We construct the equation of state (EOS) of QCD based on the finite chemical potential information from the functional QCD approaches with the results at zero chemical potential being calibrated by lattice QCD simulations. Specifically, the construction is achieved by parametrizing the order parameters from the functional QCD data into the EOS, namely, the dynamical quark mass for the chiral phase transition and the Polyakov loop for the deconfinement phase transition. The obtained EOS is consistent with the up-to-date results of the QCD phase diagram, including a phase transition temperature at zero chemical potential of <math><mi>T</mi><mo>=</mo><mn>155</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, the curvature of the transition line <math><mi>κ</mi><mo>=</mo><mn>0.016</mn></math>, and also a critical end point at <math><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo><mo>=</mo><mo>(</mo><mn>118</mn><mo>,</mo><mn>600</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. We also implement the EOS in hydrodynamic simulations to compute the particle yields and ratios and find that our obtained EOS agrees well with the commonly used one based on the combination of lattice QCD simulation and hadron resonance gas model.

China

Yi Lu

Fei Gao

Baochi Fu

Huichao Song

Yu-xin Liu

Constructing the equation of state of QCD in a functional QCD based scheme

The global spin alignment of the vector meson has been observed in relativistic heavy ion collisions, but its theoretical origin is still on hot debates. Here we propose to apply the light front framework to explain this phenomenon since the light front form explicitly describes the hadron spin including both the quark spin and the orbital angular momentum. After applying the light front spinor, we find that the spin alignment in the polarization of vector mesons with <math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>00</mn></mrow></msub><mo>&gt;</mo><mn>1</mn><mo>/</mo><mn>3</mn></math> can be naturally manifested and in particular, the obtained spin alignment for ϕ meson is in good agreement with the experimental data. This implies that to explain the spin alignment it is important to properly include the contribution from the gluon interactions that are presented in terms of the orbital angular momentum of the hadron bound state.

Yu-Xin Liu

The spin alignment of vector mesons with light front quarks

<math><msubsup><mrow></mrow><mrow><mn>44</mn></mrow><mrow><mn>96</mn></mrow></msubsup></math>Ru+<math><msubsup><mrow></mrow><mrow><mn>44</mn></mrow><mrow><mn>96</mn></mrow></msubsup></math>Ru and <math><msubsup><mrow></mrow><mrow><mn>40</mn></mrow><mrow><mn>96</mn></mrow></msubsup></math>Zr+<math><msubsup><mrow></mrow><mrow><mn>40</mn></mrow><mrow><mn>96</mn></mrow></msubsup></math>Zr collisions at <math><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>NN</mi></mrow></msub></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></math> GeV provide unique opportunities to study the geometry and fluctuations raised from the deformation of the colliding nuclei. Using iEBE-VISHNU hybrid model, we predict <math><msub><mrow><mi>ac</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>{</mo><mn>3</mn><mo>}</mo></math> ratios between these two collision systems and demonstrate that the ratios of <math><msub><mrow><mi>ac</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>{</mo><mn>3</mn><mo>}</mo></math>, as well as the ratios of the involving flow harmonics and event-plane correlations, are sensitive to quadrupole and octupole deformations, which could provide strong constrains on the shape differences between $^{96}$Ru and $^{96}$Zr. We also study the nonlinear response coefficients <math><msub><mrow><mi>χ</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>22</mn></mrow></msub></math>, which show insensitivity to the deformation effect.

Shujun Zhao

Hao-jie Xu

Probing the nuclear deformation with three-particle asymmetric cumulant in RHIC isobar runs

We study the defect groups of   <math> <msubsup> <mi>D</mi> <mi>p</mi> <mi>b</mi> </msubsup> </math>  $$ {D}_p^b $$ (G) theories using geometric engineering and BPS quivers. In the simple case when b = h ∨(G), we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where b ≠ h ∨(G), we use a similar Maruyoshi-Song flow to conjecture that the defect groups of   <math> <msubsup> <mi>D</mi> <mi>p</mi> <mi>b</mi> </msubsup> </math>  $$ {D}_p^b $$ (G) theories are given by those of G (b)[k] theories. In the cases of G = A  n , E 6, E 8 we cross check our result by calculating the BPS quivers of the G (b)[k] theories and looking at the cokernel of their intersection matrix.

United Kingdom

Saghar Hosseini

Sweden

Robert Moscrop

Maruyoshi-Song flows and defect groups of   <math> <msubsup> <mi>D</mi> <mi>p</mi> <mi>b</mi> </msubsup> </math>  $$ {\mathrm{D}}_{\mathrm{p}}^{\mathrm{b}} $$ (G) theories

The reconstruction of the spectral function from the correlation function in Euclidean space is a challenging task. In this paper, we employ the machine learning techniques in terms of the radial basis functions networks to reconstruct the spectral function from a finite number of correlation data. To test our method, we first generate one type of correlation data using a mock spectral function by mixing several Breit-Wigner propagators. We found that compared with other traditional methods, e.g., truncated singular value decomposition, Tikhonov, and maximum entropy methods, our approach gives a continuous and unified reconstruction for both positive definite and negative spectral functions, which is especially useful for studying the QCD phase transition. Moreover, our approach has considerably better performance in the low-frequency region. This has advantages for the extraction of transport coefficients which are related to the zero frequency limit of the spectral function. With the mock data generated through a model spectral function of stress energy tensor, we find our method gives a precise and stable extraction of the transport coefficients.

Meng Zhou

Germany

Jingyi Chao

Application of radial basis functions neutral networks in spectral functions

We study the spin polarization generated by the hydrodynamic gradients. In addition to the widely studied thermal vorticity effects, we identify an undiscovered contribution from the fluid shear. This shear-induced polarization (SIP) can be viewed as the fluid analog of strain-induced polarization observed in elastic and nematic materials. We obtain the explicit expression for SIP using the quantum kinetic equation and linear response theory. Based on a realistic hydrodynamic model, we compute the differential spin polarization along both the beam direction <math><mover><mi>z</mi><mo>^</mo></mover></math> and the out-plane direction <math><mover><mi>y</mi><mo>^</mo></mover></math> in noncentral heavy-ion collisions at <math><mrow><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>, including both SIP and thermal vorticity effects. We find that SIP contribution always shows the same azimuthal angle dependence as experimental data and competes with thermal vorticity effects. In the scenario that <math><mi>Λ</mi></math> inherits and memorizes the spin polarization of a strange quark, SIP wins the competition, and the resulting azimuthal angle dependent spin polarization <math><msub><mi>P</mi><mi>y</mi></msub></math> and <math><msub><mi>P</mi><mi>z</mi></msub></math> agree qualitatively with the experimental data.

Shuai Y. F. Liu

Longgang Pang

Yi Yin

Shear-Induced Spin Polarization in Heavy-Ion Collisions

In this paper, we investigate the hydrodynamic collectivity in proton-proton collisions at  $$\sqrt{s}=$$  <math> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> </mrow> </math>   13 TeV, using iEBE-VISHNU with three different initial conditions, namely, HIJING, super-MC and TRENTo. With properly tuned parameters, hydrodynamics gives reasonable descriptions of the measured two-particle correlations, including the integrated and  $$p_\mathrm{T}$$  <math> <msub> <mi>p</mi> <mi>T</mi> </msub> </math>  -differential flow. However, the hydrodynamic simulations fail to describe the negative four-particle cumulant  $$c_2^v\{4\}$$  <math> <mrow> <msubsup> <mi>c</mi> <mn>2</mn> <mi>v</mi> </msubsup> <mrow> <mo>{</mo> <mn>4</mn> <mo>}</mo> </mrow> </mrow> </math>   as measured in experiments. The four-particle cumulant  $$c_2^v\{4\}$$  <math> <mrow> <msubsup> <mi>c</mi> <mn>2</mn> <mi>v</mi> </msubsup> <mrow> <mo>{</mo> <mn>4</mn> <mo>}</mo> </mrow> </mrow> </math>   is always positive after hydrodynamic evolutions. Further investigations show that the non-linear response between the final  $$v_2$$  <math> <msub> <mi>v</mi> <mn>2</mn> </msub> </math>   and the initial  $$\varepsilon _2$$  <math> <msub> <mi>ε</mi> <mn>2</mn> </msub> </math>   becomes significant in p-p systems. This leads to a large deviation from linear eccentricity scaling and generates additional flow fluctuations, which results in a positive  $$c_2^v\{4\}$$  <math> <mrow> <msubsup> <mi>c</mi> <mn>2</mn> <mi>v</mi> </msubsup> <mrow> <mo>{</mo> <mn>4</mn> <mo>}</mo> </mrow> </mrow> </math>   even with a negative  $$c_2^\varepsilon \{4\}$$  <math> <mrow> <msubsup> <mi>c</mi> <mn>2</mn> <mi>ε</mi> </msubsup> <mrow> <mo>{</mo> <mn>4</mn> <mo>}</mo> </mrow> </mrow> </math>   from the initial state. We also presented the first hydrodynamic calculations of mixed harmonic azimuthal correlations in p-p collisions. Although many qualitative features are reproduced by the hydrodynamic simulations, the measured negative normalized Symmetric-Cumulant  $$nsc_{2,3}\{4\}$$  <math> <mrow> <mi>n</mi> <mi>s</mi> <msub> <mi>c</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo>{</mo> <mn>4</mn> <mo>}</mo> </mrow> </mrow> </math>   cannot be reproduced. Obviously hydrodynamic calculations have a general difficulty to describe the data. It triggers that whether hydrodynamics with a new initial state could solve this puzzle, or hydrodynamics itself is not the appreciated mechanism of the observed collectivity, and the non-hydrodynamic modes become important in p-p collisions at the LHC.

Wenbin Zhao

Denmark

You Zhou

Koichi Murase

Searching for small droplets of hydrodynamic fluid in proton–proton collisions at the LHC

In this paper, we implement principal component analysis (PCA) to study the single particle distributions generated from thousands of $$\mathtt {VISH2+1}$$ <math><mrow><mi>VISH</mi><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>  hydrodynamic simulations with an aim to explore if a machine could directly discover flow from the huge amount of data without explicit instructions from human-beings. We found that the obtained PCA eigenvectors are similar to but not identical with the traditional Fourier bases. Correspondingly, the PCA defined flow harmonics $$v_n^\prime $$ <math><msubsup><mi>v</mi><mi>n</mi><mo>′</mo></msubsup></math>  are also similar to the traditional $$v_n$$ <math><msub><mi>v</mi><mi>n</mi></msub></math>  for $$n=2$$ <math><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math>  and 3, but largely deviated from the Fourier ones for $$n\ge 4$$ <math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math> . A further study on the symmetric cumulants and the Pearson coefficients indicates that mode-coupling effects are reduced for these flow harmonics defined by PCA.

Ziming Liu

Principal component analysis of collective flow in relativistic heavy-ion collisions

In this paper, we investigate the hydrodynamic collectivity in proton–proton (p–p) collisions at 13 TeV, using iEBE-VISHNU hybrid model with HIJING initial conditions. With properly tuned parameters, our model simulations can remarkably describe all the measured 2-particle correlations, including integrated and differential elliptic flow coefficients for all charged and identified hadrons (<math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math>, Λ). However, our model calculations show positive 4-particle cumulant <math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>{</mo><mn>4</mn><mo>}</mo></math> in high multiplicity pp collisions, and can not reproduce the negative <math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>{</mo><mn>4</mn><mo>}</mo></math> measured in experiment. Further investigations on the HIJING initial conditions show that the fluctuations of the second order anisotropy coefficient <math><msub><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msub></math> increases with the increase of its mean value, which leads to a similar trend of the flow fluctuations. For a simultaneous description of the 2- and 4- particle cumulants within the hydrodynamic framework, it is required to have significant improvements on initial condition for pp collisions, which is still lacking of knowledge at the moment.

Weitian Deng

Hydrodynamic collectivity in proton–proton collisions at 13 TeV

In this paper, we study and predict flow observables in 2.76 and 5.02 A TeV PbÂ +Â Pb collisions, using the iEBE-VISHNU hybrid model with TRENTo and AMPT initial conditions and with different forms of the QGP transport coefficients. With properly chosen and tuned parameter sets, our model calculations can nicely describe various flow observables in 2.76 A TeV PbÂ +Â Pb collisions, as well as the measured flow harmonics of all charged hadrons in 5.02 A TeV PbÂ +Â Pb collisions. We also predict other flow observables, including vn(pT) of identified particles, event-by-event vn distributions, event-plane correlations, (normalized) symmetric cumulants, non-linear response coefficients and pT -dependent factorization ratios, in 5.02 A TeV PbÂ +Â Pb collisions. We find many of these observables to remain approximately the same values as the ones in 2.76 A TeV PbÂ +Â Pb collisions. Our theoretical studies and predictions could shed light to the experimental investigations in the near future.