Low bound on a magnetic field strength in the hot universe
It is assumed that the intergalactic magnetic fields were spontaneously generated in the early Universe due to vacuum polarization of non-Abelian gauge fields at high temperature T. Here, a procedure to estimate the field strengths B(T) at different T is developed and the value of B(Tew) ~ 10⁻¹⁴ G,...
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| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2012 |
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| Мова: | Англійська |
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Institute for Space Science, ICE-CSIC and IEEC
2012
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| Цитувати: | Low bound on a magnetic field strength in the hot universe / E. Elizalde, V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 143-146. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860200927313526784 |
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| author | Elizalde, E. Skalozub, V. |
| author_facet | Elizalde, E. Skalozub, V. |
| citation_txt | Low bound on a magnetic field strength in the hot universe / E. Elizalde, V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 143-146. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | It is assumed that the intergalactic magnetic fields were spontaneously generated in the early Universe due to vacuum polarization of non-Abelian gauge fields at high temperature T. Here, a procedure to estimate the field strengths B(T) at different T is developed and the value of B(Tew) ~ 10⁻¹⁴ G, at the electroweak phase transition temperature, is derived by taking into consideration the present intergalactic magnetic field strength B0 ~ 10⁻¹⁵ G.
Высказано предположение о том, что межгалактические магнитные поля произошли в ранней вселенной при высокой температуре T вследствие спонтанного намагничения вакуума неабелевых калибровочных полей. Развита процедура, позволяющая оценить напряжённости поля B(T) при различных T, и получено значение напряжённости поля B(Tew) ~ 10⁻¹⁴ G при температуре электрослабого фазового перехода, принимая значение существующего в настоящее время межгалактического магнитного поля B0 ~ 10⁻¹⁵ G.
Висловлено припущення про те, що міжгалактичні магнітні поля виникають у ранньому всесвіті при високій температурі T внаслідок спонтанного намагнічування вакууму неабелевих калібрувальних полів. Розроблено процедуру, яка дає змогу оцінити напруженість поля B(T) при різних T, і обчислено значення напруженості поля B(Tew) ~ 10⁻¹⁴ G при температурах електрослабкого фазового переходу, приймаючи значення існуючого в наш час міжгалактичного магнітного поля порядку B0 ~ 10⁻¹⁵G.
|
| first_indexed | 2025-12-07T18:10:23Z |
| format | Article |
| fulltext |
LOW BOUND ON A MAGNETIC FIELD STRENGTH
IN THE HOT UNIVERSE
E. Elizalde 1 and V. Skalozub 2∗
1Institute for Space Science, ICE-CSIC and IEEC, Campus UAB, 08193, Bellaterra, Barcelona, Spain
2Dnipropetrovsk National University, 49010, Dnipropetrovsk, Ukraine
(Received October 24, 2011)
It is assumed that the intergalactic magnetic fields were spontaneously generated in the early Universe due to vacuum
polarization of non-Abelian gauge fields at high temperature T . Here, a procedure to estimate the field strengths
B(T ) at different T is developed and the value of B(Tew) ∼ 1014 G, at the electroweak phase transition temperature,
is derived by taking into consideration the present intergalactic magnetic field strength B0 ∼ 10−15 G.
PACS: 98.62.En, 12.38.-t, 11.15.Ex
1. INTRODUCTION
Recent experimental discovery of intergalactic mag-
netic fields having the field strength of the order
B ∼ 10−15 G [1, 2] is one of the most interesting
events of modern cosmology. In Ref. [3] a model-
independent 95 per sent CL interval 1 × 10−17 G ≤
B ≤ 3 × 10−14 G is determined. This discovery, in
particular, restricts the possible processes resulting
in the generation of fields in the hot Universe [2, 4],
and stimulates further investigations.
In the present report we discuss a mechanism
based on non-Abelian magnetic fields. As it was
shown recently, a spontaneous magnetization hap-
pens in non-Abelian gauge theories at high temper-
ature T . This phenomenon is the extension of the
Savvidy [9] vacuum B(T ) = const �= 0 to the finite
temperature case. In contrast to the zero temper-
ature, the state B(T ) appears to be stable due to
a magnetic mass of the color charged gluon and a
A0-condensate. Its energy is below the perturbative
vacuum one, and the minimum of the effective po-
tential is reached for a field of order gB ∼ g4T 2.
Although this phenomenon was discovered in SU(2)
gluodynamics, it is common for other SU(N) gauge
fields.
An important property of such magnetic fields is
the vanishing of their magnetic mass, mmagn. = 0
[8, 11]. The mass parameter describes the inverse
spatial scales of the transverse field components.The
absence of the screening mass means that the sponta-
neously generated Abelian chromomagnetic fields are
long range at high temperature. Hence, it is reason-
able to believe that, in the hot Universe, at each stage
of its evolution spontaneously created, strong, long-
range magnetic fields of different types have been
present. Since they are constant ones, their scale
is coinciding with the horizon scale at a particular
temperature.
In what follows, in the frameworks of the standard
model (SM), we estimate the strength of the magnetic
field at the temperature T ew
c of the electroweak phase
transition (EWPT), assuming the mechanism as de-
scribed above. We carry out an actual calculation
in the frame of a consistent effective potential (EP)
accounting for the one-loop, V (1), and the daisy (or
ring), V ring, diagram contributions. In Sec. 2 the EP
of an Abelian constant electromagnetic B field at fi-
nite temperature is obtained. It is used, in Sec. 3, to
estimate the magnetic field strength at the (EWPT).
In Sec. 4 the discussion of the results is given.
2. EFFECTIVE POTENTIAL AT HIGH
TEMPERATURE
The complete EP for the standard model is given in
the review [10]. In the present investigation we are
interested in two limits:
1. Weak magnetic field and large scalar field con-
densate, h = eB/M2
w < φ2, φ = φc/φ0, β =
1/T ;
2. Case of the restored symmetry, φ = 0, gB �=
0, T �= 0.
For the former case we show the absence of sponta-
neous vacuum magnetization at finite temperature.
For the latter one we estimate the field strength at
high temperature. Here Mw is the W -boson mass at
zero temperature, φc is a scalar field condensate, and
φ0 its value at zero temperature.
To demonstrate the first property we consider the
one-loop contribution of W -bosons:
∗Corresponding author E-mail address: skalozubv@daad-alumni.de
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N 1.
Series: Nuclear Physics Investigations (57), p. 143-146.
143
V (1)
w (T, h, φ) =
h
π2β2
∞∑
n=1
[ (φ2 − h)1/2β
n
K1(nβ(φ2 − h)1/2) − (φ2 + h)1/2β
n
K1(nβ(φ2 + h)1/2)
]
, (1)
where n labels discrete energy values and K1(z) is
the MacDonald function.
The main goal of our investigation is the restored
phase of the SM. So, we adduce the high temperature
contribution of the complete effective potential rele-
vant for this case using the results in Ref. [10]. First
we write down the one-loop W -boson contribution
as the sum of the pure Yang-Mills weak-isospin part
(B̃ ≡ B(3)),
V (1)
w (B̃, T ) =
B̃2
2
+
11
48
g2
π2
B̃2 log
T 2
τ2
− 1
3
(gB̃)3/2T
π
− i
(gB̃)3/2T
2π
+ O(g2B̃2), (2)
where τ is a temperature normalization point, and
the charged scalars,
V (1)
sc (B̃, T ) = − 1
96
g2
π2
B̃2 log
T 2
τ2
+
1
12
(gB̃)3/2T
π
+ O(g2B̃2), (3)
describing the contribution of longitudinal vector
components. This representation is convenient for
the case of extended models including other gauge
and scalar fields. In the SM, the contribution of
Eq. (3) has to be taken with a factor 2, in the case
of the Two-Higgs-Doublet SM, this factor must be 4,
etc. The imaginary part is canceled by the term ap-
pearing in the contribution of the daisy diagrams for
the unstable mode [6],
Vunstable =
gB̃T
2π
[Π(B̃, T )−gB̃]1/2+i
(gB̃)3/2T
2π
. (4)
Here Π(B̃, T ) is the mean value for the charged gluon
polarization tensor taken in the ground state of the
spectrum. If this value is sufficiently large, spec-
trum stabilization due to radiation correction takes
place. This possibility formally follows from the tem-
perature and field dependence of the polarization
tensor in the high temperature limit T → ∞ [12]:
Π(B̃, T ) = c g2T
√
gB̃, where c > 0 is a constant
which must be calculated explicitly. At high temper-
ature the first term can be larger then gB̃.
The high temperature limit of the fermion contri-
bution looks as follows:
Vfermion = −α
π
∑
f
1
6
q2
f B̃2 log
T
τ
, (5)
where the sum is extended to all leptons and quarks,
and qf is the fermion electric charge in positron units.
We observed the stable vacuum state in the lattice
simulations [7]. Therefore, we believe that this prob-
lem has a positive solution.
3. MAGNETIC FIELD STRENGTH AT Tew
Let us now show that the spontaneous vacuum mag-
netization does not happen for non-small values of
φ �= 0. To do that we notice that the magnetization
is produced by the gauge field contribution, given
in Eq. (1). We consider the limit of gB
T 2 � 1 and
φ2 > h. For this case we use the asymptotic expan-
sion of K1(z),
K1(z) ∼
√
π
2z
e−z
(
1 +
3
8z
− 15
128z2
+ · · · ), (6)
where z = nβ(φ2 ± h)1/2. Now, we investigate the
limit of β → ∞, T
φ � 1. We can also substitute
(φ2 ± h)1/2 = φ(1 ± h
2φ2 ). In this approximation, the
sum of the tree level energy and (1) reads
V =
h2
2
− h2
π3/2
T 1/2
φ1/2
(
1 − T
2φ
)
e− φ
T . (7)
The stationary equation ∂V
∂h = 0 has the solution
h = 0. Hence we conclude, after symmetry break-
ing the spontaneous vacuum magnetization does not
happen.
To estimate the magnetic field strength in the re-
stored phase at the EWPT temperature the total
EP deduced in the previous section must be used.
This can be best done numerically. To explain the
procedure, we consider here a part of this potential
accounting for the one-loop W -boson contributions
given in Eq. (2). The value of the spontaneously gen-
erated magnetic weak isospin field is calculated from
Eqs. (2) and (3):
B̃(T ) =
1
16
g3
π2
T 2
(1 + 5
12
g2
π2 log T
τ )2
. (8)
We relate this expression with the intergalactic mag-
netic field B0.
Let us introduce the standard parameters and de-
finitions, g2
4π = αw, α = αw sin θ2
w, (g′)2
4π = αY and
tan2 θw(T ) = αY (T )
αw(T ) , where α is the fine structure
constant. Here, instead to find the temperature de-
pendence of the Weinberg angle, we, for a rough
estimate, substitute the zero temperature number:
sin2 θw(T ) = sin2 θw(0) = 0.23.
Other point – re-scaling – must be taken into ac-
count in the expanding Universe. As is well known,
the temperature dependence B(T0) = B(T )( a(T )
a(T0) )
2
takes place, where a(t) is a metric scale factor. At
the same time, for magnetic fields after symmetry
breaking (as for relic photons) the scaling behavior
T (t) ∼ 1/a(t) is usually assumed. That results in the
temperature dependence of B ∼ (T/T0)2. Hence, the
possibility to relate B0 with B(Tew) is in order.
144
For the given temperature of the EWPT, Tew, the
magnetic field is
B(Tew) = B0
T 2
ew
T 2
0
= sin θw(Tew)B̃(Tew). (9)
Assuming Tew = 100 GeV = 1011 eV and T0 =
2.7K = 2.3267 · 10−4 eV, we obtain
B(Tew) ∼ 1.85 1014 G. (10)
To take into consideration the fermion contribu-
tion Eq.(5) we have to substitute the expression
5
12
g2
π2 log T
τ in Eq.(8) by the value
(
5
3
−
∑
f
1
6
q2
f )
αs
π
log
T
τ
. (11)
In the above estimate, we have taken into account the
one-loop part of the EP of order ∼ g2 in the coupling
constant. The ring diagrams have order ∼ g3 and
give a small numeric correction to this result in the
high temperature approximation. Note, had we taken
into account all the terms listed in the previous sec-
tion, the results not changed essentially as compared
to given in Eq.(10).
4. DISCUSSION
We here summarize our main results. In the prob-
lem under investigation, the key point is the spon-
taneous vacuum magnetization, which eliminates the
magnetic flux conservation principle at high tempera-
ture. Vacuum polarization is responsible for the value
of the field strength B(T ) at each temperature and
serves as a source of it. We also have shown that,
at finite temperature and after symmetry breaking, a
scalar field condensate suppresses the magnetization.
At Tew the magnetization is stopped and the frozen
in of the magnetic field lines into the plasma happens.
Due to this property the field strengths at different
temperatures can be estimated and related to B0 in
various models.
Hence it follows that the actual nature of the ex-
tended model is not essential at sufficiently low tem-
peratures when the decoupling of heavy gauge fields
has happened. In particular, from this one can con-
clude that the vacuum polarization “washed out” the
relics of the magnetic fields generated at very high
temperature or at inflation.
The present value of the intergalactic magnetic
field is related in our model with the field strengths
at high temperatures in the restored phase. Because
of the zero magnetic mass for Abelian magnetic fields,
as discovered recently [8], there is no problem in the
generation of fields having a large coherence length.
In fact, we have assumed that it is of the order of the
horizon scale, λB ∼ RH(T ), in our estimate. This is
reasonable because at a given temperature the field
B(T ) = const, generated due to vacuum polariza-
tion, occupies all space. In this scenario, a large scale
domain structure is also permissible that requires an
addition consideration. Knowing the particular prop-
erties of the extended model it is possible to estimate
the field strengths at any temperature. This can be
done for different schemes of spontaneous symme-
try breaking (restoration) by taking into account the
fact that, after the decoupling of some massive gauge
fields, the corresponding magnetic fields are screened.
Thus, the higher the temperature is the larger num-
ber of strong long range magnetic fields of different
types will be generated in the early Universe.
One of us (VS) was supported by the ESF
CASIMIR Networking program. This work has been
also partly supported by MICINN (Spain), projects
FIS2006-02842 and FIS2010-15640, by the CPAN
Consolider Ingenio Project, and by AGAUR (Gen-
eralitat de Catalunya), contract 2009SGR-994.
References
1. S. Ando, A. Kusenko. Evidence for Gamma-
Ray Halos Around Active Galactic Nuclei and
the First Measurement of Intergalactic Magnetic
Fields // Astrophys. J. Lett. 2010, v. 722, p. L39-
L44.
2. A. Neronov, E. Vovk. Evidence for Strong Ex-
tragalactic Magnetic Fields from Fermi Obser-
vations of TeV Blazars // Science. 2010, v. 328,
p. 73-75.
3. W. Essey, S. Ando, A. Kusenko. Determination
of intergalactic magnetic fields from gamma ray
data // Astropart. Phys.. 2011, v. 35, p. 135-139.
4. D. Grasso and H.R. Rubinstein. Magnetic Fields
In The Early Universe // Phys. Rept. 2001,
v. 348, p. 163-266.
5. A.O. Starinets, A.S. Vshivtsev and V.C. Zhu-
kovsky. Color ferromagnetic state in SU(2) gauge
theory at finite temperature // Phys. Lett. 1994,
v. B322, p. 403.
6. V. Skalozub and M. Bordag. Once more on a
colour ferromagnetic vacuum state at finite tem-
perature // Nucl. Phys. 2000, v. B576, p. 430.
7. V.I. Demchik and V.V. Skalozub. On the Spon-
taneous Creation of Chromomagnetic Fields at
High Temperature // Phys. Atom. Nucl. 2008,
v. 71, p. 180.
8. S. Antropov, M. Bordag, V. Demchik and
V.V. Skalozub. Long range chromomagnetic
fields at high temperature // arXiv: 1011.314/v1
[hep-ph], 13 Nov. 2010.
9. G.K. Savvidy. Infrared Instability of the Vacuum
State of Gauge Theories and Asymptotic Free-
dom // Phys. Lett. 1977, v. B71, p. 133.
145
10. V. Skalozub and V. Demchik. Electroweak
Phase Transition in Strong Magnetic Fields in
the Standard Model of Elementary Particles //
arXiv: hep-th/9912071, 8 Dec. 1999.
11. M. Bordag and V. Skalozub. Neutral gluon po-
larization tensor in color magnetic background
at finite temperature // Phys. Rev. 2007, v. D75,
125003.
12. M. Bordag, V.V. Skalozub. Polarization tensor
of charged gluons in color magnetic background
field at finite temperature // Phys. Rev. 2008,
v. D77, 105013.
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|
| id | nasplib_isofts_kiev_ua-123456789-107014 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:10:23Z |
| publishDate | 2012 |
| publisher | Institute for Space Science, ICE-CSIC and IEEC |
| record_format | dspace |
| spelling | Elizalde, E. Skalozub, V. 2016-10-11T08:13:49Z 2016-10-11T08:13:49Z 2012 Low bound on a magnetic field strength in the hot universe / E. Elizalde, V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 143-146. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 98.62.En, 12.38.-t, 11.15.Ex https://nasplib.isofts.kiev.ua/handle/123456789/107014 It is assumed that the intergalactic magnetic fields were spontaneously generated in the early Universe due to vacuum polarization of non-Abelian gauge fields at high temperature T. Here, a procedure to estimate the field strengths B(T) at different T is developed and the value of B(Tew) ~ 10⁻¹⁴ G, at the electroweak phase transition temperature, is derived by taking into consideration the present intergalactic magnetic field strength B0 ~ 10⁻¹⁵ G. Высказано предположение о том, что межгалактические магнитные поля произошли в ранней вселенной при высокой температуре T вследствие спонтанного намагничения вакуума неабелевых калибровочных полей. Развита процедура, позволяющая оценить напряжённости поля B(T) при различных T, и получено значение напряжённости поля B(Tew) ~ 10⁻¹⁴ G при температуре электрослабого фазового перехода, принимая значение существующего в настоящее время межгалактического магнитного поля B0 ~ 10⁻¹⁵ G. Висловлено припущення про те, що міжгалактичні магнітні поля виникають у ранньому всесвіті при високій температурі T внаслідок спонтанного намагнічування вакууму неабелевих калібрувальних полів. Розроблено процедуру, яка дає змогу оцінити напруженість поля B(T) при різних T, і обчислено значення напруженості поля B(Tew) ~ 10⁻¹⁴ G при температурах електрослабкого фазового переходу, приймаючи значення існуючого в наш час міжгалактичного магнітного поля порядку B0 ~ 10⁻¹⁵G. One of us (VS) was supported by the ESF
 CASIMIR Networking program. This work has been
 also partly supported by MICINN (Spain), projects
 FIS2006-02842 and FIS2010-15640, by the CPAN
 Consolider Ingenio Project, and by AGAUR (Generalitat
 de Catalunya), contract 2009SGR-994. en Institute for Space Science, ICE-CSIC and IEEC Вопросы атомной науки и техники Section C. Theory of Elementary Particles. Cosmology Low bound on a magnetic field strength in the hot universe Граница снизу на напряженность магнитного поля в горячей вселенной Границя знизу на напруженість магнітного поля в гарячому всесвіті Article published earlier |
| spellingShingle | Low bound on a magnetic field strength in the hot universe Elizalde, E. Skalozub, V. Section C. Theory of Elementary Particles. Cosmology |
| title | Low bound on a magnetic field strength in the hot universe |
| title_alt | Граница снизу на напряженность магнитного поля в горячей вселенной Границя знизу на напруженість магнітного поля в гарячому всесвіті |
| title_full | Low bound on a magnetic field strength in the hot universe |
| title_fullStr | Low bound on a magnetic field strength in the hot universe |
| title_full_unstemmed | Low bound on a magnetic field strength in the hot universe |
| title_short | Low bound on a magnetic field strength in the hot universe |
| title_sort | low bound on a magnetic field strength in the hot universe |
| topic | Section C. Theory of Elementary Particles. Cosmology |
| topic_facet | Section C. Theory of Elementary Particles. Cosmology |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/107014 |
| work_keys_str_mv | AT elizaldee lowboundonamagneticfieldstrengthinthehotuniverse AT skalozubv lowboundonamagneticfieldstrengthinthehotuniverse AT elizaldee granicasnizunanaprâžennostʹmagnitnogopolâvgorâčeivselennoi AT skalozubv granicasnizunanaprâžennostʹmagnitnogopolâvgorâčeivselennoi AT elizaldee granicâznizunanapruženístʹmagnítnogopolâvgarâčomuvsesvítí AT skalozubv granicâznizunanapruženístʹmagnítnogopolâvgarâčomuvsesvítí |