- Email: [email protected]
Magnetic monopole condensation in the monopole Nambu-Jona-Lasinio model
3 October
1996
PHYSICS
ELSEMER
LETTERS B
Physics Letters B 386 ( 1996) 198-206
Magnetic monopole condensation in the monopole Nambu- Jona-Lasinio model M. Faber I, A.N. Ivanov2, W. Kainz, N.I. Troitskaya 2 Institut fiir Kernphysik. Technische Universifiit Wien. Wiedner Hauptstr. 8-10, A-1040 Vienna. Austria Received 22 March 1996; revised manuscript Editor: l?V. Landshoff
received
I July 1996
Abstract We propose a model of monopoles where a condensate of monopole-antimonopole pairs leads to confinement. We investigate the condensation of these monopole-antimonopole pairs around a dual Dirac string.
1. Introduction
Compact Quantum Electrodynamics (CQED), defined for lattices as nonlinear U( 1) gauge theory, possesses a confined phase [ 11. There is evidence [ 21 that the vacuum of CQED behaves like an effective dual superconductor with magnetic monopoles. The magnetic monopoles rearrange the electric fields so that there is a net flux tube between quarks which looks like a dual Dirac string. The energy per unit length of this flux is the string tension. Monopole condensation is able to realize the electric charge confinement in CQED. Lattice calculations of inter-quark potentials, field and charge-density distributions, and chiral symmetry breaking show a very close similarity between CQED and QCD. However the mechanism of quark confinement realized by CQED and being most likely very similar to that of QCD could be carried over to the description of quark confinement in experimentally measured processes of low-energy interactions of hadrons only via the employment of a continuum model embodying the main non-perturbative properties of CQED. In our recent investigation [ 31 we have suggested a Nambu-Jona-Lasinio approach to the description of magnetic monopoles interacting with dual Dirac strings, called the Monopole Nambu-Jona-Lasinio (MNJL) model. In this paper we apply the MNJL model to the investigation of the magnetic monopole condensate around a dual Dirac string and compare the obtained results with those predicted within CQED. The symmetry group of the magnetic monopole system in the MNJL model is the Abelian U( 1) group of magnetic monopole charge g conservation. By analogy with CQED the MNJL model should have a confining vacuum, where only ’ E-mail: [email protected]. 2 E-mail: [email protected]. Permanent Address: State Technical 0370-2693/%/$12.00 Copyright PfI SO370-2693(96)00959-S
University, Department
of Theoretical
Physics,
0 1996 Elsevier Science B.V. All rights reserved.
195251 St. Petersburg,
Russian Federation
M. Faber et al./ Physics Letrers B 386 (1996) 198-206
199
monopole-antimonopole pairs can be condensed, created and destroyed. This does not violate magnetic charge conservation and leaves the magnetic U( 1) symmetry unbroken as well. Such an unbroken U( 1) symmetry in the condensed phase distinguishes the MNJL model from the BCS-theory of superconductivity, where the group of U( 1) electric charge transformations gets broken due to condensation of electron-electron Cooper pairs. In the MNIL model we assume that the massless magnetic monopoles, described by the fermion field x(x) , interact via a local four-monopole interaction of the NIL kind given by [ 31 (see also [ 41) CMNJL(X)
[x(x)r~x(x)l[X(x)r~X(x)l,
=C[f,(d,+)12-G1
(1)
where G and Gt are positive phenomenological constants responsible for the magnetic monopole condensation and a dual-vector field mass, respectively. The effective Lagrangian ( 1) is invariant under the U( 1) group connected with magnetic monopole charge conservation. Due to strong attraction in the lx-channels produced by local four-monopole interactions (1) the monopoles and antimonopoles become unstable under monopole-antimonopole condensation, i.e., (Xx) #O. Indeed, in the condensed phase the energy of the ground state is negative, i.e.,
[(Xx)12<09
W=-(~~NJL(X))=-(%G+G,)
(2)
whereas in the non-condensed phase, when (xx) = 0, we have W = 0. This means that the condensed phase is much more advantageous to the monopole-antimonopole system, described by the Lagrangian ( 1) . Monopole-antimonopole pairs become condensed without breaking of magnetic U( 1) symmetry as well. In the condensed phase magnetic monopoles acquire a mass M that satisfies the gap-equation [ 3,4]
M = -2G(x(O),y(O)) =
-$J, (W
-
z 1 $!A
=-z[h2--M21n(l+$)],
(3)
where A is the ultra-violet cut-off defining a quadratically divergent integral. The magnetic monopole condensation accompanies the creation of collective Xx-excitations with the quantum numbers of a scalar Higgs meson field (+ and a dual-vector field C, [ 3,4]. The dual-vector field C, is massive, it is created by the four-monopole interaction proportional to the coupling constant Gt and acquires the mass the main contribution to which is of order O( l/G,) [ 3,4]. Since the C,-field is created massive, it possesses both longitudinal and transverse components from the very beginning. Therefore, the C,-field does not need magnetic U( 1) symmetry breaking to acquire a longitudinal component in terms of a Goldstone boson. The effective Lagrangian obtained after the integration over monopole degrees of freedom and involving quarks, antiquarks and the fields of collective scalar u and dual-vector C, excitations reads [ 3,4] C,,(X)
= &(x)P”(x)
+ $4~C,(x)Cc”(x)
+ $@(X)d%(X)
where M, = 2M and MC arethe masses of the IJ and C, fields [3,4], such as MC is given by [3,4]
M: = g
-&M(M)
where 52 (M) is a logarithmically J2(M)
=
(5)
+M2J2(M)],
I
J
d”k v2i
1 ( M2 - 12~)~
divergent
integral A2 M2+h2’
(6)
M. Faber et al. /Physics Letters B 386 (1996) 198-206
200
The coupling
constants
g and K obey the constraint
[ 3,4]
&J*(M)=$Jz(M) =I giving the relation
K’
=
&,, quark(X)= -
2g2/3
c
[
mi
i=q.rj
(7)
1] . Cfreequark(x) is a kinetic term for quark and antiquark
dXY(r) dXT(r)
J( dr
-$-y
l/2
---&--g~Y
aC4’(x-
X,(r)) I
(8)
’
We consider quark and antiquark as classical point-like particles with masses mq = rnq = m, electric charges Qq = -Qa = Q, and trajectories Xi(r) and Xi(r), respectively. The electric quark current J”(x) is given by
J”(X)
-Xi(T)).
= CQiJdTyG’“‘(.X
(9)
i=q,q
The field strength F@“(x) is defined [3,5] Fp”(x) = fp”(x) - *dCfi”(x), where dCfiCLY(x)= ~PC”(~) P’Cp(x), and *dCfiLY(x) is a dual version, i.e., *dCpY(x) = ~.Y’a~dCap(x)(~0123 = 1). The electric field strength &p”(x), induced by a dual Dirac string, is defined following [3,5-71 I@“(x)
=
JJ
d2u8(4)(x - X)ap’(X),
Q
-
(10)
where we have denoted d2v = dada and gw(x)
-
dXpax” dX”axp, a(T
a7
a7
(11)
au
Here Xp = X+( r, (+) represents the position of a point on the world sheet swept by the string. The sheet is parametrized by the internal coordinates -cc < r < oc and 0 I u I 7~, so that X@(r, 0) = X/le (7) and Xp(r, n-) = Xc(r) represent the world lines of an antiquark and a quark [5]. Within the definition (5) the tensor field Z@“(x) satisfies identically the equation of motion, i.e., d,PV( x) = J’(x), the first pair of equations of motion of Dirac’s extension of Maxwell’s Electrodynamics. This means that the inclusion of a dual Dirac string in terms of 8@““(x) defined by ( 10) saturates completely the electric Gauss law. In the presence of a dual Dirac string ended by a quark and an antiquark, which are considered as classical point-like particles with masses my and mg, respectively, the massive dual-vector field C, has the shape of a dual Abrikosov flux line [3,5-71 C’[E(x)]
= -
J
d4x’A(x
(12)
- x’)J;&p”“(x’),
where A(x - x’) is the Green function A(x-x’)=
J
&k
e-ik,(x-x’)
(2~)~ h4; - k2 - i0’
The vacuum expectation values (v.e.v.) of time-ordered monopole field, i.e., the monopole Green function G(Xl) . . . ,A)
=
(13) products of densities expressed in terms of the massless-
(OlT(,V(xl)~l~(x~).. .R(~n>r~~(xn))IO)conn.,
(14)
M. Faber et al./Physics
Letters B 384 (1996) 198-206
201
where Ii(i = 1,. . . , n) are the Dirac matrices, taken in the tree-approximation depending on the string shape are given by [ 3,4]
x
exp i .1’d4x{
- g~M(X)r”XM(X)CY[&(X)l
- KjddXdddx)})
of the a-field
exchange
lo)::;,.
and
(15)
Then 10)‘M) is the wave-function of the nonperturbative vacuum in the condensed phase. The NJL model [ 81 as well as the BCS-theory of superconductivity [9] admits the exact solution of IO)(M) in the form [ 8,101
JO)(M) =,E, where fiP = p/E,
[@+ = p/dm
A~~u’o’t(p,A)b~o)‘(-p,*)] IO), is the velocity of massive monopoles
t
with the mass M, and a(“+(p,
16)
A) (or
b’O)+( -p, A)) denotes the creation operator of a massless monopole (or antimonopole) with the momentum p and helicity A; IO) E [O)(O) is the wave-function of the vacuum in the non-condensed phase. It is seen that the wave-function IO)(M) is invariant under magnetic U( 1) symmetry transformations. This means that in the condensed phase magnetic (I( 1) symmetry is not broken. Eq. (16) contains the factor (-A) [ 101 that has been lost in [ 81. The availability of this factor is very important for the correct behaviour of the wave-function of the nonperturbative vacuum (0)cM) under parity transformations. Indeed, since parity is conserved the wave-function IO)(W should be invariant under parity transformations P, i.e., PIO)““” = IO)‘M’ [ 111. By using the operators a”‘+(p, A) and bco)+( -p. A) one can construct two operators 0,
=2x
(3_ =2x
c Ab’“‘+(-p,A)a’o’t(p,A), p A=*1
c b’“)+(-p,A)a’o’+(p,A), p A=fl
(17)
possessing different properties under parity transformations, i.e., PO+Pt = Cl, and PO-P+ = --CL. Following the BCS-theory of superconductivity [9] we can identify the v.e.v. of the O+-operator per unit volume with an order parameter (0,)
=;
(“)(OIO+IO)(M) = -;
c
c p
d4p -4M, / =
(2r)4i
-$w)
A2dq=
-4Mx
A=il
2EPV
= -4M
J d3p (27r)32Ep
=
1 M2 -p2 - i0
=(/f(o)xto)),
(18)
where V is a normalization volume. Thus we have got (0,) the parity conservation in the MNJL model.
2. Magnetic monopole
1 p
condensate
= (j(O)x(O)).
In turn (0_)
= 0. This implies
around a dual Dirac string
In order to calculate the magnetic monopole condensate turn to Eq. (10) and consider the following v.e.v.
depending
on the dual Dirac string shape we should
M. Faber et al./Physics
202
Letters B 386 (1996) 198-206
b Fig. I. One-monopole loop diagrams describing a magnetic monopole condensate around a dual Dirac string. (X(x)x(x);E)
=
‘“)(OIT(X~W~~(X)
-g~‘M(Z)YYX~M(Z)C,[E(Z)l -
x expi/d4Z(
KX~(Z)~~(Z)~(Z)})lO)~~d.
The matrix element (T( X)X( x) ; &), calculated in the tree-approximation of the g-field described by the diagrams depicted in Fig. 1. The contribution of the diagram in Fig. la coincides with (1(0)x(O)), i.e., (f(X>,A’(X);
The contribution
&)Fig. la = (X(o>~(o)).
of the diagrams
(19) exchange,
can be
(20)
in Fig. lb is defined by the expression
(21) By using the relations 1
/
K2JI(M) - 2MJ2(W = Mi
479 Therefore,
M, = 2M and Eq. (7) one can show that
we get
--$(xmx(o,)
(22)
M. Faber et al./Physics
J { 4
%r
x (-1)”
X
r2i
Letters B 386 (1996) 198-206
I
-&dm)l
&?m(X2)1
M-L-L,
203
. . .
1 M-I;-,&,
(23)
-...-I,
Neglecting the momenta of the dual-vector fields with respect to the magnetic monopole where e = 1,. . . , n, one can reduce the r.h.s. of (23) to the following closed expression:
(,?(x)x(x);
_ l;i2
&)Fig. lb = &
(X(0)x(O)) M2
J $tr{
(x(o~‘o))
J-i d4k r2i
J-t
(24)
M2 - (k - gC[&(x)])2
d”k
=
- L} M-i
1
The integration over k we should perform by applying suggested by Gertsein and Jackiw [ 121 we get
n-*i
M _ R + :C,E(x)l
mass, i.e., M >> jktl
the cut-off regularization
[3,4].
Using
the technique
1 M2 - (k - gC[E(x)])2
- M2
1 l-~ -gC,[&(x)]-& J$!{ev(' p M2 - k2 M2-k2
= ;g2c,[E(x)]cp[E(x)].
(25)
For the more detailed computation of integrals like (25) we refer readers to Ref. [ 121. The contribution of the diagram in Fig. lb reads
(26) Summing the contributions of the diagrams in Fig. 1 we obtain the total expression condensate depending on the shape of a dual Dirac string
(X(x)x(x);E) = ~/i3o)xto))
[
I+
~~g2c~,E~x~lc~,t(x),].
of the magnetic
monopole
(27)
Since C, [E(x) ] Cp [ I( n) ] < 0 it is seen that at distances close to a dual Dirac string the magnetic monopole condensate is suppressed. On the other hand, at large distances far from a string where the influence of a string is exponentially suppressed due to the Meissner effect, the magnetic monopole condensate (x(x)x(x) ; E) tends to the magnitude of the order parameter, i.e., @(x)x(x);&) -+ (jj(O>x(O)). As an example, let us consider the magnetic monopole condensate around a static infinitely-long dual Dirac string strained along the z-axis. In this case the induced electric field reads [ 131: E(r) = e,Q$‘)(r), where r is the radius-vector in the plane perpendicular to the z-axis and e, is the unit vector directed along the z-axis. A dual-vector potential, taken in the temporal gauge CO(X) = 0, acquires the form [ 131 C(r)
= L$L
C,(r).
(28)
M. Faber et al./Physics
204
The azimuthal Ca(r)
component
Pp(r)
=
where K, (Mcr)
=
is given by
QMc
,K1(Mcr),
2n
is the McDonald
s
d2k
9(r)
= --
dr
2~
C,(r)
GM2
&k.r C
+k2 =
Letters B 386 (1996) 198-206
function. co
The scalar function
dkkJo( kr)
s 0
M;+k2
(29) po(r) is defined by
= Ko(Mcr),
(30)
where Jo( kr) and KO(Mcr) are Bessel and McDonald functions, respectively, The prime stands for the derivative with respect to the argument. Substituting (29) in (27) and using the relation Qg = 2~ we obtain
(t(r)A”(r);E) = (~
1-
(
If
= -K1 ( Mcr).
&$f(M,r)].
(31)
This is the magnetic monopole condensate around a static infinitely-long can be calculated by analogy with (27). The v.e.v. (j,(x)jp(x>;E) expounded above one obtains
x
and KA( Mcr)
dual Dirac string. Following the procedure
having
been
2
&g2cp1EwlcP[a(+)l
>
.
(32)
It is seen that the v.e.v. (j,(x) jb( x) ; E) is suppressed for distances close to a dual Dirac string. This also agrees with the results obtained within CQED [ 141. For the static infinitely-long dual Dirac string, directed along the z-axis, Eq. (32) reads
(j,(r).?‘(r);E) = -s2[(X(0)x(O))12
1+
e sK:(Mcr)
::
For A 2 2M the last factor can be neglected according to Eq. (3). For a more detailed comparison with CQED we should adduce separately components of (j,(r) jP(r); E)
-(jO(r)jO(r);E)=g2[(~.(0),y(O))12~
1 + g#(&r))
2
1 - ~K,(M,r))2.
(33)
the v.e.v. of the time and spatial
(1- $$Kf(Mcr))2, 2
i(j
.j(r>;E)
=g2[(X(0)x(O))12~
1 + fsKf(Mcr)
>(
GM: 2 1 - i#,(Mcr)
>
.
One can see that the v.e.v. of the time components of the magnetic current has the same suppression in the .j(r);&)as a function of r is vicinity of a Dirac string that the v.e.v. of the spatial ones. The v.e.v. i(j(r) smaller than -(j’(r) j”( r) ; E) for finite values of r and goes to the same magnitude at r --+ m. This agrees qualitatively with the numerical data obtained on the lattice within CQED [ 141. Unfortunately, one cannot carry out a quantitative comparison of - (j” ( r ) j” ( r) ; E) and i (j(r) .j( r) ; E) with the numerical data obtained on the lattice [ 141. First, this is connected with a singular behaviour of the McDonald function Kf (Mcr) in the region close to the string, and the necessity to fit three parameters MC, it4 and G. The former is hardly unambiguous
M. Faber et al./Physics
Letters B 386 (1996) 198-206
205
procedure. Second, in addition to this reason there is another one concerning the principle distinction of our calculations from the lattice numerical data. Indeed, the v.e.v. -(j”( r)j”( r) ; E) and i(j( r) . j(r) ; E) have been computed on the lattice by taking into account automatically string fluctuations. This is not done here and goes beyond the scope of this letter. Therefore, we can restrict ourselves by the discussion of a qualitative comparison only. We can compute the v.e.v. of any operator constructed from massless monopole fields. For example, the v.e.v. computed with analogy to those given above reads (X(X)~/SX(X)X(X)y5X(X);&)
(35) The v.e.v. (~(x)r5~(x)~(x)yS~(x);I) h as not been calculated within CQED up to now. Therefore, our result (35) should be considered as a prediction to be compared with numerical experiments on lattice within CQED.
3. Conclusion We have suggested a consistent procedure for the computation of vacuum expectation values of any magnetic monopole densities in dependence on the shape of a dual Dirac string. The investigation of the magnetic monopole condensate around a dual Dirac string has shown that the influence of a dual Dirac string is reducing to the suppression of the magnetic monopole condensate at distances close to the string. In turn, at distances far from the string, where the influence of the string is exponentially suppressed due to the Meissner effect, the contribution of the string decreases, and at infinitely-large distances the magnitude of the magnetic monopole condensate tends to the magnitude of the order parameter, i.e., (0,) = (X(0)x(O)). A similar influence of a dual Dirac string on the magnitude of the magnetic monopole condensate has been observed within Compact Quantum Electrodynamics (CQED) [ 141.
Acknowledgements We thank the Fonds zur Fiirderung der wissenschaftlichen Forschung in Gsterreich (project PI 1156-PHY) for their support. We acknowledge fruitful discussions with Prof. G. E. Rutkovsky. The discussions with Dr. M. Zach concerning comparison of our results with numerical data on lattice are appreciated.
References [ I] P. Becher and H. Joos. Z. Phys. C 15 (1982) 343. [2] V. Singh, D. Browne and R. Haymaker, Phys. Rev. D 47 ( 1993) 1715. 131M. Faber, A.N. lvanov, W. Kainz and N.I. Troitskaya, Nambu-Jona-Lasinio approach to magnetic monopole physics with dual Dirac strings, (submitted to Z. Phys. C). 141A.N. Ivanov, M. Nagy and N.I. Troitskaya, Int. J. Mod. Phys. A 7 (1992) 7305; A.N. Ivanov, Int. J. Mod. Phys. A 8 (1993) 853; A.N. lvanov, N.I. Troitskaya and M. Nagy, Int. J. Mod. Phys. A 8 (1993) 2027; A 8 (1993) 3425; B 308 (1993) 111; B 295 (1992) 308; A.N. Ivanov and N.I. Troitskaya, Nuovo Cim. A 108 (1995) 555. 151 M. Faber, W. Kainz, A.N. Ivanov and N.I. Troitskaya, Phys. Lett. B 344 ( 1995) 143, and references therein. [6 1 P.A.M. Dirac, Proc. Roy. Sot. A 133 (1931) 60; Phys. Rev. 74 (1948) 817. [ 71 Y. Nambu. Phys. Rev. D 10 (1974) 4246.
206 [ 8 1 Y. Nambu and Cl. Jona-Lasinio,
M. Faber et al./Physics
Letters B 386 (1996) 198-206
Phys. Rev. 122 ( 1961) 345; 124 (1961)
246.
[ 91 J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162; 108 ( 1957) 1175. [ IO] A.N. Ivanov, in Lectures on Nonperturbative Phenomena of QCD, Part 2, Institut fur Kemphysik, Technische Universmit Wien, October 1995-February 1996. [II] C. Itzykson and J.-B. Zuber, Quantum Field Theory ( Mcgraw-Hill), 1980. I121 13. Gertsein and R. Jackiw, Phys. Rev. 181 (1969) 1955. [ 131 M. Faber, A.N. Ivanov, W. Kainz and N.I. Troitskaya. Dual Higgs model with dual Dirac strings, ESI-Preprint No. 286, November 1995 (to be published in Nucl. Phys. B). 1141 M. Zach, M. Faber, W. Kainz and P. Skala, Phys. Lett. B 358 (1995) 325.
1996
PHYSICS
ELSEMER
LETTERS B
Physics Letters B 386 ( 1996) 198-206
Magnetic monopole condensation in the monopole Nambu- Jona-Lasinio model M. Faber I, A.N. Ivanov2, W. Kainz, N.I. Troitskaya 2 Institut fiir Kernphysik. Technische Universifiit Wien. Wiedner Hauptstr. 8-10, A-1040 Vienna. Austria Received 22 March 1996; revised manuscript Editor: l?V. Landshoff
received
I July 1996
Abstract We propose a model of monopoles where a condensate of monopole-antimonopole pairs leads to confinement. We investigate the condensation of these monopole-antimonopole pairs around a dual Dirac string.
1. Introduction
Compact Quantum Electrodynamics (CQED), defined for lattices as nonlinear U( 1) gauge theory, possesses a confined phase [ 11. There is evidence [ 21 that the vacuum of CQED behaves like an effective dual superconductor with magnetic monopoles. The magnetic monopoles rearrange the electric fields so that there is a net flux tube between quarks which looks like a dual Dirac string. The energy per unit length of this flux is the string tension. Monopole condensation is able to realize the electric charge confinement in CQED. Lattice calculations of inter-quark potentials, field and charge-density distributions, and chiral symmetry breaking show a very close similarity between CQED and QCD. However the mechanism of quark confinement realized by CQED and being most likely very similar to that of QCD could be carried over to the description of quark confinement in experimentally measured processes of low-energy interactions of hadrons only via the employment of a continuum model embodying the main non-perturbative properties of CQED. In our recent investigation [ 31 we have suggested a Nambu-Jona-Lasinio approach to the description of magnetic monopoles interacting with dual Dirac strings, called the Monopole Nambu-Jona-Lasinio (MNJL) model. In this paper we apply the MNJL model to the investigation of the magnetic monopole condensate around a dual Dirac string and compare the obtained results with those predicted within CQED. The symmetry group of the magnetic monopole system in the MNJL model is the Abelian U( 1) group of magnetic monopole charge g conservation. By analogy with CQED the MNJL model should have a confining vacuum, where only ’ E-mail: [email protected]. 2 E-mail: [email protected]. Permanent Address: State Technical 0370-2693/%/$12.00 Copyright PfI SO370-2693(96)00959-S
University, Department
of Theoretical
Physics,
0 1996 Elsevier Science B.V. All rights reserved.
195251 St. Petersburg,
Russian Federation
M. Faber et al./ Physics Letrers B 386 (1996) 198-206
199
monopole-antimonopole pairs can be condensed, created and destroyed. This does not violate magnetic charge conservation and leaves the magnetic U( 1) symmetry unbroken as well. Such an unbroken U( 1) symmetry in the condensed phase distinguishes the MNJL model from the BCS-theory of superconductivity, where the group of U( 1) electric charge transformations gets broken due to condensation of electron-electron Cooper pairs. In the MNIL model we assume that the massless magnetic monopoles, described by the fermion field x(x) , interact via a local four-monopole interaction of the NIL kind given by [ 31 (see also [ 41) CMNJL(X)
[x(x)r~x(x)l[X(x)r~X(x)l,
=C[f,(d,+)12-G1
(1)
where G and Gt are positive phenomenological constants responsible for the magnetic monopole condensation and a dual-vector field mass, respectively. The effective Lagrangian ( 1) is invariant under the U( 1) group connected with magnetic monopole charge conservation. Due to strong attraction in the lx-channels produced by local four-monopole interactions (1) the monopoles and antimonopoles become unstable under monopole-antimonopole condensation, i.e., (Xx) #O. Indeed, in the condensed phase the energy of the ground state is negative, i.e.,
[(Xx)12<09
W=-(~~NJL(X))=-(%G+G,)
(2)
whereas in the non-condensed phase, when (xx) = 0, we have W = 0. This means that the condensed phase is much more advantageous to the monopole-antimonopole system, described by the Lagrangian ( 1) . Monopole-antimonopole pairs become condensed without breaking of magnetic U( 1) symmetry as well. In the condensed phase magnetic monopoles acquire a mass M that satisfies the gap-equation [ 3,4]
M = -2G(x(O),y(O)) =
-$J, (W
-
z 1 $!A
=-z[h2--M21n(l+$)],
(3)
where A is the ultra-violet cut-off defining a quadratically divergent integral. The magnetic monopole condensation accompanies the creation of collective Xx-excitations with the quantum numbers of a scalar Higgs meson field (+ and a dual-vector field C, [ 3,4]. The dual-vector field C, is massive, it is created by the four-monopole interaction proportional to the coupling constant Gt and acquires the mass the main contribution to which is of order O( l/G,) [ 3,4]. Since the C,-field is created massive, it possesses both longitudinal and transverse components from the very beginning. Therefore, the C,-field does not need magnetic U( 1) symmetry breaking to acquire a longitudinal component in terms of a Goldstone boson. The effective Lagrangian obtained after the integration over monopole degrees of freedom and involving quarks, antiquarks and the fields of collective scalar u and dual-vector C, excitations reads [ 3,4] C,,(X)
= &(x)P”(x)
+ $4~C,(x)Cc”(x)
+ $@(X)d%(X)
where M, = 2M and MC arethe masses of the IJ and C, fields [3,4], such as MC is given by [3,4]
M: = g
-&M(M)
where 52 (M) is a logarithmically J2(M)
=
(5)
+M2J2(M)],
I
J
d”k v2i
1 ( M2 - 12~)~
divergent
integral A2 M2+h2’
(6)
M. Faber et al. /Physics Letters B 386 (1996) 198-206
200
The coupling
constants
g and K obey the constraint
[ 3,4]
&J*(M)=$Jz(M) =I giving the relation
K’
=
&,, quark(X)= -
2g2/3
c
[
mi
i=q.rj
(7)
1] . Cfreequark(x) is a kinetic term for quark and antiquark
dXY(r) dXT(r)
J( dr
-$-y
l/2
---&--g~Y
aC4’(x-
X,(r)) I
(8)
’
We consider quark and antiquark as classical point-like particles with masses mq = rnq = m, electric charges Qq = -Qa = Q, and trajectories Xi(r) and Xi(r), respectively. The electric quark current J”(x) is given by
J”(X)
-Xi(T)).
= CQiJdTyG’“‘(.X
(9)
i=q,q
The field strength F@“(x) is defined [3,5] Fp”(x) = fp”(x) - *dCfi”(x), where dCfiCLY(x)= ~PC”(~) P’Cp(x), and *dCfiLY(x) is a dual version, i.e., *dCpY(x) = ~.Y’a~dCap(x)(~0123 = 1). The electric field strength &p”(x), induced by a dual Dirac string, is defined following [3,5-71 I@“(x)
=
JJ
d2u8(4)(x - X)ap’(X),
Q
-
(10)
where we have denoted d2v = dada and gw(x)
-
dXpax” dX”axp, a(T
a7
a7
(11)
au
Here Xp = X+( r, (+) represents the position of a point on the world sheet swept by the string. The sheet is parametrized by the internal coordinates -cc < r < oc and 0 I u I 7~, so that X@(r, 0) = X/le (7) and Xp(r, n-) = Xc(r) represent the world lines of an antiquark and a quark [5]. Within the definition (5) the tensor field Z@“(x) satisfies identically the equation of motion, i.e., d,PV( x) = J’(x), the first pair of equations of motion of Dirac’s extension of Maxwell’s Electrodynamics. This means that the inclusion of a dual Dirac string in terms of 8@““(x) defined by ( 10) saturates completely the electric Gauss law. In the presence of a dual Dirac string ended by a quark and an antiquark, which are considered as classical point-like particles with masses my and mg, respectively, the massive dual-vector field C, has the shape of a dual Abrikosov flux line [3,5-71 C’[E(x)]
= -
J
d4x’A(x
(12)
- x’)J;&p”“(x’),
where A(x - x’) is the Green function A(x-x’)=
J
&k
e-ik,(x-x’)
(2~)~ h4; - k2 - i0’
The vacuum expectation values (v.e.v.) of time-ordered monopole field, i.e., the monopole Green function G(Xl) . . . ,A)
=
(13) products of densities expressed in terms of the massless-
(OlT(,V(xl)~l~(x~).. .R(~n>r~~(xn))IO)conn.,
(14)
M. Faber et al./Physics
Letters B 384 (1996) 198-206
201
where Ii(i = 1,. . . , n) are the Dirac matrices, taken in the tree-approximation depending on the string shape are given by [ 3,4]
x
exp i .1’d4x{
- g~M(X)r”XM(X)CY[&(X)l
- KjddXdddx)})
of the a-field
exchange
lo)::;,.
and
(15)
Then 10)‘M) is the wave-function of the nonperturbative vacuum in the condensed phase. The NJL model [ 81 as well as the BCS-theory of superconductivity [9] admits the exact solution of IO)(M) in the form [ 8,101
JO)(M) =,E, where fiP = p/E,
[@+ = p/dm
A~~u’o’t(p,A)b~o)‘(-p,*)] IO), is the velocity of massive monopoles
t
with the mass M, and a(“+(p,
16)
A) (or
b’O)+( -p, A)) denotes the creation operator of a massless monopole (or antimonopole) with the momentum p and helicity A; IO) E [O)(O) is the wave-function of the vacuum in the non-condensed phase. It is seen that the wave-function IO)(M) is invariant under magnetic U( 1) symmetry transformations. This means that in the condensed phase magnetic (I( 1) symmetry is not broken. Eq. (16) contains the factor (-A) [ 101 that has been lost in [ 81. The availability of this factor is very important for the correct behaviour of the wave-function of the nonperturbative vacuum (0)cM) under parity transformations. Indeed, since parity is conserved the wave-function IO)(W should be invariant under parity transformations P, i.e., PIO)““” = IO)‘M’ [ 111. By using the operators a”‘+(p, A) and bco)+( -p. A) one can construct two operators 0,
=2x
(3_ =2x
c Ab’“‘+(-p,A)a’o’t(p,A), p A=*1
c b’“)+(-p,A)a’o’+(p,A), p A=fl
(17)
possessing different properties under parity transformations, i.e., PO+Pt = Cl, and PO-P+ = --CL. Following the BCS-theory of superconductivity [9] we can identify the v.e.v. of the O+-operator per unit volume with an order parameter (0,)
=;
(“)(OIO+IO)(M) = -;
c
c p
d4p -4M, / =
(2r)4i
-$w)
A2dq=
-4Mx
A=il
2EPV
= -4M
J d3p (27r)32Ep
=
1 M2 -p2 - i0
=(/f(o)xto)),
(18)
where V is a normalization volume. Thus we have got (0,) the parity conservation in the MNJL model.
2. Magnetic monopole
1 p
condensate
= (j(O)x(O)).
In turn (0_)
= 0. This implies
around a dual Dirac string
In order to calculate the magnetic monopole condensate turn to Eq. (10) and consider the following v.e.v.
depending
on the dual Dirac string shape we should
M. Faber et al./Physics
202
Letters B 386 (1996) 198-206
b Fig. I. One-monopole loop diagrams describing a magnetic monopole condensate around a dual Dirac string. (X(x)x(x);E)
=
‘“)(OIT(X~W~~(X)
-g~‘M(Z)YYX~M(Z)C,[E(Z)l -
x expi/d4Z(
KX~(Z)~~(Z)~(Z)})lO)~~d.
The matrix element (T( X)X( x) ; &), calculated in the tree-approximation of the g-field described by the diagrams depicted in Fig. 1. The contribution of the diagram in Fig. la coincides with (1(0)x(O)), i.e., (f(X>,A’(X);
The contribution
&)Fig. la = (X(o>~(o)).
of the diagrams
(19) exchange,
can be
(20)
in Fig. lb is defined by the expression
(21) By using the relations 1
/
K2JI(M) - 2MJ2(W = Mi
479 Therefore,
M, = 2M and Eq. (7) one can show that
we get
--$(xmx(o,)
(22)
M. Faber et al./Physics
J { 4
%r
x (-1)”
X
r2i
Letters B 386 (1996) 198-206
I
-&dm)l
&?m(X2)1
M-L-L,
203
. . .
1 M-I;-,&,
(23)
-...-I,
Neglecting the momenta of the dual-vector fields with respect to the magnetic monopole where e = 1,. . . , n, one can reduce the r.h.s. of (23) to the following closed expression:
(,?(x)x(x);
_ l;i2
&)Fig. lb = &
(X(0)x(O)) M2
J $tr{
(x(o~‘o))
J-i d4k r2i
J-t
(24)
M2 - (k - gC[&(x)])2
d”k
=
- L} M-i
1
The integration over k we should perform by applying suggested by Gertsein and Jackiw [ 121 we get
n-*i
M _ R + :C,E(x)l
mass, i.e., M >> jktl
the cut-off regularization
[3,4].
Using
the technique
1 M2 - (k - gC[E(x)])2
- M2
1 l-~ -gC,[&(x)]-& J$!{ev(' p M2 - k2 M2-k2
= ;g2c,[E(x)]cp[E(x)].
(25)
For the more detailed computation of integrals like (25) we refer readers to Ref. [ 121. The contribution of the diagram in Fig. lb reads
(26) Summing the contributions of the diagrams in Fig. 1 we obtain the total expression condensate depending on the shape of a dual Dirac string
(X(x)x(x);E) = ~/i3o)xto))
[
I+
~~g2c~,E~x~lc~,t(x),].
of the magnetic
monopole
(27)
Since C, [E(x) ] Cp [ I( n) ] < 0 it is seen that at distances close to a dual Dirac string the magnetic monopole condensate is suppressed. On the other hand, at large distances far from a string where the influence of a string is exponentially suppressed due to the Meissner effect, the magnetic monopole condensate (x(x)x(x) ; E) tends to the magnitude of the order parameter, i.e., @(x)x(x);&) -+ (jj(O>x(O)). As an example, let us consider the magnetic monopole condensate around a static infinitely-long dual Dirac string strained along the z-axis. In this case the induced electric field reads [ 131: E(r) = e,Q$‘)(r), where r is the radius-vector in the plane perpendicular to the z-axis and e, is the unit vector directed along the z-axis. A dual-vector potential, taken in the temporal gauge CO(X) = 0, acquires the form [ 131 C(r)
= L$L
C,(r).
(28)
M. Faber et al./Physics
204
The azimuthal Ca(r)
component
Pp(r)
=
where K, (Mcr)
=
is given by
QMc
,K1(Mcr),
2n
is the McDonald
s
d2k
9(r)
= --
dr
2~
C,(r)
GM2
&k.r C
+k2 =
Letters B 386 (1996) 198-206
function. co
The scalar function
dkkJo( kr)
s 0
M;+k2
(29) po(r) is defined by
= Ko(Mcr),
(30)
where Jo( kr) and KO(Mcr) are Bessel and McDonald functions, respectively, The prime stands for the derivative with respect to the argument. Substituting (29) in (27) and using the relation Qg = 2~ we obtain
(t(r)A”(r);E) = (~
1-
(
If
= -K1 ( Mcr).
&$f(M,r)].
(31)
This is the magnetic monopole condensate around a static infinitely-long can be calculated by analogy with (27). The v.e.v. (j,(x)jp(x>;E) expounded above one obtains
x
and KA( Mcr)
dual Dirac string. Following the procedure
having
been
2
&g2cp1EwlcP[a(+)l
>
.
(32)
It is seen that the v.e.v. (j,(x) jb( x) ; E) is suppressed for distances close to a dual Dirac string. This also agrees with the results obtained within CQED [ 141. For the static infinitely-long dual Dirac string, directed along the z-axis, Eq. (32) reads
(j,(r).?‘(r);E) = -s2[(X(0)x(O))12
1+
e sK:(Mcr)
::
For A 2 2M the last factor can be neglected according to Eq. (3). For a more detailed comparison with CQED we should adduce separately components of (j,(r) jP(r); E)
-(jO(r)jO(r);E)=g2[(~.(0),y(O))12~
1 + g#(&r))
2
1 - ~K,(M,r))2.
(33)
the v.e.v. of the time and spatial
(1- $$Kf(Mcr))2, 2
i(j
.j(r>;E)
=g2[(X(0)x(O))12~
1 + fsKf(Mcr)
>(
GM: 2 1 - i#,(Mcr)
>
.
One can see that the v.e.v. of the time components of the magnetic current has the same suppression in the .j(r);&)as a function of r is vicinity of a Dirac string that the v.e.v. of the spatial ones. The v.e.v. i(j(r) smaller than -(j’(r) j”( r) ; E) for finite values of r and goes to the same magnitude at r --+ m. This agrees qualitatively with the numerical data obtained on the lattice within CQED [ 141. Unfortunately, one cannot carry out a quantitative comparison of - (j” ( r ) j” ( r) ; E) and i (j(r) .j( r) ; E) with the numerical data obtained on the lattice [ 141. First, this is connected with a singular behaviour of the McDonald function Kf (Mcr) in the region close to the string, and the necessity to fit three parameters MC, it4 and G. The former is hardly unambiguous
M. Faber et al./Physics
Letters B 386 (1996) 198-206
205
procedure. Second, in addition to this reason there is another one concerning the principle distinction of our calculations from the lattice numerical data. Indeed, the v.e.v. -(j”( r)j”( r) ; E) and i(j( r) . j(r) ; E) have been computed on the lattice by taking into account automatically string fluctuations. This is not done here and goes beyond the scope of this letter. Therefore, we can restrict ourselves by the discussion of a qualitative comparison only. We can compute the v.e.v. of any operator constructed from massless monopole fields. For example, the v.e.v. computed with analogy to those given above reads (X(X)~/SX(X)X(X)y5X(X);&)
(35) The v.e.v. (~(x)r5~(x)~(x)yS~(x);I) h as not been calculated within CQED up to now. Therefore, our result (35) should be considered as a prediction to be compared with numerical experiments on lattice within CQED.
3. Conclusion We have suggested a consistent procedure for the computation of vacuum expectation values of any magnetic monopole densities in dependence on the shape of a dual Dirac string. The investigation of the magnetic monopole condensate around a dual Dirac string has shown that the influence of a dual Dirac string is reducing to the suppression of the magnetic monopole condensate at distances close to the string. In turn, at distances far from the string, where the influence of the string is exponentially suppressed due to the Meissner effect, the contribution of the string decreases, and at infinitely-large distances the magnitude of the magnetic monopole condensate tends to the magnitude of the order parameter, i.e., (0,) = (X(0)x(O)). A similar influence of a dual Dirac string on the magnitude of the magnetic monopole condensate has been observed within Compact Quantum Electrodynamics (CQED) [ 141.
Acknowledgements We thank the Fonds zur Fiirderung der wissenschaftlichen Forschung in Gsterreich (project PI 1156-PHY) for their support. We acknowledge fruitful discussions with Prof. G. E. Rutkovsky. The discussions with Dr. M. Zach concerning comparison of our results with numerical data on lattice are appreciated.
References [ I] P. Becher and H. Joos. Z. Phys. C 15 (1982) 343. [2] V. Singh, D. Browne and R. Haymaker, Phys. Rev. D 47 ( 1993) 1715. 131M. Faber, A.N. lvanov, W. Kainz and N.I. Troitskaya, Nambu-Jona-Lasinio approach to magnetic monopole physics with dual Dirac strings, (submitted to Z. Phys. C). 141A.N. Ivanov, M. Nagy and N.I. Troitskaya, Int. J. Mod. Phys. A 7 (1992) 7305; A.N. Ivanov, Int. J. Mod. Phys. A 8 (1993) 853; A.N. lvanov, N.I. Troitskaya and M. Nagy, Int. J. Mod. Phys. A 8 (1993) 2027; A 8 (1993) 3425; B 308 (1993) 111; B 295 (1992) 308; A.N. Ivanov and N.I. Troitskaya, Nuovo Cim. A 108 (1995) 555. 151 M. Faber, W. Kainz, A.N. Ivanov and N.I. Troitskaya, Phys. Lett. B 344 ( 1995) 143, and references therein. [6 1 P.A.M. Dirac, Proc. Roy. Sot. A 133 (1931) 60; Phys. Rev. 74 (1948) 817. [ 71 Y. Nambu. Phys. Rev. D 10 (1974) 4246.
206 [ 8 1 Y. Nambu and Cl. Jona-Lasinio,
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Letters B 386 (1996) 198-206
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246.
[ 91 J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162; 108 ( 1957) 1175. [ IO] A.N. Ivanov, in Lectures on Nonperturbative Phenomena of QCD, Part 2, Institut fur Kemphysik, Technische Universmit Wien, October 1995-February 1996. [II] C. Itzykson and J.-B. Zuber, Quantum Field Theory ( Mcgraw-Hill), 1980. I121 13. Gertsein and R. Jackiw, Phys. Rev. 181 (1969) 1955. [ 131 M. Faber, A.N. Ivanov, W. Kainz and N.I. Troitskaya. Dual Higgs model with dual Dirac strings, ESI-Preprint No. 286, November 1995 (to be published in Nucl. Phys. B). 1141 M. Zach, M. Faber, W. Kainz and P. Skala, Phys. Lett. B 358 (1995) 325.