Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group
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Paul <strong>Hoyer</strong> Mugla 2010<br />
<strong>Introduction</strong> <strong>to</strong> <strong>QCD</strong><br />
Paul <strong>Hoyer</strong><br />
University of Helsinki<br />
Akyaka - Mugla, Turkey<br />
27.8 - 4.9 2010<br />
1
As simple as 1-2-3?<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
The Standard Model<br />
SU(3) x SU(2)L x U(1)<br />
<strong>QCD</strong><br />
Electroweak<br />
Not exactly: The strong, electromagnetic and weak interactions<br />
manifest themselves very differently in Nature<br />
SU(2)L x U(1): Where does the lagrangian come from?<br />
SU(3): What does the lagrangian do?<br />
2
Paul <strong>Hoyer</strong> Mugla 2010<br />
The Discovery of the Strong Interaction<br />
Rutherford’s experiment 1911:<br />
The positive charges in matter are<br />
located in a tiny nucleus, whose radius<br />
is ~ 10 –5 of the a<strong>to</strong>mic size.<br />
⇒ There must be a strong, short-ranged<br />
force <strong>to</strong> counteract the Coulomb repulsion<br />
Thomson’s a<strong>to</strong>m: Rutherford’s a<strong>to</strong>m:<br />
F = α<br />
r 2<br />
Rnucleus<br />
Ra<strong>to</strong>m<br />
� 10 −5<br />
3
Molecules<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Matter<br />
-<br />
e<br />
-<br />
e<br />
-<br />
e<br />
A<strong>to</strong>m<br />
The Standard Model A. Pich - CERN Summer Lectures 2008<br />
-<br />
e<br />
Electron<br />
Nucleus<br />
p<br />
n n<br />
n<br />
pp<br />
p<br />
n<br />
p<br />
n p n<br />
p<br />
n n<br />
n p<br />
p<br />
Pro<strong>to</strong>n Neutron<br />
Quarks<br />
4
Paul <strong>Hoyer</strong> Mugla 2010<br />
The Pion as a carrier of the strong force<br />
In 1935 Hideki Yukawa suggested the existence of a new, strongly interacting<br />
particle “U” (later named the pion). The strength and range of the strong<br />
interaction could be unders<strong>to</strong>od as arising from pion exchange.<br />
• • •<br />
π<br />
• • •<br />
π<br />
π<br />
N<br />
• • •<br />
R < ∼ 1/Mπ<br />
Postulating a new particle was considered very bold in those days, when<br />
only a handful of elementary particles were known: pho<strong>to</strong>n, pro<strong>to</strong>n, neutron,<br />
electron, (neutrino).<br />
It has been suggested that “social pressure” may have kept physicists in the<br />
West from a similar proposal.<br />
Even <strong>to</strong>day in <strong>QCD</strong>, the pion is special: It is a Golds<strong>to</strong>ne boson.<br />
5
Paul <strong>Hoyer</strong> Mugla 2010<br />
Birth of Yang-Mills Theory (1954)<br />
Quantum ElectroDynamics (QED) is invariant under local (space and time -<br />
dependent) gauge transformations:<br />
Electron field:<br />
Pho<strong>to</strong>n field:<br />
ψ(x) → e ieΛ(x) ψ(x)<br />
Aµ(x) → Aµ(x) − ∂µΛ(x)<br />
Λ(x) may be any<br />
regular function<br />
In 1954, Yang and Mills generalized this local U(1) gauge symmetry <strong>to</strong> the<br />
SU(2) group of isospin, with the pro<strong>to</strong>n and neutron forming an SU(2)<br />
� �<br />
p<br />
isospin doublet just as in Yukawa’s theory:<br />
n<br />
This established the structure of non-abelian gauge symmetry.<br />
Nature found, however, different uses of YM theories.<br />
⇐ As in classical ED!<br />
6
Matter field:<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
ψ(x) → U(x)ψ(x)<br />
Physical Review 96 (1954) 191<br />
For a local gauge transformation defined by an SU(2) matrix U(x), Yang and<br />
Mills found that the theory is symmetric provided the fields transform as:<br />
Gauge field: Aµ(x) → U(x)Aµ(x)U † (x) − i<br />
g U(x)∂µU † (x)<br />
The same rule holds for any group, such as SU(3). In QED,<br />
g is the same<br />
for all ψ!<br />
U(x) =e ieΛ(x) Then U1U2 = U2U1, i.e., all group elements commute,<br />
hence the U(1) gauge symmetry is said <strong>to</strong> be abelian.<br />
7
† †<br />
2 * 2 matrices: " " " " " " # ; det " " 1<br />
"<br />
" exp !<br />
a 1 a<br />
Fundamental Representation: ! F " !<br />
2 Pauli<br />
The Standard Model A. Pich - CERN Summer Lectures 2008<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
+ i - ,<br />
a a<br />
Commutation Relation:<br />
a a†<br />
a<br />
; ! " ! ; Tr ( ! ) " 0 ; a = 1 , ! , 3<br />
a b abc c<br />
/ , 0<br />
1<br />
! !<br />
2<br />
" i . !<br />
1 $ 0 1% 2 $ 0 # i % 3 $ 1 0 %<br />
! " & ' ; ! " & ' ; ! " & '<br />
( 1 0) ( i 0 ) ( 0 # 1)<br />
8
a a<br />
Fundamental Representation: ! F ! # Gell-Mann<br />
$ 0 1 0% $ 0 " i 0% $ 1 0 0% $ 0 0 1 %<br />
1 & ' 2 & ' 3 & ' 4 & '<br />
# !<br />
&<br />
1 0 0<br />
'<br />
; # !<br />
&<br />
i 0 0<br />
'<br />
; # !<br />
&<br />
0 " 1 0<br />
'<br />
; # !<br />
&<br />
0 0 0<br />
'<br />
& 0 0 0' & 0 0 0' & 0 0 0' & 1 0 0'<br />
( ) ( ) ( ) ( )<br />
$ 0 0 " i % $ 0 0 0% $ 0 0 0 % $ 1 0 0 %<br />
5 & ' 6 & ' 7 & ' 8 1 & '<br />
# !<br />
&<br />
0 0 0<br />
'<br />
; # !<br />
&<br />
0 0 1<br />
'<br />
; # !<br />
&<br />
0 0 " i<br />
'<br />
; # ! 0 1 0<br />
3<br />
& '<br />
& i 0 0 ' &0 1 0' &0 i 0 ' &0 0 " 2'<br />
( ) ( ) ( ) ( )<br />
1 123 147 156 246 257 345 367 1 458 1 678 1<br />
f ! f ! " f ! f ! f ! f ! " f ! f ! f !<br />
2 3 3 2<br />
The Standard Model A. Pich - CERN Summer Lectures 2008<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
a b abc c<br />
[ ! , ! ] ! if !<br />
1<br />
2<br />
9
Yang-Mills<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Particles found in Experiments<br />
Pauli ν<br />
using particle accelera<strong>to</strong>rs<br />
Dirac e +<br />
Einstein γ<br />
Yukawa π ±<br />
using cosmic rays<br />
10<br />
http://fafnir.phyast.pitt.edu/particles/
Mesons<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Quark Model classification of Hadrons<br />
q¯q<br />
Hadrons may be formed<br />
with any combination of<br />
q = u,d,s,c,b,t<br />
π +<br />
Free quarks have never been seen:<br />
p<br />
Baryons<br />
qqq<br />
Were quarks mere mathematical rules, or true particles? (Gell-Mann)<br />
12
Paul <strong>Hoyer</strong> Mugla 2010<br />
Quarks have Color<br />
Three quark colors were introduced by O.W. Greenberg in 1964, <strong>to</strong> make the<br />
quark model of the pro<strong>to</strong>n compatible with the Pauli exclusion principle:<br />
Baryon wave functions must be antisymmetric under the interchange of<br />
quarks. In the Quark Model, the space – spin wf is symmetric. The color wf is<br />
antisymmetric, rescuing the Pauli principle.<br />
The antisymmetric color wave function<br />
(A,B,C = red, blue, yellow) means that<br />
the pro<strong>to</strong>n is a color singlet (does not<br />
change under gauge transformations).<br />
Mesons are also color singlets:<br />
|π〉 = �<br />
A<br />
q A ¯q A<br />
|p〉 = �<br />
A,B,C<br />
U|p〉 = |p〉<br />
U|π〉 = |π〉<br />
εABCq A q B q C<br />
13
Quarks are for real: Pointlike scattering of electrons<br />
<strong>High</strong> energy electrons scatter from pointlike quarks inside the pro<strong>to</strong>n:<br />
e +q → e +q (in analogy <strong>to</strong> Rutherford’s experiment)<br />
The struck quark flies out of the pro<strong>to</strong>n and “hadronizes” in<strong>to</strong> a spray (jet) of<br />
hadrons (mostly pions).<br />
At relativistic energies quark-antiquark pairs are created <strong>to</strong> ensure that all<br />
quarks end up as constituents of mesons or baryons.<br />
Deep Inelastic Scattering (DIS):<br />
e + p → e + anything<br />
SLAC 1969: Ee = 20 GeV<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
14
Paul <strong>Hoyer</strong> Mugla 2010<br />
The search for asymp<strong>to</strong>tic freedom<br />
The SLAC data (1969) showed that the pro<strong>to</strong>n<br />
charge is located in apparently free, pointlike<br />
constituents, presumably the quarks proposed<br />
earlier based on the hadron spectrum.<br />
This raised the question:<br />
“How can the quarks be nearly free inside the pro<strong>to</strong>n,<br />
yet bind so strongly <strong>to</strong>gether that they do not exist as free particles?”<br />
It was known that the coupling<br />
strength of QED “runs”, ie.,<br />
α = α(Q 2 ) increases as the<br />
distance scale r = 1/Q decreases.<br />
If there were theories where α = α(Q 2 )<br />
decreases with r this might explain<br />
the SLAC data.<br />
α(r)<br />
e<br />
p<br />
QED<br />
γ *<br />
e<br />
q<br />
π<br />
...<br />
p<br />
15<br />
log(r)
Paul <strong>Hoyer</strong> Mugla 2010<br />
Quantum Chromodynamics (1972)<br />
The <strong>QCD</strong> Lagrangian defines the interactions of quarks and gluons:<br />
L<strong>QCD</strong> = �<br />
f<br />
Looks like QED! But the gauge symmetry is<br />
SU(3) of color (Greenberg, 1964)<br />
ψ → Uψ U ⊂ SU(3)<br />
All quarks u,d,s,c,b,t have<br />
the same strong coupling g<br />
¯ψf (i/∂ − g /A − mf )ψf − 1<br />
4 F µν Fµν<br />
αs = g2<br />
4π<br />
Note: In QED the abelian U(1) symmetry allows<br />
different electric charges e for different particles<br />
It remains a mystery why:<br />
ψ =<br />
⎛<br />
⎝ q<br />
q<br />
q<br />
⎞<br />
⎠<br />
eu = − 2<br />
3 ee<br />
|ep + ee|/e < 1.0 · 10 −21<br />
20
The Standard Model A. Pich - CERN Summer Lectures 2008<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
#<br />
!<br />
"<br />
S. Bethke<br />
2006<br />
! $% & ' ()**+, - ()((*(<br />
!<br />
"<br />
! $% & ' ()**+! - ()((!.<br />
#<br />
2004<br />
21
Paul <strong>Hoyer</strong> Mugla 2010<br />
Asymp<strong>to</strong>tic Freedom<br />
Asymp<strong>to</strong>tic Freedom<br />
“What this year's Laureates<br />
discovered was something that, at<br />
first sight, seemed completely<br />
contradic<strong>to</strong>ry. The interpretation of<br />
their mathematical result was that the<br />
closer the quarks are <strong>to</strong> each other,<br />
the weaker is the 'colour charge'.<br />
When the quarks are really close <strong>to</strong><br />
each other, the force is so weak that<br />
they behave almost as free particles.<br />
This phenomenon is called<br />
‘asymp<strong>to</strong>tic freedom’. The converse<br />
is true when the quarks move apart:<br />
the force becomes stronger when the<br />
distance increases.”<br />
! S(r)<br />
1/r<br />
22<br />
From P. Skands
Paul <strong>Hoyer</strong> Mugla 2010<br />
Data creates Theory<br />
Data<br />
Discriminating<br />
Observables<br />
Fits<br />
Phenomenolog<br />
ical Models<br />
Individual<br />
Essential<br />
Features<br />
Quark Model,<br />
Eightfold<br />
Way, …<br />
(Solvable)<br />
Theory<br />
Complete<br />
description<br />
Quantum<br />
Chromo-<br />
Dynamics<br />
23<br />
Peter Skands, ESHEP-10
Paul <strong>Hoyer</strong> Mugla 2010<br />
Puzzles of <strong>QCD</strong><br />
<strong>QCD</strong> must have Color Confinement: Only color singlet mesons and baryons<br />
can propagate over long distances.<br />
Color confinement is verified numerically in numerical lattice simulations<br />
and modelled phenomenologically, but the mechanism is still poorly unders<strong>to</strong>od.<br />
Baryons and mesons are bound states of quarks. What are the wave functions?<br />
The quarks and gluons bound in hadrons are highly relativistic.<br />
Data and models exist, but no calculations comparable <strong>to</strong> QED a<strong>to</strong>ms.<br />
How can we compare data on hadron final states with perturbative <strong>QCD</strong>?<br />
Fac<strong>to</strong>rization theorems allow <strong>to</strong> express physical measurements in terms of hard,<br />
perturbative subprocesses and universal, measurable quark and gluon distributions.<br />
24
Paul <strong>Hoyer</strong> Mugla 2010<br />
Perspective: The divisibility of matter<br />
Since ancient times we have wondered whether matter can be divided in<strong>to</strong><br />
smaller parts ad infinitum, or whether there is a smallest constituent.<br />
Democritus, ~ 400 BC<br />
Vaisheshika school<br />
Common sense suggest that these are the two possible alternatives.<br />
However, physics requires us <strong>to</strong> refine our intuition.<br />
Quantum mechanics shows that a<strong>to</strong>ms (or molecules) are the identical<br />
smallest constituents of a given substance<br />
– yet they can be taken apart in<strong>to</strong> electrons, pro<strong>to</strong>ns and neutrons.<br />
Hadron physics gives a new twist <strong>to</strong> this age-old puzzle: Quarks can be<br />
removed from the pro<strong>to</strong>n, but cannot be isolated. Relativity – the creation of<br />
matter from energy – is the new feature which makes this possible.<br />
We are fortunate <strong>to</strong> be here <strong>to</strong> address – and hopefully develop an<br />
understanding of – this essentially novel phenomenon!<br />
25
Instantaneous Coulomb interaction<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Understanding charge screening in <strong>QCD</strong><br />
What went wrong with the nice physical argument of QED?<br />
= N _1<br />
* n 2_<br />
c *<br />
3 f 3<br />
Transverse gluons (and quarks)<br />
Instantaneous Coulomb interaction<br />
= +N 4<br />
*<br />
c<br />
Vacuum fluctuations of transverse fields<br />
Yu. L. Dokshitzer hep-ph/0306287<br />
26<br />
Physical, transverse<br />
gluons screen as QED<br />
Coulomb gluons give<br />
the opposite sign!
In <strong>QCD</strong>, mass terms can be directly introduced in the lagrangian.<br />
In the SM all masses result from “Yukawa interactions” involving the Higgs<br />
field, but they are unconstrained by the theory.<br />
The quark masses inferred<br />
from experiment are:<br />
mu = 1.5 ... 3.3 MeV<br />
md = 3.5 ... 6.0 MeV<br />
ms = 104 ± 30 MeV<br />
mc = 1.27 ± .10 GeV<br />
mb = 4.20 ± .15 GeV<br />
mt = 171.2 ± 2.1 GeV<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Quark masses: Nature’s gift<br />
} m > Λ<strong>QCD</strong> , mass effects large<br />
Non-relativistic bound states<br />
Scale of strong<br />
interactions<br />
27
Binding energy<br />
[meV]<br />
0<br />
-1000<br />
-3000<br />
-5000<br />
-7000<br />
3 1 S 0<br />
2 1 S 0<br />
1 1 S 0<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Charmonium – the Positronium of <strong>QCD</strong><br />
Since mc >> Λ<strong>QCD</strong>, the c¯c mesons are nearly non-relativistic<br />
3 3 S 1<br />
2 3 S 1<br />
1 3 S 1<br />
2 1 P 1<br />
8·10 -4 eV<br />
Ionisationsenergie<br />
2 3 P2<br />
2 3 P1<br />
2 3 P0<br />
3 1 D 2<br />
10 -4 eV<br />
e + e −<br />
3 3 D2<br />
3 3 D1<br />
3 3 D2<br />
~ 600 meV<br />
e + e- 0.1 nm<br />
Mass [MeV]<br />
4100<br />
3900<br />
3700<br />
3500<br />
3300<br />
3100<br />
2900<br />
c(3590)<br />
c(2980)<br />
(4040)<br />
(3770)<br />
(3686)<br />
(3097)<br />
3 D1<br />
hc(3525)<br />
3 P2 (~ 3940)<br />
3 P1 (~ 3880)<br />
3 P0 (~ 3800)<br />
2(3556)<br />
1(3510)<br />
0(3415)<br />
C<br />
1 D 2<br />
DD<br />
3 D 3<br />
(~ 3800)<br />
3 D2<br />
1 fm<br />
c¯c<br />
C<br />
28
e<br />
p<br />
γ *<br />
e<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Quarks move relativistically inside the pro<strong>to</strong>n<br />
q<br />
DIS measures the<br />
fraction x of the<br />
pro<strong>to</strong>n energy<br />
which is carried<br />
by the quarks,<br />
anti-quarks and<br />
gluons.<br />
For non-relativistic<br />
internal motion the<br />
x-distribution would<br />
be sharply peaked<br />
Non-relativistic uud state<br />
29
Paul <strong>Hoyer</strong> Mugla 2010<br />
Origin of the pro<strong>to</strong>n mass<br />
The u, d quarks in the pro<strong>to</strong>n have small masses<br />
2mu + md<br />
mp<br />
�<br />
10 MeV<br />
938 MeV<br />
� 1%<br />
99% of the pro<strong>to</strong>n mass<br />
is due <strong>to</strong> interactions!<br />
1% is due <strong>to</strong> Higgs.<br />
⇒ Ultra-relativistic state<br />
Compare this with positronium (e + e – ), the lightest QED a<strong>to</strong>m:<br />
2me<br />
mpos<br />
� 100.00067%<br />
Binding energy is tiny wrt mc 2<br />
⇒ Nonrelativistic state<br />
The compatibility of the non-relativistic⎥p〉 = ⎥uud〉 quark model description<br />
of the pro<strong>to</strong>n with its ultra-relativistic par<strong>to</strong>n model picture remains a mystery<br />
– but a mystery that we can address within <strong>QCD</strong>.<br />
Both are supported by data: = ??<br />
⎥p〉<br />
uud<br />
30<br />
gluon<br />
q¯q
Unlike a<strong>to</strong>ms, hadrons have<br />
no ionization threshold,<br />
where the quark constituents<br />
would be liberated.<br />
Hadron masses are<br />
generated by the<br />
potential and kinetic<br />
energies of the<br />
constituents<br />
Not yet explained<br />
by <strong>QCD</strong><br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Hadron mass spectrum is relativistic<br />
R e ! (m )<br />
2<br />
8.00<br />
7.00<br />
6.00<br />
5.00<br />
4.00<br />
3.00<br />
2.00<br />
1.00<br />
P.Desgrolard, M.Giffon, E.Martynov, E.Predazzi, hep-ph/0006244<br />
Spin<br />
" (770)<br />
# (782)<br />
Regge trajec<strong>to</strong>ry<br />
f 2(1270)<br />
a 2 (1318)<br />
" 3(1700)<br />
# (1670)<br />
3<br />
f 4 (2050)<br />
a (2020)<br />
4<br />
0.00<br />
0 .00 2.00 4.0 0 6.00 8.00 10.00<br />
m 2 (G eV )<br />
2<br />
(2350)<br />
" 5<br />
f (2510)<br />
6<br />
a (2 450)<br />
6<br />
For unknown<br />
reasons, hadron<br />
spins are<br />
proportional <strong>to</strong><br />
their mass 2<br />
31
√+%&'(<br />
!"#$$%$&'()#*%+,-.<br />
!" #<br />
!"<br />
!" $!<br />
π-<br />
π,<br />
!" $!<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
! "#$ %&'()*<br />
! !" !" #<br />
./0 0 1 2 3 4 5 6 7 .8 08 18 28<br />
⇓<br />
! !" !" #<br />
⇓<br />
elastic<br />
Total and Elastic Hadron Cross Sections<br />
0/0 1 2 3 4 5 6 7 .8 08 18 28 38 48<br />
π ∓ d <strong>to</strong>tal<br />
π − p elastic<br />
π − p <strong>to</strong>tal<br />
Not yet explained<br />
by <strong>QCD</strong><br />
! "#$ %&'()*<br />
Figure 40.13: Total and elastic cross sections for π ± p and π ± d (<strong>to</strong>tal only) collisions as a function of labora<strong>to</strong>ry beam momentum and <strong>to</strong>tal<br />
center-of-mass energy. Corresponding computer-readable data files may be found at http://pdg.lbl.gov/current/xsect/. (Courtesy of the<br />
COMPAS <strong>Group</strong>, IHEP, Protvino, August 2005)<br />
32
Paul <strong>Hoyer</strong> Mugla 2010<br />
Heavy Ion Collisions<br />
The quest for a new phase of matter at RHIC, LHC,...<br />
33
Paul <strong>Hoyer</strong> Mugla 2010<br />
34<br />
P. Skands
Paul <strong>Hoyer</strong> Mugla 2010<br />
Hadron masses from Lattice calculations<br />
35<br />
P. Skands
Paul <strong>Hoyer</strong> Mugla 2010<br />
The accuracy of QED perturbation theory<br />
Many of our most accurate predictions come from QED a<strong>to</strong>ms.<br />
For example, the 2S1/2 – 8S1/2 splitting in Hydrogen:<br />
Δ(2S1/2 – 8S1/2)H = 770 649 350 012.0(8.6) kHz EXP<br />
= 770 649 350 016.1(2.8) kHz QED<br />
The QED result is based on perturbation theory:<br />
– an expansion in α = e 2 /4π ≈ 1/137.035 999 11(46)<br />
U.D. Jentschura et al,<br />
PRL 95 (2005) 163003<br />
However, the series must diverge since for any α = e2/4π < 0 the electron<br />
charge e is imaginary: The Hamil<strong>to</strong>nian is not hermitian and probability not<br />
conserved. F. Dyson<br />
The perturbative expansion is believed <strong>to</strong> be divergent (asymp<strong>to</strong>tic).<br />
The good agreement of data with QED is fortuituous, from a<br />
theoretical point of view. For a discussion of the truncation effects in asymp<strong>to</strong>tic<br />
expansions see Y. Meurice, hep-th/0608097<br />
<strong>QCD</strong> perturbation theory has many features in common with that of QED.<br />
36
Paul <strong>Hoyer</strong> Mugla 2010<br />
Applications of perturbative <strong>QCD</strong> <strong>to</strong> data<br />
The <strong>QCD</strong> perturbative expansion successfully describes short distance<br />
processes, which involve high virtualities Q 2 and small αs(Q 2 ).<br />
Long distance dynamics is “universal”, i.e., independent of the hard process.<br />
Measuring the soft quark distribution and fragmentation functions in one<br />
process one can then predict measurable hadron cross sections.<br />
The applications of P<strong>QCD</strong> depends on Fac<strong>to</strong>rization Theorems, which hold <strong>to</strong><br />
all orders in αs at “Leading Twist”, for sufficiently inclusive processes.<br />
The “<strong>High</strong>er Twist” corrections are power-suppressed in the hard scale,<br />
∝ 1/Q 2 and can usually not be predicted.<br />
37
dσ<br />
dX<br />
Fac<strong>to</strong>rization Theorem<br />
Fac<strong>to</strong>rization: expresses the independence of long-wavelength (soft)<br />
Fac<strong>to</strong>rization emission on the nature of the hard (short-distance) process.<br />
� �<br />
�<br />
=<br />
a,b<br />
f<br />
ˆX f<br />
f!x,Q i "<br />
fa(xa, Q 2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q 2 i , Q2 f )<br />
d ˆ Xf<br />
!<br />
ˆ<br />
"<br />
! sum over long!wavelength his<strong>to</strong>ries<br />
leading <strong>to</strong> a with xa at the scale Qi 2<br />
�<br />
�<br />
!<br />
Paul <strong>Hoyer</strong> Mugla 2010 + (At H.O. each of these defined in a specific scheme, usually MS) P. Skands<br />
ˆ<br />
X<br />
D<br />
!<br />
p = x j<br />
!<br />
!<br />
P pro<strong>to</strong>n<br />
Par<strong>to</strong>n distribution<br />
fa(xa, Q<br />
functions (PDF)<br />
2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f )<br />
b, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f )<br />
rization<br />
a,b<br />
f<br />
ˆX f<br />
d ˆ Xf<br />
fa(xa, Q (ISR)<br />
2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f )<br />
d ˆ Xf<br />
d ˆ Xf<br />
D( ˆ Xf → X, Q 2 i , Q 2 f)<br />
Illustration by M. Mangano<br />
X<br />
D( ˆ Xf → X, Q 2 i , Q 2 f)<br />
38<br />
D( ˆ Fragmentation<br />
Xf → X<br />
Function (FF)<br />
! Sum over long!wavelength his<strong>to</strong>ries<br />
from at Qf 2 D( <strong>to</strong> X<br />
ˆ Xf → X, Q 2 i , Q 2 f) (FSR and Hadronization)
Paul <strong>Hoyer</strong> Mugla 2010<br />
e !<br />
# q<br />
e " q<br />
" #<br />
" #<br />
$ ( ee & hadrons) $ ( ee & qq)<br />
q<br />
R '<br />
! !<br />
NC<br />
Q<br />
" # " # " # " # -<br />
$ ( ee & % % ) $ ( e e & % % )<br />
,<br />
(<br />
)<br />
)<br />
*<br />
)<br />
)<br />
)<br />
+<br />
-<br />
, ( u, d, s)<br />
, ( u, d, s, c)<br />
, ( u, d, s, c,<br />
b)<br />
The Standard Model A. Pich - CERN Summer Lectures 2008<br />
2<br />
3<br />
10<br />
9<br />
11<br />
9<br />
N<br />
N<br />
N<br />
C<br />
C<br />
C<br />
q<br />
2<br />
q<br />
40
ξ =<br />
1 +<br />
= log E + p �<br />
� m 2 + p 2 ⊥<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Q + M X<br />
2x<br />
�<br />
1 + 4m2 px2 /Q2 c¯c b¯b s¯s<br />
� − log tan(θ/2)<br />
R = σ(e+ e − → hadrons)<br />
σ(e + e − → µ + µ − )<br />
=3 �<br />
q<br />
e 2 q<br />
�<br />
1+ αs<br />
π<br />
�<br />
41
Paul <strong>Hoyer</strong> Mugla 2010<br />
LEP determination of neutrino number<br />
e + e − → hadrons<br />
at Z peak<br />
42
:02 PM<br />
e +<br />
e –<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Traces of Quarks and Gluons in the final state?!<br />
e + e – → 2 jets e + e – → 3 jets<br />
Z<br />
q<br />
_<br />
q<br />
h’s<br />
h’s<br />
Q 2 = m 2 Z<br />
e +<br />
e –<br />
Z<br />
q<br />
_<br />
q<br />
g<br />
h’s<br />
h’s<br />
43<br />
h’s
No exclusive amplitudes for charged particles (I)<br />
The identification of the data on e + e – → 2 jets with the <strong>QCD</strong> process<br />
e + e − → q¯q<br />
Gauge invariance dictates that amplitudes<br />
with external charged particles vanish:<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
requires care, since the latter does not exist!<br />
This is a general feature of gauge theories, and as such is best illustrated by<br />
the QED process e + e − → µ + µ −<br />
A(e + e − → µ + µ − ) = 0<br />
This is because the amplitude must be invariant under local U(1) gauge<br />
transformations. Multiplying one of the external fermions by<br />
U = e iπ = – 1 we get A → – A.<br />
44
No exclusive amplitudes for charged particles (II)<br />
In the perturbation expansion the Born<br />
term is well-defined and ≠ 0:<br />
This problem shows up at order α 2 as an<br />
infrared singularity in the loop integral for k → 0:<br />
This may be seen without calculation:<br />
The two fermion propaga<strong>to</strong>rs ∝ k, e.g.:<br />
(p1 − k) 2 − m 2 µ = −2p1 · k + k 2 ∝ k<br />
The pho<strong>to</strong>n propaga<strong>to</strong>r ∝ k 2 , giving a log singularity at k = 0<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
e +<br />
e –<br />
e +<br />
e –<br />
γ*<br />
γ*<br />
q<br />
�<br />
0<br />
p1<br />
p2<br />
d 4 k<br />
k 4<br />
⇒ The exclusive process e + e – → µ + µ – is ill defined.<br />
µ +<br />
µ –<br />
µ +<br />
k<br />
µ –<br />
45
No exclusive amplitudes for charged particles (III)<br />
Two charged particles at different positions x, y must be connected by a<br />
pho<strong>to</strong>n string <strong>to</strong> be gauge invariant:<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
¯ψ(y)exp<br />
�<br />
ie<br />
� y<br />
x<br />
dzνA ν �<br />
(z) ψ(x)<br />
The pho<strong>to</strong>n (gauge) field serves as a connection, which “informs” about the<br />
choice of gauge at each point in space.<br />
In perturbation theory, the missing string causes an infrared singularity at the<br />
one pho<strong>to</strong>n correction level.<br />
But we previously saw that there is no problem with the <strong>to</strong>tal e + e – cross<br />
section, which includes the µ + µ – final state?!<br />
To see how the IR singularity cancels in the <strong>to</strong>tal cross section we may use<br />
the optical theorem.<br />
46
Unitarity (white):<br />
σ<strong>to</strong>t(s) = �<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Some Theorems<br />
X<br />
�<br />
Before <strong>QCD</strong><br />
E.g., Lorentz inv., unitarity and the optical theorem<br />
SS † = 1<br />
“something will happen”<br />
note: includes “no” scattering<br />
The Lagrangian of <strong>QCD</strong><br />
Optical Theorem<br />
As a consequence of the unitarity of the scattering matrix:<br />
the <strong>to</strong>tal cross section may be expressed in terms of the<br />
Unitarity (white)<br />
imaginary part of the forward elastic amplitude:<br />
SS † =1<br />
dΦX |MX| 2 = 8π<br />
Optical Theorem (white) √ Im [Mel(θ = 0)]<br />
s<br />
σ<strong>to</strong>t(s) =<br />
µ<br />
)( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1<br />
4 F a µνF aµν<br />
The Lagrangian of <strong>QCD</strong> in white<br />
µ<br />
)( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1<br />
X ~ XX<br />
2<br />
X<br />
10<br />
4 F a µνF aµν<br />
SS † = 1<br />
Total = The Lagrangian that can of happen <strong>QCD</strong><br />
dΦX |M X | 2 = 8π<br />
√ Im [M<br />
s<br />
el(θ = 0) ]<br />
Sum over everything<br />
“Square Root” of<br />
nothing happening<br />
L = ¯ ψ i q(iγ µ )( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1<br />
4 F a µνF aµν<br />
The Lagrangian of <strong>QCD</strong> in white<br />
L = ¯ ψ i q(iγ µ )( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1<br />
4 F a µνF aµν<br />
The sum over all states X becomes a completeness sum on the rhs.<br />
QED satisfies unitarity at each order of α.<br />
=<br />
47<br />
P. Skands
µ +<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
µ +<br />
γ* γ*<br />
µ –<br />
Optical Theorem for σ<strong>to</strong>t(e in QED (I)<br />
+ e − )<br />
O (α) O � α 2�<br />
µ –<br />
γ ∗ → µ + µ −<br />
µ +<br />
µ –<br />
γ<br />
µ +<br />
µ –<br />
γ ∗ → µ + µ − γ<br />
µ +<br />
µ –<br />
µ +<br />
µ –<br />
γ ∗ → µ + µ −<br />
At O(α 2 ) there are two contributions <strong>to</strong> the imaginary part (dashed line).<br />
The IR singularity cancels between them, i.e., between different final states!<br />
Since the γ*→ γ* amplitude does not have external charges, it “has <strong>to</strong> be” regular.<br />
It means that even in QED we must define cross sections such that they include<br />
(arbitrarily soft) pho<strong>to</strong>ns. There are no free, “bare electrons”.<br />
48
Paul <strong>Hoyer</strong> Mugla 2010<br />
Optical Theorem for σ<strong>to</strong>t(e in QED (II)<br />
+ e − )<br />
γ* γ*<br />
k k<br />
Since the IR singularity is only at k = 0, it suffices <strong>to</strong> include only pho<strong>to</strong>ns with<br />
k < k0, for arbitrarily small k0.<br />
The criterion is <strong>to</strong> sum over all states that are degenerate in energy,<br />
since these have an asymp<strong>to</strong>tically long formation time Δt ∼ 1/ΔE<br />
In QED, collinear IR singularities are regulated by the electron mass:<br />
49<br />
Kinoshita-<br />
Lee-Nauenberg<br />
(KLN) theorem<br />
p<br />
Ee =<br />
xp<br />
(1–x)p<br />
+<br />
� p2 + m2 e Ee+γ = � (xp) 2 + m2 e + (1 − x)|p|<br />
For |p| >> me the<br />
collinear e + γ state has<br />
nearly the same energy<br />
as the single electron e.<br />
⇒ In QED, collinear bremsstrahlung is enhanced by a fac<strong>to</strong>r log(|p|/me).
Paul <strong>Hoyer</strong> Mugla 2010<br />
Removing IR sensitivity in <strong>QCD</strong><br />
In <strong>QCD</strong> even soft gluons can change the color of the quark<br />
⇒ we can never measure the color of a quark!<br />
Also: Want <strong>to</strong> sum over all soft and collinear divergences up <strong>to</strong> a<br />
“Fac<strong>to</strong>rization scale” µ, large enough <strong>to</strong> apply P<strong>QCD</strong> for Q > µ.<br />
Define IR safe cross sections, which can be calculated in P<strong>QCD</strong><br />
using the fac<strong>to</strong>rization theorem, and measured in experiments.<br />
50
e�nition<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Infrared Safe observables<br />
An obser vable is infrared safe if it is insensitive <strong>to</strong><br />
SOFT radiation:<br />
Adding any number of infinitely soft particles should not<br />
change the value of the obser vable<br />
COLLINEAR radiation:<br />
Splitting an existing particle up in<strong>to</strong> two comoving particles<br />
each with half the original momentum should not change<br />
the value of the obser vable<br />
51<br />
P. Skands
LO:<br />
σ NLO<br />
�<br />
1 (R) =<br />
� LO, NLO, etc<br />
|M (0)<br />
2Re[M (1)<br />
52<br />
σ Z → 2 1-loop:<br />
qk<br />
LO<br />
(R) = |M X+1 (0)<br />
X+1 |2<br />
Cross sections<br />
�<br />
at NLO<br />
σBorn = |M<br />
Z → 3 jets:<br />
(0)<br />
X |2<br />
Z → 2 1-loop:<br />
qk<br />
g<br />
qi �<br />
ik<br />
σ<br />
qk<br />
qi<br />
a<br />
qk<br />
LO<br />
X+1 (R) = |M<br />
R<br />
(0)<br />
X+1 |2<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
Z decay:<br />
q q<br />
q q<br />
�<br />
|M|<br />
colours<br />
2 Z → 3 jets:<br />
qk<br />
qi<br />
g<br />
qi<br />
=<br />
jk<br />
qk<br />
qi<br />
a g<br />
qi<br />
ik<br />
Z → 2 1-loop:<br />
qk<br />
a<br />
g<br />
qi �<br />
ik<br />
|M<br />
R<br />
qk<br />
qi<br />
a<br />
(0)<br />
X+1 |2<br />
Z → 3 jets:<br />
qk �<br />
qi<br />
g<br />
qi<br />
jk<br />
qk<br />
qi<br />
a g<br />
qi<br />
ik<br />
Removing IR sensitivity in <strong>QCD</strong> at NLO<br />
a<br />
g<br />
qi<br />
qk<br />
orem (Kinoshita-Lee-Nauenberg)<br />
ities cancel at complete order (only finite te<br />
ik<br />
�<br />
qk<br />
qi<br />
a<br />
σBorn + |M qk<br />
(note: Not the 1-loop diagram sq<br />
(0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
M X X ]<br />
|M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
M X X ]<br />
��<br />
rn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M<br />
σ NLO<br />
O, etc<br />
�<br />
σBorn = |M<br />
= X σBorn(1 + K)<br />
(0)<br />
X |2<br />
σ LO<br />
�<br />
(R) = |M X+1<br />
R<br />
(0)<br />
X+1 |2<br />
�<br />
σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
M X X ]<br />
Z → 2 1-loop:<br />
LO:<br />
g<br />
NL Theorem (Kinoshita-Lee-Nauenberg)<br />
Singularities cancel at complete order (only finite terms left<br />
Lemma: only � after some hard �work<br />
ik<br />
σ<br />
(note: Not the 1-loop diagram squared)<br />
LO<br />
X+1 (R) = |M<br />
R<br />
(0)<br />
X+1 |2<br />
σ NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
NLO<br />
X = σBorn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0<br />
X M X<br />
σ NLO<br />
LO, NLO, etc<br />
�<br />
σBorn = |M<br />
X = σBorn(1 + K)<br />
� �<br />
� �<br />
(0)<br />
X |2<br />
σ LO<br />
�<br />
X+1 (R) = |M<br />
R<br />
(0)<br />
X+1 |2<br />
σ NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
(note: Not the 1-loop diagram squared)<br />
NL Theorem (Kinoshita-Lee-Nauenberg)<br />
Singularities cancel at complete order (only finite terms left over)<br />
Lemma: only after some hard work<br />
�� � ��<br />
NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
NLO<br />
X = σBorn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
σ NLO<br />
X = σBorn(1 + K)<br />
NNLO<br />
X = σ NLO<br />
X +<br />
� �<br />
|M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
� �<br />
+ 2Re[M (1)<br />
σBorn = |M<br />
(0)<br />
X+1M X+<br />
(0)<br />
X |2<br />
σ LO<br />
�<br />
X+1 (R) = |M<br />
R<br />
(0)<br />
X+1 |2<br />
σ NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
��<br />
X = σBorn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
σ NLO<br />
X = σBorn(1 + K)<br />
!1(e + e − → q ¯q(g)) = !0(e + e − �<br />
→ q ¯q) 1 + "s(ECM)<br />
+ O("<br />
#<br />
2 X<br />
�<br />
+ |M<br />
(note: Not the 1-loop diagram squared)<br />
(Kinoshita-Lee-Nauenberg)<br />
ncel at complete order (only finite terms left over)<br />
after some hard work<br />
�<br />
s)<br />
29<br />
(0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
te |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
LO<br />
= σBorn(1 + K)<br />
�<br />
|M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
� �<br />
+ 2Re[M (1) (0)∗<br />
X+1M X+1 ]+<br />
�<br />
|M (0)<br />
X+2 |2<br />
σBorn = |M (0)<br />
X |2<br />
�<br />
LO<br />
X+1 (R) = |M<br />
R<br />
(0)<br />
X+1 |2<br />
�<br />
|M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
| 2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
nite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
NLO<br />
X = σBorn(1 + K)<br />
)) = !0(e<br />
�<br />
� �<br />
�<br />
+ e − �<br />
→ q ¯q) 1 + "s(ECM)<br />
+ O("<br />
#<br />
2 ...<br />
�<br />
s)<br />
Z decay:<br />
q<br />
X+1 |2 +<br />
q q<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
q<br />
�<br />
|M| 2 =<br />
colours<br />
�<br />
∝ δijδ ∗ ji<br />
= Tr[δij]<br />
= NC<br />
Z decay:<br />
NNLO = σ NLO +<br />
8<br />
q<br />
∝ δijδ ∗ ji<br />
= Tr[δij]<br />
= NC<br />
q<br />
q q<br />
�<br />
|M| 2 =<br />
colours<br />
∝ δijδ ∗ ji<br />
= Tr[δij]<br />
= NC<br />
qi<br />
qi<br />
qk qk<br />
g jk<br />
a<br />
�<br />
(1) 2 (2) (0)∗<br />
M (0)∗<br />
R<br />
X qk<br />
qk<br />
]<br />
g ik<br />
Z → 3 jets: a<br />
qi<br />
qi<br />
qk<br />
g jk<br />
8<br />
a<br />
qi<br />
16<br />
8<br />
qi<br />
qk<br />
g ik<br />
a<br />
8<br />
qi<br />
qi<br />
� � � �<br />
|M (1) | 2 + 2Re[M (2) M (0)∗ (1) (0)∗ ]<br />
29<br />
qk<br />
qi<br />
qi<br />
qk<br />
qk<br />
a<br />
16<br />
+<br />
qk<br />
qk<br />
16<br />
2Re[M (1)<br />
P. Skands<br />
(0) 2
Cross<br />
σσ Sections at NNLO<br />
X+1 (R)<br />
Sections<br />
= |MX+1 |<br />
at NNLO<br />
X+1 (R) = |MX+1 |<br />
R<br />
NLO<br />
X = σBorn + |M<br />
NLO<br />
(0)<br />
X+1 |2 + 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
O<br />
= |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
O<br />
= σBorn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
�<br />
qk qk<br />
qj qj<br />
NNLO<br />
qi<br />
qi<br />
qi<br />
qk<br />
NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
NNLO �� � ��<br />
�<br />
qk qk<br />
qj qj<br />
LO<br />
X+1 (R) = X+1<br />
R<br />
|2<br />
σ NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
NNLO �� � ��<br />
�<br />
qk qk<br />
qj qj<br />
NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
Removing IR sensitivity in <strong>QCD</strong> at NNLO<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
�� � ��<br />
�<br />
qk qk<br />
qj qj<br />
σ NLO<br />
σ = σBorn+Finite<br />
NLO σ = σBorn+Finite<br />
NLO<br />
= σBorn+Finite<br />
X X<br />
LO = σ NLO<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
g ik<br />
σ NLOg<br />
qi<br />
ik<br />
a<br />
σX NLO qi qi<br />
qi<br />
X<br />
qk qk qk<br />
X +<br />
� �<br />
X +<br />
� �<br />
X + X +<br />
σ NNLO<br />
= σ NLO<br />
σ NNLO<br />
= σ NLO<br />
σ NNLO<br />
= σ NLO<br />
X X<br />
Cross Sections at NNLO<br />
qi qi<br />
qi<br />
a<br />
gik a<br />
gik |M (0)<br />
X+1 |2 |M qi<br />
(0)<br />
X+1 |2 |M qi<br />
(0)<br />
X+1 |2 qi<br />
a<br />
g<br />
= σBorn(1 qi + K)<br />
ik<br />
ga ik g<br />
a<br />
ik<br />
a<br />
= σBorn(1 qi<br />
σ + K)<br />
NLO<br />
a<br />
σ = σBorn(1 qi + K)<br />
NLO<br />
= σBorn(1 qi + K)<br />
qk<br />
�� ��<br />
g ik<br />
qk qk qk<br />
qk<br />
|M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
|M<br />
X M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
|M +<br />
(1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
|M +<br />
(1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ] +<br />
Z → 2 2-loop:<br />
15<br />
15<br />
15<br />
Z → 2 2-loop:<br />
qk<br />
qk<br />
qj<br />
qj qk<br />
qi<br />
qi qj<br />
gij g jk gij a g jk<br />
g b<br />
ij<br />
a g jk<br />
+Finite<br />
g<br />
qi<br />
qj<br />
ij<br />
a g<br />
qk<br />
jk<br />
qi a b b<br />
qi<br />
qi<br />
qi<br />
qj<br />
qj<br />
qj<br />
qk<br />
qk<br />
qk<br />
b<br />
qk<br />
qk<br />
qk<br />
qk<br />
Two-Loop ! Born Interference<br />
Two-Loop<br />
Two-Loop<br />
!<br />
!<br />
Born<br />
Born<br />
Interference<br />
Interference<br />
18<br />
18<br />
18<br />
|M (0)<br />
gjk g ij<br />
gjk 2Re[M (1) (0)∗<br />
2Re[M M (1) (0)∗<br />
2Re[M M (1) (0)∗<br />
M<br />
qi g<br />
qk<br />
ik<br />
a<br />
b X<br />
c<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jkX<br />
a<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jk<br />
X<br />
a<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jk<br />
a<br />
qk<br />
qk<br />
X ]<br />
��<br />
� ��<br />
��<br />
�<br />
+<br />
qk<br />
Two-Loop ! Born 30<br />
30<br />
30 Interference<br />
X ] X ]<br />
X a ]<br />
2Re[M (1)<br />
2Re[M (1)<br />
2Re[M (1)<br />
18<br />
18<br />
18<br />
and so on, at each fixed order in αs<br />
qk<br />
2Re[M (1) (0)∗<br />
(0)∗<br />
X+1 M M<br />
X+1<br />
18<br />
X+1 ]+<br />
�<br />
X+1 ]+<br />
�<br />
X+1 ]+<br />
�<br />
(0)∗<br />
X+1M Z Z Z→ → →4: 4:<br />
qi<br />
qi<br />
qi<br />
qi qi qi<br />
qj qj<br />
qj<br />
gik g ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
qk qk qk<br />
X+1 ]+<br />
�<br />
|M (0)<br />
X+2 |2 X+2 |2 |M (0)<br />
X+2 |2<br />
|M (0)<br />
a<br />
qj Z → 4:<br />
qi<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
qj<br />
qj<br />
qk qk<br />
qi<br />
qiqj<br />
qi<br />
Real Real ! ! Real Real (X+2) (X+2) qk<br />
Real ! Real (X+2)<br />
g ij<br />
qi qi<br />
b<br />
g ik<br />
a<br />
53<br />
|M (<br />
X<br />
Real ! Real<br />
P. Skands
Paul <strong>Hoyer</strong> Mugla 2010<br />
Event genera<strong>to</strong>rs<br />
At high energies many gluons are radiated. Then one needs <strong>to</strong> include<br />
high orders in αs for which complete, IR regulated matrix elements are not<br />
available.<br />
⇒ Shower Monte Carlo methods:<br />
Include only the log enhanced terms of tree digrams (no loops)<br />
Use phenomenological hadronization model for Q < µ.<br />
This gives approximate results for hadron<br />
distributions in the final state.<br />
The reliability is tested by including<br />
next-<strong>to</strong>-leading logarithms and comparing<br />
several hadronization models.<br />
Cluster Fragmentation<br />
55
Paul <strong>Hoyer</strong> Mugla 2010<br />
<strong>QCD</strong> is very successful in comparisons with data<br />
Rate of 3-jet events<br />
in e + e – annihilations<br />
e + e – → q q g<br />
→ 3 jets<br />
Ex: Estimate the CM<br />
energy in e+ e– annihilations<br />
at which 2-jet structure<br />
emerges. In quark<br />
fragmentation, pions get an<br />
average fraction ≈ 0.1<br />
of the quark energy, and<br />
≈ 350 MeV.<br />
S. Bethke, hep-ex/0606035<br />
56
Angular distribution<br />
of 4-jet events in<br />
e + e – annihilations<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
S. Bethke, hep-ex/0606035<br />
57
Measurement of quark<br />
and gluon color charges<br />
in e + e – annihilations<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
quark<br />
color<br />
charge<br />
e + e –<br />
S. Bethke, hep-ex/0606035<br />
gluon charge<br />
58
(Nonperturbative)<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
Jet production in hadron collisions<br />
(Inclusive sum)<br />
(log. scaling<br />
violations)<br />
(Perturbative)<br />
(Nonperturbative)<br />
(log. scaling<br />
violations)<br />
arXiv:1002.1708<br />
(or: h + X)<br />
(<strong>High</strong>er<br />
order in αs)<br />
59
Paul <strong>Hoyer</strong> Mugla 2010<br />
The CDF Detec<strong>to</strong>r<br />
60
Fermilab:<br />
〉 pp<br />
– �= → 0 jet and + X 〈0|F a µνF µν<br />
ECM = 1.96 TeV<br />
Quarks and gluons<br />
are pointlike down<br />
q<br />
<strong>to</strong> the best resolution<br />
that has been reached<br />
Ex: Estimate the maximum<br />
radius of quarks and gluons,<br />
given the agreement of<br />
<strong>QCD</strong> with the Fermilab jet<br />
2x<br />
data.<br />
ξ =<br />
Rapidity:<br />
F2(x) = �<br />
x =<br />
1 +<br />
y = log E + p �<br />
� m 2 + p 2 ⊥<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
e 2 q xfq(x)<br />
Q 2<br />
Q 2 + M 2 X<br />
�<br />
1 + 4m 2 px 2 /Q 2<br />
� − log tan(θ/2)<br />
ECM = 1960 GeV<br />
a |0〉 �= 0<br />
CDF Collab., hep-ex/0701051<br />
61
Paul <strong>Hoyer</strong> Mugla 2010<br />
pp<br />
–<br />
→ jet + X: Jet mass distribution<br />
62
Paul <strong>Hoyer</strong> Mugla 2010<br />
Concluding remarks<br />
Perturbative <strong>QCD</strong> has been successfully applied <strong>to</strong> hard collision data.<br />
!<br />
"<br />
! $% & ' ()**+, - ()((*(<br />
#<br />
<strong>QCD</strong> effects constitutes the major background in the search for new physics<br />
at the Tevatron and the LHC. Hence much effort is expended on making the<br />
calculations as accurate as possible.<br />
e Standard Model A. Pich - CER<br />
<strong>High</strong> intensity electron beams at Jefferson Lab, Mainz,... are mapping out<br />
hadron structure through Form fac<strong>to</strong>rs and Generalized Par<strong>to</strong>n Distributions.<br />
Lattice <strong>QCD</strong> methods allow fast progress in the calculation of nonperturbative<br />
quantities, such as the hadron spectrum.<br />
The simple Quark Model systematics of the hadron spectrum remains an<br />
encouraging mystery.<br />
63