BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 68) Bled, Slovenia, July 8-15, 2000 Vacuum properties in the presence of quantum fluctuations of the quark condensate Georges Ripka ? Service de Physique Théorique, Centre d’Etudes de Saclay F-91191 Gif-sur-Yvette Cedex, France Abstract. The quantum fluctuations of the quark condensate are calculated using a regu- lated Nambu Jona-Lasinio model. The corresponding quantum fluctuations of the chiral fields are compared to those which are predicted by an ”equivalent” sigma model. They are found to be large and comparable in size but they do not restore chiral symmetry. The restoration of chiral symmetry is prevented by an ”exchange term” of the pion field which does not appear in the equivalent sigmamodel. A vacuum instability is found to be dangerously close when the model is regulated with a sharp 4-momentum cut-off. 1 Introduction. This lecture discusses the modifications of vacuum properties which could arise due to quantum fluctuations of the chiral field, more specifically, due to the quan- tum fluctuations of the quark condensate. The latter is found to be surprisingly large, the root mean square deviation of the quark condensate attaining and ex- ceeding 50% of the condensate itself. We shall discuss two distinct modifications of the vacuum: restoration of chiral symmetry due to quantum fluctuations of the chiral field, as heralded by Kleinert and Van den Boosche [1], and a vacuum instability not related to chiral symmetry restoration [2]. 2 Chiral symmetry restoration due to quantum fluctuations of the chiral field. 2.1 The linear sigma model argument. The physical vacuum with a spontaneously broken chiral symmetry is often de- scribed by the linear sigma model, which, in the chiral limit (m = 0), has a eu- clidean action of the form:I = Z d4x12 ()2 + 12 (i)2 + 28 2 + 2i - f22 (1)? E-mail: ripka@cea.fr Vacuum properties in the presence of quantum fluctuations... 69 Classically, we have (for translationally invariant fields):2 + 2i = f2 (2) and the vacuum stationary point is: = f i = 0 (3) We assume that 2 is large enough (and the -meson is heavy enough) not to have to worry about the quantum fluctuations of the  field. So we quantize the pion fieldwhile the  field remains classical.Wemay then say that: 2 = f2- 2i . Classically, 2i = 0 but the quantum fluctuations of the pion fieldmake 2i > 0 and therefore 2 < f2. Let us estimate the fluctuation 2i of the pion field. A system of free pions of massm is described by the partition function:Z = Z D () e- 12 R d4xi(-2+m2)i = e- 12 tr ln(-2+m2) (4) It follows that:12 Z d4x 2i (x) = - lnZm2 = m2 12tr ln -2 +m2 = 12 N2f - 1Xk< 1k2 +m2 (5) where the sum is regularizedusing a 4-momentum cut-off andwhere = R d4x.is the euclidean space-time volume. In the chiral limitm = 0, we have: 2i (x) = 12 N2f - 1Xk< 1k2 = N2f - 1 2162 (6) so that: 2 = f2 - N2f - 1 2162 (7) If we had evaluated this quantity with a 3-momentum cut-off, we would have obtained 2i = N2f - 1 282 . Let us pursue with a 4-momentum cut-off. We have: 2i f2 = N2f - 1 2162f2 (8) We deduce that chiral symmetry restoration will occur when  = 0, that is, whenh2i if2 > 1: 2i f2 = N2f - 1 2162f2 > 1 (9) With f = 93 MeV and withN2f - 1 = 3 pions, the condition reads:2 > 12:20  10-6  > 674 MeV (10) In most calculations which use the Nambu Jona-Lasinio model, this condition is fulfilled. We conclude that the quantum fluctuations of the pion do indeed restore chiral symmetry. If we had used a 3-momentum cut-off, chiral symmetry would be restored when  > 477 MeV. 70 G. Ripka 2.2 The non-linear sigma model argument. We now argue that this is precisely what is claimed by Kleinert and Van den Boosche [1], although it is said in a considerably different language. They argue as follows. If 2 (and therefore the  mass) is large enough, the action can be thought of as the action of the non-linear sigma model, which in turn can be viewed as an action with N2f fields, namely (; i), subject to the constraint:2 + 2i = f2 (11) The way to treat the non-linear sigma model is in the textbooks [3]. We work with the action:I (; ) = Z d4x12 ()2 + 12 (i)2 +  2 + 2i - f2 (12) in which we add a constraining parameter . The action is made stationary with respect to variations of . We integrate out the  field, to get the effective action:I () = Z d4x12 ()2 +  2 - f2+ 12tr ln -2 +  (13) The action is stationary with respect to variations of  and  if: = 0 2 = f2 - 12 N2f - 1Xk 1k2 +  (14) So either  = 0 and  6= 0, in which case we have:2 = f2 - 12 N2f - 1Xk 1k2 (15) or  6= 0 and  = 0. The condition (15) is exactly the same as the condition (7). Thus, the ”stiffness factor”, discussed in Ref.[1], is nothing but a measure of h2i if2 . 3 Quantum fluctuations of the quark condensate calculated in the Nambu Jona-Lasinio model. We now show that the quantum fluctuations of the chiral field are indeed large in the Nambu Jona-Lasinio model, but that chiral symmetry is far from being restored. The regularized Nambu Jona-Lasinio model is defined in section 4. We begin by giving some results. In the Nambu Jona-Lasinio model, the chiral field is composed of a scalar field S andN2f - 1 pseudoscalar fields Pi. They are related to the quark bilinears:S = V ̄  Pi = V ̄i 5i  (16) Vacuum properties in the presence of quantum fluctuations... 71 where V = - g2N is the 4-quark interaction strength. The quark propagator in the vacuum is: 1k  +M0r2k (17) and the model is regularized using either a sharp 4-momentum cut-off or a soft gaussian cut-off function:rk = 1 if k2 < 2 rk = 0 if k >  (sharp cut-off) (18)rk = e- k222 (gaussian regulator) : Let '0 = M0 be the strength of the scalar field in the physical vacuum. We shall show results obtained with typical parameters. If we chooseM0 = 300 MeV and = 750 MeV, then M0 = 0:4. We then obtain f = 94:6 MeVwith a sharp cut-off and f = 92:4 MeV with a gaussian cut-off (in the chiral limit). The interaction strengths are:V = -9:53 -2 (sharp cut-off) V = -18:4 -2 (gaussian cut-off) (19) and the squared pseudo-scalar field has the expectation value P2i = V2D ̄i 5i 2E (20) At low q we identify the pion field as:i =pZPi f =pZM0 (21) so that, in the Nambu Jona-Lasinio model: 2i f2 = V2D ̄i 5i 2EM20 (22) where D ̄i 5i 2E is the pion contribution to the squared condensate. 3.1 Results obtained for the quark condensate and for the quantum fluctuations of the chiral field. Let us examine the values of the quark condensates and of the quantum fluctua- tions of the chiral field calculated in the chiral limit. The quark condensate calculated with a sharp cut off is: ̄ 13 = -0:352  = 263 MeV (sharp cut-off) (23) is about 25 % smaller when it is calculated with a soft gaussian regulator: ̄ 12 = -0:280  = 210 MeV (gaussian regulator) (24) 72 G. Ripka ̄ -contribution -contribution total classical -0.04187 0 -0.04187 exchange term 0.00158 -0.00475 -0.00317 ring graphs 0.00014 0.00134 0.00148 total contribution -0.04015 -0.00341 -0.04356 Table 1. Various contributions to the quark condensate calculated with a sharp 4- momentum cut-off and with M0= = 0:4. The quark condensate is expressed in units of 3. ̄ -contribution -contribution total classical -0.02178 0 -0.02178 exchange term 0.00162 -0.00486 -0.00324 ring graphs 0.00077 0.00228 0.00305 total contribution -0.0193 -0.00258 -0.02197 Table 2. Various contributions to the quark condensate calculated with a gaussian cut-off function and withM0= = 0:4. The quark condensate is expressed in units of 3. The magnitude of the quantum fluctuations of the pion field can bemeasured by the mean square deviation 2 of the condensate from its classical value:2 = D ̄a 2E- ̄ 2 (25) The relative root mean square fluctuation of the condensate  is: ̄ = 0:41 (sharp cut-off)  ̄ = 0:77 (gaussian regulator) (26) These are surprisingly large numbers, certainly larger than 1=N . The linear sigma model estimate did give us a fair warning that this might occur. This feature also applies to the ratio h2i if2 = V2 ( ̄i 5i )2 M20 which was so cru- cial for the linear sigma model estimate of the restoration of chiral symmetry. We find: 2i f2 = 0:38 (sharp cut-off) 2i f2 = 0:85 (gaussian regulator) (27) In spite of these large quantum fluctuations of the chiral field, the quark con- densates change by barely a few percent. This is shown in tables 1 and 2where various contributions to the quark condensate are given in units of 3. The change in the quark condensate is much smaller than 1=N . 3.2 The effect and meaning of the exchange terms. The tables 1 and 2 show that, among the 1=N corrections, the exchange terms dominate. The exchange and ring graphs are illustrated on figures 1 and 2. The Vacuum properties in the presence of quantum fluctuations... 73 way in which they arise is explained in section 4.1. The exchange graphs con- tribute 2-3 times more than the remaining ring graphs. Furthermore, the pion contributes about three times more to the condensate than the sigma, so that the sigma field contributes about as much to the exchange term as any one of the pions. However, the exchange term in the pion channel enhances the quark con- densate instead of reducing it. As a result of this there is a very strong cancellation between the exchange terms and the ring graphs. This is why the sigma and pion loops contribute so little to the quark condensate. They increase the condensate by 4%when a sharp cut-off is used, and by 1%when a gaussian regulator is used. This is about ten times less than 1=N . The ring graphs reduce the condensate (in absolute value) in both the sigma and pion channels. This can be expected. Indeed, the ring graphs promote quarks from the Dirac sea negative energy orbits (which contribute negative values to the condensate) to the positive energy orbits (which contribute positive values to the condensate). The net result is a positive contribution to the condensate which reduces the negative classical value. What then is the meaning of the exchange terms? The exchange terms have the special feature of belonging to first order perturbation theory (see figures 1 and 2). Their contribution to the energy is not due to a modification of the Dirac sea. It is simply the exchange term arising in the expectation value of the quark- quark interaction in the Dirac sea. However, the contribution of the exchange term to the quark condensate does involve qq̄ excitations. These excitations are due to a modification of the constituent quark mass which is expressed in terms of quark-antiquark excita- tions of the Dirac sea. The exchange term is modifying (increasing in fact) the constituent quark mass and therefore the value of f. These results suggest that, in order to reduce the Nambu Jona-Lasinio model to an equivalent sigma model, it might be better to include the exchange term in the constituent quark mass, which is another way of saying that, in spite of the 1=N counting rule, it may be better to do Hartree-Fock theory than Hartree theory. The exchange (Fock) term should be included in the gap equation. The direct (Hartree) term is, of course, included in the classical bosonized action. In the equivalent sigma model, f is proportional to the constituent quark mass. Failure to notice that that the constituent quark mass is altered by the ex- change term is what lead Kleinert and Van den Boosche to conclude erroneously in Ref.[1] that chiral symmetry would be restored in the Nambu Jona-Lasinio model. They were right however in expecting large quantum fluctuations of the quark condensate. 4 The regularized Nambu Jona-Lasinio model. The condensates quoted in section 3.1 were calculated with a regularized Nambu Jona-Lasinio model which is defined by the euclidean action:Im (q; q̄) = Z d4x q̄ (-i )q +m ̄ - g22N + j ̄a 2 (28) 74 G. Ripka The euclidean Dirac matrices are  =  = (i ; ). The matrices a = (1; i 5) are defined in terms of the N2f - 1 generators  of flavor rotations. Results are given for Nf = 2 flavors. The coupling constant g2N is taken to be inversely pro- portional to N in order to reproduce the N counting rules. The current quark mass m is introduced as a source term used to calculate the regularized quark condensate ̄ . We have also introduced a source term 12 j ̄a 2 which is used to calculate the squared quark condensate D ̄a 2E. The quark field is q (x) and the (x) fields are delocalized quark fields, which are defined in terms of a regulator r as follows: (x) = Z d4x hx jrjyi q (y) (29) The regulator r is diagonal in k-space: hk jrjk0i = Ækk0r (k) and its explicit form in given in Eq.(18). The use of a sharp cut-off function is tantamount to the cal- culation of Feynman graphs in which the quark propagators are cut off at a 4- momentum  - a most usual practice. The regulator r, introduced by the delo- calized fields, makes all the Feynman graphs converge. A regularization of this type results when quarks propagate in a vacuum described by in the instanton liquid model of the QCD (see Ref.[4] and further references therein). A Nambu Jona-Lasinio model regulated in this manner with a gaussian regulator was first used in Ref.[5], and further elaborated and applied in both the meson and the soliton sectors [5],[6],[7],[8], [9],[2]. Its properties are also discussed in [10]. With one exception. In this work, as in Ref.[2], the regulator multiplies the current quark mass. The introduction of the regulator in the current quark mass term m ̄ = mq̄r2q requires some explanation. The current quark mass m is used as a source term to calculate the quark condensate ̄ which, admittedly, would be finite (by reason of symmetry) even in the absence of a regulator - and, indeed, values of quark condensates are usually calculated with an unregular- ized source term in the Nambu Jona-Lasinio action. However, when we calcu- late the fluctuation D ̄a 2E - ̄ 2 of the quark condensate, the expecta- tion value D ̄a 2E diverges. It would be inconsistent and difficult to inter- pret the fluctuation D ̄a 2E - ̄ 2 if ̄ were evaluated using a bare source term and D ̄a 2E using a regulator. When a regularized source termm ̄ = mq̄r2q is used, the current quark mass m can no longer be identified with the current quark mass term appearing in the QCD lagrangian. Of course, when a sharp cut-off is used, it makes no difference if the current quarkmass term is multiplied by the regulator or not. We have seen in section 3.1 that the lead- ing order contribution to the quark condensate ̄ 1=3 diminishes by only 20% when the sharp cut-off is replaced by a gaussian regulator. (This statement may be misleading because when the sharp cut-off is replaced by a gaussian regulator, the interaction strength V is also modified so as to fit f. If we use a gaussian regulator, the quark condensate calculated with a regulated source term mr2 is ̄ = -0:0218 3 whereas the quark condensate calculated with a bare source termm is ̄ = -0:0505 3.) Vacuum properties in the presence of quantum fluctuations... 75 The way in which the current quark mass of the QCD lagrangian appears in the low energy effective theory is model dependent and it has been studied in some detail in Ref.[11] within the instanton liquid model of the QCD vacuum [12],[13],[4]. An equivalent bosonized form of the Nambu Jona-Lasinio action (28) is:Ij;m (') = -Tr ln(-i  + r'aar) - 12 (' -m) (V- j)-1 (' -m) (30) The first term is the quark loop expressed in terms of the chiral field ', which is a chiral 4-vector 'a = (S; Pi) so that 'aa = S + i 5iPi. In the second term, the chiral 4-vector ma  (m; 0; 0; 0) is the current quark mass and V is the local interaction: hxa jV jybi = - g2N ÆabÆ (x- y) (31) The partition function of the system, in the presence of the sources j and m is given by the expression:e-W(j;m) = Z D (') e-Ij;m(')- 12 tr ln(V-j) (32) The quark condensate ̄ and the squared quark condensates D ̄a 2E can be calculated from the partition functionW (j;m) using the expressions: ̄ = 1 W (j;m)m 12 D ̄a 2E = - 1 W (j;m)j (33) where is the space-time volume R d4x = . The stationary point 'a = (M;0; 0; 0) of the action is given by the gap equa- tion: (V- j)-1 = -4N Nf MM-mgM (34) This equation relates the constituent quark mass M to the interaction strengthV- j. 4.1 The exchange and ring contributions. The second order expansion of the action Ijm (') around the stationary point reads: Ijm (') = Ijm (M) + 12'+ (V- j)-1' (35) where Ijm (M) is the action calculated at the stationary point ' = (M;0; 0; 0) and where  is the polarization function (often referred to as the Lindhardt function):hxa jjybi = - ÆÆ'a (x) Æ'b (y)Tr ln(-i  + r'aar) (36) Substituting this expansion into the partition function (32), we can calculate the partition functionW (j;m) using gaussian integration with the result:W (j;m) = Ijm (M) + 12tr ln (1 - (V- j)) (37) 76 G. Ripka The first term of the action (37) is what we refer to as the ”classical” action. The values labelled ”classical” in the tables displayed in section 3.1 are obtained by calculating the condensates (33) while retaining only the term Ijm in the parti- tion function (37). The logarithm in (37) is what we refer to as the loop contribu- tion. The expansion of the logarithm expresses the loop contribution in terms of the Feynman graphs shown on figure 1. + + + ... Fig. 1. The Feynman graphs which represent the meson loop contribution to the partition function. The first graph is the exchange graph and the remaining graphs are the ring graphs. The first term of the loop expansion is what we call the ”exchange term”, also referred to as the Fock term:1:Wex h = -12tr (V- j) (38) The remaining terms are what we call the ring graphs. + + Fig. 2. The contribution to the quark condensate of the Feynman graphs shown on figure 1. The black blob represents the operator ̄ . The first graph (which is the dominating contribution) is the contribution of the exchange term. It represents qq̄ excitations which describe a change in mass of the Dirac sea quarks. This exchange graph would not appear in a Hartree-Fock approximation, which would include the exchange graph in the gap equation. It is simple to show that the inverse meson propagators are given by:K-1 = + (V- j)-1 (39) They are diagonal inmomentum and flavor space: qa K-1 k0q = ÆabÆkk0Ka (q) and a straightforward calculation yields the following explicit expressions for theS (sigma) and P (pion) inverse propagators: 1 The direct (Hartree) term is included in the ”classical” action Ij;m. Vacuum properties in the presence of quantum fluctuations... 77 0.0 0.5 1.0 M 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 Gaussian regulator Fig. 3. The effective potential plotted against M when a soft gaussian cut-off function is used. The potential is expressed units of 4.K-1S (q) = 4N Nf12q2f22M (q) +M2 f26M (q) + f44M (q)- gM (q) + MM-mgM (40)K-1P (q) = 4N Nf12q2f22M (q) +M2 f26M (q) - f44M (q)- gM (q) + MM-mgM where the loop integrals are:fnpM (q) = 1 Xk rnk-q2 rpk+q2k - q2 2 + r4k-q2M2k + q2 2 + r4k+q2M2 (41) and: gM (q) = 1 Xk r2k-q2k - q2 2 + r4k-q2M2 r2k+q2 gM  gM (q = 0) (42) These are the expressions which are obtained from the second order expansion of the action (30) retaining the regulators from the outset and throughout. Innumerable papers have been published (including some of my own) in which the meson propagators are derived from the unregulated Nambu Jona- Lasinio action:Ij;m (') = -Tr ln(-i  +'aa) - 12 (' -m) (V- j)-1 (' -m) (43) 78 G. Ripka 0.0 0.5 1.0 M −0.002 0.000 0.002 0.004 0.006 0.8 0.6 0.4 0.2 0.2 0.4 Sharp cut−off Fig. 4. The effective potential plotted againstM when a sharp cut-off is used.The effective potential is expressed in units of 4. The expressions obtained for the propagators are then:K-1S (q) = 4N Nf12 q2 + 4M2 fM (q) + mM-mgM (44)K-1P (q) = 4N Nf12q2fM (q) + mM -mgM where the loop integrals are:fM (q) = 1 Xk< 1k- q2 2 +M2k + q2 2 +M2 (45) and: gM = 1 Xk< 1k - q2 2 +M2 (46) The table 3 shows the low q behaviour of the S and P inverse propagators in various approximations. They are calculated in the chiral limit. 5 An instability of the vacuum. The partition function (37) can also be used to calculate the effective potential: =W (j;m) - jW (j;m)j =W (j;m) + 12jD ̄a 2E (47) Vacuum properties in the presence of quantum fluctuations... 79 0.0 0.2 0.4 0.6 0.8 1.0 1.2 M −0.005 −0.003 −0.001 0.001 0.003 0.005 0.007 0.009 Classical (Hartree) Exchange (Fock) Ring diagrams Total Fig. 5. Various contributions to the effective potential calculated with a sharp cut-off andM0= = 0:8. The contributions are expressed in units of 4. inverse propagators K-1P (q = 0) Z = dK-1Pdq2 q=0 K-1S (q = 0) dK-1Sdq2 q=0 regulated action 0 0.0995 0.0546 0.0592 regulated f (q) 0 0.0850 0.0544 0.0448f (q) = f (0) 0 0.0850 0.0544 0.0850 Table 3. Three approximations to the inverse S and P propagators, calculated with a sharp 4-momentum cut-off and withM0= = 0:4. The first row gives the values obtained from an regularized action (30). The second row gives the values obtained from a unregularized action and by subsequently regularizing the loop integrals. The last row gives the results obtained by neglecting the q dependence of the loop integral f (q). The inverse quark propagators are given in units of 2 and dK-1dq2 is dimensionless. As we vary j, the squared condensate D ̄a 2E changes. Thus, when we plot the effective potential against j, we discover how the energy of the system varies when the system is forced to modify the squared condensate D ̄a 2E. The effective potential has a stationary point at j = 0, that is, in the absence of a con- straint. If the stationary point of the effective potential is a minimum, the system is (at least locally) stable against fluctuations of D ̄a 2E. If it is an inflection point, it is unstable and we shall indeed find that this can easily occur when a sharp cut-off is used. When j is varied, the constituent quark mass M also changes, according to the gap equation (34). One finds thatM is a monotonically increasing function of j so that the effective potential can be plotted againstM equally well. The vacuum constituent quark mass is the mass M0 obtained with j = 0. The contribution 80 G. Ripka 0.0 0.5 1.0 M 0.000 0.002 0.004 0.006 0.008 0.010 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Fig. 6. The effective potential calculated with a sharp 3-momentum cut-off plotted againstM. It is expressed in units of 4. of each Feynman graph to the effective potential is stationary at the pointM =M0 and this is why plots of the the effective potential against M are nicer to look at than plots against j. The vacuum constituent quark massM0 is a measure of the interaction strength V, to which it is related by the gap equation. For a given shape of the regulator, the occurrence of an instability depends on only one parameter, namelyM0=. Figure 3 shows the effective potential calculated with a gaussian cut-off for various values ofM0=. The ground state appears to be stable within the range of reasonable values ofM0=. Figure 4 shows the effective potential plotted againstMwhen a sharp cut-off is used. WhenM0= > 0:74 the ground state develops an instability with respect to increasing values ofM. This instability is not related to the restoration of chiral symmetry and, indeed, the pion remains a Goldstone boson for all values ofM: As shown on Fig.5, the instability is due to the classical action and the meson loop contributions do not modify it. Figure 6 shows the effective potential calculated with a sharp 3-momentum cut-off. No instability appears. This provides a clue as to the cause of the in- stability which arises when a sharp 4-momentum cut-off is used. Indeed, when a 3-momentum cut-off is used, the Nambu Jona-Lasinio model defines a time- independent hamiltonian and the 3-momentum cut-off simply restricts theHilbert space available to the quarks. This allows a quantum mechanical interpretation of the results. If H is the Nambu Jona-Lasinio hamiltonian, then the ground state Vacuum properties in the presence of quantum fluctuations... 81 wavefunction jji is calculated with the hamiltonianH̄j = H- j Z d3x ̄a 2 (48) containing the constraint proportional to j. The effective potential is then equal to the energy E (j) = hj jHj ji of the system and it displays a stationary point whenj = 0 or, equivalently, whenM = M0. The Nambu Jona-Lasinio model, regular- ized with a 3-momentum cut-off, has been used in Refs.[14] and [15] for example. The use of a 3-momentum cut-off has another important feature. The meson propagators have only poles on the imaginary axis where they should. When a 4- momentum cut-off is used, unphysical poles appear in the complex energy plane, as they do when proper-time regularization is used for the quark loop [16]. The fact that the instability occurs when the model is regularized with a 4- momentum cut-off and not when a 3-momentum cut-off is used, strongly sug- gests that the instability is due to the unphysical poles introduced by the regula- tor. This conclusion is corroborated by the observation that the instability also oc- curs when a gaussian cut-off is used, but at the much higher valuesM0=  2:93 where the cut-off is too small to be physically meaningful. With a gaussian reg- ulator and in the relevant range of parameters 0:4 < M0= < 0:8, one needs to probe the system with values as high asM= > 4 before it becomes apparent that the energy is not bounded from below. The instability is an unpleasant feature of effective theories which use relatively low cut-offs. However, the low value of the cut-off is dictated by the vacuum properties and we need to learn to work with it. Further details are found in Ref.[2]. We conclude from this analysis that it is much safer to use a soft regulator, such as a gaussian, than a sharp cut-off. References 1. H. Kleinert and B. Van Den Boosche, Phys.Lett. B474, page 336, 2000. 2. G.Ripka,Quantum fluctuations of the quark condensate, hep-ph/0003201, 2000. 3. J.Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 4. C. Weiss D.I. Diakonov, M.V. Polyakov,Nucl. Phys. B461, page 539, 1996. 5. R.D.Bowler and M.C.Birse,Nucl.Phys. A582, page 655, 1995. 6. R.S.Plant and M.C.Birse,Nucl.Phys. A628, page 607, 1998. 7. W.Broniowski B.Golli and G.Ripka, Phys.Lett. B437, page 24, 1998. 8. Wojciech Broniowski,Mesons in non-local chiral quark models, hep-ph/9911204, 1999. 9. B.Szczerebinska and W.Broniowski, Acta Polonica 31, page 835, 2000. 10. Georges Ripka, Quarks Bound by Chiral Fields, Oxford University Press, Oxford, 1997. 11. M.Musakhanov, Europ.Phys.Journal C9, page 235, 1999. 12. E.Shuryak,Nucl.Phys. B203, pages 93,116,140, 1982. 13. D.I.Diakonov and V.Y.Petrov,Nucl.Phys. B272, page 457, 1986. 14. D.Blaschke S.Schmidt and Y.Kalinovsky. Phys.Rev. C50, page 435, 1994. 15. S.P. Klevansky Y.B. He, J. Hfner and P. Rehberg, Nucl. Phys. A630, page 719, 1998. 16. E.N.NikolovW.Broniowski, G.Ripka and K.Goeke, Zeit.Phys. A354, page 421, 1996. BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 82) Bled, Slovenia, July 8-15, 2000 Nucleon-Nucleon Scattering in a Chiral Constituent Quark Model Floarea Stancu? Institute of Physics, B.5, University of Liege, Sart Tilman, B-4000 Liege 1, Belgium Abstract. We study the nucleon-nucleon interaction in the chiral constituent quark model of Refs. [1,2] by using the resonating group method, convenient for treating the interac- tion between composite particles. The calculated phase shifts for the 3S1 and 1S0 channels show the presence of a strong repulsive core due to the combined effect of the quark in- terchange and the spin-flavour structure of the effective quark-quark interaction. Such a structure stems from the pseudoscalar meson exchange between quarks and is a conse- quence of the spontaneous breaking of the chiral symmetry. We perform single and cou- pled channel calculations and show the role of coupling of the  and hidden colour CC channels on the behaviour of the phase shifts. The addition of a -meson exchange quark- quark interaction brings the 1S0 phase shift closer to the experimental data. We intend to include a tensor quark-quark interaction to improve the description of the 3S1 phase shift. In this talk I shall mainly present results obtained in collaboration with Daniel Bartz [3,4] for the nucleon-nucleon (NN) scattering phase shifts calculated in the resonating group method. The study of the NN interaction in the framework of quark models has al- ready some history. Twenty years ago Oka and Yazaki [5] published the first L = 0 phase shifts with the resonating group method. Those results were obtained from models based on one-gluon exchange (OGE) interaction between quarks. Based on such models one could explain the short-range repulsion of the NN interaction potential as due to the chromomagnetic spin-spin interaction, com- bined with quark interchanges between 3q clusters. In order to describe the data, long- and medium-range interactions were added at the nucleon level. During the same period, using a cluster model basis as well, Harvey [6] gave a classi- fication of the six-quark states including the orbital symmetries [6℄O and [42℄O. Mitja Rosina, Bojan Golli and collaborators [7] discussed the relation between the resonating group method and the generator coordinate method and introduced effective local NN potentials. Here we employ a constituent quark model where the short-range quark- quark interaction is entirely due to pseudoscalar meson exchange, instead of one-gluon exchange. This is the chiral constituent quark model of Ref. [1], para- metrized in a nonrelativistic version in Ref. [2]. The origin of thismodel is thought to lie in the spontaneous breaking of chiral symmetry in QCD which implies the existence of Goldstone bosons (pseudoscalar mesons) and constituent quarks? E-mail: fstancu@ulg.ac.be