Perturbative Approach to Effective Shell-Model Hamiltonians and Operators
PPerturbative Approach to EffectiveShell-Model Hamiltonians and Operators
Luigi Coraggio , ∗ and Nunzio Itaco , , ∗ Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Napoli, Italy Dipartimento di Matematica e Fisica, Universit `a degli Studi della Campania “LuigiVanvitelli”, Caserta, Italy
Correspondence*:Luigi CoraggioSezione INFN di Napoli, Complesso Universitario di Monte Sant’Angelo, Via Cintia -I-80126 Napoli, [email protected] ItacoDipartimento di Matematica e Fisica, Universit `a degli Studi della Campania “LuigiVanvitelli”, Viale A. Lincoln 5, Caserta, I-81100, [email protected]
ABSTRACT
The aim of this work is to present an overview of the derivation of the effective shell-modelHamiltonian and decay operators within many-body perturbation theory, and to show the results ofselected shell-model studies based on their utilisation. More precisely, we report some technicaldetails that are needed by non-experts to approach the derivation of shell-model Hamiltoniansand operators starting from realistic nuclear potentials, in order to provide some guidance to shell-model calculations where the single-particle energies, two-body matrix elements of the residualinteraction, effective charges and decay matrix elements, are all obtained without resorting toempirical adjustments. On the above grounds, we will present results of studies of double- β decay of heavy-mass nuclei where shell-model ingredients are derived from theory, so to assessthe reliability of such a way to shell-model investigations. Attention will be also focussed onthe relevant aspects that are connected to the behavior of the perturbative expansion, whoseknowledge is needed to establish limits and perspectives of this approach to nuclear structurecalculations. Keywords: Nuclear shell model, effective interactions, many-body perturbation theory, nuclear forces
The present paper is devoted to the presentation of the formal details of the derivation of effective shell-model Hamiltonians ( H eff ) and decay operators by way of a perturbative approach, and to review alarge sample of its most recent applications to the study of spectroscopic properties of atomic nuclei.The goal of our work is to provide a useful tool for those practitioners who are interested in employingshell-model single-particle energies, two-body matrix elements, effective charges, magnetic-dipole and β -decay operators, which are produced by way of many-body theory, without resorting to parameters thatare empirically adjusted to reproduce a selection of observables. a r X i v : . [ nu c l - t h ] J u l . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators
As is well known, the nuclear shell model (SM) is widely considered the basic theoretical tool for themicroscopic description of nuclear structure properties. Nuclear shell model is based on the ansatz thateach nucleon inside the nucleus moves independently from the others, in a spherically symmetric meanfield plus a strong spin-orbit term. This first-approximation depiction of a nucleus is supported by theobservation of “magic numbers” of protons and/or neutrons, corresponding to nuclei which are more tightlybound than their neighbors.These considerations lead to depict the nucleons as arranging themselves into groups of energy levels,the “shells”, well separated from each other. The main product of the shell-model scheme is the reductionof the complex nuclear many-body problem to a very simplified one, where only a few valence nucleonsinteract in a reduced model space spanned by a single major shell above an inert core.The cost that has to be paid for such a simplification is that shell-model wave functions, describing theindependent motion of individual nucleons, do not include the correlations which are induced by the strongshort-range bare interaction, and therefore could be very different from the real wave functions of thenuclei. The shell-model Hamiltonian, which will be introduced in the following section, contains one-and two-body components whose characterizing parameters, namely the single-particle (SP) energies andtwo-body matrix elements (TBMEs) of the residual interaction, account for the degrees of freedom that arenot explicitly included in the truncated Hilbert space of the configurations. As a matter of fact, SP energiesand TBMEs should be determined to include, in an effective way, the excitations both of core nucleons andof the valence nucleons into the shells above the model space.The way to the effective SM Hamiltonian may follow two distinct paths.One approach is phenomenological, that is the one- and two-body components of the Hamiltonian areadjusted to reproduce a selected set of experimental data. This can be done either using an analyticalexpression for the residual interaction with adjustable parameters, or treating the Hamiltonian matrixelements directly as free parameters (see [1, 2]).This has been, during seventy years and more of SM calculations, a very successful tool to reproduce ahuge amount of data and to describe some of the most fundamental physical properties of the structureof atomic nuclei. In this regard, it is worth to mention the review by Caurier et al. [3] for an interestingdiscussion about the properties of the effective SM Hamiltonian; a few more references and discussion willbe reported in the following section.The alternative way to the construction of H eff is to start from realistic nuclear forces - two- and three-body potentials (if possible) - and derive the effective Hamiltonian in the framework of the many-bodytheory, namely a H eff whose eigenvalues belong to the set of eigenvalues of the full nuclear Hamiltonian,defined in the whole Hilbert space.To this end, we need a similarity transformation which arranges, within the full Hilbert space of theconfigurations, a decoupling of the model space P where the valence nucleons are constrained from itscomplement Q = 1 − P .Nowadays, this may achieved within the framework of the ab initio methods, which aim to solvethe full Hamiltonian of A nucleons by employing controlled truncations of the accessible degrees offreedom. However, this approach is strictly constrained by the advance in computational power, and, evenif successful, is currently confined to few nuclear mass regions. A comprehensive report of possible waysto tackle the problem of the derivation of H eff starting from ab initio methods can be found in Ref. [4],where the authors review also some SM applications and results. This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators
Our work will be focused on the perturbative expansion of the effective SM Hamiltonian, that is groundedin the energy-independent linked-diagram perturbation theory [5], which has been extensively used inshell-model calculations during the last fifty years (see also review papers [6, 7]).The earlier attempt along this line has been made by Bertsch [8], who employed as interaction verticesthe matrix elements of the reaction matrix G derived from the Kallio-Kolltveit potential [9] to studythe role played by the core-polarization diagram at second order in perturbation theory, accounting forone-particle-one-hole ( p − h ) excitations above the Fermi level of the core nucleons. The results of thiswork evidenced that the contribution of such a diagram to H eff was about of the first-order two-bodymatrix element, when considering the open-shell nuclei O and Sc outside doubly-closed cores O and Ca, respectively.Then, it came the seminal paper by Tom Kuo and Gerry Brown [10], which is a true turning point innuclear structure theory. It has indeed been the first successful attempt to perform a shell-model calculationstarting from the free nucleon-nucleon (
N N ) Hamada-Johnston potential (HJ) [11], and resulted in aquantitative description of the spectroscopic properties of sd -shell nuclei.The TBMEs of the sd -shell effective interaction in Ref. [10] were derived starting from the HJ potential,the hard-core component being renormalized via the calculation of the reaction-matrix G . The matrixelements of G were then employed as interaction vertices of the perturbative expansion of H eff , includingterms up to second order in G .The TBMEs obtained within this approach were used to calculate the energy spectra of O and Fand provided good agreement with experiment. Moreover, these matrix elements, as well as those derivedtwo years later for SM calculations in the f p shell [12], have become the backbone of the fine tuning ofsuccessful empirical H eff s such as the USD [13] and the KB3G potentials [3, 14].The theoretical framework has evolved between the end of 1960s and beginning of 1970s, thanks to theintroduction of the folded-diagrams expansion which has formally defined the correct procedure for theperturbative expansion of effective shell-model Hamiltonians [15, 16].In the forthcoming sections we are going to present in detail the derivation of H eff and of consistenteffective SM decay operators, according to the theoretical framework of the many-body perturbation theory.The core of our approach is the perturbative expansion of two vertex functions, the so-called ˆ Q -box and ˆΘ -box, in terms of irreducible valence-linked Goldstone diagrams. The ˆ Q -box is then employed to solvenon-linear matrix equations to obtain H eff by way of iterative techniques [17], while the latter togetherwith the ˆΘ -box are the main ingredients to derive the effective decay operators [18].Our paper is organized as follows.In the next section we will present a general overview of the SM eigenvalue problem, and of the derivationof the effective SM Hamiltonian.In section 3 we will tackle the problem on the grounds of the Lee-Suzuki similarity transformation[17, 19], and in its sections we will introduce the iterative procedures to solve the decoupling equationwhich provide this similarity transformation into H eff , both for degenerate and non-degenerate modelspaces. Two sections will be devoted also to the perturbative expansion of the ˆ Q -box vertex function and tothe derivation of effective SM decay operators.In section 4 we will show the results of investigations about the double- β decay of Te and
Xe, anddiscuss the perturbative properties of H eff and effective shell-model decay operators. Frontiers 3 . Coraggio and N. Itaco
Perturbative Approach to Effective Shell-Model Hamiltonians and Operators
In the last section, a summary of our present work will be reported.
As mentioned in the Introduction, the shell model, introduced into nuclear physics seventy years ago[20, 21], is based on the assumption that, as a first approximation, each nucleon (proton or neutron) insidethe nucleus moves independently in a spherically symmetric potential representing the average interactionwith the other nucleons. This potential is usually described by a Woods-Saxon or harmonic oscillator(HO) potential including a strong spin-orbit term. The inclusion of the latter term is crucial providingsingle-particle states clustered in groups of orbits lying close in energy (shells). Each shell is well separatedin energy from the others, and this enables to schematize the nucleus as an inert core, made up by shellsfilled up with neutrons and protons paired to a total angular momentum J = 0 + , plus a certain number ofexternal nucleons, the so-called “valence” nucleons. This extreme single-particle shell model is able tosuccessfully describe various nuclear properties [22], as, for instance, the angular momentum and parityof the ground-states in odd-mass nuclei. However, it is clear that to describe the low-energy structure ofnuclei with two or more valence nucleons the “residual” interaction between the valence nucleons hasto be considered explicitly, the term residual meaning that part of the interaction which is not taken intoaccount by the central potential. The inclusion of the residual interaction removes the degeneracy of thestates belonging to the same configuration and produces a mixing of different configurations.Let us now use the simple nucleus O to introduce some common terminologies used in effectiveinteraction theories.Suppose we want to calculate the properties of the low-lying states in O. Then, we must solve theSchr¨odinger equation H | Ψ ν (cid:105) = E ν | Ψ ν (cid:105) , (1)where H = H + H , (2)and H = A (cid:88) i =1 (cid:18) p i m + U i (cid:19) , (3) H = A (cid:88) i In Fig. 1 we report the portion of the H spectrum relevant for O. valencefilledshellsemptyshellsshells f p gd s p s d s Figure 1. Energy shells which characterize the core, valence space, and empty orbitals for OWe expect that the wave functions of the low-lying states in O are dominated by components with aclosed O core (i.e. the s and p orbits are filled) and two neutrons in the valence orbits s and d .Thus, we choose a model space which is spanned by the vectors | Φ i (cid:105) = (cid:88) αβ ∈ valence space C iαβ [ a † α a † β ] i | c (cid:105) , i = 1 , ..., d, (6)where | c (cid:105) represents the unperturbed O core, as obtained by completely filling the s and p orbits | c (cid:105) = (cid:89) α ∈ filled shells a † α | (cid:105) , (7)and the index i stands for all the other quantum numbers needed to specify the state (e.g. the total angularmomentum).To sketch pictorially the situation, we report in Fig. 2 some SM configurations labeled in terms of particlesand holes with respect to the O core.To solve Eq. (1) using basis vectors like those shown in Fig. 2 amounts to diagonalizing the infinite matrix H in Fig. 3. This is unfeasible, so we want to reduce this huge matrix to a smaller one, H eff , requiringthat the eigenvalues of the latter belong to the set of the eigenvalues of the former. The notation | (cid:48) (cid:105) represents a configuration with a closed O core plus 2 particles constrained interact in the sd shell.More formally, it is convenient to introduce the projection operators P and Q = 1 − P that projectfrom the complete Hilbert space onto the model space and its complementary space (excluded space),respectively. P can be expressed in terms of the vectors in Eq. (6) as follows P = d (cid:88) i =1 | Φ i (cid:105)(cid:104) Φ i | , (8) Frontiers 5 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators |2p 0h > |3p 1h > |4p 2h > pn np n ps p s d p f Figure 2. Some O shell-model configurations ( β ) |2p 0h > |3p 1h > |4p 2h > ... H eff |2p’ 0h > |2p’ 0h >0h|2p<<< 3p4p 1h|2h|... H ( α ) Figure 3. Representation of the matrices H and H eff for OThe projection operators P and Q satisfy the properties P = P, Q = Q, P Q = QP = 0 . (9)The scope of the effective SM interaction theory is to transform the eigenvalue problem of Eq. (1) into areduced model-space eigenvalue problem P H eff P | Ψ α (cid:105) = ( E α − E C ) P | Ψ α (cid:105) , (10)where E C is the true energy of the core; i.e. in the present case, the true ground-state energy of O.As mentioned in the Introduction, there are two main lines of attack to derive H eff : • using a phenomenological approach • starting from the bare nuclear interactions by means of a well-suited many-body theory.In the phenomenological approach, empirical effective interactions containing adjustable parametersare introduced and modified to fit a certain set of experimental data or the two body-matrix elementsthemselves are treated as free parameters. This approach is very successful and we refer to several excellentreviews [2, 3, 25, 26, 27] for a complete discussion on the topic.Nowadays, there are several approaches to derive an effective SM Hamiltonian starting from the bareinteraction acting among nucleons. As a matter of fact, aside the well-established approaches based on the This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators many-body perturbation theory [5] or on the Lee-Suzuki transformation [17, 19], novel non-perturbativemethods like valence-space in-medium SRG (VS-IMSRG) [28], shell-model coupled cluster (SMCC)[29], or no-core shell-model (NCSM) with a core [30, 31, 32, 33] based on the Lee-Suzuki similaritytransformation, are now available. These non-perturbative approaches are firmly rooted in many-bodytheory and provide somehow different paths to H eff . They can be derived in the same general theoreticalframework expressing H eff as the result of a similarity transformation acting on the original Hamiltonian H eff = e G He −G , (11)where the transformation is parametrized as the exponential of a generator G , and is such that the decouplingcondition is satisfied QH eff P = 0 . (12)In Ref. [4], it can be found a very detailed discussion showing how the different methods (perturbativeand non-perturbative) can be derived in such a general framework and describing the correspondingapproximation schemes employed in each approach.As written in the Introduction, the aim of present review is to describe in detail the perturbative approachto the derivation of H eff , topic that will be discussed in the next section. We refer to the already cited reviewpaper by Stroberg et al [4] for an exhaustive description of the alternative methods. Here, we will present the formalism of the derivation of the effective SM Hamiltonian, according tothe similarity transformation introduced by Lee and Suzuki [19]. It is worth noting that this approachhas been very successful since it is amenable of a straightforward perturbative expansion of H eff foropen-shell systems outside a closed core, whereas in other approaches - such as, for example, the oscillatorbased effective theory (HOBET) proposed by Haxton and Song [34] or the coupled-cluster similaritytransformation [35] - the factorization of the core configurations with respect to the valence nucleons is farmore complicated to perform.We start from the Schr¨odinger equation for the A -nucleon system, defined in the whole Hilbert space: H | Ψ ν (cid:105) = E ν | Ψ ν (cid:105) . (13)As already mentioned, within the SM framework an auxiliary one-body potential U is introduced toexpress the nuclear Hamiltonian as the sum of an unperturbed one-body mean-field term H , plus theresidual interaction Hamiltonian H . The full Hamiltonian H is then rewritten in terms of H , H , asreported in Eqs. (2-4).According to the nuclear SM that we have introduced in the previous section, the nucleus may be depictedas a frozen core, composed by a number of nucleons which fill a certain number of energy shells generatedby the spectrum of the one-body Hamiltonian H , plus a remainder of n interacting valence nucleonsmoving in the mean field H . Frontiers 7 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators The large energy gap between the shells allows to consider the A − n core nucleons, filling completelythe shells which are the lowest in energy, as inert. The SP states accessible to the valence nucleons arethose belonging to the major shell just placed (in energy) above the closed core. The configurations allowedby the valence nucleons within this major shell define a reduced Hilbert space, the model space, in terms ofa finite subset of d eigenvectors of H , as expressed in Eq. (6).We then consider the projection operators P (see Eq.(8)) and Q = 1 − P , which project from the completeHilbert space onto the model space and its complementary space, respectively, and satisfy the properties inEq. (9).The goal of a SM calculation is to reduce the eigenvalue problem of Eq. (13) to the model-spaceeigenvalue problem H eff P | Ψ α (cid:105) = E α P | Ψ α (cid:105) , (14)where α = 1 , .., d and H eff is defined only in the model space.This means that we are looking for a new Hamiltonian H whose eigenvalues are the same of theHamiltonian H for the A -nucleon system, but satisfies the decoupling equation between the model space P and its complement Q : Q H P = 0 , (15)which guarantees that the desired effective Hamiltonian is H eff = P H P .The Hamiltonian H should be obtained by way of a similarity transformation defined in the whole Hilbertspace: H = X − HX . (16)Of course, the class of transformation operators X that satisfy the decoupling equation (15) is infinite,and Lee and Suzuki [17, 19] have proposed an operator X defined as X = e ω . Without loss of generality, ω can be chosen to satisfy the following properties: ω = QωP , (17) P ωP = QωQ = P ωQ = 0 . (18)Eq. (17) implies that ω = ω = ... = 0 . (19)According to the above equation, X may be written as X = 1 + ω , and consequently we have thefollowing expression for H eff : This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators H eff = P H P = P HP + P HQω . (20)The operator ω may be calculated by solving the decoupling equation (15), and the latter may be rewrittenas QHP + QHQω − ωP HP − ωP HQω = 0 . (21)The above matrix equation is non-linear and, once the Hamiltonian H is expressed explicitly in the wholeHilbert space, it can be easily solved. Actually, this is not an easy task for nuclei with mass A > , and,as mentioned in the previous section, this approach has been employed only for light nuclei within the ab-initio framework.A successful way to the solution of Eq. (21) for SM calculations is the introduction of a vertex function,the ˆ Q -box, which is suitable of a perturbative expansion.We proceed now to explicit the ˆ Q -box approach towards the derivation of H eff , and it is important topoint out that in the following we assume our model space to be degenerate: P H P = (cid:15) P . (22)Then, thanks to the decoupling equation (15), the effective Hamiltonian H eff1 = H eff − P H P can beexpressed as a function of ω H eff1 = P H P − P H P = P H P + P H Qω . (23)The above identity, the decoupling equation (21), and the properties of H and H allow to definerecursively the effective Hamiltonian H eff1 .First, since H is diagonal, we can write the following identity: QHP = QH P + QH P = QH P . (24)Then, the decoupling equation (21) can be rewritten in the following form: QH P + QHQω − ω ( P H P + P H P + P H Qω ) = QH P + QHQω − ω ( (cid:15) P + H eff1 ) = 0 . (25)Using this expression of the decoupling equation, we can write a new identity for the operator ω : ω = Q (cid:15) − QHQ QH P − Q (cid:15) − QHQ ωH eff1 . (26)Finally, we obtain a recursive equation by inserting Eq. (26) into the identity (23) which defines H eff1 : Frontiers 9 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators H eff1 ( ω ) = P H P + P H Q (cid:15) − QHQ QH P − P H Q (cid:15) − QHQ ωH eff1 ( ω ) . (27)We define now the vertex function ˆ Q -box as follows: ˆ Q ( (cid:15) ) = P H P + P H Q (cid:15) − QHQ QH P , (28)that allows to express the recursive equation (27) as H eff1 ( ω ) = ˆ Q ( (cid:15) ) − P H Q (cid:15) − QHQ ωH eff1 ( ω ) . (29)As can be seen from both Eqs. (28,29), configurations belonging to the Q space that are close in energyto the unperturbed energy of model-space configurations (intruder states) may provide unstable solutionsof Eq. (29). This is the so-called “intruder-state problem” as introduced in Ref. [36, 37] by Schucan andWeidenm ¨uller. In the following sections we first show two possible iterative techniques to solve Eq. (29),as suggested by Lee and Suzuki[17]. These methods, which are based on the calculation of the ˆ Q -box andits derivatives, are known as the Krenciglowa-Kuo (KK) and the Lee-Suzuki (LS) techniques. In particular,we point out that in Ref. [17] the authors have shown that the Lee-Suzuki iterative procedure is convergenteven when there are some intruder states.Then, we will present other approaches that generalize the derivation of H eff , based on the calculation ofthe ˆ Q -box, to unperturbed Hamiltonians H which provide non-degenerate model spaces. The Krenciglowa-Kuo (KK) iterative technique for solving the recursive equation (29) traces back to thecoupling of Eqs. (29) and (26), which provides the iterative equation: H eff1 ( ω n ) = ∞ (cid:88) m =0 (cid:34) − P H Q (cid:18) − (cid:15) − QHQ (cid:19) m +1 QH P (cid:35) (cid:104) H eff1 ( ω n − ) (cid:105) m . (30)The quantity inside the square brackets of Eq. (30), that will be dubbed from now on as ˆ Q m ( (cid:15) ) , isproportional to the m -th derivative of the ˆ Q -box calculated in (cid:15) = (cid:15) : ˆ Q m ( (cid:15) ) = − P H Q (cid:18) − (cid:15) − QHQ (cid:19) m +1 QH P = 1 m ! (cid:34) d m ˆ Q ( (cid:15) ) d(cid:15) m (cid:35) (cid:15) = (cid:15) . (31)We may then rewrite Eq. (30), according to the above identity, as: H eff1 ( ω n ) = ∞ (cid:88) m =0 m ! (cid:34) d m ˆ Q ( (cid:15) ) d(cid:15) m (cid:35) (cid:15) = (cid:15) (cid:104) H eff1 ( ω n − ) (cid:105) m = ∞ (cid:88) m =0 ˆ Q m ( (cid:15) ) (cid:104) H eff1 ( ω n − ) (cid:105) m . (32) This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators The starting point of the KK iterative method is the assumption that H eff1 ( ω ) = ˆ Q ( (cid:15) ) , which leads torewrite Eq. (32) in the following form: H eff = ∞ (cid:88) i =0 F i , (33)where F = ˆ Q ( (cid:15) ) F = ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) F = ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) + ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) ... (34)The above expression represent the well-known folded-diagram expansion of the effective Hamiltonian asintroduced by Kuo and Krenciglowa, since in Ref. [38] the authors demonstrated the following operatorialidentity: ˆ Q ˆ Q = − ˆ Q (cid:90) ˆ Q , (35)where the integral sign corresponds to the so-called folding operation as introduced by Brandow in Ref.[15]. The Lee-Suzuki (LS) technique is another iterative procedure which can be carried out by rearranging Eq.(29) in order to obtain an explicit expression of the effective Hamiltonian H eff1 in terms of the operators ω and ˆ Q [17]: H eff1 ( ω ) = (cid:18) P H Q (cid:15) − QHQ ω (cid:19) − ˆ Q ( (cid:15) ) . (36)The iterative form of the above equation is the following: H eff1 ( ω n ) = (cid:18) P H Q (cid:15) − QHQ ω n − (cid:19) − ˆ Q ( (cid:15) ) , (37)and we may also write an iterative expression of Eq. (26): ω n = Q (cid:15) − QHQ QH P − Q (cid:15) − QHQ ω n − H eff1 ( ω n ) . (38)The standard procedure is to start the iterative procedure by choosing ω = 0 , so that we may write: Frontiers 11 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators H eff1 ( ω ) = ˆ Q ( (cid:15) ) ω = Q (cid:15) − QHQ QH P . It can be demonstrated, by performing some algebra, the following identity: ˆ Q ( (cid:15) ) = − P H Q (cid:15) − QHQ Q (cid:15) − QHQ QH P = − P H Q (cid:15) − QHQ ω , (39)then, for the following iteration n = 2 we have: H eff1 ( ω ) = (cid:18) P H (cid:15) − QHQ ω (cid:19) − ˆ Q ( (cid:15) ) == 11 − ˆ Q ( (cid:15) ) ˆ Q ( (cid:15) ) ω = Q (cid:15) − QHQ QH P − Q (cid:15) − QHQ ω H eff1 ( ω ) . (40)Finally, the LS iterative expression of H eff is the following: H eff1 ( ω n ) = (cid:34) − ˆ Q ( (cid:15) ) n − (cid:88) m =2 ˆ Q m ( (cid:15) ) n − (cid:89) k = n − m +1 H eff1 ( ω k ) (cid:35) − ˆ Q ( (cid:15) ) . (41)It is important to point out that KK and LS iterative techniques, which allow the solution of the decouplingequation 25, in principle do not provide the same H eff . Suzuki and Lee have shown that the KK iterativeapproach provides an effective Hamiltonian whose eigenstates have the largest overlap with the modelspace ones, and that H eff obtained employing the LS technique has eigenvalues that are the lowest in energyamong those belonging to the set of the full Hamiltonian H [17].Both procedures we have presented are are limited to employ an unperturbed Hamiltonian H whosemodel-space eigenstates are degenerate in energy. However, in Ref. [39] the authors have introduced analternative approach to the standard KK and LS techniques, whose goal is to extend these methods to thenon-degenerate case by introducing multi-energy ˆ Q -boxes. Actually, this approach is quite involved forpractical applications, the only one existing in the literature being that in Ref. [40].In the following sections, we outline two methods [41, 42] to derive effective SM Hamiltonians whichmay be implemented straightforwardly to employ H s that are non-degenerate within the model space. The extended Kuo-Krenciglowa (EKK) method is an extension of the KK iterative technique to derive a H eff within non-degenerate model spaces [41, 43]. This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators We will now summarize the EKK method as follows.First, a shifted Hamiltonian ˜ H is introduced in terms of an energy parameter E ˜ H = H − E . (42)Then, we rewrite Eq. (25) in terms of ˜ H : ( E − QHQ ) ω = QH P − ωP ˜ HP − ωP H Qω = QH P − ω ˜ H eff . (43)Eq. (43) may be solved by way of an iterative procedure, analogously to the KK technique, in terms of the ˆ Q -box and its derivatives, as defined in Eqs. (28) and (31), respectively.The effective Hamiltonian ˜ H eff at the n -step of the iterative procedure may be then expressed as follows[41]: ˜ H ( n )eff = ˜ H BH (0) + ∞ (cid:88) k =1 ˆ Q k (0) (cid:104) ˜ H ( n − (cid:105) k , (44)where ˜ H BH is the solution of the Bloch-Horowitz equation [44]: ˜ H BH ( E ) = P ˜ HP + P H Q E − QHQ QH P . (45)We observe that the EKK method does not require H to be degenerate within the model space, and hasbeen therefore applied to derive H eff in a multi-shell valence space [45, 46] and in Gamow SM calculationswith realistic N N potentials [47, 48].It is worth pointing out that, since ˜ H eff = lim n →∞ ˜ H ( n )eff , we can write ˜ H eff = ˜ H BH (0) + ∞ (cid:88) k =1 ˆ Q k (0) (cid:104) ˜ H eff (cid:105) k , (46)Eq. (46) may be interpreted as a Taylor series expansion of ˜ H eff around ˜ H BH , and the parameter E corresponds to a shift of the origin of the expansion, and a resummation of the series [45]. As a matter offact, because of Eq. (42) we may express H eff as H eff = ˜ H eff + E = H BH (0) + ∞ (cid:88) k =1 ˆ Q k (0) (cid:104) ˜ H eff (cid:105) k , (47)Now, both sides of the above equation will be independent of E , providing that the summation is carriedout at infinity, and the parameter E may be tuned to accelerate the convergence of the series, when inpractical applications a numerical partial summation needs to be employed and a perturbative expansion ofthe ˆ Q -box is carried out [45]. Frontiers 13 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators ˆ Z ( (cid:15) ) vertex function Suzuki and coworkers in Ref. [42] have proposed an approach to the derivation of H eff that aims to avoidthe divergencies of the ˆ Q -box vertex function, if a non-degenerate model space in considered. In fact, thedefinition of the ˆ Q -box in Eq. (28) evidences that if (cid:15) approaches one of the eigenvalues of QHQ , theninstabilities may arise if one employs a numerical derivation, since these eigenvalues are poles for ˆ Q ( (cid:15) ) .We now sketch out the procedure of Ref. [42] and, for the sake of simplicity, consider the case of adegenerate unperturbed model space (i.e., P H P = (cid:15) P ).A new vertex function ˆ Z ( (cid:15) ) is introduced and defined in terms of ˆ Q ( (cid:15) ) and its first derivative as follows: ˆ Z ( (cid:15) ) ≡ − ˆ Q ( (cid:15) ) (cid:104) ˆ Q ( (cid:15) ) − ˆ Q ( (cid:15) )( (cid:15) − (cid:15) ) P (cid:105) . (48)It can be demonstrated that ˆ Z ( (cid:15) ) satisfies the following equation [42]: (cid:104) (cid:15) + ˆ Z ( E α ) (cid:105) P | Ψ α (cid:105) = E α P | Ψ α (cid:105) ( α = 1 , .., d ) . (49)Consequently, H eff1 may be obtained by calculating the ˆ Z -box for those values of the energy, determinedself-consistently, that correspond to the “true” eigenvalues E α .To calculate E α , we solve the following eigenvalue problem (cid:104) (cid:15) + ˆ Z ( (cid:15) ) (cid:105) | φ k (cid:105) = F k ( (cid:15) ) | φ k (cid:105) , ( k = 1 , , · · · , d ) , (50)where F k ( (cid:15) ) are d eigenvalues that depend on (cid:15) . Then, the true eigenvalues E α can be obtained through thesolution of the d equations (cid:15) = F k ( (cid:15) ) , ( k = 1 , , · · · , d ) . (51)First, it is worth pointing out some fundamental properties of ˆ Z ( (cid:15) ) and of the associated functions F k ( (cid:15) ) and then we proceed to discuss the solution of the equations (50,51).The behavior of ˆ Z ( (cid:15) ) in proximity of the poles of ˆ Q ( (cid:15) ) is dominated by ˆ Q ( (cid:15) ) , and we may write ˆ Z ( (cid:15) ) ≈ ( (cid:15) − (cid:15) ) P . This means that ˆ Z ( (cid:15) ) has no poles and therefore F k ( (cid:15) ) are continuous and differentiablefunctions for any value of (cid:15) .Eqs. (51) may have solutions that do not correspond to the true eigenvalues E α , namely spurious solutions.In Ref. [42] it has been shown that, since the energy derivative of F k ( (cid:15) ) approaches to zero at (cid:15) = E α , thestudy of this derivative provides a criterion to locate and reject spurious solutions. The solution of Eqs. (50)and (51), that are necessary to derive the effective interaction, may be obtained employing both iterativeand non-iterative methods.We describe here a graphical non-iterative method to solve Eqs. (51). This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators As mentioned before, the F k ( (cid:15) ) ’s are continuous functions of the energy, therefore the solutions of Eqs.(51) may be determined as the intersections of the graphs y = (cid:15) and y = F k ( (cid:15) ) employing one of thewell-known algorithms to solve nonlinear equations.More precisely, if we define the functions f k ( (cid:15) ) as f k ( (cid:15) ) = F k ( (cid:15) ) − (cid:15) , the solutions of Eqs. (51) canobtained by finding the roots of the equations f k ( (cid:15) ) = 0 . From inspection of the graphs y = (cid:15) and y = F k ( (cid:15) ) , we can locate for each intersection a small surrounding interval [ (cid:15) a , (cid:15) b ] where f k ( (cid:15) a ) f k ( (cid:15) b ) < .The assumption that f k ( (cid:15) ) is a monotone function within this interval implies the existence of a unique root,which can be accurately determined by means of the secant method algorithm (see for instance Ref. [49]).After we have determined the true eigenvalues E α , the effective Hamiltonian H eff1 is constructed as H eff1 = d (cid:88) α =1 ˆ Z ( E α ) | φ α (cid:105)(cid:104) ˜ φ α | , (52)where | φ α (cid:105) is the eigenvector obtained from Eq. (50) while (cid:104) ˜ φ α | is the correspondent biorthogonal state( (cid:104) ˜ φ α | φ α (cid:48) (cid:105) = δ αα (cid:48) ).As we have mentioned at the beginning of this section, we have considered the case of a degenerateunperturbed model space (i.e., P H P = (cid:15) P ), but the above formalism can be easily generalized to thenon-degenerate case replacing (cid:15) P with P H P in Eqs. (48-50). ˆ Q -box vertex function The methods to derive H eff , which have been presented in the previous sections, need the calculation ofthe ˆ Q -box function vertex function: ˆ Q ( (cid:15) ) = P H P + P H Q (cid:15) − QHQ QH P . For our purpose, the term / ( (cid:15) − QHQ ) should be expanded as a power series (cid:15) − QHQ = ∞ (cid:88) n =0 (cid:15) − QH Q (cid:18) QH Q(cid:15) − QH Q (cid:19) n , (53)leading to a perturbative expansion of the ˆ Q -box. It is useful to employ a diagrammatic representation ofthis perturbative expansion, which is a collection of Goldstone diagrams that have at least one H -vertex,are irreducible - namely at least one line between two successive vertices does not belong to the modelspace - and are linked to at least one external valence line (valence linked) [16].The standard procedure for most perturbative derivations of H eff is to deal with systems with one and twovalence nucleons, but later we will also show our way to include contributions from three-body diagramsthat come into play when more than two valence nucleons are considered. H b eff of single valence-nucleonnuclei provides the theoretical effective SP energies, while TBMEs of the residual interaction V eff areobtained from the H b eff for systems with two valence nucleons. This can be achieved by a subtractionprocedure [50], namely removing from H b eff the diagonal component of the effective SP energies, derivedfrom the H b eff of the one valence-nucleon systems. Frontiers 15 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators A useful tool, for those who want acquire a sufficient knowledge about the calculation of ˆ Q -box diagramsin an angular momentum coupled representation, is the paper by Kuo and coworkers in Ref. [51].It is worth pointing out that in current literature the effective SM Hamiltonians are derived accountingfor ˆ Q -box diagrams at most up to the third order in perturbation theory, since it is computationally highlydemanding to perform calculations including complete higher-order sets of diagrams. A complete list canbe found in Ref. [52], Appendix B, and it is a collection of 43 one-body and 135 two-body diagrams.it should be pointed out that lists of diagrams may be easily obtained using algorithms which generateorder-by-order Hugenholtz diagrams for perturbation theory applications (see, for example, Ref. [53]). Ja bc dp pp p 213 4 J p p a bp p p p p p c dJJ J JJJ Figure 4. Two-body ladder diagram at third order in perturbation theory. Arrow lines representincoming/outcoming and intermediate particle states. Wavy lines indicate interaction vertices.Since the aim of present work is to support practitioners with useful tips to derive effective SMHamiltonians within the perturbative approach, we are going to show some selected examples of ˆ Q -box diagrams and their analytical expression. Our first example is the third-order ladder diagram V ladder shown in Fig. 4, and to explicit its expression we will use the proton-neutron angular-momentum coupledrepresentation for the TBMEs of the input V NN : (cid:104) , J | V NN | , J (cid:105) ≡ (cid:104) n l j t z , n l j t z ; J | V NN | n l j t z , n l j t z ; J (cid:105) . (54)The TBMEs elements of the input potential V NN are antisymmetrized but not normalized to ease thecalculation of the ˆ Q -box diagrams, n m , l m , j m , t z m indicate the orbital and isospin quantum numbers ofthe SP state m .The analytical expression of V ladder is: (cid:104) a, b ; J | V ladder | c, d ; J (cid:105) = + 14 (cid:88) p p p p (cid:104) a, b ; J | V NN | p , p ; J (cid:105)(cid:104) p , p ; J | V NN | p , p ; J (cid:105)(cid:104) p , p ; J | V NN | c, d ; J (cid:105) [ (cid:15) − ( (cid:15) p + (cid:15) p )][ (cid:15) − ( (cid:15) p + (cid:15) p )] , (55)where (cid:15) m denotes the unperturbed single-particle energy of the orbital j m , (cid:15) is the so-called startingenergy, namely the unperturbed energy of the incoming particles (cid:15) = (cid:15) c + (cid:15) d . This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators We point out that the factor +1 / is related to the rules that characterize the calculation of overall factorsin ˆ Q -box Goldstone diagrams; for any diagram we have a phase factor ( − ( n h + n l + n c + n exh ) whose value is determined by the total number of hole lines ( n h ), the total number of closed loops ( n l ), thetotal number of crossings of different external lines as they trace through the diagrams ( n c ), and the totalnumber of external hole lines which continuously trace through the diagrams ( n exh ) [51]. There is alsoa factor (1 / n ep , that accounts of the number of pairs of lines which start together from one interactionvertex and end together to another one ( n ep ).The diagram in Fig. 4 exhibits n h = n l = n c = n exh = 0 , consequently the phase is positive. Thenumber of pairs of particles starting and ending together in the same vertices is n ep = 2 , and consequentlythe overall factor is +1 / . ca p p h pJ’ dbp hJ’ a p p p p p c hJ’’J’’J’’J’’ bd J’J’J’J’J’dbp acp J’d p pa bc hhp p p hJJ AA A A Figure 5. Two-body p - h diagram at third order in perturbation theory. Arrow lines representincoming/outcoming and intermediate particle/hole states. Wavy lines indicate interaction vertices.The factorization of Goldstone diagrams such as the ladder one in Fig. 4 is quite simple in terms of itsinteraction vertices. There is a large class of diagrams, as for example the three-particle-one-hole diagram( p - h ) in Fig. 5, which require some considerations in order to provide a straightforward factorization.The factorization may be easily performed by taking into account that the interaction operator V NN transforms as a scalar under rotation, and so introducing the following cross-coupling transformation of theTBMEs: (cid:104) a, b ; J | V NN | c, d ; J (cid:105) CC = 1ˆ J (cid:88) J (cid:48) ˆ J (cid:48) X j c j a Jj d j b JJ (cid:48) J (cid:48) (cid:104) a, b ; J (cid:48) | V NN | c, d ; J (cid:48) (cid:105) , (56)where ˆ x = (2 x + 1) / . X is the so-called standard normalized 9- j symbol, expressed as follows in termsof the Wigner - j symbol [54]: X r s tu v wx y z = ˆ t ˆ w ˆ x ˆ y r s tu v wx y z . Frontiers 17 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators The orthonormalization properties of the X symbol allow then to write the direct-coupled TBMEs interms of the cross-coupled TBMEs: (cid:104) a, b ; J | V NN | c, d ; J (cid:105) = 1ˆ J (cid:88) J (cid:48) ˆ J (cid:48) X j c j d Jj a j b JJ (cid:48) J (cid:48) (cid:104) a, b ; J (cid:48) | V NN | c, d ; J (cid:48) (cid:105) CC (57)Eqs. (56,57) help to perform the factorization of diagram in Fig. 5; first a rotation according Eq. (57)transforms the direct coupling to the total angular momentum J into the cross-coupled one J (cid:48) (diagramA going into diagram A in Fig. 5). This allows to cut the inner loop and factorize the diagram into twoterms, a ladder component ( α ) and a cross-coupled matrix element ( β ) (diagram A in Fig. 5): ( α ) = (cid:104) a, p ; J (cid:48) | A | c, h ; J (cid:48) (cid:105) CC ( β ) = (cid:104) h, b ; J (cid:48) | V NN | p , d ; J (cid:48) (cid:105) CC Then, we transform the ladder diagram (A) back to a direct coupling to J (cid:48)(cid:48) by way of Eq. (56), andfactorize it into the TBMEs (I) and (II) (diagram A in Fig. 5): (I) = (cid:104) a, p ; J (cid:48)(cid:48) | V NN | p , p ; J (cid:48)(cid:48) (cid:105) (II) = (cid:104) p , p ; J (cid:48)(cid:48) | V NN | c, h ; J (cid:48)(cid:48) (cid:105) The analytical expression of the diagram in Fig. 5 is the following: (cid:104) a, b ; J | V p h | c, d ; J (cid:105) = − 12 1ˆ J (cid:88) hp p p (cid:88) J (cid:48) J (cid:48)(cid:48) ˆ J (cid:48)(cid:48) X j c j d Jj a j b JJ (cid:48) J (cid:48) X j c j a J (cid:48) j h j p J (cid:48) J (cid:48)(cid:48) J (cid:48)(cid:48) (58) × (cid:104) h, b ; J (cid:48) | V NN | p , d ; J (cid:48) (cid:105) CC (cid:104) a, p ; J (cid:48)(cid:48) | V NN | p , p ; J (cid:48)(cid:48) (cid:105)(cid:104) p , p ; J (cid:48)(cid:48) | V NN | c, h ; J (cid:48)(cid:48) (cid:105) [ (cid:15) − ( (cid:15) p + (cid:15) p )][ (cid:15) − ( (cid:15) p + (cid:15) p )] , The factor ( − / accounts the fact that n ep = 1 , n h = n l = 1 , and that an extra-phase factor ( − n ph isneeded for the total number of cut of particle-hole pairs ( n ph ) [51], since in order to factorize the diagramwe have cut the inner loop.It is important pointing out that there are other three diagrams with the same topology as the one in Fig.5, which corresponds to the exchange of the external incoming and outcoming particles.Let us now turn our attention to one-body diagrams.First of all, we consider the contribution of diagrams such as the one in Fig. 6.The diagram in Fig. 6 is the so-called ( V - U ) -insertion diagram, and is composed of the self-energydiagram ( V -insertion diagram) minus the auxiliary potential U -insertion. The U -insertion diagrams aredue to the presence of the - U term in H . The analytical expression of this diagrams is the following: This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators (B)ab abhba (A) Figure 6. ( V - U )-insertion diagram. Graph ( A ) is the self-energy diagram. Graph ( B ) represents the matrixelement of the harmonic oscillator potential U = mω r . (cid:104) a || ( V - U ) || b (cid:105) = δ j a j b j a + 1 (cid:88) Jh (2 J + 1) (cid:104) j a , h ; J | V | j b , h ; J (cid:105) − (cid:104) a || U || b (cid:105) (59) = δ j a j b j a + 1 (cid:88) Jh (2 J + 1) (cid:104) j a , h ; J | V | j b , h ; J (cid:105) − (cid:104) a || mω r || b (cid:105) The calculation of the self-energy diagram A has been performed by coupling the external lines to ascalar, which leads the SP total angular momentum and parity j a , j b being identical. Then, we cut theinner hole line and, since SP states a, b are coupled to J = 0 + , we apply the transformation in Eq. (56) for J = 0 + .Since the standard choice for the auxiliary potential is the harmonic-oscillator (HO) one, it appears alsothe reduced matrix element of U = mω r between SP states a and b (graph B ).It is worth pointing out that the diagonal contributions of ( V - U )-insertion diagrams, for SP statesbelonging to the model space, correspond to the first order contribution of the perturbative expansion of theeffective SM Hamiltonian of single valence-nucleon systems H b eff .Moreover, ( V - U ) -insertion diagrams turn out to be identically zero when employing a self-consistentHartree-Fock (HF) auxiliary potential [40], and in Ref. [52] it has been discussed the important role playedby these terms, comparing different effective Hamiltonians derived starting from ˆ Q -boxes with and withoutcontributions from ( V - U ) -insertion diagrams.Now, we will show an example of one-body diagram and comment briefly its analytical calculation.We consider the diagram as reported in Fig. 7, while the complete list of third-order one-body diagramscan be found in Ref. [52] Fig. B.19.We dub this diagram V p h , since between the upper interaction vertices they appear two particles and 1hole as intermediate states. This belongs to the group of non-symmetric diagrams, which occur always inpairs giving equal contributions. Its analytical expression is: Frontiers 19 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators p p j hh jjj p h p h h j h h p p h p p h j JJJJJ’=0 + J’=0 + J’=0 + AAA Figure 7. An example of one-body diagram (see text for details). (cid:104) j || V p h || j (cid:105) = − 12 12 j + 1 (cid:88) Jp p h h (2 J + 1) (cid:104) j, h ; J | V NN | p , p ; J (cid:105)(cid:104) p , p ; J | V NN | h , h ; J (cid:105)(cid:104) h || V - U || j (cid:105) [ (cid:15) − ( (cid:15) p + (cid:15) p − (cid:15) h )][ (cid:15) − ( (cid:15) j + (cid:15) p + (cid:15) p − (cid:15) h − (cid:15) h )] , (60) (cid:15) = (cid:15) j being the unperturbed SP energy of the incoming particle j .In order to factorize the diagram, we have first cross-coupled the incoming and outcoming model-spacestates j to J (cid:48) = 0 + (diagram A in Fig. 7). Then we cut the hole-line h and, by way of Eq. (56), we obtaina sum of two-body diagrams which are direct-coupled to the total angular momentum J [51] (diagram A in Fig. 7). These operations are responsible of the factors / (2 j + 1) and (2 J + 1) , the overall factor / is due to the pair of particle lines p , p starting and ending at same vertices, and the minus sign comesfrom the 2 hole lines and 1 loop appearing in the diagram. The factorization accounts also of the ( V - U ) insertion (cid:104) h || V - U | j (cid:105) .As mentioned before, this diagrammatics is valid to derive H eff for one- and two-valence nucleon systems,and things are different and more complicated if one would like to derive H eff for system with three or morevalence nucleons.Actually, none of available SM codes can perform the diagonalization of SM Hamiltonians with three-body components, apart from BIGSTICK SM code [55] but only for light nuclei.In order to include the contribution to H eff from ˆ Q -box diagrams with at least three incoming andoutcoming valence particles, we resort to the so-called normal-ordering decomposition of the three-bodycomponent of a many-body Hamiltonian [56]. To this end, we include in the calculation of the ˆ Q -box alsosecond-order three-body diagrams, which, for those nuclei with more than 2 valence nucleons, account forthe interaction via the two-body force of the valence nucleons with core excitations as well as with virtualintermediate nucleons scattered above the model space (see Fig. 8).For each topology reported in Fig. 8, there are nine diagrams, corresponding to the possible permutationsof the external lines. The analytical expressions of the second-order three-body contributions is reported inRef. [57], and we derive from those expressions a density-dependent two-body term.To this end, we calculate, for each ( A, B ) topology, nine one-loop diagrams, namely the graph ( α ) in Fig.9. Their explicit form, in terms of the three-body graphs ( A, B ) , is: This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators cba p f fa bh A B dc e ed Figure 8. Second-order three-body diagrams. The sum over the intermediate lines runs over particle andhole states outside the model space, shown by A and B, respectively. For the sake of simplicity, for eachtopology we report only one of the diagrams which correspond to the permutations of the external lines. α dca b efJ’J’JJ cdab a bc d mJJ A,B Figure 9. Density-dependent two-body contribution that is obtained from a three-body one. α is obtainedby summing over one incoming and outgoing particle of the three-body graphs A reported in Fig. 8. (cid:104) ( j a j b ) J | V α | ( j c j d ) J (cid:105) = (cid:88) m,J (cid:48) ρ m ˆ J (cid:48) ˆ J (cid:104) [( j a j b ) J , j m ] J (cid:48) | V A,B | [( j c j d ) J , j m ] J (cid:48) (cid:105) , (61)where the summation over m -index runs in the model space. ρ m is the unperturbed occupation density ofthe orbital m according to the number of valence nucleons.Finally, the perturbative expansion of the ˆ Q -box contains one- and two-body diagrams up to third orderin V NN , and a density-dependent two-body contribution accounting for three-body second-order diagrams[58, 57].It should be pointed out that the latter term depends on the number of valence protons and neutrons, thusleading to the derivation of specific effective shell-model Hamiltonians, that differ only for the two-bodymatrix elements. In the shell-model approach, we are interested not only in calculating energies, but also the matrixelements of operators Θ representing physical observables (e.g. e.m. transition rates, multipole moments,...).Since the wave-functions | ψ α (cid:105) obtained diagonalizing H eff are not the true ones | Ψ α (cid:105) , but their projectionsonto the chosen model space ( | ψ α (cid:105) = P | Ψ α (cid:105) ), it is obvious that one has to renormalize Θ to take into Frontiers 21 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators account the neglected degrees of freedom corresponding to the Q -space. In other words, one has to considerthe short range correlation “wounds” inflicted by the bare interaction on the SM wave functions. Formally,one wants to derive an effective operator Θ eff such that (cid:104) ˜Ψ α | Θ | Ψ β (cid:105) = (cid:104) ˜ ψ α | Θ eff | ψ β (cid:105) . (62)The perturbative expansion of effective operators has been approached since the earliest attempts toemploy realistic potentials for SM calculations, and among many they should be mentioned the fundamentaland pioneering studies carried out by L. Zamick for the problematics of electromagnetic transitions[59, 60, 61] and I. S. Towner for the study of the quenching of spin-operator matrix elements [62, 63].In present section we discuss the formal structure of non-Hermitian effective operators, as introducedby Suzuki and Okamoto in Ref. [18]. More precisely, we provide an expansion formula for the effectiveoperators in terms of the ˆΘ -box, that analogously to the ˆ Q -box in the effective interaction theory (see Sec,3), is the building block for constructing effective operators.According to Eq. (20) (and keeping in mind that ω ≡ QωP ), we may write H eff as H eff = P H ( P + ω ) , (63)so that we can express the true eigenstates | Ψ α (cid:105) and their orthonormal counterparts (cid:104) ˜Ψ α | as | Ψ α (cid:105) = ( P + ω ) | ψ α (cid:105) (cid:104) ˜Ψ α | = (cid:104) ˜ ψ α | ( P + ω † ω )( P + ω † ) . (64)On the other hand, a general effective operator expression in the bra-ket representation is given by Θ eff = (cid:88) αβ | ψ α (cid:105)(cid:104) ˜Ψ α | Θ | Ψ β (cid:105)(cid:104) ˜ ψ β | , (65)where Θ is a general time-independent Hermitian operator. Therefore we can write Θ eff in an operatorform as Θ eff = ( P + ω † ω ) − ( P + ω † )Θ( P + ω ) . (66)It is worth noting that Eq. (62) holds independently of the normalization of | Ψ α (cid:105) and | ψ α (cid:105) , but if the trueeigenvectors are normalized, then (cid:104) ˜Ψ α | = (cid:104) Ψ α | and the | ψ α (cid:105) should be normalized in the following way (cid:104) ˜ ψ α | ( P + ω † ω ) | ψ α (cid:105) = 1 . (67)To explicitly calculate Θ eff , we introduce the ˆΘ -box defined as ˆΘ = ( P + ω † )Θ( P + ω ) , (68)so that Θ eff can be factorized as Θ eff = ( P + ω † ω ) − ˆΘ . (69)The derivation of Θ eff is divided in two parts: the calculation of ˆΘ and the one of ω † ω . This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators According to Eq. (68) and taking into account the expression of ω in terms of H eff ω = ∞ (cid:88) n =0 ( − n (cid:18) (cid:15) − QHQ (cid:19) n +1 QH P ( H eff1 ) n , (70)we can write ˆΘ = ˆΘ P P + ( ˆΘ P Q + h.c. ) + ˆΘ QQ , (71)where ˆΘ P P = P Θ P , (72) ˆΘ P Q = P Θ ωP = ∞ (cid:88) n =0 ˆΘ n ( H eff1 ) n , (73) ˆΘ QQ = P ω † Θ ωP = ∞ (cid:88) n,m =0 ( H eff1 ) n ˆΘ nm ( H eff1 ) m , (74)and ˆΘ m , ˆΘ mn have the following expressions: ˆΘ m = 1 m ! d m ˆΘ( (cid:15) ) d(cid:15) m (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = (cid:15) , (75) ˆΘ mn = 1 m ! n ! d m d(cid:15) m d n d(cid:15) n ˆΘ( (cid:15) ; (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = (cid:15) ,(cid:15) = (cid:15) , (76)with ˆΘ( (cid:15) ) = P Θ P + P Θ Q (cid:15) − QHQ QH P , (77) ˆΘ( (cid:15) ; (cid:15) ) = P H Q (cid:15) − QHQ Q Θ Q (cid:15) − QHQ QH P . (78)As regards the product ω † ω , using the definition (31), we can write ω † ω = − ∞ (cid:88) n =1 ∞ (cid:88) m =1 (( H eff1 ) † ) n − ˆ Q ( (cid:15) ) n + m − ( H eff1 ) m − . (79)Expressing H eff1 in in terms of the ˆ Q -box and its derivatives (see Eqs. (33,34)), the above quantity may berewritten as ω † ω = − ˆ Q + ( ˆ Q ˆ Q + h.c. ) + ( ˆ Q ˆ Q ˆ Q + h.c. ) + ( ˆ Q ˆ Q ˆ Q + h.c. ) + · · · (80)Putting together Eq. (77) and (80), we can write the final perturbative expansion form of the effectiveoperator Θ eff Θ eff = ( P + ˆ Q + ˆ Q ˆ Q + ˆ Q ˆ Q + ˆ Q ˆ Q + · · · ) × ( χ + χ + χ + · · · ) , (81) Frontiers 23 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators where χ = ( ˆΘ + h.c. ) + ˆΘ , (82) χ = ( ˆΘ ˆ Q + h.c. ) + ( ˆΘ ˆ Q + h.c. ) , (83) χ = ( ˆΘ ˆ Q ˆ Q + h.c. ) + ( ˆΘ ˆ Q ˆ Q + h.c. ) + ( ˆΘ ˆ Q ˆ Q + h.c. ) + ˆ Q ˆΘ ˆ Q . (84) · · · It is worth enlightening the strong link existing between H eff and any effective operator. This is achievedby inserting the identity ˆ Q ˆ Q − = in Eq. (81), so to obtain the following expression: Θ eff = ( P + ˆ Q + ˆ Q ˆ Q + ˆ Q ˆ Q + ˆ Q ˆ Q + · · · ) ˆ Q ˆ Q − × ( χ + χ + χ + · · · ) == H eff ˆ Q − ( χ + χ + χ + · · · ) . (85)In actual calculations the χ n series is arrested to a finite order and the starting point is the derivation of aperturbative expansion of ˆΘ ≡ ˆΘ( (cid:15) ) and ˆΘ ≡ ˆΘ( (cid:15) ; (cid:15) ) , including diagrams up to a finite order in theperturbation theory, consistently with the expansion of the ˆ Q -box. The issue of the convergence of the χ n series and of the perturbative expansion of ˆΘ and ˆΘ will be extensively treated in section 4.1.In Fig. 10 we report all the diagrams up to second order appearing in the ˆΘ expansion for a one-bodyoperator Θ . hab h pab + + + apb ab Figure 10. One-body second-order diagrams included in the perturbative expansion of ˆΘ . The asteriskindicates the bare operator Θ .The evaluation of the diagrams involved in the derivation of Θ eff follows the same rules described in theprevious section. In the following, therefore, we will just outline the procedure to calculate such diagramswith one Θ vertex.Let us suppose that the operator Θ transforms like a spherical tensor of rank λ , component µ : Θ ≡ T λµ , (86)with ( T λµ ) † = ( − λ − µ T λ − µ . (87) This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators hab h p λ λ ab λλ pph Figure 11. One-body second-order 2p-1h diagram included in the perturbative expansion of ˆΘ . The crossindicates the bare operator Θ .By using the Wigner-Eckart theorem, it is possible to express any transition matrix element in terms of areduced transition element (cid:104) j a || T λ || j b (cid:105) = ( − λ − µ (cid:104) j a | T λµ | j b (cid:105) , (88)where in the r.h.s. of Eq. (88) j b is coupled to j a to a total angular momentum and projection equal to λ and − µ , respectively, and we have assumed, without any lack of generality, that we are dealing withsingle-particle states.Therefore, we evaluate each diagram as a contribution to the reduced matrix element of the effectiveoperator. To be more explicit, we consider as an example the calculation of the following second-orderdiagram that takes into account the renormalization of the operator due to p - h core excitations.The first step is to couple j b and j a to a total angular momentum equal to λ . This enables to factorize thediagram as the product of a cross-coupled matrix element of the interaction and the reduced matrix elementof the operator (see right-hand side of Fig. 11).Explicitly, we can evaluate the diagram as (cid:104) j a || Θ p h || j b (cid:105) = − (cid:88) p,h ( − j p + j h − λ (cid:104) j a , p ; λ | V NN | j b , h ; λ (cid:105) CC (cid:104) h || T λ || p (cid:105) (cid:15) − ( (cid:15) a + (cid:15) b − (cid:15) h ) . (89)The minus sign in front of the value is due to the fact that n h = n l = 1 , and that an extra-phase factor ( − n ph is needed for the total number of cut of particle-hole pairs ( n ph ) [51], since we have cut the innerloop to factorize the diagram. We present in this section a specific example of SM calculations which have been performed by employingeffective SM Hamiltonian a decay operators derived from realistic nuclear potentials within the many-bodyperturbation theory.It should be noted that this kind of calculations have been carried out since the middle of 1960s, but thespirit has been mostly to retain only the TBMEs, since the single-body component of H eff has not beenconsidered enough accurate to provide SM results in a good agreement with experiment. A large sample ofcalculations which are framed in such a successful approach can be found in previous reviews on this topic[6, 7]. Frontiers 25 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators We have sampled in this work the results of a calculation where both SP energies and TBMEs, thatare needed to diagonalize the SM Hamiltonian, have been obtained by deriving H eff according to theprocedures that have been reported in the previous section. Besides H eff , the many-body perturbationtheory has been employed to derive consistently effective operators to calculate electromagnetic transitionrates and Gamow-Teller (GT) strengths without resorting to empirical effective charges or quenchingfactors for the axial coupling constant g A .There are some motivations that lead to perform SM calculations by deriving and employing all SMparameters - SP energies, TBMEs, effective transition and decay operators - starting from realistic nuclearforces: • the need to study the soundness of many-body perturbation theory in order to provide reliable SMparameters; • to investigate the ability of classes of nuclear potentials to describe nuclear structure observables; • the opportunity to compare and benchmark SM calculations with other nuclear structure methodswhich employ realistic potentials.The goal of these studies is to test the reliability of such an approach to the nuclear shell model, especiallyits predictiveness that is crucial to describe physical phenomena that are not yet at hand experimentally. β decay around doubly-closed Sn Neutrinoless double- β ( νββ ) decay is an exotic second-order electroweak process predicted byextensions of the Standard Model of particle physics. Observation of such a process would show thenon-conservation of lepton number, and evidence that neutrinos have a Majorana mass component (seeRefs. [64, 65] and references therein).In the framework of light-neutrino exchange, the half life for the νββ decay is inversely proportional tothe square of the effective Majorana neutrino mass (cid:104) m ν (cid:105) : (cid:104) T ν / (cid:105) − = G ν (cid:12)(cid:12) M ν (cid:12)(cid:12) g A (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) m ν (cid:105) m e (cid:12)(cid:12)(cid:12)(cid:12) , (90)where g A is the axial coupling constant, m e is the electron mass, G ν is the so-called phase-space factor (orkinematic factor), and M ν is the nuclear matrix element (NME), that is connected to the wave functionsof the nuclei involved in the decay.At present, the phase-space factors for nuclei that are possible candidates for νββ decay may becalculated with great accuracy [66, 67]. Therefore, it is crucial to have precise values of the NME, both toimprove the reliability of the νββ lifetime predictions - fundamental ingredient to design new experiments- and to extract neutrino properties from the experimental results, when they will become available.Several nuclear structure models are exploited to provide NME values as precise as possible, the mostlargely employed being, at present, the Interacting Boson Model (IBM) [68, 69, 70], the QuasiparticleRandom-Phase Approximation (QRPA) [71, 72, 73, 74], Energy Density Functional methods [75], theCovariant Density Functional Theory [76, 77, 78], the Generator-Coordinate Method (GCM) [79, 80, 81,82], and the Shell Model (SM) [83, 84, 85, 86, 87]. This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators All the above models use a truncated Hilbert space to overcome the computational complexity, and eachof them can be more efficient than another for a specific class of nuclei. However, when comparing thecalculated NMEs obtained with different approaches, it can be shown that, at present, the results can differby a factor of two or three (see for instance the review in Ref. [88]).In Ref. [89], it has been reported on the calculation of the νββ -decay NME for Ca, Ge, Se, Te,and Xe in the framework of the realistic SM, where H eff s and νββ -decay effective operators areconsistently derived starting from a realistic N N potential, the high-precision CD-Bonn potential [90].It is worth mentioning that the above work is not the first example of such an approach, since it has beenpioneered by Kuo and coworkers [91, 92], and more recently pursued by Holt and Engel [93].In present work, we will restrict ourselves to present the results obtained in Ref. [89] for the heavy-massnuclei around Sn, Te and Xe. At present, these nuclei are under investigation as νββ -decaycandidates by some large experimental collaborations. The possible νββ decay of Te is explored bythe CUORE collaboration at the INFN Laboratori Nazionali del Gran Sasso in Italy [94], while Xeis investigated both by the EXO-200 collaboration at the Waste Isolation Pilot Plant in Carlsbad, NewMexico, [95], and by the KamLAND-Zen collaboration in the Kamioka mine in Japan [96].The starting point of the SM calculation is the high-precision CD-Bonn N N potential [90], whosenon-perturbative behavior induced by its repulsive high-momentum components is healed resorting to theso-called V low - k approach [97].This provides a smooth potential which preserves exactly the onshell properties of the original N N potential up to a chosen cutoff momentum Λ . As in other SM studies [98, 99, 100, 101], the value of thecutoff has been chosen as Λ = 2 . fm − , since the role of missing three-nucleon force (3NF) decreases byenlarging the V low - k cutoff [99]. In fact, in Ref. [99] it has been shown that H eff s derived from V low - k s withsmall cutoffs ( Λ = 2 . fm − ) own SP energies in a worse agreement with experiment, and also anunrealistic shell-evolution behavior. This characteristic may be ascribed to larger impact of the induced3NF, which becomes less important by enlarging the cutoff.We have experienced that Λ = 2 . fm − , within a perturbative expansion of the ˆ Q -box, is an upper limit,since a larger cutoff worsens the order-by-order behavior of the perturbative expansion, and at the endof present section it is reported a study of the perturbative properties of H eff and of the effective decayoperators derived using this V low - k potential.The Coulomb potential is explicitly taken into account in the proton-proton channel.The shell-model effective hamiltonian H eff has been derived within the framework of the many-bodyperturbation theory as described in section 3, including diagrams up to third order in H in the ˆ Q -box-expansion, while all the effective operators, both one- and two-body, have been obtained consistently usingthe approach described in section 3.3, including diagrams up to third order in perturbation theory in theevaluation of the ˆΘ -box and arresting the χ n series in Eq. (81) to χ .The effective hamiltonian and operators are defined in a model space spanned by the five g / , d / , d / , s / , h / proton and neutron orbitals outside the doubly-closed Sn core. Thesingle-particle (SP) energies, and the two-body matrix elements (TBMEs) of H eff can be found in Ref.[101].Before showing the results for the νββ NME obtained in Ref. [89], it is worth checking the reliability ofthe adopted approach. To this end, we present here some results obtained for the calculation of quantities Frontiers 27 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators for which there exist an experimental conterpart to compare with. Namely, we show some selected resultsfor the electromagnetic properties, GT strength distributions and νββ decays in Te and Xe, that arereported in Refs. [101, 102].In Figs. 12 and 13 they are shown the experimental [103, 104] and calculated low-energy spectraand B ( E s of parent and grand-daughter nuclei involved in double-beta-decay of Te and Xe,respectively. E ( M e V ) Exp Th + + ,4 + + + + + + + + + + + + + Te + E ( M e V ) Exp Th + (0 + )2 + + + + + + + + + + + + + Xe + + + Figure 12. Experimental and calculated spectra of Te and Xe. The arrows are proportional to the B ( E strengths, whose values are reported in e fm . Reproduced from Ref. [102]. E ( M e V ) Exp Th + (3 + ,4 + )4 + + + + + + + + Xe 420 6 E ( M e V ) Exp Th + + + + + - + + + + + + - + Ba + + Figure 13. Same as in Fig. 12, but for Xe and Ba. Reproduced from Ref. [102].From the inspection of Figs. 12 and 13, it can be seen that the comparison between theory and experiment,as regards the low-lying excited states and the B ( E transition rates, is quite good for Te, Xe, and Ba, while it is less satisfactory for Xe, whose theoretical spectrum is expanded when compared withthe observed one. As regards the electromagnetic properties, in Ref. [102] they are calculated also some B ( M strengths and magnetic dipole moments using an effective spin-dependent M operator, and thecomparison with the available data (see Tables VII and IX in Ref. [102]) evidences a good agreement.Two different kinds of experimental data related to GT decay are available in Te and Xe: GT-strengthdistributions, and the NMEs involved in νββ decays.The GT strength B (GT) can be extracted from the GT component of the cross section at zero degreeof intermediate energy charge-exchange reactions, following the standard approach in the distorted-waveBorn approximation (DWBA) [105, 106]: This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators dσ GT (0 ◦ ) d Ω = (cid:16) µπ (cid:126) (cid:17) k f k i N στD | J στ | B ( GT ) , (91)where N στD is the distortion factor, | J στ | is the volume integral of the effective N N interaction, k i and k f are the initial and final momenta, respectively, and µ is the reduced mass.On the other hand, the experimental νββ NME M ν GT can be extracted from the observed half life T ν / of the parent nucleus (cid:104) T ν / (cid:105) − = G ν (cid:12)(cid:12) M ν GT (cid:12)(cid:12) . (92)Both the above quantities can be calculated in terms of the matrix elements of the GT − operator (cid:126)στ − B ( GT ) = (cid:12)(cid:12)(cid:12) (cid:104) Φ f || (cid:80) j (cid:126)σ j τ − j || Φ i (cid:105) (cid:12)(cid:12)(cid:12) J i + 1 , (93) M ν GT = (cid:88) n (cid:104) + f || (cid:126)στ − || + n (cid:105)(cid:104) + n || (cid:126)στ − || + i (cid:105) E n + E , (94)where E n is the excitation energy of the J π = 1 + n intermediate state, E = Q ββ (0 + ) + ∆ M , Q ββ (0 + ) and ∆ M being the Q value of the ββ decay and the mass difference between the daughter and parent nuclei,respectively. The nuclear matrix elements in Eqs. (93) and (94) are calculated within the long-wavelengthapproximation, including only the leading order of the Gamow-Teller operator in a nonrelativistic reductionof the hadronic current.In Ref. [102], the GT strength distributions and the νββ NMEs have been calculated for Te and Xe using an effective spin-isospin dependent GT operator, derived consistently with H eff by followingthe procedure described in section 3.3.Fig. 14 shows the theoretical running sums of the GT strengths Σ B (GT) , calculated with both bare andeffective GT operators, reported as a function of the excitation energy, and compared with the available dataextracted from ( He, t ) charge-exchange experiments [107, 108], for Te and Xe. From its inspection,it can be seen that, in both nuclei, the GT-strength distributions calculated using the bare GT operatoroverestimate the experimental ones by more than a factor of two. Including the many-body renormalizationof the GT operator brings the predicted GT strength distribution into much better agreement with thatextracted from experimental data.In Ref. [102] the NMEs M ν GT involved in the decay of Te and Xe are calculated using the definitionin Eq. (94), by means of the Lanczos strength-function method as in Ref. [3]. The results obtained with thebare GT operator and with the effective one are reported in Table 1, and compared with the experimentalvalues [109].The effective operator induces a relevant quenching of the calculated NME, 47% for Te and 37%for Xe decay, leading to a quite good agreement with the experimental value in both nuclei, that is ofthe same quality as that of other shell-model calculations where all parameters (SP energies and TBMEs)have been fitted to experiment and a quenching factor q has been introduced to reproduce GT data (see for Frontiers 29 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators x (MeV)00,511,522,53 Σ B ( G T ) bare effectiveexp( He,t) Te GT strength x (MeV)00,511,522,53 Σ B ( G T ) bare effectiveexp( He,t) Xe GT strength Figure 14. Running sums of the B ( GT ) strengths as a function of the excitation energy E x up to 3 MeV,and 4.5 MeV respectively for Te and Xe. Reproduced from Ref. [102]. Table 1. Experimental [109] and calculated NME of the νββ decay (in MeV − ) for Te and Xe.Decay NME Expt bare effective Te → Xe . ± . Xe → Ba . ± . g A . This supports thereliability of this approach to calculate the NME involved in νββ , whose results have been reported inRef. [89], and are shortly recollected in the following.The νββ two-body operator for the light-neutrino scenario can be expressed in the closure approximation(see for instance Refs. [111, 112]) in terms of the neutrino potentials H α and form functions h α ( q ) ( α = F ,GT, or T ) as follows Θ GT = (cid:126)σ · (cid:126)σ H GT ( r ) τ − τ − (95) Θ F = H F ( r ) τ − τ − (96) Θ T = [3 ( (cid:126)σ · ˆ r ) ( (cid:126)σ · ˆ r ) − (cid:126)σ · (cid:126)σ ] H T ( r ) τ − τ − , (97) H α ( r ) = 2 Rπ (cid:90) ∞ j n α ( qr ) h α ( q ) qdqq + (cid:104) E (cid:105) . (98)The value of the parameter R is R = 1 . A / fm, the j n α ( qr ) are the spherical Bessel functions, n α = 0 for Fermi and Gamow-Teller components, n α = 2 for the tensor one. The explicit expression of neutrino This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators form functions h α ( q ) may be found in Ref. [89], and the average energies (cid:104) E (cid:105) are evaluated as in Refs.[111, 112].Apart from effects related to sub-nucleonic degrees of freedom, that have not been taken into accountin Ref. [89], the νββ -decay operator has to be renormalized to take into account both the degrees offreedom that are neglected in the adopted model space and the contribution of the short-range correlations(SRC). The latter arise since the action of a two-body decay operator on an unperturbed (uncorrelated)wave function, as the one used in the perturbative expansion of Θ eff , differs from the action of the sameoperator on the real (correlated) nuclear wave function.It is worth pointing out that the calculations for νββ decay are not affected by this renormalization,since, as mentioned before, we retain only the leading order of the long-wavelength approximation whichcorresponds to a zero-momentum-exchange ( q = 0 ) process. On the other hand, the inclusion of higher-order contributions or corrections due to the sub-nucleonic structure of the nucleons [113, 114, 115, 116]would connect high- and low-momentum configurations and this renormalization should be carried out forthe two-neutrino emission decay too.In Ref. [117] the inclusion of SRC has been realized by means of an original approach [117] thatis consistent with the V low - k procedure. The νββ operator Θ , expressed in the momentum space, isrenormalized by way of the same similarity transformation operator Ω low - k that defines the V low - k potential.This enables to consider effectively the high-momentum (short range) components of the N N potential, ina framework where their direct contribution is not explicitly considered above a cutoff Λ . The resulting Θ low - k vertices are then employed in the perturbative expansion of the ˆΘ -box to calculate Θ eff using Eq.(85). More precisely, the perturbative expansion has considered diagrams up to third order in perturbationtheory, including the ones related to the so-called Pauli blocking effect (see Fig. 2 in Ref. [89]), and the χ n series has been arrested to χ .In Ref. [89] the contribution of the tensor component of the neutrino potential (Eq. (97)) is neglected,and therefore the total nuclear matrix element M ν is expressed as M ν = M ν GT − (cid:18) g V g A (cid:19) M ν F , (99)where g A = 1 . , g V = 1 [118], and the matrix elements between the initial and final states M να arecalculated within the closure approximation M να = (cid:88) j n j n (cid:48) j p j p (cid:48) (cid:104) f | a † p a n a † p (cid:48) a n (cid:48) | i (cid:105) × (cid:10) j p j p (cid:48) | Θ α | j n j n (cid:48) (cid:11) . (100)The calculated NMEs using the νββ -decay effective operator are reported in Table 2 and compared withthe values obtained with the bare operator without any renormalization.The most striking feature that can be inferred from the inspection of Table 2 is that the effects of therenormalization of the νββ -decay operator are far less relevant than those observed in the νββ -decaysector.A long standing issue related with the calculation of M ν is the possible interplay between the derivationof the effective one-body GT operator and the renormalization of the two-body GT component of the νββ operator, some authors presuming that the same empirical quenching introduced to reproduce the Frontiers 31 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators Table 2. Calculated values of M ν for Te and Xe decay. The first column corresponds to the resultsontained employing the bare νββ -decay operator, the second one to the calculations performed with Θ eff .Decay bare operator Θ eff130 Te → Xe 3.27 3.16 Xe → Ba 2.47 2.39observed GT-decay properties (single- β decay strengths, M ν GT s, etc.) should be also employed to calculate M ν (see for instance Refs. [119, 120]). Actually, the comparison of the results in Tables 1 and 2 showsthat the mechanisms which rule the microscopic derivation of the one-body single- β and the two-body νββ decay effective operators lead to a considerably different renormalization, at variance with the abovehypothesis.SM calculations of this section have been performed by employing, as interaction vertices of theperturbative expansion of the ˆ Q -box, a realistic potential derived from the high-precision CD-Bonn N N potential [90]. This potential is characterized by a strong repulsive behavior in the high-momentum regime,so, as mentioned before, it has been renormalized by deriving a low-momentum N N potential using the V low - k approach [97].As in other SM studies [98, 99, 100, 101], the value of the cutoff has been chosen as Λ = 2 . fm − ,since the role of missing three-nucleon force (3NF) decreases by enlarging the V low - k cutoff [99]. Thisvalue, within a perturbative expansion of the ˆ Q -box, is an upper limit, since a larger cutoff worsens theorder-by-order behavior of the perturbative expansion. Here, we report some considerations about theproperties of the perturbative expansion of H eff and the SM effective transition operator, when this “hard” V low - k is employed to derive SM Hamiltonian and operators.Studies of the perturbative properties of the SP energy spacings and TBMEs are reported in Refs. [99]and [121], where H eff has been derived within the model space outside Sn and starting from the “hard” V low - k . However, in Ref. [122] it can be found a systematic investigation of the convergence propertiesof theoretical SP energy spectra, TBMEs, and νββ NMEs as a function both of the dimension of theintermediate-state space and the order of the perturbative expansion. Moreover, in Ref. [89] they are alsoreported the convergence properties of the perturbative expansion of the effective νββ -decay operatorwith respect to the number of intermediate states, and the truncation both of the order of χ n operators andthe perturbative order of the diagrams.Here, we sketch out briefly these results in order to assess the reliability of realistic SM calculationperformed starting from a “hard” V low - k .The model space employed for the SM calculations reported in Ref. [122] is spanned by the five protonand neutron orbitals g / , d / , d / , s / , h / outside the doubly-closed Sn, in order to studythe νββ decay of Te and Xe.In Fig. 15, it can be found the behavior of the calculated SP spectrum of Sn, with respect to the g / SP energy, as a function of the maximum allowed excitation energy of the intermediate states expressed interms of the oscillator quanta N max .From the inspection of Fig. 15, it is clear that the results achieve convergence at N max = 14 , whichjustifies the choice, for the perturbative expansion of the effective SM Hamiltonian and decay operators, to This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators Max ene r g y [ M e V ] ’ perturbative order–10–9–8–7–6–5–4–3 ene r g y [ M e V ] Figure 15. Neutron SP energies as a function of N max (left-hand side) of the perturbative order (right-handside). Reproduced from Ref. [122] under the Creative Commons CC BY license.include intermediate states with an unperturbed excitation energy up to E max = N max (cid:126) ω with N max = 16 [101, 122, 102, 89].As regards the order-by-order convergence of the SP energies, in Fig. 15, the calculated neutron SPenergies, using a number of intermediate states corresponding to N max = 16 , are reported as a function ofthe order of the perturbative expansion up to the third order. They are also and compared with the Pad´eapproximant [2 | of the ˆ Q -box, which estimates the value to which the perturbative series may converge.The results at third order are very close to those obtained with the Pad´e approximant, indicating that thetruncation at third order should be a reasonable estimate of the sum of the series.As regards the TBMEs, we report in Fig. 16 and the neutron-neutron diagonal J π = 0 + TBMEs as afunction both of N max and of the perturbative order. These TBMEs, which contain the pairing properties ofthe effective Hamiltonian, are the largest in size of the calculated matrix elements and the most sensitive tothe behavior of the perturbative expansion.From the inspection of Fig. 16 the convergence with respect to N max appears to be very fast for diagonalmatrix elements (1 d / ) , (1 d / ) , and (2 s / ) , while those corresponding to orbitals lacking of their ownspin-orbit partner, (0 g / ) and (0 h / ) , show a slower convergence. The order-by-order convergence, asshown in Fig. 16, is quite satisfactory, and again the results at third order are very close to those obtainedwith the Pad´e approximant. Therefore, the conclusion is that H eff , calculated from a V low - k with a cutoffequal to 2.6 fm − by way of a perturbative expansion arrested at third order, is a good estimate of the sumof its perturbative expansion, both for the one- and two-body components.Now, we focus the attention on the perturbative expansion of the GT effective operator GT eff .The selection rules of the GT operator, that characterize a spin-isospin-dependent decay, drive a fastconvergence of the matrix elements of its SM effective operator with respect to N max . In fact, if theperturbative expansion is arrested at second order, their values do not change from N max = 2 on [62], andat third order in perturbation theory their third decimal digit values do not change from N max = 12 on.In Table 3 the results of the calculated NME of the νββ decays Te g . s . → Xe g . s . and Xe g . s . → Ba g . s . , obtained with effective operators at first, second, and third order in perturbation Frontiers 33 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators max –1.0–0.8–0.6–0.4–0.2–0.0 V e ff m a t r i x e l e m en t s [ M e V ] (g7/2) (d5/2) (d3/2) (s1/2) (h11/2) ’ Perturbative order–1.0–0.8–0.6–0.4–0.2–0.0 V e ff m a t r i x e l e m en t s [ M e V ] (g7/2) (d5/2) (d3/2) (s1/2) (h11/2) Figure 16. Neutron-neutron diagonal J π = 0 + TBMEs as a function of N max (left-hand side) of theperturbative order (right-hand side). Reproduced from Ref. [122] under the Creative Commons CC BYlicense.theory ( χ n series in Eq. (85) is arrested to χ ), are reported and compared with the experimental results[109]. Table 3. Order-by-order M ν GT s (in MeV − ) for Te and Xe [122].Decay 1st ord M ν GT M ν GT M ν GT Expt. Te → Xe 0.142 0.040 0.044 . ± . Xe → Ba 0.0975 0.0272 0.0285 . ± . As can be seen, also the order-by-order convergence of the M ν GT s is very satisfactory, since the resultschange for both transitions about 260 % from the first- to second-order calculations, while the change is 9 % and 5 % from the second- to third-order results for Te and Xe decays, respectively. This suppressionof the third-order with respect to the second-order contribution is favoured by the mutual cancelation ofthird-order diagrams.In Ref. [89] it has been performed also a study about the convergence properties of the effective decay-operator Θ eff for the νββ decay, with respect the truncation of the χ n operators, the number of intermediatestates which have been accounted for the perturbative expansion, as well as the order-by-order behavior upto third order in perturbation theory.In Fig. 17 the results of the calculated values of M ν for the Ge → Se decay are drawn as afunction of the maximum allowed excitation energy of the intermediate states expressed in terms of theoscillator quanta N max , and they are reported including χ n contributions up to n = 2 . We can see that the M ν s values are convergent from N max = 12 on and that contributions from χ are quite relevant, thosefrom χ being almost negligible. This is a provisional file, not the final typeset article . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators N max M ν Ge χ χ + χ χ + χ + χ Figure 17. M ν for the Ge → Se decay as a function of N max . The red diamonds correspond to atruncation of χ n expansion up to χ , blue squares up to χ , and black dots up to χ . Reproduced from Ref.[89].It is worth pointing out that, according to expressions (84), χ is defined in terms of the first, second, andthird derivatives of ˆΘ and ˆΘ , as well as on the first and second derivatives of the ˆ Q -box. This meansthat one could estimate χ being about one order of magnitude smaller than χ contribution.On the above grounds, in Ref. [89] the effective SM νββ -decay operator has been obtained including inthe perturbative expansion up to third-order diagrams, whose number of intermediate states corresponds tooscillator quanta up to N max = 14 , and up to χ contributions.Now, in order to consider the order-by-order convergence behavior in Fig. 18 are reported the calculatedvalues of M ν , M ν GT , and M ν F for Te, and Xe νββ decay, respectively, at first, second-, andthird-order in perturbation theory. We compare the order-by-order results also with their Pad´e approximant [2 | , as an indicator of the quality of the perturbative behavior [123]. M (cid:105) (cid:95) Te Pade` [2|1] M (cid:105) M (cid:105) GT M (cid:105) F M (cid:105) (cid:95) Xe Pade` [2|1] M (cid:105) M (cid:105) GT M (cid:105) F Figure 18. M ν for the Te → Xe and the Xe → Ba decay as a function of the perturbativeorder. The green triangles correspond to M ν F , the blue squares to M ν GT , and the black dots to the full M ν .Reproduced from Ref. [89].It is worth pointing out that the perturbative behavior is ruled by the Gamow-Teller component, theFermi matrix element M ν F being only slightly affected by the renormalization procedure. Moreover, if theorder-by-order perturbative behavior of the effective SM νββ -decay operator is compared with the single Frontiers 35 . Coraggio and N. Itaco Perturbative Approach to Effective Shell-Model Hamiltonians and Operators β -decay one, we observe a less satisfactory perturbative behavior for the calculation of M ν , the differencebetween second- and third-order results being about for Te, Xe νββ decays. This paper has been devoted to a general presentation of the perturbative approach for deriving effectiveshell-model operators, namely the SM Hamiltonian and decay operators.First, we have presented the theoretical framework, which is essentialy based on the perturbative expansionof a vertex function, the ˆ Q -box for the effective Hamiltonian and the ˆΘ -box for effective decay operators,whose calculation is pivotal within the Lee-Suzuki similarity transformation. 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