High-energy neutrino conversion into electron-W pair in magnetic field and its contribution

a r X i v :h e p -p h /0208111v 2 17 F e

b 2003INFNCA-TH0205

High-energy neutrino conversion into electron-W pair in magnetic

?eld and its contribution to neutrino absorption

Andrea Erdas 1,2,3,?and Marcello Lissia 2,1,?

1Dipart.di Fisica dell’Universit`a di Cagliari,S.P.Sestu Km 1,I-09042Monserrato (CA),Italy 2Ist.Naz.Fisica Nucleare (I.N.F.N.)Cagliari,S.P.Sestu Km 1,I-09042Monserrato (CA),Italy 3Department of Physics and Astronomy,The Johns Hopkins University,Baltimore,MD 21218(August 9,2002;revised November 22,2002)Abstract We calculate the conversion rate of high-energy neutrinos propagating in constant magnetic ?eld into an electron-W pair (ν→W +e )from the imagi-nary part of the neutrino https://www.360docs.net/doc/0516410496.html,ing the exact propagators in constant magnetic ?eld,the neutrino self-energy has been calculated to all orders in the ?eld within the Weinberg-Salam model.We obtain a compact formula in the limit of B ?B cr ≡m 2/e .We ?nd that above the process threshold

E (th )≈2.2·1016eV ×(B cr /B )this contribution to the absorption of neutrinos

yields an asymptotic absorption length ≈1.1m ×(B cr /B )2×(1016eV /E).

14.60.Lm,95.30.Cq

Typeset using REVT E X

I.INTRODUCTION

The study of creation,propagation,energy loss,and absorption of neutrinos in magnetic ?eld is important in several astrophysical contexts and in the early cosmology[1].Neutrino self-energy and dispersion relation is modi?ed in magnetized media[2–4],and processes where neutrinos radiate electron-positron pairs(ν→ν+e++e?)[5–7]or gammas(ν→ν+γ)[8,9,7]have been investigated in the range of energies where it is possible to use an e?ective four-fermion interaction and rates have been obtained in the limits of weak and strong magnetic?elds.As an example of the importance of macroscopic magnetic?eld as an e?ective source of energy loss for energetic neutrinos,we recall that the estimate[5]for the rateν→ν+e++e?in the strong magnetic?eld near the surface of a neutron star is about ten times the rate of pair production in the Coulomb?eld near the nucleus of metallic iron.

Conversion of neutrinos in W-lepton pairs in the presence of magnetic?elds(νl→W+l) should be considered when studying the propagation of neutrinos of su?ciently high energies: we shall show that in this limit this process gives an important contribution to neutrino absorption.A similar process,νγ→l W+,has been studied by Seckel[10],who shows that at energies above the threshold for W production this process is competitive withννscattering at the same center of mass energies.

The process we are considering,where an extremely energetic neutrino creates a real W, is a second order process in the weak coupling constant g,while the radiation of a e+e?pair through a virtual W or a virtual Z is a fourth order process.Therefore,there is an energy above which the conversion into an electron-W pair becomes the dominant process. In addition,since we use the electro-weak lagrangian and not an e?ective low-energy theory, our result is valid also at very high energies,much higher than the W mass;actually,the W-electron decay rate contributes signi?cantly to neutrino absorption only in this limit. Notice that eventually the real W decays and that in about10.5%of cases the?nal state contains a neutrino of the same?avor(e.g.,νe→W+e→νe e+e?),so that the process can

be thought of as the radiation of a lepton pair;in about21%of cases the?nal state contains a neutrino of di?erent?avor(e.g.,νe→W+e→νμμe),and in the remaining68.5%of cases the?nal state contains hadrons.

In this paper we use Schwinger’s proper time method[11]to calculate the neutrino self-energy in homogeneous magnetic?elds and then we obtain the probability of decay into W-electron pair by extracting the imaginary part of the self-energy.A similar strategy was used by Tsai and Erber[12]to extract the photon pair creation probability from the vacuum polarization in intense homogeneous magnetic?elds.

In Section2we brie?y review the notation and derive the one-loop neutrino self-energy in constant magnetic?eld[2],in Section3we obtain the imaginary part of the self-energy and the rate of W-electron pairs creation in magnetic?eld.The ensuing discussion and conclusions are in Section4.

II.NEUTRINO SELF-ENERGY IN A CONSTANT MAGNETIC FIELD

In this section we review the calculation of the one-loop neutrino self-energy in a homo-geneous magnetic?eld,using the exact fermion and gauge boson propagators in a constant magnetic?eld[2].We consider a magnetic?eld with magnitude B pointing along the pos-itive z-direction.The only non-vanishing components of the electromagnetic?eld strength tensor Fμνare F12=?F21=B.The exact expression for the electron S(x′,x′′)[11,13] and W boson Gμν(x′,x′′)[2]propagators in a constant magnetic?eld are obtained using Schwinger’s proper time method:

S(x′,x′′)=φ?(x′,x′′) d4k

e ik·(x′?x′′)Gμν(k),(2)

(2π)4

where the translationally invariant parts of the propagators are

S(k)=i ∞0ds

×exp ?is m2?i?+k2 +k2⊥tan eBs cos eBs , and

Gμν(k)=i ∞0ds eBs e?is(M2?i?)[gμν +(e2eF s)μν⊥] + kμ+kλFμλtan eBs B +i e

M2 .(4) In the rest of the paper we shall drop the in?nitesimal imaginary contribution to the masses ?i?,which determines the correct boundary conditions;if necessary it can be easily reintro-duced:m2→m2?i?and M2→M2?i?.In our notation,?e and m are the charge and mass of the electron,M the W-mass,x the gauge parameter,σ3=σ12=i

Fμνxνand,therefore,the phase factor in Eqs.(1)and(2),which is independent of 2

the integration path,is[13]

φ(x′,x′′)=exp ie x′x′′dxμAμ(x) =exp i e

sin2(eBs).(6)

B

Note that the W and Goldstone scalar propagator were obtained in Ref.[2]by introducing a new gauge?xing term(EGF)which is manifestly invariant under electromagnetic gauge transformations.The advantage of the EGF gauge is that the electromagnetic potential has no cross couplings with the W and Goldstone?elds and,therefore,these two?elds do not mix in the presence of a magnetic?eld.

In the remainder of this paper we focus our attention on electron-type neutrinos,the generalization toμandτ-neutrino is straightforward.For the purpose of this work,it would seem convenient to work in the unitary gauge(x→∞),where the unphysical scalars

disappear.However,the W-propagator is quite cumbersome in this gauge.We prefer to work in the Feynman gauge(x=1),where the expression of the propagator is much simpler. In principle the choice of the Feynman gauge carries the price of calculating an additional bubble diagram:the one with the Goldstone scalar.But this scalar bubble diagram does not contribute to leading order,since it is suppressed by a factor of m2/M2≈4.04·10?11,and can be neglected.Therefore,we only need to calculate the bubble diagram with a W-boson:

ΣW(x′,x′′)=

ig2

2

γL=

1?γ5

(2π)4

e ip·(x′?x′′)ΣW(p)(9)

as

ΣW(p)=?ig2

(2π)4 ∞0ds1cos z2e?is1(m2+q2 +q2⊥tan z1z2)×

γRγμ (m?q )e?iz1σ3?q⊥

z1+z2p ,k⊥+

tan z1

ΣW(p)=

g2

(z1+z2)sin(z1+z2)

e?i(m2/eB)[z1+z2M2/m2+(z1+z2)?0]× z2sin(z

1

+z2)p⊥

γL+(c.t.)(13)

where

?0=z1z2

m2

+

sin z1sin z2

m2

.(14)

and the appropriate counter-terms(c.t.)are de?ned such that

(c.t.)=?ΣW(p) B=0,p=0?p ?ΣW(p)

eB (z1+z2)≡

1

m2

z2

η

z2

(4π)2 eB sinβz

M2/m20du e?iz[1?ηu+u+?0]×

ηu p e iσ3βz(1+ηu)+sin[βzηu]

M2

+

sin[βz(1?ηu)]sin[βzηu]

M2

.(18) III.RATE OF W-ELECTRON PAIR CREATION

Because of the oscillating phase exp(?iz),the main contribution to the integral over z comes from the region where z<～1.If we are only interested in neutrinos that travel through

a“moderate”magnetic?eld eB?m2=eB cr,this meansβ?1,we can expand the terms in the integrand in power series ofβz?1.Furthermore,sinceΣW is quite small,of order g2eB/M2,as it can be inferred from the expression(17)or from the explicit calculation in Ref.[2],we can set p=0inΣW.

After this is all done,we obtain

ΣW(p)?

g2

M2 2p⊥γL

∞0dzz M2/m20duu 23(ηu)2

×e?iz[1+u?ηu+1m2(eB

mM2

,(20)

which can be read from the ratio of the z and z3term in the exponential of Eq.(19)or inferred from kinematical considerations and the momentum change of a virtual e?W pair in a magnetic?eld.

At low energies,ξ?1,the z3term in the exponential can be dropped,the self-energy is real and we obtain the result of Ref.[2];at high energies,ξ?1,the self-energy acquires an imaginary part that we shall calculate in the following.In this last case we writeΣW as

ΣW(p)=2

(4π)2 eB3ξ2z2u2](21)

where we have dropped all nonleading terms inηu and extended the integration in du to ∞,due to the fact that the main contribution to the integral in du comes from the region u<～1because of the oscillating phase exp(?izu)and thatη=m2/M2is extremely small.

The integral in dz of the imaginary part of the self-energy can be performed with the substitution z=y

√

3 ∞0y sin 33y3) dy(22)

obtaining

?ΣW(p)=?2(4π)2 eB√u K2/3 √ξ23 3/2 .(23) The?nal integration in du yields

?ΣW(p)=?g2M2 2 1+√eBp⊥ e?√eBp⊥,(24) and,therefore,the absorption coe?cient,α=?2p⊥?ΣW(p),is

α=g2

ˉh c m B cr

2 1+√m Mc B e?√m Mc B.(25)

From the exponential in Eq.(25)we can read the threshold for the process1:if we de?ne ξ(th)=

√

3Mc2M

B≈2.2·1016eV

B cr

12πˉh c p⊥c

M2 2=g2ˉh c

m B cr 2=0.935 p⊥c B cr 2m?1,(27)

where we have used the numerical value G F/(ˉh c)3=g2/[

√

1The kinematical threshold is obviously given by the sum of the masses M+m,below which the rate is rigorously zero:this threshold is lost in our expansion,but it is unimportant as long as the e?ective threshold,below which the process is exponentially suppressed,is very much larger than M+m.

2In principle the full result would have additional dependences on the lepton mass coming from terms that containη=(m/M)2and that we have disregarded being extremely small for electrons; these corrections are larger for muons and especially taus,but still small.

IV.DISCUSSION AND CONCLUSIONS

In Figure1we show the absorption coe?cient forνe→W+e as function of the neutrino transverse energy E≡p⊥c for several values of the magnetic?eld B:B=10?1B cr,B= 10?2B cr,B=10?3B cr,B=10?4B cr,and B=10?5B cr with B cr=m2/e=4.4·1010Gauss. It is evident that the process has an energy threshold that grows for smaller values of B, as quantitatively described by Eq.(26).If we consider the angular coe?cient,which is one, and the spacing between the curves above the threshold,we?nd thatαgrows linearly with E and quadratically with B,in agreement with Eq.(27).The same Figure1is valid forνμ(ντ),if one multiplies both the horizonal and the vertical scale times(mμ/m e)=206.768266 ((mτ/m e)=3477.6).

In Figure2we compare the processνe→W+e with the processν→ν+e++e?that has a much lower threshold;this threshold can be estimated from the dimensionless?eld dynamical parameter characteristic to this processκ≡eBp⊥/m3,which is analogous toξof Eq.(20)with the substitution M→m:the threshold is smaller by a factor of about m2/M2≈4·10?11.The curves plotted as function ofκshow clearly that the thresholds of both processes are function only of the product Bp⊥,see Eq.(26).For the process ν→ν+e++e?we use the result shown in Eq.(8)of Ref.[6]:

α(ν→νe+e?)=G2F(g2V+g2A)

B cr 2

ln(κ)?ln(3)24 ,(28)

which the authors claim valid for E?M3/eB(κ?(M/m)3≈4·1015):we plot Eq.(8) of Ref.[6],see Eq.(28),up toκ=1015.We show results only for B=0.1B cr(top curves) and B=0.001B cr(bottom curves).We see that the processν→ν+e++e?(dashed curves)dominates below the threshold ofν→W+e(solid curves),but above this threshold ν→W+e is almost two orders of magnitude larger(about a factor50).Therefore,above the threshold E≈(mM2)/eB(κ≈(M/m)2≈2·1010),the e?ective lagrangian(four fermion interaction)cannot be used.The rate ofν→ν+e++e?can instead be estimated using the rate ofν→W+e,which gives the total adsorption,times the branching ratio of W→νe+e,which is(10.66±0.20)%.

In conclusion,we have calculated the absorption rate of very high energy neutrinos in magnetic?eld.Our main result is given by the compact formula in Eq.(25)valid for electron neutrinos.The result for muon or tau neutrinos can be obtained by substituting the electron mass m with the muon or tau mass remembering that the electron mass m appears also in B cr=m2/e.

This process is exponentially suppressed,and,therefore,it can be disregarded,for en-ergies below a threshold energy inversely proportional to the magnetic?eld;for a?eld one tenth of the critical?eld this energy is of the order of1017eV,see Eq.(26)and Fig.1.

Above this threshold the absorption coe?cient grows linearly with energy and quadrat-ically with the?eld as shown in Eq.(27):for a?eld one tenth of the critical?eld and an energy of1018eV the absorption coe?cient is about one inverse meter(see Fig.1).

Above the threshold this process substitutes the radiation of e+e?pairs as dominant mechanism forνabsorption in magnetic?eld(see Fig.2).

ACKNOWLEDGMENTS

A.Erdas wishes to thank Gordon Feldman for helpful discussions and the High Energy Theory Group of the Johns Hopkins University for the hospitality extended to him during his several visits.

This work is partially supported by M.I.U.R.(Ministero dell’Istruzione,dell’Universit`a e della Ricerca)under Co?nanziamento P.R.I.N.2001.

REFERENCES

?Electronic address:andrea.erdas@ca.infn.it

?Electronic address:marcello.lissia@ca.infn.it

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FIG.1.Neutrino absorption coe?cient for the processνe→W+e in inverse meters as func-tion of the neutrino transverse energy(E≡p⊥c)in eV for?ve values of the magnetic?eld:

B=10?1B cr,B=10?2B cr,B=10?3B cr,B=10?4B cr,B=10?5B cr going from top to bottom,

with B cr=4.4·1010Gauss.The same curves apply forνμ(ντ),if both the horizonal and the vertical scale are multiplied times(mμ/m e)=206.768266((mτ/m e)=3477.6).

FIG.2.Neutrino absorption coe?cients for the processesνe→W+e,(solid curves)and ν→ν+e++e?(dashed curves)in inverse meters as function of the dimensionless characteristic

parameterκ≡eBp⊥/m3for two values of the magnetic?eld:B=10?1B cr and B=10?3B cr

going from top to bottom.