March 2011 1

Structure of subatomic particles

• Introduction• Rutherford and Mott scattering • Sizes of particles• Form factors• Deep inelastic scattering

March 2011 2

Introduction

This chapter describes what led to the present picture of partons in the proton, the result of the SLAC deep inelastic scattering experiment. To get there we first study the elastic scattering case of two point-like particles, like e- - scattering, through the kinematic variables, inclusion of spin and other experimental facts.

We go from simple Rutherford scattering, to Mott scattering, to the inclusions of form factors in the Rosenbluth formula. We also discuss the important measurement by Hofstadter of the radius of the proton.

All the above will be an introduction to the actual Deep Inelastic Scattering (DIS) formalism and experimental results, with their physical interpretation.

March 2011 3

e- - elastic scattering

e-

-

e-

-

( , )k E k,,,,,,,,,,,,,,

' ( ', ')k E k,,,,,,,,,,,,,,

( , )q q,,,,,,,,,,,,,,

( ,0)p Mp ’

Treat as spinless particles. Use the laboratory frame (initial at rest). The four momentum transfer between initial and final electrons carried by the exchanged photon: q = k – k’. Neglecting the electron mass (k2=k’2≈0)

2 22 ' 2 '(1 cos ) 4 'sin ( / 2)q k k EE EE If p ’ is the four-vector of the scattered , q+p=p’.2 2 2 2 2 2

2

( ) ' 2 ' 2 2

'2

p q p p p q q p q p q M

qE E

M

March 2011 4

Order of cross section

According to the Feynman rules, amplitude will have a factor of from the couplings at the two vertexes, and a propagator. Since a real photon has mass zero, the propagator has the form of 1/q2.

2

2 2

4 4sin ( / 2)

Aq

q

March 2011 5

Rutherford Scattering Formula

2

2 44 sin ( / 2)R

d

d E

Rutherford assumed a scattering of two point-like spinless particles, where the target is infinitely heavy and thus doesn’t recoil. In this case the energy of the scattered electron, E ’, is the same as that of the initial electron, E.

March 2011 6

Mott Scattering Formula

2 2

2 4

cos ( / 2)

4 sin ( / 2)Mott

d

d E

One can use the Dirac equation to take into account the spin of the electron. If one still considers the target to be point-like, spinless and infinitely heavy, one gets the Mott formula:

(Actually, the factor is 1-2sin2(/2), which, for 1, turns into cos2(/2))

March 2011 7

Form factorsSo far, particles treated as point-like particles. Charge was concentrated in one point. If the particle is an extended object, one can assign a charge distribution to that object. Usually one talks about a normalized charge distribution, or a probability density (r), so that a particle with charge Q has a charge density of Q(r), with

3( ) 1r d r The probability density enters in the theoretical calculations through the scattering potential V(x), and the outcome of the calculation is

22( )Mott

d dF q

d d

F(q2) is the form factor, obtained from the Fourier transform of the probability density.

2 / 3( ) ( ) iq rF q r e d r

March 2011 8

Form factors (2)

22

exp

/ ( )Mott

d dF q

d d

Note, form factor is function of q2 only because assumed spherical symmetric system. In that case one can write

2 24 sin(| | / )

( ) ( ) | | /q r

F q r r drq r

March 2011 9

Form factors (3)In principle, the radial charge distribution (r) could be determined from the inverse Fourier transform if the q2 dependence of the form factor is known:

2 / 33

1( ) ( )

(2 )iqrr F q e d q

Dashed line: plane wave scattering from homogenous sphere with diffuse surface.

Hofstadter measured electron scattering from 12C in 1957.

March 2011 10

proton charge radius• Low-q2 behaviour F(q2):

• Measure F(q2) at low q2

0

)(2222

2

6

qdq

qdFr

2 24 sin(| | / )

( ) ( ) | | /q r

F q r r drq r

22 24 1

( / )( ) ( )

6

qrF q r r dr

)...0cos()0sin(

)0cos()/()0sin()/sin(

!3)/(

!2)/( 32

qrqr

qrqr

2 2 3( )r r r d r• Fourier expansion:

• Yielding

2 22

21

1( ) ....

6

q rF q

• mean square radius:

March 2011 11

Experimental Results

• McAllister and Hofstadter obtained a first approximation to the structure and size of a proton – a mean squared radius of the proton was found to be: – (0.78±0.20)x10-13cm at

236MeV.– (0.70±0.24)x10-13cm at

188MeV.• Together, the best result was:

– <r2>1/2=(0.74±0.24)x10-

13cm.

March 2011 12

Nuclear charge distributions

acrer

/)(1

)0()(

Using measurements of form factors, get information on charge density. The density is described by a two-parameters Fermi function:

c=1.07 fm A1/3; a=0.54 fm

Almost constant charge density in interior. c is the radius at which density decreases by one half.

March 2011 13

Dirac Scattering Formula

2

'2

1 sin ( / 2)

EE

EM

2 22 2

2 4 2

'. [cos ( / 2) sin ( / 2)]

4 sin ( / 2) 2Dirac

d E q

d E E M

2

e

mc

If one takes into account that the target has a finite mass and thus recoils, the energy of the elastically scattered electron, E ’, is:

In addition, one treats both particles as Dirac particles, namely as point-like particles with spin ½, having magnetic moments:

The terms in front of the square brackets are from the Rutherford formula, with a recoiling target. The first term in the square brackets is from the Mott formula, while the second terms includes the magnetic spin flip.

March 2011 14

Nucleon magnetic moment

(Dirac) (observed)

Proton 1 n.m. +2.79n.m.

Neutron 0 -1.91n.m.

Dirac formula assumes that the nucleon has a magnetic moment corresponding to the nuclear magneton (n.m.):

182 3.1525 10 /

p

eN m c MeV G

March 2011 15

Rosenbluth Scattering Formula

• All the previous equations assume point-like particles.

• As protons are not point-like and the magnetic moments differ in reality to the Dirac , we have to include the Form Factors GE and GM and a factor to correct for .

• Rosenbluth:

2 22 2

2

( )[ 2 tan ( )]

1E M

MMott

G bGd dbG

d d b

2 2

2 24 4

q Qb

M M

March 2011 16

Experimental Results

• Three theoretical curves:a) Mott cross-sec – a

spinless point charge proton.

b) Dirac cross-sec – a point-like proton with mass M, and (Dirac).

c) Rosenbluth cross-sec – a point-like proton with mass M and corrected .

• Results deviate from theory due to a “structure factor” – the proton is not point-like but has finite size. Form Factors introduce this deviation.

March 2011 17

Test of Rosenbluth formula2 2

2 22

( )[ 2 tan ( )]

1E M

MMott

G bGd dbG

d d b

The form factors are function of Q2 only, with the normalizations

(0) 1; (0) 0; (0) 2.79; (0) 1.91p n p nE E M MG G G G

The formula can be tested by doing an experiment at fixed Q2 and varying the angle :

2 2 2/ ( ) ( ) tan ( / 2)Ros Mott

d dA Q B Q

d d

March 2011 18

Dipole form factor

Form factors have been measured to higher energies for protons and for neutrons. The latter have been obtained from scattering on deuterium target, from which the measured cross sections on the proton were subtracted (+ some corrections). The resulting form factors obey a simple law:

2 22 2 2( ) ( )

( ) ( ), ( ) 0p n

p nM ME E

np

G Q G QG Q G Q G Q

The universal form factor, called the dipole form factor, can be described by an empirical formula:

22

2

1( )

1V

G QQM

MV2=(0.84GeV)2=0.71

GeV2

March 2011 19

Nucleon form-factor data

March 2011 20

High Q2 f.f. data

What is the physical meaning of a form factor? It measures the amplitude that under an impact the proton remains intact. As the impact gets larger, the probability that the proton remains a proton gets smaller. The elastic scattering cross section gets small. For example, for Q2=20 GeV2, the elastic cross section is reduced by 6 orders of magnitude.

March 2011 21

Deep inelastic scattering (DIS)

At high energies, elastic scattering becomes relatively unlikely (elastic form factor falls rapidly with q2). Instead: proton breaks up into hadrons: e- p e- X

‘Deep’ – high q2, ‘Inelastic’ – proton breaks up.

At HERA, reached already Q2=40,000 GeV2, probing structures down to ~ 10-18 m.

March 2011 22

Review of elastic scattering

e p e p

1 3E E

2 2q M

E3 and are related:

2 21 3 1 324 sin 2 ( )q E E M E E

only have to measure one of the (say ) 2 22

2 2 232 22 4

1 12

( )cos 2 sin

4 sin 1E M

M

E G bGdbG

d E E b

March 2011 23

Bjorken xIn DIS, mass of system X not fixed:

2 2 22 Xq M M M

• q2 and no longer related G(q2) replaced by F(,q2)

• equivalently: E3 and no longer related have to measure both E3 and Final state contains at least one baryon 2 2 2 22 ( 0)XM M q M q

Bjorken x:2

(0 1)2

qx x

M

(x=1 for elastic scattering e-pe-

p)

( )XW M

2

2

Qx

M

March 2011 24

SLAC DIS experimentMost general form of DIS cross section is:

Measured DIS cross section by detecting the scattered electron and measuring its energy E3 for different scattering angles

2 222 23 2 1

2 22 41 12

( , ) 2 ( , )cos sin

4 sin

E F q F qd

d E E M

March 2011 25

Result of SLAC DIS experiment

Contrary to the elastic case, the cross section is found to depend very weakly on q2, reminiscent of the Rutherford experiment.

Measurements presented in the figure are for two different values of the invariant mass W of the hadronic final state X.

From cross section can obtain the structure functions F1

and F2, which can be functions of and q2, or alternatively, functions of x and Q2: Fi(x,Q2).

March 2011 26

Bjorken scaling

F1 and F2 are found to be approximetly independent of Q2 for fixed x. This suggests that the virtual photon is scattering off pointlike constituents within the nucleon.

March 2011 27

Bjorken scaling (2)

March 2011 28

Latest measurements from HERA presented at the Moriond Electroweak workshop, March 2004

Scaling violation

March 2011 29

Callan-Gross relation

F1 and F2 are found to be closely related:

2 12F xF

known as Callan-Gross relation.

March 2011 30

The Parton ModelCan obtain Bjorken scaling and Callan-Gross relation by assuming that the virtual photon scatters elastically from a point-like object within the proton, named by Feynman as partons. The other partons are spectators.

Assume interaction with parton takes place sufficiently fast that can be treated as free particle of mass m:

22 2 2 2 2 2

2

( , ) 2 2

02

m m q m m q m m q

q

m

relates E ’ and

2 22

'

4 'sin

E E

q EE

March 2011 31

e-parton elastic scatteringelastic scattering cross section for e-partone-parton can be written as: 2 2 2

2 2 22 22 4 2

2

cos sin' 4 sin 2 2

d q qz

dE d E m m

2 2

02 2

q q mx x

m M M

2 2 22 2 2

2 22 4 2 22

cos sin' 4 sin 2 2

d q qz

dE d E M x Mx

z – charge of partons in units of e. Assume from here that partons are quarks. proton made up of 3 quarks: uud. Can extract F1 and F2:

22 22

2 22 21

2 2

22

22

2 2

u d

u d

F qz z

Mx

F q qz z

M M x Mx

March 2011 32

Consequensces of parton model

F1 and F2 are related as follows:

2 2 2 22

21

2 1

/ 2 2

2 / 2

2

F M x M x Mx

F M q M x

F xF

Callan-Gross

relationusing:

2 22 2

1( ) ( ) (2 ) ( )u d

max x F z z x x F x

a M

thus F2 is only function of x, independent of q2 Bjorken scaling • F1, F2 are functions of one variable

only because underlying scattering is elastic and pointlike, giving extra constraint between E ’ and .

• partons are quarks with spin ½. Only one unknown function, because charge and magnetic moment are related for Dirac particles: = e/2m.

But measured F2 not delta function!

March 2011 33

Expectations for F2

proton is pointlike

proton consists of only 3 free quarks

proton consists of only 3 bound quarks (Fermi motion)

qu

ark

-p

art

on

m

od

el

(QP

M)

proton consists of 3 bound ‘valence’ quarks + other partons (QCD)

March 2011 34

Physical meaning of Bjorken x

q

( , )e E k( ', ')e E k

,,,,,,,,,,,,,,

( , )pP E P,,,,,,,,,,,,,,

zP

(1 )z P

( , )p p,,,,,,,,,,,,,,

use frame where proton momentum is very large (‘infinite momentum frame’). In this frame, momenta of partons collinear with parent proton. Each parton carries a fraction z of the proton’s momentum, and are approximately massless (p2≈0).

In addition, Q2-q2»M2

( ')

( ') p

p k k zP

E E zE

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

22 22 2

22 2 2 2

2 2 2

( ') ( ')

( ') ( ') ( ) 2 ( ') 2 ( ')

2 ( ') 2 ( ') 0

p

p p

p

p p E E zE k k zP

E E k k z E P zE E E zP k k

q z M zE E E zP k k

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

March 2011 35

Bjorken x (2)

neglecting z2M2 : 222 0

2

QM z Q z x

M

Thus, in the infinite momentum frame, x is the fraction of the proton momentum carried by the (massless) parton. This also means that measured cross section at a given x is proportional to the probability of finding a parton with a fraction x of the proton momentum.

since22

24 'sin

2 2 ( ')

EEQx

M M E E

can measure parton’s momentum from scattered electron alone (E’,)

March 2011 36

Quark distribution functions

F2 is not a delta function, since not all quarks have x=1/3. We thus define quark distribution functions as follows:

( )pu x probability that u quark in proton has momentum x

1

0

( ) 2pu x dx

( )pd x probability that d quark in proton has momentum x

1

0

( ) 1pd x dx 1 2

2 22

0

( ) ( ) ( )2

4 1( ) ( )

9 9

ep p pu d

p p

QF x z xu x z xd x x dx

M

xu x xd x

proton structure

2epF

March 2011 37

qdfs (2)

neutron structure

2enF

2

4 1( ) ( ) ( )

9 9en n nF x xu x xd x

isospin symmetry (see later):

( ) ( ) ( )

( ) ( ) ( )

n p

n p

u x d x d x

d x u x u x

2

2

4 1( ) ( ) ( )

9 94 1

( ) ( ) ( )9 9

ep

en

F x xu x xd x

F x xd x xu x

can extract u(x) and d(x) from measurement of the proton and neutron structure function. These function can not yet be predicted from theory which needs a better understanding of the non-perturbative regime.

March 2011 38

inclusion of ‘sea’ quarksneed to extend picture by inclusion of ‘sea’ quarks in addition to the valence quarks :

( ) , ( ) ( , , )u x d x neglect ss cc etc

2

2

4 1 4 1

9 9 9 9

4 1 4 1

9 9 9 9

ep

en

F x u d u d

F x d u d u

In order to find the fraction of the proton’s momentum carried by u and anti-u quarks, denoted fu, and the one of fd, need to integrate F2 over x:

1 1 1

2

0 0 0

1 1 1

2

0 0 0

4 1 4 1( ) ( ) 0.18

9 9 9 9

4 1 4 1( ) ( ) 0.12

9 9 9 9

0.36 , 0.18

epu d

enu d

u d

F dx x u u dx x d d dx f f

F dx x u u dx x d d dx f f

f f

from experimental measurements (eg page 26)

Only 54% of proton’s momentum is carried by quarks rest, by gluons.

March 2011 39

Picture of the proton

qq pair

gluon

valence quark

valence quark

valence quark

March 2011 40

Wavelength of probe

• q² klein lange WW: nur Valenz-Quarks sichtbar (fermi-verschmiert)

• q² groß kurze WW: See-Quarks sichtbar (Stroboskop-Aufnahme)

•:

March 2011 41

QCD and scaling violationExistence of gluon proven experimentally (3-jets events). Opened field of Quantum Chromodynamics (QCD – see later). Closer picture of F2 vs Q2 showed scaling violation, which can be explained as follows: As the momentum of the probe increases and the distance it resolves decreases, it begins to see the detailed quantum mechanical subprocesses of QCD in the environment of the struck quark. What may have appeared to be a quark with a given x at low Q2, may be revealed as a quark and a gluon, at higher Q2, with the quark having lower x.

March 2011 42

Scaling violation (2)

As the momentum of the probe increases, the average fraction of the total proton momemtum carried by the quarks appears to decrease. As the momentum of the probe increases still further, it may see the gluon radiated be the valence quark dissociating into a quark-antiquark pair. So there will appear to be even more quarks carrying very low fractions of the total proton momentum.

Number of low x partons increases as q2 increases, while at high x, the number decreases as q2 increases.

March 2011 43

Latest measurements from HERA presented at the Moriond Electroweak workshop, March 2004

Scaling violation

So after all, F2 is not just function of x, but also of Q2:

22 ( , )F x Q

Curves in the figure: result of a perturbative QCD (pQCD) calculation.