# Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering

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.