FYS4560 ¡
In ep elastic scattering §  One parameter: scattering angle θ
¡
In ep deep-­‐inelastic scattering §  2 independent variables: (x, Q2), … ¡
In hadron-­‐hadron collisions §  3 independent variables: (x1,x2,Q2) ▪  related to 3 experimentally well measured quantities ▪  ppàjet1+jet2+X: differential cross-­‐section described by θ1, θ2 and pT §  Transverse momentum: pT=sqrt(px2+py2)=p sinθ ▪  Azimuthal angle φ : px=pT cosφ and py=pT sinφ ; pz=p cosθ ¡
In hadron-­‐hadron collider §  collisions take place in CoM (pp for LHC) which is not the com of colliding partons (q, antiquark, gluon) §  Net longitudinal momentum: (x1-­‐x2)/Ep §  Final-­‐state objects ( jets and leptons) boosted along beam direction §  Jet angles expressed in terms of rapidity 1 " E + pz %
Rapidity :y = ln $
'
2 # E − pz &
¡
Rapidity y §  Rapidity differences are invariant under boosts along beam direction 1 " E '+ p'z % 1 " γ ( E − β pz ) + γ ( pz − β E ) %
''
y' = ln $
' = ln $$
2 # E '− p'z & 2 # γ ( E − β pz ) − γ ( pz − β E ) &
1 " (1− β ) ( E + pz ) %
1 " 1− β %
'' = y + ln $
= ln $$
'
2 # (1+ β ) ( E − pz ) &
2 # 1+ β &
⇒ Δy' = Δy
¡
Jet Mass M §  High energy jets: M<pz~Ecosθ 1 # 1+ cosθ & 1 # 2 θ &
y ≈ ln %
( = ln % cot (
§  Approximation for y 2 $ 1− cosθ ' 2 $
2'
§  Define pseudo-­‐rapidity η * θ-
η ≡ − ln ,tan /
+ 2.
R-measure
R=
2
(η1 − η2 ) + (φ1 − φ2 )
2
!
"⇒
cosh y = 1 / sin θ ; sinh y = 1 / tan θ ; tanh y = cosθ #
px = pT cos φ ; py = pT sin φ ; pz = p cosθ
E 2 = p2 + m2 !
# 2
2
2
2
p = pz + pT " E = pz2 + mT2
2
2
2#
mT = m + pT $
"
E = mT cosh y
$
#⇒
pz = mT sinh y = E tanh y
$
2
2
cosh y − sinh y = 1%
E 2 − pz2
=1
2
mT
⇒ p µ = ( E, px , py , pz ) = ( mT cosh y, pT cos φ, pT sin φ, mT sinh y )
β −>1
pz mγβ cosθ
tanh y = =
= β cosθ = cosθ E
mγ
dy =
dpz
E
(E → m; dy → dvz )
Massless particles : mT
≈ pT ⇒ y ≈ η
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Kinematics of produced, or secondary particles One Body Phase Space §
§
¡
NR Relativistic Kinematically allowed range in y of a proton with PT=0 §
Max y : Beam direction y max
" s%
= ln$$ ''
# mp &
!
dp = p 2 dpdΩ = dpz pT dpT dφ
!
!
d
p
1
d 4 pδ ( E 2 − p 2 − m 2 ) =
= pT dpT dφ dy = dpT2 dφ dy ≡ d 2 pT dy
E
2
2
1 " E + pz % 1 " ( E + pz ) ( E + pz ) % 1 "$ ( E + pz ) %'
'' = ln
y = ln $
' = ln $$
2 # E − pz & 2 # ( E − pz ) ( E + pz ) & 2 $# E 2 − pz2 '&
2
" (2 p ) %
" s%
1 "$ ( 2 pbeam ) %'
beam
'' = ln $$ ''
ymax = ln $
= ln $$
2
'
2 # mp &
# mp &
# mp &
Tevatron : s = 2TeV ymax = 7.7
LHC :
14TeV
ymax = 9.6
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LHC ¡
Plateau height and width depend on √s Region around y=0 (90o) has a “plateau” with width Δy ~ 6 for LHC Detector coverage @LHC ¡
¡
¡
Width depends on produced particle’s mass and pT , but only logarithmically E = mT cosh y
mT2 = m 2 + PT2
y~ -­‐5à+5 Regions of high y correspond to beam fragmentation Jet-­‐cross section as function of jet ET at various η ¡
For ET small wrt √s there is a rapidity plateau at the Tevatron with Δy ~ 4 at ET < 100 GeV. ¡
Δy Plateau shrinks at fixed CM enery as ET increases ¡
Δy increases at high energies (6 at LHC, 4 at Tevatron) Rapidity plateau increasing with Energy Particle density raises slowly with Energy ¡
¡
Residual fragments of p/pbar evolve into soft (pT ~ 0.5 GeV) pions with a density ~ 5 per unit of rapidity at Tevatron §  equal numbers of π+ πo π-­‐. At higher pT, “minijets” become a prominent feature ¡
¡
Energy-­‐dependence for pT < 1 GeV is small pT distribution fitted to power law dσ / πdydpT2 ~ A /( pT + po ) n
A ~ 450mb / GeV 2 , po ~ 1.3 GeV , n ~ 8.2
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Total cross section of order 100 mb à mostly low pT particles make the bulk ¡
pp at 14 TeV, subprocess ggàgg, ▪  cut on pT(g)> 5 GeV. ▪  Note scale is mb/GeV ▪  No fragmentation of gluons ¡
Gluon jets produced at LHC at low pT §  Low pt mini-­‐jets with cross section ~ mb at 10 GeV ~1/PT3 Comphep: ggàgg J. High Energy Phys. 02 (2010) 041 Transverse-­‐
momentum and pseudo-­‐rapidity distributions of charged hadrons in pp collisions ¡
http://hep.uchicago.edu/~pilcher/p463/Old/Lecture03%20Kinematics%20MB%20UE%20MPI.bw.pdf ¡
¡
Gluons belong to the octet obtained by combining color and ant color 3 ⊗ 3 = 1⊕ 8
Similar situation as in quark model with 3 flavors, hence SU(3) §  Singlet completely symmetric. Does not interact with quarks à no single gluon g0 =
1
(RR + BB + GG )
3
§  By analogy with meson octet: g1 = RG ; g 2 = RB ; g 3 = GR ; g 4 = GB ; g 5 = BR ; g 6 = BG
15/04/15 1
1
(RR − GG ) ; g8 = (RR − 2BB + GG )
g7 =
2
6
17 g1 = RG
g2 = RB
3 ⊗ 3 = 8a ⊕1s
g3 = GR
g4 = GB
g5 = BR
1
g0 =
RR + BB + GG )
(
3
15/04/15 g6 = BG
1
g7 =
RR − GG )
(
2
1
g8 =
(RR + GG − 2BB )
6
18 ¡
Gluons are colored §  can interact coupled by continuous color lines (a) §  Gauge symmetry of strong interactions: SU(3)c ¡
+ self-­‐gluon interactions! Gluon-­‐gluon scattering g + g → g + g can happen by exchanging another gluon (b) §  further contribution is the 4-­‐gluon vertex (c) 15/04/15 F. Ould-­‐Saada 19 ß QED
Interaction amplitude
proportional to z1z2α
ß QCD
¡
Incoming and outgoing fermions may be different §  gluon has color of one quark and the opposite of the color of the other quark ¡
Vertex contains coupling constant AND color factor §  ci and cj : colors of 2 quarks -­‐ λ : gluon type (color lines continuous) 15/04/15 20 g1 = RG
g2 = RB
g3 = GR
g4 = GB
κ 1RG = 1
κ 1RG = −1
κ 2RB = 1
κ 2RB = −1
κ 3GR = 1
κ 3GR = −1
κ 4GB = 1
..................
κ 5BR = 1
g5 = BR
g6 = BG
1
g7 =
RR − GG )
(
2
1
g8 =
RR + GG − 2BB )
(
6
κ 6BG = 1
Colour factors of antiquark are opposite to those of quarks κ 6RG = 1
1
1
; κ 7GG = −
2
2
1
1
2
=
; κ 8GG =
; κ 8BB = −
6
6
6
κ 7RR =
κ 8RR
g1 à g6 have colour and anti-­‐colour g7 has 2 colours and 2 anti-­‐coulours 15/04/15 g8 has 3 colours and 3 anti-­‐coulours 21 BB→BB
B
B
B
B
q+ q→ q+ q
κ 8BB = −
2
6
1 BB 1 BB 1 ⎛ −2 ⎞ ⎛ −2 ⎞ 1
κ8
κ 8 = ⎜
⎟ ⎜
⎟ =
2 ⎝ 6 ⎠ ⎝ 6 ⎠ 3
2
2
RR→RR
R
¡
q + R q →R q + R q
1
2
1
=
6
§
κ 7RR =
κ 8RR
1 RR 1 RR 1 RR 1 RR 1 ⎛ 1 ⎞ ⎛ 1 ⎞ 1 ⎛ 1 ⎞ ⎛ 1 ⎞ 1
κ7
κ7 +
κ8
κ 8 = ⎜
⎟ ⎜
⎟ + ⎜
⎟ ⎜
⎟ =
2 ⎝ 2 ⎠ ⎝ 2 ⎠ 2 ⎝ 6 ⎠ ⎝ 6 ⎠ 3
2
2
2
2
15/04/15 As expected from symmetry, force between R and R is the same as between B and B. ¡
verify intensity of force between R and G + sign à repulsive force: as in QED same-­‐sign color charges repel each other 22 ¡
Hadrons have no color charge, but are made of colored quarks §  ⇒ Color charges of quarks form “neutral” combination §  In QED: analogue to atom (neutral because it has as many + charges as – charges) §  QCD ⇒ neutrality is color singlet state ¡
3 ⊗ 3=8A ⊕1S
Mesons: quark-­‐antiquark bound states §  Binding because their product contains singlet (qq )singoletto
singlet =
¡
¡
1
3
( q q+
R
R
B
q B q + G qG q
)
Mesons: quark-­‐antiquark “interaction” à factor -­‐4/3 αs à attractive force è Baryons: qqq “interaction” à factor -­‐2/3 αs à attractive force 15/04/15 23 B
B
q q
κ 2RB = 1;
κ 2RB = −1
κ 4GB = 1;
κ 4GB = −1
κ 8BB = −
2
;
6
κ 8BB = +
Symmetry ⇒ interaction between 2 pairs equal ⇒ calculate for one and multiply by 3
2
1 ⎛ 4
4 α
⎛ 1 ⎞ 1 BB BB
⎞
RB RB
GB GB
⎡
⎤
3⎜
κ
κ
+
κ
κ
+
κ
κ
=
−
−1−1
=
−
s
⎜⎝
⎟⎠
5
5
6
6 ⎦
⎝ 3 ⎟⎠ 2 ⎣ 8 8
2
6
3
αs
Attractive force!
15/04/15 24 2
6