ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
MONOPOLES AND VORTICES IN YANG-MILLS PLASMA
© 2009 M. N. Chernodub1),2)*, V. I. Zakharov2),3)
Received October 24, 2008; in final form, May 12, 2009
We discuss the role of magnetic degrees of freedom in Yang—Mills plasma at temperatures above and of order of the critical temperature Tc. While at zero temperature the magnetic degrees of freedom are condensed and electric degrees of freedom are confined, at the point of the phase transition both magnetic and electric degrees of freedom are released into the thermal vacuum. This phenomenon might explain the observed unusual properties of the plasma.
PACS:12.38.Aw, 25.75.Nq, 11.15.Tk
1. INTRODUCTION
At vanishing temperature, the Yang—Mills theories exhibit the phenomenon of the color confinement. In case of pure Yang—Mills theories, on which we concentrate below, the criterion of the confinement is the area law for large enough Wilson loops:
W(C))~exp(-a • Amin) Vqq(R) ^ a • R.
(1)
or
Here, C is the rectangular contour with time and space dimensions T and R, respectively, Amin = R ■ T is the minimal area of a surface spanned on the contour C, a = 0 is the confining string tension, and Vqq(R) is the heavy-quark potential at large distances R between quark and antiquark.
While there is yet no understanding of the confinement from the first principles in the non-Abelian case, it can be modeled in Abelian theories. In particular, in ordinary superconductor the potential between external magnetic monopoles grows linearly at large distances R,
V
M M
ам • R,
mimicking the heavy-quark potential (1). The microscopical mechanism behind this example is the condensation of the Cooper pairs. In the effective-theory language the mechanism is a condensation of the electrically charged scalar field:
<0el) = 0.
By analogy with this well understood case it was speculated long time ago [1] that in the non-Abelian
"LMPT, CNRS UMR 6083, Fee dee ration Denis Poisson, Université de Tours, France.
2)ITEP, Moscow, Russia.
3)INFN, Sezione di Pisa, Italy.
E-mail: maxim.chernodub@lmpt.univ-tours.fr
theories it is the condensate of magnetically charged field,
<0magn ) =0, (2)
that ensures confinement of color charges (1).
Within this general framework of the dual-superconductor model (2) the main question is: what is the microscopical mechanism behind (2). In other words, the question is what is a Yang—Mills analog of the Cooper pairs of the ordinary superconductor? (Remember that we are considering pure Yang—Mills theories without fundamental scalar fields.)
A clue to the answer to this question might be provided by the example of the so-called compact U(1) theory [2] where the magnetic degrees of freedom are identified as topological excitations of the original theory. In more detail, the Lagrangian is the same as for a free electromagnetic field:
1
C.
U (1)
=
4e2'
¡¡V 1
supplemented, however, by the condition that the Dirac string carries no action. The condition is automatically satisfied in the lattice, or compact version of the theory.
Since the Dirac string is not seen (it costs no action) there appear particle-like excitations, or magnetic monopoles with magnetic field
(3)
e r3
where e is the electric charge and the constant is determined by the Dirac quantization condition. Equation (3) corresponds to the static monopole located at the origin of the coordinate system.
Admitting singular fields, or monopoles into the theory violates Bianchi identities and modifies the equations of motion:
d^v = 0,
д f = jmon
¡IV — Jv ,
2197
dv jmon = 0,
where jmon is the monopole current. Moreover, the nonvanishing, conserved current jmon can be traded for a magnetically charged scalar field ^magn, which appeared previously in Eq. (2). In order to prove this fact, one uses the so-called polymer representation of field theory in the Euclidean space—time (for review of this representation see, e.g., [3] while specific applications to the lattice monopoles are discussed, in particular, in [4]).
Thus, in case of the compact U(1) theory we have both microscopical description of the magnetic degrees of freedom in terms or the topological excitations, or monopoles, and macroscopical description in terms of the magnetic condensate (2). In the most interesting case of non-Abelian theories we are not far now from completing a similar program. A specific feature here is that knowledge on the topo-logical excitations emerged mainly from the lattice studies. The topological excitations appear to be the magnetic monopoles and the magnetic vortices. The observed percolating properties of the monopoles and the vortices are revealing the nature of the magnetic condensation in the non-Abelian case.
Reviewing properties of the topological excitations in the Yang—Mills case is a subject of the present notes. Such kind of a review could be written, however, a few years ago as well. What we are adding, is a discussion of much more recent development which is related to the role of the magnetic degrees of freedom at temperatures above the temperature Tc of the confinement—deconfinement phase transition. Namely, it was conjectured in [5—7] that above Tc the magnetic degrees of freedom constitute an important fraction of the Yang—Mills plasma responsible for the unusual properties of the plasma which appears to be rather similar to an ideal liquid than to an ideal gas of gluons (for review see [8]).
In our presentation, we follow the logic of [5]. It is natural to expect that at the point of the phase transition the condensate (2) is destroyed:
^magn )(Tc) — 0.
(4)
this symmetry is indeed restored, then one should not think about the Yang—Mills plasma as a plasma of gluons alone but as of plasma with a magnetic component. Thus, restoration of symmetry could be a general phenomenon behind observation of unusual properties of the plasma. All these speculations can be checked, and partially have already been checked on the lattice.
2. PHENOMENOLOGY OF MONOPOLES IN YANG-MILLS THEORY
Historically, the ideas about the nature of confinement in the non-Abelian case were strongly influenced by the well-understood Abelian examples mentioned above. However, in Yang-Mills theories the effective coupling is large at large distances and no reliable analytical tools exist. As a result, numerical simulations on the lattice acquired central role in exploring the monopole mechanism of the color confinement.
The first, and apparently serious obstacle on this path is difficulty to define the monopoles. Indeed, the monopoles are intrinsically Abelian objects. In particular, in the absence of scalar fields there are no monopole-like solutions with finite energy in the non-Abelian case.
A way out was found with the help of the Abelian projection method [9]. The basic idea behind Abelian projections is to fix partially the non-Abelian gauge symmetry up to an Abelian subgroup. In the physically relevant SU(N) Yang-Mills theories the residual Abelian subgroup is compact due to the compactness of the original non-Abelian group. It is the compact nature of the residual gauge subgroup which leads to emergence of the Abelian monopoles, see the Introduction.
More specifically, one first utilizes the local gauge invariance to rotate the non-Abelian gauge field as close as possible to the Cartan direction in the color space. Specifically, in the SU(2) gauge theory one minimizes the lattice analog of the functional
Moreover, it is not a mere speculation but a hypothesis supported by the lattice data (as we review later). The main question is: which symmetry is restored when the condensate (2) is destroyed? For example, in case of the Standard Model one is commonly assuming that at high temperatures all the particles become massless, as in the original Lagrangian. In our case of non-Abelian field theories one can speculate that, in some sense, it is symmetry between electric and magnetic degrees of freedom which is restored upon the destruction (4) of the magnetic condensate by the thermal fluctuations. Moreover, if
d4x{A (x)]2 + [A2(x)]2 }
with respect to the local gauge transformations. This minimization makes the offdiagonal components small and, thus, this procedure maximizes the role of the diagonal (Cartan) elements of the gauge field. Next, one replaces the original non-Abelian fields by their projections on the Abelian direction:
Äß(x) = taÄaß(x) ^ t3Al(x)
(5)
(here ta are generators of the gauge group), and then one defines the monopoles in terms of the projected
gauge fields A^(x) as if they were the fields of a compact U(1) theory. This step is very crucial.
Analytically, it is impossible to evaluate the effect of the projection (5). However, the numerical simulations of non-Abelian gauge theories on the lattice provide us with a powerful method to probe the dual superconductor hypothesis. This "experimental" method allows us to investigate features of the Abelian monopoles numerically from the first principles of the theory. In numerical simulations the formation of the chromoelectric string can be observed in a very straightforward way. The contribution of the monopole-like gluonic configurations to the energy of the string — probed by quantum averages of time-oriented Wilson loops — turns out to dominate over the contribution from other ("irrelevant") fluctuations of the gluonic fields. This property of the confining configurations — which is also known as the "monopole dominance" — was clearly demonstrated in [10] following the original study of [11]. The monopole dominance confirms that the color confinement in Yang—Mills theory originates from a specific dynamics of the Abelian monopoles.
In addition to the monopole dominance one can also observe numerically that the Abelian monopole currents circulate around the confining chromoelec-tric flux tube in the same way as the Cooper pairs in the ordinary superconductor circulate around the Abrikosov tube
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