PHYSICS LETTERS
Volume 31B, number 9
CONFIRMATION
27Apml 1970
OF A NEW THEORETICAL
FOR THE LAMB SHIFT
B. E. LAUTRUP,
VALUE
A. PETERMAN and E. DE R A F A E L
CERN-Geneva, S w i t z e r l a n d
Received 16 March 1970
We have performed a recalculation of the contribution to the Lamb shift from two fourth order vertex
graphs. We agree with a recent calculation by Appelqmst and Brodsky.
T h e r e has been a r e c e n t change in the theor e t i c a l value for the Lamb shift. A new c a l c u l a hon [1] of the fourth o r d e r c o n t r i b u t i o n to the
slope of the D l r a c form factor of the e l e c t r o n
m s a g r e e s with the p r e v i o u s r e s u l t s [2, 3]. The
new value for the slope, which c o n t r i b u t e s to
o r d e r ~ 2 (Z, ~)4 m c 2 to the L a m b shift, when
added to the other r a d m t l v e c o r r e c t m n s [4] gives
a 2Sj/2 - 2P1/2 s e p a r a t i o n in hydrogen and deut e r i u m , which is in e x c e l l e n t a g r e e m e n t with the
r e s u l t s of r e c e n t e x p e m m e n t s [5].
The c o n t r i b u t m n to Lamb shift is
AE =
5/o
4(Z~ )4mc2
n3
(~
ref. [2] c a l c u l a t e d the i n f r a - r e d d i v e r g e n t t e r m s
and p a r t of the covergent t e r m s a n a l y U c a l l y and
e s t i m a t e d the r e m a i n d e r n u m e m c a l l y , while Soto
[3] p e r f o r m e d the whole calculation a n a l y t i c a l l y .
The two r e s u l t s a g r e e with each other (for the
c o r n e r graphs) and the quoted value is
= (_~_)2 {_~ iog2 _2 + ~ iog;_2 + 2.432}
The new value by Appelquist and B r o d s k y is the
r e s u l t of a completely n u m e m c a l computation.
They find (for the c o r n e r graphs)
g=\~-]
~ l o g 2 X -2 -Tt~ logX-2
1.91 ± 0.02}
where the quantity a is defined as
(~ = m 2 ~Fl(q)2 q
~q2
2 =0
It is i m p o r t a n t here to r e m a r k that we use a
timehke metmc.
The difference between the p r e v i o u s c a l c u l a h o n s [2, 3] and the r e c e n t one by Appelqmst
and Brodsky [1] lies f i r s t of all in an o v e r a l - a l l
change of mgn for all t e r m s , and secondly in a
n u m e m c a l d i s a g r e e m e n t between the n o n - i n f r a red c o n t r i b u t m n s from the graphs a and b in
fig. 1. (the " c o r n e r graphs"). The a u t h o r s of
In view of the c r u c i a l n a t u r e of thls d l s c r e p a n c y
we have c o n s l d e r e d It worth while to p e r f o r m yet
a n o t h e r e v a l u a t i o n of this quantity.
We reduced the graphs in figs. l a and Ib to
p a r a m e t r i c form by hand, i n s e r t i n g p a r a m e t r i z e d
v e r t e x part, and a f t e r w a r d s p a r a m e t r i z i n g the
outer loop by s t a n d a r d methods. In this way we
obtained a d i v i s i o n of g into t h r e e p a r t s
(7 = (~1 + Cr2 + (:r3
where (~1 s t e m s from the r e n o r m a l i z a t i o n c o u n t e r
t e r m s in the i n s e r t i o n and is e a s i l y c a l c u l a b l e
analytically
: @)2 { 1og2;2- log;2+ }
a~
b|
Fig. I.
¢)
The r e m a i n d e r ((~2+ (~3) is o r i g i n a l l y a f i v e - d i m e n s i o n a l i n t e g r a l but a ]uchcious choice of par a m e t e r s allows one i n t e g r a t i o n to be p e r f o r m e d
tmvially. This i n t e g r a l was then divided into two
p a r t s , g2 and g3 of which cr2 was i n f r a - r e d d i v e r g e n t , but c o m p a r i t i v e l y s i m p l e , and cr3 was lnfra577
Volume 31B, number 9
PHYSICS
r e d c o n v e r g e n t , b u t e x t r e m e l y c o m p l i c a t e d . It
was p o s s i b l e to e x t r a c t the i n f r a - r e d d~vergence
a n a l y t i c a l l y f r o m ~2 with the r e s u l t
cr2 = ( ~ - ) 2 { - ~o1 o g 2 ~ - 2 +%I~ l
og k-2
LETTERS
27Apml 1970
c(~)
2D
18
+C2}
@
a
o
o
w h e r e C 2 ~s c o n s t a n t for ~
0. By adding a l
and ~2 we o b t a i n the ~ n f r a - r e d d i v e r g e n t p a r t s
g~ven by A p p e l q u l s t and B r o d s k y . The c o n t r l b u h o n ~3 was e v a l u a t e d n u m e m c a l l y w~thout t r o u b l e
a n d the r e s u l t ~s
~3 = (~_)2{_ 0 . 1 7 6 ± 0.003}
o
o
o
1l,
o
tZ
o
o
10
The only r e m a l m n g q u a n t i t y to be d e t e r m i n e d is
C 2. As it ms d~fflcult to keep t r a c k of the c o n v e r g e n t p a r t s left out d u r i n g the e x t r a c t i o n of
the i n f r a - r e d p a r t we d e t e r m i n e d C 2 by c a l c u l a t i n g the full i n t e g r a l a2(X) n u m e r i c a l l y as a
f u n c t i o n of X and a f t e r w a r d s c a n c e l l i n g out the
chvergent t e r m s . P r e c i s e l y , we h t t e d the q u a n hty
+
{i- o.18}
to t h e f o r m
C()0 = C + d l ~ log2X-2+d2X l o g ~ - 2 + d3X
+ d4 x2 log2X -2 + d5X2 log k-2 + d6X2
T h i s q u a n t i t y will a p p r o a c h the n o n - i n f r a - r e d
d~vergent t e r m , C ~n (~ for X - - 0. O u r r e s u l t was
(see a l s o fig. 2)
C = - 1.95 + 0.05
dl =0.6±1.1
d4=-6 :t:9
d2 = - 1 1 + 22
d 5 = -17 • 34
d 3 = 73 ± 128
d 6 = - 7 7 ± 134
T h i s is in e x c e l l e n t a g r e e m e n t with the r e s u l t
of ref. [1]. We r e m a r k that the u n c e r t a i n t y in C
ms s o m e w h a t l a r g e r t h a n the u n c e r t a i n t y quoted
in ref. [1] (± 0.02) m a i n l y b e c a u s e we f i t t e d with a
l a r g e n u m b e r of b a c k g r o u n d t e r m s . We i n c l u d e d
t h e s e xn o r d e r to e x p l o r e the p o s s i b l e i n f l u e n c e
of o t h e r b a c k g r o u n d t e r m s t h a n those t a k e n in
ref. [1] in the d e t e r m i n a t i o n of C. As c a n be
578
16
o
I
l
2
I
3
~
~
Log
~
~
7
t_
8
;-2
Fig. 2.
s e e n f r o m h g . 2, C(X) ms r o u g h l y c o n s t a n t o v e r
four o r d e r s of m a g n i t u d e i n k 2.
We should h k e to m e n h o n that this c a l c u l a h o n is i n d e p e n d e n t of the c a l c u l a t i o n of ref. [1]
as f a r a s the a n a l y t i c a l e v a l u a t i o n goes. U n f o r t u n a t e l y , it w a s n e c e s s a r y to u s e the s a m e i n t e g r a t i o n s u b r o u t i n e for the n u m e r i c a l i n t e g r a t i o n as was u s e d in ref. [1]. We a t t e m p t e d f i r s t
to u s e s t a n d a r d G a u s s l a n m e t h o d s , b u t they_
t u r n e d out to be r e h a b l e only for ~+2 >~ 10-3 and
did f u r t h e r m o r e not give any i d e a about the u n c e r
t a l n t y on the r e s u l t . A p r o g r a m w r i t t e n by G.
Sheppey which w a s u s e d i n ref. [1] and w h m h is
d e s c m b e d in ref. [6], gave r e l i a b l e r e s u l t s as
f a r down a s ~2 >/ 10-9, and f u r t h e r m o r e y i e l d e d
a n e s t i m a t e of the u n c e r t a i n t y in the v a l u e s . It
has b e e n t h o r o u g h l y t e s t e d by the a u t h o r s of ref.
[6] a n d the m e r e c o n s t a n c y of C (X) for s m a l l
a m p h h e s the c o n f i d e n c e we have in i t s use.
F i n a l l y , we a g r e e with A p p e l q u i s t and B r o d sky i n the o v e r - a l l s i g n d i f f e r e n c e . One of u s
(B. E. L. ) h a s f u r t h e r m o r e c h e c k e d the c o n t m b u tlon f r o m the v a c u u m p o l a r i z a t i o n t e r m (fig. l c )
a n a l y t i c a l l y with the r e s u l t .
o"
vacpol
=
{~oz~2t 777r2
\'~-] ~ 8 ~
1099~
- 1296
w h i c h h a s the o p p o s i t e s i g n of the Soto v a l u e [3].
The a u t h o r s a r e g r a t e f u l to Dr. S. J. B r o d s k y
for c o m m u n i c a t i o n of the r e s u l t s of ref. [1]
p r i o r to p u b l i c a t i o n .
V o l u m e 31B, n u m b e r 9
PHYSICS
References
1. Th. A p p e l q u i s t and S. J. B r o d s k y , SLAC P r e p r i n t
(1970).
2. J. W e n e s e r . R. B e r s o h n a n d N. M. K r o l l , P h y s . Rev.
91 (1953) 1257.
3. M . F . S o t o , J r , , P h y s . Rev. L e t t e r s 17 (1966) 1153.
4. G . W . E m e k s o n and D R. Y e n m e , Ann. P h y s . (N.Y.)
35 {1965) 217, 35 (1965) 447.
LETTERS
27 A p m l 1970
5, F o r a r e v i e w , s e e :
S. J B r o d s k y . S t a t u s q u a n t u m e l e c t r o d v n a m l c s .
P r o c . 4th I n t e r n . S,~,mp. on E l e c t r o n and photon
i n t e r a c t i o n s at hlgh e n e r g i e s ( D a r e s b u r y N u c l e a r
P h y s i c s L a b o r a t o r y , 1969).
6. J. A l d i n s S . J . B r o d s k y A . J . D u f n e r and T . K m o s h l t a , SLAC P r e p r m t (1970).
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