- Email: [email protected]
QCD instantons at finite temperature (I). Gluonic interactions and the fermion determinant
Nuclear Physics B364 (1991) 255-282 North-Holland
QCD INSTANTONS AT FINITE TEMPERATURE (I). Gluonic interactions and the fermion determinant E.V. SHURYAK*‘**
AND J.J.M. VERBAARSCHOT**
Department
SUNy
of Physics,
Stony
Brook,
NY I1 794, USA
Received 28 January 1991
This is the first in a series of papers devoted to instanton-induced effects at non-zero temperatures. The main points will be the restoration of the chiral symmetry and the modification of the various correlation functions with increasing temperature. In this paper we investigate the temperature dependence of the gluonic and the quark-induced pseudo-particle interaction. We find that the gluonic interaction becomes much more long ranged, whereas the quark-induced interaction becomes exponentially short ranged.
1. Introduction Since their discovery in 1975 [l], instantons are generally believed to be an important ingredient of hadronic physics. The semiclassical theory of gauge field tunneling, its mathematical structure and its phenomenological implications challenged many theorists in the late seventies. However, due to complexity of the problem, only recently it became possible to formulate a consistent theory of interacting instantons, including all orders in the ‘t Hooft effective interaction and, at the same time, being simple enough to make practical calculations feasible. This is also the framework used in the present paper. Most people agree that the so-called U(1) problem, related to the axial anomaly, can only be understood with the use of instantons. However, it is far from being the only instanton-induced effect, or even the most important one. In fact, it is clear that instanton-induced forces between light quarks are very strong, so that they are probably the dominant element in the formation of non-perturbative vacuum “condensates” and even in the constitution of the hadronic masses themselves. If the experimental fact that the n’ mass is not of the order of 400 MeV (as chiral perturbation theory suggests) but nearly 1 GeV, can indeed be explained by * Supported in part by the US Department of Energy under grant DE-AC02-76CH00016. **Supported in part by the US Department of Energy under grant DE-FG02-88-ER40388. 0550-3213/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
256
E.V. Shutyak,
MM.
Verbaarschot
/
QCD
instantons
instantons, it is clear that such huge effect cannot be completely unimportant for other hadrons. Looking at the correlation functions, it is easy to see that the same ‘t Hooft effective interaction [2], which is repulsive in 7)’ channel, is also present in all scalar and pseudoscalar channels. In particular, it is strongly attractive in the scalar isospin 0 and in the pion channel, which has led to the idea [3,4], that instantons are also related to SU(N,) chiral symmetry breaking. If so, they also should provide the “effective masses” of light quarks, which in turn are the main building blocks of the hadronic masses. Let us also briefly mention the main steps in the development of this theory. Phenomenological connections outlined above were analyzed, starting from papers [5,6], and then led to refs. [7, S], where the qualitative features of the ensemble of interacting pseudoparticles or the “instanton liquid” have been pointed out. Surprisingly enough, essentially the same features emerged from a pioneering paper on the variational approach to theory of interacting instantons [9]. Encouraged by this agreement, many efforts were devoted to the study of the approximations involved in this analysis and to develop a more quantitative theory. This goal was essentially reached in ref. [lo], where it was demonstrated that the problem allows for straightforward numerical solution. Recently, it was applied to calculation of various mesonic correlation functions with surprisingly good results [ll]. With this paper we start a new series of papers, in which several questions will be addressed. First of all, we are planning to do .much more accurate calculations than have been performed previously, reducing both the statistical and the systematical errors (the latter because we are working with larger systems). Second, we want to increase substantially the number of calculated quantities, in particular we want to investigate off-diagonal correlators (like the w-4 one, etc.) and the baryonic correlators. Third, we want to study in a systematic way the temperature dependence of all these quantities. The technical obstacle which prevented the study of the last point in earlier work is that, at non-zero T, the instanton solution (called “caloron”) no longer has the four-dimensional spherical symmetry. Therefore, the interaction between these objects depends on many more variables, and its understanding becomes much more involved. The investigation of these, mainly technical problems is the main topic of the present paper. Several other works that study instanton-induced effects at finite temperature should be mentioned. At the one-instanton level there is the well-known statement that instantons are suppressed at pT < 1 [12,13], which was analyzed quantitatively in ref. [14]. This by itself shows that, at high T, the role of these effects is strongly diminished, and they are present only in the form of a “molecular vacuum” [lo] in which the chiral symmetry is restored. The temperature dependence of instanton-induced effects was investigated with the help of a mean-field approximation both without quarks [l&16] and with light quarks included [17,18]. The former one [17] compares the thermodynamical functions for “liquid” and
E.V. Shuryak, J.J.M. Verbaarschot / QCD instantons
257
“molecular” phases, indicating that chiral restoration is to take place at T, - 200 MeV, a conclusion which was also reached in ref. [18]. In connection with the present paper let us note that in ref. [16] the pseudoparticle interaction averaged over all orientations and positions, was evaluated at nonzero temperatures. The conclusion was that the temperature dependence is very weak (a few percent). However, our findings are qualitatively different. Most probably, their “temperature independence” is due to the use of the sum ansatz with its artificial divergence of the field strength at the centers of the pseudoparticles, as well as due to the averaging over the positions. Finally, we would like to make some comments related to lattice simulations of QCD. Instanton-like configurations have been found in the vacuum fields by different methods (the “cooling method”, the “geometric method” and the “fermionic method”, see refs. [19-211 and references therein), and with still large uncertainties the density is roughly consistent with that suggested by the “instanton liquid” model. It is not yet clear in which way these configurations are related to many other lattice results, e.g. the quark condensate value or the hadronic masses. (However, see ref. 1221 for progress in this direction.) Attempts to get the correct n’ mass on the lattice are not yet successful, so something is still missing. An interesting branch of lattice calculations, which is strongly related to the instanton-induced phenomena we are going to study, is the investigation of chiral symmetry restoration at finite temperatures. In this framework, it was found that “quenching” of the fermionic determinant is not a good thing to do, and, in fact, the phase transition parameters are very sensitive to the presence of light quarks and their masses. In particular, according to ref. [23], the cases of Nr = 2 and 3 are very different. Also the value of the strange quark mass below which the strong first-order transition takes place is very small, of the order 50 MeV. These observations are in interesting correlation with our results 1241 for the numerical simulation of the ensemble of instantons. We also see a large difference between 2 and 3 flavors, and the transitions is at about at the same mass. We hope to clarify this correspondence in our next papers. However, from general considerations it is clear, that in spite of these correlations, the lattice data cannot really describe the phenomena we are discussing. The reason is, that in order to speak about phases in the ensemble of pseudoparticles, one should have many of them. For example, we are typically working with an ensemble of 32 pseudoparticles, while present lattice calculations with dynamical fermions have only room for a few pseudoparticles. Therefore, they can only see some traces of the instanton-induced phase transition, at best. Let us now outline the contents of the first paper of this series. In sect. 2 we introduce known formulae for finite-temperature instantons (calorons) and generalize the trial functions used for multi-pseudoparticle configurations to the non-zero temperature case. A detailed discussion of the classical interaction between a pair of pseudoparticles is presented in sect. 3. Analytical results are obtained for the
258
E. K Shuryak, J.J.M. Verbaarschot
/
QCD
instantons
high-temperature limit (details are given in appendices A and B) and for a wide range of parameters in between this limit and the well-known zero-temperature limit the interaction is computed numerically. Sect. 4 is devoted to the so-called overlap matrix elements of the Dirac operator between fermionic zero modes of different pseudoparticles. Again, we obtain, analytical results for the high-temperature limit (see appendix C for details), and we perform extensive numerical calculations in order to cover the range of parameters between this limit and zero temperature. The results of both the gluonic and the quark-induced interaction are summarized by a fitted expression which is an approximate but closed form suitable for further studies of the statistical mechanics of the system of interacting instantons.
2. Calorons and trial functions As is well known, in the euclidean formulation of quantum field theory, a finite temperature is formally introduced via periodic (anti-periodic) boundary conditions for all bosonic (fermionic) fields on the strip [O, p = l/T]. The topologically non-trivial solutions of the classical Yang-Mills equations satisfying these periodicity conditions were obtained in ref. [25] as a special case of the ‘t Hooft multiinstanton solutions. The explicit expression for a gauge field has the form A;( x) = fi;,a,n-
’)
(2.1)
where sinh( 2rr/j3) n(x)=l+* /3r cosh(2rrr//3) - cos(2at//3)
*
(2.2)
Here, p denotes the size of the caloron and xP = (r, f&. Of course, at T + 0 this field reduces to the field of a single instanton. However, in the opposite limit, Tp Z+ 1, this solution has a quite different form. As was pointed out in ref. [14] in this case the calorons develop a “core” of the B(p) where the field is very strong, G,, = &‘(p-*). For distances between 8’(p) and @(p*/fl> the caloron solution can be approximated by a p-independent dyon field, given by the very simple formula A:(x)
= F&rk/r2.
(2.3)
In the third region, at distances larger than 6’(p*//3> from the center, “the caloron periphery”, the field strength is of @(pm*). For illustration, we show in fig. 1 the dependence of the square of the field strength on the spatial distance from the center for several values of the temperature.
E. V: Shwyak,
J.J.M.
-4
-2
Verbaarschot
0
/
QCD
instantons
r/P
259
4
2
Fig. 1. The distribution of the field strength squared as a function of the spatial distance from the caloron center r, in units of the caloron radius p. The total action is normalized to one. Five curves from top to bottom correspond to values of the temperature Tp = 8,2,1/2,1/4,1/g, in this order. At high temperatures one observes that a core develops, as well as a universal dyon region between the core and periphery.
Now we arrive at the discussion of multi-pseudoparticle configurations. In this case explicit solutions of the classical Yang-Mills equations are known. It is also a direct consequence of a general theorem that the classical interaction between instantons is zero, so that the action is proportional to the topological charge. Configurations containing both instantons and anti-instantons are much more complicated. The discussion of the so-called “streamline” set of configurations, which are most suitable for their description, can be found in refs. [26,27]. However, due to technical difficulties these “streamlines” were only found for some toy models like the quantum mechanical double-well problem. For the description of instanton-anti-instanton configurations in a four-dimensional Yang-Mills theory one still has to rely on the use of trial functions. In particular, in ref. [9] the so-called “sum-ansatz” was used, in which A:(x) is given just by the sum of one-pseudoparticle solutions (in the singular gauge). However, it has some artifacts, and an improved “ratio ansatz” was suggested in ref. [lo] which for an instanton-anti-instanton pair is 2 oFbT;,YlyP:/Yf A”,(x)=i
1+ P:/Y:
+ O:brl;,YA,P:/Y: + Pi/Y&?
,
(2.4)
where y, =x -RI and y,, =x -R,, and R,, R, are the positions of the centers of the instanton and the anti-instanton, respectively. The orientation of the pseudoparticles in color space is given by the O(3) matrices 0, and 0,. In the case of
260
E.K
Shutyak,
J.J.h4. Verbaarschot
/ QCD
instantons
one instanton-anti-instanton pair, a very interesting trial function was suggested in ref. [28]. It has interesting properties and is probably the best one so far suggested. Results related to this ansatz will be presented in a separate paper [32]. Unfortunately, it is unclear how to generalize it to many pseudoparticles and to non-zero temperature. Our first new step in this paper is a straightforward generalization of the ratio ansatz to the case of finite temperatures, which is based on the observation that a one-caloron solution can be considered as an infinite sum of instantons, with centers located equidistantly along the “time”-axis with the spacing p. The ratio ansatz for this configuration can be rewritten as 1 o$$, Axx)
= -ii
where L!, =n(y,) and Ir, high-temperature limit the modified by the presence of p .=x YA -K P2/& and p + 0, AL(x)
aJ7, + o;bq;,a”17* n,+rr,-1
(2.5)
’
=Il(y,+,) and Kx) is defined in eq. (2.2). In the field in the core region of the instanton will be an anti-instanton and vice versa. In the limit yr -ZEp, the gauge field is given by
= 2 g y;( ,O; + y;( 1 + r2p:/3b2
+
~ii/~YA))
’
P-6)
In this limit yA = r % B(p), and therefore, eq. (2.6) describes the gauge field of an instanton with renormalized size pR given by 1 -25 PZR
3 P Pk I+----+3 P2 i
r
r P[
3P2 ?T’pf I *
(2-V
Here and below we will denote the spatial distance between a pair of pseudo-particles by r, their temporal distance by 7 and their distance in euclidean space-time by R. In sect. 3 we will need this limit in order to estimate the contribution of the core region to the pseudoparticle interaction. In the limit j3 -+ 0 for fixed distance r between the centers of the pseudoparticles, the total gauge field in eq. (2.5) up to a distance @(p2//?> simplifies to the following expression: 1
1
Y,(Y,+yAP:/Pi)
--
1
Y:
1 YA(YA+YIP~/P:)
I'
(2.8)
E.K
Shutyak,
J.J.M.
Verbaarschot
/
QCD
instantons
261
where the indices k run from 1 till 3, and the overall scale of pr, pA has dropped out. In sect. 3. and in the appendix B we will evaluate the instanton-anti-instanton interaction resulting from this gauge field.
3. Classical pseudoparticle
interaction for the ratio ansatz
The classical interaction is defined as the difference of the action of a pair of pseudo-particles at finite separation and their action at infinite separation. In the limit where the four-dimensional distance R between the center of the pseudopartitles is much larger than their size p (R Z+ p), the classical instanton-anti-instanton interaction S,, at zero temperature is given by the well-known dipole formula
m 301, 4PfPi S,, = d7
>
d=1-4(u*R)2/R2.
(3.1)
Here, the action is given in units of the action of one instanton S, = (8rr2/g2). The orientation-dependent factor d depends on the four-component vector uP related to the spinor representation of the relative orientation matrix U as U = iuPrl CT,‘= (7, -i) and 7; = (T, i)). The relative orientation matrix can also be written in terms of the vector representation of U which will be denoted by 0 and is given by
If one deals with the SU(2) color group, U is unitary and one has real up and u: = 1. The matrix 0 is real and orthogonal in this case. Note that the group average of d over orientations vanishes. For arbitrary NC, d generalizes to d=lu12-41(u.R)12/R2,
P-3)
where U = iuprl is now the upper 2 X 2 corner of the NC X NC relative orientation matrix. It is not unitary and therefore the ug are complex with 1u12# 1. In this case, the matrix 0 is complex. On general grounds, one can make the following statement about the angular dependence of this interaction at zero T. As the pseudoparticle interaction is the same for all directions perpendicular to R,, and as our expression for AZ has only quadratic terms in u, the general orientation dependence in the case of W(2)-,color for arbitrary values of R is given by S,=s,(R)
+~,(R)(u~~)~+s,(R)(u*~)~,
(3.4)
where 2, is a unit vector in the direction of R,. From now on we will choose the
262
E. V. Shrlryak, J./M Verbaarschot / QCD instantons
l-axis in the direction of R, and write u, instead of (u *a>. For the sum ansatz the pseudo-particle interaction was evaluated in ref. [9] at large R up to order 6’(l/R6> and in the limit R + 0. In both cases it was found that s,(R) = 0. This statement also holds true for the ratio ansatz, and therefore we expect only a small correction proportional to (u * &I4 at intermediate distances R. At non-zero temperature the “time’‘-axis is special, leading to a caloron interaction with less symmetry. There is only symmetry in the plane orthogonal to time axis and r (the three-dimensional distance is denoted by small r). As a result, the interaction depends on the following variables: 6) the spatial distance r between two centers; (ii) the “temporal” distance 7; (iii) the sizes p,, pA; (iv) relative orientation matrix U, and (v) the temperature T. The orientation dependence of the interaction for the case of SU(2)-color can be parameterized as follows: S,,=s,+s,u:+s,u:+s,(u:+u:)+s4(u:+U:)2+S~U:(U:+U:),
(3.5)
where the l-axis has been chosen along the line connecting the two pseudoparticles. We have finished the general considerations and now start with the discussion of the high-temperature limit. As mentioned above, the caloron field has a different form in three different regions: (i> a core of radius 5Q>; (ii) a dyon region outside the core up to a distance @‘(p2/p>; and (iii) a peripheral region outside. It is natural to split the interaction of two calorons into the contributions from these regions. In the core region the field of the other pseudoparticle has the net effect that it leads to a renormalized size given by eq. (2.7). The core contribution to the total action is proportional to the product of the square of the field strength at the core and the volume of the core region,
(3.6) leading to an orientation independent repulsive correction to the interaction of the order P/R. (Of course, our crude argument is not sufficient to provide us with the coefficient, but this result is supported by the numerical data discussed below.) For the sum ansatz this additional renormalization is absent, and the leading contribution of the core region to the dipole is given by the product of dyon field of the other pseudoparticle and the &p-21 field strength of the core, resulting in a contribution of &Q2>. In the limit of high temperatures and fixed distance r there is another leadingorder contribution to the pseudoparticle interaction: the contribution from the
E.V. Shuryak,
dyon-dyon
J.J.M.
Verbaarschot
/
QCD
instantons
263
region given by SDD=/d’x[F:.(A:,o)-~-~]. IA
(3.7)
Here, A;,, is the dyon-dyon gauge field for the ratio ansatz given in eq. (2.8). In order to obtain the interaction we have subtracted the field strength of the free dyon fields. Since the field strength of the dyon at distance x from the center is of 6’(xV2>, integration over the 4-volume results in a factor of &Q/R). For pr/pA = 1 the orientation dependence of the coefficient of P/R can be obtained analytically (see appendix A). The final result for arbitrary value of u (u is real and 1~1 = 1 for SU(2)) is given by $$+-
P (-0.2732 R
+ 0.0354 Iu14 - 0.0963 lu12 + 0.2165(1u,12 + lu312) + 0.0317
x [IL+ x
X [lu214
+
flU,12(1+ lul’) - f(lu212+ lu,12)] + 0.0317
[~Iu,I~(Iu~I~ lu314
+
+
~1~~1~1~~1~
lu312)
+ f(
+ $KT’(K$
u2uz
+ KS)
+ K~K:)~
+
+uf(u;”
+ KS’)]
+
0.0174
- $(1+l’+ IK~I~)IKI~]). (3.8)
Note that the interaction is the same for uO = 1 and u, = 1 and that the coefficients of the b(u4)-terms are numerically small. Because the size of the core is not well defined the value of the core contribution is not well defined either. However, since the core contribution to the interaction is due to a size renormalization, it is the same for the II-case and the IA-case. In the former case we know that for d = 1 the ratio ansatz represents an exact ‘t Hooft multi-instanton solution, and the total interaction is equal to zero. Therefore, the @‘(p/R) contribution from the core has to be cancelled by the only other &T/?/R) contribution from the dyon-dyon region. In appendix B we show that for pA/p, = 1 the coefficient of the latter term is equal to l/r. We thus conclude that in the p + 0 limit and for pseudoparticles of equal size the core contribution is
which is orientation independent. For z~a= 1 and K, = 1 it is nearly cancelled by the dyon-dyon part of the IA-interaction. For the sum ansatz there is no size renormalization and, in the high-temperature limit, the &‘(p) contribution of the core is absent. In this limit the B(p) contribu-
E. K Shurynk,
J.J.M.
Verbaarschot
/
QCD
instantons
-0.4 tl
0.0
’ I I
I I I I I I I ’ ’ I I I I ’ ’ I 1I ’
1.0
2.0
3.0
4.0
l-l
5.0
r/P Fig. 2. The interaction between an instanton and an anti-instanton in units of So, the action of a single instanton, as a function of the spatial distance between their centers r, given in units of the geometric mean p = (p,pA)l12. The four-component quantity u given in the figure defines the relative orientation of two pseudoparticles (see text). Four set of points and four curves (solid, dotted, dashed and dash-dotted) correspond to values of the temperature Tp = 0,1/2,1,8, respectively.
tion solely originates from the dyon-dyon analytically. The result for arbitrary u is sp,D=- 3; ;(8(lul’-
interaction
which also can be obtained
lu12)+ 81u,12(1- ;lul’) + 12(lu212+ bs12)(1 - $I”)
+41U,12(lU212+ lU,12)+2U:2(U:+U:)+2U:(U:2+
U:')),
(3.10)
showing that the sum ansatz leads to a much stronger interaction. In this case the interaction does not depend on the ratio pl/pA. We have made extensive numerical studies of this interaction for the case of the SU(2)-color group, using a Monte Carlo integration with importance sampling. At zero temperature we chose the sum of the field strength squared of two instantons (see ref. [lo]> as a weight function. At non-zero temperatures we tried several weight functions but the one given by the sum of the field strength squared of two calorons of that temperature appeared to give the best results. Our results at T = 0 agree with those given in this paper inside the error bars. In figs. 2-4 we show some samples from our data, together with the curves obtained from a fit to all our
E. K Shutyak, J.J.M. Verbaarschot / QCD instantons
265
o 0.6 ?
z vl 0.4
0.2
0.0
‘;;Q&& -J-. -.-.-.~-.-.-.2.-.-.-.-.-.-. 0.0
1.0
2.0
3.0
4.0
5.0
r/P Fig. 3. The same as in fig. 2, but for a different relative orientation (see figure).
1.0
I I I I I I I I I ’ I I I l1I I ’ ’ l1I I I I-L,)
0.8
u = (O,O,O,l)
-
o 0.6 ? 2 fn 0.4
0.2
0.0 0.0
1.0
3.0
2.0
4.0
5.0
r/P Fig. 4. The same as in fig. 3, but for different relative orientation (see figure).
266
E. K Shutyak,
J.J.M.
Verbaarschot
/
QCD
inslantons
data which is given by
4.03
s,* =
(R*+2.10)* P*+5.21 16.16 u;
-
(1+1.68R2)3 2.73 uf
+ - (R*+ 2.03)* + (1+0.33R*)~
+ (l+0.42R2)4
1
P*+O.75
,6*+0.24+ 11.50 R*/( 1+1.14 R*)
1 (u; + us)
P2 1 P2
0.721og R*
1.66
P2
(1+0.013
1
R2)4 P*+l-73 '
(3.11)
Here, R is understood as four-dimensional distance between the centers of the pseudoparticles, R* = Y* + (t, - t,)* and all the lengths are in units of pmean = (p,pA)‘/* (the bulk of the calculations was performed for p, =pA = 1). We have shown numerically that the dependence of the interaction through other combinations of r, t, p, and pA is very weak. This corroborates with the analytical results for the low- and high-temperature limits. Note that terms of fourth order in uP are absent. From calculated values of the interaction for a wide range of p and R and the orientations ua = 1, u, = 1, u3 = 1, u,, = u, = l/a, u0 = u3 = l/a and u, = u3 = l/ fi we have determined the coefficients sc, . . . , ss in eq. (3.5) for many values of /3 and R, with the result that s2 = s4 = ss = 0 inside the error bars. At large distances and at zero T our results are consistent with the asymptotic dipole formula (3.1). In the high-temperature limit, only the terms proportional to p survive, and from the last term in eq. (3.11) we obtain 0.208 P/R, as opposed to 0.206 p/R for the analytical result in this limit (see eq. (3.8)). Note that for R + 0 the interaction becomes large. In fact, it can be shown that it diverges logarithmically in this limit, a result which has also been found for the sum ansatz [9]. Certainly this is an artifact of the trial function. Whether or not it is important for the statistical mechanics of interacting instantons, will be discussed in the next papers of this series. We have also studied the instanton-instanton (II) case. Of course, for exact multi-instanton configurations there should be no classical interaction, and the action should not depend on the collective variables. However, as discussed in ref. DO], this interaction is non-zero for the ratio ansatz. In the high-temperature limit we again find contributions from the core and from the dyon-dyon interaction. The former only involves a renormalization of the size, and we find the same result as in the instanton-anti-instanton case. For the
E.V. Slwyak, J.J.M. Verbaarschot / QCD instantons
dyon-dyon
261
interaction we now find (the uP are arbitrary complex numbers)
s,~D= -P ( -0.2732 R
+ 0.0744 lu14 - 0.1195 lul*+
+0.0780 (u*u)’
O.l805u*u
- 0.1171 lul*u*u
- 0.0317 (lu212 + lu312)
+0.00793[(u~u,+u;uO)*+
(u;u,+U~uO)*]).
(3.12)
When the orientation matrix U = 1, the II gauge field is an exact ‘t Hooft multi-instanton solution at any temperature. Therefore, if one adds to eq. (3.12) the contribution of the core the interaction should be exactly zero. In our numerical calculations we indeed find this to happen. Surprisingly, in our numerical calculations we also find that the II-interaction is the same (inside error bars) for all three other orientations. For the terms linear in 0 (or quadratic in u) this is obvious. The only invariant is given by
Rhq$O"bq~KRK=TrO(R*+
Ri).
(3.13)
At non-zero temperatures the coefficients of R* and Ri are different, but the relative orientation only enters in the combinations Tr 0. For terms quadratic in 0 one can write down other invariants, and the simple dependence on Tr 0 gets spoilt (see e.g. the result for the high-temperature limit). In order to analyze the orientation dependence we calculated the interaction for cl,, = 1, u, = 1, ug = 1, ua=u,=l/\/z, z+,=z+=l/fi and u,=+=l/fi enablingustocalculatethe coefficients of sa, . . . , ss in eq. (3.5). However, in all cases considered, we found that s2 = s4 and sg = 0 within our numerical accuracy. Apparently, the coefficients of the terms that are not proportional to u* or (u*>* are too small to be seen within the error bars of our numerical results. Our numerical results for the case of the SU(2)-color group can be summarized by the following fit: S,, = [0.63u2 + 0.071 (u*)*]
X
’ (1+0.43R2)3
log R*
P2
(1 + 1.17R2)4
P* + la17
+ [o.o7u’+
P2 P*+5.33
[0.05u*+0.47(u*)*] -
o.o5(u’)*]
log (3.14)
Again the bulk of the calculations was performed
for p, =P+, = 1. For a selective
E. K Shuryak,
0.0
J.J.M.
Verbaarschot
1.0
2.0
/
3.0
QCD
instantons
4.0
5.0
r/P Fig. 5. The same as in fig. 2, but for a pair of instantons, with a relative orientation as indicated in the legend of the figure.
number of cases it was shown that the size dependence is well taken into account by expressing all lengths in units of (or pA) ‘I*. The coefficients of the last term agree within error bars with our analytical result for the dyon-dyon interaction. [In fact, the x*-value of the fit is not sensitive to the ratio of the coefficients of u2 and (u*)* in the last term of eq. (3.14). Therefore, we fixed this ratio at the value given by the high-temperature limit. For the other terms the coefficients of both u2 and (u*>* are well determined by the fit.] Some of our numerical results together with graphs of the above fit are shown in fig. 5.
4. Overlap matrix elements In the semiclassical limit the leading part of the fermion-induced interaction arises in the form of the determinant of the Dirac operator in the space of zero modes. In this section we calculate the elements Tij of this determinant at finite temperatures. The fermionic zero mode of the massless Dirac equation in the background field of a caloron was first obtained in ref. [31]. It has the following form:
E. V. Shuryak, J.J.M. Verbaarschot / QCD instantons
269
where the constant spinor l a is a 2 X 4 (color X Dirac) given by
E
=- 10-l a@ fi ( 1
01 0
-1
0 1 .c1’
(4.2)
The function &, is given by
where II has been defined in eq. (2.2) and cp= (17 - 1) cos(rrt/~)/cosh(2ar/B>. The overlap matrix of the fermionic zero modes is defined by
(4.4) where covariant derivative b in the Dirac operator contains the caloron field. We make two assumptions: (i) the gauge field Ai is just the sum of a caloron and an anti-caloron, and (ii) for the calculation of T,* one may consider only the fields of this particular pair of pseudoparticles, ignoring all others. The former assumption needs some comments. Above we have used a much more complicated expression for the gauge field, the ratio ansatz, but now we do not use it. The reasons are: (i) the classical interaction considered above stands in the exponent requiring a high accuracy, whereas the overlap matrix elements under considerations only enter through the fermion determinant which is a quantum effect; (ii) the simple sum ansatz produces artifacts in the field strength, making it infinite at the centers, but there is no such problem for the ferrnionic zero modes. These assumptions enable us, using the fact that both I& satisfy the corresponding Dirac equation, to get rid of gluon field, resulting in an expression obtained from (4.4) by replacing 6 by 8. Let us first make several general comments on this definition. The matrix element depends on the same variables as the interaction discussed above. Since the zero modes are normalized to 1, the dimension of this matrix is that of the Dirac operator itself, namely that of an inverse length. The relative orientation dependence of T,, is linear in U = iuwrl, and thus can be parameterized in terms of two functions,
The functions f, and fi can be obtained by carrying out the Dirac and the color
270
E. V. Shuryak,
JJM
Verbaarschot
/
QCD
instantons
traces. The result is
The large-distance behaviour of matrix elements T,, can be obtained analytically both in the zero-temperature limit and in the high-temperature limit. For T = 0 the asymptotic result is known, (4.8) where U&T: is decomposition of the relative orientation matrix (see sect. 3). At r s /3 the behavior of & is dominated by the hyperbolic functions in eq. (4.31, which lead to an exponential decay. This is a general consequence of anti-periodic boundary conditions for fermions along the “time” axis. As we will see below due to this fact the overlap matrix at high temperatures will be qualitatively different from the T = 0 case. From the structure of the zero modes it is also clear that in the limit r XDj3 its time dependence in form of COS(~T/~> factorizes from its r-dependence (the temporal distance between the pseudoparticles is denoted by 7). In the integrand of the overlap matrix elements, the zero modes give rise to the exponential factor exp(-rr/P((p* + (z -R)*)‘/* + (p* + z*)‘/~), where z is the coordinate in the direction of the line connecting the two centers and p is the distance to this line. It is clear that the exponent is constant on ellipsis surrounding the two centers, and in particular, it is constant on the line segment connecting the two centers. The remaining part of the integrand is an algebraic function, and in the limit of large distances the integral can be performed analytically (see appendix C for details). As result for f, and f2 in the limit (p + 0, R + a~)we obtain fy=i$sin(y)exp(-F),
(4.9a)
(4.9b) which does not depend on the sizes of the pseudoparticles. We made an elaborate numerical study of the integrals f, and f2. By using the axial symmetry about the line connecting the two centers, the space-time integral
E.V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
271
imtantons
can be trivially reduced to a three-dimensional integration, which can be performed by a standard gaussian integration. We evaluated the integrals for a wide range of points in the (R, T)-plane and for different orientations. The results of these studies are fitted by the following formulae which interpolate between the analytically obtained high-temperature limits and low-temperature limits,
(r/p)sin(rT/P) cosh(ar/P) “l’ = (cosh( rr/j?)
- cos( rr//I)
+ K:)’ 1
’ (p2/rr2)(
exp( -rr/2p)
+ rrr/2p)
+ (2/~r)(
quantity
0.69 exp( - 1.75 r/P))
2)(l+(~i’l+o;~~2r2).
(0.82 r2 + 1)
The temperature-dependent
I-
(4.10)
is given by
K:(T)
1.07 K T(T) = /P/d
‘\
-6
0.0
’ ’ “I
(4.11)
+ 0.53 *
-.
\
-*.
’ ’ ’ ’ ’ ’ ’ ’” ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’
1.0
2.0
3.0
4.0
5.0
r/P Fig. 6. A logarithmic plot of the overlap matrix element of the fermionic zero modes of an instanton and an anti-instanton T,A (times the geometric average p of their sizes, to make it dimensionless) versus the spatial distance between their centers r (in the same units). The relative orientation is such that the vector u is parallel to r (in this case in the z-direction). Five sets of points and five curves (solid, dotted, dashed, long-dashed and dash-dotted) correspond to temperatures Tp = 0,1/2,1,2,8, respectively.
272
E.V. Shutyak,
2
J.J.M.
III1
Verbaarschot
III1
/
IIII
QCD
instantons
IIII
l-
III1
u = (O,O,O,l) -
\4 \
-4 i i
III!1
-6
0.0
IIll
‘. .I \\
l .
\
II11
1.0
-. -.
\
III1
2.0
*.
-.
llll.
3.0
4.0
*.
_ ‘*.
5.0
r/P Fig. 7. The same as fig. 6, but for the orientation vector u pointing into the time direction. In this case the temporal distance between the centers is not zero, as in fig. 6, but p/2, and one more difference is that the solid curve now corresponds to a temperature of l/8 instead of zero as in fig. 6.
For the absolute value of the coefficient
If*1=
f2 we find
(d@cOsh/~> SiWNP) (cosh( rr/p)
x
i
- cos( ‘TTT/p)
+ K;)’
1+0.069
(4.12)
where K:(T) is given by 1.00 K;(T)
=
0.69
+
@*/T*
(4.13) *
Again, ah quantities are expressed in units of the geometric average of pI and PA. We have checked numerically that the dependence of fl and f2 on pr and PA through other combinations than (pr pA)‘/* is weak. The accuracy of these fitted formulae can be seen from figs. 6 and 7, where we have used eq. (4.5) to relate fr and f2 to the overlap matrix elements TIA’
E.K Shutyak, J.J.M. Verbaarschot / QCD instantons
273
5. Summary In this paper we present extensive studies of both the classical gluonic and the quark-induced interaction between pseudoparticles. In particular, we have obtained the following results: (i) Analytical expressions for the high-T limit, valid for any SU(N,) color group. (ii) Numerical data for SU(2) color covering a wide range of distances, temperatures and relative orientations. We have shown that these data agree with all known asymptotic expressions. (iii) These data were fitted numerically by some multi-parameter trial functions, resulting in closed expressions to be used in future calculations of the statistical mechanics of the system of interacting instantons at non-zero temperatures. As the lengthy formulae are not very transparent, it is worth to emphasize the main qualitative differences between our results for non-zero T and the known interaction at T = 0. Contrary to some claims in the literature (and even contrary to our original expectations) both types of the interaction are found to be quite different at high and low T. At low T the classical interaction at large R is a dipole-dipole one of 8(1/R4>. At small values of R it is repulsive. On the contrary, at large T we find an B@/R) interaction in a wide range of intermediate distances p * r < p2//?, which is more long ranged and may lead to stronger correlations between neighboring pseudoparticles. In contrast to this, the quark-induced interaction is becoming very short range at high T. Instead of an @(l/R31 behaviour of the overlap matrix element at zero T, at high temperatures we obtain an exponentially decaying function exp(-rrTr). There is a general reason for that: fermions obey anti-periodic boundary conditions in euclidean time, and therefore the corresponding Matsubara frequency is never zero. As a result, at high temperature the quark zero modes are concentrated in a small core region, so that the instanton-anti-instanton molecules become strongly bound. We would like to thank P. van Baa1 for useful discussions.
Note added in proof For most attractive orientation a numerical solution of the streamline equation was found in ref. [32]. This provides us with an ansatz-independent gluonic instanton-anti-instanton interaction. However, the method used in ref. [321 is based on conformal symmetry, and therefore cannot be generalized to non-zero temperatures. After completion of this manuscript a paper [33] in which the high-temperature limit of the pseudoparticle interaction is considered came to our attention. The authors of this paper obtain the same asymptotic expressions for the ‘overlap
214
E.V. Shwyak, J.J.M. Verbaarschot / QCD instantons
matrix elements of the zero modes. However, we do not agree with their result for the gluonic part of the interaction which does not include the dyon-dyon contribution.
Appendix A In this appendix we evaluate the high-temperature limit of the instanton-antiinstanton interaction for the sum ansatz. In this case the gauge field is given by
(A-1) where z =x -R/2, y =x + R/2, and Oab is the relative orientation of the pseudoparticles in color space. The temporal component of the 4-vectors x and R is equal to zero. The field (A.0 corresponds to the superposition of two dyon configurations. The total field strength can be written as the sum of the following three terms (see ref. [9]):
where (A.3a)
F;J 1,2) = eubc$$
In terms of these combinations
s,D,D = &14( d x
[ $k,$Ocd
+ ?j:k?,;,O”“]
the total dyon-dyon
0 0
2F 1 F 2 +2F(l)F(1,2)
.
(A.3c)
interaction can be written as +2F(2)F(1,2)
+F*(1,2)), (A.41
where color and Dirac indices have been suppressed. The evaluation of the different terms contributing to this integral is straightforward but tedious. The
E. V. Shutyak,
JIM.
Verbaarschot
/
QCD
instantons
275
result is given by -TrO+F+-
+ TrO(y*z)* J&2 Y2
Z0.Z
YOY
(ASa) -
TrO(y*z)(y*+z*)
+
y*2* (ASb) P(1,2)
2
= JW
4-
(y.z)* JW
+ ((y*z)TrO-yOt-zOy)* )A* (A.5c)
Note that FWF(2) vanishes for 0 = 1. Also the other contributions greatly simplify in this case. After changing to cylindrical coordinates the interaction can be expressed in terms of the following integrals: /d+:-+)=f,
jd,u(+t)*=+
I dpp*=
+,
/dpz*(++)=O,
I
dpp’=
/dpz2y2=l,
$,
/dpz*p*=+,
I
dpp*x:=$,
/dp(x;-$+p’)‘=+,
where = -$~-pdpj--=/q
/ Gf(X,,p)
f2;:’
(A.7a)
,
y*= (x, ++)*+p*,
(A.7b)
2* = (x, - +)‘+ p*.
(A.7c)
The matrix 0 can be expressed in terms of the 2 X 2 complex matrix iz.$~i in the left-upper corner of the IV,, X NC relative orientation matrix as oah = (lu()l* - luI*)s,,
+ U~Ub + u,u;E: + 2E,&( 4&
+ u~u,)
.
(A.8)
276
E.V. Shutyak,
L3.M.
Verbaarschot
/
QCD
instontons
Substitution of these results into eq. (A.4) yields the high-temperature instanton-anti-instanton interaction for the sum ansatz,
SDD=$$(8(lu14lu12j+81u,12(1-~lu12)+12(1~212+
limit of the
1~~1~)(1-~lul~)
IA
+ 41U,12(lU212 + 1U312 + 2U72(4 + us) + 244’
+ u:“)).
(A-9)
Appendix B
In this appendix we calculate the high-temperature limit of the instanton-antiinstanton interaction and the instanton-instanton interaction for the ratio ansatz in the case that the sizes of the pseudoparticles are equal. The gauge field is given by (see eq. (2.8))
for the instanton-anti-instanton At(x)
for the instanton-instanton
case and by
=7iKYK4Y,z)
+ Oab7~AzAa(z, y),
case. The scalar function 1 a(y,z)=;-i.-
a(y, z> is defined by
1 Y(Y +z)
(B.2)
P-3)
’
and z=x-R/2, y=x+R/2 and 0”’ are defined as in the case of the sum ansatz. We decompose the total field strength as for the sum ansatz (see eq. (A.2)). Using the definition of the ‘t Hooft symbols the field strength F$l) of the instanton can be written as F;v(l)
= %/wf,
+
h~,kYkYp
-
EopkYkYv)f3 - (EavkykZv
-
%pkykZp)f4
E. V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
instantons
277
The functions fk are defined by (BSa)
f2=a(y,z)
f3=---
-
Y*,:;z,2
U(Y,Z>
2
Y2
Y4
zy(Y
L +d2 *
f4=-
(BSb)
’
2
+
(BSc)
+z12 ’
Y2(Y
(BSd)
The field strength F32) of the anti-instanton is obtained from F$l) by interchanging y and z, replacing aP4 and 6,, by -a,, and -aV4, respectively, and by multiplying the field strength by the relative orientation matrix 0. In the case that both pseudoparticles are instantons we do not change the sign of the delta functions. Because of the non-linear term in the field strength, also the combination I$( 1,2) = he”bcy,z,(7j;,p,40cd + T&T$~O~~) P.6) enters in the total gluonic action. The function h is defined by h=~(y,z)~(z,y).
P-7)
In the II case all n-symbols are barred. In order to obtain the pseudo-particle interaction we have to subtract the action of the system at infinite separation. For the ratio ansatz, F2(1) and F2(2> do not coincide with the action of an isolated dyon, and therefore the total interaction is given by F2(1) - -$ + P(2)
+2F(l)F(1,2)
- 5 + 2F( l)F(2)
+2F(2)F(1,2)
+P(1,2)
, (B.8) 1
where 4/y2 and 4/z2 represent the action density of a free dyon. The different terms contributing to SDD can be written as
F2(1) =6[f:+f,2+y4f:+y2z2f2+ 4Y2(f*f3
+f2f3)
-
4(Y
2y2f,f, *zHf*f4
+Z(Y *z)flf4-2~*(~
+f2f4)
-
qz2y2-
*z)f3f.t] OJ *z12)f42* (B-9)
E. K Shutyak,
278
J.J. M. Verbaarschot
/
QCD
instantons
The field strength F*(2) is obtained from F*(l) by interchanging combination F(lP(2) we find F(1)F(2)=2TrO[f1f;‘5f,f;-
z2f*L
-Y2f3f,
+ (Y *z,‘(f&
z and y. For the
+f&)]
+2TrO[-z*(y.z)f,f;-y*(y.z)f~~+(~.z)(f~f,+f,~)] + 2YOY(f,f;
- 5f3f;)
+ XYOzf,J5
+ 2lZOY
+ 2zOz(f,L (LA
-fLfl
-
u2f-J
-f&)
+2(~Y0z+z0Y)[-(Y~z)f&~Y*z~f3L+z2f‘&+Y2f3L]
(B.lO)
+2f3f;zxyOzxy.
In order to distinguish between the IA case and the II case we have introduced the sign 5 defined by [=l
for the II-case,
5 = - 1 for the IA-case. For F(l)F(l,
(B.ll)
2) we obtain
J’(l)F(L2)
=2TrO(y*z)h(f,
-f2>
-2zOyh(f,
- 2yOyz2hf4 + 2[zOzy*hf,
+ 2(1-
-f2)
+45yOzhf,
S)yOz( y*z)hf4.
(B.12)
By interchanging z and y in the above equation we obtain the combination F(2)F(l, 2). Finally, for the combination F*(l, 2) we find F2(1,2)
= @Wy2z*-
2h(y~z)2)fTr06+2h2((y~z)TrO-~yOz-zOy)2. (B.13)
The terms that do not depend on 0 are evaluated by first transforming to cylindrical coordinates, and then the remaining two-dimensional integrals are calculated numerically. Since Tr 0 and R,OabRb are the only invariants linear in 0, we can obtain the coefficients of the terms linear in 0 by calculating the integrals numerically for two different orientations. The coefficients of the terms
E. V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
instantons
279
quadratic in 0 are evaluated explicitly. This requires the following integrals: 4 jp dp dx, h’( X: - t)’ = 0.02019, 7 4 7jp 4 7 /
(B.14a)
dp dx, h2p2x; = 0.010095,
(B.14b)
p dp dx, h2p4 = 0.02949,
(B.14~)
4 ;;5jpdpdx,h2p2(x+;)=0.
(B.14d)
Collecting all terms and using the parametrization of 0 given in eq. (A.@ we find for the dyon-dyon contribution to the instanton-anti-instanton interaction for the ratio ansatz: SDD=- P (-0.2732 IA R x[IK,14-
+ 0.0354 lu14 - 0.0963 lu12+ 0.2165 (lu212 + flK,12(1
x[#u,12(Iu212
x [1U2i4 +
+
+
lK314
IKI')-
lu,12)+
f(1K2i2+
lK312)]
$KT~(K~+K~)+
+ $IK212b312
+ f(
+ K~K~)'
+
0.0317
0.0317
+
+K:(K~~+
u2uf
lK312)
uT2)]
-
+
0.0174
$(If.d212 + b312)l~12]). (B.15)
The dyon-dyon contribution of the instanton-instanton the same way. The result is SDD = g(-0.2732 II
+ 0.0744
+ 0.0780 (u*n)’
-
jKi4-
o.0317(lK212+
+0.00793((~~~,+u~u,)~+(~~K,+u~K,)~).
0.1195
1~1~
interaction
+ 0.1805~*~
is obtained in
- 0.1171
lu12K*K
lK312)
(B.16)
Note that for SU(2) instantons we have Iu12 = 1, and in this case the first terms can be identified as - l/r (with an accuracy of six digits for the calculated results, but of course not for the displayed numbers).
E.V. Shutyak,
280
J.J.M.
Verbaarschot
/
QCD
instantons
Appendix C In this appendix we study the overlap matrix elements in the limit p + 0. For p +Z r -X p*/p the function 4,(x, t) defined in eq. (4.3) is to leading order in /3 given by (6,(w)
-‘i2$(-$f
+ Wlxl)
exp( - F)),
(C.1)
where (x, t) denotes the distance to the pseudoparticle. We take the instanton to be localized at (O,O, r,r) and the anti-instanton at (O,O,O,O), and denote their coordinates by (r,, t,) and (tZ, t,), respectively. First, we study the function fr defined in eq. (4.6), which in the above limit reduces to fi=
-~/rd4zsin~~,cos$f2-$&exp[-$(r,+r,)]
+;p
4~cosTtp
-n
1
sin-tp
(r,*r*) 2’f/2’25/2
ew
I
-35
+r*)
1 (C.2) *
For r, G p or r2 Q p the approximation (C.l) used to obtain the integrand is no longer valid. One can estimate that the contribution to the integral from this region is of B(J3- ‘12), whereas the contribution obtained by using the r -=xfi limit for r& is only of 8(l). Therefore, only those terms in the asymptotic expansion of fi that exceed HP-‘I*) can be trusted. In eq. (C.2) the temporal integrals factorize from the spatial ones and can be evaluated to give 7r B sinT(t-r)cos--Idf= /0 P P
P --sinTr. 2
P
(C.3)
The spatial integral can be rewritten in terms of cylindrical coordinates z and p. Because the exponent depends only on (p2 + z2)l12 + (p* + (z - r)2)‘/2, the asymptotic expansion is more readily obtained after transforming to the “elliptic” variables u and u, defined by
24 = ((p*+z*)“*+(p*+(z-r)2)1’2)/r,
(C.4a)
u=((p*+zy*- (p’+(z-r)y*/r.
(C.4b)
Using that the integration
measure transforms as p dp dz + fr3( u* - u*) du du ,
(C-5)
E.V. Shuryak, J.J.M. Verbaarschot / QCD itwantons
281
we obtain
For rr//3 = 0 this integral can be evaluated analytically. However, in this limit the contribution from the region around zero where the approximation to the integrand is not valid is also of @(l/p). The situation is better for ar/p z=-1. Then the contribution from the regions near the centers of the pseudoparticles is only of &‘(/3-‘/2>, and the leading-order approximation to fi can be trusted. In this limit the integral over u is located near its lower limit, and the leading order term in P/rr is obtained by putting u = 1 + 6u and keeping only the leading order terms in 6u from pre-exponential factors. The result is
(C-7) The calculation of the asymptotic result for (C.11, f2 can be written as
f2=
f2 proceeds analogously. Using eq.
-~/Td4xcos$(1-r)cos$fr&2
- $1
exp[-i(r,+r,)]
d4xsin T
The contributions to the integral from the regions near the centers of the pseudoparticles where the approximation (C.l) is not valid is of ~9(/3-‘/~), and also is this case only terms that exceed this magnitude can be trusted. Again, the temporal integral factorizes from the spatial integral, and can be performed trivially. As before, the spatial integral is obtained by transforming to (u,u> variables (see eq. (C.4)). The final result for f2 is given by
x ;(uu-l)+~(uu+l) ( By putting
u = 1 in the pre-exponential
(~))ew(-+). factors
and performing
(C.9) the u- and
282
E. K Shwyak,
J.J. M. Verbaarschot
/
QCD
instantons
u-integrals we find that the asymptotic result for f2 is given by a2i fp=+-coszexp P In order elements center of no longer
-P
(
rrr P 1
.
(C.10)
to obtain the next to leading-order contributions to the overlap matrix one has to take into account the contributions from the region near the the pseudoparticles where the present approximation to the integrand is valid. References
[l] [2] [3] [4] [5] [6] [7] [a] [9] [lo] [ll] [I21 [I31 [14] [15] [16] [17] [la] [19] [20]
A.A. Belavin, A.M. Polyakov, A.A. Schwartz and Yu.S. Tyupkin, Phys. Lett. B59 (1975) 85 G. ‘t Hooft, Phys. Rev. D14 (1976) 3432 C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Rev. D19 (1979) 1826 R.D. Carlitz and D.B. Creamer, Ann. Phys. (N.Y.) 118 (1979) 429 B.V. Geshkenbein and B.L. Ioffe, Nucl. Phys. B166 (1980) 340 V.A. Novikov, M.A. Shifman, AI. Vainshtein and V.I. Zakharov, Nucl. Phys. B191 (1981) 301 E.V. Shuryak, Nucl. Phys. B203 (1982) 93, 116, 140 E.V. Shut-yak, Nucl. Phys. B214 (1983) 237 D.I. Diakonov and V.Yu. Petrov, Nucl. Phys. B245 (1984) 259 E.V. Shuryak, Nucl. Phys. B302 (1988) 559, 574,599 E.V. Shutyak, Nucl. Phys. B328 (1989) 85, 102 E.V. Shut-yak, Phys. Lett. B79 (1978) 135 E.V. Shuryak, Phys. Rep. 61 (1980) 71 D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43 T. Kanki, Osaka University preprint OSGE 86-24 (1988) D.I. Diakonov and A.D. Mirlin, Phys. Lett. B203 (1988) 299 E.M. Ilgenfrih and E.V. Shuryak, Nucl. Phys. B319 (1989) 511 M.A. Nowak, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B325 (1989) 581 J. Hoek, M. Teper and J. Waterhouse, Nucl. Phys. B288 (1987) 589 M. Kremer, A. Kronfeld, M. Laursen, G. Schierholtz, C. Schleiermacher and U.J. Wiese, Nucl. Phys. B305 (1988) 109 1211 J. Smit and J.C. Vink, Nucl. Phys. B298 (1988) 557 [22] S. Hands and M. Teper, Nucl. Phys. B347 (1990) 819 [U] F.R. Brown, F.P. Butler, H. Chen, N.H. Christ, Z. Dong, W. Schaffer, L. Unger and A. Vaccarino, Phys. Rev. Lett. 65 (1990) 2491 [24] E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. B341 (1990) 1 [25] B. Harrington and H. Shepard, Phys. Rev. D17 (1978) 2122 [26] 1.1. Balitsky and A.V. Yung, Phys. Lett. B168 (1986) 113 [27] E.V. Shut-yak, Nucl. Phys. B302 (1988) 621 [281 A.V. Yung, Nucl. Phys. B297 (1988) 47 [29] D. Foerster, Phys. Lett. B66 (1977) 279 1301 C.G. Callan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717 [31] B. Grossman, Phys. Lett. A61 (1977) 86 1321 J.J.M. Verbaarschot, Nucl. Phys. B362 (1991) 33 [33] V.V. Khoze and A.V. Yung, Z. Phys. C30 (1991) 155
QCD INSTANTONS AT FINITE TEMPERATURE (I). Gluonic interactions and the fermion determinant E.V. SHURYAK*‘**
AND J.J.M. VERBAARSCHOT**
Department
SUNy
of Physics,
Stony
Brook,
NY I1 794, USA
Received 28 January 1991
This is the first in a series of papers devoted to instanton-induced effects at non-zero temperatures. The main points will be the restoration of the chiral symmetry and the modification of the various correlation functions with increasing temperature. In this paper we investigate the temperature dependence of the gluonic and the quark-induced pseudo-particle interaction. We find that the gluonic interaction becomes much more long ranged, whereas the quark-induced interaction becomes exponentially short ranged.
1. Introduction Since their discovery in 1975 [l], instantons are generally believed to be an important ingredient of hadronic physics. The semiclassical theory of gauge field tunneling, its mathematical structure and its phenomenological implications challenged many theorists in the late seventies. However, due to complexity of the problem, only recently it became possible to formulate a consistent theory of interacting instantons, including all orders in the ‘t Hooft effective interaction and, at the same time, being simple enough to make practical calculations feasible. This is also the framework used in the present paper. Most people agree that the so-called U(1) problem, related to the axial anomaly, can only be understood with the use of instantons. However, it is far from being the only instanton-induced effect, or even the most important one. In fact, it is clear that instanton-induced forces between light quarks are very strong, so that they are probably the dominant element in the formation of non-perturbative vacuum “condensates” and even in the constitution of the hadronic masses themselves. If the experimental fact that the n’ mass is not of the order of 400 MeV (as chiral perturbation theory suggests) but nearly 1 GeV, can indeed be explained by * Supported in part by the US Department of Energy under grant DE-AC02-76CH00016. **Supported in part by the US Department of Energy under grant DE-FG02-88-ER40388. 0550-3213/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
256
E.V. Shutyak,
MM.
Verbaarschot
/
QCD
instantons
instantons, it is clear that such huge effect cannot be completely unimportant for other hadrons. Looking at the correlation functions, it is easy to see that the same ‘t Hooft effective interaction [2], which is repulsive in 7)’ channel, is also present in all scalar and pseudoscalar channels. In particular, it is strongly attractive in the scalar isospin 0 and in the pion channel, which has led to the idea [3,4], that instantons are also related to SU(N,) chiral symmetry breaking. If so, they also should provide the “effective masses” of light quarks, which in turn are the main building blocks of the hadronic masses. Let us also briefly mention the main steps in the development of this theory. Phenomenological connections outlined above were analyzed, starting from papers [5,6], and then led to refs. [7, S], where the qualitative features of the ensemble of interacting pseudoparticles or the “instanton liquid” have been pointed out. Surprisingly enough, essentially the same features emerged from a pioneering paper on the variational approach to theory of interacting instantons [9]. Encouraged by this agreement, many efforts were devoted to the study of the approximations involved in this analysis and to develop a more quantitative theory. This goal was essentially reached in ref. [lo], where it was demonstrated that the problem allows for straightforward numerical solution. Recently, it was applied to calculation of various mesonic correlation functions with surprisingly good results [ll]. With this paper we start a new series of papers, in which several questions will be addressed. First of all, we are planning to do .much more accurate calculations than have been performed previously, reducing both the statistical and the systematical errors (the latter because we are working with larger systems). Second, we want to increase substantially the number of calculated quantities, in particular we want to investigate off-diagonal correlators (like the w-4 one, etc.) and the baryonic correlators. Third, we want to study in a systematic way the temperature dependence of all these quantities. The technical obstacle which prevented the study of the last point in earlier work is that, at non-zero T, the instanton solution (called “caloron”) no longer has the four-dimensional spherical symmetry. Therefore, the interaction between these objects depends on many more variables, and its understanding becomes much more involved. The investigation of these, mainly technical problems is the main topic of the present paper. Several other works that study instanton-induced effects at finite temperature should be mentioned. At the one-instanton level there is the well-known statement that instantons are suppressed at pT < 1 [12,13], which was analyzed quantitatively in ref. [14]. This by itself shows that, at high T, the role of these effects is strongly diminished, and they are present only in the form of a “molecular vacuum” [lo] in which the chiral symmetry is restored. The temperature dependence of instanton-induced effects was investigated with the help of a mean-field approximation both without quarks [l&16] and with light quarks included [17,18]. The former one [17] compares the thermodynamical functions for “liquid” and
E.V. Shuryak, J.J.M. Verbaarschot / QCD instantons
257
“molecular” phases, indicating that chiral restoration is to take place at T, - 200 MeV, a conclusion which was also reached in ref. [18]. In connection with the present paper let us note that in ref. [16] the pseudoparticle interaction averaged over all orientations and positions, was evaluated at nonzero temperatures. The conclusion was that the temperature dependence is very weak (a few percent). However, our findings are qualitatively different. Most probably, their “temperature independence” is due to the use of the sum ansatz with its artificial divergence of the field strength at the centers of the pseudoparticles, as well as due to the averaging over the positions. Finally, we would like to make some comments related to lattice simulations of QCD. Instanton-like configurations have been found in the vacuum fields by different methods (the “cooling method”, the “geometric method” and the “fermionic method”, see refs. [19-211 and references therein), and with still large uncertainties the density is roughly consistent with that suggested by the “instanton liquid” model. It is not yet clear in which way these configurations are related to many other lattice results, e.g. the quark condensate value or the hadronic masses. (However, see ref. 1221 for progress in this direction.) Attempts to get the correct n’ mass on the lattice are not yet successful, so something is still missing. An interesting branch of lattice calculations, which is strongly related to the instanton-induced phenomena we are going to study, is the investigation of chiral symmetry restoration at finite temperatures. In this framework, it was found that “quenching” of the fermionic determinant is not a good thing to do, and, in fact, the phase transition parameters are very sensitive to the presence of light quarks and their masses. In particular, according to ref. [23], the cases of Nr = 2 and 3 are very different. Also the value of the strange quark mass below which the strong first-order transition takes place is very small, of the order 50 MeV. These observations are in interesting correlation with our results 1241 for the numerical simulation of the ensemble of instantons. We also see a large difference between 2 and 3 flavors, and the transitions is at about at the same mass. We hope to clarify this correspondence in our next papers. However, from general considerations it is clear, that in spite of these correlations, the lattice data cannot really describe the phenomena we are discussing. The reason is, that in order to speak about phases in the ensemble of pseudoparticles, one should have many of them. For example, we are typically working with an ensemble of 32 pseudoparticles, while present lattice calculations with dynamical fermions have only room for a few pseudoparticles. Therefore, they can only see some traces of the instanton-induced phase transition, at best. Let us now outline the contents of the first paper of this series. In sect. 2 we introduce known formulae for finite-temperature instantons (calorons) and generalize the trial functions used for multi-pseudoparticle configurations to the non-zero temperature case. A detailed discussion of the classical interaction between a pair of pseudoparticles is presented in sect. 3. Analytical results are obtained for the
258
E. K Shuryak, J.J.M. Verbaarschot
/
QCD
instantons
high-temperature limit (details are given in appendices A and B) and for a wide range of parameters in between this limit and the well-known zero-temperature limit the interaction is computed numerically. Sect. 4 is devoted to the so-called overlap matrix elements of the Dirac operator between fermionic zero modes of different pseudoparticles. Again, we obtain, analytical results for the high-temperature limit (see appendix C for details), and we perform extensive numerical calculations in order to cover the range of parameters between this limit and zero temperature. The results of both the gluonic and the quark-induced interaction are summarized by a fitted expression which is an approximate but closed form suitable for further studies of the statistical mechanics of the system of interacting instantons.
2. Calorons and trial functions As is well known, in the euclidean formulation of quantum field theory, a finite temperature is formally introduced via periodic (anti-periodic) boundary conditions for all bosonic (fermionic) fields on the strip [O, p = l/T]. The topologically non-trivial solutions of the classical Yang-Mills equations satisfying these periodicity conditions were obtained in ref. [25] as a special case of the ‘t Hooft multiinstanton solutions. The explicit expression for a gauge field has the form A;( x) = fi;,a,n-
’)
(2.1)
where sinh( 2rr/j3) n(x)=l+* /3r cosh(2rrr//3) - cos(2at//3)
*
(2.2)
Here, p denotes the size of the caloron and xP = (r, f&. Of course, at T + 0 this field reduces to the field of a single instanton. However, in the opposite limit, Tp Z+ 1, this solution has a quite different form. As was pointed out in ref. [14] in this case the calorons develop a “core” of the B(p) where the field is very strong, G,, = &‘(p-*). For distances between 8’(p) and @(p*/fl> the caloron solution can be approximated by a p-independent dyon field, given by the very simple formula A:(x)
= F&rk/r2.
(2.3)
In the third region, at distances larger than 6’(p*//3> from the center, “the caloron periphery”, the field strength is of @(pm*). For illustration, we show in fig. 1 the dependence of the square of the field strength on the spatial distance from the center for several values of the temperature.
E. V: Shwyak,
J.J.M.
-4
-2
Verbaarschot
0
/
QCD
instantons
r/P
259
4
2
Fig. 1. The distribution of the field strength squared as a function of the spatial distance from the caloron center r, in units of the caloron radius p. The total action is normalized to one. Five curves from top to bottom correspond to values of the temperature Tp = 8,2,1/2,1/4,1/g, in this order. At high temperatures one observes that a core develops, as well as a universal dyon region between the core and periphery.
Now we arrive at the discussion of multi-pseudoparticle configurations. In this case explicit solutions of the classical Yang-Mills equations are known. It is also a direct consequence of a general theorem that the classical interaction between instantons is zero, so that the action is proportional to the topological charge. Configurations containing both instantons and anti-instantons are much more complicated. The discussion of the so-called “streamline” set of configurations, which are most suitable for their description, can be found in refs. [26,27]. However, due to technical difficulties these “streamlines” were only found for some toy models like the quantum mechanical double-well problem. For the description of instanton-anti-instanton configurations in a four-dimensional Yang-Mills theory one still has to rely on the use of trial functions. In particular, in ref. [9] the so-called “sum-ansatz” was used, in which A:(x) is given just by the sum of one-pseudoparticle solutions (in the singular gauge). However, it has some artifacts, and an improved “ratio ansatz” was suggested in ref. [lo] which for an instanton-anti-instanton pair is 2 oFbT;,YlyP:/Yf A”,(x)=i
1+ P:/Y:
+ O:brl;,YA,P:/Y: + Pi/Y&?
,
(2.4)
where y, =x -RI and y,, =x -R,, and R,, R, are the positions of the centers of the instanton and the anti-instanton, respectively. The orientation of the pseudoparticles in color space is given by the O(3) matrices 0, and 0,. In the case of
260
E.K
Shutyak,
J.J.h4. Verbaarschot
/ QCD
instantons
one instanton-anti-instanton pair, a very interesting trial function was suggested in ref. [28]. It has interesting properties and is probably the best one so far suggested. Results related to this ansatz will be presented in a separate paper [32]. Unfortunately, it is unclear how to generalize it to many pseudoparticles and to non-zero temperature. Our first new step in this paper is a straightforward generalization of the ratio ansatz to the case of finite temperatures, which is based on the observation that a one-caloron solution can be considered as an infinite sum of instantons, with centers located equidistantly along the “time”-axis with the spacing p. The ratio ansatz for this configuration can be rewritten as 1 o$$, Axx)
= -ii
where L!, =n(y,) and Ir, high-temperature limit the modified by the presence of p .=x YA -K P2/& and p + 0, AL(x)
aJ7, + o;bq;,a”17* n,+rr,-1
(2.5)
’
=Il(y,+,) and Kx) is defined in eq. (2.2). In the field in the core region of the instanton will be an anti-instanton and vice versa. In the limit yr -ZEp, the gauge field is given by
= 2 g y;( ,O; + y;( 1 + r2p:/3b2
+
~ii/~YA))
’
P-6)
In this limit yA = r % B(p), and therefore, eq. (2.6) describes the gauge field of an instanton with renormalized size pR given by 1 -25 PZR
3 P Pk I+----+3 P2 i
r
r P[
3P2 ?T’pf I *
(2-V
Here and below we will denote the spatial distance between a pair of pseudo-particles by r, their temporal distance by 7 and their distance in euclidean space-time by R. In sect. 3 we will need this limit in order to estimate the contribution of the core region to the pseudoparticle interaction. In the limit j3 -+ 0 for fixed distance r between the centers of the pseudoparticles, the total gauge field in eq. (2.5) up to a distance @(p2//?> simplifies to the following expression: 1
1
Y,(Y,+yAP:/Pi)
--
1
Y:
1 YA(YA+YIP~/P:)
I'
(2.8)
E.K
Shutyak,
J.J.M.
Verbaarschot
/
QCD
instantons
261
where the indices k run from 1 till 3, and the overall scale of pr, pA has dropped out. In sect. 3. and in the appendix B we will evaluate the instanton-anti-instanton interaction resulting from this gauge field.
3. Classical pseudoparticle
interaction for the ratio ansatz
The classical interaction is defined as the difference of the action of a pair of pseudo-particles at finite separation and their action at infinite separation. In the limit where the four-dimensional distance R between the center of the pseudopartitles is much larger than their size p (R Z+ p), the classical instanton-anti-instanton interaction S,, at zero temperature is given by the well-known dipole formula
m 301, 4PfPi S,, = d7
>
d=1-4(u*R)2/R2.
(3.1)
Here, the action is given in units of the action of one instanton S, = (8rr2/g2). The orientation-dependent factor d depends on the four-component vector uP related to the spinor representation of the relative orientation matrix U as U = iuPrl CT,‘= (7, -i) and 7; = (T, i)). The relative orientation matrix can also be written in terms of the vector representation of U which will be denoted by 0 and is given by
If one deals with the SU(2) color group, U is unitary and one has real up and u: = 1. The matrix 0 is real and orthogonal in this case. Note that the group average of d over orientations vanishes. For arbitrary NC, d generalizes to d=lu12-41(u.R)12/R2,
P-3)
where U = iuprl is now the upper 2 X 2 corner of the NC X NC relative orientation matrix. It is not unitary and therefore the ug are complex with 1u12# 1. In this case, the matrix 0 is complex. On general grounds, one can make the following statement about the angular dependence of this interaction at zero T. As the pseudoparticle interaction is the same for all directions perpendicular to R,, and as our expression for AZ has only quadratic terms in u, the general orientation dependence in the case of W(2)-,color for arbitrary values of R is given by S,=s,(R)
+~,(R)(u~~)~+s,(R)(u*~)~,
(3.4)
where 2, is a unit vector in the direction of R,. From now on we will choose the
262
E. V. Shrlryak, J./M Verbaarschot / QCD instantons
l-axis in the direction of R, and write u, instead of (u *a>. For the sum ansatz the pseudo-particle interaction was evaluated in ref. [9] at large R up to order 6’(l/R6> and in the limit R + 0. In both cases it was found that s,(R) = 0. This statement also holds true for the ratio ansatz, and therefore we expect only a small correction proportional to (u * &I4 at intermediate distances R. At non-zero temperature the “time’‘-axis is special, leading to a caloron interaction with less symmetry. There is only symmetry in the plane orthogonal to time axis and r (the three-dimensional distance is denoted by small r). As a result, the interaction depends on the following variables: 6) the spatial distance r between two centers; (ii) the “temporal” distance 7; (iii) the sizes p,, pA; (iv) relative orientation matrix U, and (v) the temperature T. The orientation dependence of the interaction for the case of SU(2)-color can be parameterized as follows: S,,=s,+s,u:+s,u:+s,(u:+u:)+s4(u:+U:)2+S~U:(U:+U:),
(3.5)
where the l-axis has been chosen along the line connecting the two pseudoparticles. We have finished the general considerations and now start with the discussion of the high-temperature limit. As mentioned above, the caloron field has a different form in three different regions: (i> a core of radius 5Q>; (ii) a dyon region outside the core up to a distance @‘(p2/p>; and (iii) a peripheral region outside. It is natural to split the interaction of two calorons into the contributions from these regions. In the core region the field of the other pseudoparticle has the net effect that it leads to a renormalized size given by eq. (2.7). The core contribution to the total action is proportional to the product of the square of the field strength at the core and the volume of the core region,
(3.6) leading to an orientation independent repulsive correction to the interaction of the order P/R. (Of course, our crude argument is not sufficient to provide us with the coefficient, but this result is supported by the numerical data discussed below.) For the sum ansatz this additional renormalization is absent, and the leading contribution of the core region to the dipole is given by the product of dyon field of the other pseudoparticle and the &p-21 field strength of the core, resulting in a contribution of &Q2>. In the limit of high temperatures and fixed distance r there is another leadingorder contribution to the pseudoparticle interaction: the contribution from the
E.V. Shuryak,
dyon-dyon
J.J.M.
Verbaarschot
/
QCD
instantons
263
region given by SDD=/d’x[F:.(A:,o)-~-~]. IA
(3.7)
Here, A;,, is the dyon-dyon gauge field for the ratio ansatz given in eq. (2.8). In order to obtain the interaction we have subtracted the field strength of the free dyon fields. Since the field strength of the dyon at distance x from the center is of 6’(xV2>, integration over the 4-volume results in a factor of &Q/R). For pr/pA = 1 the orientation dependence of the coefficient of P/R can be obtained analytically (see appendix A). The final result for arbitrary value of u (u is real and 1~1 = 1 for SU(2)) is given by $$+-
P (-0.2732 R
+ 0.0354 Iu14 - 0.0963 lu12 + 0.2165(1u,12 + lu312) + 0.0317
x [IL+ x
X [lu214
+
flU,12(1+ lul’) - f(lu212+ lu,12)] + 0.0317
[~Iu,I~(Iu~I~ lu314
+
+
~1~~1~1~~1~
lu312)
+ f(
+ $KT’(K$
u2uz
+ KS)
+ K~K:)~
+
+uf(u;”
+ KS’)]
+
0.0174
- $(1+l’+ IK~I~)IKI~]). (3.8)
Note that the interaction is the same for uO = 1 and u, = 1 and that the coefficients of the b(u4)-terms are numerically small. Because the size of the core is not well defined the value of the core contribution is not well defined either. However, since the core contribution to the interaction is due to a size renormalization, it is the same for the II-case and the IA-case. In the former case we know that for d = 1 the ratio ansatz represents an exact ‘t Hooft multi-instanton solution, and the total interaction is equal to zero. Therefore, the @‘(p/R) contribution from the core has to be cancelled by the only other &T/?/R) contribution from the dyon-dyon region. In appendix B we show that for pA/p, = 1 the coefficient of the latter term is equal to l/r. We thus conclude that in the p + 0 limit and for pseudoparticles of equal size the core contribution is
which is orientation independent. For z~a= 1 and K, = 1 it is nearly cancelled by the dyon-dyon part of the IA-interaction. For the sum ansatz there is no size renormalization and, in the high-temperature limit, the &‘(p) contribution of the core is absent. In this limit the B(p) contribu-
E. K Shurynk,
J.J.M.
Verbaarschot
/
QCD
instantons
-0.4 tl
0.0
’ I I
I I I I I I I ’ ’ I I I I ’ ’ I 1I ’
1.0
2.0
3.0
4.0
l-l
5.0
r/P Fig. 2. The interaction between an instanton and an anti-instanton in units of So, the action of a single instanton, as a function of the spatial distance between their centers r, given in units of the geometric mean p = (p,pA)l12. The four-component quantity u given in the figure defines the relative orientation of two pseudoparticles (see text). Four set of points and four curves (solid, dotted, dashed and dash-dotted) correspond to values of the temperature Tp = 0,1/2,1,8, respectively.
tion solely originates from the dyon-dyon analytically. The result for arbitrary u is sp,D=- 3; ;(8(lul’-
interaction
which also can be obtained
lu12)+ 81u,12(1- ;lul’) + 12(lu212+ bs12)(1 - $I”)
+41U,12(lU212+ lU,12)+2U:2(U:+U:)+2U:(U:2+
U:')),
(3.10)
showing that the sum ansatz leads to a much stronger interaction. In this case the interaction does not depend on the ratio pl/pA. We have made extensive numerical studies of this interaction for the case of the SU(2)-color group, using a Monte Carlo integration with importance sampling. At zero temperature we chose the sum of the field strength squared of two instantons (see ref. [lo]> as a weight function. At non-zero temperatures we tried several weight functions but the one given by the sum of the field strength squared of two calorons of that temperature appeared to give the best results. Our results at T = 0 agree with those given in this paper inside the error bars. In figs. 2-4 we show some samples from our data, together with the curves obtained from a fit to all our
E. K Shutyak, J.J.M. Verbaarschot / QCD instantons
265
o 0.6 ?
z vl 0.4
0.2
0.0
‘;;Q&& -J-. -.-.-.~-.-.-.2.-.-.-.-.-.-. 0.0
1.0
2.0
3.0
4.0
5.0
r/P Fig. 3. The same as in fig. 2, but for a different relative orientation (see figure).
1.0
I I I I I I I I I ’ I I I l1I I ’ ’ l1I I I I-L,)
0.8
u = (O,O,O,l)
-
o 0.6 ? 2 fn 0.4
0.2
0.0 0.0
1.0
3.0
2.0
4.0
5.0
r/P Fig. 4. The same as in fig. 3, but for different relative orientation (see figure).
266
E. K Shutyak,
J.J.M.
Verbaarschot
/
QCD
inslantons
data which is given by
4.03
s,* =
(R*+2.10)* P*+5.21 16.16 u;
-
(1+1.68R2)3 2.73 uf
+ - (R*+ 2.03)* + (1+0.33R*)~
+ (l+0.42R2)4
1
P*+O.75
,6*+0.24+ 11.50 R*/( 1+1.14 R*)
1 (u; + us)
P2 1 P2
0.721og R*
1.66
P2
(1+0.013
1
R2)4 P*+l-73 '
(3.11)
Here, R is understood as four-dimensional distance between the centers of the pseudoparticles, R* = Y* + (t, - t,)* and all the lengths are in units of pmean = (p,pA)‘/* (the bulk of the calculations was performed for p, =pA = 1). We have shown numerically that the dependence of the interaction through other combinations of r, t, p, and pA is very weak. This corroborates with the analytical results for the low- and high-temperature limits. Note that terms of fourth order in uP are absent. From calculated values of the interaction for a wide range of p and R and the orientations ua = 1, u, = 1, u3 = 1, u,, = u, = l/a, u0 = u3 = l/a and u, = u3 = l/ fi we have determined the coefficients sc, . . . , ss in eq. (3.5) for many values of /3 and R, with the result that s2 = s4 = ss = 0 inside the error bars. At large distances and at zero T our results are consistent with the asymptotic dipole formula (3.1). In the high-temperature limit, only the terms proportional to p survive, and from the last term in eq. (3.11) we obtain 0.208 P/R, as opposed to 0.206 p/R for the analytical result in this limit (see eq. (3.8)). Note that for R + 0 the interaction becomes large. In fact, it can be shown that it diverges logarithmically in this limit, a result which has also been found for the sum ansatz [9]. Certainly this is an artifact of the trial function. Whether or not it is important for the statistical mechanics of interacting instantons, will be discussed in the next papers of this series. We have also studied the instanton-instanton (II) case. Of course, for exact multi-instanton configurations there should be no classical interaction, and the action should not depend on the collective variables. However, as discussed in ref. DO], this interaction is non-zero for the ratio ansatz. In the high-temperature limit we again find contributions from the core and from the dyon-dyon interaction. The former only involves a renormalization of the size, and we find the same result as in the instanton-anti-instanton case. For the
E.V. Slwyak, J.J.M. Verbaarschot / QCD instantons
dyon-dyon
261
interaction we now find (the uP are arbitrary complex numbers)
s,~D= -P ( -0.2732 R
+ 0.0744 lu14 - 0.1195 lul*+
+0.0780 (u*u)’
O.l805u*u
- 0.1171 lul*u*u
- 0.0317 (lu212 + lu312)
+0.00793[(u~u,+u;uO)*+
(u;u,+U~uO)*]).
(3.12)
When the orientation matrix U = 1, the II gauge field is an exact ‘t Hooft multi-instanton solution at any temperature. Therefore, if one adds to eq. (3.12) the contribution of the core the interaction should be exactly zero. In our numerical calculations we indeed find this to happen. Surprisingly, in our numerical calculations we also find that the II-interaction is the same (inside error bars) for all three other orientations. For the terms linear in 0 (or quadratic in u) this is obvious. The only invariant is given by
Rhq$O"bq~KRK=TrO(R*+
Ri).
(3.13)
At non-zero temperatures the coefficients of R* and Ri are different, but the relative orientation only enters in the combinations Tr 0. For terms quadratic in 0 one can write down other invariants, and the simple dependence on Tr 0 gets spoilt (see e.g. the result for the high-temperature limit). In order to analyze the orientation dependence we calculated the interaction for cl,, = 1, u, = 1, ug = 1, ua=u,=l/\/z, z+,=z+=l/fi and u,=+=l/fi enablingustocalculatethe coefficients of sa, . . . , ss in eq. (3.5). However, in all cases considered, we found that s2 = s4 and sg = 0 within our numerical accuracy. Apparently, the coefficients of the terms that are not proportional to u* or (u*>* are too small to be seen within the error bars of our numerical results. Our numerical results for the case of the SU(2)-color group can be summarized by the following fit: S,, = [0.63u2 + 0.071 (u*)*]
X
’ (1+0.43R2)3
log R*
P2
(1 + 1.17R2)4
P* + la17
+ [o.o7u’+
P2 P*+5.33
[0.05u*+0.47(u*)*] -
o.o5(u’)*]
log (3.14)
Again the bulk of the calculations was performed
for p, =P+, = 1. For a selective
E. K Shuryak,
0.0
J.J.M.
Verbaarschot
1.0
2.0
/
3.0
QCD
instantons
4.0
5.0
r/P Fig. 5. The same as in fig. 2, but for a pair of instantons, with a relative orientation as indicated in the legend of the figure.
number of cases it was shown that the size dependence is well taken into account by expressing all lengths in units of (or pA) ‘I*. The coefficients of the last term agree within error bars with our analytical result for the dyon-dyon interaction. [In fact, the x*-value of the fit is not sensitive to the ratio of the coefficients of u2 and (u*)* in the last term of eq. (3.14). Therefore, we fixed this ratio at the value given by the high-temperature limit. For the other terms the coefficients of both u2 and (u*>* are well determined by the fit.] Some of our numerical results together with graphs of the above fit are shown in fig. 5.
4. Overlap matrix elements In the semiclassical limit the leading part of the fermion-induced interaction arises in the form of the determinant of the Dirac operator in the space of zero modes. In this section we calculate the elements Tij of this determinant at finite temperatures. The fermionic zero mode of the massless Dirac equation in the background field of a caloron was first obtained in ref. [31]. It has the following form:
E. V. Shuryak, J.J.M. Verbaarschot / QCD instantons
269
where the constant spinor l a is a 2 X 4 (color X Dirac) given by
E
=- 10-l a@ fi ( 1
01 0
-1
0 1 .c1’
(4.2)
The function &, is given by
where II has been defined in eq. (2.2) and cp= (17 - 1) cos(rrt/~)/cosh(2ar/B>. The overlap matrix of the fermionic zero modes is defined by
(4.4) where covariant derivative b in the Dirac operator contains the caloron field. We make two assumptions: (i) the gauge field Ai is just the sum of a caloron and an anti-caloron, and (ii) for the calculation of T,* one may consider only the fields of this particular pair of pseudoparticles, ignoring all others. The former assumption needs some comments. Above we have used a much more complicated expression for the gauge field, the ratio ansatz, but now we do not use it. The reasons are: (i) the classical interaction considered above stands in the exponent requiring a high accuracy, whereas the overlap matrix elements under considerations only enter through the fermion determinant which is a quantum effect; (ii) the simple sum ansatz produces artifacts in the field strength, making it infinite at the centers, but there is no such problem for the ferrnionic zero modes. These assumptions enable us, using the fact that both I& satisfy the corresponding Dirac equation, to get rid of gluon field, resulting in an expression obtained from (4.4) by replacing 6 by 8. Let us first make several general comments on this definition. The matrix element depends on the same variables as the interaction discussed above. Since the zero modes are normalized to 1, the dimension of this matrix is that of the Dirac operator itself, namely that of an inverse length. The relative orientation dependence of T,, is linear in U = iuwrl, and thus can be parameterized in terms of two functions,
The functions f, and fi can be obtained by carrying out the Dirac and the color
270
E. V. Shuryak,
JJM
Verbaarschot
/
QCD
instantons
traces. The result is
The large-distance behaviour of matrix elements T,, can be obtained analytically both in the zero-temperature limit and in the high-temperature limit. For T = 0 the asymptotic result is known, (4.8) where U&T: is decomposition of the relative orientation matrix (see sect. 3). At r s /3 the behavior of & is dominated by the hyperbolic functions in eq. (4.31, which lead to an exponential decay. This is a general consequence of anti-periodic boundary conditions for fermions along the “time” axis. As we will see below due to this fact the overlap matrix at high temperatures will be qualitatively different from the T = 0 case. From the structure of the zero modes it is also clear that in the limit r XDj3 its time dependence in form of COS(~T/~> factorizes from its r-dependence (the temporal distance between the pseudoparticles is denoted by 7). In the integrand of the overlap matrix elements, the zero modes give rise to the exponential factor exp(-rr/P((p* + (z -R)*)‘/* + (p* + z*)‘/~), where z is the coordinate in the direction of the line connecting the two centers and p is the distance to this line. It is clear that the exponent is constant on ellipsis surrounding the two centers, and in particular, it is constant on the line segment connecting the two centers. The remaining part of the integrand is an algebraic function, and in the limit of large distances the integral can be performed analytically (see appendix C for details). As result for f, and f2 in the limit (p + 0, R + a~)we obtain fy=i$sin(y)exp(-F),
(4.9a)
(4.9b) which does not depend on the sizes of the pseudoparticles. We made an elaborate numerical study of the integrals f, and f2. By using the axial symmetry about the line connecting the two centers, the space-time integral
E.V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
271
imtantons
can be trivially reduced to a three-dimensional integration, which can be performed by a standard gaussian integration. We evaluated the integrals for a wide range of points in the (R, T)-plane and for different orientations. The results of these studies are fitted by the following formulae which interpolate between the analytically obtained high-temperature limits and low-temperature limits,
(r/p)sin(rT/P) cosh(ar/P) “l’ = (cosh( rr/j?)
- cos( rr//I)
+ K:)’ 1
’ (p2/rr2)(
exp( -rr/2p)
+ rrr/2p)
+ (2/~r)(
quantity
0.69 exp( - 1.75 r/P))
2)(l+(~i’l+o;~~2r2).
(0.82 r2 + 1)
The temperature-dependent
I-
(4.10)
is given by
K:(T)
1.07 K T(T) = /P/d
‘\
-6
0.0
’ ’ “I
(4.11)
+ 0.53 *
-.
\
-*.
’ ’ ’ ’ ’ ’ ’ ’” ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’
1.0
2.0
3.0
4.0
5.0
r/P Fig. 6. A logarithmic plot of the overlap matrix element of the fermionic zero modes of an instanton and an anti-instanton T,A (times the geometric average p of their sizes, to make it dimensionless) versus the spatial distance between their centers r (in the same units). The relative orientation is such that the vector u is parallel to r (in this case in the z-direction). Five sets of points and five curves (solid, dotted, dashed, long-dashed and dash-dotted) correspond to temperatures Tp = 0,1/2,1,2,8, respectively.
272
E.V. Shutyak,
2
J.J.M.
III1
Verbaarschot
III1
/
IIII
QCD
instantons
IIII
l-
III1
u = (O,O,O,l) -
\4 \
-4 i i
III!1
-6
0.0
IIll
‘. .I \\
l .
\
II11
1.0
-. -.
\
III1
2.0
*.
-.
llll.
3.0
4.0
*.
_ ‘*.
5.0
r/P Fig. 7. The same as fig. 6, but for the orientation vector u pointing into the time direction. In this case the temporal distance between the centers is not zero, as in fig. 6, but p/2, and one more difference is that the solid curve now corresponds to a temperature of l/8 instead of zero as in fig. 6.
For the absolute value of the coefficient
If*1=
f2 we find
(d@cOsh/~> SiWNP) (cosh( rr/p)
x
i
- cos( ‘TTT/p)
+ K;)’
1+0.069
(4.12)
where K:(T) is given by 1.00 K;(T)
=
0.69
+
@*/T*
(4.13) *
Again, ah quantities are expressed in units of the geometric average of pI and PA. We have checked numerically that the dependence of fl and f2 on pr and PA through other combinations than (pr pA)‘/* is weak. The accuracy of these fitted formulae can be seen from figs. 6 and 7, where we have used eq. (4.5) to relate fr and f2 to the overlap matrix elements TIA’
E.K Shutyak, J.J.M. Verbaarschot / QCD instantons
273
5. Summary In this paper we present extensive studies of both the classical gluonic and the quark-induced interaction between pseudoparticles. In particular, we have obtained the following results: (i) Analytical expressions for the high-T limit, valid for any SU(N,) color group. (ii) Numerical data for SU(2) color covering a wide range of distances, temperatures and relative orientations. We have shown that these data agree with all known asymptotic expressions. (iii) These data were fitted numerically by some multi-parameter trial functions, resulting in closed expressions to be used in future calculations of the statistical mechanics of the system of interacting instantons at non-zero temperatures. As the lengthy formulae are not very transparent, it is worth to emphasize the main qualitative differences between our results for non-zero T and the known interaction at T = 0. Contrary to some claims in the literature (and even contrary to our original expectations) both types of the interaction are found to be quite different at high and low T. At low T the classical interaction at large R is a dipole-dipole one of 8(1/R4>. At small values of R it is repulsive. On the contrary, at large T we find an B@/R) interaction in a wide range of intermediate distances p * r < p2//?, which is more long ranged and may lead to stronger correlations between neighboring pseudoparticles. In contrast to this, the quark-induced interaction is becoming very short range at high T. Instead of an @(l/R31 behaviour of the overlap matrix element at zero T, at high temperatures we obtain an exponentially decaying function exp(-rrTr). There is a general reason for that: fermions obey anti-periodic boundary conditions in euclidean time, and therefore the corresponding Matsubara frequency is never zero. As a result, at high temperature the quark zero modes are concentrated in a small core region, so that the instanton-anti-instanton molecules become strongly bound. We would like to thank P. van Baa1 for useful discussions.
Note added in proof For most attractive orientation a numerical solution of the streamline equation was found in ref. [32]. This provides us with an ansatz-independent gluonic instanton-anti-instanton interaction. However, the method used in ref. [321 is based on conformal symmetry, and therefore cannot be generalized to non-zero temperatures. After completion of this manuscript a paper [33] in which the high-temperature limit of the pseudoparticle interaction is considered came to our attention. The authors of this paper obtain the same asymptotic expressions for the ‘overlap
214
E.V. Shwyak, J.J.M. Verbaarschot / QCD instantons
matrix elements of the zero modes. However, we do not agree with their result for the gluonic part of the interaction which does not include the dyon-dyon contribution.
Appendix A In this appendix we evaluate the high-temperature limit of the instanton-antiinstanton interaction for the sum ansatz. In this case the gauge field is given by
(A-1) where z =x -R/2, y =x + R/2, and Oab is the relative orientation of the pseudoparticles in color space. The temporal component of the 4-vectors x and R is equal to zero. The field (A.0 corresponds to the superposition of two dyon configurations. The total field strength can be written as the sum of the following three terms (see ref. [9]):
where (A.3a)
F;J 1,2) = eubc$$
In terms of these combinations
s,D,D = &14( d x
[ $k,$Ocd
+ ?j:k?,;,O”“]
the total dyon-dyon
0 0
2F 1 F 2 +2F(l)F(1,2)
.
(A.3c)
interaction can be written as +2F(2)F(1,2)
+F*(1,2)), (A.41
where color and Dirac indices have been suppressed. The evaluation of the different terms contributing to this integral is straightforward but tedious. The
E. V. Shutyak,
JIM.
Verbaarschot
/
QCD
instantons
275
result is given by -TrO+F+-
+ TrO(y*z)* J&2 Y2
Z0.Z
YOY
(ASa) -
TrO(y*z)(y*+z*)
+
y*2* (ASb) P(1,2)
2
= JW
4-
(y.z)* JW
+ ((y*z)TrO-yOt-zOy)* )A* (A.5c)
Note that FWF(2) vanishes for 0 = 1. Also the other contributions greatly simplify in this case. After changing to cylindrical coordinates the interaction can be expressed in terms of the following integrals: /d+:-+)=f,
jd,u(+t)*=+
I dpp*=
+,
/dpz*(++)=O,
I
dpp’=
/dpz2y2=l,
$,
/dpz*p*=+,
I
dpp*x:=$,
/dp(x;-$+p’)‘=+,
where = -$~-pdpj--=/q
/ Gf(X,,p)
f2;:’
(A.7a)
,
y*= (x, ++)*+p*,
(A.7b)
2* = (x, - +)‘+ p*.
(A.7c)
The matrix 0 can be expressed in terms of the 2 X 2 complex matrix iz.$~i in the left-upper corner of the IV,, X NC relative orientation matrix as oah = (lu()l* - luI*)s,,
+ U~Ub + u,u;E: + 2E,&( 4&
+ u~u,)
.
(A.8)
276
E.V. Shutyak,
L3.M.
Verbaarschot
/
QCD
instontons
Substitution of these results into eq. (A.4) yields the high-temperature instanton-anti-instanton interaction for the sum ansatz,
SDD=$$(8(lu14lu12j+81u,12(1-~lu12)+12(1~212+
limit of the
1~~1~)(1-~lul~)
IA
+ 41U,12(lU212 + 1U312 + 2U72(4 + us) + 244’
+ u:“)).
(A-9)
Appendix B
In this appendix we calculate the high-temperature limit of the instanton-antiinstanton interaction and the instanton-instanton interaction for the ratio ansatz in the case that the sizes of the pseudoparticles are equal. The gauge field is given by (see eq. (2.8))
for the instanton-anti-instanton At(x)
for the instanton-instanton
case and by
=7iKYK4Y,z)
+ Oab7~AzAa(z, y),
case. The scalar function 1 a(y,z)=;-i.-
a(y, z> is defined by
1 Y(Y +z)
(B.2)
P-3)
’
and z=x-R/2, y=x+R/2 and 0”’ are defined as in the case of the sum ansatz. We decompose the total field strength as for the sum ansatz (see eq. (A.2)). Using the definition of the ‘t Hooft symbols the field strength F$l) of the instanton can be written as F;v(l)
= %/wf,
+
h~,kYkYp
-
EopkYkYv)f3 - (EavkykZv
-
%pkykZp)f4
E. V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
instantons
277
The functions fk are defined by (BSa)
f2=a(y,z)
f3=---
-
Y*,:;z,2
U(Y,Z>
2
Y2
Y4
zy(Y
L +d2 *
f4=-
(BSb)
’
2
+
(BSc)
+z12 ’
Y2(Y
(BSd)
The field strength F32) of the anti-instanton is obtained from F$l) by interchanging y and z, replacing aP4 and 6,, by -a,, and -aV4, respectively, and by multiplying the field strength by the relative orientation matrix 0. In the case that both pseudoparticles are instantons we do not change the sign of the delta functions. Because of the non-linear term in the field strength, also the combination I$( 1,2) = he”bcy,z,(7j;,p,40cd + T&T$~O~~) P.6) enters in the total gluonic action. The function h is defined by h=~(y,z)~(z,y).
P-7)
In the II case all n-symbols are barred. In order to obtain the pseudo-particle interaction we have to subtract the action of the system at infinite separation. For the ratio ansatz, F2(1) and F2(2> do not coincide with the action of an isolated dyon, and therefore the total interaction is given by F2(1) - -$ + P(2)
+2F(l)F(1,2)
- 5 + 2F( l)F(2)
+2F(2)F(1,2)
+P(1,2)
, (B.8) 1
where 4/y2 and 4/z2 represent the action density of a free dyon. The different terms contributing to SDD can be written as
F2(1) =6[f:+f,2+y4f:+y2z2f2+ 4Y2(f*f3
+f2f3)
-
4(Y
2y2f,f, *zHf*f4
+Z(Y *z)flf4-2~*(~
+f2f4)
-
qz2y2-
*z)f3f.t] OJ *z12)f42* (B-9)
E. K Shutyak,
278
J.J. M. Verbaarschot
/
QCD
instantons
The field strength F*(2) is obtained from F*(l) by interchanging combination F(lP(2) we find F(1)F(2)=2TrO[f1f;‘5f,f;-
z2f*L
-Y2f3f,
+ (Y *z,‘(f&
z and y. For the
+f&)]
+2TrO[-z*(y.z)f,f;-y*(y.z)f~~+(~.z)(f~f,+f,~)] + 2YOY(f,f;
- 5f3f;)
+ XYOzf,J5
+ 2lZOY
+ 2zOz(f,L (LA
-fLfl
-
u2f-J
-f&)
+2(~Y0z+z0Y)[-(Y~z)f&~Y*z~f3L+z2f‘&+Y2f3L]
(B.lO)
+2f3f;zxyOzxy.
In order to distinguish between the IA case and the II case we have introduced the sign 5 defined by [=l
for the II-case,
5 = - 1 for the IA-case. For F(l)F(l,
(B.ll)
2) we obtain
J’(l)F(L2)
=2TrO(y*z)h(f,
-f2>
-2zOyh(f,
- 2yOyz2hf4 + 2[zOzy*hf,
+ 2(1-
-f2)
+45yOzhf,
S)yOz( y*z)hf4.
(B.12)
By interchanging z and y in the above equation we obtain the combination F(2)F(l, 2). Finally, for the combination F*(l, 2) we find F2(1,2)
= @Wy2z*-
2h(y~z)2)fTr06+2h2((y~z)TrO-~yOz-zOy)2. (B.13)
The terms that do not depend on 0 are evaluated by first transforming to cylindrical coordinates, and then the remaining two-dimensional integrals are calculated numerically. Since Tr 0 and R,OabRb are the only invariants linear in 0, we can obtain the coefficients of the terms linear in 0 by calculating the integrals numerically for two different orientations. The coefficients of the terms
E. V. Shuryak,
J.J.M.
Verbaarschot
/
QCD
instantons
279
quadratic in 0 are evaluated explicitly. This requires the following integrals: 4 jp dp dx, h’( X: - t)’ = 0.02019, 7 4 7jp 4 7 /
(B.14a)
dp dx, h2p2x; = 0.010095,
(B.14b)
p dp dx, h2p4 = 0.02949,
(B.14~)
4 ;;5jpdpdx,h2p2(x+;)=0.
(B.14d)
Collecting all terms and using the parametrization of 0 given in eq. (A.@ we find for the dyon-dyon contribution to the instanton-anti-instanton interaction for the ratio ansatz: SDD=- P (-0.2732 IA R x[IK,14-
+ 0.0354 lu14 - 0.0963 lu12+ 0.2165 (lu212 + flK,12(1
x[#u,12(Iu212
x [1U2i4 +
+
+
lK314
IKI')-
lu,12)+
f(1K2i2+
lK312)]
$KT~(K~+K~)+
+ $IK212b312
+ f(
+ K~K~)'
+
0.0317
0.0317
+
+K:(K~~+
u2uf
lK312)
uT2)]
-
+
0.0174
$(If.d212 + b312)l~12]). (B.15)
The dyon-dyon contribution of the instanton-instanton the same way. The result is SDD = g(-0.2732 II
+ 0.0744
+ 0.0780 (u*n)’
-
jKi4-
o.0317(lK212+
+0.00793((~~~,+u~u,)~+(~~K,+u~K,)~).
0.1195
1~1~
interaction
+ 0.1805~*~
is obtained in
- 0.1171
lu12K*K
lK312)
(B.16)
Note that for SU(2) instantons we have Iu12 = 1, and in this case the first terms can be identified as - l/r (with an accuracy of six digits for the calculated results, but of course not for the displayed numbers).
E.V. Shutyak,
280
J.J.M.
Verbaarschot
/
QCD
instantons
Appendix C In this appendix we study the overlap matrix elements in the limit p + 0. For p +Z r -X p*/p the function 4,(x, t) defined in eq. (4.3) is to leading order in /3 given by (6,(w)
-‘i2$(-$f
+ Wlxl)
exp( - F)),
(C.1)
where (x, t) denotes the distance to the pseudoparticle. We take the instanton to be localized at (O,O, r,r) and the anti-instanton at (O,O,O,O), and denote their coordinates by (r,, t,) and (tZ, t,), respectively. First, we study the function fr defined in eq. (4.6), which in the above limit reduces to fi=
-~/rd4zsin~~,cos$f2-$&exp[-$(r,+r,)]
+;p
4~cosTtp
-n
1
sin-tp
(r,*r*) 2’f/2’25/2
ew
I
-35
+r*)
1 (C.2) *
For r, G p or r2 Q p the approximation (C.l) used to obtain the integrand is no longer valid. One can estimate that the contribution to the integral from this region is of B(J3- ‘12), whereas the contribution obtained by using the r -=xfi limit for r& is only of 8(l). Therefore, only those terms in the asymptotic expansion of fi that exceed HP-‘I*) can be trusted. In eq. (C.2) the temporal integrals factorize from the spatial ones and can be evaluated to give 7r B sinT(t-r)cos--Idf= /0 P P
P --sinTr. 2
P
(C.3)
The spatial integral can be rewritten in terms of cylindrical coordinates z and p. Because the exponent depends only on (p2 + z2)l12 + (p* + (z - r)2)‘/2, the asymptotic expansion is more readily obtained after transforming to the “elliptic” variables u and u, defined by
24 = ((p*+z*)“*+(p*+(z-r)2)1’2)/r,
(C.4a)
u=((p*+zy*- (p’+(z-r)y*/r.
(C.4b)
Using that the integration
measure transforms as p dp dz + fr3( u* - u*) du du ,
(C-5)
E.V. Shuryak, J.J.M. Verbaarschot / QCD itwantons
281
we obtain
For rr//3 = 0 this integral can be evaluated analytically. However, in this limit the contribution from the region around zero where the approximation to the integrand is not valid is also of @(l/p). The situation is better for ar/p z=-1. Then the contribution from the regions near the centers of the pseudoparticles is only of &‘(/3-‘/2>, and the leading-order approximation to fi can be trusted. In this limit the integral over u is located near its lower limit, and the leading order term in P/rr is obtained by putting u = 1 + 6u and keeping only the leading order terms in 6u from pre-exponential factors. The result is
(C-7) The calculation of the asymptotic result for (C.11, f2 can be written as
f2=
f2 proceeds analogously. Using eq.
-~/Td4xcos$(1-r)cos$fr&2
- $1
exp[-i(r,+r,)]
d4xsin T
The contributions to the integral from the regions near the centers of the pseudoparticles where the approximation (C.l) is not valid is of ~9(/3-‘/~), and also is this case only terms that exceed this magnitude can be trusted. Again, the temporal integral factorizes from the spatial integral, and can be performed trivially. As before, the spatial integral is obtained by transforming to (u,u> variables (see eq. (C.4)). The final result for f2 is given by
x ;(uu-l)+~(uu+l) ( By putting
u = 1 in the pre-exponential
(~))ew(-+). factors
and performing
(C.9) the u- and
282
E. K Shwyak,
J.J. M. Verbaarschot
/
QCD
instantons
u-integrals we find that the asymptotic result for f2 is given by a2i fp=+-coszexp P In order elements center of no longer
-P
(
rrr P 1
.
(C.10)
to obtain the next to leading-order contributions to the overlap matrix one has to take into account the contributions from the region near the the pseudoparticles where the present approximation to the integrand is valid. References
[l] [2] [3] [4] [5] [6] [7] [a] [9] [lo] [ll] [I21 [I31 [14] [15] [16] [17] [la] [19] [20]
A.A. Belavin, A.M. Polyakov, A.A. Schwartz and Yu.S. Tyupkin, Phys. Lett. B59 (1975) 85 G. ‘t Hooft, Phys. Rev. D14 (1976) 3432 C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Rev. D19 (1979) 1826 R.D. Carlitz and D.B. Creamer, Ann. Phys. (N.Y.) 118 (1979) 429 B.V. Geshkenbein and B.L. Ioffe, Nucl. Phys. B166 (1980) 340 V.A. Novikov, M.A. Shifman, AI. Vainshtein and V.I. Zakharov, Nucl. Phys. B191 (1981) 301 E.V. Shuryak, Nucl. Phys. B203 (1982) 93, 116, 140 E.V. Shut-yak, Nucl. Phys. B214 (1983) 237 D.I. Diakonov and V.Yu. Petrov, Nucl. Phys. B245 (1984) 259 E.V. Shuryak, Nucl. Phys. B302 (1988) 559, 574,599 E.V. Shutyak, Nucl. Phys. B328 (1989) 85, 102 E.V. Shut-yak, Phys. Lett. B79 (1978) 135 E.V. Shuryak, Phys. Rep. 61 (1980) 71 D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43 T. Kanki, Osaka University preprint OSGE 86-24 (1988) D.I. Diakonov and A.D. Mirlin, Phys. Lett. B203 (1988) 299 E.M. Ilgenfrih and E.V. Shuryak, Nucl. Phys. B319 (1989) 511 M.A. Nowak, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B325 (1989) 581 J. Hoek, M. Teper and J. Waterhouse, Nucl. Phys. B288 (1987) 589 M. Kremer, A. Kronfeld, M. Laursen, G. Schierholtz, C. Schleiermacher and U.J. Wiese, Nucl. Phys. B305 (1988) 109 1211 J. Smit and J.C. Vink, Nucl. Phys. B298 (1988) 557 [22] S. Hands and M. Teper, Nucl. Phys. B347 (1990) 819 [U] F.R. Brown, F.P. Butler, H. Chen, N.H. Christ, Z. Dong, W. Schaffer, L. Unger and A. Vaccarino, Phys. Rev. Lett. 65 (1990) 2491 [24] E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. B341 (1990) 1 [25] B. Harrington and H. Shepard, Phys. Rev. D17 (1978) 2122 [26] 1.1. Balitsky and A.V. Yung, Phys. Lett. B168 (1986) 113 [27] E.V. Shut-yak, Nucl. Phys. B302 (1988) 621 [281 A.V. Yung, Nucl. Phys. B297 (1988) 47 [29] D. Foerster, Phys. Lett. B66 (1977) 279 1301 C.G. Callan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717 [31] B. Grossman, Phys. Lett. A61 (1977) 86 1321 J.J.M. Verbaarschot, Nucl. Phys. B362 (1991) 33 [33] V.V. Khoze and A.V. Yung, Z. Phys. C30 (1991) 155