The CFL Phase and m(strange): An Effective Field Theory Approach

The CFL Phase and m(strange): An Effective Field Theory Approach Thomas Schaefer North Carolina State

1

µ → ∞: CFL Phase F C L L

Consider Nf = 3 (mi = 0) hqia qjb i = φ (δia δjb − δja δib )

L L

C F R R

R R

hudi = husi = hdsi hrbi = hrgi = hbgi Symmetry breaking pattern:

Rotate left flavor

Compensate by rotating color

SU (3)L × SU (3)R × [SU (3)]C × U (1) → SU (3)C+F

... have to rotate right flavor also !

All quarks and gluons acquire a gap

2

hψL ψL i = −hψR ψR i

The Role of the Strange Quark Mass Include strange quark mass in gap equations main parameter x =

m2s pF ∆

x ∼ 1: transition CFL → 2SC But: Problems turns out to be much more difficult (even if the coupling is weak!) additional scales electric neutrality, gauge invariance many gap parameters, how to find the right ansatz 3

Effective Field Theories quarks, gluons

QCD 2p F HDET 2∆

quasi−particles, holes, gluons CFLChTh p=p

Goldstone bosons Fermi surface

F

4

High Density Effective Theory Quasi-particles (holes)

ε

L=

v

ψv† (iv

p

v

p =µv + l + l

v’

ψ

1 a a · D)ψv − Gµν Gµν + O(1/µ) 4 5




p

Effective lagrangian for ψv+ X




1±α ~ · ~v 2

¶






















ψv± = e

−iµv·x

µ

Λ

particles F

Effective field theory on v-patches




















































holes



































































p E± = −µ ± p~2 + m2 ' −µ ± |~ p|

Mass Terms: Match HDET to QCD

L=

† MM ψR

2µ

†

ψR +

† M ψL

R

†

M ψL 2µ

R

R

+ MM

C † † + 2 (ψR M λa ψL )(ψR M λ a ψL ) µ

R + M

M

R R

L

g

L

L M

g2MM R

L

R

L g

mass corrections to FL parameters µ ˆ, vF and V0++−−

6

M

EFT in the CFL Phase Consider HDET with a CFL gap term ³ ´ ∆n ¡ o ¢ £ ¡ ¢¤ 2 † L = Tr ψL (iv · D)ψL + Tr X † ψL X † ψL − κ Tr X † ψL 2 + (L ↔ R, X ↔ Y ) ψL → LψL C T , X → LXC T , hXi = hY i = 11 Quark loops generate a kinetic term for X, Y ¢ª fπ2 © ¡ † 2 † 2 Tr (X D0 X) + (Y D0 Y ) + ... L=− 2 Integrate out gluons, identify low energy fields (ξ = Σ1/2 ) Σ = XY †

NL = ξ(ψL X † )ξ †

[8]+[1] GBs

[8]+[1] Baryons 7

Effective chiral theory ¡ ¢ ¢ª fπ2 © ¡ † 2 † Tr ∇0 Σ∇0 Σ − vπ Tr ∇i Σ∇i Σ L = 4 ¢ ¡ † µ ¢ ¡ † µ + Tr N iv Dµ N − DTr N v γ5 {Aµ , N } o ¢ ∆n ¡ † µ 2 Tr (N N ) − [Tr (N )] − F Tr N v γ5 [Aµ , N ] + 2 with Dµ N = ∂µ N + i[Vµ , N ] Vµ Aµ

¢ i¡ † † = − ξ∂µ ξ + ξ ∂µ ξ 2 ¢ i ¡ † = − ξ ∂µ Σ ξ 2

2 µ 21 − 8 log 2 fπ2 = 18 2π 2

vπ2 8

1 = 3

1 D=F = 2

Mass Terms: Match HDET to CFLχTh Kinetic term:

† † ψL X L ψL + ψ R X R ψR

¢ 1¡ † † ξXR ξ + ξ XL ξ D0 N = ∂0 N + i[Γ0 , N ], Γ 0 = V0 + 2 ∇0 Σ = ∂0 Σ + iXL Σ − iΣXR vector (axial) potentials Contact term:

† † (ψR M ψL )(ψR M ψL )

ª 3∆2 © 2 L= [Tr(M Σ)] − Tr(M ΣM Σ) 2 4π meson mass terms

9

Phase Structure and Spectrum Phase structure determined by effective potential ¤ ¢ £ fπ2 ¡ † † † 2 Tr XL ΣXR Σ − ATr(M Σ ) − B1 Tr(M Σ ) + . . . V (Σ) = 2 V (Σ0 ) ≡ min Fermion spectrum determined by o n ∆ 2 L = Tr N † iv µ Dµ N + Tr N † γ5 ρA N + Tr (N N ) − [Tr (N )] , 2 ¡

¢

ρV,A

¡

¢

¾ ½ † † 1 M M † † MM = ξ ±ξ ξ ξ 2 2pF 2pF

10

p ξ = Σ0

Phase Structure and Spectrum 20



+

 

  

40

-20



µQ (MeV)

  



  

 

60

0

CFLπ

CFLK

−

 



-10

  

80

CFL

 

0

CFLK

 

10

 

20





-30

 

-40 10

-8

10

-6

10

-4

10

-2

2 10

0

10

20

20

30

40

80

60
















ms /2µ (MeV)

gapless modes? (gCFLK)

meson condensation: CFLK 1/3

µs (crit) ∼

ms (crit) ∼ mu ∆2/3

Figures: Kaplan & Reddy (2002)

4∆ 3

Kryjevski & Sch¨ afer (2005)

11

100

Instabilities Consider meson current Σ(x) = UY (x)ΣK UY (x)†

UY (x) = exp(iφK (x)λ8 )

~ K ∇φ ~ ~ ~ K (eiφK u (−2Iˆ3 + 3Yˆ ) A(x) V(x) = = ∇φ ˆ+ + e−iφK u ˆ− ) 4 Gradient energy fπ2 2 2 ~ K vπ K ~k = ∇φ E= 2 Fermion spectrum 4µs 1 l2 − − ~v · ~K ωl = ∆ + 2∆ 3 4 2

µ E= 2π 2

Z

dl

Z

ˆ ωl Θ(−ωl ) dΩ 12

Ω

j

Stability lost 1




 65

2

¯ ∂ E ¯¯ 2 mV = 2 ¯ ∂ =0

70

75

80

-1

-2

-3

E = Cfh (x)

jk x= a∆

3µs − 4∆ h= a∆

h i 1 (h + x)5/2 Θ(h + x) − (h − x)5/2 Θ(h − x) fh (x) = x2 − x see also: Son & Stephanov cond-mat/0507586, Kryjevski hep-ph/0508180

13

Energy Functional #$

20

40

60

80

100



0.2

-5

!

 





  



"

  



  










-10

0.1  






-15 40

50

60

70

'

30

&%

20

(

10

)

   

  '

&%

-20

-0.1 

'

&%

(

  





¯ 3µs − 4∆ ¯¯ ¯ ∆

= ahcrit

hcrit = −0.067

crit

2 a= 2 2 4 15 cπ vπ

[Figures include baryon current jB = αB /αK jK ]

14

Stability found

 

1


  

2

55

60

70

75

80

-1



2

¯ ∂ E ¯¯ 2 mV = 2 ¯ ∂ =0

65



¯ ∂ E ¯¯ 2 mV = 2 ¯ ∂ 0

-2

-3

E = Cfh (x)

jk x= a∆

15

3µs − 4∆ h= . a∆

85

Notes No net current, meson current canceled by backflow of gapless modes (δE)/(δ∇φ) = 0 Instability related to “chromomagnetic instability” CFL phase: gluons carry SU (3)F quantum numbers Meson current equivalent to a color gauge field P-wave meson condensate continuously connected to LOFF? Additional currents? Higher order corrections to µs |crit = (4 + ahcrit )∆/3 ? 16