We present a unified free field realization of the Ding-Iohara algebra at arbitrary levels, which satisfies a generalized form of the Serre relations. This realization, constructed using six free boson fields, arises from a specialized factorization of the structure function in the defining relations of the algebra. Based on this construction, we then develop intertwining operators for the Ding-Iohara algebra.

China

Zitao Chen

Xiang-Mao Ding

Free field realization of the Ding-Iohara algebra at general levels

We investigated the impact of a constant external magnetic field on the dressed propagators of up-, down-, and strange quarks. In the weak field limit, we derive a general momentum-space representation for the propagator and numerically solve the corresponding gap equation. Our analysis reveals that the vector term of the propagator can be decomposed into components parallel and perpendicular to the magnetic field, resulting in anisotropic effective masses, with the transverse mass consistently exceeding the longitudinal mass. This mass disparity exhibits a power law dependence on the magnetic field strength and is less pronounced for the strange quark compared to up and down quarks. Additionally, the magnetic field induces axial-vector and tensor terms, highlighting the Zeeman effect, resulting from quark interactions with the magnetic field. These findings have important implications for (inverse) magnetic catalysis and phenomena such as vector meson and pion condensations.

Germany

Minghui Ding

Fei Gao

Sebastian M. Schmidt

Anisotropic quark propagation and Zeeman effect in an external magnetic field * Fei Gao is supported by the National Science Foundation of China (12305134). Minghui Ding appreciates the support received from the Helmholtz-Zentrum Dresden-Rossendorf High Potential Programme

We present some lattice QCD results on first (<math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>i</mi></mrow></msubsup></mrow></math>) and second (<math><msubsup><mi>χ</mi><mn>2</mn><mi>i</mi></msubsup></math>) cumulants of and correlations (<math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msubsup></mrow></math>) among net baryon-number (<math><mi>B</mi></math>), strangeness (<math><mi>S</mi></math>) and electric charge (<math><mi>Q</mi></math>) along the pseudocritical line [<math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math>] in the temperature (<math><mi>T</mi></math>)–baryon chemical potential (<math><msub><mi>μ</mi><mi>B</mi></msub></math>) phase diagram of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD. We point out that violations of sum rules among second order cumulants, which hold in the isospin symmetric limit of vanishing electric charge chemical potential, are small along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> for the entire range of <math><msub><mi>μ</mi><mi>B</mi></msub></math> covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup></math> and <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>Q</mi><mi>S</mi></mrow></msubsup></math>. We compare lattice QCD results for <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> line with preliminary experimental measurements of <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> for collision energies <math><mrow><mn>7.7</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>≤</mo><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo>≤</mo><mn>62.4</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. While we find good agreements for <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>≥</mo><mn>39</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, differences are sizeable at smaller values of <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt></math>. Moreover, we compare lattice QCD results for the ratio of the strangeness (<math><msub><mi>μ</mi><mi>S</mi></msub></math>) to baryon (<math><msub><mi>μ</mi><mi>B</mi></msub></math>) chemical potentials, which define a strangeness neutral system with fixed electric charge to baryon number density, with experimental results obtained by the STAR collaboration for <math><msub><mi>μ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math> using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out (<math><msubsup><mi>μ</mi><mi>B</mi><mi>f</mi></msubsup></math>) by comparing <math><msubsup><mi>χ</mi><mn>1</mn><mi>B</mi></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>B</mi></msubsup></math> along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> with the experimentally measured net-proton cumulants <math><msubsup><mi>χ</mi><mn>1</mn><mi>p</mi></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>p</mi></msubsup></math>. We find that <math><mrow><mo>{</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>)</mo><mo>}</mo></mrow></math> are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>≥</mo><mn>11.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.

United States

D. Bollweg

H.T. Ding

Japan

J. Goswami

F. Karsch

Swagato Mukherjee

P. Petreczky

C. Schmidt

Strangeness-correlations on the pseudocritical line in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD

We analyze the dependence of the chiral phase transition temperature on baryon number and strangeness chemical potentials by calculating the leading order curvature coefficients in the light and strange quark flavor basis as well as in the conserved charge (<math><mrow><mi>B</mi></mrow></math>, <math><mrow><mi>S</mi></mrow></math>) basis. Making use of scaling properties of the magnetic equation of state (MEoS) and including diagonal as well as off-diagonal contributions in the expansion of the energylike scaling variable that enters the parametrization of the MEoS, allows to explore the variation of <math><msub><mi>T</mi><mi>c</mi></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>,</mo><msub><mi>μ</mi><mi>S</mi></msub><mo>)</mo><mo>=</mo><msub><mi>T</mi><mi>c</mi></msub><mo>(</mo><mn>1</mn><mo>−</mo><mo>(</mo><msubsup><mi>κ</mi><mn>2</mn><mi>B</mi></msubsup><msubsup><mover><mi>μ</mi><mo>^</mo></mover><mi>B</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>κ</mi><mn>2</mn><mi>S</mi></msubsup><msubsup><mover><mi>μ</mi><mo>^</mo></mover><mi>S</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msubsup><mi>κ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><msub><mover><mi>μ</mi><mo>^</mo></mover><mi>B</mi></msub><msub><mover><mi>μ</mi><mo>^</mo></mover><mi>S</mi></msub><mo>)</mo><mo>)</mo></math> along different lines in the <math><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>,</mo><msub><mi>μ</mi><mi>S</mi></msub><mo>)</mo></math> plane. On lattices with fixed cutoff in units of temperature, <math><mi>a</mi><mi>T</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>8</mn></math>, we find <math><msubsup><mi>κ</mi><mn>2</mn><mi>B</mi></msubsup><mo>=</mo><mn>0.015</mn><mo>(</mo><mn>1</mn><mo>)</mo></math>, <math><msubsup><mi>κ</mi><mn>2</mn><mi>S</mi></msubsup><mo>=</mo><mn>0.0124</mn><mo>(</mo><mn>5</mn><mo>)</mo></math> and <math><msubsup><mi>κ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><mo>=</mo><mo>−</mo><mn>0.0050</mn><mo>(</mo><mn>7</mn><mo>)</mo></math>. We show that the chemical potential dependence along the line of vanishing strangeness chemical potential is about 10% larger than along the strangeness neutral line. The latter differs only by about 3% from the curvature on a line of vanishing strange quark chemical potential, <math><msub><mi>μ</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn></math>. We also show that close to the chiral limit the strange quark mass contributes like an energylike variable in scaling relations for pseudocritical temperatures. The chiral phase transition temperature decreases with decreasing strange quark mass, <math><msub><mi>T</mi><mi>c</mi></msub><mo>(</mo><msub><mi>m</mi><mi>s</mi></msub><mo>)</mo><mo>=</mo><msub><mi>T</mi><mi>c</mi></msub><mo>(</mo><msubsup><mi>m</mi><mi>s</mi><mrow><mi>phy</mi></mrow></msubsup><mo>)</mo><mo>(</mo><mn>1</mn><mo>−</mo><mn>0.097</mn><mo>(</mo><mn>2</mn><mo>)</mo><mo>(</mo><msub><mi>m</mi><mi>s</mi></msub><mo>−</mo><msubsup><mi>m</mi><mi>s</mi><mrow><mtext>phys</mtext></mrow></msubsup><mo>)</mo><mo>/</mo><msubsup><mi>m</mi><mi>s</mi><mrow><mi>phy</mi></mrow></msubsup><mo>+</mo><mi>O</mi><mo>(</mo><mo>(</mo><mi>Δ</mi><msub><mi>m</mi><mi>s</mi></msub><msup><mo>)</mo><mn>2</mn></msup><mo>)</mo></math>.

O. Kaczmarek

Taiwan

Mugdha Sarkar

Sipaz Sharma

Curvature of the chiral phase transition line from the magnetic equation of state of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD

The correlation between net baryon number and electric charge, <math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow><mrow><mi>BQ</mi></mrow></msubsup></mrow></math>, can serve as a magnetometer of QCD. This is demonstrated by lattice QCD computations using the highly improved staggered quarks with physical pion mass of <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>135</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> on <math><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>8</mn></math> and 12 lattices. We find that <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>BQ</mi></mrow></msubsup></math> along the transition line starts to increase rapidly with magnetic field strength <math><mrow><mi>e</mi><mi>B</mi><mo>≳</mo><mn>2</mn><msubsup><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math> and by a factor 2 at <math><mi>e</mi><mi>B</mi><mo>≃</mo><mn>8</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup></math>. Furthermore, the ratio of electric charge chemical potential to baryon chemical potential, <math><msub><mi>μ</mi><mi>Q</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math>, shows significant dependence on the magnetic field strength and varies from the ratio of electric charge to baryon number in the colliding nuclei in heavy ion collisions. These results can provide baselines for effective theory and model studies, and both <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>BQ</mi></mrow></msubsup></math> and <math><msub><mi>μ</mi><mi>Q</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math> could be useful probes for the detection of magnetic fields in relativistic heavy ion collision experiments as compared with corresponding results from the hadron resonance gas model.

J.B. Gu

A. Kumar

S.T. Li

J.H. Liu

Baryon Electric Charge Correlation as a Magnetometer of QCD

Macroscopic properties of the strong interaction near its chiral phase transition exhibit scaling behaviors, which are the same as those observed close to the magnetic transition in a three-dimensional classical spin system with O(4) symmetry. We show that the universal scaling properties of the chiral phase transition in quantum chromodynamics (QCD) at the macroscale are, in fact, encoded within the microscopic energy levels of its fundamental constituents, the quarks. We establish a connection between the cumulants of the chiral order parameter, i.e., the chiral condensate, and the correlations among the energy levels of quarks, i.e., the eigenspectra of the massless QCD Dirac operator. This relation elucidates how the fluctuations of the chiral condensate arise from the correlations within the infrared part of the energy spectra of quarks, and naturally leads to a generalization of the Banks-Casher relation for the cumulants of the chiral condensate. Then, through (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD calculations with varying light quark masses near the QCD chiral transition, we demonstrate that the correlations among the infrared part of the Dirac eigenvalue spectra exhibit same universal scaling behaviors as expected of the cumulants of the chiral condensate. We find that these universal scaling behaviors extend up to the physical values of the up and down quark masses. Our study reveals how the hidden scaling features at the microscale give rise to the macroscopic universal properties of QCD.

W.P. Huang

Microscopic Encoding of Macroscopic Universality: Scaling Properties of Dirac Eigenspectra near QCD Chiral Phase Transition

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general “master pentagon identity” for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We prove this curious identity and discuss its implications.

Italy

Russia

Yegor Zenkevich

On pentagon identity in Ding-Iohara-Miki algebra

We point out that chiral condensates at nonzero temperature and magnetic fields are in strict connection to the space-time integral of corresponding two-point neutral meson correlation functions in the pseudoscalar channel via the Ward-Takahashi identity. Screening masses of neutral pseudoscalar mesons, which are defined as the exponential decay of the corresponding spatial correlation functions in the long distance, thus are intrinsically connected to (inverse) magnetic catalysis of chiral condensates. To study this we performed lattice simulations of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD on <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><msub><mi>N</mi><mi>t</mi></msub></math> lattices with pion mass <math><msub><mi>M</mi><mi>π</mi></msub><mo>≃</mo><mn>220</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> in a fixed scale approach having temperature <math><mi>T</mi><mo>∈</mo><mo>[</mo><mn>17</mn><mo>,</mo><mn>281</mn><mo>]</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and magnetic field strength <math><mi>e</mi><mi>B</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2.5</mn><mo>]</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. We find that screening lengths, i.e., inverses of screening masses of <math><msup><mi>π</mi><mn>0</mn></msup></math>, <math><msup><mi>K</mi><mn>0</mn></msup></math> and <math><msubsup><mi>η</mi><mrow><mi>s</mi><mover><mi>s</mi><mo>¯</mo></mover></mrow><mn>0</mn></msubsup></math>, turn out to have the similar complex <math><mi>e</mi><mi>B</mi></math> and <math><mi>T</mi></math> dependences of the corresponding chiral condensates. Although the transition temperature is found to always decrease as <math><mi>e</mi><mi>B</mi></math> grows, we show that the suppression due to magnetic fields becomes less significant for hadron screening length and chiral condensates with heavier quarks involved, and ceases to occur for <math><msubsup><mi>η</mi><mrow><mi>s</mi><mover><mi>s</mi><mo>¯</mo></mover></mrow><mn>0</mn></msubsup></math> and strange quark chiral condensate. The complex <math><mi>e</mi><mi>B</mi></math> and <math><mi>T</mi></math> dependences of both screening masses and chiral condensates, reflecting the crossover nature of the QCD transition, are attributed to the competition between sea and valence quark effects. These findings could be useful to guide low-energy models and effective theories of QCD.

X.D. Wang

Chiral condensates and screening masses of neutral pseudoscalar mesons in thermomagnetic QCD medium

We present continuum extrapolated results of charmonium and bottomonium correlators in the vector channel at several temperatures below and above <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>. The continuum extrapolation jointly performed with the interpolations to have physical values of <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math><mi>ϒ</mi></math> masses in the confined phase is based on calculations on several large quenched isotropic lattices using clover-improved Wilson valence fermions carrying different quark masses. The extrapolated lattice correlators are confronted with perturbation theory results incorporating resummed thermal effects around the threshold from potential nonrelativistic QCD (pNRQCD) and vacuum asymptotics above the threshold. An additional transport peak is modeled below the threshold allowing for an estimate of the diffusion coefficients for charm and bottom quarks. We find that charmonium correlators in the vector channel can be well reproduced by perturbative spectral functions above <math><msub><mi>T</mi><mi>c</mi></msub></math> where no resonance peaks for <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> are needed at and above <math><mrow><mn>1.1</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>, while for bottomonium correlators a resonance peak for <math><mi>ϒ</mi></math> is still needed up to <math><mrow><mn>1.5</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>. By analyzing the transport contribution to the correlators we find that the drag coefficient of a charm quark is larger than that of a bottom quark.

A.L. Lorenz

H. Ohno

H. Sandmeyer

H.T. Shu

Charm and beauty in the deconfined plasma from quenched lattice QCD

We present lattice QCD results for masses and magnetic polarizabilities of light and strange pseudoscalar mesons, chiral condensates, decay constants of neutral pion, and neutral kaon in the presence of background magnetic fields with <math><mi>e</mi><mi>B</mi></math> ranging up to around <math><mrow><mn>3.35</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> (<math><mrow><mo>∼</mo><mn>70</mn><msubsup><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math>) in the vacuum. The computations were carried out in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD mostly on <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>96</mn></math> lattices using the highly improved staggered quark action with <math><msub><mi>M</mi><mi>π</mi></msub><mo>≈</mo><mn>220</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> at zero temperature. We find that the masses of neutral pseudoscalar mesons monotonously decrease as the magnetic field strength grows and then saturate at a nonzero value, while there exists a nonmonotonous behavior of charged pion and kaon masses in the magnetic field. We observe a <math><mi>q</mi><mi>B</mi></math> scaling of the up and down quark flavor components of neutral pion mass, neutral pion decay constant as well as the quark chiral condensates at <math><mrow><mn>0.05</mn><mo>≲</mo><mi>e</mi><mi>B</mi><mo>≲</mo><mn>3.35</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. We show that the correction to the Gell-Mann-Oakes-Renner relation involving the neutral pion is less than 6% and the correction for the relation involving neutral kaon is less than 30% at <math><mrow><mi>e</mi><mi>B</mi><mo>≲</mo><mn>3.35</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. We also derive the Ward-Takahashi identities for QCD in the magnetic field in the continuum formulation including the relation between integrated neutral pseudoscalar meson correlators and chiral condensates.

A. Tomiya

Y. Zhang

Chiral properties of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD in strong magnetic fields at zero temperature