Thermal effects on warm chromoinflation

Warm inflation [1, 2, 3] has recently seen successful implementations in models featuring pseudo-Nambu-Goldstone scalar fields [4, 5, 6, 7, 8]. These models often involve axion-like fields directly coupled to non-Abelian gauge fields. The dissipation of the axion-inflaton into gauge fields can naturally lead to a thermal radiation bath, even from initial vacuum conditions [9, 7, 8]. This thermal bath is a hallmark of the warm inflationary regime.

The explicit realization of consistent warm inflation dynamics in these models is a significant achievement. Warm inflation has emerged as a promising inflation model that aligns with effective field theory and potentially finds a ultraviolet (UV) completion in quantum gravity [10, 11, 12, 13, 14, 15, 16, 17, 18]. Therefore, constructing explicit models of warm inflation and exploring their implications has become increasingly important.

In this work, we investigate an axion warm inflation model incorporating a Chern-Simons interaction between the axion inflaton field and a $SU(2)$ gauge field. It is well-established [19, 20, 21, 22] that, in addition to the background inflaton field, an isotropic solution for the $SU(2)$ gauge field is also permissible (for a review, see, e.g. ref. [23]). This background gauge field can substantially influence the evolutionary dynamics of the expanding system at both the background and perturbation levels.

Here, we demonstrate that the presence of a thermal bath naturally induces a plasma mass for the background gauge field. This thermal mass contribution to the gauge background field, similar to the thermal mass affecting the inflaton equation in warm inflation [24, 25], can potentially disrupt the slow-roll conditions for the gauge field, driving it towards a vanishing value. We investigate the impact of these thermal effects on the background dynamics and illustrate our results using a commonly employed axion potential.

This paper is organized as follows. In section 2, we briefly review the background dynamics of chromoinflation in the context of warm inflation. In section 3, we compute the leading order thermal contributions to the background dynamics. In section 4, we illustrate our results by explicitly studying the dynamics in a case of axion-like potential. In section 5, we give our concluding remarks.

Throughout this paper, we work with the natural units, in which the speed of light, Planck’s constant and Boltzmann’s constant are all set to $1$, $c=\hbar=k_{B}=1$. We work in the context of a spatially flat homogeneous and isotropic background metric with scale factor $a(t)$, where $t$ is physical time. We also work with the reduced Planck mass, defined as $M_{\rm Pl}=(8\pi G)^{-\frac{1}{2}}\simeq 2.44\times 10^{18}$ GeV and where $G$ is Newton’s gravitational constant. The Hubble expansion rate is $H(t)={\dot{a}(t)}/a(t)$, where an overdot denotes the derivative with respect to time.

We work with a minimal setting for an axion-like field $\phi$ making the role of the inflaton and coupled to a non-Abelian $SU(2)$ gauge field $A_{\mu}$ with the standard dimension five interaction, with action in a FLRW metric,

	$\displaystyle S$	$\displaystyle=$	$\displaystyle\int d^{4}xa^{3}(t)\left[\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}^{c}F^{c\,\mu\nu}\right.$		(2.1)
		$\displaystyle-$	$\displaystyle\left.\frac{\lambda}{4f}\phi F_{\mu\nu}^{c}\tilde{F}^{c\,\mu\nu}\right],$

where $c=1,\ldots,N_{c}^{2}-1$ is the group index, with $N_{c}=2$ for $SU(2)$, $f$ is the axion decay constant, $F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}$ is the gauge field tensor, $\tilde{F}^{c\,\mu\nu}$ is its dual,

$$\tilde{F}^{c\,\mu\nu}=\frac{\epsilon^{\mu\nu\rho\sigma}}{2a^{3}(t)}F_{\rho\sigma}^{c},$$

(2.2)

and $a(t)$ is the scale factor.

The $SU(2)$ gauge field admits a homogeneous vacuum expectation value (VEV), defined as [19, 20, 21]

$\displaystyle\bar{A}^{c}_{i}=a(t)\psi(t)\delta^{c}_{i},\;\;\;\;\bar{A}^{c}_{0}=0,$

(2.3)

which leads to the gauge field strength components,

	$\displaystyle\bar{F}^{c\,0i}=-\bar{F}^{c,i0}=-\frac{1}{a}(H\psi+\dot{\psi})\delta^{ci},\;\;\;\bar{F}^{c\,ij}=\frac{g}{a^{2}}\psi^{2}\epsilon^{cij},$
	$\displaystyle\bar{F}_{0i}^{c}=-\bar{F}_{i0}^{c}=a(H\psi+\dot{\psi})\delta_{i}^{c},\;\;\;\bar{F}_{ij}^{c}=ga^{2}\psi^{2}\epsilon_{ij}^{c}.$		(2.4)

Note that in chromoinflation [19, 20, 21] the coupling $\lambda$ between the inflaton $\phi$ and the gauge fields in the interaction term in eq. (2.1) is assumed to be an independent constant. Typically, a consistent background evolution for the inflaton and gauge field is achieved for large couplings, or more generally, $\lambda M_{\rm Pl}/f\gg 1$. How to achieve such large couplings in effective theories has been discussed recently for example in refs. [26, 27]. Note that in the axion dynamics case, the coupling $\lambda$ is simply related to the gauge coupling $g$ by $\lambda=g^{2}/(8\pi^{2})$.

The inflaton (axion) field has been shown to dissipate into radiation bath gauge fields, which forms a thermal bath with temperature $T$ and whose dissipation coefficient is given by [6, 7]

$$\Upsilon(T)=\kappa\frac{T^{3}}{f^{2}},$$

(2.5)

where $\kappa$ is given by

$$\kappa\simeq 1.2\pi\frac{g^{4}(g^{2}N_{c})^{3}(N_{c}^{2}-1)}{(64\pi^{3})^{2}}\left[\ln\left(\frac{m_{D}}{\gamma}\right)+3.041\right],$$

(2.6)

where $m_{D}^{2}=g^{2}N_{c}T^{2}/3$ is the Debye mass squared of the Yang-Mills plasma and $\gamma$ is given by the solution of [28]

$$\gamma=\frac{g^{2}N_{c}T}{4\pi}\left[\ln\left(\frac{m_{D}}{\gamma}\right)+3.041\right].$$

(2.7)

Explicitly, this gives for $\kappa$ the result

$\displaystyle\kappa$

$\displaystyle=$

$\displaystyle 0.3\alpha_{g}^{5}N_{c}^{3}(N_{c}^{2}-1)W\left(e^{3.041}\sqrt{\frac{4\pi}{3\alpha_{g}N_{c}}}\right),$

(2.8)

where $\alpha_{g}=g^{2}/(4\pi)$ is the fine structure Yang-Mills coupling and $W(x)$ is the Lambert function, given by the principal solution of $x=we^{w}$. For $N_{c}=2$ and assuming e.g. $\alpha_{g}=0.1$, we obtain that $\kappa\simeq 2.4\times 10^{-4}$.

To take into account the effects of the thermal bath on the background dynamics, we will work analogously to the loop expansion in quantum field theory. This starts by expanding the fields around their vacuum background expectation values. In this work in particular, we will be looking how the fluctuations around the gauge field background $\psi$ will affect the dynamics. Hence, the gauge field in (2.1) is expanded like $A_{\mu}\to A^{\prime}_{\mu}=\bar{A}_{\mu}(t)+A_{\mu}$, where $\langle A^{\prime}_{\mu}\rangle=\bar{A}_{\mu}(t)$, with $\bar{A}_{\mu}(t)$ given by eq. (2.3) and $\langle A_{\mu}\rangle=0$. Hence, we have for instance that

$\displaystyle F^{a}_{\mu\nu}=\bar{F}_{\mu\nu}^{a}+\left(\bar{D}_{\mu}A_{\nu}\right)^{a}-\left(\bar{D}_{\nu}A_{\mu}\right)^{a}+g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c},$

(3.1)

where $\bar{F}_{\mu\nu}^{a}$ is given by eq. (2.4) and

$$\bar{D}_{\mu}=\partial_{\mu}-igJ^{a}\bar{A}_{\mu}^{a},$$

(3.2)

is the covariant derivative expressed in terms of the background gauge field, with $J^{a}$ the generators of the $SU(2)$ group. Working in the adjoint representation of the $SU(2)$ gauge group, $(J_{a})_{bc}=\epsilon_{abc}$.

Using eq. (3.1) in (2.1), the action, up to second order in the gauge field fluctuations, then takes the form

$$S\simeq S_{0}+\delta^{2}S,$$

(3.3)

where

	$\displaystyle S_{0}$	$\displaystyle=$	$\displaystyle\int d^{4}xa^{3}(t)\left[\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)-\frac{1}{4}\bar{F}_{\mu\nu}^{c}\bar{F}^{c\,\mu\nu}\right.$		(3.4)
		$\displaystyle-$	$\displaystyle\left.\frac{\lambda}{4f}\phi\frac{\epsilon^{\mu\nu\rho\sigma}}{2a^{3}(t)}\bar{F}_{\mu\nu}^{c}\bar{F}_{\rho\sigma}^{c}\right]$
		$\displaystyle=$	$\displaystyle\int d^{4}x\,a^{3}\left[\frac{\dot{\phi}^{2}}{2}-V(\phi)+\frac{3}{2}\left(\dot{\psi}+H\psi\right)^{2}\right.$
		$\displaystyle-$	$\displaystyle\left.\frac{3}{2}g^{2}\psi^{4}-3\frac{\lambda}{f}g\phi\left(\dot{\psi}+H\psi\right)\psi^{2}\right],$

is the zeroth order action in the fluctuations and $\delta^{2}S$ is the action when expanding the fields up to the second-order in the fluctuations (see also ref. [31] for a similar approach for deriving the perturbation equations in chromoinflation). The Yang-Mills and Chern-Simons contributions can then be expressed as

	$\displaystyle\delta^{2}S_{\rm YM}$	$\displaystyle=$	$\displaystyle\int d^{4}x\,a^{3}\left\{-\frac{1}{2}(\bar{D}^{\mu}A^{\nu})^{a}(\bar{D}_{\mu}A_{\nu})^{a}\right.$
		$\displaystyle+$	$\displaystyle\left.\frac{1}{2}(\bar{D}^{\mu}A^{\nu})^{a}(\bar{D}_{\nu}A_{\mu})^{a}-\frac{1}{2}g\epsilon^{abc}\bar{F}^{a\,\mu\nu}A_{\mu}^{b}A_{\nu}^{c}\right.$
		$\displaystyle-$	$\displaystyle\left.\frac{\lambda}{2f}\phi\;\frac{\epsilon^{\mu\nu\rho\sigma}}{a^{3}}\left[(\bar{D}_{\mu}A_{\nu})^{a}(\bar{D}_{\rho}A_{\sigma})^{a}\right.\right.$
		$\displaystyle+$	$\displaystyle\left.\left.\frac{g}{2}\epsilon^{abc}\bar{F}^{a}_{\mu\nu}A_{\rho}^{b}A_{\sigma}^{c}\right]\right\}.$

We still need to choose a gauge fixing term. Here, it becomes convenient to fix a gauge that depends explicitly on the background gauge field, such that (see, e.g. [34])

$$\delta^{2}S_{\rm gf}=\int d^{4}x\,a^{3}\left[-\frac{1}{2\alpha}(\bar{D}^{\mu}A^{\mu})^{a}(\bar{D}_{\nu}A_{\nu})^{a}\right],$$

(3.6)

and with the corresponding Faddeev-Popov ghost term,

$$\delta^{2}S_{\rm ghost}=\int d^{4}x\,a^{3}\left[(\bar{D}^{\mu}\bar{\eta})^{a}(\bar{D}_{\mu}\eta)^{a}\right],$$

(3.7)

where $\eta$ and $\bar{\eta}$ are the Faddeev-Popov ghost fields. Hence, by choosing the gauge $\alpha=1$ and after some manipulation of indexes in eq. (LABEL:S2YM) and integration by parts, we obtain the result for the quadratic term in the fluctuations in the action (note that in arriving at the expression below, a term coming from the integration by parts, but not depending on the background gauge field, was neglected)

	$\displaystyle\delta^{2}S$	$\displaystyle=$	$\displaystyle\int d^{4}x\,a^{3}\left\{-\frac{1}{2}(\bar{D}^{\mu}A^{\nu})^{a}(\bar{D}_{\mu}A_{\nu})^{a}\right.$
		$\displaystyle+$	$\displaystyle\left.(\bar{D}^{\mu}\bar{\eta})^{a}(\bar{D}_{\mu}\eta)^{a}-g\epsilon^{abc}\bar{F}^{a\,\mu\nu}A_{\mu}^{b}A_{\nu}^{c}\right.$
		$\displaystyle-$	$\displaystyle\left.\frac{\lambda}{2f}\phi\;\frac{\epsilon^{\mu\nu\rho\sigma}}{a^{3}}\left[(\bar{D}_{\mu}A_{\nu})^{a}(\bar{D}_{\rho}A_{\sigma})^{a}\right.\right.$
		$\displaystyle+$	$\displaystyle\left.\left.\frac{g}{2}\epsilon^{abc}\bar{F}^{a}_{\mu\nu}A_{\rho}^{b}A_{\sigma}^{c}\right]\right\}.$

By substituting in eq. (LABEL:S2YM2) the equations (2.3) and (2.4), we have many terms depending on the background field $\psi$. Many of them vanish when performing the functional integration over the fluctuations (e.g. the Chern-Simons dependent terms which are total derivatives). For the terms effectively contributing, one notices that in the presence of a nonvanishing background gauge field, both the gauge field and the ghosts acquire a mass square, $m_{A}^{2}=m_{\eta}^{2}=2g^{2}\psi^{2}$. Their functional integration lead to an effective potential for $\psi$ which can be expressed as³³3Note that curvature effects are expected to be negligible for masses larger than the Hubble scale, e.g. for $2g^{2}\psi^{2}>H^{2}$ and, furthermore, at finite temperature, the relevant momentum modes for the gauge fluctuations contributing for the loop integrals are hard momenta, $k\sim T$. In particular, this later condition is well satisfied in warm inflation by definition, where $T>H$. This justifies a Minskowski like derivation of the loop terms used to derive the result given by eq. (3.9).

$\displaystyle\!\!\!\!\!\Delta V_{\rm eff}(\psi)$

$\displaystyle=$

$\displaystyle(N_{c}^{2}-1)\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\ln\left(k_{E}^{2}+2g^{2}\psi^{2}\right),$

(3.9)

where $k_{E}$ refers to the (physical) momentum expressed in Euclidean spacetime. Diagrammatically, the terms contributing to the effective potential at leading order are displayed in figure 1.

From the usual rules of finite temperature quantum field theory [35], $k_{E}^{2}=k_{4}^{2}+{\bf k}^{2}$ and $k_{4}\equiv\omega_{n}=2\pi nT$, $n\in\mathbb{Z}$ and $\omega_{n}$ are the Matsubara’s frequencies for bosons. The integrals over momenta are expressed as

$$\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\to T\sum_{k_{4}=\omega_{n}}\int\frac{d^{3}k}{\left(2\pi\right)^{3}}.$$

(3.10)

At zero temperature, the momentum integral in eq. (3.9) is divergent. The divergent terms can be handled as usual in perturbation theory in quantum field theory by adding the appropriate renormalization counterterms in the tree-level action eq. (3.4). The relevant contributions for us are the finite temperature ones, which, after performing the Matsubara’s frequency sum in eq. (3.9), we are left with the thermal contribution,

$\displaystyle\Delta V_{\rm eff}(\psi,T)$

$\displaystyle=$

$\displaystyle 2(N_{c}^{2}-1)\frac{T^{4}}{2\pi^{2}}J_{B}(y),$

(3.11)

where [35]

	$\displaystyle J_{B}(y)$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}dx\,x^{2}\ln\left(1-e^{-\sqrt{x^{2}+y^{2}}}\right)$		(3.12)
		$\displaystyle\simeq$	$\displaystyle-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}y^{2}+{\cal O}(y^{3}),$

with $y^{2}=2g^{2}\psi^{2}/T^{2}$ and the last line in eq. (3.12) follows by using an expansion for $y\ll 1$. We could as well consider the additional terms in the expansion in eq. (3.12). However, for our purposes of demonstrating the effects of the thermal contributions, the truncation of the series up to quadratic order for the thermal integral $J_{B}(y)$ suffices for us⁴⁴4We have nevertheless explicitly verified in our numerical analysis that the higher order terms remain subdominant compared to the quadratic term in eq. (3.12).. We then obtain for eq. (3.9) the result

$\displaystyle\Delta V_{\rm eff}(\psi,T)$

$\displaystyle\simeq$

$\displaystyle-2(N_{c}^{2}-1)\frac{\pi^{2}}{90}T^{4}+(N_{c}^{2}-1)g^{2}\frac{T^{2}}{6}\psi^{2}.$

The total energy density is then modified by the thermal effects, such that now we have that

	$\displaystyle\rho_{T}$	$\displaystyle=$	$\displaystyle\frac{\dot{\phi}^{2}}{2}+V(\phi)$		(3.14)
		$\displaystyle+$	$\displaystyle\frac{3}{2}(\dot{\psi}+H\psi)^{2}+\frac{3}{2}g^{2}\psi^{4}$
		$\displaystyle+$	$\displaystyle\Delta V_{\rm eff}(\psi,T)+Ts,$

where we opted from now on to work for convenience with the entropy density $s$,

$\displaystyle s$

$\displaystyle=$

$\displaystyle-\frac{\partial\Delta V_{\rm eff}}{\partial T}\simeq\frac{4\pi^{2}(N_{c}^{2}-1)T^{3}}{45}-\frac{(N_{c}^{2}-1)g^{2}T\psi^{2}}{3}.$

We thus see that the presence of the thermal bath in warm chromoinflation produces not only a friction term due to the nonperturbative spharelon effects in the gauge vacua, but also leads to a thermal plasma mass for the background gauge field $\psi$,

$$m_{\psi}^{2}(T)=(N_{c}^{2}-1)\frac{g^{2}T^{2}}{3}.$$

(3.16)

The background field equations in warm chromoinflation then get modified such that now we have that

	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ddot{\phi}+3H\dot{\phi}+V_{,\phi}+3\frac{\lambda}{f}g\psi^{2}(\dot{\psi}+H\psi)+\Upsilon\dot{\phi}=0,$		(3.17)
	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ddot{\psi}+3H\dot{\psi}+(2H^{2}+\dot{H})\psi+2g^{2}\psi^{3}$
	$\displaystyle+m_{\psi}^{2}(T)\psi-\frac{\lambda g}{f}\psi^{2}\dot{\phi}=0,$		(3.18)
	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!T\dot{s}+3HTs=\Upsilon\dot{\phi}^{2},$		(3.19)

and with the Hubble parameter given by

	$\displaystyle H^{2}$	$\displaystyle=$	$\displaystyle\frac{1}{3M_{\rm Pl}^{2}}\left[\frac{\dot{\phi}^{2}}{2}+V(\phi)+\frac{3}{2}(\dot{\psi}+H\psi)^{2}\right.$		(3.20)
		$\displaystyle+$	$\displaystyle\left.\frac{3}{2}g^{2}\psi^{4}+\Delta V_{\rm eff}(\psi,T)+Ts\right].$

We can qualitatively already investigate the effect of the thermal mass term in the background equation for $\psi$. In the slow-roll approximation, from eq. (3.18), we obtain that

$\displaystyle 3H\dot{\psi}\simeq-\left[2H^{2}+m_{\psi}^{2}(T)\right]\psi-2g^{2}\psi^{3}+\frac{\lambda g}{f}\psi^{2}\dot{\phi}.$

(3.21)

If we associate the right-hand side in eq. (3.21) with an analogous of the field derivative of an effective potential for $\psi$ (see, e.g. ref. [36] for a similar approach in cold chromoinflation), then we can define the effective potential (and assuming $\dot{\phi}$ as constant),

$$V_{\rm eff}(\psi,T)\simeq\left[2H^{2}+m_{\psi}^{2}(T)\right]\frac{\psi^{2}}{2}-\frac{\lambda g\dot{\phi}}{f}\frac{\psi^{3}}{3}+g^{2}\frac{\psi^{4}}{2}.$$

(3.22)

By defining the dimensionless quantities,

	$\displaystyle\Psi$	$\displaystyle=$	$\displaystyle\frac{g\psi}{H},$
	$\displaystyle\xi$	$\displaystyle=$	$\displaystyle\frac{\lambda\dot{\phi}}{2fH},$		(3.23)
	$\displaystyle M_{\psi}$	$\displaystyle=$	$\displaystyle\frac{m_{\psi}(T)}{H},$

we are left with

$$\bar{V}_{\rm eff}(\Psi)=\left(2+M_{\psi}^{2}\right)\frac{\Psi^{2}}{2}-\frac{2\xi}{3}\Psi^{3}+\frac{\Psi^{4}}{2},$$

(3.24)

where $\bar{V}_{\rm eff}(\Psi)=g^{2}V_{\rm eff}(\psi,T)/H^{4}$. The potential $\bar{V}_{\rm eff}(\Psi)$ has in general three extrema: at $\Psi=0$ and at

$$\Psi_{\pm}=\frac{1}{2}\left[\xi\pm\sqrt{\xi^{2}-2\left(2+M_{\psi}^{2}\right)}\right].$$

(3.25)

It can be easily checked that $\Psi=0$ is always a minimum of $\bar{V}_{\rm eff}(\Psi)$, while $\Psi_{-}$ is a maximum and $\Psi_{+}$ is another minimum provided that $\xi^{2}>2\left(2+M_{\psi}^{2}\right)$. At $\xi^{2}=2\left(2+M_{\psi}^{2}\right)$, $\Psi_{\pm}$ coalesces to an inflection point of $\bar{V}_{\rm eff}(\Psi)$ located at $\Psi=\xi/2$. For $\xi^{2}<2\left(2+M_{\psi}^{2}\right)$, we have that $\Psi=0$ is the only solution. Finally, there is a value for $\xi$, where the minimum at the origin is degenerate with the minimum at $\Psi_{+}$, i.e., $\bar{V}_{\rm eff}(0)=\bar{V}_{\rm eff}(\Psi_{+})$, which happens at

$$\xi^{2}=\frac{9}{4}\left(2+M_{\psi}^{2}\right).$$

(3.26)

This behavior of the effective potential, as seen here as function of the values of the parameters, resembles exactly that one for a first-order phase transition as a function of the temperature. The value of the temperature for which $\bar{V}_{\rm eff}(0)=\bar{V}_{\rm eff}(\Psi_{+})$ is satisfied would correspond to the critical temperature of the phase transition, as conventionally considered in the literature of Ginzburg-Landau phase transition when it is first order [37]. We can here estimate this critical temperature if we use the slow-roll approximation obtained from the entropy evolution equation, eq. (3.19),

$$Ts\simeq\frac{\Upsilon}{3H}\dot{\phi}^{2}=\frac{\kappa T^{3}}{3fH}\dot{\phi}^{2},$$

(3.27)

where we have used the expression (2.5) for the dissipation coefficient. Finally, from eq. (LABEL:entropy), considering $g\psi\ll T$, and from the definitions given in eq. (3.23), we obtain the result for the critical temperature for phase transition in terms of the parameters of the model,

$\displaystyle\frac{T_{c}}{H}$

$\displaystyle\sim$

$\displaystyle\frac{2\pi^{2}\lambda^{2}}{45g^{2}\kappa}\left[1+\sqrt{1-\frac{6075g^{2}\kappa^{2}}{2(N_{c}^{2}-1)\pi^{4}\lambda^{4}}}\,\right]\sim\frac{4\pi^{2}\lambda^{2}}{45g^{2}\kappa}.$

In warm inflation we have in general that $T/H$ increases during inflation [18, 38]. Thus, after the critical temperature eq. (LABEL:Tc) is reached, $\Psi_{+}$ becomes a local minimum. Hence, above $T_{c}$ it becomes energetically favored for the system to be driven to $\psi=0$. This happens rather fast, in less than one e-fold, as indicated by our numerical results, after which the gauge vacuum expectation value no longer plays a role in the dynamics. The above qualitative analytical analysis is confirmed by our numerical results that are presented in the next section.

To illustrate our results, let us consider an axion-like potential for the inflaton and given by

$$V(\phi)=\Lambda^{4}\left[1+\cos(\phi/f_{D})\right],$$

(4.1)

where $\Lambda$ is the normalization of the potential (typically $\Lambda^{4}=m_{\phi}^{2}f_{D}^{2}$, where $m_{\phi}$ is the mass of the axion-like particle and $f_{D}$ its decay constant). Note that in standard cold natural inflation one typically requires $f_{D}\gtrsim M_{\rm Pl}/\sqrt{2}$ to have a slowly rolling regime for the inflaton [39]. To be concrete, we assume $\Lambda$ to correspond to some confinement energy scale of some UV complete theory. Inflation here would then happen at energy scales below $\Lambda$ but still larger than some IR energy scale for the confinement of the $SU(2)$ gauge field considered here, such that the presence of a thermal bath and the use of the dissipation coefficient eq. (2.5) are justified.

In warm inflation with a dissipation term like eq. (2.5) it was also been shown [40] that consistency of the natural inflation potential with the observations still requires $f_{D}\sim M_{\rm Pl}$. In particular, a more recent analysis [41] has also shown that this model can only sustain inflation consistent with the observations in the weak regime of warm inflation, in which case $Q=\Upsilon/(3H)\ll 1$. Motivated by these previous references, here we will not assume that $f$, which appears in the dissipation and chromoinflation expressions and that comes from the Chern-Simons interaction, and $f_{D}$, which appears in the potential are the same⁵⁵5This can also be interpreted in terms of an effective value for the coupling $\lambda$, such that it would be analogous to consider a modified $\bar{\lambda}$ given by $\bar{\lambda}=\lambda f_{D}/f$.. This will give us some more freedom when playing with the parameters of the model⁶⁶6It can be assumed in particular that $f_{D}$ is a much larger scale than $f$, which is motivated from recent works based on clockwork models [42, 43].. By fixing $f_{D}$, $\lambda$ and $g$, we can obtain for instance $f$ by demanding to inflation to last a specific number of efolds for a given dissipation ratio $Q$.

Specific examples are given in figures 2 and 3, where we show some of the relevant background quantities. In both examples shown in those figures the total number of e-folds of inflation was taken to be $N_{\rm infl}=55$ and for illustration purposes the parameters taken were $g=0.6$, $\lambda=100$, $f_{D}/M_{\rm Pl}=3/\sqrt{2}$ (as motivated e.g. from ref. [40]). The initial dissipation ratio in figure 2 is $Q_{0}=1.2\times 10^{-3}$, from which the values of $f$ and $\Lambda$ are found to be $f\simeq 0.05M_{\rm Pl}$ and $\Lambda\simeq 0.003M_{\rm Pl}$, respectively. In figure 3 we have $Q_{0}\simeq 1.3\times 10^{-2}$, with $f\simeq 0.035M_{\rm Pl}$ and $\Lambda\simeq 0.0016M_{\rm Pl}$.

The results displayed in figures 2 and 3 show that the most important effect of including the thermal corrections to the gauge field background is on the evolution of $\psi$ itself. In the absence of a thermal mass correction, the gauge field background is sustained throughout the inflationary evolution. However, in the presence of the thermal correction it is eventually driven to zero well before inflation ends. Furthermore, the larger is the initial value for the dissipation ratio $Q$, the sooner $\psi$ is driven to zero. For the parameters considered, this suppression of $\psi$ already happens in the weak regime of warm inflation, $Q\ll 1$. For the parameters considered and for a dissipation ratio $Q\gtrsim 5\times 10^{-2}$, we find that the gauge field background already vanishes at the onset and remains null throughout the inflationary evolution.

In principle, we could believe that the results could be changed by making $g$ very small, thus suppressing the thermal effects and allowing a nonvanishing value for $\psi$ to be sustained throughout inflation. However, this also affects the dissipation coefficient through its dependence on the coupling $g$, forcing either $\lambda$ to be larger or $f$ to be smaller. In general we find that larger values of $Q$ becomes hard to be obtained (we also recall the results of refs. [40, 41] of the difficulties of having warm inflation in the strong regime, $Q>1$, in axion-like inflaton potentials). In special we find that the larger is the dissipation ratio $Q$, one requires smaller couplings to support a gauge background $\psi\neq 0$.

Another approach to favor chromoinflation in the current scenario would be to increase the gauge background field. This would require a gauge field mass, $m_{A}$, much larger than the temperature. This would suppress thermal effects from eq. (3.11), as they would be Boltzmann-suppressed. However, this introduces a new challenge. A larger gauge field mass, resulting from a non-zero $\psi$, would also strongly suppress the dissipation coefficient in eq. (2.5). This is analogous to what is expected to happen in the electroweak phase transition [44]: above the electroweak scale, massless gauge bosons allow rapid sphaleron processes; below the scale, mass acquisition suppresses these processes. Similarly, here, a larger gauge field mass would suppress the dissipation necessary for warm inflation. While cold chromoinflation could still occur, a warm inflation regime would be precluded.

As discussed at the end of the last section, we can interpret the vanishing of $\psi$ as a true phase transition. This is explicitly illustrated in figure 4, where we show snapshots of the effective potential eq. (3.22) as a function of $\psi$ at different values of e-folds for the case of $Q_{0}\simeq 1.3\times 10^{-2}$, which is the case shown in figure 3. From the results of figures 2 and 3 (e.g. from the panels (b) in those figures), this behavior of the minimum of the effective potential can be interpreted in terms of an increasing value of the temperature at the corresponding values of e-folds.

The corresponding critical temperature for the (first-order) phase transition seen in figure 4(b) is $T_{c}\simeq 1.6\times 10^{-4}M_{\rm Pl}$, while for the case of the parameters shown in figure 2(d), i.e. for $Q_{0}=1.2\times 10^{-3}$ corresponds to $T_{c}\simeq 4.9\times 10^{-4}M_{\rm Pl}$. These values agree well with the simpler estimate given by eq. (LABEL:Tc). Hence, the smaller is $Q_{0}$, the larger becomes $T_{c}$.

The minimum of the effective potential, corresponding to the solution $\Psi_{+}$ in eq. (3.25), as a function of the temperature, is displayed in figure 5. It shows the behavior typical of a order parameter as a function of the temperature when the phase transition is first order. The order parameter (which is here represented by $\psi_{+}$) jumps discontinuously from a nonnull to null value across the phase transition point.

A pseudo-Nambu-Goldstone scalar field coupled to non-Abelian gauge fields via a Chern-Simons term provides a successful model of warm inflation, known as minimal warm inflation. This model arises from the natural dissipation of the axion-like inflaton field due to sphaleron transitions in a thermal bath, ensuring a warm inflationary regime where the temperature $T$ exceeds the Hubble parameter $H$.

Chromoinflation, another model involving axion-like fields coupled to non-Abelian gauge fields, allows for a homogeneous background gauge field. A natural extension is to incorporate this into the minimal warm inflation framework. The thermalized bath of gauge field fluctuations is expected to induce thermal corrections, including a thermal plasma mass, for the background gauge field.

In this paper, we investigated the impact of this thermal mass correction on the background gauge field $\psi$. Our findings reveal that the evolution of $\psi$ can be significantly influenced by thermal effects, particularly at larger dissipation ratios $Q$ characteristic of warm inflation. A non-vanishing initial value of $\psi$ can be rapidly driven to zero by these thermal effects. Subsequently, the dynamics effectively reduces to that of minimal warm inflation without a background gauge field. Conversely, a non-zero gauge field mass, induced by a non-vanishing $\psi$, suppresses sphaleron processes responsible for the dissipation term in eq. (2.5). This suppression hinders warm inflation. Therefore, a successful warm inflation scenario in chromoinflation would be prevented.

We have demonstrated that the process by which the background gauge field transitions from a non-zero value to zero closely resembles a phase transition triggered by temperature variations. At a specific critical temperature, the gauge field undergoes a first-order phase transition. We have analytically characterized the properties of this phase transition. We have also illustrated it by an explicit numerical example.

In this work, we have primarily focused on the background dynamics. It is crucial to extend this analysis to perturbations to understand how they differ from standard warm inflation scenarios without a background gauge field. The interplay between the background gauge field and thermal effects could lead to significant deviations. Furthermore, the backreaction effects, commonly studied in cold chromoinflation [20, 45, 46], should be considered. These additional factors are essential for comparing the model’s predictions with observational data. We plan to address these aspects in a future work.

The authors would like to thank A. Berera and A. Maleknejad for discussions in the early stages of developing this work. V.K. would like to acknowledge the McGill University Physics Department and Trottier Space Institute for hospitality and partial financial support. R.O.R. acknowledges financial support by research grants from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant No. 307286/2021-5, and from Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Grant No. E-26/201.150/2021.