a color magnetic vortex condensate in qcd

Nuclear Physics BI70[FSI] (1980) 265-282 © North-Hol land Publishing C ompany

A C OLOR MAGNETIC VORTEX CONDENSATE IN QCD

J. AMBJI3RN and P. OLESEN

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 and 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 magnet ic 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 has 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 rom 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 follow-

ing: (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

~.<,,, = - '-~ r),w,2,:~ . , , - O : ~ * ) ( n T , - n , ~ )

_ D;W.~ D).W). I~f~.2 _ 5( 2

-- ig( O~A: - O ~ A ~ ) W ; W :

1 2 , + ~g ( w . w ; - w . w ; ) ' ,

D~, = O~- igA~,,

f~, = a~A~ - 0~A~. (2.2)

We now consider a homogeneous color magnetic field H in the 3-direction. We choose a gauge such that A 2 = Hx~. Apart from gauge terms, the linearized equations of motion for W: then become

[ - ( Ox - igAx )2 8~,, + 2igf~,, ] W,, = O . (2.3)

The term 2igf~,W, is due to the magnetic moment of the gluon and is of crucial importance for the following. In the external field eq. (2.3) can be seen to induce an imaginary part in the vacuum energy [3]. Using the eigenvalue equation

as well as

- E ( 3 l ~ - i g A l ~ ) 2 W , ~ = 2 g H ( n + ½ ) W , ~ , (2.4) f l= l ,2

2igf~ = - 2 g H ( S3)~, p , (2.5)

where $3 is the third component of the gluon spin, one finds [3] that the vacuum energy can be written as a sum over modes,

The unstable mode comes from the second term with n -- 0. For k 2 < gH, Eva ¢ has an imaginary part. The question as to how to remove Im Eva ¢ then arises.

For the static problem the unstable mode can be projected out of W~ by requiring the unstable mode field W~ (°) to satisfy

[ - (3x - igAx)26~, + 2igF~,]W~ (°)= - g H W ~ (°) , (2.7)

268 J. Ambjern, P. Olesen / Magnetic vortex condensate

corresponding to eq. (2.4) with n = 0. The total Yang-Mills field is then

w.= + y. (2.8) n , S 3 ~ ~ 1

n=O, $3~ - 1

where W~ TM s3) are the other modes. Diagonalizing eq. (2.7) one sees that

W I <°) = - i W 2 ~°) ~ W '°) • (2.9)

The eigenfunction W ~°) is a harmonic oscillator wave function in the 1-direction and can be written [4, 6, 7]

W < ° ) ( x l , x 2 ) = e - t g H / 2 ) X ~ F ( z ) , z = x I + i x 2 , (2.10)

where a possible x3,x o dependence has been ignored. In eq. (2.10), F ( z ) is an arbitrary analytic function.

The meaning of eq. (2.10) is thus that it projects out the unstable mode responsible for the imaginary part of Eva c.

Because of eq. (2.10) it turns out by use of the Cauchy-Riemann relations that the only non-vanishing field strength is [7]

Fi 3 = H - 2gl W~°) 12. (2.11)

The classical energy is therefore

= d x~(F,2 ) f3, 3z

= f d3x[ ½H 2 - 2 g H ] W < ° ) 1 2 + 2g21W<°)14 ] . (2.12)

It was pointed out in ref. [6] that this expression has a remarkable similarity with the Higgs potential. This effect is due to the gluons magnetic moment, which produces the term

- 2 g n l W ~ ° ) 1 2 . (2.13)

Thus, the magnetic moment produces a "Higgs mass". In the following the higher modes are ignored. It is therefore of much impor-

tance to understand that these modes [i.e., W t"'s3) in eq. (2.8)] do n o t produce a pseudo-Higgs mechanism like the one exhibited in eq. (2.12) simply because their eigenvalues are always positive, in contrast to W ~°) [see eq. (2.7)]. Therefore, although the higher modes contribute to the classical energy in general, we do not expect c l a s s i c a l contributions from the higher modes in the lowest state (quantum mechanically they do contribute, see, e.g., sect. 4 in ref. [8]).

J. Ambjern, P. Olesen / Magnetic vortex condensate 269

The last point of this section concerns the removal of the instability. The "pseudo-Higgs" form of the energy (2.12) strongly indicates that this should be possible. To understand in more physical terms the mechanism behind the instability, one should notice that eq. (2.3) shows by means of a Fourier transform that the instability occurs for momenta satisfying

k 2 < g H . (2.14)

In other words, if one has a homogeneous color magnetic field exceeding a critical

distance*

dcr i t = _ _ C , (2.15) Vgn

then the vacuum energy develops an imaginary part. The way to remove the instability is therefore to ensure that the magnetic field is never homogeneous over distances larger than dcrit. It is enough that this is true in a plane, which we take to be the (XI,X2) plane. If the magnetic field is not homogeneous over distances larger than den t in the (x I , x 2 ) plane, it means that domains are necessarily formed in the (XI,X2)plane. Keeping homogeneity in the 3-direction (for fixed values of x I and x 2), this implies that magnetic flux tubes running in the 3-direction exist.

The domains in the ( x l , x 2 ) plane are only determined subject to the require- ment that their linear dimensions should never exceed d¢~it. The domains can therefore form any irregular lattice in the (x 1, x2) plane. To simplify calculations, and also because energy minimization usually prefers a rather symmetric lattice (or crystal), we studied the consequences of assuming a regular lattice of domains [4]. Thus, IW<°)l is required to be periodic in the ( x l , x 2 ) plane, and using the eigenfunction (2.10) we found that the analytic function F ( z ) could be written [4]

-t-oo

F ( z ) - - E c. e2 ' '* /b , (2.16) n m - - o 0

where the expansion coefficients satisfy

On+ K ~--- e - 2 ~ r a K / b C n , (2.17)

where a and ib are the lattice vectors**. The two equations then lead to the O-functions discussed in refs. [4, 9]. Here we

shall not repeat these technical details. Instead, in fig. 1 we show the beautiful geometric patterns of the contour lines for constant values of W t°) I-

* Here c is a c o n s t a n t of o rde r one. I t tu rns ou t t ha t c s ~ .

** [a[ a n d [b[ m u s t thus not exceed the cr i t ica l d i s t ance den t.


7.2

6.q

S.6

4.S

>. 4.0

8 .2

2.4

1.4

0.0

0.0 0.0 0,6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 S.4 6.0

X

Fig. I. Contour lines for [W(°)[ 2, which is normalized to be 1 at the maximum. In the center of the circular regions ] W (°) [ 2 = 0. The circle-like contour lines from the center and outwards correspond to the following values of I w(°)12 : 0.2, 0.4, 0.6, 0.8. The nearly hexagonal curve corresponds to [W(°)12 = 0.91. The "center" of the triangular region has [W(°)[ 2 = I, and proceeding outwards one has [W(°)12 = 0.999, 0.98, 0.95, 0.92, and 0.91. These contour lines are also contour lines for the Higgs field [epl2 in the

superconducting case.

T h e c o n t o u r l i ne s s h o w n i n fig. 1 c o r r e s p o n d to m i n i m i z i n g t h e e n e r g y * in -

t e g r a t e d o v e r a f u n d a m e n t a l l a t t i ce . T h u s , t h e e n e r g y is m i n i m i z e d in a n o n - t r i v i a l

m a n n e r : the e q u a t i o n s of m o t i o n a r e n o t s a t i s f i e d loca l ly , b u t t h e y a r e s a t i s f i e d

w h e n i n t e g r a t e d o v e r a f u n d a m e n t a l l a t t i ce . T h e m e c h a n i s m b e h i n d th i s h a s b e e n

* A straightforward Higgs mlnlmalization of eq. (2.12) gives I W¢°)l 2 =gH/282, which is not in accordance with the general form (2.10) of W t°).

J. Ambjcrn, P. Olesen / Magnetic vortex condensate 271

extensively discussed in the previous literature [4, 6, 9]. The point is that because of eqs. (2.10), (2.16) and (2.17), W (°) depends on a number of parameters (lattice size, magnetic field . . . . ), which we can denote collectively by p, W= W(x l, x2,p). The energy minimization then gives (static case)

0E 0 - ~ - - - ~p

=--lone d3x0~(W)0w OW(Xl'X2'P)op (2.18)

lattice cell

The integral vanishes, not because 8E/OW= O, but because the integral of (8E/OW)(OW/Op) vanishes [6] when one integrates over one domain (one lattice).

When the energy minimization is carried out we find [4] the energy density averaged over one lattice:

=½ "0 .14H 2 . (2.19)

The classical energy density is thus lowered relative to the classical energy density ½ H 2 of a homogeneous magnetic field.

The state considered in this section is ordered, because the "Higgs" field has an expectation value. The disordered quantum liquid state discussed in ref. [8] arises because of quantum fluctuations of the higher modes.

3. Comparison with superconductivity and identification of a dynamical Higgs field in QCD

In this section we shall compare the results mentioned above with a lattice of vortex lines in a superconductor.

3.1. THE ABELIAN SUPERCONDUCTOR

A lattice of vortex lines is described by the Landau-Ginzburg equation (see, e.g., ref. [5]). To make the equations more similar to those in sect. 2 we shall, however, start from the Higgs lagrangian,

~ = -I(0~ - igh~)ep[ 2 + m21~[ 2

_ / & [ ~ [ 4 , 2 - ~F.~. (3.1)

272 J. Ambjcrn, P. Olesen / Magnetic vortex condensate

The equat ions of mot ion are

- - ( 0 ~ - - i g A ~ )2 t~ - - m 2t~ -t- )k I~'1% = o , (3.2)

- - O ~ C , ' = j ,

= - i g e p * ~ , e p - 2 g 2 A ~ I,~12 . (3.3)

We consider an external magne t ic field n ext point ing in the 3-direction. Further , let H ext be very close to the critical field H¢2, so that the superconduc tor is close to the transit ion point to the normal state, where vortices are not formed. This means that

I~1 = is very small, and consequent ly we can expand,

_ _ e x t + o(I, 12), (3.4)

~ = F I X ' + O(1~,12). (3.5)

In the equations of mot ion (3.2) and (3.3) we then drop all terms of order ]~]3 or higher. Hence we have

- ( O k - i g A ~ X t ) 2 q ~ - m2~ = 0, (3.6)

- 0iF. k = 2g2lqa] 0kX

where we in t roduced the phase X by

-- [e/, [e ix , (3.8)

and where the only non-vanish ing c o m p o n e n t of F~, is

6 = -- n°xt + cl~l 2 + O ( l ~ l ' ) . (3.9)

F r o m eq. (3.6) we see that the largest value of n ext which is al lowed in order to get a b o u n d e d solution, is given by

m a x ( g H ext) = m 2. (3.10)

In this case the solution is

ep = e - S H ° * ' * U 2 F ( z ) , z = x I + i x 2 . (3.11)

I t thus follows that when g n ext ~ m 2, eq. (3.11) provides a solution to the equat ion

J. Ambj¢rn, P. Olesen / Magnetic vortex condensate

of motion (3.6). Inserting eq. (3.9) in eq. (3.7) we obtain

cOll~' 2--- -2g21~b'2( lg 02X - - X, Hext )

273

(3.12)

!

c0=l~12 = 2g21~l=g olX. (3.13)

Using eq. (3.12) in the form

--gH~tx~/2 ¢ -- e IF(z ) le ;X , (3.14)

one finds by performing the differentiation, that

011t~' 2 = 2gl~b'2( 1 ) g OEX -- x l H cxt , (3.15)

and

02[t~l 2 = - - 2 [ ~ b [ E O l x • (3.16)

These two equations are consistent with eqs. (3.12) and (3.13) if and only if

c = - g . (3.17)

Hence, eq. (3.9) gives

Fi2 = n e X ' - g l e p l 2 . (3.18)

When the Gibbs free energy is minimized, it is then seen that one gains energy relative to a homogeneous field by having a periodic lattice of vortex lines. The geometric structure of the lattice is precisely the same as the corresponding structure in the non-abelian case. Consequently fig. 1 also represents the abelian case! The analytical expression for the function F ( z ) is the same as in the non-abelian case, and therefore eqs. (2.16) and (2.17) are valid in the abelian case too.

3.2. F L U X Q U A N T I Z A T I O N

A priori one might think that flux quantization cannot be seen in a lattice of flux tubes. The reason for this is that there could be a considerable amount of overlap between the vortices, and hence an individual vortex flux cannot be identified.


This reasoning is, however, totally incorrect. To see what goes on, let us write eq. (3.3) in the form

A = l_ O~x - L / ( 2 1 ~l z) . (3.19) g

In the lattice we have periodicity for I¢ I, F~,, and j , . Thus, let us consider two parallel sides of the fundamental lattice, and let us take a point 1 on one of the sides and let 2 denote the corresponding point on the parallel side. Due to periodicity I¢(1)l 2 = ITS(2) I 2, andj~(1) =j~(2). Consequently we have the relation

A~(I) =A~(2) + 1 8 ~ [ X(1) - X(2)] . (3.20)

It therefore follows that if we do the line integral of A~(x) over one lattice, we obtain

~oo A~(x) d x . - AX - 2vrn n = 0, 1, (3.21) ne la t t ice g g

cel l

where - A X is the difference between the phase at the point from which we start and the phase at the same point after one round trip along the sides of the lattice cell. By the usual argument the phase of 4, at any point is only determined up to a multiple of 2~r, and hence flux quantization follows in the usual way. The reader can easily convince himself, that the reasoning given above applies no matter how the lattice cell is placed. It is enough that the parallel sides over which the line integral are performed are separated by the corresponding lattice vectors.

It should be noticed that eq. (3.20) is the abelian version of 't Hooft ' s quasi- periodic boundary conditions [1]. The only difference concerns the interpretation: in the present case the lattice does indeed have a physical meaning*, and is not merely introduced with the purpose of letting the lattice size go to infinity in the end of the calculations.

3.3. IDENTIFICATION OF A DYNAMICAL HIGGS FIELD

The abelian case discussed above can now be compared to the non-abelian case discussed in sect. 2. For the moment we leave out the comparison of fluxes in the two cases. This point will be treated separately in sect. 5.

In table 1 we have given a brief comparison of the two cases. The main difference is that in the non-abelian case we project out the unstable mode, which is only part of the total field W~, and hence the equations of motion are not

* The lattice vectors are determined by minimizing the energy. Their size is ~ 0.6 fm [4].

J. Ambjcrn , P . Olesen / Magnetic vortex condensate

TAaLE 1 Comparison of a superconductor with the unstable mode

275

Superconductor The unstable mode

Only non-vanishing field component FI2= H - gl¢~[ 2

Potential -m2[~[2 +½~.l~p14

Higgs m a s s - ?n 2

Higgs field

I¢1 ] ¢ [4 coupling

½x Form of the field a)

,~ e -gHx~/2 F( z ) , 7. -~ X 1 + ix 2 Quantity which is mlnimalized: the Gibbs free energy (including a magnetization term)

Only non-vanishing field component F~f f i H - 2gIW(°)I 2

Potential - 2 g H I W ( ° ) 1 2 + 2g21W(°)14

Pseudo-Higgs mass - g H

Pseudo-Higgs field

v~ tw(O) I I W(°) 14 coupling

½g2

Form of the field a) W (0) ffi e -gHx~/2F(z) , z ffi x I + ix 2

Quantity which is minimalized: the energy (no magnetization term)

a)F(z) is the same on the left and on the right.

satisfied. Nevertheless, the parameters can be chosen in such a way that the energy

is minimized as described in sect. 2. The main similarity is perhaps that a Higgs field can be identified as

I~ 12_~ 21W(0) 12. (3.22)

The Higgs-like character of W (°) should be understood physically as due to a condensation of the gluons. This is, however, only possible when quantum fluctuations are included, since they presumably makes it energetically favorable to have a color magnetic field (see sect. 6).

The behavior of W °) under gauge transformations will be discussed in sect. 5.

4. "QCD" with an anomalous magnetic moment

In this section we shall briefly discuss the case when the gluon has an anomalous magnetic moment. This spoils renormalizability a n d / o r unitarity, and hence such a model has no realistic value. However, as one varies the moment, it is possible to see on the classical level (where renormalizability is not needed) that a phase transition occurs. In the new phase it does not pay energetically to have vortices. This behavior is in complete analogy with the superconducting case near the

critical field.

276 J. AmbjOrn, P. Olesen / Magnetic vortex condensate

Let a denote the anomalous magnetic moment. The lagrangian is thus

E = EVM - 2iga(3~,A~ - 3~A~,)W~*W.. (4.1)

The linearized field equation is

[ _ (0 x _ . z + a)F~.]W,, 0. ,gAx) 8~,~ + 2ig(1 = (4.2)

The eigenvalues of the modes are

E 2 = 2 g H ( n + ½ ) + k 2 ¥ 2 g H ( 1 +a) , (4.3)

similar to eq. (2.6). Thus, for a < - 1 , there is no instability, and Eva c remains real.

For a = - ~ + e(e > 0) the instability is very weak, and the pseudo-Higgs mass m E is very small,

- m 2 = - g H ( 1 + 2a) . (4.4)

It therefore follows that the pseudo-Higgs field is small, and hence we can ignore 1 the I w(°)l 4 term relative to the ]W(°)12 term in the lagrangian. For a = - 5 + e the

calculations therefore proceed exactly like in superconductivity close to the critical

point, and

W ~°) = e -gHx~/2F( g ) (4.5)

now becomes an approximate solution to the equations of motion. The solution becomes more and more accurate as e becomes smaller (a = - 5 + l e). For a <

W (°) - -0 in the lowest state, and hence it is now more favorable to have a - - 5 '

homogeneous field than it is to have vortices. This is like in a superconductor. For a = 0 one has QCD, and the expression (4.5) is now no longer an approxi-

mate solution to the equation of motion in a local sense, but it still minimizes the energy. For a > 2, the first "stable" mode becomes unstable. In the following we shall only discuss the true QCD case where a = 0.

5. Flux quantization in the non-abelian case

' t Hoof t [1] has recently shown how the non-trivial topology of the torus leads to topologically stable magnetic fluxes. This was done by imposing periodic boundary conditions in a box. Physical variables should then be periodic. The vector potential A~ is, however, not an observable and needs only to be periodic up to a

gauge transformation. If we take for simplicity a rectangular box in the x 1, x 2 plane (one may trivially

generalize to an arbitrary parallelogram) and take the x 3 direction as the direction

J. AmbjCrn, P. Olesen / Magnetic vortex condensate 277

of magnet ic flux, then the b o u n d a r y condit ions are (unlike ' t H o o f t [1] we thus do

not consider periodici ty in the x 3 direct ion*)

At,( x , , a2) -- a2( Xl )At , ( x , , O)a 2 '( x , )

i az(X,)O.af l(xO ' g

(5.1)

At~(a I , X2) = a,(xE)A.(O, x 2 ) a ? l (x E )

i - - l -~a , (x~)~a , (XE). (5.2)

where a l, a 2 are the lengths of the box sides and where fll and ~'~2 have to satisfy

f~E(a,)a,(O) = a , ( a 2 ) a E ( O ) Z , , , . ( 5 . 3 )

In (5.3) Z m is a c e n t e r element of S U ( N ) * * :

Z m = e E " m / N I , m = 0 . . . . . N - 1. (5.4)

By a regular (non-per iodic) gauge t r ans fo rmat ion f~(Xl, x 2), the gauge t r ans fo rmed t t t field A~ satisfies (5.1) and (5.2) with new ~2 l, f~2 given by

a t l ( X 2 ) ~--- a( al,x2)a,( XE)a-'(O, xE), (5.5)

a 2 ( X I ) = a ( X i :, a E ) a 2 ( x I ) a - - l ( X l , 0 ) . (5.6)

We can choose ~'~(Xi,XE) such that f ~ ( x l ) = I . The new f~'l, f ~ still satisfy (5.3) and ' t Hoof t identifies the gauge- invar iant n u m b e r m with a new topological flux.

We have a l ready argued in sect. 3 that the v a c u u m structure of Q C D m a y have a lattice structure similar to the one present in a type II superconduc tor in an external magnet ic field. As the quot ient space RE/lat t ice is a torus, the investiga- t ion of gauge field conf igurat ions having lattice structure is equivalent with the invest igat ion of gauge field conf igurat ions on a torus. There fore it is perhaps not so surprising that the field conf igurat ions explicitly cons t ruc ted using the unstable m o d e satisfy ' t Hoof t ' s b o u n d a r y condi t ions and au tomat ica l ly have flux quant i -

zat ion in the sense def ined by ' t Hoof t .

* The following can easily be extended to cover this case by introducing a lattice vector a 3 in the 3 direction and letting a3--. ec in the end.

** From a mathematical point of view (5.1)-(5.3) define the topological inequivalent principal fiber bundles on the torus with fiber SU(N)/Zjv.

278 J. Ambjcrn , P. Olesen / Magnetic vortex condensate

It is interesting, however, that the mechanism generating this flux quantization seems to be rather analogous to the one operating in the abelian case. In the abelian Higgs model the A t fields on two opposite lattice sides are connected by a gauge transformation [see (3.20)] just as in (5.1), (5.2). For the topological flux quantization it is essential that X in (3.20) is the phase of a Higgs field ~. Analogously we will show that our field configuration satisfy (5.1)-(5.3), because the phase of ~l(X2) is closely connected to the phase of the unstable mode W (°) which in our case plays the role of a dynamical Higgs field.

We start by investigating the SU(2) case in some detail. The A~ fields are written

A. = A~:-", (5.7)

where a _ 1 oa oa r - 3 and are the Pauli matrices. The field configuration determined by the unstable mode (2.10) and the external color magnetic field A 3 = (0, x lH, O, O) may be written

A , ( x , , x 2 ) -- V ~ (Re F(z) ,r I + I m F ( z ) r 2 )

× e-gnx'~/2 (5.8)

A 2 ( x , , x 2 ) --- V'2 ( - I m F ( z ) ' r , + Re F(z),r2)

X e -g//x~/2 + xlH,r3, (5.9)

A3(Xl, x2) = -4o(xl, x2) = 0. (5.10)

We may use the known properties of the function F ( z ) [4] to show that AI ,A 2 satisfy (5.1), (5.2) with

~ , ( x 2) = e-iSu . . . . . 3 (5.11)

~2 (x , ) -- I . (5.12)

It is, however, more instructive instead to go the opposite way and determine F ( z ) by using (5.1), (5.2) and (5.11), (5.12).

The gauge transformation (5.11) is chosen so that the term XlH % in (5.9) satisfies

(x, + al)n~3 = U,(x~)x, m 3 u ? '(x2) - ~ U,(x:)O~u? '(x2). (5.13)

Eqs. (5.1) and (5.2) reduce to

W(°)(Xl + al ,x2)( ' r , + i'r2) = ~, (x2 )W(° ) (x , , x2 ) (~ ", +-i 'r2)~-l(x2) , (5.14)

w(°)( x , , x ~ + a2) = w(°)( x , , x ~ ) , (5.15)

J. Ambj¢rn, P. Olesen / Magnetic vortex condensate

where W(°)(xl,x2) is given by (2.10). Eqs. (5.14) and (5.15) give

279

F(z + a i ) = e-Sn'#12 +gtlalZF( g ), (5.16)

r ( + ia ) = r ( z ) . (5.17)

This is the functional equation derived in ref. [4]. It has the generalized Jacobi 0-functions as solutions. These #-functions have a fundamental lattice area

2~1¢ lattice area = - - x = 1,2, 3 . . . . . (5.18)

gH '

From (5.18) it follows that (5.3) is satisfied with m = 1 for x = 1 , 3 , 5 . . . and m - - 0 for x = 2 ,4 ,6 . . . . .

All the above arguments may easily be generalized to an arbitrary lattice (not necessarily rectangular). The fact that (5.3) is satisfied with

f ~ l ( a 2 ) f ~ { I(0) ----. e 21ri Zig (5.19)

may be understood in terms of the phase of the "Higgs field" W (°) just as flux quantization in the abelian case.

In the abelian case the difference A X in the phase of the Higgs field one picks up when going round the lattice boundary gives the quantized magnetic flux

~ = l A x = 2~rn/g g

[see (3.21)]. If we write W (°) = I W(°)leeX we have from (5.14) and (5.15)

X(x I + a l ,x2) = X(Xl,X2) + algHx 2, (5.20)

X(x l , x2 +a2) = X ( x l , x 2 ) . (5.21)

Therefore the phase difference of X one picks up when going round the lattice boundary is 2rrx, where x is defined in (5.18). This is analogous to the abelian case and is just a reflection of the fact that W (°) has ~ simple zeros in a fundamental lattice. The interpretation of ~ as a number describing the magnetic flux is, however, different in the two cases, because of the different group structure of U(1) and SU(2). In the abelian case ~ represents the magnetic flux (except for a factor 2~r/g). In the SU(2) case it is r modulus 2 which is the magnetic flux as defined by 't Hooft.

The generalization of the above ideas to S U ( N ) is straightforward if we choose as color direction for the external magnetic field the (N 2 - D-direction* [8 direction for SU(3)]. In this case we have N - 1 unstable modes [6] which couple to the

* Other directions are likely to be unstable. We thank B. Felsager for a discussion on this point.

280 J. Ambj¢rn, P. Olesen / Magnetic vortex condensate

external magnet ic field with a gauge coupl ing

gN = ~/2(NN__ 1) g . (5.21)

In par t icular (5.1 l), (5.18) and (5.19) are replaced by

~'~l(X2) = e-isH . . . . F1~2-1, (5.22)

2~'K lattice area -- - - x = 1,2, 3, (5.23)

g N H . . . . ,

~l (a2 )~11(0) = ei2"gr"2- ' /g~ (5.24)

l

1 1 0 = . ( 5 . 2 5 )

FN~-' f f 2 N ( N - - l) ""

0 -- ( N - 1)

This means that ~ l ( a 2 ) ~ / l ( 0 ) for • = 1 . . . . . N gives the different center e lements in S U ( N ) as required by (5.3).

It is to be emphasized that in the case discussed above the lattice is physical, with side lengths fixed by energy minimizat ion*.

6. Condensation and disordering as quantum phenomena

So far we have considered the classical energy in an external field. F o r m a t i o n of vortices lowered this energy, but the min imal classical energy (3.19), i.e.,

( ~ )on© lattice ---- 0 .14(½H 2 ) (6.1)

is higher than the energy 0 of the per turbat ive " g round state". On the classical level there is thus no real condensat ion.

The energy densi ty of a homogeneous magnet ic field including one- loop quan- t um correct ions has been c o m p u t e d by several authors [10], and the result is**

gq = 11N ( ln g H _ 1 ~ (6.2) 96~.2 g2H2 A 2 2]"

* If a lattice is introduced also in the 3-direction, the lattice vector a 3 should be send to oo at the end of the calculation.

** This expression is valid for fields homogeneous over distances larger than ~ d ~ t . Hence it is also valid for slowly varying fields.

J. Ambjern, P. Olesen / Magnetic vortex condensate

The effective coupling as a function of gH is, in the same notation,

g ( H ) 2 = 1 (11N/96 ~r 2 ) l n ( g H / A 2 )

281

(6.3)

For gH >> 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 = A 2. 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).

i H 2. We now take into account Eq. (5.2) includes the classical energy density* 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 . 4 3 H 2 + l lNg2H2( ln g n _ ± ] (6.4) 96tr~ ~ A 2 2]"

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 ln (gH/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 (A2 / # 2) = 48~r2/1 I N g 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 reali- zation 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 phenome- non 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. One 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 Hoof t 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] G. 't Hooft, Nu¢l. Phys. B138 (1978) 1; B153 (1979) 141 [2] G. Mack, Carg~se lectures 0979) [3] N.K. Nielsen and P. Olesen, NucL Phys. B144 0975) 376 [4] J. Ambj~rn and P. Ole~en, Nu¢l. Phys. BI70[FS1] 0980) 60 [5] D. Saint-Jam~, G. Sarma and E.J. Thomas, Type II sup~-condietivity (Pergamon Press, 1969) [6] 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 [7] H. Arod~ private communication 0978) [8] H.B. Nielsen and P. Olesen, Nucl. Phys. Bl60 (1979) 380 [9] H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 0979) 1

[10] 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

a color magnetic vortex condensate in qcd

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