Asymmetric allylic substitution by chiral palladium catalysts: Which is more reactive, major π-allyl Pd(II) species or minor π-allyl species?

Article abstract of DOI:10.1016/j.mcat.2020.111221

The reactivity difference of major and minor π-allyl species was examined for two typed asymmetric allylic substitutions via linear symmetrical π-allyl and linear unsymmetrical π-allyl intermediates. ³¹P NMR observation of the stoichiometric re

Full text of DOI:10.1016/j.mcat.2020.111221

Physics Letters B 659 (2008) 252–259
www.elsevier.com/locate/physletb
The lattice free energy of QCD with clover fermions, up to three-loops
A. Athenodorou a, H. Panagopoulos b,∗, A. Tsapalis c
a
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UK
b
Department of Physics, University of Cyprus, PO Box 20537, Nicosia CY-1678, Cyprus
c
Institute of Accelerating Systems and Applications, University of Athens, Greece
Received 26 October 2007; accepted 9 November 2007
Available online 4 December 2007
Editor: J.-P. Blaizot
Abstract
We calculate the perturbative value of the free energy in Lattice QCD, up to three loops. Our calculation is performed using Wilson gluons and
the Sheikholeslami–Wohlert (clover) improved action for fermions. The free energy is directly related to the average plaquette. To carry out the
calculation, we compute all relevant Feynman diagrams up to 3 loops, using a set of automated procedures in Mathematica; numerical evaluation
of the resulting loop integrals is performed on finite lattices, with subsequent extrapolation to infinite size. The results are presented as a function
of the fermion mass m, for any SU(N ) gauge group, and for an arbitrary number of fermion flavors. In order to enable independent comparisons,
c
we also provide the results on a per diagram basis, for a specific mass value.
© 2007 Elsevier B.V. All rights reserved.
PACS: 11.15.Ha; 12.38.Gc; 12.38.Bx
Keywords: Lattice perturbation theory; Free energy; Average plaquette; Clover action
1. Formulation of the problem
In this work we calculate the free energy of QCD on the lattice, up to three loops in perturbation theory. We employ Wilson
gluons and the O(a) improved Sheikholeslami–Wohlert (clover) [1] action for fermions. The purpose of this action is to reduce
finite lattice spacing effects, leading to a faster approach to the continuum. Dynamical simulations employing the clover action
are currently in progress by the CP-PACS/JLQCD [2] and UKQCDSF [3] Collaborations and therefore perturbative studies of
properties of the QCD action with clover quarks are worthy of being undertaken. The free energy, in the simpler case of Wilson
fermions, was studied in [4].
The free energy in QCD on the lattice can be related to the average plaquette. The results find several applications, for example:
(a) in improved scaling schemes, using an appropriately defined effective coupling which depends on the average plaquette (see,
e.g., [5,6]); (b) in long standing efforts, starting with [7], to determine the value of the gluon condensate; (c) in studies of the
interquark potential [8]; and (d) as a test of perturbation theory, at its limits of applicability.
Indeed, regarding point (d) above, the plaquette expectation value is a prototype for additive renormalization of a composite,
dimensionful operator. The vacuum diagrams contributing to such a calculation are power divergent in the lattice spacing and may
well dominate over any nonperturbative signal in a numerical simulation.
Starting from the Wilson formulation of QCD on the lattice, with the addition of the clover (SW) fermion term, the action reads
in standard notation:
*
Corresponding author.
E-mail addresses: a.athenodorou1@physics.ox.ac.uk (A. Athenodorou), haris@ucy.ac.cy (H. Panagopoulos), a.tsapalis@iasa.gr (A. Tsapalis).
0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.physletb.2007.11.064

A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
253
SL ≡ SG + SF ,
ꢀ
ꢁ
ꢂ
1
SG =
Tr 1 − Uμ,ν(x) ,
g2
x,μ,ν
ꢀꢀ
ꢀꢀ
ꢁ
ꢂ
1
2
†
¯
¯ ¯
ψf (x)(r − γμ)Uμ(x)ψf (x + μˆ ) + ψf (x + μˆ )(r + γμ)Uμ(x) ψf (x)
SF =
(4r + mB)ψf (x)ψf (x) −
x
x,μ
f
f
ꢀ ꢀ
i
¯
ˆ
ψf (x)σμνFμν(x)ψf (x),
+
cSW
(1)
(2)
4
x,μ,ν
f
1
ˆ
where: Fμν
≡
(Qμν − Qνμ), Qμν = Uμ,ν + Uν,−μ + U−μ,−ν + U
.
−ν,μ
8
Here Uμ,ν(x) is the usual product of SU(Nc) link variables Uμ(x) along the perimeter of a plaquette in the μ–ν directions,
originating at x; g denotes the bare coupling constant; r is the Wilson parameter, which will be assigned its standard value r = 1;
f is a flavor index; σμν = (i/2)[γμ, γν]. Powers of the lattice spacing a have been omitted and may be directly reinserted by
dimensional counting.
The clover coefficient cSW is a free parameter for the purposes of the present calculation and our results will be presented as a
polynomial in cSW, with coefficients which we compute. Preferred values for cSW have been suggested by both perturbative (1-loop)
[1] and non-perturbative [9] studies.
We use the standard covariant gauge-fixing term [10]; in terms of the vector field Qμ(x) [Uμ(x) = exp(igQμ(x))], it reads:
ꢀꢀ
Sgf = λ0
TrΔ−μ Qμ(x)Δ−ν Qν(x), Δμ−Qν(x) ≡ Qν(x − μˆ ) − Qν(x).
x
(3)
μ,ν
Having to compute a gauge invariant quantity, we chose to work in the Feynman gauge, λ0 = 1. Covariant gauge fixing produces
the following action for the ghost fields ω and ω
ꢃ
ꢀ
ꢄ
ꢅ ꢄ
ꢁ
ꢂ
ꢁ
ꢂ
ig0
†
Sgh = 2
Tr Δ+μ ω(x) Δ+μ ω(x) + ig0 Qμ(x), ω(x) +
Qμ(x), Δ+μ ω(x)
2
x,μ
ꢆ
g02
g04
ꢁ
ꢁ
ꢂꢂ
ꢁ
ꢁ
ꢁ
ꢁ
ꢂꢂꢂꢂ
ꢅ
−
Qμ(x), Qμ(x), Δ+μ ω(x)
−
Qμ(x), Qμ(x), Qμ(x), Qμ(x), Δ+μ ω(x)
+ ···
,
12
720
Δ+μ ω(x) ≡ ω(x + μˆ ) − ω(x).
(4)
Finally the change of integration variables from links to vector fields yields a Jacobian that can be rewritten as the usual measure
term Sm in the action:
ꢃ
ꢆ
Ncg02
Ncg04
g04
ꢀ
ꢄ
ꢅ ꢇ
ꢄ
ꢅ ꢇ
ꢄ
ꢄ
ꢅ ꢇꢅ
2
4
2
2
Sm =
Tr Qμ(x)
+
Tr Qμ(x)
+
Tr Qμ(x)
+ ··· .
(5)
12
1440
480
x,μ
In Sgh and Sm we have written out only terms relevant to our computation. The full action is:
S = SL + Sgf + Sgh + Sm.
(6)
(7)
(8)
The average value of the action density, S/V , is directly related to the average plaquette. For the gluon part we have:
ꢈ
ꢉ
1
Nc
ꢀSG/V ꢁ = 6βEG, EG ≡ 1 −
Tr Uμ,ν(x) , β = 2Nc/g2.
As for ꢀSF /V ꢁ, it is trivial in any action which is bilinear in the fermion fields, and leads to:
ꢀSF /V ꢁ = −4NcNf
(Nf : number of fermion flavors).
We will calculate EG in perturbation theory:
EG = c1g2 + c2g4 + c3g6 + ···.
(9)
The n-loop coefficient can be written as cn = cnG + cnF where cnG is the contribution of diagrams without fermion loops and
cnF comes from diagrams containing fermions. The coefficients cnG have been known for some time up to 3 loops [11] (also in
3 dimensions [12], where they are applied to “magnetostatic” QCD [13] and to dimensionally reduced QCD [14,15]). Independent
estimates of higher loop coefficients have also been obtained using stochastic perturbation theory [16]. The fermionic coefficients cnF
are known to 2 loops for overlap fermions [17] and up to 3 loops for Wilson fermions [4]; in the present work we extend this
computation to the clover action.

254
A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
The calculation of cn proceeds most conveniently by computing first the free energy −(lnZ)/V , where Z is the full partition
function:
ꢊ
¯
Z ≡ [DUDψiDψi]exp(−S).
(10)
Then, EG is extracted through
ꢋ
ꢌ
1 ∂ lnZ
EG = −
.
(11)
6 ∂β
V
In particular, the perturbative expansion of (lnZ)/V :
3(Nc2 − 1)
d1
β
d2
β2
(lnZ)/V = d0 −
lnβ +
+
+ ···.
(12)
(13)
2
leads immediately to the relations:
ꢄ
ꢅ
ꢄ
ꢅ
c2 = d1/ 24Nc2
,
c3 = d2/ 24Nc3
.
2. Calculation and results
A total of 62 Feynman diagrams contribute to the present calculation, up to three loops. The first 36 diagrams are totally
gluonic, and the others have both gluon and fermion contributions; these are shown in Appendix A. The involved algebra of
lattice perturbation theory was carried out using our computer package in mathematica. The value for each diagram is computed
numerically for a sequence of finite lattices, with typical size L ꢀ 36.
Certain diagrams must be grouped into infrared-finite sets, before extrapolating their values to infinite lattice size (diagrams
11+12+13, 22 through 36, 43+53, 44+52+58, 46+56, 51+57, 55+60, 61+62). Extrapolation leads to a (small) systematic
error, which is estimated quite accurately; a concise description of the procedure is provided in Ref. [12].
Diagrams in the shape of a triangular pyramid (18, 19, 20, 49, 50) are the most CPU demanding, since integration over the 3
loop momenta cannot be factorized; these diagrams were necessarily evaluated for smaller L, but fortunately L ∼ 16 was already
sufficient for a very stable extrapolation in these cases. Diagram 40 vanishes identically by color antisymmetry.
The pure gluon contributions are already known [11]:
ꢋ
ꢌ
Nc2 − 1
1
ꢄ
ꢅ
c1G
=
,
c2G = Nc2 − 1 0.0051069297 −
,
8Nc
128 Nc2
ꢋ
ꢌ
ꢄ
ꢅ
0.0023152583(50) 0.002265487(17)
c3G = Nc2 − 1
−
+ 0.000794223(19)Nc
.
(14)
Nc3
Nc
Fermion contributions take the form:
ꢋ
ꢌ
Nf2
Nc
ꢄ
ꢅ
ꢄ
ꢅ
Nf
Nc
Nf
Nc2
c1F = 0,
c2F = Nc2 − 1 h2
,
c3F = Nc2 − 1 h30Nf + h31
+ h32
.
(15)
(16)
The coefficients h2, h30, h31, h32 depend polynomially on the clover parameter cSW
:
h2 = h(0) + h(21)cSW + h(2)cS2W
,
h3i = h(0) + h(1)cSW + h(2)cS2W + h(3)cS3W + h(4)cS4W
.
2
2
3i
3i
3i
3i
3i
We have calculated h2(j), h(j) for typical values of the Lagrangian (unrenormalized) fermion mass parameter m, which is con-
nected to the familiar hopping3eixpansion parameter κ = 1/(2m + 8). Our results are listed in Appendix B. A complete per diagram
breakdown of the results would be far too lengthy to present; instead, for potential comparisons, we provide in Appendix B a
breakdown only for a particular value of m.
In Figs. 1 and 2 we present the dependence of c2 and c3, respectively, on m, for three typical values of cSW
We list below some examples of values for EG, setting Nc = 3. For Nf = 0 we have:
.
EG = (1/3)g2 + 0.0339109931(3)g4 + 0.0137063(2)g6.
(17)
(18)
For two degenerate flavors (Nf = 2) and m = −0.518106 (corresponding to κ = (8 + 2m)−1 = 0.1436):
cSW = 0.0: EG = (1/3)g2 + 0.026185200(3)g4 + 0.0119649(3)g6,
cSW = 2.0: EG = (1/3)g2 + 0.013663456(3)g4 + 0.0110200(13)g6.
For Nf = 2 and m = 0.038:
cSW = 0.0: EG = (1/3)g2 + 0.030438866(3)g4 + 0.0138181(2)g6,

A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
255
Fig. 1. The dependence of c on the fermion mass m, for some standard values of c . N = 3, N = 2.
c
2
SW
f
Fig. 2. The dependence of c on the fermion mass m, for some standard values of c . N = 3, N = 2.
c
3
SW
f
Fig. 3. E as a function of β, for N = 3, N = 2, and specific mass values.
c
G
f
cSW = 1.3: EG = (1/3)g2 + 0.025219798(9)g4 + 0.0129659(5)g6,
cSW = 2.0: EG = (1/3)g2 + 0.01800170(1)g4 + 0.012948(1)g6.
(19)
It is seen that the 3-loop coefficients are quite pronounced for typical values of g used in numerical simulations. For convenience,
the behaviour of EG versus β is also presented in Fig. 3, for the same parameter values as in Eqs. (18) and (19).
The detailed results, tabulated in Appendix B for arbitrary values of Nc, Nf , cSW, show a very smooth behaviour as a function
of m; consequently, one is able to reconstruct EG also for arbitrary values of m by naive interpolation, to excellent precision.

256
A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
Appendix A
Fig. 4 depicts all diagrams contributing to the free energy at 1 loop (diagram 1), 2 loops (2–6, 37–38), and 3 loops (7–36, 39–62).
Solid (curly, dashed) lines represent fermions (gluons, ghosts), and the filled square is the contribution from the measure part of the
action. The filled circle, corresponding to the non-fermionic part of the 1-loop gluon self-energy, is given in Fig. 5.
Fig. 4. Feynman diagrams contributing to the free energy, at one, two, and three loops.

A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
257
Fig. 5. Non-fermionic diagrams renormalizing the gluon propagator at 1 loop.
Appendix B
Tables 1–4 provide a per diagram breakdown of our results, at a given value of m (m = 0.038), in order to allow for potential
comparisons and cross checks. The total results for the coefficients h(2j), h(3j0), h3(j1), h(j) are listed in Tables 5–8, respectively,
32
for a wide selection of m values which are used in the literature. Given the smooth dependence of all these coefficients on m,
interpolations to other intermediate values of m can be performed with great accuracy.
Table 1
(j)
2
Per-diagram contributions to h . m = 0.038
(0)
2
(1)
2
(2)
2
4
4
4
Diagram
h
· 10
h
· 10
h
· 10
37
38
0.932189(6)
−7.442427121(2)
0.14684(1)
0
−5.903340(3)
0
Table 2
(j)
30
Per-diagram contributions to h . m = 0.038
Diagram
(0)
30
(1)
30
(2)
30
(3)
30
(4)
30
4
4
4
4
4
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
39
41
42
0.19218007757(5)
−0.096284(2)
0.00427805(2)
−1.49693(1)
1.574326(1)
0.102791(5)
−0.0006435(4)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−0.047087(2)
3.176923(5)
0
−1.303406(1)
−1.343093(3)
0
43 + 53
0.031601(2)
0
45
46 + 56
47
48
−0.005620(1)
−0.0369073(4)
−0.0000025(1)
−0.01919(3)
0.023672(5)
−0.3910(2)
−0.0017428(5)
0.01395(1)
−0.011239(2)
0.1231742(3)
−0.0637549(3)
−0.0395366(6)
−0.017117(8)
−0.4837191(5)
−0.41414(8)
0.00356648(1)
0
49
0.12905(6)
−0.0078566(7)
51 + 57
−0.37601(2)
−0.24831(4)
0.028358(1)
0.012859(2)
0.20558(1)
0
0
0
54
55 + 60
59
61 + 62
0.03556619(6)
0.07042(2)
0.05069(3)
0
−0.020929(8)
0.2463485(6)
0
Table 3
(j)
31
Per-diagram contributions to h . m = 0.038
Diagram
(0)
31
(1)
31
(2)
31
(3)
31
(4)
31
4
4
4
4
4
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
39
41
−0.2882701164(1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.144427(2)
0.048077(2)
0
−2.951676(8)
42
−0.00855609(4)
1.62756(1)
0
43 + 53
−0.036704(2)
1.475838(3)
0
0.0042472(4)
0.01081(4)
0.41414(8)
−0.07042(2)
45
−1.574326(1)
0.0012870(5)
−0.0096(1)
0
47
50
54
0.0073718(6)
−0.02767(1)
0.3910(2)
−0.01395(2)
0.000361(3)
0
−0.05069(3)
−0.007025(2)
0.24831(5)
0
59
−0.012859(2)
0.020929(8)
Table 4
(j)
32
Per-diagram contributions to h . m = 0.038
Diagram
(0)
32
(1)
32
(2)
32
(3)
32
(4)
32
6
6
6
6
6
h
· 10
3.65818(5)
h
· 10
h
· 10
h
· 10
h
· 10
44 + 52 + 58
−0.24532(8)
7.62942(7)
−0.41017(3)
5.742740(1)

258
A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
Table 5
(j)
2
Total values of h , for various masses
(0)
2
(1)
2
(2)
2
3
3
3
m
h
· 10
h
· 10
h
· 10
−1.014925
−0.9512196
−0.8749999
−0.8253968
−0.7948719
−0.5181059
−0.423462
−0.4028777
−0.3140433
−0.301775
−0.2962964
−0.2852897
−0.2769916
−0.2686568
−0.1482168
0
−2.4083635(4)
−2.2738528(3)
−2.1173032(2)
−2.0179859(2)
−1.9578750(2)
−1.4485862(6)
−1.2897989(2)
−1.2563603(3)
−1.1167334(5)
−1.0980649(6)
−1.0897779(5)
−1.0732218(2)
−1.0608234(1)
−1.0484419(4)
−0.8779913(3)
−0.6929202(1)
−0.6510238(6)
−0.615948(1)
−0.5956994(5)
0.395934967(3)
0.359466059(5)
0.318089891(2)
0.292493808(2)
0.277266040(2)
0.15739842(6)
0.1238792(6)
0.1170884(9)
0.089818(2)
0.086312(1)
0.084766(2)
0.081700(5)
0.079422(6)
−0.71409763(8)
−0.70919125(2)
−0.70272042(1)
−0.698193815(6)
−0.695294722(3)
−0.66565598(2)
−0.6543190(2)
−0.6517790(3)
−0.6405147(6)
−0.6389201(5)
−0.6382049(7)
−0.636761(3)
−0.635665(4)
0.077161(3)
0.04774(1)
0.02010061(2)
0.014684(1)
0.010488(4)
−0.634562(4)
−0.618118(2)
−0.59630769(1)
−0.5903340(3)
−0.584812(2)
0.038
0.072
0.0927
0.008214(2)
−0.5813703(7)
Table 6
(j)
30
Total values of h , for various masses
(0)
30
(1)
30
(2)
30
(3)
30
(4)
30
4
4
4
4
4
m
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
−1.014925
−3.61593(9)
−3.30186(3)
−2.93686(3)
−2.70587(4)
−2.56636(4)
−1.4032(2)
−1.0538(3)
−0.9817(2)
−0.6879(1)
−0.6497(1)
−0.6329(1)
−0.5994(1)
−0.5746(1)
−0.54989(8)
−0.23130(6)
0.01867(6)
0.03126(8)
0.03095(9)
0.02828(7)
−0.5205(4)
−0.5660(4)
−0.6119(4)
−0.6368(5)
−0.6503(5)
−0.7097(3)
−0.7039(1)
−0.7010(1)
−0.6810(1)
−0.6770(3)
−0.6751(2)
−0.6719(4)
−0.6686(5)
−0.6653(5)
−0.6101(4)
−0.49191(1)
−0.4436(2)
−0.3997(2)
−0.3744(3)
−0.16938(2)
−0.16042(2)
−0.15115(2)
−0.14582(2)
−0.14275(1)
−0.1185(1)
−0.1097(1)
−0.10756(6)
−0.0971(1)
−0.0956(1)
−0.0948(1)
−0.0934(1)
−0.0921(2)
−0.0907(2)
−0.0704(2)
−0.02955(1)
−0.01234(8)
0.00431(7)
0.0144(1)
0.18044(3)
0.17452(3)
0.16699(4)
0.16186(4)
0.15863(4)
0.126761(7)
0.11495(2)
0.11233(2)
0.100645(8)
0.09905(3)
0.09828(5)
0.09675(5)
0.09565(5)
0.09458(3)
0.07775(9)
0.053914(1)
0.04640(3)
0.03982(2)
0.03610(3)
−0.01301953(3)
−0.01362776(1)
−0.014206930(3)
−0.014512957(7)
−0.01467848(1)
−0.01575561(2)
−0.016111(1)
−0.016198(2)
−0.016638(4)
−0.016708(4)
−0.016735(8)
−0.01681(1)
−0.9512196
−0.8749999
−0.8253968
−0.7948719
−0.5181059
−0.423462
−0.4028777
−0.3140433
−0.301775
−0.2962964
−0.2852897
−0.2769916
−0.2686568
−0.1482168
0
−0.016868(9)
−0.016911(8)
−0.01785(2)
−0.01998581(2)
−0.020929(8)
−0.021807(5)
−0.022300(6)
0.038
0.072
0.0927
Table 7
(j)
31
Total values of h , for various masses
(0)
31
(1)
31
(2)
31
(3)
31
(4)
31
4
4
4
4
4
m
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
−1.014925
4.2718(2)
3.9078(2)
3.4867(2)
3.2212(2)
3.0614(2)
1.7413(2)
1.3494(2)
1.2686(2)
0.9409(2)
0.8984(2)
0.8797(2)
0.8424(2)
0.8147(2)
0.61383(4)
0.636745(4)
0.65829(6)
0.6689(1)
0.6741(1)
0.6750(2)
0.6559(1)
0.6503(1)
0.6207(1)
0.6156(3)
0.6132(1)
0.6090(3)
0.6050(4)
−1.1331(2)
−1.1391(2)
−1.1434(2)
−1.1447(2)
−1.1450(2)
−1.1343(1)
−1.12732(6)
−1.12575(6)
−1.11893(5)
−1.1180(1)
−1.1176(1)
−1.1168(1)
−1.1163(2)
−0.18127(4)
−0.17467(4)
−0.16655(3)
−0.16119(3)
−0.15783(3)
−0.12639(1)
−0.11512(2)
−0.11264(1)
−0.10162(2)
−0.10013(4)
−0.09941(6)
−0.09798(5)
−0.09695(6)
0.002886(2)
0.003690(2)
0.004505(1)
0.004966(1)
0.005226(1)
0.007141(2)
0.007770(1)
0.007917(2)
0.008611(6)
0.008715(5)
0.008759(8)
0.00887(1)
0.00895(1)
−0.9512196
−0.8749999
−0.8253968
−0.7948719
−0.5181059
−0.423462
−0.4028777
−0.3140433
−0.301775
−0.2962964
−0.2852897
−0.2769916

A. Athenodorou et al. / Physics Letters B 659 (2008) 252–259
259
Table 7 (continued)
(0)
31
(1)
31
(2)
31
(3)
31
(4)
31
4
4
4
4
4
m
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
−0.2686568
−0.1482168
0
0.7872(2)
0.4324(2)
0.1479(2)
0.1279(1)
0.1221(1)
0.1211(1)
0.6011(4)
0.5382(4)
0.41633(5)
0.3681(2)
0.3248(2)
0.3000(3)
−1.1159(2)
−1.1090(2)
−1.11100(1)
−1.1171(1)
−1.1232(1)
−1.1268(1)
−0.09594(3)
0.009012(8)
0.01029(2)
0.01284418(4)
0.013904(9)
0.014891(5)
0.015451(6)
−0.0801(1)
−0.057537(3)
−0.05033(3)
−0.04400(2)
−0.04044(3)
0.038
0.072
0.0927
Table 8
(j)
32
Total values of h , for various masses
(0)
32
(1)
32
(2)
32
(3)
32
(4)
32
4
4
4
4
4
m
h
· 10
h
· 10
h
· 10
h
· 10
h
· 10
−1.014925
0.49011557(4)
0.43836868(2)
0.38167686(2)
0.34767753(1)
0.327840761(5)
0.18222240(4)
0.1450777(2)
0.1377609(1)
0.1091234(3)
0.1055301(5)
0.1039530(7)
0.1008352(9)
0.0985290(7)
0.0962507(4)
0.0674197(9)
0.04159747(5)
0.0365818(5)
0.0326353(4)
0.0304618(8)
−0.19966299(3)
−0.17353447(1)
−0.14550571(1)
−0.129038814(4)
−0.119564143(4)
−0.0539075(2)
−0.0386899(4)
−0.0357991(4)
−0.0249053(4)
−0.023593(2)
−0.023021(2)
−0.021900(2)
−0.021075(1)
−0.020271(6)
−0.010680(4)
−0.00356316(1)
−0.0024532(8)
−0.001675(1)
−0.0012878(2)
0.36828835(3)
0.343436025(6)
0.31500822(1)
0.297262047(4)
0.286632961(4)
0.20003143(6)
0.1742885(2)
0.1689417(2)
0.1468864(3)
0.1439703(7)
0.1426783(9)
0.1401015(9)
0.1381755(8)
0.1362553(5)
0.1101470(8)
0.08245984(4)
0.0762942(7)
0.071174(1)
−0.11015360(1)
−0.10025167(1)
−0.088919069(7)
−0.081862310(7)
−0.077648919(7)
−0.04416150(5)
−0.0347205(4)
−0.0328053(6)
−0.025111(1)
−0.024122(3)
−0.023686(3)
−0.022821(1)
−0.022178(4)
−0.021542(4)
−0.013260(4)
−0.005574256(7)
−0.0041017(3)
−0.0029808(5)
−0.002377(2)
0.08360193(2)
0.082274351(7)
0.080600027(9)
0.079469961(8)
0.078761444(7)
0.072038437(4)
0.06966149(6)
0.0691407(1)
0.0668770(3)
0.0665614(9)
0.0664209(8)
0.0661380(3)
0.0659243(2)
0.0657091(3)
0.062561(1)
−0.9512196
−0.8749999
−0.8253968
−0.7948719
−0.5181059
−0.423462
−0.4028777
−0.3140433
−0.301775
−0.2962964
−0.2852897
−0.2769916
−0.2686568
−0.1482168
0
0.058518838(6)
0.05742740(1)
0.0564209(3)
0.0557953(1)
0.038
0.072
0.0927
0.0682377(9)
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