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Issue 3 (September) PROGRESS IN PHYSICS Volume 16 (2020)
The Unpublished Feynman Diagram IIc
Oliver Consa
Department of Physics and Nuclear Engineering, Universitat Politècnica de Catalunya
Campus Nord, C. Jordi Girona, 1-3, 08034 Barcelona, Spain
E-mail: oliver.consa@gmail.com
Quantum Electrodynamics (QED) is considered the most accurate theory in the history
of science. However, this precision is limited to a single experimental value: the anoma-
lous magnetic moment of the electron (g-factor). The calculation of the electron g-factor
was carried out in 1950 by Karplus and Kroll. Seven years later, Petermann detected
and corrected a serious error in the calculation of a Feynman diagram; however, neither
the original calculation nor the subsequent correction was ever published. Therefore,
the entire prestige of the QED depends on the calculation of a single Feynman diagram
(IIc) that has never been published and cannot be independently verified.
1 Introduction
According to the Dirac equation, the value of the magnetic
moment of the electron should be exactly one Bohr magne-
ton. In 1947 it was discovered that the experimental value of
the magnetic moment of the electron presented an anomaly
of 0.1% with respect to the theoretical value [1] [2]. This
anomaly was called the electron g-factor.
µe=gµB=ge~
2me
(1)
Schwinger carried out the first theoretical calculation of
the electron g-factor obtaining a value very similar to the ex-
perimental value. This value is known as the Schwinger fac-
tor [3].
g=1+α
2π=1.001162 (2)
According to Quantum Electrodynamics (QED), the theo-
retical value of the electron g-factor is obtained by calculating
the coefficients of a number series called the Dyson series [4].
When Feynman, Schwinger, and Tomonaga received the 1965
Nobel Prize for the development of QED, only the first two
coefficients in the series had been calculated. The rest of the
coefficients in the Dyson series were calculated many years
later with the help of supercomputers.
g=C1α
π+C2α
π2
+C3α
π3
+C4α
π4
+C5α
π5
... (3)
Each coefficient in the series requires the calculation of an
increasing number of Feynman diagrams. The first coefficient
in the Dyson series is the Schwinger factor and has an exact
value of 0.5. The second coefficient was calculated in 1950
by Karplus and Kroll [6], who obtained a result of -2.973.
This result was corrected seven years later by Petermann [8],
who obtained a result of -0.328, almost 10 times lower than
the previous calculation.
g=1+1
2α
π−0,328 α
π2
=1,0011596 (4)
The error was found in the calculation of the Feynman
diagram IIc. According to the Karplus and Kroll’s, original
calculation, the value of diagram IIc was -3.178 while in the
Petermann correction the value of diagram IIc was -0.564.
Fig. 1: Feynman diagram IIc.
The entire prestige of the QED is based on its impressive
level of precision of the electron g-factor. Currently the QED
allows the achievement of the electron g-factor with a preci-
sion of 12 decimal places of the theoretical value with respect
to the experimental value.
•2008 Gabrielse’s experimental value [13]:
∗1.001,159,652,180,73(28)
•2018 Kinoshita’s theoretical value [14]:
∗1.001,159,652,182,032(720)
The calculation of the electron g-factor is based on the
calculation of the second coefficient of the Dyson Series. The
second coefficient of the Dyson series is based on the calcu-
lation of the Feynman diagram IIc. Therefore, the calculation
of the Feynman diagram IIc performed by Karplus and Kroll
in 1950 [6] can be considered the most important calculation
in the history of modern physics.
Surprisingly, the original calculation of this diagram IIc
turned out to be completely wrong and was corrected seven
Oliver Consa. The Unpublished Feynman Diagram IIc 1
arXiv:2010.10345v1 [physics.hist-ph] 18 Oct 2020
Volume 16 (2020) PROGRESS IN PHYSICS Issue 3 (September)
years after its publication. Inexplicably, both the original
Feynman diagram IIc calculation and the subsequent correc-
tion have never been published, so the most important cal-
culation in the history of modern physics cannot be indepen-
dently verified.
2 Original calculation
2.1 Karplus and Kroll’s paper
In 1949, Gardner and Purcell [5] published an new experi-
mental result for the electron g-factor of 1.001146. In re-
sponse, Karplus and Kroll performed the necessary calcula-
tions to obtain the second coefficient in the Dyson series.
In 1950, Karplus and Kroll [6] published a value of -2.973
for the second Dyson series coefficient and a new theoretical
value of 1.001147 for the electron g-factor. In good agree-
ment with the experimental data.
g=1+α
2π
−2.973 α
π2
=1.001147 (5)
The paper, published the February 14 in the Physical Re-
view Journal 77, consists of 14 pages full of complex mathe-
matical calculations.
On the second page of the document, the authors indicate
that to obtain the coefficient it is necessary to calculate 18
Feynman diagrams grouped in five groups (I, II, III, IV and
V). However, on pages 3 and 4, they argue that groups III,
IV and V are not necessary. Therefore, it is only necessary
to calculate seven Feynman diagrams, identified as I, IIa, IIb,
IIc, IId, IIe, IIf. A lot of calculations are done between pages
4 and 11 that only serve to show that diagrams IIb and IIf
are not necessary either. Therefore, it is only necessary to
calculate five Feynman diagrams (I, IIa, IIc, IId, IIe).
Fig. 2: Feynman Diagrams.
The calculation of diagrams IIe (0.016) and IId (-0.090)
are performed on pages 11 and 12 respectively. It follows that
[6] “The expressions for I, IIa and IIc become successively
more complicated and very much more tedious to evaluate
and cannot be given in detail here.”. In other words, the
complete calculation of three of the five diagrams was never
published. On page 13, the results of the three remaining di-
agrams are shown (I =-0.499, IIa =0.778 and IIc =-3.178).
Finally, page 14 of the paper presents the "Summary of Re-
sults" with the results of each of the five diagrams.
C2=I+IIa +I Ic +IId +IIe =−2,973 (6)
I IIa IIc IId IIe Total
-0.499 0.778 -3.178 -0.090 0.016 -2.973
Table 1: Values of the five Feynman diagrams.
From the analysis of the results it is evident that diagram
IIc is the dominant diagram. Diagrams I and IIa are less rele-
vant and practically cancel each other out. Diagrams IId and
IIe are the only two diagrams whose calculations are included
in the paper; however, their values are completely irrelevant.
The calculation of Feynman diagram IIc is made up of
four components:
IIc=−
323
24 +31
9π2−
49
6π2ln(2) +107
4ζ(3) (7)
Constant π2π2ln2ζ(3) Total
-13,458 33,995 -55,868 32,153 -3,178
Table 2: Value of the four components of Feynman diagram IIc.
The four components of IIc have abnormally high values
(-13, 34, -55 and 32) which surprisingly compensate for each
other, resulting in -3,178, an order of magnitude lower. It
is not possible to say anything more about the calculation of
diagram IIc because the complete calculation was never pub-
lished.
The authors indicate that [6]: “The details of two inde-
pendent calculations which were performed so as to provide
some check of the final result are available from the au-
thors”. That is, the authors affirm that the calculations were
carried out independently by two teams of mathematicians
who obtained the same result, as a guarantee that the calcula-
tions were correct.
2.2 New experimental value
Six years after the publication of the Karplus and Kroll’s pa-
per, Franken and Liebes [7] published new and more precise
experimental data that showed a very different value for the
2 Oliver Consa. The Unpublished Feynman Diagram IIc
Issue 3 (September) PROGRESS IN PHYSICS Volume 16 (2020)
electron g-factor (1.001165). This value was higher than the
Schwinger factor, so the value of the second coefficient cal-
culated by Karplus and Kroll not only did not improve the
Schwinger factor, but made it worse. With the new experi-
mental data, the value of the second coefficient in the series
should have been +0.7 instead of -2.973.
Karplus and Kroll admitted that it was not true that two
independent calculations had been carried out, so it was pos-
sible that there were errors in the calculations.
According to Kroll [15]: “Karplus and I carried out the
first major application of that program, to calculate the
fourth order magnetic moment, which calculation subse-
quently turned out to have some errors in it, which has been
a perpetual source of embarrassment to me, but neverthe-
less the paper I believe was quite influential. (...) The errors
were arithmetic (...) We had some internal checks but not
nearly enough. (...) it was referred and published and was
a famous paper and now it’s an infamous paper.”.
The history of this correction is complex and confusing.
We will now try to reconstruct this story from the published
papers and quotes from its protagonists.
3 The history of the correction
3.1 Petermann’s numerical calculation
Petermann was the first person to identify an error in the orig-
inal calculation of Karplus and Kroll. He performed a numer-
ical analysis of the five Feynman diagrams and he found that
the solution of diagram IIc was clearly wrong, since its value
was outside the limits. The rest of the diagrams were within
limits [9]: “The numerical results for the terms I, IIa, IIc,
IId, IIe in the work by Karplus and Kroll have been checked
by rigorous upper and lower bounds. Whereas every other
term fell well between these bounds, agreement could not be
obtained for diagram IIc. (...) The numerical value for this
term has been found to satisfy IIc =-1.02 +/- 0.53.”.
Petermann published a second paper where he adjusted
his calculations [10]: ”the diagram IIc is found to satisfy IIc
=-0.60 +/- 0.11 in contradiction with the value -3.18 given
by the previous authors.”.
Between the publication of these two papers, Petermann
communicated privately to Sommerfield the result of another
calculation [11]: "Note added its proof. Petermann has
placed upper and lower bounds on the separate terms of
Karplus and Kroll. He finds that their value for IIc does
not lie within the appropriate bounds. Assuming the other
terms to be correct, he concludes that the result is -0.53 +/-
0.37.”.
Petermann worked for three months following a numer-
ical methodology that allowed him to narrow the margin of
error in diagram IIc. Surprisingly, 14 days after his third
numerical calculation, he made an unexpected change in his
methodology and published the exact analytical calculation,
with no margins of error.
The articles published by Petermann on the calculation
of the Feynman diagram IIc are summarized in the following
table:
Date IIc Method Publication
28/5 -1.02 +/- 0.53 Numerical Nuclear Phys. 3
1/7 - 0.53 +/- 0.37 Numerical Phys. Rev. 107,
Note added in
proof.
Private comm.
with
Sommerfield
3/8 -0.60 +/- 0.11 Numerical Nuclear Phys. 5
17/8 -0,564 Analytical Helvetica Physica
Acta 30
Table 3: Petermann’s publications.
3.2 Sommerfield and the Green’s functions
After the publication of the new experimental value by
Franken and Liebes [7], Schwinger commissioned a 22-year-
old student named Sommerfield to redo the Kroll and Karplus
calculations. Schwinger proposed using his own method
based on Green’s functions instead of using Feynman dia-
grams.
According to Sommerfield’s testimony [16]: "Julian as-
signed us three problems, one of which involved the anoma-
lous magnetic moment (...). At my meeting with him, he
suggested that I continue the calculation of the anomalous
magnetic moment to the next fourth order (...). Schwinger
wanted me to use the other method, while respecting gauge
invariance at every step. Many years later Roy Glauber told
me that the faculty was not entirely happy that a graduate
student had been given such a problem.".
In May 1957, Sommerfield sent a two-page paper to the
Physical Review Journal where he published his results [12]:
“The fourth-order contribution to the moment is found to
be -0.328 (..) Thus the result is 1.0011596.“. This new the-
oretical value of the electron g-factor was in good agreement
with the new experimental value of Franken and Liebes.
As Schwinger states [18]: “Interestingly enough,
although Feynman-Dyson methods were applied early [by
Karplus and Kroll], the first correct higher order calcula-
tion was done by Sommerfield using [my] methods.”.
The second coefficient of the Dyson series calculated by
Sommerfield consisted of four components, the same as the
original result for Karplus and Kroll, but with very different
values:
[K&K]
C2=−
2687
288 +125
36 π2−9π2ln(2) +28ζ(3) =−2,973 (8)
Oliver Consa. The Unpublished Feynman Diagram IIc 3
Volume 16 (2020) PROGRESS IN PHYSICS Issue 3 (September)
[Sommerfield]
C2=197
144 +1
12 π2−
1
2π2ln(2) +3
4ζ(3) =−0,328 (9)
Const. π2π2ln(2) ζ(3) Total
K&K -9,329 34,269 -61,569 33,656 -2,973
Pet. 1,368 0,822 -3,421 0,901 -0,328
Diff. 10,697 -33,447 58,148 -32,754 2,645
Table 4: Comparative components of C2.
Sommerfield’s paper does not include the calculations
performed, but the author states that [11]: “The present cal-
culation has been checked several times and all of the auxil-
iary integrals have been done in at least two different ways.”.
As a guarantee that the calculations were correct.
In 1958 Sommerfield published his g-factor calculations
in the Annals of Physics [12] as part of his doctoral thesis.
If we analyze his extensive 32-page paper, we verify that he
used Green’s functions instead of Feynman diagrams. For this
reason, the calculation of the enigmatic Feynman diagram IIc
does not appear in this paper.
In the third volume of "Particles, Sources, and Fields"
published in 1989 [3], Schwinger devoted more than 60 pages
to a detailed calculation of the second coefficient of Dyson
series getting exactly the same result. But, once again, using
Green’s functions instead of Feynman diagrams.
In his 1957 paper Sommerfield also states that [11]: “The
discrepancy has been traced to the term I y IIc of Karplus
and Kroll.”. This statement about the origin of the error can-
not be deduced from Sommerfield’s calculations,
since he used Green’s functions instead of Feynman
diagrams. So Sommerfield had to receive this information
from other sources (Petermann, Karplus or Kroll).
3.3 Petermann’s definitive correction
The definitive solution to the problem was presented in 1957
by Petermann in a paper published in the Swiss journal Hel-
vetica Physica Acta [8]. Although the paper was signed by a
single author, actually the result was obtained by consensus
between the results of the Petermann’s numerical analysis,
the Sommerfield calculation of C2using Green’s functions
and the correction of the Feynman diagrams carried out by
Kroll himself. Petermann acknowledges that the result was
obtained by consensus [8]: “The new fourth order correc-
tion given here is in agreement with: (a) The upper and
lower bounds given by the author. (b) A calculation using a
different method, performed by C. Sommerfield. (c) A recal-
culation done by N. M. Kroll and collaborators.”.
The article was signed by a single author due to an in-
ternal conflict between the researchers. As Sommerfied re-
calls [16]: "In the meantime Schwingerian Paul Martin had
gone to the Niels Bohr Institute in Copenhagen and had
spoken to Andre Petermann, a postdoc with the Swedish the-
oretician Gunnar Kallen. Martin told Petermann about my
work (...) In the end, however, after both of our calcula-
tions were completely finished they were in agreement with
each other but not with Karplus and Kroll. We agreed to
cite each other’s work when published. However, Schwinger
and Kallen had had a somewhat acrimonious discussion (...)
and Kallen had forbidden Petermann to mention my work.
Petermann’s apology to me was profuse.".
The Petermann’s final result for the electron g-factor was
identical to the Sommerfield’s result published three
months earlier.
C2=197
144 +1
12 π2−
1
2π2ln(2) +3
4ζ(3) =−0,328 (10)
In the paper, Petermann states that: “We have performed
an analytic evaluation of the five independent diagrams
contributing to this moment in fourth order. The results
are the following (I =-0.467, IIa =0.778, IIc =-0.564, IId
=-0.090, IIe =0.016, Total =-0.328). Compared with the
values in their original paper by Karplus and Kroll, one can
see that two terms were in error: I differs by 0.031 and IIc
differs by 2.614.”.
I IIa IIc IId IIe Total
-0.467 0.778 -0.564 -0.090 0.016 -0.328
Table 5: Corrected values of the five Feynman diagrams.
Comparing the results of the calculations of the Feynman
IIc diagram carried out by Karplus and Kroll with the Peter-
mann calculations we observe the following:
[K&K]
IIc=−
323
24 +31
9π2−
49
6π2ln(2) +107
4ζ(3) (11)
[Petermann]
IIc=−
67
24 +1
18 π2+1
3π2ln(2) −
1
2ζ(3) (12)
The calculation of each of the four factors in diagram IIc
is shown in the following table:
Const. π2π2ln(2) ζ(3) Total
K&K -13,458 33,995 -55,868 32,153 -3,178
Pet. -2,791 0,548 2,280 -0,601 -0,564
Diff. 10,667 -33,447 58,148 -32,754 2,614
Table 6: Comparative components of Feynman diagram IIc.
4 Oliver Consa. The Unpublished Feynman Diagram IIc
Issue 3 (September) PROGRESS IN PHYSICS Volume 16 (2020)
The corrections are huge, one or two orders of magnitude
for each component of diagram IIc. We cannot know the ori-
gin of these discrepancies because the correction calculations
were also not published.
4 Summary
The calculation of the Feynman diagram IIc can be consid-
ered the most important calculation in the history of mod-
ern physics. However, the history of this calculation is sur-
rounded by big errors and inexplicable coincidences.
•The original calculation of the Feynman diagram IIc
published in 1950 was completely wrong.
•Karplus and Kroll stated that the calculation had been
performed by two teams independently. This statement
was made to give guarantees about the validity of the
calculations, and yet it turned out to be false.
•Despite having published a completely wrong result,
the prestige of Karplus and Kroll was not affected at all.
On the contrary, both enjoyed brilliant careers full of
awards and recognition for their professional achieve-
ments.
•The Karplus and Kroll miscalculation was consistent
with the experimental value previously published by
Gardner and Purcell, even though that experimental
value was also wrong.
•The error in the calculation was not reported until seven
years after its publication.
•The error in the calculation was detected just when a
new experimental value was published by Franken and
Liebes. The corrected theoretical value also coincided
with the new experimental value.
•Neither the original calculation of the Feynman dia-
gram IIc nor its subsequent correction has been pub-
lished to date.
1 September 2020
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