aa r X i v : . [ nu c l - t h ] J a n Coulomb corrections to Fermi beta decay in nuclei
Naftali Auerbach ∗ and Bui Minh Loc , † School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel. Division of Nuclear Physics, Advanced Institute of Materials Science,Ton Duc Thang University, Ho Chi Minh City, Vietnam. Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam. (Dated: January 18, 2021)We study the influence of the Coulomb force on the Fermi beta-decays in nuclei. This work iscomposed of two main parts. In the first part, we calculate the Coulomb corrections to super-allowed beta decay. We use the notion of the isovector monopole state and the self-consistentcharge-exchange Random Phase Approximation to compute the correction. In the second part ofthis work, we examine the influence of the anti-analog state on isospin mixing in the isobaric analogstate and the correction to the beta-decay Fermi transition.
I. INTRODUCTION
In a number of studies attempts are made to determinecorrections one has to introduce in the evaluation of thebeta-decay matrix elements. In particular super-allowedtransitions in T = 1 , T z = +1 (or T z = −
1) nuclei [1–4]are extensively studied theoretically and experimentally.These corrections are important because using the mea-sured f t values one can relate these to the u -quark to d -quark transition matrix element V ud in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the StandardModel (SM) this matrix fulfils the unitarity condition,the sum of squares of the matrix elements in each row(column) is equal to one, as for example: V ud + V us + V ub = 1 (1)In order to determine the V ud term using the exper-imental f t values in the super-allowed beta-decay, onemust introduce a number of corrections [1]. In this paper,similarly to reference [2] we will consider one importantaspect of it, namely the Coulomb correction. There havebeen a number of works which dealt with this problemusing different methods [1–4] and more. In particular,the authors of [1, 4] have devoted a considerable amountof work to study the influence of the Coulomb interactionon the f t values. Of course in any of these studies, thereare some approximations involved. One of the main is-sues is the way the Coulomb force introduces admixturesof higher excitations into the parent and its analog state.This was the main topic of reference [2]. The Coulombforce admixes particle-hole ( ph ) states, mostly of 2 ~ ω atunperturbed energy positions. There is, however, a ph in-teraction that changes the situation, creating a collectivestate. In the case of a one-body Coulomb potential, theexcitation caused by it leads to J = 0 + , T = 1 ph states.In the isovector channel, the ph interaction is repulsiveand therefore there is an upward energy shift. The re-sulting collective state is the isovector monopole (IVM) ∗ [email protected] (Corresponding author) † [email protected] giant resonance. The shift is substantial as many Ran-dom Phase Approximation (RPA) studies indicate [5–7]about 1 ~ ω , and in some other studies even higher [8, 9],2 ~ ω shifts. In the RPA, the energy weighted sum rule isconserved, and therefore the upward energy shift will re-duce the strength. The amount of Coulomb mixing is de-termined by the strength divided by the energy squared.Therefore in a hand waving argument this amount will bereduced by a factor (2 / = 8 /
27 in the RPA, comparedto the calculation in which unperturbed 2 ~ ω ph excita-tions are used. As we will see in the next sections theactual calculations confirm this rough estimate. Thereare additional drawbacks in the shell model approaches[1] as pointed out in reference [10] which were avoidedin [2]. We will now briefly outline the main steps in thetheory given in [2]. II. COULOMB MIXING AND THE ISOVECTORMONOPOLE
The Fermi beta decay matrix element between theground state and its isobaric analog state (IAS) we writein the form: | M F | = | M F | (1 − δ C ) (2)where M F is the physical Fermi matrix element: M F = h Ψ | T + | Ψ i (3) | Ψ i and | Ψ i are the parent and daughter physicalstates. The symbol M F stands for the Fermi matrix el-ement obtained in the limit when in the Hamiltonianall the charge-dependent parts are put to zero, and thewave functions are eigenstates of the charge-independentHamiltonian H . In this case M F = √ T . The symbol δ C is the Coulomb correction. The eigenstates of thisHamiltonian with isospin T and T z will be denoted as | T, T z i and: H | T, T z i = E T | T, T z i . (4)The 2 T + 1, components with different T z values are de-generate, the action of the isospin lowering and raisingoperators, T − , T + gives: T − | T, T i = √ T | T, T − i ,T + | T, T − i = √ T | T, T i . (5)We introduce now a charge-dependent part V CD . Thedominant part of the charge dependent interaction isthe charge asymmetric one-body Coulomb potential V C (While the charge-dependent components of the two-body nuclear force might be important in changing therelative spacing of levels in the analog nucleus its influ-ence in isospin mixing in the ground state or isobaricanalog state (IAS) is expected to be small).The one-body Coulomb potential will now admix intothe ground state and its IAS the IVM [2, 6]. In per-turbation theory the effect of the charge-dependent parton the wave functions of the two members of the isospinmultiplet, | T, T i and | T, T − i will be:Ψ = N − ( | T, T i + ε T | M T,T i + ε T +1 | M T +1 ,T i ) , (6a)Ψ = N − ( | T, T − i + η T − | M T − ,T − i + η T | M T,T − i + η T +1 | M T +1 ,T − i ) , (6b)where | M T ′ ,T ′ z i , are the T ′ , T ′ z components of the IVM,and where N = q ε T + ε T +1 , and N = q η T − + η T + η T +1 , with ε i = h T, T | V C | M T + i,T i E M T + i,T − E , i = 0 , , (7)where E is the g.s. energy in this nucleus, η i = h T, T − | V C | M T + i,T − i E M T + i,T − − E , i = − , , . (8)Here E is the energy of the analog state. One can writethese as: ε i = h T, T, , | T + i, T i h T + i || V C || T i E M T + i,T − E , (9) η i = h T, T, , | T + i, T − i h T + i || V C || T i E M T + i,T − − E . (10)Introducing the Clebsch-Gordan (C.G.) coefficients andassuming that the reduced matrix elements are equal,one arrives (2) at the simple expression: h Ψ | T + | Ψ i = 2 T (cid:20) − T + 1) V ξ ~ ωA ε (cid:21) (11)and δ C = 4( T + 1) V ξ ~ ωA ε (12) Here ξ ~ ω is the energy of the IVM in the parent nucleus, V is the symmetry energy parameter determined fromthe equation E M T +1 ,T − E M T,T ≈ V N − ZA (13)and ε is the admixture of the T + 1 component of theIVM in the parent nucleus. We should emphasize thatthe result in eq. (11) is dependent implicitly on all thevarious admixture in eq. (6a, 6b) and (9, 10).The assumption of equal reduced matrix elements inderiving eq. (11) is approximate. The differences betweenthe reduced matrix elements for different isospin compo-nents increase with the increasing number of excess neu-trons. See [6, 7] and references therein. For nuclei withlow neutron excess, in particular, for super-allowed de-cays ( N − Z = 2), this is a very good approximation. Weapply here eq. (11, 12) to super-allowed transitions only. III. RESULTS OF THE COULOMBCORRECTIONS TO SUPER-ALLOWED BETADECAY δ C In reference [2], the calculation of δ C were based onvalues of isospin mixing derived from some general sumrules and not on detailed microscopic computations ofisospin impurities. One calculation presented there hasrelied on a schematic microscopic model [11] which wasintroduced in the 1970s. We return to the subject ofCoulomb corrections because at present new, more ad-vanced methods to calculate isospin mixing in low-lyingnuclear states are available. We mainly rely on the re-cently published article [7] about isospin impurities cal-culated using microscopic theories and new types of in-teractions. Using the formalism described in the previoussection we apply equations (11, 12) to compute the valuesof δ C for a number of nuclei through the periodic table.We concentrate on super-allowed beta-decay transitions.The calculations are performed using the Hartree-Fock(HF) RPA. For open-shell nuclei one should take into ac-count the pairing correlations and so one has to use theQRPA. However in the case of only two nucleons out-side the closed shells, one can limit ourselves to RPA.So for the super-allowed transitions (in T = 1 nuclei) weproceed our calculation the following way. We calculatein the charge-exchange HF-RPA [7], the distribution ofthe IVM strength in the N = Z closed shell nuclei Ca,
Ni,
Zr, and
Sn. In these cases, the IVM has onlya T = 1 isospin. We compute the Coulomb mixing of theIVM into the ground states of these nuclei, (see for de-tails ref. [7]) and denote the amount of isospin admixtureas ¯ ε . The results are presented in Table I.In the neighboring T = 1 nuclei, Ca,
Ni,
Zr, and
Sn the admixture of the T + 1 (in this case T + 1 = 2)can be approximated by introducing the C.G. squaredcoefficient 1 / ( T + 1) = 1 / ε ≈
12 ¯ ε (14)The error here is very small. We now apply eq. (12).We use the above relation and instead of ξ ~ ω , we use theenergy of the IVM determined in the RPA calculations¯ E . We note that the value of ξ is between 3 to 4. For V we take the results of ref. [7] to determine the valueof V by using eq. (13). For this purpose, we utilizethe RPA results [7] for nuclei that have a neutron excess.For example in the case of Ca we use the
Ca results.For illustration purposes we show in Table II the
CaRPA results. The isospin mixing of the T + 1 states tothe ground state is denoted as ε T +1 (see ref. [7]). Whenaveraging the values obtained with different Skyrme in-teractions we find the value of V for the Ca region tobe 90 MeV, for Ni 120 MeV, for Zr 60 MeV, and Sn90 MeV. Except for Zr, the values of V are around 100MeV. This is the value we used in ref. [7]. The Zr regionis exceptional, the symmetry energy potential is weakeras noticed a long time ago [12]. This point will be men-tioned later in the article. The Coulomb potential V C iscomputed using the Hartree-Fock calculation. Introduc-ing all mentioned above quantities into eq. (12) we findthe total Coulomb corrections δ C for the super-allowedbeta transitions in T = 1 nuclei (see Table III). TABLE I. The Coulomb mixing ¯ ε (%) and the IVM deter-mined in the RPA calculation ¯ E (MeV) for N = Z nuclei. Ca NiSkyrme ¯ ε (%) ¯ E (MeV) Skyrme ¯ ε (%) ¯ E (MeV)SIII 0.68 35.08 SIII 1.22 36.52SKM* 0.78 32.51 SKM* 1.42 34.57SLy4 0.77 31.13 SLy4 1.43 32.70BSK17 0.70 32.79 BSK17 1.23 34.74SAMi0 0.74 33.66 SAMi0 1.36 34.57 Zr SnSkyrme ¯ ε (%) ¯ E (MeV) Skyrme ¯ ε (%) ¯ E (MeV)SIII 3.63 32.06 SIII 4.54 34.35SKM* 4.07 30.18 SKM* 5.34 32.45SLy4 3.96 28.93 SLy4 5.27 30.83BSK17 3.72 30.21 BSK17 4.75 32.40SAMi0 3.96 30.75 SAMi0 5.15 32.58TABLE II. Results for Ca ( T = 4).Skyrme int. ε T +1 (%) ¯ E (MeV) V (MeV) δ C (%)SIII 0.10 34.79 106.14 0.13SKM* 0.12 32.54 92.17 0.14SLy4 0.12 30.57 97.80 0.16BSK17 0.10 32.83 108.63 0.13SAMi0 0.11 32.28 77.25 0.11 We should mention that our calculations of δ C ex-presses the global features of this quantity over the pe-riodic table and do not attempt to fit the small fluctua- TABLE III. The Coulomb correction δ C (%) for T = 1 nuclei. Ca Ni Zr SnSkyrme δ C (%) δ C (%) δ C (%) δ C (%)SIII 0.20 0.27 0.35 0.49SKM* 0.21 0.30 0.39 0.53SLy4 0.23 0.34 0.49 0.65BSK17 0.22 0.31 0.39 0.57SAMi0 0.16 0.26 0.34 0.49 tions of this quantity for different nuclei. Our main con-clusion is that the δ C is smaller by factor 3-4 comparedwith references [1, 4]. The main reason was explained inthe Introduction. In our approach there is no divisionof the correction into two parts (overlap corrections andthe rest) all is taken into account in the single expressioneq. (12). In references [1, 4] the small changes of δ C fordifferent nuclei are accounted for by changing the charge-dependent interaction to fit in each case the isobaric mul-tiplet mass equation. We of course do not do that andtherefore the Conserved Vector Current (CVC) relationdoes not apply here [1]. Our result for the correction δ C iscloser to some other computations in reference [3] (Liang et al. , Rodin) because in these works some correctionsof collective nature of Coulomb strength are taken intoconsideration. However our results for the Coulomb cor-rection are still about 50% or more, smaller than these,see Table IV. In Table II we have included also the valueof δ C . Since in Ca the isospin is T = 4, the assumptionabout the equality of the reduced matrix elements forthe IVM components with isospins T + 1, T , T − δ C is approximate. Arough estimate would assign an uncertainty of 10 − δ C , meaning that for nuclei with a largeneutron excess the values of the Coulomb correction aresmaller than in the case of super-allowed transitions. TABLE IV. Results of δ C (%) in various approaches. A ≈ A ≈ A ≈ IV. FERMI BETA TRANSITIONS, ISOSPINMIXING, AND THE ROLE OF THEANTI-ANALOG STATE
So far we have discussed the role of the IVM in induc-ing isospin impurities into the low-lying nuclear states.The energy of the IVM is high and is distant from thelevel it admixes. The amount of mixing changes smoothlywhen going from one nucleus to the next. The IVM in-volves 2 ~ ω ph excitations and cannot be properly de-scribed in a small space shell-model calculation. Whenwe pass from the parent state | π i to the analog nucleusthat is the one where one of the neutrons was changedto a proton ( N − Z + 1 protons) stateswith isospin T − | A i ) but are constructed tobe orthogonal to the IAS. Of course, they are not eigen-states of the Hamiltonian but are mixed with other T − V. COULOMB MIXING OF THEANTI-ANALOG AND ANALOG
Consider a simple parent state in which n excess neu-trons occupy orbit j ( n ) and n neutrons orbit j ( n ).In the parenthesis, we put n , or p for neutrons or pro-tons. (In some light nuclei the role of excess neu-trons is interchanged with excess protons). Of course, n + n ≡ N − Z = 2 T . The parent state is: | π i = | j n ( n ) j n ( n ) i (15)has isospin T . The analog is: | A i = 1 √ T (cid:2) √ n (cid:12)(cid:12) j n − ( n ) j ( p ) j n ( n ) (cid:11) + √ n (cid:12)(cid:12) j n ( n ) j n − ( n ) j ( p ) (cid:11) (cid:3) (16)has isospin T . The anti-analog | ¯ A i is then: | ¯ A i = 1 √ T (cid:2) √ n (cid:12)(cid:12) j n − ( n ) j ( p ) j n ( n ) (cid:11) −√ n (cid:12)(cid:12) j n ( n ) j n − ( n ) j ( p ) (cid:11) (cid:3) . (17)We consider here parent nuclei with simple configura-tions: for even-even nuclei, the n and n are even andin each orbit the excess nucleons are coupled to J = 0 + and in odd-even nuclei n is odd and n is even. Theone-body Coulomb matrix element between the analogand anti-analog is then [6, 9, 12]: h ¯ A | V C | A i = √ n n T [ h j | V C | j i − h j | V C | j i ] , (18)where V C is the Coulomb potential. If the excess neu-trons occupy orbits belonging to different major shells,this matrix element is sizable. The energy splitting be-tween the analog and anti-analog is often given by thesymmetry potential V : E ¯ A − E A = V ( N − Z ) A . (19) The value of V is smaller than in the splitting of the IVMand it about 50 MeV (see reference [12] and experimentaldata quoted in this reference).The coupling between the analog and anti-analog suc-cessfully explained [6, 13] isospin forbidden decays inlight nuclei [14, 15]. As one goes to heavy nuclei alongthe stability line the number of excess neutrons increaseswhich leads to reductions in the matrix element (18) andthe increase in the energy splitting (19), causing the mix-ing of T − VI. THE ANTI-ANALOG AND THE COULOMBCORRECTION δ C We discuss now the contribution of the anti-analog tothe Coulomb corrections for Femi beta-decay transitions.Using the definitions in eq. (3) | Ψ i = | π i , (20)and | Ψ i = p − ε | A i + ε | ¯ A i , (21)with ε = h ¯ A | V C | A i E ¯ A − E A . (22)Note that this is the isospin mixing of the anti-analoginto the analog. With these expressions one immediatelysees that: δ C = ε . (23) VII. HARMONIC OSCILLATOR ESTIMATE
For a uniform charge distribution with radius R theinner part of the Coulomb potential is V C = 12 Ze R r . (24)For R = 1 . A / fm one can write the matrix element inequation (18) as: h ¯ A | V C | A i ≈ . √ n n T ZA (cid:2) h j | r | j i − h j | r | j i (cid:3) . (25)If j and j belong to two major shells differing by onenode then the difference in the radii square in a harmonicoscillator well becomes:∆( r ) = ~ mω (26) m is the mass of the nucleon and ω the oscillator fre-quency. Taking ~ ω = 41 A − / MeV, we obtain: h ¯ A | V C | A i = 0 . √ n n T ZA / MeV . (27)Taking V ≈
50 MeV in eq. (19), N − Z = 2 Tδ C = 5 . × − Z A / n n (2 T ) . (28) VIII. NUMERICAL ESTIMATES FOR THEANTI-ANALOG MIXING
Using the self-consistent HF potential we computed thedifference of the Coulomb matrix elements in eq.(18) forthe orbits 2 p / and 1 g / for Sr. We use the SkyrmeHF for the five different forces given previously in the pa-per [7]. The difference in the Coulomb matrix elementsand δ C for the above two orbits are shown in Table V.For the harmonic oscillator the difference in the matrixelements was 0.250 MeV and δ C = 0 . j and j but both orbits belonging to the samemajor harmonic oscillator shell, formula (26) is not appli-cable. However, it was shown in ref. [6, 12, 13] that dueto the different binding energies, and different angularmomentum of the two orbits, in a finite well potential,the difference of the two Coulomb matrix elements in eq.(25) is comparable (within a factor of 2) to the resultsof the harmonic oscillator. See for example Table 3.2 inreference [13]. TABLE V. h ¯ A | V C | A i is calculated for Sr. From the har-monic oscillator estimate h ¯ A | V C | A i = 0.25 (MeV), and δ C =0 . h ¯ A | V C | A i δ C int. (MeV) (%)SIII 0.293 0.18SKM* 0.257 0.14SLy4 0.281 0.17BSK17 0.289 0.18SAMi0 0.331 0.23 In ref. [12] in Table 1 are listed a number of Coulombenergy differences for orbits that are within the samemajor shell or in different major shells. The values arequite similar. One can get an estimate by comparingthe relative shifts of states in mirror nuclei. Comparingthe low-lying spectra of F and O one finds that theCoulomb energy difference in the parenthesis of eq. (18)for the s / and d / is about 400 keV. From the spec-tra of Sc and Ca one finds that the difference in theCoulomb energies for the orbits p / and f / is about220 keV and from the spectra of Cu and Ni the dif-ference in Coulomb energies for the p / and f / is 260 keV. These differences are about half of the Coulomb en-ergy differences found for harmonic oscillator orbits indifferent major shells.It is interesting to mention in this respect that largeisospin impurities in the analog have been measured[16, 17] in the A = 32 isobars. In ref. [16] the exper-iment involved the Fermi transitions within the T = 1isotriplet. The analysis of the experiment indicated alarge impurity and a δ C correction of 5.3 %. In the same A = 32 nuclei members of the T = 2 multiplet were alsomeasured. (The parent state is Ar with 4 excess pro-tons). A large isospin impurity of about 1-2% was foundin the analog state [17]. The shell-model calculation in arestricted space, finds the isospin admixture to be 0.43%[18]. It is remarkable that in these nuclei the primaryconfiguration populated by the excess protons involvestwo different orbits, the s / and d / , thus allowing forthe formation of the anti-analog. If we use equation (28)for the isospin quintet in the A = 32, we find δ C = 0 . N and N + 1nodes, however as already discussed above the differencein the Coulomb matrix elements is affected by the bindingenergies and angular momentum, and sizable matrix ele-ments between the analog and anti-analog are produced[6, 12, 13]. Although the mixing with anti-analog mightcontribute to δ C it will not reach the large percentagefound in the experiment.As already remarked proceeding along the stability lineto heavier nuclei the number of excess neutrons increasesand the isospin admixture caused by the anti-analog de-creases. However, presently, (and even more in the fu-ture) it will be possible to study, proton-rich, heavy ex-otic nuclei, with a small neutron (or proton) excess (thuslow T ). In such nuclei the isospin admixtures, as onecan see from formula (28), will strongly increase. Choos-ing nuclei in which the excess protons (neutrons) occupyorbits in different major shells, and have low isospin wecan point out some examples (which are not necessarilyall feasible for experimental studies). For T = 3 / N and find fromeq. (28) δ C = 0 . Ti , δ C = 0 .
7% and for Sr , δ C = 3 . T = 2 nuclei one can look atthe example of Sr , here δ C = 2 . δ C with mass number A , for low isospin states and excess neutrons occupyingdifferent major shells, is seen in eq. (28). For T = 2 andthe mass A = 40, δ C = 0 . A = 60, δ C = 0 .
9% andfor A = 100, δ C = 3 . IX. δ C AND THE SPREADING WIDTH ANDENERGY SHIFTS OF THE ANALOG
In the doorway state approximation [6, 9] the spread-ing width of the IAS is given byΓ ↓ A = X d |h A | V C | d i| | E A − E d | Γ ↓ d , (29)where | d i denotes doorway states and Γ ↓ d their spreadingwidth.For the analog, the important doorways are the anti-analog in lighter nuclei and the IVM in the heavier. Herewe limit ourselves to the anti-analog | A i . Eq. (29) canbe written as Γ ↓ A = δ C Γ ↓ ¯ A . (30)The spreading width of the anti-analog Γ ↓ ¯ A is due to thestrong interaction and therefore is of the order of a single-particle spreading widths, thus several MeV.The practicality of the above equation is quite limited.The total width of the analog state is in general com-posed of the escape width [6, 9, 13] and spreading width.It is usually difficult to separate the two. And then thespreading width of the analog gets a contribution fromthe IVM and anti-analog and again it is not easy to sepa-rate the two. As we just mentioned above, the spreadingwidth of the anti-analog is not well known. One canget only a rough idea about the contribution of the anti-analog to the width of the analog. For example, if we usea 3 MeV spreading width for the anti-analog in Sr andthe estimated value for δ C , we conclude that the contri-bution of the anti-analog to the spreading width of theanalog is only a few keV. However, it is worth noticingthat in some medium mass nuclei, in low-isospin exoticnuclei, the mixing with the anti-analog can produce rel-atively large spreading widths of the analog, of the orderof a few tens of keV. For example, in Sr the contri-bution of the anti-analog to the spreading width of theanalog is of the order of 100 keV. In experiments, onewould observe broadened analog resonances. The mixingdiscussed here affects also the energies of the IAS. Thismixing might produce shifts of the order:∆ E = δ C ( E ¯ A − E A ) (31)The values of ∆ E are typical of the order of several tensof keV. For example in Sr this equals 70 keV. Theshifts may vary for different states in the analog nucleusand the spectrum maybe somewhat distorted comparedto the parent nucleus. For example, the three excessneutrons may occupy states with J = 5 / , J = 1 / p / f / , p / f / while another configuration with quantum number J = 5 / f / . The firsttwo states will have anti-analogs, the third will not haveand therefore the first two states will be shifted accordingto eq. (31) while the third will not.A short account of this work was discussed at a con-ference in 2014 in section 4 of ref. [19]. X. DISCUSSION AND CONCLUSION OF THEANTI-ANALOG PART
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