Transcript Deep Inelastic Scattering

Electron-nucleon scattering
Rutherford scattering: non relativistic a scatters off a nucleus without
penetrating in it (no spin involved).
Mott scattering: 2 ultra-relativistic point-like fermion scattering off each
others
Most of the figures of this talk are from Henley and Garcia’s book titled “Subatomic Physics”
Some of the slides are from B. Meadow (U of Cincinatti) and from G van der
Steenhoven(NIKHEF/RuG)
Henley & Garcia, Subatomic Physics
R. Hofstadter
Nobel prize 1961
Form factors and charge distributions
F(q2 ) =
r(r)
Henley & Garcia, Subatomic Physics
3
d
ò r r(r )exp(iq× r )
Henley & Garcia, Subatomic Physics
• Minima of cross-section are comparable
to diffraction minima
• These kind of data where used as basis
for establishing
< r 2 > = 0.94 ´ A1/ 3 fm
Elastic e--m Scattering

e--p scattering is like e-m scattering if p is point-like
e-
p1
p3
e-
e-
e-
q
m

p2
q
p4
m
p
p form-factor
p
For e-m scattering the steps to obtain the cross section are
 Use the Feynman’s rule for one helicity state (initial and final) Eq 7.106
 Apply the Casimir trick to take into account all spin configurations. Eq 7.126
 Compute the traces. Eq 7.129
Elastic e--p Scattering

e--p scattering is like e-m scattering if p is point-like
e-
p1
p3
e-
e-
e-
q
m

p2
q
p4
m
p
p form-factor
p
For e-m scattering we obtained:
a function of p1 and p3 but it could also be p1 and q
Eq. 7.129
Proton Form-Factor

e - ’s (or m ’s) can be used to “probe” inside the proton
e-
p1
p3
e-
e-
e-
q
m

p2
q
p4
m
p
p form-factor
p
What do we know about Kmn?
 depend on p2=p and q (with q=p2-p4)
K3 is reserved to neutrino scattering

As a (virtual)  does the probing, we anticipate two form
factors => K1(Q2) and K2(Q2)
Note Q2=-q2>0
Rosenbluth formula

Evaluate the cross-section in the lab frame where
and we neglect m (<< M)

Traditionally, a different definition of K1 and K2 is used.
electric (GE) and magnetic (GM) form factors are used to
obtain the Rosenbluth formula
Strategy to measure nucleon form factors




Scatter electron off a hydrogen target
Count the number of scattered electron of energy E’ at angle q
Change E’ and q at least three times.
Perform a Rosenbluth separation.
Henley & Garcia, Subatomic Physics
Effective function for nucleon Form-Factor
It turns out that GEp, GMp and GMn have the same functional form
(up to a certain Q2)
Dipole function for form factors yields an exponential charge distribution
Deep Inelastic e-p scattering (DIS)
invariant energy of virtual-photon proton system:
E’(scattered electron energy)
W 2 = m 2p + 2m pn - Q 2
For a given E
Elastic
Inelastic
q (electron scattering angle)
In inelastic scattering, the energy (E’)
of the scattered electron is not uniquely
determined by E and q.
DIS cross-section
Start like for the elastic scattering
The cross section is for observing the scattered electron only. Need to
integrate over the complete hadronic systems.
DIS cross-section
Again just like for elastic scattering where
Wmn can be defined in term of Wi
W1 and W2 are functions of q2 and q.p (or Q2 and x)
Bjorken scaling
variable
GE and GM are functions of Q2 only.
ep cross-section summary
Non relativistic and no spin
Ultra relativistic point like fermions
Point like fermion (one light, one heavy)
ep elastic
DIS ep
Looking deep inside the proton

First SLAC experiment (‘69):
 expected from proton form factor:
2
ö
ds / dE ' dW æ
1
÷ µ Q -8
= çç
2
2 ÷
(ds / dW) Mott è (1 + Q / 0.71) ø

First data show big surprise:
 very weak Q2-dependence:
W1,2 (Q2, x) Þ W1,2 (x)
 scattering off point-like objects?

How to proceed:
 Find more suitable variable
 What is the meaning of
s ep / s Mott µ "structure function"
Nobel prize ’90
Friedman, Kendall and Taylor
As often at such a moment….
…. introduce a clever model!
Looking deep inside the proton

With a larger momentum transfer, the
probing wavelength gets smaller and
looks “deeper” inside the proton

Therefore :
Consider the case now where the
Electron scatters on quarks/partons
Particles of spin ½
The Quark-Parton Model

Assumptions (infinite momentum frame):
e’
 Neglect masses and pT’’s
e
 Proton constituent = Parton
 Impulse Approximation:
P
ignore the binding of quarks between each others
Quasi-elastic scattering off partons

parton
Lets assume: pquark = xPproton
2
( xP + q) 2 = p'2quark = mquark
»0

®1
W 2 = M p2 + 2M pn - Q 2 ¾x¾®
¾
M p2
 if |x2P2 |=x2M2 <<q2 it follows:
2
2
Q
Q
2xP × q+ q2 » 0 Þ x =
=
2Pq 2Mn
Definition Bjorken scaling variable
Check limiting case:

Therefore:
x = 1: elastic scattering
and 0 < x < 1
Structure Functions F1, F2


Instead of W1 and W2 use F1 and F2:
ds
2n
æ ds ö 1 é
ù
2
(
)
F1 = MW1 and F2 = nW2 Þ
=ç
F
(
x
)
+
F
(
x
)
tan
q
/
2
÷
2
1
úû
dE ' dW è dW ø M n êë
M
2
2
t
=
Q
/
4
m
Rewrite this in terms of :
quark
(elastic e-q scatt.: 2mqn = Q2 )
2
2
é
ù
4
m
ds
d
s
1
Q
æ
ö
q n
2
F1 ( x) tan (q / 2 )ú =
ç
÷ = ê F2 ( x) + 2
2
2
dE ' dW è dW ø M n êë
4mq Q M
ûú
1é
= F2 ( x) + 2t × 2 xF1 ( x) tan 2 (q / 2)ù
úû
n êë
if F2 ( x) = 2 xF1 ( x)
1
é1 + 2t tan 2 (q / 2)ù
=
F
(
x
)
2
Callan-Gross relation
êë
úû
n
/

Experimental data for 2xF1(x) / F2(x)
→ quarks have spin 1/2 and are point-like
(if bosons: no spin-flip  F1(x) = 0)
Structure Functions F1, F2

From the Callan-Gross relationship:
F1 = MW1 and F2 = nW2 = 2 x F1

Introduce the concept of density function
f i (x) dx
is the number of quark of flavor I that carry a
fractional momentum in the range

Such that :
f i (x)
1
F1 =
2
åQ
2
i
i
fi
Interpretation of F1(x) and F2(x)
In the quark-parton model:
F1 ( x) = å f 12 z 2f [q f ( x) + q f ( x)]
[and F2 = 2xF1 analogously]
Quark momentum distribution
Valence quark vs Sea quark
p = u p u p d p + uu + dd + ss + ...
n = d n d n u n + uu + dd + ss + ...
isospin symmetry
u p = d n = uv
d p = u n = dv
1 ep 1
F2 = [ 4uv + dv ] + S
x
9
1 en 1
F2 = [ uv + 4dv ] + S
x
9
F2N
F2N
Momentum of the proton

Do quark account for the momentum of the proton?
1
1
e u = ò dx x (u + u)
e d = ò dx x (d + d)
0

0
Integrating over F2ep(x) and F2en(x)
4
1
ò dx F (x) = 9 e u + 9 e d = 0.18
ep
2

Therefore:
e u » 0.36
e d » 0.18
eu + eu + eg = 1
eg =
1
ò dx x g » 0.46
0
1
4
ò dx F (x) = 9 e u + 9 e d = 0.12
en
2
Momentum sum rule
Gluons carry about 50% of
the proton’s momentum:
Indirect evidence for gluons.
Quarks in protons & neutrons

If qsp(x) = qsn(x) and x  0:
F2n x[ 19 (d sn + d sn ) + 94 (usn + usn ) + 19 ( ss + ss )]
= 1 p
®1
p
p
p
p
4
1
F2
x[ 9 (d s + d s ) + 9 (us + us ) + 9 ( ss + ss )]

In the limit x  1:
assume isospin symmetry
 assume same high-x tail:
p
p
 assume uv >> dv
F
u + 4d
1
=
Þ
F
d + 4u
4
n
2
p
2
p
v
p
v
p
v
p
v
→ u-quark dominance
Modern data

First data (1980):

“Scaling violations”:
 weak Q2 dependence
 rise at low x
 what physics??
….. QCD
PDG 2002