# Thermal contribution of unstable states

## Abstract

Within the framework of the Lee model, we analyze in detail the difference between the energy derivative of the phase shift and the standard spectral function of the unstable state. The fact that the model is exactly solvable allows us to demonstrate the construction of these observables from various exact Green functions. The connection to a formula due to Krein, Friedal, and Lloyd is also examined. We also directly demonstrate how the derivative of the phase shift correctly identifies the relevant interaction contributions for consistently including an unstable state in describing the thermodynamics.

## 1 Introduction

Formal treatment of the interactions in a gas of particles at finite temperature is an important topic in thermal field theory [1, 2, 3]. In particular, a consistent description of the unstable states is imperative for understanding the hadron gas, see, e.g., Refs. [3, 4, 5, 6]. Some questions of interest include: How are the bulk properties of the medium, such as pressure and energy density, affected by unstable particles? What are the effective ways to take these into consideration? What insights can be gained from comparing different approaches, e.g., the standard imaginary time formalism and those based on a virial expansion? [5, 7, 8, 9, 10, 11, 12, 13] In this work we tackle these issues using an effective Hamiltonian approach. Besides an intuitive modification driven by the spectral function of the unstable particle (the resonant contribution), we shall demonstrate how the very presence of an interaction modifies the 2-body states composed of the stable particles (the nonresonant contribution). The sum of these modifications recovers a well-known result [5, 14, 15], according to which the derivative of the scattering phase shift can be identified as the density of state for computing the partition function.

Many thermal models have the widths of the resonances implemented but the nonresonant interactions are neglected. This can lead to misleading results when interpreting the contribution from an interaction channel [16, 17]. An illustrative example is the pion-pion scattering in the \(I=0\) channel, in which the famous \(f_0(500)\) resonance, a.k.a. the \(\sigma \)-meson, is involved. The empirical phase shift, analyzed by the chiral perturbation theory [18], reveals that there are substantial (effective) repulsive corrections coming from the exchange interactions in the t- and u-channels. Additional cancellation also comes from the \(I=2\) channel. A model that corrects only for the width of the resonance is incapable of handling these effects. Some models try to remedy this by introducing extraneous repulsive forces, e.g., via an excluded volume. This, however, will generally lead to a model which contradicts the known phase shift [19].

In this study we consider a system of stable particles “\(\pi \)” (the pions) and unstable particle “\(\rho \)” (the \(\rho \)-mesons).^{1} Each \(\rho \) can decay into two \(\pi \)’s via the interaction \(\rho \rightarrow \pi \pi \). Chains of interactions, e.g., \(\pi \pi \rightarrow \rho \rightarrow \pi \pi \), are also included. The fact that only two types of particles are considered drastically simplifies the discussion, but the main non-trivial features are kept.

*E*and \(E+dE\). The spectral function can be calculated from the imaginary part of the \(\rho \)-propagator. See Sect. 3 for details. For a narrow-width state this can be approximated by the Breit-Wigner (BW) formula [38, 39],

*a*, which depends on its mass (\(m_a\)) and degeneracy. The dependence on temperature

*T*is understood. Even if the interaction is switched on, Eq. (2) can still be used as an estimate of the pressure for a narrow-width \(\rho \). This is the fundamental premise of the hadron resonance gas (HRG) model [45, 46]: contribution of resonances to the thermodynamics is approximated by an uncorrelated gas of zero-width particles.

*g*(or \(\Gamma _{\rho \pi \pi }\)) is given by (S-matrix scheme):

- (i)
Observe that there is no explicit \(\rho \) contribution in Eq. (5): The pressure is determined based on the scattering information of the asymptotic (stable) states alone. In fact, it is not compulsory to introduce the \(\rho \) state as an explicit degree of freedom. Its presence is encoded in the phase shift. This point will be made clear by direct model calculations.

- (ii)Equations (4) and (5) reduce to the free gas result (2) in the limit of \(g \rightarrow 0\) ( or \(\Gamma _{\rho \pi \pi } \rightarrow 0\) ).
^{2}For the latter, we have$$\begin{aligned} \delta _{\pi \pi }(E)&\rightarrow \pi \, \theta (E-m_\rho ) \nonumber \\ B(E)&\rightarrow 2 \pi \, \delta (E-m_\rho ). \end{aligned}$$(6) - (iii)
Generally, \(B(E) \ne A_\rho (E)\), and Eqs. (4) and (5) are thus different. Systems which show substantial deviation are plenty: In addition to the case of \(f_0(500)\) mentioned, nonresonant contribution is found to be important in the study of \(\kappa (700)\) [16, 17], the \(N^*\) and \(\varDelta \) resonances [53, 54], the \(S=-1\) hyperons [55], etc. It is also the case for the recently discovered

*X*,*Y*,*Z*states [56, 57, 58]. As shown in a recent work [59], the state*X*(3872) makes only a small contribution to the thermodynamics due to nonresonant effects. For what concerns other states, future studies based on Eq. (5) are needed.

The paper is organized as follows: In Sect. 2 the details of the Lee model are presented. Then, in Sect. 3, the spectral functions for both \(\rho \) and \(\pi \pi \), and the phase shift are introduced. Here the important Eq. (7) is derived. In Sect. 4 the thermodynamic properties of the system are determined analytically, with special focus on the pressure with its various contributions. A numerical example shows that \(\varDelta P_{2\pi }\) can be sizable and in general should not be neglected. Finally, discussions and conclusions are given in Sect. 5.

## 2 The Lee model

^{3}Introducing the basis states in the center-of-mass (CM) frame:

*q*is the momentum label for the two-pion state

*V*describes the coupling of \(\rho \) with the \(\vert q \rangle \) states

*L*being the size of the box. The coupling \(g_\mathrm{eff}\) is generally

*q*-dependent. The full Hamiltonian then reads

*E*dependence unless there is a chance for confusion.

*S*is extracted from the (lower-right) \(N_q \times N_q\) block of \(\hat{S}\). In addition, we introduce an operator \(\mathscr {K}\), due to Krein, Friedal and Lloyd (KFL) [63, 64], defined as the difference of the spectral functions

*B*. The latter is defined as

## 3 Phase shift and effective spectral function

In the Lee model various theoretical quantities, e.g. the propagator and the self-energy of \(\rho \), can be analytically computed. It is a useful exercise to revisit these formulas as it helps to build an understanding the physical content of the KFL operator \(\mathscr {K}\) and the derivative of the phase shift.

### 3.1 The \(\rho \)-propagator

*G*(

*E*), as

### 3.2 The 2-pion propagator

*G*(

*E*):

*V*is off-diagonal. The second term gives

### 3.3 Effective spectral function

We are now ready to examine the expressions of the phase shift \(\mathscr {Q}\), the effective spectral function *B* and the operator \(\mathscr {K}\) in the context of the Lee model.

^{4}

*B*is known to be related to the interacting part of the density of state. In the Lee model, it is

*B*extracts the physical content, including the contribution from the unstable state \(\rho \), of the system. From the first line of Eq. (40), we see that

*B*includes the contribution from the full spectral function \(A_\rho \) and the 2-pion nonresonant interaction \(\sum _q \varDelta A_{2 \pi }(E;q) \). The second line offers an alternative, but equivalent interpretation:

*B*includes the contribution from the bare-\(\rho \), together with the interaction contribution contained in \(\mathrm{tr} \, \mathscr {K}\). The latter includes contributions from the change in the energy spectra of both the \(\rho \) and the 2-pion states.

*B*would give

*B*is used.

*l*is the relative orbital angular momentum between the pions. Note that terms that are proportional to \(\mathrm{Im} (\Sigma _\rho )\) are subleading relative to \(\frac{\partial }{\partial E}\mathrm{Im} (\Sigma _\rho )\), as the latter is of \(\mathscr {O}(q^{2l-1})\). This should be distinguished from the residual effect of the resonance width at threshold, which is of \(\mathscr {O}(q^{2l+1})\). Thus, even an energy dependent Breit-Wigner model can not capture the effect of this term. In addition, the scattering lengths are well constrained by the chiral perturbation theory, and indeed the stated form of

*B*was derived [65].

### 3.4 Numerical results

- (i)
Construct the matrices

*H*and \(H_0\). - (ii)For each
*E*, construct the matricesand invert$$\begin{aligned} M_1(E)&= E \, I - H_0 + i \varepsilon \, I \nonumber \\ M_2(E)&= E \, I - H + i \varepsilon \, I; \end{aligned}$$(43)$$\begin{aligned} \hat{G}_0(E)&= \{ M_1(E)\}^{-1} \nonumber \\ \hat{G}(E)&= \{ M_2(E)\}^{-1}. \end{aligned}$$(44) - (iii)Extract the quantities of interest. For example, the propagators are obtained fromwhere \(q_i\) is the discrete momentum of the$$\begin{aligned} G_\rho&= \hat{G}(1,1) \nonumber \\ G^0_\rho&= \hat{G}_0(1,1) \nonumber \\ G_{2\pi }(q_i)&= \hat{G}(i,i) \nonumber \\ G^0_{2\pi }(q_i)&= \hat{G}_0(i,i). \end{aligned}$$(45)
*i*-th grid. - (iv)The spectral functions
*A*’s can be obtained from the propagators by simply taking the imaginary part. To calculate the*B*function, one possible method is$$\begin{aligned} B(E)&= -2 \, \mathrm{Im} \, \mathrm{tr} \, \left( G(E) - G_0(E) \right) \nonumber \\&\quad + 2 \, \mathrm{Im} \, \hat{G}_0(1,1). \end{aligned}$$(46)

^{5}It can be motivated in a quantum field theory with non-local interactions or dressing of vertex [66, 67, 68, 69]. For what concerns us here it parametrizes the finite-size effects of the hadrons. See Ref. [70] for a rigorous treatment within the QFT framework.

The numerical results for \(\mathscr {Q}\) and various spectral functions are shown in Fig. 1. We have computed the phase shift on the momentum grid, (\(1+N_q=800\); \(L\approx 800\) fm; \(\varepsilon =0.01\) GeV), in two ways: one uses \(G_\rho \) via Eq. (36) (grey squares), the other uses the scattering matrix *S* via Eq. (48) (black circles). Both results agree quite well with the continuum limit, although they appear to have different convergence property.^{6}

A key feature to note is the apparent shift of the strength of *B*, compared to \(A_\rho \), towards lower energies. This effect originates from the nonresonant scattering term \(\sum _q \varDelta A_{2 \pi }(E;q)\), and is needed, in addition to \(A_\rho \), for a complete description of the interacting system.

### 3.5 Phase shift from S-matrix

*I*with 1 in the above.

Furthermore, we add that from Cayley Hamilton theorem, the determinant of \(\hat{t}\) essentially vanishes. In fact, all eigenvalues of \(\hat{t}\) are zero, except one, which is \(\mathrm{tr} \, \hat{t}\). This is another way to understand why the approximation made in Eq. (48) is justified. The full implication of this result is not yet completely clear, and will be explored in a future work.

## 4 Thermodynamics

The change in the density of state due to interactions, as revealed by the KFL \(\mathscr {K}\) operator or the *B* function, is the key input for the S-matrix formulation of statistical mechanics [5, 6]. The approach is based on the method of cluster expansions, and for the second virial coefficient the result is exact. We retrace a few basic steps in relating the scattering phase shift to the thermal partition function.

*Z*reads

^{7}

*E*is the energy of the relative motion

*D*(

*E*), which is the change in the density of state due to interactions.

^{8}

*D*(

*E*) to use. The S-matrix formulation of statistical mechanics by Dashen

*et al.*[5] dictates the choice of \(D(E) \rightarrow B(E) = 2 \frac{\partial }{\partial E} \mathscr {Q}(E)\). This means that we compute the thermodynamic pressure via

*D*(

*E*), though with a different interpretation. For example, one can choose instead \(D(E) \rightarrow \mathrm{tr} \mathscr {K}\), and in this case the contribution of \(A^0_\rho \) needs to be added separately as

The various partial pressures are shown in Fig. 2. Due to the contribution from the nonresonant \(\pi \pi \) state, the pressure based on *B* is substantially larger than the one based on \(A_\rho \) alone. We stress that only the former one gives a consistent description of the thermodynamics, as can be verified by the direct construction of the partition function from the eigenvalues of the Hamiltonian (black circles). Therefore Eq. (5) should be used instead of Eq. (4).

These observations are in accord with the previous analysis based on a \(\Phi \)-derivable approach [15]. The *B*-function-based description requires only the input from the scattering of asymptotic states. This underlines an important concept in the formulation: In computing the density of states it is not mandatory to introduce the unstable state as an explicit degree of freedom. For approaches that use only stable states as degrees of freedom, such as an effective field theory where resonances are dynamically generated [74], the same density of states would be obtained as long as the phase shifts agree. And when an empirical phase shift \(\mathscr {Q}(E)\) is used, the function *B*(*E*) becomes model independent, while the splitting into \(A_\rho (E)\) and \(\sum _q \varDelta A_{2 \pi }(E;q)\) is model dependent. The Lee model studied here provides a clear picture of such a splitting, and demonstrates how an unstable state should be included in the description of the thermodynamics.

## 5 Conclusion

In the context of the Lee model we have clarified the relation between the energy derivative of the phase shift and the spectral functions of the degrees of freedom composing the system. We have also illustrated how these quantities enter the thermal description of the system via the S-matrix formulation of the statistical mechanics. This consolidates our understanding of the connection between this and the standard approach based on thermal Green functions. In particular, we have shown that the thermodynamic trace requires the inclusion of the nonresonant contribution (\(\varDelta A_{2 \pi }\)), in addition to, and independent of, the effect coming from the width of the unstable state \(\varDelta A_{\rho }\).

Besides acting as an effective density of state, an alternative interpretation of the energy derivative of the phase shift is the concept of time delay [75, 76]: particles spend longer or shorter in the interaction region due to the attractive or repulsive nature of the interaction. In the contexts of transport models and resonance identification, it was argued [77, 79, 80, 81, 82] that such a time delay, instead of the inverse width \(1/\Gamma (E)\), should be used to measure the life-time of a resonance. A related problem is the study of the survival probability of an unstable state. According to Fonda et al. [40], the standard exponential decay law is valid only in the limited case of an energy-independent Breit-Wigner distribution. Re-scattering effect, apparently related to \(\varDelta A_{2 \pi }\), would lead to non-exponential behavior [83]. A clearer theoretical understanding of *B* and \(A_\rho \) could provide further insights into these topics.

So far we have restricted our discussion to Fock space up to two body. It will be extremely interesting to extend the scheme to include multi-channel and multi-body scatterings [13, 21, 55, 84, 85], and understand how these interactions would influence thermodynamic quantities. This can be a useful framework to analyze the observables in Heavy Ion Collision experiments, such as hadron yields and the momentum distributions of light hadrons [60, 86, 87, 88]. We defer this more challenging problem to future research.

## Footnotes

- 1.
These notions obviously mirror the case of a thermal hadron gas with pions and \(\rho (770)\)’s. However, our discussion generally applies to any thermal system with unstable states.

- 2.
The \(g=0\) limits of \(A_\rho (E)\) and

*B*(*E*) are not well-defined. - 3.
In the following we measure energy with respect to \(2 m_\pi \), and the nonrelativistic dispersion relation \(\epsilon (q) = {q^2}/{m_\pi }\) is used. We also choose to present our model in a discretized form. This prepares for the later numerical treatment of solving the system on a momentum grid. [61]. It is easy to go to the continuum by taking \( \sum _q C(q)^2 \, (\cdots ) \rightarrow \int \frac{d^3 q}{(2 \pi )^3} \, (\cdots )\).

- 4.
One can obtain the phase shift directly from the S-matrix. See Eq. (48).

- 5.
The imaginary part is finite even without a regulator. We checked that its value is only slightly modified by the form factor.

- 6.
- 7.
For simplicity we neglect corrections due to quantum statistics and focus on the \(\mu =0\) case.

- 8.
In fact, computations involving the difference between the fully interacting system and the free case, e.g., \(G(z)-G_0(z)\), are numerically more stable [72]. This also opens up the possibility of solving the system with other numerical techniques, such as the use of a harmonic oscillator basis in the expansion [73].

## Notes

### Acknowledgements

PML thanks Eric Swanson for stimulating discussions. He also acknowledges fruitful discussions with Hans Feldmeier, Bengt Friman, and Piotr Bozek. The authors are also grateful for the constructive conversations with Wojciech Broniowski, Wojciech Florkowski, and Stanislaw Mrowczynski. PML was partly supported by the Polish National Science Center (NCN), under Maestro Grant No. DEC-2013/10/A/ST2/00106 and by the Short Term Scientific Mission (STSM) program under COST Action CA15213 (reference number: 41977). FG acknowledges financial support from the Polish National Science Centre (NCN) through the OPUS project no. 2015/17/B/ST2/01625.

## Supplementary material

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