Eur. Phys. J. C manuscript No. (will be inserted by the editor) The cosmic QCD transition for large lepton flavour asymmetries Mandy M. Middeldorf-Wygasa,1, Isabel M. Oldengottb,2, Dietrich B¨odekerc,1, Dominik J. Schwarzd,1 1Fakult¨atf¨urPhysik, Universit¨atBielefeld, Postfach 100131, 33501 Bielefeld, Germany 2Departament de Fisica Te`oricaand IFIC, Universitat de Val`encia,46100 Burjassot, Spain Received: date / Accepted: date Abstract We study the impact of large lepton flavour are however much fewer mechanisms known that could asymmetries on the cosmic QCD transition. Scenar- provide a first-order QCD transition (e.g. [5,6]), which ios of unequal lepton flavour asymmetries are observa- is why this possibility has obtained considerably less tionally almost unconstrained and therefore open up attention. The softening of the equation of state dur- a whole new parameter space in order to study the ing the QCD transition, be it first order or a crossover, nature of the cosmic QCD transition. For very large leads to an enhancement of the production of primor- asymmetries, we point out two limitations to our cur- dial black holes [7,8,9], one of the prime candidates for rent method, namely the possible formation of a Bose- dark matter, see e.g. [10]. Einstein condensate of pions and the reliability of the In this work, we present a simple and observation- Taylor expansion applied in this work. ally unconstrained mechanism that has an impact on the cosmic trajectory at temperatures around the cos- Keywords early Universe · cosmic QCD transition · mic QCD transition. We extend the work of [11] and lepton asymmetry discuss how large lepton flavour asymmetries could pos- sibly affect the cosmic QCD transition [12,13]. Besides the aforementioned imprint in the GW spectrum, a 1 Introduction first-order QCD phase transition could possibly lead to the formation of other observable relics of the early The recent direct detection of gravitational waves (GWs) Universe [14,15]. by the LIGO/Virgo collaboration [1] has revived the in- Our current understanding of the QCD phase dia- terest in phase transitions in the early Universe [2]. In gram is as follows. Whether strongly interacting par- general, first-order phase transitions can be accompa- ticles exist in the form of quarks and gluons or in the nied by processes that lead to the emission of GWs, form of hadrons (i.e. confined quarks and gluons) de- while crossovers do not lead to a strong enhancement pends in general on the temperature T and the baryon over the primordial GW spectrum. Within the Stan- chemical potential µB (and all other potentials associ- dard Model of particle physics (SM) both the electroweak arXiv:2009.00036v1 [hep-ph] 31 Aug 2020 ated with relevant conserved quantum numbers) of the transition at Tew ∼ 130 GeV as well as the transition of system. It is known from lattice QCD [16,17,18] that quantum chromodynamics (QCD) at TQCD ∼ 150 MeV at vanishing chemical potentials the transition between are expected to be crossovers. However, many exten- these two phases is a crossover, where a pseudocritical sions of the scalar sector of the Standard Model can give temperature is calculated to be TQCD = 156:5 ± 1:5 rise to a first order elelectroweak phase transition [3]. MeV [19] (TQCD = 147(2) − 165(5) MeV [20]). At large Even the simplest extension, i.e. by a real singlet, is baryon chemical potential and vanishing temperatures difficult to exclude at the LHC (see, e.g. [4]). There in contrast, effective models of QCD (like the Nambu ae-mail: [email protected] Jona-Lasinio model) predict a first-order chiral transi- be-mail: [email protected] tion. This leads to the speculation that there exists a ce-mail: [email protected] critical line in the (µB;T )-diagram which separates the de-mail: [email protected] two phases by a first-order transition and which is sup- 2 posed to end in a second-order critical end point. Func- volume lepton flavour number lα, baryon number b and tional QCD methods predict this critical end point to electric charge q in comoving volume can be written as CEP CEP be located around (µ ;T ) = (672; 93) MeV [21]. nα + nν B l = α (α = e; µ, τ); (1) The standard trajectory of the Universe in the QCD α s diagram is based on the assumption of tiny matter- X Bini b = ; (2) antimatter asymmetries and is expected to start at large s i temperatures and low baryon chemical potentials. Due X Qini q = : (3) to the expansion of space the temperature decreases, s but the baryon chemical potential is expected to re- i mπ The sum in the last two equations goes over all con- main small roughly until pion annihilation at T ≈ 3 , tributing particles species with baryon number Bi and when µB starts to approach the nucleon mass. There- fore, as mentioned above, the transition is expected to electric charge Qi. s denotes the total entropy density be a crossover. This idea of the standard cosmic trajec- and nj the net number densities of particle species j tory is based on the observation that the Universe has (i.e. the number density of particles minus the number an extremely small (and well measured) baryon asym- density of anti-particles). metry, i.e., b = (8:70±0:06)×10−11 (inferred from [22]). The basic idea of this work and [11,23] can be sum- If the lepton asymmetry is assumed to be on the same marized as follows: The cosmic trajectory in the QCD order of magnitude, this indeed implies a small baryon phase diagram is determined by the conservation of lα, mπ b and q, i.e., by the requirement that eqs. (1){(3) are chemical potential µB at T & 3 [11]. However, obser- vational constraints on the lepton asymmetry are much fulfilled throughout the evolution of the Universe af- weaker, allowing in principle a lepton asymmetry that ter the electroweak transition and before the onset of is many orders of magnitude larger than the baryon neutrino oscillations (Tosc < T < Tew). The comov- asymmetry. As shown in [13,11], for increasing values ing baryon number (from here on referred to as baryon −11 of the lepton asymmetry the cosmic trajectory is shifted asymmetry) can be fixed to b = 8:70×10 [22] and it is reasonable to assume an electric charge neutral Uni- towards larger values of µB. While [13,11] assumed the verse, q = 0 [25]. This leaves us with the three comov- lepton flavour asymmetries lα to have the same values l ing lepton flavour numbers lα as free input parameters, (i.e. lα = 3 where l is the total asymmetry), in this work we show that scenarios of unequal lepton flavour asym- which we refer to as lepton flavour asymmetries in the metries are even less constrained and therefore can have rest of this work. Each conserved charge can be assigned with a chemical potential, i.e., µ , µ and µ . Those an even larger impact on the cosmic trajectory. The im- B Q Lα pact of lepton flavour asymmetries on the abundance of chemical potentials can be related to the chemical po- weakly interacting dark matter has been studied in [23]. tentials of the different particle species on the RHS of This article is organized as follows. In section 2 eqs. (1){(3), for which we furthermore obtain relations we summarize our method to calculate the cosmic tra- by assuming chemical equilibrium. In general, large lep- ton asymmetries l induce not only large µ but also jectory. Section 3 discusses the situation of unequal α Lα lepton flavour asymmetries. Our main results are pre- large µB and µQ through relations (1){(3). This implies sented in section 4, where we also discuss the limita- that our choice for the lepton flavour asymmetries has tions of our current method. Further details are pro- a possible impact on the nature of the cosmic QCD vided in two appendices. We conclude in section 5. All transition. expressions of this work are provided in natural units, By numerically solving eqs. (1){(3) for 10 MeV < T < 500 MeV and for a given set of lα we obtain the i.e., c = ~ = kB = 1. Universe's trajectory in six-dimensional (µB; µQ; µLα ;T ) space. Due to the confinement of quarks into hadrons 2 Method at TQCD ∼ 150 MeV we divide this temperature range into three different regimes that we will describe in the In this section, we summarize the method of [11] and following. Leptons on the other can be treated at all also reveal an improvement to it in the high tempera- times in the same way, i.e., we calculate their ther- ture regime. For more details we refer the reader to [24, modynamic quantities assuming Fermi-Dirac distribu- 11,13]. tions (including chemical potentials). In the tempera- After the electroweak transition (Tew ∼ 100 GeV) ture regime of interest it is entirely sufficient to treat and before the onset of neutrino oscillations (Tosc ∼ 10 neutrinos as massless particles. MeV) electric charge, baryon number and lepton flavour number are conserved in a comoving volume. Assuming Quark-gluon plasma (QGP, T & TQCD) In [11] we made furthermore the conservation of entropy in a comoving the assumption that quarks and gluons behave as an 3 40 pressure, QCD QCD 1 4 p (T; µ)=p (T; 0) + µaχab(T )µb + O(µ ) 30 2 (4) ● QCD QCD ● ≡ p0 (T ) + p2 (T; µ); ● ● 3 ● ● 20 ● ● with an implicit sum over a; b 2 fB; Qg.