Semi-Microscopic PHDOM Framework

• The semi-microscopic PHDOM is a Green’s-function framework that describes particle–hole excitations by combining Landau damping, continuum coupling, and many-quasiparticle spreading effects.
• It utilizes a dispersive relation to link the real and imaginary parts of the complex energy-dependent self-energy, ensuring causality and accurate energy shifts in resonance calculations.
• The model has been applied to giant multipole resonances in closed-shell and one-closed-shell nuclei, successfully reproducing experimental energies, widths, and decay branching ratios.

The semi-microscopic particle-hole dispersive optical model (PHDOM) is a Green’s-function framework for particle–hole ($p$–$h$) excitations, especially giant resonances, that combines three relaxation mechanisms in a single energy-dependent description: Landau damping, coupling to the single-particle continuum, and coupling to many-quasiparticle configurations. In practical implementations, the mean field and the residual Landau–Migdal $p$–$h$ interaction are treated microscopically or partially self-consistently, while the spreading effect is represented by an optical-model-like, complex $p$–$h$ self-energy whose real and imaginary parts are linked by a dispersive relation. This structure has been used for isoscalar giant multipole resonances, their overtones, and charge-exchange spin-isospin modes in medium-heavy, closed-shell, and one-closed-shell nuclei (Gorelik et al., 23 Aug 2025).

1. Conceptual basis and scope

An attempt to formulate the optical model of particle-hole-type excitations, including giant resonances, was undertaken through the Bethe–Goldstone equation for the $p$–$h$ Green function, with a specific energy-dependent $p$–$h$ interaction responsible for the spreading effect (Urin, 2010). In that formulation, the central analogy is with the single-quasiparticle dispersive optical model: the one-body self-energy $h$0 is replaced by a $h$1–$h$2 polarization operator $h$3, and the spreading width is generated by the imaginary part of this effective interaction.

In its later implementations, PHDOM is explicitly presented as an extension of continuum RPA. The microscopic part consists of a Woods–Saxon mean field, single-particle spectra and wave functions, the Landau–Migdal residual interaction, and an exact treatment of the single-particle continuum. The phenomenological part is the energy-dependent, complex $h$4–$h$5 self-energy that represents coupling of doorway $h$6–$h$7 configurations to many-quasiparticle states and produces the spreading width (Gorelik et al., 2020).

A persistent source of confusion is the relation between PHDOM and the nucleonic dispersive optical model. They share the same dispersive logic, causality constraints, and Green’s-function language, but they are not the same construction. In the single-particle DOM, the basic object is the nucleon self-energy governing bound and scattering propagation across the Fermi energy; in PHDOM, the basic object is the energy-averaged $h$8–$h$9 propagator in the excitation channel of interest. This distinction is explicit in the many-body and optical-model overviews of dispersive methods (Dickhoff et al., 2018).

2. Green’s-function formulation

For an isoscalar multipole of rank $p$0, the external field is written as

$p$1

and the effective-field method leads to an energy-dependent effective single-particle field that accounts for core polarization. The strength function can then be written either through the full $p$2–$p$3 propagator $p$4,

$p$5

or through the double transition density,

$p$6

These two forms are operationally equivalent in the PHDOM formulation (Gorelik et al., 23 Aug 2025).

The propagator obeys a Dyson-type equation,

$p$7

where $p$8 is the “free” $p$9–$h$0 propagator, $h$1 is the static spinless Landau–Migdal interaction, and $h$2 is the complex, energy-dependent $h$3–$h$4 self-energy. Escape width is generated microscopically through $h$5 by coupling to the single-particle continuum with scattering boundary conditions, while the spreading width is generated by $h$6 (Gorelik et al., 23 Aug 2025).

Causality is enforced by a subtracted dispersive relation. In the isoscalar multipole applications,

$h$7

with subtraction point $h$8. The same causal structure is central in the earlier optical-model formulation of $h$9–$p$0 excitations, where the imaginary part broadens and redistributes the RPA poles and the dispersive real part produces the corresponding energy shifts (Urin, 2010).

3. Microscopic ingredients and calibration strategy

The mean field used in PHDOM is a partially self-consistent Woods–Saxon potential with isoscalar central and spin–orbit terms, a symmetry potential $p$1, and the Coulomb potential $p$2, both computed self-consistently from proton and neutron-excess densities. In the one-closed-shell calculations for $p$3Ni, $p$4Sn, and $p$5Nd, the parameter sets were specified as follows: for $p$6Ni, $p$7 MeV, $p$8 MeV·fm$p$9, $h$0 fm, $h$1 fm, $h$2, $h$3; for $h$4Sn, $h$5 MeV, $h$6 MeV·fm$h$7, $h$8 fm, $h$9 fm, $p$0, $p$1; for $p$2Nd, $p$3 MeV, $p$4 MeV·fm$p$5, $p$6 fm, $p$7 fm, $p$8, $p$9 (Gorelik et al., 23 Aug 2025).

The residual interaction is the spinless Landau–Migdal form

$h$0

with $h$1 MeV·fm$h$2. The isoscalar strength is parameterized as

$h$3

where $h$4 is universal, while $h$5 is adjusted nucleus by nucleus to place the $h$6 spurious center-of-mass state near zero energy and make it exhaust almost the full EWSR of the $h$7 translation operator. In this way, translational invariance is enforced operationally (Gorelik et al., 23 Aug 2025).

The spreading self-energy is represented phenomenologically by a smooth energy-dependent function. For the $h$8Pb closed-shell implementation, the imaginary part is written as

$h$9

with $p$0 MeV$p$1, $p$2 MeV, and $p$3 MeV. The real part $p$4 is then obtained numerically from the dispersive relation. In the one-closed-shell multipole study, a universal three-parameter form for $p$5 was retained across $p$6–3 (Gorelik et al., 2020).

The treatment of pairing is selective. For one-closed-shell nuclei, pairing was neglected for the “collective” characteristics of $p$7 multipole giant resonances with $p$8, because its effect was taken to be of order $p$9. At the same time, pairing-induced Fermi-surface smearing was recognized as relevant for decay probabilities and was delegated to a recalculation procedure in decay analyses (Gorelik et al., 23 Aug 2025).

4. External operators, transition densities, and observables

For isoscalar giant multipole resonances, the standard radial operators are

$h$0

$h$1

$h$2

The monopole and dipole forms are chosen to suppress spurious components, while $h$3 and $h$4 are taken as simple powers for the main tones. Overtones of $h$5 and $h$6 are generated with

$h$7

where $h$8 is fixed by orthogonality to the main-tone transition density at the peak energy (Gorelik et al., 23 Aug 2025).

The model provides both the double transition density and a projected one-body transition density. The latter is related to the external field by convolution and normalized so that

$h$9

An approximate factorization,

$h$00

is then available for DWBA applications. In the detailed $h$01Pb analysis of isoscalar monopole excitations, the exact energy-averaged double transition density was compared with factorized forms built from projected microscopic densities and with classical collective-model densities. The projected microscopic factorization reproduced the Born-approximation strength near the ISGMR and ISGMR2 maxima better than the microscopically corrected classical densities, especially in the vicinity of the peaks (Gorelik et al., 2015).

Moments and sum-rule fractions are central observables. With

$h$02

the centroid in a specified interval is $h$03, while the relative energy-weighted strength is

$h$04

The EWSR fraction in an interval $h$05 is

$h$06

This representation is used systematically in the multipole studies to compare theory with experimental strength distributions over broad energy windows (Gorelik et al., 23 Aug 2025).

PHDOM also calculates direct one-nucleon decay. For channel $h$07 or $h$08, populating a specific hole state,

$h$09

In the pure continuum-RPA limit, $h$10; the deficit $h$11 measures the spreading contribution (Gorelik et al., 23 Aug 2025).

5. Principal applications and empirical systematics

For the closed-shell nucleus $h$12Pb, PHDOM was implemented for the ISGMR, ISGDR, ISGQR, ISGOR, and selected overtones. In the 5–35 MeV window, the calculated ISGMR had peak $h$13 MeV, centroid $h$14 MeV, and FWHM $h$15 MeV; the high-energy ISGDR component had peak $h$16 MeV, centroid $h$17 MeV, and FWHM $h$18 MeV; the ISGQR had peak $h$19 MeV, centroid $h$20 MeV, and FWHM $h$21 MeV; the ISGOR had peak $h$22 MeV, centroid $h$23 MeV, and FWHM $h$24 MeV. The overtone strengths ISGMR2 and ISGQR2 were concentrated near 26–28 MeV. Projected transition densities at the resonance peaks followed the expected node patterns: one node for ISGMR and high-energy ISGDR, nodeless for ISGQR and ISGOR, and higher-node structure for the overtones (Gorelik et al., 2020).

In one-closed-shell nuclei, the model was applied to $h$25Ni, $h$26Sn, and $h$27Nd. For the ISGMR, the calculations gave a broad single component centered at about 15–20 MeV depending on $h$28, with generally good agreement with experiment across a wider set including $h$29Ca, $h$30Zr, $h$31Pb, and the three one-closed-shell systems. In $h$32Sn, the PHDOM centroid was $h$33 MeV in 7–25 MeV, the peak was $h$34 MeV, and the FWHM was $h$35 MeV; in $h$36Nd, the centroid was about 16.1 MeV in 10–30 MeV and $h$37 MeV. The ISGDR displayed distinct low-energy and high-energy components, with the high-energy component near 25 MeV in both $h$38Sn and $h$39Nd, while the low-energy component was fragmented. The ISGOR strength appeared around 22–28 MeV with FWHM of about 3–4 MeV. The principal unresolved discrepancy concerned the ISGQR channel, particularly in $h$40Nd, where the calculated main $h$41 mode and the overtone ISGQR2 suggest that experimental “quadrupole strength” extracted by $h$42-scattering may mix the main tone and overtone when DWBA uses an overtone-insensitive transition density (Gorelik et al., 23 Aug 2025).

The charge-exchange extension of the model preserved the same semi-microscopic structure while replacing the spinless residual interaction by the spin–isovector Landau–Migdal interaction. For the Gamow–Teller resonance in $h$43Ca, $h$44Zr, $h$45Sn, and $h$46Pb, the calculated peak energies and widths were $h$47 MeV and $h$48 MeV, $h$49 MeV and $h$50 MeV, $h$51 MeV and $h$52 MeV, and $h$53 MeV and $h$54 MeV, respectively. The model also described the isovector giant spin-monopole resonance as the overtone of GT, with the $h$55 peak in $h$56Pb calculated at $h$57 MeV with $h$58 MeV. In this charge-exchange sector, the non-energy-weighted sum rule was enforced, the GT projected transition density was node-less, the spin-monopole density had one node, and the calculated total direct one-proton decay branching ratio for the $h$59Pb GT resonance, $h$60, agreed well with the measured $h$61 (Bondarenko et al., 2022).

Direct one-nucleon decay is one of the model’s distinctive outputs. In the closed-shell $h$62Pb implementation, PHDOM tended to overestimate direct neutron decay: for example, total neutron branching ratios were about $h$63 for ISGMR and $h$64 for ISGDR, compared with much smaller experimental values. By contrast, in the charge-exchange GT application the channel-resolved branching ratios were more successful. This pattern was attributed to the single-hole assumption in the decay treatment and to the reduction of direct decay by more complex configuration mixing in reality (Gorelik et al., 2020).

6. Unitarity, relation to dispersive optical methods, and open problems

Because the spreading effect is introduced phenomenologically through an energy-dependent complex term, weak violations of unitarity arise in the original PHDOM. In the isoscalar monopole sector these appeared as a spurious response to the unit operator, small negative values of the strength function at high excitation energy, and slight deviations from exact sum-rule exhaustion. A restoration procedure was proposed that combines a renormalization removing the spurious isoscalar monopole response and a metric modification compensating for the energy dependence of the spreading term. In $h$65Pb, the restored version brought the integrated ISGMR strength from $h$66 to $h$67 and the ISGMR2 strength from $h$68 to $h$69, while eliminating the negative high-energy tails and leaving the resonance region essentially unchanged (Gorelik et al., 2017).

The broader dispersive optical-model literature clarifies the relationship between PHDOM and nucleonic DOM. In the latter, the same nonlocal, energy-dependent self-energy is used below and above the Fermi energy to constrain bound-state properties, elastic scattering, particle number, and charge density through a subtracted dispersion relation. PHDOM transfers that dispersive optical logic to the $h$70–$h$71 channel: the object governed by causality is no longer the single-particle propagator but the $h$72–$h$73 propagator, and the optical-model-like ingredient acts in $h$74–$h$75 space rather than in the one-body space of nucleon addition and removal (Dickhoff et al., 2016).

The main limitations are explicitly identified in the applications. The function $h$76 is phenomenological; explicit microscopic $h$77–$h$78 and more complex couplings are not constructed but are effectively absorbed into $h$79. Pairing is neglected in collective observables, though it is relevant for decay analyses. Tensor correlations are neglected for $h$80 in the isoscalar multipole calculations. In the quadrupole channel, the extraction of experimental strength from $h$81-scattering remains sensitive to the transition density used in DWBA, especially when overtone admixtures are present (Gorelik et al., 23 Aug 2025).

Microscopic self-energy calculations also indicate directions for refinement. FRPA studies of nucleon self-energies in calcium isotopes found substantial orbital-angular-momentum dependence in the imaginary self-energy and suggested that future dispersive optical fits should consider parity and angular-momentum dependence explicitly. A plausible implication is that analogous nonlocality and $h$82-dependence may ultimately be required in more microscopic versions of PHDOM as well, especially if one seeks tighter links between spreading parameterizations and underlying many-body dynamics (Waldecker et al., 2011).