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Abrikosov Gluon Vortices in Color Superconductors
Efrain J. Ferrer The University of Texas at El Paso
OUTLINE
EJF & de la Incera, PRL 97 (2006) 122301 EJF & de la Incera, PRD 76 (2007) 045011 EJF & de la Incera, PRD 76 (2007) 114012
XII-MWPF 2009.
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Boson: Zero Spin and opposite momenta
Electric Charge
Broken Symmetry : U(1)em
u d
Electric Superconductivity Color Superconductivity
e e
Color Charge
Barrois ‘77; Frautschi ’78; Bailin and Love’84; Alford, Rajagopal and Wilczek ’98; Rapp, Schafer, Shuryak and Velkovsky ‘98
Broken Symmetry : SU(3)C, U(1)em
J. Bardeen, L. N. Cooper and J. R. Schrieffer
L. D. Landau V. L. Ginzburg
Cooper Pair Condensation
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d
s
ud d
u
u
s
s
In-Medium Magnetic Field
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Diameter:
Mass:
Magnetic fields:
Temperature:
pulsar’s surface: B~ 1012–1014G magnetar’s surface: B~ 1015–1016G
Neutron Stars
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Sketch of vs phases of a Color Superconductor with Three-Quark Flavors
EJF & Incera, PRD 76 (2007)045011
* #
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Because of the modified electromagnetism, gluons are charged in the color superconductor
0 0 0 1 -1 1 -1 0
Gluon Mean-Field-Effective Action in the CFL Phase:
Magnetic Effects on the Gluon Sector EJF & de la Incera, PRL 97 (2006) 122301
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where
0 0 0 1 -1 1 -1 0
Effective action for the charged gluons within CFL at asymptotic densities
Charged Gluons Effective Action
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Assuming that there is an external magnetic field in the z-direction, one mode becomes unstable when
with corresponding eigenvector:
“Zero-mode problem” for non-Abelian gauge fields whose solution is the formation of a vortex condensate of charged spin-1 fields.
Nielsen & Olesen NPB 144 (1978)
Skalozub, Sov.JNP23 (1978);ibid 43 (1986)
Ambjorn & Olesen, NPB315 (1989)
EJF & Incera, IJMP 11 (1996)
Magnetic Instability for Charged Spin-1 Fields
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Gibbs Free-Energy:
Minimum Equations:
In the approximation:
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Minimum Equations:
=0
Magnetic Antiscreening
+
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Since we have that which is the required condition for vortex nucleation.
Surface Energy Density and Vortex Nucleation
Condensed Phase
CFL Normal Phase
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H < Hc H ≥ Hc
H < Hc H ≥ Hc
Color Superconductor
Conventional Superconductor vs Color Superconductor
Conventional Superconductor
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B
H Hc
Variation of internal magnetic field (B) with applied magnetic field (H) for Type I, Tipe II and Color Superconductors
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Linearizing the minimum equation for G* around the critical field
where
to find the solution
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The vortex lattice induces a magnetic field that forms a fluxoid along the z-direction. The magnetic flux through each periodicity cell in the vortex lattice is quantized
Vortex lattice, First image 1967
From the experience with conventional type II superconductivity, it is known that the inhomogeneous condensate solutions prefer periodic lattice domains to minimize the energy. Then, putting on periodicity in the y-direction:
The periodicity is also transferred to the x-direction:
Then, the general solution is given by the superposition:
Vortex Solution
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Neutrality Conditions
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• The optimal Cooper pairing occurs when
• Matter inside a star should be electrically neutral to guarantee the star stability and to minimize the system energy
• There are no enough electrons in β equilibrium to balance the charge deficit
Cooper pairing is consequently distorted by the Fermi sphere mismatch.
Cooper Pairing and Neutrality Conditions
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Color Neutrality and beta equilibrium
Unstable Gapped 2SC
a=1,2,3 massless a=4,5,6,7 negative a=8 positive
Gapless 2SC
a=1,2,3 massless a=4,5,6,7 negative a=8 negative
Stable Gapped 2SC
a=1,2,3 massless a=4,5,6,7 positive a=8 positive
Gluons Masses
Huang/Shovkovy, PRD 70 (2004) 051501
Chromomagnetic Instabilities in 2SC
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Some Suggestions
Crystalline Superconductivity Alford, Bowers & Rajagopal, PRD 63 (2001) 074016
Phases with Additional Bose Condensates Bedaque & Schäfer, NPA 697 (2002) 802
Homogeneous Gluon Condensate Gorbar, Hashimoto & Miransky, PLB 632 (2006) 305
Inhomogeneous Gluon Condensate with an Induced Magnetic Field
Ferrer& Incera, PRD 76 (2007) 114012
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EJF & de la Incera, PRD 76 (2007) 114012 Chromomagnetic Instabilities & G-B Condensates in 2SC
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Neutrality Conditions
Stable Phase:
Huang/Shovkovy, PLB 564 (2003) 205
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Effective Action
-
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At Tachyonic Mode of Charged Gluons
µ8
Weakly First-Order Phase transition
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Free-Energy:
Minimum Equations:
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Linear Equations
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Gluon-Condensate Solution
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Condensate Free-Energy
Inhomogeneous Condensate:
Homogeneous Condensate:
0.1ξ
EJF & de la Incera, PRD 76 (2007) 114012
Gorbar/Hashimoto/Miransky, PLB 632 (2006) 305
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Fukushima, PRD 72 (2005) 074002; Casalbouni et al, PLB 605 (2005) 362; Alford/Wang JPG 31 (2005) 719.
Meissner Masses & Chromomagnetic Instabilities in Gapless-Three Flavor Quark Matter
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Origin of Stellar Magnetic Fields
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Conclusions and Future Directions
Magnetism in Color Superconductivity is totally different from magnetism in Conventional Superconductivity
Magnetism is reinforced in Color Superconductivity
Magnetars CS Cores (?)
Numerically solving the nonlinear equation, looking for the realization of the vortex state
Exploring the possibility to induce a magnetic field in a three-flavor system: vortex state in gCFL