ik ) ( f(ρ ) = k 2 ie ikρ (k + ik ) 1) (22) k=ik f(ρ ) = k 2 e k ρ (23) Exponential (interesting), normalized to 1 2. Using the oppositely oriented contour, we get a different distribution: f(ρ ) = k 2 ( ie ikρ (k ik ) 1) = k k= ik 2 ek ρ (24) We must choose the contour that follows Jordan s lemma - (see below). 39

40 cauchy integral theorem What about a form factor like: f(ρ ) = k exp( k z ), ρ > (25) Fourier transform gives the form factor: F JCH (k) = dz e ikρ k e k ρ = k k ik (26) Only one pole, at k = ik, and still cuts off the high-frequency spectrum. So in the 1D case, just remove a pole, or, take the contour that converges. 4

41 cauchy integral theorem Taking the inverse Fourier transform of F JCH requires closing the contour around the one pole, and using Cauchy s formula. f(ρ )/k = 1 2π f(ρ )/k = 1 2πi e ikρ dk (27) k ik ie ikρ k ik dk = 1 2πi e ikρ k ik dk (28) f(ρ ) = k e k ρ, ρ > (29) Notice that R(F JCH ) = F ZHS, and F JCH = F ZHS (for same k ). This means that my form factor also cuts off the spectrum at high frequencies, and reduces to ZHS if we ignore imaginary E before taking the magnitude. Interesting that arg(f JCH ) k/k, so k should be large, to avoid adding extraneous phases. 41

42 result -3 E ν = 17 ev, RB EM shower (1 km), no LPM E-field [V/m/MHz] Form factor f(z) = b exp(bz), z< b -1 1 cm Frequency [MHz] 42

43 3d case - ρ -form factor dependence For the general, 3D case, I propose a ρ -dependence as follows: f(x ) = f δ(z ) exp( 2πρ ρ ), dz d 2 ρ f(x ) = 1, f = ρ 2 (3) F(q) = π π dz ρ dρ dϕ e iq x f(x ) (31) γ = k sin θ (m 1 ) (32) σ = γ 2πρ (33) So σ is the ratio of the lateral projection of the wave-vector and the lateral charge extent. Perform z -integration and substitute: F(q) = ρ 2 π ρ dρ dϕ exp{ (iγ cos ϕ + γ/σ)ρ } (34) π 43

44 ρ -form factor dependence Shift ϕ ϕ π/2 (cylindrical symmetry), and perform ϕ-integration: F(q) = ρ 2 F(q) = ρ 2 F(q) = 2πρ 2 π ρ dρ dϕ exp{ (iγ cos ϕ + γ/σ)ρ } (35) π ρ dρ exp { γ π σ ρ } dϕ exp{ iγρ sin ϕ} (36) π { dρ ρ exp γ σ ρ } J (γρ ) (37) F(q) = σ 2 du u exp{ u /σ}j (u ) (38) Table of integrals...and finally: F(k, θ) = ( ( ) 2 ( ) ) 2 3/2 1 k sin θ (1 + σ 2 ) = 1 + (39) 3/2 ρ 2π 44

45 combined results

46 ρ -form factor dependence - monte carlo fit parameters Shower energy: 17 ev. 3 (2π) 1/2 ρ [m -1 ] t [ns] 46

47 complete field - θ polarization 6 16 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 47

48 complete field - θ polarization 6 17 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 48

49 complete field - θ polarization 6 18 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 49

50 complete field - θ polarization 6 16 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 5

51 complete field - θ polarization 6 17 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 51

52 complete field - θ polarization 6 18 ev, (2π) 1/2 ρ = 25 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 52

53 complete field - θ polarization 6 16 ev, (2π) 1/2 ρ = 2.5 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 53

54 complete field - θ polarization 6 17 ev, (2π) 1/2 ρ = 2.5 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 54

55 complete field - θ polarization 6 18 ev, (2π) 1/2 ρ = 2.5 m -1, LPM, 1 km 1 Viewing Angle (deg) Time (ns) -1 55

56 spectra - θ polarization ev, (2π) 1/2 ρ = 25. m -1, LPM, 1 km E-field [V/m/MHz] Frequency [MHz] 56

57 new results - purely analytic time-domain fields

58 analytic time-domain fields At the Cerenkov cone, with ideal form factor (F(ω, θ) = 1), the E-field takes a convenient form: RE(ω, θ C ) = iωe sin θ C e iωr/c (1 iη) 1/2 ê θ [V/Hz] (4) E : Energy scaling-normalization (goes as n max a, or the area under the Gaussian n(z ), i.e. total charged particles). η = k(a 2 sin 2 θ)/r, same parameter from RB, η = ω/ω C. ê θ is the spherical unit vector (field is linearly polarized orthogonal to viewing direction). RE(t r, θ C ) iω CE sin θ C d ê θ π dt r C e itrω dω (41) ω + 2iω C 58

59 jordan s lemma, inverse fourier transform 2 Phase/π, t=-2. ns 2 2 Phase/π, t=2. ns I(ω) I(ω) R(ω) R(ω) -2 log (Norm), t=-2. ns log (Norm), t=2. ns I(ω) I(ω) R(ω) R(ω) -2 59

60 analytic time-domain fields RE(t r, θ C ) 4E sin(θ C )ω 2 Cêθ { exp(2ωc t r ) [V] t r exp( 2ω C t r ) [V] t r > (42) Under the Lorentz gauge condition for Maxwell s equations, in the absence of any static potentials, the negative derivative of the vector potential yields the electric field: A/ t = E. RA(t r, θ C ) 2E ω C sin θ C ê θ { exp(ωc t r ) [V s] t r exp( ω C t r ) [V s] t r > (43) 6

61 the coherence limiting frequency 5 log(ν C [GHz]), θ C = 55.8 deg 4 a [m] R [m] -2 ν C = cr 2πa 2 sin 2 θ C (44) 61

62 including form factor Let σ = ω/ω CF. iωe sin θ C e iωr/c RE(ω, θ C ) = (1 iω/ω C ) 1/2 (1 + (ω/ω CF ) 2 ) 3/2 êθ [V/Hz] (45) In the limit σ < 1, and η < 1, ω = 2/3 ω CF. RE(t r, θ C ) 2i 3π êθ d dt r E sin θ C ωcf 2 dω ω Ce itrω (ω + 2iω C )(ω + iω )(ω iω ) [V] (46) Key figure of merit: ratio of form factor limiting frequency, and the coherence limiting frequency: ϵ = ( ( a ) 2 2πρ ρ) (47) R 62

63 analytic time-domain fields log (ω CF /ω C ), R = 5 m 8 a [m] (2π) 1/2 ρ [m -1 ] -2 ϵ = ( ( a ) 2 2πρ ρ) (48) R 63

64 final results (vector potential) Units of (E ω C ) [V s] R A(t r,θ C ) 2 a=(1,2,...,) m t r (ns) Units of (E ω ) [V s] R A(t r,θ C ), (2π) 1/2 ρ = 1. m -1 2 a=(1,2,...,) m t r (ns) Units of (E ω ) [V s] R A(t r,θ C ), (2π) 1/2 ρ = 5. m -1 2 a=(1,2,...,) m t r (ns) Units of (E ω ) [V s] R A(t r,θ C ), (2π) 1/2 ρ =. m -1 2 a=(1,2,...,) m t r (ns) 64

65 for comparison to arvz (211) Our model reproduces the shape, width, and asymmetry, of the semi-analytical approach. (Taylor expansion of ω C exponential gets the form of ARVZ A(t, θ C )). 65

66 conclusions

67 conclusions I. Developed a fully analytic RB-model of Askaryan radiation A. Accounts for LPM effect, and F(ω, θ). B. F(ω, θ) was derived with the help of the Greisen parameterization, Geant4, and the OSC II. Results in the Fourier domain, LPM and form factor together III. Newest work: analytic equations in the time-domain IV. This work will be published in a forthcoming paper, and posted on arxiv by April 23 The code is on github: git clone https : //github.com/918particle/arasim2 AraSim2 67

68 conclusions - developers needed for arasim2 68