Preprint[pdf] - HU Berlin

The primary determination of the electron and muon masses come from measuring the ratio with respect
to the mass of a nucleus and the masses are obtained in atomic mass units (amu). Therefore the ratios
are known more precisely, than the numbers we get by inserting lepton masses given in MeV. In fact, the
conversion factor to MeV is more uncertain than the mass of the electron and muon in amu.
Note that the mass–dependent contributions in fact differ for ae, aµ and aτ, such that lepton universality
is broken: ae �= aµ �= aτ.
More SM parameters will be needed for the evaluation of weak and hadronic contributions. We have
collected them in Appendix A together with known polylogarithmic functions needed for the representation
of analytic results of the QED calculations.
Until recently the electron anomaly ae and, until before the advent of the Brookhaven muon g−2 measurements,
also aµ were considered to provide the most clean and precise tests of QED. In fact the by far largest
contribution to the anomalous magnetic moment is of pure QED origin, and with the new determination of
ae by the Harvard electron g−2 experiment [105,106] ae together with its QED prediction [108] allows for the
most precise determination of the electromagnetic fine structure constant. The dominance of just one type of
interaction in the electromagnetic vertex of the leptons, historically, was very important for the development
of QFT and QED, as it allowed one to test QED as a model theory under simple unambiguous conditions.
How important such experimental tests were we may learn from the fact that it took about 20 years from
the invention of QED (Dirac 1928 [ge = 2]) until the first reliable results could be established (Schwinger
1948 [a (2)
e = α/2π]) after a covariant formulation and renormalization was understood and settled in its
main aspects.
When the precision of experiments improved, the QED part by itself became a big challenge for theorists,
because higher order corrections are sizable, and as the order of perturbation theory increases, the complexity
of the calculations grows dramatically. Thus experimental tests were able to check QED up to 7 digits in the
prediction which requires one to evaluate the perturbation expansion up to 5 terms (5 loops). The anomalous
magnetic moment as a dimensionless quantity exhibits contributions which are just numbers expanded in
powers of α, what one would get in QED with just one species of lepton, and contributions depending
on the mass ratios if different leptons come into play. Thus taking into account all three leptons we obtain
functions of the ratios of the lepton masses me, mµ and mτ. Considering aµ, we can cast it into the following
form [109,24]
a QED
µ = A1 + A2(mµ/me) + A2(mµ/mτ) + A3(mµ/me, mµ/mτ) . (45)
Here A1 denotes the universal term common for all leptons. It also includes those diagrams with closed
lepton loops that have the same mass as the external lepton. The term A2 depends on one scale and
gets contributions from diagrams with closed fermion loops where the fermion differs from the external
one. Such contributions start at the two loop level: for the muon as the external lepton we have two
possibilities: an additional electron–loop (light–in–heavy) A2(mµ/me) or an additional τ–loop (heavy–in–
light) A2(mµ/mτ) two contributions of quite different character. The first produces large logarithms ∝
ln(mµ/me) 2 and accordingly large effects while the second, because of the decoupling of heavy particles in
QED like theories 6 , produces only small effects of order ∝ (mµ/mτ) 2 . The two–scale contribution requires
a light as well as a heavy extra loop and hence starts at the three loop order. We will discuss the different
types of contribution in the following. Each of the terms is given in renormalized perturbation theory by an
appropriate expansion in α:
A1 = A (2)
1
�
α
�
π
+ A (4)
1
A2 = A (4)
2
�
α
�π α
π
� 2
� 2
+ A (6)
1
+ A (6)
2
A3 = A (6)
3
�
α
�π α
�π α
π
� 3
� 3
� 3
+ A (8)
1
+ A (8)
2
+ A (8)
3
�
α
�π α
�π α
π
� 4
� 4
� 4
+ A (10)
1
+ A (10)
2
+ A (10)
3
6 The Appelquist-Carrazone decoupling–theorem [110] implies that in theories like QED or QCD, where couplings and masses
are independent parameters of the Lagrangian, a heavy particle of mass M decouples from physics at lower scales E0 as E0/M
for M → ∞.
20
�
α
�π α
�π α
π
� 5
� 5
� 5
+ · · ·
+ · · ·
+ · · ·