The Structure of the ProtonA.M.Cooper-Sarkar

Feb 6th 2003

RSE•Parton Model

•QCD as the theory of strong interactions

•Parton Distribution Functions

•Extending QCD calculations across the kinematic plane – understanding small-x, high density, non-perturbative regions

d~

2

LW

Ee

E

Ep

q = k – k,Q2 = -q2

Px = p + q , W2 = (p + q)2

s= (p + k)2

x = Q2 / (2p.q)

y = (p.q)/(p.k)

W2 = Q2 (1/x – 1)

Q2 = s x y

s = 4 Ee Ep

Q2 = 4 Ee Esin2e/2y = (1 – E/Ee cos2e/2)x = Q2/sy

The kinematic variables are measurable

Leptonic tensor - calculable

Hadronic tensor- constrained by

Lorentz invariance

d2(e±N) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) Y_xF3(x,Q2)], Y = 1 (1-y)2

dxdy

F2, FL and xF3 are structure functions –

The Quark Parton Model interprets themd = ei

2 xs [ 1 + (1-y)2] , for elastic eqQ4 dy

d2= s [ 1 + (1-y)2] i ei2(xq(x) + xq(x))

dxdy Q4

for eNisotropic

non-isotropic

Now compare the general equation to the QPM prediction

F2(x,Q2) = i ei2(xq(x) + xq(x)) – Bjorken scaling

FL(x,Q2) = 0 - spin ½ quarks

xF3(x,Q2) = 0 - only exchange

(xP+q)2=x2p2+q2+2xp.q ~ 0

for massless quarks and p2~0 so

x = Q2/(2p.q)

The FRACTIONAL momentum of the incoming nucleon taken by the struck

quark is the MEASURABLE quantity x

4

22

Q

sfor charged lepton hadron scattering

Compare to the general form of the cross-section for / scattering via W+/-

FL (x,Q2) = 0

xF3(x,Q2) = 2ix(qi(x) - qi(x))

Valence

F2(x,Q2) = 2ix(qi(x) + qi(x))

Valence and Sea

And there will be a relationship between F2

eN and F2

NOTE scattering is FLAVOUR sensitive

-

d

uW+

W+ can only hit quarks of charge -e/3 or antiquarks -2e/3

p)~ (d + s) + (1- y)2 (u + c)

p) ~ (u + c) (1- y)2 + (d + s)

Consider , scattering: neutrinos are handed

d()= GF2 x s d() = GF

2 x s (1-y)2

dy dy For q (left-left) For q (left-right)

d2() = GF2 s i [xqi(x) +(1-y)2xqi(x)]

dxdy For N

d2() = GF2 s i [xqi(x) +(1-y)2xqi(x)]

dxdy For N

Clearly there are antiquarks in the

nucleon

3 Valence quarks plus a flavourless

qq Sea

q = qvalence +qsea q = qsea qsea= qsea

)(9

4)(

9

1)(

9

1)(

9

4()(2 ccssdduuxplF

)()(),(3 vv duxdduuxNxF

So in scattering the sums over q, q ONLY contain the appropriate flavours BUT-

high statistics data are taken on isoscalar targets e.g. Fe (p + n)/2=N

d in proton = u in neutron

u in proton = d in neutron

A TRIUMPH(and 20 years of understanding the c c contribution)

3)(1

0

3 dxdudxx

xFvv

)(),(2 ccssdduuxNF

1

0 2 ?1),( dxNF

)(

5

8)(

5

2

18

5)(2 ccssdduuxlNF

)(18

5)( 22 NFlNF

GLS sum rule

Total momentum of quarks

QCD improves the Quark Parton Model

What if

or

Before the quark is struck?

F2(N) dx ~ 0.5 where did the momentum go?

Pqq Pgq

Pqg Pgg

Note q(x,Q2) ~ s lnQ2, but s(Q2)~1/lnQ2, so s lnQ2 is O(1), so we must sum all terms

sn lnQ2n

Leading Log

Approximation

x decreases fromtarget to probe

xi-1> xi > xi+1….pt

2 of quark relative to proton increases from target to probe

pt2

i-1 < pt2

i < pt2 i+1

Dominant diagrams have STRONG pt ordering

ss(Q2)

The DGLAP equations

xi+1

xi

xi-1

x xy y

y > x, z = x/y

Terrific expansion in measured range across the x, Q2 plane throughout the 90’s

HERA data

Pre HERA fixed target p,D NMC,BDCMS, E665 and , Fe CCFR

Bjorken scaling is broken – ln(Q2)N

ote

stro

ng r

ise

at s

mal

l x

•Valence distributions evolve slowly

•Sea/Gluon distributions evolve fast

•Parton Distribution Functions PDFs are extracted by MRST, CTEQ, ZEUS, H1

•Parametrise the PDFs at Q20 (low-scale)

xuv(x) =Auxau (1-x)bu (1+ u √x + u x)

xdv(x) =Adxad (1-x)bd (1+ d √x + d x)

xS(x) =Asx-s (1-x)bs (1+ s √x + s x)

xg(x) =Agx-g(1-x)bg (1+ g √x + g x)

Some parameters are fixed through sum rules-

others by model choices- typically ~15 parameters

•Use QCD to evolve these PDFs to Q2 > Q20

•Construct the measurable structure functions in terms of PDFs for ~1500 data points across the x,Q2 plane

•Perform 2 fit

Note scale of xg,

xS

Not

e er

ror

band

s on

PD

Fs

The fact that so few parameters allows us to fit so many data points established QCD as the THEORY OF THE STRONG INTERACTION and provided the first measurements of s (as one of the fit parameters)

These days we assume the validity of the picture to measure PDFs which are transportable to other hadronic processes

But where is the information coming from?

F2(e/p)~ 4/9 x(u +u) +1/9x(d+d)

F2(e/D)~5/18 x(u+u+d+d)

u is dominant , valence dv, uv only accessible at high x

(d and u in the sea are NOT equal, dv/uv 0 as x 1)Valence information at small x only from xF3(Fe) xF3(N) = x(uv + dv) - BUT Beware Fe target!

HERA data is just ep: xS, xg at small x

xS directly from F2

xg indirectly from scaling violations dF2 /dlnQ2

Fixed target p/D data- Valence and Sea

HERA at high Q2 Z0 and W+/- become as important as

exchange NC and CC cross-sections comparable

for NC processes

F2 = i Ai(Q2) [xqi(x,Q2) + xqi(x,Q2)]

xF3= i Bi(Q2) [xqi(x,Q2) - xqi(x,Q2)]

Ai(Q2) = ei2 – 2 ei vi ve PZ + (ve

2+ae2)(vi

2+ai2) PZ

2

Bi(Q2) = – 2 ei ai ae PZ + 4ai ae vi ve PZ2

PZ2 = Q2/(Q2 + M2

Z) 1/sin2W

a new valence structure function xF3 measurable from

low to high x- on a pure proton target

sensitivity to sin2W, MZ and electroweak couplings vi, ai for u and d type quarks- (with electron beam polarization)

CC processes give flavour information

d2(e-p) = GF2 M4

W [x (u+c) + (1-y)2x (d+s)] dxdy 2x(Q2+M2

W)2

d2(e+p) = GF2 M4

W [x (u+c) + (1-y)2x (d+s)] dxdy 2x(Q2+M2

W)2

MW informationuv at high x dv at high x

Measurement of high x dv on a pure proton target

(even Deuterium needs corrections, does dv/uv 0, as x 1? )

Valence PDFs from ZEUS data alone-

NC and CC e+ and e- beams high x

valence dv from CC e+, uv from CC e-

and NC e+/-

Valence PDFs from a GLOBAL fit to all DIS data high x valence from CCFR xF3(,Fe) data and NMC F2(p)/F2(D) ratio

Parton distributions are transportable to other processes

Accurate knowledge of them is essential for calculations of cross-sections of any process involving hadrons. Conversely, some processes have been used to get further information on the PDFs E.G

DRELL YAN – p N +- X, via q q * +-, gives information on the Sea

Asymmetry between pp +- X and pn +- X gives more information on d - u difference

W PRODUCTION- p p W+(W-) X, via u d W+, d u W- gives more information on u, d differences

PROMPT - p N X, via g q q gives more information on the gluon

(but there are current problems concerning intrinsic pt of initial partons)

HIGH ET INCLUSIVE JET PRODUCTION – p p jet + X, via g g, g q, g q subprocesses gives more information on the gluon –

for ET > 200 GeV an excess of jets in CDF data appeared to indicate new physics beyond the Standard Model BUT a modification of the u PDF which still gave a ‘reasonable fit’ to other data could explain it

Cannot search for physics within (Higgs) or beyond (Supersymmetry) the Standard Model without knowing EXACTLY what the Standard Model predicts – Need estimates of the PDF uncertainties

2 = i [ FiQCD – Fi

MEAS]2

(iSTAT)2+(i

SYS)2 = i2

Errors on the fit parameters evaluated from 2 = 1, can be propagated back to the PDF shapes to give uncertainty bands on the predictions for structure functions and cross-sections

THIS IS NOT GOOD ENOUGH

Experimental errors can be correlated between data points- e.g. Normalisations

BUT there are more subtle cases- e.g. Calorimeter energy scale/angular resolutions can move events between x,Q2 bins and thus change the shape of experimental distributions

2 = i j [ FiQCD – Fi MEAS] Vij

-1 [ FjQCD – Fj

MEAS]

Vij = ij(iSTAT)2 +

SYS jSYS

Where iSYS is the correlated error on point i due to systematic error source

2 = i [ FiQCD – si

SYS- Fi MEAS]2 + s2

(iSTAT) 2

s are fit parameters of zero mean and unit variance

modify the measurement/prediction by each source of systematic uncertainty

HOW to APPLY this ?

OFFSET method

1. Perform fit without correlated errors

2. Shift measurement to upper limit of one of its systematic uncertainties (s = +1)

3. Redo fit, record differences of parameters from those of step 1

4. Go back to 2, shift measurement to lower limit (s= -1)

5. Go back to 2, repeat 2-4 for next source of systematic uncertainty

6. Add all deviations from central fit in quadrature

HESSIAN method

Allow fit to determine the optimal values of s

If we believe the theory why not let it calibrate the detector?

In a global fit the systematic uncertainties of one experiment will correlate to those of another through the fit

We must be very confident of the theory/model

xg(x)O

ffset method

Hessian m

ethod

Q2=2.5 Q2=7

Q2=20 Q2=200

Model Assumptions – theoretical assumptions later!

•Value of Q20, form of the parametrization

•Kinematic cuts on Q2, W2, x

•Data sets included …….

Changing model assumptions changes parameters

model error

MODEL EXPERIMENTAL - OFFSET

MODEL EXPERIMENTAL - HESSIAN

You win some and you lose some!

The change in parameters under model changes is frequently outside the 2=1 criterion of the central fit

e.g. the effect of using different data sets on the value of s

Are some data sets incompatible? PDF fitting is a compromise, CTEQ suggest 2=50 may be a more reasonable criterion for error estimation – size of errors determined by the Hessian method rises to the size of errors determined by the Offset method

Comparison of ZEUS (offset) and H1(Hessian) gluon distributions –

Yellow band (total error) of H1 comparable to red band (total error) of ZEUS

Comparison of ZEUS and H1 valence distributions. ZEUS plus fixed target data H1 data alone- Experimental errors alone by OFFSET and HESSIAN methods resp.

Value of s and shape of gluon are correlated

s increases harder gluon

dF2 = s(Q2) [ Pqq F2 + 2 i ei2 Pqg xg]

dlnQ2 2

NLOQCD fit results

Determinations of s

Pqq(z) = P0qq(z) + s P1qq(z) +s2 P2qq(z)

LO NLO NNLO

Theoretical Assumptions- Need to extend the formalism?

What if

Optical theorem2 Im

The handbag diagram- QPM

QCD at LL(Q2)

Ordered gluon ladders (s

n lnQ2 n)

NLL(Q2) one rung disordered s

n lnQ2 n-1

Or higher twist diagrams?

low Q2, high x

Eliminate with a W2 cut

BUT what about completely disordered

Ladders?

Ways to measure the gluon distribution Knowledge increased dramatically in the 90’s

Scaling violations dF2/dlnQ2 in DIS

High ET jets in hadroproduction- Tevatron

BGF jets in DIS * g q q

Prompt

HERA charm production * g c c

For small x scaling violation data from HERA are most accurate

Pre HERAPost HERA

xg(x,Q2) ~ x -g

At small x,

small z=x/y

Gluon splitting functions become singular

t = ln Q2/2

s ~ 1/ln Q2/2

Gluon becomes very steep at small x AND F2 becomes gluon dominated

F2(x,Q2) ~ x -s, s=g -

,)/1ln(

)/ln(12 2

1

0

0

x

ttg

Still it was a surprise to see F2 still steep at small x - even for Q2 ~ 1 GeV2

should perturbative QCD work? s is becoming large - s at Q2 ~ 1 GeV2 is ~ 0.32

The steep behaviour of the gluon is deduced

from the DGLAP QCD formalism – BUT the

steep behaviour of the Sea is measured from

F2 ~ x -s, s = d ln F2

MRST PDF fit

xg(x) ~ x -g

xS(x) ~ x -s

at low x

For Q2 >~ 5 GeV2

g > s

g

s

Perhaps one is only surprised that the onset of the QCD generated rise appears to happen at Q2 ~ 1 GeV2

not Q2 ~ 5 GeV2 d ln 1/x

Need to extend formalism at small x?

The splitting functions Pn(x),

n= 0,1,2……for LO, NLO, NNLO etc

Have contributions

Pn(x) = 1/x [ an ln n (1/x) + bn ln n-1 (1/x) ….

These splitting functions are used in

dq/dlnQ2 ~ s dy/y P(z) q(y,Q2)

And thus give rise to contributions to the PDF

s p (Q2) (ln Q2)q (ln 1/x) r

Conventionally we sum

p = q ≥ r ≥ 0 at Leading Log Q2 - (LL(Q2))

p = q+1 ≥ r ≥ 0 at Next to Leading log Q2 (NLLQ2) – DGLAP summations

But if ln(1/x) is large we should consider

p = r ≥ q ≥ 1 at Leading Log 1/x (LL(1/x))

p = r+1 ≥ q ≥ 1 at Next to Leading Log (NLL(1/x)) - BFKL summations

LL(Q2) is STRONGLY ordered in pt.

At small x it is also STRONGLY ordered in x – Double Leading Log Approximation

s p (Q2) (ln Q2)q(ln 1/x)r, p=q=r

LL(1/x) is STRONGLY ordered in ln(1/x) and can be disordered in pt

s p(Q2)(ln 1/x)r, p=r

BFKL summation at LL(1/x)

xg(x) ~ x -

= s CA ln2 ~ 0.5, for s ~ 0.25

steep gluon even at moderate Q2

But this is considerable softened at NLL(1/x)

( way beyond scope !)

Furthermore if the gluon density becomes large there maybe non-linear effects

Gluon recombination g g g

~ s22/Q2

may compete with gluon evolution g g g

~ s

where is the gluon density

~ xg(x,Q2) –no.of gluons per ln(1/x)R2 nucleon size

Non-linear evolution equations – GLR

d2xg(x,Q2) = 3s xg(x,Q2) – s2 81 [xg(x,Q2)]2

dlnQ2dln1/x 16Q2R2

s s2 2/Q2

The non-linear term slows down the evolution of xg and thus tames the rise at

small x

The gluon density may even saturate

(-respecting the Froissart bound)

Extending the conventional DGLAP equations across the x, Q2 plane

Plenty of debate about the positions of these lines!

Colour Glass Condensate, JIMWLK, BK

Higher twist

Do the data NEED unconventional explanations ?

In practice the NLO DGLAP formalism works well down to Q2 ~ 1 GeV2

BUT below Q2 ~ 5 GeV2 the gluon is no longer steep at small x – in fact its becoming NEGATIVE!

We only measure F2 ~ xq

dF2/dlnQ2 ~ Pqg xg

Unusual behaviour of dF2/dlnQ2 may come from

unusual gluon or from unusual Pqg- alternative evolution?

We need other gluon sensitive measurements at low x

Like FL, or F2charm

`Valence-like’ gluon shape

Current measurements of FL and F2charm at small x are not yet accurate enough to

distinguish different approaches

FLF2 charm

xg(x)

Q2 = 2GeV2

The negative gluon predicted at low x, low Q2 from NLO DGLAP remains at NNLO (worse)

The corresponding FL is NOT negative at Q2 ~ 2 GeV2 – but has peculiar shape

Including ln(1/x) resummation in the calculation of the splitting functions (BFKL `inspired’) can improve the shape - and the 2 of the global fit improves

Are there more defnitive signals for `BFKL’ behaviour?

In principle yes, in the hadron final state, from the lack of pt ordering

However, there have been many suggestions and no definitive observations-

We need to improve the conventional calculations of jet production

The use of non-linear evolution equations also improves the shape of the gluon at low x, Q2

The gluon becomes steeper (high density) and the sea quarks less steep

Non-linear effects gg g involve the summation of FAN diagrams –

Q2 = 1.4 GeV2

Q2 = 2 Q2=10 Q2=100 GeV2

Non linear

DGLAP

xg

xu

xs

xuv

xd

xc

Such diagrams form part of possible higher twist contributions at low x there maybe further clues from lower Q2 data?

xg

Linear DGLAP evolution doesn’t work for Q2 < 1 GeV2, WHAT does? – REGGE ideas?

Reg

ge r

egio

npQ

CD

reg

ion

Small x is high W2, x=Q2/2p.q Q2/W2

(*p) ~ (W2) -1 – Regge prediction for high energy cross-sections

is the intercept of the Regge trajectory =1.08 for the SOFT POMERON

Such energy dependence is well established from the SLOW RISE of all hadron-hadron cross-sections - including (p) ~ (W2) 0.08 for real photon- proton scattering

For virtual photons, at small x (*p) = 42 F2

Q2

~ (W2)-1 F2 ~ x 1- = x - so a SOFT POMERON would imply = 0.08 only a very gentle rise of F2 at

small x

For Q2 > 1 GeV2 we have observed a much stronger rise

px2 = W2

q

p

The slope of F2 at small x , F2 ~x - , is equivalent to a rise of (*p) ~ (W2)which is only gentle for Q2 < 1 GeV2

(*p) gent

le r

ise

muc

h st

eepe

r ri

se

GBW dipole

QCD improved dipole

Regge region pQCD generated slope

So is there a HARD POMERON corresponding to this steep rise?

A QCD POMERON, (Q2) – 1 = (Q2)

A BFKL POMERON, – 1 = = 0.5

A mixture of HARD and SOFT Pomerons to explain the transition Q2 = 0 to high Q2?

What about the Froissart bound ? – the rise MUST be tamed – non-linear effects?

Dipole models provide a way to model the transition Q2=0 to high Q2

At low x, * qq and the LONG LIVED (qq) dipole scatters from the proton

The dipole-proton cross section depends on the relative size of the dipole r~1/Q to the separation of gluons in the target R0

=0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)~1/xg(x)

r/R0 small large Q2, x ~ r2~ 1/Q2

r/R0 large small Q2, x ~ 0 saturation of the dipole cross-section

GBW dipole model

(*p)

But(*p) = 42 F2 is generalQ2

(p) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and (*p) ~ 1/Q2

(at small x)

Now

there is HE

RA

data right across the transition region

is a new scaling variable, applicable at small x

It can be used to define a `saturation scale’ , Q2s = 1/R0

2(x) x -~ x g(x), gluon density

- such that saturation extends to higher Q2 as x decreases

Some understanding of this scaling, of saturation and of dipole models is coming from work on non-linear evolution equations applicable at high density– Colour Glass Condensate, JIMWLK, Balitsky-Kovchegov. There can be very significant consequences for high energy cross-sections e.g. neutrino cross-sections – also predictions for heavy ions- RHIC, diffractive interactions – Tevatron and HERA, even some understanding of soft hadronic physics

= 0 (1 – exp(-1/))

Involves only

=Q2R02(x)

= Q2/Q02 (x/x0)

And INDEED, for x<0.01, (*p) depends only on , not on x, Q2

separately

x < 0.01

x > 0.01

Q2 < Q2s

Q2 > Q2s

Summary

Measurements of Nucleon Structure Functions are interesting in their own right- telling us about the behaviour of the partons – which must eventually be calculated by non-perturbative techniques- lattice gauge theory etc.

They are also vital for the calculation of all hadronic processes- and thus accurate knowledge of them and their uncertainties is vital to investigate all NEW PHYSICS

Historicallythese data established the Quark-Parton Model and the Theory of QCD, providing measurements of the value of s(MZ

2) and evidence for the running of s(Q2)

There is a wealth of data available now over 6 orders of magnitude in x and Q2 such that conventional calculations must be extended as we move into new kinematic regimes – at small x, at high density and into the non-perturbative region. The HERA data has stimulated new theoretical approaches in all these areas.