Towards neutrino transport with flavor mixing in supernovae: theLiouville operator

C. Y. Cardallab∗

aPhysics Division, Oak Ridge National Laboratory,Oak Ridge, TN 37831-6354, United States of America

bDepartment of Physics and Astronomy, University of Tennessee,Knoxville, TN 37996-1200, United States of America

The calculation of neutrino decoupling from nuclear matter requires a transport formalism capable of handlingboth collisions and flavor mixing. The first steps towards such a formalism are the construction of neutrino andantineutrino ‘distribution matrices,’ and a determination of the Liouville equations they satisfy in the noninter-acting case. These steps are accomplished through study of a Wigner-transformed ‘density function,’ the meanvalue of paired neutrino quantum field operators.

A massive star supports itself against gravitythrough the energy generated by a succession ofnuclear fusion reactions; but once the core burnsto iron-group elements it can extract no furtherenergy from fusion, and when electron degeneracycan no longer support it, catastrophic collapse re-sults.

The halt of collapse results in a shock wavethat will eventually disrupt the entire star. Theinner part of the core collapses subsonically untilit reaches nuclear density or so; but the outer partof the core collapses supersonically, and it doesn’treceive word that the collapse of the inner corehas halted until it slams into the inner core andforms a shock wave. The shock moves outwardsbefore stalling due to energy losses from neutrinoemission and the breakup of heavy nuclei fallingthrough the shock.

The mechanism of shock revival is still un-known, but simulations exploring the mechanismhave to take account of the neutrino heating andcooling rates behind the shock. These rates de-pend on the energy and angle distributions of neu-trinos, which must be tracked as the neutrinosdecouple from the newly-born neutron star.

∗Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the DoE under contract DE-AC05-00OR22725.

Microscopic quantum mechanical effects can beincluded if the tracking of neutrino energy and an-gle distributions is handled with classical trans-port. A classical distribution function f(t,x,p)gives the differential number of neutrinos withmomenta within d3p of p at positions within d3xof x:

dN = f(t,x,p)g d3p(2π)3

d3x. (1)

(For neutrinos the spin degeneracy g is 1). Theevolution of f is given by the Boltzmann equation

pμ ∂f

∂xμ− Γi

νρ pνpρ ∂f

∂pi= C(f). (2)

The left-hand side is the Liouville operator. It isclosely related to the geodesic equations describ-ing classical particle trajectories in phase space.The first term gives the spacetime dependence,and the second term includes (for instance) par-ticle redshifts and trajectory bending due to gen-eral relativity. The collision integral on the right-hand side describes particle interactions that addor remove particles from a particular trajectory.Rates computed with (for instance) quantum fieldtheory can be used here, so that inherently quan-tum mechanical—but microscopic—phenomenasuch as particle creation and annihilation, andphase space blocking, can be included.

Nuclear Physics B (Proc. Suppl.) 188 (2009) 264–266

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doi:10.1016/j.nuclphysbps.2009.02.060

However, neutrino flavor mixing is a macro-scopic quantum effect that cannot be treated withclassical transport, which refers only to particlenumber and not amplitudes or phases.

What about the evolution of quantum occupa-tion numbers rather than classical transport? Anoccupation number n(t,q) gives the average num-ber of particles occupying momentum eigenstateq in the (effectively infinite) quantization volumeV . This is most naturally useful in the case ofspatial homogeneity. The time derivative dn/dtis given by quantum transition rates.

Noting that neutrino interaction rates haveoff-diagonal elements when written in terms of‘physical states’ (or mass eigenstates), Dolgov[1] expanded the occupation number to a matrixρ(t, |q|) in flavor/mass space in a study of neu-trinos in the (homogeneous and isotropic) earlyuniverse.

Keep in mind the differences between classicaldistribution functions f(t,x,p) and quantum oc-cupation numbers n(t,q). The momentum of aclassical particle is here labeled by p, while a mo-mentum eigenstate is labeled by q. In the classi-cal limit of a quantum description, p might be thecentroid of a wave packet superposition of eigen-states q.

However, in evolving the neutrino occupationmatrix in the context of the early universe, theclassical Liouville operator was employed to ac-count for the redshift associated with cosmic ex-pansion. That is, Dolgov tacitly assumed the sub-stitution

dρ(t, |q|)dt

→ ∂ρ(t, |p|)∂t

− H|p|∂ρ(t, |p|)∂|p| , (3)

in which the time derivative of the quantum occu-pation matrix goes over to the Liouville operatoracting on a distribution matrix whose momentumdependence has its classical meaning.

Unlike the early universe case, neutrino trans-port in supernovae requires treatment of spatialinhomogeneity. The ‘Liouville replacement’ ofEq. (3) seems natural enough in the homogeneouscase, but the introduction of spatial dependenceshould give one pause, because of the quantummechanical incompatibility of position and mo-mentum. Nevertheless, in the limit in which the

neutrino trajectories are classical and only the fla-vor evolution is quantum mechanical, one expectsthe same terms as in the classical Liouville oper-ator to emerge, including the spatial derivativeterm.

The Liouville operators for neutrino and an-tineutrino distribution matrices also contain ef-fects of flavor mixing. In flat spacetime, and inthe absence of interactions, the distribution ma-trices are governed by

pμ ∂

∂xμρ(t,x,p) +

i2

[Δ, ρ(t,x,p)] = 0, (4)

pμ ∂

∂xμρ(t,x,p) − i

2[Δ, ρ(t,x,p)] = 0. (5)

Species indices are suppressed. Only squaredmass differences affect observable oscillation phe-nomena, and so a squared mass matrix M2 in thecommutator has been replaced by

Δ = M2 − 13Tr

(M2

). (6)

The commutator enters with opposite sign in thecase of antineutrinos if one wants the neutrinoand antineutrino distribution matrices to trans-form in the same way under a flavor/mass basischange.

How can the Liouville equations of Eqs. (4) and(5)—in particular, their spatial dependence—bederived rather than just assumed?

One way is through a covariant version of thefamiliar simple model of flavor mixing [2]. Inthis version a Schrodinger equation for a neu-trino state is written down that involves not evo-lution in time or space alone, but along a world-line with affine parameter λ. One can build adensity operator out of the neutrino states; theLiouville equation then follows from applicationof the Schrodinger equation to the kets and brasout of which the density operator is constructed,together with the relation d/dλ = pμ∂/∂xμ.

But we will ultimately want a formalism inwhich collisions can be included, and as a firststep towards this, the Liouville equations can beconstructed from the equations of motion for aquantum ‘density function.’ the mean value ofpaired neutrino quantum field operators [2].

Define a density function Γ�mij (y, z) as the mean

C.Y. Cardall / Nuclear Physics B (Proc. Suppl.) 188 (2009) 264–266 265

value of a pair of normal-ordered neutrino quan-tum field operators:

i Γ�mij (y, z) =

⟨Nν�

i (y) νmj (z)

⟩. (7)

It depends on spacetime positions y and z, andhas mass indices i and j and spinor indices � andm. The N denotes normal ordering. Writing thequantum field operator as ν(y) = A(y) + B(y) interms of its positive- and negative-frequency partsA(y) and B(y), the density operator becomes

i Γ�mij (y, z)=− ⟨

Amj (z)A�

i(y)⟩+

⟨B�

i (y)Bmj (z)

⟩,(8)

in which we have separate neutrino and an-tineutrino terms (rapidly oscillating cross termsnot relevant to the macroscopic limit have beendropped).

The mixed representation of the density func-tion begins to look more like a classical distri-bution function. Introduce the average positionvariable x = (y + z)/2 and the spatial differencevariable Ξ = y − z. The mixed representationis obtained with a Wigner transformation, whichis a Fourier transformation with respect to thedifference variable:

G�mij (x, P ) =

∫d4Ξ eiP ·Ξ Γ�m

ij

(x +

Ξ2

, x − Ξ2

).(9)

Separate neutrino and antineutrino distribu-tions G�m

ij (x, p) and G�mij (x, p) are obtained from

G�mij (x, P ) by projecting out its positive- and

negative-frequency parts.In general circumstances the mixed represen-

tation provides only complementary position ormomentum distributions, when integrated overmomentum space or the volume of the system re-spectively. This is in accordance with the quan-tum mechanical incompatibility of position andmomentum.

However, a joint position/momentum distribu-tion does emerge under classical conditions. Thiscan be shown by explicitly evaluating the mixedrepresentation with respect to a state of N uncor-related wave packets [2]. After calculation, andarguments about coarse graining, and applicationof classicality conditions of weak inhomogeneityand slow time variation, a neutrino distribution

matrix built on classical trajectories emerges:

ρij(t,x,p) =N∑

a=1

(2π)3 δ3(x − xa(t)) δ3(p − pa)

× e−i(Epa,i−Epa,j)t Uαaj U∗αai. (10)

(The antineutrino distribution matrix is the sameexcept for a sign reversal in the exponential.)Here xa(t) is the classical trajectory followed bythe ath neutrino, whose wave packet centroid ispa. This expression corresponds to a system ofneutrinos born in flavors αa at t = 0; the ener-gies corresponding to the superposed mass eigen-state wave packet centroids are Epa,i. The trans-formation to a flavor-basis distribution matrixfollows directly from Eq. (7) and the relationνα(y) = Uαi νi(y) between flavor and mass fields.The diagonal elements of the flavor-basis distribu-tion matrix are just as expected for a collection ofneutrinos whose flavor content varies along classi-cal trajectories according to the familiar vacuumoscillation probabilities.

The Liouville equations follow from the differ-ence of Klein-Gordon equations

(�y − �z) Γ�m(y, z) +[Δ,Γ�m(y, z)

]= 0 (11)

obeyed by the density function. This can be seenby noting that �y − �z = 2 ∂

∂Ξ · ∂∂x in the co-

ordinates of the mixed representation: when Eq.(11) is transformed by the inverse of Eq. (9) andits positive- and negative-frequency parts are pro-jected out, Eqs. (4) and (5) are the result.

The calculations outlined here serve as a firststep towards the handling of neutrino interac-tions with a diagrammatic approach based on anonequilibrium Green’s function [3]. In additionto a Green’s function, this approach—which issometimes called ‘Keldysh theory’—involves thedensity function considered in this paper and twoother types of field operator pairings.

REFERENCES

1. A. D. Dolgov, Sov. J. Nucl. Phys. 33 (1981)700.

2. C. Y. Cardall, Phys. Rev. D 78 (2008) 085017.3. E. M. Lifshitz, L. P. Pitaevskiı, Physical Ki-

netics, Pergamon, Oxford, 1981.

C.Y. Cardall / Nuclear Physics B (Proc. Suppl.) 188 (2009) 264–266266