HEP Lecture 9 1

High Energy Physics

Lecture 9

Deep Inelastic ScatteringScaling Violation

HEP Lecture 9 2

The reaction equation of DIS is written

e p e X+ → +

where X is a system of outgoing hadrons (mostly pions).Observed is only the scattered electron. The unobservedhadronic system is the missing mass.

The energy of the incident electron beam is accurately knownThe proton is the target particle; in the SLAC experiments(and many later experiments at CERN) the target is at restin the laboratory. This defines the LAB frame.

Deep Inelastic Scattering:

HEP Lecture 9 3

DIS Kinematics:

or, since E’ and θ are measured:

( )2 2 20 02 ' 4 'sin

2W M M E E E E θ

= + − −

Note: for elastic scattering W=M

HEP Lecture 9 4

Feynman diagram of inelastic electron-proton scattering:(from H.W. Kendall, “Deep Inelastic Scattering”, Nobel Lecture, December 8, 1990)

HEP Lecture 9 5

Differential Cross Section of Inelastic ep Scattering:

( )21,2 1,2 ,W W Q ν= structure functions

On general grounds the structure functions are functionsof two kinematical invariantsThey were expected to drop with increasing Q2 asrapidly as the form factors

The surprising experimental result was different!

HEP Lecture 9 6

W = 2 GeV

W = 3 GeV

W = 3.5 GeV

HEP Lecture 9 7

up to about W = 1.8 GeV there is structure corresponding tothe production of resonances (excited nucleon states);there is no structure above 1.8 GeV: this is the region of DIS.

HEP Lecture 9 8

Scaling:

Bjorken (1969): for

2 2and , such that 2 / is fixedQ M Qν ω ν→∞ →∞ =

(“Bjorken limit”), the structure functions depend only on ω:

( ) ( )( ) ( )

21 1

22 2

2 ,

,

MW Q F

W Q F

ν ω

ν ν ω

→

→

this behaviour is called scaling; ω is the scaling variable

HEP Lecture 9 9

0 2 4 6 80

2

4

6

8

SLAC data: x=0.275

Fep 2

Q2 (GeV2)

Q2 dependence of structure function F2(x,Q2)SLAC data at x=0.275, Q2= 0.9, … , 8.22 GeV2

Later, a new scaling variablewas defined by Feynman:x=1/ω. This is now useduniversally and calledBjorken-x

HEP Lecture 9 10

Q2 dependence of structure function F2(x,Q2)SLAC data at x=0.35, Q2 = 1.1, … , 10.95 GeV2

0 5 10 15 20 250

2

4

6

8

SLAC data: x=0.35

Fep 2

Q2 (GeV2)

HEP Lecture 9 11

Q2 dependence of structure function F2(x,Q2)SLAC data at x=0.45, Q2 = 1.7, … , 16.36 GeV2

0 2 4 6 8 10 12 14 16 180

2

4

6

8

SLAC data: x=0.45

Fep 2

Q2 (GeV2)

HEP Lecture 9 12

Scaling found a natural explanation in the parton model(Feynman).Partons are constituents of the proton(more generally of hadrons).

They are quarks and gluons:quarks are point-like spin-1/2 fermions like the leptons,gluons are massless spin-1 bosons: they are the carriers ofthe strong interaction.

Unlike leptons, quarks take part in strong as well aselectromagnetic and weak interactions.

Recall:charged leptons take part only in electromagnetic and weak interactions,neutrinos, i.e. neutral leptons, only in weak interactions.

HEP Lecture 9 13

Parton model picture of DIS:

HEP Lecture 9 14

The electron collides elastically with a parton thatcarries a fraction x of the proton momentum.

At high momentum (“infinite” momentum) the partons are free.Therefore the collision of one parton with the electron does notaffect the other partons. This leads to scaling in x

[ ]21 2 0,1x Q Mω ν= = ∈

DIS experiments have also been done with muonsand with neutrinos.

Since 1992 DIS experiments are also done on theelectron-proton collider HERA.

HEP Lecture 9 15

Proton structure functionF2(x,Q2).Open triangles: SLAC dataSolid squares:a) BCDMS datax values (from below):

0.275, 0.225, 0.180, 0.140,0.100, 0.070

b) EMC datax values (from below):0.250, 0.175, 0.125, 0.080

the individual x ranges arescaled by a factor of 1.5 toseparate them from each other.

Note the slight Q2 dependence:scaling violation!

HEP Lecture 9 16

Nucleon structure functionF2

νN measured inneutrino-nucleon DIS

Data of CDHSW collaboration,1989

The individual x ranges arescaled by the factors shownto separate them from eachother.

Clearly seen is the scaling violationat the lowest and highest x values

HEP Lecture 9 17

Compilation of data on theproton structure functionF2

p(x,Q2)

Experiments:E666 (Fermilab)H1 (DESY)BCDMS (CERN)NMC (CERN)ZEUS (DESY)

an amount C(x) is added“by hand” to F2 of eachx range to separate thedata sets.

Note the scaling violationat the lowest values of x

HEP Lecture 9 18

The H1 detector at the ep collider HERA (DESY):

Electrons of 30 GeV arecolliding with protons of820 GeV, giving

E_cm=313 GeVandQ2_max=98 400 GeV2

DIS by W and Z exchangeare comparable to DIS byphoton exchange at thisenergy.

The asymmetry of the colliding system leads toan asymmetric design ofthe detector.

HEP Lecture 9 19

In a neutral current event the transverse momentum is balanced,indicating that the hard electron-quark collision proceeds by photonor Z boson exchange:

In a charged current event, the electron-quark collision proceeds byW boson exchange; the outgoing lepton is a neutrino and is notdetected, therefore the measured transverse momentum is not balanced:

The following pictures are computer reconstructions of DIS eventsas seen by the H1 detector at DESY.

HEP Lecture 9 20Neutral current event in the H1 detector

HEP Lecture 9 21Neutral current event in the H1 detector

HEP Lecture 9 22

HEP Lecture 9 23

Charged current event: note the imbalanceof transverse momentum.

HEP Lecture 9 24Charged current event in the H1 detector

HEP Lecture 9 25

Scaling violation; gluons

Exact scaling holds if the quarks are free. In reality theyare not free: quarks are held together in hadrons by gluons.

Gluons are the carriers of the strong interaction; they coupleto the quarks and to each other. This gives rise to the followingelementary processes:

HEP Lecture 9 26

Elementary Parton Processes:

a) Quark emitting gluon:

b) Gluon splitting into 2 gluons:

c) Gluon splitting intoquark-antiquark pair:

HEP Lecture 9 27

Build-up of gluon – quark – antiquark sea from elementary processes:

(ii) Gluon splitting intoquark-antiquark pair

followed by recombination:

(ii) Gluon exchange betweenquark lines

(iii) Gluon exchange with gluonsplitting into quark-antiquark pairfollowed by recombination:

and these combine into more and more complicated processes

HEP Lecture 9 28

Revised Parton Picture of the Protonproton = valence quarks (uud) + gluon–quark–antiquark sea

The proton charge is the sum of the valence quark charges:the charge of the u quark is +2/3ethe charge of the d quark is -1/3ethe parton sea is neutral

A theoretical framework to describe the proton structure(and the structure of all hadrons) is

Quantum ChromoDynamics (QCD)

But there are great computational difficulties ...

If the structure function is known at some value of Q2, then onecan reasonably accurately calculate the SF at all Q2 using theDokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations.

(this is what is shown in the figure of compilation of structure function data)

HEP Lecture 9 29

Cross sections for DIS processes, i = NC or CC:

NC structure functions:

+(-) for u (d) –type quarks

HEP Lecture 9 30

θW is the weak mixing angle: sin We g θ=

2 2 2sin 1 0.23W W ZM Mθ = − ≅

, , ,...q u d s=5 21.166 10 GeVFG −= × Fermi constant

In the “naïve” quark-parton model, the parton densities are functionsof the Bjorken-Feynman scaling variable x only. The observed scaling violation means that they are functions of x and Q2. The Q2 dependencecannot be calculated, but if the x dependence is taken from experimentat some value of Q2, then the x dependence at other Q2 values canbe calculated using the QCD evolution equations of Dokshitzer, Gribov,Lipatov, Altarelli and Parisi (DGLAP equations).

HEP Lecture 9 31

DGLAP QCD evolution equation for parton densities:

P(y) are splitting functions which can be calculated within theframework of perturbative QCD (pQCD).

HEP Lecture 9 32

In lowest order pQCD the splitting functions are:

where

HEP Lecture 9 33

Diagram of deep inelastic p-anti p collision:

a parton from the proton, thatcarries a fractional momentum x1,makes a hard collision with aparton from the anti proton, thatcarries a fractional momentum x2.The final state of this collisionare fermions f1 and f2.

The remaining partonic system of the proton and anti protonhadronise into systems X and Y

A particular case of the hard collision is the reaction

qq Z e e+ −→ →first seen in the UA1 detector at CERN in 1982. Such collisions arealso seen at the Tevatron (Fermilab) and will be seen at the LHC (CERN).

HEP Lecture 9 34

q1

q2

f1

f2

A process of the type

where q1 and q2 are quarks from different hadrons,and f1 and f2 are final state fermions, are called Drell-Yanprocesses. They are particularly important in proton-protonand proton-antiproton collisions.

HEP Lecture 9 35

The data from all DIS experiments taken togetherin “Global Fits” using aQCD based theoreticalframework yield distributionsof the individual partons.

Shown here are the momentumdistributions of the valenceu and d quarks, quark-antiquarksea and gluons.

Note the scaling of the gluonand sea distributions!

The fits of three differentgroups are in good agreement