structure of hadrons and the parton model - uzh · v. chiochia (uni. zürich) – phenomenology of...

V. Chiochia (Uni. Zürich) – Phenomenology of Particle Physics, FS2010

Vincenzo ChiochiaUniversity of Zürich - Physik InstitutPhenomenology of Particle Physics - FS2010

Lecture: 23/2/2010

Structure of Hadrons and the

parton model


Topics in this lecture

How do we study the structure of composite particles?Is the proton an elementary particles?If not, what do we see inside the proton?Are there only charged partons inside the proton?

2

dσ

dΩ=

dσ

dΩ

point

|F (q)|2

q = ki − kf

dσ

dΩ

point

=(Zα)2E2

4k4 sin4(θ/2)

1− k2

E2sin2 θ/2


Probing a charge distributionTo probe a charge distribution in a target we can scatter electrons on it and measure their angular distributionThe measurement can be compared with the expectation for a point charge distribution

3

ki kfe- e-

Chargedistribution

Momentum transfer:

Form factor

Structureless target:


Proton is not a “point”

4

Ł“The Discovery of the Point-Like Structure of Matter” 4 presented by Professor R.E. Taylor on May 24, 2000 The Royal Society Discussion Meeting – The Quark Structure of Matter

The first electron scattering measurements at Stanford were made using carbon targets. Scattering from hydrogen was first observed using a CH2 target, although the first published data on hydrogen came from measurements using high-pressure gas targets. The early data indicated that the “proton was not a point”, that there were contributions from magnetic scattering, and also that the magnetic and electrical sizes were comparable.

7030 110 150Scattering Angle (deg)

Experimental

AnomalousMoment

Electron Scatteringfrom Hydrogen188 MeV (LAB)

Mott

Cro

ss S

ectio

n (c

m /

ster

ad)

2

10-31

10-32

10-30

10 29

Figure 3 First published results on electron-proton elastic scattering measured at the Mark III accelerator at Stanford.

These very direct measurements of the proton’s extended charge and magnetic moment distributions were a major event in high energy physics in the mid-1950s. Measurements on hydrogen (and deuterium) targets continued at energies up to 1 Gev. The electron community began to consider electron accelerators with even higher electron energies. By this time, it had been demonstrated (at Cornell and elsewhere) that scattering experiments could be performed at electron synchrotrons using internal targets or external beams. CEA, DESY and SLAC were soon under construction.

In the original proposal for SLAC (1957), electron scattering was mentioned as an extension of the successful experiments at Stanford’s Mark III linac, and as experiments where violations of QED might be observed. In a 1960 summer study at SLAC, J. Cassels produced a more sophisticated analysis of electron scattering, finding that there might still be lots to learn from elastic scattering at SLAC energies.

In 1963 a collaboration of physicists from MIT, Caltech and SLAC began to think seriously about scattering experiments at SLAC, and the equipment that would be needed to make such measurements.

dσ

dΩ

point

=(Zα)2E2

4k4 sin4(θ/2)

1− k2

E2sin2 θ/2

Structureless point-like target does not describe the data!

¯|M |2 =8e4

q42M2EE

cos2

θ

2− q2

2M2sin2 θ

2

dσ

dΩ=

α2

4E2 sin4 θ2

E

E

cos2

θ

2− q2

2M2sin2 θ

2

¯|M |2 =e4

q4Lµν

e Lmuonµν

q2 −2k · k −4EE sin2 θ

2


e--m scattering in the lab frame

5

k=(E,k)

k’=(E’,k’)

u

p=(M,0)

p’

1) Matrix element

q=(n,q)2) Transferred momentum

Target recoil

g


Electron-proton scatteringThe scattering picture used so far needs to be extended for a composite objectThe invariant mass spectrum shows the elastic peak, excited baryons followed by an inelastic smooth distribution

6

ki kfe- e-

...

12

n

q

e-

p

Invariantmass W Elastic

peakD resonance

ep→eD+→epp0Inelasticregion

Wµν = −W1gµν +

W2

M2pµpν +

W4

M2qµqν +

W5

M2(pµqν + qµpν)

W5 = −p · q

q2W2

W4 =

p · q

q2

2

W2 +M2

q2W1


Hadronic tensor - 1

The most general form of the tensor W depends on gmn and on the momenta p and q

7

dσ ∼ LeµνLµν

muon → dσ ∼ LeµνWµν

proton

parametrizesthe current at the proton

vertex

qn

pm

Wmn

From current conservation ∂mJm=0

electron-muonscattering

p’=p+q

Wµν = −W1

−gµν +

qµqν

q2

+ W2

1M2

pµ − p · q

q2qµ

pν − p · q

q2qν

q2

ν ≡ p · q

M

x =−q2

2p · q=−q2

2Mν

y =p · q

p · k

[ 0 ≤ x ≤ 1

0 ≤ y ≤ 1

W 2 = (p + q)2 = M2 + 2Mν + q2


Hadronic tensor - 2Only two independent inelastic structure functions (W1 and W2)

8

Each structure function has two independent variables

The invariant mass of the hadronic system in the final state is

Dimensionless variables

[Bjorkenscaling variable

square of transferred four-momentum

energy transferred to the nucleonby the scattering electron


Kinematic phase-space

9

x = 1→ −q2 = 2Mν →W 2 = M2

Elastic scattering

Line of constant invariant mass

Line of constant momentum fraction

dq

dEdΩ=

α2

4E2sin4 θ2

cos2

θ

2W2(ν, q2) + sin2 θ

22W1(ν, q2)

dq

dΩ=

α2

4E2sin4 θ2

E

E

cos2

θ

2W2(ν, q2) + sin2 θ

22W1(ν, q2)


Cross sectionWe can now use the hadronic tensor to calculate the matrix elementIn the laboratory frame:

10

(Le)µνWµν = 4EEcos2

θ

2W2(ν, q2) + sin2 θ

22W1(ν, q2)

Using the flux factor and phase-space factor

Integrating on the outgoing electron energy

λ ∼ 1−q2

∼ 1 Fermi

λ ∼ 1−q2

1 Fermi


Increasing spatial resolutionThe key factor for understanding the proton substructure is the wavelength of the probing photon

11

limQ2→∞,ν/Q2 fixed

νW2(Q2, ν) = F2(x)

limQ2→∞,ν/Q2 fixed

MW1(Q2, ν) = F1(x)


Bjorken ScalingIn 1968 J.Bjorken proposed that in the structure functions should depend only on the ratio n/q2 (proportional to x) in the limit q2→∞ and n→∞In other words: at large Q2≡-q2 the inelastic e-p scattering is viewed as elastic scattering of the electron on free “partons” within the proton

12

quark

g* g*

Bjorken - 1969


SLAC-MIT experiment

13


SLAC-MIT experiment

14


There were two common ways of defining the inelastic form factors in the 1950s, and that history has left us using a mixture of parameters from the two expressions for the cross-section. In the end the virtual photon approach did not simplify the physics, but we still talk about R, the ratio of longitudinal to transverse virtual photons, along with the structure functions W1 and W2 .

In the initial planning of the SLAC experimental facilities, the kinematics and the estimates of cross-sections for elastic scattering and for photoproduction of pions provided the main design guidelines for the equipment. It was important to have sufficient resolution to cleanly separate states that differed by a single pion mass in photoproduction or to separate the excited states in inelastic scattering.

I will not describe the spectrometer facility we built in End Station A, since that has been done many times and details are available in the literature. We built three spectrometers capable of analyzing singly charged particles with moment of 20, 8, and 1.6 Gev/c. The solid angle acceptances were 0.1, 1, and 5 milli-steradians respectively. The scale of the devices was quite impressive for its day.

TARGET POSITION

8 Gev SPECTROMETER

20 Gev SPECTROMETER

1.6 GevSPECTROMETER

MONITORSBEAM

Figure 5 Spectrometer facility at the Stanford Linear Accelerator. Each of the Spectrometers can be rotated about the target position to vary the angle of scattering.

The basic design philosophy was conservative – SLAC was a very visible project, and it was important to the laboratory that reliable results be generated in the early running. It seemed unlikely that the experiments could be reproduced soon at another accelerator, so any wrong answers were likely to mislead for a long time. Beam time would be very costly at SLAC, so we tried hard to make the spectrometer complex efficient. This led us to incorporate a mid-sized computer dedicated to our data acquisition and on-line analysis. Our computer system became a model for those who could afford such things.

ω =2q · p

Q2=

2Mν

Q2


First hints of Bjorken’s scaling

15


0.6

0.5

0.4

0.3W2

! = 6O

W2 >> 2W, tan2 !/2

0.2

0.1

00 2 4

" (Gev)6 8

4 6

Q2 (Gev/c)2

– 0.8– 1.0– 1.2– 1.6– 2.0– 2.2

5-20008374A14

Figure 8 The measured form factor W2 plotted against the energy loss ". The magnified portion shows how W2 depends on Q2 over a range in ".

For a small range of values of ", the value of W2 doesn’t seem to vary with Q2 (an even stronger constraint than scaling). Kendall prepared a second plot, this time of " W2 for two cases, corresponding to the two extreme cases of the value of R.

Figure 9 The measured proton structure function, F2, plotted against "/Q2, showing approximate scaling beyond the resonance region (as predicted by Bjorken).


! = 4

4 6 820Q2

[(GeV/c)2]

0

0.2

0.4

"W2

6O 18O

10OooOOO

O 26O

7145A54-92

Figure 13 Values of "W2 vs. Q2 at ! = 4, showing that "W2 does not vary with Q2, (ie. "W2 “scales”)

So things were in pretty good shape, but nothing is ever perfect. As the data improved, it became clear that scaling was not working over the full range of our data (it turns out that ! = 4 is not a good place to look for scale breaking). At first, the scale breaking was observed only below W = 2.6 Gev, and we wondered if the "resonant region" was more extensive than we had assumed, even though we were seeing no visible peaks between 2 and 2.6 Gev. So for a while we made “scaling plots” including only data having W >2.6 Gev. Another solution was to use a slightly different scaling variable, !’= W2 /Q2 – (this variable is equivalent to 2M"/Q2 in the limit as ", Q2, and W go to infinity, so Bjorken’s hypothesis was still valid). We could detect only minor deviations from scaling in !’, and used !’ in our presentations for a couple of years.

By the summer of 1971, most people were at least aware of our results and the quark-parton interpretation of our data. A growing number of theorists were hard at work, and soon the concepts of “asymptotic freedom” and “confinement” (sometimes called “infra-red slavery”) led on to Quantum Chromodynamics. These advances actually predicted scale breaking, so we could go back to ! (or x) as the scaling variable.

It was in 1971 that the original SLAC-MIT collaboration split into two independent groups. There was, at that time, some disagreement about what to do next. Most of the SLAC contingent worked on an experiment at 4º (and at 58 – 60°), while all the MIT scientists and a couple of people from SLAC repeated the hydrogen measurements at 18º, 26º and 34º along with new measurements on deuterium. The breakup was so friendly that many people don't realize that it ever happened and make no distinction between SLAC-MIT and MIT-SLAC.

Late in 1969 we had heard rumors that an analysis of CERN neutrino data was indicating that the neutrino scattering cross-sections were proportional to the neutrino energy, as expected in the quark-parton picture. By 1972 a major independent confirmation of the quark model was announced by Don Perkins at the ICHEP conference in Batavia. The Gargamelle data showed

First data at 6°: F2 plotted as function of the scaling variable n/Q2 is roughlyindependent on Q2

More data at different angles:F2 for a fixed v does not depend on thetransferred momentum Q2

Friedman, Kendal and Taylor - 1969

=

i

dxe2

i [ ]

i

dx xfi(x) = 1fi(x) =

dPi

dx


Parton distributionsPartons carry a different fraction x of the proton’s momentum and energy

16

g* g*

E,p

xE,xpi

(1-x)p

Sumover

partons

Integrateover

momentumfractions

The probability that the struck parton carries a fraction x of the proton momentum is usually called parton distribution or parton density functionTotal probability must be equal to one:

R.Feynman - 1969

νW2(ν, Q2)→ F2(x) =

i

e2i xfi(x)

MW1(ν, Q2)→ F1(x) =12x

F2(x)

2W1

W2=

Q2

2m2, W1 = F1/M, W2 = F2/ν

[m = xM, Q2 = 2mν]→ F1

F2=

12x


Structure function revisitedIn the Feynman’s parton model the structure functions are sums of the parton densities constituting the proton

17

The result 2xF1=F2 is known as Callan-Gross relation and is a consequence of quarks having spin 1/2Comparing e-p with e-m scattering cross sections (with m≡quark mass):

The parton carries a fraction x of the proton mass M

1x

F ep2 =

23

2

[up + up] +

13

2

[dp + dp] +

13

2

[sp + sp].

up(x) = dn(x) ≡ u(x)

dp(x) = un(x) ≡ d(x)

sp(x) = sn(x) ≡ s(x)


Proton/Neutron parton densities

We write the equivalent structure function for the neutron as

18

Valencequarks

Gluons

Quark-antiquarkpairs from gluonsplitting (“sea”)

1x

F en2 =

23

2

[un + un] +

13

2

[dn + dn] +

13

2

[sn + sn].

Proton and neutron parton densities are correlated

1x

F ep2 =

19[4uv + dv] +

43S

1x

F en2 =

19[uv + 4dv] +

43S

uv + 4dv

4uv + dv∼ 1

4


Constraints to parton densitiesWe assume the three lightest quark flavours (u,d,s) occur with equal probability in the sea

19

us = us = ds = ds = ss = ss = S(x)u(x) = uv(x) + us(x)d(x) = dv(x) + ds(x)

Combining all constraints:

At small momenta (x~0) the structure function is dominated by low-momentum quark pairs constituting the “sea”. For x~1 the valence quarks dominate and the ratio F2en/F2ep becomes


Ratio of structure functions

20

low momentumfractions

(sea)

Partons carrymost of the hadron

momentum(valence)


Summary of F2 proton

21

1

0dx x(u + u + d + d + s + s) = 1− pg

p= 1− g

u ≡ 1

0dx x(u + u) g 1− u − d = 1− 0.36− 0.18 = 0.46


How about gluons?Summing over the momenta of all partons we should reconstruct the total proton momentum:

22

Neglecting the small fraction carried by the strange quarks we have and using the results of experimental data

Experimental data indicate that about 50%of the proton momentum is carried by neutral partons,

not by quarks!


Gluons and the parton model

23

g*

E,p

xE,xpi

(1-x)p

g*

g*

O(aem)

O(as aem)

gluon

quark

quark

antiquark

+ Initial Gluon emission

+ crossed diagram

F2(x,Q2)x

=

i

e2i

1

x

dy

yfi(y)δ(1− x/y) =

i

e2i fi(x)

F γ∗q→qg2 (x,Q2)

x=

i

e2i

1

x

dy

yfi(y)

αs

2πPqq(x/y) log

Q2

µ2

Pqq(z) =43

1 + z2

1− z


Gluon emission: contribution to F2

24

g*

Protonp pi=yp zpi=xp

g*

i

g*

gluon

quark

2

+ Initial Gluon emission

2

(z=x/y)

Splitting function:probability of a quark to emita gluon and reduce momentum by a fraction zN.B.: divergent for soft gluons (z→1)

Scalingviolation!

Cut-offfor soft gluons

Scaling:no Q2

dependence


Experimental techniques

25


HERA accelerator complex

26


Kinematic region

27

Larger sensitivity to sea quarks (gluons)

Larger momentum transfers

(large electronscattering angles)


HERA experiments

28

ZEUS Experiment


Typical DIS event

29

Electron (30 GeV) Proton (820 GeV)

Scattered electron

Quark jet

Scattered electron

Scattered electron

Quark jet

Energy (GeV)

Phi (degrees)

Theta (degrees)

Electromagnetic calorimeter

Hadronic calorimeter

ue

E’


Events with two jets

30

Transverse view Longitudinal view

Jet 1

Jet 2

Muon

Jet 1

Jet 2

Muon

e + p → e + q + anti-q + proton-remnantmuon from decay of heavy quark?


Kinematic reconstructionTo fully characterize a deep inelastic scattering event kinematics both Q2 and x (or y) have to be measured

31

4.4 Hadronic energy determination

The hadronic energies, unlike the case of the positron energy, are not corrected for energy loss inthe passive material. Instead, the transverse momentum of the positron, pTe, calculated usingthe positron energy corrected as described in the last section, is compared to the pTh of thehadrons in both the detector simulation and data. From this comparison, uncertainties in thedetermination of the hadronic energy are estimated. The mean pTh

pTeas a function of !

Hagrees

within 3% between MC simulation and data for the entire range of kinematics covered in thispaper. The pTh

pTedistributions in bins of !

Hare compared (see Sect. 6) for MC simulation and

data, and are in good agreement. An uncertainty of ±3% is assigned to the hadronic energymeasurement, based on these comparisons.

5 Kinematic reconstruction

In deep inelastic scattering, e(k) + p(P ) ! e(k!) + X, the proton structure functions areexpressed in terms of the negative of the four-momentum transfer squared, Q2, and Bjorken x.In the absence of QED radiation,

Q2 = "q2 = "(k " k!)2, (5)

x =Q2

2P · q, (6)

where k and P are the four-momenta of the incoming particles and k! is the four-momentumof the scattered lepton. The fractional energy transferred to the proton in its rest frame isy = Q2/(sx) where s is the square of the total center of mass energy of the lepton-protoncollision (s = 90200 GeV2).

The ZEUS detector measures both the scattered positron and the hadronic system. The fourindependent measured quantities E !

e, "e, #h and pTh, as described in the previous section, over-constrain the kinematic variables x and Q2 (or equivalently, y and Q2).

In order to optimise the reconstruction of the kinematic variables, both the resolution androbustness against possible systematic shifts (stability) of each measured quantity must beconsidered.

For the present analysis, a new method (PT) is used to reconstruct the kinematic variables.The PT method achieves both superior resolution and stability in x and Q2 in the full kine-matic range covered, in comparison with reconstruction methods used in our previous structurefunction measurements.

5.1 Characteristics of standard reconstruction methods

As discussed in Sect. 4, the positron variables, E !e and "e are measured with high precision, and

the systematic uncertainties are small. The kinematic variables calculated from these quantitiesare given by:

ye = 1 "E !

e

2Ee

(1 " cos "e), (7)

Q2e = 2EeE

!

e(1 + cos "e). (8)

8




Hagrees







Q2 = "q2 = "(k " k!)2, (5)

x =Q2

2P · q, (6)









ye = 1 "E !

e

2Ee

(1 " cos "e), (7)

Q2e = 2EeE

!

e(1 + cos "e). (8)

8




Hagrees







Q2 = "q2 = "(k " k!)2, (5)

x =Q2

2P · q, (6)









ye = 1 "E !

e

2Ee

(1 " cos "e), (7)

Q2e = 2EeE

!

e(1 + cos "e). (8)

8

Both variables can be measured e.g. detecting only the scattered electron




Hagrees







Q2 = "q2 = "(k " k!)2, (5)

x =Q2

2P · q, (6)









ye = 1 "E !

e

2Ee

(1 " cos "e), (7)

Q2e = 2EeE

!

e(1 + cos "e). (8)

8More precise methods combine the measurements (energy and polar angle) of both electron and hadronic system


Measured kinematic range

32

x

Q2 (G

eV2 )

b)

a)

ZEUS 1994

Figure 7: a)The distribution of the events in the (x ,Q2) plane. b) The (x ,Q2)-bins used in thestructure function determination. Also indicated are lines of constant y and of constant !

H.

The !H

values for this figure are calculated directly from x and Q2.

35

x

Q2 (G

eV2 )

b)

a)

ZEUS 1994

Figure 7: a)The distribution of the events in the (x ,Q2) plane. b) The (x ,Q2)-bins used in thestructure function determination. Also indicated are lines of constant y and of constant !

H.

The !H

values for this figure are calculated directly from x and Q2.

35


F2 results from HERA

33

x=0.65

x=0.40

x=0.25

x=0.18

x=0.13

x=0.08

x=0.05

x=0.032

x=0.02

x=0.013

x=0.008

x=0.005

x=0.0032

x=0.002

x=0.0013

x=0.0008

x=0.0005

x=0.00032

x=0.0002

x=0.00013

x=0.00008x=0.00005

x=0.000032x=0.00002

(i=1)

(i=10)

(i=20)

(i=24)

Q2 /GeV2

Fp 2+c i(x

)

NMC BCDMS SLAC H1

H1 96 Preliminary(ISR)

H1 97 Preliminary(low Q2)

H1 94-97 Preliminary(high Q2)

NLO QCD FitH1 Preliminary

ci(x)= 0.6 • (i(x)-0.4)

0

2

4

6

8

10

12

14

16

1 10 10 2 10 3 10 4 10 5

g*

i

g*

Approximatescaling, x~10-1

Scaling violation (logarithmic dependence)x<0.01


Parton distributions from HERA

34

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1

HERAPDF1.0 exp. uncert.

model uncert. parametrization uncert.

x

xf 2 = 1.9 GeV2Q

vxu

vxd

0.05)!xS ( 0.05)!xg (

H1 and ZEUS

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1



x

xf 2 = 10 GeV2Q

vxu

vxd 0.05)!xS (

0.05)!xg (

H1 and ZEUS

0.2

0.4

0.6

0.8

1

Figure 18: The parton distribution functions from HERAPDF1.0, xuv, xdv, xS = 2x(U+ D), xg,at Q2 = 1.9 GeV2 (top) and Q2 = 10 GeV2 (bottom). The gluon and sea distributions are scaleddown by a factor 20. The experimental, model and parametrisation uncertainties are shownseparately.

57

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1



x

xf 2 = 1.9 GeV2Q

vxu

vxd

0.05)!xS ( 0.05)!xg (

H1 and ZEUS

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1



xxf 2 = 10 GeV2Q

vxu

vxd 0.05)!xS (

0.05)!xg (

H1 and ZEUS

0.2

0.4

0.6

0.8

1

Figure 18: The parton distribution functions from HERAPDF1.0, xuv, xdv, xS = 2x(U+ D), xg,at Q2 = 1.9 GeV2 (top) and Q2 = 10 GeV2 (bottom). The gluon and sea distributions are scaleddown by a factor 20. The experimental, model and parametrisation uncertainties are shownseparately.

57


References• F.Halzen, A.Martin, Quarks and Leptons, Wiley, Sections 8/9/10• C.Amsler, Kern- und Teilchenphysik, UTB

35

structure of hadrons and the parton model - uzh · v. chiochia (uni. zürich) – phenomenology of...

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