# Longitudinal DIS Structure functions

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• CAF123
In summary: Q##.In summary, the conversation discusses the relationship between DIS observables and the structure functions F1, F2, and FL. It is mentioned that F1 and F2 can be expressed as a sum over x and an integral over y, with coefficients C1 and C2 and a function f. The conversation also introduces the relation ##F_L = F_2 - 2xF_1## and discusses how this relates to the coefficient function ##C_L##, which is not just a function of y and Q. The question is raised about the possibility of extracting the longitudinal coefficient function for FL from knowledge of the coefficient functions for F1 and F2. Ultimately,
CAF123
Gold Member
DIS observables can be expressed in terms of structure functions F1, F2 and FL. There exists the relation ##F_L = F_2 - 2xF_1##.

We can write $$F_L = \sum_a x \int_x^1 \frac{dy}{y} C_{a,L}(y,Q) f_a (\frac{x}{y},Q)$$ and similarly for ##F_1## and ##F_2##:

$$F_1 = \sum_a x \int_x^1 \frac{dy}{y} C_{a,1}(y,Q) f_a (\frac{x}{y},Q)$$

$$F_2 = \sum_a x \int_x^1 \frac{dy}{y} C_{a,2}(y,Q) f_a (\frac{x}{y},Q)$$

Then ##F_L = F_2 - 2xF_1## means that also

$$F_L = \sum_a x \int_x^1 \frac{dy}{y} \left( C_{a,2}(y,Q) - 2x C_{a,1}(y,Q) \right) f_a (\frac{x}{y},Q).$$

Comparing this with above eqn for ##F_L## means that ##C_{a,L}## is not just a function of y and Q. Is it possible to extract the longitudinal coefficient function for ##F_L## from knowledge of the coefficient function for ##F_1## and ##F_2##?

As far as I can tell it really depends on what the function f_a's and constants C_{a,i}'s are, to tell if C_{a,L} depends on x.

CAF123 said:
DIS observables can be expressed in terms of structure functions F1, F2 and FL. There exists the relation ##F_L = F_2 - 2xF_1##.

We can write $$F_L = \sum_a x \int_x^1 \frac{dy}{y} C_{a,L}(y,Q) f_a (\frac{x}{y},Q)$$ and similarly for ##F_1## and ##F_2##:

$$F_1 = \sum_a x \int_x^1 \frac{dy}{y} C_{a,1}(y,Q) f_a (\frac{x}{y},Q)$$

$$F_2 = \sum_a x \int_x^1 \frac{dy}{y} C_{a,2}(y,Q) f_a (\frac{x}{y},Q)$$

Then ##F_L = F_2 - 2xF_1## means that also

$$F_L = \sum_a x \int_x^1 \frac{dy}{y} \left( C_{a,2}(y,Q) - 2x C_{a,1}(y,Q) \right) f_a (\frac{x}{y},Q).$$

Comparing this with above eqn for ##F_L## means that ##C_{a,L}## is not just a function of y and Q. Is it possible to extract the longitudinal coefficient function for ##F_L## from knowledge of the coefficient function for ##F_1## and ##F_2##?
You write the relation for ##C_L## there, which depends also on the hadronic variable ##x## (in addition to ##y## and ##Q^2##).

Maybe it would make sense to write: ##C_L(x,Q,y)##

## What are longitudinal DIS structure functions?

Longitudinal DIS structure functions are mathematical quantities used to describe the behavior of particles in deep inelastic scattering (DIS) experiments. They provide information about the internal structure of particles and their interactions with other particles.

## What is the purpose of studying longitudinal DIS structure functions?

The study of longitudinal DIS structure functions allows scientists to gain a better understanding of the fundamental building blocks of matter and the forces that govern their interactions. This information can help us to develop more accurate models of the universe and improve our understanding of the laws of physics.

## How are longitudinal DIS structure functions measured?

Longitudinal DIS structure functions are typically measured through experiments involving the scattering of high-energy particles off of a target particle. These experiments involve analyzing the energy and momentum of the scattered particles to determine the structure functions.

## What are the main types of longitudinal DIS structure functions?

There are three main types of longitudinal DIS structure functions: F1, F2, and FL. F1 and F2 describe the distribution of quarks and gluons within a particle, while FL provides information about the longitudinal polarization of the particle.

## How do longitudinal DIS structure functions relate to other types of structure functions?

Longitudinal DIS structure functions are closely related to other types of structure functions, such as transverse DIS structure functions and fragmentation functions. These different types of structure functions provide complementary information about the internal structure of particles and their interactions.

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