Nuclear Electromagnetic Currents to Fourth Order in Chiral Effective Field Theory

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.

Unitary Transformation to Exactly Generate the Off-Shell Single-Nucleon Current in Eq. (2.16)

In this appendix we construct a unitary transformation, which generates the off-shell single-nucleon longitudinal current in Eq. (2.16).

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

$$\begin{aligned} U (t)= & {} \exp \big (J(t)\big ), \quad J(t)\,=\,\int d^3 x \,v_\mu (\mathbf {x}, t) \, X^\mu (\mathbf {x}\,). \end{aligned}$$

(1.1)

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

$$\begin{aligned} W [v] \rightarrow U^\dagger (t) W[v] U(t) + \bigg (i\frac{\partial }{\partial t}U^\dagger (t)\bigg ) U(t)=W[v]+\big [\hat{H}, \, \hat{J}(t)\big ] - i\frac{\partial }{\partial t} J(t) + \mathcal{O}(v^2), \end{aligned}$$

(1.2)

with $H:=W[0]$. For the vector current operator one obtains⁴

$$\begin{aligned} V_\mu (\mathbf {k})= & {} -\frac{\delta W[v]}{\delta \tilde{v}^\mu (\mathbf {k},k_0)}\Bigg |_{v=0}, \quad \tilde{v}^\mu (\mathbf {k},k_0) :=\int \frac{d^4 x}{(2\pi )^4} e^{i k\cdot x} v^\mu (x), \end{aligned}$$

(1.3)

see [33] for more details, where the operator $W[v]$ is taken at $t=0$:

$$\begin{aligned} W[v]=H+\int d^3 x \,v_\mu (\mathbf {x},0)\, \tilde{V}^\mu (\mathbf {x}\,) \,=\,H + \int d^4 k\, \tilde{v}_\mu (\mathbf {k},k_0)\, V^\mu (\mathbf {k}\,). \end{aligned}$$

(1.4)

Using

$$\begin{aligned}&J(0)=\int d^3 x\, v^\mu (\mathbf {x},0) \, X_\mu (\mathbf {x}\,)\,=\,\int d^4 k \,\tilde{v}^\mu (\mathbf {k},k_0)\, \tilde{X}_\mu (\mathbf {k}\,), \nonumber \\&i\frac{\partial }{\partial t} J(t)\bigg |_{t=0}=\int d^3 x\, i\,\dot{v}^\mu (\mathbf {x},0) \, X_\mu (\mathbf {x}\,)\,=\,\int d^4 k\, k_0 \tilde{v}^\mu (\mathbf {k},k_0)\, \tilde{X}_\mu (\mathbf {k}\,), \end{aligned}$$

(1.5)

where

$$\begin{aligned} \tilde{X}^\mu (\mathbf {k}\,):=\int d^3 x \,e^{i \mathbf {k}\cdot \mathbf {x}}\, X^\mu (\mathbf {x}\,), \end{aligned}$$

(1.6)

we obtain

$$\begin{aligned} V^\mu (\mathbf {k}\,)\rightarrow V^\mu (\mathbf {k}\,) - \bigg (k_0 \tilde{X}^\mu (\mathbf {k}\,) - \big [\hat{H}, \, \hat{\tilde{X}}^\mu (\mathbf {k}\,)\big ]\bigg ). \end{aligned}$$

(1.7)

Thus, choosing

$$\begin{aligned} \langle p^\prime |\tilde{X}^0(\mathbf {k}\,)|p\rangle= & {} 0,\nonumber \\ \langle p^\prime |\tilde{\mathbf {X}}(\mathbf {k}\,) |p\rangle= & {} \chi ^{\prime \,\dagger }\bigg (-\mathbf {k} \frac{e}{\mathbf {k}^2}\bigg [\big ({G}_E(\mathbf {k}^2)-{G}_E(0)\big )+\frac{i}{2m^2}\mathbf {k}\cdot (\mathbf {k}_1\times \varvec{\sigma })\big ({G}_M(\mathbf {k}^2) -\,{G}_M(0)\big )\bigg ]\bigg )\chi ,\nonumber \\ \end{aligned}$$

(1.8)

we obtain the off-shell term given in Eq. (2.16). 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).