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Colour confinement as dual Meissner effect: SU(2) gauge theory
27 July 1995 PHYSICS ELSl3VIER
LETTERS
B
Physics Letters B 355 (1995) 255-259
Colour confinement as dual Meissner effect: W(2) theory * L. Del Debbio, A. Di Giacomo,
gauge
G. Paffuti, P. Pieri
Dipartimento di Fisica dell’Universitti and LNXN., I-56126 Piss, Italy
Received 11 May 1995 Editor: R. Gatto
Abstract We demonstrate that confinement in SU(2) gauge theory is produced by dual superconductivity of the vacuum. We show that for T < Tc (temperature of the deconfining phase transition) the U( 1) symmetry related to monopole charge conservation is spontaneously broken; for T > K the symmetry is restored.
1. Introduction Dual superconductivity of the vacuum has been advocated as the mechanism for confinement of colour [ l-31 : the chromoelectric field is channeled into Abrikosov [4] flux tubes, producing a static potential proportional to the distance between 44 pairs. Magnetic charges, defined as Dirac monopoles of a residual U( 1) symmetry selected by a suitable gauge fixing (abelian projection), should accordingly condense in the vacuum, in the same way as Cooper pairs do in an ordinary superconductor. Evidence has been collected by numerical simulations of the theory on the lattice, that such monopoles do exist, and that their number density is correlated with the deconfining phase transition: we refer to [ 51 for a review of these results. For a recent updating we refer to [ 61. However a direct demonstration that confinement is produced by monopole condensation is still lacking, “Partially supported by MURST and by EC Contract CHEXCT92-0051. Elsevier Science B.V. SSDI 0370-2693(95)00702-4
In fact monopole condensation means that the ground state of the system is a superposition of states with different magnetic charges, or that the dual (magnetic) U( 1) symmetry is spontaneously broken, in the same way as the electric U( 1) .symmetry is spontaneously broken in a ordinary superconductor [ 71. Such a breaking is monitored by the nonvanishing of the vacuum expectation value (VEV) of any operator ,X with nontrivial magnetic charge. (p) is called a disorder parameter: it is nonzero in the broken phase, and vanishes in the ordered phase, at least in the thermodynamical limit V --+ cm. The construction of a disorder parameter for dual superconductivity has been presented in [ 81 where it has been successfully tested on compact U( 1) gauge theory. In this paper we use the same construction to probe the vacuum condensation of the monopoles defined by abelian projection in SU(2) gauge theory. We find that the abelian projection which diagonalizes the Polyakov line [9] identifies monopoles which condense in the confined phase and do not in the deconfined one.
256
L. Del Debbio et al. /Physics Letters B 355 (1995) 255-259
Monopoles defined by the abelian projection which diagonalizes a component of the field strength [ 91 do not show any signal of condensation correlated with confinement. We conclude that: 1) Confinement of colour is related to dual superconductivity of gauge theory vacuum. 2) Not all the abelian projections are equivalent, or define monopoles which condense in the vacuum to confine colour. In Section 2 we recall the basic ideas of the abelian projection, the definition of the corresponding monopoles, and their role in confinement. In Section 3 we present our results, in Section 4 the conclusions.
2. Monopoles in QCD Stable monopoles configurations in gauge theories are related to the first homotopy group II 1 of the gauge group [ lo]. Since IIt [ SU( N) ] is trivial, the symmetry has to break down to some nonsimply connected subgroup, in order to define magnetic charges. In the Georgi-Glashow [ 1 l] model with gauge group SU(2) coupled to a scalar field @J’ in the adjoint representation a spontaneous breaking SU( 2) -+ U( 1) allows to define an abelian field strength [ 121 fp,, = &‘G;,
-
-$ab,&a(D,&)b(D,8)C
(1)
which admits stable monopole configuration [ 12,131, behaving as Dirac monopoles at large distances. In Eq. (1) 6a = @‘/]@I. Putting up =
&At
Eq. (5) is the usual expression of the electromagnetic field strength in terms of the (abelian) potential up. The choice of the gauge Eq. (4)) which is defined up to an arbitrary gauge rotation around the third axis, is called an abelian projection. The strategy for relating confinement of colour to dual superconductivity in QCD is to make a guess of a possible effective Higgs field 6, belonging to the adjoint representation, and then perform the abelian projection (4) and look for condensation of the corresponding Dirac monopoles. In most of the lattice investigations on the problem, the density of monopoles, or quantities related to it, have been studied: of course the density of magnetic charges is not a disorder parameter for dual superconductivity, in the same way as the number density of electrons is not for ordinary superconductivity: a nonvanishing VEV of an operator with nontrivial charge is needed, while the density of charge commutes with total charge operator (which is in fact neutral). We will instead make use of a genuine disorder parameter that we have constructed and checked on the U( 1) gauge theory [ 81. Another question is if all abelian projections are physically equivalent, a possibility suggested in [ 91. We will study the projection in which Q, is the Polyakov line, and the one in which 4 is any component of the field strength G,, [ 91. For reasons which will be explained in Section 3, we have technical difficulties (computing power) to explore the so-called maximal abelian gauge [ 51.
3. The disorder parameter:
results.
As for the U( 1) case [ 81 we define the operator which creates a monopole at the point z and time za
one has [ 141
d3yfoi(y, fpu in Eq. ( 1) is gauge invariant, since both 6 and GcL,,are gauge covariant. up in Eq. (2) is not. In the gauge in which &,a = s(;
numerical
(2)
(4)
zo)bi(Y-
1
z>
(6)
where fci is the electric field strength (1) and hi/g is the vector potential produced by a Dirac monopole, with the Dirac string subtracted. b( r ) is given by b(r)
=
i-An r(r - r. n)
(7)
Eq. (3) becomes (5)
if the gauge is chosen in such a way that the string singularity is in the direction n. The equal time com-
L. Del Debbio et al./Physics
mutator between the vector potential ai (Eq. (2) ) and
Letters B 355 (1995) 255-259
257
and g is now subjected to the constraint
.fOi is
=ii%j~33(x -y>
[ak(x,~~)~.foj(y,x~)]
(8)
as in the U( 1) gauge theory fai is the conjugate momentum to ai, and as a consequence ,u (Eq. (6) ) is an operator which translates the field ai ( X) by bi ( x 2)/g. A proper definition
DA, exp [-PSI
J’
(pu) =
of the VEV of p is [ 81
I
d3yg2(y)
=
s
d3y MY - d) - MY) I*
Instead of (p) we will measure P =
$ ln(pu>= (S + S,)S+S, -
(S + &)s+&
(17)
In terms of p
~(z, z”) (18)
(9)
s
(16)
DA, exp [-PSI
r(z’) For b6
where y( z”) is a translation of the field ai by a time independent g(x) with V A g = 0
d3xfidy, subjected
d3Xb2(X)
=
(11)
d3Xg2(X)
After Wick rotation,
s
DA,exp
&)I
(12) DA, exp [-KS
+ s,)]
with
Sb(X,XO)= &(x,
s s
d3yMy -
x0)
x)foi(y,
x0> x0>= d3Ygi(Y)foi(Y,
(134 (13b)
Similarly the correlation function can be defined of any number of monopoles and antimonopoles: for example for a pair m, 5 at equal time and distance d
s s
DA, exP [-PV + %J]
b44Pu(O)) =
( 14)
DA,exp
sbl;=
J
d3yfoi(wo)
condensation
(14 # 0 or by
[-PCS+
[My-d)
s,)]
-My)1
In terms of p the cluster property, Eq. (20) reads
(21)
Eq. (9) can be written [-P(S+
If there is monopole cluster property
(20)
s
s
(10)
to the constraint
J
(lu> =
1
z')a(y)
(19)
(15)
We have measured p for a single monopole and for an m , ti pair at different distances, across the deconfining phase transition of an SU( 2) gauge theory. The abelian projection in the gauge which diagonalizes the Polyakov line gives a clear signal of condensation: Fig. 1 shows p for a 123 x 4 lattice; Fig. 2 for a 163 x 6 lattice. A clear signal is observed of transition from superconductivity to normal vacuum at /3 = PC_ The (known) deconfining /3= for the two lattices (NT = 4, NT = 6) is indicated by the vertical lines in Figs. 1 and 2. In Fig. 3 pb6 of a miiz pair at distance d = 10 lattice spacing is compared to 2. pb, corresponding to a single monopole, checking successfully Eq. (21). No signal is observed in the abelian projection which diagonalizes a component (say Fi2) of the field strength. A few points about the lattice version of the approach. fci of Eq. (6) is defined by Eq. ( 1). For the abelian projection in which & is the direction of log of the Polyakov loop,L, the second term in Eq. ( 1) is absent when p or v take the value 0, since DOL. = 0.
L. Del Debbie et al. /Physics
258
Letters B 355 (1995) 25.5-259
0.0
.
0.0
*
: . .
-100.0
i /
-100.0
-200.0
+j P
P
+ I*
-200.0
-300.0
-300.0 -400.0
-400.0
1.0
2.0
3.0
-500.0
4.0
i 1.0
2.0
Fig. 1. p versus p on a 123 x 4 lattice. The vertical line denotes & corresponding to the deconfining transition.
4.0
3.0
P
5.0
P
Fie. 2. D versus L3 on a 163 x 6 lattice. The vertical line denotes PC corresponding to the deconfining transition.
Then constructing fei is simply a projection of Gsi on the direction of L: of course on the lattice Gai can be taken as the imaginary part of the plaquette IIai. For the abelian projection in which @ = lnIIt2 the second term of Eq. ( 1) is not zero, but is computable. In the case of the so-called “maximal abelian gauge” [ 51, in which the gauge is fixed by maximizing the quantity
-I----------
0.0
-200.0
-
-400.0
-
-600.0
-
-800.0
’ 0.0
1
P
the effective Higgs 6 to introduce in Eq. (1) is not known explicitly, but must be determined by the maximization on each configuration. This is a serious problem from the numerical point of view, since at each change of the configuration in the updating procedure to compute p by Eq. (19) the maximization must be repeated to determine foi and sb. A detailed finite size scaling analysis to extract the thermodynamical limit from our data is under study.
’ 2.;
’ 4.0
’ 6.0
’ 8.0
’ 10.0
B
Fig. 3. pb$(d) at d = 10 (circles) and their difference (squares).
compared
to 2p (triangles)
L. DeZ Debbio et al. /Physics
Letters 3 355 (1995) 255-259
259
Fig. 4. p for the abelian projection diagonalising Fra on g3 x 4 (dots) and 123 x 4 (triangles) lattices.
4. Conclusions We conclude
that:
1) gauge theory vacuum is a dual superconductor:
2)
the monopoles defined by the abelian projection diagonalizing the Polyakov line do condense in the confined phase, and the corresponding dual U( 1) symmetry is restored in the deconfined phase. Not all abelian projections are physically equivalent: the monopoles in the gauge in which the field strength is diagonal are irrelevant to confinement.
References [ 1 J G. ‘t Hooft, in: High Energy Physics, EPS International Conference, Palermo 1975, ed. A. Zichichi. [2] S. Mandelstam, Phys. Rep. C 23 (1976) 245. [ 31 G. Parisi, Phys. Rev. D 11 ( 1975) 970.
[4] A.B. Abrikosov, JETP 5 (1957) 1174. [S] T. Suzuki, NucI. Phys. B 30 (Proc. Suppl.) (1993) 176. 161 T. Suzuki, S. Ryan, Y. Matsubara, T. Okuda and K. Yotsuji, Phys. I.&t. B 347 (1995) 375. [7] S. Weinberg, Progr. of Theor. Phys. Suppl. No. 86 (1986) 43. [S] L. Del Debbio, A. Di Giacomo and G. Paffuti, Phys. Lett. B; Proceedings LATTICE 94, Bielefeld Sept. 94. [9] G. ‘t Hooft, Nucl. Phys. B 190 (1981) 455. [lo] S. Coleman, in: Erice Lectures 1975, ed. A. Zichichi. [ 1 l] H. Georgi and S. Glashow, Phys. Rev. Lett. 28 (1972) 1494. [12] G. ‘t HooR, Nucl. Phys. B 79 (1974) 276. [ 131 A.M. Polyakov, JEPT Lett. 20 (1974) 894. [ 141 J. Arafune, PG.0. Freund and G.J. Goebel J. Math. Phys. 16 (1974) 433.
LETTERS
B
Physics Letters B 355 (1995) 255-259
Colour confinement as dual Meissner effect: W(2) theory * L. Del Debbio, A. Di Giacomo,
gauge
G. Paffuti, P. Pieri
Dipartimento di Fisica dell’Universitti and LNXN., I-56126 Piss, Italy
Received 11 May 1995 Editor: R. Gatto
Abstract We demonstrate that confinement in SU(2) gauge theory is produced by dual superconductivity of the vacuum. We show that for T < Tc (temperature of the deconfining phase transition) the U( 1) symmetry related to monopole charge conservation is spontaneously broken; for T > K the symmetry is restored.
1. Introduction Dual superconductivity of the vacuum has been advocated as the mechanism for confinement of colour [ l-31 : the chromoelectric field is channeled into Abrikosov [4] flux tubes, producing a static potential proportional to the distance between 44 pairs. Magnetic charges, defined as Dirac monopoles of a residual U( 1) symmetry selected by a suitable gauge fixing (abelian projection), should accordingly condense in the vacuum, in the same way as Cooper pairs do in an ordinary superconductor. Evidence has been collected by numerical simulations of the theory on the lattice, that such monopoles do exist, and that their number density is correlated with the deconfining phase transition: we refer to [ 51 for a review of these results. For a recent updating we refer to [ 61. However a direct demonstration that confinement is produced by monopole condensation is still lacking, “Partially supported by MURST and by EC Contract CHEXCT92-0051. Elsevier Science B.V. SSDI 0370-2693(95)00702-4
In fact monopole condensation means that the ground state of the system is a superposition of states with different magnetic charges, or that the dual (magnetic) U( 1) symmetry is spontaneously broken, in the same way as the electric U( 1) .symmetry is spontaneously broken in a ordinary superconductor [ 71. Such a breaking is monitored by the nonvanishing of the vacuum expectation value (VEV) of any operator ,X with nontrivial magnetic charge. (p) is called a disorder parameter: it is nonzero in the broken phase, and vanishes in the ordered phase, at least in the thermodynamical limit V --+ cm. The construction of a disorder parameter for dual superconductivity has been presented in [ 81 where it has been successfully tested on compact U( 1) gauge theory. In this paper we use the same construction to probe the vacuum condensation of the monopoles defined by abelian projection in SU(2) gauge theory. We find that the abelian projection which diagonalizes the Polyakov line [9] identifies monopoles which condense in the confined phase and do not in the deconfined one.
256
L. Del Debbio et al. /Physics Letters B 355 (1995) 255-259
Monopoles defined by the abelian projection which diagonalizes a component of the field strength [ 91 do not show any signal of condensation correlated with confinement. We conclude that: 1) Confinement of colour is related to dual superconductivity of gauge theory vacuum. 2) Not all the abelian projections are equivalent, or define monopoles which condense in the vacuum to confine colour. In Section 2 we recall the basic ideas of the abelian projection, the definition of the corresponding monopoles, and their role in confinement. In Section 3 we present our results, in Section 4 the conclusions.
2. Monopoles in QCD Stable monopoles configurations in gauge theories are related to the first homotopy group II 1 of the gauge group [ lo]. Since IIt [ SU( N) ] is trivial, the symmetry has to break down to some nonsimply connected subgroup, in order to define magnetic charges. In the Georgi-Glashow [ 1 l] model with gauge group SU(2) coupled to a scalar field @J’ in the adjoint representation a spontaneous breaking SU( 2) -+ U( 1) allows to define an abelian field strength [ 121 fp,, = &‘G;,
-
-$ab,&a(D,&)b(D,8)C
(1)
which admits stable monopole configuration [ 12,131, behaving as Dirac monopoles at large distances. In Eq. (1) 6a = @‘/]@I. Putting up =
&At
Eq. (5) is the usual expression of the electromagnetic field strength in terms of the (abelian) potential up. The choice of the gauge Eq. (4)) which is defined up to an arbitrary gauge rotation around the third axis, is called an abelian projection. The strategy for relating confinement of colour to dual superconductivity in QCD is to make a guess of a possible effective Higgs field 6, belonging to the adjoint representation, and then perform the abelian projection (4) and look for condensation of the corresponding Dirac monopoles. In most of the lattice investigations on the problem, the density of monopoles, or quantities related to it, have been studied: of course the density of magnetic charges is not a disorder parameter for dual superconductivity, in the same way as the number density of electrons is not for ordinary superconductivity: a nonvanishing VEV of an operator with nontrivial charge is needed, while the density of charge commutes with total charge operator (which is in fact neutral). We will instead make use of a genuine disorder parameter that we have constructed and checked on the U( 1) gauge theory [ 81. Another question is if all abelian projections are physically equivalent, a possibility suggested in [ 91. We will study the projection in which Q, is the Polyakov line, and the one in which 4 is any component of the field strength G,, [ 91. For reasons which will be explained in Section 3, we have technical difficulties (computing power) to explore the so-called maximal abelian gauge [ 51.
3. The disorder parameter:
results.
As for the U( 1) case [ 81 we define the operator which creates a monopole at the point z and time za
one has [ 141
d3yfoi(y, fpu in Eq. ( 1) is gauge invariant, since both 6 and GcL,,are gauge covariant. up in Eq. (2) is not. In the gauge in which &,a = s(;
numerical
(2)
(4)
zo)bi(Y-
1
z>
(6)
where fci is the electric field strength (1) and hi/g is the vector potential produced by a Dirac monopole, with the Dirac string subtracted. b( r ) is given by b(r)
=
i-An r(r - r. n)
(7)
Eq. (3) becomes (5)
if the gauge is chosen in such a way that the string singularity is in the direction n. The equal time com-
L. Del Debbio et al./Physics
mutator between the vector potential ai (Eq. (2) ) and
Letters B 355 (1995) 255-259
257
and g is now subjected to the constraint
.fOi is
=ii%j~33(x -y>
[ak(x,~~)~.foj(y,x~)]
(8)
as in the U( 1) gauge theory fai is the conjugate momentum to ai, and as a consequence ,u (Eq. (6) ) is an operator which translates the field ai ( X) by bi ( x 2)/g. A proper definition
DA, exp [-PSI
J’
(pu) =
of the VEV of p is [ 81
I
d3yg2(y)
=
s
d3y MY - d) - MY) I*
Instead of (p) we will measure P =
$ ln(pu>= (S + S,)S+S, -
(S + &)s+&
(17)
In terms of p
~(z, z”) (18)
(9)
s
(16)
DA, exp [-PSI
r(z’) For b6
where y( z”) is a translation of the field ai by a time independent g(x) with V A g = 0
d3xfidy, subjected
d3Xb2(X)
=
(11)
d3Xg2(X)
After Wick rotation,
s
DA,exp
&)I
(12) DA, exp [-KS
+ s,)]
with
Sb(X,XO)= &(x,
s s
d3yMy -
x0)
x)foi(y,
x0> x0>= d3Ygi(Y)foi(Y,
(134 (13b)
Similarly the correlation function can be defined of any number of monopoles and antimonopoles: for example for a pair m, 5 at equal time and distance d
s s
DA, exP [-PV + %J]
b44Pu(O)) =
( 14)
DA,exp
sbl;=
J
d3yfoi(wo)
condensation
(14 # 0 or by
[-PCS+
[My-d)
s,)]
-My)1
In terms of p the cluster property, Eq. (20) reads
(21)
Eq. (9) can be written [-P(S+
If there is monopole cluster property
(20)
s
s
(10)
to the constraint
J
(lu> =
1
z')a(y)
(19)
(15)
We have measured p for a single monopole and for an m , ti pair at different distances, across the deconfining phase transition of an SU( 2) gauge theory. The abelian projection in the gauge which diagonalizes the Polyakov line gives a clear signal of condensation: Fig. 1 shows p for a 123 x 4 lattice; Fig. 2 for a 163 x 6 lattice. A clear signal is observed of transition from superconductivity to normal vacuum at /3 = PC_ The (known) deconfining /3= for the two lattices (NT = 4, NT = 6) is indicated by the vertical lines in Figs. 1 and 2. In Fig. 3 pb6 of a miiz pair at distance d = 10 lattice spacing is compared to 2. pb, corresponding to a single monopole, checking successfully Eq. (21). No signal is observed in the abelian projection which diagonalizes a component (say Fi2) of the field strength. A few points about the lattice version of the approach. fci of Eq. (6) is defined by Eq. ( 1). For the abelian projection in which & is the direction of log of the Polyakov loop,L, the second term in Eq. ( 1) is absent when p or v take the value 0, since DOL. = 0.
L. Del Debbie et al. /Physics
258
Letters B 355 (1995) 25.5-259
0.0
.
0.0
*
: . .
-100.0
i /
-100.0
-200.0
+j P
P
+ I*
-200.0
-300.0
-300.0 -400.0
-400.0
1.0
2.0
3.0
-500.0
4.0
i 1.0
2.0
Fig. 1. p versus p on a 123 x 4 lattice. The vertical line denotes & corresponding to the deconfining transition.
4.0
3.0
P
5.0
P
Fie. 2. D versus L3 on a 163 x 6 lattice. The vertical line denotes PC corresponding to the deconfining transition.
Then constructing fei is simply a projection of Gsi on the direction of L: of course on the lattice Gai can be taken as the imaginary part of the plaquette IIai. For the abelian projection in which @ = lnIIt2 the second term of Eq. ( 1) is not zero, but is computable. In the case of the so-called “maximal abelian gauge” [ 51, in which the gauge is fixed by maximizing the quantity
-I----------
0.0
-200.0
-
-400.0
-
-600.0
-
-800.0
’ 0.0
1
P
the effective Higgs 6 to introduce in Eq. (1) is not known explicitly, but must be determined by the maximization on each configuration. This is a serious problem from the numerical point of view, since at each change of the configuration in the updating procedure to compute p by Eq. (19) the maximization must be repeated to determine foi and sb. A detailed finite size scaling analysis to extract the thermodynamical limit from our data is under study.
’ 2.;
’ 4.0
’ 6.0
’ 8.0
’ 10.0
B
Fig. 3. pb$(d) at d = 10 (circles) and their difference (squares).
compared
to 2p (triangles)
L. DeZ Debbio et al. /Physics
Letters 3 355 (1995) 255-259
259
Fig. 4. p for the abelian projection diagonalising Fra on g3 x 4 (dots) and 123 x 4 (triangles) lattices.
4. Conclusions We conclude
that:
1) gauge theory vacuum is a dual superconductor:
2)
the monopoles defined by the abelian projection diagonalizing the Polyakov line do condense in the confined phase, and the corresponding dual U( 1) symmetry is restored in the deconfined phase. Not all abelian projections are physically equivalent: the monopoles in the gauge in which the field strength is diagonal are irrelevant to confinement.
References [ 1 J G. ‘t Hooft, in: High Energy Physics, EPS International Conference, Palermo 1975, ed. A. Zichichi. [2] S. Mandelstam, Phys. Rep. C 23 (1976) 245. [ 31 G. Parisi, Phys. Rev. D 11 ( 1975) 970.
[4] A.B. Abrikosov, JETP 5 (1957) 1174. [S] T. Suzuki, NucI. Phys. B 30 (Proc. Suppl.) (1993) 176. 161 T. Suzuki, S. Ryan, Y. Matsubara, T. Okuda and K. Yotsuji, Phys. I.&t. B 347 (1995) 375. [7] S. Weinberg, Progr. of Theor. Phys. Suppl. No. 86 (1986) 43. [S] L. Del Debbio, A. Di Giacomo and G. Paffuti, Phys. Lett. B; Proceedings LATTICE 94, Bielefeld Sept. 94. [9] G. ‘t Hooft, Nucl. Phys. B 190 (1981) 455. [lo] S. Coleman, in: Erice Lectures 1975, ed. A. Zichichi. [ 1 l] H. Georgi and S. Glashow, Phys. Rev. Lett. 28 (1972) 1494. [12] G. ‘t HooR, Nucl. Phys. B 79 (1974) 276. [ 131 A.M. Polyakov, JEPT Lett. 20 (1974) 894. [ 141 J. Arafune, PG.0. Freund and G.J. Goebel J. Math. Phys. 16 (1974) 433.