QCD theta angle and Neutron electric dipole moment

[ansgar]

Dear all

I have done some studies trying to understand the relation between the QCD theta angle and neutron electric dipole moment.

General, the QCD vacuum produces the term

[tex] L_{\theta} = \theta g_s^2\: G_a^{\mu\nu} \, G^a_{\mu\nu} [/tex]

this I can derive! I have studied Srednicki ch 93, Ramond (journeys beyond SM) ch. 5.6 and Axions : theory, cosmology, and experimental searches ch 1

Now we will generate a similar term with the inclusion of massive quarks, with "paramter"

[tex] \text{Arg}\, \text{Det}\,M [/tex]

where M is a (in general) complex mass matrix for the quarks, thus the "total theta" reads:

[tex] \tilde{\theta} = \theta _{QCD} + \text{Arg}\, \text{Det}\,M [/tex]

I have also understood that if the quarks are massless, then the QCD theta is not a physical parameter due to redefinition of dummy variable in the Path Integral (thanks to the anomalous U(1)_A symmetry).

Now my quest is to understand how this QCD-theta affects the Electric Dipole moment of the Neutron.

From reading Srednicki, the angle which gives a contribution to that is the angle from the complex mass matrix! See eq. 94.10, and not the total theta!

I mean, WHY should the theta in eq. 94.10 be the same as in the Path Integral eq. 94.1?

So my question is, how does the QCD theta affect el-dip-mom of the neutron?

the "other books", (the ones listed above) and Burgess and Moore (standard model - a primer) says that it should be [tex] \tilde{\theta} = \theta _{QCD} + \text{Arg}\, \text{Det}\,M [/tex] that comes into the electric dipole moment...

Thank you in advance

 

[ansgar]

bump: humanino, blechman... help :)

 

[humanino]

In the meantime of more competent people than me stepping in (such as blechman), I'll report on my own findings.

I found the best explanation in Weinberg's second volume.

A mere change of path integration variables cannot have any physical effect, so observable quantities cannot depend separately on [itex]\theta[/itex] or the phases of the mass parameters [itex]{\cal M}_f[/itex], but only on the combination
[tex]\tilde{\theta}=\theta _{QCD} + \text{Arg}\, \text{Det}\,{\cal M}[/tex] (23.6.12)

He comes to this conclusion after discussing the general open-minded definition for the expectation value of an observable [itex]{\cal O}[/itex] in a spacetime volume [itex]\Omega[/itex]
[tex]\langle{\cal O}\rangle_\Omega = \frac{\sum_\nu f(\nu)\int_\nu [d\phi]\exp(I_\Omega[\phi]){\cal O}[\phi]}{\sum_\nu f(\nu)\int_\nu [d\phi]\exp(I_\Omega[\phi])}[/tex]
with an arbitrary weight function [itex]f(\nu)[/itex] for configuration with winding number [itex]\nu[/itex]. He thereupon argues from his "cluster decomposition principle" that the form of this weight must be [itex]f(\nu)=\exp(i\theta\nu)[/itex] for his principle to hold (in other places, this is discussed in terms of stability of the vacuum). As you already know,
[tex]\nu\propto \int\tilde{G}G=\int G_a^{\mu\nu} \, G^a_{\mu\nu}=\int \epsilon^{\kappa\lambda\mu\nu}G^a_{\kappa\lambda}G^{a}_{\mu\nu}[/tex]

After establishing (23.6.12) he explicitly decides to redefine the phases for fermion fields such that [itex]\theta=0[/itex].

The procedure in Srednicki is equivalent, but I find significantly less clear (I admit that this may be personal bias). He constructs "by hand" the modified mass term implicitly in (94.15) so it takes thinking on part of the reader to realize how this could be linked with more general principles (such as stability of the vacuum or cluster decomposition)

[tex]{\cal L}_m=\bar{\Psi}\left[M+i
\tilde{\theta}\tilde{m}\gamma_5+O(\theta^2)\right]\Psi[/tex]

Also, I find it desirable to include the strange quark to clearly see why this is dominated by light quark effects :
[tex]\tilde{m}=\frac{m_um_dm_s}{m_um_d+m_um_s+m_dm_s}\approx\frac{m_um_d}{m_u+m_d}[/tex]

So the short form of my answer : either one redefines the phases of the quarks, or equivalently sums appropriately over different winding numbers for the gluon vacuum, but at the end of the day, only [tex]\tilde{\theta}[/tex] is physical.

edit
Also, the evaluation of the neutron EDM is difficult and varies by about an order of magnitude (eventually, we still have an angle of at most a billionth anyway). From this point of view, Srednicki actually goes more into details. If you can, you may also want to check "Dynamics of the standard model" by Donoghue et al (Cambridge monograph)

 

[ansgar]

Ah ok thank you!

I have access to both those books, Weinberg 2 and Donoghue I'll have a look in them :)

So basically Weinberg (23.6.12) is what I have read everywhere and agreed upon since I stumbled on Srednicki's explanation, which seemed to just ignored to stress that fact enough and didn't took explicit "tilde" notation seriously.

Thank you again, much appreciated!