- Email: [email protected]
On soft singular infrared behaviour of the gluon propagator
26 January 1995 PHYSICS
LETTERS
B
Physics Letters B 344 (1995) 325-328
On soft singular infrared behaviour of the gluon propagator A.I. Alekseev
’
Institute for High Energy Physics, Protvino 142284, Moscow Region, Russia
Received 20 September 1994; revised manuscript received 28 November 1994 Editor: PV. Landshoff
Abstract The Schwinger-Dyson equation for the gluon propagator in the axial gauge is considered with account of the gauge identities for the gluon Green’s functions. A possibility of soft singular power type infrared behaviour for the gluon propagator
is studied, D(q) - (q*) -‘, q* + 0 where c is a small positive number. Under certain assumptions a characteristic equation for the exponent is obtained. This equation is shown not to have solutions in the interval 0 < c < 1.
Clarification of the gluon propagator behaviour in the infrared region is one of the most important problems for understanding nonperturbative aspects of QCD. Recently a possibility of soft singular power type behaviour for full gluon propagator at the origin is being discussed [ 11. Under soft singularity one implies the behaviour which is softer then the pole of bare propagator. The possibility of power infrared behaviour of the full gluon propagator was considered in Refs. [ 21, but the soft singular behaviour was not found. The interest to the soft singular behaviour is connected with the success [ 31 of phenomenological models of nonperturbative exchange of quarks and gluons when describing diffractive scattering, gluon structure function, elastic scattering and total cross section. On the one hand, the soft singular behaviour of the gluon propagator provides the fulfillment of the gauge condition rrFp (0) = 0, on the other hand in phenomenological applications it allows one to consider the gluon propagator at the origin practically as a constant.
’ E-mail address: [email protected]. Elsevier Science B.V. KSDI 0370-2693( 94)01570-8
Various investigations of the field equations do not lead to the unique result for the infrared gluon propagator behaviour. The widespread inference (e.g., Refs. [4-61) is that D(q) N (q*)-*,q* + 0. This behaviour gives the linear qq potential at large distances in one dressed gluon exchange approximation. Moreover this infrared behaviour serves as a basis in physically attractive picture of the QCD vacuum as dual superconductive medium [7]. Refs. [ 81 give grounds for the constant behaviour of the gluon propagator at the origin. The behaviour D(q) l/q* ln*(q*/p*), 9* + 0 was obtained [9] using the developed gauge technique. As noted in Refs. [l] the fact that the equation for the propagator is highly nonlinear can lead to the existence of different solutions. In a more wide sense the question arises if non-uniqueness of the infrared behaviour of the gluon propagator is connected with the truncation methods of the equations for Green’s functions or it reflects the complicated structure of the QCD vacuum. Here we consider the equation whose investigation allowed the authors of Refs. [ 1 ] to come to a conclu-
326
A.I. Aleliseev / Physics Letters 6 344 (1995) 325-328
sion on the existence of the solution having soft power behaviour at the origin. We shall briefly describe the way of obtaining the main closed integral equation. Let us consider the Schwinger-Dyson equation for the gluon propagator for the pure Yang-Mills theory in ghost-free axial gauge [lo] defined via the condition A:qcL = 0, vp is a fixed axial gauge vector. Following the approach of [ 51 we assume that the transverse part of the triple gluon vertex does not play an important role in determining low momentum behaviour of the gluon propagator. The longitudinal part of the triple gluon vertex can be expressed [ 111 in terms of scalar functions characterizing the propagator via the Ward-SlavnovTaylor identity. In the axial gauge the propagator depends, in general, on two scalar functions. Suppose also that the gluon propagator is proportional [ 5 ] to the bare one, i.e., D,,(q)
= Z(q2PC0)(q) PV
-ml*)
1
4prlv + &V/J +
=-- cl* gw-
4rwl*
(ml
(m)*
1
’ (1)
where Z(q*) does not depend on the gauge parameter y(q) = (qv)*/q*v*. The identity r),D,,(k) =0 allows one to go to the equation contracted with vector 71~ which do not contain quadrupole gluon vertex. The tadpole term is independent of the external momentum, and angular integration turns it to zero. After approximate angular integration [ 121 and ultraviolet subtractions the equation can be obtained for renormalized function ZR(q*) = Z(q’)/Z(p*),p* is a renormalization point. This equation has the form [ I] : 1 --=1++++&), zR(q*) Y2 F
=
P=
m.l*,yb.iy
-
s 0
+
00
J
F2(q*vY) -
92
J
F’(~*~y)dY~
2
T2 =
F&q2>y)dy s
0
JF&*4& Ir2
‘I2 -
0
+
co
J Fdq*>y>- JFdp*,y)dy.
92
(2)
P2
The functions FI,~(x, y) in Eq. (2) are linear in ZR(X), ZR(Y), ZR(X + y), and the functions F4,5( x, y) are quadratic. It is assumed that Eq. (2) obtained as a result of the approximations described above gives correct infrared behaviour of the propagator and does not claim to describe the ultraviolet behaviour of the propagator. After numerical investigation of Eq. (2) in Refs. [ l] the statement is made on the presence of the solution ZR(q*) which has soft singular power asymptotics for low q*. The integration in Eq. (2) is carried out over all values of q*. Refs. [ l] consider two variants of the behaviour of ZR(q*) at q” -+ co which differ from unity of the bare propagator by the power of the logarithm of q*. As it turns out the exact behaviour of ZR(q*) at q* -+ 03 does not strongly influence the behaviour of the solution at zero. Smooth matching of the behaviour at zero and at infinity was realized by introducing a number of parameters. Fitting the parameters has shown that ZR(~*) behaves as (q*) I-’ at q* + 0 with quite sufficient accuracy. In this the value of the parameter c is within the interval from 0.01 to 0.3 and more singular behaviour of the propagator is excluded. We will show that for non-integer and non-half-integer values of the exponent c the behaviour of the propagator at intermediate and large values of the momentum is really unessential and we will obtain the characteristic equation for the determination of the parameter c. Suppose that ZR(q*) has not any singularities at 0 < q* < cm and the behaviour of Z,( q2) at infinity is adequate to the subtraction procedure of Eq. (2) (e.g., power of the logarithm of q*). Then let the infrared behaviour be described by the series of the form: ZR(x) = (x/p*)‘-’
mp*,y)dy s 0
co
co
x (~o+cuix/~*+
(Y*(X/p*)*+...)
(3)
with a nonzero radius of convergence. We will take the advantage of the circumstance that we can extract the terms of the integrals in the quantities TI and T2 with power dependence on q* at q’ -+ 0 for the infrared behaviour Eq. (3) with non-integer c and for any non-singular behaviour of Za ( X) at finite and positive values of q*.
A.I. Aleheev / PhysicsLettersB 344 (I 995) 325-328
Eq. (2) includes
the integrals
over y from zero to over y from q” to infinity which we will denote as J with different indexes. Dealing with separate integrals we will introduce an auxiliary constant AL instead of the infinite limit of integration. This constant will not enter the final expressions for Tr and T2 which do not contain the ultraviolet divergences. The integrals of type 1 display power behaviour for ZR (q2) described by Eq. (2) at q” + 0. The analysis of the type J integrals shows that apart from the non-integer power terms they contain the integer power series and terms behaving as l/( q2)“:, m = 1,2. We can calculate the terms of type I integrals which behave like ( q2) ‘-’ at q2 ----t 0 and the terms of type J integrals with the behaviour (q2)2-2c, l/q2, l/(q2)2 at q2 -_) 0. Substitute the expressions for the type I, J integrals into 7’1 and T2 of Eq. (2). The terms behaving as 1/q2, l/( q2)2 are mutually canceled and we obtain q2, which we will denote as I and the integrals
Tl = PI (s2) + ~&72/~2)1-c T2 = P2(q2)
+
f
{AI (c) -t O(q2,‘p2)},
(4)
that Eq. (7) is the equation for function Z,(X) with assumed infrared behaviour rather then for the values of the variable x. If we substitute the solution ZR( x) of Eq. (2) into Eq. (6), which has the structure of Eq. (7), then Eqs. (6)) (7) turn into identities. We will apply the theorem of uniqueness for the power series. Then, rounding the branch point x = 0 and subtracting the initial identity from the obtained one we conclude that pn = q,, = r, = 0,~ = 0, 1,2, . . . for noninteger and non-half-integer values of the parameter c. Although the terms of Eq. (6) behaving as ( q2) 1-c at q2 --f 0 are not the leading terms, they are unambiguously defined by the leading term of the infrared asymptotics, Eq. (3). Then the value of the exponent of the infrared asymptotics should be determined by the characteristic equation A(c) = 0. Let us consider the characteristic function A(c) at 0 < c 5 1. This function has the poles at c = 0, l/2,1. We will extract explicitly the pole terms and obtain the integral representation which is convenient for the numerical study. After somewhat lengthy calculations which will not be presented here the characteristic function takes the form:
zR(q2>Q2(q2>
~6(q2/p2)2-2c {b(c) + 0(q2/p2)}.
(5)
The functions PI (q2), P2(q2), Q2(q2) in Eqs. (4), (5) are regular at q’ = 0 and they depend on the behaviour of ZR(~~) at all values of q2 > 0. We do not determine these functions here. Then Eq. (2) takes the form:
327
A(c)
=X+2- 12 1 -
G>
= ;{I
1
2c
2 1 - 5I_c+&c),
(8)
dt, (1 + t)1-c(28t-3+2c
+ 28t2-C
0
_ gt-2+c - 16( 1 + t-3+2c >( 1 - P)/(
1 - t))
- 12t-3+2c + 4( 1+ 3c) t-2+2c I
= 1 + 3%(P2) -{PI
zR(q2)
(q2>
IT
+
+ 2( 16 - 5c - 3~~)t-r~~~
Qh2>
- (23 - 7~)t-‘+~]
1 +
--P2Gz2) zR(q2)
+ wdq2/~2~‘-c[AW
- lly(4 +
Wq2/p2>1},
(6)
where A (c) = A 1(c) -!-A2 (c) is a dimensionless characteristic function. According to our assumption on the analytical structure of the solution ZR ( X) Eq. (6) has the following structure: P(x)
+ x1-‘Q(x)
+ @R(x)
= 0,
(7)
where P(X), Q(x), R(x) are integer power series with nonzero radii of convergence and with some coefficients pn, qn, r,, n = 0, 1,2, . . . correspondingly. Note
+ 6/(3
- 7r-2+c
- 23y(2 - c) + 41y(3
- c) + ~(5 - c) - 16/(2 - c) + 6/(4
- 8 + 3~).
- c) - l/(5
- c)
- c) - c) (9)
wherey(x) = (2X-1)/x,andA(c) =Ar(c)+A2(c) is regular at 0 I c 5 1. The subtractions in the integral representation for A(c) provide finiteness of the function under integral. The signs of the residues in the poles c = 0, l/2,1 are +, -, + respectively. So, one can expect an even number of roots of the characteristic equation in the interval [ 0, 1] of the parameter c. In Fig. 1 we show the characteristic function A(c) as are-
A.I. Alekseev / Physics Letters B 344 (1995) 325-328
328
2T
I wish to thank B.A. Arbuzov, V.E. Rochev and S.N. Sokolov for useful discussions. This work is supported by the Russian Foundation of Fundamental Investigations, Grant No. 94-02-03545-a.
10 5 0
References
-5
‘;‘
-10 15 -15
. v
[ 1] J.R. Cudell and D.A. Ross, NucI. Phys. B 359 (1991) 247;
I‘\ 0
0.25 0.5 0.75
1 c
Fig. 1. The characteristic function A(c) for the power infrared behaviour of the gluon propagator D(q) N (q2) --c, q2 -+ 0.
sult of numerical integration of Eqs. (8)) (9). It can be seen that the characteristic equation has no solutions for the soft-singular behaviour of the gluon propagator. The module of the characteristic function has the local minima at cl N 0.28, c2 11 0.68. Weak dependence of the results of the numerical analysis [ 1] on the exact behaviour of Za(q2) at large q” allows one to suppose that the leading terms of Eq. (6) at q2 4 0 are numerically small. In this case the parameters fitting to minimize the difference between Zpt ( q2) and ZF(q2) will give the value of the parameter c corresponding to the minimum of IA(c) I. It explains the results of the numerical investigation of Refs. [ 11. Note that the consideration of the particular case of the infrared behaviour Eq. (3) does not exclude the possibility of a more general soft singular power infrared behaviour. The method of extracting nonleading terms at q2 -+ 0 of the integral equation used in this paper can be generalized to investigate this case. Having in mind the problem of finding the selfconsistent solution for the gluon propagator in the infrared region it is worth considering the approximations leading to Eq. (2) more carefully. One of the possibilities is to include the transverse part of the triple gluon vertex into the consideration [ 61.
J.R. Cudell, CERN preprint CERN-TH 6192/91 (1991). [2] A.I. Alekseev, Teor. Mat. Fiz. 48 ( 1981) 324; A.N. Vassiliev, Yu.M. Pis’mak, Yu.R. Honkonen, Teor. Mat. Fiz. 48 (1981) 284. [3] F!V. Landshoff and 0. Nachtmann, Z. Phys. C 35 (1987) 405; J.R. Cudell, A. Donnachie and PV. Landshoff, Nucl. Phys. B 322 (1989) 55. [4] S. Mandelstam, Phys. Rev. D 20 (1979) 3223; H. Pagels, Phys. Rev. D 15 ( 1977) 2991; N. Brown, R. Pennington, Phys. Rev. D 38 ( 1988) 2266. [5] M. Baker, J.S. Ball and E Zachariasen, Nucl. Phys. B 186 (1981) 531, 560. [6] AI. Alekseev, Yad. Fiz. 33 (1981) 516; AI. Alekseev, V.F. Edneral, Yad. Fiz. 45 (1987) 1105. [7] Cl.? Hooft, Phys. Scripta 25 (1982) 133; V.P. Nair, C. Rosenzweig, Phys. Rev. D 31 ( 1985) 401; M. Baker, J.S. Ball and E Zachariasen, Phys. Rep. 209 (1991) 73. [8] J.M. Cornwall, Phys. Rev. D 26 (1982) 1453; J.M. Namyslowski, Warsaw Univ. preprint IFI’/13/90 ( 1990) ; V.V. Kiselev, IHEP preprint IHEP 92-5 1 ( 1992). [9] R. Delbourgo, J. Phys. G: Nucl. Phys. 5 (1979) 603. [IO] W. Kummer, Acta Phys. Austr. 41 (1975) 315. [ll] S.K. Kim, M. Baker, Nucl. Phys. B 164 (1980) 152. [ 121 W.J. Schoenmaker, Nucl. Phys. B 191 (1982) 535.
LETTERS
B
Physics Letters B 344 (1995) 325-328
On soft singular infrared behaviour of the gluon propagator A.I. Alekseev
’
Institute for High Energy Physics, Protvino 142284, Moscow Region, Russia
Received 20 September 1994; revised manuscript received 28 November 1994 Editor: PV. Landshoff
Abstract The Schwinger-Dyson equation for the gluon propagator in the axial gauge is considered with account of the gauge identities for the gluon Green’s functions. A possibility of soft singular power type infrared behaviour for the gluon propagator
is studied, D(q) - (q*) -‘, q* + 0 where c is a small positive number. Under certain assumptions a characteristic equation for the exponent is obtained. This equation is shown not to have solutions in the interval 0 < c < 1.
Clarification of the gluon propagator behaviour in the infrared region is one of the most important problems for understanding nonperturbative aspects of QCD. Recently a possibility of soft singular power type behaviour for full gluon propagator at the origin is being discussed [ 11. Under soft singularity one implies the behaviour which is softer then the pole of bare propagator. The possibility of power infrared behaviour of the full gluon propagator was considered in Refs. [ 21, but the soft singular behaviour was not found. The interest to the soft singular behaviour is connected with the success [ 31 of phenomenological models of nonperturbative exchange of quarks and gluons when describing diffractive scattering, gluon structure function, elastic scattering and total cross section. On the one hand, the soft singular behaviour of the gluon propagator provides the fulfillment of the gauge condition rrFp (0) = 0, on the other hand in phenomenological applications it allows one to consider the gluon propagator at the origin practically as a constant.
’ E-mail address: [email protected]. Elsevier Science B.V. KSDI 0370-2693( 94)01570-8
Various investigations of the field equations do not lead to the unique result for the infrared gluon propagator behaviour. The widespread inference (e.g., Refs. [4-61) is that D(q) N (q*)-*,q* + 0. This behaviour gives the linear qq potential at large distances in one dressed gluon exchange approximation. Moreover this infrared behaviour serves as a basis in physically attractive picture of the QCD vacuum as dual superconductive medium [7]. Refs. [ 81 give grounds for the constant behaviour of the gluon propagator at the origin. The behaviour D(q) l/q* ln*(q*/p*), 9* + 0 was obtained [9] using the developed gauge technique. As noted in Refs. [l] the fact that the equation for the propagator is highly nonlinear can lead to the existence of different solutions. In a more wide sense the question arises if non-uniqueness of the infrared behaviour of the gluon propagator is connected with the truncation methods of the equations for Green’s functions or it reflects the complicated structure of the QCD vacuum. Here we consider the equation whose investigation allowed the authors of Refs. [ 1 ] to come to a conclu-
326
A.I. Aleliseev / Physics Letters 6 344 (1995) 325-328
sion on the existence of the solution having soft power behaviour at the origin. We shall briefly describe the way of obtaining the main closed integral equation. Let us consider the Schwinger-Dyson equation for the gluon propagator for the pure Yang-Mills theory in ghost-free axial gauge [lo] defined via the condition A:qcL = 0, vp is a fixed axial gauge vector. Following the approach of [ 51 we assume that the transverse part of the triple gluon vertex does not play an important role in determining low momentum behaviour of the gluon propagator. The longitudinal part of the triple gluon vertex can be expressed [ 111 in terms of scalar functions characterizing the propagator via the Ward-SlavnovTaylor identity. In the axial gauge the propagator depends, in general, on two scalar functions. Suppose also that the gluon propagator is proportional [ 5 ] to the bare one, i.e., D,,(q)
= Z(q2PC0)(q) PV
-ml*)
1
4prlv + &V/J +
=-- cl* gw-
4rwl*
(ml
(m)*
1
’ (1)
where Z(q*) does not depend on the gauge parameter y(q) = (qv)*/q*v*. The identity r),D,,(k) =0 allows one to go to the equation contracted with vector 71~ which do not contain quadrupole gluon vertex. The tadpole term is independent of the external momentum, and angular integration turns it to zero. After approximate angular integration [ 121 and ultraviolet subtractions the equation can be obtained for renormalized function ZR(q*) = Z(q’)/Z(p*),p* is a renormalization point. This equation has the form [ I] : 1 --=1++++&), zR(q*) Y2 F
=
P=
m.l*,yb.iy
-
s 0
+
00
J
F2(q*vY) -
92
J
F’(~*~y)dY~
2
T2 =
F&q2>y)dy s
0
JF&*4& Ir2
‘I2 -
0
+
co
J Fdq*>y>- JFdp*,y)dy.
92
(2)
P2
The functions FI,~(x, y) in Eq. (2) are linear in ZR(X), ZR(Y), ZR(X + y), and the functions F4,5( x, y) are quadratic. It is assumed that Eq. (2) obtained as a result of the approximations described above gives correct infrared behaviour of the propagator and does not claim to describe the ultraviolet behaviour of the propagator. After numerical investigation of Eq. (2) in Refs. [ l] the statement is made on the presence of the solution ZR(q*) which has soft singular power asymptotics for low q*. The integration in Eq. (2) is carried out over all values of q*. Refs. [ l] consider two variants of the behaviour of ZR(q*) at q” -+ co which differ from unity of the bare propagator by the power of the logarithm of q*. As it turns out the exact behaviour of ZR(q*) at q* -+ 03 does not strongly influence the behaviour of the solution at zero. Smooth matching of the behaviour at zero and at infinity was realized by introducing a number of parameters. Fitting the parameters has shown that ZR(~*) behaves as (q*) I-’ at q* + 0 with quite sufficient accuracy. In this the value of the parameter c is within the interval from 0.01 to 0.3 and more singular behaviour of the propagator is excluded. We will show that for non-integer and non-half-integer values of the exponent c the behaviour of the propagator at intermediate and large values of the momentum is really unessential and we will obtain the characteristic equation for the determination of the parameter c. Suppose that ZR(q*) has not any singularities at 0 < q* < cm and the behaviour of Z,( q2) at infinity is adequate to the subtraction procedure of Eq. (2) (e.g., power of the logarithm of q*). Then let the infrared behaviour be described by the series of the form: ZR(x) = (x/p*)‘-’
mp*,y)dy s 0
co
co
x (~o+cuix/~*+
(Y*(X/p*)*+...)
(3)
with a nonzero radius of convergence. We will take the advantage of the circumstance that we can extract the terms of the integrals in the quantities TI and T2 with power dependence on q* at q’ -+ 0 for the infrared behaviour Eq. (3) with non-integer c and for any non-singular behaviour of Za ( X) at finite and positive values of q*.
A.I. Aleheev / PhysicsLettersB 344 (I 995) 325-328
Eq. (2) includes
the integrals
over y from zero to over y from q” to infinity which we will denote as J with different indexes. Dealing with separate integrals we will introduce an auxiliary constant AL instead of the infinite limit of integration. This constant will not enter the final expressions for Tr and T2 which do not contain the ultraviolet divergences. The integrals of type 1 display power behaviour for ZR (q2) described by Eq. (2) at q” + 0. The analysis of the type J integrals shows that apart from the non-integer power terms they contain the integer power series and terms behaving as l/( q2)“:, m = 1,2. We can calculate the terms of type I integrals which behave like ( q2) ‘-’ at q2 ----t 0 and the terms of type J integrals with the behaviour (q2)2-2c, l/q2, l/(q2)2 at q2 -_) 0. Substitute the expressions for the type I, J integrals into 7’1 and T2 of Eq. (2). The terms behaving as 1/q2, l/( q2)2 are mutually canceled and we obtain q2, which we will denote as I and the integrals
Tl = PI (s2) + ~&72/~2)1-c T2 = P2(q2)
+
f
{AI (c) -t O(q2,‘p2)},
(4)
that Eq. (7) is the equation for function Z,(X) with assumed infrared behaviour rather then for the values of the variable x. If we substitute the solution ZR( x) of Eq. (2) into Eq. (6), which has the structure of Eq. (7), then Eqs. (6)) (7) turn into identities. We will apply the theorem of uniqueness for the power series. Then, rounding the branch point x = 0 and subtracting the initial identity from the obtained one we conclude that pn = q,, = r, = 0,~ = 0, 1,2, . . . for noninteger and non-half-integer values of the parameter c. Although the terms of Eq. (6) behaving as ( q2) 1-c at q2 --f 0 are not the leading terms, they are unambiguously defined by the leading term of the infrared asymptotics, Eq. (3). Then the value of the exponent of the infrared asymptotics should be determined by the characteristic equation A(c) = 0. Let us consider the characteristic function A(c) at 0 < c 5 1. This function has the poles at c = 0, l/2,1. We will extract explicitly the pole terms and obtain the integral representation which is convenient for the numerical study. After somewhat lengthy calculations which will not be presented here the characteristic function takes the form:
zR(q2>Q2(q2>
~6(q2/p2)2-2c {b(c) + 0(q2/p2)}.
(5)
The functions PI (q2), P2(q2), Q2(q2) in Eqs. (4), (5) are regular at q’ = 0 and they depend on the behaviour of ZR(~~) at all values of q2 > 0. We do not determine these functions here. Then Eq. (2) takes the form:
327
A(c)
=X+2- 12 1 -
G>
= ;{I
1
2c
2 1 - 5I_c+&c),
(8)
dt, (1 + t)1-c(28t-3+2c
+ 28t2-C
0
_ gt-2+c - 16( 1 + t-3+2c >( 1 - P)/(
1 - t))
- 12t-3+2c + 4( 1+ 3c) t-2+2c I
= 1 + 3%(P2) -{PI
zR(q2)
(q2>
IT
+
+ 2( 16 - 5c - 3~~)t-r~~~
Qh2>
- (23 - 7~)t-‘+~]
1 +
--P2Gz2) zR(q2)
+ wdq2/~2~‘-c[AW
- lly(4 +
Wq2/p2>1},
(6)
where A (c) = A 1(c) -!-A2 (c) is a dimensionless characteristic function. According to our assumption on the analytical structure of the solution ZR ( X) Eq. (6) has the following structure: P(x)
+ x1-‘Q(x)
+ @R(x)
= 0,
(7)
where P(X), Q(x), R(x) are integer power series with nonzero radii of convergence and with some coefficients pn, qn, r,, n = 0, 1,2, . . . correspondingly. Note
+ 6/(3
- 7r-2+c
- 23y(2 - c) + 41y(3
- c) + ~(5 - c) - 16/(2 - c) + 6/(4
- 8 + 3~).
- c) - l/(5
- c)
- c) - c) (9)
wherey(x) = (2X-1)/x,andA(c) =Ar(c)+A2(c) is regular at 0 I c 5 1. The subtractions in the integral representation for A(c) provide finiteness of the function under integral. The signs of the residues in the poles c = 0, l/2,1 are +, -, + respectively. So, one can expect an even number of roots of the characteristic equation in the interval [ 0, 1] of the parameter c. In Fig. 1 we show the characteristic function A(c) as are-
A.I. Alekseev / Physics Letters B 344 (1995) 325-328
328
2T
I wish to thank B.A. Arbuzov, V.E. Rochev and S.N. Sokolov for useful discussions. This work is supported by the Russian Foundation of Fundamental Investigations, Grant No. 94-02-03545-a.
10 5 0
References
-5
‘;‘
-10 15 -15
. v
[ 1] J.R. Cudell and D.A. Ross, NucI. Phys. B 359 (1991) 247;
I‘\ 0
0.25 0.5 0.75
1 c
Fig. 1. The characteristic function A(c) for the power infrared behaviour of the gluon propagator D(q) N (q2) --c, q2 -+ 0.
sult of numerical integration of Eqs. (8)) (9). It can be seen that the characteristic equation has no solutions for the soft-singular behaviour of the gluon propagator. The module of the characteristic function has the local minima at cl N 0.28, c2 11 0.68. Weak dependence of the results of the numerical analysis [ 1] on the exact behaviour of Za(q2) at large q” allows one to suppose that the leading terms of Eq. (6) at q2 4 0 are numerically small. In this case the parameters fitting to minimize the difference between Zpt ( q2) and ZF(q2) will give the value of the parameter c corresponding to the minimum of IA(c) I. It explains the results of the numerical investigation of Refs. [ 11. Note that the consideration of the particular case of the infrared behaviour Eq. (3) does not exclude the possibility of a more general soft singular power infrared behaviour. The method of extracting nonleading terms at q2 -+ 0 of the integral equation used in this paper can be generalized to investigate this case. Having in mind the problem of finding the selfconsistent solution for the gluon propagator in the infrared region it is worth considering the approximations leading to Eq. (2) more carefully. One of the possibilities is to include the transverse part of the triple gluon vertex into the consideration [ 61.
J.R. Cudell, CERN preprint CERN-TH 6192/91 (1991). [2] A.I. Alekseev, Teor. Mat. Fiz. 48 ( 1981) 324; A.N. Vassiliev, Yu.M. Pis’mak, Yu.R. Honkonen, Teor. Mat. Fiz. 48 (1981) 284. [3] F!V. Landshoff and 0. Nachtmann, Z. Phys. C 35 (1987) 405; J.R. Cudell, A. Donnachie and PV. Landshoff, Nucl. Phys. B 322 (1989) 55. [4] S. Mandelstam, Phys. Rev. D 20 (1979) 3223; H. Pagels, Phys. Rev. D 15 ( 1977) 2991; N. Brown, R. Pennington, Phys. Rev. D 38 ( 1988) 2266. [5] M. Baker, J.S. Ball and E Zachariasen, Nucl. Phys. B 186 (1981) 531, 560. [6] AI. Alekseev, Yad. Fiz. 33 (1981) 516; AI. Alekseev, V.F. Edneral, Yad. Fiz. 45 (1987) 1105. [7] Cl.? Hooft, Phys. Scripta 25 (1982) 133; V.P. Nair, C. Rosenzweig, Phys. Rev. D 31 ( 1985) 401; M. Baker, J.S. Ball and E Zachariasen, Phys. Rep. 209 (1991) 73. [8] J.M. Cornwall, Phys. Rev. D 26 (1982) 1453; J.M. Namyslowski, Warsaw Univ. preprint IFI’/13/90 ( 1990) ; V.V. Kiselev, IHEP preprint IHEP 92-5 1 ( 1992). [9] R. Delbourgo, J. Phys. G: Nucl. Phys. 5 (1979) 603. [IO] W. Kummer, Acta Phys. Austr. 41 (1975) 315. [ll] S.K. Kim, M. Baker, Nucl. Phys. B 164 (1980) 152. [ 121 W.J. Schoenmaker, Nucl. Phys. B 191 (1982) 535.