Final States in Electron-Nucleus Scattering
Kajetan Niewczas
Institute of Theoretical Physics University of Wrocław
October 24th 2016
Motivation
Ratio of MiniBooNE νµCCQE data versus a RFG simulation as a function of reconstructed muon angle and kinetic energy. The prediction is prior to any CCQE model adjustments. The effective axial
e
−( E
k0, k
0)
e
−( E
k, k )
γ (ω, q )
I ( M
A, 0 )
F ( E
F, p
F)
Electron-nucleus interaction
Initial state:
|Ψ
ii = | k , s i
e⊗ | I i
A(1) Final state:
|Ψ
fi = k
0, s
0e
⊗ | F , p
Fi
A(2)
T-matrix element:
D Ψ
fi T ˆ
Ψ
iE
= Z
d
4q e
2q
2δ
(4)Ω( q + k
0− k ) ¯ u (
k0, s
0) γ
µu (
k, s )
× ( 2 π)δ
T( E
F− M
A+ ω)
F ,
pFZ
V
d
3x e−iq·xJ
µ(
x) I
.
(3)
Initial state:
|Ψ
ii = | k , s i
e⊗ | I i
A(1) Final state:
|Ψ
fi = k
0, s
0e
⊗ | F , p
Fi
A(2)
T-matrix element:
D Ψ
fi T ˆ
Ψ
iE
= Z
d
4q e
2q
2δ
(4)Ω( q + k
0− k ) ¯ u ( k
0, s
0) γ
µu ( k , s )
× ( 2 π)δ
T( E
F− M
A+ ω)
F , p
FZ
V
d
3x e
−iq·xJ
µ( x ) I
.
(3)
Electron-nucleus interaction
Cross section:
d σ
Fd
3k
0d
3p
F= 1 4
1 E
kM
AE
k0E
Fα
2q
4L
µνW
µν(4)
Leptonic tensor:
L
µν≡ 1 2
X
s,s0
¯
u (
k0, s
0) γ
µu (
k, s ) ¯ u (
k, s ) γ
νu (
k0, s
0) (5)
Cross section:
d σ
Fd
3k
0d
3p
F= 1 4
1 E
kM
AE
k0E
Fα
2q
4L
µνW
µν(4)
Leptonic tensor:
L
µν≡ 1 2
X
s,s0
¯
u ( k
0, s
0) γ
µu ( k , s ) ¯ u ( k , s ) γ
νu ( k
0, s
0) (5)
Electron-nucleus interaction
Hadronic tensor:
W
µν= X
σI
F , p
FZ
V
d
3x e
−iq·xJ
µ( x ) I
×
I
Z
V
d
3x e
iq·xJ
ν†( x )
F , p
F× 1
( 2 π)
3V δ( E
F− M
A− ω)
q=k −k0(6)
Inclusive cross section:
d σ = X
F
d σ
F(7)
Hadronic tensor:
W
µν= X
F ,σI
Z d
3pFF ,
pFZ
V
d
3x e−iq·xJ
µ(
x) I
×
I
Z
V
d
3x eiq·xJ
ν†(
x)
F ,
pF× 1
( 2 π)
3V δ( E
F− M
A− ω)
q=k −k0(8)
Electron-nucleus interaction
Inclusive cross section:
d σ = X
F
d σ
F(7)
Hadronic tensor:
W
µν= X
F ,σI
Z d
3p
FF , p
FZ
V
d
3x e
−iq·xJ
µ( x ) I
×
I
Z
V
d
3x e
iq·xJ
ν†( x )
F , p
F× 1
( 2 π)
3V δ( E
F− M
A− ω)
q=k −k0(8)
e − ( E k0, k 0 )
e − ( E k , k )
γ (ω, q )
N ( E p , p )
N ( E p0, p 0 )
IA
One-body current:
J
µ( x ) ≈
A
X
N=p,n
X
σN0
Z d
3p
N0( 2 π)
3√
2E
N0d
3p
N( 2 π)
3√
2E
NN
0, p
N0j
µ( x )
N , p
N× a
†N0( p
N0) a
N( p
N)
τN=τN0(9)
Matrix element:
F ,
pFZ
V
d
3x e−iq·xJ
µ(
x) I
=
= ( 2 π)
3A
X
N=p,n
X
σN0
Z d
3pN0( 2 π)
3√
2E
N0d
3pN( 2 π)
3√
2E
Nδ
V(3)(
pN0−
pN−
q)
×
N
0,
pN0j
µ( 0 )
N ,
pND F ,
pFa
†
N0
(
pN0) a
N(
pN) I
E
(10)
One-body current:
J
µ( x ) ≈
A
X
N=p,n
X
σN0
Z d
3p
N0( 2 π)
3√
2E
N0d
3p
N( 2 π)
3√
2E
NN
0, p
N0j
µ( x )
N , p
N× a
†N0( p
N0) a
N( p
N)
τN=τN0(9)
Matrix element:
F , p
FZ
V
d
3x e
−iq·xJ
µ( x ) I
=
= ( 2 π)
3A
X
N=p,n
X
σN0
Z d
3p
N0( 2 π)
3√
2E
N0d
3p
N( 2 π)
3√
2E
Nδ
V(3)( p
N0− p
N− q )
×
N
0, p
N0j
µ( 0 )
N , p
ND F , p
Fa
†
N0
( p
N0) a
N( p
N) I
E
(10)
IA
Hadronic tensor:
W
µν= X
F ,σI
Z
d
3p
F( 2 π)
3V δ( E
F− M
A− ω)
×
A
X
N=p,n
X
σN0
Z d
3p
N0( 2 π)
3√
2E
N0d
3p
N( 2 π)
3√
2E
Nδ
(3)V( p
N0− p
N− q )
×
N
0, p
N0j
µ( 0 )
N , p
ND F , p
Fa
†
N0
( p
N0) a
N( p
N) I
E
×
A
X
M=p,n
X
σM0
Z d
3p
M0
( 2 π)
3√ 2E
M0d
3p
M( 2 π)
3√
2E
Mδ
(3)V( p
M− p
M0+ q )
× D M , p
Mj
ν†
( 0 ) M
0
, p
M0E D I
a
†
M
( p
M) a
M0( p
M0) F , p
FE
.
(11)
Elementary cross section:
d σ d Ω
k0∼
N
0, p
N0j
µ( 0 )
N , p
N2
(12)
Spectral function:
P ( E ,
p) ∼ D
F ,
pFa
†
N0
(
pN0) a
N(
pN) I
E
2
(13)
Different matrix elements, hence no factorization in IA!
Factorization
Elementary cross section:
d σ d Ω
k0∼
N
0, p
N0j
µ( 0 )
N , p
N2
(12)
Spectral function:
P ( E , p ) ∼ D
F , p
Fa
†
N0
( p
N0) a
N( p
N) I
E
2
(13)
Different matrix elements, hence no factorization in IA!
Elementary cross section:
d σ d Ω
k0∼
N
0, p
N0j
µ( 0 )
N , p
N2
(12)
Spectral function:
P ( E , p ) ∼ D
F , p
Fa
†
N0
( p
N0) a
N( p
N) I
E
2
(13)
Different matrix elements, hence no factorization in IA!
Plane wave IA (PWIA)
Final state factorization:
| F , p
Fi
A→ | X , p
Xi ⊗ | R , p
Ri
A−1(14) The inclusive cross section:
d σ = X
X ,R
d σ
X ,R, (15)
d σ
d Ω
k0dE
k0= 1 8 ( 2 π)
3E
k0E
k1 M
AE
XE
Rα
2q
4L
µνW
µν(16)
Final state factorization:
| F , p
Fi
A→ | X , p
Xi ⊗ | R , p
Ri
A−1(14) The inclusive cross section:
d σ = X
X ,R
d σ
X ,R, (15)
d σ
d Ω
k0dE
k0= 1 8 ( 2 π)
3E
k0E
k1 M
AE
XE
Rα
2q
4L
µνW
µν(16)
Plane wave IA (PWIA)
Final state factorization:
| F , p
Fi
A→ | X , p
Xi ⊗ | R , p
Ri
A−1(14) The inclusive cross section:
d σ = X
X ,R
d σ
X ,R, (15)
d σ
d Ω
k0dE
k0= 1 8 ( 2 π)
3E
k0E
k1 M
AE
XE
Rα
2q
4L
µνW
µν(16)
Hadronic tensor:
W
µν= X
X ,R,σI
Z
d
3p
Xd
3p
R( 2 π)
3V δ( E
F− M
A− ω)
×
A
X
N=p,n
X
σN0
Z d
3p
N0( 2 π)
3√
2E
N0d
3p
N( 2 π)
3√
2E
Nδ
(3)V( p
N0− p
N− q )
×
N
0, p
N0j
µ( 0 )
N , p
ND
X , p
X; R , p
Ra
†
N0
( p
N0) a
N( p
N) I
E
×
A
X
M=p,n
X
σM0
Z d
3p
M0
( 2 π)
3√ 2E
M0d
3p
M( 2 π)
3√
2E
Mδ
(3)V( p
M− p
M0+ q )
× D M , p
Mj
ν†
( 0 ) M
0
, p
M0E D I
a
†
M
( p
M) a
M0( p
M0)
X , p
X; R , p
RE
(17)
PWIA
One-particle states annihilation:
D
X , p
X; R , p
Ra
†
N0
( p
N0) a
N( p
N) I
E D I
a
†
M
( p
M) a
M0( p
M0)
X , p
X; R , p
RE
= ( 2 π)
3p
E
Xδ
(3)( p
X− p
N0)δ
X ,N0h R , p
R| a
N( p
N)| I i
× ( 2 π)
3p
E
Xδ
(3)( p
X− p
M0)δ
X ,M0D I
a
† M
( p
M)
R , p
RE
(18) Identification of N and M [2]:
W
µν= 1 2
X
σX,R A
X
N=p,n
Z
d
3pXd
3pδ( E
F− M
A− ω) δ
(3)(
pX−
p−
q)
× 1
( 2 π)
62E
N|h X ,
pX| j
µ( 0 )| N ,
pi|
2|h R , −
p| a
N(
p)| I i|
2(19)
One-particle states annihilation:
D
X , p
X; R , p
Ra
†
N0
( p
N0) a
N( p
N) I
E D I
a
†
M
( p
M) a
M0( p
M0)
X , p
X; R , p
RE
= ( 2 π)
3p
E
Xδ
(3)( p
X− p
N0)δ
X ,N0h R , p
R| a
N( p
N)| I i
× ( 2 π)
3p
E
Xδ
(3)( p
X− p
M0)δ
X ,M0D I
a
† M
( p
M)
R , p
RE
(18) Identification of N and M [2]:
W
µν= 1 2
X
σX,R A
X
N=p,n
Z
d
3p
Xd
3p δ( E
F− M
A− ω) δ
(3)( p
X− p − q )
× 1
( 2 π)
62E
N|h X , p
X| j
µ( 0 )| N , p i|
2|h R , − p | a
N( p )| I i|
2(19)
PWIA
Spectral function:
P
N( p , E ) = 1 ( 2 π)
62E
NX
R
|h R , − p | a
N( p )| I i|
2δ( E − M + M
A− E
R) (20)
Hadronic tensor:
W
µν= X
σX
A
X
N=p,n
Z
d
3pXd
3pdE PN(
p, E )
× |h X ,
pX| j
µ( 0 )| N ,
pi|
2δ( M − E − E
X+ ω) δ
(3)(
pX−
p−
q),
(21)
Spectral function:
P
N( p , E ) = 1 ( 2 π)
62E
NX
R
|h R , − p | a
N( p )| I i|
2δ( E − M + M
A− E
R) (20)
Hadronic tensor:
W
µν= X
σX
A
X
N=p,n
Z
d
3p
Xd
3pdE P
N( p , E )
× |h X , p
X| j
µ( 0 )| N , p i|
2δ( M − E − E
X+ ω) δ
(3)( p
X− p − q ),
(21)
PWIA
Elementary hadronic tensor:
ω
Nµν≡ 1 2
X
σN0,σN
N
0, p
0j
µ( 0 )
N , p
2
δ( M − E − E
p0+ ω) δ
(3)( p
0− p − q ) (22) Effective energy transfer:
˜
ω ≡ ω − B = ω + M − E − p
p2
+ M
2(23)
˜
q ≡ (˜ ω,
q) (24)
ω
Nµν= 1 2
X
σN0,σN
N
0,
p0j
µ( 0 )
N ,
p2
δ
(4)( p
0− p − ˜ q ) (25)
Elementary hadronic tensor:
ω
Nµν≡ 1 2
X
σN0,σN
N
0, p
0j
µ( 0 )
N , p
2
δ( M − E − E
p0+ ω) δ
(3)( p
0− p − q ) (22) Effective energy transfer:
˜
ω ≡ ω − B = ω + M − E − p
p
2+ M
2(23)
˜
q ≡ (˜ ω, q ) (24)
ω
Nµν= 1 2
X
σN0,σN
N
0,
p0j
µ( 0 )
N ,
p2
δ
(4)( p
0− p − ˜ q ) (25)
PWIA
Elementary hadronic tensor:
ω
Nµν≡ 1 2
X
σN0,σN
N
0, p
0j
µ( 0 )
N , p
2
δ( M − E − E
p0+ ω) δ
(3)( p
0− p − q ) (22) Effective energy transfer:
˜
ω ≡ ω − B = ω + M − E − p
p
2+ M
2(23)
˜
q ≡ (˜ ω, q ) (24)
ω
Nµν= 1 2
X
σN0,σN
N
0, p
0j
µ( 0 )
N , p
2
δ
(4)( p
0− p − ˜ q ) (25)
W
µν= Z
d
3p
0d
3pdE ZP
p( p , E ) ω
µνp+ ( A − Z ) P
n( p , E ) ω
nµν(26)
Integration:
d σ = d Ω
k0dE
k0d
3p0d
3pdE... (27)
P (
p, E ) : δ(...) (28)
ω
µν: δ
(4)(...) (29)
→ d σ = d Ω
k0d
3pdE... (30)
Factorized cross section
W
µν= Z
d
3p
0d
3pdE ZP
p( p , E ) ω
µνp+ ( A − Z ) P
n( p , E ) ω
nµν(26)
Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (27)
P (
p, E ) : δ(...) (28)
ω
µν: δ
(4)(...) (29)
→ d σ = d Ω
k0d
3pdE... (30)
W
µν= Z
d
3p
0d
3pdE ZP
p( p , E ) ω
µνp+ ( A − Z ) P
n( p , E ) ω
nµν(26)
Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (27)
P ( p , E ) : δ(...) (28)
ω
µν: δ
(4)(...) (29)
→ d σ = d Ω
k0d
3pdE... (30)
Factorized cross section
W
µν= Z
d
3p
0d
3pdE ZP
p( p , E ) ω
µνp+ ( A − Z ) P
n( p , E ) ω
nµν(26)
Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (27)
P ( p , E ) : δ(...) (28)
ω
µν: δ
(4)(...) (29)
→ d σ = d Ω
k0d
3pdE ... (30)
d σ d Ω
k0A
= Z
d
3pdE χ
ZP
p( p , E )
d σ
pd Ω
k0+ ( A − Z ) P
n( p , E )
d σ
nd Ω
k0(31) Kinematical factor:
χ = 1 2 ( 2 π)
3ME
pM
A2E
R(32)
Elementary cross section:
d σ
Nd Ω
k0= 1 4
E
k20E
k21 ME
pα
2q
4L
µνω
Nµν(33)
J.A. Caballero et al. [3]
d
5σ
d Ω
k0dE
k0d Ω
p0= 2 α
2q
4E
k0E
k| p
0| MM
RM
Af
rec2 X
|M|
2(34) where
M = j
µeJ
Nµ(35)
j
µe= ¯ u
σk 0
(
k0)γ
µu
σk(
k) (36)
J
Nµ= ¯ u
σp0
(
p0) ˆ J
µΨ
mbb(
p) (37)
Using completeness:
J
Nµ= ¯ u
σp0
(
p0) ˆ J
µu
σp(
p)[¯ u
σp(
p)Ψ
mbb(
p)] (38)
[¯ u
σp(
p)Ψ
mbb(
p)] ∼ α
b(
p) (39)
d
5σ
d Ω
k0dE
k0d Ω
p0= 2 α
2q
4E
k0E
k| p
0| MM
RM
Af
rec2 X
|M|
2(34) where
M = j
µeJ
Nµ(35)
j
µe= ¯ u
σk 0
( k
0)γ
µu
σk( k ) (36)
J
Nµ= ¯ u
σp0
( p
0) ˆ J
µΨ
mbb( p ) (37)
Using completeness:
J
Nµ= ¯ u
σp0
(
p0) ˆ J
µu
σp(
p)[¯ u
σp(
p)Ψ
mbb(
p)] (38)
[¯ u
σp(
p)Ψ
mbb(
p)] ∼ α
b(
p) (39)
J.A. Caballero et al. [3]
d
5σ
d Ω
k0dE
k0d Ω
p0= 2 α
2q
4E
k0E
k| p
0| MM
RM
Af
rec2 X
|M|
2(34) where
M = j
µeJ
Nµ(35)
j
µe= ¯ u
σk 0
( k
0)γ
µu
σk( k ) (36)
J
Nµ= ¯ u
σp0
( p
0) ˆ J
µΨ
mbb( p ) (37)
Using completeness:
J
Nµ= ¯ u
σp0
( p
0) ˆ J
µu
σp( p )[¯ u
σp( p )Ψ
mbb( p )] (38)
[¯ u
σp( p )Ψ
mbb( p )] ∼ α
b( p ) (39)
Initial state factorization (mean-field):
| I i → X
b
Z
d
3p
b(| R
b, − p
bi ⊗ | b , p
bi) α
b( p
b) (40) Matrix element:
h R , −
p| a
N(
p)| I i = X
b
( 2 π)
3p
2E
Nδ
R,Rbδ
N,bα
b(
p) (41) Integration:
d σ = d Ω
k0dE
k0d
3p0d
3pdE... (42)
P (
p, E ) : δ(...) (43)
ω
µν: δ
(4)(...) (44)
→ d σ = d Ω
k0dE
k0d Ω
p0... (45)
Specific nucleon solution
Initial state factorization (mean-field):
| I i → X
b
Z
d
3p
b(| R
b, − p
bi ⊗ | b , p
bi) α
b( p
b) (40) Matrix element:
h R , − p | a
N( p )| I i = X
b
( 2 π)
3p
2E
Nδ
R,Rbδ
N,bα
b( p ) (41) Integration:
d σ = d Ω
k0dE
k0d
3p0d
3pdE... (42)
P (
p, E ) : δ(...) (43)
ω
µν: δ
(4)(...) (44)
→ d σ = d Ω
k0dE
k0d Ω
p0... (45)
Initial state factorization (mean-field):
| I i → X
b
Z
d
3p
b(| R
b, − p
bi ⊗ | b , p
bi) α
b( p
b) (40) Matrix element:
h R , − p | a
N( p )| I i = X
b
( 2 π)
3p
2E
Nδ
R,Rbδ
N,bα
b( p ) (41) Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (42)
P (
p, E ) : δ(...) (43)
ω
µν: δ
(4)(...) (44)
→ d σ = d Ω
k0dE
k0d Ω
p0... (45)
Specific nucleon solution
Initial state factorization (mean-field):
| I i → X
b
Z
d
3p
b(| R
b, − p
bi ⊗ | b , p
bi) α
b( p
b) (40) Matrix element:
h R , − p | a
N( p )| I i = X
b
( 2 π)
3p
2E
Nδ
R,Rbδ
N,bα
b( p ) (41) Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (42)
P ( p , E ) : δ(...) (43)
ω
µν: δ
(4)(...) (44)
→ d σ = d Ω
k0dE
k0d Ω
p0... (45)
Initial state factorization (mean-field):
| I i → X
b
Z
d
3p
b(| R
b, − p
bi ⊗ | b , p
bi) α
b( p
b) (40) Matrix element:
h R , − p | a
N( p )| I i = X
b
( 2 π)
3p
2E
Nδ
R,Rbδ
N,bα
b( p ) (41) Integration:
d σ = d Ω
k0dE
k0d
3p
0d
3pdE ... (42)
P ( p , E ) : δ(...) (43)
ω
µν: δ
(4)(...) (44)
→ d σ = d Ω
k0dE
k0d Ω
p0... (45)
Specific nucleon solution
Cross section:
d σ
bd Ω
k0dE
k0d Ω
p0A
= X
b
χ
d σ d Ω
k0α
2b( p ) (46)
where
χ = 1 ( 2 π)
3E
kE
k0ME
p| p
0|
M
AE
R(47)
Completeness:
X
s
u
α( p , s )¯ u
β( p , s ) − v
α( p , s )¯ v
β( p = δ
αβ(48)
Now
J
Nµ= J
uµ− J
vµ(49) where
J
uµ= ¯ u
σp0
(
p0) ˆ J
µu
σp(
p)[¯ u
σp(
p)Ψ
mbb(
p)] (50) [¯ u
σp(
p)Ψ
mbb(
p)] ∼ α
b(
p) (51) and
J
vµ= ¯ u
σp0
(
p0) ˆ J
µv
σp(
p)[¯ v
σp(
p)Ψ
mbb(
p)] (52)
[¯ v
σp(
p)Ψ
mbb(
p)] ∼ β
b(
p) (53)
J.A. Caballero et al. [3] - Relativistic PWIA (RPWIA)
Completeness:
X
s
u
α( p , s )¯ u
β( p , s ) − v
α( p , s )¯ v
β( p = δ
αβ(48)
Now
J
Nµ= J
uµ− J
vµ(49) where
J
uµ= ¯ u
σp0
(
p0) ˆ J
µu
σp(
p)[¯ u
σp(
p)Ψ
mbb(
p)] (50) [¯ u
σp(
p)Ψ
mbb(
p)] ∼ α
b(
p) (51) and
J
vµ= ¯ u
σp0
(
p0) ˆ J
µv
σp(
p)[¯ v
σp(
p)Ψ
mbb(
p)] (52)
[¯ v
σp(
p)Ψ
mbb(
p)] ∼ β
b(
p) (53)
Completeness:
X
s
u
α( p , s )¯ u
β( p , s ) − v
α( p , s )¯ v
β( p = δ
αβ(48)
Now
J
Nµ= J
uµ− J
vµ(49) where
J
uµ= ¯ u
σp0
( p
0) ˆ J
µu
σp( p )[¯ u
σp( p )Ψ
mbb( p )] (50) [¯ u
σp( p )Ψ
mbb( p )] ∼ α
b( p ) (51) and
J
vµ= ¯ u
σp0
(
p0) ˆ J
µv
σp(
p)[¯ v
σp(
p)Ψ
mbb(
p)] (52)
[¯ v
σp(
p)Ψ
mbb(
p)] ∼ β
b(
p) (53)
J.A. Caballero et al. [3] - Relativistic PWIA (RPWIA)
Completeness:
X
s
u
α( p , s )¯ u
β( p , s ) − v
α( p , s )¯ v
β( p = δ
αβ(48)
Now
J
Nµ= J
uµ− J
vµ(49) where
J
uµ= ¯ u
σp0
( p
0) ˆ J
µu
σp( p )[¯ u
σp( p )Ψ
mbb( p )] (50) [¯ u
σp( p )Ψ
mbb( p )] ∼ α
b( p ) (51) and
J
vµ= ¯ u
σp0
