2. TWO DYNAMO MECHANISMS OF MAGNETIC FIELD AMPLIFICATION
From the Maxwell equation dH/dt = —V x E and the Ohm's law E = —v x H + j/a = —v x H + + (Vx H)/(4^a) one obtains the Faradey equation (vm = (4^acond)_1)
dH
~dt
-— = Vx v x H + vmAH
dt
dPr.
dt
Vpr
+ qr(E + [Vr x B]) + f
weak r .
where the weak force includes two terms: fweak =
= fi-y) + fi-A). The weak vector force fi-y) derived independently by another way (Lagrangian method) in the work [8]:
fr(y) = —sign(a)GF x
(3)
» cV
— Vn. (x,t) — +
dt
for the total magnetic field H = B + b and velocity field v = V + u, which include small-scale (random) components (b) = (u) = 0.
Krause and Radler [3] define the ponderomotive force as the mean one e = (u x b) = vturbV x B — — aB, where the second term is stipulated by the fluid vortices in a turbulent medium, that leads to the evolution equation for large-scale magnetic field (aQ-dynamo):
<9B
— = VxV x B — Vx vturbV x B + Vx aB.
+ Vr x Vx j (x.t)
and the weak axial vector force fi-A) [6] appearing in magnetized plasma:
f(A) =
GFV2ör eSign(a)
n
(4)
E <
a=e,ß,r
(A)
nor b
d5nv ~dt
+ NorV(b • j)
(1)
One can notice that hydrodynamic diffusion is much bigger than the microscopic one caused by the plasma conductivity, vturb » vm. The first term in the r.h.s. of (1) is called dynamo term which depends on the differential rotation of a medium and vanishes, e.g., for rigid rotation of a magnetized body. Namely, the differential rotation plus a effect given by last term in (1) means the standard aQ-dynamo mechanism. Nevertheless, even in the absence of such rotation the existence of a term can lead to so-called a2-dynamo, or to the a effect (compare with (9) below). The hydrodynamic helicity a = —t(u • (V x u))/3 is the pseudoscalar in standard MHD (without neutrino!), or obviously, (1) preserves parity as it should be in QED. The detailed analysis of the dynamo mechanisms in macroscopic electrodynamics is done in the book [5].
3. MHD IN POLARIZED MEDIUM AND PARITY NONCONSERVATION
We refer here to the work [6], where MHD derived from the kinetic equations [7] is generalized in SM of particle physics both for an isotropic plasma and a magnetized medium. The Euler equation for a component of plasma (accounting for collisions in t approximation) takes the form
Here, j = (5nVa ,5jVa) = (x, t) — j£a (x, t) is the neutrino four-current density asymmetry, n = = 0,1,2,3; B is the usual (axial-vector) magnetic field which can include the self-consistent magnetic
field; b = B/B is the unit vector along the magnetic field; n0a = \e\BTln2/(2^2) is the charged lepton density at the main Landau level in hot plasma with the temperature T; N0a is the relativistic correction [6] to that density, N0a — n0a in a non-
relativistic plasma; V = 2{ ± 0.5, cOVl = ^0.5 are vector and axial-vector couplings in SM (upper sign for electron neutrinos), £ = sin2 dW = 0.23 is the Weinberg parameter.
The physical origin of macroscopic weak forces (3) and (4) is connected with the non-forward neutrino (antineutrino) scattering off charged leptons, vae± — — vae±, with the change of neutrino momentum causing the recoil of charge particles. In the case of a polarized plasma, for which the magnetization
M(
(r)
^b(VV Yj ) = №sign(a)b noa is given by the lepton population at the main Landau level, ~n0a, such force f(A) = —VV(A) comes from the parity violation part of the averaged weak interaction
potential, VA) = (GF/V2)cV^)(Mja)/^B)jjVa)(x,t), or for a large-scale magnetization this force should be proportional to the gradients of the neutrino current,
f (A) fk
—dkjjVa) (x. t), as it is seen in (4).
—vem5Pr — (Vrv + Vrv )Pr — (2)
Multiplying Euler equation by the electric charge qr, summing over a and dividing by Q2 = ^ r q2 , we find the electric field E = — £r(q2/Q2)Va x B + ...
The electric field E derived from Euler equations for plasma components takes the form [6] which
x
x
a
x
rv
a
158
SEMIKOZ, SOKOLOFF
includes the contribution of weak interactions Eweak taken in the collisionless Vlasov approximation. We do not consider other known terms which describe weak interaction collisions [4], Biermann battery effects, etc., and which do not play substantially in favor of helicity generation. In turn, we keep in Eweak only the axial-vector term which violates the parity:
For small neutrino chemical potentials (T) = = !va (T)/T < 1, the neutrino density asymmetry entering (7) is of the form
— = V c(A) ^va =
nu ^ eVa nUa
E
(A) weak
Gf
V2\e\
e\nf
E^W + no+ )b X (5)
2n
9Z (3)
[U (T) + U (T) - U (T)],
m-nvg(x,t) dt
+ (No- + No+ )V(b • j (x,t)). \ < 0.07 [9].
where one can use the bound on the electron-neutrino chemical potential coming from the BBN constraints,
Accounting for the first line in (5) we obtain the axial-vector term E^k = —aB, where the helicity coefficient a is the scalar in the SM with neutrinos. Thus, from the Maxwell equation dtB = —V x E one obtains the Faradey equation for evolution of mean (large-scale) magnetic field in the SM which leads to a2-dynamo [5] while with the new helicity a (see below (7)):
4. AMPLIFICATION OF LARGE-SCALE MAGNETIC FIELDS IN THE EARLY UNIVERSE
The spatial scale of the mean magnetic field, A = = n/a, obeying the evolution equation (6) is given
by[1]:
dtB = V x aB + nV2B.
(6)
A_
In
= 1.6 x 10
T \ -5 ( \(v) T \ I \
fluid
MeVy [lv (T )
\£ve (T )\
-i
Here, diffusion coefficient n = 1/(4^ • 137T) is given by the conductivity of relativistic plasma.
We neglected above the neutrino vorticity, V x x j = 0 (neutrino gas rotation is absent), or the weak vector force (3) does not contribute to (6). From the axial-vector weak force given by (4) that implies parity violation using a5ij term and without any fluid rotation and corresponding action of Coriolis force that leads in standard MHD to mirror asymmetry of left-handed and right-handed fluid vortices we derive the scalar a parameter [ 1] entering Faradey equation (6):
(8)
The growth rate, a2/(4n), defines the mean magnetic field amplitude
B (t)= Bmax exp I J
a2 (tQ 4n(t' )
dt'
(9)
Gf
a
2V2\e\B
(7)
Accounting (7), the expansion time (in s) t = = 2.4(T/MeV)-2/^g* and the change of the variable T/(2 x 104 MeV) ^ x < xmax = 1, such amplitude takes the form
' (X) ^ (X)10dX
B(x) = Bmax exp
25
0.07
(A)
n0- + n0+ \ dön:
ln 2
ne
4^2
n2
10-5T
,mp fluid/
dt f
V nv
Here, Aflu)d is the scale of neutrino gas inhomogeneity. ^ 100 GeV
It is remarkable that due to the change of the weak force for neutrino scattering off electrons and
positrons (fjAA ~ M(a) ~ sign(^) = ±1) the electric field (hence, the magnetic field) does not depend on the sign of electric charge qa, or the sum of electron and positron contributions enters the helicity parameter a (7).
(10)
Thus, during the cooling of Universe the scale of mean magnetic field (8) overcomes horizon, A > > lH, somewhere at the temperature T < 100 MeV, while its amplitude (10) increases approximately by 10 orders of magnitude from a small initial value Bmax at the high temperature T ~ 20 GeV < TEW ~
5. COMPARISON WITH WEAK COLLISION MECHANISM
Accounting to the weak collision terms in Euler equation (2) one can find that the difference of weak cross sections ove- — ove+ = 7GFT2 leads to the friction force separating electrons and positrons, or
x
t
x
V
a
x
ev
a
to the electric current caused by the neutrino density asymmetry [4]:
Tcollision ^ Text —
eneönv
3-
(ave- - ave+ )Te =
(11
= 4 x 10-20 eT ^ T) Ï ~ \MeV J \nv
Such current generates field dtB = ... + ^^¿V x
collision x J ext .
Let us compare the collision current (11) with its analogue caused by collective mechanism, e.g., with the weak vector current originated by the force (3)
J
collective ext
eGF\[2cV
a
^cond [V xVx Vv 5nv].
One obtains the ratio
T
collision T
ext =2 x 10s -— VMeV J
-3
Tcollective Text
' \(v) '
\fluid
lv (T )
(12)
or at high temperatures, T > 1 GeV, and for the neutrino fluid inhomogeneity scale Afl^id — (T), collective mechanism is more efficient. And vice versa, near the decoupling time (low temperatures T ~ ~ MeV) collision mechanism is more important.
generalized helicity evolution equation (14) obey CP invariance as it should be for the electroweak interactions in SM since the new coefficient a (7) is CP-odd, (CP)a(CP)-1 = —a, as well as operator V. This is due to the changes n0- <—► n0+ and 5nVa — —SnVa in (7), provided by the well-known properties, that particle helicities are P-odd and particles become antiparticles under the charge conjugation operation C. In particular, active left-handed neutrinos convert to the active right-handed antineutrinos under CP operation, va — Va. The product (E • B) entering the helicity evolution (13) is CP-odd as well because both electric and magnetic fields are C-odd and have opposite P parities. The first term in the r.h.s. of (14) gives conventional ohmic losses for magnetic helicity and usually is neglected in helicity balance.
7. SEED MAGNETIC HELICITY IN COSMOLOGY
Neglecting in evolution equation (14) the first diffusion term, we can calculate the magnetic helicity H(t) using (10) to yield
t
H(t) =2Bmax / d3r f dt'a(t') x (15)
6. MAGNETIC HELICITY GENERATION BY COLLECTIVE NEUTRINO-PLASMA INTERACTIONS
Let us consider how the collisionless neutrino interaction with charged leptons can produce the primordial magnetic helicity H = /(A • B)d3x, where v
v
is the volume that encloses the magnetic field lines. To this end we should substitute into the derivative,
x exp I 2
f = -2/(E ^ B)d3x,
a2(t )
4n(t" )
dt I + H (tmax ),
(13)
the electric field E given by (5). Neglecting any rotation of primordial plasma given by the first dynamo term, or retaining the resistive term and the
weak interaction term E^k given by (5) that is the main one in the absence of any vorticities, one finds from (13)
dH
— = —2n d3x(V x B) • B + 2a d3xB2.
v v
(14)
Note that th
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