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Can a neutron star be a giant MIT bag?
Volume 62B, number 2
PHYSICS LETTERS
CAN A NEUTRON
24 May 1976
STAR BE A GIANT MIT BAG?*
G. BAYM and S.A. CHIN
Department of Physics, Universityof Illinois at Urbana-Champaign, Urbana,Illinois 61801, USA Received 30 March 1976 We show, on the basis of the M.I.T. bag model of hadrons, that a neutron matter-quark matter phase transition is energetically favorable at densities around ten to twenty times nuclear matter density. It is unlikely, however, that quark matter can be found within stable neutron stars, or that it may form a third family of dense stellar objects.
The success of the quark model in classifying the hadronic spectrum [ 1] and in interpreting leptonnucleon scattering data [2,3] strongly suggests that hadrons are bound states of quarks. This picture of hadrons as composite particles is potentially important to our understanding of dense matter inside neutron stars; in particular it suggest a possible phase transition from neutron matter to a uniform quark liquid at high density [4]. Intuitively one expects such a detocalization transition (analogous to a Mott metal-insulator transition**) to take place when the nucleons overlap sufficiently that the individual quarks fail to recognize to which nucleons they belong. In this letter we shall estimate, using the M.I.T. bag model of hadrons [5,6] the density at which this transition occurs. As we shall see, it is unlikely that the transition density is sufficiently low for quark matter to exist in stable neutron stars. In the prevailing picture of hadronic structure [3,7], hadrons are composed of spin-I/2, fractionally charged, point-like quarks - three per baryon, and one quark plus one antiquark per meson• There are four (possibly more) types or flavors of quarks (u, d, s and c), and each comes in three colors. Quarks are generally light [3] (with the possible exception of the c-quark), with masses < 1 GeV. They interact weakly with each other at short (i.e., intra-nucleon), distances but are not separable into solitary, free states. The mechanism by which quarks are confined within hadrons remains conjectural at present. * Research supported in part by National Science Foundation Grants MPS74-22148, GP 40395 and DMR75-22241. ** Only here the transition is characterized by a change in the "color conductivity" of the matter from zero in the nucleon phase to a finite value in the quark phase.
Among the various models incorporating these features [ 5 - 8 ] , the M.I.T. bag model provides one of the simpler descriptions of quark confinement, adequate for a first estimate of the phase transition. In this model the quarks in a hadron are assumed confined to a finite region of space, known as the "bag"; the bag volume is limited by the introduction o f a term in the hadron energy equal to the volume of the bag times a constant B > 0. Quarks inside the bag are coupled via the exchange of eight massless color SU(3) vector gluons in the usual Yang-Mills fashion: Z?int = ffgc/2);kab~ai7 ~biAu where ~baiis the quark field of color a and flavor i, the A m are the gluon fields (a = 1 .... 8), Xab are the SU(3) generators and gc is the color coupling constant. The parameters of the theory, B, gc, a certain zero-point energy term (irrelevant for our consideration), and the mass of the strange quark, are determined by fitting the masses of four known hadrons [6]. The u, d quarks are taken to be massless. The resultant values for B and gc, as taken from ref. [6], are B 1/4 = 145 MeV and (gc/2)2/47r = 0.55. To determine a phase transition from nucleonic to quark matter one must compare the energies per baryon at fixed volume per baryon in the two phases. Let us consider N nucleons in a volume V. The corresponding quark phase has N i quarks of flavor i(= u, d) with energy density/9 = E/V given by •
~
-"
12
Ot
19 = B + ~ (3/4~r 2 +g2/8rr4)p}i i
(1)
where pfi = Or2fiN/V)I/3, fi = Ni/N and ~i fi = 3. The three terms above are the bag energy, the massless quarks' kinetic energy, and the lowest order color interaction (or exchange) energy [9] respectively. For the quark phase of neutron matter: fu = 1,fd = 2; for 241
Volume 62B, number 2
PHYSICS LETTERS
2.4
MEAN IELD
>. 2.0 1.6 i
z w
1.2 0.8 0.4 0.14
o,
0.27
0.49
0.83 I
I
I
I
o!,'o'.2 ;3 o14 ;'5 o16 o!7 o B 09
,o
V/N (frnS/boryon)
Fig. 1. Energy per baryon as a function of specific volume for the quark phase of neutron matter ("Bag") and two bracketing calculations of neutron matter ("Mean Field" and "Reid"). For details, consult text. nuclear matter: fu = fd = 3/2. The energy per baryon in the quark phase is thus E / N = B V/N + D ( N / V ) 1/3 ,
(2)
with D - 3 z r 2 ( l + g2/6rr2) Z i f4/3. The corresponding equation of state is P = - ~ E / ~ V = 5(0 : - 4B).
(3)
Eq. (2) has a minimum, corresponding to zero pressure, 104~
___
//~REID
g
,,oo
,;5
,o6
,;,
MASS DENSITY (g/cm 3)
Fig. 2. Equation of state for the quark phase of neutron matter ("Bag") and two representative calculations of neutron matter ("Mean Field" and "Reid"). For details, consult text.
242
24 May 1976
at baryon density n - N / V = (3B/D) 3/4 with E / N = 4B1/4(D/3) 3/4. For the quark phase of neutron matter, using the above values of B and gc" this minimum occurs at n = 0.19 baryon/fm 3 with E / N - Mnc2 = 306 MeV, where M n is the nucleon mass. Eq. (2) is given in fig. 1, for densities n > 1 fm -3, by the curve "bag". The corresponding equation of state is given in fig. 2. The various calculations in the literature of the neutron matter energy and equation of state differ somewhat in their predictions of the high density behaviour. For the present study, we select four as being representative: 1) Pandharipande's variational calculation employing Reid's potential [10], 2) Bethe and Johnson's calculation using their own model V neutron potential [ 11,12], 3)Pandharipande and Smith's neutron solid calculation with tensor interaction [ 13], and 4) Walecka's calculation based on a relativistic mean theory [14,15]. Bethe and Jonhson's potential is an imporvement over Reid's in that its repulsive core is adjusted to agree more closely with the observed properties of the co-meson. Pandharipande and Smith's calculation is of interest in case neutron matter turns solid before dissolving into quarks. Walecka's calculation is noteworthy in that he did not use static poten{ials, but rather obtained the energy by linearizing a fully relativistic field theory in which neutrons are assumed to interact via a scalar and a vector meson exchange. The results of the first and the last of the above four calculations are shown in figs. 1 and 2, labelled "Reid" and "Mean Field" respectively. The results of the remaining two, BJVN and PS Solid, generally lie between these two, and are not shown for clarity. As we see in fig. 1 the quark phase has lower energy than the neutron phase at higher density. The density discontinuity at the first order phase transition between the two phases is determined by making the standard "double tangent" construction, shown in the two cases in fig. 1 as a dashed line. At the tangent points (indicated by the numbers above the axis) the pressures (negative of the slope) are equal in the two phases, as are the baryon chemical potentials (y axis intercept of the tangent). A neutron-quark phase transition seems inescapable since all neutron matter calculations yield E / N o: n as n ~ oo (the expected result if the short range nucleon repulsion is dominated by an co-meson type exchange), while for quark matter, eq. (2) gives E / N o: nl/3 in the same limit. (A similar conclusion holds for nuclear matter.) In fig. 2 the phase transition
Volume 62B, number 2
PttYSICS LETTERS
24 May 1976
Table 1 Neutron matter-quark matter phase transition for various models of pure neutron matter. The final two colums are the mass and central density of the maximum mass neutron star, for the given equations of state. The references are to the neutron star calculations. Eqs. of state
PT ( 103 MeV fm -a)
n T (bary. fm -3)
PT ( 10ts gcm-3)
Mmax/M.~
Pc ( 1015 gcm-3 )
Meand field [15] PS Solid [13] Bag [181 BJVN[12] Reid [10]
1.3 1.9 5.8 7.5
1.2 - 2.0 1.7 - 2.6 3.3--5.9 3.7 - 7.1
3.5 -- 7.5 5.4 - 11 13 - 3 1 14 - 40
2.57 2.28 1.93 1.76 1.66
1.6 1.1 2.1 3.3 4.1
is shown by the dotted line connecting the equation of state o f the two phases. Note that the phase transition does not occur when the pressure versus mass density curves (fig. 2) cross (a condition used for equation of state matching in ref. [ 16] ) but rather it takes place at considerably higher density, particulady in the Reid potential case. The details o f the phase transition for all four neutron matter equations o f state are summarized in table 1. The second columns gives the pressure PT at the transition; columns 3 and 4 delineate the transition region in terms of baryon and mass density respectively. Columns 5 and 6 give the maximum neutron star mass and its central density when the corresponding equation of state of column 1 is used to integrate the TOV equations. By comparing columns 4 and 6 we observe that in all cases considered, the neutron-quark phase transition lies above the maximum density obtainable in a stable neutron star. Hence quark matter is unlikely to be expected within these dense stellar objects. Furthermore, since the slope of the equation of state decreases from P/pc 2 ~ 1 to P/oc 2 ~ 1/3 during the phase transition, Gerlach's analysis [19] would imply that such a transition will also not yield a third regime of dense "quarkstars", a possibility discussed in refs. [20, 2 1 ] . In general, one may not assume that nonstrange quarks are strictly massless. When they are massive, eq. (2) is replaced by
E/N = B V/N + (V/N) (3/8rr 2) X ~. m 4 {xir/i(2xi
+
1 ) - ln(xi + r / i ) } - (V/N)(g2/4rr 4)
l
X ~m4{x4-a[xir/i-ln(xi+r/i)]2 } i
(4)
where m i is the i flavor quark mass, x i =Pfilmi, r/i =(1 + x 2 ) l / 2 , and each term has the same interpretation as before [e.g. 9 ] . For fixed values of B, gc, and density, and for m i not too massive (m i
243
Volume 62B, number 2
PHYSICS LETTERS
References [1] J.J.J. Kokkedee, The quark model (Benjamin, New York, 1969). [2] F.J. Gilman, Proc. XVII Int. Conf. High Energy Physics, London, IV-149 (1974). [3] D.H. Perkins, Contemp. Phys. 16 (1975) 173. [4] J.C. Collins and M.J. Perry, Phys. Rev. Letters 34 (1975) 1353. [5] A. Chodos et al. Phys. Rev. D9 (1974) 3471. [6] T. De Grand et al., Phys. Rev. D12 (1975) 2060. [7] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [8] W.A. Bardeen et al., Phys. Rev. D l l (1975) 1094. [9] G. Baym and S.A. Chin, Nucl. Phys. A (in press).
244
24 May 1976
[10] V.R. Pandharipande, Nucl. Phys. A178 (1971) 123. [11] H.A. Bethe and M.B. Johnson, Nucl. Phys. A230 (1974) 1. [12] R.C. Malone, M.B. Johnson and H.A. Bethe, Ap. J. 199 (1975) 741. [13] V.R. Pandharipande and R.A. Smith, Nucl. Phys. A237 (1975) 507. [14] J.D. Walecka, Ann. Phys. 83 (1974)491. [15] S.A. Chin and J.D. Walecka, Phys. Lett. 52B (1974) 24. [16] K. Brecher and G. Caporaso, Nature 259 (1976) 377. [17] G. Baym, C. Pethick and P. Sutherland, Ap. J. 170 (1971) 299. [ 18] John Neatrour, private communication. [19] U.H. Gerlach, Phys. Rev. 172 (1968) 172. [20] J.R. Ipser, M.B. Kislinger and P.D. Morley, EFI preprint 75-38 (University of Chicago). [21] N. Itoh, Prog. Theor. Phys. 44 (1970) 291.
PHYSICS LETTERS
CAN A NEUTRON
24 May 1976
STAR BE A GIANT MIT BAG?*
G. BAYM and S.A. CHIN
Department of Physics, Universityof Illinois at Urbana-Champaign, Urbana,Illinois 61801, USA Received 30 March 1976 We show, on the basis of the M.I.T. bag model of hadrons, that a neutron matter-quark matter phase transition is energetically favorable at densities around ten to twenty times nuclear matter density. It is unlikely, however, that quark matter can be found within stable neutron stars, or that it may form a third family of dense stellar objects.
The success of the quark model in classifying the hadronic spectrum [ 1] and in interpreting leptonnucleon scattering data [2,3] strongly suggests that hadrons are bound states of quarks. This picture of hadrons as composite particles is potentially important to our understanding of dense matter inside neutron stars; in particular it suggest a possible phase transition from neutron matter to a uniform quark liquid at high density [4]. Intuitively one expects such a detocalization transition (analogous to a Mott metal-insulator transition**) to take place when the nucleons overlap sufficiently that the individual quarks fail to recognize to which nucleons they belong. In this letter we shall estimate, using the M.I.T. bag model of hadrons [5,6] the density at which this transition occurs. As we shall see, it is unlikely that the transition density is sufficiently low for quark matter to exist in stable neutron stars. In the prevailing picture of hadronic structure [3,7], hadrons are composed of spin-I/2, fractionally charged, point-like quarks - three per baryon, and one quark plus one antiquark per meson• There are four (possibly more) types or flavors of quarks (u, d, s and c), and each comes in three colors. Quarks are generally light [3] (with the possible exception of the c-quark), with masses < 1 GeV. They interact weakly with each other at short (i.e., intra-nucleon), distances but are not separable into solitary, free states. The mechanism by which quarks are confined within hadrons remains conjectural at present. * Research supported in part by National Science Foundation Grants MPS74-22148, GP 40395 and DMR75-22241. ** Only here the transition is characterized by a change in the "color conductivity" of the matter from zero in the nucleon phase to a finite value in the quark phase.
Among the various models incorporating these features [ 5 - 8 ] , the M.I.T. bag model provides one of the simpler descriptions of quark confinement, adequate for a first estimate of the phase transition. In this model the quarks in a hadron are assumed confined to a finite region of space, known as the "bag"; the bag volume is limited by the introduction o f a term in the hadron energy equal to the volume of the bag times a constant B > 0. Quarks inside the bag are coupled via the exchange of eight massless color SU(3) vector gluons in the usual Yang-Mills fashion: Z?int = ffgc/2);kab~ai7 ~biAu where ~baiis the quark field of color a and flavor i, the A m are the gluon fields (a = 1 .... 8), Xab are the SU(3) generators and gc is the color coupling constant. The parameters of the theory, B, gc, a certain zero-point energy term (irrelevant for our consideration), and the mass of the strange quark, are determined by fitting the masses of four known hadrons [6]. The u, d quarks are taken to be massless. The resultant values for B and gc, as taken from ref. [6], are B 1/4 = 145 MeV and (gc/2)2/47r = 0.55. To determine a phase transition from nucleonic to quark matter one must compare the energies per baryon at fixed volume per baryon in the two phases. Let us consider N nucleons in a volume V. The corresponding quark phase has N i quarks of flavor i(= u, d) with energy density/9 = E/V given by •
~
-"
12
Ot
19 = B + ~ (3/4~r 2 +g2/8rr4)p}i i
(1)
where pfi = Or2fiN/V)I/3, fi = Ni/N and ~i fi = 3. The three terms above are the bag energy, the massless quarks' kinetic energy, and the lowest order color interaction (or exchange) energy [9] respectively. For the quark phase of neutron matter: fu = 1,fd = 2; for 241
Volume 62B, number 2
PHYSICS LETTERS
2.4
MEAN IELD
>. 2.0 1.6 i
z w
1.2 0.8 0.4 0.14
o,
0.27
0.49
0.83 I
I
I
I
o!,'o'.2 ;3 o14 ;'5 o16 o!7 o B 09
,o
V/N (frnS/boryon)
Fig. 1. Energy per baryon as a function of specific volume for the quark phase of neutron matter ("Bag") and two bracketing calculations of neutron matter ("Mean Field" and "Reid"). For details, consult text. nuclear matter: fu = fd = 3/2. The energy per baryon in the quark phase is thus E / N = B V/N + D ( N / V ) 1/3 ,
(2)
with D - 3 z r 2 ( l + g2/6rr2) Z i f4/3. The corresponding equation of state is P = - ~ E / ~ V = 5(0 : - 4B).
(3)
Eq. (2) has a minimum, corresponding to zero pressure, 104~
___
//~REID
g
,,oo
,;5
,o6
,;,
MASS DENSITY (g/cm 3)
Fig. 2. Equation of state for the quark phase of neutron matter ("Bag") and two representative calculations of neutron matter ("Mean Field" and "Reid"). For details, consult text.
242
24 May 1976
at baryon density n - N / V = (3B/D) 3/4 with E / N = 4B1/4(D/3) 3/4. For the quark phase of neutron matter, using the above values of B and gc" this minimum occurs at n = 0.19 baryon/fm 3 with E / N - Mnc2 = 306 MeV, where M n is the nucleon mass. Eq. (2) is given in fig. 1, for densities n > 1 fm -3, by the curve "bag". The corresponding equation of state is given in fig. 2. The various calculations in the literature of the neutron matter energy and equation of state differ somewhat in their predictions of the high density behaviour. For the present study, we select four as being representative: 1) Pandharipande's variational calculation employing Reid's potential [10], 2) Bethe and Johnson's calculation using their own model V neutron potential [ 11,12], 3)Pandharipande and Smith's neutron solid calculation with tensor interaction [ 13], and 4) Walecka's calculation based on a relativistic mean theory [14,15]. Bethe and Jonhson's potential is an imporvement over Reid's in that its repulsive core is adjusted to agree more closely with the observed properties of the co-meson. Pandharipande and Smith's calculation is of interest in case neutron matter turns solid before dissolving into quarks. Walecka's calculation is noteworthy in that he did not use static poten{ials, but rather obtained the energy by linearizing a fully relativistic field theory in which neutrons are assumed to interact via a scalar and a vector meson exchange. The results of the first and the last of the above four calculations are shown in figs. 1 and 2, labelled "Reid" and "Mean Field" respectively. The results of the remaining two, BJVN and PS Solid, generally lie between these two, and are not shown for clarity. As we see in fig. 1 the quark phase has lower energy than the neutron phase at higher density. The density discontinuity at the first order phase transition between the two phases is determined by making the standard "double tangent" construction, shown in the two cases in fig. 1 as a dashed line. At the tangent points (indicated by the numbers above the axis) the pressures (negative of the slope) are equal in the two phases, as are the baryon chemical potentials (y axis intercept of the tangent). A neutron-quark phase transition seems inescapable since all neutron matter calculations yield E / N o: n as n ~ oo (the expected result if the short range nucleon repulsion is dominated by an co-meson type exchange), while for quark matter, eq. (2) gives E / N o: nl/3 in the same limit. (A similar conclusion holds for nuclear matter.) In fig. 2 the phase transition
Volume 62B, number 2
PttYSICS LETTERS
24 May 1976
Table 1 Neutron matter-quark matter phase transition for various models of pure neutron matter. The final two colums are the mass and central density of the maximum mass neutron star, for the given equations of state. The references are to the neutron star calculations. Eqs. of state
PT ( 103 MeV fm -a)
n T (bary. fm -3)
PT ( 10ts gcm-3)
Mmax/M.~
Pc ( 1015 gcm-3 )
Meand field [15] PS Solid [13] Bag [181 BJVN[12] Reid [10]
1.3 1.9 5.8 7.5
1.2 - 2.0 1.7 - 2.6 3.3--5.9 3.7 - 7.1
3.5 -- 7.5 5.4 - 11 13 - 3 1 14 - 40
2.57 2.28 1.93 1.76 1.66
1.6 1.1 2.1 3.3 4.1
is shown by the dotted line connecting the equation of state o f the two phases. Note that the phase transition does not occur when the pressure versus mass density curves (fig. 2) cross (a condition used for equation of state matching in ref. [ 16] ) but rather it takes place at considerably higher density, particulady in the Reid potential case. The details o f the phase transition for all four neutron matter equations o f state are summarized in table 1. The second columns gives the pressure PT at the transition; columns 3 and 4 delineate the transition region in terms of baryon and mass density respectively. Columns 5 and 6 give the maximum neutron star mass and its central density when the corresponding equation of state of column 1 is used to integrate the TOV equations. By comparing columns 4 and 6 we observe that in all cases considered, the neutron-quark phase transition lies above the maximum density obtainable in a stable neutron star. Hence quark matter is unlikely to be expected within these dense stellar objects. Furthermore, since the slope of the equation of state decreases from P/pc 2 ~ 1 to P/oc 2 ~ 1/3 during the phase transition, Gerlach's analysis [19] would imply that such a transition will also not yield a third regime of dense "quarkstars", a possibility discussed in refs. [20, 2 1 ] . In general, one may not assume that nonstrange quarks are strictly massless. When they are massive, eq. (2) is replaced by
E/N = B V/N + (V/N) (3/8rr 2) X ~. m 4 {xir/i(2xi
+
1 ) - ln(xi + r / i ) } - (V/N)(g2/4rr 4)
l
X ~m4{x4-a[xir/i-ln(xi+r/i)]2 } i
(4)
where m i is the i flavor quark mass, x i =Pfilmi, r/i =(1 + x 2 ) l / 2 , and each term has the same interpretation as before [e.g. 9 ] . For fixed values of B, gc, and density, and for m i not too massive (m i
243
Volume 62B, number 2
PHYSICS LETTERS
References [1] J.J.J. Kokkedee, The quark model (Benjamin, New York, 1969). [2] F.J. Gilman, Proc. XVII Int. Conf. High Energy Physics, London, IV-149 (1974). [3] D.H. Perkins, Contemp. Phys. 16 (1975) 173. [4] J.C. Collins and M.J. Perry, Phys. Rev. Letters 34 (1975) 1353. [5] A. Chodos et al. Phys. Rev. D9 (1974) 3471. [6] T. De Grand et al., Phys. Rev. D12 (1975) 2060. [7] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [8] W.A. Bardeen et al., Phys. Rev. D l l (1975) 1094. [9] G. Baym and S.A. Chin, Nucl. Phys. A (in press).
244
24 May 1976
[10] V.R. Pandharipande, Nucl. Phys. A178 (1971) 123. [11] H.A. Bethe and M.B. Johnson, Nucl. Phys. A230 (1974) 1. [12] R.C. Malone, M.B. Johnson and H.A. Bethe, Ap. J. 199 (1975) 741. [13] V.R. Pandharipande and R.A. Smith, Nucl. Phys. A237 (1975) 507. [14] J.D. Walecka, Ann. Phys. 83 (1974)491. [15] S.A. Chin and J.D. Walecka, Phys. Lett. 52B (1974) 24. [16] K. Brecher and G. Caporaso, Nature 259 (1976) 377. [17] G. Baym, C. Pethick and P. Sutherland, Ap. J. 170 (1971) 299. [ 18] John Neatrour, private communication. [19] U.H. Gerlach, Phys. Rev. 172 (1968) 172. [20] J.R. Ipser, M.B. Kislinger and P.D. Morley, EFI preprint 75-38 (University of Chicago). [21] N. Itoh, Prog. Theor. Phys. 44 (1970) 291.