Higgs inflation still alive
Abstract
The observed value of the Higgs mass indicates that the Higgs potential becomes small and flat at the scale around . Having this fact in mind, we reconsider the Higgs inflation scenario proposed by Bezrukov and Shaposhnikov. It turns out that the nonminimal coupling of the Higgssquared to the Ricci scalar can be smaller than ten. For example, corresponds to the tensortoscalar ratio , which is consistent with the recent observation by BICEP2.
The observed value of the Higgs mass [1] indicates that the Standard Model (SM) Higgs potential becomes small and flat at the scale around ; see e.g. [2, 3, 4, 5, 6, 7, 8, 9] for latest analyses.^{1}^{1}1 It is an intriguing fact that the bare Higgs mass also becomes small at the same scale [7, 10, 11]; see also Refs. [12, 13, 14]. The running Higgs mass after the subtraction of the quadratic divergence is considered e.g. in Ref. [15]. See Fig. 1 for the Higgs potential around that scale for various values of the top quark mass [10]. We see that by tuning the top quark mass, we can make the first derivative at the inflection point arbitrarily small as shown by the blue (center) line. Note that the required tuning of the top quark mass is rather strict. The values of are given to show the amount of tuning and should not be taken literally.^{2}^{2}2 The latest combined result for the top quark mass is [16]. Note that there can be a discrepancy between the pole mass and the one measured at the hadron colliders; see e.g. Refs. [5, 10]. The latter is obtained as an invariant mass of the color singlet final states, whereas the former is a pole of a colored quark. At the hadron colliders, the observed pair is dominantly color octet, and there may be discrepancy of order 12 GeV in drawing extra lines to make the singlet final states. We thank Yukinari Sumino on this point. See also Ref. [17]. There are several arguments that this tuning is required by a principle such as the multiple point principle [18, 19, 20], the maximum entropy principle [21, 22], the classical conformality [23, 24, 25, 26, 27, 28, 29, 30, 31], and the asymptotic safety [32].
It is known that this inflection point cannot be used to achieve a successful inflation [33, 34].^{3}^{3}3 See e.g. Refs. [35, 36] for attempts of the inflection point inflation. Slowroll condition restricts the field value to be very close to the inflection point. To earn a sufficient folding within this range of , the first derivative at the inflection point must be very small, and hence cannot yield the right amount of the amplitude at .
In Ref. [34], we have discussed a possibility that a new physics, such as string theory, modifies the Higgs potential above the scale . In this Letter, we pursue another possibility that the nonminimal coupling of the Higgssquared to the Ricci scalar, , leads to a successful inflection point inflation.
The main differences from the ordinary Higgs inflation scenario [37, 38, 39, 40, 41] are the following two points:^{4}^{4}4 For the other attempts, see Refs. [42, 43, 44, 45]. See also Refs. [46, 47].

The folding is earned in passing the inflection point, and hence the relation no longer holds. Therefore, the scalartotensor ratio can be sizable to match the recent BICEP2 result [48]:
(1) at the 68% CL.

can be smaller than ten, since the Higgs quartic coupling is small at due to the tuning mentioned above.
We start from the same Lagrangian as the ordinary Higgs inflation [37, 39, 40].^{5}^{5}5 Here we are treating the Higgs inflation as a singlefield model, as in Refs. [37, 39, 40]. Even when the NambuGoldstone bosons are explicitly included, the multifield version of Higgs inflation rapidly evolves as an effectively singlefield model due to the dynamics of the system’s evolution [49, 50], rather than by tacitly assuming unitary gauge. The analysis in Ref. [49] did not incorporate quantum corrections as the present paper does, but nonetheless, based on Ref. [49] as well as Ref. [50], it should be clear that the singlefield (dynamical) attractor behavior holds whenever the nonminimal coupling is sufficiently large. The potential in the Einstein frame can be obtained from the effective potential
(2) 
in the flat space, by setting with
(3) 
where is the Higgs field in the Jordan frame.^{6}^{6}6 This choice corresponds to the prescription I in Ref. [39], which minimizes the oneloop logarithmic correction to the effective potential in the Einstein frame.
The running coupling has a minimum at , depending on the Higgs mass [2, 3, 4, 5, 6, 7, 8, 9].^{7}^{7}7 The Higgs quartic coupling grows above the minimum due to the contribution of the growing coupling. Qualitatively, the position and height of the minimum depend on the Higgs and top masses, respectively. Around the minimum, can be expanded as
(4) 
where in the SM [34]. The term proportional to and higher are small in the region of our interest, and we will neglect them hereafter. The value of depends on the top quark mass, and we can set it arbitrarily small by tuning the top quark mass within the current experimental bound.
For the potential to be monotonically increasing around the inflection point, it is necessary and sufficient that
(5) 
The equality holds when the potential has a plateau. That is, when we put , the point becomes a saddle point with vanishing first and second derivatives.^{8}^{8}8 There appears another inflection point at too.
We set the value of slightly larger than to realize an inflection point inflation, while keeping the potential above sufficiently small by the introduction of in order to evade the problem described above. The three cases , , and corresponds to the red (upper), blue (middle), and green (lower) curves in Fig. 1, respectively. An important point here is that the value of in Eq. (3) is saturated to for large values of (), and therefore the potential does not grow rapidly. In order for this saturation to work to avoid too large , we need , that is, .
As concrete examples, we show our results for several benchmark points with the parameter choice , and 1000 with , , and in the left panel in Fig. 2; the same figure is drawn in linear plot for in the right panel.
To fit the cosmological data, we can e.g. take , , , to get , , and , where
(6) 
with
(7) 
For the same parameters, the Einsteinframe time evolution of the Higgs field is plotted in Fig. 3. We see that substantial time is spent around the inflection point.
Once the tensortoscalar ratio is fixed to be , the slowroll parameter becomes , and the amplitude fixes the potential height . The potential height is determined in our case to be , which is the same as the Higgs inflation. The difference is the value of that allows us to take .
In this Letter, we have matched the renormalization scale in the Einstein frame, as in Eq. (3). If we instead match it in the Jordan frame,^{9}^{9}9 It is argued in Ref. [51] that these two choices correspond to different theories. The choice corresponds to the prescription II in Ref. [39], which minimizes the oneloop logarithmic correction in the Jordan frame. We note however that there is a criticism on this choice [52]. Whether calculations involving quantum corrections are or are not framedependent is a point of ongoing research, and we leave it open for further discussion. i.e. if we set in Eq. (2), we obtain the chaotic inflation at . In this region, the canonically normalized field is in the Einstein frame. The potential for becomes quadratic:
(8) 
We see that by taking , we get the right amount of the inflaton mass .
Finally we comment on the unitarity issue in the Higgs inflation due to the large nonminimal coupling, which requires a new physics above the scale in order to cure the scattering being strongly coupled on the electroweak vacuum () [53, 54, 55, 56, 57].^{10}^{10}10 We note that the cutoff scale depends on the background field value , and for , it becomes [40]. That is, inflation takes place below the cutoff scale. It is an implicit assumption of the Higgs inflation that such an extension does not affect the result qualitatively, that is, the Wilson coefficients of the higher order terms are sufficiently smaller than . Note that our is greatly reduced from the value in the ordinary scenario.
Note added:
Acknowledgement
K.O. and S.C.P. thank Takuya Kakuda and Jinsu Kim for useful discussions. We thank Yukinari Sumino for the useful comments. The work of Y.H. is partly supported by a GrantinAid from the Japan Society for the Promotion of Science (JSPS) Fellows No. 25.1107. S.C.P. is supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (20110010294) and (20110029758) and (NRF2013R1A1A2064120). The work of K.O. is in part supported by the GrantinAid for Scientific Research Nos. 23104009, 20244028, and 23740192.
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