Magnetic field generation from bubble collisions during first-order phase transition

Though the existence of the cosmological magnetic fields has been established by observations, its origin is still a long-standing unsolved problem, which may generate during inflation Turner:1987bw , and electroweak phase transition Vachaspati:1991nm . The magnetic fields from the electroweak first-order phase transition (FOPT) may seed the Intergalactic Magnetic FieldsVachaspati:2001nb . Due to the phase transition in the Standard Model is a cross-over DOnofrio:2014rug , and the electroweak FOPT is a general prediction of many models beyond the Standard Model, e.g., the SM extended by dimensional-six operator $(\Phi^{\dagger}\Phi)^{3}/\Lambda^{2}$ Grojean:2004xa ; Grojean:2006bp , singlet extension of the SM Profumo:2014opa ; Zhou:2019uzq ; Zhou:2020idp ; Alves:2018jsw ; Profumo:2007wc ; Espinosa:2011ax ; Jiang:2015cwa ; Xie:2020wzn ; Liu:2021jyc , two-Higgs-doublet models  Cline:2011mm ; Dorsch:2013wja ; Dorsch:2014qja ; Bernon:2017jgv ; Andersen:2017ika ; Kainulainen:2019kyp , George-Macheck model Zhou:2018zli , and Next-to minimal supuersymmetry model Bian:2017wfv ; Huber:2015znp . Therefore, measurements of Intergalactic Magnetic Fields may provide an additional way to probe physics beyond the Standard Model Durrer:2013pga ; Vachaspati:2016xji . For previous reviews on the primordial magnetic field, we refer to Ref. Grasso:2000wj ; Durrer:2013pga ; Yamazaki:2012pg . For the status of the observation of magnetic fields in the Galaxy, we refer to Ref. jlh .

A FOPT proceeds with bubble nucleations and collisions. In analogy with the Kibble and Vilenkin Kibble:1995aa , J.Ahonen and K.Enqvist Ahonen:1997wh studied the ring-like magnetic fields generation in collisions of bubbles of broken phase in an abelian Higgs model, and evaluated the root-mean-square magnetic field to be around $10^{-21}G$ at the comoving scale of 10 Mpc today after including the turbulent enhancement. T. Stevens et al studied the magnetic field creation from the currents induced by the charged W fields when two bubble collide in Ref. Stevens:2007ep , and they further considered the wall thickness effects in Ref. Stevens:2009ty for two bubble collisons. Recently, they utilized the thermal erasure principle to solve the equation of motions (EOMs) of electromagnetic fields in the Non-Abelian Higgs model and found the strength of the magnetic field are comparable to those found in the Abelian Higgs model for two bubbles collision, see Ref. Stevens:2012zz . Different from previous studies, in this work, we take into account the effects of the bubble dynamics during the FOPT, i.e., the dynamics of the bubble walls in the intersecting regions of bubbles and other regions induced by the thermal frictions, see Ref. Ellis:2019oqb ; Cai:2020djd . We consider the magnetic field generation by bubble collisions during the electroweak FOPT. For concreteness and simplicity, we consider the magnetic field generation for two- and three- bubble collisions.

This work is organized as follows. In Sec.II, we consider the dynamics of general bubble collision. In Sec.III, we solve the EOMs for the W and Z fields, with which, we derive the electromagnetic current by solving the Higgs phase equation and obtain the formula for estimation of the magnetic field. With these preparations, in Sec. IV, we calculate the magnetic field of electroweak bubbles collision in ideal and revised situations by considering both equal and unequal bubbles collision. In Sec.V, we evaluate the root-mean-squared magnetic field at correlation length after taking into account hydromagnetic turbulent effect. At last, we conclude with Sec.VI.

At the thin-wall limit, the Lagrangian as a function of the bubble size $R$ can be written asDarme:2017wvu :

$$L=-4\pi\sigma R^{2}\sqrt{1-\dot{R}^{2}}+\frac{4\pi}{3}R^{3}p\;,$$

(1)

where $\sigma$ is the bubble wall tension and $p$ is the pressure acting on the bubble wall. The smallest bubble size of the case where bubbles would expand instead of collapsing after nucleating is $R=2\sigma/p\equiv R_{c}$. The EOM to describe bubble growth is given by

$$\ddot{R}+2\frac{1-\dot{R}^{2}}{R}=\frac{p}{\sigma}(1-\dot{R}^{2})^{\frac{3}{2}}\;.$$

(2)

For an expanding bubble, the initial size must be larger than critical radius $R_{c}$. We can rewrite Eq. 2 in terms of Lorentz factor $\gamma$:

$$\frac{d\gamma}{dR}+\frac{2\gamma}{R}=\frac{p}{\sigma}\;,$$

(3)

where $\gamma\equiv 1/\sqrt{1-\dot{R}^{2}}$. It can be solved analytically by giving an initial condition of $\gamma$ and R.

When the bubbles are expanding in the plasma background, the friction force can be exerted by the surrounding plasma. In the case where the bubble wall is very relativistic, the leading-order friction is caused by the change of the effective mass during the $1\rightarrow 1$ particle transmission and reflection in the vicinity of the bubble wall, which is independent of the Lorentz factor $\gamma$, and is estimated to be Bodeker:2009qy ,

$$\Delta P_{LO}\approx\frac{\Delta m^{2}T^{2}}{24}.$$

(4)

The next-to-leading order term arising from the particle splitting and transition radiation at the bubble wall is proportional to $\gamma$ Bodeker:2017cim :

$$\gamma\Delta P_{NLO}\approx g^{2}\Delta m_{V}T^{3},$$

(5)

where, the squared masses differences between true and false vacuum are given by

$$\Delta m^{2}\equiv\sum_{i}c_{i}N_{i}\Delta m_{i}^{2},\ \ \ \ g^{2}\Delta m_{V}\equiv\sum_{i\in V}g_{i}^{2}N_{i}\Delta m_{i}\;,$$

(6)

where, the sum running over all gauge bosons with its masses changing across the wall, $\Delta m_{i}=m_{i,t}-m_{i,f}$, $c_{i}=1(1/2)$ for bosons(fermions), $N_{i}$ is the number of internal degrees of freedom of particles, and the $g_{i}$ are their gauge couplings. After these frictions are included, the total pressure can be written as,

$$p\equiv\Delta V-\Delta P_{LO}-\gamma\Delta P_{NLO}\;,$$

(7)

and Eq. 3 becomes Ellis:2019oqb

$$\frac{d\gamma}{dR}+\frac{2\gamma}{R}=\frac{\Delta P_{NLO}}{\sigma}(\gamma_{eq}-\gamma).$$

(8)

where $\gamma_{eq}=(\Delta V-\Delta P_{LO})/(\Delta P_{NLO})$ and $\mathop{lim}\limits_{R\rightarrow\infty}\gamma(R)=\gamma_{eq}$ (We note that the equation is revised in Ref. Cai:2020djd with an additional correction for $\gamma$-dependent friction). We assume the Lorentz factor of the bubble wall $\gamma=\gamma_{eq}$ when the bubble collisions take place. After the collisions, we have $p\equiv 0$ in the intersection region. Assuming Eqs. (3,8) still hold when $p=0$ for the intersection regions of different bubbles, we get

$$\frac{d\gamma}{dR}+\frac{2\gamma}{R}=0\;,$$

(9)

with the solution being

$$\gamma=\gamma_{eq}\frac{R_{col}^{2}}{R^{2}}\;,$$

(10)

where $R_{col}$ is the radius of the bubble at the collision time $t_{col}$. And the rest of the bubbles outside the intersection regions are still described by the Eq. 3 with $p=\Delta V-\Delta P_{LO}-\gamma\Delta P_{NLO}$, so bubbles still expand with a velocity where $\gamma=\gamma_{eq}$.

At the time of $t^{\prime}\equiv t-t_{col}=t_{m}$, the points on the bubble wall along the collision axis reach the maximum distance away from the bubble center, we show the shape of the bubble walls with solid lines in the left panel of Fig. 1 (which corresponds to the revised situation in the right panel). Where we introduce an angle to describe the position of the points on the wall, the distance between wall and bubble center is related to the angel down from the collision axis, i.e., $\phi$. The bubble wall with different $\phi$ begins to intersect with the other bubble at different time. Therefore, the distance between the points on bubble walls in the intersection region and the bubble center depends on the angel ($\phi$) and the time after collision. The Eq. 9 applies until the moment when the points on the bubble wall with a angle $\phi$ reach the maximum distance. Meanwhile, the bubble wall outside the intersection region still expand with a velocity $\gamma=\gamma_{eq}$. The dotted lines are plotted to describe the ideal case which is adopted to evaluate the magnetic field in the previous study of Ref. Stevens:2012zz . In the right panel, we plot the time evolution of bubble walls for revised situation and ideal situation.

In this section, we review the derivation of the magnetic field from EOMs of gauge bosons. The relevant Lagrangian of electroweak bosonic fields is

$$L_{EW}=L_{1}+L_{2}-V(\phi)\;,$$

(11)

where,

	$\displaystyle L_{1}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}W_{\mu\nu}^{i}W^{i\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\;,$
	$\displaystyle W_{\mu\nu}^{i}$	$\displaystyle=$	$\displaystyle\partial_{\mu}W_{\nu}^{i}-\partial_{\nu}W_{\mu}^{i}-g\epsilon_{ijk}W_{\mu}^{j}W_{\nu}^{k}\;,$
	$\displaystyle B_{\mu\nu}$	$\displaystyle=$	$\displaystyle\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}\;,$		(12)

and

$$L_{2}=|(i\partial_{\mu}-\frac{g}{2}\tau\cdot W_{\mu}-\frac{g^{\prime}}{2}B_{\mu})\Phi|^{2}\;,$$

(13)

where $\tau_{i}$ is the $SU(2)$ generator and $V(\Phi)$ is the Higgs potential. Here, the Higgs potential $V(\phi)$ with proper barrier for quantum tunneling at finite temperature around $\mathcal{O}(10^{2})$ GeV can feasible an electroweak FOPT proceeding with bubble nucleations and collisions Grojean:2004xa , and therefore yields production of the magnetic fields Durrer:2013pga ; Kandus:2010nw ; Subramanian:2015lua . The physical $Z$ and $A^{em}_{\mu}$ fields are

	$\displaystyle A^{em}_{\mu}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{g^{2}+g^{\prime 2}}}(g^{\prime}W^{3}_{\mu}+gB_{\mu})\;,$
	$\displaystyle Z_{\mu}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{g^{2}+g^{\prime 2}}}(gW^{3}_{\mu}-g^{\prime}B_{\mu})\;,$		(14)

and Higgs doublet takes the form of

$$\Phi(x)=\left(\begin{array}[]{ccc}0\\ \rho(x)exp(i\Theta(x))\end{array}\right)\;,$$

(15)

where $\Theta(x)$ is the phase of the Higgs field and $\rho(x)$ is its magnitude. For this choice of gauge, the EOM for $B$ field is

$$\partial^{2}B_{\nu}-\partial_{\nu}\partial\cdot B+g^{\prime}\rho(x)^{2}\psi_{\nu}(x)=0\;,$$

(16)

where the $\psi_{\nu}$ is

$$\psi_{\nu}(x)\equiv\partial_{\nu}\Theta-\frac{\sqrt{g^{2}+g^{\prime 2}}}{2}Z_{\nu}\;,$$

(17)

and satisfies

$$\partial_{\nu}\left(\rho(x)^{2}\psi_{\nu}(x)\right)=0\;.$$

(18)

For $i=3$, gauge field $W^{i}$ satisfies the following equation

$$\partial^{2}W^{3}_{\nu}-\partial_{\nu}\partial\cdot W^{3}-g\rho(x)^{2}\psi_{\nu}(x)=j^{3}_{\nu}(x)\;,$$

(19)

and, for $i=1,2$, we have

$$\partial^{2}W^{i}_{\nu}-\partial_{\nu}\partial\cdot W^{i}+m_{W}(x)^{2}W^{i}_{\nu}=j^{i}_{\nu}(x)\;,$$

(20)

where $m_{W}(x)^{2}=g^{2}\rho(x)^{2}/2$ and $j^{i}_{\nu}(x)$ is,

	$\displaystyle j^{i}_{\nu}(x)$	$\displaystyle\equiv$	$\displaystyle g\epsilon_{ijk}(W^{k}_{\nu}\partial\cdot W^{j}+2W^{j}\cdot\partial W^{k}_{\nu}-W^{j}_{\mu}\partial_{\nu}W^{k\mu})$		(21)
			$\displaystyle-g^{2}\epsilon_{klm}\epsilon_{ijk}W^{j}_{\mu}W^{l\mu}W^{m}_{\nu}\;.$

The EOM for $A^{em}$ casts the form of,

$$\partial^{2}A^{em}_{\nu}-\partial_{\nu}\partial\cdot A^{em}=j^{em}_{\nu}(x)\;,\\$$

with

$$j^{em}_{\nu}(x)=\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime 2}}}j^{3}_{\nu}(x)\;.$$

(22)

And, the EOM for the Z field is obtained as,

$$\partial^{2}Z_{\nu}-\partial_{\nu}\partial\cdot Z-\rho(x)^{2}\sqrt{g^{2}+g^{\prime 2}}\psi_{\nu}(x)=\frac{g}{g^{\prime}}j^{em}_{\nu}(x)\;.$$

(23)

Utilizing the thermal erasure Stevens:2012zz of $\langle Z\rangle=0$, and suppose $\rho(x)=\rho_{0}$, which applies to the thin-wall limit for bubble collisions. Applying the ensemble averaging to Eq. (18,23), we get

	$\displaystyle\langle j^{em}_{\nu}\rangle$	$\displaystyle=$	$\displaystyle-\frac{g^{\prime}}{g}\sqrt{g^{2}+g^{\prime 2}}\rho_{0}^{2}\times\partial_{\nu}\Theta(x)\;,$		(24)
	$\displaystyle\partial^{2}\Theta(x)$	$\displaystyle=$	$\displaystyle 0\;.$		(25)

Consequently, the Eq. (III) recasts the form of the Maxwell equation,

	$\displaystyle\partial^{2}A_{\nu}-\partial_{\nu}\partial\cdot A$	$\displaystyle=$	$\displaystyle j^{em}_{\nu}(x)$		(26)
		$\displaystyle=$	$\displaystyle-\frac{g^{\prime}}{g}\sqrt{g^{2}+g^{\prime 2}}\rho_{0}^{2}\times\partial_{\nu}\Theta(x)\;.$

Due to the magnetic field $\vec{B}=\vec{\nabla}\times\vec{A}^{em}$, we can calculate the strength of the magnetic field after obtaining the electromagnetic current through,

$$\nabla^{2}\vec{B}=\vec{\nabla}\times\vec{j}^{em}\;.$$

(27)

Eq. 24 suggests that when bubbles collide there would be a large gradient of the Higgs phase, and consequently a large electromagnetic current and create large magnetic field through Eq (26,27).

In this section, we calculate magnetic field generated when two bubbles and three bubbles collide.

The comoving Hubble length at the electroweak phase transition temperature $T_{*}$ is given byKahniashvili:2012uj ,

$$\lambda_{H_{*}}=5.8\times 10^{-10}{\rm Mpc(100{\rm GeV}/T_{*})}(100/g_{*})^{1/6}\;,$$

(53)

where $g_{*}$ is the number of relativistic degrees of freedom at the moment when the primordial magnetic field is generated. The comoving correlation length for a primordial magnetic field at the generation time can be evaluated to be

$$\xi_{\star}=\Gamma\lambda_{H_{*}}\;.$$

(54)

Here, $\Gamma$ is the factor to account for bubble numbers inside one Hubble radius at the FOPT, and $\Gamma\simeq 0.01$ for the electroweak FOPT. For recent simulations, see Ref. Zhang:2019vsb ; Di:2020nny .

The physical magnetic field amplitudes scale with the expansion of the universe at the generation time as

$\displaystyle B_{*}$

$\displaystyle=$

$\displaystyle(\frac{a_{*}}{a_{0}})^{2}B\;,$

(55)

with the time-temperate relation being,

$\displaystyle\frac{a_{*}}{a_{0}}$

$\displaystyle\simeq$

$\displaystyle 8\times 10^{-16}(100{\rm GeV}/T_{*})(100/g_{*})^{1/3}\;.$

(56)

The simulation of the evolution of hydromagnetic turbulence from the electroweak phase transition suggests that the root-mean-squared non-helical magnetic field amplitude and the correlation length satisfies the following relation Brandenburg:2017neh ,

$$B_{rms}=B_{*}(\frac{\xi}{\xi_{*}})^{-(\beta+1)/2}\;,$$

(57)

where $\beta=1,2$ and 4 for non-helical case, with $\xi$ being the magnetic correlation length. For illustration, we show in Fig. 10 the bounds of blazars spectra on the magnetic field at variant correlation lengths, which depicts that the case of $\beta=1$ case is allowed by the data. Where, we consider the Equal bubbles-Ideal situation and the Equal bubbles-Revised situation for two bubbles collision, the magnetic fields are generated at $z=0$ with the largest $r$ at $t$($=t_{m}+t_{col}$), where we have the $r\sim 3R_{col}$, where we have the ring-like distribution of the magnetic field. The magnetic field strength here is almost the same with the three bubbles collision situations. Primordial magnetic field suffer bounds from the Big-Bang Nucleosynthesis Kahniashvili:2010wm ; Kawasaki:2012va and the measurements of the spectrum and anisotropies of the cosmic microwave background Seshadri:2009sy ; Ade:2015cva ; Jedamzik:1999bm ; Barrow:1997mj ; Durrer:1999bk ; Trivedi:2010gi , these limits are not shown in the figure since they are not relevant for the parameter space under study in this work.

We use the EOMs of gauge fields to get the magnetic field generated during bubble collisions at the electroweak FOPT. After obtaining the Higgs phases when bubbles collide, we calculate the magnetic field strength after obtaining the electromagnetic current, and apply the approach to the situations of two bubbles and three bubbles collisions, equal and unequal bubbles, ideal and revised situations. We found the electroweak bubble collisions produce the ring-like magnetic field even when we consider the revised situation with bubble walls deviating from the spherical shape. For that situation, we get a smaller magnetic field strength because the electromagnetic current distributed in a smaller area. The scaling law resulting from the hydromagnetic turbulence after the electroweak FOPT suggests that this kind of magnetic field under study can be probed by the observation of the Intergalactic Magnetic Fields. The magnetic field strength calculated here is comparable with the magnetic field generated from the bubble collisions simulation performed in Ref. Zhang:2019vsb ; Di:2020nny .

We note that, in the electroweak baryogenesis, the Chern-Simons connects the helicity of the magnetic fields produced during bubble collisions and the baryon asymmetry of the early Universe Copi:2008he ; Vachaspati:2001nb . Therefore, the observation of the helicity of the primordial magnetic fields may serve as a test of the electroweak baryogenesis. Ref. Ellis:2019tjf studied the primordial magnetic field from first-order phase transition in $B-L$ model and the SM extended by dimensional-six operator $(\Phi^{\dagger}\Phi)^{3}/\Lambda^{2}$, with the physical implication that the observable gravitational waves and collider signatures would be complementary to the magnetic field observation from the first-order phase transitions. when the inverse cascade process are taken into account for helical magnetic fields  Cornwall:1997ms ; Giovannini:1997gp ; Ji:2001rx .

We thank Yi-Zen Chu, Francesc Ferrer, Jinlin Han, Marek Lewicki, Shao-Jiang Wang, Ke-Pan Xie, and Yiyang Zhang for useful communications and discussions. This work is supported by the National Natural Science Foundation of China under the grants Nos.12075041, 11605016, and 11947406, and Chongqing Natural Science Foundation (Grants No.cstc2020jcyj-msxmX0814), and the Fundamental Research Funds for the Central Universities of China (No. 2019CDXYWL0029).