We show that the color magnetic flux tubes are quantized in terms of the center ... moment term in QCD is of crucial importance for condensate formation: as one.
Nuclear Physics BI70[FSI] (1980) 2 6 5 - 2 8 2 © North-Holland Publishing C o m p a n y
A C O L O R M A G N E T I C VORTEX C O N D E N S A T E IN QCD J. A M B J I 3 R N a n d P. O L E S E N
The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen t~, Denmark Received 1 April 1980
It is shown that there exists a very close analogy between a lattice of vortices in a superconductor near the critical field a n d a condensate of color magnetic flux tubes due to the unstable mode in QCD. This analogy makes it possible to identify a dynamical Higgs field in QCD. We show that the color magnetic flux tubes are quantized in terms of the center group Z(2) in the SU(2) case. In the case of S U ( N ) it is possible to select a color direction of the field such that o n e h a s Z ( N ) quantization.
1. Introduction
It has been argued by 't Hooft [1] and by Mack [2] that a confining ground state in QCD (i.e., the QCD vacuum) should be a condensate of color magnetic vortices with a flux quantized in terms of the center group Z ( N ) . F r o m this point of view it thus becomes very interesting to investigate whether any sign of a vortex condensation can be found in QCD. Some time ago Nielsen and one of us [3] investigated the behavior of the vacuum energy as a function of a homogeneous color magnetic field. It was found that this situation was unstable because of the gluon magnetic moment: for distances larger than of the order of the inverse square root of the color magnetic field, the vacuum energy develops an imaginary part. Therefore, by dividing space into color magnetic flux tubes with a cross section of the order of the inverse of the magnetic field, the instability disappears. In a recent paper [4] we have investigated these flux tubes in some detail. The present paper contains some further results, the most interesting being the following: (a) The condensate of flux tubes studied in ref. [4] has almost the same properties (geometric structure, equations of motion, ... ) as the lattice of vortices occurring in ordinary superconductors (see, e.g., ref. [5]). This is remarkable, since we do not have any Higgs field. This close analogy allows us to identify a dynamical Higgs field in QCD. That this should be possible in a vortex condensate has already been emphasized by Mack [2]. Our identification has the special advantage that it is very 265
266
j . Ambj~rn , P. Olesen / Magnetic vortex condensate
simple and almost self-evident if one compares our equations to the corresponding equations in an abelian superconductor. (b) In the previous paper [4] it was not clear that the flux tubes resulting from the removal of the instability [3] were quantized in accordance with the center group. Without making any new assumptions, we show that in the case of SU(2) the flux is quantized in terms of the center Z(2). For the case of S U ( N ) i t is possible to choose the initial direction of the color magnetic field in such a way that Z ( N ) quantization emerges. The center quantization is a rather "local" property: the condensate consists of a lattice of flux tubes, and the flux quantization is associated with each fundamental lattice cell. Again, this is in very close analogy to what happens in the abelian superconductor. An important difference between the abelian and non-abelian cases is that in the former case, the external field and the vortices require energy to be produced (i.e., energy per unit length of a vortex is positive), whereas in the latter case it appears reasonable that asymptotic freedom makes these fields be produced spontaneously (i.e., the energy per unit length of a vortex is negative relative to the perturbative ground state). The plan of this paper is the following: in sect. 2 we discuss in some consequences of the instability discussed in ref. [3]. In particular, we emphasize features which are analogous to those that appear in a lattice of vortices in a superconductor. Sect. 3 contains a brief summary of the abelian case, with emphasis on the equations which are similar (and, in some cases, identical) to the non-abelian equations. We also compare with the results in sect. 2. In sect. 4 we discuss an illustrative example, where QCD is modified in such a way that the gluon has an anomalous color magnetic moment. This example is in many ways unrealistic (e.g., renormalizability a n d / o r unitarity is lost), but it emphasizes that the magnetic moment term in QCD is of crucial importance for condensate formation: as one varies the magnetic moment of the gluon, a critical value is reached, and the condensate ceases to exist. Near the critical value of the magnetic moment the similarity with ordinary superconductors becomes very clear. In sect. 5 we discuss the center flux quantization in the case of QCD, whereas sect. 6 contains some discussions of the results, as well as a discussion of the important role of quantum fluctuations. 2. The instability of a homogeneous color magnetic field In this section we shall discuss the instability [3] of a homogeneous color magnetic field in QCD with special emphasis on those features which are relevant for comparison with an abelian superconductor. We consider the case of SU(2), where the field is denoted A,. We introduce the notation -
(2.1)
J. Ambjern, P. Olesen / Magnetic vortex condensate
267
The lagrangian can then be written
~.> A 2 it follows that g ( H ) is small, and hence the expression (6.2) is valid for strong fields. Eq. (6.2) has a minimum [10] for gH = A2. This minimum is, however, outside the range of validity of eq. (6.2) and can therefore not be trusted (it corresponds to the usual Landau singularity in the one-loop effective coupling). Eq. (5.2) includes the classical energy density* i H 2. We now take into account that this expression is modified by the vortex formation. Adding the contribution from the unstable mode one gets for the average energy density E= -0.43H 2 + llNg2H2(ln gn_±] 96tr~ ~ A2 2]"
(6.4)
This expression has minimum for
gH 96~r 2 In - ~ = 0.43 1 INg 2 >> 1
for g2 small,
(6.5)
and M i n e < 0. Since we have to be close to the classical situation, g2 must be small. This means, however, that 1/g 2 is large, and hence the logarithm l n ( g H / A 2) is large. The effective coupling (6.3) is therefore small, and it appears reasonable to use the expression (6.4) for the energy. On the basis of the above argument it thus appears reasonable to expect vortex condensation to occur in QCD because of quantum corrections, i.e., essentially because of asymptotic freedom. The above argument is incomplete because we have not included quantum fluctuations in the unstable mode W (°) as well as in the higher modes W tn's3). In principle it is straightforward to compute these corrections, but in practice it involves solving very complicated Schr6dinger-like equations with potentials given in terms of Jacobi's 0-function. So far very little progress has been made on this problem, which presumably can only be solved by a computer. The quantum corrections have not yet been computed. One may hope that they are small relative to the classical energy density of the unstable mode, provided the coupling g2 is sufficiently small. The magnitude of the classical energy density of the unstable mode is 0.43 H 2, and hence the question is whether the corrections are * A is related to the normalization point # by I n ( A 2 / # 2) = 48~r2/1
INg 2 .
282
J. Ambjern, P. Olesen / Magnetic cortex condensate
smaller than 0.43 H 2 for g2 small. If so, these corrections will not upset the lowering of the energy in eq. (6.4). Of course, to be absolutely sure, the calculations have to be performed explicitly. If the expression (6.4) is a reliable representative of the vacuum energy for small coupling, it follows that we have shown that a condensate of vortices is formed. These vortices have beautiful topological properties, and provide an explicit realization of the ideas of 't Hooft [1] and Mack [2]. The field configuration studied in the present paper is hardly stable, and is only valid for very weak coupling, so it could be a short-distance approximation. In general the higher modes also contribute. It has been estimated [8] that the effect of the quantum fluctuations in these modes is to produce a quantum liquid, where the flux tubes move around so as to produce rotational invariance. A similar phenomenon does not happen in a superconductor, because the relevant coupling is too small to produce a liquid (the Lindemann ratio [8] becomes far too small to reach the critical magnitude). A full treatment of the liquid state presumably requires strong coupling. However, it is natural to expect that the topological flux quantization remains valid. The quantum liquid state is a disordered state. In this state the dynamical Higgs field has no expectation value, because the expectation value is to be taken as an average over all directions, which occur with equal probability in the liquid state. Thus, in the liquid state confinement is very likely [ 1, 2]. We thank B. Felsager, J. Leinaas and A. Patkos for instructive discussions on the center Z ( N ) and its role. O n e of us (P.O.) would like to thank G. Mack for very interesting discussions, and for his insistence that somehow Z ( N ) ought to show up in our calculations, and G. 't Hooft for an interesting question on the same subject. The comments of Mack and 't Hooft very much stimulated the work reported in this paper. We also thank M. Ninomiya for interesting discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. 't Hooft, Nu¢l. Phys. B138 (1978) 1; B153 (1979) 141 G. Mack, Carg~se lectures 0979) N.K. Nielsen and P. Olesen, NucL Phys. B144 0975) 376 J. Ambj~rn and P. Ole~en, Nu¢l. Phys. BI70[FS1] 0980) 60 D. Saint-Jam~, G. Sarma and E.J. Thomas, Type II sup~-condietivity (Pergamon Press, 1969) N.K. Nielsen and P. Olesen. Phys. Left. 79B 0978) 304; J. Ambjzrn, N.K. Nielsen and P. O l d , a, NueL Phys. B152 (1979) 75 H. Arod~ private communication 0978) H.B. Nielsen and P. Olesen, Nucl. Phys. Bl60 (1979) 380 H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 0979) 1 S.G. Matinyan and G.K. Savvidy, Nucl. Phys. B134 0978) 539; G.K. Savvidy, Phys. Lett. 71B 0977) 133; M.J. Duff and M. Ram~n-Medrano, Phys. Rev. Dl2 (1976) 3357; H. Pagels and E. Tomboulis, Nu¢l. Phys. B143 0978) 485
