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Infrared stability of the Yang-Mills theory in the zero-instanton sector
Volume 66B, number 4
PHYSICS LETTERS
INFRARED
14 February 1977
STABILITY OF THE YANG-MILLS THEORY
IN THE ZERO-INSTANTON
SECTOR
Poul OLESEN
The Nwls Bohr InsHtute, University of Copenhagen, DK-21 O0 Copenhagen 0 , Denmark Received 17 December 1976 Abstracting the decoupling theorem of Appelqmst and Carazzone from perturbation theory we show that the Yang-MlUs theory ISinfrared stable in the zero-mstanton sector. We point out that our argument is not valid when mstantons axe present.
It has been shown by Pogglo and Qulnn [1] that the Yang-Mills theory does not imprison quarks and gluons to any finite order of perturbation theory. In this note we wish to point out the existence of a simple argument which strongly indicates that even if one sum over all orders the Yang-Mtlls theory is Infrared stable. However, as we shall show later, our argument breaks down when instantons [2] are taken into account. Therefore there is hope that confinement sets in because of the existence of lnstantons in YangMills theories. The main assumption in the following is that the decoupling theorem of Appelquist and Carazzone [3] can be abstracted from perturbation theory. From this it follows that the pure Yang-Mills Greens function UyM can be represented in terms of a zero-mass renormallzed [4] Greens function Pw, which in addition to the vector gluons also contains virtual quark pairs [5, 6] (since P w does not decouple the quarks to any finite order) PvM(kp ' g2,/a) = qbr(mw/P, gw)
(1) 2 mw, u,M) [1 + "00, 2, U2/m2)"]. x rw(No,g;v, Possible logarithms have been ignored in the subdomlnant terms. Here m w is the quark mass in Weinberg's zero mass renormalization and gw is the coupling constant. The symbol p stands for a set of fixed momenta. The "O"-symbol indicates that the order of magnitude estimate of the neglected terms is valid to any finite order of perturbation theory, but may not be valid when summed over all orders. In any case, the abstraction of the decoupling means that we assume that the terms contained in the "O" symbol can be ignored when summation over all orders is performed. 370
The quantity M (flavor) is the number of fermion multlplets, and is explicitly written in Pw in order to indicate that Pw still depends on M [5]. Due to the independence of the left-hand side of (1) on M, the right-hand side of (1) must have a complete degeneracy with respect to M. The quantity qb is a finite multiplicative renormallzatlon constant. Because of this, with n external gluons qb is the (n/2)th power of the finite renormalization constant which connects the inverse propagator in the Y-M case and in the Weinberg case. As discussed previously [5, 6], because of the complete degeneracy with respect to flavor, we can utilize the observation of Gross and Wilczek [7] that for M large, M ~ 16 for SU(3), there is a non-trivial infrared stable fixed point, g()02 = g~ +A~V,
(2)
where 7 is a small positive power and g2 is small, g2/l&r2 = 1/302 f o r m = 16 [8]. Applying Weinberg's renormallzation group on the right-hand side of eq. (1) we obtain r y M (No, g2-~U)~ CI'r(mw//a, gw) ~D r -'rw (g l)
(3) X rw(p,gO02 , mw/;k 1+'/°(gl), I.t,M) for ), -~ 0. Here D r is the canonical dimension of P, 7W(gl) is the anomalous dimension in Weinberg's renormalization, 70(gl) is the mass dimension, and mw(;~) = mw/Xl+~o (gD,
)~ -~ O,
(4)
is the lnfini_te effective muss of the quarks. We shall now analyze the contents of eq. (3). To simplify the notation, we leave out variables which
Volume 66B, number 4
PHYSICS LETTERS
14 l-ebruary 1977
are inessential in the following, and rewrite eq. (3) as
of the type
PyM(~.) = qbr(mw)Fw(g(~) 2, mw/Xl+'ro(ga) )
g l 2n ,x (M 0 _ M')-n,
(5)
(10)
ix
is regular for g2 -+ g21. Eq. (5) then becomes the functional equation
with M 0 = T N . However, we know that in perturbation theory the dependence on M comes about through the Caslmlr operators, which are regular polynomials m M Consequently the behavior (8) can be ruled out because such a singular behaviour is not found in perturbation theory. In the case where the perturbative expansion is a divergent asymptotic expansion around g2, perturbation theory should look like
PyM (X) = ~ r (mw) ~Dr - ~ (gl)
~ C n ( g 2 _ g2)n,
X ~.ol'-')'w(gl) .
We can now distinguish two posslblhtles, namely the following: (a) The function
(gl)),
pw(g(~,)2 ' m w / ~ 1 +~'o
x rwfa 2, mw/X 1 +'tO~1)).
(6)
This equation has as the soluUon [6] a power behavior for all the functions involved, Due to the fact that (I) is the ratio of multiplicative renormahzation constants one easily sees that [6] r v M (xp, g2,/a) ~ ~,D r - n # ,
(7)
where/3 is a constant. From eq. (7) one concludes that the theory is infrared stable (the effectwe coupling approaches a constant m X for ~, -+ 0). (b) The function pw (g(X)2, mw/~,1 +'r0 (gl)), has a singularity* for g(X)2 _+ g2. In this case the behavior (7) cannot be concluded from eq. (5). On the contrary, we can say that PYM can have any behavior as a functmn of ~. Let us now analyze the case (b) in the zero-instanton sector. Since PW has a singularity when g(X) 2 approaches g2, we expect that in sufficiently high orders o f perturbation theory one encounters terms proportional to
g2m g l 2 n (m >I n),
(8)
corresponding to a radius of convergence g2 m the g 2 plane. For M sufficiently close to ~ N (for SU(N)), gl2 is given by [8] g~l
llN-
2M
16n 2 - - 3 4 N 2 + 1 3 N M - 3M/N "
(9)
(11)
where the coefficients c n are such that the series ~Cn xn has zero radius of convergence. This case also leads to a violent clash with perturbation theory. For instance, in third order one has (gl2 - g2)3 which for example gives rise to a term g4g2, which is proportional to (M 0 - M)2g 2. However, to order g2 the Casimir operators contain at most the first power of M. Tins objection becomes worse as one goes to higher orders in gl2 - g2, where terms of the type (M 0 - M)ng 2 are produced with n an arbitrarily large integer. This is in violent disagreement with perturbation theory (unless some of the c n are singular in M 0 - M, which re-introduces the singularities in (10)). Hence our argument does not depend essentially on the meaning of the perturbative expansion. We therefore conclude that the zero-instanton sec-
tor o f the Yang-Mills theory is infrared stable (power behaved Greens functions], and there is no reason for quark-gluon confinement. When we include instantons, the above argument breaks down. Presumably the decouphng and renormalizability still hold, so eq. (5) is still valid. However, now we have no argument to rule out the existence of a singular behavior in the coupling constant for g2w _~ g2, since mstantons are basically non-perturbative. Of course, it does not follow from our arguments that when instantons are present, quarks are confined. However, one may hope that coupling constant singularities occur in this case.
The perturbatwe coefficients g{2n are thus singular * This possibility was not discussed m a previous work.
The author thanks D. Gross, A. Polyakov, and K. Symanzik for interesting and stimulating discussions. 371
Volume 66B, number 4
PHYSICS LETTERS
He also thanks N.K. Nielsen for pointing out an obscurity m the original version o f the manuscript.
References [1] E.C. Pogglo and H.R. Qumn, Phys. Rev. D14 (1976) 578. [2] A.A. Belavin et al., Phys. Lett. 59B (1975) 85. [3] T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856.
372
14 February 1977
[4] S. Weinberg, Phys. Rev. D8 (1973) 3497. [5] P. Olesen, Nucl. Phys. B104 (1976) 125. [6] P. Olesen, to be published in the Proc. of the Workshop on quark binding at the Umverslty of Rochester, 1976 (John Wiley, New York). [7] D. Gross and F. Wllczek, Phys. Rev. D8 (1973) 3633. [8] W.E. CasweU, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, Nucl. Phys. B75 (1974) 531; A.A. Belavm and A.A. Mlgdal, unpubhshed (1974).
PHYSICS LETTERS
INFRARED
14 February 1977
STABILITY OF THE YANG-MILLS THEORY
IN THE ZERO-INSTANTON
SECTOR
Poul OLESEN
The Nwls Bohr InsHtute, University of Copenhagen, DK-21 O0 Copenhagen 0 , Denmark Received 17 December 1976 Abstracting the decoupling theorem of Appelqmst and Carazzone from perturbation theory we show that the Yang-MlUs theory ISinfrared stable in the zero-mstanton sector. We point out that our argument is not valid when mstantons axe present.
It has been shown by Pogglo and Qulnn [1] that the Yang-Mills theory does not imprison quarks and gluons to any finite order of perturbation theory. In this note we wish to point out the existence of a simple argument which strongly indicates that even if one sum over all orders the Yang-Mtlls theory is Infrared stable. However, as we shall show later, our argument breaks down when instantons [2] are taken into account. Therefore there is hope that confinement sets in because of the existence of lnstantons in YangMills theories. The main assumption in the following is that the decoupling theorem of Appelquist and Carazzone [3] can be abstracted from perturbation theory. From this it follows that the pure Yang-Mills Greens function UyM can be represented in terms of a zero-mass renormallzed [4] Greens function Pw, which in addition to the vector gluons also contains virtual quark pairs [5, 6] (since P w does not decouple the quarks to any finite order) PvM(kp ' g2,/a) = qbr(mw/P, gw)
(1) 2 mw, u,M) [1 + "00, 2, U2/m2)"]. x rw(No,g;v, Possible logarithms have been ignored in the subdomlnant terms. Here m w is the quark mass in Weinberg's zero mass renormalization and gw is the coupling constant. The symbol p stands for a set of fixed momenta. The "O"-symbol indicates that the order of magnitude estimate of the neglected terms is valid to any finite order of perturbation theory, but may not be valid when summed over all orders. In any case, the abstraction of the decoupling means that we assume that the terms contained in the "O" symbol can be ignored when summation over all orders is performed. 370
The quantity M (flavor) is the number of fermion multlplets, and is explicitly written in Pw in order to indicate that Pw still depends on M [5]. Due to the independence of the left-hand side of (1) on M, the right-hand side of (1) must have a complete degeneracy with respect to M. The quantity qb is a finite multiplicative renormallzatlon constant. Because of this, with n external gluons qb is the (n/2)th power of the finite renormalization constant which connects the inverse propagator in the Y-M case and in the Weinberg case. As discussed previously [5, 6], because of the complete degeneracy with respect to flavor, we can utilize the observation of Gross and Wilczek [7] that for M large, M ~ 16 for SU(3), there is a non-trivial infrared stable fixed point, g()02 = g~ +A~V,
(2)
where 7 is a small positive power and g2 is small, g2/l&r2 = 1/302 f o r m = 16 [8]. Applying Weinberg's renormallzation group on the right-hand side of eq. (1) we obtain r y M (No, g2-~U)~ CI'r(mw//a, gw) ~D r -'rw (g l)
(3) X rw(p,gO02 , mw/;k 1+'/°(gl), I.t,M) for ), -~ 0. Here D r is the canonical dimension of P, 7W(gl) is the anomalous dimension in Weinberg's renormalization, 70(gl) is the mass dimension, and mw(;~) = mw/Xl+~o (gD,
)~ -~ O,
(4)
is the lnfini_te effective muss of the quarks. We shall now analyze the contents of eq. (3). To simplify the notation, we leave out variables which
Volume 66B, number 4
PHYSICS LETTERS
14 l-ebruary 1977
are inessential in the following, and rewrite eq. (3) as
of the type
PyM(~.) = qbr(mw)Fw(g(~) 2, mw/Xl+'ro(ga) )
g l 2n ,x (M 0 _ M')-n,
(5)
(10)
ix
is regular for g2 -+ g21. Eq. (5) then becomes the functional equation
with M 0 = T N . However, we know that in perturbation theory the dependence on M comes about through the Caslmlr operators, which are regular polynomials m M Consequently the behavior (8) can be ruled out because such a singular behaviour is not found in perturbation theory. In the case where the perturbative expansion is a divergent asymptotic expansion around g2, perturbation theory should look like
PyM (X) = ~ r (mw) ~Dr - ~ (gl)
~ C n ( g 2 _ g2)n,
X ~.ol'-')'w(gl) .
We can now distinguish two posslblhtles, namely the following: (a) The function
(gl)),
pw(g(~,)2 ' m w / ~ 1 +~'o
x rwfa 2, mw/X 1 +'tO~1)).
(6)
This equation has as the soluUon [6] a power behavior for all the functions involved, Due to the fact that (I) is the ratio of multiplicative renormahzation constants one easily sees that [6] r v M (xp, g2,/a) ~ ~,D r - n # ,
(7)
where/3 is a constant. From eq. (7) one concludes that the theory is infrared stable (the effectwe coupling approaches a constant m X for ~, -+ 0). (b) The function pw (g(X)2, mw/~,1 +'r0 (gl)), has a singularity* for g(X)2 _+ g2. In this case the behavior (7) cannot be concluded from eq. (5). On the contrary, we can say that PYM can have any behavior as a functmn of ~. Let us now analyze the case (b) in the zero-instanton sector. Since PW has a singularity when g(X) 2 approaches g2, we expect that in sufficiently high orders o f perturbation theory one encounters terms proportional to
g2m g l 2 n (m >I n),
(8)
corresponding to a radius of convergence g2 m the g 2 plane. For M sufficiently close to ~ N (for SU(N)), gl2 is given by [8] g~l
llN-
2M
16n 2 - - 3 4 N 2 + 1 3 N M - 3M/N "
(9)
(11)
where the coefficients c n are such that the series ~Cn xn has zero radius of convergence. This case also leads to a violent clash with perturbation theory. For instance, in third order one has (gl2 - g2)3 which for example gives rise to a term g4g2, which is proportional to (M 0 - M)2g 2. However, to order g2 the Casimir operators contain at most the first power of M. Tins objection becomes worse as one goes to higher orders in gl2 - g2, where terms of the type (M 0 - M)ng 2 are produced with n an arbitrarily large integer. This is in violent disagreement with perturbation theory (unless some of the c n are singular in M 0 - M, which re-introduces the singularities in (10)). Hence our argument does not depend essentially on the meaning of the perturbative expansion. We therefore conclude that the zero-instanton sec-
tor o f the Yang-Mills theory is infrared stable (power behaved Greens functions], and there is no reason for quark-gluon confinement. When we include instantons, the above argument breaks down. Presumably the decouphng and renormalizability still hold, so eq. (5) is still valid. However, now we have no argument to rule out the existence of a singular behavior in the coupling constant for g2w _~ g2, since mstantons are basically non-perturbative. Of course, it does not follow from our arguments that when instantons are present, quarks are confined. However, one may hope that coupling constant singularities occur in this case.
The perturbatwe coefficients g{2n are thus singular * This possibility was not discussed m a previous work.
The author thanks D. Gross, A. Polyakov, and K. Symanzik for interesting and stimulating discussions. 371
Volume 66B, number 4
PHYSICS LETTERS
He also thanks N.K. Nielsen for pointing out an obscurity m the original version o f the manuscript.
References [1] E.C. Pogglo and H.R. Qumn, Phys. Rev. D14 (1976) 578. [2] A.A. Belavin et al., Phys. Lett. 59B (1975) 85. [3] T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856.
372
14 February 1977
[4] S. Weinberg, Phys. Rev. D8 (1973) 3497. [5] P. Olesen, Nucl. Phys. B104 (1976) 125. [6] P. Olesen, to be published in the Proc. of the Workshop on quark binding at the Umverslty of Rochester, 1976 (John Wiley, New York). [7] D. Gross and F. Wllczek, Phys. Rev. D8 (1973) 3633. [8] W.E. CasweU, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, Nucl. Phys. B75 (1974) 531; A.A. Belavm and A.A. Mlgdal, unpubhshed (1974).