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Thermodynamical theory of pion condensation
Nuclear Physics A404 (1983) 525-550 @ North-Holland Publishing Company
THERMODYNAMICAL
THEORY
OF PION CONDENSATION
G.G. BUNATIAN Joint Institutefor P&clear Research, Dubtta, USSR and 1.N. ~IS~USTIN ~~rc~atov roseate ofAtomic Energy (IA@, Moscow, USSR Received 21 December 1981 (Revised 6 January 1983) Abstract: A tbe~modynam~c~ approach to the description of the pion field and ~-condensation in nuclear matter of high density and nonzero temperature is developed. It has been shown that the quantum and thermal fluctuations of pion fields play a decisive role near the critical point; they are the reason why the equation D-‘&J, R, p, T) = 0 has no solutions 02(k) ==z 0 at any density 6 and temperature T, i.e. the criterion earlier used for the formation of the rr-condensate is not satisfied. In order that the g-condensate may arise in thesystem with given p, T the thermodynamical potentiaI should decrease when the ?r-condensate field is switched on: A@p) 4 0. The character as well as the critical parameters of the phase transition are strongIy dependent on the effective VW interaction. In the framework of the &I” model, the critical density and the jumps of the condensate field amplitude, along with the renormatized gap of the pion excitation spectrum and of the thermodyn~i~al quantities, are estimated. The area of appIicab~Iity of our approach as well as possible consequences of the theory are discussed.
1. Introduction Recently the behaviour of nuclear matter has been intensively studied under extreme conditions: high density, high temperature, anomalous ratio Z/N, etc. Interest was aroused by the works of Migdal lm3)who indicated the possibility of phase transitions in nuclear matter caused by pion condensate formation. Both theory and experiment have shown 4, that there seems to be no rr-condensate in ordinary nuclei, i.e. at normal density ~0. To create the v-condensate, one should significantly increase the density of the nuclear matter p IS (2-3)p0, since this follows from different estimates made with realistic parameters of particle interaction in nuclear matter 5,6). This can be achieved in the laboratory in collisions of heavy nuclei wifh energies higher than E/A -0.5-l GeV per nucleon ‘). Certainly, the energy brought about by such collisions in the nucleus will only partly serve to increase the density. A considerable part of it will be spent on the thermal excitation of the dense nuclear matter created 7>.The excited state can be studied within the statistical approach characterized by a temperature T. Calculations have shown that at E/A a 1 GeV the temperature appears to be sufficiently high, i.e. up to 525
526
G.G. Bunatian
and I.N. Mishustin / Pion condensation
T - mmcL,where m, is the mass of the r-meson. The r-condensate may also exist in the high-dense matter of a neutron star. The dynamical effects associated with the phase transition in the neutron star formation stage following the supernova explosion when T - O.lm,c’ [ref. “)I are most interesting. Therefore it is important to study the properties of the highly dense nuclear matter at nonzero temperatures (see refs. 6*g-11)),It is usually assumed that the formation of the rr-condensate field in the medium is a conventional second-order phase transition. In ref. rr) it is noted that one should take into consideration the occurrence of a large number of interacting g-meson excitations in the nuclear medium at T # 0 in thermal equilibrium with the matter, i.e. “black-body radiation of r-mesons”. The corresponding general formulas for the polarization operator #and the pion .Green function are derived. The basic characteristic feature of precritical nuclear matter is the softening of the pion degree of freedom with the increase of density precisely at the high pion-field momenta k,- md. Large k, implies a large phase volume of the pion-field fluctuations near the rr-condensation critical point. As a result, the amplitude of the fluctuations increases greatly when the system parameters achieve critical values r2,r3). This phenomenon is of prime importance. If one takes into consideration the above-said, it will change the conventional physical picture, thus leading to a number of qualitatively new effects as it will be shown below. Sect. 2 is devoted to a study of the properties of the polarization operator n, the pion Green function D and the ~TT~T interaction in the matter. Sect. 3 contains a calculation of the correlation function N(p, T) of the pion field fluctuations and their main contribution to I?. The general formulas for these quantities are derived and their behaviour in the vicinity of the critical point is studied. In sect. 4 the relative contribution of various diagrams to 17, D is estimated. Conditions are outlined under which it is possible to obtain approximate close equations for these quantities. Using the said equations the physical picture of the behaviour of the system near the critical point in the absence of the condensate field is given in sect. 5. Sect. 6 presents the equations of the r-condensate transition derived from the requirement that the thermodynamical potential 0 be a minimum in the nonzero condensate field. In sect. 7 the physical picture arising in the first-order phase transition is described. Finally, sect. 8 deals with the main features of the general theory developed here as well as with some physical conclusions drawn from it.
2. Polarization
operator and n-meson
Green function in the medium
at T # 0
Let us first discuss the principal characteristics of pion excitation in the medium in the absence of TZ- interaction. The isotopically symmetric equilibrium system of interacting nucleons, A,, isobars and r-mesons at T # 0, is considered. The polarization operator 17”(0, k ; p, T) and meson Green function W’(W, k ; p, T) = [co2 - m,+z2- k2 -i7”(w, k; p ; T)]-l calculated in ref. 11) for this medium taking no
G.G. Bunatian and I.N. Mishustin J Pion condensation
521
account for the rir interaction and pion field ~uctuations seem satisfactory and we shall use them in the present paper. Note that the D(w, k ; p, T) function in ref. ‘I> is essentially the analytic continuation of the Matzubara temperature Green function ti((wn, k ; p, T) from discrete values w, = 2drzT, m = 0, A1 . . . , to the complex plane w. Thus, in the upper half of the latter (Im w > 0) the function 13 (w) coincides with the retarded Green function D&W), while in the lower half (Im w C 0), with the advanced one DA(u) [ref. ‘“)I. In all the earlier works the critical density p,(T) was defined as the density p =pc(T) by which the equation D-‘(w, k,; p, T) = 0 had the solution w’(k,) = 0 for the one pion field momentum k = k&,(T), T). For p >p,(T) the solution o’(k) < 0 exists within the certain interval kl< k < k2, kl
T)]-l = w2 -
l-k2-Re.U”(o,k;p,T)-iImB”(w,k;p,T)
-o’a(T)+iwp(T)sign -&(P,
T)-y(k
-ko)‘,
(Imo)
Imu+O;
The calculation, performed as described in refs. 6S11),yields the values of the parameters in eq. (1). The values n = 1, 2, x = 1.3, k: = 3, y =0.8 are practically independent of T; at T =O, fi(k,)=OS, a(ko)=1.6 The value P(T) increases, P(T = 1) = 0.8, cy(T) decreases with the increase of 7’. Hereinafter, the R = c = pn,.,c’= 1 system of units is used, for example, T = 1 means T = mg2 = 140 MeV, p. iz=0.5 = 0.17 fmd3. At T = 0 and o s 2 the Im 17” becomes zero at large k e.g. at p = 1.5-2 it is zero at k 2 8. At T - 1 ImIl”(w, k; p, 2”) is not exactly zero at such values of o and k, but becomes infinitely small with the increase of k [see figs. 2 and 3 in refs. “)I. The approximation (1) holds true within fairly wide limits, at least one can use it at T s 1, w 6 k+ Ik - kol c 1, lw:Is 1, which is valid for further consideration. Note that, generally speaking, the expression (1) for the D” function may be considered as a semiphenomenological one and its parameters could be obtained from the presently available experimental data concerning the study of the nuclei
528
G.G. Bunatian
and I.N. Mishustin / Pion condensation
degree of closeness to the r-condenstate instability “). The proximity of the system without the 7r~rrinteraction to the rr-condensate instability at given p, T is evidently characterized by the degree of the og(p, T) = min [D”(O, k; p, T)]-’ proximity to zero. This minimum is achieved at a high critical momentum of the pion field ko. Therefore near the critical point the phase volume of soft pion field modes is large and the fluctuation amplitude increases fast with wg(p, T) + 0 [refs. 12,1’)]. Note that for the system with the roton-type spectrum, which may be obtained from (1) at Im 17” = 0,~: is a gap in the excitation spectrum. For an example of the spectrum to be obtained from the equation [D”(w, k)]-’ = 0 at Im I7”= 0, refer to ref. ‘l), fig. 2. Now it is necessary to include the r-meson interactions. The rr~~ interaction lagrangian in the medium is presented here as follows: *
2&*(x) =
x9
-&i(~(x))2(tr(x))2 =
(24
where the isotopic vector & is the operator of the r-meson field, (I^is the effective amplitude of the ~7r interaction in the medium, which depends on the kinematical parameters and properties of the medium in a complicated way. It is not possible to calculate it yet. Within our approach A is assumed to be constant and, indeed, is a free parameter of the theory. In order to make our approach valid, the value of A should not exceed 0.5. The purpose of the calculation using (2a) is only to qualitatively estimate the effect made by the account of the nr interaction on the properties of r-meson field in the medium. The influence of the medium on A is discussed in refs. 293).I n a 11our calculations the rr7r interaction is assumed to be repulsive for A > 0, which is essential for the stability of the system. The case of A < 0 will be discussed in brief in the last section of the paper. The self-energy part of the pion IZ T, induced by the interaction lagrangian (2a), is an infinite series of skeleton graphs:
KY(W, k;p, T)= (3a) where wavy lines correspond to the exact pion Green functions in the medium with account taken of 17” and fll”(v = 0, f or v = 1,2,3). Further we shall consider the system with a m-condensate field q(x). Then, instead of (2a), we shall have: (2b) P, = -&4(+(X) +lp(x))4. This interaction lagrangian induces an addition to the polarization
operator:
(3b)
529
G.G. 3anat~a~ and I.N. ~is~ustin f Pion condensation
AS we shall see in sect. 7, with the interaction lagrangian @a, b) the first-order phase transition is possible in the system: the condensate field originates from (cp(x))” f 0. However, it is clear that such interaction lagrangians may be chosen that there would be no Ir-condensate phase transition at any p and T within our approach. But, in any case, the fluctuations of the pion field near the “old” critical point affect the properties of the system considerably. When the pion excitations interaction and the classical field are taken into account, the pion Green function takes the form
D;‘(&&k;p,
r)=12-l-k2-17~(0,k;p,
T)
,-n~(U,k;p,T)-fl~(o,k;p,T).
(la)
The expression (la) is the equation for D, since P, II”, 17” depend on the total pion~-function. In the presence of the o- and ~-independent 17”, If” the approximation (1) for D-‘(w, k; p, T) apparently contains, instead of w%(p, T), the following quantity: &‘I$, T)=&(p,
T)+IF”(O,k;p,
T)+If”(O,k;p,
T),
(lb)
which plays the same role as 0: without the VQTinteraction (2a), (2b): &“(p, T) = min L)-l(O, k; p, T). The softening of the pion degree of freedom in nuclear matter would mean proximity to w-condensate instability for cS’- 0 and complete instability for &‘@, YZ’) < 0. But this (see sect. 5) cannot be achieved due to increasing pion field fluctuations near the criticdl point, It seems impossible to solve eq. (la) in the general form. An approximate solution is possible in a number of cases when the account of only the first simplest graphs in (3a), (3b) can be justified. Their contribution to the II”, ff” will be calculated in sect. 3. In sect. 4 we shall make estimates of the more complicated diagrams and find the conditions under which their contribution may be neglected.
3. Calculation
of the ~o~eIation function and of the main terms of the polarization
operators
II*, IIs.
It seems useful to introduce the correlation function of pion field ~uctuations here: N&Y
Pt
Tl = ~~~~(r;
P, %p=T,fO
= f%.&h,
Mb(~21~2~~
k; p, T) eik’r-mm:O,
r =r2-rl.
(4)
The averaging is performed here over the equilib~um state of the whole system at a temperature T. Notice that at r = 0 N is not the density of pion excitations in the medium, since they are absolutely unstable.
530
G.G. Bunatim and LN. ~is~astin / pion condensa~on
Fig. 1. The integration contour for eq. (5).
As can be seen below, the equation D-‘(w, k;p, T) = 0 does not have the solution w’(k) < 0 at any p, l” if the interacting pion excitations in the medium are taken into account. Then, using the conventional method of calculation 14’16)we find *l) for the isotopically symmetrical medium
x (5) = (e”’ - 1)-l , The integration contour is shown in fig. 1. As has already been noted, D(l -I-i0) = I&(t) is over the integration axis and D(t - iO>=Dn(E) below it. Using the well-known properties of the L&.+ for the Bose system 14), O,(~+io)=D;(G$-iO),
Re
&&)
= Re
OR(+)
Im OR(<)
,
=
-Em Dn(-0
(in the isosymmetric medium lTI. = 0), the expression (5) can be transformed
x {[e2- 1 - k2 - Re LX’,&k)]‘+ [Im lr,([, k)j2}-’ . The self-energy part corresponding medium
is easily expressed
into
(6)
to the first graph in (3) for the isosymmetric
in terms of the correlation
function:
H: (P, T) = 5~N”(O; P, T) .
(7)
In this case (LI = const) it is independent of w, k ; v = 0, It(l, 2,3). As can be easily seen, the expression (7) does not become equal to zero at T = 0, x(p) = 0. The expression for the n”(p, 0) contains also a trivial vacuum term iI”(0, 0) which has already entered into the observed pion mass. It should be subtracted from (7) and this is implied in the further consideration. The rest of the II” at T = 0, II”@, 0) D”(O, 0) is the contribution into the LIT of the pion field quantum fluctuations in the medium at T = 0. Note that formally the whole part of the 17” without ~(5) can be excluded In the medium
by writing down the lagrangian (2) in the normal product form 11’12). it may lead to a loss of the physical contribution of the quantum
G.G. ~una~ian and I.N. ~i~hus~~n / Pion condensation
531
fluctuations to the fT”, though that is not very essential in comparison with the contribution of the thermal fluctuations near the critical point. Let us consider only T 6 1, then 6% 1 is essential far the integral (6) and at k - k. the expressions (l), (lb) may be used to write down the expression for the denominator in (6) as follows:
Large k do not contribute to (6) due to the described above equation properties of Im n”(& k ; p, T). As will be seen below, the main con~ibution is given by k - ko, if ~5’4 1, since under a condition like that the integrand has a sharp maximum. Then, for the value of N(O; p, T) determining according to (7) the II” value we approximately have N(0; p, T) =
7T-3
Jk:
k’dk
o~d5B-i&-x(5))5 J
x ([b?+ y(k - kof2]2@-2 i- c$2>-1,
ewe0 kl=ckockz.
(8)
Developing the above consideration and omitting x(c) we have the following estimate for the contribution to N(0; p, 0) of the quantum fluctuations near the critical point:
This expression does not diverge at G2+0, i.e. the quantum fluctuations do not grow near the critical point. Making a series expansion at &’ + 0 one has N(O;~,O)=SA-B&.?+O(&~),
fj=-.----.ki 2$3~”
(9)
where the constants A and B are of the order unity in pionic units. With 4 = 1 and the other parameters the same as in eq. (1) A and B can be estimated as 0.25 and 0.9, respectively. Note that in the recent paper 17) the logarithmically diverging expression -In IS] for the quantum fluctuations was obtained by mistake. The author of ref. 17)made a mistake because he had ascribed wrong analytical properties to the pion Green function in the medium. Namely, his expression for I) would correspond to one having a real pole of the D-function at k - ko, w - 0, whereas, in reality, the D-function has branch points and is determined on the complex plane w with a cut aIong the real axis ‘I). From (1) one can conclude that ImD is not small at k - ko co =SkoF and changes its sign in crossing the real axis. A jump of the Im D on the cut determines the value of N, 17” [ref. ‘“)I.
532
G.G. Bunatian
and I.N. Mishustin / Pion condensation
expressions (S), (6), (8) a part of the integral containing x(t) is evidently the correlation function of the pion field thermal fluctuations. At T = 0 it surely becomes zero. From eq. (8) we obtain In
N(O;p, T)-N(O;p,0)=&\~:2k2dk~-1
x{ln($)-f-P(&)},
Y
=[~2+y(~-hd211PT,
where q is the Euler function r8). The integrand in (9) depends only on y, which is the value characterizing the “softening” of the pion degree of freedom divided by the temperature T. At y
C = 0.57721 . (10)
As is clear from eq. (10) and has been stated above, the integrand as a function of k has a sharp maximum at k - ko. For the estimates it is convenient to choose kI, k2 as symmetric about ko. The integration of the first two terms in (10) yields finite expressions at &I’+ 0, and the integral of ?r/y contains the term 4-l which evidently determines N and ILP at &_J2 K mT: N(0; p, T) = CT/&?,
Il”(p,
T) = 5ACT/&?,
C = k;/2&.
(11)
Eq. (9) may be written in a simpler form in another extreme case, y >>1, while keeping the condition T <<6’ < 1. We find 18) from eq. (10)
N(O; P, T) -NCR
P, 0)
=~~k~k2dk[,2~+,,~+...].
(12)
(Bernouilly numbers B2 =& B4 =&. Again the main contribution is made by k - k,. In the case of y >>1 only the first term should be left in (12). And once more it is convenient to take kI and k2 as symmetrical about ko. One can easily see that the integration in (12) yields the terms containing no singularities at ; -+ 0, or -K1, _; -2, -$ -3. For the estimates it suffices to take into account only the most “singular” terms -K3. Then at T + 0 N(O;p, T)=A-B
2 ,c3’T2 a-“---- 12 ~~2~3/2+’
* *
12 rr2 /3’T2 I7”(p, T) = 54A - 5AB dw - -l2 -((;2)3/2
1 7
+
* ’ ’
1 .
(13)
G.G. Bunatian and I.N. Mishustin / Pion condensation
533
While calculating further, one should keep in mind that eqs. (11) and (13) can be used only if the constraints under which they have been obtained are followed. One may move from p = p,(T) (i.e. w$(p, T) = 0) by increasing p until the pion D-function in the medium reaches a maximum at w - 0, k -kc, &‘< 1. At p appreciably exceeding p,(T) (i.e. w’, B 1) the parameters in (1) surely change, while the D(w, k ; p, T) character of eq. (1) remains basically the same. We shall employ eqs. (9), (ll), (13) derived in this section on the basis of eq. (1) to qualitatively study the physical picture for rather large p-values too. The correlation function N(r; p, Tj at Y # 0, necessary for the further calculation, can be easily expressed through iV(0; p, T) for the two cases considered above: N(r;p,
T)=N(O;p,
sink,, . r _1,1
T)ke
=.
(14)
0.
It is apparently characterized by the two lengths: the oscillation wavelength 2r/ko and the correlation radius rc = dy/G '@I, T). As usual, in phase transitions I, grows near the critical point. The main contribution to 17” has been made by the first graph in (3b) as can be seen below: n”, =~(&)>+2~(&))
*
(7b)
This expression will be used in the following. Thus, we have estimated the contribution of the simplest graphs to (3a), (3b). Now, let us deal with the others.
4. Estimation
of complicated
diagrams;
equations
for JIq N
In general all the complicated diagrams in (3a), (3b) are not small as compared with those calculated in the previous section. Only under certain conditions to be clarified here, it suffices to take into account just the first simplest diagrams in (3a), (3b) for calculating HT”, 17”. Let us consider first the contribution of the second graph into (3a) at w -0, k - ko. In the case of the quantum fluctuations (T = 0) this diagram, as well as the first one, yields no singular contribution at (3 + 0. Taking into account the fact that the second diagram has an additional vertex factor A, as compared with the first one, it can be assumed that its contribution is small if A CC1. At low temperatures, T <
534
G.G. 3~n~t~~n and IA? ~~shust~~ / piopl conversation
At high temperatures the contribution eq. (14), and can be estimated as:
-$ J dr eik”~3(r;p,
of this diagram is expressed through N(r),
T)Y~‘T*/W”~(~,
T>
.
Comparison with exp. (11) shows that this contribution may also be neglected if the parameter AT/G2 is small. At first glance one can conclude that the second diagram is even more important near the critical point. However, as we shall see later, 4’ is never too near to zero due to the influence of thermal fluctuations. At the phase transition point, provided that a transition takes place at all, (cz is fairly large, while the parameter AT/&E is actually small (see sect. 7, fig. 6) at n <<1. Using the same considerations, one can show that the other more complicated diagrams in (3a), as compared with the first one, may also be neglected. The contribution of the second diagram to (3b) is maximum for the components of the fields q(x), 2(x) having momenta equal in value but opposite in direction and frequencies equal to zero, i.e. the longitudinal channel zero momentum and frequency are transferred. In this case the contribution of the second diagram to (3b) is as follows:
It is easy to obtain the following approximate
estimate for this value:
-(cp2(x))(i12/&)[const + (T/4z)2] at low temperatures
T ccG* < 1, and -(cp”(x,>A”T/;”
at temperatures T >>c32. In the former case this estimate is small as compared with If” from (7b), since (T/&z)<< 1 and A/&E - 1 are near the critica point as will be shown in sect. 7. In the latter case the relative contribution of this diagram is proportional to the above-discussed parameter AT/ (32 < 1. Under this condition the relative contribution of more complicated diagrams is even less. Note that the contribution of such diagrams at T = Clwas discussed in ref. I’). Thus, there are reasons to take into account only the first diagrams in (3a), (3b). Notice that the same conclusions were made in ref. *‘) i.n the study of systems with the roton-type excitation spectra. To be exact, the additions to 17 may be other than those of (3a), (3b). It is clear that pion fluctuations and the occurrence of Q change the self-energy part of a
G.G. ~unutjan and I.N. ~sk~s~n
535
j Pion condensation
nucleon, which in its turn leads to an additional cont~bution
to R of the following
One can see that it is not as important to take this process into consideration near the critical point as that described in (3a), (3b). Since, according to the conclusions of the present section, only the con~butions of the first diagrams in (3a), (3b) are left in If”, 17”, eqs. (6)-(13) are transformed into closed equations for N(0; p, T) or If”@, T). The self-energy part 17”(w, k; p, T) entering eqs. (6), (7) was calculated in ref. ll). L!“(p, T) can be calculated numerically at given A, p, T without simplifying approximate transformations leading to (Q-(13). Having found TT”, & one can specify at which A, p, T the parameter AT/&~l’p, T) determining, as has been shown, the relative cont~bution of various diagrams to (3a), (3b), is actually small. In the present paper the behaviour of the system near the critical point is described by eqs. (1 I), (13), bearing in mind that one should remain within the field of application of these equations. Namely, the following relations should be fulfilled: (3z(p, T)K =T, AT/G: (p, T) <<1. At A K ?r it is possible to satisfy both. The values &(p, T) obtained in our calculations for various n will be presented below in figs. 4-6. It is also convenient to rewrite 17” in terms of & Using (lb) one obtains from (9), (1% (13) ~~=“~~~a~5A~T~~~,
(114
for high and low temperatures, respectively. In these equations 00 is different from that of eq. (1) by the constant 5AA ori~nating from the quantum fluctuations. This renormalization slightly changes the bar critical density p,(O) which is a rather uncertain quantity. Having analyzed various diagrams con~buting to the polarization operator and having established the main equations for lP and N, we proceed to the description of the system behaviour near the critical point first in the absence of a condensate field (sect. 5), and then in its possible presence (sects. 6, 7). 5. A physical picture near the critical point ~thout the condensate tieid Now let us consider the solution of eqs. (lla), (13a) without the condensate field (17” = 0). In case of zero temperature when the quantum fluctuations are taken into account, eq. (13a) has a simple solution:
536
G.G. Bunatian and I.N. Mishitin
Go
is positive at & > 0 and tends to zero at 0% + 0. In this limit 40 = ~~/~~~,
fl”(p,
/ Pion condensation
0) = -c&p,
0) *
It is easy to show that cj + 0 at wz < 0 and T + 0. To prove it let us consider the limit T+ 0 more carefully. At wxp,(T}, the limiting behaviour of (;t2 is described by the high temperature expansion of eq. (lla). Indeed, the solution of this equation at T + 0 can be obtained by dropping tj’ in the 1.h.s. Then taking into account that p,(T) = p,(O) +p2T2 [refs. 6,10)]we have
&p,
T-+0)=
25A2c2T2
(16)
G%P, 0)
where w &, 0) = q [p,(O) -p]. In this limit a”@,
25A2c2
T+O)=-w;fP,0)+T2
m-qp2
I .
The solution (16) is valid if G2
-o:fp,0)
(.~AcT)~‘~ <<
.
f18a)
On the other hand, eq. (1 la) itself holds at T + 0 if G2 CCVT or
It is clear that at s~~ciently low temperatures (T < (Z5/~3)A2cz), if eq. (lgb) holds true, eq. (18a) is also true. Hence, at any wg< 0 and T-+ 0 one has G2-+ 0 and Lrw + -w z(p,0). In a similar way one can see that at 0: > 0 and T + 0 ~2’ tends to the limiting solution (3 (eq. (15)). From eq. (13a) one has (19) The dependence discussed above of G.7’on 0: (i.e. on p) at T =+O (quantum fluctuations) is presented by the curve in fig. 2. At finite temperatures ~7’ is always positive and higher than &* at T = 0 due to an additional contribution of the thermal fluctuations. For instance, at p = p,(T), i.e. to; = 0, eq. (lla) has the solution G2(pc(T), T) = ~~~~{T),
T) = (~AcT)~‘“,
(20)
which lies within the region of applicability of eq. (lla), if T >>(25/7r3)A2c2. In this case eq. @la> can be used to calculate 13’ and 1T” in the whole region wg < 0, where c.$’ is even smaller than at ox= 0. With the increase of T eq. (lla) is applicable to larger G2 in an expanded region of 0:. To describe G2 outside this region, one should use eq. (13a).
G.G. Bunaiian and I.N. Mishusrin f Ron candensatim
537
-2
tii
Fig. 2. Dependence of &’ on w$ at A = 0.1 and7 = 0 (a), 0.1 (b) and 0.5 (c), The region of intermediate temperatures (dashed portion of curves) is interpolated by hand.
In fig. 2 the dependence of (3’ on ~“0 is ihustrated at two temperatures: 1’ = 0.1 and 0.5 (the curves b and c, respectively). These curves were obtained by solving eqs. (lla) and (13a) in low and high G* limits, respectively, and by approximately joining these solutions at ~5’- T. Taking into consideration the discussion above and fig. 2, we have reached the conclusion that the pion field fluctuations near the critical point stabilize the system so that at any p and T the equation D-l@, k; p, T) = 0 should have no solutions w’(k)
of pion condensation
Pion condensation may be presented lU3)as the formation of the mean classical w-meson field cp(r, t) in the nuclear medium due to the rearrangement of the ground state of the system. In our case the mean isovector pion field cp(r, b) is the order parameter of the phase ~ansition. The necessary condition for the formation of such a field is the appearance of the minimum of the thermodyn~ical potential a at cp#O:
where the averaging is performed over the equlibrium state of the system, which includes the condensate held. The effective hamiltonian needs to be constructed for the system of the interacting quantum and classical In-meson fields in the
538
G.G. Bunatian and I.h? Mishustin / Pion condensation
medium. Without the rrrr interaction, A = 0, the r-meson field in the nuclear medium satisfies the equation 6;’ (r, t)(~~(r, t) = 0. The interaction of a-meson fields is described by eq. (Zb). The classical field f~~(r, t) = opt exp (i&) [refs. 2*3)], where p,, is the pion chemical potential. In the isosymmetrical medium ,cI,~= 0. The rr-condensate instability should be expected for modes with k - k. -pF. Therefore in the Fourier expansion of q(r) in plane waves only the most important terms with k = k. [refs. “‘“)I can be left. Let the amplitude a of the field q(r) be defined so that (${r)> = a2. At present some most simple spa&l and isotopical configurations of the condensate field are being discussed in the literature 2*3).For example, one may choose cpl(r) = (p2(r) = p3(r) = d$a
sin koz
(204
(the isotopicahy symmetrical solution) or q&r)
= J&ql(r)
f icp2(r)) = @a e*:iko+,
qzg = q3 4 0
(2Ob)
(the electrically neutral condensate
of charged mesons). Going on the above assumptions, one can write the effective lagrangian follows 2*3): 2(x) = $ C D-‘(0, w,k
k; p, T)(%&l)2+~D-1@,
as
ko; p, W~o,&))~+%,rw (21)
and the effective hamiltonian
Within our approximation the classical field contains w = F,, = 0 and only the wave vectors *t-k,-,. The condition field amplitude a, x2
a”
aa=(“3 aa=0.
only the frequency (19) determines the
(194
The existence of the nontrivial solution of (19), (19a) a # 0, while being a necessary condition of the phase transition, is not always a sufficient one. To obtain the su~cient condition, let us consider a change of the thermodynamical potential AL!(a) by the inclusion of the condensate field:
(22) The dependence of A&?(a ; p, T) on a may take qu~itatively different shapes relative to the kind of the interaction p& as is shown in fig. 3. In the case of curve 1, when eq. (19a) has only one solution 8(p, T) # 0 at p 2jTC(T), it is this d that
539
0
Fig. 3. An illustration of the qualitative dependence of Afin on the squared amplitude of the condensate field a2 for different variants of the theory. Curve 1 -second-order phase transition, 2 -first-order phase transition, 3 -no phase transition. Curve 2a has the bending point corresponding to the single solution of eqs. (23), (25). Curve 2c corresponds to the critical density.
corresponds to a stable equilib~um. This solution d comes into existence at p = bC(T) and then increases beginning from zero with the increase of p. The above-mentioned corresponds to the second-order phase transitions. But a qualitatively different dependence of AJ2 (a ; p, T) on a given by the curves 2a-2d in fig. 3 is possible as will be shown further. In this case at a given T eq. (19a) has two solutions for p >pb(T) (curve Za), corresponding to the two extrema of Af.2(a2). If the curve 2b has a minimum of Af2(a2) 2 0, then, evidently, the condensate field is not formed yet, since it is not energetically profitable. It is necessary for the formation of the field that min AR(a; p, T) ~0 (curves 2c, 2d). The critical density b,(T) is determined from the condition min AJ’~(u ; & T) = 0 (curve 2~); at p >&i,(T) min Af2(u ; p, T) -C0. At p = &(2”) the condensate field appears due to a jump of the amplitude a, = n&(T)) which corresponds to the first order phase transition. The _Z’Zmwritten in the form (2b), as shown in sect. 6, brings about such a picture. Naturally, one can choose pTW so that AR has no extrema at any p, T and no condensation takes place. 7. The first order phase transition to the pion condensed
state
Let us assume that the ground state of the system contains the pion condensate field cp(r) of the form (20a). After calculating the average
540
G.G. Bunatian
and I.N. Mishustin
/ Pion condensation
eq. (19a) for the equilibrium amplitude d of the condensate field reads an ~=wfja+5AaN(O;p, The contribution
to the polarization
T)+$4a3=0.
(23)
operator induced by cp is
II”, =Aa’+2A(cpZ(r)).
(24)
For the isosymmetric medium and condensate field (20a) it is reasonable to assume that (cpZ(r)) =$(cp”(r)) = $a’. Thus in eqs. (ll), (13) IIa=$la2
and
(3(p, T)=&(p,
T)+5flN(O;p,
T)+&ia’.
(25)
From eqs. (23), (25) we find generally that
(26) where d is the true solution of eq. (23). From eqs (22), (23) we can write An(a) = Af2c,(a) +A&(u) a A&(a)
=$oifia2+$Aa4,
Al&(a) = 5A
I0
N(a)a da.
(27)
This expression is valid for an arbitrary a and, in particular, for a = ii. Now let us discuss separately the situation at zero, low and high temperatures. Zero temperature. At T = 0 quantum fluctuations of the pion field contribute positively to iV(0; p, O), eq. (9), stabilizing (3*. After absorbing the constant 5AA into the renormalized gap w& one obtains from eqs. (23), (25) an -=m~a-5AB&?u+~Aa3=0,
(284
&.I
L3*=co~-5ABh?+~Aa2.
(28b)
Eliminating & * one has la\ =~(~A)“*B*[~AB*-~~~/A]“*.
(2%
From this result it is clear that the solution d # 0 appears first at the point where the square root vanishes, i.e. w~~‘w~GO~,O)=~)~~(O)-P~]=~*B*.
(30)
This solution corresponds to a bending point on the AL!(u*) curve (see fig. 3, curve 2a). At pb
G.G. Bunatian and I.N. Mishustin / Pion condensation
541
one corresponding to the maximum of A~((a2) and the second (larger) one, to the minimum of Aa (see fig. 3, curves 2b, 2~). These solutions take place first for 40 > 0 and should be interpreted as metastable states with cp# 0. With the density increasing above Pb the minimum of AL!(a*) decreases and at a certain critical density & the nonhomogeneous state with cp# 0 becomes stable, i.e. energetically more favourable than the homogeneous state with cp= 0. To evaluate An(a) we should know (3* as a function of a. This function is given by the solution of eq. (28b). Taking into account the discussion below eq. (15) we have (31) at C,J~ + z/la* > 0 and &* = 0 at wg + $Aa* < 0. The latter inequality can be realized at og< 0 only. Eq. (31) is a generalization of eq. (15) for a2 # 0. Now we can perform the integration in eq. (27) that gives A&(a)
= -B[w~+~A2B2+~Aa2]3’2+~A2B2
+A~I,
m~+$Aa*>O,
(32)
where 25
B(w;+~rl Ai& =
2
2 312
B )
yA3~4+yA~2&j,
,
w&o W&O.
(33)
Using eq. (29) one can easily express Ah!@) in terms of “‘0 only, but we do not present these rather long expressions here. Equating AiT! to zero we can find the value of wg corresponding to the new critical density GC.Our analysis shows that this value, o& = w:&, O), is extremely close to that given by eq. (30) and corresponding to the bending point. Indeed, AR at the bending point is extremely close to zero, precisely, Af2(~: = cd&,) = 0.0012A3B4. Thus, we can conclude that the first-order phase transition to the nonhomogeneous state with cp# 0 has to occur at w;, =q(pc-P”+$$A*B*,
(3; s;*(&
0)
-$&
a,-$(iA)“*B,
+&*B*.
(34)
Notice that the renormalized gap (3: of the pion spectrum has a rather large jump at the critical point. Using eq. (31), one has ;;;
(a2= ii,“)
$(a2=0)
=
4.
When the density becomes equal to the bare critical point, p = pc, of = 0, the two solutions (29), corresponding to the maximum and minimum of AR, are 0 and %A)““B, respectively. The barrier between a nonhomogeneous state and normal
542
G.G. Bunatian
and I.N. Mishustin
/ Pion condensation
one disappears at this point. The energy gained condensed state is equal to 0.043A3B4. At higher a plus sign in eq. (29) has to be used. It is clear Iw$ >>A2B2 the amplitude and the energy of the values calculated in the mean field approximation: n:, =-$&A,
by the transition to the pion densities only the solution with from eqs (29) and (32) that at condensate field approach the
A~cl=-&&i.
(32a)
Low temperatures. Taking into consideration the above-said the characteristic energy scale at zero temperature has been established as AB. Thus, a temperature is assumed to be low if T<
(35)
Now let us consider low temperature properties of the pion condensation. To calculate the leading correction terms to the zero-temperature results, we include the low-temperature contribution to the correlation function, eq. (13), in the form N(O;p, T)=A-B[&+&r2~2T2/~;],
(36)
where
and is generally given by eq. (31). Substituting eq. (36) in eqs (23), (25)and using eq. (26) we find for the low-temperature case
Jtz2(a =a)=
(&162)1’2=$lB
+$&i2B2-&j-&~2AB~2T2/&]1'2.
(37)
By definition &g is given by the same formula with T = 0.As at T = 0 the solutions for d appear first when the square root in eq. (37) vanishes. These solutions correspond to metastable states up to the critical density p",(T) when An(a)= 0. But, as well as in the zero-temperature case, these two conditions almost coincide and the critical density can be determined roughly by zero of the square root in eq. (37). Keeping only the main terms one has
(38) where p”,(O)=p“ is given by eq. (34). This relation can be rewritten transparent form: p”,(T) = ,&(O)(l+ T’/T:)
,
T<
in a more
(38')
where To and TI are expressed in terms of parametes B, p"c and A. Using the estimate B = 0.7, p = 0.5,v = 1.2and p”== 2p,, consistent with the parameters of eq. (l), one has numerically To- 0.1A2and Tl- O.lA. If A<< 1, then the lowtemperature region is very short and the temperature dependence of p”Cin this region is very weak (To<< TI). Using eqs. (27), (36) and (37), one can calculate AR for the low-temperature case. Formally it differs from that of eq. (32) by an
G.G. Bunatian and I.N. Mishustin / Pion condensation
543
additional T2 term:
-[(~A2B2+W~+&z
*) 112-&31-l}
.
Induced by the phase transition, a jump of the entropy density at the critical point is given by the expression AS = -;r*p*T/A
= -4S(T/A)m:.
(39)
As expected, AS < 0 and AS + 0 at T + 0. If A 6 1, the entropy jump is small over the whole region of low temperatures T < To. The negative sign of the AS is quite natural when entropy is assumed to be a function of the excitation energy E”. It means that the entropy of an inhomogeneous phase grows more rapidly with E” than that of a normal phase. According to the Gibbs criterion of the maximum entropy the pion condensed state will be realized when its entropy exceeds that of the normal state. High temperatures. As it has been shown above, the pion field quantum fluctuations are valid at zero and very low temperatures T c To=0.1A2. In a more interesting case with the temperatures being high enough, T >>To, the physical picture is determined by the thermal fluctuations. The correlation function N(O;p, T) is given by eq. (11) in this case. Substituting into eqs. (23), (29, one can see that nontrivial solutions a # 0 are possible only at rather large negative values of wi. Further it will be convenient to use new variables: 2
(d!?-wiy
5AcT & = (_213/2
;
a&co.
(40)
One will see that all critical parameters are scaled- with E. Eliminating (3’ from eqs. (23), (29, one can easily get an explicit equation for 8 =Aa’/(-co;): fi(l-;@=&&.
(41)
Evidently, at small E eq. (41) has nontrivial solutions in the interval 0<8<$.
(42)
There is no solution s= 0 and the condensate field cp can appear with a finite amplitude d # 0 only. With decreasing E (decreasing T or increasing p) the solution of eq. (41) appears first when E reaches the value of Q, =@ corresponding to the maximum of the 1.h.s. at 8 =$. This solution corresponds to a bending point of the Ati curve (see fig. 3, curve 2a). At E C&b eq. (41) has two solutions corresponding to the maximum and to the minimum of Afi(a*), As long as min AL! (a”) > 0, these solutions have to be interpreted as metastable states of the system with cpZ 0. At a certain critical value cc the nonhomogeneous state with
544
G.G. Bunatian and I.N. Mishustin / Pion condensation
0 becomes stable. To find this critical point, we must know the explicit dependence dR(a*). Using eqs (25), (27) with N taken from eq. (ll), we get @#
~n(a2)=~~~a2+~A~4+3cT[J~-2)~~/32(a2)]~z, where the function CG “(a “) is determined
(43)
from the cubic equation
&*-(5A~T/&?)-o&$la~=O.
(44)
At the critical point, E = E, or p =6,(T), the curve Ati is tangential to the horizontal axis at ti*(p,(T), T) =az(T) (or f?(.sJ = 8,), as is curve 2c in fig. 3. It is clear that such a situation corresponds to the first-order phase transition. The simplified model. Now let us consider a simplified model illustrating the situation ‘near the critical point fairly well. It will be seen later that in the case of thermal fluctuations a very good (10% accuracy) approximation is E
(45)
eq. (44) has a simple solution
~=J+-g[l-j&2+*-~]
3 3
(46)
3
with E and 8 defined by eq. (41). Taking into account eq. (45) we immediately get the equation for the equilibrium value 8 = s X&l-$t?)=&,
0<8<$.
(47)
In comparison with eq. (42) the coefficient ? is replaced here by 2. Actually the relative difference between these two numbers depending on the condensate field structure is a really small parameter of the model. According to eq. (47) the metastable states appear first at &b= J& with & = $. These values differ by 10% 1 from the exact numbers of 2Jm and 3, respectively. Substituting solution (46) in eq. (43) and keeping the main terms, one has An(e)
-2 [aBZEa$$+.. . ,
=
3
(48)
I
From the condition da(@) = 0 one can easily find the parameters of the critical point EC=+/; 2
BE=&.
In terms of temperature a change of the critical density Ap,(T) =p”,(T) -p,(T) induced by the thermal fluctuations is Ap,(T) = -w,,,(T)/q
= (1/77)(5A~T/~,)~‘~ =7.5(~lT)“~.
(50)
The squared amplitude of the condensate field at the critical point is a;(T)
= &(-&,(T))/A
=2.7A-1’3T2’3.
(51)
G.G. ~~~~tian and I.N.
M~s~~~i~ / Ron ~~~den~a~~o~
545
The numerical coefficients are obtained here for c = 0.6 which is consistent with the parameter values used above (see eq. (1)). It follows from eq. (46) that the renormal~~ed gap G2 of the pion excitation spectrum has a large jump at the critical point: A0“: =Uj2(8’0&c)-(;2(0,E,)=~~~, where 6: =&(&,
EJ = -&a& =0.45(1IT)~‘~.
From eq. (48) it is easy to calculate the jump of the entropy density at the critical point: AS = -~(25~~~2}2’3(~T)~‘3 = -O.~(JIT)~‘~ o
(52)
This value is rather small even at AT - 1. It is interesting to note that the function Ati( eq. (48), besides the rn~njrn~ at 8 = s always has a maxims at 8 = 6~‘. It means that at finite temperatures the pion condensed state is separated from the normal one by a potential barrier of the height: maxAfi(fB, E = aJ =0.02A1’3T4’3,
WW
Having the explicit expression for G2@, T), eq. (51), one can reformulate the constraints discussed in sects. 3 and 4 in a more transparent way. Namely, the high-temperature expansion can be used if ?rT >G2, but the neglection of complicated diagrams may be justified if AT<< G2ip, T). Using eq. (51), one finds that both the conditions will be satisfied at the critical point if 1
ic2A2<
1 2=o.09. ZC
(53)
Note that the 1.h.s. of this inequality is very close to TO= 0.l.A” which determines the region of quantum ~uctuations* Inequality (53) is easily satisfied for A =G0.1. But we hope that our theory yields quaiitatively correct results even for A -0.5. We also expect that a picture being qualitatively the same will take place when the pion condensate field has the space and isospin structure other than that of (20a). Concluding this section we refer to the recent paper 21) by Dyugaev where a similar problem is considered within the framework of the comas-Fermi model. ~~~eric~l results. Computational results are represented in figs. 4-6. They show the increment A@,(T) of the critical density, the jump of the squared amplitude of the condensate field cxz (T) at p = &(T) and the renorma~zed gap G ‘$(T) (in units of T) above (a2 = 0) and below (a2 = a:) the critical point. These results correspond to high temperature~T >>T0 = 0.1A2) and are obtained by solving exact equations f43), (44) of sect. 4. The corresponding results for the low temperature case are not given in figs. 4-6 because the low temperature region is too narrow, T < To
546
G.G. Bunatian and I.N. Mishustin / Pion condensation
0
0.5
’
T
Fig. 4. Temperature dependence of the critical density increment Ap, induced by thermal pion field fluctuations for first-order phase transition. Curves a and b are drawn for A = 0.1 and 0.5, respectively.
(To = 0.001 for A 4) that only the considered if the that this condition
= 0.1 and To = 0.025 for A = 0.5). It has been proven above (sect. simplest diagrams induced by pion field fluctuations can be condition AT/&: K 1 is satisfied. From fig. 6 we may conclude is fulfilled only by order of magnitude even at A = 0.1.
0
0.5
’
T
Fig. 5. The jump of the squared amplitude of the condensate field a:(T) in the critical function of T. Curves a and b are drawn for A = 0.1 and 0.5, respectively.
point
as a
G.G. ~~nat~a~and IN. Mishasti~j Pion condensation
547
Fig. 6. Temperature dependence of the renormalized gap G;(T) on the temperature ratio above (a, b) and below (a’, b’f the critical point. Curves a, a’ correspond to A = 0.1 and b, b’, to A = OS.
Fig. 4 shows that thermal ~uctuations induce a considerable increment of the critical density Ap,(T) which grows faster than it has been predicted by the Tzi3 law derived in the framework of the simplified model of sect. 7. This difference appears mainly because in the numerical calculation the density dependence of the condensate field wave number kg in accordance with eq. (1) has been taken into account. The critical amplitude of the condensate field a:(T) (fig. 5) is also a more rapidly increasing function of T than T2’3. 8. ~iscussjon of the basic features of the theory and its eousequenees The basic conclusion of the present paper is that strong pion field fluctuations must arise near the critical point of pion condensation due to the soft mode with a large momentum k,. That is the main difference of the investigated phenomenon as compared with other phase transitions, e.g. with the transition in a superconductor where, due to k,= 0, the ~uctuations of the order parameter have no singularities near the critical point and lead only to minor modifications of p. T,, 4:. The fluctuations begin to grow well before reaching the “old” critical point (calculated without taking them into account) and reach the maximum at p -p,(T), From the consideration above we can easily find the domain of parameters where fluctuations are small At zero and low temperat~es 7’ < TOthe quantum fluctuations contribution may be considered as small perturbation (6’ = ~“0) if “; >>25A2B2=20A2.
CW
In the high-temperature case (Trr >>03 the condition of small thermal ~uctuations of the pion field is 5AcT/oo<< & or
548
G.G. Bunatian and I.N. Mishustin / Pion condensation
At T - 1 this condition can be satisfied only if A = 0.1-0.2. Thus, the mean field approximation which up till now has been used to calculate the critical parameters of the pion condensation becomes qualitatively wrong well before the critical point. This qualitative conclusion has been confirmed by our numerical calculations. As is seen in figs. 4, 5, the area of strong fluctuations (with the Ap,(T) serving as their measure) expands promptly with the increasing temperature. So, well before the critical point the thermal fluctuations become important and decisively change the character of the phase transition as well as the thermodynamical character of the system. That is what makes the pion condensation essentially different from a phase transition proceeding at k, = 0 (e.g. superconductivity) where the thermal fluctuations are vital only in the nearest vicinity of the critical point. Strong fluctuations lead to some important physical consequences. First of all they affect the character of a transition. From general considerations it is clear that a phase transition must start, if at all, in the area of not very strong fluctuations (the “old” critical point), but they should not be very weak either, otherwise the Landau theory would be valid in this area (54). Going away from the critical point the formulas for the condensate field amplitude and for the energy gain must be transformed into those of the Landau theory. In this paper two approaches of two different types have been employed. First, closed eqs. (6), (7) are used to obtain N, 17” (or (3’) either in the region of T cc 3’ < 1 or in that of AT/G2 <<1, since only there it suffices to consider just the first simplest graphs of an infinite series of graphs (3a), ‘(3b). Second, eqs. (6), (7) are not solved exactly, though it is possible in principle, but transformed into the form of (ll), (lla) for G2<
G.G. Bunatian and LN. Mishustin / Pion condensafim
549
Further study of the ~-condensation would require use of all the knowledge that has already been obtained on phase transitions including the critical indices, scale invariance, etc. The results obtained would allow us to change our view of the methods of the pion condensation investigation in high-energy heavy ion cohisions, Until now the following has been used as a test for the presence of pion condensation: (a) the occurrence in the phase transition region of two shock waves “) instead of one in a normal phase; (b) a threshold growth of the thermal energy and temperature of the matter and, consequently, of the multiplicity of “thermal” n--mesons due to the new phase which is energetically more profitable 23); (c) a reduction of the sidewards matter flow in heavy-ion collisions due to the softening of the equations of state 24). U~ortunately, the first two methods do not seem to be realistic. First, because the dimensions of nuclei are not large enough for the evidently shaped shock waves to propagate in the latter. Second, because within the area of densities and temperatures achieved in a collision the energy gain of pion condensation appears to be very small, less than 10 MeV per nucleon, as follows from calculations presented in sect. 7 [eqs. (32), (48), and see also ref. ““)f. It seems more promising to search for a phase transition with the help of the pion field fluctuations following it. They arise due to the formation of the ‘“soft mode” of pion excitations and are always strong near the critical point regardless of the energy gain of the new phase. This must bring about several observable effects. First of all it is clear that due to the ~uctuations near the critical point the behaviour of all the thermod~~ical parameters changes, in particular, approaching the critical point, the heat capacity of the system must increase sharply. If during the collision of heavy ions the matter is compressed enough, that is the system approaches the critical point, the system temperature increases slower than in the case of the ideal gas and comes to saturation. Thus, during the collision of heavy ions in a certain collision energy interval corresponding to the vicinity of the phase transition, a slow-down of the temperature growth and of the thermal r-mesons multiplicity is expected in contrast with the statements of ref. 2z). Moreover, the strong ~uctuations must lead to an increase of the effective interaction of particles with the nuclear matter resulting in a considerable reduction of the nucleon mean free path 25). An appearance of maxima in two-particle correlation functions of secondary particles with relative momenta -kO is also predicted. All these questions require further study. In conclusion the authors would like to express their gratitude to S.T. Beiyaev, A.M. Dyugaev, N.E. Zein, D.N. Voskresensky; E.E. Saperstein, V.A. Khodel for
550
G.C. Bunatian and IN. Mishustin / Pion condensation
critical remarks and usefu1 discussions and to N.N. Serebryakov preparation of this paper.
for his help in the
References 1) A.B. Migdal, ZhETF (USSR) 61(1971); 63 (1972) 1993 A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF (USSR) 66 (1974) 443 2) A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF (USSR) 70 j1376j 1592 3) A.B. Migdal, Fermion and bosons in strong fields (Nauka, Moscow, 1978) 4) MA. Troitsky, M.B. Roldaev and N.I. Cbekunaev, ZhETF (USSR) 73 (1977) 1258; V.S. Butsev and D. Chultem, Phys. Lett. 67B (1977) 33 5) 0. Baym, in Proc. of the 7th Int. Conf. on high energy physics and nuclear structure, Zurich, 1977, ed. M.P. Lecher (Birkh;iuser, Basel, 1977) p. 309; S.-0-Backman and W. Weise, in Mesons in nuclei, vol. III, ed M. Rho and D.H. Wilkinson gosh-Holland, Amsterdam, 1979) p. 1095 6) G.G. Bunatian, JINR, P4-11755, Dubna, 1978; Yad. Fiz. (USSR) 30 (1979) 258 7) K.K. Gudyma, H. Iwe and V.D. Toneev, J. of Phys. 95 (1979) 229 8) A& MigdaI, A.I. Chernoutsan and I.N. Mishustin, in Proc. Int. Conf. on extreme states in nucfear systems, Dresden, 1980, ed. H. Prade, S. Tesch (Rossendorf, 1980) vol. 2, p. 45 9) V. Ruek, M. Gyulassy and W. Greiner, Z. Phys. A277 (1976) 391; P. Necking, Nucl. Phys. A348 (1980) 493; M. Wakamatsu and A. Hagashi, Prog. Theor. Phys. 63 (1980) 1688; G. Baym, Nucl. Phys. A352 (1981) 365; K. Ko~ehmainer and G. Baym, Nucl. Phys. A382 (1982) 882; G. Baym, B.L. Friman and G. Grinstein, Nucl. Phys. B210 (1982) 193 10) D.N. Voskresensky, I.N. Mishustin, ZhETF Pis’ma (USSR) 28 (1978) 486 11) G.G. Bunatian, JINR, P2-12518, Dubna, 1979; Yad. Fiz. (USSR) 31(1980) 1186; Abstr. of Contr. Papers of the Int. Conf. on extreme states in Nuclear Systems, Dresden,‘1980, ed. R. Azlet, B. Kiihn (Rossendorf, 1980) vol. 1, p. 14 12) G.G. Bunatian, I.N. Mishustin, JINR, PZ-81-291, Dubna, 1981; JINR, P2-81-500, Dubna, 1981 13) D,N. Voskresensky and I.N. M~shustin, ZhETF Pis’ma (USSR) 34 (1981) 317 14) A.A. Abrikosov, L.P. Gorlcov and I.E. DzyaIoshinsky, The methods of quantum field theory in statistical physics (FM, Moscow, 1962) 15) I.N. Borsov, E.E. Saperstein, S.V. Tolokonnikov and S.A. Fayans, Particles and Nucleus 12 (1981) 848 16) I.M. Luttinger and J.C. Ward, Phys. Rev. 118(1960) 1417 17) H, Kleinert, Phys. Lett. 102B (1981) 5 18) L.S. Gradstein and I.M. Ryzhik, The tables of integrals, sums, series and products (FM, Moscow, 1963) 19) A.M. Dyugaev, ZhETF Pis’ma (USSR) 22 (1975) 181 20) S.A. Brazovsky, ZhETF (USSR) 68 (1975) 175 21) A.M. Dyugaev, ZhETF Pis’ma (USSR) 35 (1982) 341 22) V.M. Galitsky and I.N. Mishustin, Phys.Lett. 72B (1978) 285 23) G, Buchwald et aZ., in Proc. Int. Conf. on Extreme States in Nuclear Systems, Dresden, 1980, ed. H. Prade, S. Tesch (Rossendorf, 1980) vol. 2, p. 23 24. H. StiJcker etaf., Phys. Rev. Lett. 47 (1981) 1807 25) M. Gyulassy, Nucl. Phys. A354 (1981) 395
THERMODYNAMICAL
THEORY
OF PION CONDENSATION
G.G. BUNATIAN Joint Institutefor P&clear Research, Dubtta, USSR and 1.N. ~IS~USTIN ~~rc~atov roseate ofAtomic Energy (IA@, Moscow, USSR Received 21 December 1981 (Revised 6 January 1983) Abstract: A tbe~modynam~c~ approach to the description of the pion field and ~-condensation in nuclear matter of high density and nonzero temperature is developed. It has been shown that the quantum and thermal fluctuations of pion fields play a decisive role near the critical point; they are the reason why the equation D-‘&J, R, p, T) = 0 has no solutions 02(k) ==z 0 at any density 6 and temperature T, i.e. the criterion earlier used for the formation of the rr-condensate is not satisfied. In order that the g-condensate may arise in thesystem with given p, T the thermodynamical potentiaI should decrease when the ?r-condensate field is switched on: A@p) 4 0. The character as well as the critical parameters of the phase transition are strongIy dependent on the effective VW interaction. In the framework of the &I” model, the critical density and the jumps of the condensate field amplitude, along with the renormatized gap of the pion excitation spectrum and of the thermodyn~i~al quantities, are estimated. The area of appIicab~Iity of our approach as well as possible consequences of the theory are discussed.
1. Introduction Recently the behaviour of nuclear matter has been intensively studied under extreme conditions: high density, high temperature, anomalous ratio Z/N, etc. Interest was aroused by the works of Migdal lm3)who indicated the possibility of phase transitions in nuclear matter caused by pion condensate formation. Both theory and experiment have shown 4, that there seems to be no rr-condensate in ordinary nuclei, i.e. at normal density ~0. To create the v-condensate, one should significantly increase the density of the nuclear matter p IS (2-3)p0, since this follows from different estimates made with realistic parameters of particle interaction in nuclear matter 5,6). This can be achieved in the laboratory in collisions of heavy nuclei wifh energies higher than E/A -0.5-l GeV per nucleon ‘). Certainly, the energy brought about by such collisions in the nucleus will only partly serve to increase the density. A considerable part of it will be spent on the thermal excitation of the dense nuclear matter created 7>.The excited state can be studied within the statistical approach characterized by a temperature T. Calculations have shown that at E/A a 1 GeV the temperature appears to be sufficiently high, i.e. up to 525
526
G.G. Bunatian
and I.N. Mishustin / Pion condensation
T - mmcL,where m, is the mass of the r-meson. The r-condensate may also exist in the high-dense matter of a neutron star. The dynamical effects associated with the phase transition in the neutron star formation stage following the supernova explosion when T - O.lm,c’ [ref. “)I are most interesting. Therefore it is important to study the properties of the highly dense nuclear matter at nonzero temperatures (see refs. 6*g-11)),It is usually assumed that the formation of the rr-condensate field in the medium is a conventional second-order phase transition. In ref. rr) it is noted that one should take into consideration the occurrence of a large number of interacting g-meson excitations in the nuclear medium at T # 0 in thermal equilibrium with the matter, i.e. “black-body radiation of r-mesons”. The corresponding general formulas for the polarization operator #and the pion .Green function are derived. The basic characteristic feature of precritical nuclear matter is the softening of the pion degree of freedom with the increase of density precisely at the high pion-field momenta k,- md. Large k, implies a large phase volume of the pion-field fluctuations near the rr-condensation critical point. As a result, the amplitude of the fluctuations increases greatly when the system parameters achieve critical values r2,r3). This phenomenon is of prime importance. If one takes into consideration the above-said, it will change the conventional physical picture, thus leading to a number of qualitatively new effects as it will be shown below. Sect. 2 is devoted to a study of the properties of the polarization operator n, the pion Green function D and the ~TT~T interaction in the matter. Sect. 3 contains a calculation of the correlation function N(p, T) of the pion field fluctuations and their main contribution to I?. The general formulas for these quantities are derived and their behaviour in the vicinity of the critical point is studied. In sect. 4 the relative contribution of various diagrams to 17, D is estimated. Conditions are outlined under which it is possible to obtain approximate close equations for these quantities. Using the said equations the physical picture of the behaviour of the system near the critical point in the absence of the condensate field is given in sect. 5. Sect. 6 presents the equations of the r-condensate transition derived from the requirement that the thermodynamical potential 0 be a minimum in the nonzero condensate field. In sect. 7 the physical picture arising in the first-order phase transition is described. Finally, sect. 8 deals with the main features of the general theory developed here as well as with some physical conclusions drawn from it.
2. Polarization
operator and n-meson
Green function in the medium
at T # 0
Let us first discuss the principal characteristics of pion excitation in the medium in the absence of TZ- interaction. The isotopically symmetric equilibrium system of interacting nucleons, A,, isobars and r-mesons at T # 0, is considered. The polarization operator 17”(0, k ; p, T) and meson Green function W’(W, k ; p, T) = [co2 - m,+z2- k2 -i7”(w, k; p ; T)]-l calculated in ref. 11) for this medium taking no
G.G. Bunatian and I.N. Mishustin J Pion condensation
521
account for the rir interaction and pion field ~uctuations seem satisfactory and we shall use them in the present paper. Note that the D(w, k ; p, T) function in ref. ‘I> is essentially the analytic continuation of the Matzubara temperature Green function ti((wn, k ; p, T) from discrete values w, = 2drzT, m = 0, A1 . . . , to the complex plane w. Thus, in the upper half of the latter (Im w > 0) the function 13 (w) coincides with the retarded Green function D&W), while in the lower half (Im w C 0), with the advanced one DA(u) [ref. ‘“)I. In all the earlier works the critical density p,(T) was defined as the density p =pc(T) by which the equation D-‘(w, k,; p, T) = 0 had the solution w’(k,) = 0 for the one pion field momentum k = k&,(T), T). For p >p,(T) the solution o’(k) < 0 exists within the certain interval kl< k < k2, kl
T)]-l = w2 -
l-k2-Re.U”(o,k;p,T)-iImB”(w,k;p,T)
-o’a(T)+iwp(T)sign -&(P,
T)-y(k
-ko)‘,
(Imo)
Imu+O;
The calculation, performed as described in refs. 6S11),yields the values of the parameters in eq. (1). The values n = 1, 2, x = 1.3, k: = 3, y =0.8 are practically independent of T; at T =O, fi(k,)=OS, a(ko)=1.6 The value P(T) increases, P(T = 1) = 0.8, cy(T) decreases with the increase of 7’. Hereinafter, the R = c = pn,.,c’= 1 system of units is used, for example, T = 1 means T = mg2 = 140 MeV, p. iz=0.5 = 0.17 fmd3. At T = 0 and o s 2 the Im 17” becomes zero at large k e.g. at p = 1.5-2 it is zero at k 2 8. At T - 1 ImIl”(w, k; p, 2”) is not exactly zero at such values of o and k, but becomes infinitely small with the increase of k [see figs. 2 and 3 in refs. “)I. The approximation (1) holds true within fairly wide limits, at least one can use it at T s 1, w 6 k+ Ik - kol c 1, lw:Is 1, which is valid for further consideration. Note that, generally speaking, the expression (1) for the D” function may be considered as a semiphenomenological one and its parameters could be obtained from the presently available experimental data concerning the study of the nuclei
528
G.G. Bunatian
and I.N. Mishustin / Pion condensation
degree of closeness to the r-condenstate instability “). The proximity of the system without the 7r~rrinteraction to the rr-condensate instability at given p, T is evidently characterized by the degree of the og(p, T) = min [D”(O, k; p, T)]-’ proximity to zero. This minimum is achieved at a high critical momentum of the pion field ko. Therefore near the critical point the phase volume of soft pion field modes is large and the fluctuation amplitude increases fast with wg(p, T) + 0 [refs. 12,1’)]. Note that for the system with the roton-type spectrum, which may be obtained from (1) at Im 17” = 0,~: is a gap in the excitation spectrum. For an example of the spectrum to be obtained from the equation [D”(w, k)]-’ = 0 at Im I7”= 0, refer to ref. ‘l), fig. 2. Now it is necessary to include the r-meson interactions. The rr~~ interaction lagrangian in the medium is presented here as follows: *
2&*(x) =
x9
-&i(~(x))2(tr(x))2 =
(24
where the isotopic vector & is the operator of the r-meson field, (I^is the effective amplitude of the ~7r interaction in the medium, which depends on the kinematical parameters and properties of the medium in a complicated way. It is not possible to calculate it yet. Within our approach A is assumed to be constant and, indeed, is a free parameter of the theory. In order to make our approach valid, the value of A should not exceed 0.5. The purpose of the calculation using (2a) is only to qualitatively estimate the effect made by the account of the nr interaction on the properties of r-meson field in the medium. The influence of the medium on A is discussed in refs. 293).I n a 11our calculations the rr7r interaction is assumed to be repulsive for A > 0, which is essential for the stability of the system. The case of A < 0 will be discussed in brief in the last section of the paper. The self-energy part of the pion IZ T, induced by the interaction lagrangian (2a), is an infinite series of skeleton graphs:
KY(W, k;p, T)= (3a) where wavy lines correspond to the exact pion Green functions in the medium with account taken of 17” and fll”(v = 0, f or v = 1,2,3). Further we shall consider the system with a m-condensate field q(x). Then, instead of (2a), we shall have: (2b) P, = -&4(+(X) +lp(x))4. This interaction lagrangian induces an addition to the polarization
operator:
(3b)
529
G.G. 3anat~a~ and I.N. ~is~ustin f Pion condensation
AS we shall see in sect. 7, with the interaction lagrangian @a, b) the first-order phase transition is possible in the system: the condensate field originates from (cp(x))” f 0. However, it is clear that such interaction lagrangians may be chosen that there would be no Ir-condensate phase transition at any p and T within our approach. But, in any case, the fluctuations of the pion field near the “old” critical point affect the properties of the system considerably. When the pion excitations interaction and the classical field are taken into account, the pion Green function takes the form
D;‘(&&k;p,
r)=12-l-k2-17~(0,k;p,
T)
,-n~(U,k;p,T)-fl~(o,k;p,T).
(la)
The expression (la) is the equation for D, since P, II”, 17” depend on the total pion~-function. In the presence of the o- and ~-independent 17”, If” the approximation (1) for D-‘(w, k; p, T) apparently contains, instead of w%(p, T), the following quantity: &‘I$, T)=&(p,
T)+IF”(O,k;p,
T)+If”(O,k;p,
T),
(lb)
which plays the same role as 0: without the VQTinteraction (2a), (2b): &“(p, T) = min L)-l(O, k; p, T). The softening of the pion degree of freedom in nuclear matter would mean proximity to w-condensate instability for cS’- 0 and complete instability for &‘@, YZ’) < 0. But this (see sect. 5) cannot be achieved due to increasing pion field fluctuations near the criticdl point, It seems impossible to solve eq. (la) in the general form. An approximate solution is possible in a number of cases when the account of only the first simplest graphs in (3a), (3b) can be justified. Their contribution to the II”, ff” will be calculated in sect. 3. In sect. 4 we shall make estimates of the more complicated diagrams and find the conditions under which their contribution may be neglected.
3. Calculation
of the ~o~eIation function and of the main terms of the polarization
operators
II*, IIs.
It seems useful to introduce the correlation function of pion field ~uctuations here: N&Y
Pt
Tl = ~~~~(r;
P, %p=T,fO
= f%.&h,
Mb(~21~2~~
k; p, T) eik’r-mm:O,
r =r2-rl.
(4)
The averaging is performed here over the equilib~um state of the whole system at a temperature T. Notice that at r = 0 N is not the density of pion excitations in the medium, since they are absolutely unstable.
530
G.G. Bunatim and LN. ~is~astin / pion condensa~on
Fig. 1. The integration contour for eq. (5).
As can be seen below, the equation D-‘(w, k;p, T) = 0 does not have the solution w’(k) < 0 at any p, l” if the interacting pion excitations in the medium are taken into account. Then, using the conventional method of calculation 14’16)we find *l) for the isotopically symmetrical medium
x (5) = (e”’ - 1)-l , The integration contour is shown in fig. 1. As has already been noted, D(l -I-i0) = I&(t) is over the integration axis and D(t - iO>=Dn(E) below it. Using the well-known properties of the L&.+ for the Bose system 14), O,(~+io)=D;(G$-iO),
Re
&&)
= Re
OR(+)
Im OR(<)
,
=
-Em Dn(-0
(in the isosymmetric medium lTI. = 0), the expression (5) can be transformed
x {[e2- 1 - k2 - Re LX’,&k)]‘+ [Im lr,([, k)j2}-’ . The self-energy part corresponding medium
is easily expressed
into
(6)
to the first graph in (3) for the isosymmetric
in terms of the correlation
function:
H: (P, T) = 5~N”(O; P, T) .
(7)
In this case (LI = const) it is independent of w, k ; v = 0, It(l, 2,3). As can be easily seen, the expression (7) does not become equal to zero at T = 0, x(p) = 0. The expression for the n”(p, 0) contains also a trivial vacuum term iI”(0, 0) which has already entered into the observed pion mass. It should be subtracted from (7) and this is implied in the further consideration. The rest of the II” at T = 0, II”@, 0) D”(O, 0) is the contribution into the LIT of the pion field quantum fluctuations in the medium at T = 0. Note that formally the whole part of the 17” without ~(5) can be excluded In the medium
by writing down the lagrangian (2) in the normal product form 11’12). it may lead to a loss of the physical contribution of the quantum
G.G. ~una~ian and I.N. ~i~hus~~n / Pion condensation
531
fluctuations to the fT”, though that is not very essential in comparison with the contribution of the thermal fluctuations near the critical point. Let us consider only T 6 1, then 6% 1 is essential far the integral (6) and at k - k. the expressions (l), (lb) may be used to write down the expression for the denominator in (6) as follows:
Large k do not contribute to (6) due to the described above equation properties of Im n”(& k ; p, T). As will be seen below, the main con~ibution is given by k - ko, if ~5’4 1, since under a condition like that the integrand has a sharp maximum. Then, for the value of N(O; p, T) determining according to (7) the II” value we approximately have N(0; p, T) =
7T-3
Jk:
k’dk
o~d5B-i&-x(5))5 J
x ([b?+ y(k - kof2]2@-2 i- c$2>-1,
ewe0 kl=ckockz.
(8)
Developing the above consideration and omitting x(c) we have the following estimate for the contribution to N(0; p, 0) of the quantum fluctuations near the critical point:
This expression does not diverge at G2+0, i.e. the quantum fluctuations do not grow near the critical point. Making a series expansion at &’ + 0 one has N(O;~,O)=SA-B&.?+O(&~),
fj=-.----.ki 2$3~”
(9)
where the constants A and B are of the order unity in pionic units. With 4 = 1 and the other parameters the same as in eq. (1) A and B can be estimated as 0.25 and 0.9, respectively. Note that in the recent paper 17) the logarithmically diverging expression -In IS] for the quantum fluctuations was obtained by mistake. The author of ref. 17)made a mistake because he had ascribed wrong analytical properties to the pion Green function in the medium. Namely, his expression for I) would correspond to one having a real pole of the D-function at k - ko, w - 0, whereas, in reality, the D-function has branch points and is determined on the complex plane w with a cut aIong the real axis ‘I). From (1) one can conclude that ImD is not small at k - ko co =SkoF and changes its sign in crossing the real axis. A jump of the Im D on the cut determines the value of N, 17” [ref. ‘“)I.
532
G.G. Bunatian
and I.N. Mishustin / Pion condensation
expressions (S), (6), (8) a part of the integral containing x(t) is evidently the correlation function of the pion field thermal fluctuations. At T = 0 it surely becomes zero. From eq. (8) we obtain In
N(O;p, T)-N(O;p,0)=&\~:2k2dk~-1
x{ln($)-f-P(&)},
Y
=[~2+y(~-hd211PT,
where q is the Euler function r8). The integrand in (9) depends only on y, which is the value characterizing the “softening” of the pion degree of freedom divided by the temperature T. At y
C = 0.57721 . (10)
As is clear from eq. (10) and has been stated above, the integrand as a function of k has a sharp maximum at k - ko. For the estimates it is convenient to choose kI, k2 as symmetric about ko. The integration of the first two terms in (10) yields finite expressions at &I’+ 0, and the integral of ?r/y contains the term 4-l which evidently determines N and ILP at &_J2 K mT: N(0; p, T) = CT/&?,
Il”(p,
T) = 5ACT/&?,
C = k;/2&.
(11)
Eq. (9) may be written in a simpler form in another extreme case, y >>1, while keeping the condition T <<6’ < 1. We find 18) from eq. (10)
N(O; P, T) -NCR
P, 0)
=~~k~k2dk[,2~+,,~+...].
(12)
(Bernouilly numbers B2 =& B4 =&. Again the main contribution is made by k - k,. In the case of y >>1 only the first term should be left in (12). And once more it is convenient to take kI and k2 as symmetrical about ko. One can easily see that the integration in (12) yields the terms containing no singularities at ; -+ 0, or -K1, _; -2, -$ -3. For the estimates it suffices to take into account only the most “singular” terms -K3. Then at T + 0 N(O;p, T)=A-B
2 ,c3’T2 a-“---- 12 ~~2~3/2+’
* *
12 rr2 /3’T2 I7”(p, T) = 54A - 5AB dw - -l2 -((;2)3/2
1 7
+
* ’ ’
1 .
(13)
G.G. Bunatian and I.N. Mishustin / Pion condensation
533
While calculating further, one should keep in mind that eqs. (11) and (13) can be used only if the constraints under which they have been obtained are followed. One may move from p = p,(T) (i.e. w$(p, T) = 0) by increasing p until the pion D-function in the medium reaches a maximum at w - 0, k -kc, &‘< 1. At p appreciably exceeding p,(T) (i.e. w’, B 1) the parameters in (1) surely change, while the D(w, k ; p, T) character of eq. (1) remains basically the same. We shall employ eqs. (9), (ll), (13) derived in this section on the basis of eq. (1) to qualitatively study the physical picture for rather large p-values too. The correlation function N(r; p, Tj at Y # 0, necessary for the further calculation, can be easily expressed through iV(0; p, T) for the two cases considered above: N(r;p,
T)=N(O;p,
sink,, . r _1,1
T)ke
=.
(14)
0.
It is apparently characterized by the two lengths: the oscillation wavelength 2r/ko and the correlation radius rc = dy/G '@I, T). As usual, in phase transitions I, grows near the critical point. The main contribution to 17” has been made by the first graph in (3b) as can be seen below: n”, =~(&)>+2~(&))
*
(7b)
This expression will be used in the following. Thus, we have estimated the contribution of the simplest graphs to (3a), (3b). Now, let us deal with the others.
4. Estimation
of complicated
diagrams;
equations
for JIq N
In general all the complicated diagrams in (3a), (3b) are not small as compared with those calculated in the previous section. Only under certain conditions to be clarified here, it suffices to take into account just the first simplest diagrams in (3a), (3b) for calculating HT”, 17”. Let us consider first the contribution of the second graph into (3a) at w -0, k - ko. In the case of the quantum fluctuations (T = 0) this diagram, as well as the first one, yields no singular contribution at (3 + 0. Taking into account the fact that the second diagram has an additional vertex factor A, as compared with the first one, it can be assumed that its contribution is small if A CC1. At low temperatures, T <
534
G.G. 3~n~t~~n and IA? ~~shust~~ / piopl conversation
At high temperatures the contribution eq. (14), and can be estimated as:
-$ J dr eik”~3(r;p,
of this diagram is expressed through N(r),
T)Y~‘T*/W”~(~,
T>
.
Comparison with exp. (11) shows that this contribution may also be neglected if the parameter AT/G2 is small. At first glance one can conclude that the second diagram is even more important near the critical point. However, as we shall see later, 4’ is never too near to zero due to the influence of thermal fluctuations. At the phase transition point, provided that a transition takes place at all, (cz is fairly large, while the parameter AT/&E is actually small (see sect. 7, fig. 6) at n <<1. Using the same considerations, one can show that the other more complicated diagrams in (3a), as compared with the first one, may also be neglected. The contribution of the second diagram to (3b) is maximum for the components of the fields q(x), 2(x) having momenta equal in value but opposite in direction and frequencies equal to zero, i.e. the longitudinal channel zero momentum and frequency are transferred. In this case the contribution of the second diagram to (3b) is as follows:
It is easy to obtain the following approximate
estimate for this value:
-(cp2(x))(i12/&)[const + (T/4z)2] at low temperatures
T ccG* < 1, and -(cp”(x,>A”T/;”
at temperatures T >>c32. In the former case this estimate is small as compared with If” from (7b), since (T/&z)<< 1 and A/&E - 1 are near the critica point as will be shown in sect. 7. In the latter case the relative contribution of this diagram is proportional to the above-discussed parameter AT/ (32 < 1. Under this condition the relative contribution of more complicated diagrams is even less. Note that the contribution of such diagrams at T = Clwas discussed in ref. I’). Thus, there are reasons to take into account only the first diagrams in (3a), (3b). Notice that the same conclusions were made in ref. *‘) i.n the study of systems with the roton-type excitation spectra. To be exact, the additions to 17 may be other than those of (3a), (3b). It is clear that pion fluctuations and the occurrence of Q change the self-energy part of a
G.G. ~unutjan and I.N. ~sk~s~n
535
j Pion condensation
nucleon, which in its turn leads to an additional cont~bution
to R of the following
One can see that it is not as important to take this process into consideration near the critical point as that described in (3a), (3b). Since, according to the conclusions of the present section, only the con~butions of the first diagrams in (3a), (3b) are left in If”, 17”, eqs. (6)-(13) are transformed into closed equations for N(0; p, T) or If”@, T). The self-energy part 17”(w, k; p, T) entering eqs. (6), (7) was calculated in ref. ll). L!“(p, T) can be calculated numerically at given A, p, T without simplifying approximate transformations leading to (Q-(13). Having found TT”, & one can specify at which A, p, T the parameter AT/&~l’p, T) determining, as has been shown, the relative cont~bution of various diagrams to (3a), (3b), is actually small. In the present paper the behaviour of the system near the critical point is described by eqs. (1 I), (13), bearing in mind that one should remain within the field of application of these equations. Namely, the following relations should be fulfilled: (3z(p, T)K =T, AT/G: (p, T) <<1. At A K ?r it is possible to satisfy both. The values &(p, T) obtained in our calculations for various n will be presented below in figs. 4-6. It is also convenient to rewrite 17” in terms of & Using (lb) one obtains from (9), (1% (13) ~~=“~~~a~5A~T~~~,
(114
for high and low temperatures, respectively. In these equations 00 is different from that of eq. (1) by the constant 5AA ori~nating from the quantum fluctuations. This renormalization slightly changes the bar critical density p,(O) which is a rather uncertain quantity. Having analyzed various diagrams con~buting to the polarization operator and having established the main equations for lP and N, we proceed to the description of the system behaviour near the critical point first in the absence of a condensate field (sect. 5), and then in its possible presence (sects. 6, 7). 5. A physical picture near the critical point ~thout the condensate tieid Now let us consider the solution of eqs. (lla), (13a) without the condensate field (17” = 0). In case of zero temperature when the quantum fluctuations are taken into account, eq. (13a) has a simple solution:
536
G.G. Bunatian and I.N. Mishitin
Go
is positive at & > 0 and tends to zero at 0% + 0. In this limit 40 = ~~/~~~,
fl”(p,
/ Pion condensation
0) = -c&p,
0) *
It is easy to show that cj + 0 at wz < 0 and T + 0. To prove it let us consider the limit T+ 0 more carefully. At wx
&p,
T-+0)=
25A2c2T2
(16)
G%P, 0)
where w &, 0) = q [p,(O) -p]. In this limit a”@,
25A2c2
T+O)=-w;fP,0)+T2
m-qp2
I .
The solution (16) is valid if G2
-o:fp,0)
(.~AcT)~‘~ <<
.
f18a)
On the other hand, eq. (1 la) itself holds at T + 0 if G2 CCVT or
It is clear that at s~~ciently low temperatures (T < (Z5/~3)A2cz), if eq. (lgb) holds true, eq. (18a) is also true. Hence, at any wg< 0 and T-+ 0 one has G2-+ 0 and Lrw + -w z(p,0). In a similar way one can see that at 0: > 0 and T + 0 ~2’ tends to the limiting solution (3 (eq. (15)). From eq. (13a) one has (19) The dependence discussed above of G.7’on 0: (i.e. on p) at T =+O (quantum fluctuations) is presented by the curve in fig. 2. At finite temperatures ~7’ is always positive and higher than &* at T = 0 due to an additional contribution of the thermal fluctuations. For instance, at p = p,(T), i.e. to; = 0, eq. (lla) has the solution G2(pc(T), T) = ~~~~{T),
T) = (~AcT)~‘“,
(20)
which lies within the region of applicability of eq. (lla), if T >>(25/7r3)A2c2. In this case eq. @la> can be used to calculate 13’ and 1T” in the whole region wg < 0, where c.$’ is even smaller than at ox= 0. With the increase of T eq. (lla) is applicable to larger G2 in an expanded region of 0:. To describe G2 outside this region, one should use eq. (13a).
G.G. Bunaiian and I.N. Mishusrin f Ron candensatim
537
-2
tii
Fig. 2. Dependence of &’ on w$ at A = 0.1 and7 = 0 (a), 0.1 (b) and 0.5 (c), The region of intermediate temperatures (dashed portion of curves) is interpolated by hand.
In fig. 2 the dependence of (3’ on ~“0 is ihustrated at two temperatures: 1’ = 0.1 and 0.5 (the curves b and c, respectively). These curves were obtained by solving eqs. (lla) and (13a) in low and high G* limits, respectively, and by approximately joining these solutions at ~5’- T. Taking into consideration the discussion above and fig. 2, we have reached the conclusion that the pion field fluctuations near the critical point stabilize the system so that at any p and T the equation D-l@, k; p, T) = 0 should have no solutions w’(k)
of pion condensation
Pion condensation may be presented lU3)as the formation of the mean classical w-meson field cp(r, t) in the nuclear medium due to the rearrangement of the ground state of the system. In our case the mean isovector pion field cp(r, b) is the order parameter of the phase ~ansition. The necessary condition for the formation of such a field is the appearance of the minimum of the thermodyn~ical potential a at cp#O:
where the averaging is performed over the equlibrium state of the system, which includes the condensate held. The effective hamiltonian needs to be constructed for the system of the interacting quantum and classical In-meson fields in the
538
G.G. Bunatian and I.h? Mishustin / Pion condensation
medium. Without the rrrr interaction, A = 0, the r-meson field in the nuclear medium satisfies the equation 6;’ (r, t)(~~(r, t) = 0. The interaction of a-meson fields is described by eq. (Zb). The classical field f~~(r, t) = opt exp (i&) [refs. 2*3)], where p,, is the pion chemical potential. In the isosymmetrical medium ,cI,~= 0. The rr-condensate instability should be expected for modes with k - k. -pF. Therefore in the Fourier expansion of q(r) in plane waves only the most important terms with k = k. [refs. “‘“)I can be left. Let the amplitude a of the field q(r) be defined so that (${r)> = a2. At present some most simple spa&l and isotopical configurations of the condensate field are being discussed in the literature 2*3).For example, one may choose cpl(r) = (p2(r) = p3(r) = d$a
sin koz
(204
(the isotopicahy symmetrical solution) or q&r)
= J&ql(r)
f icp2(r)) = @a e*:iko+,
qzg = q3 4 0
(2Ob)
(the electrically neutral condensate
of charged mesons). Going on the above assumptions, one can write the effective lagrangian follows 2*3): 2(x) = $ C D-‘(0, w,k
k; p, T)(%&l)2+~D-1@,
as
ko; p, W~o,&))~+%,rw (21)
and the effective hamiltonian
Within our approximation the classical field contains w = F,, = 0 and only the wave vectors *t-k,-,. The condition field amplitude a, x2
a”
aa=(“3 aa=0.
only the frequency (19) determines the
(194
The existence of the nontrivial solution of (19), (19a) a # 0, while being a necessary condition of the phase transition, is not always a sufficient one. To obtain the su~cient condition, let us consider a change of the thermodynamical potential AL!(a) by the inclusion of the condensate field:
(22) The dependence of A&?(a ; p, T) on a may take qu~itatively different shapes relative to the kind of the interaction p& as is shown in fig. 3. In the case of curve 1, when eq. (19a) has only one solution 8(p, T) # 0 at p 2jTC(T), it is this d that
539
0
Fig. 3. An illustration of the qualitative dependence of Afin on the squared amplitude of the condensate field a2 for different variants of the theory. Curve 1 -second-order phase transition, 2 -first-order phase transition, 3 -no phase transition. Curve 2a has the bending point corresponding to the single solution of eqs. (23), (25). Curve 2c corresponds to the critical density.
corresponds to a stable equilib~um. This solution d comes into existence at p = bC(T) and then increases beginning from zero with the increase of p. The above-mentioned corresponds to the second-order phase transitions. But a qualitatively different dependence of AJ2 (a ; p, T) on a given by the curves 2a-2d in fig. 3 is possible as will be shown further. In this case at a given T eq. (19a) has two solutions for p >pb(T) (curve Za), corresponding to the two extrema of Af.2(a2). If the curve 2b has a minimum of Af2(a2) 2 0, then, evidently, the condensate field is not formed yet, since it is not energetically profitable. It is necessary for the formation of the field that min AR(a; p, T) ~0 (curves 2c, 2d). The critical density b,(T) is determined from the condition min AJ’~(u ; & T) = 0 (curve 2~); at p >&i,(T) min Af2(u ; p, T) -C0. At p = &(2”) the condensate field appears due to a jump of the amplitude a, = n&(T)) which corresponds to the first order phase transition. The _Z’Zmwritten in the form (2b), as shown in sect. 6, brings about such a picture. Naturally, one can choose pTW so that AR has no extrema at any p, T and no condensation takes place. 7. The first order phase transition to the pion condensed
state
Let us assume that the ground state of the system contains the pion condensate field cp(r) of the form (20a). After calculating the average
540
G.G. Bunatian
and I.N. Mishustin
/ Pion condensation
eq. (19a) for the equilibrium amplitude d of the condensate field reads an ~=wfja+5AaN(O;p, The contribution
to the polarization
T)+$4a3=0.
(23)
operator induced by cp is
II”, =Aa’+2A(cpZ(r)).
(24)
For the isosymmetric medium and condensate field (20a) it is reasonable to assume that (cpZ(r)) =$(cp”(r)) = $a’. Thus in eqs. (ll), (13) IIa=$la2
and
(3(p, T)=&(p,
T)+5flN(O;p,
T)+&ia’.
(25)
From eqs. (23), (25) we find generally that
(26) where d is the true solution of eq. (23). From eqs (22), (23) we can write An(a) = Af2c,(a) +A&(u) a A&(a)
=$oifia2+$Aa4,
Al&(a) = 5A
I0
N(a)a da.
(27)
This expression is valid for an arbitrary a and, in particular, for a = ii. Now let us discuss separately the situation at zero, low and high temperatures. Zero temperature. At T = 0 quantum fluctuations of the pion field contribute positively to iV(0; p, O), eq. (9), stabilizing (3*. After absorbing the constant 5AA into the renormalized gap w& one obtains from eqs. (23), (25) an -=m~a-5AB&?u+~Aa3=0,
(284
&.I
L3*=co~-5ABh?+~Aa2.
(28b)
Eliminating & * one has la\ =~(~A)“*B*[~AB*-~~~/A]“*.
(2%
From this result it is clear that the solution d # 0 appears first at the point where the square root vanishes, i.e. w~~‘w~GO~,O)=~)~~(O)-P~]=~*B*.
(30)
This solution corresponds to a bending point on the AL!(u*) curve (see fig. 3, curve 2a). At pb
G.G. Bunatian and I.N. Mishustin / Pion condensation
541
one corresponding to the maximum of A~((a2) and the second (larger) one, to the minimum of Aa (see fig. 3, curves 2b, 2~). These solutions take place first for 40 > 0 and should be interpreted as metastable states with cp# 0. With the density increasing above Pb the minimum of AL!(a*) decreases and at a certain critical density & the nonhomogeneous state with cp# 0 becomes stable, i.e. energetically more favourable than the homogeneous state with cp= 0. To evaluate An(a) we should know (3* as a function of a. This function is given by the solution of eq. (28b). Taking into account the discussion below eq. (15) we have (31) at C,J~ + z/la* > 0 and &* = 0 at wg + $Aa* < 0. The latter inequality can be realized at og< 0 only. Eq. (31) is a generalization of eq. (15) for a2 # 0. Now we can perform the integration in eq. (27) that gives A&(a)
= -B[w~+~A2B2+~Aa2]3’2+~A2B2
+A~I,
m~+$Aa*>O,
(32)
where 25
B(w;+~rl Ai& =
2
2 312
B )
yA3~4+yA~2&j,
,
w&o W&O.
(33)
Using eq. (29) one can easily express Ah!@) in terms of “‘0 only, but we do not present these rather long expressions here. Equating AiT! to zero we can find the value of wg corresponding to the new critical density GC.Our analysis shows that this value, o& = w:&, O), is extremely close to that given by eq. (30) and corresponding to the bending point. Indeed, AR at the bending point is extremely close to zero, precisely, Af2(~: = cd&,) = 0.0012A3B4. Thus, we can conclude that the first-order phase transition to the nonhomogeneous state with cp# 0 has to occur at w;, =q(pc-P”+$$A*B*,
(3; s;*(&
0)
-$&
a,-$(iA)“*B,
+&*B*.
(34)
Notice that the renormalized gap (3: of the pion spectrum has a rather large jump at the critical point. Using eq. (31), one has ;;;
(a2= ii,“)
$(a2=0)
=
4.
When the density becomes equal to the bare critical point, p = pc, of = 0, the two solutions (29), corresponding to the maximum and minimum of AR, are 0 and %A)““B, respectively. The barrier between a nonhomogeneous state and normal
542
G.G. Bunatian
and I.N. Mishustin
/ Pion condensation
one disappears at this point. The energy gained condensed state is equal to 0.043A3B4. At higher a plus sign in eq. (29) has to be used. It is clear Iw$ >>A2B2 the amplitude and the energy of the values calculated in the mean field approximation: n:, =-$&A,
by the transition to the pion densities only the solution with from eqs (29) and (32) that at condensate field approach the
A~cl=-&&i.
(32a)
Low temperatures. Taking into consideration the above-said the characteristic energy scale at zero temperature has been established as AB. Thus, a temperature is assumed to be low if T<
(35)
Now let us consider low temperature properties of the pion condensation. To calculate the leading correction terms to the zero-temperature results, we include the low-temperature contribution to the correlation function, eq. (13), in the form N(O;p, T)=A-B[&+&r2~2T2/~;],
(36)
where
and is generally given by eq. (31). Substituting eq. (36) in eqs (23), (25)and using eq. (26) we find for the low-temperature case
Jtz2(a =a)=
(&162)1’2=$lB
+$&i2B2-&j-&~2AB~2T2/&]1'2.
(37)
By definition &g is given by the same formula with T = 0.As at T = 0 the solutions for d appear first when the square root in eq. (37) vanishes. These solutions correspond to metastable states up to the critical density p",(T) when An(a)= 0. But, as well as in the zero-temperature case, these two conditions almost coincide and the critical density can be determined roughly by zero of the square root in eq. (37). Keeping only the main terms one has
(38) where p”,(O)=p“ is given by eq. (34). This relation can be rewritten transparent form: p”,(T) = ,&(O)(l+ T’/T:)
,
T<
in a more
(38')
where To and TI are expressed in terms of parametes B, p"c and A. Using the estimate B = 0.7, p = 0.5,v = 1.2and p”== 2p,, consistent with the parameters of eq. (l), one has numerically To- 0.1A2and Tl- O.lA. If A<< 1, then the lowtemperature region is very short and the temperature dependence of p”Cin this region is very weak (To<< TI). Using eqs. (27), (36) and (37), one can calculate AR for the low-temperature case. Formally it differs from that of eq. (32) by an
G.G. Bunatian and I.N. Mishustin / Pion condensation
543
additional T2 term:
-[(~A2B2+W~+&z
*) 112-&31-l}
.
Induced by the phase transition, a jump of the entropy density at the critical point is given by the expression AS = -;r*p*T/A
= -4S(T/A)m:.
(39)
As expected, AS < 0 and AS + 0 at T + 0. If A 6 1, the entropy jump is small over the whole region of low temperatures T < To. The negative sign of the AS is quite natural when entropy is assumed to be a function of the excitation energy E”. It means that the entropy of an inhomogeneous phase grows more rapidly with E” than that of a normal phase. According to the Gibbs criterion of the maximum entropy the pion condensed state will be realized when its entropy exceeds that of the normal state. High temperatures. As it has been shown above, the pion field quantum fluctuations are valid at zero and very low temperatures T c To=0.1A2. In a more interesting case with the temperatures being high enough, T >>To, the physical picture is determined by the thermal fluctuations. The correlation function N(O;p, T) is given by eq. (11) in this case. Substituting into eqs. (23), (29, one can see that nontrivial solutions a # 0 are possible only at rather large negative values of wi. Further it will be convenient to use new variables: 2
(d!?-wiy
5AcT & = (_213/2
;
a&co.
(40)
One will see that all critical parameters are scaled- with E. Eliminating (3’ from eqs. (23), (29, one can easily get an explicit equation for 8 =Aa’/(-co;): fi(l-;@=&&.
(41)
Evidently, at small E eq. (41) has nontrivial solutions in the interval 0<8<$.
(42)
There is no solution s= 0 and the condensate field cp can appear with a finite amplitude d # 0 only. With decreasing E (decreasing T or increasing p) the solution of eq. (41) appears first when E reaches the value of Q, =@ corresponding to the maximum of the 1.h.s. at 8 =$. This solution corresponds to a bending point of the Ati curve (see fig. 3, curve 2a). At E C&b eq. (41) has two solutions corresponding to the maximum and to the minimum of Afi(a*), As long as min AL! (a”) > 0, these solutions have to be interpreted as metastable states of the system with cpZ 0. At a certain critical value cc the nonhomogeneous state with
544
G.G. Bunatian and I.N. Mishustin / Pion condensation
0 becomes stable. To find this critical point, we must know the explicit dependence dR(a*). Using eqs (25), (27) with N taken from eq. (ll), we get @#
~n(a2)=~~~a2+~A~4+3cT[J~-2)~~/32(a2)]~z, where the function CG “(a “) is determined
(43)
from the cubic equation
&*-(5A~T/&?)-o&$la~=O.
(44)
At the critical point, E = E, or p =6,(T), the curve Ati is tangential to the horizontal axis at ti*(p,(T), T) =az(T) (or f?(.sJ = 8,), as is curve 2c in fig. 3. It is clear that such a situation corresponds to the first-order phase transition. The simplified model. Now let us consider a simplified model illustrating the situation ‘near the critical point fairly well. It will be seen later that in the case of thermal fluctuations a very good (10% accuracy) approximation is E
(45)
eq. (44) has a simple solution
~=J+-g[l-j&2+*-~]
3 3
(46)
3
with E and 8 defined by eq. (41). Taking into account eq. (45) we immediately get the equation for the equilibrium value 8 = s X&l-$t?)=&,
0<8<$.
(47)
In comparison with eq. (42) the coefficient ? is replaced here by 2. Actually the relative difference between these two numbers depending on the condensate field structure is a really small parameter of the model. According to eq. (47) the metastable states appear first at &b= J& with & = $. These values differ by 10% 1 from the exact numbers of 2Jm and 3, respectively. Substituting solution (46) in eq. (43) and keeping the main terms, one has An(e)
-2 [aBZEa$$+.. . ,
=
3
(48)
I
From the condition da(@) = 0 one can easily find the parameters of the critical point EC=+/; 2
BE=&.
In terms of temperature a change of the critical density Ap,(T) =p”,(T) -p,(T) induced by the thermal fluctuations is Ap,(T) = -w,,,(T)/q
= (1/77)(5A~T/~,)~‘~ =7.5(~lT)“~.
(50)
The squared amplitude of the condensate field at the critical point is a;(T)
= &(-&,(T))/A
=2.7A-1’3T2’3.
(51)
G.G. ~~~~tian and I.N.
M~s~~~i~ / Ron ~~~den~a~~o~
545
The numerical coefficients are obtained here for c = 0.6 which is consistent with the parameter values used above (see eq. (1)). It follows from eq. (46) that the renormal~~ed gap G2 of the pion excitation spectrum has a large jump at the critical point: A0“: =Uj2(8’0&c)-(;2(0,E,)=~~~, where 6: =&(&,
EJ = -&a& =0.45(1IT)~‘~.
From eq. (48) it is easy to calculate the jump of the entropy density at the critical point: AS = -~(25~~~2}2’3(~T)~‘3 = -O.~(JIT)~‘~ o
(52)
This value is rather small even at AT - 1. It is interesting to note that the function Ati( eq. (48), besides the rn~njrn~ at 8 = s always has a maxims at 8 = 6~‘. It means that at finite temperatures the pion condensed state is separated from the normal one by a potential barrier of the height: maxAfi(fB, E = aJ =0.02A1’3T4’3,
WW
Having the explicit expression for G2@, T), eq. (51), one can reformulate the constraints discussed in sects. 3 and 4 in a more transparent way. Namely, the high-temperature expansion can be used if ?rT >G2, but the neglection of complicated diagrams may be justified if AT<< G2ip, T). Using eq. (51), one finds that both the conditions will be satisfied at the critical point if 1
ic2A2<
1 2=o.09. ZC
(53)
Note that the 1.h.s. of this inequality is very close to TO= 0.l.A” which determines the region of quantum ~uctuations* Inequality (53) is easily satisfied for A =G0.1. But we hope that our theory yields quaiitatively correct results even for A -0.5. We also expect that a picture being qualitatively the same will take place when the pion condensate field has the space and isospin structure other than that of (20a). Concluding this section we refer to the recent paper 21) by Dyugaev where a similar problem is considered within the framework of the comas-Fermi model. ~~~eric~l results. Computational results are represented in figs. 4-6. They show the increment A@,(T) of the critical density, the jump of the squared amplitude of the condensate field cxz (T) at p = &(T) and the renorma~zed gap G ‘$(T) (in units of T) above (a2 = 0) and below (a2 = a:) the critical point. These results correspond to high temperature~T >>T0 = 0.1A2) and are obtained by solving exact equations f43), (44) of sect. 4. The corresponding results for the low temperature case are not given in figs. 4-6 because the low temperature region is too narrow, T < To
546
G.G. Bunatian and I.N. Mishustin / Pion condensation
0
0.5
’
T
Fig. 4. Temperature dependence of the critical density increment Ap, induced by thermal pion field fluctuations for first-order phase transition. Curves a and b are drawn for A = 0.1 and 0.5, respectively.
(To = 0.001 for A 4) that only the considered if the that this condition
= 0.1 and To = 0.025 for A = 0.5). It has been proven above (sect. simplest diagrams induced by pion field fluctuations can be condition AT/&: K 1 is satisfied. From fig. 6 we may conclude is fulfilled only by order of magnitude even at A = 0.1.
0
0.5
’
T
Fig. 5. The jump of the squared amplitude of the condensate field a:(T) in the critical function of T. Curves a and b are drawn for A = 0.1 and 0.5, respectively.
point
as a
G.G. ~~nat~a~and IN. Mishasti~j Pion condensation
547
Fig. 6. Temperature dependence of the renormalized gap G;(T) on the temperature ratio above (a, b) and below (a’, b’f the critical point. Curves a, a’ correspond to A = 0.1 and b, b’, to A = OS.
Fig. 4 shows that thermal ~uctuations induce a considerable increment of the critical density Ap,(T) which grows faster than it has been predicted by the Tzi3 law derived in the framework of the simplified model of sect. 7. This difference appears mainly because in the numerical calculation the density dependence of the condensate field wave number kg in accordance with eq. (1) has been taken into account. The critical amplitude of the condensate field a:(T) (fig. 5) is also a more rapidly increasing function of T than T2’3. 8. ~iscussjon of the basic features of the theory and its eousequenees The basic conclusion of the present paper is that strong pion field fluctuations must arise near the critical point of pion condensation due to the soft mode with a large momentum k,. That is the main difference of the investigated phenomenon as compared with other phase transitions, e.g. with the transition in a superconductor where, due to k,= 0, the ~uctuations of the order parameter have no singularities near the critical point and lead only to minor modifications of p. T,, 4:. The fluctuations begin to grow well before reaching the “old” critical point (calculated without taking them into account) and reach the maximum at p -p,(T), From the consideration above we can easily find the domain of parameters where fluctuations are small At zero and low temperat~es 7’ < TOthe quantum fluctuations contribution may be considered as small perturbation (6’ = ~“0) if “; >>25A2B2=20A2.
CW
In the high-temperature case (Trr >>03 the condition of small thermal ~uctuations of the pion field is 5AcT/oo<< & or
548
G.G. Bunatian and I.N. Mishustin / Pion condensation
At T - 1 this condition can be satisfied only if A = 0.1-0.2. Thus, the mean field approximation which up till now has been used to calculate the critical parameters of the pion condensation becomes qualitatively wrong well before the critical point. This qualitative conclusion has been confirmed by our numerical calculations. As is seen in figs. 4, 5, the area of strong fluctuations (with the Ap,(T) serving as their measure) expands promptly with the increasing temperature. So, well before the critical point the thermal fluctuations become important and decisively change the character of the phase transition as well as the thermodynamical character of the system. That is what makes the pion condensation essentially different from a phase transition proceeding at k, = 0 (e.g. superconductivity) where the thermal fluctuations are vital only in the nearest vicinity of the critical point. Strong fluctuations lead to some important physical consequences. First of all they affect the character of a transition. From general considerations it is clear that a phase transition must start, if at all, in the area of not very strong fluctuations (the “old” critical point), but they should not be very weak either, otherwise the Landau theory would be valid in this area (54). Going away from the critical point the formulas for the condensate field amplitude and for the energy gain must be transformed into those of the Landau theory. In this paper two approaches of two different types have been employed. First, closed eqs. (6), (7) are used to obtain N, 17” (or (3’) either in the region of T cc 3’ < 1 or in that of AT/G2 <<1, since only there it suffices to consider just the first simplest graphs of an infinite series of graphs (3a), ‘(3b). Second, eqs. (6), (7) are not solved exactly, though it is possible in principle, but transformed into the form of (ll), (lla) for G2<
G.G. Bunatian and LN. Mishustin / Pion condensafim
549
Further study of the ~-condensation would require use of all the knowledge that has already been obtained on phase transitions including the critical indices, scale invariance, etc. The results obtained would allow us to change our view of the methods of the pion condensation investigation in high-energy heavy ion cohisions, Until now the following has been used as a test for the presence of pion condensation: (a) the occurrence in the phase transition region of two shock waves “) instead of one in a normal phase; (b) a threshold growth of the thermal energy and temperature of the matter and, consequently, of the multiplicity of “thermal” n--mesons due to the new phase which is energetically more profitable 23); (c) a reduction of the sidewards matter flow in heavy-ion collisions due to the softening of the equations of state 24). U~ortunately, the first two methods do not seem to be realistic. First, because the dimensions of nuclei are not large enough for the evidently shaped shock waves to propagate in the latter. Second, because within the area of densities and temperatures achieved in a collision the energy gain of pion condensation appears to be very small, less than 10 MeV per nucleon, as follows from calculations presented in sect. 7 [eqs. (32), (48), and see also ref. ““)f. It seems more promising to search for a phase transition with the help of the pion field fluctuations following it. They arise due to the formation of the ‘“soft mode” of pion excitations and are always strong near the critical point regardless of the energy gain of the new phase. This must bring about several observable effects. First of all it is clear that due to the ~uctuations near the critical point the behaviour of all the thermod~~ical parameters changes, in particular, approaching the critical point, the heat capacity of the system must increase sharply. If during the collision of heavy ions the matter is compressed enough, that is the system approaches the critical point, the system temperature increases slower than in the case of the ideal gas and comes to saturation. Thus, during the collision of heavy ions in a certain collision energy interval corresponding to the vicinity of the phase transition, a slow-down of the temperature growth and of the thermal r-mesons multiplicity is expected in contrast with the statements of ref. 2z). Moreover, the strong ~uctuations must lead to an increase of the effective interaction of particles with the nuclear matter resulting in a considerable reduction of the nucleon mean free path 25). An appearance of maxima in two-particle correlation functions of secondary particles with relative momenta -kO is also predicted. All these questions require further study. In conclusion the authors would like to express their gratitude to S.T. Beiyaev, A.M. Dyugaev, N.E. Zein, D.N. Voskresensky; E.E. Saperstein, V.A. Khodel for
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critical remarks and usefu1 discussions and to N.N. Serebryakov preparation of this paper.
for his help in the
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