We calculate the Cherenkov radiation from an $e^+e^-$ pair at small
separations, as occurs shortly after a pair conversion. The radiation is
reduced (compared to that from two independent particles) when the pair
separation is smaller than the wavelength of the emitted light. We estimate the
reduction in light in large electromagnetic showers, and discuss the
implications for detectors that observe Cherenkov radiation from showers in the
Earth's atmosphere, as well as in oceans and Antarctic ice.Comment: Final version, with minor changes, to appear in PRD. 5 pages with 4
  figure

D.  H. Perkins

G. A. Askaryan

H.  A. Bethe

I.  M. Frank

J. David Jackson

J.  D. Jackson

J.  V. Jelley

P.  A. Cherenkov

S.  I. Vavilov

Sourav K. Mandal

Spencer R. Klein

English

arXiv

Crossref

Cherenkov radiation frome+e−pairs and its effect onνeinduced showers

Mandal, Sourav K.

Klein, Spencer R.

Jackson, J. David

eScholarship - University of California

Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleCherenkov Radiation from e+e- Pairs and Its Effect on nu e Induced ShowersPermalinkhttps://escholarship.org/uc/item/2z04522wAuthorsMandal, Sourav K.Klein, Spencer R.Jackson, J. DavidPublication Date2005-06-08 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaarXiv:physics/0506078 v1   8 Jun 2005Cherenkov Radiation from e+e−Pairs and Its Effect on νeInduced ShowersSourav K. Mandal, Spencer R. Klein, J. David JacksonLawrence Berkeley National LaboratoryBerkeley, CA 94720We calculate the Cherenkov radiation from an e+e− pairat small separations, as occurs shortly after a pair conversion.The radiation is reduced (compared to that from two indepen-dent particles) when the pair separation is smaller than thewavelength of the emitted light. We estimate the reduction inlight in large electromagnetic showers, and discuss the impli-cations for detectors that observe Cherenkov radiation fromshowers in the Earth’s atmosphere, as well as in oceans andAntarctic ice.I. INTRODUCTIONCherenkov radiation from relativistic particles hasbeen known for over 70 years [1]. However, to date, al-most all studies have concentrated on the radiation fromindividual particles. Frank [2], Eidman [3] and Balazs[4] considered the Cherenkov radiation from electric andmagnetic dipoles, but only in the limit of vanishing sep-arations d. Their work was nicely reviewed by Jelley [5].Several more recent calculations have consideredCherenkov radiation from entire electromagnetic show-ers, in the coherent or almost coherent limit [6]. Thefields from the e+ and e− largely cancel, and the bulk ofthe coherent radiation is due to the net excess of e− overe+ (the Askaryan effect) [7]. Hadronic showers produceradiation through the same mechanism [8]. Coherent ra-diation occurs when the wavelength of the radiation islarge compared to the radial extent of the shower; forreal materials, this only occurs for radio waves.Here, we consider another case, the reduction of ra-diation from slightly-separated oppositely-charged co-moving pairs. This includes e+e− pairs produced byphoton conversion. When high-energy photons convertto e+e− pairs, the pair opening angle is small and the e+and e− separate slowly.Near the pair, the electric and magnetic fields from thee+ and e− must be considered separately. However, foran observer far away from the pair (compared to the pairseparation d), the electric and magnetic fields from the e+and e− largely cancel. Cherenkov radiation is producedat a distance of the order of the photon wavelength Λfrom the charged particle trajectory. So, for d < Λ, can-cellation reduces the Cherenkov radiation from a pair tobelow that for two independent particles. For a typicalpair opening angle m/k, where k is the photon energyand m the electron mass, without multiple scattering,Λ > d for a distance kΛ/m. For blue light (Λ = 400nm) from a 1 TeV pair, the radiation is reduced untilthe pair travels a distance of 40 cm (neglecting multiplescattering).In this paper, after a more detailed calculation of theCherenkov radiation from e+e− pairs, we consider co-herent optical radiation from pairs that follow realistictrajectories, and from electromagnetic showers. We con-sider two classes of experiments: underwater/in-ice neu-trino observatories and air Cherenkov telescopes.II. CHERENKOV RADIATION FROM PAIRSCherenkov radiation from closely spaced e+e− pairscan be derived by extending the derivation for pointcharges, by replacing a point charge with an oppositelycharged, separated pair. We sketch the derivation forradiation from point charges, review previous work onradiation from infinitesimal dipoles, and derive the ex-pression for Cherenkov radiation from a closely-spacedco-moving pair.We follow the notation and derivation from Ref. [9]. InFourier space, the charge density ρ and current ~J froma point charge ze propagating with speed v in the x1direction can be written asρ(~k, ω) =ze2πδ(ω − k1v)~J(~k, ω) = ~vρ(~k, ω)(1)where ~k is the wave vector and ω the photon energy. Thiscurrent deposits energy into the medium through elec-tromagnetic interactions. We use Maxwell’s equationsbeyond a radius a around the particle track, where a iscomparable to the average atomic separation. Then, byconservation of energy, the Cherenkov radiation power isequal to the the energy flow through a cylinder of thisradius, giving(dEdx)= −caRe∫ ∞0B∗3(ω)E1(ω)dω . (2)E1 is the component of ~E parallel to the particle track,and B3 is the component of ~B in the x3 direction, eval-uated at an impact parameter b at a point with x2 = b,x3 = 0.Using the wave equations in a dielectric medium andthe definition of fields, then integrating over momenta,one findsE1(ω) = −izeωv2(2π)1/2 [1ǫ(ω)− β2]K0(λb) (3)1whereλ2 =ω2v2[1 − β2ǫ(ω)] .Similarly,E2(ω) =zev(2π)λǫ(ω)K1(λb)B3(ω) = ǫ(ω)βE2(ω) .(4)Since we are ultimately interested in far-field radiationwe take the asymptotic form of the energy depositionat |λa| ≫ 1. Taking the case β > 1/√ǫ(ω) for realǫ(ω), λ becomes completely imaginary. The asymptoticcontribution of the Bessel functions in the integrand ofdE/dx is finite, giving the well-known expression for theCherenkov radiation(dEdx)=(ze)2c2×∫ǫ(ω)>1/β2ω(1 − 1β2ǫ(ω))dω .(5)Note how a has dropped out [9, Ch. 13]. The derivationof this Cherenkov radiation may be expanded to give thefield from an pair.The radiation from an e+e− pairs depends on two pa-rameters: the separation d and the angle between thedirection of motion and the orientation of the pair. Forrelativistic pairs created by photon conversion, the trans-verse (to the direction of motion) separation is important;the longitudinal separation of a highly relativistic paircan be neglected, due to Lorentz length contraction.Balazs [4] provided an expression for Cherenkov radi-ation from an infinitesimal dipole D oriented transverseto its momentum. These fields are well approximated byby a linear Taylor expansion of the corresponding point-charge fields:E(D)1 (ω) = −d∂E1(ω)∂x2B(D)3 (ω) = −d∂B3(ω)∂x2,where d is the effective pair separation, so D = zed.Then, following the same steps as in the point-chargecase, Balazs finds(dEdx)=12D2c4×∫ǫ(ω)>1/β2ǫ(ω)ω3(1 − 1β2ǫ(ω))2dω .(6)Jelley [5] also provides an expression for a point dipoleoriented parallel to its direction of motion,(dEdx)=D2c4(1 − β2)×∫ǫ(ω)>1/β2ω3(1 − 1β2ǫ(ω))dω .The dE/dx falls to zero for β → 1, whereas in the perpen-dicular case the radiation is finite (for ǫ(ω) > 1). Again,this is due to Lorentz contraction.To compute the Cherenkov radiation for finite separa-tions d, let us consider a pair moving in the +x direction.The pair lies entirely in the transverse plane y-z, with theline between them making an angle α with respect to they-axis. Then, generalizing Eq. (1), the charge densityfrom the pair isρ(~k, ω) =ze2πδ(ω − k1v)[e−i(k2y+−k3z+) − e−i(k2y−ik3z−)].The two charges have positions, relative to the center ofmassy+ =d2cosα z+ = −d2sin αy− = −d2cosα z− =d2sin α .The angle α is then the relative azimuth between the lineconnecting the two charges and the azimuth of observa-tion.We proceed by analogy to Eq. (3). Eq. (13.60) ofRef. [9] gives the electric field from a single particle. Wegeneralize that equation to a pair and integrate over mo-menta successively, so thatE1(ω) = −2izeω(2π)3/2v2(1ǫ(ω)− β2)∫dk2eibk2×∫dk3e−ik2d2cos αeik3d2sin α − c.c.λ2 + k22 + k23= − izeω(2π)1/2v2(1ǫ(ω)− β2)×∫dk2e−d2sin α√λ2+k22√λ2 + k22×(ei(b−d2cos α)k2 − ei(b+ d2 cos α)k2). (7)Then,E1(ω) =−izeωv2√π2(1ǫ(ω)− β2)× [K0(λb−) − K0(λb+)] (8)whereb± =√d24sin2 α + (b ± d2cosα)2 .As before, we take |λa| ≫ 1 and a < b, so we need onlyto consider d ≪ b; there is little interference for d < b.Therefore, we can simplify usingb± ≃ b ±d2cosα .2Then, as before, considering completely imaginary λ and|λa| ≫ 1,E1(ω) =2zeωc2(1 − 1β2ǫ(ω))×√i|λ|ei|λ|bbsin[d2|λ| cos α](9)where we have taken b− ≃ b+ ≃ b in the denominator.At α = ±π/2 E1(ω) = 0. The Cherenkov radiation isno longer symmetric about the direction of motion, andvanishes at right angles to the direction of the dipole. Asthe charge separation increases (or the wavelength de-creases), the angular distribution evolves from two widelobes into a many-lobed structure, as shown in Fig. 1.After integration over even a narrow range of ω or d, theangular distribution becomes an almost-complete disk,with two narrow zeroes remaining at a direction perpen-dicular to the dipole vector.y-axis-0.3 -0.2 -0.1 0 0.1 0.2 0.3z-axis-0.3-0.2-0.100.10.20.3z-axis+-FIG. 1. The azimuthal angular distribution (transverse tothe direction of motion) of Cherenkov radiation for 500 nmphotons from a pair of charges oriented as shown in the Fig-ure. Distributions are shown for pair separations 100 nm(solid line), 1 µm (dashed line) and 5 µm (dotted line), with√ǫ(ω) = n = 1.3 and β = 1.After assembling the pieces, and averaging over α, wefind the generalization of Eq. (5),(dEdx)=(ze)2c2∫ǫ(ω)>1/β2dωω(1 − 1βǫ(ω))× 2 [1 − J0(λd)] .(10)For λd ≪ 1, this reproduces Eq. (6). For λd ≫ 1, thedE/dx is twice that expected for an independent particle(Eq. (5)), as expected. The transition is shown in Fig.2. As the emission wavelength Λ approaches d, the pairspectrum converges to the point-charge spectrum in anoscillatory fashion, characteristic of the Bessel function.For certain values of λd, the radiation exceeds that oftwo independent charged particles. [nm]ΛPhoton wavelength 300 400 500 600 700 800]-1 nm-1 [cmΛN/dxd2Photon rate d00.511.522.533.544.5]-1 nm-1 [cmΛN/dxd2Photon rate dFIG. 2. The spectrum of Cherenkov radiation at β = 1,√ǫ(ω) = n = 1.3. Solid line is for e+e− with the particlesconsidered independently, and the dashed lines are for pairstreated coherently, with separations 100 nm, 1 µm and 5 µm.For the remainder of the paper, we assume that mediasatisfy√ǫ(ω) = n, where n is independent of frequency.In realistic detection media, any variation of n with fre-quency is small, and would have little effect on Cherenkovradiation from relativistic particles.With real e+e− pairs, two effects should be consid-ered. Radiation is not emitted instantaneously, but overa period known as the formation time, τ = 1/cλ. If thepair separation varies significantly during this time, lossof longitudinal coherence may reduce the radiation. Thismay be significant when the change in λd is of order one.Where the coherence is important, cτ and d are compa-rable, and the e+ and e− follow roughly parallel tracks,so d will not change significantly during the formationtime.Second, the Cherenkov radiation produced at a point(x-coordinate) depends on the fields emitted by thecharged particles at earlier times, when d may be dif-ferent than at the point of radiation. For full rigor, theseretarded separations should be used in the calculation.Again, this has a negligible effect on the results.III. RADIATION FROM e+e− PAIRS INSHOWERSMany experiments study Cherenkov radiation fromlarge electromagnetic showers. The radiation from ashower may be less than would be expected if every parti-cle were treated as independent. We use a simple simula-tion to consider 300 to 800 nm radiation from electromag-3netic showers. This frequency range is typical for pho-tomultiplier based Cherenkov detectors; at longer wave-length, there is little radiation, while shorter wavelengthlight is absorbed by the glass in the phototube.We simulated 1000 γ conversions to e+e− pairs withtotal energies from 108 to 1020 eV. Pairs were pro-duced with the energy partitioned between the e+ ande− following the Bethe-Heitler differential cross sectiondσ ≈ E±(1−E±), where E± is the electron (or positronenergy) [11]. At high energies in dense media (above 1016eV in water or ice), the LPM effect becomes important,and more asymmetric pairs predominate [10]. The pairsare generated with initial opening angle of m/k; the fixedangle is a simplification, but the pair separation is domi-nated by multiple scattering, so it has little effect on ourresults.The e− and e+ are tracked through a water medium(with n =√ǫ = 1.3) in steps of 0.02X0, where X0 is theradiation length. At each step, the particles multiple-scatter, following a Gaussian approximation [12, Ch. 27].The particles radiate bremsstrahlung photons, using asimplified model where photon emission follows a Pois-son distribution, with mean free path X0. Although thismodel has almost no soft bremsstrahlung, soft emissionhas little effect on Cherenkov radiation, since the electronor positron velocity is only slightly affected.At each step, we compute the Cherenkov radiation foreach pair. They are treated coherently when the d < 2Λ;at larger separations the particles radiate independently.Path length [X0]0 0.5 1 1.5 2Dipole suppression00.20.40.60.81Dipole suppressionFIG. 3. Average Cherenkov photon emission rate for pairswith energies from 1010 (leftmost, dot-dashed curve) to 1015eV (rightmost, solid curve) vs. the distance travelled by thepair in water, relative to emission from two independent par-ticles.As shown in Fig. 3, the particles in lower energy pairs(< 1010 eV) radiate almost independently. In contrast,the radiation from very high energy pairs (> 1015 eV) islargely suppressed. The broad excursions slightly aboveunity occur when J0(λd) > 1 for many of the scatteredpairs.IV. IMPLICATIONS FOR EXPERIMENTSAt least two types of astrophysical observatories de-pend on Cherenkov radiation. Water and ice based neu-trino observatories observe Cherenkov radiation from thecharged particles produced in neutrino interactions, andair Cherenkov telescopes look for γ-ray induced electro-magnetic showers in the Earth’s atmosphere.Current neutrino observatories can search for electronneutrinos with energies above 50 TeV (for νµ, the thresh-old is much lower) [14]. They use large arrays of photo-multiplier tubes to observe the Cherenkov radiation fromνe induced showers. For water, n ≈ 1.3, and Fig. 3 showsthat λd < 1 while the pair travels significant distances.Ice is similar to water, with a slightly lower density; n ofice depends on its structure, and is typically ≈ 1.29 [16].To quantify the effect of Cherenkov radiation from νeinteractions, we use a toy model of an electromagneticshower. The shower evolves through generations, witheach generation having twice as many particles as thepreceding generation, with half the energy. Each gen-eration evolves over a distance of X0; other simulationshave evolved generations over a shorter distance (ln 2)X0,leading to a more compact shower [15]. In these showers,most of the particles are produced in the last radiationlengths.Fig. 4 shows the Cherenkov radiation expected from amodel 1020 eV shower with coherent Cherenkov radiation(solid line) and in a model where all particles radiate in-dependently (dotted line). This model does not includethe LPM effect, so it should be considered only illustra-tive. The LPM effect lengthens the high-energy (above1017 eV) portion of the shower. By spreading out theshower longitudinally, the LPM effect will give the elec-trons and positrons more time to separate, and so willsomewhat lessen the difference between the two results.However, it is clear from Fig. 4 that coherence has a sig-nificant effect for the first ≈ 22X0. Since the front of theshower contains relatively few particles, it will not affectthe measured energy. However, by suppressing radiationfrom the front of the shower, it could affect the measure-ment of the shower development and the reconstructionof the shower position. This effect can only be assessedwith a detailed model of a particular experiment.Atmospheric Cherenkov telescopes like the Whippleobservatory study astrophysical γ-rays with energiesfrom 100 GeV to 10 TeV. These telescopes observeCherenkov radiation from pairs in the upper atmosphere;for a 1 TeV shower, the maximum particle density occursat an altitude of 8 km above sea level (asl) [13], where thedensity is about 1/3 that at sea level. Since n−1 dependslinearly on the density, at 8 km asl n−1 ≈ 1×10−4, so for500 nm photons radiated from ultra-relativistic particles,λd < 1 only for d < 6 µm. In this low-density medium,the effect of the pair opening angle is significant and mul-tiple scattering is less important. Pairs with k < 1 TeVwill separate by 30 µm in a distance less than 30 meters;4at 8 km asl, this is 3% of a radiation length. This dis-tance is too short to affect the radiation pattern from theshower.Cherenkov radiation is also studied in lead-glass blockcalorimetry, and in Cherenkov counters for particle iden-tification; their response to photon conversions may beaffected by this coherence.Depth [X0]0 5 10 15 20 25 30 35 40]-1Photon rate [cm-410-21012104106108101010121014101610]-1Photon rate [cmIndependentPairFIG. 4. Cherenkov radiation from a 1020 eV shower inwater, using the Heitler toy model, versus shower depth(smoothed). The two curves compare the radiation for e+e−calculated as independent particles and as coherent pairs.Although this calculation applies for Cherenkov radi-ation, a similar effect should occur for e+e− pair energyloss through dE/dx. When d is smaller than the typicalimpact parameter to cause ionization of a medium, theelectric fields from the e+ and e− will largely cancel, andthe energy loss will be reduced.V. CONCLUSIONWe have calculated the Cherenkov radiation from e+e−pairs as a function of the pair separation d. When d2 <v2/(ω2[1−β2ǫ(ω)]), the radiation is suppressed comparedto that from two independent particles.This suppression affects the radiation from electromag-netic showers in dense media. Although the total radi-ation from a shower is not affected, emission from thefront part of the shower is greatly reduced; this will af-fect studies of the shower development, and may affectmeasurements of the position of the shower.This work was funded by the U.S. National ScienceFoundation under Grant number OPP-0236449 and theU.S. Department of Energy under contract number DE-AC-76SF00098.[1] P. A. Cherenkov, Dokl. Akad. Nauk. SSR 2, 451 (1934);S. I. Vavilov, Dokl. Akad. Nauk. SSR 2, 457 (1934); I.M. Frank and Ig. Tamm, Dokl. Akad. Nauk. SSR 14, 109(1937).[2] I. M. Frank, Zh. fiz. SSR 7, 49 (1943).[3] V. Eidman, Transactions of the Gorky Research Phys-iotechnical Institute and Radiophysical Faculty of theGorky State University, Scientific Notes 30 (1956).[4] N. L. Balazs, Phys. Rev. 104, 1220 (1956).[5] J. V. Jelley, Cherenkov Radiation and its applications(Pergamon Press, 1958).[6] E. Zas, F. Halzen and T. Stanev, Phys. Rev. D45, 362(1992); J. Alvarez-Muniz et al., Phys. Rev. D68, 043001(2003); S. Razzaque et al., Phys. Rev. D69, 047101(2004); A. R. Beresnyak, astro-ph/0310295.[7] G. A. Askaryan and B. A. Dolgoshein, JETP Lett. 25,213 (1977).[8] J. Alvarez-Muniz and E. Zas, PHys. Lett. B434, 396(1998).[9] J.D. Jackson, Classical Electrodynamics, 3rd edition(John Wiley & Sons, New York, 1998).[10] S. Klein, Rev. Mod. Phys. 71, 1501 (1999); S. Klein,hep-ex/0402028.[11] H. A. Bethe and W. Heitler, Proc. R. Soc. London, SerA 146, 83 (1934).[12] S. Eidelman et al., Phys. Lett. B592, 1 (2004).[13] C. M. Hoffman, C. Sinnis, P. Fleury and M. Punch, Rev.Mod. Phys. 71, 897 (1999).[14] M. Ackermann, Astropart. Phys. 22, 127 (2004); the Ice-cube collaboration, IceCube Preliminary Design Docu-ment, Oct., 2001. Available at www.icecube.wisc.edu[15] S. Klein, astro-ph/0412546.[16] P. B. Price, K. Woschnagg, Astropart. Phys. 22, 97(2001).5

Cherenkov Radiation from e+e- Pairs and Its Effect on nu e Induced Showers

Cherenkov radiation from
                    
                      
                        e

We calculate the Cherenkov radiation from an e{sup +}e{sup -} pair at small separations, as occurs shortly after a pair conversion. The radiation is reduced (compared to that from two independent particles) when the pair separation is smaller than the wavelength of the emitted light. We estimate the reduction in light in large electromagnetic showers, and discuss the implications for detectors that observe Cherenkov radiation from showers in the Earth's atmosphere, as well as in oceans and Antarctic ice

UNT (University of North Texas) Digital Library

arXiv:physics/0506078 v1   8 Jun 2005Cherenkov Radiation from e+e−Pairs and Its Effect on νeInduced ShowersSourav K. Mandal, Spencer R. Klein, J. David JacksonLawrence Berkeley National LaboratoryBerkeley, CA 94720We calculate the Cherenkov radiation from an e+e− pairat small separations, as occurs shortly after a pair conversion.The radiation is reduced (compared to that from two indepen-dent particles) when the pair separation is smaller than thewavelength of the emitted light. We estimate the reduction inlight in large electromagnetic showers, and discuss the impli-cations for detectors that observe Cherenkov radiation fromshowers in the Earth’s atmosphere, as well as in oceans andAntarctic ice.I. INTRODUCTIONCherenkov radiation from relativistic particles hasbeen known for over 70 years [1]. However, to date, al-most all studies have concentrated on the radiation fromindividual particles. Frank [2], Eidman [3] and Balazs[4] considered the Cherenkov radiation from electric andmagnetic dipoles, but only in the limit of vanishing sep-arations d. Their work was nicely reviewed by Jelley [5].Several more recent calculations have consideredCherenkov radiation from entire electromagnetic show-ers, in the coherent or almost coherent limit [6]. Thefields from the e+ and e− largely cancel, and the bulk ofthe coherent radiation is due to the net excess of e− overe+ (the Askaryan effect) [7]. Hadronic showers produceradiation through the same mechanism [8]. Coherent ra-diation occurs when the wavelength of the radiation islarge compared to the radial extent of the shower; forreal materials, this only occurs for radio waves.Here, we consider another case, the reduction of ra-diation from slightly-separated oppositely-charged co-moving pairs. This includes e+e− pairs produced byphoton conversion. When high-energy photons convertto e+e− pairs, the pair opening angle is small and the e+and e− separate slowly.Near the pair, the electric and magnetic fields from thee+ and e− must be considered separately. However, foran observer far away from the pair (compared to the pairseparation d), the electric and magnetic fields from the e+and e− largely cancel. Cherenkov radiation is producedat a distance of the order of the photon wavelength Λfrom the charged particle trajectory. So, for d < Λ, can-cellation reduces the Cherenkov radiation from a pair tobelow that for two independent particles. For a typicalpair opening angle m/k, where k is the photon energyand m the electron mass, without multiple scattering,Λ > d for a distance kΛ/m. For blue light (Λ = 400nm) from a 1 TeV pair, the radiation is reduced untilthe pair travels a distance of 40 cm (neglecting multiplescattering).In this paper, after a more detailed calculation of theCherenkov radiation from e+e− pairs, we consider co-herent optical radiation from pairs that follow realistictrajectories, and from electromagnetic showers. We con-sider two classes of experiments: underwater/in-ice neu-trino observatories and air Cherenkov telescopes.II. CHERENKOV RADIATION FROM PAIRSCherenkov radiation from closely spaced e+e− pairscan be derived by extending the derivation for pointcharges, by replacing a point charge with an oppositelycharged, separated pair. We sketch the derivation forradiation from point charges, review previous work onradiation from infinitesimal dipoles, and derive the ex-pression for Cherenkov radiation from a closely-spacedco-moving pair.We follow the notation and derivation from Ref. [9]. InFourier space, the charge density ρ and current ~J froma point charge ze propagating with speed v in the x1direction can be written asρ(~k, ω) =ze2πδ(ω − k1v)~J(~k, ω) = ~vρ(~k, ω)(1)where ~k is the wave vector and ω the photon energy. Thiscurrent deposits energy into the medium through elec-tromagnetic interactions. We use Maxwell’s equationsbeyond a radius a around the particle track, where a iscomparable to the average atomic separation. Then, byconservation of energy, the Cherenkov radiation power isequal to the the energy flow through a cylinder of thisradius, giving(dEdx)= −caRe∫ ∞0B∗3(ω)E1(ω)dω . (2)E1 is the component of ~E parallel to the particle track,and B3 is the component of ~B in the x3 direction, eval-uated at an impact parameter b at a point with x2 = b,x3 = 0.Using the wave equations in a dielectric medium andthe definition of fields, then integrating over momenta,one findsE1(ω) = −izeωv2(2π)1/2 [1ǫ(ω)− β2]K0(λb) (3)1whereλ2 =ω2v2[1 − β2ǫ(ω)] .Similarly,E2(ω) =zev(2π)λǫ(ω)K1(λb)B3(ω) = ǫ(ω)βE2(ω) .(4)Since we are ultimately interested in far-field radiationwe take the asymptotic form of the energy depositionat |λa| ≫ 1. Taking the case β > 1/√ǫ(ω) for realǫ(ω), λ becomes completely imaginary. The asymptoticcontribution of the Bessel functions in the integrand ofdE/dx is finite, giving the well-known expression for theCherenkov radiation(dEdx)=(ze)2c2×∫ǫ(ω)>1/β2ω(1 − 1β2ǫ(ω))dω .(5)Note how a has dropped out [9, Ch. 13]. The derivationof this Cherenkov radiation may be expanded to give thefield from an pair.The radiation from an e+e− pairs depends on two pa-rameters: the separation d and the angle between thedirection of motion and the orientation of the pair. Forrelativistic pairs created by photon conversion, the trans-verse (to the direction of motion) separation is important;the longitudinal separation of a highly relativistic paircan be neglected, due to Lorentz length contraction.Balazs [4] provided an expression for Cherenkov radi-ation from an infinitesimal dipole D oriented transverseto its momentum. These fields are well approximated byby a linear Taylor expansion of the corresponding point-charge fields:E(D)1 (ω) = −d∂E1(ω)∂x2B(D)3 (ω) = −d∂B3(ω)∂x2,where d is the effective pair separation, so D = zed.Then, following the same steps as in the point-chargecase, Balazs finds(dEdx)=12D2c4×∫ǫ(ω)>1/β2ǫ(ω)ω3(1 − 1β2ǫ(ω))2dω .(6)Jelley [5] also provides an expression for a point dipoleoriented parallel to its direction of motion,(dEdx)=D2c4(1 − β2)×∫ǫ(ω)>1/β2ω3(1 − 1β2ǫ(ω))dω .The dE/dx falls to zero for β → 1, whereas in the perpen-dicular case the radiation is finite (for ǫ(ω) > 1). Again,this is due to Lorentz contraction.To compute the Cherenkov radiation for finite separa-tions d, let us consider a pair moving in the +x direction.The pair lies entirely in the transverse plane y-z, with theline between them making an angle α with respect to they-axis. Then, generalizing Eq. (1), the charge densityfrom the pair isρ(~k, ω) =ze2πδ(ω − k1v)[e−i(k2y+−k3z+) − e−i(k2y−ik3z−)].The two charges have positions, relative to the center ofmassy+ =d2cosα z+ = −d2sin αy− = −d2cosα z− =d2sin α .The angle α is then the relative azimuth between the lineconnecting the two charges and the azimuth of observa-tion.We proceed by analogy to Eq. (3). Eq. (13.60) ofRef. [9] gives the electric field from a single particle. Wegeneralize that equation to a pair and integrate over mo-menta successively, so thatE1(ω) = −2izeω(2π)3/2v2(1ǫ(ω)− β2)∫dk2eibk2×∫dk3e−ik2d2cos αeik3d2sin α − c.c.λ2 + k22 + k23= − izeω(2π)1/2v2(1ǫ(ω)− β2)×∫dk2e−d2sin α√λ2+k22√λ2 + k22×(ei(b−d2cos α)k2 − ei(b+ d2 cos α)k2). (7)Then,E1(ω) =−izeωv2√π2(1ǫ(ω)− β2)× [K0(λb−) − K0(λb+)] (8)whereb± =√d24sin2 α + (b ± d2cosα)2 .As before, we take |λa| ≫ 1 and a < b, so we need onlyto consider d ≪ b; there is little interference for d < b.Therefore, we can simplify usingb± ≃ b ±d2cosα .2Then, as before, considering completely imaginary λ and|λa| ≫ 1,E1(ω) =2zeωc2(1 − 1β2ǫ(ω))×√i|λ|ei|λ|bbsin[d2|λ| cos α](9)where we have taken b− ≃ b+ ≃ b in the denominator.At α = ±π/2 E1(ω) = 0. The Cherenkov radiation isno longer symmetric about the direction of motion, andvanishes at right angles to the direction of the dipole. Asthe charge separation increases (or the wavelength de-creases), the angular distribution evolves from two widelobes into a many-lobed structure, as shown in Fig. 1.After integration over even a narrow range of ω or d, theangular distribution becomes an almost-complete disk,with two narrow zeroes remaining at a direction perpen-dicular to the dipole vector.y-axis-0.3 -0.2 -0.1 0 0.1 0.2 0.3z-axis-0.3-0.2-0.100.10.20.3z-axis+-FIG. 1. The azimuthal angular distribution (transverse tothe direction of motion) of Cherenkov radiation for 500 nmphotons from a pair of charges oriented as shown in the Fig-ure. Distributions are shown for pair separations 100 nm(solid line), 1 µm (dashed line) and 5 µm (dotted line), with√ǫ(ω) = n = 1.3 and β = 1.After assembling the pieces, and averaging over α, wefind the generalization of Eq. (5),(dEdx)=(ze)2c2∫ǫ(ω)>1/β2dωω(1 − 1βǫ(ω))× 2 [1 − J0(λd)] .(10)For λd ≪ 1, this reproduces Eq. (6). For λd ≫ 1, thedE/dx is twice that expected for an independent particle(Eq. (5)), as expected. The transition is shown in Fig.2. As the emission wavelength Λ approaches d, the pairspectrum converges to the point-charge spectrum in anoscillatory fashion, characteristic of the Bessel function.For certain values of λd, the radiation exceeds that oftwo independent charged particles. [nm]ΛPhoton wavelength 300 400 500 600 700 800]-1 nm-1 [cmΛN/dxd2Photon rate d00.511.522.533.544.5]-1 nm-1 [cmΛN/dxd2Photon rate dFIG. 2. The spectrum of Cherenkov radiation at β = 1,√ǫ(ω) = n = 1.3. Solid line is for e+e− with the particlesconsidered independently, and the dashed lines are for pairstreated coherently, with separations 100 nm, 1 µm and 5 µm.For the remainder of the paper, we assume that mediasatisfy√ǫ(ω) = n, where n is independent of frequency.In realistic detection media, any variation of n with fre-quency is small, and would have little effect on Cherenkovradiation from relativistic particles.With real e+e− pairs, two effects should be consid-ered. Radiation is not emitted instantaneously, but overa period known as the formation time, τ = 1/cλ. If thepair separation varies significantly during this time, lossof longitudinal coherence may reduce the radiation. Thismay be significant when the change in λd is of order one.Where the coherence is important, cτ and d are compa-rable, and the e+ and e− follow roughly parallel tracks,so d will not change significantly during the formationtime.Second, the Cherenkov radiation produced at a point(x-coordinate) depends on the fields emitted by thecharged particles at earlier times, when d may be dif-ferent than at the point of radiation. For full rigor, theseretarded separations should be used in the calculation.Again, this has a negligible effect on the results.III. RADIATION FROM e+e− PAIRS INSHOWERSMany experiments study Cherenkov radiation fromlarge electromagnetic showers. The radiation from ashower may be less than would be expected if every parti-cle were treated as independent. We use a simple simula-tion to consider 300 to 800 nm radiation from electromag-3netic showers. This frequency range is typical for pho-tomultiplier based Cherenkov detectors; at longer wave-length, there is little radiation, while shorter wavelengthlight is absorbed by the glass in the phototube.We simulated 1000 γ conversions to e+e− pairs withtotal energies from 108 to 1020 eV. Pairs were pro-duced with the energy partitioned between the e+ ande− following the Bethe-Heitler differential cross sectiondσ ≈ E±(1−E±), where E± is the electron (or positronenergy) [11]. At high energies in dense media (above 1016eV in water or ice), the LPM effect becomes important,and more asymmetric pairs predominate [10]. The pairsare generated with initial opening angle of m/k; the fixedangle is a simplification, but the pair separation is domi-nated by multiple scattering, so it has little effect on ourresults.The e− and e+ are tracked through a water medium(with n =√ǫ = 1.3) in steps of 0.02X0, where X0 is theradiation length. At each step, the particles multiple-scatter, following a Gaussian approximation [12, Ch. 27].The particles radiate bremsstrahlung photons, using asimplified model where photon emission follows a Pois-son distribution, with mean free path X0. Although thismodel has almost no soft bremsstrahlung, soft emissionhas little effect on Cherenkov radiation, since the electronor positron velocity is only slightly affected.At each step, we compute the Cherenkov radiation foreach pair. They are treated coherently when the d < 2Λ;at larger separations the particles radiate independently.Path length [X0]0 0.5 1 1.5 2Dipole suppression00.20.40.60.81Dipole suppressionFIG. 3. Average Cherenkov photon emission rate for pairswith energies from 1010 (leftmost, dot-dashed curve) to 1015eV (rightmost, solid curve) vs. the distance travelled by thepair in water, relative to emission from two independent par-ticles.As shown in Fig. 3, the particles in lower energy pairs(< 1010 eV) radiate almost independently. In contrast,the radiation from very high energy pairs (> 1015 eV) islargely suppressed. The broad excursions slightly aboveunity occur when J0(λd) > 1 for many of the scatteredpairs.IV. IMPLICATIONS FOR EXPERIMENTSAt least two types of astrophysical observatories de-pend on Cherenkov radiation. Water and ice based neu-trino observatories observe Cherenkov radiation from thecharged particles produced in neutrino interactions, andair Cherenkov telescopes look for γ-ray induced electro-magnetic showers in the Earth’s atmosphere.Current neutrino observatories can search for electronneutrinos with energies above 50 TeV (for νµ, the thresh-old is much lower) [14]. They use large arrays of photo-multiplier tubes to observe the Cherenkov radiation fromνe induced showers. For water, n ≈ 1.3, and Fig. 3 showsthat λd < 1 while the pair travels significant distances.Ice is similar to water, with a slightly lower density; n ofice depends on its structure, and is typically ≈ 1.29 [16].To quantify the effect of Cherenkov radiation from νeinteractions, we use a toy model of an electromagneticshower. The shower evolves through generations, witheach generation having twice as many particles as thepreceding generation, with half the energy. Each gen-eration evolves over a distance of X0; other simulationshave evolved generations over a shorter distance (ln 2)X0,leading to a more compact shower [15]. In these showers,most of the particles are produced in the last radiationlengths.Fig. 4 shows the Cherenkov radiation expected from amodel 1020 eV shower with coherent Cherenkov radiation(solid line) and in a model where all particles radiate in-dependently (dotted line). This model does not includethe LPM effect, so it should be considered only illustra-tive. The LPM effect lengthens the high-energy (above1017 eV) portion of the shower. By spreading out theshower longitudinally, the LPM effect will give the elec-trons and positrons more time to separate, and so willsomewhat lessen the difference between the two results.However, it is clear from Fig. 4 that coherence has a sig-nificant effect for the first ≈ 22X0. Since the front of theshower contains relatively few particles, it will not affectthe measured energy. However, by suppressing radiationfrom the front of the shower, it could affect the measure-ment of the shower development and the reconstructionof the shower position. This effect can only be assessedwith a detailed model of a particular experiment.Atmospheric Cherenkov telescopes like the Whippleobservatory study astrophysical γ-rays with energiesfrom 100 GeV to 10 TeV. These telescopes observeCherenkov radiation from pairs in the upper atmosphere;for a 1 TeV shower, the maximum particle density occursat an altitude of 8 km above sea level (asl) [13], where thedensity is about 1/3 that at sea level. Since n−1 dependslinearly on the density, at 8 km asl n−1 ≈ 1×10−4, so for500 nm photons radiated from ultra-relativistic particles,λd < 1 only for d < 6 µm. In this low-density medium,the effect of the pair opening angle is significant and mul-tiple scattering is less important. Pairs with k < 1 TeVwill separate by 30 µm in a distance less than 30 meters;4at 8 km asl, this is 3% of a radiation length. This dis-tance is too short to affect the radiation pattern from theshower.Cherenkov radiation is also studied in lead-glass blockcalorimetry, and in Cherenkov counters for particle iden-tification; their response to photon conversions may beaffected by this coherence.Depth [X0]0 5 10 15 20 25 30 35 40]-1Photon rate [cm-410-21012104106108101010121014101610]-1Photon rate [cmIndependentPairFIG. 4. Cherenkov radiation from a 1020 eV shower inwater, using the Heitler toy model, versus shower depth(smoothed). The two curves compare the radiation for e+e−calculated as independent particles and as coherent pairs.Although this calculation applies for Cherenkov radi-ation, a similar effect should occur for e+e− pair energyloss through dE/dx. When d is smaller than the typicalimpact parameter to cause ionization of a medium, theelectric fields from the e+ and e− will largely cancel, andthe energy loss will be reduced.V. CONCLUSIONWe have calculated the Cherenkov radiation from e+e−pairs as a function of the pair separation d. When d2 <v2/(ω2[1−β2ǫ(ω)]), the radiation is suppressed comparedto that from two independent particles.This suppression affects the radiation from electromag-netic showers in dense media. Although the total radi-ation from a shower is not affected, emission from thefront part of the shower is greatly reduced; this will af-fect studies of the shower development, and may affectmeasurements of the position of the shower.This work was funded by the U.S. National ScienceFoundation under Grant number OPP-0236449 and theU.S. Department of Energy under contract number DE-AC-76SF00098.[1] P. A. Cherenkov, Dokl. Akad. Nauk. SSR 2, 451 (1934);S. I. Vavilov, Dokl. Akad. Nauk. SSR 2, 457 (1934); I.M. Frank and Ig. Tamm, Dokl. Akad. Nauk. SSR 14, 109(1937).[2] I. M. Frank, Zh. fiz. SSR 7, 49 (1943).[3] V. Eidman, Transactions of the Gorky Research Phys-iotechnical Institute and Radiophysical Faculty of theGorky State University, Scientific Notes 30 (1956).[4] N. L. Balazs, Phys. Rev. 104, 1220 (1956).[5] J. V. Jelley, Cherenkov Radiation and its applications(Pergamon Press, 1958).[6] E. Zas, F. Halzen and T. Stanev, Phys. Rev. D45, 362(1992); J. Alvarez-Muniz et al., Phys. Rev. D68, 043001(2003); S. Razzaque et al., Phys. Rev. D69, 047101(2004); A. R. Beresnyak, astro-ph/0310295.[7] G. A. Askaryan and B. A. Dolgoshein, JETP Lett. 25,213 (1977).[8] J. Alvarez-Muniz and E. Zas, PHys. Lett. B434, 396(1998).[9] J.D. Jackson, Classical Electrodynamics, 3rd edition(John Wiley & Sons, New York, 1998).[10] S. Klein, Rev. Mod. Phys. 71, 1501 (1999); S. Klein,hep-ex/0402028.[11] H. A. Bethe and W. Heitler, Proc. R. Soc. London, SerA 146, 83 (1934).[12] S. Eidelman et al., Phys. Lett. B592, 1 (2004).[13] C. M. Hoffman, C. Sinnis, P. Fleury and M. Punch, Rev.Mod. Phys. 71, 897 (1999).[14] M. Ackermann, Astropart. Phys. 22, 127 (2004); the Ice-cube collaboration, IceCube Preliminary Design Docu-ment, Oct., 2001. Available at www.icecube.wisc.edu[15] S. Klein, astro-ph/0412546.[16] P. B. Price, K. Woschnagg, Astropart. Phys. 22, 97(2001).5

Cherenkov Radiation from e+e- Pairs and Its Effect on nu e InducedShowers

Cherenkov Radiation from $e^+e^-$ Pairs and Its Effect on $\nu_e$
  Induced Showers

Cherenkov Radiation from $e^+e^-$ Pairs and Its Effect on $\nu_e$ Induced Showers

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Cherenkov Radiation from e+e−e^+e^-e+e− Pairs and Its Effect on νe\nu_eνe​ Induced Showers

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Cherenkov Radiation from $e^+e^-$ Pairs and Its Effect on $\nu_e$ Induced Showers