The astrophysical factor and its implications for Big Bang Nucleosynthesis
Abstract
The radiative capture is studied in order to predict the Li primordial abundance. Within a twobody framework, the particle and the deuteron are considered the structureless constituents of Li . Five potentials are used to solve the twobody problem: four of them are taken from the literature, only one having also a tensor component. A fifth model is here constructed in order to reproduce, besides the Li static properties as binding energy, magnetic dipole and electric quadrupole moments, also the state asymptotic normalization coefficient (ANC). The twobody bound and scattering problem is solved with different techniques, in order to minimize the numerical uncertainty of the present results. The longwavelength approximation is used, and therefore only the electric dipole and quadrupole operators are retained. The astrophysical factor is found to be significantly sensitive to the ANC, but in all the cases in good agreement with the available experimental data. The theoretical uncertainty has been estimated of the order of few % when the potentials which reproduce the ANC are considered, but increases up to % when all the five potential models are retained. The effect of this factor prediction on the Li primordial abundance is studied, using the public code PArthENoPE. For the five models considered here we find H, with the baryon density parameter in the 3 range of Planck 2015 analysis, .
pacs:
98.80.Ft, 26.35.+c,98.80.Ft, 26.35.+c, 98.80.kI Introduction
The radiative capture
(1) 
has recently received quite some interest, triggered by the socalled Li problem. In the theory of Big Bang Nucleosynthesis (BBN), even if it is a weak electric quadrupole transition, this reaction is important as represents the main Li production process. In 2006 Asplund et al. performed high resolution observations of Li absorption lines in old halo stars asp06 . The Li/Li ratio was found to be of about , more than two orders of magnitude larger than the expected BBN prediction. Since the analysis is performed on old stars, the quantity of the present Li should be a good estimate of the one at the star formation, i.e. the same after BBN. This great discrepancy is the socalled second Lithium problem. However, recent analyses with threedimensional modelling of stellar atmosphere, which do not assume local thermodynamical equilibrium and include surface convection effects, show that these can explain the observed line asymmetry. The problem, therefore, would be weakened cayrel ; perez ; steffen ; lind .
We recall that the BBN relevant energy window is located between 50 and 400 keV, and experimental studies of Eq. (1) at these energies are very difficult, due to the exponential drop of the reaction cross section as a consequence of the Coulomb barrier. Furthermore, this reaction is affected by the isotopic suppression of the electric dipole operator, as it will be discussed in Sec. II.2. The reaction (1) was first studied experimentally in the early 1980s rob81 and then thorough the 1990s kie91 ; moh94 ; cec96 ; iga00 . However the data in the BBN energy range were affected by large uncertainties. The latest measurement is that performed by the LUNA Collaboration and14 ; tre17 .
The theoretical study of this reaction is also very difficult, since, in principle, we should solve a sixbody problem, i.e. we should consider the six nucleons contained in the and Li particles, and their interaction with the photon. Such an approach is known as the abinitio method, and it has been used only by Nollett et al. in Ref. nol01 . However, the numerical techniques used in Ref. nol01 to solve the sixbody problem, i.e. the variational Monte Carlo method, provide solutions for the initial and final state wave functions with uncertainties at the 1020% level. Since abinitio methods are still nowadays hardly implemented for radiative captures, the study of the reaction has been done using a simplified model, where Li is seen as an system and the problem is reduced to a twobody problem. Then a crucial input for the calculation is represented by the potential model, which describes the interaction. Five different potential models have been considered in this work, four of them taken from Refs. ham10 ; tur15 ; muk11 ; dub98 , and a last one constructed here starting from the model of Ref. dub98 , and then modifying it in order to reproduce the asymptotic normalization coefficient (ANC), i.e. the ratio between the relative radial wave function in Li and the Whittaker function for large distances. It describes the boundstate wave function in the asymptotic region. To be noticed that only the potential of Ref. dub98 and this last model have a tensor component, necessary to describe the experimental values for the Li magnetic dipole and electric quadrupole moments. Our calculations have been performed using two methods to solve the twobody Schrödinger equation, both for the bound and the scattering states, in order to verify that our results are not affected by significant numerical uncertainties.
The paper is organized as follows: in Sec. II we introduce all the main ingredients of the present calculation for the astrophysical factor and we present in Sec. II.3 our results. In Sec. III we discuss the implications of the present calculated factor for the BBN prediction of Li abundance. We give our final remarks in Sec. IV.
Ii The Sfactor
The astrophysical factor , being the initial centerofmass energy, is defined as
(2) 
where is the capture cross section, and is the Sommerfeld parameter, being the fine structure constant and the relative velocity. With this definition, the factor has a smooth dependence on and can be easily extrapolated at low energies of astrophysical interest. The reaction cross section is given by
(3) 
where the differential cross section can be written as
(4) 
Here is the Li mass, is the photon momentum and its polarization vector, is the Fourier transform of the nuclear electromagnetic current, and and are the initial and final Li wave functions, with spin projection and . In Eq. (4), we have averaged over the initial spin projections and summed over the final ones.
In order to calculate the cross section, it is necessary to evaluate the Li and wave functions. This point is described in the next Subsection.
ii.1 The Li and systems
A crucial input for our calculation is represented by the Li and wave functions. We consider first the bound state. The nucleus of Li has , a binding energy respect to the threshold of 1.475 MeV dub98 , a nonnull electric quadrupole moment fm and a magnetic dipole moment dub98 . As it was shown in Ref. muk11 , the astrophysical factor at low energies is highly sensitive not only to the Li binding energy and the scattering phase shifts, but also to the Li state asymptotic normalization coefficient (ANC). This quantity is crucial due to the peripheral nature of the reaction at low energies, where only the tail of the Li wave function gives most of the contribution in the matrix element of Eq. (4). The state ANC is defined as
(5) 
where is the state Li reduced wave function, is the Whittaker function, is the Sommerfeld factor and for the state ANC. Its experimental value for Li is ANC fm tur15 .
In the present study we consider the Li nucleus as a compound system, made of an particle and a deuteron. In fact, as it was shown in Ref. dub94 , the clusterization percentage in Li can be up to about 6080%. Therefore, we solve in this work a twobody problem, including both and states in the bound system. The first observable that we will try to reproduce is the binding energy, but we will consider also the above mentioned observables of Li .
At this point, an important input for the calculation is represented by the potential. The different models considered in this work will be discussed below.
ii.1.1 The Potentials
For our calculation we consider five different potential models. The use of so many models allows us to get a hint on the theoretical uncertainty arising from the description of the Li nucleus and the scattering system. Four of these potentials are taken from Refs. ham10 ; muk11 ; tur15 ; dub98 , while the last one has been constructed in the present work as described below. The physical constants present in each potential as listed on the original references are summarized in Table 1.
units  and  , and  

  2.01411  2  
  4.00260  4  
MeV  931.494043  938.973881  
MeV  1248.09137  1251.96518  
MeV fm  15.5989911176  15.5507250000  
  7.29735256810  7.29740599910  
MeV fm  1.4399644567  1.4399750000  
MeV  1.474  1.475 ()  
1.4735 ( and ) 
The first potential used in our study has been taken from Ref. ham10 and has the form
(6) 
It contains a spinindependent WoodSaxon component, a spinorbit interaction term, and a modified Coulomb potential, which is written as
(7) 
The values for all the parameters present in Eqs. (6) and (7), as well as those of the following potentials, are listed in Table 2, apart from , which is fm, with . To be noticed that this potential does not reproduce the experimental value of the ANC, as it has been noticed in Ref. muk11 , and we have ourselves verified by calculating the Li properties (see below).
The second potential is taken from Ref. tur15 , and can be written as
(8) 
It is therefore the sum of a Gaussian function and a Coulomb pointlike interaction . It reproduces the experimental ANC for the Li (see below).
The third potential is obtained by adding to the potential of Ref. ham10 , a new term , such that the new potential
(9) 
reproduces the experimental ANC muk11 . The procedure to obtain is discussed at length in Ref. muk11 . Here we have generalized it to the coupledchannel case, and it will be discussed below.
The potentials , and considered so far are central potentials, which have, at maximum, a spinorbit term. Therefore, these potentials are unable to give rise to the component in the Li wave function. The nonzero Li quadrupole moment has induced us to consider also potentials which include a tensor term. In this study, we have used the potential of Ref. dub98 , which can be written as
(10) 
where is the spin operator acting on Li. The coefficients and have been taken from Ref. dub98 . However, in Ref. dub98 this potential was used only for the boundstate problem. Therefore, we have modified the potential in order to reproduce also the scattering phaseshifts up to . In order to do so, the depth has been fitted to the experimental scattering phaseshifts for every initial channel, minimizing the of the calculated phase shifts with respect to the available experimental data taken from Refs. jen83 ; mci67 ; gru75 ; bru82 ; kel70 . In this procedure, we minimized the changing the value of . We have used both the bisection and the Newton’s method, finding no difference between the calculated values of . These have been listed in Table 2.
As in the case of , also the potential, does not reproduce the Li ANC. Therefore, we have constructed a new model generalizing the procedure of Ref. muk11 to the coupledchannel case. We start from a generic Hamiltonian operator , for which we know the bound state radial eigenfunction , the corresponding binding energy and the ANC for the state. We have defined to be the vector containing the  and state bound wave functions, i.e. and normalized it to unity, i.e.
(11) 
We want to find a potential part of an Hamiltonian which has the same binding energy, but the correct value for , which we will call . As an Ansatz, we assume that our new solution has the form
(12) 
with
(13) 
where is a parameter to be fitted to the experimental ANC value. This solution is correctly normalized and the new ANC is given by
(14) 
It is then enough to choose , so that . For the potential, muk11 , while for this coupledchannel case .
In order to obtain the new wave function , we define a new Hamiltonian operator as
(15) 
and we impose
(16) 
knowing that
(17) 
Subtracting Eq. (17) from Eq. (16), we obtain
(18) 
which can be rewritten as
(19) 
Writing explicitly and , and assuming for simplicity to be diagonal, we get
(20)  
(21) 
Note that if we consider only central potentials, then the potential acts only on the Li S state, and reduces to
(22) 
as obtained in Ref. muk11 . This is the term added in Eq. (9). As we have seen, this new term should give rise to no changes in the binding energy, nor in the scattering phaseshifts with respect to those evaluated with the . This has been verified with a direct calculation.
Potential  Parameters  



60.712  56.7  2.4  2.271  2  0.65  
92.44  0.25  68.0  0.22  79.0  0.22  85.0  0.22  63.0  0.19  69.0  0.19  80.88  0.19  


71.979  0.2  27.0  1.12  77.4  73.08  78.42  72.979  86.139 
ii.1.2 Numerical methods
In order to solve the Schrödinger equation, both for the initial and final states, two methods have been adopted, the Numerov’s and the variational method. In particular, we have used the Numerov’s method for the boundstate problem, and the variational one for both the bound and the scatteringstate problem. The convergence of the two methods has been tested, proving that both methods give the same numerical results for the factor, with a very good accuracy. The choice of the variational method for the scatteringstate is related to the fact that this method is simpler to be extended to the coupledchannel case. In fact, the Numerov’s method, even for the boundstate problem, needs some improvement respect to the singlechannel case. Here we have proceeded as follows. The reduced radial waves solutions for the () and () states of Li must satisfy the coupled equations
(24)  
(25) 
We solve this system of equations iteratively. First we consider to be zero. Eq.(25) then becomes
(26) 
which we solve with the standard Numerov’s algorithm, obtaining and . Then we calculate the solution of Eq. (24) giving an initial value for the normalization ratio , defined as
(27) 
and applying again the Numerov’s algorithm to obtain . With the evaluated , we calculate again with Eq. (25), and so on, until we converge both for and for within the required accuracy.
The method for the singlechannel scattering problem is straightforward, as the Numerov’s outgoing solution from to the final grid point is matched to the function
(28) 
which is normalized to the unitary flux. Here and are the regular and irregular Coulomb functions, and the relative momentum. The scattering phaseshift is then easily obtained.
The variational method has been used for both the bound and the scattering states. For the bound state, we expand the wave function as
(29) 
where , are orthonormal functions and are unknown coefficients. We use the RayleighRitz variational principle to reduce the problem to an eigenvalueeigenvector problem, which can be solved with standard techniques (see Ref. Mar11 for more details). Here we use a basis function defined as
(30) 
with fm for each , and are Laguerre polynomials. Note that so defined, these functions are orthonormal.
For the scattering problem the wave function is decomposed as
(31) 
where are unknown coefficients, has the same form as in Eq. (29), while and are regular and (regularized for ) irregular Coulomb functions, with a nonlinear parameter of the order of 4 fm. The Kohn variational principle is used to obtain the unknown coefficients and of Eq. (29), with a standard procedure as outlined in Ref. Mar11 .
In Fig. 1 we show a comparison for the Li reduced radial wave functions obtained, using the potential, with the Numerov’s or variational method. Similar results can be found for the other potentials. As it can be seen by inspection of the figure, the variational method is unable to reproduce the Li wave function at large distances, of the order of 3040 fm. In this case the reduced wave function has been cured in order to get the correct asymptotic behaviour. Within the Numerov’s method, the long range wave function is constructed by hand. The agreement between the two methods is much nicer for the scattering problem, although the Numerov’s method has been used for the single channels. In these cases, the agreement between the two methods is at the order of 0.1%.
ii.1.3 The Li nucleus and the scattering state
The Li static properties, i.e. the binding energy respect to the threshold, the state ANC, the magnetic dipole moment and the electric quadrupole moment are given in Table 3. By inspection of the table we can conclude that each potential nicely reproduces the experimental binding energy, while only , and give good values for the ANC. Also, the and potentials are the only ones which include the state contributions in the Li wave function. Therefore, the values of and obtained with these potentials are closer to the experimental values, while and calculated with the , , and potentials are simply those of the deuterium. Finally, we show in Fig. 2 the Li reduced wave function evaluated with each potential. The differences between the various potentials are quite pronounced for fm. However, this is not too relevant for our reaction, which is peripheral and therefore most sensitive to the tail of the wave function and to the state ANC.
EXP.  

1.474  1.475  1.474  1.4735  1.4735  1.474  
2.70  2.31  2.30  2.50  2.30  2.30  
0.857  0.857  0.857  0.848  0.848  0.822  
0.286  0.286  0.286  0.066  0.051  0.082 
For the initial scattering state, the scattering phase shifts obtained with each potential are in good agreement with the experimental data, as it can be seen in Fig. 3 for the single channels and in Fig. 4 for the coupled channels. In particular, the results obtained with the () and () potentials coincide.
ii.2 The transition operator
To evaluate the reaction cross section, we need to write down the nuclear electromagnetic current operator of Eq. (4). This can be written as
(32) 
with
(33) 
where , , and are respectively the momentum, the mass, the position and the charge of the ith particle. The matrix element appearing in Eq. (4), , can be rewritten expressing and as
(34)  
(35)  
where and are the Li and reduced radial functions discussed in Sec. II.1. In the partial wave decomposition of Eq. (35), we have retained all the contributions up to . By then performing a multipole expansion of the operator, we obtain
(36) 
where is the multipole index, while and are the socalled electric and magnetic multipoles of order . They are defined as
(37)  
(38) 
where is the spherical Bessel function of order and is the vector spherical harmonic of order .
In this work we adopt the socalled long wavelength approximation (LWA), since, for the energy range of interest, the momentum of the emitted photon is much smaller than the Li dimension. This means that we can expand the multipoles in powers of . Furthermore, in the present calculation, we include only electric dipole and quadrupole multipoles, since it has been shown in Ref. nol01 that the magnetic multipoles are expected to give small contributions to the factor.
With this approximation, can be written as
(39) 
where muk16
(40)  
(41) 
and is the socalled effective charge, and is given by
(42) 
Note that when only the first order contribution in the LWA is retained, reduces to
(43) 
The use of Eqs. (40) and (41) instead of Eq. (43) leads to an increase in the factor of the order of 1 %. This has been shown in Ref. muk16 and has been confirmed in the present work.
In the formalism of the LWA the total cross section of Eq. (3) can be written as
(44) 
where is the cross section evaluated with the electric multipole and the initial state with orbital (total) angular momentum (). It can be written as
(45) 
For simplicity we define the partial factor as
(46) 
The results for these quantities evaluated with the potential are shown in Fig. 5. The ones for the other potentials have the same shapes and properties. The only difference comes for , and , where the contribution to the factor for the initial state are zero, being the transition forbidden for each multipole term. Due to the nature of the LWA, the largest contribution to the total cross section, and therefore to the astrophysical factor, should be given by the transition, but, as we can see from Fig. 5, the transition dominates only at energies of the order of few keV. This is due to the socalled isotopic suppression. As we have seen the multipole expansion at th order for the electric terms depends on the square of the effective charge and, for our reaction, and . Therefore the contribution to the factor is strongly suppressed, except for very low energies, where the other multipoles are reduced due to their energy dependence.
ii.3 The theoretical astrophysical factor
The calculated astrophysical factor is compared in Fig. 6 with the available experimental data from Refs. rob81 ; kie91 ; moh94 ; cec96 ; iga00 ; and14 ; tre17 . By inspection of the figure we can conclude that the tail of the factor at low energies has a strong dependence with respect to the ANC value. In fact, the three potentials which reproduce the ANC give very close results. The and potentials, giving a larger value for the ANC than the other potentials, predict higher values for the factor. Thanks to the relatively large number of considered potentials, we can give a rough estimate of the theoretical uncertainty of our predictions. Therefore, in Fig. 7 we show the same results of Fig. 6 as two bands, one obtained using all the five potentials and a much narrower one calculated with only the three potentials which reproduce the correct ANC value. As we can see from the figure, the theoretical uncertainty for the factor is much smaller in this second case: at centerofmass energies keV, it is of the order of 2%, but it becomes at the 1% level at the LUNA available energies, i.e. for keV. On the other hand, if we consider all of the potentials, the previous estimates grow to 25% and 24% at and 100 keV, respectively. The available experimental data, though, are not accurate enough in order to discriminate between the results obtained with these five potentials. Therefore, in the following Section, where the primordial Li abundance is discussed, we consider conservatively the results for the astrophysical factor obtained with all the five potentials.
Iii The Li primordial abundance
Li is expected to be produced during BBN with a rather low number density, Li/H , for the baryon density as obtained by the 2015 Planck results Ade:2015xua . This result still holds using the factor described in the previous Section (see below), and it is too small to be detectable at present. Actually, some positive measurements in old halo stars at the level of Li/Li were obtained in the last decade asp06 , but they may reflect the postprimordial production of this nuclide in Cosmic Ray spallation nucleosynthesis. Moreover, as we mentioned already, a more precise treatment of stellar atmosphere, including convection, shows that stellar convective motions can generate asymmetries in the line shape that mimic the presence of Li, so that the value 0.05 should be rather understood as a robust upper limit on Li primordial abundance. This does not mean that the issue is irrelevant for BBN studies since the study of the chemical evolution of the fragile isotopes of Li, Be and B could constraint the Li primordial abundance, and clarify the observational situation of Spite Plateau, see e.g. Ref. Olive:2016xmw .
The whole Li is basically produced via the process, which is thus the leading reaction affecting the final yield of this isotope. The new theoretical factors detailed so far have been used to compute the thermal rate in the BBN temperature range, by folding the cross section with the MaxwellBoltzmann distribution of involved nuclides. We have then changed the PArthENoPE code Pisanti:2007hk accordingly, and analyzed the effect of each different factor on the final abundance of Li, as function of the baryon density. For comparison, we also consider the value of the factor as obtained from fitting experimental data from Refs. RO81 ; MO94 ; IG00 ; AN14
(47)  
as well as the NACRE 1999 fit nacre99 , which is used as benchmark rate in PArthENoPE public code. The results are shown in Fig. 8, normalized to NACRE 1999.
As we can see, the change is in the 1020 % range. If we adopt the Planck 2015 best fit for the baryon density parameter Ade:2015xua , we obtain values for the Li/H density ratio in the range , slightly smaller than what would be the result if the experimental data fit is used, as it can be seen in Table 4.
bench  data  H  T  M  D  G  

Li/H 
Notice that, at least with present sensitivity on Li yields, the dependence on the baryon density, or equivalently, the baryon to photon density ratio , is quite mild, as shown in Fig. 9. The lower band in this plot cover the range of values obtained when the five potential models are used, and we can conservatively say that standard BBN predicts Li/H. This range is also in good agreement with the results of other studies ham10 ; Cyburt:2015mya . In Fig. 9 we also show the final abundance of Be+Li (upper band), which remains in the range , and it is, as expected, almost independent of the potential model adopted for the radiative capture reaction considered here.
Iv Conclusions
The radiative capture has been studied within a twobody framework, where the particle and the deuteron are considered as structureless constituent of Li. The longwavelength approximation (LWA) has been used, and the electric and multipoles have been retained. In order to study the accuracy that the present theoretical framework can reach, we have used five different models for the interaction, among which also, for the first time, potential models with a tensor term, able to reproduce the magnetic dipole and electric quadrupole moments of Li , as well the state ANC and the scattering phase shifts. The theoretical uncertainty to the astrophysical factor, the observable of interest, is of the order of 20% if all the five potential models are retained, but reduces to few % if only those potentials which reproduce the state ANC are considered. The experimental data, however, are affectd by an uncertainty much larger than the theoretical one.
The calculated values for the astrophysical factor have been used in the PArthENoPE public code in order to estimate the Li and Li+Be primordial abundances. The Li abundance is predicted to be slightly smaller than what would result from the available experimental data and from the NACRE 1999 compilation, but still in the range of . We conclude that this result of standard BBN is thus quite robust. Further studies about Li astrophysical measurement may be needed to check the claim of a much larger ratio Li/Li obtained in Ref. asp06 . On the other hand, the final Li+Be abundance is almost independent on the result for the astrophysical factor presented here, and is found to be in the range of .
Finally, we would like to notice that the present calculation for the astrophysical factor is, to our knowledge, the most uptodate one working within a twobody framework. However, the assumption that the deuteron is a structureless constituent of Li can be considered rather weak, and the present study could be improved if the sixbody systems are viewed as a core of an particle and two nucleons, i.e. as a threebody systems. The first steps within this threebody framework have been done in Ref. Tur16 , and further work along this line is currently underway.
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