Magnetogenesis from rotating scalar: \`a la scalar chiral magnetic effect

The chiral magnetic effect is a phenomenon that an electric current parallel to the magnetic fields is induced in the presence of the chiral asymmetry in the fermionic system. In this article we point out that the electric current induced by the dynamics of a pseudo scalar field that anomalously couples to electromagnetic fields can be interpreted as a similar effect in the scalar system. Noting that the velocity of the pseudo scalar field, which is the phase of a complex scalar, represents that the system carries a global U(1) number asymmetry, we see that the induced current is proportional to the asymmetry and parallel to the magnetic field, which is the same to the chiral magnetic effect. We discuss that in a mechanism like the Affleck-Dine mechanism an asymmetry carried by the Affleck-Dine field can induce the electric current and give rise to the instability in the (electro)magnetic field if it is unbroken by the expectation value of the Affleck-Dine field. Cosmological consequences of this mechanism, which is similar to the chiral plasma instability, is investigated.


I. INTRODUCTION
The chiral magnetic effect (CME) [1,2], from which electric currents are induced by magnetic fields in the presence of chiral asymmetry, has recently received strong interests in broad range of fields of study. Since it is originated from quantum anomalies [3,4], which are ubiquitous in quantum systems regardless of their energy scales, it has been noticed that it can play important roles in relativistic heavy-ion collisions [2,[5][6][7][8][9][10][11][12], Weyl semimetals [13][14][15][16][17][18][19], astrophysical objects such as neutron stars [20][21][22][23][24] and supernovae [25,26], and so on. Moreover, in the hot early Universe when the chiral asymmetry is a good conserved quantity [27], it has been argued that the CME can cause a tachyonic instability in the (hyper)magnetic fields [28,29], now known as the chiral plasma instability, which is studied very recently with a full magnetohydrodynamic simulations [30][31][32][33]. In these studies, it has been identified that the maximal transfer from the chiral asymmetry to the magnetic helicity is likely to be accomplished. The generated (hyper)magnetic fields are maximally helical and hence they can be the source of the baryon asymmetry of the Universe [34][35][36][37][38][39][40]. 1 In these phenomena, the chiral asymmetry carried by light fermions is essential since it is the origin of the CME. On the other hand, it has been noticed that the chiral anomaly originated from heavy fermions leaves its traces in the low energy effective theory of axions [49][50][51][52] in the form of anomalous coupling between the axion and gauge bosons [53]. 2 Then, the background dynamics of axion field also induces an electric current, similar to the CME. It has also been argued that the chiral asymmetry is interpreted effectively as an axion-like scalar degree of freedom [55][56][57]. Indeed, the cosmological coherent dynamics of axion-like fields are applied for the generation of cosmological magnetic fields during [58][59][60][61] and after inflation [62][63][64], or later times [65].
Once we would like to interpret the magnetic field amplification from the axion-like fields in a similar way to the CME, the dynamics of the axion-like fields can be identified as the non-vanishing chemical potential of the global U(1) PQ symmetry, which is similar to the chiral chemical potential. Then the difference between the chiral plasma instability and magnetic field amplification from the axion-like fields in the literatures [58][59][60][61][62][63][64][65] lies in the 1 Due to the baryon overproduction [39,40], it is impossible for this mechanism to be responsible for the intergalactic magnetic fields suggested by the blazar observations [41][42][43][44][45][46][47][48]. 2 See also the recent discussions in Ref. [54].
conservation of chirality/global U(1) PQ charge. In the former case, after the generation of chiral asymmetry, it is assumed that the chiral symmetry gets back a good symmetry of the system in the absence of chiral anomaly so that the maximal transfer from the chiral asymmetry to the magnetic helicity is possible. On the contrary, in the latter case, the axion dynamics is induced by the global U(1) PQ symmetry breaking potential and hence U(1) PQ charge is not a conserved charge when the magnetic fields are amplified. Thus, in a sense, the magnetic field amplification from the axion-like fields in the literatures is not maximally efficient.
We here note that the anomalous coupling of the axion-like fields is not limited to the QCD axions or axion-like particles appeared in the string theory but is common to pseudo scalar fields in general. One example that takes advantage of the pseudo scalar dynamics in cosmology is the Affleck-Dine (AD) mechanism for baryogenesis [66,67]. In this mechanism, a complex scalar field with baryonic (or leptonic) charge acquires an expectation value in the early Universe, and an explicit baryon (or lepton) number violating interaction gives once the non-vanishing velocity in the phase direction (or the Nambu-Goldstone mode) or the baryon (or lepton) asymmetry. It is implemented that the baryon (or lepton) number violating interaction gets ineffective quickly so that the baryon (or lepton) number becomes a good conserved quantity after its generation. As a consequence, the complex scalar exhibits a coherent rotation in the field space with a constant angular velocity. In this article, we point out that if a complex scalar field charged underà la global U(1) PQ symmetry whose phase direction has an anomalous coupling to the U(1) gauge fields evolves in a similar way to the AD mechanism, the magnetic fields are amplified through the induced current from a constant global U(1) asymmetry, in which the maximal transfer from the asymmetry to the magnetic helicity is possible. Thus, the mechanism is more efficient than the previous realizations of the magnetogenesis through the axion-like field dynamics and is similar to the chiral plasma instability. This mechanism has several important messages on the model building of the early Universe cosmology. On the one hand, if the axion-like fields including the Peccei-Quinn-Weinberg-Wilczek (PQWW) axions [49][50][51][52] experiences the cosmological evolution like the AD mechanism, it leads to a new mechanism of magnetogenesis. On the other hand, if the phase direction of the AD field has an anomalous coupling to the unbroken U(1) gauge fields, the magnetic fields amplification can occur even in the usual AD mechanism in the minimal supersymmetric standard model (MSSM) and other supersymmetric extensions of the Standard Model of particle physics (SM), which has not been noticed before. Indeed, we will show that in some flat directions in the supersymmetric SM, the phase direction of the AD fields has the anomalous coupling to unbroken U(1) gauge symmetry and this new mechanism can be realized along the flat directions. This may change the cosmological consequences of the AD mechanism such as the Q-ball formation [68][69][70][71][72][73][74]. One may wonder if it may spoil the AD mechanism as the baryogenesis mechanism. Indeed, the baryon or lepton asymmetry carried by the AD field is first transferred to the magnetic helicity and it once gets smaller. But the baryon asymmetry is regenerated at the electroweak symmetry breaking through the transfer from the magnetic helicity, as is shown in Ref. [39]. Thus the AD mechanism can still be responsible for the baryon asymmetry, but it is somehow indirect, like the case discussed in Ref. [40]. This paper is organized as follows. In the next section we will study the cosmological consequences of the complex scalar fields with the anomalous coupling that experiences the evolution like the AD mechanism and determine the resultant magnetic field properties generated by this new mechanism in terms of the model parameters. In Sec. III we discuss the realization and embedding of the system for this mechanism in well-motivated models beyond the SM. Sec. IV is devoted for our concluding remarks and future prospects of this mechanism.

II. MAGNETOGENESIS FROM ROTATING SCALAR IN THE FIELD SPACE
A. Axion-induced current as the scalar chiral magnetic effect First we study a toy model as a low energy effective theory and investigate its cosmological consequences. In the next section we will discuss the realizations of the scenario in the realistic models of physics beyond the SM. Let us consider a simple model of a complex scalar field (à la AD field) with an approximate global U(1) A symmetry and a massless U(1) gauge field, motivated by the AD mechanism [66,67], which catches the essence of our idea, Note that we take that the scalar field φ is neutral under the U(1) gauge interaction. Here we adopt the metric convention g µν = (−, +, +, +) and consider the Friedmann background ds 2 = −dt 2 + a 2 (t)dx 2 with H =ȧ/a being the Hubble parameter. We use the dot as the derivative with respect to the physical time t. m 0 is the zero-temperature mass, c H is a numerical coefficient of the order of the unity that parameterizes the negative Hubble induced mass, b φ and a φ parameterizes the small global U(1) A symmetry breaking terms (b φ and a φ -terms, respectively), and M is the cutoff scale of the higher-order operators. We assume that the scalar field receives the negative Hubble induced mass during and after inflation before reheating and the value of c H does not change significantly. b φ is taken to be a real while a φ is taken to be complex without loss of generality.F µν = ǫ µνρσ F ρσ /2 √ −g is the dual tensor with ǫ µνρσ being the Levi-Civita symbol, ǫ 0123 = 1, θ is the phase of the scalar field φ, e is the gauge coupling constant, and c F is the numerical coefficient of the order of the unity for the anomalous coupling. As the phase θ is the (pseudo) Nambu Goldstone boson associated with spontaneous symmetry breaking of U(1) A symmetry, it can also be regarded as an axion-like field.
When the Hubble parameter is much larger than the zero-temperature mass, the net mass term is negative and the scalar field gets an expectation value, Once the phase of the scalar field acquire the non-zero velocity,θ = 0, or the scalar field rotates in the field space, due to e.g., the U(1) A breaking a φ -term, like in the case of the AD mechanism [66,67], this represents the U(1) A asymmetry is induced in the system, Taking this configuration as the background, it can be easily seen that the equation of motion for the gauge field reads Thus we can identify that we have an induced current by the number density of the U(1) A asymmetry, so that this induced current mimics the chiral magnetic effect [1,2] with a correspondence Here the physical electric and magnetic fields are defined as This induced electric current is nothing but the axion-induced current in the axion electromagnetism. We should also note that it has been argued that the chiral magnetic effect is understood as an effective axion field in literatures [55][56][57]. We here just simply emphasize that by relating the axion velocityθ to the number density of the U(1) A asymmetry, the correspondence between the chiral magnetic effect and the axion-induced current is clearer.
Note that the number density of the chiral asymmetry at high temperature T is given in terms of the chiral chemical potential by n 5 = µ 5 T 2 /6.

B. Generation of U(1) A asymmetry and magnetogenesis in the early Universe
The axion-induced current causes the tachyonic instability on the gauge fields, which is the essence of the axionic inflationary magnetogenesis [58][59][60]. In that case, the nonzeroθ is induced by the potential that strongly breaks the U(1) A symmetry and hence the corresponding asymmetry n A is not a constant during the course of the magnetic field amplification. Especially during the axion oscillation, subsequent to the slow-roll inflation, θ changes its sign constantly. Thus the magnetic field amplification is less efficient, and the process is somehow different from the chiral plasma instability [28][29][30][31][32]40]. In contrast, if the scalar field rotation in the field space is induced by a U(1) A breaking term that gets ineffective just after its onset, the U(1) A asymmetry is an approximate conserved quantity and n A orθ can be taken as a constant, until the backreaction becomes important. In that case, the process is quite similar to the chiral plasma instability. In the following, we investigate the mechanism to generate the U(1) A asymmetry in the similar way to the AD mechanism [66,67], and the magnetogenesis from that.
Suppose the Universe has experienced inflation and the inflaton oscillation dominated era with the matter domination like evolution of the scale factor, a ∝ t 2/3 , follows. Here we adopt the model with Eq. (1) 3 , assuming m 0 ≃ |a φ | ≫ b φ . When the Hubble parameter is large during inflation and during the inflaton oscillation dominated era, H > m 0 / √ c H , φ field follows the (time-dependent) potential minimum generated by the balance between the negative Hubble induced mass term and the |φ| 2n−6 term, ϕ ≃ (HM n−3 ) 1/(n−2) , with a spatially homogeneous distribution. Thanks to inflation, we also suppose that the phase direction θ also distributes spatially homogeneously and is taken to be a constant. As the Hubble parameter decreases, eventually the potential minimum disappear at H osc ≃ m 0 / √ c H and the φ field starts oscillating coherently around the origin. At the onset of oscillation, the a φ -term gives the kick in the phase direction so that non-zero number density of U(1) A charge, is generated and the trajectory of the scalar fields in the complex field space is an ellipse with a small eccentricity for a φ ∼ m 0 [67]. Here the subscript "osc" indicates that the quantity is evaluated at the onset of the scalar field oscillation. Soon after the onset of oscillation, U(1)-breaking a φ -term gets ineffective quickly and n A becomes a good conserved quantity.
Then the scalar field evolve as as long as the b φ -term is negligible. The former comes from the fact that both the real and imaginary part of the scalar fields are the harmonic oscillator in the matter dominated Universe and damp in proportion to t −1 and the latter is derived from the comoving number density conservation, a 3 n A = a 3θ ϕ 2 = const. Now let us examine how the gauge fields are amplified due to the tachyonic instability and how they backreact to the scalar field dynamics. The equations of motion for the phase 3 Here we do not take into account the thermal effects [75,76] to show our idea simply and clearly. In principle, there should be thermal corrections to the scalar potential even before the completion of reheating, since the partial decay of inflaton quanta generates a high temperature plasma as a subdominant component of the Universe. The absence of such thermal corrections are valid if, e.g., the inflaton decays mainly into a hidden sector and the SM particles are not produced. If thermal corrections to the scalar field potential exist, they induce an early onset of the scalar field oscillation, whose eccentricity is larger.
direction of the scalar field and gauge fields are given by The latter exhibits the instability of the gauge fields for non-zero backgroundθ. It can be explicitly seen as follows. As long as the phase direction evolves with a homogeneous constant velocity, ∂ µ θ ≃ (θ, 0, 0, 0), with a negligible backreaction, we can take them as a background for the evolution of the gauge fields. Switching from the physical time to the conformal time so that the metric is ds 2 = a 2 (τ )(−dτ 2 + dx 2 ), the equation of motion for the gauge fields reads where we work in the radiation gauge ∇ · A = 0, A 0 = 0. To solve the equation of motion, it is convenient to work in the momentum space with performing a Fourier transformation, with ǫ i,h (k) being the circular polarization tensor that satisfies With these decomposition the equation motion for the Fourier modes is rewritten as We can see that the last term acts as a tachyonic mass term forθλ > 0 and triggers the instability of gauge fields. For the inflaton oscillation dominated Universe with a(t) ∝ t 2/3 , a(τ ) ∝ τ 2 , the ± mode of the gauge field feels instability forθ ≷ 0 at k ins /a(τ ins ) ≃ c F e 2θ /4π 2 around a(τ ins )τ ins ≃ 4π 2 /(c F e 2θ ) or H ins ≃ c F e 2θ /8π 2 ≃ c F e 2 m 0 /8π 2 and grows exponentially. As a result, maximally helical gauge fields are produced. Here the subscript "ins" indicates that the quantity is evaluated at the time when the instability starts to grow.
We here assumed that there are no light charged degrees of freedom. If they exist, electric currents are induced like the Schwinger effect. Then the magnetic field amplification would get less efficient [61] and the light particles may be thermalized [77]. Hence the process of gauge field amplification gets more involved, which is beyond the scope of the present article. See also the discussion at the end of Sec. III B.
The amplification of gauge fields stops when the backreaction gets non-negligible. Since by taking the spatial average, Eq. (10) can be understood as the conservation law between the asymmetry and the magnetic helicity, we can estimate that when the amplification of gauge fields get saturated. In other words, magnetic field amplification stops when the maximal transfer from the chiral asymmetry to the magnetic helicity is completed. Here the subscript "sat" indicates that the quantity is evaluated at the time when the gauge field amplification gets saturated. Note that since the instability is an exponential grow, we can approximate that τ sat ≃ τ ins . Focusing on the magnetic fields, by we obtain the physical magnetic field properties, the magnetic field strength B and coherence length λ B , at the time when the gauge field amplification gets saturated as Here we take e ≃ 0.3. It is noted that B sat is independent of c F while λ B,sat is inversely proportional to c F . Note also that in the absence of thermal plasma, the electric fields with a similar amount to the magnetic fields are produced at the same time.

C. Cosmological evolution of magnetic fields
Thus far we have not specified the relationship between the fields in the model to the particle contents of the SM and the arguments are also applicable for the hidden U(1) gauge fields. Let us investigate the cosmological consequences in the case if the gauge fields are those of the U(1) gauge symmetry in the SM. After the saturation of gauge field amplification, the physical magnetic field (as well as the electric field) evolves adiabatically, B ∝ a −2 and λ B ∝ a, until the SM particles are thermalized and the magnetohydrodynamics becomes important for their evolution [37,80]. Once the SM particles are thermalized, the electric fields are screened due to the thermal effect while magnetic fields keep their properties. The magnetic fields induces the fluid dynamics and the fluid develops a turbulence. Then both magnetic fields and velocity fields start to co-evolve according to the magnetohydrodynamic equations and follow the inverse cascade process once the eddy turnover scale of the fluid catches up the magnetic field coherence length, [78,79]. The magnetic field further evolve until today according to the magnetohydrodynamics, which determines the linear relation between the magnetic field strength and coherence length today as [78] λ where t 0 is the present physical time. On the other hand, thermal plasma induces a large electric conductivity, which makes the comoving magnetic helicity is a good conserved quantity. Since during the adiabatic evolution it is also conserved, we have the relation Then we have where we have used g * s = 3.91, T 0 = 2.3 × 10 −13 GeV, 1pc = 1.56 × 10 32 GeV −1 , and 1 G = 1.95 × 10 −20 GeV 2 (in natural Lorentz-Heaviside units) and assumed that at H = H RH , the Universe is filled with relativistic particles with effective temperature T RH with the energy density and entropy is given by ρ = (π 2 g RH * s /30)T 4 RH , s = (2π 2 g RH * s /45)T 3 RH . We have also assumed that the Universe is eventually filled with the SM radiation without additional entropy production. Combining it with Eq. (21), and assuming c F ≃ 1, we obtain the present magnetic field properties, Thus the detection of intergalactic magnetic fields with maximal helicity can be a trace of this scenario.
Moreover, we note that the set of fiducial values is suitable for baryogenesis [39]. This is not surprising because if there is not a magnetic field amplification and the asymmetry is conserved, the asymmetry-to-entropy ratio is for the fiducial values. In this scenario, if the generated magnetic fields are those of hypergauge interaction, the asymmetry produced by the scalar field dynamics is first transferred to the hypermagnetic helicity, and it is eventually transferred back to the baryon asymmetry at the electroweak symmetry breaking without large loss in the sum of magnetic helicity and U(1) A asymmetry, as is similar to the case studied in Ref. [40]. Even if the electroweak symmetry is broken down to the electromagnetism by the expectation values of the scalar field and the electromagnetic fields are produced in this scenario, they transform into the hypermagnetic fields once when the scalar field decays. Then the same process in the above follows.

D. Comment on the b φ -term
Thus far we completely omitted the effect of b φ -term to avoid the time variation of the U(1) A asymmetry. However, in the phenomenological point of view, this term is unavoidable in some realizations as we discuss in the next section. We here discuss how small this term should be for this mechanism to work.
Let us examine the evolution of the scalar fields in more depth after the onset of oscillation.
Taking into account the b φ -term, the mass of the scalar field in the real and imaginary part differs as When b φ is hierarchically smaller than m 2 0 , ∆m ≃ b φ /m 0 ≪ m 0 . Since the evolution of the scalar fields is given by thenθ is given bẏ for the criteria for the duration during whenθ can be regarded as a constant.
Since the magnetic field amplification occurs with the time scale Requiring that this is much shorter than ∆t c , we obtain the constraint on ∆m as This gives a constraint on the b φ -term in the phenomenological model building, which is discussed in the next section.

III. REALIZATION
In this section, we describe how the low energy effective Lagrangian of the form of Eq (1) is realized in the well-motivated models. The idea is completely analogous to the axions.
Namely Let us see in more depth how to realize the situation of our interest in the type-II 2HDM, especially how to realize the coherent motion of the Higgs fields and the vanishingly small b-term as discussed in Sec. II D.

Scalar potential
Let us first investigate how to construct the scalar potential in the type-II 2HDM that allows the Higgs field to develop expectation value during inflation with |H 1 | = |H 2 | = ϕ/2 and allows us to identify the ϕ field as the AD(-like) field. The SM gauge charges as well as PQ charges for the SM fields in the type-II 2HDM are given as Table I, which allows us to determine the Yukawa couplings as For the Lagrangian of the Higgs sector the form of the scalar potential V (H 1 , H 2 ) is crucial to realize our setup. Note that the PQ symmetry is anomalous under the hypergauge interaction, which is essential for our scenario, as we will see.
There are eight degrees of freedom of the Higgs fields in total, which we characterize in terms of the four complex scalars as Indeed, we can construct a scalar potential with a flat direction along a complex scalar degree of freedom among the four, while other six degrees of freedom are heavy enough along the flat direction. To realize such a feature, the scalar potential inspired by the supersymmetric theories can provide a good example. We can borrow some key ideas of the form of scalar potential in the supersymmetric SM for illustration. In the softly-broken supersymmetric SM, two Higgs doubles are naturally introduced. There are three contributions to the Higgs potential, namely, from the D-term, F -term, and soft breaking terms. Assuming that the PQ symmetry of the Higgs sector is broken only by the following higher dimensional superpotential, where M is the cutoff scale, the scalar potential of H 1 and H 2 is obtained as where the first three terms are the soft supersymmetry (SUSY)-breaking terms with m 1 ∼ m 2 ∼ a H being soft breaking parameters of O(0.1 − 1TeV), and the last term in the first line is the F -term contribution. The quartic potential in the second line is the D-term potential, which gives the approximate flat direction in the unitary gauge: spontaneously broken to U(1) em , and denoting the fields along the flat direction as δH, three scalar degrees, G 0 ≡ Im(δH 0 1 − δH 0 2 ) and G + ≡ δH + 1 − δH − * 2 , G − ≡ G + * are eaten by Z 0 and W ± , and become massive with masses gϕ/2 and g 2 + g ′2 ϕ/2, respectively. One of the CP-even Higgs degrees of freedom, H 0 ≡ Re(δH 0 1 −δH 0 2 ) and the charged Higgs components, and H − ≡ H + * , are also heavy with masses g 2 + g ′2 ϕ/2 and gϕ/2 at the leading order. The scalar fields ϕ and θ get masses only from soft terms and a higher dimensional operator and hence they are much lighter than the above six scalar degrees of freedom. Note that the θ field is nothing but the CP-odd Higgs or the PQWW axion.
The negative Hubble induced mass terms for H 1 and H 2 can be added as by supposing, e.g., the non-minimal couplings to gravity, −ξ 1 R|H 1 | 2 − ξ 2 R|H 2 | 2 with R being the Ricci scalar, or non-trivial Köhler potential between the inflaton and the Higgs doublets in the supersymmetric case [67,81] (32)), the value of |H 0 2 | at the present Universe is too small to be realistic because |H 0 2 | / |H 0 1 | ≃ |b H |/(m 1 + m 2 ) 2 . This leads to non-perturbatively large Yukawa couplings to obtain correct masses of down quarks and charged leptons, m d/e = y d/e |H 0 2 | . One way to avoid this problem and give more freedom to the b H -term is to consider the case where the b H -term in the present Universe is dominated by the vacuum expectation value of a scalar field as b H ∼ S 2 = O(m 2 1 + m 2 2 ), by introducing a gauge singlet PQ charged complex scalar field, S, while the soft-breaking b H -term contribution is vanishingly small. Let us consider the following potential for the S field, Here |m S | ∼ |a S | = O(m 2 1 + m 2 2 ), κ, κ 1 , κ 2 , and λ S are parameters of the order of the unity, and a S is the soft PQ breaking parameter, which allows S field develops an expectation value of order of 0.1 − 1 TeV in the vacuum to give the b H -term to the Higgs doublets.
When the Higgs field develops the expectation values along the flat direction, H 1 ≃ H 2 ∼ ϕ, S becomes heavy with a mass of O(ϕ), and its vacuum value shifted by a S -term is quite suppressed as S ∼ a 3 S /ϕ 2 ≪ m 0 = (m 2 1 + m 2 2 )/2. The resulting b H = κS 2 ∼ a 6 S /ϕ 4 is much smaller than m 2 0 so that an effective magnetic field generation is allowed. As the ϕ field value decreases and becomes O(m 0 ), then S ∼ m 0 , and b H ∼ m 2 0 , so the PQWW axion becomes heavy with a mass of O(m 0 ), and can be safe from various constraints at the present Universe.
We would like to emphasize that the scalar potential we suggest in this section is a proof of concept, in which a flat direction (|H 1 | = |H 2 |) exists and b H -term is dynamical, which is suitable for our magnetogenesis scenario. Clever ideas are welcome and desirable in order to provide more natural set-up for our mechanism. See App. A for a concrete example to realize the H 1 H 2 flat direction without a bare b H -term in a supersymmetric extension of the SM (H 1 → H u , and H 2 → H d ).

Effective action with light degrees of freedom
Let us now see how the anomalous coupling ∼ (e 2 /16π 2 )θF µνF µν is obtained in the low energy effective Lagrangian. Here we focus on the non-supersymmetric theory although we use the SUSY-inspired potential. When the Higgs fields obtain large field values along the flat direction ϕ ≫ m 0 , we can divide the fields, not only the Higgs field described in the above but also the matter and gauge fields, into the heavy fields whose masses are proportional to ϕ, and the light fields which are massless or obtain masses at most with the soft breaking scales. The former includes the quarks, leptons except for the neutrinos, weak gauge bosons, and heavy Higgs fields, as well as the singlet scalar S, if any, and the latter includes the gluons, (electromagnetic) photon, neutrinos, and light Higgs field (theà la AD field). In the unitary gauge, the Lagrangian density for the light fields is where can be naturally positive even if m 2 1 m 2 2 < 0 in order for electroweak symmetry breaking at the present Universe. The Lagrangian density for the heavy fields up to quadratic order is given as where Dirac fermions are constructed as with using the chiral representation for the Dirac matrices, S = (S R + iS I )/ √ 2, and for simplicity κ and a S are taken to be real. The unbroken gauge group is SU(3) C ×U(1) em , and the corresponding covariant derivative is given by where T a ψ is the generator of SU(3) C , and q ψ is the EM charge for a given fermion ψ. For quarks, T a u,d = λ a /2, where λ a ij are Gell-Mann matrices, and for charged leptons, T a e = 0. The EM charges (q ψ ) are q u = 2/3, q d = −1/3, and q e = −1. We ignore the interaction between θ and S because it does not cause any effects of our interest.
For the low energy scale much less than ϕ, the effective action can be obtained by integrating out heavy fields. Since the expectation values of heavy fields vanish due to the heavy mass from the flat direction, basically they do not leave any traces, but expect for the anomalous coupling and the threshold correction. While the latter can be absorbed by the redefinition of model parameters, the former should be added explicitly to the Lagrangian.

It is derived by calculating one-loop triangle diagrams mediated by heavy fermions (ψ
Here N f is the number of heavy families, and we take N f = 3 for the SM. The appearance of such anomalous terms can be understood by noting that the flat direction is charged under PQ symmetry, which is anomalous under the U(1) em and SU ( In the previous section, we utilize some of the properties of supersymmetric SM just to justify a part of the form of the scalar potential in the type-II 2HDM but do not take into account any SUSY fields. In this section, we shall consider the supersymmetric extension of the SM more seriously, as is adopted in the AD mechanism. In the MSSM, or extended supersymmetric SMs, there are many scalar fields, namely, the SUSY partners of the SM fermions such as squarks and sleptons, which exhibit many flat directions [82], along which the scalar potential vanishes except for the SUSY-breaking effects and contributions from non-renormalizable operators. Scalar fields can develop expectation values along a flat direction to cause the AD mechanism. Let us focus on the LH u flat direction as a proof of concept, which has been often used for the AD leptogenesis [83]. In order to make the scalar dynamics simpler, we will consider a flat direction only governed by a slepton with a single flavor f ,L Lf , and H u 4 , while H d and other scalar fields do not develop non-zero field values. Hereafter we use the tilde for supersymmetric partners. Such a condition can be easily realized, e.g., in next-to-minimalsupersymmetric Standard Model (NMSSM) with a superpotential, It can be easily seen that with the configuration L Lf fields other than the ϕ and θ fields are absorbed by vector bosons. Similarly, one CP even and one complex field become also heavy from D-term potentials, ≃ gϕ. As a result their field values can be safely set to be zero, and, again, they can be integrated out from the low-energy effective theory. The low energy effective scalar potential along the D-flat direction, parameterized by the ϕ and θ fields, is same as that of Eq. (41). 5 The difference compared to the non-supersymmetric type-II 2HDM studied in the previ- In the unitary gauge, the corresponding Lagrangian density for the heavy fermions can be written as where the Dirac fermions ψ are defined as Note that ψ H d and ψ W become heavy due to the non-zero L f = ϕ/2. They have same electromagnetic charges (q e = −1), but couple to the axion oppositely, so integrating them out does not yield the low energy coupling between the axion and photons. This is consistent with the fact that lepton number is not anomalous under U(1) em . There is no such a kind of cancellation between ψ ui and ψ Hu (q u = 2/3, q Hu = q e = −1), which yields the low energy couplings as Once more, we have used N f = 3. Since H d = 0 during evolution of the ϕ field, three dquark pairs ψ di=1,2,3 = (d Li d c † Ri ), and two charged lepton pairs ψ ei =f = (e Li e c † Ri ) are massless. Because in our field basis those light charged fermions only couple to H d , not H u andL f , there is no coupling between the axion and massless fermions. This can be seen by assigning U(1) A ′ charges to the fermion fields including the axion θ as so that the charge of axion is one, Q θ = 1, but the electromagnetic charged massless fermions are neutral. Here the PQ charge assignments q PQ are given in Table II. Since this U(1) A ′ contains the PQ charge, it is clear that it is anomalous under SU(3) C ×U(1) em .
where · · · denotes the higher dimensional operators for other lepton flavors (but with omitting them) and the negative Hubble induced mass term forL f and H u in the same way to the 2HDM case, the final low energy Lagrangian density for the light fields is given by where we have imposed the a φ -term while the b φ -term is absent since lepton number breaking is prohibited at the renormalizable level. Thus we reach at the effective Lagrangian of the form of Eq. (1), but massless U(1) em charged particles also exist.
We can take different field basis, by θ dependent chiral transformation of d-quarks and charged leptons. Then the axion photon couplings can be removed through the chiral anomaly. Instead, the axion-current interactions are generated. 6 Therefore this coupling is important for both generation of gauge fields and helicity of the fermions, which has also been discussed in the context of inflationary magnetogenesis in Ref. [61]. Note that the fermion production through the axion-current interaction has also been studied recently in Refs. [89][90][91][92]. Let us comment on the effects of the existence of massless charged particles on magnetogenesis. Through the chiral anomaly, once helical magnetic fields are generated from the dynamics of the rotating scalar, fermions with chiral asymmetry will be also generated, by satisfying ∆h ≃ (e 2 /16π 2 )∆n 5 , with n 5 being the number density of the chiral asymmetry.
Moreover, through the Schwinger effect, non-chiral particles can be also generated, which can lead to thermalization of the charged particles [77]. As is discussed in Ref. [61], these effects will suppress the efficiency of magnetogenesis. Thus we might not have magnetic helicity as much as evaluated in Sec. II. However, in the case of standard chiral plasma instability, the numerical MHD studies have shown that the full transfer of chiral asymmetry to the magnetic helicity is possible even in the fully thermalized system [30][31][32][33]. From these observations, we expect that the full transfer of the scalar asymmetry to the magnetic helicity can be accomplished even in our case in the existence of the light particles as well as the thermal plasma. For the concrete conclusion, nevertheless further investigation is needed, which is left for the future study.
In this subsection we have focused on the LH u flat direction just for a concrete example as a proof of concept, but we expect that similar effects can be seen in other flat direction in the supersymmetric SM including the MSSM because it is often the case that there remain an unbroken U(1) gauge symmetry along a flat direction. For example, in the case of udd flat direction, a linear combination of the hyper gauge field, and the third and eighth gluons is unbroken and its anomalous coupling to the phase direction of the flat direction is expected.
In this section, we show that the new mechanism of magnetogenesis studied in Sec. II can be easily realized in the PQWW axion dynamics as well as the usual AD mechanism.
As described in the introduction, our findings have two important messages. Namely, 1) by supposing a cosmic history like the AD mechanism, axions can generate magnetic fields efficiently. 2) In some cases the AD mechanism experiences the magnetic field generation, which requires to consider the scenario carefully. Since we have studied only some of simplified situations to show the proof of concept of this idea, further studies are needed to give precise and quantitative consequences of this effect.

IV. DISCUSSION
In this work, we studied the evolution of the U(1) gauge fields that have an anomalous coupling to the phase of a rotating complex scalar field, which is often realized in cosmology in the context of the AD mechanism. The existence of such an anomalous coupling is not surprising since the phase of the AD field can be identified as an axion. Compared to the magnetogenesis from axion dynamics, where the axion oscillates around the CP-violating potential, our new mechanism of magnetogenesis is remarkable in a sense that the CP-violating effect are important only at the onset of the dynamics in the phase direction and are absent during most of its dynamics. As a result, only one helicity mode of the gauge fields receives tachyonic instability continuously, which is the source of efficient magnetogenesis so that the complete transfer from the asymmetry carried by the scalar fields to the magnetic helicity is possible. This is in contrast to the magnetogenesis from the oscillating axions, where the asymmetry carried by the axions are not conserved and hence the complete transfer from the asymmetry to the magnetic helicity is not possible. The similarity between the chiral magnetic effect and the axion-photon coupling has been pointed out, but the mechanism studied in this work has a much closer analogy to the chiral plasma instability, where the chiral magnetic effect induces the instability of the magnetic fields.
It is not trivial if such a situation can be realized in the well motivated models of physics beyond the SM. As a proof of concept, we identified that the PQWW axion in the type-II 2HDM as well as the phase of the LH u flat direction, often adopted in the AD leptogenesis, can play the role of the phase of the rotating scalar for this new magnetogenesis scenario.
In order to avoid the problems caused by the b φ -term, we adopted a singlet extension of the (MS)SM, but we expect that the magnetogenesis from the phase of a rotating scalar field is unavoidable general phenomena of the AD mechanism even in the MSSM and other similar mechanisms, which has not been recognized before.
In order to evaluate the consequences of magnetogenesis, we employed relatively simplified setup, namely, we assumed that there is no thermal plasma during the scalar field dynamics and omitted the effects of possibly existing light charged particles. The former triggers the early onset of the scalar field rotation, which makesθ is not a constant during the oscillation.
The latter implies the induction of the electric current, which correspond to the Schwinger effect in the vacuum and just the Ohm's current in the thermal plasma. It will screen the electric field and suppresses the efficiency of the magnetogenesis. Due to the chiral anomaly the estimate of the induced current would be quite involved. Since the purpose of the present work is demonstrate the existence of such a magnetogenesis process in the AD mechanism, a popular scenario in the early Universe, here we do not go into the detail but postpone them for the future study.
One may wonder if the anomalous coupling of the AD field can play important role in later times. Especially, one may expect it can cause a new channel of Q-ball decay, since this process breaks the global U(1) symmetry that guarantees the stability of Q balls. However, while the size of a Q ball is inverse of the phase velocity of the AD field, the instability scale is larger than that by a factor of inverse of the fine structure constant. Therefore we conclude the Q-ball decay triggered by the anomalous coupling is not so efficient, but it may be interesting to explore in depth.
supported by IBS under the project code, IBS-R018-D1. In supersymmetric extension of the SM, µ-term can be given by the vacuum value of the scalar field, S. Let us consider the following superpotential: where we have introduced two gauge singlets S and S c . Here M 1 M 2 , and M 3 are very large values compared to the weak scales. The PQ charge assignment of S and S c are given by  Then the scalar potential from D-terms, F -terms, soft breaking terms, and the Hubble induced mass term is where |c u |, |c d |, |c L | = O(1). We can easily see that the field configuration of the pure LH u flat direction (|h 0 u | = |ν|) with |h 0 d | = 0 is impossible since there is the tadpole potential for h 0 d , induced by SUSY breaking (Bµ h 0 u h 0 d ) and supersymmetric (µ ν 2 h 0 u * h 0 d /M) contributions. Therefore we have to estimate how h d can be large along the LH u direction.
For the large values of |h 0 u | and |ν| compared to soft SUSY breaking masses, the D-term potential makes one scalar degree heavy, so we can integrate out the corresponding field through the equations of motion. Parameterizing the scalar field amplitudes as for m 2 , H 2 ≪ ϕ 2 d ≪ ϕ 2 u ∼ ϕ 2 l ≪ M 2 , which is realized for the negative Hubble induced mass for L and H u , ϕ l is determined by the D-flat condition, By imposing this D-flat condition, the potential for ϕ u and ϕ d as well as the gauge invariant phase fields, θ H and θ L , defined as is given by Here we have assumed that all constant parameters are real for simplicity.
For m ≪ H, with a reasonable assumption: ϕ u gets a finite vacuum value as with c = O(1) whereas ϕ d ≪ ϕ u . By inserting this to the potential, supposing c d −c L −12c 4 > 0, the dominant contribution for the vacuum value of ϕ d is given by Note that the contributions from the µ-term is stronger than those from the Bµ-term. The angular field, θ H also get a mass squared of O(µH ϕ u / ϕ d ) ∼ H 2 , so that θ H is also heavy and follows the slow-rolling θ L as θ H = 2θ L + π.
As H decreases and crosses the value of O(µ), the field value of ϕ u gets around √ µM.
Then the contribution of Bµ-term is no longer negligible for the potential of the ϕ d field so Thus we find that ϕ d becomes same order of ϕ u . Now the dynamics of θ H is governed by the Bµ-term, which gives a constant heavy mass of O( √ Bµ). Then the θ H will exhibits the damped oscillation around π. Therefore while the phase of LH u rotates in the same way as the usual AD leptogenesis, H u H d rotation will be quickly damped away. Since all the electromagnetic charged fermions that are massless in the pure LH u flat direction case, such as d quarks, acquire heavy masses from the H d field value, the anomalous coupling between the phase of LH u flat direction and photons is cancelled in the low energy effective theory. Now we have found that the dynamical phase θ L does not have the anomalous coupling to photons and another phase θ H , which has the anomalous coupling, no longer shows the constant velocity, we conclude that in the MSSM with a bare Bµ-term the magnetogenesis does not happen unless the Bµ-term is sufficiently suppressed as discussed in Sec. II D. Note that in Ref. [85] the Bµ-term is not taken into account. This is the reason whyθ H becomes constant but not is damped after the onset of scalar field oscillations around the origin there.