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Quantum
Chromodynamics,
Colliders & Jets Stephen D. Ellis
University of Washington
Maria Laach September 2008
Lecture 1: The Big Picture – How do we Think About and Calculate processes at Colliders?The Parton Model + QCD
ASIDE: I am here to provide historical context, i.e.,
I can remember particle physics before QCD, jets
or colliders!
S. D. Ellis Maria Laach 2008 Lecture 1 2
Historical art
work by Siggi
© S. Bethke June 1993
S. D. Ellis Maria Laach 2008 Lecture 1 3
Outline
1. Introduction – The Big Picture
pQCD - e+e- Physics and Perturbation Theory
(the Improved Parton Model);
pQCD - Hadrons in the Initial State and PDFs
2. pQCD - Hadrons and Jets in the Final State
3. Colliders & Jets at Work
S. D. Ellis Maria Laach 2008 Lecture 1 4
Concepts/Vocabulary*
• Matter – quarks & leptons, quark model of states and resonances
• Parton Model – parton distribution functions (pdf’s), fragmentation functions
• Symmetries – global and local –SU(3) of QCD (local, unbroken), U(1) of E&M (local unbroken), SU(2)L of Weak (local, broken), SU(2) (to SU(6)) of Flavor (global, approximate)
• Interactions – mediated by gauge bosons (local symmetry)Strong – gluons (massless)Electromagnetic – photons (massless)Weak – Z0, W+, W- (massive)
* Of course, in 3 hours we won’t really cover everything – actually nearly nothing in detail!
S. D. Ellis Maria Laach 2008 Lecture 1 5
Concepts/Vocabulary II
• Quantum Field Theories – local non-Abelian gauge symmetries, UV singularities, running couplings UV freedom & IR slavery, perturbative expansions, IR & collinear singularities, (leading to)renormalization (scale and scheme, e.g., MSbar) of PDFs factorization (scale), power corrections, log resummation
• Experimental quantities – exclusive cross sections, inclusive cross sections, IR safe quantities, jets
• Experimental processes – e+e- hadrons, e()pe+hadrons, pphadrons (jets), pp+X, pp+-+X, ppB(eyond the)SM
SMSM
S. D. Ellis Maria Laach 2008 Lecture 1 6
(Incomplete) References: (I’ll focus on concepts/images)
• “The Pink Book” – QCD and Collider Physics, R.K. Ellis, W.J. Stirling and B.R. Webber (Cambridge University Press, 1996)
• My PiTP 2007 Lectureshttp://www.phys.washington.edu/users/ellis/PiTP%20July%2007.htm
• My TSI 2006 lectures -http://www.phys.washington.edu/users/ellis/TSI%20July%2006.htm
• “Jets in Hadron-Hadron Collisions” by S.D. Ellis, J. Huston, K. Hatakeyama, P. Loch, M. Toennesmann, arXiv:hep-ph/0712.2447v1
• The “Primer for LHC Physics” by J.M. Campbell, J.W. Huston, W.J. Stirling, arXiv:hep-ph/0611148v1
• Lectures by George Sterman, et al. – (for more references in formal details) arXiv:hep-ph/0807.5118v1, arXiv:hep-ph/0412013v1, arXiv:hep-ph/0409313v1
S. D. Ellis Maria Laach 2008 Lecture 1 7
(more) References:
• QCD Summary on the Web at the Particle Data Group site: http://pdg.lbl.gov/2008/reviews/qcdrpp.pdf
• The CTEQ Handbook in Rev. Mod. Phys. Volume 67, Number 1, January 1995, (pp. 157-248) and on the Web: http://www.phys.psu.edu/~cteq/#Handbook
Simple Initial Picture – The Naive Parton Model
(~1970, BSM = Before the SM)Imagine a “theory” of hadrons composed of (nearly massless) quarks and
(massless) gluons which (based on experiment – no YM theory)
• Interact via scale invariant, perturbative (weak) interactions dominated by
exchanges with momenta k ≤ m ~ 1 GeV (typical hadron mass scale)
• Are never seen as isolated states
• Interact with Electro-weak currents in expected way (quark model charges)
• Inside (relativistic) hadrons are described by (scale invariant) parton distribution
functions: q(x) = Fq/h(x) = probability to find quark (of flavor q) with (collinear)
momentum fraction x in hadron (and little transverse momentum ~1/hadron size)
• When isolated in phase space, fragment into hadrons as described by (scale
invariant) fragmentation functions: Dh/q(z) = probability to find hadron in
(collinear) debris of quark with momentum fraction zS. D. Ellis Maria Laach 2008 Lecture 1 8
S. D. Ellis Maria Laach 2008 Lecture 1 9
Parton Model –
partons are the building blocks of hadrons and
play a role in the dynamics (even if we didn’t
understand it!).
Consider the inclusive deeply inelastic scattering of electrons from protons – DIS, (e.g., SLAC). e
e’
S. D. Ellis Maria Laach 2008 Lecture 1 10
Recall the EXclusive case: ep → ep
= p’s anomalous magnetic moment,
2 2
1 2
22 2 2 2
22
2
2
1 1
4 21
4
p
E M M E
p p
p
J ieu p F q i q F q u pm
qieu p G q G q i q G q G q u p
qm m
m
General EM vertex (see HW) symmetries 2 functions
22 20
22 2 2 2 2
0 0~ 0.7 GeV
0 1, 0 1, 1
,
i E M
i j q q
F G G
F q G q q q q
Form Factors – The harder you hit a proton, the more likely it is to fall apart !
Electric Form Factor Magnetic Form Factor
S. D. Ellis Maria Laach 2008
Lecture 1
11
Now the INclusive case: ep→eX
In the proton rest frame (Lab) the
kinematics look like:
22 2
2 2
2 2
2
2 2
2
2
2 1
1
X
p p
p p
p
M W p q
q p q p
m m Q
m x m
Q m
2 2
,0 , , , ,
1
2 2
1
p
p
p p
p m k E k k E k
p qE E
m
q Qx
m m E E
q p Ey
k p E
A new degree of freedom, x (MX or ), but still two (dimensionful) functions (allowed by symmetries) describing the scattering (x→1 = elastic) (See the HW)
2 2
22 2 2 2
1 22 4
2sin , cos ,2 2
4 sin2
L L
Q qLL ep ep
dW Q W Q
dE d E
S. D. Ellis Maria Laach 2008
Lecture 1
12
2 new dimensionless functions (see the
HW) in Relativistic Notation
2
2 2 2 2
1 2 12 4
4 11 1 , , 2 ,
d yy F x Q F x Q xF x Q
dxdQ Q x
F1 absorption of transversely polarized photons
F2 – 2x F1 longitudinally polarized photons (in the high energy limit)
Recall: for an elementary fermion (onshell ), electric charge ef
2 2p q p
22
2
2 4
4| 1 1 1
2
f
f
edy x
dxdQ Q
2 2 2 2
1 1 2 2, , , , ,F x Q mW Q F x Q W Q
S. D. Ellis Maria Laach 2008 Lecture 1 13
Scaling, the bj limit• Limit fixed (the “bj” or scaling limit), if there is no large
hadronic scale (the hadronic physics is soft or “slow”), we naively
expect , , i.e., scaling.
2, ;Q x
2 2 2,i i iF x Q F x m Q F x
• Interpretation – the proton is composed of essentially free, point-like charged partons = quarks (?) with x as the fraction of the proton’s moment carried by the scattered parton - what could be simpler!
Plot versus x for different Q2,
Not falling off rapidly with Q2 like form factors
The proton is not filled with mush!
S. D. Ellis Maria Laach 2008 Lecture 1 14
Sum of individual (incoherent) quark (parton)
contributions (the parton model)
xF = the momentum fraction carried by the quark,
For the proton we have
2
2 1ˆ ˆ2bj bj q bj bj FF x xF x e x x x
2
2 1 /
,
2 q q p
q q
F x xF x e xF x
/q pF x q x distribution of quarks within the proton (factored from rest of the event).
1 22xF x F x Callan-Gross relation (spin ½).
Experimentally (approximately) true - evidence that partons are quarks (or at least fermions).
,q Fp x p 2 2bjx Q m
S. D. Ellis Maria Laach 2008 Lecture 1 15
Flavors: SU(3) hadrons in 8’s, 10’s and 1’s
Define distributions for each flavor, with valence quarks and a flavor
neutral sea:
/ /;u p V d p VF x u x u x S x F x d x d x S x
S x u x d x s x s x
So that
2
4 1
9 9F x x u x u x d x d x s x s x
with (experimentally correct)
1 1
0 0
2; 1V Vdxu x dxd x
Valence quarks
“Ocean” quarks
S. D. Ellis Maria Laach 2008 Lecture 1 16
Momentum:
But total momentum (in DIS) -
1 1
0 0
6 0.5V V dataq
dxx q x q x dxx u x d x S x
• Only 50% of the momentum is carried by quarks, the rest is glue!
• Typical parton distribution functions look like
PDFs
Note factors of x
Note that the sea is NOT SU(3) or even SU(2) symmetric.
Collider Lessons
• pp collisions really parton-parton
• Small x is dominated by glue
• SM (< 1 TeV) Physics at the LHC is dominantly
from gluon-gluon collisions
-not like the Tevatron!
S. D. Ellis Maria Laach 2008 Lecture 1 17
S. D. Ellis Maria Laach 2008 Lecture 1 18
Other Processes: e+e- hadrons (final state)
Think of this inclusive process in terms of . The total cross
section is thus
e e qq
22 2
0 2
4 2;
3 3q qe e
q q
e e qqe R e
Q e e
The picture looks like –
again factor short and
long distances
At “long distances” the scattered quarks pull further quarks and anti-quarks out of the vacuum that somehow reassemble into hadrons.
< data !
S. D. Ellis Maria Laach 2008 Lecture 1 19
New “long Distance” Concepts:
• Jets - A jet is a “spray” of essentially collinear hadrons whose total momentum and even flavor quantum numbers track (but don’t equal) those of the fragmenting quark – the “footprint” of the quark. Based on observation, e.g., in high energy cosmic ray collisions, that hadron-hadron collisions produce mostly hadrons in longitudinal direction, low relative kT (reason for parton model).
■ Expect 2 jets in electron-positron annihilation. ■ One “current” jet (from scattered quark) and remnant of target in DIS.
• Fragmentation - the fragmentation function Dh/q(z) describes the probability to find a hadron h in the collinear debris of the fragmenting quark q with momentum fraction z of the original quark, assuming cutoff in transverse momentum, kT < 500 MeV/c.
1
0
1h q
h
dz zD z 1
~
h qn
h q h q
zD z N
z
Hard hadrons unlikely
Soft hadrons likelyMomentum conservation
S. D. Ellis Maria Laach 2008 Lecture 1 20
Jets in e+e- Physics
• Study the kinematics of the produced quarks by studying the kinematics of the leading hadrons – forming 2 jets.
• The angular cross section for electrons to quarks, i.e., spin ½ fermions, should track the angular distribution of the jets (or at least the leading hadrons) –
2
2 2
21 cos
cos 2q
q
de
d Q
and it (approximately) agrees with the data! This is another indication that the charged partons are really quarks!
,
ˆ ˆ2 2
h q h q
qh
h h h h h
q jet jet
de e hX e e qq D z D z
dz
E E p j E p jz
E Q Q E p
h = hadron, not Higgs
1 2 1 22 2 2
2
2
4 0
2
ˆ~ , , , ,
1
T
T T T
T
T
T
d d sF x F x p x x z D z
dp dp p
m
p
dp pp X
dp
S. D. Ellis Maria Laach 2008 Lecture 1 21
Hadron-Hadron collisions Large Transverse
Momentum (Large PT) Inclusive Cross section
• Treat as FACTORING into 4
independent components!!
[Factor short distance/large
momentum from long
distance/low momentum]
e.g., pp→0 +X
[As observed (incorrectly) in 1972]
x1
x2
z
{Dimensional analysis}
S. D. Ellis Maria Laach 2008 Lecture 1 22
For Example: (More Later!)
Unfortunately no (known) quantum field theory has
all of these properties exactly!• Fortunately QCD, an SU(3) non-Abelian gauge theory, has approximately
these properties (and thus explains the observations)!
• Unfortunately proving this resemblance requires the calculation of (too)
many Feynman diagrams, a careful choices of gauges, a thorough
understanding of the Renormalization Group, etc., the proof has taken 30
years and still needs work.
• Fortunately there are many smart people doing the hard work (including
string theorists)!
Big issue is that in a gauge theory there are (sometimes) relevant
interactions at all momentum scales!
• Fortunately the dominant dynamics is (approximately) local in momentum
space and FACTORIZATION still works (for the right questions); we can
approximate full dynamics as a convolution of several factors involving
different momentum scales.
S. D. Ellis Maria Laach 2008 Lecture 1 23
Same form – more Details – be explicit about scale
dependence
S. D. Ellis Maria Laach 2008 Lecture 1 24
2
1 2 1 22 2 2 2
2
1 22 2
2
/
ˆ~ , , , , 1
,
, ,
ˆ 1,, , , , ,
pp
a p a F b p b F
T
ab
T
T T T T
T TT
T
c c F
F T
dF x F x
d d s mF x F x p x x z D z
dp dp p p
p ps mp x x z
p
dp
D zp
Short distance, UV physics, k > running coupling s() in perturbative calculation of ̂
Long distance, IR physics, kT < F (collinear) scale dependent, universal PDFs and Fragmentation functions
NOTE: Full Physics is independent of scale choices, scale dependences must match (order-by-order in PertThy)
20
T
d d
d dp
S. D. Ellis Maria Laach 2008 Lecture 1 25
The (Classical) QCD Lagrangian (+ gauge fix + counter
terms)
Acting on the triplet and octet, respectively, the covariant derivative is
,
, ,
1
4
B
QCD B f a f f babf
B B B BCD C D
L F F q iD m q
F A A gf A A
;C C B B
ab CDab CDab CDD ig t A D ig T A
The matrices for the fundamental (tabB) and adjoint (TCD
B) representations carry the information about the Lie algebra
, ; , ; ;
4; ;
2 3
3
B C BCD D B C BCD D B BCD
CD
BCB C BC B B
R ab bc ac F ac
B C BC BC
A
t t if t T T if T T if
Tr t t T t t C
Tr T T C
(fBCD is the structure constant of the group)
S. D. Ellis Maria Laach 2008 Lecture 1 26
Yang-Mills – first easy (algebraic) improvements
to parton model
Quarks have a previously hidden quantum number, COLOR, that comes in 3 values (quarks are a 3 under the corresponding SU(3)) so that
• Color singlet ground state - meson - baryon with 3 quarks is anti-symmetric ( )
• Must sum over colors in e+e- final state factor of 3, Re+e-2!
• Extra partons holding proton together gluons, carrying the rest of the momentum (a LOCAL SU(3) symmetry), but only “small” corrections to parton model
• QCD Dynamics – look at where perturbation theory is large (divergences) – UV, soft and/or collinear configurations (propagators ~ on-shell) – what is in the Monte Carlos
a aq q
abc a b cq q q
S. D. Ellis Maria Laach 2008 Lecture 1 27
Feynman Rules:
Propagators – (in a general gauge represented by the parameter ,
Feynman gauge is = 1; this form does not include axial gauges)
Vertices –
Quark – gluon 3 gluons
S. D. Ellis Maria Laach 2008 Lecture 1 28
Feynman Rules II:
4 gluons
S. D. Ellis Maria Laach 2008
Lecture 1
29
pQCD I - Use QCD Lagrangian to Correct the
Parton Model
• Naïve QCD Feynman diagrams exhibit infinities at nearly every turn, as they must in a conformal theory with no “bare” dimensionfulscales (ignore quark masses for now).***
First consider life in the Ultra-Violet – short distance/times or large momenta (the Renormalization Group at work):
• The singular UV behavior means that the theory
does not specify the strength of the coupling in terms of the “bare” coupling in the Lagrangian
does specify how the coupling varies with scale [s() measures the “charge inside” a sphere of radius 1/]
*** Typical of any renormalizable gauge field theory. We will not discuss the issue of choice of gauge. Typically axial gauges ( ) yield diagrams that are more parton-model-like, so-called physical gauges.
ˆ 0n A
S. D. Ellis Maria Laach 2008 Lecture 1 30
Consider a range of distance/time scales – 1/
• use the renormalization group below some (distance) scale 1/m (perhaps down to a GUT scale 1/M where theory changes?) to sum large logarithms ln[M/] and ln[/m]
• use fixed order perturbation theory around the physical scale 1/ ~ 1/Q (at hadronic scale 1/m things become non-perturbative, above the scale M the theory may change)
Short distance Long distance
(non-perturbative)(perturbative)(new)
S. D. Ellis Maria Laach 2008 Lecture 1 31
Diagrammatically
• Corrections to the parton model come from
adding gluon interactions, including
LO
1 Loop2 Loop
Loops are UV divergent like dk4/k4 - keep (logarithmic) contributions from the range to M as a “formal” series for the effective coupling in terms of the initial coupling.
22 2
2 2 30 0
2 2
0
~ ln ln4 4
11 2 211
3 3
s s s s
A f
f
M M MM M
C nn
S. D. Ellis Maria Laach 2008 Lecture 1 32
Interpret as screening/anti-screening of color
charge in volume (1/)3
2/ + +
+
S. D. Ellis Maria Laach 2008 Lecture 1 33
This result is more compactly specified by the renormalization
group equation, which can be evaluated order-by-order in
perturbation theory (PDG notation)
Sum (reorganize) the (cutoff) calculation results as an effective (renormalized) coupling**
02
02
11 2 2; 11
3 31 ln
4
A fs
s f
s
C nMn
MM
**Masses and wave functions also exhibit renormalization.
2 3 4
0 1 22 3
1
2
2
22 4 64
1951
3
5033 3252857
9 27
s s s ss
f
f f
d
d
n
n n
S. D. Ellis Maria Laach 2008 Lecture 1 34
Lesson: Can Sum Large Logarithms
The “running” coupling illustrates typical features of QCD –
• expanding to a fixed power of s is often not enough*
• large logarithms (the remnants of the infinities) must be resumed to all orders by some technique
• By measuring s at some scale 0 can define a dimensionful parameter QCD
0 0
2
0
0
2
ln
s
s QCD
QCD
e
Dimensional transmutation !!
* In any case is an asymptotic expansion, not convergent series
S. D. Ellis Maria Laach 2008
Lecture 1
35
The first form above is the “one-loop” solution for s (keeping only the 0 term). pQCD allows one to systematically include the higher loop corrections, as expansion in inverse powers of ln[].
2
2
1
22 20
2 20
22 2
1 2 02 22
4 2 120
ln ln4 2
1
ln ln
4 1 5ln ln
2 8 4ln
QCD
s
QCD QCD
QCD
QCD
Beyond 1-Loop
S. D. Ellis Maria Laach 2008 Lecture 1 36
Asymptotic Freedom/Infrared Slavery
Our knowledge of the behavior in the UV is now encoded in QCD. Note that the precise value of QCD will to depend on the order of the function used (1-loop, 2-loop, etc.) and the scheme. The data does not change, only the internal theoretical parameters.
• Experimentally QCD ~ 21625 MeV (using 5 “active” flavors at the Z pole)
• The running of the coupling is clear in the data, as is the precision of our knowledge of s, e.g., s(mZ) = 0.1176 0.002.
Look at the (amazing!) behavior of running s –
As increases, s decreases – asymptotic freedom!
As decreases, s increases – infrared slavery!
Just what we wanted in the parton model!!!!!
S. D. Ellis Maria Laach 2008 Lecture 1 37
• s(Q)
• EM(Q) – only fermion
loops contribute, runs the
other way (0EM < 0!)
NOTE -1
S. D. Ellis Maria Laach 2008
Lecture 1
38
But Note!!
• Physical quantities, (Q), cannot depend on
• This is essential to QCD engineering!
, , 0s
d Q
d m
S. D. Ellis Maria Laach 2008
Lecture 1
39
Other Potential Singularities – Infrared (after
renormalizing, i.e., removing, the UV singularities, formally
with counter terms)
• Soft & Collinear! (Massless) Propagators can go on shell due to
emission of soft & collinear gluons – See SCET (Effective Theories)
• Infrared (soft) familiar from QED – e.g., since the photon is zero
mass (in the gauge symmetric theory), the theory wants to emit an
infinite number of zero energy photons and the exclusive (electron)
cross section diverges. Fix with inclusive cross section that sums
over soft photons over 0 < E < E leading to Ln[E/Q] dependence
• Collinear, mq 0, still gives ln[Q/ mq] which is large for mq Q, and
here we will think about mq → 0
2 21 2
22 1 or 2
3 1 2 0
1 2 1 2
0, soft0
, 0, collineark k
kk k k
k k k k
S. D. Ellis Maria Laach 2008 Lecture 1 40
pQCD II - Perturbative Corrections to Parton
Model – e+e- Annihilation (illustrative example)
• Revisit e+e- scattering (massless partons!) – real emission
• Define handy variables (q2=Q2=s)
3
221
2
2 20 2
2 11 cos
2
i ii i i
i
i j k k
ij
i j i j i j
E p qx x x
p p q p x
E E E E x x
S. D. Ellis Maria Laach 2008 Lecture 1 41
Sum & Square -
• Sum amplitudes and square, visualized (ignoring the lepton part) as
the (3-body) imaginary (absorptive) parts of the following loop
diagrams (the vertical dashed line identifies the particles that are put
on the mass shell, i.e., that are the “real” particles in the final state)
+ + +
2 2
1 20
1 2 2 3 1 3
2 2 221 2
0 0
1 2
2
43
2 1 1 3
sF
sF f
f
E EdC s
dx dx p p p p
x xC e
x x s
S. D. Ellis Maria Laach 2008 Lecture 1 42
e+e- Annihilation cont’d
• The cross section looks like (see
HW)
Hence the singular regions are:
• Collinear gluon -- 130, x21
230, x11
• Soft gluon – x30 (x11 and
x21) with (1- x1)/(1- x2) fixed
2 2
1 20
1 2 1 22 1 1
sF
d x xC
dx dx x x
Phase Space
The red singularities arise from a propagator above going on-shell – either 1+3 or 2+3 !
S. D. Ellis Maria Laach 2008 Lecture 1 43
Long Distance Collinear/Soft Singularities
• On-shell propagators – long distance propagation – perturbative
expansion fails – 1+ s x big + s2 x bigger … (No Surprise)
• Still parton model-like picture – short distance/time simple, long
distance complex.
How do we proceed?
• Ask questions that are insensitive to long distance structure (IR Safe)
e.g., TOT which receives contributions from all states – details cannot
matter – in detail the singularities in the virtual graph (interfering with
LO) cancel with those above
+
S. D. Ellis Maria Laach 2008 Lecture 1 44
ASIDE: Dim(emsional) Reg(ulation)
• Say we want –
• Consider – [44-2, Wick rotate to Euclidean space]
• Calculate –
4
4 22 2
1
2
d kI
k m
4 4 22 2 3 24 2
4 4 2 4 22 2 2
E E
d k d k ddk k
4 2
4 2 2
2 13 22 11
2 22 2
0 0
2
2 1
22 4
12
21
2 2
EE
E
d
kdk dzz z
mk m
m
S. D. Ellis Maria Laach 2008 Lecture 1 45
Simplify
• Using –
• Find –
singular bits, plus finite bits 0, plus log singularity as m0
• Define Scheme – subtract (absorb) 1/ , E and ln(4) bits****
1 ; 1 0.5772Ez z z
2
2 20
1 1ln 4 2ln
4 4EI O
mm
MS
**** You can hid anything in infinity!
S. D. Ellis Maria Laach 2008 Lecture 1 46
pQCD Calculation III: Apply Dim-Reg to total
e+e- cross section ( )
• Real emission
2 2
1 2 1 2
0 1 2 1 1
1 2 1 2
2
0 2
2
1 2 12
2 1 1 1
2 3 19
2 2
3 1 41
3 2 2 2
qqg F s
F s
x x x xCH dx dx
x x x x
CH O
H O
Dependence of Born
Numerator - Dependence of matrix element
Denominator - Dependence of phase space
4 2D
S. D. Ellis Maria Laach 2008
Lecture 1
47
Virtual -
• Virtual emission (interference, < 0!)
• Sum and set 0,
R = (e+e-hadrons)/(e+e-+-)
2
0 2
2 38
2
qq g F sCH O
0
2
2 20
2
33
3
4
1
1
f
f
qq gqqg
F s
f s
f
s
s
CR e
e
O
O
Parton Model (with color)
NLO QCD Correction
S. D. Ellis Maria Laach 2008 Lecture 1 48
Well behaved as promised !
• Finite and well behaved – more work, higher order corrections
22
2 2
32 2
2
2 2
4
1 1.4092 1.9167ln3
12.805 7.8179ln 3.674ln
s s
f
f
s
s
RK
e Q
Q Q
O
S. D. Ellis Maria Laach 2008 Lecture 1 49
Higher Orders Typical Behavior When -n Cancel -
• Physical quantity is INdependent !!Fixed Order pQCD is NOT!!
• pQCD higher orders exhibit explicit ln(/Q) factors
• of higher orders exhibits reduced dependence on unphysical parameter
• at order sn the residual
explicit ln() dependence is order s
n+1
• dependence is an artifact of the truncation of the perturbative expansion
S. D. Ellis Maria Laach 2008 Lecture 1 50
Standardize the Real – Virtual Cancellation with
Concept - InfraRed Safety!!
• Define InfraRed Safe (IRS) quantities – insensitive to collinear and
soft emissions, i.e., real and virtual emissions contribute to same
value of quantity and the infinites can cancel! (can really set quark
masses to zero here)
• Powerful tools exist to study the appearance of infrared poles (in dim
reg) in complicated momentum integrals viewed as contour integrals
in the complex (momentum) plane. For a true singularity the contour
must be “pinched” between (at least) 2 such poles (else Cauchy will
allow us to avoid the issue). We will not review these tools in detail
here.
• See Lecture 2
S. D. Ellis Maria Laach 2008 Lecture 1 51
Summary:
• Fix issues of Parton Model with QCD!
• “Physical” picture remains the same – partons at short distances,
hadrons at long distances
• Some changes – both coupling and distributions now vary with scale
in predictable way - must be measured experimentally at some scale
• Physics still factorizes into convolution of factors depending on
different scales
• Corrections to Parton Model are “small” for the “right” (IRS)
quantities!
• Next Lecture – More about calculating in QCD, IRS & Jets
S. D. Ellis Maria Laach 2008 Lecture 1 52
Extra Detail Slides
S. D. Ellis Maria Laach 2008
Lecture 1
53
The Big Picture of the Big Four (except Dark
Energy):
Interaction Observed Strength Range Carrier Theory
Strong Nuclear
Forces
~1
<1
~10-15 m
1/r2 (< 10-15 m)
pion
Gluon
SM
EM Atomic
Systems
~10-2 1/r2 Photon SM
Weak Decays ~10-5 ~10-18 m W, Z SM
Gravity Astronomy ~10-39 1/r2 Graviton SUSY
Strings
S. D. Ellis Maria Laach 2008
Lecture 1
54
First a Bit of History
• PreQCD I (< ~1968) Quarks “algebraic” with no dynamics (of
course, we had dynamics in the form of Regge theory, the
bootstrap picture and dual models but …). Hadron quantum
numbers correct if baryon, meson,
approximate FLAVOR SU(3)
qqq qq
Property/Quark u d s
Electric Charge +2/3 -1/3 -1/3
Isospin, I3 +1/2 -1/2 0
Strangeness 0 0 -1
S. D. Ellis Maria Laach 2008 Lecture 1 55
Hadrons appear in multiplets of SU(2),
e.g.,mesons
1 1 1 12 3 2 3
23
.
, ,
0,
z
z
Y S B I Y
d u
I
s
3
23
1 1 1 12 3 2 3
,
0,
, ,
z
z
Y I Y
s
I
u d
3
0 0 1 12 2
0 01 12 2
1
,1 ,1
2 0,0
11,0 1,02 6
00,02
, 1 , 1
z
Y
K K ds K K us
uu dd
Idu ud Iuu dd ss
I
uu dd
K K su K K sd
-8 0
3uu dd ss
ss
1
S. D. Ellis Maria Laach 2008 Lecture 1 56
1970s Include c quark - SU(4) Representations
Mesons Baryons
S = 0- (↑↓)
S = 1- (↑↑)
S = 1/2+(↑↑↓)
S = 3/2+(↑↑↑)
S. D. Ellis Maria Laach 2008 Lecture 1 57
ASIDE: Early Duality
If no fundamental particles,
• then particles are made out of themselves = “bootstrap”
• resonances “made” in the s-channel are equivalent (dual) to resonances exchanged in the t-channel
t-channels-channel
=
Topologically equivalent, like stretching an elastic sheet String Theory (eventually)
S. D. Ellis Maria Laach 2008
Lecture 1
58
Review Elastic Scattering of “elementary”
fermions: e → e (see HW)
• Kinematics
2 2 2 2 2 2
2 2
2 22
2 2
2 2
2 2 2 2
2 2
,
2 2
1
e
e
e e
e
p p m k k m
s p k p k
t q k k p p
s m m
u k p k p
u
y
ym m s m m
s t u m m
2 2
2 2 2 22
2 22 2, 2 2
2 41 1
es m m
e e
d s uy
dq sq q
S. D. Ellis Maria Laach 2008
Lecture 1
59
ASIDE: Viewed in Rest Frame
2 2
rest frame: ,0 , , , , ,
, 4 sin2
L
p m k E k k E k
E Ey q EE
E
2 2
22 4
2,2 4
rest frame
2 2 22
22 4
2cos sin
2 24 sin
2
cos2 1 tan
224 sin2
e
L Ls m m
LLab
L
L
L
d E EE
d E mE
E q
E mE
spin flipRutherford spin nonfliprecoil
Mott (electron on scalar)
S. D. Ellis Maria Laach 2008 Lecture 1 60
ep → ep Scattering
2 2
2 2 222 2 2
1 2 1 22 22 4
22 2 2 2
22 22 2 2 2
2 22 4
2
cos2 tan
24 24 sin2
4cos sin
2 224 sin 124
L
L
LL p pep ep
E M
p L LM
L p
pQ q
d E t tF t F t F t F t
d E m mE
QG Q G Q
mE QG Q
QE mEm
Kinematics (in p rest frame)
22 2
22 2 2
,0 , , , ,
4 sin ,2
p
L
p
p m k E k k E k
E Eq k k EE y
E
m p p p q
S. D. Ellis Maria Laach 2008 Lecture 1 61
Impact Scenario in Space-Time
• electron and quark interaction on short time scale (~1/Q); quarks interact on long time scale (~1/mass).
Essentially free during scattering
• Aside Also useful to consider the “infinite momentum frame” where Pinfinity, the proton is “mostly” contracted (except wee partons ~dx/x), internal interactions are “frozen” (dilated)
S. D. Ellis Maria Laach 2008
Lecture 1
62
QCD:
• QCD Field Theory – unbroken SU(3)
• quarks in the fundamental representation – the triplet,
• anti-quarks in the complex conjugate representation,
• Recall that we now have 6 quarks (with masses spread over an
enormous range – NOT explained by QCD – why not?)
quark u d s c b t
charge 2/3 -1/3 -1/3 2/3 -1/3 2/3
mass ~ 4 MeV ~ 7 MeV ~ 135
MeV
~1.5 GeV ~ 5 GeV ~ 178 GeV
S. D. Ellis Maria Laach 2008 Lecture 1 63
Local Symmetry vector gluons in the Adjoint
Representation, the 8
• Gauge Transformation coupling
• Quarks have both a flavor and a color index (and spin) - ,f aq
• Color singlet states (1) with no indices - just what we wanted!!
- mesons
- baryons
a aq q
abc a b cq q q