If cosmic background neutrinos interact very weakly with each other, through
spin-spin interactions, then they may have experienced a phase transition,
leading to a ferromagnetic ordering. The small magnetic field resulting from
ferromagnetic ordering -- if present before galaxy formation -- could act as a
primordial seed of the magnetic fields observed in several galaxies. Our
findings suggest that the magnetization could occur in the right epoch, if the
exchange boson of neutrino-neutrino interaction is a massless boson beyond the
Standard Model, with a coupling constant of $2.2\times 10^{-13}
\left(\frac{m_\nu}{10^{-4}\,\rm{eV}}\right)^2<g<2.3\times 10^{-7}$. The
estimation of the magnetic seed is $2.3 \times 10^{-27}\,\rm{G}\lesssim B_{\rm
CNB}\lesssim 6.8\times 10^{-10}G$.Comment: 4 pages. Accepted for publication in Physics Letters 

Arias, Paola

Gamboa, Jorge

Lopez-Sarrion, Justo

Physics Letters B

English

arXiv

AbstractIf cosmic background neutrinos interact very weakly with each other, through spin–spin interactions, then they may have experienced a phase transition, leading to a ferromagnetic ordering. The small magnetic field resulting from ferromagnetic ordering – if present before galaxy formation – could act as a primordial seed of the magnetic fields observed in several galaxies. Our findings suggest that the magnetization could occur in the right epoch, if the exchange boson of neutrino–neutrino interaction is a massless boson beyond the Standard Model, with a coupling constant of 2.2×10−13(mν10−4eV)2<g<2.3×10−7. The estimation of the magnetic seed is 2.3×10−27G≲BCNB≲6.8×10−10G

López-Sarrión, Justo

Elsevier - Publisher Connector 

Physics Letters B 735 (2014) 173–175Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Cosmic neutrino background as a ferromagnet
Paola Arias a,∗, Jorge Gamboa b, Justo López-Sarrión b
a Departamento de Física, Pontiﬁcia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
b Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile
a r t i c l e i n f o a b s t r a c t
Article history:
Received 31 December 2013
Received in revised form 12 June 2014
Accepted 12 June 2014
Available online 17 June 2014
Editor: M. Trodden
If cosmic background neutrinos interact very weakly with each other, through spin–spin interactions, 
then they may have experienced a phase transition, leading to a ferromagnetic ordering. The small 
magnetic ﬁeld resulting from ferromagnetic ordering – if present before galaxy formation – could act 
as a primordial seed of the magnetic ﬁelds observed in several galaxies. Our ﬁndings suggest that the 
magnetization could occur in the right epoch, if the exchange boson of neutrino–neutrino interaction is 
a massless boson beyond the Standard Model, with a coupling constant of 2.2 × 10−13 ( mν
10−4 eV )
2 < g <
2.3 × 10−7. The estimation of the magnetic seed is 2.3 × 10−27 G BCNB  6.8 × 10−10 G.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.1. Introduction
There are two untested hypothesis, that if conﬁrmed would 
allow us to better understand the ﬁrst moments of the early uni-
verse: they are the detection of the theorized Cosmic Neutrino 
Background (CNB) [1] and the generation mechanism of a primor-
dial magnetic seed [2–5].
The detection of the CNB is hard because, besides the well 
known weak interactions of neutrinos, the CNB is decoupled from 
the matter content of the universe and has a temperature of about 
T0ν ∼ 1.9 K. Therefore, observing the CNB seems like an impossi-
ble quest.
In order to look for indirect consequences of the CNB, it is 
needed to identify the key processes involved. One possibility is 
that neutrino–neutrino interactions [6] could lead to an observable 
phenomenon, but it is expectable that – in order for them to be 
effective enough to leave an observable trace – the exchange par-
ticle of the process should not be the usual W or Z boson, but 
a particle beyond the Standard Model [7]. These interactions can 
indeed play an important role, at some epoch after the neutrino 
decoupling.
If the latter is true, we ﬁnd spin–spin interaction as the domi-
nant one in the CNB, provided that the momentum transferred to 
the new boson is small. This last fact means that the processes 
νν → νν – at this low energy limit – induce a spin effective in-
teraction S · S, such as the spin models in statistical mechanics [8]. 
* Corresponding author.
E-mail addresses: paola.arias@ﬁs.puc.cl (P. Arias), jgamboa55@gmail.com
(J. Gamboa), justinux75@gmail.com (J. López-Sarrión).http://dx.doi.org/10.1016/j.physletb.2014.06.034
0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article
SCOAP3.In other words, the CNB undergoes a phase transition which is re-
sponsible of the magnetization of the background.
On the other hand, magnetic ﬁelds up to ∼ μG have been ob-
served in galaxies. Their unknown origin has prompt several ideas 
about generation mechanisms [9]. Some scenarios assume that a 
primordial magnetic seed gets adiabatically compressed when pro-
togalactic cloud collapse [2–5]. Other proposals point to an astro-
physical mechanism for the generation of these ﬁelds, a Biermann 
battery effect [10,11]. It is unknown whether one of these mecha-
nisms is actually responsible of the observed magnetic ﬁelds. Inde-
pendently of the mechanism, the existence of primordial ﬁelds has 
an impact in the development of cosmology, such as the success 
of Big Bang Nucleosynthesis, structure formation, etc. [2–5].
In this paper we propose that neutrinos from the CNB inter-
act with each other via the exchange of an intermediate X bo-
son, much lighter than Z0. These neutrinos acquire a spontaneous 
magnetization from their spin–spin interaction and if this magne-
tization was present before the galaxy formation, would provide a 
mechanism for a primordial magnetic seed.
The article is organized as follows: in Section 2 we present an 
effective model for the interaction νν → νν and we write the cor-
responding effective Hamiltonian, using a Breit approximation for 
small transferred momentum between neutrinos and the new bo-
son. We argue why spin–spin interaction should be the dominant 
one among the neutrinos in the CNB, and we write the critical 
temperature of the magnetic phase transition. Further, we impose 
constraints to this new interaction, ﬁrstly for a low mass scale and 
secondly for a large mass scale. In Section 3 we estimate the mag-
nitude of the magnetic ﬁeld generated in the ferromagnetic phase 
transition and Section 4 contains the conclusions. under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by 
174 P. Arias et al. / Physics Letters B 735 (2014) 173–1752. Effective interaction
Let us start by considering the coupling between Dirac neutri-
nos to a generic light neutral boson (Xμ)
L= −gν¯γμνXμ + 1
2
M2X X
2, (1)
where a sum over neutrino species is assumed. Formally, we will 
refer to MX as the mass of the exchange boson, and g the effective 
coupling constant of a secret sector with neutrinos. In this sense, 
the vector ﬁeld Xμ parametrizes the secret neutrino interactions, 
but does not necessarily represent a fundamental particle.
Deﬁning the neutral current as Jμ = ν¯γμν , the effective Hamil-
tonian density can be written as a four-neutrino interaction, 
namely
H= g
2
M2X
Jμ J
μ. (2)
The next step is to study the process νν → νν by using the Breit 
approximation and assuming an almost vanishing transferred mo-
mentum to the new vector boson, namely q = pini − pﬁn. Thus, the 
two-body Breit potential [12] is
HI = − g
2
M2X
nν
∑
{i, j}
S(i) · S( j), (3)
where indices i, j are particles ﬁxed indices and the bracket no-
tation {i, j} stands for nearest neighbors particle interaction. The 
parameter nν is the neutrino number density.
In (3) we have not written the Coulomb, dipolar, quadrupolar 
and spin–orbit interactions because they are negligibly small com-
pared with spin–spin interaction.
The Hamiltonian (3) is obtained by computing at tree-level the 
scattering amplitude in 1/c powers, although, it is not diﬃcult to 
show that this term is the leading effect even in a relativistic case. 
In this quantum ﬁeld theory context we get
g2
M2X
∑
i
δ(xi − xi+1)S(i) · S(i+1) → g
2
M2X
nν
∑
i
S(i) · S(i+1),
and therefore the delta-function corresponds to the neutrino den-
sity. Note that the minus sign in the RHS (3) appears naturally, 
implying the interaction is ferromagnetic.
We stress spin–spin interaction is the dominant one, because of 
the transferred momentum to the new vector boson is small. This 
turns the interaction exclusively as a contact one.
The case of interest to us is when there is a phase transition 
(at some critical temperature Tc) and then CNB acquires a sponta-
neous magnetization.
In order to investigate this, we use the mean ﬁeld approxima-
tion, i.e. the interaction Hamiltonian is replaced by
HI = − J 〈S〉
∑
i
Si, (4)
where J = g2
M2X
nν , 〈S〉 is the mean value of the spin, and the con-
dition for a phase transition is
J
T
 1. (5)
By identifying J = Tc , the critical temperature, the phase transition 
condition reads Tc > T .
Replacing the value of J and using that the neutrino density of 
the CNB scales with the redshift as nν = n0(1 + z)3, where n0 is the present neutrino density, n0 ∼ 56 cm−3, and the temperature 
scales as T = T0ν(1 + z), with T0ν ∼ 1.9 K, we ﬁnd
(1+ z)2  T0νM
2
X
g2n0
. (6)
The above equation can be used if an upper bound for the red-
shift is applied.
On cosmological grounds, it is natural to require that the new 
interaction gets enhanced at an epoch, zi , after the CNB decou-
pling, or if present before that, should not be the predominant one. 
This reasoning leads us to set an upper limit of zi < zν dec, where 
the decoupling of neutrinos from the thermal bath is known to be 
around zν dec ∼ 109 [13].
On the other hand, has been found in Ref. [14], that a re-
coupling of the CNB – if existing – should not be present at recom-
bination era (zrec ∼ 1500), where neutrinos are free-streaming [15]. 
We will assume that at recombination epoch, neutrinos free-
stream, namely, neutrino–neutrino interaction rate, Γνν→νν , does 
not exceed the cosmic expansion rate, Hrec at this redshift
Γνν→νν(zrec) < Hrec. (7)
From these two requirements we can ﬁnd if a new vector boson, 
interacting only with neutrinos, could lead to a ferromagnetic or-
dering of the latter, which is not in conﬂict with observations.
2.1. Light vector boson exchange (MX mν )
Let us ﬁrst consider the case of a very light boson, such that 
MX is of the order of neutrino mass, i.e. MX ∼ mν . In this case 
Eq. (6) becomes
(1+ z)2 > m
2
ν T0ν
n0g2
. (8)
By imposing that the interaction neutrino-hidden boson turns on 
at an epoch after neutrino decoupling, zi < 109, we get
2.2× 10−13
(
mν
10−4 eV
)2
≤ g. (9)
On the other hand, to respect the free-streaming of neutrinos at 
the epoch of recombination, we require that at zi = 1500, the in-
teraction should be negligible.
At recombination era, the universe was matter dominated, 
so the Hubble parameter is given by Hrec =
100 km−1 Mpc−1(ΩMh2)1/2(zrec + 1)3/2. Here ΩMh2 is the cos-
mic matter density and is given by ΩMh2 = 0.134. The Hubble 
parameter at this redshift takes the value Hrec = 4.5 × 10−29 eV.
On the other hand, neutrino–neutrino interaction rate is given 
in this (nearly massless) case by Γνν→νν = nν〈σ v〉, where 〈σ v〉 ∼
g4nν/〈s〉, and the free-streaming requirement, Eq. (7), reads
g < 2.3× 10−7. (10)
Combining both limits we ﬁnd that spontaneous magnetization of 
the CNB is possible, at a redshift in between, 1500 < zi < 109, if 
the exchange boson of neutrino–neutrino interactions is a nearly 
massless vectorial boson, with a constant coupling of
2.2× 10−13
(
mν
10−4 eV
)2
≤ g ≤ 2.3× 10−7. (11)
2.2. Massive vector boson exchange (MX mν )
Secondly, we will consider a massive vector boson, such 
MX  mν . In the massive case, the constrained parameter is the 
P. Arias et al. / Physics Letters B 735 (2014) 173–175 175Fermi-like constant GX ≡ g2M2X . From the requirement that the in-
teraction turns on at a temperature, Tc , after neutrino decoupling 
we get, using Eq. (6)
GX ≥ 4.8× 1013GF , (12)
where, GF ∼ 10−23 eV−2, is the Fermi constant.
Meanwhile, the requirement that the interaction should be 
weak enough to let neutrinos free-streaming at recombination era, 
Γνν→νν/Hrec < 1, gives
GX ≤ 6.4× 1010GF , (13)
where we have used that the interaction rate is Γνν→νν = G2X T 2nν .
Putting both region of interest together, we ﬁnd that sponta-
neous magnetization of the CNB mediated by a massive boson is 
not possible at the epoch before recombination, because the two 
required constraints do not complement each other
4.8× 1013GF ≤ GX ≤ 6.4× 1010GF . (14)
Also, a vector boson, in any of these two limits, has been ruled out 
from several astrophysical tests [16].
3. Estimating the magnetic seed ﬁeld
A rough estimation of the magnetic ﬁeld generated at the phase 
transition can be made by approximating each neutrino as a spin. 
An upper limit in the magnetic ﬁeld would be to assume that all 
species of neutrinos undergo the ferromagnetic phase transition. 
With this assumption, the magnetic seed ﬁeld is
BCNB = μnnν . (15)
Here μν is the neutrino magnetic moment, predicted to be μν ∼
10−19μB [17,18]. Replacing in the above equation, and using our 
redshift window of 1500 < zi < 109, we get an estimate of
2.3× 10−27 G BCNB  6.8× 10−10 G. (16)
The maximal correlation length for the magnetic ﬁeld would be 
the Hubble radius at the time of generation, H−1(zi). Therefore, 
the maximal correlation length of the ﬁeld generated by the spon-
taneous magnetization ranges from λdec ∼ 3.4 npc at neutrino de-
coupling era, to λrec ∼ 100 kpc at recombination.
The cosmological magnetic ﬁeld damps with the scale factor 
as a−2, so a quick check tell us that an observed ﬁeld of B ∼ 1 μG
in galaxy clusters, corresponds to a primordial magnetic ﬁeld of 
Bprim ∼ 1 nG at the epoch z ∼ 1100 [19], which seems in agree-
ment with our result.
4. Discussion and summary
In this paper we have proposed a mechanism by which the 
Cosmic Neutrino Background could become observable in a new 
physics framework. Such a mechanism is produced by an enhance-
ment of neutrino–neutrino interactions after their decoupling from 
the thermal bath, in the early universe.
We have found that if the scattering has a small momentum 
transfer, |q|, the dominant interaction in the CNB is a ferromag-
netic spin–spin coupling. Such interaction – if existing – should 
have a coupling strength given by (11), and a low mass mediator.
As a result of the coupling, a ferromagnetic phase in the CNB 
appears at a temperature before structure formation (z ∼ 10), and 
thus, the resulting magnetization can act as a primordial magnetic 
seed, and later be ampliﬁed by a dynamo mechanism.Our results are not in contradiction with limits found in sev-
eral related proposals [7,20–25]. Although the idea proposed here 
is the result of observing CNB neutrinos as a spin system, this in-
terpretation is also consistent and complementary with the secret 
interactions studied in the above mentioned papers. It is remark-
able that the behavior of neutrinos at very low scale of energy can 
be very unconventional and a detailed study, very likely, should 
provide unexpected phenomena.
Finally we have made an estimation of the magnitude of the 
magnetic ﬁeld generated by the CNB, assuming all neutrinos from 
the three species undergo the ferromagnetic phase transition. The 
strength of the magnetic ﬁeld depends on the epoch at which 
the transition took place. For a redshift between 1500 < zi < 109, 
(corresponding to a temperature in the range of MeV–eV), the 
magnetic ﬁeld could be in the range of 2.3 × 10−27 G  BCNB 
6.8 × 10−10 G.
Acknowledgements
We would like to thank Professors C. A, Garcia-Canal, F. A. Scha-
posnik, A. Ringwald, A. Reisenegger and A. Wipf for valuable com-
ments. This work was supported by grants from FONDECYT-Chile 
11121403 and Anillo ACT 1102 (P.A.), 1130020 (J.G.) and 1100777, 
1140243 (J.L-S.) and DICYT 041131LS.
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Cosmic neutrino background as a ferromagnet 

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Cosmic Neutrino Background as a Ferromagnet

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