Nuclear electromagnetic currents to fourth order in chiral effective field theory

TL;DR

This paper derives electromagnetic nuclear currents up to fourth order in chiral effective field theory, including new three-nucleon charge operators, ensuring consistency with the modified continuity equation.

Contribution

It provides the first derivation of the leading three-nucleon charge operator at this order, completing the electroweak current derivation in chiral EFT.

Findings

Validated the modified continuity equation for all contributions

Derived the leading three-nucleon charge operator

Completed the fourth-order electroweak current derivation

Abstract

Recently, we have shown that the continuity equation for the nuclear vector and axial current operators acquires additional terms if the latter depend on the energy transfer. We analyze in detail the electromagnetic single-nucleon four-current operators and verify the validity of the modified continuity equation for all one- and two-nucleon contributions up to fourth order in the chiral expansion. We also derive, for the first time, the leading contribution to the three-nucleon charge operator which appears at this order. Our study completes the derivation of the electroweak nuclear currents to fourth order in the chiral expansion.

Tables

Table 1. Table 1: Chiral expansion of the nuclear electromagnetic current operator up to N 3 LO. LO, NLO, N 2 LO and N 3 LO refer to chiral orders Q − 3 superscript 𝑄 3 Q^{-3} , Q − 1 superscript 𝑄 1 Q^{-1} , Q 0 superscript 𝑄 0 Q^{0} and Q 𝑄 Q , respectively. The single-nucleon contributions are given in Eqs. ( 2.1 ) and ( 2.1 ).

order		single-nucleon		two-nucleon		three-nucleon
LO		—		—		—
NLO		${\overrightarrow{V}}_{1N:\mathrm{static}}$		${\overrightarrow{V}}_{2N:\mathrm{\hspace{0.17em}1}\pi }$, Eq. (4.16) of Kolling:2011mt		—
		$+{\overrightarrow{V}}_{1N:\mathrm{\hspace{0.17em}1}/m}$
N2LO		${\overrightarrow{V}}_{1N:\mathrm{static}}$		—		—
N3LO		${\overrightarrow{V}}_{1N:\mathrm{static}}$		${\overrightarrow{V}}_{2N:\mathrm{\hspace{0.17em}1}\pi }$, Eq. (4.28) of Kolling:2011mt		—
		$+{\overrightarrow{V}}_{1N:\mathrm{\hspace{0.17em}1}/m}$		$+{\overrightarrow{V}}_{2N:\mathrm{\hspace{0.17em}2}\pi }$, Eq. (2.18) of Kolling:2009iq
		$+{\overrightarrow{V}}_{1N:\mathrm{off}-\mathrm{shell}}$		$+{\overrightarrow{V}}_{2N:\mathrm{cont}}$, Eq. (5.3) of Kolling:2011mt

Table 2. Table 2: Chiral expansion of the nuclear electromagnetic charge operator up to N 3 LO. LO, NLO, N 2 LO and N 3 LO refer to chiral orders Q − 3 superscript 𝑄 3 Q^{-3} , Q − 1 superscript 𝑄 1 Q^{-1} , Q 0 superscript 𝑄 0 Q^{0} and Q 𝑄 Q , respectively. The single-nucleon contributions are given in Eq. ( 2.6 ).

order		single-nucleon	two-nucleon	three-nucleon
LO		$V_{1N:\mathrm{static}}^0$	—	—
NLO		$V_{1N:\mathrm{static}}^0$	—	—
N2LO		$V_{1N:\mathrm{static}}^0$	—	—
N3LO		$V_{1N:\mathrm{static}}^0$	$V_{2N:\mathrm{\hspace{0.17em}1}\pi }^0$, Eq. (4.30) of Kolling:2011mt	$V_{3N:\pi }^0$, Eq. (4.1)
		$+V_{1N:\mathrm{\hspace{0.17em}1}/m}^0$	$+V_{2N:\mathrm{\hspace{0.17em}2}\pi }^0$, Eq. (2.19) of Kolling:2009iq	$+V_{3N:\pi }^0$, Eq. (4.2)
		$+V_{1N:\mathrm{\hspace{0.17em}1}/m^2}^0$	$+V_{2N:\mathrm{cont}}^0$, Eq. (5.6) of Kolling:2011mt †	$+V_{3N:\mathrm{cont}}^0$, Eq. (4.3)
			$+V_{2N:\mathrm{\hspace{0.17em}1}\pi ,\mathrm{\hspace{0.17em}1}/m}^0$, Eq. (4.30) of Kolling:2011mt
† Different conventions are being used in the literature for the leading-order  two-nucleon
contact interactions $\propto C_{S,T}$. To match the convention of Refs. Epelbaum:2014sza ; Reinert:2017usi ; Epelbaum:2014efa , the factors of $32F_{\pi }^2$
in Eq. (5.7) of Kolling:2011mt should be replaced by $16F_{\pi }^2$.

Equations

$$S=-\sum _{i=1}^7{\bar{\beta }}_i{S}_i,$$

$$S_1$$

$$S_2$$

$$S_3$$

$$S_4$$

$$S_5$$

$$S_6$$

$$S_7$$

$$S^{\prime }={\bar{\beta }}_8{S}_8+{\bar{\beta }}_9{S}_9,$$

$$S_8$$

$$S_9$$

$${\bar{\beta }}_1=1,{\bar{\beta }}_4-3{\bar{\beta }}_5-3{\bar{\beta }}_6=2,$$

$$\displaystyle\langle\vec{p}\mkern 2.0mu\vphantom{p}^{\prime}|\hat{\mbox{$V$}}^{\mu}_{\rm 1N}|\vec{p}\,\rangle$$

$$\displaystyle\langle\vec{p}\mkern 2.0mu\vphantom{p}_{1}^{\prime}\vec{p}\mkern 2.0mu\vphantom{p}_{2}^{\prime}|\hat{\mbox{$V$}}^{\mu}_{\rm 2N}|\vec{p}_{1}\vec{p}_{2}\rangle$$

$$\displaystyle\langle\vec{p}\mkern 2.0mu\vphantom{p}_{1}^{\prime}\vec{p}\mkern 2.0mu\vphantom{p}_{2}^{\prime}\vec{p}\mkern 2.0mu\vphantom{p}_{3}^{\prime}|\hat{\mbox{$V$}}^{\mu}_{\rm 3N}|\vec{p}_{1}\vec{p}_{2}\vec{p}_{3}\rangle$$

$$\displaystyle\mbox{$\mathcal{V}$}_{{\rm 1N}}^{\mu}$$

$$\bar{u}(p)u(p)$$

$$\displaystyle{\mbox{$G$}}_{E}(Q^{2})$$

$$\displaystyle{\mbox{$G$}}_{M}(Q^{2})$$

$$\sqrt{\frac{m}{E_{p^{\prime}}}}\mbox{$\mathcal{V}$}_{{\rm 1N}}^{\mu}\sqrt{\frac{m}{E_{p}}}=\chi^{\prime\dagger}\bigg{(}{\mbox{$V$}}_{{\rm 1N:\,static}}^{\mu}+{\mbox{$V$}}_{{\rm 1N:}\,1/m}^{\mu}+{\mbox{$V$}}_{{\rm 1N:}\,1/m^{2}}^{\mu}+{\cal O}\bigg{(}\frac{1}{m^{3}}\bigg{)}\bigg{)}\chi,$$

$$\displaystyle{\mbox{$V$}}_{{\rm 1N:\,static}}^{0}$$

$$\displaystyle{\mbox{$V$}}_{{\rm 1N:}\,1/m}^{0}$$

$$\displaystyle{\mbox{$V$}}_{{\rm 1N:}\,1/m^{2}}^{0}$$

$$\displaystyle\vec{{\mbox{$V$}}}_{{\rm 1N:\,static}}$$

$$\displaystyle\vec{{\mbox{$V$}}}_{{\rm 1N:}\,1/m}$$

$$\displaystyle\vec{{\mbox{$V$}}}_{{\rm 1N:}\,1/m^{2}}$$

$$\vec{k}\cdot{\hat{\vec{\mbox{$V$}}}}(\vec{k},0)-\bigg{[}\hat{{\mbox{$H$}}},\,\hat{\mbox{$V$}}_{0}(\vec{k},0)-\frac{\partial}{\partial k_{0}}\bigg{(}\vec{k}\cdot{\hat{\vec{\mbox{$V$}}}}(\vec{k},k_{0})-\big{[}\hat{{\mbox{$H$}}},\,\hat{\mbox{$V$}}_{0}(\vec{k},k_{0})\big{]}\bigg{)}\bigg{]}=0,$$

$$\displaystyle\vec{k}\cdot{\vec{\mbox{$V$}}}_{{\rm 1N:\,static}}$$

$$\displaystyle\vec{k}\cdot{\vec{\mbox{$V$}}}_{{\rm 1N:}\,1/m}$$

$$\displaystyle\vec{k}\cdot{\vec{\mbox{$V$}}}_{{\rm 1N:}\,1/m^{2}}$$

$$\displaystyle\bigg{[}\frac{\hat{\vec{p}}^{\,2}}{2m},\,\hat{\mbox{$V$}}_{{\rm 1N:\,static}}^{0}\bigg{]}$$

$$\displaystyle\bigg{[}\frac{\hat{\vec{p}}^{\,2}}{2m},\,\hat{\mbox{$V$}}_{{\rm 1N:}\,1/m}^{0}\bigg{]}$$

$$\displaystyle\vec{\mbox{$V$}}_{{\rm 1N:\,off-shell}}$$

$$\displaystyle{\mbox{$V$}}_{{\rm 1N}}^{0}$$

$$\displaystyle\vec{{\mbox{$V$}}}_{{\rm 1N}}$$

$$\displaystyle\vec{k}\cdot{\vec{\mbox{$V$}}}_{{\rm 1N:\,off-shell}}(\vec{k},0)$$

$$\displaystyle\bigg{[}\frac{\hat{\vec{p}}^{\,2}}{2m},\,\frac{\partial}{\partial k_{0}}\vec{k}\cdot\hat{\vec{{\mbox{$V$}}}}_{{\rm 1N:\,off-shell}}(\vec{k},k_{0})\bigg{]}$$

$$\displaystyle\vec{{\mbox{$V$}}}_{{\rm 1N:\,singular}}^{(Q)}$$

$$\displaystyle\vec{\mbox{$V$}}_{{\rm 1N:\,off-shell}}$$

Full text

1 Introduction

Chiral effective field theory has been extensively used in the past decades to derive nuclear forces and the corresponding vector and axial charge and current operators. Presently, the two-nucleon force contributions have been worked out completely up through fifth order $Q^{5}$ of the chiral expansion, i.e. up to N4LO Entem:2014msa ; Epelbaum:2014sza ; Entem:2015xwa ; Reinert:2017usi ; Entem:2017gor . Here and in what follows, the expansion parameter is defined as $Q\in\{M_{\pi}/\Lambda_{b},\;p/\Lambda_{b}\}$, where $p$ refers to four- (three-)momenta of external pions (nucleons), while $\Lambda_{b}$ is the breakdown scale of the chiral expansion. In the Goldstone boson and single-baryon sectors one usually assumes $\Lambda_{b}=\Lambda_{\chi}\sim M_{\rho}\sim 4\pi F_{\pi}\sim 1$ GeV Manohar:1983md . In the few-nucleon sector, the breakdown scale is estimated to be of the order of $\Lambda_{b}\sim 600$ MeV Epelbaum:2014efa ; Furnstahl:2015rha . The expressions for the three- and four-nucleon forces up to fourth order, i.e. $Q^{4}$ or N3LO, calculated using dimensional regularization can be found in Refs. vanKolck:1994yi ; Epelbaum:2004fk ; Epelbaum:2006eu ; Ishikawa:2007zz ; Bernard:2007sp ; Bernard:2011zr . Most of the N4LO contributions to the three-nucleon force have also been worked out Krebs:2012yv ; Krebs:2013kha ; Epelbaum:2014sea ; Girlanda:2011fh .

Nuclear electroweak current operators have also attracted considerable attention. Starting from the pioneering (but incomplete and kinematically restricted) studies by Park et al. Park:1993jf ; Park:1995pn , the exchange electromagnetic currents have been worked out to the leading one-loop order $Q$, i.e. N3LO111The dominant contributions to the single- and two-nucleon operators appear at orders $Q^{-3}$ and $Q^{-1}$, respectively. When the nucleon mass $m$ is counted according to $m\sim\Lambda_{b}^{2}/Q$ Weinberg:1991um as done here and commonly employed in few-nucleon studies, see e.g. Refs. Epelbaum:2014sza ; Reinert:2017usi ; Epelbaum:2014efa ; Bernard:2007sp ; Bernard:2011zr ; Krebs:2012yv ; Krebs:2013kha ; Epelbaum:2014sea , no corrections to the charge and current operators appear at order $Q^{-2}$. Thus, the terms LO, NLO, N2LO and N3LO refer to the contributions at orders $Q^{-3}$, $Q^{-1}$, $Q^{0}$ and $Q$, respectively. Notice that the authors of Refs. Pastore:2008ui ; Pastore:2009is ; Pastore:2011ip employ the counting scheme with $m\sim\Lambda_{b}$ leading to a different expansion pattern. , using the method of unitary transformation (UT) Epelbaum:1998ka ; Epelbaum:2007us ; Epelbaum:2008ga in Refs. Kolling:2009iq ; Kolling:2011mt and time-ordered perturbation theory in Refs. Pastore:2008ui ; Pastore:2009is ; Pastore:2011ip , see also Ref. Hoferichter:2015ipa for a related discussion. In fact, the UT method was first used in this context in Ref. Walzl:2001vb . The main idea behind the method of UT is to find a unitary operator in the pion-nucleon Fock space which decouples its purely nucleonic subspace, called $\eta$-space, from the rest of the Fock space. Nuclear forces are then identified with the resulting decoupled Hamiltonian which acts on the nucleonic subspace. Notice that it is always possible to perform additional UTs on the $\eta$-space which affect the off-shell behavior of the nuclear potentials but do not change observable quantities. This unitary ambiguity was found to be considerably reduced if one demands renormalizability of the nuclear potentials Epelbaum:2006eu ; Epelbaum:2007us . While the on-shell scattering amplitude calculated in chiral effective field theory is defined unambiguously and guaranteed to be renormalizable, this does not hold anymore for irreducible (i.e. non-iterative) contributions alone, which are associated with nuclear forces and current operators. Renormalizability of nuclear potentials can, however, be enforced by performing suitable UTs on the $\eta$-space. For a broad class of UTs considered in Refs. Epelbaum:2006eu ; Epelbaum:2007us ; Bernard:2007sp , the resulting renormalized static contributions to the nuclear forces up to N4LO were found to be determined unambiguously, while the leading relativistic corrections depend on two arbitrary phases $\bar{\beta}_{8}$ and $\bar{\beta}_{9}$ Bernard:2011zr ; Epelbaum:2014efa , see also Friar:1999sj and references therein for a related earlier discussion of this kind of unitary ambiguity.

In the presence of external classical sources, the UTs determined in the strong sector as outlined above allow one to derive the corresponding charge and current operators. To enforce renormalizability also for the four-current operators one, however, needs to employ a more general class of $\eta$-space UTs, whose generators involve a single insertion of an external classical sources. At the order considered they are parametrized via Kolling:2011mt 222The minus sign is missing in Eq. (4.3) of Ref. Kolling:2011mt and for the quantities $\delta J_{c2,\,c5,\,c6,\,c7}$ in Eq. (4.6) of that paper.

[TABLE]

where $\bar{\beta}_{i}\in\mathbb{R}$ are the transformation “angles” while the corresponding anti-hermitian generators $S_{i}$ have the form

[TABLE]

Here, $\eta$ and $\lambda^{i}$ denote the projection operators onto the purely nucleonic part of the Fock space and Fock-space components with $i$ pions, respectively. Further, $E_{\pi}=\sum_{i}\sqrt{\vec{p}_{i}^{\,2}+M_{\pi}^{2}}$ is the total energy of pions with three-momenta $\vec{p}_{i}$ in the intermediate $\lambda^{i}$-state while $H_{ab}^{(\kappa)}$ ($J_{ab}^{(\kappa)}$) refer to the vertices in the effective chiral Hamiltonian with $a$ nucleon and $b$ pion fields without (with a single insertion of) external electromagnetic sources. The superscript $\kappa$ is related to the canonical field dimension of the fields via $\kappa=d+(3/2)n+p-4$, where $d$, $n$ and $p$ denote the number of derivatives or $M_{\pi}$-insertions and the nucleon and pion field operators, respectively. In the method of UT, all contributions to the nuclear forces and currents are written as time-ordered sequences of vertices and energy denominators. A contribution involving $N$ vertices of type $\kappa_{i}$, $i=1,\ldots N$, appears at order $Q^{\nu}$ with $\nu=\sum_{i}\kappa_{i}-2$ ($\nu=\sum_{i}\kappa_{i}-3$) for the forces (currents). More details on the notation and the method of UT can be found in Ref. Epelbaum:2006eu ; Epelbaum:2007us , while the explicit form of the relevant terms in the effective Lagrangian and Hamiltonian is given in Ref. Kolling:2011mt . For the sake of completeness, we also give the expressions for the generators of the already mentioned $\eta$-space UTs which parametrize the unitary ambiguity of the leading relativistic corrections to the nuclear forces and currents:

[TABLE]

with the operators $S_{8,9}$ given by

[TABLE]

Here, $\tilde{H}_{ab}^{(\kappa)}$ denote the $1/m$-corrections to the corresponding vertices.

The UTs listed in Eqs. (1.1,1.2) induce contributions to the electromagnetic charge and current operators at N3LO and higher chiral orders which restore their renormalizability upon a suitable choice of $\bar{\beta}_{i}$. More precisely, it was found in Ref. Kolling:2011mt that the expressions for the two-nucleon charge and current operators do not depend on the parameters $\bar{\beta}_{2}$, $\bar{\beta}_{7}$ and $\bar{\beta}_{5}-\bar{\beta}_{6}$. Further, renormalizability of the one-loop contributions to the one-pion exchange current operator, calculated in dimensional regularization, was found to require the choice

[TABLE]

while the parameter $\bar{\beta}_{3}$ was set to $\bar{\beta}_{3}=0$. With this choice, the one-loop contributions to the electromagnetic charge and current operators still depend on an arbitrary phase $\bar{\beta}_{4}$, for which the value of $\bar{\beta}_{4}=-1$ was adopted in Ref. Kolling:2011mt .333Requiring, in addition to renormalizability of the single- and two-nucleon current operators, also factorizability of the exchanged pions in the three-nucleon charge operator, see Krebs:2016rqz for more details, leads to the stronger constraints on the phases, namely $\bar{\beta}_{1}=1$, $\bar{\beta}_{4}=-1$ and $\bar{\beta}_{5}=\bar{\beta}_{6}=-1/2$.

In Ref. Krebs:2016rqz , the framework outlined above was extended to derive the nuclear axial charge and current operators. Similarly to the vector currents, renormalized expressions could only be obtained by taking into account additional $\eta$-space UTs whose generators depend on the external axial sources and have been parametrized in terms of the real phases $\alpha_{i}^{ax}$, $i=1,2,\ldots,33$. While renormalizability alone does not completely eliminate the resulting unitary ambiguity, the requirement of the pion-pole contributions to the axial current to match the corresponding terms in the nuclear potentials was shown to result in unique expressions. Ref. Krebs:2016rqz also provides a detailed discussion of the subtleties associated with the explicit time dependence of the additional UTs due to the dependence of their generators on the external classical sources. It is shown in that paper that the explicit time dependence of the UTs is, in general, expected to yield contributions which depend on the energy transfer $k_{0}$ of the external source. Also the continuity equations, which are manifestations of gauge invariance and the chiral symmetry, get modified and take a more general form that involves the energy-dependent contributions to the charge and current operators, see Eq. (2.42) of Ref. Krebs:2016rqz . The validity of the continuity equation for the axial current was explicitly verified in that paper. For a related recent work on the nuclear axial currents in the framework of time-ordered perturbation theory see Refs. Baroni:2015uza .

The purpose of this study is to complete the derivation of the electromagnetic currents at N3LO initiated in Kolling:2009iq ; Kolling:2011mt and to investigate the implications of the findings of Ref. Krebs:2016rqz . In particular, we discuss in detail the expressions for the single-nucleon charge and current operators including the relevant relativistic corrections and provide their parametrization in terms of the corresponding nucleon form factors. We also work out the three-nucleon contributions to the charge operator, which completes the derivation of the nuclear electromagnetic currents at N3LO. Last but not least, we explicitly verify the validity of the continuity equation for all considered classes of diagrams. Notice that since single-nucleon contributions were not considered in our earlier papers Kolling:2009iq ; Kolling:2011mt , the continuity equation could so far only be verified explicitly for the static two-pion exchange terms.

Our paper is organized as follows. In Sec. 2 we discuss the single-nucleon charge and current operators. Special attention is paid to the energy-transfer-dependent contributions, which lead to the already mentioned modified form of the continuity equation. In Sec. 3 we use the derived single-nucleon current to explicitly verify the validity of the continuity equation for the one-pion exchange (OPE) and short-range two-nucleon operatiors. Next, in Sec. 4 we derive the leading three-nucleon contributions to the electromagnetic charge operator. We observe that there are no three-nucleon current contributions up to the considered order. The main results of our paper are summarized in Sec. 5.

2 Single nucleon electromagnetic current

Here and in what follows, we stay as close as possible to the notation employed in Ref. Krebs:2016rqz . In particular, the one-, two- and three-nucleon four-current operators are defined according to

[TABLE]

where $\vec{p}_{i}$ ($\vec{p}\mkern 2.0mu\vphantom{p}_{i}^{\prime}$) denotes the incoming (outgoing) momentum of nucleon $i$ while $\vec{k}$ is the momentum of the external electromagnetic source. The electromagnetic current operators we are interested in here are linear combinations of isoscalar and isovector quantities. For the sake of compactness, we refrain from explicitly showing isospin indices except for the isospin Pauli matrices $\tau$. Further, $\hat{X}$ means that the quantity $X$ is to be regarded as an operator rather than matrix element with respect to the nucleon momenta. Finally, we employ throughout our work the same choice of the unitary phases $\bar{\beta}_{i}$ as adopted in Ref. Kolling:2011mt and explained in the previous section.

2.1 Parametrization of the single nucleon current in terms of

the form factors

Single-nucleon current operators can be conveniently parametrized in terms of the Dirac and Pauli electromagnetic form factors of the nucleon $F_{1}(Q^{2})$ and $F_{2}(Q^{2})$, respectively. In the relativistic kinematics, the on-shell single-nucleon current is given by

[TABLE]

where the $\gamma^{\mu}$ are Dirac matrices, $\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$, $k=p^{\prime}-p$ is the four-momentum transfer while $Q^{2}=-k_{\mu}k^{\mu}$. The Dirac spinors are normalized according to

[TABLE]

As already pointed out above, we suppress the isospin indices to simplify the notation. In the following, we will express all our results in terms of the Sachs electric and magnetic form factors

[TABLE]

The $1/m$-expansion of the non-relativistically normalized electromagnetic current is defined according to

[TABLE]

where $E(p)=\sqrt{m^{2}+\vec{p}^{\,2}}$, while $\chi^{\prime}$ and $\chi$ refer to Pauli spinors. In terms of the Sachs form factors, the charge operator is parametrized via

[TABLE]

while the current operator is given by

[TABLE]

where $\vec{k}_{1}=(\vec{p}^{\,\prime}+\vec{p}\,)/2$ and $\vec{\sigma}$ refers to the Pauli matrices in spin space. The powers of the nucleon mass in Eqs. (2.6,2.1) can be traced back to the convention, according to which the nonrelativistic expansion of the electric and magnetic Sachs form factors starts from terms of orders $m^{0}$ and $m^{1}$, respectively. Notice further that replacing $Q^{2}$ by $\vec{k}^{2}$ does not affect Eqs. (2.6,2.1) since $Q^{2}=\vec{k}^{2}+{\cal O}(1/m^{2})$.

Before discussing the results for the charge and current operators in the method of UT, which may generally be expected to differ from the on-shell expressions given in Eqs. (2.6) and (2.1) by off-shell terms, it is instructive to address the validity of the continuity equation for the on-shell contributions alone. The general form of the continuity equation for the vector current at the considered order in the chiral expansion can be written as Krebs:2016rqz

[TABLE]

where $H$ denotes the strong part of the nuclear Hamiltonian. The longitudinal part of the on-shell single nucleon current operator is given by

[TABLE]

while the commutator of the kinetic energy with the on-shell charge operator reads

[TABLE]

Thus, if there is no dependence on the energy transfer, the parametrized on-shell current satisfies the continuity equation in the single-nucleon sector. This is, however, not the only possibility. If, for example, the current operator has an off-shell longitudinal component given by

[TABLE]

where $X$ is an arbitrary function of momenta, the single-nucleon electromagnetic current

[TABLE]

still satisfies the continuity equation (2.8) since

[TABLE]

and

[TABLE]

We now turn to the method of UT. Performing explicit calculations, we find that the on-shell expressions for the nuclear charge and current operators to have the same form as given in Eqs. (2.6) and (2.1). Using the same choice of the additional $\eta$-space UTs as adopted in Ref. Kolling:2011mt and described in Sec. 1, we, however, find an additional off-shell contribution to the single-nucleon current operator which depends on the phase $\propto\bar{\beta}_{6}$, see Eq. (1.2). Moreover, using dimensional regularization to calculate the loop integrals at order $Q$, we obtain a divergent contribution to the current of the form

[TABLE]

where $d$ denotes the number of space-time dimensions. Thus, renormalizability of the current operators requires taking $\bar{\beta}_{6}=-1/2$, and the finite pieces of single-nucleon current operator does explicitly depend on the energy transfer $k_{0}$. The corresponding off-shell contribution to the current operator can be expressed in terms of the form factors as

[TABLE]

Clearly, the unitary transformation $\propto\bar{\beta}_{6}$ in Eq. (1.2) only induces terms in \vec{\mbox{V}}_{{\rm 1N:\,off-shell}} up to order $Q$. In Appendix A we, however, provide an explicit form of the unitary transformation in the Hilbert space, which exactly generates the off-shell terms given in Eq. (2.1).

2.2 Chiral expansion of the Sachs form factors

In order to verify the validity of the continuity equation we provide in this section explicit expressions for the single-nucleon form factors at lowest orders in the chiral expansion, as already employed in Ref. Walzl:2001vb . We refer the reader to Ref. Kubis:2000zd ; Schindler:2005ke and references therein for a detailed analysis of the electromagnetic form factors of the nucleon in relativistic formulations of baryon chiral perturbation theory. We employ in this section the standard power counting used in the single-nucleon heavy baryon approach, where the nucleon mass is treated on the same footing as the chiral symmetry breaking scale $m\sim\Lambda_{\chi}$. The corresponding powers of the soft scale relative to the dominant contribution will be written in square brackets to avoid a possible confusion with the counting scheme employed throughout the rest of this paper as explained in Sec. 1. For what concerns the relativistic corrections, we only list below the contributions which are relevant to the derivation of the current operators at the considered order. For the sake of simplicity, we do note differentiate between the various quantities in the chiral limit and their physical values here.

• •

The leading contributions to the form factors {\mbox{G}}_{E}(Q^{2}) and {\mbox{G}}_{M}(Q^{2}) have the form

[TABLE]

where $\kappa_{s}$ and $\kappa_{v}$ refer to the isoscalar and isovector anomalous magnetic moments of the nucleon, respectively.

• •

At order $Q^{1}$, there are only loop contributions to the magnetic form factor

[TABLE]

where $g_{A}$, $F_{\pi}$ and $M_{\pi}$ refer to the nucleon axial-vector coupling, pion decay constant and pion mass, respectively. Further, the loop function $A(|\vec{k}\,|)$ is given by

[TABLE]

• •

At order $Q^{2}$ one has further contributions to both electromagnetic form factors. These read

[TABLE]

with the loop function $L(|\vec{k}\,|)$ given by

[TABLE]

Here, $c_{i}$, $\bar{d}_{i}$ and $\bar{e}_{i}$ are the low-energy constants (LECs) from the second-, third- and fourth-order pion-nucleon effective chiral Lagrangian, see Ref. Fettes:2000gb for details.

• •

At order $Q^{3}$ one has to take into account the leading two-loop contributions, whose calculation goes beyond the scope of our study, as well as the additional relativistic corrections at the one-loop level given by

[TABLE]

Notice that the chiral expansion of the electromagnetic form factors of the nucleon is known to converge rather slowly. The form factors are known to be largely driven by vector mesons Kubis:2000zd ; Schindler:2005ke , which are not included as explicit degrees of freedom in the approach we are using. In this sense, it is certainly more advantageous from the phenomenological point of view to employ empirical parametrizations of the single-nucleon form factors in the single-nucleon charge and current operators when performing few-nucleon calculations, rather then to strictly rely on their chiral expansion. Such an approach is, in fact, adopted in most of the studies in the few-nucleon sector, see e.g. the early work in Ref. Phillips:1999am .

3 Current conservation

Having derived the single-nucleon contributions we are now in the position to explicitly demonstrate current conservation by verifying the validity of the continuity equation (2.8).

The single-nucleon contributions to the current operator have already been shown to fulfill the continuity equation in Sec. 2.1. The leading two-nucleon contributions to the current operator at order $Q^{-1}$, stemming from the OPE at tree level, are also well known to fulfill the continuity equation. Further, as already pointed out in the introduction, the two-pion exchange contributions to the current operator at order $Q$ are shown to be conserved in Ref. Kolling:2009iq . Thus, it remains to verify the validity of the continuity equation for the order-$Q$ short-range pieces and corrections to the OPE.

The OPE contributions at the leading one-loop level have been calculated in Kolling:2011mt without considering possible contributions induced by the explicit time dependence of the imposed UTs in Eqs. (1.1,1.2). We have verified that using the choice of unitary phases $\bar{\beta}_{i}$ from Ref. Kolling:2011mt described in Sec. 1, the OPE contributions to the charge and current operators up to order $Q^{1}$ do not depend on the energy transfer $k_{0}$. Thus, the complete expressions for the OPE contributions to the charge and current operators at order $Q$ are given in Eqs. (4.28)-(4.31) of Ref. Kolling:2011mt . We now verify the validity of the continuity equation at order $Q^{2}$ for the OPE contributions and begin with the longitudinal part of the current operator

[TABLE]

where the order-$Q^{2}$ correction to OPE potential is given by

[TABLE]

with $\vec{q}$ denoting the three-momentum transfer of the nucleons, and the LEC $\bar{d}_{18}$ parameterizes the deviation from the Goldberger-Treiman relation Fettes:1998ud . The leading-order single-nucleon charge operator has the form

[TABLE]

Here and in what follows, $\vec{\sigma}^{(i)}$ and \mbox{\boldmath\tau}^{(i)} refer to the Pauli spin and isospin matrices of nucleon $i$. On the other hand, the longitudinal component of the energy-transfer-dependent current is given by

[TABLE]

so that

[TABLE]

The double commutator in the continuity equation (2.8) contributes at orders higher than $Q^{2}$. Thus, we conclude that the continuity equation for the contributions of the OPE-range is fulfilled at order $Q^{2}$ in the static limit of $m\to\infty$. Notice that without the \frac{\partial}{\partial k_{0}}\vec{k}\cdot\hat{\vec{{\mbox{V}}}}_{{\rm 1N:\,off-shell}}^{(Q)} contribution, the traditional form of the continuity equation \vec{k}\cdot\hat{\vec{{\mbox{V}}}}=[\hat{{\mbox{H}}},\,\hat{\mbox{V}}^{0}] would not be satisfied. The validity of the continuity equation for the leading relativistic corrections at the same chiral order follows trivially from \vec{{\mbox{V}}}_{{\rm 2N:}\,1\pi,\,1/m}^{(Q)}=0 along with the observation

[TABLE]

which implies that

[TABLE]

We now turn to the short-range terms. The leading static contribution to the current operator is generated by tree-level diagrams at order $Q^{1}$ and given in Eq. (5.3) of Kolling:2011mt . Clearly, the terms in \vec{k}\cdot\vec{{\mbox{V}}}_{{\rm 2N:\,cont}}^{(Q)} proportional to the LECs $C_{i}$ coincide with the corresponding contributions in [\hat{{\mbox{H}}}_{{\rm 2N:\,cont}}^{(Q^{2})},\,\hat{\mbox{V}}_{{\rm 1N:\,static}}^{0\,(Q^{-3})}], while the ones proportional to the LECs $L_{i}$ are purely transversal. Further, due to Eq. (3.4), we obtain

[TABLE]

Thus, the continuity equation for the short-range current is fulfilled at order $Q^{2}$ in the static limit. Since

[TABLE]

the continuity equation is trivially fulfilled for the short-range relativistic corrections at order $Q^{2}$ as well.

4 Three-nucleon charge operator

While the leading contributions to the three-nucleon current appear at order $Q^{2}$ which is beyond the accuracy of our study, the three-nucleon charge operator receives the leading contributions already at N3LO. Using the method of UT, we obtain the following result for the diagrams $\sim g_{A}^{4}$ shown in Fig. 1

[TABLE]

while the $\sim g_{A}^{2}$ contributions visualized in Fig. 2 have the form

[TABLE]

There are also diagrams involving a single insertion of the leading-order contact interactions shown in Fig. 3, which lead to the following result:

[TABLE]

To the best of our knowledge, the three-nucleon contributions to the charge operator have not been considered before in the framework of chiral effective field theory.

5 Summary and conclusions

In this paper, we have completed the derivation of the nuclear electromagnetic current and charge operators to fourth order (N3LO) in the chiral expansion. The corresponding contributions, obtained using dimensional regularization, are summarized in Tables 1 and 2.

The main results of our study can be summarized as follows:

• •

We have analyzed the single-nucleon contributions to the electromagnetic four-current at the one-loop level using the method of UT and identified the choice of the unitary transformations leading to a renormalized result. We also provide a parametrization of the single-nucleon operators including the leading and subleading relativistic corrections in terms of the electromagnetic form factors of the nucleon. Apart from the well known on-shell pieces, see e.g. Marcucci:2015rca , we found an additional energy-dependent off-shell contribution to the current operator whose form is dictated by the renormalizability constraint.

• •

We have verified the completeness of the expressions for the two-nucleon contributions derived in Refs. Kolling:2009iq ; Kolling:2011mt without considering possible energy-dependent pieces due to the explicit time-dependence of the employed unitary transformations, see Ref. Krebs:2016rqz for more details. For the choice of unitary phases $\bar{\beta}_{i}$ adopted in Refs. Kolling:2011mt , the two-nucleon current operator \mbox{V}_{\rm 2N}^{\mu} is found to be $k_{0}$-independent.

• •

We have worked out the leading three-nucleon contributions to the charge operator which appears at N3LO.

• •

With the complete N3LO result for the electromagnetic current \mbox{V}^{\mu} at hand, we were able to explicitly verify the validity of the modified form of the continuity equation derived in Ref. Krebs:2016rqz at order $Q^{2}$ for all one- and two-nucleon contributions.

The obtained results can be used to study electromagnetic properties of nuclei such as the form factors, photo- and electrodisintegration reactions and the radiative capture processes with light nuclei in the framework of chiral effective field theory with fully consistent nuclear forces and currents. This, however, would require the development and implementation of the regularization procedure, which is consistent with the one employed in the high-precision semilocal chiral nuclear potentials of Ref. Reinert:2017usi . Work along these lines is in progress.

Acknowledgments

This work was supported by DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”) and the BMBF (Grant No. 05P15PCFN1). Further support was provided by the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (grant no. 2018DM0034) and by VolkswagenStiftung (grant no. 93562).

Appendix A Unitary transformation to exactly generate the off-shell

single-nucleon current in Eq. (2.1)

In this appendix we construct a unitary transformation, which generates the off-shell single-nucleon longitudinal current in Eq. (2.1).

In order to see how the nuclear operators are affected by time-dependent unitary transformations, we perform an additional unitary transformation of the kind

[TABLE]

where $v_{\mu}(x)$ is an external vector source and \mbox{X}_{\mu}(\vec{x}\,) is some antihermitean operator. The unitary transformation changes the Hamilton operator in the presence of the vector source to

[TABLE]

with \mbox{H}:=\mbox{W}[0]. For the vector current operator one obtains444We adopt here the commonly accepted sign convention for the electromagnetic current, which differs from the one used in our paper Krebs:2016rqz .

[TABLE]

see Krebs:2016rqz for more details, where the operator \mbox{W}[v] is taken at $t=0$:

[TABLE]

Using

[TABLE]

where

[TABLE]

we obtain

[TABLE]

Thus, choosing

[TABLE]

we obtain the off-shell term given in Eq. (2.1). It is important to emphasize that the derivation of Eq. (1.7) does not rely on chiral perturbation theory. No approximations were made to derive this equation (except for neglecting the contributions involving more than a single insertion of the external vector source).