A place in the Sun (of cosmological inflation) for the big brother of the Higgs boson

Looking for a non-supersymmetric cosmological B-L inflation...

Committed to the heuristics consisting to explore the landscape of high energy physics speculations along the perspective of the extrapolation of the Higgs mechanism and gauge unification but disregarding supersymmetry as a solution to the naturalness problem in the scalar sector of Yang-Mills-Higgs quantum field theories, I have decided to look today at the following work:

In this article we will deal with a minimal extension of the SM which can explain the vacuum stability of the SM as well as the inflationary dynamics in the light of PLANCK and BICEP2 results. Furthermore, it is well known that in non-supersymmetric theories the quadratic divergence of the Higgs mass remains an open problem and few attempts have been made to deal with such difficulties [48]. In this article we would also try to address whether the Higgs mass can be kept light at the inflationary scale so that reheating at the end of inflation can take place through the channel of inflaton decay into SM Higgs field. There are few other very important outstanding issues with the present Standard Models of particle physics and cosmology, such as the origin of Dark Energy and Dark Matter, which are beyond the scope of the present article. 

To serve our purpose, we extend the SM by an additional U(1)_B-L gauge symmetry. Phenomenological aspects and the issues related to the vacuum stability of this U(1)_B-L extended SM have been extensively analyzed in [26,27,49,50,51,52,53,54]. Such extension of SM has also been discussed previously in several cosmological contexts such as in inflationary scenarios [55,56], to explain the origin of dark matter [57,58,59,60,61,62,63], baryogenesis and leptogenesis [63,64,65] and production of gravitational waves [66]. Here we consider the spontaneous breaking of the U(1)_B-L  symmetry and the electro-weak symmetry to take place at very different energy scales. While the former takes place above the inflationary scale (∼ 10¹⁶ GeV) and the real part of the scalar of this symmetry group plays the role of inflaton, the SM electroweak symmetry breaking takes place around 246 GeV. Also, to our advantage, couplings between the scalar of the U(1)_B-L and the SM particles help reheat the universe at the end of inflation... 

...with a (the?) minimal non-supersymmetric extension of the standard model... 

The U(1)_B-L gauged extended SM, where the full symmetry group is depicted as SU(3)_C⊗SU(2)_L⊗U(1)_Y⊗U(1)_B-L, (2.1) contains three extra right handed neutrinos to cancel all the gauge as well as gravitational anomalies, one extra gauge boson and one extra heavy scalar field (Φ) along with the SM particles. Here the complex scalar field Φ, which is singlet under SM but carries a nonzero B−L charge, is required to break the U(1)_B-L symmetry just above the inflation scale, and after the symmetry breaking the real part of Φ is identified as the inflaton in our scenario... 

Just above the inflationary scale, spontaneous breaking of the U(1)_B-L symmetry yields Φ = 1/√2 (v_φ+φ(t,x)) where v_φ ≡ √(m_φ²/λ²) is the vev acquired by Φ and we have not written the phase part which yields the Goldstone mode. The real part, φ(t,x), of Φ, apart from the vev, can be written as a background field φ₀(t) which plays the role of inflaton and fluctuations δφ(t,x) which give rise to the primordial perturbations during inflation. After the spontaneous breaking of B − L symmetry, the scalar potential V(S,Φ) of this extended theory can be written as : 

V(S,Φ) = λ₁(S^†S)² − m_S²(S^†S) + λ₂(Φ^†Φ − 1/2 v_φ²)² + λ₃(S^†S) (Φ^†Φ − 1/2 v_φ²). (2.5) 

[where S represents the SM Higgs field] We see that various possible terms are generated in the scalar potential part of the Lagrangian, like λ₃v_φ² S^†S, λ₃v_φ S^†Sφ₀, λ₃S^†Sφ₀φ_0. The first term redefines the mass parameter of the S field, the second term opens up the possibility of decay of inflaton into two SM Higgs fields during reheating. The third term introduces scattering of the light Higgs and the inflaton during the inflationary regime. We will concentrate on the importance of the second term later while discussing the decay of inflaton during reheating. 

After the electroweak symmetry is broken at 246 GeV the Higgs field is redefined as S=(0 1/√2(v_S + s))^T and below this scale both the scalar fields get mixed and the physical fields (φ_l and φ_h, where the subscripts l and h stands for ‘light’ and ‘heavy’ respectively) are achieved by diagonalizing the scalar mass matrix. The physical masses of these scalars are given as 

M²φ_l,φ_h = 1/2 { λ₁v_S² + λ₂v_φ²  ± √[(λ₁v_S² - λ₂v_φ²)² + λ₃v_S²v_φ²]} ,  (2.6) 

 where the mixing angle is  tan(2α) = λ₃ v_Sv_φ / (λ₁v_S² - λ₂v_φ²).    (2.7). 

We have set no Abelian mixing at tree level, i.e., g' = 0 at electroweak scale which can be done without any loss of generality. But this mixing will arise through the renormalisation group evolutions [68,69,70] and that has been taken into account in our analysis...

... free from an unstable electroweak vacuum up to the Planck scale ... 

As has been proposed in [72, 73], presence of a heavy scalar, besides the SM particles, eventually leads to a threshold correction to the SM Higgs quartic coupling and helps stabilize the electroweak vacuum as long as the mass of the heavy scalar lies below the instability scale of electroweak vacuum which is around 10¹⁰ GeV. This is the key feature of our model and we would show that though one requires to break the U(1)_B-L symmetry at very high scale (∼ 10¹⁶ GeV) to have successful inflation at GUT scale, the mass of the inflaton can lie below the electroweak instability scale as the quartic coupling of the inflaton has to be fine tuned to yield the correct amplitude of scalar power spectrum, as we show below.  

To show how the threshold correction, due to presence of a heavy scalar, modifies the evolution of Higgs quartic coupling λ1 at lower scale [72, 73], let us consider the scalar potential after U(1)_B-L symmetry breaking given in Eqn. (2.5). At lower energy scales, when the heavy scalar Φ has reached its minima, its equation of motion yields 

Φ^†Φ = 1/2 v_φ² −  λ₃/2λ₂ S^†S. (2.9) 

Below the mass scale mφ of the inflaton, one can thus integrate out the heavy field Φ using the above equation of motion and the potential given in Eqn. (2.5) becomes ...

V(S)|_SM  = λ_S(S^†S)²− m_S²(S^†S.).  (2.11)

Here λ_S is the SM Higgs quartic coupling related to the electroweak symmetry breaking scale and the SM Higgs mass only. At m_φ scale the impact of heavy inflaton field redefines the Higgs quartic coupling as λ_S = λ₁ − λ₃²/4λ₂. This is a pure tree-level effect by which the heavy scalar of the extended theory affects the stability bound of the low energy effective theory even when these two theories are effectively decoupled. The Higgs quartic coupling λ_S of the low energy effective theory receives a positive shift at the mass scale of the inflaton which thus helps avoid the instability which might have occurred above m_φ scale. 

 ... accommodating a large field chaotic type inflation paradigm... 

In this extended model under consideration the real part of the U(1)_B-L breaking scalar field, i.e., φ₀ , plays the role of inflaton. Such a scenario has previously been considered in [56]... . The amplitude of the two-point correlation function or the power spectrum of primordial scalar perturbations are measured through the two-point correlation of the temperature fluctuations in the CMBR. PLANCK has measured this value as [31] P_R ∼ 2.215×10^-9. (2.12) The ratio of the tensor (P_T) and the scalar (P_R) power spectrum is represented as r = P_T/P_R , (2.13) where r is conventionally called the tensor-to-scalar ratio. This ratio r has recently been measured by the BICEP2 experiment to be 0.20_-0.05^+0.07 [3]. But after the release of PLANCK’s recent dust data [32] the observation of BICEP2 has been put under serious scrutiny. Though for the time being, before PLANCK and BICEP2 combine their observations, the upper-limit on r set by PLANCK [32] still survives, i.e., r < 0.11 (95% CL)... 

Assuming that r would not change much during inflation, and {the number of e-foldings} ∆N≈65 to solve the issues with Big Bang scenario, we have 

∆φ₀ /M_Pl=(√530)×r.               (2.22)  

{∆φ₀ is the excursion of the inflaton field during inflation} Hence, for r ≥ O(10^-2) the field excursion during inflation would be super-Planckian (large- field inflationary models), and for r < O(10^-2) it would be sub-Planckian (small-field inflationary models).

 In the present model the inflaton potential can be written as [56] 

V(φ₀) = 1/4 λ₂ (φ₀²− v_φ²)² + aλ₂ log(φ₀/v_φ ) φ₀⁴ , (2.23) 

where we have  

a ≡ 1/(16π²λ₂) {20λ₂² + 2λ₃² + 2λ₂ [Σ(Y_i^N_R)² − 24g²_B-L]+ 96g⁴_B-L − Σ(Y_i^N_R)⁴}.  (2.24) 

The above potential contains the radiative correction added to the tree-level one. Here Y_i^N_R stand for the right handed neutrino Yukawa couplings. The value of ‘a’ determines whether the U(1)_B-L symmetry is broken through the tree-level potential or the radiatively generated logarithmic term. As the value of ‘a’ mostly depends on the value of g_B-L and Y_i^N_R , it can either be positive or negative depending upon the values of the couplings at inflationary scale. At tree level one can then identify the mass term of the inflaton as m_φ=(√λ₂)v_φ. (2.25) In large-field inflationary models one would naturally expect the quartic term with radiative corrections to dominate over the mass term in the inflaton potential.

...compatible with PLANCK and BICEP2 data... 

... if one assumes that the quartic self-interacting term without the radiative correction in the inflaton potential drives inflation... If the pivot scale set by PLANCK, i.e., k∗=0.002 Mpc^-1, crosses the horizon during inflation when N∗∼65 then it generates large tensor-to-scalar ratio as r∗∼0.25 which is also large enough even for BICEP2 observations. This corresponds to the field excursion during inflation to be ∆φ∼12M_Pl. Hence, our aim would be to generate lower values of r while keeping the scenario consistent with the observations of ns and PR by PLANCK. It has been pointed out in [75] that the radiative corrections to the quartic potential play an important role to lower the tensor-to-scalar ratio. Hence, for our inflationary scenario we consider the inflaton potential including radiative correction for inflation... 

{Defining the useful parameter} u = [1+a+4a×ln(φ₀/v_φ)]/a {then} the tensor-to-scalar ratio, the scalar spectral index and the running of the scalar spectral index can be written {respectively} as: 

r = 128M_Pl²/φ₀² × u²/(u − 1)² , 

n_S = 1 − 8M_Pl²/φ₀² × (3u² − u + 4)/(u − 1)² , 

dn_S /dlnk = − 64M_Pl⁴/φ₀⁴ ×  [u(3u³ − 4u² + 15u + 10)/(u − 1)]⁴  ,    (2.37)

... the power spectrum for the inflaton potential including radiative correction turns out to be 

P_R = λ₂ /(768π²) ×(φ₀ M_Pl) ⁶× a(u − 1)³ /u².  (2.39)

In the limit u ≫ 1, one can have |a|≪1, then the radiative corrections become negligible. In such a case the standard results for φ⁴ potential should be retrieved. The other branch known as hilltop solution is important when u≈1 leading to a∼ −(4ln(φ₀/v_φ))^-1. 

We also require to determine the reheat temperature in order to compute the number of e-foldings which corresponds to the pivot scale ... We notice that, apart from the self-interaction term, the inflaton field is also coupled to the SM Higgs field via the mixing term λ₃ which allows it to decay into a pair of SM Higgs during inflation. The decay rate of such an interaction is given as [56]: 

Γ_S(φ₀ → SS) = λ₃²v_φ²/(32πm_φ).            (2.40) 

This decay of inflaton field into SM Higgs would make inflaton unstable for larger values of λ₃. Thus one requires to restrain the decay width of the inflaton during inflation. This requirement can be met if one demands that Γ_S < m_φ which yields 

 λ₃ < √(32π λ₂ ).       (2.41)  

From Eqn. (2.40) we can also roughly estimate the order of reheating temperature T_RH  if the reheating phase is dominated by the Higgs decay. If during the reheating phase the inflaton and its decay products are just in equilibrium then Γ_S∼H where H is the Hubble parameter during the radiation dominated reheating phase. This condition yields   

λ₃²v_φ²/(32πm_φ) =√(π²/90×g∗) × T_RH²/M_Pl ,     (2.42) 

where g∗∼100. Now, let us determine the parameters for a large-field inflationary scenario and take φ_0k∗∼23M_Pl. Putting the central value of scalar spectral index as n_S=0.9603 we find two solutions (u∗) for u at the pivot scale: −0.333 and −11.001. The first solution indicates a hilltop branch inflation whereas the second one gives rise to a φ⁴−branch inflation:

...thanks to a (fine-tuned) radiatively corrected hilltop inflationary scenario... 

Hilltop inflation : If one sets the vev that breaks U(1)_B-L, i.e., the scale of inflation inflation as 10¹⁶ GeV, then for u∗=−0.333 one finds a∗∼ −0.028. This indicates the field value at the end of inflation would be φ_0end ∼ 0.71M_Pl .... This value of u∗ yields the tensor-to-scalar ratio as r∗=0.015 and the inflaton quartic coupling, from the observation of the scalar power amplitude by PLANCK, as λ₂∼1.89×10^-13 . This yields the tree-level mass of the inflaton as m_φ∼4.3×10⁹ GeV. The evolution of the spectral index in such a scenario would be dns dlnk |k∗∼1.07×10^-4. In this scenario the inflaton-Higgs coupling can be of the order of ∼10^-6, which yields the reheating temperature as T_RH∼1.29×10¹³ GeV. This reheating temperature and the energyscale of inflation yield the e-folding at which pivot scale would have exited the horizon as N∗∼ 67... 

 φ⁴−branch inflation : If one sets the scale of inflation to be 10¹⁶  GeV like the hilltop case, one gets a∗ ∼ −0.022 for u∗ = −11.001. This indicates that the field value at the end of inflation, when V ≈ 1, would be φ_0end∼2.6 M_Pl. This u∗ yields the tensor-toscalar ratio as r∗ = 0.203. 

...we have shown that to achieve successful inflation, both the inflaton quartic coupling and the interaction quartic coupling have to be fine tuned. Fine-tuning of inflaton quartic coupling evidently brings down the mass scale of the inflaton field which turns out to be below the instability scale of the electroweak vacuum. Following [72], one can integrate out the heavy inflaton field below its mass scale which then adds a tree-level threshold correction to the low energy effective Higgs quartic coupling λ_S as (see Eqn. (2.10)) 

λ₁ = λ_S + λ₃²/4λ₂       (3.1)

 ... Hence below the inflaton mass scale the stability condition (λ_S>0) for the SM Higgs quartic coupling would get shifted upwards λ₁> δλ ≡ λ₃²/4λ₂. The other two quartic couplings λ₂ and λ₃ would start evolving at energies above this mass scale...

This plot shows the running of the SM quartic coupling as a function of energy scale. The discrete jump at scale ∼ 10⁹GeV is because of the presence of the inflaton having mass ∼10⁹ GeV.

 Apart from the SM fermions this model also contains three right handed neutrinos, N_Ri, which appear in the Lagrangian {and give} rise to the coupling of the inflaton to heavy right handed neutrinos and also masses for N_R. It is important to note that when the (B−L) symmetry is broken at the TeV scale the masses of the right handed neutrinos are less compared to the present scenario. In case of TeV scale breaking the Yukawa couplings (Y ^ν_L) giving rise to the Dirac mass of light neutrinos have to be vanishingly small unless some special textures are considered. Thus in such cases, impact of Y^ν_L in the evolutions of the quartic and other necessary couplings is negligible. But in the present case the right handed neutrino masses are very heavy ∼10^11-13 GeV, due to high U(1)_B-L breaking scale. Thus the light neutrino masses are still light ∼O(eV) even with Y^ν_L∼O(1). Hence unlike the cases, where U(1)B−L symmetry is broken at TeV scale, one can not ignore the contributions of light neutrino Yukawa couplings Y^ν_L in the RGEs in our scenario... 

Looking at the threshold correction, given in Eqn. (3.1), which is essential for electroweak vacuum stability, it may seem that λ₃<0 can still be retained as a possible condition. But, in our analysis this opportunity of achieving larger parameter space for λ₃ is restricted as here λ₂ is very small ∼10^-14 due to inflationary constraints. The absolute value of λ₃ can never be too large as it affects the running of λ₂ by driving its value to a much larger value which might not be able to explain inflationary dynamics. Thus λ₃ is constrained from above by the requirement of inflation. The smallness of |λ₃| ensures that the two scalars present in the theory are basically decoupled from each other as the mixing angle between then becomes too small, see Eqn. (2.7). This confirms that the ‘decoupling theorem’ holds good in our scenario.

...that guarantees a light Higgs at the inflation scale

We would like to note in passing that the stabilization of the SM Higgs boson mass under the quadratic divergences specially in high scale theory is an unavoidable issue. The generic problem with any high scale non-supersymmetric models is related to the stabilization of scalar masses, specially the SM Higgs mass. Due to the quadratic divergences the SM Higgs mass acquires a correction proportional to Λ², where Λ is the scale of new physics. In supersymmetric theory these corrections automatically get cancelled out with the ones coming from their supersymmetric partners. To avoid such large contributions to scalar mass in non-supersymmetric models one needs to impose the Veltman condition. This prescription, as suggested in [76, 77], confirms the removal of quadratic divergences of the scalar masses to stabilize them. We note here that in our model the scalar masses might obtain a large radiative correction. To avoid such catastrophe in scalar masses we need to satisfy the Veltman criteria (VC), which for this scenario would be [78] 

δm_s² ∝ v_φ² [(2λ₁ + λ₃/3) cos⁴α + (2M_W²+M_Z²)/v²)cos²α + 4g_B-L²sin²α − 4(Y_t²cos²α + (Y^ν_L)²cos²α + (Y^ν_R)² sin² α)],     (3.4)

with cos²α∼ 0.99879, where α is the mixing angle as given in Eqn. (2.7). Since  v_φ is very large for our case, Z_B-L and ν_R will not affect this criteria much. Also λ₃ is very small. But the light neutrino Yukawa coupling can be large here and thus its impact can be sizeable. This has been added with top quark contribution. We find that within the available parameter space in our model it is indeed possible to satisfy VC either at the inflation scale or at Planck scale. With light neutrino Yukawa to be 0.1462 and 0.2413 the VC can be satisfied at the inflation and the Planck scales respectively. This implies that the light Higgs remains light at the inflation scale and the decay of the inflaton, considered in this paper for explaining the reheating, does not suffer any catastrophe under the impact of radiative corrections. Thus we can stabilize the Higgs mass at the inflation scale but perhaps this mechanism does not solve the stabilization at other scales. 

To summarize 

In this work, we have adopted a gauge extended SM scenario which contains a SM singlet scalar field with a new U(1)_B-L gauge charge. This field acquires vev at a very high scale and breaks the U(1)_B-L symmetry spontaneously and also couples to the SM particles. Apart from this SM singlet scalar, there are three right handed neutrinos which successfully generates the light neutrino masses through type-I seesaw without fine-tuning the Dirac Yukawa coupling. 

The electroweak breaking scale is around 246 GeV whereas in this case the U(1)_B-L breaking scale should lie near the GUT scale so that such high-scale inflation can take place as has been demanded by BICEP2. But, as the U(1)_B-L  and electroweak breaking scales lie far apart in our scenario, these two theories are basically decoupled from each other, as has been demanded by the ‘decoupling theorem’. Then it might imply that extending the SM by such high scale U(1)_B-L gauge theory fails to serve its purpose of taking care of the stability of electroweak vacuum. But the advantage of introducing such high scale U(1)_B-L symmetry is that it provides a heavy scalar Φ in the theory, whose real part plays the role of the inflaton in such a scenario. Presence of a heavy scalar yields a threshold correction to the Higgs quartic coupling, if one integrates out this heavy scalar below its mass scale. Hence if the mass of this heavy scalar lies below the electroweak instability scale (∼ 10¹⁰ GeV), the threshold correction eventually helps avoid the instability of the vacuum by correctly uplifting the value of the SM Higgs quartic coupling at this mass scale. Hence the key point is to keep the mass scale of the heavy scalar in the theory below the electroweak scale if one wants the threshold corrections to help stabilize the vacuum, even though the explicit value of the mass of the heavy scalar does not play any important role.

Higgs vacuum stability and inflationary dynamics after BICEP2 and PLANCK dust polarisation data

Kaushik Bhattacharya, Joydeep Chakrabortty, Suratna Das, Tanmoy Mondal

(Submitted on 18 Aug 2014 (v1), last revised 17 Nov 2014 (this version, v2))