Alternative to axions

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>C</e:mi> <e:mi>P</e:mi> </e:math> -violating phase which arises below chiral symmetry breaking in a non-Abelian gauge theory is “secretly” the magnetic dual of the flux of a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>U</g:mi> <g:mo stretchy="false">(</g:mo> <g:mn>1</g:mn> <g:mo stretchy="false">)</g:mo> </g:math> 4-form gauge field. Thus, the discharge of this 4-form flux by Schwinger production of charged membranes reduces the total <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>C</k:mi> <k:mi>P</k:mi> </k:math> -violating phase toward zero. This can safely restore <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>C</m:mi> <m:mi>P</m:mi> </m:math> symmetry to within the observational limits, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mrow> <o:msub> <o:mrow> <o:mi>θ</o:mi> </o:mrow> <o:mrow> <o:menclose notation="updiagonalstrike" other="updiag6"> <o:mrow> <o:mi mathvariant="monospace">CP</o:mi> </o:mrow> </o:menclose> </o:mrow> </o:msub> <o:mo>≲</o:mo> <o:msup> <o:mrow> <o:mn>10</o:mn> </o:mrow> <o:mrow> <o:mo>−</o:mo> <o:mn>10</o:mn> </o:mrow> </o:msup> </o:mrow> </o:math> , while also satisfying all limits from cosmology, when the charge and the tension of the gauge field sources are in the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mrow> <t:mo>≲</t:mo> <t:mi>keV</t:mi> </t:mrow> </t:math> range.