The Standard Model of particle physics is minimally extended by a dilaton field. This field expresses the spontaneous breaking of scale invariance of an unspecified ultraviolet complete theory. While being manifestly scale invariant at the classical level, the low energy theory admits quantum scale anomaly responsible for the generation of mass scales. In addition to predicting a very small dilaton mass, this model has important consequences for the cosmology of the early universe. The electroweak phase transition can only be triggered by the QCD chiral phase transition occurring around 132 MeV. Since the QCD transition is expected to be first order in this picture, such a scenario gives rise to the production of stochastic gravitational waves with a peak frequency around ∼ 10−8 Hz

Lagger, C.

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DOI 10.1393/ncc/i2018-18133-5Colloquia: LaThuile 2018IL NUOVO CIMENTO 41 C (2018) 133Cosmological phase transitions in the Standard Model withhidden scale invarianceC. Lagger(∗)ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics,The University of Sydney - Sydney, NSW 2006, Australiareceived 6 September 2018Summary. — The Standard Model of particle physics is minimally extended by adilaton field. This field expresses the spontaneous breaking of scale invariance of anunspecified ultraviolet complete theory. While being manifestly scale invariant at theclassical level, the low energy theory admits quantum scale anomaly responsible forthe generation of mass scales. In addition to predicting a very small dilaton mass,this model has important consequences for the cosmology of the early universe.The electroweak phase transition can only be triggered by the QCD chiral phasetransition occurring around 132 MeV. Since the QCD transition is expected to befirst order in this picture, such a scenario gives rise to the production of stochasticgravitational waves with a peak frequency around ∼ 10−8 Hz.1. – IntroductionScale invariance is an appealing concept in field theory to accommodate both theorigin of mass as well as the hierarchy between mass scales. Starting from a classicalLagrangian containing only dimensionless couplings, quantum fluctuations are knownto break classical scale invariance and to generate an overall scale through dimensionaltransmutation [1]. Additional mass scales and the hierarchy between them are theninduced from various combinations of the initial couplings with the overall scale.Among different approaches, one concrete realisation proposed in [2] consists in as-suming the existence of an ultraviolet complete and scale invariant theory. Withoutspecifying the details of such a completion, it is assumed that spontaneously brokenscale invariance manifests itself in the low energy regime of the theory through a dilatonfield. In practice, this low energy theory can be described as a minimal extension ofthe Standard Model with all the dimensionfull parameters promoted to the dynamicaldilaton. Phenomenological and cosmological properties of this model are presented.(∗) E-mail: cyril.lagger@sydney.edu.auCreative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 C. LAGGER2. – Standard Model with hidden scale invarianceThe Standard Model is considered here as an effective low energy limit of an unknownUV complete theory. The energy scale Λ below which the approximation is valid rep-resents a physical parameter encompassing the effects of the ultraviolet sector. In thisway, the Higgs potential is written as [2](1) V (Φ†Φ) = V0(Λ) + λ(Λ)[Φ†Φ − v2ew(Λ)]2+ · · · ,where Φ is the Higgs electroweak doublet and V0 the bare cosmological constant. Theellipsis indicates that higher order nonrenormalisable operators are irrelevant at lowenergy. In order to express the scale invariance of the fundamental theory in this regime,the mass parameters (including the cut-off Λ) are replaced by a dynamical dilaton fieldχ as follows:(2) Λ → Λ χfχ, v2ew(Λ) →v2ew(χ)f2χχ2 ≡ ξ(χ)2χ2, V0(Λ) →V0(χ)f4χχ4 ≡ ρ(χ)4χ4,where fχ is the dilaton decay constant associated with the spontaneous breaking of scalesymmetry. The new Higgs-dilaton potential is therefore explicitly scale invariant at theclassical level(1)(3) V (Φ†Φ, χ) = λ(χ)[Φ†Φ − ξ(χ)2χ2]2+ρ(χ)4χ4.In this approach, mass scales originate from quantum scale anomaly which is incor-porated in the potential (3) through the χ-dependence of the dimensionless couplings:λ(χ) = λ(μ) + βλ(μ) ln(χ/μ) + β′λ(μ) ln2(χ/μ) + · · · (idem for ξ(χ) and ρ(χ)). For con-venience, the renormalisation scale μ is fixed at the dilaton VEV defined itself at thecut-off scale: vχ = Λ. The existence of this dilaton VEV actually relies on the presence,at the classical level, of a flat direction in the potential (3), along which the Coleman-Weinberg mechanism [1] takes place. This is mathematically expressed by solving thetwo minimisation conditions dVdχ |Φ=vew,χ=vχ =dVdΦ |Φ=vew,χ=vχ = 0. Taking also into ac-count the phenomenological constraint on the cosmological constant, V (vew, vχ) = 0,this leads to [2]ρ(Λ) = 0, βρ(Λ) = 0, ξ(Λ) =v2ewv2χ.(4)These equations provide a relation between dimensionless and dimensionfull parameters(dimensional transmutation) as well as the hierarchy between the electroweak and thehigh-energy scales. Note that the coupling ξ(Λ) can be arbitrarily small in the technicalsense [3-5].The running mass of the dilaton is as well induced by the quantum anomaly, at twoloop level [2]: m2χ(Λ) β′ρ(Λ)4ξ(Λ) v2ew. In particular, the dilaton appears to be very light(1) For simplicity, the dilaton decay constant is assumed here to be similar to the cut-off scale:fχ ∼ Λ.PHASE TRANSITIONS IN THE STANDARD MODEL WITH SCALE INVARIANCE 3169 170 171 172 173 174 175101910201021mt/GeVΛ/GeVFig. 1. – Plot of the allowed range of parameters (shaded region) with m2χ(vew) > 0, i.e., theelectroweak vacuum being a minimum. The solid line displays the cut-off scale Λ as a functionof the top-quark mass mt for which the conditions in eq. (4) are satisfied.compared to the Higgs mass m2h  2λ(Λ)vew, namely m2χ/m2h  ξ(Λ). An estimate forthe physical dilaton mass can then be evaluated by solving the (one-loop) renormalisationgroup equations starting from the high-energy scale Λ down to the electroweak scale [2,6].Such a solution has to satisfy the conditions (4) as well as m2χ(vew) > 0 to ensure thatthe field configuration corresponds to a local minimum of the potential at low energy.The results are shown in fig. 1 which displays the allowed values of Λ and top quarkmass mt satisfying these conditions. For a cut-off at the Planck scale the dilaton massis predicted to be mχ ≈ 10−8 eV.3. – Cosmological phase transitionsThe behaviour of the Higgs and dilaton fields during the early universe is describedby the free energy density of the system, given by the finite-temperature Higgs-dilatonpotential VT (Φ, χ). This potential can be evaluated as the sum of the zero-temperatureexpression (3) and one-loop thermal corrections (see review [7]). It is actually sufficientto evaluate it only along the flat direction of the classical potential, namely where themain dynamics takes place. In this case, the thermal potential can be expressed as afunction of the Higgs field only. The main effect of thermal corrections can then beunderstood by looking at the high-temperature expansion of the potential [6]VT (h, χ(h)) =[c(h)π2 − λ(Λ)576]T 4+148[4λ(Λ) + 6y2t (Λ) +92g2(Λ) +32g′2(Λ)]h2T 2 + · · · ,(5)where h is the neutral component of the Higgs doublet Φ = (0, h/√2)T and c(h) isthe number of relativistic degrees of freedom at temperature T . In (5), the temperatureindependent terms have vanished as a consequence of the flatness of the classical potential4 C. LAGGER0.0 0.1 0.2 0.3 0.4 0.5–0.000050.00000.000050.0001h / GeVV / GeV4 T=132 MeVT=130 MeVT=128 MeVT=126 MeVFig. 2. – VT (h) − VT (0) for different temperatures below the chiral phase transition.in this direction(2). Since the coefficient 4λ(Λ) + 6y2t (Λ) +92g2(Λ) + 32g′2(Λ) is positive(as confirmed numerically), the term proportional to T 2 produces a barrier between theminimum at h = 0 and the minimum corresponding to electroweak symmetry breaking.Taking into account that these two minima are degenerate at the classical level, V (0, 0) ≈V (vew, vχ) (see eqs. (3), (4)), and that the barrier persists until T = 0, the electroweakphase transition will not occur unless the universe reaches a very low temperature.The previous analysis indicates that the temperature at which QCD interactionsbecome strong will be reached before the electroweak transition happens. Therefore,all quarks are still massless at this epoch (since h = 0) and the chiral symmetry is givenby SU(6)L × SU(6)R. Once this symmetry gets spontaneously broken, quark-antiquarkcondensates form, according to [8]〈q̄q〉T = 〈q̄q〉[1 − (N2 − 1) T212Nf2π− 12(N2 − 1)(T 212Nf2π)2+ · · ·],(6)where N is the number of massless quarks, 〈q̄q〉 ≈ −(250MeV)3 is the zero temperaturecondensate and fπ ≈ 93MeV is the pion decay constant. As pointed out by Witten [9] andothers [10], these condensates contribute linearly to the Higgs-dilaton potential throughthe quark-Higgs Yukawa interactionsVT (h, χ(h)) → VT (h, χ(h)) + yq〈q̄q〉Th√2,(7)where yq is the Yukawa coupling of the q-type quark. As the temperature decreasesand 〈q̄q〉T increases, this new term starts to dominate over the thermal barrier until thebarrier completely disappears. The Higgs field will then smoothly rolls from h = 0 toh = vew. Namely, the chiral phase transition triggers the electroweak symmetry breaking.(2) Zero-temperature one-loop corrections are not included in this expression as they are sub-dominant compared to thermal corrections.PHASE TRANSITIONS IN THE STANDARD MODEL WITH SCALE INVARIANCE 5This scenario has been numerically investigated with the use of the full one-loopeffective thermal potential [6]. In summary, the QCD phase transition occurs at thecritical temperature Tc ≈ 132MeV defined by 〈q̄q〉Tc = 0, for N = 6. Then, as shown infig. 2, the barrier in the potential (7) persists for 127 < T < 132MeV, but it disappearsonce T ≈ 127MeV and the electroweak phase transition completes at that temperature.One typical prediction of this scenario is the production of a stochastic background ofgravitational waves. Although the Higgs transition is homogeneous and does not proceedthrough bubble nucleation, there are theoretical arguments [11] supporting that QCDchiral phase transition with N ≥ 3 massless quark is first-order, with the production ofQCD vacuum bubbles. Therefore, bubble collisions and bubble-plasma interactions willgenerate gravitational waves with a peak frequency fp ≈ R−1c where Rc is the bubble sizeat collision time tc. Taking into account the redshift, the peak frequency today becomesf0 ≈ fpa(tc)/a(t0) ≈ 10−8 Hz for typical values of Rc [6]. Such a signal can potentiallybe detected by pulsar timing arrays like the SKA telescope [12].4. – ConclusionBoth phenomenological and cosmological properties of a minimal scale invariant ex-tension of the Standard Model have been presented. Motivated by the origin of massand hierarchy problem, this model introduces a new dilaton field in order to dynam-ically generate all the mass scales of the theory. This dilaton is predicted to be verylight. Moreover, the evolution of the thermal Higgs-dilaton potential shows that theelectroweak symmetry breaking can only be triggered by the QCD chiral phase transi-tion occurring at T ≈ 132 MeV. This QCD transition is expected to be first order andto produce a gravitational wave background with a peak frequency ∼ 10−8 Hz.∗ ∗ ∗I acknowledge S. Arunasalam, A. Kobakhidze, S. Liang and A. Zhou for a fruitful andenjoyable collaboration. The work was supported in part by the Australian ResearchCouncil.REFERENCES[1] Coleman S. and Weinberg E., Phys. Rev. D, 7 (1973) 1888.[2] Kobakhidze A. and Liang S., arXiv:1701.04927 [hep-ph].[3] Wetterich C., Phys. Lett. B, 140 (1984) 215.[4] Bardeen W., FERMILAB-CONF-95-391-T.[5] Kobakhidze A. and McDonald K., JHEP, 07 (2014) 155, arXiv:1404.5823 [hep-ph].[6] Arunasalam S., Kobakhidze A., Lagger C., Liang S. and Zhou A., Phys. Lett. B,776 (2018) 48, arXiv:1709.10322 [hep-ph].[7] Quiros M., arXiv:hep-ph/9901312.[8] Gasser J. and Leutwyler H., Phys. Lett. B, 184 (1987) 83.[9] Witten E., Nucl. Phys. B, 177 (1981) 477.[10] Buchmuller W. and Wyler D., Phys. Lett. B, 249 (1990) 281.[11] Pisarski R. and Wilczek F., Phys. Rev. D, 29 (1984) 338.[12] Dewdney P., Hall P., Schilizzi R. and Lazio T., Proc. IEEE, 97 (2009) 1482.

Cosmological phase transitions in the Standard Model with

hidden scale invariance

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Cosmological phase transitions in the Standard Model with hidden scale invariance

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