Contribution of Kaon component in viscosity and conductivity of hadronic medium

# Contribution of Kaon component in viscosity and conductivity of hadronic medium

## Abstract

The two-point correlation functions of kaonic viscous stress tensors and electro-magnetic currents have been calculated respectively to estimate the contributions of the strange sector in the shear viscosity and electrical conductivity of hadronic medium. In the one-loop correlators, kaon propagators contain a non-zero thermal width that leads to non-divergent values of transport coefficients. With the help of effective Lagrangian densities of strange hadrons, we have calculated in-medium self-energy of kaon for different possible mesonic loops, whose imaginary part provide the estimation of kaon thermal width. It is observed that near the quark-hadron transition temperature, the contribution of kaons to shear viscosity is larger and increases faster with temperature than pionic contribution. In case of electrical conductivity the trend due kaon component appears to be opposite in nature with respect to pion component.

## I Introduction

The relativistic heavy ion collision at Relativistic Heavy Ion Collider (RHIC) (1); (2); (3); (4) and Large Hadron Collider (LHC) (5); (6); (7) produce a very hot and dense matter with quarks and gluons as its elementary constituents, this state of matter is known as quark-gluon plasma (QGP). It behaves like perfect fluid having very low shear viscosity to entropy ratio (1); (2); (3); (4). Within the framework of ADS/CFT correspondence (8) Kovtun, Son and Starinets (KSS) (9), conjectured the lowest bound of shear viscosity () as, , where is the entropy density. One of the major objectives of these experiments is to understand the quark-hadron transition in the early universe.

In this context it is important to understand how the matter, created in the experiments of heavy ion collision, evolves in space and time. The relativistic viscous hydrodynamics is an efficient tool to simulate this evolution, however, various transport coefficients such as shear and bulk viscosities are required as inputs to these simulations in addition to initial conditions and equation of state (EoS). These transport coefficients of quark and hadronic matter can be calculated microscopically by using effective interaction models and this is one of the active field of contemporary research in the heavy ion physics. It is important to investigate the dependence of these transport coefficients on temperature of the medium to characterize the state of the matter. In order to describe the elliptic flow of hadrons in ultra-relativistic heavy-ion collisions at RHIC and LHC (10); (11), it is crucial to know the temperature dependence of .

There is a large number of literature on calculation of the shear viscosity of hadronic matter (12); (13); (14); (15); (16); (17); (18); (19); (20); (21). Some effective QCD model calculations (22); (23); (24); (25); (26); (27); (28); (29); (30) gave the estimation of in both hadronic and quark temperature domain. There are also results, based on numerical simulation, addressed in Refs (31); (32); (33); (34).

In this work we also estimate another transport coefficient, the electrical conductivity , which has rich phenomenological and theoretical implications in heavy ion physics. The electrical conductivity can be associated to the rate of soft dilepton production (35) and photon multiplicity near zero transverse momentum (14); (50). It also controls how rapidly the magnetic fields, which may produced in heavy ion collisions, will decay with time (36).

Several authors (37); (38); (24); (39); (40); (41); (42); (43); (44); (45); (46); (47); (48); (49); (50); (51); (52); (53); (54); (55) have calculated in different ways such as by using - chiral perturbation theory (50); (14), numerical solution of Boltzmann transport equation (37); (38), dynamical quasi-particle model (24), off-shell transport model (39); (40), techniques of holography (41), Dyson - Schwinger approach (42) and lattice gauge theory (43); (44); (45); (46); (47); (48); (49). These literature covers a wide range of numerical values of . Among them Refs (39); (24); (50); (14); (51); (52) have observed the decreasing nature of in temperature domain relevant for hadronic phase and increasing trend in the temperature domain relevant for QGP phase (24); (37); (38); (39); (41); (49); (53). However, it has shown in the Refs. (41); (54); (55) that increases with in hadronic phase. The Uncertainty appears not only in nature but also in their numerical values, with a range to for hadronic phase.

The estimates from earlier investigations on transport coefficients of hadronic matter do not converge which pave way for further investigation. In this work we have mainly focused on the contribution of K meson component to shear viscosity and electrical conductivity of hadronic matter. These transport coefficients are estimated by using their standard expressions, obtained from relaxation time approximation (RTA) or Kubo formalism with one-loop diagram, where the relaxation time or thermal width is determined from the interaction Lagrangian density based on effective hadronic model. Owing to the optical theorem of thermal field theory, we have calculated kaon thermal width from the imaginary part of kaon self-energy for different possible mesonic loops.

The paper is organized as follows. In the next section, we have addressed the formalism part, which reveals the expressions for shear viscosity and electrical conductivity of kaon component in term of its thermal width. This section also includes the derivation of kaon thermal width from the in-medium kaon self-energy. The numerical results on these transport coefficients for kaon component have been discussed in section III. Section IV is devoted to summary and conclusion.

## Ii Formalism

Let us consider a hot mesonic matter, where pion and kaon are main constituents of the medium. Here, our main focus is on the constituent of strange sector meson - kaons. First we concentrate on shear viscosity.

### ii.1 Shear viscosity

Let us start with the viscous-stress tensor for kaonic medium to construct the spectral density

 AηK=∫d4xeiq⋅x⟨[πij(x),πij(0)]⟩β , (1)

where for stands for the thermal ensemble average of ; . With the help of the Kubo formalism (56); (57), one can relate this spectral density with the shear viscosity () of kaonic medium as (14)

 η=120limq0,q→0AηKq0 . (2)

Following traditional quasi-particle method of Kubo framework (14); (15); (58); (59), the simplest one-loop expressions of Eq. (2) for kaon () is:

 ηK=βIK30π2∫∞0dkk6ωKk2ΓKnk(ωKk){1+nk(ωKk)} , (3)

where is the Bose-Einstein (BE) distribution of kaon with energy . Fig. 1(a) represents the schematic one-loop diagram, derived from two point function of viscous stress tensor in the kaonic medium. in Eq. (3) is the thermal width of kaon in the medium, which is considered in the kaon internal lines of Fig. 1(a). Hence, these internal lines are drawn in double line pattern. This adoption of finite thermal width is a very well established technique (14); (15); (58); (59), which is generally used in Kubo approach to get a non-divergent value of the shear viscosity coefficient. This treatment is equivalent to quasi-particle approximation. Again, this one-loop expression of shear viscosity from Kubo approach (14); (15); (58); (59) exactly coincides with the expression originating from the relaxation-time approximation of the kinetic theory approach (16); (60); (61), which can be derived as follows. The dissipative part of energy momentum tensor, responsible for shear viscosity coefficient, can be expressed as

 TμνD = η(Dμuν+Dνuμ+23Δμν∂σuσ) (4) = ∫d3k(2π)3kμkνωKkδn ,

where

 Dμ=∂μ−uμuσ∂σ, and Δμν=uμuν−gμν . (5)

It is interesting to note that Eq. (4) relates the collective fluid four velocity, to the four momentum of elementary constituent, through the non-equilibrium distribution function , which is assumed to be slightly shifted from equilibrium distribution function . can be taken as,

 δn=Ckμkν(Dμuν+Dνuμ+23Δμν∂σuσ)nk(1+nk) , (6)

The Boltzmann equation in RTA can be written as,

 kμ∂μnk=ωKkτKδn , (7)

which fixes the value of as . Using this expression of in Eq. (6) one can obtain Eq. (3) through Eq. (4) in RTA approach where we have to identify the kaon relaxation time . Hence, the thermal width or relaxation time of kaon plays a vital role in determining the numerical strength of shear viscosity for kaon component.

Owing to the Cutkosky rule, we will estimate the thermal width from the imaginary part of kaon self-energy at finite temperature. Here, we have considered and loops for calculating kaon self-energy, which is shown in Fig. 1(b). We can write total thermal width of kaon as

 ΓK = −ImΠRK(πK∗)(k0=ωKk,k)/mK −ImΠRK(Kϕ)(k0=ωKk,k)/mK ,

where and are kaon self-energy for and loops respectively. The superscripts stands for retarded component of self-energy and indicates the external mesons. The mesons within parenthesis stand for those present in the internal lines of the kaon self-energy graphs as shown in Fig. 1(b).

The imaginary part of kaon self-energy for and loops are respectively given as:

 ImΠRK(πK∗)(k) = ∫d3l32π2ωπlωK∗uLK(πK∗)(k,l)|l0=−ωπl (9) {nl(ωπl)−nu(ωK∗u=k0+ωπl)} δ(k0+ωπl−ωK∗u) ,

and

 ImΠRK(Kϕ)(k) = ∫d3l32π2ωKlωϕuLK(Kϕ)(k,l)|l0=−ωKl (10) {nl(ωKl)−nu(ωϕu=k0+ωKl)} δ(k0+ωKl−ωϕu) ,

where , , , , are BE distribution functions of , and mesons respectively.

Using the interaction Lagrangian densities (62) :

 LintKπK∗=igKπK∗[¯¯¯¯¯K∗μ⋅→τ{K(∂μπ)−(∂μK)π}]+h.c. , (11)

and

 LintKKϕ=gKKϕ[ϕμ{¯¯¯¯¯K(∂μK)−(∂μ¯¯¯¯¯K)K}] , (12)

we obtain the and vertices as:

 L(k,l)KπK∗ = g2KπK∗[{k2+l2+2(k⋅l)} (13) −(k2−l2)2m2K∗] , for πK∗ loop ,

and

 L(k,l)KKϕ = g2KKϕ[{k2+l2+2(k⋅l)} (14) −(k2−l2)2m2ϕ] , for Kϕ loop .

The effective coupling constants and are fixed (62) from the experimental decay widths of the processes and respectively.

Now we would like to estimate the effects of kaon-nucleon interaction on the width of kaons in the thermal bath. For this purpose, our methodology will naturally proceed to calculate different possible baryon loops for Kaon self-energy. The and can be considered as possible candidates in the internal baryonic loops of kaon self energy. The Landau and unitary cut contributions to kaon self-energy can be calculated using effective and interaction Lagrangian densities. However, we don’t get any on-shell value for kaon relaxation time because kaon pole () is located neither in its Landau cut region ( to ) nor in its unitary cut region ( to ), where GeV, GeV and GeV are the masses of , and baryons respectively. Therefore, instead of this methodology of loop calculation, we have followed the following alternative way.

Let us take an experimental values of scattering length of interaction, where stands for different isospin states of system. From Refs. (63); (64); (65), using the scattering lengths fm and fm, we obtain an isospin average cross sections,

 σKN=4π∑I=0,1(2I+1)|aIKN|2/∑I=0,1(2I+1)≈ 4.7 mb , (15)

which can be used to calculate the relaxation time or thermal width of from the relation,

 ΓK=1/τK=∫d3l(2π)3 σKN vKN nN , (16)

where relative velocity of system is given by,

 vKN = [{s−(mK+mN)2}{s−(mK−mN)2}]1/22ωKωN , with s = (ωK+ωN)2, ωK={k2+mK}1/2, ωN = {l2+mN}1/2, (17)

and is the Fermi-Dirac distribution function for nucleon. Finally, for total thermal width of , we have to add contribution from Eqs. LABEL:Gam_K and 16.

### ii.2 Electrical conductivity

Similar to shear viscosity, one can derive expression for electrical conductivity, for kaon component, using the spectral density of current-current correlator

 AσK=∫d4xeiq⋅x⟨[Ji(x),Ji(0)]⟩β . (18)

 σK=16limq0,q→0AσKq0 . (19)

Within the one loop approximation the expression for is given by (50); (52)

 σK=geK3T∫d3k(2π)3ΓK(kωkK)2[nK(ωkK){1+nK(ωkK)}] , (20)

where is isospin factor for charged kaon - and . Same expression for can be obtained in RTA approach.

## Iii Results and Discussion

At first, let us consider the individual contributions from and loops to the (off-shell) thermal width of kaon as a function of the invariant mass for two different values of momentum GeV and GeV at temperature GeV. The results are shown in Fig. (2), where we have used GeV, GeV, =0.890 GeV, GeV. It is clear that the Landau cuts end sharply at GeV for the loop and GeV for loop. The red vertical dashed line in Fig. (2) denotes the physical pole mass of kaon and its corresponding contributions will provide on-shell values of kaon thermal widths for and loops respectively. One should keep in mind that these Landau cut contributions are completely originated in presence of medium only. In vacuum, we can’t get any kind of (on-shell) decay like or because of kinematic restriction. Here we get a non-zero thermal width of kaon because of in-medium and scatterings, which were absent in vacuum picture.

In Fig. 3 we display the temperature dependence of total (on-shell) thermal width and mean free path of kaon for GeV (solid line) and GeV (dashed line). The total (on-shell) thermal width of kaon is the summation of and loop contributions, where a dominating nature of former than latter is already revealed in Fig. (2). We observe that the thermal width increases with the temperature. As the mean free path is inversely proportional to the thermal width (), the decreases rapidly as temperature increases. We also notice that the kaon thermal width is not too sensitive to the three momenta as opposed to the mean free path which is strongly affected by the change in momentum. We have seen that the nucleonic contribution due to interaction to the kaon thermal width or mean free path is insignificant compared to the contributions from mesonic loops, therefore, the nucleonic part has not been shown separately. The horizontal red dashed line indicates an approximate dimension of fireball, produced in heavy ion experiments like RHIC and LHC. One can notice that dashed and solid lines in Fig. 3(b) have different temperature ranges, where their mean free paths remain within the fireball dimension. We expect that kaon dissipation within these temperature ranges will contribute to the transport coefficients of the fireball. The kaon, having mean free path greater than this fireball dimension, will leave the system without facing any interaction and hence will not contribute to the transport coefficients.

In Fig. (4) this fact is explored by plotting the total mean free path of as a function of momentum at different values of temperatures. Total thermal width is basically obtained by adding the contributions from Eqs. (LABEL:Gam_K) and (16). We see that as the temperature decreases, kaon mean free path increases rapidly.

Now for a particular temperature, let us define a threshold momentum or cutoff momentum (say), beyond which the mean free path of kaon exceeds the fireball dimension. So, we will get different momentum cutoff for different temperatures as we notice in Fig. (4).

We find the average thermal width of kaon, defined as

 ΓavK=∫∞0d3k(2π)3ΓK(k,T)nK(ωkK)/∫∞0d3k(2π)3nK(ωkK) , (21)

and average mean free path, which is defined as

 λav = vavK/ΓavK , where vavK = ∫∞0d3k(2π)3kωkKnK(ωkK)/∫∞0d3k(2π)3nK(ωkK) .

Using these relations, we have obtained and for kaon, drawn by solid lines in Fig. (5) they are also compared with those for pion component (dotted line), taken from Ref. (28). We see that remain larger than the fireball dimension in the entire temperature domain, relevant for hadronic phase. Such results has hardly any phenomenological relevance for matter formed at the RHIC or LHC collisions. As a possible phenomenological approach, we use temperature dependent as upper limits of integration in Eqs. (21) and (III). These results of average thermal width and mean free path with momentum cutoff (MC) for kaon are displayed by solid line with circles in Fig. (5). We note that the values of with MC becomes lower than the fireball dimension, in such a scenario the kaons will have important role in transport coefficients of RHIC and LHC matter.

In Fig. 6, we present the temperature dependence of shear viscosities of kaon (solid lines) and compare with pion (dotted lines) for constant relaxation time (a) and for temperature dependent relaxation time (b). For constant relaxation time, shear viscosity for kaon component is smaller than that of pion but for temperature dependent relaxation time, their roles become opposite. Since the former case reflects the dependence of phase space in , so the massive kaons are exposed to larger thermal suppression than the lighter pion. On the other hand, our calculated relaxation time or mean free path for kaon is much larger than that for pion, therefore, in that case, the kaon component dominates over the pion component. In Fig. 6(b), dash line indicates shear viscosity of kaon component when thermal width of kaon is evaluated with the mesonic loop only. The inclusion of nucleonic part to the mesonic loop leads to the viscosity shown by solid line. A mild suppression of shear viscosity at high temperature range is noticed and we realize that the nucleonic part of Kaon thermal width is responsible for this fact.

The temperature dependence of electrical conductivities for kaon (solid lines) and pion (dotted line) (Fig. 7) exhibit approximately same pattern as in Fig. (6). One of the major difference between the expressions of and , given by Eqs. (20) and (3), is different power of momentum in their integrand. As a result of it we get little different dependence of and .

We have parametrized the threshold momentum or cutoff momentum as a function of as, . The accuracy of the parameterization is displayed in Fig. 8(a). When we use this as upper limit of integration in Eq. (3), the kaon contribution to the shear viscosity gets a modified as shown by dashed line in Fig. 8(b). This curve shows that up to GeV, the shear viscosity is very small and beyond that temperature, it rises rapidly. Shear viscosity of pion component (dotted line) shows very little dependence on temperature. Adding these two component, the total shear viscosity (solid line) is controlled by the pion component up to GeV, beyond which it is highly influenced by kaon component. Hence, kaon component enhances the shear viscosity in the high domain close to the transition temperature. This fact agrees with the results of Ref. (66); (61).

Lastly in Fig. 9, we have shown the temperature dependence of (a) and (b) for kaon component with momentum cutoff (dashed lines), for pion component (dashed lines) and the sum of the contributions (solid lines). The upper panel shows that electrical conductivity of kaon (with momentum cutoff) is zero up to temperature 0.14 GeV and then rises up, but for pion, it slowly decreases with temperature. The total value decreases slowly, up to GeV and then increases significantly owing to the contributions from kaons. Therefore, we may conclude that the kaon component act mainly on high domain, close to the transition temperature and its action is to enhance the transport coefficients in that high domain.

## Iv Summary and Conclusion

In this work we have estimated the contributions of strange hadrons to shear viscosity and electrical conductivity of hadronic matter. One loop self energies of kaon for and interactions have been calculated to extract the thermal width of kaons by using effective Lagrangian density. These results have been used to evaluate shear viscosity and electrical conductivity. The effects of kaon nucleon interaction has also been included. Since mean free path of Kaon with high momentum exceeds the dimension of fireball, we have introduced momentum cutoff approximation in the calculation. By this process, we may get some idea on the contribution of Kaon component in shear viscosity and electrical conductivity for RHIC and LHC matter. We observe that in the vicinity of the quark-hadron transition temperature, kaonic degrees of freedom contribute more to the shear viscosity and electrical conductivity than the pions.

Acknowledgment : Sabyasachi Ghosh is supported from UGC Dr. D. S. Kothari Post Doctoral Fellowship under grant No. F.4-2/2006 (BSR)/PH/15-16/0060. Snigdha Ghosh (CNT project No. 3/2/3012/VECC/R&D-I/1928) and MR are grateful to Department of Atomic Energy, Govt of India for financial support.

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