Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Why is there no theta-angle (topological term) for the weak interactions?

+ 12 like - 0 dislike
2846 views

Why is there no analog for $\Theta_\text{QCD}$ for the weak interaction? Is this topological term generated? If not, why not? Is this related to the fact that $SU(2)_L$ is broken?

This post has been migrated from (A51.SE)
asked Jan 26, 2012 in Theoretical Physics by Henry (115 points) [ no revision ]
Good question, and looking forward to the answers if any. ;-)

This post has been migrated from (A51.SE)

2 Answers

+ 9 like - 0 dislike

In the presence of massless chiral fermions, a $\theta$ term in can be rotated away by an appropriate chiral transformation of the fermion fields, because due to the chiral anomaly, this transformation induces a contribution to the fermion path integral measure proportional to the $\theta$ term Lagrangian.

$$\psi_L \rightarrow e^{i\alpha }\psi_L$$

$${\mathcal D}\psi_L {\mathcal D}\overline{\psi_L}\rightarrow {\mathcal D} \psi_L {\mathcal D}\overline{\psi_L} \exp\left(\frac{i\alpha g N_f}{64 \pi^2}\int F \wedge F\right)$$

So the transformation changes $\theta$ by $C \alpha g N_f $ ($g$ is the coupling constant, $N_f$ the number of flavors).

The gluons have the same coupling to the right and left handed quarks, and a chiral rotation does not leave the mass matrix invariant. Thus the QCD $\theta$ term cannot be rotated away.

The $SU(2)_L$ fields however, are coupled to the left handed components of the fermions only, thus both the left and right handed components can be rotated with the same angle, rotating away the $\theta$ term without altering the mass matrix.

This post has been migrated from (A51.SE)
answered Jan 26, 2012 by David Bar Moshe (4,355 points) [ no revision ]
Nice, would you add one or two formulae? What is the parameter of the transformation (and which one) to remove the $\theta\cdot F\wedge F$ term? And a related question: is there some simple way to add some chiral couplings of new fermions to $SU(3)_{color}$ to solve the strong CP-problem?

This post has been migrated from (A51.SE)
@Luboš I am not an expert, from reading only, I think that your suggestion is quite close to one solution to the strong CP problem assuming the mass of the u-quark is exactly zero,though not widely accepted.

This post has been migrated from (A51.SE)
Thanks a lot, David!

This post has been migrated from (A51.SE)
What about the Yukawa couplings? You absorb the phase into the Higgs? Or into right handed fermions?

This post has been migrated from (A51.SE)
@Thomas: To the right handed fermions. They are not coupled to the gauge fields so their transformation does not change the path integral measure

This post has been migrated from (A51.SE)
+ 0 like - 0 dislike

See CHIRAL ANOMALIES IN FIELD THEORIES (H. Banerjee), section "Global Chiral Anomaly and the Strong CP Problem".

answered May 8, 2017 by illuminates (15 points) [ no revision ]

This appears to be Section 5 in 

H. Banerjee, Chiral anomalies in field theories, https://arxiv.org/abs/hep-th/9907162

Why does this answer the question? 

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...