Table Of ContentCherenkov radiation by neutrinos
Ara N. Ioannisian1,2, Georg G. Raffelt2
1
Yerevan Physics Institute, Yerevan 375036, Armenia
2
Max-Planck-Institutfu¨rPhysik (Werner-Heisenberg-Institut), F¨ohringerRing6, 80805
Mu¨nchen, Germany
9
9
9 Abstract
1
n We discuss the Cherenkov process ν νγ in the presence of a homogeneous magnetic field.
→
a The neutrinos are taken to be massless with only standard-model couplings. The magnetic
J
field fulfillsthe dualpurposeof inducingan effective neutrino-photon vertex and of modifying
6
2 the photon dispersion relation such that the Cherenkov condition ω < k is fulfilled. For a
field strength Bcrit = m2e/e = 4.41 1013 Gauss and for E = 2me the Che|re|nkov rate is about
1 6 10−11 s−1. ×
v ×
4
2 In many astrophysical environments the absorption, emission, or scattering of neutrinos
4
occursindensemediaorinthepresenceofstrongmagneticfields[1]. Ofparticularconceptual
1
0 interest are those reactions which have no counterpart in vacuum, notably the decay γ ν¯ν
9 →
andthe Cherenkov process ν νγ. Thesereactions donotoccur in vacuumbecausethey are
9 →
kinematically forbidden and because neutrinos do not couple to photons. In the presence of a
/
h
mediumorB-field,neutrinosacquireaneffectivecouplingtophotonsbyvirtueofintermediate
p
- charged particles. In addition, media or external fields modify the dispersion relations of all
p
particles so that phase space is opened for neutrino-photon reactions of the type 1 2+3.
e
→
h If neutrinos are exactly massless as we will always assume, and if medium-induced modifi-
:
v cations of their dispersion relation can beneglected, the Cherenkov decay ν νγ is kinemat-
i ically possible whenever the photon four momentum k = (ω,k) is space-like,→i.e. k2 ω2 > 0.
X
−
Often the dispersion relation is expressed by k = nω in terms of the refractive index n. In
r | |
a this language the Cherenkov decay is kinematically possible whenever n > 1.
Aroundpulsars field strengths around thecritical value Bcrit = m2e/e = 4.41 1013 Gauss.
×
The Cherenkov condition is satisfied for significant ranges of photon frequencies. In addition,
the magnetic field itself causes an effective ν-γ-vertex by standard-model neutrino couplings
to virtual electrons and positrons. Therefore, we study the Cherenkov effect entirely within
the particle-physics standard model.
This process has been calculated earlier in [2]. However, we do not agree with their
results.
Our work is closely related to a recent series of papers [3] who studied the neutrino
radiative decay ν ν′γ in the presence of magnetic fields.
→
Our work is also related to the process of photon splitting that may occur in magnetic
fields as discussed, for example, in Refs. [4, 5].
Photons couple to neutrinos by the amplitudes shown in Figs. 1(a) and (b). We limit our
discussion to field strengths not very much larger than Bcrit = m2e/e. Therefore, we keep only
electron in the loop. Moreover, we are interested in neutrino energies very much smaller than
the W- and Z-boson masses, allowing us to use the limit of infinitely heavy gauge bosons and
thus an effective four-fermion interaction (Fig. 1(c)). The matrix element has the form
2
@
@
@@I Z γ @I(cid:3)@@@@ γ @@I γ
@@ (cid:23)(cid:19)(cid:16)(cid:20) W(cid:2)(cid:0)@@@@ @@(cid:23)(cid:19)(cid:16)(cid:20)
(cid:18)ν(cid:0)(cid:0)(cid:3)(cid:0)(cid:2)(cid:3)(cid:1)(cid:0)(cid:2)(cid:3)(cid:1)(cid:0)(cid:2)(cid:22)(cid:1)(cid:18)e (cid:17)(cid:3)(cid:21)(cid:0)(cid:2)(a(cid:3)(cid:1))(cid:0)(cid:2)(cid:3)(cid:1)(cid:0)(cid:2)(cid:1) (cid:3)(cid:3)(cid:2)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:3)(cid:0)(cid:2)(cid:3)(cid:1)(cid:0)(cid:2)(b(cid:3)(cid:1))(cid:0)(cid:2)(cid:1) (cid:18)(cid:0)ν(cid:0)(cid:22)(cid:18)e (cid:17)(cid:3)(cid:21)(cid:0)(cid:2)(cid:3)(cid:1)(c(cid:0)(cid:2))(cid:3)(cid:1)(cid:0)(cid:2)(cid:1)
ν(cid:0)(cid:18)(cid:0)(cid:2)
(cid:0)(cid:0)
Figure 1: Neutrino-photon coupling in an external magnetic field. The double line represents
the electron propagator in the presence of a B-field. (a) Z-A-mixing. (b) Penguin diagram
(only for ν ). (c) Effective coupling in the limit of infinite gauge-boson masses.
e
G
= F Zεµν¯γν(1 γ5)ν(gVΠµν gAΠµ5ν), (1)
M −√2e − −
where ε is the photon polarization vector and Z its wave-function renormalization factor. For
the physical circumstances of interest to us, the photon refractive index will be very close to
2 1
unity so that we will be able to use the vacuum approximation Z = 1. g = 2sin θ +
V W 2
1 2 1 1
and g = for ν , and g = 2sin θ and g = for ν .
FoAllowi2ng Refes. [4, 6,V7, 8] Πµν aWnd−Π2µν are A −2 µ,τ
5
e3B
Πµν(k) = (4π)2h(gµνk2−kµkν)N0− (gkµνkk2−kkµkkν)Nk+(g⊥µνk⊥2 −k⊥µk⊥ν)N⊥i,
e3
Πµν(k) = C kν(Fk)µ + C kν(kF)µ +kµ(kF)ν k2Fµν , (2)
5 (4π)2m2en− k k ⊥h ⊥ ⊥ − ⊥ io
e e e e
here Fµν = 21ǫµνρσFρσ, where F12 = −F21 = B. The k and ⊥ decomposition of the metric
is g = diag( ,0,0,+) and g = g g = diag(0,+,+,0). k is the four momentum of the
phoktoen. N0, −N⊥,Nk, C⊥ and⊥Ck are−funkctions on B,kk2 and k⊥2. They are real for ω < 2me,
i.e. below the pair-production threshold.
The four-momenta conservation constrains the photon emission angle to have the value
1 ω
2
cosθ = 1+(n 1) , (3)
n (cid:20) − 2E(cid:21)
where θ is the angle between the emitted photon and incoming neutrino. It turns out that for
all situations of practical interest we have n 1 1 [4, 9]. This reveals that the outgoing
| − | ≪
photon propagates parallel to the original neutrino direction.
Itiseasytoseethattheparity-conservingpartoftheeffective vertex(Πµν)isproportional
tothesmallparameter(n 1)2 1andtheparity-violatingpart(Πµν)isnot. Itisinteresting
5
− ≪
to compare this finding with the standard plasma decay process γ ν¯ν which is dominated
by the Πµν. Therefore, in the approximation sin2θ = 1 only the e→lectron flavor contributes
W 4
to plasmon decay. Here the Cherenkov rate is equal for (anti)neutrinos of all flavors.
We consider at first neutrino energies below the pair-production threshold E < 2m . For
e
ω < 2m the photon refractive index always obeys the Cherenkov condition n > 1 [4, 9].
e
Further, it turns out that in the range 0 < ω < 2m C ,C depend only weakly on ω so that
e k ⊥
it is well approximated by its value at ω = 0. For neutrinos which propagate perpendicular
to the magnetic field, a Cherenkov emission rate can be written in the form
4αG2E5 B 2 E 5 B 2
Γ F h(B) = 2.0 10−9 s−1 h(B), (4)
≈ 135(4π)4 (cid:18)Bcrit(cid:19) × (cid:18)2me(cid:19) (cid:18)Bcrit(cid:19)
3
where
(4/25)(B/Bcrit)4 for B Bcrit,
h(B) = ≪ (5)
(cid:26)1 for B Bcrit.
≫
Turning next to the case E > 2m we note that in the presence of a magnetic field the
e
electronandpositronwavefunctionsareLandaustatessothattheprocessν νe+e− becomes
→
kinematically allowed. Therefore, neutrinoswithsuchlargeenergieswillloseenergyprimarily
by pair production rather than by Cherenkov radiation (for recent calculations see [10]).
The strongest magnetic fields known in nature are near pulsars. However, they have a
spatial extent of only tens of kilometers. Therefore, even if the field strength is as large as the
critical one, most neutrinos escaping from the pulsar or passing through its magnetosphere
will not emit Cherenkov photons. Thus, the magnetosphere of a pulsar is quite transparent
to neutrinos as one might have expected.
Acknowledgments
It is pleasure to thanks the organizers of the Neutrino Workshop at the Ringberg Castle for
organizing a very interesting and enjoyable workshop.
References
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