Dr M.A. Thomson e+ q √α γ S g √α √α Qq 1 e- q2 q
Part II, Lent Term 2004 HANDOUT III
Dr M.A. Thomson Lent 2004 2 QCD
QUANTUM ELECTRODYNAMICS: is the quantum theory of the electromagnetic interaction. ★ mediated by massless photons
★ photon couples to electric charge,
Ô
★ À » «
Strength of interaction : .
¾
«
QUANTUM CHROMO-DYNAMICS: is the quantum
theory of the strong interaction.
¾ Õ ★ mediated by massless gluons, i.e. ½ propagator ★ gluon couples to “strong” charge ★ Only quarks have non-zero “strong” charge, therefore only quarks feel strong interaction
Basic QCD interaction looks like a stronger
« « Å version of QED, Ë QED q QCD q √α Q √α S q q q
α 2 π α 2 π
= e /4 ~ 1/137 γ S = gS /4 ~ 1 g «
( subscript em is sometimes used to distinguish the Ñ «
of electromagnetism from Ë ).
Dr M.A. Thomson Lent 2004 3 COLOUR
In QED: ★ Charge of QED is electric charge. ★ Electric charge - conserved quantum number. In QCD: ★ Charge of QCD is called “COLOUR” ★ COLOUR is a conserved quantum number with 3 VALUES labelled “red”, “green” and
“blue”
Quarks carry “COLOUR” Ö
Anti-quarks carry “ANTI-COLOUR” Ö
¦ ¼
Ï Leptons, , , DO NOT carry colour, i.e.
“have colour charge zero” DO NOT participate in STRONG interaction. Note: Colour is just a label for states in a
non-examinable SU(3) representation
¼ ¼ ½
¼ ½ ¼
Ö
½ ¼ ¼
Dr M.A. Thomson Lent 2004 4 GLUONS
In QCD: ★ Gluons are MASSLESS spin-1 bosons
Consider a red quark scattering off a green quark. Colour is exchanged but always conserved.
q q r g gluon r g q q g r
UNLIKE QED: ★ Gluons carry the charge of the interaction. ★ Gluons come in different colours.
Expect 9 gluons (3 colours ¢ 3 anti-colours)
Ö Ö Ö
Ö , , , , ,
Ö Ö , , However: Real gluons are orthogonal linear
combinations of the above states. The
½
Ô
´Ö · · µ
combination Ö is colourless ¿ and does not take part in the strong interaction.
Dr M.A. Thomson Lent 2004 5
Colour at Work Õ EXAMPLE: Õ Annihilation q q rr g
q q g g
Normally do not show colour on Feynman diagrams - colour is conserved.
QED POTENTIAL:
«
V É Ö
QCD POTENTIAL:
At short distances QCD potential looks similar
«
Ë
V É
¿ Ö
apart from colour factor. ¿
Note: the colour factor (4/3) arises because more than one
Õ gluon can participate in the process Õ . Obtain colour factor from averaging over initial colour states and summing over final/intermediate colour states
Dr M.A. Thomson Lent 2004 6 SELF-INTERACTIONS
At this point, QCD looks like a stronger version of QED. This is true up to a point. However, in practice QCD behaves very differently to QED. The similarities arise from the fact that both involve the exchange of MASSLESS spin-1 bosons. The big difference is that GLUONS carry colour “charge”. GLUONS CAN INTERACT WITH OTHER GLUONS: g g g g g g g
3 GLUON VERTEX 4 GLUON VERTEX
EXAMPLE: Gluon-Gluon Scattering
+ +
e.g.rrrrgg + bb→ +
Dr M.A. Thomson Lent 2004 7 CONFINEMENT
NEVER OBSERVE: single FREE quarks/gluons ★ quarks are always confined within hadrons ★ This is a consequence of the strong self-interactions of gluons. Qualitatively, picture the colour field between two quarks. The gluons mediating the force act as additional sources of the colour field - they attract each other. The gluon-gluon interaction pulls the lines of colour force into a narrow tube or STRING. In this model the string has a ‘tension’ and as the quarks separate the string stores potential energy.
Energy stored per unit length constant.
Î ´Ö µ » Ö ★ Requires infinite energy to separate two quarks. Quarks always come in combinations with zero net colour charge: CONFINEMENT.
Dr M.A. Thomson Lent 2004 8 How Strong is Strong ?
QCD Potential between quarks has two
components:
« Ë
★ “COULOMB”-LIKE TERM :
¿ Ö
Ö ★ LINEAR TERM : ·
α 1 -4 s V = 3r +kr (GeV) 0 -4α V = s
QCD 3r -1 V α =0.2 -2 s k=1 GeV/fm -3 0 0.2 0.4 0.6 0.8 1 r(fm)
Force between two quarks at separated by 10 m:
«
Ë
Î · Ö
É
Ö
½ Î
with fm
« Î
Ë
·
¾
Ö Ö
½¼
½ ¢ ½¼
Æ
at large r
½
½¼
½ ¼¼¼¼ Æ
Equivalent to the weight of approximately 65 Widdecombes.
Dr M.A. Thomson Lent 2004 9
JETS
·
Õ ÕÕ Consider the Õ pair produced in : e+ q γ qq e- q
1 -4α V = s+kr As the quarks separate, the 3r (GeV) 0 energy stored in the colour -4α V = s
QCD 3r field (‘string’) starts to in- -1 crease linearly with separa- V
α =0.2 tion. When ×ØÓÖ -2 s
k=1 GeV/fm
¾Ñ ÕÕ
Õ new pairs can be created. -3 0 0.2 0.4 0.6 0.8 1 r(fm)
q q
q q
q q q q
as energy decreases... hadrons freeze out
Dr M.A. Thomson Lent 2004
10 Õ As quarks separate, more Õ pairs are produced from the potential energy of the colour field. This process is called HADRONIZATION. Start out with quarks and end up with narrowly collimated JETS of HADRONS
q π+ q (ud) q q etc... q q SPACE q q π+ q q 0 - -π +π q π K π+ q π0π0 π0 TIME p
e+ q q γ e- e+ -
e q q
·
ÕÕ Typical Event
The hadrons in a quark(anti-quark) jet follow the direction of the original quark(anti-
quark). Consequently
·
ÕÕ is observed as a pair of back-to-back jets of hadrons
Dr M.A. Thomson Lent 2004 11 aside.....
DALI Run=56698 Evt=7455 ALEPH 3 Gev EC 6 Gev HC
Y" 1cm 0 −1cm
−1cm 0 1cm X" Filename: DC056698_007455_000830_1723.PS Made on 30-Aug-2000 17:24:02 by konstant with DALI_F1. Z0<5 D0<2 RO TPC (φ −175)*SIN(θ) x xo o o−xo−oxx−− o x o x
o o − − o x − −− o xo−x x xoo− − x x o − o x o x x o − − x o − − x − o xo − o − − oo− o − x o o x − x − x o x ox − x −−o − x
15 GeV θ=180 θ=0
★ You will now recognize the “Higgs” event from
the cover of Handout I as
·
ÕÕÕÕ something
Dr M.A. Thomson Lent 2004
12 «
Running of Ë ★ «
Ë specifies the strength of the strong
interaction ★ «
BUT just as in QED, Ë isn’t a constant, it “runs” ★ In QED the bare electron charge is screened by a cloud of virtual electron-positron pairs. ★ In QCD a similar effect occurs.
In QCD quantum fluctuations lead to a ‘cloud’ of Õ virtual Õ pairs q
one of many (an infinite set) q q q such diagrams analogous to q those for QED. q q
In QCD the gluon self-interactions ALSO lead to a ‘cloud’ of virtual gluons g one of many (an infinite set) such diagrams. Here there is no g g q analogy in QED, photons don’t have self-interactions since they g q don’t carry the charge of the in- q teraction.
Dr M.A. Thomson Lent 2004 13 Colour Anti-Screening ★ Due to the gluon self-interactions bare colour charge is screened by both virtual quarks and virtual gluons ★ The cloud of virtual gluons carries colour charge and
the effective colour charge INCREASES with distance !
★ «
At low energies (large distances) Ë becomes large can’t use perturbation theory (not a weak perturbation) α α EM QED S QCD
1/128 M 1/137 Z Mp
High EnergyMZ Low Energy High Energy Low Energy
s 1 s 1 α α
MZ
Mp 0 0 0246-20 -18 -16 2 2
log10(q /GeV ) log10(r/m) ★ «
At High energies (short distances) Ë is small. In this regime treat quarks as free particles and can use perturbation theory
ASYMPTOTIC FREEDOM
Ô
★ × ½¼¼ Î « ¼ ½¾
At , Ë .
Dr M.A. Thomson Lent 2004 14 Scattering in QCD
EXAMPLE: High energy proton-antiproton scattering
Jet along u √α u beam direction p S
g g 1 q2
p √α Visible jet u S u
in direction of q
¾
½ ´« µ
Ô Ô
Ë
Å « « µ
Ë Ë
¾
Õ ª ×Ò ¾
The upper points are the Geiger and Marsden data (1911) for the elastic scatter-
ing of « particles as they traverse thin gold and silver foils. The lower points show the angular distribution of the quark jets observed in
proton-antiproton scattering
¾ ¾
¾¼¼¼Î at Õ Both follow the Rutherford
formula for elastic scatter-
ing: ×Ò . ¾
Dr M.A. Thomson Lent 2004
15
·
Ô EXAMPLE: ÔÔ vs scattering
p π+
g g
p p
·
´ÔÔµ ´ Ôµ Calculate ratio of total to total ★ QCD does not distinguish between quark flavours, only COLOUR charge of quarks matters.
At high energy (E bind- ing energy of quarks within
hadrons) ratio of ÔÔ and
Ô total cross sections de- pends on number of pos- sible quark-quark combina- tions.
Predict
´Ôµ
¾¢¿ ¾
´ÔÔ µ ¿¢¿ ¿
Experiment
´Ôµ
¾ Ñ
´ÔÔµ ¿ Ñ
¼ ¿
Dr M.A. Thomson Lent 2004
16
· QCD in Annihilation
Direct evidence for the existence of colour comes
·
from Annihilation.
· · ·
ÕÕ
★ Compare , :
·
´ ÖÓÒ× µ
Ê
· ·
´ µ
e+ µ+ e+ q γ γ √α √α √α √α Qq 1 1 e- q2 µ- e- q2 q
If we neglect the masses of the final state
quarks/muons then the ONLY difference is the
É ½ É Õ
charge of the final state particles ( ,
¾ ½
= · or )
¿ ¿
Start by calculating the cross section for the
·
´ µ
process .( represent a
·
ÕÕ
fermion-antifermion pair e.g. or ).
·
see Handout II for the case where
Dr M.A. Thomson Lent 2004 17 + f e µ f p2 γ θ √α √α - + Qf e e µ 1 p - 1 q2 e f f
Electron/Positron beams along Þ -axis
Ô ´ Ô Ô Ô µ
Ü Ý Þ
½
Ô ´ ¼ ¼ µ Ò Ð ØÒ Ñ
½
´ ¼ ¼ µ Ô
¾
Õ Ô · Ô
½ ¾
´¾ ¼ ¼ ¼µ
¾ ¾
Õ ×
¾ µ where × is (centre-of-mass energy .
Fermi’s Golden rule and Born Approximation
´ µ
¾
¾ Å
ª ª
Matrix element Å :
½
É Ù Å Ú Ú É Ù
·
¾
Õ
¾
«É É
ÛØ «
¾
Õ
¾ ¾
´ «É É µ ½
¾
¾ ´½ · Ó× µ
¾
ª Õ ´¾ µ
¾ ¾
« É
¾
µ ´½ · Ó×
×
Dr M.A. Thomson Lent 2004
18
¾
· Ó× µ ★ ´½ comes from spin-1 photon “decaying” to
two spin-half fermions. see lecture on Dirac equation
·
Total cross section for
ª
ª
¾ ¾
¾
« É
¾
´½ · Ó× µ×Ò
×
¼ ¼
¾ ¾
·½
« É
¾
´½ · Ý µÝ ´Ý Ó× µ
¾×
½
¾ ¾
« É
¿×
¾
«
· ·
´ µ
¿×
· ·
´ µ
· for collider data at centre-of-mass ener- gies 8-36 GeV
Dr M.A. Thomson Lent 2004 19
Back to
·
´ ÖÓÒ×µ
Ê
· ·
´ µ
For a single quark flavour of a given colour
¾
Ê É
Õ
· ·
! Ø× ÙÙ However, we measure not .A
jet from a u-quark looks just like a jet from a d-quark... Need
to sum over flavours (u,d,c,s,t,b) and colours (Ö ).
¾
Ê ¿ É ´¿ÓÐÓÙÖ× µ
where the sum is over all quark flavours kinematically Ô
accessible at centre-of-mass energy, × , of collider.
Energy Ratio R
Ô
½ ½
× ¾Ñ ¿´ · · µ ¾
× 1 GeV =
u,d,s
Ô
½ ½ ½
× ¾Ñ ¿´ ¿ · · · µ
4 GeV =
¿
u,d,s,c
Ô
½ ¾
× ¾Ñ ¿´ · ¿ µ
10 GeV =
¿
u,d,s,c,b
Ô
µ × ¾Ñ ¿´ · #
Ø 350 GeV = u,d,s,c,b,t
Dr M.A. Thomson Lent 2004
20
·
´ ÖÓÒ×µ
Ê
· ·
´ µ
Ô
¼ ¼
Data: × from GeV
Ô
★ Ê ×
increases in steps with
Ô
½½ !Î
★ × region complicated by resonances:
$
%
charmonium (c$) and bottomonium (b ). ★ Ê
Data exclude ‘no colour’ hypothesis. STRONG EVIDENCE for COLOUR
Dr M.A. Thomson Lent 2004 21
Experimental Evidence for Colour
★ Ê
´×××µ
★ The existence of the ª
¿
´×××µ
The ª is a (L=0) spin- baryon consisting ¾
of 3 strange-quarks. The wave-function
× × ×
is SYMMETRIC under particle interchange. However quarks are FERMIONS, therefore require an ANTI-SYMMETRIC wave-function, i.e. need
another degree of freedom, namely COLOUR.
´× × × µ
ÓÐÓÙÖ
½
Ö Ö Ö Ö Ö µ ´Ö Ô
ÓÐÓÙÖ + + - - -
¼
★ decay rate
¼
Need colour to explain decay rate. u γ π0 u
u γ
¼ ¾
´ µ » Æ
ÓÐÓÙÖ
Æ ¾ ¦ ¼ ½¾
EXPT ÓÐÓÙÖ
Dr M.A. Thomson Lent 2004 22 Evidence for Gluons
In QED, electrons can radiate photons. In QCD
quarks can radiate gluons.
·
ÕÕ e+ q √α γ S g √α √α Qq 1
e- q2 q
Ô «
giving an extra factor of Ë in the matrix «
element, i.e. an extra factor of Ë in cross section. In QED we can detect the photons. In QCD we never see free gluons due to confinement. Experimentally detect gluons as an additional jet: 3-Jet Events. g q q ★ Angular distribution of gluon jet depends on gluon spin
Dr M.A. Thomson Lent 2004 23
3-Jet Events and Gluon Spin
Ô
¿½ JADE × GeV Direct Evidence for Glu-
ons (1978)
Ô
½ OPAL × GeV (1990)
Distribution of the angle, be- tween the highest energy jet (assumed to be one of the quarks) relative to the flight di- rection of the other two (in their
cms frame). depends on the spin of the gluon.
µ GLUON is SPIN-1
Dr M.A. Thomson Lent 2004 24 Gluon Self-Interactions
Direct Evidence for the exis- tence of the the gluon self- interactions from 4-JET events.
g g g g g g g Y 3 GLUON VERTEX 4 GLUON VERTEX
X Z
e+ q e+ q g g g g e- q e- q
e+ q g e+ q q
g q e- q e- q ★ Angular distribution of jets is sensitive to
existence triple gluon vertex:
½
★ ÕÕ
¯ vertex consists of 2 spin- quarks ¾
and a spin-1 gluon. ★ ¯ vertex consists of 3 spin-1 gluons,
µ different angular distribution.
Dr M.A. Thomson Lent 2004 25 Experimentally:
★ Define the two lowest energy jets as the gluons. (gluon jets are more likely to be low energy than quark jets) ★ Measure angle between the plane containing
the ‘quark’ jets and the plane containing the
‘gluon’ jets, q q
g g χ χ BZ BZ
g g
q q
Gluon self- interactions are required to describe the experimental data. Theory with- out self-interactions (ABELIAN) is in- consistent with observations
Dr M.A. Thomson Lent 2004
26 «
Measuring Ë «
Ë can be measured in many ways. The cleanest is from Ê
: In practice measure
+ +....
i.e. don’t distinguish 2/3 jets. So measure
·
´ µ
hadrons
Ê
· ·
´ µ
·
´ Õ Õµ
ÒÓØ Ê
· ·
´ µ
When gluon radiation is included :
«
Ë
¾
Ê ¿ É ½ ·
Õ
È
¾
¾
É ¿
Õ
Õ
¿
« ¿
Ë
´½ ·
Therefore µ
¿
¾ ¾
« ´Õ ¾# µ ¼ ¾¼
giving Ë
Dr M.A. Thomson Lent 2004
27 «
Many other ways to measure Ë ....
·
ÕÕ e.g. 3 jet rate : e+ q √α γ S g √α √α Qq 1
e- q2 q
´Õ ´ µ Õ µ
3 jet
´ µ ´ÕÕµ
2 jet
» «
Ë «
Ë Summary «
Summary of Ë
measurements «
Ë RUNS !
Dr M.A. Thomson Lent 2004 28 Nucleon-Nucleon Interactions
★ Bound ÕÕÕ states (e.g. Protons and Neutrons) are COLOURLESS (COLOUR SINGLETS).
★ They can only interact via COLOURLESS intermediate states - i.e. not by single gluons. (conservation of colour charge)
★ Interact by exchange of PIONS
★ One possible diagram shown below : d d u u pp u u
uu π0
u u u u d d pp
★ Nuclear potential is YUKAWA potential with
¾
Ñ Ö
Î ´&Öµ
Ö
½
★ ´Ñ µ
Short range force : range
½
´¼ ½¼ Î µ
Range Ê
½
' Î
½¼
' "´½ ¢ ½¼ µ Ñ
½ #Ñ
Dr M.A. Thomson Lent 2004