Magnetogenesis for Baryogenesis
from Axion Inﬂation
based on: T. Fujita (Geneva) & KK, PRD93 (2016) 083520 [arXiv:1602.02109 (hep-ph)],KK & A.J.Long (Michigan), PRD94 (2016) 063501 [arXiv:1606.08891 (astro-ph.CO)],PRD94 (2016) 123509 [arXiv:1610.03074 (hep-ph)],D. Jiménez (MPIK), KK, K. Schmitz (Padova) & X.J. Xu (MPIK),
JCAP12 (2017) 011[arXiv:1707.07943[hep-ph]].

Kohei Kamada(RESCEU, U Tokyo)
mini-workshop: Axion Cosmology
14/05/2019 @ Yukawa Institute for Theoretical Physics, Kyoto University

Courtesy H.Oide1. Baryogenesis from decaying hypermagnetic
helicity
2. Axion inﬂation can be the source ofmagnetic ﬁelds for baryogenesis
3. Summary

Courtesy H.Oide

Baryogenesis from helical hypermagnetic ﬁelds
T. Fujita (Kyoto) & KK, PRD93 (2016) 083520 [arXiv:1602.02109 (hep-ph)],KK & A.J.Long (Chicago), PRD94 (2016) 063501 [arXiv:1606.08891 (astro-ph.CO)],PRD94 (2016) 123509 [arXiv:1610.03074 (hep-ph)]Courtesy H.Oide

Baryon Asymmetry of the Universe- We live in a matter-antimatter asymmetric Universe.
- BBN and CMB can evaluate it quantitatively.

23. Big-Bang nucleosynthesis 3

12

CMB

G. Steigman: Primordial Nucleosynthesis

bh

2

BBN

= 0.018, 0.023, 0.028

ʼ03 Steigman
Figure 6. The CBR temperature fluctuation anisotropy spectra for three choices of the baryon
density parameter ωB = 0.018, 0.023, 0.028, in order of increasing height of the first peak. Also
shown are the WMAP data points.

nB= (6.09 ± 0.06)
n

10 10

(Planck 2015)

Particle Data Group 2015
Figure 23.1: The abundances of 4 He, D, 3 He, and 7 Li as predicted by the standard
model of Big-Bang nucleosynthesis — the bands show the 95% CL range [5].
Boxes indicate the observed light element abundances. The narrow vertical band
indicates the CMB measure of the cosmic baryon density, while the wider band
indicates the BBN concordance range (both at 95% CL).

= (6.180 ± 0.195)

10 10

BBN+D ; ʼ15 Cyburt+

February 8, 2016

19:55

Courtesy H.Oide

Inﬂation dilutes the preexisting asymmetry.

WMAP teamAfter inﬂation before BBN, asymmetry must be generated.

{{matter :~10000000000+1
annihilation
anti matter : ~10000000000
T 100 MeV
photon :~10000000000

matter :~1
anti matter : ~0
photon :~10000000000
Courtesy H.OideIn order to generate baryon asymmetry...

Sakharovʼs condition is required. (ʼ67 Sakharov)
1. B-violation
2. C & CP-violation
3. Deviation from thermal equilibrium

Courtesy H.OideIn order to generate baryon asymmetry...

Sakharovʼs condition is required. (ʼ67 Sakharov)
1. B-violation
2. C & CP-violation
3. Deviation from thermal equilibrium
BSM is required!?
- Leptogenesis (ʼ85 Fukigita&Yanagida)- Aﬄeck-Dine: RH neutrinos: SUSY with B and CP op.

(ʼ85 Aﬄeck&Dine, ʼ95 Dine,Randall&Thomas)- EW baryogenesis: 1st order EWPT + CP op.

(ʼ85 Kuzmin, Rubakov&Shaposhnikov)

Courtesy H.OideIn order to generate baryon asymmetry...

Sakharovʼs condition is required. (ʼ67 Sakharov)
1. B-violation
2. C & CP-violation
3. Deviation from thermal equilibrium
BSM is required!?
- Leptogenesis (ʼ85 Fukigita&Yanagida)- Aﬄeck-Dine: RH neutrinos: SUSY with B and CP op.

(ʼ85 Aﬄeck&Dine, ʼ95 Dine,Randall&Thomas)- EW baryogenesis: 1st order EWPT + CP op.

(ʼ85 Kuzmin, Rubakov&Shaposhnikov)

If magnetic ﬁelds existed in the early Universe,
a new story can be imagined !!
Giovannini&Shaposhnikov (ʼ98), KK& Fujita, KK&Long (ʼ16)
Courtesy H.Oide

Key elements
Hypermagnetic helicity
H==V

d3 x ijk Yi j Yk
d3 k
R 2
k
|Y
k |
3
(2 )

|YkL |2(The gauge ﬁeld is the one of U(1)Y in
the SM. )solar.physics.montana.edu

Courtesy H.Oide

Key elements
Hypermagnetic helicity
H==V

Chiral anomaly in the SM (ʼ76 ʻt Hooft)

d3 x ijk Yi j Yk
d3 k
R 2
k
|Y
k |
3
(2 )

µ
j
µ B =

|YkL |2µjµ L = Ng

g2
µ
tr
W
W̃
µ
16 2

g2
µ
Y
Ỹ
µ
32 2

SU(2) part:“EW sphaleron”
U(1) part: often neglectedsolar.physics.montana.edu

Courtesy H.Oide

Key elements
Hypermagnetic helicity
H=
=V

3

d x

ijk

Chiral anomaly in the SM (ʼ76 ʻt Hooft)

Yi j Yk

d3 k
R 2
k
|Y
k |
3
(2 )

QB =

QL = Ng

NCS

g2
HY
2
16

|YkL |2
SU(2) part:“EW sphaleron”
U(1) part: often neglected

Change of BG hypermagnetic
helicity can be the source of BAU
solar.physics.montana.edu

Courtesy H.Oide

Key elements
Hypermagnetic helicity
H=

3

d x

ijk

Yi j Yk

Chiral anomaly in the SM (ʼ76 ʻt Hooft)

QB =QL = Ng

NCS

g2
HY
2
16

d3 k
R 2
L 2
=V
k
|Y
|
|Y
k
k |
3
(2 were)If there
BG large-scale helical hyperMFs,
SU(2) part:“EW sphaleron”

U(1) part: often neglected
B-violation: SM chiral anomaly
Change of BG hypermagnetic
C&CP-violation/non-equilibrium
helicity can be the source of BAU:existence of large-scale magnetic helicity
solar.physics.montana.edu

Sakharovʼs conditions are satisﬁed!

Courtesy H.Oide

Once there are large-scale helical hyperMFs before EWSB,
decay of hypermagnetic helicity occurs automatically.

Courtesy H.Oide

Once there are large-scale helical hyperMFs before EWSB,
decay of hypermagnetic helicity occurs automatically.

1. Decay due to MHD with ﬁnite conductivity (ʼ98 Giovannini&Shaposhnikov)

1
V

tH =

2 EY · B Y =

2

BY ·

BY

100T

2 Bp2
B
(ʼ97 Baym+, ʼ00 Arnold+)

Courtesy H.Oide

Once there are large-scale helical hyperMFs before EWSB,
decay of hypermagnetic helicity occurs automatically.

1. Decay due to MHD with ﬁnite conductivity (ʼ98 Giovannini&Shaposhnikov)

1
V

tH =

2 EY · B Y =

2

BY ·

BY

100T

2 Bp2
B
(ʼ97 Baym+, ʼ00 Arnold+)

2. Electroweak symmetry breaking (ʼ98 Giovannini&Shaposhnikov,ʼ16 KK&Long)

Courtesy H.Oide

Electroweak symmetry breaking
Gauge group
SU (2)W U (1)Y

Bem

BY

BA

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U (1)em

Large-scale (massless) MFs

Bem = cos W BY + sin w BW 3

BY
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Magnetic helicity
BW 3

after
Hem
= HYbefore

HYbefore<latexit sha1_base64="iM+h1joxSyfAPw/k2e6TCqnWyyE=">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</latexit>
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after
HYafter = cos2 W Hem= cos2 W HYbefore

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sha1_base64="ELzl6Kg5Saj335WnpPvjhM+9GA8=">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</latexit>

afterN
ure 1: A graphical representation of the conversion from hypermagnetic field BY into elec- CS,W3

after
sin2 W Hem= sin2 W HYbefore<latexit sha1_base64="qCZqWejWXYr8jA3CqEZSMH4mssA=">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</latexit>
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sha1_base64="YGCU86pm7GrJrP+gIVz5pAznaD4=">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</latexit>

magnetic field Bem during the EW crossover. The (blue) parabolas indicate the curvature ofthermal e↵ective potential. The weak mixing angle ✓W (t) measures the separation of the flat
ection (massless field degree of freedom) and the U(1)Y axis.

lution of the massless field degree of freedom, we have reduced the problem to a single degree
reedom as represented by the classical vector field Aµ (x).

The Ansatz (2.3) is represented graphically in Fig. 1, which illustrates the conversion frompermagnetic field to electromagnetic field. Here we denote the magnetic field of a gauge field Y

Courtesy H.Oide

Electroweak symmetry breaking
Gauge group
SU (2)W U (1)Y

Bem

BY

BA

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sha1_base64="pxeBIPcxH3KQsxldTorKwvrVNF4=">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</latexit>✓W (t)

U (1)em

Large-scale (massless) MFs

Bem = cos W BY + sin w BW 3

BY
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sha1_base64="CdsEJGaD5+oNtrK6ebmGvxEfum4=">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</latexit>

Magnetic helicity
BW 3

after
Hem
= HYbefore

HYbefore<latexit sha1_base64="iM+h1joxSyfAPw/k2e6TCqnWyyE=">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</latexit>
sha1_base64="ukVAXzeknIe/vfwtzrtLijh+Wfk=">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</latexit>
sha1_base64="ssV8+1OZW+7YfFz46zzVdE48+kc=">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</latexit>

after
HYafter = cos2 W Hem= cos2 W HYbefore

BZ<latexit sha1_base64="GmMgkdVEFAIgyMjqsMUGSolECjI=">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</latexit>
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afterN
ure 1: A graphical representation of the conversion from hypermagnetic field BY into elec- CS,W3

BAU
induction
magnetic field
B during
the EW crossover. The (blue) parabolas indicate the curvature ofafter
sin2 W Hem= sin2 W HYbefore<latexit sha1_base64="qCZqWejWXYr8jA3CqEZSMH4mssA=">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</latexit>
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thermal e↵ective potential. The weak mixing
angle✓W (t) measures the separation of the flat2
before=sin
H
H
Y of freedom) and W
Y Y axis.
ection (massless field degree
the U(1)<latexit sha1_base64="cad3FnJLB8lKXZT0X8ObuMQCfAg=">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</latexit>
sha1_base64="LkvF7ZMXWVBi987jxWVmAbFEE/Q=">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</latexit>
sha1_base64="6oZYXpPiCcwKyv23wyVdpCdvGVM=">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</latexit>

QB = # NCS

before
sin
N
H
CS
W
Y
lution of the massless field degree of freedom, we have reduced the problem to a single degree2<latexit sha1_base64="8c/JTxjDjCvzGBxG64hI8xrNg+U=">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</latexit>
sha1_base64="P/SPtjas3YVQXEBa3F4ACoQAP1w=">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</latexit>
sha1_base64="NCed7iK9PwYk4r6JW0f8//1LTc0=">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</latexit># HY

sin2 W HYbefore<latexit sha1_base64="imvshEYkDn3rmh55kSLQKfkLDvk=">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</latexit>
sha1_base64="4ZDUNQnOMMIRA96ghJD101iTRcQ=">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</latexit>
sha1_base64="qb/vDvadk0vgR+kQAsfKiCRbSy8=">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</latexit>

reedom as represented by the classical vector field Aµ (x).

The Ansatz (2.3) is represented graphically in Fig. 1, which illustrates the conversion frompermagnetic field to electromagnetic field. Here we denote the magnetic field of a gauge field Y

Courtesy H.Oide

Back reactions/washout eﬀects?
EW sphalerons+chirality ﬂip by electron Yukawa

Chiral Magnetic Eﬀect (ʼ80 Vilenkin, ʼ08 Fukushima, Kharzeev, &Warringa)

Courtesy H.Oide

Back reactions/washout eﬀects?
Maximal Temperature of the Early Universe

9

EW sphalerons+chirality ﬂip by electron Yukawa
sL

sL

tL

cL

Thermal ﬂuctuations of W-boson inducesbL

Sphaleron

dL

bL

NCS

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Together with Yukawa int., try to washout QB<latexit sha1_base64="mcf3USQripUUYWZaTqae91chaGo=">AAACD3icbVC7TgJBFL2LL8QXamkzkZhYkV0bLQk2lhAFTGBDZmcvMGFmdzMza0I2fIKFjX6KnbH1E/wSWwfYQsCTTHJyzr25Z06QCK6N6347hY3Nre2d4m5pb//g8Kh8fNLWcaoYtlgsYvUYUI2CR9gy3Ah8TBRSGQjsBOPbmd95QqV5HD2YSYK+pMOIDzijxkr3zX69X664VXcOsk68nFQgR6Nf/umFMUslRoYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa2lEJWo/m0edkgurhGQQK/siQ+bq342MSq0nMrCTkpqRXvVm4r9eIJcuZ9pGG2G4EscMbvyMR0lqMGKLNINUEBOTWTkk5AqZERNLKFPcfoiwEVWUGVthyTblrfayTtpXVc+tek23UqvnnRXhDM7hEjy4hhrcQQNawGAIz/AKb86L8+58OJ+L0YKT75zCEpyvX6HHnQU=</latexit>
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sha1_base64="GP1qIWHYX1V2Xdalw0gYOOk9HWA=">AAACD3icbVDLSgMxFM34rPVVdalIsAiuyowbXZa6cdmifUA7lEzmtg1NMkOSEcrQpUsXbvRT7Erc+gl+gx/g1vSxsK0HAodz7uWenCDmTBvX/XJWVtfWNzYzW9ntnd29/dzBYU1HiaJQpRGPVCMgGjiTUDXMcGjECogIONSD/s3Yrz+A0iyS92YQgy9IV7IOo8RY6a7SLrVzebfgToCXiTcj+eLJqPL9eDoqt3M/rTCiiQBpKCdaNz03Nn5KlGGUwzDbSjTEhPZJF5qWSiJA++kk6hCfWyXEnUjZJw2eqH83UiK0HojATgpienrRG4v/eoGYu5xqG60H4UIc07n2UybjxICk0zSdhGMT4XE5OGQKqOEDSwhVzH4I0x5RhBpbYdY25S32skxqlwXPLXgVW1kJTZFBx+gMXSAPXaEiukVlVEUUddETekGvzrPz5rw7H9PRFWe2c4Tm4Hz+AhfMoLA=</latexit>

ντ

dL
uL
νe

νµ

W. Buchmüller, 1212.3554

5. One of the 12-fermion processes which are in thermal equilibrium in thetemperature phase of the standard model.

Chiral Magnetic Eﬀect (ʼ80 Vilenkin, ʼ08 Fukushima, Kharzeev, &Warringa)

sition.
The second crucial nonperturbative aspect of baryogenesis is the connecbetween baryon number and lepton number in the high-temperature,
metric phase of the Standard Model. Due to the chiral nature of theinteractions B and L are not conserved [23]. At zero temperature
has no observable e↵ect due to the smallness of the weak coupling.
ever, as the temperature reaches the critical temperature Tc of the eleceak phase transition, B and L violating processes come into thermal
ibrium [24]. The rate of these processes is related to the free energy
phaleron-type field configurations which carry topological charge. In
standard model they lead to an e↵ective interaction of all left-handedions [23] (cf. Fig. 5),
Y
OB+L =(qLi qLi qLi lLi ) ,(3.3)
ih violates baryon and lepton number by three units,

Courtesy H.Oide

Back reactions/washout eﬀects?
Maximal Temperature of the Early Universe

9

EW sphalerons+chirality ﬂip by electron Yukawa
sL

sL

tL

cL

Thermal ﬂuctuations of W-boson inducesbL

Sphaleron

dL

bL

NCS

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sha1_base64="G9RT+0RkWDay5C4JDix5UVoCMAg=">AAACH3icbVC7SgNBFJ2NrxhfGwUbm8EgWIVdGy1DYmElCZoHJEuYndwkQ2Z2l5lZJSz5FAsbrfwLwU5s0/gbtk4ehUk8MHA4517umeNHnCntOGMrtba+sbmV3s7s7O7tH9jZw5oKY0mhSkMeyoZPFHAWQFUzzaERSSDC51D3B6WJX38AqVgY3OthBJ4gvYB1GSXaSG0727oGrgm+bSctKXDpbtS2c07emQKvEndOcoXjyjd7K76X2/ZPqxPSWECgKSdKNV0n0l5CpGaUwyjTihVEhA5ID5qGBkSA8pJp9BE+M0oHd0NpXqDxVP27kRCh1FD4ZlIQ3VfL3kT81/PFwuVEmWh96CzF0d0rL2FBFGsI6CxNN+ZYh3hSFu4wCVTzoSGESmY+hGmfSEK1qTRjmnKXe1kltYu86+TdiqmsiGZIoxN0is6Riy5RAd2gMqoiih7RE3pBr9az9WF9Wl+z0ZQ13zlCC7DGvxkWpm0=</latexit>

Together with Yukawa int., try to washout QB<latexit sha1_base64="mcf3USQripUUYWZaTqae91chaGo=">AAACD3icbVC7TgJBFL2LL8QXamkzkZhYkV0bLQk2lhAFTGBDZmcvMGFmdzMza0I2fIKFjX6KnbH1E/wSWwfYQsCTTHJyzr25Z06QCK6N6347hY3Nre2d4m5pb//g8Kh8fNLWcaoYtlgsYvUYUI2CR9gy3Ah8TBRSGQjsBOPbmd95QqV5HD2YSYK+pMOIDzijxkr3zX69X664VXcOsk68nFQgR6Nf/umFMUslRoYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa2lEJWo/m0edkgurhGQQK/siQ+bq342MSq0nMrCTkpqRXvVm4r9eIJcuZ9pGG2G4EscMbvyMR0lqMGKLNINUEBOTWTkk5AqZERNLKFPcfoiwEVWUGVthyTblrfayTtpXVc+tek23UqvnnRXhDM7hEjy4hhrcQQNawGAIz/AKb86L8+58OJ+L0YKT75zCEpyvX6HHnQU=</latexit>
sha1_base64="ni4jNcWseCfpPhE9wlvqPupjiio=">AAACD3icbVC7TsMwFL0pr1JeBUYYLCokpiphgbEqC2Mr6ENqo8pxnNaq7US2g1RF/QQGFvgUmBArn8DGX7DiPgbaciRLR+fcq3t8goQzbVz3y8mtrW9sbuW3Czu7e/sHxcOjpo5TRWiDxDxW7QBrypmkDcMMp+1EUSwCTlvB8Gbitx6o0iyW92aUUF/gvmQRI9hY6a7eq/aKJbfsToFWiTcnpcrpW/0bAGq94k83jEkqqDSEY607npsYP8PKMMLpuNBNNU0wGeI+7VgqsaDaz6ZRx+jcKiGKYmWfNGiq/t3IsNB6JAI7KbAZ6GVvIv7rBWLhcqZttAENl+KY6NrPmExSQyWZpYlSjkyMJuWgkClKDB9Zgoli9kOIDLDCxNgKC7Ypb7mXVdK8LHtu2avbyqowQx5O4AwuwIMrqMAt1KABBPrwCM/w4jw5r8678zEbzTnznWNYgPP5CwXJn0o=</latexit>
sha1_base64="GP1qIWHYX1V2Xdalw0gYOOk9HWA=">AAACD3icbVDLSgMxFM34rPVVdalIsAiuyowbXZa6cdmifUA7lEzmtg1NMkOSEcrQpUsXbvRT7Erc+gl+gx/g1vSxsK0HAodz7uWenCDmTBvX/XJWVtfWNzYzW9ntnd29/dzBYU1HiaJQpRGPVCMgGjiTUDXMcGjECogIONSD/s3Yrz+A0iyS92YQgy9IV7IOo8RY6a7SLrVzebfgToCXiTcj+eLJqPL9eDoqt3M/rTCiiQBpKCdaNz03Nn5KlGGUwzDbSjTEhPZJF5qWSiJA++kk6hCfWyXEnUjZJw2eqH83UiK0HojATgpienrRG4v/eoGYu5xqG60H4UIc07n2UybjxICk0zSdhGMT4XE5OGQKqOEDSwhVzH4I0x5RhBpbYdY25S32skxqlwXPLXgVW1kJTZFBx+gMXSAPXaEiukVlVEUUddETekGvzrPz5rw7H9PRFWe2c4Tm4Hz+AhfMoLA=</latexit>

ντ

dL
uL
νe

νµ

W. Buchmüller, 1212.3554

5. One of the 12-fermion processes which are in thermal equilibrium in thetemperature phase of the standard model.

Chiral Magnetic Eﬀect (ʼ80 Vilenkin, ʼ08 Fukushima, Kharzeev, &Warringa)

sition.
The second crucial nonperturbative aspect of baryogenesis is the connecbetween baryon number and lepton number in the high-temperature,
metric phase of the Standard Model. Due to the chiral nature of theinteractions B and L are not conserved [23]. At zero temperatureY
has no observable e↵ect due to the smallness of the weak coupling.
ever, as the temperature reaches the critical temperature Tc of the eleceak phase transition, B and L violating processes come into thermal
ibrium [24]. The rate of these processes is related to the free energy
phaleron-type field configurations which carry topological charge. In
standard model they lead to an e↵ective interaction of all left-handedions [23] (cf. Fig. 5),
Y
OB+L =(qLi qLi qLi lLi ) ,(3.3)

Ampereʼs law
B = J = (EY + v

BY ) +

Ohmʼs current=> EY =

v

BY +

2 Y

µ5 BY

Chiral magnetic current
1

BY

2 Y

µ5 BY

i

h violates baryon and lepton number by three units,

Courtesy H.Oide

Back reactions/washout eﬀects?
Maximal Temperature of the Early Universe

9

EW sphalerons+chirality ﬂip by electron Yukawa
sL

sL

tL

cL

Thermal ﬂuctuations of W-boson inducesbL

Sphaleron

dL

bL

NCS

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sha1_base64="G9RT+0RkWDay5C4JDix5UVoCMAg=">AAACH3icbVC7SgNBFJ2NrxhfGwUbm8EgWIVdGy1DYmElCZoHJEuYndwkQ2Z2l5lZJSz5FAsbrfwLwU5s0/gbtk4ehUk8MHA4517umeNHnCntOGMrtba+sbmV3s7s7O7tH9jZw5oKY0mhSkMeyoZPFHAWQFUzzaERSSDC51D3B6WJX38AqVgY3OthBJ4gvYB1GSXaSG0727oGrgm+bSctKXDpbtS2c07emQKvEndOcoXjyjd7K76X2/ZPqxPSWECgKSdKNV0n0l5CpGaUwyjTihVEhA5ID5qGBkSA8pJp9BE+M0oHd0NpXqDxVP27kRCh1FD4ZlIQ3VfL3kT81/PFwuVEmWh96CzF0d0rL2FBFGsI6CxNN+ZYh3hSFu4wCVTzoSGESmY+hGmfSEK1qTRjmnKXe1kltYu86+TdiqmsiGZIoxN0is6Riy5RAd2gMqoiih7RE3pBr9az9WF9Wl+z0ZQ13zlCC7DGvxkWpm0=</latexit>

Together with Yukawa int., try to washout QB<latexit sha1_base64="mcf3USQripUUYWZaTqae91chaGo=">AAACD3icbVC7TgJBFL2LL8QXamkzkZhYkV0bLQk2lhAFTGBDZmcvMGFmdzMza0I2fIKFjX6KnbH1E/wSWwfYQsCTTHJyzr25Z06QCK6N6347hY3Nre2d4m5pb//g8Kh8fNLWcaoYtlgsYvUYUI2CR9gy3Ah8TBRSGQjsBOPbmd95QqV5HD2YSYK+pMOIDzijxkr3zX69X664VXcOsk68nFQgR6Nf/umFMUslRoYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa2lEJWo/m0edkgurhGQQK/siQ+bq342MSq0nMrCTkpqRXvVm4r9eIJcuZ9pGG2G4EscMbvyMR0lqMGKLNINUEBOTWTkk5AqZERNLKFPcfoiwEVWUGVthyTblrfayTtpXVc+tek23UqvnnRXhDM7hEjy4hhrcQQNawGAIz/AKb86L8+58OJ+L0YKT75zCEpyvX6HHnQU=</latexit>
sha1_base64="ni4jNcWseCfpPhE9wlvqPupjiio=">AAACD3icbVC7TsMwFL0pr1JeBUYYLCokpiphgbEqC2Mr6ENqo8pxnNaq7US2g1RF/QQGFvgUmBArn8DGX7DiPgbaciRLR+fcq3t8goQzbVz3y8mtrW9sbuW3Czu7e/sHxcOjpo5TRWiDxDxW7QBrypmkDcMMp+1EUSwCTlvB8Gbitx6o0iyW92aUUF/gvmQRI9hY6a7eq/aKJbfsToFWiTcnpcrpW/0bAGq94k83jEkqqDSEY607npsYP8PKMMLpuNBNNU0wGeI+7VgqsaDaz6ZRx+jcKiGKYmWfNGiq/t3IsNB6JAI7KbAZ6GVvIv7rBWLhcqZttAENl+KY6NrPmExSQyWZpYlSjkyMJuWgkClKDB9Zgoli9kOIDLDCxNgKC7Ypb7mXVdK8LHtu2avbyqowQx5O4AwuwIMrqMAt1KABBPrwCM/w4jw5r8678zEbzTnznWNYgPP5CwXJn0o=</latexit>
sha1_base64="GP1qIWHYX1V2Xdalw0gYOOk9HWA=">AAACD3icbVDLSgMxFM34rPVVdalIsAiuyowbXZa6cdmifUA7lEzmtg1NMkOSEcrQpUsXbvRT7Erc+gl+gx/g1vSxsK0HAodz7uWenCDmTBvX/XJWVtfWNzYzW9ntnd29/dzBYU1HiaJQpRGPVCMgGjiTUDXMcGjECogIONSD/s3Yrz+A0iyS92YQgy9IV7IOo8RY6a7SLrVzebfgToCXiTcj+eLJqPL9eDoqt3M/rTCiiQBpKCdaNz03Nn5KlGGUwzDbSjTEhPZJF5qWSiJA++kk6hCfWyXEnUjZJw2eqH83UiK0HojATgpienrRG4v/eoGYu5xqG60H4UIc07n2UybjxICk0zSdhGMT4XE5OGQKqOEDSwhVzH4I0x5RhBpbYdY25S32skxqlwXPLXgVW1kJTZFBx+gMXSAPXaEiukVlVEUUddETekGvzrPz5rw7H9PRFWe2c4Tm4Hz+AhfMoLA=</latexit>

ντ

dL
uL
νe

νµ

W. Buchmüller, 1212.3554

5. One of the 12-fermion processes which are in thermal equilibrium in thetemperature phase of the standard model.

Chiral Magnetic Eﬀect (ʼ80 Vilenkin, ʼ08 Fukushima, Kharzeev, &Warringa)dndt

sition.
f
The second crucial nonperturbative aspect of baryogenesis is the connecbetween baryon number and lepton number in the high-temperature,
metric phase of the Standard Model. Due to the chiral nature of theinteractions B and L are not conserved [23]. At zero temperature
has no observable e↵ect due to the smallness of the weak coupling.
ever, as the temperature reaches the critical temperature
qR/LTc of 2the elec- 2µ=()
6yfthermal
nf /T5
eak phase transition, B and L violatingprocesses come into
ibrium [24]. The rate of these processes is frelated to the free energy
phaleron-type field configurations which carry topological charge. In
standard model they lead to an e↵ective interaction of all left-handedions [23] (cf. Fig. 5),
Y
OB+L =(qLi qLi qLi lLi ) ,(3.3)# Yµ Ỹ µ
=#
=#

1
1

(=

BY · (
Bp2
B

2

4 EY · B Y )BY )2µ5 |BY |2µ5 Bp2ih violates baryon and lepton number by three units,

Courtesy H.Oide

Schematically evolution equation is given by
Source term

dnB
=
dt

#

B2

+ # ˙W B 2

washout termw.o. nB<latexit sha1_base64="C3sLmR0URZPbic9B77CouHNN76Q=">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</latexit>
sha1_base64="HT/Quc64oHKcWeyZaGcYqCq6Id0=">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</latexit>
sha1_base64="e7XAbmXV357U36/X2srca0D4Gpk=">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</latexit>

MHD decay

EWSB

EW sphaleron chirality-ﬂip CME

Courtesy H.Oide

Schematically evolution equation is given by
Source term

dnB
=
dt

#

B2

washout term

+ # ˙W B 2w.o. nB<latexit sha1_base64="C3sLmR0URZPbic9B77CouHNN76Q=">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</latexit>
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sha1_base64="e7XAbmXV357U36/X2srca0D4Gpk=">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</latexit>

MHD decay

EWSB

EW sphaleron chirality-ﬂip CME

Reaches at “terminal” asymmetry…

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Courtesy H.Oideij dxij ⇣ ⌘ i⇣ ⌘ui j=1⌘ +
⌘dj ⌘
⌘dj ⌘
⌘ 0
dL
ij
uhd
L
R
dhd
R
,
S
⌘
,
(2.11b)dhd
Ng2d⌘ ikui Xk⇣+
kdj
2
kdi
k 0
k dj
⌘
dR L
L
R ji
R
ji
ji
2
bkg
Sdhd
+
S
+
S
+
S
N
c yd R S y ⌘ j ⌘s,sphij ⇣ ⌘ i=
ij
dd
uhd
⌘
⇣
⌘
j
⌘
i
dx
⌘ 0
⌘ +
⌫L j=1
eR
eL
eR
ij
ij
S⌫he
⌘ ⌫he
, Sehe⌘ ehe,(2.11c)2 k⌫ i Nkg ⇣+
kej
2
k
k
k
0
i
j
e
e
⌘
L
R
d⌘ei L X ji R ji
ji
R=
Sehe + See + S⌫heye2R Sybkg
dxinteractions. We estimate the transport coefficients in Eq. (2.26). After thearise from the Yukawa
j=1
ij
Suhd
⌘

We
solved…
SMreally
Boltzmann
eqns. w/ anomaly terms

electroweak phase transition, ⇣the gauge and⌘ Yukawa
Ng ⇣ interactions mediate ⌘scattering withi the Higgs
X
arise from the weak gauge
the corresponding
coefficients, udw ,d⌘ + interactions. We estimate
ij
ij transportij
condensate.
This
gives
rise
to
the
additionalsourceterms,
=
S
+
S
+
S
+
S
+
S
hhw
hw
i are
dhu
uhd
⌫he
The Standard Model transport equations used in our analysis⌫ewsummarized
below:, and
hhw in Eq.
dx(2.24). The source terms,
i,j=1⇣ ⌘ ui⌘ j ⌘Ng ⇣ij ⇣ ⌘ i
ijuR⇣ ⌘ i
ij
ij
L
⌘
j ⌘
N
⌘
g
X
⌘ 0⌘ ⌘ujR ⌘
⌘
+
d⌘ui
⇣
Suuu
(2.12a)ij
Nc
uu ij
dhud⌘ 0dLR ⌘X
uhu ij u,L
ij
ij
ij
i
L
ij
ij
S
⌘
+
,
S
⌘
+
,
(2.11a)kuSi
kuSj= Sudw
Suhu + Suu + Suhd
Ss,sphSw,sph
dhu
uhu
=
S
S
+
+
+
S
L
hhw
h
2
k
k
k
2
k
k
k
+
0
i
j
i
j
Rdhd
uhu
ehe
uL
uR
uR
dx
2
dx dL
⇣ ⌘di
⌘
j ⌘
j=1
i,j=1
d
ij ⇣ ⌘ i
ij R⇣ ⌘ i
ij
ij
⌘
⌘
L
⌘
⌘
j
⌘ 0
⌘ +
⇣
uL
d, L
djR
Sddd
(2.12b)
ij
R ⌘
dhd
dd S
Nij
yQL bkg ⌘Nc bkg
g
Suhd
⌘ uhd
,
⌘
,
(2.11b)
⇣
⌘
⇣
⌘
2
bkg
k
k
i
j
X
dhd
d
+ N c yQ L S y +S+ Nc
S2d⌘Wk+ui (2.9a)
k +
k 0
k dj
i 2dR kidi
Shhw +kdSjRhw + ⇣ LSudw+⌘S⌫ew
.
L=
L
2 w
2 yw
R
⌘
j
⌘
i
eL
eR⇣ ⌘
ij dx
ij
ij
ij
⇣
⌘
⌘
⌘
⌘
j
j
Ng ⇣
⌘
i
i
⌘ 0
⌘ +
⌘
⌫L
e,L
eR
SeRee ⌘ ee i=1(2.12c)
X ij
d⌘di
ij
ij
ehe
Nc
ij
ij
S⌫he
⌘ ⌫he
, Sehe⌘,(2.11c)
i
L
k
k
i
j
e
= Sudw
Sdhd + Sdd + Sdhu
Ss,sphSw,sph
2 kterms
kfall
kej the following
2 eR kei
k 0 source
kej terms,+
iL⌫L
The source
into
categories.
The
L◆
✓
dx
2
R
R
arise
from
the
weak
gauge
interactions.
Weestimatethecorresponding transport coeffi
j=1
⌘ +
⌘W +
!
S
⌘
,
i
⇣
⌘
hw(2.24).
hw The
⌘
⌘
arise
from
the
Yukawainteractions.
WeestimatethetransportinEq.
(2.26). After the(2.12d)i coefficients
i
,
and
in
Eq.
sourceterms,
y
N
hhw
⌘
+
u
d
⌫ew
Q
c
k
k
+
+
2
bkg
bkg
bkg
L
W
i
i
L
L
W
+ N c yQ L S y +
S
Nc
S
⌘ udw
, with the Higgs
udw
electroweak phase transition, (2.9b)
the gauge and SYukawa
mediatescattering⌘ 0 interactions
2 w
2 yw
k
k
k
+
i
i
W
u
d
ij S⇣ ⌘⌘i
ij
⌘
⇣
⌘ uj
⌘u(2.12e)
j ⌘
L
L
⌘ ui
⌘ 0
⌘ +. terms,
h dL h!
Ng
ij
ij
condensate.
This
gives
rise
to
the
additionalsource
L
dhu
R
uhu
R
⇣
⌘
k
0
X ij
d⌘⌫ i
Sdhu ⌘+⌘+,
1
1 bkg yLL bkg⌘⌫ i , S⌘euhu
i
i
2
bkg
L
⌘
+
2
k
k
k
2
k
k
k
+
0
i
j
i
j
W
i
i
L
L
= S⌫ewS⌫he
Sw,sph + yLL Sy + Sw +Syw
(2.9c)
dLS⇣ ⌘ i ⌘ ⌘ j u⌘
u
u
,L
R
⌫ew
uL
uR R
dx
2
2
2
ij
ij ⌫ewk⌫ ikei
kWij+ ⇣
We estimate
these transport coefficientsinEq.
(2.29).
S⌘⌘ uu,(2.12a)j=1
ijuu ⇣
⌘
⌘
⌘
⌘
j
j
L
⌘i
⌘ 0
uiL ku⌘i + kuj dR
dR
✓ L
ij
ij
uhd
dhd◆ dL
L
R
Suhdterms⌘ are associated with the⌘ Standard
S⌘dhd
,
Ng ⇣
+,
0 ⌘
⌘W2+ chiralModel⌘ source
⌘ 1
⇣
X
d⌘ei
2ij kuiijS⇣hhw
k,di anomalies.
k 0
kdjTher+ ⌘ jk ⌘j1 bkgThe yremaining⌘dki ⌘
LL bkg
ij
ij
d
i
2
bkg
hhw
L
d
L
L
L
R k +
= S⌫ew
Sehe + See
Sw,sph + yLL Sy +malSS
(2.9d)Sddgauge
k 0
kW +
⌘
, the terms(2.12b) R
fieldsleadR todx
2
2 wfluctuations
2 ywof the non-Abelian
ij ⇣ ⌘ idd kdi
ij ⇣ ⌘ ikj j ⌘⌘⌘e j ⌘⌘ 0⌘L + dReR
j=1
⌫L
eL
ij
ij
ehe
R
!
S⌫he
⌘ ⌫he
,
S
⌘
,
⇣ ⌘ei
⌘ej ⌘
Ng
ehe
Ng ⇣
X
⌘
⌘
⌘
⌘
i
i
i
i
ij
ij
2
k
k
k
2
k
k
k
+
0
i
j
i
j
L d
R
⌘
u
u
d
⌫
e
X
d⌘ui
eR
See ⌘ LeeL + L eR ,R 8 R(2.12c)Lji
jiji
Ss,sph ⌘(2.9e),(2.13a)
R
s,sph
kei
kj
=
Suhu
+ Suu
+ Sdhu+ Ss,sph Nc yu2R Sybkgkk
L k i eR k i
i
i
u
d
u
d
dx
◆
i=1
L✓
L
R
R
arise from the Yukawa
interactions.
j=1
⌘ +
⌘We
+ estimate the transport! coefficients in Eq. (2.
W
N
g⌘ hw
Shw
,
(2.12d)X Nc ⌘the
NkcW⌘+and
1 ⌘⌫Li interactions
1 ⌘eiLNg ⇣uiL+ gauge
diL Yukawa
electroweak phase transition,
mediate scattering w
k
⌘
X
d⌘di
Sw,sph ⌘ w,sph+++,(2.13b)
ji
ji
ji
R
0
kuthe
2 kdi
2 k⌫ i terms,
2 kei
i
=
Sdhd
+ Sdd
+ Suhd+Ss,sph Nc yd2R&Sybkg(2.9f)gives
condensate.
rise2⌘ toadditionalsourceHypercharge
Weak Isospin Source
Terms This
i=1
L
L
L
L
S
⌘
.
(2.12e)
h
h
dx
k 0
j=1
⇣ ⌘ ui⌘uj ⌘the transport coefficients
which
are
known
as
the
strongandelectroweak
sphalerons.
extract
ij
ij We
The
three
remainingsourceterms
correspondtobackgroundcontributions
fromthehypercharge
L
R
Ng ⇣
⌘
,
X ji
d⌘ei
We estimate these transport coefficients in Eq. (2.29).Suu ⌘ uu k i
ji
ji
2
bkg
R
k
j presence of a backgroundfromtheresultsoflatticesimulations
inEqs.
(2.43a)and(2.43b).
In
the
u
weak
isospingauge
fields.
TranscribingfromEq.
(2.34),
theyarewrittenas=
Sehe + See + S⌫he andye R S(2.9g)uLyR
dx
The remaining source terms are associated with the Standard
anomalies. Ther⇣ ⌘dModel⌘dchiral
j ⌘
i
j=1
ij
ij
L
R
↵
1
1
y
µ⌫⇢
mal fluctuationsof the ✏non-Abelian
toddthe⌘ terms,
dd k
Sybkg =
hYµ⌫ ihY⇢gaugei fields lead S(2.44a)
Ng ⇣
⇣
⌘ X
⌘
kdj
i
d
sT
4⇡
2
d⌘ +
ij
ij
ij
R
! L
Ng
=
Shhw + Shw +Sdhu+ Suhd+ S⌫he(2.9h)
1
1
↵
1
X
⌘
⌘
⌘
⌘
i
i
i
i ⇣⌘ i
⌘
j ⌘
w
bkg
µ⌫⇢
u
d
u
d
a ihW a Li
eL (2.44b)
eR
dx
R
ij
ijR
Sw =
hWµ⌫
Ss,sph✏ ⌘ s,sph+ L,(2.13a)⇢i,j=1⌘ k ee,sT 2 4⇡
2
k ui
kdi
kSuee
i
i
d
k
k
i
j
i=1
L
L
R
R
eL
eR
Ng ⇣
1 gg 0 /4⇡ µ⌫⇢
⌘
!
X
bkg
✓
◆
3 i .
d⌘ 0
N
g
ij
ij
ij
S
=
✏
hY
ihW
(2.44c)µ⌫X ⇢N ⌘ui⌘di⌘⌫ i ⌘ 1+⌘ei ⌘W +yw
N1
= Shhw Sh +Suhu + Sdhd + Sehe
(2.9i)
c
c
sT 4⇡
L
L
,
Sw,sph ⌘ w,sph+ 9 ShwL ⌘+ hwL +,
(2.13b)dx
2 k ui2 kdi
2 k⌫ i k 2+kei kW +
i,j=1
i=1
L
It is appropriate to discuss Related
these contributions
together,
theyL become Lentangled
afterLelecwork: Giovannini& because
Shaposhnikov;
⌘ 0
Ng ⇣
⇣
⌘ X
⌘
Fujita&are
Kamada;AL,
Sabancilar,
VachaspaT;
Sh ⌘ inWe.
In
this
section,first
calculatethese source
terms
symmetric
d⌘W +
h the
which
known
as we
the
strongand&electroweak
sphalerons.
extract
the transport coefficients
i troweak
i symmetry breaking.
k0= Shhw + Shw +Sudw + S⌫ew .
(2.9j)
Semikoz,
Dvornikov,
Smirnov,
Sokoloﬀ,Vallephase,
andthenconsiderthebrokenphase.
dx
from the results of lattice simulations in Eqs. (2.43a) and (2.43b). In the presence of a backgroundi=1
We estimate
these transport
Eq.Eq.
(2.29).
In the symmetric phase, the non-Abelian
iso-magnetic
field W 3coefficientsis screenedin(cf.
(2.37)),
Courtesy:
A. J. Long
The source terms fall into the following categories. The source terms,
and the corresponding source terms vanish The remaining source terms are associated with the Standard Model chiral ano
!
⌘ ui
⌘di
mal bkg
fluctuations of the non-Abelian gauge fields lead to the termsCourtesy H.Oide
⌘
+
W bkg
i
i
L
L
Sw , = 0
and
Syw = 0(2.10a)(symmetric phase) .
(2.45)Sudw⌘ udw

Evolution of baryon asymmetry
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T
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







 =  -  =  = - 









 -  =  - =  -


E
C

 -

D

 -

 

-  =  / 

 =  - 

-

TEWSB
   ()  =  / 

T

-

B
 -
A
 -

-








 = /









 (ʼ16 KK&Long)  - /   ()(ʼ16 KK&Long)   =  / 

Figure 2: Evolution of the baryon asymmetry in the presence of 
decayingfield.
helicalmagnetic
We take B0 = 10 16 G for the field strength today, 0 = 10 2 pc for = the - coherence length today, -  =  -
and fh$ee = fflip = 1 for the spin-flip fudge factor. The magnetic field is injected at a temperature
Tini , which ranges from 108 to 103 GeV by factors of 10. The-dashed lines shown the analytic
approximations in Eqs. (3.6) and (3.10).
 -

E

D
in some previous studies [24, 25].) In the symmetric phase, the equilibrium
solution
scales as
B ACCourtesy H.Oide

What determines ﬁnal BAU?
Early EWSB (crossover) completion, late sphaleron freeze out=> Net BAU is suppressed(ʼ98 Giovannini&Shaposhnikov)

Early sphaleron freeze out, late EWSB (crossover) completion=> Net BAU is eﬃciently remained

Courtesy H.Oide

What determines ﬁnal BAU?
Freezeout of EW sphalerons vs3
exp[ 145 + 0.8(T /GeV)]T

-10

cal

pure gauge
becomes sensitive to the near-criticalbehaviour and themass becomes small but still remains non-zero. Even at
its largest, the Higgs correlation length is smaller than
BY the largest lat- Bem
10/T , which is substantially smaller thantice sizes ∼ 70-80/T .
In the symmetric phase, the non-abelian gauge
BA fields
are confining, and the operators couple to bound statesW (t)of two scalars. The correlation functions ✓becomenoisy
and the screening masses increase rapidly.
The U(1) gauge field correlation function can be usede /T
to measure the γ-Z mixing,standard
i.e. the eﬀective Weinberg
multicanonicalangle. We define the operator
fit! perturbative
Op (z) =α12 (x1 , x2 , z)eip·x ,
(22)

1.1

Lattice results

log Γ/Τ

4

-20

-25

-30

-35

x1 ,x2-40

180

of tem-ial realal prefis thendetails
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sha1_base64="7sEsULW1SX8xaMkyygs7uDws5ws=">AAACIHicbVC7TgJBFJ31ifhCLG0mEBOMCdml0ZJoY4kJr4RFMjtcYMLsIzN3jWTDX1hb2Oin2BlL/BFbh0ch4EkmOTnn3twzx4uk0GjbE2tjc2t7Zze1l94/ODw6zpxk6zqMFYcaD2Womh7TIEUANRQooRkpYL4noeENb6d+4xGUFmFQxVEEbZ/1A9ETnKGROpmsy0P9UKIuDgBZp1GoXnQyebtoz0DXibMg+XLOvXyelEeVTubH7YY89iFALpnWLceOsJ0whYJLGKfdWEPE+JD1oWVowHzQ7WSWfUzPjdKlvVCZFyCdqX83EuZrPfI9M+kzHOhVbyr+63n+0uVEm2gD6K7Ewd51OxFBFCMEfJ6mF0uKIZ22RbtCAUc5MoRxJcyHKB8wxTiaTtOmKWe1l3VSLxUdu+jcm8puyBwpckZypEAcckXK5I5USI1w8kReyBt5t16tD+vT+pqPbliLnVOyBOv7F6TCpgc=</latexit>-15

8

weak mixing angle cos2 W (T )W

Completion of EWSB

1

0.9

0.8

BW 3

Analytic (1-loop)

where the sum is taken over the plane (x1 , x2 ), αij is2cos θWthe (non-compact) hypercharge U(1) plaquette (9)BZand pis a transverse momentum vector compatible with peri-45
0.7
130
140
150 1: A (p160, p , p
170
Figuregraphical
representationof the conversion from hypermagnetic
field BY into elecodic boundary
conditions:
1 2 3 ) = 2π/N (n1 , n2 , 0)
140
145
150
155
160
165
170
Ttromagnetic
/ GeV
field
B
duringtheEWcrossover.
The(blue)
parabolasindicatethecurvature
of
em
T/GeV
with integer n1 and n2 . In our measurements we use
the thermal e↵ective(ʼ14potential.
The weak mixing angle ✓W (t) measures the separation of the flat
DʼOnofrio)(ʼ16 DʼOnofrio)
the smallest non-vanishing
momentum,with|p| = 2π/N .
FIG. 3: The measured sphalerondirection
rate andthe fitthe broken(masslessfieldtodegreeof freedom) and the U(1)Y axis.
FIG. 9: The eﬀective γ−Z mixing as a function of the temperAt p = 0 the operator Op vanishes, due to the periodic
phase rate, Eq. (7), shown with a shaded
error band. The
ature. The dashed lines show the 1-loop perturbative results.
boundary conditions. The correlation function
perturbative result is from Burnieret ofal.the[11]withthenonevolution
massless
field
degreeof freedom, we have reduced the problem to a single degree!
of freedom
as represented
by the classicalperturbative correction used theresee ∗maintext. vector field Aµ (x).
1removed;
G(z)
=
⟨O
(t)O
(z
+
t)⟩
(23)
p
Ansatz(2.3) isp
represented
graphically in Fig. 1, which illustrates the conversion from
3SU(2)
Pure gauge refers to the rate in hot
gauge
theory[19].
NTheVII. CONCLUSIONS
field toelectromagnetic
The freeze-out temperature T∗ hypermagnetic
is solvedt fromthecrossing offield. Here we denote the magnetic field of a gauge field Yas
BY ⌘ r ⇥Y. Weshown
have drawn
the thefigure so as to suggest that |BA | does not decrease appreciably
Γ and the appropriately
Hubblerate,
has the longscaled
distance
behaviour[30] withduringtheEWcrossover.
As
we
will
explain later, this is the
becauseAµ evolvesdeterminedslowlyWecasehave
accurately
the Higgs field expecalmost horizontal line.
√
accordingand the inverse cascade. tation value and its susceptibility across the Standard
ap2cosmic expansion
Aγ z to the−z p2 +m2γ"eG(z) →
Having generalized the gauge field Ansatz(24)from our Model
earlier work,
we are now
prepared
to simulations of an eﬀective
cross-overusinglattice
2βG
2 + m2
Courtesy H.Oideprevisit the calculationγof source terms (2.2). Using the Ansatz
in Eq. (2.3), thetheory.
source terms
can23-dimensional
Defining
the cross-over temper-

lattice simulations for the EW crossover with 125 GeV Higgs

BAU remains! Quantitative results are sensitive to<latexit sha1_base64="kojk5pA6EEQCHQqxu2vidpTg8pM=">AAACF3icbVC7TsNAEDyHVwivACWNRYQUmsimgTKChjJI5CElVnQ+r+NT7mzrbo0UWfkMChr4FDpES8mX0HJJXJCEkU4azexq58ZPBdfoON9WaWNza3unvFvZ2z84PKoen3R0kikGbZaIRPV8qkHwGNrIUUAvVUClL6Drj+9mfvcJlOZJ/IiTFDxJRzEPOaNopP4AI0A67NbxclitOQ1nDnuduAWpkQKtYfVnECQskxAjE1Trvuuk6OVUIWcCppVBpiGlbExH0Dc0phK0l88jT+0LowR2mCjzYrTn6t+NnEqtJ9I3k5JipFe9mfiv58uly7k20SIIVuJgeOPlPE4zhJgt0oSZsDGxZyXZAVfAUEwMoUxx8yGbRVRRhqbKimnKXe1lnXSuGq7TcB+cWvO26KxMzsg5qROXXJMmuSct0iaMJOSZvJI368V6tz6sz8VoySp2TskSrK9f0gSgUA==</latexit>
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W (t)Since the lattice results and one-loop calculation for the SM
EW crossover diﬀer each other signiﬁcantly for this purpose,
we choose several ﬁtting functions of temperaturedependence of the weak mixing angle.
   ()  =     =  =     =    =  /   

  =     =    =     =     =     =  

 
 
 












 -  =  - =  -

EC

 -

D

 -

    

 






B


-

A
 -






   [   ]   - /   ()EvolutionofB-asymmetry 






Parameterization of the evolution of weakangle
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The time-dependent weak mixing angle, expressed as cos ✓W (t). Results of numericalulations [61] appear as (gray) data points, and results of one-loop perturbative analyticO(103-)  == --
ns [63] appear as a (black) dashed line. The other curves correspond to the “smoothedpolating function from Eq. (4.3), which we use for our analysis.

From B to E, uncertainty is



 -

(ʼ16 KK&Long)

Courtesy H.Oide

What determines ﬁnal BAU?
B

10

10

f( W , T

135GeV)<latexit sha1_base64="JmB1kZW9ApD2BvHU5k0bmbgKNz0=">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</latexit>
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sha1_base64="851+IL1tc1yc7gv3ulxEE5tSIWk=">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</latexit>f( W , T )<latexit sha1_base64="oUcchEdDd2ZS1GqLuNcDPLRDwK4=">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</latexit>
sha1_base64="h0wz2BcaJCwWyteXjIJh1VbYjWc=">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</latexit>
sha1_base64="11Zouxyme1YP6KW6eypc0wkOS0I=">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</latexit>

BEW
10 3 GeV2

EW

106 GeV 1

d W
sin(2 W )T
dT

(

1

@T

2

135GeV)(ʼ16 KK&Long)

Magnetogenesis with positive helicity before EWSB.
With appropriate properties of hyper MFs,
present BAU can be explained.
※Since helicity is just the diﬀerence between the right and left helicity modes,
the sign of helicity can be the same beyond the coherence length of MFs.

Courtesy H.Oide

Helical magnetic ﬁeld generationfrom Axion inﬂation
D. Jiménez, KK, K. Schmitz & X.J. Xu (MPIK), JCAP12 (2017) 011[arXiv:1707.07943[hep-ph]].

Courtesy H.Oide

Pseudoscalar inﬂation (ʼ90 Freese+, ʼ93 Adams, … )Inﬂation (ʼ80 Starobinsky, ʼ81 Sato, ʼ81 Guth, ʼ82 Linde, ʼ82 Albrecht & Steinhardt)… an era with accelerated cosmic expansion before the Big Bang.
- often driven by the potential energy of a scalar ﬁeld (inﬂaton)- solution to the ﬂatness/horizon/monopole problems and seed
for the CMB ﬂuctuation and large scale structures.

(from WMAP team)Courtesy H.Oide

Pseudoscalar inﬂation (ʼ90 Freese+, ʼ93 Adams, … )If inﬂaton is a pseudoscalar, e.g. axions,
it can have an anomalous coupling to hypergauge ﬁelds:fDuring inﬂation, inﬂaton
slow-rolls homogeneously.
˙ = 0,<latexit sha1_base64="FIhPg5R0X78fvrfPWnXV6lam40c=">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</latexit>
sha1_base64="b2j+Hdkx2wPRA2qsplDFQK82lVE=">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</latexit>

| i |

| ˙|

V( )

Yµ Ỹ µThe dynamics of the gauge ﬁelds
is aﬀected by inﬂaton motion.
2
2

+k

2

4 ˙ /(Hf )1±k

Y± = 0.

One polarization mode feels instability,
especially around the end of inﬂation(ʼ06 Anber+)

Generation of maximally helical hypermagnetic (and electric) Fields!
Courtesy H.Oide

Estimated magnetic ﬁeld properties just after inﬂation
B

10

2 e
2
H
5/2

with

H

<latexit sha1_base64="JavjJo+cwAZuBIFery1OBaG5txw=">AAACKHicbVDLSsNAFJ3UV62vqAtRN4NFcFWSurDLopsuK9gHNKFMJpN26EwSZiZiCQG/xoUb/ZTupG4EV/6BW6ePha0eGDiccy5z7/FiRqWyrLGRW1ldW9/Ibxa2tnd298z9g6aMEoFJA0csEm0PScJoSBqKKkbasSCIe4y0vMHNxG/dEyFpFN6pYUxcjnohDShGSktd88RhOuwj6EjKoRMIhFPngWZpLeuaRatkTQH/EntOitXKx+jo/eu43jW/HT/CCSehwgxJ2bGtWLkpEopiRrKCk0gSIzxAPdLRNEScSDed3pDBc634MIiEfqGCU/X3RIq4lEPu6SRHqi+XvYn4r+fxhZ9TqVfrE39pHRVU3JSGcaJIiGfbBAmDKoKT1qBPBcGKDTVBWFB9EMR9pMtSutuCbspe7uUvaZZL9mWpfKsruwYz5MEpOAMXwAZXoApqoA4aAINH8ARewKvxbIyMN2M8i+aM+cwhWIDx+QMsQ6tW</latexit><latexit sha1_base64="3SCDKGSOEeFIzqKuNsSIjg7pZoc=">AAACPnicbVBNSwMxEM36bf2qehQkKIIX6+6K6LHUgz1WsCq4bcmmszY02V2SrFjC3v0F/hoPXvQ/eBdv4lGPpq0HrQ4kPN6bYd68MOVMadd9dsbGJyanpmdmC3PzC4tLxeWVM5VkkkKdJjyRFyFRwFkMdc00h4tUAhEhh/Owe9TXz69BKpbEp7qXQkOQq5hFjBJtqVbRreBAMYE9t2l2/BwHkSTUQNMEKcPBDctzY/+m2d/18xxXm36ruOmW3EHhv8D7BpvlcuV2/XjlpdYqfgTthGYCYk05UerSc1PdMERqRjnkhSBTkBLaJVdwaWFMBKiGGVyW4y3LtHGUSPtijQfszwlDhFI9EdpOQXRHjWp98l8tFL82G2WtdaA9YkdHhw3D4jTTENOhmyjjWCe4nyVuMwlU854FhEpmD8K0Q2yA2iZesEl5o7n8BWd+ydsr+Sc2sgoa1gxaQxtoG3noAJVRFdVQHVF0h+7RI3pyHpxX5815H7aOOd8zq+hXOZ9fRfuxYg==</latexit>

˙
2f H<latexit sha1_base64="emq9HYH8u+2blwBOYbJvXZoriZ0=">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</latexit>(ʼ06 Anber+)Hinf & : parameters that characterizes magnetogenesis<latexit sha1_base64="28DORFKnmiP4d55flhzC80b3wi4=">AAACGnicbVC7TgJBFL2LL8QXamJjM5GYWJFdLLQk2FBCIo8EkMzOzsKEmd3NzKwJ2fAfFjba+xPGxtja2Pgbts4ChYAnmeTknHsz5x434kxp2/6yMmvrG5tb2e3czu7e/kH+8KipwlgS2iAhD2XbxYpyFtCGZprTdiQpFi6nLXd0k/qteyoVC4NbPY5oT+BBwHxGsDbSXbXfFVgPpUhY4E/6+YJdtKdAq8SZk0L5pP7NXipvtX7+p+uFJBY00IRjpTqOHelegqVmhNNJrhsrGmEywgPaMTTAgqpeMk09QedG8ZAfSvMCjabq340EC6XGwjWTaUa17KXiv54rFn5OlIk2pN5SHO1f98zVUaxpQGZp/JgjHaK0J+QxSYnmY0MwkcwchMgQS0y0aTNnmnKWe1klzVLRuSyW6qayCsyQhVM4gwtw4ArKUIUaNICAhAd4gmfr0Xq13q2P2WjGmu8cwwKsz1+/5KXn</latexit><latexit sha1_base64="7x6cEa4cKDAB9rkLe5AmsPAePkQ=">AAACD3icbVC7TsMwFL3hWcqrwAhCERUSU5WUAcYKFsZW0IfURpXjOK1V24lsB1FFHRkZWOBT6IRY+QS+gQ9gxX0MtOVIlo7OuVf3+Pgxo0o7zpe1tLyyurae2chubm3v7Ob29msqSiQmVRyxSDZ8pAijglQ11Yw0YkkQ9xmp+73rkV+/J1LRSNzpfkw8jjqChhQjbaTb1gNt5/JOwRnDXiTulORLR8PK9+PxsNzO/bSCCCecCI0ZUqrpOrH2UiQ1xYwMsq1EkRjhHuqQpqECcaK8dBx1YJ8aJbDDSJontD1W/26kiCvV576Z5Eh31bw3Ev/1fD5zOVUmWpcEc3F0eOmlVMSJJgJP0oQJs3Vkj8qxAyoJ1qxvCMKSmg/ZuIskwtpUmDVNufO9LJJaseCeF4oVU9kVTJCBQziBM3DhAkpwA2WoAoYOPMELvFrP1pv1bn1MRpes6c4BzMD6/AWVzKD/</latexit>

Conditions to avoid too much backreaction;<latexit sha1_base64="v+Dc5LshST3t7kGR9U39PhHeuq4=">AAACMXicbVDLSsNAFJ3UV62vqktFBosgCCWpC12WiuCyBfuAJpTJZNIOnUnCzEQooUsXfo0LN/VTuhO3/QC3TtIubOuFYQ7n3ss957gRo1KZ5tTIbWxube/kdwt7+weHR8Xjk5YMY4FJE4csFB0XScJoQJqKKkY6kSCIu4y03eFD2m+/ECFpGDyrUUQcjvoB9SlGSlO9YskWg7BXu8m+R2gzBjNoc6QGgic08Md6yiybWcF1YC1AqXo+acxeLyb1XvHH9kIccxIozJCUXcuMlJMgoShmZFywY0kihIeoT7oaBogT6SSZmTG80owH/VDoFyiYsX83EsSlHHFXT6Ya5WovJf/tuXzpciK1tAHxVuQo/97RrqNYkQDP1fgxgyqEaXzQo4JgxUYaICyoNgTxAAmElQ65oJOyVnNZB61K2botVxo6shqYVx6cgUtwDSxwB6rgCdRBE2DwBt7BBHwaH8bU+DK+56M5Y7FzCpbKmP0Cly2ufw==</latexit>

B +

E

inf

Backreaction parameter F
10

15

Exact
Fit

r > 0.11

Hubble rate H [GeV]

1014
1013

10 -5

1

F > 11012

10 -101011
1010

10 -15109
10 -20

108
2

4

6

8

Instability parameter Figure 2: Backreaction parameters10(ʼ17 Jiménez+)

F (left panel) andinstabilityparam
KG (right panel) as functions of the
Courtesy
H.OideMagnetic ﬁeld evolution after inﬂation# instant reheating is assumed

Eddy turn-over scale
T0

Trec

1040

TEW
1020

1020

100
(rh , Hrh )(8, 1013 GeV)

100

10-20

(4, 1013 GeV)
(4, 106 GeV)

10-20
10-10

10-5

100

105

1010

10-40

Field strength Bp [GeV2 ]

1040
TQCD

Field strength Bp [Gauss]1060

ed = vt<latexit sha1_base64="lFxSiDc7OlcXoxrqph/1EVcSBf0=">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</latexit>

initially<latexit sha1_base64="Nh83GJYBXVMM2TJ4ieemTIS32XQ=">AAACK3icbVDLSsNAFJ34rPUVFdyIMFgEVyWpC11JqRuXLdgHNCFMJpN26EwSZiZCCV35NS660V/wD1wp4s5vcOuk7cK2Hhg4nHsvc87xE0alsqx3Y2V1bX1js7BV3N7Z3ds3Dw5bMk4FJk0cs1h0fCQJoxFpKqoY6SSCIO4z0vYHt/m8/UCEpHF0r4YJcTnqRTSkGCkteeapw/RygOANnDHP4Uj1Bc9IMPLMklW2JoDLxJ6RUvW48UXHtde6Z/44QYxTTiKFGZKya1uJcjMkFMWMjIpOKkmC8AD1SFfTCHEi3WwSYwTPtRLAMBb6RQpO1L8XGeJSDrmvN3OLcnGWi//OfD73cya1tb6ON29HhdduRqMkVSTCUzdhyqCKYV4cDKggWLGhJggLqgNB3EcCYaXrLeqm7MVelkmrUrYvy5WGrqwGpiiAE3AGLoANrkAV3IE6aAIMHsETeAYvxth4Mz6Mz+nqijG7OQJzML5/AYufq+o=</latexit>>edadiabatic evolution

1015

a 2,

B

Temperature T [GeV](rh , Hrh )10-10(8, 1013 GeV)
(4, 1013 GeV)

1030

10-20

(4, 106 GeV)

1020

10-30

1010

10-40

100

T0

Trec

10-10

10-5

100

10-10

TEW
105

1010

1015

Correlation length p [GeV-1 ]

1040

100

TQCD

Correlation length p [Mpc]<latexit sha1_base64="YhJIeISGpxnM2cEN8BJRqF5NvCw=">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</latexit>

10-50

B
H

eventuallyTemperature T [GeV](ʼ17 Jiménez+)

Figure 3: Physical field strength Bp (upper panel) and physical correlation lengthp(lower panel) of thehypermagnetic B field as functions of the radiation temperature T for representative values of H and ⇠ at the endof inflation. The vertical dotted lines mark the respective temperatures at which the adiabatic regime transitionsaed<latexit sha1_base64="PDvDhHFarbhKRoXjdp8QU5VnhJ4=">AAACMHicbVC7TgJBFJ31ifhCTWxsJhATK7KLhZYEG0tI5JGwhMzOXmDCzO46M2tCNvT+jBY2+gF+hDYaW3/A1lmgEPAkk5w5997cc48Xcaa0bb9bK6tr6xubma3s9s7u3n7u4LChwlhSqNOQh7LlEQWcBVDXTHNoRRKI8Dg0veFVWm/egVQsDG70KIKOIP2A9Rgl2kjdXN7lptkn2FVMwC2efbuuIHogRQL+uJsr2EV7ArxMnBkplI9rH+yh8lrt5n5cP6SxgEBTTpRqO3akOwmRmlEO46wbK4gIHZI+tA0NiADVSSa3jPGpUXzcC6V5gcYT9e9EQoRSI+GZztSiWqyl4r81T8xtTpSxNjDnzdvRvctOwoIo1hDQqZtezLEOcZoe9pkEqvnIEEIlMwdhOiCSUG0yzpqknMVclkmjVHTOi6WaiayCpsigE5RHZ8hBF6iMrlEV1RFF9+gRPaMX68l6sz6tr2nrijWbOUJzsL5/AT9Krlk=</latexit>

MHD evolution; “inverse cascade”
B<latexit sha1_base64="1pzHZSFayG0A9vPMNMwztfLxtqg=">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</latexit>

a 7/3 ,

a5/3
(ʼ04 banerjee&Jedamzik)(approximate) helicity conservation
prevents MFs from complete decay.

into the inverse cascade regime. Both plots account for the decrease in the e↵ective number of DOFs in thecourse of the expansion. The kinks around T ⇠ 100 MeV correspond, e.g., to the QCD phase transition. For
T < 10 MeV, damping e↵ects might become important (see [49, 73, 75]) and our description of the magnetic fieldevolution becomes less accurate. For this reason, we only draw dashed lines in the low-temperature regime.

Courtesy H.Oide

Resultant baryon asymmetry
(10 3

B

1)

10 16

e2

Hrh
1013 GeVrh

4rh<latexit sha1_base64="i2O79kI+D8niDVPSaSPSmwApGOU=">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</latexit>

3/2

Baryon asymmetry B
10

15

Max
Min

r > 0.11

MFs that predict B
remains until today atintergalactic void with<latexit sha1_base64="sAN2lsGfg3VrAJ4acQj96QC8itA=">AAACInicbVDLTgIxFO3gC/E1PnYa00hM3EhmcKFLghuXkMgjASSdcoGGdmbSdkxwwtL/cOFGv4BvcGdcmfgHfoBby2Mh4EmanJxzb+7p8ULOlHacTyuxtLyyupZcT21sbm3v2Lt7ZRVEkkKJBjyQVY8o4MyHkmaaQzWUQITHoeL1rkd+5R6kYoF/q/shNATp+KzNKNFGatoHddCkmcd1xQR2nbv43HUGTTvtZJwx8CJxpySdOxoWvx+Ph4Wm/VNvBTQS4GvKiVI11wl1IyZSM8phkKpHCkJCe6QDNUN9IkA14nH6AT41Sgu3A2mer/FY/bsRE6FUX3hmUhDdVfPeSPzX88TM5ViZaF1ozcXR7atGzPww0uDTSZp2xLEO8Kgv3GISqOZ9QwiVzHwI0y6RhGrTaso05c73skjK2Yx7kckWTWV5NEESHaITdIZcdIly6AYVUAlR9ICe0At6tZ6tN+vd+piMJqzpzj6agfX1C1HNpv4=</latexit>

1013
F > 1
10

12

10

1011

-10

B
<latexit sha1_base64="Sc9RQxKi0Dly8/e7AVvq3RsSsWg=">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</latexit>

10

10 10

B ~

Hubble rate H [GeV]

1014

10 17 G,

10 2 pc

10

109
10-22

10-18

10-14

10-10

10-6

6

8

10-2

108
2

4

Instability parameter 10(ʼ17 Jiménez+)e 5: Baryon asymmetry ⌘B = nB /s as a function of the instability parameter ⇠ and the Hubble rate H;

q. (62). Here, both ⇠ and H are understood to correspond to the respective values at the end of inflation,

Courtesy H.Oide

One beneﬁt: Generation of stochastic GWs
2

GW h

H
( GW h2 )inf(fL ( ) + fR ( ))e4Mpl
H
22
2.3 10exp[0.91 4 (4.61)]
1011 GeV

2<latexit sha1_base64="rmIqhRun1P5jvn6A4g9GDthXwyU=">AAACoXicbVFdb9MwFHXC1yhfBR55wKJCSoRUJQVpeyzwwEBDlImuk+ouctybxpqdBPsGUUX5FUj8N34DPwDxhttOE+t2JUvH59xr33tuWilpMYp+ef616zdu3tq53blz9979B92Hj45sWRsBY1Gq0hyn3IKSBYxRooLjygDXqYJJevp2pU++gbGyLL7gsoKZ5otCZlJwdFTS/cE+aVjwhGmOudHNu0lL85MBZVZq+EqDq+XwnJBF1lKGLtlSpiDDgGWGi2a/bT6eJ1WqbZmRixxD93SQJQcB+y7DF1lyuAYhhZPmFWWVpO7aJt1e1I/WQS+D+Az0hq9/Hgw/mN+jpPuHzUtRayhQKG7tNI4qnDXcoBQK2g6rLVRcnPIFTB0suOt21qzNa+lzx8xpVhp3CqRr9v+Khmtrlzp1matx7La2Iq/UUn3h58a61nKYb7WD2d7MuVjVCIXYdJPVimJJV+uic2lAoFo6wIWRbiAqcu4cRrfUjnMq3vblMjga9OOX/cFnZ9kbsokd8oQ8IwGJyS4Zkn0yImMiyF/vqRd4od/z3/sj/3CT6ntnNY/JhfCn/wCVSNKD</latexit>

4<latexit sha1_base64="DbQIViu7CpdPP3fJcW4mWJ97QKE=">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</latexit>f HHzL
106

-4

103

log10 WGW h2

-6

100

10-3

10-6

(ʼ12 Cook+)
10-9

10-12

10-15

10-18

LIGO
LISA

-8
-10

C

-12
-14
-16

A

-18

B

Wedding20170709

0

10

20

30
Ne

40

50

60

(ʼ17 Jiménez+)

But appropriate parameter choice gives the GW signature
at small amplitude GW h2 10 17 and high frequency f 1 102 MHz<latexit sha1_base64="K3lZLqV3D191tjqvcyhqe78phnI=">AAACJHicbVC7TgJBFJ3FF+Jr1cJCYyYSExvJLo3aEW1oTCBxhQSQzA6zMGFmdjMza4IbSr/EwkZ7fsLKWNj4A36ArcOjEPAkk5ycc2/m3ONHjCrtOJ9WamFxaXklvZpZW9/Y3LK3d25VGEtMPByyUFZ9pAijgniaakaqkSSI+4xU/O7V0K/cE6loKG50LyINjtqCBhQjbaSmvRfAuqIcuqeuc5eHSV1yeF186DftrJNzRoDzxJ2QbOFgUP5+PByUmvZPvRXimBOhMUNK1Vwn0o0ESU0xI/1MPVYkQriL2qRmqECcqEYyOqAPj43SgkEozRMajtS/GwniSvW4byY50h016w3Ffz2fT/2cKBOtQ1ozcXRw3kioiGJNBB6nCWIGdQiHlcEWlQRr1jMEYUnNQRB3kERYm2Izpil3tpd54uVzFzmnbBq7BGOkwT44AifABWegAIqgBDyAQR88gRfwaj1bb9a79TEeTVmTnV0wBevrF1ZAp6Y=</latexit>
sha1_base64="cT8fTe4zANWy8CCBx0S7kYXDMy0=">AAACJHicbVC7SgNBFL0bXzG+ohYWWgwGwcawm0btgjZphASMCSRrmJ2dTYbM7C4zs0Jc8jUWNtr7E1ZiYSP+ha2TR2GiBwYO59zLnHu8mDOlbfvDyiwsLi2vZFdza+sbm1v57Z0bFSWS0DqJeCSbHlaUs5DWNdOcNmNJsfA4bXj9y5HfuKNSsSi81oOYugJ3QxYwgrWROvm9ALUVE8g5cezbEkrbUqCryv2wky/YRXsM9Jc4U1IoH7zUvgCg2sl/t/2IJIKGmnCsVMuxY+2mWGpGOB3m2omiMSZ93KUtQ0MsqHLT8QFDdGQUHwWRNC/UaKz+3kixUGogPDMpsO6peW8k/ut5YubnVJloPerPxdHBmZuyME40DckkTZBwpCM0qgz5TFKi+cAQTCQzByHSwxITbYrNmaac+V7+knqpeF60a6axC5ggC/twCMfgwCmUoQJVqAOBITzAEzxbj9ar9Wa9T0Yz1nRnF2Zgff4ARaOmQA==</latexit><latexit sha1_base64="J3pMzgeYwST7JeRUd2Oz4zaqEAk=">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</latexit>
sha1_base64="rjK3vVdoq+UHvCmpEznI3yS8QVI=">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</latexit>

One beneﬁt: Generation of stochastic GWs
f HHzL
10-9

10-12

10-15

10-16

LIGO
LISA

-8
-10

C

-12
-14
-16

A

-18

B

Wedding20170709

0

10

10-14

10-18

20

30

40

50

60

10-15

10-18
10-20
10-22
10

10

- 10

-16

ηB

0
∼1

r > 0.11

10-6

co
nt
rib
ut
io
n

10-3

GW peak strength [Ω0GW h2 ]peaklog10 WGW h2

-6

100

um

-4

103

106 107 108 109 1010 1011 1012 1013 1014 1015

Va
cu

106

Hubble rate at the end of inflation Hrh [GeV]

-24

10-26
Present-day

10-28

magnetic field

10-17

B0p [Gauss]

10-30

Ne
(ʼ17 Jiménez+)

10-1
100
101
102
GW peak frequency fpeak [MHz]103

But appropriate parameter choice gives the GW signature10
at small amplitude GW h
and high frequency f 1 10 MHz

Figure 7: Present-day magnetic field strength Bp0 as a function of the peak frequency fpeak and the⇥the peak in the GW spectrum associated with the gauge field production at the end of inflation, ⌦2 Hubble rate a
2 In the approximation17
see Eq. (77).
of instant reheating, fpeak is directly related to the

inflation; see Eq. (76). The green band illustrates the region in parameter space where ⌘B ⇠ 10 10 ; s<latexit sha1_base64="K3lZLqV3D191tjqvcyhqe78phnI=">AAACJHicbVC7TgJBFJ3FF+Jr1cJCYyYSExvJLo3aEW1oTCBxhQSQzA6zMGFmdjMza4IbSr/EwkZ7fsLKWNj4A36ArcOjEPAkk5ycc2/m3ONHjCrtOJ9WamFxaXklvZpZW9/Y3LK3d25VGEtMPByyUFZ9pAijgniaakaqkSSI+4xU/O7V0K/cE6loKG50LyINjtqCBhQjbaSmvRfAuqIcuqeuc5eHSV1yeF186DftrJNzRoDzxJ2QbOFgUP5+PByUmvZPvRXimBOhMUNK1Vwn0o0ESU0xI/1MPVYkQriL2qRmqECcqEYyOqAPj43SgkEozRMajtS/GwniSvW4byY50h016w3Ffz2fT/2cKBOtQ1ozcXRw3kioiGJNBB6nCWIGdQiHlcEWlQRr1jMEYUnNQRB3kERYm2Izpil3tpd54uVzFzmnbBq7BGOkwT44AifABWegAIqgBDyAQR88gRfwaj1bb9a79TEeTVmTnV0wBevrF1ZAp6Y=</latexit>
sha1_base64="cT8fTe4zANWy8CCBx0S7kYXDMy0=">AAACJHicbVC7SgNBFL0bXzG+ohYWWgwGwcawm0btgjZphASMCSRrmJ2dTYbM7C4zs0Jc8jUWNtr7E1ZiYSP+ha2TR2GiBwYO59zLnHu8mDOlbfvDyiwsLi2vZFdza+sbm1v57Z0bFSWS0DqJeCSbHlaUs5DWNdOcNmNJsfA4bXj9y5HfuKNSsSi81oOYugJ3QxYwgrWROvm9ALUVE8g5cezbEkrbUqCryv2wky/YRXsM9Jc4U1IoH7zUvgCg2sl/t/2IJIKGmnCsVMuxY+2mWGpGOB3m2omiMSZ93KUtQ0MsqHLT8QFDdGQUHwWRNC/UaKz+3kixUGogPDMpsO6peW8k/ut5YubnVJloPerPxdHBmZuyME40DckkTZBwpCM0qgz5TFKi+cAQTCQzByHSwxITbYrNmaac+V7+knqpeF60a6axC5ggC/twCMfgwCmUoQJVqAOBITzAEzxbj9ar9Wa9T0Yz1nRnF2Zgff4ARaOmQA==</latexit><latexit sha1_base64="J3pMzgeYwST7JeRUd2Oz4zaqEAk=">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</latexit>
sha1_base64="rjK3vVdoq+UHvCmpEznI3yS8QVI=">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</latexit>

Especially, detectable GWs predicts baryon overproduction.
the strengthof the observedisGW
signal turns out to be consistent with an inflaton c=> New U(1) gaugeinteractionneeded.

the hypercharge gauge field — or whether this assumption would lead to baryon overp(ii) Along the diagonal line in Fig. 7, the GW spectrum at the end of inflation is dom

Remaining issues
1. Reheating…at oscillating stage, | ˙ | is larger.
Larger MFs can be generated?
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3. Other backreaction problem? Schwinger eﬀect?
(ʼ18 Domcke&Mukaida)Summary

Courtesy H.Oide

1.

2.

3.

4.

Helical hypermagnetic ﬁelds, if existed before the EWSB, canbe the source of BAU through the SM chiral anomaly. No BSM
physics is involved in this mechanism.
Axion inﬂation is one of the well-motivated models for thesource of helical hypermagnetic ﬁelds (but not limited to).
In the point of view of axion inﬂation, this mechanism iscomplimentary to the one in Peterʼs talk.
GWs from the gauge ﬁelds from axionic inﬂation is also ofinterest, but detectable ones predict baryon overproduction, if
they are the SM gauge ﬁelds.
In other words, detection of such GWs at LIGO/Virgo/KAGRA
shows the existence of hidden gauge interaction.

Courtesy H.Oide

Appendix

Courtesy H.Oide

IGMF Constraints from Gamma
The Astrophysical Journal, 814:20 (14pp), 2015 November 20
Present
properties of MFs in the B0-λ0 plane…

log(B/Gauss)lengths. Thi
has been poi
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
and Nerono-6
CMB constraint contours at-8
Banerjee+ (ʼ04)
the coarsene
B0
1pc
-10
both B and0
10 14 G
Low mag-12>5σ. We co-14
many auth1804.08035-16
et al. 2011
Baryon overproduction
component i
KK&Long (ʼ16)-18
implicitly ru-20
magnetic ﬁeconstraining-10 -8
-6
-4
-2
0
2
4
to be 1ES 0log(LB/Mpc)
Finke et al. (ʼ15)
of which ind
coherent
Figure 4. Values of parameterMF
spaceof B and Llength
B ruled out for the combined
Fig. 4.— The values of parameter space of B and LB ruled out
Fig.
Our
lowe
conservative results of
4.1 for allof our objects.ﬁelds
The contours
represent
inSection
the casemagneticaremaximally
helical.
for thecombinedconservative
resultsofSection
4.1
for
all
of
our
Kneis
authorshave
the signiﬁcance a particular region of parameter space is ruled out, in number
objects.
Theascontours
represent
the significancea particularregionofsigma,
indicated
by
the
bar.
These
constraints
assume
the
Finke
et
al.
H0 (e.g., Ne
MFs that explain BAU cannot explain the blazar observation simultaneously.
of parameteris ruled
rad. in number of sigma, as indicated Dolag et al(2010) EBLspacemodel and
θj = 0.1out,by the bar. These constraints assume the Finke et al. (2010) EBL
conservativemodel and θ = 0.1 rad.
Courtesy H.Oide