Generalized isobaric multiplet mass equation and its application to the Nolen-Schiffer anomaly
J. M. Dong Y. H. Zhang W. Zuo J. Z. Gu L. J. Wang Y. Sun
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          =0pt#1###2∼GBKsongInstitute of Modern Physics, Chinese
        Academy of Sciences, Lanzhou 730000, ChinaInstitute of Modern Physics, Chinese Academy of
        Sciences, Lanzhou 730000, ChinaInstitute of Modern Physics, Chinese Academy of
        Sciences, Lanzhou 730000, ChinaSchool of Physics,
        University of Chinese Academy of Sciences, Beijing 100049, ChinaChina Institute of Atomic Energy, P. O. Box 275(10),
        Beijing 102413, ChinaSchool of Physics and Astronomy, Shanghai Jiao Tong
        University, Shanghai 200240, China[ ]sunyang@sjtu.edu.cn Institute of Modern Physics, Chinese Academy of
        Sciences, Lanzhou 730000, ChinaSchool of Physics and
        Astronomy, Shanghai Jiao Tong University, Shanghai 200240, ChinaCollaborative Innovation Center of IFSA, Shanghai Jiao
        Tong University, Shanghai 200240, China
        
        The Wigner Isobaric Multiplet Mass Equation (IMME) is the most
        fundamental prediction in nuclear physics with the concept of
        isospin. However, it was deduced based on the Wigner-Eckart theorem
        with the assumption that all charge-violating interactions can be
        written as tensors of rank two. In the present work, the
        charge-symmetry breaking (CSB) and charge-independent breaking (CIB)
        components of the nucleon-nucleon force, which contribute to the
        effective interaction in nuclear medium, are established in the
        framework of Brueckner theory with AV18 and AV14 bare interactions.
        Because such charge-violating components can no longer be expressed
        as an irreducible tensor due to density dependence, its matrix
        element cannot be analytically reduced by the Wigner-Eckart theorem.
        With an alternative approach, we derive a generalized IMME (GIMME)
        that modifies the coefficients of the original IMME. As the first
        application of GIMME, we study the long-standing question for the
        origin of the Nolen-Schiffer anomaly found in the Coulomb
        displacement energy of mirror nuclei. We find that the
        naturally-emerged CSB term in GIMME is largely responsible for
        explaining the Nolen-Schiffer anomaly.
        
        24.80.+y, 13.75.Cs, 21.65.Ef, 21.10.Dr
        
        
        
        
        
        Generalized isobaric multiplet mass equation and
        its application to the Nolen-Schiffer anomaly
            Y. Sun
            December 30, 2023
        ==============================================================================================
        
         Introduction. The similarity of proton and neutron masses and
        approximate symmetry of nucleon-nucleon interactions under the
        exchange of the two kinds of nucleons lead to the concept of isospin
        <cit.>. At the isospin-symmetry limit, the
        charge-symmetry requires that the free proton-proton interaction
        v_pp excluding the Coulomb force is equal to the neutron-neutron
        v_nn, while the charge-independence requires that the
        neutron-proton interaction v_np = (v_nn+v_pp)/2 <cit.>.
        However, the nucleon-nucleon scattering data suggested that v_nn
        is slightly more attractive than v_pp, and v_np is stronger
        than (v_nn+v_pp)/2 <cit.>. In real nuclear
        systems where many-body effects are important <cit.>,
        isospin symmetry breaking has long been an active research theme
        connected to different subfields, for examples, in understanding the
        precise values of the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix
        elements between the u and d quarks <cit.>,
        the changes in nuclear structure near the N=Z line due to
        charge-violating nuclear force <cit.>, and
        the influence in nova nucleosynthesis <cit.>.
        
        Isobaric nuclei with the same mass number A, total isospin T,
        and spin-parity J^π, but different T_z=(N-Z)/2, form an
        isobaric multiplet. The Wigner isobaric multiplet mass equation
        (IMME) <cit.>
            ME(A,T,T_z)=a+bT_z+cT_z^2
        
        provides a relationship for mass excesses of an isobaric multiplet,
        where a, b and c are the coefficients depending on T and
        reduced matrix elements. This quadratic form of IMME turns out to
        work remarkably well for almost all isobaric multiplets where data
        exist <cit.>. Hence it becomes a powerful
        tool to predict unknown masses, particularly those of very
        neutron-deficient nuclei important for the astrophysical
        rp-process <cit.>. Modern radioactive beam facilities can
        provide the testing grounds of the validity of the
        IMME <cit.>, from which one may learn about the
        effective forces for nuclear many-body systems
        <cit.>.
        
        The IMME is regarded to be valid for  any charge-violating
        interactions, with the Coulomb interaction to be the dominant
        contributor. The values of b and c in Eq. (<ref>), which are
        determined experimentally, can potentially yield individual
        information on violations of the charge symmetry and
        charge-independence <cit.>. However, the proven validity of the
        IMME does not in itself provide any direct information on the nature
        of the charge-violating nuclear interaction. In shell-model
        calculations, such interaction <cit.> are 
        added to an isospin-conserving Hamiltonian, with the
        charge-symmetry breaking (CSB) or charge-independent breaking (CIB)
        components in the strong nuclear force fitted to data. In this Rapid
        Communication, we consider the contributions of CSB and CIB derived
        from nuclear medium in the effective nucleon-nucleon interaction.
        Due to density dependence of the charge-violating components,
        additional terms emerge as compared to the Wigner original IMME,
        leading to a generalized isobaric multiplet mass equation (GIMME).
        As the first application of GIMME, the binding-energy difference
        between two members of a multiplet, defined as the Coulomb
        displacement energy (CDE), is examined. The long-standing problem of
        the Nolen-Schiffer anomaly <cit.> in CDE is addressed by using
        our new formulae, without the need of involving any empirical terms.
        
        
         The effective CSB and CIB interactions in nuclear matter. In
        the study of nuclear matter with the assumption of isospin
        conservation in nuclear forces, the energy per nucleon is generally
        given as a function of density ρ =ρ _n+ρ _p and
        isospin asymmetry β =(ρ _n-ρ _p)/ρ, via E(ρ
        ,β )= E(ρ ,0)+S_2(ρ )β ^2+𝒪(β
        ^4) <cit.>, where the
        density-dependent S_2(ρ ) is the widely-studied 2nd-order
        symmetry energy coefficient. If one does not neglect the CSB and CIB
        components, additional terms appear
        
            E(ρ ,β )    =   E(ρ ,0)+S_0^(CIB)(ρ )+S_1^(CSB)
            (ρ )β
                  +[ S_2(ρ )+S_2^(CIB)(ρ )] β ^2+
            𝒪(β ^3).
        
        Specifically, the effective CSB interaction, namely the CSB
        component of the effective nucleon-nucleon interaction, gives rise
        to the 1st-order symmetry energy coefficient, defined as S_1(ρ )=∂ E(ρ ,β )/∂β|_β
        =0, while the CIB interaction solely contributes to even-order
        ones. In other words, S_1^(CSB)
        (S_2^(CIB)) measures the CSB (CIB) effect in nuclear
        medium.
        
        
        
        
        Contributions of the CSB and CIB components in a bare potential to
        the effective two-body interaction in nuclear matter can be obtained
        by solving the Bethe-Goldstone equation in the Brueckner theory with
        the AV18 interaction. AV18 contains explicit charge-dependence and
        charge-asymmetry supplemented to the AV14 potential <cit.>. To
        achieve a reliable accuracy, we determine the S_1^(CSB)
        term with the formula
        
            E(ρ ,β )-E(ρ ,-β )/2β|_AV18=S_1^(CSB)(ρ ),
        
        in order to cancel out the systematical uncertainty effectively. In
        addition, the n-p mass difference in nucleonic kinetic energy
        leads to a small part of 1st-order symmetry energy <cit.>,
        and is incorporated into the CSB effect in the present discussion.
        Similarly, the even-order symmetry energy coefficients originating
        from the CIB interaction are extracted by adopting both the AV18 and
        AV14 potentials via
        
            E(ρ ,β )+E(ρ ,-β )/2|_AV18-E(ρ ,β )|_
            AV14=S_0^(CIB)(ρ )+S_2^(CIB)(ρ
            )β ^2.
        S_0^(CIB) is an additional energy induced by the CIB
        interactions in symmetric nuclear matter, referred to as the
        zeroth-order symmetry energy coefficient. As a constant for an
        isobaric multiplet, S_0^(CIB) can be absorbed into
        E(ρ ,0), playing no role in the present discussion.
        Figure <ref> illustrates the density-dependent
        S_1^(CSB)(ρ ) and S_2^(CIB)(ρ ),
        which are found to be much smaller than the widely-investigated
        2nd-order one S_2(ρ ), and therefore, have been completely
        neglected in the study of nuclear matter. For the discussion below,
        we perform polynomial fittings for S_1^(CSB)(ρ ) and
        S_2^(CIB)(ρ ) obtained from the Brueckner theory
        
            S_1^(CSB)(ρ )    =   a_0ρ , 
            
            S_2^(CIB)(ρ )    =   a_1ρ +a_2ρ ^2+a_3ρ
            ^3,
        
        with the resulting coefficients listed in Table <ref>.
        
        
         The CSB and CIB effects in finite nuclei. With the above
        results derived for nuclear matter, we now build a Skyrme energy
        density functional for the effective CSB and CIB interactions.
        Considering the above Eqs. (<ref>,<ref>) obtained, we
        construct the effective two-body CSB and CIB interactions by
        
            v_CSB   =   -2a_0P_ij^σ(τ_3,i+τ_3,j)δ (
            r_i-r_j) 
            
            v_CIB   =   -4(a_1 +a_2ρ +a_3ρ ^2)P_ij^σδ (r_i-r_j),
        
        where the P_ij^σ is the spin exchange operator and
        τ_3 is the third-component of the Pauli operator. The local
        density ρ is evaluated at (r_i +
        r_j)/2, with r_i and
        r_j being, respectively, the spacial
        coordinates of the i-th and j-th nucleons. Accordingly, the
        expressions for the corresponding energy density are given as
        
            ℋ_CSB   =   a_0(ρ _n^2-ρ _p^2), 
            ℋ_CIB   =   (a_1+a_2ρ +a_3ρ ^2)(2ρ
            _n^2+2ρ _p^2-ρ ^2),
        
        and hence the energy per nucleon ℋ_CSB/ρ(ℋ_CIB/ρ) is exactly the
        symmetry energy term S_1^(CSB)(ρ )β(S_2^(CIB)(ρ )β ^2) in
        Eq. (<ref>). Note that the isospin exchange operator
        P_12^q=δ _q1,q2 is assumed since the charge-mixing is
        quite weak. Therefore, the 1st- and 2nd-order symmetry energy
        coefficients, a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) for finite nuclei, can be
        calculated as corresponding density functionals
        
            a_sym,1^(CSB)(A,T_z)
               =   1/IA∫_0^∞4π r^2ρ (r)S_1^(CSB)(ρ )β (r)dr,  
            
            a_sym,2^(CIB)(A,T_z)
               =   1/I^2A∫_0^∞4π r^2ρ
            (r)S_2^(CIB)(ρ )β ^2 (r)dr.
        
        In the above equations, I=(N-Z)/A=2T_z/A denotes isospin asymmetry
        of a nucleus, and β(r) =(ρ _n(r)-ρ _p(r))/ρ(r) is
        the local isospin asymmetry, with ρ _p(r) and ρ_n(r)
        being the proton and neutron density distribution, respectively.
        
        We comment on how excited states in a given multiplet are calculated
        in our theory, although these states do not appear in the discussion
        of the present work. For an isobaric analog state (IAS) with N-1
        neutrons and Z+1 protons (N>Z) whose T is greater than
        |T_z|, its wave function can be obtained by <cit.>|IAS⟩
        =|T,T_z=T-1⟩=1/√(2T)T_-|0⟩,
        where T_- is the isospin lowering operator and |0⟩
        is the ground state of the parent nucleus belonging to a multiplet
        with T = T_z (N neutrons and Z protons). Due to the above
        isospin-symmetry conserving operation, it naturally leads to (ρ
        _n+ρ _p)_IAS=(ρ _n+ρ_p)_parent.
        However, T_-|0⟩/√(2T) describes the IAS with
        the zeroth-order approximation only which conserves isospin. Because
        of the core polarization induced by the charge-violating
        interactions, corresponding corrections should be
        introduced <cit.>. Consequently, (ρ_n-ρ_p)
        _IAS-(ρ_n-ρ_p)
        _parent=-1/Tρ_n,parent ^exc.
        is obtained, where ρ _n,parent^exc. is the
        density of the excess neutrons in the parent nucleus. Thus with the
        obtained nucleonic density distributions, the symmetry energies of
        the IAS can be also computed by the above density functionals.
        
        Since a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) are related solely to the
        nuclear force, one should perform many-body calculations excluding
        the Coulomb force, which leads to
        a^(CSB)_sym,1(A,T_z)=a^(CSB)_sym,1(A,-T_z)
        and
        a^(CIB)_sym,2(A,T_z)=a^(CIB)_sym,2(A,-T_z)
        for mirror nuclei within an isobaric multiplet. Furthermore,
        considering the fact that the CSB and CIB effects are small, we
        treat them as perturbations. Consequently, both
        a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) are completely isolated
        from the rest of the energy, and thus can be reliably extracted.
        
        
        
        
        We now briefly discuss the calculated
        a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) for finite nuclei. The
        Skyrme-Hartree-Fock-BCS approach with three interactions studied in
        our previous work <cit.>, i.e., the SLy4, SLy5 and KDE
        interactions <cit.>, are employed to calculate the quantities of
        Eqs. (<ref>, <ref>), in which the empirical gaps from
        Ref. <cit.> are applied. Table <ref> lists the calculated
        results, taking ^20O (a member of A=20 quintet), ^53Ni (a
        member of A=53 quartet), and a heavy nucleus ^208Pb as
        examples. Both a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) are found to be weakly
        model-dependent because different interactions generate nearly
        identical nucleonic density profiles. For the members of isobaric
        multiplets, such as ^53Ni, the values of the 1st-order symmetry
        energy term E^(CSB)_sym,1(A,T_z)=a^(CSB)_sym,1(A,T_z)I
        A are very small due to their low isospin asymmetries I and the
        undersized S_1^(CSB)(ρ ). On the other hand,
        E^(CSB)_sym,1(A,T_z) for ^208Pb can be as
        large as -5 MeV. Apparently, the 2nd-order ones,
        E^(CIB)_sym,2(A,T_z)=a^(CIB)_sym,2(A,T_z)I^2
        A, are smaller.
        
        We thus conclude that the 1st-order symmetry energy term should not
        always be neglected in the calculations for neutron-rich nuclei. We
        note, for example, that nuclear masses can be presently predicted by
        employing macroscopic-microscopic mass models <cit.>
        with an accuracy of several hundred keV. Furthermore, CSB
        interaction has been shown to play an important role in nuclear
        structure <cit.>. Our obtained effective interactions
        including the symmetry-breaking components could be employed to
        explore the relevant problems such as the charge-exchange reactions,
        Gamow-Teller transitions, and β-decays. Up to now, shell-model
        calculations for these quantities can only be performed by
        introducing phenomenological symmetry-breaking terms with the
        strengths fitted to data <cit.>.
        
        
         A generalized IMME including effective CSB and CIB
        interactions. In his derivation of Eq. (<ref>), Wigner assumed
        |α T T_z⟩ to be the eigenstate of the
        charge-independent Hamiltonian H_0, with α for all
        additional quantum numbers to specify this state. All
        charge-violating two-body interactions, including the Coulomb
        interaction H_C among protons and H_CSB+CIB of
        CSB and CIB interactions, are treated by the first-order
        perturbation. The total negative binding energy is given by
        
            -BE(α TT_z)=⟨α TT_z|H_0+H_C+H_CSB+CIB
            |α TT_z⟩,
        
        where H_C and H_CSB+CIB are assumed to be
        written as tensors of rank two. With help of the Wigner-Eckart
        Theorem for irreducible tensor, the perturbing terms can be neatly
        expressed as reduced matrix elements and the coefficients involving
        only T and T_z.
        
        However, in nuclear medium, H_CSB+CIB becomes 
        density-dependent effective interaction. As a
        result, it can no longer be expressed as an irreducible tensor, and
        the corresponding perturbation energy ⟨α
        TT_z|H_CSB+CIB|α TT_z⟩ does not have
        analytic forms as in the case of the Coulomb interaction. When the
        effective CSB and CIB interactions are present, the perturbation
        energy in the present work is expressed as the symmetry energy terms
        
            ⟨α TT_z|H_CSB+CIB|α TT_z⟩ =a_sym,1
            ^(CSB)(A,T_z)IA+a_sym,2^(CIB)(A,T_z)I^2A,
        
        with the zeroth-order symmetry energy coefficient absorbed into a.
        One thus ends up with a generalized IMME (GIMME) in the form of
        
            ME(A,T,T_z)    =   a+( b_c+Δ _nH+2a_sym,1^(CSB)
            (A,T_z)) T_z
                  +(
            c_c+4/Aa_sym,2^(CIB)(A,T_z))
            T_z^2,
        
        with Δ _nH=0.782 MeV being the neutron-hydrogen mass
        difference. As a mass equation beyond the original IMME, the
        contribution from the effective charge-violating nuclear
        interactions is now completely separated from that of the Coulomb
        force, while the T_z-independent b_c and c_c in
        Eq. (<ref>) are induced solely by the Coulomb interaction. The coefficients of T_z
        and T_z^2 are no longer constants for a given multiplet. The
        T_z-dependence of the new
        a^(CSB)_sym,1(A,T_z) and
        a^(CIB)_sym,2(A,T_z) terms, originating from the
        CSB and CIB components of nuclear medium, are an explicit indication
        of the breakdown of the original IMME. We remark that this
        T_z-dependence is quite weak, supporting the general validity of
        the original IMME <cit.> that has been tested against many
        experimental data. Yet, under certain circumstances, the quadratic
        form of the IMME may break down, and the underlying mechanism will
        be discussed in further detail in a forthcoming paper.
        
        
        
        
        
        We now discuss how much the corrections are actually introduced by
        the CSB and CIB effects, and examine their systematic behavior. With
        the assumption that the nucleus is treated as a non-uniformly
        charged sphere <cit.>, the Coulomb energy E_c can be written as
        
            E_c=3e^2/5r_0A^1/3(1+Δ )[ Z(Z-1)-0.25(
            1-(-1)^Z) ] ,
        
        with r_0=1.2 fm, where the correction due to the last unpaired
        proton <cit.> is supplemented. The parameter Δ=5π^2
        d^2/(6r_0^2A^2/3) with d ≈ 0.55 fm <cit.> is
        introduced to describe the effect of the surface diffuseness on the
        Coulomb energy, which is a correction to the uniformly charge sphere
        model <cit.>, and the Coulomb interaction on the surface
        asymmetry is ignorable for the N≈ Z nuclei. Hence the
        contributions of the Coulomb energy to the coefficients of T_z and
        T_z^2 are simply derived as
        
            b_c   =   3e^2/5r_0A^1/3(1+Δ )[ (1-A)+
            (-1)^A/2-T-(-1)^A/2+T/8T] ,   
               c_c   =   3e^2/5r_0A^1/3(1+Δ )[ 1+
            (-1)^A/2-T+(-1)^A/2+T/4(2T-1)] .
        
        
        Figure <ref> illustrates the coefficients of the T_z and of
        T_z^2 terms extracted from the masses of the T=3/2 isobaric
        quartets <cit.>, and compares them with those given by a
        non-uniformly charged sphere <cit.>. The contributions of the CSB
        and CIB effects in Eq. (15), taking T_z= T nuclei as examples, are
        also presented in Fig. 2 for comparison. The contribution of the CSB
        effect to the coefficient of T_z term increases roughly from -80
        keV to -220 keV when A goes up from 17 to 53, which is found to
        be consistent with the estimations for the T=1 multiplets given in
        Table 5.4  of Ref. <cit.>. In general, the CSB effect results in
        a reduction of the coefficient of T_z term by 2.0%-3.1%, and
        the CIB effect enhances the coefficient of T_z^2 term by
        1.6%-4.4%. Note that, while the energy splitting among the
        isobaric multiplet is predominately attributed to the Coulomb
        interaction, clearly the corrections to the IMME have the CSB and
        CIB origin.
        
        
         On the Nolen-Schiffer anomaly. The Nolen-Schiffer anomaly
        <cit.> (NSA) is a long-standing historical problem. The
        Coulomb displacement energy (CDE) – the difference in binding
        energy between two members of a multiplet – is directly related to
        the IMME coefficients in Eq. (<ref>): for adjacent members of a
        multiplet one has CDE(A,T,T_z)=-b-c(2T_z+1)+Δ
        _nH<cit.>, where T_z is taken for the isobar with
        the larger proton number. It is an anomaly because when all the
        corrections were taken into account, there remained a consistent
        under-estimate of the CDE by about a few to ten percents
        <cit.>. Now with our GIMME, the CDE
        expression is modified as
        
            CDE(A,T,T_z)=-b_c-c_c(2T_z+1)+Δ _NSA,
        
        where the new last term arising from the CSB and CIB components of
        the nuclear medium is given by
        
            Δ _NSA   =   -2a_sym,1^(CSB)(A,T_z>)-
            4(2T_z+1)/Aa_sym,2^(CIB)(A,T_z>)  
                  +2T_z[ a_sym,1^(CSB)(A,T_z)-a_sym,1^
            (CSB)(A,T_z>)]  
                  +4T_z^2/A[ a_sym,2^(CIB)(A,T_z)-a_
            sym,2^(CIB)(A,T_z>)] ,  
               ≃   -2a_sym,1^(CSB)(A,T_z>)-4(2T_z+1)/Aa_
            sym,2^(CIB)(A,T_z>),
        
        with T_z>=T_z+1. With Δ _NSA, it becomes clear
        that the CDE has contributions from CSB and CIB, in addition to the
        Coulomb force. The CSB effect contributes
        predominately in Eq. (19), whereas the CIB effect is much smaller,
        particularly for heavier masses due to the 1/A dependence.
        According to Fig. <ref>, Δ _NSA accounts
        for 2%-3% of the CDE for isobaric quartets, which, according to our calculation,
        can add to CDE with 100-200 keV for T_z=±1/2 and 300-600 keV for
        T_z=±3/2 mirror pairs. These amounts are qualitatively consistent with what
        is needed to account for the Nolen-Schiffer anomaly, as discussed in Ref. <cit.>.
        
        
        
        
        The CDE for a T=1/2 pair of mirror nuclei,
        defined as CDE(A,T=1/2)=BE(A,T_z=1/2)-BE(A,T_z=-1/2),
        has been widely used to study the Nolen-Schiffer anomaly. In our method, the CDE is given by
        CDE(A)=-b_c-2a^(CSB)_sym,1, and the CSB effect is simply
        obtained with Δ
        _NSA=-2a^(CSB)_sym,1. Our calculated Δ
        _NSA for nuclei near the closed shells with the magic numbers 8 and 20, compared with those based on the SIII Skyrme interaction <cit.> and a
        calibrated independent-particle model <cit.> with
        inclusion of many corrections for some extensively studied mirror
        pairs, are listed in Table <ref>. It should be noted
        that the results from the dominant Coulomb term and the
        small corrections, such as the finite size of nucleons and
        short-range correlation, exhibit considerable differences
        between Refs. <cit.> and <cit.>, suggesting a
        model-dependence character in the results. Moreover, the core-polarization correction, even
        in its sign, presents a strong model-dependence <cit.>. Interestingly,
        our results are found consistent with those in Ref. <cit.> ( <cit.>)
        for particle (hole) nuclei. We emphasize, however, that our Δ
        _NSA is completely separated from the Coulomb energy, and
        in addition, our results are obtained without tuning any particular parameters.
        
        
        
        
        From our derivation, the CDE of a pair of mirror nuclei with T=1 becomes
        
            CDE(A, T=1)=-2b_c-4a^(CSB)_sym,1(A,T_z=1).
        
        In the above expression, the contribution to the CSB effect is
        directly obtained as Δ
        _NSA=-4a^(CSB)_sym,1(A,T_z=1). In order
        to compare our results with the CDE data, we compute the CDE with
        the non-uniformly charged sphere model, with and without the second
        term in Eq. (<ref>). The results together with experimental data are presented in Fig. <ref>. The difference between the two calculations is obvious. It can be seen that overall, the calculated CDE with inclusion of the CSB effect tends to describe the experimental data, where the CSB effect contributes an amount of 2%-3%.
        
        The origin of the Nolen-Schiffer anomaly has been studied by many
        works (see, for example, Ref. <cit.>) and is generally expected
        to result mainly from the CSB effect. The (isospin)
        symmetry-breaking terms are usually not included in normal
        shell-model Hamiltonians, and therefore the answer to the anomaly
        lies likely in a deeper level <cit.>. In contrast to the
        applied models based on the effects of the nucleon mass splitting or
        meson mixing <cit.>, in our
        framework, the CSB and CIB effects starting from nuclear medium are
        established by employing the microscopic Brueckner theory without
        any adjustable parameter.
        
        
        
        Incidently, the triplet displacement energy (TDE) <cit.> is
        related to our new coefficients through
        TDE(A)=2c_c+8a_sym,2^(CIB)(T_z=1)/A.
        However, the c_c coefficient cannot be well achieved with the
        charged sphere model, as that in Fig. <ref>. Different from CDE
        of a mirror pair discussed above, the TDE originates from the
        Coulomb force together with the CIB effect, where the latter
        contributes about 3%. As the Nolen-Schiffer anomaly for CDE, the
        Coulomb interaction alone cannot account for TDE, a conclusion
        consistent with the shell model studies <cit.>.
        
        
         Summary. We have generalized the Wigner IMME by considering
        the contributions of CSB and CIB derived in nuclear medium to the
        effective nucleon-nucleon interaction, and used it to study the
        Nolen-Schiffer anomaly. The main conclusions are as follows. i) The
        density-dependent CSB and CIB interactions in nuclear matter,
        characterized respectively by the symmetry energy coefficients
        S_1^(CSB) and S_2^(CIB), were built within
        the Brueckner theory with the bare interactions as inputs.
        Therefore, our work bridges the charge-violating nuclear force in
        free nucleons and that in nuclear medium. ii) With these results as
        calibrations, we established the effective CSB and CIB interactions
        in the Skyrme functions, and carried out the calculations of their
        effects in finite nuclei. For neutron-rich nuclei, we found that the
        1st-order symmetry energy term E_sym,1(A) induced by the
        CSB effect, which is generally dropped in nuclear mass calculations,
        should not be neglected. iii) The perturbative Hamiltonian with the
        density-dependent effective CSB and CIB interactions is no longer an
        irreducible tensor, hence its matrix element cannot be analytically
        reduced via the Wigner-Eckart theorem, as Wigner did <cit.>.
        We derived the GIMME which presents new corrections to the original
        Wigner IMME, where the contribution of the effective CSB and CIB
        interactions is clearly separated from that of Coulomb force. iv) As
        the first application of GIMME, the Nolen-Schiffer anomaly, which
        has been a long-standing challenge to nuclear physics, was naturally
        elucidated to a large extent to originate from the CSB effect, with
        the needed correction of several hundreds keV being reproduced.
        
        Finally, we note that our obtained CIB interaction in an effective Skyrme energy density functional describes only the ground-state properties for finite nuclei. The J-dependence of CIB (see, for example Refs. <cit.>) cannot be discussed here. This is however an important aspect of CIB, and should be investigated in the future within the present theory.
        
        J. M. D. would like to thank Yu. A. Litvinov for helpful comments
        and suggestions, and gratefully acknowledge the support of K. C.
        Wong Education Foundation. Y. S. acknowledges the discussion with J.
        Schiffer in his visit to the Argonne National Laboratory in July
        2017. This work was supported by the National Natural Science
        Foundation of China under Grants Nos. 11435014, 11775276, 11405223,
        11675265, and 11575112, by the 973 Program of China under Grant No.
        2013CB834401 and No. 2013CB834405, by the National Key Program for
        S&T Research and Development (No. 2016YFA0400501), by the Knowledge
        Innovation Project (KJCX2-EW-N01) of Chinese Academy of Sciences, by
        the Funds for Creative Research Groups of China under Grant No.
        11321064, and by the Youth Innovation Promotion Association of
        Chinese Academy of Sciences.
        
        
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