Transcript QCD and the origin of proton

Deep Inelastic Scattering
CTEQ Summer School
Madison, WI, July 2011
Cynthia Keppel
Hampton University / Jefferson Lab
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40 years of physics
Maybe 100 experiments
...in an hour…..
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How to probe the nucleon / quarks?
• Scatter high-energy
lepton off a proton:
Deep-Inelastic
Scattering (DIS)
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large momentum -> short distance
(Uncertainty Principle at work!)
• In DIS experiments pointlike leptons + EM
interactions which are well
understood are used to
probe hadronic structure
(which isn’t).
• Relevant scales:

d probed     1018 m
p
DIS Kinematics
• Four-momentum transfer:
 
 
q  ( E  E ' )  (k  k ' )  (k  k ' ) 
 
2
2
 me  me '  2( EE' | k | | k ' | cos ) 
2
2
  4 EE' sin 2   Q 2
• Mott Cross Section (c=1):
d
d Mott
( )
L : lepton ten
sor
W : hadron tensor
a virtual photon of fourmomentum q is able to resolve
structures of the order /√q2



4 2 E '2
Q4
4 2 E '2
16 E 2 E '2 sin 4 2
 2 cos2 2
4E
2
sin 4 2
2 

cos
2 

cos
 EE'
 1 E (11cos )
M
 1 E ( 21sin 2  )
M
2
Electron scattering of a spinless point particle
Electron-Proton Scattering
• Effect of proton spin:
• Nucleon form factors:
– Mott cross section:
 Mott 
4 2 E '2
Q4
2
cos 

E'
E

2 
cos
Ruth

– Effect proton spin 
2
with:
GE2  GM2
2
2
A(Q ) 
and B(Q )  2GM
1
2
• helicity conservation
• 0 deg.: ep(magnetic)  0
• 180 deg.: spin-flip!
magn ~ Ruth
 ep   Mott [ A(Q 2 )  B(Q 2 ) tan2  ]
sin2(/2)
GEp (0)  1 and GMp (0) 
gp
2
 N  2.79 N
GEn (0)  0 and GMn (0) 
gn
2
 N  -1.91 N
~ Mott tan2(/2)
 espin    Mott  [1  2 tan2  ]
1
2
• with
2

Q2
• The proton form factors
have a substantial Q2
dependence.
4 M 2c 2
Mass of target = proton
Measurement kinematics…
ep collision
Final electron energy
Initial electron energy
Q2=-q2=-(k-k’)2=2EeE’e(1+cosθe)
Electron scattering angle
W2=(q + Pp)2= M2 + 2M(Ee-E’e) - Q2
= invariant mass
of final state hadronic system
Everything we need can be reconstructed from the
measurement of Ee, E’e and θe. (in principle) ->
try a measurement!….
Excited states of the nucleon
D(1232)
• Scatter 4.9 GeV electrons
from a hydrogen target. At 10
degrees, measure ENERGY
of scattered electrons
• Evaluate invariant energy of
virtual-photon proton system:
W2 = 10.06 - 2.03E’e *
• In the lab-frame: P = (mp,0) 
W 2  (Pp  q)2  P2  2Pq  q2
W 2  m2p  2m p  Q 2
* Convince yourself of this!
• Observe excited resonance
states:
Nucleons are composite
→ What do we see in the
data for W > 2 GeV ?
• First SLAC experiment (‘69):
– expected from proton form factor:
2

d / dE ' d 
1
  Q 8
 
2
2 
(d / d) Mott  (1  Q / 0.71) 
• First data show big surprise:
– very weak Q2-dependence
– form factor -> 1!
– scattering off point-like objects?
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…. introduce a clever
model!
The Quark-Parton Model
• Assumptions:
– Proton constituent = Parton
– Elastic scattering from a quasifree spin-1/2 quark in the
proton
– Neglect masses and pT’’s,
“infinite momentum frame”
• Lets assume: pquark = xPproton
2
( xP  q)2  p'2quark  mquark
0
– Since xP2  M2 <<Q2 it follows:
Q2
Q2
2 xP  q  q  0  x 

2Pq 2M
2
= (q.p)/M = Ee-Ee’
Definition Bjorken scaling variable
e’
e
P
parton
• Check limiting case:
1
W 2  M p2  2M p  Q2 x

M p2
• Therefore:
x = 1: elastic scattering
and 0 < x < 1
Structure Functions F1, F2
• Introduce dimensionless structure functions:
F1  MW1 and F2  W2 
d
2
 d  1 

2



F
(
x
)

F
(
x
)
tan

/
2

2
1

dE' d  d  M  
M
2
• Rewrite this in terms of :   Q2 / 4mquark
(elastic e-q scatt.: 2mq = Q2 )
2

d
1
Q 2 4mq 
 d 
2
F1 ( x) tan  / 2 

   F2 ( x)  2 2 2
dE' d  d  M  
4mq Q M

1
  F2 ( x)  2  2 xF1 ( x) tan2  / 2

 
if F2 ( x)  2 xF1 ( x)
1
 F2 ( x) 1  2 tan2  / 2



/
• Experimental data for 2xF1(x) / F2(x)
→ quarks have spin 1/2
(if bosons: no spin-flip  F1(x) = 0)
Interpretation of F1(x) and F2(x)
• In the quark-parton model:
F1 ( x)   f 12 z 2f [q f ( x)  q f ( x)]
Quark momentum
distribution
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The quark structure of nucleons
• Quark quantum numbers:
• Structure functions:
– Spin: ½  Sp,n = () = ½
F2p  x[ 19 (d vp  d sp  d sp )  94 (uvp  usp  u sp )  19 ( ss  ss )]
– Isopin: ½  Ip,n = () = ½
F2n  x[ 19 (d vn  d sn  d sn )  94 (uvn  u sn  usn )  19 ( ss  ss )]
– Isospin symmetry:
• Why fractional charges?
uvn  dvp , dvn  uvp , usn  dsn  usp  dsp
– Extreme baryons: Z = (1,2)
1  3zq  2  -  zq  
1
3
2
3
– Assign: zup =+ 2/3, zdown = - 1/3
• Three families:
– ‘Average’ nucleon F2(x)
with q(x) = qv(x) + qs(x) etc.
F2N  12 ( F2p  F2n )
 185 x   (q( x)  q ( x))  19 x  [ ss ( x)  ss ( x)]
u ,d
 u  c  t 
   
 d  s  b 
 z   23 ; mu  mc (  1.5 GeV)  mt
 z   13 ; md  ms (  0.3 GeV)  mb
– mc,b,t >> mu,d,s : no role in p,n
• Neutrinos:
F2  x[(d v  d s  d s )  (uv  us  us )  ( ss  ss )]
 x[(d  u  s)  (d s  us  ss )]  x  (q( x)  q ( x))
u ,d , s
Fractional quark charges
• Neglect strange quarks 
F2e, N
5

F2 , N 18
– Data confirm factor 5/18:
Evidence for fractional charges
• Fraction of proton momentum
carried by quarks:
1
F
 ,N
2
0
1
( x)dx  185  F2e, N ( x)dx  0.5
0
– 50% of momentum due to nonelectro-weak particles:
Evidence for gluons


IF, proton was made of 3 quarks each with 1/3 of proton’s
momentum:
no anti-quark!
F2 = x∑(q(x) + q(x)) eq2
q(x)=δ(x-1/3)
F2
or with some
smearing
1/3
x
The partons are point-like and incoherent
then Q2 shouldn’t matter.
 Bjorken scaling: F2 has no Q2 dependence.
Thus far, we’ve covered:
 Some history
 Some key results
 Basic predictions of the parton model
The parton model assumes:
 Non-interacting point-like particles
→ Bjorken scaling, i.e. F2(x,Q2)=F2(x)
 Fractional charges (if partons=quarks)
 Spin 1/2
 Valence and sea quark structure (sum rules)
Makes key predictions that can be tested by
experiment…..
Let’s look at some data
Proton Structure Function F2
F2
Seems to be…. …uh oh…
Lovely movies are from R. Yoshida, CTEQ Summer School 2007
Deep Inelastic Scattering experiments
HERAtarget
collider:
H1at
and
ZEUS
experiments
Fixed
DIS
SLAC,
FNAL,
CERN,
1992JLab
– 2007
now
Modern data
• First data (1980):
• Now..“Scaling violations”:
– weak Q2 dependence
– rise at low x
– what physics??
….. QCD
PDG 2002
Quantum Chromodynamics (QCD)
• Field theory for strong interaction:
– quarks interact by gluon exchange
– quarks carry a ‘colour’ charge
– exchange bosons (gluons) carry
colour  self-interactions (cf. QED!)
• Hadrons are colour neutral:
q
q
gg
s
q
q
– RR, BB, GG or RGB
– leads to confinement:
| q, | qq or | qqq  forbidden
• Effective strength ~ #gluons exch.
– low Q2: more g’s: large eff. coupling
– high Q2: few g’s: small eff. coupling
s
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The QCD Lagrangian
Lqcd  i qj  (D ) jk qk   mq qj qk  14 Ga Ga
q
q
(j,k = 1,2,3 refer to colour; q = u,d,s refers to flavour; a = 1,..,8 to gluon fields)
Covariant derivative:
D     i gsaG
1
2
qg-interactions
SU(3) generators:
([a , b ]  i 12 f abc c )
a

Free
quarks
 0 1 0
0  i



1   1 0 0  2   i 0
 0 0 0
0 0



0 0  i
0 0



5   0 0 0  6   0 0
i 0 0 
0 1



Gμνa    Ga   Ga  g s f abc GaGa

 
 
Gluon kinetic
energy term
Gluon selfinteraction
0
 1 0 0
0 0




0  3   0  1 0  4   0 0
 0 0 0
1 0
0 



0
0 0 0 
1



1 
1  7   0 0  i  8   0
3
0 i 0 
0 


0
1

0
0 
0 0

1 0
0  2 
So what does this mean..?
QCD brings new possibilities:
q
q
q
quarks can radiate gluons
gluons can produce qq pairs
q
gluons can radiate gluons!
Proton
e’
e
γ*(Q2)
~1.6 fm (McAllister & Hofstadter ’56)
r
Virtuality (4-momentum transfer) Q gives the distance
scale r at which the proton is probed.
r≈ hc/Q = 0.2fm/Q[GeV]
CERN, FNAL fixed target DIS:
HERA ep collider DIS:
HERA: Ee=27.5 GeV, EP=920 GeV
rmin≈ 1/100 proton dia.
rmin≈ 1/1000 proton dia.
(Uncertainty Principle again)
F2
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Higher the resolution
(i.e. higher the Q2)
more low x partons we
“see”.
So what do we expect F2 as a function of x at
a fixed Q2 to look like?
F2(x)
F2(x)
1/3
Three quarks
with 1/3 of
total
proton
momentum each.
x
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1/3
x
F2(x)
1/3
x
Three quarks
with some
momentum
smearing.
The three quarks
radiate partons
at low x.
….The answer depends on the Q2!
Proton Structure Function F2
How this
change with
Q2 happens
quantitatively
described by
the:
DokshitzerGribovLipatovAltarelli-Parisi
(DGLAP)
equations
QCD predictions: scaling violations
• Originally: F2 = F2(x)
– but also Q2-dependence
• Why scaling violations?
– if Q2 increases:
 more resolution (~1/ Q2)
 more sea quarks +gluons
• QCD improved QPM:
F2 ( x, Q 2 )
x

2
2
+
+
• Officially known as: Altarelli-Parisi Equations (“DGLAP”)
DGLAP equations are easy to “understand” intuitively..
First we have four “splitting functions”
z
1-z
z
z
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.
These additional contributions to F2(x,Q2) can be calculated.
Now DGLAP equations (schematically)
convolution
dqf(x,Q2)
= αs [qf × Pqq + g × Pgq]
2
d ln Q
strong coupling constant
qf is the quark density summed over all active flavors
Change of quark distribution q with Q2
is given by the probability that q and g radiate q.
Same for gluons:
dg(x,Q2)
d ln
Q2
= αs [∑qf × Pqg + g × Pgg]
Violation of Bjorken scaling predicted by QCD logarithmic dependence, not dramatic
DGLAP fit (or QCD fit) extracts the parton
distributions from measurements.
(CTEQ, for instance :) )
Basically, this is accomplished in two steps:
Step 1: parametrise the parton momentum density
f(x) at some Q2. e.g. f(x)=p1xp2(1-x)p3(1+p4√x+p5x)
uv(x)
dv(x)
g(x)
S(x)
u-valence
“The original three quarks”
d-valence
gluon
“sea” (i.e. non valence) quarks
Step 2: find the parameters by fitting to DIS (and
other) data using DGLAP equations to evolve f(x) in
Q2.
QCD fits of
F2(x,Q2)
data
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The DGLAP evolution
equations are extremely
useful as they allow
structure functions
measured by one
experiment to be compared
to other measurements and to be extrapolated to
predict what will happen in
regions where no
measurements exist, e.g.
LHC.
Q2
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x
Evolving PDFs up to MW,Z scale
xg(x)
xS(x)
xg(x)
xS(x)
“evolved”
“measured”
“measured”
xf(x)
Finally…
Valence quarks
maximum around
x=0.2; q(x) →0 for
x→1 and x→0
Sea quarks
and gluons contribute at
low values of
x
QCD predictions: the running of s
• pQCD valid if s << 1:

Q2
> 1.0
(GeV/c)2
CERN 2004
PDG 2002
• pQCD calculation:
 s (Q 2 ) 
12
33  2n f )  ln(Q 2 / L2 )
– with Lexp = 250 MeV/c:
Q2     s  0
 asymptotic freedom
Q2  0   s  
 confinement
Running coupling constant is
quantitative test of QCD.
QCD fits of F2(x,Q2) data
• Free parameters:
– coupling constant:
s 
Quarks
12
 0.16
 n f ) ln(Q 2 / L )
– quark distribution q(x,Q2)
– gluon distribution g(x,Q2)
• Successful fit:
Corner stone of QCD
Gluons
What’s still to do?
LOTS still to do!
• Large pdf uncertainties still at large x, low x
• pdfs in nuclei
• FL structure function - unique sensitivity to the glue
•
•
•
•
•
•
•
•
EIC
(F2 = 2xF1 only true at leading order)
Spin-dependent structure functions and transversity
Generalized parton distributions
Quark-hadron duality, transition to pQCD
Neutrino measurements - nuclear effects different? F3
structure function (Dave Schmitz talks next week)
Parity violation, charged current,….
NLO, NNLO, and beyond
Semi-inclusive (flavor tagging)
BFKL evolution, Renormalization
Large x (x > 0.1) -> Large PDF Uncertainties
u(x)
d(x)
d(x)
g(x)
Typical W, Q cuts are VERY restrictive….
Current Q2 > 4 GeV2, W2 > 12.25
GeV2, cuts
Recent CTEQ-Jlab effort to
reduce cuts
Essentially leave no data below x~0.75
What large x data there is has large uncertainty
(Ignore red mEIC
proposed data
points.)
Nuclear medium modifications, pdfs
The moon at nuclear densities
(Amoon ≈ 5x1049)
The deuteron is a nucleus, and
corrections at large x matter….
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The extremes of variation of
the u,d, gluon PDFs, relative to
reference PDFs using different
deuterium nuclear corrections
Differential parton
luminosities for fixed
rapidity y = 1, 2, 3, as a
function of τ = Q2/S,
variations due to the
choice of deuterium
nucleon corrections.
The gg, gd, du luminosities control the “standard candle”
cross section for Higgs, jet W- production, respectively.
QCD and the Parton-Hadron Transition
Hadrons
Nucleons
Quarks and
Gluons
Quark-Hadron Duality
• At high energies: interactions between quarks and gluons
become weak
(“asymptotic freedom”)
 efficient description of phenomena afforded in terms of
quarks
• At low energies: effects of confinement make strongly-coupled
QCD highly non-perturbative
 collective degrees of freedom (mesons and baryons) more
efficient
• Duality between quark and hadron descriptions
– reflects relationship between confinement and asymptotic
freedom
– intimately related to nature and transition from nonperturbative to perturbative QCD
Duality defines the transition from soft to hard QCD.
Duality observed (but not understood) in
inelastic (DIS) structure functions
First observed in F2 ~1970 by
Bloom and Gilman at SLAC
• Bjorken Limit: Q2,  
• Empirically, DIS region is
where logarithmic scaling
is observed: Q2 > 5 GeV2,
W2 > 4 GeV2
• Duality: Averaged over W,
logarithmic scaling
observed to work also for
Q2 > 0.5 GeV2, W2 < 4
GeV2, resonance regime
(CERN Courier, December 2004)
Beyond form factors and quark distributions –
Generalized Parton Distributions (GPDs)
X. Ji, D. Mueller, A. Radyushkin (1994-1997)
Proton form
factors, transverse
charge & current
densities
Correlated quark momentum
and helicity distributions in
transverse space - GPDs
Structure functions,
quark longitudinal
momentum & helicity
distributions
Again, LOTS still to do!
• Large pdf uncertainties still at large x, low x
• pdfs in nuclei
• FL structure function - unique sensitivity to the glue
•
•
•
•
•
•
•
•
EIC
(F2 = 2xF1 only true at leading order)
Spin-dependent structure functions and transversity
Generalized parton distributions
Quark-hadron duality, transition to pQCD
Neutrino measurements - nuclear effects different? F3
structure function (Dave Schmitz talks next week)
Parity violation, charged current,….
NLO, NNLO, and beyond
Semi-inclusive (flavor tagging)
BFKL evolution, Renormalization
More challenges….
• Extrapolate s to the size
of the proton, 10-15 m:
• If s >1 perturbative
expansions fail…
 Non-perturbative QCD:
– Proton structure & spin
– Confinement
– Nucleon-Nucleon forces
– Higher twist effects
– Target mass corrections
l  rproton   s  1