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Ground state of high-density matter at finite temperature
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
G R O U N D STATE O F H I G H - D E N S I T Y M A T I ' E R AT F I N I T E T E M P E R A T U R E Esteban R O U L E T and Daniele T O M M A S I N I International School for Advanced Studies, 1-34014 lrieste, Italy Received 31 July 1989
The possibility that the ground state of high-density matter is a phase in which the electroweak symmetry is restored, rather than ordinary nuclear matter or strange matter, is analysed at finite temperatures. For this to be possible, the temperature should be lower than a critical value 7"~_~17 MeV and the SU (2) × U ( 1) symmetric vacuum should be nearly degenerate in energy with the ordinary (Uc,,,( 1) symmetric) vacuum. In the minimal version of the standard model, this yields a stringent bound on the Higgs mass, similar to the one already obtained for zero temperature.
Nuclear matter, at temperatures 7"< 20 MeV and densities o f the order o f the nuclear saturation density no'--0.17 fm -3, is expected to show a phase transition between the v a p o r phase (gas o f nucleons, deuterons, He particles . . . . ) and the liquid phase ( o r d i n a r y nuclear m a t t e r ) [ I ]. It has also been suggested [2] that strange m a t t e r might be stable with respect to nuclear m a t t e r for virtually any value o f the baryon number, since the increase in energy due to the strange quark mass could be c o m p e n s a t e d by a decrease on the Fermi level o f up and down quarks, due to the Pauli exclusion principle. Further studies o f strange matter at high densities [3] and temperalures [4] support this idea. Also the possibility that the true ground state o f high-density m a t t e r corresponds to a phase in which the electroweak s y m m e t r y is restored has been considered [ 5,6 ]. In this case, all (massless) quark flavors contribute to reduce the Fermi level in the s y m m e t r i c phase, and this gain in energy could be sufficient to c o m p e n s a t e the increase in the Higgs potential energy required to go from the a s y m m e t r i c to the s y m m e t r i c vacuum, p r o v i d e d this increase is small enough. Our aim here is to extend this study to finite temperatures. To analyse the stability o f the electroweak s y m m e t r i c phase (hereafter EWS phase), we shall c o m p a r e its properties with those o f nuclear m a t t e r and strange matter, computed as in ref. [4]. If the EWS phase were stable, the formation o f macroscopic objects (stars) in this phase could be 444
expected. These objects can be described as non-topological solitons ( N T S ) corresponding to regions of space with high baryonic density matter, where the Higgs field takes a vanishing expectation value ( EWS p h a s e ) , i m m c r s e d in the o r d i n a r y (only U~m( 1 ) inv a r i a n t ) vacuum. Wc will take the EWS phase as c o m p o s e d o f quark matter, described as a gas o f unconfined quarks whose interactions arc taken into account phenomenologically following the bag model picture o f Q C D [7]. This phase was studied at zero t e m p e r a t u r e by Copeland, Kolb and Lec [6] *~ Inside the NTS, all fcrmions and gaugc bosuns are massless and the chemical equilibrium through elcctroweak interactions is established. The chemical equilibrium in processes such as e + c *--, c + c + y (or b °) implies that the chemical potentials u ( b °) and # ( y ) o f the neutral S U ( 2 ) and U ( I ) gauge bosuns vanish. Since a particle and its antiparticle annihilate into these neutral bosuns, their chemical potentials are opposite. In particular, for the charged gauge bosuns l L ( b + ) = - l z ( b - ) , so that the Bose-Einstein distribution for masslcss particles requires /t(b + ) = i t ( b - ) = 0 . Similarly, the process q + q *--' g leads to a vanishing chemical potential for the gluons g. Hence #(b +)=,uIb-)=!t(b
u) = / t ( y ) =/L(g) = 0 .
(1)
~,t Another possibility is that the massless quarks in the EWS phase remain confined into baryons [ 5 ]. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume 231, number 4
PHYSICS LETTERS B
We can obtain also the further constraints ll( VeL ) : ,tt( eL ) : O ,
(2a)
It (UL) =//(dr.) .
(2b)
It(UL) =II(UR) +/.I(eR) ,
(2C)
~(UL) =fl(dR) --l/(eR) ,
(2d)
where eq. (2a) expresses the fact that the neutrinos are able to escape from the NTS and the condition of chemical equilibrium in the process VL+b- '--' Ct. Similarly, the process UL+ b - ~ d , yields (2b), while the Higgs mediated processes Uc+eL ~ UR+eR and UL+dR '--' eR+VL yield (2c), (2d). Similar relations also hold for the fermions of the remaining generations. One crucial requirement for the stability o f a NTS is the existence of a non-zero conserved charge, the baryonic number in this case. The baryonic density inside the NTS is na=-~ Y. hq.
q
(3)
where the sum runs over all quark flavors of the three generations and h , - n q - n~. with the fermion densities given by
nf = :X"ff ( -d3k ~ n )~g k ,
(4)
with Nf a degeneracy factor (colour and spin ) and gk -='l I + e x p [ f l ( e - - l t ) 1 } - ' ,
(5)
w h e r e / / - ~, It and e are the temperature, chemical potential and fermion energy respectively. One may think that the particles whose mass outside the NTS is smaller than ~ 7" would escape from it. The escape o f quarks is supressed at the temperalures we are considering (T~< 100 MeV) since outside the NTS they have to be confined into hadrons. The evaporation ore* and light mesons from the surface will contribute, as the escape of neutrinos and photons, to the cooling of an isolated NTS. However, a non-vanishing fi~, is needed to ensure charge neutrality. If eR were to escape yielding to a net electric charge in the NTS, a large Coulomb energy would be generated. The only massive particles inside the NTS are the Higgs bosons. In the minimal version of the standard
16 November 1989
model there is only one Higgs doublet 45-r= ( 45+, 45o). Its mass in the EWS phase is given by M ~ = d 2 l'/d02 [o=o, where 0 is associated to the real part of the neutral Higgs component and the scalar potential including one-loop contributions [8] can be casted in the form (using that d V / d 0 = 0 in the true vacuum 0 = v - 2 4 6 GeV)
v(o)=(2A-C)v202-..lO4+('O41n(O2/v 2) .
(6)
The constant C can be expressed in terms of the masses m~ and m f o f t h e bosons and fermions (in the true vacuum ) with multiplicities ga.f:
C=
gBm~-- ~f gfm~
,
(7)
and A can be determined as a function of the ordinary Higgs mass
In~1 =
( d 2 V / d O 2 ) o = , , = 4 t ' 2 ( 3 ( ' - 2A ) .
(8)
Consequently, the Higgs mass in the EWS phase (45=0) is ,.V/~ =2v2(2A - C ) .
(9)
As it will be shown later, in the case of interest here. i.e.. when the scalar potential I" is such that the EWS phase is the ground state of high-density matter, the expression (9) is positive and it is possible to treat the 45 excitations as particles with mass M . . Moreover. we will obtain that typically M . ~ few GeV's. so that for temperatures T<< GeV the Higgs densities can be consistently ignored. Since the electroweak bosons are massless inside the NTS, they generate Coulomb like energy terms unless the net SUt.(2 ) and Uy( 1 ) charges vanish. To avoid these contributions it is then necessary that 7"3= ) ' = 0 . The T s = 0 condition is automatically satisfied, since all members of each S U ( 2 ) multiplet have the same chemical potential and hence, the contributions to "/~s cancel within each multiplet. Instead, the Y = 0 requirement gives the last constraint to the chemical potentials ( Y= 2 ( Q - T3) ) Y= ~[fi(u..)+fi(dt.)] +~fi(UR)-- ifi(dR)--2fi(eu) - [h(v~) +t~(e,.) ] = 0 .
(10)
We will study the properties of high-density matter in bulk (neglecting surface effects from the boundary of the NTS) at zero external pressure. In the ther445
Volume 231. number 4
PHYSICS LETTERS B
m o d y n a m i c a l limit, the density o f the t h e r m o d y n amical potential is to=B+
l,'o-P,
( 1I )
where B is the energy density that accounts for the non-perturbative Q C D structure o f the vacuum, following the M I T bag-model picture: I"o is the price to pay to restore the clectroweak s y m m e t r y inside the NTS, i.e.. the difference between the Higgs potential at O= 0 and at the absolute m i n i m u m ¢~= v where only the electromagnetic s y m m e t r y survives, i.e., Vo= v 4 ( ( ' - , - 1 ) . Finally, the pressure P gives the temperature dependent part. It is a sum o f contributions from all fcrmions in the three generations and all vector bosonsin SU(3)×SU(2)×U(I )
Njf / ' = 7a ~ T
d3k k 2 (2n') 3 c
1000
I
16 November 1989 [
,
[
'
i
B=(t50 MeV)"
8oo
~eV)"
~" ~"~~~..w(tso
......... \ \
6OO
',..
v=O0o ue).]"
%
"'...
"....~ "STRANGE_
'~\~.
400 "NUCLEAR MA'ITER" 200
l
gk ( ,tts ) ,
(12)
where for the bosons gk(.u) = ~ l - e x p [ / 3 ( ¢ - / t ) ] }
-t .
(13)
In the case o f massless particles with zero chemical potential, eq. ( 12 ) gives the well known results
40
r (u,v)
80
120
Fig. 1. Free energy per unit baryon as a function of temperature. Dolled lines correspond to the EWS phase, for B'/4= 150 MeV and values of I'o~/4 = 100 and 160 MeV. For comparison, the continuous line is the expectation tbr nuclear matter in Walecka's model and the dashed line corresponds to strange matter with B ~/4= 150 MeV, computed as in ref. [ 4 ].
7/-27"4
l ' ~ s ' (l~= 0 ) =,'v, 90 ~27"4
l,~r~"'(,u=0)=~N,
~
.
(14)
From the thermodynamical potential, the free energy density is obtained as 0 = co + ~ llj,§.
( 15 )
I
The equilibrium configuration is d e t e r m i n e d by m i n i m i z i n g the free energy per unit baryon number, i.e., 6 ( 0 / n , ) = 0 . This equation, together with eqs. ( 4 ) , ( 5 ) and (10), were solved numerically. Fig. 1 is a plot o f ~ / n B as a function o f the temperature for B Z / 4 = 150 MeV and values o f Vg/4= 100 and 160 MeV ( d o t t e d lines). (Typical values o f B ~/4, obtained by fitting known properties o f hadrons with MIT bag model predictions, range from 140 to 200 MeV. ) Also the predictions o f Walecka's model [ I ] for the free energy per b a D o n n u m b e r o f nuclear matter, c o m p u t e d as in ref. [4], are shown in fig. I ( c o n t i n u o u s lines). This curve takes a value = 9 2 2 MeV for small t e m p e r a t u r e s and shows an abrupt change n e a r a critical temperature 7~-~ 17 MeV, where the fall down in the free energy corresponds to the
446
transition to the nucleonic gaseous phase (the scale o f T,. is given by the binding energy per nucleon, E l ,,I= - 16 McV at T = 0 ) . It is a p p a r e n t from fig. I that for the values o f Vo and B considcrcd the EWS phase is energetically favoured for temperatures less than ~ "1"~..However, it is possible that strange matter bc energetically favoured with respect to ordinary nuclear matter. This could be the case [4] at temperaturcs below 7"~. if Bn/4<~ 155 MeV (assuming m s = 150 M c V ) . In this case, to find the ground state o f high density matter one should c o m p a r e the frcc energy per baryon o f the EWS phase with that of strange matter, so that an interplay appears between the three phases. In fig. 1 wc also present (dashed lines) the expectations for strange matter, computed as in ref. [ 4 ], for B ~/4 = 150 McV. For values o f Vo than ~ ( 160 McV )4, there is however no t e m p e r a t u r e for which the EWS phase has lower free energy than nuclear matter and strange matter. In fig. 2 we show the values o f I/g/4 (the ones ~ below the curve [.'1/4 . m . . . . as a function o f B ~/4 that yield to a stable EWS phase for T = 10 MeV. These region varies only slightly with temperature, pro-
Volume 231. number 4
PHYSICS LETTERS B T=10 MeV
200
~
I
'
I
t/,*
"NUCLEAR MATTER"
"STRANGE klATrER" 160 ~
,.o:
0/
'
\
,
I
140
160
,
B'/" (MeV)
I
180
200
Fig. 2. "'Phase diagram" for T= 10 MeV. For the values of V~/4 below the continuous line I"~/,~ the ground state of high-density matter is the EWS phase. This range diminishes for increasing values of the bag constant B. for I.o> 1/,,~, strange matter is favoured for B~/4< 155 MeV or otherwise, nuclear matter is the stable phase• vided T < 7"~, as can bc deduced from fig. I. Also the region where the strange matter phasc is stable is indicated in fig. 2. As discussed in refs. [5,6], the requirement Vo < Vma~ for the stability o f the EWS phase implies a fine tuning o f the parameters entering in the Higgs potential, and though, strong restrictions for the Higgs boson mass. For instance, in the minimal version with only one Higgs doublet that we are considering 8
i ~ 1 ~ 1 = 4 / , ' 2 ( " ' t - / ~ I' 0 .
anticipated before, in the EWS phase the Higgs mass results MH ~--m..,,JVf2 = few GeV's. Although in the m i n i m a l standard model this solution seems contrived, there is more room available in models with more Higgs fields and, also, some supergravity induced extensions o f the electroweak theory [ 10] predict that the ~ = 0 and 0 = v m i n i m a are necessarily degenerate, leading to a value o f Vo naturally small. Finally, fig. 3 shows the baryonic densities (in units of the nuclear saturation density no) o f nuclear matter (continuous line), of strange matter (dashed line) for H ~/4 = 150 MeV and o f the EWS phase ( d o t t e d lines) also for BI/4= 150 MeW and values o f V~/4= 100 and 160 MeV. The higher densities arise in the SU (2) × U ( I ) symmetric case, implying that the size o f a NTS is smaller than the corresponding one (for the same value of the total baryonic n u m b e r ) for nuclear matter or strange matter. At temperatures ~ T~, the nuclcar matter density goes to zero since in the gaseous phase nuclear matter at zero external pressure is unbounded. This is a consequence o f the Van dcr Waals like behaviour o f nucleonic interactions. In conclusion, we have considered the possibility that at finite temperatures, the ground state of high-
'
I
I
..............................................
"..., V=(100 MeV)'
2
=--2
.............
"STRANGE MATTER~ ' - ~
where, for
'
B=(150 MeV)4 .................."....• V=(180 lleV) 4
d' (16)
16 November 1989
I
,
"..
DI H <<7. U,
C_~2× 1 0 - ' [ I - (mtoo/79 G e V ) ' ] .
(17)
As a consequence, the EWS phase stability constraint I,'o< Vmax~O( (0.2 G e V ) 4) requires thc Higgs mass to be a few eV above the m i n i m u m allowed Higgs mass value [9] (corresponding to V o = 0 ) that is obtained from m ~ m = 4 v 2 C . For instance, mtop=60 GeV yields mini, -~ 6 GeV. However, for m t > 80 GeV thcre are no values o f the Higgs mass satisfying ( 15 ) for so small values o f Vo and simultaneously giving a positive value of C to make the potential V stable. As
MATTER"
\
\ L
,
40
T (ldeV)
[ BO
\
120
Fig. 3. Baryonic density in units of the nuclear saturation density no=0.17 fm -~, as a function of the temperature, for the three phases: electroweak symmetric (dotted lines), nuclear matter (continuous line) and strange matter (dashed lines) with the same values of Voand B as in fig. I. 447
Volume 231, number 4
PHYSICS LETTERS B
density m a t t e r be a phase w h e r e the e l e c t r o w e a k symm e t r y is restored, with all s t a n d a r d f e r m i o n s and gaugc bosons being massless and being in t h e r m a l and c h e m i c a l e q u i l i b r i u m . If the p o t e n t i a l cnergy differe n c c b e t w e e n the s y m m e t r i c and a s y m m e t r i c v a c u a is small c n o u g h ( V o < O ( (0.2 G e V ) 4 ) ), this phase is energetically f a v o u r e d for T~< 17 M e V and for m o s t r e a s o n a b l e values o f the bag energy B. T h e e x i s t e n c e o f such a phase c o u l d h a v e i m p o r t a n t c o s m o l o g i c a l and astrophysical c o n s e q u e n c e s . F o r instance, neutron stars m i g h t c o n v e r t into " c l e c t r o w e a k s o l i t o n s ' . T h e finite t e m p e r a t u r e b e h a v i o u r o f this phase c o u l d be r e l e v a n t in the study o f the early stages o f n e u t r o n star f o r m a t i o n or in the t h e o r y o f s u p e r n o v a e explosions. We w o u l d like to t h a n k G r a c i e l a G e l m i n i for m a n y useful d i s c u s s i o n s and Ed C o p e l a n d for help.
448
16 November 1989
References [ 1 ] J.D. Walecka, Ann. Phys. 83 (1974) 491- Phys. Lett. B 59 (1975) 109: H. Reinhardt and H. Schulz, Nuc. Phys. A 432 ( 1985 ) 630, and references therein. [21 E. Witten, Phys. Rev. D 30 (1984) 272. [3] E. Farhi and R.L. Jaffe, l'hys. Rev. D 30 (1984) 2379. [4 ] H. Reinhardt and B.V. Dang, Phys. Lett. B 202 ( 1988 ) 133. [ 5 ] S.Y. Khlebnikov and M.E. Shaposhnikov, Phys. Len. B 180 (1986) 93; Sov. J. Nucl. Phys. 45 (1987) 747. [ 6 ] E. Copeland, E.W. Kolb and K. Lee, Fcrmilab preprint 88/ 84-A (October 1988). [7] A. Chodos, R.J. Jaffe. K. Johnson, C.B. Thorn and V.F. Weiskopf, Phys. Re','. D 9 (1974) 3471; W.A. Bardeen, M.S. Chanowitz, S.D. Drell, M. Weinstein and T.M. Yan, Phys. Rev. D I1 (1974) 1094. [8] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888: S. Weinberg, Phys. Rev. I) 7 (1973) 2887: E. Gildener and S. Weinberg, Phys. Rev. D 13 (1976) 3333. [ 9 ] S. Weinberg, Phys. Rev. Lelt. 36 ( 1976 ) 294; A.D. Linde, Phys. Lett. B 70 (1977) 306. [ 10] E. Cremmer, P. Fayet and L. Girardello, Phys. Left. B 122 (1983)41.
PHYSICS LETTERS B
16 November 1989
G R O U N D STATE O F H I G H - D E N S I T Y M A T I ' E R AT F I N I T E T E M P E R A T U R E Esteban R O U L E T and Daniele T O M M A S I N I International School for Advanced Studies, 1-34014 lrieste, Italy Received 31 July 1989
The possibility that the ground state of high-density matter is a phase in which the electroweak symmetry is restored, rather than ordinary nuclear matter or strange matter, is analysed at finite temperatures. For this to be possible, the temperature should be lower than a critical value 7"~_~17 MeV and the SU (2) × U ( 1) symmetric vacuum should be nearly degenerate in energy with the ordinary (Uc,,,( 1) symmetric) vacuum. In the minimal version of the standard model, this yields a stringent bound on the Higgs mass, similar to the one already obtained for zero temperature.
Nuclear matter, at temperatures 7"< 20 MeV and densities o f the order o f the nuclear saturation density no'--0.17 fm -3, is expected to show a phase transition between the v a p o r phase (gas o f nucleons, deuterons, He particles . . . . ) and the liquid phase ( o r d i n a r y nuclear m a t t e r ) [ I ]. It has also been suggested [2] that strange m a t t e r might be stable with respect to nuclear m a t t e r for virtually any value o f the baryon number, since the increase in energy due to the strange quark mass could be c o m p e n s a t e d by a decrease on the Fermi level o f up and down quarks, due to the Pauli exclusion principle. Further studies o f strange matter at high densities [3] and temperalures [4] support this idea. Also the possibility that the true ground state o f high-density m a t t e r corresponds to a phase in which the electroweak s y m m e t r y is restored has been considered [ 5,6 ]. In this case, all (massless) quark flavors contribute to reduce the Fermi level in the s y m m e t r i c phase, and this gain in energy could be sufficient to c o m p e n s a t e the increase in the Higgs potential energy required to go from the a s y m m e t r i c to the s y m m e t r i c vacuum, p r o v i d e d this increase is small enough. Our aim here is to extend this study to finite temperatures. To analyse the stability o f the electroweak s y m m e t r i c phase (hereafter EWS phase), we shall c o m p a r e its properties with those o f nuclear m a t t e r and strange matter, computed as in ref. [4]. If the EWS phase were stable, the formation o f macroscopic objects (stars) in this phase could be 444
expected. These objects can be described as non-topological solitons ( N T S ) corresponding to regions of space with high baryonic density matter, where the Higgs field takes a vanishing expectation value ( EWS p h a s e ) , i m m c r s e d in the o r d i n a r y (only U~m( 1 ) inv a r i a n t ) vacuum. Wc will take the EWS phase as c o m p o s e d o f quark matter, described as a gas o f unconfined quarks whose interactions arc taken into account phenomenologically following the bag model picture o f Q C D [7]. This phase was studied at zero t e m p e r a t u r e by Copeland, Kolb and Lec [6] *~ Inside the NTS, all fcrmions and gaugc bosuns are massless and the chemical equilibrium through elcctroweak interactions is established. The chemical equilibrium in processes such as e + c *--, c + c + y (or b °) implies that the chemical potentials u ( b °) and # ( y ) o f the neutral S U ( 2 ) and U ( I ) gauge bosuns vanish. Since a particle and its antiparticle annihilate into these neutral bosuns, their chemical potentials are opposite. In particular, for the charged gauge bosuns l L ( b + ) = - l z ( b - ) , so that the Bose-Einstein distribution for masslcss particles requires /t(b + ) = i t ( b - ) = 0 . Similarly, the process q + q *--' g leads to a vanishing chemical potential for the gluons g. Hence #(b +)=,uIb-)=!t(b
u) = / t ( y ) =/L(g) = 0 .
(1)
~,t Another possibility is that the massless quarks in the EWS phase remain confined into baryons [ 5 ]. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume 231, number 4
PHYSICS LETTERS B
We can obtain also the further constraints ll( VeL ) : ,tt( eL ) : O ,
(2a)
It (UL) =//(dr.) .
(2b)
It(UL) =II(UR) +/.I(eR) ,
(2C)
~(UL) =fl(dR) --l/(eR) ,
(2d)
where eq. (2a) expresses the fact that the neutrinos are able to escape from the NTS and the condition of chemical equilibrium in the process VL+b- '--' Ct. Similarly, the process UL+ b - ~ d , yields (2b), while the Higgs mediated processes Uc+eL ~ UR+eR and UL+dR '--' eR+VL yield (2c), (2d). Similar relations also hold for the fermions of the remaining generations. One crucial requirement for the stability o f a NTS is the existence of a non-zero conserved charge, the baryonic number in this case. The baryonic density inside the NTS is na=-~ Y. hq.
q
(3)
where the sum runs over all quark flavors of the three generations and h , - n q - n~. with the fermion densities given by
nf = :X"ff ( -d3k ~ n )~g k ,
(4)
with Nf a degeneracy factor (colour and spin ) and gk -='l I + e x p [ f l ( e - - l t ) 1 } - ' ,
(5)
w h e r e / / - ~, It and e are the temperature, chemical potential and fermion energy respectively. One may think that the particles whose mass outside the NTS is smaller than ~ 7" would escape from it. The escape o f quarks is supressed at the temperalures we are considering (T~< 100 MeV) since outside the NTS they have to be confined into hadrons. The evaporation ore* and light mesons from the surface will contribute, as the escape of neutrinos and photons, to the cooling of an isolated NTS. However, a non-vanishing fi~, is needed to ensure charge neutrality. If eR were to escape yielding to a net electric charge in the NTS, a large Coulomb energy would be generated. The only massive particles inside the NTS are the Higgs bosons. In the minimal version of the standard
16 November 1989
model there is only one Higgs doublet 45-r= ( 45+, 45o). Its mass in the EWS phase is given by M ~ = d 2 l'/d02 [o=o, where 0 is associated to the real part of the neutral Higgs component and the scalar potential including one-loop contributions [8] can be casted in the form (using that d V / d 0 = 0 in the true vacuum 0 = v - 2 4 6 GeV)
v(o)=(2A-C)v202-..lO4+('O41n(O2/v 2) .
(6)
The constant C can be expressed in terms of the masses m~ and m f o f t h e bosons and fermions (in the true vacuum ) with multiplicities ga.f:
C=
gBm~-- ~f gfm~
,
(7)
and A can be determined as a function of the ordinary Higgs mass
In~1 =
( d 2 V / d O 2 ) o = , , = 4 t ' 2 ( 3 ( ' - 2A ) .
(8)
Consequently, the Higgs mass in the EWS phase (45=0) is ,.V/~ =2v2(2A - C ) .
(9)
As it will be shown later, in the case of interest here. i.e.. when the scalar potential I" is such that the EWS phase is the ground state of high-density matter, the expression (9) is positive and it is possible to treat the 45 excitations as particles with mass M . . Moreover. we will obtain that typically M . ~ few GeV's. so that for temperatures T<< GeV the Higgs densities can be consistently ignored. Since the electroweak bosons are massless inside the NTS, they generate Coulomb like energy terms unless the net SUt.(2 ) and Uy( 1 ) charges vanish. To avoid these contributions it is then necessary that 7"3= ) ' = 0 . The T s = 0 condition is automatically satisfied, since all members of each S U ( 2 ) multiplet have the same chemical potential and hence, the contributions to "/~s cancel within each multiplet. Instead, the Y = 0 requirement gives the last constraint to the chemical potentials ( Y= 2 ( Q - T3) ) Y= ~[fi(u..)+fi(dt.)] +~fi(UR)-- ifi(dR)--2fi(eu) - [h(v~) +t~(e,.) ] = 0 .
(10)
We will study the properties of high-density matter in bulk (neglecting surface effects from the boundary of the NTS) at zero external pressure. In the ther445
Volume 231. number 4
PHYSICS LETTERS B
m o d y n a m i c a l limit, the density o f the t h e r m o d y n amical potential is to=B+
l,'o-P,
( 1I )
where B is the energy density that accounts for the non-perturbative Q C D structure o f the vacuum, following the M I T bag-model picture: I"o is the price to pay to restore the clectroweak s y m m e t r y inside the NTS, i.e.. the difference between the Higgs potential at O= 0 and at the absolute m i n i m u m ¢~= v where only the electromagnetic s y m m e t r y survives, i.e., Vo= v 4 ( ( ' - , - 1 ) . Finally, the pressure P gives the temperature dependent part. It is a sum o f contributions from all fcrmions in the three generations and all vector bosonsin SU(3)×SU(2)×U(I )
Njf / ' = 7a ~ T
d3k k 2 (2n') 3 c
1000
I
16 November 1989 [
,
[
'
i
B=(t50 MeV)"
8oo
~eV)"
~" ~"~~~..w(tso
......... \ \
6OO
',..
v=O0o ue).]"
%
"'...
"....~ "STRANGE_
'~\~.
400 "NUCLEAR MA'ITER" 200
l
gk ( ,tts ) ,
(12)
where for the bosons gk(.u) = ~ l - e x p [ / 3 ( ¢ - / t ) ] }
-t .
(13)
In the case o f massless particles with zero chemical potential, eq. ( 12 ) gives the well known results
40
r (u,v)
80
120
Fig. 1. Free energy per unit baryon as a function of temperature. Dolled lines correspond to the EWS phase, for B'/4= 150 MeV and values of I'o~/4 = 100 and 160 MeV. For comparison, the continuous line is the expectation tbr nuclear matter in Walecka's model and the dashed line corresponds to strange matter with B ~/4= 150 MeV, computed as in ref. [ 4 ].
7/-27"4
l ' ~ s ' (l~= 0 ) =,'v, 90 ~27"4
l,~r~"'(,u=0)=~N,
~
.
(14)
From the thermodynamical potential, the free energy density is obtained as 0 = co + ~ llj,§.
( 15 )
I
The equilibrium configuration is d e t e r m i n e d by m i n i m i z i n g the free energy per unit baryon number, i.e., 6 ( 0 / n , ) = 0 . This equation, together with eqs. ( 4 ) , ( 5 ) and (10), were solved numerically. Fig. 1 is a plot o f ~ / n B as a function o f the temperature for B Z / 4 = 150 MeV and values o f Vg/4= 100 and 160 MeV ( d o t t e d lines). (Typical values o f B ~/4, obtained by fitting known properties o f hadrons with MIT bag model predictions, range from 140 to 200 MeV. ) Also the predictions o f Walecka's model [ I ] for the free energy per b a D o n n u m b e r o f nuclear matter, c o m p u t e d as in ref. [4], are shown in fig. I ( c o n t i n u o u s lines). This curve takes a value = 9 2 2 MeV for small t e m p e r a t u r e s and shows an abrupt change n e a r a critical temperature 7~-~ 17 MeV, where the fall down in the free energy corresponds to the
446
transition to the nucleonic gaseous phase (the scale o f T,. is given by the binding energy per nucleon, E l ,,I= - 16 McV at T = 0 ) . It is a p p a r e n t from fig. I that for the values o f Vo and B considcrcd the EWS phase is energetically favoured for temperatures less than ~ "1"~..However, it is possible that strange matter bc energetically favoured with respect to ordinary nuclear matter. This could be the case [4] at temperaturcs below 7"~. if Bn/4<~ 155 MeV (assuming m s = 150 M c V ) . In this case, to find the ground state o f high density matter one should c o m p a r e the frcc energy per baryon o f the EWS phase with that of strange matter, so that an interplay appears between the three phases. In fig. 1 wc also present (dashed lines) the expectations for strange matter, computed as in ref. [ 4 ], for B ~/4 = 150 McV. For values o f Vo than ~ ( 160 McV )4, there is however no t e m p e r a t u r e for which the EWS phase has lower free energy than nuclear matter and strange matter. In fig. 2 we show the values o f I/g/4 (the ones ~ below the curve [.'1/4 . m . . . . as a function o f B ~/4 that yield to a stable EWS phase for T = 10 MeV. These region varies only slightly with temperature, pro-
Volume 231. number 4
PHYSICS LETTERS B T=10 MeV
200
~
I
'
I
t/,*
"NUCLEAR MATTER"
"STRANGE klATrER" 160 ~
,.o:
0/
'
\
,
I
140
160
,
B'/" (MeV)
I
180
200
Fig. 2. "'Phase diagram" for T= 10 MeV. For the values of V~/4 below the continuous line I"~/,~ the ground state of high-density matter is the EWS phase. This range diminishes for increasing values of the bag constant B. for I.o> 1/,,~, strange matter is favoured for B~/4< 155 MeV or otherwise, nuclear matter is the stable phase• vided T < 7"~, as can bc deduced from fig. I. Also the region where the strange matter phasc is stable is indicated in fig. 2. As discussed in refs. [5,6], the requirement Vo < Vma~ for the stability o f the EWS phase implies a fine tuning o f the parameters entering in the Higgs potential, and though, strong restrictions for the Higgs boson mass. For instance, in the minimal version with only one Higgs doublet that we are considering 8
i ~ 1 ~ 1 = 4 / , ' 2 ( " ' t - / ~ I' 0 .
anticipated before, in the EWS phase the Higgs mass results MH ~--m..,,JVf2 = few GeV's. Although in the m i n i m a l standard model this solution seems contrived, there is more room available in models with more Higgs fields and, also, some supergravity induced extensions o f the electroweak theory [ 10] predict that the ~ = 0 and 0 = v m i n i m a are necessarily degenerate, leading to a value o f Vo naturally small. Finally, fig. 3 shows the baryonic densities (in units of the nuclear saturation density no) o f nuclear matter (continuous line), of strange matter (dashed line) for H ~/4 = 150 MeV and o f the EWS phase ( d o t t e d lines) also for BI/4= 150 MeW and values o f V~/4= 100 and 160 MeV. The higher densities arise in the SU (2) × U ( I ) symmetric case, implying that the size o f a NTS is smaller than the corresponding one (for the same value of the total baryonic n u m b e r ) for nuclear matter or strange matter. At temperatures ~ T~, the nuclcar matter density goes to zero since in the gaseous phase nuclear matter at zero external pressure is unbounded. This is a consequence o f the Van dcr Waals like behaviour o f nucleonic interactions. In conclusion, we have considered the possibility that at finite temperatures, the ground state of high-
'
I
I
..............................................
"..., V=(100 MeV)'
2
=--2
.............
"STRANGE MATTER~ ' - ~
where, for
'
B=(150 MeV)4 .................."....• V=(180 lleV) 4
d' (16)
16 November 1989
I
,
"..
DI H <<7. U,
C_~2× 1 0 - ' [ I - (mtoo/79 G e V ) ' ] .
(17)
As a consequence, the EWS phase stability constraint I,'o< Vmax~O( (0.2 G e V ) 4) requires thc Higgs mass to be a few eV above the m i n i m u m allowed Higgs mass value [9] (corresponding to V o = 0 ) that is obtained from m ~ m = 4 v 2 C . For instance, mtop=60 GeV yields mini, -~ 6 GeV. However, for m t > 80 GeV thcre are no values o f the Higgs mass satisfying ( 15 ) for so small values o f Vo and simultaneously giving a positive value of C to make the potential V stable. As
MATTER"
\
\ L
,
40
T (ldeV)
[ BO
\
120
Fig. 3. Baryonic density in units of the nuclear saturation density no=0.17 fm -~, as a function of the temperature, for the three phases: electroweak symmetric (dotted lines), nuclear matter (continuous line) and strange matter (dashed lines) with the same values of Voand B as in fig. I. 447
Volume 231, number 4
PHYSICS LETTERS B
density m a t t e r be a phase w h e r e the e l e c t r o w e a k symm e t r y is restored, with all s t a n d a r d f e r m i o n s and gaugc bosons being massless and being in t h e r m a l and c h e m i c a l e q u i l i b r i u m . If the p o t e n t i a l cnergy differe n c c b e t w e e n the s y m m e t r i c and a s y m m e t r i c v a c u a is small c n o u g h ( V o < O ( (0.2 G e V ) 4 ) ), this phase is energetically f a v o u r e d for T~< 17 M e V and for m o s t r e a s o n a b l e values o f the bag energy B. T h e e x i s t e n c e o f such a phase c o u l d h a v e i m p o r t a n t c o s m o l o g i c a l and astrophysical c o n s e q u e n c e s . F o r instance, neutron stars m i g h t c o n v e r t into " c l e c t r o w e a k s o l i t o n s ' . T h e finite t e m p e r a t u r e b e h a v i o u r o f this phase c o u l d be r e l e v a n t in the study o f the early stages o f n e u t r o n star f o r m a t i o n or in the t h e o r y o f s u p e r n o v a e explosions. We w o u l d like to t h a n k G r a c i e l a G e l m i n i for m a n y useful d i s c u s s i o n s and Ed C o p e l a n d for help.
448
16 November 1989
References [ 1 ] J.D. Walecka, Ann. Phys. 83 (1974) 491- Phys. Lett. B 59 (1975) 109: H. Reinhardt and H. Schulz, Nuc. Phys. A 432 ( 1985 ) 630, and references therein. [21 E. Witten, Phys. Rev. D 30 (1984) 272. [3] E. Farhi and R.L. Jaffe, l'hys. Rev. D 30 (1984) 2379. [4 ] H. Reinhardt and B.V. Dang, Phys. Lett. B 202 ( 1988 ) 133. [ 5 ] S.Y. Khlebnikov and M.E. Shaposhnikov, Phys. Len. B 180 (1986) 93; Sov. J. Nucl. Phys. 45 (1987) 747. [ 6 ] E. Copeland, E.W. Kolb and K. Lee, Fcrmilab preprint 88/ 84-A (October 1988). [7] A. Chodos, R.J. Jaffe. K. Johnson, C.B. Thorn and V.F. Weiskopf, Phys. Re','. D 9 (1974) 3471; W.A. Bardeen, M.S. Chanowitz, S.D. Drell, M. Weinstein and T.M. Yan, Phys. Rev. D I1 (1974) 1094. [8] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888: S. Weinberg, Phys. Rev. I) 7 (1973) 2887: E. Gildener and S. Weinberg, Phys. Rev. D 13 (1976) 3333. [ 9 ] S. Weinberg, Phys. Rev. Lelt. 36 ( 1976 ) 294; A.D. Linde, Phys. Lett. B 70 (1977) 306. [ 10] E. Cremmer, P. Fayet and L. Girardello, Phys. Left. B 122 (1983)41.