We study three-flavor QCD in a uniform magnetic field using chiral perturbation theory (χPT). We construct the vacuum free energy density, quark condensate shifts induced by the magnetic field and the renormalized magnetization to <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> in the chiral expansion. We find that the calculation of the free energy is greatly simplified by cancellations among two-loop diagrams involving charged mesons. In comparing our results with recent <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor lattice QCD data, we find that the light quark condensate shift at <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> is in better agreement than the shift at <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math>. We also find that the renormalized magnetization, due to its small-ness, possesses large uncertainties at <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> due to the uncertainties in the low-energy constants.

United States

Prabal Adhikari

Norway

Inga Strümke

Vacuum free energy, quark condensate shifts and magnetization in three-flavor chiral perturbation theory to In a Uniform Magnetic Field

We address finite volume effects of lattice QCD calculations in background magnetic fields. Using chiral perturbation theory at next-to-leading order, volume effects are calculated for thermodynamic quantities: the chiral condensate, pressure anisotropy, and magnetization. The neutral pion effective action in a finite volume is additionally derived. For these charge neutral observables, volume and source averaging are shown to capitalize on magnetic periodicity, which is the remnant translational invariance of the finite-volume theory. For a fixed magnetic field strength, certain volume and source averaged quantities are independent of the size of the lattice transverse to the magnetic field. Despite this simplifying feature, finite volume corrections to the magnetic field dependence of the chiral condensate and neutral pion magnetic polarizability can be non-negligible. The pressure anisotropy at fixed magnetic flux, moreover, appears acutely sensitive to the lattice volume in the chiral regime.

Brian C. Tiburzi

QCD thermodynamics and neutral pion in a uniform magnetic field: Finite volume effects

We study the topological susceptibility and fourth cumulant of the QCD vacuum in a background magnetic field (H) using three-flavor chiral perturbation theory (χPT) for arbitrary quark masses and n-flavor χPT with degenerate quark masses. We find that the enhancement by the magnetic field of the topological susceptibility relative to the <math><mi>H</mi><mo>=</mo><mn>0</mn></math> vacuum value is larger in the three-flavor χPT compared to two-flavor χPT. Additionally, in comparing the fourth cumulant, we find that its suppression is comparable for magnetic fields, <math><mi>e</mi><mi>H</mi><mo>≲</mo><mn>0.8</mn><msubsup><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math>, and weaker for larger magnetic fields in three-flavor χPT with its enhancement beginning at a significantly lower critical magnetic field compared to two-flavor χPT. We also find that the enhancement of the topological susceptibility in n-flavor χPT with degenerate quarks is significantly larger and the suppression of the topological cumulant significantly greater at weak fields with the critical magnetic field pushed out to larger magnetic fields compared to both two and three-flavor χPT.

Topological susceptibility and fourth cumulant in a uniform magnetic field

We study the topological susceptibility and the fourth cumulant of the QCD vacuum in the presence of a uniform, background magnetic field in two-and-three flavor QCD finding novel, model-independent sum rules relating the shift in the topological susceptibility due to the background field to the shift in the quark condensates, and the shift in the fourth cumulant to the shifts in the quark condensates and susceptibilities.

Topological susceptibility in a uniform magnetic field

We study the θ-vacuum of QCD using two-flavor chiral perturbation theory (χPT) in the presence of a uniform, background magnetic field calculating the magnetic field-dependent free energy density, the topological density, the topological susceptibility and the fourth cumulant at one-loop order. We find that the topological susceptibility is enhanced by the magnetic field while the fourth topological cumulant is diminished at weak fields and enhanced at larger fields when <math><mi>θ</mi><mo>=</mo><mn>0</mn></math>. However, in the QCD vacuum with <math><mi>θ</mi><mo>≠</mo><mn>0</mn></math>, the topological susceptibility can be either monotonically enhanced or diminished relative to their θ-vacuum values. The fourth cumulant also exhibits monotonic enhancement or suppression except for regions of θ near 0 and 2π, where it is both diminished and enhanced. Finally, the topological density is enhanced for all magnetic fields with its relative shift being identical to the relative shift of the up and down quark condensates in the θ-vacuum.

Qcd -vacuum in a Uniform Magnetic Field

We calculate the light-quark condensate, the strange-quark condensate, the pion condensate, and the axial condensate in three-flavor chiral perturbation theory ( $$\chi $$  <math> <mi>χ</mi> </math>  PT) in the presence of an isospin chemical potential at next-to-leading order at zero temperature. It is shown that the three-flavor  $$\chi $$  <math> <mi>χ</mi> </math>  PT effective potential and condensates can be mapped onto two-flavor  $$\chi $$  <math> <mi>χ</mi> </math>  PT ones by integrating out mesons with strange-quark content (kaons and eta), with renormalized couplings. We compare the results for the light-quark and pion condensates at finite pseudoscalar source with ( $$2+1$$  <math> <mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </math>  )-flavor lattice QCD, and we also compare the axial condensate at zero pseudoscalar and axial sources with lattice QCD data. We find that the light-quark, pion, and axial condensates are in very good agreement with lattice data. There is an overall improvement by including NLO effects.

Jens Andersen

Martin Mojahed

Quark, pion and axial condensates in three-flavor finite isospin chiral perturbation theory

We consider two-flavor chiral perturbation theory ( $$\chi $$  <math> <mi>χ</mi> </math>  PT) at finite isospin chemical potential  $$\mu _I$$  <math> <msub> <mi>μ</mi> <mi>I</mi> </msub> </math>   and finite temperature T. We calculate the effective potential and the quark and pion condensates as functions of T and  $$\mu _I$$  <math> <msub> <mi>μ</mi> <mi>I</mi> </msub> </math>   to next-to-leading order in the low-energy expansion in the presence of a pionic source. We map out the phase diagram in the  $$\mu _I$$  <math> <msub> <mi>μ</mi> <mi>I</mi> </msub> </math>  –T plane. Numerically, we find that the transition to the pion-condensed phase is second order in the region of validity of  $$\chi $$  <math> <mi>χ</mi> </math>  PT, which is in agreement with model calculations and lattice simulations. Finally, we calculate the pressure to two-loop order in the symmetric phase for nonzero  $$\mu _I$$  <math> <msub> <mi>μ</mi> <mi>I</mi> </msub> </math>   and find that  $$\chi $$  <math> <mi>χ</mi> </math>  PT seems to be converging very well.

Condensates and pressure of two-flavor chiral perturbation theory at nonzero isospin and temperature

In this paper, we consider two-flavor QCD at zero temperature and finite isospin chemical potential  $$\mu _I$$  <math> <msub> <mi>μ</mi> <mi>I</mi> </msub> </math>   using a model-independent analysis within chiral perturbation theory at next-to-leading order. We calculate the effective potential, the chiral condensate and the pion condensate in the pion-condensed phase at both zero and nonzero pionic source. We compare our finite pionic source results for the chiral condensate and the pion condensate with recent (2+1)-flavor lattice QCD results. Agreement with lattice results generally improves as one goes from leading order to next-to-leading order.

Quark and pion condensates at finite isospin density in chiral perturbation theory

We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential (<math><msub><mrow><mi>μ</mi></mrow><mrow><mi>I</mi></mrow></msub></math>) and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at <math><msub><mrow><mi>μ</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>=</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub></math>. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for <math><msub><mrow><mi>μ</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>&lt;</mo><mn>200</mn></math> MeV. For larger values of <math><msub><mrow><mi>μ</mi></mrow><mrow><mi>I</mi></mrow></msub></math>, the agreement between lattice data and the two-flavor predictions is surprisingly good and better than with the three-flavor predictions. Finally, in the limit <math><msub><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>≫</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>=</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msub></math>, we show that the three-flavor observables reduce to the two-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor χPT can largely be explained by the differences in the measured low-energy constants.

Denmark

Jens O. Andersen

QCD at finite isospin density: Chiral perturbation theory confronts lattice data

We consider three-flavor chiral perturbation theory ( PT) at zero temperature and nonzero isospin ( ) and strange ( ) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the -condensed phase, the -condensed phase, and the $$ {K}^0/{\overline{K}}^0 $$ -condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, $$ \left|{\mu}_I\right|={m}_{\pi },\left|\frac{1}{2}{\mu}_I+{\mu}_S\right|={m}_K $$ , and $$ \left|-\frac{1}{2}{\mu}_I+{\mu}_S\right|={m}_K $$ , respectively at tree level and remains unchanged at NLO. The transition between the two condensed phases is first order. The effective potential in the pion-condensed phase is independent of and in the kaon-condensed phases, it only depends on the combinations $$ \pm \frac{1}{2}{\mu}_I+{\mu}_S $$ and not separately on and . We calculate the pressure, isospin density and the equation of state in the pion-condensed phase and compare our results with recent (2 + 1)-flavor lattice QCD data. We find that the three-flavor PT results are in good agreement with lattice QCD for 200 MeV, however for larger values PT produces values for observables that are consistently above lattice results. For 200 MeV, the two-flavor results are in better agreement with lattice data. Finally, we consider the observables in the limit of very heavy -quark, where they reduce to their two-flavor counterparts with renormalized couplings. The disagreement between the predictions of two and three flavor PT can largely be explained by the differences in the experimental values of the low-energy constants.