Lecture 11 – Elastic and inelastic lepton-nucleon scattering

●

Elastic scattering

 Form factors  Evidence for nucleon substructure ●

Deep-inelastic scattering

  Structure functions ”Seeing the quarks” within the proton  Proton and neutron structure functions FK7003 1

Lepton nucleon scattering

Objective is to study a special type of interactions which have traditionally been used to study nucleon lepton-nucleon scattering. structure.

Generally, "clean interactions" in which lepton scat tering angle and energy can be measured/calculated.

e e

q

p

Take 

e p

scattering as our main process. Results/formulae valid with small modifications to 

e n

,  

p

,  

n

..

and (for deep-inelastic scattering)

e p e n

,  

p

,..

 scatterings involving antileptons and antinucleons.

FK7003 2

Rutherford scattering cross section

Rutherford scattering: scattering of an electron off a point-like static proton.

C

 

e

2 4  0 1

r

  

r

(2.27)

e

Born approximation:

d



d

 

m

2 4  2 2 (11.01)

e

q   3 ( )

C

  3

d r

 

r



e

;

q p

' (2.29)

p

 lim   0  3

d r

   

e

 

r r



e

     4 

q

2 (11.02) (in. upp. 1) Scattering off a static potential:

q

2   

p



p

'  2

d



d

 

R

 2

p

2  2

p

2 cos q 

m

2 4  2 

p

2

p

2  

p

' 

p

' 

p

2 cos q ( 11.03) q   4

p

2 sin 2 q 2 (11.04) 2 =

m

2  2 4

p

4 sin 4 q 2 (11.05) FK7003 3

Form factors

elastic 

e p

scattering given by Rutherford scattering:

d



d

   

R m

2  2 4

p

4 sin 4 q 2 (11.05) N on-point like

p p

charge densit y

e



e

:

V

 

d



d



d



d



R G E

(11.06)

G E

 

V

 

r

=charge density.

 electric form factor (11.07) Valid for: electron energy << proton mass .

The for m facto r can be measured and shows how much the cross section changes due to the structure

p e r

2 = 

V r

2  ( )  6

dG E dq

2 |

q

2  0 (11.08) (inl.upp.)  Measurement of a form factor allows the measurement of t he of

p

Formul a relevant for re lativistic elastic scattering.

FK7003

p p

 1 4

Qualitative but valid interpretation of what is going on

The photon interacts with the proton. The form factor is just an expression of our ignorance regarding the content of the proton.

The form factor

G E

depends on

q

2

e

(or wavelength of the probing photon  1 ).

q

Each interaction will give a different

q

2 .

We can measure

G E



d



d



d



d

 observed  point-like(calculated) For increasing

q

2 clearly depends on th e structure of the proton. Scattering If the proton has no structure

G E

 1 = constant!

FK7003

p e

q 5

Charge distributions and form factors

Form factor 1 1 1 1 1

e

  12

C



e

  12

C

Form factor measurements reveal size and charge distribution in probed object .

As pro be

q

    ) the form factor  1 (probed object appea rs point-like) FK7003 6

Relativistic treatment

Still consider elastic scattering

ep



ep

Electron energies > proton mass use

Q

2 instead of

q

2 .

Proton recoil effects can't be ignored.

Q

2   

P e



P e

'  2 (11.09) (4-vectors)

e (P e ) e (P e ’)

q

q

"Resolving" power of the photon pro be.

Wavelength  1

Q

(11.10)

Q

 1 GeV    1  10  15 fm (11.11)

Q

 Sensitive to structure within the proton.

FK7003 7

Rosenbluth-Formula

Relativistic and spin effects taken into account.

G E G M

     Electric form factor (internal charge distribution) = Magnetic form factor (dipole moment)

d



d

 

G

1 2  2

E

' 4

E

2 sin 4   

G E

q

E



G

1 2 2     1  

G M

cos 2 q 2  2 

G

2 2   ;

G

2 2 

G M

sin 2 q 2 ;  

Q

2 4

m p

(11.12) Rosenbluth formula.

Form factors normalised to:

G E

 1 ;

G M

 

P

(proton magnetic moment, units of FK7003

e

2

m p

) (11.13) 8

Form factors from elastic electron-proton scattering

Size of the proton: Using (10.43) :

r E

 0.85

fm (10.49)  proton "radius".

point-like proton Measurements of Fitted to

G E



G M



p E M

     1  1

Q

2 0.84

2     (11.14) So-called dipole fit.

Elastic scattering experiments reveal t

Q 2 /GeV 2

hat the proton is not a point-like structureless object.

It remains to show that scattering experiments reveal quark structure.

 Deep-inelastic scattering needed!

FK7003 9

Question

An experiment measures an elastic reaction between an electron and a proton. Of course it measures the energies and angles of the particles after the collision.

Sometimes its better to think about the collision with variables other than energies and angles. Show that the variable

Q

2 can be used to characterise the reaction i.e.

determine the scattering angle of the electron and the proton and their energies after scattering. The incoming electron and proton energies are known. Assume that rest masses can be neglected.

e e -

q Elastic  q

p p

electron and proton before and after collision. Work in the centre-of mass, electron and proton go off back-to-back. Energies are unchanged. Any energies, angles in the c.o.m are easily transformed to any other frame, eg lab frame.

P e



P p



P e

' 

P p

' 

P e



P e

' 

P p

' 

P p

 

P e



P e

'

Q

2 2

P p

'   

P e



P p

 

P e

'  2 2 

E E p

'

p

   

p p P p

'  

p

'

p P p

   2   2

E

2

p

 

m

2

p

2

p

2

p



m

2

p

cos q  2

P p

'  

P p

2

E p

2   

P p

' 

P p

 2  2

P p

' 

P p

q  .



Q

2 gives proton (and  electron) scattering angle. FK7003 10

Strategy for understanding deep-inelastic scattering

Aim is to test the hypothesis that objects exist in the proton (partons) and that they are quasi-free, point-like charged particles with charges expected by our quark model. Outline how a parton model o f proton structure is needed to describe DIS data.

(1) Define deep-inelastic scattering and choose a reference frame.

Need for two variables, not just

Q

2 to characterise structure.

(2) Structure func tion - characterises our ignorance about proton structure in DIS as form factors do in elastic scattering.

Structure function show point-like quasi-free quarks in the proton.

(3) Show that the quark picture of the proton (

uud

) and neutron (

udd

) is confirmed by direct scattering measurements. FK7003 11

(a)

Deep-inelastic scattering

(b)

Proton breaks up (  inelastic) eg

e



e

   (a) hard interaction between photon and a parton Photon penetrates deeply to probe proton structure:   0   1

Q

15 <

had

  int .

FK7003 12

Deep-inelastic scattering

Mainly focus on

e



e

 

X

and consider photon exchange Proceeds via electromagnetic and weak neutral current processes.

Weak contribution is negligible unless

Q

2

e e e -

g + 

M Z

2 .

Ignore here.

e Z 0

Weak charged current DIS processes also possible: Eg

e

 

e p



e e



X



X

Charged current interaction gives sensitivity to charge of quark struck by exchanged boson (räkneövning).

e -

FK7003 

e W -

13

Fast proton frame of reference

Physics process is invariant to a change of inertial frame. It helps, however, to use a frame in which the interpretation is most straightforward to make.

Partons within a proton interact with each other.

At any given time the partons in a virtual state with lifetime 

v

 0 Fast proton moving along

z

 axis.

v



c

State is "frozen" - no interactions.

g .

Incoming electron flies by and has hard interaction

e -

with a "frozen" parton. Interaction time  int 1

Q

 g .

 The parton is free during the hard interaction.

(argument by Feynman, 1 FK7003 969) 14

Kinematics of DIS

Proton breaks up. If an experiment studies an inelastic collision then two variables are needed since the proton is "changed" into a system of mass

W

.

P

e

e

.-

P

e

’

e

.-

P

g g Experimentalists would choose: scattered electron e nergy and angle.

Often better to use a combination of:

Q

2

W

2    

P e



P

g 

P e

'  2 (11.09)     

P p

 2 (11.15) 1

Q

   + ma ny more.

2

P p Q

2 

P

g

P

p

p

(11.16)  fraction of proton's momentum carried by struck parton.

}

W

Q

2 .

if you know the scattered electron energy and angle then you can calculate if you know 2 then you can calculate scattered electron ene rgy and angle .

FK7003 2 .

15

Question

An electron and a proton move towards each other and interact. In fact, the electron scatters off an object in the proton. Show that the fraction of the proton's momentum carried by the object is  2

P P

 •

P e

 

P e

 

P e

'   

P e

 ' 2  where

P e

 ,

P e

 ' ,

P P

are the four-momenta of the incident electron, scattered electron and incoming proton, respectively. Assume a high energy collision in which the proton moves quickly.

P

g 

P e

'  

P e

 ;

z



E q



p q



E p p p



zP p P q

'2  

m q

2

P e

  

P e

'   2  0 ;

m

2

p P q

'2 ;  0  

zP p

 

zP p P e

 

P q



P e

 ' 

P e

'  

zP p

;

P q

 2

P e

  

P e

 2

p



P q

' 

P e

'  

zP p



P e



P e

   

P e



P e

 '   2

P e

'   

P e

'   2  2

zP p

2

zP p

 

P e

  

P e

 

P e

 '  

P e

'   

P q

'  

P e

 

P e

 '  2  2

zP p

 

P e

 

P e

 '   0

z Q

2 2

P p

•

P

g how much momentum the struck quark possessed! FK7003 16

DIS kinematics

Inelastic collisions will give very different values of scattered electron energy therefore different values of and angle

Q 2

and

x.

and

Eg

an experiment colliding electrons and protons at fixed energies gives rise to a certain range of

Q 2

and

x.

Q

2

=data point FK7003

x

17

Some interpretation

An experiment measures and

Q

2 for each DIS interaction.

 for each interaction (if our thinking is correct) we know the fraction of momentum carried by the struck parton and the wavelength of the pro bing photon.

Q

2

=data point FK7003

x

18

Strategy for understanding deep-inelastic scattering

Aim is to test the hypothesis that objects exist in the proton (partons) and that they are quasi-free, point-like quarks. Explore the parton model.

 (1) Define deep-inelastic scattering and choose a refe rence frame.

Need for two variables, not just

Q

2 to characterise structure.

(2) Structure function - characterises our ignorance about proton structure in DIS as form factors do in elastic sc attering.

Structure function show point-like quasi-free quarks in the proton.

(3) Show that the quark picture of the proton (

uud

) and neutron (

udd

) is confirmed by direct scattering measureme nts. FK7003 19

Testing this assumption

Cross section for DIS interactions: 1 

d

   , 2 4

E

2  2 sin 4

F x Q

2 , 2 q  1   cos 2 2  , 2   sin 2

Q

2

xm

2

p

1   2 are dimensionless structure functions. , Analogous to form factors in el astic scattering

G Q

1 ( 2 ),

G Q

2 ( 2 ).

They contain information on proton structure.

Spin 1 2 objects: 2 1  , 2   2  , 2  2    (11.17) Qualitatively a measurement of 1  , to find a parton in 2

F x Q

2 , 2  reveals how likely it is 1 momentum when a probing photon of wavelength

Q

was used. FK7003 20

Electron-proton to electron-parton interactions!

e

-

Q

2

e

-

P

e

=data point

p P

p

e

-

P

e

e

-

parton xP x

If our model of partons in the proton is correct then

p

 Electron - proton colli sion becomes electron - parton collision.

We fixed and can study the influence of changing one variable:

Q

2 .

FK7003 21

Quarks in the proton

Measurement at SLAC, California 1969, electron-proton DIS.

F

2  doesn't depend on

Q

2 at a fixed

x

.

increase the resolution but see no difference.

 point-like charged objects: consistent with quarks but w e still need to check their charges are consistent with what we expect.

This is known as Bjorken scaling.

FK7003 22

● ●

DIS experiments

Early DIS experiments   1960s/1970s at SLAC, California Jerome Friedman, Henry Kendall, Richard Taylor et al.  Nobel prize 1990 More recently, electron and proton collisions at HERA accelerator at DESY in Hamburg. Measured by H1 experiment/collaboration.

HERA accelerator H1 Collaboration 23

HERA

Protons at 820 GeV energy collide with electrons (or positrons) at 27.5 GeV. Particles were accelerated along a 6.3km circumference ring underneath Hamburg at the DESY laboratory.

10 13 protons and electrons pass any point on the ring each second.

FK7003 24

A DIS Interaction at HERA

An electron interacts with a proton and recoils at a large angle after scattering off substructure within the proton.

Modern day Rutherford scattering.

Further description in accelerators/detectors lectures at the end of the course.

FK7003 25

Strategy for understanding deep-inelastic scattering

Aim is to test the hypothesis that objects exist in the proton (partons) and that they are quasi-free, point-like quarks. Explore the parton model.

 (1) Define deep-inelastic scattering and choose a refe rence frame.

Need for two variables, not just

Q

2 to characterise structure.

(2) Structure function - characterises our ignorance about proton  structure in DIS as form factors do in elastic sc attering.

Structure function show point-like quasi-free quarks in the proton.

(3) Show that the quark picture of the proton (

uud

) and neutron (

udd

) is confirmed by direct scattering measureme nts. FK7003 26

A more sophisticated view of proton structure

We've shown that the proton consists of point-like charged objects. Are they the same quarks needed to understand the hadron masses and properties.

Using this information and the uncertainty principle what do we expect a photon to scatter off in a DIS interaction i.e. which quarks live in the proton ? Traditionally view the proton as comprising: 2 up and 1 down quark. This is naive.

The up and down q uarks are "valence quarks".

From the uncertainty principle exchanged gluons can fluctuate into

qq

pairs of any flavour (sea quarks): , , , , Lifetime  1 (11.21)  heavy quark content: ,

tt

negligible!

m q

 Expect that up, down and strange quarks form the structure of the proton as seen in DIS interactions.

FK7003 Valence quarks Non-strange sea quarks Strange sea quarks 27

More on

F 2

Regard electron-proton DIS interactions as the incoherent sum of electron-quark scatters i.e. probability of hitting an up quark is independent of probability of hitting a down.  1 0  2    

q q

sum extends over all flavours of quarks in the proton.

Eg

q

      parton distribution functions (pdfs); eg up quarks in proton;   

u p

Interpretation - consider that we have a large sample of protons then:  in range to

x



dx

(11.19)  in range to

x



dx

(11.20

) Two up and 1 down in the proton 1 0  

u p

Similarly can define pdfs for neutron:

n



u n



u p

  

q dx



dx

   ( 11.18) 1 0 

d n

1 0 

d p



d n



d p d x



dx

 FK7003 1 0  1 0 

n s p



s n



s p dx

 0 (11.21)

dx

 0 (11.22) .

28

F

2

2  

for different scenarios of proton structure

 

q

2

e xq x q

   2

e xq x q

(11.18) - depends on valence/sea quark distributions.

F 2

H1 experiment at HERA FK7003 Observed

x

29

Quark pdfs in the proton

1.0

0.8

0.6

0.4

0.2

q v

 average number of valence quarks

x dx q s

(  average number of sea quarks ) between

x dx

0

x

Use structure function data to extract pdfs. We can’t calculate them from first principles.

Weak dependence on value at which they’re measured

Q 2

– scaling violations/next lecture.

FK7003 30

Seeing the valence quarks

The amount of sea quarks in the neutron and proton should be the same (isospin symmetry).

Differences between

F

2

p

and

F

2

n

should arise only from difference in valence quarks.

Define pdfs for proton and ne utron:

u u d v v p p

 

u s p p d s p



u v p



u s p

(11.23)  proton "sea" up pdf  proton "sea" down pdf Similarly for neutron

v n s n

( ),

v n

(

x

),

d s n F

2

p F

2

n

  

q

2

e xq x q

   2

e xq x q

  

x

  4 9

u v p



q

2    2   

x

  1 9

d v n

 1 9

d v p

 4 9

v n

  FK7003 31

u v p



F

2

n



d v n

 ;

x

  1 9

u v p v n



F

2

p



F

2

n

 

d v

4 9

p d v p



x

  1 3

u v p

(11.26) (isospin symmetry)   1 3  sea quark contribution (11.27)  expect a broad peak at

x

 1 3 Obs erved!

d v p

  (11.28)

F

2

p



F

2

n

There are three valence quarks with the charges we expect!

x

FK7003 32

Strategy for deep-inelastic scattering

Aim is to test the hypothesis that objects exist in the proton (partons) and that they are quasi-free, point-like quarks. Explore the parton model.

 (1) Define deep-inelastic scattering and choose a refe rence frame.

Need for two variables, not just

Q

2 to characterise structure.

(2) Structure function - characterises our ignorance about proton   structure in DIS as form factors do in elastic sc attering.

Structure function show point-like quasi-free quarks in the proton.

(3) Show that the quark picture of the proton (

uud

) and neutron (

udd

) is confirmed by direct scattering measureme nts. FK7003 33

Summary

● ● ● ● ● Lepton nucleon scattering experiments provide direct evidence for the existence of partons in nucleons.

Example of electron-proton scattering to develop formalism Elastic scattering   Form factors Fall with

Q 2

: structure!

Deep-inelastic scattering  Evidence for charged point-like objects (Bjorken scaling)  Structure function analysis reveals sea quarks and consistency with our picture of neutron and proton quark contents The quark model needed to describe hadron masses and properties is confirmed by direct scattering measurements.

FK7003 34