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Perturbative QCD in a non-perturbative vacuum
Volume 101B, number 1,2
PHYSICS LETTERS
30 April 1981
PERTURBATIVE QCD IN A NON-PERTURBATIVE VACUUM R.J. CANT Department of Theoretical Physics, The University, Manchester M13 9PL, UK Received 6 February 1981
We propose a procedure for interpreting the presence of a background gluon field in the context of perturbative QCD. We expect our method to be useful when non-perturbative effects are significant but not completely dominant.
There has been a lot of interest lately in the possibility that the QCD vacuum has a non-trivial structure [ 1 4 ] . Much of this work has been explicitly aimed at providing a theoretical basis in QCD for the MIT bag model. The idea has been to show that the vacua inside and outside the bag can be identified with different phases of the field theory. Whilst some authors [1 ] have invoked the effects of instantons and merons to provide this structure, others have relied on the infrared instability of the system. In particular Savvidy [2] has shown that, in the one-loop approximation, a lowered energy density can be obtained by giving the colour magnetic field a vacuum expectation value. These computations were extended in a series of papers [3] by Nielsen, Olesen and others, who showed that the energy could be further lowered by giving the field some space-time dependence. These authors have postulated that the true vacuum, outside the bag, should consist of "random colour magnetic spaghetti". More recently, Fukuda and Kazama [4] have shown that the true vacuum state has (G 2) :/: 0. This does seem to explain the origin of the bag, at least qualitatively. There is, however, a snag. If the region outside the bag is nonuniform, the bag model can only be a good approximation when
(A being the usual QCD scale parameter) if the vacuum fluctuations are to be a negligible correction to perturbative QCD. Putting these requirements together we obtain 1/rbag ,~ 1/Xvae < A ,
i.e. 200 MeV ,< 1/Xvae < 5 00 MeV, taking typical values. There is not a lot of room for the vacuum fluctuations. Hence either (i) they provide a large correction to perturbative QCD or (ii) the bag shape is determined from without by the vacuum rather than from within by the confined fields. In the present work we will follow (ii) and propose an alternative approximation scheme based on a fluctuating background field with which perturbative fields will interact. We expect this scheme to be valid as a means of calculating non-perturbative corrections to perturbative QCD since we still rely on the smallness'. of the running coupling constant. The determination of the form of the background field is a non-perturbative problem, however, and work on this topic will be published elsewhere. Our approach has certain similarities with that of Mandelstam [5], especially in motivation. The main difference is that whereas he moved directly to a trunrbag >> Xvac cation of the Schwinger-Dyson equation for the gluon propagator, we will retain a background field explicitly. (rbag being the radius of the bag, and Xvac the typical We will show that results similar to Mandelstam's can length scale associated with the vacuum nonuniformity). be obtained in such a scheme and comment on the form However, the nonuniformity associated with the vacsuch a field would have to take. If this form could be uum is a non-perturbative effect, hence we expect deduced from QCD then our method would have the l/X~ac < A, advantage of automatically reproducing perturbation theory in the ultraviolet limit. 0 031-9163/81/0000-0000/$ 02.50 © North-HoUand Publishing Company 108
Volume 101B, number 1,2
PHYSICS LETTERS
We begin with the QCD lagrangian with a source H~v representing the background field, coupled to the field tensor G gv. .~=
_~-(a~)2 + '-,-:-<, ,-,o " 2"-'uv':uv
(1)
The generating functional for the disconnected Green's functions is: W[H~v, i~]
=-ilnlfel)A~exp(ifd4x(~+l~A~)t],
(2)
where we have introduced a further source ]~ which we will use to extract gluon Green's functions in the usual way. We have: --
*ab 2 Duv ( P )
(4)
=D~b',(p2) ~_,(p2)(p2g#v, -- Pu'Pv') Db'b v'v(P 2 ) , where Z (p2) is the Fourier transform of the correlation function A(x) introduced in eq. (3). There are now extra diagrams in the model in which ordinary propagators are rephced by expression (4) in all possible ways, subject to the constraint that the diagram cannot be made disconnected by cutting the D*. If the Landau gauge is selected for the calculation then the modified propaga. tor simplifies to (5)
'
and C/)A~ is taken to include whatever gauge fixing and ghost factors are necessary. We will take HJv to be a quenched random source and average W rather than exp(iW). This has the important effect of preventing any energy transfer between the two systems ensuring that the perturbative one is conservative. We write: WR [/g]= fc-l)H~v exp [-¼
nection between pairs of sources. The resulting new propagators are of the form
*ab (p 2 ) --D ~ ab D.v ( p 2 )]~(p2).
a
GJv - OuA v - ~vA~ + gfabcAb].1.A c
30 April 1981
fHg=(x)A-'(x- y )
X H~v(y ) d4x d4y] W[H~v , j ~ ] .
(3)
This procedure is well known in solid state physics [6] but we have introduced a novelty in the form of a correlation function A(x --y). There are two physical motivations for doing this. Firstly we do not see macroscopic vacuum fluctuations and hence A(x - y) must be damped at large distances. Secondly we expect perturbation theory to be correct at large momenta and so the effective A(x - y ) should decrease rapidly at short distances. This could reflect either a real damping of the random field or the tendency for a high momentum particle to push it out of the way without being affected. Now clearly W[H~z,, jg] consists of connected diagrams with the sourcesH~ coupled via a term of the form:
(Dab,(p2"~ a uu" , p v , - D u ab v ' ( P 2 )P~')Hu'v' [Dab(p2] being the usual gluon propagator]. The next integration, over the H~,, introduces a con-
Our methods differ from those employed to analyse the random field problem in solid state physics [6] in that whereas they were interested in the leading infrared behaviour and thus kept only those diagrams which dominated in that limit, we include all possible diagrams. This is necessary because we are still using perturbation theory and relying on the coupling being small in the high energy limit. Our diagrams give the leading nonperturbative contribution in that limit. The modified Green's function (5) also differs from that used in the usual random field problem in that the extra power of p - 2 from the doubling up of the propagator has been cancelled through the coupling. The only modification is the presence of ~(p2). The form of N(p2) must be input for the model to be useful. The theoretical determination of Y~(p2) can be approached in several different ways, and the details will be discussed elsewhere. However, we can deduce some of the properties of N (p2) without a fuUy-fledged investigation. Fukuda and Kazama [4] find a unique expectation value of (G~v)2 which minimises the energy in their approximation. To leading order in the renormalised coupling
~(fd4x(G~v)2)~2e-21[Jog2oefp2d(p2) ~(p2) , (6)
where ~2 is the total volume of space-time and the renormalisation group/~ function has the form #(g) = u og/au = -t3093 + O(gS).
It is interesting to note that we can expect Y.(p2) to 109
Volume 101B, number 1,2
PHYSICS LETTERS
behave like e x p ( - 1 / g 2) giving it a typical non-perturbative behaviour in the small coupling limit. Note also that if, as Fukuda's work suggests, (G~,) 2 has a finite expectation value then the integral in eq. (6) should converge. We do not trust our approximation scheme in the infrared limit because of the possible divergence of the effective coupling but convergence in the ultraviolet limit requires that Z (p2) must decrease faster than p - 4 . This should correspond to a gluon propagator which contains terms like p - 6 or p - 7 and terms like r 3 or r 4 in the quark-antiquark potential. We might expect such terms to be the first non-Coulomb forces to appear in the potential as r increases. It is interesting to note that a potential going like r 4 has recently been deduced via the techniques of the ITEP group [7] by Bell et al. [8]. To produce results similar to Mandelstam's we would require:
30 April 1981
We have seen how to incorporate the effects of a non-zero expectation value of (G~w)2 into perturbation theory. Such a technique may be useful for the comparison of non-perturbative phenomena such as the nonCoulomb parts of heavy quark potentials and higher twist effects. If a non-perturbative approach to QCD can be used to calculate ~ ( p 2 ) then direct predictions of these effects can be made. It may well be easier to make such predictions using the technique outlined than by the direct route. It may be possible to determine Z (p2) usinglattice methods, instantons, or monopoles as well as by the approach used in refs. [2-4]. Further results will be reported elsewhere. The author is grateful to Professor M.A. Moore and Dr. A.J. Bray for useful discussions. Financial support from the S.R.C. is gratefully acknowledged.
~(p2) = F(p2) _ 1 ,
References
F ( p 2) being as defined in [5]. However, we do not expect Mandelstam's F ( p 2) to be correct at very low p2 (below his P0), because his approximation scheme is based on a truncation of the Schwinger-Dyson equations and such approximations usually break down for very large coupling (see, for example ref. [9]). In the intermediate momentum range, however, (P0 toP~ ) his F ( p 2) is probably a good approximation and we would expect (p2) to follow eq. (6) in this region. The form of ~ (p2) would then be
[ 1 ] D.J. Gross, Phys. R ep. 49 (1979) 143, and references therein. [2] G.K. Savvidy,Phys. Lett. 71B (1977) 133; S.G. Matinyan and G.K. Savvidy,Nucl. Phys. B134 (1978) 539. [3] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 485; Phys. Lett. 79B (1978) 304; J. Ambj¢rn, N.K. Nielsen and P. Olesen, Nucl. Phys. B152 (1979) 75; H.B. Nielsen, Phys. Lett. 80B (1978) 133; H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1979) 1. [4 ] R. Fukuda and Y. Kazama, Phys. Rev. Lett. 45 (1980) 1142; R. Fukuda, Phys. Rev. D21 (1980) 485. [5] S. Mandelstam, Phys. Rev. D20 (1979) 3223. [6] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744; Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35 (1975) 1399; A.P. Young, J. Phys. C10 (1977) L257. [7] M.A. Shffman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385. [8] J.S. Bell and R. Bertimann, CERN preprint TH.2986. [9] R.J. Cant and R.J. Rivers, J. Phys. A13 (1980) 1623.
~ ( p 2 ) = aA2]p2 + bpZ/A2 , the constant term in F ( p 2) being produced by the ordinary gluons. Around Mandelstam's P~ we would expect ~ ( p 2 ) to decay rapidly as mentioned above - making way for ordinary perturbative behaviour in the ultraviolet limit.
110
PHYSICS LETTERS
30 April 1981
PERTURBATIVE QCD IN A NON-PERTURBATIVE VACUUM R.J. CANT Department of Theoretical Physics, The University, Manchester M13 9PL, UK Received 6 February 1981
We propose a procedure for interpreting the presence of a background gluon field in the context of perturbative QCD. We expect our method to be useful when non-perturbative effects are significant but not completely dominant.
There has been a lot of interest lately in the possibility that the QCD vacuum has a non-trivial structure [ 1 4 ] . Much of this work has been explicitly aimed at providing a theoretical basis in QCD for the MIT bag model. The idea has been to show that the vacua inside and outside the bag can be identified with different phases of the field theory. Whilst some authors [1 ] have invoked the effects of instantons and merons to provide this structure, others have relied on the infrared instability of the system. In particular Savvidy [2] has shown that, in the one-loop approximation, a lowered energy density can be obtained by giving the colour magnetic field a vacuum expectation value. These computations were extended in a series of papers [3] by Nielsen, Olesen and others, who showed that the energy could be further lowered by giving the field some space-time dependence. These authors have postulated that the true vacuum, outside the bag, should consist of "random colour magnetic spaghetti". More recently, Fukuda and Kazama [4] have shown that the true vacuum state has (G 2) :/: 0. This does seem to explain the origin of the bag, at least qualitatively. There is, however, a snag. If the region outside the bag is nonuniform, the bag model can only be a good approximation when
(A being the usual QCD scale parameter) if the vacuum fluctuations are to be a negligible correction to perturbative QCD. Putting these requirements together we obtain 1/rbag ,~ 1/Xvae < A ,
i.e. 200 MeV ,< 1/Xvae < 5 00 MeV, taking typical values. There is not a lot of room for the vacuum fluctuations. Hence either (i) they provide a large correction to perturbative QCD or (ii) the bag shape is determined from without by the vacuum rather than from within by the confined fields. In the present work we will follow (ii) and propose an alternative approximation scheme based on a fluctuating background field with which perturbative fields will interact. We expect this scheme to be valid as a means of calculating non-perturbative corrections to perturbative QCD since we still rely on the smallness'. of the running coupling constant. The determination of the form of the background field is a non-perturbative problem, however, and work on this topic will be published elsewhere. Our approach has certain similarities with that of Mandelstam [5], especially in motivation. The main difference is that whereas he moved directly to a trunrbag >> Xvac cation of the Schwinger-Dyson equation for the gluon propagator, we will retain a background field explicitly. (rbag being the radius of the bag, and Xvac the typical We will show that results similar to Mandelstam's can length scale associated with the vacuum nonuniformity). be obtained in such a scheme and comment on the form However, the nonuniformity associated with the vacsuch a field would have to take. If this form could be uum is a non-perturbative effect, hence we expect deduced from QCD then our method would have the l/X~ac < A, advantage of automatically reproducing perturbation theory in the ultraviolet limit. 0 031-9163/81/0000-0000/$ 02.50 © North-HoUand Publishing Company 108
Volume 101B, number 1,2
PHYSICS LETTERS
We begin with the QCD lagrangian with a source H~v representing the background field, coupled to the field tensor G gv. .~=
_~-(a~)2 + '-,-:-<, ,-,o " 2"-'uv':uv
(1)
The generating functional for the disconnected Green's functions is: W[H~v, i~]
=-ilnlfel)A~exp(ifd4x(~+l~A~)t],
(2)
where we have introduced a further source ]~ which we will use to extract gluon Green's functions in the usual way. We have: --
*ab 2 Duv ( P )
(4)
=D~b',(p2) ~_,(p2)(p2g#v, -- Pu'Pv') Db'b v'v(P 2 ) , where Z (p2) is the Fourier transform of the correlation function A(x) introduced in eq. (3). There are now extra diagrams in the model in which ordinary propagators are rephced by expression (4) in all possible ways, subject to the constraint that the diagram cannot be made disconnected by cutting the D*. If the Landau gauge is selected for the calculation then the modified propaga. tor simplifies to (5)
'
and C/)A~ is taken to include whatever gauge fixing and ghost factors are necessary. We will take HJv to be a quenched random source and average W rather than exp(iW). This has the important effect of preventing any energy transfer between the two systems ensuring that the perturbative one is conservative. We write: WR [/g]= fc-l)H~v exp [-¼
nection between pairs of sources. The resulting new propagators are of the form
*ab (p 2 ) --D ~ ab D.v ( p 2 )]~(p2).
a
GJv - OuA v - ~vA~ + gfabcAb].1.A c
30 April 1981
fHg=(x)A-'(x- y )
X H~v(y ) d4x d4y] W[H~v , j ~ ] .
(3)
This procedure is well known in solid state physics [6] but we have introduced a novelty in the form of a correlation function A(x --y). There are two physical motivations for doing this. Firstly we do not see macroscopic vacuum fluctuations and hence A(x - y) must be damped at large distances. Secondly we expect perturbation theory to be correct at large momenta and so the effective A(x - y ) should decrease rapidly at short distances. This could reflect either a real damping of the random field or the tendency for a high momentum particle to push it out of the way without being affected. Now clearly W[H~z,, jg] consists of connected diagrams with the sourcesH~ coupled via a term of the form:
(Dab,(p2"~ a uu" , p v , - D u ab v ' ( P 2 )P~')Hu'v' [Dab(p2] being the usual gluon propagator]. The next integration, over the H~,, introduces a con-
Our methods differ from those employed to analyse the random field problem in solid state physics [6] in that whereas they were interested in the leading infrared behaviour and thus kept only those diagrams which dominated in that limit, we include all possible diagrams. This is necessary because we are still using perturbation theory and relying on the coupling being small in the high energy limit. Our diagrams give the leading nonperturbative contribution in that limit. The modified Green's function (5) also differs from that used in the usual random field problem in that the extra power of p - 2 from the doubling up of the propagator has been cancelled through the coupling. The only modification is the presence of ~(p2). The form of N(p2) must be input for the model to be useful. The theoretical determination of Y~(p2) can be approached in several different ways, and the details will be discussed elsewhere. However, we can deduce some of the properties of N (p2) without a fuUy-fledged investigation. Fukuda and Kazama [4] find a unique expectation value of (G~v)2 which minimises the energy in their approximation. To leading order in the renormalised coupling
~(fd4x(G~v)2)~2e-21[Jog2oefp2d(p2) ~(p2) , (6)
where ~2 is the total volume of space-time and the renormalisation group/~ function has the form #(g) = u og/au = -t3093 + O(gS).
It is interesting to note that we can expect Y.(p2) to 109
Volume 101B, number 1,2
PHYSICS LETTERS
behave like e x p ( - 1 / g 2) giving it a typical non-perturbative behaviour in the small coupling limit. Note also that if, as Fukuda's work suggests, (G~,) 2 has a finite expectation value then the integral in eq. (6) should converge. We do not trust our approximation scheme in the infrared limit because of the possible divergence of the effective coupling but convergence in the ultraviolet limit requires that Z (p2) must decrease faster than p - 4 . This should correspond to a gluon propagator which contains terms like p - 6 or p - 7 and terms like r 3 or r 4 in the quark-antiquark potential. We might expect such terms to be the first non-Coulomb forces to appear in the potential as r increases. It is interesting to note that a potential going like r 4 has recently been deduced via the techniques of the ITEP group [7] by Bell et al. [8]. To produce results similar to Mandelstam's we would require:
30 April 1981
We have seen how to incorporate the effects of a non-zero expectation value of (G~w)2 into perturbation theory. Such a technique may be useful for the comparison of non-perturbative phenomena such as the nonCoulomb parts of heavy quark potentials and higher twist effects. If a non-perturbative approach to QCD can be used to calculate ~ ( p 2 ) then direct predictions of these effects can be made. It may well be easier to make such predictions using the technique outlined than by the direct route. It may be possible to determine Z (p2) usinglattice methods, instantons, or monopoles as well as by the approach used in refs. [2-4]. Further results will be reported elsewhere. The author is grateful to Professor M.A. Moore and Dr. A.J. Bray for useful discussions. Financial support from the S.R.C. is gratefully acknowledged.
~(p2) = F(p2) _ 1 ,
References
F ( p 2) being as defined in [5]. However, we do not expect Mandelstam's F ( p 2) to be correct at very low p2 (below his P0), because his approximation scheme is based on a truncation of the Schwinger-Dyson equations and such approximations usually break down for very large coupling (see, for example ref. [9]). In the intermediate momentum range, however, (P0 toP~ ) his F ( p 2) is probably a good approximation and we would expect (p2) to follow eq. (6) in this region. The form of ~ (p2) would then be
[ 1 ] D.J. Gross, Phys. R ep. 49 (1979) 143, and references therein. [2] G.K. Savvidy,Phys. Lett. 71B (1977) 133; S.G. Matinyan and G.K. Savvidy,Nucl. Phys. B134 (1978) 539. [3] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 485; Phys. Lett. 79B (1978) 304; J. Ambj¢rn, N.K. Nielsen and P. Olesen, Nucl. Phys. B152 (1979) 75; H.B. Nielsen, Phys. Lett. 80B (1978) 133; H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1979) 1. [4 ] R. Fukuda and Y. Kazama, Phys. Rev. Lett. 45 (1980) 1142; R. Fukuda, Phys. Rev. D21 (1980) 485. [5] S. Mandelstam, Phys. Rev. D20 (1979) 3223. [6] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744; Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35 (1975) 1399; A.P. Young, J. Phys. C10 (1977) L257. [7] M.A. Shffman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385. [8] J.S. Bell and R. Bertimann, CERN preprint TH.2986. [9] R.J. Cant and R.J. Rivers, J. Phys. A13 (1980) 1623.
~ ( p 2 ) = aA2]p2 + bpZ/A2 , the constant term in F ( p 2) being produced by the ordinary gluons. Around Mandelstam's P~ we would expect ~ ( p 2 ) to decay rapidly as mentioned above - making way for ordinary perturbative behaviour in the ultraviolet limit.
110