Joseph Kapusta University of MinnesotaNeutrino Superfluids Joseph Kapusta University of Minnesota...
Transcript of Joseph Kapusta University of MinnesotaNeutrino Superfluids Joseph Kapusta University of Minnesota...
Neutrino Superfluids
Joseph KapustaUniversity of Minnesota
Subal Festschrift!4 December 2004
Superfluids/Superconductors
• Conventional superconductivity of elements• Conventional superconductivity of compounds• Heavy fermion superconductivity• High temperature superconductivity• Superconductivity in double-walled carbon nanotubes• Superfluid He-3• Dilute neutron matter 1S0 superfluidity• Dense neutron matter 3P2-3F2 superfluidity• Color superconductivity in quark matter
Neutrino Superfluids?
LL νν − pairing cannot occur due to repulsive Z0-exchange.(Caldi and Chodos 1999, unpublished)
Neutrino Superfluids?
LL νν − pairing cannot occur due to repulsive Z0-exchange.(Caldi and Chodos 1999, unpublished)
RL νν − pairing might occur due to due to attractive Higgs exchange; right-handed neutrinos, if they exist, do not couple to anything else in the standard model. This assumes that neutrinos are Dirac particles and that they obtain their mass via the usual Higgs mechanism.
+
=Φσ0
02
1v
GeV 246210 == FGvHiggs field
ννσν ννν
+=+Φ=
2.. hmchlhL RcLYukawa 20vhm νν =
( )( )ννννσ
ν2
2
4mhHI −=
Low energy contact interaction is attractive!
Express the neutrino field in Dirac representation as
( )
( )
=+=
−
=−=
R
RR
L
LL
ψψ
νγν
ψψ
νγν
211
21
2
1121
5
5
spinors.component - twoare and where RL ψψ
[ ]LaLbRbRaRaRbLbLaRaLbRbLaI mh
H ψψψψψψψψψψψψσ
ν ++++++ ++−= 24 2
2
Allow for condensation of spin-0 Cooper pairs of the form
DabbR
aL εψψ =
DabbR
aL εψψ =
[ ] abRaLbRbLa
MF DDmhH
Iεψψψψ
σ
ν *2
2
2+= ++
and making the mean-field approximationUsing
In terms of the usual creation and annihilation operators
[ ][ ]
−+−−+−−+
+−−−−−+−=
−
++++
∑),(),(),(),(e),(),(),(),(e
4 2*
2
2
2
pbpbpbpbDpbpbpbpbDm
mhH
ti
tiMFI ε
εν
σ
ν
εp
[ ]),(),(),(),( −−+++= ++∑ pbpbpbpbH freepε
. isn Hamiltonia full The . where 22 MFIfree HHHmp +=+= νε
[ ]22 )()(
),(),(),(),(
εµε νKmE
pcpcpcpcEH
+−=
−−+++=∑ ++
p
The Hamiltonian can be diagonalized to the form
by making the canonical transformation
),(esin),(ecos),(),(esin),(ecos),(
)()(
)()(
+−+−=−
−−−+=++++−
+++−
pbpbpcpbpbpc
titi
titi
εαεα
εαεα
θθ
θθ
)(tan
µεεθ ν
−=
Km
αiDD 2e||=
The gap equation is derived by demanding self-consistencybetween the assumed value of the condensate and the valueobtained via the canonical transformation of creation andannihilation operators. One finds that either D=0 or α=π/2with the magnitude of the gap determined by
∫ =+−
1)()(
1)2(8 222
2
3
3
2
2
εµεεπν
ν
σ
ν
Kmmpd
mh
µνKm=∆In the limit of weak coupling the gap is
.4factor -form eor with th order of
limitupper an with off-cut be couldit divergent; is integral The222
+Λ pmmm σσσ
Put the gap equation in a form similar to that ofconventional condensed matter superconductivity:
∫ =∆+
max
min
1)0(21
22
ξ
ξ ξξdgNGap equation
2
2
2
2
2
2
22)0(
πεεπν
µε
ν FvmmddppN =
=
=
(Relativistic) density of states
µξµξ νν σ−+=Λ=−−= 22
maxmin ,)( mmm
)0(/1maxmin e||2 gN−=∆ ξξSolution
In terms of neutrino parametersthe solution to the gap equations is
[ ]Fvmhmm 22228exp)(2 ννσν πµ −Λ−=∆
Since this is a typical BCS-type theory the critical temperature is
∆≈∆= 57.0eπ
γ
cT
These formulae do not assume any connection betweenthe neutrino mass and the neutrino-Higgs coupling.
Light Neutrinos
( ) ! 10exp)(2
GeV 110 and eV 1For
46Fvm
mm
−Λ−=∆
==
ν
σν
µ
Heavy NonrelativisticNeutrinos
( )( )
2
3
3/1222
20
3/12
3
32
e21
e32
πρ
ρππ
ππξ
ρπ
νννν
νν
σ
ν
ν
ν
ν
ν
F
xFLondon
x
pmnm
mmvmx
mvmm
==
=
Λ=
∆=
Λ=∆ −Gap
London coherence length
Mass density
Cosmology
?
Cosmology
3keV/cm 5 <νρ 3keV/cm 5 <νρ
K 7.2<<cT
Neutron Stars
?
Neutron Star
Choose a reference mass of 10 TeV anda reference energy density of 10 MeV/fm3.
3/83/13
4London
3/43/1
3
TeV 10MeV/fm 1073.4
fm e1071.5
keV eTeV 10MeV/fm 10
2.67
=
×=
=∆
−
−
νν
ν
ν
ρ
ξ
ρ
mx
mx
x
fm 065.0<ξ
Neutron Stars
TeV 10>νm
K 109.3fm 065.0
6×>
>
cTξ
Supernovae
?
Conclusion
• Neutrino superfluidity is a possibility if Dirac neutrinos exist with nonzero mass.
• If the neutrino coupled to a much lighter scalar boson than the Higgs, or if superheavy neutrinos exist, then neutrino superfluidity could conceivably be realized in nature.