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# Pattern of global spin alignment of ϕ and K*0 mesons in heavy-ion collisions

## Abstract

Notwithstanding decades of progress since Yukawa first developed a description of the force between nucleons in terms of meson exchange1, a full understanding of the strong interaction remains a considerable challenge in modern science. One remaining difficulty arises from the non-perturbative nature of the strong force, which leads to the phenomenon of quark confinement at distances on the order of the size of the proton. Here we show that, in relativistic heavy-ion collisions, in which quarks and gluons are set free over an extended volume, two species of produced vector (spin-1) mesons, namely ϕ and K*0, emerge with a surprising pattern of global spin alignment. In particular, the global spin alignment for ϕ is unexpectedly large, whereas that for K*0 is consistent with zero. The observed spin-alignment pattern and magnitude for ϕ cannot be explained by conventional mechanisms, whereas a model with a connection to strong force fields2,3,4,5,6, that is, an effective proxy description within the standard model and quantum chromodynamics, accommodates the current data. This connection, if fully established, will open a potential new avenue for studying the behaviour of strong force fields.

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## Data availability

All raw data for this study were collected using the STAR detector at Brookhaven National Laboratory and are not available to the public. Derived data supporting the findings of this study are publicly available in the HEPData repository (https://www.hepdata.net/record/129067) or from the corresponding author on request.

## Code availability

Codes to process raw data collected by the STAR detector and codes to analyse the produced data are not available to the public.

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## Acknowledgements

We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL and the Open Science Grid Consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. National Science Foundation, National Natural Science Foundation of China, Chinese Academy of Sciences, the Ministry of Science and Technology of China and the Chinese Ministry of Education, the Higher Education Sprout Project by Ministry of Education at NCKU, the National Research Foundation of Korea, the Czech Science Foundation and Ministry of Education, Youth and Sports of the Czech Republic, the Hungarian National Research, Development and Innovation Office, New National Excellency Programme of the Hungarian Ministry of Human Capacities, Department of Atomic Energy and Department of Science and Technology of the Government of India, the National Science Centre of Poland, the Ministry of Science, Education and Sports of the Republic of Croatia, German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF), Helmholtz Association, Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS).

## Author information

### Contributions

All authors contributed extensively.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Andrea Dainese, Ilya Selyuzhenkov and Xin-Nian Wang for their contribution to the peer review of this work.

### Extended data

is available for this paper at https://doi.org/10.1038/s41586-022-05557-5.

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## Extended data figures and tables

### Extended Data Fig. 1 Example of combinatorial-background-subtracted invariant-mass distributions and the extracted yields as a function of cosθ* for ϕ and K*0 mesons.

a, Example of ϕ → K+ + K invariant-mass distributions, with combinatorial background subtracted, integrated over cosθ*. b, Example of $${{\rm{K}}}^{* 0}\,(\bar{{{\rm{K}}}^{* 0}})\to {{\rm{K}}}^{-}{{\rm{\pi }}}^{+}\,({{\rm{K}}}^{+}{{\rm{\pi }}}^{-})$$ invariant-mass distributions, with combinatorial background subtracted, integrated over cosθ*. c, Extracted yields of ϕ as a function of cosθ*. d, Extracted yields of K*0 as a function of cosθ*.

### Extended Data Fig. 2 Efficiency-corrected ϕ-meson yields as a function of cosθ* and corresponding fits with equation (2) in Methods.

The red stars are efficiency-corrected yields for ϕ mesons with |y| < 1.0 and 1.2 < pT < 1.8 GeV c−1, for 20–60% centrality at $$\sqrt{{s}_{{\rm{NN}}}}=27\,{\rm{GeV}}$$.

### Extended Data Fig. 3 Efficiency-corrected and acceptance-corrected K*0-meson yields as a function of cosθ* and corresponding fits with equation (5) in Methods.

The blue circles are efficiency-corrected and acceptance-corrected yields for K*0 mesons with |y| < 1.0 and 2.0 < pT < 2.5 GeV c−1, for 20–60% centrality at $$\sqrt{{s}_{{\rm{NN}}}}=54.4\,{\rm{GeV}}$$.

### Extended Data Fig. 4 ϕ-meson ρ00 obtained from first-order and second-order event planes.

The red stars (grey squares) show the ϕ-meson ρ00 as a function of beam energy, obtained with the second-order (first-order) event plane.

### Extended Data Fig. 5 ϕ-meson ρ00 with respect to different quantization axes.

ϕ-meson ρ00 as a function of beam energy, for the out-of-plane direction (stars) and the in-plane direction (diamonds). Curves are fits based on theoretical calculations with a ϕ-meson field2. The corresponding Gs values obtained from the fits are shown in the legend.

### Extended Data Fig. 6 ρ00 as a function of transverse momentum for ϕ for different collision energies.

The grey squares and red stars are results obtained with the first-order and second-order event planes, respectively.

### Extended Data Fig. 7 ρ00 as a function of transverse momentum for K*0 for different collision energies.

The solid circles are results obtained with the second-order event plane.

### Extended Data Fig. 8 ρ00 as a function of centrality for ϕ (upper panels) and K*0 (lower panels).

The solid squares and stars are results for the ϕ meson, obtained with the first-order and second-order event planes, respectively. The solid circles are results for the K*0 meson, obtained with the second-order event plane.

### Extended Data Fig. 9 Global spin alignment measurement of ϕ and K*0 vector mesons in Au+Au collisions at 0–20% centrality.

The solid squares and stars are results for the ϕ meson, obtained with the first-order and second-order event planes, respectively. The solid circles are results for the K*0 meson, obtained with the second-order event plane.

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STAR Collaboration. Pattern of global spin alignment of ϕ and K*0 mesons in heavy-ion collisions. Nature (2023). https://doi.org/10.1038/s41586-022-05557-5

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• DOI: https://doi.org/10.1038/s41586-022-05557-5

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