l6 Neutrino
Transcript of l6 Neutrino
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12004, Torino Aram Kotzinian
Neutrino Scattering
Neutrino interactions
Neutrino-electron scattering
Neutrino-nucleon quasi-elastic scattering
Neutrino-nucleon deep inelastic scattering
Variables
Charged current
Quark content of nucleons
Sum rules
Neutral current
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22004, Torino Aram Kotzinian
NeutrinoNeutrino--electron scatteringelectron scattering
Tree level Feynman diagrams:
0 Z
eR eR
e
e
W
eR
eR
e
e
p ee ee R R
Eff ective Hamiltonian:
? A? A _ ae g g eG
AV ee F ))1(1()1(2
55 K K R K K R Q Q!
(through a Fierz transf ormation)
? A? A ? A? A _ ae g g eeeG
H AV eeee F
eff )()1()1()1(
2
5555 K K R K K R R K K K K R Q Q
Q Q!
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32004, Torino Aram Kotzinian
Only charged current:
W
QR
eR e
Q
p QR R Q ee
)()(2)()( 2 L ABin E me p p s e Q Q R R !!
22)()( QR Q p pqt !!
)(
)(
)()(
)()(
)()()( LABin
pe p
p pe p y
Q
Q
Q
Q
R
QR
R
QR
)()(2)( 2
22
22
LABin
q
s
dy
ed e
W
W CC
Q Q R
T T
R W }
Inelasticity variable (0<y<1)
243
2
10104.0)( cm
MeV
E sGe F
¹¹ º
¸©©ª
¨v!!
T R QTotal cross-section:
(cross-section proportional to energy!)
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42004, Torino Aram Kotzinian
Only neutral current:
p ee Q Q R R )()(
¼¼½
»
¬¬-
«¹
º
¸©ª
¨
24
2
2
22
22
)1(sinsin2
1)( y
q
s
dy
ed W W
NC UU
T
R W Q
0 Z
QR
)(
e
QR
)(
e
¼¼½
»
¬¬-
« ¹ º ¸©
ª¨
W W
Z
Z F NC y
q
s
dy
ed UU
T R W Q 42
22
22
22
sin)1(sin2
1)(
ee g ee g e g g e R L AV )1()1()( 555 K K K K K K Q Q Q
W AV Lg g g U2sin
2
1)(
2
1!!
W AV g g g U2sin)(2
1!!
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52004, Torino Aram Kotzinian
Only neutral current (total cross-section):
2434
2
2
2
101015.0sin
31sin
21)( cm
M eV
E
se W W F
NC ¹¹ º
¸©©ª
¨!
¼¼½
»
¬¬-
«
¹ º ¸©ª̈ ! R Q UUT R W
2434
2
2
2
101014.0sinsin
2
1
3
1)( cm
M eV
E se W W
F
NC ¹¹ º
¸©©ª
¨!
¼¼½
»
¬¬-
«¹
º
¸©ª
¨ ! R Q UU
T R W
Can obtain value of sin2 UW
f rom neutrino electron
scattering (CHARM II):
0059.00058.02324.0sin 2 ss!W U
p ee Q Q R R )()(
)1(22 ym E ee !5
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62004, Torino Aram Kotzinian
Back to (charged and neutral currents)
Then:
p ee ee R R
W W AV L g g g UU
22
sin2
1
1sin2
1
)11(2
1
!!!W AV g g g U2sin))1(1(
2
1!!
¼¼½
»
¬¬-
«¹
º
¸©ª
¨!
24
2
2
2
1sinsin2
1)( y
sG
dy
ed
W W
F e UUT
R
This cross-section is a consequence of the inter f erence of the
charged and neutral current diagrams.
2434
2
2
2
10109.0sin
3
1sin
2
1)( cm
M eV
E se W W
F
e ¹¹ º
¸©©ª
¨!
¼¼½
»
¬¬-
«¹
º
¸©ª
¨ ! R UUT
R W
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72004, Torino Aram Kotzinian
Neutrino pair production:
Then:
eeee R R p
¼¼½
»
¬¬-
«¹
º
¸©ª
¨!p
4
1sin2
2
1
12)(
2
2
2
W F
ee
see U
T R R W
Contribution f rom both W and Z graphs.
W
e
e
eR
eR Z
e
e
eR
eR
Only neutral current contribution to: Q Q R R p
ee
¼¼½
»
¬¬-
«¹
º ¸
©ª¨ !p
4
1sin2
2
1
12)(
2
2
2
W F sG
ee UT
R R Q Q
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82004, Torino Aram Kotzinian
NeutrinoNeutrino--electron scatteringelectron scattering Summary neutrino electron scattering processes:
p ee Q Q R R
¼½
»¬-
« W W
F sGUU
T 422
2
sin3
41sin2
4
Process Total cross-section
p ee Q Q R R
p ee ee R R
ee R QR Q p
p ee ee R R
eeee R R p
Q Q R R p ee
¼½
»¬-
« W W
F sGUU
T 422
2
sin41sin23
1
4
¼
½
»¬
-
« W W F s
UU
T
422
2
sin
3
41sin2
4
¼½
»¬-
« W W
F sGUU
T 422
2
sin41sin23
1
4
T
sG F
2
¼½»¬
-« W W
F sG UUT 42
2
sin4sin221
12
¼½
»¬-
« W W
F sGUU
T 42
2
sin4sin22
1
12
)()(2 frame LABthein E m s e QR !
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92004, Torino Aram Kotzinian
Neutrino-nucleon quasi-elastic scattering
Quasi-elastic neutrino-nucleon scattering reactions (small q2
):
W
QR
pn
Q
pn p QR
Q
!! n H p M eff ,,
QR Q
p p p
Q Q R R )()(
n p p QR Q
W
QR
n
Q
0 Z
QR
)(
p p
QR
)(
factor formv ctor q F V !)( 2
factor formvector axial q F A )( 2
)(975.0cos ang l eC abbiboC !U
? A ? AnqqG
V 5225 )()()1(2
K K R K K QU
Q Q Q
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102004, Torino Aram Kotzinian
Ne tri - cle asi-elastic scatteri
F r l w e er y e tri s ( E
R <<m
N ):
.73.1)( s!! A A F
? A22
22
)0(3)0(cos
)()( AV C F
ee F F E G
pn !!T
UR W R W
R
2
2
42
101075.9 cm
M e ¹¹ º
¸©©ª
¨v} R
Form f actors introduced since proton, neutron not elementary.
Depend on vector and axial weak charges of the proton and neutron.
Two hypotheses:
- Conservation of Vector Current (CVC):
- Partial conservation of Axial Current (PC AC):
22
2
1/1
)(
)( qqV
V
!
1)0( !V F
22
2
65.1/1
)()(
q
F q F A
A
!
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112004, Torino Aram Kotzinian
Inelastic neutrino-nucleon scattering
Since parity is not conserved in weak interactions, there are
more structure functions for weak processes, like neutrino
scattering, than for electromagnetic processes, like electron
scattering.Again the variables x = Q2 /2 M R and y = R / E can be used.
R Q nucleon p Q
X
Parton model is used to make predictions for deep inelasticneutrino-nucleon scattering.
Neutrino beams from pion and kaon decays, dominated by
muon neutrinos are used to study this process.
Q nucleon p Q
X
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122004, Torino Aram Kotzinian
Weak structure functions
General form for the neutrino-nucleon deep inelastic scatteringcross-section, neglecting lepton masses and corrections of the
order of M / E :
d W R ,R
d xdy! F
2 M E
T 1 y F 2
R N y2 x F 1
R N m y y2
2
¨ª© ¸
º¹ x F 3
R N «
-¬»
½¼
The functions F 1
, F 2
and F 3
are the functions of Q2 and R . In the scaling
limit they are the functions of x only.
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132004, Torino Aram Kotzinian
Scaling behaviour
Compilation of the
data on structure
functions in deepinelastic neutrino
scattering (1983)
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142004, Torino Aram Kotzinian
Neutrino proton CC scattering:
= number of u-quarks in proton between x and x+dx
Some of the quarks are from sea:
For proton (uud):
X p p p dp )()( QR Q
? A´´ ! 00 2)()()(dx xu xudx xu
V
Scattering off quarks:
dx xu )(
)()()( x x xS V
! )()()( x x xS V
!
)()( x xS
! )()( xd xd S
!
? A´´ !!1
0
1
01)()()( dx xd xd dx xd
V
T
R W R W Q Q E mG
dy
qd
dy
qd q F CC CC
22)()(
!!
22
2)()( y
E mG
dy
qd
dy
qd q F CC CC !!
T
R W R W ¡ ¡
Ucos1
2
11 !
d
! E
E ywith
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152004, Torino Aram Kotzinian
Scattering off proton:
? A ? A_ a22
)1()()()()(2)(
y xc xu x s xd x ME G
dxdy
pd F CC
R W Q
? A)()()()(2)(2 xc x s xu xd x x F p !R
? A ? A_ a)()()()()(2)(
2
2
x s xd y xc xu x E
d xd y
pd F !
T R Q
Structure f unctions:
Callan-Gross relationship:
? A)()()()()(3 xc x s xu xd x x x F p !R
)()(2 21 x F x x F !
? A)()()()(2)(2 x s xd xc xu x x F p !R
? A)()()()(2)( x s x xc x x x x F !¢
Neutron (isospin symmetry):
? A)()()()(2)(2 xc x s xd xu x x F n !R
)()()()(2)(3 xc x s xd xu x x xF n !R
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162004, Torino Aram Kotzinian
Scattering off isoscalar target (equal number neutrons and protons):
_ a2
2
)()()(
)(
y xq xq x
ME G
x y
N d F CC
! T
R W Q
c sd uq | c sd uq |
? A)()()(2 xq xq x x F
N
!R
? A)()()()()(3 xc x s xq xq x x x F N !R
)()(2)()()(3 xc x xq xq x x xF N !R
_ a)())(()(
2
2
xq y xq x M E G
d xdy
d F CC
!T
R W Q
Total cross-section:
GeV cmQQG
E N F
CC /1067.0
3
1/)( 238
2v¼½
»¬-« R W Q
GeV cQQG
E N F
£ £
/1034.03
1/)( 238
2
v!¼½»
¬-« !
T W Q
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172004, Torino Aram Kotzinian
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182004, Torino Aram Kotzinian
R ise of mean q2 with energy
Mean q2 was found to be linear function in neutrino (antineutrino) energy.
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192004, Torino Aram Kotzinian
Quark content of nucleons from CC cross-sections
Define:
Experimental values from y distribution of cross-sections yields:
If
.,)(1
0etcd uU ´
03.015.0 s! QQ
Q03.000.0 s!
S 01.016.0 s!
S Q
)(495.0
)(
)(
mea s r ed N
N
r CC
CC
!| R
R
19.03
13
}
! r
r
Q
Q
33.0}! QQQV 08.0}!! QQQ
S S
49.0)(1
2 }!´ QQd x x F ¤
Quarks and antiquarks carry 49% of proton momentum, valencequarks only 33% and sea quarks only 16%.
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202004, Torino Aram Kotzinian
Some details
Note that for right-handed incident anti-neutrinos the term changes
sign. Note also that the term is orthogonal to the asymmetric hadronic
term that is proportional to since q = l ± l¶ and gives zerowhen dotted into
where both signs for the last term appear in the literature.
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212004, Torino Aram Kotzinian
To obtain these expressions we have used
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222004, Torino Aram Kotzinian
Finally we can put the pieces together to obtain the corresponding cross
sections(in the limit )
We recognize this to be similar to the EM result but with replacements
, an extra factor of 4 and the (new) term.
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232004, Torino Aram Kotzinian
We now consider the scaling limit
Substituting in terms of the scaling variables
we find the result
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242004, Torino Aram Kotzinian
For scattering on structureless fermions/antifermions (e.g ., point
particle quarks) we have
Thus measures the difference between quarks and antiquarks.
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252004, Torino Aram Kotzinian
For elastic neutrino scattering from quark and antiquark we have:
and
Working the details out explicitly in terms of the parton momentum and
mass, we find
Thus for pointlike quarks we have