The existence of the isovector neutron-proton pairing -- only in asymmetric nuclear matter -- is an open question in Nuclear Physics. Motivated by the results of 12^C(e,e pN) reaction at Jefferson Lab [see Sci 320, 2008, 1476] we used symmetries of Dirac-Hartree-Fock-Bogolyubov to evaluate possibles constraints in order to estimate neutron-proton gap

1. Transformations of the pairing fields ∆k in
nuclear matter
Alex Quadrosa and Brett Vern Carlsonb
a
Departamento de Astronomia, Observat´orio Nacional-ON, Rio de Janeiro, Brazil
b
Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica-ITA, S˜ao Jos´e dos Campos, Brazil
I - Nuclear pairing II - DHFB formalism
III - Transformations IV - Results

2. I - Nuclear pairing (motivation & main points)
• Pairing of nucleons: The very simple idea is that each nucleon binds with
another one to form a pair.
• Hypothesis: When the nucleus has an even number of nucleons, each one of
them finds a partner.
• Evidence: Recent experimental results show that protons and neutrons have
the tendency to form pairs strongly correlated at short distances Sci 320 (2008) 1476
• Relevance: Although small (≈ 1.5 MeV), the pairing energy contributes
significantly to the stability of nuclei where Z is equal or close to N Phy. Rev. C56
(1997) 3097

3. • Neutrons and protons in Nuclei and Neutron Stars (NS) tend to form pairs
that are strongly correlated at short distances.
• In summary
Particles Pairing Type Nuclear Matter Nuclei
proton-proton ∆pp standard a
symmetric-asymmetric yes
neutron-neutron ∆nn standard a
symmetric-asymmetric yes
neutron-proton ∆np n-p (T=1) b
symmetric- ? yes
a Nucl. Phys. A788 (2007), 316c-321c
b Nucl. Phys. A790 (2007), 588c-592c, Alex Quadros, Msc thesis (2009)

4. • 12
C(e, e pN) reaction at Jefferson Lab†‡
†Results from Science Maganize (july of 2008).
‡
JLab data: incident electron beam 4.627 GeV, current between 5 and 40µA, and target 0.25-nm-thick pure 12
C

5. • Nuclear pairing → Neutron Star is possible too†‡
†See Dr. Sanjay REDDY’s talk at CompStar 2009: The crust of compact stars and beyond (http://nautilus.fis.uc.pt/compstar/)
‡See Achim Schwenk’s talk at The New Physics of Compact Stars, ECT 2005: Superfluidity in neutron stars (http://www.esf.org/index)

6. II - Dirac-Hartree-Fock-Bogolyubov in Nuclear Matter
• Taking the hamiltonian form of the HFB approximation as


hk − µ ¯∆†
k
¯∆k −hTk + µT




ψk
ψTk

 = εkl


ψk
ψTk


hk = α · k + βM + βΣk
, hTk = α · k + βM + βΣTk
• The relativistic description of the ∆k , Σk and hk is done selecting the γ and
τ operators ones
{1γ, γµ
, σµν
, γ5
, γµ
γ5
} ⊗ {1τ , τ}

7. • The general form to self–energy field one is
βΣk
= βΣs0(k) + Σ00(k) + α · ˆkΣv0(k) ⊗1τ + βΣsi (k) + Σ0i (k) + α · ˆkΣvi (k) ⊗τi .
• The general form to pairing field one is
¯∆k
= ¯∆si (k) + β ¯∆0i (k) + α · ˆk ¯∆Ti (k) ⊗ τi .
• The hamiltonian hk expanded in Dirac space is given by
hk = [γ0h00 + γ3h03 + 1γh0s ] ⊗ 1τ + [γ0h0i + γ3h3i + 1γhsi ] ⊗ τi
It‘s called Dirac‘s decomposition of the ¯∆k , Σk and hk

8. III - Transformations
• Here, any combinations of the γ’s and τ’s matrices can represent the basis
vector in decomposition of the hk , ∆k , and Σk . But, in fact, we want to reduce
this representation (it is a very nice work to do)
• In sumary
¯∆k
= ¯∆si (k) + β ¯∆0i (k) + α · ˆk ¯∆Ti (k) ⊗ τi .
hk = [γ0h00 + γ3h03 + 1γh0s ] ⊗ 1τ + [γ0h0i + γ3h3i + 1γhsi ] ⊗ τi
βΣk
= βΣs0(k) + Σ00(k) + α · ˆkΣv0(k) ⊗1τ + βΣsi (k) + Σ0i (k) + α · ˆkΣvi (k) ⊗τi .

9. • We suppose H = ¯Ψk HΨk
¯ψk , ¯ψTk


hk − µ ¯∆†
k
¯∆k −hTk + µT




ψk
ψTk


. : ¯ψk hk ψk = hk , ¯ψk ∆k ψk = ∆k , ¯ψk Σk ψk = Σk
. : ψk → Uψk , ¯ψk → ψ†
k U†
γ0, U = exp [iαΛ]
Λ ∈ {1γ, γµ
, σµν
, γ5
, γµ
γ5
} ⊗ {1τ , τ}
e−iαΛ
hk eiαΛ
= hk + iα [hk , Λ] +
(iα)2
2!
[Λ, [hk , Λ]] + . . .

10. IV - Results
We can reduce the two sets of matrices above to:
• which only transform Hartree-Fock hamiltonian unitarily
Oα ∈ 1τ ⊗ {1γ} ⊕ {τ3} ⊗ {Σ12, γ5γ1, γ5γ2}
• which transform both Hartree-Fock hamiltonian and pairing field unitarily
Oβ ∈ τ3 ⊗ {1γ} ⊕ {1τ } ⊗ {Σ12, γ5γ1, γ5γ2}
In this cases, the self-energy Σk transforms unitarily.

11. • So what comes next?
- Question: Why the existence of the isovector n–p pairing (only in
asymmetric nuclear matter!) is an open question in Nuclear Physics?
- Is it necessary more constrains to perform numerical calculations? B. Funkee
Haas, Ph.D. Thesis (2004)
- First shot: The answer to our question can be on the Oα or Oβ probably.

12. EXTRA SLIDES
(to show after the talk if necessary!)

13. • Nuclear Matter (basic ingredients) (??)
• Perfect fluid (in medium).
• With no geometric, A → ∞, Z = N (symmetric matter) and Z = N
(asymmetric matter).
• Turn off Coulomb interaction.

14. • Dirac-Hartree-Fock-Bogolyubov (DHFB) approximation means
- Nucleons like puntiform particles (both particle-hole).
- Self-consistent solutions.
- Particle-hole (and hole-particle) transformation.
- 2 nucleons (p, n) and 6 mesons (σ, ω, ρ, δ, η, π).
- 2 mean-fields – Σk describes the long–range particle–hole correlations
between the nucleons, while ∆k describes short–range correlation.
- Gorkov propagators – describes the propagations of both particle and holes
in the nuclear medium.

15. Density of lagrangian including pairing terms (HFB approximation)
Lagrangian density (L = L0 + Leff )
L0 = ¯ψ(x) iγµ∂
µ
− M ψ(x) +
1
2
∂
µ
φ(x)∂µφ(x) − m
2
σφ
2
(x)
+
1
2
∂
µ
δ(x)∂µδ(x) − m
2
δδ
2
(x) +
1
2
∂
µ
η(x)∂µη(x) − m
2
ηη
2
(x)
+
1
2
∂
µ
π(x)∂µπ(x) − m
2
ππ
2
(x) +
1
2
m
2
ωVµ(x)V
µ
(x)
+
1
2
m
2
ρρ
µ
(x) · ρ
µ
(x) −
1
4
Gµν · G
µν
−
1
4
Fµν F
µν
Fµν = ∂µVν − ∂ν Vµ, Gµν = ∂µρν − ∂ν ρµ
Leff = ¯ψ (x) [i /∂ − M + µγ0]δ(x − x )ψ(x ) − ¯ψΣ x − x ψ x
+
1
2
¯ψ(x)∆(x − x )ψT (x ) +
1
2
¯ψT (x) ¯∆(x − x )ψ(x ),
Σ (x) = γ0Σ
†
(−x) γ0, ∆ (x) = γ0
¯∆
†
(−x) γ0.
∆ (x) = −B
T
∆
T
(−x) B
−1
, ¯∆ (x) = −B ¯∆
T
(−x) B
∗
.

16. • By definition, the hole wave function is
ψT = B ¯ψT
, ¯ψT = ψT
B†
where ψT
denotes the transpose of the wave function ψ, and the matrix
B = τ2 ⊗ γ5C (B is the operator that transform particle-hole and vice-versa).
The isospin doublet ψT is time-reverse of ψ.
Ψk =


ψk
ψTk

 , ¯Ψk = Ψ†
k


γ0 0
0 γ0


Under an arbitrary transformations, we find
ψk → Uψk ψTk → B(Uγ0U†γ0)T B†ψTk
¯ψk → ¯ψk γ0U†γ0
¯ψTk → ¯ψTk BUT B†
which define an extended transformation
Ψk →


U 0
0 −B(Uγ0U†γ0)T B†

 Ψk , ¯Ψk → ¯Ψk


γ0U†γ0 0
0 BUT B†


In general, ∆k , Σk transforms in the same manner if
γ0U†
γ0 = BUT
B†
, U = B(γ0U†
γ0)T
B†

17. • Transformations of the Σk and ∆k
The set of matrices γ, τ and our combinations which has the property
γ0Λ†
γ0 = Λ, BΛ†
B†
= Λ
is given by
Λα ∈ 1τ ⊗ {Σµν , γµγ5} ⊕ {τi } ⊗ {1γ, γµ}, ¯∆k → U ¯∆k U
Another set of matrices which has the property
γ0Λ†
γ0 = Λ, BΛ†
B†
= −Λ
is given by
Λβ ∈ 1τ ⊗ {1γ, γµ} ⊕ {τi } ⊗ {Σµν , γµγ5}, ¯∆k → U−1 ¯∆k U
In both cases, the self-energy Σk transforms unitarily.

18. • Transformations of the hk
The subset of matrices that commute with the Dirac matrices of the basis of
hk is
O ∈ {1τ , τ3} ⊗ {1γ, Σ12, γ5γ1, γ5γ2}
The subset above can be divided into
Oα ∈ 1τ ⊗ {1γ} ⊕ {τ3} ⊗ {Σ12, γ5γ1, γ5γ2}
which transform
hk → U−1
hk U, ¯∆k → U ¯∆k U.
Also, we have other subset
Oβ ∈ τ3 ⊗ {1γ} ⊕ {1τ } ⊗ {Σ12, γ5γ1, γ5γ2}
which transform
hk → U−1
hk U, ¯∆k → U−1 ¯∆k U.
In both cases, the self-energy Σk as defined in transforms unitarily

19. • Results of ∆pp and ∆n in the nuclear matter (both symmetric and
asymmetric)
∆pp ≈ 2.5MeV, kF ≈ 150 MeV, α = 0
Alex Quadros, Msc thesis (2009)