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I've learned in statistical mechanics that $\beta = \frac{1}{kT}$, which, in natural units, simplifies to $\beta = \frac{1}{T}$. When I started studying thermal field theory, it seemed that this relationship still holds. However, I became confused about when to use $\beta$ and when to use $T$. For example, in the paper I'm reading arXiv:1701.01554, these two quantities are used interchangeably from time to time.

From my understanding, $\beta$ is more closely associated with statistical mechanics, while $T$ might be more essential in cosmology (especially on topics like EW transition which I haven't met yet), but I still haven't grasped the general principle for choosing between these two quantities in thermal field theory. Could someone clarify when each should be used and explain the reasoning behind it?

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  • $\begingroup$ As you said, $\beta = 1/T$ so they can be used interchangeably all the time. You can use $\beta$ and $T$ in the same formula at the same time if you want. There is no "rule". $\endgroup$ Commented Apr 1 at 9:32

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It seems that in the paper you are referring to, the authors are consistent in their use of $\beta=\frac{1}{T}$ after setting $k_B=1$.

They actually emphasis this on page 2:

We employ the canonical ensemble, whereby $Z$ is a function of $T$ ; introducing units in which $k_B = 1$ (i.e., $T_{here} ≡ k_BT_{SI-units}$)

They key point there is that the temperatures in which they carry out their calculations relate to temperatures in SI units via a proportionality factor being $k_B$.

Overall, it is not about when one should use which convention, one may use either of them but certain conventions make up for a "cleaner" derivation if I may speak (rather than keeping track of certain fundamental constants, a smarter unit system choice may allow to avoid carrying them all along the entire calculation). It is about being consistent in the notation along the entire calculation and being aware of the dimensionality of the results obtained. The results can then be casted back to the SI units using the proper conversion factors if needed.

Typically each community tends to favor certain units over others (ex: eV often used in semiconductor physics or high energy physics rather than Joules to express energies) as it makes it easier to keep track of various numerical values (for the record: $1eV = 1.60218$e-19 Joules).

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  • $\begingroup$ Thanks for the clarification! I understand that $\beta = \frac{1}{T}$ holds consistently. However, in Eq. (2.44), both $T$ and $\beta$ are used in the same expression, and that's what makes me wonder if there's some conceptual nuance or subtlety behind this choice. $\endgroup$ Commented Apr 1 at 8:47
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    $\begingroup$ @Photon : it is a fair question to which I do not have a direct answer. An educated guess would be that it is common to find $e^{-\beta \epsilon}$ like terms throughout many branches of physics, hence the fact that their keep this notation apparent in their calculation. As for $T$ appearing as a pre-factor, remember that $\frac{1}{\beta}$ is an energy, therefore in this unit system $T$ is an energy, which is consistent with the fact that the second term is also an energy, namely $\frac{\epsilon_k}{2}$. Sorry not the best insight but I still think the convention they use holds. $\endgroup$ Commented Apr 1 at 9:21
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It is mostly about niceness of the resulting expression. The Equation (2.44) that you mention in a comment on another answer, is clearly written to make it so that there are no term appearing in denominators.

There are plenty of equations that have some emphasis, familiarity, or whatnot, that predisposes us to use one form v.s. another. For example, familiarity with the ideal gas law suggests for us to use $$pV=Nk_BT\tag1$$ which emphasises the linearity in $T$, even though we could have equally easily have used $$\beta\,p\,V=N\tag2$$ if we wanted.

Other quantities might be defined in terms of derivatives of various thermodynamics potentials or Massieu functions. Some may look easier either by derivatives w.r.t. $T$ or w.r.t. $\beta$ respectively. For example, the volumetric coefficient of thermal expansion at constant pressure is defined as $$\tag3\alpha_p=\left(\frac{\partial\ln V}{\partial\,T}\right)_p$$ which looks nice in terms of $T$ but looks ugly in terms of $\beta$, whereas the canonical ensemble average energy $$\tag4\left<E\right>=-\frac{\partial\ln Z}{\partial\beta}$$ and the Gibbs–Helmholtz equation $$\tag5H=\left(\frac{\partial(\beta\,G)}{\partial\beta}\right)_p$$ looks much better in terms of $\beta$ than in terms of $T$

The one case in which $\beta$ is vastly preferred over $T$ is when we go to negative temperatures. That is, the temperature scale of $T$ goes from coldest at absolute zero, up to a very hot temperature at $T=\infty$, and going past that into negative temperature, it is even hotter, and the hottest is at $T=-0$; this is much more intuitive if you consider $\beta$, which is coldest at $\beta=+\infty$, going hotter towards $\beta=-\infty$, with smooth and sensible continuity all the way.

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