CTEQ-Fermilab School on QCD and Electroweak …053101,2008. 14 Parton Model 15 Proton as a bag of...
Transcript of CTEQ-Fermilab School on QCD and Electroweak …053101,2008. 14 Parton Model 15 Proton as a bag of...
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CTEQ-Fermilab School on CTEQ-Fermilab School on QCD and Electroweak Phenomenology QCD and Electroweak Phenomenology
Lima, Peru
30 July - 9 August 2012
Deeply Inelastic Scattering (DIS)
Fred Olness (SMU)
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HOW TO CHARACTERIZE THE PROTON
Deeply Inelastic Scattering
(DIS)
… also see lectures by George Sterman
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Inclusive Deeply Inelastic Scattering (DIS)
Analogue of Rutherford scattering
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Inclusive Deeply Inelastic Scattering (DIS)
Other common DIS variables
x: partonic* momentum fractionQ: characteristic energy scale
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Lepton Tensor (L) and Hadronic Tensor (W)
lepton current
hadron current
interaction
Current Interactions
Lµν
Wµν
Leptonic Tensor
Hadronic Tensor
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W and F Structure Functions
q
p
Convert to “Scaling” Structure Functions
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What's all this talk about
Scaling????
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The Scaling of the Proton Structure Function
Data lies along a universal curve
It is (relatively) independent of energy
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What do we expect for a point like particle
Dimensional considerations
Structure Function
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Is this a point like particle ???
We found the Higgs
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Structure of the Proton
of order of the proton mass scale
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The Scaling of the Proton Structure Function
Data is (relatively) independent of energy
Scaling Violations observed at extreme x
values
Varies with energy
Varies with energy
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… not yet at the
Parton Model
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Use Helicity Basis
I have not yet mentioned the parton model!!!A Review of Target Mass Corrections.Ingo Schienbein et al. J.Phys.G35:053101,2008.
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Parton Model
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Proton as a bag of free Quarks
Phred
's
PDFs
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Quarks are not quite free
Q
Hard Interaction time ~ T
I ~ 1/Q
Soft Interaction time ~ T
S ~ 1/
This picture is valid IF:
TI << T
S thus Q >>
Corrections to this picture (non-factorizable/ higher twist) terms are suppressed by powers of /Q
Proton
Electron
“Higher Twist” contributions
Q: characteristic interaction scale : Scale of the proton
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fP→ a
σa→ c
Parton Distribution Functions
(PDFs) fP→ a
are the key to calculations
involving hadrons!!!a
σP γ→c = f P→a ⊗ σ̂aγ→c
Electron
Proton
The Parton Model and Factorization
c
ScalarAnti-QuarkQuark
calculable from theoretical model
must extract from experiment
Corrections of order (2/Q2)
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Part 1) Show these 3 definitions are equivalent; work out the limits of integration.
f ⊗g=∫0
1
∫0
1f ( x) g ( y )δ (z−x∗y )dx dy
f ⊗g=∫ f ( x) g (zx)dxx
f ⊗g=∫ f (zy) g ( y)
dyy
Part 2) Show convolutions are the ``natural'' way to multiply probabilities.
If f represents the heads/tails probability distribution for a single coin flip, show that the distribution of 2 coins is f ⊕ f and 3 coins is: f ⊕ f ⊕ f.
BONUS: How many processes can you think of that don't factorize?
f ⊕g=∫ f ( x) g ( y)δ( z−( x+ y))dx dy
f ( x)=12(δ (1− x)+δ(1+ x))
Homework Problem: Convolutions
Careful: convolutions
involve + and *
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Structure Function & PDF Correspondence at Leading Order
Callan-Gross Relation
Computein Parton
Model
Compute with
Hadronic Tensor
Scalar
Scalar
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FL
and
Callan-Gross
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Masses are important
Why is FL special ???
Higher ordersare important
Callan-Gross
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TOY
PDFs
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Proton as a bag of free Quarks: Part 2
Perfect Scaling PDFsQ independent
Quark Number Sum Rule
Quark Momentum Sum Rule
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Problem #1: The proton does not add up???
SOLUTION:
Gluons carry half the momentum,
but don't couple to the photons
Momentum Sum Rule
Substitute F
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Gluons smear out PDF momentum
Gluons allow partons to exchange momentum fraction
Momentum Fraction x
Part
on D
istr
ibut
ion
Momentum Fraction x
Part
on D
istr
ibut
ion
Prot
on
S is large at low Q, so it is easy to emit soft gluons
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Problem #2: Infinitely many quarks
Reconsider the Quark Number Sum Rule
SOLUTION: Infinite number of u quarks in proton, because they can be pair produced:
(We neglect saturation ....)
Quark Number Sum Rule: More Precisely
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PDFs
cf., lectures by Dan Stump
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Sample PDFs: The rich structure of the proton
Scaling violations are essential feature of PDFs
2=10 GeV22=10,000 GeV2
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Calculable from theoretical model
σP γ→c = f P→a ⊗ σaγ→c
Where do PDFs come from???? Universality!!!
Must extract from experiment
pdg.lbl.gov
Note we can combine different experiments.
FACTORIZATION!!!
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How do we distinguish flavors???
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How to Distinguish Flavors???
u
u 4/9
d
d 1/9
d
u
W-
u
d
W+
u
u
Z
d
d
Z
PHOTON W-BOSON Z-BOSON
Use different boson probes
Take Cabibbo angle =0
Note: equal weight
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HOMEWORK
Sum Rules &
Structure Functions
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Homework: Part 1 Structure Functions & PDFs
Verify: i.e., Check for typos ...
In the limit
We use these different observables to
dis-entangle the flavor structure of the PDfs
See talks by Alberto Gago (Neutrinos)
& Dan Stump (PDFs)
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Adler(1966)
Gottfried(1967)
Gross Llewellyn-Smith(1969)
Bjorken(1967)
Homework(19??)
Homework: Part 2 Sum Rules
Verify: i.e., Check for typos ...
Before the parton model was invented, these
relations were observed. Can you understand
them in the context of the parton model?
This one has been particularly important/controversial
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How do we distinguish flavors???
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Different Targets
Possible Targets: p Protonn NeutronN Nucleon ~ Z p + (A-Z) n
… for isoscalar: N= ½ (p+n)
Relate p and n Proton Neutron
up
= dn
dp
= un
qp
= qn
ComplicationsN Z p + (A-Z) n
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Nuclear Corrections
???
4141
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Discovered by the French in 1799 at Rosetta, a harbor on the Mediterranean coast in Egypt. Comparative translation of the stone assisted in understanding many previously undecipherable examples of hieroglyphics.
Independent of Q, A, F123
, ...
Where do Nuclear Corrections
come from ???
Proton
Lead
Iron
carved in stone
4242
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Start with Neutrino data
on heavy targets (Fe, Pb, ...)
“Corrected” data for isoscalar
“Corrected” data for proton
Traditional Input to Global PDF Fit
Global PDF Fit
NuclearCorrection
IsospinSymmetry
NuclearCorrection“Frozen”
No Feedback!!!
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Include Nuclear Dimension Dynamically
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Oooooops!
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Nuclear Corrections: Compare Neutrino and Charged Lepton DIS
Neutrino DIS
Charged Lepton DIS
/Z
W
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✔ CTEQ style global fit extended
handle various nuclear targets
✔ CTEQ Data + nuclear DIS & DY
[~15 targets; ~2000+ data]
✔ A-dependence modeled;
NLO fits work well
Nuclear PDFs from neutrino deep inelastic scattering. I. Schienbein, J.Y. Yu, C. Keppel, J.G. Morfin, F. Olness, J.F. Owens. Phys.Rev.D77:054013,2008.
ak=ak ,0+ak ,1(1−A−ak ,2)
A-Dependent PDFs
nCTEQ Nuclear PDF's 46
proton
lead
proton
lead
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K Kovarik, I. Schienbein, J.Y. Yu,T. Stavreva, T Jezo, C. Keppel,J.G. Morfin, F. Olness,J.F. Owens.
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Per ton per 10^20 protons
Per ton per w/ 4-year run
Experiment
CCFR 1.03 M 0.179 M
NuTeV 1.3 M 0.4 M
DIS events
anti- DIS events
Target Events
Fe 0.70 2 MPb 0.85 2.5 MHe 0.40 600 KC 0.15 430 K
CH 3 9 M
Mass (Tons)
MINERA Comparisons
DIS Comparisons: Charged Current
MINERvA
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Evolution
What does the proton look like???
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`Cheshire Puss,' ...
`Would you tell me, please, which way I ought to go from here?'
`That depends a good deal on where you want to get to,' said the Cat.
`I don't much care where--' said Alice.
`Then it doesn't matter which way you go,' said the Cat.
`--so long as I get somewhere,' Alice added as an explanation.
`Oh, you're sure to do that,' said the Cat, `if you only walk long enough.'
The answer is dependent upon
the question
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g
g
q
q
UT B C S D
Evolution: What you see depends upon what you ask
G
Proton is a complex object
... leads to scaling violation
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Evolution of the PDFs
Large Medium Small
How does f change with scale ???
dependence must balance
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Evolution of the PDFs
Up
Momentum Fraction x
x f(
x)
Down
Momentum Fraction x
x f(
x)
Gluon
Momentum Fraction x
x f(
x) Strange
Momentum Fraction x
x f(
x)
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u-bar
strangestrange
d-bar
down
up
charm
gluon
bottom
Scale
Momentum Fraction
PDF Momentum Fractions vs. scale
SU(3)
Scaling violations are essential feature of PDFs
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Renormalization Group Equation
Anomalous Dimension
Not physical! Poor notation
Take Mellin Transform
It is the dimension of the mass scaling
If “f” scaled, would vanish
DGLAPEquation
Parton Model
RenormalizationGroup Equation
Separation of variables
DGLAP
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DETAILS
Evolution and such
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Homework: Mellin Transform
A useful reference:Courant, Richard and Hilbert, David. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989. 561 p.
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Evolution of the PDFs
Up
Momentum Fraction xx
f(x)
1
1-x
xfP
P
P
P
Large
Medium
Small
Evolution (generally) shifts partons from hi-x to low-x
DGLAP Equation
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Up
Momentum Fraction x
x f(
x)
Evolution of the PDFs
1
1-x
x
fP
P
P
P
Large
Medium
Small
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HOMEWORK
Sum Rules &
Structure Functions
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1 1-x
x
1 1-x
x
1 1-x
x
1
1-x
x
The Splitting Functions:
Read backwards
Note singularities
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Homework: Part 1 The Plus Function
Definition of the Plus prescription:
1) Compute:
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Homework: Part 2 P(qq)
2) Verify:
1 1-x
x
1 1-x=0
x=1
1
1-x
x
Observe
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HOMEWORK: Part 3: Symmetries & Limits
For the regular part show:
For the regular part show:
Verify the following relation among the regular parts (from the real graphs)
Verify, in the soft limit:
11-x
x1 1-x
x
1 1-x
x
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HOMEWORK: Part 4: Conservation Rules
Verify conservation of momentum fraction
Verify conservation of fermion number
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Use conservation of fermion number to compute the delta function term in P(qq)
Homework: Part 5: Using the Real to guess the Virtual
This term only starts at NNLO
1 1-x
x
Powerful tool: Since we know real and virtual must balance, we can use to our advantage!!!
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Recap so far
● Rutherford Scattering Deeply Inelastic Scattering (DIS)
● Works for protons as well as nuclei
● Compute Lepton-Hadron Scattering 2 ways
● Use Leptonic/Hadronic Tensors to extract Structure Functions
● Use Parton Model; relate PDFs to F123
● Parton Model Factorizes Problem:
● PDFs are independent of process
● Thus, we can combine different experiments. ESSENTIAL!!!
● PDFs are not truly scale invariant; they evolve
● We use evolution to “resum” an important set of graphs
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N3LO68
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LO NLO N2LO N3LO
Full ACOT
ACOT Extension to Higher Orders 69
Distinguish:“phase space” mass
“dynamic” mass
Demonstrate:1) PS mass dominates2) Estimated Error small
Extensible to any order
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F2,L
@ N3LO 70
x=10-1 x=10-5
LONLON2LON3LO
Q Q Q
x=10-3
F2
FL
x=10-1 x=10-5x=10-3
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Strange Production
the LHC and PDFs
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What constrains the Strange???
72
0.02 0.05 0.10 0.20 0.50 1.00
0.1
0.2
0.3
0.4
0.5
0.6
U
D
SC
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Di-muon production Extract s(x) Parton Distribution
N
sc
XN
sc
X
73
Extract s(x) Extract s(x)
s(x) and s(x) are essential in extraction of SinW
Used in CTEQ6 Fits
W-
g
s
c
CDF: PRL 100:091803,2008.D0: PLB666:23,2008.
s gWc at the Tevatron
CDF & D0
Consistent with SM
Also a challenge at LHC
Depends on nuclear
corrections
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Heavy quark PDFs are essential ingredient
W Production at LHC: A Benchmark Cross Section
Heavy Quark components play anincreasingly important role at the LHC
tot
cs
us
ud
cd
d/d
y(W
+)
at L
HC
y
74
tot
cs
us
ud
cd
y
d/d
y(W
+)
at T
eva t
ron
Tevatron LHC
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75PDF Uncertainties S(x) PDF W/Z at LHC
PDF Uncertainties will feed into
LHC “Benchmark” processes
W+ at LHC
Z at LHC
Anastasiou, Dixon, Melnikov, Petriello, Phys.Rev.D69:094008,2004.
VRAP Code
strangecontribution
strangecontribution
rapidity
rapidity
Cro
ss S
e cti
onC
ross
Se c
tion
Comparison with new NNPDF sets: Les Houches 2009
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76SUMMARY 76
●Rutherford Scattering Deeply Inelastic Scattering (DIS)
● Works for protons as well as nuclei
●Compute Lepton-Hadron Scattering 2 ways
● Use Leptonic/Hadronic Tensors to extract Structure Functions
● Use Parton Model; relate PDFs to F123
●Parton Model Factorizes Problem:
● PDFs are independent of process
● Thus, we can combine different experiments. ESSENTIAL!!!
●PDFs are not truly scale invariant; they evolve
● We use evolution to “resum” an important set of graphs
●NLO Calculations, and beyond
●New Era: Strong constraints on PDF from LHC
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END OF LECTURE
77