Magnetic moment of an electron (LO)

In this section we derive the leading order (LO) order-$\alpha$ correction to the magnetic moment of an electron. This has been first computed in 1948 by Schwinger.

We start by computing the electron vertex function for the process $\gamma(q)\to e^+(p) + e^-(p')$:

where $k$ corresponds to some heavy target. If $A_\mu^{\mathrm{cl}}$ is a fixed classical potential, we get:

Using general arguments (Lorentz invariance, parity-conservation, Ward identity) we can always write $\Gamma^\mu$ as:

where $F_1$ and $F_2$ ar unknown functions of $q^2 = (p'-p)^2 = -2p'\cdot p + 2m^2$ called form factors. As we will see below, in the lowest order we get $F_1 = 1$ and $F_2 = 0$.

We can calculate the amplitude for elastic Coulomb scattering of a nonrelativistic electron from a region of nonzero electrostatic potential by setting $A_\mu^{\mathrm{cl}}(x)=(\phi(\mathbf{x}), 0)$, then:

If the electrostatic field is very slowly varying over a large (even macroscopic) region, $\tilde\phi(\mathbf{q})$ will be concentrated about $\mathbf{q} = 0$, then we can take the limit $\mathbf{q}\to 0$:

This corresponds to the Born approximation for scattering from a potential

Thus $F_1(0)$ is the electric charge of the electron, in units of $e$. Since $F_1(0) = 1$ already in the first order of perturbation theory, radiative corrections to $F_1(q^2)$ must vanish at $q^2=0$.

Now we calculate the scattering from a static vector potential by setting $A_\mu^{\mathrm{cl}}(x)=(0, \mathbf{A}^{\mathrm{cl}}(\mathbf{x}))$, then:

In the limit $q\to 0$ this becomes:

where

is the Fourier transform of the magnetic field produced by $\mathbf{A}^{\mathrm{cl}}(\mathbf{x})$.

This corresponds to the Born approximation for scattering from a potential

where

where

The coefficient $g$ is called the Landé g-factor, and since the leading order of perturbation theory gives $F_2(0)=0$ (and we know that $F_1(0)=1$ to all orders), we get:

This is the standard prediction of the Dirac equation. The anomalous magnetic moment is then:

To calculate that, we need to evaluate the one-loop correction to the vertex function, so we start by deriving the appropriate Green function for the process $\gamma(q) + e^+(p) \to e^+(p')$:

where:

is the interacting Green function for the Lagrangian $-\lambda \bar e \gamma^\mu e A_\mu$. In the first order:

so the amplitude is:

and we got $\Gamma^\mu = \gamma^\mu$, so $F_1=1$ and $F_2=0$ in the lowest order. In the next order we get:

Now we can write:

where

So the expressions for the form factors are:

$F_1$ contains both ultraviolet and infrared divergencies. To cure the infrared divergence, we add a term $\mu^2 z$ to $\Delta$. To cure the ultraviolet divergence, we make the substitution:

where $\delta F_1$ is the first order (in $\alpha$) correction to $F_1$ (i.e. $F_1 = 1 + \delta F_1 + O(\alpha^2)$):

so the corrected $F_1$ is:

Neither the ultraviolet nor the infrared divergence affects $F_2(q^2)$, so we just set $q=0$:

Thus we get the correction to the $g$-factor of the electron:

Code:

>>> from math import pi
>>> alpha = 1/137.035999049
>>> a_e = alpha / (2*pi)
>>> a_e
0.0011614097331824923

Experiments give $a_e = 0.00115965218073\pm 0.00000000000028$ (arXiv:1412.8284, eq. (1)).

Higher order corrections from QED can also be calculated:

we already know that $A_1 = \frac{1}{2}$. See for example hep-ph/9410248 for the expression for $A_2$:

Code:

>>> from sympy import zeta, S, log
>>> A_2 = S(197)/144 + zeta(2)/2 + 3*zeta(3)/4 - 3*zeta(2) * log(2)
>>> A_2.n()
-0.328478965579194

See hep-ph/9602417 for the $A_3$ term:

Code:

>>> from sympy import pi, zeta, S, log, sum, var, oo
>>> var("n")
n
>>> a4 = sum(1/(2**n * n**4), (n, 1, oo))
>>> A_3 = 83*pi**2*zeta(3)/72 - 215*zeta(5)/24 + 100*(a4 + log(2)**4/24 - \
...         pi**2*log(2)**2/24)/3 - \
...         239*pi**4/2160 + 139*zeta(3)/18 - 298 * pi**2 * log(2)/9 + \
...         17101 * pi**2 / 810 + S(28259)/5184
>>> A_3.n()
1.18124145658720

Higher terms are only known numerically. The $A_4$ and $A_5$ terms can be found in arXiv:1412.8284:

We can now sum $a_e$ up to a given order by the following script:

from sympy import pi, zeta, S, log, summation, var, oo
var("n")
a4 = summation(1/(2**n * n**4), (n, 1, oo))
A1 = S(1)/2
A2 = S(197)/144 + zeta(2)/2 + 3*zeta(3)/4 - 3*zeta(2) * log(2)
A3 = 83*pi**2*zeta(3)/72 - 215*zeta(5)/24 + 100*(a4 + log(2)**4/24 - \
        pi**2*log(2)**2/24)/3 - \
        239*pi**4/2160 + 139*zeta(3)/18 - 298 * pi**2 * log(2)/9 + \
        17101 * pi**2 / 810 + S(28259)/5184
A4 = -1.91298
A5 = 7.795
alpha = 1/137.035999049
a_e_exp = 0.00115965218073
a_e_exp_err = 0.00000000000028
a_e_other = 0.00000000000448
A = [A1, A2, A3, A4, A5]
a_e= []
for i in range(len(A)):
    a_e.append((A[i]*(alpha/pi)**(i+1)).n())
print "========== ================"
print "Order      $a_e$"
print "========== ================"
for i in range(len(A)):
    print "%d          %16.14f" % (i+1, sum(a_e[:i+1]))
print "Other      %16.14f" % a_e_other
print "Total      %16.14f" % (sum(a_e) + a_e_other)
print "Experiment %16.14f" % a_e_exp
print "Difference %16.14f" % abs(sum(a_e) + a_e_other - a_e_exp)
print "Exp. err   %16.14f" % a_e_exp_err
print "========== ================"

and obtain the following table:

The "Other" line are contributions from the dependence on the muon and tau particle masses, the hadronic vacuum-polarization, the hadronic light-by-light-scattering and the electroweak contribution (see arXiv:1412.8284). The "Difference" line is the difference from the theory (the "Total" line) and experiment. The "Exp. err" line is the experimental error.

At this level of accuracy, the uncertainty of the exact value of $\alpha$ is the primary cause of the difference from experiment, and one can use this result to predict a more accurate value for $\alpha$, assuming that QED and the standard model are valid.