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Mechanisms of colour confinement
IUCL[ Ar~ PItYSICS
Nuclear PhysicsB (Proc. Suppl.) 64 (1998) 322-328
ELSEVIER
Mechanisms
PROCEEDINGS SUPPLEMENTS
of c o l o u r c o n f i n e m e n t .
A. Di Giacomo Dipartimento di Fisica and INFN, 2 Piazza Torricelli 56100 Pisa, Italy Evidence is reviewed for dual superconductivity of QCD vacuum as a mechanism for confinement of colour via Meissner effect.
©
1. I n t r o d u c t i o n Colour is confined. The expected abundance of quarks nq in the cosmological standard model is of the order[l] nq ~ 10 -x2 np
(1)
if there is no confinement, np in eq.(1) is the abundance of protons. The experimental upper-limit is presently
nq < 10-27 np
(2)
which results from the analysis of ~ 102 gr of matter by Millikan-like experiments. If QCD is the theory of hadrons its vacuum cannot be the Fock space vacuum of quarks and gluons. In fact QCD vacuum mimics Fock vacuum at short distances. Infrared renormalons in the perturbative expansion, however[2], signal instability. At large distances the coupling constant g becomes large anyway, and only non perturbative formulations, like lattice, are reliable tools of investigation. The word "mechanism" means describing the degrees of freedom relevant for confinement in terms of the known behaviour of some other physical system. Many mechanisms have been proposed for confinement. A class of them rely on stochastic behaviour of the gauge fields at large distances[3]. The most attractive mechanism, however, is based on the idea that QCD vacuum behaves as a dual superconductor of type II[4,5]. By this mechanism the chromoelectric field existing in the space between a heavy q@ pair is channeled into Abrikosov[6] flux tubes by dual Meissner effect (fig, l) 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All fights reserved. PII S0920-5632(97)01082-7
Fig.1 Flux tube between q@pair. The energy is proportional to the distance R
E = aR
(a = string tension)
(3)
and the particles are confined: an infinite amount of energy is necessary to pull them apart from each other. Dual means that the role of electric and magnetic quantities is interchanged with respect to usual superconductors. Latt.~c~e simulations of QCD show, from first principles, that 1) String tension exists. Large Wilson lops W ( R , T) obey area law[7]
W ( R , T)
"~
R,T--+~
exp(-aRT)
(4)
W(R,T) is the parallel transport along a rectangle in Euclidean space time of size T in the time direction and R in a space direction. A theorem states that, when R, T go large with respect the correlation length
W ( R , T) ,.~ e x p ( - V ( R ) T )
(5)
with V(R) the static potential between the particles transported along the loop. Eq.(4), (5) imply confinement (3).
2) Flux tubes exist in the region of space between q and ~[8,9]. Their transverse size is
~ 0.5 3) Flux tubes behave like strings. modes can be studied[12].
Higher
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
4) Not only quark pairs but also particles in higher colour representations are confined[13]. For SU(2) gauge group particles with colour J experience a string tension ad which is related to the value o'1/2 for the fundamental representation by the relation J ( J + 1) a j = 1/2(1/2 + 1) al/2
If #2 > 0 the ground state has #2 _ <3> = T e iqO # 0
Ordinary superconductivity is the spontaneous breaking of the U(1) symmetry related to charge conservation. The ground state of a superconductor is a superposition of states with different charge ( = different numbers of Cooper pairs), and is not U(1) invariant. A non zero vacuum expectation value (vev) (3) of the effective, charged, field which creates a Cooper pair signals charge condensation in the ground state[14]. Dual superconductivity is condensation of monopole charges, and the spontaneous breaking of the U(1) symmetry related to monopole charge conservation will be signalled by vev (3M) of an operator 3M carrying non zero magnetic charge[15]. The questions we shall address are:
D , 3 = e iqe lot` + iq(A# - O,/~)] p
For the benefit of non experts we recall the basic facs of superconductivity. We will refer to the Higgs lagrangian as an effective lagrangian. It is a relativistic version of the Ginzburg-Landau free energy[16] f-. = -1Ft`vFt`v + (Di,3) t (Dt`3) - V ( 3 )
(7)
with Dt` the covariant derivative Dt` = Ot` + iqAt`
At` = At` - 0t`/~
Ft`~ = Ot`A~ - O~At` = Ot`fi~ - O v A .
v(3) =
[(3"3) 2
(9)
(14)
The effective lagrangean reads E% --~1 Ft`vFt`v + V (p) + (0t`p) 2 + q2 p2 ~t`~t` (15) .... and the equations of motion (16)
As a consequence of the spontaneous symetry breaking a current has appeared, jt` = q21312.4t`, which is known as London current. For a stationary configuration E = 0 and eq.(16) implies that ~ ~ 0: since j' --- a/~, a must be infinity, or the resistivity must be zero. Eq.(16) also implies a finite penetration depht 1
"~2 = iq~l
(17)
for the magnetic field, and this is nothing but Meissner effect. The key parameter is (~ = (3) 0: the key physical fact the spontaneous breaking
v(x). There are two scales of length 1
= -#
and
(13)
which appears in eq.(12) is gauge invariant. Moreover, by gauge invariance
of
(8)
(12)
Under a gauge transformation 3 -+ 3 e iqa or 0 --+ 8 + a, and A t` ~ At` + Ot`a. The quantity
Ot`Ft`v + q2[3[2At` = 0
2. Basic s u p e r c o n d u c t i v i t y .
(11)
With this parametrization the covariant derivative reads
1) Is QCD vacuum a dual superconductor? 2) If so, what monopoles condense to break magnetic U(1) ?
(10)
and the Higgs phenomenon takes place. A convenient parametrization for the field 3 is 3 = pe iqO
(6)
323
(18)
and A2, eq.(17). If )~2 >> A1 the superconductor is called type II, otherwise it is called type I.
324
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
For A_~ _> v~A1 (type II), formation of flux tubes is energetically favoured for the penetration of the magnetic field in the superconductor[6]. For type I instead by increasing the magnetic field there is an abrupt penetration in the bulk of the system and superconductivity is destroyed.
T~ = i ¢iaj being the generators of SU(2) in the adjoint representation. As a consequence of eq.(27) [D,, Dr] ~ = 0
(28)
or
3. M o n o p o l e s in QCD. We will now inquire on the monopoles which could condense in QCD vacuum and give dual superconductivity. We will consider SU(2) gauge group for simplicity: in going to the more realistic group SU(3) physics is the same, with the addition of formal complications in the argument. Let ~(x) be any field in the adjoint representation and ~(x) _= g~(x)/[~(x)[ its direction in colour space. A transformation of the gauge group exists such that ~(x) = R -1 (x)~ °
(19)
~,o = (0, 0, 1) being a fixed vector. More generally three unit orthonormal vectors can be defined ~(x), (i = 1,2,3) with (20)
such that C(x) = R - x ( ~ ) g
(21)
~o =
(1,0,0)
(22)
=
(0,1,0)
(23)
=
CO, O, 1)
(24)
with
The gauge transformation R(x) brings (~(x) to ~o = ~o, and is called abelian projection. R is defined up to a U(1) rotation around ¢~, which parametrizes the arbitrariness in choosing ~1 and ~2. Eq.(21) implies 6qt,~~ = w,(x) -. A~
(25)
and, since ~ is a complete basis, F~v(w) -- 0. Here F,,.(,,,) = 0 , , ~
- 0~,,
(30)
- ~,, ^ ~
is the field strength associated to ~ . ~ , is a pure gauge. The solution of eq.(25), with b.c. = = is ~ ( x ) - Pexp
(f2 i
w,(x)Tdx" ,c
)
~o
(31)
The path C along with the parallel transport is performed in eq.(31) is irrelevant, as long as eq.(30) is valid, or ~ ( x ) is a pure gauge. In fa¢~ this is not true in general, because R(x) is sing~ilar at the points where ~(x) = 0
(32)
The field ~ has then non trivial connectdness. The sites where eq.(32) is satisfied are world lines of Dirac monopoles. If .4~ is the gauge field, the gauge transformation R which performs the abelian projection, can modify the field strength G~,,(A~,) because of the above singularities. The field strength becomes
~ . ( ~ ) is a singular field parallel to ~ in colour space, and describes Dirac strings attached to the monopoles. U(1) symmetry is related to the arbitrariness in the choice of ~ , ~ in the definition of the abelian projection. Monopoles are Dirac monopoles interacting with the abelian field
with
(26)
= ~ . (~,v - l ~ ( D , g2 A D , ~ )
(34)
which, after abelian projection is nothing but
Eq.(25) can also be read (27)
F,~, = O, av - Ova,
(35)
A. Di Giacomo/Nuclear PhysicsB (Proc. Suppl.) 64 (1998) 322-328 a~ = ~.,4~,. Monopoles are U(1) monopoles. The existence of monopoles is not a gauge artifact, but reflects the topology of the configurations of the field (~, which can be dynamically relevant. For each choice of (~ one can inquire if the corresponding monopoles do condense in QCD vacuum. The problem is thus shifted to construct a reliable tool to investigate condensation of abelian monopoles, or spontaneous breaking of the U(1) symmetry related to conserx~tion of magnetic charge. Popular choices for the operator (I) are[17] 1) The Polyakov line, which is the parallel transport from a point to itself along the time axis, closed by periodic boundary conditions. 2) Any component of the gauge field . ~ . 3) The operator (~ which is implicitely defined by the maximization with respect to gauge transformations of the quantity[17,19]
~Tr
{fl(n)U~,(n)f~t(n
+
+
1)a3
}
(maximal abelian gauge). 4) ( Ga, ~ T a) 2 for SU(3) gauge theory, for which the d algebra is not trivial. 4. T h e d i s o r d e r p a r a m e t e r for dual s u p e r conductivity. For any choice of ~ one can look for monopole condensation: a creation operator for a monopole can be defined, which carries nonzero magnetic charge[15]. The v.e.v. (#) is then a disorder parameter for dual superconductivity: if (#) ¢ 0 magnetic U(1) is spontaneously broken, and dual Meissner effect takes place. The construction of # can in fact be extended to all the phenomena in which excitations with nontrivial topology are expected to condense in the vacuum. Compact 4-d U(1) gauge theory is a prototype, and the corresponding construction can be easily adapted to non abelian theory, since, after abelian projection, monopoles are
325
U(1) monopoles. Condensation of vortices in 3-d x - y model, which describes superfluid He4, can be studied by the same techniques[20] and also the Heisenberg model[21], where 2 dimensional structures with nontrivial topology condense in the vacuum in the disordered phase. All these models have been successfully investigated. The correlation is measured[22]
7)(x) - (#(x)#(0)>
(36)
I)(x) ~ Aexp(-Mlxl) + (#)2
(37)
As
A numerical experiment can determine from first principles M and I/z)2. : M is the lowest mass with magnetic charge equal to the one carried by the operator #: if a dual Higgs phenomenon takes place, the mass of the Higgs is thus larger or equal to M. (/z) ~ 0 signals dual superconductivity. The deconfining phase transition shows up as a sharp negative peak in the quantity p = ~ In #. Fig.2 shows it for SU(3) deconfining transition. 200.0
0.0
:? -200.0
•e
-400.0
-600.0 4.0
slo
90
Fig.2 p vs/~ for SU(3) at finite T. The peak is at the deconfining transition.
A similar behaviour is observed at the deconfining transition of SU(2) (fig.3), compact U(1) gauge theory (rigA) 3d X - Y model (fig.5)
A. Di Giacomo /Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
326
,~).0
0.0
0.0 . * * . * . . * e
P
t
-,~0.0
-100.0
•
L¢ttice 20 ~
"Laltice 30 ~ • Lattice 401
P -200.0
L i
,i0
,:s
Fig.5 p vs/~ for X - Y model. The peak is at the transition to superfluid.
t: i IO
*I -300.0
-400.0
-
i
1.0
2.0
3.0
4.0
5.0
Fig.3 p vs fl for SU(2) at finite T. The peak is at the deconfining transition.
500.0 •
°
•
°
•
**~
-500.0 -1000.0
The critical index 5, as well as Tc and the critical index u for the correlation length can be determined[221 by a finite size scaling technique. The penetration depth of the U(1) projected electric field ca~"be measured Ell (x) = Eli (0) exp(-md)
(39)
1) QCD vacuum shows, in the confined phase,
-1500.0 -2000.0 0,4
(38)
M/m determines the type of the S.C., as discussed in sect.2. M/m > v~ gives type II. An extensive investigation is on the way on these problems. At the present stage the following preliminary results have been obtained
t000.0
0.0,
1 -
,! o16
o18
1.o
1:2
1,
Fig.4 p vs fl for U(1). The peak is at the deconfining transition. If dual superconductivity is the mechanism of confinement one expects (#) ~ 0 in the confined phase, (#) -- 0 in the deconfmed phase. At the decomCining transition
condensation of monopoles with respect to all the abelian projections (Polyakov line, field strength, F~v, maximal abelian) both for SU(2) and for SU(3). 2) For U(1) superconductor is type II. Work is in progress to determine the type of superconductor for SU(2) and SU(3) 3) The criticalindices for SU(2) agree with 3d Ising model as expected[23] A systematic analysis is in progress.
A. Di Giacomo/NuclearPhysics B (Proc. Suppl.) 64 (1998) 322-328 5. C o n c l u s i o n s Dual superconductivity works as a mechanism for colour confinement. Many different abelian projections define monopoles which condense in the confining phase, and not in the deconfined one. The question if all abelian projections show condensation, or some of them is better than the others is an open question. The point of view of t'Hooft[17] is that alll monopoles, whatever projection is used to define them, are physically equivalent. In favour of this possibility, besides the observed evidence from the lattice, the following arguments can be produced. a) Argument of continuity. Any field ~ in the adjoint representation defines monopoles by abelian projection. Their location correspond to the zeros of ~ in any field configuration. Inside the functional infinity of possibilities in defining kg, many can be deformed continuously into each other, e.g. by a continuous displacement of the zeros. If they were not equivalent physics would change drastically under infinitesimal changes. b) In a single abelian projection one gluon for SU(2), two of them for SU(3) have zero charge with respect to the abelian projected U(1). They cannot then be confined by Abrikosov flux tubes. A cultivated way of saying the same fact is that the Wilson loop in the adjoint representation does not obey the area law. If e.g. both the Polyakov line abelian projection and the max abelian are at work the gluons which are neutral in one of them are charged in the other and will be confined. As mentioned in sect 1 area law is indeed obeyed also by Wilson loops in the adjoint representation[13]. c) If a flux tube between a q ~ pair is produced by monopole condensation after abelian projection, the chromoelectric field in it
327
should be ailigned along the residual U(1). Lattice observations[10,24] show that this is not true: both in the Polyakov line abelian projection and in the max abelian the field in flux tubes is isotropically distributed in colour space. In conclusion QCD vacuum looks as a dual superconductor in the confined phase with respect to the magnetic U(1) identified by many different abelian projections. This indicates that a mechanism of dual superconductivity with respect to a single U(1) is inadequate to describe physics. Maybe a genuine non abelian superconductivity can be identified as a new mechanism, wich works as U(1) superconductivity in infinitely many abelian projections. REFERENCES 1. L.B.Okun: Leptons and quarks North Holland - Amsterdam 1982. 2. A.H. Miiller: Nucl. Phys. B250 (1985) 327. 3..*.:~H.G.Dosch, Yu. A. Simonov: Phys. Lett. B205 (1988) 339. 4. G. 't Hooft, in "High Energy Physics", EPS International Conference, Palermo 1975, ed. A. Zichichi, Bologna 1976, p 1225. 5. S. Mandelstam: Phys. Rep. 23C (1976) 245. 6. A.B. Abrikosov JETP 5 (1957) 1174. 7. M. Creutz: Phys. Rev. D21 (1980) 2308. 8. R.W. Haymaker, J. Wosiek: Phys. Rev. D 36 (1987) 3297. 9. A. Di Giacomo, M. Maggiore and S. Olejnik: Phys. Left. B236 (1990) 199; 10. A. Di Giacomo, M. Maggiore and S. Olejm~: Nucl. Phys. B 3 4 7 (1990) 441. 11. L.Del Debbio, A.Di Giacomo, Yu. A. Simonov: Phys. Let~. B332 (1994) 111. 12. M.Caselle, R.Fiore, F.Gliozzi, M.Hasenbusch, P.Provero: Nucl. Phys. B486 (1997) 245. 13. L.Del Debbio, M. Faber, J. Greensite, S. Olejn~: Phys. Rev. D53 (1996) 5891. 14. N.N. Bogolubov: Physica 26 1 (1960); for a recent review see: S. Weinberg: Prog Th. Phys. Suppl. 86 (1986) 43. 15. L.Del Debbio, A.Di Giacomo, G.Paffuti, Phys. Lett.B 349 (1995) 513; L.Del Debbio,
328
16. 17. 18. 19. 20. 21. 22. 23. 24.
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
A.Di Giacomo, G.Paffuti and P.Pieri, Phys. Lett.B 355 (1995) 255. L.D. Landau,V.L. Ginzburg Z. Exp. Th. Fiz. 20 (1950) 1064. G. 't Hooft, Nucl. Phys. B190 (1981) 455. A.Di Giacomo, M.Mathur: Phys. Lett. B400 (1997) 129. A.S. Kronfeld, G. Schierholz, U.J. Wiese: Nucl. Phys. B 293 (1987) 461. G. Di Cecio, A. Di Giacomo, G. Paffuti, M. Trigiante: Nucl. Phys. B 489 (1997) 739. A. Di Giacomo, D. Martelli, G. Paffuti: in preparation. A.Di Giacomo, G. Paffuti : hep-lat 9707003, to appear in phys. Rev. D. B. Svetitsky, L.G.Jaffe: Nucl. Phys. B210 (1982) 423. J.Greensite, J.Winchester: Phys. Rev. D40 (1989) 4167.
Discussions
V. Z a k h a r o v , Univ. of Michigan Ann Arbor I have two questions: a) Do the monopoles you are talking about have eventually size of order (AQcD) -1 ? b) Do I understood correctly that you have a condensate like (¢2), i.e. of dimension mass squared?
A. Di G i a c o m o If the size is defined as the correlation length of operators with monopole charge, preliminary indications are that this length is O(GeV-1).
b)
True: if the picture is correct in the dual description there should exist a local operator, ¢ acting as a Higgs, which gives rise to dual superconductivity by acquiring a non vanishing (¢2). The explicit construction of ¢ does not exist yet.
Nuclear PhysicsB (Proc. Suppl.) 64 (1998) 322-328
ELSEVIER
Mechanisms
PROCEEDINGS SUPPLEMENTS
of c o l o u r c o n f i n e m e n t .
A. Di Giacomo Dipartimento di Fisica and INFN, 2 Piazza Torricelli 56100 Pisa, Italy Evidence is reviewed for dual superconductivity of QCD vacuum as a mechanism for confinement of colour via Meissner effect.
©
1. I n t r o d u c t i o n Colour is confined. The expected abundance of quarks nq in the cosmological standard model is of the order[l] nq ~ 10 -x2 np
(1)
if there is no confinement, np in eq.(1) is the abundance of protons. The experimental upper-limit is presently
nq < 10-27 np
(2)
which results from the analysis of ~ 102 gr of matter by Millikan-like experiments. If QCD is the theory of hadrons its vacuum cannot be the Fock space vacuum of quarks and gluons. In fact QCD vacuum mimics Fock vacuum at short distances. Infrared renormalons in the perturbative expansion, however[2], signal instability. At large distances the coupling constant g becomes large anyway, and only non perturbative formulations, like lattice, are reliable tools of investigation. The word "mechanism" means describing the degrees of freedom relevant for confinement in terms of the known behaviour of some other physical system. Many mechanisms have been proposed for confinement. A class of them rely on stochastic behaviour of the gauge fields at large distances[3]. The most attractive mechanism, however, is based on the idea that QCD vacuum behaves as a dual superconductor of type II[4,5]. By this mechanism the chromoelectric field existing in the space between a heavy q@ pair is channeled into Abrikosov[6] flux tubes by dual Meissner effect (fig, l) 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All fights reserved. PII S0920-5632(97)01082-7
Fig.1 Flux tube between q@pair. The energy is proportional to the distance R
E = aR
(a = string tension)
(3)
and the particles are confined: an infinite amount of energy is necessary to pull them apart from each other. Dual means that the role of electric and magnetic quantities is interchanged with respect to usual superconductors. Latt.~c~e simulations of QCD show, from first principles, that 1) String tension exists. Large Wilson lops W ( R , T) obey area law[7]
W ( R , T)
"~
R,T--+~
exp(-aRT)
(4)
W(R,T) is the parallel transport along a rectangle in Euclidean space time of size T in the time direction and R in a space direction. A theorem states that, when R, T go large with respect the correlation length
W ( R , T) ,.~ e x p ( - V ( R ) T )
(5)
with V(R) the static potential between the particles transported along the loop. Eq.(4), (5) imply confinement (3).
2) Flux tubes exist in the region of space between q and ~[8,9]. Their transverse size is
~ 0.5 3) Flux tubes behave like strings. modes can be studied[12].
Higher
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
4) Not only quark pairs but also particles in higher colour representations are confined[13]. For SU(2) gauge group particles with colour J experience a string tension ad which is related to the value o'1/2 for the fundamental representation by the relation J ( J + 1) a j = 1/2(1/2 + 1) al/2
If #2 > 0 the ground state has #2 _ <3> = T e iqO # 0
Ordinary superconductivity is the spontaneous breaking of the U(1) symmetry related to charge conservation. The ground state of a superconductor is a superposition of states with different charge ( = different numbers of Cooper pairs), and is not U(1) invariant. A non zero vacuum expectation value (vev) (3) of the effective, charged, field which creates a Cooper pair signals charge condensation in the ground state[14]. Dual superconductivity is condensation of monopole charges, and the spontaneous breaking of the U(1) symmetry related to monopole charge conservation will be signalled by vev (3M) of an operator 3M carrying non zero magnetic charge[15]. The questions we shall address are:
D , 3 = e iqe lot` + iq(A# - O,/~)] p
For the benefit of non experts we recall the basic facs of superconductivity. We will refer to the Higgs lagrangian as an effective lagrangian. It is a relativistic version of the Ginzburg-Landau free energy[16] f-. = -1Ft`vFt`v + (Di,3) t (Dt`3) - V ( 3 )
(7)
with Dt` the covariant derivative Dt` = Ot` + iqAt`
At` = At` - 0t`/~
Ft`~ = Ot`A~ - O~At` = Ot`fi~ - O v A .
v(3) =
[(3"3) 2
(9)
(14)
The effective lagrangean reads E% --~1 Ft`vFt`v + V (p) + (0t`p) 2 + q2 p2 ~t`~t` (15) .... and the equations of motion (16)
As a consequence of the spontaneous symetry breaking a current has appeared, jt` = q21312.4t`, which is known as London current. For a stationary configuration E = 0 and eq.(16) implies that ~ ~ 0: since j' --- a/~, a must be infinity, or the resistivity must be zero. Eq.(16) also implies a finite penetration depht 1
"~2 = iq~l
(17)
for the magnetic field, and this is nothing but Meissner effect. The key parameter is (~ = (3) 0: the key physical fact the spontaneous breaking
v(x). There are two scales of length 1
= -#
and
(13)
which appears in eq.(12) is gauge invariant. Moreover, by gauge invariance
of
(8)
(12)
Under a gauge transformation 3 -+ 3 e iqa or 0 --+ 8 + a, and A t` ~ At` + Ot`a. The quantity
Ot`Ft`v + q2[3[2At` = 0
2. Basic s u p e r c o n d u c t i v i t y .
(11)
With this parametrization the covariant derivative reads
1) Is QCD vacuum a dual superconductor? 2) If so, what monopoles condense to break magnetic U(1) ?
(10)
and the Higgs phenomenon takes place. A convenient parametrization for the field 3 is 3 = pe iqO
(6)
323
(18)
and A2, eq.(17). If )~2 >> A1 the superconductor is called type II, otherwise it is called type I.
324
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
For A_~ _> v~A1 (type II), formation of flux tubes is energetically favoured for the penetration of the magnetic field in the superconductor[6]. For type I instead by increasing the magnetic field there is an abrupt penetration in the bulk of the system and superconductivity is destroyed.
T~ = i ¢iaj being the generators of SU(2) in the adjoint representation. As a consequence of eq.(27) [D,, Dr] ~ = 0
(28)
or
3. M o n o p o l e s in QCD. We will now inquire on the monopoles which could condense in QCD vacuum and give dual superconductivity. We will consider SU(2) gauge group for simplicity: in going to the more realistic group SU(3) physics is the same, with the addition of formal complications in the argument. Let ~(x) be any field in the adjoint representation and ~(x) _= g~(x)/[~(x)[ its direction in colour space. A transformation of the gauge group exists such that ~(x) = R -1 (x)~ °
(19)
~,o = (0, 0, 1) being a fixed vector. More generally three unit orthonormal vectors can be defined ~(x), (i = 1,2,3) with (20)
such that C(x) = R - x ( ~ ) g
(21)
~o =
(1,0,0)
(22)
=
(0,1,0)
(23)
=
CO, O, 1)
(24)
with
The gauge transformation R(x) brings (~(x) to ~o = ~o, and is called abelian projection. R is defined up to a U(1) rotation around ¢~, which parametrizes the arbitrariness in choosing ~1 and ~2. Eq.(21) implies 6qt,~~ = w,(x) -. A~
(25)
and, since ~ is a complete basis, F~v(w) -- 0. Here F,,.(,,,) = 0 , , ~
- 0~,,
(30)
- ~,, ^ ~
is the field strength associated to ~ . ~ , is a pure gauge. The solution of eq.(25), with b.c. = = is ~ ( x ) - Pexp
(f2 i
w,(x)Tdx" ,c
)
~o
(31)
The path C along with the parallel transport is performed in eq.(31) is irrelevant, as long as eq.(30) is valid, or ~ ( x ) is a pure gauge. In fa¢~ this is not true in general, because R(x) is sing~ilar at the points where ~(x) = 0
(32)
The field ~ has then non trivial connectdness. The sites where eq.(32) is satisfied are world lines of Dirac monopoles. If .4~ is the gauge field, the gauge transformation R which performs the abelian projection, can modify the field strength G~,,(A~,) because of the above singularities. The field strength becomes
~ . ( ~ ) is a singular field parallel to ~ in colour space, and describes Dirac strings attached to the monopoles. U(1) symmetry is related to the arbitrariness in the choice of ~ , ~ in the definition of the abelian projection. Monopoles are Dirac monopoles interacting with the abelian field
with
(26)
= ~ . (~,v - l ~ ( D , g2 A D , ~ )
(34)
which, after abelian projection is nothing but
Eq.(25) can also be read (27)
F,~, = O, av - Ova,
(35)
A. Di Giacomo/Nuclear PhysicsB (Proc. Suppl.) 64 (1998) 322-328 a~ = ~.,4~,. Monopoles are U(1) monopoles. The existence of monopoles is not a gauge artifact, but reflects the topology of the configurations of the field (~, which can be dynamically relevant. For each choice of (~ one can inquire if the corresponding monopoles do condense in QCD vacuum. The problem is thus shifted to construct a reliable tool to investigate condensation of abelian monopoles, or spontaneous breaking of the U(1) symmetry related to conserx~tion of magnetic charge. Popular choices for the operator (I) are[17] 1) The Polyakov line, which is the parallel transport from a point to itself along the time axis, closed by periodic boundary conditions. 2) Any component of the gauge field . ~ . 3) The operator (~ which is implicitely defined by the maximization with respect to gauge transformations of the quantity[17,19]
~Tr
{fl(n)U~,(n)f~t(n
+
+
1)a3
}
(maximal abelian gauge). 4) ( Ga, ~ T a) 2 for SU(3) gauge theory, for which the d algebra is not trivial. 4. T h e d i s o r d e r p a r a m e t e r for dual s u p e r conductivity. For any choice of ~ one can look for monopole condensation: a creation operator for a monopole can be defined, which carries nonzero magnetic charge[15]. The v.e.v. (#) is then a disorder parameter for dual superconductivity: if (#) ¢ 0 magnetic U(1) is spontaneously broken, and dual Meissner effect takes place. The construction of # can in fact be extended to all the phenomena in which excitations with nontrivial topology are expected to condense in the vacuum. Compact 4-d U(1) gauge theory is a prototype, and the corresponding construction can be easily adapted to non abelian theory, since, after abelian projection, monopoles are
325
U(1) monopoles. Condensation of vortices in 3-d x - y model, which describes superfluid He4, can be studied by the same techniques[20] and also the Heisenberg model[21], where 2 dimensional structures with nontrivial topology condense in the vacuum in the disordered phase. All these models have been successfully investigated. The correlation is measured[22]
7)(x) - (#(x)#(0)>
(36)
I)(x) ~ Aexp(-Mlxl) + (#)2
(37)
As
A numerical experiment can determine from first principles M and I/z)2. : M is the lowest mass with magnetic charge equal to the one carried by the operator #: if a dual Higgs phenomenon takes place, the mass of the Higgs is thus larger or equal to M. (/z) ~ 0 signals dual superconductivity. The deconfining phase transition shows up as a sharp negative peak in the quantity p = ~ In #. Fig.2 shows it for SU(3) deconfining transition. 200.0
0.0
:? -200.0
•e
-400.0
-600.0 4.0
slo
90
Fig.2 p vs/~ for SU(3) at finite T. The peak is at the deconfining transition.
A similar behaviour is observed at the deconfining transition of SU(2) (fig.3), compact U(1) gauge theory (rigA) 3d X - Y model (fig.5)
A. Di Giacomo /Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
326
,~).0
0.0
0.0 . * * . * . . * e
P
t
-,~0.0
-100.0
•
L¢ttice 20 ~
"Laltice 30 ~ • Lattice 401
P -200.0
L i
,i0
,:s
Fig.5 p vs/~ for X - Y model. The peak is at the transition to superfluid.
t: i IO
*I -300.0
-400.0
-
i
1.0
2.0
3.0
4.0
5.0
Fig.3 p vs fl for SU(2) at finite T. The peak is at the deconfining transition.
500.0 •
°
•
°
•
**~
-500.0 -1000.0
The critical index 5, as well as Tc and the critical index u for the correlation length can be determined[221 by a finite size scaling technique. The penetration depth of the U(1) projected electric field ca~"be measured Ell (x) = Eli (0) exp(-md)
(39)
1) QCD vacuum shows, in the confined phase,
-1500.0 -2000.0 0,4
(38)
M/m determines the type of the S.C., as discussed in sect.2. M/m > v~ gives type II. An extensive investigation is on the way on these problems. At the present stage the following preliminary results have been obtained
t000.0
0.0,
1 -
,! o16
o18
1.o
1:2
1,
Fig.4 p vs fl for U(1). The peak is at the deconfining transition. If dual superconductivity is the mechanism of confinement one expects (#) ~ 0 in the confined phase, (#) -- 0 in the deconfmed phase. At the decomCining transition
condensation of monopoles with respect to all the abelian projections (Polyakov line, field strength, F~v, maximal abelian) both for SU(2) and for SU(3). 2) For U(1) superconductor is type II. Work is in progress to determine the type of superconductor for SU(2) and SU(3) 3) The criticalindices for SU(2) agree with 3d Ising model as expected[23] A systematic analysis is in progress.
A. Di Giacomo/NuclearPhysics B (Proc. Suppl.) 64 (1998) 322-328 5. C o n c l u s i o n s Dual superconductivity works as a mechanism for colour confinement. Many different abelian projections define monopoles which condense in the confining phase, and not in the deconfined one. The question if all abelian projections show condensation, or some of them is better than the others is an open question. The point of view of t'Hooft[17] is that alll monopoles, whatever projection is used to define them, are physically equivalent. In favour of this possibility, besides the observed evidence from the lattice, the following arguments can be produced. a) Argument of continuity. Any field ~ in the adjoint representation defines monopoles by abelian projection. Their location correspond to the zeros of ~ in any field configuration. Inside the functional infinity of possibilities in defining kg, many can be deformed continuously into each other, e.g. by a continuous displacement of the zeros. If they were not equivalent physics would change drastically under infinitesimal changes. b) In a single abelian projection one gluon for SU(2), two of them for SU(3) have zero charge with respect to the abelian projected U(1). They cannot then be confined by Abrikosov flux tubes. A cultivated way of saying the same fact is that the Wilson loop in the adjoint representation does not obey the area law. If e.g. both the Polyakov line abelian projection and the max abelian are at work the gluons which are neutral in one of them are charged in the other and will be confined. As mentioned in sect 1 area law is indeed obeyed also by Wilson loops in the adjoint representation[13]. c) If a flux tube between a q ~ pair is produced by monopole condensation after abelian projection, the chromoelectric field in it
327
should be ailigned along the residual U(1). Lattice observations[10,24] show that this is not true: both in the Polyakov line abelian projection and in the max abelian the field in flux tubes is isotropically distributed in colour space. In conclusion QCD vacuum looks as a dual superconductor in the confined phase with respect to the magnetic U(1) identified by many different abelian projections. This indicates that a mechanism of dual superconductivity with respect to a single U(1) is inadequate to describe physics. Maybe a genuine non abelian superconductivity can be identified as a new mechanism, wich works as U(1) superconductivity in infinitely many abelian projections. REFERENCES 1. L.B.Okun: Leptons and quarks North Holland - Amsterdam 1982. 2. A.H. Miiller: Nucl. Phys. B250 (1985) 327. 3..*.:~H.G.Dosch, Yu. A. Simonov: Phys. Lett. B205 (1988) 339. 4. G. 't Hooft, in "High Energy Physics", EPS International Conference, Palermo 1975, ed. A. Zichichi, Bologna 1976, p 1225. 5. S. Mandelstam: Phys. Rep. 23C (1976) 245. 6. A.B. Abrikosov JETP 5 (1957) 1174. 7. M. Creutz: Phys. Rev. D21 (1980) 2308. 8. R.W. Haymaker, J. Wosiek: Phys. Rev. D 36 (1987) 3297. 9. A. Di Giacomo, M. Maggiore and S. Olejnik: Phys. Left. B236 (1990) 199; 10. A. Di Giacomo, M. Maggiore and S. Olejm~: Nucl. Phys. B 3 4 7 (1990) 441. 11. L.Del Debbio, A.Di Giacomo, Yu. A. Simonov: Phys. Let~. B332 (1994) 111. 12. M.Caselle, R.Fiore, F.Gliozzi, M.Hasenbusch, P.Provero: Nucl. Phys. B486 (1997) 245. 13. L.Del Debbio, M. Faber, J. Greensite, S. Olejn~: Phys. Rev. D53 (1996) 5891. 14. N.N. Bogolubov: Physica 26 1 (1960); for a recent review see: S. Weinberg: Prog Th. Phys. Suppl. 86 (1986) 43. 15. L.Del Debbio, A.Di Giacomo, G.Paffuti, Phys. Lett.B 349 (1995) 513; L.Del Debbio,
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16. 17. 18. 19. 20. 21. 22. 23. 24.
A. Di Giacomo/Nuclear Physics B (Proc. Suppl.) 64 (1998) 322-328
A.Di Giacomo, G.Paffuti and P.Pieri, Phys. Lett.B 355 (1995) 255. L.D. Landau,V.L. Ginzburg Z. Exp. Th. Fiz. 20 (1950) 1064. G. 't Hooft, Nucl. Phys. B190 (1981) 455. A.Di Giacomo, M.Mathur: Phys. Lett. B400 (1997) 129. A.S. Kronfeld, G. Schierholz, U.J. Wiese: Nucl. Phys. B 293 (1987) 461. G. Di Cecio, A. Di Giacomo, G. Paffuti, M. Trigiante: Nucl. Phys. B 489 (1997) 739. A. Di Giacomo, D. Martelli, G. Paffuti: in preparation. A.Di Giacomo, G. Paffuti : hep-lat 9707003, to appear in phys. Rev. D. B. Svetitsky, L.G.Jaffe: Nucl. Phys. B210 (1982) 423. J.Greensite, J.Winchester: Phys. Rev. D40 (1989) 4167.
Discussions
V. Z a k h a r o v , Univ. of Michigan Ann Arbor I have two questions: a) Do the monopoles you are talking about have eventually size of order (AQcD) -1 ? b) Do I understood correctly that you have a condensate like (¢2), i.e. of dimension mass squared?
A. Di G i a c o m o If the size is defined as the correlation length of operators with monopole charge, preliminary indications are that this length is O(GeV-1).
b)
True: if the picture is correct in the dual description there should exist a local operator, ¢ acting as a Higgs, which gives rise to dual superconductivity by acquiring a non vanishing (¢2). The explicit construction of ¢ does not exist yet.