week ii: introduction to deep inelastic scattering
TRANSCRIPT
What is DIS ?●Let's start with elastic lepton scattering on a spinless target at E ≈ 10 - 100 MeV
e(E,k)e(E',k')
γ* -q2 = Q2
d d
=d
d point
∣F q ∣2
●F(q) is the Fourier transform of the spatial charge distribution●increasing the Q2, results e- seeingthe proton excited states and a continuum
●At high energies, elastic collisions become very unlikely- elastic scattering form factor decreases rapidly with q2
The regime of Deep Inelastic Scattering
Instead: the proton breaks up into hadron fragments
Probe a proton structure to q2 up to 10000 GeV2 - (i.e down to 10-18 m)
e p → e X
Cross-section for inelastic e(μ)p scattering
Inclusive cross-section (we observe the final lepton andsum over all possible hadronic final states):
E' Electron energy transfer ν = E – E'
q(ν,q)
θE
p
p1
p2
p3
p4
p2q= p4 p2 q=M ,0 , q
q 2M 22 p2 q=W 2 p2 q=M
q2 = -2Mν
Furthermore, E' and θ are related:
q2 = -4EE' sin2θ/2 = -2M(E – E')
Invariant mass W
Only have to measure one of them , e.g. (θ)
This defines your scattering angle, from which you can deduceinformation and calculate the cross-section.
if M = W
d ≃2
Q4LW q , p
starting from our lepton-hadron scattering cross section:
In the regime of DIS, mass of system W is not fixed q2 and ν are not related <=> E' and θ are not related
- have to measure both E' and θ
The two functions W1 and W
2 contain information on the structure of the proton.
Final state contains at least one baryon :
W2 > M2 => – q2 < 2Mν (q2 < 0)
Most general form of DIS cross-section is:
which in the lab frame gives:
d 2dE ' d
LAB
= 2
4 E 2 sin4 2
F 2 , Q 2
cos 2 2
2 F 1 , Q 2M
sin 2 2
d 2dE ' d
LAB
= 2
4 E 2 sin4 2
W 2 , Q 2cos 2 22 W 1 , Q 2sin 2
2
Bjorken scalingFunctions F
1 and F
2 are known as the structure functions
They are not directly related to F.T of charge distribution (at q2 = 0) because of energy transfer ν.
Give info on the number and properties of quark and gluon constituents in the proton
- “quark-parton model”
Cannot (yet) be predicted from first principles (QCD)
- must be measuredIn 1969, Bjorken proposed the following scaling properties:
In the limit Q2 → ∞, W2 → ∞, x is fixed; x= Q2
2 p.qAt fixed x (scaling variable), the scattering is independent of q2 This suggests that the probing “virtual photon” scatters against something pointlike
Cross-section independent of both λ and Q2 - a point is a point irrespective of wavelengthwhereas for sub-sctructure with length scale 1/q
0
- cross-section would depend on q2/q0
2
~2∣q∣
The Quark parton model1969: the point like constituents of the nucleon were termed partons by Feynman – well before quarks and gluons became established
At high Q2 the electron sees pointlike objects called partons and makesat elastic e – parton scattering:
q(ν, p)xp
∑p
=
In the infinite momentun frame p >> Mp, m
q, due to time dilation:
the virtual photon γ* “sees” free partons. This is the limitof asymptotic freedom – very small values of QCD coupling:
DIS ep scattering is the sum of elastic e-parton scatterings
∗−parton≪ parton− parton
Asymptotic freedom & Quark parton model● pertubative QCD approach to hadronic physics applies to inclusive
hard-scattering processes
- based on { Asymptotic freedon (AF)Parton model
1) Hard scattering: at leat one momentum scale Q >> Mhadron
≈ 1 GeV
- At this scale QCD coupling α(Q2) can be sufficiently small
s Q2=
40 ln Q 2/ QCD
2 =
4 11−2/3 N f ln Q
2/ 2
α(Q2)
Q 2
2) Inclusive: Factorization of long and short distance physics σ ~ (f
1 f
2) × σ
hard× d + O(1/p
┴)p), where
d = final-state decay (fragmentation) function; O( p┴)
is a QCD power correction (higher-order )
as afunction of transverse mom.
The scaling variable xBJ
What is the physical meaning of x (xBJ
)?
q
p
e ,μ
zp
zp+q
mi
2 = (q + zp)2 ≈ 0
(zp)2 – Q2 + 2zpq = 0
z = Q2/2pq ≡ x
Defn: scaling variable x ≡ fraction of proton momentum carried by the interacting parton (0 < x < 1).
Can measure quarks' momentum from scattered leptons alone: (E', θ)
Partons were later indentified with quaks (Gell-Mann in 1964): They had a more mathematical model at the time.
Structure functions F1 and F
2
Extracted by comparing with the general expression for cross-section
F 2
=2 z u
2 z d2⋅[ q 2
2 Mx ]2 F 1
M=2 z u
2 z d2⋅[ −q 2
2 M 2 x 2 ]⋅ [ q 2
2 Mx ]Summing over uud quarks in the proton
F 2/ 2 F 1/M
=−2 M 2 x 2
q 2 =Mx
F 2=2 xF 1➔ Callan-Gross relation
F2 =
2xF
1
xm/M 0 1
Elastic scattering (e-q →e-q) cross-section in QPM
d 2dE ' d
LAB
=d 2
dx d Q 2=4 2
xQ4 y 2 xF 1 x 1− y F 2 x
where the variables x and Q 2 are
Q2 = – q2 = – (k – k')2 and x= Q2
2 p.q
y= 2 p.qp.k
inelasticity, is the fraction of lepton energy transferred to the proton in its rest frame
Q2=xys
Can also show that x= mM
, where m and M are parton and protonmasses, respectively
Using the Feynman hypothesis: d 2dx d Q 2=∑
i∫dx f i x
d 2 i
dxdQ 2
d2σi /dxdQ 2 is the cross-section for elastic electron-quark scattering
● Putting it all together we get: F 2=2 xF 1=∑i
e i2 xf i x
What is F1(x) ?
From the data: F2 (x) - 2xF
1 (x) ≡ F
L α σ
L
Where σL
= cross-section for absorption of longitudinally polarised
photons
FL
= longitudinal structure function
Due to helicity conservation; only objects with spin ½ can collidehead on with photons of helicity +1 or -1, objects with spin 0 cannot absorb a photon
- quarks have spin 1/2
What we've learnt thus far about the QPM➢Bjorken scaling:
F1 and F
2 are functions of one variable, not two, because the
underlying scattering is elastic and pointlike
➢ Callan-Gross relation:
However, in reality these structure functions are not delta functions
- must take into account quark-gluon coupling!
x 1/3 1
F2
What is F2(x) (cont'd)?
- can use this to look at proton structure in detail
up(x) = probability that a u quark in a proton has momentum fraction xdp(x) = probability that a d quark in a proton has momentum fraction x
∫0
1
u p x dx=2 and ∫0
1
d p x dx=1, normalisation
F 2ep x =∫
0
1
[z u2 xu p x z d
2 xd p x ]⋅ [ x q 2
2 M ]=49
xu p x 19
xd p x
Similarly neutron structure can be worked out:
F 2e n x =4
9xun x 1
9xd n x
For isospin invariance:(p=uud; n=udd ) &{
F 2ep x =4
9xu p x 1
9xd p x
F 2e n x =4
9xd p x 1
9xu p x {
un x =d p x ≡u x d n x =u p x ≡d x
From now on we drop the suffix and include the sea quark contribution
F 2ep x =x [ 49 u x 1
9d x 4
9u x 1
9d x 1
9s x 1
9s x ... ]
F 2e n x =x [ 49 d x 4
9d x 1
9u x 1
9u x 1
9s x 1
9s x ... ]
Area under F2, neglecting the strange quark sea:
∫0
1
F 2ep x dx=4
9∫01
x u x u x dx19∫0
1
x d x d x dx=49
f u19
f d
∫0
1
F 2e n x dx=4
9∫01
x du x d x dx19∫0
1
x u x u x dx=49
f d19
f u
Where fu(f
n) is the fraction of proton(neutron) momentum carried
by all quarks and anti-quarks
The above momentum sum rule fu = 0.36 and f
n = 0.18
- quarks (anti-quarks) only carry half the momentumfirst indirect evidence of GLUONS!
First estimate on parton densities were done by comparing F2
ep & F2
en
- take separate contributions of the valence and the “sea”
u x =uv x u s x & d x =d v x d s x
All sea components are “equal” → u s x =d s x =u x =d x ≡S x
F 2ep x =x [ 49 u v x
19
d v x 109
S ] F 2e n x =x [ 49 d v x
19
u v x 109
S ]F 2
ep x −F 2e n x =x [ 13 u v−
13
d v ]∫0
1
uv x dx=2 ∫0
1
d v x dx=1& ...normalisation
∫0
11xF ep− F e ndx= 1
3∫0
1
uv−d vdx= 13
∫0
11xF ep− F e ndx= 1
3∫0
1
uv−d vdx= 1N v
Gottfried sum rule
Consistent with experiment
&
What we have learned until now
●Nucleons are very complicated objects●To study them we need to make deep inelastic scattering with leptons
- this probes a distance proportional to 1/Q
●The differential cross-section depends on 2 structure functions F1 and F
2
●From the scaling properties of two structure funcions, the QPM model was developed – i.e. the probe sees putative objects (free pointlike) partons inside the proton.
Scaling Violations
In late 70's: experiments on DIS at CERN and Fermilab showed that,at higher Q2 values, deviations from Bjorken scaling appeared.
F2(x) → F
2(x,Q2)
- F2 grows with Q2 at low x (sea region)
- F2 decreases with Q2at high x (valence region)
Good news because we can apply pertubative QCD - Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations.
- QPM is the 0th order of pertubative expansion
F 2 x , Q 2=∑q
xe 2q x q x , q 2
q x , Q 2= s
2∫
x
1dx '
xq x ' Pq q
xx
' ln Q 2
k 2...
Pqq
is probability that a quark emits a gluon radiating a fraction x/x'
of its momentum, when Q2 changes by dlnQ2
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations
●F2 increases with Q2 at small x as the number of soft gluons
increases – inferred from data
●F2 decreases with Q2 as the valence quarks emit gluons and q(x)
decreases
The proton, according to HERA measurements: It is densely filled with quarks, antiquarks and gluons.
New view of the proton – as a result of scaling violations
- Number of quark – anti-quark pairs is unexpectedly large
F2 increases steeply at low x
At low x, the proton is made of many partons with low momentum
- so far, no parton recombinations have been seen!
Other topics not covered in this series
●Neutrino Deep Inelastic scattering● Charged current (cc) – has only electro-weak contribution
ν(Eν ) l(e,μ)
W
P●Neutral current (NC) – has both electro-weak and electromagnetic contributions
ν(Eν ) l(e,μ)
γ*, Z*
PCan use the above to look for :●new Physics – (squarks, lepto-quarks and study of quark-substructure!)