All the lepton masses from G, pi, e

[arivero]

http://prola.aps.org/abstract/PRD/v17/i7/p1854_1
$$m_e / m_\mu=N \alpha / \pi \sin^2 \theta_W$$ where N is a pure number of order 1 which depends on the specific model

 

[Hans de Vries]

http://prola.aps.org/abstract/PRD/v17/i7/p1854_1
$$m_e / m_\mu=N \alpha / \pi \sin^2 \theta_W$$ where N is a pure number of order 1 which depends on the specific model

"Calculating the electron mass in terms of measured quantities"
December 1977 Barr, S. M.; and Zee, A.

is this the A.Zee as the A.Zee in "QFT in a Nutshell"?

I don't have access to APS unfortunately.

Regards, Hans.

 

[arivero]

is this the A.Zee as the A.Zee in "QFT in a Nutshell"?

I guess so. But from a glance to preprints and bibliography, it seems that the model builder is Barr.

In any case, it is just one of the older (pre-GUT, practically) articles on extended symmetries, and its value here is only to point out that a good bunch group theoretical models contain predictions for mass quotients.

 

[arivero]

Let me add that a possible next term is 1/2 (mu/W)^2
$$\ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} + \ \ \frac12 ({m_\mu \over m_W})^2 \ \ = \ \ .00116590899$$
to be compared with current experimental value .0011659208(6)

This travel to madness goes from surprise to surprise. Considering that in this sum we had first order term for the anomalous moment of electron, then a first order term for the difference between an. m. of electron and muon, and then a second order term for the an. m. of electron; thus, it seems, it is natural to look for another second order term. I have been trying terms with $$m_e^2$$ with no success. But, to my astonishment, it is possible to use terms on $$m_e m_\tau$$!

(examples: mm/mz+me/mw+.5*(mm*mm/mw/mw)-.5*(me*mt/mz/mw) = .00116584708, the pure QED value. Or mm/mz+me/mw+(1/2)*(mm^2/mw^2)+(1/6)*(me*mt/mz^2)= .0011659272,
nearer to the experimental value than the uncorrected term)

A minor problem with all the second order terms is that it is not clear when to use W and when to use Z; the election at so small corrections is almost an aesthetic issue.

Also other adjustments are possible if we consider separately the difference between moments and the electron moment.

 

[Hans de Vries]

A minor problem with all the second order terms is that it is not clear when to use W and when to use Z; the election at so small corrections is almost an aesthetic issue.

I see, It's the m_Z in the first term which ultimately determines the precision.

Regards, Hans

 

[arivero]

I see, It's the m_Z in the first term which ultimately determines the precision.

Yep Hans. Even if your angular momenta devices could have a role somewhere. Your hat is always surprising.

Let me -marcus style - restate the equations as a pair. First, our experimental input is
mw=80525 (+-38)
mz=91187.6 (+-2.1)
me=0.51099892 (+-0.00000004)
mm=105.658369 (+-0.000009)
mt=1776.99(+0.29-0.26)
ae=0.001159652187 (+-0.000000000004) (ie +-4 10^-12)
am=0.0011659208 (6)

The pdg 2004 has 0.0011659203 (+-0.0000000007), surely the previous run of the g-2 experiment.

Lets take separately ae, and the difference d=am-ae=.0000062686; so at first order our comparisions are
0,00115869=mm/mz
0,001159652187=ae
with a quotient 0.99917
and
0,000006345=me/mw
0,0000062686=d
quotient 1.01219

Note I am shortening decimal precision to follow approximately the experimntal precision of W and Z measuremente. Wel, now let's enter our second order corrections:

(1/2) (mm/mw)^2 =.000000861 is to be added to the first comparation, s that now we have

.00115955=mm/mz+(1/2) (mm/mw)^2
.001159652187=ae

and now put for instance (1/2) (me*mt/mw^2)=.00000007001 to be substracted, so that we compare

.000006275=me/mw - (1/2) (me*mt/mw^2)
.0000062686=d=am-ae

The quotients are respectively .999911 and 1.00102. It is possible to reverse the pair and to use the experimental anomalous magnetic moments to calculate mass of W and mass of Z.

The total sum .001165825 fails the experimental point 0.0011659208 (6) by a quotient .999918, ie less than 0.01%. This total formula, although, is already a bit ugly:
$$a_\mu= {m_\mu \over m_Z}+ {m_e \over m_W}+ \frac12 {m_\mu^2-m_e m_\tau \over m_W^2}=.001165825$$

 

[arivero]

On the other hand, the theoretical electroweak value (some quark loops?, but no hadronic correction) is .0011658487 and the pure QED value is 0011658471. The corresponding quotients are .999979 and .999981

For the first order approximation, the quotient was in any case (including against the experimental) about .9993, which was already less than 0.1%

 

[arivero]

What about the anomalous magnetic momentum of the tau lepton? Well, it is not a measured quantity, but at least the QED value has been calculated by Samuel, Li and Mendel to be 0.0011732 (a previous calculation by Narison, giving 0.0011696, was retired by its author, in favour of the Samuel et al. result). The electroweak contribution, scaled from the muon calculation, should increase the value up to 0.0011737. This info comes from Phys Rev Letters v67 p 668 and a further erratum at v 69 p 995.

To look for a "first order" correction, we should find a <0.1% match for the QED differences $$a_\tau-a_\mu=.0000073$$ or $$a_\tau-a_e=.0000135$$, or to the corresponding EW values, .0000005 higher.

I will not try to land on these values using Hans' generation mechanism. Instead, let me see if we can do something from phenomenological values. A first inspection shows us that any quotient using the mass of muon, or using the mass of tau, will ask for a denominator mass fairly above 10 TeV. So we are left with the electron mass.

Now, to fit the electron mass to values .0000073 .0000078 .0000135 .0000140, we need respective masses of 69.9, 65.5, 37.8, 36.4 GeV. Looking at the catalogue of forgotten experimental deviations, we had a mass around 40-45 GeV back in 1984 (Nature 12 July 1984, p. 310) and, hey, a mass around 68-70 GeV in 1999 from the L3 collaboration: hep-ex/9909044, hep-ex/0009010, hep-ex/0105057. Nowadays, the statistics of this L3 excess has been reevaluated so the final deviation is under three sigmas; but it could be said that it joins the 115 GeV event in waiting further clarification at LHC.

The putative assignment for the 68 GeV particle was a charged scalar. If we have also faith in the 115 GeV range for the Higgs, we can give -speculate- a prediction:

$$a_\tau= {m_\mu \over m_Z}+ {m_e \over m_{W^+}}+{m_e \over m_{H_{L3}^+}} +{m_\mu^2-m_e m_\tau \over m^2_H_0}$$

 

[arivero]

references

hep-ph/9810512 is already an up-to-date source for the tau data, and besides it has an interesting remarck for the muon: "In models where the muon mass is generated by quantum loops... under very general asumptions, the induced \delta a_\mu is given by $$\delta a_\mu= C m_\mu^2 / \Lambda^2$$ where Lambda is the scale of 'New Physics' responsible for generating m_\mu". (C is a constant of order unity)

So our square terms are not so rare, after all.

According Brodsky et al. hep-ph/0406325, we are meeting a challenge from Feynman in the 12th Solvay conference, where he asked: "Is there any method of computing the anomalous moment of the electron which, on first approximation, gives a fair approximation to the \alpha term and a crude one to \alpha^2; and when improved, increases the accuracy of the \alpha^2 term, yielding a rough estimate to \alpha^3 and beyond"

 

[Kea]

question re formula

$$A = 1+{\alpha \over (2\pi)^0 }+{\alpha^2 \over (2\pi)^1 }+{\alpha^3 \over (2\pi)^3 } + ...$$

The series converges straightforward to reproduce the value
of the fine structure constant exact in all its digits:

after term 0: 0.0071918833558268
after term 1: 0.0072972279174862
after term 2: 0.0072973525456204
after term 3: 0.0072973525686533
f.s. constant 0.007297352568.(+/-24)

This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?

Cheers
Kea

 

[arivero]

This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?

(Just to fix attributions, and lacking Bourbaki here, the trunc. formula is also from Hans). I supposse the idea was to enter inside the 1sigma level, +-24, as a proof of concept. 568-24 is 544, so the previous iteration is only in the border. It has sense if you think that nowadays a lot of people translates sigmas to "Confidence Levels".

 

[arivero]

I am also asking myself what happens with alpha. More exactly, why our mass-based formulae have lost any reference to the coupling constant alpha, while Schwinger first estimate was just alpha/2pi.

But even if coincidental or unrelated (or perhaps related!), it could be time to name the recent numerology (well, halfbaked theory) of Jay R. Yablon, ...

Lets keep calm. The whole Web Page.htm]webpage of Yablon[/url] is about a method to get the masses of the three leptons as a perturbative expansion in alpha. Really a methodology of this kind could explain our results: For some reason, the alpha expansion generating lepton mass comes to coincide with the expansion of the vertex diagram, and we can absorb Schwinger-like terms (or whole series of them!) into lepton masses.

From the seventies, there is a whole market of theories having perturbative lepton masses, usually via the so called "horizontal symmetries". Here we are looking for a theory based on expansion of the vertex, using the symmetry breaking scale of the electroweak group, and having no new coupling constants. Pity there is not a catalogue of "beyond SM theories".

 

[Hans de Vries]

A bit about Jay's numerical series.

Jay, in his enthusiasm, is bringing this out in the open at the moment he
discovers something. Leaving the reflections for later. If he plays a little bit
more with this then he will find out how careful one must be with numerical
coincidences.

At one hand it makes sense in a QFT world to develop a series based on a
coupling constant. On the other hand it's always possible to transform a decimal
number to any other number system. For instance a number system based on
1/e = 11.72... instead of 10.

If one needs 5 terms to approximate the 0.511.. MeV mass of the electron
to 0.509 MeV then there's no more "numerical coincidence" value left in it.
Jay will find out that there are many other combinations that will lead to
equal or better results.


Still the approach makes sense and the most intriguing hint may be the anomalies.
To bring them up one more time:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly


Regards, Hans.

 

[Hans de Vries]

This actually agrees after term 2 already, no? Why not develop Alejandro's truncated formula then?

Cheers
Kea

Hi Kea,

What I looked for was a series with a single generating principle. In a sense
such a series is simpler than a formula with three terms.

Regards, Hans

 

[Hans de Vries]

Hi, Alejandro

first this:

Mass = inversely proportional to Square Root of the Giro Magnetic Ratio

Simply from the classical magnetic moment: $$\mu \ = \ I A \ = \ e f_0 \pi r^2$$ where:
I is the current, A the area enclosed by the orbit, e the charge and for f₀ we
take the rest-frequency of the particle. Masses are inversely proportional
to size so the r² factor leads to the inverse square root dependency. We
get fractions which are about half the size:

g_e = 1.001159652187 --> 1/√g_e = 0.9994206777
g_μ = 1.0011659208__ --> 1/√g_μ = 0.9994175489


Now Look at this:

$$\begin{array}{lcl} 1 - \frac{1}{2} \frac{m_{\mu}}{m_W} & = & 0.999420654 \\ 1/\sqrt{g_e} & = & 0.999420678 \\ \mbox{the absolute error} & = & 0.000000024 \\ \end{array}$$


$$\begin{array}{lcl} 1-\frac{1}{2} \frac{m_{\mu}}{m_W}-\frac{1}{2} \frac{m_e}{m_Z} & = & 0.999417477 \\ 1/\sqrt{g_{\mu}} & = & 0.999417549 \\ \mbox{the absolute error} & = & 0.000000072 \\ \end{array}$$


The absolute error becomes 40 times smaller for the electron case!


Regards, Hans

 

[Jay R. Yablon]

If one needs 5 terms to approximate the 0.511.. MeV mass of the electron
to 0.509 MeV then there's no more "numerical coincidence" value left in it.
Jay will find out that there are many other combinations that will lead to
equal or better results.

Hans, this is Jay. I am not fixed for sure on the 5 terms for the electron, but what is very nice about this result is that the electron mass is then characterized in leading order by only four-loop terms and that all five of the four loop terms go into the electron mass. So, the fact that every one of these terms has a common Feynman diagram interpretation as a four-loop term seems to be based on some physics beyond coincidence and beyond picking and choosing terms

Still the approach makes sense and the most intriguing hint may be the anomalies.
To bring them up one more time:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly


Our friend arivero has now got me looking at these anomalies using the terms I have developed. I will let you know what I find.

Jay

 

[arivero]

A small corrected improvement. While mm/mZ is 99.76% of the Schwinger correction, the following approximation
$${m_\mu\over m_Z}={\alpha \over 2 \pi} - 0.5 ({\alpha\over\pi})^2$$
is 99.9982% accurate

Still I hope the 0.5 comes from addition of multiple Feynman diagrams, then being approximate itself. I am afraid than an exact -1/2 coefficient in second order QED comes not from a finite term, but associated to the infrared divergence.

 

[arivero]

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order $\alpha/2\pi$ term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons.

Hmm but at second order of alpha, there is a vacuum polarisation term due to pairs electron positron, and it is not negligible. It is $$0.015687 \alpha^2/\pi^2$$, ie .0000000846.

Er, wait... let's supposse this term is not in your ratio. Add it!
0.0011586922+.0000000846=.0011587768

Ok it does not seem very much of an improvement. But if we add also my term (1/2) (mm/mW)^2 we have
0.0011586922+.0000000846+.0000008608=.0011596376
to be compared with experimental 0.0011596521. No bad.

I would conclude that our series (well, two terms) on masses does not approach to the experimental magnetic moment, but to the two loop QED (or full electroweak, does not matter) vertex correction, excluding the vacuum polarisation loop. On first examination, it seems that this loop is recovered when we introduce the product of electron and tau masses, but I have not examined the expansion for this anomalous moment.

Just for the record, the two loop correction for electron is 0.5 a/pi - 0.3284794 (a/pi)^2. In our case, disregarding the vacuum polarisation amounts to a second term coefficient -.3441636 We have
$$a_e^{\mbox{2qed-v.p.}}=.00115955280$$
$${m_\mu \over m_Z}+ \frac12 {m_\mu^2 \over m_W^2}=.001159553$$
agreement about, well, dammn, it is already inside the experimental error for Z0... if you want, respective to central values it is of Z and W, it is 99.99997%.

If one feels bad about having the 2 m_W^2 in the denominator, you can always use the square of an unknown neutral mass about 114 GeV... the LHC start will wait some time yet.

 

[Hans de Vries]

Hmm but at second order of alpha, there is a vacuum polarisation term due to pairs electron positron, and it is not negligible. It is $$0.015687 \alpha^2/\pi^2$$, ie .0000000846.

That's just the value you want it to have! Do you have you any more
data like that. Like the explicit value of the same term in the Muon?
Really interesting would also be the relation with the muon/electron
mass with these terms in the formula's (which are known analytically.
I suppose that we can ignore the same second order terms for virtual
muon-anti muon pairs for the time being.


0.000 006 263 813 : Difference between electron and muon anomaly
0.000 000 084 639 : First vacuum polarization term of the electron
----------------------------------------------------------------------
0.000 006 348 452 : Sum of the vacuum polarization terms (the above)

0.000 006 353 732 : (+/-3 000) mass ratio of electron and the W boson.
0.000 000 002 998 : uncertainty from the Z mass


Regards, Hans.

 

[arivero]

That was the point. We have hit experimental precision in both expressions.

Sunday night I mailed you a separate email, the bibliography tells where to find such terms. You would not like me to write the closed expression for the "vacuum polarised" terms of the muon, it is about four lines in the Phys Rev. Moreover, the next order terms are also relevant.

Alejandro

 

[arivero]

0.000 006 263 813 : Difference between electron and muon anomaly
0.000 000 084 639 : First vacuum polarization term of the electron
----------------------------------------------------------------------
0.000 006 348 452 : Sum of the vacuum polarization terms (the above)

0.000 006 353 732 : (+/-3 000) mass ratio of electron and the W boson.
0.000 000 002 998 : uncertainty from the Z mass

Hmm, I had tried the same with the data truncated at two loops for electron and three for the muon and I got a somehow weaker result. It seems one needs to add the three loop data for the electron, but it is a mess because then we have three new diagrams to exclude.

 

[Hans de Vries]

The other interesting observation is that the missing term
becomes equal in both cases to within experimental value.


0.00115877693 : mu/mZ + VP2
0.00115965218 : electron anomaly
0.00000087525 : missing

0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2)
0.0011659208_ : muon anomaly
0.00000087478 : missing

0.00000000047 : missing1 - missing2
0.00000002668 : uncertainty due to Z (cancels if missings are subtracted)
0.00000000299 : uncertainty due to W mass


So there may be a single missing term.


Regards, Hans

PS: VP2 = 0.00000008464 : First vacuum polarization term of the electron
which is a second order term.

PPS: I'm not entirely sure if the term 0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2)
should indeed not include VP2.

me/mW - VP2 = difference between muon and electron anomaly.
mu/mZ + VP2 = all self energy terms + first vacuum polarization term of the electron anomaly

 

[arivero]

Making it in a reverse way: as the missing term is already in the electron anomalous moment, assume it is sort of square of the first term. So it is mu^2/X^2. Solve for X:
X=sqrt(mu^2/.000000875015)=112.95 GeV

The quantity is interesting in two ways. We suspect of a neutral scalar H0 at 115 GeV, and it could have this role. But also mw*sqrt(2) is 113.87 GeV, so we can use the W particle, which was my first attempt for the missing term. Lacking of more theory, both are equally suitable: values up to 114.5 GeV are covered by the Z indeterminacy. The first has the advantage of not using an arbitrary 1/2 coefficient and it is neutral as the Z, but it has not been discovered (yet?), the second is an already discovered particle but we have used it for the "vacuum polarised graphs", and it is surprising to have it here too, even if squared.

 

[arivero]

References

For the sake of completeness, references. The industry of calculation of the anomalous moment seems to be based in Cornell, around a veteran named T. Kinoshi-ta. Other group does exist in North America around A. Czarnecki

http://arxiv.org/abs/hep-ph/9810512, from Czarnecki and Marciano, is the main entry point for the calculation up to order alpha^4. It is regretly a short preprint and it does not separate loop by loop, so one is referred to more detailed bibliography, which is not in the arxiv

The five diagrams for order alpha^2 appear well separated in Levine and Wright Phys. Rev. D 8, 3171-3179 (1973) http://prola.aps.org/abstract/PRD/v8/i9/p3171_1. I got from here the specific value we were using above.

Also some sums for 40 diagrams of order alpha^3 are presented there. Note that of these, 12 diagrams are vacuum polarisation loops, amounting perhaps to a contribution of 0.37 (alpha/pi)^3

The alpha^2 "polarisation loop", depending of the mass quotient of the external and internal lepton, is studied both analytic and numerically by Li Mendel and Samuel, Phys. Rev. D 47, 1723-1725 (1993) http://prola.aps.org/abstract/PRD/v47/i4/p1723_1

Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.

 

[Jay R. Yablon]

Do you habe any ideas about the tau lepton?

Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.

Alejandro"

Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?

Jay.

 

[arivero]

Alejandro"
Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?

Hmm the answer past yesterday was yes, the answer today is more towards no. On one side the quantity $$m_e m_\tau/m_W^2$$ in of the right order of magnitude to do further corrections in our calculations, but we do not need it anymore, giving the uncertainty in the mass of Z. On other hand, and more concretely answering your question, the difference between anomalous moment of mu and tau can only be covered with a new quotient $$m_e/m_{X^+}$$, and the mass of the new X+ particle should be around 68-70 GeV. At these energy range, the LEP2 presented an slight "statistical" deviation, but no particle

 

[arivero]

mw=80525 (+-38)

80425

So I was getting a discrepance between calculations at home and calculations at work.

 

[Jay R. Yablon]

Mass Term Gordon-Like Magnetic Moment Decomposition

Dear Alejandro and Hans:

I just posted to my website http://home.nycap.rr.com/jry/FermionMass.htm , a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft.

I hope this can help you in your efforts by providing a covariant field theory context for your efforts to characterize the magnetic moments. I know that your efforts have helped me recognize that consideration of magnetic moments is likely to be a critical aspect of what I am attempting to do.



Jay.

 

[arivero]

I just posted to my website http://home.nycap.rr.com/jry/FermionMass.htm , a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft.

Probably Hans is also exploring this way, but I am not so optimistic about a direct connection; perhaps a semiclassical effect, could be. But even that is strange to manage. To me, it seems more as if the symmetry breaking mechanism of the electroweak group (and its vacuum value) were needing of the lepton masses is some misterious way.

From our quadratic formulae we can get rather intriguing equations. For instance this one:

$${m_\tau\over m_Z} + {m_\mu\over m_W}= {m_\tau\over m_\mu} a_\mu^{s.e.} + {m_\mu\over m_e} a_\mu^{v.p.}$$

Where $$a_\mu^{s.e.},a_\mu^{v.p.}$$ are the self-energy and vacuum polarisation parts of the muon anomalous magnetic moment; note that the v.p. part depends internally of lepton mass quotients, while the s.e. is mass independent, in QED (in the full electroweak theory new dependences appear).

 

[arivero]

Some of the development of the thread uploaded at http://arxiv.org/abs/hep-ph/0503104

 

[Hans de Vries]

Hmm,

I was doing something else and ran just incidently into this one:

$$\sqrt{ \ 2 \ \frac{m_V}{m_Z} \ \frac{m_{\tau}}{m_e}} \ = \ 137.038 (12)$$

m_V is the vacuum expectation value of 246.22046 GeV (according to Jay)
The biggest uncertainty is from the tau mass.


Regards, Hans


$\ \ \alpha \ \ \$ = 1/137.03599911
m_V = 246220.46
m_Z = 91187.6 (+-2.1)
m_τ = 1776.99(+0.29-0.26)
m_e = 0.51099892 (+-0.00000004)

 

[arivero]

vacuum expectation value of 246.22046 GeV (according to Jay)
The biggest uncertainty is from the tau mass.

Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.

 

[Hans de Vries]

Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.

It's basically "one over the square of" : The Fermi Coupling constant 1.16637 (1) times sqrt(2)

So it should be 246.2206 (11)

Regards, Hans.

 

[arivero]

That's just the value you want it to have! Do you have you any more data like that.

Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of
http://prola.aps.org/abstract/PR/v95/i5/p1300_1
which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS".

The authors are a M.Gell-Mann and a F.E.Low, from Illinois.

 

[Hans de Vries]

Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of
http://prola.aps.org/abstract/PR/v95/i5/p1300_1
which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS".

The authors are a M.Gell-Mann and a F.E.Low, from Illinois.

There are related follow-ups:

Quantum Electrodynamics at Small Distances,
Baker, Johnson, 1969.
http://prola.aps.org/abstract/PR/v183/i5/p1292_1


Quantum Electrodynamics Without Photon Self-Energy Parts
S. L. Adler and W. A. Bardeen 1971
http://prola.aps.org/abstract/PRD/v4/i10/p3045_1


Regards, Hans