An update on fine-tunings in the triple-alpha process

The triple-alpha process, whereby evolved stars create carbon and oxygen, is believed to be fine-tuned to a high degree. Such fine-tuning is suggested by the unusually strong temperature dependence of the triple-alpha reaction rate at stellar temperatures. This sensitivity is due to the resonant character of the triple-alpha process, which proceeds through the so-called “Hoyle state” of C with spin-parity 0. The question of fine-tuning can be studied within the ab initio framework of nuclear lattice effective field theory, which makes it possible to relate ad hoc changes in the energy of the Hoyle state to changes in the fundamental parameters of the nuclear Hamiltonian, which are the light quark mass mq and the electromagnetic fine-structure constant. Here, we update the effective field theory calculation of the sensitivity of the triple-alpha process to small changes in the fundamental parameters. In particular, we consider recent high-precision lattice QCD calculations of the nucleon axial coupling gA, as well as new and more comprehensive results from stellar simulations of the production of carbon and oxygen. While the updated stellar simulations allow for much larger ad hoc shifts in the Hoyle state energy than previously thought, recent lattice QCD results for the nucleon S-wave singlet and triplet scattering lengths now disfavor the scenario of no fine-tuning in the light quark mass mq. PACS. 21.10.Dr – 21.30.-x – 21.45.-v – 21.60.De – 26.20.Fj


Introduction
The production of carbon and heavier elements in stars is complicated by the instability of the 8 Be nucleus. The way this bottleneck is circumvented in nature is by means of the triple-alpha process, where the production rate of 12 C is strongly enhanced by a fortuitously placed 0 + resonance, known as the Hoyle state [1]. As small ad hoc changes in the excitation energy of the Hoyle state relative to the triple-alpha threshold can lead to large changes in the relative abundance of carbon and oxygen, the question arises whether the universe should be regarded as fine-tuned with respect to the likelihood of carbon-oxygen based life to arise, for a recent review on fine-tunings see [2]. The physics of the triple-alpha process has recently been studied using nuclear lattice effective field theory (NLEFT). The ground state energies of 4 He, 8 Be and 12 C, and of the energy of the Hoyle state in 12 C, were all found to be strongly correlated with respect to small changes in the fundamental constants of nature, an effect of the clustering of alpha particles in the respective nuclei. We review here how the sensitivity of the triple-alpha reaction rate with respect to small changes in the light quark mass and the electromagnetic fine-structure constant is treated in the effective field theory (EFT) framework. The main source of uncertainty is due to the shortrange part of the nucleon-nucleon interaction, and we discuss recent progress in narrowing down this uncertainty using updated theoretical knowledge of the quark mass dependence of the two-nucleon S-wave scattering parameters, including the results of recent lattice QCD work. We also contrast this theoretical treatment with recent highprecision calculations of stellar nucleosynthesis, which find that the allowable range of Hoyle state energies is larger than previously thought [3]. This paper is structured as follows. In Sec. 2 we update the pion (quark) mass dependence of the nuclear Hamiltonian, which is central for the following discussion. In Sec. 3 we review the current status of stellar nucleosynthesis calculations, with focus on the resulting abundances of carbon and oxygen under ad hoc shifts in the Hoyle state resonance. In particular, we pay attention to recent new results in this field. In Sec. 4, we revisit the theoretical status of EFT calculations of the sensitivity of the Hoyle state energy to small changes in the light quark mass m q and the electromagnetic fine-structure constant α em . Finally, in Sec. 5 we discuss how the EFT treatment of the triple-alpha process could be improved with regards to recent progress in the nuclear lattice EFT description of the nuclear forces.

Quark mass dependence of the nuclear Hamiltonian
The ground-state energies and spectra of light and mediummass nuclei can be calculated to a good precision in the framework of NLEFT, as described in detail in the monograph [4]. Variations of the fundamendat parameters like the average light quark mass or the electromagnetic finestructure constant can also be investigated within this approach. In what follows, we update our knowledge of the quark mass dependence of the nuclear Hamiltonian, which is central to the study of fine-tunings in the triple-alpha process. Note that to high accuracy the Gell-Mann-Oakes-Renner relation M 2 π ∼ m q , with m q = (m u + m d )/2 the average light quark mass, is fulfilled in QCD and we thus can use the notions "quark mass dependence" and "pion mass dependence" synonymously. This update concerns in particular the hadronic parameters x 1 and x 2 and the nuclear parametersĀ s andĀ t . For details, the reader is referred to Refs. [5,6,4].
We first discuss x 1 , which describes the dependence of the nucleon mass m N on the pion mass M π . This is related to the pion-nucleon σ-term σ πN , via and we note that the best determinations of σ πN are from the recent Roy-Steiner-equation analyses of pion-nucleon scattering, leading to σ πN = (59.1 ± 3.5) MeV [7] (with the inclusion of pionic hydrogen and deuterium data) and σ πN = (58 ± 5) MeV [8] (pion-nucleon scattering data only). We take the central value of Ref. [7] and the uncertainty of Ref. [8], to be on the conservative side. While this gives we note that lattice QCD determinations give systematically smaller values for σ πN and thus x 1 . For reasons explained in Ref. [9], such as the inconsistency of the lattice QCD values with the precisely determined S-wave pionnucleon scattering lengths, we do not consider the lattice QCD results here. Next, we turn to x 2 , which describes the dependence of the strength of the one-pion exchange (OPE) g A /(2F π ) on M π . This is given by with M ph π denoting the physical value of the pion mass. Noe that x 2 turned out to be small and of indeterminate sign in Ref. [6]. This was largely due to the inconclusive situation of lattice QCD calculations of g A . Such problems have recently been overcome by high-precision lattice QCD calculations with close-to-physical quark masses [10]. In particular, consistent values of g A with minimal model dependence were obtained for a range of polynomial and chiral perturbation theory (ChPT) extrapolations in M π . In order to make use of the analysis of Ref. [10], we define where and in terms of which were obtained from an extrapolation using the complete NNLO chiral expression, with and without inclusion of the N3LO contact terms [11]. It should be noted that unlike the determination of g A itself, the value of ∂g A /∂M * does depend significantly on the choice of extrapolation of the lattice QCD data. For instance, significantly larger values can be obtained by means of linear or quadratic extrapolations in M * . However, we shall here rely on the chiral NNLO result (7), in particular as it was found to show good convergence of the chiral expansion [10]. The dependence of F π on M π was not yet obtained in Ref. [10]. We recall that Ref. [12] provided ∂F π ∂M π M ph π = 0.066 (16), (8) which was used in Ref. [6]. This should be compared with the sub-leading order ChPT result where F 86.2 MeV denotes F π in the chiral limit, and 4 = 4.3(3) from the review [13]. We note that Eq. (9) gives a number comparable with (though slightly larger than) Eq. (8). These values are also consistent with the most recent FLAG lattice QCD determination [14].
Having fixed the hadronic input parameters, we now consider the leading order four-nucleon contact interactions that can be mapped on the derivatives of the inverse singlet and triplet neutron-proton scattering lengths, Earlier, modelling based on resonance saturation [15] was used to get a handle on these quantities, as discussed in detail in Ref. [12]. Here, we attempt to fixĀ s,t from available lattice QCD data, which should be the method of choice. Before doing so, some words of caution are in order. The situation with lattice QCD simulations in the nucleon-nucleon (NN) sector is at present highly controversial. Fully dynamical simulations at unphysically heavy pion masses, carried out by the NPLQCD Collaboration and Yamazaki et al. and based on the standard approach to extract the ground state energy by fitting plateaus of the correlation functions find more attraction in both the 1 S 0 and 3 S 1 channels at heavy pion masses than for physical pion masses, see [16,17,18,19,20]. These results contradict the findings of the HAL QCD Collaboration using a (scheme-dependent) potential at the intermediate stage of extracting NN observables. This group finds no bound states in both S-wave channels for pion masses ranging from 469 to 1171 MeV [21]. The HAL QCD Collaboration has already carried out simulations at the physical point, but the results for the nonstrange channels have, as far as we know, not been released yet. The HAL QCD Colalboration has criticized the direct method by pointing out the danger of observing fake plateaus [22,23], see, however, the response of the NPLQCD Collaboration in Ref. [24]. The HAL QCD approach has also been criticized e.g. in Refs. [25,26]. The weakest point of this method seems to be its reliance on the derivative expansion, whose convergence is not clear a priori. Interestingly, the recent lattice QCD study in the strangeness S = −2 two-baryon sector by the CERN-Mainz group [27] using a superior distillation method finds for M π = 960 MeV the H-dibaryon energy perfectly consistent with HAL QCD, but in a strong disagreement with the NPLQCD result.
Recently, in Ref. [28] the use of low-energy theorems (LETs) to reconstruct the energy dependence of the NN scattering amplitude in a large kinematical domain from a single observable (e.g. binding energy, scattering length, effective range) at a given fixed value of the pion mass was proposed. The method relies on the dominance of the one-pion exchange (OPE) at large distances, which governs the near-threshold energy dependence of the scattering amplitude. At the physical point, LETs are known to work accurately in the 3 S 1 channel in line with the strong tensor interaction induced by the OPE, while less accurately in the 1 S 0 partial wave, where the OPE potential is very weak [29]. Notice that this approach employs the lattice QCD results to determine the strength of the OPE potential at unphysical values of M π and does not rely on the chiral expansion. At heavy pion masses M π ∼ M ρ , the LETs loose their predictive power, and the approach becomes equivalent to the effective range expansion. In [28], the LETs were used to test the linear interpolation of M π r as function of M π between M ph π and M π 800 MeV conjectured by the NPLQCD Collaboration [19]. These study was restricted to the 3 S 1 channel. The assumed linear interpolation was indeed found to be consistent with the available lattice QCD results for the deuteron binding energy obtained using the plateau method, see Fig. 6 of Ref. [28]. In [30] the LETs were applied to test the consistency between the bound state energies and phase shifts obtained using the Lüscher method by the NPLQCD Collaboration at M π = 450 MeV [18] in both the 1 S 0 and 3 S 1 channels. It was found that the NPLQCD phase shifts are inconsistent with their own results for the deuteron and dineutron energies. The inconsistency was later reemphasized by the HAL QCD Collaboration using the effective range expansion [22,23].
With these drawbacks and inconsistencies in mind, we nevertheless go forward and analyze the available lattice-QCD results for the deuteron and dineutron binding energy of Refs. [16,17,18,19,20], which seem to be mutually consistent, in order to extract the quantitiesĀ s andĀ t . The best way to extractĀ s,t is to use the LETs to compute the inverse scattering lengths from the binding energies at the corresponding values of M π , and to perform a subsequent interpolation. In table 1, we collect the binding energies of the deuteron and dineutron states from the calculations of Refs. [17,16,18,20] and the resulting values of the inverse scattering lengths. In addition to these results, we also include the direct NPLQCD determination of the scattering lengths at M π 806 MeV from Ref. [19] To perform the interpolation between these five points and the experimental values of the inverse scattering lengths, see Fig. 1, we use a simple quadratic ansatz where the coefficients a, b are determined from a least square fit to the available values of a −1 s,t at heavier-thanphysical pion masses. The results of the fits are shown by the solid (black) lines in Fig. 1. With χ 2 /N DOF = 4.3 and 2.4 in the singlet and triplet channels, respectively, the quality of the fit is not really good. Using a thirddegree polynomial leads to the results shown by the dotted Table 1. Available lattice QCD results for the deuteron and dineutron binding energies obtained from the plateau method along with the resulting values of the inverse scattering lengths calculated from the LETs at NLO. The uncertainties in the energies are taken from the correponding papers. The first error of a −1 s,t reflects the uncertainty of the lattice results for the binding energies used as input (for Mπ = 300, 390 and 510 MeV, the different lattice errors for the binding energies have been added in quadrature), while the second one corresponds to the uncertainty of the LETs estimated as explained in Ref. [30].
The LO chiral EFT result in Eq. (18) corresponds to an renormalizable expression for the the scattering amplitude with the static one-pion exchange, and the uncertainty is estimated from the cutoff variation from Λ = 600 MeV to infinity. The positive sign ofĀ t in Eq. (19) is consis-tent with a stronger attraction in the deuteron channel at heavy M π . HAL QCD results would presumably yield a negative value ofĀ t . In what follows, we will use the values collected in Eq. (19).

The Hoyle state in stellar nucleosynthesis
The stellar synthesis of elements heavier than 4 He is complicated by the fact that no stable nucleus exists for A = 8, at least for the physical values of the fundamental constants. In the absence of stable 8 Be nuclei, helium fusion instead takes place through the triple-alpha reaction 3( 4 He) → 12 C + γ, which requires a number of intermediate steps. The first step is 4 He + 4 He ↔ 8 Be, whereby a transient equilibrium population of 8 Be is maintained in the stellar core. It should be noted that the unstable 8 Be resonance decays back into two alpha particles with a half-life of ∼ 10 −16 s. The reaction rate for the formation of 8 Be is controlled by the energy difference where E 4 and E 8 denote the ground states of 4 He and 8 Be, respectively. Though 8 Be is short-lived, a sufficiently large transient 8 Be population in stellar cores is formed to allow for the second step 8 Be + 4 He → 12 C in the triple-alpha process.
In stellar cores composed primarily of helium, the nonresonant reaction proceeds too slowly to explain the observed abundances of carbon and oxygen in the universe. However, as the 12 C nucleus possesses an excited 12 C(0 + 2 ) state (known as the Hoyle state) with an empirical excitation energy of 7.6444 MeV, the reaction can also proceed in a resonant manner, which greatly enhances the triplealpha reaction rate. We define which controls the reaction rate for the second step 8 Be + 4 He ↔ 12 C(0 + 2 ), where E 12 is the (total) energy of the Hoyle state resonance. The energy scale E R which controls the resonant triple-alpha reaction is then which is empirically known (in our universe) to be E ph R = 379.47 (18) keV. Note that this is much smaller than the binding energies of the nuclei participating in the triplealpha reaction, which are 28 MeV for 4 He and 92 MeV for 12 C.
For a stellar plasma at temperature T , the reaction rate r 3α for fusion of three alpha particles via the ground state of 8 Be and the Hoyle state of 12 C is where N α is the number density of alpha particles, and k B is the Boltzmann constant. It should be noted how the exponential dependence on E R and T −1 arises, as the observation that the rate of stellar carbon production is exponentially sensitive to E R is central to the anthropic picture of the triple-alpha process, for discussions on this issue see [32,33]. For non-resonant reactions, the corresponding factor is given by the convolution of Coulomb barrier penetration with a thermal distribution of particle velocities, which gives a ∼ T −1/3 dependence on temperature. For resonant reactions a fixed energy is singled out, in this case given by Eq. (22), which leads to the exponential dependence of Eq. (23). It should be noted that E R is clearly the dominant control parameter of the triplealpha process, in comparison with the linear dependence on the radiative width Γ γ 0.0037 eV of the Hoyle state. Still, Γ γ should be sufficiently large to allow for the radiative decay of the Hoyle state to be competitive with fragmentation into 8 Be and 4 He. The radiative decay proceeds either through 12 C(0 + 2 ) → 12 C(0 + 1 ) + γ, or 12 C(0 + 2 ) → 12 C(2 + 1 ) + γ, after which the 12 C(2 + 1 ) decays to the ground state 12 C(0 + 1 ) by emission of a second photon. In practice, this two-step E2 process is more efficient than the direct M 0 decay, which is highly suppressed. Interestingly, the channel 12 C(0 + 2 ) → 12 C(2 + 1 ) + γ is strongly enhanced compared to the single-particle Weisskopf rate (often referred to as "strongly collective behavior"), which is correctly predicted by recent lattice EFT calculations. We define when the energy of the Hoyle state is shifted from its physical value. Clearly, for δE R > 0 (the Hoyle state energy is increased), the rate of carbon production (at constant stellar temperature T ) is decreased. In order to generate sufficient energy to counteract gravitation, the stellar core must increase T in order to compensate for the reduction in r 3α . It should be noted that the production of 12 C via the triple-alpha reaction competes with the destruction of 12 C by the formation of 16  On a phenomenological level, a variation of E R thus leads to a change in the relative importance of the competitive processes by which 12 C is produced, and destroyed by further processing into 16 O (and heavier alpha nuclei). Hence, for a sufficiently large positive δE R , a regime is encountered in which little 12 C and 16 O remains after stellar nucleosynthesis, with most material having been processed into 24 Mg and 28 Si. Conversely, for negative δE R , stellar core temperatures during helium burning are substantially lower, leading to end products with plentiful 12 C but relatively little 16 O. However, the latter point turns out to be sensitive to the initial stellar metallicity. Also, when the Hoyle state energy is lowered (δE R < 0), one should also consider the sensitivity where the stability of the star requires that the triplealpha reaction rate (and hence the energy production) increase as T increase, such that Ξ T > 0. Hence, E R should satisfy which places a lower bound on the permissible values of the Hoyle state energy. However, as stellar cores have roughly k B T 10 keV during helium burning, this bound is an order to magnitude smaller than the observed value of E R 380 keV.
Recently, comprehensive simulations of stellar nucleosynthesis in massive stars that eventually explode as supernovae have become available [3]. These studies follow stars ranging from 15 to 40 solar masses up to the stage where a degenerate iron core is formed. The yields of various isotopes are then weighted according to the stellar mass distribution function, taken to be dN/dM ∝ M −2.3 (for the range of stellar masses M considered), which accounts for the relative scarcity of heavier stars. For the stellar metallicity, Ref. [3] considered two cases. Firstly, the low-metallicity simulations used Z = 10 −4 , which is representative of the currently known stars with the lowest observed metallicity. Secondly, the case of Z = Z sun was studied, where Z sun = 0.02 denotes the observed solar metallicity. The main effect of Z is to alter the relative importance of the p-p chain and the CNO cycle, such that stars can enter the helium burning phase with different configurations.
We are now in a position to summarize the findings of Ref. [3] for the range in δE R , for which the final abundances of 12 C and 16 O exceed their initial values. These can be expressed as where the boundaries for each nucleus depend on the chosen initial stellar metallicity. For Z = 10 −4 (low metallicity), the ranges compatible with carbon-oxygen based life are 12 C(Z = 10 −4 ) : −300 keV ≤ δE R ≤ 500 keV, (28) and 16 O(Z = 10 −4 ) : −300 keV ≤ δE R ≤ 300 keV, (29) where for negative δE R , sufficient 12 C and 16 O were produced for all values of the Hoyle state energy compatible with the constraint (26). For Z = 0.02 (solar metallicity), the corresponding ranges were found to be significantly narrower. Specifically, where significant carbon production could still be maintained for all negative δE R . In stars of solar metallicity, the production of 16 O appears to be the limiting factor, as it is only possible in a roughly symmetric (though rather broad) envelope centered on the physical value of E R . Previously, the stellar simulations of Refs. [34,35] indicated that sufficient abundances of both carbon and oxygen are only possible for δE R ±100 keV around the empirical value E R = 379.47 (18) keV. From the present results, we conclude that the energy of the Hoyle state is likely to be less fine-tuned than previously thought. However, if the Hoyle state is raised by more than 300 keV, the generation of sufficient oxygen would encounter difficulties. Were the Hoyle state located more than 500 keV above its physical energy, the universe would also be unlikely to contain a sufficient amount of carbon.

Sensitivity to small changes in the fundamental parameters
We shall now update the ranges of variation of the light quark mass δm q and the fine-structure constant δα em , which are compatible with the formation of sufficient amounts of carbon and oxygen in our universe, and thus with the existence of carbon-oxygen based life. As in Ref. [6], we express the shift in the Hoyle state δE R as for |δm q /m q | 1 and |δα em /α em | 1. We shall first consider the effects of varying m q , for which we have and we recall that K q Mπ = 0.494 +0.009 −0.013 [12]. As in the NLEFT calculation of Ref. [6], we find where the only change from Ref. [6] is in the constant term, which has been recalculated using the updated values of x 1 and x 2 . The numbers in parentheses denote Monte Carlo uncertainties, and as in Ref. [6], the relatively small additional errors due to the uncertainties of x 1 and x 2 have been neglected. As such uncertainties are much reduced here, this simplification is better justified.
The boundaries of the envelope where a sufficient abundance of carbon or oxygen is maintained are then where the earlier stellar nucleosynthesis calculations of Refs. [34,35] indicated an overall bound of |δE (−) R | = 100 keV from ad hoc variations of the Hoyle state energy. The present situation is slightly more involved, as the upper and lower boundaries are not symmetric, and moreover the boundaries for 12 C and 16 O are different, an additional factor being the (initial) metallicity of the star under consideration. On the one hand, for E R to not increase beyond the permissible range, we require that for positive shifts in m q such that δm q → |δm q |, and ≥ − δE for negative shifts in m q such that δm q → −|δm q |. On the other hand, for E R to not decrease too much, we should ≥ δE for positive shifts in m q , and − 0.572(19)Ā s − 0.933(15)Ā t + 0.068 (7) ≤ − δE for negative shifts in m q . Hence, the values ofĀ s andĀ t compatible with a given variation in m q are given by the regions enclosed by Eqs. (36) and (38) for δm q > 0, and by Eqs. (37) and (39) for δm q < 0. The constraints onĀ s andĀ t due to the conditions (36) through (39) are illustrated by the shaded bands in Fig. 2. These bands cover the values ofĀ s andĀ t consistent with the ability of stars to produce 12 C and 16 O, when m q is varied by 0.5%, 1% and 5%. As an example, given the boundaries for 12 C production in stars with solar metallicity (Z = 0.02), the interpolated lattice QCD result is compatible with a 0.8% increase in m q , beyond which E R is decreased too much. Conversely, beyond a 0.4% decrease in m q , E R is increased too much. For stars with solar metallicity, the production of 16 O is clearly the most heavily constraining factor, although low-metallicity stars (Z = 10 −4 ) allow for much greater variation of m q . This value of the metallicity corresponds to the metal-poorest stars observed in the universe. In such metal-poor stars, the chiral EFT determinations ofĀ s andĀ t suggest that changes of ∼ 5% in m q are permissible. However, from the interpolated lattice QCD results, the allowed range in m q is strongly reduced to a mere 0.8%, largely because of the positive value ofĀ t .
Finally, we note that the effect of shifts in the electromagnetic fine-structure constant lead to the constraint where Q em (E R ) = 3.99(9) MeV was determined in the NLEFT calculation of Ref. [6]. With the bound |δE R | = 100 keV [34,35], this is compatible with a 2.5% shift in α em . With the much more relaxed bound |δE R | = 300 keV due to the production of 16 O in stars with Z = 10 −4 , this tolerance is significantly increased to 7.5%.

Discussion
We have reconsidered the sensitivity of the triple-alpha process with respect to shifts in the fundamental parameters of nature, especially the light quark mass m q and the electromagnetic fine-structure constant α em . Our knowledge of the quark-mass dependence of the hadronic parameters in the nuclear Hamiltonian has improved significantly, in particular with respect to the nucleon axialvector coupling g A . There, new lattice QCD data allow for an accurate chiral extrapolation which shows good convergence. Much more detailed predictions of the effects of ad hoc variation of the position of the Hoyle state resonance on the stellar yields of 12 C and 16 O have also become available. These show that the production of 16 O in lowmetallicity stars is likely to be the limiting factor, although the bounds on carbon-oxygen based life have in general become much less stringent. At the same time, much more lattice QCD data on the singlet and triplet S-wave nucleon scattering lengths at unphysical quark masses have been produced. The current lattice QCD data appear to exclude the no-fine-tuning scenario, to the extent that a relatively small 0.5% shift in m q would eliminate carbon-oxygen based life from the universe. On the other hand, such life could possibly persist up to 7.5% shifts in α em . Clearly, more reliable lattice QCD data at close-to-physical pion masses are required to overcome the remaining uncertainties discussed in detail in Sec. 2. While we have here mostly focused on updating the nuclear, hadronic and astrophysical inputs to the EFT calculation of the fine-tuning of the triple-alpha process, it is also of interest to perform improved 12 C simulations in NLEFT. Apart from effects of smearing of the LO operators, the EFT calculation of the triple-alpha process is essentially a LO calculation. Extending this to higher orders would require much more detailed information on the nuclear force at unphysical pion masses, for higher orders in the EFT expansion. However, as modern NLEFT potentials, see e.g. [36], use a combination of local and nonlocal smearing at LO, the description of 12 C including the Hoyle state is nevertheless expected to be improved.