Diffusion of neutrinos in proto-neutron star matter with quarks

7 June 2001

Physics Letters B 509 (2001) 10–18www.elsevier.nl/locate/npe

Diffusion of neutrinos in proto-neutron star matter with quarks

Andrew W. Steiner, Madappa Prakash, James M. LattimerDepartment of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY 11794-3800, USA

Received 31 January 2001; received in revised form 9 March 2001; accepted 13 March 2001Editor: W. Haxton

Abstract

Neutrino opacities important in the evolution of a proto-neutron star containing quark matter are studied. The results forpure quark matter are compared with limiting expressions previously derived, and are generalized to the temperatures, neutrinodegeneracies and lepton contents encountered in a proto-neutron star’s evolution. We find that the appearance of quarks inbaryonic matter drastically reduces the neutrino opacity for a given entropy, the reduction being sensitive to the thermodynamicconditions in the mixed quark–hadron phase. 2001 Elsevier Science B.V. All rights reserved.

PACS: 97.60.Jd; 21.65.+f; 13.15.+g; 26.60.+c

A general picture of the early evolution of a proto-neutron star (PNS) is becoming well established [1–6].Neutrinos are produced in large quantities by electroncapture as the progenitor star collapses, but most aretemporarily prevented from escaping because theirmean free paths are considerably smaller than theradius of the star. During this trapped-neutrino era,the entropy per baryons is about 1 through mostof the star and the total number of leptons perbaryonYL = Ye + Yνe 0.4. The neutrinos trappedin the core strongly inhibit the appearance of exoticmatter, whether in the form of hyperons, a Bose(pion or kaon) condensate or quarks, due to the largevalues of the electron chemical potential. As the starcools, the neutrino mean free path increases, andthe neutrinos eventually leak out of the star, on atimescale of 20–60 s. During deleptonization, neutrinodiffusion heats the matter to an approximately uniform

E-mail address: [email protected](M. Prakash).

entropy per baryon of 2. If the strongly interactingcomponents consist only of nucleons, the maximumsupportable mass increases. In the case that hyperons,a Bose condensate (pion, kaon) or quarks appear inthe core as the neutrinos leave, the maximum massdecreases with decreasing leptonic content. In thiscase, neutron stars which have masses above themaximum mass for completely deleptonized matterare metastable, and will collapse into a black holeduring deleptonization. Alternatively, if the mass ofthe neutron star is sufficiently small, the star remainsstable and cools within a minute or so to temperaturesbelow 1 MeV as the neutrinos continue to carry energyaway from the star.

The way in which this picture is modified whenthe core of a PNS contains deconfined quark mat-ter is only beginning to be investigated [2,7–9]. Inhis seminal paper, Iwamoto [10] noted that the non-degenerateν mean free path in cold quark matter isabout ten times larger than in nucleonic matter. Wefind that in PNS matter, in which quarks appear to-wards the end of deleptonization, similarly large en-

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00434-8

A.W. Steiner et al. / Physics Letters B 509 (2001) 10–18 11

hancements persist even up to the largest relevant tem-peratures (∼ 30–40 MeV[9]), inasmuch as quarks re-main largely degenerate. On this basis, it can be antic-ipated that the presence of quark matter increases theneutrino fluxes while simultaneously decreasing thedeleptonization time, relative to matter without quarks.In work to be reported elsewhere, we explore the pos-sibility that such a change might be detected from aGalactic supernova in current and planned neutrino de-tectors. This would have direct implications for thetheoretical understanding of the high-density regimeof QCD which is inaccessible to high energy Relativis-tic Heavy-Ion Collider experiments, and, currently, tolattice QCD calculations at finite baryon density.

To perform detailed simulations of the neutrinosignal from a PNS containing quark matter, as hasalready been done for matter containing nucleons,hyperons and/or a kaon condensate [1–6], consistentcalculations of neutrino interactions in hot lepton-rich matter containing quarks are required. It is mostlikely that quarks exist in a mixed phase with hadrons[7,9,11]. Steiner et al. [9] recently showed that thetemperature of an adiabate decreases as a functionof density in a mixed phase of quarks and nucleons.Becauseν-cross sections usually scale withT 2, thissuggests that the presence of quark matter mightinfluence the neutrino signal of a PNS with quarks.

In this work, we calculate the diffusion coefficientsof neutrinos in a mixed phase of hadrons and quarksfor the temperatures, neutrino degeneracies, and lep-ton contents likely to be encountered in the evolutionof a PNS with quarks. We demonstrate that the crosssections for scattering and absorption of neutrinos bynucleons, leptons, and quarks are reduced to two inte-grals, whose integrands are products of simple polyno-mials and thermal distribution functions. The limitingbehaviors of the cross sections, for non-degenerate anddegenerate neutrinos, respectively, are compared withprevious calculations [10] in the case of pure quarkmatter. For simulations of PNS evolution using thediffusion approximation, diffusion coefficients, whichare energy weighted averages of neutrino cross sec-tions, are required for matter in which quarks exist ina mixed phase with hadrons. We examine the relevantdiffusion coefficients for two thermodynamic condi-tions especially germane to PNS evolution. The firstsituation is when neutrinos are trapped,s ≈ 1, andthe total lepton content of the matterYL = Ye + Yνe

(which measures the concentrations of the leptons perbaryon) is approximately 0.4. We also consider the sit-uation when neutrinos have mostly left the star and thematter has been diffusively heated (Yν ≈ 0, s ≈ 2). Wediscuss the impact these results might have upon theevolution of a PNS which contains quarks in a mixedphase.

For the ν-energies of interest, the neutral andcharged currentν-interactions with the matter in aPNS are well described by a current–current La-grangian [12]

L= G2F√2

(ψν(1)(1− γ5)ψp3(3))

(1)× (ψp2(2)(V −Aγ5)ψp4(4)) + H.C.,

whereV andA are the vector and axial-vector cou-pling constants (see Table 1) andGF 1.17 GeV−2

is the Fermi weak coupling constant. The subscriptsi = 1,2,3, and 4 on the four-momentapi denote theincoming neutrino, the incoming lepton, baryon orquark, the outgoing neutrino (or electron), and theoutgoing lepton, baryon or quark, respectively. Thecharged current reactions contribute to absorption ofneutrinos by baryons or quarks and scattering of neu-trinos with leptons of the same generation. The neutralcurrent interactions contribute to scattering of neutri-nos with leptons and baryons or quarks. The chargedcurrent contribution to neutrino–lepton scattering inthe same generation can be transformed into the neu-tral current form, which modifies the constantsV andA for that case. For completeness, the values ofVandA for electron neutrinos are given in Table 1. Thecorresponding values for reactions with electron anti-neutrinos are obtained by the replacementA → −A.

From Fermi’s golden rule, the cross section per unitvolume (or inverse mean free path) is

σ

V= g

∫d3p2

(2π)3

∫d3p3

(2π)3

∫d3p4

(2π)3

×Wf if2(1− f3)(1− f4)(2π)4

(2)× δ4(p1 + p2 − p3 − p4),

where the degeneracy factorg is 6 (3 colors× 2spins) for reactions involving quarks while it is 2(2 spins) for baryons of a single species. The Fermi–Dirac distribution functions are denoted byfi =[1 + exp(Ei−µi

T)]−1, whereEi andµi are the energy

12 A.W. Steiner et al. / Physics Letters B 509 (2001) 10–18

Table 1The standard model charged and neutral current vector and axial-vector couplings of neutrinos to leptons, baryons, and quarks;θC is theCabbibo angle (cosθC = 0.973),θW is the weak mixing angle (sin2 θW = 0.231), andgA = 1.23 is the baryon axial-vector coupling constant

1+ 2→ 3+ 4 V A

Charged current νe +µ− → νµ + e− 1 1

νl + n→ l− +p 12 cosθC

12gA cosθC

νl + d → l− + u cosθC cosθC

νl + s → l− + u sinθC sinθC

Neutral current νe + e− → νe + e− 12 + 2sin2 θW

12

νe +µ− → νe +µ− − 12 + 2sin2 θW

12

νe + n→ νe + n − 12 − 1

2gA

νe + p → νe + p 12 + 2sin2 θW

12gA

νe + u→ νe + u 12 − 4

3 sin2 θW12

νe + d → νe + d 12 + 2

3 sin2 θW − 12

νe + s → νe + s 12 + 2

3 sin2 θW − 12

and chemical potential of particlei. The transitionprobability Wf i , summed over the initial states andaveraged over the final states, is

Wf i = G2F

E1E2E3E4

[(V +A)2(p1 · p2)(p3 · p4)

+ (V −A)2(p1 · p4)(p3 · p2)

(3)− (V2 −A2)(p1 · p3)(M2M4)

].

Utilizing d3pi = p2i dpi dΩi = piEi dEi dΩi and

integrating overE4, Eq. (2) may be cast in the form

σ

V= gG2

F

32π5

∞∫M2

dE2

∞∫0

dE3SE3

E1| p2|| p4|

(4)

× [(V +A)2Ia + (V −A)2Ib − (

V2 −A2)Ic],

where S = f2(1 − f3)(1 − f4), Mi is the mass ofparticlei, and

Ia =∫

dΩ2dΩ3dΩ4

(5)× δ3(p1 + p2 − p3 − p4)(p1 · p2)(p3 · p4).

The integralsIb andIc are defined similarly toIa , andall can be performed analytically. Explicitly,

Ia = π2

15p1p2p3p4

(6)

× [3(P 5

max− P 5min

) − 10(A+B)(P 3

max− P 3min

)+ 60AB(Pmax− Pmin)

],

where

2A= 2E1E2 +p21 + p2

2,

2B = 2E3E4 + p23 + p2

4,

Pmin = max(|p1 − p2|, |p3 − p4|

),

(7)Pmax= min(p1 + p2,p3 + p4).

In the above expression,pi ≡ pi . Ib is defined to bethe same asIa , but with appropriate replacements

(8)Ib ≡ Ia(p2 ↔ p4,E2 ↔ −E4) ,

andIc is given by

(9)

Ic = 2π2M2M4

3p1p2p3p4

[(Q3

max−Q3min

)− 6C

(Qmax−Qmin

)],

where

2C = −2E1E3 + p21 + p2

3,


Qmin = |p1 − p3|,(10)Qmax = min(p1 + p3,p2 + p4).

In Eqs. (7), (8) and (9),E4 = E1 + E2 − E3 and

| p4| =√(E1 +E2 −E3)2 −M2

4. Eqs. (4) through (9)are the principal results of this work and enable us tocompute, for arbitrary conditions of neutrino degen-eracy and matter’s temperature, the neutrino diffusioncoefficients required in simulations of PNSs contain-ing quark matter. We note that similar techniques wereemployed to calculateν-emissivities in cold catalyzedneutron stars in Refs. [13,14].

In limiting cases when the neutrinos are eitherdegenerate or non-degenerate, and the quarks, whichare always degenerate in PNSs, are massless, simpleanalytical expressions for the cross section may beobtained by replacing momenta by Fermi momentaand energies by chemical potentials in the integralsIa , Ib, and Ic . For the sake of comparing suchlimits with the general results obtained from Eq. (4),we record various limiting forms obtained earlier inRefs. [10,15].

(1) Scattering of degenerate neutrinos. The result isthe same as for neutrino–electron scattering:

σS

V= G2

Fµ32

5π3

[(E1 −µ1)

2 + π2T 2](xE1

µ2

)1/2

(11)× [(V2 +A2)(10+ x2) + 5(2VA)x

],

wherex = min(E1,µ2)/max(E1,µ2). Here, and inthe following, we have removed the factor 1− f1 =(1 + e−(E1−µ1)/T )−1 from [10,15] to obtain transportmean free paths.

(2) Scattering of non-degenerate neutrinos. Theinverse scattering mean free path is

(12)σS

V= G2

FE31µ

22

5π3,

when it is additionally assumed thatµ2 is largecompared toE1.

(3) Absorption of degenerate neutrinos.

σA

V= 2G2

Fµ33

5π3µ21

(10µ2

4 + 5µ4µ3 +µ24

)

(13)× [(E1 −µ1)

2 + π2T 2].(4) Absorption of non-degenerate neutrinos. The

general result is greatly simplified by additionally as-suming that the quark chemical potentials are modified

by perturbative gluon exchange:

(14)σA

V= 16

π4αcG2FpF2pF3pF4

[E2

1 + π2T 2].Representative cross sections from Eq. (4) are com-

pared with the limiting forms in Eqs. (11)–(14) inFig. 1. Degenerate neutrinos are assumed in this fig-ure to haveµν T and non-degenerate neutrinosare assumed to haveµν ≈ 0. In the regions wherethey were expected to be valid, namelyEν/T 1for degenerate absorption and scattering, and also non-degenerate scattering, andEν ≈ T for non-degenerateabsorption, the limiting forms give adequate represen-tations of the general results. However, significant de-viations occur in the cases of non-degenerate absorp-tion whenEν = T , and for degenerate absorption andnon-degenerate scattering whenEν T . The devia-tion for non-degenerate absorption is due to the ne-glect ofpν in the momentum conservation conditionin Eq. (2) in Ref. [10] (which is appropriate for coldcatalyzed stars, but not for hot matter in PNSs), whichlimits its applicability to the regionEν ≈ T . The othertwo deviations are simply due to the assumption inRef. [10] thatEν T .

The weak interaction timescales of neutrinos aremuch smaller than the dynamical timescale of PNSevolution, which is on the order of seconds. Thus,until neutrinos enter the semi-transparent region, theyremain close to thermal equilibrium in matter. Hence,neutrino propagation may be treated in the diffusionapproximation with the differential equations for theflux of energy (Fν) and lepton number (Hν) [5]:

Hν = −T 2e−Λ−φ

6π2

[D4

∂(T eφ)

∂r+ (

T eφ)D3

∂η(r)

∂r

],

Fν = −T 3e−Λ−φ

6π2

[D3

∂(T eφ)

∂r+ (

T eφ)D2

∂η(r)

∂r

],

whereΛ and φ are general relativistic metric func-tions, η = µν/T andD2, D3, andD4 are diffusioncoefficients decomposed as

D2 =Dνe2 +D

νe2 , D3 =D

νe3 −D

νe3 ,

(15)D4 =Dνe4 +D

νe4 + 4D

νµ4 .

The transport ofµ andτ neutrinos and anti-neutrinosare well approximated [1] by assuming that theycontribute equally toD4 and are represented byDνµ .These coefficients are defined in terms of the energy


Fig. 1.νe cross sections per unit volume in pure quark matter, forT = 5 MeV. Solid lines show complete results from Eq. (4), and dashed linesindicate limiting forms from [10]. The labels “Degenerate” and “Non-Degenerate” refer to neutrinos. Upper panels show results for two timesnuclear matter density, and the lower panels show results for a neutrino energy of 50 MeV (µ is the chemical potential of the incoming quark).

dependent diffusion coefficientDp(Eν) by

(16)Dpn =

∞∫0

dx xnDp(E1)f (E1)[1− f (E1)

],

wherex =E1/T , and the superscriptp denotes eitherthe electron neutrino, the anti-electron neutrino, orthe µ and τ neutrinos and their antineutrinos. Inturn, the energy dependent diffusion coefficient isobtained directly from the cross sections per unitvolume through

(17)

(Dp(E1)

)−1 = 1

1− f1

[ ∑r=(p,L)

σr

V+ χ

∑r=(p,H)

σr

V

+ (1− χ)∑

r=(p,Q)

σr

V

],

where the(p,L), (p,H), and (p,Q) represent thesum over all the reactions of particlep with leptons,hadrons, or quarks, respectively. The factor(1−f1)

−1

ensures detailed balance, andχ is the volume fractionof matter in the hadronic phase.

Note that in the case of scattering, the Pauli block-ing factor corresponding to the outgoing neutrino isomitted, since the neutrino distribution function is notalways known a priori unless a full transport schemeis employed. It is possible, however, to devise a sim-plified scheme [5] in which the dependence on theneutrino distribution function is minimized. Such ascheme is valid only when scattering from light parti-cles (either electrons or quarks) does not dominate theopacity. Our results below show that this requirementis indeed met, because absorption dominates over scat-tering by a factor of 2 to 5 at all densities interior to thecentral densities of PNSs investigated here.


We describe neutron star matter at finite densityand temperature using the Gibbs phase rules [9,11].The conditions of baryon number density and chargeconservation in the mixed phase are

nB = χnHB + (1− χ)nQB ,

0 = χnHc + (1− χ)nQc + nLc ,

wherenB andnc are the baryon number and chargedensities, respectively;H , Q, andL denote hadrons,quarks, and leptons. Since the dynamical time scaleis much longer than the weak interaction time scale,beta equilibrium implies that the various chemicalpotentials satisfy the relations

µe −µνe = µµ −µνµ,

(18)µB = biµn − qi(µe −µνe ),

wherebi and qi are the baryon number and electriccharge of the hadron or quark of speciesi. Whenthe neutrinos are trapped, the electron lepton numberYL = (ne + nνe )/nB is initially fixed at a value 0.4as suggested by collapse calculations.

Hadronic matter is described using a field-theoreti-cal description in which nucleons interact via the ex-change ofσ , ω, andρ mesons. The meson–nucleoncouplings and the couplings of theσ self-interactionterms are determined by reproducing the empiri-cal properties of nuclear matter,EB = −16.0 MeV,M∗/M = 0.6, K = 250 MeV,asym = 32.5 MeV, andn0 = 0.16 fm−3. Quark matter is described usingthe MIT Bag model, with a bag constant ofB =200 MeV/fm3. (Similar results are obtained with four-quark interactions in the Nambu–Jona-Lasinio model[9].) The mixed phase is assumed to be homogeneous.For more details of the calculation of the EOS, seeRef. [9].

The cross sections per unit volume (or inverse meanfree paths) forνe scattering and absorption are shownin Fig. 2 for the two stages of PNS evolution describedbefore. It is important to recall that these curves aredrawn under conditions of fixed entropy. (Constant en-tropy adiabates shown in Fig. 3 of Ref. [9] are help-ful to gain insights into the behavior of the cross sec-tions shown here.) The individual contributions fromthe different reactions in the pure nucleon and quarkphases (thin lines) and in the mixed phase (thick lines)are marked in this figure. The labels “νs trapped” and“ν-poor” refer to the stage where the neutrino mean

free path is much less than the radius and the leptonfraction is constant, and the stage when the neutrinosfreely leave the star and have zero chemical potential,respectively. The vertical dashed lines show the centraldensities of 1.4M and the maximum mass configura-tions, respectively. Notice that quarks exist only in themixed phase; the pure quark phase occurs at densitiesabove the central densities of maximum mass stars inall cases shown here.

In general, for a given density and temperature,the pure quark-phase opacity (or equivalently thecross section per unit volume) is less than that ofhadrons due to the former’s smaller matrix elementssampled by a relativistic phase space. In addition, fora given entropy and density, pure quark matter favorsa lower temperature than hadronic matter [9]. It isnatural, therefore, that within the mixed phase regionof hadrons and quarks, the net cross section per unitvolume either flattens or decreases with increasingdensity. The reduction of opacities from that of thepure hadronic phase is enhanced in theν-poor case,reflecting the more extreme decrease of temperatureacross the mixed phase region in that case [9]. Theprecise density dependence of the net neutrino opacitydepends upon the details of the mixed phase.

Note that the total absorption cross section is largerthan the scattering cross section for both degenerateand non-degenerate situations. As discussed above,this justifies our approximate treatment of scatteringin the calculation of the diffusion coefficients.

The diffusion coefficients most relevant for the PNSsimulations, in matter with and without a mixed phaseof hadrons and quarks, areD2 andD4, and these areshown in Fig. 3. Insofar as absorption dominates scat-tering, the behavior of these coefficients can be under-stood qualitatively by utilizing the limiting forms forneutrino absorption in the degenerate (Eq. (13)) andnon-degenerate (Eq. (14)) cases, respectively. The ac-tual behavior is somewhat more complicated, but thisassumption will suffice for a qualitative interpretationof Fig. 3. In this case, the leading behaviors may beextracted to be

(19)

D2 ∝ λ(µν/T )(µν/T )2 and D4 ∝ λ(µν/T = 0),

where the mean free pathλ = (σ/V )−1. In theseequations,D2 is evaluated under conditions of extreme


Fig. 2. νe cross sections with various particles in matter containing a mixed phase of quarks and hadrons (n0 = 0.16 fm−3). The left panelsshow scattering cross sections for neutrinos with the indicated incoming hadrons, quarks, or leptons. Thick lines show the extent of the mixedphase region. The right panels show absorption cross sections on nucleons and quarks. The upper panels correspond to the neutrino-trappedera whens = 1 andYL = 0.4, and the lower panels to the time following deleptonization whens = 2 andYν = 0. The vertical dashed lineslabelledu1.4 andumax indicate the central densities of aMG = 1.4M star and the maximum mass star (MG = 2.22M for the upper panelsandMG = 1.89M for the lower panels), respectively.

neutrino degeneracy andD4 is evaluated assumingthat µν = 0. Thus, bothD2 and D4 should simplyreflect the inverse behavior of the cross section per unitvolume, which decreases with increasing density inthe pure phases, but increases within the mixed phaseregion.

Concerning the evolution of a PNS, we expect thatthe initial star, which is lepton rich, will not havean extensive mixed phase region. Only after severalseconds of evolution will quark matter appear. Inthe newly-formed mixed phase region, the neutrinoopacity will be substantially smaller than in the case inwhich a mixed phase region does not appear. However,due to the largeν-optical depth of the PNS, neutrinosremain trapped, and no significant effect on emergent

neutrino luminosities is expected at early times. Asthe star evolves, however, the relatively larger increasein opacity (note the increases inD4 relative toD2)and the growing extent of the mixed phase regioneventually allows a larger flux of neutrinos, andthereby a more rapid evolution.

In summary, we have calculated neutrino opacitiesfor matter containing quarks for the temperatures, neu-trino degeneracies and lepton contents relevant forPNS simulations, employing Gibbs phase rules to con-struct a mixed hadron–quark phase. We find that, inthe presence of quarks, neutrinos have a significantlysmaller opacity and hence larger diffusion coefficientsthan those in purely hadronic matter at similar densi-ties. These differences may have an observable impact


Fig. 3. Diffusion coefficients for the neutrino-trapped era (left panel) and hot deleptonized era (right panel). Thick lines show the extent of themixed phase. Solid lines correspond to matter with a mixed phase, and dashed lines to matter containing only nucleons. The vertical dashedlines have the same meaning as in Fig. 2.

on the neutrino flux from PNSs containing quark mat-ter, but these differences are not expected to becomeapparent until the PNS is 10–20 s old. Simulations ofPNSs with a mixed phase of hadrons and quarks areunder investigation [16]. The influence of heteroge-neous structures [17,18] and superfluidity [8] in matterwill be addressed in future work.

Acknowledgements

We acknowledge research support from the US De-partment of Energy under contracts DOE/DE-FG02-88ER-40388 (AWS and MP) and DOE/DE-FG02-87ER-40317 (JML). We thank Jose A. Pons, SanjayReddy, and Prashanth Jaikumar for useful discussions.

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Diffusion of neutrinos in proto-neutron star matter with quarks