# Shu-yi Wei

## Abstract

Most of my scientific works do not involve numbers when I was a Ph.D. student in Shandong University. I am not going to bother you with the terrible handwritten notes.

Blessed or cursed, I started to dance with fortran during the postdoc phase of my life. She is so pretty that I almost fall in love. This webpage is a romance that tells most of our interesting dates. While my recollection might be declining, the internet remembers. Hopefully these short notes will demonstrate their importance in the future.

If you are interested in any code, please feel free to contact me for any piece of the fortran code. My contract information can be found on the mainpage.

## Codes

Most of my numerical calculations are about integrations. I rely heavily on the VEGAS algarithm to achieve this goal.

### Transverse Polarization of $\Lambda$ in $e^+e^-$ annihilations

This paper was published on PLB (10.1016/j.physletb.2021.136217) and the arXiv number is: 2102.00658. This code is available upon request.

Belle experiment [arXiv:1808.05000] measured the transverse polarization of $\Lambda$ in the $e^+e^-$ annihilations collisions. Two observables are presented. (1) Transverse polarization in $\Lambda + h$ production process, where $\Lambda$ and $h$ locate in different semispheres (almost back-to-back) and $h$ is a light hadron. The transverse direction is defined with respect to the production plane of these two hadrons. (2) Transverse polarization in the inclusive $\Lambda$ production process. The transverse direction plane is defined with repect to the plane composed by the momentum of $\Lambda$ and the thrust axis.

While the provious works employed parameterization forms that break the isospin symmetry, we believe the isospin symmetry should be kept in QCD. Our work shows that the Belle data can still be well described under this presumption and therefore it does not signal the isospin symmetry broken.

Therefore, in our paper, we require that $D_{1T}^{\perp, u} (z, p_T) = D_{1T}^{\perp, d} (z, p_T)$, $D_{1T}^{\perp, \bar u} (z, p_T) = D_{1T}^{\perp, \bar d} (z, p_T)$ and release all the other constraints. Well, of course, the positivity boundary condition still applies.

Fig. 1 shows the flavor component in the $\Lambda +\pi^+$ production process, which is defined in Eq. (21). The FFs of $\pi^+$ and $K^+$ are given by DHESS [arXiv:1410.6027; arXiv:1702.06353] and those of $\Lambda$ are given by DSV [arXiv: hep-ph/9711387]. Two NOTES here: (1) DSV provides only the FFs of $q \to \Lambda + \bar \Lambda$. We have employed the simple assumption that $D_{1}^{u \to \Lambda} (z) = \frac{1+z}{2} D_{1}^{u \to \Lambda +\bar \Lambda} (z)$ and $D_{1}^{\bar u \to \Lambda} (z) = \frac{1-z}{2} D_{1}^{u \to \Lambda +\bar \Lambda} (z)$. This has been explained in Ref. [22] in our paper. (2) DHESS does not provide numerical datasets to the public. We evolved their parameterization at the initial scale to $\mu_f$ = 10 GeV with the LO DGLAP evolution eqution. We believe there is a typo in Table II of arXiv:1702.06353. $N_{c+\bar c} = 0.1255$ should be $N_{c} = N_{\bar c} = 0.1255$.

Figs. 2 and 3 show the transverse polarization of $\Lambda$ calculated with Eqs. (19-20) and parameters in Tab. I.

The fortran code is given in three folders. (1) DGLAP_Evolution_of_Kaons_DHESS: This code can evolve the FFs at the initial scale to those at higher factorization scales. The default setups is for the $K^+$ production. The FFs at the initial scale are defined in the ini.f file. (2) DSV+DHESS_Polarization_Flavor_Prism: This code calculates theoretical curves for DSV+DHESS in Figs. 1-3. (3) pythia: This code generates FFs from a Pythia simulation, which I do not even want to describe.

The useage is quite straightforward. (1) Enter to the centain folder with your terminal. (2) Type "make" to generate the excuable file. (3) Use "./generated excuable file" to run the code. (4) After everything is done, you can use "make clean" command to remove the ".o" files and the results are stored in the "results.dat" file.

### Elliptic Flow at the Electron-Ion Collider

This paper was published on PRD (10.1103/PhysRevD.103.054017) and the arXiv number is: 2008.03569. A copy of this code can be requested by email.

ATLAS measured the elliptic flow of light hadrons among the high-multiplicity events in the ultra-peripheral AA collisions. The underlining reaction is actually the $\gamma^* + A$ collisions, where $\gamma^*$ is almost on-shell. We argue that there is a partonic Fock state in the wave function of the almost on shell photon, due to the rare QCD fluctuation. The high-multiplicity events are dominated by these partonic states. The strong resemblance between $\gamma^* + A$ collisions and pA collisions natually explains the significant elliptic flow measured in UPC.

In the UPC folder, the code calculates the integrated $\kappa_2$ and the integrated $\kappa_0$ first. The integrated $v_2$ is given by $v_2^{\rm int} \equiv \sqrt{\kappa_2/\kappa_0}$. Then the code calculates the differential $\kappa_2 (p_T)$ and the differential $\kappa_0 (p_T)$. The differential $v_2 = \frac{1}{v_2^{\rm int}} \frac{\kappa_2(p_T)}{\kappa_0 (p_T)}$. This is the so-called two-particle method to measure the elliptic flow.

In the EIC folder, please visit the parton_level_ratios folder. The code returns the integrated $v_2$, which is presented in our paper.

### Vector Meson Spin Alignmet in pp Collisions

This paper was published on PRD (10.1103/PhysRevD.102.034001) and the arXiv number is: 2002.09890. A copy of this code can be requested by email.

In this paper, we extracted the spin alignment dependent FFs from LEP data in two scenarios of parameterization. Then, we made predictions for the pp collisions.

The unpolarized FF of $K^{*+}$ is related to that of $K^+$ by: $D^{K*+}(z) = c_0 (1+2z) D^{K+}(z)$. The unpolarized FF of $\rho^0$ can be calculated in a much complicated way (see our paper for details).

There are four folders in the ZIP file. The common_files folder stores the unpolarized and the polarized FFs of $K^{*+}$ and $\rho^0$ based on the AKK08 and DHESS parameterizations. The data files are in the binary format to save disk space. The flavor_componet folder reproduces the plots in Sec. IV.A. The spin_alignment_pT folder reproduces Figs. 13-14 and The spin_alignment_xF folder reproduces Figs. 15-16.

### Solving BK evolution equation

This paper will be published in JHEP and the arXiv number is: 2012.08562. A copy of this code can be requested by email.

The BK evolution equation is quite simple. However, the devil is in the details.

We present three codes to solve the BK equation. In the "08.solving.BK.equations" folder, there are three subfolders.

(1) BK_FixedCoupling;
(2) Running_coupling_BK_LO;
(3) Collinearly-improved_rcBK_with_bstar_prescription.


The BK_FixedCoupling folder is the code to solve the BK evolution equation with fixed $\alpha_s = 0.3$. Fixed-Coupling BK equation evolves the scattering amplitude in a very fast speed. Therefore, one has to redifne the coordinate grid at large rapidities. For instance, at Y=0, there are 800 grids between $\ln(r_{\rm max} = 50)$ and $\ln(r_{\rm min} = 10^{-9})$. However, at Y=2, there are 800 grids between $\ln(r_{\rm max} = 50)$ and $\ln(r_{\rm min} = 10^{-10})$.

The Running_coupling_BK_LO folder is the code to solve the BK evolution equation with running coupling. In this case, we do not need to redefine the grids at large rapidities. However, different prescriptions for the $\alpha_s$ in the coordinate space exist. Pls make sure the correction prescription according to your MODEL is implimented. For the αs at the IR region, we have provided two simple functions. [1] Hard Cut-off. $\alpha_s (r) = 0.7$ at large $r$. [2] $b_*$-prescription. Replace $\alpha_s (r)$ with $\alpha_s (r_*)$ where $r_* = r/\sqrt{1+r^2/r_{\rm max}^2}$.

The hard cut-off in the running coupling might result in the oscillation problem after the Fourier Transform of the scattering amplitude at large $k_\perp$. The $b_*$-prescription can mitigate this problem.

The Collinearly-improved_rcBK_with_bstar_prescription folder solves the rcBK evolution equation described in our paper. The Collinearly-Improved evolution equation takes into account the NLO corrections.

### Z0-boson production in forward pp and pA collisions

This paper was published on PLB (10.1016/j.physletb.2020.135253) and the arXiv number is: 1909.08572. A copy of this code can be requested by email.

In this paper, we calculate the transverse momentum spectrum of $Z^0$-boson produced in forward pp and pA collisions. We have employed two frameworks to make that happen. (1) Sudakov resummation in the collinear factorization framework; (2) Sudakov resummation in the CGC framework.

The Sudakov resummation in the C.F. framework was established by Collins, Soper and Sterman in 1985. All the details has been presented in that paper. A copy of my code at the NLL accuracy is presented in the Collinear.Factorization.phistar (which calculates the $\phi^*$ distribution) and Collinear.Factorization.qperp (which calculates the $q_\perp$ distribution) folders. Experimental configurations are already encoded in the main.f file.

CME: center of mass energy of pp collision;
Q2Min and Q2Max: Useless variables;
ymin and ymax: Rapidity range of the dilepton in the Lab-frame; yshift: rapidity shift from the Lab frame to the Center-of-Mass frame.
iTarget: Collisional process: 1: ppbar, 2: pp, 3: pA; iTP: Number of protons; iTN: Number of neutrons. For the pA collisions, we have adopted the so-called EPPS16 nPDF. By default, the code currently loads the nPDF of Pb(208).

PLEASE NOTE: Lines 241 to 253 in the main.f file translates the rapidity of Z0-boson to that of the dileptons. In case you want to calculate the cross section of Z0-boson instead of dilepton, you shall re-define wFactor to be 1. (See Eq. (11) and the related discussion in our paper.)

The formalism of Sudakov resummation in the dilute-dense factorization is presented in this paper. A copy of the fortran code at the LL accuracy is given in the CGC.phistar (which calculates the $\phi^*$ distribution) and CGC.qperp (which calculates the $q_\perp$ distribution) folders.

In the main.f file, iTarget specifies the collisional process. 1: pp, 2: minimal bias pA, 3: central pA. area = 15 mb * $A^{2/3}$ is the cross section of elastic scatterings.

### Dijet azimuthal angular correlation in the forward pp, and AA collisions (Jet Quenching)

This paper was published on PRD (10.1103/PhysRevD.101.094008) and the arXiv number is: 1910.05290. A copy of this code can be requested by email.

PLEASE NOTE: Please change iTarget=1. The public version of this code is for pp, ppbar and pA collisions. Normally we can ignore the cold nuclear effect in AA collisions. IF you want to be more sophisticate, you can modify partondistributions subroutine in the hard.f file.

### Dihadron azimuthal angular correlation in the forward pp, and pA collisions (GBW Model)

This paper was published on PLB (10.1016/j.physletb.2018.08.011) and the arXiv number is: 1805.05712. A copy of this code can be requested by email.

In this code, we have implemented Sudakov resummation in the dilute-dense factorization framework. This factorization framework employs the collinear PDF to describe the parton density of the projectile proton and employs CGC theory to describe the parton density of the target proton or nucleus. Two folders are presented in the ZIP file. pp and pA calculate the dihadron angular correlations in pp and pA collisions respectively. The ONLY difference is the saturation scale.

PLEASE NOTE: The scattering amplitude is given by the GBW model instead of the rcBK solution.