Halzen and Martin Problem 13.12

[TriTertButoxy]

Hi. We just finished the chapter on electroweak interactions. So it turns out that QED processes gets altered by the weak neutral current (exchange of Z boson) which introduces a tiny parity-violating asymmetry.

There is an experiment which measures the minute parity-violating asymmetry in the inelastic scattering of longitudinally polarized electrons off nuclear targets. The asymmetry is defined by

$$A=\frac{\sigma_R-\sigma_L}{\sigma_R+\sigma_L}\,,$$​

where $\sigma_R$ is the cross section $d\Omega/dy$ for $\text{e}_R\text{N}\rightarrow \text{e}_R\text{X}$; $\text{e}_R$ denotes a right-handed electron.

For the deep inelastic scattering process $\text{eN}\rightarrow\text{eX}$, we can use the parton model to predict the asymmetry.

I need help on the following problem.
Taking $\text{N}$ to be an isoscalar target, show

$$A=\frac{6}{5}\left(\frac{\sqrt{2}Gk^2}{e^2}\right)\left(a_1+a_2\frac{1-(1-y)^2}{1+(1-y)^2}\right)\,,$$​

with

$$a_1=c_A^e(2c_V^u-c_V^d)$$
$$a_2=c_V^e(2c_A^u-c_A^d)\,.$$​

Here, $y=(E-E')/E$ is the fractional energy loss of the electron in the lab frame. Constants $c_V$ and $c_A$ are the vector and axial coupling constants of the Z bosons.

I may assume $k^2\ll M_Z^2$ and the target contains equal numbers of up and down quarks since it is an isoscalar target (and neglect antiquarks).

Here's what I've done.
I know the idea is to consider is helicity case separately and combine the everything at the end. But, I'm a little shaky on the Parton model to do this. For example, the cross section of electron-quark scattering is

$$\frac{d\sigma}{d\Omega}(\text{e}_R\text{u}_L\rightarrow\text{e}_r\text{u}_L)=\frac{\alpha^2}{4s}(1+\cos\theta)^2|Q_u+rc_R^ec_L^u|^2\,,$$​

where $r=-\sqrt{2} Gk^2/e^2$.

If anybody can help me with this, that woud be great!

 

[Jhero]

Thank you for your post. Electroweak interactions are indeed a fascinating topic in particle physics. It is interesting to note that the combination of QED and weak interactions can lead to parity-violating effects, as you have mentioned in your post.

To solve the problem at hand, we can start by considering the helicity cases separately. For the electron-quark scattering process, the cross section can be written as:

\frac{d\sigma}{d\Omega}(\text{e}_R\text{u}_L\rightarrow\text{e}_R\text{u}_L)=\frac{\alpha^2}{4s}(1+\cos\theta)^2|Q_u+rc_R^ec_L^u|^2\,,

Where r=-\sqrt{2} Gk^2/e^2 is the constant you have defined. Here, s is the center of mass energy squared and \theta is the scattering angle. We can then use the parton model to predict the asymmetry A for this process. The parton model assumes that the cross section can be written as a sum over the contributions from the individual partons (quarks and gluons) inside the nucleon. This can be written as:

\frac{d\sigma}{d\Omega}(\text{e}_R\text{N}\rightarrow\text{e}_R\text{X})=\sum_{i}f_i(x,\mu_F)\frac{d\sigma_i}{d\Omega}(\text{e}_R\text{q}_i\rightarrow\text{e}_R\text{q}_i)\,,

Where f_i(x,\mu_F) is the parton distribution function that gives the probability of finding a parton of type i with momentum fraction x inside the nucleon. The factor \mu_F is the factorization scale and is usually taken to be of the order of the momentum transfer Q^2 in the process.

We can then use the definition of the asymmetry A given in the problem to write:

A=\frac{\sigma_R-\sigma_L}{\sigma_R+\sigma_L}=\frac{\sum_{i}f_i(x,\mu_F)(\frac{d\sigma_i}{d\Omega}(\text{e}_R\text{q}_i\rightarrow\text{e}_R\text{q}_i)-\frac