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LL νν − pairing cannot occur due to repulsive Z0-exchange.(Caldi and Chodos 1999, unpublished)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LL νν − pairing cannot occur due to repulsive Z0-exchange.(Caldi and Chodos 1999, unpublished)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.