CFTs with $\mathbf{U(m)\times U(n)}$ global symmetry in 3D and the chiral phase transition of QCD

<jats:p>Conformal field theories (CFTs) with <jats:inline-formula><jats:alternatives><jats:tex-math>U(m)\times U(n)</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> global symmetry in <jats:inline-formula><jats:alternatives><jats:tex-math>d=3</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> dimensions have been studied for years due to their potential relevance to the chiral phase transition of quantum chromodynamics (QCD). In this work such CFTs are analyzed in <jats:inline-formula><jats:alternatives><jats:tex-math>d=4-\varepsilon</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>d=3</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. This includes perturbative computations in the <jats:inline-formula><jats:alternatives><jats:tex-math>\varepsilon</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>ε</mml:mi></mml:math></jats:alternatives></jats:inline-formula> and large-<jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> expansions as well as non-perturbative ones with the numerical conformal bootstrap. New perturbative results are presented and a variety of non-perturbative bootstrap bounds are obtained in <jats:inline-formula><jats:alternatives><jats:tex-math>d=3</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. Various features of the bounds obtained for large values of <jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> disappear for low values of <jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> (keeping <jats:inline-formula><jats:alternatives><jats:tex-math>m<n</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>m</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> fixed), a phenomenon which is attributed to a transition of the corresponding fixed points to the non-unitary regime. Numerous bootstrap bounds are found that are saturated by large-<jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> results, even in the absence of any features in the bounds. A double scaling limit is also observed, for <jats:inline-formula><jats:alternatives><jats:tex-math>m</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>m</mml:mi></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> large with <jats:inline-formula><jats:alternatives><jats:tex-math>m/n</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>m</mml:mi><mml:mi>/</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> fixed, both in perturbation theory as well as in the numerical bootstrap. For the case of two-flavor massless QCD existing bootstrap evidence is reproduced that the chiral phase transition may be second order, albeit associated to a universality class unrelated to the one usually discussed in the <jats:inline-formula><jats:alternatives><jats:tex-math>\varepsilon</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>ε</mml:mi></mml:math></jats:alternatives></jats:inline-formula> expansion. Similar evidence is found for the case of three-flavor massless QCD, where we observe a pronounced kink. </jats:p>