Intro. Relativistic Heavy
Ion Collisions
pQCD : The Drell-Yan process at Leading Order
Following slides from Fred Olness, SMU
Manuel Calderón de la Barca Sánchez
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History:
Discovery of J/ψ, Upsilon, W/Z, and “New Physics”
???
Calculation of q q →μ+μ- in the Parton Model
Scaling form of the cross section
Rapidity, longitudinal momentum, and xF
Comparison with data:
NLO QCD corrections essential (the K-factor)
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Alternating Gradient Synchrotron
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The Goal: p + N → W + X
They found: p + N →μ+ μ− + X
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The Process: p + Be → e+ e- X
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very narrow width
⇒ long lifetime
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related by crossing
Drell-Yan
Brookhaven AGS
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e+e- Production
SLAC SPEAR
Frascati ADONE
8
a
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Calculation:
First, we'll compute the partonic sˆ in the partonic
CMS
Born Process
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q + q ® e+ + e-
Diagram:
Gather factors and contract with metric tensor, we
e2
get:
-iM = iQi 2 { v ( p2 )g m u( p1 )} {v( p4 )g m u( p3 )}
q
Squaring and averaging over spin and color:
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p1 =
p2 =
p3 =
p4 =
Define the Mandelstam Variables
ŝ = ( p1 + p2 )2 = ( p3 + p4 )2
tˆ = ( p1 - p3 )2 = ( p2 - p4 )2
û = ( p1 - p4 )2 = ( p2 - p3 )2
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ŝ
(1,0, 0,+1)
2
ŝ
(1,0, 0,-1)
2
ŝ
(1,+sin q , 0,+cosq )
2
ŝ
(1,-sinq ,0,-cosq )
2
ˆt = - ŝ (1- cosq )
2
ŝ
û = - (1+ cosq )
2
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With Dirac algebra for the g matrices and Traces:
Tr[ p2 g m p1 g n ]Tr[ p3 g m p4 g n ]
= 4 éë p1m p2n + p2m p1n - gmn ( p1 × p2 )ùû ´ 4 éë p3,m p4,n + p4,m p3,n - gmn ( p3 × p4 )ùû
= 2 5 [( p1 × p3 )( p2 × p4 )+ ( p1 × p4 )( p2 × p3 )]
= 2 3[tˆ 2 + û 2 ]
We used:
ŝ = 2( p1 × p2 ) = 2( p3 × p4 )
tˆ = 2( p1 × p3 ) = 2( p2 × p4 )
, i.e. massless fermions.
û = 2( p1 × p4 ) = 2( p2 × p3 )
And q 2 = ŝ . So we have:
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In the partonic CMS:
where
d 3 p3
d 3 p4
d cosq
4
dG =
(2p ) d ( p1 + p2 - p3 - p4 ) =
3
3
(2p ) 2E3 (2p ) 2E4
16 p
ŝ
ŝ
ˆ
Since t = - (1- cosq ) and û = - (1+ cosq ) , we have:
2
2
dŝ
2 2 p 1
2
= Qi a
1+ cos q )
(
d cosq
6 ŝ
The total cross section is :
2 p 1
2
4
pa
2
2
sˆ = Qi2a
dcos
q
1+
cos
q
=
Q
(
)
ò
i º ŝ 0
6 ŝ -1
9 ŝ
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dŝ
2 2 p 1
2
= Qi a
1+ cos q )
(
d cosq
6 ŝ
Characteristic of scattering of spin ½ constituents by
a spin 1 vector boson.
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s
P1 =
(1, 0, 0,+1)
2
s
P1 =
(1, 0, 0, -1)
2
s
s
s = (P1 + P2 ) =
=
x1 x2 t
2
P12 = 0
P22 = 0
Fraction of Energy2 present in
partonic interaction out of the one
present in the hadronic system.
Where:
ŝ M 2
t = x1 x2 = º
s
s
ds
2
= å ò dx1 ò dx2 { q(x1 )q(x2 )+ q(x2 )q(x1 )} ŝ 0d (Q - ŝ)
2
dM
0
q,q 0
1
1
Hadronic Cross Section
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Parton
Distribution
Function
Partonic
Cross
Section
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4 pa 2 2
Qi and
Using sˆ 0 =
9 ŝ
d (M 2 - ŝ) =
1
t
d (x2 - )
sx1
x1
We can write the cross section in the scaling form:
1
ds 8pa 2
2
M
=
åQi ò dx1 {q(x1 )q(t / x1 )+ q(x1 )q(t / x1 )}
dM
9 q,q
t
3
Note that the RHS is a function
of only t, not of M alone.
Therefore, this quantity should
lie on a universal scaling curve.
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Partonic CMS has longitudinal momentum in hadron frame
p12 = ( p1 + p2 ) = ( E12 ,0,0, pL )
s
(x1 + x2 )
2
s
s
pL =
(x1 - x2 ) º
xF
2
2
E12 =
We can use xF as a measure of the longitudinal momentum
1 æ E12 + pL ö 1 æ x1 ö
y = ln ç
÷ = ln ç ÷
2 è E12 - pL ø 2 è x2 ø
x1,2 = t e
±1
dx1dx2 = dt dy
dQ 2 dxF = dydt s
ds
4 pa 2
=
2
dM dxF 9M 4
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xF2 + 4t
xF2 + 4t
t åQi2 {q(x1 )q(t / x1 )+ q(x1 )q(t / x1 )}
q,q
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2 pa s
Need QCD corrections: K = 1+
(...)+
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800 GeV proton beam, on fixed target Cu and d.
pp & pN processes sensitive to
anti-quark distributions
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A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne,
Eur. Phys. J. C23, 73 (2002);
Eur. Phys. J. C14, 133 (2000);
Eur. Phys. J. C4, 463 (1998)
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