Form Factors and Structure Functionsmoller.physics.berkeley.edu/.../Phys226/hadronStructure.pdfStructure functions are a charge-weighted sum of the quark momentum distributions. Have

Form Factors and Structure FunctionsYury Kolomensky

Physics 226, Fall 2010

Phys226 YGK, Hadronic Structure

Running of Coupling ConstantsGeneric property of any field theory: higher order corrections (loops) induce momentum (distance) dependence of coupling constants.

This is known as an effect of “vacuum polarization”

The magnitude and the direction of the change depends on the type of the interaction

αS(

Q)

Q (GeV) = ℏ/Δx


Running of Coupling Constants• QED (electromagnetic interactions): photons have no

charge, so to first order include only fermion loops• Coupling constant depends on momentum transfer

Q2 ~ ℏ2/Δx2, the number of active fermions nf (with

mass (2mf)2<Q2), and the value of the coupling measured at some physical scale μ (typically, an electron mass).

α(Q2) =α(µ2)

1− nfα(µ2)

3π log (Q2/µ2)

α(M2Z)−1 = 127.916(15)

α−1 = 137.035999084(51)

e+e− → γe+e−


Fine Structure Constant

10log(-Q2/GeV2)

!-1

(Q2 )

L3

small anglelarge angle

128

130

132

134

136

-1 0 1 2 3 4

Figure 3: Measurements of !!1(Q2) for Q2 < 0. The results of the small-angle andlarge-angle Bhabha scattering measurements described in this article are shown as asolid circle and square, respectively. The corresponding reference Q2 values at whichthe value of !!1(Q2) is fixed to its expectation are shown as open symbols. The errorbar on the large-angle point indicates the experimental and the total uncertainty.

10

High Energies:

Low energies:

Radiative Bhabha scattering data


Strong Interactions• Both quarks and gluons contribute to loops

Boson and fermion loops have opposite signs (statistics)

There exists a value of momentum Λ~300 MeV where the coupling constant is ~infinitePerturbative expansion breaks down

α(1 GeV2)≈0.5α(MZ2)=0.1184±0.0007

αS(Q2) =12π

(33− 2nf ) log(Q2/Λ2)


Strong Coupling

9. Quantum chromodynamics 25

The central value is determined as the weighted average of the individual measurements.For the error an overall, a-priori unknown, correlation coe!cient is introduced anddetermined by requiring that the total !2 of the combination equals the number ofdegrees of freedom. The world average quoted in Ref. 172 is

"s(M2Z) = 0.1184 ± 0.0007 ,

with an astonishing precision of 0.6%. It is worth noting that a cross check performed inRef. 172, consisting in excluding each of the single measurements from the combination,resulted in variations of the central value well below the quoted uncertainty, and in amaximal increase of the combined error up to 0.0012. Most notably, excluding the mostprecise determination from lattice QCD gives only a marginally di"erent average value.Nevertheless, there remains an apparent and long-standing systematic di"erence betweenthe results from structure functions and other determinations of similar accuracy. Thisis evidenced in Fig. 9.2 (left), where the various inputs to this combination, evolved tothe Z mass scale, are shown. Fig. 9.2 (right) provides strongest evidence for the correctprediction by QCD of the scale dependence of the strong coupling.

!"## !"#$ !"#%! (" )& '

!"#$%&'(#)*+#,,(-./

012))34)*5678/)

#9:.-#;<)*5678/

012))=.,<)*578/

.>.?)=.,<)@)<AB<)*5578/)

.+.-,$&C.#%)D(,<)*5678/)

.>.?)=.,<)@)<A#B.<)*5578/)

$):.-#;<)*578/

!E0 ! (" ) = 0.1184 ± 0.0007< F

!"#

!"$

!"%

!"(

!")

!&*+,-

# #! #!!,*./012

G.#H;)!"#$%&'(#.>.?))I''(A(+#,(&'0..B)1'.+#<,(-)2-#,,.$('J

K"+;)4LLM

Figure 9.2: Left: Summary of measurements of "s(M2Z), used as input for the

world average value; Right: Summary of measurements of "s as a function of therespective energy scale Q. Both plots are taken from Ref. 172.

July 30, 2010 14:57

Quark confinement

Asymptotic freedom

Two regimes: quarks are asymptotically free at short distances (high energies), but are confined to hadrons when pulled apart (low energies)


Hadronic Structure

• Goal: quantitative understanding of hadronic structureHow does proton mass arise from its constituents (quarks and gluons) How does proton spin arise from its constituents ?

• Complications: Strong interactions, quark confinement Main tool: lepton (electron, muon) scattering High energy probe, no strong interactions between target and

e(k) e(k’)


Accelerator Probes of Matter• “A better magnifying glass”

Need to look deeper into the structure of matter Spatial resolution is limited by wavelength of the

probe Microscope: visible light λ ∼ 1 µm → cell structure Higher energy particles: λ = 2π/p X-rays: λ ∼ 0.01−10 nm → atomic, crystal structure

Charged particles: Rutherford experiment: p ~ 10 keV, λ ∼ 10−10 m Discovery of quarks: p ~ 10 GeV, λ ∼ 0.1 fm LHC: p ~ 1−10 TeV, λ ∼ 10−16−10−17 m → quark


Inclusive Fixed-Target Scattering

Scatter lepton off nucleon (proton or neutron) and measure only the momentum of the scattered lepton2 kinematic variables: scattered momentum magnitude k’ and angle θDefine Lorentz-invariant quantities:

Q2 ≡− q2 = −(k − k)2 = 4EE sin2(θ/2)

ν ≡p · q

M= E − E

W 2 ≡(p + q)2 = M2 + 2Mν −Q2

W2


Elastic Scattering• No extra particles created, target is not broken

apart: W2=M2

Only one free variable left: Q2

Can express the cross section in terms of the cross section for a point-like particle times a form-factor, which says something about the structure:

dσ

dΩ=

dσ

dΩ

point

F (Q2)2


Meaning of the Form Factor• For purely Coulomb interaction (spinless

target, e.g. a 12C nucleus) the form-factor is a Fourier transform of charge distribution:

• Angular dependence is trivial. Can expand in terms of small values of Q2=-q2:

F (q) =

d3rρ(r)eiq·r

F (Q2) =

d3rρ(r)

1 + iq · r − 12(q · r)2 + · · ·

≈1− 16q2r2 Charge radius


Nucleon Form Factors• For nucleons, have to take into account electric

(Coulomb) and magnetic (spin-flip) interactions:

point-like target

Electric form-factor

Magnetic form-factor

τ ≡ Q2

4M2

Measure form factors by measuring dependence of the cross section on the scattering angle θ

dσ

dΩ=

α2

4E2 sin4 θ/2E

E

G2

E + τG2M

1 + τcos2 θ/2 + 2τG2

M sin2 θ/2

r2 = 6

dGE(q2)dq2

q2=0

= (0.81 fm)2


Elastic Form Factor

Determine the average charge radius of the proton:

dσ ∼ Lµν(k, k)Wµν(p, p)Lµν(k, k) = 2

k

µkν + kνkµ − (k · k −m2)gµν


Inelastic Scattering

W2>M2

Since the lepton and the hadronic system do not interact after scattering, can factorize the cross section into leptonic and hadronic tensors (c.f. calculation for the eµ scattering cross section)

Wµµ(p, q = p − p) = −W1gµν +

W2

M2pµpν +

W4

M2qµqν +

W5

M2(pµqν + qµpν)

qµWµν = qνWµν = 0


Inelastic Scattering (cont.)Summing over spins, and assuming parity conservation, we can write the most generic form of the hadronic tensor

Conservation of the EM current leads to two conditions

which means only two functions, W1 and W2, survive. These functions describe the internal structure of the proton and its possible excitations.


Kinematics

Q2 ≡− q2 = −(k − k)2 = 4EE sin2(θ/2)

ν ≡p · q

M= E − E

W 2 ≡(p + q)2 = M2 + 2Mν −Q2

Reminder: two independent Lorentz-invariant variables

d2σ

dEdΩ=

α2

4E2 sin4 θ2

W2(ν, Q2) cos2

θ

2+ 2W1(ν, Q2) sin2 θ

2

Inelastic cross section depends on two structure functions:


Stanford Linear Accelerator Center


End Station AEnd Station A


End Station A: How It All Started

SLAC-R-090 (08/1968)

Experiment E-4: SLAC-MIT-CIT (precursor to discovery of quarks)First high-power LH2 target at SLAC !

Phys226 YGK, Hadronic Structure20

P.N. Kirk et al, PRD8, 63 (1973)SLAC-PUB-656

SLAC 8-GeV Spectrometer


Inelastic Scattering Cross SectionJ.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972).

Elasticpeak

Δ→pπ resonanceDeep Inelastic Scattering regime


Structure Functions in DISJ.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972).

x=1/ω=0.25

p =p + q

E =E − Q2

2M

ν =E − E =Q2

2M

νW point2 =δ

1− Q2

2M

2MW point1 =

Q2

2Mνδ

1− Q2

2M


Scaling in DIS• Consider scattering off point-like target (e.g.

muon) Scattering is elastic, so

νW point2 =δ (1− x)

2MW point1 =xδ (1− x)


Bjorken Scaling

If the structure functions W1,2 do not depend on Q2, but only on Bjorken x, it hints at scattering off point-like objects

x =Q2

2p · q=

Q2

2M(E − E)

y =p · q

p · k=

E − E

E= 1− E

E

Define two dimensionless variables:

Bjorken x (scaling var)

Fraction of incident energy carried away by hadronic remnants

For point-like targets,


Scaling In DISJ.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972).

x=1/ω=0.25


Nobel Prize in Physics, 1990

MW1 →F1(x)νW2 →F2(x)


Parton Model of DIS

dxiq22dσ ≡

i

x’p

q

DIS cross section is an incoherent sum of elastic scattering cross sections off individual constituents (partons).We now these partons as asymptotically free quarks.Parton model should work perfectly for Q2→∞ where the strong coupling constant is zero. In this limit


Parton Distribution Functions• Define parton momentum distribution

probability to find a parton with momentum fraction x of the proton

Then

fi(x) ≡ dPi

dx

F2(x) =

i

q2i xfi(x)

F1(x) =12x

F2(x) Callan-Gross relation

i

1

0dxfi(x) = 1where


Proton Structure Functions

Valence quarks

sea quarks

gluons

Structure functions are a charge-weighted sum of the quark momentum distributions. Have to account for “valence” quarks (which are always in the proton) and “sea” quarks (which pop out of the gluon field)

F p2 (x)x

=

23

2

[u(x) + u(x)]

+

13

2

[d(x) + d(x)]

+

13

2

[s(x) + s(x)]

Fn2 (x)x

=

23

2

[d(x) + d(x)]

+

13

2

[u(x) + u(x)]

+

13

2

[s(x) + s(x)]


Proton and Neutron Structure Functions

(ignore charm and bottom quarks, which do not contribute much, except at very high energies, e.g. LHC)

Isospin symmetry implies interchange up(x)↔dn(x):

uV (x) ≡u(x)− u(x)dV (x) ≡d(x)− d(x)

1

0dxuV (x) ≡

1

0dxu(x)− u(x) = 2

1

0dxdV (x) ≡

1

0dxd(x)− d(x) = 1

1

0dxsV (x) ≡

1

0dxs(x)− s(x) = 0


Valence Quark Distributions

Sum rules:

Expect maximum to be near x~1/3


Valence Quark DistributionJ.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972).


Proton vs Neutron

J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972).

F 2n /F

2p

Sea dominates

uV dominates

Fermi motion correction


Sea Quarks

Sea quarks are produced by splitting of the gluons into quark-antiquark pairs. Gluons are in turn radiated off valence quarks… Both of these processes make sea quarks very soft (small momentum, which means small x). Expect probability to find a sea quark to increase steeply at low x


Photon-Nucleon Cross Section


Unpolarized Structure FunctionsFrom PDG08

valence quarksdominate

sea quarksdominate

Scal

ing

viol

atio

ns !


Parton Distributions


Perturbative QCD• QCD is based on the SU(3) symmetry of color

Non-abelian group: gauge carriers charged under group 3 quark colors, 8 colored gluons Predicts asymptotic freedom Gross, Wilczek, Politzer (1973)

• Perturbative QCD: perturbative expansion in terms of αS Log dependence on Q2: scaling violations in DIS Confirmed by: Measuring running of αS(Q2) QCD sum rules Q2 evolution of parton distributions e+e– annihilation into hadrons, jets, etc.


Nobel Prize in Physics, 2004

εq + εg = 1

1

0F p

2 (x)dx =49εu +

19εd ≈ 0.18

1

0Fn

2 (x)dx =49εd +

19εu ≈ 0.12


Momentum Sum Rule

Momentum fraction carried by all quarks

Basic momentum sum rule:

Naively, would expect εq≈1. However, DIS measures

εq=0.48 predicted in 1974 by Gross and Wilczek from perturbative QCD (PRD 9, 980 (1974)]: consequence of the color structure of SU(3). First indirect evidence for the gluon

1

0dx x[u + u + d + d + s + s] ≡ εq

εq=0.54 at low Q2 (ignoring strange quarks) with error of about 10%.

1

0

F p2 (x)− Fn

2 (x)x

dx =13

+23

1

0[u(x)− d(x)]dx =

13

0.8

0.004

F p2 (x)− Fn

2 (x)x

dx = 0.2281± 0.0065

u(x) = d(x)

u(x) = d(x) = s(x)


Gottfried Sum Rule

Valid for symmetric sea, i.e.

Experimentally:

(NMC, Q2=4 GeV2)

Interpretation: sea quark distributions are asymmetric, i.e.

Most likely consequence of small quark mass difference


Scaling Violations• At finite Q2, the strong coupling constant αS is not

small Corrections of order O(αS) can be important For example, here are some contributions to DIS

The virtual diagrams typically lead to O(1+αS(Q2)/π) factor in the cross section (log dependence of F2 on Q2)

First term is qualitatively different. It changes the momentum fraction x of the parton. In other words, it evolves parton PDF fi(x)

To find the distribution sampled by the photon, need to integrate over all y, or equivalently, all gluon momenta

Proton Structure, Partons, QCD, DGLAP and Beyond 2041

to ensure the continuity6 of !s. For example at the b-quark threshold wewill require !s(m2

b , 4) = !s(m2b , 5). Finally, recall that perturbative QCD

tells us how the coupling varies with scale, but not the absolute value itself.The latter is obtained from experiment. Traditionally the value is quoted atQ = MZ in the MS scheme for nf = 5; the current value, determined frommany independent experiments is

!s!

M2Z

"

= 0.1176 ± 0.002 . (44)

Lattice QCD can also predict the value of the coupling. It is encouragingthat these very di!erent determinations are found to be consistent with eachother.

6. Running parton densities: DGLAP equations

In terms of QCD perturbation theory, the QPM formula (20) may beregarded as the zeroth-order term in the expansion of F2 as a power seriesin !s. To include the O(!s) QCD corrections, we have to calculate thephoton–parton subprocess diagrams shown in Fig. 12. The QPM of Section 3is the first diagram in the second | . . . |2. Due to the propagator of the virtualquark, the first diagram in the first | . . . |2 is proportional to (yp ! k)!2 "(2p · k)!1, for massless quarks. It therefore has a collinear divergence whenthe gluon of 4-momentum k is emitted parallel to the incoming quark of4-momentum yp.

Fig. 12. The O(!s) diagrams which contribute to the proton structure function.On the first diagram we show new variables, the longitudinal momentum fractiony carried by the quark and the transverse momentum kt of the emitted gluon,which must be integrated over.

The partonic approach is only valid if the second diagram in the first| . . . |2 can be neglected. Then, the emitted gluon can be considered as part ofthe proton structure. It turns out that both diagrams are required to ensuregauge invariance, but that the second only plays the role of cancelling thecontributions from the unphysical polarisation states of the gluon. Adoptinga physical gauge, in which we sum only over transverse gluons, only the firstdiagram remains.

6 Beyond two-loops there are discontinuities in !s at the flavour thresholds. Of course,the observables are continuous, since the above discontinuities are cancelled by onesoccurring in the coe!cients functions, see (75) below.

A.D. MartinActa Physica Polonica B39, 2025 (2008)

δfi(x,Q2) =αS

2πlog

Q2

µ2

1

x

dy

yfi(y)Pqq(x/y)


Evolution of Parton PDFs

where

Taking into account gluon radiation, we can writeF2(x,Q2)

x=

i

q2i [fi(x) + δfi(x,Q2)]

some arbitrary momentum scale quark-quark

“splitting function”PDF measured at Q2=µ2

dfi(x,Q2)d log Q2

=αS

2π

1

x

dy

yfi(y)Pqq(x/y)

Can rewrite it as a differential equation:

DGLAP equation


DGLAP Equations

Nuclear Physics B126 (1977) 298-318 © North-Holland Publishing Company

ASYMPTOTIC FREEDOM IN PARTON LANGUAGE

G. ALTARELLI * Laboratoire de Physique Th~orique de l'Ecole Normale Sup~rieure **, Paris, France

G. PARISI *** Institut des Hautes Etudes Scientifiques, Bures-sur- Yvette, France

Received 12 April 1977

A novel derivation of the Q2 dependence of quark and gluon densities (of given helicity) as predicted by quantum chromodynamies is presented. The main body of predictions of the theory for deep-inleastic scattering on either unpolarizedor polarized targets is re-obtained by a method which only makes use of the simplest tree diagrams and is entirely phrased in parton language with no reference to the conventional operator formalism.

I . Introduction

The quark parton model [1 ] provides us with a very useful and simple description of the physics of deep inelastic phenomena [2]. The theoretical framework which justifies the parton model is given by the asymptotically free gauge theory of strong interactions based on the color degrees of freedom [3] (quantum chromodynamics, QCD). Although scaling is predicted to be broken by logarithms (a fact which appears to be well consistent with present experiments), the deviations from scaling can be and have been computed for deep inelastic structure functions for either unpolarized [4,5] or polarized targets [6,7]. In the leading logarithmic approximation, the results can again be phrased in the parton language by assigning a well determined Q2 depend. ence to the parton densities. In spite of the relative simplicity of the final results, their derivation, although theoretically rigorous, is somewhat abstract and formal, being formulated in the language of renormalization group equations for the coeffi- cient functions of the local operators which appear in the light cone expansion for the product of two currents.

* On leave of absence from the Istituto di Fisica dell'Universit~ di Roma. ** Laboratoire propre du CNRS associ6 ~ l'Ecole Normale Sup&ieure et ~ l'Universit~ de Paris-

Sud. Postal address: 24, rue Lhomond, 75231 Paris Cedex 05, France. *** On leave of absence from the Laboratori Nazionali di Frascati.

298

Yuri Dokshitzer Vladimir Gribov Lev Lipatov Guido Altarelli Giorgio ParisiNucl. Phys. B126, 298 (1977)Sov.J.Nucl.Phys. 15, 438 (1972)

Яд. Физ. 15, 781(1972)Sov.Phys. JETP 46, 641 (1977)ЖЭТФ 73, 1216 (1977)

Describe evolution of the quarkand gluon distribution functionswith Q2

Σ ≡

q

q + q

qV ≡ q − q

P ⊗ q ≡ 1

x

dy

yP (x/y)q(y)


DGLAP Equations

Proton Structure, Partons, QCD, DGLAP and Beyond 2045

It is convenient to introduce flavour singlet (! ) and non-singlet (qNS)quark distributions:

! =!

i

(qi + qi) . (61)

An example of a non-singlet is the up valence distribution

uv = u ! u . (62)

Non-singlet evolution satisfies (50) and decouples from the singlet and gluonevolution equations, which are coupled together as follows

!

!logQ2

"

!

g

#

="s

2#

"

Pqq 2nfPqg

Pgq Pgg

#

""

!

g

#

. (63)

In general the splitting functions can be expressed as a power series in "s

Pab("s, z) = PLOab (z) + "sP

NLOab (z) + "2

sPNNLOab (z) + . . . , (64)

where the NLO expressions were computed in the period 1977–80 andthe NNLO in the period ending 2004. Leading order (LO) DGLAP evolution,which we have outlined, sums up the leading log contributions("slogQ2)n, and next-to-leading order evolution includes the summation ofthe "s("slogQ2)n!1 terms.

If we are given the x dependence of the parton densities at some inputscale Q2

0 then we may solve the evolution equations to determine them athigher Q2. Frequently this is performed simply by step-by-step integrationup in Q2.

An alternative procedure is to rewrite the equations in terms of moments,which for an arbitrary function f(z) are defined as

f (n) =

1$

0

dz

zznf(z) . (65)

If we now multiply the DGLAP equation (50) by xn!1, and integrate over x,we obtain

!

!logQ2

1$

0

xn!1qNS%

x,Q2&

dx ="s

2#

1$

0

yn!1qNS%

y,Q2&

dy

1$

0

zn!1Pqq(z)dz

(66)using x = yz. That is, the evolution equation then turns into an ordinarylinear di!erential equation for the moments,

!q(n)NS

!logQ2=

"s

2#P (n)

qq q(n)NS . (67)

2044 A.D. Martin

where (1 ! z)+ = (1 ! z) for z < 1. Ensuring the constraint (51) gives

Pqq =4

3

1 + z2

(1 ! z)++ 2!(1 ! z) . (53)

Our O("s) treatment is still not complete. In addition to the #q " gqsubprocesses shown in Fig. 12, at O("s), we need to include the #g " qqprocesses. Then the DGLAP evolution equation (50) for the quark densityq # fq becomes

$q!

x,Q2"

$logQ2=

"s

2%(Pqq $ q + Pqg $ g) , (54)

where g # fg is the gluon density, and Pqq # P is the q " q(g) splittingfunction of (53). It can be shown that the g " q splitting function

Pqg =1

2

!

z2 + (1 ! z)2"

. (55)

In general Pab describes the b " a parton splitting. Also in (54) we haveused $ to abbreviate the convolution integral

P $ f #1

#

x

dy

yfq(y) P

$

x

y

%

. (56)

Clearly we must also consider the evolution of the gluon density

$g!

x,Q2"

$logQ2=

"s

2%

&

'

i

Pgq $ (qi + qi) + Pgg $ g

(

, (57)

where the sum is over the i quark flavours, and where the q " g and g " gsplitting functions can be shown to be

Pgq = Pqq(1 ! z) =4

3

1 + (1 ! z)2

z, (58)

Pgg = 6

$

1 ! z

z+

z

(1 ! z)++ z(1 ! z)

%

+

$

11

2!

nf

3

%

!(1 ! z) . (59)

Here the coe!cient of the !(1! z) term can be obtained from the constraintthat all of the momentum of the proton must be carried by its constituents

1#

0

dz z

&

'

i

!

qi!

z,Q2"

+ qi!

z,Q2""

+ g!

z,Q2"

(

= 1 (60)

for all Q2.

VV

where and

Interpretation: quark and gluon PDFs feed off each other, increasing at larger Q2 and smaller x. This introduces logarithmic scaling violations to DIS. DIS is sensitive to gluon distributions at O(αS). 2048 A.D. Martin

Fig. 13. Schematic picture of the factorisation theorem for a deep inelastic structurefunction of the proton.

where H denotes the triggered hard system, such as a weak boson, a pairof jets, a Higgs boson etc. The typical hard scale Q could be the invariantmass of H or the transverse momentum of a jet. Then according to thefactorisation theorem the cross-section is of the form

! =!

i,j

1"

xmin

dx1dx2 fi#

x1, µ2F

$

fj#

x2, µ2F

$

!ij#

x1p1, x2p2, Q . . . ;µ2F

$

, (77)

where typically xmin>! Q2/s where s = (p1 + p2)2. For the production of

a system H of invariant mass M and rapidity y, the momentum fractionsx1,2 = Me±y/

"s. The fi and ! depend on the renormalisation scale µR via

"s(µ2R). For instance

!ij = "ks

n!

m=0

C(m)ij "m

s , (78)

where LO, NLO. . . correspond to n=1,2. . . ; note that, for example, k=0,2,. . .for W , dijet,. . . production. We should work to the same order in the seriesexpansion of the splitting functions. In practical applications it is usual tochoose µF = µR ! Q and to use variations about this value to estimate theuncertainty in the predictions. Of course the physical cross-section ! doesnot depend on the scales, but the truncation of the perturbative series bringsin scale dependence. If we truncate at order "n

s , then the uncertainty is oforder "n+1

s .

9. Global parton analyses

Two groups (CTEQ [10] and MRST [11]) have used all available deepinelastic and related hard scattering data involving incoming protons (andantiprotons) to determine the parton densities, fi, of the proton. The pro-cedure is to parametrise the x dependence of fi(x,Q2

0) at some low, yetperturbative, scale Q2

0. Then to use the DGLAP equations to evolve the fi

up in Q2, and to fit to all the available data (DIS structure functions, Drell–Yan production, Tevatron jet and W production. . . ) to determine the val-ues of the input parameters. In principle there are 11 parton distributions

Non-perturbative PDFscannot be computed

“Hard-scattering” cross sections:can be computed perturbatively

A.D. MartinActa Physica Polonica B39, 2025 (2008)

Can factorize the cross section:


Parton Distributions


Spin (Polarized) Structure FunctionsScattering of polarized beam off polarized targets

Spin sum rule:

Experiment:

The rest (70%) of the spin has to come from gluons and orbital angular momentum

(this used to be known as “Spin Crisis” of parton model)


Polarized Parton Distributions12 16. Structure functions

!

!"#

!"$

%!"#

!

%!"&

!

!"&

%!"&

!

!"&

'()*(++*,,-*./++01

!2304

01!5304

01!2304

01!5304

0

01!6304

!

!"&

&!%#

&!%&

Figure 16.5: Distributions of x times the polarized parton distributions !q(x)(where q = u, d, u, d, s) using the LSS2006 [45], AAC2008 [46], and DSSV2008 [47]parameterizations at a scale µ2 = 2.5 GeV2, showing the error corridor of the latterset (corresponding to a one-unit increase in !2). Points represent data from semi-inclusive positron (HERMES [51,52]) and muon (SMC [53] and COMPASS [54])deep inelastic scattering given at Q2 = 2.5 GeV2. SMC results are extracted underthe assumption that !u(x) = !d(x).

The hadronic photon structure function, F !2 , evolves with increasing Q2 from

the ‘hadron-like’ behavior, calculable via the vector-meson-dominance model, to thedominating ‘point-like’ behaviour, calculable in perturbative QCD. Due to the point-likecoupling, the logarithmic evolution of F !

2 with Q2 has a positive slope for all values of x,see Fig. 16.14. The ‘loss’ of quarks at large x due to gluon radiation is over-compensatedby the ‘creation’ of quarks via the point-like " ! qq coupling. The logarithmic evolutionwas first predicted in the quark–parton model ("!" ! qq) [63,64], and then in QCD inthe limit of large Q2 [65]. The evolution is now known to NLO [66–68]. NLO data

July 30, 2010 14:36



e+e- Annihilation into HadronsRecall from QED

What is the difference for quarks ? Charge, and the fact that they are confined (not free). If they were free, cross section would be:

e–

e+

µ–

µ+e e

e–

e+

q

qe q _ count 3 colors

How do we account for hadronization, i.e. production of hadrons from quarks ? Quark-hadron duality theorem: the sum over all possible hadronic final states away from bound states is equivalent to quark cross section. Basically, it means that probability for a quark to hadronize into something is unity.

σ(e+e− → µ+µ−) =4πα

3s≈ 87 nb

s (GeV)22

σ(e+e− → qq) =4πα

3s· q2 · 3

2

R ≡ σ(e+e− → qq)σ(e+e− → µ+µ−)

= 3

i

q2i

1 +

αS

π


e+e- Annihilation into Hadrons

e–

e+e q

hadrons

hadrons

Taking into account corrections of O(αS) due to gluon radiation,

Here the sum is over all quarks that can be produced at the CM energy of the collision, i.e. (2mq)2 < s

Ratio R measures the number of colors, and quark masses, i.e. fundamental parameters of QCD

σ(e+e− → qq) =4πα

3s

i

3q2i

2


Hadronic R Ratio

(u,d,s): R≈2(u,d,s,c): R≈10/3 (u,d,s,c,b): R≈11/3

Documents

Form Factors and Structure Functionsmoller.physics.berkeley.edu/.../Phys226/hadronStructure.pdfStructure functions are a charge-weighted sum of the quark momentum distributions. Have