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EMC effect in neutrino DIS
Ian CloetCollaborators
Wolfgang Bentz and Tony Thomas
EMC effect in neutrino DIS – p.1/25
Outline
Nuclear structure functions
Convolution formalism
Nambu−Jona-Lasinio (NJL) modelQuark distributions
Nucleon distributions
ResultsQuark distributions in nucleiEMC effect for neutrino DIS
Conclusion
EMC effect in neutrino DIS – p.2/25
Parton Model Structure Functions
The Isoscalar Parton model expressions
F(ν)JH2 (x) =
∑
q
x[
qJH(x) + qJH(x)]
,
F(ν)JH3 (x) =
∑
q
[
qJH(x) − qJH(x)]
,
F(ν)i (x) ≡ 1
2J + 1
J∑
H=−J
F(ν)JHi (x).
2J + 1 quark distributions
J integer =⇒ 2J + 2 structure functionsJ half-integer =⇒ 2J + 1 structure functions
EMC effect in neutrino DIS – p.3/25
The Calculation
Definition: Nuclear quark distribution functions
qJHA (xA) =
P+
A
∫
dξ−
2πeiP
+ xA ξ−/A
〈A,P,H|ψ(0) γ+ ψ(ξ−)|A,P,H〉.
Using Convolution formalism
qJHA (xA) =
∑
κ,m
∫
dyA
∫
dx δ(xA − yA x) f(JH)κ,m (yA) qκ(x) ,
EMC effect in neutrino DIS – p.4/25
Diagrammatically
P PA-1
p p
EMC effect in neutrino DIS – p.5/25
The NJL Model
Investigate the role of quark degrees of freedom.
Low energy effective theory
⇒G
Lagrangian has same flavour symmetries as QCD:Importantly chiral symmetry and CSB,→ Dynamically generated quark masses,→ Non-zero chiral condensate.
EMC effect in neutrino DIS – p.6/25
The NJL Model
Lagrangian
LNJL = ψ (i 6∂ −m)ψ +G(
ψΓψ)2,
Γ = Dirac, colour, isospin matrices
Using Fierz transformation can decompose LI intosum of qq interaction channels
LI,s = Gs
(
ψ γ5C τ2 βAψ
T)(
ψT C−1 γ5 τ2 βAψ)
,
LI,a = Ga
(
ψ γµC ~ττ2 βAψ
T)(
ψTC−1γµ ~ττ2 βAψ)
.
EMC effect in neutrino DIS – p.7/25
Regularization
Proper-time regularization
1
Xn=
1
(n− 1)!
∫ ∞
0dτ τn−1 e−τ X
−→ 1
(n− 1)!
∫ 1/(ΛIR)2
1/(ΛUV )2dτ τn−1 e−τ X .
IR-cutoff eliminates unphysical thresholds forhadrons decaying into quarks and mesons.→ simulates confinement.
Need this to obtain nuclear matter saturation.W. Bentz, A.W. Thomas, Nucl. Phys. A 696, 138 (2001)
EMC effect in neutrino DIS – p.8/25
The Nucleon in the NJL model
Nucleon is approximated as a quark-diquark boundstate.
We use a relativistic Faddeev approach to describethis bound state.
First diquark - bound state of two quarks:
Solve Bethe-Salpeter equation for diquark.
= +τ K K τ
Here we include scalar and axial-vector diquarks.
EMC effect in neutrino DIS – p.9/25
Nucleon quark distributions
q(x) associated with a Feynman diagram calculation.
p p
k k
p-k
+p p
q q
k k
p-q
q-k
q(x) → X = γ+ δ(x− k+
p+ )
∆q(x) → X = γ+γ5 δ(x− k+
p+ )
EMC effect in neutrino DIS – p.10/25
Quark distributions in the ProtonSpin-independent
uv(x) = f sq/P (x) +
1
2f sq(D)/P (x) +
1
3faq/P (x) +
5
6faq(D)/P (x),
dv(x) =1
2f sq(D)/P (x) +
2
3faq/P (x) +
1
6faq(D)/P (x),
Spin-dependent
∆uv(x) = f sq/P (x) +
1
2f sq(D)/P (x) +
1
3faq/P (x)
+5
6faq(D)/P (x) +
1
2√
3fmq(D)/P (x),
∆ dv(x) =1
2f sq(D)/P (x) +
2
3faq/P (x)
+1
6faq(D)/P (x) − 1
2√
3fmq(D)/P (x),
EMC effect in neutrino DIS – p.11/25
uv(x) and dv(x) distributions
0
0.4
0.8
1.2
1.6xdv(x
)an
dxuv(x
)xdv(x
)an
dxuv(x
)
0 0.2 0.4 0.6 0.8 1xx
Q20 = 0.16 GeV2
Q2 = 5.0 GeV2
MRST (5.0 GeV2)
MRST, Phys. Lett. B 531, 216 (2002).
EMC effect in neutrino DIS – p.12/25
∆uv(x) and ∆dv(x) distributions
−0.2
0
0.2
0.4
0.6
0.8x
∆dv(x
)an
dx
∆uv(x
)x
∆dv(x
)an
dx
∆uv(x
)
0 0.2 0.4 0.6 0.8 1xx
Q20 = 0.16 GeV2
Q2 = 5.0 GeV2
AAC
M. Hirai, S. Kumano and N. Saito, Phys. Rev. D 69, 054021 (2004).
EMC effect in neutrino DIS – p.13/25
NJL Model at Finite Density
Re-calculate diagrams
L = ψ (i 6∂ −M∗− 6V )ψ − (M∗ −m)2
4Gπ+VµV
µ
4Gω+ LI
Equivalent to:Scalar field: via effective massesVector field: via scale transformation
Nuclear Matter (εF = EF + 3V0)
qA (xA) =εFEF
qA0
(
εFEF
xA − V0
EF
)
,
Finite Nuclei (MNκ = MN − 3Vκ)
qA,κ(xA) =MN
MN
qA0,κ
(
MNκ
MNκ
xA − Vκ
MNκ
)
.
EMC effect in neutrino DIS – p.14/25
Nucleon distribution functions
Definition
fκm(yA) =
√2MN
A
∫
d3p
(2π)3
δ(
p3 + εκ −MN yA
)
Ψκm(~p ) γ+ Ψκ m(~p ) ,
Central Potential Dirac eigenfunctions
Ψκm(~p ) = i`
(
Fκ(p) Ωκm(θ, φ)
−Gκ(p) Ω−κm(θ, φ)
)
,
spherical two-spinor has the form
Ωκm(θ, φ) =∑
m`,ms
〈`m` sms|j m〉Y`m`(θ, φ) χsms
,
EMC effect in neutrino DIS – p.15/25
Nucleon distributions: Results
Spin-independent nucleon distribution
fκ,m(yA)=P
k=0,2,...,2j(−1)j−m
√2k+1
“
j j km −m 0
”
(2j+1)√
2k+1MN16π3
R
∞
Λdp p
2√
6(−1)`Fκ(p)Gκ(p)
P
L=k±1(2L+1)PL
“
MN yA−εκ
p
”√(2`+1)(2˜+1)
“
` L ˜0 0 0
”“
L 1 k0 0 0
”
(
˜ s jL 1 k` s j
)
+(−1)j+12 Pk
“
MN yA−εκ
p
”h
Fκ(p)2(2`+1)“
` k `0 0 0
”n
` k `j s j
o
+Gκ(p)2(2˜+1)“
˜ k ˜0 0 0
”n
˜ k ˜j s j
oi
.Infinite nuclear matter
f(yA)= 3
4
“
εFpF
”3»
“
pFεF
”2
−(1−yA)2–
.
EMC effect in neutrino DIS – p.16/25
Nucleon distributions: 12C
0
1
2
3
4
5
6
7N
ucl
eon
Dis
trib
uti
ons
Nu
cleo
nD
istr
ibu
tion
s
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2yAyA
κ = −1
κ = −2
κ = −2, m = 12
κ = −2, m = 32
EMC effect in neutrino DIS – p.17/25
Results: Quark distributions
Putting it all together, an example
uJHA (xA) =
∑
κ,m
[up,κ(x) ⊗ fκm(yA)] +∑
κ,m
[un,κ ⊗ fκm(yA)]
EMC effect in neutrino DIS – p.18/25
Quark distribution in 12C
12C
0
0.2
0.4
0.6
0.8
1
1.2x Aq A
(xA
)x Aq A
(xA
)
0 0.2 0.4 0.6 0.8 1 1.2xAxA
freescalar+ Fermi+ vector
EMC effect in neutrino DIS – p.19/25
The EMC effect
F2 EMC ratio
R2(x) =F2A(x)/A
F2N (x)
F3 EMC ratio
R3(x) =F3A(x)/A
F3N (x)
Ratios equal 1 in non-relativistic and no-mediummodification limit.
EMC effect in neutrino DIS – p.20/25
EMC ratios 12C
12C
Q2 = 5 GeV2
0.6
0.7
0.8
0.9
1
1.1
1.2E
MC
Rat
ios
EM
CR
atio
s
0 0.2 0.4 0.6 0.8 1xx
R(e)2 EMC effect
R(ν)3 EMC effect
12C – J. Gomez et al., Phys. Rev. D 49, 4348 (1994).
EMC effect in neutrino DIS – p.21/25
EMC ratios 16O
16O
Q2 = 5 GeV2
0.6
0.7
0.8
0.9
1
1.1
1.2E
MC
Rat
ios
EM
CR
atio
s
0 0.2 0.4 0.6 0.8 1xx
R(e)2 EMC effect
R(ν)3 EMC effect
12C – J. Gomez et al., Phys. Rev. D 49, 4348 (1994).
EMC effect in neutrino DIS – p.22/25
EMC ratios 28Si
28Si
Q2 = 5 GeV2
0.6
0.7
0.8
0.9
1
1.1
1.2E
MC
Rat
ios
EM
CR
atio
s
0 0.2 0.4 0.6 0.8 1xx
R(e)2 EMC effect
R(ν)3 EMC effect
27Al – J. Gomez et al., Phys. Rev. D 49, 4348 (1994).
EMC effect in neutrino DIS – p.23/25
Nuclear Matter
Nucl. Matt.
Q2 = 10 GeV2
0.6
0.7
0.8
0.9
1
1.1
1.2E
MC
Rat
ios
EM
CR
atio
s
0 0.2 0.4 0.6 0.8 1xx
R(e)2 EMC effect
R(ν)3 EMC effect
I. Sick and D. Day, Phys. Lett. B 274, 16 (1992).
EMC effect in neutrino DIS – p.24/25
Conclusions
Effective chiral quark theories can be used toincorporate quarks into many-body physics.
Calculated nuclear quark distributions where thequarks bind to mean scalar and vector fields.
Reproduced EMC effect.
Determined medium modifications to F3.Compariable with F2, although increased Adependence.
Future Workinclude ρ mean field to calculate N 6= Z.Solve self-consistently for nuclear potentials.incorporate pions.
EMC effect in neutrino DIS – p.25/25
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