Analogous Gamow-Teller and M1 Transitions in Tz = ±1⁄2 Mirror Nuclei and in Tz = ±1, 0 Triplet Nuclei relevant to Low-energy Super GT state

Nuclei have spinand isospin-degrees of freedom. Therefore, Gamow-Teller (GT) transitions caused by the στ operator (spin-isospin operator) are unique tools for the studies of nuclear structure as well as nuclear interactions. They can be studied in β decays as well as charge-exchange (CE) reactions. Similarly, M1 γ decays are mainly caused by the στ operator. Combined studies of these transitions caused by Weak, Strong, and Electro-Magnetic interactions provide us a deeper understanding of nuclear spin-isospin-type transitions. We first compare the strengths of analogous GT and M1 transitions in the A = 27, Tz = ±1/2 mirror nuclei 27Al and 27Si. The comparison is extended to the Tz = ±1, 0 nuclei. The strength of GT transition from the ground state (g.s.) of 42Ca to the 0.611 MeV first J = 1 state in 42Sc is compared with that of the analogous M1 transition from the 0.611 MeV state to the T = 1, 0 g.s. (isobaric analog state: IAS) in 42Sc. The 0.611 MeV state has the property of Low-energy Super GT (LeSGT) state, because it carries the main part of the GT strength of all available transitions from the g.s. of 42Ca (and 42Ti) to the J = 1 GT states in 42Sc.

The M1 transitions studied in γ decays or (e, e ′ ) reactions are the simplest "magnetic-type transitions" caused by the electro-magnetic (EM) interaction [1,4,6]. In spite of the name M1, the operator that causes M1 transition is similar to that of GT in the sense that the main contribution is expected from the στ operator.
Our aim here is to compare the strengths of analogous GT and M1 transitions and derive one-step deeper structure information that are not obtained in individual studies of GT and M1 transitions.

Reduced GT and M1 transition strengths
The strengths of GT and M1 transitions can be compared by means of the "reduced transition strength (probability)" B(GT) and B(M1). The expression for these values reduced in spin (J) can be found in text books (see e.g. [6,7]). In order to compare transition strengths for analogous GT and/or M1 transitions, however, it is important to reduce the matrix element in terms of isospin as ⋆ e-mail: fujita@rcnp.osaka-u.ac.jp well [8,9]. Therefore, we start here with the reduced matrix elements in spin (J) but not in isospin and follow the convention of Edmonds [10].

Reduced GT transition strength
The reduced GT transition strength B(GT) for the transition from the initial state with spin J i , isospin T i , and zcomponent of isospin T zi to the final state with J f , T f , and T z f is given by [11] where τ ±1 = ∓(1/ √ 2)(τ x ± iτ y ) and transforms as a tensor of rank one, and T z = (N − Z)/2. By applying the Wigner-Eckart theorem in isospin space, we get where C GT is the isospin Clebsch-Gordan (CG) coefficient (T i T zi 1 ± 1|T f T z f ), and the M GT (στ) is the isovector (IV) spin-type GT matrix element. From this expression for the "reduced" GT transition strength, we see that B(GT) consists of the squared value of the matrix element of the IV spin operator M GT (στ) and spin and isospin geometrical factors. Therefore, even if the initial and final states are common, transitions in reversed directions have different B(GT) values. For example, a GT transition from a state having |J T T z ⟩ of |0 T 0 T 0 ⟩ to the |1 T 0 − 1 T 0 − 1⟩ state has three times larger B(GT) than that in the reverse direction.

Reduced M1 transition strength
The operator µ for M1 transitions and magnetic moments consists of an orbital part g ℓ ℓ and a spin part g s s [= (1/2)g s σ]. It can be rewritten as the sum of isoscalar (IS) and IV terms (for example, see [6]) as, where µ N is the nuclear magneton, the z-component of the isospin operator τ z j = 1 for neutrons and −1 for protons, and τ z j is τ 0 j . The coefficients g IS and g IV are the IS and IV combinations of gyromagnetic factors (g factors): , and g IV s = 1 2 (g π s − g ν s ). For bare protons and neutrons, the orbital and spin g factors are g π l = 1 and g ν l = 0, and g π s = 5.586 and g ν s = −3.826, respectively. Therefore, we get g IS ℓ = 0.5, g IS s = 0.880, g IV ℓ = 0.5, and g IV s = 4.706. Starting from the reduced matrix elements in spin but not in isospin, and following the convention of Edmonds [10], the reduced M1 transition strength B(M1) can be written as [11] where T z = T z f = T zi for M1 transitions. By again applying the Wigner-Eckart theorem in isospin space, we get Since the coefficient g IV s (= 4.706) is the largest, the contribution from the term of IV spin matrix M M1 (στ) is often the largest [7,12]. On the other hand, it is expected that the contributions of the IS terms are small [12].

Meson exchange current
In the comparison of GT transitions with the analogous M1 transitions, a simple relationship is obtained for the transition strengths if the IV spin term is dominant in the M1 transitions. From the comparison of Eq. (2) and Eq. (5), the "quasi proportionality" between B(GT) and B(M1) is expressed as However, it is known that the contributions from mesonexchange currents (MEC) enhance B(M1) strength over the corresponding B(GT) strength [8,9]. The enhancement is traced back to larger and additive contributions of the vector MEC over the axial-vector MEC which are active in M1 and GT transitions, respectively [13]. Taking the enhancement factor R MEC into account, we have now the relationship In order to remove the constructive and destructive contributions of both IS and orbital terms, it was proposed to sum up the strengths over a wide range of excitation energy [14] and such cancellation of orbital contributions was predicted in shell-model calculations [15]. The analyses performed for the even-even self-conjugate nuclei 24 Mg and 28 Si suggested that the R MEC value is in the range of 1.20 -1.85 [14,16,17].
Further detailed analysis has been performed for the analogous B(M1) and B(GT) values in the A = 27 mirror nuclei 27 Al and 27 Si [8,9] (also see Sec. 3). The cumulative sums of B(M1) and B(GT) values were calculated for the states with reliable B(GT) values and good correspondence in the region up to the excitation energy E x = 8.2 MeV. From the ratio of these cumulative sums, an enhancement factor R MEC = 1.4 has been derived [8,9]. Since the sum is for a limited region in excitation energy, it is not appropriate to extract a definite conclusion for the value of R MEC . We simply mention that this value 1.4 is consistent with the previous results.

Isoscalar and orbital contributions in M1
transitions From Eq. (7), we now see that the value is the στ contribution expected in the M1 transition. The interference of IS and IV orbital terms with the IV spin term in an M1 transition can be studied for the jth analogous M1 and GT transitions in isobars by examining the ratio defined by 2 EPJ Web of Conferences 178, 05001 (2018) https://doi.org/10.1051/epjconf/201817805001 CGS16 By comparing Eq. (2) and Eq. (5), it is seen that R ISO > 1 usually indicates that the IS term and/or the IV orbital term make a constructive contribution to the IV spin term, while R ISO < 1 shows a destructive contribution. As discussed in [9,18], the contribution of the IS term is usually minor. Therefore, it is expected that the deviation of R ISO from unity mainly shows a contribution of the IV orbital term in each M1 transition.

Analog states and analogous transitions
A pair of isospin T = 1/2 mirror nuclei is characterized by T z = ±1/2. All other quantum numbers of corresponding states are the same. Thus, with the assumption that isospin is a good quantum number, for every state in one of the mirror nuclei, an analog state should be found in the other nucleus (see Fig. 1). The energy levels should be almost identical in the pair nuclei, although small differences are expected from the state-dependent differences in the Coulomb displacement energies. In addition, the Coulomb displacement energy itself allows the g.s. of the T z = −1/2 nucleus to undergo β decay to the g.s. as well as to several low-lying states of the T z = +1/2 nucleus. As a result of the analogous nature of corresponding states in the mirror nuclei, following analogous transitions are expected. They are transitions from an initial to a final state in the T z = +1/2 and -1/2 nuclei and all other transitions in which initial and/or the final state is replaced by the respective analog state. Transitions reversing the initial and final states are also possible. Analogous M1 and GT transitions from or to the g.s. with spin value J g.s. 0 and with isospin T = 1/2 are schematically shown in Fig. 1.

Comparison of M1 and GT transition strengths
By comparing the analogous B(M1) and B(GT) strengths in T = 1/2 mirror nuclei, it is possible to extract the combined contribution of the IS term and the IV orbital term in the B(M1) strength using Eq. (9). Since the effect of MEC should be independent of the wave function of the individual state, the value R MEC = 1.4 derived from the analysis of the A = 27 system is used (for details, see Ref. [8]). The R ISO ratios were calculated for the fourteen pairs of M1 and GT transitions up to the excitation energy of 9.7 MeV in 27 Al, where pairs well corresponding in E x values were selected.
The results are shown in Fig. 2 for the M1 transitions in the T z = +1/2 27 Al nucleus as a function of B(M1)↑ value [the B(M1) value from the g.s. to the jth final excited state]. It is interesting to note that the R ISO value tends to deviate from unity by more than a factor of two when the B(M1)↑ is less than approximately 0.1. This shows that the "combined IS and orbital contribution" is rather large in weaker transitions and the quasi proportionality [Eq. (7)] of the B(M1) values for ∆T = 0 M1 transitions and the analogous B(GT) values is lost. This finding is interpreted as follows; since the IS term is small and the IV orbital term cannot be large for the s or d orbits with smaller orbital angular momentum ℓ, the dominance of the IV spin term of the M1 operator is guaranteed if the transitions are at least of average strength. However, the contribution of the IV spin term can also be small. Then the relative contribution of the IS term and the IV orbital term becomes significant although the transition itself is weak [8,12,18,19]. A similar discussion applies to the M1 transitions in T z = −1/2 mirror nuclei [9].

Comparison of analogous M1 and GT transition strengths in A = 42 nuclei
In a simple shell model (SM) picture, GT transitions are allowed among the LS -partner orbits j > and j < . As a result, GT excitations are expected in the low-energy region (nominally at E x = 0) and the region around E x = 3 − 6 MeV (the energy difference of the LS partner orbits).  [23] using the GXPF1J interaction [27]. The notation f 7 → f 7, for example, stands for the transition with the ν f 7/2 → π f 7/2 type. The summed value of the matrix elements is denoted by ΣM GT and its squared value is the B(GT), where the B(GT) values do not include the quenching factor of the SM calculation.

States in 42 Sc
Configurations Note that the overview of GT responses can be studied by CE reactions [3,4]. Against the above expectation for the GT strength distribution, these studies showed that the distribution in each nucleus can be largely different and dependent on the specific nuclear structure.

Low-energy Super Gamow-Teller transition observed in 42 Ca( 3 He, t) 42 Sc reaction
The most famous structure formed by GT excitation is the GT resonance (GTR) situated in high E x regions of 9-15 MeV. In 1980s, GTRs were intensively studied by pioneering (p, n) reactions [3]. They were observed in almost all nuclei with mass A > 60 and N > Z. It was found that GTRs consume ≈ 60% of the GT strength predicted by the Ikeda sum rule [20].
Various random-phase-approximation (RPA) calculations can reproduce the high E x values and the concentration of the available GT strengths in GTRs by introducing IV-type effective residual interactions (ERIs). The "repulsive" IV-type ERIs push GT strengths up in energy than is expected in a simple SM picture and concentrate the GT strengths in GTRs. In addition, available configurations contributing to the excitation of GTRs are "in phase" [21], showing the collective nature of them. It is known that IVtype ERIs are active in configurations with particle−hole (p−h) nature in GT excitations. Note that this condition is always realized in N >> Z nuclei.
On the other hand, high resolution ( 3 He, t) study on 42 Ca [22,23], and recently on 18 O [24], showed that the GT strength can also be concentrated in the lowest-energy GT state. As we see in  [22,23]. In RPA calculations, it was found that LeSGT states are formed by the contribution of IS-type ERIs [22,23,25] that are active in proton particle−neutron particle (πp−νp) configurations [25,26] on top of the LS -closed core of 40 Ca. It was found that the SM calculation for 42 Ca → 42 Sc GT transitions using the GXPF1J interaction [23,27] reproduces the concentration of the GT transition strength to the lowest T = 0, 1 + state (see Table 1). Note that not only the main configurations, i.e., ν f 7/2 → π f 7/2 and ν f 7/2 → π f 5/2 , but also p-shell configurations contribute all in phase to the excitation of the lowest 1 + state at 0.33 MeV, i.e., the LeSGT state. On the other hand, contributions of the configurations are out of phase in the other two 1 + states.

Analogous M1 and GT transitions from and to
the LeSGT state  is obtained from the 42 Ti β + decay and the same value is expected in the 42 Ca( 3 He, t) 42 Sc reaction [23] on the basis of isospin symmetry between the mirror transitions. This large B(GT) value, as discussed, is due to the in-phase nature of all available f -and p-shell configurations contributing to this GT transition. We also notice that the M1 γ decay from the 1 + , 0.611 MeV state in 42 Sc to the 0 + g.s. is analogous to the GT transitions (see Fig. 4). For this M1 transition, a large B(M1) of 6.2 (26) (15)] has been derived [28]. Large uncertainty is due to that in the halflife measurement of the 0.611 MeV state [T 1/2 = 28 (12) fs]. From the B(GT) = 2.17 (5) of the analogous GT transition and using Eq. (8), a value B(M1 στ )↑ = 8.1 (24) µ 2 N , i.e., the B(M1 στ ) value from the T = 1 g.s. to the 1 + , 0.611 MeV state, is obtained, where R MEC = 1.4 is assumed. The value B(M1 στ )↓ that can be directly compared with the γ-decay B(M1) is derived taking the differences of spin and isospin CG coefficients into account. We obtained a rather small value of 2.7 (8) µ 2 N . Relatively large uncertainty comes from that of the value of R MEC = 1.2 -1.85 that is not well known for p f -shell nuclei [14].
As discussed, under the assumption that strong M1 transitions are mainly caused by the στ-part of the M1 operator, it is expected that analogous GT and M1 transitions have corresponding transition strengths. However, here, we hesitate to come to this simplified conclusion. Note that the contribution of the IV orbital term in Eq. (4) can be larger for larger ℓ orbits (ℓ = 3 for the f 7/2 and f 5/2 orbits). Looking for a clue to answer this question, we performed a three-body model calculation assuming valence p and n particles on top of a 40 Ca core [29], where both IS and IV pairing correlations were included. The calculation showed a B(M1) value of 6.8 µ 2 N . It also suggested that contributions of the spin and orbital terms are constructive. The ratio of these contributions to the transition amplitude is approximately 1 : 0.47, which suggests that the B(M1) can be approximately twice larger than the B(M1 στ ) value.
As we have seen, the combined studies of GT and M1 transitions for the analogous transitions allow us to access "one-step deeper fields" in nuclear physics such as orbital and isoscalar contributions in M1 transitions or mesonexchange currents in M1 and GT transitions. Steady efforts for quantitative measurements, such as life-time measurements of excited states [30], and combined efforts of γand β-decay studies as well as studies using CE reactions are needed to approach such quantities.