Connecting Theory to Heavy Ion Experiment

Abstract

Only a fraction of all $\Lambda$ and $\bar{\Lambda }$ hyperons detected in heavy ion collisions are produced from the hot and dense matter directly at the hadronization. These hyperons are called the primary hyperons. The rest of the hyperons are products of the decays of heavier hyperon states, which in turn are produced at the hadronization. As such, the polarization of only primary hyperons can be described with the formulae introduced in Sect. 8. For the rest of the hyperons, the polarization transfer in the decays has to be computed, and convoluted with the polarization of the mother hyperon. In this chapter, a derivation of the polarization transfer coefficients, as well as the computation of the mean polarization of all $\Lambda$ hyperons detected in the experiment, is presented. The chapter is concluded with the calculation of the resonance contributions to the global and local $\Lambda$ polarizations.

Appendices

Appendix 1 Lorentz Boost and Jacobian Determinant

In this Appendix, we demonstrate details to derive (10.79) and (10.81) shown in Sect. 10.5. As mentioned in the context, $p^\mu _\Lambda =(\varepsilon _\Lambda , \mathbf{p }_\Lambda )$ and $p_*^\mu =(\varepsilon _{\Lambda *} , \mathbf{p }_*)$ are the four-momenta of $\Lambda$ in the QGP frame and Mother’s rest frame, respectively, and $p_H^\mu =(\varepsilon _H, \mathbf{p }_H)$ the four-momentum of the Mother in QGPF. The pure Lorentz boost transforming the momentum of $\Lambda$ from QGPF to MRF reads

$$\begin{aligned} \varepsilon _{\Lambda *}={}&\gamma _H(\varepsilon _\Lambda -\mathbf{v }_H\cdot \mathbf{p }_\Lambda ), \end{aligned}$$

(10.103)

$$\begin{aligned} \mathbf{p }_* = {}&\mathbf{p }_\Lambda + \left( \frac{\gamma _H -1}{\mathbf{v }_H^2}\mathbf{v }_H \cdot \mathbf{p }_\Lambda - \gamma _H~\varepsilon _\Lambda \right) \mathbf{v }_H , \end{aligned}$$

(10.104)

where $\mathbf{v }_H=\mathbf{p }_H/\varepsilon _H$ is the velocity of the Mother and $\gamma _H=\varepsilon _H/m_H$ the corresponding Lorentz factor. Hence, the explicit forms of (10.103) and (10.104) are

$$\begin{aligned} \varepsilon _{\Lambda *}={}&{1\over m_H} (\varepsilon _H\varepsilon _\Lambda - \mathbf{p }_H\cdot \mathbf{p }_\Lambda ), \end{aligned}$$

(10.105)

$$\begin{aligned} \mathbf{p }_* = {}&\mathbf{p }_\Lambda + \left[ \frac{\mathbf{p }_H\cdot \mathbf{p }_\Lambda }{m_H (\varepsilon _H + m_H)} -\frac{\varepsilon _\Lambda }{m_H} \right] \mathbf{p }_H, \end{aligned}$$

(10.106)

then the expression of $\mathbf{p }_H\cdot \mathbf{p }_\Lambda$ from (10.105) can be substituted into (10.106) to get

$$\begin{aligned} \mathbf{p }_* = \mathbf{p }_\Lambda + \left[ \frac{\varepsilon _H\varepsilon _\Lambda - m_H \varepsilon _{\Lambda *}}{m_H(\varepsilon _H+m_H)} -\frac{\varepsilon _\Lambda }{m_H} \right] \mathbf{p }_H = \mathbf{p }_\Lambda - \frac{\varepsilon _{\Lambda *} + \varepsilon _\Lambda }{\varepsilon _H+m_H}\mathbf{p }_H. \end{aligned}$$

(10.107)

Moving $\mathbf{p }_\Lambda$ to the left-hand side of (10.107) and take square of both sides, we have

$$\begin{aligned} (\mathbf{p }_* - \mathbf{p }_\Lambda )^2=\frac{(\varepsilon _{\Lambda *} + \varepsilon _\Lambda )^2}{(\varepsilon _H+m_H)^2}\mathbf{p }_H^2 = \frac{\varepsilon _H-m_H}{\varepsilon _H+m_H}(\varepsilon _{\Lambda *} + \varepsilon _\Lambda )^2, \end{aligned}$$

(10.108)

which then gives the energy of the Mother in terms of the energy-momenta of the Daughter as

$$\begin{aligned} \varepsilon _H=m_H \frac{(\varepsilon _{\Lambda *} + \varepsilon _\Lambda )^2 + (\mathbf{p }_* - \mathbf{p }_\Lambda )^2}{(\varepsilon _{\Lambda *} + \varepsilon _\Lambda )^2 - (\mathbf{p }_* - \mathbf{p }_\Lambda )^2}. \end{aligned}$$

(10.109)

By substituting (10.109) back into (10.107), the final expression for the momentum of the Mother follows directly

$$\begin{aligned} \mathbf{p }_H= 2m_H \frac{\varepsilon _+\mathbf{p }_-}{\varepsilon _+^2- \mathbf{p }_-^2} \quad \text {with}\quad \varepsilon _+=\varepsilon _\Lambda +\varepsilon _{\Lambda *}, \quad \mathbf{p }_-=\mathbf{p }_\Lambda -\mathbf{p }_*. \end{aligned}$$

(10.110)

Now, the above equation (10.110) can be easily adopted to alter the integration variable involved in (10.78) from $\mathbf{p }_H$ to $\mathbf{p }_*$ by fixing $\mathbf{p }_\Lambda$. The Jacobian matrix of the transformation can be evaluated as

$$\begin{aligned} \frac{\partial \mathrm{p}_{Hi}}{\partial \mathrm{p}_{*j}}=\frac{2m_H}{\varepsilon _+^2-\mathbf{p }_-^2} \left\{ \left[ \mathrm{p}_{- ,i}\frac{\mathrm{p}_{*j}}{\varepsilon _{\Lambda *}} - \varepsilon _+\delta _{ij} \right] -{2\varepsilon _+\mathrm{p}_{- ,i}\over \varepsilon _+^2-\mathbf{p }_-^2} \left[ \varepsilon _+\frac{\mathrm{p}_{*j}}{\varepsilon _{\Lambda *}} +\mathrm{p}_{- ,j} \right] \right\} \end{aligned}$$

(10.111)

for $i,j=x,y,z$, and the determinant follows directly after some algebraic manipulations:

$$\begin{aligned} \left| \frac{\partial \mathbf{p }_H}{\partial \mathbf{p }_*} \right| = \frac{4m_H^3\varepsilon _+^2(\varepsilon _+^2+\mathbf{p }_-^2)}{\varepsilon _{\Lambda *}(\varepsilon _+^2-\mathbf{p }_-^2)^3}. \end{aligned}$$

(10.112)

Appendix 2 Integrands for the Transverse and Longitudinal Polarizations

Herein, we work out the integrands for the evaluations of the transverse and longitudinal components of the mean spin vector, fed down from the strong and EM decays. Taking the most complicated component ${S}_{\Lambda y}^{PC}(\mathbf{p }_*)$, along the total angular momentum, for example, inserting (10.82) into the second equation of (10.92) gives

$$\begin{aligned} {S}_{\Lambda y}^{PC}(\mathbf{p }_*)= & {} 2(g_0 + g_1 \cos \phi _H + g_2 \cos 2 \phi _H) \Big ( A + B \sin ^2 \phi _* \sin ^2 \theta _* \Big ) + B \Big ( f_2\sin 2 \phi _H\nonumber \\&\sin \phi _* \sin 2 \theta _* + (h_1 \sin \phi _H + h_2 \sin 2 \phi _H) \sin 2 \phi _* \sin ^2 \theta _* \Big ). \end{aligned}$$

(10.113)

Because $h_1(\mathrm{P}_T,Y_H)$ and $g_1(\mathrm{P}_T,Y_H)$ are odd functions of $Y_H$ thus also of “$\cos \theta _*$” and all the trigonometric functions of the Mother in (10.85) are even functions of “$\cos \theta _*$”, the terms proportional to $h_1$ and $g_1$ do not contribute at all after integrating over $\theta _*$. Likewise, the term proportional to $f_2(\mathrm{P}_T,Y_H)$, which is an even function of “$\cos \theta _*$”, vanishes upon integration over $\theta _*$ because the function $\sin 2 \theta _*$ is odd. So we are left with

$$\begin{aligned} {S}_{\Lambda y}^{PC}(\mathbf{p }_*)=(g_0 + g_2 \cos 2 \phi _H) \left( F - B\cos 2\phi _*\sin ^2 \theta _* \right) + B\, h_2 \sin 2 \phi _H \sin 2 \phi _* \sin ^2 \theta _*, \end{aligned}$$

(10.114)

where $F=2A+B\sin ^2 \theta _*$.

Inserting (10.85) and replacing $\phi _*$ by $\phi _\Lambda + \psi$, (10.114) becomes explicitly

$$\begin{aligned}&[g_0 + g_2 ({\mathcal {A}}\cos 2\phi _\Lambda - {\mathcal {B}}\sin 2\phi _\Lambda )] \left[ F - B(\cos 2\phi _\Lambda \cos 2\psi -\sin 2\phi _\Lambda \sin 2\psi )\sin ^2 \theta _* \right] \nonumber \\&+ B\, h_2({\mathcal {A}}\sin 2\phi _\Lambda +{\mathcal {B}}\cos 2\phi _\Lambda ) (\cos 2\phi _\Lambda \sin 2\psi +\sin 2\phi _\Lambda \cos 2\psi ) \sin ^2 \theta _*. \end{aligned}$$

(10.115)

Remember that any terms that are odd functions of “$\cos \theta _*$” or $\psi$ vanish after solid angle integrations. Thus, by taking into account the even-oddness of the relevant functions listed in Table 10.2, the following terms are left:

$$\begin{aligned}&(g_0 + g_2 {\mathcal {A}}\cos 2\phi _\Lambda )(F - B\cos 2\phi _\Lambda \cos 2\psi \sin ^2 \theta _*) -g_2 {\mathcal {B}}B\sin ^22\phi _\Lambda \sin 2\psi \sin ^2 \theta _*\nonumber \\&+ {B} h_2({\mathcal {A}}\sin ^22\phi _\Lambda \cos 2\psi +{\mathcal {B}}\cos ^2 2\phi _\Lambda \sin 2\psi )\sin ^2 \theta _*. \end{aligned}$$

(10.116)

Finally, we adopt the double-angle relationships for the trigonometric functions:

$$\cos ^2x={1\over 2}(\cos 2x+1),\qquad \sin ^2x={1\over 2}(-\cos 2x+1)$$

to put the result (10.116) in harmonics of $\phi _\Lambda$:

$$\begin{aligned} {S}_{\Lambda y}^{PC}(\mathbf{p }_*)= & {} \left[ g_0F+{B\over 2} (h_2-g_2) ({\mathcal {A}}\cos 2\psi +{\mathcal {B}}\sin 2\psi ) \sin ^2 \theta _*\right] \nonumber \\&-(g_0B\cos 2\psi \sin ^2 \theta _*-g_2F {\mathcal {A}})\cos 2\phi _\Lambda \nonumber \\&- {B\over 2} (h_2+g_2)({\mathcal {A}}\cos 2\psi -{\mathcal {B}}\sin 2\psi )\sin ^2 \theta _*\cos 4\phi _\Lambda . \end{aligned}$$

(10.117)

One finds that $h_2$ and $g_2$ terms give rise to contributions to both global and $4\phi _\Lambda$ harmonic modes for the TLP $P_y$.

Similarly, $h_1,g_1$ and $f_2$ do not contribute to the TLP $P_x$ because the relevant terms in the integrand ${S}_{\Lambda x}^{PC}(\mathbf{p }_*)$ are also odd functions of “$\cos \theta _*$”. So by combining (10.82) and (10.85) with the first equation in (10.92), the integrand is explicitly

$$\begin{aligned} {S}_{\Lambda x}^{PC}(\mathbf{p }_*)= & {} h_2({\mathcal {A}} \sin 2 \phi _\Lambda + {\mathcal {B}} \cos 2 \phi _\Lambda )\Big ( F + {B} \cos 2 \phi _* \sin ^2 \theta _* \Big )\nonumber \\&+ B [g_0+g_2({\mathcal {A}} \cos 2 \phi _\Lambda - {\mathcal {B}} \sin 2 \phi _\Lambda )] \sin 2 \phi _* \sin ^2 \theta _*, \end{aligned}$$

(10.118)

which becomes

$$\begin{aligned}&h_2\left[ {\mathcal {A}} \sin 2 \phi _\Lambda \Big ( F + {B} \cos 2 \psi \cos 2 \phi _\Lambda \sin ^2 \theta _* \Big )-{B\over 2} {\mathcal {B}}\sin 2 \psi \sin 4 \phi _\Lambda \sin ^2 \theta _* \right] \nonumber \\&+ B \left[ (g_0+g_2{\mathcal {A}} \cos 2 \phi _\Lambda )\cos 2 \psi \sin 2 \phi _\Lambda -g_2 {{\mathcal {B}}\over 2}\sin 4\phi _\Lambda \sin 2 \psi \right] \sin ^2 \theta _* \end{aligned}$$

(10.119)

after replacing $\phi _*$ by $\phi _\Lambda + \psi$. And the double-angle relationships give

$$\begin{aligned} {S}_{\Lambda x}^{PC}(\mathbf{p }_*)= & {} (h_2F{\mathcal {A}}+g_0B\cos 2 \psi \sin ^2 \theta _*)\sin 2 \phi _\Lambda \nonumber \\&+{B\over 2}(h_2+g_2)({\mathcal {A}}\cos 2 \psi -{\mathcal {B}}\sin 2 \psi )\sin ^2 \theta _*\sin 4 \phi _\Lambda , \end{aligned}$$

(10.120)

where we recognize that the coefficient of the $4\phi _\Lambda$ harmonic is opposite to that of ${S}_{\Lambda y}^{PC}(\mathbf{p }_*)$.

For the longitudinal component, $g_0,g_2$, and $h_2$ do not contribute because the relevant terms in the integrand ${S}_{\Lambda z}^{PC}(\mathbf{p }_*)$ are also odd functions of “$\cos \theta _*$”. So by combining (10.82) and (10.85) with the third equation in (10.92), the integrand is explicitly

$$\begin{aligned} {S}_{\Lambda z}^{PC}(\mathbf{p }_*)= & {} 2 f_2({\mathcal {A}} \sin 2 \phi _\Lambda + {\mathcal {B}} \cos 2 \phi _\Lambda )\Big (A + {B} \cos ^2 \theta _* \Big )+ B [h_1({\mathcal {C}} \sin \phi _\Lambda + {\mathcal {D}} \cos \phi _\Lambda )\nonumber \\&\cos \phi _*+g_1({\mathcal {C}} \cos \phi _\Lambda - {\mathcal {D}} \sin \phi _\Lambda )\sin \phi _*] \sin 2 \theta _*, \end{aligned}$$

(10.121)

which becomes

$$\begin{aligned} {S}_{\Lambda z}^{PC}(\mathbf{p }_*)=\left[ 2 f_2{\mathcal {A}}\Big (A + {B} \cos ^2 \theta _* \Big )+ {B\over 2} (h_1+g_1)({\mathcal {C}}\cos \psi - {\mathcal {D}} \sin \psi ) \sin 2 \theta _*\right] \sin 2 \phi _\Lambda \end{aligned}$$

(10.122)

after replacing $\phi _*$ by $\phi _\Lambda + \psi$. Note that the LLP keeps the same harmonic as the primary one without any other mixing, that is, $\sim \sin 2 \phi _\Lambda$.