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Introduction to Deep Inelastic Scattering (DIS)

Rik YoshidaArgonne National Laboratory

CTEQ summer school 07May 30, 2007

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Some preliminary remarks• This is not a historical review

– for a very nice historical review see EnricoTassi’s lectures from 2003:

• http:://www-zeus.desy.de/~tassi/cteq2003.ppt• Nor a review of experimental status

• Enrico’s second lecture (same place)• Max Klein’s DIS lecture from CTEQ 2006

• Nor a theoretical discussion– Morning lectures from George Sterman

• Aim: to leave you with some intuitive feeling for what is happening in Deep Inelastic Scattering (DIS). Going to stick to electron- (positron-) proton DIS

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Partons in the protonFeynman’s parton model: the nucleon is made up of point-like constituents (later identified with quarks and gluons)which behave incoherently.The probability f(x) for the parton f to carry the fractionx of the proton momentum is an intrinsic property of thenucleon and is process independent.

If I were thinking about an experiment where wecollide protons with protons at, say, 14 TeV: then this is great! Because:

-Protons are just a “beam of partons” (incoherent)-The f(x)s, the “beam parameters”, could be measured in some other process. (process independent)

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Quarks and Gluons as partons

∫x[u(x)+u(x)+d(x)+d(x)+s(x)+s(x)+….]dx = 1

u(x) : up quark distributionu(x) : up anti-quark distributionetc.

Momentum has to add up to 1 (“momentum sum rule”)

Quantum numbers of the nucleon has to be right

∫[u(x)-u(x)]dx=2 ∫[d(x)-d(x)]dx=1

∫[s(x)-s(x)+……]dx=0

So for a proton:

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DIS kinematicsep collision proton in “∞” momentum frame

√s = ep cms energyQ2=-q2= 4-momentum transfer squared (or virtuality of the “photon”)

No transversemomentum

x = fractional longitudinal momentum carried by the struck parton

0 ≤ x ≤ 1

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DIS kinematicsep collision

Q2=-q2=-(k-k’)2=2EeE’e(1+cosθe)

x =Q2/2P•q = Ee E’e (1+cosθe)EP 2Ee-E’e(1-cosθe)

Initial electron energyFinal electron energy

Initial proton energy

Electron scattering angle

Everything we need can be reconstructed from themeasurement of E’e and θe. (in principle)

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Deep Inelastic Scattering experiments

Fixed target DIS at SLAC, FNAL and CERN completed ~ 10-20 years agoHERA collider: H1 and ZEUS experiments 1992 – 2007 (will complete July 2, 2007)

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e-p Neutral Current (NC) cross-section:

d2σ 2πα2

dxdQ2 xQ4 [Y+F2(x,Q2)-y2 FL(x,Q2)+Y-xF3(x,Q2)]=

y=Q2/xs 0 ≤ y ≤ 1 “inelasticity” Y±=1±(1-y)

Has to do withZ0 exchange:small for Q<<MZ

Has to do withlong. photon. Only large at largest y

We’ll come backto these

d2σ 2πα2

dxdQ2 xQ4= Y+F2(x,Q2)

So for now:

F2 = x∑(q + q) eq + Z-exchangequark charge

quark and anti-quark distributions

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The partons are point-like and incoherentthen Q2 shouldn’t matter. Bjorken scaling: F2 has no Q2 dependence.

IF, proton was made of 3 quarks each with 1/3 of proton’smomentum:

F2 = x∑(q(x) + q(x)) eq

no anti-quark!

F2

1/3 x

q(x)=δ(x-1/3)

or with some smearing

Let’s look at some data

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Proton Structure Function F2

F2

Seems to be….NOT

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So what does this mean..?

QCD, of course:

quarks radiate gluons

q

q

qq

gluons can produce qq pairs

gluons can radiate gluons!

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r≈ hc/Q = 0.2fm/Q[GeV]

rγ*(Q2)

Virtuality (4-momentum transfer) Q gives the distancescale r at which the proton is probed.

~1.6 fm (McAllister & Hofstadter ’56)

CERN, FNAL fixed target DIS: rmin≈ 1/100 proton dia.HERA ep collider DIS: rmin≈ 1/1000 proton dia.

e

e’Proton

HERA: Ee=27.5 GeV, EP=920 GeV

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Higher the resolution (i.e. higher the Q2) more branchingsto lower x we “see”.

So what do we expect F2 as a function of x ata fixed Q2 to look like?

F2

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1/3

1/3

1/3

F2(x)

F2(x)

F2(x)

x

x

x

Three quarks with 1/3 of total proton momentum each.

Three quarks with some momentumsmearing.

The three quarks radiate partons at low x.

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Proton Structure Function F2

How this change with Q2 happens quantitatively described by the:

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations

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DGLAP equations are easy to “understand” intuitively

First we have the four “splitting functions”

z z z z

1-z 1-z 1-z 1-z

Pab(z) : the probability that parton a will radiate a parton b with the fraction z of the original momentum carried by a.

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= αs [qf × Pqq + g × Pgq]

Now DGLAP equations (schematically)

dqf(x,Q2)d ln Q2

convolution

strong coupling constant

Change of quark distribution q with Q2 is given by the probability that q and g radiate q.

dg(x,Q2)= αs [∑qf × Pqg + g × Pgg]d ln Q2

Same for gluons:

o o

o o

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DGLAP fit (or QCD fit) extracts the partondistributions from measurements. (Lectures by Jeff Owens next week)

Here’s a 1 min description: Step 1: parametrise the parton momentum desity f(x) at some Q2. e.g.

uv(x) u-valence dv(x) d-valence g(x) gluon S(x) sum of all “sea” (i.e. non valence) quarks

Step 2: find the parameters by fitting to DIS (andother) data using DGLAP equations to evolve f(x) inQ2.

“The orginal three quarks”

f(x)=p1xp2(1-x)p3(1+p4√x+p5x)

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Sea PDF

x

xS

At x<<1/3, quarks and (antiquarks) are all “sea”.Since F2 = eq ∑x(q + q), xS is very much like F2

Fractionaluncertainty

2

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Gluon PDF

x

xg

Gluons, on the other hand, are determined fromthe scaling violations dF2/dlnQ2 via the DGLAP equations.

Uncertainties are larger.Scaling violations couple αs and gluon g

Fit with αs alsoa free param.

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So far:

F2 ~ ∑(q+q) ≈ S (sea quarks) measured directly in NC DIS

Scaling violations

dF2/dlnQ2 ~ αs•g Scaling violations gives gluons (times αs). DGLAP equations.

What about valence quarks?

∑(q-q) = uv + dv can we determine them separately?

Can we decouple αs and g ?

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Return to Neutral Current (NC) cross-section:

d2σ(e±p) 2πα2

dxdQ2 xQ4 [Y+F2(x,Q2) Y-xF3(x,Q2)]=Y±=1±(1-y)

±

Now write out the e+p and e-p separately

xF3 = ∑(q(x,Q2)-q(x,Q2)) xBq ~The valence quarks!

Bq = -2eqaqaeχZ+ 4vqaqveaeχZ2

χZ= ( ) Keeps xF3 small if Q<MZ1 Q2

sin2θW MZ+Q22

(keep ignoring FL for now..)

Bq = -2eqaqaeχZ+ 4vqaqveaeχZ2

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Return to Neutral Current (NC) cross-section:

d2σ(e±p) 2πα2

dxdQ2 xQ4 [Y+F2(x,Q2) Y-xF3(x,Q2)]=Y±=1±(1-y)

±

Now write out the e+p and e-p separately

xF3 = ∑(q(x,Q2)-q(x,Q2)) xBq ~The valence quarks!

Bq = -2eqaqaeχZ+ 4vqaqveaeχZ2

(keep ignoring FL for now..)

Bq = -2eqaqaeχZ+ 4vqaqveaeχZ2

eq: electric charge of a quarkaqvq: axial-vector and vector couplings of a quarkaeve: axial-vector and vector couplings of an electron

γ-Z interference Z-exchange

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Return to Neutral Current (NC) cross-section:

d2σ(e±p) 2πα2

dxdQ2 xQ4 [Y+F2(x,Q2) Y-xF3(x,Q2)]=Y±=1±(1-y)

±

Now write out the e+p and e-p separately

xF3 = ∑(q(x,Q2)-q(x,Q2)) xBq ~The valence quarks!

(keep ignoring FL for now..)

Let’s look at the “reduced NC cross-section”

σNC± = F2(x,Q2) (Y-/Y+)•xF3(x,Q2)±

Note the change of sign from e+p to e-p

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σNC±

xMeasurements are at relatively high x

Reduced Neutral Current Cross-section

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Recent (Spring 07) preliminary result from HERA

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Charged Current Cross-Sections

dσCC(e±p) GF MW dxdQ2 2πx MW+Q2= [ ]2σCC±

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2 2

Skip a few steps….

σCC+ = x [u + c + (1 - y)2(d + s)] ~ d

σCC- = x [u + c + (1 – y)2(d + s)] ~ ucharm

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σCC±

Reduced Charged-Current Cross-Section

x

σCC+ ~ d σCC- ~ u

Now let’s look at the valence quarks from the QCD fits

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Valence PDFs

x

xf

The momenta from valence quarks are producinggluons and sea quarks at low x

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Jet production in DIS (HERA)

Sensitive to αs

Sensitive to gluon ~10-3 < x < ~10-2Sensitive to quarks

~10-2 < x < ~10-1complementaryto gluon from F2

Same range as NC and CC

σjet ~ αs•f(x)

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No ET in Breit Frame

Jet production cross-section used in QCD fit

Jet measurements in Breit frame

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Gluon distributions

x xUsing only HERA (ZEUS)data including NC,CC and jets

Using HERA (ZEUS) F2 dataand FNAL, CERN fixed tgt

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Finally…

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Proton Structure Function F2

F2

Now we understand what is happening here.

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Some remarks about DGLAP equations:

But now parton densities must be “evolved” in Q2.

What does this mean?

The “incoherence” of the original parton modelis preserved. i.e. a parton doesn’t know anythingabout its neighbor.

never happens

The “process independent” partons also survive.

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A parton at x at Q2 is a source of partons atx’ < x at Q’2 > Q2.

x

Q2

Q’2

x’

In fact, any parton atx > x’ at Q2 is a source.

To know the parton densityat x’, Q’2 it’s necessary(and sufficient) toknow the parton densityin the range: x’ ≤ x ≤ 1at some lower Q2.

1

measured

known

unkn

own

What does this mean for the LHC?

If you know the partons in range x’ ≤ x ≤ 1 at some Q2,then you know the partons in the range x’ ≤ x ≤ 1 for allQ’2 > Q2.

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Fixed targetDIS

HERA DIS

Tevatron jets

~safe Q2

“known”

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2 4

parton1(x1) + parton2(x2) State with mass M

LHC (or hadron-hadron) parton kinematics

x1= (M/√s) exp (y) x2= (M/√s) exp (-y)

y= ln( )1 E+PZ2 E-PZ

rapidity:

pseudo-rapidity:

η=-ln tan(θ/2)

angle wrt beam

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2 4

So if I want to predict Z or W productioncross-section at LHC at some rapidity y, say, -4:

q,q(x1=10-4,Q2=MW,Z) q,q(x2=0.3,Q2=MW,Z)

need

2 2

and

σ(ppW,Z+X) ~ q,q(x1,MW,Z) × q,q(x2,MW,Z) × σ(qqW,Z)22

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xf(x)

xS(x)xg(x) “measured”

“mea

sure

d”“evolved”xg(x)xS(x)

Evolving PDFs up to MW,Z scale

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-6 6

σ(ar

b. sc

ale)

η

Z production at LHC

Uncertainty ~5%

A. Cooper-Sarkar (HERA-LHC workshop 2007)

Jet production at LHCExamples of predictions for LHC using partons from DIS

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Final remarks I• We’ve just gone through an informal tour of QCD-improved

parton model and its application to data from ep Deep Inelastic Scattering.

• Some health warnings:– Most of what I talked about is a leading-order picture. In

practice, most things are done at least to next-to-leading order. At NLO, the interpretation of the results are not as straight-forward.

– Many people worry about whether we are not missing something fundamentally with the picture of DGLAP equations.

• Much of the data are at very low x: DGLAP is a lnQ2 approximation. Why aren’t ln(1/x) terms important…or are they? BFKL equations.

• The density of the partons, especially that of the gluons is getting very high. When and where should we worry about “shadowing”, “gluon recombination” etc.

• The idea of incoherence of partons may be breaking down in some kinematic regions: phenomenon of “hard diffraction” is difficult to understand in terms of partons without correlations to each other.

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Final remarks II• There are many other DIS physics topics I did not cover

here.• Electoweak physics• Heavy quark production• Diffraction, Vector Meson production, low Q2 physics• Beyond the SM searches.• Polarized DIS• …

• I hope I have refreshed your memory about some familiar DIS physics, and got you ready for the rest of the school.

• Thanks to the organizers for their kind invitation. Thanks to Claire Gwenlan for preparing some of the plots animation for me.

– You can find the animated gifs in:• http://www.hep.anl.gov/ryoshida/animated_proton.htm

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Extras (FL)

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FL=(Q2/4π2α) σL

Longitudinal cross-section

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QCD predicts a relationshipbetween scaling violationsand FL through the gluondensity.

increasing y

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You can determineFL from a NLO DGLAPfit to NC cross-section.

x

Indeed, we also only determineF2 the same way, in principle:

We measure this only

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C. Diaconu: DIS 07 conference April 07

HERA measurement of FL on-going nowNormally 920 GeV