Quantum Monte Carlo calculations of electromagnetic transitions in 8Be with meson-exchange currents derived from chiral effective field theory
aa r X i v : . [ nu c l - t h ] J un Quantum Monte Carlo calculations of electromagnetic transitions in Be withmeson-exchange currents derived from chiral effective field theory
S. Pastore , ∗ R. B. Wiringa , † Steven C. Pieper , ‡ and R. Schiavilla , § Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208 Physics Division, Argonne National Laboratory, Argonne, Illinois 60439 Theory Center, Jefferson Laboratory, Newport News, Virginia 23606 Department of Physics, Old Dominion University, Norfolk, Virginia 23529 (Dated: November 11, 2018)We report quantum Monte Carlo calculations of electromagnetic transitions in Be. The realisticArgonne v two-nucleon and Illinois-7 three-nucleon potentials are used to generate the ground stateand nine excited states, with energies that are in excellent agreement with experiment. A dozen M E E M PACS numbers: 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 electro-magnetic (EM) transitions in A ≤ M χ EFT) formulation of Refs. [4–6]. Nuclear wave func-tions (w.f.’s) were obtained from a Hamiltonian consist-ing of the non-relativistic nucleon kinetic energy plus theArgonne v (AV18) two-nucleon [7] and Illinois-7 (IL7)three-nucleon [8] potentials. The SNPA MEC were con-structed to obey current conservation with this Hamil-tonian, while the use of χ EFT MEC constitutes a hy-brid calculation. The two methods are in substantialagreement, producing a theoretical microscopic descrip-tion of nuclear dynamics that successfully reproduces theavailable experimental data, although the χ EFT MECgive somewhat better results. Two-body components inthe current operators provide significant corrections tosingle-nucleon impulse-approximation (IA) calculations.For example, they contribute up to ∼
40% of the totalpredicted value for the C magnetic moment [1].In this work, we implement the framework described ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] above for twenty EM transitions in the Be nucleus us-ing only the χ EFT MEC. The experimental spectrumand EM transitions we consider are illustrated in Fig. 1.This even-even nucleus exhibits a strong two- α clusterstructure in its ground state, characterized by angularmomentum, parity, and isospin ( J π ; T ) = (0 + ; 0), and aYoung diagram spatial symmetry that is predominantly[44]. The ground state lies ∼ . α ’s, while the (2 + ; 0) state at ∼ + ; 0) state at ∼
11 MeV, are[44] rotational states with large ( ∼ + ; 0+1) states lying below the threshold for breakupinto Li+ p and having α + α decay widths of ∼
100 keV.The isospin mixing is due to the interplay between T = 0states and T = 1 states that are the isobaric analogs ofthe lowest three states in Li and B, all with the samedominant [431] spatial symmetry. There are many ad-ditional broad excited states above these isospin-mixeddoublets that are not shown before the final state weconsider, the (0 + ; 2) isobaric analog of the He groundstate at 27 MeV excitation, with dominant [422] spatialsymmetry and a very narrow 5 keV decay width.A comprehensive set of QMC calculations of A = 8nuclei was carried out in Ref. [9] for a Hamiltonian withAV18 and the older Urbana IX three-nucleon poten-tial [13]. More recently, energies, radii, and quadrupolemoments of this nucleus have been recalculated for the[44] symmetry states [10], and for the isospin-mixedstates [11], using the newer Illinois-7 potential. Thepresent work complements these studies by calculatingmany EM transitions between the low-lying states, which -60-56-52-48-44-40-36-32-28 E n e r gy ( M e V ) S[422] P[431] D[431] G[44] D[44] S[44] 2 + ;00 + ;04 + ;02 + ;0+11 + ;0+13 + ;0+10 + ;2 α+α Li+p Be FIG. 1: (Color online) Experimental spectrum of Be: hor-izontal lines denote energy levels, with blue for T =0 states,magenta for mixed T =0+1 states, and violet for T =2; blackdash-dot lines indicate thresholds for breakup as indicatedand shaded areas denote the large widths of the Be rotationalstates. Vertical lines with arrows indicate the electromagnetictransitions studied: blue short-dash for E
2, red long-dash for M
1, and magenta dash-dot for combined E M are also illustrated in Fig. 1. The M χ EFT currents, whichprovide important corrections of order 20–30%. The two-body current corrections to the E χ EFT expansion [6] and are not computedhere.QMC techniques and χ EFT EM currents were pre-sented in Ref. [1] and references therein. We refer to thatwork for more details on the calculational scheme whichis here briefly summarized in Sec. II. From there on, wefocus on providing and discussing the results. In par-ticular, the calculated Be energy spectrum is presentedin Sec. III, while results for E M II. QMC METHOD, NUCLEAR HAMILTONIANAND χ EFT EM CURRENTS
EM transition matrix elements are evaluated betweenw.f.’s which are solutions of the Schr¨odinger equation: H Ψ( J π ; T, T z ) = E Ψ( J π ; T, T z ) , (1)where Ψ( J π ; T, T z ) is a nuclear w.f. with specific spin-parity J π , isospin T , and charge state T z . The nuclearHamiltonian used in the calculations consists of a kineticterm plus two- and three-body interaction terms, namelythe AV18 [7] and the IL7 [8], respectively: H = X i K i + X i 12 n.m. and κ V = 3 . 706 n.m. are theisoscalar (IS) and isovector (IV) combinations of theanomalous magnetic moments of the proton and neutron,and e is the electric charge.Two-body EM currents are constructed from a χ EFTwhich retains as explicit degrees of freedom both pionsand nucleons. The resulting operators are expressed asan expansion in nucleon and pion momenta, genericallydesignated as Q . The leading-order (LO) contribution inEq. (11) is of order e Q − and contributions up to N3LOor e Q are retained in the expansion. These contribu-tions were first calculated by Park et al. in Ref. [18] us-ing covariant perturbation theory. More recently, K¨ollingand collaborators [19], as well as some of the present au-thors [4–6, 20], derived them using two different imple-mentations of time-ordered perturbation theory. In thiswork, we use the operators developed in Refs. [4–6, 20],where details on the derivation and a complete listing ofthe formal expressions may be found.The two-body χ EFT EM currents consist of long-and intermediate-range components described in terms ofone-pion exchange (OPE) and two-pion exchange (TPE)contributions, respectively, as well as contact currents en-coding short-range dynamics. In particular, OPE seagulland pion-in-flight currents appear at next-to-leading or-der (NLO) ( e Q − ) in the Q expansion, while TPE cur-rents occur at N3LO. The LO and N2LO ( e Q ) contribu-tions are given by the single-nucleon operators describedabove, i.e. , the IA operator and its relativistic correction,respectively.At N3LO, the current operators involve a number ofunknown low energy constants (LECs) which are fixed toexperimental data. The LECs multiplying four-nucleoncontact operators are of two kinds, namely minimal andnon-minimal. The former also enter the χ EFT nucleon-nucleon potential at order Q and are therefore fixed byreproducing the np and pp elastic scattering data, alongwith the deuteron binding energy. For these, we take thevalues resulting from the fitting procedure implementedin Refs. [21, 22]. Non-minimal LECs (there are two ofthem, one multiplying an isoscalar operator and the otheran isovector operator) need to be fixed to EM observ-ables.At N3LO, there is also an additional current of one-pion range which involves three LECs. One of thesemultiplies an isoscalar structure, while the remainingtwo multiply isovector structures. As first observed inRef. [18], the isovector component of this current hasthe same operator structure as that associated with a∆-resonance transition current involving a one-pion ex-change. In this type of two-body contribution, the ex-ternal photon couples with a nucleon to excite a ∆-resonance state. The latter decays emitting a pion whichis then reabsorbed by a second nucleon. Given this theo-retical insight, one can impose the condition that the twoisovector LECs are in fact given by the couplings of the∆-resonance current. This mechanism is referred to as∆-resonance saturation and has been utilized in variousstudies of EM observables of light nuclei (see for exam-ple [1, 6, 23–26]). Once the ∆-saturation mechanism isinvoked to fix two of the unknown LECs, the resultingthree LECs are fit to the deuteron and the trinucleonmagnetic moments.The values of the LECs are not unique, in that they de-pend on the particular momentum cutoff used to regular-ize the configuration-space singularities of the EM oper-ators. In momentum space, these operators have a powerlaw behavior for large momenta, k , which is regularizedby a momentum cutoff of the form C ( k ) = exp( − k / Λ ).For a list of the numerical values of the LECs for Λ = 600MeV, which is the cutoff utilized in these calculations, werefer to Ref. [1].The N2LO relativistic correction to the one-body IAoperator involves two derivatives acting on the nucleonfield. In the GFMC calculation we do not explicitly eval-uate this p i term, but instead approximate it with itsaverage value, that is p i ∼ (cid:10) p i (cid:11) , as determined from theexpectation value of the kinetic energy operator in Be,from which we obtain (cid:10) p i (cid:11) = 1 . 375 fm − . This term isa small fraction of the total MEC (see, e.g., Table IVbelow) so the approximation has little practical effect.To be consistent with the nomenclature utilized inRef. [1], we denote with ‘MEC’ components in the EMcurrents beyond the IA one-body operator at LO. How-ever, we stress that the N2LO contribution is a one-bodyoperator, which does not involve meson-exchange mech-anisms. III. BE ENERGY SPECTRUM The experimental [27] and calculated GFMC energiesfor the Be spectrum are presented in Table I, along withthe GFMC point proton radii. The calculations weredone by propagating up to some τ max with an evaluationof observables after every 40 propagation steps, i.e. , atintervals of τ = 0 . 02 MeV − , and averaging in the in-terval τ =[(0.1 MeV − )– τ max ]; τ max is typically 0.3 to 0.4MeV − .The calculation of the spectrum is rather involved [9],with two main challenges to face. The first originatesfrom the resonant nature of the first two excited states(gray shaded states in Fig. 1), and the ensuing difficulty TABLE I: GFMC ground state energy and excitations in MeVfor the AV18+IL7 Hamiltonian compared to experiment [27]for the Be spectrum. Empirical energies are obtained by un-folding the isospin-mixed experimental energies using inferredmixing coefficients (see text for explanation). Also given arethe GFMC point proton (= neutron) radii in fm. Theoreti-cal or experimental errors ≥ J π ; T GFMC Empirical Experiment r p + –56.3(1) –56.50 2.402 + + 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) 2.282 + ; 1 +16.8(2) +16.802(3) +16.922(3) 2.331 + ; 1 +17.5(2) +17.66(1) +17.640(1) 2.391 + ; 0 +18.0(2) +18.13(1) +18.150(4) 2.363 + ; 1 +19.4(2) +19.10(3) +19.07(3) 2.313 + ; 0 +19.9(2) +19.21(2) +19.235(10) 2.350 + ; 2 +27.7(2) +27.494(2) 2.58 of extracting a stable resonance energy from the calcu-lated energies which are evolving to the energy of twoseparated α ’s. This issue was addressed in Ref. [9], andmore recently, however succinctly, in Ref. [10]. The lastreference reported an updated measurement of the E Bemeasured via the α + α radiative capture with an un-certainty of ∼ 10% (as opposed to the estimated ∼ E + ; 0) state and theground state. We reprise this calculation in more detailbelow.The second non-trivial issue is encountered when deal-ing with the spectrum of the isospin-mixed states at 16–19 MeV (magenta states in Fig. 1). These excited stateshave been extensively discussed in Ref. [11]. We computeunmixed T = 0 or T = 1 states but experimental valuesare of course for the mixed states. The isospin-mixingcoefficients can be extracted from experimental decaywidths [29]. For the 2 + multiplet this is unambiguous,but for the 1 + and 3 + multiplets theoretical decay widthsbased on shell-model calculations have been used. This isdiscussed further below. In Table I we use the mixing pa-rameters to unfold the “empirical” pure-isospin energiesfor comparison with our calculations, while in subsequenttables we fold the computed EM matrix elements to gen-erate mixed matrix elements to compare to the data.We studied the convergence of the GFMC calculationswith respect to variations in the number of unconstrainedsteps ( n u =20 and 50) followed after the path constraint isrelaxed, and found that energies, magnetic moments, andrms radii converge at n u = 20, which is what is used for τ (MeV -1 ) E ( M e V ) + + + FIG. 2: (Color online) GFMC propagation in imaginary time τ of the energy expectation values of the first three states inthe Be spectrum. Black dots are GFMC propagation pointsfor the ground state, blue dots refer to the (2 + ; 0) rotationalstate at ∼ + ; 0) stateat ∼ 11 MeV excitation. Solid lines represent a linear fit tothe GFMC points in the indicated time interval. the final results reported here. Most of the calculationswe present are obtained by averaging two calculations,each using 50,000 walkers. For the physically narrow,nonresonant states, the energy expectation value is seento stabilize at τ ∼ . − .For the physically wide, resonant states, the bindingenergy, magnitude of the quadrupole moment, and pointproton radius all increase monotonically as τ increases.We interpret this as an indication that the system isdissolving into two separated α ’s. In Fig. 2, we showthe GFMC propagation points for the energy expecta-tion values of the first three states of Be. In particu-lar, the ground state energy is obtained with n u = 20and 20,000 walkers, while the resonant state energies areobtained using n u = 20 and averaging two calculationswith 50,000 walkers each. From the figure, we see thatthe ground state initial VMC energy expectation value at τ = 0 quickly drops and reaches stability around τ = 0 . − (this point is indicated in the figure with an openstar). The energies of the two resonant states, instead,keep falling with time: the (2 + ; 0) state decreases 0.25MeV over the interval τ = [0 . , . 3] MeV − , while the(4 + ; 0) states falls by 1 MeV. With this declining energythere is a corresponding increase of the point proton ra-dius expectation values, as shown in Fig. 3 and in themagnitude of the (negative) electric quadrupole moment.Quantities associated with the resonant states havebeen calculated assuming that, also for these states, τ ∼ . − is the point at which spurious contami-nation in the nuclear w.f.’s have been eliminated by theGFMC propagation. Thus, we make a linear fit to theGFMC values in the interval τ = [0 . , . 3] MeV − , andextrapolate to τ = 0 . − for the reported values.The choice of τ = 0 . − is somewhat arbitrary. τ (MeV -1 ) r (f m ) + + + FIG. 3: (Color online) GFMC propagation in imaginary time τ of the point proton radius expectation values of the firstthree states in Be spectrum; notation is the same as in Fig. 2. To account for this uncertainty we increase the GFMCstatistical error by a systematic error that is obtainedby studying the sensitivity of the results with respect tofitting procedures implemented in two different intervals,namely τ = [0 . , . 3] MeV − and τ = [0 . , . 3] MeV − ,while keeping the same extrapolating point. The total er-ror is represented in the figures by the dashed lines.For the six states at 16–19 MeV excitation, the GFMCcalculations are done for pure isospin states of either T =0 or 1. The w.f.’s of the isospin-mixed states are writtenas Ψ aJ = α J Ψ J,T =0 + β J Ψ J,T =1 , Ψ bJ = β J Ψ J,T =0 − α J Ψ J,T =1 , (13)where the mixing angles satisfy α J + β J = 1. As onecan see from Fig. 1 and Table I, experimentally there aretwo J π = 2 + isospin-mixed states at 16.626 and 16.922MeV excitation energies, two J π = 1 + states at 17.64and 18.15 MeV, and two J π = 3 + states at 19.07 and19.235 MeV. The mixing angles are inferred from theexperimental values of the decay widths. We follow theanalysis carried out by Barker in Ref. [29] and update theexperimental widths with more recent values to obtainthe following mixing coefficients [11]: α = 0 . , β = 0 . ,α = 0 . , β = 0 . , (14) α = 0 . , β = 0 . . Mixing coefficients for the 2 + states are well known be-cause for these states there is only one decay channelenergetically open, that is the 2 α emission channel, forwhich the experimental widths are known with ∼ . M + pair, computed using the M E a,b = H + H ± s(cid:18) H − H (cid:19) + ( H ) (15)where H is the diagonal energy expectation in the pure T =0 state, H is the expectation value in the T =1 state,and H is the off-diagonal isospin-mixing (IM) matrixelement that connects T =0 and 1. The inferred H and H are the empirical values given in Table I.Finally, the narrow (0 + ; 2) state at 27 MeV excita-tion, which has a dominant [422] spatial symmetry, is astraightforward GFMC calculation. There could in prin-ciple be isospin-mixing with the third (0 +3 ; 0) state in the p -shell construction of Be, which also has [422] symme-try, via the EM and charge-dependent parts of AV18.No such state has been identified experimentally. A firstVMC calculation places this state 0.7(1) MeV higher inexcitation with a 125 keV IM matrix element, which pre-dicts α = 0 . β = 0 . + isospin-mixed doublet is a littletoo high in excitation and a little too spread out com-pared to the measured values. IV. ELECTROMAGNETIC TRANSITIONS IN BE We present our results in terms of reduced matrix el-ements (using Edmonds’ convention) of the E M B ( E 2) and B ( M λ ( X designates E or M ), B ( Xλ ) = (cid:10) Ψ J f || Xλ || Ψ J i (cid:11) / (2 J i + 1) (16)is in units of e λ fm λ for electric transitions and (n.m.) λ for magnetic transitions. The widths are given byΓ Xλ = 8 π ( λ + 1) λ [(2 λ + 1)!!] α ¯ hc (cid:18) ∆ E ¯ hc (cid:19) λ +1 B ( Xλ ) , (17)where ∆ E is the difference in MeV between the exper-imental initial and final state energies, E i and E f ; α isthe fine-structure constant; and ¯ hc is in units of MeV fm.The calculations of electromagnetic matrix elementshave been described in detail in Refs. [2, 17]. Our presentresults for E Be are given in Table IIwhere the initial and final ( J π ; T ) states and the domi-nant associated spatial symmetries are shown in the first τ (MeV -1 ) 〈 E 〉 ( e f m ) (2 + → + )(4 + → + ) FIG. 4: (Color online) GFMC propagation in imaginary time τ of the reduced E Be spectrum; upper red dots are for the (4 + ; 0) → (2 + ; 0) transition, lower blue dots are for the (2 + ; 0) → (0 + ; 0)transition and open stars denote the extrapolated values. column and the reduced matrix elements between statesof pure isospin are given in the second column. The ex-perimental energies for the physical states are given inthe third column, and the corresponding theoretical andexperimental widths are shown in the fourth and fifthcolumns. We use the IA operator E e X k (cid:2) r k Y (ˆ r k ) (cid:3) (1 + τ kz ) (18)without any MEC corrections.In previous calculations [1, 17, 30] of nuclei in the A = 6–10 range, we have found that E Li (2 + ; 0) → (0 + ; 0) decay, show a signifi-cant evolution as a function of τ . This is also true for thefirst two transitions in Be from the broad rotational 2 + and 4 + states. The matrix element grows monotonicallyas the GFMC solution evolves in τ toward a separated α + α configuration, as illustrated in Fig. 4. This growthis slow for the lower (2 + ; 0) → (0 + ; 0) transition, butmuch more pronounced for the upper (4 + ; 0) → (2 + ; 0)transition. Consequently, we make an extrapolation backto τ = 0 . ± . 02 MeV − to obtain our best estimate forthe matrix element, just as we did for the energy andpoint proton radius discussed above in conjunction withFigs. 2 and 3. Our error estimate combines both theMonte Carlo statistical error and the uncertainty in theextrapolation point. The numerical results for these twomatrix elements and corresponding decay widths Γ E arereported at the top of Table II. The transitions, which arebetween states of the same dominant [44] spatial symme-try, are very large and consistent with a rotor picture for Be. TABLE II: Calculated reduced E J π ; T ) states and the dominant associated spatial symmetries, 2) the GFMCmatrix 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 πi ; T i ) → ( J πf ; T f ) E2[ e fm ] E i [MeV] → E f [MeV] Γ E [eV][s.s.] i → [s.s.] f IA Expt.(2 + ; 0) → (0 + ; 0) 10.0(2) 3 . → . + ; 0) → (2 + ; 0) 15.6(4) 11 . → . 03 0.87(5) 0.67(7)[44] → [44](2 +2 ; 0) → (0 + ; 0) 0.55(11) 16 . → . + ; 1) → (0 + ; 0) –0.23(2) 16 . → . → [44](2 +2 ; 0) → (2 + ; 0) 0.26(7) 16 . → . 03 3.6(2.2)[–3] –(2 + ; 1) → (2 + ; 0) 0.03(2) 16 . → . 03 1.7(1.4)[–3] –[431] → [44](1 + ; 1) → (2 + ; 0) 1.93(6) 17 . → . 03 0.63(5) 0.12(5)(1 + ; 0) → (2 + ; 0) –0.03(5) 18 . → . 03 4.0(1.1)[–2] –[431] → [44] We have also calculated an additional six E + and 1 + doublets withdominant [431] spatial symmetry, to the T = 0 groundstate or first 2 + state. We denote the isospin-pure matrixelements by E T f T i = (cid:10) Ψ J f ,T f || E || Ψ J i ,T i (cid:11) and then usethe definitions given in Eq. (13) to combine them via < Ψ J f , || E || Ψ aJ i > = α J i E + β J i E ,< Ψ J f , || E || Ψ bJ i > = β J i E − α J i E , (19)to evaluate the widths of the physical transitions for com-parison to experiment. Because the E α - α rotational band.This makes accurate calculations of these transitions sig-nificantly more difficult.As an example, we can compare the two E + states to the 0 + groundstate. As discussed in Refs. [31, 32], the 0 + state has fivecontributing LS -coupled symmetry components: S [44], P [431], D [422], S [422], and P [4211], with the firstcomponent having an amplitude in the present VMCstarting w.f. of 0.996. The 2 + states are linear com-binations of eight components: D [44], P [431], D [431], F [431], S [422], D [422], D [422], and P [4211]. Thefirst 2 + state also has an amplitude of 0.996 from the D [44] component, while the second 2 + state is dom-inated by the P [431] component with an amplitude of0.902. Consequently, 99% of the large E D [44] and S [44] compo-nents. However, for the much smaller E + state, this pair of components contributes1.65 times the final result, canceled by the matrix elementbetween the two P [431] components, which gives − . 44 times the final result. The remaining 38 smaller terms,among which there is much additional cancellation, give80% of the total.Changes in these small components, which may havelittle effect on the energy of a given state and hence arenot highly constrained by the GFMC propagation, canhave a significant effect on the E E A = 10 nuclei [30]. Thisis also true for many of the M + state because the GFMC propagation isnot guaranteed to preserve the orthogonality of the w.f.relative to the first 2 + state. In practice, GFMC propa-gation starting from orthogonal VMC w.f.’s preserves theorthogonality to a high degree [31]; in this case the am-plitude h Ψ +2 ( τ ) | Ψ V i increases from 0.0010(7) for τ =0to 0.040(6) averaged over 0 . ≤ τ ≤ . 3. This small ad-mixture leaves the energy and point proton radius of thesecond 2 + state as stable functions of τ , as expected for anarrow state. However, for the E +2 state to states of dominant [44] symmetry, thereare the large cancellations discussed above and a smalladmixture of the the 2 + state with its large E +2 ( τ ) to Ψ V ,Ψ + ′ ( τ ) = Ψ +2 ( τ ) − h Ψ +2 ( τ ) | Ψ V i Ψ + ( τ ) . (20)This reduces the mixed estimates h Ψ +2 ( τ ) | E | Ψ V i by50% and h Ψ +2 ( τ ) | E | Ψ V i by 20%. This correction is TABLE III: Calculated reduced M J π ; T ) states and the dominant associated spatial symmetries, 2) the GFMC matrixelements 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. Inthe width values we use the notation [ − x ] = 10 − x . The results marked with a * or † are extra VMC calculations discussed inthe text. ( J i ; T i ) → ( J f ; T f ) M1[n.m.] E i [MeV] → E f [MeV] Γ M [eV][s.s.] i → [s.s.] f IA Total z IA Total Expt.(2 +2 ; 0) → (2 + ; 0) 0.014(6) 0.013(6) 16 . → . 03 0.23(3) 0.51(6)(2 + ; 1) → (2 + ; 0) 0.297(12) 0.447(18) 33% 16 . → . 03 0.30(4) 0.70(7)[431] → [44] 16 . 626 + 16 . → . 03 0.53(5) 1.21(9) 2.80(18)(1 + ; 1) → (0 + ; 0) 0.551(7) 0.767(9) 28% 17 . → . 00 6.2(2) 12.0(3) 15.0(1.8)(1 + ; 1) → (2 + ; 0) 0.398(6) 0.567(11) 30% 17 . → . 03 1.9(1) 3.8(2) 6.7(1.3)(1 + ; 0) → (0 + ; 0) 0.012(1) 0.014(1) 18 . → . 00 0.25(1) 0.50(2) 1.9(0.4)(1 + ; 0) → (2 + ; 0) 0.018(3) 0.021(3) 18 . → . 03 0.06(1) 0.13(2) 4.3(1.2)[431] → [44](1 + ; 1) → (2 +2 ; 0) 2.287(10) 2.910(13) 21% 17 . → . 626 1.92(2)[–2] 2.97(3)[–2] 3.2(3)[–2](1 + ; 1) → (2 + ; 1) 0.139(2) 0.176(3) 21% 17 . → . 922 1.22(3)[–3] 2.20(5)[–3] 1.3(3)[–3](1 + ; 0) → (2 +2 ; 0) 0.167(3) 0.189(3) 12% 18 . → . 626 2.52(3)[–2] 2.87(3)[–2] 7.7(1.9)[–2](1 + ; 0) → (2 + ; 1) 2.596(11) 2.887(13) 10% 18 . → . 922 3.26(3)[–2] 4.18(3)[–2] 6.2(7)[–2][431] → [431](3 + ; 1) → (2 + ; 0) 0.386(13) 0.622(22) 38% 19 . → . 03 0.87(6) 2.3(2) 10.5(3 + ; 0) → (2 + ; 0) 0.015(1)* 0.030(1)* 19 . → . 03 0.15(2) 0.37(4) –[431] → [44](0 + ; 2) → (1 + ; 1) 0.793(7) 1.095(8) 28% 27 . → . 64 6.7(1) 12.7(2) 21.9(3.9)(0 +3 ; 0) → (1 + ; 1) 0.553(3) † † 21% 8.3(3) † † (0 +3 ; 0) → (1 + ; 0) 0.073(1) † † 11% 27 . → . 15 0.28(1) † † –[422] → [431] also made for corresponding M M M µ IA = A X i =1 ( e N,i L i + µ N,i σ i ) , (21)while the one-body current at N2LO generates the fol-lowing additional M1 operator terms [4] µ N2LO = − e m N A X i =1 " (cid:8) p i , e N,i L i + µ N,i σ i (cid:9) + e N,i p i × ( σ i × p i ) , (22)where p i = − i ∇ i and L i are the linear momentum andangular momentum operators of particle i , and { . . . , . . . } denotes the anticommutator.The matrix element associated with the contributionof two-body currents is h J πf , M f | µ MEC z | J πi , M i i = − i lim q → m N q h J πf , M f | j MEC y ( q ˆ x ) | J πi , M i i , (23)where the spin-quantization axis and momentum transfer q are, respectively, along the ˆ z and ˆ x axes, and M f = M i .The various contributions are evaluated for two small val-ues of q < . 02 fm − and then extrapolated linearly tothe limit q =0. The error due to extrapolation is muchsmaller than the statistical error in the Monte Carlo sam-pling.In Table III, we report the results for the M M between the low-lying excited states. The first columnspecifies the initial and final states of pure isospin. Thesecond column, labeled ‘IA’, shows the IA results ob-tained with the transition operator of Eq. (21), while thethird column labeled with ‘Total’ shows results obtainedwith the complete EM current operator, Eqs. (21–23).The percentage of the total matrix element given by theMEC contributions is shown in the fourth column. Thefifth column shows the energies of the physical states,while the last three columns compare the correspondingwidths with the experimental data from Ref. [27].As observed in Ref. [1], IA matrix elements are found tohave larger statistical fluctuations than the MEC matrix ρ M (r) ( µ N f m - ) Be(1 + ;1 → ;0) Be(1 + ;0 → + ;1) Be(1 + ;1 → + ;0) ρ M (r) ( µ N f m - ) Be(1 + ;1 → + ;1) Be(1 + ;0 → ;0) Be(1 + ;0 → + ;0) p L p S n S M1(IA) FIG. 5: (Color online) One-body (IA) M for selected M1 transitions (see textfor explanation). elements. We separately compute IA and MEC matrixelements, and then sum the resulting values to obtain thetotal numbers.It is worthwhile noting that M1 transitions involvingthe resonant states do not monotonically change as τ in-creases, a behavior unlike the quadrupole moments, pointproton radii, and energies of these states. This stabilityis understood by observing that the (2 + ;0) and (4 + ;0) ro-tational states in Be are ∼ 99% pure D [44] and G [44]states, so they are quantized with L=2 and L=4, respec-tively. The orbital contribution to the magnetic momentis just L/2 nuclear magnetons because only protons con-tribute, i.e. , it is equal to 1.00 n.m. in the (2 + ;0) state and2.00 n.m. in the (4 + ;0) state. Because it is quantized, themagnetic moment should not vary as the nucleus startsto break up in the GFMC propagation, unlike the pointproton radius where r is growing as τ increases. Dueto this stability, we can safely propagate M1 matrix ele-ments involving resonant states to larger values of τ andaverage the GFMC result in larger τ intervals.As for the E M T = 0 or 1. We denote these matrix elements as M T f T i = (cid:10) Ψ J f ,T f || µ || Ψ J i ,T i (cid:11) , with T f and T i equal to 0or 1. For transitions involving isospin-mixing in the ini-tial or final state, we use expressions similar to Eq.(19) togenerate the physical transition rates. For transitions inwhich both the initial and final states are isospin-mixed, using the definitions given in Eq. (13), we obtain the fol-lowing expressions for the isospin-mixed M1 transitionmatrix elements: < Ψ aJ f || M || Ψ aJ i > = α J f α J i M + α J f β J i M + β J f α J i M + β J f β J i M ,< Ψ bJ f || M || Ψ aJ i > = β J f α J i M + β J f β J i M − α J f α J i M − α J f β J i M ,< Ψ aJ f || M || Ψ bJ i > = α J f β J i M − α J f α J i M (24)+ β J f β J i M − β J f α J i M ,< Ψ bJ f || M || Ψ bJ i > = β J f β J i M − β J f α J i M − α J f β J i M + α J f α J i M . The isospin-mixed M1 matrix elements are used to eval-uate the widths as given in Eq. (17) for comparison toexperiment. The IA and total values are reported in thesixth and seventh columns of Table III, and the exper-imental widths (where available) are given in the lastcolumn of the table.Three extra transitions that were calculated only inVMC are marked by a * or † in Table III; they may af-fect the physical decay widths through isospin mixing.The (3 + ; 0) → (2 + ; 0) transition marked by a * is tinyand its isospin mixing has little effect on the transitionfrom the physical 19.07 MeV state. The correspondingtransition from the 19.235 MeV state is predicted to bemuch smaller and has not been reported experimentally.0 ρ M (r) ( µ N f m - ) Be(1 + ;1 → ;0) (fm) ρ M (r) ( µ N f m - ) Be(1 + ;1 → + ;1) TotalNLO-OPEN2LO-RCN3LO-CTN3LO-TPEN3LO- ∆ Be(1 + ;0 → + ;1) (fm) TotalN2LO-RCN3LO-CTN3LO- ρπγ Be(1 + ;0 → ;0) Be(1 + ;1 → + ;0) (fm) Be(1 + ;0 → + ;0) FIG. 6: (Color online) Two-body (IA) M for selected M1 transitions (seetext for explanation). Perhaps more interesting and important, although specu-lative, is the isospin mixing of the proposed (0 +3 ; 0) state,discussed at the end of Sec. III, into the physical 27.49MeV state, as shown in the last two lines in Table IIImarked by a † . The line above these gives the result as-suming the physical state is pure T = 2, and even withMEC contributions, the theoretical width is noticeablyunderpredicted. The first line marked with a † showsthat mixing in the (0 +3 ; 0) state, using α = 0 . + ; 2) → (1 + ; 0), has ∆ T = 2 and vanishes in IA andalso for the MEC considered in this paper.The M + ; 1) → (2 +2 ; 0)and (1 + ; 0) → (2 + ; 1). All four states involved have pre-dominant [431] spatial symmetry, so there is maximumoverlap between the w.f.’s. Further, because ∆ T = 1,the spin-magnetization terms of the protons and neu-trons add constructively. This feature is illustrated inthe top left and center panels of Fig. 5, where we plot theIA contributions to the magnetic transition density from Eq.(21), evaluated with the starting VMC w.f.’s. In thefigure, the red upward-pointing triangles show the protonspin contribution, the blue downward-pointing trianglesshow the neutron spin contribution, the green diamondsare the proton orbital term, and the black circles givethe total transition density. In both these transitions,the spin contributions are large and the proton orbitalpiece is very small, resulting in a total matrix element of ∼ . i.e. ,∆ T = 0, which results in small matrix elements:(1 + ; 1) → (2 + ; 1) and (1 + ; 0) → (2 +2 ; 0). These are il-lustrated in the bottom left and center panels of Fig. 5.The magnitudes of the proton spin and neutron spin con-tributions are very similar to the ∆ T = 1 case, but theyhave opposite signs and cancel against each other, andthere is a more substantial proton orbital term whichfurther reduces the total, leading to matrix elements of ∼ . d r are given in Table IV for the transitions shownin the upper and lower left panels of Fig.5.Next, there are five matrix elements which are betweenstates of different spatial symmetry, and are ∆ T = 1transitions, such as the (1 + ; 1) → (0 + ; 0) transition illus-trated in the top right panel of Fig. 5. These transitionshave proton and neutron spin contributions that add co-1 TABLE IV: Individual IA and MEC contributions to oneisovector and one isoscalar M1 transition matrix elements inunits of n.m. calculated in VMC, corresponding to the twoleft-hand panels of Figs. 5 and 6.( J i , T i ) → ( J f , T f ) (1 + ; 1) → (2 +2 ; 0) (1 + ; 1) → (2 + ; 1)IA- p L . − . p S . . n S . − . . . . − . − . . − . . . ρπγ − . . . herently, but are small because of the small overlap of theinitial and final w.f.’s. However, they have larger protonorbital pieces, which also add coherently and dominatethe total, leading to medium-size matrix elements in therange 0.5–1.0 n.m.Finally, there are three matrix elements between statesof different spatial symmetry that have ∆ T = 0, andthese are tiny. An example is the (1 + ; 0) → (2 + ; 0) tran-sition in the lower right panel of Fig. 5. In these casesthe proton and neutron spin terms are small in magni-tude and of opposite sign, and the proton orbital piece isalso very small, resulting in matrix elements < . 03 n.m.The net contribution of MEC EM currents (whereMEC = Total - IA) is best appreciated by looking atmatrix elements between states with well-defined isospin,as given in the second to fourth columns of Table III.The quantity z in the fourth column is the percentagecontribution of the MEC to the total; it is not given, ifthe MEC is less than the statistical error of the total.MEC contributions to ∆ T = 0 transitions are generallysmaller than ∆ T = 1 transitions. This is due to the factthat the major MEC correction, given by the OPE seag-ull and pion-in-flight terms at NLO, is purely isovector,and cannot contribute to ∆ T = 0 transitions. There-fore, only higher order terms, i.e. , terms at N2LO andN3LO, contribute to these matrix elements, for which wefind z ∼ T i = T f , are instead characterized by a z factor spanningthe interval ∼ − ∼ − 70% of the total MECcorrection. From Table III, we see that the contributiongiven by the MEC currents (with only one exception)improves the IA values, bringing the theory into betteragreement with the experimental data.It is also interesting to examine the transition mag-netic densities due to MEC. As examples, we discussthe same six transitions whose IA densities are given in TABLE V: Effect of alternate isospin mixing coefficient α onΓ M ; the notation [ − x ] = 10 − x . E i → E f [MeV] Γ M [eV] α = 0 . α = 0 . 31 Expt.17 . → . 00 12.0(3) 11.4(3) 15.0(1.8)17 . → . 03 3.8(2) 3.6(2) 6.7(1.3)18 . → . 00 0.50(2) 1.16(4) 1.9(0.4)18 . → . 03 0.13(2) 0.32(3) 4.3(1.2)[431] → [44]17 . → . 626 2.97(3)[–2] 3.28(3)[–2] 3.2(3)[–2]17 . → . 922 2.20(5)[–3] 1.39(4)[–2] 1.3(3)[–3]18 . → . 626 2.87(3)[–2] 1.84(2)[–2] 7.7(1.9)[–2]18 . → . 922 4.18(3)[–2] 4.59(3)[–2] 6.2(7)[–2][431] → [431] Fig. 5. The associated two-body magnetic densities ob-tained from MEC terms are shown in Fig. 6, again ascalculated with the starting VMC w.f.’s. For the upperpanels, which are isovector transitions, the red circleslabeled ‘NLO-OPE’ show the density due to the long-ranged OPE currents, while corrections associated withTPE currents at N3LO are given by the cyan squareslabeled ‘N3LO-TPE’. Contact current contributions, ofboth minimal and non-minimal nature, are representedby the green fort symbols labeled ‘N3LO-CT’, while thecontribution due to the current of one-pion-range, whichis been saturated by the ∆-resonance, is represented bythe magenta triangles labeled ‘N3LO-∆’. In the figure,we also show with blue stars labeled ‘N2LO-RC’ the one-body relativistic correction given in Eq. (22). The blackdiamonds give the sum of the various contributions. Thetail of the magnetic distribution is dominated by the long-range OPE contribution, followed by the N3LO-∆ one; atintermediate- to short-range TPE contributions becomeimportant. The integrated values of the individual MECcontributions to the (1 + ; 1) → (2 +2 ; 0) isovector transition(upper left panel of Fig. 6) are listed in Table IV.Two-body magnetic densities for the isoscalar tran-sitions are shown in the lower panels of Fig. 6. Theisoscalar component of the M1 operator has a rather dif-ferent structure in comparison with that of its isovectorcomponent; it has no contributions at NLO, thereforeisoscalar transitions are suppressed with respect to theisovector ones. The first correction beyond the IA pictureenters at N2LO and is given by the one-body relativis-tic correction of Eq. (22), shown by the blue stars labeledwith ‘N2LO-RC’. There are two isoscalar contributions atN3LO. The first is associated with the tree-level currentof one-pion range represented by the cyan squares labeled‘N3LO- ρπγ ’. This isoscalar tree-level current can, inprinciple, be saturated by the ρπγ transition current [20],however, we fix its associated LEC so as to reproducethe magnetic moments of the deuteron and the isoscalar2combination of the trinucleon magnetic moments, as ex-plained in Sec. II. The second is the contact current atN3LO, 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) → (2 + ; 1) transition(lower left panel of Fig. 6) are also given in Table IV. V. DISCUSSION The spatial symmetry-conserving M + and 1 + doublets, so thecomparison with experimental widths requires both thematrix elements between isospin-pure states and the α J and β J parameters of Eqs. (13,14) as input. We con-sider the α and β to be well-determined by the Γ α measurements for the 2 + doublet. However, the α and β were first estimated by Barker [29] by looking at theratio of the Γ M ’s for the 1 + doublet and comparing toshell-model calculations. Instead, we could use our moresophisticated calculations to determine the best isospin-mixing parameters.If we minimize the χ with respect to experiment forthe four spatial symmetry-conserving transitions, i.e.,those given in the third block of Table III, we find α =0.31(4), compared to the “experimental” value of 0.21(3)used above in Table III and discussed in Ref. [11]. Thepredicted widths for these two isospin-mixing parame-ters are compared in Table V, along with the four othersymmetry-changing transitions from the 1 + doublet tothe ground or first excited state; the χ comparison withexperiment for these cases is also improved. However,this alternate value for α implies a significantly largerIM matrix element H = − − − E → [44] transitions andshow reasonable agreement with the recently remeasured(4 + ; 0) → (2 + ; 0) width. The calculations underpredictthe [44] → [431] transitions from the isospin-mixed 2 + doublet to the ground state, although here both the-ory and experiment have large error bars. The predictedtransitions to the first 2 + are smaller, and perhaps notsurprisingly unobserved to date. For the E + at 17.64 MeV, we significantly overpre-dict the width, due to a surprisingly large ∆ T = 1 matrix element between D [44] and P [431] symmetry compo-nents. The unobserved transition from the 1 + state at18.15 MeV is tiny, due to a vanishing ∆ T = 0 matrixelement. The larger value of α discussed above wouldreduce the discrepancy with experiment slightly.The QMC results for M → [431]transitions are in fair agreement with experiment, onceMEC contributions are included. The agreement can beimproved further by searching for better isospin-mixingparameters, α J and β J , as discussed above. Seven ofthe eight symmetry-changing M + ; 0) → (2 + ; 0) transition that also vanishes in E ab initio calculations of EM transitions in A > M M Acknowledgments The many-body calculations were performed on theparallel computers of the Laboratory Computing Re-source Center, Argonne National Laboratory. This workis supported by the National Science Foundation, GrantNo. PHY-1068305 (S.P.), and by the U.S. Depart-ment of Energy, Office of Nuclear Physics, under con-tracts 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-3grant. [1] S. Pastore, S. C. Pieper, R. Schiavilla, and R. B. Wiringa,Phys. Rev. C , 035503 (2013). [2] L. E. Marcucci, M. Pervin, S. C. Pieper, R. Schiavilla, and R. B. Wiringa, Phys. Rev. C , 065501 (2008).[3] L. E. Marcucci, M. Viviani, R. Schiavilla, A. Kievsky,and S. Rosati, Phys. Rev. C , 014001 (2005).[4] S. Pastore, R. Schiavilla, and J.L. Goity, Phys. Rev. C , 064002 (2008).[5] S. Pastore, L. Girlanda, R. Schiavilla, M. Viviani, and R.B. Wiringa, Phys. Rev. C , 034004 (2009).[6] M. Piarulli, L. Girlanda, L.E. Marcucci, S. Pastore, R.Schiavilla, and M. Viviani, Phys. Rev. C , 014006(2013).[7] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys.Rev. C , 38 (1995).[8] S. C. Pieper, AIP Conf. Proc. , 143 (2008).[9] R. B. Wiringa, S. C. Pieper, J. Carlson, and V. R. Pand-haripande, Phys. Rev. C , 014001 (2000).[10] V. M. Datar et al. , Phys. Rev. Lett. , 062502 (2013).[11] R. B. Wiringa, S. Pastore, Steven C. Pieper, Gerald A.Miller, Phys. Rev. C , 044333 (2013).[12] R. B. Wiringa, Phys. Rev. C , 1585 (1991).[13] B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C.Pieper, and R. B. Wiringa, Phys. Rev. C , 1720 (1997).[14] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A.H. Teller, and E. Teller, J. Chem. Phys. , 1087 (1953).[15] J. Carlson, Phys. Rev. C , 2026 (1987); Phys. Rev. C , 1879 (1988).[16] G. P. Kamuntaviˇcius, P. Navr´atil, B. R. Barrett, G.Sapragonaite, and R. K. Kalinauskas, Phys. Rev. C ,044304 (1999).[17] M. Pervin, S. C. Pieper, and R. B. Wiringa, Phys. Rev.C , 064319 (2007). [18] T.-S. Park, D.-P. Min, and M. Rho, Nucl. Phys. A596 ,515 (1996).[19] S. K¨olling, E. Epelbaum, H. Krebs, and U.-G. Meissner,Phys. Rev. C , 045502 (2009); Phys. Rev. C , 054008(2011).[20] S. Pastore, L. Girlanda, R. Schiavilla, and M. Viviani,Phys. Rev. C , 024001 (2011).[21] D.R. Entem and R. Machleidt, Phys. Rev. C , 041001(2003).[22] R. Machleidt and D.R. Entem, Phys. Rep. , 1 (2011).[23] Y.-H. Song, R. Lazauskas, T.-S. Park, and D.-P. MinPhys. Lett. B 656, 174 (2007).[24] Y.-H. Song, R. Lazauskas, and T.-S. Park, Phys. Rev. C79, 064002 (2009).[25] R. Lazauskas, Y.-H. Song, and T.-S. Park, Phys. Rev. C83, 034006 (2011).[26] L. Girlanda, A. Kievsky, L.E. Marcucci, S. Pastore, R.Schiavilla, and M. Viviani, Phys. Rev. Lett. 105, 232502(2010).[27] D. R. Tilley, J. H. Kelley, J. L. Godwin,D. J. Millener,J. E. Purcell, C. G. Sheu, and H. R. Weller, Nucl. Phys.A , 155 (2004).[28] V. M. Datar et al. , Phys. Rev. Lett. , 122502 (2005).[29] F. C. Barker, Nucl. Phys. , 418 (1966).[30] E. A. McCutchan, et al. , Phys. Rev. C , 014312 (2012).[31] S. C. Pieper, R. B. Wiringa, and J. Carlson, Phys. Rev.C , 054325 (2004).[32] R. B. Wiringa, Phys. Rev. C73