Xin Zhang

Collaborators: Yi Li, Giuliano Giacalone, Jiangming Yao

IWND2026 , Taiyuan, 2026.8.24

Supervisor: K. Hagino

We need to perform correlation analysis and uncertainty quantification for each model to understand this discrepancy.

The matrix element (NME) is:

Various nuclear models predict NMEs that differ by three to more!

J, M, Yao et al., PPNP. 126, 103965 (2022)

The CDFT starts from the following energy density functional (EDF) under the mean-field approximation:

The interaction part with low-energy constants

The local densities and currents are :

Covariant Density Functional Theory (CDFT)

The variational principle yields the Hill-Wheeler-Griffin (HWG) equation :

Energy density functionals : Rodríguez 2010 PRL; Vaquero 2013 PRL; Song 2014 PRC; Yao 2015 PRC

Shell model Hamiltonian : Jiao 2017 PRC; Jiao 2018 PRC

Chiral EFT interaction : Yao 2020 PRL

The GCM wave function is expanded on a set of non-orthogonal basis for a certain EDF:

P. Ring and P. Schuck, The nuclear many-body problem (1980)

where are symmetry-projected quasiparticle vacuum.

NME calculated by GCM show explicit effect of deformations:

Tomás R. Rodríguez et al., PRL 105, 252503(2010)

The STAR Collaboration, Rep. Prog. Phys. 88 108601 (2025)

The initial shape of the nuclei can be inferred from the observables of the collision’s final state. �

high-energy heavy-ion collisions

Giuliano Giacalone, PRL. 124, 202301 (2020) Giuliano Giacalone et al., PRL. 127, 242301 (2021) B. Bally, et al., PRL. 128, 082301 (2022)

Chunjian Zhang et al., PRL. 128, 022301 (2022) Jiangyong Jia et al., PRL. 131, 022301 (2023) M. I. Abdulhamid et al., Nature 635, 67–72 (2024)

G. Giacalone et al., PRL. 135, 012302 (2025) G. Giacalone et al., PRL. 134, 082301 (2025) Y. Li et al., PRL. 135, 022301 (2025)

Initial state quantity: eccentricity

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​

​

​

​

​

​

​

​

​

G. Giacalone, PRC 102, 024901 (2020)

Ultracentral collisions

Final state quantities:

Transverse momentum �

elliptic flow

relates to

G. Giacalone, PRC 102, 024901 (2020)

�

Do these observables have correlations with the NME? How to implement the correlation analysis?

The initial shape of the nuclei can be inferred from the observables of the collision’s final state. �

Eigenvector Continuation(EC)

Dillon Frame et al., PRL 121，032501 (2018)

C. Drischler et al., Front. Phys. 10, 1092931 (2023)

Here, we define the norm and Hamiltonian kernels of the EC method for a target EDF,

The mixing coefficient is determined by the following equation,

FIG. The ratio of the computation time of MR-CDFT calculation to that of SP-CDFT calculation for ¹⁵⁰Nd

The SP-CDFT allows us within half an hour using a PC, to predict nuclear low-lying states for 1.3 millions of EDF samples which would otherwise take years with the MR-CDFT.

Benchmark calculation using 64 testing set (¹⁵⁰Nd)

The relative error for the ground state properties are all within 0.3%

The relative error for the low-lying states properties are all within 13%, due to the overlap of errors between the ground state and the lower-lying state.

P. W. Zhao et al., Phys. Rev. C 82, 054319 (2010)

T. Bürvenich et al., Phys. Rev. C 65, 044308 (2002)

1.3 million samples using quasi Monte-Carlo (MC) sampling with a uniform distribution.

Samples of EDFs: 1310720

(in blue). �

The correlations of the

quantities for nuclear matter

with M^0ν are weak. �

Samples of EDFs refined with

the nuclear matter properties:

457380 (in red).�

There are rather strong

correlations between the M0ν

and nuclear low-lying states,

but these correlations vary with

nucleus. �

X. Zhang, C. C. Wang, C. R. Ding, and J. M. Yao, Phys. Rev. C 112, L021302 (2025)

The collectivity of the single-magic nucleus ¹³⁶Xe is overestimated due to the lack of particle-hole excitation configurations in the model space.

C. R. Ding et al., PRC 108, 054304 (2023)

The final mean value and statistical uncertainty (68% confidence level) of the NME are 5.51(14), and 4.33(5) for ¹⁵⁰Nd and ¹³⁶Xe, respectively, in line with the values 5.60 and 4.32 by the PC-PK1.

FIG. Collider experiments with isobars provide robust probes of the nuclear geometry.

Y.Li, X. Zhang, G. Giacalone, J. M. Yao, Phys. Rev. Lett. 135, 022301 (2025)

SP-CDFT

All of these variations are strongly correlated with the value of the NME.

By extending the high-energy data to the Bayesian analysis should effectively improve the extraction of the nuclear deformation.

Summary

The high-energy collision observables investigated in this study exhibit a strong correlation with NME, offering a promising way for constraining NME in the future.

By using SP-CDFT, We have conducted the first statistical analysis of the correlation between NMEs of 0νββ decay and observables from nuclear reactions using covariant density functional theory.

Prospect

The method is readily extendable to more candidate nuclei and we can combine with experimental data to constrain the value of NME. �

We have developed SP-CDFT to emulate MR-CDFT.

Thank you!

Initial state eccentricity :

a background fluctuation splitting is introduced,

Insert fluctuation into eccentricity, one obtains :

With :

Thomas Duguet et al., PRL 135, 182301 (2025)�

One-point and connected two-point functions are expressed by :

One obtains the mean squared anisotropy with two-body density : �

In the Lab system

(1)

One should use the concept of conditional probability, which is related to two-body density : �

It has an axial shape : �

r₁

r₂

It can be reflected with the expectation of : �

r₂

θ

Variance:

Connection with Intrinsic density

where

Intrinsic density in the Quadrupole Rigid-rotor model (RRM) :

Spherical !

Proportional to deformation parameter

If we only consider one-body density in the lab system, the eccentricity will be trivial!

We must consider two-body density

Mean-field approximation of RRM

Why in the TRENTO/Glauber models, from intrinsic one-body density one can have non-trivial eccentricity?

The input of TRENTO/Glauber models : Joint probability distribution

Under independent nucleon approximation,

Sampling nucleons positions from the joint probability distribution

Construct the entropy density from the nucleon samples, making dynamical evolutions, obtain

Because of the limitations of the independent nucleon approximation and RRM, we need to consider real two body and even higher many-body correlations.

Large-scale sampling

Relativistic EDF

SP-CDFT

nuclear structure

fusion reaction

CCCC

Target nuclei

CC-Full

J. M. Yao, K. Hagino, X. Zhang, in preparation