Günter Sigl II. Institut theoretische Physik, Universität Hamburg

Chiral Electrodynamics and Cosmic Magnetic Fields

1. Basics on Magnetic Field Evolution2. A general approach to the chiral magnetic effect3. The chiral effect in hot neutron stars and in the

early Universe4. Conclusions and outlook

1

Observations and simulations of the non-thermal Universe

4

Structured Extragalactic Magnetic Fields

Kotera, Olinto, Ann.Rev.Astron.Astrophys. 49 (2011) 119

Filling factors of extragalactic magnetic fields are not well known, depend on initial conditions and come out different in different large scale structure simulations

Miniati

5

EGMF - OriginThe origin of EGMF is still uncertain - mainly two di�erent seedmechanisms:

I Astrophysical scenario: Seed magnetic fields are generatedduring structure formation (e.g. by a Biermann Battery[Biermann, 1950]) and are then amplified by the dynamoe�ect [Zeldovich et al., 1980]

I Cosmological scenario: Strong seed magnetic fields aregenerated in the Early Universe, e.g. at a phase transition(QCD, electroweak) [Sigl et al., 1997] or during inflation[Turner and Widrow, 1988], and some of the initial energycontent is transfered to larger scales.

The latter are the so-called primordial magnetic fields and will befocused on in the following.

I Basics for the time evolution: Homogeneous and isotropicmagnetohydrodynamics in an expanding Universe.

6 Günter Sigl

6

Primordial Magnetic fields - Simple Estimates

The main problem is that the comoving horizon at the temperatureTg of creation is very small,

lH,0 ≥ TgT0

1H(Tg)

ƒ 0.2A

100 MeVTg

B

pc ,

so that length scales of interest today are far in the tail.A magnetic field in equipartition with radiation corresponds toB ƒ 3 ◊ 10≠6 G.

7 Günter Sigl

7

On the other hand, if there is rough equipartition between kinetic and magnetic turbulence, vrms ~ vA = B/(4𝝿)1/2 , and coherence length is comparable to size of eddy which turns once in a Hubble time, one gets a relation between field amplitude B0 and coherence length lc,0,

if magnetic fields are close to maximally helical, i.e. <A B> ~ +- lcB2, helicity conservation yields lc,0(T)B0(T)2 ~ const.

8

Summary of Current Constraints

partly based on A. Neronov, I. Vovk, Science 328, 73 (2010)

helical

non-helical

G. Sigl, book “Astroparticle Physics: Theory and Phenomenology”, Atlantis Press 2016

9

Primordial Magnetic fields - Basic MHD

Magnetohydrodynamics (MHD)I Maxwell’s equations:

Ò · B = 0, Ò ◊ E = ≠ˆtB, Ò ◊ B = 4fij

I Continuity equation for mass density fl: ˆtfl + Ò(flv) = 0I Navier-Stokes equations:

fl (ˆtv + (vÒ) v) = ≠Òp + µ�v + (⁄ + µ)Ò (Òv) + f

For the magnetic field and the turbulent fluid it follows therefore

ˆtB =1

4fi‡�B + Ò ◊ (v ◊ B)

ˆtv = ≠ (vÒ) v +(Ò ◊ B) ◊ B

4fifl+ fv .

8 Günter Sigl

10

Primordial Magnetic fields - Basic MHD

I Switch to Fourier (k-)space: B(x) æ B(q), v(x) æ v(q)

ˆtB(q) = ≠ 14fi‡

q2B(q) +

iV 12

(2fi)32q ◊

5⁄d3k

1v(q ≠ k) ◊ B(k)

26

ˆt v(q) = ≠ iV 12

(2fi)32

⁄d3k [(v(q ≠ k) · k) v(k)]

+iV 1

2

(2fi)32

14fifl

⁄d3k

Ë1k ◊ B(k)

2◊ B(q ≠ k)

È.

(1)

Terms of the type v(q ≠ k) ◊ B(k) describe mode-mode couplingsuch that power from small length scales 1/k can be transportedto large length scales 1/q.

10 Günter Sigl

based on Saveliev, Jedamzik, Sigl, PRD 86, 103010 (2012), PRD 87, 123001 (2013)

11

Primordial Magnetic Fields - Correlation Function

Aim: Computation of the correlation function for B and vI Homogeneity: The correlation function cannot depend on the

position in spaceI Isotropy: The correlation function only depends on the

magnitude of the spatial separationIn Fourier space this means that the most general Ansatz is[de Kármán and Howarth, 1938]

ÈB(k)B(kÕ)Í ≥ ”(k ≠ k

Õ)[(”lm ≠ klkmk2 )

Mkk2 + i‘lmj

kjk Hm

k ]

Èv(k)v(kÕ)Í ≥ ”(k ≠ k

Õ)[(”lm ≠ klkmk2 )

Ukk2 + i‘lmj

kjk Hv

k ]

11 Günter Sigl

12

Master Equations for the Power Spectra

In the absence of helicity, Hmk = Hv

k = 0, the master equation forthe magnetic field power spectrum then reads

ȈtMqÍ =⁄ Œ

0dk

I

�t⁄ fi

0d◊

C

≠ 12

q2k4

k41

sin3 ◊ÈMqÍÈUk1Í+

+12

q4

k41

1q2 + k2 ≠ qk cos ◊

2sin3 ◊ÈMkÍÈUk1Í

≠ 14q2

13 ≠ cos2 ◊

2sin ◊ÈMkÍÈMqÍ

DJ

,

where ◊ is the angle between q and k.

12 Günter Sigl

13

Primordial Magnetic Fields: Full-Blown Numerical MHD Simulations versus semi-analytical methods based on transport equations

Andrey Saveliev, PhD thesis

magnetic fields

turbulent velocity

magnetic helicity

dotted = initial condition

dashed = final state without helicity

solid = final state with maximal helicity

< 10-9 G

< 10-11 G

QCD horizon scale

normalized to turbulence energy, < 10-6 G

14

Some literature: M. E. Shaposhnikov, Nucl.Phys. B299 797-817 (1988)

A. Vilenkin, Phys. Rev. D 22, 3080 (1980)

N. Yamamoto, Phys.Rev. D93, 065017 (2016)

Boyarsky, J. Fröhlich, and O. Ruchayskiy, Phys. Rev. Lett. 108, 031301 (2012)

A. Boyarsky, J. Fröhlich, O. Ruchayskiy, Phys. Rev. D 92, 043004 (2015)

H. Tashiro, T. Vachaspati, A. Vilenkin, PRD 86, 105033 (2012)

L. Campanelli, Phys.Rev.Lett. 98 251302 (2007)

papers by Dvornikov and Semikoz, in particular Phys. Rev. D 95, 043538 (2016)

papers by Leite, Pavlovic and Sigl

We here use Gaussian natural units, ε0 = 1/(4π) and µ0 = 4π, respectively.

Note that also often used in the literature are Lorentz-Heaviside units for which ε0 = µ0 = 1.

A General Approach to the Chiral Magnetic Effect as a Possible Source of Magnetic Helicity

15

The current of left-chiral minus right-chiral charges satisfies

In thermal equilibrium a magnetic field leads to a preferential alignment of magnetic moment and thus spin with respect to the magnetic field. If one chirality is preferred this leads to a preferential alignment of momentum with respect to the magnetic field, and thus a current proportional to B and the chiral asymmetry.

@µ (jµL � jµR) =

e2

8⇡2Fµ⌫ F

µ⌫ = � e2

2⇡2E ·B .

16

If in addition an electric field aligned with the magnetic field is present, momentum and thus chiral asymmetry changes which is described by anomaly equation above.

The following slides give technical details and can be skipped if only interested in the idea. They assume a cosmological context, but can easily be translated to other cases such as hot neutron stars.

Tashiro, Vachaspati, Vilenkin, PRD 86, 105033 (2012)

17

For the electron chiral asymmetry N5

⌘ NL � NR and the magnetic helicityH ⌘ R d3rB ·A the electromagnetic chiral anomaly gives

d

dt

N

5

� e2

4⇡2

H!

= 0 , (1)

and e2H/(4⇡2) is just the Chern-Simons number of the electromagnetic field. Thegeneralized Maxwell-Ampere law

r⇥B =@E

@t+ µ

0

(jem

+ jcB) , with jcB = � e2

2⇡2

µ5

B , (2)

and in the absence of external currents Ohm’s law j

em

= �E gives

E ' �v ⇥B + ⌘

r⇥B +

2e2

⇡µ5

B

!, (3)

where ⌘ = 1/(µ0

�) is the resistivity and the e↵ective chemical potential is givenby

µ5

=µL � µR

2� V

5

=µL � VL � µR + VR

2, (4)

where V5

is a possible e↵ective potential due to a di↵erent forward scatteringamplitude for left- and right-chiral electrons. Inserting this into the induction

18

equation the MHD is modified to

@tB = r⇥ (v ⇥B) + ⌘�B� 2e2

⇡⌘µ

5

r⇥B . (5)

This equation is similar to the mean field dynamo equation which also has growingsolutions. Neglecting the velocity term the evolution equations for the powerspectra Mk and Hk [note UB =

Rd ln kMk and H =

Rd ln kHk] now become

@tMk = �⌘k2

2Mk +

e2

2⇡2

µ5

Hk

!,

@tHk = �⌘⇣2k2Hk + 32e2µ

5

Mk

⌘. (6)

Integrating over ln k gives

@tH = �⌘

Zd ln k

⇣2k2Hk + 32e2µ

5

Mk

⌘. (7)

In an FLRW metric these are comoving quantities and conformal time.Now express N

5

in terms of µ5

,

N5

= c(T, µe)V µ5

, with c(T, µe) =µ2

e

⇡2

+T 2

3for µ2

e + T 2 � m2

e , (8)

19

where the second expression holds for relativistic electrons. Applying this toEq. (1) we get

dH =4⇡2

e2dN

5

=4⇡2V c(T, µe)

e2dµ

5

. (9)

We now take into account damping of µ5

from chirality flips. In the cosmologicalcontext after electroweak breaking at T = T

ew

' 160GeV the rate is

Rf '✓me

6T

◆2

R ⇠✓me

6T

◆2

400

✓↵em

6T

◆2

10T 3 ⇠ 10�4

m2

e

T, T <⇠ T

ew

, (10)

which is larger than the Hubble rate. Before electroweak breaking

Rf ' T5

TH(T

5

) , T >⇠ Tew

, T5

' 10TeV . (11)

Inserting Eq. (7) into Eq. (9) then yields

@tµ5

= � e2⌘

2⇡2V c(T, µe)

Zd ln k

⇣k2Hk + 16e2µ

5

Mk

⌘� 2Rf

�µ5

� µ5,b

�. (12)

Here µ5,b = �V

5

+ µs is the equilibrium value of the e↵ective chemical potentialµ5

for ⌘ ! 0. Other processes such as electroweak interactions with other speciesas for example neutrinos can be taken into account by the term µ

5,b and thus thesource term 2Rfµs.

20

From Eq. (6) growing solutions exist for wavenumbers

k < k5

⌘ k5

(µ5

) ⌘ 2e2

⇡|µ

5

| . (13)

This follows from using helicity modes in Eq. (5) which gives

@tb±k

= ⌘k

⌥2e2

⇡µ5

� k

!b±k

, (14)

Thus if the condition Eq. (13) is fulfilled, the helicity with the opposite sign as µ5

will grow whereas the same sign helicity will decay and the absolute value of thehelicity will be close to the maximal value given by

|Hk| 8⇡Mk

k. (15)

In contrast, for k >⇠ k5

both helicities will decay with roughly the resistive rate.For the helicity with opposite sign to µ

5

the first term in Eq. (14) corresponds toa growth rate in the cosmological context

Rc(k) =2e2

⇡⌘k|µ

5

| ' 2⇥ 1010✓TeV

T

◆✓k

k5

◆✓µ5

T

◆2

H(T ) , (16)

21

The total rate Rc � Rr reaches its maximum value Rmax

= ⌘k25

/4 at k = k5

/2which for

µ5

T>⇠ 10�5

✓T

TeV

◆1/2

(17)

is larger than the Hubble rate. Furthermore, Eq. (12) shows that for growingmodes |µ

5

| shrinks for either sign of µ5

. Therefore, the chiral magnetic insta-

bility transforms energy in the electron asymmetry N5

into mag-

netic energy. This is because by definition of the chemical potential µ5

theenergy U

5

associated with the chiral lepton asymmetry is given by

dU5

= µ5

dN5

= V c(T, µe)µ5

dµ5

, U5

=V c(T, µe)µ2

5

2. (18)

Now assume that initially µ5

= µ5,i and UB = 0. Since the sign of dµ

5

isopposite to the sign of µ

5,i, Eq. (9) implies that the magnetic helicity will havethe opposite sign as µ

5,i. The growth rate peaks at wavenumber k = k5

/2 givenby Eq. (13) and for a given mode k growth stops once |µ

5

| has decreased to thepoint that Eq. (13) is violated. Since the instability produces maximally helical

22

fields saturating Eq. (15), with Eq. (9) we obtain

dUB ' dMk5 ' k5

|dHk5|/(8⇡) ' k5

|dH|/(8⇡) = V c(T, µe)µ5

dµ5

,

�Em ' V c(T, µe)(µ2

5,i � µ2

5

)

2. (19)

Adding Eqs. (18) and (19) gives a total energy Utot

= U5

+UB ' V c(T, µe)µ2

5,i/2which only depends on µ

5,i. The maximal magnetic energy density then becomes

�UB

V<⇠

c(T, µe)µ2

5,i

2' µ2

5,iT2

6, (20)

where the last expression follows from Eq. (8). Eq. (12) implies that @tµ5

= 0 if

µ5

=Rfµ5,b � 2e2⌘

⇡c(T,µe)

Rd ln kkMk

V

⇣Hk

8⇡Mk/k

⌘

Rf +4e4⌘

⇡2c(T,µe)UBV

, (21)

where Hk has again be normalized to its maximal value given by Eq. (15). Fornegligible magnetic fields µ

5

' µ5,b, as expected and magnetic field modes with

k < k5

(µ5,b) are growing exponentially with rate Rc(k) � Rr given by Eq. (16).

The magnetic field terms start to dominate for

UB

V>⇠

c(T, µe)Rf

4e4⌘' 10⇡

3e4T 2m2

e ' 2⇥ 105T 2m2

e , (22)

23

In this case Eq. (21) gives

µ5

' � ⇡

2e2UB

Zd ln kkMk

✓Hk

8⇡Mk/k

◆. (23)

This is what Ruchayskiy et al call tracking solution. Note that µ5

from Eq. (21)varies with rates in general much slower than Rf and Rc. Also, since in generalµ5

6= µ5,b, the two terms in Eq. (12) do not vanish separately but only tend to

compensate each other and are both roughly constant since µ5

is approximatelyconstant. Since at saturation, Eq. (23), the first term in Eq. (12) vanishes. Thusµ5

and due to Eqs. (9) also the magnetic helicity change linearly in time with

@tH ' 8⇡2V c(T, µe)

e2Rf(µ5

� µ5,b) . (24)

Since helicity is nearly maximal this also implies that the magnetic energy alsoroughly grows or decreases linearly with time, depending on the sign of (µ

5

�µ5,b)/H.

24

Combining Eqs. (6), (12) and (18) the rate of change of the total energy is

@tUtot

= @tUB + @tU5

= (25)

= �2⌘

Zd ln kMk

8<

:(k � k5

)2 + 2k5

k

"✓Hk

8⇡Mk/k

◆sign(µ

5

) + 1

#9=

;

�2RfV c(T, µe)µ5

�µ5

� µ5,b

�,

where k5

= k5

(µ5

) is given by Eq. (13). Since the expression in large braces isnon-negative due to Eq. (15), up to the term proportional to µ

5,b which describesa possible energy exchange with external particles, the total energy can only de-crease due to the finite resistivity and the chirality-flip rate. The only equilibriumstate with @tUtot

= 0 is given by µ5

= µ5,b and a magnetic energy concentrated

in the mode k = k0

= k5

(µ5,b) with maximal magnetic helicity with the opposite

sign as µ5,b, Hk0 = sign(µ

5,b)8⇡Mk0/k0.The evolution of µ

5,b due to energy exchange with the background matter canbe modeled as follows: In absence of magnetic fields multiplying Eq. (12) withc(T, µe) and using Eq. (8) gives

@tn5

= �2Rf [n5

� c(T, µe)µ5,b] = ±Rwnb � 2Rfn5

, (26)

where the gain term was written as a parity breaking electroweak rate Rw times

25

the number density nb of the background lepton species. This implies

nb = 2c(T, µe)Rf

Rw|µ

5,b| , |µ5,b|T

' 0.1gbRw

Rf, (27)

where the second expression holds for gb non-degenerate relativistic fermionicdegrees of freedom. The energy Ub associated with these background particles isthus given by

Ub

V=

Z µ5,b

0

µ05,bdnb =

Rf

Rwc(T, µe)µ

2

5,b ⇠ 3⇥ 10�3g2bRw

RfT 4 , (28)

where the last expression again holds in the non-degenerate relativistic case. Notethat for µ

5,i ⇠ µ5,b ⇠ (Rw/Rf)T Eq. (28) is of order (Rw/Rf)T 4 whereas U

5

from Eq. (18) is of order (Rw/Rf)2T 4. Both energies vanish in the limit of parityconservation, Rw ! 0, as it should be. In terms of initial equilibrium chiralpotential µ

5,bi and for Rw<⇠ Rf the maximal magnetic energy is then

�UB

V<⇠

Rf

Rwc(T, µe)µ

2

5,bi ⇠ 3⇥ 10�3g2bRw

RfT 4 . (29)

Setting @tUb = �@tU5

to conserve energy and equating @tU5

with the last termin Eq. (25) yields an equation for the evolution of µ

5,b,

@tµ5,b = Rwµ5

µ5,b(µ

5

� µ5,b) . (30)

26

When Ub is included in Utot

the second term in Eq. (25) is absent in the timederivative of U

tot

. The total energy is then dissipated exclusively through resis-tive magnetic field damping. Eq. (30) indicates that µ

5,b typically changes withthe rate Rw. A stationary state is reached if µ

5,b = µ5

and the magnetic fieldconcentrates in k

5

= k5

(µ5,b) with maximal helicity of sign opposite to µ

5,b. Thisrequires magnetic field growth rates larger than the Hubble rate, see Eq. (17).

µ5

is continuously recreated with a rate Rfµ5,b ⇠ 0.1gbRwT , see Eq. (27), anda time-independent or slowly varying µ

5

can be established which is given byEq. (21). This can be the case, for example, in a supernova or a neutron star dueto URCA processes which absorb left-chiral electrons with a rate Rw and turnthem into neutrinos that subsequently escape the star.

Due to Eq. (6) amplification stops and resistive damping sets in when 2⌘k25

t ⇠1, thus

k5

⇠ k05

✓t0

t

◆1/2

, µ5

⇠ µ0

5

✓t0

t

◆1/2

. (31)

27

The Chiral Magnetic Effect in Hot Supernova Cores

The spin flip rate is dominated by the modified URCA rate

The resistivity η=1/(4πσ) is given by the conductivity

Comparing the velocity and chiral magnetic term for a velocity spectrum v(l)~(l/L)n/2 for integral scale L at the length scale of maximal growth l=2π/k5=(π/e)2/|µ5| gives

For vrms ~ 10-2 in a supernova this is ≲ 1 if n ≳ 4/3.

based on Sigl and Leite, JCAP 1601 (2016) 025 [arXiv:1507.04983]

28

1 10 100T (MeV)

10�1210�1110�1010�910�810�710�610�510�410�310�210�1100

�w/�

f

1 10 100T (MeV)

105106107108109

101010111012101310141015101610171018

Bm

ax(G

)

URCA to spin flip rateResulting maximal field in hot neutron star within our formalism

29

Sigl, Leite, arXiv:1507:04983

30

31

The Chiral Magnetic Effect around the Electroweak Transition

based on Pavlovic, Leite, Sigl, JCAP 06 (2016) 044 [arXiv:1602.08419]

We here assume that one starts with a finite 𝞵5 but we neglect possible contributions from a background medium, 𝞵5,b=0, because contrarily to stars there is no “exterior” medium.

Conductivity σ ~ 70 T, chirality flip rates in the symmetric and broken phase; less well known in the symmetric phase

Notation for following plots: 𝞨m=𝞺m/𝞺tot=B2/(8𝞹𝞺tot), H=hV, hmax=𝞺m lc = 𝞺m /kc

32

above electroweak scale (symmetric phase)

take 1-loop Standard Model electroweak results as example for 1st order phase transition

33

below electroweak scale (broken phase)

34

chemical potential 𝞵5

chemical potential 𝞵5

varying uncertain chirality flip rates for initial maximal magnetic helicity

35

chemical potential 𝞵5

36

magnetic energy density

magnetic helicity

37

Main results for magnetic field evolution around electroweak transition:

No significant magnetic field enhancement under realistic conditions; only resistive damping rate is somewhat slowed down; magnetic fields tend to be helical

Open questions:

Turbulence could play a more important role than in hot neutron stars: chiral/turbulent term > 1 only for vrms < 10-5, 𝞵5 > 0.01 T

Role of spatially varying chiral chemical potential.

38

Extension to Turbulent Velocity Fields

In the drag time approximation the Lorentz force induces a velocity

v ⇠ ⌧dj⇥B

p + ⇢= ⌧d

(r⇥B)⇥B

p + ⇢, (32)

where ⌧d is the response or drag time. Assuming Gaussian closure and performingthe relevant Wick contractions following Campanelli this generalizes Eq. (6) to

@tMk = �2⌘e↵

k2Mk +↵+

4⇡k2Hk ,

@tHk = �2⌘e↵

k2Hk + 16⇡↵�Mk , (33)

where

⌘e↵

= ⌘ +4

3⌧dUB/V

p + ⇢

↵± =2⌘e2µ

5

⇡⌥ 1

6

⌧dp + ⇢

Zd ln kk2

Hk

V, (34)

with V the volume of the system, see also Dvornikov and Semikoz.Problem: this only holds for unpolarized magnetic fields, whereas in the limit

of maximally helical fields Eq. (32) vanishes ! Thus Eqs. (33) are only applicableif chiral e↵ect is sub-dominant.

39

Relation to Baryon and Lepton Number

@µJµB(x) = @µJ

µL(x) = nf@µK

µew =

nf

32⇡2

⇣g

2W

↵µ⌫W

↵,µ⌫ � g

02Bµ⌫B

µ⌫⌘,

There is a strong connection between gauge fields with helicity and baryon and lepton number:

This violates B and L separately but conserves B-L.

40

@t⌘B ' �nf↵em

4⇡n�@tH �Re⌘B , Re ' |hl

e|2T/(8⇡) ' 2⇥ 10�13T .

Relation to Baryon and Lepton Number

lepton number damping rate is dominated by electron Yukawa coupling hel

based on Fujita and Kamada, Phys. Rev. D93, 083520 (2016) [arXiv:1602.02109], Phys. Rev. D94, 063501 (2016) [arXiv:1606.08891]

⌘B(T ) ⇠ 16⇡ng↵em

⌘(T )

|hle|2n�(T )T

✓T

T0

◆5

✓H

Hmax

◆B2

0

(T )

lc,0(T )

⇠ 2.4⇥ 10�9

✓H

Hmax

◆✓B

0

10�14 G

◆✓T

163GeV

◆4/3

,

In a stationary situation this gives

where the subscript 0 refers to comoving units and T0 is the CMB temperature today More detailed calculations give correct baryon abundance for B0 ~ 10-17 G, lc,0 ~ 10-3 pc and H ~ +Hmax

Conclusions 1

41

1.) For homogeneous and isotropic two-point correlation functions the evolution of primordial magnetic fields can be efficiently modelled within a Gaussian closure approximation.

2.) Evolution in particular of coherence scale strongly depends on helicity of magnetic fields: inverse cascades for helical fields.

3.) Helical magnetic fields may be connected to baryon and lepton numbers.

Conclusions 2

42

1.) The chiral magnetic effect can lead to growing, helical magnetic fields in the presence of a chiral asymmetry in the lepton sector.

2.) However, spin flip interactions can damp the chiral asymmetry faster than the magnetic field growth rate.

3.) In hot supernova cores the chiral magnetic effect could play a significant role. This is less likely in the early Universe.

4.) Still, for 𝞵5/T > 10-9 one could obtain almost maximally helical field and for B0~10-16 G one obtains right order of magnitude baryon number.

43

Outlook/Open Questions1.) Role of turbulence unclear: If velocity term > chiral term the chiral magnetic effect could be considerably modified. Suppression if magnetic fields transported toward smaller scale ? Enhancement if transported toward larger scales (inverse cascade) ?

2.) Role of fermion mass: strictly speaking in thermodynamic equilibrium one can define 𝞵5 only if m=0 identically which is not the case. Is there a discontinuous change of physics at m=0 ?

3.) Spatially varying chiral potential should be discussed quantitatively