Towards a matrix element definition of PDF

[CAF123]

In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.696, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. Most of the derivation is clear to me, except a couple of things at the start and midway. The derivation begins by saying that the probability for finding a quark within a proton with a given momentum fraction (=PDF) is given by $$f(\zeta) = \sum_X \int \text{d}\Pi_X | \langle X|\psi|P \rangle |^2 \delta( \zeta n \cdot P - n \cdot p),$$ with $P = p + p_X$. Why is this the correct mathematical representation for a PDF? I understand the delta function constrains the parton to take a fraction $\zeta$ of the proton's momentum in the $n$ direction and that probabilities are squared matrix elements but what does the sum over X mean here (we have no scattering taking place) and what is the meaning of $\langle X | \psi | P \rangle$?

The other thing was he states that since the proton moves mostly in the $\bar n = (1,0,0,-1)$ direction, $\not \bar n \psi \approx 0$. Why does this approximation hold?

Thanks!

 

[ChrisVer]

I think the approach holds by the assumptions you take, i.e. that the parton does not have exactly the a fraction of the proton's momentum but it has mostly the proton's momentum fraction + small corrections that come from the transverse motion and the opposite direction. It derives from 32.111

As for your first question, as is mentioned in the book that's the probability of the proton to have momentum $P= p + p_X$. X is summed over because you have several other partons inside the proton- which will have some fraction of the proton...

 

[CAF123]

Hi ChrisVer,

I think the approach holds by the assumptions you take, i.e. that the parton does not have exactly the a fraction of the proton's momentum but it has mostly the proton's momentum fraction + small corrections that come from the transverse motion and the opposite direction. It derives from 32.111

As for your first question, as is mentioned in the book that's the probability of the proton to have momentum $P= p + p_X$. X is summed over because you have several other partons inside the proton- which will have some fraction of the proton...

Yes, the concepts are clear - what's bothering me is why the concepts are represented mathematically in the way written down. E.g what does $\langle X | \psi | P \rangle$ mean? Expressions such as $\langle \psi_j | H | \psi_i \rangle = \delta_{ij}$ in QM make sense as saying the Hamiltonian H of some theory is diagonal in the basis spanned by its eigenstates $\left\{\psi \right\}$. So what is the equivalent meaning of $\langle X | \psi | P \rangle$ ?

Thanks!

 

[ChrisVer]

Let me try to rephrase how I'd interpret the expression of QM you gave: It's the amplitude of an interacting [via the Hamiltonian] state to fall from the state $\psi_i$ to $\psi_j$... I think in a similar way you can interpret the final expression [psi is a dirac operator]- it's an amplitude. Unfortunately I won't have access to the book before 9th of January to recheck :(