What can be subtracted from the loop terms $\Pi_{LOOP}^*(q^2)$ in 10.3 p.441 of Weinberg's QFT book?

Question

I'm studying the basics of renormalization, and I want to know the nature of renormalization.

In 10.3 p.441 of Weinberg's QFT books,it states that

In actual calculations it is simplest just to say that the loop terms $\Pi_{LOOP}^*(q^2)$ we must subtract a first-order polynomial in $q^2$ with coefficients chosen so that  the difference satisfies Eqs. (10.3.17) and (10.3.18). As we shall see, this subtraction procedure incidentally cancels  the infinities that arise form the momentum space integrals in $\Pi^*_{LOOP}$.

However, as this discussion should make clear， the renormalization of masses and fields has nothing to do with the presence of infinities, and would be necessary even in a theory in which all momentum space integrals were convergent.

Eq. 10.3.17:\(\Pi^*(-m^2)=0\)

Eq. 10.3.18:\(\bigg[\frac{d}{dq^2}\Pi^*(q^2)\bigg]_{q^2=-m^2}=0\)

If this is true, the first-order polynomial must be equal to zero. Nothing could be subtracted!

What is wrong? What can be subtracted from the loop terms $\Pi_{LOOP}^*(q^2)$ ?

And another question, what infinities are canceled in this procedure?

Comments

Not sure to understand the question. Equations 10.3.17 and 10.3.18 must be satisfied by $\Pi^*$ so there is no first order polynomial in $\Pi^*$. But it is not the case for $\Pi_{LOOP}$, $\Pi_{LOOP}$ will have a first order  polynomial part and we need to substract it to recover $\Pi^*$ from $\Pi_{LOOP}$.

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@40227

Now I understand a little, the difference means just the $\Pi^*$ but not what is subtracted from $\Pi^*_{LOOP}$. At first \(\Pi_{LOOP}^*(q^2)\) have infinity, then subtracted by other terms it becomes $\Pi^*$, which have no infinity.

Do you agree?

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I think you were confusing $\Pi^*$ with $\Pi^*_\text{LOOP}$, these two differ precisely by a first order polynomial of $q^2$, which is what Weinberg was referring to.