aa r X i v : . [ h e p - t h ] J a n Non-Abelian bootstrap of primordial magnetism
Poul Olesen ∗ The Niels Bohr InstituteBlegdamsvej 17, Copenhagen Ø, Denmark
October 7, 2018
Abstract
We point out that a primordial magnetic field can be generated in the electroweakphase transition by a non-Abelian bootstrap, where the field is generated by currents of W ′ s, which in turn are extracted from the vacuum by the magnetic field. This magneticfield is produced as a vortex condensate at the electroweak phase transition. It becomesstringy as a consequence of the dynamical evolution due to magnetohydrodynamics. There is much evidence for the existence of a primordial magnetic field. There are severalproposals for how these fields are generated, as reviewed for example in the paper [1]. Onepossibility is genesis at the electroweak phase transition, as first discussed by Vachaspati[2]. In his case the magnetic field is generated from properties of the Higgs field. In thisnote we shall discuss another possible mechanism for generation at the electroweak phasetransition, namely a non-Abelian bootstrap mechanism whereby a magnetic field is generatedfrom currents coming from charged W ′ s which in turn are generated from the magnetic field.This self organized mechanism relies heavily on properties that are generic for non-Abelianvector fields. The proposed mechanism is therefore also of interest in principle, since it maygive direct observational information on non-Abelian field theory of vectors.We start by considering a simple model with an SU(2) massive vector field, L = − F µν − m W ( T ) W † µ W µ . W µ = 1 √ A µ + iA µ ) , (1)where F µν = ∂ µ A ν − ∂ ν A µ + ig [ A µ , A ν ] , A µ = A aµ σ a / . (2)This is the same model as considered some time ago by Ambjørn and the author in [3],except that the mass depends on the temperature T . It is assumed that there is a phasetransition, m W ( T ) = 0 for T > T c , m W ( T ) = 0 otherwise . (3) ∗ email: [email protected] he magnetic field f µν = ∂ µ A ν − ∂ ν A µ (4)is given by [3] ef = m W ( T ) + 2 e | W | , (5)where we used the ansatz [3] W = W , W = iW ≡ iW, W = W = 0 , W = W ( x , x ).The result (5) arizes from minimizing the energy written in the formEnergy density = | ( D + iD ) W | + 12 f − m W ( T ) e − e | W | ! + m W ( T ) e f − m W ( T ) e . (6)In this equation D a = ∂ a − ieA a . (7)We carry out the minimization by requiring that the two positive quadratic terms vanishlike in the Bogomol’nyi limit. We immediately obtain Eq. (5). The equation of motion for W = | W | exp( iχ ) can be obtained from the vanishing of the first term,( D + iD ) W = 0 , (8)from which we get by use also of Eq. (5) − ( ∂ + ∂ ) ln | W | = m W ( T ) + 2 e | W | − ǫ ij ∂ i ∂ j χ, (9)where χ is the phase of W . The relative plus sign between the two terms on the right handside of Eq. (5) reflects the antiscreening of this solution. Because of this sign there is nosingle vortex solution. Instead Eq. (9) has periodic solutions, corresponding to a latticeof vortices. As usual in each periodicity domain the term ǫ ij ∂ i ∂ j χ gives a delta functioncorresponding to the delta function coming from the zero of | W | on the left hand side of (9).The result (5) is now considered as a self organized solution of the equations of motionwhich shows the possibility of creating a magnetic field from the non-Abelian dynamics. Theenergy is taken from the expansion energy of the universe.The order of magnitude of the field below the temperature T c is f ∼ m W /e ∼ G, (10)which is a large field, of the same order of magnitude as the one found by Vachaspati [2].Each flux tube has a dimension of order 1 /m W . For T > T c the solution of the equationof motion (9) can be found explicitly in terms Weierstrass’ p-function, and it can be shownthat it corresponds to a zero energy solution [4]. By a nonperturbative gauge transformationone can transform this solution to the perturbative ground state A aµ = 0.The solution of Eq. (9) is a bootstrap type of solution, because the magnetic field isinherent in the vacuum and is extracted from “emptiness” by the appearance of the mass in This is the anti-Lenz’ law according to which the magnetic field will be enhanced by the current. Thenecessary energy is produced by the W − condensate. W ′ s. In other words, the magneticfield is generated by W − currents, ∂ f = 2 e∂ | W | = − j and ∂ f = 2 e∂ | W | = j , (11)and these W ′ s in turn are generated by the magnetic field, because of the non-Abelianinstability discussed a long time ago [5], according to which the magnetic field exceeding themagnitude m W /e is unstable unless stabilized by W ′ s from the vacuum. Thus the magneticfield and the vector bosons are interwoven in the structure of the solution of Eq. (9) andonly exist because of one another.The energy density is given by E = m W ( T ) e f − m W ( T ) e (12)We see that this energy is smaller than the no condensate energy f / W condensate.Considering the vortices as strings we can compute the string tension by integratingthe energy density over a single quadratic domain with area c/m w , where c is a numericalconstant. The result is string tension ≡ σ ( T ) = (2 π − c/ m W ( T ) e , (13)where we used the quantization of the fluxFlux = Z domain f d x = 2 π/e. (14)Thus we see that the string tension vanishes above the critical temperature, where the fieldcontents of the solution becomes non-perturbative pure gauge fields, as discussed in [4].As the universe expands the strings develop according to the magnetohydrodynamic(MHD) field equations. A long time ago we showed [6] that in the limit of infinite con-ductivity (“ideal” MHD) these equations are satisfied by Nambu-Goto strings. Later thiswas discussed including dissipative effects by Schubring [7]. Also, numerically the turbu-lent plasma governed by the MHD equations has been found to be extremely intermittentwith the vorticity concentrated in thin vortex types with the magnetic field concentratedalso in thin vortex types [8]. Therefore the stringy initial behavior exhibited by the vortexcondensate discussed above fits well with the subsequent MHD governed develpment of theuniverse. In the string picture the magnetic field is given by [6],[7] B i ( x , t ) = X strings b Z dσ ∂z i ( σ, y ) ∂σ δ ( x − z ( σ, t )) , (15) For simplicity we consider a quadratic domain. Energy may be minimized by other types of domains.The constant c is close to 2 π if m W >> e | W | . In a certain non-perturbatively defined gauge the strings still exist [4] with zero string tension abovethe critical temperature. This vacuum string configuration is, however, degenerate with the perturbativevacuum. f = B etc. and b is the magnetic flux. In this equation there should be a sum overall the strings in the vortex lattice. The string coordinates satisfy ∂z i ∂t ∂z i ∂σ = 0 , and ∂ z i ∂σ = 1 v ∂ z i ∂t . (16)Here v is a maximum transverse velocity.The string tension from the gauge theory is temperature dependent and vanishes abovethe critical temperature. This phenomenon was found in string theory for the Nambu-Goto string long time ago by Pisarski and Alvarez [9], where the critical temperature is thedeconfinement (Hagedorn) temperature. Their result would be obtained from Eq. (13) if m W ( T ) ∝ q − ( T /T c ) (17)In our case, the exitence of the critical temperature indicates that the vortex/string picturebreaks down above this temperature. Of course, even in the field theory case this temperaturewould also correspond to deconfinement if monopoles exist. They would be confined belowthe critical temperature, and released above the critical temperature because of zero stringtension.So far we have consideed the simple model (1) with a temperature dependent mass. Weshall now consider the standard electroweak theory with a Higgs field φ , where the magneticfield turns out to be given by [10] f = g φ ( T ) θ + 2 g sin θ | W | (18)in the Bogomol’nyi limit where the Higgs mass equals the Z mass. For the realistic masscase a much more complicated perturbative treatment is necessary. For simplicity we shalltherefore stick to the Bogomol’nyi limit. The equations of motion are [10] − ( ∂ + ∂ ) ln | W | = g φ + 2 g | W | − ǫ ij ∂ i ∂ j χ, (19)which is analogous to Eq. (9), and( ∂ + ∂ ) ln φ = g θ ( φ − φ ( T ) ) + g | W | . (20)It has been proven mathematically that these coupled equations have periodic solutions[11]-[15]. Here φ ( T ) vanishes above the critical temperature and the solution then becomesdegenerate with the perturbative vacuum [4].The string tension can again be obtained by integrating the energy density over onedomain in the plane. The result is the same as in Eq. (13). Thus the previous discussioncan be repeated for the electroweak theory, at least in the Bogomol’nyi limit. Again thestring tension vanishes above the critical temperature.We end this discussion with some remarks on the chiral anomaly effect on the evolutionof the primordial magnetic field [16]-[21]. The inclusion of this effect will modify the MHD4quations by adding an effective electric current. Also, hyperfields are relevant above theelectroweak phase transition, and there may be magnetic helicity above and below thistransition. In ideal MHD helicity is conserved, but this is not valid when the Ohmic resistanceis included, and the helicity will ultimately decay. It is clear that our solution is not bornwith helicity, since for this solution AB = 0, but due to fluctuations from the full MHDequations there will always be some helicity [22].It is always a problem for primordial magnetic fields generated from particle physics thatthe initial scale is small. Even though the expansion of the universe increases this scale ingeneral this is not enough for the generation of realistic scales. Therefore the phenomenonof inverse cascading, i.e. the drift of energy towards larger scales, is important . Oftenthis phenomenon is linked with (conserved) helicity [23]. However, with vanishing helicitythere is still an inverse cascade in freely decaying MHD, moving energy from smaller tolarger scales, as discussed recently [24]-[26]. Thus, helicity is not a necessary condition foran inverse cascade to occur. More explicitly it was found numerically by Zrake [24] that theenergy scales in a self-similar manner, which was shown by the author [26] to be an exact consequence of the standard MHD equations for freely decaying turbulence when dissipationis included. The energy density should satisfy E ( k, t ) = s t t E k s tt , t ! (21)According to this formula (which is one of the few known exact results in HD and MHD)energy is moved from smaller to larger scales as time passes. This is obviously important inorder to increase the scale of the magnetic field on the top of the expansion of the universe.These results have to be modified if chiral MHD turbulence is taken into account, as discussedrecently in [21].For completeness we display the magnetic energy for an expanding flat universe with themetric dτ = dt − a ( t ) d x = a (˜ t ) (cid:16) d ˜ t − d x (cid:17) . (22)Here t is the Hubble time and ˜ t = Z dt/a ( t ) (23)is the conformal time corresponding to the expansion parameter a ( t ). Eq. (21) is thenreplaced by E B ( k, ˜ t ) = a ( ˜ t ) a (˜ t ) ! s ˜ t ˜ t E B k vuut ˜ t ˜ t , ˜ t . (24)Again we see a drift towards large distances as the univese expands.In conclusion we have shown that a primordial magnetic field can be generated in theelectroweak phase transition by a non-Abelian vector bootstrap. The resulting field consistsof a set of antiscreening vortices which, because they have to follow the MHD equations,develop in a stringy manner. Due to the inverse cascade the field may survive at large A more appropriate term is perhaps drift towards the infrared, i.e. small k . References [1] D. Grasso and H. R. Rubinstein, Phys. Rep. 348 (2001) 163[2] T. Vachaspati, Phys. Lett. B 265 (1991) 258[3] J. Ambjørn and P. Olesen, Phys. Lett. B 214 (1988) 565[4] P. Olesen, Phys. Lett. B 268 (1991) 389; arXiv:1605.00603[5] N. K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376[6] P. Olesen, Phys. Lett. B 366 (1996) 117[7] D. Schubring, Phys. Rev. D 91 (2015) 043518[8] A. 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