Electromagnetic reactions on light nuclei

Nuclear χEFT potentials are expressed in terms of multiple-pion exchange contributions, which describe the long- and intermediate-range components of the interaction, along with many-nucleons contact terms (CT) that encode short-range dynamics [31–33]. The chiral nuclear force involves a number of unknown LECs which need to be determined. LECs entering the NN chiral potential are fitted to the available NN scattering data, as it is done for conventional potentials. The chiral NN potential at LO (${{Q}^{\nu =0}}$) consists of the static OPE potential given in equation (3) plus two four-nucleon contact interaction terms involving two unknown LECs, C_S and C_T. The contact contribution of order $\nu =0$ is represented by the diagram illustrated in panel $({\rm a})$ of figure 2, and the complete chiral potential at LO ($\nu =0$) reads

$$\begin{eqnarray}&&{{v}^{\nu =0}}={{v}_{\pi }}+{{v}_{{\rm CT}}}=-\frac{g_{A}^{2}}{F_{\pi }^{2}}\;{{{\boldsymbol{ \tau }} }_{i}}\cdot {{{\boldsymbol{ \tau }} }_{j}}\;\frac{{{{\boldsymbol{ \sigma }} }_{i}}\cdot {\bf k}\;{{{\boldsymbol{ \sigma }} }_{j}}\cdot {\bf k}}{\omega _{k}^{2}}+{{C}_{S}}+{{C}_{T}}\;{{{\boldsymbol{ \sigma }} }_{i}}\cdot {{{\boldsymbol{ \sigma }} }_{j}}.\end{eqnarray} \tag{ 14 }$$

The NLO⁷ contribution scales as ${{Q}^{\nu =2}}$ (there are no $\nu =1$ contributions to the NN interaction) and, referring to figure 2, it includes the one-loop diagrams illustrated in panels $({\rm d})$–$({\rm i})$, plus seven additional contact terms, illustrated in panel $({\rm c})$. The latter involve two nucleon momenta and are written in terms of seven unknown LECs. At this order, there are also 3N contributions of one-pion range which, however, are seen to vanish (see, e.g., [28]). At NLO ($\nu =2$), all the spin-isospin operator structures necessary to describe the NN interaction have been generated, however one needs to incorporate higher order contributions in order to accurately fit NN data. The N2LO ($\nu =3$) nuclear force consists of additional one-loop terms contributing to the NN sector of the interaction, along with the first non-vanishing contribution to the 3N interaction [31–33]. We note that the leading 3N forces ($\nu =3$) are suppressed by two orders with respect to the leading NN potential ($\nu =0$). Currently, phase-shift equivalent chiral NN potentials include up to N3LO ($\nu =4$) corrections consisting of two-loop contributions and 15 additional contact terms associated with 15 unknown LECs to be fitted to NN scattering data. Chiral NN potentials up to N3LO ($\nu =4$) have been been developed by Epelbaum and collaborators in [94–96], and by Entem and Machleidt in [97, 98].

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So far, we discussed NN potentials that are constructed in the center-of-mass frame of the two nucleons. However, in [99] it has been shown that, by requiring Poincaré covariance of the theory, one is forced to introduce additional terms with fixed LECs, in order to describe the two-nucleon potential in reference frames other than the center-of-mass frame. These terms depend on the total momentum of a nucleon pair which can be set to be equal to zero in the center-of-mass frame of a two-nucleon system, but it is in general different from zero in systems with mass number $A\gt 2$, and the effects of these corrections have not yet been quantitatively studied within χEFT formulations [99].

The 3N forces [31–33] enter at N2LO ($\nu =3$) and involve two unknown LECs. Strong observables that can be used to pin them down are, for example, the binding energies of A = 3 and 4 nuclei. An alternative strategy, implemented by Gazit and collaborators in [100], is based on the observation that one of the strong LECs enters also the chiral contact two-nucleon weak current at N2LO ($\nu =2$) [44]. Therefore, experimental data of electroweak processes induced by this weak operator can also be used to fix the LEC. In [100], the 3N force at N2LO ($\nu =3$) is constrained so as to reproduce the binding energies of the A = 3 nuclei and the empirical value of the Gamow–Teller matrix element in triton β-decay. Of course, the values of the LECs entering the 3N potential are obtained in combination with the chiral NN interaction utilized in the calculations. 3N forces at N3LO ($\nu =4$) are presently being investigated by the theoretical community [101]. Chiral 3N forces, if included, are taken at N2LO in the results presented in this review.

Once the chiral potentials are constructed, standard ab initio techniques may be used to solve the many-body Schrödinger equation. Calculations of the energy spectrum of light nuclei that use NN interactions at N3LO and 3N forces at N2LO compare very well with the experimental data (see, e.g., [70, 102]). More recently, computational techniques developed in the field of lattice QCD have been implemented to solve the many-body problem of nuclear physics in combination with chiral potentials. The method is known as lattice EFT and currently it has been applied to structure studies of nuclei with mass number up to A = 28 [103–105], while selected e.m. transitions have been investigated in ¹²C [106] and ¹⁶O [107].

Few comments are now in order. Nuclear potentials and current operators present ultraviolet divergences which need to be removed by a proper regularization procedure. There are two kinds of regularizations which are implemented. The first one concerns the regularization of the loop integrals entering loop contributions (for example, those illustrated in diagrams $({\rm d})$–$({\rm i})$ of figure 2). This is accomplished via one of the usual schemes commonly adopted in quantum filed theory, e.g., dimensional regularization, and it is followed by a renormalization procedure in which divergences, isolated by the regularization scheme, are reabsorbed by the LECs of the theory. The second kind of regularization follows from the fact that nuclear operators, obtained by evaluating transition amplitudes, are then used in the Schrödinger equation to construct nuclear wave functions and to evaluate matrix elements. Chiral operators have a power law behavior at large momenta that needs to be cut off to avoid infinities. This is accomplished by multiplying nuclear operators with a short-range momentum cutoff. The latter is usually taken to be of the form ${\rm exp} [-{{(Q/\Lambda )}^{2\,\nu }}]$, where the choice of ν is contingent to the accuracy one aims to reach. That is, for small values of Q, where the cutoff behaves as $\sim 1-{{(Q/\Lambda )}^{2\,\nu }}+...$, spurious contributions $\propto {{(Q/\Lambda )}^{2\,\nu }}$ generated by the insertion of the cutoff need to be much smaller than the order at which the calculation is performed. In a theory that explicitly includes up to three-pion-exchange contributions, as it is done in the case of the chiral NN potential at N3LO ($\nu =4$), it is reasonable to set $\Lambda \sim 500$ MeV $\sim 3\;{{m}_{\pi }}$. This way, the cutoff eliminates short-range contributions of four-pion range and beyond, that are not explicitly included in the theory, but are rather subsumed in the LECs. We remark that the discussion on what is the proper renormalization scheme (and power counting) to be adopted is still open [31–33, 108–112]. We will not address this issue further, and simply keep in mind that calculations with chiral operators depend on an additional parameter, i.e., the momentum cutoff Λ defined above. For comprehensive reviews on χEFT nuclear forces we refer to the review articles by Epelbaum et al and by Machleidt and Entem given in [31, 32], and [33], respectively.

One of the great advantages of the χEFT formulation is that e.m. currents are naturally constructed with nuclear forces in a consistent way. Gauge invariance is one among the fundamental symmetries the theory is required to satisfy. Hence, e.m. fields are coupled to nuclear currents which satisfy the continuity equation, order by order, with chiral potentials. Chiral two-body currents ${\bf j}$ were first investigated within χEFT by Park, Min, and Rho in [45] and constructed up to include one-loop corrections by using covariant perturbation theory. In recent years, the two-body operators for both ρ and ${\bf j}$ have been derived within two different implementation of time-ordered perturbation theory up to include TPE corrections. The JLab–Pisa group [37, 46–48] used standard time-ordered perturbation theory, while the Bochum–Bonn group [49, 50] used the method of the unitary transformation, which was also utilized to construct the N3LO ($\nu =4$) two-body potential developed in [94–96]. Differences between these e.m. operators have been discussed at length in [37, 46–48, 50]. Here, we will qualitatively describe the hierarchy of the e.m. currents and charge operators that emerges from the chiral expansion, and refer to the aforementioned references for details and formal expressions of the operators. We point out that a proper renormalization of the e.m. OPE operators has been carried out only within the unitary transformation formalism (see [50]).

We start off with the e.m. current operator ${\bf j}$, whose contributions are diagrammatically listed in figure 3. In table 1, we also report the scaling in $e{{Q}^{\nu }}$ (where e is the electric charge brought in by the e.m. coupling) and the range of the operators at each order. Note that the LO term here scales as $e{{Q}^{\nu =-2}}$, which follows from direct application of the power counting [45] to the disconnected diagram illustrated in panel $({\rm a})$ of figure 3. In particular, the γNN vertex scales as the single nucleon e.m. current given in equation (8), that is $eQ$. In addition, disconnected contributions, such as those illustrated in panels $({\rm a})$ and $({\rm d})$ of this figure, involve an extra δ-function in the initial and final momenta of the spectator nucleon—that is $\delta ({\bf p}_{j}^{\prime }-{{{\bf p}}_{j}})$, and they are thus enhanced by a factor ${{Q}^{-3}}$ with respect to connected diagrams. Then, the power counting of diagram $({\rm a})$ is $eQ\times {{Q}^{-3}}=e{{Q}^{-2}}$. Other conventions for the LOʼs scaling have also been utilized in the literature. Ultimately, what matters is the suppression factor of a given operator $\propto e{{Q}^{\nu }}$ with respect to the LO term $\propto e{{Q}^{{{\nu }_{{\rm LO}}}}}$, that is $e{{Q}^{(\nu -{{\nu }_{{\rm LO}}})}}$, which is independent of the choice made for the LO term scaling. Translation from one notation to the other is easily achieved keeping that in mind⁸ .

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Table 1.  Scaling in $e{{Q}^{\nu }}$ up to $\nu =1$ and ranges of the chiral e.m. current and charge operators. The ${\bf j}$ operator at N4LO $(\nu =2)$ are have not been derived yet. The acronyms stand for OPE = one-pion-exchange, TPE = two-pion-exchange, CT = contact term, IA = impulse approximation, NR = non-relativistic, and RC = relativistic correction.

Operator	LO	NLO	N2LO	N3LO	N4LO
${\bf j}$	$\nu =-2$	$\nu =-1$	$\nu =0$	$\nu =1$
	IA(NR)	OPE	IA(RC)	OPE(LECs)
				TPE
				CT(LECs)
$\rho$	$\nu =-3$	$\nu =-2$	$\nu =-1$	$\nu =0$	$\nu =1$
	IA(NR)	—	IA(RC)	OPE	TPE

Referring to figure 3, the LO ($\nu =-2$) operator is represented by the diagram illustrated in panel $({\rm a})$ and corresponds to the standard non-relativistic IA current given in equation (8). A strict chiral expansion of the nuclear current operators does not guarantee the convergence of the calculated observables to the experimental data. One has to account for nucleonic structure effects via suited form factors. The latter can also be derived within χEFT, however, as it has been observed in [113–115] in the case of the deuteron charge form factor, the description of nuclear observables is limited by the difficulty of χEFT in describing nucleonic form factors for values of momentum transfer larger than $q\sim 1.5$ fm⁻¹. Therefore, it is customary also within χEFT formulations to account for nucleonic structure effects via form factors taken from fits to elastic electron scattering data on deuteron and proton. As discussed above, relativistic corrections to the LO one-body current generate a one-body operator which is suppressed by a ${{(Q/m)}^{2}}$ factor with respect to the LO one. This single-nucleon term, also denoted with IA(RC), is represented by the disconnected diagram illustrated in panel $({\rm d})$ and scales as $e{{Q}^{\nu =0}}$ (N2LO). Nuclear two-body effects appear at NLO ($\nu =-1$) with the OPE currents represented by the diagrams illustrated in panels $({\rm b})$ and $({\rm c})$. These are the well known seagull and pion-in-flight currents of equation (10). At N3LO ($\nu =1$), there are currents of one- and two-pion range as well as contact currents. In particular, pure TPE one-loop contributions are illustrated in panels $({\rm e})$–$({\rm i})$, while one-loop short-range currents are represented in panels $({\rm l})$–$({\rm o})$. One-loop corrections at N3LO ($\nu =1$)—that is those illustrated in panels $({\rm e})$–$({\rm i})$ and $({\rm l})$–$({\rm o})$—and OPE currents at NLO ($\nu =-1$) lead to purely isovector operators.

We note that, so far, the e.m. operators involve known LECs, namely the axial coupling constant g_A and the pion decay amplitude ${{F}_{\pi }}$. Unknown LECs enter the N3LO ($\nu =1$) tree-level and contact currents illustrated in panels $({\rm j})$ and $({\rm k})$, respectively. We distinguish between 'minimal' and 'non-minimal' contact currents. The former are linked to the contact potential at NLO ($\nu =2$)—schematically illustrated in panel $({\rm c})$ of figure 2—via the continuity equation. That is, the diagram of panel $({\rm j})$ in figure 3 is determined from the nucleon vertex in panel $({\rm c})$ of figure 2 via minimal substitution in the nucleon momentum, ${\bf p}\to {\bf p}-ie{\bf A}$, where ${\bf A}$ is the vector photon field. Therefore 'minimal' currents involve the same LECs that enter the contact NN potential at NLO ($\nu =2$), and these can be constrained by fitting $np$ and $pp$ elastic scattering data and the deuteron binding energy. Additional contact currents at N3LO ($\nu =1$) are of 'non-minimal' nature and follow from the coupling of the e.m. field tensor ${{F}_{\mu \nu }}=({{\partial }_{\mu }}{{A}_{\nu }}-{{\partial }_{\nu }}{{A}_{\mu }})$. In particular, there are two 'non-minimal' contact currents, one is isoscalar and the other one isovector. These contact currents involve unknown LECs, which need to be fixed so as to reproduce e.m. observables.

The tree-level current at N3LO ($\nu =1$), represented by the diagram illustrated in panel $({\rm j}),$ results from a $\gamma \pi N$ coupling of order $e{{Q}^{2}}$ (indicated by a solid circle) and involves three unknown LECs. Two of them multiply isovector operators and the remaining one multiplies an isoscalar operator. In the present χEFT formulation, with pions and nucleons as relevant degrees of freedom, LECs subsume interactions involving heavy-mesons or nucleon resonances integrated out from the theory. In fact, the isovector part of the tree-level current at N3LO ($\nu =1$) has the same structure as the current involving the $N\Delta$-excitation that is illustrated in panel $({\rm a})$ of figure 1. The latter is known to provide the major contribution to the 'model dependent' conventional currents [23]. Similarly, the isoscalar component of the tree-level current at N3LO ($\nu =1$) simulates the $\rho \pi \gamma$ transition current [23], illustrated in panel $({\rm b})$ of figure 1. A strategy often implemented to reduce the number of unknown LECs is to 'saturate' them with the couplings entering the Δ-resonance and/or the $\rho \pi \gamma$ transition currents illustrated in panels $({\rm a})$ and $({\rm b})$ of figure 1, respectively (see [45]). Once the five LECs listed so far, two associated with contact currents and three belonging to the tree-level current at N3LO ($\nu =1$), are determined (via saturation and/or via fits to e.m. observables) the current operator can then be used predictively.

As an example, we briefly discuss calculations [48] of the $n+p\to d+\gamma$ cross section at thermal neutron energy $\sigma _{{\rm np}}^{\gamma }$ (left panel of figure 4) and the isovector combination of the A = 3 nucleiʼs magnetic moments ${{\mu }^{{\rm V}}}$(³H/³He) (right panel of figure 4). These observables are induced by the magnetic dipole operator associated with the e.m. nuclear currents. In these calculations, 'minimal' LECs associated with the contact currents at N3LO—see panel $({\rm k})$ of figure 3—have been fixed to NN scattering data [33, 98]. The saturation of the Δ-resonance has been exploited to fix the two LECs entering the isovector component of the tree-level current at N3LO ($\nu =1$)—see panel $({\rm j})$ of figure 3, while the two isoscalar LECs, entering the tree-level and contact currents at N3LO ($\nu =1$)—see panels $({\rm j})$ and $({\rm k})$ of figure 3, respectively—are determined so as to reproduce the deuteron as well as the isoscalar combination of the trinucleon magnetic moments. Then, only one unknown isovector LEC is left. For the calculation in the left (right) panel this LEC is obtained by reproducing the experimental value of the isovector combination of the trinucleon magnetic moments ($n+p\to d+\gamma$ cross section at thermal neutron energy) for each value of the momentum cutoff $\Lambda =500,600$ MeV. The results are obtained including cumulatively contributions at LO ($\nu =-2$), NLO ($\nu =-1$), N2LO ($\nu =0$), and at N3LO ($\nu =1$). The cumulative contribution at N3LO ($\nu =1$) is separated into the results labeled 'N3LO(no-LECs)' and those labeled 'N3LO(full)'. The former include only contributions of minimal nature. That is the loop-corrections illustrated in panels $({\rm e})$–$({\rm i})$ and $({\rm l})$–$({\rm o})$ of figure 3, and the 'minimal' contact currents represented in panel $(k)$ of the same figure. The magenta band labeled 'N3LO(full)' is obtained using the full e.m. current operator up to N3LO ($\nu =1$). The thickness of the bands represents variations due to the use of different nuclear Hamiltonians to generate the nuclear wave functions, namely the AV18+UIX or the NN(N3LO)+3N(N2LO), where the two– and three–body chiral interactions are form [98] and [100], respectively. The predictions indicated by the magenta bands are within 1% for $\sigma _{{\rm np}}^{\gamma }$ and 3% for ${{\mu }^{{\rm V}}}$(³H/³He) of the experimental values [48], which are represented by the black bands in the figure. We note that the leading two-body correction at NLO ($\nu =-1$) given by the long-ranged OPE seagull and pion-in-flight currents enhances the LO ($\nu =0$) prediction (or IA prediction) for $\sigma _{{\rm np}}^{\gamma }$ [${{\mu }^{{\rm V}}}$(³H/³He)] by ∼4% (∼9%). We also note that the N3LO ($\nu =1$) contributions due to the tree-level and 'non-minimal' contact currents are 'large' and crucial to bring the theory in agreement with the experimental data. This is an indication that explicit inclusion of the Δ-resonance as a fundamental degree of freedom of the theory should improve the convergence of the chiral expansion.

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An important check on the chiral e.m. currents is to verify that they satisfy the continuity equation with the chiral potential order by order in the chiral expansion. Of course, up to NLO ($\nu =-1$) the current is conserved since it just involves the non-relativistic IA and OPE seagull and pion-in-flight operators of equations (8) and (10), respectively, which are known to satisfy the continuity equation with kinetic term and OPE potential entering the many-body nuclear Hamiltonian. A simple counting of the powers of momentum entering the lhs and rhs of equation (13)⁹ , indicates that 'minimal' N3LO ($\nu =1$) e.m. currents, illustrated in figure 3, must satisfy the continuity equation with the NN potential of order Q², illustrated in figure 2. This statement has been explicitly verified in [37, 49].

A natural question to ask is whether at the order we are considering, that is N3LO $(\nu =1),$ there are three-body currents. Above, we mentioned that 3N forces appear for the first time at NLO ($\nu =2$), however they are seen to vanish. Therefore, in order to satisfy the continuity equation, three-body currents at N3LO ($\nu =1$) must either vanish, or be transverse to the photon field. They are, in fact, found to vanish [116], thus the current operator up to N3LO ($\nu =1$) includes only one- and two-body terms. Leading three-body e.m. currents appear at N4LO ($\nu =2$) [45], and they have not been derived yet.

The e.m. charge operator has been first investigated from a χEFT perspective by Walzl et al in [113] and Phillips in [114, 115]. Phillips derived it up to N3LO (that is, up to $\nu =0$ in our counting)¹⁰ , while the first derivation of one-loop corrections, entering at N4LO ($\nu =1$), has been carried out in [49] by Kölling et al using the unitary transformation method. A time-ordered perturbation theory calculation has subsequently appeared in [47]. Within this framework, the construction of the charge operator up to one-loop necessarily requires the study of non-static contributions to the chiral OPE and TPE potentials. These corrections that go beyond the static limit are not uniquely determined off-the-energy-shell, therefore the specific form of the N3LO ($\nu =0$) and N4LO ($\nu =1$) corrections of one- and two-pion range are found to depend on the off-the-energy-shell prescriptions adopted for non-static terms in the OPE and TPE potentials, respectively [47]. The ambiguity in the non-static potential and charge operators is of no consequence, since different forms are related to each other by a unitary transformation [47], a finding that was first unraveled by Friar [117] for non-static potentials and charge operators of one-pion range.

In what follows, we refer to figure 5 and list the various contributions to the charge operator up to corrections of order $e{{Q}^{(\nu =1)}}$. The LO ($\nu =-3$) contribution is represented by the diagram illustrated in panel $({\rm a})$ and corresponds to the non-relativistic IA operator given in equation (6). In principle, at NLO ($\nu =-2$) there are contributions of one-pion range which, however, are seen to vanish in the static limit. At N2LO ($\nu =-1$) there is the one-body operator illustrated in panel $(b)$ corresponding to the relativistic correction to the leading IA operator [or IA(RC)]. Two-body contributions appear at N3LO ($\nu =0$) and are illustrated in panels $({\rm c})$, $({\rm d})$ and $({\rm e})$. In particular, the contribution of panel $({\rm c})$ involves a $\gamma \pi$ NN vertex—denoted by a solid circle—obtained from a chiral Lagrangian of order $eQ$ [114, 115], while operators associated with the diagrams of panels $({\rm d})$ and $({\rm e})$ result from accounting for non-static corrections, and are found to be suppressed with respect to the contribution of panel $({\rm c})$ [48].An operator that has the same structure as the operator resulting from the evaluation of the transition amplitude associated with the diagram of panel $({\rm c})$, has been reported in the late eighties in the review article by Riska [20], however its derivation within a χEFT context has been carried out in recent years by Phillips [114]. Finally, one-loop corrections enter at N4LO ($\nu =1$) and are represented by the diagrams of panels $({\rm f})$–$({\rm o})$. Of course, at N3LO ($\nu =0$) and N4LO ($\nu =1$) one should also include relativistic corrections to the OPE at NLO ($\nu =-2$) and IA at N2LO ($\nu =-1$), respectively. Note that, due to charge conservation, contributions beyond the IA or LO term vanish at $|{\bf q}|=0$.

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We emphasize that the structure of the charge operator is quite different from that of the current operator. To begin with, the e.m. current operator at LO ($\nu =-2$) is suppressed by a factor of Q with respect to the LO ($\nu =-3$) charge operator. Secondly, two-body effects of one-pion range enter the charge operator at N3LO ($\nu =0$). They instead constitute the NLO ($\nu =-1$) and therefore the major correction to the IA picture in the case of the e.m. current operator. In addition, OPE corrections to the charge operator are $1/m$ terms that vanish in the static limit of $m\to \infty$. Finally, the charge operator does not involve unknown LECs.

Electron-deuteron elastic scattering reactions have been intensively investigated both from an experimental and a theoretical point of view. There exists a vast literature on this topic, for which we refer to the review articles of [8, 121, 134]. Conventional calculations based on realistic NN interactions and consistent e.m. two-body MEC provide a satisfactory description of the available experimental data [23, 76, 135, 136]. Currently, there are also quite a few calculations of deuteron e.m. properties that are based on χEFT formulations [48, 113–115, 137–140]. Within the χEFT framework, a general good agreement between the calculated and experimental deuteron form factors is observed, provided that the e.m. structure of the nucleons is accounted for via suited nucleonic form factors.

Traditionally, the charge and magnetic spatial distributions of the deuteron are studied in terms of the charge (${{G}_{C}}$), quadrupole (${{G}_{{\rm Q}}}$), and magnetic (G_M) form factors, and they are related to the Coulomb, T^C_J, and magnetic transverse, T^M_J, multipole operators discussed in section 3.1 via [23]

$$\begin{eqnarray}&&\sqrt{\frac{4\pi }{3}}T_{0}^{C}(q)=(1+\eta ){{G}_{C}}(q),\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\sqrt{\frac{4\pi }{3}}T_{2}^{C}(q)=\frac{2\sqrt{2}}{3}\eta (1+\eta ){{G}_{Q}}(q),\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&-\;i\;\sqrt{\frac{4\pi }{3}}T_{1}^{M}(Q)=\frac{2}{\sqrt{3}}\sqrt{\eta (1+\eta )}{{G}_{C}}(q),\end{eqnarray} \tag{ 29 }$$

where $\eta ={{(q/2\;{{M}_{d}})}^{2}}$ and M_d is the deuteron mass. These form factors are normalized as

$$\begin{eqnarray}&&{{G}_{C}}(0)=1,\;{{G}_{M}}(0)=({{M}_{d}}/{{m}_{N}}){{\mu }_{d}},\;{{G}_{Q}}(0)=M_{d}^{2}\;{{Q}_{d}},\end{eqnarray} \tag{ 30 }$$

where μ_d and Q_d are the deuteron magnetic moment (in units of ${{\mu }_{N}}$) and quadrupole moment, respectively. The deuteron form factors are extracted from the measured structure functions A and B, and tensor polarization T₂₀. The expressions relating these measured quantities with the form factors can be found in [23].

In figure 10, the calculated deuteron charge and quadrupole form factors from [48] are compared with the experimental data in panels $({\rm a})$ and $({\rm b})$, respectively. In these figures, results from both a χEFT and a hybrid calculation based on the χEFT charge operator at LO ($\nu =-3$) and N3LO ($\nu =0$) are shown. The cyan dotted band (blue dashed line) and magenta hatched (red solid) band are obtained using deuteron wave functions from the N3LO ($\nu =4$) chiral NN potential by Entem and Machleidt [98] (AV18 potential), in combination with the charge operator at LO ($\nu =-3$) and N3LO ($\nu =0$), respectively. The thickness of the bands indicates the sensitivity of the results with respect to two values of the regularization cutoff corresponding to $\Lambda =500$ and 600 MeV (used consistently in the chiral NN potential and e.m. currents, for the χEFT calculation). The IA and IA(RC) operators at LO and N2LO—see panels $({\rm a})$ and $({\rm b})$ of figure 5—are cutoff independent, therefore hybrid results at LO are represented by a line, while the thickness of the cyan dotted band associated with the χEFT results at LO is due to variations in the cutoff utilized to regularize the chiral NN potential.

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The charge and quadrupole form factors are determined by the isoscalar component of the charge operator. Therefore, the N4LO ($\nu =1$) loop-corrections, illustrated in panels $({\rm f})$–$({\rm o})$ of figure 5, do not contribute to these observables, as they lead to isovector operator structures.

We emphasize that the results discussed here include e.m. nucleonic form factors—that is the $G_{E/M}^{S/V}$ introduced in equations (7) and (9). As noted in [113], a simple chiral expansion of the charge operator provides poor agreement with the experimental data for $q\gtrsim 1.5$ fm⁻¹. The inclusion of e.m. nucleonic form factors taken from experimental data ensures a reasonable fall-off of the calculated deuteron form factors at increasing values of q. In doing so, a very good agreement between the full theory and experiment is observed up to values of $q\sim 3$ fm⁻¹ in the case of the charge form factor, and of $q\sim 6$ fm⁻¹ in the case of the quadrupole form factor. Hybrid and χEFT results are in very good agreement for low values of momentum transfered ($q\lesssim 3$ fm⁻¹), however, as q increases these observables become sensitive to the wave functions utilized in the computations, which display some differences (see, e.g., figure 16 of [33]). This can be appreciated by comparing the calculations based on the N3LO ($\nu =0$) e.m. charge operators, with the chiral N3LO ($\nu =4$) [98] (magenta hatched band) and the AV18 [76] (solid red band) NN potentials. The diffraction region of G_C is better reproduced by the AV18 calculation, while the χEFT results for G_Q are in better agreement with the experimental data for $q\gtrsim 3$ fm⁻¹.

In the bottom panels of figure 10, we compare the full χEFT results by Piarulli et al [48] (magenta hatched band) with those obtained by Phillips in [114, 115] (purple solid band). The latter employ the chiral N2LO ($\nu =3$) NN interaction from [96] and the charge operator at N3LO ($\nu =0$).¹⁵ These calculations are in good agreement at low values of momentum transfer and exhibit a similar cutoff dependence. However, as q increases the predicted form factors exhibit a more noticeable sensitivity to the particular NN potential utilized to solve the Schrödinger equation (a finding that confirms the conclusions of previous studies by Phillips [114, 115]).

We also report that the χEFT (AV18) calculated deuteron quadrupole moment, Q_d, inclusive of MEC, is found to be within 1% (2%) of the experimental value, and to show a $\sim 1\%$ (negligible) variation in going from $\Lambda =500$ to 600 MeV [48].

We now turn our attention to the deuteron magnetic form factor. Because the deuteron is isoscalar, only isoscalar components in the chiral e.m. current contribute to its magnetic form factor. Therefore, OPE and TPE currents at NLO ($\nu =-1$)—see panels $({\rm b})$ and $({\rm c})$ of figure 3—and N3LO ($\nu =1$)—see panels (e)–(i) and (l)–(o) of the same figure—do not contribute to this observable. The isoscalar e.m. current operator involves only two e.m. LECs both entering at N3LO ($\nu =1$); one accompanies a short-ranged contact interaction—see panel $(k)$ of figure 3—and the other one multiplies the isoscalar part of the tree-level current illustrated in panel $({\rm j})$ of figure 3. We note that isoscalar contributions of one-pion range are suppressed by three powers of Q with respect to the LO ($\nu =-2$) or IA term.

In panel $(a)$ of figure 11, we show the calculated deuteron magnetic form factor from [48], with the same color code as in figure 10. Here, the e.m. LECs entering the isoscalar current at N3LO ($\nu =1$) have been constrained so as to reproduce the deuteron magnetic moment, μ_d, as well as the isoscalar combination of the trinucleon magnetic moments, ${{\mu }^{S}}$, defined in equation (31). Therefore, there is no prediction for the deuteron static magnetic moment with this chiral e.m. current. LECs of minimal nature that multiply contact operators at N3LO ($\nu =1$) have been taken from the phase-shift analysis carried out in [33, 98]. We observe a rather good agreement with the experimental data for values of $q\simeq 2$ fm⁻¹. However, at larger values of momentum transferred the hybrid and χEFT results are quite different. In particular, calculations based on the AV18 interaction and e.m. operators at LO ($\nu =-2$) do not show the diffraction pattern observed in the χEFT results at LO ($\nu =-2$). This is due to the different deuteron wave functions generated by the two corresponding NN potentials. We remark that the N3LO ($\nu =1$) e.m. current is conserved with the NN potential at NLO ($\nu =2$), therefore, the χEFT calculation based on the NN potential at N3LO ($\nu =4$) would require an e.m. current of order $\nu =3$ in order to strictly fulfill the continuity equation.

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In panel $(b)$ of figure 11, we compare the results by Piarulli et al [48] (hatched magenta band) with the fully consistent χEFT calculations by Kölling et al [140] (solid purple band) based on the chiral NN potential at N2LO ($\nu =3$) [96] and chiral e.m. currents at N3LO ($\nu =1$) [49, 50], both derived within the unitary transformation method. In that work, the short-ranged e.m. LEC—see panel (k) of figure 3—is fixed to the deuteron magnetic moment, while the isoscalar OPE e.m. LEC—see panel (j) of figure 3—is fitted to the measured B structure function [140]. The theoretical results for the deuteron magnetic form factor are in very good agreement with each other and with the experimental data for values of momentum transferred $q\simeq 3$ fm⁻¹, and present a comparable cutoff dependence.

Finally, we report on a very recent evaluation of deuteron G_C, G_Q, and G_M form factors carried out in a fully consistent and cutoff independent χEFT formulation [165]. The calculation, based on the renormalizable approach of [166], uses e.m. charge, e.m. currents, and NN operators at LO, and predicts deuteron form factors which are in good agreement with the experimental data, provided that the nucleonic e.m. structure is folded in the evaluation via appropriate nucleonic form factors.

Currently, the only complete study on elastic form factors of nuclei with mass number A = 3 that uses χEFT potentials at N3LO ($\nu =4$), N3LO ($\nu =1$) e.m. currents and N4LO ($\nu =1$) charge operators has been carried out by Piarulli et al in [48]. In that work, the calculations of the relevant matrix elements have been performed using wave functions obtained with the HH expansion method carried out in momentum space. The e.m. current and charge operators of [37, 46–48] have been sandwiched in between wave functions obtained from the NN(N3LO)+3N(N2LO) nuclear Hamiltonian, where the two-body interaction is from [98], while the 3N force has been taken at N2LO ($\nu =3$) with corresponding LECs constrained as in [100].

In figure 12, the charge form factors (F_C) of ³He and ³H are represented in the upper panels, while the bottom panels show the isoscalar (F_C^S) and isovector (F_C^V) combinations defined as

$$\begin{eqnarray}&&F_{C}^{S,V}(q)=\frac{1}{2}[2{{F}_{C}}(q{{,}^{3}}{\rm He})\pm {{F}_{C}}(q{{,}^{3}}{\rm H})],\end{eqnarray} \tag{ 31 }$$

and normalized, respectively, to 3/2 and 1/2 at q = 0. The χEFT charge operator, schematically illustrated in figure 5, does not involve unknown LECs up to N4LO. Results obtained with the charge operator at LO (N4LO) are represented by the cyan dotted (magenta hatched) band, and their thickness represent the spread in the calculations corresponding to a variation of the regularization cutoff (Λ = 500–600 MeV). The latter has been varied consistently in both the potentials and the currents. Hybrid results, obtained with wave functions from the AV18+UIX nuclear Hamiltonian, are also given in figure 12. In particular, those based on the charge operator at LO (N4LO) are illustrated by the blue dashed line (red solid band). Note that one-body operators at LO are cutoff independent, which is why hybrid results at LO are given by a line. The calculated charge form factors are in excellent agreement with the experimental data for values of $q\lt 3$ fm⁻¹. However, the positions of the zeros, as well as those of the maxima at $q\sim 4$ fm⁻¹, are not well reproduced by the theory. The results displayed in the bottom panels of figure 12 suggest that larger (in magnitude) two-body isovector contributions to the e.m. charge operator are needed to reproduce the first zero in the isovector form factor, F_C^V.

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The trinucleon magnetic form factors are given in figure 13. Minimal LECs entering the N3LO contact currents—see panel (k) of figure 3—have been taken from fits to NN scattering data [33, 98]. The LECs entering the isovector component of the χEFT e.m. current at N3LO, illustrated in panel $({\rm j})$ of figure 3, have been saturated by the Δ-isobar. The LEC associated with the isoscalar component of the same current, along with the LEC associated with the isoscalar N3LO contact current of non-minimal nature (see diagram in panel $({\rm k})$ of figure 3) have been determined by reproducing the deuteron magnetic moment as well as the isoscalar combination of the trinucleon magnetic moments, ${{\mu }_{S}}=[\mu {{(}^{3}}{\rm He})+\mu {{(}^{3}}{\rm H})]/2$. The remaining LEC, which multiplies an isovector non-minimal contact current at N3LO (see panel $({\rm k})$ of figure 3) is fixed so as to reproduce the isovector combination of the trinucleon magnetic moments, ${{\mu }_{V}}=[\mu {{(}^{3}}{\rm He})-\mu {{(}^{3}}{\rm H})]/2$. As before, in figure 13, the top panels show the ³He and ³H magnetic form factors, while the bottom ones show the isoscalar(F_M^S) and isovector (F_M^V) combinations given by

$$\begin{eqnarray}&&F_{M}^{S,V}(q)=\frac{1}{2}[\mu {{(}^{3}}{\rm He}){{F}_{M}}(q{{,}^{3}}{\rm He})\pm \mu {{(}^{3}}{\rm H}){{F}_{M}}(q{{,}^{3}}{\rm H})],\end{eqnarray} \tag{ 32 }$$

and normalized, respectively, to ${{\mu }_{S}}$ and ${{\mu }_{V}}$ at q = 0. In figure 13, the color code is as in figure 12, the only difference being that the vector e.m. currents include up to N3LO ($\nu =1$) corrections. As it is well know from studies based on the conventional approach (see [23]), two-body e.m. currents are crucial to improve the agreement between the observed positions of the zeros and the predicted ones at LO (or IA). Despite the excellent agreement between theory and experiment for $q\leqslant 2$ fm⁻¹, the theory underpredicts the data at higher momentum transfers, while the zeros are found at lower values of q than observed. The theoretical description of the first diffraction region is still incomplete. This is a finding that confirms the conclusions of previous studies based on the conventional approach [21, 23, 90, 168]. It is interesting to point out that, in [48], different LEC parameterizations have been investigated. In particular, it has been found that if one fixes the isovector LEC to the $np$ radiative capture cross section at thermal neutron energies, as opposed to ${{\mu }_{V}}$, the χEFT calculations of the magnetic form factors would lead to significantly better agreement with data over the whole range of momentum transfers, while overestimating the observed ${{\mu }_{V}}$ by $\simeq 3$% (see figure 4 and associated discussion).

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Ab initio calculations of the ⁴He elastic charge form factor have been performed by Schiavilla and collaborators [169] and reported in the review article of [23]. Since then, improved nuclear wave functions from conventional nuclear interactions, as well as calculations from chiral potentials, have appeared in the literature.

Results from Viviani et al [170] are presented in figure 14 along with the experimental data by Frosch et al [171]. Wave functions have been obtained using the HH method with the AV18+UIX nuclear Hamiltonian and consistent conventional MEC. The calculation in IA (blue dashed line) deviates from the experimental data for values of momenta larger than $q\sim 2$ fm⁻¹, while the agreement with the experiment is restored once MEC are included (red solid line). The effect of two-body currents becomes more pronounced as q increases to larger values. In particular, two-body corrections lead to a very different diffraction minimum than that observed in the IA results, and in agreement with experiment.

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In [172], an IA calculation with the AV18+UIX Hamiltonian has been performed using the EIHH method, a different formulation of hyperspherical harmonics. A perfect agreement between results from these calculations (not shown in figure 14, but displayed in [172]) and the IA results by Viviani et al [170] has been obtained, indicating that a high level of accuracy has been reached in benchmarking few-body nuclei calculations, especially those of ground-state properties.

Also shown in figure 14 is an IA (or LO) calculation (cyan band) from [172] obtained from χEFT potentials within the EIHH method. The chiral Hamiltonians have been taken to consist of an NN force at N3LO ($\nu =4$) [98] and a 3N force at N2LO ($\nu =3$), with LECs parameterized as in [173] (choosing one LEC of natural value and fitting the other so as to reproduce the binding energy in A = 3 nuclei) and as in [100] (fitting on the triton binding energy and empirical Gamow–Teller matrix element) at fixed cutoff $\Lambda =500$ MeV. The curves corresponding to the two above mentioned chiral Hamiltonians overlap in the figure and give rise to the thin band. At very low momentum, the conventional and χEFT approaches agree with each other. This is not very surprising, because they both use the same charge operator. Furthermore, conventional and chiral Hamiltonians give very similar results for the ⁴He radius, which determines the shape of the elastic form factor at low-q. In fact, the point-proton radius is 1.432(2) fm [174] (1.464(2) fm [172]) when the AV18+UIX (NN(N3LO)+3N(N2LO)) nuclear Hamiltonian is used. These calculated values are both close to the charged radius measured from elastic electron scattering, once the finite proton and neutron size are subtracted, yielding to $1.463(6)$ fm (see also [175] for more details).

Finally, we remind that new JLab data have been recently published in [176], where both experiment and conventional theory have been extended up to higher values of momentum transfer.

There are few ab initio calculations of elastic form factors for $A\gt 4$ nuclear systems. Elastic charge and magnetic form factors of ⁶Li have been calculated in [177] from VMC nuclear wave functions obtained using the AV18+UIX nuclear Hamiltonian, and conventional MEC. In particular, the charge form factor has been found to be in excellent agreement with the experimental data, with conventional two-body corrections from the π-like charge operator improving on the IA results. The calculated transverse form factor, instead, has been found to deviate from the experimental data beyond values of momentum transfer $q\sim 1$ fm⁻¹. Appreciable MEC effects for this isoscalar observable have been found for values of momentum transferred larger than 3 fm⁻¹, that is beyond the range of the available experimental data [23, 177]. The discrepancy with the experimental data has been attributed to a possible insufficient accuracy of the VMC nuclear wave functions, which could be resolved using improved GFMC evolutions [23, 177].

A comprehensive and computational demanding GFMC calculation of the ¹²C elastic charge form factor has recently appeared in the literature [24]. In that work, Lovato et al have used the AV18+IL7 nuclear Hamiltonian and conventional MEC. The calculated ¹²C values of the ground-state energy and rms charge radius obtained with this Hamiltonian are -93.3(4) MeV and 2.46(2) fm, respectively, to be compared with the experimental values of −92.16 MeV and 2.471(5) fm. In figure 15, the calculated ¹²C charge form factor [24] is shown along with the experimental data from [178]. Here, the one-body charge operator includes relativistic corrections (RC) and the calculated form factor obtained from it is indicated by the blue circles, which are in very good agreement with the experimental data for values of momentum transferred $q\lesssim 3$ fm⁻¹. Two-body charge operators used in the calculations include π-like and ρ-like exchange terms, as well as a $\rho \pi \gamma$ transition current (see section 2.1 for more details on conventional MEC operators). The full calculation with MEC (red triangles) provides an improved description of the data in the $q\geqslant 3$ fm⁻¹ region.

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Since the review by Carlson and Schiavilla [23], thanks to the tremendous advancement of many-body computational techniques, accurate nuclear wave functions are now available for up to $A\sim 20$ nuclear systems. This opens up the possibility of studying the charge and magnetic spatial distributions of more complex nuclear systems. In addition, chiral potentials and e.m. currents are now available and in the future studies, such as those of [24, 177], can be performed with χEFT and extended to other nuclei. Effects of two-body e.m. currents have been proved to be significant and crucial for an accurate theoretical description of the available experimental data. It would also be interesting to study their contributions in neutron-rich nuclei that display halo features, such as, e.g., the ⁶He nucleus. Electron scattering experiments on unstable nuclei have not been carried out yet. However, plans to measure charge distributions of neutron-rich nuclei have been presented at both European (FAIR) and Japanese (RIKEN) facilities. Such electron scattering reactions on unstable nuclei, to be performed in storage rings, can potentially reveal new interesting nuclear structure properties, for which the community can provide theoretical guidance due to the high level of reliability that is being reached by ab initio calculations.

In the case of A = 2 and 3 nuclei, the final states $|{{\Psi }_{f}}\rangle$ can be exactly evaluated with computational few-body techniques also in the continuum, thus the longitudinal and transverse response functions, R_L and R_T of equation (34), are obtained by summing over all the final states. Alternatively, when specific break-up channels are selected, exclusive processes can be studied.

Results for the deuteron obtained within the conventional approach have been already discussed in earlier review articles (see, e.g., [23, 180]). Recently, Yang and Phillips performed a χEFT calculation of the deuteron longitudinal response function in [181]. They have employed N2LO ($\nu =3$) wave functions from χEFT power counting and their subtractive renormalization method [182]. By using the LO¹⁷ charge operator they obtained results which, at low energy and low momenta, are in agreement with conventional calculations by Arenhövel et al carried out in [183] using the Bonn potential [184].

The inelastic responses of ³H and ³He with conventional potentials and currents in IA have been discussed in the review by Carlson and Schiavilla [23]. Conventional two-body currents have been recently included in the studies of [185, 186], and reviewed by Golak et al in [168], while Δ-isobar excitations below pion threshold have been studied in [187, 188]. For the inelastic electron scattering of the three-body nuclei not much has been done using the χEFT approach.

From the available calculations of A = 2 and 3 nuclei, it is generally observed that the effect of two-body operators is much larger on the transverse response, R_T, than in the longitudinal one, R_L. This can be easily understood in the χEFT language. In fact, two-body corrections appear at NLO and at N3LO for the current and charge operator, respectively (see table 1).

Because the longitudinal response R_L is not very sensitive to two-body operators, for $A\geqslant 3$ nuclei one can focus on studying the sensitivity of this observable to 3N forces, while keeping the charge operator in IA. The three-body nuclei have been studied for example in [191] employing the LIT method. Conventional 3N forces were found, e.g., to reduce R_L in the quasi-elastic peak by 5–10% in the momentum transfer regime between 174 MeV and about 400 MeV. However, 3N forces can also have much larger effects. In figure 16(a), the threshold behavior of R_L is shown in the case of ³He at q = 174 MeV: 3N forces (red solid curve with the AV18+UIX potential) lead to a quenching of the response function with respect to the calculation without 3N forces (blue dashed curve with the AV18 potential) and they were proven to be necessary to obtain a satisfactory description of the experiment. In figure 16(c), a comparison with another calculation by Viviani et al [186] is shown, where the same potentials were used and charge operators included MEC (green dots connected by red line). In [186], it was found that MEC have a negligible effect in this kinematical region, thus, the difference between the two calculations with the AV18+UIX force is likely coming from the different numerical methods used. It is worth noticing that such theoretical difference is much smaller than the experimental error. Furthermore, an even better agreement between the LIT calculations by Efros et al [191] and the variational calculations by Viviani et al [186] is found for larger momentum transfers, as shown in [191].

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In case of the transverse response function, R_T, it is generally observed that two-body currents have a substantial contribution on the total strength. For the three-body nuclei, R_T was calculated, for example, in [168, 186, 192] with the AV18+UIX force and MEC and in [193] with the Bonn potential [184], and the Tucson-Melbourne [194] 3N force and corresponding MEC. In the latter calculations, it has been shown that, for momentum transfer values of 174 MeV and 400 MeV, the two-body currents enhance the quasi-elastic peak by about 10% and 6%, respectively. The largest MEC effects have been found away from the quasi-elastic peak and particularly close to threshold, where two-body currents can enhance the strength by up to 200%. Generally, exchange currents contributions relative to the IA are enhanced when the difference $|\omega -{{\omega }_{q.e.}}|$ is larger. This is not unexpected as the quasi-elastic peaks correspond to the maximum contributions of the one-body current.

In figures 16(b) and (d), we show the case of the ³He transverse response at q = 487 MeV and discuss the threshold results obtained in calculations that utilized the AV18+UIX Hamiltonian. In panel (b), calculations by Leidemann et al from [192] obtained with the LIT method point to the importance of MEC. The curve including MEC (red solid) leads to a considerably improved agreement with experimental data with respect to the IA calculation (blue dashed curve). In figure 16(d), results by Leidemann et al from [192] are compared to those by Viviani et al from [186] (green dots connected by red line) and by Golak et al from [168] (dash-dotted curve). Besides using the same Hamiltonian, those calculations adopt slightly different implementations of the current operator. Calculations by Leidemann et al include relativistic corrections to the one-body operator and use the method devised by Arenhövel and Schwamb [195] in coordinate space to derive the MEC. Calculations by Golak et al do not include Coulomb interactions in the final state, do not include relativistic corrections to the current operator and implement MEC obtained according to the momentum-space method of Riska [196] (both methods for the MEC derivation lead to the same MEC contributions). Calculations by Viviani et al instead employ more sophisticated MEC, which also include contribution from the Δ-excitation. One can observe that all the three calculations nicely agree with data and with each other, proving that the dynamic of the few-body system is well under control.

At momentum transfer values of the order of 500 MeV and higher, relativity starts to play a role. Relativistic effects in A = 3 due to the frame dependence were studied, e.g., in [197, 198].

While inclusive processes can help planning for future exclusive experiments, exclusive reactions are much richer in the information they provide about the nuclear dynamics. In the case of three-body nuclei, the exclusive $p-d$ break-up has been studied within the Faddeev approach and using conventional forces and currents [185]. It has been observed that the final state interaction (FSI) plays an important role, however not many experimental data are available to compare.

For four-body systems the explicit calculation of all final states in the continuum becomes more cumbersome, thus often approximations of various kinds are introduced. The simplest model for the response function is obtained under the assumption that the photon is absorbed by one nucleon only (via a one-body operator) and that the hadronic final state is just a free propagation (plane wave) of the knocked out nucleon that does not interact with the remaining spectator (A-1)-nucleus. Such an approximation is named plane wave impulse approximation (PWIA). Under this assumption and neglecting excitations of the residual nucleus¹⁸ , the longitudinal response function can be obtained from the ground-state nucleon momentum distribution $N({\bf p})$ as [23]

$$\begin{eqnarray}&&R_{L}^{{\rm PWIA}}(\omega ,q)=\int {\rm d}{\boldsymbol{p}} \;N({\boldsymbol{p}} )\delta \left( \omega -\frac{{{({\boldsymbol{p}} +{\boldsymbol{q}} )}^{2}}}{2\;m}-\frac{{{{\boldsymbol{p}} }^{2}}}{2(A-1)m}-{{E}_{s}} \right),\end{eqnarray} \tag{ 35 }$$

with m being the mass of the struck nucleon (dominantly the proton) and E_s the proton separation energy. The PWIA does not consider the antisymmetrization of the final state and ignores the effect of the FSI. It is thus expected to be better at high-momentum transfer, where the FSI is small. It also neglects the effect of two-body currents. In the quasi-elastic regime, corresponding to momentum transfers of a few hundreds MeV and energies around ${{\omega }_{q.e.}}\approx {{q}^{2}}/2\;m$, one can envision that the electron has scattered elastically with a single nucleon. Thus, the PWIA can be considered the base line in a comparison with more sophisticated calculations.

For mass number A = 2 and 3 nuclei, the PWIA fails to reproduce experimental data for R_L below momentum transfer $q\sim 500$ MeV, see, e.g., [23]. In particular, it typically yields to too much strength near the quasi-elastic peak and too little strength at energies below and above this peak. When the FSI is included, the agreement with experiment is restored. The same conclusions were drawn for the ⁴He nucleus, in the first calculation of R_L with the Laplace transform approach [203, 204] (already reported in the review [23]) and in a LIT calculation with a central NN potential model [205] and then also confirmed in a more recent realistic calculation [200].

In figure 17, we show the longitudinal response function of ⁴He from [200] calculated with the AV18+UIX potential for two values of momentum transfer q = 300 and 400 MeV. In these calculations the charge density operator is used in IA. The latter is expanded in multipoles and then each multipole response function is separately calculated using the LIT method in conjunction with an EIHH expansion. For these values of momentum transfer a numerical precision of 1% is reached when multipoles up to J = 6 are summed up. What is evident from figure 17 is that the PWIA (black dotted curve) is rather poor. When the FSI is included using the LIT method (red solid curve) the agreement with the experimental data is restored. The advantage of the LIT method over the Laplace transform used in [204] is that the first method allows for a stable inversion of the transform [206], so a direct comparison of the theoretical response function to the experimental data is possible. This is preferable with respect to applying the integral transform to the experimental data, as the latter usually requires an extrapolation of the data in the high energy tail. Ultimately, the advantage of a direct comparison of the theoretical and experimental R_L is that one obtains the ω-dependence of the FSI effects as an additional information, as shown in figure 17. For the chosen kinematical values, the FSI is important also in the region of the quasi-elastic peak.

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It is interesting to study the sensitivity of the longitudinal response functions to different Hamiltonians, with emphasis on 3N force effects. In figure 18, results for ⁴He obtained with the LIT method in [200, 207] are shown. In those calculations, conventional potentials have been used and the charge operator has been kept in IA. One observes a rather strong effect of 3N forces: the difference between calculations with the AV18 only (blue dashed curve) and those comprehensive of the UIX three-body force (red solid curve) is increasing at low q. At q = 350 MeV, as shown in panel (a), the UIX leads to about a 10% reduction of the strength and improves the description of the data. At q = 150 and 50 MeV, presented in panel (b) and (c), where no experimental data exist, the UIX quenches the response function by to up to 50%. This fact is confirmed when using a different three-body force like the TM$^{\prime }$ potential [194] (magenta dash-dotted curve). With different three-body Hamiltonians a 10% variation in R_L is observed. Calculations with the χEFT approach are in progress [208] and it is worth recalling that the TM$^{\prime }$ force is basically the long range part of the χEFT three-body force at N2LO ($\nu =3$). Because of the sensitivity of R_L to 3N forces and because of such 10% variation with different 3N forces, precise future experiments can help to better constrain realistic three-body Hamiltonians. Thus, the longitudinal response function of ⁴He is an e.m. observable, complementary to the hadronic ones, that can potentially allow to study and constrain 3N forces.

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In case of the transverse response function, R_T, where two-body currents contribute already at NLO ($\nu =-1$) as suggested by the χEFT power counting, one can speculate that their strength enhancing effect will in part cancel the quenching due to the 3N forces that was observed in R_L. However, no calculation studying both the effect of switching on and off 3N forces and MEC exists. The first calculation of the transverse response function of ⁴He was performed by Carlson and Schiavilla using the Laplace transform [204]. Conventional three-body Hamiltonians, always comprehensive of 3N forces, and MEC have been implemented. A large effect of two-body currents has been pointed out at momentum transfer q = 300 and 400 MeV, leading to an enhancement of the strength by 20%, with respect to the IA. The major contribution was shown to come from OPE and proven necessary to describe the data. A comparison to the experiment could, though, only be done by Laplace transforming the data, so it was not possible to resolve the ω dependence of the MEC. A direct comparison to data was done instead in [209] using the LIT method. There, however, a semirealistic NN potential was utilized and approximations were introduced in the two-body currents. With this simplified model, two-body currents were not found to be very important, probably due to the missing one-pion exchange character both in the NN force and corresponding MEC.

As already mentioned, from inelastic electron scattering experiments other observables can be extracted. For example, the inelastic charge form factor, $F_{L}^{tr}(q)$, pertaining to the transition from the ground state to the resonant first ${{0}^{+}}$ excited state of ⁴He was recently studied with realistic Hamiltonians in [172]. The transition form factor corresponds to the inelastic longitudinal response function ${{R}_{L}}(\omega ,q)$ integrated around the energy of the ${{0}^{+}}$ resonance. Because in this case ${{J}_{f}}={{J}_{0}}=0$, effectively only the monopole part of the charge operator contributes to this transition. Several experimental data sets are available [210–212], where the background of other multipoles and of the monopole high energy tail has been subtracted. The first ab initio calculation was performed by Hiyama et al in [213] with the AV8$^{\prime }$ two-body potential [65], a reduction of the AV18 force, in combination with a simple central 3N force. The two free parameters of the latter have been calibrated to reproduce the ³H and ⁴He binding energies. As shown in figure 19, a rather good description of the data for $F_{L}^{tr}(q)$ has been achieved (black dash-dotted curve). More recent calculations of $F_{L}^{tr}(q)$ have been performed with the LIT method, using realistic Hamiltonians both from conventional theory as well as from χEFT. As shown in figure 19, the following Hamiltonians have been used: the AV18+UIX (red solid curve) and the NN(N3LO)+3N(N2LO) (magenta band) chiral forces. For the latter, the NN potential was used at N3LO ($\nu =4$) [98] and the 3N force at N2LO ($\nu =3$). The bandwidth is obtained by using two different parameterization of the LECs in the 3N force, following [100] and [173], respectively. A dramatic dependence of the results on the starting three-body Hamiltonian is observed. Three different Hamiltonians which describe the ⁴He ground-state energy within 1% from experiment show large differences in their predictions for $F_{L}^{tr}(q)$. This is quite surprising and highlights the richness of the dynamical information provided by inelastic observables. The failure of the realistic forces to reproduce the available experimental data for $F_{L}^{tr}(q)$ is not believed to be cured by two-body charge operators, as they appear only at N3LO ($\nu =0$) in χEFT and thus are not expected to play a major role. This example of disagreement between state-of-the-art theoretical calculations and experiment will hopefully stimulate further theoretical and experimental activity. From the theoretical point of view, the first excited state of ⁴He can be tackled with alternative few-body approaches. Interesting other features, such as its relation to collective modes, can also be explored, see, e.g., [214]. From the experimental point of view, the failure of the realistic forces to reproduce the available experimental data for $F_{L}^{tr}(q)$ has motivated new proposals to measure the monopole form factor via higher accuracy electron scattering at the S-DALINAC [215] and via the ⁴He(⁴He,⁴He)⁴He$^{*}$ reaction at LNS in Catania with the spectrometer MAGNEX [216].

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Regarding exclusive electron scattering reactions, the ⁴He($e,{{e}^{\prime }}p$)³H process was studied, e.g., by Quaglioni et al in [217]. Because often ($e,e^{\prime} p$) experiments are performed to study the structure of target nucleus [218] and spectroscopic factors are extracted under the assumption of direct knock-out of the proton neglecting the FSI, a check on such assumptions for A = 4 is very instructive. In [219], by using the LIT method and a simple semirealistic potential, it was shown that FSI effects are rather large, especially at low q, amounting up to 40%. It was concluded that the extraction of spectroscopic factors is more viable in the kinematic regions with high momentum transfer and small missing momentum. Very recently, a study on exclusive processes including the ³He($e,e^{\prime} p$)²H and ⁴He($e,e^{\prime} p$)³H reactions has been carried out in [220], extending the applications of MEC to high energies. It is found that FSI are of utmost importance and necessary for a satisfactory description of data.

The ($e,e^{\prime} NN$) reaction, in which two nucleons are knocked out and detected, is also considered an important tool to investigate NN correlations in nuclei. The ⁴He($e,e^{\prime} d$)d process was studied theoretically in [221] using a semirealistic interaction and an exact and consistent treatment of the FSI. The latter was found to be substantial, pointing out the importance of having all relevant effects under control if one wants to study NN correlations. Although several model studies are available in the literature, Unfortunately, fully ab initio exclusive calculations, where the Schrödinger equation is solved exactly, are not yet available in more complex nuclei, mostly because of the major stumbling block of dealing with the final states in the continuum.

There exist few calculations of transition form factors in $A\gt 4$ nuclei. Inelastic transition form factors of ⁶Li have been studied in [177] using VMC wave functions for the ground and excited states obtained from the AV18+UIX potential. In that work, conventional and consistent MEC have been used and they have been found to significantly improve the agreement between theory and experiment for the calculated longitudinal and transverse transition form factors to a number of low-lying excited states of ⁶Li.

Very recently, GFMC calculations of the charge form factors and sum rules of e.m. response functions in ¹²C have been carried out by Lovato et al in [24]. The study of e.m. inelastic responses in ¹²C is particularly interesting in view of the recent anomaly observed in the neutrino quasi-elastic charge-changing scattering data on ¹²C measured at MiniBooNE [4]. In particular, the measured cross section has been found to be larger than predicted by previous theoretical calculations. The latter, however, were not carried out within strictly ab initio frameworks.

The ab initio study of [24] has aimed at quantitatively evaluating, albeit in processes induced by e.m. probes, the effect of MEC on calculated ¹²C response functions. In particular, Lovato et al calculated the ¹²C e.m. sum rules [222]

$$\begin{eqnarray}&&{{S}_{L/T}}={{C}_{L/T}}\int _{{{\omega }_{{\rm th}}}}^{\infty }{\rm d}\omega \frac{{{R}_{L/T}}(q,\omega )}{G{{_{E}^{p}}^{2}}({{Q}^{2}})}\;,\end{eqnarray} \tag{ 36 }$$

where ${{R}_{L/T}}$ are the longitudinal and transverse response functions of equation (34), and the ${{C}_{L/T}}$ factors are defined as

$$\begin{eqnarray}&&{{C}_{L}}=\frac{1}{Z},\;\;\;\;{{C}_{T}}=\frac{2}{Z\mu _{p}^{2}+N\mu _{n}^{2}}\frac{{{m}^{2}}}{{{q}^{2}}}.\end{eqnarray} \tag{ 37 }$$

The non-energy weighted sum rules given in equation (36) can be expressed as ground-state expectation values by using closure relations, i.e.,

$$\begin{eqnarray}&&{{S}_{L/T}}={{C}_{L/T}}[\langle {{\Psi }_{0}}|O_{L/T}^{\dagger }({\bf q}){{O}_{L/T}}({\bf q})|{{\Psi }_{0}}\rangle -|\langle {{\Psi }_{0}};{\bf q}|{{O}_{L/T}}({\bf q})|{{\Psi }_{0}}\rangle {{|}^{2}}],\end{eqnarray} \tag{ 38 }$$

where O_L and O_T are the charge and current operators, respectively, and $|{{\Psi }_{0}};{\bf q}\rangle$ denotes the ground state of the nucleus recoiling with total momentum ${\bf q}$.

The GFMC results for ¹²C, based on the AV18+IL7 nuclear Hamiltonian and consistent conventional MEC, are shown in figure 20. In the figure, blue dots are obtained in IA (and IA+IA(RC) for the one-body charge operator), while red triangles include MEC corrections due to π-like and ρ-like exchanges, to the $\rho \pi \gamma$ transition current, and to OPE currents involving the excitations of Δ intermediate states. The comparison to the experimental data is not straightforward. In fact, in electron scattering, space-like virtual photons are exchanged for which $|{\bf q}|\gt \omega$. Therefore, in order to compare the theoretical results with the data one has to estimate the strengths in the energy region that is not accessed experimentally. In figure 20, the experimental points (open squares in the figures) have been obtained by integrating the available data from [223] over the accessible energy interval. The tail contribution, necessary to perform the integral appearing in equation (36) up to infinity, has been estimated by assuming that at these energies the ¹²C response is proportional to that of the deuteron, which can be calculated exactly (see [24] for details). MEC contributions are found to be small in S_L, and rather large in S_T where they can increase the IA results by up to a 50%. The study of [24], and that of [224] reporting on GFMC calculations of ¹²C sum rules of responses induced by neutral weak currents, have found that calculated MEC effects are large and should not be ignored in a careful interpretation of available experimental data. These findings may have implications for the above mentioned anomaly observed at MiniBooNE, and other quasi-elastic scattering data on nuclei.

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The photodissociation of the deuteron has been extensively studied for a long time both theoretically and experimentally (see, e.g., [8, 225]). The deuteron is the simplest nuclear system available in which one can explore the role of two-body physics coming from two-body current operators. A χEFT study along these lines has been presented in [226], where OPE currents (diagrams $({\rm b})$ and $({\rm c})$ in figure 3) and TPE currents (diagrams $({\rm e})-({\rm H})$ in figure 3) have been included. This calculation is not complete in that contact two-body currents also entering at N3LO (see diagrams $({\rm j})-({\rm o})$ in figure 3) have not been included. Nevertheless, it is interesting to discuss the main findings in comparison to experimental data and to calculations based on the conventional approach.

In panel (a) of figure 22, the calculated differential cross sections from [226] are shown. Calculations have been performed by introducing the explicit form of the current operator in the relevant matrix elements entering equations (40) and (41). It is apparent that predictions with the one-body current only (cyan dotted band) strongly underestimate the data, while, when two-body currents are included (magenta hatched band), an agreement with the data is established. As expected, most of the effect of two-body currents comes from the OPE term. The band is obtained by using five different parameterizations for the chiral potential at N2LO [96, 229], and its thickness can be interpreted as an estimate of the theoretical uncertainty. Such a band is expected to decrease with the addition of the contact two-body currents and when the chiral order of both potential and currents is increased. The χEFT approach also agrees fairly well with the conventional approach (red solid curve), where the AV18 potential was used and MEC were included (see [226] for details).

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In panel (b) of figure 22, the total photoabsorption cross section is displayed as a function of the photon energy. The χEFT calculation from [226] explicitly includes the effect of two-body currents from the one- and two-pion exchanges and clearly leads to a nice agreement with data. The conventional calculation with the AV18 potential (red solid curve) from [228] is instead based on the use of the Siegert theorem, and is related to an $E1$ response as in equations (42) and (43). Thus, the main part of the MEC is included. It is well known from the conventional approach that the IA strongly underestimates the total cross section. To highlight this fact, in figure 22(b) we report on the conventional calculation by Marcucci et al from [21], where the IA (blue dashed curve) is used for energies below 20 MeV. In [21], it was also shown that a perfect agreement between conventional calculations that explicitly include MEC and those obtained with the Siegert theorem is obtained below 20 MeV photon energies. Previous studies, reported in [225], have shown that retardation effects and higher order multipoles contribute to an enhancement of the cross section of about 10% at $\omega =80$ MeV, while further contributions due to exchange currents and relativistic effects cancel each other in the energy range from 5 to 100 MeV.

It is interesting to note that the χEFT and the conventional Siegert calculations are consistent with each other. The main difference in percentage is found in the high energy tail, where, in fact, corrections to the unretarded dipole approximation may start to be relevant. Nevertheless, the Siegert/AV18 curve falls into the error band of the χEFT calculation.

Finally, we would like to mention that polarization observables have been studied in the two-body case, both in the conventional approach (see, e.g., [21, 230]) and in the χEFT approach (see, e.g., [226]), where important effects of exchange currents have been found.

In moving from two- to three-body nuclei, the dynamics becomes richer due to the appearance of 3N forces. Because of the present incomplete theoretical underpinning of 3N forces, a possible way to search for effects of 3N forces is to use high precision NN potentials in three-body calculations of hadronic observables and look for differences between theoretical predictions and experimental data. This kind of studies has been performed for bound-state energies of ³H, ³He (and ⁴He) [231, 232], as well as for nucleon-deuteron collisions, see, e.g., [233, 234]. Complementary information can be accessed by studying photodisintegration processes, involving the interaction between photons and many-body nuclear currents. Thus, e.m. reactions with A = 3 nuclei can be crucial observables to test, refine and possibly even discriminate among different theoretical approaches for Hamiltonians and currents.

The photodisintegration of three-body nuclei has been investigated by a number of groups using different techniques. Calculations using conventional potentials and currents have been performed with the HH method in [21, 186], with the Faddeev approach in [235–237] and with the LIT method in [236, 238, 239]. Details on the different few-body methods can be found in the recent review of [51]. Regarding the effect of conventional two-body currents, Faddeev and HH calculations have shown that for the two-body channel at low photon energy MEC enhance the differential photodisintegration cross section by 50% in the peak region. The first study of three-body photodisintegration processes carried out within χEFT is found in [226], where the two- and three-body photodisintegration of ³He have been studied with Faddeev-type calculations. In that study, 3N forces were omitted and Coulomb effects were neglected in the scattering states. When looking for effects of one- and two-pion exchange currents in the differential cross section, the situation resembled very much that one of the deuteron presented in figure 22. Two features are worth stressing: (i) the addition of two-body currents strongly enhances the strength in the differential cross section, and (ii) the χEFT approach agrees rather well with the conventional calculations with the AV18 potential and corresponding MEC.

We now turn our attention to study the sensitivity of the calculated cross sections to different Hamiltonians used to generate the nuclear wave functions. Specifically, we are interested to see what happens when 3N forces are switched on and off. In [237], it has been shown that cross sections calculated with explicit two-body currents or with the Siegert theorem nicely agree with each other at low-energies. Thus, in a Siegert calculation of the photodisintegration cross section the only input that can be changed is the nuclear Hamiltonian. In figure 23, we show results from [236] obtained with the LIT method using the AV18 (blue dashed curve) and AV18+UIX (red solid curve) potentials for ³H $({\rm a})$ and ³He $({\rm b})$, in comparison with available experimental data. Similar results were obtained with slightly different three-body Hamiltonians in [238]. We emphasize that, due to the Siegert theorem, these results also include 3N currents effects in the electric dipole response, that otherwise should have been included explicitly as it has been done in [21]. In figure 23 $({\rm a})$, we also show the calculation for ³H performed within the Faddeev approach [236] (red dots) using the same interaction, i.e., the AV18+UIX, and dipole transition operator. One can appreciate that the agreement between the two different methods is very good. The small difference between the two approaches can be interpreted as the numerical error of a theoretical calculation with this interaction. The latter is smaller than the theoretical deviation obtained when using a Hamiltonian without 3N forces and smaller than the experimental error bar indicated by the gray band.

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In [236, 238], it has been found that the addition of the 3N forces reduces the strength of the calculated cross sections by roughly 10% at the peak region, leading to a better agreement with experimental data, as can be especially appreciated in the ³H case illustrated in figure 23 $({\rm a})$. This behavior observed in presence of the 3N forces is related to the additional binding they provide in the three-body system, which, in a naïve picture, reduces the probability to disintegrate the nucleus. In fact, for the triton, the binding energy is $B.E.=7.60(8.47)$ MeV with the AV18(AV18+UIX) potential. In [238], the isospin $T=3/2$ channel (which is exclusively a three-body break-up channel) was found to be more sensitive to 3N forces than the $T=1/2$, their main effect being an enhancement of the tail of the cross section, amounting to 15% at 70 MeV.

Other non-local two-body Hamiltonians have been used to calculate the total cross section in the unretarded dipole approximation. Among these are the the potential obtained within the unitary correlation operator method (UCOM) [242] used in [239], and the J-matrix inverse scattering potential (JISP) [243] used in [244]. It is interesting to see how these potentials compare to other interactions and to experimental data, as they may induce strong exchange currents due to their non-locality. In case of the UCOM potential, the results lie between the AV18 and AV18+UIX curve, while for the JISP potential, the cross section has a higher tail with respect to the AV18+UIX curve, indicating that strong exchange currents are induced with this potential. Due to the large experimental error bar, it is not quite possible to discard such a potential model based solely on the three-body data.

The contribution of two- and three-body break-up channels to the total cross section has also been studied with the Faddeev approach. The three-body break-up cross section becomes larger than the two-body break-up one at 14 MeV and at higher energies it is the dominant channel. Similarly to the inclusive case, in the exclusive two-body channel, e.g., $\gamma {{+}^{3}}{\rm H}$ $\to {\mkern 1mu} d+n$, the addition of 3N forces leads to a lower peak in the cross section and to a better agreement with experiment. However, the data are quite scattered, making it hard to draw final conclusions on different potentials. The photodisintegration of ³He into three-nucleons was extensively studied in [235] with conventional Hamiltonians and currents. Specifically, effects due to 3N forces have been sought for by comparing results with and without 3N forces in various kinematic regions. Very large effects of 3N forces amounting up to a 85% were found, a finding that could help planning for future experiments.

We recall that studies on polarization observables on ³He have been performed, e.g., in [21, 186, 245], where a sensitivity to both 3N forces and to the detail of the many-body contributions to the nuclear current operator has been observed. In [226], polarization observables have been studied within the χEFT approach and very strong effects of exchange currents have been found in the photon analyzing power, leading even to a completely different shape of the angle-dependent curves when going from the simple IA picture to the one that includes two-body e.m. currents. Furthermore, recent studies on neutral pion production off three-body nuclei carried out in χEFT have also pointed out to large effects of two-body currents [246].

Particular attention has been recently devoted to the study of the ⁴He photodisintegration reaction. This process is particularly interesting, from both a theoretical and an experimental point of view, due to a number of reasons. First, with a binding energy per nucleon of about 8 MeV/A, ⁴He can be considered the link between the very light nuclei,²H, ³H, and ³He, and heavier systems. Second, because NN and 3N forces are typically calibrated on two- and three-body nuclei, ⁴He is the most natural testing ground for microscopic nuclear forces. Third, due to its larger density with respect to the lighter nuclei, intermediate and short range physics effects, such as those generated by 3N forces, are expected to be more important. In fact, conventional 3N forces contribute 17% of additional binding energy, leading to 24.27 (28.42) MeV with the AV18 (AV18+UIX) potential. This correction due to 3N forces is much larger than the 10% effect found in ³H. Finally, because of gauge invariance, nuclear forces also manifest themselves as exchange currents which are very important in photonuclear reactions, making the photodisintegration of ⁴He particularly interesting to study further effects of 3N forces.

The theoretical computation of such process is complicated by the difficulty of calculating the four-body continuum states. For the ground state of ⁴He, several few-body methods can be applied to precisely calculate its properties using realistic forces (as it has been shown in the famous benchmark study reported in [247]). The same is not true for break-up observables where many channels in the final state are open. However, one can avoid the complications of the explicit calculations of continuum final states by using a bound-state formulation of the continuum problem. This is what is achieved for, example, with the LIT method, and with the complex scaling method, see, e.g., the review articles of [51] and [248].

From the experimental point of view, most of the efforts have been devoted to the measurements of exclusive channels, like ⁴He(γ, n)³He and ⁴He(γ, p)³H. For the ⁴He(γ, p)³H reaction, the available data are scattered in the energy range between 20 and 35 MeV. The current experimental status is nicely summarized in [249], where data from both photon beams and as well as those deduced from the time reversed radiative-capture reaction using the principle of detailed balance are used. For the ⁴He(γ, n)³He reaction, the experimental situation is unsettled (see [250] for a summary on this topic).

In figure 24, we present ab initio theoretical results in comparison with the most recent experimental data (from 2005 or later). On the left, we show the total cross section and on the right we present the ⁴He $(\gamma ,p)$³H exclusive reaction. For the total cross section, the first realistic calculations have been performed in [174] with the AV18+UIX potential (red solid band), using the LIT method in conjunction with the EIHH expansion. The χEFT approach has been utilized, instead, in [254] (magenta hatched band), where the NN force was taken at N3LO ($\nu =4$) and the 3N force at N2LO ($\nu =3$) with a fit of the LECs from [173]. There, the LIT method has been used in combination with the NCSM [255]. For these realistic three-body Hamiltonians the results are shown with a band, corresponding to the theoretical uncertainty of the used computational method. An other recent calculation has been performed by Horiuchi et al [256] (dash dotted curve) using the complex scaling method with the two-body AV8$^{\prime }$ potential [65] (a simplified version of the AV18 NN potential) supplemented by a simple central 3N force from [213]. These three theoretical calculations are shown in figure 24(a). They are all based on the use of the Siegert theorem, albeit with different Hamiltonians, which all reproduce the experimental binding energy of ⁴He within one percent. Because they form a kind of theoretical band, with a variation of the order of 10% at the peak, we can be quite confident in saying that there is a solid understanding of nuclear dynamics in this regime. The large differences at threshold are due to the inaccuracy of the complex scaling method (see [256]) and should not to be considered as an effect of the potential.

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When we compare against recent experimental data, the most striking disagreement between theory and experiments, amounting up to a factor of 2, is with the data from Shima et al from [253]. They have been obtained with a quasi-monoenergetic photon beam and a time projection chamber, where a simultaneous measurement of both the ⁴He(γ, n)³He and ⁴He(γ, p)³H reactions has been performed. The data in the energy range below 30 MeV, where the large disagreement exist, have been remeasured by the same group, confirming their findings [257]. The latter are, however, in disagreement with previous data from Nilsson et al on the neutron channel [252] (for the total cross section we show them below the three-body disintegration threshold, where ${{\sigma }_{\gamma }}\simeq 2\sigma (\gamma ,n)$) and with data from Nakayama et al [251], measured via the ⁴He(⁷Li,⁷Be) reaction. To clarify the situation, an experiment has been carried out at the High Intensity Gamma-Ray source [249, 250]. Both the exclusive two-body channels have been measured (in figure 24(a) we sum the two channels and show just the points in the energy range below the three-body break-up). This last experiment has confirmed that the peak of the cross section is at around 25–26 MeV, in nice agreement with the theory and with the data from Nilsson et al and Nakayama et al, and in strong disagreement with the data from Shima et al. Discarding this specific set of data, it can be said that there is nice agreement between theory and experiment.

For the exclusive ⁴He(γ, p)³H, in figure 24(b) we show the calculations with the MTI-III potential from [219] (grey solid curve), performed with the LIT method, and those with the AV8$^{\prime }$ with a central 3N force from [256] (dash-dotted curve), performed with the microscopic R-matrix method. We compare them to the exclusive measurements from Tornow et al [250] and from Shima et al [253]. Also in this case, the theory supports a higher cross section in disagreement with the data from Shima et al. One can also see that the calculation with the more realistic AV8$^{\prime }$ potential and the central 3N force is actually in better agreement with the data from Tornow et al. Efforts are being directed towards a calculation of this cross section based on the AV18+UIX with the LIT method [258].

Finally, we would like to comment on the role of 3N forces. In [174], they were found to lead to a 6% decrease of the cross section peak, but also to a large enhancement, up to 35%, of the strength at energies above 50 MeV. Such 3N force dependence cannot be simply interpreted as a binding effect. The only inclusive experimental data that cover a wide range in energy are from Arkatov et al [259]. When the UIX potential is added to the AV18 two-body force, the agreement with data is actually improved in the energy range from $\omega =50$ to 80 MeV. Beyond that, and up to pion threshold, the AV18 potential does provide the best description of the data. Finally, we remind that other non-local two-body potentials, like the JISP, actually do lead to a cross section which is higher and not in agreement with the experimental data, as pointed out in [244]. This is presumably due to large exchange currents induced by the non-locality in the potential.

We now consider slightly heavier nuclei. Traditionally, nuclei with a number of nucleons between 4 and about 12 are considered as a bridge between the few- and the many-body systems. Furthermore, with increasing mass numbers, nuclear spectra become more complex and new interesting phenomena arise. For example, within the isobar nuclei with mass number A = 6, we have a two neutron-halo nucleus, i.e., ⁶He [260]. This happens also to be the lightest halo nucleus in the nuclear chart, and, as such, has received quite a bit of attention both from the theoretical and experimental point of view (see, e.g., [175]). ⁶He is a very short lived nucleus (half life of ${{t}_{1/2}}$ = 807 ms) that can be compared to the stable ⁶Li nucleus. An interesting question to ask is: does the interaction of a real photon with these two isobar analog nuclei lead to different structures in the photodisintegration cross section? Obviously, to answer this question one has to face two challenges, one theoretical and one experimental: (i) the exact calculation of six-body final state wave functions with various open channels is currently out of reach, and (ii) experiments with real photons cannot be done on unstable nuclei, so alternative techniques need to be looked for.

The theoretical problem has been circumvented by using the LIT method. The application of this method in conjunction with the EIHH expansion has allowed to study the photodisintegration of the six-body nuclei in [261, 262]. Simple semi-realistic interactions have been used, that just fit the S-wave or (partially) the P-wave phase-shifts in the NN collisions. Even though they do miss some of the complexity of realistic potentials, such as the tensor and spin-orbit force, they decently reproduce (within 10%) the photodisintegration of lighter nuclei, as discussed earlier. The studies on the six-body nuclei have shown that the halo structure of the rare ⁶He isotope leads to considerable differences from the stable ⁶Li nucleus in photodisintegration. As shown in figure 25, for ⁶He the dipole cross section exhibits two well separated peaks, while a single resonant shape is observed for ⁶Li. The first peak corresponds to the break-up of the neutron halo, while the second peak corresponds to the break-up of the α-particle. These results do not depend on the employed NN interaction: three different semi-realistic interactions have been used, and they have been found to lead to very similar structures. The curves in figure 25 have been obtained with the AV4$^{\prime }$ potential [263], which is a truncated version of the AV18 potential, that leads to a reasonable description (10–15% from experiment) of the binding energy of these nuclei and includes some P-wave components, which are found to be important in p-shell nuclei [261].

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Photonuclear experiments on unstable system cannot be done. However, the low-energy photodisintegration cross section, which is related to the dipole response function via the Siegert theorem as in equation (42), can be inferred from Coulomb excitation experiments, see, e.g., the review [5]. In this case, a beam of rare isotopes is used as projectile in collisions with heavier targets. In specific circumstances, the Coulomb interaction between the projectile and the target either dominates or can be separated from the nuclear part. From the Coulomb interaction one can extract the $E1$ strength and then reconstruct the photoabsorption cross section. For ⁶He, this kind of experiments has been done at GSI by Aumann et al [264] with high projectile energy (240 MeV/nucleon) and at NCSL by Wang et al [265] with lower projectile energy (25.2 MeV/nucleon). In figure 26(a), we show these experimental results in comparison to the theoretical calculations from [261]. As one can see, the theory decently describes the data for $\omega \gt 5$ MeV, but underpredicts the strength very close to threshold. The spin-orbit force, which is missing in the AV4' potential, could possibly enhance the strength in this energy region.

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For ⁶Li, which is a stable isotope, several experiments on photonuclear reactions exist, however none of them corresponds to an inclusive measurement. Because summing all the different channels coming from different experiments is delicate and because the experimental situation for the exclusive processes is not settled, we prefer, instead, to show the comparison of theory with experiment for ⁷Li. Calculations on A = 7 nuclei have been performed in [267] using the LIT with an EIHH expansion and the AV4' potential. Inclusive data for ⁷Li have been measured by Ahrens et al [266] and are shown in figure 26(b). One can observe that, despite the simple potential used, the theoretical curve nicely describes the gross feature of the experimental data, i.e., a steep rise, a broad maximum and a slow fall-off. ⁷Li, as ⁶Li, is a stable nucleus and show only one large bump, very differently from the two separated peaks found in ⁶He by theory. These theoretical calculations have stimulated new experimental efforts devoted to measure, e.g., the photoabsorption of $^{6,7}$Li at the High Intensity Gamma-Ray Source [268] and at MAX-lab in Lund [269, 270]. The predicted presence of the second peak in the ⁶He photodisintegration cross section is still waiting for an experimental observation.

First principle calculations of photonuclear reactions, where one starts from realistic NN forces that reproduce (some) NN phase-shifts, have been available only for the light systems, in the recent past. For the medium-mass nuclei, instead, several studies exist in the literature, where the photodisintegration is investigated using Skyrme-functionals (see, e.g., [271, 272]), which are typically calibrated on finite nuclei. Alternative (macroscopic) approaches, like, e.g., Halo EFT have also been recently investigated, see [273]. The reason for the lack of ab initio calculations in the medium-mass regime is that one has to account for the very strong correlations induced by realistic interactions in both the ground state and in the final states in the continuum. The difficulty of theoretically describing the continuum states can be circumvented with the LIT method which reduces the problem to a bound-state-like equation. As shown above, the LIT method has been used together with HH expansions and with the NCSM. Both these methods do not lend themselves to a straightforward application to the medium-mass nuclei. In a recent paper [274], it has been shown that the LIT method can be used in conjunction to CC theory, a many-body approach well suited for the medium-mass and heavy nuclei [73]. The combination of the LIT with CC theory [274, 275] has allowed to extend the previous mass limits of the theory and address the inclusive photodisintegration in ¹⁶O. The method has been first benchmarked with the exact EIHH on ⁴He using the same nuclear Hamiltonian. Despite the approximations introduced in the CC calculations, a very nice agreement has been found between the two-methods for ⁴He.

In figure 27, we present the theoretical results for the dipole response function of ¹⁶O in comparison to the available data from Ahrens et al [266]. The experimental cross section is transformed into a dipole response function using equation (42). For the starting Hamiltonian, the χEFT approach has been taken, where the NN interaction at N3LO ($\nu =4$) from Entem and Machleidt [98] (inclusive of Coulomb force) has been used, but 3N forces have been omitted. The band of the cyan curve is obtained by inverting the LIT with a slightly different regularization procedure, see, e.g., [206, 274] for more details. It is apparent that the experimental location of the peak is correctly reproduced by the calculation. Also the total experimental dipole strength is reproduced, while the width of the theoretical resonance is broader than the experimental one. The investigation of the impact of the neglected 3N forces and higher order correlations on the photonuclear cross section of medium-mass nuclei is currently underway. This first case study paves the way for many future investigations of continuum responses in medium-mass nuclei for both stable and unstable isotopes, which will be accessible using the LIT method in conjunction with CC theory.

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Low-energy proton and neutron captures on few-nucleon systems are particularly interesting from the standpoint of nuclear structure studies. Indeed, MEC effects were at first quantified as a correction to the calculated thermal neutron radiative capture on proton cross section, and were found to provide the 10% contribution needed to resolve the discrepancy with the experimental datum [13].

Thermal neutron radiative captures are induced by the $M1$ transition operator that connects the initial two-cluster state in relative S-wave and the final bound state. The experimental values of the cross sections for thermal neutron captures on ¹H, d, and ³He in mb are: ($332.6\pm 0.7$) [284], ($0.508\pm 0.015$) [285], and ($0.055\pm 0.003$) [286, 287], respectively. The drop in the values in going from A = 2 to A = 3,4 is due to the so called 'pseudo-orthogonality' between the initial and final wave functions in the reactions involving A = 3,4 nuclei. In fact, the ³H and ⁴He wave functions, Ψ₃ and Ψ₄ respectively, are approximately eigenfunctions of the LO (or IA) one-body $M1$ operator ${\boldsymbol{ \mu }} ({\rm IA})$ (see equation (26)), namely $\mu {{({\rm IA})}_{z}}{{\Psi }_{3}}\simeq {{\mu }_{p}}{{\Psi }_{3}}$ and $\mu {{({\rm IA})}_{z}}{{\Psi }_{4}}\simeq 0$, where μ_p = 2.793 n.m. is the proton magnetic moment—the experimental value of the ³H magnetic moment is 2.979 n.m, while ⁴He has no magnetic moment. If small components in wave functions, generated by tensor components in the nuclear potentials, are neglected, then the matrix elements $\langle {{\Psi }_{3}}\mid \mu {{({\rm IA})}_{z}}\mid {{\Psi }_{1+2}}\rangle$ and $\langle {{\Psi }_{4}}\mid \mu {{({\rm IA})}_{z}}\mid {{\Psi }_{1+3}}\rangle$ vanish due to orthogonality between the initial and final states. In the case of the deuteron, instead, the $M1$ operator can connect the large deuteron S-wave component to the T = 1¹S₀ $np$ scattering state.

Due to this suppression at the IA level, radiative capture cross sections are particularly sensitive both to small components in the wave functions (and, therefore, indirectly to the nuclear Hamiltonians utilized to generate them), and to many-body components in the e.m. current operators. Early calculations of the $nd$ and n³He radiative capture cross sections in IA predicted only $\sim 50\%$ [288] and $\sim 10\%$ [89], respectively, of the corresponding experimental values. Studies that account for corrections due to conventional MEC reported a calculated value of the $nd$ radiative capture within 15% of the experimental value [87], while the calculated value of the n³He cross section was found to be 86 μb [289], to be compared to the experimental value of 55 ± 3 μb.

In recent years, methods for solving the A = 3 and 4 Schrödinger equations have been refined [290–292], and highly accurate nuclear wave functions for these systems are now available. Theoretical calculations of the $nd$ and n³He radiative capture cross sections [293] based on nuclear wave functions obtained from the HH techniques [58] are shown in figure 28. The bands represent results from hybrid and chiral calculations that use the χEFT currents developed in [37, 46]. The thickness of the bands represents the spread in the calculated values corresponding to the two considered nuclear Hamiltonians, namely the AV18+UIX and the NN(N3LO)+3N(N2LO), where the two– and three–body chiral interactions are form [98] and [100], respectively. The sensitivity to the regularization cutoff $\Lambda$ = 500–700 MeV is shown on the x-axis. The procedure implemented to fix the LECs entering the e.m. currents is different from that one utilized for the calculations of [48, 123] that have been discussed in sections 3.2 and 3.3.2, respectively. In particular, here, the Δ-resonant saturation argument has been exploited to infer the ratio between the isovector LECs entering the tree-level current at N3LO ($\nu =1$)—see panel $({\rm k})$ of figure 3. Therefore, one is left with four e.m. LECs. Namely, an isoscalar and an isovector LEC associated with the tree-level current at N3LO, see panel $({\rm k})$ of figure 3, plus an isoscalar and an isovector LEC associated with a contact-like current at N3LO, see panel $({\rm j})$ of figure 3. These four LECs have been fixed so as to reproduce the deuteron magnetic moment, the isoscalar and isovector combinations of the trinucleon magnetic moments, and the $np\to d\gamma$ cross section at thermal neutron energy [293]. In addition, LECs of minimal nature entering the N3LO contact currents have been taken from a NLO ($\nu =2$) χEFT potential [46] rather than from the N3LO ($\nu =4$) [98] χEFT potential.

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In figure 28, results obtained with the complete N3LO e.m. operator are shown by the magenta (light) band labeled N3LO(full), and are in very satisfactory agreement with data (black band). Their sensitivity to the cutoff is negligible ($\sim 10$%) for the $nd$ ($n{{\;}^{3}}$He) capture. As discussed above, these processes are strongly suppressed at IA or LO ($\nu =-2$): the calculated $\sigma _{nd}^{\gamma }$(LO) and $\sigma _{n{{\,}^{3}}{\rm He}}^{\gamma }$(LO) are less than half and a factor of five smaller than the measured values, respectively. The IA(RC) correction at N2LO ($\nu =0$), corresponding to the relativistic correction to the LO e.m. current illustrated in panel $({\rm d})$ of figure 3, along with the correction labeled N3LO(no-LECs), corresponding to the one-loop and minimal contact currents at N3LO, is found to have opposite sign with respect to the LO contribution. The N3LO(full) contributions, corresponding to the tree-level and non-minimal contact currents at N3LO—see panels $({\rm j})$ and $({\rm k})$ of figure 3—are large and crucially important for bringing theory into agreement with experiment. Results obtained within the conventional model are also shown in figure 28. These calculations use the MEC of [21] that have been constructed from the AV18+UIX nuclear Hamiltonian, along with the A = 3, 4 improved nuclear wave functions [292]. In particular, empty red squares represent calculations obtained with the non-relativistic IA operator plus conventional MEC corrections. The full red squares add to the empty ones the IA(RC) contribution (or, the N2LO current contribution illustrated in panel (d) of figure 3). These relativistic corrections have been neglected in previous conventional calculations [21, 87, 89, 288, 289], however, they are found to be significant and, at least in the case of the $nd$ capture, they bring the prediction of the conventional approach within 4% of the experimental data, as opposed to the 15% estimate from previous studies [87]. Despite this satisfactory result, the description of the n³He remains problematic. We note that hybrid studies based on the χEFT currents by Park et al [45] have reported values for the $nd$ and n³He capture cross sections about 6% and 15% smaller than measured, with a cutoff sensitivity of about 15% [129, 130, 294].

Proton capture reactions on neutron or few-body nuclei have been calculated in the past with conventional potentials and currents by Viviani et al [186], by Golak et al [237], and by Marcucci et al [21]. At energies of the order of a few MeV and up, where this process is dominated by an $E1$ transition, conclusions similar to those obtained from studies on photodisintegration reactions are drawn. In particular, the one-body current alone does not suffice to describe the data, however, when MEC contributions are included, explicitly or implicitly via Siegert theorem, an improved description of the data is obtained. At the low energies relevant for astrophysics, the capture process is instead dominated by the $M1$ multipole. Because there is no Siegert theorem for magnetic transitions, MEC have to be included explicitly. This has been accomplished in [21], where MEC contributions have been found to be large and necessary for a satisfactory description of the LUNA data for the ²H(p, γ)³He reaction.

The d(d, γ)⁴He reaction has been recently investigated with ab initio methods by Arai et al [295]. This radiative capture can impact studies on the abundances of primordial elements. This process occurs predominantly via an $E2$ transition at low-energy. In [295], calculations with a realistic potential and with simple semirealistic central forces, have been compared, and it has been found that the S-factor for the d(d, γ)⁴He reaction is very sensitive to the tensor forces.

Many of the relevant nuclear reactions in primordial nucleosynthesis and in solar neutrino production involve light nuclei with mass number $A\gt 4$ [277, 278]. The first steps towards ab initio calculations of a few key reactions have been taken using Quantum Monte–Carlo methods. For example, the α capture reactions ²H(α, γ)⁶Li, ³H(α, γ)⁷Li, and ³He(α, γ)⁷Be have been evaluated in [296, 297] using VMC wave functions for the initial clusters and the final nucleus. However, a phenomenological relative wave function between the two initial clusters was used. Similarly, a first attempt to calculate the ⁷Be(p, γ)⁸B reaction was proposed by Navratil et al [298] using the interior overlap functions obtained from NCSM to constrain the tail of a phenomenological Wood-Saxton potential.

A full ab initio calculation requires the use of the same Hamiltonian for both the initial continuum state and the bound final state. This has been only recently achieved for nuclei with $A\gt 5$. Below, we highlight two examples.

The ⁷Be(p, γ)⁸B S-factor is an important input in modeling the neutrino flux coming from our Sun [278]. An ab initio calculation of this process has been performed by Navratil et al [299] using the NCSM with the resonating group method [300, 301], which allows to describe continuum states. The used starting interaction is a χEFT two-body potential at N3LO ($\nu =4$) [98], which has then been softened by a Similarity Renormalization Group (SRG) transformation [302]. The latter is characterized by an evolution parameter, ${{\Lambda }_{{\rm SRG}}}$, which has been chosen by tuning the calculated ⁸B separation energy close to the experimental value. In fact, the low-energy behavior of the S-factor is very sensitive to the threshold energy.

This process has been investigated using different experimental techniques, including direct measurements with proton beams on ⁷Be targets and indirect Coulomb excitations with ⁸B beams hitting on a heavy target and breaking into proton and ⁷Be. Data are shown in figure 29 for the astrophysical S-factor. Even though they are a bit scattered, they clearly display a resonant feature due to the ${{1}^{+}}$ excitation, which however is very narrow and does not affect much the S-factor at zero energy, where the process is dominated by a dipole transition. The theoretical curve shown in figure 29 does not include any $M1$ transitions, thus it does not show the resonant feature. However, it nicely reproduces the shape of the $E1$ contribution to the S-factor. Because the dipole operator is protected by the Siegert theorem, the leading part of the MEC is implicitly included. SRG renormalizations of the dipole operator and overall 3N forces have been conjectured to be small, and omitted for the time being [299].

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The ³He(α, γ)⁷Be radiative capture takes place in the solar Hydrogen burning reaction chains. It is important in determining the high energy solar neutrino flux and in understanding the abundance of primordial ⁷Li. It has been measured by several groups, but the difficulty of reaching the very low solar energies makes it hard to extract the needed S-factor at zero energy. This reaction has been recently calculated by Neff [303] within the Fermionic Molecular Dynamics (FMD) [74], which allows to consistently obtain bound and scattering states from a soft effective interaction. The starting potential used in the calculations is the AV18 two-body force, which has been then softened by using the UCOM [242, 310]. Here, the evolution parameter, ${{\Lambda }_{{\rm UCOM}}}$, is tuned so as to reproduce the ³He+⁴He threshold in ⁷Be. The calculation carried out with a dipole transition operator is shown in figure 30 in comparison to recent measurements. As one can see, both the energy dependence and the absolute normalization are in good agreement with experiment. The calculation provides a zero-energy S-factor of 0.592 keVb. Also in this case 3N forces and renormalizations to the dipole operator have been omitted.

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The two examples above represent a break-through in the development of calculational techniques that can tackle radiative capture reactions in larger nuclei. However, further studies are needed to assess the theoretical sensitivity of these processes to intermediate and short range physics, entering, e.g., in the form of 3N forces or MEC. In the future, we can expect more investigations along these lines also from the newly developed method to calculate radiative captures from lattice EFT introduced in [311], and firstly applied to the $p(n,\gamma )d$ reaction.

Finally, we would like to mention that alternative low-energy effective field theories, such as Halo EFTs, can be applied to the study of radiative capture reactions. This has been recently accomplished, e.g., in [312–314] for A = 8 systems.