Why the Feynman diagram with loops attached to external legs is irrelevant to the $T$-matrix?

Question

Hello friends I was stumbled when I learnt the scattering theory from textbook titled "Quantum Field Theory for the Gifted Amateur", which has related the scattering probability to the matrix elements of S-matrix: $$<p_{i1}p_{i2}|\hat{S}|p_{f1}p_{f2}>$$ where $p_i$ and $p_f$ are the initial moments and final moments after scattering.

And in p.g.192, the author defined the T-matrix as: $\hat{S}=1+i\hat{T}$, and the issue was converted to the evaluation of eq. (20.19): $$<p_{i1}p_{i2}|i\hat{T}|p_{f1}p_{f2}>=(2\pi)^4\delta^{(4)}(p_{i1}+p_{i2}-p_{f1}-p_{f2})iM$$

And it was announced that the computation of T-matrix involves the summation over

All connected, amputated Feynman diagrams with incoming momentum $p_{i}$ and out-going momentum $p_{f}$.

Amputation here means the the diagrams with loops on external legs are irrelevant to the $T$-matrix. I cannot understand why loops on external legs are not permitted. Frankly, I think loops contribute infinitely via terms like $\int dp\Delta(0)$, in which $\Delta$ corresponds to the free propagator.

Answer 1

A proper way to determine S-matrix elements is through the LSZ reduction formula. Take e.g. a theory with one self interacting scalar. First recall that the full two-point function is given by $$ \Pi(p^2) = \frac{i}{p^2 - m_0^2 - \Sigma(p^2)} = \frac{iR}{p^2 - m_P^2} + \text{terms finite as $p^2 \rightarrow m_P^2$}, $$ where $\Sigma(p)$ is the sum over the all 1PI diagrams with two external legs, the two external propagators not included. $m^2$ is defined to be the value of $p^2$ where the $\Pi(p^2)$ diverges, so it is the formal solution of $m_P^2 = m_0^2 + \Sigma(m_P^2)$ and $$ R = \frac{1}{1-\Sigma'(m_P^2)} $$ is the Residue of $\Pi(p^2)$ at this pole.

Now to the LSZ formula, which reads $$ \langle p_1,...,p_n| S |q_1,...,q_m \rangle = \lim_{p_1^2,...,p_n^2,q_1^2,...,q_m^2 \rightarrow m_P^2} \prod_{i=1}^n \Big{[}- \frac{i (p_i^2 - m_P^2)}{R^{1/2}} \Big{]} \prod_{j=1}^m \Big{[}- \frac{i (q_j^2 - m_P^2)}{R^{1/2}} \Big{]} \Gamma^{(n+m)}(p_1,...,p_n;-q_1,...,-q_n), $$ where $\Gamma^{(n+m)}(p_1,...,p_n;-q_1,...,-q_m)$ is the connected momentum space $n+m$ point function with $n$ ingoing and $m$ outgoing particles.
And there you have it: $\Gamma^{(n+m)}$ contains the full propagators at the external legs that is the sum of all diagrams with loop at the external legs. These get cancelled by the formula the other terms vanish when taking the limit. Fermions and Bosons are more complicated but similar.

In the renormalized theory with an on-shell subtraction scheme, where $\phi_0 = Z^{1/2} \phi_R$ and $m_0 = Z_m m_R$, one chooses the counterterms such that $Z = R, m_R = m_P$. Because $\Gamma^{(n+m)} = Z^{(n+m)/2} \Gamma_R^{(n+m)}$ the factors of $R^{1/2}$ in the LSZ formula above formula cancel.

Details can be found in most QFT text books.