Is Feynman diagram all about S-matrix? This is what I am concluding after reading texts about Feynman diagram: that Feynman diagram is all about finding S-matrix or scattering amplitudes. Is there anything more to Feynman diagram?
 A: While scattering amplitudes (corresponding to terms in the S-matrix) may be computed by Feynman diagrams, there are other quantities of interest that may be computed, and generally they may be represented in terms of correlation functions.

Anomalous Magnetic Moment
The magnetic moment $g$ of an electron determines the torque it experiences in a magnetic field, and so this is characterised by diagrams involving photons and spinors. It turns out the correction,
$$g = 2 + 2F_2(0)$$
is given in terms of a form factor $F_2$, related to scattering amplitudes. At one loop order, it is a famous result by Schwinger that the correction is $\alpha/2\pi$. Further corrections can be computed by considering higher order diagrams.

Interaction Potentials
According to the Born approximation, corrections to an interaction potential $V(r)$ are given by the Fourier transform of the two-point function. As an example, in linearised quantum gravity, we have an action,
$$\mathcal L = \sqrt{-\det(g)} \left( M^2_{\mathrm{pl}} R + L_1 R^2 + L_2 R_{\mu\nu}R^{\mu\nu} + L_3 R_{\mu\nu\sigma \rho}R^{\mu\nu\sigma\rho} + \dots\right)$$
where the higher order curvature terms arise as counter-terms to absorb divergences (as the theory is an effective field theory and non-renormalisable).
In this theory, computing contributions the graviton propagator, one can find corrections to the potential. For example, the potential around the Sun is corrected and given by,
$$h(r) = \frac{M_{\mathrm{sun}}}{M_{\mathrm{pl}}} \frac{1}{r}\left[ 1 -
 \frac{M_{\mathrm{sun}}}{M^2_{\mathrm{pl}} r} - \frac{127}{30\pi^2} \frac{1}{M^2_{\mathrm{pl}} r^2} - 128 \pi^2 \frac{L_1+L_2}{M^2_{\mathrm{pl}}} \delta^{(3)}(\vec{r}) + \dots\right].$$
To find the $L_1$ and $L_2$ one must compute the appropriate diagrams which give rise to divergences that must be absorbed by the $L_1$ and $L_2$ terms respectively.

Anomalies
Gauge anomalies correspond to gauge symmetries which are not preserved after quantisation. One method to show this is a lack of invariance under the appropriate symmetry in the path integral measure. However, naturally, they can also be shown to arise from loop corrections.
In the case of the theory of massless fermions, the chiral anomaly does not preserve the conservation law, $\partial_\mu J^\mu = \partial_\mu J^{\mu 5} = 0$. The correlation function to consider is a triangle diagram, for which we would like to verify,
$$\frac{\partial}{\partial x^\mu} \langle J^{\alpha 5}(x)J^\mu(y)J^\nu(z) \rangle = 0.$$
In terms of the Ward identity, we would like $q^1_\mu M_5^{\alpha\mu\nu} = 0$ for this process. It turns out,
$$q^1_\mu M_5^{\alpha\mu\nu} = \frac{1}{4\pi^2} \epsilon^{\alpha\nu \rho \sigma} q^\rho_1 q^\sigma_2 \neq 0.$$
(There is a deep connection between anomalies and characteristic classes of fibre bundles that I should point out for the interested reader. See also the Fujikawa method for the path integral derivation.)
