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Are quarks short range solitons?
Volume 64B, number 1
PHYSICS LETTERS
30 August 1976
A R E Q U A R K S S H O R T R A N G E SOLITONS? P. HASENFRATZt and D.A. ROSS Instituut voor Theoretische Fysica, Sorbonnelaan 4, De Uithof, Utrecht, The Netherlands Received 21 April 1976 It is shown that in order to have half integer spin states in a gauge theory without fermions, which possesses a soliton solution, the soliton solution must have at least one long range component.
The fact that quarks form symmetric representations of SU(6) despite the fact that they carry spin is usually explained by assuming that all hadrons are singlets of another internal SU(3), known as colour, so that the total wavefunction is antisymmetric. If, further, this colour SU(3) is an unbroken gauge symmetry it has been suggested [1 ] that the infrared behaviour of the massless Yang-MiUs fields forbids the creation of free quarks or other colour non singlet states. However, recent advances on the infrared behaviour of non Abelian Yang-Mills fields [2] suggest that the infrared divergences can be handled as in Q.E.D. so that they cannot be responsible for quark confinement. Since the relationship between spin and statistics is based on the assumption of locality (i.e., that particles may be considered as point-like objects) another possible explanation of the experimentally observed violation of spin and statistics is to assume that quarks are extended particles so that they may obey Bose Einstein statistics when bound in a hadron, when their separation would be of the same order of magnitude as their size. In this case no colour SU(3) would be needed. It has been shown [3, 4] that the total angular momentum of a system consisting of a magnetic monopole soliton and an isodoublet scalar particle is half odd integer. Furthermore Goldhaber [5] has shown that although the part of the wavefunction relevant to physically measurable quantities is antisymmetric when such systems are sufficiently separated, they obey Bose Einstein statistics when close together. It is obviously unattractive to suggest that quarks I" On leave of absence from the Central Research Institute'for Physics, Budapest, Hungary.
78
are magnetic monopoles unless we can formulate a mechanism for confining the magnetic flux [6], but it is tempting to search for short range soliton solutions which have the same property with respect to angular momentum. In order to have such short range soliton solutions in a gauge theory it is necessary to break completely the gauge symmetry, so that all the vector mesons are massive and all the forces are short range. Coleman [7] has shown that in this case these are no topological conservation laws proving that short range solitons exist, and although this certainly casts doubt on their existence they have not been explicitly disproved. However we show in this that the following two statements must be true for anomalous terms to occur in the angular momentum operator. (1) The soliton solution must leave a subgroup of the gauge symmetry unbroken. (2) In the gauge where the classical solution for the scalar fields tends to a constant far from the centre of the soliton and the longitudinal component of the massless gauge fields vanishes, the soliton solution for at least one of the massless gauge fields must be singular in some region. The second condition means that at least one of the massless gauge fields must have a non-zero soliton solution and since in the gauge mentioned the equations of motion for these fields has the form of a Poisson equation, the soliton solution must be long range (i.e., have a power behaviour as r ~ ~), unless some mechanism can be found for confining the measurable components of the massless field [6]. To prove these conditions we first define the Hamiltonian for a gauge theory of a group with n generators and an r-dimensional (possibly reducible) repre sentation of real scalar fields
Volume 64B, number 1
PHYSICS LETTERS
H_l~.a~.a 1 a~ q~a)2 + U(¢), (1) -'~UkUk ~_irar, ~ 4~ij~ija +l/r~2 +.$(Di where
G = a;A -
0:A," +
30 August 1976
theory. Since C o are constants the term 7ra/)k¢a in (3b) becomes rfakC, a + ~'~'ak~~' (a' =n + 1 ...r),
fo A Af
b,
= 1 ...,),
where r f are the momenta canonical to ~a and ¢~', ffa' are the other linear combinations of Ca and their canonical momenta respectively. We now choose our gauge condition to be
fabc are the structure constants of the group = Oi< a +
(,,,
= l
. . .
ra~ are the r X r representations of the generators of the group and U(¢) is a non-negative, ringlet, renormalisable potential, b~ and 7ra are the momenta conjugate to the gauge fields, A a, and scalar fields, ¢% respectively. Because of the gauge invariance not all of the variables in (I) are independent and we have the n constraints x~ = ok b~ - ~ 8 = 0,
(2)
Ca
=
0.
(5)
Since the gauge group is completely broken this completely fixes the gauge choice. By imposing the constraints (2) and the gauge conditions (5) we arrive at an angular momentum operat'or in terms of the remaining independent field variables A a, & , and their canomcal momenta b i , ~a' --
t
•
Ji= f d3rei/kr)tb~OkA~-
•
a
Ol(bfA~)+~'~'akca'].
(6)
where
/ 8 -_fa b~b ibA ic-
r~orr.~o.
Now we assume that the equations of motion have a solution which completely breaks the gauge invariance, i.e., it is not invariant under any of the generators of the group. If this is so there exists a gauge in which the solutions for the gauge fields, A'f, are short range (i.e., an exponentially decreasing function of r for sufficiently large r). Since we require that the energy be finite, D~O¢t~ must go to zero sufficiently rapidly at large distances and if the gauge field is short range this means that the solution for the scalar fields, ~t3, tends to a constant, C#, at large distances. The angular momentum operator may be written
Ji =
f d3reijk r/[b~Gfc I + I r ~ D ~ ] ,
(3a)
but to get it into a useful form we must express it in terms of the independent variables. To do this we note that (3a) may be immediately rewritten as
Ji =f d3reijkr/[b~OkA~ - Ol(bTAak) + Irc'Ok(9a + A ~ , a ] . (3b) We now make a set of orthogonal linear combinations of ¢ a, the first n of which are
~ka = Ca~f3CO , which we immediately recognise as the Goldstone bosons of a usual spontaneous symmetry breaking
(4)
This angular momentum operator has the usual form and it can be seen by inspection that the commutator of this operator with the physical fields gives the usual angular momentum assignments. This completes the proof of the first statement. If, on the other hand, we had a soliton solution which leaves an n - m dimensional subgroup invariant (we still choose to work in the gauge in which ~# tends to a constant at large distances) then we have n - m remaining massless Yang-Mills fields. Eq. (4) now defines the m Goldstone bosons and eq. (5) gives m of the gauge conditions. The remaining n - m gauge conditions must be obtained by removing one of the three degrees of freedom of the massless A a. A suitable choice would be
~iA~=O
( 5 = m + l ...n).
(7)
This means that the canonical conjugates to (7),
Oib~i are not independent variables and it can be seen from (3b) that unlike the case of a completely broken symmetry, imposing the conditions (7) does not elimi.hate ai ba from the expression for the angular momentum operator. This must be achieved by imposing the constraints X~. We now show that when this is done extra terms in the angular momentum operator only occur if at least one of the A~ is singular in some region of space. We write 79
PHYSICS LETTERS
Volume 64B, number 1
bg =
- azv
(8)
where
atb~ = O, and V~ is given (from eq. (2)) by at(6gKa t + gI~GVAf)V ~ = -glO.,~ ,
where I0 Is equal to j~ with b~ replaced by b~Tl, which are the independent momenta conjugate to the transverse components of A~. By partial integration it can be shown that the contribution to Ji from alV~ vanishes unless the surface term is non-zero. Usually we take this surface to be the sphere at inf'mity. If, however, ~ is singular in a certain region, this region must be excluded from the volume of integration and this gives extra contributions to the surface term. In general the angular momentum operator in terms of the independent variables is
Jt=fd3r{ei/kr/[b~akA~ + bTlakA a ~l + Sa'ak~a' ] a
a
a
+ eUk [b~Ak + bnAk] }
(10)
v akAD-~ V Ak] / ....
rk\
4nj['dsei/k[ra 1 jtrA~-rlA~r)a/Ak] gQa , (a = l...m,~=m+l ...n,~'=m+1 ...r) where Q~ is the volume integral of/g, dFg is an element of the surface surrounding the region where is singular, n~ is a unit vector normal to that surface and ds is a surface element on the sphere at infinity. Using Stokes theorem it can be shown that this last surface term vanishes unless A'~ is singular at some point on the sphere at infinity. It is these surface terms which give rise to anomalous angular momenta. In the case of the magnetic monopole of 't Hooft [8], if we choose the gauge where only the third component of the isotriplet of scalar fields has a non-zero
80
classical solution which tends to a constant at large distances from the monopole, the soliton solution for the photon field has a singularity on the negative zaxis and d F is a thih semi-infinite cylinder surrounding this axis. It can be easily shown that by performing the surface integral in this case we get an extra term in the angular momentum operator
(9)
• t~" •
a
30 August 1976
f d3r jo(r)
(ll) where/0(r ) is the electric charge density and A D is the vector potential for a Dirac monopole [9]. Using the technique of Jackiw and Rebbi [4] it can be shown that this gives an expression for the angular momentum operator which has the form given in refs. [3] and [4]
Ji =jo +Si,
(12)
where j o is the angular momentum operator of a usual theory and S i is the isospin operator. The authors are greatly indebted to M.J.G. Veltman and G.P. Passarino for valuable discussions and useful suggestions. This work is partly supported by the "Stichting voor Fundamenteel Onderzoek der Materie" (F.O •M.).
References [1 ] S. Weinberg, Phys. Rev. D8 (1973) 4482. [2] T. Appeiquist et al., Yale Report No. C00-3075-130 (1976); C.P. Korthais-Altes and E. de Rafael, Marseilles preprint, 76, p. 820. [3] P. Hasenfratz and D.A. Ross, Utrecht I~eprint, 1976; P. Hasenfratz and G. 't Hooft, Phys. Rev. Lett., to be published. [4] R. Jackiw and C. Rebbi, Phys. Rev. Lett., to be published• [5] A.S. Goldhaber, Stony-Brook preprint No. ITP-SB-76-20. [6] M. Creutz, Phys. Rev. D10 (1974) 2696. [7] S. Coleman "Ettore Majorana" Lectures, 1975. [8] G. 't Hooft, Nucl. Phys. B79 (1974) 279. [9] P.A.M. Ditac, Ptoc. Roy. Soc. A133 (1931) 60.
PHYSICS LETTERS
30 August 1976
A R E Q U A R K S S H O R T R A N G E SOLITONS? P. HASENFRATZt and D.A. ROSS Instituut voor Theoretische Fysica, Sorbonnelaan 4, De Uithof, Utrecht, The Netherlands Received 21 April 1976 It is shown that in order to have half integer spin states in a gauge theory without fermions, which possesses a soliton solution, the soliton solution must have at least one long range component.
The fact that quarks form symmetric representations of SU(6) despite the fact that they carry spin is usually explained by assuming that all hadrons are singlets of another internal SU(3), known as colour, so that the total wavefunction is antisymmetric. If, further, this colour SU(3) is an unbroken gauge symmetry it has been suggested [1 ] that the infrared behaviour of the massless Yang-MiUs fields forbids the creation of free quarks or other colour non singlet states. However, recent advances on the infrared behaviour of non Abelian Yang-Mills fields [2] suggest that the infrared divergences can be handled as in Q.E.D. so that they cannot be responsible for quark confinement. Since the relationship between spin and statistics is based on the assumption of locality (i.e., that particles may be considered as point-like objects) another possible explanation of the experimentally observed violation of spin and statistics is to assume that quarks are extended particles so that they may obey Bose Einstein statistics when bound in a hadron, when their separation would be of the same order of magnitude as their size. In this case no colour SU(3) would be needed. It has been shown [3, 4] that the total angular momentum of a system consisting of a magnetic monopole soliton and an isodoublet scalar particle is half odd integer. Furthermore Goldhaber [5] has shown that although the part of the wavefunction relevant to physically measurable quantities is antisymmetric when such systems are sufficiently separated, they obey Bose Einstein statistics when close together. It is obviously unattractive to suggest that quarks I" On leave of absence from the Central Research Institute'for Physics, Budapest, Hungary.
78
are magnetic monopoles unless we can formulate a mechanism for confining the magnetic flux [6], but it is tempting to search for short range soliton solutions which have the same property with respect to angular momentum. In order to have such short range soliton solutions in a gauge theory it is necessary to break completely the gauge symmetry, so that all the vector mesons are massive and all the forces are short range. Coleman [7] has shown that in this case these are no topological conservation laws proving that short range solitons exist, and although this certainly casts doubt on their existence they have not been explicitly disproved. However we show in this that the following two statements must be true for anomalous terms to occur in the angular momentum operator. (1) The soliton solution must leave a subgroup of the gauge symmetry unbroken. (2) In the gauge where the classical solution for the scalar fields tends to a constant far from the centre of the soliton and the longitudinal component of the massless gauge fields vanishes, the soliton solution for at least one of the massless gauge fields must be singular in some region. The second condition means that at least one of the massless gauge fields must have a non-zero soliton solution and since in the gauge mentioned the equations of motion for these fields has the form of a Poisson equation, the soliton solution must be long range (i.e., have a power behaviour as r ~ ~), unless some mechanism can be found for confining the measurable components of the massless field [6]. To prove these conditions we first define the Hamiltonian for a gauge theory of a group with n generators and an r-dimensional (possibly reducible) repre sentation of real scalar fields
Volume 64B, number 1
PHYSICS LETTERS
H_l~.a~.a 1 a~ q~a)2 + U(¢), (1) -'~UkUk ~_irar, ~ 4~ij~ija +l/r~2 +.$(Di where
G = a;A -
0:A," +
30 August 1976
theory. Since C o are constants the term 7ra/)k¢a in (3b) becomes rfakC, a + ~'~'ak~~' (a' =n + 1 ...r),
fo A Af
b,
= 1 ...,),
where r f are the momenta canonical to ~a and ¢~', ffa' are the other linear combinations of Ca and their canonical momenta respectively. We now choose our gauge condition to be
fabc are the structure constants of the group = Oi< a +
(,,,
= l
. . .
ra~ are the r X r representations of the generators of the group and U(¢) is a non-negative, ringlet, renormalisable potential, b~ and 7ra are the momenta conjugate to the gauge fields, A a, and scalar fields, ¢% respectively. Because of the gauge invariance not all of the variables in (I) are independent and we have the n constraints x~ = ok b~ - ~ 8 = 0,
(2)
Ca
=
0.
(5)
Since the gauge group is completely broken this completely fixes the gauge choice. By imposing the constraints (2) and the gauge conditions (5) we arrive at an angular momentum operat'or in terms of the remaining independent field variables A a, & , and their canomcal momenta b i , ~a' --
t
•
Ji= f d3rei/kr)tb~OkA~-
•
a
Ol(bfA~)+~'~'akca'].
(6)
where
/ 8 -_fa b~b ibA ic-
r~orr.~o.
Now we assume that the equations of motion have a solution which completely breaks the gauge invariance, i.e., it is not invariant under any of the generators of the group. If this is so there exists a gauge in which the solutions for the gauge fields, A'f, are short range (i.e., an exponentially decreasing function of r for sufficiently large r). Since we require that the energy be finite, D~O¢t~ must go to zero sufficiently rapidly at large distances and if the gauge field is short range this means that the solution for the scalar fields, ~t3, tends to a constant, C#, at large distances. The angular momentum operator may be written
Ji =
f d3reijk r/[b~Gfc I + I r ~ D ~ ] ,
(3a)
but to get it into a useful form we must express it in terms of the independent variables. To do this we note that (3a) may be immediately rewritten as
Ji =f d3reijkr/[b~OkA~ - Ol(bTAak) + Irc'Ok(9a + A ~ , a ] . (3b) We now make a set of orthogonal linear combinations of ¢ a, the first n of which are
~ka = Ca~f3CO , which we immediately recognise as the Goldstone bosons of a usual spontaneous symmetry breaking
(4)
This angular momentum operator has the usual form and it can be seen by inspection that the commutator of this operator with the physical fields gives the usual angular momentum assignments. This completes the proof of the first statement. If, on the other hand, we had a soliton solution which leaves an n - m dimensional subgroup invariant (we still choose to work in the gauge in which ~# tends to a constant at large distances) then we have n - m remaining massless Yang-Mills fields. Eq. (4) now defines the m Goldstone bosons and eq. (5) gives m of the gauge conditions. The remaining n - m gauge conditions must be obtained by removing one of the three degrees of freedom of the massless A a. A suitable choice would be
~iA~=O
( 5 = m + l ...n).
(7)
This means that the canonical conjugates to (7),
Oib~i are not independent variables and it can be seen from (3b) that unlike the case of a completely broken symmetry, imposing the conditions (7) does not elimi.hate ai ba from the expression for the angular momentum operator. This must be achieved by imposing the constraints X~. We now show that when this is done extra terms in the angular momentum operator only occur if at least one of the A~ is singular in some region of space. We write 79
PHYSICS LETTERS
Volume 64B, number 1
bg =
- azv
(8)
where
atb~ = O, and V~ is given (from eq. (2)) by at(6gKa t + gI~GVAf)V ~ = -glO.,~ ,
where I0 Is equal to j~ with b~ replaced by b~Tl, which are the independent momenta conjugate to the transverse components of A~. By partial integration it can be shown that the contribution to Ji from alV~ vanishes unless the surface term is non-zero. Usually we take this surface to be the sphere at inf'mity. If, however, ~ is singular in a certain region, this region must be excluded from the volume of integration and this gives extra contributions to the surface term. In general the angular momentum operator in terms of the independent variables is
Jt=fd3r{ei/kr/[b~akA~ + bTlakA a ~l + Sa'ak~a' ] a
a
a
+ eUk [b~Ak + bnAk] }
(10)
v akAD-~ V Ak] / ....
rk\
4nj['dsei/k[ra 1 jtrA~-rlA~r)a/Ak] gQa , (a = l...m,~=m+l ...n,~'=m+1 ...r) where Q~ is the volume integral of/g, dFg is an element of the surface surrounding the region where is singular, n~ is a unit vector normal to that surface and ds is a surface element on the sphere at infinity. Using Stokes theorem it can be shown that this last surface term vanishes unless A'~ is singular at some point on the sphere at infinity. It is these surface terms which give rise to anomalous angular momenta. In the case of the magnetic monopole of 't Hooft [8], if we choose the gauge where only the third component of the isotriplet of scalar fields has a non-zero
80
classical solution which tends to a constant at large distances from the monopole, the soliton solution for the photon field has a singularity on the negative zaxis and d F is a thih semi-infinite cylinder surrounding this axis. It can be easily shown that by performing the surface integral in this case we get an extra term in the angular momentum operator
(9)
• t~" •
a
30 August 1976
f d3r jo(r)
(ll) where/0(r ) is the electric charge density and A D is the vector potential for a Dirac monopole [9]. Using the technique of Jackiw and Rebbi [4] it can be shown that this gives an expression for the angular momentum operator which has the form given in refs. [3] and [4]
Ji =jo +Si,
(12)
where j o is the angular momentum operator of a usual theory and S i is the isospin operator. The authors are greatly indebted to M.J.G. Veltman and G.P. Passarino for valuable discussions and useful suggestions. This work is partly supported by the "Stichting voor Fundamenteel Onderzoek der Materie" (F.O •M.).
References [1 ] S. Weinberg, Phys. Rev. D8 (1973) 4482. [2] T. Appeiquist et al., Yale Report No. C00-3075-130 (1976); C.P. Korthais-Altes and E. de Rafael, Marseilles preprint, 76, p. 820. [3] P. Hasenfratz and D.A. Ross, Utrecht I~eprint, 1976; P. Hasenfratz and G. 't Hooft, Phys. Rev. Lett., to be published. [4] R. Jackiw and C. Rebbi, Phys. Rev. Lett., to be published• [5] A.S. Goldhaber, Stony-Brook preprint No. ITP-SB-76-20. [6] M. Creutz, Phys. Rev. D10 (1974) 2696. [7] S. Coleman "Ettore Majorana" Lectures, 1975. [8] G. 't Hooft, Nucl. Phys. B79 (1974) 279. [9] P.A.M. Ditac, Ptoc. Roy. Soc. A133 (1931) 60.