Parton Shower Monte Carlos

Parton Shower Monte Carlos
Peter Skands (CERN Theoretical Physics Dept)
LHCphenonet Summer School
Cracow, Poland, September 2013
Scattering Experiments
source
⇤⌅
Z
LHC detector
Cosmic-Ray detector
Neutrino detector
X-ray telescope
…
d
Ncount(⇤⌅) /
d⌅
d⌅
⇥⌅
→ Integrate differential cross sections
over specific phase-space regions
⇤⌅
Predicted number of counts
= integral over solid angle
Z
d
Ncount(⇤⌅) /
d⌅
d⌅
⇥⌅
In particle physics:
Integrate over all quantum histories
(+ interferences)
P. S k a n d s
Lots of dimensions?
Complicated integrands?
→ Use Monte Carlo
2
General-Purpose Event Generators
Reality is more complicated
Calculate Everything ≈ solve QCD → requires compromise!
Improve lowest-order perturbation theory,
by including the ‘most significant’ corrections
→ complete events (can evaluate any observable you want)
The Workhorses
PYTHIA : Successor to JETSET (begun in 1978). Originated in hadronization studies: Lund String.
HERWIG : Successor to EARWIG (begun in 1984). Originated in coherence studies: angular ordering.
SHERPA : Begun in 2000. Originated in “matching” of matrix elements to showers: CKKW-L.
+ MORE SPECIALIZED: ALPGEN, MADGRAPH, ARIADNE, VINCIA, WHIZARD, MC@NLO, POWHEG, …
P. S k a n d s
3
Divide and Conquer
Factorization → Split the problem into many (nested) pieces
+ Quantum mechanics → Probabilities → Random Numbers
Pevent = Phard ⌦ Pdec ⌦ PISR ⌦ PFSR ⌦ PMPI ⌦ PHad ⌦ . . .
Hard Process & Decays:
Use (N)LO matrix elements
→ Sets “hard” resolution scale for process: QMAX
ISR & FSR (Initial & Final-State Radiation):
Altarelli-Parisi equations → differential evolution, dP/dQ2, as
function of resolution scale; run from QMAX to ~ 1 GeV (More later)
MPI (Multi-Parton Interactions)
Additional (soft) parton-parton interactions: LO matrix elements
→ Additional (soft) “Underlying-Event” activity (Not the topic for today)
Hadronization
Non-perturbative model of color-singlet parton systems → hadrons
P. S k a n d s
4
(PYTHIA)
PYTHIA anno 1978
(then called JETSET)
LU TP 78-18
November, 1978
A Monte Carlo Program for Quark Jet
Generation
T. Sjöstrand, B. Söderberg
A Monte Carlo computer program is
presented, that simulates the
fragmentation of a fast parton into a
jet of mesons. It uses an iterative
scaling scheme and is compatible with
the jet model of Field and Feynman.
Note:
Field-Feynman was an early fragmentation model
Now superseded by the String (in PYTHIA) and
Cluster (in HERWIG & SHERPA) models.
P. S k a n d s
5
(PYTHIA)
PYTHIA anno 2013
(now called PYTHIA 8)
LU TP 07-28 (CPC 178 (2008) 852)
October, 2007
A Brief Introduction to PYTHIA 8.1
T. Sjöstrand, S. Mrenna, P. Skands
The Pythia program is a standard tool
for the generation of high-energy
collisions, comprising a coherent set
of physics models for the evolution
from a few-body hard process to a
complex multihadronic final state. It
contains a library of hard processes
and models for initial- and final-state
parton showers, multiple parton-parton
interactions, beam remnants, string
fragmentation and particle decays. It
also has a set of utilities and
interfaces to external programs. […]
P. S k a n d s
~ 100,000 lines of C++
What a modern MC generator has inside:
• Hard Processes (internal, interfaced, or via Les Houches events)
• BSM (internal or via interfaces)
• PDFs (internal or via interfaces)
• Showers (internal or inherited)
• Multiple parton interactions
• Beam Remnants
• String Fragmentation
• Decays (internal or via interfaces)
• Examples and Tutorial
• Online HTML / PHP Manual
• Utilities and interfaces to
external programs
6
(some) Physics
cf. equivalent-photon
approximation
Weiszäcker, Williams
~ 1934
Charges Stopped
or kicked
Radiation
Radiation
a.k.a.
Bremsstrahlung
Synchrotron Radiation
The harder Associated
they stop, the
harder the
field
fluctations (fluctuations)
that continue tocontinues
become radiation
7
Jets = Fractals
Most bremsstrahlung is
driven by divergent
propagators → simple structure
/
1
2(pa · pb )
a
b
Amplitudes factorize in
singular limits (→ universal
i
j
k
“conformal” or “fractal” structure)
Partons ab →
P(z) = Altarelli-Parisi splitting kernels, with z = energy fraction = Ea/(Ea+Eb)
“collinear”:
P (z)
2 a||b 2
|MF +1 (. . . , a, b, . . . )| ! gs C
|MF (. . . , a + b, . . . )|2
2(pa · pb )
Gluon j
→ “soft”:
Coherence → Parton j really emitted by (i,k) “colour antenna”
|MF +1 (. . . , i, j, k. . . )|
+ scaling violation: gs2 →
2 jg !0
!
gs2 C
4παs(Q2)
See: PS, Introduction to QCD, TASI 2012, arXiv:1207.2389
P. S k a n d s
(pi · pk )
|MF (. . . , i, k, . . . )|2
(pi · pj )(pj · pk )
Can apply this many times
→ nested factorizations
8
Bremsstrahlung
dσ X
ddσσ
X+X
For any basic process
d
d
= ✓
X+1
$
1+&2&&
X
d
d
X+2
X+3
(calculated process by process)
⇠
2 dsi1
NC 2gs
si1
ds1j
d
s1j
⇠
2 dsi2
NC 2gs
si2
ds2j
d
s2j
X+1
✓
⇠
2 dsi3
NC 2gs
si3
ds3j
d
s3j
X+2
...
X
✓
Factorization in Soft and Collinear Limits
P(z) : “Altarelli-Parisi Splitting Functions” (more later)
|M (. . . , pi , pj . . .)|
2 i||j
!
|M (. . . , pi , pj , pk . . .)|
gs2 C
2 jg !0
!
P (z)
2
|M (. . . , pi + pj , . . .)|
sij
gs2 C
2sik
2
|M (. . . , pi , pk , . . .)|
sij sjk
“Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later)
P. S k a n d s
9
Bremsstrahlung
For any basic process
dσ X
ddσσ
X+X
d
d
= ✓
X+1
$
1+&2&&
X
d
d
X+2
X+3
(calculated process by process)
⇠
2 dsi1
NC 2gs
si1
ds1j
d
s1j
⇠
2 dsi2
NC 2gs
si2
ds2j
d
s2j
X+1
✓
⇠
2 dsi3
NC 2gs
si3
ds3j
d
s3j
X+2
...
✓
X
Singularities: mandated by gauge theory
Non-singular terms: process-dependent
SOFT
COLLINEAR

✓
◆
0
2
|M(Z ! qi gj q̄k )|
2sik
1
sij
sjk
2
= gs 2CF
+
+
0
2
|M(Z ! qI q̄K )|
sij sjk
sIK sjk
sij
0
2
|M(H ! qi gj q̄k )|
2
=
g
s 2CF
0
2
|M(H ! qI q̄K )|
P. S k a n d s

✓
◆
2sik
1
sij
sjk
+
+
+2
sij sjk
sIK sjk
sij
SOFT
COLLINEAR+F
10
Bremsstrahlung
dσ X
ddσσ
X+X
For any basic process
d
X
= ✓
X+1
$
1+&2&&
d
d
d
X+2
X+3
(calculated process by process)
⇠
2 dsi1
NC 2gs
si1
ds1j
d
s1j
⇠
2 dsi2
NC 2gs
si2
ds2j
d
s2j
X+1
✓
⇠
2 dsi3
NC 2gs
si3
ds3j
d
s3j
X+2
...
X
✓
Iterated factorization
Gives us a universal approximation to ∞-order tree-level cross sections.
Exact in singular (strongly ordered) limit.
Finite terms (non-universal) → Uncertainties for non-singular (hard) radiation
But something is not right … Total σ would be infinite …
P. S k a n d s
11
Loops and Legs
Coefficients of the Perturbative Series
Loops
X(2) X+1(2)
…
The corrections from
Quantum Loops are
missing
X(1) X+1(1) X+2(1) X+3(1)
Born X+1(0) X+2(0) X+3(0)
…
…
Universality
(scaling)
Jet-within-a-jet-within-a-jet-...
Legs
P. S k a n d s
12
Evolution
Q ⇠ QX
Leading Order
%
of LO
“Experiment”
100
100
75
75
%
50
of σtot
50
25
25
0
0
Born
+1
+2
Born (exc) +1 (exc)
+2 (inc)
Exclusive = n and only n jets
Inclusive = n or more jets
P. S k a n d s
13
Evolution
QX
Q⇠
“A few”
Leading Order
%
of LO
“Experiment”
100
100
75
75
%
50
of σtot
50
25
25
0
0
Born
+1
+2
Born (exc) +1 (exc)
+2 (inc)
Exclusive = n and only n jets
Inclusive = n or more jets
P. S k a n d s
14
Evolution
Q ⌧ QX
Leading Order
400
100
300
%
of LO
%
of σtot
N
U
100
0
A
T
I
75
200
Born
+1
50
25
+2
Cross Section Diverges
P. S k a n d s
I
R
Y
T
“Experiment”
0
✓
Born (exc) + 1 (exc)
+ 2 (inc)
Cross Section Remains = Born (IR safe)
Number of Partons Diverges (IR unsafe)
15
Unitarity → Evolution
Unitarity
Kinoshita-Lee-Nauenberg:
Imposed by Event evolution:
When (X) branches to (X+1):
Gain one (X+1). Loose one (X).
(sum over degenerate quantum states = finite)
Loop = - Int(Tree) + F
Parton Showers neglect F
→ Leading-Logarithmic (LL) Approximation
→ evolution equation with kernel
d X+1
d X
Evolve in some measure of resolution
~ hardness, 1/time … ~ fractal scale
→ includes both real (tree) and virtual (loop) corrections
►  Interpretation: the structure evolves! (example: X = 2-jets)
• 
• 
• 
• 
• 
Take a jet algorithm, with resolution measure “Q”, apply it to your events
At a very crude resolution, you find that everything is 2-jets
At finer resolutions  some 2-jets migrate  3-jets = σX+1(Q) = σX;incl– σX;excl(Q)
Later, some 3-jets migrate further, etc  σX+n(Q) = σX;incl– ∑σX+m<n;excl(Q)
This evolution takes place between two scales, Qin ~ s and Qend ~ Qhad
►  σX;tot = Sum (σX+0,1,2,3,…;excl ) = int(dσX)
P. S k a n d s
16
hich defines the probability for a radiator not to have any emissions between two scales,
the total probability for branching of the event as a whole is obtained as a sum of such
dP ( ) An equally
object in both analytical resummations
and in parton showe
= gs2 C A(fundamental
) ,!
(9)
!
Z Q
Z Q2 for a radiator not to have any emissions betw
d form
2
factor,
defines
the
probability
2 dP (which
2
)
2
2
2
Q
,
Q
)
=
exp
d
=
exp
g
,
(10)
and
point.
(If
there
are
several
partons/dipoles/antennae,
Q1 and
Q
,
1 to2denote a phase-space
s C A( ) d
2
d
Q21
Q21
branching of the event as a whole is obtained as a sum ofZ such
terms.) !
Z Q2
2
Q2
2
dP
(
)
2
2
2
ntal
object
in
both
analytical
resummations
and
in
parton
showers
is
the
Sudakov
derstood that the integral boundaries
must
be
imposed
either
as
step
functions
on
the
(Q
,
Q
)
=
exp
d
=
exp
g
1 a2 differential equation
s C A( ) d
What
we
need
is
d by
probability
for a radiator
not to havevariables,
any emissions
two
scales, factors. Q21
Q21between
yesa the
suitable
transformation
of integration
accompanied
Jacobian
Boundary
condition:
a few
partons
defined
at a
scale (Q
very close analogue
in the simple
process
of nuclear
decay,
in which
thehigh
probability
forF)a
where it !
is understood that the integral boundaries
must be imposed either as step
!
evolves
(or “runs”)
that
parton
system down to a low scale
Z Q
Znuclear
2Then
2
ergo a decay,
per
unit
time,
is
given
by
the
decay
constant,
Q
2 dP
2
integrand
or by a suitable
transformation
of integration variables, accompanied by Jaco
( hadronization
)
2 GeV) → It’s an evolution equation in QF
(the
cutoff
~
1
= exp
d has =
exp close analogue
gs C A(
) dsimple, process(10)
This
a
very
in
the
of nuclear decay, in which the
dP
(t)
d
Q21
Q21
=
cN . per unit time, is given by the nuclear decay
(11) constant,
nucleus
to
undergo
a
decay,
Close analogue:
dt nuclear decay
hat the integral boundaries must be imposed either as step functions on the
Evolve at
antime
unstable
nucleus.
Checkbefore
if it decays
dP
(t)t2 ,+is follow chains of
ye transformation
for a nucleus existing
t1 to remain
undecayed
time
of integration
variables,
accompanied
by Jacobian
factors.
= cN .
decays. ✓ Z
dt
analogue in the simple process oft2nuclear◆
decay, in which the probability
for a
)The
=
exp
cN adt
=constant,
exp ( to
cat
t) t.1 toundecayed
(12)
1 , t2is
N remain
Probability
in the
timetime t2 , is
probability
for
nucleus
existing
time
remain undecayed
before
ay, per unit (t
time,
given
by the nuclear
decay
Decay constant
t1
interval [t1,t2] ✓ Z t
◆
2
dP (t)
pecially simple, since the decay
per
cN , is constant.
By conservation
= cNprobability
.
(t1unit
, t2 )time,
= exp
cN dt(11)
= exp ( cN t) .
dt the activity per nucleon at time t, equivalent
t1
mber of nuclei (unitarity),
to the “resummed”
cleus
existing
at time
t1 to
remain
undecayed
before
timethe
t2decay
, is probability per unit time, c , is constant.
ity per
unit time,
isThis
minus
the
derivative
of
, since
case
is
especially
simple,
= 1 cN t N+ O(c2N )
Decay probability per unit time
✓ ofZthe
◆
t2 total number
dPres (t)
d of nuclei (unitarity), the activity per nucleon at time t, equivalent to
(t1 , t2 ) = exp decay cprobability
exp
( cNctime,
. . the derivative (12)
= = per
=
t)
(13)
N dt
N (tt)
1 , minus
unit
is
of ,
dt
tdt
1
∆(t1,t2) : “Sudakov Factor”
(requires that the nucleus did not already decay)
dPthe
d for an atennna not to
res (t)
e
emission
probability
varies over
space,cNhence
probability
mple, since the
decay probability
perphase
unit time,
, is constant.
By
conservation
=
= cN (t1 , t) .
dt
ore
formper
of eq.
(10).atBy
unitarity,
thedtresummed
branching probability
clei elaborate
(unitarity),integral
the activity
nucleon
time
t, equivalent
to the “resummed”
P.derivative
S kis
a nminus
ds
ofthe
thederivative
Sudakov
factor,
itthe
time,
In QCD,of
the ,emission probability varies over phase space, hence the probability 17
for
Evolution Equations
PDGLAP =
dQ
dz Pi(z)
2
Q
"
Nuclear
Decay
ds ds |M (s , s
P
=
i
2
,
s)|
3 ij jk
ij
jk
Antenna
16π 2s
|M2(s)|2
! " t2
#
dP
Nuclei remaining undecayed =
∆(t1, t2) = exp −
dt
after time t
dt
t1
100 %
Second
Order
50 %
All Orders
Exponential
0%
Early
Times
Time
Third Order
Late
Times
First Order
-50 %
-100 %
P. S k a n d s
18
1
2
s
2
d
hich defines the probability for a radiator not to have any emissionsQbetween
two scales,
1
Q21
The Sudakov Factor
where it is understood that the integral boundaries must be imposed either as step functions on the
!
!
Z
Z
2
2
ntegrand or by a suitableQtransformation
of integration variables,
accompanied by Jacobian factors.
Q2
2 dP ( )
2
Q21 This
, Q22 )has
= aexp
d
=
exp
g
) d in which
,
very close analogue in the simple process of nuclear
decay,
the (10)
probability for a
s C A(
2
2
d
Q1
1
ucleus to undergo a decay,
per unit time, is given by theQnuclear
decay constant,
derstood that
integral boundaries
mustdP
be(t)imposed
either as step
functions
on the
Inthenuclear
decay,
the
Sudakov
factor
counts:
=accompanied
cN .
(11)
y a suitable transformation of integration variables,
by
Jacobian
factors.
dt undecayed after a time t
How
many
nuclei
remain
very close analogue in the simple process of nuclear decay, in which the probability for a
The probability for a nucleus existing at time t1 to remain undecayed before time t2 , is
ergo a decay, per unit Probability
time, is given
by theundecayed
nuclear decay
to remain
in the constant,
time interval [t1,t2]
✓ Z t2
◆
(t)
(t1 , tdP
(12)
2 ) = exp
= cN .t cN dt = exp ( cN t) .
(11)
1
dt
This
is especially
simple,
since
decayundecayed
probabilitybefore
per unittime
time,
y forcase
a nucleus
existing
at time
t1 tothe
remain
t2c, N
is, is constant. By conservation
f the total number
nuclei✓(unitarity),
the◆activity
per a
nucleon
at time t,system
equivalent to the “resummed”
The ofSudakov
for
parton
Z t2 factor
ecay probability
the derivative
(t1 , t2per
) =unit
exptime, is minus
cN dt
= exp (ofcN , t) .
(12)
counts:
t1
dPresthat
(t) thed parton system doesn’t evolve
The probability
=per unit time,
= cN
(t1constant.
, t) .
(13)
pecially simple, since the decay probability
c
,
is
By
conservation
N
dt run the
dt factorization
(branch) when we
scale (~1/time)
mber of nuclei (unitarity), the activity per nucleon at time t, equivalent to the “resummed”
In QCD, the emission
phase space, hence the probability for an atennna not to
from a probability
high to varies
a lowover
scale
ity per unit time, isEvolution
minus the
derivative
of
,
probability
perof
unit
mit has the more elaborate
integral
form
eq.“time”
(10). By unitarity, the resummed branching probability
s again minus the derivative
factor,
dPres (t) of the Sudakov
d
=
= cN (t1 , t) .
(13)
dt
dPresdt
( ) (replace
cN by proper shower
evolution kernels)
= gs2 C A(
) (Q21 , Q2 ( )) ,
(14)
(replace
by shower
evolution
scale)
e emission probability varies overd phase space,
hencet the
probability
for an
atennna not to
oreP.elaborate
integral form of eq. (10). By unitarity, the resummed branching probability
S k2
where
Q a (n d s) gives the value of the shower evolution scale (typically chosen as a measure of invariant19
h parton b taking a fraction z and parton c a fraction 1 z. To specify the
Q
azimuthal angle ⇧ of the b around the a direction is needed in addition;
is the QCD scale in s . Of course, this choice is more direct
iscussions ⇧ is chosen to bewhere
isotropically
distributed,
although
options
for
QCD parts of the shower, but it can be used just as well for the QED on
istributions currently are the
thedefaults.
two variables t and z, the di⇥erential probability dP for parton a to
ility for a parton to branch is given by the evolution equations (also called
abc
arelli–Parisi [Gri72, Alt77]). It is convenient to introduce
dPa =
Pa bc (z) dt dz .
What’s the evolution kernel?
Altarelli-Parisi splitting functions
2
dQ
2
2
2
b,c
2⌅
t = ln(Q
/ ) ⇤ Here
dt (in
=
ln(Q )supposed
= 2 limit)
,to run over
(162)
thedsum
all allowed
branchings, for a qua
Can be derived
the iscollinear
from
requiring
Q
q ⇥ q⇥, and so on. The abc factor is em for QED branchings and
invariance ofbethe
physical
result
with
respect
to QF → RGE
evaluated
at some
suitable
scale,
see below).
QCD scale in s . Of course,
choice
is more
Thethis
splitting
kernels
Pa bcdirected
(z) are towards the
he shower, but it can
be used just as well for the QED ones. In terms of
Altarelli-Parisi
1+
z2
es t and z, the di⇥erential
probability
dP
for
parton
a
to
branch
is
now
(E.g., PYTHIA)
Pq qg (z) = CF
,
1 z
2
abc
(1
z(1
z))
dPa =
Pa bc (z) dt dz .
(163)
P
(z)
=
N
,
g
gg
C
2⌅
z(1 z)
b,c
2
2
P
(z)
=
T
(z
+
(1
z)
),
g
qq
R
c
s supposed to run over all allowed branchings, for a quark q ⇥1 +
qgz 2and
b
Pq q s (z)
e2q ones (to
,
on. The abc factorais em for QED
branchings and
for =
QCD
1 z
= z pa
some suitable scale, seepbbelow).
2
1
+
z
P⇥ ⇥ (z) = e2⇥
,
g kernels Pa bc (z) arepc = (1-z) pa
1 z
Pq
Pg
P. S k a n d s
s
fo
2
1
+
z
with
C
3, TQR2 some
= nf measure
/2 (i.e. of
TR“hardness”
receives a contribution
F = 4/3, NC =
2
…
with
dQ Callowed
,
2
2
qg (z) =
F
2flavour),
electric charge (4/9 for u-t
qq
and
e
and
e
= event/jet
resolution
⇥ the squared
q
dt = 2 = d
1 ln Q
z
Q for d-type ones, and
measuring
parton virtualities / formation time / …
2 1 for leptons).
(1
z(1familiar
z)) with analytical calculations may wonder why the ‘+
Persons
,
gg (z) = NC
z) of the splitting kernels in eq. (164) are missing. Thes
and ⇤(1z(1z) terms
20
fulfil the task of ensuring flavour and energy conservation in the analytical
Coherence
Coherence
QED: Chudakov effect (mid-fifties)
e+
e−
cosmic ray γ atom
Approximations to
Coherence:
Angular Ordering (HERWIG)
Illustration by T. Sjöstrand
reduced
ionization
emulsion plate
normal
ionization
Angular Vetos (PYTHIA)
Coherent Dipoles/Antennae
(ARIADNE, Catani-Seymour, VINCIA)
QCD: colour coherence for soft gluon emission
2
+
2
=
by • ofrequiring
emission angles
to that
be decreasing
→ ansolved
example
an interference
effect
can be treated probabilistically
or
•
requiring transverse momenta to be decreasing
More interference effects can be included by matching to full matrix elements
P. S k a n d s
21
rat
0.02
10-4
Coherence at Work
Figure 3: Di↵erent color flows and corresponding emission
Pythia
patterns in qq ! qq scattering. The straight (black) lines are
0.01
quarks with arrows denoting the direction of motion in the iniVincia (default)
tial or final states, and the curved (colored) lines indicating the
Vincia (enh. antennae)
color flow. The beam axis is horizontal, and the vertical axis
Example taken from: Ritzmann, Kosower, PS, PLB718 (2013) 1345
0
is transverse
to the beam. The initial-state momenta would be
0
5
10
15
20
reversed in a Feynman diagram, so that the gluon emissions
p
Example: quark-quark scattering
in byhadron
collisions
symbolically indicated
curly lines would
be inside the corresponding color antennæ. Forward flow isoshown on the left,
Consider
one
specific
phase-space
point (eg scattering at 45 )
igure 2: The Drell-Yan pT spectrum. The dashed red curve
and backward flow on the right.
hows the value 2
computed
using colour
Vincia withflows:
default antennæ
possible
a and b
unctions, while the dotted green curve shows the Vincia preicted with an enhanced antenna function. The solid blue
urve gives the Pythia 8 prediction.
inset shows the higha)The
“forward”
2
180°
T tail.
colour flow
remit
2
ertainty due to the shower function and in particuar higher-order terms in the shower. The di↵ernce shown here is illustrative only; a more exensive Figure
exploration
of possible
variations
3: Di↵erent
colorantenna
flows and
corresponding emission
0°
45°
90°
135°
180°
would be
required
the“backward”
spread
as a (black) lines
b)
patterns
in qqbefore
! qqtaking
scattering.
The straight
are
q Hgluon, beamL
colour
flow
uantitative
estimate
of
the
uncertainty.
We
may
quarks with arrows denoting the direction of motion in the inionetheless
thatand
thethe
Pythia
referencelines indicating
tial orobserve
final states,
curved8(colored)
the
Figure 4: Angular
distribution of the first gluon emission in
alculation
the Vincia
(with de! qq axis
scattering at 45 , for the two di↵erent color flows.
colordi↵ers
flow. from
The beam
axis is one
horizontal,
and the qq
vertical
Thewould
light (red)
ault antenna)
by roughly
the same
in the
is transverse
to the beam.
Theamount
initial-state
momenta
be histogram shows the emission density for the
forward flow, and the dark (blue) histogram shows the emiseak region
as does
the enhanced
Vincia
reversed
in a Feynman
diagram,
so predicthat the gluon
emissions
sion density for the backward flow.
on.
This
illustrates
aindicated
tradeo↵ by
between
a more
ac- be inside the corsymbolically
curly
lines
would
nt color
flows
and corresponding
emission
responding
color
antennæ.
Forward
is shown on the left,
ve
strategy
and
a more
active
radiqq recoil
scattering.
The(Pythia)
straight
(black)
lines
areflow
Another good recent example is the SM contribution to the Tevatron top-quark forwardand backward
flow
theasymmetry
right.
tion
pattern
(enhanced
Vincia),
which
in- showers,
ws
denoting
the
direction
ofon
motion
in thewill
inibackward
frombe
coherent
see: PS,
Webber, should
Winter, JHEP
1207 (2012)
151
plies that
radiation
be directed
primarily
inP. S k a n d s
and the curved
(colored)
lines indicating
thep , all
eresting
to study
more closely.
At large
3
1
180°
4
22
S({p}X , O) = (O ⇤ O({p}X ))
⇧
⇤
⇥ thad
⇥ thad
dP
dP
(O⇤O({p}X )) +
dt
dtX+1
(O⇤O({p
S({p}X , O) = 1 ⇤
dt
dtX+1
tstart
tstart
⇥ thad
d⇧(tstart, t)
dt
S({p}X+1, O)
S({p}X , O) = ⇧(tstart, thad) (O⇤O({p}X )) ⇤
dt
tstart
Observation: the
⇥ evolution ⌅kernel is responsible
d
X+1 wX+1 ⌅⌅
for generating
P = real radiation.
d X wX ⌅PS
→ Choose it to be the ratio
⇥ of 2the real-emission matrix element
dQ
to the Born-level matrix element
PDGLAP =
dz Pi(z)
2
Q
→ AP in coll limit, buti also includes
the Eikonal for soft radiation.
⇥
2
|M
ds
ds
(s
,
s
,
s)|
ij
jk
3
ij
jk
Dipole-Antennae
PAntenna =
2→3 instead of 1→2
2
2
16⌦ s
|M2(s)|
(E.g., ARIADNE, VINCIA)
(→ all partons on shell)
Antennae
dPIK
s
ijk
I
K
k
(sij,sjk)
P. S k a n d s
a(sij , sjk )
⇤ ⇥
⇧(t1, t2) = exp ⇤
=
i
j
dsij dsjk
16⌃ 2 s
(…)
(…)
t2
t1
Antenna functions of invariants
⇥
2CF
2
2
aqq̄ qgq̄ = sij s 2sik s + sij + sjk
⇧
dP
aqg qgg =
dt
dt
agg
ggg
=
jk
CA
sij sjk
CA
sij sjk
TR
sjk
aqg
q q̄ 0 q 0
=
agg
g q̄ 0 q 0
= aqg
2sik s +
s2ij
2sik s +
s2ij
s
2sij +
q q̄ 0 q 0
+
s2jk
s3ij
+
s2jk
s3ij
2s2ij
⇥
⇥
s3jk
⇥
… + non-singular terms
23
Bootstrapped Perturbation Theory
Start from an arbitrary lowest-order process (green = QFT amplitude squared)
(virtual corrections)
No. of Quantum Loops
Parton showers generate the bremsstrahlung terms of the rest of the
perturbative series (approximate infinite-order resummation)
+0(2)
+1(2)
Universality (scaling)
…
Jet-within-a-jet-within-a-jet-...
+0(1)
+1(1)
+2(1)
+3(1)
Un
ita
rit
y
Cancellation of real & virtual singularities
Lowest
+1(0)
Order
+2(0)
+3(0)
Exponentiation
fluctuations within fluctuations
But No.
≠ full
QCD!
Only
LLEmis
Approximation
(→ matching)
of Brem
sstra
hlung
sions
(real corrections)
P. S k a n d s
24
Perturbation
shower
formalism. Theory with Markov Chains
The Shower Operator
nsider the Born-level cross section for an arbitrary hard process, H, differentially in an arbitrary
ared-safe
observable Theory
O,
2.1
Perturbation
with Markov Chains
!
"
Consider the Born-level
dσcross
(0) 2 hard process, H, differentially in an arbitrary
H !! section for an arbitrary
(1)
infrared-safe observabledO
O, !Born = dΦH |MH | δ(O − O({p}H )) ,
!
"
H = Hard process
!
dσ
(0)
H
2
here the integration runs over !the full
phase space of H (this expression and
= final-state
dΦH |Mon-shell
(1)
Born dO !
H | δ(O − O({p}H )) ,
{p} : partons
se below would also apply to Born
hadron collisions were we to include integrations over the parton
ribution
functions
in the
initial
andfinal-state
the δ function
projects
a 1-dimensional
slice defined
where the
integration
runs
overstate),
the full
on-shell
phaseout
space
of H (this expression
and
O
evaluated
on the also
set ofapply
final-state
momenta
which
we we
denote
{p}H (without
the over
δ function,
the
those
below would
to hadron
collisions
were
to include
integrations
the parton
instead
of
evaluating
O
on not
the
finalone).
state,
grationBut
overfunctions
phase space
would
just
giveand
thethe
total
cross section,
differential
distribution
in the
initial
state),
δdirectly
function
projects
outthe
aBorn
1-dimensional
slice defined
ToO
make
the connection
parton
showers,
to
discuss
all-orders
in that context,
insert
aand
showering
operator
by
evaluated
on the settooffirst
final-state
momenta
which
we denote
{p}resummations
H (without the δ function, the
may insert over
an operator,
S, that
actsjust
on the
finalsection,
state before
thedifferential
observableone).
is evaluated,
integration
phase space
would
giveBorn-level
the total cross
not the
To make the connection to parton
showers,
and to discuss all-orders resummations in that context,
!
"
{p} : partons
!
dσ
Born
(0)
H
2 final state before the observable is evaluated,
we may insert an operator, S, that! acts
on dΦ
the HBorn-level
|
S({p}H , O) . S : showering operator (2)
=
|M
!
H
+ shower dO S!
i.e.,
"
!
dσH !
(0) 2be responsible for generating all (real and
rmally, this operator — the evolution
operator
—
will
(2)
=
dΦ
|M
H
!
H | S({p}H , O) .
ual) higher-order corrections dO
to theS Born-level expression. The measurement δ function appearUnitarity:
to evolution
first order,
does
Formally, this
operator — the
operatorS—
will benothing
responsible for generating all (real and
virtual) higher-order corrections to the Born-level
The measurement δ function appear3 expression.
S({p} , O) = (O
O({p} )) + O(↵ )
H
H
s
3
P. S k a n d s
25
The Shower Operator
S({p}
,
O)
=
δ(O
−
O({p}
))
X
S({p}X
,
O)
=
δ(O
−
O({p}
))
X
X
!
#
"
"t thad
!
#
" tthad
"
had
had
dP
dP
dPd
S({p}
dt
δ(O−O({p}XX))))++
S({p}XX,,O)
O) =
= 1−
dt
δ(O−O({p}
dtdt
X+1
X+1
dt
(Markov Chain)
dt
dtdt
To ALL Orders
tstart
tstart
X
tstart
start
""thad
X , O) = δ(O − O({p}X ))
thad
d∆(t
d∆(t
, t), t)
#
start
" thad
" thad
start
)δ(O−O({p}
S(S
S({p}XX,,O)
O) =
= ∆(t
∆(tstart, thad
dtdt
had)δ(O−O({p}
dPS({p}
dP XX))))−−
−
dt
δ(O−O({p}“Nothing
dt→X+1
δ(O−O({p}
tstart
tstart
X )) + Happens”
X+1)) dtdt
“Evaluate$Observable”
$
" tstart
"
dt
dtX+1
tstart
$
$
dΦ
w
dΦ
w
X+1
X+1
X+1
X+1
"
$$
thad
P=
P
d∆(t
$$
start, t)
dΦ
w
X
X
tstart, thad)δ(O−O({p}X )) −
dt dΦ"X wS({p}
X PS
X+1 , O)
PS
"dt dQ22
tstart
%
%
$
dQ
"
“Something
Shower”
PDGLAP
= Happens” → 2“Continue
dz
PPi(z)
DGLAP
P
=
dz
(z)
dΦX+1 wX+1 $$
i
Q
2
Q
=
ii
$
dΦX wX PS
"
2
"
|M
(s
,
s
,
s)|
ds
ds
"
3 (sij , happens
jk , s)|2
ij dsjknothing
All-orders
Probability
that
2
|M
s
ds
%
3 ij jk
ij jk
P
=
dQ
Antenna
2s
2
P
=
Antenna
16π
|M
(s)|
=
dz
P
(z)
2
2s
2
AP
i
16π
|M
(s)|
2
2
! " t2
#
Q
i
(Exponentiation)
dP
"
Analogous to nuclear decay
∆(t
dt
2 1, t2) = exp −
dsij dsjk |M3(sij , sjk , s)|
N(t) ≈ N(0) exp(-ct)
dt
t1
16π 2s
|M2(s)|2
P. S k a n d s
26
A Shower Algorithm
1.0
Solve equation R =
(t1 , t) for t
yjk ! sjk !sijk ! 1"xi
1. Generate Random Number, R ∈ [0,1]
(with starting scale t1)
Analytically for simple splitting kernels,
else numerically (or by trial+veto)
→ t scale for next branching
t
0.8
t1
0.6
(t,z)
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
yij ! sij !sijk ! 1"xk
1.0
Figure 1: Contours of constant value of the antenna function, ā0ijk for q q̄ → qg q̄ d
as function of the two phase-space invariants, with an arbitrary normalization an
scale. Larger values are shown in lighter shades. The (single) collinear diverge
while the (double) soft divergence sits at the origin.
2. Generate another Random Number, Rz ∈ [0,1]
To find second (linearly independent) phase-space
invariant
factor, and ā is a generic color- and coupling-stripped dipole-antenna function,
0
ijk
Iz (z, t)
Solve equation Rz =
Iz (zmax (t), t)
With the “primitive function”
for z
Z
z
denote a tree-level quantity. The three-particle matrix element is averaged azimu
that our use of lower-case letters for the antenna function is intended to signify t
whatscale
is called a sub-antenna
in ref. [36] for which lower-case letters are likewise
(at
)
For illustration, contours of constant value of ā0qgq̄ (s, sqg , sgq̄ ) as derived from
in fig. 1, over the 2 → 3 phase space, with an arbitrary normalization and a log
This function is called A03 in ref. [36] and is identical to the radiation function
splittings in A RIADNE. One clearly sees the large enhancements towards the e
with a double pole (the overlap of two singularities, usually called soft and co
origin, and single singularities (soft or collinear) localized on the axes.
0 as g2 = 4παs and combining it with the phase s
Writing the coupling factor
we have the following antenna function normalization
t
d (t0 )
Iz (z, t) =
dz
dt0
zmin (t)
t =t
3. Generate a third Random Number, Rφ ∈ [0,1]
1
αs
0
a0IK→ijk (s, sij , sjk ) ≡ ! "
# 4π Cijk āijk (s, sij , sj
2
2
λ s, mI , mK
is, we use the notation ā for the coupling- and color-stripped antenna funct
Solve equation R' = '/2⇡ for φ → Can nowaThat
3Dantenna
branching
fordo
the “dressed”
function, i.e., including its coupling, color, and phase
P. S k a n d s
Note that g 2 ×(phase-space normalization) leads to a factor αs /(4π) indepen
branching. As we believe that the formalism becomes more transparent if the
is kept clear throughout, we shall therefore use this factor for all branchings,
traditional convention of using αs /(2π) for some branchings and αs /(4π) for ot
convention choice will be compensated by our conventions for the color factors a
normalizations, such that the final result remains independent of this choice.
27
combined result should contain exactly the correct leading soft and collinear logarithms with no
under-counting. Non-logarithmic (‘finite’) terms are in constrast arbitrary. They correspond to
around inside the leading-logarithmic uncertainty envelope. The renormalization scale µR
principle be a constant (fixed coupling) or running. Again, the point here is not to impose a
choice but just to ensure that the language is sufficiently general to explore the ambiguity.
Together, eqs. (2), (4), and (11) can be used as a framework for defining more concret
The
final
states
generated
by based
a shower
showers.
An explicit
evolution
algorithm (whether
on partons, dipoles, or other objec
algorithm
will depend on
specify:
Perturbative Ambiguities
1. The choice of perturbative evolution variable(s) t[i] .
[i]
2. The choice of phase-space mapping dΦn+1 /dΦn .
Ordering & Evolutionscale choices
Recoils, kinematics
3. The choice of radiation functions ai , as a function of the phase-space variables.
4. The choice of renormalization scale function µR .
5. Choices of starting and ending scales.
P.
Non-singular terms,
Reparametrizations,
Subleading Colour
Phase-space limits / suppressions for
hard radiation and choice of
hadronization scale
The definitions above are already sufficient to describe how such an algorithm can be ma
→ gives
additional
handles
forshall
uncertainty
beyond
just μR of these
fixedus
order
perturbation
theory. We
later presentestimates,
several explicit
implementations
+ ambiguities
be reduced
by including
more pQCD → matching!
the form of thecan
V INCIA
code, see section
5.
Let us begin by seeing what contributions the pure parton shower gives at each order in pert
theory. Since ∆ is the probability of no branching between two scales, 1 − ∆ is the integrated b
Skands
28
probability P
. Its rate of change gives the instantaneous branching probability over a dif
Jack of All Orders, Master of None?
Nice to have all-orders solution
But it is only exact in the singular (soft & collinear) limits
→ gets the bulk of bremsstrahlung corrections right, but
fails equally spectacularly: for hard wide-angle radiation:
visible, extra jets
… which is exactly where fixed-order calculations work!
Skands
P. P.
Skands
Introduction
QCD
Introduction
toto
QCD
So combine them!
(2) (2)
1 1
. . .. . .
(1)
1 1 0(1) 0
(1) (1)
1 1
(1) (1)
2 2
(0)
0 0 0(0) 0
0 0
(0) (0)
1 1
(0) (0)
2 2
1 1
2 2
k (legs)
k (legs)
. . .. . .
++
(0) (0) . . .
...
3 3
3 3 . . .. . .
(2)
2 2 0(2) 0
(2) (2)
1 1
?
& F+1
LO⇥LL
F &F F+1
@@
LO⇥LL
. . .. . .
(1)
1 1 0(1) 0
(1) (1)
1 1
(1) (1)
2 2
(0)
0 0 0(0) 0
0 0
(0) (0)
1 1
(0) (0)
2 2
1 1
2 2
k (legs)
k (legs)
. . .. . .
==
(0) (0) . . .
...
3 3
3 3 . . .. . .
(2)
2 2 0(2) 0
` (loops)
` (loops)
` (loops)
` (loops)
(2)
2 2 0(2) 0
F+1
LO⇥LL
F+1
@@
LO⇥LL
` (loops)
` (loops)
F@
LO⇥LL
F@
LO⇥LL
(2) (2)
1 1
. . .. . .
(1)
1 1 0(1) 0
(1) (1)
1 1
(1) (1)
2 2
(0)
0 0 0(0) 0
0 0
(0) (0)
1 1
(0) (0)
2 2
1 1
2 2
k (legs)
k (legs)
. . .. . .
(0) (0) . . .
...
3 3
3 3 . . .. . .
FAIL!involving
Figure22:
22:The
Thedouble-counting
double-countingproblem
problemcaused
causedbybynaively
naivelyadding
addingcross
crosssections
sections
involving
Figure
See: PS, Introduction to QCD, TASI 2012, arXiv:1207.2389
matrix
elements
with
different
numbers
legs.
matrix
elements
with
different
numbers
ofof
legs.
P. S k a n d s
29
Matching 1: Slicing
Examples: MLM, CKKW, CKKW-L
First emission: “the HERWIG correction”
Use the fact that the angular-ordered HERWIG parton
shower to
has
a
P. Skands
Introduction
QCD
“dead zone” for hard wide-angle radiation
(2)
0
(2)
1
1
(1)
0
(1)
1
(1)
2
0
(0)
0
(0)
1
(0)
2
0
1
+
...
2
k (legs)
(0)
3
...
3
...
F @ LO1 ⇥LL (H ERWIG Matched)
...
2
(2)
0
(2)
1
1
(1)
0
(1)
1
(1)
2
0
(0)
0
(0)
1
(0)
2
0
1
...
2
k (legs)
=
(0)
3
...
3
...
` (loops)
...
2
P. Skands
F+1 @ LO⇥LL (H ERWIG Corrections)
` (loops)
` (loops)
F @ LO⇥LL-Soft (H ERWIG Shower)
(Seymour, 1995)
...
2
(2)
0
(2)
1
1
(1)
0
(1)
1
(1)
2
0
(0)
0
(0)
1
(0)
2
0
1
2
k (legs)
...
(0)
3
...
3
...
Introduction to QCD
P.
1
(1)
0
...
(1)
1
...
+
2
(2)
0
1
(1)
0
...
(1)
1
...
+
2
(2)
0
1
(1)
0
...
(1)
1
...
=
` (loops)
(2)
0
` (loops)
2
` (loops)
` (loops)
Figure
23: Hemissions:
ERWIG ’s original matching
[112,
in which the
dead zone of the
Many
thescheme
MLM
&113],
CKKW-L
prescriptions
H ERWIG shower was used as an effective “matching scale” for one emission beyond a basic
F @ LO⇥LL-Soft (excl)
F+1 @ LO⇥LL-Soft (excl)
F+2 @ LO⇥LL (incl)
F @ LO2 ⇥LL (MLM & (L)-CKKW)
hard process.
2
(2)
0
1
(1)
0
...
(1)
1
...
(0)
(0)
(0)
(0) independent
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
since
generalized
by several
groups
to include
arbitrary
of addi0 been
0
0
0numbers
0
1
2
0
1
2
0
1
2
0
1
2
tional legs,
the
most
well-known
of 1these 2being the CKKW
[114],
CKKW-L [115,
116],
and
0
1
2
0
0
1
2
0
1
2
k
(legs)
k
(legs)
k
(legs)
k
(legs)
MLM [117, 118] approaches.
(Mangano, 2002)
(CKKW
& Lönnblad,the
2001)
(+many
more
recent;
see
Alwallscale,
et al., EPJC53(2008)473)
Effectively,
shower
approximation
is
set
to
zero
above
some
either due
to the
Figure 24: Slicing, with up to two additional emissions beyond the basic process.
showers
Image The
Credits:
istockphoto
presence
of
explicit
dead
zones
in
the
shower,
as
in
H
ERWIG , or by vetoing any emissions
S k a noff
d s F and F + 1 are set to zero above a specific “matching scale”. (The number of coefficients
above a certain matching scale, as in the (L)-CKKW and MLM approaches. The empty part of
30
The “CKKW” Prescription
Catani, Krauss, Kuhn, Webber, JHEP11(2001)063
Lönnblad, JHEP05(2002)046
Start from a set of fixed-order MEs
Separate Phase-Space Integrations
inc
F
inc
F +1 (Qcut )
inc
F +2 (Qcut )
Wish to add showers while eliminating Double Counting:
Transform inclusive cross sections, for “X or more”, to exclusive ones, for “X and only X”
Jet Algorithm (CKKW) → Recluster back to F → “fake” brems history
Or use statistical showers (Lönnblad), now done in all implementations
Reweight each internal line by shower Sudakov factor & each vertex by αs(µPS)
exc
F +1 (QF +1 )
exc
F +2 (QF +2 )
Reweight each external line by shower Sudakov factor
exc
F (Qcut )
exc
F +1 (Qcut )
Now add a genuine parton shower → remaining evolution down to confinement scale
Start from Qcut
P. S k a n d s
Start from QF+2
31
Slicing: The Cost
1. Initialization time
2. Time to generate 1000 events
(to pre-compute cross sections
and warm up phase-space grids)
(Z → partons, fully showered &
matched. No hadronization.)
10000s
1000s
C
+
A
RP
1000 SHOWERS
IX
M
O
1000s
rt)
a
the
f
o
e
t
sta
f
o
le
p
am
x
e
(
E
SH
100s
100s
10s
10s
1s
E
H
S
(
A
RP
L)
KW
K
C
1s
2
3
4
5
6
Z→n : Number of Matched Emissions
0.1s
2
3
4
5
6
Z→n : Number of Matched Emissions
Z→udscb ; Hadronization OFF ; ISR OFF ; udsc MASSLESS ; b MASSIVE ; ECM = 91.2 GeV ; Qmatch = 5 GeV
SHERPA 1.4.0 (+COMIX) ; PYTHIA 8.1.65 ; VINCIA 1.0.29 (+MADGRAPH 4.4.26) ;
gcc/gfortran v 4.7.1 -O2 ; single 3.06 GHz core (4GB RAM)
P. S k a n d s
32
Matching 2: Subtraction
Examples: MC@NLO, aMC@NLO
LO × Shower
P. S k a n d s
X(2)
X+1(2)
X(1)
X+1(1) X+2(1) X+3(1)
Born X+1(0) X+2(0) X+3(0)
NLO
…
X(2)
X+1(2)
…
X(1)
X+1(1) X+2(1) X+3(1)
…
…
Born X+1(0) X+2(0) X+3(0)
…
…
Fixed-Order Matrix Element
…
Shower Approximation
…
33
Matching 2: Subtraction
Examples: MC@NLO, aMC@NLO
LO × Shower
P. S k a n d s
X(2)
X+1(2)
X(1)
X+1(1) X+2(1) X+3(1)
Born X+1(0) X+2(0) X+3(0)
NLO - Shower NLO
…
X(2)
X+1(2)
…
X(1)
X+1(1) X+2(1) X+3(1)
…
…
Born X+1(0) X+2(0) X+3(0)
…
…
Fixed-Order Matrix Element
…
Shower Approximation
…
Expand shower approximation to
NLO analytically, then subtract:
…
Fixed-Order ME minus Shower
Approximation (NOTE: can be < 0!)
34
Matching 2: Subtraction
Examples: MC@NLO, aMC@NLO
LO × Shower
P. S k a n d s
X(2)
X+1(2)
X(1)
X+1(1) X+2(1) X+3(1)
Born X+1(0) X+2(0) X+3(0)
(NLO - Shower NLO )
× Shower
…
X(1)
X(1)
…
…
X(1)
X(1)
X(1)
X(1)
…
…
Born X+1(0)
X(1)
X(1)
…
…
Fixed-Order Matrix Element
…
Fixed-Order ME minus Shower
Approximation (NOTE: can be < 0!)
…
Shower Approximation
…
Subleading corrections generated by
shower off subtracted ME
35
Matching 2: Subtraction
Examples: MC@NLO, aMC@NLO
Combine → MC@NLO
Frixione, Webber, JHEP 0206 (2002) 029
Consistent NLO + parton shower
(though correction events can have w<0)
Recently, has been almost fully automated in aMC@NLO
Frederix, Frixione, Hirschi, Maltoni, Pittau, Torrielli, JHEP 1202 (2012) 048
NLO: for X inclusive
LO for X+1
LL: for everything else
X(2)
X+1(2)
…
X(1)
X+1(1) X+2(1) X+3(1)
…
Born X+1(0) X+2(0) X+3(0)
…
Note 1: NOT NLO for X+1
Note 2: Multijet tree-level
matching still superior for X+2
NB: w < 0 are a problem because they kill efficiency:
Extreme example: 1000 positive-weight - 999 negative-weight events → statistical precision of 1
event, for 2000 generated (for comparison, normal MC@NLO has ~ 10% neg-weights)
P. S k a n d s
36
Matching 3: ME Corrections
Standard Paradigm:
Have ME for X, X+1,…, X+n;
Want to combine and add showers
Double counting, IR
divergences, multiscale logs
→
“The Soft Stuff”
Works pretty well at low multiplicities
Still, only corrected for “hard” scales; Soft still pure LL.
At high multiplicities:
Efficiency problems: slowdown from need to compute and
generate phase space from dσX+n, and from unweighting
(efficiency also reduced by negative weights, if present)
Scale hierarchies: smaller single-scale phase-space region
Powers of alphaS pile up
Better Starting Point: a QCD fractal?
P. S k a n d s
37
(shameless VINCIA promo)
(plug-in to PYTHIA 8 for ME-improved final-state showers, uses helicity matrix elements from MadGraph)
Interleaved Paradigm:
Have shower; want to improve it using ME for X, X+1, …, X+n.
Interpret all-orders shower structure as a trial
distribution
Quasi-scale-invariant: intrinsically multi-scale (resums logs)
Unitary: automatically unweighted (& IR divergences → multiplicities)
More precise expressions imprinted via veto algorithm: ME
corrections at LO, NLO, and more? → soft and hard
No additional phase-space generator or σX+n calculations → fast
Automated Theory Uncertainties
For each event: vector of output weights (central value = 1)
+ Uncertainty variations. Faster than N separate samples; only
one sample to analyse, pass through detector simulations, etc.
LO: Giele, Kosower, Skands, PRD84(2011)054003
P. S k a n d s
NLO: Hartgring, Laenen, Skands, arXiv:1303.4974
38
Matching 3: ME Corrections
Examples: PYTHIA, POWHEG, VINCIA
|MF |2
|MF |2
Generate “shower”
|MF |2 emission
|MF |2
2 LL
Virtues:
No “matching scale”
No negative-weight events
Can be very fast
Loops
Start at Born level
X
+2
Repeat
|MF +1 | ⇠
ai |MF |2
X
2
+1 2
i2ant
|MX
2 LL
22 LL
F|
|MF +1 | ⇠
a|M
ai |MF |
F +1
i |M
F || ⇠
2
|M
|
i2ant
F
+1
X Element i2ant
Correctato
Matrix
LL
2!
+0
|MF +1 | ⇠ |M2Fa|2i2|MF |2
2
|MFF|+1 |
|M
|MF +1 |
i2ant
ai ! P
ai ! P
2
aX
ai |MF |2
i |MF2|
+0
+1
+2
+3
Legs
2 LL
2
|M
|
F
+1
|M
|
⇠
a
|M
|
i
F
aFi +1
!P
2
i2ant
ai |MF |
First Order
Unitarity of|MShower
2
PYTHIA: LO1 corrections to most SM and BSM decay
FZ+1 |
ai ! P
processes, and for pp → Z/W/H (Sjöstrand 1987)
|2
Virtual = ai |MFReal
Virtual =
Z
POWHEG (& POWHEG BOX): LO1 + NLO0 corrections for
generic processes (Frixione, Nason, Oleari, 2007)
Real
Correct to Matrix Element
Z
|MF |2 ! |MF |2 + 2Re[MF1 MF0 ] +
Real
Illustrations from: PS, TASI Lectures, arXiv:1207.2389
P. S k a n d s
Multileg NLO:
VINCIA: LO1,2,3,4 + NLO0,1 (shower plugin to PYTHIA 8;
formalism for pp soon to appear) (see previous slide)
MiNLO-merged POWHEG: LO1,2 + NLO0,1 for pp → Z/W/H
UNLOPS: for generic processes (in PYTHIA 8, based on
POWHEG input) (Lönnblad & Prestel, 2013)
39
Speed
Larkoski, Lopez-Villarejo, Skands, PRD 87 (2013) 054033
1. Initialization time
2. Time to generate 1000 events
(to pre-compute cross sections
and Initialization
warm up phase-space
grids)
Time (seconds)
(Z → partons, fully showered &
No 1000
hadronization.)
Time matched.
to generate
showers
(seconds)
1000 SHOWERS
10000
SHERPA 1.4.0
VINCIA 1.029
seconds
1000
1000
IX
M
O
rt)
a
e
h
t
of
E
e
t
H
S
sta
f
o
e
l
p
m
a
(ex
+C
A
RP
100
10
100
SH
Sector
Old Sector
L)
KW
K
(C
2
3
4
5
unpolarized
VINCIA (GKS)
sector
polarized
1
global
0.1
SHERPA
PA
R
E
10
PYTHIA+VINCIA
1
Global
Old Global
6
Z→n : Number of Matched Legs
0.1
2
Hadronization
Time (LEP)
3
4
5
6
Z→n : Number of Matched Legs
Z→udscb ; Hadronization OFF ; ISR OFF ; udsc MASSLESS ; b MASSIVE ; ECM = 91.2 GeV ; Qmatch = 5 GeV
SHERPA
1.4.0 (+COMIX)
; PYTHIA
8.1.65 ; SHERPA
VINCIA 1.0.29version
+ MADGRAPH
Comparison of
computation
speeds
between
1.4.0 4.4.26
[27] ; and VINCIA 1.029
gcc/gfortran
v 4.7.1
-O2
; single
3.06 GHz ;core
(4GB RAM)
Z→udscb ; Hadronization
OFF ; ISR
OFF
; udsc
MASSLESS
b MASSIVE
; ECM = 91.2 GeV ; Qmatch = 5 GeV
Figure 7:
+
PYTHIA 8.171, as a function of the number of legs that are matched to matrix elements, for hadronic Z
SHERPA 1.4.0 (+COMIX) ; PYTHIA 8.1.65 ; VINCIA 1.0.29 (+MADGRAPH 4.4.26) ;
P. S k a n d s
decays.
Left: initialization time (to precompute cross sections, warm up phase-space grids, etc, before event 40
Confinement
Potential between a quark and an
antiquark as function of distance, R
Long Distances ~
Linear Potential
Quarks (and
gluons) confined
inside hadrons
Short Distances ~
“Coulomb”
What physical
system has a
linear potential?
Partons
~ Force required to lift a 16-ton truck
P. S k a n d s
41
String Breaks
In “unquenched” QCD
g→qq → The strings would break
String Breaks:
via Quantum Tunneling
(simplified colour representation)
P / exp
m2q

p2?q
!
Illustrations by T. Sjöstrand
P. S k a n d s
42
The (Lund) String Model
Pedagogical Review: B. Andersson, The Lund model.
Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol., 1997.
Map:
• Quarks → String
Endpoints
• Gluons → Transverse
Excitations (kinks)
• Physics then in terms of
string worldsheet
evolving in spacetime
• Probability of string
break (by quantum
tunneling) constant per
unit area → AREA LAW
Simple space-time picture
Details of string breaks more complicated
P. S k a n d s
43
H a d ro n i z a t i o n : S u m m a r y
The problem:
Given a set of coloured partons resolved at a scale of ~ 1 GeV, need a
(physical) mapping to a new set of degrees of freedom = colourneutral hadronic states.
Numerical models do this in three steps
1. Map partons onto endpoints/kinks of continuum of strings ~ highly
excited hadronic states (evolves as string worldsheet)
2. Iteratively map strings/clusters onto discrete set of primary hadrons
(string breaks, via quantum tunneling)
3. Sequential decays into secondary hadrons (e.g., ρ→ππ , Λ 0 →nπ 0 , π 0 →γγ, ...)
D i s t a n c e S c a l e s ~ 1 0 -15 m = 1 f e r m i
What is Tuning?
FSR pQCD Parameters
αs(mZ)
The value of the strong coupling at the Z pole
Governs overall amount of radiation
αs Running
Renormalization Scheme and Scale for αs
1- vs 2-loop running, MSbar / CMW scheme, µR ~ pT2
Matching
Additional Matrix Elements included?
At tree level / one-loop level? Using what scheme?
Ordering variable, coherence treatment, effective
Subleading Logs 1→3 (or 2→4), recoil strategy, …
Branching Kinematics (z definitions, local vs global momentum
conservation), hard parton starting scales / phase-space cutoffs,
masses, non-singular terms, …
P. S k a n d s
45
tates).
e event shapes requiring three-particle final states, six variables were studied
2 /s [20], the wide
l: the thrust T [19], the normalised heavy jet mass MH
broadenings BW and BT [21], the C-parameter [22] and the transition from
wo-jet final states in the Durham jet algorithm Y3 [23].
Need IR Corrections?
T [19]
ust variable for a hadronic final state in e+ e− annihilation is defined as [19]
!"
#
|!
p
·
!
n
|
1
i i
"
T = max
,
1 T !(2.1)
2
!
n
pi |
i |!
1 T !0
denotes the three-momentum of particle i, with the sum running over all
1-Thrust (udsc)
Major
Minor
. The
unit
vector
!
n
is
varied
to
find
the
thrust
direction
!
n
which
maximises
T
Delphi
Delphi
L3
10
10
10
Pythia
Pythia
Pythia
ession in parentheses.
Major
1/N dN/d(O)
Minor
1/N dN/d(Minor)
1
1
Oblateness
1
1
ximum value of thrust, T → 1, is obtained in the limit where there are only
10
10
10
icles in the
event. For a three-particle
event the minimum value
of thrust is
Minor
1-T
Major
10
10
. 10
-1
10
-2
-2
-2
10
-3
10
Data from CERN-PPE-96-120
Pythia 8.165
-3
10
-2
Data from CERN-PPE-96-120
Pythia 8.165
0.5
Theory/Data
Theory/Data
2 /s0.4[20]0.5 0
0.1 mass,
0.2
0.2
0.4
0.6
0.1
0.2
0.3
0.4
emisphere
M0.3H
1.40
1.4
1.40
1.2
1.2
1.2
riginal
definition [20] one divides
the event into two hemispheres.
In each
1
1
1
0.8
0.8
0.8
ere, 0.6Hi , one also computes the0.6hemisphere invariant mass
0.6 as:
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0
0.1
0.2
0.3
0.4

2
1 & 
2
pk
,
(2.2)
Mi /s = 2
Evis
Significant Discrepancies (>10%)
Theory/Data
V INCIARO OT
Data from Phys.Rept. 399 (2004) 71
Pythia 8.165
-1
1-T (udsc)
Major
Theory/Data
-3
V INCIARO OT
-1
V INCIARO OT
-1
10
Delphi
Pythia
10
0.5
Oblateness
= Major - Minor
-3
10
1.40
V INCIARO OT
1/N dN/d(Major)
1/N dN/d(1-T)
PYTHIA 8 (hadronization off) vs LEP: Thrust
Data from CERN-PPE-96-120
Pythia 8.165
0.2
0.4
0.2
0.4
0.6
1.2
1
0.8
0.6
0
Minor
0.6
O
k∈Hi
for T < 0.05, Major < 0.15, Minor < 0.2, and for all values of Oblateness
Evis is the total energy visible in the event. In the original definition, the
ere is chosen such that M12 +M22 is minimised. We follow the more customary
P. S k a n d s
n whereby the hemispheres are separated by the plane orthogonal to the
46
tates).
e event shapes requiring three-particle final states, six variables were studied
2 /s [20], the wide
l: the thrust T [19], the normalised heavy jet mass MH
broadenings BW and BT [21], the C-parameter [22] and the transition from
wo-jet final states in the Durham jet algorithm Y3 [23].
Need IR Corrections?
T [19]
ust variable for a hadronic final state in e+ e− annihilation is defined as [19]
!"
#
|!
p
·
!
n
|
1
i i
"
T = max
,
1 T !(2.1)
2
!
n
pi |
i |!
1 T !0
denotes the three-momentum of particle i, with the sum running over all
1-Thrust (udsc)
Minor
Major
. The
unit
vector
!
n
is
varied
to
find
the
thrust
direction
!
n
which
maximises
T
Delphi
L3
Delphi
10
10
10
Pythia
Pythia
Pythia
ession in parentheses.
Major
Minor
1
-2
-2
-2
-3
10
Data from CERN-PPE-96-120
Pythia 8.165
-3
10
-2
Data from CERN-PPE-96-120
Pythia 8.165
0.5
Theory/Data
Theory/Data
Theory/Data
2 /s0.4[20]0.5
0.1mass,
0.2
0.1
0.2
0.3
0.4
emisphere
M0.3H
0.2
0.4
0.6
1.40
1.40
1.40
1.2
1.2
1.2
riginal
definition [20] one divides
the event into two hemispheres.
In each
1
1
1
0.8
0.8
0.8
ere, 0.6Hi , one also computes the0.6hemisphere invariant mass
0.6 as:
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0
0.2
0.4
0.6

2
1 & 
2
pk
,
(2.2)
Mi /s = 2
Note:
Value
of
Strong
coupling
is
Evis
1-T (udsc)
Major
k∈Hi
Theory/Data
10
Data from Phys.Rept. 399 (2004) 71
Pythia 8.165
-1
10
V INCIARO OT
-1
V INCIARO OT
-1
V INCIARO OT
-1
Delphi
Pythia
10
1
1
ximum value of thrust, T → 1, is obtained in the limit where there are only
10
10
10
icles in the event. For a three-particle event the minimum value of thrust is
10
10
. 10
-3
Oblateness
0.5
Minor
10
-3
10
1.40
V INCIARO OT
1
1/N dN/d(O)
1/N dN/d(Minor)
1/N dN/d(Major)
1/N dN/d(1-T)
PYTHIA 8 (hadronization on) vs LEP: Thrust
Data from CERN-PPE-96-120
Pythia 8.165
0.2
0.4
0.2
0.4
0.6
1.2
1
0.8
0.6
0
0.6
O
αs(MZ) = 0.14
Evis is the total energy visible in the event. In the original definition, the
ere is chosen such that M12 +M22 is minimised. We follow the more customary
P. S k a n d s
n whereby the hemispheres are separated by the plane orthogonal to the
47
tates).
e event shapes requiring three-particle final states, six variables were studied
2 /s [20], the wide
l: the thrust T [19], the normalised heavy jet mass MH
broadenings BW and BT [21], the C-parameter [22] and the transition from
wo-jet final states in the Durham jet algorithm Y3 [23].
Value of Strong Coupling
T [19]
ust variable for a hadronic final state in e+ e− annihilation is defined as [19]
!"
#
|!
p
·
!
n
|
1
i i
"
T = max
,
1 T !(2.1)
2
!
n
pi |
i |!
1 T !0
denotes the three-momentum of particle i, with the sum running over all
Minor
1-Thrust (udsc)
Major
. The
unit
vector
!
n
is
varied
to
find
the
thrust
direction
!
n
which
maximises
Delphi
Delphi
10T
L3
10
10
Pythia
Pythia
Pythia
ession in parentheses.
Major
Minor
1
-2
-2
-2
-3
10
Data from CERN-PPE-96-120
Pythia 8.165
-3
10
-2
Data from CERN-PPE-96-120
Pythia 8.165
0.5
Theory/Data
Theory/Data
Theory/Data
2 /s0.4[20]0.5 0
0.1
0.2
0.3
0.4
0.1 mass,
0.2
0.2
0.4
0.6
emisphere
M0.3H
1.40
1.40
1.4
1.2
1.2
1.2
riginal
definition [20] one divides
the event into two hemispheres.
In each
1
1
1
0.8
0.8
0.8
0.6 as:
ere, 0.6Hi , one also computes the0.6hemisphere invariant mass
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6

2
1 & 
2
pk
,
(2.2)
Mi /s = 2
Note:
Value
of
Strong
coupling
is
Evis
1-T (udsc)
Major
k∈Hi
Theory/Data
10
Data from Phys.Rept. 399 (2004) 71
Pythia 8.165
-1
10
V INCIARO OT
-1
V INCIARO OT
-1
V INCIARO OT
-1
Delphi
Pythia
10
1
1
ximum value of thrust, T → 1, is obtained in the limit where there are only
10
10
10
icles in the event. For a three-particle event the minimum value of thrust is
10
10
. 10
-3
Oblateness
0.5
Minor
10
-3
10
1.40
V INCIARO OT
1
1/N dN/d(O)
1/N dN/d(Minor)
1/N dN/d(Major)
1/N dN/d(1-T)
PYTHIA 8 (hadronization on) vs LEP: Thrust
Data from CERN-PPE-96-120
Pythia 8.165
0.2
0.4
0.2
0.4
0.6
1.2
1
0.8
0.6
0
0.6
O
αs(MZ) = 0.12
Evis is the total energy visible in the event. In the original definition, the
ere is chosen such that M12 +M22 is minimised. We follow the more customary
P. S k a n d s
n whereby the hemispheres are separated by the plane orthogonal to the
48
Wait … is this Crazy?
Best result
Obtained with αs(MZ) ≈ 0.14
≠ World Average = 0.1176 ± 0.0020
Value of α s depends on the order and scheme
MC ≈ Leading Order + LL resummation
Other leading-Order extractions of αs ≈ 0.13 - 0.14
Effective scheme interpreted as “CMW” → 0.13;
2-loop running → 0.127; NLO → 0.12 ?
Not so crazy
Tune/measure even pQCD parameters with the actual generator.
Sanity check = consistency with other determinations at a
similar formal order, within the uncertainty at that order
(including a CMW-like scheme redefinition to go to ‘MC scheme’)
Improve → Matching at LO and NLO
P. S k a n d s
49
Sneak Preview:
Multijet NLO Corrections with VINCIA
Hartgring, Laenen, Skands, arXiv:1303.4974
First LEP tune with NLO 3-jet corrections
(1-loop running, MSbar)
1/N dN/dC
L3
10
C Parameter (udsc)
102
Vincia (NLO)
L3
1
-1
10-2
10-2
Data from Phys.Rept. 399 (2004) 71
1.40
Vincia 1.030 + MadGraph 4.426 + Pythia 8.175
0.1
0.2
0.3
0.4
10
0.5
1.2
1
0.8
0.6
0
Vincia (LO tune)
10-1
Data from Phys.Rept. 399 (2004) 71
-3
Theory/Data
Theory/Data
10
Vincia (LO tune)
1.40
Vincia 1.030 + MadGraph 4.426 + Pythia 8.175
0.2
0.4
0.6
0.8
0.2
0.3
0.4
0.5
1.2
1
0.8
0
(a)
10
1.40
Vincia 1.030 + MadGraph 4.426 + Pythia 8.175
0.2
0.4
0.6
0.2
0.4
0.6
0.8
1.2
1
0.8
0.6
0.2
0.4
1-T (udsc)
P. S k a n d s
Data from Phys.Rept. 399 (2004) 71
-3
1
0.6
0.1
10-2
Theory/Data
VINCIAROOT
10
Vincia (NLO off)
1
-1
10
-3
Vincia (NLO)
Vincia (NLO off)
Vincia (LO tune)
L3
10
Vincia (NLO)
10
Vincia (NLO off)
1
D Parameter (udsc)
VINCIAROOT
1-Thrust (udsc)
102
(2-loop running, CMW)
VINCIAROOT
1/N dN/d(1-T)
NLO tune: αs(MZ) = 0.122
1/N dN/dD
LO tune: αs(MZ) = 0.139
0.6
0.8
1
0
C (udsc)
(b)
0.8
D (udsc)
(c)
50
Summary
Parton Shower Monte Carlos
Improve lowest-order perturbation theory by including
‘most significant’ corrections
Resonance decays, soft- and collinear radiation,
hadronization, … → complete events
Coherence
→ Angular ordering or Coherent Dipoles/Antennae
Hard Wide-Angle Radiation: Matching
Slicing (Qcut), Subtraction (w<0), or ME Corrections
Next big step: showers with multileg NLO corrections
MCnet Review: Phys.Rept. 504 (2011) 145-233
PS, TASI Lectures: arXiv:1207.2389
P. S k a n d s
51
MCnet
Studentships
MCnet
MCnet projects:
• PYTHIA (+ VINCIA)
• HERWIG
• SHERPA
• MadGraph
• Ariadne (+ DIPSY)
• Cedar (Rivet/Professor)
Activities include
• summer schools
(2014: Manchester?)
• short-term studentships
• graduate students
• postdocs
• meetings (open/closed)
Torbjörn Sjöstrand
P. S k a n d s
Monte Carlo
training studentships
m
Lund
Durha
ster
e
h
uvain
c
o
n
L
a
M
ngen
i
t
t
ö
n
o
G
Lond
uhe
r
s
l
r
a
K
N
CER
3-6 month fully funded studentships for current PhD
students at one of the MCnet nodes. An excellent opportunity
to really understand and improve the Monte Carlos you use!
Application rounds every 3 months.
funded by:
MARIE CURIE ACTIONS
Monte Carlo Generators and Soft QCD 1
for details go to:
www.montecarlonet.org
slide 7/40
52
Factorization
Fixed Order requirements:
All resolved scales >> ΛQCD AND no large hierarchies
Trivially untrue for QCD
We’re colliding, and observing, hadrons → small scales
We want to consider high-scale processes → large scale differences
Factorization
2
2
⇥ ⇥ → A Priori, no perturbatively
dˆ
⇤
(x
,
x
,
f,
Q
,
Q
calculable
d⇤
ab
f
a
b
i
f)
2
2
=
fa(xa, Qi )fb(xb, Qi )
D(X̂f
X, Q2i , Q2f )
observables
in hadron-hadron
collisions
dX
dX̂f
X̂f
a,b
f
PDFs: needed to compute
inclusive cross sections
FFs: needed to compute
(semi-)exclusive cross
sections
Resummed: All resolved scales >> ΛQCD AND X Infrared Safe
P. S k a n d s
53
Jets and Showers
Infrared Safety: Jet clustering algorithms
Map event from low resolution scale (i.e., with many
partons/hadrons, most of which are soft) to a higher
resolution scale (with fewer, hard, jets)
Many soft particles
Q ~ Λ ~ mπ
~ 150 MeV
Jet Clustering
(Deterministic*)
(Winner-takes-all)
A few hard jets
Q~ Ecm
~ MX
Q ~ Qhad
~ 1 GeV
Parton Showering
(Probabilistic)
Hadronization
Born-level ME
Parton shower algorithms
Map a few hard partons to many softer ones
Probabilistic → closer to nature. Not uniquely invertible by
any jet algorithm*
(* See “Qjets” for a probabilistic jet algorithm, arXiv:1201.1914)
(* See “Sector Showers” for a deterministic shower, arXiv:1109.3608)
P. S k a n d s
54
Matching 1: Slicing
Examples: MLM, CKKW, CKKW-L
LO 0 × PS (pT>pTcut)
+
Std: veto shower above some pTcut
LO 1(pT1>pTcut) × PS (pT<pT1)
Highest n: veto shower above pTn
X(2)
X+1(2)
…
X+1(2)
X(1)
X+1(1) X+2(1) X+3(1)
…
X+1(1) X+2(1) X+3(1)
…
Born X+1(0) X+2(0) X+3(0)
…
X+1(0) X+2(0) X+3(0)
…
…
Fixed-Order Matrix Element
…
Shower Approximation
…
…
Fixed-Order ME above pT cut
& nothing below
Illustrations from: PS, TASI Lectures, arXiv:1207.2389
P. S k a n d s
55
Matching 1: Slicing
Examples: MLM, CKKW, CKKW-L
LO 0 × PS (pT>pTcut)
+
LO 1(pT1>pTcut) × PS (pT<pT1)
Std: veto shower above pTcut
X(2)
X+1 now LO
correct for hard
radiation and still LL
correct for soft
X(1)
X+1(2)
Highest n: veto shower above pTn
…
X+1(1) X+2(1) X+3(1)
…
Born X+1(0) X+2(0) X+3(0)
…
…
Fixed-Order Matrix Element
…
Shower Approximation
+ Generalizes to
arbitrary numbers of
jets (at LO)
Much work on
extensions to NLO
…
Fixed-Order ME above pT cut
& nothing below
…
Fixed-Order ME above pT cut
& Shower Approximation below
Illustrations from: PS, TASI Lectures, arXiv:1207.2389
P. S k a n d s
56
Matching: Classic Example
W + Jets
Number of jets in
pp→W+X at the LHC
mcplots.cern.ch
Number of Jets
From 0 (W inclusive)
to W+3 jets
PYTHIA includes
matching up to W+1
jet + shower
With ALPGEN (MLM),
also the LO matrix
elements for 2 and 3
jets are included
But Normalization
still only LO
P. S k a n d s
W
LHC 7 TeV
W+Jets
ETj > 20 GeV
|ηj| < 2.8
W
ith
ith
ou
t
Ma
tch
Ma
tch
ing
ing
RATIO
57
QCD Jets
Matching not
always needed.
Even at 6 jets, there
is almost always at
least one strongly
ordered path
→ showers work!
(In W+jets, that is
not the case)
But note that spin
correlations between
the jets will still be
absent
P. S k a n d s
58