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Infrared asymptotics of gluon and quark propagators in QCD
Volume 125B, number 6
PHYSICS LETTERS
16 June 1983
INFRARED ASYMPTOTICS OF GLUON AND QUARK PROPAGATORS IN QCD B.A. ARBUZOV Institute for High Energy Physics, Serpukhov, USSR
Received 3 February 1983 Revised manuscript received 29 March 1983
An arbitrary parameter in the definition of the infrared asymptotics of the gluon propagator D (k) ~ M2/(k2) 2 in an axial gauge is shown to be fixed by boundary conditions in a coordinate space. The presence of the singularity of a quark propagator in the point p2 = rn 2 follows from the properties of the obtained gluon propagator. The last result forces us to consider seriously the possibility that coloured objects, first of all quarks, are in principle observable.
It xs well known that an investigation of the infrared region in QCD encounters great difficulties in this case due to a failure of perturbation theory. In order to go beyond perturbation theory, in several papers devoted to the infrared region [1-3] a method was developed based on an asymptotic solution of the Schwinger-Dyson set of equations using constraints imposed by Ward-Slavnov-Taylor identities. An axial gauge is mostly used in these references because in this case the equations and identities take the simplest form. The central problem of the approach concerns infrared asymptotics of the gluon propagator. Arguments were repeatedly expressed in favour of the following infrared behavlour of the gluon propagator [1 ] : D~b(k)
=
6ab[M2/(k2) 2]
gluon propagator and of multigluon vertices are described by the following effective lagrangian 2 ab b ac c Left = (1/4/14)Dp F~vD o F~v
- ( g / 6 M )2f a bc F ~av F vbo F ;cu ,
(2)
where we use the standard notations of a non-abelian theory. In particular, the first term in the lagrangian (2) gwes the propagator and the longitudinal part of the three-gluon vertex, whereas the second one gives the transverse part of the vertex. In ref. [2] the Schwinger-Dyson equation for the gluon propagator is shown to be satisfied not only by asymptotlcs (1), but by a more general form of asymptotacs. D ~ = 6ab [M2/(k2)2 l
k2-*O
X [guy - (kurlv + kvrlu)/krl + kukvrl2/(krl) 2] ,
(1)
where a, b are colour indices, flu is the constant fourvector, which defines an axial gauge by the condition (rlA a) = O, M is a constant having dmaenslon mass. In the framework of the approach mentioned, it is shown [2] that asymptotics (1) is actually valid provided the three-gluon vertex comprises not only the longitudinal part, which is defined by the Ward identity, but also the transverse part, which has not been taken into account in previous papers [1 ]. According to these results [2] the infrared asymptotics of the 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
X [guy - (kurlv + kvrl~z)/krl + kukvrl2/(krl) 2
+ a(4 - n)(guv - rlurlv/r~2)] ,
(3)
where n is the space-time dimensionality in the framework of a dimensional regularization, n -+ 4 + 2e, e ~ +0, and a is an arbitrary constant. An expression of the form (3) independently arises under investxgation of infrared asymptotics of a quark propagator with the aid of a Schwinger-Dyson equation. In partmular in ref. [3] is shown that, provided the gluon propagator has the form (3) with a = 1, a quark propagator has the quasi-free form G(p)= A l P - m ~ = 497
Volume 125 B, number 6
PHYSICS LETTERS
%~p'~) near the mass shell. A natural question arises concerning the sense o f the second term In expression (3), whmh evidently vanishes in the limit n ~ 4. In the present note we show that its sense is clarified by studying the gluon Green function in a coordinate space. Let us perform a Fourier transformation of expression (3). While doing this we use a dimensional regularization and take principal values of the denominators (kr/)a. A limit for n ~ 4 exists and is equal to
bgb(x)= lira ~
n~4 1, )
fdnk Dg~(k)exp(ikx)
= 6ab(1M2/4rr 2) X ([guy - (xurlv + xvrlu)/xrl + xuxvrl2/(xn) 2] F ( y ) + (g.~ - rlurlv/rl2)[-FO,)/y + ½(1 - a ) l } ,
(4)
where
16 June 1983
A~)(x) = fla' f dt dz DO0 ~a'a (t, x - z) p (z)
=
i~aM2f dz O(z) f art(1-
t2~2/(x
-
z,
- 1 ] y ) F ( y ) + ½(1 - a)] ,
r/) 2
(7)
y = -(rl, x - z)2/[t 2 - (x - z)21 772 . In as much as y -+ - e 2 / t 2, t -+ oo and according to eq. (5) F ( v ) ~ y / 2 , y -+ - 0 , the time integral converges then and only then if a = 1. So the boundary condition fixes the value a = 1, and the gluon Green's function in a coordinate space takes the following form (in an arbitrary axial gauge because a is gauge independent).
JDgb (x ) = 6ab ( iM2 / 4 rr2) X [guy -- (xur~v + xuxu)/xrl + xlaxvrl2/(xrl)2
y = (xrl)2/x2rl 2 ' - (1/y)(guu - rturlu/r/2)] F ( y ) . F ( y ) = [y2/2(1 - y ) ] F ( 1 , 1 ; - ~ ; y ) + y / 2 ( 1 - y ) ,
As for the field of a point-like source, it can conveniently be obtained from calculations in a m o m e n t u m space
y~<0, F ( y ) = - Lv2/2 (1 - y ) ] F ( 1 , 1, ~; 1 - y ) +y/2 (1 - y ) ,
t
~
Aao(X)= J3a j y > 0,
~
r
dt D g ~ ( t , x )
(5)
F(a, b; c ; y ) is the Gauss h y p e r g e o m e m c function. By a direct substitution one is convinced that the Green's function defined by relations (4), (5) satisfies the equation
apapa o b
(8)
(x - z) - apao o.a
b
-- lim
n--,4 (2
-1
dnkS(ko)D~(k)exp(-ikx)
M2flarr (n-l)/2 = hm n--~4
(2 rr)n - 1
1+a(4-n) P(½(n-5))
2n- 3
(x - z)
X (x2) (n-3)/2 . = 8ab(guv -- 8vn~/Orl ) 6(x - z ) M 2 ,
(6)
where b o = O/ax;. This equation follows also from lagrangian (2) in an axial gauge. Solution (4) of eq. (6) has an ambigmty due to an arbitrary constant a. One has to use boundary conditions to fix this constant. We formulate boundary conditions demanding the existence o f a field o f a static source. Let the statm source be given by the relatmn ]if, = 13aSuoO(x). In this case one should choose the space-like gauge vector r/u = (0, r/). Then by definition
498
Here we have not yet put a = 1, as it follows from the boundary condition, with a view to demonstrate once more qualitatively the necessity of this demand. Indeed, if a 4: 1, it is impossible to continue the result from the region n > 5 where the integral undoubtedly exists, to the physical value n = 4, because there is a pole m n = 5 between the two regions. However l f a = 1 the pole vanishes, continuation becomes possible, and we obtain
Volume 125B, number 6
PHYSICS LETTERS
Aao(x) = -(3a/8rr) M2r.
(9)
So, by considering boundary conditmns in coordinate space we have proved that the infrared asymptotics of the gluon propagator of the type D(k) M2/(k2) 2 has the umque form (8) in any axial gauge. In momentum space this corresponds to expression (3) with a = 1. The result (8) leads to a very important consequence. We have quoted the result [3] according to which the condition a = 1 leads to quasi-free infrared asymptotics of a quark propagator. One can obtain the same result using the functional integration method in the approximation of soft gluons. Really, following the method developed in refs. [ 4 - 6 ] one easily obtains the following representation for a fermion propagator m this approximation which gives the main infrared term G(p) = P + m 1
j
dr exp [ir(p 2 - m2)]
0
×~
(10) dr
0
t
tr ~ a ~
t
¢!
dr Duu(2p(r - r ))PuPv , 0
where q~(0) = 1 and the form of function ~ depends on the theory under consideration. For example in QED q~is an exponential function. Here the form of function q~does not matter because expression (8) for the infrared asymptotics of/)gb(x) has a wonderful property: ~ac Duv(bx ) xux v = 0 ,
(11)
for arbitrary b and for any value of the gauge parametery. Note that this property is fulfilled exclusively due to the boundary c o n d m o n a = 1. So from eqs. (10) and (11) we obtain the leading infrared term of a quark propagator in any axial gauge 1
G ~ pZmr~--P _ m
16 June 1983
complete analogy with the situation In QED. Properties of the corrections need of course additional investigation. So we are convinced that a ~uark propagator has a singularity in the point p2 = m ~ at least in an arbitrary axial gauge. Thus it can by no means be an entire function o f p ~ as is often assumed for the justification of the confinement hypothesis. An analogy with confining two-dimensional QCD is also often used to support this hypothesis. The result of the present work differ positively from the situation m two dimensions. In the latter case the infrared asymptotlcs of/)~b(x) coincides with the free one and the argument of ¢ in eq. (10) m an axial gauge is proportional to g272p2(1 - y ) instead of zero (11) for any y. So a quark propagator in two dimensions has the form (12) only f o r y = 1, and for other values o f y it is apparently an entire function o f p 2. Note one more distinction of fourdimensional QCD from two-dimensional QCD. Even for the general asymEtotics (3), (4) the double loop integral f dx u f dyVDgb(x - y) is equal to zero, whereas in two dimensions this integral is proportional to the area of the loop which corresponds to fulfilment of the Wilson criterion. So, the results of the present work, as well as of several previous ones [3,8, 9], do not support the hypothesis of absolute confinement. As for the linearly rising potential (9), one has to bear in mind of course that this expression is valid only in one full gluon exchange approximation. It is not excluded that taking into account all gluon exchanges one can obtain a change of the potential behaviour at very large distances, in particular a decrease for r -+ oo. Thus the potential may have the form of a very high hill, which predicts a very high energy threshold for free quark production (>35 GeV [10] ; see, however, ref. [11 ]). To conclude, let us emphasize that further searches for quarks quite make sense.
References (12)
Further corrections for the main approximation are apparently similar to the ones in electrodynamics. For example, account in eq. (10) for possible additional terms with infrared asymptohcs c]k 2 in propagator (3) only may lead to replacement of the pole (12) with a branch point i n p 2 = m 2 (see e.g. ref. [7]), in
[1] H. Pagels, Phys. Rev. D15 (1977) 2991; R. Anishetty et al., Phys. Lett. 86B (1979) 52; J.S. Ball et al., Nucl. Phys. B186 (1981) 531. [2] A.I. Alekseev, B.A. Arbuzov and V.A. Baykov, Teor. Mat. Fiz. 52 (1982) 187 [3] A.I. Alekseev, B.A. Arbuzov and V.A. Baykov, Yad. Fiz. 34 (1981) 1347. [4] A.A. Logunov, JETP 29 (1955) 828. 499
Volume 125B, number 6
PHYSICS LETTERS
[5] B.M. Barbashov, JETP 48 (1965) 607. [6] E.S. Fradkin, Method of Green functions in quantum field theory and in quantum statistics. Proc. Lebedev Physical Institute (Moscow, 1965) Vol. 29. [7] V.N. Popov and T.T. Wu, Phys. Lett. 85B (1979) 395.
500
16 June 1983
[8] B.A. Arbuzov and S.S. Kurennoj, Yad. Fiz. 36 (1982) 1314. [9] B.A. Arbuzov et al., preprint IHEP, 82-128 (Serpukhov, 1982). [10] W. Bartel et al., preprint DESY 80/71 (1980). [11] G.S. La Rue, D.P. James and W.M. Fairbank, Phys. Rev. Lett. 46 (1981) 967.
PHYSICS LETTERS
16 June 1983
INFRARED ASYMPTOTICS OF GLUON AND QUARK PROPAGATORS IN QCD B.A. ARBUZOV Institute for High Energy Physics, Serpukhov, USSR
Received 3 February 1983 Revised manuscript received 29 March 1983
An arbitrary parameter in the definition of the infrared asymptotics of the gluon propagator D (k) ~ M2/(k2) 2 in an axial gauge is shown to be fixed by boundary conditions in a coordinate space. The presence of the singularity of a quark propagator in the point p2 = rn 2 follows from the properties of the obtained gluon propagator. The last result forces us to consider seriously the possibility that coloured objects, first of all quarks, are in principle observable.
It xs well known that an investigation of the infrared region in QCD encounters great difficulties in this case due to a failure of perturbation theory. In order to go beyond perturbation theory, in several papers devoted to the infrared region [1-3] a method was developed based on an asymptotic solution of the Schwinger-Dyson set of equations using constraints imposed by Ward-Slavnov-Taylor identities. An axial gauge is mostly used in these references because in this case the equations and identities take the simplest form. The central problem of the approach concerns infrared asymptotics of the gluon propagator. Arguments were repeatedly expressed in favour of the following infrared behavlour of the gluon propagator [1 ] : D~b(k)
=
6ab[M2/(k2) 2]
gluon propagator and of multigluon vertices are described by the following effective lagrangian 2 ab b ac c Left = (1/4/14)Dp F~vD o F~v
- ( g / 6 M )2f a bc F ~av F vbo F ;cu ,
(2)
where we use the standard notations of a non-abelian theory. In particular, the first term in the lagrangian (2) gwes the propagator and the longitudinal part of the three-gluon vertex, whereas the second one gives the transverse part of the vertex. In ref. [2] the Schwinger-Dyson equation for the gluon propagator is shown to be satisfied not only by asymptotlcs (1), but by a more general form of asymptotacs. D ~ = 6ab [M2/(k2)2 l
k2-*O
X [guy - (kurlv + kvrlu)/krl + kukvrl2/(krl) 2] ,
(1)
where a, b are colour indices, flu is the constant fourvector, which defines an axial gauge by the condition (rlA a) = O, M is a constant having dmaenslon mass. In the framework of the approach mentioned, it is shown [2] that asymptotics (1) is actually valid provided the three-gluon vertex comprises not only the longitudinal part, which is defined by the Ward identity, but also the transverse part, which has not been taken into account in previous papers [1 ]. According to these results [2] the infrared asymptotics of the 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
X [guy - (kurlv + kvrl~z)/krl + kukvrl2/(krl) 2
+ a(4 - n)(guv - rlurlv/r~2)] ,
(3)
where n is the space-time dimensionality in the framework of a dimensional regularization, n -+ 4 + 2e, e ~ +0, and a is an arbitrary constant. An expression of the form (3) independently arises under investxgation of infrared asymptotics of a quark propagator with the aid of a Schwinger-Dyson equation. In partmular in ref. [3] is shown that, provided the gluon propagator has the form (3) with a = 1, a quark propagator has the quasi-free form G(p)= A l P - m ~ = 497
Volume 125 B, number 6
PHYSICS LETTERS
%~p'~) near the mass shell. A natural question arises concerning the sense o f the second term In expression (3), whmh evidently vanishes in the limit n ~ 4. In the present note we show that its sense is clarified by studying the gluon Green function in a coordinate space. Let us perform a Fourier transformation of expression (3). While doing this we use a dimensional regularization and take principal values of the denominators (kr/)a. A limit for n ~ 4 exists and is equal to
bgb(x)= lira ~
n~4 1, )
fdnk Dg~(k)exp(ikx)
= 6ab(1M2/4rr 2) X ([guy - (xurlv + xvrlu)/xrl + xuxvrl2/(xn) 2] F ( y ) + (g.~ - rlurlv/rl2)[-FO,)/y + ½(1 - a ) l } ,
(4)
where
16 June 1983
A~)(x) = fla' f dt dz DO0 ~a'a (t, x - z) p (z)
=
i~aM2f dz O(z) f art(1-
t2~2/(x
-
z,
- 1 ] y ) F ( y ) + ½(1 - a)] ,
r/) 2
(7)
y = -(rl, x - z)2/[t 2 - (x - z)21 772 . In as much as y -+ - e 2 / t 2, t -+ oo and according to eq. (5) F ( v ) ~ y / 2 , y -+ - 0 , the time integral converges then and only then if a = 1. So the boundary condition fixes the value a = 1, and the gluon Green's function in a coordinate space takes the following form (in an arbitrary axial gauge because a is gauge independent).
JDgb (x ) = 6ab ( iM2 / 4 rr2) X [guy -- (xur~v + xuxu)/xrl + xlaxvrl2/(xrl)2
y = (xrl)2/x2rl 2 ' - (1/y)(guu - rturlu/r/2)] F ( y ) . F ( y ) = [y2/2(1 - y ) ] F ( 1 , 1 ; - ~ ; y ) + y / 2 ( 1 - y ) ,
As for the field of a point-like source, it can conveniently be obtained from calculations in a m o m e n t u m space
y~<0, F ( y ) = - Lv2/2 (1 - y ) ] F ( 1 , 1, ~; 1 - y ) +y/2 (1 - y ) ,
t
~
Aao(X)= J3a j y > 0,
~
r
dt D g ~ ( t , x )
(5)
F(a, b; c ; y ) is the Gauss h y p e r g e o m e m c function. By a direct substitution one is convinced that the Green's function defined by relations (4), (5) satisfies the equation
apapa o b
(8)
(x - z) - apao o.a
b
-- lim
n--,4 (2
-1
dnkS(ko)D~(k)exp(-ikx)
M2flarr (n-l)/2 = hm n--~4
(2 rr)n - 1
1+a(4-n) P(½(n-5))
2n- 3
(x - z)
X (x2) (n-3)/2 . = 8ab(guv -- 8vn~/Orl ) 6(x - z ) M 2 ,
(6)
where b o = O/ax;. This equation follows also from lagrangian (2) in an axial gauge. Solution (4) of eq. (6) has an ambigmty due to an arbitrary constant a. One has to use boundary conditions to fix this constant. We formulate boundary conditions demanding the existence o f a field o f a static source. Let the statm source be given by the relatmn ]if, = 13aSuoO(x). In this case one should choose the space-like gauge vector r/u = (0, r/). Then by definition
498
Here we have not yet put a = 1, as it follows from the boundary condition, with a view to demonstrate once more qualitatively the necessity of this demand. Indeed, if a 4: 1, it is impossible to continue the result from the region n > 5 where the integral undoubtedly exists, to the physical value n = 4, because there is a pole m n = 5 between the two regions. However l f a = 1 the pole vanishes, continuation becomes possible, and we obtain
Volume 125B, number 6
PHYSICS LETTERS
Aao(x) = -(3a/8rr) M2r.
(9)
So, by considering boundary conditmns in coordinate space we have proved that the infrared asymptotics of the gluon propagator of the type D(k) M2/(k2) 2 has the umque form (8) in any axial gauge. In momentum space this corresponds to expression (3) with a = 1. The result (8) leads to a very important consequence. We have quoted the result [3] according to which the condition a = 1 leads to quasi-free infrared asymptotics of a quark propagator. One can obtain the same result using the functional integration method in the approximation of soft gluons. Really, following the method developed in refs. [ 4 - 6 ] one easily obtains the following representation for a fermion propagator m this approximation which gives the main infrared term G(p) = P + m 1
j
dr exp [ir(p 2 - m2)]
0
×~
(10) dr
0
t
tr ~ a ~
t
¢!
dr Duu(2p(r - r ))PuPv , 0
where q~(0) = 1 and the form of function ~ depends on the theory under consideration. For example in QED q~is an exponential function. Here the form of function q~does not matter because expression (8) for the infrared asymptotics of/)gb(x) has a wonderful property: ~ac Duv(bx ) xux v = 0 ,
(11)
for arbitrary b and for any value of the gauge parametery. Note that this property is fulfilled exclusively due to the boundary c o n d m o n a = 1. So from eqs. (10) and (11) we obtain the leading infrared term of a quark propagator in any axial gauge 1
G ~ pZmr~--P _ m
16 June 1983
complete analogy with the situation In QED. Properties of the corrections need of course additional investigation. So we are convinced that a ~uark propagator has a singularity in the point p2 = m ~ at least in an arbitrary axial gauge. Thus it can by no means be an entire function o f p ~ as is often assumed for the justification of the confinement hypothesis. An analogy with confining two-dimensional QCD is also often used to support this hypothesis. The result of the present work differ positively from the situation m two dimensions. In the latter case the infrared asymptotlcs of/)~b(x) coincides with the free one and the argument of ¢ in eq. (10) m an axial gauge is proportional to g272p2(1 - y ) instead of zero (11) for any y. So a quark propagator in two dimensions has the form (12) only f o r y = 1, and for other values o f y it is apparently an entire function o f p 2. Note one more distinction of fourdimensional QCD from two-dimensional QCD. Even for the general asymEtotics (3), (4) the double loop integral f dx u f dyVDgb(x - y) is equal to zero, whereas in two dimensions this integral is proportional to the area of the loop which corresponds to fulfilment of the Wilson criterion. So, the results of the present work, as well as of several previous ones [3,8, 9], do not support the hypothesis of absolute confinement. As for the linearly rising potential (9), one has to bear in mind of course that this expression is valid only in one full gluon exchange approximation. It is not excluded that taking into account all gluon exchanges one can obtain a change of the potential behaviour at very large distances, in particular a decrease for r -+ oo. Thus the potential may have the form of a very high hill, which predicts a very high energy threshold for free quark production (>35 GeV [10] ; see, however, ref. [11 ]). To conclude, let us emphasize that further searches for quarks quite make sense.
References (12)
Further corrections for the main approximation are apparently similar to the ones in electrodynamics. For example, account in eq. (10) for possible additional terms with infrared asymptohcs c]k 2 in propagator (3) only may lead to replacement of the pole (12) with a branch point i n p 2 = m 2 (see e.g. ref. [7]), in
[1] H. Pagels, Phys. Rev. D15 (1977) 2991; R. Anishetty et al., Phys. Lett. 86B (1979) 52; J.S. Ball et al., Nucl. Phys. B186 (1981) 531. [2] A.I. Alekseev, B.A. Arbuzov and V.A. Baykov, Teor. Mat. Fiz. 52 (1982) 187 [3] A.I. Alekseev, B.A. Arbuzov and V.A. Baykov, Yad. Fiz. 34 (1981) 1347. [4] A.A. Logunov, JETP 29 (1955) 828. 499
Volume 125B, number 6
PHYSICS LETTERS
[5] B.M. Barbashov, JETP 48 (1965) 607. [6] E.S. Fradkin, Method of Green functions in quantum field theory and in quantum statistics. Proc. Lebedev Physical Institute (Moscow, 1965) Vol. 29. [7] V.N. Popov and T.T. Wu, Phys. Lett. 85B (1979) 395.
500
16 June 1983
[8] B.A. Arbuzov and S.S. Kurennoj, Yad. Fiz. 36 (1982) 1314. [9] B.A. Arbuzov et al., preprint IHEP, 82-128 (Serpukhov, 1982). [10] W. Bartel et al., preprint DESY 80/71 (1980). [11] G.S. La Rue, D.P. James and W.M. Fairbank, Phys. Rev. Lett. 46 (1981) 967.