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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0704.0001 | Pavel Nadolsky | C. Bal\'azs, E. L. Berger, P. M. Nadolsky, C.-P. Yuan | Calculation of prompt diphoton production cross sections at Tevatron and
LHC energies | 37 pages, 15 figures; published version | Phys.Rev.D76:013009,2007 | 10.1103/PhysRevD.76.013009 | ANL-HEP-PR-07-12 | hep-ph | null | A fully differential calculation in perturbative quantum chromodynamics is
presented for the production of massive photon pairs at hadron colliders. All
next-to-leading order perturbative contributions from quark-antiquark,
gluon-(anti)quark, and gluon-gluon subprocesses are included, as well as
all-orders resummation of initial-state gluon radiation valid at
next-to-next-to-leading logarithmic accuracy. The region of phase space is
specified in which the calculation is most reliable. Good agreement is
demonstrated with data from the Fermilab Tevatron, and predictions are made for
more detailed tests with CDF and DO data. Predictions are shown for
distributions of diphoton pairs produced at the energy of the Large Hadron
Collider (LHC). Distributions of the diphoton pairs from the decay of a Higgs
boson are contrasted with those produced from QCD processes at the LHC, showing
that enhanced sensitivity to the signal can be obtained with judicious
selection of events.
| [
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] | 2008-11-26 | [
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] | arXiv:0704.0001v2 [hep-ph] 24 Jul 2007
ANL-HEP-PR-07-12,
arXiv:0704.0001
Cal ulation
of
prompt
diphoton
pro
du tion
ross
se tions
at
T
ev
atron
and
LHC
energies
C.
Balázs1
,∗
E.
L.
Berger1
,†
P
.
Nadolsky1
,‡
and
C.-P
.
Y
uan2§
1
High
Ener
gy
Physi s
Division,
A
r
gonne
National
L
ab
or
atory,
A
r
gonne,
IL
60439
2
Dep
artment
of
Physi s
and
Astr
onomy,
Mi higan
State
University,
East
L
ansing,
MI
48824
(Dated:
Ma
y
3,
2007)
Abstra t
A
fully
dieren
tial
al ulation
in
p
erturbativ
e
quan
tum
hromo
dynami s
is
presen
ted
for
the
pro
du tion
of
massiv
e
photon
pairs
at
hadron
olliders.
All
next-to-leading
order
p
erturbativ
e
on
tributions
from
quark-an
tiquark,
gluon-(an
ti)quark,
and
gluon-gluon
subpro
esses
are
in luded,
as
w
ell
as
all-orders
resummation
of
initial-state
gluon
radiation
v
alid
at
next-to-next-to-leading
logarithmi
a ura y
.
The
region
of
phase
spa e
is
sp
e i ed
in
whi
h
the
al ulation
is
most
reliable.
Go
o
d
agreemen
t
is
demonstrated
with
data
from
the
F
ermilab
T
ev
atron,
and
predi tions
are
made
for
more
detailed
tests
with
CDF
and
DØ
data.
Predi tions
are
sho
wn
for
distributions
of
diphoton
pairs
pro
du ed
at
the
energy
of
the
Large
Hadron
Collider
(LHC).
Distributions
of
the
diphoton
pairs
from
the
de a
y
of
a
Higgs
b
oson
are
on
trasted
with
those
pro
du ed
from
QCD
pro
esses
at
the
LHC,
sho
wing
that
enhan ed
sensitivit
y
to
the
signal
an
b
e
obtained
with
judi ious
sele tion
of
ev
en
ts.
P
A
CS
n
um
b
ers:
12.15.Ji,
12.38
Cy
,
13.85.Qk
Keyw
ords:
prompt
photons;
all-orders
resummation;
hadron
ollider
phenomenology;
Higgs
b
oson;
LHC
∗
balazs hep.anl.go
v;
Curren
t
address:
S
ho
ol
of
Ph
ysi s,
Monash
Univ
ersit
y
,
Melb
ourne
VIC
3800,
Australia
†
b
erger anl.go
v
‡
nadolsky hep.anl.go
v
§
yuan pa.msu.edu
1
I.
INTR
ODUCTION
The
long-sough
t
Higgs
b
oson(s) h
of
ele tro
w
eak
symmetry
breaking
in
parti le
ph
ysi s
ma
y
so
on
b
e
observ
ed
at
the
CERN
Large
Hadron
Collider
(LHC)
through
the
diphoton
de a
y
mo
de
(h →γγ
).
Purely
hadroni
standard
mo
del
pro
esses
are
a
opious
sour e
of
diphotons,
and
a
narro
w
Higgs
b
oson
signal
at
relativ
ely
lo
w
masses
will
app
ear
as
a
small
p
eak
ab
o
v
e
this
onsiderable
ba
kground.
A
pre ise
theoreti al
understanding
of
the
kinemati
distributions
for
diphoton
pro
du tion
in
the
standard
mo
del
ould
pro
vide
v
aluable
guidan e
in
the
sear
h
for
the
Higgs
b
oson
signal
and
assist
in
the
imp
ortan
t
measuremen
t
of
Higgs
b
oson
oupling
strengths.
In
this
pap
er
w
e
address
the
theoreti al
al ulation
of
the
in
v
arian
t
mass,
transv
erse
mo-
men
tum,
rapidit
y
,
and
angular
distributions
of
on
tin
uum
diphoton
pro
du tion
in
proton-
an
tiproton
and
proton-proton
in
tera tions
at
hadron
ollider
energies.
W
e
ompute
all
on-
tributions
to
diphoton
pro
du tion
from
parton-parton
subpro
esses
through
next-to-leading
order
(NLO)
in
p
erturbativ
e
quan
tum
hromo
dynami s
(QCD).
These
higher-order
on
tri-
butions
are
large
at
the
LHC,
and
their
in lusion
is
mandatory
for
quan
titativ
ely
trust-
w
orth
y
predi tions.
W
e
resum
initial-state
soft
and
ollinear
logarithmi
terms
asso
iated
with
gluon
radiation
to
all
orders
in
the
strong
oupling
strength αs
.
This
resummation
is
essen
tial
for
ph
ysi ally
meaningful
predi tions
of
the
transv
erse
momen
tum
(QT
)
distri-
bution
of
the
diphotons
at
small
and
in
termediate
v
alues
of QT
,
where
the
ross
se tion
is
large.
In
addition,
w
e
analyze
the
nal-state
ollinearly-enhan ed
on
tributions,
also
kno
wn
as
`fragmen
tation'
on
tributions,
in
whi
h
one
or
b
oth
photons
are
radiated
from
nal-state
partoni
onstituen
ts.
W
e
ompare
the
results
of
our
al ulations
with
data
on
isolated
diphoton
pro
du tion
from
the
F
ermilab
T
ev
atron
[1℄.
The
go
o
d
agreemen
t
w
e
obtain
with
the
T
ev
atron
data
adds
on den e
to
our
predi tions
at
the
energy
of
the
LHC.
The
presen
t
w
ork
expands
on
our
re en
t
abbreviated
rep
ort
[2℄,
and
it
ma
y
b
e
read
in
onjun tion
with
our
detailed
treatmen
t
of
the
on
tributions
from
the
gluon-gluon
subpro
ess
[3℄.
Our
atten
tion
is
fo
used
on
the
pro
du tion
of
isolated
photons,
i.e.,
high-energy
photons
observ
ed
at
some
distan e
from
appre iable
hadroni
remnan
ts
in
the
parti le
dete tor.
The
rare
isolated
photons
tend
to
originate
dire tly
in
hard
QCD
s attering,
in
on
trast
to
opiously
pro
du ed
non-isolated
photons
that
arise
from
nonp
erturbativ
e
pro
esses
su
h
as
π
and η
de a
ys,
or
from
via
quasi- ollinear
radiation
o
nal-state
quarks
and
gluons.
W
e
ev
aluate
on
tributions
to
on
tin
uum
diphoton
pro
du tion
from
the
basi
short-
distan e
hannels
for γγ
pro
du tion
initiated
b
y
quark-an
tiquark
and
(an
ti)quark-gluon
s attering,
as
w
ell
as
b
y
gluon-gluon
and
gluon-(an
ti)quark
s attering
pro
eeding
through
a
fermion-lo
op
diagram.
A
t
lo
w
est
order
in
QCD,
a
photon
pair
is
pro
du ed
from q ̄
q
anni-
hilation
[Fig.
1
(a)℄.
Represen
tativ
e
next-to-leading
order
(NLO)
on
tributions
to q ̄
q + qg
s attering
are
sho
wn
in
Fig.
1
(b)-(e).
They
are
of O(αs)
in
the
strong
oupling
strength
[4,
5℄.
Pro
du tion
of γγ
pairs
via
a
b
o
x
diagram
in gg
s attering
[Fig.
1
(h)℄
is
suppressed
b
y
t
w
o
p
o
w
ers
of αs
ompared
to
the
lo
w
est-order q ̄
q
on
tribution,
but
it
is
enhan ed
b
y
a
pro
du t
of
t
w
o
large
gluon
parton
distribution
fun tions
(PDF
s)
if
t
ypi al
momen
tum
fra tions x
are
small
[6℄.
The O(α3
s)
or
NLO
orre tions
to gg
s attering
in lude
one-lo
op gg →γγg
diagrams
(i)
and
(j)
deriv
ed
in
Ref.
[7,
8℄,
as
w
ell
as
4-leg
t
w
o-lo
op
diagrams
(l)
omputed
in
Ref.
[9,
10
℄.
In
this
study
w
e
also
in lude
subleading
on
tributions
from
the
pro
ess
(k),
gqS →γγqS
via
the
quark
lo
op,
where qS = P
i=u,d,s,...(qi + ̄
qi)
denotes
the
a
v
or-singlet
om
bination
of
quark
s attering
hannels.
2
Direct γγ production
Single−photon
fragmentation
+...
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Figure
1:
Represen
tativ
e
partoni
subpro
esses
that
on
tribute
to
on
tin
uum
diphoton
pro
du tion.
All
leading-order
and
next-to-leading
order
dire t
pro
du tion
subpro
esses,
i.e.,
on
tributions
(a)-
(e)
and
(h)-(l),
are
in luded
in
this
study
.
Diagrams
(f
)
and
(g)
are
examples
of
single-photon
one-
and
t
w
o-fragmen
tation.
F
a torization
is
a
en
tral
prin iple
of
hadroni
al ulations
in
p
erturbativ
e
QCD,
in
whi
h
a
high-energy
s attering
ross
se tion
is
expressed
as
a
on
v
olution
of
a
p
erturbativ
e
partoni
ross
se tion
with
nonp
erturbativ
e
parton
distribution
fun tions
(PDF
s),
th
us
separating
short-distan e
from
long-distan e
ph
ysi s.
The
ommon
fa torization
is
a
longitudinal
no-
tion,
in
the
sense
that
the
on
v
olution
is
an
in
tegral
o
v
er
longitudinal
momen
tum
fra tions,
ev
en
if
some
partons
in
the
hard-s attering
pro
ess
ha
v
e
transv
erse
momen
ta
that
b
order
the
nonp
erturbativ
e
regime.
Unph
ysi al
features
ma
y
then
arise
in
the
transv
erse
momen
tum
(QT
)
distribution
of
a
olor-neutral
ob
je t
with
high
in
v
arian
t
mass
(Q
),
su
h
as
a
pair
of
photons
pro
du ed
in
hadron-hadron
ollisions.
When
al ulated
in
the
ommon
fa torization
approa
h
at
an
y
nite
order
in
p
erturbation
theory
,
this
distribution
div
erges
as QT →0 ,
signaling
that
infrared
singularities
asso
iated
with QT →0
ha
v
e
not
b
een
prop
erly
iso-
lated
and
regulated.
These
singularities
are
asso
iated
with
soft
and
ollinear
radiation
from
initial-state
partons
sho
wn
b
y
the
diagrams
in
Figs.
1
(b),
(d),
and
(i).
A
generalized
fa torization
approa
h
that
orre tly
des rib
es
the
small-QT
region
w
as
dev
elop
ed
b
y
Collins,
Sop
er,
and
Sterman
(CSS)
[11℄
and
applied
to
photon
pair
pro
du -
tion
[7,
12
,
13
℄.
In
this
approa
h
the
hadroni
ross
se tion
is
expressed
as
an
in
tegral
o
v
er
the
transv
erse
o
ordinate
(impa t
parameter).
The
in
tegrable
singular
fun tions
presen
t
in
the
nite-order
dieren
tial
distribution
as QT →0
are
resummed,
to
all
orders
in
the
strong
oupling αs
,
in
to
a
Sudak
o
v
exp
onen
t,
and
a
w
ell-b
eha
v
ed
ross
se tion
is
obtained
for
all QT
v
alues.
As
explained
in
Se .
I
I,
our
resummed
al ulation
is
a urate
to
next-to-
next-to-leading-logarithmi
(NNLL)
order.
It
is
appli able
for
v
alues
of
diphoton
transv
erse
momen
tum
that
are
less
than
the
diphoton
mass,
i.e.,
for QT < Q
.
When QT ∼Q
,
terms
of
the
form lnn(QT/Q)
b
e ome
small.
A
p
erturbativ
e
expansion
with
a
single
hard
s ale
is
then
appli able,
and
the
ross
se tion
an
b
e
obtained
from
nite-order
p
erturbation
theory
.
3
In
addition
to
the
initial-state
logarithmi
singularities,
there
is
a
set
of
imp
ortan
t
nal-
state
singularities
whi
h
arise
in
the
matrix
elemen
ts
when
at
least
one
photon's
momen
tum
is
ollinear
to
the
momen
tum
of
a
nal-state
parton.
They
are
sometimes
referred
to
as
`fragmen
tation'
singularities.
A
t
lo
w
est
order
in αs
,
the
nal-state
singularit
y
app
ears
only
in
the qg →γγq
diagrams,
as
in
Fig.
1
(e).
There
are
v
arious
metho
ds
used
in
the
literature
to
deal
with
the
nal-state
singularit
y
,
in luding
the
in
tro
du tion
of
expli it
fragmen
tation
fun tions Dγ(z)
for
hard
photon
pro
du tion,
where z
is
the
ligh
t- one
fra tion
of
the
in
terme-
diate
parton's
momen
tum
arried
b
y
the
photon.
These
single-photon
one-fragmen
tation
and
t
w
o-fragmen
tation
on
tributions,
orresp
onding
to
one
or
b
oth
photons
pro
du ed
in
indep
enden
t
fragmen
tation
pro
esses,
are
illustrated
b
y
the
diagrams
in
Figs.
1
(f
)
and
(g).
In
addition,
a
fragmen
tation
on
tribution
of
en
tirely
dieren
t
nature
arises
when
the γγ
pair
is
relativ
ely
ligh
t
and
pro
du ed
from
fragmen
tation
of
one
parton,
as
dis ussed
in
Se s.
I
I
C
2
and
I
I
I
A
3.
A
full
and
onsisten
t
treatmen
t
of
the
nal-state
logarithms
b
ey
ond
lo
w
est
order
w
ould
require
a
join
t
resummation
of
the
initial-
and
nal-state
logarithmi
singularities.
In
the
w
ork
rep
orted
here,
w
e
are
guided
b
y
our
in
terest
in
des ribing
the
ross
se tion
for
isolate
d
photons,
in
whi
h
the
fragmen
tation
on
tributions
are
largely
suppressed.
A
t
ypi al
isolation
ondition
requires
the
hadroni
a tivit
y
to
b
e
minimal
(e.g.,
omparable
to
the
underlying
ev
en
t)
in
the
immediate
neigh
b
orho
o
d
of
ea
h
andidate
photon.
Candidate
photons
an
b
e
reje ted
b
y
energy
dep
osit
nearb
y
in
the
hadroni
alorimeter
or
the
presen e
of
hadroni
tra
ks
near
the
photons.
A
theory
al ulation
ma
y
appro
ximate
the
exp
erimen
tal
isolation
b
y
requiring
the
full
energy
of
the
hadroni
remnan
ts
to
b
e
less
than
a
threshold
isolation
energy Eiso
T
in
a
one
of
size ∆R
around
ea
h
photon.
The
t
w
o
photons
m
ust
b
e
also
separated
in
the
plane
of
the
rapidit
y η
and
azim
uthal
angle φ
b
y
an
amoun
t
ex eeding
the
resolution ∆Rγγ
of
the
dete tor.
The
v
alues
of Eiso
T , ∆R ,
and ∆Rγγ
serv
e
as
rude
hara teristi s
of
the
a tual
measuremen
t.
The
magnitude
of
the
nal-state
fragmen
tation
on
tribution
dep
ends
on
the
assumed
v
alues
of Eiso
T , ∆R ,
and ∆Rγγ
.
An
additional
ompli ation
arises
when
the
fragmen
tation
radiation
is
assumed
to
b
e
exa tly
ollinear
to
the
photon's
momen
tum,
as
implied
b
y
the
photon
fragmen
tation
fun -
tions Dγ(z).
The
ollinear
appro
ximation
onstrains
from
b
elo
w
the
v
alues
of z
a essible
to Dγ(z): z > zmin
.
The
size
of
the
fragmen
tation
on
tribution
ma
y
dep
end
strongly
on
the
v
alues
of Eiso
T
and zmin
as
a
onsequen e
of
rapid
v
ariation
of Dγ(z)
with z
.
In
our
w
ork
w
e
treat
the
nal-state
singularit
y
using
a
pres ription
that
repro
du es
desir-
able
features
of
the
isolated
ross
se tions
while
b
ypassing
some
of
the
te
hni al
di ulties
alluded
to
ab
o
v
e.
F
or QT > Eiso
T
,
w
e
a
v
oid
the
nal-state
ollinear
singularit
y
in
the qg
s attering
hannel
b
y
applying
quasi-exp
erimen
tal
isolation.
When QT < Eiso
T
,
w
e
apply
an
auxiliary
regulator
whi
h
appro
ximates
on
a
v
erage
the
full
NLO
rate
from
dire t qg
and
fragmen
tation
ross
se tions
in
this QT
range.
T
w
o
pres riptions
for
the
auxiliary
regulator
(subtra tion
and
smo
oth- one
isolation
inside
the
photon's
isolation
one)
are
onsidered
and
lead
to
similar
predi tions
at
the
T
ev
atron
and
the
LHC.
W
e
b
egin
with
our
notation
in
Se .
I
I
A
,
follo
w
ed
b
y
an
o
v
erview
of
the
pro
edure
for
resummation
of
initial-state
m
ultiple
parton
radiation
in
Se .
I
I
B
.
The
issue
of
the
nal-
state
fragmen
tation
singularit
y
is
dis ussed
in
Se .
I
I
C
.
Our
approa
h
is
ompared
with
that
of
the
DIPHO
X
al ulation
[14℄,
in
whi
h
expli it
fragmen
tation
fun tion
on
tributions
are
in luded
at
NLO,
but
all-orders
resummation
is
not
p
erformed.
Our
theoreti al
framew
ork
is
summarized
in
Se .
I
I
D.
In
Se .
I
I
I
w
e
ompare
the
predi tions
of
our
resummation
al ulation
with
T
ev
atron
4
data.
Resummation
is
sho
wn
to
b
e
imp
ortan
t
for
the
su essful
des ription
of
ph
ysi al QT
distributions,
as
w
ell
as
for
stable
estimates
of
the
ee ts
of
exp
erimen
tal
a eptan e
on
distributions
in
the
diphoton
in
v
arian
t
mass.
W
e
ompare
our
results
with
the
DIPHO
X
al ulation
[14℄
and
demonstrate
that
the
requiremen
t QT < Q
further
suppresses
the
ee ts
of
the
nal-state
fragmen
tation
on
tribution,
b
ey
ond
the
redu tion
asso
iated
with
isolation.
Next,
w
e
presen
t
our
predi tions
for
distributions
of
diphoton
pairs
pro
du ed
at
the
energy
of
the
LHC.
V
arious
distributions
of
the
diphoton
pairs
pro
du ed
from
the
de a
y
of
a
Higgs
b
oson
are
on
trasted
with
those
pro
du ed
from
QCD
on
tin
uum
pro
esses
at
the
LHC,
sho
wing
that
enhan ed
sensitivit
y
to
the
signal
an
b
e
obtained
with
judi ious
ev
en
t
sele tion.
Our
on lusions
are
presen
ted
in
Se .
IV
.
I
I.
THEOR
Y
O
VER
VIEW
A.
Notation
W
e
onsider
the
s attering
pro
ess h1(P1) + h2(P2) →γ(P3) + γ(P4) + X
,
where h1
and
h2
are
the
initial-state
hadrons.
In
terms
of
the
en
ter-of-mass
ollision
energy
√
S
,
the
in
v
arian
t
mass Q,
transv
erse
momen
tum QT
,
and
rapidit
y y
of
the γγ
pair,
the
lab
oratory
frame
momen
ta P μ
1
and P μ
2
of
the
initial
hadrons
and qμ ≡P μ
3 + P μ
4
of
the γγ
pair
are
P μ
1 =
√
S
2 {1, 0, 0, 1};
(1)
P μ
2 =
√
S
2 {1, 0, 0, −1} ;
(2)
qμ =
q
Q2 + Q2
T cosh y, QT, 0,
q
Q2 + Q2
T sinh y
.
(3)
The
ligh
t- one
momen
tum
fra tions
for
the
b
o
osted 2 →2
s attering
system
are
x1,2 ≡2(P2,1 * q)
S
=
p
Q2 + Q2
Te±y
√
S
.
(4)
De a
y
of
the γγ
pairs
is
des rib
ed
in
the
hadroni
Collins-Sop
er
frame
[15℄.
The
Collins-
Sop
er
frame
is
a
rest
frame
of
the γγ
pair
(with qμ = {Q, 0, 0, 0}
in
this
frame),
hosen
so
that
(a)
the
momen
ta ⃗
P1
and ⃗
P2
of
the
initial
hadrons
lie
in
the Oxz
plane
(with
zero
azim
uthal
angle),
and
(b)
the z
axis
bise ts
the
angle
b
et
w
een ⃗
P1
and −⃗
P2
.
The
photon
momen
ta
are
an
tiparallel
in
the
Collins-Sop
er
frame:
P μ
3 = Q
2 {0, sin θ∗cos φ∗, sin θ∗sin φ∗, cos θ∗} ,
(5)
P μ
4 = Q
2 {0, −sin θ∗cos φ∗, −sin θ∗sin φ∗, −cos θ∗} ,
(6)
where θ∗
and φ∗
are
the
photon's
p
olar
and
azim
uthal
angles.
In
this
se tion,
w
e
deriv
e
resummed
predi tions
for
the
fully
dieren
tial γγ
ross
se tion dσ/(dQ2dydQ2
TdΩ∗),
where
dΩ∗= d cos θ∗dφ∗
is
a
solid
angle
elemen
t
around
the
dire tion
of ⃗
P3
in
the
Collins-Sop
er
frame
de ned
in
Eq.
(5).
The
angles
in
the
Collins-Sop
er
frame
are
denoted
b
y
a
∗
subs ript,
in
on
trast
to
angles
in
the
lab
frame,
whi
h
do
not
ha
v
e
su
h
a
subs ript.
The
parton
momen
ta
and
heli ities
are
denoted
b
y
lo
w
er ase pi
and λi
,
resp
e tiv
ely
.
5
B.
Resummation
of
the
initial-state
QCD
radiation
F
or
ompleteness,
w
e
presen
t
an
o
v
erview
of
the
nite-order
and
resummed
on
tributions
asso
iated
with
the
dire t
pro
du tion
of
diphotons.
A
t
the
lo
w
est
order
in
the
strong
oupling
strength αs
,
photon
pairs
are
pro
du ed
with
zero
transv
erse
momen
tum QT
.
The
Born
q ̄
q →γγ
ross
se tion
orresp
onding
to
Fig.
1
(a)
is
dσq ̄
q
dQ2dy dQ2
TdΩ∗
Born
= δ( ⃗
QT)
X
i=u, ̄
u,d, ̄
d,...
Σi(θ∗)
S
fqi/h1(x1, μF)f ̄
qi/h2(x2, μF),
(7)
where fqi/h(x, μF)
denotes
the
parton
distribution
fun tion
(PDF)
for
a
quark
of
a
a
v
or i
,
ev
aluated
at
a
fa torization
s ale μF
of
order Q
.
The
prefa tor
Σi(θ∗) ≡σ(0)
i
1 + cos2 θ∗
1 −cos2 θ∗
,
(8)
with
σ(0)
i
≡α2(Q)e4
i π
2NcQ2 ,
(9)
is
omp
osed
of
the
running
ele tromagneti
oupling
strength α ≡e2/4π
ev
aluated
at
the
s ale Q
,
fra tional
quark
harge ei = 2/3
or −1/3 ,
and
n
um
b
er
of
QCD
olors Nc = 3.
The
lo
w
est-order gg →γγ
s attering
pro
eeds
through
an
amplitude
with
a
virtual
quark
lo
op
(a
b
o
x
diagram)
sho
wn
in
Fig.
1
(h).
Its
ross
se tion
tak
es
the
form
dσgg
dQ2dy dQ2
TdΩ∗
Born
= δ( ⃗
QT)Σg(θ∗)
S
fg/h1(x1, μF)fg/h2(x2, μF),
(10)
where
the
prefa tor
Σg(θ∗) ≡σ(0)
g Lg(θ∗)
(11)
dep
ends
on
the
p
olar
angle θ∗
through
a
fun tion Lg(θ∗)
presen
ted
expli itly
in
Ref.
[3℄.
The
o
v
erall
normalization
o
e ien
t
σ(0)
g
=
α2(Q)α2
s(Q)
32πQ2(N2
c −1)
X
i
e2
i
!2
(12)
in
v
olv
es
the
sum
of
the
squared
harges e2
i
of
the
quarks
ir ulating
in
the
lo
op.
The
NLO
dire t
on
tributions,
represen
ted
b
y
Figs.
1
(b)-(e),
(i)-(l)
and
denoted
as
P(Q, QT, y, Ω∗),
are
omputed
in
Refs.
[3,
4
,
5,
7
,
8,
9,
10
℄.
The
NLO 2 →3
dieren-
tial
ross
se tion
gro
ws
logarithmi ally
if
the
nal-state
parton
is
soft
or
ollinear
to
the
initial-state
quark
or
gluon,
i.e.,
when QT
of
the γγ
pair
is
m
u
h
smaller
than Q
.
These
initial-state
logarithmi
on
tributions
are
summed
to
all
orders
later
in
this
subse tion.
The
NLO qg
ross
se tion
also
on
tains
a
large
logarithm
when
one
of
the
photons
is
pro-
du ed
from
a
ollinear q
(−) →q
(−)γ
splitting
in
the
nal
state.
This
nal-state
ollinear
limit
is
dis ussed
in
Se tion
I
I
C
.
6
With
on
tributions
from
the
initial-state
soft
or
ollinear
radiation
in luded,
the
NLO
ross
se tion
is
appro
ximated
in
the
small-QT
asymptoti
limit
b
y
Aq ̄
q(Q, QT, y, Ω∗) =
X
i=u, ̄
u,d, ̄
d,...
Σi(θ∗)
S
n
δ( ⃗
QT)Fi,δ(Q, y, θ∗) + Fi,+(Q, y, QT)
o
(13)
in
the q ̄
q + qg
s attering
hannel,
and
b
y
Agg(Q, QT, y, Ω∗) = 1
S
Σg(θ∗)
h
δ( ⃗
QT)Fg,δ(Q, y, θ∗) + Fg,+(Q, y, QT)
i
+Σ′
g(θ∗, φ∗)F ′
g(Q, y, QT)
(14)
in
the gg+gqS
s attering
hannel.
The
fun tions Fa,δ(Q, y, θ∗)
and F (′)
a,+(Q, y, QT)
for
relev
an
t
parton
a
v
ors a
are
listed
in
App
endix
B
.
They
in lude
`plus
fun tion'
on
tributions
of
the
t
yp
e
Q−2
T lnp (Q2/Q2
T)
+
with p ≥0 ,
univ
ersal
fun tions
des ribing
soft
and
ollinear
s attering,
and
pro
ess-dep
enden
t
orre tions
from
NLO
virtual
diagrams.
The q ̄
q + qg
asymptoti
ross
se tion Aq ̄
q(Q, QT, y, Ω∗)
is
prop
ortional
to
the
angular
fun tion Σi(θ∗),
the
same
as
in
the
Born q ̄
q →γγ
ross
se tion,
f.
Eq.
(7
).
Similarly
,
the
gg +gqS
asymptoti
ross
se tion Agg(Q, QT, y, Ω∗)
in ludes
a
term
prop
ortional
to
the
Born
angular
fun tion Σg(θ∗).
In
addition, Agg(Q, QT, y, Ω∗)
on
tains
another
term
prop
ortional
to Σ′
g(θ∗, φ∗) ≡L′
g(θ∗) cos 2φ∗
,
where L′
g(θ∗)
is
deriv
ed
in
Ref.
[3℄.
This
term
arises
due
to
the
in
terferen e
of
Born
amplitudes
with
in oming
gluons
of
opp
osite
p
olarizations
and
ae ts
the
azim
uthal
angle
(φ∗
)
distribution
of
the
photons
in
the
Collins-Sop
er
frame
[3℄.
The
small-QT
represen
tations
in
Eqs.
(13)
and
(14)
an
b
e
used
to
ompute
xed-order
parti le
distributions
in
the
phase-spa e
sli ing
metho
d.
In
this
metho
d,
w
e
ho
ose
a
small
QT
v
alue Qsep
T
in
the
range
of
v
alidit
y
of
Eqs.
(13)
and
(14).
If
the
a tual QT
in
the
omputation
ex eeds Qsep
T
,
w
e
al ulate
the
dieren
tial
ross
se tion
using
the
full 2 →3
matrix
elemen
t.
When QT
is
smaller
than Qsep
T
,
w
e
al ulate
the
ev
en
t
rate
using
the
small-
QT
asymptoti
appro
ximation A(Q, QT, y, Ω∗)
and 2 →2
phase
spa e.
Hen e,
the
lo
w
est
bin
of
the QT
distribution
is
appro
ximated
in
the
NLO
predi tion
b
y
its
aver
age
v
alue
in
the
in
terv
al 0 ≤QT ≤Qsep
T
,
omputed
b
y
in
tegration
of
the
asymptoti
appro
ximations.
The
phase-spa e
sli ing
pro
edure
is
su ien
t
for
predi tions
of
observ
ables
in lusiv
e
in
QT
,
but
not
of
the
shap
e
of dσ/dQT
distributions.
The
latter
goal
is
met
b
y
all-orders
summa-
tion
of
singular
asymptoti
on
tributions
with
the
help
of
the
Collins-Sop
er-Sterman
(CSS)
metho
d
[11,
16
,
17℄.
The
small-QT
resummed
ross
se tion
is
denoted
as W(Q, QT, y, Ω∗)
and
giv
en
b
y
a
t
w
o-dimensional
F
ourier
transform
of
a
fun tion f
W(Q, b, y, Ω∗)
that
dep
ends
on
the
impa t
parameter ⃗
b :
W(Q, QT, y, Ω∗) =
Z
d⃗
b
(2π)2ei ⃗
QT *⃗
bf
W(Q, b, y, Ω∗)
≡
Z
d⃗
b
(2π)2ei ⃗
QT *⃗
bf
Wpert(Q, b∗, y, Ω∗)e−FNP (Q,b).
(15)
In
this
equation,
f
W(Q, b, y, Ω∗)
is
written
as
a
pro
du t
of
the
p
erturbativ
e
part
f
Wpert(Q, b∗, y, Ω∗)
and
the
nonp
erturbativ
e
exp
onen
t exp (−FNP(Q, b)) ,
whi
h
des rib
e
the
7
dynami s
at
small
(b ≲1
Ge
V−1
)
and
large
(b ≳1
Ge
V−1
)
impa t
parameters,
resp
e tiv
ely
.
The
purp
ose
of
the
v
ariable b∗
is
review
ed
b
elo
w.
If Q
is
large,
the
p
erturbativ
e
form
fa tor f
Wpert
dominates
the
in
tegral
in
Eq.
(15
).
It
is
omputed
at
small b
as
f
Wpert(Q, b, y, θ∗) =
X
a
Σa(θ∗)
S
h2
a(Q, θ∗)e−Sa(Q,b)
×
Ca/a1 ⊗fa1/h1
(x1, b; μ)
C ̄
a/a2 ⊗fa2/h2
(x2, b; μ).
(16)
The
hard-v
ertex
fun tion Σa(θ∗)h2
a(Q, θ∗)
is
the
normalized
ross
se tion
for
the
Born
s at-
tering a ̄
a →γγ
,
with a = u, ̄
u, d, ̄
d, ...
in q ̄
q →γγ
,
and a = ̄
a = g
in gg →γγ
.
The
Sudak
o
v
exp
onen
t
Sa(Q, b) =
Z C2
2Q2
C2
1/b2
d ̄
μ2
̄
μ2
Aa (C1, ̄
μ) ln
C2
2Q2
̄
μ2
+ Ba (C1, C2, ̄
μ)
(17)
is
an
in
tegral
of
t
w
o
fun tions Aa (C1, ̄
μ)
and Ba (C1, C2, ̄
μ)
b
et
w
een
momen
tum
s ales C1/b
and C2Q
,
and C1
and C2
are
onstan
ts
of
order c0 ≡2e−γE = 1.123...
and 1 ,
resp
e tiv
ely
.
The
sym
b
ol
Ca/a1 ⊗fa1/h
(x, b; μ)
stands
for
a
on
v
olution
of
the kT−
in
tegrated
PDF fa1/h(x, μ)
and
Wilson
o
e ien
t
fun tion Ca/a1(x, b; C1/C2, μ),
ev
aluated
at
a
fa torization
s ale μ
and
summed
o
v
er
in
termediate
parton
a
v
ors a1
:
Ca/a1 ⊗fa1/h
(x, b; μ) ≡
X
a1
Z 1
x
dξ
ξ Ca/a1
x
ξ , b; C1
C2
, μ
fa1/h(ξ, μ)
.
(18)
W
e
ompute
the
fun tions ha, Aa
, Ba
and Ca/a1
up
to
orders αs, α3
s, α2
s,
and αs,
resp
e tiv
ely
,
orresp
onding
to
the
NNLL
a ura y
of
resummation.
The
p
erturbativ
e
o
e ien
ts
at
these
orders
in αs
are
listed
in
App
endix
A.
The
subleading
on
tribution
from
the
nonp
erturbativ
e
region b ≳1
Ge
V−1
is
in luded
in
our
al ulation
using
a
revised
b∗
mo
del
[18℄,
whi
h
pro
vides
ex ellen
t
agreemen
t
with pT
-
dep
enden
t
data
on
Drell-Y
an
pair
and Z
b
oson
pro
du tion.
In
this
mo
del,
the
p
erturbativ
e
form
fa tor f
Wpert(Q, b∗, y, Ω∗)
in
Eq.
(15)
is
ev
aluated
as
a
fun tion
of b∗≡b/(1+b2/b2
max)1/2,
with bmax = 1.5
Ge
V−1
.
The
fa torization
s ale μ
in [C ⊗f]
is
set
equal
to c0
p
b−2 + Q2
ini
,
where Qini
is
the
initial
s ale
of
order
1
Ge
V
in
the
parameterization
emplo
y
ed
for fa/h(x, μ),
for
instan e,
1.3
Ge
V
for
the
CTEQ6
PDF
s
[19℄.
W
e
ha
v
e f
Wpert(b∗) = f
Wpert(b)
at b2 ≪b2
max,
and f
Wpert(b∗) = f
Wpert(bmax)
at b2 ≫b2
max
.
Hen e,
this
ansatz
preserv
es
the
exa t
form
of
the
p
erturbativ
e
form
fa tor f
Wpert(Q, b, y, Ω∗)
in
the
p
erturbativ
e
region
of
small b
,
while
also
in orp
orating
the
leading
nonp
erturbativ
e
on
tributions
(des rib
ed
b
y
a
phenomenologi al
fun tion FNP(Q, b))
at
large b
.
The
form
of FNP(Q, b)
found
in
the
global pT
t
in
Ref.
[18
℄
suggests
appro
ximate
inde-
p
enden e
of FNP(Q, b)
from
the
t
yp
e
of q ̄
q
s attering
pro
ess.
It
is
used
here
to
des rib
e
the
nonp
erturbativ
e
terms
in
the
leading q ̄
q →γγ
hannel.
W
e
negle t
p
ossible
orre tions
to
the
nonp
erturbativ
e
on
tributions
arising
from
the
nal-state
soft
radiation
in
the qg
han-
nel
and
additional
√
S
dep
enden e
ae ting
Drell-Y
an-lik
e
pro
esses
at x ≲10−2
[20℄,
as
these
ex eed
the
a ura y
of
the
presen
t
measuremen
ts
at
the
T
ev
atron.
The
exp
erimen
tally
unkno
wn FNP(Q, b)
in
the gg
hannel
is
appro
ximated
b
y FNP(Q, b)
for
the q ̄
q
hannel,
8
m
ultiplied
b
y
the
ratio CA/CF = 9/4 .
This
hoi e
is
motiv
ated
b
y
the
fa t
that
the
lead-
ing
Sudak
o
v
olor
fa tors A(k)
a
in
the gg
and q ̄
q
hannels
are
prop
ortional
to CA = 3
and
CF = 4/3 ,
resp
e tiv
ely
.
The
un ertain
ties
in
the γγ
ross
se tions
asso
iated
with FNP(Q, b)
are
in
v
estigated
n
umeri ally
in
Ref.
[3℄.
In
the
region QT ∼Q
,
ollinear
QCD
fa torization
at
a
nite
xed
order
in αs
is
ap-
pli able.
In
order
to
in lude
non-singular
on
tributions
imp
ortan
t
in
this
region,
w
e
add
to
W(Q, QT, y, Ω∗)
the
regular
pie e Y (Q, QT, y, Ω∗),
de ned
as
the
dieren e
b
et
w
een
the
NLO
ross
se tion P(Q, QT, y, Ω∗)
and
its
small-QT
asymptoti
appro
ximation A(Q, QT, y, Ω∗):
dσ(h1h2 →γγ)
dQ dQ2
T dy dΩ∗
= W(Q, QT, y, Ω∗) + P(Q, QT, y, Ω∗) −A(Q, QT, y, Ω∗)
≡W(Q, QT, y, Ω∗) + Y (Q, QT, y, Ω∗).
(19)
A
t
small QT,
subtra tion
of A(Q, QT, y, Ω∗)
in
Eq.
(19
)
an els
large
initial-state
radia-
tiv
e
orre tions
in P(Q, QT, y, Ω∗),
whi
h
are
in orp
orated
in
their
resummed
form
within
W(Q, QT, y, Ω∗).
A
t QT
omparable
to Q
, A(Q, QT, y, Ω∗)
an els
the
leading
terms
in
W(Q, QT, y, Ω∗),
but
higher-order
on
tributions
remain
from
the
in nite
to
w
er
of
logarith-
mi
terms
that
are
resummed
in W
.
In
this
situation
the W + Y
ross
se tion
drops
b
elo
w
the
nite-order
result P(Q, QT, y, Ω∗)
at
some
v
alue
of QT
(referred
to
as
the
r
ossing
p
oint)
in
b
oth
the q ̄
q +qg
and gg +gqS
hannels,
for
ea
h Q
and y
.
W
e
use
the W +Y
ross
se tion
as
our
nal
predi tion
at QT
v
alues
b
elo
w
the
rossing
p
oin
t,
and
the
NLO
ross
se tion P
at QT
v
alues
ab
o
v
e
the
rossing
p
oin
t.
A
few
ommen
ts
are
in
order
ab
out
our
resummation
al ulation.
The
hard-v
ertex
on
tri-
bution Σa(θ∗)h2
a(Q, θ∗)
and
the
fun tions Ba (C1, C2, ̄
μ)
and Ca/a1(x, b; C/C2, μ)
an
b
e
v
aried
in
a
m
utually
omp
ensating
w
a
y
while
preserving
the
same
v
alue
of
the
form
fa tor W
up
to
higher-order
orre tions
in αs
.
This
am
biguit
y
,
or
dep
enden e
on
the
hosen
resummation
s
heme
[21℄
within
the
CSS
formalism,
an
b
e
emplo
y
ed
to
explore
the
sensitivit
y
of
the-
oreti al
predi tions
to
further
next-to-next-to-next-to-leading
logarithmi
(NNNLL)
ee ts
that
are
not
a oun
ted
for
expli itly
.
The
p
erturbativ
e
o
e ien
ts
in
App
endix
A
are
presen
ted
in
the
CSS
resummation
s
heme
[11℄,
our
default
hoi e
in
n
umeri al
al ulations,
and
in
an
alternativ
e
s
heme
b
y
Catani,
de
Florian
and
Grazzini
(CF
G)
[21℄.
In
the
original
CSS
resummation
s
heme,
the B
and
C
fun tions
on
tain
the
nite
virtual
NLO
orre tions
to
the 2 →2
s attering
pro
ess,
whereas
in
the
CF
G
s
heme
the
univ
ersal B
and C
dep
end
only
on
the
t
yp
e
of
in iden
t
partons,
and
the
pro
ess-dep
enden
t
virtual
orre tion
is
in luded
in
the
fun tion ha
.
The
dieren e
b
et
w
een
the
CSS
and
CF
G
s
hemes
is
n
umeri ally
small
in γγ
pro
du tion
at
b
oth
the
T
ev
atron
and
the
LHC
[3℄.
In
the gg + gqS
s attering
hannel,
the
unp
olarized
resummed
ross
se tion
in ludes
an
additional
on
tribution
from
elemen
ts
of kT
-dep
enden
t
PDF
spin
matri es
with
opp
osite
heli ities
of
outgoing
gluons
[3
℄.
The
NLO
expansion
of
this
spin- ip
resummed
ross
se tion
generates
the
term
prop
ortional
to Σ′
g(θ∗, φ∗) ∝cos 2φ∗
in
the
small-QT
asymptoti
ross
se tion,
f.
Eq.
(14).
Although
the
logarithmi
spin- ip
on
tribution
m
ust
b
e
resummed
in
prin iple
to
all
orders
to
predi t
the φ∗
dep
enden e
in
the gg + gqS
hannel,
it
is
negle ted
in
the
presen
t
w
ork
in
view
of
its
small
ee t
on
the
full γγ
ross
se tion.
When
in
tegrated
o
v
er QT
from
0
to
s ales
of
order Q
,
the
resummed
ross
se tion
b
e-
omes
appro
ximately
equal
to
the
nite-order
(NLO)
ross
se tion,
augmen
ted
t
ypi ally
b
y
a
few-p
er en
t
orre tion
from
in
tegrated
higher-order
terms
logarithmi
in QT
.
In lusiv
e
9
observ
ables
that
allo
w
su
h
in
tegration
(e.g.,
the
large-Q
region
of
the γγ
in
v
arian
t
mass
distribution)
are
appro
ximated
w
ell
b
oth
b
y
the
resummed
and
NLO
al ulations.
Ho
w-
ev
er,
the
exp
erimen
tal
a eptan e
onstrains
the
range
of
the
in
tegration
o
v
er QT
in
parts
of
phase
spa e
and
ma
y
break
deli ate
an ellations
b
et
w
een
in
tegrable
singularities
presen
t
in
the
nite-order
dieren
tial
distribution.
In
this
situation
(e.g.,
in
the
vi init
y
of
the
kine-
mati
uto
in dσ/dQ
dis ussed
in
Se .
I
I
I
)
the
NLO
ross
se tion
b
e omes
unstable,
while
the
resummed
ross
se tion
(free
of
dis on
tin
uities)
on
tin
ues
to
dep
end
smo
othly
on
kine-
mati
onstrain
ts.
W
e
see
that
the
resummation
is
essen
tial
not
only
for
the
predi tion
of
ph
ysi al QT
distributions
in γγ
pro
du tion,
but
also
for
redible
estimates
of
the
ee ts
of
exp
erimen
tal
a eptan e
on
distributions
in
the
diphoton
in
v
arian
t
mass
and
other
v
ariables.
C.
Final-state
photon
fragmen
tation
1.
Single-photon
fr
agmentation
In
addition
to
the
QCD
singularities
asso
iated
with
initial-state
radiation
[des rib
ed
b
y
the
asymptoti
terms
in
Eqs.
(13)
and
(14
)℄,
other
singularities
arise
in
the O(αs)
pro
ess
q(p1) + g(p2) →γ(p3) + γ(p4) + q(p5)
[Fig.
1
(e)℄
when
a
photon
is
ollinear
to
the
nal-state
quark.
In
this
limit,
the qg →qγγ
squared
matrix
elemen
t
gro
ws
as 1/sγ5
,
when sγ5 →0 ,
where sγ5
is
the
squared
in
v
arian
t
mass
of
the
ollinear γq
pair.
In
this
limit,
the
squared
matrix
elemen
t
fa tors
as
|M(qg →qγγ)|2 ≈2e2e2
i
sγ5
Pγ←q(z)|M(qg →qγ)|2
(20)
in
to
the
pro
du t
of
the
squared
matrix
elemen
t |M(qg →qγ)|2
for
the
pro
du tion
of
a
photon
and
an
in
termediate
quark,
and
a
splitting
fun tion Pγ←q(z) = (1 + (1 −z)2)/z
for
fragmen
tation
of
the
in
termediate
quark
in
to
a
ollinear γq
pair.
In
Eq.
(20
) z
is
the
ligh
t-
one
fra tion
of
the
in
termediate
quark's
momen
tum
arried
b
y
the
fragmen
tation
photon,
and eei
is
the
harge
of
the
in
termediate
quark.
When
the
photon-quark
separation ∆r =
p
(η5 −ηγ)2 + (φ5 −φγ)2
in
the
plane
of
pseudorapidit
y η = −log(tan(θ/2))
and
azim
uthal
angle φ
in
the
lab
frame
is
small,
as
in
the
ollinear
limit, sγ5 ≈ETγET5∆r2,
where ETγ
and ET5
are
the
transv
erse
energies
of
the
photon
and
quark,
with ET ≡E sin θ
.
Note
that
ET5 = QT
at
the
order
in αs
at
whi
h
w
e
are
w
orking.
Therefore,
on
tributions
from
the
nal-state
ollinear,
or
fragmen
tation,
region
are
most
pronoun ed
at
small ∆r
and
relativ
ely
small QT.
1
A
fully
onsisten
t
treatmen
t
of
the
initial-
and
nal-state
singularities
w
ould
require
a
join
t
initial-
and
nal-state
resummation.
In
the
approa
hes
tak
en
to
date,
the
fragmen
tation
singularit
y
ma
y
b
e
subtra ted
from
the
dire t
ross
se tion
and
repla ed
b
y
a
single-photon
one-fragmen
tation
on
tribution q + g →(q
frag
−
→γ) + γ
,
where
(
frag
q −
→γ)
denotes
ollinear
pro
du tion
of
one
hard
photon
from
a
quark,
des rib
ed
b
y
a
fun tion Dγ(z, μ)
at
a
ligh
t-
one
momen
tum
fra tion z
and
fa torization
s ale μ .
Single-photon
t
w
o-fragmen
tation
1
In
the
soft,
or E5 →0,
limit,
the
nal-state
ollinear
on
tribution
is
suppressed,
re e ting
the
absen e
of
the
soft
singularit
y
in
the qg →qγγ
ross
se tion.
10
on
tributions
arise
in
pro
esses
lik
e g + g →(q
frag
−
→γ) + ( ̄
q
frag
−
→γ)
and
in
v
olv
e
on
v
olutions
with
t
w
o
fun tions Dγ(z, μ)
(one
p
er
photon).
The
lo
w
est-order
F
eynman
diagrams
for
the
one-
and
t
w
o-fragmen
tation
on
tributions
are
sho
wn
in
Figs.
1
(f
)
and
1
(g),
resp
e tiv
ely
.
P
arameterizations
m
ust
b
e
adopted
for
the
nonp
erturbativ
e
fun tions Dγ(z, μ)
at
an
initial
s ale μ = μ0
.
This
is
the
approa
h
follo
w
ed
in
the
DIPHO
X
al ulation
[14℄,
in
whi
h
the
sum
of
real
and
virtual
NLO
orre tions
to
dire t
and
single-γ
fragmen
tation
ross
se tions
is
in luded.
When
expli it
fragmen
tation
fun tion
on
tributions
are
in luded,
the
in lusiv
e
rate
is
in reased
b
y
higher-order
on
tributions
from
photon
pro
du tion
within
hadroni
jets.
Ho
w
ev
er,
m
u
h
of
the
enhan emen
t
is
suppressed
b
y
isolation
onstrain
ts
imp
osed
on
the
in lusiv
e
photon
ross
se tions
b
efore
the
omparison
with
data.
Nev
ertheless,
fragmen
tation
on
tributions
surviving
isolation
ma
y
b
e
mo
derately
imp
ortan
t
in
parts
of
phase
spa e.
An
infrared-safe
pro
edure
an
b
e
form
ulated
to
apply
isolation
uts
at
ea
h
order
of
αs
[22
,
23,
24℄.
This
pro
edure
en oun
ters
di ulties
in
repro
du ing
the
ee ts
of
isolation
on
fragmen
tation
on
tributions,
b
e ause
theoreti al
mo
dels
re e t
only
basi
features
of
the
exp
erimen
tal
isolation
and
ma
y
in
tro
du e
new
logarithmi
singularities
near
the
edges
of
the
isolation
ones.
As
men
tioned
in
the
In
tro
du tion,
the
magnitude
of
the
fragmen
tation
on
tribution
de-
p
ends
on
the
v
alues
of
isolation
parameters Eiso
T , ∆R ,
and ∆Rγγ
,
mo
deled
only
appro
ximately
in
a
theoreti al
al ulation.
The
ollinear
appro
ximation
onstrains
from
b
elo
w
the
v
alues
of z
a essible
to Dγ(z, μ): z > zmin ≡(1 + Eiso
T5/ETγ)−1
.
If Dγ(z, μ)
v
aries
rapidly
with z
,
the
fragmen
tation
ross
se tion
is
parti ularly
sensitiv
e
to
the
assumed
v
alues
of Eiso
T
and
zmin
.
F
or
instan e,
if Dγ(z, μ) ∼1/z
,
the
fragmen
tation
ross
se tion
is
roughly
prop
ortional
to Eiso
T
under
a
t
ypi al
ondition Eiso
T /ETγ ≪1 .
Su
h
nearly
linear
dep
enden e
on Eiso
T
of
the
fragmen
tation
ross
se tion dσ/dQT
is
indeed
observ
ed
in
the
DIPHO
X
al ulation,
as
review
ed
in
Se .
I
I
I.
In
realit
y
,
some
spread
of
the
parton
radiation
in
the
dire tion
trans-
v
erse
to
the
photon's
motion
is
exp
e ted.
The
treatmen
t
of
kinemati s
in
parton
sho
w
ering
programs
lik
e
PYTHIA
results
in
somewhat
dieren
t
dep
enden e
on z
[12℄
ompared
to
the
ollinear
appro
ximation,
hen e
in
a
dieren
t
magnitude
of
the
fragmen
tation
ross
se tion.
In
this
w
ork
w
e
adopt
a
pro
edure
that
repro
du es
desirable
features
of
the
isolated
ross
se tions,
while
b
ypassing
some
of
the
di ulties
summarized
ab
o
v
e.
T
o
sim
ulate
exp
erimen-
tal
isolation,
w
e
reje t
an
ev
en
t
if
(a)
the
separation ∆r
b
et
w
een
the
nal-state
parton
and
one
of
the
photons
is
less
than ∆R ,
and
(b) ET5
of
the
parton
is
larger
than Eiso
T
.
This
ondition
is
applied
to
the
NLO
ross
se tion P(Q, QT, y, Ω∗),
but
not
to W(Q, QT, y, Ω∗)
and A(Q, QT, y, θ∗),
as
these
orresp
ond
to
initial-state
QCD
radiation
and
are
free
of
the
nal-state
ollinear
singularit
y
.
This
quasi-exp
erimen
tal
isolation
ex ludes
the
singular
nal-state
dire t
on
tributions
at
ET5 > Eiso
T
and ∆r < ∆R
(or sγ5 < ETγET5∆R2
).
It
is
ee tiv
e
for QT > Eiso
T
,
but
the
ollinear
dire t
on
tributions
surviv
e
when QT < Eiso
T
.
The
in
tegrated
(but
not
the
dieren
tial)
fragmen
tation
rate
in
the
region QT < Eiso
T
ma
y
b
e
estimated
from
a
al ulation
with
expli it
fragmen
tation
fun tions.
In
our
approa
h,
w
e
do
not
in
tro
du e
fragmen
tation
fun tions,
but
w
e
apply
an
auxiliary
regulator
to
the
dire t qg
ross
se tion
at QT < Eiso
T
and ∆r < ∆R .
In
our
n
umeri al
study
w
e
nd
that
this
pres ription
preserv
es
a
on
tin
uous
dieren
tial
distribution
ex ept
for
a
small
nite
dis on
tin
uit
y
at QT = Eiso
T
.
It
appro
ximately
repro
du es
the
in
tegrated qg
rate
obtained
in
the
DIPHO
X
al ulation
at
small QT
,
for
the
nominal Eiso
T
.
11
Figure
2:
Lo
w
est-order
F
eynman
diagrams
des ribing
fragmen
tation
of
the
nal-state
partons
in
to
photon
pairs
with
relativ
ely
small
mass Q.
T
w
o
forms
of
the
auxiliary
regulator
are
onsidered
b
elo
w,
based
on
subtra tion
of
the
leading
ollinear
on
tribution
and
smo
oth- one
isolation
[25
℄.
In
the
rst
ase,
w
e
subtra t
the
leading
part
Eq.
(20
)
of
the
dire t qg
matrix
elemen
t
when ET5 < Eiso
T
and ∆r < ∆R.
W
e
tak
e z = 1 −ps * p5/(ps * pf + ps * p5 + pf * p5),
where pμ
f, pμ
5,
and pμ
s
are
the
four-momen
ta
of
the
fragmen
tation
photon,
fragmen
tation
quark,
and
sp
e tator
photon,
resp
e tiv
ely
[26℄.
This
pres ription
is
used
in
most
of
the
n
umeri al
results
in
this
pap
er.
In
the
se ond
ase,
w
e
suppress
fragmen
tation
on
tributions
at ∆r < ∆R
and ET5 < Eiso
T
b
y
reje ting
ev
en
ts
in
the ∆R
one
that
satisfy ET5 < χ(∆r),
where χ(∆r)
is
a
smo
oth
fun tion
satisfying χ(0) = 0, χ(∆R) = Eiso
T
.
This
smo
oth- one
isolation
[25℄
transforms
the
fragmen
tation
singularit
y
asso
iated
with Dγ(z, μ)
in
to
an
in
tegrable
singularit
y
,
whi
h
dep
ends
on
the
assumed
fun tional
form
of χ(∆r).
The
ross
se tion
for
dire t
on
tributions
is
rendered
nite
b
y
this
pres ription
without
expli it
in
tro
du tion
of
fragmen
tation
fun tions
Dγ(z, μ).
F
or
our
smo
oth
fun tion,
w
e
ho
ose χ(∆r) = Eiso
T (1−cos ∆r)2/(1−cos ∆R)2
,
whi
h
diers
from
the
sp
e i
form
onsidered
in
Ref.
[25℄,
but
still
satis es
the
ondition χ(0) = 0.
Our
earlier
results
in
Ref.
[2℄
are
omputed
with
this
pres ription.
Here
w
e
emplo
y
it
only
in
a
few
instan es
for
omparison
with
the
subtra tion
metho
d
and
obtain
similar
results.
Dieren es
b
et
w
een
the
t
w
o
pres riptions
an
b
e
used
to
quan
tify
sensitivit
y
of
the
pre-
di tions
to
the
treatmen
t
of
the QT < Eiso
T
and ∆r < ∆R
region.
The
t
w
o
pres riptions
yield
iden
ti al
predi tions
outside
of
this
restri ted
region,
notably
at QT > Eiso
T
,
where
our
NLO
p
erturbativ
e
expression P(Q, QT, y, Ω∗)
in
the q ̄
q + qg
hannel
is
on
trolled
only
b
y
quasi-exp
erimen
tal
isolation
and
oin ides
with
the
orresp
onding
dire t
ross
se tion
in
DIPHO
X.
The
default
subtra tion
pres ription
predi ts
a
v
anishing dσ/dQT
in
the
extreme
QT →0
limit,
while
the
smo
oth- one
pres ription
has
an
in
tegrable
singularit
y
in
this
limit,
a
v
oided
b
y
an
expli it
small-QT
uto
in
the
al ulation
of
our Y
-pie e.
Both
pres riptions
are
free
of
the
logarithmi
singularit
y
at QT = Eiso
T
arising
in
the
xed-order
(DIPHO
X)
al ulation.
2.
L
ow-Q
diphoton
fr
agmentation
Another
lass
of
large
radiativ
e
orre tions
arises
when
the γγ
in
v
arian
t
mass Q
is
smaller
than
the γγ
transv
erse
momen
tum QT
.
In
this
ase,
one
nal-state
quark
or
gluon
fragmen
ts
in
to
a
lo
w-mass γγ
pair,
e.g.
as q + g →(q
frag
−
→γγ) + g
.
The
lo
w
est-order
on
tributions
of
this
kind
are
sho
wn
in
Fig.
2
.
The
pro
ess
is
des rib
ed
b
y
a γγ
-fragmen
tation
fun -
tion Dγγ(z1, z2, μ),
dieren
t
from
the
single-photon
fragmen
tation
fun tion Dγ(z, μ).
This
12
new
t
w
o-photons
from
one-fragmen
tation
on
tribution
is
not
in luded
y
et
in
existing
al-
ulations,
ev
en
though
similar
fragmen
tation
me
hanisms
ha
v
e
b
een
studied
in
large-QT
Drell-Y
an
pair
pro
du tion
[27,
28
℄.
The
imp
ortan e
of
lo
w-Q γγ
-fragmen
tation
ma
y
b
e
el-
ev
ated
in
some
kinemati
regions
for
t
ypi al
exp
erimen
tal
uts.
They
an
b
e
remo
v
ed
b
y
adjustmen
ts
in
the
exp
erimen
tal
uts,
as
dis ussed
in
Se .
I
I
I.
D.
Summary
of
the
al ulation
W
e
on lude
this
se tion
b
y
summarizing
the
main
features
of
our
al ulation.
F
ull
dire t
NLO
ross
se tions,
represen
ted
b
y
the
graphs
(a)-(e),
(h)-(l)
in
Fig.
1,
are
omputed,
and
their
initial-state
soft/ ollinear
logarithmi
singularities
are
resummed
at
small QT
in
b
oth
the q ̄
q +qg
and gg +gqS
hannels.
The
p
erturbativ
e
Sudak
o
v
fun tions A
and B
and
Wilson
o
e ien
t
fun tions C
in
the
resummed
ross
se tion W
are
omputed
up
to
orders α3
s, α2
s
,
and αs
,
resp
e tiv
ely
,
orresp
onding
to
resummation
at
NNLL
a ura y
.
Our
resummation
al ulation
requires
an
in
tegration
o
v
er
all
v
alues
of
impa t
parameter
b
,
in luding
the
nonp
erturbativ
e
region
of
large b
.
In
our
default
al ulation
of
the
resummed
ross
se tion,
w
e
adopt
the
nonp
erturbativ
e
fun tions
in
tro
du ed
in
Ref.
[18℄.
W
e
onsider
t
w
o
resummation
s
hemes,
the
traditional
s
heme
in
tro
du ed
in
the
CSS
pap
er
as
w
ell
as
an
alternativ
e
s
heme
[21℄.
The
omparison
allo
ws
us
to
estimate
the
magnitude
of
y
et
higher-
order
orre tions
that
are
not
in luded.
The
size
of
these
ee ts
is
dieren
t
in
the q ̄
q + qg
and gg + gqS
hannels
but
not
parti ularly
signi an
t
in
either
[3℄.
The
nal-state
ollinear
singularit
y
in
the qg
s attering
hannel
is
a
v
oided
b
y
applying
quasi-exp
erimen
tal
isolation
when QT > Eiso
T
and
an
auxiliary
regulator
when QT < Eiso
T
to
appro
ximate
on
a
v
erage
the
full
NLO
rate
from
dire t qg
and
fragmen
tation
ross
se tions
in
this QT
range.
T
w
o
pres riptions
for
the
auxiliary
regulator
(subtra tion
and
smo
oth
isolation
inside
the
photon's
isolation
one)
are
onsidered
and
lead
to
similar
predi tions
at
the
T
ev
atron
and
LHC.
The
singular
logarithmi
on
tributions
asso
iated
with
initial-state
radiation
are
sub-
tra ted
from
the
NLO
ross
se tion P
to
form
a
regular
pie e Y,
whi
h
is
added
to
the
small-QT
resummed
ross
se tion W
to
predi t
the
pro
du tion
rate
for
small
and
in
termedi-
ate
v
alues
of QT
.
In
the gg + gqS
hannel,
w
e
also
subtra t
from P
a
new
singular
spin- ip
on
tribution
that
ae ts
azim
uthal
angle
(φ∗)
dep
enden e
in
the
Collins-Sop
er
referen e
frame.
W
e
swit
h
our
predi tion
to
the
xed-order
p
erturbativ
e
result P
at
the
p
oin
t
in QT
where
the
ross
se tion W + Y
drops
b
elo
w P
.
This
rossing
p
oin
t
is
lo
ated
at QT
of
order
Q
in
b
oth q ̄
q + qg
and gg + gqS
hannels.
I
I
I.
COMP
ARISONS
WITH
D
A
T
A
AND
PREDICTIONS
Our
al ulation
of
the
dieren
tial
ross
se tion dσ/(dQdQTdydΩ∗)
is
esp
e ially
p
ertinen
t
for
the
transv
erse
momen
tum QT
distribution
in
the
region QT ≲Q
,
for
xed
v
alues
of
diphoton
mass Q
( f.
Se tion
I
I
I
A
1).
It
w
ould
b
e
b
est
to
ompare
our
multiple
dieren
tial
distribution
with
exp
erimen
t,
but
published
ollider
data
tend
to
b
e
presen
ted
in
the
form
of
singly
dieren
tial
distributions
in Q
, QT
,
and ∆φ ≡φ3 −φ4
in
the
lab
frame,
after
in
tegration
o
v
er
the
other
indep
enden
t
kinemati
v
ariables.
W
e
follo
w
suit
in
order
to
mak
e
omparisons
with
T
ev
atron
ollider
data,
but
w
e
re ommend
that
more
dieren
tial
studies
13
b
e
made,
and
w
e
ommen
t
on
the
features
that
an
b
e
explored.
W
e
sho
w
results
at
the
energy
of
the
T
ev
atron
ollider
and
then
mak
e
predi tions
for
the
Large
Hadron
Collider.
The
analyti al
results
of
Se .
I
I
are
implemen
ted
in
our
omputer
o
de.
As
a
rst
step,
resummed
and
NLO γγ
ross
se tions
are
omputed
on
a
grid
of
dis rete
v
alues
of Q
, QT
,
and y
b
y
using
the
resummation
program
Lega
y
des rib
ed
in
Refs.
[29
,
30℄.
A
t
the
se ond
stage,
mat
hing
of
the
resummed
and
NLO
ross
se tions
is
p
erformed,
and
fully
dieren
tial
ross
se tions
are
ev
aluated
b
y
Mon
te-Carlo
in
tegration
of
the
mat
hed
grids
in
the
latest
v
ersion
of
the
program
ResBos
[31,
32
℄.
The
al ulation
is
done
for Nf = 5
a tiv
e
quark
a
v
ors
and
the
follo
wing
v
alues
of
the
ele tro
w
eak
and
strong
in
tera tion
parameters
[33℄:
GF = 1.16639 × 10−5
Ge
V−2,
mZ = 91.1882
Ge
V,
(21)
α(mZ) = 1/128.937,
αs(mZ) = 0.1187.
(22)
The
follo
wing
hoi es
of
the
fa torization
onstan
ts
are
used: C1 = C3 = 2e−γE ≈1.123...,
and C2 = C4 = 1.
The
hoi e C4 = 1
implies
that
w
e
equate
the
renormalization
and
fa torization
s ales
to
the
in
v
arian
t
mass
of
the
photon
pair, μR = μF = Q
,
in
the
xed-
order
and
asymptoti
on
tributions P(Q, QT, y, Ω∗)
and A(Q, QT, y, Ω∗).
W
e
use
t
w
o-lo
op
expressions
for
the
running
ele tromagneti
and
strong
ouplings α(μ)
and αS(μ),
as
w
ell
as
the
NLO
parton
distribution
fun tion
set
CTEQ6M
[19
℄
with Qini = 1.3
Ge
V.
F
or
al ulations
with
expli it
nal-state
fragmen
tation
fun tions
in luded,
w
e
use
set
1
of
the
NLO
photon
fragmen
tation
fun tions
from
Ref.
[34
℄.
A.
Results
for
Run
2
at
the
T
ev
atron
1.
Kinemati
onstr
aints
In
this
se tion,
w
e
presen
t
our
results
for
the
T
ev
atron p ̄
p
ollider
op
erating
at
√
S =
1.96
T
e
V.
In
order
to
ompare
with
the
data
from
the
Collider
Dete tor
at
F
ermilab
(CDF)
ollab
oration
[1℄,
w
e
mak
e
the
same
restri tions
on
the
nal-state
photons
as
those
used
in
the
exp
erimen
tal
measuremen
t
(unless
stated
otherwise):
transv
erse
momen
tum pγ
T > pγ
T min = 14 (13)
Ge
V
for
the
harder
(softer)
photon,
(23)
and
rapidit
y |yγ| < 0.9
for
ea
h
photon.
(24)
W
e
imp
ose
isolation
onditions
des rib
ed
in
Se tion
I
I
C,
assuming
the
nominal
isolation
energy Eiso
T
= 1
Ge
V
sp
e i ed
in
the
CDF
publi ation,
along
with ∆R = 0.4,
and ∆Rγγ =
0.3 .
W
e
also
sho
w
predi tions
for
the
onstrain
ts
that
appro
ximate
ev
en
t
sele tion
onditions
used
b
y
the
F
ermilab
DØ
Collab
oration
[35
℄: pγ
T > pγ
T min = 21 (20)
Ge
V
for
the
harder
(softer)
photon, |yγ| < 1.1 ,
and Eiso
T /Eγ
T = 0.07
for
ea
h
photon,
for
the
same ∆R
and ∆Rγγ
v
alues
as
in
the
CDF
ase.
A
s atter
plot
of
ev
en
t
distributions
from
our
theoreti al
sim
ulation
for
CDF
kinemati
uts
and
arbitrary
luminosit
y
is
sho
wn
in
Fig.
3
.
The
ev
en
ts
are
plotted
v
ersus
the
in
v
arian
t
mass Q
,
transv
erse
momen
tum QT
,
rapidit
y
separation |∆y| ≡|yhard −ysoft|
,
and
azim
uthal
separation ∆φ ≡|φhard −φsoft|
(with 0 ≤∆φ ≤π)
b
et
w
een
the
harder
and
softer
photon
in
the
lab
frame,
as
w
ell
as
the
osine
of
the
p
olar
angle θ∗
in
the
Collins-Sop
er
frame.
It
an
b
e
seen
from
the
gure
that ∆φ
is
orrelated
with
the
dieren e QT −Q
.
Ev
en
ts
with
14
20
0
0
0
1
0
1.5
2
3
'
(rad)
Q
T
>
Q
Q
T
<
Q
Kinemati al
distributions
in
the
T
evatron
Run-2
(CDF)
1
-1
2
40
100
80
60
40
20
0
100
80
60
0.5
1
2.5
Q
T
(Ge
V)
jy
j
os
?
Q
(Ge
V)
p
T
uts
R
ut
Figure
3:
The
diphoton
ev
en
t
distribution
from
the
theoreti al
sim
ulation
for
√
S = 1.96
Ge
V,
with
the
sele tion
riteria
imp
osed
in
the
CDF
measuremen
t,
as
a
fun tion
of
the
v
arious
kinemati
v
ariables
des rib
ed
in
the
text,
sho
wn
for QT < Q
and QT > Q
separately
.
QT < Q
(QT > Q
)
tend
to
p
opulate
regions
with ∆φ > π/2
(∆φ < π/2 ).
The
extreme
ase
QT = 0
relev
an
t
to
the
Born
appro
ximation
orresp
onds
to ∆φ = π
.
The pγ
T
uts
suppress
the
mass
region Q ≲2
p
pγ3
Tminpγ4
Tmin ≈27
Ge
V
at ∆φ ≈π
and QT ≲
25
Ge
V
at ∆φ ≈0 ,
leading
to
the
app
earan e
of
a
kinemati
uto
in
the
in
v
arian
t
mass
distribution
and
a
shoulder
in
the
transv
erse
momen
tum
distribution,
as
sho
wn
in
later
se tions.
Our
theoreti al
framew
ork
is
appli able
in
the
region QT ≲Q
(large ∆φ),
where
the
dominan
t
fra tion
of
ev
en
ts
o
urs.
The
app
earan e
of
singularities
in
the
NLO
al ulation
at QT →0
and
the
fa t
that
there
are
t
w
o
dieren
t
hard
s ales, QT
and Q
,
relev
an
t
for
the
ev
en
t
distributions
in
the
lo
w-QT
region
require
that
w
e
address
and
resum
large
logarithmi
terms
of
the
form log(Q/QT).
Dieren
t
and
in
teresting
ph
ysi s
b
e omes
imp
ortan
t
in
the
omplemen
tary
region QT > Q
(small ∆φ),
a
topi
w
e
address
in
Se .
I
I
I
A
3
.
2.
T
evatr
on
r
oss
se
tions
W
e
ompare
our
resummed
and
nite-order
predi tions
for
the
in
v
arian
t
mass
(Q
)
dis-
tribution
of
photon
pairs,
sho
wn
in
Fig.
4
as
solid
and
dashed
lines,
resp
e tiv
ely
.
The
nite-order
ross
se tion
is
ev
aluated
at O(αs)
a ura y
in
the q ̄
q + qg
hannel
and
at O(α3
s)
a ura y
in
the gg + gqS
hannel.
These
nite-order
al ulations
are
p
erformed
with
the
15
pp
_ → γγX, √S = 1.96 TeV
Q (GeV)
dσ/dQ (pb/GeV)
Resummed (NNLL)
Fixed-order (NLO)
CDF, 207 pb-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10
20
30
40
50
60
70
80
90
100
Figure
4:
In
v
arian
t
mass
distributions
of
photon
pairs
in p ̄
p →γγX
at
√
S = 1.96
T
e
V
with
QCD
on
tributions
al ulated
in
the
softgluon
resummation
formalism
(red
solid)
and
at
NLO
(blue
dashed).
The
al ulations
in lude
the
uts
used
b
y
the
CDF
ollab
oration
whose
data
are
sho
wn
[1
℄.
phase-spa e
sli ing
metho
d
des rib
ed
in
Se .
I
I
B
.
When
in
tegrated
o
v
er
all QT
,
as
in
the
dσ/dQ
distribution
at
large Q
,
the
resummed
logarithmi
terms
from
higher
orders
in αs
pro
du e
a
relativ
ely
small
NNLO
orre tion,
su
h
that
the
resummed
and
nite-order
mass
distributions
in
Fig.
4
are
lose
to
one
another
in
normalization
and
shap
e.
Both
distributions
also
agree
with
the
CDF
data
in
this Q
range
within
exp
erimen
tal
un ertain
ties.
The
shap
e
of dσ/dQ
at
small Q
is
ae ted
b
y
the
uts
in
Eq.
(23)
on
the
transv
erse
momen
ta pγ
T
of
the
t
w
o
photons.
In
addition
to
b
eing
resp
onsible
for
the
hara teristi
uto
at Q ≈27
Ge
V
explained
in
the
previous
subse tion,
the
uts
on
the
individual
transv
erse
momen
ta pγ
T
also
in
tro
du e
a
dep
enden e
of
the
in
v
arian
t
mass
distribution
on
the
shap
e
of
the QT
sp
e trum
of
the γγ
pairs.
Be ause
of
this
orrelation
b
et
w
een
the Q
and QT
distributions,
the
dis on
tin
uities
in dσ/dQT
as QT →0 ,
when
omputed
at
nite
order,
mak
e
nite-order
predi tions
for dσ/dQ
somewhat
unstable.
The
nite-order
exp
e tation
for
the
transv
erse
momen
tum
distribution dσ/dQT
(i.e.,
the
in
tegral
of P(Q, QT, y, Ω∗)
o
v
er Q
, y
,
and Ω∗
,
or P
for
brevit
y)
is
sho
wn
as
a
dashed
urv
e
in
Fig.
5(a).
It
exhibits
an
in
tegrable
singularit
y
in
the
small-QT
limit.
T
erms
with
in
v
erse
p
o
w
er
and
logarithmi
dep
enden e
on QT
,
asso
iated
with
initial-state
radiation
as QT →0 ,
are
extra ted
from P
and
form
the
asymptoti
on
tribution,
denoted
as A
(dotted
urv
e).
In
the
gure,
b
oth P
and A
are
trun ated
at
a
small
v
alue
of QT
,
that
is,
not
dra
wn
all
the
w
a
y
to QT = 0.
The
urv
es
for P
and A
are
lose
at
small
v
alues
of QT
,
signaling
that
the
initial-state
logarithmi
singularities
dominate
the
NLO
distribution.
The
dieren e Y
16
10
-1
1
10
0
5
10
15
20
25
30
35
40
pp
_ → γγX, √S = 1.96 TeV
QT (GeV)
dσ/dQT (pb/GeV)
Fixed order (NLO)
Asymptotic
CDF, 207 pb-1
pp
_ → γγX, √S = 1.96 TeV
QT (GeV)
dσ/dQT (pb/GeV)
Resummed (NNLL)
Finite-order (NLO)
CDF, 207 pb-1
10
-1
1
0
5
10
15
20
25
30
35
40
(a)
(b)
Figure
5:
T
ransv
erse
momen
tum
distributions
in p ̄
p →γγX
at
√
S = 1.96
T
e
V
along
with
the
CDF
data:
(a)
the
xed-order
predi tion P
(dashes)
and
its
asymptoti
appro
ximation A
(dots);
(b)
the
full
resummed
ross
se tion
(solid),
obtained
b
y
mat
hing
the
resummed W + Y
to
the
xed-order
predi tion P
(dashed,
same
as
in
(a))
at
large QT
.
b
et
w
een
the P
and A
distributions
in ludes
the
nite
regular
terms
not
in luded
in A
and
logarithmi
terms
from
the
nal-state
fragmen
tation
singularities,
with
the
latter
subtra ted
when QT < Eiso
T
,
as
des rib
ed
in
Se .
I
I
C
.
The
data
learly
disfa
v
or
the
xed-order
predi tion
in
the
region
of
lo
w QT
.
Figure
5(b)
features
the
resummed W + Y
on
tribution
(solid
urv
e).
Resummation
of
the
initial-state
logarithmi
terms
renders W
nite
in
the
region
of
small QT
.
The
sum
of W
and Y
in ludes
the
resummed
initial-state
singular
on
tributions
plus
the
remaining
relev
an
t
terms
in P
.
Sin e P
pro
vides
a
reliable
xed-order
estimate
at
large QT
,
w
e
presen
t
our
nal
resummed
predi tion
b
y
swit
hing
from W + Y
to P
at
the
p
oin
t
at
whi
h
the
t
w
o
dieren
tial
ross
se tions
(as
fun tions
of Q
, QT
and y
)
ross
ea
h
other.
In
on
trast
to
the
xed-order
(dashed)
urv
e P
in
Fig.
5
(b),
the
agreemen
t
with
data
is
impro
v
ed
at
the
lo
w
est
v
alues
of QT
,
where
resummation
brings
the
rate
do
wn,
and
for QT = 12 −32
Ge
V,
where
the
resummed
logarithmi
terms
in rease
the
rate.
The
resummed
predi tions
for
the
T
ev
atron
exp
erimen
ts
are
pra ti ally
insensitiv
e
to
the
hoi e
of
the
resummation
s
heme
and
the
nonp
erturbativ
e
mo
del
[3℄.
Ab
out
75%
(25%)
of
the
total
rate
at
the
T
ev
atron
with
CDF
uts
imp
osed
omes
from
the q ̄
q+qg + ̄
qg
(gg +gqS
)
initial
state.
The
fra tions
for
the
uts
used
b
y
DØ
are
84%
and
16%.
The gg + gqS
on
tribution
falls
steeply
after QT > 22
Ge
V,
b
e ause
the
gluon
PDF
de reases
rapidly
with
parton
fra tional
momen
tum x
[3℄.
The
distribution
in
the
dieren e ∆φ
of
the
azim
uthal
angles
of
the
photons
is
sho
wn
in
Fig.
6
.
As
is
true
for
the
transv
erse
momen
tum
distribution
in
the
limit QT →0 ,
the
distribution
omputed
at
xed
order
is
ill-de ned
at ∆φ = π
.
The
resummed
distribution
sho
ws
a
larger
ross
se tion
near ∆φ = 2.5
rad,
in
b
etter
agreemen
t
with
the
data.
In
the
region
of
small ∆φ ≲π/2 ,
the
xed-order
and
the
resummed
predi tions
are
the
same,
a
17
pp
_ → γγX, √S = 1.96 TeV
∆φ (rad)
dσ/d∆φ (pb/rad)
Resummed (NNLL)
Fixed-order (NLO)
CDF, 207 pb-1
10
-1
1
10
0
0.5
1
1.5
2
2.5
3
Figure
6:
The
dieren e ∆φ
in
the
azim
uthal
angles
of
the
t
w
o
photons
in
the
lab
oratory
frame
predi ted
b
y
the
resummed
(solid)
and
xed-order
(dashed)
al ulations,
ompared
to
the
CDF
data.
result
of
our
mat
hing
of
the
resummed
and
xed-order
distributions
at
mid
to
high
v
alues
of QT
.
Although
the
ross
se tion
is
not
large
in
the
region ∆φ < π/2 ,
there
is
an
indi ation
of
a
dieren e
b
et
w
een
our
predi tions
and
data
in
this
region,
a
topi
w
e
address
b
elo
w.
3.
The
r
e
gion QT > Q
It
is
eviden
t
from
Fig.
3
that
the ∆φ < π/2
region
is
p
opulated
mostly
b
y
ev
en
ts
with
QT > Q
.
New
t
yp
es
of
radiativ
e
on
tributions
ma
y
b
e
presen
t
in
this
region,
in luding
v
arious
fragmen
tation
on
tributions
des rib
ed
in
Se .
I
I
C
and
enhan emen
ts
at
large | cos θ∗|
in
the
dire t
pro
du tion
rate.
While
exp
erimen
tal
isolation
generally
suppresses
long-distan e
fragmen
tation,
a
greater
fra tion
of
fragmen
tation
photons
are
exp
e ted
to
surviv
e
isolation
when ∆φ < π/2 .
Besides
single-photon
`one-fragmen
tation'
and
`t
w
o-fragmen
tation'
on
tributions
(with
one
photon
p
er
fragmen
ting
parton),
one
en oun
ters
additional
logarithmi
singularities
of
the
form
log(Q/QT).
W
e
noted
in
Se .
I
I
C
that
these
logarithms
are
asso
iated
with
the
fragmen
tation
of
a
parton
arrying
large
transv
erse
momen
tum QT
in
to
a
system
of
small
in
v
arian
t
mass Q
[27,
28℄,
a
ligh
t γγ
pair
in
our
ase.
Small-Q γγ
fragmen
tation
of
this
kind
is
not
implemen
ted
y
et
in
theoreti al
mo
dels.
Therefore,
w
e
are
prepared
for
the
p
ossibilit
y
that
b
oth
the
xed-order
al ulation
and
our
resummed
al ulation
ma
y
b
e
de ien
t
in
the
region QT ≫
Q
.
A
detailed
exp
erimen
tal
study
of
the
region QT > Q
ma
y
oer
the
opp
ortunit
y
to
measure
the
parton
to
t
w
o-photon
fragmen
tation
fun tion Dγγ(z1, z2),
pro
vided
that
the
18
single-photon
`one-fragmen
tation'
fun tion Dγ(z)
is
determined
b
y
single-photon
data,
and
the
lo
w-Q
logarithmi
terms
are
prop
erly
resummed
theoreti ally
.
In
addition
to
the
lo
w-Q
fragmen
tation,
the
small-∆φ
region
ma
y
b
e
sensitiv
e
to
large
higher-order
on
tributions
asso
iated
with b
t -
or b
u -
hannel
ex
hanges
in
the q ̄
q →γγ
and
gg →γγ
subpro
esses.
In
the
Born
pro
esses
in
Figs.
1
(a)
and
(h),
the b
t
-
and b
u
-
hannel
singularities
arise
at cos θ∗≈±1
and ∆φ ≈π
.
These
singularities
are
ex luded
b
y
the pγ
T
uts
in
Eq.
(23
),
but
related
residual
enhan emen
ts
in
the
NLO
on
tributions
ma
y
still
p
ersist
at | cos θ∗| ≈1
and ∆φ →0 ,
not
ex luded
b
y
the
uts
( f.
Fig.
3).
Be ause |cos θ∗|
is
large
in
su
h
ev
en
ts,
they
tend
to
ha
v
e
substan
tial |∆y| ,
so
they
are
retained
b
y
the ∆Rγγ > 0.3
ut.
In
on
trast,
the
lo
w-Q
fragmen
tation
on
tributions
tend
to
b
e
abundan
t
at
small |∆y| .
It
ma
y
b
e
therefore
p
ossible
to
distinguish
b
et
w
een
the
large-| cosθ∗|
and
fragmen
tation
ev
en
ts
at
small ∆φ
based
on
the
distribution
in |∆y| .
W
e
exp
e t
m
u
h
b
etter
agreemen
t
of
our
predi tions
with
data
if
the
sele tion QT < Q
is
made.
This
sele tion
preserv
es
the
bulk
of
the
ross
se tion
and
assures
that
a
fair
omparison
is
made
in
the
region
of
phase
spa e
where
the
predi tions
are
most
v
alid.
4.
F
r
agmentation
and
omp
arison
with
the
DIPHO
X
o
de
One
w
a
y
to
obtain
an
estimate
of
theoreti al
un ertain
t
y
is
to
ompare
theoreti al
ap-
proa
hes
in
v
arious
parts
of
phase
spa e,
in luding
small ∆φ.
W
e
handle
the
ollinear
nal-state
photon
singularities
in
the
manner
des rib
ed
in
Se .
I
I,
without
in luding
photon
fragmen
tation
fun tions
expli itly
.
An
alternativ
e
al ulation
implemen
ted
in
the
DIPHO
X
o
de
[14℄
in ludes
NLO
ross
se tions
for
single-photon
fragmen
tation
pro
esses.
Neither
o
de
in ludes
a
term
in
whi
h
b
oth
photons
are
fragmen
tation
pro
du ts
of
the
same
nal-
state
parton,
i.e.,
the
diphoton
fragmen
tation
fun tion Dγγ(z1, z2).
In
Ref.
[2℄
w
e
sho
w
omparisons
of
our
predi tions
with
those
of
DIPHO
X
along
with
the
CDF
data.
Here
in
Fig.
7
,
w
e
sho
w
analogous
plots
of
the
in
v
arian
t
mass
and
transv
erse
mo-
men
tum
distributions
for
DØ
uts.
W
e
note
that
our
xed-order q ̄
q + qg
on
tribution
agrees
w
ell
with
the
dire t
on
tribution
in
DIPHO
X.
This
agreemen
t
is
parti ularly
impressiv
e
in
the
region
of
large QT
,
where
b
oth
o
des
use
the
same
xed-order
formalism
to
handle
dire t
on
tributions.
A
on
tribution
from
the gg
hannel
is
also
presen
t
in
b
oth
o
des,
omputed
at
LO
in
DIPHO
X
but
at
NLO+NNLL
in
our
ase.
Sin e
the gg + gqS
on
tribution
is
not
dominan
t
(esp
e ially
in
the
high QT
region),
this
dieren e
do
es
not
ha
v
e
a
signi an
t
impa t
on
the
omparison.
The
expli it
single-photon
fragmen
tation
on
tributions
in
DIPHO
X
(mostly
`one-
fragmen
tation'
on
tribution)
are
quite
small
for
the
nominal
hadroni
energy Eiso
T
∼1
Ge
V
in
the
one
around
ea
h
photon.
Ex eptions
o
ur
in
the
region QT ≤Eiso
T
,
where
the
fragmen
tation
on
tributions
to dσ/dQT
ha
v
e
logarithmi
singularities,
and
in
the ∆φ →0
region,
where
fragmen
tation
is
omparable
to
the
dire t
on
tributions.
Our
isolation
pre-
s ription
repro
du es
the
in
tegrated
DIPHO
X
rate
w
ell
for 0 ≤QT ≤Eiso
T
,
leading
to
lose
agreemen
t
b
et
w
een
the
resummed
and
DIPHO
X
in lusiv
e
rates
for
most Q
v
alues.
Returning
to
the
CDF
measuremen
t,
w
e
remark
that
the
resummed
and
DIPHO
X
ross
se tions
for
the
same Eiso
T
= 1
Ge
V
underestimate
the
data
within
t
w
o
standard
deviations
for Q ≲27
Ge
V, QT > 25
Ge
V,
and ∆φ < 1
rad
( f.
the
relev
an
t
gures
in
Ref.
[2℄).
The
DIPHO
X
ross
se tion
an
b
e
raised
to
agree
with
data
in
this
shoulder
region,
if
a
m
u
h
larger
isolation
energy
(Eiso
T
= 4
Ge
V)
is
hosen,
and
smaller
fa torization
and
19
pp
_ → γγX, √S =1.96 GeV
Q (GeV)
dσ/dQ (pb/GeV)
ET
iso
/pTγ = 0.07, ∆Rcone = 0.4, ∆Rγγ > 0.3
(pTγhard, pTγsoft) > (21, 20) GeV
Resummed (NNLL)
Resummed (qq
_+qg+q
-_g)
DIPHOX (qq
_+qg+q
_g)
10
-3
10
-2
10
-1
50
100
150
200
250
300
pp
_ → γγX, √S =1.96 GeV
QT (GeV)
dσ/dQT (pb/GeV)
ET
iso
/ET
γ = 0.07, ∆Rcone = 0.4, ∆Rγγ > 0.3
(pTγhard, pTγsoft) > (21, 20) GeV
Resummed (NNLL)
Resummed (qq
_+qg+q
-_g)
DIPHOX (qq
_+qg+q
_g)
10
-4
10
-3
10
-2
10
-1
0
20
40
60
80
100
120
140
0
0.4
0.8
0
5
10
(a)
(b)
Figure
7:
Comparison
of
our
resummed
and
DIPHO
X
predi tions
for
(a)
the
in
v
arian
t
mass
and
(b)
transv
erse
momen
tum
distributions
of γγ
pairs
for
DØ
kinemati
uts.
The
solid
urv
es
sho
w
our
resummed
distributions
with
all
hannels
in luded.
The
dashed
and
dotted
urv
es
illustrate
the
resummed
and
DIPHO
X
distributions
in
the q ̄
q + qg
hannel.
renormalization
s ales
are
used
(μF = μR = Q/2 ).
These
are
the
hoi es
made
in
the
CDF
study
[1℄.
Sin e Eiso
T
is
an
appro
ximate
hara teristi
of
the
exp
erimen
tal
isolation,
one
migh
t
argue
that
b
oth Eiso
T
= 1
and
4
Ge
V
an
b
e
appropriate
in
a
al ulation
to
mat
h
the
onditions
of
the
CDF
measuremen
t.
The
dire t
on
tribution
is
w
eakly
sensitiv
e
to Eiso
T
,
while
the
one-fragmen
tation
part
of dσ/dQT
is
roughly
prop
ortional
to Eiso
T
( f.
Se tion
I
I
C
).
The
one-fragmen
tation
on
tribution
is
enhan ed
on
a
v
erage
b
y
400%
if Eiso
T
is
in reased
in
the
al ulation
from
1
to
4
Ge
V.
The
rate
in
the
shoulder
region
is
enhan ed
further
if
the
fa torization
s ale μF
is
redu ed.
Sin e
the
theoreti al
sp
e i ations
for
isolation
and
for
the
fragmen
tation
on
tribution
are
admittedly
appro
ximate,
w
e
question
whether
great
imp
ortan e
should
b
e
pla ed
on
the
agreemen
t
of
theory
and
exp
erimen
t
in
the
region
of
small ∆φ
or
in
the
shoulder
region
in
the QT
distribution.
A
straigh
tforw
ard
w
a
y
to
redu e
sensitivit
y
to
fragmen
tation
is
to
require Q > 27
Ge
V
or QT < Q
,
as
dis ussed
ab
o
v
e.
The
t
w
o
uts
ha
v
e
similar
ee ts
on
the
ev
en
t
distributions.
Figure
8
sho
ws
the
ee ts
of
the QT < Q
ut
on
the QT
and ∆φ
distributions.
The
ut QT < Q
is
parti ularly
e ien
t
at
suppressing
the
fragmen
tation QT
shoulder
(and
the
region
of
small ∆φ
altogether),
while
only
a
small
p
ortion
of
the
ev
en
t
sample
is
lost.
This
ut
is
esp
e ially
fa
v
orable,
sin e
it
onstrains
the
omparison
with
data
to
a
region
where
the
theory
is
w
ell
understo
o
d
and
has
a
small
un ertain
t
y
.
F
urthermore,
with
the
requiremen
t
of QT < Q
,
the
dep
enden e
of
dieren
tial
ross
se tions
on
the
hoi es
of
isolation
energy Eiso
T
and
fa torization
s ale μF
is
greatly
redu ed
to
the
t
ypi al
size
of
higher-order
orre tions.
W
e
predi t
that
if
a QT < Q
ut,
or
a Q > 27
Ge
V
ut,
is
applied
to
the
T
ev
atron
data,
the
enhan emen
t
at
lo
w ∆φ
and
in
termediate QT
asso
iated
with
the
fragmen
tation
on
tribution
will
disapp
ear.
This
is
an
imp
ortan
t
on lusion
of
our
study
,
and
w
e
urge
the
CDF
and
DØ
ollab
orations
to
apply
these
uts
in
their
future
analyses
of
the
20
pp
_ → γγX, √ S = 1.96 TeV
QT (GeV)
dσ/dQT (pb/GeV)
Resummed (NNLL)
DIPHOX, ET
iso
= 1 GeV
DIPHOX, ET
iso
= 4 GeV,
μF=Q/2
QT < Q
10
-2
10
-1
1
0
5
10
15
20
25
30
35
40
pp
_ → γγX, √S = 1.96 TeV
∆φ (rad)
dσ/d∆φ (pb/rad)
Resummed (NNLL),ET
iso
= 1 GeV
μF = Q (lower), Q/2 (upper)
DIPHOX, ET
iso
= 1 GeV
DIPHOX, ET
iso
= 4 GeV, μF = Q/2
QT < Q
10
-1
1
10
0
0.5
1
1.5
2
2.5
3
(a)
(b)
Figure
8:
Predi ted
ross
se tions
for
diphoton
pro
du tion
in p ̄
p →γγX
at
√
S = 1.96
T
e
V
as
a
fun tion
of
(a)
the γγ
pair
transv
erse
momen
tum QT
and
(b)
the
dieren e ∆φ
in
the
azim
uthal
angles
of
the
t
w
o
photons.
Our
resummed
predi tions
(solid)
are
sho
wn
together
with
DIPHO
X
predi tions
for
the
default
isolation
energy Eiso
T
= 1
Ge
V
and
fa torization
s ale μF = Q
(dashed),
and
for Eiso
T
= 4
Ge
V, μF = Q/2
(dotted).
W
e
imp
ose
the
ondition QT < Q
to
redu e
theoreti al
un ertain
ties
asso
iated
with
fragmen
tation.
diphoton
data.
5.
A
ver
age
tr
ansverse
momentum
An
imp
ortan
t
predi tion
of
the
resummation
formalism
is
the
hange
of
the
transv
erse
momen
tum
distribution
with
the
diphoton
in
v
arian
t
mass.
This
dep
enden e
omes,
in
part,
from
the ln Q2
dep
enden e
in
the
Sudak
o
v
exp
onen
t,
Eq.
(17
),
and
it
is
desirable
to
iden
tify
this
feature
amid
other
in uen es.
In
Fig.
9(a),
w
e
sho
w
normalized
resummed
transv
erse
momen
tum
distributions
for
v
arious
sele tions
of
the
in
v
arian
t
mass
of
the
photon
pairs.
Without
kinemati al
onstrain
ts
on
the
de a
y
photons,
the QT
distribution
is
exp
e ted
to
broaden
with
in reasing Q
,
and
the
p
osition
of
the
p
eak
in dσ/dQT
to
shift
to
larger QT
v
alues.
The
shift
of
the
p
eak
ma
y
or
ma
y
not
b
e
observ
ed
in
the
data
dep
ending
on
the
hosen
lo
w
er
uts
on pT
of
the
photons,
whi
h
suppress
the
ev
en
t
rate
at
lo
w Q
and QT
.
The
in
terpla
y
of
the
Sudak
o
v
broadening
of
the QT
distribution
and
kinemati al
suppression
b
y
the
photon pT
uts
is
re e ted
in
the
shap
e
of dσ/dQT
in
dieren
t Q
bins.
A
ording
to
dimensional
analysis,
the
a
v
erage ⟨QT⟩
in
the
in
terv
al QT ≤Q
ma
y
b
e
exp
e ted
to
b
eha
v
e
as
⟨QT⟩QT ≤Q = Qf(Q/
√
S),
(25)
21
10
-3
10
-2
10
-1
0
5
10
15
20
25
30
35
40
pp
_ → γγX, √S = 1.96 TeV
QT (GeV)
σ-1 dσ/dQT (1/GeV)
30 GeV < Q < 35 GeV
35 GeV < Q < 45 GeV
45 GeV < Q < 60 GeV
60 GeV < Q < 100 GeV
pp
_ → γγX, √ S = 1.96 TeV
Q (GeV)
〈QT〉 (GeV)
Resummed (NNLL); QT < Q
0
5
10
15
20
25
30
25
50
75
100
125
150
175
200
(a)
(b)
Figure
9:
(a)
Resummed
transv
erse
momen
tum
distributions
of
photon
pairs
in
v
arious
in
v
arian
t
mass
bins
used
in
the
CDF
measuremen
t,
normalized
to
the
total
ross
se tion
in
ea
h Q
bin.
(b)
The
a
v
erage QT
as
a
fun tion
of
the γγ
in
v
arian
t
mass,
omputed
for QT < Q.
where
the
s aling
fun tion f(Q/
√
S
)
re e ts
phase
spa e
onstrain
ts,
dep
enden e
on
the
Sudak
o
v
logarithm,
and
the x
dep
enden e
of
the
PDF
s.
Figure
9(b)
sho
ws
our
al ulated
diphoton
mass
dep
enden e
of ⟨QT⟩.
The
linear
in rease
sho
wn
in
Eq.
(25
)
is
observ
ed
o
v
er
the
range 30 < Q < 80
Ge
V.
F
or
v
alues
of Q
b
elo
w
the
kinemati
uto
at
ab
out
30
Ge
V,
the
uts
sho
wn
in
Fig.
3
suppress
diphoton
pro
du tion
at
small QT
,
and ⟨QT/Q⟩
gro
ws
to
w
ard
1
as Q
de reases
( orresp
onding
to
pro
du tion
only
at QT
lose
to Q
).
F
or Q ∼80
Ge
V
and
ab
o
v
e,
w
e
see
a
saturation
of
the
gro
wth
of ⟨QT⟩,
a
re e tion
of
the
in uen es
of
the
x
dep
enden e
of
the
PDF
s
and
other
fa tors.
Similar
saturation
b
eha
vior
is
observ
ed
in
al ulations
of ⟨QT⟩
in
other
pro
esses
[36℄.
It
w
ould
b
e
in
teresting
to
see
a
omparison
of
our
predi tion
with
data
from
the
CDF
and
DØ
ollab
orations.
B.
Results
for
the
LHC
1.
Event
sele
tion
T
o
obtain
predi tions
for pp
ollisions
at
the
LHC
at
√
S = 14
T
e
V,
w
e
emplo
y
the
uts
on
the
individual
photons
used
b
y
the
A
TLAS
ollab
oration
in
their
sim
ulations
of
Higgs
b
oson
de a
y
, h →γγ
[37℄.
W
e
require
transv
erse
momen
tum pγ
T > 40 (25)
Ge
V
for
the
harder
(softer)
photon,
(26)
and
rapidit
y |yγ| < 2.5
for
ea
h
photon.
(27)
In
a ord
with
A
TLAS
sp
e i ations,
w
e
imp
ose
a
lo
oser
isolation
restri tion
than
for
our
T
ev
atron
study
,
requiring
less
than Eiso
T
= 15
Ge
V
of
hadroni
and
extra
ele tromagneti
22
pp → γγX, √S = 14 TeV
QT (GeV)
dσ/dQT (pb/GeV)
70 < Q < 115 GeV
115 < Q < 140 GeV
140 < Q < 250 GeV
70 < Q < 115 GeV,
QT < Q
1
0.01
0.003
0.1
0
20
40
60
80
100
120
Figure
10:
Resummed
transv
erse
momen
tum
distributions
of
photon
pairs
in
v
arious
in
v
arian
t
mass
bins
at
the
LHC.
The
uts
listed
in
Eqs.
(26
)
and
(27)
are
imp
osed.The QT
distribution
for
70 < Q < 115
Ge
V
with
an
additional
onstrain
t QT < Q
is
sho
wn
as
a
dotted
line.
transv
erse
energy
inside
a ∆R = 0.4
one
around
ea
h
photon.
W
e
also
require
the
separation
∆Rγγ
b
et
w
een
the
t
w
o
isolated
photons
to
b
e
ab
o
v
e
0.4.
The
uts
listed
ab
o
v
e,
optimized
for
the
Higgs
b
oson
sear
h,
ma
y
require
adjustmen
ts
in
order
to
test
p
erturbativ
e
QCD
predi tions
in
the
full γγ
in
v
arian
t
mass
range
a essible
at
the
LHC.
The
v
alues
of
the pγ
T
uts
on
the
photons
in
Eq.
(26
)
preserv
e
a
large
fra tion
of
Higgs
b
oson
ev
en
ts
with Q > 115
Ge
V.
These
uts
ma
y
b
e
to
o
restri tiv
e
in
studies
of γγ
pro
du tion
at
smaller Q
,
onsidering
that
the
t
w
o
nal-state
photons
most
lik
ely
originate
from
a γγ
pair
with
small QT
and
ha
v
e
similar
v
alues
of pγ
T
of
ab
out Q/2 .
The pT
uts
in
terfere
with
the
exp
e ted
Sudak
o
v
broadening
of QT
distributions
with
in reasing
diphoton
in
v
arian
t
mass,
as
dis ussed
in
Se tion
I
I
I
A
5
.
W
e
further
note
that
the
asymmetry
b
et
w
een
the pγ
T
uts
on
the
harder
and
softer
photons
is
ne essary
in
a
xed-order
p
erturbativ
e
QCD
al ulation,
but
it
is
not
required
in
the
resummed
al ulation.
A
t
a
xed
order
of αs
,
asymmetry
in
the pγ
T
uts
prev
en
ts
instabilities
in dσ/dQ
aused
b
y
logarithmi
div
ergen es
in dσ/dQT
at
small QT
.
Su
h
instabilities
are
eliminated
altogether
on e
the
small-QT
logarithmi
terms
are
resummed
to
all
orders
of αs
.
Here
w
e
do
not
onsider
alternativ
e
pγ
T
uts,
although
exp
erimen
tal
ollab
orations
are
en ouraged
to
emplo
y
relaxed
and/or
symmetri
uts
to
in rease
the γγ
ev
en
t
sample
in
their
data
analysis.
2.
R
esumme
d QT
distributions
and
aver
age
tr
ansverse
momentum
Figure
10
sho
ws
transv
erse
momen
tum
distributions
of
the
photon
pairs
for
v
arious
in-
v
arian
t
masses.
The
a
v
erage γγ
transv
erse
momen
tum
gro
ws
with Q
,
as
demonstrated
b
y
Fig.
11.
Ho
w
ev
er,
the
rate
of
the
gro
wth
de reases
monotoni ally
with Q,
for
similar
reasons
as
at
the
T
ev
atron.
The γγ
distributions
in Q
and ∆φ
for
dieren
t
om
binations
of
s attering
sub
hannels
and
hoi es
of
theoreti al
parameters
are
dis ussed
in
Refs.
[2,
3℄.
In
all
ranges
of Q
,
the γγ
pro
du tion
rate
is
dominated
b
y
a
large qg
on
tribution,
a oun
ting
for
ab
out
50%
of
the
23
pp → γγX, √S = 14 TeV
Q (GeV)
〈QT〉 (GeV)
Resummed (NNLL); QT < Q
0
10
20
30
40
50
50
100
150
200
250
Figure
11:
The
a
v
erage QT
at
the
LHC
as
a
fun tion
of
the γγ
in
v
arian
t
mass Q.
xed-order
(NLO)
rate.
Although
this
n
um
b
er
dep
ends
on
the
hoi e
of
the
fa torization
s
heme
and
s ale,
and,
on
the
other
hand,
separate
treatmen
t
of
the q ̄
q
and qg
ross
se tions
is
not
meaningful
in
the
resummation
al ulation
[3℄,
it
nonetheless
re e ts,
in
a
rude
w
a
y
,
the
in reased
relativ
e
imp
ortan e
of
the qg
ross
se tion.
The gg + gqS
hannel
on
tributes
ab
out
25%
at Q ∼80
Ge
V
(the
lo
ation
of
the
uto
in dσ/dQ
due
to
the
uts
on pγ
T
)
and
less
at
larger Q.
As
at
the
T
ev
atron,
the
dep
enden e
of
the
ross
se tions
on
the
resummation
s
heme
is
small
[3℄.
The
dep
enden e
on
the
nonp
erturbativ
e
mo
del
an
also
b
e
negle ted,
as
long
as
the
nonp
erturbativ
e
fun tion
do
es
not
v
ary
strongly
with x
[3℄.
3.
Final-state
fr
agmentation
and
omp
arison
with
DIPHO
X
The
impa t
of
the
nal-state
fragmen
tation
at
the
LHC
an
b
e
ev
aluated
if
w
e
ompare
our
results
with
DIPHO
X
predi tions.
The
transv
erse
momen
tum
and
in
v
arian
t
mass
dis-
tributions
in
the q ̄
q + qg
hannel
from
the
t
w
o
approa
hes
are
sho
wn
in
Fig.
12
.
In
b
oth
al ulations,
quasi-exp
erimen
tal
isolation
remo
v
es
dire t
NLO
ev
en
ts
with
ollinear
nal-
state
photons
and
partons
when QT > Eiso
T
= 15
Ge
V,
but
not
when QT
is
b
elo
w Eiso
T
.
Con en
trating
rst
on γγ
ev
en
ts
with QT > Eiso
T
,
w
e
observ
e
that,
at QT > 80
Ge
V,
the
resummed q ̄
q+qg
ross
se tion
redu es
to
the
dire t
xed-order
ross
se tion,
ev
aluated
in
the
same
w
a
y
as
in
the
DIPHO
X
o
de.
Our
resummed
and
the
dire t
DIPHO
X
ross
se tions,
sho
wn
as
solid
and
dashed
urv
es,
resp
e tiv
ely
,
in
Fig.
12(a)
onsequen
tly
agree
w
ell
at
large QT
.
A
t
smaller QT,
the
resummed
ross
se tion
is
enhan ed
b
y
to
w
ers
of
higher-order
logarithmi
on
tributions.
On
the
other
hand,
the
full q ̄
q + qg
DIPHO
X
rate
(sho
wn
as
a
dotted
line)
also
in ludes
single-photon
fragmen
tation
on
tributions,
whi
h
add
to
the
dire t
pro
du tion
ross
se tion.
F
or
the
nominal
isolation
parameters,
the
expli it
fragmen
tation
on
tribution
onstitutes
ab
out
25%
of
the
full
DIPHO
X
rate
for 60 < QT < 120
Ge
V.
Its
magnitude
in reases
appro
ximately
linearly
with
the
assumed Eiso
T
v
alue.
F
or QT < Eiso
T
,
the
nal-state
ollinear
region
of
the
dire t
on
tribution
is
regulated
b
y
the
ollinear
subtra tion
pres ription
adopted
in
the
resummation
al ulation,
whereas
24
ET
iso
pp → γγX, √S = 14 TeV
QT (GeV)
dσ/dQT (pb/GeV)
qq
_ + qg only
Resummed
DIPHOX (direct)
DIPHOX (direct+frag.)
10
-2
10
-1
1
0
20
40
60
80
100
120
pp → γγX, √S = 14 TeV
Q (GeV)
dσ/dQ (pb/GeV)
qq
_ + qg only
Resummed
DIPHOX (direct+fragm.)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
50
75
100
125
150
175
200
225
250
(a)
(b)
Figure
12:
T
ransv
erse
momen
tum
and
in
v
arian
t
mass
distributions dσ/dQ
in
the q ̄
q + qg
hannel
obtained
in
the
resummation
(solid)
and
DIPHO
X
(dotted)
al ulations.
the
fragmen
tation
singularit
y
is
subtra ted
from
the
dire t
on
tribution
and
repla ed
b
y
photon
fragmen
tation
fun tions
in
the
DIPHO
X
al ulation.
Subtra tion
of
singularities
in
DIPHO
X
in
tro
du es
in
tegrable
singularities
in dσ/dQT
at
dieren
t
v
alues
of QT
b
elo
w Eiso
T
.
The
origin
of
the
nal-state
logarithmi
singularities
at
v
alues
of QT
b
elo
w Eiso
T
is
dis ussed
in
Refs.
[22,
23
,
24℄.
F
or QT < Eiso
T
,
the
DIPHO
X
urv
es
represen
t
the
a
v
erage
o
v
er
singular
on
tributions
in
this QT
in
terv
al.
After
the
fragmen
tation
singularit
y
is
subtra ted,
the
DIPHO
X
dire t
on
tribution
(dashed
line)
is
on
a
v
erage
b
elo
w
our
resummed q ̄
q + qg
rate
(solid
line)
o
v
er
most
of
the
range
of QT
sho
wn
in
Fig.
12(a).
After
in
tegration
o
v
er
all QT
,
our
resummed
and
DIPHO
X
q ̄
q + qg
ross
se tions
agree
within
10-20%
at
most
v
alues
of Q
( f.
Fig.
12
(b)),
with
our
resummed
rate
b
eing
b
elo
w
the
DIPHO
X
rate
at
all Q
.
The
largest
dieren e
o
urs
at
the
lo
w
est
v
alues
of Q
(b
elo
w
the
uto
),
where
the
rates
an
dier
b
y
a
fa tor
of
2.
In
this
region,
orresp
onding
to
diphoton
ev
en
ts
with
small ∆φ
and QT
larger
than Q
,
the
photon
fragmen
tation
on
tributions
in luded
in
the
DIPHO
X
al ulation
are
large
in
omparison
to
the
dire t
rate.
Finally
,
w
e
note
that
the
in
tegrated
rate
in
DIPHO
X
is
more
stable
with
resp
e t
to
v
ariations
in Eiso
T
than
the
dieren
tial
distributions
in
DIPHO
X,
b
e ause Eiso
T
dep
enden e
for QT > Eiso
T
is
an eled
to
a
go
o
d
degree
b
y Eiso
T
dep
enden e
for QT < Eiso
T .
T
o
obtain
the
nal γγ
pro
du tion
ross
se tions,
after
in lusion
of
all
hannels,
w
e
om
bine
the
resp
e tiv
e q ̄
q +qg
results
with
the
resummed
NLO gg +gqS
ross
se tion
in
our
ase
and
with
the
LO gg
ross
se tion
in
the
DIPHO
X
ase.
The
distributions
in
the γγ
in
v
arian
t
mass Q,
the
transv
erse
momen
tum QT
,
and
the
azim
uthal
angle
separation ∆φ
in
the
lab
frame
are
sho
wn
in
Fig.
13
.
F
or
the
uts
hosen,
the
LO gg
and
the
resummed gg +gqS
total
rates
onstitute
ab
out
9%
and
20%
of
the
total
rate.
The
resummed
and
DIPHO
X
in
v
arian
t
mass
distributions
(Fig.
13(a))
are
brough
t
loser
to
one
another
as
a
result
of
the
in lusion
25
pp → γγX, √S = 14 TeV
Q (GeV)
dσ/dQ (pb/GeV)
Resummed (NNLL)
Fixed-order (NLO)
DIPHOX (direct+frag.)
10
-2
10
-1
1
50
75
100
125
150
175
200
225
250
pp → γγX, √S = 14 TeV
QT (GeV)
dσ/dQT (pb/GeV)
∆Rcone = 0.4, ET
iso
= 15 GeV, ∆Rγγ > 0.4
pTγ
hard > 40 GeV, pTγ
soft > 25 GeV
Resummed (NNLL)
Fixed-order (NLO)
DIPHOX (direct+frag.)
10
-2
10
-1
1
0
20
40
60
80
100
120
(a)
(b)
pp → γγX, √S = 14 TeV
∆φ (rad)
dσ/d∆φ (pb/rad)
Resummed (NNLL)
Fixed-order (NLO)
DIPHOX
1
10
10 2
0
0.5
1
1.5
2
2.5
3
( )
Figure
13:
In
v
arian
t
mass,
transv
erse
momen
tum,
and ∆φ
distributions
from
our
resummed
al-
ulation
and
from
DIPHO
X
at
the
LHC.
W
e
sho
w
our
xed-order
(dashed)
and
resummed
(solid)
distributions.
All
initial
states
are
in luded
in
b
oth
al ulations,
and
the
single-γ
fragmen
tation
on
tributions
are
in luded
in
DIPHO
X.
26
10
-2
10
-1
1
50
75
100
125
150
175
200
225
250
pp → γγX, √S = 14 TeV
Q (GeV)
dσ/dQ (pb/GeV)
Resummed (NNLL)
Fixed order (NLO)
DIPHOX (direct+frag.)
10
-2
10
-1
1
0
20
40
60
80
100
120
pp → γγX, √S = 14 TeV
QT (GeV)
dσ/dQT (pb/GeV)
Resummed (NNLL)
Fixed order (NLO)
DIPHOX (direct+frag.)
(a)
(b)
Figure
14:
In
v
arian
t
mass
and
transv
erse
momen
tum
distributions
from
our
resummed,
NLO,
and
DIPHO
X
al ulations
at
the
LHC,
with
the QT < Q
onstrain
t
imp
osed.
of
the gg + gqS
on
tribution
in
the
resummed
al ulation.
F
or QT ̸= 0,
the
full
DIPHO
X QT
distribution
in
Fig.
13(b)
is
determined
en
tirely
b
y
dire t
plus
fragmen
tation
on
tributions
(the
same
as
in
Fig.
12
(a)),
b
e ause
the
LO gg
ross
se tion
on
tributes
at QT = 0
only
.
In
on
trast,
our
resummed gg + gqS
on
tribution
mo
di es
the
ev
en
t
rate
at
all QT
.
The
resummed
and
DIPHO
X
rates
are
in
a
reasonable
agreemen
t
for 1.5 ≲∆φ ≲2.5 ,
as
sho
wn
in
Fig.
13( ).
In
the ∆φ →π
limit,
the
xed-order
rates
in
DIPHO
X
div
erge
b
e ause
of
the
singularities
at
small QT
,
while
our
resummed
rate
yields
a
nite
v
alue.
F
or ∆φ < 1.5 ,
the
DIPHO
X
ross
se tion
is
enhan ed
b
y
photon
fragmen
tation
on
tributions.
As
at
the
energy
of
the
T
ev
atron,
theoreti al
un ertain
ties
are
greater
at
small ∆φ.
Predi tions
are
most
reliable
when QT < Q
(and
the
angles θ∗
and φ∗
are
a
w
a
y
from
0
or π
).
With
the QT < Q
ut
imp
osed,
the
un ertain
large-QT
photon
fragmen
tation
on
tributions
are
suppressed,
and
the
resummed
and
DIPHO
X
ross
se tions
agree
w
ell
at
large QT
( f.
Fig.
14
(b)).
The QT
distribution
in
the
in
terv
al 70 < Q < 115
Ge
V
with
the
QT < Q
onstrain
t
is
sho
wn
in
Fig.
10
b
y
a
dotted
urv
e.
Distributions
in
the
other
t
w
o
mass
bins
in
Fig.
10
are
essen
tially
not
ae ted
b
y
this
ut
in
the QT
range
presen
ted.
Our
al ulation
aptures
the
dominan
t
on
tributions
to γγ
pro
du tion
at
the
LHC.
Ho
w-
ev
er,
as
w
e
noted,
dire t qg
s attering,
ev
aluated
at
order O(αs)
in
our
al ulation,
is
the
leading
s attering
hannel
in
the
region
relev
an
t
for
the
Higgs
b
oson
sear
h
at
the
LHC.
It
is
imp
ortan
t
to
emphasize
that
the
nal-state
ollinear
radiation
is
not
the
main
reason
b
ehind
the
enhan emen
t
of
the qg
rate,
whi
h
is
in reased
predominan
tly
b
y
on
tributions
from
non-singular
phase
spa e
regions.
Consequen
tly
,
the q ̄
q + qg
dire t
rate
is
only
w
eakly
sensitiv
e
to
adjustmen
ts
in
the
isolation
parameters Eiso
T
and ∆R
[10
℄.
The
unkno
wn O(α2
s)
on
tributions
to qg
s attering
ma
y
b
e
non-negligible,
and
it
w
ould
b
e
v
aluable
to
ompute
them
in
the
future
when
LHC
data
are
a
v
ailable.
27
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0
10
20
30
40
50
60
70
pp → hX → γγX, √S = 14 TeV
QT (GeV)
σ-1 dσ/dQT (1/GeV)
Signal
Background
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
0.5
1
1.5
2
2.5
3
pp → hX → γγX, √S = 14 TeV
φ* (rad)
σ-1 dσ/dφ* (1/rad)
Signal
Background
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3
-2
-1
0
1
2
3
pp → hX → γγX, √S = 14 TeV
∆y
σ-1 dσ/d∆y
Signal
Background
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
pp → hX → γγX, √S = 14 TeV
cos θ
*
σ-1 dσ/d(cos θ
*)
Signal
Background
Figure
15:
Comparison
of
the
normalized
Higgs
b
oson
signal
and
diphoton
ba
kground
distributions
at
the
LHC,
b
oth
omputed
at
NNLL
a ura y
.
The
Higgs
b
oson
mass
is
tak
en
to
b
e mH = 130
Ge
V,
and
the
ba
kground
is
al ulated
for 128 < Q < 132
Ge
V.
C.
Comparison
with
Higgs
b
oson
signal
distributions
W
e
highligh
t
some
similarities
and
dieren es
b
et
w
een
the
pro
du tion
sp
e tra
for
the
Higgs
b
oson
signal
and
the
QCD
ba
kground
dis ussed
in
this
pap
er.
W
e
fo
us
on
the
diphoton
de a
y
mo
de
of
a
SM
Higgs
b
oson
pro
du ed
from
the
dominan
t
gluon-fusion
me
h-
anism, gg →h0 →γγ
,
where
the
Higgs
b
oson
pro
du tion
ross
se tion
is
al ulated
at
the
same
order
of
pre ision
as
the
QCD
on
tin
uum
ba
kground.
W
e
in lude
initial-state
QCD
on
tributions
at O(α3
s)
(NLO)
and
resummed
on
tributions
at
NNLL
a ura y
.
These
on-
28
tributions
are
also
o
ded
in
ResBos
[38℄,
and
w
e
an
apply
the
same
uts
on
the
momen
ta
of
the
photons
to
the
signal
and
ba
kground.
Our
ndings
should
remain
broadly
appli able
after
the
NNLO
orre tions
to
Higgs
b
oson
pro
du tion
[39,
40℄
are
in luded.
W
e
ompute
the
ba
kground
in
the
range 128 < Q < 132
Ge
V,
and
the
signal
at
a
xed
Higgs
b
oson
mass mH = 130
Ge
V.
W
e
imp
ose
the
kinemati
sele tion QT < Q
,
but
its
in uen e
is
not
imp
ortan
t
at
the
large
v
alues
of
diphoton
mass
of
in
terest
here.
The
ross
se tion
times
bran
hing
ratio
for
the
Higgs
b
oson
signal
is
substan
tially
smaller
than
the
QCD
on
tin
uum.
T
o
b
etter
illustrate
their
dieren es,
Fig.
15
presen
ts
distributions
normalized
to
the
resp
e tiv
e
total
rates.
The
top-left
panel
sho
ws
normalized
transv
erse
momen
tum
distributions
of
photon
pairs.
The
signal
and
ba
kground
p
eak
at
ab
out
12
and
5
Ge
V,
resp
e tiv
ely
.
The
a
v
erage
v
alues
of QT
are
26
and
23
Ge
V,
omputed
o
v
er
the
range
0
to
75
Ge
V.
Dieren es
in
the
shap
es
of
these QT
sp
e tra
an
b
e
attributed
to
the
dieren
t
stru ture
of
the
leading
terms
in
the
initial-state
Sudak
o
v
exp
onen
ts
and
to
the
ee ts
of
nal-state
photon
fragmen
tation.
The
Higgs
b
oson
signal
is
on
trolled
b
y
the
hara teristi s
of
the
gg+gqS
initial
state,
whereas
the
on
tin
uum
is
on
trolled
primarily
b
y
the q ̄
q+qg
initial
state.
Be ause
the
dominan
t
Sudak
o
v
o
e ien
t A(k)
q
∝CF
in
the q ̄
q
ase
is
smaller
than A(k)
g
=
(CA/CF)A(k)
q
in
the gg
ase,
the
resummed q ̄
q + qg
initial-state
radiation
pro
du es
narro
w
er
QT
distributions
than gg + gqS
initial-state
radiation.
Ab
out
80%
of
the
diphoton
rate
is
pro
vided
b
y
the q ̄
q + qg
hannel,
implying
a
narro
w
er QT
distribution
of
the
ba
kground,
if
based
on
the
v
alue
of A(k)
alone.
The
on
tin
uum
ba
kground
on
tribution
is
also
enhan ed
b
y
nal-state
radiation
in qg
s attering.
The QT
pro le
of
the
nal-state
ollinear
terms
dep
ends
more
on
the
isolation
mo
del
(in luding Eiso
T
and ∆R )
than
on
the
initial-state
Sudak
o
v
exp
onen
t.
F
or
the
nominal
A
TLAS
uts,
the
nal-state
ollinear
on
tribution
in
our
al ulation
hardens
the
ba
kground
QT
distribution,
diminishing
its
dieren e
from
the
Higgs
b
oson
signal
distribution.
More
ee tiv
e
isolation
ma
y
redu e
the
impa t
of
the
nal-state
radiation
on QT
distributions.
Another
dieren e
b
et
w
een
the
signal
and
on
tin
uum
is
observ
ed
in
the
distribution
in
the
azim
uthal
angle
of
the
photons,
su
h
as
the
angle φ∗
in
the
Collins-Sop
er
frame
sho
wn
in
the
top-righ
t
panel
of
Fig.
15.
This
distribution
is
qualitativ
ely
the
same
if
in
tegrated
o
v
er
all QT
,
as
in
Fig.
15
,
or
in
tegrated
ab
o
v
e
some
minimal QT
v
alue,
as
in
an
exp
erimen
tal
measuremen
t.
Without
isolation
imp
osed,
the
spin-0
Higgs
b
oson
signal
m
ust
b
e
at
in φ∗,
but
the
QCD
ba
kground
p
eaks
to
w
ard φ∗= 0
and π
(i.e., sin φ∗= 0)
as
a
result
of
the
nal-state qg
singularit
y
.
2,
3
Isolation
uts
suppress
b
oth
the
signal
and
the
ba
kground
for
sin φ∗< sin ∆R .
The
result
is
a
signal
distribution
with
a
broad
p
eak
near φ∗= π/2 ,
while
the
ba
kground
fa
v
ors
v
alues
of φ∗
near 0
and π
.
A
sele tion
of
ev
en
ts
with φ∗
in
the
vi init
y
of π/2 ,
and QT
large
enough,
helps
to
redu e
the
impa t
of
the qg
ba
kground.In
the
lab
frame,
a
related
distribution
is
in
the
v
ariable |φ3T −φ4T| ,where φiT
is
the
azim
uthal
angle
b
et
w
een ⃗
pγi
T
and ⃗
QT
.
The
signal
(ba
kground)
pro
esses
tend
to
ha
v
e
more
ev
en
ts
with
large
2
By
de nition,
the
re oil
parton
5
alw
a
ys
lies
in
the Oxz
plane
(has
zero
azim
uthal
angle)
in
the
Collins-
Sop
er
frame.
F
or
the
nal-state
singularit
y
to
o
ur
at
NLO,
the
photons
should
b
e
in
the
same
plane
with
5,
i.e.,
ha
v
e sin φ∗= 0 .
3
One
of
the
resummed
stru ture
fun tions
for
the gg
ba
kground
is
mo
dulated
b
y cos 2φ∗
(see
Se .
I
I
B),
but
w
e
negle t
this
mo
dulation
in
our
presen
t
al ulation.
29
(small)
magnitude
of |φ3T −φ4T| .
A
third
p
oten
tial
dis riminator
b
et
w
een
the
signal
and
ba
kground
is
the
dieren e
in
the
rapidities ∆y = yhard −ysoft
of
the
photons
with
harder
and
softer
v
alues
of pγ
T
in
the
lab
frame,
al ulated
on
an
ev
en
t
b
y
ev
en
t
basis.
This
distribution
is
displa
y
ed
in
the
lo
w
er-
left
frame
of
Fig.
15.
The
ba
kground
distribution
p
eaks
at
the
origin,
while
the
signal
is
almost
at
o
v
er
a
wide
range
of ∆y
.
Dieren
t
spin
orrelations
in
the
de a
y
of
a
spin-0
Higgs
b
oson
from
those
hara teristi
of
QCD
ba
kground
pro
esses
are
the
sour e
of
this
distin tion.
Dis rimination
based
on
this
dieren e
an
impro
v
e
the
statisti al
signi an e
of
the
signal
[10
℄.
W
e
note
that
our
resummed
al ulation
do
es
not
exhibit
the
kinemati
singularit
y
at ∆y ≈2
presen
t
in
the
nite-order
ross
se tion
and
ob
vious
in
Fig.
10
of
Ref.
[10℄,
where
the
distribution
with
resp
e t
to y∗≡∆y/2
is
sho
wn.
The
dis on
tin
uit
y
in
dσ/dQT
aused
b
y
the
nite-order
appro
ximation
is
resummed
in
our
al ulation,
yielding
a
smo
oth
result.
The
rapidit
y
dieren e
is
related
to
the
s attering
angle
in
the
Collins-Sop
er
frame:
tanh(∆y/2) = cos θ∗
when QT
is
zero.
The cos θ∗
distribution
is
sho
wn
in
the
lo
w
er-righ
t
frame
of
Fig.
15.
The
dieren e
b
et
w
een
the
signal
and
ba
kground
rates
is
ev
en
more
pronoun ed
in
this
v
ariable,
learly
expressing
the
dieren e
in
the
spin
orrelations
of
the
systems
pro
du ing
the
photons.
A
omparison
of QT
distributions
in
the
top-left
panel
of
Fig.
15
suggests
that
the
signal
v
ersus
ba
kground
ratio
w
ould
b
e
enhan ed
if
a
ut
is
made
to
restri t QT > 10
Ge
V.
After
applying
this
ut,
w
e
ma
y
again
examine
the
distributions
in
the
rapidit
y
dieren e
of
the
t
w
o
photons,
the
s attering
angle
in
the
Collins-Sop
er
frame,
and
the
azim
uthal
angle
distribution
of
the
photons
in
the
Collins-Sop
er
frame.
The
results
are
qualitativ
ely
similar
to
those
in
Fig.
15
and
are
not
sho
wn
here.
A
more
e ien
t
pro
edure
to
in rease
the
Higgs
b
oson
dis o
v
ery
signi an e
is
to
apply
a
sim
ultaneous
lik
eliho
o
d
analysis
to
sev
eral
kinemati
distributions.
Based
on
the
presen
t
dis ussion,
w
e
w
ould
argue
that
the
resummed
QT
, φ∗
,
and cos θ∗
distributions
are
go
o
d
dis riminators
b
et
w
een
the
Higgs
b
oson
signal
and
ba
kground
in
su
h
an
analysis.
IV.
CONCLUSIONS
The
theoreti al
study
of
on
tin
uum
diphoton
pro
du tion
in
hadron
ollisions
is
in
teresting
and
v
aluable
for
sev
eral
reasons:
there
are
data
from
the
CDF
and
DØ
ollab
orations
at
F
ermilab
with
the
promise
of
larger
ev
en
t
samples;
there
are
new
theoreti al
hallenges
asso
iated
with
all-orders
soft-gluon
resummation
of
t
w
o-lo
op
amplitudes;
and
on
tin
uum
diphotons
are
a
large
standard-mo
del
ba
kground
ab
o
v
e
whi
h
one
ma
y
observ
e
the
pro
du ts
of
Higgs
b
oson
de a
y
in
to
a
pair
of
photons
at
the
LHC.
In
this
pap
er
and
Refs.
[2,
3℄,
w
e
presen
t
our
al ulation
of
the
fully
dieren
tial
ross
se tion dσ/(dQdQTdydΩ∗)
as
a
fun tion
of
the
mass Q
,
transv
erse
momen
tum QT
,
and
rapidit
y y
of
the
diphoton
system,
and
of
the
p
olar
and
azim
uthal
angles
of
the
individual
photons
in
the
diphoton
rest
frame.
Our
basi
QCD
hard-s attering
subpro
esses
are
all
omputed
at
next-to-leading
order
(NLO)
in
the
strong
oupling
strength αs
,
and
w
e
in lude
the
state-of-art
resummation
of
initial-state
gluon
radiation
to
all
orders
in αs
,
v
alid
to
next-
to-next-to-leading
logarithmi
a ura y
(NNLL).
Resummation
is
essen
tial
for
a
realisti
and
reliable
al ulation
of
the QT
dep
enden e
in
the
region
of
small
and
in
termediate
v
alues
of
QT
,
where
the
ross
se tion
is
greatest.
It
is
also
needed
for
stable
estimates
of
the
ee ts
of
30
exp
erimen
tal
a eptan e
on
distributions
in
the
diphoton
in
v
arian
t
mass
and
other
v
ariables.
Our
analyti al
results
are
in luded
in
a
fully
up
dated
ResBos
o
de
[31,
32℄.
This
n
u-
meri al
program
allo
ws
us
to
imp
ose
sele tions
on
the
transv
erse
momen
ta
and
angles
of
the
nal
photons,
in
order
to
mat
h
those
emplo
y
ed
b
y
the
CDF
and
DØ
ollab
orations,
as
w
ell
as
those
an
ti ipated
in
exp
erimen
ts
at
the
LHC.
Our
predi tions
are
esp
e ially
p
ertinen
t
in
the
region QT ≲Q
.
W
e
sho
w
that
our
results
at
the
T
ev
atron
and
at
the
LHC
are
insensitiv
e
to
the
hoi e
of
the
resummation
s
heme
and
of
the
nonp
erturbativ
e
fun tions
required
b
y
the
in
tegration
in
to
the
region
of
large
impa t
parameter.
The
published
ollider
data
are
presen
ted
in
the
form
of
singly
dieren
tial
distributions.
W
e
follo
w
suit
in
order
to
mak
e
omparisons,
and
w
e
nd
ex ellen
t
agreemen
t
with
data,
as
sho
wn
in
Se .
I
I
I.
W
e
re ommend
that
more
dieren
tial
studies
b
e
made,
and,
to
motiv
ate
these,
w
e
presen
t
predi tions
for
the
hanges
exp
e ted
in
the QT
distribution
as
a
fun tion
of
mass Q
,
and
for
the
dep
enden e
of
the
mean
transv
erse
momen
tum
on Q
.
W
e
mak
e
predi tions
for
on
tin
uum
diphoton
mass,
transv
erse
momen
tum,
and
angular
distributions
at
the
energy
of
the
LHC.
Moreo
v
er,
w
e
on
trast
in
Fig.
15
the
shap
es
of
some
of
these
distributions
with
those
exp
e ted
from
the
de a
y
of
a
Higgs
b
oson.
The
distin t
features
of
the
signal
and
ba
kground
suggest
that
that
the
Higgs
b
oson
dis o
v
ery
signi an e
an
b
e
in reased
via
a
sim
ultaneous
lik
eliho
o
d
analysis
of
sev
eral
kinemati
distributions,
parti ularly
the
resummed QT
, φ∗
,
and cos θ∗
distributions.
Another
approa
h
to
the
omputation
of
on
tin
uum
diphoton
pro
du tion
is
presen
ted
b
y
the
DIPHO
X
ollab
oration
[14℄.
This
al ulation
in ludes
b
oth
the
dire t
pro
du tion
of
photons
from
hard-s attering
pro
esses
and
the
photons
pro
du ed
from
fragmen
tation
of
(an
ti-)quarks
or
gluons.
It
is
v
alid
at
NLO,
ex ept
for
the gg
subpro
ess,
whi
h
is
in luded
at
leading
order
only
.
The
DIPHO
X
o
de
is
useful
for
rates
in
tegrated
o
v
er
transv
erse
mo-
men
tum,
but
it
is
not
designed
to
predi t
the
transv
erse
momen
tum
distribution
or
other
distributions
sensitiv
e
to
the
region
in
whi
h
the
transv
erse
momen
tum
of
the
diphoton
pair
is
small.
Compared
to
a
xed-order
al ulation,
su
h
as
dire t
photon
pair
al ulation
in
DIPHO
X,
our
al ulation
impro
v
es
the
theoreti al
predi tion
for
ev
en
t
distributions
whi
h
are
sensitiv
e
to
the
region
of
lo
w QT
.
F
urthermore,
our
al ulation
in ludes
the
NLO
on-
tribution
from
the
om
bined gg + gqS
hannel,
leading
to
more
a urate
predi tions
at
the
LHC,
where
the gg + gqS
on
tribution
is
generally
not
small.
Only
isolate
d,
not
in lusiv
e,
photons
are
iden
ti ed
exp
erimen
tally
.
While
it
is
straigh
tfor-
w
ard
to
de ne
an
isolated
photon
in
a
giv
en
exp
erimen
t,
it
is
hallenging
to
devise
a
theoret-
i al
pres ription
that
an
mat
h
the
exp
erimen
tal
de nition,
short
of
rst
understanding
the
long-distan e
dynami s
of
QCD.
The
isolated
diphoton
pro
du tion
rate
is
mo
deled
in
the
DIPHO
X
o
de
b
y
in luding
expli it
photon
fragmen
tation
fun tion
on
tributions
at
NLO
a ura y
.
A
short oming
of
this
approa
h
(as
w
ell
as
of
our
metho
d
for
treating
isolation)
is
that
one
annot
a urately
represen
t
photon
fragmen
tation
without
in luding
nal-state
parton
sho
w
ering
in
the
presen e
of
isolation
onstrain
ts.
There
is
inevitable
am
biguit
y
and
un ertain
t
y
in
the
hoi e
of
the
isolation
energy
used
to
de ne
an
isolated
photon
the-
oreti ally
for
omparison
with
the
isolated
photon
measured
exp
erimen
tally
.
As
sho
wn
in
Se .
I
I
I,
the
DIPHO
X
ross
se tion
an
v
ary
b
y
a
large
fa tor
in
some
regions
of
phase
spa e
at
the
T
ev
atron
when Eiso
T
is
hanged
from
1
Ge
V
to
4
Ge
V.
Our
approa
h
is
to
on en
trate
on
ph
ysi al
observ
ables
whi
h
are
less
sensitiv
e
to
the
frag-
men
tation
on
tributions.
W
e
apply
the
ollinear
subtra tion
pres ription
or
the
smo
oth-
one
isolation
pres ription
to
de ne
an
isolated
photon
in
our
al ulation.
W
e
nd
go
o
d
31
agreemen
t
with
the
data,
ex ept
in
the
region
with
small Q
and ∆φ < π/2 ,
onsisten
t
with
our
theoreti al
exp
e tation
that
higher-order
dire t
photon
pro
du tion
and
photon
fragmen-
tation
on
tributions
an
strongly
mo
dify
the
rate
of
diphoton
pairs
in
this
region.
W
e
suggest
that
m
u
h
b
etter
agreemen
t
with
urren
t
and
future
data
will
b
e
obtained
if
an
addition
requiremen
t
of QT < Q
is
applied.
With
this
ut,
the
fragmen
tation
on
tributions
are
largely
suppressed.
With
the
ut QT < Q
ut
applied
to
the
T
ev
atron
data,
the
enhan emen
t
at
lo
w ∆φ
and
in
termediate QT
(the
shoulder
region)
should
disapp
ear.
W
e
urge
the
CDF
and
DØ
ollab
orations
to
apply
these
uts
in
future
analyses
of
the
diphoton
data.
In
our
al ulation,
w
e
iden
tify
an
in
teresting
spin- ip
on
tribution
(with cos 2φ∗
dep
en-
den e)
in
the gg
hannel,
f.
Ref.
[3℄,
and
w
e
suggest
that
measuremen
ts
b
e
made
of
the
distribution
of φ∗
as
a
fun tion
of QT
.
All-orders
resummation
of
the
gluon
spin- ip
on
tri-
bution
ma
y
b
e
needed
when
a
larger
statisti al
sample
of
diphoton
data
is
a
v
ailable.
The
on
tributions
from qg + ̄
qg
pro
esses
b
e ome
more
imp
ortan
t
at
the
LHC
than
at
the
T
ev
atron,
and
al ulations
at
a
higher
order
of
pre ision
ma
y
b
e
w
arran
ted
ev
en
tually
.
T
o
impro
v
e
the
theoreti al
predi tion
in
the
region
of
phase
spa e
with QT < Eiso
T
and φ∗∼0
or π
,
a
join
t
resummation
al ulation
is
needed
in
whi
h
the
ee ts
of
b
oth
the
initial-
and
nal-state
m
ultiple
parton
emissions
are
treated
sim
ultaneously
.
Although
w
e
emphasize
that
b
etter
agreemen
t
of
our
predi tions
with
data
should
b
e
apparen
t
if
the
sele tion QT < Q
is
made,
w
e
also
p
oin
t
out
that
the
region QT > Q
should
manifest
v
ery
in
teresting
ph
ysi s
of
a
dieren
t
sort.
A
dditional
logarithmi
singularities
of
the
form log(Q/QT)
are
en oun
tered
in
the
region QT ≫Q
.
These
logarithms
are
asso
iated
with
the
fragmen
tation
of
a
parton
arrying
large
transv
erse
momen
tum QT
in
to
a
system
of
small
in
v
arian
t
mass Q
[27,
28
℄,
a
ligh
t γγ
pair
in
our
ase.
Small-Q γγ
fragmen
tation
of
this
kind
is
not
implemen
ted
y
et
in
theoreti al
mo
dels.
Exp
erimen
tal
study
of
the
region
QT ≫Q
ma
y
oer
the
opp
ortunit
y
to
measure
the
parton
to
t
w
o-photon
fragmen
tation
fun tion Dγγ(z1, z2).
A
kno
wledgmen
ts
Resear
h
in
the
High
Energy
Ph
ysi s
Division
at
Argonne
is
supp
orted
in
part
b
y
US
Departmen
t
of
Energy
,
Division
of
High
Energy
Ph
ysi s,
Con
tra t
DE-A
C02-06CH11357.
The
w
ork
of
C.-P
.
Y.
is
supp
orted
b
y
the
U.
S.
National
S ien e
F
oundation
under
gran
t
PHY-0555545.
C.
B.
thanks
the
F
ermilab
Theoreti al
Ph
ysi s
Departmen
t,
where
a
part
of
this
w
ork
w
as
done,
for
its
hospitalit
y
and
nan ial
supp
ort.
The
diagrams
in
Figs.
1
and
2
w
ere
dra
wn
with
aid
of
the
program
Jax
oDra
w
[41℄.
App
endix
A:
SUMMAR
Y
OF
PER
TURBA
TIVE
COEFFICIENTS
In
this
app
endix
w
e
presen
t
an
o
v
erview
of
the
p
erturbativ
e
QCD
expressions
for
the
resummed
and
asymptoti
ross
se tions
used
in
our
omputation.
The
fun tions Aa(C1, ̄
μ), Ba(C1, C2, ̄
μ), Ca/a1(x, b; C1/C2, μ),
and ha(Q, θ∗)
are
in
tro
du ed
in
Se .
I
I.
These
fun tions
are
deriv
ed
as
p
erturbativ
e
expansions
in
the
small
parameter
32
αs/π
:
Aa(C1, ̄
μ) =
∞
X
n=1
A(n)
a (C1)
αs( ̄
μ)
π
n
; Ba(C1, C2, ̄
μ) =
∞
X
n=1
B(n)
a (C1, C2)
αs( ̄
μ)
π
n
;
Ca/a1
x, b; C1
C2
, μ
=
∞
X
n=0
C(n)
a/a1(x, bμ, C1
C2
)
αs(μ)
π
n
; ha(Q, θ∗) =
∞
X
n=0
h(n)
a (θ∗)
αs(Q)
π
n
.
The
v
alue
of
a
p
erturbativ
e
o
e ien
t F (n)
for
a
set
of
s ales C1/b
and C2Q
an
b
e
expressed
in
terms
of
its
v
alue F (n,c)
obtained
for
the
anoni al
om
bination C1 = c0
and C2 = 1.
Here c0 ≡2e−γE ≈1.123,
where γE = 0.5772 . . .
is
the
Euler
onstan
t.
The
relationships
b
et
w
een F (n)
and F (n,c)
tak
e
the
form
A(1)
a (C1) = A(1,c)
a
;
(A1)
A(2)
a (C1) = A(2,c)
a
−A(1,c)
a
β0 ln c0
C1
;
(A2)
A(3)
a (C1) = A(3,c)
a
−2A(2,c)
a
β0 ln c0
C1
−A(1,c)
a
2
β1 ln c0
C1
+ A(1,c)
a
β2
0
ln c0
C1
2
;
(A3)
B(1)
a (C1, C2) = B(1,c)
a
−A(1,c)
a
ln c2
0C2
2
C2
1
;
(A4)
B(2)
a (C1, C2) = B(2,c)
a
−A(2,c)
a
ln c2
0C2
2
C2
1
+ β0
A(1,c)
a
ln2 c0
C1
+ B(1,c)
a
ln C2 −A(1,c)
a
ln2 C2
;
(A5)
C(1)
a/a1(x, bμ, C1
C2
) = C(1,c)
a/a1(x) + δaa1δ(1 −x)
B(1,c)
a
2
ln c2
0C2
2
C2
1
−A(1,c)
a
4
ln c2
0C2
2
C2
1
2!
−Pa/a1(x) ln μb
c0
.
(A6)
They
dep
end
on
the
QCD
b
eta-fun tion
o
e ien
ts β0 = (11Nc −2Nf)/6 , β1 = (17N2
c −
5NcNf −3CFNf)/6
for Nc
olors
and Nf
a tiv
e
quark
a
v
ors,
with CF = (N2
c −1)/(2Nc) =
4/3
for Nc = 3.
The
relev
an
t O(αs)
splitting
fun tions Pa/a1(x)
are
Pq/q = CF
1 + z2
1 −x
+
; Pq/g = 1
2(1 + 2x + 2x2); Pg/qS = CF
(1 −x)2 + 1
x
;
(A7)
Pg/g = 2CA
x
(1 −x)+
+ 1 −x
x
+ x(1 −x)
+ β0δ(1 −x).
(A8)
The
o
e ien
ts h(1)(θ∗), B(2)
,
and C(1)
dep
end
on
the
resummation
s
heme.
The
hard-
s attering
fun tion
is
ha(Q, θ∗) = 1 + δs
αs(Q)
π
Va(θ∗)
4
+ ...,
(A9)
where δs = 0
in
the
CSS
s
heme
and δs = 1
in
the
CF
G
s
heme.
The
fun tions Vq(θ∗)
for
q ̄
q →γγ
s attering
and Vg(θ∗)
for gg →γγ
s attering
are
deriv
ed
in
Refs.
[12℄
and
[13℄,
resp
e tiv
ely
.
33
F
or
the q ̄
q + qg
initial
state,
w
e
obtain
the
follo
wing
expressions
for
the
o
e ien
ts A , B
,
and C
:
A(1,c)
q
= CF;
A(2,c)
q
= CF
67
36 −π2
12
CA −5
9TRNf
;
(A10)
A(3,c)
q
= C2
FNf
2
ζ(3) −55
48
−CFN2
f
108
+ C2
ACF
11ζ(3)
24
+ 11π4
720 −67π2
216 + 245
96
+ CACFNf
−7ζ(3)
12
+ 5π2
108 −209
432
;
B(1,c)
q
= −3
2CF;
B(2,c)
q
= −1
2
CF
2
3
8 −π2
2 + 6ζ(3)
+ CFCA
17
24 + 11π2
18
−3ζ(3)
−CFNfTR
1
6 + 2π2
9
+ β0
CFπ2
12
+ (1 −δs)Vq(θ∗)
4
;
C(0)
j/k(x) = δjkδ(1 −x); C(0)
j/g(x) = 0;
C(1,c)
j/k (x) = δjk
CF
2 (1 −x) + δ(1 −x)(1 −δs)Vq(θ∗)
4
;
C(1,c)
j/g (x) = 1
2x(1 −x).
(A11)
Here CA = Nc, TR = 1/2 ,
and
the
Riemann
onstan
t ζ(3) = 1.202 . . .
.
The C
fun tions
are
giv
en
for j, k = u, ̄
u, d, ̄
d, . . .
.
These
o
e ien
ts
are
tak
en
from
[12
,
42
,
43
℄.
Similarly
,
the A , B
,
and C
o
e ien
ts
in
the gg + gqS
hannel
are
A(k,c)
g
= (CA/CF)A(k,c)
q
,
for k = 1, 2, 3;
B(1,c)
g
= −β0;
B(2,c)
g
= −1
2
C2
A
8
3 + 3ζ(3)
−CFTRNf −4
3CATRNf
+ β0
CAπ2
12
+ (1 −δs)Vg(θ∗)
4
;
C(0)
g/a (x) = δgaδ(1 −x); C(1,c)
g/g (x) = δ(1 −x)(1 −δs)Vg(θ∗)
4
; C(1,c)
g/qS (x) = CF
2 x.
(A12)
These
o
e ien
ts
are
tak
en
from
Refs.
[12,
13,
44
,
45
℄.
App
endix
B:
COMPONENTS
OF
THE
ASYMPTOTIC
CR
OSS
SECTIONS
In
Se .
I
I
B
w
e
in
tro
du e
asymptoti
small-QT
appro
ximations
for
the q ̄
q+qg
and gg+gqS
NLO
ross
se tions,
Aq ̄
q(Q, QT, y, Ω∗) =
X
i=u, ̄
u,d, ̄
d,...
Σi(θ∗)
S
n
δ( ⃗
QT)Fi,δ(Q, y, θ∗) + Fi,+(Q, y, QT)
o
,
(B1)
34
and
Agg(Q, QT, y, Ω∗) = 1
S
Σg(θ∗)
h
δ( ⃗
QT)Fg,δ(Q, y, θ∗) + Fg,+(Q, y, QT)
i
+Σ′
g(θ∗, φ∗)F ′
g(Q, y, QT)
.
(B2)
The
fun tions F
in
these
equations
are
de ned
as
Fi,δ(Q, y, θ∗) ≡fqi/h1(x1, μF)f ̄
qi/h2(x2, μF)
1 + 2αs
π h(1)
q (θ∗)
+ αs
π
h
C(1,c)
qi/a ⊗fa/h1
i
(x1, μF) −
Pqi/a ⊗fa/h1
(x1, μF) ln μF
Q
f ̄
qi/h2(x2, μF)
+fqi/h1(x1, μF)
h
C(1,c)
̄
qi/a ⊗fa/h2
i
(x2, μF) −
P ̄
qi/a ⊗fa/h2
(x2, μF) ln μF
Q
;
(B3)
Fq,+ =
1
2π
αs
π
fqi/h1(x1, μF)f ̄
qi/h2(x2, μF)
A(1,c)
q
1
Q2
T
ln Q2
Q2
T
+
+ B(1,c)
q
1
Q2
T
+
+
1
Q2
T
+
Pqi/a ⊗fa/h1
(x1, μF) f ̄
qi/h2(x2, μF)
+fqi/h1(x1, μF)
P ̄
qi/a ⊗fa/h2
(x2, μF)
;
(B4)
Fg,δ ≡fg/h1(x1, μF)fg/h2(x2, μF)
1 + 2αs
π h(1)
g (θ∗)
+ αs
π
h
C(1,c)
g/a ⊗fa/h1
i
(x1, μF) −
Pg/a ⊗fa/h1
(x1, μF) ln μF
Q
fg/h2(x2, μF)
+fg/h1(x1, μF)
h
C(1,c)
g/a ⊗fa/h2
i
(x2, μF) −
Pg/a ⊗fa/h2
(x2, μF) ln μF
Q
;
(B5)
Fg,+ =
1
2π
αs
π
fg/h1(x1, μF)fg/h2(x2, μF)
A(1,c)
g
1
Q2
T
ln Q2
Q2
T
+
+ B(1,c)
g
1
Q2
T
+
+
1
Q2
T
+
Pg/a ⊗fa/h1
(x1, μF) fg/h2(x2, μF)
+fg/h1(x1, μF)
Pg/a ⊗fa/h2
(x2, μF)
;
(B6)
and
F ′
g,+ =
1
2π
αs
π
1
Q2
T
+
P ′
g/g ⊗fg/h1
(x1, μF) fg/h2(x2, μF)
+fg/h1(x1, μF)
P ′
g/g ⊗fg/h2
(x2, μF)
.
(B7)
35
Expressions
for
the
o
e ien
ts A(1,c)
a
, B(1,c)
a
, h(1)
a (θ∗), C(1,c)
a/a′ (x),
and
splitting
fun tions Pa/c(x),
are
listed
in
App
endix
A.
Summation
o
v
er
all
relev
an
t
parton
a
v
ors a′ = g, u, ̄
u,d, ̄
d, ...
for a = q
and a′ = g, qS
for a = g
is
assumed.
In
addition,
the φ∗
-dep
enden
t
part
Σ′
g(θ∗, φ∗)F ′
g(Q, y, QT)
of
the gg + gqS
asymptoti
ross
se tion Agg
on
tains
a
splitting
fun tion
P ′
gg(x) = 2CA(1 −x)/x,
(B8)
on
tributed
b
y
the
in
terferen e
of
splitting
amplitudes
with
opp
osite
gluon
p
olarizations
in
the
heli it
y
amplitude
formalism
[46,
47
,
48
,
49℄.
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origin
and
b
eha
vior
of
this
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37
|
0704.0002 | Louis Theran | Ileana Streinu and Louis Theran | Sparsity-certifying Graph Decompositions | To appear in Graphs and Combinatorics | null | null | null | math.CO cs.CG | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use
it obtain a characterization of the family of $(k,\ell)$-sparse graphs and
algorithmic solutions to a family of problems concerning tree decompositions of
graphs. Special instances of sparse graphs appear in rigidity theory and have
received increased attention in recent years. In particular, our colored
pebbles generalize and strengthen the previous results of Lee and Streinu and
give a new proof of the Tutte-Nash-Williams characterization of arboricity. We
also present a new decomposition that certifies sparsity based on the
$(k,\ell)$-pebble game with colors. Our work also exposes connections between
pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and
Westermann and Hendrickson.
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] | Sparsity-certifying Graph Decompositions
Ileana Streinu1∗, Louis Theran2
1 Department of Computer Science, Smith College, Northampton, MA. e-mail: streinu@cs.smith.edu
2 Department of Computer Science, University of Massachusetts Amherst. e-mail: theran@cs.umass.edu
Abstract. We describe a new algorithm, the (k,l)-pebble game with colors, and use it to obtain a charac-
terization of the family of (k,l)-sparse graphs and algorithmic solutions to a family of problems concern-
ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have
received increased attention in recent years. In particular, our colored pebbles generalize and strengthen
the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri-
zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k,l)-pebble
game with colors. Our work also exposes connections between pebble game algorithms and previous
sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].
1. Introduction and preliminaries
The focus of this paper is decompositions of (k,l)-sparse graphs into edge-disjoint subgraphs
that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a
graph is (k,l)-sparse if no subset of n′ vertices spans more than kn′ −ledges in the graph; a
(k,l)-sparse graph with kn′ −ledges is (k,l)-tight. We call the range k ≤l≤2k −1 the upper
range of sparse graphs and 0 ≤l≤k the lower range.
In this paper, we present efficient algorithms for finding decompositions that certify sparsity
in the upper range of l. Our algorithms also apply in the lower range, which was already ad-
dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs
and graphs admitting the decomposition coincide.
Our algorithms are based on a new characterization of sparse graphs, which we call the
pebble game with colors. The pebble game with colors is a simple graph construction rule that
produces a sparse graph along with a sparsity-certifying decomposition.
We define and study a canonical class of pebble game constructions, which correspond to
previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide
a unifying framework for all the previously known special cases, including Nash-Williams-
Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the
properties of the augmenting paths used in matroid union and intersection algorithms[5, 6].
Since the sparse graphs in the upper range are not known to be unions or intersections of the
matroids for which there are efficient augmenting path algorithms, these do not easily apply in
∗Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO
CCR-0310661 to the first author.
arXiv:0704.0002v2 [math.CO] 13 Dec 2008
2
Ileana Streinu, Louis Theran
Term
Meaning
Sparse graph G
Every non-empty subgraph on n′ vertices has ≤kn′ −ledges
Tight graph G
G = (V,E) is sparse and |V| = n, |E| = kn−l
Block H in G
G is sparse, and H is a tight subgraph
Component H of G
G is sparse and H is a maximal block
Map-graph
Graph that admits an out-degree-exactly-one orientation
(k,l)-maps-and-trees
Edge-disjoint union of ltrees and (k −l) map-grpahs
lTk
Union of ltrees, each vertex is in exactly k of them
Set of tree-pieces of an lTk induced on V ′ ⊂V
Pieces of trees in the lTk spanned by E(V ′)
Proper lTk
Every V ′ ⊂V contains ≥lpieces of trees from the lTk
Table 1. Sparse graph and decomposition terminology used in this paper.
the upper range. Pebble game with colors constructions may thus be considered a strengthening
of augmenting paths to the upper range of matroidal sparse graphs.
1.1. Sparse graphs
A graph is (k,l)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤
kn′ −l. We observe that this condition implies that 0 ≤l≤2k −1, and from now on in this
paper we will make this assumption. A sparse graph that has n vertices and exactly kn−ledges
is called tight.
For a graph G = (V,E), and V ′ ⊂V, we use the notation span(V ′) for the number of edges
in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail
in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge.
There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of
a sparse graph. A component is a maximal block.
Table 1 summarizes the sparse graph terminology used in this paper.
1.2. Sparsity-certifying decompositions
A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees.
Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described
by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight
graphs.
A map-graph is a graph that admits an orientation such that the out-degree of each vertex is
exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map-
graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible
configuration certifying that each color forms a map-graph. Map-graphs may be equivalently
defined (see, e.g., [18]) as having exactly one cycle per connected component.1
A (k,l)-maps-and-trees is a graph that admits a decomposition into k −ledge-disjoint
map-graphs and lspanning trees.
Another characterization of map-graphs, which we will use extensively in this paper, is as
the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that
the converse holds as well.
1 Our terminology follows Lov ́
asz in [16]. In the matroid literature map-graphs are sometimes known as bases
of the bicycle matroid or spanning pseudoforests.
Sparsity-certifying Graph Decompositions
3
a
c
b
e
d
(a)
1
2
3
4
(b)
(c)
Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a
(2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is
shown with a certifying orientation.
A lTk is a decomposition into ledge-disjoint (not necessarily spanning) trees such that each
vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2.
Given a subgraph G′ of a lTk graph G, the set of tree-pieces in G′ is the collection of the
components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute
multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come
from the same tree or be single-vertex "empty trees." It is also helpful to note that the definition
of a tree-piece is relative to a specific subgraph. An lTk decomposition is proper if the set of
tree-pieces in any subgraph G′ has size at least l.
Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an
isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree-
pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges.
These count as three tree-pieces, even though they come from the same back tree when the
whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three
gray tree-pieces and one black one.
Table 1 contains the decomposition terminology used in this paper.
The decomposition problem.
We define the decomposition problem for sparse graphs as tak-
ing a graph as its input and producing as output, a decomposition that can be used to certify spar-
sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper lTk decompositions;
and the pebble-game-with-colors decomposition, which is defined in the next section.
2. Historical background
The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to
the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint,
4
Ileana Streinu, Louis Theran
0
1
2
3
4
5
(a)
0
1
2
3
4
5
(b)
0
1
2
3
4
5
(c)
Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right
corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray
tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a
single vertex) and one black tree-piece.
Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps-
and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and
matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19].
In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman)
graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay
[21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight
graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the
equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a
direct proof of Laman's theorem and generalized the 3T2 condition to all lTk for k ≤l≤2k−1.
Haas [7] studied lTk decompositions in detail and proved the equivalence of tight graphs and
proper lTk graphs for the general upper range. We observe that aside from our new pebble-
game-with-colors decomposition, all the combinatorial characterizations of the upper range of
sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24].
A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick-
son's Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the
pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and
Streinu [12] generalized the pebble game to the entire range of parameters 0 ≤l≤2k −1, and
left as an open problem using the pebble game to find sparsity certifying decompositions.
3. The pebble game with colors
Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative
integers k and l. We will use the pebble game with colors as the basis of an efficient algorithm
for the decomposition problem later in this paper. Since the phrase "with colors" is necessary
only for comparison to [12], we will omit it in the rest of the paper when the context is clear.
Sparsity-certifying Graph Decompositions
5
We now present the pebble game with colors. The game is played by a single player on a
fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the
addition and/or orientation of an edge. At any moment of time, the state of the game is captured
by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored
by the pebbles on them. While playing the pebble game all edges are directed, and we use the
notation vw to indicate a directed edge from v to w.
We describe the pebble game with colors in terms of its initial configuration and the allowed
moves.
⇒
⇒
(a)
⇒
⇒
(b)
Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices
are shown as black or gray dots. Edges are colored with the color of the pebble on them.
Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start
by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2,...,k.
Add-edge-with-colors: Let v and w be vertices with at least l+1 pebbles on them. Assume
(w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw
to E(H) and put the pebble picked up from v on the new edge.
Figure 3(a) shows examples of the add-edge move.
Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace
vw with wv in E(H); put the pebble that was on vw on v; and put p on wv.
Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows
examples. The convention in these figures, and throughout this paper, is that pebbles on vertices
are represented as colored dots, and that edges are shown in the color of the pebble on them.
From the definition of the pebble-slide move, it is easy to see that a particular pebble is
always either on the vertex where it started or on an edge that has this vertex as the tail. However,
when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is
sometimes convenient to think of this path reversal sequence as bringing a pebble from the end
of the path to the beginning.
The output of playing the pebble game is its complete configuration.
Output: At the end of the game, we obtain the directed graph H, along with the location
and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble
game configuration colors the edges.
We say that the underlying undirected graph G of H is constructed by the (k,l)-pebble game
or that H is a pebble-game graph.
Since each edge of H has exactly one pebble on it, the pebble game's configuration partitions
the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble-
game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a
pebble-game decomposition.
Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges,
and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con-
6
Ileana Streinu, Louis Theran
(a)
(b)
(c)
Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to
show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an
empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two
black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph.
(c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges
contain a cycle and do not contribute a piece of tree to the subgraph.
Notation
Meaning
span(V ′)
Number of edges spanned in H by V ′ ⊂V; i.e. |EH(V ′)|
peb(V ′)
Number of pebbles on V ′ ⊂V
out(V ′)
Number of edges vw in H with v ∈V ′ and w ∈V −V ′
pebi(v)
Number of pebbles of color ci on v ∈V
outi(v)
Number of edges vw colored ci for v ∈V
Table 2. Pebble game notation used in this paper.
nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic
subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′
otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with
the corresponding definition for lTks, the set of tree-pieces is defined relative to a specific sub-
graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned
by G′.
The properties of pebble-game decompositions are studied in Section 6, and Theorem 2
shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows
this.
For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom-
position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated
vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black
tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not
contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees.
In the following discussion, we use the notation peb(v) for the number of pebbles on v and
pebi(v) to indicate the number of pebbles of colors i on v.
Table 2 lists the pebble game notation used in this paper.
4. Our Results
We describe our results in this section. The rest of the paper provides the proofs.
Sparsity-certifying Graph Decompositions
7
Our first result is a strengthening of the pebble games of [12] to include colors. It says
that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games
discussed in this paper are our pebble game with colors unless noted explicitly.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k,l)-sparse
with 0 ≤l≤2k −1 if and only if G is a pebble-game graph.
Next we consider pebble-game decompositions, showing that they are a generalization of
proper lTk decompositions that extend to the entire matroidal range of sparse graphs.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ltree-pieces of the (1,0)-sparse
graphs in the decomposition.
The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus
Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained
by playing the pebble game defined in the previous section. Notice the similarity between the
requirement that the set of tree-pieces have size at least lin Theorem 2 and the definition of a
proper lTk.
Our next results show that for any pebble-game graph, we can specialize its pebble game
construction to generate a decomposition that is a maps-and-trees or proper lTk . We call these
specialized pebble game constructions canonical, and using canonical pebble game construc-
tions, we obtain new direct proofs of existing arboricity results.
We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo-
sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning
trees contributes at least one piece of tree to every subgraph.
The case of proper lTk graphs is more subtle; if each color in a pebble-game decomposition
is a forest, then we have found a proper lTk , but this class is a subset of all possible proper
lTk decompositions of a tight graph. We show that this class of proper lTk decompositions is
sufficient to certify sparsity.
We now state the main theorem for the upper and lower range.
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤l≤k. A graph G is a tight pebble-game graph if and only if G is a (k,l)-
maps-and-trees.
Theorem 4 (Main Theorem (Upper Range): Proper lTk graphs coincide with pebble-game
graphs). Let k ≤l≤2k−1. A graph G is a tight pebble-game graph if and only if it is a proper
lTk with kn−ledges.
As corollaries, we obtain the existing decomposition results for sparse graphs.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let l≤k. A graph
G is tight if and only if has a (k,l)-maps-and-trees decomposition.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤l≤2k −1. A graph G is tight if and only if it is a
proper lTk .
Efficiently finding canonical pebble game constructions.
The proofs of Theorem 3 and Theo-
rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem.
Our last result improves on this, showing that a canonical pebble game construction, and thus
8
Ileana Streinu, Louis Theran
a maps-and-trees or proper lTk decomposition can be found using a pebble game algorithm in
O(n2) time and space.
These time and space bounds mean that our algorithm can be combined with those of [12]
without any change in complexity.
5. Pebble game graphs
In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game
with colors. Since many of the relevant properties of the pebble game with colors carry over
directly from the pebble games of [12], we refer the reader there for the proofs.
We begin by establishing some invariants that hold during the execution of the pebble game.
Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following
invariants are maintained in H:
(I1) There are at least lpebbles on V. [12]
(I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12]
(I3) For each V ′ ⊂V, span(V ′)+out(V ′)+peb(V ′) = kn′. [12]
(I4) For every vertex v ∈V, outi(v)+pebi(v) = 1.
(I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with
a pebble of color ci or a cycle.
Proof. (I1), (I2), and (I3) come directly from [12].
(I4) This invariant clearly holds at the initialization phase of the pebble game with colors.
That add-edge and pebble-slide moves preserve (I4) is clear from inspection.
(I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of
the same color on it. If there is no pebble of that color reachable, then the path must eventually
visit some vertex twice.
From these invariants, we can show that the pebble game constructible graphs are sparse.
Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the
pebble game. Then H is sparse. If there are exactly lpebbles on V(H), then H is tight.
The main step in proving that every sparse graph is a pebble-game graph is the following.
Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce
the out degree of v by one.
Lemma 9 (The l+1 pebble condition [12]). Let vw be an edge such that H +vw is sparse. If
peb({v,w}) < l+1, then a pebble not on {v,w} can be brought to either v or w.
It follows that any sparse graph has a pebble game construction.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k,l)-sparse
with 0 ≤l≤2k −1 if and only if G is a pebble-game graph.
6. The pebble-game-with-colors decomposition
In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We
start with the following lemmas about the structure of monochromatic connected components
in H, the directed graph maintained during the pebble game.
Sparsity-certifying Graph Decompositions
9
Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub-
graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for
i = 1,...,k.
Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex.
Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H
in a pebble game construction contains at least lmonochromatic tree-pieces, and each of these
is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge.
Recall that an out-edge from a subgraph H′ = (V ′,E′) is an edge vw with v ∈V ′ and vw /
∈E′.
Proof. Let H′ = (V ′,E′) be a non-empty subgraph of H, and assume without loss of generality
that H′ is induced by V ′. By (I3), out(V ′) + peb(V ′) ≥l. We will show that each pebble and
out-edge tail is the root of a tree-piece.
Consider a vertex v ∈V ′ and a color ci. By (I4) there is a unique monochromatic directed
path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle.
Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the
monochromatic path from v leaves V ′), then the path cannot have a cycle in H′.
Since this argument works for any vertex in any color, for each color there is a partitioning
of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each
pebble and out-edge tail is the root of a monochromatic tree, as desired.
Applied to the whole graph Lemma 11 gives us the following.
Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of
color ci is the root of a (possibly empty) monochromatic tree-piece of color ci.
Remark: Haas showed in [7] that in a lTk , a subgraph induced by n′ ≥2 vertices with m′
edges has exactly kn′ −m′ tree-pieces in it. Lemma 11 strengthens Haas' result by extending it
to the lower range and giving a construction that finds the tree-pieces, showing the connection
between the l+1 pebble condition and the hereditary condition on proper lTk .
We conclude our investigation of arbitrary pebble game constructions with a description of
the decomposition induced by the pebble game with colors.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ltree-pieces of the (1,0)-sparse
graphs in the decomposition.
Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub-
graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs.
For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can
span at most n −ti edges; summing over all the colors shows that a graph with a pebble-game
decomposition must be sparse. Apply Theorem 1 to complete the proof.
Remark: We observe that a pebble-game decomposition for a Laman graph may be read out
of the bipartite matching used in Hendrickson's Laman graph extraction algorithm [9]. Indeed,
pebble game orientations have a natural correspondence with the bipartite matchings used in
[9].
10
Ileana Streinu, Louis Theran
Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there
are no cycles in lof the colors, then the trees rooted at the corresponding lpebbles must be
spanning, since they have n −1 edges. Also, if each color forms a forest in an upper range
pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de-
composition is a proper lTk.
In the next section, we show that the pebble game can be specialized to correspond to maps-
and-trees and proper lTk decompositions.
7. Canonical Pebble Game Constructions
In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves-
tigation of decompositions induced by pebble game constructions by studying the case where a
minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15
and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that
this is always possible, implying that monochromatic map-graphs are created only when we
add more than k(n′ −1) edges to some set of n′ vertices. For the lower range, this implies that
every color is a forest. Every decomposition characterization of tight graphs discussed above
follows immediately from the main theorem, giving new proofs of the previous results in a
unified framework.
In the proof, we will use two specializations of the pebble game moves. The first is a modi-
fication of the add-edge move.
Canonical add-edge: When performing an add-edge move, cover the new edge with a color
that is on both vertices if possible. If not, then take the highest numbered color present.
The second is a restriction on which pebble-slide moves we allow.
Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a
monochromatic cycle.
We call a pebble game construction that uses only these moves canonical. In this section
we will show that every pebble-game graph has a canonical pebble game construction (Lemma
14 and Lemma 15) and that canonical pebble game constructions correspond to proper lTk and
maps-and-trees decompositions (Theorem 3 and Theorem 4).
We begin with a technical lemma that motivates the definition of canonical pebble game
constructions. It shows that the situations disallowed by the canonical moves are all the ways
for cycles to form in the lowest lcolors.
Lemma 13 (Monochromatic cycle creation). Let v ∈V have a pebble p of color ci on it and
let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created
in exactly one of the following ways:
(M1) The edge vw is added with an add-edge move.
(M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse
edge vw.
Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7.
By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a
connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble
game construction, since the color of an edge only changes when it is inserted the first time or
a new pebble is put on it by a pebble-slide move.
Sparsity-certifying Graph Decompositions
11
⇒
v
w
v
w
(a)
⇒
v
w
v
w
(b)
Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by
adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are
labeled according to their role in the definition of the moves.
Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves,
respectively, in a (2,0)-pebble game construction.
We next show that if a graph has a pebble game construction, then it has a canonical peb-
ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa-
rately. The proof gives two constructions that implement the canonical add-edge and canonical
pebble-slide moves.
Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc-
tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤i ≤l′, where
l′ = min{k,l}.
Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If
this is not possible, then there are l+1 distinct colors present. Use the highest numbered color
to cover the new edge.
Remark: We note that in the upper range, there is always a repeated color, so no canonical
add-edge moves create cycles in the upper range.
The canonical pebble-slide move is defined by a global condition. To prove that we obtain
the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma
9 to only canonical moves. The main step is to show that if there is any sequence of moves that
reorients a path from v to w, then there is a sequence of canonical moves that does the same
thing.
Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading
to an add-edge move can be replaced with one that has no (M2) steps and allows the same
add-edge move.
In other words, if it is possible to collect l+ 1 pebbles on the ends of an edge to be added,
then it is possible to do this without creating any monochromatic cycles.
12
Ileana Streinu, Louis Theran
Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this
the shortcut construction by analogy to matroid union and intersection augmenting paths used
in previous work on the lower range.
Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step
at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one
application of the shortcut construction reorients a simple path from a vertex w′ to w, and a
path from v to w′ is preserved, the shortcut construction can be applied inductively to find the
sequence of moves we want.
w
v
(a)
v
w
(b)
w
v
w'
(c)
Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines
indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle,
shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part
of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray
tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is
simple, and the shortcut construction can be applied inductively to it.
Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple
path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble
of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v
and w are contained in a maximal monochromatic tree of color ci. Call this tree H′
i, and observe
that it is rooted at w.
Now consider the edges reversed in our sequence of moves. As noted above, before we make
any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on
this path in H′
i. We modify our sequence of moves as follows: delete, from the beginning, every
move before the one that reverses some edge yz; prepend onto what is left a sequence of moves
that moves the pebble on w to z in H′
i.
Sparsity-certifying Graph Decompositions
13
⇒
(a)
⇒
(b)
Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path.
The path where the pebbles move is indicated by doubled lines.
⇒
(a)
⇒
(b)
Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is
(M2); (b) avoiding the (M2) and simplifying the path.
Since no edges change color in the beginning of the new sequence, we have eliminated
the (M2) move. Because our construction does not change any of the edges involved in the
remaining tail of the original sequence, the part of the original path that is left in the new
sequence will still be a simple path in H, meeting our initial hypothesis.
The rest of the lemma follows by induction.
Together Lemma 14 and Lemma 15 prove the following.
Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction.
Using canonical pebble game constructions, we can identify the tight pebble-game graphs
with maps-and-trees and lTk graphs.
14
Ileana Streinu, Louis Theran
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤l≤k. A graph G is a tight pebble-game graph if and only if G is a (k,l)-
maps-and-trees.
Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game
decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game
graph.
For the reverse direction, consider a canonical pebble game construction of a tight graph.
From Lemma 8, we see that there are lpebbles left on G at the end of the construction. The
definition of the canonical add-edge move implies that there must be at least one pebble of
each ci for i = 1,2,...,l. It follows that there is exactly one of each of these colors. By Lemma
12, each of these pebbles is the root of a monochromatic tree-piece with n −1 edges, yielding
the required ledge-disjoint spanning trees.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let l≤k. A graph
G is tight if and only if has a (k,l)-maps-and-trees decomposition.
We next consider the decompositions induced by canonical pebble game constructions when
l≥k +1.
Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb-
ble-game graphs). Let k ≤l≤2k−1. A graph G is a tight pebble-game graph if and only if it
is a proper lTk with kn−ledges.
Proof. As observed above, a proper lTk decomposition must be sparse. What we need to show
is that a canonical pebble game construction of a tight graph produces a proper lTk .
By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom-
position into ledge-disjoint trees. Finally, an application of (I4), shows that every vertex must
in in exactly k of the trees, as required.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤l≤2k −1. A graph G is tight if and only if it is a
proper lTk .
8. Pebble game algorithms for finding decompositions
A na ̈
ıve implementation of the constructions in the previous section leads to an algorithm re-
quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n)
applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running
time of Θ(n3) for the decomposition problem.
In this section, we describe algorithms for the decomposition problem that run in time
O(n2). We begin with the overall structure of the algorithm.
Algorithm 17 (The canonical pebble game with colors).
Input: A graph G.
Output: A pebble-game graph H.
Method:
– Set V(H) = V(G) and place one pebble of each color on the vertices of H.
– For each edge vw ∈E(G) try to collect at least l+1 pebbles on v and w using pebble-slide
moves as described by Lemma 15.
Sparsity-certifying Graph Decompositions
15
– If at least l+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma
14, otherwise discard vw.
– Finally, return H, and the locations of the pebbles.
Correctness.
Theorem 1 and the result from [24] that the sparse graphs are the independent
sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction
found is canonical, the main theorem shows that the coloring of the edges in H gives a maps-
and-trees or proper lTk decomposition.
Complexity.
We start by observing that the running time of Algorithm 17 is the time taken to
process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an
edge of G that is added to H.
Each of the pebble game moves can be implemented in constant time. What remains is to
describe an efficient way to find and move the pebbles. We use the following algorithm as a
subroutine of Algorithm 17 to do this.
Algorithm 18 (Finding a canonical path to a pebble.).
Input: Vertices v and w, and a pebble game configuration on a directed graph H.
Output: If a pebble was found, 'yes', and 'no' otherwise. The configuration of H is updated.
Method:
– Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and
return 'no.'
– Otherwise a pebble was found. We now have a path v = v1,e1,...,ep−1,vp = u, where the vi
are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use
the array c[] to keep track of the colors of pebbles on vertices and edges after we move them
and the array s[] to sketch out a canonical path from v to u by finding a successor for each
edge.
– Set s[u] = 'end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in
reverse order: vp,ep−1,ep−2,...,e1,v1. For each i, check to see if c[vi] is set; if so, go on to
the next i. Otherwise, check to see if c[vi+1] = c[ei].
– If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge.
– Otherwise c[vi+1] ̸= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If
a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2,..., fq−1,xq = x
that is monochromatic in the color of the edges; set c[xi] = c[fi] and s[xi] = fi for i =
1,2,...,q−1. If c[x] = c[fq−1], stop. Otherwise, recursively check that there is not a monochro-
matic c[x] path from xq−1 to x using this same procedure.
– Finally, slide pebbles along the path from the original endpoints v to u specified by the
successor array s[v], s[s[v]], ...
The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut
construction. Efficiency comes from the fact that instead of potentially moving the pebble back
and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three
times: once in the initial depth-first search, and twice while converting the initial path to a
canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time
spent processing edges in H.
Although we have not discussed this explicity, for the algorithm to be efficient we need to
maintain components as in [12]. After each accepted edge, the components of H can be updated
in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1)
time each.
16
Ileana Streinu, Louis Theran
Summarizing, we have shown that the canonical pebble game with colors solves the decom-
position problem in time O(n2).
9. An important special case: Rigidity in dimension 2 and slider-pinning
In this short section we present a new application for the special case of practical importance,
k = 2, l= 3. As discussed in the introduction, Laman's theorem [11] characterizes minimally
rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the
current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com-
binatorially, we model the bar-slider frameworks as simple graphs together with some loops
placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each
color.
We characterize the minimally rigid bar-slider graphs [20] as graphs that are:
1. (2,3)-sparse for subgraphs containing no loops.
2. (2,0)-tight when loops are included.
We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse
graphs studied in our paper [14].
The connection with the pebble games in this paper is the following.
Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we
replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph.
Proof. Follows from invariant (I3) of Lemma 7.
In [15], we study a special case of slider pinning where every slider is either vertical or
horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction.
For this axis parallel slider case, the minimally rigid graphs are characterized by:
1. (2,3)-sparse for subgraphs containing no loops.
2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each
monochromatic tree spans exactly one loop of its color.
This also has an interpretation in terms of colored pebble games.
Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)-
pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the
graph of a minimally pinned axis-parallel bar-slider framework.
Proof. Follows from Theorem 4, and Lemma 12.
10. Conclusions and open problems
We presented a new characterization of (k,l)-sparse graphs, the pebble game with colors, and
used it to give an efficient algorithm for finding decompositions of sparse graphs into edge-
disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range
and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the
upper range from [12].
We also used the pebble game with colors to describe a new sparsity-certifying decomposi-
tion that applies to the entire matroidal range of sparse graphs.
Sparsity-certifying Graph Decompositions
17
We defined and studied a class of canonical pebble game constructions that correspond to
either a maps-and-trees or proper lTk decomposition. This gives a new proof of the Tutte-Nash-
Williams arboricity theorem and a unified proof of the previously studied decomposition cer-
tificates of sparsity. Canonical pebble game constructions also show the relationship between
the l+1 pebble condition, which applies to the upper range of l, to matroid union augmenting
paths, which do not apply in the upper range.
Algorithmic consequences and open problems.
In [6], Gabow and Westermann give an O(n3/2)
algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from
dense ones. Their technique is based on efficiently finding matroid union augmenting paths,
which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to
find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch
scanning, which finds groups of disjoint augmenting paths.
We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester-
mann's algorithm without changing the running time. The data structures used in the implemen-
tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those
used to support cyclic scanning.
The two major open algorithmic problems related to the pebble game are then:
Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain
an implementable O(n3/2) algorithm for the lower range.
Problem 2. Extend batch scanning to the l+1 pebble condition and derive an O(n3/2) pebble
game algorithm for the upper range.
In particular, it would be of practical importance to find an implementable O(n3/2) algorithm
for decompositions into edge-disjoint spanning trees.
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|
0704.0003 | Hongjun Pan | Hongjun Pan | The evolution of the Earth-Moon system based on the dark matter field
fluid model | 23 pages, 3 figures | null | null | null | physics.gen-ph | null | " The evolution of Earth-Moon system is described by the dark matter field\nfluid model proposed in(...TRUNCATED) | "W3sidmVyc2lvbiI6InYxIiwiY3JlYXRlZCI6IlN1biwgMSBBcHIgMjAwNyAyMDo0Njo1NCBHTVQifSx7InZlcnNpb24iOiJ2MiI(...TRUNCATED) | 2008-01-13 | [
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0704.0004 | David Callan | David Callan | A determinant of Stirling cycle numbers counts unlabeled acyclic
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0704.0005 | Alberto Torchinsky | Wael Abu-Shammala and Alberto Torchinsky | From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$ | null | Illinois J. Math. 52 (2008) no.2, 681-689 | null | null | math.CA math.FA | null | " In this paper we show how to compute the $\\Lambda_{\\alpha}$ norm, $\\alpha\\ge\n0$, using the d(...TRUNCATED) | [
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0704.0006 | Yue Hin Pong | Y. H. Pong and C. K. Law | Bosonic characters of atomic Cooper pairs across resonance | 6 pages, 4 figures, accepted by PRA | null | 10.1103/PhysRevA.75.043613 | null | cond-mat.mes-hall | null | " We study the two-particle wave function of paired atoms in a Fermi gas with\ntunable interaction (...TRUNCATED) | [
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0704.0007 | Alejandro Corichi | Alejandro Corichi, Tatjana Vukasinac and Jose A. Zapata | Polymer Quantum Mechanics and its Continuum Limit | 16 pages, no figures. Typos corrected to match published version | Phys.Rev.D76:044016,2007 | 10.1103/PhysRevD.76.044016 | IGPG-07/03-2 | gr-qc | null | " A rather non-standard quantum representation of the canonical commutation\nrelations of quantum m(...TRUNCATED) | "W3sidmVyc2lvbiI6InYxIiwiY3JlYXRlZCI6IlNhdCwgMzEgTWFyIDIwMDcgMDQ6Mjc6MjIgR01UIn0seyJ2ZXJzaW9uIjoidjI(...TRUNCATED) | 2008-11-26 | "W1siQ29yaWNoaSIsIkFsZWphbmRybyIsIiJdLFsiVnVrYXNpbmFjIiwiVGF0amFuYSIsIiJdLFsiWmFwYXRhIiwiSm9zZSBBLiI(...TRUNCATED) | "arXiv:0704.0007v2 [gr-qc] 22 Aug 2007\nPolymer Quantum Mechanics and its Continuum Limit\nAlejand(...TRUNCATED) |
0704.0008 | Damian Swift | Damian C. Swift | Numerical solution of shock and ramp compression for general material
properties | Minor corrections | Journal of Applied Physics, vol 104, 073536 (2008) | 10.1063/1.2975338 | LA-UR-07-2051, LLNL-JRNL-410358 | cond-mat.mtrl-sci | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | " A general formulation was developed to represent material models for\napplications in dynamic loa(...TRUNCATED) | "W3sidmVyc2lvbiI6InYxIiwiY3JlYXRlZCI6IlNhdCwgMzEgTWFyIDIwMDcgMDQ6NDc6MjAgR01UIn0seyJ2ZXJzaW9uIjoidjI(...TRUNCATED) | 2009-02-05 | [
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0704.0010 | Sergei Ovchinnikov | Sergei Ovchinnikov | Partial cubes: structures, characterizations, and constructions | 36 pages, 17 figures | null | null | null | math.CO | null | " Partial cubes are isometric subgraphs of hypercubes. Structures on a graph\ndefined by means of s(...TRUNCATED) | [
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