Z. Phys. A 357, 333–338 (1997) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1997

Parametrization of realistic Bethe-Salpeter amplitudefor the deuteronA.Yu. Umnikov?

Department of Physics, University of Perugia, and INFN, Sezione di Perugia, via A. Pascoli, I-06100 Perugia, Italy

Received: 1 August 1996 / Revised version: 16 October 1996Communicated by W. Weise

Abstract. The parametrization of the realistic Bethe-Salpeteramplitude for the deuteron is given. Eight components of theamplitude in the Euclidean space are presented as an analyt-ical fit to the numerical solution of the Bethe-Salpeter equa-tion in the ladder approximation. The applicability of theparametrization to the observables of the deuteron is brieflydiscussed.

PACS: 25.30.−c; 13.60.Hb; 13.40.−f

1 Motivations

Since the first solution of the realistic Bethe-Salpeter (BS)equation for the deuteron [1], the BS-amplitudes have beenapplied to describe various processes with the deuterons [1–5]. The obvious advantages of the approaches based on theBS formalism are the explicit covariance and connectionto the covariant dynamical (field) theory. In spite of this,practical use of the BS amplitudes is not as popular as ofnonrelativistic wave functions [6, 7]. A more complicatedphysical interpretation and technical complexity of the ap-proaches based on the BS amplitudes are the main reasonsfor that.

In a series of recent papers it has been argued that anew intuition can be developed in working with the BS am-plitudes [3–5]. It has been shown that calculations of manyobservables for the deuteron are reduced to calculations sim-ilar to those in field theory. It has also been stressed that theusage of the BS amplitude for the deuteron in the Diracmatrix basis can be more convenient than in the spinor ba-sis [8]. In this case, computations are formalized enoughto extensively apply analytical computing software, such asthe Mathematica [9], to calculate the matrix elements of ob-servables in terms of the components of the BS amplitude.However, the promotion of a new technique should assumethat all basic ingredients of the calculation are available tothe potential user. The BS formalism still lacks this feature,since there is no simple parametrization available for the

? INFN Postdoctoral Fellow

realistic deuteron amplitude, such as is available for nonrel-ativistic wave functions [7, 10] or relativistic wave functionsof the spectator equation [11].

This paper presents the analytical parametrization of theBethe-Salpeter amplitude for the deuteron. The parametersare fixed by fitting to the recent numerical solution of thehomogeneous BS equation using the Dirac matrix basis [8,5]. The one-boson exchange potential from [1, 12] was usedwith a minor adjustment of its parameters [3], so in thissense the solution does not contain new physics (or differentphysics) then the pioneer paper.

2 Definitions and kinematics

The realistic BS amplitude of the deuteron,Φ, can be ob-tained as a solution to the homogeneous BS equation withthe effective one-boson-exchange kernel [1, 3]:

Φ(p, PD) = iS(p1, p2)∑B

∫d4p′

(2π)4

g2BΓ

(1) ⊗ Γ (2)

(p− p′)2 − µ2B + iε

Φ(p′, PD), (1)

wherePD is the deuteron momentum,µB is the mass ofthe mesonB, Γ (k)

B is the meson-nucleon vertex, correspond-ing to the mesonB and connected to thek-th nucleon. Thetensor product in the r.h.s. of (1) is for a possible intercon-nection of the quantum numbers in two vertices (such as inthe case of the vector or isovector mesons). The two-nucleonpropagator,S(p1, p2), is defined as:

S(p1, p2) ≡ p1 +mp2

1 −m2 + iε· p2 +mp2

2 −m2 + iε= (p1 +m)(p2 +m)D(p, PD), (2)

p1,2 =PD2± p, (3)

wherem is the nucleon mass,p1,2 are nucleon momenta,pk = γµp

µk andD(p, PD) is “scalar two particle propagator”.

The (nontruncated) BS amplitude(Φ)αβ is a 4×4-matrixin the space of spinor indices. Let its first index,α, corre-spond to the spinor index of the first nucleon, andβ to the

334

spinor index of the second nucleon. In this notation the sym-bolic equation (1) has the following explicit spinor structure:

(Φ(p, PD))αβ = i(p1 +m)αα′ (p2 +m)ββ′D(p, PD) (4)∑B

∫d4p′

(2π)4

g2B

(Γ (1)

)τα′α′′

(Γ (2)

)τβ′β′′

(p− p′)2 − µ2B + iε

(Φ(p′, PD)

)α′′β′′ ,

where we replaced the tensor product by the explicit indexτ . By interchanging indices we can rewrite r.h.s. of (4) inthe form of the product of matrices, with the matrices corre-sponding to one of the nucleons transposed. Transpositionswould also occur when the deuteron observables are calcu-lated in terms of BS amplitude,Φ. To simplify calculations,it is convenient to use the amplitude,Ψ , conjugated withrespect to one of the nucleon lines [8]:

Φ = Ψγc or Ψ = −Φγc, (5)

whereγc = γ3γ1, with the propertyγcγTµ = γµγc. In terms

of the amplitudeΨ , the Eq. (1) and the matrix elements ofobservables are written as products of matrices without trans-positions, e.g.

Ψ (p, PD) = iD(p, PD)∑B

∫d4p′

(2π)4

g2B(p1 +m)Γ τΨ (p′, PD)Γ τ (p2 +m)

(p− p′)2 − µ2B + iε

. (6)

Examples of observables in terms of amplitudeΨ are pre-sented in Sect. 4.

The amplitudeΨ can be decomposed in terms of a com-plete set of the Dirac matrices, their bilinear combinationsand the 4× 4-identity matrix, 1, with sixteen componentsbeing the coefficients of decomposition:

Ψ = 1/υs + γ5/υp + γµ/υµv + γ5γµ/υ

µa + σµν/υ

µνt . (7)

In the deuteron rest frame the notions are used:

/υµv ≡ (/υ0v, /υv), /υµa ≡ (/υ0

a, /υa), (8)

/υ0it ≡ /υ0

t , /υijt ≡ εijk/υkt[/υkt ≡ /υt

], (9)

where i, j, k = 1, 2, 3 and other tensor components of/υµνtare equal to zero.

In order to separate the amplitude with the deuteron’squantum numbers, a partial wave decomposition of the fourvector and four scalar functions, (7)-(8), is performed. Fixingthe total momentumJ = 1, we get:

/υJM |J=1 = /υ(p0, |p|; 1)Y1M (Ωp) (10)

/υJM |J=1 =∑

L=0,1,2

/υ(p0, |p|;L 1)Y L1M (Ωp), (11)

whereYJM andY LJM are the spherical harmonics and vector

spherical harmonics respectively. Then the parity invarianceof the amplitude is exploited:

P Ψ (p0,p) = ηγ0Ψ (p0,−p)γ0, (12)

where for the positive parity deuteronη = 1. Separating thecomponents with positive parity, the deuteron’s amplitudecomponents read as:

/υp(1), /υ0a(1), /υv(11), /υa(01),

/υa(21), /υ0t(11), /υt(01), /υt(21). (13)

Components with different quantum numbers, such asJ /= 1and/or negative parity, do not mix with the state defined by(13). The components/υ0

a(1) and/υ0t(11) are odd functions of

p0 and the rest of components (13) are even.Often, instead of theγ-matrices representation, (7), al-

ternative the two-spinor basis is used [1, 13], which meansan outer product of two spinors representing solutions ofthe free Dirac equation with positive and negative energies.This basis is labeled by the helicities,λ1,2, the “energy-spin”,ρ1,2, and the relative momentum of the nucleons, sometimesalso called (J, λ1, λ2, ρ1, ρ2)-representation. In this case thespectroscopic notation,2S+1Lρ1ρ2

J , can be used similarly tothe traditional nuclear physics symbolics. Then an approxi-mate correspondence between the set of amplitudes (13) andthe two-spinor representation reads [14]:

3S++1 ∼ 2/υt(01) +/υa(01),

3S−−1 ∼ 2/υt(01)− /υa(01), (14)

3D++1 ∼ 2/υt(21) +/υa(21),

3D−−1 ∼ 2/υt(21)− /υa(21), (15)

3P +−1 +3 P−+

1 ∼ /υ0t(11),

3P +−1 −3 P−+

1 ∼ /υv(11), (16)1P +−

1 +1 P−+1 ∼ /υp(11),

1P +−1 −1 P−+

1 ∼ /υ0a(11). (17)

It can be shown that the waves3S++1 and3D++

1 directly corre-spond to theS andD waves in the deuteron wave function,while others vanish in the nonrelativistic limit. A more de-tailed discussion about the connection of the Dirac matricesbasis to other representations, and about the relation of theBS components to the nonrelativistic wave functions is givenin [14].

The BS equation with the realistic one-boson exchangepotential is solved using the Wick rotation, which corre-sponds to replacep0 → ip4 and /υ0

a(1) → i/υ0a(1) [8]. This

procedure removes singularities from the exchange mesonpropagators in (1) and from the scalar propagator,D, whichin the deuteron rest frame takes the form:

D(p, PD)−1 = D(p4, |p|)−1

= [m2 + p2 + p24 −

14M2

D]2 + p24M

2D, (18)

whereMD = 2m + εD is the deuteron mass.After the Wick rotation, the components of the deuteron

amplitude arecomputedalong the imaginary axe in thecomplexp0-plane. The parameters of the ladder kernel arepresented in Table 1, with coupling constants correspond-ing to the following definition of the cut-off formfactors:FB(k) = (Λ2 − µ2

B)/(Λ2 − k2). The meson parameters(masses, coupling constants and cut-off parameters) havebeen taken to be the same as in [1, 12], except for the cou-pling constant of the scalarσ-meson, which has been ad-justed to provide a numerical solution of the homogeneousBS equation. The chosen set of meson parameters requiressome adjustments in light of two circumstances. First, herewe use a simplified form of the propagator for of the vec-tor mesons, omittingkµkν/µ2

B-term. Second, the differentnumerical procedures in solving the eigenvalue problem forthe BS equation can also affect the value of the parameters.

335

Since the inverse Wick rotation of thenumericallyknown amplitude is an ill-defined operation, the parametriza-tion for the components is obtained in the Wick rotated case.The possibility to analytically continue those amplitudes intoa physical region is discussed in Sect. 4.

3 The parametrization

The parametrization of all components has the form (indexJ = 1 is omitted):

/υ(p4, |p|;L) = f (p4, |p|;L)

Expg(|p|;L)p2

4

D(p4, |p|)m4, (19)

whereL = 0, 1, 2, depending on the quantum numbers ofthe component and functionsf andg are given by:

f (p4, |p|;L) =Nf∑i=0

Ai(p4)|p|L

α2i + p2

,

g(|p|;L) =Ng∑i=1

Bi|p|L

β2i + p2

. (20)

The form of parametrization (20) and the scale of theparametersαi are prompted by the previous works withparametrizations of the wave functions [7, 10, 11]:

α0 = µ0/√

2, αi = iµ0, i = 1 . . . Nf ,

βi = iµ1, i = 1 . . . Ng,

µ0 = 0.139 GeV, µ1 = 2µ0,

m = 0.939 GeV, MD = 2m + εD,

εD = −2.2246 MeV. (21)

The numberNf is equal to 11 for all components, whereasNg differs for different components. The small|p| behaviorof the form∝ |p|L for components with the angular mo-mentumL is exactly obtained by analyzing the partial com-ponents of the kernel of the BS equation at|p| → 0. Theform of p4-dependence in (19) is found empirically. Afteran explicit separation of the factorD(p4, |p|) the remainingpart of all components has very weakp4-dependence, whichat any given|p| is nicely approximated by the Gaussian ex-ponent,∼ Exp[−gp2

4]. The coefficientg is of the order of∼ 1 GeV2 and has a weak|p|-dependence. An additionalp4-dependence in coefficientsAi is not important, and it isintroduced to account for minor deviations from the Gaus-sian form:

Ai(p4) = Ai + p24A

′i, for

/υp(1), /υv(1), /υa(0), /υa(2), /υt(0), /υt(2);

Ai(p4) = p4(Ai + p24A

′i), for

/υ0a(1), /υ0

t(1). (22)

The explicit extra factorsp4 in the two last coefficients aredue to the oddity of these two functions.

CoefficientsAi, A′i andBi for all components are given

in Appendix A, Tables 2-9. Obviously, a normalization con-dition for the BS amplitude can be considered as a constrainton the set of parameters used in our parametrization. How-ever, we choose a different approach, we perform aninde-pendentfit of each of the components, without envoking the

D

1p

2p

O(q=0)^

P

Fig. 1. The diagram for the matrix element of operatorO over the deuteronstate in the impulse approximation

requirement that all together these components must satisfya normalisation condition. Instead, in the next section, weuse the normalization on the vector charge to test the qualityof the obtained set of parameters.

The presented parametrization contains a seemingly largenumber of parameters, 27 to 29 per every of eight compo-nents plus three parameters common for all of them, includ-ing the nucleon mass,m, deuteron binding energy,εD, andan additional mass scale parameter,µ0. This looks ratherunusual for such type of parametrizations. The parametriza-tions of wave functions [7, 10, 11], for instance, contain onlyn ∼ 10 parameters per component. The reason for this dif-ference is that the components of the BS amplitude dependupon two independent variables, the relative momentum,p,and “relative energy”,p4. It is clear now, that the presentedparametrization contains quite a modest number of parame-ters; it is not evenn2 compared to the one dimensional fitof the wave functions.

4 The applicability of the parametrization

The parametrization (19)-(22) is obtained by fitting the nu-merical solution to the BS equation, using the least-squaresprocedure [9]. The validity domain of the parametrization inrelative momentum is 0< |p| < 3 GeV, which means thatthe solution of the BS equation was fitted up to this point.The domain of validity in relative energyp4 (which is ac-tually ip0) is defined as follows. First, the singular structureof the BS amplitude in the Minkowski space is governed bythe singularities of the propagatorD(PD, p), Eqs. (2) and(18), where the closest nucleon pole is most important forthe physical applications. Thus, the parametrization is validat least up top4 ∼ MD/2−√

m2 + |p|2, corresponding tothe nucleon pole at a given|p|. Second, the parametrizationallows for the integration in the matrix elements overp4 withinfinite limits,(−∞,+∞), in the Euclidean space.

The starting point for calculating any quantity with theBS amplitude is therelativistic impulse approximation. Inmany cases the relativistic impulse approximation is pre-sented by the Feynman “triangle diagram” with zero transferof the momentum [3–5, 8],q = 0 (Fig. 1):

〈O〉 =∫

d4p

(2π)4Tr

Ψ (p0,p)OΨ (p0,p)(p2 −m)

, (23)

whereΨ = γ0Ψ†γ0. Two important examples are the matrix

elements of the vector and axial currents,O = γµ, γ5γµ.

336

Fig. 2. Densities of the vector (solid line) and axial (dashed line) chargescalculated with the presented parametrization

The matrix element〈γ0〉, the vector charge, is used to nor-malize the BS amplitude:

1 =1

2MD

∫d4p

(2π)4Tr

Ψ (p0,p)γ0Ψ (p0,p)(p2 −m)

(24)

= − 1MD

∫dp4d|p|p2

(2π)4

−8m (/υa(0)/υt(0) + /υa(2)/υt(2))

+4p√

3

(−2/υp(1)/υt(0) + 2

√2/υp(1)/υt(2)

+√

2/υa(0)/υv(1) + /υa(2)/υv(1))

+(Md − 2p4)(/υ0a(1)

2+ /υa(0)2 + /υa(2)2 + /υp(1)2

+/υv(1)2 + 4/υt(0)2 + 4/υt(2)2 + 4/υ0t(1)

2)

. (25)

The components in (25) are parametrized by Eqs. (19)-(22).(Note that factor 2π from the integration over angleφ isabsorbed by the parametrization.) By integrating overp4 in(25) but keeping|p|-dependence, one gets the charge den-sity in the momentum space, analogous to the square of thedeuteron wave function in a nonrelativistic approach. Thischarge density is shown in Fig. 2 together with the densityof the 3-rd component of the axial current (omitting terms,vanishing after integration overθ):

〈γ5γ3〉 =1

2MD

∫d4p

(2π)4Tr

×Ψ (p0,p)γ5γ3Ψ (p0,p)(p2 −m)

(26)

=1

MD

∫dp4d|p|p2

(2π)4

2m (4/υa(0)/υt(0)− 2/υa(2)/υt(2)

−2√

2/υ0a(1)/υ0

t(1) +√

2/υp(1)/υv(1))

+2p√

3

(4/υp(1)/υt(0) + 2

√2/υp(1)/υt(2)

−2√

2/υa(0)/υv(1) + /υa(2)/υv(1))

+(Md − 2p4)

(−/υa(0)2 +

12/υa(2)2 − 1

2/υv(1)2

−4/υt(0)2 + 2/υt(2)2 − 2/υ0t(1)

2)

. (27)

The quality of the parametrization has been checked by acomparison of the charge and axial densities, as well as theirintegrals, computed using the parametrization and originalnumerical components. The original amplitude is normalizedby (24) and is exactly equal to 1.0, whereas the parametriza-tion results in the normalization equal to 0.9997. This is nota trivial result, since all components were fitted indepen-dently. One can use this number for the “renormalization”of observables. The original amplitude gives an axial chargevalue of 0.9215, while the parametrization yields the same.The error of the parametrization describing the densities is∼ 0.01% at small|p| to ∼ 1− 2% at |p| ∼ 1-3 GeV.

As usual, when the observables of the deuteron are dis-cussed, the question about a comparison with other modelsand approaches should be raised. Since there is no directway to meaningfully compare the BS amplitudes and non-relativistic wave functions, more sophisticated methods mustbe employed. For instance, we can compare the relativisticand nonrelativistic approaches by comparing the densities ofvarious charges and currents calculated in both approaches.In particular, it was found that the popular models [6, 7] leadto densities very similar to the ones, presented in Fig. 2 forthe BS approach [5, 14]. Another way to make a comparisonis to use approximate methods, which allow to “compute”wave function from the BS amplitude [14]. These “wavefunctions” are also quite similar to the realistic wave func-tion [6, 7]. For a more detailed discussion, we readdress thereader to [14], which is essentially devoted to the compari-son of the BS approach to the nonrelativistic approaches.

Finally, the issue of an “inverse Wick rotation” shouldbe addressed. The analysis of the singular structure of the“triangle graph” and behavior of the BS amplitude result inthe conclusion that, perhaps, the presented parametrizationcan be used for an analytical continuation,p4 → −ip0, ofthe BS amplitude up to the closest nucleon pole ofp0 =MD/2−√m2 + |p|2. However, such a procedure works well(with accuracy∼ 10%) only up to|p| ∼ m. The accuracywas estimated by calculating the vector and axial densitiesfor the processes with the second nucleon on mass-shell. Ifit is used further, the procedure gives an accuracy of∼ 50%at |p| = 1.5 GeV and it should not be used beyond this point.

337

5 Summary

The parametrization of the realistic Bethe-Salpeter amplitudefor the deuteron has been presented. All eight components ofthe amplitude are given in the Wick rotated case in the formof analytical functions. Simple examples of the use of theparametrization are presented and the applicability domainis discussed.

It is a pleasure to acknowledge stimulating conversations with L. Kaptariand F. Khanna. I would like to thank D. White for reading the manuscriptand comments.

References

1. M.J. Zuilhof and J.A. Tjon, Phys. Rev. C22 (1980) 23692. B.D. Keister and J.A. Tjon, Phys. Rev. C26 (1982) 5783. A.Yu. Umnikov, L.P. Kaptari, K.Yu. Kazakov and F. Khanna, Phys.

Lett. B334 (1994) 1634. L.P. Kaptari, A.Yu. Umnikov, B. Kampfer and F. Khanna, Phys. Lett.

B351 (1995) 4005. A.Yu. Umnikov, L.P. Kaptari and F. Khanna, e-print archives hep-

ph/96084596. M. Lacombe et al., Phys. Rev. C21 (1980) 8617. R. Machleid, K. Holinde and Ch. Elster, Phys. Rep.149 (1987) 18. A.Yu. Umnikov and F. Khanna, Phys. Rev. C49 (1994) 23119. S. Wolfram, Mathematica, Addison-Wesley (Reading, Massachusetts,

1993)10. M. Lacombe et al., Phys. Lett.B101 (1981) 13911. W.W. Buck and F. Gross, Phys. Rev. C20 (1979) 236112. J. Fleischer and J.A. Tjon, Nucl. Phys.B84 (1975) 375; Phys. Rev. D

15 (1977) 253713. J.J. Kubis, Phys. Rev. D6 (1972) 54714. L.P. Kaptari, A.Yu. Umnikov, S.G. Bondarenko, K.Yu. Kazakov, F.C.

Khanna and B. Kampfer, Phys. Rev. C54 (1996) 986

Appendix A. Tables of parameters

Table 1. Parameters of the ladder kernel

Meson Coupling constants Mass Cut-off IsospinB g2/(4π); [gt/gv ] µB , GeV Λ, GeVσ 12.2 0.571 1.29 0δ 1.6 0.961 1.29 1π 14.5 0.139 1.29 1η 4.5 0.549 1.29 0ω 27.0 [0] 0.783 1.29 0ρ 1.0 [6] 0.764 1.29 1

Table 2. Parameters for the/υp(1) component

L i Ai A′i Bi

1 0 -0.0108242476 -34.3742135 –1 0.0576888290 119.943051 02 -3.00806368 -1257.32744 03 52.5918636 13865.0551 -222.0894944 -549.793133 -101125.416 890.1028915 3325.75711 472699.731 -1210.297496 -11944.6774 -1417223.72 548.3008187 26383.0580 2743193.30 –8 -34689.7448 -3406750.03 –9 24977.5913 2621309.83 –10 -8159.59409 -1137897.83 –11 607.051002 213102.794 –

Table 3. Parameters for the/υ0a(1) component

L i Ai A′i Bi

1 0 0.113315679 15.2791178 –1 -0.565717405 -42.6551270 02 12.3795242 292.530922 03 -211.521767 -2058.70255 04 2025.71260 8992.57139 -48.37359895 -11769.4095 -20393.3149 91.95306716 43037.5609 10486.7684 -44.74385237 -101273.551 55592.5995 –8 151268.732 -141733.721 –9 -136889.618 152338.654 –10 67943.7391 -80778.8808 –11 -14143.4907 17289.2804 –

Table 4. Parameters for the/υv(1) component

L i Ai A′i Bi

1 0 0.00545480918 0.452976149 –1 -0.0413469671 3.58498830 02 -0.626170875 49.3118036 03 -12.5904324 -435.917083 04 144.575745 3114.24317 6.149539945 -1128.73423 -14069.8986 -25.21611926 4901.27381 41906.4275 19.98447727 -13706.4378 -85174.0821 –8 24829.8901 114516.487 –9 -26877.9006 -95556.5511 –10 15516.3123 44352.4015 –11 -3665.63223 -8706.31030 –

Table 5. Parameters for the/υa(0) component

L i Ai A′i Bi

0 0 -0.00198690442 -2.91350897 –1 0.01915782253 12.2001594 -0.8538057932 -0.154455300 -214.362824 17.75852823 25.2282913 3289.28876 -89.39458644 -361.376166 -30582.8134 155.5784115 2946.98165 166962.671 -92.04113856 -13477.6590 -553839.568 –7 37613.2426 1152552.96 –8 -63648.9139 -1515457.41 –9 62158.0044 1223772.96 –10 -31778.3109 -554471.928 –11 6523.12231 107982.421 –

338

Table 6. Parameters for the/υa(2) component

L i Ai A′i Bi

2 0 0.0906145663 -54.7995054 –1 -0.712220945 256.049408 -57.81957682 -20.3136219 470.222091 211.6302533 81.4188182 -8294.9022 -385.3189644 -544.595070 33870.9853 367.4618615 2355.01212 -77139.6959 -137.1874056 -6864.79033 118585.336 –7 13318.8325 -138016.983 1 –8 -16303.3461 126552.418 –9 11929.7419 -84889.6419 –10 -4708.14062 35162.7881 –11 756.740280 -6501.81818 –

Table 7. Parameters for the/υ0t(1) component

L i Ai A′i Bi

1 0 0.0126177910 0.547674806 –1 -0.0622969395 -3.75788455 02 1.64026310 30.8407337 03 -25.8788379 -220.882836 04 241.662795 1370.42460 7.639381445 -1410.95190 -5648.39611 -31.18365356 5126.52517 15857.5305 24.37387237 -12077.1617 -30611.8247 –8 18231.6691 39147.3503 –9 -16701.2532 -31118.3495 –10 8364.33221 13774.6556 –11 -1750.41655 -2578.09116 –

Table 8. Parameters for the/υt(0) component

L i Ai A′i Bi

0 0 -0.000764090007 -0.909552636 –1 0.00774111957 3.53114793 0.02667578512 0.081210854 -79.1547771 4.169479453 8.49620518 1058.66511 -3.266852124 -122.622299 -8772.66567 -36.12069065 1056.75816 43565.4328 36.13898586 -5058.22423 -132777.961 –7 15091.7728 258802.623 –8 -28171.6682 -327964.927 –9 31347.0373 263416.014 –10 -18926.2139 -122064.787 –11 4783.25176 24814.8930 –

Table 9. Parameters for the/υt(2) component

L i Ai A′i Bi

2 0 0.0927963894 6.63409360 –1 -0.566806578 27.0403716 -26.44808412 -6.23310249 628.891454 39.09793953 -22.0317803 -5172.06340 -12.49597844 299.860807 22767.0450 –5 -2053.92122 -74280.5598 –6 8124.81954 177690.047 –7 -20191.0789 -294391.655 –8 31928.1425 323661.633 –9 -30700.6358 -224618.740 –10 16220.4293 89089.3349 –11 -3598.88935 -15407.7777 –