Baryogenesis via gauge field production from a relaxing Higgs
Yann Cado, Benedict von Harling, Eduard Masso, Mariano Quiros
PPrepared for submission to JHEP
Baryogenesis via gauge field production from arelaxing Higgs
Yann Cado, a Benedict von Harling, a Eduard Mass´o, a,b
Mariano Quir´os a a Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, CampusUAB, 08193 Bellaterra (Barcelona), Spain b Grup de F´ısica Te`orica, Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra,Spain
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We show that the baryon asymmetry of the universe can be explained in models wherethe Higgs couples to the Chern-Simons term of the hypercharge group and is away from the late-time minimum of its potential during inflation. The Higgs then relaxes toward this minimum onceinflation ends which leads to the production of (hyper)magnetic helicity. We discuss the conditionsunder which this helicity can be approximately conserved during its joint evolution with the thermalplasma. At the electroweak phase transition the helicity is then converted into a baryon asymmetryby virtue of the chiral anomaly in the standard model. We propose a simple model which realizesthis mechanism and show that the observed baryon asymmetry of the universe can be reproduced.
Keywords:
Higgs, baryogenesis, magnetic helicity, chiral anomaly a r X i v : . [ h e p - ph ] F e b ontents Electroweak (EW) baryogenesis is an appealing mechanism which can in principle generate thebaryon asymmetry of the universe during the EW phase transition. However, it requires this phasetransition to be first-order, while in the standard model (SM) it is a smooth crossover. New physicsbeyond the SM is then necessary to make it first-order. In this paper, we will consider a scenarioin which the baryon asymmetry is also generated during the EW phase transition but the lattercan be a crossover like in the SM. It has another similarity with EW baryogenesis in the sense thatthe dynamics of the Higgs is important for the mechanism. Compared to EW baryogenesis though,part of this dynamics takes place at much earlier times, after the end of inflation.We assume that the Higgs Φ couples to the Chern-Simons term of the hypercharge gauge group, | Φ | Y µν ˜ Y µν . If the Higgs is elongated away from the late-time minimum of its potential duringinflation, it will start to relax toward this minimum after inflation ends. During this stage, the EWsymmetry is broken. The evolution of the Higgs results in the copious production of photons viathe above coupling. As we will see, these photons are predominantly produced with one helicity,leading to a helical background of photons. Once the universe reheats and the EW symmetryis restored, they are transformed into helical hypermagnetic fields. The latter subsequently startto interact and evolve jointly with the thermal plasma. Under certain conditions, which we will– 1 –iscuss in some detail, the helical hypermagnetic fields can survive until the EW phase transition.The EW symmetry is then once again broken and the hypermagnetic fields are transformed intoordinary magnetic fields. The hypercharge gauge boson contributes to the anomaly of baryon pluslepton number, B + L , and changes in the hypermagnetic helicity therefore result in changes ofthe B + L -charge [1]. Since the photon does not contribute to the anomaly, the transformation ofthe hypermagnetic fields into ordinary magnetic fields leads to a compensating B + L -asymmetry.While it is being generated, EW sphalerons start to erase this asymmetry but they freeze outtoward the end of the phase transition when the source for the asymmetry is still active. Bycarefully analyzing the dynamics of the EW phase transition in the SM, it was shown in [2] that asizeable B + L -asymmetry can survive which can reproduce the observed baryon asymmetry of theuniverse.This baryogenesis mechanism has previously been considered in the context of axion inflation,where the hypermagnetic helicity is produced during inflation via the coupling of the axion-likeinflaton to the Chern-Simons term of the hypercharge gauge group [3–6]. Other related scenariosinvolving the decay of hypermagnetic helicity to generate the baryon asymmetry have been studiedin [7–13]. The relaxation of the Higgs toward the minimum of its potential after inflation, on theother hand, has been used in [14–16] to implement spontaneous baryogenesis via a coupling of theHiggs to the B + L -current. This is related to our scenario by a chiral rotation of the SM fermions(but also differs in important aspects).To induce a large vacuum expectation value (VEV) for the Higgs during inflation, we consider acoupling to the Ricci scalar. The sign of this coupling is chosen such that it leads to a tachyonic massterm for the Higgs. This mass term and thus the minimum of the Higgs potential are approximatelyconstant during inflation but they decrease rapidly once inflation ends. The Higgs then startsmoving from its large initial VEV toward the origin of its potential. This leads to the productionof a helical background of photons as mentioned before. Let us emphasize that the coupling to theRicci scalar is only one option to obtain a large initial VEV for the Higgs. It has the advantagethat it allows us to consider only the time evolution of the Higgs. We also comment on anotheroption, a coupling of the Higgs to the inflaton which also results in a tachyonic mass term, butleave a more detailed study to future work.As we will discuss, the magnetic fields which are produced after inflation need to be sufficientlystrong to guarantee their survival until the EW phase transition via a process known as the inversecascade. We find that this in turn means that the initial Higgs VEV needs to be quite large, (cid:29) GeV. At such large VEVs, the Higgs quartic coupling is expected to run to negative valuesin the SM. This is dangerous since the Higgs could be driven into the resulting unphysical deeperminimum during inflation. In order to avoid this, we will couple the Higgs to a scalar singlet whosequantum corrections ensure that the Higgs quartic coupling stays always positive. Again, this isonly one option and of course different new physics could be introduced to achieve the same effect.Armed with a model for the Higgs potential, we numerically solve for the time evolution of theHiggs and from this calculate the produced helicity and magnetic field strength. For definiteness,we study three benchmark points which span a representative set of possibilities. Our main resultis that the helicity can survive until the EW phase transition and that its conversion can reproducethe observed baryon asymmetry of the universe.This paper is organized as follows. We derive the relevant equations of motion (EOMs) for theHiggs and the photon in sec. 2 and comment on the backreaction of photon production on the Higgs,finite-temperature effects and the simultaneous generation of asymmetries in SM fermions. In sec. 3,we then discuss the evolution of the helicity and the asymmetries between reheating and the EWphase transition and derive conditions on the survival of the helicity. Furthermore, we estimate thebaryon asymmetry that is generated during the EW phase transition from the conversion of thehelicity. We specify the Higgs potential in sec. 4 and present our numerical results for the three– 2 –enchmark points. Finally, we conclude in sec. 5. Three appendices complement the paper. Inappendix A, we give an example of a ultraviolet (UV) completion for the required coupling of theHiggs to the hypercharge gauge boson. The renormalization group (RG) equations for the modelwith the added singlet scalar are presented in appendix B and the initial conditions of the Higgsafter inflation are derived in appendix C.
We consider the action S = (cid:90) d x (cid:20) √− g (cid:18) − g µν D µ Φ D ν Φ † − V (Φ) − g µν g ρσ Y µρ Y νσ (cid:19) + 12 | Φ | M Y µν ˜ Y µν (cid:21) , (2.1)where Φ is the Higgs doublet, V its potential, Y µν the field strength of the hypercharge gaugefield A Y µ and ˜ Y µν = (cid:15) µνρσ Y ρσ / (cid:15) = 1. We will use the conformally-flat Robertson-Walker metric g µν = a η µν , where a is the scale factor and η µν = diag( − , , , M will be specified later.We will be interested in the relaxation of the Higgs from some initial value after inflation to theminimum of its potential. The VEV of the Higgs is then large (except for the brief moments whenit crosses zero in case it oscillates) and EW symmetry is broken. The higher-dimensional couplingin eq. (2.1) is accordingly replaced by couplings to the photon and the Z boson. Since the Z bosonis massive, we will focus on the coupling to the photon. Furthermore, all but one degree of freedomof the Higgs doublet are eaten and we get S = (cid:90) d x (cid:20) − a ∂ µ h ∂ µ h − a V ( h ) − F µν F µν + cos θ W h M F µν ˜ F µν + . . . (cid:21) , (2.2)where h is the real Higgs field, F µν the field strength of the photon A µ and θ W the EW angle. Fromhere onwards, all spacetime indices are contracted with η µν .We will consider the evolution of the Higgs and the photon starting at the end of inflation. Weassume that the inflaton initially oscillates in its potential, leading to a matter-dominated phasebefore reheating. Furthermore, we assume that perturbative and non-perturbative decays of theinflaton during this period are negligible and set the temperature to zero. The validity of the latterassumption will be discussed in sec. 2.3. The EOMs for the Higgs and the photon read (cid:3) h − τ ∂∂τ h − τ τ ddh V ( h ) + cos θ W τ τ hM F µν ˜ F µν = 0 (2.3) ∂ µ F µν − cos θ W (cid:15) µνρσ ∂ µ h M ∂ ρ A σ = 0 , (2.4)where τ is the conformal time. Here and below we fix a = 1 at the onset of matter dominationwhich corresponds to the conformal time τ md = 2 /H inf , where H inf is the Hubble rate at the endof inflation.If the Higgs is away from its late-time minimum after inflation, it subsequently relaxes towardthis minimum. As we will momentarily see, the coupling to the Higgs in eq. (2.4) then leads tothe production of photons. The latter can in turn backreact on the Higgs via the coupling ineq. (2.3). As discussed in [17, 18], if this backreaction is important, it leads to the excitation of– 3 –igher-momentum modes of the Higgs and it is no longer sufficient to consider the evolution of thezero mode. In our numerical simulations, we will always ensure that the backreaction is sufficientlysmall to be neglected which will yield a condition on the ratio h/M at the end of inflation. We canthen restrict ourselves to the zero mode of the Higgs, i.e. h ( τ, x ) = h ( τ ), and neglect the couplingto the gauge field in its EOM which simplifies to ∂ ∂τ h + 4 τ ∂∂τ h + τ τ ddh V ( h ) = 0 . (2.5)For a given potential and initial condition, we can now solve for the time evolution of the Higgs. In order to see how the relaxing Higgs leads to the production of photons, we next quantize thephoton. We define A µ = ( A , A ) and work in radiation gauge A = 0 and ∇ · A = 0. Going tomomentum space, we then get A ( τ, x ) = (cid:88) λ = ± (cid:90) d k (2 π ) (cid:2) (cid:15) λ ( k ) a λ ( k ) A λ ( τ, k ) e i k · x + h.c. (cid:3) , (2.6)where λ = ± is the helicity of the photon and the a λ ( k ) are annihilation operators that fulfill thecanonical commutation relations. The polarization vectors satisfy k · (cid:15) λ ( k ) = 0 , k × (cid:15) λ ( k ) = − iλk (cid:15) λ ( k ) , (cid:15) ∗ λ (cid:48) ( k ) · (cid:15) λ ( k ) = δ λλ (cid:48) , (cid:15) ∗ λ ( k ) = (cid:15) λ ( − k ) , (2.7)where k ≡ | k | . From eq. (2.4), the EOM for the mode functions reads ∂ ∂τ A λ ( τ, k ) + k ( k − λ ξ ( τ )) A λ ( τ, k ) = 0 , (2.8)where ξ ( τ ) ≡ − cos θ W ∂ τ h M . (2.9)We solve eq. (2.8) using the evolution of the Higgs that follows from eq. (2.5), starting from the endof inflation. Assuming ξ ≈ τ ≤ τ md , the initial conditions for the mode functions at τ = τ md are given by A λ ( τ md , k ) = 1 √ k (2.10a) ∂A λ ( τ md , k ) ∂τ = − i (cid:114) k . (2.10b)As follows from eq. (2.8), modes with momenta k < | ξ | and helicity λ = sign( ξ ) have a tachyonicinstability. This leads to the copious production of these modes. In addition, also modes withmomenta k > | ξ | can be produced via parametric resonance due to the time-dependence of theeffective mass in eq. (2.8). We will be interested in having an imbalance in the amplitudes of modeswith positive and negative helicity that are produced. The net helicity density is given by H ≡ lim V →∞ V (cid:90) V d x (cid:104) A · B (cid:105) = (cid:90) dk k π (cid:0) | A + | − | A − | (cid:1) , (2.11) To be more precise, we choose Coulomb gauge ∇ · A = 0. The ν = 0-component of eq. (2.4) together with ∇ h ( τ ) = 0 then allows us to set A = 0. Note that if ξ (cid:54) = 0 for τ ≤ τ md , modes with momenta k < | ξ | may already be excited above the vacuum. Usingeq. (2.10) for the initial conditions then corresponds to neglecting this contribution. – 4 –here the integral over V averages the quantity over space and (cid:104) ... (cid:105) denotes the expectation value ofthe operators. The magnetic field in eq. (2.11) is given by B = ∇ × A . Similarly, in the radiationgauge, the electric field is given by E = − ∂ τ A . These are comoving quantities from which thephysical electric and magnetic fields follow as ˆ E = E /a and ˆ B = B /a . Hereafter we will denotephysical quantities, as opposed to comoving ones, with a hat. The energy densities in the electricand magnetic field read ρ E ≡ lim V →∞ V (cid:90) V d x (cid:104) E (cid:105) = (cid:90) k c dk k π (cid:0) | ∂ τ A + | + | ∂ τ A − | (cid:1) (2.12a) ρ B ≡ lim V →∞ V (cid:90) V d x (cid:104) B (cid:105) = (cid:90) k c dk k π (cid:0) | A + | + | A − | (cid:1) . (2.12b)The integral over momentum space diverges as k for large momenta since | ∂ τ A λ | = (cid:112) k/ | A λ | = 1 / √ k in the vacuum. We therefore impose a momentum cutoff k c which we choose asthe largest momentum for which the corresponding modes are excited above the vacuum afterproduction has shut off, | A λ | > c/ √ k with c an O (1) constant. We have checked that our resultsdo not depend sensitively on the value of this constant c . We will also need the correlation lengthof the magnetic field which can be estimated as [19] λ B = 2 πρ B (cid:90) dk k π (cid:0) | A + | + | A − | (cid:1) . (2.13)Note that the helicity H , the energy densities ρ E,B and the correlation length λ B are again comovingquantities. The corresponding physical quantities are given by ˆ H = H /a , ˆ ρ E,B = ρ E,B /a andˆ λ B = aλ B . We now comment on three issues that could affect the evolution of the Higgs and the production ofphotons. Firstly, the Higgs of course couples not only to the photon via the term in eq. (2.2) but alsoto the other SM particles. The time evolution of the Higgs in particular leads to time-dependentmasses for the latter which in turn results in the non-perturbative production of SM particles [20].The backreaction from this process can then affect the time evolution of the Higgs. Due to thePauli exclusion principle, the production of fermions is suppressed. We can therefore focus on themassive gauge bosons, W ± and the Z . For a Higgs potential and initial conditions similar to theones that we will consider, it was found in [16] that the production of W ± and Z bosons via thisprocess begins to affect the time evolution of the Higgs only after it has oscillated several times inits potential. This can be understood from the fact that the production mechanism is active onlywhile the Higgs crosses zero and the SM particles become light. The Higgs therefore has to crosszero several times before enough energy is dumped into W ± and Z bosons to affect its evolution.This is in contrast to the production of photons via the coupling in eq. (2.2) which is active also forlarge Higgs VEVs and is thus much more efficient. We will find that the majority of the helicity isproduced during the first few oscillations and will therefore use eq. (2.5) for the Higgs, neglectingthe backreaction from the production of W ± and Z bosons at early stages (while we check that thebackreaction from photon production is small as discussed in sec. 2). Related to this, the Higgs canalso decay perturbatively which leads to an additional damping term in eq. (2.5). The Higgs decayrate is dominated by decays into b quarks also for large Higgs VEVs [20]. The corresponding decayrate is given by Γ( h → b ¯ b ) = 3 / √ λ h y b π h (cid:18) − y b λ h (cid:19) / , (2.14)– 5 –here h is the Higgs VEV, λ h the (field-dependent) Higgs quartic coupling and y b ∼ − theYukawa coupling to the b quark. For h (cid:28) · H/ √ λ h , where H is the Hubble rate, the dampingterm from perturbative decays is negligible compared to the Hubble-induced damping term ineq. (2.5). This will always be fulfilled in the cases that we consider.Secondly, the presence of a strong electromagnetic field can lead to the pair production ofcharged particles via the Schwinger effect [21]. The particles are subsequently accelerated in theelectromagnetic field, leading to a current [22–27]. This current drains energy from the electromag-netic field and thereby backreacts on its production. In our case, we have nonvanishing electric andmagnetic fields and all charged particles are heavy while the Higgs rolls down its potential. Forparallel and constant physical electric and magnetic fields, the resulting current was calculated in[27]. From eq. (2.12) the strengths of the electric and magnetic field are ˆ E ≈ √ ρ E and ˆ B ≈ √ ρ B ,respectively. The induced physical current ˆ J ind of a particle with mass m and electric charge eQ along the direction of parallel electric and magnetic fields is then determined by [27] eQ ∂∂τ (cid:16) a ˆ J ind (cid:17) = a ( e | Q | ) π ˆ E ˆ B coth (cid:32) π ˆ B ˆ E (cid:33) exp (cid:32) − πm e | Q | ˆ E (cid:33) . (2.15)The EOM for the gauge field including the contribution from the current leads to the conservationequation [27] a − ∂∂τ (ˆ ρ E + ˆ ρ B ) = − H (ˆ ρ E + ˆ ρ B ) + a − | ξ | ˆ E ˆ B − eQ ˆ E ˆ J ind . (2.16)The second-last term on the RHS arises from the production of gauge fields, while the last term is dueto the backreaction from the current. Jumping ahead, we have applied eq. (2.15) to the benchmarkpoints studied in sec. 4. We find that the backreaction term in eq. (2.16) becomes comparableto the production term toward the end of the time evolution, when the production mechanismbecomes inefficient because the Higgs evolves too slowly. This means that the backreaction couldbecome important. However, note that the physical electric and magnetic field in our case are farfrom constant but change quickly during the production process. The current may therefore differfrom the one estimated from eq. (2.15). Furthermore, once the production mechanism switches off,the electric and magnetic field become free fields and orthogonal to each other. Subsequently, pairproduction no longer happens [21]. We leave a more thorough analysis of the backreaction fromthe current in our scenario for future work.A related consequence of the Schwinger effect in a strong electromagnetic field is the productionof particle asymmetries. For definiteness, let us focus on the right-handed electron. The currentassociated with U (1) rotations of the right-handed electron is anomalous and its divergence is givenby ∂ µ J µe R = − α π F µν ˜ F µν + . . . , (2.17)where α = e / π is the fine-structure constant and the ellipsis denotes the contribution from themass term of the electron. We typically have ˆ E (cid:29) m e , where m e is the electron mass, in whichcase this contribution can be neglected [27]. Defining the charge q e R ≡ lim V →∞ V (cid:90) V d x (cid:104) J e R (cid:105) (2.18)corresponding to the asymmetry in the number densities of the right-handed electron and its an-tiparticle, the anomaly equation then gives ∂ τ q e R (cid:39) − α π ∂ τ H . (2.19)– 6 –rom this, we expect that the asymmetry q e R (cid:39) − α π H (2.20)is generated together with the helicity [25, 27]. Similarly, asymmetries are also produced for theother SM fermions. We will discuss the subsequent evolution of these asymmetries in sec. 3.Finally, we comment on our assumption that the temperature vanishes while the magnetichelicity is produced and that reheating takes place only afterwards. The inflaton can produceparticles nonperturbatively while it oscillates in its potential in a process known as preheating.This process can be strongly suppressed, however, if the inflaton couples dominantly to fermionsinstead of bosons or if its couplings are small. In this case, the reheating of the universe proceedsvia perturbative decays of the inflaton. Reheating is usually defined to happen when the Hubblerate has fallen sufficiently so that it equals the inflaton decay rate. The temperature at this time(assuming thermalization) is obtained from equating the two rates,ˆ T rh = (cid:18) π g ∗ (cid:19) / (cid:112) Γ inf M Pl , (2.21)where Γ inf is the inflaton decay rate, M Pl the reduced Planck mass and g ∗ the number of relativisticdegrees of freedom at reheating. As has been pointed out in [28, 29], the temperature beforereheating does not necessarily vanish though. Indeed for a constant decay rate, the inflaton alreadyslowly decays during inflaton oscillations, leading to a plasma of particles. Whether and when thisplasma thermalizes depends on the particles into which the inflaton decays and the strengths oftheir couplings. Assuming that it does thermalize, the temperature of the plasma quickly reachesa maximum after inflation [28, 29]. For the case that the plasma consists of SM particles, thismaximum temperature readsˆ T max ≈ . (cid:18) π g ∗ (cid:19) / ( H inf M Pl ) / (cid:113) ˆ T rh = 0 . (cid:113) ˆ T insrh ˆ T rh , (2.22)where ˆ T insrh is the temperature after instant reheating, i.e. eq. (2.21) for Γ inf = H inf . This maximaltemperature is thus larger than the reheating temperature eq. (2.21). The temperature subsequentlydecreases and equals eq. (2.21) at the time of reheating. The presence of a plasma of SM particles(whether it is thermalized or not) before reheating could affect both the evolution of the Higgsand the production of magnetic helicity. Indeed the resulting thermal corrections in the Higgspotential could dominate due to the large temperature. Furthermore, charged particles in theplasma would react to the generated electric field with a current which would backreact and suppressthe production of electromagnetic fields. We should therefore ensure the absence of a plasma of SMparticles while the magnetic helicity is produced. Jumping ahead again, we will have Higgs VEVs (cid:29) H inf during the production process and the SM particles are thus very heavy. Inflaton decays canbe kinematically forbidden if the inflaton is lighter than the SM particles that it couples to. Suchdecays and reheating then become possible only at later times when the Higgs VEV has sufficientlydecreased. If the inflaton decay rate is larger than the Hubble rate once decays are kinematicallyallowed, the universe reheats quickly at this time. We then assume this process happens after themagnetic helicity has been produced. Alternatively, the inflaton could couple to and decay intosome new particles which couple to the SM sector sufficiently weakly that they reach equilibriumwith the SM particles only after the magnetic helicity has been produced. See also the relateddiscussion in [30]. – 7 – Evolution of the helicity after reheating
During the rolling and the oscillations of the Higgs in its potential, EW symmetry is broken exceptfor the brief moments when the Higgs crosses zero. In sec. 2, we have therefore focused on thephoton. W ± and Z bosons may also be produced whenever the Higgs crosses zero (either throughthe coupling in eq. (2.1) or through the gauge couplings) but we expect their contribution to besmall and to decay quickly once the Higgs is again away from zero. In this section, we will considerthe relevant quantities after reheating. At reheating the EW symmetry is permanently restored by the thermal corrections (until the EWphase transition) and the helicity in photons is transformed into helicity in hypercharge gauge fields.Setting A Y µ = ( A Y , A Y ) and using the radiation gauge, the helicity in hypercharge gauge fields H Y is defined analogously to eq. (2.11). We then expect H Y = cos θ W H (3.1)after reheating, where H is the helicity in photons that was produced from the rolling and theoscillations of the Higgs. Note that part of the helicity is also converted into helicity in W bosonsbut we expect this do be washed out quickly in the thermal plasma due to the thermal mass andself-interaction of the W boson.As we have discussed in sec. 2.3, we expect that asymmetries in the SM fermions are generatedtogether with the helicity. In particular, the asymmetry in right-handed electrons is given ineq. (2.20). Ignoring additional contributions from the conversion of the helicity (which we expectto be small), using eq. (3.1) this can be written as q e R (cid:39) − α Y π H Y , (3.2)where α Y = g Y / π is the hypercharge fine-structure constant. At temperatures below 10 GeV,EW sphalerons come into thermal equilibrium and lead to the rapid erasure of the asymmetriesstored in left-handed fermions. Similarly, the asymmetries in right-handed fermions are driven tozero by sphalerons once right- and left-handed particles reach chemical equilibrium via their Yukawacouplings. Due to its small Yukawa coupling, the right-handed electron is the last species to reachchemical equilibrium, at temperatures ∼ GeV. Below this temperature, all the asymmetries areerased. We will see in sec. 3.3, though, that a process called chiral plasma instability can potentiallyconvert the asymmetries into hypercharge helicity before they can be erased by sphalerons. Beforeaddressing this, in the next section we will discuss the joint evolution of the hypercharge gaugefields with the thermal plasma.
The hypercharge gauge fields interact with the thermal plasma of SM particles after reheating andset it into motion. This in turn backreacts on the hypercharge gauge fields. The combined systemis described by Maxwell’s equations and the Navier-Stokes equation (see [19, 31] for reviews). Therelevant Maxwell’s equations are ∂ B Y ∂τ = − ∇ × E Y ∂ E Y ∂τ = ∇ × B Y − J Y , (3.3)where the hypercharge electric and magnetic fields are given in terms of the hypercharge gauge fieldin radiation gauge by E Y = − ∂ τ A Y and B Y = ∇ × A Y , respectively. Furthermore, the current– 8 –an be estimated from a generalized Ohm’s law J Y = σ ( E Y + v × B Y ) + 2 α Y π µ B Y , (3.4)where v is the fluid velocity of the thermal plasma and σ (cid:39) c σ T / ( α Y log( α − Y )) with c σ ≈ . T = ˆ T a is a comoving reference temperature, where ˆ T is the physical temperature. Up to changesin the number of degrees of freedom, T stays constant during radiation domination. Furthermore,the last term in eq. (3.4) is due to the chiral magnetic effect [35–37] with µ = (cid:88) α (cid:15) α N α Y α µ α , (3.5)where α runs over all SM species, with multiplicity N α and hypercharge Y α , and (cid:15) α = ± for right-/left-handed particles. Defining asymmetries q α for the SM species in analogy with eq. (2.18), µ α = 6 q α / ( N α T ) for µ α (cid:28) | v | (cid:28)
1, we can neglect the displacement current ∂ τ E Y in the Amp`ere-Maxwell equation in eq. (3.3). Combining this with eq. (3.4), we can then solve for the hyperelectricfield which gives E Y = 1 σ ∇ × B Y − α Y σ π µ B Y − v × B Y . (3.6)Together with the Maxwell-Faraday equation in eq. (3.3), this yields the magnetohydrodynamics(MHD) equation for the hypermagnetic field ∂∂τ B Y = 1 σ ∇ B Y + ∇ × ( v × B Y ) + 2 α Y π µ σ ∇ × B Y . (3.7)This is supplemented by the Navier-Stokes equation for the velocity field of an incompressible fluidinteracting with the hypermagnetic field ∂∂τ v = ν ∇ v − ( v · ∇ ) v + 1 ρ + p ( ∇ × B Y ) × B Y , (3.8)where ρ and p are respectively the energy and pressure density of the plasma. For radiationdomination, we have p = ρ/
3. Furthermore, ν (cid:39) c ν / ( α Y log( α − Y ) T ) is the kinematic viscosity with c ν ≈ .
01 for temperatures above the EW scale [34].The MHD equations (3.7) and (3.8) determine the coevolution of the hypermagnetic field andthe fluid velocity of the thermal plasma. We are in particular interested in the evolution of thehypermagnetic helicity that is generated after inflation. By taking the time derivative of the hy-permagnetic helicity defined analogously to eq. (2.11) and using eq. (3.6), we find ∂∂τ H Y = lim V →∞ − V (cid:90) V d x E Y · B Y = lim V →∞ V (cid:90) V d x (cid:18) σ B Y · ∇ A Y + 4 α Y π µ σ B Y (cid:19) . (3.9) Note that the µ -dependent term is only applicable for µ /T (cid:28) α Y . This will always be fulfilled in our case.See [32] and references therein for other regimes. This can be seen as follows: Let us denote the characteristic electric and magnetic field and time and length scaleof a gauge field configuration with E Y , B Y , τ Y and λ Y , respectively. We can estimate the terms in the Amp`ere-Maxwell equation as | ∂ τ E Y | ∼ E Y /τ Y and | ∇ × B Y | ∼ B Y /λ Y . Using the estimate E Y /B Y ∼ λ Y /τ Y that followsfrom the Maxwell-Faraday equation, we then find | ∂ τ E Y | / | ∇ × B Y | ∼ ( λ Y /τ Y ) ∼ | v | (cid:28) Note that the viscous-damping term ν ∇ v is only present if the correlation length of the hypermagnetic field islarger than the mean free path of the particles in the plasma. This will always be fulfilled in our case. See [38, 39]for the damping term in the opposite regime. – 9 –et us for the moment ignore the µ -dependent terms in eqs. (3.7) and (3.9). The hypermagneticfield can decay due to magnetic diffusion and induction from the plasma motion, correspondingto the first and second term on the RHS of eq. (3.7), respectively. Denoting the characteristicstrength and correlation length of the magnetic field with B Y and λ B Y , respectively, and the typicalvelocity of the plasma at the length scale λ B Y with v , we can estimate | ∇ B Y | /σ ∼ B Y / ( λ B Y σ )and | ∇ × ( v × B Y ) | ∼ B Y v/λ B Y . Induction then dominates over magnetic diffusion if the magneticReynolds number satisfies R m ≡ σ v λ B Y (cid:38) . (3.10)Next notice that eq. (3.9) does not depend on the plasma velocity which has dropped out. Inductionfrom the plasma motion therefore does not lead to the decay of the helicity. We then expect thatif the magnetic Reynolds number is larger than unity and the dynamics of the hypermagnetic fieldis dominated by the plasma motion, the helicity is preserved.In order to discuss this in more detail, let us estimate the typical velocity v . To this end, wenote that the last term on the RHS of eq. (3.8) acts as a source term that sets the plasma intomotion. A steady velocity is obtained by balancing the first and second term with this source term.If the kinetic Reynolds number satisfies R e ≡ vλ B Y ν (cid:38) , (3.11)the second term | ( v · ∇ ) v | ∼ v /λ B Y dominates over the first term ν | ∇ v | ∼ ν v/λ B Y and thetypical velocity v can be estimated as [38, 39] v ∼ B Y √ ρ ⇐⇒ ρ v ∼ B Y . (3.12)This corresponds to an equipartition between the kinetic energy in the plasma and the magneticenergy. If also the magnetic Reynolds number is larger than unity and the helicity is conserved,one finds the scaling relations (see e.g. appendix B in [6] for a derivation) B Y ∝ τ − , λ B Y ∝ τ , v ∝ τ − . (3.13)From this, we see that the Reynolds numbers grow with time and thus remain larger than one. Theconservation of the helicity, the equipartition of energy densities and the above scaling relations inthis regime have been verified in numerical MHD simulations [38–40]. In particular, the conserva-tion of the helicity can be understood as being due to an inverse cascade during which helicity istransferred from smaller to larger length scales. This is reflected in the growth of the characteristiclength scale λ B Y with time in eq. (3.13) and leads to the diffusion term in eq. (3.9) being more andmore suppressed over time. It thus never becomes important.However, we find that in our scenario the kinetic Reynolds number is typically smaller thanone. In this case, we can estimate the typical velocity v by balancing the first and the last term onthe RHS of eq. (3.8). This gives [38, 39] v ∼ λ B Y B Y νρ ⇐⇒ ρ v ∼ R e B Y (3.14)and the kinetic energy and velocity are thus smaller than for the case R e (cid:38)
1. Using this in– 10 –q. (3.10), we expect that the helicity will be conserved at reheating if R m ∼ σλ B Y ν B Y ρ ∼ (cid:18) π g ∗ (cid:19) / c σ α Y c ν ρ B Y λ B Y M Pl H inf (cid:32) ˆ T rh ˆ T insrh (cid:33) / (cid:38) . (3.15)In the last step, we have used the estimate B Y ≈ ρ B Y , where ρ B Y is the energy density calculatedin analogy to eq. (2.12b). The correlation length of the hypermagnetic field λ B Y can be calculatedanalogously to eq. (2.13). In the regime of small kinetic Reynolds number, one finds the scalingrelations (see again appendix B in [6]) B Y ∝ τ − , λ B Y ∝ τ , v ∼ const. (3.16)As before, we see that the magnetic Reynolds number grows with time and thus stays larger thanone. Correspondingly, we expect that the helicity is protected from diffusion and remains conservedif eq. (3.15) is fulfilled at reheating. The estimate for the velocity in eq. (3.14) and the scalingrelations in eq. (3.16) were previously derived in [38, 39] together with corresponding relationsfor the case that the velocity dissipates due to free streaming of particles instead of diffusion. Anumerical MHD simulation was performed only for the latter case but the relations were verifiedwith very good accuracy. We expect that the relations in our case of diffusion damping describe theevolution of the system similarly well but a numerical verification of this is clearly desirable. Notethat also the kinetic Reynolds number grows with time in the scaling regime eq. (3.16). Eventually itmay therefore become larger than unity and the quantities could subsequently scale as in eq. (3.13).Alternatively, we can derive an upper bound on the magnetic Reynolds number without referringto the underlying MHD dynamics. Since the hypermagnetic field is the source of the velocity field,the kinetic energy of the latter is limited by the energy in the former, ρ v (cid:46) B Y . Plugging thevelocity saturating this bound into eq. (3.10), we obtain the more conservative criterion R max m ∼ σ λ B Y B Y √ ρ ∼ (cid:18) π g ∗ (cid:19) / c σ α Y log( α − Y ) λ B Y √ ρ B Y √ H inf M Pl (cid:32) ˆ T rh ˆ T insrh (cid:33) / (cid:38) R e < Let us next discuss the effect of the µ -dependent terms in eqs. (3.7) and (3.9) which we haveso far ignored. To this end, recall that asymmetries in the number densities of particles andtheir antiparticles are generated together with the helicity as discussed in sec. 2.3. Since theseasymmetries are related to the helicity via the chiral anomaly, they can be transformed back intohelical gauge fields as we will now explain. Let us focus on the right-handed electron since it is thelast species to come into chemical equilibrium, at temperatures ∼ GeV, and its asymmetry thussurvives the longest. The anomaly equation for the current corresponding to U (1) rotations of theright-handed electron in the symmetric phase gives (cf. eq. (2.19)) ∂ τ q e R (cid:39) − α Y π ∂ τ H Y , (3.18)where we have assumed temperatures above 10 GeV and dropped the contribution from the Yukawacoupling. From this, we see that if the asymmetry q e R is driven to zero, a gauge field configurationwith helicity H Y ∼ q e R /α Y is generated. Denoting the characteristic magnetic field of this config-uration with B Y and its characteristic size with λ Y , we can estimate its helicity in terms of these– 11 –uantities using eq. (2.11) as H Y ∼ λ Y B Y . The energy densities in right-handed electrons and thegauge field configuration are ∼ q e R /T and ∼ B Y , respectively. The gauge field configuration thenhas lower energy density than the equivalent asymmetry of right-handed electrons for [35] λ Y (cid:38) T α Y q e R . (3.19)The formation of such an energetically favoured helicity configuration from an asymmetry is calledthe chiral plasma instability (CPI) [13, 35, 41–45]. In eq. (3.9) for the helicity evolution, it arisesfrom the µ -dependent term. The fastest growing mode has a length scale saturating eq. (3.19). Us-ing that µ ∼ µ e R ∼ q e R /T in kinetic equilibrium if only right-handed electrons have an asymmetryand eqs. (3.9) and (3.19), we can estimate the time scale of the CPI as τ CPI ∼ σα Y µ , (3.20)where we have ignored all numerical prefactors. A more careful analysis of eqs. (3.7) and (3.9) inmomentum space shows that eq. (3.20) also applies to the more general situation where severalspecies have asymmetries and gives π / GeV before it can happen. Todetermine the parameter µ in eq. (3.20), let us consider temperatures somewhat above 10 GeVwhere all SM species except for the right-handed electron are in chemical equilibrium. Imposingconstraints from sphalerons, Yukawa interactions and conserved quantities, the asymmetries andchemical potentials of all SM species can then be expressed in terms of those for the right-handedelectron. Using eq. (3.2), we get [6] µ = − α Y π H Y T . (3.21)On the other hand, at temperatures below 10 GeV, the electron Yukawa coupling is in equilib-rium and all charge asymmetries are erased. This gives µ = 0 and the CPI is no longer possible.Relating eq. (3.20) to the temperature of the universe at that time and using eq. (3.21), we thendemand thatˆ T CPI ∼ α Y µ π σH inf ˆ T / ( ˆ T insrh ) / ∼ g ∗ α Y log( α − Y ) π c σ H Y M H (cid:32) ˆ T rh ˆ T insrh (cid:33) (cid:46) GeV (3.22)to avoid the erasure of the hypermagnetic helicity by the CPI. Let us emphasize though that thederivation of this condition was necessarily approximate and that a numerical MHD simulationtaking into account the charge asymmetries would be required to establish the condition for helicitysurvival in more detail.
At temperatures around the EW scale, the Higgs again obtains a VEV at h (cid:39)
246 GeV and EWsymmetry is broken. We assume no new physics which could affect this phase transition and it isthus a crossover. During the phase transition, the hypermagnetic helicity is converted back into– 12 –agnetic helicity. The hypercharge gauge boson contributes to the B + L -anomaly, while the photondoes not. The anomaly equation for the B + L -current then yields a relation similar to eq. (2.19)for the B + L -charge which in the unbroken phase depends on the hypermagnetic helicity but in thebroken phase does not depend on the magnetic helicity. The conversion of the helicity at the EWphase transition therefore generates a B + L -asymmetry. Sphalerons erase part of this asymmetry.As we will now explain though, since sphalerons switch off due to the broken EW symmetry at thesame time as the asymmetry is being produced, a net asymmetry survives [2]. The conversion of the hypermagnetic fields into ordinary magnetic fields during the EW phasetransition is governed by the EW angle θ W , i.e. the angle of the SO (2) rotation that diagonalizesthe mass matrix for the gauge bosons A Y µ and W µ . The crucial point is that the EW angle changessmoothly since the thermal (magnetic) mass for (the transverse modes of) W µ on the diagonal of themass matrix initially dominates over the off-diagonal mass from the Higgs VEV which graduallydevelops during the crossover. The EW angle thus becomes a function of the temperature andchanges from θ W = 0 at high temperatures to θ W = arctan g Y /g W somewhat below the EW scale,where g Y and g W are the gauge couplings of respectively U (1) Y and SU (2) L . This gives rise to asmooth source term for the B + L asymmetry which is controlled by the changing EW angle. Abovetemperatures ˆ T (cid:39)
130 GeV [49], on the other hand, EW sphalerons are in thermal equilibrium andtend to erase the asymmetry. Including both contributions, the Boltzmann equation for the baryon-to-entropy ratio η B reads [2] dη B dx = − γ W sph η B + 316 π ( g Y + g W ) sin(2 θ W ) dθ W dx H Y s , (3.23)where x = ˆ T /H ( ˆ T ) with H ( ˆ T ) being the Hubble rate at temperature ˆ T , H Y is the hyperchargehelicity that is initially present and s = (2 π / g ∗ T is the comoving entropy density of the SMplasma. Furthermore, γ W sph is the dimensionless transport coefficient for the EW sphaleron whichfor temperatures ˆ T <
161 GeV is found from lattice simulations as [49] γ W sph (cid:39) exp (cid:32) − . . T
130 GeV (cid:33) . (3.24)The temperature-dependence of the EW angle θ W has been determined analytically and fromlattice simulations but is subject to significant uncertainties [47, 48]. We follow [2, 5] and model itwith a smooth step functioncos θ W = g W g Y + g W + 12 g Y g Y + g W (cid:32) (cid:34) ˆ T − ˆ T step ∆ ˆ T (cid:35)(cid:33) (3.25)which for 155 GeV (cid:46) ˆ T step (cid:46)
160 GeV and 5 GeV (cid:46) ∆ ˆ T (cid:46)
20 GeV describes the analytical andlattice results for the temperature dependence reasonably well.The Boltzmann equation (3.23) has been numerically solved in [2] and the baryon-to-entropyratio η B was found to become frozen, i.e. ∂ τ η B = 0, at a temperature ˆ T (cid:39)
135 GeV. As expected,this is close to the temperature ˆ T (cid:39)
130 GeV at which EW sphalerons freeze out. Setting the RHS We expect that a B + L -asymmetry is similarly produced at reheating where the inverse conversion of the helicitytakes place. This asymmetry is subsequently completely erased by sphalerons and Yukawa interactions though asdiscussed in sec. 3.1. On the other hand, no (magnetic) mass arises for the hypercharge gauge boson or the photon [46–48]. – 13 –f eq. (3.23) to zero, the observed baryon asymmetry of the universe is reproduced if η B (cid:39) π ( g Y + g W ) H Y ˆ T rh M H (cid:34) f θ W γ W sph H ( ˆ T )ˆ T (cid:35) ˆ T =135 GeV (cid:39) · − , (3.26)where f θ W ≡ − sin(2 θ W ) dθ W /d log ˆ T . Varying ˆ T step and ∆ ˆ T in the ranges given below eq. (3.25),one finds 5 . · − < f θ W < .
32 at ˆ T = 135 GeV. Let us summarize the constraints which we have derived. We need to generate gauge fields withhelicity H Y , energy density ρ B Y and correlation length λ B Y which satisfy the condition on themagnetic Reynolds number eq. (3.15) (or at least eq. (3.17)) and the condition from the CPIeq. (3.22) to survive until the EW phase transition and which fulfill eq. (3.26) to reproduce theobserved baryon asymmetry of the universe. These conditions can be rewritten as η B (cid:39) · − f θ W H Y H (cid:18) H inf GeV (cid:19) / (cid:32) ˆ T rh ˆ T insrh (cid:33) (cid:39) · − , (3.27)where 5 . · − < f θ W < .
32 is evaluated at ˆ T = 135 GeV, andˆ T CPI ∼ · − GeV H Y H (cid:18) H inf GeV (cid:19) (cid:32) ˆ T rh ˆ T insrh (cid:33) (cid:46) GeV , (3.28a) R m ∼ · − ρ B Y λ B Y H (cid:18) H inf GeV (cid:19) (cid:32) ˆ T rh ˆ T insrh (cid:33) / (cid:38) , (3.28b)or alternatively R max m ∼ · − λ B Y √ ρ B Y H inf (cid:18) H inf GeV (cid:19) / (cid:32) ˆ T rh ˆ T insrh (cid:33) / (cid:38) . (3.29)Note that the dimensionless combinations H Y /H , ρ B Y /H and λ B Y H inf in the conditions areindependent of H inf if we keep M /H and h ( τ md ) /H inf fixed, where h ( τ md ) is the initial HiggsVEV. To see this, rescale the Higgs, the photon, conformal time and momentum by powers of H inf to make them dimensionless. The solutions to the EOMs of the Higgs and the photon, eqs. (2.5) and(2.8), after this rescaling depend on H inf only via the dimensionless ratios H /M , cf. eq. (2.9),and h ( τ md ) /H inf . This is therefore also the case for the rescaled helicity, magnetic energy densityand correlation length which are derived from these solutions.The constraints all depend on the reheating temperature ˆ T rh and we need to obtain a rangefor ˆ T rh for which they are all simultaneously fulfilled. In the next section, we will discuss a modelwhich achieves that. Note that the estimates for the magnetic Reynolds number, the CPI temperature and the baryon asymmetrydepend on the gauge couplings which in turn depend on the RG scale µ . For the magnetic Reynolds number, thehypercharge gauge coupling should be evaluated near the reheating temperature. For concreteness, we have chosenthe renormalization scale µ = 10 GeV in the estimates. Similarly, we have set µ = 10 GeV in the estimate for theCPI temperature. In both cases, the dependence on the precise value of µ is quite weak. – 14 – Higgs potential and numerical results
We are now in a position to study examples for the evolution of the Higgs after the end of inflationand to see whether this can generate the baryon asymmetry of the universe. We have seen in thelast section that three conditions need to be fulfilled. These are eq. (3.28a) from the CPI andeq. (3.28b) (or at least eq. (3.29)) on the magnetic Reynolds number to ensure that the helicitysurvives until the EW phase transition. The observed baryon asymmetry is then reproduced ifeq. (3.27) is satisfied. The baryon-to-entropy ratio, the magnetic Reynolds number and the CPItemperature all grow with the inflation scale. In order to increase the first two parameters, we set H inf = 10 GeV for the Hubble rate at the end of inflation. This leaves some margin to satisfy theupper limit on the Hubble rate from bounds on the tensor-to-scalar ratio (i.e. roughly 60 e-foldsbefore the end of inflation), H inf < . · GeV [50].Let us first assume no new physics beyond the SM (except for the inflaton). During inflation,the Higgs is driven to large VEVs due to quantum fluctuations in de Sitter space. Depending onthe precise values of SM parameters (in particular the top Yukawa coupling), the Higgs quarticcoupling runs to negative values at energies as low as ∼ GeV [51]. For inflation scales largerthan this instability scale, the Higgs is driven into the resulting AdS minimum, with catastrophicconsequences [52].We will therefore assume new physics which causes the Higgs quartic coupling to either notrun negative or only at much larger energy scales than in the SM. Quantum fluctuations duringinflation are then expected to drive the average Higgs VEV to values ∼ λ − / h H inf , where λ h is the Higgs quartic coupling (see e.g. [20, 52]). Starting from this VEV, the Higgs rolls downits potential toward the minimum once inflation ends. Assuming the coupling in eq. (2.1), thisleads to the production of electromagnetic fields. Let us consider the constraint on the magneticReynolds number to ensure their survival until the EW phase transition. The energy density of thehypermagnetic fields is bounded by the initial potential energy density of the Higgs, ρ B Y (cid:46) H .We then see from eq. (3.28b) that even for instant reheating, ˆ T rh = ˆ T insrh , the correlation lengthof the hypermagnetic fields has to satisfy λ B Y (cid:38) H − in order to obtain a magnetic Reynoldsnumber above one. For consistency though, we need ˆ T rh < ˆ T insrh since we have assumed a phaseof matter domination and negligible temperature in the SM sector while the Higgs rolls down itspotential and produces the magnetic fields. This makes the bound on the correlation length evenstronger. We find, however, that typically λ B Y (cid:46) H − and this bound cannot be satisfied with theabove initial conditions for the Higgs. Instead we will arrange for the Higgs to have a VEV (cid:29) H inf directly after inflation. This raises the energy density in the Higgs which can be converted intoelectromagnetic fields and thereby allows us to satisfy the constraint from the magnetic Reynoldsnumber. There are of course many ways to affect the running of the Higgs quartic coupling. For definiteness,we focus on the simple possibility of a real scalar φ which couples to the Higgs doublet. Imposingthe Z symmetry φ → − φ , the most general renormalizable potential for φ and the Higgs doubletreads V ⊃ − m h | Φ | + λ h | Φ | + 12 m φ φ + λ φ φ + λ φh φ | Φ | . (4.1)We next include the coupling V ⊃ ξ R R | Φ | , (4.2) If the top mass is near the lower value of its measured range at 3 σ , the Higgs quartic coupling can remain positiveup to the Planck scale [51]. In this case, we would not need new physics for our scenario. Since it is experimentallydisfavoured, however, we will not investigate this case further. – 15 –here R is the Ricci scalar. The background value of R leads to a time-dependent contribution tothe Higgs mass. As we review in appendix C, R = − − (cid:15) ( τ )] H ( τ ) during inflation, where H ( τ )is the Hubble rate and (cid:15) ( τ ) the slow-roll parameter during inflation.We choose m φ , λ φ , λ φh > φ is always at the origin. We will arrange for theHiggs to have a VEV (cid:29) H during inflation. This yields an effective mass (cid:29) H for φ . De Sitterfluctuations of φ are then suppressed and it is anchored at the origin during inflation [52]. It istherefore a spectator field which enters the dynamics only via its loop corrections to the Higgs. Wecan take the loop corrections into account by calculating the RG-improved effective potential. To agood approximation this amounts to replacing the parameters in the tree-level potential by runningparameters evaluated at the scale µ = h . Setting φ = 0, the effective potential for the Higgs duringinflation reads V eff (cid:39) − − (cid:15) ( τ )] ξ R H ( τ ) h + λ h ( h )4 h . (4.3)We have neglected the Higgs mass parameter m H since it is very small compared to the massinduced by the Ricci scalar. The coupling ξ R runs very slowly and changes by at most 20% betweenthe Planck scale and the EW scale (see e.g. [53]). We therefore neglect its running and use a fixedvalue. For the Higgs quartic coupling, we use the RGEs at one-loop for the SM plus the singletwhich we summarize in appendix B. We choose values for the singlet mass m φ and the couplings λ φ and λ φh such that the Higgs quartic coupling remains always positive when running from theEW scale up. An example is shown in the upper left panel of fig. 1.For ξ R >
0, the Higgs potential has a minimum away from the origin which is induced by thetachyonic mass term due to the Ricci scalar. In most inflation models, (cid:15) ∼ (cid:15) ( τ inf ) = 1 can be taken to define the conformal time τ inf when inflation ends. Thisgives R = − H at the end of inflation, where we have used that by definition H inf = H ( τ inf ).In hybrid inflation models, on the other hand, the inflaton stays in the slow-roll regime and theend of inflation is instead triggered by a second, waterfall field. In this case, we expect (cid:15) (cid:28) R = − H at the end of inflation. Using this, the minimum of the Higgs potential at the end ofinflation is determined by h min (cid:39) (cid:115) c R ξ R λ h + β λ h / H inf , (4.4)where β λ h is the β -function of λ h given in eq. (B.3) in appendix B and c R = 12 for hybrid inflationor c R = 6 otherwise. All couplings on the RHS of eq. (4.4) have to be evaluated at the scale µ = h min . We assume that the Higgs sits in this minimum at the end of inflation, h ( τ inf ) = h min .For h min (cid:29) H inf , the resulting Higgs mass is much larger than H inf and its fluctuations aroundthe minimum are suppressed. This ensures that our baryogenesis mechanism does not lead tounacceptably large isocurvature perturbations [14–16]. Furthermore, as discussed in appendix C,we find that h (cid:48) ( τ inf ) (cid:39) h (cid:48) ( τ inf ) (cid:39) − h min H inf otherwise.Except for hybrid inflation, the velocity of the Higgs at the end of inflation is thus sizeable. Thiscould lead to an additional contribution to photon production from the last e-folds of inflation whichwould enhance the helicity. However, the velocity of the Higgs during inflation is very suppressedand grows with the slow-roll parameter (cid:15) only toward the end (see eq. (C.9) in appendix C). Thesize of this additional contribution therefore depends on how fast the slow-roll parameter increasesfrom (cid:15) (cid:28) (cid:15) = 1 at the end of inflation and thus on the inflaton potential. In the following, weignore the contribution to photon production during inflation and leave it to future work. In regions of the potential with h < m φ , one has to appropriately decouple the scalar in the RGEs. Since wechoose m φ ∼ few TeV, while h (cid:29) H inf = 10 GeV during most of the evolution that is relevant for us, we can ignorethis in practice. – 16 –oint init. cond. ξ R M/ GeV h ( τ md ) /H inf . · .
72 B 150 1 . · .
73 C 25 1 . · . Table 1 : Definition of three benchmark points, setting m φ = 4 TeV, λ φh = 0 . and λ φ = 0 . at theEW scale and H inf = 10 GeV. For point 1 and 2, we have fixed ˆ T rh = 0 . · GeV and for point3, we have chosen ˆ T rh = 1 . · GeV. This corresponds to reheating happening at τ ≈ H − and τ ≈ H − , respectively. During the subsequent phase of inflaton oscillations, the Ricci scalar briefly changes its signeach time the inflaton crosses the origin of its potential (see e.g. [54]). If the oscillation frequencyof the inflaton is larger than the Hubble rate at the end of inflation, it oscillates many times duringone Hubble time. We can then use the averaged value R = − H ( τ ), where H ( τ ) = H inf ( τ md /τ ) isthe Hubble rate during matter domination. We will focus on this case. Note that in the oppositecase of slow inflaton oscillations, the sign changes of the Ricci scalar could induce additional motionin the Higgs, potentially increasing gauge field production. This case is left for future work. Theeffective potential for the Higgs during matter domination then reads V eff (cid:39) − ξ R H (cid:16) τ md τ (cid:17) h + λ h ( h )4 h . (4.5)The minimum of this potential at the onset of matter domination follows from eq. (4.4) with c R = 3.At later times, the minimum scales approximately like ( τ md /τ ) .Notice that the minima of the Higgs potential at the end of inflation and the onset of matterdomination differ. This happens since R = − H or R = − H at the former time, while R = − H at the latter time. From this, we expect three extremal possibilities for the initialconditions of the Higgs at the onset of matter domination: If the Ricci scalar changes slowlybetween its values at the end of inflation and at the onset of matter domination, the Higgs cantrack the minimum of its potential and h ( τ md ) = h min with h min determined by eq. (4.4) for c R = 3.We then expect that the initial velocity of the Higgs is small and set h (cid:48) ( τ md ) = 0. If the Ricci scalarchanges quickly, on the other hand, the Higgs is not able to follow the minimum and its initial value h ( τ md ) = h min is determined by eq. (4.4) with c R = 12 for hydrid inflation or c R = 6 otherwise.The initial velocity then is h (cid:48) ( τ md ) (cid:39) h (cid:48) ( τ md ) (cid:39) − h min H inf , respectively. To summarize weconsider the three different initial conditionsA: h ( τ md ) = h min for c R = 3 and h (cid:48) ( τ md ) = 0 ,B: h ( τ md ) = h min for c R = 6 and h (cid:48) ( τ md ) = − h min H inf ,C: h ( τ md ) = h min for c R = 12 and h (cid:48) ( τ md ) = 0 .These initial conditions are clearly idealizations but we expect that they cover the range of possi-bilities relevant for baryogenesis. In particular, keeping the parameters which determine the Higgspotential eq. (4.5) fixed, initial condition C allows for significantly stronger gauge field productionthan A in the regime where the backreaction can be neglected. Initial condition B, on the otherhand, lies in the middle. Notice that for simplicity we identify the Hubble rates at the end of inflation and the onset of matter domination, H ( τ md ) = H ( τ inf ) = H inf . Depending on how much time is spent until the Ricci scalar evolves as in matterdomination, the Hubble rate at the latter time may be somewhat smaller. – 17 – - - μ ( H inf ) λ h τ ( H inf - ) h ( H i n f ) ξ ( H i n f ) - - τ ( H inf - ) h ( H i n f ) ξ ( H i n f ) - - τ ( H inf - ) h ( H i n f ) ξ ( H i n f ) Figure 1 : Upper left panel: The running of the Higgs quartic coupling for m φ = 4 TeV, λ φh = 0 . and λ φ = 0 . at the EW scale. The scale µ is shown in units of H inf = 10 GeV. Upper rightpanel and lower panels: The time evolution of the Higgs h (blue) and the instability parameter ξ (purple) in the EOM of the gauge field (2.8) . The upper right panel is for point 1 in table 1, thelower left panel for point 2 and the lower right panel for point 3. We find that λ h + β λ h / h min (cid:29) H inf with ξ R (cid:46)
1. Therefore we need ξ R (cid:29)
1. As has been pointed out in [55], the coupling betweenthe Higgs and the Ricci scalar lowers the cutoff of the theory to Λ c ∼ M Pl /ξ R for ξ R (cid:29)
1. Wewill therefore demand that h min (cid:46) M Pl /ξ R during inflation. The Higgs VEV contributes to thePlanck mass via its coupling to the Ricci scalar. The resulting effective Planck mass is given by M , eff = M + ξ R h . For h (cid:46) M Pl /ξ R and ξ R (cid:29)
1, we then have M Pl , eff (cid:39) M Pl . We have calculated the evolution of the Higgs and the produced gauge fields for the threebenchmark points defined in table 1. Notice that the initial value of the Higgs is a factor ∼ − M . One may be worried about the validity of the effective field theory (EFT).As we show in appendix A, however, UV completions exits where higher-dimensional operators ofthe form h n F µν ˜ F µν for n > n = 1 by powers ofa small parameter ( λ Sh in our example). Choosing this parameter small enough ensures the validityof the EFT. We plot the time evolution of the Higgs h and the instability parameter ξ in the EOMof the gauge field (2.8) in fig. 1. For point 1 (upper-right panel), it is clear from the plot that anet helicity is produced since the instability parameter almost always stays positive, leading to thedominant production of modes with positive helicity. For point 2 and 3 (lower-left and lower-rightpanel), on the other hand, the instability parameter frequently changes sign but decreases over The parameter ξ R can also be constrained from measurements of the Higgs couplings but the resulting limit isquite weak, | ξ R | < . · [56]. We are working in the Jordan frame. Going to the Einstein frame removes the coupling of the Higgs to theRicci scalar and transforms the inflaton potential V inf to V inf / (1 + ξ R h /M ) . Expanding this for h (cid:28) M Pl / √ ξ R and using V inf = 3 M H gives a contribution to the Higgs mass parameter of ∆ m h = − ξ R H during inflation,consistent with what we have obtained in the Jordan frame. – 18 – τ ( H inf - ) ℌ ( H i n f ) ( H inf ) ℌ k ( H i n f ) × × × × × × τ ( H inf - ) ℌ ( H i n f ) × × × × k ( H inf ) ℌ k ( H i n f ) τ ( H inf - ) ℌ ( H i n f ) × × × k ( H inf ) ℌ k ( H i n f ) Figure 2 : The left panels show the time evolution of the helicity H and the right panels the helicityspectrum H k , i.e. the integrand in eq. (2.11) , evaluated at the maximal τ shown in the left panel.The first, second and third row is for point 1, 2 and 3 in table 1, respectively. time. This results in an excess of modes with positive helicity over those with negative helicity.We show the time evolution of the helicity H in fig. 2, together with the helicity spectrum H k ,i.e. the integrand in eq. (2.11), evaluated at the endpoint of the shown time evolution. That the(comoving) helicity H becomes constant follows from the fact that the produced gauge fields evolveas free fields once the production switches off for ξ ≈
0. Then E · B = 0 and ∂ τ H = 0 as followsfrom eq. (3.9).The results for several quantities of interest are summarized in table 2. The baryon asymmetryis calculated assuming that the helicity survives until the EW phase transition. Its range arisesfrom the uncertainties in the dynamics of the EW phase transition. We have fixed the reheatingtemperature for the different points such that the observed value η B, obs (cid:39) · − is the upperlimit for point 1 and the lower limit for point 2 and 3. The baryon asymmetry increases withthe reheating temperature. Therefore the chosen reheating temperature is the lowest one whichis consistent with the observed baryon asymmetry (given the uncertainties from the EW phasetransition) for point 1 and the highest one for point 2 and 3. The criterion on the CPI temperature– 19 –oint H /H ρ B /H λ B H inf R m R max m ˆ T CPI / GeV η B . · . · . · − · − (2 · − , · − )2 3 . · . · . · − , · − )3 1 . · . · . · − , · − ) Table 2 : Results for the three benchmark points defined in table 1. The range for the baryonasymmetry arises due to the uncertainties in the dynamics of the EW phase transition. Due to theuncertainties in our estimates for the correlation length, the magnetic Reynolds number, the CPItemperature and the baryon-to-entropy ratio, we have rounded these quantities to the first significantdigit. is satisfied for all points too. We find that the criterion on the magnetic Reynolds number is moredifficult to fulfill though. For point 2 and 3, we find R m ∼ ξ R . Decreasing the reheatingtemperature, on the other hand, would lower the magnetic Reynolds number. For point 1, themagnetic Reynolds number as calculated using the estimate for the velocity eq. (3.14) is very small, R m ∼ · − . This likely means that the inverse cascade does not develop and that the helicitydoes not survive. Note, however, that eq. (3.14) has to our knowledge not been numerically verifiedfor our case of diffusion damping. We therefore also give the upper bound R max m on the magneticReynolds number following from energy conservation. We find that R max m ∼ ξ R for point 2 and 3. A dedicated MHDsimulation is clearly desirable to settle this question.The magnetic field that arises after the EW phase transition may survive until late times andcontribute to the magnetic field which permeates the voids of the universe. The observed γ -rayspectra of blazars are suppressed in the range of 1 −
100 GeV compared to expectations (see [19] fora review). This part of the γ -ray spectra receives secondary contributions from high-energy γ -rayswhich scatter on low-energy photons in the intergalactic voids and produce electron-positron pairs.The latter subsequently upscatter CMB photons via inverse Compton scattering. The flux near theearth from this secondary emission can be suppressed if the intermediate electron-positron pairs aredeflected by a magnetic field in the intergalactic voids [57, 58]. It is then an interesting question if thelate-time remnant of the magnetic field that was produced from the relaxing Higgs could explain theobserved suppression in the γ -ray spectra of blazars. To answer this, we need to evolve the strength B of the magnetic field (or B Y between reheating and the EW phase transition) and its correlation λ B (respectively λ B Y ) in time until today. The magnetic field initially scales adiabatically and thecomoving quantities B and λ B are constant. Once enough time has passed for the thermal plasmato affect the magnetic field over its correlation length, τ ∼ λ B /v with v being the characteristicplasma velocity, the inverse cascade sets in. We expect that subsequently the magnetic field scaleseither as in eq. (3.12) or eq. (3.16), depending on the value of the kinetic Reynolds number. Weassume that the inverse cascade continues until recombination and that the magnetic field scalesadiabatically afterwards until today, i.e. B and λ B again being constant. Since we find that thekinetic Reynolds number at reheating is much smaller than unity for all benchmark points in table 1,we expect that the appropriate scaling regime after reheating is given by eq. (3.16). As one option,we have used this scaling regime for the whole period between reheating and recombination. Sincethe kinetic Reynolds number grows in time, we have considered as a second option the possibility– 20 –hat the scaling regime changes to eq. (3.12) once the kinetic Reynolds number becomes biggerthan unity. For completeness, we have as a third option assumed that the scaling regime eq. (3.12)applies during the whole period between reheating and recombination. This gives three values forthe magnetic field strength and its correlation length for each benchmark point in table 1. Wefind that for all three options and for each benchmark point the magnetic field strength for thegiven correlation length is a few orders of magnitude too small to explain the blazar observations[57, 58]. As before, we emphasize, however, that this is only a rough estimate for the magnetic fieldtoday and that a dedicated MHD simulation would be necessary to establish its value with moreconfidence. Let us also note that the magnetic field strength that we find is at least 8 orders ofmagnitude below the upper bound from CMB measurements (see [19]). Finally, let us comment on another option to induce the tachyonic mass term for the Higgs duringinflation. Instead of the coupling of the Higgs to the Ricci scalar in eq. (4.2), we could consider acoupling to the inflaton. The combined potential then reads V (Φ , χ ) = − λ χh χ | Φ | + λ h | Φ | + V inf ( χ ) , (4.6)where χ is the inflaton and V inf ( χ ) the inflationary potential. This leads to a minimum of the Higgspotential which in terms of the inflaton is given by h min ( χ ) = (cid:115) λ χh λ h + β λ h / χ . (4.7)Assuming that the Higgs sits in this minimum, the potential for the inflaton is V ( h min , χ ) (cid:39) − λ h h ( χ ) + V inf ( χ ) , (4.8)where we are neglecting β λ h / λ h . From this, we find the condition V inf ( χ ) > λ h h ( χ ) (4.9)to ensure that the inflationary potential is not perturbed.In particular, condition (4.9) should be fulfilled at the end of inflation. Denoting the value ofthe inflaton at this time by χ , we have V inf ( χ ) = 3 M H , where H inf is the Hubble parameter atthe end of inflation. Taking, as we are doing in this paper, H inf = 10 GeV and λ h (cid:39) . − . h min ( χ ) (cid:46) × GeV (4.10)on the Higgs value at the end of inflation and thus the initial value h ini = h min ( χ ) for the later timeevolution. Moreover, from eqs. (4.7) and (C.7), and assuming χ ≈ M Pl , we get ˙ h ini (cid:39) − h min H inf as the initial condition for the time derivative.Of course condition (4.9) should be fulfilled for the whole inflationary period. In addition weshould also ensure that the joint potential for the Higgs and inflaton has no runaway directions.Both requirements translate into an upper bound on the coupling λ χh . However this bound is modeldependent. Just to illustrate this point we will consider here the simplest inflationary model ofchaotic inflation based on the potential V inf = V ( χ )+ V ( χ ) where V ( χ ) = m χ χ and V ( χ ) = λ χ χ .We will assume that the inflaton potential is dominated during inflation by the quartic potential– 21 – inf (cid:39) V ( χ ), i.e. for χ ∈ (cid:104) √ , (cid:112) N ) (cid:105) M Pl , in which case λ χ (cid:39) . × − as follows fromthe Planck measurement of the primordial amplitude A s = 2 . × − [50] at N = 60 e-folds beforethe end of inflation. From eq. (4.9), the upper bound on λ χh then reads λ χh (cid:46) (cid:112) λ χ λ h ≈ × − . (4.11)This condition also guarantees the absence of runaway directions. At the end of inflation, definedas the time when the slow-roll parameter (cid:15) = 1, the inflaton value in this model is χ = 2 √ M Pl .This gives H inf (cid:39) . × for the Hubble rate at the end of inflation, i.e. somewhat lower thanwhat we have considered in this paper. From this, we get a slightly more stringent condition h min ( χ ) (cid:46) GeV (4.12)on the Higgs value at the end of inflation.After inflation ends, the inflaton oscillates in its potential. We assume that at this stage theinflaton potential is dominated by the quadratic term, V inf ( χ ) (cid:39) V ( χ ), in which case one finds that m χ (cid:39) . × GeV. The amplitude of the inflaton then redshifts like a − / and the time-averageof the Higgs mass term m h = − λ χh χ decreases as a − . This is similar to what happens in thecase of the coupling to the Ricci scalar and induces a movement of the Higgs. Note, however, thatusing the time-averaged value for the Higgs mass term may not be a good approximation here sinceit changes on a similar timescale ∼ H − as the Higgs. It may therefore be necessary to solve forthe joint time-evolution of the Higgs and the inflaton. Note also that in this model the Higgs massand thus the minimum of the Higgs potential already change significantly during inflation. Theresulting movement of the Higgs during inflation may lead to an important additional contributionto photon production. We leave a further study of this model for future work. In this paper, we have considered a scenario where the Higgs couples to the Chern-Simons term ofthe hypercharge gauge group, ∼ | Φ | Y µν ˜ Y µν . If the Higgs is away from the late-time minimum of itspotential during inflation, it relaxes toward this minimum after inflation ends. The time-dependentHiggs value then yields a source term for the photon and can result in the production of a netmagnetic helicity. At reheating this helicity is converted into hypermagnetic helicity which undercertain conditions can survive until the EW phase transition. At this point, the hypermagnetichelicity is converted back into ordinary magnetic helicity. The former enters into the anomalyequation of the current for baryon plus lepton number, B + L , while the latter does not. Theconversion of the helicity during the EW phase transition therefore leads to the production of acompensating B + L -asymmetry. This is partly washed out by EW sphalerons but since the latterfreeze-out while the EW phase transition progresses, a sizeable net B + L -asymmetry can remain.We have first presented the EOMs for the Higgs and the photon which need to be solved inorder to determine the produced magnetic helicity. For simplicity, we have neglected the backre-action of photon production onto the Higgs and finite-temperature effects and have discussed theconditions under which this is justified (conditions which we have checked in all presented cases).After reheating, the resulting hypermagnetic fields start to interact with the thermal plasma andboth evolve jointly afterwards. The comoving helicity can be approximately conserved during thisevolution if the magnetic Reynolds number is sufficiently large. Furthermore, fermion asymmetriesare produced together with the helicity and can partly cancel the helicity by a chiral plasma insta-bility if they are not erased by EW sphalerons before. We have estimated the resulting conditionson the helicity, energy density and correlation length of the produced magnetic fields and on the re-– 22 –eating temperature. Subsequently, we have presented an estimate of the baryon asymmetry whichis generated from the helicity conversion during the EW crossover. The theoretical uncertaintieson these estimates are, however, quite large. Firstly, this is due to the fact that to our knowledgeno numerical MHD simulation exists in the regime which is relevant for us, with small kinetic andlarge magnetic Reynolds number and in the presence of fermion asymmetries. We have instead usedanalytical estimates which are less reliable than a full numerical simulation. Additional theoreticaluncertainties arise from the dynamics of the EW phase transition. Both types of uncertainties willhopefully shrink in the future though, with dedicated MHD simulations and more lattice studies ofthe EW crossover.We have assumed a simple model to obtain a large, initial Higgs value. To this end, wehave included a coupling of the Higgs to the Ricci scalar. The sign was chosen such that it yields atachyonic mass term for the Higgs during inflation. We have also added a singlet scalar which couplesto the Higgs and prevents the Higgs quartic coupling from running negative. During inflation, theHiggs potential then has a minimum at large field values. After inflation ends, this minimum andwith it the Higgs move toward the origin. Via the coupling to the Chern-Simons term of thehypercharge gauge group, this movement leads to the production of a net helicity. We have studiedthree benchmark points to show that the helicity can survive until the EW phase transition andcan reproduce the observed baryon asymmetry of the universe. Other models are conceivable toinduce a large, initial Higgs value and it would be interesting to explore them. In particular, wehave commented on another possibility to obtain a large tachyonic Higgs mass during inflation, bycoupling the Higgs to the inflaton. We leave a more detailed study of this case for future workthough. Acknowledgements
We would like to thank Valerie Domcke and Evangelos Sfakianakis for very useful discussions andcomments. This work is supported by the Secretaria d’Universitats i Recerca del Departamentd’Empresa i Coneixement de la Generalitat de Catalunya under the grant 2017SGR1069, by theMinisterio de Econom´ıa, Industria y Competitividad under the grant FPA2017-88915-P and fromthe Centro de Excelencia Severo Ochoa under the grant SEV-2016-0588. IFAE is partially fundedby the CERCA program of the Generalitat de Catalunya. YC is supported by the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grantagreement No. 754558.
A Ultraviolet completion
In this appendix, we present a possible UV completion which gives rise to the dimension-six operator L = 12 | Φ | M Y µν ˜ Y µν (A.1)which we are using to generate the hypermagnetic field after the end of the inflationary period.A very simple model consists of a complex scalar field S , a singlet under the SM gauge group,which interacts with Y µν ˜ Y µν via the dimension-five operator L = 12 f S (cid:0) e iα S + h.c. (cid:1) Y µν ˜ Y µν , (A.2)where α is an arbitrary phase and f S is a mass scale. After decomposition of the complex field intoits real and imaginary parts, S = r + ia , this gives rise to the usual axial coupling ( a/f S ) Y µν ˜ Y µν for α = π/
2, but of course the coupling can be much more general.– 23 –e will now consider a general renormalizable potential for the field S , with a coupling to theHiggs doublet Φ as V ( S, Φ) = − µ ( e iα S + h.c.) | Φ | + m S | S | + λ Sh | S | | Φ | + 12 λ S | S | + V SM (Φ) , (A.3)where µ ≥ m S ≥ S , and λ Sh ≥ , λ S are real couplings. The global invariance S → e iθ S S , where θ S is an arbitrary phase, is explicitly broken by the first term in the potential (A.3) which thenprevents the appearance of a massless Goldstone boson if S acquires a VEV once EW symmetry isbroken.For momenta much smaller than m S , the field S is decoupled from the theory and can beintegrated out neglecting its kinetic term and simply using its potential. Minimization of thepotential (A.3) yields for S = e iθ | S | (A.4)the equations determining the minimum θ = − αµ | Φ | = ( m S + λ Sh | Φ | + λ S | S | ) | S | . (A.5)A quick glance at eq. (A.5) shows that for µ = 0 the only solution is | S | = 0 as there can be nomassless Goldstone boson. For µ (cid:54) = 0 and λ S , m S (cid:54) = 0 the solution, as can be seen from eq. (A.5),depends on two parameters λ S | S | m S = f ( x , y ) , y ≡ λ S µm S | Φ | m S , x ≡ λ Sh | Φ | m S (A.6)which can be solved analytically. For the validity of the EFT expansion, the parameters x and y should be small. A power series expansion in x and y then gives | S | = µ | Φ | m S (cid:20)
11 + x − y (1 + x ) + 3 y (1 + x ) + · · · (cid:21) = µ | Φ | m S (cid:20) − λ Sh | Φ | m S + O ( | Φ | /m S ) (cid:21) , (A.7)and the term in eq. (A.2) at the minimum yields L = µf S | Φ | m S (cid:20)
11 + x − y (1 + x ) + 3 y (1 + x ) + · · · (cid:21) Y µν ˜ Y µν = µf S | Φ | m S (cid:20) − λ Sh | Φ | m S + O ( | Φ | /m S ) (cid:21) Y µν ˜ Y µν . (A.8)Matching the leading term with eq. (A.1) we find M = (cid:115) f S µ m S . (A.9)Similarly, the potential (A.3) at the minimum is given by V (Φ) = µ m S | Φ | (cid:20) −
11 + x + y x ) − y (1 + x ) + · · · (cid:21) + V SM (Φ)– 24 – µ m S | Φ | (cid:20) − λ Sh | Φ | m S + O ( | Φ | /m S ) (cid:21) + V SM (Φ) . (A.10)Consistently with the condition x , y (cid:28)
1, we will consider field configurations such that λ Sh f S µ | Φ | (cid:28) M , λ S f S µ | Φ | (cid:28) M , (A.11)where we have used eq. (A.9). We are interested in field values up to | Φ | ∼ M . The conditions canthen be fulfilled for example for λ Sh (cid:28) µ ≈ f S (cid:28) M . This ensures that higher-dimensionaloperators in eq. (A.8) of the form ( | Φ | n /M n ) Y µν ˜ Y µν for n ≥ V SM (Φ) ineq. (A.10) are greatly suppressed.Finally, note that a simple way of generating a term like that in eq. (A.2) is through theintroduction of a massive hypercharged vector-like (Dirac) fermion χ with Yukawa coupling [59] λ = | λ | e iθ λ , where θ λ is an arbitrary phase. The corresponding term reads L = λ ¯ χ L Sχ R + h.c. = | λ || S | [cos( θ λ − α ) ¯ χχ + sin( θ λ − α ) ¯ χiγ χ ] , (A.12)where the EOMs (A.5) for the field S have been used. For the phase values θ λ = α ± π/
2, eq. (A.12)yields L = ±| λ || S | ¯ χiγ χ (A.13)Through one-loop diagrams where a loop of χ -fermions is exchanged and emits two photons, thisgives rise to the interaction in eq. (A.2) evaluated in the minimum in eq. (A.5). B Solving the electroweak vacuum instability
In this appendix, we will present a possible SM completion to avoid the instability of the SMvacuum. The simplest possibility is to introduce a massive real singlet φ which couples to the Higgsby a quartic coupling. The joint potential reads V ( φ, Φ) = 12 m φ φ + λ φh | Φ | φ + λ φ φ + V SM (Φ) , (B.1)where we have introduced the symmetry φ → − φ . The coupling λ φh will contribute positively tothe β -function of the Higgs quartic coupling for scales µ > m φ and can avoid the SM instability, aphenomenon which mainly depends on the values of m φ and of λ φh .The RG equations for all dimensionless parameters ( X ), including those of the SM, are dXdt ≡ π ) β X (B.2)where t = log µ and µ is the renormalization scale. With t = log m φ , the β -functions are givenby [60] β g = 4110 g , β g = − g , β g = − g ,β y t = y t (cid:18) y t − g − g − g (cid:19) ,β λ h = λ h (cid:18) λ h + 12 y t − g − g (cid:19) − y t + 27200 g + 920 g g + 98 g + 2 λ φh θ ( t − t ) , For arbitrary values of the phase θ λ , the coefficient of the term ¯ χχ in eq. (A.12) does not vanish, and thecorresponding interactions would also give rise to the Lagrangian term | S | Y µν Y µν . – 25 – λ φh = (cid:20) λ h λ φh + 8 λ φh + 24 λ φh λ φ − λ φh (cid:18) g + 910 g − y t (cid:19)(cid:21) θ ( t − t ) ,β λ φ = (cid:2) λ φh + 72 λ φ (cid:3) θ ( t − t ) . (B.3) C The Higgs potential during inflation
In this appendix, we provide some details about the Higgs potential during the inflationary period.We assume a coupling of the Higgs h to the Ricci scalar R : L = − ξ R R h . (C.1)The Ricci scalar for a flat universe is given in terms of the scale factor a by R = − (cid:34) ¨ aa + (cid:18) ˙ aa (cid:19) (cid:35) , (C.2)where ˙ a ≡ da/dt with t being the cosmological time. During inflation, the slow-roll parameter (cid:15) isgiven by ¨ aa = [1 − (cid:15) ( χ )] H , (C.3)where χ is the inflaton and H = ˙ a/a the Hubble parameter during inflation. Using this, we canwrite R = − − (cid:15) ( χ )] H . (C.4)Let us denote the value of the inflaton (Hubble parameter) for N e-folds before the end ofinflation as χ N ( H N ). At N = 60, one has (cid:15) ( χ ) (cid:28) R (cid:39) − H . The nominal end ofinflation, i.e. N = 0, occurs for (cid:15) ( χ ) ≡
1, which gives R (cid:39) − H . For the subsequent period ofmatter domination, a ∝ t / and R = − H /a . Note that we denote the Hubble rate at the endof inflation by H inf in the rest of the paper, i.e. H = H inf .The Higgs potential during inflation is given by V = − − (cid:15) ( χ N )]2 ξ R H N h N + 14 λ h h N , (C.5)where h N is the value of the Higgs at N e-folds before the end of inflation. The Higgs potential hasa minimum at h N = (cid:115) − (cid:15) ( χ N )] ξ R λ h + β λ h / H N . (C.6)Using the slow-roll equations for the inflaton (cid:15) ( χ ) = 12 M (cid:18) ˙ χH (cid:19) = M (cid:18) V (cid:48) inf ( χ ) V inf ( χ ) (cid:19) , (C.7)and that the inflationary potential is V inf ( χ ) = 3 H M , we find˙ H N = − (cid:15) ( χ N ) H N . (C.8)Taking the time derivative of h N in eq. (C.6), neglecting the field dependence in the square-rootand using eq. (C.8) we then get ˙ h N = − (cid:15) ( χ N ) h N H N . (C.9)– 26 –n models with a single inflaton, the rolling of the inflaton triggers the breakdown of the slow-roll conditions at some value χ = χ such that (cid:15) ( χ ) = 1 and the value and velocity of the Higgs atthe end of inflation are h = (cid:115) ξ R λ h + β λ h / H , ˙ h = − h H . (C.10)If the transition from the end of the inflationary period to matter domination is instantaneous,then the initial value of the Higgs field is h ini ≈ √ h min , where h min is the minimum value of theHiggs field at the onset of matter domination, as given in eq. (4.7), with H inf = H , and its initialvelocity is given by eq. (C.9), ˙ h ini = − h ini H inf .However in multi-field models, or hybrid inflation models [61], there is an instability of the extra(waterfall) field which is triggered when the inflaton reaches a critical value χ c , with correspondingHubble parameter H c . If such a value is reached when (cid:15) (cid:28)
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