mercredi 13 juillet 2016

The Inflation-Higgs connection...

... made quantitative (at the state of the art?)
Here is a recent preprint that claims to derive a rigorous correlation between a future measurement of the tensor-to-scalar ratio and the scale at which the Higgs potential must receive stabilizing corrections in order for the Universe to have survived inflation until today:

A striking feature of the Standard Model (SM) is that, in the absence of stabilizing corrections, the Higgs potential develops an instability, with the maximum of the potential occurring at V(Λmax)1/41010 GeV. This leads to the existence of a “true vacuum” at large Higgs field values, which may carry important consequences for our Universe [1–9]. Our present existence does not necessarily demand physics beyond the SM, since current measurements of the Higgs boson and top quark masses indicate that the electroweak (EW) vacuum is metastable, i.e., long-lived relative to the age of the Universe. The scenario is different, however, if our Universe underwent an early period of cosmic inflation with substantial energy density. The inflaton energy density, parametrized by the Hubble parameter H, produces large local fluctuations in the Higgs field, δh ∼ H/2π . As such, when H is sufficiently large during inflation, the Higgs field may sample the unstable part of the potential. If sampling this part of the potential can be shown to be catastrophic for the surrounding spacetime, the eventual survival of our Universe in the EW vacuum would consequently imply constraints on the nature of the inflationary epoch that gave rise to our Universe. Conversely, near-future Cosmic Microwave Background (CMB) experiments will probe tensor-to-scalar ratios of r>0.002 [10], corresponding to inflationary scales H > 1013 GeV. If it can be shown that the SM Higgs potential is inconsistent with such high-scale inflation, a measurement of non-zero r provides evidence for the existence of stabilizing corrections to the Higgs potential... 
The main goal of this paper is a definitive study of [two important aspects: first, the evolution of the Higgs field under a combination of (inflation-induced) quantum fluctuations and the classical potential and, secondly, the evolution of spacetime responding to the Higgs vacuum].
Limits on the inflaton energy density, parametrized by the Hubble parameter H [GeV] (black contours) in the (mhmt) plane requiring Nmax ≥ 60 (solid) or 50 (dashed). Central values are taken to be mh= 125.09 ± 0.24 GeV [40] and mt= 172.44 ± 0.70 GeV [41], with contours corresponding to 1-, 2-, and 3-σ regions as for two parameters. The shaded regions represent: the Higgs potential is stable up to MP (green); the Higgs potential is unstable, but current limits r<0.07 [45] permit required amount of inflation (blue); and instability would preclude the combination of Nmax > 60 and r > 0.002, to be probed by near-future experiments [10] (red).

We have studied the dynamical response of inflating spacetime to unstable fluctuations in the Higgs field with numerical simulations of Einstein gravity. Our results offer, for the first time, an in-depth understanding of how spacetime evolves as a Higgs fluctuation falls towards, and eventually reaches, the true, negative energy, vacuum. We find that when true vacuum patches stop inflating and create a crunching region, the energy liberated creates a black hole surrounded by a shell of negative energy density. This region of true vacuum persists and grows throughout inflation, with more and more energy being locked behind the black hole horizon. In contrast to the naïve expectation that this growth is due to the boundary between true and metastable vacua sweeping outward in space, in an exponentially expanding spacetime the growth occurs in a causally-disconnected manner. Spatial points fall to the true vacuum independent of the fact that neighboring points have also reached the true vacuum. Hence, under most circumstances, this process is insensitive to the behavior in the interior region, and to the exact shape of the potential close to the true minimum. 
We also extended the numerical solution of the Fokker-Planck equation to resolve the field distribution in the exponentially suppressed tails. This is necessary to extract the tiny probabilities associated with a single true vacuum patch in our past light-cone, while simultaneously incorporating the effects from renormalization group running of the quartic in the Higgs potential on the evolution of the probability distribution. Using this solution, in conjunction with the result from our classical General Relativity simulations that a single true vacuum patch in our past light-cone destroys the Universe, we derived a bound H/Λmax<0.07 on the scale of inflation. This bound is the most accurate available to date, and we compared it to bounds derived previously. We also found, as shown in Fig. 8, that a future measurement of the tensor to scalar ratio with r > 0.002 would imply the need for a stabilizing correction to the Higgs potential at a scale < 1014 GeV supposing mt> 171.4 GeV. We are thus able to correlate a cosmological quantity with the necessity of stabilizing corrections to the Higgs potential 
Finally, we re-emphasize that the results in this paper are of wider interest than the SM Higgs potential, as they are applicable to the inflationary dynamics of any scalar field with a negative energy true vacuum.
(Submitted on 1 Jul 2016)

mardi 12 juillet 2016

My own private ostinato

A tentative articulate answer to some recent Lubos Motl comments that appeared here and there about physics on non-commutative spaces

In a recent post* Lubos Motl appears to me as reducing the noncommutative geometrization of physics by Alain Connes and his collaborators as a fancy formalism to rewrite quantum field theory in an abstract way but it is much more than that. I am going to skip through his most easily proved wrong assertion
Connes' "fix" that reduced the prediction to 125 GeV was largely ignored by the later pro-Connes literature that kept on insisting that 170 GeV is indeed what the theory predicts.
just providing the two following informative examples:
NCG made a prediction for the SM Higgs at approximately 170 GeV [CIS99,KS06,CCM07], which is now ruled out by experiment [A+12, C+12b]. Since then a number of solutions to the ‘Higgs mass problem’ have been proposed. Notably by Estrada and Marcolli who include gravitation corrections to obtain a 125 GeV Higgs [EM13a], while Chamseddine and Connes propose an alternative solution in which they add an extra scalar field σ into the model by hand [CC12]. Later papers by Devastato et al. [DLM14a] and Chamseddine et al. [CCvS13] construct standard model extensions which naturally include the σ field as an output. Our fused algebra formulation of the NCG SM {FB15b} offers a natural solution to the problem which I will outline in full detail...  

The recent discovery of the Higgs boson with a mass mH≃126 Gev suggests the big desert hypothesis should be questioned. There is indeed an instability in the electroweak vacuum which is meta-stable rather than stable (see [5] for the most recent update). There does not seem to be a consensus in the community whether this is an important problem or not: on the one hand the mean time of this meta-stable state is longer than the age of the universe, on the other hand in some cosmological scenario the meta-stability may be problematic [26, 27]. Still, the fact that mH is almost at the boundary value between the stable and meta-stable phases of the electroweak vacuum suggests that “something may be going on”. In particular, particle physicists have shown how a new scalar field suitably coupled to the Higgs - usually denoted σ - can cure the instability (e.g. [13, 25]). 
Taking into account this extra field in the NCG description of the SM induces a modification of the flow of the Higgs mass, governed by the parameter r = kν kwhich is the ratio of the Dirac mass of the neutrino and of the Yukawa coupling of the quark top. Remarkably, for any value of Λ between 1012 and 1017 GeV, there exists a realistic value r ≃ 1 which brings back the computed value of mH to 126 Gev [8]. 
The question is then to generate the extra field σ in agreement with the tools of noncommutative geometry. Early attempts in this direction have been done in [34], but they require the adjunction of new fermions (see [35] for a recent state of the art). In [8], a scalar σ correctly coupled to the Higgs is obtained without touching the fermionic content of the model, simply by turning the Majorana mass kR of the neutrino into a field kR → kRσ.
(Submitted on 5 Nov 2014 (v1), last revised 29 Jan 2015 (this version, v2))

I prefer to insist first on a problem of terminology that Lubos skips completely in his post and comments:

We shall make the distinction between noncommutative spacetimes intended as spaces whose coordinates no longer commute, and spectral geometries intended as a space whose algebra of functions is nonnecessarily commutative. 
Noncommutative spacetimes, ... , are obtained as a deformation of a usual space by trading the (commutative) coordinate functions xµ, xν of a manifold with coordinate operators qµ, qν satisfying non-trivial commutation relations. Besides the seminal quantum spacetime model of Doplicher, Fredenhagen, Roberts [2] treated in L. Tomassini talk [3], such spaces are present in many - if not all - approaches to quantum gravity, including loop quantum gravity, string theory... as well as in more phenomenology-oriented models like doubly special relativity...
Noncommutative spacetimes also emerged very early as a possible solution to ultraviolet divergencies in quantum field theory, especially in the work of Snyder [6]. Quantum field and gauge theories on noncommutative spacetimes have thus developed as a theory on their own, independently of any consideration on quantum gravity...  
Spectral geometries [11]... consists in a generalization of Gelfand duality between locally compact spaces and C∗-algebras, so that to encompass all the aspects of Riemannian geometry [12] beyond topology. It furnishes a geometrical interpretation of the Lagrangian of the standard model of elementary particles [13, 14], as well as some possibilities to go beyond [15, 16]...

There is a second aspect completely skipped by the author of the Reference Frame blog. It is the following. To quote the physicist Thomas Schucker :
... noncommutative {spectral geometries} are close enough to Riemannian spaces such that Einstein’s derivation of gravity from Riemannian geometry carries over to noncommutative {spectral geometries}. In Connes’ derivation, the entire Yang-Mills-Higgs action pops up as a companion to the Einstein-Hilbert action, just like the magnetic field pops up as a companion to the electric field, when the latter is generalized to Minkowskian geometry, i.e. special relativity. 
(Submitted on 29 Mar 2010)
I think Lubos completely misses or eludes this point in his paragraph (and the followings) starting with :
There are many detailed questions that Connes can't quite answer that show that he doesn't really know what he's doing. One of these questions is really elementary: Is gravity supposed to be a part of his picture?
The understanding (checking by computation) of what Schucker claims requires time and skill (that I do not claim to have). Some accomplished physicists did. Lubos and all string theorists have skill. The current paucity of empirical evidence for TeV scale susy/wim-particles may give time to some young bold ones to appreciate the fact that Connes noncommutative geometry has been extended since 2006 and might have already uncovered at least one really nontrivial fact in our quest to quantum gravity: 
...any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of {some proposed} two-sided commutation relations in dimension 4 and ... th{is} representation give{s} a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics 
(Submitted on 4 Nov 2014)
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions. The two algebras M2() and M4() are obtained which are the exact constituents of the Standard Model. Using the two maps from M4 to S2 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter and area quantization of black holes. 
(Submitted on 8 Sep 2014 (v1), last revised 11 Feb 2015 (this version, v4))

These higher degree Heisenberg-like new commutation relations with the help of the spectral action principle might select the very specific effective quantum field theory beyond the Standard Model viable up to the seesaw scale and further to grand unification. Moreover the proposed quanta of geometry might provide non trivial insights into the black hole microscopic degrees of freedom and the dark sector of the lambda-CDM cosmological standard model. This is my tentative answer to Lubos, in particular when he writes:

Now, Connes and collaborators claim to have something clearly different from the usual rules of quantum field theory (or string theory). The discovery of a new framework that would be "on par" with quantum field theory or string theory would surely be a huge one, just like the discovery of additional dimensions of the spacetime of any kind. Except that we have never been shown what the Connes' framework actually is, how to decide whether a paper describing a model of this kind belongs to Connes' framework or not. And we haven't been given any genuine evidence that the additional dimensions of Connes' type exist... 
Connes et al. basically want to have a non-singular compactification without branes and they still want to claim that they may decouple some ordinary standard-model-like physics from everything else – like the excited strings or (even if you decided that those don't exist) the black hole microstates which surely have to exist. But that's almost certainly not possible. I don't have a totally rock-solid proof but it seems to follow from what we know from many lines of research and it's a good enough reason to ignore Connes' research direction as a wrong one unless he finds something that is really nontrivial, which he hasn't done yet...

Lubos finished his post with the following sentence:
"The principles producing theories that seem to work should be taken very seriously" 
I could not say it better indeed! 

jeudi 7 juillet 2016

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 (vφ+φ(t,x)) where vφ ≡ √(mφ22) 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(SS)2 − mS2(SS) + λ2Φ − 1/2 vφ2)2 + λ3(SS) (ΦΦ − 1/2 vφ2). (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 λ3vφSS, λ3vφ S0, λ3S0φ0. 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(vS + s))T and below this scale both the scalar fields get mixed and the physical fields (φl and φh, 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 
M2φl= 1/2 { λ1vS2 + λ2vφ2  ± √[(λ1vS2 - λ2vφ2)2 + λ3vS2vφ2]} ,  (2.6) 
 where the mixing angle is  tan(2α) = λvSvφ / (λ1vS2 - λ2vφ2).    (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 [7273], 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 
ΦΦ = 1/2 vφ2 −  λ3/2λSS. (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  = λS(SS)2− mS2(SS.).  (2.11)
Here λS is the SM Higgs quartic coupling related to the electroweak symmetry breaking scale and the SM Higgs mass only. At mφ scale the impact of heavy inflaton field redefines the Higgs quartic coupling as λS = 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 λS 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 mφ 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 [31PR ∼ 2.215×10-9. (2.12) The ratio of the tensor (PT) and the scalar (PR) power spectrum is represented as r = PT/PR , (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 /MPl=(√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 λ(φ02− vφ2)2 + aλ2 log(φ0/vφφ04 , (2.23) 
where we have  
a ≡ 1/(16π2λ2) {20λ22 + 2λ32 + 2λ[Σ(YiNR)2 − 24g2B-L]+ 96g4B-L − Σ(YiNR)4}.  (2.24) 
The above potential contains the radiative correction added to the tree-level one. Here YiNR 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 gB-L and YiNR , 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 mφ=(λ2)vφ. (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∼12MPl. 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/vφ)]/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 = 128MPl202 × u2/(u − 1)2 , 
nS = 1 − 8MPl20× (3u2 − u + 4)/(u − 1)2 , 
dnS /dlnk = − 64MPl404 ×  [u(3u3 − 4u2 + 15u + 10)/(u − 1)]4  ,    (2.37)
... the power spectrum for the inflaton potential including radiative correction turns out to be 
PR = λ2 /(768π2×(φ0 MPl) 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/vφ))-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]: 
ΓS0 → SS) = λ32vφ2/(32πmφ).            (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 ΓS < mφ which yields 
 λ3 < √(32π λ2 ).       (2.41)  
From Eqn. (2.40) we can also roughly estimate the order of reheating temperature TRH  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 ΓS∼H where H is the Hubble parameter during the radiation dominated reheating phase. This condition yields   
λ32vφ2/(32πmφ) =√(π2/90×g× TRH2/MPl ,     (2.42) 
where g∼100. Now, let us determine the parameters for a large-field inflationary scenario and take φ0k∼23MPl. Putting the central value of scalar spectral index as nS=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 φ0end ∼ 0.71MPl .... 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 mφ∼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 TRH∼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 φ0end∼2.6 MPl. 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 λS as (see Eqn. (2.10)) 
λ1 = λS + λ32/4λ2       (3.1)
 ... Hence below the inflaton mass scale the stability condition (λS>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, NRi, which appear in the Lagrangian {and give} rise to the coupling of the inflaton to heavy right handed neutrinos and also masses for NR. 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 νL) 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 YνL 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 ∼1011-13 GeV, due to high U(1)B-L breaking scale. Thus the light neutrino masses are still light ∼O(eV) even with YνL∼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 YνL 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] 
δms2 ∝ vφ2 [(2λ1 + λ3/3) cos4α + (2MW2+MZ2)/v2)cos2α + 4gB-L2sin2α − 4(Yt2cos2α + (YνL)2cos2α + (YνR)2 sin2 α)],     (3.4)
with cos2α∼ 0.99879, where α is the mixing angle as given in Eqn. (2.7). Since  vφ is very large for our case, ZB-L and νR 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.
(Submitted on 18 Aug 2014 (v1), last revised 17 Nov 2014 (this version, v2))