Searching for the 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<EMAIL_ADDRESS>
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. Although it’s hard to believe,
neither the original calculation nor the subsequent correction was ever
published. Therefore, the entire prestige of QED and the Standard Model
depends on the calculation of a single Feynman diagram (IIc) that has never
been published and cannot be independently verified. In this article we begin
the search for any published recalculation of this Feynman diagram IIc that
allows us to independently validate the theoretical calculation.

##  1 The big problem

###  1.1 Renormalization

The Standard Model of Particle Physics brings together two different physical
theories: Electroweak Theory (EWT) and Quantum Chromodynamics (QCD). For
decades a "Grand Unification Theory (GUT)" has been unsuccessfully sought to
integrate both theories into one unified theory.

Both QCD and EWT are mainly mathematical theories. The aim is to identify a
set of gauge symmetries for each theory that allows a concrete mathematical
formulation to be obtained. EWT forms a SU(2) x U(1) symmetry gauge group
while QCD forms a SU(3) symmetry gauge group. The theory is considered correct
if the theoretical values obtained with these mathematical formulas coincide
with the experimental values obtained with particle colliders.

Both QCD and EWT are based on and completely dependent on the validity of
quantum electrodynamics (QED), developed by Feynman, Schwinger, and Dyson. QED
in turn is a quantum field theory (QFT). QFT emerged in the 1930s in an
attempt to quantify the electromagnetic field itself. But QFT has a serious
problem. All calculations give the same result: Infinity.

In the 1940s, QED developers managed to solve the infinities problem using a
technique called "Renormalization.". Many methods can be used to eliminate
these infinities, but the main ones are:

  * •

Substitution: replacing a divergent series with a specific finite value that
has been arbitrarily chosen (for example, the energy of an electron).

  * •

Separation: separating an infinite series into two components, one that
diverges to infinity and another that converges to a finite value. Eventually,
the infinite component is ignored and only the finite part remains.

  * •

Cut-off: focusing on an arbitrary term in the evolution of a series that
diverges to infinity and ignoring the rest of the terms of the series.

Fig. 1: Layers of logical dependencies.

As an example of the use of these Renormalization techniques we can look at
the calculation of the Casimir effect [4]. The equation of the Casimir effect
depends on the Riemann function. However, the Riemann function is defined only
for positive values, since for negative values the Riemann function diverges
to infinity.

$\frac{F_{c}}{A}=\frac{d}{da}\frac{<E>}{A}=-\frac{\hbar c\pi^{2}}{2a^{4}}\
\zeta(-3)=\frac{\hbar c\pi^{2}}{20a^{4}}\ \zeta(-1)$ (1)

In particular the Riemann function of -1 corresponds to the value of the sum
of all positive integers. Applying a Renormalization technique, the Indian
mathematician Ramanujan came to the conclusion that the sum of all positive
integers is not infinity but -1/12 [3]. And this is precisely the value that
is used in the equation of the Casimir effect.

$\zeta(-1)=\sum_{n=1}^{\infty}{n}=1+2+3+4+5+...=\frac{-1}{12}$ (2)

$\frac{F_{c}}{A}=\frac{\hbar
c\pi^{2}}{20a^{4}}\left(\frac{-1}{12}\right)=-\frac{\hbar c\pi^{2}}{240a^{4}}$
(3)

Despite being one of the main creators of QED, Feynman was not very convinced
about Renormalization:“The shell game that we play is technically called
’renormalization’. But no matter how clever the word, it is still what I would
call a dippy process! Having to resort to such hocus-pocus has prevented us
from proving that the theory of quantum electrodynamics is mathematically
self-consistent. It’s surprising that the theory still hasn’t been proved
self-consistent one way or the other by now; I suspect that renormalization is
not mathematically legitimate.” [1]

For his part, Dirac was always clearly against these techniques: “I must say
that I am very dissatisfied with the situation because this so-called ’good
theory’ does involve neglecting infinities which appear in its equations,
ignoring them in an arbitrary way. This is just not sensible mathematics.
Sensible mathematics involves disregarding a quantity when it is small – not
neglecting it just because it is infinitely great and you do not want it!.”
[2]

Today the scientific community accepts these renormalization techniques as
fully legitimate. But if Dirac was right and renormalization is not a
legitimate mathematical technique, then the Standard Model, EWT, QCD, QED and
all theories based on QFT would be incorrect and worthless.

###  1.2 QED precision

The entire credibility of the renormalization techniques is based on its level
of precision of the theoretical value with respect to the experimental value.
As an example, the electron g-factor offers an impressive level of precision
of 12 decimal places:

  * •

Experimental value [12]: $1.001,159,652,180,73(28)$

  * •

Theoretical value [13]: $1.001,159,652,182,032(720)$

In 1970 Brodsky and Drell summarised the situation in their paper "The present
status of the Quantum Electrodynamics" as follows: “The renormalization
constants are infinite so that each calculation of a physical quantity has an
infinity buried in it. Whether this infinity is a disease of the mathematical
techniques of perturbation expansions, or whether it is symptomatic of the
ills accompanying the idealization of a continuum theory, we don’t know.
Perhaps there is a "fundamental length" at small distances that regularizes
these divergences (…). Quantum electrodynamics has never been more successful
in its confrontation with experiment than it is now. There is really no
outstanding discrepancy despite our pursuing the limits of the theory to
higher accuracy and smaller (…) however, and despite its phenomenal success,
the fundamental problems of renormalization in local field theory and the
nature of the exact solutions of quantum electrodynamics are still to be
resolved.” [14]

It seems inconceivable that using an incorrect theory we can obtain the
correct results with an unprecedented level of precision. And it is extremely
unlikely that this finite theoretical value coincides with the experimental
value by pure chance. Therefore, the only reasonable explanation is that
renormalization techniques must be mathematically legitimate even though we
cannot prove it at the moment.

Fig. 2: Layers of logical dependencies.

###  1.3 Dyson series

Quantum Electrodynamics (QED) is considered the most accurate theory in the
history of science. However, this impressive precision is limited to a single
experimental value: the anomalous magnetic moment of the electron (g-factor).

According to Quantum Electrodynamics (QED), the theoretical value of the
electron g-factor is obtained by calculating the coefficients of a number
series called the Dyson series [4]. Each coefficient in the series requires
the calculation of an increasing number of Feynman diagrams.

$g=C_{1}\left(\frac{\alpha}{\pi}\right)+C_{2}\left(\frac{\alpha}{\pi}\right)^{2}+C_{3}\left(\frac{\alpha}{\pi}\right)^{3}+C_{4}\left(\frac{\alpha}{\pi}\right)^{4}+C_{5}\left(\frac{\alpha}{\pi}\right)^{5}...$
(4)

The first coefficient in the Dyson series is the Schwinger factor and has an
exact value of 0.5. The second coefficient was initially calculated in 1950 by
Karplus and Kroll [5],and it was corrected in 1957 by Petermann [6], who
obtained a result of -0.328. The rest of the coefficients in the Dyson series
were calculated many decades later with the help of supercomputers.

$g=1+\frac{1}{2}\left(\frac{\alpha}{\pi}\right)-0,328\left(\frac{\alpha}{\pi}\right)^{2}=1,0011596$
(5)

This result of the $C_{2}$ coefficient of the Dyson series (fourth-order
coefficient) was decisive for the acceptance of the renormalization techniques
proposed by Feynman, Schwinger, and Tomonaga, who received the Nobel Prize in
1965 for the development of QED. It can therefore be considered the most
relevant theoretical calculation in modern physics.

###  1.4 Feynman Diagram IIc

The error in the calculation of $C_{2}$ discovered by Petermann was found in
the calculation of the Feynman diagram IIc.

Fig. 3: 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.

[Karplus & Kroll]

$II_{c}=-\frac{323}{24}+\frac{31}{9}\pi^{2}-\frac{49}{6}\pi^{2}ln(2)+\frac{107}{4}\zeta(3)=-3.178$
(6)

[Petermann]

$II_{c}=-\frac{67}{24}+\frac{1}{18}\pi^{2}+\frac{1}{3}\pi^{2}ln(2)-\frac{1}{2}\zeta(3)=-0.564$
(7)

However, hard to believe, neither the original calculation carried out in 1950
by Karplus and Kroll nor the subsequent correction of Petermann was ever
published. Therefore, the entire legitimacy of the Standard Model and the QED
depends on the calculation of a single Feynman diagram (IIc) that has never
been published and cannot be independently verified. [11]

##  2 Searching for the missing calculation

###  2.1 Barbieri & Remiddi

At this point, we set out on a mission to find the missing calculus of the
Feynman diagram IIc. We assume that given the seriousness of the situation,
someone must have recalculated previously this Feynman diagram and published
it years ago.

After a long search we believe we found the paper we were looking for. It is a
paper published in 1972 and written by Remiddi among other authors [8].
Remiddi is one of the most prestigious researchers in the calculation of the
electron g-factor because in 1996 he published the definitive analytical value
of the $C_{3}$ coefficient (sixth-order coefficient).

The paper is a long 93-page document entitled "Electron form factors up to
fourth order". It was published in 1972 by Barbieri and Remiddi. According to
the authors: “This paper is devoted to the analytic evaluation of the two form
factors of the electromagnetic vertex of the electron in quantum
electrodynamics, up to fourth order of the perturbative expansion (…)
[Calculation] of the fourth-order form factors can also be found in the
literature. They are the famous fourth-order anomalous magnetic moment
evaluated by Petermann and Sommerfield (…). Such values are obviously
reproduced in this paper. (…) Calculations are done in the framework of the
usual Feynman-graph expansion of the S-matrix in the interaction
representation, using the Feynman gauge for the photon propagator. The
relevant graphs for second-order and fourth-order radiative corrections are
shown (..). The approach is dispersive, and the discontinuities of the various
Feynman graphs are obtained by means of the Cutkosky rules”. [8]

From this introduction we understand that Barbieri and Remiddi performed a
recalculation of the Feynman diagrams corresponding to the Fourth-order
coefficient ($C_{2}$) and they confirmed the results obtained by Petermann.

Fig. 4: $C_{2}$ Feynman diagrams.

The authors identify the Feynman IIc diagram as the "c" diagram in Figure 4,
divide it into two symmetrical diagrams and give it the descriptive name of
"Corner Graphs". The result shown in the paper is identical to that published
by Petermann in 1957.

In the 93 pages of the paper, the authors describe several of the techniques
they have used to renormalize the divergences that appear in the calculations
and how they have overcome the problems they have encountered. On the specific
calculations, the authors state the following: “Once these problems are
mastered, a very long and complicated algebra is also needed to do in practice
the calculation. Fortunately, the major part of it, like traces,
straightforward algebraic manipulations, book-keeping of analytic formulae,
integrations by parts, differentiations and so on, was done by computer, using
the program SCHOONSCHIP of VELTMAN, without whose continuous and determinant
help the present work could hardly have been accomplished.” [8]

That is, they used a computer program to perform the mathematical
calculations, but they did not publish the code of the programs used, so,
again, it is not possible to replicate the calculations.

Considering the date of the paper (1972) it is quite plausible to assume that
there are no more calculations, since it was considered unnecessary to carry
out more checks of the $C_{2}$ coefficient. Fortunately, in the paper itself
the authors identify two other independent calculations of the $C_{2}$
coefficient. One published in 1960 by Smrz and Uleha and the other published
in 1962 by Terentiev.

###  2.2 Smrz & Uleha

We obtain the paper published in 1960 by these two Czech researchers [9]. It
is a short paper of 2 pages where the situation generated in 1957 by
Petermann’s correction is explained. The paper indicates that the difference
between the original Feynman IIc diagram calculation of Karplus and Kroll with
respect to the one performed by Petermann is excessive. The authors states
that they performed an independent calculation of the Feynman IIc diagram and
obtained exactly the same result as Petermann.

“Since the considerable difference between the original value of the magnetic
moment (Karplus & Kroll [5]) and the values calculated later (Petermann [6])
originates in the calculation of the contribution from the third diagram, only
the value of this contribution was determined by the standard technique and
the above regularization in the infra-red region. The contribution from the
third diagram (-0.564) is in complete agreement with Petermann’s value.” [9]

Unfortunately, when searching for the reference of the work we note that it
has not been published either: "Smrz P.: Diploma thesis, Faculty of Tech. and
Nucl. Physics, Prague 1960, unpublished."

Just another unpublished paper claiming to have calculated the Feynman IIc
diagram but with no one to review it.

###  2.3 Terentiev

We obtained a copy of the paper published by Terentiev in 1962. The paper
contains about 50 pages [10]. The paper is only in Russian and there is no
English translation. We identify the equation "60" of the paper as the $C_{2}$
coefficient of the Dyson series, with the same expression and value obtained
by Petermann and Sommerfield.

$g=1+\frac{1}{2}\left(\frac{\alpha}{\pi}\right)-0,328\left(\frac{\alpha}{\pi}\right)^{2}=1,0011596$
(8)

Analyzing the document, we interpret that this equation is the result of the
sum of nine other equations identified as equations 22, 24, 27, 31, 33, 47,
51, 58 and 59. There are nine equations instead of the five Feynman diagrams
of Karplus and Kroll and none of the these equations correspond to the Feynman
Diagram IIc.

However, it is not necessary to carry out a more in-depth analysis of the
paper. On the first page of the Barbieri-Remiddi paper we can read a reference
to Terentiev’s paper: “Actually, dispersion relations are used in the
Terentiev work only to write down suitable multiple integral representations,
which are in general manipulated to get the final result, without explicitly
evaluating the discontinuities. The problem of infra-red divergences has been
further overlooked, and many of the intermediate results are wrong, even if
somewhat ad hoc compensations make the final result correct.”

Year | Author |  Status
---|---|---
1950 | Karplus & Kroll |  Wrong and Unpublished
1957 | Petermann |  Right but Unpublished
1957 | Sommerfield |  Right but using Green Functions instead of Feynman Diagrams
1960 | Smrz & Uleha |  Right but Unpublished
1962 | Terentiev |  Wrong intermediate results with ad hoc compensations to make the final result correct
1972 | Remiddi |  Right but Unpublished Computer calculation
Table 1:  fourth-order coefficient calculation.

##  3 Summary

Incredible as it may seem to believe, the most important calculation in the
history of modern physics was published in 1950 by Karplus and Kroll and
turned out to be completely incorrect. The error was not detected until 7
years later by Petermann and Sommerfield. Neither the original calculation nor
the subsequent correction was ever published. Therefore, the entire legitimacy
of the Standard Model and the QED depends on the calculation of a single
Feynman diagram (IIc) that has never been published and cannot be
independently verified.

In this paper we have detected three other published recalculations of the
fourth-order coefficient of the g-factor. The detailed calculations of two of
them were also not published (Barbieri-Remiddi and Smrz-Uleha). In the third
calculation performed by Terentiev, serious errors were detected 10 years
after the original publication. Erroneous intermediate results manipulated
with ad hoc compensations to obtain the correct final result.

Our search has been extensive, so we believe that there are no other published
calculations of the Feynman IIc diagram. The only line of investigation that
remains open would be to find the source code of the computer programs that
are currently used to carry out this type of calculation.

1 September 2021

## References

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  * 5. Karplus R., Kroll N. Fourth-order corrections in quantum electrodynamics and the magnetic moment of the electron. Phys. Rev., 1950. v. 77(4), 536–549.
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  * 13. Aoyama T., Kinoshita T., Nio M. Revised and improved value of the QED tenth-order electron anomalous magnetic moment Phys. Rev. D., 2018. v. 97(3), 036001.
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