arXiv:2108.11966v1 [hep-ph] 26 Aug 2021 oma rtcltmeaueo bu 5MV[ MeV 85 about of temperature critical a at form evberlc uha h aynaymtyo h nvre[ universe the of asymmetry the as such relics servable eylwtmeaue,s htteuies uecoswl below well supercools phas universe the high-temperature [ that ture metastable so the temperatures, of low rate very decay it The furthermore, tures: and, transition, phase [ electroweak crossover first-order smooth a in broken get QCD of symmetries approximate [ fields magnetic scale large or eprtr eoe slwas low th as for becomes mass temperature thermal The field(s). Higgs the for mass (negative) no s electroweak both transition: phase - or QCD, a nor ohv rtodrcia hs rniin [ transitions massles phase 6 chiral with QCD first-order massless. a still have are to flavors quark 6 all and broken, tv,s httesaewith state the that so itive, 1 is-re hs rniin r neetn o omlg beca cosmology for interesting are transitions phase First-order oee,i erysaeivratetnin fteSadr Mod Standard the of extensions scale-invariant nearly in However, eako h C-lcrwa hs transition phase QCD-electroweak the on Remark [email protected] 9 hnhpe nascn-re hs rniin symmetry- sector. a light-quark transition, the in phase occur second-order could c a transition top-quark in breaking negligible happen symmetry be A not then QCD. may massless transition of phase chiral symmetries the on effect orde its first large is non-perturbatively transition is phase interaction Yukawa chiral top-Higgs the symmet that the argued on been Based has transition. through massl electroweak potential 6 the Higgs with triggering the proceed o destabilizes could of condensate QCD temperatures of quark transition to phase delayed chiral be the can transition phase troweak – 12 netnin fteSadr oe ihn iesoflpar dimensionful no with Model Standard the of extensions In auta ¨rPyi,Uiesta ilfl,351Biel 33501 Fakult¨at Universit¨at Bielefeld, f¨ur Physik, .Iprateape r oea-enegtp oes[ models type Coleman-Weinberg are examples Important ]. nasproldUniverse supercooled a in H 4 .I h tnadMdlteei ete nelectroweak, an neither is there Model Standard the In ]. ean eatbeuntil metastable remains 0 = O 10 e hl h lcrwa ymtyi tl un- still is symmetry electroweak the while MeV (100) itihB¨odekerDietrich Abstract 1 14 ,i hc ur condensates quark which in ], 15 .I hsas apn ntestrongly the in happens also this If ]. 1 tteQDsae n that and scale, QCD the at .W on u htthe that out point We r. h uaainteraction, Yukawa the 1 iso asesQD it QCD, massless of ries a aevr euirfea- peculiar very have can , iltn oeo the of some violating , T 2 fl,Germany efeld, dr10MV Then MeV. 100 rder s urs h top- The . ess ,gaiainlwvs[ waves gravitational ], raigfirst-order breaking .I oecssthe cases some In 0. = a etn onto down tiny be can e nesto would ondensation mer [ ymmetry ig field Higgs e s hycnlaeob- leave can they use mtr h elec- the ameters h rtcltempera- critical the 7 ursi expected is quarks s , 8 13 lteecnb a be can there el ] ,wihcontain which ], 5 , 6 H n the and ] h qq spos- is 6 i 0 = 3 ], supercooled universe, the top condensate would give rise to a linear term in the effective Higgs potential through the top Yukawa interaction

L t-Yukawa = ytQ3HtR + H.c., (1) e where Q3 = (tL, bL) is the doublet of third-family left-handed quarks, tR is the right- handed top quark, and H = iσ2H∗. The linear term destabilizes the minimum of the e effective potential at H = 0 and triggers electroweak symmetry breaking [9]. After the top condensation the universe either evolves into a metastable state, followed by another phase transition, or it could directly roll to the minimum of the effective potential. Similarly patterns arise in other scale invariant SM extension, such as Randall-Sundrum models [16]. At a QCD phase transition the QCD coupling is large, and perturbation theory cannot be applied. Non-perturbative lattice computations with light quarks are very difficult (see [17] for simulations with large number of flavors). Insights into the nature of a phase transition can be gained from symmetry considerations. The Lagrangian of massless

QCD with 6 quarks is invariant under chiral SU(6)L × SU(6)R flavor-transformations. A quark condensate breaks the axial subgroup, leaving a vector SU(6) symmetry. If this happens in a second-order phase transition, then at the critical temperature some fields become effectively massless,2 and the corresponding effective theory becomes scale invariant. Then the group for the couplings of this effective theory has an infrared-stable fixed point. The non-existence of an infrared-stable fixed point, on the other hand, is an indication that the phase transition is first order. Near a second- order phase transition, the effective long-distance degrees of freedom are scalar fields describing the quark condensates. The effective Lagrangian and thus the beta functions are determined by the symmetries. The beta functions can be computed in the so-called ǫ expansion: They are treated perturbatively near d = 4 dimensions by writing d =4 − ǫ, and expanding in powers of ǫ. The effective theory describing the long-distance degrees of freedom near a second order is 3-dimensional, so the ǫ expansion is applied to ǫ = 1. For massless QCD with Nf ≥ 3 quark flavors one finds no infrared-stable fixed point, which hints at a first-order chiral phase transition [14]. At nonzero but small quark masses a first order phase transition is expected, even though it has not been observed in lattice simulations so far [18–20]. Previous discussions of an electroweak phase transition triggered by top-quark con- densation [11] were based on the symmetries of QCD alone. However, neither Yukawa nor electroweak gauge interactions respect the SU(6)L × SU(6)R symmetry. One may as- sume that electroweak interactions can be neglected, but the top Yukawa coupling become nonperturbatively large near the QCD scale [21, 22]. Therefore it is likely that the top

2Or, equivalently, some correlations lenghts diverge.

2 Yukawa interaction in Eq. (1) would affect the chiral phase transition. When the Higgs field and the top Yukawa interaction are included, the remaining symmetry is

G = U(1)B × U(4)L ×U(5)R ×Gt , (2) where U(1)B represents baryon number, U(4)L acts on uL,...,cL, U(5)R acts on uR,...,bR, and Gt =SU(2)×U(1) acts like the electroweak gauge group, but only on Q3, tR, and H. In a chiral phase transition G is broken by expectation values of

Φij ∼ qLiqRj. (3)

If such a phase transition was second order, some components of Φ would become effec- tively massless at the critical temperature. Unlike in the SU(6)L × SU(6)R symmetric case [14], not all Φij become massless at the same temperature, but only those related by the symmetry G. Thus the breaking of G might proceed in a sequence of phase transitions.

When the top condensate forms, Φi6 with i = 5, 6 (corresponding to b and t), gets a nonzero expectation value. It transforms as a doublet under SU(2).3 The long-distance effective theory is equivalent to an O(4) model, which is known to have a second-order phase transition (see e.g. [23]). Thus one would conclude that the breaking of Gt through top-quark condensation happens in a second-order phase transition. However, this is not the only possibility. There could also first be a phase transition giving an expectation value to Φij with i =1,..., 4 or with i =5, 6, and j =1,..., 5, or † to Φi6 with i =1,..., 4. These fields transform like Φ → U ΦV with U ∈ SU(4) or SU(2), and V ∈ SU(5), or like Φ → U †Φ with U ∈ SU(4). In [24] the fixed points for complex scalar fields Φ = (Φij), i = 1,...,M, j = 1,...,M with the U(M)×U(N)-symmetric Lagrangian

† µ † u † 2 v † 2 L = Tr(∂µΦ )(∂ Φ) + rTr Φ Φ+ Tr Φ Φ + Tr Φ Φ (4) 4 4 were studied at leading order in the ǫ expansion. It was found that there is an infrared- stable fixed point for MN < 2, and that there are additional fixed points when (N+M)2 ≥ 2 1/2 12(MN − 2), i.e., for N > N+ or N < N− with N± =5M ± [24(M − 1)] . Then there should be no infrared-stable fixed points for (M, N) = (4, 5) or (M, N) = (2, 5). One would therefore conclude that in these two cases such a transition is first order, while it is second order for (M, N) = (4, 1). The analysis of [24] was extended to five [25] and recently to six loops [26]. The ǫ expansion turns out to be not particularly well behaved for ǫ = 1. The result of a resummation performed in Ref. [25] indicate that N+ could be slightly less than 5 when M = 2, so there might be an infrared-stable fixed point

3It can mix with H, which has the same quantum numbers. e

3 for (M, N) = (2, 5). This was confirmed in [26]. On the other hand, Refs. [24–26] all

find that N+ > 5 for M = 4, indicating that there is a first-order phase transition for (M, N)=(4, 5). If there is a first-order phase transition as described in the previous paragraph, there are two possibilities. The first is that in this phase transition a top condensate emerges as well, not for symmetry reasons, but because there is a discontinuous change of the properties of the system. That would mean that the top condensate forms in a first- order phase transition. Another possibility is that the top condensate forms later in a second-order transition as described above. Which of these two possibilities is realized can probably not be inferred from symmetry considerations, but only from a detailed QCD computation.

If electroweak interactions cannot be neglected, the Φij are not gauge invariant, and cannot serve as an order parameter indicating the breaking of a global symmetry. In particular, there is no global symmetry which gets broken when tt¯ gets an expectation value from which one could infer information about the existence or the order of a phase transition.4 Baryon number is violated due to electroweak sphalerons. A remaining global symmetry is SU(2)Q ×SU(2)u ×SU(3)d acting on the left-handed quark doublets and the right-handed up-type quarks of the first two families, and on all right-handed down- type quarks. This symmetry can be broken by expectation values of the gauge invariant operators Q Hu (α = 1, 2; j = 1, 2), or Q Hd (α = 1, 2; j = 1, 2, 3). In a phase α e j α j transition in which these operators acquire an expectation value the Higgs field would also have to become non-zero. Then it would still have to overcome the barrier of the effective potential, without assistance from the top quark. Acknowledgments I would like to thank Frithjof Karsch, Rob Pisarski, Laura Sagunski, Wolfgang Unger and Clemens Werthmann for useful discussions or remarks. This work was funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 315477589 – TRR 211.

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