Thermoelectric transport coefficients of hot and dense QCD matter

Abstract

The presence of a nonvanishing thermal gradient and/or a chemical potential gradient in a conducting medium can lead to an electric field—an effect known as thermoelectric effect or Seebeck effect. We discuss here the thermoelectric effects for hot and dense strongly interacting matter within the framework of relativistic Boltzmann equation in the relaxation time approximation. In the context of heavy-ion collisions, the Seebeck coefficients for the quark matter as well as for the hadronic matter are estimated within this approach. The quark matter is modeled by the two flavor Nambu–Jona–Lassinio (NJL) model and the hadronic medium is modeled by the hadron resonance gas (HRG) model with hadrons and their resonances up to a mass cutoff ${\tilde{\varLambda }}\sim 2.6$ GeV. For the estimation of thermoelectric transport coefficients, for the quark matter, we calculate the relaxation times for the quarks and antiquarks from the quark–quark and quark–antiquark scattering through meson exchange within the NJL model. On the other hand, for the hadronic medium, the relaxation times of hadrons and their resonances are estimated within a hard sphere scattering approximation. We also discuss the formalism of the thermoelectric effect in the presence of a nonvanishing external magnetic field. We give an estimation of the associated magneto-Seebeck coefficient and the Nernst coefficient for the hot and dense QCD matter.

Appendix A: Conservation of energy momentum tensor and baryon current in a quasiparticle approach

Here, we show the conservation of energy momentum tensor and the baryon number conservation for quasiparticles. Similar calculation can also be performed for free particles. Let us note that the energy, momentum, and baryon number etc. are conserved during the collisions between the quasi particles—here, the constituent quarks. Let $\chi _a$ be any such conserved quantity (e.g., baryon number, energy, momentum, etc.) associated with the particle $`a'$. For any two body scattering, e.g., $a+b\rightarrow c+d$ as considered here, we will have $\chi _a+\chi _b=\chi _c+\chi _d$. It can be shown that for any conserved quantity $\chi _a$, we have [106, 107, 147]:

$$\begin{aligned} \sum _a \int d\Gamma _a \chi _a{{{\mathcal {C}}}}_a = 0, \end{aligned}$$

(A.1)

where ${{{\mathcal {C}}}}_a$ is the collision integral in the Boltzmann equation and $d\Gamma \equiv \gamma _g d^3 p/(2 \pi )^3$. $\gamma _g$ is the degeneracy factor including color degree of freedom, flavor degree of freedom, and spin degree of freedom. Using the Boltzmann equation, the collision integral for the quasiparticle can be given by:

$$\begin{aligned} {{{\mathcal {C}}}}_a=\frac{p^\mu _a}{\epsilon _a}\partial _\mu f_a+(\partial ^i \epsilon _a)\frac{\partial f_a}{\partial p^i_a}, \end{aligned}$$

(A.2)

where $\epsilon =\sqrt{\mathbf {p}^2+M^2}$ is single (quasi) particle energy which, in general, depends upon the mean field/the quark condensate. Here, we have not included the electromagnetic field contribution for simplicity, but can be included in general. Next, one can substitute the expression for ${{{\mathcal {C}}}}_a$ as given in Eq. (A.2) into Eq. (A.1) to simplify further. Of the two terms in ${{{\mathcal {C}}}}_a$, one can integrate by parts with respect to $x^{\mu }\equiv (t,\mathbf {x})$ in the first term and with respect to $\mathbf {p}$ in the second term after substituting in Eq. (A.1). After some manipulations, this leads to:

$$\begin{aligned}&\partial _\mu \bigg (\sum _a\int d\Gamma _a f_a\frac{p^\mu _a\chi _a}{\epsilon _a}\bigg )\nonumber \\&-\sum _a\int d\Gamma _a f_a\bigg [\frac{ p^\mu _a}{\epsilon _a}\partial _\mu \chi _a +\partial ^i \epsilon _a \frac{\partial \chi _a}{\partial p_a^i}\bigg ]=0. \end{aligned}$$

(A.3)

The baryonic current can be expressed as:

$$\begin{aligned} J^\mu _b=\sum _a b_a\int d\Gamma _a\frac{p^\mu _a}{\epsilon _a} f_a; \end{aligned}$$

(A.4)

$b^a$ is the baryon number and $b^a =\pm 1$ for particles and antiparticles, respectively. Let us first discuss the conservation of baryon current. In Eq. (A.3), let $\chi _a=b_a$, the baryon number. Then, all the derivatives of $\chi _a$ vanish and we have from Eq. (A.3):

$$\begin{aligned} \partial _\mu \left( \sum _ab_a\int d\Gamma _a f_a \frac{p^\mu }{\epsilon _a}\right) =\partial _\mu J^\mu _b = 0. \end{aligned}$$

(A.5)

Next, we write down the energy momentum tensor in terms of the distribution functions and the mean fields as was done in Ref. [117] for the two flavor NJL model:

$$\begin{aligned} T^{\mu \nu }=\sum _a \int d\Gamma _a \frac{p_a^\mu p_a^\nu }{\epsilon _a} f_a+g^{\mu \nu } V, \end{aligned}$$

(A.6)

where V represents the vacuum energy density contribution from the mean fields or equivalently the constituent quark mass, that is:

$$\begin{aligned} V=-\int d \Gamma \sqrt{ \mathbf {p}^2+M^2}+\frac{(M-m_0)^2}{4G}. \end{aligned}$$

(A.7)

Here, M is the consistent quark mass and G is the scalar coupling of the NJL model. To get the conservation of the energy momentum tensor, one can take $\chi _a=\epsilon _a$ and $\chi _a=p^i_a$ in Eq. (A.3) to write down the conservation equation of energy, and momentum, respectively, and can be combined to get energy momentum conservation equation as:

$$\begin{aligned}&\partial _\mu \left( \sum _a\int d\Gamma _a f_a\frac{p^\mu p^\nu }{\epsilon _a}\right) \nonumber \\&\quad -\sum _a \int d\Gamma _a f_a\partial ^\nu \epsilon _a=0. \end{aligned}$$

(A.8)

Next, let us note that from Eq. (A.7), we have:

$$\begin{aligned} \partial ^\nu V&=\frac{\partial V}{\partial M}\partial ^\nu M \nonumber \\&=\bigg [-\int d\Gamma _a \frac{M}{\epsilon _a} +\frac{M-m_0}{2G}\bigg ]\partial ^\nu M. \end{aligned}$$

(A.9)

Using the gap equation:

$$\begin{aligned} \frac{M-m_0}{2 G}=\int d\Gamma _a \frac{M}{\epsilon _a}-\sum _a \int d\Gamma _a \frac{M}{\epsilon _a}f_a; \end{aligned}$$

(A.10)

for the constituent quarks in the right-hand side of Eq. (A.9), we obtain:

$$\begin{aligned} \partial ^\nu V&=-\sum _a\int d\Gamma _a \frac{M}{\epsilon _a} f_a \partial ^\nu M \nonumber \\&=-\sum _a\int d\Gamma _a f_a \partial ^\nu \epsilon ^a. \end{aligned}$$

(A.11)

Using Eq. (A.11) for the second term in Eq. (A.8), we have:

$$\begin{aligned} 0=\partial _\mu \left( \sum _a\int d\Gamma _a f_a\frac{p^\mu p^\nu }{\epsilon _a}\right) +\partial ^\nu V\equiv \partial _\mu T^{\mu \nu }, \end{aligned}$$

(A.12)

where we have used the definition of $T^{\mu \nu }$ in terms of the microscopic distribution functions and mean field as given in Eq. (A.6). Equations  (A.5) and  (A.12) thus demonstrate the macroscopic current and energy momentum conservation, where $J^\mu _b$ and $T^{\mu \nu }$ defined in terms of microscopic distribution functions and mean fields. Furthermore, for free particles (in the absence of a medium-dependent mass), $\partial _{\mu }\epsilon = 0$. Therefore, for free particles, Eq. (A.8) gives us the conservation of energy momentum tensor.