Higgs Inflation as Nonlinear Sigma Model and Scalaron as its $\sigma$-meson

We point out that a model with scalar fields with a large nonminimal coupling to the Ricci scalar, such as Higgs inflation, can be regarded as a nonlinear sigma model (NLSM). The $\sigma$-meson, which is induced by quantum corrections, is identified as the scalaron in this model. Our understanding provides a novel alternative picture for the emergence of the scalaron, which was previously studied in the Jordan frame by computing the running of the term quadratic in the Ricci scalar. We demonstrate that quantum corrections inevitably induce other operators on top of this NLSM, which give rise to the $\sigma$-meson. We confirm that the $\sigma$-meson indeed corresponds to the scalaron. With the help of the $\sigma$-meson, the NLSM is UV-completed to be a linear sigma model (LSM). We show that the LSM only involves renormalizable interactions and hence its perturbative unitarity holds up to the Planck scale unless it hits a Landau pole, which is in agreement with the renormalizability of quadratic gravity.


Introduction
Cosmic inflation is a successful paradigm for the description of the very early Universe. While solving the flatness and horizon problems in Big Bang cosmology, its accelerated expansion of the Universe provides an origin of anisotropies in the cosmic microwave background (CMB) and gives rise to primordial gravitational waves. Such quasi-de Sitter phase is realized once we have a scalar field, the so-called inflaton, slowly rolling down its potential during inflation. The combined bounds from the current observations in the (n s , r )-plane [1] imply a concave potential for the inflaton.
Inflation caused by the Standard Model Higgs stands out as an attractive candidate model among many others because of its minimality. To have successful inflation, a nonminimal coupling to gravity is introduced in Refs. [2][3][4]: where H is the Standard Model Higgs doublet and ξ is the nonminimal coupling of H to the Ricci scalar R. This term modifies the Higgs quartic potential for a large field value of the Higgs |H | M P /ξ, surprisingly in perfect agreement with the aforementioned observational bound [1]. To produce a curvature perturbation of the right magnitude, the Higgs quartic coupling λ and the nonminimal coupling ξ should fulfill ξ 2 2 × 10 9 λ, implying ξ 1 unless λ is extremely small. 1 Classically, such a large coupling is just a choice of a parameter. However, quantum corrections induce other operators associated with this large coupling via a renormalization group (RG) flow. By computing scalar one-loop diagrams in the Jordan frame, one finds an enhancement of the R 2 term for ξ 1 [7][8][9][10][11][12] L α = αR 2 , dα d ln µ = − N 1152π 2 (6ξ + 1) 2 , (1.2) where N counts the number of real scalar fields, i.e., N = 4 for Higgs-inflation. This R 2 term makes the spin-0 part of the metric dynamical [13][14][15][16], corresponding to the introduction of the the so-called scalaron, whose mass is m 2 s ∼ M 2 P /α. Although one may choose α to be small at a particular scale, this never holds for the entire range of energy scales, implying its typical value is α ∼ ξ 2 1. 2 On the other hand, the Renormalization Group Equation (RGE) of the other operator appearing at the same loop level, L α 2 = α 2 (R µν R µν − R 2 /3), does not depend on ξ. 3 Therefore, as depicted in Fig. 1, the scalaron typically shows up much below the cutoff scale where the problematic spin-2 ghost associated 1 See Refs. [5,6] for critical Higgs inflation where the top quark mass is tuned such that the Higgs four-point coupling becomes tiny via the running. 2 Strictly speaking, the running coupling at a scale µ involves a numerical factor and a log term as α µ ∼ 10 −2 ξ 2 ln Λ/µ (for N = 4). Here Λ is the UV cutoff scale and it is chosen so that α Λ is small. Throughout this paper, we omit this numerical factor in the estimation because it is translated into at most an order one factor in the scalaron mass, m s ∼ M P /α 1/2 . 3 The beta functions of α and α 2 are computed in, e.g., Refs. [12,[17][18][19][20][21][22][23][24][25][26], although the sign of the beta function of α is wrong in some references. Eq. (1.2) agrees with, e.g., Refs. [12, 17, 20-22, 25, 26]. M m s ∼ M P /ξ ξR |H | 2 , αR 2 m 2 ∼ M P α 2 R µν R µν Figure 1: A schematic picture of the typical scale M of the spin-0 operators ξR |H | 2 and αR 2 , and the spin-2 operator α 2 R µν R µν for ξ 1. Quantum corrections induce the R 2 term with its coefficient typically α ∼ O (ξ 2 ), while the R µν R µν term is not affected by ξ and hence typically α 2 ∼ O (1). As a result, there is a large hierarchy between the scalaron mass m 2 s = M 2 P /12α and the spin-2 ghost mass m 2 2 = −M 2 P /2α 2 , which justifies to ignore the spin-2 sector for ξ 1.
to R µν R µν [27] appears, i.e., m s ∼ M P /ξ m 2 ∼ M P for ξ 1 [12]. This also guarantees the appearance of the scalaron within the validity of effective field theory.
Although physics should be frame independent, all these observations for the appearance of the light scalaron at ξ 1 rely on the Jordan frame analysis. To illustrate this question, let us go to a conformal frame by performing the Weyl transformation, where the Higgs is decoupled from the scalar part of gravity (i.e., conformal coupling). 4 In this frame the appearance of the scalaron cannot be seen as an enhancement of the R 2 term because there is no large running for α. The large nonminimal coupling in the defining frame (1.1) is now converted to the curvature of the field space metric in the conformal frame. Namely, Higgs inflation becomes a nonlinear sigma model (NLSM) with Notice that, since the Higgs field contains four degrees of freedom, one cannot flatten away this curvature by using field redefinitions. 5 Hence, in this frame, the light scalaron should solely stem from the . In Refs. [31,32], this model was investigated in the large-N analysis in 2 + dimensions, and a σ-meson as a new scalar degree of freedom was found. In Refs. [33,34] it is suggested that the standard model Higgs boson could be regarded as a σ-meson by identifying longitudinal modes of standard model gauge bosons as NG bosons on O(4)/O (3). Another well-known example is a NLSM on CP N −1 which can be written as a linear sigma model (LSM) by introducing an auxiliary gauge field that makes the hidden local symmetry [35,36] apparent. As shown in Refs. [36][37][38][39][40][41], quantum corrections make this auxiliary gauge field dynamical, the so-called ρ-meson. The Nambu-Jona-Lasinio model [42,43] and the Gross-Neveu model [44], although 4 See Eq. (2.15) for its precise definition. 5 One should not naively think that the kinetic term can be flattened by just looking at the radial part of Higgs because there are NG bosons. For a theory with one real scalar field, one can indeed flatten the kinetic term because its target space is a one-dimensional manifold whose curvature always vanishes. In the gauged case, the longitudinal modes of the gauge bosons play the same role as the NG bosons in the unitary gauge because of the NG boson equivalence theorem. Refs. [28][29][30] provide an understanding of the cut-off scale of Higgs inflation in terms of the target space curvature in the Einstein frame, including the above points, in detail. not NLSMs, have a similar property, where mesons absent classically get generated by resummation of four-Fermi interactions in the large-N limit.
Our main purpose is to provide a better understanding of the emergent new scalar degree of freedom in Higgs inflation. To this end, we show that Higgs inflation can be written as a NLSM on a specific target space embedded in R 1,N +1 which is composed of not only the Higgs field but the scalar part of the metric, as far as the scalar sector of gravity is concerned. The frame transformation is then regarded as a subset of coordinate transformations in R 1,N +1 , which clarifies the frame independence. By computing quantum corrections on this NLSM, we show that a new scalar degree of freedom shows up in the spectrum which can be identified as a σ-meson. In particular, if one goes to a field basis corresponding to the conformal frame (1.3), this identification becomes even more transparent because the structure of the model is quite similar to the O(N ) NLSM. 6 We confirm that this σ-meson exactly corresponds to the scalaron. As expected, the σ-meson UV-completes the NLSM to be a LSM with renormalizable interactions, which is consistent with the renormalizability of quadratic gravity [22,[45][46][47][48]. 7

Organization of this paper
The goal of this paper is to obtain a better understanding on Higgs inflation, in particular its quantum properties, by reinterpreting it in terms of the NLSM. As a first step, in the next Sec. 2, we show that Higgs inflation at the classical level (i.e., without any counter terms) can be mapped to a NLSM, as far as only the scalar sector of the gravity is concerned. As we have argued in the introduction, it is justified to focus on the scalar sector of the gravity in the limit ξ 1 (which we assume to be valid), since the scalar sector of gravity is controlled by M P /ξ, while the spin-2 sector is controlled by M P . We thus discuss the physics above M P /ξ but below M P and hence decouple the spin-2 sector of gravity in the following. 8 Once we map Higgs inflation to the NLSM, we can understand several properties of Higgs inflation in the language of the NLSM. For instance, since the Higgs is now regarded as a pion, this NLSM implies the existence of a new particle, i.e., the σ-meson. In Sec. 3, by quoting the result of Sec. 4, we show that the σ-meson of Higgs inflation is nothing but the scalaron (which comes from the R 2 term in the original Jordan frame language). As we will see, the NLSM viewpoint clarifies the reason why Higgs inflation with the scalaron becomes renormalizable as far as the scalar sector is concerned.
In Sec. 4, we explicitly show how quantum corrections give rise to the σ-meson in the large-N analysis. 6 Recall that the O(N ) NLSM is given by .
(1.4) 7 Other UV-completions of Higgs inflation are discussed in Refs. [49][50][51][52]. In particular, Refs. [50,51] add an additional scalar degree of freedom to UV-complete Higgs inflation which they call σ-field in analogy with the LSM. Their model is, however, not a LSM in a strict sense since the target space with the additional field is still not completely flat. On the other hand, the scalaron makes the target space including the scalar part of gravity completely flat (see Eqs. (3.3) and (3.12)), and hence is identified as the σ-meson in a more strict sense. 8 Even if we include the spin-2 sector of the gravity, we may regard Higgs inflaiton as a NLSM conformally coupled to gravity with the help of the conformal compensator field, as demonstrated in App. C.
We see that the scalaron, or the σ-meson, naturally becomes light due to quantum corrections for ξ 1, in agreement with the result of Ref. [12]. We also clarify that the frame-transformation invariance of the results is simply understood as the invariance under a field redefinition among the pions, or a coordinate transformation in the target space. Sec. 5 is devoted to summary and discussion.

Higgs inflation as NLSM
This section is composed of two parts. In Sec. 2.1, we show that a model with scalar fields nonminimally coupled to gravity, such as Higgs inflation, can be written as a NLSM, starting from the action in the Jordan frame. The Weyl transformation in the original language is understood as a field redefinition among scalar fields, or a coordinate transformation of the target space. In Sec. 2.2, we show by a field redefinition that the target space of Higgs inflation takes the following simple form where φ i is a real scalar field, Φ is the scalar part of the metric (which we will define below), and ξ is a nonminimal coupling. The summation over the index i ranging from 1 to N is implied. For Higgs inflation N = 4. Note that the standard model Higgs potential has a global symmetry under O(4) SU(2) L × SU(2) R , which leads to the custodial symmetry.
We emphasize here that we do not consider any quantum effects in this section. Thus "Higgs inflation" in this section always indicates the theory without any counter terms. Quantum corrections of Higgs inflation are studied in the large-N limit in Sec. 4, where we see that a light σ-meson (or scalaron) emerges as a dynamical degree of freedom, as we have already advertised in the introduction.

Higgs inflation as NLSM
We start from the action for Higgs inflation in the Jordan frame: where M P is the reduced Planck mass, g µν is the spacetime metric with g its determinant, R is the Ricci scalar, ξ is a nonminimal coupling, λ is a quartic coupling, and i = 1, ..., N with N = 4 for the standard model Higgs. As mentioned in the introduction, we focus on the scalar sector of this model by assuming ξ 1 in this paper. The metric contains two scalar degrees of freedom before gauge fixing, and we may kill one of them by fixing the gauge such that the metric takes the form The Ricci scalar for the metric (2.3) is given by the action is rewritten as where the fieldΦ is defined asΦ Thus, Higgs inflation is now written in the form of the NLSM with the curvature of its target space controlled by the parameter 6ξ + 1. In particular, if the scalar fields are conformally coupled to gravity, ξ = −1/6, the target space is flat and the action reduces to a LSM as expected.
Two comments are in order. First, the kinetic term ofΦ has the wrong sign, and henceΦ is a ghostlike mode. In fact, such a ghost exists even in pure Einstein gravity (see, e.g., Ref. [53]), and it resembles the time-like component of the U(1) gauge field in the Lorenz gauge. Although ghost-like, it is unphysical and hence does not spoil the theory thanks to a residual gauge symmetry (see App. B for more details on these points). Second, the Weyl transformation is now equivalent to a field redefinition ofΦ.
One sometimes performs the Weyl transformation with some function Ω to move, e.g., from the Jordan frame to the Einstein frame. Since the trace part of the metric is now entirely contained in the fieldΦ, the Weyl transformation is equivalent to the field redefinition,Φ → Ω −1Φ . (2.9) In general, the physics of the NLSM is invariant under a coordinate transformation of the target space, which in our case is a field redefinition amongΦ and φ i . Since the Weyl transformation is merely one particular form of such a transformation, physical results are not affected. In the next Sec. 2.2, we perform a coordinate transformation of the target space so that one may easily find the target space of Higgs inflation.

Target space of Higgs inflation
In the previous Sec. 2.1, we have seen that Higgs inflation is written in the form of the NLSM (2.6). In this subsection, we study the target space of Higgs inflation in detail. For this purpose, let us go to a 9 The right arrow hereafter means that we replace quantities on the left-hand side by those on the right-hand side. particular field basis by performing a field redefinition ofΦ as The new field Φ satisfies and hence the action (2.6) is written in terms of Φ as where the scalar function h = h(Φ, φ) is given by It is clear from this expression that the target space of Higgs inflation is given by The target space of Higgs inflation takes the rather simple form (2.14). In particular, if we take ξ = 1/6 and completely ignore gravity by taking Φ constant, it reduces to the O(N + 1) NLSM. In this sense, Higgs inflation can be understood as a modification of the O(N +1) NLSM. This simplicity explains why Higgs inflation is unitarized by a σ-meson (or scalaron) induced by quantum effects, as we discuss in Secs. 3 and 4 (see also Refs. [10,12,54]), or more generally, why the scalar sector of the quadratic gravity is renormalizable [22,[45][46][47][48]. There is, however, an important difference between the ordinary NLSMs and our NLSM (2.14): we have a ghost-like direction Φ originating from the metric. It is crucial to keep track of Φ to correctly compute the scalaron potential as we see in Sec. 4. It also enables us to recover the action with a general spacetime metric from the relation in Eq. (2.3). For instance, since Φ always appears in place of 6 M P , it is easy to see that the curved spacetime action corresponding to Eq. (2.12) is where the scalar functionh =h(φ) is given bỹ

Scalaron as σ-meson
In the previous Sec. 2, we have shown that Higgs inflation at the classical level can be regarded as a NLSM on an N +1-dimensional hypersurface spanned by the Higgs φ i and the scalar part of the metric Φ in R 1,N +1 . This structure can be seen easily in a particular basis as shown in Eq. (2.12): Its target space is given by Eq. (2.14).
As in the next Sec. 4. Higgs inflation with quantum corrections in the large-N limit is given by The target space is now completely flat and the theory has become a LSM of Φ, φ i , and σ. One can recover Eq. (3.1) by taking the limit of α → 0. Although one may fix the α parameter to be constant classically, quantum corrections force it to run as Eq. (4.12), implying its typical value to be α ∼ N ξ 2 .
In the following, we first show that this σ-meson is nothing but the scalaron. Then, we discuss the unitarity and renormalizability of the Higgs-scalaron system. We see that the expression Eq. (3.3) clarifies these issues.

Scalaron as σ-meson
To avoid unnecessary confusion, let us first recover the information of the Ricci scalar so that the gravity sector of (3.3) has the conventional form. By identifying Φ to be 6M P e ϕ , recalling that g µν = e 2ϕ η µν and rescaling φ i and σ, we obtain Eq. (3.3) in the conformal frame where both the Higgs and the σ-meson are conformally coupled. This form of the potential is anal- A non-zero finite value of α generated quantum mechanically allows fluctuations perpendicular to the target space, corresponding to the σ-meson. Now it is clear that such an enhanced coupling, α ∼ N ξ 2 , leads to a light σ-meson below the cutoff, i.e., m s ∼ M P / α M P . Below, we show explicitly that this LSM is equivalent to Higgs inflation with an R 2 term in the Jordan frame and hence the σ-meson is nothing but the scalaron.
As mentioned in the introduction (1.2), it is well known that a large ξ induces the R 2 term by quantum corrections in the Jordan frame analysis: We rewrite it by introducing an auxiliary field X as usual: Since X does not have any kinetic term, its equation of motion is a constraint, and one recovers the original action by substituting its solution (which gives X = R). We now transform the action to the conformal frame, where the scalaron and the Higgs are conformally coupled to gravity as in Eq. (3.4).
We perform the Weyl transformation as and then the action is rewritten as (3.8) In order to end up in the conformal frame, we specify the conformal factor Ω as 10 1 We can think of σ as a fundamental field instead of X , and then X is given by As a result, the action can be expressed in terms of σ as which coincides with Eq. (3.4) as expected. This clarifies that, once Higgs inflation is viewed as a NLSM, the scalaron that originated from the R 2 term UV-completes the NLSM to be a LSM. The scalaron in this language is nothing but the σ-meson.

Unitarity and renormalizability
We have seen that, if we regard Higgs inflation as a NLSM (2.12), the scalaron is understood as the the unitarity and renormalizability can hold even above the Planck scale as far as the scalar sector is concerned, which is consistent with the analysis based on the scattering amplitude in Ref. [12]. In reality, of course, the renormalizability is lost by the presence of the spin-2 graviton at the Planck scale.
Still, the field basis given in Eq. izability of quadratic gravity [22,[45][46][47][48]. As far as the scalar sector is concerned, the Higgs-scalaron system is equivalent to quadratic gravity with scalar fields nonminimally coupled to gravity, since the other operator in quadratic gravity, R µν R µν (or more precisely R µν R µν − R 2 /3), only affects the tensor sector, leading to the infamous spin-2 ghost. Hence, the unitarity and renormalizability of quadratic gravity up to the Planck scale can also be understood to be a property of Eq.
Although this is equivalent to Eq. (3.3), its properties such as the unitarity scale and renormalizabitily are not apparent in this field (or coordinate) basis. The power of Eq. (3.3) comes from its appropriate field basis which makes the flatness of the target space manifest. As a result, we can easily understand the unitarity and renormalizability by just looking at its potential.

Large-N analysis of NLSM
In this section, we compute quantum corrections to the NLSM [Eq. (2.12)] in the large-N limit. We will show that an additional operator is required to renormalize the theory, which induces a new scalar degree of freedom, i.e., the σ-meson as given in Eq. (3.3). Let us start with which is a slight generalization of Eq. (2.12). Its target space is N +1) . In this way, we will discuss the UV-completion We can show that this choice of parameters is special as it allows us to have successful inflation as discussed in App. D. 11 Here we clarify the role of Φ by comparing this theory with a usual NLSM. One may decouple gravity completely by taking Φ to be constant. Then, the action becomes quite similar to the O(N ) NLSM (1.4): We can see that the information of the parameter c is now lost. As we will see in the end of Sec. 4.2, while one can reproduce the correct mass for the σ-meson starting from Eq. (4.3), the overall potential shape requires Φ and hence the full Eq. (4.1). This is intuitive because the property at the potential minimum is pure IR while its overall structure depends on the strength of the coupling to (the scalar part of) gravity.
In order to perform the large-N analysis, the field basis in Eq. (4.1) is not convenient because it involves a square root, and hence we have to deal with an infinite series of φ 2 i after Taylor expanding to keep track of the leading N consistently. This motivates us to perform a field redefinition (or coordinate transformation of the target space) so that the large-N analysis becomes more transparent: which implies By usingΦ, one may write down Eq. (4.1) as follows: where the action (4.6) now contains only a finite number of φ 2 i -interactions. 12 Note here that, since the full result does not depend on the choice of field basis, the result at each order in the large-N expansion is also independent of this choice as one may vary N arbitrarily. This is the manifestation of the frame independence in our language. The gauge independence of our results follows in the same way. Now we are in a position to study quantum corrections of this model in the large-N limit, and see that the σ-meson emerges owing to the quantum corrections. We will also see that the large-N analysis not only predicts the existence of the σ-meson, but also correctly reproduces the running of its mass solely from the NLSM that is the IR theory. In this sense, it provides a bottom-up way to estimate the property of the UV theory from the IR.
As mentioned in the introduction, let us emphasize again that the emergence of a new degree of freedom from quantum corrections is not unique to our NLSM. In Ref. [31,32], it is shown that the O(N ) NLSM in 2+ dimensions is UV-completed to be a LSM with a σ-meson emerging from quantum corrections. It is implied in Refs. [33,34]  A NLSM on CP N −1 can be expressed as a LSM with an auxiliary gauge field that makes the hidden local symmetry manifest [35,36]. Once quantum corrections are taken into account, the gauge boson develops its kinetic term and hence becomes dynamical [36][37][38][39][40][41]. The Nambu-Jona-Lasinio model [42,43] and the Gross-Neveu model [44] are also within this class of models, where mesons that are classically absent are generated dynamically due to resummation of four-fermi interactions in the large-N limit.
It is part of the virtue of mapping Higgs inflation to the NLSM that we can see the similarity between the analysis in this paper (and Refs. [12,55,56]) and the literature on the NLSM.

Emergence of σ-meson as scalaron in Higgs inflation
Let us start with the Higgs inflation model with a = c = 1/2 and b = (6ξ + 1)/2. The action (4.6) in this case is given by Here we keep the Higgs four-point interaction to clarify its effect in the large-N analysis.
Let us first focus on divergences involving the Higgs four-point interaction. Adopting dimensional regularization, we have two divergent diagrams in the large N (4.8) 12 It does not matter thatΦ appears in the denominator since we have to care only about φ i in the large-N limit.
where the solid line indicates the operator Φ /Φ and the dotted line the scalar fields φ i . The first diagram is renormalized by the Higgs four-point coupling, and the second one is renormalized by the nonminimal coupling 6ξ + 1. Hence, to cure the divergences involving the Higgs four-point interaction, we do not need to introduce any additional operators, since both of them are already present in Eq. (4.7). It is straightforward to check that they correctly reproduce the running of the Higgs four-point coupling and the nonminimal coupling.
On the other hand, a new operator is required in order to renormalize the two-point function of the operator Φ /Φ, which is diagrammatically given by (4.9) and whose corresponding counter term is (4.10) Note that the divergences at the higher loop level, which are diagrammatically given by are renormalized by the same term (4.10), and hence no other terms are required in the large-N limit.
We obtain the RG running of α dα d ln µ = − N 1152π 2 (6ξ + 1) 2 , (4.12) which coincides with the running of the R 2 term (1.2). The value of α at a specific energy scale depends on the boundary condition which is a parameter choice of the theory. Including quantum corrections at the leading order in the large-N limit, the classical action (4.7) is now modified to (4.13) One can see that this expression coincides with Eq. (3.13) as expected. Namely, the field basisΦ convenient for the large-N analysis corresponds to the Jordan frame.
Since the counter term (4.10) is a higher derivative term, it implies the existence of an additional degree of freedom. To extract it, we introduce an auxiliary field σ (4.14) After shifting σ as σ → σ + (6ξ + 1)φ 2 i /2Φ, we obtain Finally, by redefining the field as Φ ≡Φ − σ, we obtain the desired result (4.16) Thus, the additional degree of freedom introduced by the higher derivative term (4.10) is indeed the σ-meson that linearizes the original NLSM (4.7), or equivalently, the scalaron as we discussed in Sec. 3.
The striking property of Eq. (4.16) is its renormalizability. As discussed in the previous Sec. 3.2, this feature corresponds to the renormalizability of quadratic gravity as long as the scalar sector is concerned.
Moreover, this expression (4.16) is also quite useful in computing quantum corrections above the mass scale of the σ-meson or the scalaron. All one needs to do is to compute loops in this renormalizable scalar field theory, with one wrong sign in the kinetic term of Φ. See App. B for the reason why this wrong kinetic term is harmless. We emphasize that the running of the mass term computed from the LSM (4.16) agrees with the running of α (4.12) computed from the NLSM (4.7) in the large-N limit, and hence the large-N analysis indeed provides a bottom-up approach to study the properties of the UV theory.
By inserting Φ = 6M P e ϕ and g µν = e 2ϕ η µν , and rescaling φ i and σ, one may recover the conventional expression with Ricci scalar, leading to Eq. (4.16) in the conformal frame: (4.17) Here we derive this expression in a rather indirect way by killing the irrelevant degrees of freedom in the metric as g µν = e 2ϕ η µν . As outlined in App. C, it could be possible to obtain Eq. (4.17) in a more direct way with the help of the conformal compensator. We leave a detailed derivation for our future work.
Here we want to make one remark. Additional degrees of freedom that arise due to higher derivative terms are often ghost-like, known as Ostrogradsky ghosts (see e.g., Ref. [57] and references therein). In our case, however, σ has a kinetic term with the correct sign, and hence is healthy. It is because the scalar part of the metric Φ has a kinetic term with the wrong sign and is ghost-like. Thus, we may phrase this phenomenon as "minus times minus gives plus," or "the ghost of a ghost is healthy."

Emergence of σ-meson in general model
Now, we consider the general model (4.6) with arbitrary a, b and c. Although the computation is technically more involved, the heart of the analysis is the same as in the previous case; we first find divergences and counter terms in the large-N limit, and then extract the additional degree of freedom that arises from the counter term as a fundamental field.
In the general case, the action (4.6) contains two types of interactions, i.e., (4.18) We can find divergences and counter terms by taking both of these interactions into account in the large-N limit. Instead of doing so, here we introduce two vector auxiliary fields, ρ µ and A µ , to reduce the number of relevant interactions further. 13 With these fields, we rewrite the action (4.6) as The fields ρ µ and A µ are indeed the auxiliary fields since they do not have any kinetic terms, and one can recover the original action by solving the constraint equations for them. The interaction of φ i is now contained entirely in the term and hence the computation in the following is greatly simplified. We emphasize here that it is merely for convenience, and the final result should not change even if we do not introduce the vector auxiliary fields.
We now study quantum corrections to the action (4.19). As in the previous section, the Higgs fourpoint interaction just gives the running of λ and b, and hence we drop it in the following. The new divergence only appears in the two-point function of the operator ∂ µ ρ µ /Φ in the large-N limit, which at the one-loop level is diagrammatically given by (4.21) where the wavy line indicates the operator ∂ µ ρ µ /Φ and the dotted line denotes the scalar fields φ i . We have to introduce the following operator as a counter term: Note that the leading order terms at the higher loop level, which are diagrammatically given by, are also renormalized by the same term (4.22) as in the previous subsection. After the renormalization, the coupling α runs according to the beta function as in the large-N limit.
By including the term generated by the quantum correction (4.22), the action is now given by Now ρ µ has obtained a kinetic term due to quantum corrections, which implies the appearance of a new degree of freedom. In order to extract it in a simpler form, we introduce a scalar auxiliary fieldσ as One can see that the original action is recovered by integrating outσ. Performing integration by parts and shifting ρ µ as ρ µ → ρ µ + (1 + c/a)A µ , we obtain At this stage, the derivatives are not acting on ρ µ and A µ any more, and hence we can integrate them out without introducing non-local terms. By further redefining the fields as we finally obtain Thus, the additional degree of freedom is indeed the σ-meson that UV-completes the original NLSM as a LSM even for the general case with arbitrary a, b and c.
The corresponding action in the conformal frame is obtained by identifying Φ = 6M P e ϕ and recalling g µν = e 2ϕ η µν : (4.30) We can verify that the running of the mass term within the UV theory (4.29) and the RG running of α (4.24) computed within the IR theory agree with each other in the large-N limit.
Finally we comment on Eq. Once we go to the higher energy region, M P /ξ < µ < M P , it is linearized as Eq. (3.3) with the scalaron playing the role of the σ-meson. In the even higher energy region µ > M P , other operators such as R µν R µν come into play. One is probably required to fully take quantum gravity into account in this energy region, which is beyond the scope of this paper.
mass of the σ-meson remains the same in both cases, the overall potential shapes become different.
As already mentioned, while an IR property such as the mass term can be obtained by completely forgetting about gravity, we need to remember the coupling to gravity for the potential at large field values. This clarifies the role of Φ, which contains the information of how the Higgs field couples to (the scalar part of) gravity.  [10,54] depending on the boundary condition of α at the UV, this indicates that the tree-level unitarity violation mentioned above can be cured by quantum corrections [12,55,56].

Higgs inflation introduces a nonminimal coupling ξ between the Higgs
In this paper, we have attempted to provide a better understanding of the emergence of the scalaron and the unitarity restoration. In particular, we have shown that the emergence of the scalaron in Higgs inflation can be understood in the language of the nonlinear sigma model (NLSM NLSM [31,32], the CP N −1 model [36][37][38][39][40][41], the Nambu-Jona-Lasinio model [42,43], and the Gross-Neveu model [44]. The phase diagram of Higgs inflation is summarized in Fig. 2. Higgs inflation is a NLSM (2.12) below the energy scale M P /ξ, and it becomes a linear sigma model (LSM) (3.3) with the scalaron as the σ-meson above the scale M P /ξ. As depicted in Fig. 2, it is justified to ignore spin-2 operators such as R µν R µν since they are suppressed by M P , not M P /ξ.

Discussion
We conclude this paper with some remarks that are not addressed in detail in the main text.
Frame/gauge independence. Our analysis was done in the large-N expansion. Since the full result is frame and gauge independent, and we can arbitrarily vary N , each order in this expansion is by itself frame and gauge independent. Note that, in our language, a frame transformation corresponds to a coordinate transformation of the target space.
Cut-off scale in the large-N limit. In the main text, we have argued that the cut-off scale of Higgs inflation with the R 2 term is the Planck scale. Strictly speaking, the cut-off scale is ∼ M P / N once we take the large-N limit as in Sec. 4, which can be seen, e.g., from d -wave parts of scattering amplitudes or the RG running of the R µν R µν term (see also Refs. [65,66]). However, the typical scale of the R 2 term also scales in the same way [see Eq. (1. 2)], and hence the fact that the spin-2 sector can be ignored in the large-ξ limit is not affected. For this reason, we have ignored this subtlety in the main text.
Heavy scalaron during inflation, fine tuning and perturbativity. Throughout this work, we have claimed that the natural mass scale of the scalaron is M P / 12α ∼ M P /ξ. Since α runs according to Eq. (1.2), the value of the mass depends on the boundary condition, or equivalently the choice of the scale Λ at which α vanishes. 14 In this sense, we can think of Λ instead of α as a model parameter, in the same way that we can think of Λ QCD instead of the gauge coupling g 3 as a model parameter in QCD (or the "dimensional transmutation" [67]). Therefore, one might choose Λ such that the scalaron remains heavy during inflation and the inflationary dynamics is described by Higgs inflation without the R 2 term. Although possible, there are three subtleties one has to keep in mind in this scenario. First, Λ has to be tuned to be close to the inflationary scale for the scalaron to be heavy during inflation. Hence this scenario is not very attractive in the sense that it requires tuning. Second, due to the running of α, it is not possible to keep the scalaron heavy for all energy scales for ξ 1. Even if the scalaron is heavy during inflation, it becomes light after inflation and affects, e.g., reheating as we will see below. Finally, 14 One should not confuse Λ with the renormalization scale. It is rather a model parameter as we explain just below.
there is an issue related to perturbativity. As long as we rely on the large-N limit, our analysis is valid for any value of α. If one computes quantities in the standard coupling expansion in the Higgs-scalaron system, however, perturbativity requires ξ 2 /4α 4π, and hence the small value of α implies that the system is in a strong coupling regime. This may not be an issue if the energy scale of our interest is below the mass scale of the scalaron, but poses a problem above the mass scale of the scalaron.
Unitarity during preheating. A very important consequence of the emergence of the σ-meson, or the scalaron, is that the unitarity cut-off scale of Higgs inflation can be lifted to the Planck scale (where the spin-2 part of the metric becomes relevant), depending on the UV boundary condition of α. This feature is essential to follow the dynamics of Higgs inflation from inflation to reheating without ambiguity.
Although the energy scale of Higgs inflation at the classical level (i.e. without the R 2 term) lies below the cut-off scale and does not necessarily lead to a problem during inflation thanks to the finite Higgs field value [61], the story drastically changes after inflation, during (p)reheating. After inflation, the Higgs field oscillates around the bottom of its potential. When the Higgs field crosses zero, the strong curvature in the target space leads to violent production of longitudinal gauge bosons (or equivalently NG bosons), with momenta that seemingly violate the unitarity scale [62][63][64]. Moreover, a naive estimate of the reheating temperature yields a value in the strong coupling regime. On the contrary, reheating with the R 2 term was studied in [68,69], where it was shown that the presence of the scalaron generally weakens particle production and unitarity is no longer violated by the produced particles.

Sub-leading terms in the large-N expansion.
In the main text, we have relied on the large-ξ and the large-N limits. Although the large-ξ limit is expected to be good for ξ = O (10 4 ), one may wonder how sub-leading terms in the large-N expansion affect our understanding of Higgs inflation since N = 4 for Higgs inflation. In the following, we suggest that the LSM (3.3) provides a clue to answering this question.
As we have shown in the main text, the LSM (3.3) describes the system to the leading order in the large-N limit if we ignore the spin-2 sector of gravity, which is valid in the large-ξ limit. Thus, subleading order terms in the large-N limit can be obtained by computing quantum corrections of the Hence we expect that, other than generating the Higgs mass term and the cosmological constant, sub-leading order terms do not affect our understanding of Higgs inflation. In particular, we do not expect that operators such as R n , with n > 2, to be important below M P , since these higher-dimensional operators are not required to make the LSM (3.3) renormalizable, i.e., these are irrelevant operators. In other words, we expect that the large-ξ limit is sufficient for our understanding of Higgs inflation, although we have relied on the large-N limit to make our analysis simpler in this paper. Note that the R n -operators with n > 2 suppressed by M P are not expected to affect the inflationary prediction of the Higgs-R 2 system for ξ ∼ O (10 4 ) and α ∼ ξ 2 [70,71].
It is of course desirable to examine the above expectation by directly computing sub-leading order terms in the large-N limit, which we leave for future work. quadratic gravity in Refs. [20][21][22]. 15 Note that the LSM (3.3) greatly simplifies the computation since it does not contain any tensor modes. In particular, we can see that the Higgs mass term and the cosmological constant are radiatively generated even if they are absent at a specific energy scale. This is because the scalaron introduces an additional mass scale, and it can be understood as a specific form of the hierarchy problem. The existence of the flat direction in the Higgs-scalaron system is not affected by these operators, but the inflationary predictions such as the spectral index and the tensorto-scalar ratio may be affected in particular by the Higgs mass term. We will come back to these points in a separate publication.

Conformal compensator formalism.
In the main text, we have focused on the scalar part of the gravity by taking the limit ξ 1. Even if we include the spin-2 part of the gravity, however, Higgs inflation may still be regarded as a NLSM. Indeed, in App. C, we have demonstrated that Higgs inflation can be regarded as a NLSM conformally coupled to gravity and the scalaron as its σ-meson, with the help of the conformal compensator formalism. Note that a similar formulation is discussed in Ref. [21] in the context of agravity. We leave a further study in this direction, including the computation of quantum corrections, for future work. hence the existence of the flat direction is stable against the RG flow in the large-N limit. 15 Indeed, a scalar field model is discussed in [21] that correctly reproduces the running of ξ and α (or f 0 in their language), which is essentially equivalent to our LSM (3.3).
Application to Higgs EFT. Although we have focussed on Higgs inflation in this paper, our methods can be used for a wide range of models. Here we want to stress that the large-N analysis can be a useful tool to extract a new degree of freedom in a non-renormalizable theory especially when the UV-completion is not already known. Therefore, it may be interesting to apply the large-N analysis to, e.g., the Higgs effective field theory (EFT) [77][78][79][80][81][82][83], and try to extract a possible property of the UVcompletion. Note that the Higgs EFT can be formulated in terms of the target space curvature [84][85][86][87], or equivalently regarded as a NLSM, and hence it is expected to be straightforward to apply the large-N analysis to this theory.

Acknowledgement
The authors would like to thank Valerie Domcke and Ben Mares for helpful discussions and comments.

A.1 Conventions
Here we summarize our conventions. In this paper we work with the almost-minus convention for the spacetime metric. In particular, the flat spacetime metric is given by We define the Christoffel symbol as the Ricci tensor as and the Ricci scalar as This fixes the sign convention for the Ricci scalar. In particular, the Ricci scalar transforms under the Weyl transformation g µν → Ω −2 g µν as The conformal coupling corresponds to ξ = −1/6 with this convention. with some function Ω = Ω(φ) which we determine below. The action (2.2) is then given by We fix Ω as with the scalar functionh =h(φ) required to satisfy which is solved ash With the help ofh, the action is simplified as 16 and hence we have obtained the desired result.

B Gauge fixing and residual gauge symmetry
In this appendix, we discuss gauge fixing conditions and residual gauge symmetries. In the main text, we have focused on the trace part of the metric Φ and ignored the other scalar mode of the metric.
Here we show a gauge fixing condition that corresponds to this treatment; see Eqs. (B.12) and (B.17).
We also confirm that the ghost-like field Φ is indeed harmless due to the residual gauge symmetry; see

B.1 U(1) gauge theory
As a warm-up, we consider the U(1) gauge theory in this subsection. The discussion is quite parallel to the gravity case, and hence it is useful to understand this simpler case first. We consider the U(1) gauge field A µ that transforms under a U(1) gauge transformation as We may impose the Lorenz gauge condition to kill one degree of freedom. It still has a residual gauge symmetry. Indeed, one can perform the transformation (B.1) without affecting Eq. (B.2) provided θ satisfies that makes another degree of freedom unphysical. As a result, there are two physical modes in A µ that correspond to the two polarizations of the photon. Note that the Lagrangian for A µ is given after imposing Eq. (B.2) by It is important to notice that there is an ambiguity in the decomposition (B.5); we can shift A ⊥ µ and A as without spoiling the transverse property of A ⊥ µ . Due to this ambiguity, it is enough to require

B.2 Gravity
Now we consider the gravity case. We may expand the metric around flat spacetime as and treat h µν as a perturbation just for simplicity. Under the general coordinate transformation, it is transformed as at first order in perturbations. We may impose a gauge fixing condition where f ⊥ µν and f ⊥ µ satisfy the same properties as h ⊥ µν and h ⊥ µ , respectively. By acting with ∂ µ ∂ ν , we see that it is enough to require to kill the degree of freedom associated with ψ. The gauge fixing condition (B.12) reduces to this condition after acting with ∂ µ , and hence kills ψ. Thus, it is indeed Eq. (B.12) that we have imposed in the main text since we have focused only on ϕ and omitted ψ there. It is also easy to see that the residual gauge symmetry makes the ghost-like degree of freedom ϕ (or equivalently Φ) unphysical. If we write where f ⊥ µν satisfies the same properties as h ⊥ µν . As a result, it is enough to require to kill the degree of freedom associated with h ⊥ µ . The transformation (B.11) keeps Eq. (B.22) intact as long as ξ µ = ξ ⊥ µ and ξ ⊥ µ satisfies It is the residual gauge symmetry that kills unphysical modes in h ⊥ µν .

C Conformal compensator formalism
In the main text, we have shown that Higgs inflation can be written as a NLSM if we focus on the scalar part of gravity. In this appendix, we demonstrate that Higgs inflation may be regarded as a NLSM conformally coupled to gravity even if we also include the spin-2 part of the gravity. For this purpose, we introduce a conformal compensator field Φ that mimics the scalar part of gravity in the main text.

C.1 Weyl covariant Ricci scalar
First, we construct a Weyl covariant Ricci scalar R from the standard Ricci scalar R. We consider the Weyl transformation g µν → Ω −2 g µν , and assume that a vector field ω µ transforms as Since the Christoffel symbol transforms as the following combination transforms as Therefore we define the Weyl covariant Ricci scalar as which transforms indeed covariantly as Once we introduce the conformal compensator fieldΦ, the field ω µ is identified as which satisfies the desired property under the Weyl transformation ifΦ transforms asΦ → ΩΦ.

C.2 Higgs inflation as NLSM conformally coupled to gravity
With the help of the compensatorΦ and the Weyl covariant Ricci scalar R, we can promote the Higgs inflation model to a Weyl invariant form as where R is now understood as The action is invariant under the local Weyl transformation, g µν → Ω −2 g µν ,Φ → ΩΦ, and φ i → Ωφ i , (C.9) and reduces to the original action for Higgs inflation in the Jordan frame after fixingΦ = 6M P . Note that the kinetic term forΦ is ghost-like to be consistent with the Einstein-Hilbert action. Nevertheless it does not spoil the theory since it can be killed by the local Weyl transformation and hence is unphysical.
We may define a new filed Φ asΦ and also a scalar function h = h(Φ, φ) as and hence the following relation holds: Therefore, the action (C.7) is given in terms of Φ and h as It resembles the action of scalar fields conformally coupled to gravity, but now a fundamental scalar field replaced by the composite scalar function h = h(Φ, φ). In this sense, we may view Higgs inflation as a NLSM conformally coupled to gravity, with the help of the conformal compensator.

C.3 Scalaron as σ-meson
Now we see that the scalaron is identified as a σ-meson of Higgs inflation in the conformal compensator formalism. The Higgs-scalaron system is promoted to a Weyl invariant form as (C. 15) We now extract the additional degree of freedom, or the scalaron, as a fundamental field by the auxiliary field method. We introduce an auxiliary field σ as After some computation and shifting σ to σ → σ + (6ξ + 1)φ 2 i /2Φ, we obtain Φσ + 6ξ + 1 2 φ 2 i 2 . (C.17) By redefining Φ as Φ → Φ + σ, we obtain the final result: It is clear that one recovers the NLSM (C.14) in the limit α → 0, and the scalaron is indeed the σ-meson that linearizes the NLSM. Note that the kinetic term of σ originates from the kinetic mixing with Φ, and has a correct sign thanks to the ghost-like property of Φ.

D O(1, 1) transformation and flat potential
In this appendix, starting from the general model, we discuss the condition to have a flat potential suitable for inflation. Our starting point is Note that we can take a, c ≥ 0 without loss of generality. Higgs inflation with the σ-meson or the scalaron corresponds to a particular set of parameters a = c = 1/2 and b = (6ξ+1)/2. The main purpose of this appendix is to clarify why this choice of parameters yields a flat potential suitable for inflation and how special this choice is.
We have a flat direction in the potential in the Einstein frame if we find a trajectory θ on which the potential V (Φ θ , σ θ , φ 2 i ,θ ) approaches asymptotically to a constant or does not change at all. Since the potential should be finite along this trajectory, φ 2 i ,θ is bounded from above because of the λ(φ 2 i ) 2 term. A trivial example fulfilling these requirements is the NG boson directions of the Higgs. There φ 2 i ,θ , σ θ , and Φ θ are fixed to be constants. What we are interested in here is a less trivial trajectory. Namely, Φ θ and σ θ can be taken to infinity because of a non-trivial cancellation among them in the second term in Eq. (D.2), while φ 2 i ,θ is bounded from above. In order to have this trajectory, one should find a trajectory of V → const. for Φ θ , σ θ → ∞ even under with Λ being a constant. In the following, we discuss the impact of this condition on a, b, and c. In order for the potential to approach asymptotically to a constant value for θ → ±∞, the following condition should be fulfilled: As mentioned earlier, we can take a, c ≥ 0 without loss of generality, and hence we focus on the first three branches in the following. In the second and third branches, our vacuum in the current Universe φ 2 i = 0 (which is also a potential minimum) is located at infinity for a, c ≥ 0, i.e., a run-away potential. Similarly, one readily finds a run-away potential for the first branch at a = 0. For the first branch with a = 1 and c = 0, on the other hand, one ends up with an exactly massless mode which is completely decoupled from the Higgs field φ i . These cases might not be interesting in the context of Higgs inflation.
Therefore we arrive at the case with a + c = 1, a > 0, c > 0. (D.8) As we show below, this case is equivalent to Higgs inflation, a = c = 1/2 and b = (6ξ + 1)/2, after appropriately redefining the parameters.

D.2 Redundancy in parameters and O(1, 1) transformation
In this section, we point out redundancies in the parameters a, b, and c. To this end, the following O(1, 1) transformation plays a central role: One can see that, while this transformation does not alter the kinetic term of Eq. (D.1), the potential does change implying redundancies in the parameters.
The rest of this section is devoted to show that any set of parameters satisfying Eq. (D.8) is equivalent to a = c = 1/2 and b = (6ξ + 1)/2 because of this redundancy related to the O(1, 1) transformation. The second term in the potential (D.2) transforms as follows: One can see that the potential takes exactly the same form as a = c = 1/2 and b = (6ξ + 1)/2 after the following redefinition: b → a(6ξ + 1), α → 4a 2 α. (D.12) Now it is clear that the potential has a minimum at σ = φ i = 0 while it asymptotically approaches a constant value for σ , Φ → ∞ and φ 2 i → const. under Φ 2 −σ 2 −φ 2 i = 6M 2 P , which is suitable for inflation.