Drell-Yan Process: Part I Fred Olness SMU

Drell-Yan Process:
Part I
Fred Olness
SMU

Part I: Drell-Yan Process
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 x F
Comparison with data:
NLO QCD corrections essential (the K-factor)
σ(pd)/σ(pp) important for d-bar/ubar
W Rapidity Asymmetry important for slope of d/u at large x
Where are we going?
P T
Distribution
W-mass measurement
Resummation of soft gluons

Historical
Background

Our story begins in the late 1960's

Brookhaven National Lab
Alternating Gradient Synchrotron

The Goal:
They found:
An Early Experiment:
p + N → W + X
p + N → µ + µ − + X
at BNL AGS
Not the W
cross section
M µµ
GeV

What is the explanation???
In DIS, we have two choices for an interpretation:
lepton
current
L µν W µν
interaction
hadron
current
Conserved Current Interactions
The Parton Model
What about Drell-Yan???
? ? ?
The Parton Model

The Process: p + Be → e + e - X
Discovery of the J/Psi Particle
very narrow width
⇒ long lifetime
at BNL AGS

q e +
J / ψ
The November Revolution
related by crossing ...
e +
J / ψ
q
q
e -
e -
q
Drell-Yan
Brookhaven AGS
e + e - Production
SLAC SPEAR
Frascati ADONE
R e e hadrons
e e
3
i
Q i
2

More Discoveries with Drell-Yan
1974: The J/Psi (charm) discovery p+N → J/ψ
... 1976 Nobel Prize
1977: The Upsilon (bottom) discovery p+N → ϒ
1983: The W and Z discovery p + p → W/Z
... 1984 Nobel Prize

The Future of Drell-Yan
Where do we find
New Physics??
New Higgs Bosons
New W' or Z'
SUSY
... unknown...

Let's
Calculate

First, we'll compute
the partonic σ ^ in the
partonic CMS

Let's compute the Born process:
q q e e
ig
u p 1
q 2 i
u p 3
v p 2
ieQ i ie
v p 4
Gathering factors and contracting g µν , we obtain:
e 2
iMiQ i v p
q 2 2
u p 1
u p 3
v p 4
Squaring, and averaging over spin and color, ....
M 2 1 2
2
3 1 3
2
Q i
2 e 4
q 4 Tr p 2 p 1
Tr p 3
p 4

Let's work out some parton level kinematics
p
p
3
1 θ
p 2
p 4
p 1
s 2
p 2
s 2
1,0,0,1
1,0,0,1
p 3
s 2
1,sin ,0,cos
p 12
p 22
p 32
p 42
0
p 4
s 2
1,sin ,0,cos
Defining the Mandelstam variables ...
s p 1
p 2
2
p 3
p 4
2
t p 1
p 3
2
p 2
p 4
2
u p 1
p 4
2
p 2
p 3
2
t s 2 1cos
u s 2 1cos

We'll now compute the matrix element M
Manipulating the traces, we find ...
Tr p 2
p 1
Tr p 3
p 4
4 p 1
p 2
p 2
p 1
g
p 1
p 2
4 p 3
p 4
p 4
p 3
g p 3
p 4
2 5 p 1
p 3
p 2
p 4
p 1
p 4
p 2
p 3
2 3 t 2 u 2
Where we have used:
s2 p 1
p 2
2 p 3
p 4
p 12
p 22
p 32
p 42
0
Putting all the pieces together, we have:
M 2 2 2 2 5 2
Q i
3
t 2 u 2
s 2
with
t 2 p 1
p 3
2 p 2
p 4
u2 p 1
p 4
2 p 2
p 3
q 2 p 1
p 2
2
s
e 2
4

... and put it together to find the cross section
d 1
2 s
M 2
d
In the partonic
CMS system
d
d 3 p 3
d 3 p 4
2
2 3 2 E 3
2 3 4 p 1
p 2
p 3
p 4
d cos
2 E 4
16
Recall,
t s
2
1cos and u s
2
so, the differential cross section is ...
d
d cos
and the total cross section is ...
Q i
2
2
6
1
1
s 1
Q 2 2
i
6
1
s
1cos 2
1cos
d cos 1cos 2 4 2
9 s
Q i
2
0

Some Homework:
#1) Show:
d 3 p
d 4 p
2
2 3 2 E 2 4 p 2 m 2
This relation is often useful as the RHS is manifestly Lorentz invariant
#2) Show that the 2-body phase space can be expressed as:
d
d 3 p 3
d 3 p 4
2
2 3 2 E 3
2 3 4 p 1
p 2
p 3
p 4
d cos
2 E 4
16
Note, we are working with massless partons, and θ is in the partonic CMS frame

Some More Homework:
#3) Let's work out the general 2→2 kinematics for general masses.
p
p
3
1 θ
p 2
p 4
E 1,2
sm 1
a) Start with the incoming particles.
Show that these can be written in the general form:
p 1
E 1
,0 ,0 ,p p 12
m 1
2
p 2
E 2
,0 ,0 , p p 22
m 2
2
... with the following definitions:
2
m
2
2
2 s
p s,m 2, 2
1
m 2
2 s
a,b,c a 2 b 2 c 2 2 abbcca
Note that ∆(a,b,c) is symmetric with respect to its arguments,
and involves the only invariants of the initial state: s, m 12
, m 22
.
b) Next, compute the general form for the final state particles, p 3
and p 4
. Do this by first
aligning p 3
and p 4
along the z-axis (as p 1
and p 2
are), and then rotate about the y-axis by angle θ.

What does the angular dependence tell us?
Observe, the angular dependence:
q q e e
d
d cos
Q 2 2
i
6
1
s
1cos 2
Characteristic of scattering of spin ½ constitutients by a spin 1 vector
q
q
γ
e + e -
Note, for the photon, the mirror image of the above is also valid; hence the symmetric distribution.
The W has V-A couplings, so we'll find: (1+cosθ) 2

Next, we'll compute
the hadronic CMS

Kinematics in the Hadronic Frame
P 1
l +
p 1
= x 1
P 1
P 2
p 2
= x 2
P 2
q=(p 1
+p 2
)
l -
P 1 s 2
P 2 s 2
1,0,0,1
1,0,0,1
P 12 0
P 22 0
s P 1
P 2
2
s
x 1
x 2
s
Therefore
x 1 x 2 s s Q2
s
Fractional energy 2 between
partonic and hadronic system
d
dQ 2
q,q
dx 1 dx 2
q x 1
q x 2
q x 1
q x 2
0
Q 2 s
Hadronic
cross
section
Parton
distribution
functions
Partonic
cross
section

Scaling form of the Drell-Yan Cross Section
Using:
0
4 2
9 s
Q i
2
and
Q 2 s 1
sx 1
x 2
x 1
we can write the cross section in the scaling form:
Q 4
d
4 2
dQ 2 9
q,q
1
Q i
2
dx 1
x 1
q x 1 q x 1 q x 1 q x 1
Notice the RHS is a
function of only τ, not Q.
This quantity should
lie on a universal
scaling curve.
Cf., DIS case,
& scattering of
point-like constituents

Longitudinal Momentum Distributions
Partonic CMS has longitudinal momentum w.r.t. the hadron frame
p 12 p 1 p 2 E 12 ,0,0, p L
p 1
= x 1
P 1
p 2
= x 2
P 2
E 12
s 2 x 1 x 2
p 12
p L s 2 x 1 x 2 s 2 x F
x F
is a measure of the longitudinal momentum
The rapidity is defined as:
y 1 2 ln
E 12 p L
E 12
p L
1 2 ln x 1
x 2
dx 1
dx 2
d dy
x 1,2
e y dQ 2 dx F dy d s x F
2
4
d
dQ 2 dx F
4 2
9Q 4 1
x F
2
4
q,q
Q i 2 q x 1
q x 1
q x 1
q x 1

So, we're ready to
compare with data
(or so we think...)

Let's compare data and theory
ÌÐ ½º¾
ÜÔÖÑÒØРùØÓÖ׺
ÜÔÖÑÒØ ÁÒØÖØÓÒ Ñ ÅÓÑÒØÙÑ Ã Ñ׺
¾ ÃÔ ℄ ÔÈØ ¿¼¼»¼¼ Î ½
Ï¿ ÓÖ ¼℄ ¦ Ï ¿º Î ¾
¿ ËÑ ½℄ ÔÏ ¼¼ Î ½ ¦ ¼¿
´Ô ¹ ÔµÈØ ½¼ Î ¾¿ ¦ ¼
ÔÈØ ¼¼ Î ¿½ ¦ ¼ ¦ ¼¿
Æ¿ ¿℄ ¦ ÈØ ¾¼¼ Î ¾¿ ¦ ¼
ÈØ ½¼ Î ¾ ¦ ¼¿
ÈØ ¾¼ Î ¾¾¾ ¦ ¼¿¿
ƽ¼ Ø ℄ Ï ½ Î ¾ ¦ ¼½¾
¿¾ Ö ℄ Ï ¾¾ Î ¾¼ ¦ ¼¼ ¦ ¼¼
¿ Ò ℄ Ô Ï ½¾ Î ¾ ¦ ¼½¾ ¦ ¼¾¼
½ ÓÒ ℄ Ï ¾¾ Î ½ ¦ ¼¼
J. C. Webb, Measurement of continuum dimuon production in
800-GeV/c proton nucleon collisions, arXiv:hep-ex/0301031.

Oooops,
we need the
QCD corrections
K 1
2 s
3
2 3
... ...? s
e

Excellent agreement between data and theory
p + Cu at 800 GeV
E605 (p Cu →µ + µ - X) p LAB
= 800 GeV
p + d at 800 GeV
E772 (p d →µ + µ - X) p LAB
= 800 GeV
10 7
M 3 d 2 σ/dx F
dM (nb/GeV 2 /nucleon)
10 10
10 9
10 8
10 7
10 6
10 5
10 4
M 3 d 2 σ/dx F
dM (nb/GeV 2 /nucleon)
10 5
10 3
10
10 -1
10 -3
10 3
10 -5
10 2
10 -7
10
1
10 11 0.2 0.3 0.4 0.5
√τ
pp & pN processes sensitive to
anti-quark distributions
10 -9
10 9 0.08 0.09 0.1 0.2 0.3 0.4
√τ
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)

Drell-Yan can give us unique and
detailed information about PDF's.
We'll now examine two examples:
1) Ratio of pp/pd cross section
2) W Rapidity Asymmetry

A measurement of d(x)=u(x)
Antiquark asymmetry in the Nucleon Sea
FNAL E866/NuSea
ACU, ANL, FNAL, GSU, IIT, LANL, LSU,
NMSU, UNM, ORNL, TAMU, Valpo.
800 GeV p + p and p + d ! + X
800 GeV
Protons
Counts/0.1 GeV
10 4
10 3
10 2
10
Hadron
Absorber
High Mass
Low Mass
x 10 2
5000
Low Mass
4000
3000
2000
1000
0
2 4 6 8 10 12
10 5 2 4 6 8 10 12 14 16
7500
High Mass
5000
2500
0
2 4 6 8 10 12
d _ / u _
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
Fermilab E866 - Drell-Yan
CTEQ4M
NA 51
MRS(R2)
MRST
±0.032 Systematic error not shown
1
DiMuon Mass (GeV)
0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
x

Cross section ratio of pp vs. pd
Obtain the neutron PDF via isospin symmetry:
u d
u d
In the limit x 1
>> x 2
:
pp
pn
4 9
u x 1
u x 2
1 9
d x 1
d x 2
4 9
u x 1
d x 2
1 9
d x 1
u x 2
For the ratio we have:
pd
2 pp 1 2
1 1 4
1 1 4
d 1
u 1
1 d 2
1 d 1
d 2
u 2 2
u 1
u 2
1 d 2
u 2
As promised, this provides
information about the
sea-quark distributions
EXERCISE: Verify the above.
pd
2 pp 1 2
1 d 2
u 2

Does the theory match the data???
pd
2 pp 1 2
1 d 2
u 2
2.2
2
Fermilab E866 - Drell-Yan
NA 51
CTEQ4M
σ pd /2σ pp
1.3
1.2
1.1
1
0.9
Fermilab E866 - Drell-Yan
MRS(R2)
CTEQ4M
“CTEQ4M (d _ - u _ = 0)”
d _ / u _
1.8
1.6
1.4
1.2
1
0.8
0.8
1% Systematic error not shown
0.6
±0.032 Systematic error not shown
0.7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Implies R

E866 required significant changes in the hi-x sea distributions
1.3
2.25
1.2
2
1.1
1.75
σ pd /2σ pp
1
0.9
0.8
0.7
0.6
0.5
CTEQ5M
MRST
GRV98
CTEQ5M (d _ = u _ )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
x 2
CTEQ4M
MRS(r2)
Less than 1% systematic
uncertainty not shown
d _ /u _
1.5
1.25
1
0.75
0.5
0.25
E866/NuSea
NA51
CTEQ5M
MRST
GRV98
CTEQ4M
MRS(r2)
Systematic Uncertainty
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
x
With increased flexibility in the parameterization of the
sea-quark distributions, good fits are obtained
E.A. Hawker, et al. [FNAL E866/NuSea Collaboration], Measurement of the
light antiquark flavor asymmetry in the nucleon sea, PRL 80, 3715 (1998)
H. L. Lai, et al.} [CTEQ Collaboration], Global
{QCD} analysis of parton structure of the nucleon:
CTEQ5 parton distributions, EPJ C12, 375 (2000)

Next ...
2) W Rapidity Asymmetry

Where do the W's and Z's come from ???
d
dy W 2 3
u(x a
)
proton
u x u x
W + d(x b
)
For anti-proton:
Therefore
G F
2 q q
anti-proton
d x d x
V q q
2
q x a q x b q x b q x a
% of total σ LO
(W + ,W - )
100
10
1
flavour decomposition of W cross sections
W +
W -
_
ud_
du
_
sc _
cs
_
us
_
dc _
su
_
cd
d
dy W 2 3
d
dy W 2 3
G F
2 u x a d x b
G F
2 d x a u x b
0.1
pp
pp
1 10
√s (TeV)
A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne,
Eur. Phys. J. C23, 73 (2002); Eur. Phys. J. C4, 463 (1998)

A bit of calculation
u(x a
)
proton
anti-proton
W + d(x b
)
A y
d
dy W d
dy
d
dy W d
dy
W
W
With the previous approximation,
A u x a d x b d x a u x b
u x a d x b d x a u x b
R du x b R du x a
R du x b R du x a
where
R du x d x
u x
We can make Taylor expansions:
x 1,2
x 0
e y x 0
1y
R du x 1,2
R du x 0
yx 0
R' du
EXERCISE: Verify the above.
Thus, the asymmetry is:
A y yx 0
R' du x 0
R du x 0

Unfortunately,
we don't measure the W
directly since W→eν.
Still the lepton contains
important information
Charged Lepton Asymmetry
Charge Asymmetry
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
CDF 1992-1995 (110 pb -1 e+µ)
CTEQ-3M
RESBOS
MRS-R2 (DYRAD)
MRS-R2 (DYRAD)(d/u Modified)
MRST (DYRAD)
F. Abe, et al. [CDF Collaboration], PRL 81, 5754 (1998)
Measurement of the lepton charge asymmetry in W boson decays
produced in p anti-p collisions,
CTEQ-3M
DYRAD
0 0.5 1 1.5 2
⎮Lepton Rapidity⎮
A y
d
d
dy l
dy
d d
dy l
dy
l
l
d
u
W -
ν e -
1cos 2

d/u Ratio at High-x
The form of the
d/u ratio at large x
as a function of
1) Parameterization
2) Nuclear Corrections
S. Kuhlmann, et al., Large-x parton distributions, PL B476, 291 (2000)

End of Part I: Where have we been???
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 x F
Comparison with data:
NLO QCD corrections essential (the K-factor)
σ(pd)/σ(pp) important for d-bar/ubar
W Rapidity Asymmetry important for slope of d/u at large x
Where are we going?
P T
Distribution
W-mass measurement
Resummation of soft gluons

Drell-Yan Process:
Part II
Fred Olness
SMU

Part II: W Boson Production as an example
Finding the W Boson Mass:
The Jacobian Peak, and the W Boson P T
Multiple Soft Gluon Emissions
Single Hard Gluon Emission
Road map of Resummation
Summing 2 logs per loop: multi-scale problem (Q,q T
)
Correlated Gluon Emission
Non-Perturbative physics at small q T
.
Transverse Mass Distribution:
Improvement over P T
distribution
What can we expect in future?
Tevatron Run II
LHC

Side Note: From pp→γ/Z/W, we can obtain pp→γ/Z/W→l + l -
Schematically:
d q q l l g d q q g d l l
⊗
For example:
d
dQ 2 d t
q q l l g
d
d t
q q g
3 Q 2

Part II: W Boson Production as an example
How do we measure the W-boson mass?
u d W e
u
ν
e +
θ
d
Can't measure W directly
Can't measure ν directly
Can't measure longitudinal momentum
We can measure the P T
of the lepton
How can we use this to extract the W-Mass???

u
ν
e +
d
The Jacobian Peak
Suppose lepton distribution is uniform in θ
The dependence is actually (1+cosθ) 2 , but we'll take care of that later
What is the distribution in P T
?
transverse direction
Number of Events
P T
Max
P T
Min
beam direction
We find a peak at P T
max
≈ M W
/2

The Jacobian Peak
Now that we've got the picture, here's the math ... (in the W CMS frame)
p T2 s 4 sin 2
cos 4 p 2
T
1
s
d cos
dp T
2
2 s
1
cos
So we discover the P T
distribution has a singularity at cosθ=0, or θ=π/2
d
d
d cos
2 2
dp T
d cos dp T
d
d cos 1
cos
singularity!!!
zero p T
BUT !!!
finite p T
Measuring the Jacobian
peak is complicated if the
W boson has finite P T
.

1) The W-mass is important
fundamental quantity of the
Standard Model
2) P T
Distribution is important
for measuring the W-mass

The W-Mass is an important fundamental quantity
80.360 +/- 0.370
80.410 +/- 0.180
80.470 +/- 0.089
80.433 +/- 0.079
80.350 +/- 0.270
80.498 +/- 0.095
80.482 +/- 0.091
80.452 +/- 0.062
80.427 +/- 0.046
80.436 +/- 0.037
χ 2 /Nexp = 0.4/4
UA2 (W → eν)
CDF(Run 1A, W → eν,µν)
CDF(Run 1B, W → eν,µν)
CDF combined
D0(Run 1A, W → eν)
D0(Run 1B, W → eν)
D0 combined
Hadron Collider Average
LEP II (ee → WW)
World Average
79.5 79.7 79.9 80.1 80.3 80.5 80.7 80.9 81.1 81.3 81.5
Mw (GeV)

The W-Mass is an important fundamental quantity
M W
(GeV/c 2 )
80.6
80.5
80.4
80.3
LEP2
D0
CDF
100
250
500
1000
80.2
80.1
Higgs Mass (GeV/c 2 )
LEP1,SLD,νN data
M W
-M top
contours : 68% CL
130 140 150 160 170 180 190 200
M top
(GeV/c 2 )
T.Affolder, et al. [CDF Collaboration], PRD64, 052001 (2001)
Measurement of the W boson mass with the Collider Detector at Fermilab,

What gives the W
P T ???

What about the intrinsic k T
of the partons?
Assume a Gaussian form:
d 2
d 2 0
e p 2
T
p T
k T
760 MeV
problems
good
agreement

For high P T
, we need a hard parton emission
q
q
γ/Z/W
g
R209 at the ISR, 1982
annihilation
q
g
Compton
γ/Z/W
q
Gaussian
e p 2
T
Perturbative
1
p T
2
Combination of Gaussian
& QCD corrections

The complete P T
spectrum for the W boson
Perturbative contributions
+power corrections
The full P T
spectrum
for the W-boson
showing the different
theoretical regions
ÔÔ ´Ï· µ
ÌÉÅ
Perturbative
physics dominates
ÆÓÒÔÖØÙÖØÚ
ÝÒÑ× ´ÒØÖÒ× Ì µ

Road map for
Resummation
⇒
BEFORE
AFTER

NLO P T
distribution for the W boson
q
q
W
g
In the limit P T
→ 0
d
d dy dp T
2 d
d dy
4 s
Born 3
ln s p T
2
p T
2
s
0
d
d dy dp T
2
dp 2
T
d
d dy
Born
O s
finite
singular
p T
2
0
d
d dy dp T
2
dp 2
T
d
d dy Born
1
s
p T
2
4 s
3
ln s p T
2
p T
2
dp T
2
d
d dy
Parisi & Petronzio, NP B154, 427 (1979)
Dokshitzer, D'yakanov, Troyan, Phy. Rep. 58, 271 (1980)
Born
1 2 s
3 ln 2 s
p T
2
d
d dy Born
exp -2 s
3 ln 2 s
p T
2
effect of gluon
emission
assume this
exponentiates

Resummation of soft gluons: Step #1
Differentiating the previous expression for d 2 σ/dτ dy
Sudakov
Form Factor
d
d dy dp T
2
d
d dy
4 s
Born 3
ln s p T
2
p T
2
exp 2 s
3 ln 2
We just resummed (exponentiated) an infinite series of soft gluon emissions
Soft gluon emissions
treated as uncorrelated
e s L2 1 s L 2 s L 2 2
2 !
s L 2 3
3!
finite at p T
=0
...
s
p T
2
L ln s p T
2
I've skipped over some details ..
Parisi & Petronzio, NP B154, 427 (1979)
Dokshitzer, D'yakanov, Troyan, Phy. Rep. 58, 271 (1980)
Curci, Greco, Srivastava, PRL 43, 834 (1979); NP B159, 451 (1979)
Jeff Owens, 2000 CTEQ Summer School Lectures

We skipped over a few details ...
1) We summed only the leading logarithmic singularity, α s
L 2 .
We'll need to do better to ensure convergence of perturbation series
2) We assumed exponentiation; proof of this is non-trivial.
The existence of two scales (Q,p T
)≡(Q,q T
) yields 2 logs per loop
3) Gluon emission was assumed to be uncorrelated.
This leads to too strong a suppression at P T
=0.
Will need to impose momentum conservation for P T
.
4) In the limit P T
→ 0, terms of order α S
(µ=P T
) → ∞;
Must handle this Non-Perturbative region.

1) We summed only the leading logarithmic singularity
L ln s p T
2
d
dq T
2
s L
q T
2
1 s 1 L 2 s 2 L 4 ...
s L
q T
2
s 1 L 1 s 2 L 3 s 3 L 5 ...
we resum
these terms
we miss
these terms
d
The terms we are missing are suppressed by α s
L, not α s
!
If (somehow) we could
sum the sub-leading log ...
dq T
2
1 q T
2
1
q T
2
d
dq T
2
s L
q T
2
e s L2 L
s 1 L 1 1 s 2 L 3 L 2 s 3 L 5 L 4 ...
we resum
these terms
we miss
these terms
s 2 L 1 1 s 3 L 3 L 2 s 4 L 5 L 4 ...
Now, the terms we are missing are suppressed only by α s
!

2) We assumed exponentiation; proof is non-trivial
Review where the logs come from
Review one-scale problem (Q)
resummation via RGE
Review two-scale problem (Q,q T
)
resummation via RGE+ Gauge Invariance

Where do the
Logs come from?

Total Cross Section: σ(e + e - ) at 3 Loops
One mass scale: Q 2 . No logarithms!!!

Drelly-Yan at 2 Loops:
Two mass scales: {Q 2 ,M 2 }. Logarithms!!!

Renormalization Group Equation
More Differential Quantities ⇒ More Mass Scales ⇒ More Logs!!!
d
ln Q2
dQ 2 2
d
ln Q2
and ln q 2
T
dQ 2 2 2
How do we resum logs? Use the Renormalization Group Equation
For a physical observable R:
Using the chain rule:
dR
d 0
2
2 2 s 2
2
s 2 R 2, s 2 0
Solution⇒
s ln Q2
s
Q 2
2
s
2
dx
x

Renormalization Group Equation: OVER SIMPLIFIED!
2
2 s
s 2 R 2, s 2 0
If we expand R in powers of α s
, and we know β,
we then know µ dependence of R.
R ,Q, s 2 R 0
s 2 R 1
ln Q 2 2 c 1
s
2
2
R 2
ln 2 Q 2 2 ln Q 2 2 c 2
O s
3
2
Since µ is arbitrary, choose µ=Q.
R Q,Q, s Q 2 R 0
s Q 2 R 1
0c 1
s 2 Q 2 R 2
00c 2
...
We just summed the logs

Two-Scale Problems
For R(µ,Q,α s
), we could resum ln(Q 2 /µ 2 ) by taking Q=µ.
What about R(µ,Q,q T
,α s
); how do we resum ln(Q 2 /µ 2 ) and ln(q T2
/µ 2 ).
Are we stuck? Can't have µ 2 =Q 2 and µ 2 2
=q T
at the same time!
Solution: Use Gauge Invariance; cast in similar form to RGE
Use axial-gauge with axial vector ξ.
This enters the cross section in the form: (ξ•p).
d
d 0
2
d
d 0 p 2
resum
x, Q2
p 2
2
logs
RGE allows us to vary µ to
2 ,...
Gauge invariance allows us to vary (ξ•p) to resum logs
It is covenient to transform to impact parameter
space (b-space) to implement this mechanism
The details will fill multiple lectures:
See Sterman TASI 1995; Soper CTEQ 1995

d
3) We assumed gluon emission was uncorrelated
d dy dp T
2 ln s p T
2
p T
2
exp 2 s
3 ln 2
s
p T
2
This leads to too strong a suppression at P T
=0.
Need to impose momentum conservation for P T
.
A particle can receive finite k T
kicks,
yet still have P T
=0
k T 4
k T 1
k T 2
k T 3
A convenient way to impose transverse momentum conservation
is in impact parameter space (b-space) via the following relation:
2
n
i1
k iT p T
1
2 2 d 2 be i b p T
n
i1
e i bk iT

4) We encounter Non-Perturbative Physics
S b,Q
Q 2 d 2
1 b 2
A
2 s 2 ln Q2
B
2 s 2
as b →∞, α S
(~1/b) →∞. PROBLEM!!!
Solution: Use a Non-Perturbative Sudakov form factor (S NP
) in the
region of large b (small q T
)
b e S b e S b e S NP b
with
b
b
1b 2 2
b max
b*
1
0.8
0.6
b max
=1
Note, as b → ∞, b *
→ b max
.
0.4
0.2
0.5 1 1.5 2 b

A Brief (but incomplete) History of Non-Perturbative Corrections
Original CSS:
S CSS NP b h 1
b, a h 2
b, b h 3
b ln Q 2
J. Collins and D. Soper, Nucl.Phys. B193 381 (1981);
erratum: B213 545 (1983); J. Collins, D. Soper, and G. Sterman, Nucl. Phys. B250 199 (1985).
Davies, Webber, and Stirling (DWS):
S DWS NP b b 2 g 1
g 2
ln b max Q 2
C. Davies and W.J. Stirling, Nucl. Phys. B244 337 (1984);
C. Davies, B. Webber, and W.J. Stirling, Nucl. Phys. B256 413 (1985).
Ladinsky and Yuan (LY):
S LY NP b g 1
b bg 3
ln 100 a b g 2
b 2 ln b max Q
G.A. Ladinsky and C.P. Yuan, Phys. Rev. D50 4239 (1994);
F. Landry, R. Brock, G.A. Ladinsky, and C.P.Yuan, Phys. Rev. D63 013004 (2001).
“BLNY”:
S BLNY NP b b 2 g 1
g 1
g 3
ln 100 a b g 2
ln b max Q
F. Landry, “Inclusion of Tevatron Z Data into Global Non-Perturbative QCD Fitting”, Ph.D. Thesis, Michigan State University, 2001.
F. Landry, R. Brock, P. Nadolsky, and C.P.Yuan, PRD67, 073016 (2003)
“q T
resummation”:
F NP q T 1e aq 2
T
(not in b-space)
R.K. Ellis, Sinisa Veseli, Nucl.Phys. B511 (1998) 649-669
R.K. Ellis, D.A. Ross, S. Veseli, Nucl.Phys. B503 (1997) 309-338
Functional Extrapolation:
J. Qui, X. Zhang, PRD63, 114011 (2001); E. Berger, J. Qiu, PRD67, 034023 (2003)
Analytical Continuation:
A. Kulesza, G. Sterman, W. vogelsang, PRD66, 014011 (2002)

Recap: Where have we been???
1) We now summed the two leading logarithmic singularities, α s
(L 2 +L).
2) We still assumed exponentiation; but sketched ingredients of proof.
The existence of two scales (Q,p T
)≡(Q,q T
) yields 2 logs per loop
Use Renormalization Group + Gauge Invariance
Transformation to b-space
3) Gluon emission was assumed to be uncorrelated.
Impose momentum conservation for P T
. (In b-space)
4) Introduced Non-Perturbative function for small q T
(large b) region.

What do we get for the cross section
d
dydQ 2 dq T
2
1
2 2 0
d 2 b e ibqT W b,Q
e S b ,Q S NP b,Q
with
S b,Q
Q 2 d 2
1 b 2
2 A ln Q2
2 B
where we have resummed the soft gluon contributions
I've left out A LOT of material

Putting it all together: σ TOT
= σ RESUM
+σ PERT
-σ ASYM
Let's expand out the resummed expression:
d
s L
e s L2 L
1 2
2
2
dq T q T q T
s L s 2
L 3 L 2 ...
Compare the above with the perturbative and asymptotic results:
d resum s L s 2 L 3 L 2 00 s 3 L 5 L 4 ...
d pert s L s 2 L 3 L 2 L 1 1 s 3 00
d asym s L s 2 L 3 L 2 00 s 3 00
Note that σ ASYM
removes overlap between σ RESUM
and σ PERT
.
We expect:
σ RESUM
is a good representation for q T
~ 0
σ PERT
is a good representation for q T
~ M W

Putting it all together: σ TOT
= σ RESUM
+σ PERT
-σ ASYM
σ TOT
= σ RESUM
+σ PERT
-σ ASYM
TOT
σ RESUM
for q T
~ 0
σ PERT
for q T
~ M W
cross section dσ/dq T
2
RESUM
PERT
ASYM
transverse momentum q T

Putting it all together: σ TOT
= σ RESUM
+σ PERT
-σ ASYM
σ RESUM
for q T
~ 0
Perturbative contributions
+power corrections
TOT
σ PERT
for q T
~ M W
Perturbative
physics dominates
ÔÔ ´Ï· µ
ÌÉÅ
cross section dσ/dq T
2
RESUM
PERT
ASYM
ÆÓÒÔÖØÙÖØÚ
ÝÒÑ× ´ÒØÖÒ× Ì µ
Extra power of q T
dσ/dq T
1
transverse momentum q T
σ TOT
= σ RESUM
+σ PERT
-σ ASYM

Let's compare with some real results
We'll look at Z data where we can measure both leptons for Z→e + e -
different S NP
(b,Q) functions yield difference at small q T
.

Let's return
to the
measurement
of M W

Transverse Mass Distribution
We can measure dσ/dp T
and look for the Jacobian peak.
However, there is another variable that is relatively insensitive to p T
(W).
Transverse Mass
M T 2 e, p eT p T
2
p eT p T
2
Invariant Mass
M 2 e, p e p
2
p e p
2
In the limit of vanishing longitudinal momentum, M T
~ M.
M T
is invariant under longitudinal boosts.
ν
M T
can also be expressed as:
M T 2 e, 2 p eT p T
1cos e
For small values of P TW
, M T
is invariant to leading order.
Exercise:
a) Verify the above definitions of M T
are ≡.
b) For p Te
= +p*+p TW
/2 and p Tν
= -p*+p TW
/2; verify M T
is invariant to leading order in p TW
.
e
∆φ

Compare P T
and Transverse Mass Distribution
zero p T
P T
(GeV)
M T
distribution is
much less sensitive
to P T
of W
finite p T
Still, we need P T
distribution of W to
extract mass and
width with precision
zero p T
finite p T
PDF and p T
(W)
uncertainties will need
to be controlled:
currently uncertainty:
~10-15 & 5-10 MeV/c 2
Statistical precision in Run II
will be miniscule…placing an
enormous burden on control
of modeling uncertainties.

The Future:
Tevatron Run II
... happening now
LHC
... happening soon

Transverse Mass Distribution and M W
Measurement
Transverse Mass
Distribution from CDF
Combined World
Measurements of M W
Events / GeV/c 2
2000
1500
80.360 +/- 0.370
80.410 +/- 0.180
80.470 +/- 0.089
80.433 +/- 0.079
χ 2 /Nexp = 0.4/4
UA2 (W → eν)
CDF(Run 1A, W → eν,µν)
CDF(Run 1B, W → eν,µν)
CDF combined
1000
80.350 +/- 0.270
80.498 +/- 0.095
80.482 +/- 0.091
D0(Run 1A, W → eν)
D0(Run 1B, W → eν)
D0 combined
80.452 +/- 0.062
Hadron Collider Average
500
80.427 +/- 0.046
LEP II (ee → WW)
Fit region
80.436 +/- 0.037
World Average
0
50 60 70 80 90 100 110 120
Transverse Mass (GeV/c 2 )
79.5 79.7 79.9 80.1 80.3 80.5 80.7 80.9 81.1 81.3 81.5
Mw (GeV)
T.Affolder, et al. [CDF Collaboration], PRD64, 052001 (2001)
Measurement of the W boson mass with the Collider Detector at Fermilab,

¡
Preliminary
Run II
measurements
600
500
DO Run2 Preliminary
a)
600
500
b)
400
300
200
100
0
400
300
200
100
0
Events / 2 GeV
100
80
D0 Run II Preliminary
Number of Entries: 604
Peak Mass: 90.8 +- 0.2 GeV
Width: 3.6 +- 0.2 GeV
500
400
25 30 35 40 45 50 55 60 65 70 75
E T
/GeV
300
200
100
0
30 40 50 60 70 80 90 100 110 120
M T /GeV
c)
800
700
600
500
400
300
200
100
0
25 30 35 40 45 50 55 60 65 70 75
E T
/GeV
0 10 20 30 40 50 60
p T
/GeV
d)
60
40
20
0
0 20 40 60 80 100 120
Di-Electron Mass (GeV)

The W-Mass is an important fundamental quantity
M W
(GeV/c 2 )
80.6
80.5
80.4
80.3
LEP2
D0
CDF
100
250
500
1000
80.2
80.1
Higgs Mass (GeV/c 2 )
LEP1,SLD,νN data
M W
-M top
contours : 68% CL
130 140 150 160 170 180 190 200
M top
(GeV/c 2 )
T.Affolder, et al. [CDF Collaboration], PRD64, 052001 (2001)
Measurement of the W boson mass with the Collider Detector at Fermilab,

Part II: Drell-Yan Process: Where have we been???
Finding the W Boson Mass:
The Jacobian Peak, and the W Boson P T
Multiple Soft Gluon Emissions
Single Hard Gluon Emission
Road map of Resummation
Summing 2 logs per loop: multi-scale problem (Q,q T
)
Correlated Gluon Emission
Non-Perturbative physics at small q T
.
Transverse Mass Distribution:
Improvement over P T
distribution
What can we expect in future?
Tevatron Run II
LHC

Thanks to ...
Jeff Owens
Chip Brock
C.P. Yuan
Pavel Nadolsky
Randy Scalise
Wu-Ki Tung
Steve Kuhlmann
Dave Soper
and my other CTEQ colleagues
and the many web pages where I borrowed my figures
...

References:
Ellis, Webber, Stirling
Barger & Phillips, 2 nd Edition
Rick Field; Perturbative QCD
CTEQ Handbook
CTEQ Pedagogical Page:
CTEQ Lectures:
C.P. Yuan, 2002
Chip Brock, 2001
Jeff Owens, 2000

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