Magnetic field generation from bubble collisions during first-order phase transition
MMagnetic field generation from bubble collisions during first-orderphase transition
Jing Yang, and Ligong Bian ∗ Department of physics, Chongqing University, Chongqing 401331, China
Abstract
We study the magnetic fields generation from the cosmological first-order electroweak phasetransition. We calculate the magnetic field induced by the variation of the Higgs phase for twobubbles and three bubbles collisions. Our study shows that electromagnetic currents in the collisiondirection produce the ring-like magnetic field in the intersect regions of colliding bubbles, whichmay seed the primordial magnetic field that are constrained by intergalatic field observations.
PACS numbers: ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] F e b . INTRODUCTION Though the existence of the cosmological magnetic fields has been established by ob-servations, its origin is still a long-standing unsolved problem, which may generate duringinflation [1], and electroweak phase transition [2]. The magnetic fields from the electroweakfirst-order phase transition (FOPT) may seed the Intergalactic Magnetic Fields[4]. Due tothe phase transition in the Standard Model is a cross-over [5], and the electroweak FOPTis a general prediction of many models beyond the Standard Model, e.g., the SM extendedby dimensional-six operator (Φ † Φ) / Λ [3, 6], singlet extension of the SM [7–15], two-Higgs-doublet models [16–21], George-Macheck model [22], and Next-to minimal supuersymmetrymodel [23, 24]. Therefore, measurements of Intergalactic Magnetic Fields may provide anadditional way to probe physics beyond the Standard Model [26, 27]. For previous reviews onthe primordial magnetic field, we refer to Ref. [25, 26, 28]. For the status of the observationof magnetic fields in the Galaxy, we refer to Ref. [29].A FOPT proceeds with bubble nucleations and collisions. In analogy with the Kibble andVilenkin [30], J.Ahonen and K.Enqvist [31] studied the ring-like magnetic fields generationin collisions of bubbles of broken phase in an abelian Higgs model, and evaluated the root-mean-square magnetic field to be around 10 − G at the comoving scale of 10 Mpc todayafter including the turbulent enhancement. T. Stevens et al studied the magnetic fieldcreation from the currents induced by the charged W fields when two bubble collide inRef. [32], and they further considered the wall thickness effects in Ref. [33] for two bubblecollisons. Recently, they utilized the thermal erasure principle to solve the equation ofmotions (EOMs) of electromagnetic fields in the Non-Abelian Higgs model and found thestrength of the magnetic field are comparable to those found in the Abelian Higgs model fortwo bubbles collision, see Ref. [34]. Different from previous studies, in this work, we takeinto account the effects of the bubble dynamics during the FOPT, i.e., the dynamics of thebubble walls in the intersecting regions of bubbles and other regions induced by the thermalfrictions, see Ref. [42, 43]. We consider the magnetic field generation by bubble collisionsduring the electroweak FOPT. For concreteness and simplicity, we consider the magneticfield generation for two- and three- bubble collisions.This work is organized as follows. In Sec.II, we consider the dynamics of general bubblecollision. In Sec.III, we solve the EOMs for the W and Z fields, with which, we derive2he electromagnetic current by solving the Higgs phase equation and obtain the formulafor estimation of the magnetic field. With these preparations, in Sec. IV, we calculate themagnetic field of electroweak bubbles collision in ideal and revised situations by consideringboth equal and unequal bubbles collision. In Sec.V, we evaluate the root-mean-squaredmagnetic field at correlation length after taking into account hydromagnetic turbulent effect.At last, we conclude with Sec.VI. II. BUBBLES COLLISION DYNAMICS
At the thin-wall limit, the Lagrangian as a function of the bubble size R can be writtenas[39]: L = − πσR (cid:112) − ˙ R + 4 π R p , (1)where σ is the bubble wall tension and p is the pressure acting on the bubble wall. Thesmallest bubble size of the case where bubbles would expand instead of collapsing afternucleating is R = 2 σ/p ≡ R c . The EOM to describe bubble growth is given by¨ R + 2 1 − ˙ R R = pσ (1 − ˙ R ) . (2)For an expanding bubble, the initial size must be larger than critical radius R c . We canrewrite Eq. 2 in terms of Lorentz factor γ : dγdR + 2 γR = pσ , (3)where γ ≡ / (cid:112) − ˙ R . It can be solved analytically by giving an initial condition of γ andR. When the bubbles are expanding in the plasma background, the friction force can beexerted 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 → γ , and is estimated to be [40],∆ P LO ≈ ∆ m T . (4)The next-to-leading order term arising from the particle splitting and transition radiationat the bubble wall is proportional to γ [41]: γ ∆ P NLO ≈ g ∆ m V T , (5)3here, the squared masses differences between true and false vacuum are given by∆ m ≡ (cid:88) i c i N i ∆ m i , g ∆ m V ≡ (cid:88) i ∈ V g i N i ∆ m i , (6)where, the sum running over all gauge bosons with its masses changing across the wall,∆ m i = m i,t − m i,f , c i = 1(1 /
2) for bosons(fermions), N i is the number of internal degrees offreedom of particles, and the g i are their gauge couplings. After these frictions are included,the total pressure can be written as, p ≡ ∆ V − ∆ P LO − γ ∆ P NLO , (7)and Eq. 3 becomes [42] dγdR + 2 γR = ∆ P NLO σ ( γ eq − γ ) . (8)where γ eq = (∆ V − ∆ P LO ) / (∆ P NLO ) and lim R →∞ γ ( R ) = γ eq (We note that the equation isrevised in Ref. [43] with an additional correction for γ -dependent friction). We assume theLorentz factor of the bubble wall γ = γ eq when the bubble collisions take place. After thecollisions, we have p ≡ p = 0 for the intersection regions of different bubbles, we get dγdR + 2 γR = 0 , (9)with the solution being γ = γ eq R col R , (10)where R col is the radius of the bubble at the collision time t col . And the rest of the bubblesoutside the intersection regions are still described by the Eq. 3 with p = ∆ V − ∆ P LO − γ ∆ P NLO , so bubbles still expand with a velocity where γ = γ eq .At the time of t (cid:48) ≡ t − t col = t m , the points on the bubble wall along the collision axisreach the maximum distance away from the bubble center, we show the shape of the bubblewalls with solid lines in the left panel of Fig. 1 (which corresponds to the revised situationin the right panel). Where we introduce an angle to describe the position of the points onthe wall, the distance between wall and bubble center is related to the angel down from thecollision axis, i.e., φ . The bubble wall with different φ begins to intersect with the otherbubble at different time. Therefore, the distance between the points on bubble walls inthe intersection region and the bubble center depends on the angel ( φ ) and the time after4 igure 1: Left: The shape of the two bubbles after collision at time t (cid:48) = t m is shown with thicklines, the points on the bubble wall along the collision axis reaches the maximum distance with R col m W = 10. The dotted lines are the case where γ ≡ γ eq is always satisfied on the whole bubblewalls. Right: The time dependence of the distance between bubble center and the point of bubblewall along the collision axis in the intersection region is shown in the right panel. The blue linerepresents the revised situation and the orange one describes the ideal situation. collision. The Eq. 9 applies until the moment when the points on the bubble wall with aangle φ reach the maximum distance. Meanwhile, the bubble wall outside the intersectionregion still expand with a velocity γ = γ eq . The dotted lines are plotted to describe theideal case which is adopted to evaluate the magnetic field in the previous study of Ref. [34].In the right panel, we plot the time evolution of bubble walls for revised situation and idealsituation. III. MAGNETIC FIELD GENERATION
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 + L − V ( φ ) , (11)5here, L = − W iµν W iµν − B µν B µν ,W iµν = ∂ µ W iν − ∂ ν W iµ − g(cid:15) ijk W jµ W kν ,B µν = ∂ µ B ν − ∂ ν B µ , (12)and L = | ( i∂ µ − g τ · W µ − g (cid:48) B µ )Φ | , (13)where τ i is the SU (2) generator and V (Φ) is the Higgs potential. Here, the Higgs potential V ( φ ) with proper barrier for quantum tunneling at finite temperature around O (10 ) GeVcan feasible an electroweak FOPT proceeding with bubble nucleations and collisions [3], andtherefore yields production of the magnetic fields [26, 36, 37]. The physical Z and A emµ fieldsare A emµ = 1 (cid:112) g + g (cid:48) ( g (cid:48) W µ + gB µ ) ,Z µ = 1 (cid:112) g + g (cid:48) ( gW µ − g (cid:48) B µ ) , (14)and Higgs doublet takes the form ofΦ( x ) = ρ ( x ) exp ( i Θ( x )) , (15)where Θ( x ) is the phase of the Higgs field and ρ ( x ) is its magnitude. For this choice ofgauge, the EOM for B field is ∂ B ν − ∂ ν ∂ · B + g (cid:48) ρ ( x ) ψ ν ( x ) = 0 , (16)where the ψ ν is ψ ν ( x ) ≡ ∂ ν Θ − (cid:112) g + g (cid:48) Z ν , (17)and satisfies ∂ ν (cid:0) ρ ( x ) ψ ν ( x ) (cid:1) = 0 . (18)For i = 3, gauge field W i satisfies the following equation ∂ W ν − ∂ ν ∂ · W − gρ ( x ) ψ ν ( x ) = j ν ( x ) , (19)6nd, for i = 1 ,
2, we have ∂ W iν − ∂ ν ∂ · W i + m W ( x ) W iν = j iν ( x ) , (20)where m W ( x ) = g ρ ( x ) / j iν ( x ) is, j iν ( x ) ≡ g(cid:15) ijk ( W kν ∂ · W j + 2 W j · ∂W kν − W jµ ∂ ν W kµ ) − g (cid:15) klm (cid:15) ijk W jµ W lµ W mν . (21)The EOM for A em casts the form of, ∂ A emν − ∂ ν ∂ · A em = j emν ( x ) , with j emν ( x ) = g (cid:48) (cid:112) g + g (cid:48) j ν ( x ) . (22)And, the EOM for the Z field is obtained as, ∂ Z ν − ∂ ν ∂ · Z − ρ ( x ) (cid:112) g + g (cid:48) ψ ν ( x ) = gg (cid:48) j emν ( x ) . (23)Utilizing the thermal erasure [34] of (cid:104) Z (cid:105) = 0, and suppose ρ ( x ) = ρ , which applies to thethin-wall limit for bubble collisions. Applying the ensemble averaging to Eq. (18,23), we get (cid:104) j emν (cid:105) = − g (cid:48) g (cid:112) g + g (cid:48) ρ × ∂ ν Θ( x ) , (24) ∂ Θ( x ) = 0 . (25)Consequently, the Eq. (22) recasts the form of the Maxwell equation, ∂ A ν − ∂ ν ∂ · A = j emν ( x )= − g (cid:48) g (cid:112) g + g (cid:48) ρ × ∂ ν Θ( x ) . (26)Due to the magnetic field (cid:126)B = (cid:126) ∇ × (cid:126)A em , we can calculate the strength of the magnetic fieldafter obtaining the electromagnetic current through, ∇ (cid:126)B = (cid:126) ∇ × (cid:126)j em . (27)Eq. 24 suggests that when bubbles collide there would be a large gradient of the Higgsphase, and consequently a large electromagnetic current and create large magnetic fieldthrough Eq (26,27). 7 V. MAGNETIC FIELD GENERATION
In this section, we calculate magnetic field generated when two bubbles and three bubblescollide.
A. Two Bubbles collision
The simplest case is that two bubbles nucleate simultaneously, one bubble locatesat ( t, x, y, z ) = (0 , , , vt col ), and the other one locates at the position of ( t, x, y, z ) =(0 , , , − vt col ). We suppose they are expanding with a same velocity v , and thus the colli-sion time is t col . The system under study has a O (2) symmetry in the spatial coordinate,we therefore follow the analysis of Kibble and Vilenkin [30] and express the EOM in a co-ordinate ( τ, z ) which has a O (1 ,
2) symmetry when v = 1. To obtain the magnetic fieldgenerated by bubble wall collisions, we need to solve the equation of the Higgs field phase,i.e., Eq. 25. In ( τ, z ) coordinate, it is( v + 1 τ + v (1 − v ) t τ ) ∂ Θ ∂τ + (1 + v ( v − t τ ) ∂ Θ ∂τ − ∂ Θ ∂z = 0 , (28)where τ = √ v t − r with r = x + y . Assuming r (cid:28) vt , the equation recasts the form:2 τ ∂ Θ ∂τ + v ∂ Θ ∂τ − ∂ Θ ∂z = 0 . (29)We consider the boundary conditions on Θ being given byΘ( τ = t col , z ) = Θ (cid:15) ( z ) , ∂∂τ Θ( τ = t col , z ) = 0 , (30)with Θ being a constant. Expressing Θ( x ) as a Fourier transform in z, above equation givesa τ − depend ordinary differential equation, yielding,Θ( τ, z ) = 1 √ π (cid:90) ∞−∞ dke ikz ( a k τ − v v K ( − v v , ω k τv )+ b k τ − v v K ( − v v , ω k τv )) , (31)where ω k = √ k + m , a k and b k are determined by the boundary conditions on Θ. Whenwe take m →
0, the solution configuration can be obtained. Then, the j emν ( τ, z ) takes theform j emν ( τ, z ) = ( j z ( τ, z ) , x α j ( τ, z )) , (32)8ith j z = − g (cid:48) g (cid:112) g + g (cid:48) ρ ∂∂z Θ( τ, z ) , (33) j = − g (cid:48) g (cid:112) g + g (cid:48) ρ τ ∂∂τ Θ( τ, z ) . (34)and x α = ( vt, − x, − y ). It is clearly that the electromagnetic field has the same form as theelectromagnetic current, A emν ( τ, z ) = ( a z ( τ, z ) , x α a ( τ, z )) . (35)Taking the axial gauge, and Maxwell’s equation becomes − ∂ ∂z a ( τ, z ) = j ( τ, z ) . (36)Applying the boundary conditions, namely, a ( τ , z ) = 0, and ∂ z a ( τ = 0 , z ) = 0, we otain a ( τ, z ) = − (cid:90) z −∞ dz (cid:48) (cid:90) z (cid:48) −∞ j ( τ, z (cid:48)(cid:48) ) dz (cid:48)(cid:48) . (37)With which, and apply Eq. 27, we get the magnetic field, B z = 0 ,B x = − y (cid:90) z −∞ j ( τ, z (cid:48) ) dz (cid:48) ,B y = x (cid:90) z −∞ j ( τ, z (cid:48) ) dz (cid:48) . (38)When v = 1, Eq 31 reduces toΘ( τ, z ) = Θ τ θ ( T − | z | ) z + Θ (cid:15) ( z ) θ ( | z | − T ) , (39)where T = τ − t col . Then, the j takes the form, j = g (cid:48) g (cid:112) g + g (cid:48) ρ Θ τ θ ( T − | z | ) z . (40)Finally, we get (cid:126)B = ( − y, x, r B φ , (41)with B φ = r g (cid:48) g (cid:112) g + g (cid:48) ρ Θ τ θ ( T − | z | ) × | z | − T . (42)With increase of the r , τ would decreases, we therefore expect a largest value of thecurrent j and further a largest magnetic field strength at the largest r when two bubbles9ollide, which grows after the time of bubbles collision ( t col ). This reason leads to a ring-likedistribution of the created magnetic field close to the walls of the collided bubbles. Equal bubbles-Ideal situation:
We first consider the two colliding bubbles are of equalsizes. In Fig. 2, we show the configuration of the Θ for v = 0 . v = 1 as a function ofdistance z along the axis of collision for different τ m W . We find there are slightly differencebetween the cases of v = 0 . v = 1 for the same τ m W . Thus the magnetic field strengthfrom the bubble collisions for the two cases have the similar profile as shown in Fig. 3. Ata distance rm W = 1 and τ m W = 20 , ,
40 and 50, the magnitude of magnetic is nearlyorder of 0 . m W . It can be seen that near the center of the overlap region at z = r = 0,themagnetic field is much smaller than the region near τ = R , and magnetic field has a tendencyto drop at fixed r when the overlap region becomes larger. Magnitude of the magnetic fieldin x − y plane is shown in Fig. 4. The figure shows that large magnitude of the magneticfield almost distributes near the edge of the overlap region, which indicates that the shape ofthe magnetic field produced by the electroweak bubble collisions is approximately a ring-likedistribution. This feature confirms the discussions under Eq. 42. Figure 2: Higgs phase Θ is shown as a function of the distance z for τ m W = 20 , , ,
50, withΘ = 1. In the left panel, we plot the case of v = 0 . , t col m W = 20. In the right panel, we considerthe case of v = 1 , t col m W = 10. Unequal bubbles-Ideal situation:
Then, we turn to the unequal bubbles collisionsituation, where the two bubbles nucleate at two different moments. For simplicity, weconsider one bubble is nucleated at ( t , x , y , z ) = (0 , , , − d ) and the other one at( t , x , y , z ) = ( d − d , , , d ) where d > d >
0. We consider the case where theyexpand at a velocity v ≡ z = 0 , t = d . We findnucleation events has a space-like interval due to ∆ x µ ∆ x µ = ( d − d ) − ( d + d ) < igure 3: Magnitude of the magnetic field calculated for two bubble collisions. The magnetic fieldis shown as a function of the distance z along the axis of collision at a distance rm W = 1 from theaxis of collision for τ m W = 20 , ,
40 and 50. Left: we consider the case v = 0 . , t col m W = 20.Right: we consider the case v = 1 , t col m W = 10.Figure 4: Magnitude of the magnetic field calculated for two bubble collisions in the x − y planefor z = 0 at time t m W = 30 in the case of v = 1 and t col m W = 10. bubbles are nucleated simultaneously [38]. In the new frame after the boost, the coordinates( t (cid:48)(cid:48) , x (cid:48)(cid:48) , y (cid:48)(cid:48) , z (cid:48)(cid:48) ) has form t (cid:48)(cid:48) = γ ( t − ∆ v · x ) , x (cid:48)(cid:48) = x, y (cid:48)(cid:48) = y, z (cid:48)(cid:48) = γ ( z − ∆ v · t ) , (43)where ∆ v is the velocity of the new frame relative to the old one. The condition that twobubbles nucleate simultaneously requires t (cid:48)(cid:48) = t (cid:48)(cid:48) , so we get ∆ v = ( d − d ) / ( d + d ).We take d m W = 2 d m W = 20, and calculate the magnetic field in the new frameusing Eq. 42. In order to get the final result, we perform a Lorentz transformation ofmagnetic field calculated above back to the old frame. Fig. 5 shows the bubbles shape for11 igure 5: Left: Bubble shapes for unequal bubbles collisions. Right: Magnetic field generated byunequal bubbles collision. The field is shown at time t (cid:48) m W ≡ ( t − t col ) m W = 10 , , ,
40 aftercollision, in which we consider v = 1, t col m W = 20, rm W = 1. unequal bubbles collision where nucleations occurring at ( t , x , y , z ) = (0 , , , − d ) and( t , x , y , z ) = ( d, , , d ) (see the left panel) and the produced magnetic field in the oldframe (see the right panel). The Figure shows that magnetic fields for different times arepeaked at points with different coordinate z, and shows an asymmetry between left side andright side of peaks since unequal bubbles collision breaks the O (1 ,
2) and Z symmetries inspacetime. Equal bubbles-Revised situation:
So far, we have calculated the magnetic fieldgenerated by two bubble collisions in an ideal case where the velocity of the whole bubblewalls are unchanged after collision and the bubbles are perfect spherical shapes. While, inthe realistic situation, the velocity of the intersecting bubble walls may change due to thebubble tension, which lead to a deviation of the bubble shapes, see Section. II for details. Fora illustration, we suppose that the bubble velocity at the collision time is v col = v eq = 0 . R col m W = 10. To solve theEq. 29 by using the boundary conditions Eq. 30, we take an assumption that the solution ofΘ is still nearly proportional to z in the intersection region as shown in Fig. 2. It is easily tofind that z/τ is a solution to the Eq. 29 where τ = √ t − r . We can approximately take τ col = (cid:112) v col t − r ≈ τ and the solution takes the form asΘ( τ , τ ( t, r ) , z ) = C ( τ ( t, r )) ∗ zτ + C . (44)Then we use the boundary condition Eq. 30 and consider the constraint that in the region12ithout intersection one has a constant phase ± Θ . We found the solution isΘ( τ , τ ( t, r ) , z ) = Θ τ θ ( T ( t, r ) − | z | ) z + Θ (cid:15) ( z ) θ ( | z | − T ( t, r )) , (45)where T ( t, r ) = τ ( t, r ) − R col and τ ( t, r ) = (cid:112) R ( t, r ) − r with R ( t, r ) being the distancebetween bubble wall and bubble center as a function of r = (cid:112) x + y and time (t) aftercollision. Figure 6: The magnetic field strength is shown as a function of the distance z along the axis ofcollision with rm W = 1 where v col = 0 . , R col m W = 10. At the time t − t col = t m , the points onbubble walls along the collision axis reaches the maximum distance with bubble center in the revisedsituation. Blue dashed and Green dashed lines indicate revised and ideal situations respectively. To demonstrate the difference between the revised and ideal situations, in Fig. 6, weshow the magnetic field as a function of distance z along the axis of collision with rm w = 1.We consider the time t − t col = t m when the points on bubble walls along the collision axisreaches the maximum distance with bubble center in the revised situation. The magnitudeof magnetic field strength in the revised situation is nearly half of the magnitude in theideal situation, and the distribution area of magnetic field in the revised situation is smallerthan the ideal situation. The results are in accordance with the bubble shape after collisionas shown in Fig. 1, the intersection area of bubbles in the revised situation is smaller thanthe ideal situation, the electromagnetic current distributes in a smaller area and thereforecauses the smaller magnitude of the magnetic field strength.13 . Three bubbles collision Figure 7: Left:overlap regions of three bubbles in the y − z plane at t − t col = 0 . R col ; Right:overlapregions of three bubbles in the y − z plane at t − t col = 0 . R col . In this section, we consider three equal size bubbles nucleate simultaneously. We considerthey expand at a same velocity with γ col = γ eq and then collide with each other at the sametime. The simplest case is that one bubble nucleates at ( t, x, y, z ) = (0 , , , − R col ) andother two nucleate at (0 , , , R col ) and (0 , , −√ R col ,
0) respectively. At first, there wouldbe three regions where they overlap in pairs. After a period, three regions may overlap andthere will be a region (at the center of three bubbles) bounded by the intersection of threebubbles. We show the overlap regions of three bubbles collision in Fig. 7. We can imaginethat the magnetic field strength of the region IV can be represented by the superposition ofthe other three regions. For v col = 0 . ≈
1, we take τ = τ = √ t − r for simplicity. Weset the phases of the three bubbles as Θ = 0,Θ = π and Θ = π , we choose the center ofregion I to be the original point, the initial conditions can now be written in the followingforms:Region I: Θ( τ = t col , z ) = Θ − Θ (cid:15) ( z ) + Θ + Θ , Θ (cid:48) ( τ = t col , z ) = 0 ,τ = (cid:112) t − x − y , (46)14egion II: Θ( τ (cid:48) = t col ,
12 ( z − t col ) − √ y ) = Θ − Θ (cid:15) ( 12 ( z − t col ) − √ y )+ Θ + Θ , Θ (cid:48) ( τ (cid:48) = t col ,
12 ( z − t col ) − √ y ) = 0 ,τ (cid:48) = (cid:115) t − x − ( √
32 ( z + t col ) + 12 y ) , (47)Region III:Θ( τ (cid:48)(cid:48) = t col , −
12 ( z + t col ) − √ y ) = Θ − π − Θ (cid:15) ( −
12 ( z + t col ) − √ y )+ Θ + 2 π + Θ , Θ (cid:48) ( τ (cid:48)(cid:48) = t col , −
12 ( z + t col ) − √ y ) = 0 ,τ (cid:48)(cid:48) = (cid:115) t − x − ( √
32 ( z − t col ) − y ) . (48)And magnetic field of these three regions can be solved similarly, with:Region I B x = − yB ,B y = xB ,B z = 0 , (49)Region II B x = − ( y − √ z − √ t col ) B ,B y = x B ,B z = − √ xB , (50)Region III B x = − ( y + √ z − √ t col ) B ,B y = x B ,B z = √ xB , (51)15egion IV B x = − yB − ( y − √ z − √ t col ) B , − ( y + √ z − √ t col ) B ,B y = xB + x B x B ,B z = √ x ( B − B ) , (52)where B = g (cid:48) g (cid:112) g + g (cid:48) ρ Θ − Θ τ θ ( τ − t col − | z | ) | z | − ( τ − t col ) ,B = g (cid:48) g (cid:112) g + g (cid:48) ρ Θ − Θ τ (cid:48) θ ( τ (cid:48) − t col − |
12 ( z − t col ) − √ y | ) | ( z − t col ) − √ y | − ( τ (cid:48) − t col ) ,B = g (cid:48) g (cid:112) g + g (cid:48) ρ Θ − Θ − πτ (cid:48)(cid:48) θ ( τ (cid:48)(cid:48) − t col − |
12 ( z + t col ) + √ y | ) | ( z + t col ) + √ y | − ( τ (cid:48)(cid:48) − t col ) . Note that the solutions are only valid in the overlap region. After t > √ t col , three regionsmay overlap with each other, and the magnetic field in Region IV is the superposition ofthe three regions. Three equal bubbles-ideal institution:
For illustration, we show the strength of themagnetic field induced by three bubbles collision at t − t col = 0 . R col and t − t col = t m in y − z plane in Fig. 8. We can see that the magnetic field distributions of the three regionsis separated as expected (see the left panel), where the peak of magnetic field strength isdistributed on the symmetric axis of the overlap region. At a letter time of t − t col = t m , thethree regions would overlap and the magnetic field is continuously distributed (see the rightpanel). The magnitude of the magnetic field is nearly zero in the vicinity of the center ofthe overlap regions. And the strength of the magnetic field in region IV has the same orderas other three regions. 16 igure 8: Magnitude of the magnetic field (in the ideal situation) produced by three bubblescollisions for v col = 0 . , R col m w = t col m W = 10. We show the field as a function of latticenumbers N y and N z on y and z axes respectively with lattice spacing a = 0 . /m W . Left panel:Magnitude of the magnetic field at x = 0 , t − t col = 0 . R col . Right panel: Magnitude of themagnetic field at x = 0 , t − t col = t m .Figure 9: Left panel: The shape of three bubbles after collision at t − t col = t m for v col =0 . R col m W = 10. Right panel: Magnitude of the magnetic field produced by three bubblescollision. The magnetic field in the revised situation is shown as a function of lattice numbers N y and N z on y and z axes respectively with lattice spacing a = 1 /m W at x = 0 , t − t col = t m . hree equal bubbles-revised institution: While taking consideration of the realbubble collision situation, the overlap regions of three bubbles collision would be revisedas shown in the Fig. 9. The bubbles shape and magnetic field generation of three bubblescollision are calculated at ( x = 0 , t − t col = t m ) with v col = 0 .
99 and R col m W = 10. Incomparison with the ideal situation as shown in Fig. 8, the magnitude of the magnetic fieldstrength in the revised situation is shown to be nearly half of the ideal situation, and themagnetic field distributes more continuously. V. IMPLICATION FOR OBSERVATION
The comoving Hubble length at the electroweak phase transition temperature T ∗ is givenby[44], λ H ∗ = 5 . × − Mpc(100GeV / T ∗ )(100 /g ∗ ) / , (53)where g ∗ is the number of relativistic degrees of freedom at the moment when the primordialmagnetic field is generated. The comoving correlation length for a primordial magnetic fieldat the generation time can be evaluated to be ξ (cid:63) = Γ λ H ∗ . (54)Here, Γ is the factor to account for bubble numbers inside one Hubble radius at the FOPT,and Γ (cid:39) .
01 for the electroweak FOPT. For recent simulations, see Ref. [45, 46].The physical magnetic field amplitudes scale with the expansion of the universe at thegeneration time as B ∗ = ( a ∗ a ) B , (55)with the time-temperate relation being, a ∗ a (cid:39) × − (100GeV /T ∗ )(100 /g ∗ ) / . (56)The simulation of the evolution of hydromagnetic turbulence from the electroweak phasetransition suggests that the root-mean-squared non-helical magnetic field amplitude andthe correlation length satisfies the following relation [51], B rms = B ∗ ( ξξ ∗ ) − ( β +1) / , (57)18 lazars · ·· β = β = β = β = - - - - - - - - ξ [ Mpc ] B r m s [ G ] Figure 10: Magetic field strength B rms at the correlation length ξ calculated for the two bubblecollision of the Equal bubbles-Ideal situation (dashed lines) and
Equal bubbles-Revisedsituation (solid lines). Cyan and Blue regions are plotted to consider the bounds set by blazarsgiven in Ref.[47] and Ref. [48]. where β = 1 , ξ being the magnetic correlation length.For illustration, we show in Fig. 10 the bounds of blazars spectra on the magnetic fieldat variant correlation lengths, which depicts that the case of β = 1 case is allowed by thedata. 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 = 0with the largest r at t (= t m + t col ), where we have the r ∼ R col , where we have the ring-like distribution of the magnetic field. The magnetic field strength here is almost the samewith the three bubbles collision situations. Primordial magnetic field suffer bounds from theBig-Bang Nucleosynthesis [50, 52] and the measurements of the spectrum and anisotropiesof the cosmic microwave background [53–58], these limits are not shown in the figure sincethey are not relevant for the parameter space under study in this work. VI. DISCUSSIONS
We use the EOMs of gauge fields to get the magnetic field generated during bubble col-lisions at the electroweak FOPT. After obtaining the Higgs phases when bubbles collide,we calculate the magnetic field strength after obtaining the electromagnetic current, and19pply the approach to the situations of two bubbles and three bubbles collisions, equal andunequal bubbles, ideal and revised situations. We found the electroweak bubble collisionsproduce the ring-like magnetic field even when we consider the revised situation with bub-ble walls deviating from the spherical shape. For that situation, we get a smaller magneticfield strength because the electromagnetic current distributed in a smaller area. The scalinglaw resulting from the hydromagnetic turbulence after the electroweak FOPT suggests thatthis kind of magnetic field under study can be probed by the observation of the Intergalac-tic Magnetic Fields. The magnetic field strength calculated here is comparable with themagnetic field generated from the bubble collisions simulation performed in Ref. [45, 46].We note that, in the electroweak baryogenesis, the Chern-Simons connects the helicity ofthe magnetic fields produced during bubble collisions and the baryon asymmetry of the earlyUniverse [4, 59]. Therefore, the observation of the helicity of the primordial magnetic fieldsmay serve as a test of the electroweak baryogenesis. Ref. [35] studied the primordial magneticfield from first-order phase transition in B − L model and the SM extended by dimensional-sixoperator (Φ † Φ) / Λ , with the physical implication that the observable gravitational wavesand collider signatures would be complementary to the magnetic field observation from thefirst-order phase transitions. when the inverse cascade process are taken into account forhelical magnetic fields [60–62]. VII. ACKNOWLEDGEMENTS
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 workis supported by the National Natural Science Foundation of China under the grantsNos.12075041, 11605016, and 11947406, and Chongqing Natural Science Foundation (GrantsNo.cstc2020jcyj-msxmX0814), and the Fundamental Research Funds for the Central Uni-versities of China (No. 2019CDXYWL0029). [1] M. S. Turner and L. M. Widrow, Phys. Rev. D (1988), 2743 doi:10.1103/PhysRevD.37.2743[2] T. Vachaspati, Phys. Lett. B (1991), 258-261 doi:10.1016/0370-2693(91)90051-Q
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