Editing Strong CP problem (section)

==Theory==

CP-symmetry states that physics should be unchanged if particles were swapped with their antiparticles and then left-handed and right-handed particles were also interchanged. This corresponds to performing a charge conjugation transformation and then a parity transformation. The symmetry is known to be broken in the [[Standard Model]] through [[weak interaction|weak interactions]], but it is also expected to be broken through [[strong interaction|strong interactions]] which govern [[quantum chromodynamics]] (QCD), something that has not yet been observed. 

To illustrate how the CP violation can come about in QCD, consider a [[Yang–Mills theory]] with a single massive [[quark]].<ref>{{cite conference|url=https://www.osti.gov/servlets/purl/6260191|title=A Brief Introduction to the Strong CP Problem|last1=Wu|first1=D.|date=1991|publisher=|location=Austin, Texas, United States|id=SSCL-548}}</ref> The most general mass term possible for the quark is a complex mass written as <math>m e^{i\theta' \gamma_5}</math> for some arbitrary phase <math>\theta'</math>. In that case the [[Lagrangian (field theory)|Lagrangian]] describing the theory consists of four terms:

:<math>
\mathcal L = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} +\theta \frac{g^2}{32\pi^2}F_{\mu \nu}\tilde F^{\mu \nu} +\bar \psi(i\gamma^\mu D_\mu -me^{i\theta' \gamma_5})\psi.
</math>

The first and third terms are the CP-symmetric [[kinetic term]]s of the [[gauge theory|gauge]] and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases <math>\theta' \neq 0</math> while the second term is the so-called [[theta vacuum|θ-term]] or “vacuum angle”, which also violates CP-symmetry. 

Quark fields can always be redefined by performing a chiral transformation by some angle <math>\alpha</math> as

:<math>
\psi' = e^{i\alpha \gamma_5/2}\psi, \ \ \ \ \ \ \bar \psi' = \bar \psi e^{i\alpha \gamma_5/2},
</math>

which changes the complex mass phase by <math>\theta' \rightarrow \theta'-\alpha</math> while leaving the kinetic terms unchanged. The transformation also changes the θ-term as <math>\theta \rightarrow \theta + \alpha</math> due to a change in the [[path integral formulation|path integral]] measure, an effect closely connected to the [[chiral anomaly]].

The theory would be CP invariant if one could eliminate both sources of CP violation through such a field redefinition. But this cannot be done unless <math>\theta = -\theta'</math>. This is because even under such field redefinitions, the combination <math>\theta'+ \theta \rightarrow (\theta'-\alpha) + (\theta + \alpha) = \theta'+\theta</math> remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking <math>\alpha = \theta'</math>, but then all the CP violation goes to the θ-term which is now proportional to <math>\bar \theta</math>. If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase <math>\bar \theta</math>. Practically, it is usually useful to put all the CP violation into the θ-term and thus only deal with real masses.

In the Standard Model where one deals with six quarks whose masses are described by the [[Yukawa interaction|Yukawa matrices]] <math>Y_u</math> and <math>Y_d</math>, the physical CP violating angle is <math>\bar \theta = \theta - \arg \det(Y_u Y_d)</math>. Since the θ-term has no contributions to perturbation theory, all effects from strong CP violation is entirely non-perturbative. Notably, it gives rise to a [[neutron electric dipole moment]]<ref>{{cite book|first=M.D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|chapter=29|date=2014 |page=612|isbn=9781107034730}}</ref>

:<math>
d_N = (5.2 \times 10^{-16}\text{e}\cdot\text{cm}) \bar \theta.
</math>

Current experimental upper bounds on the dipole moment give an upper bound of <math>d_N < 10^{-26} \text{e}\cdot</math>cm,<ref>{{Cite journal |last1=Baker |first1=C.A. |last2=Doyle |first2=D.D. |last3=Geltenbort |first3=P. |last4=Green |first4=K. |last5=van&nbsp;der&nbsp;Grinten |first5=M.G.D. |last6=Harris |first6=P.G. |last7=Iaydjiev |first7=P. |last8=Ivanov |first8=S.N. |last9=May|first9=D.J.R. |date=2006-09-27 |df=dmy-all |title=Improved experimental limit on the electric dipole moment of the neutron |journal=Physical Review Letters |volume=97 |issue=13 |page=131801 |doi=10.1103/PhysRevLett.97.131801 |pmid=17026025 |arxiv=hep-ex/0602020|bibcode=2006PhRvL..97m1801B |s2cid=119431442 }}</ref> which requires <math>\bar \theta < 10^{-10}</math>. The angle <math>\bar \theta</math> can take any value between zero and <math>2\pi</math>, so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem.