Week II:Introduction to

Deep Inelastic Scattering

26 September 2006

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

Experimental proof of Callan-Gross relation

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)?

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

&

Example of parameterisation (parton distributions)

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

Scaling violations at low x (HERA)

Low Q 2 fixed target data HERA data

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!)

Other exciting, “new” physics to be looked at - Low x behaviour at LHC (2007 - )

e.g. Higgs production via gluon-gluon fusion