# Quantum Monte Carlo calculations of electromagnetic transitions in 8Be with
meson-exchange currents derived from chiral effective field theory

S. Pastore1<EMAIL_ADDRESS>R. B. Wiringa2<EMAIL_ADDRESS>Steven C.
Pieper2<EMAIL_ADDRESS>R. Schiavilla3,4<EMAIL_ADDRESS>1Department of
Physics and Astronomy, University of South Carolina, Columbia, South Carolina
29208
2Physics Division, Argonne National Laboratory, Argonne, Illinois 60439
3Theory Center, Jefferson Laboratory, Newport News, Virginia 23606
4Department of Physics, Old Dominion University, Norfolk, Virginia 23529

###### Abstract

We report quantum Monte Carlo calculations of electromagnetic transitions in
8Be. The realistic Argonne $v_{18}$ two-nucleon and Illinois-7 three-nucleon
potentials are used to generate the ground state and nine excited states, with
energies that are in excellent agreement with experiment. A dozen $M1$ and
eight $E2$ transition matrix elements between these states are then evaluated.
The $E2$ matrix elements are computed only in impulse approximation, with
those transitions from broad resonant states requiring special treatment. The
$M1$ matrix elements include two-body meson-exchange currents derived from
chiral effective field theory, which typically contribute 20–30% of the total
expectation value. Many of the transitions are between isospin-mixed states;
the calculations are performed for isospin-pure states and then combined with
empirical mixing coefficients to compare to experiment. Alternate mixings are
also explored. In general, we find that transitions between states that have
the same dominant spatial symmetry are in reasonable agreement with
experiment, but those transitions between different spatial symmetries are
often underpredicted.

###### pacs:

21.10.Ky, 02.70.Ss, 23.20.Js, 27.20.+n

##  I Introduction

We recently reported ab initio quantum Monte Carlo (QMC) calculations of
magnetic moments and electromagnetic (EM) transitions in $A\leq 9$ nuclei
Pastore13 . In that work, the calculated magnetic moments and $M1$ transitions
included corrections arising from EM two-body meson-exchange currents (MEC)
derived in two different approaches: 1) a standard nuclear physics
approximation (SNPA) MPPSW08 ; Marcucci05 , and 2) the chiral effective theory
($\chi$EFT) formulation of Refs. Pastore08 ; PGSVW09 ; Piarulli12 . Nuclear
wave functions (w.f.’s) were obtained from a Hamiltonian consisting of the
non-relativistic nucleon kinetic energy plus the Argonne $v_{18}$ (AV18) two-
nucleon WSS95  and Illinois-7 (IL7) three-nucleon P08b  potentials. The SNPA
MEC were constructed to obey current conservation with this Hamiltonian, while
the use of $\chi$EFT MEC constitutes a hybrid calculation. The two methods are
in substantial agreement, producing a theoretical microscopic description of
nuclear dynamics that successfully reproduces the available experimental data,
although the $\chi$EFT MEC give somewhat better results. Two-body components
in the current operators provide significant corrections to single-nucleon
impulse-approximation (IA) calculations. For example, they contribute up to
$\sim 40\%$ of the total predicted value for the 9C magnetic moment Pastore13
.

Figure 1: (Color online) Experimental spectrum of 8Be: horizontal lines denote
energy levels, with blue for $T$=0 states, magenta for mixed $T$=0+1 states,
and violet for $T$=2; black dash-dot lines indicate thresholds for breakup as
indicated and shaded areas denote the large widths of the 8Be rotational
states. Vertical lines with arrows indicate the electromagnetic transitions
studied: blue short-dash for $E2$, red long-dash for $M1$, and magenta dash-
dot for combined $E2$ and $M1$ transitions.

In this work, we implement the framework described above for twenty EM
transitions in the 8Be nucleus using only the $\chi$EFT MEC. The experimental
spectrum and EM transitions we consider are illustrated in Fig. 1. This even-
even nucleus exhibits a strong two-$\alpha$ cluster structure in its ground
state, characterized by angular momentum, parity, and isospin
$(J^{\pi};T)=(0^{+};0)$, and a Young diagram spatial symmetry that is
predominantly [44]. The ground state lies $\sim 0.1$ MeV above the threshold
for breakup into two $\alpha$’s, while the $(2^{+};0)$ state at $\sim 3$ MeV
excitation and $(4^{+};0)$ state at $\sim 11$ MeV, are [44] rotational states
with large ($\sim$ 1.5–3.5 MeV) decay widths. The next six higher states at
16–19 MeV excitation are three isospin-mixed doublets, with the first pair of
$(2^{+};0+1)$ states lying below the threshold for breakup into 7Li+$p$ and
having $\alpha+\alpha$ decay widths of $\sim 100$ keV. The isospin mixing is
due to the interplay between $T=0$ states and $T=1$ states that are the
isobaric analogs of the lowest three states in 8Li and 8B, all with the same
dominant [431] spatial symmetry. There are many additional broad excited
states above these isospin-mixed doublets that are not shown before the final
state we consider, the $(0^{+};2)$ isobaric analog of the 8He ground state at
27 MeV excitation, with dominant [422] spatial symmetry and a very narrow 5
keV decay width.

A comprehensive set of QMC calculations of $A=8$ nuclei was carried out in
Ref. WPCP00  for a Hamiltonian with AV18 and the older Urbana IX three-nucleon
potential PPCPW97 . More recently, energies, radii, and quadrupole moments of
this nucleus have been recalculated for the [44] symmetry states Datar13 , and
for the isospin-mixed states WPPM13 , using the newer Illinois-7 potential.
The present work complements these studies by calculating many EM transitions
between the low-lying states, which are also illustrated in Fig. 1. The $M1$
matrix elements include contributions from two-body $\chi$EFT currents, which
provide important corrections of order 20–30%. The two-body current
corrections to the $E2$ matrix elements are expected to be negligible because
they appear at higher order in the $\chi$EFT expansion Piarulli12  and are not
computed here.

QMC techniques and $\chi$EFT EM currents were presented in Ref. Pastore13  and
references therein. We refer to that work for more details on the
calculational scheme which is here briefly summarized in Sec. II. From there
on, we focus on providing and discussing the results. In particular, the
calculated 8Be energy spectrum is presented in Sec. III, while results for
$E2$ and $M1$ transitions are given in Sec. IV. We discuss the results in Sec.
V.

##  II QMC method, Nuclear Hamiltonian and $\chi$EFT EM Currents

EM transition matrix elements are evaluated between w.f.’s which are solutions
of the Schrödinger equation:

$H\Psi(J^{\pi};T,T_{z})=E\Psi(J^{\pi};T,T_{z})\ ,$ (1)

where $\Psi(J^{\pi};T,T_{z})$ is a nuclear w.f. with specific spin-parity
$J^{\pi}$, isospin $T$, and charge state $T_{z}$. The nuclear Hamiltonian used
in the calculations consists of a kinetic term plus two- and three-body
interaction terms, namely the AV18 WSS95  and the IL7 P08b , respectively:

$H=\sum_{i}K_{i}+\sum_{i<j}v_{ij}+\sum_{i<j<k}V_{ijk}\ .$ (2)

Nuclear w.f.’s are constructed in two steps. First, a variational Monte Carlo
(VMC) calculation is implemented to construct a trial w.f.
$\Psi_{V}(J^{\pi};T,T_{z})$ from products of two- and three-body correlation
operators acting on an antisymmetric single-particle state of the appropriate
quantum numbers. The correlation operators are designed to reflect the
influence of the interactions at short distances, while appropriate boundary
conditions are imposed at long range W91 ; PPCPW97 . The
$\Psi_{V}(J^{\pi};T,T_{z})$ has embedded variational parameters that are
adjusted to minimize the expectation value

$E_{V}=\frac{\langle\Psi_{V}|H|\Psi_{V}\rangle}{\langle\Psi_{V}|\Psi_{V}\rangle}\geq
E_{0}\ ,$ (3)

which is evaluated by Metropolis Monte Carlo integration MR2T2 . Here, $E_{0}$
is the exact lowest eigenvalue of $H$ for the specified quantum numbers. A
good variational trial function has the form

$|\Psi_{V}\rangle={\cal S}\prod_{i<j}^{A}\left[1+U_{ij}+\sum_{k\neq
i,j}^{A}\widetilde{U}^{TNI}_{ijk}\right]|\Psi_{J}\rangle\ ,$ (4)

where the ${\cal S}$ is a symmetrization operator. The Jastrow w.f. $\Psi_{J}$
is fully antisymmetric and includes all possible spatial symmetry states
within the $p$-shell that can contribute to the $(J^{\pi};T,T_{z})$ quantum
numbers of the state of interest, while $U_{ij}$ and
$\widetilde{U}^{TNI}_{ijk}$ are the noncommuting two- and three-body
correlation operators.

The second step improves on $\Psi_{V}$ by eliminating excited-state
contamination. This is accomplished by the Green’s function Monte Carlo (GFMC)
algorithm C87-88  which propagates the Schrödinger equation in imaginary time
($\tau$). The propagated w.f. $\Psi(\tau)=e^{-(H-E_{0})\tau}\Psi_{V}$, for
large values of $\tau$, converges to the exact w.f. with eigenvalue $E_{0}$.
In practice, a simplified version $H^{\prime}$ of the Hamiltonian $H$ is used
in the operator, which includes the isoscalar part of the kinetic energy, a
charge-independent eight-operator projection of AV18 called AV8′, a strength-
adjusted version of the three-nucleon potential IL7′ (adjusted so that
$\langle H^{\prime}\rangle\sim\langle H\rangle$), and an isoscalar Coulomb
term that integrates to the total charge of the given nucleus KNBSK99 . The
difference between $H$ and $H^{\prime}$ is calculated using perturbation
theory. More detail can be found in Refs. PPCPW97 ; WPCP00 .

Matrix elements of the operators of interest are evaluated in terms of a
“mixed” expectation value between $\Psi_{V}$ and $\Psi(\tau)$:

$\displaystyle\langle O(\tau)\rangle_{M}$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi(\tau)|O|\Psi_{V}\rangle}{\langle\Psi(\tau)|\Psi_{V}\rangle},$
(5)

where the operator $O$ acts on the trial function $\Psi_{V}$. The desired
expectation values, of course, have $\Psi(\tau)$ on both sides; by writing
$\Psi(\tau)=\Psi_{V}+\delta\Psi(\tau)$ and neglecting terms of order
$[\delta\Psi(\tau)]^{2}$, we obtain the approximate expression

$\displaystyle\langle O(\tau)\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi(\tau)|O|\Psi(\tau)\rangle}{\langle\Psi(\tau)|\Psi(\tau)\rangle}$
(6) $\displaystyle\approx$ $\displaystyle\langle O(\tau)\rangle_{M}+[\langle
O(\tau)\rangle_{M}-\langle O\rangle_{V}]~{},$

where $\langle O\rangle_{\rm V}$ is the variational expectation value.

For off-diagonal matrix elements relevant to this work the generalized mixed
estimate is given by the expression

$\displaystyle\frac{\langle\Psi^{f}(\tau)|O|\Psi^{i}(\tau)\rangle}{\sqrt{\langle\Psi^{f}(\tau)|\Psi^{f}(\tau)\rangle}\sqrt{\langle\Psi^{i}(\tau)|\Psi^{i}(\tau)\rangle}}$
(7) $\displaystyle\approx$ $\displaystyle\langle
O(\tau)\rangle_{M_{i}}+\langle O(\tau)\rangle_{M_{f}}-\langle O\rangle_{V}\ ,$

where

$\displaystyle\langle O(\tau)\rangle_{M_{f}}$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi^{f}(\tau)|O|\Psi^{i}_{V}\rangle}{\langle\Psi^{f}(\tau)|\Psi^{f}_{V}\rangle}\sqrt{\frac{\langle\Psi^{f}_{V}|\Psi^{f}_{V}\rangle}{\langle\Psi^{i}_{V}|\Psi^{i}_{V}\rangle}}\
,$ (8)

and $\langle O(\tau)\rangle_{M_{i}}$ is defined similarly. For more details
see Eqs. (19–24) and the accompanying discussions in Ref. PPW07 . Sources of
systematic error in the GFMC evaluation of operator expectation values (other
than $H^{\prime}$) include the use of mixed estimates and the constrained path
algorithm for controlling the Fermion sign problem in the propagation of
$\Psi(\tau)$. These are discussed in Ref. WPCP00 ; the convergence of the
current calculations is addressed at the beginning of Sec. III.

Nuclear EM currents are expressed as an expansion in many-body operators. The
current we use contains up to two-body effects, and is written as:

${\bf j}({\bf q})=\sum_{i}{\bf j}_{i}({\bf q})+\sum_{i<j}{\bf j}_{ij}({\bf
q})\ ,$ (9)

where ${\bf q}$ is the momentum associated with the external EM field. In what
follows, we use the notation

$\displaystyle{\bf k}_{i}={\bf p}_{i}^{\prime}-{\bf p}_{i}\ ,\qquad\qquad{\bf
K}_{i}=\left({\bf p}_{i}^{\prime}+{\bf p}_{i}\right)/2\ ,$ $\displaystyle{\bf
k}=\left({\bf k}_{1}-{\bf k}_{2}\right)/2\ ,\,\qquad{\bf K}={\bf K}_{1}+{\bf
K}_{2}\ ,$ (10)

where ${\bf p}_{i}$ (${\bf p}_{i}^{\prime}$) is the initial (final) momentum
of nucleon $i$, and ${\bf q}={\bf k}_{1}+{\bf k}_{2}$ by momentum
conservation.

There are two one-body operators resulting from retaining the first two terms
in the $({\bf p}_{i}/m_{N})^{2}$ expansion of the covariant single-nucleon EM
current. Of course, the leading-order term in this expansion corresponds to
the non-relativistic IA operator consisting of the convection and spin-
magnetization single-nucleon currents:

${\bf j}^{\rm IA}=\frac{e}{2\,m_{N}}\left[\,2\,e_{N,1}\,{\bf
K}_{1}+i\,\mu_{N,1}\,{\bm{\sigma}}_{1}\times{\bf q}\,\right]\ ,$ (11)

where

$e_{N}=(1+\tau_{z})/2\ ,\,\,\,\kappa_{N}=(\kappa_{S}+\kappa_{V}\tau_{z})/2\
,\,\,\,\mu_{N}=e_{N}+\kappa_{N}\ .$ (12)

Here $\kappa_{S}=-0.12$ n.m. and $\kappa_{V}=3.706$ n.m. are the isoscalar
(IS) and isovector (IV) combinations of the anomalous magnetic moments of the
proton and neutron, and $e$ is the electric charge.

Two-body EM currents are constructed from a $\chi$EFT which retains as
explicit degrees of freedom both pions and nucleons. The resulting operators
are expressed as an expansion in nucleon and pion momenta, generically
designated as $Q$. The leading-order (LO) contribution in Eq. (11) is of order
$e\,Q^{-2}$ and contributions up to N3LO or $e\,Q^{1}$ are retained in the
expansion. These contributions were first calculated by Park et al. in Ref.
Park96  using covariant perturbation theory. More recently, Kölling and
collaborators Kolling09-11 , as well as some of the present authors Pastore08
; PGSVW09 ; Pastore11 ; Piarulli12 , derived them using two different
implementations of time-ordered perturbation theory. In this work, we use the
operators developed in Refs. Pastore08 ; PGSVW09 ; Pastore11 ; Piarulli12 ,
where details on the derivation and a complete listing of the formal
expressions may be found.

The two-body $\chi$EFT EM currents consist of long- and intermediate-range
components described in terms of one-pion exchange (OPE) and two-pion exchange
(TPE) contributions, respectively, as well as contact currents encoding short-
range dynamics. In particular, OPE seagull and pion-in-flight currents appear
at next-to-leading order (NLO) ($e\,Q^{-1}$) in the $Q$ expansion, while TPE
currents occur at N3LO. The LO and N2LO ($e\,Q^{0}$) contributions are given
by the single-nucleon operators described above, i.e., the IA operator and its
relativistic correction, respectively.

At N3LO, the current operators involve a number of unknown low energy
constants (LECs) which are fixed to experimental data. The LECs multiplying
four-nucleon contact operators are of two kinds, namely minimal and non-
minimal. The former also enter the $\chi$EFT nucleon-nucleon potential at
order $Q^{2}$ and are therefore fixed by reproducing the $np$ and $pp$ elastic
scattering data, along with the deuteron binding energy. For these, we take
the values resulting from the fitting procedure implemented in Refs. Entem03 ;
Machleidt11 . Non-minimal LECs (there are two of them, one multiplying an
isoscalar operator and the other an isovector operator) need to be fixed to EM
observables.

At N3LO, there is also an additional current of one-pion range which involves
three LECs. One of these multiplies an isoscalar structure, while the
remaining two multiply isovector structures. As first observed in Ref. Park96
, the isovector component of this current has the same operator structure as
that associated with a $\Delta$-resonance transition current involving a one-
pion exchange. In this type of two-body contribution, the external photon
couples with a nucleon to excite a $\Delta$-resonance state. The latter decays
emitting a pion which is then reabsorbed by a second nucleon. Given this
theoretical insight, one can impose the condition that the two isovector LECs
are in fact given by the couplings of the $\Delta$-resonance current. This
mechanism is referred to as $\Delta$-resonance saturation and has been
utilized in various studies of EM observables of light nuclei (see for example
Song07 ; Song09 ; Lazauskas11 ; Piarulli12 ; Pastore13 ; Girlanda10 ). Once
the $\Delta$-saturation mechanism is invoked to fix two of the unknown LECs,
the resulting three LECs are fit to the deuteron and the trinucleon magnetic
moments.

The values of the LECs are not unique, in that they depend on the particular
momentum cutoff used to regularize the configuration-space singularities of
the EM operators. In momentum space, these operators have a power law behavior
for large momenta, $k$, which is regularized by a momentum cutoff of the form
$C(k)={\rm exp}(-k^{4}/\Lambda^{4})$. For a list of the numerical values of
the LECs for $\Lambda=600$ MeV, which is the cutoff utilized in these
calculations, we refer to Ref. Pastore13 .

The N2LO relativistic correction to the one-body IA operator involves two
derivatives acting on the nucleon field. In the GFMC calculation we do not
explicitly evaluate this $p_{i}^{2}$ term, but instead approximate it with its
average value, that is $p_{i}^{2}\sim\left<p_{i}^{2}\right>$, as determined
from the expectation value of the kinetic energy operator in 8Be, from which
we obtain $\left<p_{i}^{2}\right>=1.375$ fm-2. This term is a small fraction
of the total MEC (see, e.g., Table 4 below) so the approximation has little
practical effect.

To be consistent with the nomenclature utilized in Ref. Pastore13 , we denote
with ‘MEC’ components in the EM currents beyond the IA one-body operator at
LO. However, we stress that the N2LO contribution is a one-body operator,
which does not involve meson-exchange mechanisms.

##  III 8Be Energy Spectrum

The experimental Tilley04  and calculated GFMC energies for the 8Be spectrum
are presented in Table 1, along with the GFMC point proton radii. The
calculations were done by propagating up to some $\tau_{\rm max}$ with an
evaluation of observables after every 40 propagation steps, i.e., at intervals
of $\tau=0.02$ MeV-1, and averaging in the interval $\tau$=[(0.1
MeV${}^{-1})$–$\tau_{\rm max}$]; $\tau_{\rm max}$ is typically 0.3 to 0.4
MeV-1.

The calculation of the spectrum is rather involved WPCP00 , with two main
challenges to face. The first originates from the resonant nature of the first
two excited states (gray shaded states in Fig. 1), and the ensuing difficulty
of extracting a stable resonance energy from the calculated energies which are
evolving to the energy of two separated $\alpha$’s. This issue was addressed
in Ref. WPCP00 , and more recently, however succinctly, in Ref. Datar13 . The
last reference reported an updated measurement of the $E2$ transition between
the first two excited states of 8Be measured via the $\alpha+\alpha$ radiative
capture with an uncertainty of $\sim 10$% (as opposed to the estimated $\sim
30$% error of previous measurements Datar05 ). To accompany the experimental
result, a GFMC calculation was performed for the $E2$ transition matrix
element between the two rotational states, and between the $(2^{+};0)$ state
and the ground state. We reprise this calculation in more detail below.

Table 1: GFMC ground state energy and excitations in MeV for the AV18+IL7 Hamiltonian compared to experiment Tilley04  for the 8Be spectrum. Empirical energies are obtained by unfolding the isospin-mixed experimental energies using inferred mixing coefficients (see text for explanation). Also given are the GFMC point proton (= neutron) radii in fm. Theoretical or experimental errors $\geq 1$ in the last digit are shown in parentheses. $J^{\pi};T$ |  GFMC |  Empirical |  Experiment |  $r_{p}$
---|---|---|---|---
$0^{+}$ |  –56.3(1) |  |  –56.50 |  2.40
$2^{+}$ |  \+ 3.2(2) |  |  \+ 3.03(1) |  2.45(1)
$4^{+}$ |  +11.2(3) |  |  +11.35(15) |  2.48(2)
$2^{+}_{2};0$ |  +16.8(2) |  +16.746(3) |  +16.626(3) |  2.28
$2^{+};1$ |  +16.8(2) |  +16.802(3) |  +16.922(3) |  2.33
$1^{+};1$ |  +17.5(2) |  +17.66(1) |  +17.640(1) |  2.39
$1^{+};0$ |  +18.0(2) |  +18.13(1) |  +18.150(4) |  2.36
$3^{+};1$ |  +19.4(2) |  +19.10(3) |  +19.07(3) |  2.31
$3^{+};0$ |  +19.9(2) |  +19.21(2) |  +19.235(10) |  2.35
$0^{+};2$ |  +27.7(2) |  |  +27.494(2) |  2.58

The second non-trivial issue is encountered when dealing with the spectrum of
the isospin-mixed states at $16$–$19$ MeV (magenta states in Fig. 1). These
excited states have been extensively discussed in Ref. WPPM13 . We compute
unmixed $T=0$ or $T=1$ states but experimental values are of course for the
mixed states. The isospin-mixing coefficients can be extracted from
experimental decay widths Barker66 . For the $2^{+}$ multiplet this is
unambiguous, but for the $1^{+}$ and $3^{+}$ multiplets theoretical decay
widths based on shell-model calculations have been used. This is discussed
further below. In Table 1 we use the mixing parameters to unfold the
“empirical” pure-isospin energies for comparison with our calculations, while
in subsequent tables we fold the computed EM matrix elements to generate mixed
matrix elements to compare to the data.

We studied the convergence of the GFMC calculations with respect to variations
in the number of unconstrained steps ($n_{u}$=20 and 50) followed after the
path constraint is relaxed, and found that energies, magnetic moments, and rms
radii converge at $n_{u}=20$, which is what is used for the final results
reported here. Most of the calculations we present are obtained by averaging
two calculations, each using 50,000 walkers. For the physically narrow,
nonresonant states, the energy expectation value is seen to stabilize at
$\tau\sim 0.1$ MeV-1.

Figure 2: (Color online) GFMC propagation in imaginary time $\tau$ of the
energy expectation values of the first three states in the 8Be spectrum. Black
dots are GFMC propagation points for the ground state, blue dots refer to the
$(2^{+};0)$ rotational state at $\sim 3$ MeV excitation and red dots to the
$(4^{+};0)$ state at $\sim 11$ MeV excitation. Solid lines represent a linear
fit to the GFMC points in the indicated time interval.

For the physically wide, resonant states, the binding energy, magnitude of the
quadrupole moment, and point proton radius all increase monotonically as
$\tau$ increases. We interpret this as an indication that the system is
dissolving into two separated $\alpha$’s. In Fig. 2, we show the GFMC
propagation points for the energy expectation values of the first three states
of 8Be. In particular, the ground state energy is obtained with $n_{u}=20$ and
20,000 walkers, while the resonant state energies are obtained using
$n_{u}=20$ and averaging two calculations with 50,000 walkers each. From the
figure, we see that the ground state initial VMC energy expectation value at
$\tau=0$ quickly drops and reaches stability around $\tau=0.1$ MeV-1 (this
point is indicated in the figure with an open star). The energies of the two
resonant states, instead, keep falling with time: the $(2^{+};0)$ state
decreases 0.25 MeV over the interval $\tau=\left[0.1,0.3\right]$ MeV-1, while
the $(4^{+};0)$ states falls by 1 MeV. With this declining energy there is a
corresponding increase of the point proton radius expectation values, as shown
in Fig. 3 and in the magnitude of the (negative) electric quadrupole moment.

Quantities associated with the resonant states have been calculated assuming
that, also for these states, $\tau\sim 0.1$ MeV-1 is the point at which
spurious contamination in the nuclear w.f.’s have been eliminated by the GFMC
propagation. Thus, we make a linear fit to the GFMC values in the interval
$\tau=\left[0.1,0.3\right]$ MeV-1, and extrapolate to $\tau=0.1$ MeV-1 for the
reported values. The choice of $\tau=0.1$ MeV-1 is somewhat arbitrary. To
account for this uncertainty we increase the GFMC statistical error by a
systematic error that is obtained by studying the sensitivity of the results
with respect to fitting procedures implemented in two different intervals,
namely $\tau=\left[0.08,0.3\right]$ MeV-1 and $\tau=\left[0.12,0.3\right]$
MeV-1, while keeping the same extrapolating point. The total error is
represented in the figures by the dashed lines.

Figure 3: (Color online) GFMC propagation in imaginary time $\tau$ of the
point proton radius expectation values of the first three states in 8Be
spectrum; notation is the same as in Fig. 2.

For the six states at 16–19 MeV excitation, the GFMC calculations are done for
pure isospin states of either $T=0$ or $1$. The w.f.’s of the isospin-mixed
states are written as

$\displaystyle\Psi_{J}^{a}$ $\displaystyle=$
$\displaystyle\alpha_{J}\Psi_{J,T=0}+\beta_{J}\Psi_{J,T=1},$
$\displaystyle\Psi_{J}^{b}$ $\displaystyle=$
$\displaystyle\beta_{J}\Psi_{J,T=0}-\alpha_{J}\Psi_{J,T=1},$ (13)

where the mixing angles satisfy $\alpha_{J}^{2}+\beta_{J}^{2}=1$. As one can
see from Fig. 1 and Table 1, experimentally there are two $J^{\pi}=2^{+}$
isospin-mixed states at 16.626 and 16.922 MeV excitation energies, two
$J^{\pi}=1^{+}$ states at 17.64 and 18.15 MeV, and two $J^{\pi}=3^{+}$ states
at 19.07 and 19.235 MeV. The mixing angles are inferred from the experimental
values of the decay widths. We follow the analysis carried out by Barker in
Ref. Barker66  and update the experimental widths with more recent values to
obtain the following mixing coefficients WPPM13 :

$\displaystyle\alpha_{2}=0.7705(15),\,\,\,\,\,\beta_{2}=0.6375(19)\,,$
$\displaystyle\alpha_{1}=0.21(3),\,\,\qquad\beta_{1}=0.98(1)\,,$ (14)
$\displaystyle\alpha_{3}=0.41(10),\qquad\beta_{3}=0.91(5)\,.$

Mixing coefficients for the $2^{+}$ states are well known because for these
states there is only one decay channel energetically open, that is the
2$\alpha$ emission channel, for which the experimental widths are known with
$\sim 0.5$% accuracy. For the other isospin-mixed states, multiple decay
channels are available, which makes the extraction of the mixing coefficients
less direct. In addition theoretical values of $M1$ matrix elements must be
used; the values above were obtained using traditional shell-model without
two-body current contributions to the matrix elements WPPM13 ; Barker66 .
Revised mixing parameters for the $1^{+}$ pair, computed using the $M1$ matrix
elements developed here, are discussed in Sec. V.

The eigenenergies of the isospin-mixed states, in Table 1 are given by

$E_{a,b}=\frac{H_{00}+H_{11}}{2}\pm\sqrt{\left(\frac{H_{00}-H_{11}}{2}\right)^{2}+\left(H_{01}\right)^{2}}$
(15)

where $H_{00}$ is the diagonal energy expectation in the pure $T$=0 state,
$H_{11}$ is the expectation value in the $T$=1 state, and $H_{01}$ is the off-
diagonal isospin-mixing (IM) matrix element that connects $T$=0 and 1. The
inferred $H_{00}$ and $H_{11}$ are the empirical values given in Table 1.

Finally, the narrow $(0^{+};2)$ state at 27 MeV excitation, which has a
dominant [422] spatial symmetry, is a straightforward GFMC calculation. There
could in principle be isospin-mixing with the third $(0^{+}_{3};0)$ state in
the $p$-shell construction of 8Be, which also has [422] symmetry, via the EM
and charge-dependent parts of AV18. No such state has been identified
experimentally. A first VMC calculation places this state 0.7(1) MeV higher in
excitation with a 125 keV IM matrix element, which predicts
$\alpha_{0}=0.19(4)$ and $\beta_{0}=0.98(1)$. This small amount of mixing may
still have a moderate effect on the width of the physical state, as discussed
below.

The overall agreement between experiment and the calculated GFMC spectrum for
AV18+IL7 shown in Table 1 is excellent. Only the $3^{+}$ isospin-mixed doublet
is a little too high in excitation and a little too spread out compared to the
measured values.

##  IV Electromagnetic Transitions in 8Be

We present our results in terms of reduced matrix elements (using Edmonds’
convention) of the $E2$ and $M1$ operators, the associated $B(E2)$ and
$B(M1)$, and the resulting widths. For a transition of multipolarity $\lambda$
($X$ designates $E$ or $M$),

$B(X\lambda)=\left<\Psi_{J_{f}}\left|\left|X\lambda\right|\right|\Psi_{J_{i}}\right>^{2}/(2J_{i}+1)$
(16)

is in units of $e^{\lambda}$ fm2λ for electric transitions and (n.m.)2λ for
magnetic transitions. The widths are given by

$\displaystyle\Gamma_{X\lambda}=\frac{8\pi(\lambda+1)}{\lambda[(2\lambda+1)!!]^{2}}\alpha\hbar
c\left(\frac{\Delta E}{\hbar c}\right)^{2\lambda+1}B(X\lambda)\ ,$ (17)

where $\Delta E$ is the difference in MeV between the experimental initial and
final state energies, $E_{i}$ and $E_{f}$; $\alpha$ is the fine-structure
constant; and $\hbar c$ is in units of MeV fm.

The calculations of electromagnetic matrix elements have been described in
detail in Refs. PPW07 ; MPPSW08 . Our present results for $E2$ transitions in
8Be are given in Table 2 where the initial and final $(J^{\pi};T)$ states and
the dominant associated spatial symmetries are shown in the first column and
the reduced matrix elements between states of pure isospin are given in the
second column. The experimental energies for the physical states are given in
the third column, and the corresponding theoretical and experimental widths
are shown in the fourth and fifth columns. We use the IA operator

$E2=e{\sum_{k}}\frac{1}{2}\left[r_{k}^{2}Y_{2}(\hat{r}_{k})\right](1+\tau_{kz})$
(18)

without any MEC corrections.

Figure 4: (Color online) GFMC propagation in imaginary time $\tau$ of the
reduced $E2$ matrix elements among the first three states in 8Be spectrum;
upper red dots are for the $(4^{+};0)\rightarrow(2^{+};0)$ transition, lower
blue dots are for the $(2^{+};0)\rightarrow(0^{+};0)$ transition and open
stars denote the extrapolated values.

In previous calculations PPW07 ; M+PW12 ; Pastore13  of nuclei in the $A=6$–10
range, we have found that $E2$ matrix elements of narrow states are generally
quite stable under GFMC propagation, and seldom vary much from the initial VMC
estimate. However, matrix elements from wide states, e.g., for the 6Li
$(2^{+};0)\rightarrow(0^{+};0)$ decay, show a significant evolution as a
function of $\tau$. This is also true for the first two transitions in 8Be
from the broad rotational $2^{+}$ and $4^{+}$ states. The matrix element grows
monotonically as the GFMC solution evolves in $\tau$ toward a separated
$\alpha+\alpha$ configuration, as illustrated in Fig. 4. This growth is slow
for the lower $(2^{+};0)\rightarrow(0^{+};0)$ transition, but much more
pronounced for the upper $(4^{+};0)\rightarrow(2^{+};0)$ transition.
Consequently, we make an extrapolation back to $\tau=0.1\pm 0.02$ MeV-1 to
obtain our best estimate for the matrix element, just as we did for the energy
and point proton radius discussed above in conjunction with Figs. 2 and 3. Our
error estimate combines both the Monte Carlo statistical error and the
uncertainty in the extrapolation point. The numerical results for these two
matrix elements and corresponding decay widths $\Gamma_{E2}$ are reported at
the top of Table 2. The transitions, which are between states of the same
dominant [44] spatial symmetry, are very large and consistent with a rotor
picture for 8Be.

Table 2: Calculated reduced $E2$ matrix elements and the corresponding decay widths compared to experiment Tilley04 ; Datar13 . The various columns show 1) the initial and final $(J^{\pi};T)$ states and the dominant associated spatial symmetries, 2) the GFMC matrix elements between states of pure isospin, 3) the experimental energies, and 4) the isospin-mixed theoretical and 5) experimental widths. In the width values we use the notation $[-x]=10^{-x}$. $(J^{\pi}_{i};T_{i})\rightarrow(J^{\pi}_{f};T_{f})$ | E2[$e$ fm2] |  $E_{i}$[MeV]$\rightarrow E_{f}$[MeV] |  $\Gamma_{E2}$[eV]
---|---|---|---
[s.s.]${}_{i}\rightarrow$[s.s.]f |  |  | IA | Expt.
$(2^{+};0)\rightarrow(0^{+};0)$ | 10.0(2) | $3.03\rightarrow 0.$ | 4.12(16)[-3] | –
$(4^{+};0)\rightarrow(2^{+};0)$ | 15.6(4) | $11.35\rightarrow 3.03$ | 0.87(5) | 0.67(7)
$[44]\rightarrow[44]$ |  |  |  |
$(2^{+}_{2};0)\rightarrow(0^{+};0)$ | 0.55(11) | $16.626\rightarrow 0.$ | 1.6(1.0)[–2] | 7.0(2.5)[–2]
$(2^{+};1)\rightarrow(0^{+};0)$ | –0.23(2) | $16.922\rightarrow 0.$ | 6.2(2.0)[–2] | 8.4(1.4)[–2]
$[431]\rightarrow[44]$ |  |  |  |
$(2^{+}_{2};0)\rightarrow(2^{+};0)$ | 0.26(7) | $16.626\rightarrow 3.03$ | 3.6(2.2)[–3] | –
$(2^{+};1)\rightarrow(2^{+};0)$ | 0.03(2) | $16.922\rightarrow 3.03$ | 1.7(1.4)[–3] | –
$[431]\rightarrow[44]$ |  |  |  |
$(1^{+};1)\rightarrow(2^{+};0)$ | 1.93(6) | $17.64\rightarrow 3.03$ | 0.63(5) | 0.12(5)
$(1^{+};0)\rightarrow(2^{+};0)$ | –0.03(5) | $18.15\rightarrow 3.03$ | 4.0(1.1)[–2] | –
$[431]\rightarrow[44]$ |  |  |  |

We have also calculated an additional six $E2$ transitions from the isospin-
mixed $2^{+}$ and $1^{+}$ doublets with dominant [431] spatial symmetry, to
the $T=0$ ground state or first $2^{+}$ state. We denote the isospin-pure
matrix elements by
$E2_{T_{f}T_{i}}=\left<\Psi_{J_{f},T_{f}}||E2||\Psi_{J_{i},T_{i}}\right>$ and
then use the definitions given in Eq. (III) to combine them via

$\displaystyle<\Psi_{J_{f},0}||E2||\Psi^{a}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\alpha_{J_{i}}E2_{00}+\beta_{J_{i}}E2_{01}\ ,$
$\displaystyle<\Psi_{J_{f},0}||E2||\Psi^{b}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\beta_{J_{i}}E2_{00}-\alpha_{J_{i}}E2_{01}\ ,$ (19)

to evaluate the widths of the physical transitions for comparison to
experiment. Because the $E2$ operator largely preserves spatial symmetry,
these transitions are much weaker than the ones within the $\alpha$-$\alpha$
rotational band. This makes accurate calculations of these transitions
significantly more difficult.

As an example, we can compare the two $E2$ transitions from the first and
second $2^{+}$ states to the $0^{+}$ ground state. As discussed in Refs. PWC04
; W06 , the $0^{+}$ state has five contributing $LS$-coupled symmetry
components: ${}^{1}S[44]$, ${}^{3}P[431]$, ${}^{5}D[422]$, ${}^{1}S[422]$, and
${}^{3}P[4211]$, with the first component having an amplitude in the present
VMC starting w.f. of 0.996. The $2^{+}$ states are linear combinations of
eight components: ${}^{1}D[44]$, ${}^{3}P[431]$, ${}^{3}D[431]$,
${}^{3}F[431]$, ${}^{5}S[422]$, ${}^{5}D[422]$, ${}^{1}D[422]$, and
${}^{3}P[4211]$. The first $2^{+}$ state also has an amplitude of 0.996 from
the ${}^{1}D[44]$ component, while the second $2^{+}$ state is dominated by
the ${}^{3}P[431]$ component with an amplitude of 0.902. Consequently, 99% of
the large $E2$ transition from the first excited state to the ground state is
due to the matrix element between the ${}^{1}D[44]$ and ${}^{1}S[44]$
components. However, for the much smaller $E2$ transition from the second
$2^{+}$ state, this pair of components contributes 1.65 times the final
result, canceled by the matrix element between the two ${}^{3}P[431]$
components, which gives $-1.44$ times the final result. The remaining 38
smaller terms, among which there is much additional cancellation, give 80% of
the total.

Changes in these small components, which may have little effect on the energy
of a given state and hence are not highly constrained by the GFMC propagation,
can have a significant effect on the $E2$ matrix element. These small
components may also be rather sensitive to the three-body potential in the
Hamiltonian, as noted in an earlier study of $E2$ transitions in $A=10$ nuclei
M+PW12 . This is also true for many of the $M1$ transitions discussed below,
when the initial and final states have different dominant spatial symmetries.

Table 3: Calculated reduced $M1$ matrix elements and corresponding decay widths compared to experiment Tilley04 . The various columns show 1) the initial and final $(J^{\pi};T)$ states and the dominant associated spatial symmetries, 2) the GFMC matrix elements between states of pure isospin in IA and, 3) in total after adding MEC, 4) the % $z$ of the total given by the MEC, 5) the experimental energies, 6) the isospin-mixed theoretical decay widths in IA and, 7) in total, and 8) experimental values. In the width values we use the notation $[-x]=10^{-x}$. The results marked with a * or † are extra VMC calculations discussed in the text. $(J_{i};T_{i})\rightarrow(J_{f};T_{f})$ | M1[n.m.] |  $E_{i}$[MeV]$\rightarrow E_{f}$[MeV] |  $\Gamma_{M1}$[eV]
---|---|---|---
[s.s.]${}_{i}\rightarrow$[s.s.]f | IA | Total | $z$ |  | IA | Total | Expt.
$(2^{+}_{2};0)\rightarrow(2^{+};0)$ | 0.014(6) | 0.013(6) |  | $16.626\rightarrow 3.03$ | 0.23(3) | 0.51(6) |
$(2^{+};1)\rightarrow(2^{+};0)$ | 0.297(12) | 0.447(18) | 33% | $16.922\rightarrow 3.03$ | 0.30(4) | 0.70(7) |
$[431]\rightarrow[44]$ |  |  |  | $16.626+16.922\rightarrow 3.03$ | 0.53(5) | 1.21(9) | 2.80(18)
$(1^{+};1)\rightarrow(0^{+};0)$ | 0.551(7) | 0.767(9) | 28% | $17.64\rightarrow 0.00$ | 6.2(2) | 12.0(3) | 15.0(1.8)
$(1^{+};1)\rightarrow(2^{+};0)$ | 0.398(6) | 0.567(11) | 30% | $17.64\rightarrow 3.03$ | 1.9(1) | 3.8(2) | 6.7(1.3)
$(1^{+};0)\rightarrow(0^{+};0)$ | 0.012(1) | 0.014(1) |  | $18.15\rightarrow 0.00$ | 0.25(1) | 0.50(2) | 1.9(0.4)
$(1^{+};0)\rightarrow(2^{+};0)$ | 0.018(3) | 0.021(3) |  | $18.15\rightarrow 3.03$ | 0.06(1) | 0.13(2) | 4.3(1.2)
$[431]\rightarrow[44]$ |  |  |  |  |  |  |
$(1^{+};1)\rightarrow(2^{+}_{2};0)$ | 2.287(10) | 2.910(13) | 21% | $17.64\rightarrow 16.626$ | 1.92(2)[–2] | 2.97(3)[–2] | 3.2(3)[–2]
$(1^{+};1)\rightarrow(2^{+};1)$ | 0.139(2) | 0.176(3) | 21% | $17.64\rightarrow 16.922$ | 1.22(3)[–3] | 2.20(5)[–3] | 1.3(3)[–3]
$(1^{+};0)\rightarrow(2^{+}_{2};0)$ | 0.167(3) | 0.189(3) | 12% | $18.15\rightarrow 16.626$ | 2.52(3)[–2] | 2.87(3)[–2] | 7.7(1.9)[–2]
$(1^{+};0)\rightarrow(2^{+};1)$ | 2.596(11) | 2.887(13) | 10% | $18.15\rightarrow 16.922$ | 3.26(3)[–2] | 4.18(3)[–2] | 6.2(7)[–2]
$[431]\rightarrow[431]$ |  |  |  |  |  |  |
$(3^{+};1)\rightarrow(2^{+};0)$ | 0.386(13) | 0.622(22) | 38% | $19.070\rightarrow 3.03$ | 0.87(6) | 2.3(2) | 10.5
$(3^{+};0)\rightarrow(2^{+};0)$ | 0.015(1)* | 0.030(1)* |  | $19.235\rightarrow 3.03$ | 0.15(2) | 0.37(4) | –
$[431]\rightarrow[44]$ |  |  |  |  |  |  |
$(0^{+};2)\rightarrow(1^{+};1)$ | 0.793(7) | 1.095(8) | 28% | $27.49\rightarrow 17.64$ | 6.7(1) | 12.7(2) | 21.9(3.9)
$(0^{+}_{3};0)\rightarrow(1^{+};1)$ | 0.553(3)† | 0.689(3)† | 21% |  | 8.3(3)† | 15.5(5)† |
$(0^{+}_{3};0)\rightarrow(1^{+};0)$ | 0.073(1)† | 0.082(1)† | 11% | $27.49\rightarrow 18.15$ | 0.28(1)† | 0.54(1)† | –
$[422]\rightarrow[431]$ |  |  |  |  |  |  |

An additional complication arises for transitions from the second $2^{+}$
state because the GFMC propagation is not guaranteed to preserve the
orthogonality of the w.f. relative to the first $2^{+}$ state. In practice,
GFMC propagation starting from orthogonal VMC w.f.’s preserves the
orthogonality to a high degree PWC04 ; in this case the amplitude
$\langle\Psi^{2^{+}_{2}}(\tau)|\Psi^{2+}_{V}\rangle$ increases from 0.0010(7)
for $\tau$=0 to 0.040(6) averaged over $0.1\leq\tau\leq 0.3$. This small
admixture leaves the energy and point proton radius of the second $2^{+}$
state as stable functions of $\tau$, as expected for a narrow state. However,
for the $E2$ matrix element from the $2^{+}_{2}$ state to states of dominant
[44] symmetry, there are the large cancellations discussed above and a small
admixture of the the $2^{+}$ state with its large $E2$ matrix element to
states of dominant [44] symmetry can substantially affect the overlap. For
this reason we have applied a correction by orthogonalizing the
$\Psi^{2^{+}_{2}}(\tau)$ to $\Psi^{2+}_{V}$,

$\Psi^{2^{+\prime}_{2}}(\tau)=\Psi^{2^{+}_{2}}(\tau)-\langle\Psi^{2^{+}_{2}}(\tau)|\Psi^{2+}_{V}\rangle\Psi^{2^{+}}(\tau)\,.$
(20)

This reduces the mixed estimates
$\langle\Psi^{2^{+}_{2}}(\tau)|E2|\Psi^{2+}_{V}\rangle$ by 50% and
$\langle\Psi^{2^{+}_{2}}(\tau)|E2|\Psi^{0+}_{V}\rangle$ by 20%. This
correction is also made for corresponding $M1$ transitions discussed below,
but is relatively much less important.

For the $M1$ transitions the IA matrix element is evaluated using the $M1$
operator induced by the one-body current given in Eq. (11), namely

${\bm{\mu}}^{\rm IA}=\sum_{i=1}^{A}\left(e_{N,i}\,{\bf
L}_{i}+\mu_{N,i}\,{\bm{\sigma}}_{i}\right)\ ,$ (21)

while the one-body current at N2LO generates the following additional M1
operator terms Pastore08

$\displaystyle{\bm{\mu}}^{\rm N2LO}=$
$\displaystyle-\frac{e}{8\,m_{N}^{3}}\sum_{i=1}^{A}\Bigg{[}\left\\{p_{i}^{2}\,,\,e_{N,i}\,{\bf
L}_{i}+\mu_{N,i}\,{\bm{\sigma}}_{i}\right\\}$ (22)
$\displaystyle+e_{N,i}\,{\bf p}_{i}\times({\bm{\sigma}}_{i}\times{\bf
p}_{i})\Bigg{]}\ ,$

where ${\bf p}_{i}=-i\nabla_{i}$ and ${\bf L}_{i}$ are the linear momentum and
angular momentum operators of particle $i$, and $\\{\dots\,,\,\dots\\}$
denotes the anticommutator.

The matrix element associated with the contribution of two-body currents is

$\displaystyle\langle J^{\pi}_{f},M_{f}\mid\\!\mu_{z}^{\rm MEC}\\!\mid
J^{\pi}_{i},M_{i}\rangle=$ (23) $\displaystyle-i\lim_{q\to
0}\frac{2\,m_{N}}{q}\langle J^{\pi}_{f},M_{f}\mid\\!j_{y}^{\rm
MEC}(q\,\hat{\bf x})\\!\mid J^{\pi}_{i},M_{i}\rangle\ ,$

where the spin-quantization axis and momentum transfer ${\bf q}$ are,
respectively, along the $\hat{\bf z}$ and $\hat{\bf x}$ axes, and
$M_{f}=M_{i}$. The various contributions are evaluated for two small values of
$q<0.02$ fm-1 and then extrapolated linearly to the limit $q$=0. The error due
to extrapolation is much smaller than the statistical error in the Monte Carlo
sampling.

In Table 3, we report the results for the $M1$ transition matrix elements as
well as the decay widths $\Gamma_{M1}$ between the low-lying excited states.
The first column specifies the initial and final states of pure isospin. The
second column, labeled ‘IA’, shows the IA results obtained with the transition
operator of Eq. (21), while the third column labeled with ‘Total’ shows
results obtained with the complete EM current operator, Eqs. (21–23). The
percentage of the total matrix element given by the MEC contributions is shown
in the fourth column. The fifth column shows the energies of the physical
states, while the last three columns compare the corresponding widths with the
experimental data from Ref. Tilley04 .

Figure 5: (Color online) One-body (IA) $M1$ transition density in nuclear
magnetons per fm3 for selected M1 transitions (see text for explanation).

As observed in Ref. Pastore13 , IA matrix elements are found to have larger
statistical fluctuations than the MEC matrix elements. We separately compute
IA and MEC matrix elements, and then sum the resulting values to obtain the
total numbers.

It is worthwhile noting that M1 transitions involving the resonant states do
not monotonically change as $\tau$ increases, a behavior unlike the quadrupole
moments, point proton radii, and energies of these states. This stability is
understood by observing that the (2+;0) and (4+;0) rotational states in 8Be
are $\sim$99% pure 1D2[44] and 1G4[44] states, so they are quantized with L=2
and L=4, respectively. The orbital contribution to the magnetic moment is just
L/2 nuclear magnetons because only protons contribute, i.e., it is equal to
1.00 n.m. in the (2+;0) state and 2.00 n.m. in the (4+;0) state. Because it is
quantized, the magnetic moment should not vary as the nucleus starts to break
up in the GFMC propagation, unlike the point proton radius where $r$ is
growing as $\tau$ increases. Due to this stability, we can safely propagate M1
matrix elements involving resonant states to larger values of $\tau$ and
average the GFMC result in larger $\tau$ intervals.

As for the $E2$ transitions above, the $M1$ matrix elements are evaluated
between states with well defined isospin, $T=0$ or $1$. We denote these matrix
elements as
$M1_{T_{f}T_{i}}=\left<\Psi_{J_{f},T_{f}}||\mu||\Psi_{J_{i},T_{i}}\right>$,
with $T_{f}$ and $T_{i}$ equal to $0$ or $1$. For transitions involving
isospin-mixing in the initial or final state, we use expressions similar to
Eq.(IV) to generate the physical transition rates. For transitions in which
both the initial and final states are isospin-mixed, using the definitions
given in Eq. (III), we obtain the following expressions for the isospin-mixed
M1 transition matrix elements:

$\displaystyle<\Psi^{a}_{J_{f}}||M1||\Psi^{a}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\alpha_{J_{f}}\alpha_{J_{i}}M1_{00}+\alpha_{J_{f}}\beta_{J_{i}}M1_{01}$
$\displaystyle+$
$\displaystyle\beta_{J_{f}}\alpha_{J_{i}}M1_{10}+\beta_{J_{f}}\beta_{J_{i}}M1_{11}\
,$ $\displaystyle<\Psi^{b}_{J_{f}}||M1||\Psi^{a}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\beta_{J_{f}}\alpha_{J_{i}}M1_{00}+\beta_{J_{f}}\beta_{J_{i}}M1_{01}$
$\displaystyle-$
$\displaystyle\alpha_{J_{f}}\alpha_{J_{i}}M1_{10}-\alpha_{J_{f}}\beta_{J_{i}}M1_{11}\
,$ $\displaystyle<\Psi^{a}_{J_{f}}||M1||\Psi^{b}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\alpha_{J_{f}}\beta_{J_{i}}M1_{00}-\alpha_{J_{f}}\alpha_{J_{i}}M1_{01}$
$\displaystyle+$
$\displaystyle\beta_{J_{f}}\beta_{J_{i}}M1_{10}-\beta_{J_{f}}\alpha_{J_{i}}M1_{11}\
,$ $\displaystyle<\Psi^{b}_{J_{f}}||M1||\Psi^{b}_{J_{i}}>$ $\displaystyle=$
$\displaystyle\beta_{J_{f}}\beta_{J_{i}}M1_{00}-\beta_{J_{f}}\alpha_{J_{i}}M1_{01}$
$\displaystyle-$
$\displaystyle\alpha_{J_{f}}\beta_{J_{i}}M1_{10}+\alpha_{J_{f}}\alpha_{J_{i}}M1_{11}\
.$

The isospin-mixed M1 matrix elements are used to evaluate the widths as given
in Eq. (17) for comparison to experiment. The IA and total values are reported
in the sixth and seventh columns of Table 3, and the experimental widths
(where available) are given in the last column of the table.

Figure 6: (Color online) Two-body (IA) $M1$ transition density in nuclear
magnetons per fm3 for selected M1 transitions (see text for explanation).

Three extra transitions that were calculated only in VMC are marked by a * or
† in Table 3; they may affect the physical decay widths through isospin
mixing. The $(3^{+};0)\rightarrow(2^{+};0)$ transition marked by a * is tiny
and its isospin mixing has little effect on the transition from the physical
19.07 MeV state. The corresponding transition from the 19.235 MeV state is
predicted to be much smaller and has not been reported experimentally. Perhaps
more interesting and important, although speculative, is the isospin mixing of
the proposed $(0^{+}_{3};0)$ state, discussed at the end of Sec. III, into the
physical 27.49 MeV state, as shown in the last two lines in Table 3 marked by
a †. The line above these gives the result assuming the physical state is pure
$T=2$, and even with MEC contributions, the theoretical width is noticeably
underpredicted. The first line marked with a † shows that mixing in the
$(0^{+}_{3};0)$ state, using $\alpha_{0}=0.19(4)$, increases the decay width
20%, bringing it closer to experiment. The final line shows the corresponding
decay to the 18.15 MeV state as much smaller and thus unlikely to be observed.
A fourth possible transition in this group, $(0^{+};2)\rightarrow(1^{+};0)$,
has $\Delta T=2$ and vanishes in IA and also for the MEC considered in this
paper.

The $M1$ transitions shown in Table 3 can be sorted into four categories,
characterized by having large, medium, small, and tiny matrix elements. The
two largest matrix elements are between states of the same spatial symmetry
that change isospin: $(1^{+};1)\rightarrow(2^{+}_{2};0)$ and
$(1^{+};0)\rightarrow(2^{+};1)$. All four states involved have predominant
[431] spatial symmetry, so there is maximum overlap between the w.f.’s.
Further, because $\Delta T=1$, the spin-magnetization terms of the protons and
neutrons add constructively. This feature is illustrated in the top left and
center panels of Fig. 5, where we plot the IA contributions to the magnetic
transition density from Eq.(21), evaluated with the starting VMC w.f.’s. In
the figure, the red upward-pointing triangles show the proton spin
contribution, the blue downward-pointing triangles show the neutron spin
contribution, the green diamonds are the proton orbital term, and the black
circles give the total transition density. In both these transitions, the spin
contributions are large and the proton orbital piece is very small, resulting
in a total matrix element of $\sim 3.0$ n.m.

There are also two transitions between states of the same spatial symmetry
where isospin is conserved, i.e., $\Delta T=0$, which results in small matrix
elements: $(1^{+};1)\rightarrow(2^{+};1)$ and
$(1^{+};0)\rightarrow(2^{+}_{2};0)$. These are illustrated in the bottom left
and center panels of Fig. 5. The magnitudes of the proton spin and neutron
spin contributions are very similar to the $\Delta T=1$ case, but they have
opposite signs and cancel against each other, and there is a more substantial
proton orbital term which further reduces the total, leading to matrix
elements of $\sim 0.2$ n.m. The values of the VMC densities integrated over
$d^{3}r$ are given in Table 4 for the transitions shown in the upper and lower
left panels of Fig.5.

Next, there are five matrix elements which are between states of different
spatial symmetry, and are $\Delta T=1$ transitions, such as the
$(1^{+};1)\rightarrow(0^{+};0)$ transition illustrated in the top right panel
of Fig. 5. These transitions have proton and neutron spin contributions that
add coherently, but are small because of the small overlap of the initial and
final w.f.’s. However, they have larger proton orbital pieces, which also add
coherently and dominate the total, leading to medium-size matrix elements in
the range 0.5–1.0 n.m.

Table 4: Individual IA and MEC contributions to one isovector and one isoscalar M1 transition matrix elements in units of n.m. calculated in VMC, corresponding to the two left-hand panels of Figs. 5 and 6. $(J_{i},T_{i})\rightarrow(J_{f},T_{f})$ | $(1^{+};1)\rightarrow(2_{2}^{+};0)$ | $(1^{+};1)\rightarrow(2^{+};1)$
---|---|---
IA-$p_{L}$ | $0.031(1)$ | $-0.224(1)$
IA-$p_{S}$ | $1.442(7)$ | $1.267(7)$
IA-$n_{S}$ | $0.988(5)$ | $-0.867(5)$
IA Total | $2.461(13)$ | $0.176(3)$
NLO-OPE | $0.457(1)$ |
N2LO-RC | $-0.059(1)$ | $-0.001$
N3LO-TPE | $0.090(1)$ |
N3LO-CT | $-0.038(1)$ | $0.040(1)$
N3LO-$\Delta$ | $0.160(1)$ |
N3LO-$\rho\pi\gamma$ |  | $-0.008$
MEC Total | $0.610(2)$ | $0.031(1)$

Finally, there are three matrix elements between states of different spatial
symmetry that have $\Delta T=0$, and these are tiny. An example is the
$(1^{+};0)\rightarrow(2^{+};0)$ transition in the lower right panel of Fig. 5.
In these cases the proton and neutron spin terms are small in magnitude and of
opposite sign, and the proton orbital piece is also very small, resulting in
matrix elements $<0.03$ n.m.

The net contribution of MEC EM currents (where MEC = Total - IA) is best
appreciated by looking at matrix elements between states with well-defined
isospin, as given in the second to fourth columns of Table 3. The quantity $z$
in the fourth column is the percentage contribution of the MEC to the total;
it is not given, if the MEC is less than the statistical error of the total.
MEC contributions to $\Delta T=0$ transitions are generally smaller than
$\Delta T=1$ transitions. This is due to the fact that the major MEC
correction, given by the OPE seagull and pion-in-flight terms at NLO, is
purely isovector, and cannot contribute to $\Delta T=0$ transitions.
Therefore, only higher order terms, i.e., terms at N2LO and N3LO, contribute
to these matrix elements, for which we find $z\sim 10\%$. Transitions induced
by the isovector component of the M1 operator, that is transitions in which
$T_{i}\neq T_{f}$, are instead characterized by a $z$ factor spanning the
interval $\sim 20-40\%$. In general, the NLO currents of one-pion range
provide $\sim 60-70\%$ of the total MEC correction. From Table 3, we see that
the contribution given by the MEC currents (with only one exception) improves
the IA values, bringing the theory into better agreement with the experimental
data.

It is also interesting to examine the transition magnetic densities due to
MEC. As examples, we discuss the same six transitions whose IA densities are
given in Fig. 5. The associated two-body magnetic densities obtained from MEC
terms are shown in Fig. 6, again as calculated with the starting VMC w.f.’s.
For the upper panels, which are isovector transitions, the red circles labeled
‘NLO-OPE’ show the density due to the long-ranged OPE currents, while
corrections associated with TPE currents at N3LO are given by the cyan squares
labeled ‘N3LO-TPE’. Contact current contributions, of both minimal and non-
minimal nature, are represented by the green fort symbols labeled ‘N3LO-CT’,
while the contribution due to the current of one-pion-range, which is been
saturated by the $\Delta$-resonance, is represented by the magenta triangles
labeled ‘N3LO-$\Delta$’. In the figure, we also show with blue stars labeled
‘N2LO-RC’ the one-body relativistic correction given in Eq. (22). The black
diamonds give the sum of the various contributions. The tail of the magnetic
distribution is dominated by the long-range OPE contribution, followed by the
N3LO-$\Delta$ one; at intermediate- to short-range TPE contributions become
important. The integrated values of the individual MEC contributions to the
$(1^{+};1)\rightarrow(2_{2}^{+};0)$ isovector transition (upper left panel of
Fig. 6) are listed in Table 4.

Table 5: Effect of alternate isospin mixing coefficient $\alpha_{1}$ on $\Gamma_{M1}$; the notation $[-x]=10^{-x}$. $E_{i}\rightarrow E_{f}$[MeV] |  $\Gamma_{M1}$[eV]
---|---
| $\alpha_{1}=0.21$ | $\alpha_{1}=0.31$ | Expt.
$17.64\rightarrow 0.00$ | 12.0(3) | 11.4(3) | 15.0(1.8)
$17.64\rightarrow 3.03$ | 3.8(2) | 3.6(2) | 6.7(1.3)
$18.15\rightarrow 0.00$ | 0.50(2) | 1.16(4) | 1.9(0.4)
$18.15\rightarrow 3.03$ | 0.13(2) | 0.32(3) | 4.3(1.2)
$[431]\rightarrow[44]$ |  |  |
$17.64\rightarrow 16.626$ | 2.97(3)[–2] | 3.28(3)[–2] | 3.2(3)[–2]
$17.64\rightarrow 16.922$ | 2.20(5)[–3] | 1.39(4)[–2] | 1.3(3)[–3]
$18.15\rightarrow 16.626$ | 2.87(3)[–2] | 1.84(2)[–2] | 7.7(1.9)[–2]
$18.15\rightarrow 16.922$ | 4.18(3)[–2] | 4.59(3)[–2] | 6.2(7)[–2]
$[431]\rightarrow[431]$ |  |  |

Two-body magnetic densities for the isoscalar transitions are shown in the
lower panels of Fig. 6. The isoscalar component of the M1 operator has a
rather different structure in comparison with that of its isovector component;
it has no contributions at NLO, therefore isoscalar transitions are suppressed
with respect to the isovector ones. The first correction beyond the IA picture
enters at N2LO and is given by the one-body relativistic correction of Eq.
(22), shown by the blue stars labeled with ‘N2LO-RC’. There are two isoscalar
contributions at N3LO. The first is associated with the tree-level current of
one-pion range represented by the cyan squares labeled ‘N3LO-$\rho\pi\gamma$’.
This isoscalar tree-level current can, in principle, be saturated by the
$\rho\pi\gamma$ transition current Pastore11 , however, we fix its associated
LEC so as to reproduce the magnetic moments of the deuteron and the isoscalar
combination of the trinucleon magnetic moments, as explained in Sec. II. The
second is the contact current at N3LO, shown by the green fort symbols labeled
‘N3LO-CT’, and these, in fact, dominate the total isoscalar two-body MEC
contribution shown by the black diamonds. The integrated values for the
$(1^{+};1)\rightarrow(2^{+};1)$ transition (lower left panel of Fig. 6) are
also given in Table 4.

##  V Discussion

The spatial symmetry-conserving $M1$ transitions are between the isospin-mixed
$2^{+}$ and $1^{+}$ doublets, so the comparison with experimental widths
requires both the matrix elements between isospin-pure states and the
$\alpha_{J}$ and $\beta_{J}$ parameters of Eqs. (III,III) as input. We
consider the $\alpha_{2}$ and $\beta_{2}$ to be well-determined by the
$\Gamma_{\alpha}$ measurements for the $2^{+}$ doublet. However, the
$\alpha_{1}$ and $\beta_{1}$ were first estimated by Barker Barker66  by
looking at the ratio of the $\Gamma_{M1}$’s for the $1^{+}$ doublet and
comparing to shell-model calculations. Instead, we could use our more
sophisticated calculations to determine the best isospin-mixing parameters.

If we minimize the $\chi^{2}$ with respect to experiment for the four spatial
symmetry-conserving transitions, i.e., those given in the third block of Table
3, we find $\alpha_{1}$ = 0.31(4), compared to the “experimental” value of
0.21(3) used above in Table 3 and discussed in Ref. WPPM13 . The predicted
widths for these two isospin-mixing parameters are compared in Table 5, along
with the four other symmetry-changing transitions from the $1^{+}$ doublet to
the ground or first excited state; the $\chi^{2}$ comparison with experiment
for these cases is also improved. However, this alternate value for
$\alpha_{1}$ implies a significantly larger IM matrix element
$H_{01}=-150(18)$ keV, compared to the theoretical value for this Hamiltonian
of $-94(1)$ keV calculated in Ref. WPPM13 , which was in good agreement with
the earlier empirical value of $-103(14)$ keV.

The results of our QMC calculations are in fair agreement with experiment when
the transitions are between states of the same spatial symmetry. However, when
the spatial symmetry of the initial and final states is different, we
generally underpredict the reported experimental widths. The $E2$ calculations
of Table 2 give large matrix elements for the $[44]\rightarrow[44]$
transitions and show reasonable agreement with the recently remeasured
$(4^{+};0)\rightarrow(2^{+};0)$ width. The calculations underpredict the
$[44]\rightarrow[431]$ transitions from the isospin-mixed $2^{+}$ doublet to
the ground state, although here both theory and experiment have large error
bars. The predicted transitions to the first $2^{+}$ are smaller, and perhaps
not surprisingly unobserved to date. For the $E2$ transition from the first
$1^{+}$ at 17.64 MeV, we significantly overpredict the width, due to a
surprisingly large $\Delta T=1$ matrix element between ${}^{1}D[44]$ and
${}^{1}P[431]$ symmetry components. The unobserved transition from the $1^{+}$
state at 18.15 MeV is tiny, due to a vanishing $\Delta T=0$ matrix element.
The larger value of $\alpha_{1}$ discussed above would reduce the discrepancy
with experiment slightly.

The QMC results for $M1$ matrix elements are similar, in that the four
symmetry-conserving $[431]\rightarrow[431]$ transitions are in fair agreement
with experiment, once MEC contributions are included. The agreement can be
improved further by searching for better isospin-mixing parameters,
$\alpha_{J}$ and $\beta_{J}$, as discussed above. Seven of the eight symmetry-
changing $M1$ transitions are underpredicted by amounts ranging from only 25%
to factors of 2–5. The worst matrix element is the same
$(1^{+};0)\rightarrow(2^{+};0)$ transition that also vanishes in $E2$, leading
to a decay width for the 18.15 MeV state which is 15–30 times too small.

Even though many of the experimental widths considered in this work have large
errors, the serious discrepancies between some of the experimental and
calculated values highlight the challenge for theory to accurately predict
transition amplitudes between states with dominant admixtures of different
spatial symmetry or between states consisting of linear combinations of
components of different spatial symmetry and occurring with similar
probabilities.

To our knowledge, Refs. Pastore13 ; MPPSW08  and the present work are the only
ab initio calculations of EM transitions in $A>4$ nuclei that include MEC
contributions. We find that the calculated $M1$ matrix elements have
significant contributions, typically at the 20-30% level, from two-body EM
current operators, especially from those of one-pion range. The sizable MEC
corrections are found to almost always improve the IA results for $M1$
transitions. This corroborates the importance of many-body effects in nuclear
systems, and indicates that an understanding of low-energy EM transitions
requires contributions from MEC in combination with a complete treatment of
nuclear dynamics based on Hamiltonians that include two- and three-nucleon
forces.

###### Acknowledgements.

The many-body calculations were performed on the parallel computers of the
Laboratory Computing Resource Center, Argonne National Laboratory. This work
is supported by the National Science Foundation, Grant No. PHY-1068305 (S.P.),
and by the U.S. Department of Energy, Office of Nuclear Physics, under
contracts No. DE-FG02-09ER41621 (S.P.), No. DE-AC02-06CH11357 (S.C.P. and
R.B.W.) and No. DE-AC05-06OR23177 (R.S.), and under the NUCLEI SciDAC-3 grant.

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