fulltext

Variational nonperturbative expansion in QCD, momentum

renormalization scheme and e

1

_e

2

_{annihilation at low energies (*)}

D. EBERT(1)(**), I. L. SOLOVTSOV(1)(2) and O. P. SOLOVTSOVA(2)

(1_{) Institute of Physics, Humboldt University - Invalidenstrasse 110, D-10115 Berlin, Germany}

(2_{) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research}

Dubna 141980, Russia

(ricevuto il 4 Febbraio 1996; approvato il 18 Aprile 1997)

Summary. — We present a variational nonperturbative approach in QCD with

massive quarks based on a small expansion parameter, construct the corresponding b-function and derive values of the light pole quark masses. We apply this approach to the analysis of the process of the e1_e2_{annihilation into hadrons at low energies}

and compare the theoretical predictions with experimental data. PACS 12.38 – Quantum chromodynamics.

PACS 12.38.Lg – Other nonperturbative calculations.

PACS 13.65 – Hadron production by electron-positron collisions.

1. – Introduction

At present, perturbation theory is a powerful tool for performing calculations in quantum field theory. Its use in combination with a renormalization procedure in quantum electrodynamics, the theory of electroweak interactions, and in the weak-coupling regime of quantum chromodynamics makes it possible to analyze a large number of physical processes. However, there are many problems whose solutions cannot be obtained in the framework of the perturbative approach. In this paper we will apply the method of variational perturbation theory, which combines some variational procedure with the possibility to calculate higher-order corrections, and leads to a representation of physical quantities under consideration in the form of a floating series. The corresponding method has been proposed in [1, 2] for the quantum-mechanical anharmonic oscillator. Now, this idea has found many applications in quantum field theory providing a useful tool to go beyond perturbation theory (see,

e.g., [3] and references therein). Here, we further consider the recently obtained

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) Supported by Deutsche Forschungsgemeinschaft under contract 436 RUS 113/29.

nonperturbative QCD expansion based on a new small expansion parameter. Within this method, a quantity under consideration can be approximated by a series, the structure of which is different from the perturbative expansion and which can be used to go beyond the weak-coupling regime. This allows one to deal with considerably lower energies than in the case of perturbation theory. In the previous investigations this nonperturbative expansion has been considered and applied to the physical processes in the case of the MS-like renormalization scheme [4, 5]. In the present paper, we will instead use a momentum renormalization scheme and parametrize our results in terms of the so-called pole quark masses. In the case of standard perturbation theory, a corresponding renormalization prescription of such type was considered, e.g., in [6, 7]. In the framework of this scheme, the effective coupling constant depends on the quark masses which makes it possible to taken into account the quark-mass effects at low energy providing a natural way to include threshold effects without any additional matching procedure. These ideas are then applied to an investigation of the process of e1_e2_{annihilation into hadrons at low energies.}

The paper is organized as follows. In sect. 2 we give a brief introduction of the method. In sect. 3 the pole masses of light quarks are calculated. In sect. 4 we consider the renormalization procedure in the momentum renormalization scheme and construct the corresponding b-function. The application of the method to the process of e1_e2 annihilation into hadrons at low energies is analyzed in the sect. 5. Summarizing comments are finally given in sect. 6. Some details needed in our calculations are relegated to appendix A.

2. – Nonperturbative a-expansion in QCD

In this section we will briefly describe the results of the nonperturbative expansion method [8] in QCD based on the concept of variational perturbation theory (VPT) [3]. In the framework of this method, the original action functional is first rewritten by introducing auxiliary x-field and some auxiliary parameters of the variational type. Clearly, quantities like Green function should not depend on these parameters, but their finite approximation will do due to the truncation of the series. This fact allows one to use the freedom in the choice of the auxiliary parameters to construct a new small expansion parameter in QCD. Note that for small values of the coupling constant

g, this approach reproduces the standard perturbative expansion and, in principle, all

the high-energy physics is maintained by the method. However, in going to lower energies where standard perturbation theory does not work, the new arising expansion parameter remains small and the method remains still valid. It should be stressed that within this method the gauge properties of the original quantities are not violated.

The analysis of the structure of the VPT series in QCD gives the following connection between a new expansion parameter a and the coupling constant l :

l 4 g 2 ( 4 p)2 4 1 C a2 ( 1 2a)3 . (1)

It is clear that for all values of the coupling constant l F0 the expansion parameter a obeys the inequality 0 GaE1. The positive parameter C plays the role of a variational parameter, which is associated with the use of the variational series. This parameter can be found by using some further information, e.g., which comes from the potential approach to meson spectroscopy as in [8].

Consider the renormalization group b-function of the expansion parameter a in the leading order ba(a) 4m2 ¯a ¯m2 4 2 b0 C 1 f 8(a) , (2)

where b04 11 2 2 O3 Nfis the first coefficient of the standard b-function in perturbative

expansion, and Nf is the number of active quarks. By finding the renormalization

constants in the massless renormalization scheme with an accuracy O(a3), for the function f (a) we get

f (a) 4 2 a2 2 6 a 2 48 ln a 2 18 11 1 1 2a 1 624 121ln ( 1 2a)1 5184 121 ln

g

1 1 9 2a

h

. (3)

By solving the renormalization group eq. (2) we find the momentum dependence of the running expansion parameter a as a solution of the following transcendental equation: lnQ 2 Q02 4 C 2 b0 [ f (a) 2f(a0) ] . (4)

For any values of Q2_{, this equation has a unique solution a(Q}2_{) in the interval between} 0 and 1.

In fig. 1, we plot the function 2ba(a) for Nf4 3 and C 4 4.1. This choice of the value

of C taken from [8] corresponds to the assumption that the renormalization group

bl-function at large enough values of the coupling constant l behaves as blC 2 l. Such

a behaviour corresponds to the singular infrared behaviour of the running coupling constant l(Q2_{) AQ}22 _{and leads to the linear growth of the static quark-antiquark} potential (see also discussion in sect. 4). As it should be, the ba-function has an infrared

fixed point at a 41. In fig. 2, we demonstrate the behaviour of the running expansion parameter a(Q2_{). To define the normalization point a}

0we have used as input the value of as(Q02) 40.35 at the t-lepton mass scale: Q04 Mt4 1.777 GeV. The curve, in fig. 2,

shows that the running expansion parameter a(Q2) remains small (0.34–0.44) for

Fig. 2. – The running expansion parameter a(Q2_{) defined by eq. (4) vs. Q}2_.

sufficiently small Q2

C 0.01–0.1 GeV2. Actually, the real expansion parameter is associated with a2

OC which, of course, is smaller than a. This is important for the applicability of our following estimates of the light pole quark masses.

3. – Pole quark masses

In this section we will calculate the values of pole quark masses by using the above variational nonperturbative approach considered briefly in sect. 2. In the framework of standard perturbation theory, it is possible to carry out such calculations for heavy quarks only [9] because in the case of the light quarks the running coupling constant becomes too large to apply the perturbative expansion. However, within the nonperturbative expansion this task may be considered for the light quarks as well since the expansion parameter a remains small, and we do not find ourselves outside the region of applicability of the approach.

Following [9], we define the renormalization point and renormalization scheme invariant mass meff(p2) the same as in [9]. Then the pole quark mass, M, is defined as the location of the quark propagator pole and obeys the equation M 4meff(M2) . To derive the values of the pole quark masses, one can use any scheme, for example, the MS-scheme because the pole mass, M, like meff, is renormalization point and renormalization scheme invariant.

Let us first consider the anomalous mass dimension gmdefined by

gm4 2 l

¯b1 ¯l ,

(5)

where b1(l) is given from the mass renormalization constant Zm4 1 1 b1Oe 1 R in dimensional regularization with D 4422e. In an approximation to O(a5_{) we find}

that b14 2

{

g0 a2 C 1 3 g0 a3 C 1

k

6 Cg01 g1 2

l

a4 C2 1 [ 10 Cg01 3 g1] a5 C2

}

, (6) where g04 4 and g14 4( 101 O6 2 5 O9 Nf).

The running quark mass, m, as a solution of the renormalization group equation with boundary condition m(Q2

4 m2) 4m(m2_{) can then be found in the form}

m(Q2 ) 4m(m2_{) exp}

y

2



l(m2) l(Q2₎ dl 8 gm(l 8) b(l 8)

z

. (7)

Differently from the previous section, to obtain the nonperturbative b-function we now use the renormalization constants including terms O(a5_{) with modified value of} the parameter C 421.5 which has been found in [8] using an information coming from the meson spectroscopy. Then, we get

b 4m2 ¯l ¯m2 4 2 b0 a4 C2 f(a) ( 1 2a)2 ( 1 1aO2) , (8) where f(a) 411 9 2a 12(61a) a 2 1 5( 5 1 3 B) a3 (9) and B 4 b1 2 Cb0 , b04 11 2 2 3Nf, b14 102 2 38 3 Nf.

Substituting (8) into (7) and taking into account (5) and (6) we find the running mass

m(Q2 ) 4m( m2_{) exp}

y

2 g0 b0 1 5( 5 13B)

[

L

(

a(Q 2₎

₎

2 L

(

a(m2₎

_)]

z

_, (10) where

L(a) 4x1S(a , a1) 1x2S(a , a2) 1x3S(a , a3) (11) and (12) _{S(a , b) 42}

g

51 2 A 1 255 4

h

ln ( 1 2a) 1 2b 1 1

k

g

2 b 116b2 1 15 2 b 3

h

A 1 1 b 1 5 1 57 4 b 131b 2 1 25 2 b 3

l

ln (a 2b) 1 2b 2 2

g

47 2 a 1 15 4 a 2 1 15 2 ab

h

A 2 87 2 a 2 25 4 a 2 2 25 2 ab 2 ln a b with A 4g1O( 2 Cg0).

The values of ai in eq. (11) are the solutions of the equation f(ai) 40, and x14 1 (a12 a2)(a12 a3) , x24 1 (a22 a1)(a22 a3) , x34 1 (a32 a1)(a32 a2) . (13)

The equation for the pole mass reads as follows:

M 4 m(M2₎

y

1 14 a 2 (M2) C

(

1 13a(M 2₎

₎

g

4 3 1 ln 4 p 2 g

h

z

. (14)

For small values of the coupling constant eq. (14) reproduces the corresponding equation for the pole quark mass in the two-loop approximation of perturbation theory [9].

By using the expression (8) for the b-function we can find the running expansion parameter a(M2) as a solution of the equation

ln M 2 m02 4 C 2 b0 [ fA(a) 2fA(a0) ] , (15) where fA_{(a) 4} 1

5( 5 13B)[x1J(a , a1) 1x2J(a , a2) 1x3J(a , a3) ] (16) with (17) _{J(a , b) 42} 2 a2_b 2 4 ab2 2 12 ab 2 9 ( 1 2a)(12b) 1 4 112b121b2 b3 ln a 1 130 221b ( 1 2b)2 ln ( 1 2a)2 ( 2 1b)2 b3 ( 1 2b)2ln (a 2b) , and xi are defined by eq. (13).

To compute the values of the pole quark masses we will use as input the following values of the running masses at 1 GeV2_[10]

mu( 1 GeV )45.661.1 MeV, md(1 GeV)49.961.1 MeV, ms( 1 GeV )4199633 MeV . From eq. (14), we then get

Mu4 63 6 6 MeV , Md4 77 6 5 MeV , Ms4 379 6 43 MeV . (18)

The above values of the pole quark masses differ from the standard constituent quark masses, but are close to the values of the so-called effective quark masses which have been found from an analysis of e1_e2 _{experimental data [11] . For heavy quarks the} difference between the values of the pole quark masses which are computed perturbatively and in the framework of this approach is practically negligible and we will take Mc4 1.5 GeV, Mb4 4.5 GeV and Mt4 174 GeV.

So, we have calculated the values of pole quark, by using the MS procedure and as input the values of the running masses. One could give a basic argument against these considerations using pole quark masses by assuming that the full quark propagator

should not have a pole due to quark confinement. Leaving this fundamental question aside, we shall see that pole masses nevertheless provide a suitable parametrization in the following considerations.

4. – Momentum renormalization scheme and nonperturbative b-function

In the previous section we have calculated the values of pole quark masses which are renormalization point and renormalization scheme invariant, by using the MS procedure and as input the values of the running masses at 1 GeV. On the other hand, for an analysis of some physical processes, e.g. e1_e2 _{annihilation into hadrons, it is} suitable to use a momentum renormalization scheme with pole quark masses in order to simulate threshold effects. Therefore, in this section we will consider the renormaliza-tion procedure in the momentum scheme and establish a connecrenormaliza-tion between the nonperturbative b-function and the static quark-antiquark potential.

In the first order of our approximation, the renormalization constant Zl(m 8, m),

which describes the modification of the coupling constant l(m) when changing the scale parameter from m to m 8, has the following form:

Zl(m 8, m) 411leff[J(m 82) 2J(m2) ] . (19) Here leff4 1 Ca 2 ( 1 13a) (20) and J( m 82 ) 2J( m2 ) 411 lnm 8 2 m2 2 2 3

!

f

y

I

u

Mf 2 m 82

v

2 I

u

Mf 2 m2

v

z

, (21)

where the function I(M2

Om2) is the well-known one-loop integral

I

g

M 2 m2

h

4 6



0 1 dx x( 1 2x) ln

y

1 1 m 2 M2x( 1 2x)

z

. (22)

Using the definition of the b-function 2b(l) l 4 ¯ln Zl(m 8, m) ¯_{ln m 8}2

N

m 84m , (23) we find 2b(l) l 4 leff(a)

y

11 2 2 3

!

f F1

u

Mf2 m2

v

z

, (24) where F1(x) 4126x1 12 x2 k1 14xln k1 14x 11 k1 14x 21 . (25)

The renormalization scale dependence of the running expansion parameter a 4

a(m2) is now defined by the following equation:

C[U(a) 2U(a0) ] 411 ln m2 m20 2 2 3

!

f

y

I

u

Mf 2 m2

v

2 I

u

Mf2 m02

v

z

, (26)

where m0 is some normalization point, a04 a(m20) and the function U(a) has the following form: U(a) 4 1 a2 2 3 a 2 12 ln a 1 3 4 ln ( 1 2a)1 45 4 ln ( 1 13a) . (27)

Notice that the function U(a) is monotonous and changes its values from plus to minus infinity when the parameter a changes in the interval 0 EaE1. Thus, eq. (26) has a unique solution for the running expansion parameter a(m2_).

As seen from eq. (24) at m2

4 Q2E Mf2, the contribution of quarks is suppressed in

the b-function. Thus, for sufficiently small Q2 we can relate our result for the

b-function with the static potential of QCD without dynamical quarks.

The non-relativistic static quark-antiquark potential used in meson spectroscopy is associated with the running coupling constant l(Q2_{) as follows:}

V(r) A



dQ exp [iQ Q r] l(Q 2

)

Q2 , (28)

In order to imply the idea of quark confinement, one usually assumes that the running coupling constant has at small Q2 an infrared singularity of the form

l(Q2 ) A 1

Q2

(29)

leading at long distances to a linearly increasing confinement potential V(r) Ar. The infrared behaviour (29) of the running coupling constant then implies the following behaviour of the b-function at large values of the coupling constant

2b(l) l 4 2 1 l m 2 dl(m 2₎ dm2 C 1 . (30)

From eq. (26) we find that in the limit of large values of the coupling constant (m K0, aK1) 2b(l) l C 44 Casympt . (31)

This relation allows one to fix the parameter C within this approach: CasymptC 44. However, in an appropriate region of the momentum, we will choose a value of C slightly lower than the value of Casympt. Figure 3 shows the function (24) for C 439 calculated with pole quark masses defined in the previous section, as a function of 1 OQ.

5. – e1_e2 _{annihilation into hadrons at low energies}

In this section we will apply the results obtained to low-energy total cross-section to the process of e1_e2 _{annihilation into hadrons which is usually characterized by the}

R-ratio. It is known that in the low energy region, a direct comparison of the

theoretical prediction with experimental data is impossible due to the large contributions coming from higher order terms. Here, following [12], we can avoid this problem by introducing so-called smearing quantities. For the R-ratio the

corresponding smearing function can be defined as imaginary part of correlation function P

RD(s) 4 1

2 i[P(s 1iD)2P(s2iD) ] . (32)

The smeared quantity RD(s) then allows one to compare the experimental data with the theoretical predictions obtained in the framework of QCD.

Seemingly, one could try to calculate the function RD(s) by employing perturbation theory for the correlation function P(q2_{) in the Euclidean region q}2

E 0 using the perturbative running coupling constant as(Q24 2 q2), and then performing an analytic continuation to q2_{4 s 6 iD. Note that the correlation function has perfectly} well-defined analytic properties. Namely, P(q2_{) is an analytic function in the complex}

q2_{-plane with a cut along the positive part of the real axis. The parametrization of P(q}2₎ by the perturbative running coupling constant breaks these analytic properties. This

immediately becomes also clear if one rewrites (32) by using the dispersion relation for P(q2) RD(s) 4 D p



0 Q ds 8 R(s 8) (s 82s)2 1 D2 . (33)

The perturbative approximation of R(s) by as(s) indeed leads to a singularity of the integrand on the integration contour. By applying a special optimization method to the third-order of perturbative approximation of R(s), in [13] it has been shown how to calculate the smearing quantity RD(s) and compare the theoretical prediction with experimental data. Let us now for the same purpose apply the variational nonperturbative approach in the first order of our approximation. We will consider both the smeared quantity (32) and the following function:

WD(q2) 4 dRD(q2) dq2 4 2 1 2 i

y

D(q2 1 iD) q2 1 iD 2 D(q2 2 iD) q2 2 iD

z

. (34)

The D(q2)-function can be written as follows: (35) D(q2 ) 42q2 d dq2P(q 2 ) 4 3 p

!

f Qf2

y

F1

u

Mf 2 2q2

v

1 4 leff (q2_{) F} 2

u

Mf 2 2q2

v

z

, where F1is defined by eq. (25) and for the function F2we will use the result obtained on the basis of the Schwinger approximation for the imaginary part of P [14]

F2

g

M2 2q2

h

4 4 p 3 4 M2 2q2



0 1 dv v 2 ( 3 2v2₎ [ 4 M2 O(2q2) 112v2_]2 f (v) (36) with f (v) 4 p 2 v 2 3 1v 4

g

p 2 2 3 4 p

h

. (37)

Fig. 5. – The function WDvs. Q 4ks for D41, 2, and 4 GeV2.

The effective coupling constant leffin eq. (35) is defined by eq. (20) with the running expansion parameter a(2q2), which can be found as a solution of eq. (26). Some details of the calculation of the WD-function can be found in appendix A.

We should note that in contrast to saturated perturbation theory, where the running coupling constant has an infrared singularity contradicting the dispersion

Fig. 6. – The function WDvs. Q 4ks for D42 GeV2. The solid curve is our result, the dashed line

from the smeared experimental data (taken from [13]) and the dot-dashed curve has been obtained in [13] by applying the optimization procedure to the third-order calculation of Re1_e2.

relation for the D-function, the effective coupling constant leff is a finite function in the infrared region with freezing behaviour at small Q2.

To define the parameter a0we will normalize our results using experimental data at sufficiently large momentum Q04 6 GeV together with the values of the pole quark masses found in the sect. 3. In fig. 4, we have plotted the effective coupling constant

aeff4 4 pleffat small Q. Indeed, the effective coupling constant asis seen to be frozen, and, at very small Q, aeffC 1.2. It is interesting that the value aeffC 1.2 obtained in such a way is very close to the universal constant as( 0 ) 44pOb0obtained in the framework of analytic perturbation theory [15].

In fig. 5 we show our result for the function WD(s) at D 41, 2, 3, and 4 GeV2(curves with numbers 1, 2, 3, and 4, respectively). The two peaks in each of the curves are associated with the regions of the r-meson resonance and the charmonium. In fig. 6, we compare the function WD(s) for D 42 GeV2(solid line) with the smeared experimental data (dashed line) from [13]. It should be emphasized that our curve obtained in the first non-trivial order of leff is close to the theoretical predictions obtained in [13] on the basis of optimization of the third-order QCD perturbative corrections to Re1e2 (see

dot-dashed curve in fig. 6). As we can see from fig. 6, the value of D 42 GeV2 _{is not} sufficiently large to smooth the region of charm resonances. When the value of D grows up to 4 GeV2, the experimental curve approaches the theoretical prediction shown in fig. 5 (curve 4).

In fig. 7 and 8, we plot the function RD vs. Q 4ks for D41 GeV2and D 43 GeV2. The solid curve represents our theoretical prediction, the dashed curve is the smeared experimental data from [13], and the dot-dashed curve is the optimized third-order perturbative prediction [13]. As we can see from fig. 7 and 8, the solid and dot-dashed curves are similar to each other. The gap between these lines in the region of small momenta is mainly due to the difference in values of light quark

Fig. 8. – The function RDvs. Q 4ks for D43 GeV2. All lines are defined as in fig. 6.

masses. This difference becomes smaller if one suppresses the contribution of light quarks by increasing D.

Note that for small D C1–2 GeV2, experimental data for the smeared quantities RD and WDhave a resonance structure near the charmonium family. By increasing D up to 3–4 GeV2_{we smooth this region and the agreement between the theoretical predictions} (solid and dot-dashed curves in figs. 7, 8) and experimental (dashed curves) data becomes improved.

6. – Summary and conclusion

In this paper, we have further developed the approach to quantum chromodynamics based on a small expansion parameter and considered its application to the process of e1_e2_{annihilation into hadrons at low energies. An important virtue of this approach is} the fact that for sufficiently small asit reproduces the standard perturbative expansion and, in principle, all the high-energy physics is maintained by the method. In going to lower energies where standard perturbation theory ceases to be valid (asA 1), the new expansion parameter

(

really, a2_(Q2

) OC

)

remains small and we do not go beyond the range of applicability of the method. Using this variational nonperturbative method, we have calculated the values of pole quark masses taking the values of the running masses at 1 GeV2_{as input. Next, we have investigated the momentum renormalization} scheme with the pole quark masses. Within this scheme the problem of the number of active quarks does not appear. The effective coupling constant now depends on all quark masses and does not require any additional definition of the dependence on the number of active flavours. On the basis of these results we have then described the process of e1_e2 _{annihilation into hadrons at low energies using the smearing method} to smooth the resonance structure. In particular, it was shown that for a sufficiently large D the theoretical predictions obtained in the first order of our approximation

agree with experimental data up to lowest energies. Note that the full quantities RD and WDto which all quarks contribute are not very sensitive to the values of light quark masses. It will be further interesting to consider the corresponding quantities without the contribution of heavy quarks. In this case, there is a possibility to decrease the value of D up to the r-meson scale and to investigate the values of the light pole quark masses in detail. We plan also to apply this approach to the decays of t-lepton where the heavy quark contribution does not play an essential role and sensitivity to the values of light pole quark masses will become noticeable.

* * *

We would like to thank H. F. JONES, A. RITZand A. N. SISSAKIANfor interest in the work and useful comments. One of us (IS) would like to thank the particle theory group of the Humboldt University for kind hospitality.

AP P E N D I X A

In this appendix we collect some formulae used in the text. The function WD can be written as follows:

WD(s) 4 1 s2_{1 D}2[D D1(s , D) 2sD2(s , D) ] , (A.1) with

.

`

/

`

´

D1(s , D) 4 3 p

!

f Qf2

[

A1(s , Mf) 14

(

l1(s) A2(s , Mf) 2l2(s) B2(s , Mf)

)]

, D2(s , D) 4 3 p

!

f Qf2

[

B1(s , Mf) 14

(

l1(s) B2(s , Mf) 1l2(s) A2(s , Mf)

)]

, (A.2) where

l1(s) 4Re leff

(

a(s)

)

, l2(s) 4Im leff

(

a(s)

)

. (A.3)

The complex function a(s) 4a11 ia2 obeys the following system of equations:

.

`

/

`

´

C Re [U(a11 ia2) 2U(a0) ] 411 ln

k

s2 1 D2 Q02 2 2 3

!

f

y

A0(s , Mf) 2I

u

Mf2 Q02

v

z

, C Im U(a11 ia2) 42 11

g

p 2 1 a tan s D

h

2 2 3

!

f B0(s, Mf) . (A.4)

Finally, the functions Ai(s , M) and Bi(s , M) are defined by

(A.5) A0(s , M) 4



0 1 du u 2 ( 3 2u2₎ ( 1 2a2u2₎2 1 b2( 1 2a2u 2_{) ,} (A.6) B0(s , M) 42 b



0 1 du u 2 ( 3 2u2) ( 1 2a2u2₎2 1 b2 ,

(A.7) A1(s , M) 4 3 2



0 1 du ( 1 2u 2₎2 ( 1 2a2u2₎2 1 b2( 1 2a2u 2_{) ,} (A.8) B1(s , M) 42 3 2b



₀1_du ( 1 2u 2₎2 ( 1 2a2u2₎2 1 b2 , (A.9) A2(s , M) 4 4 p 3



0 1 du u 2 ( 3 2u2₎ [ ( 1 2a2u2₎2 1 b2]2 f (u) Q Q ]2a[ (12a2u2₎2 2 b2] 12b2 ( 1 2a2u2 )( , (A.10) B2(s , M) 4 4 p 3



0 1 du u 2 ( 3 2u2₎ [ ( 1 2a2u2₎2 1 b2]2 f (u) Q Q ]b[ (12a2u2₎2 2 b2] 12ab(12a2u2 )( . Here a 4 4 M 2_s s2_{1 D}2 , b 4 4 M2_D s2_{1 D}2 (A.11)

and the function f (u) is defined by eq. (37) in the text.

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