Study of EW Vacuum Stability

1 2 3 4 5

5.´10^-9 1.´10^-8 1.5´10^-8 2.´10^-8 2.5´10^-8

1 2 3 4 5

5.´10^-9 1.´10^-8 1.5´10^-8 2.´10^-8 2.5´10^-8

c₁H10^-11L ΛΧ

Figure 3.2: Allowed region ofc1(in unit of 10⁻¹¹) andλχ is indicated by the shadowed region where the constraints [(i)-(v)] are included . The black dot represents the reference value of

c1'm˜²= 3.39×10⁻¹¹ andλχ'2.43×10⁻⁸. αsatisfies 2.7×10⁻⁴< α <2.5×10⁻³.

φ˜^∗ ' 16 and ˜χ^∗ ' 0.9, we have λχ < 2.7×10⁻⁸ such that λχχ˜⁴/4 < (1/2) ˜m²φ˜². So λχ is restricted by 1.67×10⁻¹⁰< λ_χ<2.7×10⁻⁸. From this we note that a choicec₁ = ˜m² falls in the right ballpark which corresponds to λχ= (c1/2α) = 2.43×10⁻⁸. Considering the range of α as obtained, the allowed parameter space for c1 and λχ are shown in Fig.3.2 where the point corresponding to c₁= ˜m² andλ_χ= 2.43×10⁻⁸ is denoted by a dark dot.

3.4 Study of EW Vacuum Stability

We now turn our attention to the other part of the model which involves the SM Higgs doublet H and its interaction with the inflation sector. The relevant tree level potential is given by

V_II=λ_H

H^†H−v² 2

2

+λχH

2 (χ²−v_χ²)

H^†H−v² 2

+λφH

2 φ²

H^†H−v² 2

. (3.9)

As discussed earlier we expect the threshold effect on the running of the Higgs quartic coupling to appear from its interaction with the χ field through its large vev v_χ. For simplicity we drop the last term involving interaction between H and inflatonφfor the rest of our discussion by assuming λ_φH as vanishingly small. The presence of λ_χH coupling ensures that during inflation H settles at origin as it receives a large mass term proportional to φ as given by m^eff_H ∼[c1λχH/(2λχ)]^1/2φ(using Eq.(3.2)) compared to the Hubble termH_Inf (withλχH >> λχ

and neglectingv). So we can disregard any kind of fluctuation of the Higgs field during inflation.

Eventually the effective potential during inflation is described by φalone as in Eq. (3.5) (after χ being integrated out and setting H field at zero). However once the inflation is over, φfield starts to oscillate about its minimum at φ= 0 with large amplitude (initially) and hencem^eff_H

68 Chapter 3. EW vacuum stability and chaotic Inflation starts changing rapidly as well. The Higgs field is then expected not to stay at origin and could be anywhere in the field space [207, 310]. Hence the Higgs instability problem (provided λ_h turns negative in some energy scale ΛI) may appear if Higgs field takes value beyond ΛI. The true minimum of the Higgs potential at H =v develops at a later stage when oscillation of the φ is over and it settles at origin while χ stays at v_χ. Below we discuss how we can avoid this instability problem after inflation. To explore the stability of the Higgs potential after inflation we need to consider the λ_χterm from Eq.(3.1) as well. The part of the entire potential V_I+V_II relevant to discuss the vacuum stability issue is therefore given by

V₀ =λ_H

H^†H−v² 2

2

+λ_χH

2 (χ²−v²_χ)

H^†H− v² 2

+ λ_χ

4 (χ²−v_χ²)². (3.10) The minimum of V₀ is given by

hH^†Hi= v²

2,hχi=v_χ, (3.11)

where we have consideredλH,λχ>0. Note that the above minimum ofV0(the EW minimum) corresponds to vanishing vacuum energy i.e. V₀^EW= 0. Now in order to maintain the stability of the potential, V0 should remain positive (V0 > 0) even when the fields involved (χ and H) are away from their respective values at the EW minimum. Since the couplings depend on the renormalization scale µ(∼field value) we must ensure λ_χ(µ), λ_H(µ)>0 in order to avoid any deeper minimum (lower than theV₀^EW) away from the EW one.

In order to study the running of all the couplings under consideration, we consider the renormalization group (RG) equations for them. Below we provide the RG equations for λH, λ_χH and λ_χ [58] as

dλ_H

dt =β_λ^SM_H + 1 16π²

λ²_χH

2 , (3.12)

dλχH

dt = 1 16π²

n

12λ_Hλ_χH+ 8λ_χλ_χH + 4λ²_χH + 6y_t²λ_χH−3

2g₁²λ_χH −9

2g₂²λ_χHo

, (3.13) dλχ

dt = 1 16π²

n

20λ²_χ+ 2λ²_χH o

. (3.14)

β_λ^SM

H is the three loop β-function for the Higgs quartic coupling [58] in SM, which is corrected by the one-loop contribution in presence of the χ field. The RG equation for two new physics parameters λ_χH and λ_χ are kept at one loop. The presence of the χ field is therefore expected to modify the stability conditions above its mass mχ.

Apart from the modified running of the Higgs quartic coupling, vacuum stability is also affected by the presence of the threshold correction from the heavyχfield which carries a large

3.4. Study of EW Vacuum Stability 69 vev. The mass of the χ field is given by mχ = p

2λχvχ (following from Eq.(3.3)), the heavy field χ can be integrated out belowm_χ. By solving the equation of motion ofχ field we have

χ² 'v_χ²− λ_χH λχ

H^†H−v² 2

. (3.15)

Hence below the scale mχ, the effective potential ofV0 becomes V₀^eff 'λh

H^†H− v² 2

2

, withλh(mχ) = h

λH −λ²_χH 4λχ

i

mχ

, (3.16)

where Eq.(3.15) is used to replaceχinto Eq.(3.10). Therefore belowmχ,λh corresponds to the SM Higgs quartic coupling and abovem_χ it gets a positive shiftδλ= ^λ

2 χH

4λχ. This could obviously help in delaying the Higgs quartic coupling to become negative provided mχ is below the SM instability scale,i.e. m_χ<Λ^SM_I . In this analysis, we investigate the parameter space for which the Higgs quartic coupling remains positive upto the scaleM_P. This however depends uponm_χ and δλ. Note thatλχ involved in bothmχ and δλ which is somewhat restricted from inflation (see Fig.3.2). On the other hand v_χ is not restricted from inflation. We will have an estimate of vχ by requiring mχ<Λ^SM_I (using a specific λχ value corresponding to set-1 of Table 1.) We also consider δλ ∼λ_h(m_χ) to avoid un-naturalness in the amount of shift. This consideration in turn fixes λχH.

From the above discussion, it is clear that with suitable choice of parameters involved, the Higgs quartic coupling can remain positive till a very high scale beyond Λ^SM_I , say upto theH_Inf or even until MP. As we have discussed before, theH field stays at origin during inflation and after inflation (during the oscillatory phase of theφfield) it can be randomly placed. Howeverλ_h being positive untill a very large scale (even beyond the inflationary scale), the Higgs potential has only a single minimum at hHi=v. Hence once the oscillation period of theφfield is over, the H field naturally enters into the electroweak vacuum and electroweak symmetry is broken subsequently. At this stage the Higgs field can be redefined asH^T =

0 ^v+ζ√

2

using the unitary gauge. Now setting φ= 0 and χ=v_χ, we find the mass-squared matrix involvingχand ζ as

M²= 2λHv² λχHvvχ

λ_χHvvχ 2λχv²_χ

!

, (3.17)

which followed from Eq.(3.10). The diagonalization of this mass-squared matrix yields one light and one heavy scalars. The masses associated with the light and heavy eigenstates are given by

m_h,hH = h

λ_Hv²+λχv²_χ∓ r

λ_Hv²−λχv_χ² 2

+λ²_χHv²v²_χ i1/2

, (3.18)

70 Chapter 3. EW vacuum stability and chaotic Inflation wherehandhH correspond to light and heavy Higgses respectively. In the limit of small mixing angleθ=h

1

2tan⁻¹_λ^λχHvχv

Hv²−λχv²_χ

i

,m_hbecomes of order∼√ 2v

λ_H−^λ

2 χH

4λχ

withλ_χv_χ² >> λ_Hv². In order to avoid the unwanted negative value of mh, the extra stability condition 4λχλH > λ²_χH should be maintained (with λ_χH > 0). However it is pointed out in [58] that it is sufficient that this condition should be satisfied for a short interval around m_χ forλ_χH >0. However for λχH <0, this extra stability condition becomes 2p

λχ(µ)λH(µ) +λχH(µ)>0. As found in [58], it is difficult to achieve the absolute stability of V₀ till M_P in this case. We restrict ourselves into the caseλχH >0 for the present work.

Scale (µ) y_t g₁ g₂ g₃ λ_h

m_t 0.93668 0.357632 0.648228 1.166508 0.127102

Table 3.2: Values of couplings in SM atm_t= 173.3 GeV form_t= 173.3 GeV,m_h = 125.66 GeV andαs= 0.1184.

Next we proceed to analyze the RG running of all the relevant couplings present in the set up from µ= m_t to M_P energy scale. For µ < m_χ, as the χ field would be integrated out, we should only consider the RG evolution of the SM couplings. The initial values of all the SM couplings at m_t energy scale can be determined through Eqs.(1.97-1.101). As an example we tabulize the initial values of the top quark Yukawa coupling (yt), gauge couplings (gi) and Higgs quartic coupling (λ_h) consideringm_t= 173.3 GeV,m_h= 125.66 GeV andα_s= 0.1184 in Table 3.2.

Scale(µ) λ_χ λ_χH y_t g₁ g₂ g₃ λ_H

m_χ 2.38×10⁻⁸ 3.66×10⁻⁵ 0.59523 0.386864 0.58822 0.72327 0.0278 M_P 2.45×10⁻⁸ 3.56×10⁻⁵ 0.39112 0.467056 0.509155 0.49591 O(10⁻⁵)

Table 3.3: Values of couplings atm_χ and M_P for m_χ = 8×10⁷ GeV andm_t= 173.3 GeV, mh= 125.66 GeV andαs= 0.1184.

For energy scale µ > m_χ, two other couplings λ_χ and λ_χH will appear as in Eq.(3.10).

As discussed earlier, we already have an estimate of λχ to have successful results in inflation sector withc1 = ˜m². We consider the corresponding value ofλχ= 2.43×10⁻⁸ at inflation scale Λ_Inf∼V_Inf^1/4'10¹⁶ GeV. The initial value ofλ_χ should be fixed atm_χ (remains same atm_t) in such a way that it can reproduceλχ(Λ_Inf) correctly through its RG equation in Eq.(3.14). λχH

(m_χ) is chosen to achieve a natural enhancement δλ ∼ λ_h(m_χ) at m_χ. Hereafter above m_χ,

3.4. Study of EW Vacuum Stability 71

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸ -0.02

0 0.02 0.04 0.06 0.08 0.1

0.12 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

μ

λ

m_χ

λH

λh

SM SM+Inf fields

Figure 3.3: Running of Higgs quartic coupling tillMP forλχH(mχ) = 3.66×10⁻⁵, λχ(mχ) = 2.38×10⁻⁸ withmχ = 8×10⁷ GeV.

the Higgs quartic coupling λ_H is governed through the modified RG equation³ as in Eq.(3.12).

Note that even ifλχ is known it does not fixmχ(=p

2λχvχ) completely. Therefore we can vary v_χ to have m_χ<Λ^SM_I .

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸ -0.02

0 0.02 0.04 0.06 0.08 0.1

0.12 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

μ

λ

m_χ

λH

λ_h

SM SM+Inf fields

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸ -0.02

0 0.02 0.04 0.06 0.08 0.1

0.12 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

μ

λ

m_χ

λ_H λh

SM SM+Inf fields

Figure 3.4: Running of Higgs quartic coupling tillMP for [left panel:] mχ(= 5×10⁸GeV)>

mC, [right panel:] mχ(= 5×10⁶ GeV) < mC. We consider δλ= λh(mχ) which corresponds to the choice for (a) λχH(mχ) = 2.8×10⁻⁵,λχ(mχ) = 2.4×10⁻⁸ and (b)λχH(mχ) = 4.88×

10⁻⁵, λχ(mχ) = 2.35×10⁻⁸ respectively.

With the above mentioned scheme, we study the running of the Higgs quartic coupling for differentm_χsatisfyingm_χ<Λ^SM_I . We find that withm_χ = 8×10⁷GeV,λ_H becomes vanishingly small atMP (hence the new instability scale ΛI becomes∼MP) as shown in Fig.3.3. We specify

3It can be noted that such aλχHdoes not alter the running ofλχmuch.

72 Chapter 3. EW vacuum stability and chaotic Inflation

Figure 3.5: Running of the couplingsg_i=1,2,3, y_t, λ_H,λ_χH (in unit of 10⁻⁴) andλ_χ (in unit of 10⁻⁷) in SM+Inflation scenario fromm_χ=m_C∼8×10⁷GeV toM_P.

the corresponding m_χ value as m_C(= 8×10⁷ GeV). The value of all the relevant couplings at m_χ and M_P are provided in Table 3.3. It is then observed that in order to keep

Figure 3.6: Variation of instability scale ΛI with change of δλ where mχ =mC ' 8×10⁷ GeV.

the Higgs quartic coupling positive all the way upto M_P, we should ensure m_χ < m_C. For example with mχ = 5×10⁸ GeV (> mc) we see λH becomes negative at scale ∼10¹² GeV as shown in left panel of Fig.3.4. On the other hand in right panel of Fig.3.4 it is seen that λ_H remains positive till MP for mχ ∼ 5×10⁶ GeV < mC. In doing so we consider the amount of positive shift at mχ to be defined with δλ ∼λ_h(mχ). In Fig.3.6 we provide the variation of instability scale Λ_I (which is atmost∼M_P form_χ=m_C case) in SM+Inflation extension if we relax this assumption by changing δλ arbitrarily. To give a feeling about how other couplings

3.5. Reheating after Inflation 73