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How do I relate the topological $\theta_\text{QCD}$ parameter to the electric dipole moment (EDM) of the neutron?

I am very familiar with chiral perturbation theory. I just need to know how to take $\theta_\text{QCD}$ into account.

Any good sources?

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(and what does the "50% accept rate" in my name mean?) – QuantumDot Sep 6 '12 at 23:56
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It means that you've accepted an answer on 50% of the questions you've asked. The idea is that someone might not want to put lots of work in if they don't think you'll select it, but in practice it's not a big deal. – AlanSE Sep 7 '12 at 0:07
@AlanSE oh, ok. I'll need to accept more. Also what is the star button under the (+1) (-1) buttons to the side? – QuantumDot Sep 7 '12 at 0:21
@QuantumDot: That's "favorite", I use it to bookmark questions to come back to later, since they appear in the "favorites" section of your profile, but it also makes people think that you somehow like these question particularly well. – Ron Maimon Sep 8 '12 at 2:15

1 Answer

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The phenomenological relevant parameter is not the coefficient of the topological term $\theta_{QCD}$ but the combination: $$\theta\equiv\theta_{QCD}+\sum _q Arg \{ m_q\}$$ since one may transform the first term into the second through field redefinitions. I do not know the precise derivation of the relation between $\theta$ and the neutron's electric dipole moment $\mathcal{D}$, but a good estimate is the following. (Sorry if this is not enough for you but it could be interesting for other readers).

$\mathcal{D}$ must vanish if either the up quark or the down quark were massless and if $\theta$ were zero. Moreover, $\mathcal{D}$ has dimensions of electric charge times length and must be suppressed by $\Lambda_{QCD}$. An expression which verifies the previous requirements is:

$$\mathcal{D}\approx {\hbar \, \theta \,e\over \Lambda_{QCD}^2} \frac{m_u \cdot m_d }{m_u + m_d } $$

The experimental constraint on $\mathcal{D}$ is: $$| \mathcal{D}|< 6 \cdot 10^{-26}e \cdot \mbox{cm}$$ giving rise to:$$\theta \lesssim 5\cdot10^{-10}$$
which is in agreement with the constraint reported.

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Thanks for the rough sketch. But as you correctly presume, I am looking for a more detailed and thorough computation. – QuantumDot Sep 8 '12 at 4:09
@QuantumDot I have gone throw Weinberg vol II and his estimate is not better than "mine" (I don't remember where I read it), but he refers to V. Baluni PRD (1978) and Veneziano, Witten et al. PLB (1979) – drake Sep 8 '12 at 4:50
Try this arxiv.org/abs/hep-ph/0008248 – Physics Monkey Sep 8 '12 at 11:38

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