The non-linear dynamics of long-wavelength cosmological fluctuations may be phrased in terms of an effective classical, but stochastic evolution equation. The stochastic noise represents short-wavelength modes that continually redshift into the long-wavelength domain. The effective evolution may be derived from first principles quantum field theory in an expanding background, through a sequence of approximations calling for additional scrutiny. We perform such an analysis, putting particular emphasis on the amplitude of the stochastic noise, which ultimately determines the cosmological correlations and provides a non-perturbative IR regulator to the dynamics.

Norway

Jens Andersen

Magdalena Eriksson

Anders Tranberg

Stochastic inflation from quantum field theory and the parametric dependence of the effective noise amplitude

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.

United States

Prabal Adhikari

Martin Mojahed

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

Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.

Quantum corrections to slow-roll inflation: scalar and tensor modes

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 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.

Denmark

Pion and kaon condensation at zero temperature in three-flavor

In this paper, we calculate the equation of state of two-flavor finite isospin chiral perturbation theory at next-to-leading order in the pion-condensed phase at zero temperature. We show that the transition from the vacuum phase to a Bose-condensed phase is of second order. While the tree-level result has been known for some time, surprisingly quantum effects have not yet been incorporated into the equation of state. We find that the corrections to the quantities we compute, namely the isospin density, pressure, and equation of state, increase with increasing isospin chemical potential. We compare our results to recent lattice simulations of 2 + 1 flavor QCD with physical quark masses. The agreement with the lattice results is generally good and improves somewhat as we go from leading order to next-to-leading order in $$\chi $$ <math><mi>χ</mi></math> PT.

Patrick Kneschke

Two-flavor chiral perturbation theory at nonzero isospin: pion condensation at zero temperature

Motivated by recent work on inverse magnetic catalysis at finite temperature, we study the quark-meson model using both dimensional regularization and a sharp cutoff. We calculate the critical temperature for the chiral transition as a function of the Yukawa coupling in the mean-field approximation varying the renormalization scale and the value of the ultraviolet cutoff. We show that the results depend sensitively on how one treats the fermionic vacuum fluctuations in the model and in particular on the regulator used. Finally, we explore a B -dependent transition temperature for the Polyakov loop potential T 0 ( B ) using the functional renormalization group. These results show that even arbitrary freedom in the function T 0 ( B ) does not allow for a decreasing chiral transition temperature as a function of B . This is in agreement with previous mean-field calculations.

William Naylor

Inverse magnetic catalysis and regularization in the quark-meson model

We calculate the three-loop thermodynamic potential of QCD at finite temperature and chemical potential(s) using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The resulting analytic thermodynamic potential allows us to compute the pressure, energy density, and entropy density of the quark-gluon plasma. Using these we calculate the trace anomaly, speed of sound, and second-, fourth-, and sixth-order quark number susceptibilities. For all observables considered we find good agreement between our three-loop HTLpt calculations and available lattice data for temperatures above approximately 300 MeV.