Schwinger Effect by an $SU(2)$ Gauge Field during Inflation

Non-Abelian gauge fields may exist during inflation. We study the Schwinger effect by an $SU(2)$ gauge field coupled to a charged scalar doublet in a (quasi) de Sitter background and the possible backreaction of the generated charged particles on the homogeneous dynamics. Contrary to the Abelian $U(1)$ case, we find that both the Schwinger pair production and the induced current decrease as the interaction strength increases. The reason for this suppression is the isotropic vacuum expectation value of the $SU(2)$ field which generates a (three times) greater effective mass for the scalar field than the $U(1)$. In the weak interaction limit, the above effect is negligible and both the $SU(2)$ and $U(1)$ cases exhibit a linear increase of the current and a constant conductivity with the interaction strength. We conclude that the Schwinger effect does not pose a threat to the dynamics of inflationary models involving an $SU(2)$ gauge field.


Introduction
A strong classical field can provide enough energy for the pair production of particles out of the vacuum. The Schwinger effect, the pair production by a static electric field, is one such example [1]. In a curved spacetime, the effect receives contributions from the gravitational field as well. Recently, several aspects of the Schwinger effect in a (quasi) de Sitter background have been studied for scalar and fermionic matter [2][3][4][5]. In these studies, the background field is a static and homogeneous U (1) gauge field with a preferred direction which slightly breaks the spatial isotropy. It has been shown that in this setup, the Schwinger effect leads to a sizable particle production and backreaction in the strong field regime.
In this work, we study the Schwinger effect by an SU (2) gauge field coupled to a massive scalar doublet, ϕ, in a (quasi) de Sitter background. It couples to the gauge fields via the covariant derivative term, (D µ ϕ) † D µ ϕ . Such an interaction is well-motivated since it appears in the Standard Model of elementary particles and fields, although our SU (2) and scalar fields are different from those in the Standard Model. We shall study the Schwinger production of scalar particles in an inflating universe and check if their energy is comparable to the background energy of the gauge fields. If it is, one expects the homogeneous gauge field modes to disappear quickly due to backreaction (the excited charged scalar field effectively screening the SU (2) background).
The Schwinger particle production was originally studied in the context of charged fermions [1] in a non-zero gauge field background. We consider a bosonic field instead, since we expect more particle production (due to the absence of Pauli suppression of occupation numbers) and therefore a greater possibility to have backreaction. In other words, if the Schwinger excitation of a bosonic field does not lead to sizable backreaction effects, we do not anticipate to see them in a fermionic model either.
Why study SU (2) fields in a de Sitter background? As was first discovered by one of the authors (A.M.) [6,7] and the subsequent works [8][9][10][11][12][13][14], inflation with SU (2) gauge fields shows rich phenomenology that is absent in the usual single-scalar-field inflation models. An important example is the generation of stochastic backgrounds of chiral gravitational waves (GWs) [9,13,15]. In particular, some gauge field models can generate observable primordial GWs either as a large field model or by sourced perturbations through their interaction with the gauge fields. Moreover, the mixing between the SU (2) gauge field and perturbations in the scalar and tensor sectors are at the linear order and coming from different fluctuations. Hence, unlike the U (1) gauge field, the enhancement of GWs and the modification in the scalar perturbations are uncorrelated at tree level. These setups are therefore able to generate detectable (and chiral) GWs [16] and tensor non-Gaussianity [17,18] while having little sourced scalar power spectrum and non-Gaussinities. Finally, having a non-zero source of parity violation during inflation, these models provide a natural setting for inflationary leptogenesis and explain the observed matter asymmetry in the Universe [19][20][21]. 1 While the details of the models can be different, there are several things in common between these models. Among them is an (almost constant) isotropic and homogeneous field configuration for the SU (2) gauge field. That gauge field vacuum expectation value (VEV) then generates almost constant electric and magnetic fields. We emphasise that the su(2) algebraic isomorphy with so(3) makes it possible to have a gauge field background which respects the spatial isotropy.
If backreaction on the SU (2)-background by the Schwinger effect is important, the copious generation of large-scale GWs can be suppressed. This can provide additional constraints in parameter space since regions with backreaction will not be relevant if GWs are observed. The dynamics of an isolated axion-SU (2)-spectator sector (only gravitationally coupled to other fields) already constrains the parameters of the system. The combination of the SU (2) coupling constant, g A , its background value, Q, and the Hubble parameter during inflation, H, ξ A ≡ g A Q/H, is bound to be √ 2 < ξ A 5, whereas B ≡ ξ 4 A H 2 /(g 2 A m 2 pl ) > 10 −10 for models with r vac > 10 −4 , see Ref. [18]. In this work, we study in detail the Schwinger effect in this class of models and address the issue of backreaction.
This paper is organized as follows. In Section 2 we present our theory in which we study the Schwinger effect. In Section 3 we study the evolution of a charged scalar field in the external SU (2) background and the resulting Schwinger pair production. The generated scalar current is presented in Section 4, and the details of the computations are given in Appendices A and B. In Section 5 we compute the backreaction on the gauge field background. We conclude in Section 6.

The Setup
We consider a charged complex scalar doublet which is coupled to a non-Abelian gauge field with the action The covariant derivative is given as where I 2×2 is a 2 × 2 identity matrix. We write the SU (2) gauge field as in which σ a are the Pauli matrices. The field equation of ϕ is Here, we are interested in the Schwinger process for ϕ during inflation in the presence of a (almost) constant, isotropic and homogeneous SU (2) gauge field. There are several inflationary models involving SU (2) gauge fields in the literature that can generate the desired gauge field configuration [6-11, 15, 16, 20]. In the temporal gauge with A a 0 = 0, we have [6] A a j = a(τ )Q(τ )δ a j , where a(τ ) is the Friedmann-Robertson-Walker scale factor, Q(τ ) behaves as a scalar under spatial rotations and δ a j is the Kronecker delta. In this paper, we choose to work with one particular example of inflationary models involving SU (2) gauge fields to constrain the model by the Schwinger effect. However, we emphasise that our conclusions are generic for models with the isotropic and homogeneous background gauge field configuration.
The inflationary model that we choose is specified by the action of the spectator model studied in [16] S where S EH and S φ are the Einstein-Hilbert and the inflaton actions, respectively, which are responsible for the inflation of the universe [24][25][26][27]. The axion, χ, and the gauge field act as spectator fields given by the action where V (χ) is the axion potential, f is the decay constant and F µν is the field strength tensor of the SU (2) gauge field The last term in Eq. (2.8) is the Chern-Simons interaction, with λ A controlling its strength andF µν being the dual of F µν . The S spec action is invariant under the local SU (2) transformation where U = e −ig A β(x ν ) with β = β a σ a /2. The action, S matter , is also invariant under the above local SU (2) transformation, provided that the charged scalar transforms as ϕ → Uϕ.
In the (quasi) de Sitter geometry of slow-roll inflation the gauge field value, Q, is a slowly varying quantity, and it corresponds to the following (almost) constant electric and magnetic fields In this setup, there is no preferred spatial direction. The equation of motion for the gauge field background is where the spatially averaged component of the matter 3-current is The covariantly conserved matter 4-current, J µ = J a µ σ a /2, satisfying ∇ µ J µ = 0, is given by the general expression In our setup, the scalar doublet generates the matter 4-current and its explicit form is For our later convenience, we define the dimensionless quantity 18) which is equal to the ratio of the magnetic and electric fields, ξ A B E . In some papers, ξ A has been called m Q , e.g., in [16][17][18]. Avoiding instability in the scalar perturbations requires as was found for the original gauge-flation and chromo-natural models in Refs. [14] and [13], respectively. One mode in the scalar sector of these models would have a negative frequency at k a ∝ (2 − ξ 2 A ) and avoiding instability requires the condition (2.19).

Schwinger Effect by SU (2) fields in de Sitter Universe
In this section, we study the evolution of the charged scalar doublet in the presence of the SU (2) gauge field and its corresponding Schwinger effect. In Fourier space, the field equation of ϕ takes the form where, for the isotropic and homogeneous gauge field given in Eq. (2.6), we have (3.2) Since in general the last term in Eq. (3.2) is non-diagonal, the two components of the scalar doublet, ϕ 1 and ϕ 2 , are linearly coupled. However, we can easily diagonalize the term (and thereby diagonalize L 2×2 (k, τ )). 2 In particular, we can expand the field in real space as where e λ k are the polarization doublets and are given by whereas the polar coordinates are defined byk = (sin θ cos φ, sin θ sin φ, cos θ). The polarization doublets satisfy the following relations e λ † k · e λ k = δ λλ , e λ † k · k a σ a · e λ k = λδ λλ k . where k = kk andk = (sin θ cos φ, sin θ sin φ, cos θ).
More explicitly, the new pair of fields are governed by the following decoupled equations of motion We now proceed to quantize the system by defining the canonically normalized fields The corresponding canonical conjugate momenta are We then promote q λ,k and their conjugate momenta to quantum operators, obeying the canonical commutation relations with all other commutators vanishing. In terms of creation and annihilation operators we have where a λ,k and b λ,k are the annihilation operators of a particle with charge +g A and −g A with respect to the asymptotic past vacuum (see also Eq. (3.30)), respectively. If we normalize the mode functions as with all other commutators of creation and annihilation operators vanishing. The mode functions are governed by Eq. (3.7) which can be reduced to where Using Eq. (2.11) and Q = const., as well as Eq. (3.14) becomes with general solutions given by linear combinations of the Whittaker functions, W κ λ ,µ (z) and M κ λ ,µ (z). The solutions which reduce to the vacuum expressions at early times, i.e., when Im(z) → −∞, are One can explicitly check that the expression on the right hand side in Eq. (3.18) tends to , (3.19) in the limit of kτ → −∞. 3 Note that the effective frequency squared given in Eq. (3.15) can be negative for the λ = + polarization state of the scalar field, e.g., for the interval and m = 0. Therefore, when ξ A < 2, the scalar field with the λ = + polarization state can experience a small enhancement in comparison to the λ = − state around the horizon crossing. However, the gauge field generates an extra mass for the scalar doublet as The super-horizon behavior of the scalar field amplitude goes as ϕ λ,k ∝ a it implies that ϕ λ,k always decays faster than a − 3 5 . Therefore, the scalar field always damps after horizon crossing and does not contribute to the super-horizon scalar perturbations in the inflation sector.

Schwinger pair production
In Section 3 we showed that the scalar field modes coupled to the SU (2) gauge field background in de Sitter spacetime get excited. The mode functions change from the early-time (kτ → −∞) form given in Eq. (3.19) to the late-time excited form given in Eq. (3.18). In order to interpret these scalar field excitations as particles, we need to be in the adiabatic limit Only then we can have a well-defined late-time adiabatic vacuum. The field excitations about it play the role of particles. 3 The W and M Whittaker functions have the asymptotic expansions implying that W /M functions correspond to positive frequency modes in the asymptotic past/future limits, respectively (see Eq. (3.27)). 4 In this case, ω 2 λ=+,k = 0 has two real roots, kτ1,2 = −ξA/2 ± 2 − ξ 2 A /2 .
In (quasi) de Sitter spacetime with Q const., the frequencies given in Eq. (3.15) can be written as At early times, kτ → −∞, the adiabaticity conditions given in Eq. (3.22) are trivially satisfied and the adiabatic vacuum is the Bunch-Davies vacuum [28]. That is what we take as an initial condition, see Eq. (3.19). At later times the adiabaticity conditions can be violated. However, in the asymptotic future, kτ → 0, they can become again adiabatic if From the combination of the above conditions, we find that µ in Eq. (3.16) should be a pure imaginary quantity and for the rest of this section, we write 26) and assume that the condition given in Eq. (3.25) holds. Then the new asymptotic future vacuum mode functions of q λ,k are (see Eq. They agree up to an unphysical constant phase with the positive-frequency WKB solutions of Eq. (3.14) in the asymptotic future, kτ → 0, By expanding q λ,k in terms of the annihilation and creation operations of the asymptotic future vacuum we write where |0 in and |0 out are the vacuum states in the asymptotic past, kτ → −∞, and future, kτ → 0, of the (quasi) de Sitter spacetime, respectively. Using Bogoliubov transformations, we can relateã λ,k andb λ,k to a λ,k and b λ,k as in which α λ,k and β λ,k are four Bogoliubov coefficients satisfying the normalization condition Comparing Eqs. (3.18) and (3.27), we find 5 .
(3.34) Note that in the adiabatic limit these Bogoliubov coefficients become independent of the momentum and (almost) constant in a (quasi) de Sitter spacetime Of course, the full solution of β λ,k must vanish in the k → ∞ limit.

Pair production rate and vacuum-vacuum transition amplitude
Having the β λ,k coefficient, we are ready to determine the particle number density as well as the vacuum-vacuum transition amplitude. The number density of the created particles with charge ±g A and a given comoving momentum, k, in the asymptotic future is which has a k-independent spectrum for each polarization state in the regime where the solution is valid. The total particle creation from the asymptotic past to the asymptotic future, therefore, is which appears to be divergent. This makes physical sense since it expresses the number of pairs created for all times. A more physically meaningful quantity is the pair production rate, i.e., the number of pairs produced per unit time per unit physical volume To calculate the derivative we need to convert the wavenumber integral into a time integral. We know that for |µ| 2 1, the asymptotic past and future are adiabatic vacua for the field 5 When reading the Bogoliubov coefficients, we used the following relation between the W and M functions (which holds when 2µ is not an integer and − 3π 2 < |argz| < π 2 ) [29,30] Mκ,µ(z) = Γ(2µ + 1) Moreover, using the relation Wκ,−µ(z) = Wκ,µ(z) and the fact that in our case z, µ and κ are all imaginary quantities, we have W * κ,µ (z) = W−κ,µ(−z). and therefore no particle production occurs in the infinite past and future. A pair of particles of a given comoving momentum, k, is produced only at a single non-adiabatic event, when the adiabaticity conditions, Eq. (3.22), are violated. Given |µ| 2 1, an approximate estimate for the moment when this happens is Hence, yielding constant pair production rates Integrating Eq. (3.38), we find that the physical number densities of pairs created up to a time τ are also time independent i.e., gravitational and Schwinger particle production are exactly balanced by the the gravitational redshifting, in the limit given in Eq. (3.25).
The production rate of the pairs with λ = ± polarization states are different and the λ = + state is generated much more. In particular, in the |µ| 2 1 limit, we have Finally, summing over the two polarization states of the scalar field, we have Since |µ| 2 1, we can approximate the above as That is due to the fact that β λ coefficients in Eq. (3.34) have always a negative exponential power, i.e., regardless of the parameters.
The vacuum-vacuum transition can be computed as where Υ vac is the vacuum decay rate. Using Eq. (3.39), we obtain Since |µ| 2 1, the decay rate is well approximated by yielding an exponential suppression.

Minkowski limit
Let us consider the Minkowski limit. Using the electric field (see Eq. (2.12))  In the previous section, we showed that the individual comoving modes can be amplified both gravitationally and due to the existence of non-Abelian background fields. We now calculate the induced current, J a µ , given in Eq. (2.17), as a consequence of this amplification. Let us take the expectation value of J a µ with respect to the vacuum state of the asymptotic past, |0 in . Writing the Fourier transforms of ϕ in terms of q λ,k , we find that the expectation value of the charge density, J a 0 , for the background configuration in Eq. (2.6), vanishes In deriving this result, we used the relations in Eq. (3.6). The above result makes physical sense since particles are created in pairs and the net charge density should be zero. The spatial components are We used the identity {σ a , σ b } = 2δ ab I 2×2 as well as Eq. (3.6). Using Eq. (3.18), Eq. (4.2) can be further reduced to where J is defined by Eq. (2.15) and is given by In the above, Λ is the UV cutoff on the physical momentum, Λ = k UV aH , which we eventually send to infinity. We will later also calculate the conductivity defined by where ψ(z) ≡ ∂ z Γ(z)/Γ(z) and Γ(z) is the gamma function. J (Λ) has a quadratic and a logarithmic UV divergent terms. To deal with this divergent behaviour, we follow [2] and regularize the current using the method of adiabatic subtraction. We define the regularized current as where the last term is the expectation value of the current density given in Eq.
WKB and u ± k | WKB are the WKB solutions up to second adiabatic order. We present the details of computations in Appendix B and here we only report the final result: (4.8) Subtracting Eq. (4.8) from Eq. (4.6), we find the desired regularized current The conductivity is then given by Eq. (4.5) for J reg .

Results
In Fig. 1, we show the current and the conductivity as a function of ξ A for different values of the scalar doublet mass, m/H. The current has some interesting features which we summarize in the following.
• The dimensionless quantity J reg /(g A H 3 ) depends only on ξ A and m H .
• The current is an odd function of ξ A ; thus, J reg (ξ A = 0) = 0. As a result, the conductivity is an even function of ξ A . Besides, for all positive (negative) values of ξ A , the current is positive (negative). •

2.
In this limit, the current increases linearly with ξ A and has the form while J WKB given in Eq. (4.8) has the following finite terms As a result J reg in the ξ A m/H limit is J reg 401 360 (4.14) In the limit that ξ A 1 and ξ A m H , by increasing the ξ A = B E value, the current decreases like 1/ξ A and the conductivity like 1/ξ 2 A . Since this regime corresponds to the strong gauge field, one may naively expect that the induced current can be estimated by the semiclassical relation J reg ∼ 2g A n pairs v, where v is the velocity of the particles. That is, however, not the case because the occupation numbers of the created particles are small (|β λ | 1) and n pairs is exponentially suppressed by the large value of ξ A (see Eq. (3.45)), rendering classical approximations invalid. 6 Here we used the asymptotic series for the gamma function

Comparison with the Schwinger effect by a U (1) field in de Sitter space
In this section we compare the particle production and induced current in our setup with those by a U (1) gauge field in the (quasi) de Sitter limit studied in [2]. Here we summarize the similarities and differences of these two types of the Schwinger effect: • In the U (1) case, the non-zero VEV of the gauge field breaks spatial isotropy. Therefore, both the Bogoliubov coefficients and the induced current are direction dependent.
In particular, the induced current is in the direction of the Abelian Electric field and κ k ∝ kz k . On the other hand, in our setup, the SU (2) VEV is isotropic and so are our current and Bogoliubov coefficients.
• For the U (1) electric field and a scalar field with charge e, we have a significant pair production and a sizable induced current in the strong field limit ( eE H 2 1). For instance, the exponent of the β k Bogoliubov coefficient in the U (1) case is [2] which for modes in theẑ-direction and small mass can be negligible and generates an order one β k , hence a significant pair production. Moreover, in the large field regime, the current grows like ( eE H 2 ) 2 with the electric field value. On the other hand in our case, the pair production is exponentially suppressed in the ξ A 1 limit regardless of the mass. That is due to the fact that in the SU (2) case, the β λ coefficients in Eq. (3.34) always have a negative exponent, i.e., iκ λ − |µ| < 0, regardless of the parameters. In the limit of ξ A 1 and ξ A m H , by increasing the ξ A value, the current decreases like 1/ξ A and the conductivity like 1/ξ 2 A .
• In the small field limit of eE H 2 m H for the U (1) and ξ A m H for the SU (2) setup, both systems have the similar current and conductivity behaviors. In particular, in this regime, the current grows linearly with the field strength and the conductivity is almost a constant.
• Unlike for the U (1) case, in the SU (2) setup, the induced current has an absolute • Refs. [2] and [4] study the scalar and fermion Schwinger effects by a U (1) field and show that there are regions in the parameter space in which the conductivity is negative. These parameters make Ω 2 k = ω 2 k + a a and hence the physical momentum zero. On the other hand, in our SU (2) setup, Ω 2 k is positive definite throughout the parameter space and the conductivity is always positive.
• These two setups are different even in the Minkowski limit. The U (1) case leads to the standard Schwinger effect in the flat space. However, the SU (2) gauge field with an isotropic and homogeneous field configuration has zero Schwinger pair production in the Minkowski limit.

Backreaction Constraints on Parameter Space
Up to now, we studied the scalar Schwinger effect in the presence of the (isotropic) SU (2) scalar gauge field VEV. During slow-roll inflation, these results are independent of the details of the inflationary model that generates the gauge fields. In this section, we study the importance of the backreaction of the induced current on the dynamics of the background SU (2) gauge field, Q, for the model given in Eq. (2.8). Specifically, we assume that the background gauge field experiences strong backreaction when the last two terms in Eq.
(2.14) are at least comparable and is approximately twice the energy density fraction of the gauge field. Here, we have used the standard single-field slow-roll inflation relation r vac = 2H 2 /(A s π 2 m 2 pl ) to parametrize the energy scale of inflation, i.e., r vac is the vacuum contribution to the tensor-to-scalar ratio in the absence of matter. The measured amplitude of the curvature power spectrum is A s ≈ 2.2 × 10 −9 [31]. The dashed red line and the shaded area underneath in Fig. 2 show the inequality in Eq. (5.2), i.e., regions with strong backreaction due to the induced current. Since J reg decreases monotonically with the mass (see Fig. 1), m = 0 gives the lower bound on B . We also find that B scales linearly with r vac and therefore the Schwinger particle production becomes increasingly less important as the energy scale of inflation is reduced.
The SU (2) backgrounds can source tensor gauge modes, which can also lead to backreaction. Effects on the homogeneous dynamics can arise if [18] The solid magenta line and the area underneath in Fig. 2 show the inequality given in Eq. (5.3), i.e., regions with strong backreaction due to the tensor gauge field modes production. Ref. [18]. The bound from the Schwinger particle production, depicted by the red dashed line (for m = 0) and the area underneath, does not lead to additional bounds on the observationally relevant parameter space. The ξ A on the horizontal axis is the same as m Q in Refs. [16][17][18].
It fully covers the backreaction region of the Schwinger particle production. Since the strength of the tensor gauge mode backreaction also scales linearly with r vac , it remains the dominant factor for constraining parameter space for all energy scales of inflation. We can also infer for what self-coupling constants, g A , the Schwinger effect becomes important. Given the parameter relationship we can draw the line g A = 1 separating the strongly and weakly coupled regions in B − ξ A space. In Fig. 2 the orange solid line and the shaded area underneath show the strongly coupled regions, g A > 1. There we expect non-negligible radiative corrections, i.e., loop contributions, to the gauge field dynamics due to interactions with the scalar field. Such effects could alter the SU (2) dynamics. Since we find that the Schwinger production backreacts on the background dynamics only for g A 1, we cannot be certain when exactly the scalar excitations affect the SU (2) evolution significantly. However, we can say with absolute certainty that they do not affect the SU (2) dynamics in the weakly coupled regime, g A < 1. This applies for all energy scales of inflation, since the right hand side in Eq. (5.6) again scales linearly with r vac .
We also conclude that the Schwinger effect can be ignored when the sourced gravitational waves by the SU (2) field have a substantial power spectrum and/or bispectrum. The dark blue solid line in Fig. 2 is the current upper bound on the tensor power spectrum [32]. The light blue, r source = 0.01, and yellow, r vac = r source = 0.001, solid lines show different values of the total tensor-to-scalar ratio. In the region left of the yellow line, the sourced gravitational waves have lower power than the vacuum fluctuations in the metric. The light and dark blue dashed lines in Fig. 2 show the current and target constraints on the amplitude of tensor non-Gaussianities, f tens NL [18], respectively. The estimated Schwinger bound lies well below any of these observationally interesting regions.

Conclusions
In this paper, we have studied the Schwinger effect of a charged scalar doublet by an (isotropic and homogeneous) SU (2) gauge field during inflation. We analytically derived the explicit form of the induced current and Schwinger pair production. We found that the Schwinger effect by the SU (2) gauge field is very different from that by a (homogeneous) Abelian gauge field with a preferred spatial direction. We showed that these two cases are very different even in the Minkowski limit. In particular, the SU (2) gauge field VEV adds an extra mass term to the scalar field in such a way that this setup has negligible particle production even in the strong field limit. That is unlike the standard Abelian Schwinger effect in which a complex scalar field with a small mass experiences a significant pair production in the strong field limit [2]. The ξ A ≈ 2m/H point in parameter space divides the behavior of the induced scalar current into two different regimes. For ξ A < 2m/H, the current increases linearly with ξ A , while for ξ A > 2m/H it decreases like ξ −1 A . Its maximum value at ξ A ≈ 2m/H is J max ≈ 5 × 10 −3 g A H 3 /m.
Finally, we used our result to study the possible backreaction of the induced current on the non-Abelian background dynamics and to further constrain an axion-SU (2) inflation model. We found that the backreaction by the induced current is important only for cases when backreaction due to tensor gauge mode amplification is also significant and the self-coupling of the SU (2) fields is strong, g A 1. Therefore, the Schwinger production of charged scalar fields by an SU (2) field does not yield additional constraints on the observationally relevant parameter space.
We expect that our conclusions can be extended to charged spin-1/2 fields, since the effect will be even weaker due to suppression in particle production by the Pauli exclusion principle. A rigorous proof is left for future work. The isotropy of the SU (2) gauge field played an important role in suppressing the particle production and the decrease of the induced current in the strong field regime. Therefore, an important question is a possible effect of statistical anisotropy on the Schwinger process by an SU (2) gauge field during inflation which we also postpone for future work.

Acknowledgments
We are grateful to Aniket Agrawal and Takeshi Kobayashi for helpful discussions and to Giovanni Cabass for a careful proof-reading. EK thanks Raphael Flauger for the question he asked during the 'Fundamental Cosmology' meeting held at Teruel from September 11th to 13th, 2017, which led to this project. He also thanks Emanuela Dimastrogiovanni, Matteo Fasiello, and Tomohiro Fujita for useful discussions.

A Computation of the total current
In this appendix, we present the computation of the current, J . The current due to a constant electric field (with a preferred spatial direction) has been worked out in [2]. Comparing to the U (1) case, our current integrand enjoys spatial isotropy and the details of κ and µ coefficients are different and direction independent. The expectation value of the current given in (4.4) can be written as where G λ is the following integral Hereτ is a rescaled physical momentum and Λ is the UV cutoff onτ τ ≡ k aH and Λ ≡ k UV aH .
We shall send Λ to infinity in the end. In order to perform the integral in (A.2), we use the Mellin-Barnes integral representation of the Whittaker functions [30] which holds when and the contour of the integration separates the poles of Γ( The Γ(z) function has simple poles for non-positive values of z at In our setup, κ is pure imaginary and µ is either real or pure imaginary (i.e., κ * = −κ and µ * is either µ or −µ). Moreover, from (3.16) and recalling the fact that the consistency of our model requires ξ A > √ 2, we have 0 ≤ Re(µ) < Thus, the complex integrand of s -integral has singularities at s = − 1 2 ±µ−n and s = κ+n as well as at s = 3 − s and s = 2 − s. Moreover, since the integrand is proportional to Λ 2−s−s in the limit that Λ → ∞, it vanishes for Re(s ) > 3 − Re(s). Upon choosing the contour of s such that Re(s) > −1 and closing the s -contour in the right-half plane without passing through the poles, we are left with the following six poles s 1 = κ, s 2 = κ + 1, s 3 = κ + 2, s 3 = κ + 3, s 5 = 2 − s and s 6 = 3 − s. (A.8) Notice that an integral of (A.7) over a finite path along the real axis vanishes at lim Im(s ) → ±∞ (due to the Gamma functions) and therefore the added integration contour does not change the result. Using the Cauchy integral theorem, we can do the s -integral

Using (A.4), we obtain
This integral is divergent and includes terms proportional to Λ 3 , Λ 2 and Λ. However, the last two terms are finite and independent of Λ. Now we turn to compute the last complex integral with respect to s which is In doing the integral of the four Λ-dependent terms, we close the contour in the right-half plane. Here, we have the following poles in the Λ → ∞ limit and, unlike the first complex integral, some of them are not simple, but of order 2. 8 In order to compute G λ (Λ) in (A.10), first we compute the second complex integral for each line of G λ (s, Λ) in (A.9) individually. Specifically, we label the integral corresponding to the I-th line in (A.9) as G λ,I (Λ), where I = 1, 2, ...6. in the end, we sum up all the G λ,I (Λ)s and determine G λ (Λ). Doing the second complex integral in (A.10), the first line 8 Here we use the following formula in the complex analysis that if f (z) has a pole of order k at z = z0 then the residues are given as (A. 11) of G λ (s, Λ) in (A.9) leads to the following G λ,1 (Λ): This integral includes a derivative of the Γ-function which can be written as a digamma function In particular, the second order roots in (A.10) leads to the derivative terms of the form 9 when n is an integer. Using (A.13), we obtain G λ,1 as . 9 Here we used the fact that the harmonic series is n q=1 1 q = −ψ(1)+ψ(n+1), and the following relations Next, doing the s-integral and using (A.13), we find G λ,2 as Now we turn to compute the last two (Λ-independent) terms in (A.9) and find G λ,5 and G λ,6 . Using the following relation for Γ functions Γ(z)Γ(−z) = − π z sin(πz) (z / ∈ Z), (A. 15) we find that G λ,5 + G λ,6 can be written as The integrand has an infinite number of poles. Therefore, it is more convenient to express it in the following form in which g(s) and f (s) are The first two terms in integral (A.18) can be written as 10 The integrand has poles at s = s n,± and s n,0 (see (A.5) and (A.6)) as well as at s n,± = 1 2 + n ± µ ands n,0 = −κ − n, in which poles s n,0 with n = 0, 1, 2 are 2nd rank while the rest are simple poles. Therefore, it is more convenient to close the contour path of s on the left half-plane. Thus, only the simple poles below contribute to the I λ,1 integral Note that if µ is real (0 < Re(µ) < √ 3 2 ), the poles 0,− = 1 2 − µ might be on the left halfplane. However, as it has been shown in figure 3, it is always possible to choose the contour path so thats 0,− be out of the contour. 11 We obtain (A.24) 10 Here we used the fact that µ is either pure imaginary or real (i.e. µ * = ±µ) and hence the four Gamma functions in (A.24) are equal to π 2 cos(π(µ−s)) cos(π(µ+s)) which is invariant under s → s − 1. 11 For µ = 1 2 we haves0,− = s0,+ and the above contouring is not possible. However, in that case, we can assume µ = 1 2 and take the limit µ → 1 2 .

B Adiabatic Subtraction and the regularized current
We use the adiabatic subtraction technique in curved QFT to remove the divergent terms in the current. Consider the WKB approximation form of the mode function u λ,k (τ ) = 1 (2π) 3 2 2W λ,k (τ ) which is equal to the exact solution of the field equation when Notice that for all values of k, Ω 2 λ,k is positive definite, provided m 2 > 0. When W λ,k is real and positive,ū λ,k (τ ) corresponds to the (canonically normalized) positive frequency modes in the asymptotic past, i.e., the Wronskian is equal to i(2π) −3 .
To use the adiabatic subtraction technique, here we define an adiabatic parameter T −1 which in the limit T → 0 parametrizes an infinitely slow varying background geometry [33]. We, then, assign a power of T −1 to each time derivative in the last three terms in (B.2). For the adiabatic subtraction approach, one needs to only expand W λ,k up to the T −2 order In computation of the current integral, we must keep terms up to T −2 order and not any further. For more details on the adiabatic subtraction, see Ref. [33]. Using (B.4) in (4.4) and expanding the integrand up to order T −2 , we have