Collision-induced avor instability in dense neutrino gases with energy-dependent scattering Yu-Chia Lin1 2 3and Huaiyu Duan1y

2025-04-29 0 0 536.76KB 7 页 10玖币

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Collision-induced ﬂavor instability in dense neutrino gases with energy-dependent

scattering

Yu-Chia Lin1, 2, 3, ∗and Huaiyu Duan1, †

1Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA

2Department of Astronomy/Steward Observatory, University of Arizona, Tucson, Arizona 85721, USA

3Department of Physics, University of Arizona, Tucson, Arizona 85721, USA

(Dated: April 27, 2023)

We investigate the collision-induced ﬂavor instability in homogeneous, isotropic, dense neutrino

gases in the two-ﬂavor mixing scenario with energy-dependent scattering. We uncover a simple

expression of the growth rate of this instability in terms of the ﬂavor-decohering collision rates and

the electron lepton number distribution of the neutrino. This growth rate is common to the neutrinos

and antineutrinos of diﬀerent energies, and is independent of the mass-splitting and vacuum mixing

angle of the neutrino, the matter density, and the neutrino density, although the initial amplitude

of the unstable oscillation mode can be suppressed by a large matter density. Our results suggest

that neutrinos are likely to experience collision-induced ﬂavor conversions deep inside a core-collapse

supernova even when both the fast and slow collective ﬂavor oscillations are suppressed.

I. INTRODUCTION

Neutrinos help shape the physical and chemical evolu-

tion of the early universe, core-collapse supernovae, and

neutron star mergers where they are copiously produced.

Flavor oscillations, which alter the ﬂavor composition of

the neutrinos (see, e.g., Ref. [1] for a review), can have a

signiﬁcant impact on the physical conditions in these in-

teresting astrophysical environments. In addition to the

well-known vacuum oscillations [2, 3] and the Mikheyev-

Smirnov-Wolfenstein (MSW) eﬀect [4, 5], the ambient

neutrinos further change the refraction of the neutrinos in

these extreme environments [6–8]. As a result, the dense

neutrino gas can experience a collective ﬂavor transfor-

mation because of the tight coupling among the neutrinos

themselves [9–11]. (See, e.g., Ref. [12, 13] for reviews on

this topic and the references therein.) It has been shown

that the so-called fast ﬂavor conversions, a special type of

collective ﬂavor transformation that arises on the scales

of centimeters to meters, can take place near or even be-

low the neutrino-decoupling layer of a neutrino-emission

compact object [14, 15]. (See also Ref. [16] for a review

and the references therein.) An interesting recent devel-

opment in the research of collective neutrino oscillations

is that neutrino collisions, which are usually thought to

damp neutrino oscillations, are shown to be able to in-

duce ﬂavor conversions [17–26].

In this work, we focus on the collision-induced ﬂavor

conversions in homogeneous and isotropic neutrino gases.

We ﬁrst follow the pioneering work of Johns [20] and

solve the ﬂavor evolution in a mono-energetic neutrino

gas. By comparing the numerical and analytical solu-

tions, we demonstrate the dependence of the collision-

induced instability on the eﬀective mixing angle and the

density of the neutrino (Sec. II). We then consider the ef-

∗yuchialin@arizona.edu

†duan@unm.edu

fect of energy-dependent neutrino scattering and derive

a simple expression for the exponential growth rate of

the collision-induced ﬂavor instability for which we also

provide a few illustrative numerical examples (Sec. III).

We conclude by summarizing our results and discussing

their implications (Sec. IV).

II. MONO-ENERGETIC NEUTRINO GAS

A. Physics model

The ﬂavor content of a dense neutrino gas can be rep-

resented by the ﬂavor density matrix ρwhose diagonal

elements are the neutrino occupancies in diﬀerent weak-

interaction states and the oﬀ-diagonal elements are the

coherences among those states [27]. For a homogeneous

and isotropic environment, the ﬂavor evolution of the

neutrino gas can be solved from the following equation:

˙ρ=−i[H, ρ] + C,(1)

where Hand Care the ﬂavor-evolution Hamiltonian and

the collision term for the neutrino, respectively. In the

two-ﬂavor mixing scenario, say between the eand xﬂa-

vors, one can expand ρin terms of the 2 ×2 identity

matrix σ0and the Pauli matrices σi(i= 1,2,3) so that

ρ∝σ0P0+P·σ,(2)

where Pis the ﬂavor polarization (Bloch) vector, and P0

is proportional to the total density of the eand xﬂavor

neutrinos. The corresponding quantities ¯ρ,¯

P0, and ¯

can be deﬁned for the antineutrinos in a similar way.

We ﬁrst consider a mono-energetic neutrino gas of a

single vacuum oscillation frequency ω= ∆m2/2E, where

∆m2and Eare the mass-squared diﬀerence and energy

of the neutrino, respectively. We assume a minimal col-

lisional model that damps the ﬂavor coherence without

changing the particle numbers [28]. In this model, Eq. (1)

arXiv:2210.09218v3 [hep-ph] 26 Apr 2023

simpliﬁes as

P=ωB×P+µD×P−ΓP⊥,(3a)

P=−ωB×¯

P+µD×¯

P−¯

Γ¯

P⊥,(3b)

where B= (s2θ,0,−c2θ) in the ﬂavor basis with s2θ=

sin(2θ) and c2θ= cos(2θ), respectively, µ=√2GFn0

is the strength of the neutrino self-coupling potential,

D=P−¯

P,P⊥= (P1, P2,0) and ¯

P⊥= ( ¯

P1,¯

P2,0) rep-

resent the ﬂavor coherences of the neutrino and antineu-

trino, respectively, and Γ and ¯

Γ are the corresponding

ﬂavor-decohering neutrino collision rates.As in Ref. [20],

we approximate the matter suppression on collective neu-

trino oscillations by a small eﬀective mixing angle θ[29].

We also choose a nominal neutrino density n0

νto make P

and ¯

Pdimensionless. We focus on the physical environ-

ments where both the matter density and the neutrino

density are so large that θ,ω/µ, Γ/µ, and ¯

Γ/µ are all

much less than 1.

FIG. 1. The survival probabilities of the electron ﬂavor neu-

trino (upper panel) and antineutrino (middle panel) and the

ﬂavor coherence of the neutrino (bottom panel) as functions

of time in the mono-energetic neutrino gases with three com-

binations of the eﬀective mixing angle θand the neutrino

self-coupling strength µ(as labeled), where θ0= 10−5and

µ0= 105km−1. The horizontal dotted lines in the bottom

panel represent |Q0,1|in these scenarios (with two of them

overlapping with each other), and the slanted dotted lines are

|Q+,1e−iΩ+t|[see Eq. (12)].

As concrete examples, we solve Eq. (3) numerically

with the initial conditions P(t= 0) = Pini = (0,0,1) and

Pini = (0,0,0.8) and with ω= 0.6 km−1, Γ = 1 km−1,

Γ = 0.5 km−1, and three combinations of θand µ: (θ0,

µ0), (10θ0,µ0) and (θ0, 10µ0), where θ0= 10−5and µ0=

105km−1, respectively. We plot in Fig. 1 the survival

probabilities of νeand ¯νe, which are computed as

Pνeνe=1 + P3/P ini

2and P¯νe¯νe=1 + ¯

P3/¯

Pini

2,(4)

respectively, as well as the magnitude of the ﬂavor coher-

ence

S=P1−iP2(5)

of the neutrino.

In all three cases, the neutrino gases experience ﬂa-

vor conversions νe→νxand ¯νe→¯νxbetween 35 µs and

50 µs. While the gases start with an excess of νeand

¯νe, they end up with more νxand ¯νx. The exponential

growth of |S|between 20 µs and 40 µs conﬁrms that the

ﬂavor conversions are indeed due to some ﬂavor instabil-

ity [20]. The growth rate of |S|is largely independent

of θor µwhen Sis small. However, the ﬂavor conver-

sions are delayed by a smaller θand/or a larger value of

µ, which represent a larger matter density and a larger

neutrino density, respectively.

Although we plot the evolution of the neutrino gases

up to 100 µs in Fig. 1 for the completeness of the solu-

tion, one should note Eq. (3) is valid only in the linear

regime. Additional terms that change the populations of

diﬀerent neutrino species must be included for the com-

plete treatment in the nonlinear regime. When included,

these terms bring νeand ¯νeback to the equilibrium val-

ues after the collision-induced ﬂavor conversion subsides

[20].

B. Collision-induced ﬂavor instability

To understand the dependence of the collision-induced

ﬂavor instability on θand µ, we linearize Eq. (3) when S

and ¯

Sare small [20, 25]:

dtS

S≈ωs2θ−Pini

Pini

3+ Λ S

S,(6)

where

Λ = −ωc2θ−µ¯

Pini

3−iΓ µP ini

−µ¯

Pini

3ωc2θ+µP ini

3−i¯

Γ.(7)

Equation (6) has the solution

S(t)

S(t)=Q0+Q+e−iΩ+t+Q−e−iΩ−t,(8)

where

Q0=−ωs2θΛ−1−Pini

Pini

3,(9)

while Ω±and Q±are the eigenvalues and eigenvectors of

Λ, respectively. The amplitudes of Q±are constrained

by the initial condition

0 = S(0)

S(0)=Q0+Q++Q−(10)

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摘要：

Collision-inducedavorinstabilityindenseneutrinogaseswithenergy-dependentscatteringYu-ChiaLin1,2,3,andHuaiyuDuan1,y1DepartmentofPhysicsandAstronomy,UniversityofNewMexico,Albuquerque,NewMexico87131,USA2DepartmentofAstronomy/StewardObservatory,UniversityofArizona,Tucson,Arizona85721,USA3DepartmentofPh...

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Collision-induced avor instability in dense neutrino gases with energy-dependent scattering Yu-Chia Lin1 2 3and Huaiyu Duan1y

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