In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed

$$\int_0^1 dx x^{n-1}f_f^+(x,Q^2)=A_f^n\ .\tag{18.154}$$

The arguments given are based on the contour integral of $W_2$, which is one of the scalar factors of the forward matrix element of two currents in the proton state:

$$W^{\mu\nu}=i\int d^4x e^{iq\cdot x}\langle P|T\{J^\mu(x)J^\nu(x)\}|P\rangle\ ,\tag{18.102}$$

where $W^{\mu\nu}$ can be written in terms of two scalar form factors, namely, $W_1$ and $W_2$

$$W^{\mu\nu}=\left(-g^{\mu\nu}+\frac{q^\mu q^\nu}{q^2}\right)W_1+\left(P^\mu-q^\mu\frac{P\cdot q}{q^2}\right)\left(P^\nu-q^\nu\frac{P\cdot q}{q^2}\right)W_2\ ,\tag{18.113}$$ where $W_2=W_2(\nu,Q^2)$ with $\nu=2P\cdot q$.

Now, the counter integral in question is carried out by distorting the contour of an integral, with respect to $\nu$ for large and given $Q^2$, from a small circle surrounding the origin, into an integral over two branch cuts on the real axis of $\nu$, as shown in Fig.18.12 and (18.150-152).

However, the validity of the above precedure requires that $W_2$ does not posseess additional singularities on the complex plane. In particular, above (18.150), the text reads

The discontinuity across this cut gives the cross section for the u-channel process in which positive energy comes in through the second current and out through the first. Since $q^2=-Q^2<0$, there is no possible physical t-channel process; thus $W_2$ has no further singularities in the complex $\nu$ plane.

I feel completely lost concerning the discussions of t- and u-channels.

(1) What is the collision process in question? Is it $$e^-(k)+p(P)\to e^-(k')+X \ ,$$ or the corresponding forward scattering $$e^-(k)+p(P)\to e^-(k)+p(P) \ ?$$ It seems to me that only the former case explicitly involves the momentum $q=k'-k$ as the momentum transfered by the electron. While the latter seems to be the very collision process discussed in the context, I cannot understand why $q$ is associated as the momentum transfer in accordance with the definition of the $t$- or $u$-channel.

(2) I understand that one only need to include u- and t-channels when the final particles are indistinguishable. Otherwise the two channels do not contribute to the same S-matrix element, it seems to be matter of convention. So why the textbook conclude the branch cuts are related to the u-channel process while t-channel one is irrelevant.

(3)Subsequently, how is one led to the conclusion that there is no further singularities on the complex plane?

I am sorry that the above question is closely related specifically to the textbook.


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