DY PRODUCTION AT SMALL QT
AND
THE COLLINEAR ANOMALY
arXiv:1007.4005 + upcoming with Matthias Neubert
Thomas Becher
University of Bern
Seminar at Padua U., Jan. 19, 2011
1
OUTLINE
• Introduction
• Drell-Yan
process
• Soft-Collinear
• Factorization
Effective Theory
at low transverse momentum qT
• The
collinear anomaly and the definition of transverse
position dependent PDFs
• Resummation
of large log’s, relation to the Collins Soper
Sterman (CSS) formalism
• Expansions
from hell: Non-perturbative short-distance physics
at low qT. Numerical results
2
Entries/5 GeV
25
ATLAS Preliminary
20
! L = 229 nb
-1
Data 2010 ( s=7 TeV)
Z " µµ
15
10
5
0
0
20
40
60
80
100 120
pZ [GeV]
T
The production of a lepton pair with large invariant mass is the
most basic hard-scattering process at a hadron collider.
ATLAS has 45 pb-1 of data: ~ 1.5 x107 W’s and 106 Z’s !
This year ~ 1fb-1 should be accumulated.
3
7
1/!Z " d!Z/dpT (1/GeV)
due to the smooth10-1
condition imposed
DZero
rections, statistical
tion in each pZ
10-2
T bin
atistical uncertainspread in each bin
10-3
ations. As a result,
a in any fits as this
o an artificially low
10-4
ntral values of the
D%, L=1 fb-1
plicative correction
-5
10
+ ed to compare this
Z/
$
*
#
e
e
µ
factor labeled pT
> 15 GeV require10-6
Z/$ *# µ+µ- + corrections
for QED FSR; the
e measured lepton
-7
10
d finally the factor
102
1
10
d mass window to
pZT (GeV)
0 electron channel
plying only the pµT
FIG. 2: Measurements of the normalized differential cross
finition as previous
section in bins of pZ
T for the dielectron [13] and dimuon chanZ
nlike pT , the varinels. Both results are shown with combined statistical and
• DZero, arXiv:1006.0618,
0712.0803
systematic uncertainties.
rements had miniuirement, so a coractors are derived
4
3], as described in
“NNLO”), but a NLO differential distribution. The pre-
DRELL-YAN PROCESSES
The production of a single electroweak boson γ*, Z,W±, H is of
great interest for
•W
mass and width measurements,
• PDF
determinations, theory benchmarks
• Higgs
discovery
Low transverse momentum qT is particularly relevant
• to
extract of W mass
• to
reduce background for Higgs search
5
PERTURBATIVE EXPANSION
The perturbative expansion of the qT spectrum contains singular
terms of the form (M is the invariant mass of the lepton pair)
2
dσ
1 ! (1)
M2
M
(1)
(2) 2
3
=
A
α
ln
+
α
A
+
A
α
ln
+ ...
s
s 0
1
3
s
2
2
2
2
dqT
qT
qT
qT
"
2
M
(n)
+A2n−1 αsn ln2n−1 2 + . . . + . . .
qT
which ruin the perturbative 2expansion† atµ qT ≪ M and must be
ψ̄γµ ψ → CV (M , µ) χ̄hc Sn̄ γ Sn χhc
resummed to all orders.
coll. soft coll.
Classic example of an observable which needs resummation!
2 Achieved
= (p − p̄)2 by
JVµ Collins, Soper and Sterman (CSS) ’84.
6
RESUMMATION
A formula which allows for the resummation of the
logarithmically enhanced terms at small qT to arbitrary precision
was first obtained by Collins, Soper and Sterman (CSS) in ‘84,
based on earlier work of Collins and Soper.
Their is based on a factorization theorem for the cross section
at low qT, which Collins and Soper derived using diagrammatic
techniques.
We will instead use soft-collinear effective theory to analyze the
cross section.
7
SOFT-COLLINEAR EFFECTIVE THEORY
Bauer, Pirjol, Stewart et al. 2001, 2002; Beneke et al. 2002; ...
In collider processes, we have an interplay of three
momentum regions
•
Hard
} high-energy
•
Collinear
•
Soft
} low-energy part
Correspondingly, EFT for such processes has two lowenergy modes:
•
Collinear fields describing the energetic partons propagating in
each direction of large energy, and
•
soft fields which mediate long range interactions among them.
8
SOFT-COLLINEAR FACTORIZATION
Q
M1
M2
For M1 ~ M2 ≪Q the cross section factorizes:
hard function
H(Q2 , µ)
J(M12 , µ)
J(M22 , µ)
jet function
S(Λ2s , µ)
soft function
9
Λ2s
M12 M22
∼
Q2
!
DIAGRAMMATIC FACTORIZATION
The simple structure of soft and collinear
emissions forms the basis of the classic
factorization proofs, which were obtained by
analyzing Feynman diagrams.
J.C. Collins, D.E. Soper / Back-to-back lets
/.
I
Collins, Soper, Sterman 80’s ...
Advantages of the the SCET approach:
•
Simpler to exploit gauge invariance on
the Lagrangian level
•
Operator definitions for the soft and
collinear contributions
•
•
_J
I
Fig. 7.2. Dominant integration region for e+e annihilation for small wr. In both fig. 7.1 and
Collins and Soper ‘81
the soft gluon subgraphs may be disconnected.
Resummation with renormalization group
Can include power corrections
10
We begin by considering the slightly simpler process a + b ~ A + B + X,
and b are quarks with momenta k~, and k~ respectively. Let k~, be coll
defined in subsect. 4.2) in the v~ direction and let k~ be collinear in
direction. Then the dominant integration regions are as shown in fig. 7.3.
Consider a graph G for this process. A subgraph T of G will be calle
SCET APPLICATIONS
• Charmless
B decays, B→ππ, B→ Xulν, B→ Xs γ
• Threshold
resummation / soft gluon resummation
• DIS
at large x, Drell-Yan rapidity spectrum, inclusive Higgs
production, top production, direct photon production, single
top production, e+e− event shapes, ...
• Towards
higher-log resummation n-jet processes
• IR
singularities of multi-loop amplitudes
• Jet
definitions, beam jets (initial state showering)
11
FACTORIZATION
12
qT0 + A3
=π 2 2 A1 αs"ln 2 + A
2
= 1.78 GeV for M
= MZ
(28)
= q∗ = M exp
−
dq
q
q
M
T
T
T
(n)
2n−1
n# 2C
F αs (q∗ ) $
+A2n−1αs ln π 2 + . . . + . . .
−1
= 1.78 GeV for M =2MZ
(28)
$x % = q∗ = M exp −
dσ
†
2
q
µ
µ = max (q∗ , TqT )
2CF αs (q∗ )Tψ̄γµ ψ
M
CnV (M
, µ) χ̄hc S
γ=
(n)→
2n−1
!
2
2 n̄ 2
dσ
1
M αs (1)
dq
+
. .T. ...
+A
ln
(1)
(2) 2
3 M
2n−1
p
2
T1 αs ln
=
A
+
A
+
A
α
ln
+
0
3
sq
2
2
2 (29)
λq 2=
µ =•max
(q∗ , qT ) point is the factorization
dq
q
q
Starting
of
the
electroweak
current
in
T
T
T
T
T
† µ
CV , anti-coll.
soft
coll.
2 M
ψ̄γµlimit
ψ → CV (M , µ)
n2 χhc "
the Sudakov
pT χ̄hc Sn̄ γ S
λ =+A(n) αn ln2n−1 M + . . . + . . .
(29)
M
µ2n−1 s
2
†
p
2
hard-collinear
q
2
2
µ
i-coll.p soft
coll.
T
n
∼
ψ̄γ ψ →hcCV (M , µ) χ̄hc Sn̄ γ
p̄ M = (p − p̄)
p0 µ
pµψ → C (M 2 , µ) χ̄ S † γ2µ S χ
!
2 nµ ψ̄γ
µ
V
n hc
n̄
hc M
dσ CV , 1anti-coll.
Mcoll. ∼(1)
µ
soft
(2)
2 (1)
0
3
2
µ
p
CV , anti-coll.
soft coll
= (p=− p̄)C A
J, Vanti-coll.
µA p̄ + soft
α
ln
+
A
α
ln
+
.
.
.
s
1
0
3
s
n̄
∼
soft2coll.
p̄0
dqT2
qT2 V
qT
qT2
22
hc
p̄µ
2
µ
p p̄ M p2 p̄=M (p
−
p̄)
= (p − p̄)
n̄ ∼ 20
"
anti-hard-collinear
(n)
2n−1 M p̄
n
+ ... + ...
+A2n−1 αs ln
2
qT (SCET) this can be written in
• In Soft-Collinear Effective Theory
L⊥
$x−1
T %
SOFT-COLLINEAR FACTORIZATION
operator form as
t coll.
ψ̄γµ ψ → CV (M
2
†
, µ) χ̄hc Sn̄
4 13
γ µ Sn χhc
SCET hc quark field
† µ dq 2 (3) q
ψ̄γµ ψ → CV (M 2 , µ) χ̄hc Sn̄† γ µ 2Sn χhc
, anti-coll. 2 soft coll.2 µ ψ̄γµ ψ → CV (M , µ) χ̄hc Sn̄ γ Sn χThc
p p̄ M = soft
(p −coll.
p̄) JV
CV , anti-coll.
C , anti-coll. soft coll.
V 2 = (p − p̄)2 J µ
p̄The
M
Drell-Yan
cross
2
2 V µ section is obtained from the matrix
p̄ M = (p − p̄) JV
element of two currents
µ
p
p̄ M = (p µ†− p̄)ν JV
1
(−gµν ) #N1(p) N2(p̄)| JV (x) JV (0) |N1(p) N2(p̄)$ →
|CV (M 2 , µ)|2
2Nc
n̄
/n , anti-coll.
/̄n
n
n̄
C
× ŴDY (x) #N1 (p)| χ̄hc(x) χhc (0) |N1(p)$ #N2 (p̄)| χ̄hc (0) Vχhc (x) |N2(p̄)$
2
2
2
2
soft
1
µ†
ν
2
2 2
1
(x)
J
(0)
|N
(p)
N
(p̄)$
→
)
#N
(p)
N
(p̄)|
J
|C
(M
,
µ)|
1
2
1
2
V
2
2
Vν
V µ†
gν µν ) #N
1 (p) N2(p̄)| JV (x) JV (0) |N1(p) N2 (p̄)$ →2Nc |CV (M , µ)|
2Nc
n and n̄ are light-cone reference vectors along p and p̄ .M = (p −
/n
/̄
n
/̄nχhc (0)a |N
(0) /nχhc
(x) |N
× ŴThe
#Nfunction
χ̄hcWilson
1 (p)$ #Nof
2(p̄)$
DY (x)
1 (p)| χ̄hc (x)
2 (p̄)|
soft
contains
product
four
along
(x)
|N
× ŴDY (x) #N1 (p)| χ̄hc(x)2 χhc (0) |N1(p)$ #N2 (p̄)| χ̄hc (0) 2 χlines
2(p̄)$
hc
2
1
the directions of large2energy
flow
µ†
ν
# #(x)
$$2 (p̄)$ →
J
(0)
|N
(p)
N
(−gµν ) #N1(p) N21(p̄)|
J
|CV
1
†
V
†
V
1
ŴŴ
== #0|
TrTrSS
(x)
SSn̄(x)
SSn̄†(0)
SSn(0)
|0$
(4)
n
n̄
†
2N
DY (x)
n
c
(x)
#0|
(x)
(x)
(0)
(0)
|0$
(4)
DY
n
n̄
n̄
n
NN
c
c
/̄n
%
'
&
0
× ŴDY (x) #N1 (p)| χ̄hc(x) χhc (0) |N1(p)$ #N2 (p̄)| χ̄hc (0)
Sn (x) = P exp i
ds 2
n · As (x + sn)
(5)
(−gµν ) #N1(p) N2(p̄)
−∞
14
× Ŵ
(x) #N (p
hc2
DY
1
hc
n
/
/̄
n
π
µ
2
n
n̄
χ
(0)
|N
(p)$
(0)
χ
(x)
|N
(p̄)$
(x)
#N
(p)|
χ̄
(x)
#N
(p̄)|
χ̄
hc
1
2
DY
1
hc
2
hc
2
2 hc
# †
1
qT1
# †
$ŴDY (x) =
#0|
Tr
S
(x
†
n
µ†
ν
ŴDY (x) =
S)n̄(x)
n(0)J |0$(x) J (0) |N
n̄ (0)
(p) N2(p̄)$ →
#N1S(p)
N2S(p̄)|
n (x)µν
N1c(4)
V
V
LN⊥c #0| Tr S(−g
% & 0
%
'
&
n
/̄
$
#
0
Final step is to expand the
matrix
elements
in
small
momentum
χ
(0)
|N1(p)$ #N2 (p
×
Ŵ
(x)
#N
(p)|
χ̄
(x)
hc
DY
1
hc
π
−1
S
(x)
=
P
exp
i
ds
n
·
A
(x
+
sn)
(5)
S−n 2(x) = P exp =i 1.75
n
s q = M exp
components,
i.e. to perform
expansion.
$x %a=derivative
DERIVATIVE EXPANSION
−∞ T
n̄ · k, k⊥)
The
∗
2CF #αs (q∗ )
−∞
1
†
†
Ŵ
(x)
=
#0|
Tr
S
(x)
S
(x)
S
n̄
· k, k⊥) Nscale as n
n̄ (
light-cone components (n · k, n̄DY
c
µ =2 max (q∗ , qT )
% & (6)
expansion
parameter
phc ∼ M (λ , 1, λ) ,
phc ∼ M (1, λ2, λ) .
0
Sn (x) = P exp i qTds n · As (x
ps ∼ M (λ2, λ2 , λ2) .
(n · k, n̄ · k, k⊥)
λ =−∞
M
(7)
while the separation between the two currents scales as
x ∼ M −1 (1, 1, λ−1)
µ
p
nµ ∼ 0
p
Aµs (x) = Aµs (0) + x · ∂Aµsp(0)
+M
. . . (λ2 , 1, λ) ,
hc ∼
1
φq/N (z, µ) =
2π
!
power suppressed, can be dropped
phc ∼ M (
µ
p̄
µ
/̄n
2
2
2
n̄
∼
−iztn̄·p
p
∼
M
(λ
,
λ
,
λ
).
0
s
dt e
!N(p)|
χ̄(tn̄) χ(0) |N(p)"
15
p̄
2
1
(n · k,
n̄ M
· k,−1k(1,
⊥ )1, λ−1 )
x∼
x∼M
−1
FACTORIZATION
(1, 1,NAIVE
λ )
−1
Dropping
phc ∼ M (λ2 , 1, λ) ,
2ps
phc ∼ M (1, λ2, λ) ,
∼ M (λ2 , λ2 , λ2 ) .
2
p
∼
M
(λ
,
1,
λ)
,
p
∼
M
(1,
λ
, λ)to
,
hc
hc
power suppressed x-dependence
leads
ŴDY (0) #N1 (p)| χ̄hc (x+ + x⊥ )
the result
(6)
(7)
/̄ p ∼ M (λ2 , λ2 , λ2 ) .
n
/n
χhcs (0) |N1 (p)$ #N2(p̄)| χ̄hc (0) χhc (x− + x⊥ ) |N2(p̄)$
2
2
}
1
ŴDY (0)
1
x “transverse/̄n PDF”
x =“transverse
/nPDF”
(8)
χhc (0) |N1 (p)$ #N2(p̄)| χ̄hc (0) χhc (x− + x⊥ ) |N2(p̄)$
this
2/̄n spells trouble: well known /nthat2 transverse
KLN Ŵ
cancellation!
) χnot
#N2(p̄)|
χ̄hc
|N1(p)$
(0) χhc (x
DY (x0 ) #N1 (p)| χ̄hc (x+
hc (0)
− ) |N2 (p̄)$
PDF
well
defined
w/o
additional
regulators
2
2
ŴDY (0) = 1
ŴDY (0) #N1 (p)| χ̄hc (x+ + x⊥ )
For comparison: for soft-gluon resummation,
the result is
1
/̄
n
/n
ŴDY (x0 ) #N1 (p)| χ̄hc(x+ ) χhc (0) |N1(p)$ #N2(p̄)| χ̄hc (0) χhc (x− ) |N2 (p̄)$
2
2
“soft” x “standard PDF”
16
1
x “standard PDF”
c
T
$ Aµ (x) = Aµ (0) + x · ∂Aµ (0) + . . .
%
& 2 '
s
s
s
qT
2
2
2
,
×
eq Bq/N1 (ξ1 , xT , µ) Bq̄/N2 (ξ2 , xT , µ) + (q ↔ q̄) + O
2
M
q
!µ
Aµs (0) + x · ∂Aµs (0) /̄
n+ . . .
1 As (x) =−iztn̄·p
dt e
!N(p)| χ̄(tn̄) χ(0) |N(p)"
φq/N (z, µ) =
In
terms
of
the
hadronic
matrix
! elements 2& 2 2 '−Fqq̄ (x2T ,µ)
2π
!
3
2 "
"2 1
dσ
4πα "
xT M
n
21
2
−iq⊥ ·x⊥ /̄
!
"
−iztn̄·p
(−M
d !N(p)|
x⊥ e χ̄(tn̄) χ(0)
=
e
φ22q/N C
(z,V µ)
= , µ) dt
1
/̄n
2 dq 2 dy
−2γ|N(p)"
E
−iztn̄·p
3N
M
s
4π
4e
2πdt e
Bq/N (z,c xT , µ) =
!N(p)| χ̄(tn̄ + x2⊥ ) χ(0)
|N(p)"
T
2π
2 %
!
& 2 '
# $2
1
n
/̄
qT
2 dt e−iztn̄·p !N(p)|
2 χ̄(tn̄ + x⊥ )
2 x , µ) =
χ(0)
|N(p)"
T
,
B
(ξ
,
x
,
µ)
B
(ξ
,
x
,
µ)
+
(q
↔
q̄)
+
O
× Bq/N e(z,
q/N1 12π T
q̄/N2 2
T
q
2
2
on then obtains
the
DY
cross
section
M
q
#
CROSS SECTION
!
"
d3 σ
1
4πα2 ""
2
2
−iq⊥ ·x⊥
"2
C
(−M
,
µ)
=
d
x
e
V
⊥
2
dM 2 dqT√
dy y 3Nc M 2 s √ −y
4π
M 2 + qT2
m2⊥
=
.& '
ξ1 = τ e ,
ξ2 $= τ e ,
with τ =
%
#
s
s
qT2
2
2
2
,
×
eq Bq/N1 (ξ1 , xT , µ) Bq̄/N2 (ξ2 , xT , µ) + (q ↔ q̄) + O
2
M
2
2
2
√
√
q
ξ1 =
with
y
τe ,
ξ2 =
−y
τe
,
M + qT
m⊥
=
.
with τ =
s
s
& 2 2 '−Fqq̄ (x2T ,µ)
!
2
"
"
dσ √
4πα " √ 2 "2 1
xTM
M2 + qT2
2
−iq⊥m
·x⊥
y
−y
⊥
CV (−M
, µ)
d x⊥τe =
=
ξ
=
=4e−2γE
.
τ
e
,
ξ
=
τ
e
,
with
2
2
2
1
2
dM dqT dy
3Nc M s
4π
s
s
%
& 2 '
$
#
qT
2
2
2
,
×
eq Bq/N1 (ξ1 , xT , µ) Bq̄/N2 (ξ2 , xT , µ) + (q ↔ q̄) + O
2
M
q
3
(1
(1(
(1
(1
(1
2
17
(1
(1
M 2 dqT2 dy
=
CV (−M 1, µ)
d x⊥ e
/̄n −2γ
2s
−iztn̄·p
3Nc M
4π
dt e
!N(p)| χ̄(tn̄) 4eχ(0)E |N(p)"
φq/N (z, µ) =
2π
2
%
& 2 '
$
#
!
qT
2
2
2
1
n
/̄
,
(ξ2 , xT , µ) + (q ↔ q̄) + O
×
e B 1 (ξ1 , xT , µ) B−iztn̄·p
2
Bq/N (z,q x2T ,q/N
µ) =
dt e q̄/N2 !N(p)|
χ̄(tn̄ + x⊥ ) χ(0) |N(p)"
M
q
(11)
(
CROSS
SECTION2
2π
!
"
2
2
2
d3 σ √
1
4πα2 "" √
2 µ "2
2
−iq
·x⊥
M
+
q
m
⊥⊥
µ
µ
T
y
−y
CV (−M
, ,µ)
= ,
e+ x · ∂A
(x)4πwith
= Ads x(0)
= s (0) + . . .
(12)
ξs2 =
τe A
τ⊥ =
2
s
dM ξ2 1dq=T2 dyτ e 3N
M
c
s
s
%
& 2 '
(1
# $
!
qT
2
2
2
2 m2
√ y×
√ 1 (ξ−y
M
+
q
n
/̄
1
,
,
x
,
µ)
B
(ξ
,
x
,
µ)
+
(q
↔
q̄)
+
O
e2q Bq/N
1
q̄/N2 −iztn̄·p
2
T
⊥
T
T
2
=
.
ξ1 = τ e , q φq/N
ξ2 =
τ
e
,
with
τ
=
dt e
!N(p)| χ̄(tn̄) χ(0)M|N(p)"(13)
(z, µ) =
s
2π
s
2
!
√ y
√
The ξresummation
would
then
M2 + '
qT2 the /̄
m2⊥by& solving
nRG
−y 1 be!obtained
2
−iztn̄·p
= xχ̄(tn̄
(14)
τe B
, 4πα
=T , µ)
τ e= , " 1with
dt e τ = !N(p)|
x. ⊥(x) ,µ)χ(0) |N(p
13 =
2ξ2 "x
2 +−F
2
q/N (z,
d
σ
M
2
equation =
T s
2π"
2
"CV (−M 2 , µ)
d2 x⊥ e−iq s·x
q q̄
dM 2
dqT2
⊥
M 2s
⊥
4e−2γE
%
2
d
−M
(1
& 2 '
$
2
F
q !
2 %
#
C
(M
,
µ)
=
Γ
(α
)
ln
+
2γ
(α
)
C
(M
,
µ)
.
(15)
3 V
2
V
qT
"
2
2 s
2 4παcusp" s 2
1
2
d ln µd σ ×
µ
2
2
−iq
·x
,
eq Bq/N1 (ξ"1 , xT , µ) Bq̄/N2 (ξ
⊥ O
" 2 , xT , µ) + (q ↔ q̄)⊥ +
dy
3Nc
4π
$
CV (−M , µ)
=
2
2
q
dqSCET
3NcGao,
M sLi, Liu 2005; Idilbi, Ji, Yuan
4π
T dy papers:
see
dM 2
where
2
T
d x⊥ e
M2
2005; Mantry, Petriello 2009
$
%
2 √
2
2
2
2
2
√ correct!
M
+
q
m
B
(ξ
,
x
,
µ)
B
(ξ
,
x
,
µ)
+
(q
×
e
be
If
is
q/N
1
q̄/N
2
T ↔ q̄) + O
y
−y
⊥ T
1
2
T
q
This cannot
=
.
ξ1 = τ e ,
ξ2 = τ e ,
with τ =
q
s
s
independent of M, the above cross section is μ dependent!
#
3
dσ
4πα
2
18
"
"
2
"2 1
"
!
2
−iq⊥ ·x⊥
&
x2T M 2
(1
'−Fqq̄
T
2
3 c
Bdq/N
µ) 2= ""
σ (z, x$T ,4πα
−iztn̄·p
"2 1 !N
dt
e
χ̄(tn̄
+ x⊥&
) χ(0)
|N(10)
(p)"
2
2 (p)|
−iq
⊥ ·x⊥ %
'
"
CV (−M , µ)
=
d x⊥ e
2q 2
2
2 s 2π
dM 2 dq#
dy
3N
M
4π
T
,
(ξ , x2 , µ) + (q ↔ q̄) + O
× T e2 B c (ξ , x2 , µ) B
COLLINEAR ANOMALY
q/N1
q̄/N2 2
T
T
%M 2 & 2 '
$
#
! (ξ , x2 , µ) + (q ↔ q̄) + O qT ,
2 "e2 Bq/N (ξ1 , x2 ,"µ) Bq̄/N
×
2
1
4πα " q
2 T "2 1 2 2 T −iq⊥ ·x⊥
M2
C (−M , µ)
dx e
q
q
q
d3 σ
=
2
2
M dqT dy
3Nc M 2 s
1
V
4π
(10)
⊥
& 2 2 '−Fqq̄ (x2T ,µ)
!
"
"
σ
4πα "
xT M
2 1
$
%2 of & 2 '
2
2
−iq⊥ ·x⊥
"
RG
invariance
of
cross
section
implies
that
the
product
C
(−M
,
µ)
d
x
e
=
&
'
!
#
V
⊥
−2γE
3
2
2 −Fqq̄ (xT ,µ)
qT
"
dqT2 dy d3N
4π
4e
σ c M 2 s 4πα
1
x
M
22 "
2
2
2
T
2
2
−iq
·x
⊥
⊥
"
"
,
+
(q&↔contain
q̄)
+O
× =PDFs
eq 2Bq/N
transverse
must
CV (−M
d x2 ⊥(ξe2 , xT , µ)
1 (ξ1 , ,xµ)
T , µ) Bq̄/N
(11)
2
2
2
−2γ
'
%
$
E
M
dM dq#
4π
4e
2
T dy 2 q3Nc M s
q
T
2
2
hidden
(11)
Bq/N
× Me2q dependence.
#1 (ξ1$, xT , µ) Bq̄/N2 (ξ2 , xT , µ) + (q ↔ q̄) + O %M 2 &, q 2 '
T
2
2
2
q
3
2
×
eq Bq/N1 (ξ1 , xT , µ) Bq̄/N2 (ξ2 , xT , µ) + (q ↔ q̄) + O
M2
,
q
'−F (x ,µ)
Analyzing
the
relevant
diagrams,
!one finds that&an2 additional
3
2 "
2
"
dσ
4πα " √
2 1
2·x
2 xT M
2
2m2 −iq
"
√
M
+
q
regulator
is
needed
to
make
transverse
PDFs
well
In
where2
C
(−M
,
µ)
d
x
=
T 2 defined.
y
−y
⊥e
V
⊥
2
2
2
2
−2γ
= m⊥ M4e
. + qT
(12)
ξ1 dy
= τ e3N
, cM
τ=
√ξ2s y= τ e , √ with
M dq
4π
−y
T
s
s
the product
regulator
can
but
= be removed,
.
(12)
, two
ξ2 =PDFs,
τ e ,this with
τ=
ξ1 = ofτ ethe
s
s
'
%
&
$
2
2
#
anomalous M 2dependence2 remains. Can2 refactorize
qT
q q̄
⊥
2
T
⊥
E
µ)qq̄ (x
B2Tq̄/N
eq Bq/N
&1 (ξ21 ,2xT',−F
,µ) 2 (ξ2 , xT , µ) + (q ↔ q̄) + O
×
2
& 2 2 '−Fqq̄ (x2T ,µ) 2
x
q
M
T
2
)
xT M Bq/N1 (z1 , xT , µ) Bq̄/N
2 2 (z2 , xT , µ) , 2
−2γ
E
Bq/N1 (z1 , xT , µ) Bq̄/N2 (z2 , xT , µ)4eM 2 =
B
(z
,
x
q/N
1
1
T , µ) Bq̄/N2 (z2 , xT , µ) ,
4e−2γE
)
2
2
(
q
z1 , xT , µ) Bq̄/N22(z2 , xT , µ) q2 =2
ξ1 =
2
2
2
2
2
dF
(x
,
µ)
√
q
q̄
M
+
q
m
T
F
dF
(x
,
µ)
q=
q̄ 2Γ
T
⊥
T−y (αs ) F
(13)
with
=
2Γ
(α
)
cusp
=
. (13)
τ ey ,
ξ
=
τ
e
,
with
τ
=
s
2
cusp
d ln µ
d ln µ
s
s
√
Note that√M2y dependence
$
%
$√ exponentiates!
2
−y 2
2
2
% m2⊥
M
+
q
T
d ξ1 = dτ2 e ,
−M
=2 , µ) .
. (14)
e
,
τ(M
= 2 , µ)
2F ξ2 =
Fτ −M
, µ)(α
=s )Γln
(M
CV (M , µ)CV=(MΓcusp
+ 2γ q2with
(α+
CqV(α
.
(14)
s) C
V
s ) 2γ
cusp (α2s ) ln
s
s
19 µ
d ln µ
d ln µ
µ
,
Regular soft-collinear factorization:
H(Q2 , µ)
J(M12 , µ)
J(M22 , µ)
S(Λ2s , µ)
Anomalous factorization
H(Q2 , µ)
Bq̄/N2 (x⊥ , µ)
Bq/N1 (x⊥ , µ)
F (x⊥ , µ) ln(x2⊥ Q2 )
20
TRANSVERSE PDFS
What God has joined together, let no man separate...
The “operator definition of TMD PDFs is quite problematic [...]
and is nowadays under active investigation”.
quote from Cherednikov and Stefanis ’09. For reviews, see Collins ’03, ’08
Regularization of the individual transverse PDFs is delicate, but
the product is well defined, and has specific dependence on the
hard momentum transfer M2.
Anomaly: Classically, !N1 (p)| χ̄hc(x+ + x⊥ )/̄n χhc(0) |N1 (p)" is invariant
under a rescaling of the momentum of the other nucleon N2.
Quantum theory needsAµsregularization.
be
(x) = Aµs (0) + xSymmetry
· ∂Aµs (0) + .cannot
..
recovered after removing regulator.
Not an
/̄n
1
−iztn̄·p
dt
e
!N
(p)|
χ̄(tn̄)
χ(0) |N (p)"
φ
(z,
µ)
=
q/N of QCD, but of the low energy theory.
anomaly
2π
2
!
1
/̄n
2
−iztn̄·p
Bq/N (z, xT , µ) =
dt e21
!N (p)| χ̄(tn̄ + x⊥ ) χ(0) |N (p)"
2π
2
!
RESUMMATION
22
qT ≤QT
1
q
2
i=q,g j=q̄,g
q←q
s
z1 z2 ≥M 2 /s
q←g
s
3ow perform
Calculation
of
the
kernels
I
a perturbative
calculation
of the
relevantIq←q
kernels and
Ii←j entering
the facto
q←g
&
'
min(Q
,
z
z
s−M
)
$
−Fqq̄ (xT ,µ)
2 2
#
2
(
)
x
q
% M
2 2 & operat
T
2 at first non-trivial
2
2
n formula
(22)
order
in
α
.
Since
we
do
not
have
explicit
Bq/N1 (z1 , xT , µ) Bq̄/N2 (z2 , xT , µ) q2 = 2 −2γ s
B2q/N1 (z
,
x
,
µ)
B
(z
q̄/NQCD
2 , xT , µ) ,
21 T
2
E
T
× atransverse
dqcalculation
Cqq̄→ij
(z1 ,ofz2the
, qT ,relevant
µ) ffkernels
µtheenteri
+
(q, i
We
now
perform
perturbative
I,i←j
ij instead
tions
of the
(good)
distribution
functions
B
,M
we ,analyze
origin
T 4e
2
T
2
SIMPLIFICATION FOR q ' Λ
1 2
2
i/N
.
z1 z2 s
zation
formula
(22) at
order in αs . products
Since weofdo
exp
functions
Bi/N defined
in 0first
(11), non-trivial
keeping
twonot
suchhave
function
dFqq̄ (x2T , in
µ) mindF that only
= If
2Γwe
(α
(13)
s)
ingFor
to different
are
wellofddefined.
write
an operator-product
expansio
cuspperform
definitions
of thehadrons
(good)values
transverse
functions
, we analyze
instea
perturbative
qTdistribution
anBi/N
operator
ln
µ we can
gous
to
(19) expansion
bad)
functions
Bi/N defined
in (11),
keeping in mind
that
only products
of two s
product
Calculation
of the
kernels
I
and
I
q←q%
q←g
$are well defined.
#
eferring to different
hadrons
If
we
write
an
operator-prod
2
d
−M
"2 1 dz F
q
2
C=V (M , µ) = Γ
+ 2γ
(αsµ)
) C
, 2µ) . x2 ) ,
(14) (2
2
2
s )xln
V (M
cusp (α
2
analogous
to
(19)
B
(ξ,
x
,
µ)
I
(z,
,
µ)
φ
(ξ/z,
+
O(Λ
i←j
i/N
j/N
d
ln
µ
µ
e now performTa perturbativez calculation
of the relevant kernels
T
QCD I
Ti←j entering
ξ
j
tion formula (22) at first
non-trivial
in αs . Since we do not have(15)
expli
" # 1qT2dz'order
ΛQCD
2 transverse distribution functions
2
2instead
2
finitions
of
the
(good)
B
,
we
analyze
i/N
B
(ξ,
x
,
µ)
=
I
(z,
x
,
µ)
φ
(ξ/z,
µ)
+
O(Λ
x
owsAgain,
that
the
products
of
two
I
functions
are
well
defined
and
obey
a
factorizatio
only
the
product
of
two
functions
is
well
i←j
i/N
j/N
i←j
T
QCD T )
Text p, (p̄) to the power
α → 0, β → 0 z p → λp T
ξ keeping in mind that only products of two su
ad)defined.
functionstoBi/N
ula
analogous
(13).defined in (11),
j
erring to different hadrons are well
we write an operator-produc
'−F (x If
& 2defined.
,µ)
2
(
) two IxT q functions are well
2 the products
2
2
2
talogous
follows
that
of
defined
and
obey a
i←j
to
(19)
=
I
(z
,
x
,
µ)
I
(z
,
x
,
µ)
I
(z
,
x
,
µ)
I
(z
,
x
,
µ)
,
q←i
1
q̄←j
2
q←i
1
q̄←j
2
T
T
T
T
q
One-loop results
4e−2γ
ormula analogous to (13).
# 1
"
rbative expansions for
the kernels Ii←j
from a matching calculation,
dz can be derived
2
2
2
2
B
(ξ,
x
,
µ)
=
I
(z,
x
,
µ)
φ
(ξ/z,
µ)
+
O(Λ
x
Effective
theory
diagrams
for
are
not
well-defined
i←j
i/N
j/N
T using external parton states
QCDcarryin
T),
h the matrix elementsTin (10) and (11) zare evaluated
2
ξ
3.1 One-loop results
j
q q̄
2
2
T
E
2
in dim.ofreg..
Following
Smirnov
’83,tree-level
we use
additional
analytical
d fraction
the nucleon
momentum
p. The
result
is obviously
given by
regularization,
whichforisofthe
very
economical,
since
it does
notand
Perturbative
expansions
I
can
be
derived
from
a matching
i←j
2twokernels
follows that the
products
I
functions
are
well
defined
obey a(2fc
Ii←j (z, xT , µ) =i←j
δ(1 − z) δij + O(αs ) .
introduce
additional
scales into the problem.
which
matrix
elements
mula the
analogous
to
(13). in (10) and (11) are evaluated using external parton s
aelevant
fixed fraction
the nucleon
p.s )The
tree-level
result
is obviously
g
one-loop of
diagrams
giving momentum
rise to the23O(α
corrections
to the
kernels
Iq←q = Iq̄←
hown in the first row of Figure 1. There is no need to consider diagrams with external-le
2 ,µ)
'
&
−F
(x
2
q
q̄
2
2
T
gure 1: One-loop diagrams
contributing
to
the, µ)
matching
coefficients Iq←q (top row) and
)
dF
(x
x
q
q
q̄
T
F
T
2
2
=
2Γ
(α
)
(bottom
row).
The
vertical
lines
indicate
cut
propagators.
s
=
B
(z
,
x
,
µ)
B
(z
,
x
,
µ)
B
(z
←gq̄/N2
cusp
2
1
2,
q/N
q̄/N
T
1
T
2
q2
−2γ
4e E d ln µ
ANALYTICAL REGULARIZATION
fers to hc
the running coupling
evaluated
at the scale µ, unless
indicated otherwise.
Th
hc
hc
hc
hc
hc
d
2 d = 4$− 2!, and we omit O(!) terms.
hc Moreover,
%
mensional regulator isdF
defined
by
µ
denote
(x
,
µ)
2
q q̄ T
C
F
d scaleαdefined
−M
α
e renormalization
in
the
MS
scheme.
The
remaining
two
diagrams
turn
ou
2
F
q
2
=
2Γ
(α
)
µ ,
s +
β 2γ (α ) Cd ln
cusp
C
(M
,
µ)
=
Γ
(α
)
ln
+
(M
→
β
V
s
s
V
α
d lnregularization
µ β cusp
be ill-defined
in µ
dimensional
due to light-cone
2singularities. To give meanin
d ln
µ
hc
the corresponding
loop
integrals
requires
introducinghcadditional
regulators. The simples
hc
hc
hc
hc
hc
ossibility is to employ analytic regularization, as
is
common
in
the
context
of
asymptoti
2
$
%used
q
Λhas
Figure
2:
Matching
of
an
analytically-regularized
QCD
graph
onto
SCET
pansions [19, 20]. In the context of SCET this method
been
in diagrams.
[15, 44]. One start
QCD
T2 '
Text
p, (p̄) to
to the p
•
−M to
Raise 2QCD
propagators
carrying
large
F
q
reconsidering
the QCD
diagrams
contributing
the momentum
process
and raises 2all propagator
Cfractional
,the
µ)external
=
Γhard-collinear
(α
lnlimit
2γ
(αtoαs→
)a fractional
(M
, µ)
.
Vof(M
sin)the
cusp
diagrams
develop
singularities
β2→ 0 +
followed
by
0C
or Vvice
versa,
which
trough
p,
(p̄)
to
power
α,
(β)
powers
:
which
the
momentum
p
flows
power,
µ
µ
x
x
x
x
x
x
x
x
cancel in the sum of the results from both sectors.
2α diagrams in Figure 1 can now be
With the analytic regulators
1 in place, the remainingνtwo
1
→
computed and both give the same
result.
For
their
sum,
we obtain
2
1+α ,
2
−(p − !
k)2 "
−−αiε! QCD
"[−(p − k) − iε]
! 2 2 "!+α
T
2
2 −β
µ
q
2z
xT µ
C
α
Γ(−%
−
α)
F
s
b+c
e!γE
Iq←q
(z, x2T , µ) =
. (32)
2π
ν12
ν22
(1 − z)1−α+β
Γ(1 + α)
4
similarly for the anti-hard-collinear
propagators,
but with
a different
regulator
q 'Λ
(30
d
β and
•power
Limit
is
trivial
for
QCD,
but
effective
theory
he
α
→
0,
β
→
0
sociated
scale
ν2QCD,
. Forthe
QCD
diagrams,
such
first
graph
in taking
Figurethe2,limit
the% modification
Like
in full
analytic
regulators
mustasbethe
taken
to zero
before
→ 0.
have
which
the
ofascollinear
The
result
depends
thelimits
order
in
the limits
→
0 in
and
β → sum
0asarelong
performed.
trivialdiagrams
in the
sense
thatonpoles,
the
α which
→ 0only
and
βcancel
→α02
are
smooth
! is Exkept finite
panding
first analytic
in β and then
in α, the light-cone
are regulated
by the in
α parameter,
and
anti-collinear
diagrams.
owever,
with
the
regulators
in placesingularities
also the SCET
diagrams
the (anti-)hard
and we find for the sum of all four one-loop diagrams
llinear sectors are now# also well-defined.
The" %!
diagrams in2 "each sector involve
$!
& divergence
2
1
1+z
2
µ
CF αs
#
the analytical
regulator,
of2 all
If the momentum
L⊥ sum −
ln 2contributions.
δ(1 − z) +
Iq←q (z,
x2T , µ)# which
= − cancel in+ the
2π
%
α
ν1
(1 − z)+
α reg.
24
in (30) is hard-collinear, as in the first SCET!diagram in
Figure
2,
the
"
' regularization in
T
F
2Γcusp (α
ntributions to the expansion of the integral. One finds that only the hard, the=
hard-collinear
2 ,µ)
& are2evaluated
d2ln'µ−F
q q̄ (xTthen
nd anti-hard-collinear regions contribute. If the
diagrams
off-shell,
also a
)
x
q
T verifies that the contribution
2 arises in the individual2 diagrams, but one easily
ft contribution
(z1 , xTmode
, µ) scaling
Bq̄/N2as(zM(λ,
, µ) q2 =
Bq/N1
2, x
Na1semi-hard
−2γ
λ,Tλ) vanishes.
E
4e
$ regulators and is obtained
Since the original diagram is well defined without the analytical
2
d
−M
y addinghcup the contributions
hc limits
hc fromhcthe different2regions,
hc Fguaranteed that the
hc we are
C (M , µ) = Γ independent
(α
+
hc s ) ln
→ 0 and β → 0 can be taken in the sumV and that the 2result iscusp
of the regula2
d
ln
µ
µ
dF
(x
,
µ)
q
q̄
α
T
F
r. Individually, however,
in each sector involve divergences
in the analytical
β
α the diagrams
+
→
=
2Γ
(α
)
β
α
s
cusp
β
gulators. If the momentum k in (31) is hard-collinear,
as in the first SCET diagram in
d
ln
µ
hc
gure 2, hc
the regularization inhcthe effective
theory takes the
in QCD.
If,
on the
hc same
hc
hc form
hc
qT2as '
ΛQCD
her hand, the momentum k is anti-hard-collinear, then the propagator is far off-shell and in
Figure 2: Matching
of anline,
analytically-regularized
graph
onto SCET
diagrams.
CET is represented
by a Wilson
as shown in the QCD
second
diagram
in Figure
2. Using the
•
Text
p,and
(p̄)performing
to thethe
power
→
0, β →
0find that
Regulators
play
double
role.
E.g.
hc
$ α regulates
placement
rule (31)
appropriate
expansions,
we propagators
2 the Feynman
d from
diagrams
develop
singularities
followed line
by α W
→ 0 −M
or vice versa, whichq
hcemission
Wilson
line
2in the limit β → 0FWilson
le forand
aofgluon
the anti-hard-collinear
hc in the current operator
(M from
, µ)
Γcusp (αs ) ln 2 + 2γ (αs )
cancel in the sum ofC
theVresults
both=
sectors.
) gets replaced
by
With d
theln
analytic
two diagrams in Figure
µ regulators
µ 1 can now be
2α µ
µ in place, the remaining
ν
n
n̄
·
p
n
1
computed and both give the same result.
sum, we obtain
→ For their
.
(32)
1+α
! 2 2 "!+α
n · k !− 2iε"−α !(n2 "· k
−βn̄ · p − iε)
µ
q
2z
xT µ
C
α
Γ(−%
−
α)
F
s
b+c
2
e!γE
Iq←q
(z, x2T , µ) =
. (32)
ote that, as mentioned earlier
2.2,
regularized
rule
2π in Section
ν12
ν22 the (1
− z)1−α+β
Γ(1
α) QCD
4 for the anti-hardqT Feynman
'+ Λ
llinear
Wilson
is the
no analytic
longer
invariant
under
the rescaling
transformation
λp. As
• Regulator
Like
in fullline
QCD,
regulators must
taken
to zero before
takingsector
the limitp %→
→ 0.
breaks
invariance
of be
anti-hard-collinear
under
The result
depends
on the α
order
in which
the limits
αit→regularizes
0 and β → the
0 arefermion
performed.
Exin aFigure
2, the
regulator
plays
a→
double
role:
2. propagators
xtenhard-collinear
p,
(p̄)
to
the
power
α
0,
β
→
0
p
→
λp
rescaling
of
the
hard-collinear
momentum
panding first
in β andand
then the
in α,Wilson
the light-cone
singularities
are regulated diagrams.
by the α parameter,
diagrams
lines in
anti-hard-collinear
Both classes
and we find for the sum of all four one-loop diagrams
diagrams
develop singularities in the$!
limit β →
0 followed by
versa,
which
" %!
" α → 0 or vice
&
#
2
2
1sectors. 2
1+z
µ
CF αboth
#
s
2
ncel in the sum
of xthe
results
from
+
L
−
2
ln
δ(1
−
z)
+
Iq←q (z,
,
µ)
=
−
#
⊥
T
2
2π
% remaining
α two νdiagrams
(1 − z)+1 can now be
α reg.
25
1
With the analytic regulators
in place,
the
in Figure
ANALYTICAL REGULARIZATION
x
x
x
x
x
x
x
x
!
"
'
2
F C (M
q δ(1
2
I
(z,
x
,
µ)
=
−
z)
−
+
,
µ)
=
Γ
(α
)
ln
+
2γ
(α
)
q←q
V
s
s
T
(M
,
µ)
=
Γ
(α
)
ln
+
2γ
(α
)
C
(M
,
µ)
.
cusp
V
s
s
V
2 2π
2
$
!
d lncusp
µ
µ
µ "
"
#
#
1-LOOP
RESULT
2
2
qT ' ΛQCD
q ' ΛQCD
× Cqi z1 , αs (µb ) Cq̄j z2 , αs (µb ) φi/N1 (ξ1 /z1 , µb ) φj/N2 (ξ2 /z2 , µ
−γE
T µb = b0 /xT , and we adopt the standard choice b0 = 2e
+ δ(
x2T µ2
L⊥ = ln −2γ
4e E
!"
# $"
#
%
2
2
!"
#
$"
#
%
C
α
2
1
+
z
1
µ
F s
Iq←q2(z, x2T , µ) = δ(1 − z)
+ z2
+ L⊥ 2
−µ22 ln 2 δ(1 − z)
CF−
αs
1+
1
2π + L$⊥
ν1 − z) +
(1 − z)+
Iq←q (z, xT , µ) = δ(1 − z) −
− 2αln 2 δ(1
2 ,µ)
2π2 2 $ −Fqq̄ (xα2 "
ν
(1
−
z)
+
1
−F
(x
#
&
T ,µ)
q
q̄
2
2
2
T
&
"
2
22 # π
T
2
2
2
2 + δ(1 − z)2 − 22 + Lπ⊥ +T
− (1 − z)
(16) 2
←j q←i
2
1−6(1 T
q̄←j
+
− z)
(16) 2 q←i
+ δ(1
T1
q̄←j −2γ
2E
T q2
T − z) −q 2$2 +2L$ ⊥q←i
F T
s
6 −2γE
T then β acts as the analytic
If the expansions are performed in the q̄←q̄
opposite order,
regulator,
2 dep.
If the expansions are performed in the opposite order, then β acts
as the analyticMregulator,
anomalous
and we obtain
and we obtain
!"
# $"
#
2
!"
#
$"
#
1
2
q
C
α
F s
1
M22 ln 2 δ(1 − z)
+ L⊥ 2 − +
Iq̄←q̄2 (z, x2T , µ) = δ(1 − Cz)F α
−s
− + 2α
ln 2 δ(1
Iq̄←q̄ (z, xT , µ) = δ(1 − z) −
2π + L⊥$
ν1 − z)
2π
$
α
ν1
%
&
2
%
&
1+z
1 ++z 2
− (1 − z)
(17)
−
z)
+
(17)
(1 −−
z)(1
+
(1 − z)+
finds
ower
→first
0, βthe
→ power
0 , thenpα→→λp0,, one
xtTaking
p,α(p̄)
to
β→
0 ( p → λp)
If the expansions are performed in the oppo
and
'
& we obtain
'
&
)
x q
)
x
q
!µ
(z
,
x
,
µ)
=
I
(z
,
x
,
µ)
I
(z
,
x
,
I (z , x , µ) I 4e(z , x , µ) =
C Iα (
I (z, x , µ)4e= δ(1 − z) −
2π
2
2
In the product the 1/α divergences vanish, but anomalous M2
dependence remains.
26
%
&− z)+
(1
2
1/α
1+z
− (1 − z)
+
(17)
(1 − z)+
1/α
CF αs
•
L⊥ + O(αs2 )
Fqq̄ (L⊥ , αs ) =
π
$
"
CF αs
2
C
α
F s
L⊥ + IO(α(z,
)
Fqq̄ (L⊥ , αs ) =
L2⊥ +
q←q s L⊥ , αs ) = Iq̄←q̄ (z, L⊥ , αs ) = δ(1 − z) 1 +
π
CF αs
$
"
#% 4π
2
2
L⊥ + O(αs )
CF αs' 2
π
(
q̄ (L⊥ , αs ) =
Iπq←q (z, L⊥ , αs ) = Iq̄←q̄ (z, L⊥ , αs ) = δ(1 − z) 1−
+ CF αs LLP
+
3L
−
2
⊥
⊥
⊥ q←q (z) − (16− z) + O(αs )
4π
$
"
#%
2π
2
(18)
C
α
π
F
s
2
'
(
'
z, L⊥ , αs ) = Iq̄←q̄ (z, L⊥ , αs ) = δ(1 −
L⊥ + 3L⊥ −
CFz)
αs 1 + 4π
TF αs
2
6
L⊥IP
(z)
−
(1
−
z)
+
O(α
)
−
(z,
L
,
α
)
=
I
(z,
L
,
α
)
=
−
L⊥ Pq←g (z) − 2z(1
q←q
s
q←g
⊥
s
q̄←g
⊥
s
(18)
2π
2π
(
CF αs '
(
2T α '
L⊥ Pq←q (z) − (1 − z)
+
O(α
)
−
F
s
For
coefficients,
we +
obtain
Iq←g
, αsthe
) =first
−s two expansion
L⊥ Pq←g (z)
− 2z(1 − z)
O(αs2 )
2π(z, L⊥ , αs ) = Iq̄←g (z, L⊥
2π
'
(
) α *2 $ ΓF β
TF αs
αs2 F
s
0 0 2
z, L⊥ , αs ) = Iq̄←g (z, L⊥ , αs ) = −
L⊥ Pq←g (z) − 2z(1
− ⊥z), αs+
O(αs Γ
) 0 L⊥ +
L⊥ + ΓF1 L⊥
Fqq̄ (L
)=
2π
•
4π
4π
2
$ "
#
%
first two expansion coefficients, we obtain
808
224
dq2 = CF CA
− 28ζ3 −
TF nf
27
27
$
%
) α *2 ΓF β
αs F
s
0 0 2
Γ0 L⊥ +
L⊥ + ΓF1 L⊥ + dq2
(19)
Fqq̄ (L⊥ , αs ) =
4π
4π
2
Fqq̄ (L⊥ , αs )
Fgg (L⊥ , αs )
$ "
#
%
=
808
224
CF (20)
CA
dq2 = CF CA
− 28ζ3 −
TF nf
27
27
For the functions I and F, one then obtains
Solving its RG and using Davies, Stirling and Webber ’84 and
de Florian and Grazzini ’01, we extract the two-loop F
Casimir scaling
F (L , α )
F (L , α )
All the necessary
= input for NNLL resummation!
(21)
q q̄
⊥
CF
s
gg
⊥
s
CA
27
Fqq̄ (L⊥ , αs )
Fgg (L⊥ , αs )
=
CF
CA
(21)
RESUMMED RESULT
,
,
d3 σ
4πα2 + 2 + + 1 dz1 1 dz2
=
e
dM 2 dqT2 dy
3Nc M 2 s q q i=q,g j=q̄,g ξ1 z1 ξ2 z2
#
%
$
"
ξ1 ξ2 2
, , qT , M 2 , µ φi/N1 (z1 , µ) φj/N2 (z2 , µ) + (q, i ↔ q̄, j) .
× Cqq̄→ij
z1 z2
3
The hard-scattering kernel is
!
!2 1
2
2
2
!
Cqq̄→ij (z1 , z2 , qT , M , µ) = CV (−M , µ)!
4π
"
d2 x⊥ e−iq⊥ ·x⊥
#
x2T M 2
4e−2γE
$−Fqq̄ (x2T ,µ)
× Iq←i (z1 , x2T , µ) Iq̄←j (z2 , x2T , µ)
• Two
(22)
sources of M dependence: hard function and anomaly
• Fourier
transform can be evaluated numerically or
analytically, if higher-log terms are expanded out.
28
(23)
n ∼
p0
COLLINS SOPER STERMAN FORMULA
p̄µ
n̄ ∼ 0
p̄
µ
"
'
' ' " 1 dz1 " 1 dz2
d3 σ
4πα2 1
2
−iq⊥ ·x⊥
2
=
d
x
e
e
⊥
q
2
dM 2 dqT dy
3Nc M 2 s 4π
z1 ξ2 z2
q
i=q,g j=q̄,g ξ1
( " 2
)
*+
M
2
2
%
&
%
&
dµ̄
M
× exp −
ln 2 A αs (µ̄) + B αs (µ̄)
2
2
µ̄
µ̄
µb
)
*
%
&
%
&
s12 (ξ2b/z
q̄j(q,
2 j/N
qi1b/z1 ,1µb )i/N
φj/N
q̄, j)s2 , 2b
2 , αs (µ
2b ) φi/N
s 1 (ξ
1 2 , µ1b ) +
b i↔
qi× C1qi z1 ,sαs (µbb) Cq̄j zq̄j
!
!
"
#
"#
#
"
"
#
)2
(µ )z φ, α (µ
(ξ )/zC, µ )zφ, α (µ
(ξ /z
× C z , α (µ ) C z×, α C
4
−γstandard
E.
The
low
scale
is
,
and
we
set
µ
=
b
/x
,
and
we
adopt
the
cho
= b0•/x
,
and
we
adopt
the
standard
choice
b
=
2e
b
0
T
T
0
• Landau-pole
singularity in the Fourier transform. To use the
formula, one needs additional prescription to deal with this.
29
s
cusp
! s"
β(α
s ) dgd
1
F
2
A αs = Γcusp (αs ) − −γE −
d
µ = µb = 2e2 xβ⊥
! "
B! αs" = 2γ q (αs ) + g1 (αs ) − β
µq=
B αs = 2γ
(αsq)T+ g1 (αs ) −
!
" # !
"#
!
−γE
2
#
#
µ = µb = 2e /x⊥C"ijin! z,
αs (µ
CV !#− µ
µb"2#$
Ii←j
If adopt the choice
our
result
to
−F!q
b )" =reduces
#
b , CSS
2
!
!
2
x
M
#·x
# Ii←j z
2 1 Cij z,2αs (µb )−iq
=
C
−
µ
,
µ
T
2
2
2
V
b
⊥
⊥
b
!
!
we
identity
z2 ,formula,
qT , M provided
, µ) = C
d x⊥ e
where
V (−M , µ)
∞
−2γE %
&n
4π
4e$
α
where
s
anomaly
contribution
∞ dq %
&n
g
(α
)
=
F
(0,
α
)
=
$
1
s
s
n
α
! "
s
β(αs ) dg1 (αs )
2,
2= F (0, αs ) = n=1 dqn 4π
g
(α
)
A αs = ΓFcusp (αs ) −
1
s
× Iq←i2 (z1dα
,x
s T , µ) Iq̄←j (z2 , xT , µ)
4π
n=1
∞
$
! "
#
#
β(α
)
dg
(α
)
s
2
s
2
(72) #
2
#
B αs = 2γ q (αs ) + g1 (αs ) −
,
∞ e
g
(α
)
=
ln
C
(−µ
,
µ)
=
$
# V
#
2
dαs
−γE 2 −1s
2
2
#
#
µ
=
µ
=
2e
x
b
g
(α
)
=
ln
C
(−µ
,
µ)
= n=1 e
2 ⊥s
V
!
" # !
"#
!
"
RELATION TO CSS
Cij z, αs (µb) = #CV − µ2b , µb # Ii←j z, 0, αs (µb ) ,
The one-loop coefficients are dq1 = 0 and
q
µ=q
∞
The
one-loop
% α &n Tcoefficients are d1 = 0 and
$
s
n=1
' 2
(
g1these
(αs ) = Frelations
(0, αs ) = todqnderive, unknown three-loop coefficient,
Use
necessary
7π
q
−γE
'
(.
4π
e
=
C
−
16
n=1
2
e for/xNNLL
F
⊥
7π
resummation ∞
(73) q1
3
#
#2 $ q % αs &n
2
.
g2 (αs ) = ln #CV (−µ , µ)# =
en
4π
q
(3) TheF two-loop
coefficient
n=1
e1 = CF
− 16 .
3
q
Fin (49), while eq
d
has
been
given
2
A = Γ2 + 2β0 d2 = 239.2q2− 652.9 #= Γ2
q3
q
The
two-loop
coefficient
been giventoinB(α
(49),
while
efficients are d1 = 0 and
compiled
in [31],
howeverd2ithas
contributes
O(αes2
s ) at
3
( in
compiled
in
[31],
however
it
contributes
to
B(α
)
at
O(α
relations
(72)
are
compatible
with
our
perturbative
resu
s
Not equal toq the 'cusp
anom.
dim.
as
was
usually
assumed!
2
s
7π
e1 = CF relations
(74)
− 16 . in (72)
are compatible
our perturbative
resuC
according
to (72) with
the coefficient
A in the
3 Note that
30
Note that
according
to (72) the coefficient A in the C
anomalous
q
q dimension starting at three-loop order, and the
efficient d2 has been given in (49), while e2 can be extracted from the results
DIVERGENT EXPANSIONS,
AND OTHER SURPRISES
31
TRANSVERSE MOMENTUM SPECTRUM
The spectrum has a number of quite remarkable features wich
we now discuss in turn:
• Expansion
in αs : strong factorial divergence
• qT-spectrum:
• calculable, even
• expansion
around qT = 0: worst expansion, ever
• Long-distance
• small, but
near qT = 0
effects associated with ΛQCD
OPE breaks down
32
CV (−M
K(η,
0, r) =, µ)
r
d x⊥ e
2(η−1)γ
E Γ(η)
4π
e
.
4e 4eE−2γE
(33)
(23)
−2γ
&
1
1
e
2
2
2−iq⊥ ·x⊥ −ηL⊥ − 4 aL2
⊥ ≡
× Iq←i
(z
,
x
,
µ)
I
(z
,
x
,
µ)
d
x
e
e
K η,
1
q̄←j
2
T
T
⊥
LEADING
MOMENTUM DEPENDENCE
a = αs4π
(µ) × O(1)
(34)
2
µ
%
'
&
%
E
−2γE−1 ∞
−2γE
1
q
e
2
T
·x⊥
(1−η)#− 4 a#
#/2
µ
=
µ
=
2e
x
⊥
e
d%
J
e
b
e
=
≡
0 at
0
b0 Up
= 2e
, and in
the
case
hand
to corrections
suppressed
by
powers
of
α
,
the
q
2
s
T- µ 2
(24)
µ
µ
−∞
2
−ηL⊥−γ
− 14EaL
b⊥
−γ
e E
= αqT(µ) of
dependence
our formula result has the form
aµ =
× O(1)
s
(35)
)
αs (µ) ( F
2
%
a
=
Γ
+
&(M q, 2µ)
' β0 .
−2γ
E ηF
0
1
1C α 2 M−iq
e
2
2 ⊥ ·x⊥ −ηL⊥ − 2π
aL⊥
4
F ds x⊥ e
e
K η, a, T ,
≡
= O(1)
qln
F
2
= ηΓF2=4π
+ 2β
d
=
239.2
−
652.9
=
#
Γ
π0 2 µ
2
µ2
µ2 (36)
(25)
2e−γE , and in the 2case
2 at hand
% xT µ2 & K(η, 0, r) = r η−1 Γ(1 − η)
= ln
with L⊥exp
, and the two quantities
−αs cL
(26)
2(η−1)γE Γ(η)
⊥
−2γ
E
e
)
4e αs (µ) ( F
2
a=
2π
Γ0 + ηF (M , µ) β0 .
CF αs M 2
η=
ln 2 = O(1) and
π
µ
K(η, 0, r) = r
η−1
a = αs (µ) × O(1) .
Γ(1 − η)
.
2(η−1)γ
E
e
Γ(η)
33
(27)
.
4e
K
π
E
2
µ
CF αs M
η=
ln 2 = O(1)
x2T µ2
'µ∞ a = α
Ls⊥
=×
ln O(1)
(µ)
π
%
&E
%
2&
q −2γ
q
4e
PERTURBATIVE
EXPANSION
=
dL J e
b
e
η, a,
T
µ2
L⊥ /2
T
(1−η)L⊥ − 41 aL2⊥
0
0
2 2⊥ &
%
xT µ qC α
−∞
2 µ
1
2 term can
M
T
µ
≈
q
F
s
L
/2
(1−η)L
−
aL
L
=
ln
Since
is
not
large
as
long
as
,
the
T
⊥
⊥
⊥
'J e
4% ⊥ 2 &
−2γ
ln
=
O(1)
η
=
dLformally
b
e
−2γ
4e
⊥
0
0
2e
qT
1
away
π
µ
2 be expanded
−iq ·x
−ηL
− aL
e µ
d x⊥ e
K η, a, 2 ,
≡
2
!
$
E
⊥
⊥
⊥
4π
1
4
E
2
⊥
µ
µ
%
&
2
a
1
a
2
4
γE
K(η,
a,
r)
=
1
−
∂
+
∂
. . K(η, 0, r)
%
, and in the case at−2γ
hand
2 .&
η
η +
E
qT
e 4
2! 4
−ηL⊥ − 1 aL2
η,22 a,
,
≡
)
αs (µ) ( 2F K x
2
2
µ
µ
µ
a=
Γ
+
η
F (M T , µ) β0 .
0
L = the
ln integal explicitly:
and for a=0, we can perform
4
⊥
2π
hand
)(
F
Γ0
+
⊥
4e−2γE
2− 2η)
Γ(1
x
µ
η−1
T
K(η, 0, r) =Lr⊥ = ln
.
2(η−1)γ
E Γ(η)
E
e 4e−2γ
)
2
5
ηF (Ma =
, µ)
β
.
αs (µ)µ0×≈O(1)
qT
34
(36
K(η, a, r)
−
n!
the
critical4
n=0
=
η is close to
exp
−
n!
then
has 4
n=0
∂η K(η, 0, r) =
value 1. One
∂η r
e2(η−1)γE Γ(η)
,
(65)
FACTORIAL DIVERGENCE
t is not difficult to see that
series is factorially
divergent.
To illustrate
we
Γ(1the
− η)
1
2ζ
2ζ5 this point,
3
2
4
=
− to the
(1default
− η) scale
− choice
(1 −µη)
.. ,
onsider the special case2(η−1)γ
where Er = 1 (corresponding
= qT+) .and
e
Γ(η)
1−η
3
5
is close to the critical value 1. One then has
2n+1
and taking 2n Γ(1
derivatives
of1 the 2ζ
leading
term
generates
(2n)!/(1
−
η)
. A m
− η)
2ζ
3
5
2
4
=
−
(1
−
η)
−
(1
−
η)
+
.
.
.
,
(66)
Unfortunately,
this series
is 3strongly factorially
divergent:
E Γ(η)
e2(η−1)γ
1−η
5
analysis reveals
that
'
( . A$
∞
∞ the leading
and taking 2n derivatives
of
term
generates
(2n)!/(1 − η)2n+1
more careful
%
&
$
n
#
a
1
(2n)!
−2γE
n
# =
analysisK(η,
reveals
that
−
+
k
a
+ O(1 −
−
e
a, 1)
n
+
...
2n+1
exp
n! ' 4
(1 − η) ( ∞
∞
n=0
n=0
$
$ (2n)! % a &n
#
1
−2γE
n
#
−
+
k
a
+ O(1Nason,
− η) , Ridolfi
(67)‘99
−
e
K(η, a, 1) exp =
first
noted
by:
Frixione,
n
2n+1
n! k do
4 not
(1 −
η)
where the coefficients
exhibit
the strongn=0factorial growth of the terms
n=0
n
sum.
is badly
the fact
thatofitthe
hasterms
alternating
Can
Borelthisresum
it,
which
leads
nonperturbative
andfirstsign im
where
theWhile
coefficients
kseries
exhibit divergent,
the
strongmakes
factorial
growth
in the
n do not
can
be Borel-summed.
obtainthe factexplicit
um.
While
this
series is badly
divergent,
that it has alternating sign implies that it
highly
nontrivial
a We
dependence
) *
'
+
'
+
,(
,
an be Borel-summed. We obtain
2
# ) * π ' (1−η) +
1
1−η '
1
+
,(
−2γ
E + ,(#
a
a
√ +1 − e
K(η, a,
1 − Erf √
# 1) Borelπ= (1−η)2 e
11−−ηErf −2γ
1
E
#
a
K(η, a, 1) Borel =
e aa 1 − Erf √
− e a a 1 − Erf √
a
a
a
∞
(68)
$
∞
$
n
n
k
a
+
− η)
,
n
k+
a
+
O(1
−
η)
,O(1the
In practice, it+is simplest,
to
use
exact
expression and
n
+ ...
n=0
n=0
evaluate K-function
numerically.
where
Erf(x)
is theiserror
Note thatNote
the singularity
η = 1 has disappeared
where
Erf(x)
the function.
error function.
that theatsingularity
at η = 1 after
has
Borel summation. Expressions for the first few kn coefficients can be readily derived in terms of
disapp
Borel summation. Expressions for the first
35 few kn coefficients can be readily derived
exp −αs cL2⊥
CLOW
α
M qT.
VERY
η=
ln
F
2
s
µ2
π
For moderate qT, the natural scale L
choice
⊥ is μ = qT.
However, detailed analysis shows that near qT ≈ 0 the
Fourier integral is dominated by
#
$x−1
T % = q∗ = M exp −
x (qwhich
∗ , qT )
π
2CF αs (q∗ )
$
= 1.75 GeV for M = MZ
corresponds to η=1.
pT
λ=
computed
M
→ Spectrum can be
methods down to qT=0!
µ
p
nµ ∼ 0
p
36
with short-distance
INTERCEPT AT QT=0
• Dedicated
analysis of qT → 0 limit yields:
N −#/αs
dσ
∼√ e
(1 + c1 αs + . . . )
2
dqT
αs
Parisi, Petronzio 1979;
Collins, Soper, Sterman 1985; Ellis, Veseli 1998
• Were
for the first time we are able to compute the
normalization N and NLO coefficient c1
• Expression
singularity)
cannot be expanded about αs = 0 (essential
37
K(η, a, r) = 1 + ∂η +
xT µ∂η + . . . K(η, 0, r)
= ln4 −2γ
4 L⊥ 2!
4e E
x2T µ2
T
L⊥ = ln −2γ
4e E
%
' try to 1obtain
& we
∞
−2γ
−2γE
E
Given
our
for
the
intercept,
can
also
2
2
1 result
q
e
e
2
2
T
q⊥ ·x⊥ −ηL⊥ − 4 aL⊥
(1−η)#− 4 a#
x0T µe#/2 b0
2
e
d%
J
e
K
=
≡
derivatives with respect
to
q
.
Leading
term
is
obtained
by
T
L
=
ln
2
2
µ ⊥ −∞ 4e−2γE
µ
µ
expanding
SLOPE AT Q =0?
1
4π
%
µ ≈ qT
−ηL⊥ − 14 aL2⊥
d2 x⊥ e−iq⊥ ·x⊥ e
qT2 '
e−2γE &
K η, a, 2 ,
≡
2
µ
µ
η=1
= 2e , and in the case at hand
Yields extremely violently divergent series
)
αs (µ) ( F
2
a=
Γ0 + ηF (M
, µ) β,
0 . 2 -n−1
∞
n+1
+
* 2π
(−1)
qT
n2 /a
*
√
K(η = 1, a, qT ) exp ∼
e
2
a
q
∗
n=1
Γ(1 − η)
η−1
K(η, 0, r) = r
.
2(η−1)γ
E
e
Γ(η)
38
5
−γE
NUMERICAL RESULTS
• Spectrum
can be calculated numerically, even though power
expansion in qT2 is absolutely meaningless (not even Borel
summable)!
• Find
smooth behavior down to very small qT
Z production, Tevatron
10
25
8
20
6
4
LO+NLL
2
0
dqT
�pb�GeV�
NLO+NNLL
dΣ
dqT 2
dΣ
�pb�GeV�
Z production, Tevatron
15
10
5
0
1
2
3
4
5
6
qT
0
0
5
10
15
qT
39
20
25
30
INTERPRETATION
Z production, Tevatron
10
area
= total cross section
= perturbative!
non-perturbative
dqT
2
dΣ
�pb�GeV�
8
6
non-perturbative
4
perturbative
2
0
0
1
2
3
4
5
6
qT
• Spectrum
is short-distance dominated, but essential
features are non-perturbative!
40
of the W boson. We emphasize that the above relation only holds for xT & 1/M , because
otherwise the power corrections to our factorization formula become large. It can therefore
not be used to study the xT → 0 limit of the functions Fqq̄ or Bi/N .
LOW-ENERGY QCD EFFECTS
2.3
Simplifications at large qT2
For given transverse momentum qT , the Fourier integral in (18) receives important contributions from transverse separations xT ! qT−1 only. For large transverse momenta in the
•
perturbative
domain, qT2 & Λ2QCD , we therefore need the xT -dependent PDFs at transverse
separation xT # Λ−1
QCD . In this case these functions obey an operator-product expansion of
the form [4, 34, 35]
% ! 1 dz
Bi/N (ξ, x2T , µ) =
Ii←j (z, x2T , µ) φj/N (ξ/z, µ) + O(Λ2QCD x2T ) .
(20)
z
ξ
j
Related question is that about the impact of longdistance power correction in matching relation
In the context of SCET, generalized PDFs defined in terms of hadron matrix elements in
• collinear fields are separated by distances that are not light-like are referred to as beam
which
functions. For such functions an analogous expansion was considered in [17], and an expression
for the one-loop kernel of the quark beam function was derived in [18]. The evolution equations
for the new kernels
DGLAP equations
2 with the standard
−Λ2 x2T
2 Ii←j follow when we combine2(16)
T
T
% ! 1 du
d
φi/N (z, µ) =
Pi←j (z/u, µ) φj/N (u, µ) .
(21)
d ln µ
u
z
j
Find that these cannot be analyzed order by order,
but only numerically using functions that vanish at
large x , such as θ(1 − Λ x ) or e
• Fixed-order
OPE in xT2 is again extremely
9
divergent. Same as expansion
of K around qT=0
41
LONG-DISTANCE EFFECTS
• Yet
resummed behavior is smooth and rather
insensitive to the way in which the cutoff is introduced:
Z production, Tevatron, Gaussian cutoff
Z production, Tevatron, Gaussian cutoff
10
25
�NP �0
�NP �0.1GeV
�NP �0.3GeV
�NP �0.5GeV
�NP �0.7GeV
dΣ
4
2
0
dqT 2
�pb�GeV�
20
6
dqT 2
dΣ
�pb�GeV�
8
15
�NP �0
�NP �0.1GeV
�NP �0.3GeV
�NP �0.5GeV
�NP �0.7GeV
10
5
0
1
2
3
4
5
0
6
qT
0
1
2
3
4
qT
• Indications
that long-distance effects are very small
already above qT=2 GeV
42
5
6
Resulting power correction has a complicated shape,
but is approximately linear:
10
∼
�pb�GeV�
8
�
ΛNP
1 − 0.7 ×
qT
�
dqT 2 q
dΣ
T �0
6
4
2
0
0.0
0.5
1.0
1.5
2.0
�NP
Cannot be described in terms of a single operator
matrix element
43
CONCLUSION
•
Have derived resummed result for Drell-Yan qT spectrum
•
•
Reduces to the known CSS result for a special scale choice.
•
•
•
Naive factorization broken by collinear anomaly. Only the product of
two transverse PDFs is well defined, but has anomalous
dependence on the large momentum transfer
Obtain three-loop coefficient A(3) , the last missing piece needed
for NNLL accuracy.
Many surprising features:
•
emergence of nonperturbative scale q*~2GeV: spectrum is shortdistance dominated, even at very low qT
•
strongly divergent expansions in αs, qT/q*, ΛQCD/q*.
Phenomenological analysis at NNLL+NLO is in progress.
44