Chiral Cherenkov radiation in the presence of a time-dependent chiral chemical potential

The photon production process <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>f</a:mi><a:mo>→</a:mo><a:mi>f</a:mi><a:mo>+</a:mo><a:mi>γ</a:mi></a:mrow></a:math> in the presence of a time-dependent chiral chemical potential <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>μ</b:mi><b:mn>5</b:mn></b:msub></b:math> is studied. The validity of the adiabatic approximation is demonstrated in the ultrarelativistic limit. Analytical expressions for the photon spectrum are derived for two models of the chiral magnetic conductivity <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>b</c:mi><c:mn>0</c:mn></c:msub><c:mo>∝</c:mo><c:msub><c:mi>μ</c:mi><c:mn>5</c:mn></c:msub></c:mrow></c:math>: (i) <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>b</d:mi><d:mn>0</d:mn></d:msub><d:mrow><d:mo>(</d:mo><d:mi>t</d:mi><d:mo>)</d:mo></d:mrow><d:mo>=</d:mo><d:msub><d:mi>A</d:mi><d:mn>1</d:mn></d:msub><d:mo>+</d:mo><d:msub><d:mi>B</d:mi><d:mn>1</d:mn></d:msub><d:mo form="prefix">tanh</d:mo><d:mrow><d:mo>(</d:mo><d:mi>t</d:mi><d:mo>/</d:mo><d:mi>τ</d:mi><d:mo>)</d:mo></d:mrow></d:mrow></d:math> and (ii) <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:msub><f:mi>b</f:mi><f:mn>0</f:mn></f:msub><f:mrow><f:mo>(</f:mo><f:mi>t</f:mi><f:mo>)</f:mo></f:mrow><f:mo>=</f:mo><f:msub><f:mi>A</f:mi><f:mn>2</f:mn></f:msub><f:mrow><f:mo>(</f:mo><f:mi>t</f:mi><f:mo>/</f:mo><f:mi>τ</f:mi><f:mo>)</f:mo></f:mrow><f:mo>/</f:mo><f:mrow><f:mo>(</f:mo><f:mn>1</f:mn><f:mo>+</f:mo><f:msup><f:mi>t</f:mi><f:mn>2</f:mn></f:msup><f:mo>/</f:mo><f:msup><f:mi>τ</f:mi><f:mn>2</f:mn></f:msup><f:mo>)</f:mo></f:mrow></f:mrow></f:math>. It is known that, for constant <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>b</g:mi><g:mn>0</g:mn></g:msub></g:math>, photon emission may exhibit a resonance for one-photon polarization depending on the sign of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>b</h:mi><h:mn>0</h:mn></h:msub></h:math>. It is shown that, in model (i), up to two resonances may occur, depending on the sign of the asymptotic values <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>b</i:mi><i:mn>0</i:mn></i:msub><i:mrow><i:mo>(</i:mo><i:mi>∞</i:mi><i:mo>)</i:mo></i:mrow></i:mrow></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:msub><j:mi>b</j:mi><j:mn>0</j:mn></j:msub><j:mrow><j:mo>(</j:mo><j:mo>−</j:mo><j:mi>∞</j:mi><j:mo>)</j:mo></j:mrow></j:mrow></j:math>. No resonances are observed in model (ii). Numerical calculations are performed using parameters relevant to quark-gluon plasma, and the universality of the results is discussed.