Manuel Calderón de la Barca Sánchez

Intro. Relativistic Heavy Ion Collisions

pQCD : The Drell-Yan process at Leading Order Following slides from Fred Olness, SMU

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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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Drell-Yan Brookhaven AGS

e+e- Production SLAC SPEAR

Frascati ADONE

related by crossing

!  a

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!  Calculation: !  First, we'll compute the partonic in the partonic

CMS !  Born Process

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σ

!  Diagram:

!  Gather factors and contract with metric tensor, we get:

!  Squaring and averaging over spin and color:

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q+q→ e+ + e−

−iM = iQie2

q2v (p2 )γ

µu(p1){ } v(p4 )γµu (p3){ }

M

2=12!

"#$

%&2

3 13!

"#$

%&2

Qi2 e4

q4Tr p2 γ

µ p1γν)* +,Tr p3γµ p4 γν)* +,

!  Define the Mandelstam Variables

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p1 =s21,0,0,+1( )

p2 =s21,0,0,−1( )

p3 =s21,+sinθ ,0,+cosθ( )

p4 =s21,−sinθ ,0,−cosθ( )

s = (p1 + p2 )2 = (p3 + p4 )

2

t = (p1 − p3)2 = (p2 − p4 )

2

u = (p1 − p4 )2 = (p2 − p3)

2

t = − s2(1− cosθ )

u = − s2(1+ cosθ )

!  With Dirac algebra for the γ matrices and Traces:

!  We used: , i.e. massless fermions.

!  And . So we have:

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Tr[ p2 γµ p1γ

ν ]Tr[ p3γµ p4 γν ]

= 4 p1µ p2

ν + p2µ p1

ν − gµν (p1 ⋅ p2 )%& '(× 4 p3,µ p4,ν + p4,µ p3,ν − gµν (p3 ⋅ p4 )%& '(

= 25[(p1 ⋅ p3)(p2 ⋅ p4 )+ (p1 ⋅ p4 )(p2 ⋅ p3)]= 23[t 2 + u2 ]

s = 2(p1 ⋅ p2 )= 2(p3 ⋅ p4 )t = 2(p1 ⋅ p3)= 2(p2 ⋅ p4 )u = 2(p1 ⋅ p4 )= 2(p2 ⋅ p3)

q2 = s

M

2=Qi

2α 2 25π 2

3t 2 + u2

s2#

$%

&

'(

!  In the partonic CMS: where

!  Since , we have:

!  The total cross section is :

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dσ =

12s

M2dΓ

dΓ = d 3p3(2π )32E3

d 3p4(2π )32E4

(2π )4δ(p1 + p2 − p3 − p4 )=d cosθ16π

t = − s2

(1− cosθ ) and u = − s2

(1+ cosθ )

dσd cosθ

=Qi2α 2 π

61s1+ cos2θ( )

σ =Qi2α 2 π

61s

dcosθ 1+ cos2θ( )−1

1

∫ =4πα 2

9sQi2 ≡ σ 0

!  Characteristic of scattering of spin ½ constituents by a spin 1 vector boson.

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dσd cosθ

=Qi2α 2 π

61s1+ cos2θ( )

!   Where:

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P1 =s

21, 0, 0,+1( ) P1

2 = 0

P1 =s

21, 0, 0,−1( ) P2

2 = 0

s = (P1 +P2 )2 =

sx1x2

=sτ τ = x1x2 =

ss≡M 2

s

Fraction of Energy2 present in partonic interaction out of the one present in the hadronic system.

dσdM 2 = dx1

0

1

∫ dx2 q(x1)q(x2 )+q(x2 )q(x1){ }σ 0δ(Q2 − s)

0

1

∫q,q∑

Hadronic Cross Section

Parton Distribution Function

Partonic Cross Section

!  Using and !  We can write the cross section in the scaling form:

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σ 0 =4πα 2

9sQi2 δ(M 2 − s)= 1

sx1δ(x2 −

τx1)

M 3 dσdM

=8πα 2

9Qi2 dx1 q(x1)q(τ / x1)+q(x1)q(τ / x1){ }τ

1

∫q,q∑

Note that the RHS is a function of only t, not of M alone. Therefore, this quantity should lie on a universal scaling curve.

! Partonic CMS has longitudinal momentum in hadron frame

!   We can use xF as a measure of the longitudinal momentum

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p12 = (p1 + p2 )= E12,0,0, pL( )

E12 =s2(x1 + x2 )

pL =s2(x1 − x2 )≡

s2xF

y = 12ln E12 + pL

E12 − pL

"

#$

%

&'=12ln x1

x2

"

#$

%

&'

dQ2dxF = dydτ s xF2 + 4τ

dσdM 2dxF

=4πα 2

9M 41

xF2 + 4τ

τ Qi2 q(x1)q(τ / x1)+q(x1)q(τ / x1){ }

q,q∑

x1,2 = τ e±1

dx1dx2 = dτdy

!  Need QCD corrections: 4/11/12 Phy 224C 19

K =1+ 2παs

3(...)+

!  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)