2023 Derivation of transition density from the observed 4 He( e , e ′ ) 4 He(0 + 2 ) form factor raising α -particle monopole puzzle

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Introduction
The 4 He nucleus is the lightest nucleus that exhibits excited states (resonances). One of the important tasks in few-body nuclear physics is to solve the four-nucleon states as a stringent test for ab initio few-body methods and nuclear Hamiltonian. A benchmark test calculation for this purpose was reported in Ref. [1] (2001) by seven different few-body research groups. The 4 He ground state was solved using a realistic interaction, the Argonne V8 ′ potential (AV8 ′ ) [2]. Agreement between the results of the significantly different calculational schemes was essentially perfect in terms of the binding energy, the r.m.s. radius, and the two-body correlation function. The present author participated in the benchmark test together with Hiyama, using the Gaussian expansion method (GEM) for few-body systems [3][4][5].
One of the next challenging projects was to explain the properties of the second 0 + state (a resonance) of 4 He using realistic interactions, simultaneously reproducing the 0 + 1 and 0 + 2 states with significantly different spatial structures. Hiyama, Gibson, and the present author [6] (2004), employed the GEM and the AV8 ′ + phenomenological central 3N potential to reproduce the binding energies of 3 H, 3 He and 4 He(0 + 1 ), and predicted the energy of 4 He(0 + 2 ) as −8. 19 MeV measured from the four-body breakup threshold without any additional adjustable parameters on the basis of the bound-state approximation with approximately 100 four-body angular momentum channels in the isospin formalism. The calculated energy of the 0 + 2 state with isospin T = 0 explained the observed value −8.09 MeV with a 100 keV error. The calculated transition form factor of 4 He(e, e ′ ) 4 He(0 + 2 ) agreed with available data [7][8][9] within large observed errors ( Fig. 3 [6]). It was further found that (i) the percentage probabilities of the S -, P-and D-components in the 0 + 2 state are almost the same as those of 3 H and 3 He, and (ii) the overlap amplitude between the 4 He(0 + n ) wave function and the 3 H wave function (see Fig. 4 of Ref. [6]) represents that, in the ground state, the fourth nucleon is located close to the other three nucleons, but it is far away from them in the second 0+ state. These analyses indicate that the second 0 + state has well-developed 3N +N cluster structure with relative S -wave motion, not having a monopole breathing mode. This state property was soon confirmed by Horiuchi and Suzuki [10] adopting the stochastic variational method with the correlated Gaussian basis functions [11,12] which was one of the numerical methods used for the benchmark test calculation [1].
Bacca et al. [13,14] investigated the second 0 + state using the effective-interaction hyperspherical harmonic (EIHH) method [15,16], which is also one of the methods for the benchmark calculation [1]. Further, they employed the Lorentz integral transformation (LIT) approach [17,18] for the calculation of resonant states. For Hamiltonians, they used (i) Argonne V 18 (AV18) [19] poten-tial plus Urbana IX (UIX) [20] 3NF and (ii) a chiral effective field theory (χEFT) based potential (N 3 LO NN [21] + N 2 LO 3NF [22,23]). They found that the calculated results of the transition form factor are strongly dependent on the Hamiltonian and do not agree with the experimental data [7][8][9], especially in the case of the χEFT potential. The authors claimed that it was highly desirable to have a further experimental confirmation of the existing data and, in particular, with increased precision.
In response, a new experiment on the transition form factor with significantly improved precision was performed at the Maintz Microtoron by Kegel et al. [24]. All the four authors of Refs. [13,14] participated in this work [24] as co-authors. The precise results for the form factor are shown in Figs. 3 and 4 [24], and confirm previous data [7][8][9] with much higher precision. They found that the ab initio calculations [6,13] disagree with the observed form factors; for example, the χEFT result [13] is 100% too high with respect to the new data at the peak position.
The authors of Ref. [24] noticed that, in the momentum transfer range 0.2 ≤ q 2 ≤ 1 fm −2 , the simplified potential in Ref. [6] leads to agreement with the experimental data, whereas the realistic calculations [13] do not ( Fig. 4 [24]). They showed that the difference did not stem from the numerical methods but from the Hamiltonian; to examine it, they employed the same potential (AV8 ′ + central 3N) of Ref. [6] using their calculation method (EIHH) and reproduced the result of Ref. [6] as compared in Fig. 4 of Ref. [24].
Regarding the explicit information on the spatial structure of the 0 + 2 state, the two gross features of the transition density, monopole matrix element r 2 tr and transition radius R tr , were extracted based on the behavior of the form factor at q ∼ 0. It was noticed [24] that the AV8 ′ + central 3N potential is not compatible with the experimental value of r 2 tr , while the realistic AV18 + UIX fits the value, and the χEFT potential prediction deviates the most from the experiments even at low momentum values. Further discussion on r 2 tr and R tr is made in Sec. 4. In Ref. [24], it was concluded that there is a puzzle that is not caused by the applied few-body methods, but rather by the modeling of the nuclear Hamiltonian, and therfore further theoretical work is needed to resolve the α-particle monopole puzzle. On the day of publication of Ref. [24], this puzzle was introduced and discussed in a review article by Epelbaum [25]; Fig. 2 summarizes the transition form factors obsereved newly by Ref. [24] and previously by Refs. [7][8][9], and values calculated in Ref. [6] (yellow line) and Ref. [13] (blue and red lines).
More information regarding the spatial structure of the second 0 + state is required to solve this puzzle. Thus, the purpose of this letter is to extract a possible mass transition density from the newly observed 4 He(e, e ′ ) 4 He(0 + 2 ) form factor.
Monopole mass transition density ρ tr (r) is defined as where the | 0 + 1 and | 0 + 2 are the four-nucleon wave functions (cf. Eq. (2.1) in Ref. [6]), and r i is the position vector of ith nucleon with respect to the center-of-mass of the four nucleons. This definition is the same as that in Bacca et al. [14] in their Eq. (9)  because the left hand side is the overlap 0 + 2 | 0 + 1 between the 0 + 1 and 0 + 2 states. The form factor F M0 + (q 2 ) is related to the monopole transition density ρ tr (r) as follows: The density ρ tr (r) can be derived as if F M0 + (q 2 ) is provided for all q 2 -range.
We examined whether extending Eq. (2) to the range of q 2 > 5.5 fm −2 is meaningful. As a test example, we employed ρ tr (r) and F M0 + (q 2 ) calculated in Ref. [6]. The density ρ tr (r) (= ρ 00 (r)/ √ 4π [6]) is illustrated in Fig. 3a by the solid line with a node at r = 1.62 fm, whereas | F M0 + (q 2 )|/q 2 is shown in Fig. 3b as a solid line with a node at q 2 = 14.1 fm −2 ). The solid-line form factor in the range of 3 < q 2 < 5.5 fm 2 extends to 5.5 < q 2 < 9 fm 2 to maintain similar decay rates. We approximated the solid-line form factor, as shown in Fig. 3b in the range of 3 < q 2 < 9 fm −2 using the function with a 0 = 0.078 fm 2 and b 0 = 0.45 fm 2 , and extended it to the range q 2 > 9 fm −2 , as indicated by the dashed line in Fig. 3b; in the range of q 2 < 9 fm −2 , the dashed line was considered the same as the solid line. We substitute the dashed-line form factor into Eq. (6) and obtain the dashed-line density in Fig. 3a with a node at r = 1.66 fm. The difference between the solid-and dashed-line form factors in the range of q 2 > 9 fm −2 in Fig. 3b appears as a central depression at r 1 fm on the solid-line density in Fig. 3a, and generates only a small difference in the density for r 1 fm.
Subsequently, we considered that an extension of Eq. (2) to the range of q 2 > 5.5 fm −2 will be useful for studying the observed form factor.
We extend the form factor Eq. (2) in the range of q 2 > 5.5 fm −2 , as illustrated in Fig. 4a by the red solid line. Using the form factor in Eq. (2) over the entire q 2 range, we obtain Ref. [6]. The green line denotes the density calculated using the χEFT potential given in Fig. 1 of Ref. [14]. where 3/2 (9) with A 1 = 4.8 × 10 −2 fm −3 and A 2 = 4.0 × 10 −5 fm −3 . The density ρ tr (r) of Eq. (8) is shown in Fig. 4b using the solid red line with a node at r = 1.82 fm, which is almost the same node position as that of the dominant first term in Eq. (8) at r = √ 6b 1 = 1.80 fm. For comparison, we show in Fig. 4b the transition density given by Hiyama et al. [6] by the black line calculated using the AV8 ′ + central 3N potential and that given by Bacca et al. [14] by the green line calculated using the χEFT interaction (extracted from Fig. 1 in Ref. [14]).

Central depression of transition density
We focus on the central depression of the transition density ρ tr (r) in Fig. 4b in the range of r 1 fm as indicated by the black line [6] and the green line [14]. As shown above, the redsolid-line density with no central depression in Fig. 4b is generated by the red-solid-line form factor in Fig. 4a. We then attempt to provide an artificial example of a central depression to the red-solid-line density. We considered a small additional transition form factor ∆ F M0 + (q 2 ), which is related to an additional transition density ∆ρ tr (r) (note ∆ρ tr (r) dr = 0) We consider, as an example, a 3 = −0.00037 fm 2 and b 4 = 0.090 fm 2 , yielding A 4 = −0.0017 fm −3 .
∆ F M0 + (q 2 )/q 2 and ∆ρ tr (r) are presented in Fig. 4a and 4b, respectively, indicated by the red dotted line. The summed form factor ( F M0 + (q 2 ) + ∆ F M0 + (q 2 ))/q 2 is shown in Fig. 4a by a red dashed line with a node at q 2 = 12.2 fm −2 . The transition density ρ tr (r) + ∆ρ tr (r) is illustrated in Fig. 3b by the red dashed line, which is close to the red-solid line ρ tr (r) within the range of r 1 fm.
The central-depression phenomena of the transition density originates from the behavior of the form factor in the range of q 2 10 fm −2 , and the observed transition form factor [24] limited to the range of q 2 ≤ 5 fm −2 cannot provide information about the central-depression structure at r 1 fm. However, we considered that using the observed form factor can be used to derive the transition density ρ tr (r) in the range r 1 fm, as indicated in Fig. 4b by the solid and dashed red lines, which is significantly different from the black and green lines (r 1 fm) obtained in the literature [6,13].
4. Monopole matrix element r 2 tr and transition radius R tr As important information on the spatial structure of the second 0 + state, the authors of Ref. [24] extracted, from the observed form factor, the monopole matrix element r 2 tr and the transition radius R tr defined as (cf. Eq. (5) in Ref. [24]) r n tr = Z r n ρ tr (r) dr (n = 2 and 4, Z = 2), which are obtained by a q → 0 expansion of the form factor with e iq·r → 1 − 1 6 q 2 r 2 + 1 120 q 4 r 4 + · · · for the 'monopole' density ρ tr (r) in Eq. (5). They determined the numerical values of r 2 tr and R tr , as indicated by the first line in Table I. We calculate r 2 tr and R tr in Eq. (14) explicitly using the red-solid-line density, Eq (8), with no central depression and the red-dashed-line density, Eq.(8) plus Eq. (11), with central depression.
We obtain, for the red-dashed-line density, where a 3 and b 3 are omitted in the case of the red-solid-line density. The numerical values of r 2 tr and R tr are listed in Table I. The two transition densities reproduce the experimental value; however, this is natural because the form factors of Eqs. (2) and (10) are constructed to simulate the behavior of the observed form factor (blue line) at q 2 ∼ 0. As expected, the central-depression structure is not reflected in r 2 tr and R tr (that is, the contributions of a 3 and b 3 are negligible in Eq. (15)).
To investigate the range of r that contributes most to r 2 tr and R tr , we introduce the cumulative monopole matrix element r 2 tr (r) and the cumulative transition density R tr (r) as a function of r by where r 2 tr (∞) = r 2 tr and R tr (∞) = R tr . The functions r 2 tr (r) and R tr (r) are shown in Figs. 5a and 5b, respectively. In both cases, the dominant contribution comes from the range of r 3 fm.
Interestingly, this part appears to be minor in the region of Fig. 4b for the transition density. We   understand that we must derive the transition density ρ tr (r) up to r ∼ 10 fm so that r 2 tr and R tr are calculated accurately.

Discussions
We have derived a possible 0 + 1 → 0 + 2 transition density ρ tr (r) for r 1 fm as in Fig. 4b by the solid red line except for the inner central-depression region, utilizing the information obtained from the newly observed high-precision transition form factor for 0 ≤ q 2 ≤ 5 fm −2 [24]. The shape of the transition density is significantly different from those obtained theoretically in literature [6,13].
We now discuss a problem calculating the energy of the second 0 + state. Figure 6 schematically illustrates the energy values obtained using the AV8 ′ +central 3N potential [6] and by the χEFT FIG. 6: Schematic illustration of the 0 + 2 energy obtained using the AV8 ′ +central 3N potential Ref. [6] and by the AV18+UIX and χEFT potentials [13] along with the threshold energies of the 3 H + p, 3 He + n, their averaged 3N + N, and p + p + n + n configurations [26]. and AV18+UIX potentials [13] along with the threshold energies of the 3 H + p, 3 He + n and their average 3N + N configurations [26]. Note that the observed 0 + 2 state is located between the 3 H + p and 3 He + n thresholds and only 0.01 MeV above the 3N + N threshold with a rather narrow width of Γ p = 0.50 MeV [26] for a S -wave resonance.
In Ref. [6] which took the isospin-formalism, the 0 + 2 , T = 0 state was obtained only 0.1 MeV below the observed level as a bound state measured from the 3N + N threshold. On the other hand, in Ref. [13], the energy of the 0 + 2 state obtained as a resonance with the two realistic interactions is ∼ 0.7 MeV above the observed 0 + 2 level and even above the 3 He + n threshold. As mentioned by Bacca et al. [13] and Epelbaum [25], it is possible that the improper theoretical resonance position affects the form factor result. As pointed by Horiuchi and Suzuki (cf. last paragraph of Sec. IIIB of Ref. [10]), the observed 0 + 2 state is considered a Feshbach resonance embedded in the 3 H + p continuum as a bound state with respect to the 3 He + n threshold.
In the opinion of the present author, a possible strategy to attack the α-particle monopole puzzle using fully realistic interactions is as follows: i) Solve the 0 + 2 state using the bound-state approximation to search interaction parameters that reproduce the 0 + 2 energy well, ii) calculate the transition density and form factor, and then iii) solve the 3 H + p scattering to derive the wave function and width of the Feshbach resonance as well as the final results for the transition density and form factor.
As mentioned in Sec. 1, authors of Ref. [24] claimed the following: the large difference between the calculated form factors does not stem from numerical methods but from the Hamiltonian.
However, the comparison between the methods was performed only for the calculation based on the bound-state approximation. The calculation of the 0 + 2 state as a resonance was performed using only the LIT approach. For example, it would be desirable to examine this problem by the comparison with the form factor results produced by using any explicit four-body 3 H+ p scattering calculation such as Ref. [27].