Resonance decay dynamics and their effects on $p_T$-spectra of pions in heavy-ion collisions

TL;DR

This paper investigates how resonance decay dynamics influence pion momentum spectra in heavy-ion collisions, highlighting the importance of decay details often neglected in statistical models and proposing a new theoretical framework.

Contribution

The paper introduces a theoretical framework to incorporate hadron decay dynamics into the analysis of pion spectra, extending to general N-body decays.

Findings

Decay matrix element details significantly affect pion spectra.

Neglecting decay dynamics can lead to inaccuracies in physical observables.

The proposed framework allows for more accurate modeling of resonance decays.

Abstract

We examine the influence of resonance decay dynamics on momentum spectra of pions in heavy-ion collisions. Taking the decay $\omega \rightarrow 3 \pi$ as an example, we demonstrate how details of the decay matrix element can modify the physical observables. Such a dynamical effect is commonly neglected in statistical models. To remedy the situation, we formulate a theoretical framework for incorporating hadron dynamics into the analysis, which can be straightforwardly extended to describe general N-body decays.

Equations

$$E_{\pi }\frac{d{n}_{\pi }^{\mathrm{dec}}}{d^3{p}_{\pi }}=\mathrm{br}\times \int d^3{p}_{\mathrm{res}}\frac{d{n}_{\mathrm{res}}}{d^3{p}_{\mathrm{res}}}\times E_{\pi }^{\star }\times \mathrm{dPS}({\vec{p}}_{\pi }^{\star }).$$

$$\begin{array}{rl}{E}_{\pi }^{\star }& =\frac{E_{\pi }{E}_{\mathrm{res}}-{\vec{p}}_{\pi }\cdot {\vec{p}}_{\mathrm{res}}}{m_{\mathrm{res}}}\\ p_{\pi }^{\star }& =\sqrt{{E_{\pi }^{\star }}^2-m_{\pi }^2}.\end{array}$$

$$\frac{d{n}_{\mathrm{res}}}{d^3{p}_{\mathrm{res}}}\to \frac{g_{\mathrm{res}}}{(2\pi {)}^3}\frac{1}{e^{\beta E_{\mathrm{res}}}-1}$$

$$\begin{array}{rl}\mathrm{dPS}({\vec{p}}_1^{\star })& =\frac{1}{{\gamma }_{\mathrm{res}}}\frac{d{\gamma }_{\mathrm{res}}}{d^3{p}_1^{\star }}\\ & =\frac{1}{{\gamma }_{\mathrm{res}}}\frac{1}{2m_{\mathrm{res}}}\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\frac{d^3{p}_3^{\star }}{(2\pi {)}^3}\frac{1}{2E_3^{\star }}\cdots \frac{d^3{p}_N^{\star }}{(2\pi {)}^3}\frac{1}{2E_N^{\star }}\times \\ & (2\pi {)}^4{\delta }^4(P-\sum _i{p}_i)|{\Gamma }_{\mathrm{res}\to 1+2+\cdots +N}{|}^2\end{array}$$

$$\int d^3{p}_1^{\star }(\mathrm{dPS})=1.$$

$$\begin{array}{rl}{\mathrm{dPS}}^{(0)}& =\frac{1}{{\varphi }_N}\frac{d{\varphi }_N}{d^3{p}_1^{\star }}\\ & =\frac{1}{{\varphi }_N}\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\frac{d^3{p}_3^{\star }}{(2\pi {)}^3}\frac{1}{2E_3^{\star }}\cdots \frac{d^3{p}_N^{\star }}{(2\pi {)}^3}\frac{1}{2E_N^{\star }}\times \\ & (2\pi {)}^4{\delta }^4(P-\sum _i{p}_i).\end{array}$$

$$\begin{array}{rl}{\varphi }_N& =\int d{\varphi }_N\\ & =\int \frac{d^3{p}_1^{\star }}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\cdots \frac{d^3{p}_N^{\star }}{(2\pi {)}^3}\frac{1}{2E_N^{\star }}\times \\ & (2\pi {)}^4{\delta }^4(P-\sum _i{p}_i).\end{array}$$

$$\begin{array}{rl}{\mathrm{dPS}}^{(0)}& =\frac{1}{{\varphi }_2}\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\times \\ & (2\pi {)}^4\delta (m_{\mathrm{res}}-E_1^{\star }-E_2^{\star }){\delta }^3({\vec{p}}_1^{\star }+{\vec{p}}_2^{\star }).\end{array}$$

$$\begin{array}{rl}{I}_2& =\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\times \\ & (2\pi {)}^4\delta (m_{\mathrm{res}}-E_1^{\star }-E_2^{\star }){\delta }^3({\vec{p}}_1^{\star }+{\vec{p}}_2^{\star })\\ & =\frac{1}{4m_{\mathrm{res}}q}\frac{1}{(2\pi {)}^2}\delta (p_1^{\star }-q),\end{array}$$

$$\begin{array}{rl}{\varphi }_2(m_{\mathrm{res}}^2,m_1^2,m_2^2)& =\frac{1}{8\pi m_{\mathrm{res}}^2}\sqrt{\lambda (m_{\mathrm{res}}^2,m_1^2,m_2^2)}\\ & =\frac{q}{4\pi m_{\mathrm{res}}},\end{array}$$

$$\lambda (x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz.$$

$$\begin{array}{rl}q& =\frac{1}{2}\sqrt{m_{\mathrm{res}}^2}\times \\ & \sqrt{1-\frac{(m_1+m_2{)}^2}{m_{\mathrm{res}}^2}}\sqrt{1-\frac{(m_1-m_2{)}^2}{m_{\mathrm{res}}^2}}.\end{array}$$

$${\mathrm{dPS}}^{(0)}=\frac{1}{4\pi q^2}\delta (p_{\pi }^{\star }-q),$$

$$\begin{array}{rl}{\mathrm{dPS}}^{(0)}& =\frac{1}{{\varphi }_3}\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\frac{d^3{p}_3^{\star }}{(2\pi {)}^3}\frac{1}{2E_3^{\star }}\times \\ & (2\pi )\delta (m_{\mathrm{res}}-E_1^{\star }-E_2^{\star }-E_3^{\star })\\ & (2\pi {)}^3{\delta }^3({\vec{p}}_1^{\star }+{\vec{p}}_2^{\star }+{\vec{p}}_3^{\star }).\end{array}$$

$$\begin{array}{rl}{I}_3& =\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\frac{d^3{p}_3^{\star }}{(2\pi {)}^3}\frac{1}{2E_3^{\star }}\times \\ & (2\pi )\delta (m_{\mathrm{res}}-E_1^{\star }-E_2^{\star }-E_3^{\star })\\ & (2\pi {)}^3{\delta }^3({\vec{p}}_1^{\star }+{\vec{p}}_2^{\star }+{\vec{p}}_3^{\star })\\ & =\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \frac{1}{8\pi (P-p_1{)}^2}\sqrt{\lambda ((P-p_1{)}^2,m_2^2,m_3^2)}\end{array}$$

$$\begin{array}{rl}{\varphi }_3(s)& =\frac{1}{16{\pi }^2}\frac{1}{s}\times \\ & {\int }_{(m_2+m_3{)}^2}^{(\sqrt{s}-m_1{)}^2}d{s}^{\prime }\sqrt{\lambda (s,s^{\prime },m_1^2)}\times \\ & {\varphi }_2(s^{\prime },m_2^2,m_3^2).\end{array}$$

$${\mathrm{dPS}}^{(0)}=\frac{I_3}{{\varphi }_3(s=m_{\mathrm{res}}^2)},$$

$$|{\Gamma }_{\omega \to 3\pi }{|}^2=\mathcal{P}|C_{V\to 123}{|}^2$$

$$\begin{array}{rl}\mathcal{P}& =-\frac{1}{3}{ϵ}_{\mu \nu \alpha \beta }{ϵ}_{abcd}{P}^{\mu }{p}_1^{\nu }{p}_2^{\alpha }{P}^a{p}_1^b{p}_2^c{g}^{\beta d}\\ & =\frac{1}{12}\times \left(s_{12}{s}_{23}{s}_{13}-m_{\pi }^2(m_{\mathrm{res}}^2-m_{\pi }^2{)}^2\right),\end{array}$$

$$s_{ij}=(p_i+p_j{)}^2,$$

$$P^2=m_{\mathrm{res}}^2=s_{12}+s_{23}+s_{13}-m_1^2-m_2^2-m_3^2.$$

$$\begin{array}{rl}|C_{V\to 123}{|}^2& =|\mathcal{N}{|}^2(1+2\alpha z+2\beta z^{3/2}\sin (3\theta )\\ +2\gamma z^2+& 2\delta z^{5/2}\sin (3\theta )),\end{array}$$

$$\begin{array}{rl}\sqrt{z}\cos (\theta )& =\frac{\sqrt{3}(s_{23}-s_{13})}{2m_{\mathrm{res}}(m_{\mathrm{res}}-3m_{\pi })}\\ \sqrt{z}\sin (\theta )& =\frac{\sqrt{3}(s_c-s_{12})}{2m_{\mathrm{res}}(m_{\mathrm{res}}-3m_{\pi }})\\ s_c& =\frac{1}{3}(m_{\mathrm{res}}^2+3m_{\pi }^2),\end{array}$$

$$\begin{array}{rl}\alpha & =0.083\\ \beta & =0.022\\ \gamma & =0.001\\ \delta & =\mathrm{0.014.}\end{array}$$

$$\int d{\varphi }_3\cdots =\frac{1}{128{\pi }^3{M}^2}{\int }_{\mathrm{Dalitz}}d{s}_{12}d{s}_{23}\cdots .$$

$$\begin{array}{rl}\mathrm{dPS}& =\frac{1}{2m_{\mathrm{res}}{\gamma }_{\omega \to 3\pi }}\frac{1}{(2\pi {)}^3}\frac{1}{2E_1^{\star }}\times \\ & \int \frac{d^3{p}_2^{\star }}{(2\pi {)}^3}\frac{1}{2E_2^{\star }}\frac{d^3{p}_3^{\star }}{(2\pi {)}^3}\frac{1}{2E_3^{\star }}\times \\ & (2\pi )\delta (m_{\mathrm{res}}-E_1^{\star }-E_2^{\star }-E_3^{\star })\\ & (2\pi {)}^3{\delta }^3({\vec{p}}_1^{\star }+{\vec{p}}_2^{\star }+{\vec{p}}_3^{\star })\times |{\Gamma }_{\omega \to 3\pi }(s_{ij}){|}^2\end{array}$$

$$\begin{array}{rl}{\gamma }_{\omega \to 3\pi }& =\frac{1}{2m_{\mathrm{res}}}\int d{\varphi }_3|{\Gamma }_{\omega \to 3\pi }{|}^2.\end{array}$$

$$\begin{array}{rl}i\mathcal{M}& =\frac{-i{\left|\Gamma \right|}^2/\mathrm{br}}{M^2-{\bar{m}}_{\mathrm{res}}^2+iM{\gamma }_{\mathrm{tot}}}\\ {\gamma }_{\omega \to 3\pi }& =\frac{1}{2M}\int d{\varphi }_3|\Gamma {|}^2\\ {\gamma }_{\mathrm{tot}}& \approx {\gamma }_{\omega \to 3\pi }/\mathrm{br},\end{array}$$

$$\begin{array}{rl}\mathcal{Q}(M)& =\frac{1}{2}\mathrm{Im}\left[\ln (1+\int d{\varphi }_{3}i\mathcal{M})\right]\\ B(M)& =2\frac{d}{dM}\mathcal{Q}(M).\end{array}$$

$$A(M)$$

Full text

I introduction

The problem posed by heavy-ion collisions is to deduce physical properties of the created hadronic matter based on information of the observed particles. Since most of the particles detected are connected with the system after the freeze-out stage, precise modeling at different levels is required to reconstruct the cooling history of the originally produced hot and dense medium.

Pion production is a dominant feature in heavy-ion collisions. The experimental data on momentum distributions of pions and other identified particles present a handle for discerning particles of different momenta. In many cases the hadronic spectra are well reproduced by simple thermodynamical fits Sollfrank:1990qz . However the situation is more complicated for pions. In particular the data exhibit an unexpected enhancement of pions at low transverse momentum ($p_{T}$) Abelev:2013vea . Presumably multiple mechanisms contribute to the observed spectrum. These include collective flow Huovinen:2001cy , resonance decay Sollfrank:1990qz , influence of the medium Shuryak:2002kd ; Rapp:2003ar , and non-equilibrium effects Petran:2013lja ; Begun:2014rsa ; Begun:2015ifa .

Therefore it is no longer sufficient to perform data fitting within the framework of a single mechanism. Instead a detailed examination of each effect is required, with its consistency with known hadron physics and symmetries of QCD inspected. Only then one can isolate the (possibly dominant) effect from the thermal medium and non-equilibrium dynamics, and eventually arrive at an internally consistent picture for heavy ion collisions.

In this work we focus on the $p_{T}$-spectra of pions from resonance decays. It is known that a good description of resonances is essential for understanding the soft part of the spectrum. An extensive analysis based on statistical models was presented in Ref. Sollfrank:1990qz . However, in this and other studies Chojnacki:2011hb ; Torrieri:2004zz , resonance decay dynamics has been neglected.

The purpose of this paper is to formulate a theoretical framework for incorporating hadron dynamics into the analysis, applicable to a general N-body decay. The detail of this framework is discussed in Sec. 2. In Sec. 3, we apply the formalism to study the three-body decay of $\omega\rightarrow 3\pi$. In Sec. 4 we present the conclusion.

II momentum distributions of decay particles with decay dynamics

II.1 differential phase space

The first question to address is to determine the distribution $dn_{1}^{\rm dec}/d^{3}p_{1}$ of a particular decay particle $1$ (in this case a pion) from a given distribution $dn_{\rm res}/d^{3}p_{\rm res}$ of the resonance. A detailed account of this problem is given in textbooks textbook1 ; Byckling:1971vca . The application in heavy-ion collisions is discussed in Ref. Sollfrank:1990qz ; Gorenstein:1987zm . Here we briefly review the key steps of the calculation to establish our notations.

The pion momentum spectrum from resonance decay is given by

[TABLE]

Here ${\rm br}$ is the suitable branching ratio for the decay. The differential phase space function ${\rm dPS}(\vec{p}_{\pi}^{\,\star})$ is a key quantity of this study and will be addressed in detail. Throughout this work, variables in the resonance rest frame are denoted by $\star$, while those without are in the rest frame of the medium. The momentum variable $\vec{p}^{\,\star}_{\pi}$ should be understood as a function of $\vec{p}_{\rm res}$ and $\vec{p}_{\pi}$. The explicit expression is easily obtained by invoking Lorentz invariance of $p_{\pi}\cdot p_{\rm res}$, which dictates

[TABLE]

In this study we shall only consider resonances from a static thermal source. Hence we put

[TABLE]

for a mesonic resonance of degeneracy $g_{\rm res}$ at finite temperature ($T=155\,$ MeV is chosen) and vanishing chemical potentials.

The function ${\rm dPS}$ involves an integral over the phase space of other decay particles and the decay matrix element in the resonance rest frame. For an N-body decay the general definition reads Sollfrank:1990qz ; textbook1 ; Byckling:1971vca ; Kumar:1970cr ; Becattini:2009sc

[TABLE]

The normalization of this function is chosen such that

[TABLE]

A common approximation made by thermodynamical models when calculating this quantity is the assumption of isotropic (structureless) decay Sollfrank:1990qz . This amounts to replacing the decay matrix element $\Gamma$ with the identity $I$ and hence ${\rm dPS}\rightarrow{\rm dPS}^{(0)}$:

[TABLE]

Here we have introduced the N-body Lorentz invariant phase space $\phi_{N}$:

[TABLE]

To clarify the physical meaning of the differential phase space function ${\rm dPS}$, we explicitly work out the cases for isotropic two- and three-body decay. Starting with the two-body case:

[TABLE]

The integrals can be explicitly worked out:

[TABLE]

[TABLE]

where $\lambda(x,y,z)$ is the Källén triangle function Byckling:1971vca

[TABLE]

and $q$ is the three-momentum of the decay particle in the resonance rest frame

[TABLE]

Finally we arrive at the well-known result

[TABLE]

which may be alternatively obtained by inspection of the quantity $dN^{\rm dec}_{1}/d^{3}p_{1}^{\star}$ for a spherically symmetric two-body decay Sollfrank:1990qz ; Gorenstein:1987zm .

The generalization to the case of three-body decay is straightforward:

[TABLE]

Where the integrals can again be explicitly calculated:

[TABLE]

[TABLE]

Finally the differential phase space for isotropic three-body decay reads

[TABLE]

matching the result in Ref. Sollfrank:1990qz .

The assumption of isotropic decay can be justified in some cases when the matrix element is a scalar (e.g. $\sigma\rightarrow\pi\pi$ decay via $\mathcal{L}_{\rm int}=-g\,\sigma\pi\pi$) or depends only on $s=P^{2}$. However, as we shall demonstrate in the example of the 3-body decay $\omega\rightarrow 3\pi$, this approximation is problematic especially for soft pions.

III case study: $\omega\rightarrow 3\pi$

III.1 ${\rm dPS}$ and $p_{T}$-spectra

Multiple hadron models Klingl:1996by ; Leupold:2008bp ; Niecknig:2012sj ; Danilkin:2014cra are available to describe the decay of $\omega$-meson to three pions. The mechanisms involved are the Gell-Mann, Sharp and Wagner (GSW) process GellMann:1962jt ; Wess:1971yu ($\omega\rightarrow\rho\pi\rightarrow\pi\pi\pi$) and possibly a direct process Klingl:1996by . Here we employ the model of Ref. Danilkin:2014cra which is a dispersive study based on the isobar decomposition and subenergy unitarity. Accordingly, the decay matrix element is given by

[TABLE]

where

[TABLE]

and

[TABLE]

all subjected to the kinematic constraint

[TABLE]

The factor $\mathcal{P}$ due to anomalous coupling, which is common to all models of the decay, dominates the properties of the matrix element. On the other hand, differences among models are limited to the different recipe for the amplitude function $C_{V\rightarrow 123}$. We have numerically confirmed that different choices of the latter only lead to minimal changes to our subsequent results111Deviation among models becomes appreciable when we dial up the meson width.. The detail of the amplitude function employed in this work is given in Ref. Danilkin:2014cra and the expression is reproduced here for convenience:

[TABLE]

where

[TABLE]

with the model parameters

[TABLE]

The normalization $\mathcal{N}$ is chosen such that the integrated width matches the experimental value.

The decay matrix element (18) is a function of phase space variables and thus cannot be pulled out of the integral in Eq. (4). In fact, the integration over the phase space of other decay particles is equivalent to the integration over the region of Dalitz decay:

[TABLE]

To examine the effects of decay dynamics on ${\rm dPS}$, we numerically compute the integral in Eq. (4) together with the matrix element in Eq. (18). Explicitly, the integral reads

[TABLE]

with

[TABLE]

In this implementation the integral in Eq. (27) for $\gamma_{\omega\rightarrow 3\pi}$ is by construction given by the experimental value $\gamma_{\rm exp}=(7.57\pm 0.13)$ MeV Olive:2016xmw . The result for the ${\rm dPS}$ function is shown in Fig. 1. The key observation is that the full ${\rm dPS}$ function is substantially suppressed at low momenta. This is a direct consequence of the factor $\mathcal{P}$ in the decay matrix element. Note that both functions are normalized to unity when an integration over $\int d^{3}p_{1}^{\star}$ is performed. Furthermore, the cut at $p_{1}^{\star}\approx 0.33$ GeV simply reflects the kinematical situation where all three decay particles are collinear, with particles 1 going one way and the others going the opposite.

Next we study the influence of dynamics on momentum distributions. To construct the conventional $p_{T}$-spectra studied in experiments, we perform an additional integration over the rapidity range on Eq. (1). At this stage, even the kinematic cuts from a specific experimental analysis can be easily implemented. This may be essential for a realistic comparison with data Broniowski:2003ax . For simplicity we shall skip it here and consider only the mid-rapidity ($y=0$) and rapidity-integrated $p_{T}$-spectra of pions from the decay of $\omega$-meson. These are shown in Fig. 2.

It is somewhat surprising that despite the essential differences in the ${\rm dPS}$ functions (Fig. 1), the deviations in $p_{T}$-spectra yielded by different treatments of decay dynamics are rather mild (Fig. 2). In both spectra we see that the correction from dynamics is limited to the low-$p_{T}$ region. Moreover deviation from the isotropic case is more visible in the mid-rapidity spectrum than in the integrated one. This is expected as features of ${\rm dPS}$, and hence the influence of dynamics, will be washed out when all the momentum variables are integrated over.

Previous study Sollfrank:1990qz suggests that the $\omega$-meson is one of the major sources of low-$p_{T}$ pions. In this work, we find that imposing the correct decay dynamics leads to a reduced contribution. More unexpected effects can come from other resonances and other types of decay. Further work is required to revise the resonance decay contribution to various physical observables based on the input of robust hadron physics.

III.2 effects of finite width

Another consequence of hadron dynamics is the existence of resonance widths. These can be systematically included using the S-matrix formalism of Dashen, Ma and Bernstein Dashen:1969ep . In this model, the Feynman amplitude for $3\pi\rightarrow 3\pi$ scattering can be constructed from the decay matrix element in Eq. (18) via an S-channel resonance exchange process

[TABLE]

where $M$ is the invariant mass, $\bar{m}_{\rm res}=0.783$ GeV is the pole mass of $\omega$-meson and ${\rm br}=0.892$ is the branching ratio. In this model we do not explicitly calculate the other partial widths of $\omega$-meson, but simply prescribe a factor of $1/{\rm br}$ on $\gamma_{\omega\rightarrow 3\pi}$ to obtain the total width $\gamma_{\rm tot}$.

With the model scattering amplitude, the generalized phase shift function $\mathcal{Q}(M)$ and the effective spectral function $B(M)$ can be computed as follows How ; Fornal:2009xc ; Tayl ; kappa :

[TABLE]

These functions are displayed in Fig. 3. Since the width of $\omega$-meson is small, the phase shift function indeed behaves like a theta-function $\pi\times\theta(M-\bar{m}_{\omega})$, and the corresponding effective spectral function $B(M)$ is in practice well approximated by an energy-dependent Breit-Wigner function $A(M)$

[TABLE]

For the momentum spectrum, the influence of resonance width enters via rho-pt

[TABLE]

We perform analogous numerical integration on Eq. (31) as in the zero-width case. Here we pick $\Lambda=0.88\,{\rm GeV}$, which is how far we estimate the model given in Eq. (28) to hold. The results are shown in Fig. 2. Only moderate differences are found between this and the zero-width case. The vacuum width of $\omega$-meson is so narrow that the zero-width approximation is justified.

On the other hand, substantial broadening of the $\omega$-width in the medium is suggested by model studies Rapp:2000pe ; Schneider:2001zt ; Eletsky:2001bb ; Lutz:2001mi ; Riek:2004kx ; Martell:2004gt . To investigate the dependence on resonance width of our previous results, we simply scale the coupling constant in the matrix element in Eq. (18) by a factor of $10$. The resulting phase shift and the effective spectral functions are shown in Fig. 3.

In this particular model an increase of width leads not only to the broadening of the effective spectral functions, but also a reduction of their normalization although they are both normalized to unity in the limit of zero width. It also tends to reduce $\mathcal{Q}(M)$, and as $\gamma\rightarrow\infty$, the function will eventually approach the limit of $\pi/2$ at large invariant masses. As expected, this also leads to an overall drop in the magnitudes of the calculated $p_{T}$-spectra (Fig. 2).

Another important consequence of the broadening of resonance is the enhancement of low-$p_{T}$ pions from the use of the effective spectral function $B(M)$ over that from the use of $A(M)$. This can be traced to the greater value of $B(M)$ at low invariant masses, which translates into a larger contribution to the soft part of $p_{T}$-spectra due to the lesser Boltzmann suppression. A recent discussion of this effect is presented in Ref. rho-pt for the case of $\rho$-meson. In the case of $\omega$-meson, this effect becomes appreciable only if we increase the physical width by a factor of $10$.

IV conclusion

This study set out to investigate how details of hadron physics can modify heavy-ion collision observables. To this end, we formulate a theoretical framework for incorporating resonance decay dynamics into the analysis.

As an application we consider the decay $\omega\rightarrow 3\pi$, and find that imposing the anomalous coupling feature of the decay matrix element leads to a reduction of low-$p_{T}$ pions compared to the structureless decay treatment.

In many statistical models, the isotropic decay approximation is adopted instead of the full dynamics. The validity of this approximation has to be inspected case by case. Since multiple mechanisms are at work to produce the observed $p_{T}$-spectra, it is necessary to perform a detailed examination of each effect based on existing knowledge of hadron physics. In the current study, the finding of reduced low-$p_{T}$ pions from $\omega$-meson makes room for other important effects such as higher order N-body decay, influence of thermal medium and non-equilibrium effects to explain the unexpected enhancement of soft pions observed in the experiment.

It would be interesting to assess the effects of other important features of the strong interaction on these observables. In particular, coupled-channel dynamics Kaminski:2006yv ; Guo:2010gx ; Tornqvist:1995kr and the existence of complex objects like hadronic molecules and other exotics eric:exotica ; Amsler:2004ps . Such research is currently underway.

Acknowledgements.

I thank Francesco Becattini and Eric Swanson for carefully reading the manuscript and for giving constructive comments. I am also grateful to Thomas Klähn for stimulating discussions and to Bengt Friman, Pasi Huovinen, Krzysztof Redlich and Chihiro Sasaki for the supportive collaboration. This work was partly supported by the Polish National Science Center (NCN), under Maestro grant DEC-2013/10/A/ST2/00106 and by the Extreme Matter Institute EMMI, GSI.