Cosmic QCD Epoch at Nonvanishing Lepton Asymmetry

We investigate how a lepton asymmetry impacts the cosmic trajectory in the QCD phase diagram. We study the evolution of chemical potentials during the QCD epoch of the early Universe using susceptibilities from lattice QCD to interpolate between an ideal quark gas and an ideal hadron resonance gas. The lepton asymmetry affects the evolution of all chemical potentials. The standard cosmic trajectory is obtained assuming tiny lepton and baryon asymmetries. For larger lepton asymmetry, the charge chemical potential exceeds the baryon chemical potential before pion annihilation.

The excess of matter over antimatter in the Universe is one of the major puzzles of particle physics and cosmology. This asymmetry can be specified with respect to the conserved charges: baryon number B, lepton number L, and electric charge Q. The baryon asymmetry, defined as the net baryon number density per entropy density, b = n B /s, is tightly constrained to be b = (8.60 ± 0.06) × 10 −11 inferred from [1]. However, the standard model of particle physics (SM) fails to explain this asymmetry [2].
The lepton asymmetry, l = n L /s, is a key parameter to understand the origin of the matter-antimatter asymmetry. The idea of leptogenesis [3] is to create a primeval lepton asymmetry which due to electroweak sphaleron processes is partly converted to a baryon asymmetry. In the SM, the prediction in the case of efficient sphaleron processes is l = −(51/28)b [4]. However, there exist also models that predict a large lepton asymmetry nowadays, i.e. |l| ≫ b, and there is no preference for either sign of l. In general, in these models sphaleron processes are either suppressed [5][6][7] or the lepton asymmetry is produced after sphaleron processes cease to be efficient [8].
Observationally, the lepton asymmetry is only weakly constrained. While the charge neutrality of the Universe (see [9] for an upper limit) links the asymmetry in charged leptons to the tiny baryon asymmetry, a much larger lepton asymmetry could reside in a large neutrino asymmetry today. Constraints on the lepton asymmetry can be obtained from the cosmic microwave background |l| < 0.012 (95% C.L.) [10], and are in concordance with big bang nucleosynthesis analyses [11].
In this work, we investigate how the cosmic trajectory through the QCD phase diagram is influenced by the unknown lepton asymmetry. It has been shown by means of lattice QCD that the QCD transition is a crossover at small chemical potentials (see e.g. [12]). A pseudo-critical temperature can be defined and is measured on the lattice to be T QCD ≃ 154(9) MeV [13] (T QCD ≃ 147(2) − 165(5) MeV [14]). For vanishing temperatures and large baryon chemical potential µ B > m N effective models of QCD, like the Nambu-Jona-Lasinio model, predict a first-order chiral transition. It is speculated that there exists a critical line in the (µ B , T )-plane of the QCD phase diagram describing a first-order phase transition [15]. This line is expected to end in a second-order critical point [16,17]. Due to the infamous sign problem in lattice QCD, calculations for non-vanishing chemical potentials are very difficult [17][18][19][20].
We present a novel technique of determining the evolution of chemical potentials for arbitrary lepton asymmetries throughout the QCD epoch by using lattice QCD susceptibilities. First calculations of the cosmic trajectory in the QCD phase diagram have been performed in [21]. In [22] the evolution of chemical potentials at large lepton asymmetries has been studied in the approximation of an ideal quark gas and of a hadron resonance gas (HRG). In the HRG the QCD-sector is approximated as an ideal gas of hadron resonances. Here we extend and advance the approach of [22] in the following perspective: For the first time, we determine the cosmic trajectory accounting properly for strong interaction effects close to T QCD by using lattice QCD data for conserved charge susceptibilities. In the context of sterile neutrino production approximate relations between lepton asymmetries and chemical potentials have been studied in terms of susceptibilities by [23,24]. Furthermore, we improve the calculation of the entropy density in [22] by including chemical potentials of all relevant particle species, and we take a larger number of hadron resonances into account.
The trajectory of the early Universe, for conserved B, Q, and L, in the phase diagram of strongly interacting matter is commonly assumed to pass T QCD at vanishing chemical potentials and to proceed to µ B = m N and µ Q = µ L = 0 at m e ≪ T m N . In this scenario it is assumed that the lepton asymmetry is tiny, l = O(b).
Below we refer to this as the standard scenario. Our results for l = −(51/28)b present (to our knowledge) the first precise calculation of the standard cosmic trajectory.
It has already been shown that for |l| ≫ b the baryon and charge chemical potential becomes of the order of the lepton asymmetry µ B ∼ µ Q ∼ lT at T T QCD [22,25]. Assuming an overall electric charge neutrality and a fixed b, a lepton asymmetry in the electrically charged leptons induces an electric charge asymmetry in the quark sector which induces quark chemical potentials. As quarks carry not only electric charge but also baryon number, quark chemical potentials induce non-vanishing charge and baryon chemical potentials. With a sufficiently large primordial lepton asymmetry the cosmic trajectory could be shifted to higher charge and baryon chemical potential and thus the order of the QCD transition in the early Universe might be changed. This could have observable consequences, e.g. the production of relics [26,27]. Additionally, for charge chemical potential larger than the pion mass m π , pion condensation might occur in the early Universe [28,29]. Understanding the impact of a lepton asymmetry on the evolution of the Universe at various epochs is therefore of crucial importance.
For the epoch of the cosmic QCD transition, kinetic and chemical equilibrium are excellent approximations. The timescales of interest are the interaction rates and the Hubble time, t H = 1/H ≃ 10 −5 s at T QCD , which is large compared to the timescales of strong, electromagnetic and weak interactions.
Since neutrino oscillations, which take place after the QCD epoch at T osc ∼ 10 MeV [11,30,31], lead to a mixing of all lepton flavors, the observational constraints hold for the total lepton asymmetry l = α l α , α ∈ {e, µ, τ }. Thus, for T > T osc even larger but oppositely signed lepton flavor asymmetries are consistent with observational constraints. Note that sizable flavor asymmetries survive neutrino oscillations dependent on the neutrino mixing angles and the initial values of the lepton asymmetries and thus even after neutrino oscillations it is possible to have l α = l/3 [32][33][34]. In this letter we focus on equally distributed lepton flavor asymmetries l e = l µ = l τ = l/3.
After the electroweak transition at T EW ∼ 100 GeV and for T > T osc , there are five conserved charges in the early Universe, B, Q, and the three lepton flavor numbers L α , to which corresponding chemical potentials, µ B , µ Q , and µ Lα , can be assigned, respectively. Note that, in contrast to relativistic heavy ion collisions which also probe the QCD phase diagram, individual quark flavors like strangeness are not conserved due to electroweak processes.
Assuming a homogeneous Universe we obtain five conserved quantities: n Lα /s = l α , n B /s = b, and n Q /s = q. The entropy density s fulfills the relation T s(T, µ) = ǫ(T, µ) + p(T, µ) − a µ a n a (T, µ), with ǫ the total energy density, p the total pressure, and the sum over conserved charges a ∈ {B, Q, L α }. We fix b = 8.6 × 10 −11 and q = 0 in agreement with observations. The three lepton flavor asymmetries l α remain free parameters.
The net number density of a particle species is defined as the number density of a particle minus the density of its antiparticle. Assuming chemical equilibrium we find relations between the chemical potentials of different particle species. Since photons and gluons carry no conserved charges, their chemical potentials are zero. It follows that the chemical potentials of particles and antiparticles are equal in magnitude and opposite in sign, i.e. µ i = −µī. In kinetic and chemical equilibrium for an ideal gas we can express the net number densities by the integral over the Fermi-Dirac (Bose-Einstein) distribution for fermions (bosons) as for a particle with mass m i , chemical potential µ i , and with the + for fermions (− for bosons). The number of degrees of freedom g i counts particles and antiparticles separately, i.e. g = 1 for neutrinos, g = 2 for electrically charged leptons, and g = 6 for quarks. In this approximation, the five local conservation laws can be written in terms of the particle net number densities, with B i the baryon number and Q i the electric charge of particle species i. We can express the conserved charge chemical potentials in terms of particle chemical potentials, and at low temperatures with µ π , µ p and µ n the chemical potential of pions, protons and neutrons, respectively, and similarly for other hadrons and their resonances. In order to take into account strong interactions between quarks and gluons close to T QCD we expand the QCD pressure in a Taylor series in the chemical potentials up to second order, with an implicit sum over a, b ∈ {B, Q} here and in the following. The susceptibilities are defined by Such an expansion is also used in lattice QCD for circumventing the sign problem (cf. [13,35]). The conserved charge densities follow as The entropy density of the strongly interacting matter satisfies (cf. [36]) and with Eqs. (8) and (10) it follows: Only quarks contribute to the baryon asymmetry bs = n QCD

B
. The contribution to the electric charge can be divided into a part arising from the leptons n lep Q and one by quarks n QCD Q : qs = 0 = n QCD Q + n lep Q . With the QCD net number densities given by Eq. (10), our system of equations becomes For given temperature and lepton asymmetry l, we solve Eqs. In order to solve the system of coupled integralequations we modified the C-code used in [22] to take into account strong interactions between quarks according to Eqs. (12), (13a)-(13c). We use continuum extrapolated lattice QCD susceptibilities for a 2+1-flavor system [13], i.e. including the up, down and strange quark, and for a 2+1+1-flavor system (not continuum extrapolated, N τ = 8) [37,38], i.e. including also the charm quark, and the numerical temperature derivatives thereof. For the HRG we consider hadron resonances up to m Λ(2350) ≈ 2350 MeV ∼ 15T QCD , using particle properties according to the summary tables in [39]. Finally, we included chemical potentials in the calculation of the entropy density in all three temperature regimes i)-iii). Integrations like in Eq. (1) are performed using Gauss-Laguerre quadrature and the system of equations is solved by using Broydn's method [40]. Deviations from the results of [22] are due to those improvements and minor mistakes in the original code. We are free to choose arbitrarily five independent chemical potentials as free parameters according to Eqs. (3)-(7) in order to solve our system of integral-equations. However, one has to carefully choose them such that they are of different size to be able to obtain all particle chemical potentials without running into numerical problems. This is most important for the HRG at low temperatures where µ Q = µ p − µ n and µ n ≈ µ p . A good choice is {µ Q , µ B , µ Le , µ Lµ , µ Lτ }. Figure 1 shows the results for the temperature evolution of µ B (top), −µ Q (middle) and −µ Le (bottom) for different values of the lepton asymmetry. The evolution of µ Lµ and µ Lτ is not shown here, as they are of similar size as µ Le . However, despite the fact that all lepton asymmetries l α are assumed to be equal, the three lepton flavor chemical potentials evolve differently due to the lepton masses. We can see in Fig. 1 that, talking about absolute values, a larger total lepton asymmetry induces larger chemical potentials. The chemical potentials are proportional to l. This is true for l > O(b). For lepton asymmetries l O(b) the evolution of all chemical potentials is determined by the baryon asymmetry b and the contribution of l is negligible.
The chemical potentials obtained using lattice QCD susceptibilities connect the ideal quark gas with the HRG approximation quite well. Especially for µ Q they almost smoothly connect the two approximations at high and low temperature. For µ B , however, the results with 2+1flavor lattice QCD susceptibilities do not connect the two approximations smoothly for larger lepton asymmetries. At low temperatures there is a small gap between the lattice QCD and HRG results. At high temperatures the lattice QCD results do not smoothly converge to the ideal quark gas but they intersect in a single point for |l| 10 −8 . Taking into account the charm quark by using 2+1+1-flavor lattice QCD susceptibilities, they seem to converge to the ideal quark gas at high temperatures, see Fig. 1 (top). The uncertainty of the lattice QCD results is of the order of the point sizes in Fig. 1 and Fig. 2.
An important feature in the evolution of µ B is that for small temperatures T m π /3 ≈ 46 MeV, after the annihilation of pions (and muons), µ B no longer depends on the value of l and approaches the nucleon mass m N ∼ 1 GeV at low temperatures (see Fig. 1).
For l = −(51/28)b we obtain the standard cosmic trajectory. The reader should keep in mind that the cosmic trajectory follows a path in the 5 + 1 dimensional phase diagram. In Fig. 1 we are showing two-dimensional projections of the phase diagram.
It can also be seen, that for large lepton asymmetries l > O(b), see Fig. 1 and Fig. 2, the electric charge chemical potential becomes larger than the baryon chemical potential at non-vanishing temperature. This can be understood as follows. The electric charges of the three light quarks add up to zero. If their masses were degenerate (and heavier quarks are neglected), the susceptibility χ BQ would vanish, so that no µ B is induced. Thus for T m strange , µ B remains small. Furthermore, this is why the charm quark is important here, despite its large mass. Figure 2 shows the effect of the sign of a lepton asym- metries. For positive l, µ B is negative at high temperatures and proceeds to the nucleon mass for lower temperatures.
For an equally distributed lepton asymmetry we find that for |l| 0.15 we get |µ Q | m π , which might enable pion condensation in the early Universe [28,29]. Such a large lepton asymmetry would exceed the observational constraint by an order of magnitude. However, unequally distributed lepton asymmetries would admit the possibility of |µ Q | m π , while satisfying |l| < 0.012.
In this letter we have studied the evolution of chemical potentials as a function of temperature during the cosmic QCD epoch and investigated its dependence on a lepton asymmetry. For the first time we used lattice QCD results to properly account for the temperature regime around T QCD in order to connect the approximations of an ideal quark gas with the HRG. We provide the standard cosmic trajectory through the 5+1-dimensional QCD phase diagram for l = −(51/28)b and, furthermore, the cosmic trajectory in the presence of larger total lepton asymmetry.
There is no phase transition in the early Universe if the cosmic trajectory in the QCD phase diagram is smooth at all T . A kink in the trajectory would correspond to a non-equilibrium, first-order phase transition. We find that the 2+1+1-flavor lattice QCD susceptibilities allow us to interpolate between the trajectories of the ideal quark and HRG. We like to stress the importance of the charm quark contribution in our results to obtain a smooth trajectory. Unfortunately, no continuum extrapolated 2+1+1-flavor lattice QCD susceptibilities have been available at the time of this study. Gaps in our result for the cosmic trajectory, like in Fig. 2, are artifacts of our approximations. Gaps between the lattice QCD results and the ideal quark gas might be closed by taking higher order perturbative corrections into account. It has been shown, that these lead to smaller susceptibilities than in the ideal gas approximation and to better agreement with lattice QCD results [41]. Furthermore, it would be helpful to have lattice QCD susceptibilities for lower and higher temperatures available. The observed small gaps for µ B at low temperature between lattice QCD results and the HRG approximation might then be closed. However, these gaps might also be due to limitations of the HRG approximation to describe all thermodynamical aspects of QCD. The current precision of lattice susceptibilities and the ideal quark gas and HRG approximations used in this work do not allow us to make any statement on the nature of the cosmic QCD transition.
Our results might be of crucial importance for a better understanding of the evolution of the early Universe and can be used in cosmic evolution calculations, e.g. in predicting the abundance of various dark matter candidates. We would like to emphasize that before pion annihilation and for a lepton asymmetry |l| 10 −8 , the absolute value of the electric charge chemical potential |µ Q |, exceeds the baryon chemical potential and therefore µ Q might be more important for the thermal history of the Universe than µ B .
We thank Gabriela Barenboim