Quasiconformal models and the early universe
Abstract
Extensions of the Standard Model and general relativity featuring a UV fixed point can leave observable implications at accessible energies. Although mass parameters such as the Planck scale can appear through dimensional transmutation, all fundamental dimension4 operators can (at least approximately) respect Weyl invariance at finite energy. An example is the Weylsquared term, whose consistency and observational consequences are studied. This quasiconformal scenario emerges from the UV complete quadratic gravity and is a possible framework for inflation. We find two realizations. In the first one the inflaton is a fundamental scalar with a quasiconformal nonminimal coupling to the Ricci scalar. In this case the field excursion must not exceed the Planck mass by far. An example discussed in detail is hilltop inflation. In the second realization the inflaton is a pseudoGoldstone boson (natural inflation). In this case we show how to obtain an elegant UV completion within an asymptotically free QCDlike theory, in which the inflaton is a composite scalar due to new strong dynamics. We also show how efficient reheating can occur. Unlike the natural inflation based on Einstein gravity, the tensortoscalar ratio is well below the current bound set by Planck. In both realizations mentioned above, the basic inflationary formulæ are computed analytically and, therefore, these possibilities can be used as simple benchmark models.
1 Introduction
In 2015 (and with a recent update in 2018) the Planck collaboration [1, 2] was able to exclude (or set stringent bounds on) several inflationary models. Indeed, better determinations of many observables related to the cosmic microwave background (CMB) (such as the tensortoscalar ratio r, the scalar spectral index \(n_s\) and the curvature power spectrum \(P_R\)) were provided. Further improvements are expected in the next future: CMB Stage 4 (S4) will be active soon.^{1} Therefore, early universe cosmology continues to be an exciting research area.
Despite this big progress, several models of the early universe are still allowed and we have only a partial understanding of how the universe initially expanded and evolved. Such large ambiguity can be reduced by looking for theoretical reasons to discriminate among the various possibilities. One way to do so is to focus on UVcomplete theories that are applicable to the early universe.
In Ref. [3] it was shown how a relativistic field theory of all interactions (gravity included) can reach infinite energy. In order to achieve this goal, terms quadratic in the curvature are added to the Einstein–Hilbert action, so this theory is also known as quadratic gravity (QG) (see [4] for a review). The UV completion is obtained by demanding that the Weyl symmetry breaking terms vanish in the infinite energy limit and all couplings enjoy a UV fixed point. Therefore, this is a way of implementing asymptotic safety. An important difference between this possibility and the original proposal made by Weinberg in [5, 6], though, is the presence of only a finite number of terms in the fundamental action, which guarantees the theory to be predictive and calculable. Another interesting property of QG is the possibility to solve the hierarchy problem [3, 7], in the sense that the Higgs mass can be made technically natural [8]. When one lowers the energy from the UVconformal fixed point mentioned above, the dimensionless Weylsymmetry breaking operators are generated, but since they are sourced only by multiloop diagrams [9, 10, 11], they can remain small down to the inflationary energies. This corresponds to a scenario for the early universe, which we call “quasiconformal”.^{2}
One of the quadratic terms in the action is the squared of the Weyl tensor, which is necessary to keep the theory UV complete. Featuring higher time derivatives, this Weylsquared term brought a number of issues, which, however, have been addressed in the recent years. One of the purposes of this paper is to review and extend these arguments in favour of the viability of such term.
Another aspect that has not been investigated so far is the observational predictions of the quasiconformal scenario. The natural arena to study these effects is the early universe, which typically involves extremely high energies and field scales, sometimes reaching the Planck mass \({\bar{M}}_P\). This is the main topic of the present paper.^{3}
The “quasi” of “quasiconformal” is important. It reminds us that the Einstein–Hilbert term and other Weylbreaking parameters can be present in the Lagrangian, generated, for example, through dimensional transmutation and various quantum effects [3, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Note that an exactly Weyl invariant model would not be physically different from a Weylbreaking one; indeed, any exactly conformal model can be written as a nonconformal one by fixing a gauge for Weyl symmetry and, conversely, any model of gravity coupled to an arbitrary matter sector can always be made exactly Weyl invariant.^{4} What is proposed here is quasiconformality, in which Weyl symmetry is approximate at high energy, but not exact apart from the strict UV limit. This is also consistent with the presence of the Weyl anomaly, which would anyhow render exact conformality impossible (when the theory is not at the fixed point, for instance at accessible energies).
A first question one can ask is whether the Standard Model (SM) Higgs can play the role of the inflaton in this scenario. The answer is negative because the requirement of a quasiconformal nonminimal coupling (to the Ricci scalar) leads essentially to the same problem one has for a vanishing nonminimal coupling [26]: the SM Higgs cannot be responsible for inflation and the generation of the observed perturbations at the same time. We have verified the validity of this statement for a quasiconformal nonminimal coupling by using the precise 2loop effective potential as in [27, 28, 29, 30, 31]. A way to circumvent this problem is to demand a large [32, 33] (or order 10 [34, 35, 36, 37]) nonminimal coupling, which, however, would completely break Weyl symmetry in the UV.
Another apparently natural candidate for the inflaton is the effective spinless field corresponding to the square of the Ricci scalar, as proposed by Starobinsky [38]. However, in the quasiconformal scenario the coefficient of this term has to be very small because is not Weylinvariant, while the observed value of the curvature power spectrum requires a very large coefficient.
In this paper we discuss two implementations of the quasiconformal scenario. The first one is the class of models where the inflaton features a moderate field excursion: with “moderate” we mean that it does not exceed the Planck scale by far. This is because a quasiconformal nonminimal coupling prohibits, as we will discuss in Sect. 3, field excursions bigger than \(\sqrt{6}{\bar{M}}_P\). A specific realization that we study in detail here is hilltop inflation [39].
Another interesting implementation that we investigate in this paper is inflation triggered by a pseudoGoldstone boson, what is known as “natural inflation” in the context of the nonconformal Einstein gravity [40, 41]. Indeed, in this case the inflaton does not need to a have a (quasi)Weyl invariant nonminimal coupling being morally the phase of some field representation of a spontaneously broken global group. An attractive feature of natural inflation is the natural flatness of the potential, which is protected by Goldstone theorem from large quantum effects. This is preserved in (quasi)conformal realizations and we will study possible ways of distinguishing this setup from the one based on Einstein gravity.
The paper is organized as follows. In the next section we describe in full generality the quasiconformal scenario. As anticipated before, the conformal Weylsquared term is introduced and we review and extend arguments which indicate the viability of this term. In Sect. 3 we study the hilltop quasiconformal model, while in Sect. 4 we focus on the naturalinflation option, investigating how the universe inflated and subsequently evolved in this case. Finally, Sect. 5 offers our conclusions and some outlook of this work.
2 The quasiconformal scenario and the Weylsquared term
The quasiconformal scenario studied in this work is a subset of the theories with scale invariance in the UV. Therefore, we start here by reviewing this class of theories. In the most general case, the field content features gauge fields with field strength \(F_{\mu \nu }^A\), Weyl fermions \(\psi _j\) and real scalars \(\phi _a\). These fields should be considered as fundamental. Other composite states can appear below some confinement scales as usual.
 I

The clash between asymptotic freedom (or, more generally, asymptotic safety) and stability (understood as the absence of tachyons) [3, 4, 44, 45]: whenever the parameters are chosen to ensure stability, perturbation theory features a Landau pole in the RG flow of \(f_0\).
 II
 The presence of a field, with spin2 and massbut with an unusual sign of the kinetic term [42] (i.e. a ghostlike field). This is a generic issue of theories with more than two timederivatives in the Lagrangian, which have a classical Hamiltonian that is unbounded from below, as shown by Ostrogradsky [46]. The theory above has indeed more than two derivatives because of the Weylsquared term, \(W^2\).$$\begin{aligned} M_2=f_2 {\bar{M}}_P/\sqrt{2}, \end{aligned}$$(2.6)
In Ref. [3] it was pointed out that Problem I is an artefact of perturbation theory in \(f_0\): when this parameter grows, perturbativity in \(f_0\) is lost, but one can develop an expansion in \(1/f_0\) showing that no Landau poles are present, provided that all fundamental scalars have asymptotically Weylinvariant couplings (\(1/f_0\rightarrow 0\), \(\zeta _{ab}\rightarrow 0\)) and the remaining couplings approach fixed points in the UV limit. In other words, the theory flows to a conformal behavior in the infinite energy limit and, in this way, can be UV complete. Although flowing to conformal gravity at infinite energy is consistent, at finite energy conformal invariance is broken by the scale anomaly and the \(R^2\) term and nonvanishing values of \(\zeta _{ab}\) are generated. However, as discussed in [3], this is a multiloop effect (see [9, 10, 11] and references therein). Therefore, \(1/f_0\) and \(\zeta _{ab}\) can remain tiny down to the inflationary scales and the subsequent stages of the evolution of the universe. This leads to the quasiconformal scenario, which is investigated in this work.^{5}
 First, one observes that in the freefield limit the Hamiltonian of the massive spin2 field iswhere \(Q_\alpha \) and \(P_\alpha \) are the associated canonical variables and conjugate momenta and the helicity sum is over \(\alpha = \pm 2, \pm 1, 0\) because this massive particle has spin 2. The manifestation of the Ostrogradsky theorem in this case is the overall minus sign. However, despite that sign there are no instabilities in the freefield limit (that sign cancels in the EOM).$$\begin{aligned} H_\mathrm{2} \,\, ~ = \,\, ~  \sum _{\alpha =\pm 2, \pm 1, 0} \int d^3 q \left[ P_\alpha ^2 + (q^2+M_2^2) Q_\alpha ^2 \right] \end{aligned}$$

The effective field theory (EFT) approach tells us that at energies below \(M_2\) we should not find runaways even if the massive spin2 field has an orderone coupling, \(f_2 \sim 1\).

The intermediate case \(0< f_2 < 1\) must have intermediate energy thresholds (above which the runaways might be activated).

The weak coupling case \(f_2 \ll 1\) (compatible with Higgs naturalness) must have an energy thresholds much larger than \(M_2\): we could see the effect of the massive spin2 field without runaways.
Although the argument above gives a satisfactory solution of the classical Ostrogradsky problem it is still needed to address the issues raised by the quantization. One reason is the fact that quantum effects might lead to tunneling above the energy threshold even if the classical fields satisfy the bounds in (2.7) and (2.8). However, we know that renormalizability implies that the quantum Hamiltonian governing \({\hat{h}}_{\mu \nu }\) is bounded from below [4, 42] and, therefore, this dangerous tunneling should not occur. On the other hand, renormalizability also implies that the space of states is endowed with an indefinite metric (with respect to which the quantum mechanical “position” q and momentum p operators are selfadjoint) [4, 42, 52, 53] . The presence of an indefinite metric, therefore, leads to the question: how can we define probabilities consistently?
A remaining issue that has been pointed out is that higherderivative theories may generate a violation of micro causality in its decay processes [56]. However, whenever \(M_2< H\), where H is the Hubble rate during inflation,^{10} the width of the massive spin2 field, \(\Gamma _2\), (the only one that could lead to micro acausality) is always much smaller than H as \(\Gamma _2 \ll M_2\) in this case [59, 60, 61]. Therefore, these potential acausal processes are actually quickly diluted by the expansion of the universe. In the opposite case, \(M_2> H\), the Weylsquared term does not contribute to the experimentally observable quantities as it essentially decouples [47]. Therefore, the Weylsquared term effectively respects the causality principle
Having addressed the potential issues raised by the \(W^2\), we now turn to the analysis of the quasiconformal scenario.
3 Hilltopinflation realization
Figure 1 gives V as a function of \(\phi \) and \(V_E\) as a function of \(\chi \) for two values of v. V and \(V_E\) are very similar for small enough field values. However, in the opposite limit they differ.
3.1 Computing inflationary observables
In Fig. 2 it is shown that there are initial field values \(\chi \) for which we can have \(N_e \sim 60\) efolds and obtain \(n_s\) and \(r_E\) (and, therefore, r) in good agreement with the bounds in [1, 2], preserving the slowroll conditions, \(\epsilon \ll 1\), \(\eta \ll 1\). Since \(r_E< 0.1\) [50] the isocurvature power spectrum parameterized by \(r'\) also satisfies the bounds in [1, 2]. By choosing \(\lambda _\phi \approx 7 \times 10^{12}\) one obtains the measured \(P_R\) in Eq. (3.35). A numerical analysis shows that, generically, one has a good agreement with the inflationary observables if v is close enough to \(\sqrt{6} {\bar{M}}_P\).
Finally, we have explicitly checked in this hilltop implementation of the quasiconformal scenario that the bounds in (2.7) and (2.8) are satisfied during the whole cosmology and, therefore, the possible runaways due to the Ostrogradsky theorem are avoided.
3.2 Higgs naturalness
In the previous section we have seen that successful inflation can occur if the vacuum expectation value (VEV) v in the inflaton sector is around (but not exceeding much) the Planck scale.
3.3 Complexity of UVcompletions
The hilltop inflationary model provides us with an existing proof of quasiconformal models of inflation. Given that such scenario is motivated by possible UVcompletions of Einstein gravity one eventually want to embed hilltop inflation in a field theory without a highmomentum cutoff. In Refs. [66, 67, 68] it was shown that renormalizable field theories can be made asymptotically free with respect to all couplings (including the quartic couplings \(\lambda _{abcd}\)) if embedded in a theory with a (semi)simple gauge group and with a complex field content. The field content can be made simpler in asymptotically safe extensions of the SM, which, however, present some challenges [69, 70]. The scenario described in Sect. 2 is general enough to embed this SM extensions.
Such an embedding for the specific case of hilltop inflation, although interesting, goes beyond the scope of the present work because, as we will see in Sect. (4), there are simple and elegant asymptotically free theories in which the inflaton is identified with a composite scalar. In these theories all fundamental scalars feature a quasiconformal nonminimal coupling, but, as will be discussed in Sect. 4, the inflaton, being a pseudoGoldstone boson, has a negligible (and thus nonconformal) nonminimal coupling.
4 Naturalinflation realizations
Another way of implementing the quasiconformal scenario for the early universe is identifying the inflaton \(\phi \) with the pseudoGoldstone boson associated with the breaking of a global symmetry. In the context of the nonconformal Einsteingravity case, this scenario has been proposed in [40, 41] and is known as “natural inflation” because the potential of a pseudoGoldstone boson is naturally flat thanks to Goldstone theorem. This good feature is preserved when one constructs a naturalinflation model in the quasiconformal scenario, but, as we will see, some predictions differ with respect to the Einsteingravity case, such that one can distinguish between these two proposals.
First note that (pseudo)Goldstone bosons do not need to have a Weylinvariant nonminimal coupling to gravity in the (even exact) conformal scenario; their \(\xi \)couplings can be different from \(1/6\). Indeed, if \(\Phi \) is a scalar field in some representation of the spontaneously broken group \(G_S\), its nonminimal coupling is proportional to \(\xi _{\Phi }\Phi ^2R\), where \(\Phi ^2\) is invariant under \(G_S\), and the Goldstone boson (being morally some phase of \(\Phi \)) does not feature nonminimal couplings.
In Fig. 3 we show the predictions of the model for the observables \(n_s\), r and \(P_R\) together with the behavior of \(\epsilon \) and \(\eta \) to show the validity of the slowroll approximation. In the bottom plot on the left one can clearly see that the Weylsquared term allows perfect compatibility between natural inflation and the latest CMB data, including^{13} the value of r. We also see that for \(N=1\) the scale f should be around \(f\approx 6.6 {\bar{M}}_P\) while \(\Lambda \approx 6 \times 10^{3} {\bar{M}}_P\). Since \(r_E< 0.1\) the isocurvature power spectrum parameterized by \(r'\) [see Eq. (3.28)] also satisfies the most recent bounds (see Fig. 7 in [50]).
Like for the hilltop case of Sect. 3, we have also explicitly checked for the naturalinflation implementation that the bounds in (2.7) and (2.8) are satisfied during the whole cosmology and, therefore, the possible runaways due to the Ostrogradsky theorem are avoided.
Note that the nonminimal coupling between the Ricci scalar and \(\phi \) might be generated by quantum corrections. However, it should be small, of order of the scale \(N\Lambda ^2/f\) at which the shift symmetry of the \(\phi \) field is broken divided by the Planck mass: one has to divide by \({\bar{M}}_P\) because the nonminimal coupling is dimensionless and because the nonminimal coupling should disappears when gravity is decoupled. This gives a negligibly small^{14} nonminimal coupling of the inflaton, that is \(\xi _\phi \sim \Lambda ^2/(f{\bar{M}}_P) \sim 10^{5}\) for an order one N, \(\Lambda \approx 6 \times 10^{3} {\bar{M}}_P\) and \(f\approx 6.6 {\bar{M}}_P\).
4.1 Asymptotically free UVcompletion
4.2 Higgs naturalness
As long as the couplings of these visible sectors (charged under \(G_h\)) with the inflaton sector are small the Higgs mass is protected from large radiative corrections by an approximate shift symmetry acting on the Higgs field. As discussed in Sect. 2, this approximate symmetry is respected by gravitational interactions if (2.5) is enforced.
4.3 Reheating
Given that there is an asymptotically free embedding of natural inflation we study further the earlyuniverse implications of this scenario. Here we describe how reheating can occur.
Note that what allows us to heat the universe is the presence of many weakly coupled scalars, which have sizable couplings to the observed particles and undergo large quantum fluctuations.
Finally, we observe that a coupling between the inflaton sector and the SM is always generated in this class of theories, at least through gravitational dynamics. For example, below the condensation scale gravity generates a portal coupling in the Lagrangian between the composite scalar \(\phi \) representing the inflaton and the Higgs h of order \(f_2^4 h^2\phi ^2\). We find that the small value of \(f_2\) required by Higgs naturalness in (2.5) is compatible with the lower bound on the reheating temperature derived above.
4.4 Potential (meta)stability
A related issue in highenergy extensions of the SM is to understand whether the electroweak vacuum is stable, unstable or metastable.^{17} We address here this issue in a model independent way.
We also observe that total asymptotic freedom (theories where all couplings flow to zero at large energies) favours metastability: the smallness of all couplings at high energy suppresses the value of the potential and, therefore, raises the lifetime of the electroweak vacuum.
5 Conclusions and outlook
In this paper we explored the observable predictions of the quasiconformal scenario, in which the dimensionless couplings in the action respect (at least approximately) Weyl symmetry. This behavior is suggested by the UV complete QG, which can hold up to infinite energy if all couplings reach a conformal fixed point at infinite energies [3]. QG can also render the Higgs mass (technically) natural by allowing a good enough shift symmetry of the Higgs field even in the presence of gravity.
We have included in the discussion the Weylsquared term and reviewed and extended the arguments in favour of its viability in Sect. 2. Basically the discussion can be addressed at two levels. At the classical level, it was shown [50] that the possible Ostrogradsky runaways are avoided if the typical energies are below a threshold [given in (2.7) and (2.8)], which is high enough to describe the whole cosmology. At quantum level unitarity is restored by computing probabilities by an appropriate positive norm, which is singled out by an experimental approach to the definition of probabilities. Moreover, the possible acausal effects are extremely diluted by the expansion of the universe.
The predictions of the Weylsquared term and the quasiconformal nonminimal coupling of fundamental scalars have been worked out and compared with the available data of the early universe. We considered two implementations of the quasiconformal option.
In the first one, discussed in Sect. 3, the inflaton is identified with a fundamental scalar with a moderate (\(<\sqrt{6}{\bar{M}}_P\)) field excursion, such that the effective Planck mass never becomes imaginary. An example is hilltop inflation, which agrees well with the most recent data provided by the Planck collaboration. We pointed out, however, the complexity required to UV complete models of this sort.
This brought us to the second implementation, which we investigated in Sect. 4: a pseudoGoldstone boson as the inflaton (natural inflation), which provides a very good rationale for the flatness of the potential. In this case the nonminimal coupling can be nearly zero, rather than conformal, because a pseudoGoldstone boson has very small nonderivative interactions. In Einsteingravity implementations of natural inflation, the prediction of r is rather large and on the verge of being excluded. On the other hand, in the presence of the Weylsquared term, r is drastically reduced for a natural Higgs mass. This can revive natural inflation. Another advantage of this realization is the possibility to be elegantly UVcompleted. We have shown that an asymptotically free QCDlike sector can provide an inflaton of this type: it can be identified with the composite scalar analogous to the \(K^0\) meson in ordinary strong interactions. Furthermore, we have also shown that a satisfactory reheating can occur in this case.
We have checked, both in the hilltop and in the naturalinflation case, that the bounds (2.7) and (2.8) on the energies to avoid the Ostrogradsky instabilities are satisfied.
Furthermore, in both cases the main inflationary formulæ can be analytically computed, potentially providing simple benchmark models of inflation.
Reheating can occur in both cases too. We focused in this paper on the reheating in the UVcomplete natural inflation (see Sect. 4.3) but it can also be realised in hilltop inflation. Indeed, one can also consider the treelevel portal coupling \(\lambda _{h\phi }\) between the Higgs h and the inflaton \(\phi \) and the small value required by naturalness in (3.37) is compatible with a high enough reheating temperature.
We take advantage of the concluding section to point out that this scenario, appropriately implemented to include a UVcomplete sector containing all SM particles, as discussed in Sect. 4.2, may solve all phenomenological problems of the SM. For example, dark matter, baryogenesis and neutrino masses might be due to righthanded neutrinos [79, 80] (which in some models with UV fixed points are necessary [68]) and the electroweak vacuum stability may be achieved thanks to the extra scalars generically present in the theory (see Refs. [78, 81, 82, 83] for the stabilization through extra scalars). Therefore, we hope that the results of this paper will lead to scientific activities along these lines.
Footnotes
 1.
See, for example, the webpage https://cmbs4.org/overview.php.
 2.
Another independent motivation to study this scenario comes from string theory, where many scalars are (quasi)conformally coupled to gravity [12].
 3.
 4.
This can be achieved by simply writing the spacetime metric \(g_{\mu \nu }(x)\) as \(S(x) {\bar{g}}_{\mu \nu }(x)\), where S is a scalar quantity that depends on the spacetime point x and \({\bar{g}}_{\mu \nu }\) is a redefined metric. Indeed, this introduces the gauge redundancy \(S(x)\rightarrow \lambda (x) S(x)\), \({\bar{g}}_{\mu \nu }(x)\rightarrow {\bar{g}}_{\mu \nu }(x)/\lambda (x)\), where \(\lambda (x)\) is a generic function of x (in other words, it makes the model exactly Weyl invariant).
 5.
Another logical possibility is that the theory features a nonperturbative behavior that drives \(f_0\) from a very large value in the UV down to a very small value at the scales of the early universe, which we can access through CMB measurements. This case was studied in [18, 19, 47]. The purpose of the present paper is instead to explore the quasiconformal option.
 6.
We want a realistic theory of all forces, but it is not even known how to treat nonperturbatively e.g. chiral fermions (the lattice is currently unable to study fermions with chiral gauge interactions).
 7.
Furthermore, the fatal runaways above such threshold give an (anthropic) rationale for a homogeneous and isotropic universe [50].
 8.
One can show that the basic operators q and p as well as the Hamiltonian in the theory considered here have complete eigenstates at any order in perturbation theory.
 9.
A different proposal for unitarizing this theory was found in [57, 58]. In this work we do not adopt this alternative approach because it is not (currently) developed enough to provide us with predictions for the CMB observables. On the other hand, for the approach discussed here, the predictions for such observables have been found in [47].
 10.
 11.For a generic number \(N_s\) of scalar fields \(\phi _a\) (\(a=1..., N_s\)) Eq. (3.1) gets replaced by and going to the Einstein frame one finds where and Generically, the Ricci scalar of the field metric \(K_{ab}\) is not zero so \(K_{ab}\) is not flat.
 12.
As far as inflation is concerned, it would be equivalent to consider the potential with a different sign, \(\Lambda ^4 \left( 1\cos \left( \frac{N\phi }{f}\right) \right) \), and to restrict ourselves to the interval \(\phi \in [\pi f/N, 2\pi f/N]\). Therefore, the two signs (considered in [40, 41]) are both covered here.
 13.
 14.
See, however, Ref. [74] for a discussion of nonminimal couplings in natural inflation.
 15.
Recall that in our setup the lightest tildequarks are \({\tilde{d}}\) and \({\tilde{s}}\), while the mass of \({\tilde{u}}\) is taken at a much larger scale.
 16.
Such number of scalars can be obtained, for example, by considering several copies of the trinification model of [68], which features 54 real scalars.
 17.
We recall that the vacuum is stable if it does not decay, is metastable if its lifetime is finite but larger than the age of the universe and is unstable otherwise.
Notes
Acknowledgements
I thank G. Ballesteros and A. Strumia for useful discussions and CERN for the hospitality.
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