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The Structure of the Proton. Parton Model- QCD as the theory of strong interactions Parton Distribution Functions Extending QCD calculations across the kinematic plane – understanding Small-x High density Low Q 2. A.M.Cooper-Sarkar April 2003. Leptonic tensor - calculable. 2.

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The structure of the proton

The Structure of the Proton

  • Parton Model- QCD as the theory of strong interactions

  • Parton Distribution Functions

  • Extending QCD calculations across the kinematic plane – understanding

  • Small-x

  • High density

  • Low Q2

A.M.Cooper-Sarkar

April 2003


Leptonic tensor - calculable

2

L:< W:<

dF~

Hadronic tensor- constrained by Lorentz invariance

Et

q = k – kt,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

Ee

Ep

s = 4 Ee Ep

Q2 = 4 Ee Et sin2he/2

y = (1 – Et/Ee cos2he/2)

x = Q2/sy

The kinematic variables are measurable


d2F(e±N) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) K Y_xF3(x,Q2)], YK = 1 K (1-y)2

dxdy

for charged lepton hadron scattering

F2, FL and xF3 are structure functions –

The Quark Parton Model interprets them

d2F = 2pa2 s [ 1 + (1-y)2] Giei2(xq(x) + xq(x))

dxdy

Q4

Compare the general equation to the QPM prediction

F2(x,Q2) = Giei2(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 p2~0so

x = Q2/(2p.q)

The FRACTIONAL momentum of the incoming nucleon taken by the struck quark is the MEASURABLE quantity x


QCD improves the Quark Parton Model

What if

or

x

x

Pqq

Pgq

y

y

Before the quark is struck?

Pqg

Pgg

y > x, z = x/y

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 from

sYs(Q2)

xi+1

xi

xi-1

target to probe

xi-1> xi > xi+1….

pt2 of quark relative to proton increases from target to probe

pt2i-1 < pt2i < pt2i+1

Dominant diagrams have STRONG pt ordering

The DGLAP equations


Bjorken scaling is broken – ln(Q2)

Note strong rise at small x

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


How are such fits done?

Parametrise the parton distribution functions (PDFs) at Q20 (low-scale)

  • xuv(x) =Auxau (1-x)bu (1+ guox + (u x)

  • xdv(x) =Adxad (1-x)bd (1+ gdox + (dx)

  • xS(x) =Asx-8s (1-x)bs (1+ gsox + (sx)

  • xg(x) =Agx-8g(1-x)bg (1+ ggox + (gx)

  • 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 P2 fit

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 parton distribution functions PDFs which are transportable to other hadronic processes

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

  • Valence distributions evolve slowly

  • Sea/Gluon distributions evolve fast

  • 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

  • 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 Q parton distribution functions PDFs 2Y Z0 and W+/- become as important as

  • exchange Y NC and CC cross-sections comparable

    for NC processes

    F2= 3i Ai(Q2) [xqi(x,Q2) + xqi(x,Q2)]

    xF3= 3iBi(Q2) [xqi(x,Q2) - xqi(x,Q2)]

    Ai(Q2) = ei2 – 2 eivi vePZ + (ve2+ae2)(vi2+ai2) PZ2

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

    PZ2 = Q2/(Q2 + M2Z) 1/sin2hW

  • a new valence structure function xF3 measurable from

    low to high x- on a pure proton target


CC processes give flavour information parton distribution functions PDFs

d2F(e+p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)]

d2F(e-p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)]

dxdy

2Bx(Q2+M2W)2

dxdy

2Bx(Q2+M2W)2

uv at high x

dv at high x

MW information

Measurement of high-x dvon a pure proton target

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


Valence PDFs from ZEUS data alone- parton distribution functions PDFs

NC and CC e+ and e- beams Y high x

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

and NC e+/-

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

HERA –II data will enable accurate PDF extractions without need of nuclear corrections


Parton distributions are transportable to other processes parton distribution functions PDFs

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

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

But in 1996 an excess of jets with ET > 200 GeVin CDF data appeared to indicate new physics beyond the Standard Model BUT a modification of the PDFs with a harder high-$x$ gluon (which still gave a ‘reasonable fit’ to other data) could explain it

An excess of high-Q2 events in HERA data (1997) was initially taken as evidence for lepto quarks- but a modification to the high-$x$ u quark PDF could explain it

Currently the anomalous NuTeV measurement of sin2θW can be explained by dropping the assumption s = s in the sea?

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


P parton distribution functions PDFs 2 = 3i [ FiQCD – Fi MEAS]2

(FiSTAT)2+(FiSYS)2 = Fi2

Errors on the fit parameters evaluated from )P2 = 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

Must modify the formulation of the χ2

P2 = 3i 3j [ FiQCD – FiMEAS] Vij-1 [ FjQCD – FjMEAS]

Vij = *ij(FiSTAT)2 + 38)i8SYS)j8SYS

Where )i8SYS is the correlated error on point i due to systematic error source 8


  • Model Assumptions parton distribution functions PDFs –

  • Value of Q20, form of the parametrization

  • Kinematic cuts on Q2, W2, x

  • Data sets included …….

  • Are some data sets incompatible?

  • PDF fitting is a compromise

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


Comparison of parton distribution functions PDFs ZEUS and H1 gluon distributions –

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

Comparison of ZEUS and H1 valence distributions.


Theoretical Assumptions- Need to extend the formalism? parton distribution functions PDFs

What if

Optical theorem

2

Im

The handbag diagram- QPM

QCD at LL(Q2)

Ordered gluon ladders (asn lnQ2 n)

NLL(Q2) one rung disordered asn lnQ2 n-1

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

LO NLO NNLO

BUT what about

completely disordered

Ladders?

Or higher twist diagrams?

low Q2, high x

Eliminate with a W2 cut


Clues from the gluon distribution parton distribution functions PDFs

Knowledge increased dramatically in the 90’s

Post HERA

Pre HERA

Scaling violations dF2/dlnQ2 in DIS

High ET jets in hadroproduction- Tevatron

BGF jets in DIS (* g Y q q

Prompt (

HERA charm production (* g Y c c

For small x scaling violation data from HERA are most accurate


t = ln Q parton distribution functions PDFs 2/72

Gluon splitting functions become singular

s ~ 1/ln Q2/72

At small x,

small z=x/y

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

F2(x,Q2) ~ x -8s, 8s=8g -,

xg(x,Q2) ~ x -8g


Still it was a surprise to see parton distribution functions PDFs 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 parton distribution functions PDFs steep behaviour of the gluon is deduced

from the DGLAP QCD formalism –

BUT the

steep behaviour of the Sea can be measured from

F2 ~ x -8s, 8s = d ln F2

d ln 1/x

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

not Q2 ~ 5 GeV2


Need to extend formalism at small x? parton distribution functions PDFs

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 I dy/y P(z) q(y,Q2)

And thus give rise to contributions to the PDF sp(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)

These are the DGLAP summations

LL(Q2) is STRONGLY ordered in pt.

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)) These are the BFKL summations

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

BFKL summation at LL(1/x) xg(x) ~ x -λ

λ = s CA ln2 ~ 0.5

 steep gluon even at moderate Q2

 Disordered gluon ladders

But NLL(1/x) softens this somewhat

B


Colour Glass Condensate, JIMWLK, BK

nucleon size

BR2

Non-linear evolution equations – GLR

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

Higher twist

B

dlnQ2dln1/x

16Q2R2

asD

as2D2/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!


`Valence-like’ gluon shape parton distribution functions PDFs

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 ~ Pqgxg

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


F parton distribution functions PDFs L

F2charm

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


Q parton distribution functions PDFs 2 = 2GeV2

xg(x)

The negative gluon predicted at low x, low Q2 from NLO DGLAPremains 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 P2 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


Q parton distribution functions PDFs 2 = 1.4 GeV2

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 Y g involve the summation of FAN diagrams –

xg

xuv

xu

xd

xc

xs

Q2 = 2

Q2=10

Q2=100 GeV2

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

xg

Non linear

DGLAP


Small x parton distribution functions PDFs  high W2 (x=Q2/2p.q . Q2/W2 )

F((*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 F((p) ~ (W2) 0.08 for real photon- proton scattering

For virtual photons, at small x F((*p) = 4B2 F2

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

q

px2 = W2

p

Regge region

pQCD region

Q2

F~ (W2)-1 F2 ~ x 1- = x -8 so a SOFT POMERON would imply 8 = 0.08 Y only a very gentle rise of F2 at small x

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


QCD improved dipole parton distribution functions PDFs

GBW dipole

gentle rise

F((*p)

Regge region

pQCD generated slope

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

A QCD POMERON, (Q2) – 1 = 8(Q2)

A BFKL POMERON,  – 1 = 8 = 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?

much steeper rise

The slope of F2 at small x , F2 ~x -8 , is equivalent to a rise of F((*p) ~ (W2) 8which is only gentle for Q2 < 1 GeV2


Dipole models provide a way to model the transition Q parton distribution functions PDFs 2=0 to high Q2

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

F((*p)

Now there is HERA data right across the transition region

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

σ =σ0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)8~1/xg(x)

Butσ((*p) = 4B2 F2 is general

Q2

(at small x)

r/R0 small Y large Q2, x σ~ r2~ 1/Q2

r/R0 large Y small Q2, x σ ~ σ0Y saturation of the dipole cross-section

At high Q2, F2 ~flat (weak lnQ2 breaking) and σ((*p) ~ 1/Q2

As Q2 0, σ(γp) is finite for real photons scattering.

GBW dipole model


x < 0.01 parton distribution functions PDFs

F = F0 (1 – exp(-1/J))

Involves only

J =Q2R02(x) Y

J = Q2/Q02 (x/x0)8

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

Q2 > Q2s

Q2 < Q2s

x > 0.01

J is a new scaling variable, applicable at small x

It can be used to define a `saturation scale’ , Q2s = 1/R02(x) . x -8~ 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 at RHIC, diffractive interactions at the Tevatron and HERA, even some understanding of soft hadronic physics


Summary
Summary parton distribution functions PDFs

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 as(MZ2) and evidence for the running of as(Q2)

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


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