High energy neutrino, photon, and cosmic ray fluxes from VHS cosmic strings

PHYSICAL REVIEW D, VOLUME 65, 063005

High energy neutrino, photon, and cosmic ray fluxes from VHS cosmic strings

Ubi F. Wichoski*Physics Department, Brown University, Providence, Rhode Island 02912

and Depto. de Fı´sica, CENTRA-IST, Av. Rovisco Pais, 1-Lisbon 1049-001, Portugal

Jane H. MacGibbon†

Code SN3, NASA Johnson Space Center, Houston, Texas 77058

Robert H. Brandenberger‡

Physics Department, Brown University, Providence, Rhode Island 02912~Received 26 May 1998; revised manuscript received 1 October 2001; published 27 February 2002!

Decaying topological defects, in particular cosmic strings, can produce a significant flux of high energyneutrinos, photons and cosmic rays. According to the prevailing understanding of cosmic string dynamics in anexpanding Universe, the network of long strings loses its energy first into string loops, which in turn give offmost of their energy as gravitational radiation. However, it has been suggested by Vincent, Hindmarsh, andSakellariadou~VHS! that particle emission may be the dominant energy loss channel for the long stringnetwork. In this case, the predicted flux of high energy particles would be much larger. Here we calculate thepredicted flux of high energy gamma rays, neutrinos and cosmic ray antiprotons and protons as a function ofthe scale of symmetry breakingh at which the strings are produced and as a function of the initial energymJ

of the particle jets which result from the string decay. Assuming the validity of the VHS scenario, we find thatdue to the interactions with the cosmic radiation backgrounds all fluxes but the neutrino flux are suppressed atthe highest energies. This indicates that the observed events above the GZK cutoff can only be accounted forin this scenario if the primary particle is a neutrino andh is somewhat less than the GUT scale, i.e.h&1023 eV. The domain of parameter space corresponding to GUT-scale symmetry breaking is excluded alsoby the current observations below the GZK cutoff. A new aspect of this work is the calculation of the spectrumof the tau neutrinos directly produced in the decay of the X particles. This significantly increases the tauneutrino signal at high energies in all cosmic string scenarios.

DOI: 10.1103/PhysRevD.65.063005 PACS number~s!: 98.70.Sa, 96.40.Tv, 98.70.Rz, 98.80.Cq

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I. INTRODUCTION

Measurements of the spectra of high energyg rays, neu-trinos and cosmic rays have emerged as a useful conston particle physics theories which predict topological defe~for recent summaries see e.g. Refs.@1–3#!. Although it doesnot appear likely that the emission of high energy particby topological defects can explain the observed spectra,current data can be used as upper bounds to constrainries of particle physics beyond the standard model.

Of particular interest for cosmology are theories givirise to cosmic strings. Some time ago, the spectrum of cmic rays from nonsuperconducting strings was compu@4–6# under the assumption of thestandardscaling picture@7–9# emerging from studies of cosmic string dynamics inexpanding Universe~see Refs.@10–12# for recent reviews!.In the standard picture, the long string network evolves istring loops which then decay predominantly by gravitatioradiation. It was found that for strings withGm;1026 ~re-quired for strings to be relevant for cosmic structure formtion @13–15#!, wherem is the mass per unit length in thstring andG is the Newton gravitational constant, the pr

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

0556-2821/2002/65~6!/063005~18!/$20.00 65 0630

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dicted particle fluxes are substantially lower than the obsvational detections or limits.

Recently, however, Vincent, Hindmarsh and Sakellariad~VHS! @16# have challenged the standard picture of strievolution. They claim that the small-scale structure onstrings does not scale with the expansion of the Universethe VHSscenario, the long strings lose their energy direcinto particles instead of string loops. This leads to a greaproduction rate of particles and hence to the expectationgreater fluxes ofg rays, neutrinos, and cosmic rays at Ear

Expanding on these points further, detailed numerisimulations@17–19# have demonstrated that in an expandiUniverse, the network oflong cosmic strings~long meaningwith curvature radius greater than the Hubble radius! ap-proaches a scaling solution in which the number of lostring segments crossing each Hubble radius approachconstant valuen. In the VHS scenario, the string networmaintains this scaling solution by the direct radiation of slar and gauge particles. In the standard scenario, the snetwork maintains the scaling solution by continuously ging off part of its energy in the form of string loops witradius smaller than the Hubble radius. Based on certainoretical arguments~see e.g.@20#! it is expected that the distribution of loops in the standard scenario also eventuatakes on a scaling solution, i.e. when all lengths are scalethe Hubble radius, the statistical properties of the distributof string loops are independent of time. However, the re

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WICHOSKI, MACGIBBON, AND BRANDENBERGER PHYSICAL REVIEW D65 063005

lution of string network simulations~Nambu action strings!is not yet good enough to be able to verify the standscenario. Moreover, recent field theory simulations@21# pro-vide some evidence that the string defects give off theirergy not in the form of string loops, but directly by scaland gauge particle radiation, thus corroborating the VHS snario. ~These latter simulations remain controvers@22,23#.!

In the standard cosmic string scenario, the string looscillate due to the relativistic tension, and decay slowlythe emission of gravitational radiation. Only a small fractiof the energy is released in the standard scenario in the fof scalar and gauge particle radiation. This occurs viaprocess of cusp annihilation@24#, during the final loop col-lapse or by the evaporation of black holes created from stloops@25#. It has been shown@4–6,26# that for grand unifiedtheory~GUT! scale (Gm;1026) strings the flux of ultrahighenergy ~UHE! particles in the standard picture from themechanisms is at or below the relevant observational msurements or limits. The predicted flux increases, and maobservable, if the symmetry breaking scale is significanlower.

In the VHS scenario, in contrast, all of the string energyreleased directly from the long strings as scalar and gaparticle radiation. Additionally, the predicted flux increasas the symmetry breaking scale increases. Hence, the fluhigh energyg rays, neutrinos and cosmic rays in the VHscenario is expected to be much larger than the fluxes floops in the standard scenario. Indeed, it has been rema@1,21,27# that the fluxes from GUT scale strings in the VHscenario will be above the observational limits. In this papwe point out that this limit comes only from the neutrinemission at high energy and the cascades produced byelectromagnetic emission, whilst the nearest expected Vstring is too distant to produce an observable flux of cosrays org rays. In the VHS scenario the highest energy flux@above the Greisen-Zatsepin-Kuz’min~GZK! cutoff# of cos-mic rays andg rays are suppressed due to the interactionthese particles with the photons of the cosmic radiation bagrounds. The result is that no isotropic flux of cosmic raandg rays from a network of VHS strings is expected toobserved. However, the neutrino emission from lowGmVHS strings may produce an observable signal. We presedetailed calculation in this paper of the neutrino, photon acosmic ray fluxes for a range of string and decay scales.also show that the predictions depend sensitively on thetial energymJ characterizing the decay of the particle radtion, which in turn depends on the presently unknown phics in the regime between the scale of electroweak symmbreaking and the scale of string formation. We find thatcreasingmJ leads to a decrease in the upper cutoff in tpredicted spectrum and an increase in the cosmic ray fluat intermediate energies. Thus, even the additional freeof decreasingmJ cannot make GUT strings obeying VHdynamics consistent with the observational constraints. Nertheless, in the event that a segment of long string is sstantially closer than the expected average distance tonearest string, a highly anisotropic flux of cosmic rays ptons, antiprotons andg rays could be observed above th

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GZK cutoff in the VHS scenario. This possibility, howeveis not supported by the current observations of events abthe GZK cutoff which exhibit large scale isotropy andmost only small scale clustering in arrival angle@28#.

In this paper we perform for the first time the calculatioof the tau neutrinos directly produced by the hadronic decof the X particles. The calculation of the directnt spectrumhas not previously been included in the analysis of any cmic string scenario. This was because it was assumedthe nt spectrum was highly suppressed at all energies cpared with thene and nm spectra. Instead only the mucweakernt spectrum generated by the cascade of the X dene andnm off the cosmic relic neutrino background was dscribed. Recently, however, MacGibbon, Wichoski, aWebber@29,30# have shown that, in hadronic jets at acceerator energies, the tau neutrinos are a significant fractiothe total neutrino spectrum at highx, wherex is the ratio ofthe neutrino energy to the jet energy. Extrapolating thfragmentation function parametrization fornt to UHE ener-gies, we show here that, in the VHS and all other cosmstring scenarios, the directly produced tau neutrinos genea signal at Earth comparable at the highest energies withexpectedne andnm signal. This flux is orders of magnitudgreater than the contribution from the cascadent . One con-sequence of our result is that the detection of a significfraction of nt in UHE neutrino events may be due to haronic decay at the source and notnm→nt oscillation in tran-sit.

In the following sections, we present the neutrino, phoand cosmic ray spectra for VHS string emission takingmn

50 and omitting SUSY particles in the jet decay. Howevwe discuss in Sec. IV the effects of including a SUSY secnon-zero neutrino mass, neutrino oscillation or other extsions to the standard model. In the Appendix, we performfull computation of the flux of secondary muon and electrneutrinos resulting from muon decay in the particle jets.use units in which\5c5G51, unless otherwise stated, antake a Hubble constant ofh050.65.

II. JET FORMATION AND FRAGMENTATION

If the network of long strings scales, the correspondenergy densityr` at time t is

r`5nmt22, ~1!

wherem is the mass per unit length in the string. This eqution defines the constantn which counts the number ostrings per volumet3 in the scaling solution. The value ofncan be determined from simulations of cosmic string nwork evolution. Current work@17–19# gives n.13. In theVHS scenario, the scaling solution is maintained not byproduction of string loops, but by the radiation of gauge ascalar particles~which we collectively call X particles in thefollowing!. Making use of the equation of state of a strinthe continuity equation becomes

r`12Hr`52mX

dnX

dt~2!

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HIGH ENERGY NEUTRINO, PHOTON, AND COSMIC . . . PHYSICAL REVIEW D 65 063005

wherenX andmX are the number density and mass, resptively, of the X particles.mX is effectively the symmetrybreaking scale at which the string formed,

mX;h;m1/2 ~3!

~e.g. mX;1016 GeV for Gm;1026). Strictly speaking,m1/2/h;O(1) with the exact factor depending on the forof the potential of the scalar field.

The decay of the high energy X particles will lead to tproduction of jets analogous to the QCD jets seen in acerators. To simplify the discussion we assume that all jetthe X decay have the same initial energy,mJ . In this case,the decay of a single X particle will lead tomX /mJ jets, andthe number density of jets generated by the energy releaslong strings is

dnJ

dt5

2

3

nm

mJt23. ~4!

The largest uncertainty in the calculation of the fluxhigh energy particles generated by strings comes fromignorance of the structure of the jets at ultrahigh initial eergies. Jet production has been studied in detail in QCFrom such studies, the fragmentation function of jets iphotons, neutrinos and baryons is known@31,32#, at least toa good approximation, up to a few TeV. However, in our cathe particles producing the jets are superheavy scalargauge particles. Following Refs.@33,34#, the fragmentationfunctions of such jets are usually calculated by extrapolathe QCD fragmentation functions to higher energies~inwhich casemJ is taken to bemX/2). This is appropriate ifthere is no new physics which does not match this extralation between the electroweak symmetry breaking scalethe unification scale~the scale at which the defects are prduced!, but it is not justified if there is new physics whicdoes not sufficiently match the extrapolation. We will paraetrize ignorance about new physics and its effects on thefrom the X particle decay by introducing a new scalemJ~which may be the scale of the new physics! as the scale towhich we can satisfactorily extrapolate the fragmentatfunctions from the QCD regime. We will then assume ththe initial scalar or gauge X particle producesmX /mJ jetswith initial energymJ each.~In reality, there may be intermediate decay steps between themX and mJ decay scales.!While this may or may not be the precise description ofdecay, it will at least help further quantify the dependencethe final fluxes on the uncertainties in the fragmentation pcess.

The initial jet particle decays into quarks and leptons otime scale ofamJ

21 , wherea is the coupling constant associated with the physics at the energy scalemJ . The quarksthen hadronize on a strong interaction time scale. More t90% of the total energy in a QCD jet goes into pions, wthe majority of the remainder going into baryons~mainlyneutrons, antineutrons, protons and antiprotons!. On astro-physical time scales, the neutral pions decay into two ptons, the charged pions decay into electron and muon nenos and electrons or positrons, and the neutrons decay

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protons and leptons. In addition, a relatively small but ssignificant number of tau neutrinos are produced by thecay of heavy quarks and tau leptons. Non-jet leptonsl X pro-duced by the decayX→q1 l X may also influence the observableg-ray flux belowE&1011 eV ~see Sec. III!.

The distribution of energiesE of the primary QCD jetdecay products~predominantly equal numbers ofp0, p1,andp2) can be approximated by the fragmentation functi@34#

dN8

dx.

15

16x23/2~12x!2, ~5!

wherex5E/mJ is the fraction of the jet energy carried by thdecay product andE continues down to;109 eV. The twobody decays of the primaryp0, p1, or p2 decay particleslead, on the average, to two photons and two primary mneutrinos for every three pions. The energy distributionthe decay products can be obtained by integrating Eq.~5!from ax to 1 with the invariant measuredx/x,

dN

dx.

5

8aF16

322a1/2x1/224a21/2x21/21

2

3a23/2x23/2G ,

~6!

where the constanta depends on the decay process beconsidered~see below!. Equation ~6! applies to the finalspectrum of photons produced in the jet. The photons refrom p0 decay. In this case, both of the decay particles hvanishing rest mass, and it is possible for a single photocarry away the entire pion energy in the lab frame. Hena51 for the photons.

Electron and muon neutrinos result from thep1 andp2

decay. In the decay processp6→m61 nhm, one of the decayproducts ~the muon or antimuon! has nonvanishing resmass. In this case, the integration limits when integrating~5! are notx and 1~see e.g.@35# and@36#!. The correction tothe upper integration end forne andnm is negligible, but notso the change in the lower integration limit fornm whichbecomesax with

a5mp

2

mp2 2mm

2.2.34, ~7!

[email protected] MeV, the charged pion mass, andmm.106 MeV is the muon mass. Equation~6! describes thenm

and nm neutrinos immediately produced by the decay of tcharged pions. The final spectrum ofnm and nm neutrinos,however, is comprised of thenm and nm neutrinos producedby the subsequent decay of them1 andm2, together with thecontribution from those immediately produced by the decof thep1 andp2. The final spectrum ofne andne neutrinosis produced by the decay of the muons and antimuons. Ting into consideration the spin polarization of the muons,show in the Appendix that the final spectra of thene andnmneutrinos generated by jet decay are approximately

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dNnm1 nm

dEnm

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8mJF12.4810.44S Enm

mJD 23/2

26.12S Enm

mJD 21/2

27.16S Enm

mJD 1/2G , Enm

<mJ

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for the muon neutrinos produced in the first stage of the pdecay

dNnm1 nm

dEnm

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8mJF11.8310.48S Enm

mJD 23/2

25.80S Enm

mJD 21/2

27.33S Enm

mJD 1/2

10.97S Enm

mJD 2

20.15S Enm

mJD 3G , Enm

<mJ , ~9!

for the muon neutrinos produced by the decay of muonsthe pion decay, and

dNne1 ne

dEne

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8mJF12.6810.47S Ene

mJD 23/2

25.96S Ene

mJD 21/2

28.30S Ene

mJD 1/2

20.11S Enm

mJD 1

11.67S Enm

mJD 2

20.45 S Enm

mJD 3G , Ene

<mJ , ~10!

for the final electron neutrinos.Equation~5! also applies to the primary baryonic dec

products such as protons and neutrons. Assuming that 3the total energy of the jet goes into baryons and antibaryand noting that most of the energy of decaying neutronsantineutrons is transferred to daughter protons and antitons, the final distribution of protons and antiprotons is aproximately

dNp1 p

dx.0.03

15

16x23/2~12x!2. ~11!

Equation ~5! is one approximation to the numericalcomputed QCD fragmentation functions, taken from R@34#. It is derived simply by requiring that the numberdecay products scales asE1/2 and applying energy conservation and scale invariance. Further the ratio of nucleonpions is assumed at allx to equal the ratio of total nucleonto total pions per jet. Even with these reservations, we fithat Eqs.~8!–~10! match thene andnm spectra generated i10 TeV e1e2→qq events simulated by theHERWIG MonteCarlo code@32# to within a factor of 2 in the range 1024

,x,1 @30#. There are other analytical approximations of tfragmentation functions used in the literature. Another fois @33,34#

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dN8

dx.0.08e2.6Aln(1/x)~12x!2

„xAln~1/x!…21, ~12!

which can be well approximated by@37#

dN8

dx.3.322x21.324 ~13!

for values ofx between 10210,x,1022. A third formula isbased on the modified leading logarithmic approximat~MLLA ! @38# of QCD. At smallx, the expression become@39,40#

dN8

dx.

K

xexpF2

ln2x/xm

2s2 G , ~14!

where

2s251

6lnS mJ

L D 3/2

,

xm5(L/mJ)1/2, L50.234 GeV andK is found from

E0

1

xdN8

dxdx51.

Although the MLLA decently describes the shape of tfragmentation function at smallx, it has been pointed ou@41,42# that the normalizationK is unreliable because mosof the jet energy is carried off at highx. Further discussion ofextrapolation approaches and a more complex extrapolabased on the standard model Dokshitzer-Gribov-LipatAltarelli-Parisi ~DGLAP! equations can be found in@42#. Inall approximations, extrapolations and Monte Carlo simutions of the fragmentation functions at high energies,fraction of the jet energy carried by nucleons~or equivalentlythe nucleon multiplicity! lies in the range 3–10 % but thprecise value and dependence onx andmJ is unknown@41#.

Figure 1 shows a comparison of the three fragmentafunctions ~5!, ~12! and ~14!. The difference between thesfunctions is small in the energy range 1026,x,1020.5, andwill be unimportant for order of magnitude considerationHence, in the following we will use mostly approximatio~5!. We stress however that no extrapolated fragmentafunction should be regarded as accurate to more than ander of magnitude at highest and lowestx because any extrapolation includes assumptions about the evolution ofstandard model well beyond the regime for which expemental data exist. Investment in precision in those regionthe predicted spectra is unjustified and misleading.

For the first time in cosmic string emission analysis, walso include the distribution for the tau neutrinos producedthe jet decays, which had erroneously been assumed tnegligible in all previous work. While the number ofnt pro-duced permJ.1 TeV jet is less than 1023 of the number ofne andnm , the high energynt are predominantly produceby the initial decays of the heavier quarks with shorter litimes while thene and nm are produced by the final statcluster decays of the much lighter pions. This leads to snificantly greater relative contribution fromnt at highx than

5-4

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FIG. 1. Comparison of thethree jet fragmentation functiondiscussed in the text. The soliline represents the fragmentatiofunction ~5! used in this paper tocalculate all ultrahigh energy particle fluxes exceptnt , the dash-dotted line represents Eq.~12!, thedotted line is the approximation~13! of Eq. ~12!, and the dashedline represents Eq.~14!.

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previously assumed. The fragmentation distribution~5! is nolonger relevant for the tau neutrino. Instead MacGibbWichoski, and Webber@29,30# find that the fragmentationfunction for nt production in 300 GeV–100 TeVe1e2

→qq events, as simulated byHERWIG @32# can be param-etrized as

dNnt1 nt

dEnt

.1

mJF20.3610.15S Ent

mJD 21/2

10.27S Ent

mJD 1/2

20.06S Ent

mJD 3/2G . ~15!

Consistent results are obtained from jet decay events slated byPYTHIA or JETSET@31#.

The expressions for the number density of jets~4!, and forthe energy distribution of the jet decay products~6!, ~8!, ~9!,~10!, ~11! or ~15!, can be convolved to obtain the expectflux F(E) of high energy photons, neutrinos, and cosmrays with energyE from a VHS string distribution

F~E!51

4pEtc

t istrdt8

dnJ

dt8„z~ t8!11…23

dN

dE8

dE8@E,z~ t8!#

dE.

~16!

In Eq. ~16!, particles observed today (t5t0) with energyE were produced at a timet8 with energyE85E8@E,z(t8)#,where z(t8) denotes the redshift at timet8. The factor„z(t8)11…23 expresses the dilution of the particle numbdensity in an expanding universe. The lower cutoff timtc(E) corresponds to the maximal redshift from which pa

06300

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-

ticles of present-day energyE can reach us. This cutoff cabe due either to interactions with the ambient extragalaand galactic media during propagation, or to the constrathat the initial energyE8 must be less than the initial jeenergymJ . The upper cutoff timet istr corresponds to thelatest time that the emission from the VHS cosmic strinetwork can be considered isotropic~see Sec. III!. Thus ourupper cutoff time is that from which emission will have juisotropized by today.

For protons, in order to calculate the JacobidE8@E,z(t)#/dE we apply the continuous energy loss aproximation~CEL! and solve Eq.~16! numerically. The useof the CEL approximation is acceptable in this case whthe distance to the sources is much larger than the attenulength for the particle in the cosmic radiation backgroun~see Sec. III!.

For photons, if electromagnetic cascades are not takeninto account~see below!, we obtain@see Eq.~6! with a51#

E3Fg~E!.1

4p

5

2nmt0

22mJS E

mJD 3F4

3@~zmax11!

2~zmin11!#21

3@~zmax11!21/2

2~zmin11!21/2#S E

mJD 23/2

22@~zmax11!1/2

2~zmin11!1/2#S E

mJD 21/2

21

3@~zmax11!3/2

2~zmin11!3/2#S E

mJD 1/2G . ~17!

5-5


For neutrinos, if cascade off the relic neutrino background is not included, we have@see Eq.~10!#

E3Fne1 ne~E!.

1

4p

5

2nmt0

22mJS E

mJD 3F3.17@~zmax11!2~zmin11!#20.24@~zmax11!21/22~zmin11!21/2#S E

mJD 23/2

22.98@~zmax11!1/22~zmin11!1/2#S E

mJD 21/2

21.38@~zmax11!3/22~zmin11!3/2#S E

mJD 1/2

20.01@~zmax11!2

2~zmin11!2#S E

mJD10.14@~zmax11!32~zmin11!3#S E

mJD 2

20.03@~zmax11!42~zmin11!4#S E

mJD 3G , ~18!

for electron neutrinos and@see Eqs.~8! and ~9!#

~19!

for muon neutrinos.Similarly for the tau neutrinos, the flux from the string distribution neglecting cascading is@see Eq.~15!#

063005-6

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FIG. 2. The average maximaredshiftzmaxp

(E) from which pro-tons and antiprotons can reach thEarth as a function of the energat arrival. The solid lines represenfrom top to bottomGm51026,1028, 10210, 10212, 10214,10216. Note the sharp decrease othe curve above;431019 eVcorresponding to the GZK cutoff.

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2~zmin11!1/2#S E

mJD 21/2

10.18@~zmax11!3/2

2~zmin11!3/2#S E

mJD 1/2

20.02@~zmax11!5/2

2~zmin11!5/2#S E

mJD 3/2G . ~20!

In Eqs. ~17!–~20!, zmin is the redshift corresponding tt istr andzmax(E) is the redshift corresponding totc :

zmax~E!115minS zco~E!11,mJ

E D , ~21!

wherezco(E) denotes the redshift cutoff due to astrophysiinteractions. For convenience, we have presented Eqs.~17!–~20! for the casezmax(E),zeq . In our figures, we have calculated the more general result forzmax(E).zeq where ap-propriate.

The first important conclusion to draw from Eqs.~17!–~19! is that for energiesE substantially smaller than the efective initial jet energymJ , the fragmentation term proportional to (E/mJ)

23/2 similarly dominates the flux of photonselectron and muon neutrinos giving

E3Fg,ne1 ne ,nm1 nm~E!}mJ

21/2. ~22!

In the case of the tau neutrinos, the fragmentation term pportional to (E/mJ)

21/2 dominates the flux so that Eq.~20!gives

06300

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E3Fnt1 nt}mJ

23/2. ~23!

Hence, for fixed cosmic string mass per unit lengthm, thespectrum of high energy photons and neutrinos increasethe jet energymJ decreases. This occurs because for smavalues ofmJ there are more jets emitted from the string athe fragmentation function as a function ofE is more com-pressed. These two effects overcome themJ

3/2 or mJ1/2 factor

of the dominant term of the fragmentation function. We anote that the fluxes in the VHS scenario increase asGmincreases, or equivalently the symmetry breaking scalecreases, in contrast to the general decrease seen in thedard cosmic string scenario.

To complete the calculation of the photon, neutrino acosmic rays we need to know the values ofzmax(E). Forprotons and antiprotons with energy between;531018 eV and ;531019 eV, the dominant interaction ise1e2 pair production off the cosmic microwave backgrounAt higher energies, photopion production becomes the donant energy loss mechanism. The onset of photopion prodtion corresponds to the GZK cutoff energy@43#. Above thisenergy, the rate of energy loss increases dramatically,nucleons do not reach us from cosmological distances.

Taking into account Eq.~21! and applying the results o@44#, we find, in the case of protons and antiprotonzmaxp

(E) for various values ofmJ5h/2 ~whereh is the scale

of the symmetry-breaking! corresponding toGm51026,1028, 10210, 10212, 10214, and 10216 as shown in Fig. 2.

In Fig. 3 we plot the average maximum distance thaultrahigh energy proton or antiproton can travel in the cmic microwave background~CMB!. After propagating;100 Mpc the proton or antiproton energy decreases be;631019 eV irrespective of the initial jet energy. For aexpanded treatment see e.g.@45#.

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f


FIG. 3. Average energy of aproton ~antiproton! as a functionof the distance of travel throughthe CMB, for the same 6 values oGm ~determining the initial en-ergy! from top to bottom used inthe previous figure.

o-

-

thse

ringciesdayy

nt at

For photons of energies greater than about 1011 eV,zcog

(E) is determined by pair production off the cosmic ph

ton background. The resulting functionzcog(E) is shown in

Fig. 4 ~taken from Ref.@6#!. For lower energies, pair production off nuclei determines the form ofzcog

.For neutrinos, the dominant process is scattering off

relic 1.9 K cosmic background neutrinos. In this cazcon

(E) is given by@46#

zcon.8.83108S E

eVD 21/3

, E<3.131014 eV ~24!

06300

e,

and

zcon.1.83108S E

eVD 22/7

, E>3.131014 eV. ~25!

The decay of lepton pairs produced in the neutrino scatteprocess further enhances the spectrum of all neutrino speand allows neutrino emission, corresponding to presentenergies E<1021 eV, to be detected from significantlhigher redshifts than those given in Eqs.~24! and ~25! @47#.The cascade enhancement due to decay is only releva

f

FIG. 4. The maximal redshiftzcog

(E) from which photons canreach the Earth as a function othe energy at arrival@6#.

5-8

-a


FIG. 5. Average distance between long string segments asfunction of the redshift.

ffiw

s

s

if Xarthr

-in

bo

onag

ong

t

his

et-

errest

ed.ra-the

thethetheas

the

high redshift where the neutrino mean free path is suciently small. In the case of electron and muon neutrinos,will neglect the cascade enhancement ofzcon

because thedominant term in the neutrino flux from VHS strings, Eq~18! and ~19!, is proportional to @(zmin11)21/22(zmax11)21/2# and the modifications to thene , nm , andnt spec-tra expected at Earth are negligible aboveE;1023mX @47#.The cascade enhancement at low energies increases amXincreases due to the greater energy in the initial jets.

III. COMPARISON WITH OBSERVATIONS

As we mentioned above, the diffuse fluxes of particlesthe VHS scenario are produced directly by the decay oparticles. The X particles decay immediately after theyradiated from the long string segments. The result is thatparticle fluxes are produced along the strings. The numbelong strings per Hubble volumen we noted in Sec. I is approximately 13. As the Universe expands the comovinter-string distance grows as

Dints}2

An~11z!21/2Ho

21 . ~26!

Figure 5 depicts the evolution of the average distancetween long string segments. The distance between theserver and a long string segment,Dls is estimated to be

Dls;Dints

A3 2.

If the distance today between the observer and the lstring segment is larger than the maximum average proption distance for particles arriving with an energyE then the

06300

-e

.

n

eeof

g

e-b-

ga-

fluxes at Earth are exponentially suppressed@3#. Based onEq. ~26! we see that presently the average distance to a lstring segment is

D0,ls.2000 Mpc, ~27!

which corresponds to a redshiftzls.0.58.The present average redshift to a long string segmenzls

can be taken as a rough estimate ofzmin , i.e., of the redshiftcorresponding to the latest time (t istr) that the emission fromthe VHS string network can be considered isotropic. Tsets an upper limit of the integral~16! and consequently inEqs.~17!–~20! we have

zmin5zls.0.58. ~28!

Hence only the particle fluxes emitted from the VHS nwork beforet istr (z.zmin) are relevant for the diffuse flux.

A segment of long string could be substantially closthan D0,ls , the expected average distance to the neastring. In this case, a flux of cosmic rays protons~antipro-tons! and photons above the GZK cutoff might be observHowever, these fluxes would be highly anisotropic in contdiction to the present observations of UHE events aboveGZK cutoff @28#.

A. Protons

In what follows, we discuss the proton flux. Becausefragmentation functions and relevant interactions withcosmic backgrounds are to first order charge symmetric,expected antiproton flux from the VHS strings is the samethe proton flux.

The average maximum distance at the present time tosource of a proton arriving with energy;431019 eV ~theonset of the GZK cutoff! is &160 Mpc~see Fig. 3!. This isequivalent to the source being at a redshiftz&0.036, much

5-9

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smaller than the distance from which the flux can be conered isotropic (.2000 Mpc). It implies that no isotropicflux of protons from VHS strings is expected at Earth abothe GZK cutoff. On the other hand, a diffuse proton flfrom the strings should reach the Earth at arrival energE&631018 eV, well below the UHE event energies. Athough no isotropic flux of protons at UHE energies is epected at Earth, the UHE protons emitted at high redshwill have interacted with the universal radio backgrou~URB!, CMB, and the infrared-optical~IR-O! backgroundsto produce secondary fluxes ofg rays, electrons, and neutrnos. These secondary fluxes could in principle be usedconstrain the VHS model. However, in the context of the Tscenarios including the VHS scenario, these nucleon-indusecondary fluxes are negligible compared to the corresping fluxes directly produced by the decay of the X particlThis is because in the X particle decay, far more directg raysand neutrinos are produced than nucleons.

B. Photons and charged leptons

The maximum redshift at which a photon arriving todwith an energyE could have been emitted is given in Fig.If we compare this redshift to the present average distanca long string (zls.0.58) we would deduce that the isotropUHE g ray flux should also be exponentially suppressedthe case ofg rays, this is not the full story. Figure 4 describthe interaction length of the photons, i.e., the distance treled by a UHE photon before it is absorbed in the cosmbackgrounds. The UHE photons initiate electromagnetic ccades as they travel in the cosmic backgrounds. An imporconsequence is that the effective penetration length ofcascade is considerably larger than the interaction len@48#. In the VHS scenario, both the interaction length andeffective penetration length are much smaller than thetance to the nearest expected string@cf. Eq. ~26!#. The resultis that theE*1011 eV g rays are essentially subtracted frothe arriving flux: The flux aboveE;1020 eV is depleted dueto interactions with the URB; the flux in the range 1014&E&1020 eV is depleted by the CMB; and the flux in the ran1011&E&1014 eV is depleted by the IR-O backgroundThe energy lost by the photons emitted with energies ab;1011 eV at high redshifts is recycled into energies beloE*1011 eV. Analytic calculations@49# have shown that theelectromagnetic cascade spectrum has a generic shape b*1013 eV, i.e., the secondary cascade radiation spectruminsensitive to the injection spectrum. This implies that, evin the absence of the detection of direct UHE photons, thgray flux below E*1011 eV can be used to constrain thVHS scenario. The charged leptons created in the X partdecay also initiate electromagnetic cascades on the cobackgrounds. As in the case of the UHE photons, the eneof the UHE charged leptons is recycled and enhancesphoton flux below*1011 eV.

The constraints on the electromagnetic emission inmodels with a spectral indexp51, which is applicable to theVHS scenario, stem from the limits on the present diffusegray background between 33106 eV and 1011 eV @50#.These require that@51–54#

06300

-

e

s

-ts

to

edd-.

to

n

v-cs-ntethes-

ve

lowisn

leic

gyhe

QEM0 &3310223 eV cm23 sec21, ~29!

for h50.65 whereQEM0 is the present rate of total energ

injection into the electromagnetic channels.In the VHS scenario, we have from Eq.~4! that the

present rate of total energy injection into the electromagnchannel~photon and charged lepton emission! is

QEM-VHS0 ;2.3310212F1

2GF n

13G@Gm#

3F h

0.65G3

eV cm23 sec21. ~30!

Comparing to theQEM0 limit in Eq. ~29!, we obtain a con-

straint on the energy density of a string in the VHS scenaof

Gm&3310211. ~31!

Here we have assumed an extragalactic magnetic fieldBexgal&10211 G. Stronger values of the extragalactic manetic field generate greater initial synchrotron losses at ulhigh energies, decreasing the UHE spectra, enhancingcascade reprocessing into low energies, and thus lowethe upper limits onQEM

0 andGm. Because of the large distances to the sources, we have also neglected the unceties in the cosmic background URB and IR-O intensit@54#.

C. Neutrinos

The predicted flux of high energy neutrinos in the VHscenario can be obtained from Eqs.~18!–~20! using Eqs.~21!, ~24!, and~25! to determinezmaxn

. The resultingne and

nm fluxes are shown in Fig. 6 and thent flux in Fig. 7, for thecase whenmJ5h/2 whereh is the scale of symmetry breaking.

The cascading of the tau neutrinos off the relic neutrbackgrounds will have little effect on the tau neutrino spetrum shown in Fig. 7. However, a secondarynt componentpeaking at lower energies will be created in collisions of tstring-producedne and nm with the relic neutrino back-grounds. Thisne andnm cascading should increase the netntflux below x;1023. The nt signal expected at Earth fromVHS strings will be the sum of these two components,shown in Fig. 8 forGm;1028 @29,30#. In Fig. 8 we haveplotted the cascade component calculated in Ref.@47# for ap51 TD scenario. This is only an estimate because thetance distribution of strings varies from model to model. Wwill address full modeling of thent flux including cascadesin a later paper. Note however that it is the highx region,where the cascades have little effect, which is the mostservationally interesting.

Our new result is that thene , nm and nt fluxes are ofsimilar magnitude asE→mJ , unlike all previous stringemission calculations which predicted a negligiblent com-ponent at highx. In other string emission or X decay scenarios, thene , nm andnt flux should also be of comparabl

5-10

he

r


FIG. 6. Electron and muonneutrino fluxes in the VHS cosmicstring scenario~for mJ5h/2) forvarious values ofGm ~lines fromtop to bottom,Gm51026, 1028,10210, 10212, 10214, 10216). Thesolid lines represent thenm fluxand the dashed lines represent tne flux. Points with arrows repre-sent upper limits on the diffuseneutrino flux from the EAS-TOP@55#, the Frejus @56# and the Fly’sEye @57# experiments. The uppelimits stemming from the pro-jected sensitivities for both thePierre Auger project@59# and theOWL/Airwatch project @60# arealso plotted.

o-e,

hni

S

m

nal

mes

its

magnitude asx→1. Because of the uncertainty in extraplating to high emission energies, the precise ratio of the ntrino fluxes at highx is not known@also, as plotted in Fig. 8our extrapolation fornt is derived fromHERWIG Monte Carlosimulations while thene andnm functions are derived fromthe less rigorous approximation Eq.~5!#. The actual ratio willbe determined by the first step in the X decay chain whicunknown and may be influenced by, for example, leptoand SUSY decay channels~see Sec. IV!.

The most recent observational limits come from the EATOP @55#, the Frejus @56#, and the Fly’s Eye@57# experi-ments. These experiments give upper limits on the cos

06300

u-

isc

-

ic

ray neutrino flux in the energy range between 1013 eV and1020 eV. In Figs. 6 and 7 we have plotted the observatiolimits using the updatedsCC(nN) cross section@58#. Byinspection, we see that the most stringent constraint cofrom the highest energynm flux in Fig. 6 which gives anupper bound onGm of

Gm&10210. ~32!

This bound is derived at highx so it should be little affectedby the cascades which are neglected in Fig. 6.

We have also plotted on Figs. 6 and 7 the upper limstemming from the

ee

s

FIG. 7. Tau neutrino flux inthe VHS cosmic string scenario~for mJ5h/2) for various valuesof Gm ~lines from top to bottom,Gm51026, 1028, 10210, 10212,10214, 10216). For each value ofGm, the range of energies wherthe nt from decays dominates thtotal nt flux is depicted by thesolid section of the line. Pointswith arrows represent upper limiton the diffuse neutrino flux as inthe previous figure.

5-11

r

--e-


FIG. 8. Neutrino flux in theVHS cosmic string scenario foGm51028. The solid line repre-sents thenm flux, the dashed linerepresents thenes flux, and thedotted line thent flux. The dash-dotted line represents thent fromthe cascade ofne and nm due tothe interactions with the relic neutrino background. Points with arrows represent upper limits on thdiffuse neutrino flux as in the previous figures.

teinctth

n

inn

g

rv

e

cab

e

ite

Xtedes

caydfac-

t

chce-ayowthe

tra,ithasee

dwnnlyive,la-

ed-ch-

nic

planned sensitivities of the Pierre Auger project as estimain Ref. @59#, and the OWL/Airwatch project as estimatedRef. @60#. It is assumed that both experiments will colledata over a period of a few years. Their sensitivities spanenergy range between 1019 eV and 1022 eV and would al-low more stringent upper limits onGm up to

Gm&10212, ~33!

in the case of null detection by the Pierre Auger project a

Gm&10214, ~34!

in the OWL/Airwatch case. Unlike the present constra~32!, the Auger and OWL/Airwatch projects will constraithe nm flux in Fig. 6 at energiesE;1019 eV ~i.e. the lowerend of the energy range of these experiments!. Thent flux isbelow the present observational upper limits. Thent sensi-tivities of future experiments are expected to constrainGm,but these constraints are expected to be an order of matude weaker than those from thene andnm fluxes. The slopeof the nt spectrum is steeper than the slope of the obsetional upper limits for both present and future data~cf. Fig.7!. On the other hand, thent component of a TD signaturmaybe easier to distinguish from other sources than theneandnm components.

As is obvious from Eqs.~22!, ~23! and Fig. 9 the pre-dicted flux of cosmic ray neutrinos increases asmJ de-creases. Thus, the VHS scenario with GUT-scale stringsnot be made compatible with observational constraintslowering the energy scalemJ of the initial jets~which likelyhappens if new physics evolves between the electrowsymmetry breaking scale and the GUT scale!. Raising mJaboveh would alleviate the conflict, but this appears quunnatural.

06300

d

e

d

t

ni-

a-

n-y

ak

The initial decay channels and branching ratios for theparticle are unknown. The application of the extrapolaQCD-derived fragmentation functions to X decay assumimplicitly that the branching ratios forX→qi1qi are thesame as those fore1e2→qi1qi . This may not be so foreven hadronic decays at very high energies. The initial dechannelX→q1 l X , where l X is a charged lepton, shoulproduce spectra which are decreased by no more than ator of 2, compared with theX→q1q spectra~assumingequi-partition of the X particle energy between the initialqand l X), with the exception of annt spectrum enhancemenasE→mX/2 due to decay of the initialt @29,30#. The initialcharged leptonl X would also generate EM cascades whiare effectively attenuated at high energies in the VHS snario, as we remarked in Sec. II for the other EM decproducts, but which would contribute to the spectra at lenergies. This should increase the total energy going intoEM cascade channel by less than;50% @see Eq.~30!#.Purely leptonic decay channelsX→ l X1 l X8 , X→ l X1nX orX→nX1nX8 would produce substantially suppressed specbecause of the low multiplicity per decay compared whadronic jets, except for a possible neutrino spectra increasE→mX/2 due to the initial step in the decay chain. In thcase of purely leptonic decays, the nucleon andg ray spectrawould be wholly created by the collisions of the emitteleptons with the cosmic ambient media. It has been shothat in TD models generally, pure leptonic decays can obe made interesting if the cosmic relic neutrinos are massnon-relativistic and locally clustered thus enhancing the retive cascade-generated component~Ref. @53#!. However, it ishighly unlikely that the X particles~being high energy gaugeand scalar particles! would decay purely into leptons in thinitial step. Because of the much higher multiplicity of haronic jets, any scenario containing reasonable initial braning ratios for the X particle~say, to within a factor of 2–3:30% pure hadronic, 30% pure leptonic and 30% hadro

5-12

e-

e-


FIG. 9. Electron and muonneutrino fluxes in the VHS cosmicstring scenario forGm51026 andmJ;1025, 1023, 1021, 1019,1017 eV ~lines from bottom totop!. The solid lines represent thnm flux and the dashed lines represent thene flux. Points with ar-rows represent upper limits on thdiffuse neutrino flux as in the previous figures.

arevemn

atag-tri

ri

e

ata

e

e

reea

b

dgthton

illa-s tovedtheses.

nde,

o-

ea-

io-rly,ay

tou--

nt

ulds.on-

ingrs.

x-thetrino

and leptonic! would produce spectra dominated by the pticles produced via the hadronic decay channels. Hence,in the case of locally clustered relic neutrinos, the componproduced in pure leptonic decays would be essentially donated by the hadronic decays, with the possible exceptioa UHE neutrino feature asE→mX/2.

IV. EXTENSIONS TO THE MODEL

In the analysis so far, we have assumedmn50. If neutri-nos do possess a small mass, as present neutrino oscillexperiments hint, the predicted spectra from VHS stringsmodified in two ways. First, the interaction of the strinproduced neutrinos with the 1.9 K relic background neunos is strongly enhanced at the resonance forZ0 production@61#, Eres5mZ

2/2 mn;431021/(mn /eV) eV. ~If neutrinosare massless, theZ0 resonance occurs atEres;331024 eV@47#.! The present upper limits from particle physics expements on the neutrino masses aremne

&3 eV, mnm

&0.19 MeV, andmnt&18.2 MeV @62#. The cosmological

requirement thatVm<1 places a stricter upper limit on thsum of the masses of the three neutrino species&92h2 eV @63#. Taken together these limits imply thatleast oneZ0 resonance must occur for neutrinos todaysome energy in the range 1020 eV &Eres&331024 eV. Theexact value~s! of Eres will depend on whether or not thneutrinos have mass, and the value~s! of the neutrinomass~es! if nonzero. If the neutrino masses are nondegenate, theZ0 resonance will occur at two or three distinctEresin this range. In the case thatEres,mX , theZ0 resonance inthe VHS scenario will produce a slight decrease in the pdicted neutrino fluxes aroundEres and in the energy decadbelowEres , and an accompanying slight enhancement onspectra below;1022Eres due to secondaryZ0 decay prod-ucts. The amplitudes of all spectra should be modified

06300

-ennti-of

ionre

-

-

of

t

r-

-

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y

less than a factor of 2.~See for example Ref.@47# for thecase of degeneratemn51 eV neutrinos in other cosmicstring models.! If the relic neutrinos are nonrelativistic anclustered locally on a scale less than the attenuation lenfor photons and nucleons, the effect on the photon, proand antiproton spectra may be enhanced@64#.

Secondly, a nonzero neutrino mass may lead to osctions between neutrino species as the flux propagateEarth. In this case, the total number of neutrinos is preserbut the ratio of the flux in each species may depend onneutrino energy, the distance traveled and neutrino masRecent observations by the SuperKamiokande, KamiokaMACRO and Soudan experiments ofne and nm neutrinosproduced by cosmic rays colliding with the Earth’s atmsphere strongly suggest that neutrino oscillations~at leastbetween thenm and nt flavors! occur. ~For a review of thecurrent status of neutrino oscillation experiments and msurement implications see@65# and references therein.! Theevidence for zenith-angle dependence in the SuperKamkande data further strengthens this interpretation. Similathe long-standing observed deficit of solar neutrino flux mbe resolved byne oscillation intonm or nt although the solarneutrino experiments are not yet sufficiently consistentfavor a particular solution. The third set of evidence for netrino oscillations comes from the Liquid Scintillation Neutrino Detector~LSND! Collaboration accelerator experimewhich has seen evidence fornm into ne oscillations. Whencombined with the atmospheric and solar data, this worequire the existence of a fourth ‘‘sterile’’ neutrino specieThe LSND measurements have yet to be independently cfirmed. A number of new experiments are presently becommissioned with the objective of resolving these matteThe K2K, NuM-MINOS and CNGS experiments are epected to provide the definitive answer and exploration ofrelevant parameter space. In these experiments, neu

5-13

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beams will be sent from accelerators to detectors over blines of hundreds of kilometers. K2K is already taking daand the NuM-MINOS and CNGS will be operational in 200and 2005, respectively.

Ignoring for now cosmological evolution, the probabiliof a relativistic neutrino undergoing vacuum flavor transitiwhen propagating a distanceL is

Pnm→nt5sin2~2u0!sin2S Dmatm

2 L

4\cE D 5sin2~2u0!

3sin2F3.931025S Dmatm2

1023 eV2D S L

MpcD S E

eVD 21G~35!

in the 3-flavor solution. The atmospheric data to date arefit by an amplitude of sin2(u0)50.8221.0 and a mass difference ofDmatm

2 5(2 –6)31023 eV2 @65#. For models withdistant cosmological sources~e.g. TDs and AGNs! and aneutrino ratio at emission ofFne

:Fnm:Fnt

.1:2:0 ~in our

case this occurs atE!mJ), the ratio of the spectra reachinthe Earth will beFne

:Fnm:Fnt

51:1:1 in the3-flavor model;similarly, in the 4-flavor model consistent with the LSNdata, an initial emission ratio ofFne

:Fnm:Fnt

:Fns.1:2:0:0

will becomeFne:Fnm

:Fnt:Fns

51:1:1:0 atEarth@66#. Sincewe showed earlier that the current average distance tonearest long string is 2000 Mpc, even the highest eneneutrinos emitted in the present era will have undergtransition before reaching Earth. In this case, atE!mJ , boththe nm and nt flux reaching Earth should approximateequal half thenm flux expected without oscillations. In thcase of the flux emitted in previous epochs, this will onlyso provided the wavelength of the oscillation is much lethan the mean free path of thenm andnt in the Universe. Ifthe oscillation wavelength is greater than the mean free pthen the cascade structure will be determined by the origneutrino flavors at emission. The mean free path decresharply asz increases@47#. For example, a 1023 eV neutrinoat z;102 has a mean free path of;100 kpc @47#. Sincethese too are the redshifts and energies at which cascabecomes relevant, cascade development should be prednantly determined by the neutrino flavors at emission. InVHS scenario, however, the neutrino cascades have littlefluence on our regions of interest. Hence the predictedservablenm and nt VHS fluxes in the case with neutrinoscillations should be approximately half the unoscillatednmflux at E!mJ . As E→mJ/2, i.e. in the region where theinitial ne , nm , andnt VHS fluxes are of comparable magnitude, the ratio should also approachFne

:Fnm:Fnt

.1:1:1in the case with neutrino oscillations. The precise valuethe modulated neutrino fluxFn i

asE→mJ/2 will depend onthe initial absolute and relative string-produced fluxes whwe regard as uncertain by more than a factor of 2.

Additionally, one possible solution to the solar neutrideficit problem is vacuumne→nm or nt oscillations withDm(

2 ;10210 eV2 and sin22u(;0.621.0. In this case, the

06300

e-

st

heye

s

thales

ingmi-en-b-

f

h

phase of the transition probability,f(5Dm(2 L/(2\cE), is

&1 for E*1021 eV neutrinos emitted today from a lonstring at 2000 Mpc. Thus thene spectra at Earth would bedetermined byne→nm or nt oscillation belowE;1021 eVand exhibit negligiblene oscillation aboveE;1021 eV. Ifthis particular solar deficit solution is combined with the slution implied by the atmospheric dataDmatm

2 ;1023 eV2,an initial ratio at emission ofFne

:Fnm:Fnt

.1:2:0 or

Fne:Fnm

:Fnt.1:1:1 would be modulated toFne

:Fnm:Fnt

.1:1:1 reaching Earth because the wavelength fornm

→nt oscillation is much smaller than that of the oscillatiosolution associated with the atmospheric solar data. Atailed analysis of the detector signatures for various osction scenarios is presented in Refs.@67# and @68#.

Other extensions to the standard model could influethe predicted spectra. The existence of a supersymm~SUSY! sector would contribute extra decay channels infragmentation of the X particle and jets.~See@69# and refer-ences therein for possible approaches to modeling SUchannels for application to UHE decays.! The net effectwould be a skewing of the primary flux spectra to lowenergies and a decrease in the energy reprocessed inenergy cascades. For example, in the p51 TD model studiedin Ref. @53# with mX51016 GeV, including a SUSY sectodecreases the spectra of all species expected at Earth border of magnitude atE*1022mX , shifts the UHE peaks inthe spectra to lower energies by at least an order of matude and increases the amplitude of thene , nm , p and pspectra by about an order of magnitude at energies belowpeaks. The predictedE&1015 eV photon spectrum, predominantly generated by EM cascades, is unchanged thobecause the same fraction of initial X particle energy isprocessed into low energies in the EM cascades with or wout a SUSY sector in that model. In the case of the VHscenario, these remarks must be convolved with the suppsion of the UHEg ray and nucleon spectra due to the nearstring being farther than the relevant UHE attenuatlengths.

To date there is no experimental evidence for a SUsector and no unique theoretical predictions for the architure of the SUSY sector and superparticle properties sucmass. Other extensions to the standard model, for examtechnicolor or leptoquarks, which also increase the numof degrees of particle freedom would be expected to silarly skew the spectra to lower energies. As well, extensito the standard model may modify the high-energy cross stions governing the interactions of the emission with the cmic ambient media.

V. CONCLUSIONS

We have computed the ultrahigh energyg ray, neutrinoand cosmic ray fluxes in the VHS cosmic string scenariowhich the long string network loses its energy directlyparticle emission. This is in contrast to the standard cosstring model with a scale-invariant distribution of strinloops which lose energy predominantly by gravitational

5-14

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diation. The predicted particle fluxes are much larger inVHS scenario.

The predictions for the particle fluxes depend on extralating the QCD fragmentation function to energies muhigher than current accelerator energies. New physicstween the QCD scale and the scaleh5Am at which thedefects are formed could significantly affect the final partidistributions. We parametrize this uncertainty by introduca scalemJ as the energy scale to which the QCD fragmetation functions can be extrapolated and regardmJ as theinitial jet energy. The smaller the value ofmJ , the larger thenumber of jets which are generated by the initial emissfrom the cosmic string. We calculate the resulting partifluxes as a function of bothmJ andm. A new aspect of ourwork is the computation of the significant tau neutrino fluxdirectly produced by decay in the particle jets.

Our calculations show that the predicted fluxF(E) of grays, electron and muon neutrinos and cosmic rays scaleE3F(E);E3/2, as in other defect models, whereas tpresent observations show a much weaker increase witE.Hence, the most stringent constraints on topological demodels come from the highest energies for which data ex

SettingmJ5Am/2, the VHS strings withGm*10210 pro-duce electron and muon neutrinos in excess of the UHEservations. Thus, assuming that the strings evolve as inVHS scenario, models withGm*10210 or, equivalently asymmetry breaking scale ofh*1023 eV, are ruled out bythe UHE data.~This result is in accordance with the limit oGm in Ref. @27#.! At a given particle energy, the predictefluxes reaching the Earth scale asmJ

21/2 and increase withGm. Lowering mJ increases the disagreement between pdictions and observations, although the upper cutoff onpredicted spectra diminishes.

A consistent but slightly stronger constraint on the VHstring scenario comes from the cascading of the electromnetic emission off the cosmic radiation backgrounds andpotential conflict with the observed diffuse 33109 eV&E&1011 eV gamma ray background as probed by the Engetic Gamma Ray Experiment Telescope~EGRET!. This re-quiresGm&10211.

We conclude that, generically, GUT-scale strings are ruout in the VHS scenario. However, VHS models with lightGm, corresponding to lower symmetry-breaking scales, mbe contributing interestingly to the observed UHE events.reiterate that the VHS scenario is not universally acceptegiving the correct dynamics of cosmic strings.

Finally, we note that the uncertainties inherent in the cculation of the expected fluxes in any TD model mean tthe absolute and relative flux in each particle species shnot be regarded as accurate to more than an order of matude in any model. The list of uncertainties includes: tmass and other properties of the X particle~s!; the form of thepotential of the scalar field at string formation; the branchratios and decay channels of the X particle~s!; the validity ofthe extrapolated fragmentation functions; the possibilitynew particles~e.g. a SUSY sector! and other extensions tthe standard model which may influence particle productidecay and propagation; the strength of the extragalactic m

06300

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netic field; the intensity of the cosmic IR background; athe Hubble constant. Taken together, these factors havecapability to increase or decrease the absolute and relafluxes expected at Earth.

ACKNOWLEDGMENTS

We would like to thank V. Berezinsky for his commenand suggestions. We would also like to thank S. Sarkarhis comments. This work has been supported~at Brown! inpart by the U.S. Department of Energy under contract DFG0291ER40688, Task A, and J.H.M. was supported in pby NRC-NASA/JSC. U.F.W. has been supportedCENTRA-IST by ‘‘Fundac¸ao para a Cieˆncia e a Tecnologia’’~FCT! under the program ‘‘PRAXIS XXI’’ and in part byLIP-Lisbon.

APPENDIX: NEUTRINO FLUX FROM MUON DECAY

In the main text we computed the flux of particles resuing from two-body decay of charged pions in a jet. Tmuons which are produced in this decay are unstable anturn decay via a three body decay process producing etrons, muon neutrinos and anti-electron neutrinos. In thispendix we compute the full spectra neutrinos from thep6

→m6→n decay chain.Because of the finite muon rest mass, the muon neut

and muon created in the first stage of charged pion decannot carry an arbitrary fraction of the pion energy. In tlimit Ep@mm , the energies of the decay products lie btween

EP@0,a21Ep# ~A1!

for muon neutrinos and

EP@rEp ,Ep# ~A2!

for muons@35,36#. Here, r 5(mm /mp)2512a21 whereais given by Eq.~7!. The muon distribution is then

dm

dEm5aEp

21 . ~A3!

In order to obtain thenm flux, we integrate Eq.~5! overthe range~A1!, resulting in Eq.~6!. To obtain the flux ofmuon decay products, we follow the method of Ref.@70#. Inthe laboratory frame the distribution of leptons with enerE5yEm produced by the decay of a muon of energyEm isgiven by ~@70#, p. 92!

dn

dy5g0~y!2Pmg1~y! ~A4!

in the limit Em@mm ,me , where

g0~y!55

323y21

4

3y3

g1~y!51

323y21

8

3y3

for nm and

5-15

nit

dm

op-ec-

anon

xi-

d

in


g0~y!5226y214y3

g1~y!522112y218y218y3

for ne . In Eq. ~A4!, Pm is the projection of the muon spin ithe muon rest frame along the direction of the muon velocin the laboratory frame

Pm52Epr

Em~12r !2

11r

12r.

Convolving the distribution~A4! with the distribution ofmuons from pion decay~A3! and integrating over muon anpion energies, the neutrino spectrum in the laboratory frais

dN

dEn5E

En

mJdEpE

Emin

Ep dEm

Em

dN

Ep

dm

Em

dm

y,

Emin5min@rEp ,En#. ~A5!

For simplicity we will first evaluate Eq.~A5! for an initialpion distribution of the form

dN

dEp5KEp

2a , a.0.

Changing the order of integration,

EEn

mJdEpE

Emin

EpdEm→E

En

mJdEmE

Em

Em /r

dEp ,

the result is

dN

dEn5KEn

2a 12r a

a~12r ! F f 01f 1

12r F11r

22ar

a21 S 12r a21

12r a D G G ~A6!

where

f 052~a15!

a~a12!~a13!2

5

3a S En

mJD a

13

~a12! S En

mJD a12

24

3~a13! S En

mJD a13

f 152~12a!

a~a12!~a13!2

1

3a S En

mJD a

13

~a12! S En

mJD a12

28

3~a13! S En

mJD a13

for nm and

f 0512

a~a12!~a13!2

2

a S En

mJD a

16

~a12! S En

mJD a12

24

~a13! S En

mJD a13

06300

y

e

f 1512~a21!

a~a11!~a12!~a13!1

2

a S En

mJD a

212

~a11! S En

mJD a11

118

~a12! S En

mJD a12

28

~a13! S En

mJD a13

for ne . Because muons and antimuons are created withposite spin polarization in charged pion decays, this sptrum also applies to the antiparticle decay chain.

The distribution of pions produced in QCD jet decay cbe approximated by the polynomial fragmentation functi~6!. Evaluating Eq.~A6! for the appropriate values ofK anda and extending the analysis toa,0, we find the final dis-tribution of neutrinos produced by jet decay to be appromately

dNnm1 nm

dEnm

.5

8mJF12.4810.44S Enm

mJD 23/2

26.12S Enm

mJD 21/2

27.16S Enm

mJD 1/2G , Enm

<mJ

a,

wherea is given in Eq.~7!, for the muon neutrinos producein the first stage of the pion decay,

dNnm1 nm

dEnm

.5

8mJF11.8310.48S Enm

mJD 23/2

25.80S Enm

mJD 21/2

27.33S Enm

mJD 1/2

10.97S Enm

mJD 2

20.15S Enm

mJD 3G , Enm

<mJ ,

for the muon neutrinos produced by the decay of muonsthe pion decay, and

dNne1 ne

dEne

.5

8mJF12.6810.47S Ene

mJD 23/2

25.96S Ene

mJD 21/2

28.30S Ene

mJD 1/2

20.11S Enm

mJD 1

11.67S Enm

mJD 2

20.45S Enm

mJD 3G , Ene

<mJ ,

for the final electron neutrinos.

5-16

re,

ys

tt

k

Le

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-ic

v.

d,

on,’’

ev.

ky,

v. D

al

b-

art.

alY.-

ics


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Documents

High energy neutrino, photon, and cosmic ray fluxes from VHS cosmic strings