Title: Tilted emission source of pions in Au+Au collisions

PAs: Yevheniia Khyzhniak, Michael Lisa, ShinIchi Esumi

Contact information: eugenia.sh.el@gmail.com

Target journal: Physical Review C

Talks in PWG:
1) https://drupal.star.bnl.gov/STAR/system/files/asHBT_PWG.pdf
2) https://drupal.star.bnl.gov/STAR/system/files/asHBT_PWG_2.pdf
3) https://drupal.star.bnl.gov/STAR/system/files/preliminaryReqAsHBT_0.pdf
4) https://drupal.star.bnl.gov/STAR/system/files/analysisMeetingKhyzhniak.pdf
5) https://drupal.star.bnl.gov/STAR/system/files/asHBT_updateBeforePP.pdf
6) https://drupal.star.bnl.gov/STAR/system/files/paperProposalasHBT.pdf

Abstract
We present the first systematic measurement of the tilt of the pion emission source in relativistic Au+Au collisions at beam energies of √sNN = 7.7, 14.5, and 27 GeV, using data from the STAR experiment. The tilt angle is extracted through azimuthally sensitive femtoscopy of identical pion pairs. Our results reveal a strong dependence of the tilt parameter on the pair transverse momentum, indicating that the apparent source geometry is strongly coupled to collective flow and expansion dynamics. Moreover, we observe a rapid decrease of the tilt magnitude with increasing collision energy, consistent with the emission source approaching longitudinal boost invariance at higher energies. These findings demonstrate that the commonly assumed boost-invariant geometry is insufficient and highlight the necessity of exploring the spatial structure of a tilted source, which is required in hydrodynamic models to reproduce features of the longitudinally expanding system, such as the slope of the directed flow. Comparisons with the UrQMD transport model show that while the overall energy dependence of the tilt is qualitatively reproduced, the model systematically underestimates its magnitude.

Fig. 1
Resolution of the first-order event plane reconstructed with the EPD detector. Blue circles correspond to Au+Au collisions at $\sqrt{s_{NN}} = 7.7$GeV, red squares to 14.5GeV, and green triangles to 27~GeV. Closed markers show the resolution of the first-order event plane using the first-harmonic coefficients ( n = 1 , k = 1). Open markers show the effective resolution of the same first-order event plane when applied to second-harmonic coefficients ( n = 1 , k = 2).

Fig. 2
One-dimensional projections of the three-dimensional correlation function in the out (left column), side (central column), and long (right column) directions for eight ranges
in the azimuthal angle of the pion pair relative to the first-order event plane, together with the corresponding projections of the three-dimensional fit. Results are shown
for Au+Au collisions at $\sqrt{s_{NN}} = 7.7$~GeV in the 10-30% centrality class. Projections onto a given axis are obtained with the condition $|q_{\text{others}}| <
0.05$~GeV/$c$. Red circles with red fit curves correspond to $k_{T} = 0.15$--$0.20$~GeV/$c$, while blue triangles with blue fit curves correspond to $k_{T} = 0.30$--
$0.35$~GeV/$c$.

Fig. 3
Two-dimensional projections of the three-dimensional correlation function in the out–side (left column), side–long (central column), and out–long (right column) directions for
eight ranges in the azimuthal angle of the pion pair relative to the first-order event plane, together with the corresponding projections of the three-dimensional fit, shown as white concentric ellipses. The red shaded band indicate the correlation angle between the two corresponding axes. Results are presented for Au+Au collisions at $\sqrt{s_{NN}} = 7.7$~GeV in the 10--30% centrality class and for pion pairs with transverse momentum $k_{T} = 0.30$--$0.35$~GeV/$c$. Projections onto a given axis are obtained with the condition $|q_{\text{other}}| < 0.05$~GeV/$c$.

Fig. 4
Extracted femtoscopic radii as a function of the azimuthal angle relative to the event plane for four centrality classes. Black circles correspond to 0--10%, red squares to 10--30%, green triangles to 30--50%, and blue diamonds to 50--80% in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$~GeV. Results are shown for pion pairs with transverse momentum $k_{T}
= 0.3$--$0.35$GeV/$c$. Lines of the corresponding colors represent fits of the radii oscillations according to Eq. 2

Fig. 5
Extracted femtoscopic radii as a function of the azimuthal angle of the pion pair relative to the event plane for six ranges in transverse momentum $k_{T}$. Black circles correspond to $0.15$--$0.2$~GeV/$c$, red squares to $0.2$--$0.25$~GeV/$c$, green triangles to $0.25$--$0.3$~GeV/$c$, blue diamonds to $0.3$--$0.35$~GeV/$c$, violet crosses to
$0.35$--$0.45$~GeV/$c$, and brown stars to $0.45$--$0.8$~GeV/$c$. Results are shown for Au+Au collisions at $\sqrt{s_{NN}}=7.7$GeV in the 10--30%
centrality class. Lines of the corresponding colors represent fits of the radii oscillations according to Eq. 2.

Fig. 6
Illustration of the procedure for extracting the tilt parameter from the femtoscopic radii. Panel (a) shows the transverse-momentum dependence of $-4R^{2}{\text{side-long},1}$. Panel (b) shows the dependence of $R^{2}{\text{long},0} - R^{2}{\text{side},0} + 2R^{2}{\text{side},2}$ on transverse momentum. Panel (c) presents the ratio of the quantities from panels (a) and (b). Panel (d) shows the resulting tilt parameter, calculated as $\tfrac{1}{2}\arctan$ of the value shown in panel (c). In all panels, black circles correspond to 0--10%, red squares to 10--30%, green triangles to 30--50%, and blue diamonds to 50--80% centrality in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$~GeV. Shaded bands represent the
tilt obtained from UrQMD 3.4 in cascade mode for the corresponding centralities.

Fig. 7
Illustration of the procedure for extracting the tilt parameter from the femtoscopic radii. Panel (a) shows the transverse-momentum dependence of $-4R^{2}{\text{side-long},1}$. Panel (b) shows the dependence of $R^{2}{\text{long},0} - R^{2}{\text{side},0} + 2R^{2}{\text{side},2}$ on transverse momentum. Panel (c) presents the ratio
of the quantities from panels (a) and (b). Panel (d) shows the resulting tilt parameter, calculated as $\tfrac{1}{2}\arctan$ of the value shown in panel (c). In all panels, black circles correspond to 0--10%, red squares to 10--30%, green triangles to 30--50%, and blue diamonds to 50--80% centrality in Au+Au collisions at $\sqrt{s_{NN}} = 14.5$~GeV.

Fig. 8
Illustration of the procedure for extracting the tilt parameter from the femtoscopic radii. Panel (a) shows the transverse-momentum dependence of $-4R^{2}{\text{side-long},1}$. Panel (b) shows the dependence of $R^{2}{\text{long},0} - R^{2}{\text{side},0} + 2R^{2}{\text{side},2}$ on transverse momentum. Panel (c) presents the ratio of the quantities from panels (a) and (b). Panel (d) shows the resulting tilt parameter, calculated as $\tfrac{1}{2}\arctan$ of the value shown in panel (c). In all panels, black circles correspond to 0--10%, red squares to 10--30%, green triangles to 30--50%, and blue diamonds to 50--80% centrality in Au+Au collisions at $\sqrt{s_{NN}} = 27$~GeV. Shaded bands represent the tilt obtained from UrQMD 3.4 in cascade mode for the corresponding centralities.

Fig. 9
Schematic representation of the orientation of the homogeneity region with respect to the fixed STAR coordinate system. One particular homogeneity region is shown for the case where the azimuthal angle of the pion pair is $\phi_{k} = 90^\circ$. In this configuration, the beam direction is along the $z$ axis, $R_{\text{long}}$ is parallel to the $z$ axis, and the impact parameter vector is aligned with the $x$ axis. The $y$ axis is perpendicular to the reaction plane, $R_{\text{out}}$ points out of the page and $R_{\text{side}}$ is  perpendicular to both $R_{\text{long}}$ and $R_{\text{out}}$. Cases (a) and (b) illustrate situations where the correlation between $R_{\text{side}}$ and $R_{\text{long}}$ is negative, yielding $R^{2}{\text{side-long},1} < 0$, while cases (c) and (d) show the opposite sign. Additionally, in cases (a) and (c), $R^{2}{\text{long}}$ is larger than $R^{2}{\text{side}}$, whereas in cases (b) and (d)$, R^{2}{\text{side}}$ exceeds $R^{2}_{\text{long}}$.

Fig. 10
Energy dependence of the extracted tilt parameter. Red stars represent the STAR results obtained in this work, blue crosses correspond to AGS data, and the violet diamond denotes the estimate from HADES publication data. Green squares show the tilt values extracted from azimuthally sensitive femtoscopy using UrQMD 3.4 in cascade mode. Lines indicate tilt estimates obtained by fitting the distribution of freeze-out coordinates from UrQMD 3.3 Cascade, UrQMD 3.3+Hydro[HG], and UrQMD 3.3+Hydro[BM].

Summaries
1) Tilt parameter dependence on the transverse momentum of the
pion pair

• Strong dependence of the tilt parameter on the transverse momentum of the pion pair
• UrQMD 3.4 in cascade mode reproduces the overall $k_{T}$-dependence of the tilt but misses its magnitude

2) Energy dependence of the tilt parameter

• Results are consistent with the trend observed in AGS data
• Tilt decreases with energy, in line with the expectation that the system becomes increasingly boost-invariant
• STAR data lie slightly below the UrQMD 3.4 in cascade mode results
• Comparison of the data to the models indicates that the tilt is sensitive to the equation of state (EoS)