Chiral four-body interactions in nuclear matter
N. Kaiser
Physik Department T39, Technische Universität München
ECT* workshop: “Three-body forces: from matter to nuclei”, 7. May 2014
Hierarchy of nuclear forces in chiral effective field theory
Leading order chiral 4N-forces in nuclear and neutron matter
∆(1232)-excitation of two nucleons: 3-ring, 2-ring and 1-ring diagrams
Twofold ∆(1232)-excitation of one nucleon: 2-ring and 1-ring diagrams
Exact calculation of 3-body contact-interaction to second order
Publication: N. Kaiser, Eur. Phys. J. A48, 58; 135 (2012)
N. Kaiser
Chiral four-body interactions in nuclear matter
Introduction: Nuclear forces in chiral effective field theory
Chiral EFT: nuclear forces are organized hierarchically
For observables: 2-body force >> 3-body force >> 4-body force ...
2-body interaction: universal NN-potential Vlow−k , chiral N3 LO potential
3-body interaction: leading order chiral 3N-force, contact + 1π + 2π-exch.
Low-momentum 2N and 3N interactions: good results for nuclear and
neutron matter in many-body perturbation theory (→ second order)
Recent work by Darmstadt group (A. Schwenk et al.): sizeable attraction
from subleading 3N-interaction in neutron matter, -10 MeV at ρn =0.2fm−3
Chiral 4N-interaction constructed by Epelbaum via method of unitary
transformations: project πN-dynamics into purely nucleonic subspace
Here: exploratory study of long-range 4N-interaction mediated by
π-exchange in nuclear/neutron matter, include virtual ∆(1232)-isobars
N. Kaiser
Chiral four-body interactions in nuclear matter
Leading order terms related to 4π-vertex
Method of unitary transformations (Epelbaum) gives “induced” 4N-forces
from reducible diagrams, consider first “genuine” 4-body interactions
At leading order: determined by chiral 4π-vertex and NN3π-vertex
1/8
1/2
1/4
2-ring (Hartree) diagrams: integrals over four Fermi spheres factorize
2
7 1
9gA4 mπ
u 2
1
2
u
−
ln(1
+
4u
)
,
−
2u
arctan
2u
+
1
+
(4πfπ )6
2
8u 2
2
4 7
3g m u
1
1
Ēn (ρn ) = − A π 6 u 2 − − 2u arctan 2u + 1 +
ln(1 + 4u 2 )
2
2(4πfπ )
2
8u
Ē (ρ) =
u = kf /mπ for nuclear matter ρ = 2kf3 /3π 2 , u = kn /mπ for neutron matter ρn = kn3 /3π 2
N. Kaiser
Chiral four-body interactions in nuclear matter
Leading order terms related to 4π-vertex
Adding diagrams: only a constant −3mπ2 /fπ2 remains from ππ-interaction
0
Chiral four-body contribution
0.04
nuclear matter
neutron matter
0.02
0
-0.02
-0.04
energy per particle [MeV]
energy per particle [MeV]
0.08
0.06
nuclear matter
neutron matter
-0.1
-0.2
Chiral four-body contribution
-0.3
4π-vertex: 1-ring diagrams
4π-vertex: 2-ring diagrams
-0.06
-0.4
0.1
0.05
0.15
0.2
0.25
-3
density ρ or ρn [fm ]
0.3
0.05
0.1
0.15
0.2
0.3
0.25
-3
density ρ or ρn [fm ]
1-ring Fock diagrams: larger than Hartree contributions (unusual feature)
Ē(ρ) =
3gA4
16fπ6 ρ
Z
|~
pj |<kf
1
1
1
d 12 p
1
2 + (~
2 +~
2 +~
2 +~
(2π)12 mπ
q1 + ~
q2 + ~
q3 )2
q12 mπ
q22 mπ
q32 mπ
h
2
× −8[~
q1 ·(~
q2 ×~
q3 )]2 + mπ
q32 − 4~
q12 (~
q22 + ~
q1 ·~
q3 )
− 4~
q12 ~
+2~
q12 ~
q2 ·(2~
q1 − 5~
q3 ) + 8(~
q1 ·~
q2 )2 + ~
q1 ·~
q3 ~
q2 ·(8~
q1 + 3~
q2 ) − 6~
q1 ·~
q2 ~
q2 ·~
q3
i
In contrast to this: Ēn (ρn ) ∼ mπ2 vanishes in chiral limit
With less than 0.4 MeV for ρ ≤ 0.32 fm−3 this 4N-interaction is negligible
N. Kaiser
Chiral four-body interactions in nuclear matter
Reducible chiral 4N-interactions
Method of unitary transformations generates ”reducible” 4N-forces:
pion-exchanges in combination with a short-range contact-coupling CT
Represent 4N-interaction as a product of 4 vertices and 3 propagators
Vn = −
1
CT2 gA2 i
σ1 (~
σ2 ×~
qb )i τ2b
(~
σ3 ×~
qb )j τ3b σ4j ,
2 +~
fπ2
(mπ
qb2 )2
Vb = −
gA4
1
1
ǫabd τ2d 2
~
σ1 ·~
qa τ1a 2
32fπ6
mπ + ~
mπ + ~
qa2
qb2
bce e
bc
× ǫ τ3 ~
qb ·~
qc + δ ~
σ3 ·(~
qb ×~
qc )
Va =
V l,k =
CT gA2,4
8fπ4
1
c
~
σ4 ·~
qc τ4 ,
2 +~
mπ
qc2
abd d
1
gA6
qa ·~
qb + δ ab ~
σ2 ·(~
qa ×~
qb )
ǫ
τ2 ~
~
σ1 ·~
qa τ1a 2
16fπ6
mπ + ~
qa2
×
bce e
1
1
qb ·~
qc + δ bc ~
σ3 ·(~
qb ×~
qc )
ǫ τ3 ~
~
σ4 ·~
qc τ4c ,
2 +~
2 +~
(mπ
mπ
qb2 )2
qc2
N. Kaiser
Chiral four-body interactions in nuclear matter
~
σ1 ·~
qa τ1a . . .
Reducible chiral 4N-interactions
Zero if N-line is closed to itself: 4-ring and 3-ring diagrams vanish
Evaluate 2-ring and 1-ring diagrams for 5 classes of 4N-interactions
Most integrals over 4 Fermi spheres can be reduced to double-integrals
N. Kaiser
Chiral four-body interactions in nuclear matter
Reducible chiral 4N-interactions
energy per particle: E(ρ) [MeV]
Numerical results for energy per particle
1
reducible 4N-forces
in nuclear matter
0.5
0
-0.5
class IIV
class V
class IV
class II
class I
total sum
-1
-1.5
CT = 0.22 fm
-2
0.05
0.2
0.15
-3
density: ρ [fm ]
0.1
0.25
2
0.3
Cancelations between individual classes of contributions: net effect of
reducible chiral 4N-forces in nuclear/neutron matter less than 1 MeV
Only class I (2-ring) contributes to neutron matter: Ēn (2ρ0 ) = −1.18 MeV
N. Kaiser
Chiral four-body interactions in nuclear matter
Inclusion of virtual ∆(1232)-isobars
Phenomenology: virtual ∆(1232)-excitation of two N’s and π-exchanges
1/2
Combine direct + crossed ∆(1232)-excitation into a 2π-contact vertex
igA2
1
~
~
~
~
ǫ
τ
~
σ
·
(
q
×
q
)
q
·
q
−
δ
c
a
a
abc
b
b
ab
fπ2 ∆
4
√
with mass-splitting ∆ = 293 MeV and coupling const. ratio gπN∆ /gπNN = 3/ 2
Also a leading order 4N-force: if ∆ is small scale comparable to kf , mπ
3-ring diagram is easily calculated
16u 6
20u 2
70u 3
5
− 12u 4 +
+
arctan 2u − 12u 2 +
ln(1 + 4u 2 ) ,
9
3
3
3
9 3 4u 6
5u 2
35u 3
5
gA6 mπ
u
4
2
−u +
+
arctan 2u − u +
ln(1 + 4u 2 )
Ēn (ρn ) =
6
2
(2πfπ ) ∆
27
9
18
36
Ē(ρ) =
9 3
gA6 mπ
u
(2πfπ )6 ∆2
N. Kaiser
Chiral four-body interactions in nuclear matter
Inclusion of virtual ∆(1232)-isobars
Sizeable repulsion from these 4N-processes: reduction by “Fock” terms?
energy per particle [MeV]
70
Chiral four-body contribution
60
∆-excitation on two N: 3-ring diagram
50
40
30
0
20
nuclear matter
neutron matter
1/2
10
0
0.1
0.05
0.15
0.2
0
0.3
0.25
-3
density ρ or ρn [fm ]
2-ring diagrams of upper set: integrals factorize by use of tensors
Ē (ρ) = −
9
gA6 mπ
4(2πfπ
Ēn (ρn ) = −
)6 ∆2
9
gA6 mπ
Z
u
0
24(2πfπ )6 ∆2
h
i
dx 2GS (x)HS (x) + GT (x)HT (x) ,
Z
u
0
h
i
dx GS (x)HS (x) + 2GT (x)HT (x)
GS,T (x), HS,T (x) are auxiliary functions involving arctan(u ± x) and
ln
1+(u+x )2
1+(u−x )2
which arise from Fermi-sphere integral over π-propagators
N. Kaiser
Chiral four-body interactions in nuclear matter
Inclusion of virtual ∆(1232)-isobars
Other 2-ring diagram: integrate over shifted Fermi spheres
Ē (ρ) =
(
Z u Z u
9
gA6 mπ
x 3 (u 2 − y 2 )
2
2 2
(u
−
x)
(2u
+
x)
7xy (1 + 4y )
dx
dy
(2πfπ )6 ∆2 u 3 0
(1 + 4x 2 )2
0
136y 3
2
3
− 7x y − 5 arctan(2x + 2y ) + 5 arctan(2x − 2y )
+16x 10y −
3
)
h
i 1 + 4(x + y )2
1
2
2
2 2
2
2
2
+ (4x − 4y − 1) 7(1 + 4y ) + 8x (14x − 28y − 13) ln
16
1 + 4(x − y )2
2mπ x, 2mπ y are momentum transfers carried by pions,
energy per particle [MeV]
0
Ru
0
dy . . . is still solvable
nuclear matter
neutron matter
-10
-20
-30
Chiral four-body contribution
-40
∆-excitation on two N: 2-ring diagrams
-50
-60
0.05
0.1
0.15
0.2
0.25
0.3
-3
density ρ or ρn [fm ]
Attractive 2-ring diagrams nearly balance strong repulsion from 3-ring
N. Kaiser
Chiral four-body interactions in nuclear matter
Inclusion of virtual ∆(1232)-isobars
1-ring diagrams remain:
1/2
energy per particle [MeV]
1/2
20
Chiral four-body contribution
∆-excitation on two N: 1-ring diagrams
15
10
nuclear matter
neutron matter
5
0
0.05
0.1
0.15
0.2
0.25
0.3
-3
density ρ or ρn [fm ]
Z u
9
i
1h 3
gA6 mπ
8GS (x) + 9GS (x)GT2 (x) + GT3 (x) ,
dx
6
2
3
8(4πfπ ) ∆ u
x
0
Z u Z u (
9
i
h
gA6 mπ
Ē (ρ)Z =
dx
dy 2GS (x)GS (y ) 4xy − L + GS (x)GT (y )
6
2
3
(4πfπ ) ∆ u
0
0
L 3
3x
2
(1 + x ) +
(1 + x 2 )2 − 2 − 6x 2 + 3y 2
+ GT (x)GT (y )
× 5xy −
y
4 y2
)
xy
3
3
3
3 L
4
2
2
×
+
+ 2x − x 3 +
x
−
x
−
5
−
−
1
−
3x
,
8
8y 2x
32 y 2
2x 2
Z
~
d 12 p
1 + (x + y )2
q12 (~
q2 ·~
q3 )2
3g 6
L = ln
,
Ē (ρ)X = 6 A2
2 +~
2 +~
2 +~
fπ ∆ ρ
(2π)12 (mπ
1 + (x − y )2
q12 )(mπ
q22 )(mπ
q32 )
Ē (ρ)U =
|~
pj |<kf
N. Kaiser
Chiral four-body interactions in nuclear matter
Twofold ∆(1232)-excitation on one nucleon
Take into account direct coupling of pion to ∆(1232)-isobar: ∆∆π-vertex
0
~qa , a
~qb , b
~qc , c
1/2
Condense ∆∆π-dynamics into a totally symmetrized 3π-contact vertex
gA3
40fπ3 ∆2
− 75ǫabc ~qa ·(~qb ×~qc ) + ~qa ·~qb ~
σ ·~qc (18δab τc − 7δac τb − 7δbc τa )
+~qa ·~qc ~
σ ·~qb (18δac τb − 7δab τc − 7δbc τa ) + ~
qb ·~qc ~
σ ·~qa (18δbc τa − 7δac τb − 7δab τc )
~ the
using relation Ta Θb Tc† = 13 (5i ǫabc − δab τc + 4δac τb − δbc τa ) with Θ
∆-isospin operator and coupling ratio gπ∆∆ /gπNN = 1/5 of quark model
N. Kaiser
Chiral four-body interactions in nuclear matter
Twofold ∆(1232)-excitation on one nucleon
Corresponding 2-ring and 1-ring diagrams almost cancel each other
4
9
4u
5gA6 mπ
u
1
9
2
2
ln(1
+
4u
)
−
6u
+
2
+
10u
arctan
2u
−
+
3(4πfπ )6 ∆2
3
2
2u 2
4
1
16u
2
2
ln(1 + 4u ) ,
− 16u arctan 2u + 6 +
× 12u − 2 −
2
3
2u
Z u Z u (
9
h
i
5
3gA6 mπ
5
=
GS (x)GS (y ) 4xy − L + GS (x)GT (y )
dx
dy
6
2
3
(4πfπ ) ∆ u
4
8
0
0
3x
L
3
×
(1 + x 2 ) − 5xy +
2 + 6x 2 − 3y 2 − 2 (1 + x 2 )2
y
4
y
)
3x
L 3
3
2 2
2
(1 + x 2 ) +
(1
+
x
)
+
2
−
3x
+ GT (x)GT (y ) xy −
8
y
4 y2
Ē(ρ)2r =
0
6
-1
5
energy per particle [MeV]
energy per particle [MeV]
Ē(ρ)1r
-2
-3
Chiral four-body contribution
-4
∆∆-excitation on one N: 2-ring diagram
-5
nuclear matter
neutron matter
-6
-7
0.05
0.1
0.15
0.2
0.25
0.3
Chiral four-body contribution
∆∆-excitation on one N: 1-ring diagram
4
3
2
nuclear matter
neutron matter
1
0
0.05
0.1
0.15
0.2
0.25
-3
-3
density ρ or ρn [fm ]
density ρ or ρn [fm ]
N. Kaiser
Chiral four-body interactions in nuclear matter
0.3
Total chiral four-body contribution
energy per particle [MeV]
Total result for chiral four-body contribution in nuclear and neutron matter
20
Chiral four-body contributions
summed together
15
10
nuclear matter
neutron matter
5
0
0.05
0.1
0.15
0.2
0.25
0.3
-3
density ρ or ρn [fm ]
At saturation density ρ0 = 0.16 fm−3 moderate amount of Ē = 2.35 MeV,
but increases strongly with density, approximately as ρ3
In neutron matter repulsive four-body contrib. reduced by about factor 2
πN∆-system is strongly coupled with small mass-splitting ∆ = 293 MeV
Schwenk et al. find sizeable attraction from subleading chiral 3N-force
One has to find a new balance between these unexpectedly large terms
N. Kaiser
Chiral four-body interactions in nuclear matter
Three-body contact interaction to second order
Low-momentum NN-interactions (p ≤ 400 MeV, without hard core) show
good convergence properties in many-body perturbation theory
3N-force is essential to achieve saturation of nuclear matter
Beyond leading order so far approximate treatment via effective 2N-force
Exact calculation of simple 3-body contact interaction to second order
1st order contrib. to energy per particle: C3 = cE /fπ4 Λχ
Ē (kf ) = −C3
C3 kf6
3π 2 kf3 3
· 12 = −
= Γ3 kf6
3
2
6π
12π 4
2kf
Density-dependent two-body contact coupling:
δC0 (ρ) = C3
N. Kaiser
kf3
6π 2
·6=
C3 kf3
π2
Chiral four-body interactions in nuclear matter
Three-body contact interaction to second order
Three-particle rescattering in-medium, external momenta |~pj | < kf
p~1
B =
Z
p~2
p~3
h
i
3C3 M
d 3 l1 d 3 l2
1 − θ(kf − |~p + ~l1 − ~l2 /2|)
(2π)6 ~l 2 + 3~l 2 /4 − H/6 − iǫ
1
2
h
ih
i
× 1 − θ(kf − |~p + ~l2 |) 1 − θ(kf − |~p − ~l1 − ~l2 /2|)
Assign intermediate momenta: ~p + ~l1 − ~l2 /2, ~p + ~l2 , ~p − ~l1 − ~l2 /2
with ~p = (~p1 + ~p2 + ~p3 )/3, set |~p |/kf = s < 1
H = (~p1 − ~p2 )2 + (~p1 − ~p3 )2 + (~p2 − ~p3 )2 < 9kf2 is a Galilei invariant
B is real, Pauli-blocking and energy conservation forbids imaginary part
Vacuum term with no θ, 3 equal terms with one θ, 3 equal terms with θθ
Fermi-sphere integral of θθθ-term is zero due to odd energy denominator
N. Kaiser
Chiral four-body interactions in nuclear matter
Three-body contact interaction to second order
Vacuum loop using dim. regularization, 3 → d, prefactor λ6−2d : Γ(1−d)
√
i 3
−H
3C3 M H 2 h 1
,
−
γ
+
ln
4π
+
(1
+
ln
3)
−
ln
E
(12π)3
3−d
2
λ2
√
3 3C3 M kf4 2 1
kf
(ren)
Re B0
=
h
(3 − ln 3) − 2 ln
− ln h + ct(λ)
(4π)3
2
λ
B0 =
Need O(p4 ) counterterm prop. to H 2 , set H = 9kf2 h, constraint s2 +h < 1
q
θ term: Integrate Re 9~l22 − 2H over shifted Fermi sphere |~p + ~l2 | < kf
√
p
(1 + s)2 − 2h h
3 3C3 M kf4
16h2 + h(9s2 − 7s − 16)
θ(...)
Re B1 =
(4π)3
10s
p
i
1 + s + (1 + s)2 − 2h
3
2
√
+(1 + s) (4 − s) − 3h ln
+ (s → −s)
2h
θθ term Re B2 : Integrate hole-hole “bubble” funct. (with logs and arctan)
over Fermi sphere |~l2 − 2~p| < 2kf , lengthy analytical expression in (s, h)
N. Kaiser
Chiral four-body interactions in nuclear matter
Three-body contact interaction to second order
Real part of θθ term with its dependence on the variables s and h
Re B2 =
9C3 Mkf4
(
28
22h
33s4
2
2
2
2
(3
−
s
)
−
+
h(7
−
15s
)
−
3h
−
5
+
10s
−
(4π)4
5
5
5
2
2
5
1
× ln2s − h + + h2 3 +
+ h 15s2 + 7s − 7 −
3
2s
3s
8s
86
33s4
ln |2(1 + s)2 − h|
+ 2s3 − 10s2 −
+5+
+
5
3
45s
h
i
16
+
(2 − 3h − 3s2 )2 A(2 − 3h − 3s2 , 2 + 3s) − A(2 − 3h − 3s2 , 3s)
45s
i
1 h
+
2h(13 + s − 12s2 ) − 21h2 + 2(1 + s)3 (3s − 2)
10s
h
i
× A(3(1 + s)2 − 6h, 3(1 + s)) − A(3(1 + s)2 − 6h, 3s − 1)
)
Z
3h
64 1
2
2
dx x 2(s − x) − h A 3(s − x) −
, 1 − x + (s → −s)
+
3s 0
2
introduce auxiliary function:
√
N
A(Q, N) = 2 Q arctan √ , if Q > 0 ,
Q
N. Kaiser
A(Q, N) =
√
p
|N + −Q|
√
−Q ln
, if Q < 0
|N − −Q|
Chiral four-body interactions in nuclear matter
Three-body contact interaction to second order
Resulting energy per particle at 5-loop order takes the form:
Ē(kf ) = Γ3 2 M kf10
54
k
37π √
3 ln f + ζ0 + ζ1 + ζ2
(11 − 2 ln 2) +
35
175
λ
>0
Numerical values: ζ0 = −1.425, ζ1 = −5.653, ζ2 = −4.354 ± 0.014
Modulo ln(kf /λ)-term, the balance reads: {14.83 − 7.59}
2nd order effective 2-body interaction dominates, but gets reduced to
about half its size by (genuine) 2nd order three-body contribution
“Anomalous” 2nd order 3-body diagram vanishes at zero temperature
N. Kaiser
Chiral four-body interactions in nuclear matter