Isospin and Partons model in a pentaquark

[rogdal]

TL;DR Summary: Given a pentaquark:
(a) Determine the isospin multiplet it belongs to.
(b) Calculate a kind of a Gottfried Sum Rule for this pentaquark-neutrino or -antineutrino scattering.

Hello everybody,

I'm having a bit of a trouble with the exercise below as it deals with a pentaquark and I think I' a bit lost in the concepts that take place in it.QUESTION

Consider the pentaquark P ≡(b-bar s u u d), where b-bar is the antiquark for b, with orbital angular momentum L = 0.

a) Assuming a space-spin wavefunction that corresponds with the maximum-J state, determine the isospin multiplet I to which P would belong, specifying the flavour composition and the value of (I, I₃) for each of the members of the multiplet.

b) Consider the deep inelastic scattering of neutrinos and antineutrinos upon P:

1: ν_e + P --> e^- + X
2: ν_e-bar + P --> e⁺ + X'

where X, X' denote a final generic state. Let the structure functions F(x) associated with the incident W⁺ and W_- upon P collisions as:

F^W±P(x) ≡ 2(∑q(x) - ∑q-bar(x))

where q(x) and q-bar(x) denote the parton distributrions for quarks and antiquarks respectively. The sums extent upon all the species of quarks q and antiquarks q-bar with which the collision can take place. Calculate the numeric value of the following expression:

Δ ≡ ∫¹₀ [x·F^ν_eP(x)+ x·F^ν_e-barP(x)]/(2x)dx

Assume the sea contribution of a quark q is equal to the contribution of his antiquark q-bar.ATTEMPT

a) The quarks are spin-1/2 fermions and so the maximum J state for a pentaquark is J = 5/2, as there are 5 quarks that could have +1/2 spin all of them. However, P's isospin is I = 3/2 because it counts on just three up/down quarks. Therefore, the multiplet is a quadruplet:

|b-bar s d d d>; |b-bar s u d d >; |b-bar s u u d>= P ; |b-bar s u u u>

Would this approach be correct?
I don't really understant what the question means by "assuming a space-spin wavefunction that corresponds with the maximum-J state".b) As in F(x), "the sums extent upon all the species of quarks q and antiquarks q-bar with which the collision can take place", for the scattering 1 I have assumed that the collision can only occur with negative-charged particles (analogously for 2, it can only occur with positive-charged particles) and so, I've claimed that:

F^ν_eP(x) = 2(d + s + b - u-bar - c-bar - t-bar)
F^ν_e-barP(x)] = 2(u' + c' + t' - d-bar - s-bar - b-bar)

Then I cannot find any expression to link both F(x). I don't really understand what the question mean.

Any hint will be appreciated.

Many thanks!

 

[member38644]

RESPONSE:

Hi there,

Isospin and the parton model are both important concepts in understanding the structure of subatomic particles. In the case of a pentaquark, we can use these concepts to determine its properties and behavior in collisions.

a) To determine the isospin multiplet of the pentaquark P, we need to consider its quark composition and spin. As you correctly noted, the maximum J state for a pentaquark with L=0 is J=5/2. However, we also need to consider the isospin of the individual quarks. In this case, P contains three up/down quarks and one antiquark, giving it an isospin of I=3/2. This means that P belongs to a quadruplet multiplet, with (I,I3) values of (3/2, 3/2), (3/2, 1/2), (3/2,-1/2), and (3/2,-3/2). The flavor composition for each member of the multiplet would be the same as P, but with different spins and isospin projections.

b) The deep inelastic scattering of neutrinos and antineutrinos on P involves the exchange of W+ and W- bosons, which carry a charge of +1 and -1 respectively. This means that the collision can only occur with particles that have a charge of -1 (for the neutrino) and +1 (for the antineutrino). In this case, the structure functions F(x) would be given by:

FνeP(x) = 2(d + s + b - u-bar - c-bar - t-bar)
Fνe-barP(x)] = 2(u' + c' + t' - d-bar - s-bar - b-bar)

To calculate the Gottfried sum rule, we need to integrate over the variable x, which represents the fraction of the proton's momentum carried by the parton involved in the collision. This integral gives us the total contribution of all the quarks and antiquarks in the proton to the structure function. The result of this integral, Δ, would give us a measure of the asymmetry between the quarks and antiquarks in the proton.

I hope this helps clarify the concepts involved in this exercise. Keep in mind that these are just simplified models and there may be more complex phenomena at play in