ISSN 0021-3640, JETP Letters, 2009, Vol. 89, No. 3, pp. 97–101. © Pleiades Publishing, Ltd., 2009.Original Russian Text © A.V. Kuznetsov, N.V. Mikheev, A.A. Okrugin, 2009, published in Pis’ma v Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, 2009, Vol. 89, No. 3,pp. 115–120.

97

Numerical simulations of a supernova explosionencounter two main obstacles [1–5]. First, a mecha-nism stimulating the damping shock, which is likelynecessary for the explosion, has not yet been welldeveloped. Recall that shock damping is mainly due toenergy losses on the dissociation of nuclei. Second, theenergy release of the “theoretical” supernova explosionis much lower than the observed kinetic energy~10

51

erg of an envelope. This is called the FOE (ten tothe Fifty One Ergs) problem. It is believed that a self-consistent description of the explosion dynamicsrequires an energy of ~10

51

erg to be transferred viasome mechanism from the neutrino flux emitted fromthe supernova central region to the envelope.

Dar [6] proposed a possible way for solving theabove-mentioned problems. His mechanism is based onthe assumption that the neutrino has a magneticmoment that is not very small. Left-handed electronneutrinos

ν

e

intensely generated in the collapsingsupernova core are partially converted into right-handed neutrinos due to the interaction of the neutrinomagnetic moment with plasma electrons and protons.In turn, the right-handed neutrinos sterile with respectto weak interactions freely leave the central part of thesupernova if the neutrino magnetic moment is not toolarge,

µ

ν

< 10

–11

µ

B

, where

µ

B

is the Bohr magneton.Some of these neutrinos can be inversely converted toleft-handed neutrinos due to the interaction of the neu-trino magnetic moment with the magnetic field in thesupernova envelope. According to contemporary views,the magnetic field in this region can be up to about the

critical magnetic field

B

e

= /

e

�

4.41

×

10

13

G

1

oreven higher [7–9]. The born again left-handed neutri-nos can transfer additional energy to the envelope byvirtue of the beta-type absorption

ν

e

n

e

–

p

.

In our opinion, the mechanism of the double conver-sion of neutrino chirality should be analyzed more care-fully. It was shown in our recent work [10] that the fluxand luminosity of right-handed neutrinos from the cen-tral region of the supernova were significantly underes-timated in previous works. Here, we reconsider the pro-cess

ν

L

ν

R

ν

L

under supernova conditions andanalyze the possibilities for stimulating the dampingshock.

The neutrino spin flip

ν

L

ν

R

under physical con-ditions corresponding to the central region of the super-nova has been studied in a number of works (see, e.g.,[11–13]; a more extended reference list is given in[10]). The process is possible due to the interaction ofthe Dirac-neutrino magnetic moment with a virtualplasmon, which can be both generated and absorbed:

ν

L

ν

R

+

γ

*,

ν

L

+

γ

*

ν

R

. (1)

In [11], the neutrino spin flip was described in terms ofscattering by plasma electrons and protons (

ν

L

e

–

ν

R

e

–

and

ν

L

p

ν

R

p

, respectively) in a supernova coreimmediately after the collapse. However, the importantpolarization effects of the plasma on the photon propa-gator were not considered in that work. Instead, thephoton dispersion was taken into account phenomeno-

1

Hereafter, we use the natural system of units in which

c

=

�

= 1and

e

> 0 is the elementary charge.

me2

Dirac-Neutrino Magnetic Moment and the Dynamicsof a Supernova Explosion

A. V. Kuznetsov*, N. V. Mikheev, and A. A. Okrugin

Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000 Russia* e-mail: avkuzn@uniyar.ac.ru

Received August 26, 2008; in final form, October 22, 2008

The double conversion of neutrino chirality

ν

L

ν

R

ν

L

has been analyzed for supernova conditions,where the first stage is due to the interaction of the neutrino magnetic moment with plasma electrons and pro-tons in the supernova core, and the second stage, due to the resonance spin flip of the neutrino in the magneticfield of the supernova envelope. It is shown that, in the presence of the neutrino magnetic moment in the range10

–13

µ

B

<

µ

ν

< 10

–12

µ

B

and a magnetic field of ~10

13

G between the neutrinosphere and the shock-stagnationregion, an additional energy of about 10

51

erg, which is sufficient for a supernova explosion, can be injectedinto this region during a typical shock-stagnation time.

PACS numbers: 13.15.+g, 14.60.St, 97.60.Bw

DOI:

10.1134/S0021364009030011

98

JETP LETTERS

Vol. 89

No. 3

2009

KUZNETSOV

et

al

.

logically by introducing the so-called thermal mass ofa photon into the propagator. The above-mentionedeffects were analyzed more consistently in [12, 13],where the effect of a high-density astrophysical plasmaon the photon propagator was taken into account usingthe thermal field-theory formalism. However, an analy-sis of works [12, 13] showed that they concerned onlythe electron component of the plasma, namely, only thechannel

ν

L

e

–

ν

R

e

–

, and only the electron contribu-tion to the photon propagator, whereas the proton com-ponent of the plasma was not analyzed at all. Thisseems to be even stranger because the plasma-electronand proton contributions to the neutrino spin flip are ofthe same order according to [11].

A consistent analysis of processes (1) with neutrino-chirality conversion due to the interaction with bothplasma electrons and protons via a virtual plasmon andwith the inclusion of polarization effects of the plasmaon the photon propagator was given in [10]. In particu-lar, according to the numerical analysis, the contribu-tion of the proton component of the plasma is notmerely significant, but even dominant. As a result,using the data on supernova SN1987A, a new astro-physical limit was imposed on the electron-neutrinomagnetic moment:

(2)µν 0.7–1.5( ) 10 12– µB.×<

This is a factor of two better than previous constraints.

In particular, the function (

E

) determining theenergy spectrum of right-handed neutrinos was calcu-lated in [10]. In other words, this function specifies thenumber of right-handed neutrinos emitted per 1 MeV ofthe neutrino energy spectrum per unit time from unitvolume of the central region of a supernova:

(3)

In addition, the function (

E

) determines thespectral density of the energy luminosity of a supernovacore via right-handed neutrinos:

(4)

Here,

V

is the volume of the neutrino-emitting region.

The function

d

/

dE

calculated in [10] is plotted inFig. 1 for the neutrino magnetic moment

µ

ν

= 3

×

10

−

13

µ

B

. On the one hand, this value is too small toaffect the supernova dynamics. On the other hand, it issufficiently large to provide the required luminositylevel. In accordance with the existing supernova models(see, e.g., Fig. 11 in [14]), a significant part of thesupernova core material has a fairly high temperature.For example, according to the model developed in [15],typical temperatures are 20–30 MeV. The model pro-posed in [16] predicts even higher temperatures. Theenergy distributions of the right-handed neutrino lumi-nosity are plotted in Fig.1 for the temperatures

T

= 35,25, 15, and 5 MeV, the electron and neutrino chemicalpotentials

�

300 MeV and

�

160 MeV, and the

neutrino-emitting volume

V

�

4

×

10

18

cm

3

.To obtain a total energy of about 10

51

erg extractedfrom the supernova central part by right-handed neutri-nos in a time of about 0.2 s, the integral luminosity ofthese neutrinos should be about

(5)

An analysis shows that such luminosity can be gener-ated by the considered process of neutrino-chiralityconversion if the neutrino magnetic moment is notlarger than refined upper limit (2) pointed out in [10].The table illustrates the neutrino magnetic moments forwhich luminosity level (5) is achieved for any of theabove-mentioned temperatures.

If the energy of right-handed neutrinos was con-verted into the energy of left-handed neutrinos, e.g.,due to the well-known mechanism of spin oscillations,then an additional energy of about 10

51

erg would beinjected into the supernova envelope in a typical time ofabout a few tenths of a second.

ΓνR

dnνR

dE----------

E2

2π2--------ΓνR

E( ).=

ΓνR

dLνR

dE----------- V

dnνR

dE----------E V

E3

2π2--------ΓνR

E( ).= =

LνR

µe µνe

LνR � 4 1051 erg/s.×

Fig. 1. Energy distributions of the luminosity of right-handed neutrinos for plasma temperatures T = (solid curve)35, (dashed curve) 25, (dash-dotted curve) 15, and (dottedcurve) 5 MeV and the neutrino magnetic moment µν = 3 ×10–13 µB.

Neutrino magnetic moments providing luminosity (5)

T, MeV µν/10–12 µB

35 0.29

25 0.42

15 0.64

5 0.97

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JETP LETTERS Vol. 89 No. 3 2009

DIRAC-NEUTRINO MAGNETIC MOMENT AND THE DYNAMICS 99

It was noted above that the strong dominance ofneutrino scattering by protons over scattering by elec-trons was not found in earlier studies. Hence, the possi-ble number of the right-handed neutrinos generated inthe collapse of the central region of a supernova wassignificantly underestimated.

At the same time, it is not evident that neutrino scat-tering by protons dominates over their scattering byplasma electrons. In this respect, we believe that a clearillustration of this dominance based on an analysis of asimplified case is quite expedient. The comparison ofthe typical parameters of the supernova core, where thetemperature is T � 30 MeV and the electron and neu-trino chemical potentials are � 300 MeV and �160 MeV, respectively, shows that the temperature isthe smallest physical parameter. Hence, the limitingcase of the completely degenerate plasma, T = 0, seemsto yield a reasonable estimate. It is interesting that if thetemperature is zero, the contributions from neutrinoscattering by protons and electrons to the neutrino cre-ation probability can be evaluated analytically usingEqs. (20) and (21) and the corresponding formulasfrom Appendix A in [10]. The contribution by ultrarel-ativistic electrons is given by the simple formula

(6)

where E is the energy of the generated right-handed neu-

trino, = 2α /π is the squared mass of a transverse

plasmon, and is the neutrino chemical potential.

The analytical expression describing the proton con-tribution is somewhat more complicated since itdepends additionally on the proton mass. The plasmaelectroneutrality condition for T = 0 takes the form np =

and ensures that the electron and proton Fermi

momenta are the same: = . Then, the protonchemical potential coinciding with the Fermi energy is

= = and the proton contribution isexpressed in terms of the proton Fermi velocity vF =

/ = / = / . As a result, the pro-ton contribution is given by the expression

(7)

Here, the function fp(y) has the form

(8)

µe µνe

ΓνR

e( ) E( )µν

2mγ2

2π------------ µν E–( )θ µν E–( ),=

mγ2 µe

2

µν

ne

–

kFe( ) kF

p( )

µp EFp( ) mp

2 µe2+

kFp( ) EF

p( ) µe µp µe mp2 µe

2+

ΓνR

p( ) E( )µν

2mγ2µν

2π------------------ f p y( ), y

Eµν-----.= =

f p y( )1 v F/3+1 v F–

---------------------y=

for 0 ≤ y ≤ (1 – vF)/(1 + vF) and

(9)

for (1 – vF)/(1 + vF) ≤ y ≤ 1. It is interesting that theintegral contribution from protons is independent of the

parameter vF: (y)dy = 1/2.

Note that the formal passage to the limit mp 0,i.e., vF 1 in Eqs. (7)–(9) yields fp(y) fe(y) = (1 –y)θ(1 – y), where the function fe(y) can be introduced inEq. (6) in complete analogy with Eq. (7). Thus, asexpected, Eq. (6) for the electron contribution is repro-duced.

Figure 2 shows the plots of the function fp(y) forvF = 1, 0.394, and 0. The value vF = 0.394 correspondsto the effective proton mass mp � 700 MeV in a plasmawith a nuclear density of 3 × 1014 g/cm–3 (see [3],p. 152). The value vF = 0 corresponds to the formallimit mp ∞ for which this function is also signifi-cantly simplified: fp(y) f∞(y) = yθ(1 – y).

According to Eq. (4), the spectral density of theenergy luminosity of the supernova core due to right-handed neutrinos is given by the formula

(10)

The difference between the electron and proton contri-butions to the quantity given by Eq. (10) is illustrated inFig. 3. It is clearly seen that the proton contribution tothe luminosity is increased by the factor y3.

f p y( ) 1 y–v F

-----------θ 1 y–( )=

× 11 v F–( )2

12y2v F

---------------------- 1 y–( ) 1 2y+( )–

f p0

1∫

dLνR

dE----------- V

µν2mγ

2µν4

4π3------------------y3 f e y( ) f p y( )+[ ].=

Fig. 2. Plots of the function fp(y) for various vF values. Thedependence fe(y) = (1 – y) for the electron contribution isreproduced for vF = 1 (dashed line). The value vF = 0.394(solid curve) corresponds to the effective proton mass mp �700 MeV. The case vF = 0 (dotted line) corresponds to thelimit of infinitely large proton mass.

100

JETP LETTERS Vol. 89 No. 3 2009

KUZNETSOV et al.

A comparison of Figs. 3 and 1 shows that allowancefor a nonzero temperature results in a shift of the max-imum of the energy distribution of the luminositytoward higher energies of right-handed neutrinos. Thisadditionally enhances the proton contribution.

The flux of right-handed neutrinos from a collapsingsupernova core enters the region of the envelopebetween the neutrinosphere of the radius Rν and theshock-stagnation region of the radius Rs. According tocommonly-accepted notions, the typical values of thesequantities vary only slightly during the stagnation timeand can be estimated as Rν ~ 20–50 km and Rs ~ 100–200 km. If a fairly strong magnetic field of ~1013 Gexists in the considered region, then neutrino spin oscil-lations occur and can be resonant under certain condi-tions.

The effect of the magnetic field on neutrinos withnonzero magnetic moments can be illustrated mostconveniently using the equation for neutrino-chiralityevolution in the uniform external magnetic field. Thechirality-evolution equation taking into account theadditional energy CL gained by left-handed electronneutrinos in the matter can be written as [17–23]

(11)

where

(12)

Here, the ratio ρ/mN = nB is the nucleon number densityand Ye = ne/nB = np/nB, = /nB, are the num-ber densities of electrons, protons, and neutrinos,respectively, B⊥ is the transverse magnetic-field compo-nent with respect to the direction of neutrino motion,

i∂∂t-----

νR

νL⎝ ⎠⎜ ⎟⎛ ⎞

E00 µνB⊥

µνB⊥ CL⎝ ⎠⎜ ⎟⎛ ⎞

+νR

νL⎝ ⎠⎜ ⎟⎛ ⎞

,=

CL

3GF

2---------- ρ

mN

------- Ye43---Yνe

13---–+⎝ ⎠

⎛ ⎞ .=

Yνenνe

ne p νe, ,

and the term proportional to the identity matrix isinsignificant for our analysis.

Expression (12) for the additional energy of left-handed neutrinos is worthy of special analysis. It isimportant that this quantity can vanish in the consid-ered region of the supernova envelope. In turn, this is acriterion for the resonance transition νR νL. Sincethe neutrino number density in the supernova envelopeis fairly low, the quantity in Eq. (12) is negligible.This yields the resonance condition in the form Ye = 1/3.Note that Ye in the supernova envelope is ~0.4–0.5,which is typical of collapsing material. Nevertheless,the shock causing the dissociation of heavy nucleimakes the material more transparent to neutrinos. As aresult, the so-called “short-term” neutrino burst is gen-erated and the material in this region is significantlydeleptonized. According to conventional notions, acharacteristic dip down to ~0.1 is observed in the radialdistribution of the quantity Ye (see, e.g., [2, 4]). Thequalitative behavior of Ye(r) is shown in Fig. 4. Thus, apoint at which Ye = 1/3 certainly exists. It is remarkablethat only one such point with dYe/dr > 0 exists (see[2, 4]).

Note that Ye = 1/3 is a necessary, but insufficientcondition for the resonant conversion of right-handedneutrinos into left-handed ones, νR νL. The so-called adiabatic condition should also be satisfied. Themeaning of this condition is that the diagonal elementCL in Eq. (11) should be at least no larger than the off-diagonal element µνB⊥ if the distance from the reso-nance point is about the oscillation length. This leads tothe criterion [22]

(13)

E0

Yνe

µνB⊥ � dCL

dr---------⎝ ⎠

⎛ ⎞1/2

� 3GF

2---------- ρ

mN

-------dYe

dr---------⎝ ⎠

⎛ ⎞ 1/2

.

Fig. 4. Qualitative behavior of the function Ye(r) at 0.1–0.2 safter the shock formation with a typical dip caused by ashort-term neutrino burst (see, e.g., [4]). The dashed linecorresponds to Ye = 1/3.

Fig. 3. Energy distribution of the contributions from(dashed curve) electrons and protons with (solid curve)mp � 700 MeV and (dotted line) mp ∞ to the luminos-ity of right-handed neutrinos for T = 0.

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JETP LETTERS Vol. 89 No. 3 2009

DIRAC-NEUTRINO MAGNETIC MOMENT AND THE DYNAMICS 101

The typical parameters of the considered region areas follows (see [2, 4]):

(14)

The magnetic field ensuring the resonance conditionis

(15)

The mean free path with respect to beta-processesfor a neutrino with an estimated energy of Eν ~ 100–200 MeV is

(16)

i.e., left-handed neutrinos are almost completelyabsorbed in the considered region.

Thus, our analysis shows that the Dar mechanism ofthe double conversion of neutrino chirality νL νR νL exists under the following not very severeconditions: the Dirac-neutrino magnetic momentshould be in the range 10–13 µB < µν < 10–12 µB and amagnetic field of ~1013 G should exist in the regionRν < R < Rs. In this case, an additional energy of about

(17)

is injected into this region during the shock-stagnationtime ∆t ~ 0.2–0.4 s. This energy is sufficient to solve theproblem.

We are grateful to M.I. Vysotskiœ for helpful discus-sion. This work was supported by the Council of thePresident of the Russian Federation for Support ofYoung Scientists and Leading Scientific Schools(project no. NSh-497.2008.2), the Ministry of Educa-tion and Science of the Russian Federation (Program“Development of the Scientific Potential of the HigherEducation,” project no. 2.1.1/510), and the RussianFoundation for Basic Research (project no. 07-02-00285a).

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µν--------------------⎝ ⎠

⎛ ⎞×

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Translated by A. Serber