# Towards a matrix element definition of PDF

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• CAF123
In summary, Schwartz starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. He starts 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##. The equation then holds by assuming that the proton mostly moves in the ##\bar n = (1,0,0,-1)
CAF123
Gold Member
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!

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...

Hi ChrisVer,
ChrisVer said:
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!

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 :(

## 1. What is a matrix element definition of PDF?

A matrix element definition of PDF (Parton Distribution Function) is a method used in particle physics to calculate the probability of finding a specific type of particle within a proton or neutron. It involves using a matrix element, which is a mathematical function that describes the transitions between different states of a system, to determine the distribution of partons (quarks and gluons) within a hadron.

## 2. How is a matrix element definition of PDF different from other methods?

A matrix element definition of PDF is different from other methods because it takes into account the interactions between partons within a hadron. This allows for a more accurate calculation of the PDF, as it considers the complex nature of the strong force that binds the partons together.

## 3. What are the advantages of using a matrix element definition of PDF?

One advantage of using a matrix element definition of PDF is that it can be used to study the structure of hadrons at high energies, which is important for understanding the behavior of particles in particle accelerators. Additionally, it provides a more precise calculation of the PDF compared to other methods, which can lead to more accurate predictions of particle interactions.

## 4. Are there any challenges in using a matrix element definition of PDF?

Yes, there are challenges in using a matrix element definition of PDF. One challenge is that it requires complex calculations and can be computationally intensive. Additionally, it is difficult to incorporate all of the relevant factors and uncertainties, such as the effects of higher-order corrections and experimental errors, into the calculation.

## 5. How is a matrix element definition of PDF used in current research?

A matrix element definition of PDF is used in current research to study the properties of hadrons and their interactions with high-energy particles. It is also used in the development of new theoretical models and in the interpretation of experimental data from particle colliders. Additionally, it is an important tool in the search for new physics beyond the Standard Model.

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