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Accurate computation of fourth-order vacuum polarization
Computer Physics Communications 15 (1978) 153—159 © North-Holland Publishing Company
ACCURATE COMPUTATION OF FOURTH-ORDER VACUUM POLARIZATION
*
Clyde CHLOUBER and Mark A. SAMUEL Quantum Theoretical Research Group, Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74074, U.S.A. Received 30 January 1978; in revised form 22 March 1978
A method to accurately compute vacuum polarization to fourth-order in both the space-like and the time-like regions is described. It makes use of various techniques, including changes of variables and Padé approximants to accelerate convergence of sequences which occur in the computation. Various checks have been made which indicate that it is accurate to 9 significant figures. These include a sixth-order electron magnetic moment contribution and an order ~2 correction to the hyperfine structure of positronium.
1. Introduction
2. Definitions
The fourth-order vacuum polarization kernel [1] which gives the amplitude for the occurrence of the four fundamental processes depicted in fig. 1 contributes radiative corrections to the bound state energy
The general expression for the renormalized photon propagator is g / p p \ Re ir(p2) D~~(p)= ~ + i~g 1~, ~) 2 p p p with
levels of atoms, as well as muonic atoms, and the anomalous magnetic moments of the electron and the muon [2]. In the case of the electron magnetic moment an exact analytic result has been obtained for this contribution. In other calculations it is necessary to resort to numerical quadrature or semianalytic approximations to obtain these corrections. Although calculations of all 6th-order contributions to the anomalous moments of the electron and muon have magnetic been made, an accurate graph of the 4th. order kernel has not been presented. With this in mind we describe here in some detail VAC4, a subroutine we have developed, which accurately computes vacuum polarization to fourth-order, in both the space-like and the time-like regions. Furthermore, in the case of the muon, there are corrections of O (m~/m,~) which have yet to be determined. Some of these have recently been computed [3].
*
Re ir(p 2
2
)
1
=—
r i j
dt Im —
ir(t)
_______
r “
2’
o — where the imaginary part of the spectral function is given by 2) = (~j~, I zXz IJ~0). Im ~(p 3P p(z)=p .
In particular, the real and imaginary parts of the
e
Work supported by the U.S. Energy Research and Development Administration.
(a
(b)
(c)
(d)
Fig. 1. Fourth-order vacuum polarization diagrams. 1
c~
C. Chlouber and M.A. Samuel / Fourth-order vacuum polarization
154
4m2
4th-order vacuum polarization kernel including the three proper diagrams and the amplitude from the single double-bubble graph are given by [1]: Re
=
~4)~,2)
~[
~ i08
+
~72
(1n2 Ij—~ I—
—
A
ö~ 3
(6
~
—
i9 +
646~
3_62
2
I, III
ii~20(l —6)
B=
(~ 32
~23
~
12 62
2364÷6)B ~ 12
(1)
+(3_62)C+(3+26264)FGFI(62)1,
~(4)~2)
(4)
1+6
+ (19 5562+6) \2472 ~
Im
1>0, —
tan~ 77
~
II,
—
0(1_6)=~[1+~~=(~
~
6
[
III’
(5)
1 —6 / 1 —6 \ ~2 — +2~———--J+— (1+6) 1+6, 4 \
1
1-8
ln( 162)3]
—~
~2O(l + ln
1+6(~~6223 1_6\16
8
~ 2
ln
_+62_-_)
(3
~
862
(
—6\ +2~~)1+6
lnl—I 1+6 ln 1 646~ 1—6 _62I3]1
~6~+~) (1+6)~
_)(4~ ~ 2
/362
66
~2\
)
+~) 2
1
6 + 6)
tan_i
[~1(~
1 1 —ln +
0(1 —6),
(2)
tan
~(x)~~
where
t~
sin nx 1
62=1+
4m2
~-)—
2~i(2tan~)
6477~ 2)31’ (l+i~
— —f 0
ln(2 sin
1
p~pI.L =
II,
~t)
(6)
dt,
______
space-like virtual photon momentum —q~<0 time-like virtual photon
(q2 =
1,111
‘
>0
momentum
FGH(6 2) = fdt g(t) in Ii
ln(l f(t)
I,
f(t) +f(_t),
g(t)
The interval over which the function Re ir(4)(p2) is to be evaluated further subdivides for the time-like case. (We label region II for q <2m and region III for q > 2m.) For later reference the space-like region is referred to as region I. For regions I, II and III the following formulae apply to the variables in the expression for the kernel.
2 — ~
=
+ t)
3 ln[(l 2 1
+—
— ±
t)/2] ~——-——.
ln~tI 1 t +
(7)
3. Evaluation of the functions i,Li(x), 0(x) and FGH(52) The function 0(x) is easily computed to 14-place
A=ç
16
1+6 ln I —I 1 —6
,
277tan177,
I, III II,
,
accuracy For x >0 usingwetheevaluated method the given function by Clausen 0(x) by [4].
C. Chiouber and M.A. Samuel / Fourth-order vacuum polarization
direct summation of the series
2
t
‘
FGH(6 )Jg(t)lnIl (_l)fl_i
1
~
For x <0 we can use relations given in Mitchell [5] to arrive at the formulae 2 / x 6
x+1
0(x) =
—
o(__~-~_~) + ~ lnIxIln~+x)2
+
(9)
The arguments of 0 on the right hand sides of eqs. (8) and (9) lie on the interval (0,1), hence all values of 0(x) for—i ~x ~ 1 may be expressed in terms of 0(z)with0~z ‘~ 1, where
—
0
i+x i+x
—i
1
ir/2
f
g(6sin0)lnI1-~-sin20I6cos0d0.
-+2(1 _6)f
t=
(1 —6) sin2O
+
(16)
6
g[(l —6)sin20 +6]
o ~l
ml 1 —
FGH(62)
<2
The series to be evaluated are oscillating and rapidly convergent. For z ~ 0.8, a sum of 30 terms or less of the series gives 10-place accuracy. For z 0.8, the c-algorithm [5,6] was used to accelerate the convergence of the sequence of partial sums. The results obtained were found to be in agreement with those tabulated by Mitchell [7]. In the evaluation of FGH(62), changes of variables, numerical quadratures, and convergence techniques were used to obtain accurate results. To circumvent the logarithmic infinities which occur in the integrand of FGH(6 2) changes of variables are made. In regions I and II let r = sin 0, then =
ir/2
f~~f
f
(15)
~
—
~—~--—
6
\2
sin2O
+
1) sin 0 cos OdO
.
(17)
2
— —~—~
FGH(62)
fg(t) ln Ii — dt. a In the first integral let t = 6 sin 0
The integrals for all three regions have been reduced to the form where x(0) is finite and vanishing at the endpoints of the integration interval.
,
!
._~_
—~Idt 6
In the second integral let
—1
x
2
+
a (8)
155
=
x(O) dO,
f
(18)
e-
e-
e-
e-
e~
e+
e-
+
e-
e-
e+
-
•2
g(sin 0) lnl 1
— ~~-~--
I cos OdO
-
(14)
(d)
In region III there is a divergence at t = 6 as well as at t = 1 since 0 <6 < 1; consequently, we split the integration interval.
Fig. 2. Annihilation diagrams with four-order vacuum polarization insertions, contributing to the positronium ground-state hyperfine splitting in sixth-order.
C. Chlouber and M.A. Samuel / Fourth-order vacuum polarization
156
Table 1 Sequence of approximations to FGH(8 2) in each of the three regions, illustrating typical rates of convergence. k
Region I 2 = 2) FGH(a Sk(2)
2 3 4 5 6 7
—0.314280423196 —0.314280413178 —0.314280408538 —0.314280406017 —0.314280404496 —0.314280400283
=
Region II FGH(82 = —3) Sk(—3)
Region III FGH(82 = ~) Sk(2~~~)
Region I and III FGH(62 = 1) Sk(l)
—0.314280605732
0.168968686737
—0.992945439727
—0.901550793418
—0.314280451789
0.168968621712
—0.992944168587
—0.901544899344
0.168968609611 0.168968605368 0.168968603402 0.168968602333 0.168968601689 0.168968599903
—0.992943933026 —0.992943850560 —0.992943812385 —0.992943791646 —0.992943779140 —0.992943744506
—0.901543715052 —0.901543281071 —0.901543073668 —0.901542958211 —0.901542887201
For the numerical quadrature, the interval [0, 77/2] is divided into {m = 1,2,3,4, ...} subintervals. A Gauss quadrature is applied to each subinterval and the results of all subintervals are summed. Thus a sequence of partial sums Sk, (k 1,2, ...) to eq. (18) is ob-
T~2~54267639’
interval (0,1) to evaluate the integral 1
a~6~(fourth-order V.P.) = e
fdx(1
~
—
x)
77 J 2
—
Re
With the transformations defined above, the sequences of quadrature approximations to FGH are rapidly convergent. The sequences were extrapolated to k = 00 by the method of Padé type II approximants [8]. The results shown in table 1 are typical of the rates of convergence for other values of 62. The extrapolated value for FGH(1) agrees with the exact answer [which is known [9] to be —~~(3)] to 2 parts in l0~, and since the rates of convergence for other values of 62 is as good or better then that for 62 = 1, we can expect an error of approximately the same magnitude
— 2
—x me) I x The Padé extrapolation to this sequence of partial sums (using the 1/k coordinate method is 0.0554291769 which agrees to 8 figures with the exact result of Mignaco and Remiddi [10]. =
77(4)(q2
/
\~
a~6~(4th order V.P.) = (f-) {
~
~(2)
—~
77
+
2 2 + 32 2 1 2 ~ ( ) — ~( ) n ~ ~(2)l~~2 + ~ lfl~2}= 0.0554291775. ~
18 —
~( )
14
—
15
1
-~-
8
i~(~)
or better.
4. Consistency checks
Table 2 Sequence of quadrature approximations using VAC4 to compute the fourth-order vacuum polarization contribution to
As an independent check on the overall accuracy
of our routine VAC4 in the space-like region, we used it to compute the fourth-order vacuum polarization contribution to the sixth-order electron magnetic moment. Since the answer is known analytically [10], we can check it directly. Table 2 gives the results of
the sequence of quadrature approximations in which (m = 1,2,3,4,5) Gauss quadratures were applied on the
a~6~ in units of (~/7T)3. n Approximants to a~6~ e 1 0.055429509209975 2 0.055429368961553
3 4 _______
0.055429220345088 0.055429202459104 0.055429193887575
______________________________________-
C. Chiouber and MA. Samuel / Fourth-order vacuum polarization
As a check on VAC4’s accuracy in the time-like 2 region II, we numerically computed the order a correction to the hyperfine structure of positronium associated with the annthilation diagrams [101 shown Table 3 Re ~(4)(p2) in units of (a/lr)2 as a function of q bine the time-like region. Region I q
Re
0.0 0.10 0.20 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.03 5.50 6.00 6.50 7.00 7.53 8.00 8.50 9.OQ 9.50 10.00 10.50 11.00 11.5) 12.00
0.0 —0.01006449 —0. 03951076 —0.08660005 —0.14837312 —0.22171877 —0.30348929 —0.39086091 —0.48148185 —3.57350821 —0.66556644 —1.10110117 —1.48041182 —1.80803933 —2.09663776 —2.35556780 —2.59138591 —2.80869582 —3.01076697 —3.20007011 —3.37845225 —3.54736219 —3.70795512 —3.86116964 —4.00778077 —4.14843745 —4.28368974 —4.41400878 —4.53580192 —4.66142427 —4.77918765 —4.89336768 —5.00420942
12.50
—5.11193102
13.03 13.50 14.00 14.53 15.00 15.50 16.03 16.50 11.33
—5.21673194 —5.31878697 —5.41825781 —5.51529065 —5.61001883 —5.70256456 —5.79303978 —5.88154761 —5.96818345
17.50
—o.05303519
13.03 18.53 19.00 19.50 20.03
—6.13618455 —6.21170741 —6.29161443 —6.37615149 —6.45320019
=
in fig. 2. Since the level shift has already been determined analytically [11,12] to order a4R,~,we have a basis for an accurate comparison.
\/1p21 in units of 2m. Region I is the space-like region. Regions II and III com-
Region II q 0.02 3.04 0.06 0.38 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.3o 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.38 0.50 0.52 0.64 0.oS 0.68 0.10 0.72 0.74 0.78 0.To 0.83 0.32 0.84 0.36 0.88 0.00 3.92 0.94 0.96 0.06
157
Re
Region III q
0.00040503 1.02 0.031~2128 1.04 0.003D5216 1.06 0.00650346 1.08 0.31018331 1.10 0.01470193 1.12 0.02007175 1.14 0.02631619 1.16 0.03344167 1.18 0.04147574 1.20 0.35044320 1.30 0.06037237 1.40 0.07129536 1.50 0.08324842 1.60 0.09627236 1.80 0.11041299 2.00 0.12572175 2.20 0.14225631 2.40 0.16008137 2.60 0.11926959 2.80 0.19990266 3.00 0.22207257 3.50 0.24588322 4.00 0.27145219 4.50 0.293~1305 5.00 0.32841803 5.50 0.36014141 6.00 0.39428358 6.50 0.43107623 7.00 0.47078877 7.50 0.51373649 8.00 0.56029115 8.50 0.61089456 9.00 0.66607668 9.50 0.12641966 10.00 0.79289036 10.53 0.86628748 11.00 0.94790379 11.50 1.03932210 12.00 1.14281584 12.50 1.26056816 13.30 1.30702592 13.50 1. 5s730124 14.00 1.15025914 14.50 l.9~8’~4603 15.00
2.29411993
15.50
~.i1294134 3.34535314 4.~3425171
16.00 16.50 17.00
Re 3.28613947 1.81091252 1.03212839 0.53363132 0.18510175 —0.07138200 —0.2o657327 —0.41868906 —0.53935297 —0.63639077 —0.91526800 —1.03103375 —1.08505672 —1.11399129 —1.14896244 —1.18204457 —1.22214505 —1.26982195 ~-1.32376417 —1.38245092 —1.44456521 —1.60855134 —1.77734102 —1.94557489 —2.11063321 —2.27128471 —2.42701632 —2.57169267 —2. 72337749 —2.86423722 —3.00046790 —3.13236553 —3.26010957 —3.38395382 —3.50412164 —3.62062367 —3.73425697 —3.84460503 —3.95203821 —4.05671450 —4. 15878029 —4.25837126 —4. 355o1322 —4.45062292 —4.54350879 —4. 63437168 —4.72330547 —4.81039767 —4.89572998
C. Chiouber and MA. Samuel / Fourth-order vacuum polarization
158
The energy shift for the annihilation process is
~A=—477aI0(0)I2(S2)
Re ir~4~(—k2) k2
Table 4 Re ir(2)p2) in units of (c~/ir)as a function of q the time-like region. Region I q 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.53 6.00 6.50 7.00 (.50
8.00 8.50 9.03 9.53 10.00 10.50 11.03 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.33 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00
=
where k is the total C.M. positronium energy i.e. a2
k(2_~)m,c=l.
\/1p2 i in units of 2m. Region I is the space-like region. Regions II and III comprise
Region II
Re n(2)
q
0.00000000 —0.00265531 —0.01048803 —0.02312082 —0.03999127 —0.06042954 —0.08373414 —0.10923198 —3.13631716 —0.16447000 —0.19326127 —0.33714117 —0.46929734 —0.58590608 —0.68836240 —0.77894604 —0.85974837 —0.93247844 —0.99849043 —1.05885291 —1.11441399 —1.16585320 —1.21372088 —1.25846751 —1.30046571 —1.34002664 —1.37741253 —1.41284618 —1.’.4651837 —1.47359362 —1.50021472 —1.53850635 —1.56657797 —1.59352620 —1.61943673 —1.64438586 —1.66844186 —1.69166602 —1.71411358 —1.73583448 —1.75687401 —1.77127341 —1.79707027 —1.81629930 —1.83499116 —1.85317574 —1.87087947 —1.88812704 —1.90494126
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.4? 0.44 0.t6 0.46 0.50 0.52 0.54 0.5~ 0.58 0.60 0.62 0.64 0.66 0.od 0.70 0.72 0.14 0.16 0.78 0.80 0.82 0.84 0.86 0.38 0.90 0.92 0.94 3.96 0.98
Re ¶() 0.00010668 0.00042696 0.00096148 0.03171137 0.30251816 0.00386390 0.30527109 0.00690272 0.00876233 0.01085398 3. 01318233 0.01575265 0.31857086 0.02164362 3.32497833 0.02858325 0. 03246755 0.03664140 0.04111608 0.04550413 0.05131944 0.05641744 0.0s229532 0.06849222 0.07508952 0.08211120 0.08958414 0.09153869~ 0.10600919 0.11503468 3.12465978 3.13493576 0.14592193 0.15768140 3.17331333 0.18389594 ‘3.19855054 0.21441698 0.23166727 0.25051659 3.27123936 0.29419406 0.31986280 0.34891822 0.38234473 0.421679o0 0.46955875 0.53122539 0.62052428
Region III q 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.30 1.40 1.50 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.03 3.50 4.00 4.50 5.00 5.5o 6.00 6.50 7.00 7.50 8.00 3.50 9.00 9.50 10.00 10.50 11.00 11.50 12.30 12.50 13.03 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00
Re ~(2) 0.83710855 0.78825637 0.74208687 0.69838007 0.65693840 0.b1758404 0.58015657 0.54451103 0.51051612 0.47805268 0.33523530 0.21790643 0.11919773 0.03448666 —0.10486294 —0.21650036 —0. 30945050 —0.38908733 —0.45881427 —0.52089877 —0.57691650 —0.69716685 —0.79117723 —0.88305828 —0.95846941 —1.02578262 —1.08662852 —1.14217953 —1.19330945 —1.24068751 —1.28483916 —1.32618471 —1.36506605 1.40176525 —1.43651793 —1.46952299 —1.50095000 —1.53094471 —1. 55963337 —1.58712608 —1.61351947 —1.63889881 —1.66333975 —1.68690973 —1.70966909 —1.73167211 —1. 75296713 —1.77360029 —1.79361034
C. Chlouber and MA. Samuel/Fourth-order vacuum polarization
3 and (S2)
Substituting I I~,(0)I2= (1/77)(ma/2)
=
2,
i Res. II
I
the energy shift may be written Re ir~4~(—k2) = —2a2R.. (2 — ~a2)2 For Re Re
159 Reg.
II
-
ir~4~(—k2) we obtained the value
ir~4~(—k2) = 1.2302 X i0~
6
5
4
3
where upon using a~ = 137.035987 and R,,~= 3.289843 2 X 1015 Hz we obtained the frequency shift ~v—10.77
2
I
1
2
3
4
5
6
,
-
5
MHz,
which is in good agreement with the semi-analytic result [11] =
_k 2
a4R
1 ln a1
+
=2
21
—
32
Fig. 4. Re ~(2)~,2) in units of (a/ir) versus q = \/1p21 in units of 2m. Regions I, II and III are as previously defined.
13 324772
1n2 Recently, Fullerton and Rinker [13] reported on a computation of fourth-order vacuum polarization. Their method is different however, making use of the
— —
—10.76 MHz.
Chebyshev approximation. Their a potential
V(r),
result, expressed as 0 ~ r ~ Xe.
is accurate to 3 figures for
5. Tabulation of the function Re 7T(4)(p2) In table
3
we tabulate the function Re 77(4)(p2) on
the interval (20 X
2m)2
>p2
results are also graphed in fig. 3. For comparison we also tabulated Re 77(2)~92)in table 4 and plotted this function in fig. 4. 4
Req. I
References
> —(17 X 2m)2. The
Req. II
Req.
11] G. Kallén, Helv. Phys. Acta 25 (1952) 417 and G. Källén and A. Sabry, Kgl. Danske Videnskab, Selskab, Mat. Fys. Medd. 29(1955) no. 17.
[2] S. Brodsky and S. Drell, Ann. Rev. Nuci. Sci. 20 (1970) 147; B. Lautrup, A. Peterman and E. de Rafael, Physics Lett. 3C (1972) 193;J. Calmet, S. Narison, M. Perrottet and E. de Rafael, Rev, of Mod. Phys. 49 (1977) 21. [3] C. Chlouber and M.A. Samuel, Phys. Rev. D, to be [4] T.Clau:en, J.f. Math. (Crelle) 8(1832) 298.
~ 2
)
I
3
[5] P. Wynn, Chiffres 8 (1966) 23. [6] P. Wynn, Math. ofComp. 14(1960)147.
//
__________
2
3
~
-~
//
4
Fig. 3. Re ~(4)~,2) in units of (~/~)2 versus q = \/1p21 in units of 2m. Region I is the space-like region. Region II and III comprise the time-like region.
[71 K. Mitchell,
Phil. Mag. 40(1949)351. [8] A review of Padé approximants is given by J. ZinnJustin, Phys. Lett. 1C (1971) 55. [9] C.R.Hagen and MA. Samuel, Phys. Rev. Lett. 20 [10] J. Mignaco and E. Remiddi, Nuovo Cim. 60A (1969) 519. [11] Mark A. Samuel, Phys. Rev. AlO (1974) 1450. [121 D.A. Owen and W.W. Repko, Phys. Rev. A5 (1972) 1570. [13] L.W. Fullerton and G.A. Rinker, Jr., Phys. Rev. A13 (1976) 1283.
ACCURATE COMPUTATION OF FOURTH-ORDER VACUUM POLARIZATION
*
Clyde CHLOUBER and Mark A. SAMUEL Quantum Theoretical Research Group, Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74074, U.S.A. Received 30 January 1978; in revised form 22 March 1978
A method to accurately compute vacuum polarization to fourth-order in both the space-like and the time-like regions is described. It makes use of various techniques, including changes of variables and Padé approximants to accelerate convergence of sequences which occur in the computation. Various checks have been made which indicate that it is accurate to 9 significant figures. These include a sixth-order electron magnetic moment contribution and an order ~2 correction to the hyperfine structure of positronium.
1. Introduction
2. Definitions
The fourth-order vacuum polarization kernel [1] which gives the amplitude for the occurrence of the four fundamental processes depicted in fig. 1 contributes radiative corrections to the bound state energy
The general expression for the renormalized photon propagator is g / p p \ Re ir(p2) D~~(p)= ~ + i~g 1~, ~) 2 p p p with
levels of atoms, as well as muonic atoms, and the anomalous magnetic moments of the electron and the muon [2]. In the case of the electron magnetic moment an exact analytic result has been obtained for this contribution. In other calculations it is necessary to resort to numerical quadrature or semianalytic approximations to obtain these corrections. Although calculations of all 6th-order contributions to the anomalous moments of the electron and muon have magnetic been made, an accurate graph of the 4th. order kernel has not been presented. With this in mind we describe here in some detail VAC4, a subroutine we have developed, which accurately computes vacuum polarization to fourth-order, in both the space-like and the time-like regions. Furthermore, in the case of the muon, there are corrections of O (m~/m,~) which have yet to be determined. Some of these have recently been computed [3].
*
Re ir(p 2
2
)
1
=—
r i j
dt Im —
ir(t)
_______
r “
2’
o — where the imaginary part of the spectral function is given by 2) = (~j~, I zXz IJ~0). Im ~(p 3P p(z)=p .
In particular, the real and imaginary parts of the
e
Work supported by the U.S. Energy Research and Development Administration.
(a
(b)
(c)
(d)
Fig. 1. Fourth-order vacuum polarization diagrams. 1
c~
C. Chlouber and M.A. Samuel / Fourth-order vacuum polarization
154
4m2
4th-order vacuum polarization kernel including the three proper diagrams and the amplitude from the single double-bubble graph are given by [1]: Re
=
~4)~,2)
~[
~ i08
+
~72
(1n2 Ij—~ I—
—
A
ö~ 3
(6
~
—
i9 +
646~
3_62
2
I, III
ii~20(l —6)
B=
(~ 32
~23
~
12 62
2364÷6)B ~ 12
(1)
+(3_62)C+(3+26264)FGFI(62)1,
~(4)~2)
(4)
1+6
+ (19 5562+6) \2472 ~
Im
1>0, —
tan~ 77
~
II,
—
0(1_6)=~[1+~~=(~
~
6
[
III’
(5)
1 —6 / 1 —6 \ ~2 — +2~———--J+— (1+6) 1+6, 4 \
1
1-8
ln( 162)3]
—~
~2O(l + ln
1+6(~~6223 1_6\16
8
~ 2
ln
_+62_-_)
(3
~
862
(
—6\ +2~~)1+6
lnl—I 1+6 ln 1 646~ 1—6 _62I3]1
~6~+~) (1+6)~
_)(4~ ~ 2
/362
66
~2\
)
+~) 2
1
6 + 6)
tan_i
[~1(~
1 1 —ln +
0(1 —6),
(2)
tan
~(x)~~
where
t~
sin nx 1
62=1+
4m2
~-)—
2~i(2tan~)
6477~ 2)31’ (l+i~
— —f 0
ln(2 sin
1
p~pI.L =
II,
~t)
(6)
dt,
______
space-like virtual photon momentum —q~<0 time-like virtual photon
(q2 =
1,111
‘
>0
momentum
FGH(6 2) = fdt g(t) in Ii
ln(l f(t)
I,
f(t) +f(_t),
g(t)
The interval over which the function Re ir(4)(p2) is to be evaluated further subdivides for the time-like case. (We label region II for q <2m and region III for q > 2m.) For later reference the space-like region is referred to as region I. For regions I, II and III the following formulae apply to the variables in the expression for the kernel.
2 — ~
=
+ t)
3 ln[(l 2 1
+—
— ±
t)/2] ~——-——.
ln~tI 1 t +
(7)
3. Evaluation of the functions i,Li(x), 0(x) and FGH(52) The function 0(x) is easily computed to 14-place
A=ç
16
1+6 ln I —I 1 —6
,
277tan177,
I, III II,
,
accuracy For x >0 usingwetheevaluated method the given function by Clausen 0(x) by [4].
C. Chiouber and M.A. Samuel / Fourth-order vacuum polarization
direct summation of the series
2
t
‘
FGH(6 )Jg(t)lnIl (_l)fl_i
1
~
For x <0 we can use relations given in Mitchell [5] to arrive at the formulae 2 / x 6
x+1
0(x) =
—
o(__~-~_~) + ~ lnIxIln~+x)2
+
(9)
The arguments of 0 on the right hand sides of eqs. (8) and (9) lie on the interval (0,1), hence all values of 0(x) for—i ~x ~ 1 may be expressed in terms of 0(z)with0~z ‘~ 1, where
—
0
i+x i+x
—i
1
ir/2
f
g(6sin0)lnI1-~-sin20I6cos0d0.
-+2(1 _6)f
t=
(1 —6) sin2O
+
(16)
6
g[(l —6)sin20 +6]
o ~l
ml 1 —
FGH(62)
<2
The series to be evaluated are oscillating and rapidly convergent. For z ~ 0.8, a sum of 30 terms or less of the series gives 10-place accuracy. For z 0.8, the c-algorithm [5,6] was used to accelerate the convergence of the sequence of partial sums. The results obtained were found to be in agreement with those tabulated by Mitchell [7]. In the evaluation of FGH(62), changes of variables, numerical quadratures, and convergence techniques were used to obtain accurate results. To circumvent the logarithmic infinities which occur in the integrand of FGH(6 2) changes of variables are made. In regions I and II let r = sin 0, then =
ir/2
f~~f
f
(15)
~
—
~—~--—
6
\2
sin2O
+
1) sin 0 cos OdO
.
(17)
2
— —~—~
FGH(62)
fg(t) ln Ii — dt. a In the first integral let t = 6 sin 0
The integrals for all three regions have been reduced to the form where x(0) is finite and vanishing at the endpoints of the integration interval.
,
!
._~_
—~Idt 6
In the second integral let
—1
x
2
+
a (8)
155
=
x(O) dO,
f
(18)
e-
e-
e-
e-
e~
e+
e-
+
e-
e-
e+
-
•2
g(sin 0) lnl 1
— ~~-~--
I cos OdO
-
(14)
(d)
In region III there is a divergence at t = 6 as well as at t = 1 since 0 <6 < 1; consequently, we split the integration interval.
Fig. 2. Annihilation diagrams with four-order vacuum polarization insertions, contributing to the positronium ground-state hyperfine splitting in sixth-order.
C. Chlouber and M.A. Samuel / Fourth-order vacuum polarization
156
Table 1 Sequence of approximations to FGH(8 2) in each of the three regions, illustrating typical rates of convergence. k
Region I 2 = 2) FGH(a Sk(2)
2 3 4 5 6 7
—0.314280423196 —0.314280413178 —0.314280408538 —0.314280406017 —0.314280404496 —0.314280400283
=
Region II FGH(82 = —3) Sk(—3)
Region III FGH(82 = ~) Sk(2~~~)
Region I and III FGH(62 = 1) Sk(l)
—0.314280605732
0.168968686737
—0.992945439727
—0.901550793418
—0.314280451789
0.168968621712
—0.992944168587
—0.901544899344
0.168968609611 0.168968605368 0.168968603402 0.168968602333 0.168968601689 0.168968599903
—0.992943933026 —0.992943850560 —0.992943812385 —0.992943791646 —0.992943779140 —0.992943744506
—0.901543715052 —0.901543281071 —0.901543073668 —0.901542958211 —0.901542887201
For the numerical quadrature, the interval [0, 77/2] is divided into {m = 1,2,3,4, ...} subintervals. A Gauss quadrature is applied to each subinterval and the results of all subintervals are summed. Thus a sequence of partial sums Sk, (k 1,2, ...) to eq. (18) is ob-
T~2~54267639’
interval (0,1) to evaluate the integral 1
a~6~(fourth-order V.P.) = e
fdx(1
~
—
x)
77 J 2
—
Re
With the transformations defined above, the sequences of quadrature approximations to FGH are rapidly convergent. The sequences were extrapolated to k = 00 by the method of Padé type II approximants [8]. The results shown in table 1 are typical of the rates of convergence for other values of 62. The extrapolated value for FGH(1) agrees with the exact answer [which is known [9] to be —~~(3)] to 2 parts in l0~, and since the rates of convergence for other values of 62 is as good or better then that for 62 = 1, we can expect an error of approximately the same magnitude
— 2
—x me) I x The Padé extrapolation to this sequence of partial sums (using the 1/k coordinate method is 0.0554291769 which agrees to 8 figures with the exact result of Mignaco and Remiddi [10]. =
77(4)(q2
/
\~
a~6~(4th order V.P.) = (f-) {
~
~(2)
—~
77
+
2 2 + 32 2 1 2 ~ ( ) — ~( ) n ~ ~(2)l~~2 + ~ lfl~2}= 0.0554291775. ~
18 —
~( )
14
—
15
1
-~-
8
i~(~)
or better.
4. Consistency checks
Table 2 Sequence of quadrature approximations using VAC4 to compute the fourth-order vacuum polarization contribution to
As an independent check on the overall accuracy
of our routine VAC4 in the space-like region, we used it to compute the fourth-order vacuum polarization contribution to the sixth-order electron magnetic moment. Since the answer is known analytically [10], we can check it directly. Table 2 gives the results of
the sequence of quadrature approximations in which (m = 1,2,3,4,5) Gauss quadratures were applied on the
a~6~ in units of (~/7T)3. n Approximants to a~6~ e 1 0.055429509209975 2 0.055429368961553
3 4 _______
0.055429220345088 0.055429202459104 0.055429193887575
______________________________________-
C. Chiouber and MA. Samuel / Fourth-order vacuum polarization
As a check on VAC4’s accuracy in the time-like 2 region II, we numerically computed the order a correction to the hyperfine structure of positronium associated with the annthilation diagrams [101 shown Table 3 Re ~(4)(p2) in units of (a/lr)2 as a function of q bine the time-like region. Region I q
Re
0.0 0.10 0.20 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.03 5.50 6.00 6.50 7.00 7.53 8.00 8.50 9.OQ 9.50 10.00 10.50 11.00 11.5) 12.00
0.0 —0.01006449 —0. 03951076 —0.08660005 —0.14837312 —0.22171877 —0.30348929 —0.39086091 —0.48148185 —3.57350821 —0.66556644 —1.10110117 —1.48041182 —1.80803933 —2.09663776 —2.35556780 —2.59138591 —2.80869582 —3.01076697 —3.20007011 —3.37845225 —3.54736219 —3.70795512 —3.86116964 —4.00778077 —4.14843745 —4.28368974 —4.41400878 —4.53580192 —4.66142427 —4.77918765 —4.89336768 —5.00420942
12.50
—5.11193102
13.03 13.50 14.00 14.53 15.00 15.50 16.03 16.50 11.33
—5.21673194 —5.31878697 —5.41825781 —5.51529065 —5.61001883 —5.70256456 —5.79303978 —5.88154761 —5.96818345
17.50
—o.05303519
13.03 18.53 19.00 19.50 20.03
—6.13618455 —6.21170741 —6.29161443 —6.37615149 —6.45320019
=
in fig. 2. Since the level shift has already been determined analytically [11,12] to order a4R,~,we have a basis for an accurate comparison.
\/1p21 in units of 2m. Region I is the space-like region. Regions II and III com-
Region II q 0.02 3.04 0.06 0.38 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.3o 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.38 0.50 0.52 0.64 0.oS 0.68 0.10 0.72 0.74 0.78 0.To 0.83 0.32 0.84 0.36 0.88 0.00 3.92 0.94 0.96 0.06
157
Re
Region III q
0.00040503 1.02 0.031~2128 1.04 0.003D5216 1.06 0.00650346 1.08 0.31018331 1.10 0.01470193 1.12 0.02007175 1.14 0.02631619 1.16 0.03344167 1.18 0.04147574 1.20 0.35044320 1.30 0.06037237 1.40 0.07129536 1.50 0.08324842 1.60 0.09627236 1.80 0.11041299 2.00 0.12572175 2.20 0.14225631 2.40 0.16008137 2.60 0.11926959 2.80 0.19990266 3.00 0.22207257 3.50 0.24588322 4.00 0.27145219 4.50 0.293~1305 5.00 0.32841803 5.50 0.36014141 6.00 0.39428358 6.50 0.43107623 7.00 0.47078877 7.50 0.51373649 8.00 0.56029115 8.50 0.61089456 9.00 0.66607668 9.50 0.12641966 10.00 0.79289036 10.53 0.86628748 11.00 0.94790379 11.50 1.03932210 12.00 1.14281584 12.50 1.26056816 13.30 1.30702592 13.50 1. 5s730124 14.00 1.15025914 14.50 l.9~8’~4603 15.00
2.29411993
15.50
~.i1294134 3.34535314 4.~3425171
16.00 16.50 17.00
Re 3.28613947 1.81091252 1.03212839 0.53363132 0.18510175 —0.07138200 —0.2o657327 —0.41868906 —0.53935297 —0.63639077 —0.91526800 —1.03103375 —1.08505672 —1.11399129 —1.14896244 —1.18204457 —1.22214505 —1.26982195 ~-1.32376417 —1.38245092 —1.44456521 —1.60855134 —1.77734102 —1.94557489 —2.11063321 —2.27128471 —2.42701632 —2.57169267 —2. 72337749 —2.86423722 —3.00046790 —3.13236553 —3.26010957 —3.38395382 —3.50412164 —3.62062367 —3.73425697 —3.84460503 —3.95203821 —4.05671450 —4. 15878029 —4.25837126 —4. 355o1322 —4.45062292 —4.54350879 —4. 63437168 —4.72330547 —4.81039767 —4.89572998
C. Chiouber and MA. Samuel / Fourth-order vacuum polarization
158
The energy shift for the annihilation process is
~A=—477aI0(0)I2(S2)
Re ir~4~(—k2) k2
Table 4 Re ir(2)p2) in units of (c~/ir)as a function of q the time-like region. Region I q 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.53 6.00 6.50 7.00 (.50
8.00 8.50 9.03 9.53 10.00 10.50 11.03 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.33 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00
=
where k is the total C.M. positronium energy i.e. a2
k(2_~)m,c=l.
\/1p2 i in units of 2m. Region I is the space-like region. Regions II and III comprise
Region II
Re n(2)
q
0.00000000 —0.00265531 —0.01048803 —0.02312082 —0.03999127 —0.06042954 —0.08373414 —0.10923198 —3.13631716 —0.16447000 —0.19326127 —0.33714117 —0.46929734 —0.58590608 —0.68836240 —0.77894604 —0.85974837 —0.93247844 —0.99849043 —1.05885291 —1.11441399 —1.16585320 —1.21372088 —1.25846751 —1.30046571 —1.34002664 —1.37741253 —1.41284618 —1.’.4651837 —1.47359362 —1.50021472 —1.53850635 —1.56657797 —1.59352620 —1.61943673 —1.64438586 —1.66844186 —1.69166602 —1.71411358 —1.73583448 —1.75687401 —1.77127341 —1.79707027 —1.81629930 —1.83499116 —1.85317574 —1.87087947 —1.88812704 —1.90494126
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.4? 0.44 0.t6 0.46 0.50 0.52 0.54 0.5~ 0.58 0.60 0.62 0.64 0.66 0.od 0.70 0.72 0.14 0.16 0.78 0.80 0.82 0.84 0.86 0.38 0.90 0.92 0.94 3.96 0.98
Re ¶() 0.00010668 0.00042696 0.00096148 0.03171137 0.30251816 0.00386390 0.30527109 0.00690272 0.00876233 0.01085398 3. 01318233 0.01575265 0.31857086 0.02164362 3.32497833 0.02858325 0. 03246755 0.03664140 0.04111608 0.04550413 0.05131944 0.05641744 0.0s229532 0.06849222 0.07508952 0.08211120 0.08958414 0.09153869~ 0.10600919 0.11503468 3.12465978 3.13493576 0.14592193 0.15768140 3.17331333 0.18389594 ‘3.19855054 0.21441698 0.23166727 0.25051659 3.27123936 0.29419406 0.31986280 0.34891822 0.38234473 0.421679o0 0.46955875 0.53122539 0.62052428
Region III q 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.30 1.40 1.50 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.03 3.50 4.00 4.50 5.00 5.5o 6.00 6.50 7.00 7.50 8.00 3.50 9.00 9.50 10.00 10.50 11.00 11.50 12.30 12.50 13.03 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00
Re ~(2) 0.83710855 0.78825637 0.74208687 0.69838007 0.65693840 0.b1758404 0.58015657 0.54451103 0.51051612 0.47805268 0.33523530 0.21790643 0.11919773 0.03448666 —0.10486294 —0.21650036 —0. 30945050 —0.38908733 —0.45881427 —0.52089877 —0.57691650 —0.69716685 —0.79117723 —0.88305828 —0.95846941 —1.02578262 —1.08662852 —1.14217953 —1.19330945 —1.24068751 —1.28483916 —1.32618471 —1.36506605 1.40176525 —1.43651793 —1.46952299 —1.50095000 —1.53094471 —1. 55963337 —1.58712608 —1.61351947 —1.63889881 —1.66333975 —1.68690973 —1.70966909 —1.73167211 —1. 75296713 —1.77360029 —1.79361034
C. Chlouber and MA. Samuel/Fourth-order vacuum polarization
3 and (S2)
Substituting I I~,(0)I2= (1/77)(ma/2)
=
2,
i Res. II
I
the energy shift may be written Re ir~4~(—k2) = —2a2R.. (2 — ~a2)2 For Re Re
159 Reg.
II
-
ir~4~(—k2) we obtained the value
ir~4~(—k2) = 1.2302 X i0~
6
5
4
3
where upon using a~ = 137.035987 and R,,~= 3.289843 2 X 1015 Hz we obtained the frequency shift ~v—10.77
2
I
1
2
3
4
5
6
,
-
5
MHz,
which is in good agreement with the semi-analytic result [11] =
_k 2
a4R
1 ln a1
+
=2
21
—
32
Fig. 4. Re ~(2)~,2) in units of (a/ir) versus q = \/1p21 in units of 2m. Regions I, II and III are as previously defined.
13 324772
1n2 Recently, Fullerton and Rinker [13] reported on a computation of fourth-order vacuum polarization. Their method is different however, making use of the
— —
—10.76 MHz.
Chebyshev approximation. Their a potential
V(r),
result, expressed as 0 ~ r ~ Xe.
is accurate to 3 figures for
5. Tabulation of the function Re 7T(4)(p2) In table
3
we tabulate the function Re 77(4)(p2) on
the interval (20 X
2m)2
>p2
results are also graphed in fig. 3. For comparison we also tabulated Re 77(2)~92)in table 4 and plotted this function in fig. 4. 4
Req. I
References
> —(17 X 2m)2. The
Req. II
Req.
11] G. Kallén, Helv. Phys. Acta 25 (1952) 417 and G. Källén and A. Sabry, Kgl. Danske Videnskab, Selskab, Mat. Fys. Medd. 29(1955) no. 17.
[2] S. Brodsky and S. Drell, Ann. Rev. Nuci. Sci. 20 (1970) 147; B. Lautrup, A. Peterman and E. de Rafael, Physics Lett. 3C (1972) 193;J. Calmet, S. Narison, M. Perrottet and E. de Rafael, Rev, of Mod. Phys. 49 (1977) 21. [3] C. Chlouber and M.A. Samuel, Phys. Rev. D, to be [4] T.Clau:en, J.f. Math. (Crelle) 8(1832) 298.
~ 2
)
I
3
[5] P. Wynn, Chiffres 8 (1966) 23. [6] P. Wynn, Math. ofComp. 14(1960)147.
//
__________
2
3
~
-~
//
4
Fig. 3. Re ~(4)~,2) in units of (~/~)2 versus q = \/1p21 in units of 2m. Region I is the space-like region. Region II and III comprise the time-like region.
[71 K. Mitchell,
Phil. Mag. 40(1949)351. [8] A review of Padé approximants is given by J. ZinnJustin, Phys. Lett. 1C (1971) 55. [9] C.R.Hagen and MA. Samuel, Phys. Rev. Lett. 20 [10] J. Mignaco and E. Remiddi, Nuovo Cim. 60A (1969) 519. [11] Mark A. Samuel, Phys. Rev. AlO (1974) 1450. [121 D.A. Owen and W.W. Repko, Phys. Rev. A5 (1972) 1570. [13] L.W. Fullerton and G.A. Rinker, Jr., Phys. Rev. A13 (1976) 1283.