Effective Field Theories
Lecture 5
Aneesh Manohar
University of California, San Diego
May 2013 / Brazil
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
1 / 31
Wilson Lines
x
W (x, y ) = P exp −ig
=
Y
Z
x
y
Aµ (z) dz
µ
µ
e−igAµ (zi )∆zi
i
y
(t · D) W (x, y ) = δ(x − y )
W (x, y ) → U(x)W (x, y )U(y )†
show this
W (x, y )ψ(y ) → U(x) W (x, y )ψ(y )
Transports charge from y to x
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ICTP-SAIFR School
05.2013
2 / 31
Wilson Loop
WC (x) = P exp −ig
I
C
Aµ (z) dz
µ
WC (x) → U(x) WC (x) U(x)†
Tr WC (x)
is gauge invariant. Depends on the curve C, but not on the starting
point x. show this
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
3 / 31
Collinear Wilson Lines
n-collinear Wilson line:
(
Wn (x) = P exp −ig
Z
0
−∞
n̄ · An (x + sn̄) ds
)
An integral along the n̄ direction from −∞ to x.
Wn (x) is the Green’s function for i n̄ · Dn .
Wn (x)† n̄ · PWn (x) = n̄ · Dn
convert an ordinary derivative into a covariant derivative.
Cannot do this in a normal gauge theory because there is no preferred
path to x. Around closed loops
I
P exp ig Aµ (z)dz µ 6= 0
Here we have direction labels n, n̄.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
4 / 31
/ n,⊥ Wn (x)
L = ξ n,p (x) [in · (Dus + Dn )] + i D
1
/ ⊥,n
Wn† (x) i D
i n̄ · P
/̄
n
ξn,p (x)
2
You can also show that the Wilson lines are needed by collinear gauge invariance.
Under gauge transformations,
ψ → Uψ
g Aµ → U g Aµ U † + i∂ µ U U †
Under n-collinear gauge transformations, the x dependence on Un (x) has
momentum (1, λ2 , λ)Q.
ξn,p (x) → Un (x) ξn,p (x)
ξn̄,p → ξn̄,p
Aus → Aus
Wn (x) → Un (x) Wn (x)
so
Wn (x)† ξn,p (x)
is n-collinear gauge invariant.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
5 / 31
86
p̃ + k
=i
n
n̄·
n̄ p̃· p̃
n̄
/
2 n·k
++
p̃2?p̃2⊥++i✏i!
n · kn̄·
n̄p̃· p̃
= igT A nµ
p̃
p̃"
= igT
A
!
n̄
/
2
p̃" γµ⊥
γµ⊥ /
p̃⊥ /
/
p̃"⊥ p̃/⊥
+ ⊥ " −
n̄µ
nµ +
n̄ · p̃
n̄ · p̃
n̄ · p̃ n̄ · p̃"
"
n̄
/
2
Notation for fields
ure 3.1: Feynman rules involving collinear quarks in SCETI. The collinear particles ar
own with label momenta p̃, p̃" and residual momentum k.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
6 / 31
What makes EFT complicated?
HQET:
p = mQ v + k
v is constant, i.e. the label is conserved.
NRQCD/NRQED and SCET:
Can have label-changing interactions, so the label is a dynamical
variable, and not a constant.
quarks can change direction, n-collinear interactions can change p.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
7 / 31
SCET Matching for the Current
Suppose you have a current ψγ µ ψ in QCD, as in e+ e− → qq.
It matches to
i
h
ψγ µ ψ → C ξ n̄,p Wn̄ γ µ Wn† ξn,p (x)
by collinear gauge invariance.
C = 1 + O (αs )
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ICTP-SAIFR School
05.2013
8 / 31
High scale matching: µ ∼ Q
full theory:
p2
p1
(a)
EFT:
p2
p2
p1
p1
(a)
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p2
p1
(b)
(c)
ICTP-SAIFR School
05.2013
9 / 31
SCET Matching: J µ → qq
J µ = ψ γµψ
In the EFT, this matches to
h
i
J µ = C ξ n,p W γ µ W † ξn,p
Compute the on-shell matrix element at one loop.
Start with scattering kinematics q 2 = −Q 2 < 0.
p
Rψ
A Manohar (UCSD)
+
ICTP-SAIFR School
05.2013
10 / 31
iRψ
p
/
as
p→0
I1 = i p
/
Use Feynman gauge
Z
dd k α p
1
/ + k/
γ
γα 2
d
2
(2π)
(p + k )
k
1
αs
1
= CF
ip
−
/
4π
UV
IR
2
I2 = −g CF
−1
Rψ
A Manohar (UCSD)
αs
1
= 1 + CF
ip
/ −
4π
IR
ICTP-SAIFR School
05.2013
11 / 31
p2
p1
2
IV = −g CF
Z
/ 2 + k/ µ p
/ 1 + k/
dd k α p
1
γ
γ
γα
(2π)d (p2 + k )2 (p1 + k )2 k 2
We want the graph on-shell, so . . . p
/ 1 = 0, p
/ 2 . . . = 0.
Combine (p1 + k )2 and (p2 + k )2 first using x, and then k 2 using z.
A Manohar (UCSD)
ICTP-SAIFR School 05.2013
12 / 31
The entire integral is
(
Z 1 Z 1
h
i−
αs µ
(2 − d)2
2 CF
γ (4π µ̃ )
Γ() Q 2 x(1 − x)z 2
dx
dz z
4π
2
0
0
h
i−1−
+ Q 2 Γ(1 + ) Q 2 x(1 − x)z 2
×
)
h
i
2(1 − xz)((1 − x)z − 1) − (d − 4)x(1 − x)z 2
2
1
µ2
4
µ2
αs µ h 2
γ − 2 −
ln 2 +
−
− ln2 2
4π
IR Q
UV
IR
Q
IR
i
2
2
µ
π
− 3 ln 2 +
−8
6
Q
= CF
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ICTP-SAIFR School
05.2013
13 / 31
The on-shell matrix element is
1 + CF
i
2
µ2
3
µ2
µ2
π2
αs µ h 2
γ − 2 −
ln 2 −
− ln2 2 − 3 ln 2 +
−8
4π
IR Q
IR
6
Q
Q
IR
No UV divergence since the current has no anomalous dimension.
The EFT graphs are
p2
p1
A Manohar (UCSD)
p2
p1
p2
p1
p2
p1
ICTP-SAIFR School
05.2013
14 / 31
The n-collinear graph gives
Z
/̄ nα n
/ n̄ · (p2 − k ) µ 1
dd k n
1
n̄α 2
γ
d
2
−n̄ · k k
(2π) 2 2(p2 − k )
Z
d
d k n̄ · (p2 − k ) µ 1
1
γ
= −2ig 2 µ2 CF
d
2
−n̄ · k k 2
(2π) (p2 − k )
2 2
In = −ig µ CF
The only scale is p2− , so the integral vanishes.
The ultrasoft graph is
2
Is = −ig CF
Z
1
1
1
dd k α
n
n̄α 2
γµ
d
[n · (p2 − k )] [n̄ · (p1 − k )] k
(2π)
and also vanishes.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
15 / 31
QCD:
1 + CF
i
αs µ h 2
µ2
3
µ2
µ2
π2
2
γ − 2 −
ln 2 −
− ln2 2 − 3 ln 2 +
−8
4π
IR Q
IR
6
Q
Q
IR
SCET:
1 + CF
αs µ h 2
2
3
2
2
µ2
µ2
3 i
−
+
γ
−
+
ln
ln
−
4π
UV Q 2 IR Q 2 UV
IR
2UV
2IR
Matching:
C(µ) = 1 + CF
Anomalous dimension:
µ
A Manohar (UCSD)
i
αs h
µ2
µ2
π2
− ln2 2 − 3 ln 2 +
−8
4π
6
Q
Q
i
dC(µ)
αs h
µ2
= CF
−4 ln 2 − 6 C(µ)
dµ
4π
Q
ICTP-SAIFR School
05.2013
16 / 31
ln Q 2 in the anomalous dimension: Sums the Sudakov double logs.
For the time-like case,
ln Q 2 = ln Q 2 − i0+ → ln −q 2 − i0+ = ln q 2 − iπ
Matching:
C(µ) = 1 + CF
A Manohar (UCSD)
i
µ2
µ2
µ2 7π 2
αs h
− ln2 2 − 3 ln 2 − 2iπ ln 2 +
− 3iπ − 8
4π
6
q
q
q
ICTP-SAIFR School
05.2013
17 / 31
µ
dC(µ)
= γ(µ)C(µ)
dµ
Then
C(µ2 )
ln
=
C(µ1 )
Z
µ2
µ1
γ(µ) = A(αs ) ln
dµ
γ(µ)
µ
µ2
+ B(αs )
Q2
Can prove only a single log to all orders in perturbation theory
Gives the Sudakov resummation: ln F = Lf0 (αs L) + f1 (αs L) + . . ..
A(αs ) is the cusp anomalous dimension Γ(αs )
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ICTP-SAIFR School
05.2013
18 / 31
Cusp Anomalous Dimension
A smooth Wilson line has renormalization of g and proportional to its
length (like mass renormalization).
If there are kinks, additional anomalous dimension Γ that depends on
the angle between tangent vectors at the point.
A Manohar (UCSD)
ICTP-SAIFR School 05.2013
19 / 31
SCET developed to study processes such as B → ππ with energetic
hadrons.
Applied to study jets.
Can also be used to study electroweak corrections at the LHC.
At high energies, W and Z are effectively massless. Can be used to
compute electroweak radiative corrections and sum the Sudakov
double logs.
αW log2
s
2
MW
,Z
Might think that these are tiny, but they are acutally relevant.
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ICTP-SAIFR School
05.2013
20 / 31
Advantages are:
factored the scale s from MW , MZ , Mh .
Can do the calculation to NLL for all LHC scattering processes.
Much simpler than existing fixed order results
Can include the complete mass dependence on mt /MZ , MW /MZ
and Mh /MZ
Can include top-Yukawa corrections
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ICTP-SAIFR School
05.2013
21 / 31
tt Invariant Mass distribution
AM, M. Trott, PLB711 (2012) 312
Solid black lines which are the corrections for 7 and 8 TeV LHC:
Seen in the data.
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ICTP-SAIFR School
05.2013
22 / 31
Event Shapes
Thrust:
C parameter:
P
i |n · pi |
τ = 1 − T = 1 − max P
n
i |pi |
3
C=
2
P
i,j
|pi | pj sin2 θij
P
( |pi |)2
In the two jet region τ → 0, C → 0.
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ICTP-SAIFR School
05.2013
23 / 31
Gehrmann et al
Becher and Schwartz
Abbate et al.
Compute fixed order to αs3 and resummation to N 3 LL and
1/Q power correction
Slides from I. Stewart talk.
Moments:
Mn =
A Manohar (UCSD)
Z
0
1
τ n p(τ ) dτ
ICTP-SAIFR School
05.2013
24 / 31
!b1)!Q" 0.238 ± 0.070
N3 LL! (QCD+m
N LL (full)
Global Fits
0.11
N3 LL! (pure QCD)
0.252 ± 0.069
0.323 ± 0.045
0.254 ± 0.070
0.332 ± 0.045
�
Fit at "3LL for Α !m " and $
0.310 ± 0.049
s
Z
1
TABLE V: Theory errors from the parametertheory
scanscan
and
cenerror
0.09
tral values for
Ω1 defined at the reference scales R∆ = µ∆ =
2 GeV in units of GeV at various orders. The N3 LL! value
above the horizontal line is our final scan result, while the
ALEPH
N3 LL! values0.07
below the
horizontal line show the effect of leavOPAL
ing out the QED corrections,
and leaving out both the b-mass
L3
and QED respectively.
The central values are the average of
DELPHI
the maximal0.05
and minimal
values reached from the scan.
JADE
thrust
moment
TABLE VI: Theoretical uncertainties for first
αs (m
Z ) obtained at
moment
0.9 from two versions of the error band
2$!1order
N3 LL
method, and
thrust tail
first moment
0.9
from
our theory
!GeV"
2�scan
1 method. The uncertainties in the R-gap
thrust tail
scheme (first line) include renormalon subtractions, while the
1&Σ are therefore
�GeV�
ones in the0.8
MS
scheme (second line) do not and
larger. The same uncertainties
are obtained in the analysis 1�Σ
0.8
without nonperturbative function (third line). Larger uncer0.7
tainties are obtained from a pure O(α3s ) fixed-order analysis
(lowest line). Our theory scan method is more conservative
0.7
than the error band method.
0.6
AMY
τ dσ
σ dτ
thrust
tail
TASSO
1&Σ
0.6error bands for the thrust distri0.5
values determines
the
full "3LL results
bution at the different Q values.
Then, the perturbative
1�Σ
0.112 0.113
0.114 α0.115
0.116 0.117
0.4
error is determined
by varying
s (mZ ) keeping all thewith QED
FIG. 8: First moment 3
of the thrust distribution as a func0.5
ory
parameters
to
their
default
values
value3 of
’
Αand
Fit at
N LL
for
s!mZ "the full
& values
tion of the center
of mass
energy
Q, using the best-fit
� LL results
for αs (mZ ) and Ω1 in the Rgap scheme as given in Eq. (34). the moment Ω1 to its best fit value. The resulting per3
FIG. 9: Comparison
and our
Ω0.113
from
0.115 in
0.116 0.117
0.3
s (m0.112
1 determinations
The blue band represents the perturbative uncertainty deter- turbative
full N0.114
LL! analysis
errors of of
αsα(m
ZZ)) for
theory scan error
thrust first moment data (red) and thrust tail data (blue).
mined by our theory scan. Data is from ALEPH, OPAL, L3, the
R-gap scheme are given in the first line of Tab. VI.
Αs�mZ �
The plot corresponds to fits with N3 LL accuracy and in the
DELPHI, JADE, AMY and TASSO.
InRgap
the scheme.
secondThe
linetailthe
for αs (mZ )
fits corresponding
are performed witherrors
our improved
0.2
uses a newfor
nonsingular
two-loop function,
in code
the which
MS scheme
Ω̄1 are displayed.
Theand
leftthecolumn
now known
two-loop
soft function.
Dashed
lines correspond
to such
typically designed to eliminate initial state photon radi- gives
the error
when
the band
method
is applied
theory uncertainties, solid lines correspond to ∆χ2 = 1 comDELPHI
ation, while those of the TASSO, L3 and ALEPH collab- that
variation
leads
to ellipses,
curves and
strictly
inside
thetheoretical
αs (mZ )and
C�parameter
bined
experimental
error
wideALEPH
orations eliminated initial and final state photon radia- the
2
0.1
errorlines
bands
for all0.8
values.
For this error
method
it turns
= 2.3 combined
ellipses
dashed
correspond
toQ∆χ
OPAL
thrust
tion. It is straightforward to test for the effect of these
(corresponding
to 1-σ for
uncertainty
in two dimensions).
L3
out
that the band
the highest
Q value is the most
Q
=
m
differences
in
the
fits
by
using
our
theory
code
with
QED
Z
SLD
and sets the size of the error. The resulting
effects turned on or off depending
gC on the data set. Using restrictive
for the N3 LL! analysis in the R-gap scheme is more
0.0
our N3 LL order
code in the Rgap scheme we obtain the error
ing bottom quark mass and QED corrections we obtain
s0.376 GeV.
Z than
0.10
0.15!"#$*+,$$$$$$$$$$(%-$$$$$$$$$$$$$$$$$$$$."/0)#(%'+0.)1$
1 0.25
) = 0.1143
and Ω1 =0.30
central values
αs (mZ0.20
a factor of two smaller than the error obtained from
0.7
C dΣ
Comparing to our default results given in Tabs. I and II, our theory
2�
scan
which
is shown
in the(34)
right col=10.1140 ±
(0.0004)
αs (m
Z ) method,
exp
$
C�parameter
which are based on the theory
were QED effects
Σ dC
high Q data has a much lower statistical
FIG.
13: Thrust
distribution at N3 LL! code
order
and Q = mZ are umn. Since the
�GeV�
included for all data sets, we see that the central value
± (0.0013)hadr ± (0.0007)pert,
0.8
including
QED
and
m
corrections
using
the
best
fit
values
weight
than
the
data
from
Q
=
m
,
we
do
not
consider
b
0.50
Z
for αs is larger by 0.0003 �and the one for Ω1 is smaller
3
for αs (mZ ) and
Ω1 in GeV.
theFit
R-gap
given
in Eq.
(66).
to be sufficiently conservative and conclude
at �scheme
LL isfor
Αs�m
�1Thethan this method
by 0.001
This
shift
substantially
smaller
Z � and
Ω1 (R∆ , µ∆ ) = 0.377
0.6± (0.013)exp
0.45band represents
pink
the perturbative
determined
our perturbative
uncertainty.error
Hence
our choicefrom
to use that it should not be used. The middle column gives the
theory
the scan method
described
in Sec.
VI.
Datascan
fromerror
DELPHI,
the theory
code with
QED
effects
included
everywhere perturbative error
± when
(0.042)the
± (0.039)
GeV,
method
applied such
pert is
αs (mband
Z)
0.40
0.7
ALEPH, OPAL,
and for
SLD
also shown.
as theL3,
default
ourare
analysis
does not cause an observ2�that
the αs (mZ ) variation minimizes a χ2 function which
1
able bias regarding experiments which remove final state
where R∆ = µ∆ = 2 GeV and we quote individual 1-σ
0.35
puts
equal weight to all Q and thrust2 values. ThisQCD
sec- only
�GeV�
photons.
uncertainties for each 0.5
parameter. Here χ /dof = 1.33. 3
Q � mZ
ond
is more
conservative,
and for the N LL!
0.30
LLcomparison
(pure massless
QCD)
By
the N3in
Eq.band
(34) method
is the main
result of
this work. 0.113
method.
The
fit comparing
result is shown
with
dataand
0.112
0.114
DELPHI
in8the
R-gap
and
the
MS schemes
the disresulting 0.115
N3 LL (QCD + mb ) entries in Tabs. I and II we see that in- analyses
In
Fig.
we
show
the
first
moment
of
the
thrust
0.6
from
ALEPH, OPAL, L3, and SLD, and agrees
ALEPH
0.25 DELPHI,
Αenergy
cluding finite b-mass corrections causes a very mild shift errors
are only
10% smaller
scan
method
tribution
as a function
of thethan
centerinofthe
mass
s �m
Z �Q, that
very well. OPAL
that to
theα theory
values displayed are
of(Note
! +0.0004
themethod
bestQED andThe
mb corrections.
s (mZ ), and a somewhat larger shift weincluding
have adopted.
advantageWe
ofuse
thehere
scan
we
L3
0.20
actually
binned
according
to ΩQ
the
data
setshifts
and are
m
InALEPH
both
these
of ! −0.033
GeV to
Z cases
fitisvalues
given
in takes
Eq. (34).
The
band displays
the
theo1. =
C
τ
use
that
the
fit
into
account
theory
uncertainties
SLD
then
by a smooth
interpolation.)
the 1-σ theory
uncertainties. In the N3 LL (pure
retical uncertainty and,?7/'*/@8$0*/A'".0#3/'4.%%'
a scan
0.15 joinedwithin
1
1
including
correlations.has been determined with
massless
QCD)
analysis
the
b-quark
is
treated
as
a
mass0.5
9
on
the
parameters
included
in
our
theory,
as
explained
in
0.2
0.3
0.4
0.5
0.6
0.7
7B7@&%;'
C
less flavor, hence this analysis differs from that done by
Band Method
App. A. 0.112
The fit result
is shown0.114
in comparison
with data
0.113
0.115
of QED
and
the
bottom
JADE [23] where primary b quarks were removed using Effects
from ALEPH,
OPAL,
JADE,
gDELPHI,
gmass
' L3,
τ AMY and
C
It is useful to
compare our scan method to determine the
50
100
150
200
Q !GeV"
2345.#*%3/')*&"'!"
!"#$"%$#&'$#(")$
Ω
C-param
α
α (m )
τ
!"
universality? Ω ↔ Ω
generators.
A Manoharperturbative
(UCSD) MCerrors
with the error band method [26] that
Α �m � = Ω
Ω
+ O(10%)
Z
TASSO. Good agreement sis observed
for all Q values.
3
1
ICTP-SAIFR
School
' 1we can achieve
Given
the high-precision
at N05.2013
LL! orIt is interesting to compare the result of this analysis
25 / 31
α
1 dσ
σ dτ
1.4
Q=mZ
Sum Logs, with S
mod
+ gap
N3 LL’
N 3 LL
NNLL’
NNLL
NLL’
1.2
1.0
0.8
0.6
0.4
0.2
0.0
A Manohar (UCSD)
0.16
0.18
0.20
0.22
0.24
0.26
0.28
τ
0.30
ICTP-SAIFR School
05.2013
26 / 31
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
27 / 31
Jet Vetoes
Look at W + 0 jets, W + 1 jet, etc.
Similarly for the Higgs, there is a jet veto.
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ICTP-SAIFR School
05.2013
28 / 31
Results
cut
σ0 (pcut
T ) = σtotal − σ≥1 (pT )
�
� �
�
= 1 + αs + α2s + · · · − αsgreen:
(L2 + ·NLL
· · �) p+T α2s (L4 + · · · ) + · · ·
Higgs and Jet Binning
Jet Vetoes and Factorization
where L = ln(pcut
T /Q)
blue:
NLLpResummation
+NLOfor pjet
NNLL�p +NNLO
T
T
T
Resummation
Veto
Logs
orange:
NNLL�pT of
+NNLO
Summary
Jet Binning
NNLO Fixed Uncertainties
Order: FeHIP, HNNLOat Fixed Order
resummation
Logarithms are important for pT � QIncluding
∼ hard-interaction
scaleand fixed-order uncert
IS, Tackmann, Walsh, Zuberi (in prep)
Uncertainty procedure: IS, Tackmann
Same logarithms appear in the exclusive N-jet and inclusive (≥ N+1)-jet
25
cross section (and cancel in their sum)
gg → H (8 TeV)
σ0(pcut
T ) [pb]
Relevant
experimental
range
m = 125
GeV
2012-04-19
R
0.4 − 40 GeV
pcut
==20
T
H
20
Combining Helicity Amplitudes with Resummation
Frank Tackmann (DESY)
8
NNLO is available numerically
10
from
fully differential codes
NNLL +NNLO
� FEHiP [Anastasiou, Melnikov, Petriello]
5
NLL +NLO
� HNNLO [Grazzini]
NLL
0� MCFM [Campbell, Ellis, Williams]
10
30
40
50
60
70
80
0
20
Ecm = 7 TeV
mH = 165 GeV
µ = mH /2
µ = mH
µ = mH /4
4
2
10
20
30
40 50 60
[GeV]
pcut
T
70
80
�
pT
�
pT
pT
combined incl. unc.
0
0
90 100
pcut
T
30
2 / 24
15
6
40
δσ0(pcut
T ) [%]
gg → H +0 jet (NNLO)
σ0(pcut
T ) [pb]
10
20
10
0
−10
−20
−30
−40
0
[GeV]
Resummation of Veto Logs
Naive scale variation at fixed order ignores ∆cut
induced by log series
Frank Tackmann (DESY)
The Higgs Cross Section with a J
Can estimate ∆cut by FO scale variation of σ≥1 = σB (αs L2 + · · · )
∆y0 = ∆µ
total ,
∆cut = ∆µ
≥1
⇒
23
µ
2
2
∆20 = (∆µ
total ) + (∆≥1 )
⇒ Resum
to obtain improved predictions and uncertainty
estimates
A Manohar logs
(UCSD)
ICTP-SAIFR
School 05.2013
29 / 31
Jet Mass Distribution
Try and distinguish between quark and gluon jets.
A Manohar (UCSD)
ICTP-SAIFR School
05.2013
30 / 31
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Jet
Mass
Distribution
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a jet mass spectrum for inclusive pp → jets to the corresponding ATLAS data [26].
read
this
apples
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comparison
evel (dotted),
including
hadronization
(dashed), and including
hadronization carefully)
and multiple
)
a results reproduce the data well.
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48 jet mass data [26]. The peak position of our
e NNLL calculation with ATLAS inclusive
markably well with the inclusive dijet data. For the ATLAS date there is presumably a
ts which is compensated by a shift to higher values due to hadronization and multiple
an inclusive or exclusive jet sample, which Pythia seems
suggest in Figs. 11 and 12, a comparison between
jet School
ICTP-SAIFR
A Manohar (UCSD)to
05.2013
31 / 31