Chiral perturbation theory and Bose-Einstein condensation in QCD

We present recent results in three-flavor chiral perturbation theory at finite isospin <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>μ</a:mi><a:mi>I</a:mi></a:msub></a:math> and strangeness <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>μ</c:mi><c:mi>S</c:mi></c:msub></c:math> chemical potentials at zero temperature. The tree-level spectrum for the mesons and gauge bosons in the pion-condensed phase is derived. The phase diagram to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="script">O</e:mi><e:mo stretchy="false">(</e:mo><e:msup><e:mi>p</e:mi><e:mn>2</e:mn></e:msup><e:mo stretchy="false">)</e:mo></e:math> in the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mi>μ</j:mi><j:mi>I</j:mi></j:msub><j:mi>–</j:mi><j:msub><j:mi>μ</j:mi><j:mi>S</j:mi></j:msub></j:math> plane is mapped out with and without electromagnetic effects. The phase diagram consists of a vacuum phase and three Bose-condensed phases with condensates of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>π</l:mi><l:mo>±</l:mo></l:msup></l:math>, <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>K</n:mi><n:mo>±</n:mo></n:msup></n:math>, and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msup><p:mi>K</p:mi><p:mn>0</p:mn></p:msup><p:mo>/</p:mo><p:msup><p:mover accent="true"><p:mi>K</p:mi><p:mo stretchy="false">¯</p:mo></p:mover><p:mn>0</p:mn></p:msup></p:math>, respectively. Including electromagnetic interactions, the charged Bose-condensed phases become Higgs phases via the Higgs mechanism. We calculate the pressure, energy density, isospin density, and speed of sound in the pion-condensed phase to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi mathvariant="script">O</t:mi><t:mo stretchy="false">(</t:mo><t:msup><t:mi>p</t:mi><t:mn>4</t:mn></t:msup><t:mo stretchy="false">)</t:mo></t:math>. The results are compared with recent lattice simulations and the agreement is very good for isospin chemical potentials up to approximately 180 MeV. Moreover, by integrating out the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>s</y:mi></y:math>-quark, we show that the thermodynamic quantities can be mapped onto their two-flavor counterparts with renormalized parameters. The breaking of the U(1) symmetry in the Bose-condensed phases gives rise to a Goldstone boson, whose dispersion is linear for small momenta. We use Son’s prescription to construct an effective theory for the Goldstone mode in the pion-condensed phase, which is valid for momenta <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>p</ab:mi><ab:mo>≪</ab:mo><ab:msub><ab:mi>μ</ab:mi><ab:mi>I</ab:mi></ab:msub></ab:math>. It is shown that its damping rate is of order <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mi>p</cb:mi><cb:mn>5</cb:mn></cb:msup></cb:math> in the nonrelativistic limit, which is Beliaev’s result for a dilute Bose gas. It is also shown that in the nonrelativistic limit the energy density can be matched onto the classic result by Lee, Huang and Yang (LHY) for a dilute Bose, with an <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>s</eb:mi></eb:math>-wave scattering length that includes radiative corrections.