Shell model description of the 14C dating beta decay with Brown-Rho-scaled NN interactions
aa r X i v : . [ nu c l - t h ] F e b Shell model description of the C dating β decaywith Brown-Rho-scaled NN interactions J. W. Holt , G. E. Brown , T. T. S. Kuo , J. D. Holt , R. Machleidt Department of Physics, SUNY, Stony Brook, New York 11794, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3 Department of Physics, University of Idaho, Moscow, Idaho 83844, USA (Dated: November 1, 2018)We present shell model calculations for the β -decay of C to the N ground state, treatingthe states of the A = 14 multiplet as two 0 p holes in an O core. We employ low-momentumnucleon-nucleon (NN) interactions derived from the realistic Bonn-B potential and find that theGamow-Teller (GT) matrix element is too large to describe the known lifetime. By using a modifiedversion of this potential that incorporates the effects of Brown-Rho scaling medium modifications,we find that the GT matrix element vanishes for a nuclear density around 85% that of nuclearmatter. We find that the splitting between the ( J π , T ) = (1 + ,
0) and ( J π , T ) = (0 + ,
1) states in Nis improved using the medium-modified Bonn-B potential and that the transition strengths fromexcited states of C to the N ground state are compatible with recent experiments.
The beta decay of C to the N ground state has longbeen recognized as a unique problem in nuclear structure.Its connection to the radiocarbon dating method, whichhas had a significant impact across many areas of science,makes the decay of broad interest even beyond nuclearphysics. But a priori one would not expect the betadecay of C to be a good transition for radiocarbon dat-ing over archaeological times, because the quantum num-bers of the initial state ( J π , T ) = (0 + ,
1) and final state( J π , T ) = (1 + ,
0) satisfy the selection rules for an allowedGamow-Teller transition. The expected half-life wouldtherefore be on the order of hours, far from the unusuallylong value of 5730 years [1] observed in nature. The corre-sponding nuclear transition matrix element is very small( ≃ × − ) and is expected to result from an accidentalcancellation among the different components contribut-ing to the transition amplitude. This decay has thereforebeen used to investigate phenomena not normally con-sidered in studies of allowed transitions, such as mesonexchange currents [2, 3], relativistic effects [4], and con-figuration mixing [5, 6]. Of broader importance, however,is that this decay provides a very sensitive test for thein-medium nuclear interaction and in particular for thecurrent efforts to extend the microscopic description ofthe nuclear force beyond that of a static two-body poten-tial fit to the experimental data on two-nucleon systems.One such approach is to include hadronic medium mod-ifications, in which the masses of mesons and nucleonsare altered at finite density due to the partial restorationof chiral symmetry [7, 8, 9] or many-body interactionswith either intermediate nucleon-antinucleon excitations[10] or resonance-hole excitations [11]. These effects aretraditionally incorporated in models of the three-nucleonforce (3NF), which have been well-tested in ab initio nu-clear structure calculations of light nuclei [12, 13].In this Letter we suggest that a large part of theobserved C beta decay suppression arises from in-medium modifications to the nuclear interaction. Westudy the problem from the perspective of Brown-Rho scaling (BRS) [14, 15], which was the first model to makea comprehensive prediction for the masses of hadrons atfinite density. In BRS the masses of nucleons and mostlight mesons (except the pion whose mass is protectedby its Goldstone boson nature) decrease at finite densityas the ratio of the in-medium to free-space pion decayconstant: r g A g ∗ A m ∗ N m N = m ∗ σ m σ = m ∗ ρ m ρ = m ∗ ω m ω = f ∗ π f π = Φ( n ) , (1)where g A is the axial coupling constant, Φ is a func-tion of the nuclear density n with Φ( n ) ≃ . C beta decay provides a nearly idealsituation in nuclear structure physics for testing the hy-pothesis of Brown-Rho scaling. Just below a double shellclosure, the valence nucleons of C inhabit a region witha large nuclear density. But more important is the sen-sitivity of this GT matrix element to the nuclear tensorforce, which as articulated by Zamick and collaborators[18, 19] is one of the few instances in nuclear structurewhere the role of the tensor force is clearly revealed. Infact, with a residual interaction consisting of only cen-tral and spin-orbit forces it is not possible to achievea vanishing matrix element in a pure p − configuration[20]. Jancovici and Talmi [21] showed that by includinga strong tensor force one could construct an interactionwhich reproduces the lifetime of C as well as the mag-netic moment and electric quadrupole moment of N,although agreement with the known spectroscopic datawas unsatisfactory.The most important contributions to the tensor forcecome from π and ρ meson exchange, which act oppositeto each other: V Tρ ( r ) = f Nρ π m ρ τ · τ (cid:18) − S (cid:20) m ρ r ) + 1( m ρ r ) + 13 m ρ r (cid:21) e − m ρ r (cid:19) ,V Tπ ( r ) = f Nπ π m π τ · τ (cid:18) S (cid:20) m π r ) + 1( m π r ) + 13 m π r (cid:21) e − m π r (cid:19) . (2)Since the ρ meson mass is expected to decrease substan-tially at nuclear matter density while the π mass remainsrelatively constant, an unambiguous prediction of BRS isthe decreasing of the tensor force at finite density, whichshould be clearly seen in the GT matrix element. Infact, recent shell model calculations [22] performed in alarger model space consisting of p − + 2 ~ ω excitationshave shown that the β -decay suppression requires the in-medium tensor force to be weaker and the in-mediumspin-orbit force to be stronger in comparison to a typical G -matrix calculation starting with a realistic NN inter-action. We show in Fig. 1 the radial part of the tensorinteraction V T ( r ) = V Tπ ( r ) + V Tρ ( r ) at zero density andnuclear matter density assuming that m ∗ ρ ( n ) /m ρ = 0 . r [fm]-60-40-200204060 V T [ M e V ] n = n = n FIG. 1: The radial part of the nuclear tensor force given in eq.(2) from π and ρ meson exchange at zero density and nuclearmatter density under the assumption of BRS. Experiments to determine the properties of hadrons inmedium have been performed for all of the light mesonsimportant in nuclear structure physics. Studies of deeply-bound pionic atoms [23] find only a small increase in the π − mass at nuclear matter density and a related decreasein the π + mass. Experimental information on the scalarand vector particles comes from mass distribution mea-surements of in-medium decay processes. Recent pho-toproduction experiments [24] of correlated pions in the T = J = 0 channel ( σ meson) have found that the distri-bution is shifted to lower masses in medium. The vector mesons have been the most widely studied. Whereas thesituation is clear with the ω meson, the mass of whichdrops by ∼
14% at nuclear matter density [25], with the ρ meson it is still unclear [26, 27]. We believe that ourpresent study tests the decrease in ρ mass more simply.Today there are a number of high precision NN interac-tions based solely on one-boson exchange. In the presentwork we use the Bonn-B potential [28] which includes theexchange of the π , η , σ , a , ρ , and ω mesons. In [16] theconsequences of BRS on the free-space NN interactionwere incorporated into the Bonn-B potential and shownto reproduce the saturation properties of nuclear mat-ter in a Dirac-Brueckner-Hartree-Fock calculation. Themasses of the pseudoscalar mesons were unchanged, andthe vector meson masses as well as the correspondingform factor cutoffs were decreased according to m ∗ ρ m ρ = m ∗ ω m ω = Λ ∗ Λ = 1 − . nn . (3)The medium-modified (MM) Bonn-B potential is uniquein its microscopic treatment of the scalar σ parti-cle as correlated 2 π exchange. Finite density effectsarise through medium modifications to the exchanged ρ mesons in the pionic s -wave interaction as well as throughthe dressing of the in-medium pion propagator with ∆-hole excitations. These modifications to the vector me-son masses and pion propagator would traditionally beincluded in the chiral three-nucleon contact interactionand the 3NF due to intermediate ∆ states, respectively.Using realistic NN interactions in many-body pertur-bation theory is problematic due to the strong short dis-tance repulsion in relative S states. The modern solu-tion is to integrate out the high momentum componentsof the interaction in such a way that the low energyphysics is preserved. The details for constructing sucha low momentum interaction, V low − k , are described in[29, 30]. We define V low − k through the T -matrix equiva-lence T ( p ′ , p, p ) = T low − k ( p ′ , p, p ) for ( p ′ , p ) ≤ Λ, where T is given by the full-space equation T = V NN + V NN gT and T low − k by the model-space (momenta ≤ Λ) equation T low − k = V low − k + V low − k gT low − k . Here V NN representsthe Bonn-B NN potential and Λ is the decimation mo-mentum beyond which the high-momentum componentsof V NN are integrated out. Since pion production startsaround E lab ≃
300 MeV, the concept of a real NN po-tential is not valid beyond that energy. Consequently,we choose Λ ≈ . − thereby retaining only the in-formation from a given potential that is constrained byexperiment. In fact for this Λ, the V low − k derived fromvarious NN potentials are all nearly identical [30].We use the folded diagram formalism to reduce thefull-space nuclear many-body problem H Ψ n = E n Ψ n to a model space problem H eff χ m = E m χ m as de-tailed in [31]. Here H = H + V , H eff = H + V eff , E n = E n ( A = 14) − E ( A = 16 , core), and V denotesthe bare NN interaction. The effective interaction V eff is derived following closely the folded-diagram methoddetailed in [32]. A main difference is that in the presentwork the irreducible vertex function ( ˆ Q -box) is calculatedfrom the low-momentum interaction V low − k , while in [32]from the Brueckner reaction matrix ( G -matrix). In theˆ Q -box we include hole-hole irreducible diagrams of first-and second-order in V low − k . Previous studies [33, 34, 35]have found that V low − k is suitable for perturbative cal-culations; in all of these references satisfactory convergedresults were obtained including terms only up to secondorder in V low − k .Our calculation was carried out in jj -coupling wherein the basis n p − / , p − / p − / , p − / o one must diagonalize h V ij eff i + ǫ
00 0 2 ǫ , (4)to obtain the ground state of N (and a similar 2 × C). We used ǫ = E ( p − / ) − E ( p − / ) = 6 . − state in N. One can transform the wavefunctions to LS -coupling, where the C and N ground states are ψ i = x (cid:12)(cid:12) S (cid:11) + y (cid:12)(cid:12) P (cid:11) ψ f = a (cid:12)(cid:12) S (cid:11) + b (cid:12)(cid:12) P (cid:11) + c (cid:12)(cid:12) D (cid:11) (5)and the Gamow-Teller matrix element M GT is given by[21] X k h ψ f || σ ( k ) τ + ( k ) || ψ i i = −√ (cid:16) xa − yb/ √ (cid:17) . (6)Since x and y are expected to have the same sign [20], theGT matrix element can vanish only if a and b have thesame sign, which requires that the (cid:10) S (cid:12)(cid:12) V eff (cid:12)(cid:12) D (cid:11) ma-trix element furnished by the tensor force be large enough[21]. In Table I we show the ground state wavefunctionsof C and N, as well as the GT matrix element, cal-culated with the MM Bonn-B interaction. n/n x y a b c M GT n . In Fig. 2 we plot the resulting B ( GT ) = J i +1 | M GT | values for transitions between the low-lying states of Cand the N ground state for the in-medium Bonn-BNN interaction taken at several different densities. Re-cent experiments [36] have determined the GT strengthsfrom the N ground state to excited states of C and O using the charge exchange reactions N( d, He) Cand N( He , t ) O, and our theoretical calculations arein good overall agreement. The most prominent ef-fect we find is a robust inhibition of the ground stateto ground state transition for densities in the range of0 . − . n . In contrast, the other transition strengthsare more mildly influenced by the density dependencein BRS. In Fig. 3 we show the resulting half-life of Ccalculated from the MM Bonn-B potential. n/n B ( G T ) --> 1 --> 1 --> 1 --> 1 --> 1 Expt. ++++++
FIG. 2: The B ( GT ) values for transitions from the statesof C to the N ground state as a function of the nucleardensity and the experimental values from [36]. Note thatthere are three experimental low lying 2 + states compared totwo theoretical 2 + states in the p − configuration. n/n H a l f- li f e [ y r s ] MM Bonn-B
Experimental half-life
FIG. 3: The half-life of C, as a function of the nucleardensity, calculated from the MM Bonn-B potential.
We emphasize that the nuclear density experienced by p -shell nucleons is actually close to that of nuclear matter;in Fig. 4 we compare twice the charge distribution of Nobtained from electron scattering experiments [37, 38]with the radial part of the 0 p wavefunctions, indicat-ing clearly that the nuclear density for 0 p nucleons is ∼ . n . The first excited 0 + state of N together withthe ground states of O and C form an isospin triplet.We have calculated the splitting in energy between this r [fm]00.050.10.150.2 n [f m - ] n=n Expt. ϕ p ( r ) FIG. 4: Twice the charge distribution of N taken from [37,38] and the fourth power of the p -shell wavefunctions. state and the ground state of N for a range of nucleardensities. Our results are presented in Fig. 5, where theexperimental value is 2.31 MeV. E [ M e V ] n/n + + FIG. 5: The splitting between the 1 +1 and 0 +1 levels in N fordifferent values of the nuclear density. Also included is theexperimental value.
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