- Email: [email protected]
Re+e− at low energies in variational perturbation theory
26 January 1995
PHYSICS
LETTERS
B
Physics Letters B 344 (1995) 377-382
R,+,- at low energies in variational perturbation theory I.L. Solovtsov l, O.P. Solovtsova Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region 141980. Russia
Received 26 July 1994; revised manuscript received 31 October 1994 Editor: PV. Landshoff
Abstract We calculate R,+,- at low energies using method of variational perturbation theory in quantum chromodynamics. We apply the “smearing” method to compare the obtained theoretical prediction for R,+,- with experimental data down to lowest energies. The problem of the “freezing” of the QCD coupling constant is discussed.
More accurate results, obtained in Ref. [ 11 for the R( s)-ratio for the process of e+e- annihilation into hadrons stimulated further theoretical study of that ratio [ 2-51. Higher orders of perturbation theory for starting from the three-loop order become dependent on the renormalization scheme. Therefore, for the correct application of perturbative results it is necessary to use some procedure of optimization, for instance, the principle of minimal sensitivity [ 61. On the one hand, the dependence of a physical quantity on the renormalization scheme thus arising can be considered as a regrettable fact; on the other hand, there appears an extra degree of freedom that can be used for constructing an optimal expansion that has a wider range of applicability in a particular scheme as compared to perturbation theory. Along this line, as shown in Ref. [3], it has been possible to advance towards the low-energy region. In this paper, the ratio R(S) at low energies will be analyzed using the nonperturbative method proposed in Ref. [ 71. This approach to QCD is based on the concepts of variational perturbation theory (VFT) [8-l 11. According to [7], an expansion, different from a perturbative series, can be constructed in QCD on the basis of the Gaussian functional integrals (like in perturbation theory). The expansion parameter is a new small parameter connected with the initial coupling constant by some equation. For small values of the coupling constant g, the Nth-partial sum of the new series reproduces the Nth order of standard perturbation theory to within 0( gN” ). In the nonperturbative region, when the running coupling constant of QCD becomes large and perturbation theory does not work, the new expansion parameter remains small and the method remains still valid. We write the QCD action functional in the form S(A,qvp)
=S2(A)
+S2(q)
+s2CvD)
+g&(A,q,p)
’ E-mail address: so1ovtsokWeor.jinrc.dubna.s.u. 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(94)01525-2
+&%(A),
(1)
378
I.L. Solovtsov, 0. P. Solovtsova / Physics Letters B 344 (1995) 377-382
where S,(A), &(q) and S, (p> are free action functionals of the gluon, quark, and ghost fields, respectively; the term S2 (A) also contains a term fixing the covariant @o-gauge. The term Ss (A, q, 5p) describes the Yukawa interaction of gluons, gluons with quarks, and gluons with ghosts, Ss(A,q,qP)
= S3(A)
+&(A,q)
(2)
+ ~~(AsD).
The terms Ss (A), S3 (A, q) and S3 (A, cp) generate, respectively, three-line vertices, (AAA) , (qAq) and (pAp); whereas the term &(A) in ( 1) generates four-gluon vertices (AAAA). We will transform the latter term by introducing auxiliary fields ,yi, [ 71. Upon the x-transformation, the diagrams of the Green functions will consist only of diagrams of Yukawa type. In addition to the usual three-line vertices of QCD, vertices of the type AxA will appear. Thus, a certain Green function of QCD can be represented in the following functional integral form: G(...)
=
DxDQcr,(.
..) ew{i
s
[S(A,x)
+&(q)
f&(v)
+&(x)
+g&(A,q,v)]
},
where
S(A,x)
=;
J
dxdyA;(x)
with the gluon propagator
[D-‘(x,ylxfb
P
[D-1(x,~lx)];
D( x, ylx)
= [ (-
A;(Y)
(4)
in the x-field
gpvd2+ up% >Pb+ gJzfabc xfLp+ gauge
terms ] 6(x - y)
and the term ( . . . ) is a set of Y gluon, quark and ghost fields. Integration measure DQ~D in (3) defines standard integrations over gluon, quark, and ghost fields. Following the ideas of the VPT method, we introduce auxiliary parameters d and 5 and rewrite the action in (3) in the form
S(A,q,~vx) =$&QwP,x) +$L%q,w),
(6)
where
$,(A,q,cx) =5-‘LVx) $b%w,x)
s&(q)
+S2tv)l
+5-‘S2(x)
=g&(A,q,cp) - Cl-’ - 1) [ S(A,x)
(7)
1
+ k(q)
+ S2tpP)l - (5-’
- 1) S2Cx).
(8)
The exact value of the quantity under consideration, for instance, the Green function does not depend on the parameters 5 and 5. However, the approximation of that quantity with a finite number of terms of the VPT series, which results from the expansion in powers of the action Si (A, q, tp, x), does depend on those parameters. We can employ the freedom in the choice of the parameters 5 and 5 for our aim, construction of a new small parameter of the expansion. It is more convenient to rewrite $(A, q, p, ,y) in (7) by replacing 6-t with [ 1 + K (5-l 1) ] and 6-l with [ 1 + K (5-l 1) ] and putting K = 1 at the end of calculations. In this case, any power of the expression (J-’ - 1) [ S(A,x) + $(q) &(cp)] + (5-l - 1) So, appearing in the factor of the exponential upon expanding the Green function in powers of (8)) can be obtained by differentiating with respect to the parameter K as many times as required. Then, the integrand in the factor of the exponential will contain only the powers of the action g Ss( A, q, p) that generate the QCD Yukawa diagrams with modified propagators defined by appropriate quadratic forms in the new “free” action S& The VPT series for the Green function is given by DxDQcD(...)
[g&(A,q,plk
exp[i$dAq,ex)
1
(9)
I.L. Solovtsov, 0. P. Solovtsovu /Physics
with the above replacement
in ,I$( A, q, p, ,y). Further, it is convenient
C&q, PP)
(A,q,cp) =+-
J1
Letters B 344 (1995) 377-382
-
+K([-'
to rescale the fields
X
x*
9
1)
379
J1 + K (5-l -
1)
(10)
’
As a result, the propagators acquire the standard form and only the diagram vertices get modified. then over the field x we obtain for the Green function
X
I
DQCD(*..)
Integrating
+dG(A) I).
[g3S3(A,q,~)lkexp{i[So(A,q,cp)
(11)
Here Ss( A, q, 5p) does no longer contain the term describing the field x and represents a usual functional the QCD free action, whereas gs and g4 in the Yukawa and four-gluon vertices are defined as follows: g
g (12)
g4 =
g3 =
[ 1 + K(c$-’
[ 1+K(&-‘-1)]3’2’
-
1)]1’2
’
Analysis of the structure of the VPT series shows [ 71 that we will succeed in constructing parameter if we put 5 = ,J3 and if the parameter l is connected with the coupling constant
ALL_ (47r)2
1
of
a2
the small expansion by the equation
a=l-5,
C (l-a)3
(13)
’
where C is a positive constant. As follows from ( 13)) at any value of the coupling constant g, the new expansion parameter a obeys the inequality 0 5 a < 1. We present the result of the VPT expansion for the Green functions with an accuracy of O(a7) that allows us to carry out calculations at the two-loop level in this approach. Writing the Green functions in the form G(.
. .) =
I
and using (ll)-(
DQCD(.
. .> V(A q, cp> exp(i W
( 14)
13), we obtain
V=1+aA3+a2[~A32+A4+~A3] + a4 [&A34
+ ;A42
+a3[~A33+~A32+
+ $b2A4
+
A3~4+3~4+f~3]
;A33 + ;A3A4 + 3A32 + 6A4 + $A31
+a5[&,As5+;As3A4+;A3A42+;A34+3A32A4 + 3A42+;As3+;AsA4+5A32+10A4+$A3] +a6[&jA36+&A34A4+;A32A42 1 5 15 + --35+-A33A4+-A3A42+-A44+_~32~4+_~42 z3 4 4 ’
~As’+zAsA4+
where A3 =
47~(iS3)/@,
+;A43 7
21
21
8
2
2
yA32+15A4+$$A3] A4 = (4~)~
(is4)/c
+
+ o(a7),
(15)
380
I.L. Solovtsov, O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
We apply these results to describe the R(s)-ratio in the process of efe- annihilation into hadrons. We will consider the range of Q = fi from 0 to 6 GeV (like in Ref. [ 31) and compare with experiment by using the smearing method [ 121. According to it, we consider the following smeared quantity: a,
&m(Q) = ;
s
ds
R(s) (s-Q*)*+A*
(16)
*
0
We will determine the first nontrivial order defined by the VPT expansion (15). At this level, we do not run into the problem of optimization since we derive a scheme-independent result, as in the case of the first-order for QCD corrections. Thus, we have for R(s) R(s) = 3 x4;
~u,dQ>
T(ui) [1 + g(“i)--]
(17)
7
i
where the summation runs over all quark flavors that are above the threshold: Q = 2mi, qi are the quark charges, the functions Ui , T(U) , g(u) are
uf=1 -
4m?/Q*,
T(u) = ; u(3 -u*> )
(18)
g(u)+;-
and for LU,~(Q) /r from ( 15) we derive the following two expressions, respectively, with an accuracy of 0( a*) and 0(a3): a(*) eff -=-a*,
4
?T
c2
J3) eff -= n-
(19)
b(1+3a). c3
The parameters C2 and Cs do not depend on experimental R,+,-- data and have been determined from the condition that the renormalization group p-function at large enough values of the coupling constant behaves as /?(A) N -A. This behaviour corresponds to the singular infrared behaviour of the invariant charge as(Q*) N Q-* and ensures the linear growth of the nonrelativistic static quark-antiquark potential at large distances. Thus, we fix parameters Ci in (19) on the basis of data of hadron spectroscopy, which gives C:! = 0.977 and C3 = 4.1 [ 131. For the quark masses we take the standard values: m, = 5.6 MeV, md = 9.9 MeV, m, = 199 MeV, m, = 1.350 GeV and mb = 4.75 GeV. The momentum Q and the expansion parameter a are connected as follows
Q = QO
ew{$ [fi(a)
- fi(uo) ] }
9
where 9 12 l-u 2 - + 21 In f*(u) = - + l-u’ .* ; U 2 6 18 1 fs(u) = -U* - a - 48lnu - -11 I_a+gln(l-a)+
(21)
Nf is the number of favours. andba=ll-$Nf, To determine all our parameters, we use the normalization LX&Qu) = LY,-J and take Qa = M, = 1.777 GeV and a0 = 0.347. * Now, there are no free parameters in our scheme. * The most recent T-data provide the average value a~( M,) = 0.347 corresponding in Ref. [14]).
to (YS(Mz) = 0.121 (see, for instance,
the discussion
I.L. Solovtsov. O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
381
5
1L 4--
3 -
l-
0.0
I 0
1
I
I
I
2
I
I
I
4
5
6
0 0
S&V, Fig. 1. The functions
“~~““/‘~‘~““““““~~~’ 1 2
4
5
6
Q;GeV)
CY$)/?r, a,$) /n- and r~~‘-‘~~) /rr versus Q.
Fig. 2. The smeared R,,-ratio ( 16) for A = 3 GeV*: our result (solid line), the third-order optimization result (short-dashed line) and the experimental curve (long-dashed line) are taken from Ref. [3]
In Fig. 1, the functions ~r$)/r and c&)/7r are plotted, corresponding to expressions (19). The obtained curves are sufficiently close to each other. The stability of the result derived in different orders of the VPT approximation signifies the convergence of the series in the sense of induced convergence when the auxiliary cases, in Ref. [ 151, a strict parameter, C, changes with order. 3 Note that in the zero- and one-dimensional proof is given for the convergence of that series for the S-expansion. In Fig. 1 we show also the behaviour of the function of (~~t-‘~~~) (Q) /7r that appears in the one-loop approximation of the standard perturbation theory. In contrast to ~~‘-‘OOp)(Q) that has a pole at Q = hoob the functions c$ (Q) have no any singularities for Q > 0 and have finite limits at Q = 0, because the expansion parameter a --+ 1 as Q --) 0. In Fig. 2, the smeared quantity ( 18) is shown for A = 3 GeV* and a:,$’ from (21).4 The experimental curve is taken from Ref. [ 31; we also report the theoretical results from this paper. As can be seen, our result obtained in the first order of VPT quite well reproduces the experimental curve. It is of interest that the ratio we have derived almost coincides with the relevant result of [3] obtained on the basis of optimization of the third order of standard perturbation theory. Recently, the problem of “freezing” the QCD coupling constant at low energies has arisen. This freezing is required in many models based on QCD (for a detailed discussion, see [ 31 and references therein.) As the “experimental” value, for comparison it is convenient to use the integral independent of the fit of data [ 31,
(22)
In our case for ad:), this integral equals 0.239 GeV and 0.237 GeV for a$r). The authors would like to thank B.A. Arbuzov, V.V. Burov, M. Consoli, H.F. Jones, A.L. Kataev, D.I. Kazakov, A.N. Sissakian, PM. Stevenson and A.A. Vladimirov for interest in the work and useful comments. This investigation was performed under the financial support of Russian Science Foundation (Grant 93-02-3754). 3 An analogous
similarity
4 The curve corresponding
of the results is observed also for the p-function to QY~ does not practicahy
in a wide range of variation of the coupling
differ from the curve corresponding
constant.
to CY$’ and we do not show it in Fig. 2.
382
I.L. Solovtsov, O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
References [ 1] S.G. Gorishny, A.L. Kataev and S.A. Latin, Phys. Lett. B 259 (1991) 144; L.R. Surguladze and M.A. Samuel, Phys. Rev. Lett. 66 (1991) 560. [2] J. Chyla, A.L. Kataev and S.A. Larin, Phys. Lett. B 267 (1991) 269. [3] AC. Mattingly and PM. Stevenson, Phys. Rev. Lett. 69 (1992) 1320; Phys. Rev. D 49 ( 1994) 437. [4] L.R. Surguladze and M.A. Samuel, Phys. Lett. B 309 (1993) 157. [5] A.L. Kataev and V.V. Starshenko Preprint CERN -TH.7198/94. [6] PM. Stevenson, Phys. Rev. D 23 (1981) 2916; Nucl. Phys. B 231 (1984) 65. ]7] IL. Solovtsov, Phys. Lett. B 327 (1994) 335. [S] A.N. Sissakian and IL. Solovtsov, Phys. L&t. A 157 (1991) 261; Z. Phys. C 54 (1992) 263. [9] A.N. Sissakian, I.L. Solovtsov and O.Yu. Shevchenko, Phys. Lett. B 297 (1992) 305; Int. J. Mod. Phys. A 9 (1994) 1797; Int. J. Mod. Phys. A 9 (1994) 1929. [ lo] L.D. Korsun, A.N. Sissakian and 1.L. Solovtsov, Sov. J. Theor. and Math. Phys. 90 (1992) 37; Int. J. Mod. Phys. A 8 (1993) 5129 [ 111 A.N. Sissakian, I.L. Solovtsov and OP. Solovtsova, Phys. Lett. B 321 ( 1994) 381. [ 121 E.C. Poggio, H.R. Quinn and S. Weinberg, Phys. Rev. D 13 (1976) 1958. [ 131 A.N. Sissakian, IL. Solovtsov and 0.P Solovtsova, Mod. Phys. Lett. A (to be published). [ 141 S. Narison, Preprint CERN-TH.7188/94, PM 94/08 (1994). [ 151 I.R.C. Buckley, A. Duncan and HF. Jones, Phys. Rev. D 47 (1993) 2554; A. Duncan and H.F. Jones, Phys. Rev. D 47 ( 1993) 2560.
PHYSICS
LETTERS
B
Physics Letters B 344 (1995) 377-382
R,+,- at low energies in variational perturbation theory I.L. Solovtsov l, O.P. Solovtsova Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region 141980. Russia
Received 26 July 1994; revised manuscript received 31 October 1994 Editor: PV. Landshoff
Abstract We calculate R,+,- at low energies using method of variational perturbation theory in quantum chromodynamics. We apply the “smearing” method to compare the obtained theoretical prediction for R,+,- with experimental data down to lowest energies. The problem of the “freezing” of the QCD coupling constant is discussed.
More accurate results, obtained in Ref. [ 11 for the R( s)-ratio for the process of e+e- annihilation into hadrons stimulated further theoretical study of that ratio [ 2-51. Higher orders of perturbation theory for starting from the three-loop order become dependent on the renormalization scheme. Therefore, for the correct application of perturbative results it is necessary to use some procedure of optimization, for instance, the principle of minimal sensitivity [ 61. On the one hand, the dependence of a physical quantity on the renormalization scheme thus arising can be considered as a regrettable fact; on the other hand, there appears an extra degree of freedom that can be used for constructing an optimal expansion that has a wider range of applicability in a particular scheme as compared to perturbation theory. Along this line, as shown in Ref. [3], it has been possible to advance towards the low-energy region. In this paper, the ratio R(S) at low energies will be analyzed using the nonperturbative method proposed in Ref. [ 71. This approach to QCD is based on the concepts of variational perturbation theory (VFT) [8-l 11. According to [7], an expansion, different from a perturbative series, can be constructed in QCD on the basis of the Gaussian functional integrals (like in perturbation theory). The expansion parameter is a new small parameter connected with the initial coupling constant by some equation. For small values of the coupling constant g, the Nth-partial sum of the new series reproduces the Nth order of standard perturbation theory to within 0( gN” ). In the nonperturbative region, when the running coupling constant of QCD becomes large and perturbation theory does not work, the new expansion parameter remains small and the method remains still valid. We write the QCD action functional in the form S(A,qvp)
=S2(A)
+S2(q)
+s2CvD)
+g&(A,q,p)
’ E-mail address: so1ovtsokWeor.jinrc.dubna.s.u. 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(94)01525-2
+&%(A),
(1)
378
I.L. Solovtsov, 0. P. Solovtsova / Physics Letters B 344 (1995) 377-382
where S,(A), &(q) and S, (p> are free action functionals of the gluon, quark, and ghost fields, respectively; the term S2 (A) also contains a term fixing the covariant @o-gauge. The term Ss (A, q, 5p) describes the Yukawa interaction of gluons, gluons with quarks, and gluons with ghosts, Ss(A,q,qP)
= S3(A)
+&(A,q)
(2)
+ ~~(AsD).
The terms Ss (A), S3 (A, q) and S3 (A, cp) generate, respectively, three-line vertices, (AAA) , (qAq) and (pAp); whereas the term &(A) in ( 1) generates four-gluon vertices (AAAA). We will transform the latter term by introducing auxiliary fields ,yi, [ 71. Upon the x-transformation, the diagrams of the Green functions will consist only of diagrams of Yukawa type. In addition to the usual three-line vertices of QCD, vertices of the type AxA will appear. Thus, a certain Green function of QCD can be represented in the following functional integral form: G(...)
=
DxDQcr,(.
..) ew{i
s
[S(A,x)
+&(q)
f&(v)
+&(x)
+g&(A,q,v)]
},
where
S(A,x)
=;
J
dxdyA;(x)
with the gluon propagator
[D-‘(x,ylxfb
P
[D-1(x,~lx)];
D( x, ylx)
= [ (-
A;(Y)
(4)
in the x-field
gpvd2+ up% >Pb+ gJzfabc xfLp+ gauge
terms ] 6(x - y)
and the term ( . . . ) is a set of Y gluon, quark and ghost fields. Integration measure DQ~D in (3) defines standard integrations over gluon, quark, and ghost fields. Following the ideas of the VPT method, we introduce auxiliary parameters d and 5 and rewrite the action in (3) in the form
S(A,q,~vx) =$&QwP,x) +$L%q,w),
(6)
where
$,(A,q,cx) =5-‘LVx) $b%w,x)
s&(q)
+S2tv)l
+5-‘S2(x)
=g&(A,q,cp) - Cl-’ - 1) [ S(A,x)
(7)
1
+ k(q)
+ S2tpP)l - (5-’
- 1) S2Cx).
(8)
The exact value of the quantity under consideration, for instance, the Green function does not depend on the parameters 5 and 5. However, the approximation of that quantity with a finite number of terms of the VPT series, which results from the expansion in powers of the action Si (A, q, tp, x), does depend on those parameters. We can employ the freedom in the choice of the parameters 5 and 5 for our aim, construction of a new small parameter of the expansion. It is more convenient to rewrite $(A, q, p, ,y) in (7) by replacing 6-t with [ 1 + K (5-l 1) ] and 6-l with [ 1 + K (5-l 1) ] and putting K = 1 at the end of calculations. In this case, any power of the expression (J-’ - 1) [ S(A,x) + $(q) &(cp)] + (5-l - 1) So, appearing in the factor of the exponential upon expanding the Green function in powers of (8)) can be obtained by differentiating with respect to the parameter K as many times as required. Then, the integrand in the factor of the exponential will contain only the powers of the action g Ss( A, q, p) that generate the QCD Yukawa diagrams with modified propagators defined by appropriate quadratic forms in the new “free” action S& The VPT series for the Green function is given by DxDQcD(...)
[g&(A,q,plk
exp[i$dAq,ex)
1
(9)
I.L. Solovtsov, 0. P. Solovtsovu /Physics
with the above replacement
in ,I$( A, q, p, ,y). Further, it is convenient
C&q, PP)
(A,q,cp) =+-
J1
Letters B 344 (1995) 377-382
-
+K([-'
to rescale the fields
X
x*
9
1)
379
J1 + K (5-l -
1)
(10)
’
As a result, the propagators acquire the standard form and only the diagram vertices get modified. then over the field x we obtain for the Green function
X
I
DQCD(*..)
Integrating
+dG(A) I).
[g3S3(A,q,~)lkexp{i[So(A,q,cp)
(11)
Here Ss( A, q, 5p) does no longer contain the term describing the field x and represents a usual functional the QCD free action, whereas gs and g4 in the Yukawa and four-gluon vertices are defined as follows: g
g (12)
g4 =
g3 =
[ 1 + K(c$-’
[ 1+K(&-‘-1)]3’2’
-
1)]1’2
’
Analysis of the structure of the VPT series shows [ 71 that we will succeed in constructing parameter if we put 5 = ,J3 and if the parameter l is connected with the coupling constant
ALL_ (47r)2
1
of
a2
the small expansion by the equation
a=l-5,
C (l-a)3
(13)
’
where C is a positive constant. As follows from ( 13)) at any value of the coupling constant g, the new expansion parameter a obeys the inequality 0 5 a < 1. We present the result of the VPT expansion for the Green functions with an accuracy of O(a7) that allows us to carry out calculations at the two-loop level in this approach. Writing the Green functions in the form G(.
. .) =
I
and using (ll)-(
DQCD(.
. .> V(A q, cp> exp(i W
( 14)
13), we obtain
V=1+aA3+a2[~A32+A4+~A3] + a4 [&A34
+ ;A42
+a3[~A33+~A32+
+ $b2A4
+
A3~4+3~4+f~3]
;A33 + ;A3A4 + 3A32 + 6A4 + $A31
+a5[&,As5+;As3A4+;A3A42+;A34+3A32A4 + 3A42+;As3+;AsA4+5A32+10A4+$A3] +a6[&jA36+&A34A4+;A32A42 1 5 15 + --35+-A33A4+-A3A42+-A44+_~32~4+_~42 z3 4 4 ’
~As’+zAsA4+
where A3 =
47~(iS3)/@,
+;A43 7
21
21
8
2
2
yA32+15A4+$$A3] A4 = (4~)~
(is4)/c
+
+ o(a7),
(15)
380
I.L. Solovtsov, O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
We apply these results to describe the R(s)-ratio in the process of efe- annihilation into hadrons. We will consider the range of Q = fi from 0 to 6 GeV (like in Ref. [ 31) and compare with experiment by using the smearing method [ 121. According to it, we consider the following smeared quantity: a,
&m(Q) = ;
s
ds
R(s) (s-Q*)*+A*
(16)
*
0
We will determine the first nontrivial order defined by the VPT expansion (15). At this level, we do not run into the problem of optimization since we derive a scheme-independent result, as in the case of the first-order for QCD corrections. Thus, we have for R(s) R(s) = 3 x4;
~u,dQ>
T(ui) [1 + g(“i)--]
(17)
7
i
where the summation runs over all quark flavors that are above the threshold: Q = 2mi, qi are the quark charges, the functions Ui , T(U) , g(u) are
uf=1 -
4m?/Q*,
T(u) = ; u(3 -u*> )
(18)
g(u)+;-
and for LU,~(Q) /r from ( 15) we derive the following two expressions, respectively, with an accuracy of 0( a*) and 0(a3): a(*) eff -=-a*,
4
?T
c2
J3) eff -= n-
(19)
b(1+3a). c3
The parameters C2 and Cs do not depend on experimental R,+,-- data and have been determined from the condition that the renormalization group p-function at large enough values of the coupling constant behaves as /?(A) N -A. This behaviour corresponds to the singular infrared behaviour of the invariant charge as(Q*) N Q-* and ensures the linear growth of the nonrelativistic static quark-antiquark potential at large distances. Thus, we fix parameters Ci in (19) on the basis of data of hadron spectroscopy, which gives C:! = 0.977 and C3 = 4.1 [ 131. For the quark masses we take the standard values: m, = 5.6 MeV, md = 9.9 MeV, m, = 199 MeV, m, = 1.350 GeV and mb = 4.75 GeV. The momentum Q and the expansion parameter a are connected as follows
Q = QO
ew{$ [fi(a)
- fi(uo) ] }
9
where 9 12 l-u 2 - + 21 In f*(u) = - + l-u’ .* ; U 2 6 18 1 fs(u) = -U* - a - 48lnu - -11 I_a+gln(l-a)+
(21)
Nf is the number of favours. andba=ll-$Nf, To determine all our parameters, we use the normalization LX&Qu) = LY,-J and take Qa = M, = 1.777 GeV and a0 = 0.347. * Now, there are no free parameters in our scheme. * The most recent T-data provide the average value a~( M,) = 0.347 corresponding in Ref. [14]).
to (YS(Mz) = 0.121 (see, for instance,
the discussion
I.L. Solovtsov. O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
381
5
1L 4--
3 -
l-
0.0
I 0
1
I
I
I
2
I
I
I
4
5
6
0 0
S&V, Fig. 1. The functions
“~~““/‘~‘~““““““~~~’ 1 2
4
5
6
Q;GeV)
CY$)/?r, a,$) /n- and r~~‘-‘~~) /rr versus Q.
Fig. 2. The smeared R,,-ratio ( 16) for A = 3 GeV*: our result (solid line), the third-order optimization result (short-dashed line) and the experimental curve (long-dashed line) are taken from Ref. [3]
In Fig. 1, the functions ~r$)/r and c&)/7r are plotted, corresponding to expressions (19). The obtained curves are sufficiently close to each other. The stability of the result derived in different orders of the VPT approximation signifies the convergence of the series in the sense of induced convergence when the auxiliary cases, in Ref. [ 151, a strict parameter, C, changes with order. 3 Note that in the zero- and one-dimensional proof is given for the convergence of that series for the S-expansion. In Fig. 1 we show also the behaviour of the function of (~~t-‘~~~) (Q) /7r that appears in the one-loop approximation of the standard perturbation theory. In contrast to ~~‘-‘OOp)(Q) that has a pole at Q = hoob the functions c$ (Q) have no any singularities for Q > 0 and have finite limits at Q = 0, because the expansion parameter a --+ 1 as Q --) 0. In Fig. 2, the smeared quantity ( 18) is shown for A = 3 GeV* and a:,$’ from (21).4 The experimental curve is taken from Ref. [ 31; we also report the theoretical results from this paper. As can be seen, our result obtained in the first order of VPT quite well reproduces the experimental curve. It is of interest that the ratio we have derived almost coincides with the relevant result of [3] obtained on the basis of optimization of the third order of standard perturbation theory. Recently, the problem of “freezing” the QCD coupling constant at low energies has arisen. This freezing is required in many models based on QCD (for a detailed discussion, see [ 31 and references therein.) As the “experimental” value, for comparison it is convenient to use the integral independent of the fit of data [ 31,
(22)
In our case for ad:), this integral equals 0.239 GeV and 0.237 GeV for a$r). The authors would like to thank B.A. Arbuzov, V.V. Burov, M. Consoli, H.F. Jones, A.L. Kataev, D.I. Kazakov, A.N. Sissakian, PM. Stevenson and A.A. Vladimirov for interest in the work and useful comments. This investigation was performed under the financial support of Russian Science Foundation (Grant 93-02-3754). 3 An analogous
similarity
4 The curve corresponding
of the results is observed also for the p-function to QY~ does not practicahy
in a wide range of variation of the coupling
differ from the curve corresponding
constant.
to CY$’ and we do not show it in Fig. 2.
382
I.L. Solovtsov, O.P. Solovtsova / Physics Letters B 344 (1995) 377-382
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