Chiral phase transition in a magnetic
field
Kazuhiko Kamikado (RIKEN, Nishina center)collaboration with

Takuya Kanazawa (RIKEN, iTHES)

Motivation• Magnetic field order of strong interaction
Magnetars
1010 [T] = 106 [G]Non central heavy ion collision

v~c
B++

1011 [G] ~ eB = 0.1 [GeV2] ~ 5 mπ2Life time of magnetic field is very short
0.1fm ~ 10-24 s• In this talk, however, we investigate properties of thestrongly interacting matter under static magnetic field.
・Magnetism of the QCD vacuum
・Chiral phase transition under strong magnetic fieldv~c

Magnetism of electron gas
Bnet = (1 + )Bext

Landau diamagnetism (Orbit)

Pauli paramagnetism (Spin)

Bext

Bext
~s

Landau, 1930

2spin = 2µB ⇢(✏f )µ2B mkf=3~2 ⇡ 2

tot =orbit =spin +orbit > 0• Competition of spin and orbital angular momentum• Spin part provides larger contributions and Fermi gas
shows paramagnetism.

• Magnetism of QCD vacuum will be determined by thenature of charged quark and mesons.

13spinFree energy and χB ind = B ext + M = (1 + )B ext

Free energy:⌦=PMagnetisation:
Magnetic susceptibility:orP ⇠ P0 +2(eB)2 + O(eB 4 )• Magnetic susceptibility is the second order coeﬃcient of
the free energy.

π64π 22|qB|π32ϵFree quark (vacuum)

where ζ (1,0) (−1, 12 + x) is the derivative of the Hurwitz zeta function with respect to the first argument and where1− 21 x2 .
we have used that ζ(−1, 21 + x) = 24
The vacuum contributions from the quarks can be calculated in the same manner, and one finds [46, 63, 64]% γE 2 &ϵ'(+2Nce Λ
1 1−ϵvac2mq
Vq =(qf B)
Γ(−1 + ϵ) ζ(−1 + ϵ, xf ) − xfx=2f
Andersen and Khan (2011)
2π2|qf B|
2qf Bf&ϵ '%%&%&(22+Nc2(qf B)
Λ
142 (1,0)2=+m+1−8(qB)ζ(−1,x)−2|qB|mfff
qq log xf + O(ϵ) ,16π 22|qf B|
3ϵf(30)1⇠ # (qf B)2 + regular terms✏Eq. (9) [65]:where xf = m2q /2|qf B|. The divergences of the eﬀective
B squarehas Thea divergence (ε->0).
potential are givenTheby Eqs.
(24), (29),termand (30).
'2+ qf2 (divergences that depend on the magnetic field are given
q221−.
B→
B− Ncby22
χ must be renormalised by renormalisation48πofϵ electric12π ϵ••

V1div

•

(32)

fcharge.

+ (qf B)2The remaining divergences in V1 are given by(qB)2+ Nc.
(31)=2ϵ2ϵ96π
24π$
To avoid thecutoﬀ dependence,V div
the= following
1 # 4f44−m+3m−4NN
mc f q .
1σπ264π ϵrenormalisation condition is usually imposedin non-methods.
These divergences perturbative
are removed by wavefunction
renormalization of the (external) gauge field Aµ . This is doneby making the replacement in the tree-level Lagrangian

Normalised pressure
P = (P (T, B)
⇠

(T )
(eB)2
2

These are the same divergences as one encounters in vanishing magnetic field and so the usual renormalization
procedure can be used to eliminate them. This is done

Bonati et.al 2013

P (T, 0))4 (P (0, B)

P (0, 0))ˆ ⇡± = g(2)G⇡± (0)
g(4)
···
12n
12
2 Z 1 480 n2n=0111pp=dp✓◆1X12⇡ 2 0 4 p2 + m2⇡ e p2 +m2⇡
(2) 1(4.11)
showsup in M and grows with T . ⇡(4.21)

ef B|

2

22(4.12)

Thermal part< 0. R1

2 = 1/12 one obtains
Usingdx(x1/2)
4Magnetic
susceptibilityof free pions and quarks
0gn+ O(|ef B| )(4.22)
f B|
2lies n=0
that the charged
pion gas is diamagnetic. It diverges to 1 as m⇡ /T ! 0.
✓◆
12X✓◆|eB|
1
1
4.1 Free
pions
(2)
4X
=·|eB|
G|eB|(n+)
+O
|eB|
±1
⇡
2(2)
4ee quarks
Gq|ef B|(n + ) + O(|ef B| )(4.23)
24
2
f B| · |ef B|
n=02
Since the vacuum part✓does
contribute to the magnetic◆ susceptibility, we onl
Z 1notn=0 (sthe vacuumterm
andconcentrateon the thermal integral for a free quark of charge
2=1/2)Meson(s=0)
ZQuark|eB|
1thermal pressure
into account:(2)2
4=dxG(x)+O|eB|+O|eB|±2Nc :(2)4olor⇡
dx Gq (x) + O(|ef B| )(4.24)
f B|240Z01 Z 11⇣⌘⌘p2 2⇣⌘1pX⇣2|eB|T⇣ f|ef B|T X ⌘22|eB|
p+m+(2n+1)|eB|
p +mq +2|ef B|n(1)P⇡± = G ± 2(1) (1) G dp
1 +eO |eB|4 ⇡ ,= Nc
dp log 4 1 + e(4.13)
2 Pq (1)(1) ↵n=(0)± log
2B|G(1)
G(0)+O(|eB|),(4.25)⇡⇡qqff⇡0| ⇡ {z n=0} 024
n=0| {z }
=01

✓

=01
X1
X

1|eB|(n + )2◆⌘ |e B|↵ G (|e B|n)(4.14) ⌘ |eB|G⇡ ±hence (4.4) is proved.
is proved. f n=0 n q f
n=0
The contribution to ˆ from free charged pions reads
tribution to ˆ from free quarks reads
n =2n,0 andwith
1Z 1
1 ⇣ ef ⌘2 (1)
(1)
⇣⌘
p
ˆ
=G(0)Z±22
ˆq=GTq (0)(4.26)±p +mq +2x
⇡
⇡⇣
⌘
p2 21
Gq3(x) e⌘ Nc 2dp log 1 + e.
(4.15)
12TZZp+m+2x⇡⇡1G⇡± (x)1⌘
dp 1log 1 e.
Nc ⇣ ef ⌘2 101112⇡p2 2p2 2= 2
dp q> 0.
(4.27)=
dp0 p
< 0.
2
22p+m3⇡
ep+m12⇡22
qp + m⇡ e⇡
00+1p + mq e
1

It can be shown thatthat the charged pion gas is diamagnetic. It diverges toe quark gas is paramagnetic. It diverges to +1 as mqThis/T !implies0.
Z 12•–5–4.2

Free quarks

dx G⇡± (x) +P⇡± =|eB|
G⇡± (1) (0) + O(|eB|4 )24

1 as m0
Quarks show paramagnetism (χ > 0), while
pions show theWe omit the vacuum term and concentrate on the thermal integral for a free quad(1)diamagnetism (χ < 0).
withG(0)=G⇡ (x).
ef and color⇡ Nc :–6–±dx±x=0

Z 1• This may be understood as a competition of the orbital and
1|ef B|T X
fPq = N c↵n⇡2spin magnetisation.

n=0⌘ |ef B|1X

⇣dp log 1 + e0–4–p2

p +m2q +2|ef B|n↵n Gq (|ef B|n)

• Let’s see eﬀects of interaction and phase transition.
where ↵n = 2n,0 andn=0

Z 1Analyses on the quark meson model
TGq (x) ⌘ Nc

⇡2⇣

dp log 1 + e0

p2

p +m2q +2x

⌘.

⌘2.1

The Nf = 2 + 1 Quark-Meson model

11 i⇢2 =U(3)tr(⌃⌃)to⇢1 .
The three-flavor
QM⇥model[58, -model59] is [60–64]
the⇥ U(3)The three-flavor QM model[58, 59] is the U(3)U(3) linear
coupled3 linear† 2(2.3)-model [60–64] coupled to3-flavor Quark meson modelquarks:

L=hquarks:"@/ + g8
X

"rotation.8 However,# we include relevant invariants only and neglect higher
in this work,†+ tr[@X⌃@⌃]+U(⇢,⇢)
hi i c a ⇠ ,
µµ
1 2
order
potential is a(2.1)
function of the invariants up to 4th†@/ + ginvariants.
Ta ( aThe+ imesonic+ tr[@µ ⌃@generalhi i c a ⇠ ,5 ⇡a )µ ⌃ ] + U (⇢1 , ⇢2 )order of fields,
(2.1)
a=0

Ta ( a + i 5 ⇡a )L=a=0†⇠ = det ⌃ + det ⌃ ,#Generally, we can compose N independent invariants for the U(N ) ⇥ U(N ) flavor-chirala(2,0) 2U (⇢1 , ⇢2 ) = a⇢1 +⇢1 + a(0,1) ⇢2 ,⇠
=det
⌃
+det
⌃
,
where ⌃ is 3 ⇥ 3 complex matrix field and it is parametrised by the scalar ( a ) and pseudo- 2†scalar (⇡a ) meson multiplet,(1,0)

h(2.4)i†where a(1,0) , a(2,0) and a(0,1) are the free parametersof the model.

83 complex matrix field⇢and1 = tr
where ⌃ is 3 ⇥it ⌃⌃is parametrised by the scalar ( a ) and pseudoX⌃=Ta ( a + i⇡a ) ,(2.2)
2scalar (⇡a ) meson
multiplet,
1a=0⇢2 =8 tr ⌃⌃†X

3

⇢1

where Ta is generator of U(3) group normalized tr[Ta , Tb ] = 12 ab . The term proportional to⌃=T ( –+3 –i⇡ ) ,(2.2)

a ofa U (1) aca is so called the Kobayashi-Maskawa-’t Hooft term which represents the eﬀects
Aa=0
anomaly in the hadron sector. hi are the explicit symmetry breaking terms expressing theeﬀect of quark masses. Below we assume h3 , h8 0 and hi = 0 otherwise, and ca > 0.
1
whereTis
generatorofU(3)groupnormalized
tr[T,T]=aab
2 ab . The term proportional to⇢1 and ⇢2 are invariants under U(3) ⇥ U(3) rotations which is defined as• Σ is 3x3 complex matrix i.e., describes 8 scalar and 8 pseudo
scalar mesons.

•ca is so called the Kobayashi-Maskawa-’t Hooft term which represents the eﬀects of UA (1)†⇢intr[⌃⌃],
1 =theanomaly
hadronsector. hi arethe explicit
symmetrybreaking
terms expressing theρ1 and ρ2h are
invariants
undertheU(3)xU(3)
flavour-chiral(2.3)
1 i†
2eﬀect of quark
masses.
Below
we assume h3 , h8 0 and hi = 0 otherwise, and ca > 0.
⇢2 = tr (⌃⌃)
⇢1 .
3rotation.
⇢1 and ⇢2 are invariants under U(3) ⇥ U(3) rotations which is defined as

Generally, we can compose N independent invariants for the U(N ) ⇥ U(N ) flavor-chiralrotation. However, in this awork, we include relevant invariants only and neglecthigher†⇢=tr[⌃⌃],
1 invariants up to 4thorder invariants. The mesonic potential is a general function of thehA
order of fields,1† 2• C is Kobayashi-Masukawa term which represents theeﬀects of U (1) anomaly.

⇢ = tr (⌃⌃ )2a(2,0) 2(1,0)(0,1)
U (⇢1 , ⇢2 ) = a⇢1 +⇢ +a
⇢2 ,
2 1i

⇢1 .

3(2.4)(2.3)

Inclusion of external magnetic field is achieve by• Generally,
we can compose N independent invariants for the U(N ) ⇥ U(N ) flavor-chiralwhere a(1,0) , a(2,0) and a(0,1) are the free parameters of the model.

rotation. However, in this@µwork,include
relevant
invariants only and neglect higher! Dwe@µ + ieAµ =µorder invariants. The mesonic potential is a general function of the invariants up to 4thorder of fields,

Functional RG
1k k RkBk k k [ ] = Tr
(0,2)
2
RkB + k [ ]Trk k RkF
RkF +C. Wetterich (1993)

(2,0)
[k]

Γk : scale dependent eﬀective actionk=⇤ [] = S[ ]k=0 [IR: quantum

UV: classicalk[]= [ ]• Z AnzatsZ for"eﬀective action (Local potential approximation), , ⇡] =! dx4

d3 x ¯ [ µ Dµ + g(⌃ + i 5 ⇧)]0

• We solve flow equation for U

k+ Uk (⇢1 , ⇢2 )hi i + (Dµ ⌃)2 + (Dµ ⇧)2#

RG equation for Uk
8<k4@k Uk =12⇡ 2 :↵ef (k) = 6Nc

ef B
k2

1eb B X↵eb (k) = 3 2k n=0⇡,k,a, , , 0 ,⌘,⌘1+2r

1X1Xr

g2
22Eu,d = k +4p4⇢124⇢21

2X

coth 2T↵eb (k)
Eb0

ef B
ef Bn✓
12n22kkn=1
eb B
eb B
(2n+1)✓1(2n + 1)
k2
k2

Scale dependent Energiesr⇥ Eb ⇤!

hEf2Tca2rtanh↵ef (k)Skokov 2012;

Efu,d,si9=! 4Nc (ef B = 0)
! 1 (eb B = 0); .

Mitter et.al 20132Ek,⇡
= k 2 + @ ⇢1 U k +3p

24⇢2 @⇢2 Ukp

4⇢1 + 2 24⇢23• Vacuum is determined by @ U = @ U = 0• Pressure is given as P = (U (⇢ , ⇢ ) c )• Meson masses are given as a function of U or derivatives⇢1k=0k=01⇢22k=00

k=0of Uk=0

ex: M⇡2 = @⇢ Uk +1p

24⇢2 @⇢2 Ukca2rp
4⇢1 + 2 24⇢23, etc

Meson screening masses
eB = 14M⇡21400

1400

1200

1200

1000

1000
M [MeV]M [MeV]eB = 0800
600

800
600

400

400

20020000Mu,dMs

050M
MK

100 150 200 250 300 350 400
T [MeV]M
M'MaMM
Mf0Mu,dMs

050M
MK

100 150 200 250 300 350 400
T [MeV]M
M'• Chiral phase transition occurs.
• Lowest Landau level of charged mesons are given
22MLL= Mscr+ 2eB

MaMM
Mf0

Phase diagram
σ mass at varying eB

Tc at finite eB

800

1.16
1.12

600

Tc/Tc,B=0M [MeV]

700

500
400

1.08
1.04

300
200

1050

eB = 02M2

100

150
T [MeV]

6M2
10M2

200

250

300

02

4

68

10eB/M214M2

FRG 3-flavor

FRG 2-flavor• Tc increases with increasing eB. (Magnetic catalyses)

1214

Anti-Magnetic catalyses
2+1 flavor Lattice [G.S. Bali et al. (2012)]

Our results
1.16

Tc/Tc,B=0

1.12
1.08
1.04
1024681012eB/M2JHEP02(2FRG 3-flavor

FRG 2-flavor• As far as I know, no chiral eﬀective model explains the AntiMagnetic catalyses.

14

Pressure
0.012
0.01
P/M40.008
0.006
0.004
0.002
0-0.002

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

eB/M2
T = 0 [MeV]T = 100 [MeV]T = 200 [MeV]T = 300 [MeV]T = 400 [MeV]T = 500 [MeV]• Pressure vs eB for (0 < T < 500 [MeV] ~ 2.5 T )• We fit the pressure with trial function f (eB) = 2 (eB)cusing Gnuplot

2Comparison with Lattice QCD
Lattice results on 2+1 flavor

0.05

Bonati et.al 2013

lattice spacing adopted and for two diﬀerent lattice sizes,
243 × 4 and 243 × 24: the former corresponds to T ≃ 227
0.03
MeV, the latter is our reference T ∼ 0 lattice. The rangeof explored values of b spans the first 5 quanta of magnetic
0.02
field and for each quantum we have determined M on agrid of 16 equally spaced points. Such a grid turns out
0.01
to be fine enough to allow a reliable integration of M
(see Ref. [21] for a discussion of the related systematic
0uncertainties).
The integral of M over each quantum returns the elementary finite diﬀerences a4 (f (b) − f (b − 1)). Such-0.01
are -50
more convenient
the 150whole 200
diﬀerence-200 quantities-150 -100
0 50 than100250∆f (b) = f (b) − f (0), since they can be determined independently of each otherT(one does not need to performthe whole integration from 0 to b) and have therefore
Lattice
FRGthus2-flavor
independent
statistical errors,
allowing to exploit
FRG
3-flavor
standard
fit procedures for uncorrelated data.
The finite diﬀerences corresponding to the data inFig. 1 are reported in Fig. 2. We also report, after proper
rescaling, data obtained on a smaller 163 × 4 lattice andat the same value of the lattice spacing, which show theabsence of significant finite size eﬀects. Assuming thata4 ∆f (b) ≡ c2 b2 + O(b4 ) holds for integer b, then0.005

0.004

a=0.2173 fm, Ls=24a=0.1535 fm, Ls=32
a=0.1249 fm, Ls=400.003

~χ

0.04

0.002

0.001

0100

200

paramagnetism.

(15)

Data in Fig. 2 are well reproduced by such a behavior in
the whole explored range, and a number of diﬀerent tests
have been performed to check the stability of our fit. In

300

400

FIG. 3:
Susceptibility (SI units) as a function of T , fo
diﬀerent values of the lattice spacing.

T ⌘T

Tc

Ls Lt a (fm) T (MeV)103 χ̃102 χ̂
24 4 0.2173(4) 226(5) 2.648(62) 2.887(68)24 6 0.2173(4) 151(3) 0.620(96) 0.67(10)24 8 0.2173(4) 113(2) 0.03(10) 0.03(11)24 10 0.2173(4) 90(2) -0.02(13) -0.02(14)32 4 0.1535(3) 321(6) 4.11(10) 4.48(11)32 6 0.1535(3) 214(4) 2.32(12) 2.53(13)32 8 0.1535(3) 160(3) 0.86(13) 0.94(14)32 10 0.1535(3) 128(2) 0.29(15) 0.32(17)32 12 0.1535(3) 107(2) 0.17(15) 0.19(16)40 4 0.1249(3) 394(8) 4.62(12) 5.04(13)40 6 0.1249(3) 263(5) 2.97(14) 3.24(15)• At low temperature, the matter has diamagnetism.
• Beyond Tc, χ changes the sign and the matter has
a4 (f (b) − f (b − 1)) ≃ c2 (2b − 1) .

T [MeV]Comparison with Lattice QCD
0.05

1400
1200

0.04

1000

0.02

M [MeV]0.03

Meson dominant

0.01

Quark dominant

0800
600
400
2000-0.01-200 -150 -100 -50

05050 100 150 200 250Mu,dMs

TLattice
FRG 3-flavor

0100 150 200 250 300 350 400
T [MeV]M
MKM
M'MaMFRG 2-flavor

T ⌘TTc

T =0• At Hadron phase, charged mesons (especially pion) aredominant• While QGP phase, u,d quarks are dominant.

MMf0

Mean field
0.035
0.030.025
k k k[ ] =0.020.015

1k k RkB
Tr(0,2)
2
RkB + k [ ]Tr

k k RkF
RkF +(2,0)[k0.010.005

MF: no meson loop contributions

0-0.005

0100
FRG 3-flavor

200300T [MeV]

400500MF 3-flavor• If we neglect meson loop contributions (mean field•approximation), the matter is paramagnetic for almost
all region.
The origin of diamagnetism is charged mesons.

]Summary• We solve the 3-flavor quark-meson model under strong
magnetic field with Functional-RG.

• We have calculated magnetisation of the QCD matter atzero chemical potential.

• At the hadron phase, QCD vacuum shows diamagnetism,
due to light charged pions.

• At the QGP phase, the matter shows paramagnetism, dueto almost bare quarks.