Jet cross section
SCET and refactorization
Two-loop checks
Soft-collinear mode for jet cross sections in soft
collinear effective theory
Yang-Ting Chien
Los Alamos National Laboratory, Theoretical Division, T-2
February 22, 2016
National Taiwan University, Taipei, Taiwan
In collaboration with C. Lee and A. Hornig (Phys. Rev. D 93, 014033 (2016))
Y.-T. Chien
Jet cross sections using SCET
1 / 20
Conclusions
Jet cross section
SCET and refactorization
Two-loop checks
Outline
• Precision physics
– Higgs/W/Zs + jets and jet vetos
– Small radius jets and jet substructure
• Soft collinear effective theory (SCET)
– Factorization and resummation
– Renormalization group evolution
• Soft-collinear mode and refactorization
– SCET++
• Conclusion
Y.-T. Chien
Jet cross sections using SCET
2 / 20
Conclusions
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Events / 2 GeV
A 125 GeV massive boson
10000
Selected diphoton sample
Data 2011+2012
Sig+Bkg Fit (m =126.8 GeV)
H
Bkg (4th order polynomial)
8000
ATLAS Preliminary
H→γ γ
6000
4000
∫
∫
-1
s = 7 TeV, Ldt = 4.8 fb
2000
-1
Events - Fitted bkg
s = 8 TeV, Ldt = 20.7 fb
500
100
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0
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-200
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mγ γ [GeV]
•
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On July 4th, 2012, the Higgs boson discovery was announced!
Since then, we have entered into the era of precision Higgs studies
It plays a central role in the mechanism of electroweak symmetry breaking
However, is it really "the" Higgs boson of the Standard Model (SM)?
Can it teach us anything about the BSM physics?
Y.-T. Chien
Jet cross sections using SCET
3 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
1
LHC HIGGS XS WG 2013
Higgs BR + Total Uncert
Precision physics
WW
bb
10-1
ττ
gg
ZZ
cc
10-2
Zγ
γγ
10-3
µµ
-4
10 80
100
120
140
160
180
200
MH [GeV]
•
•
•
•
We want to measure the Higgs cross sections precisely
We want to measure the coupling of the Higgs to everything precisely
Huge hadronic backgrounds need to be suppressed
Hadronic activities in LHC events are usually seen as jets
Y.-T. Chien
Jet cross sections using SCET
4 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Events
Jet veto
Data
WW
ATLAS Preliminary
6000
s = 8 TeV, ∫ Ldt = 5.8 fb-1
tt
Z+jets
(*)
H→WW →eνµ ν/µ νeν
5000
BG (sys ⊕ stat)
WZ/ZZ/W γ
Single Top
W+jets
H [125 GeV]
4000
3000
2000
1000
0
0
2
4
6
8
10
Events / 10 GeV
N jets
800
ATLAS Preliminary
s = 8 TeV, ∫ Ldt = 5.8 fb-1
700
(*)
H→WW →eνµ ν/µ νeν + 0 jets
600
Data
WW
tt
Z+jets
BG (sys ⊕ stat)
WZ/ZZ/W γ
Single Top
W+jets
H [125 GeV]
500
• The H → WW ∗ → lνlν channel is important for
400
spin and coupling measurements but with large tt̄
backgrounds
300
200
100
0
0
• Jet vetoes can efficiently suppress the
20
40
60
80
100
120
140
160
P llT [GeV]
backgrounds
• We need accurate theoretical predictions and uncertainty estimations for Higgs
cross sections with jet vetos, which demands the precise understanding of jets
Y.-T. Chien
Jet cross sections using SCET
5 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Jets and QCD
• Jets are collimated particles observed
ubiquitously at high energy colliders
• They are manifestations of underlying colored
partons and defined using jet algorithms with
radius R
• Small R is used to mitigate huge underlying
event and pileup contaminations
• For hadronic boosted particles, an even
smaller R is used to resolve jet substructures
Y.-T. Chien
Jet cross sections using SCET
6 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Resummation of log R
Λ
mL
e−
e+
R
mR
τ (J) =
1 X i −ηi
m2
≈ 2J
|pT |e
2EJ i∈J
EJ
τ = τL + τR
Z τ
dσ
dτ ′ ′
σc (τ ) =
dτ
0
• We will study jet thrust τ (J) and exclusive 2-jet cross sections in e+ e− collisions
using effective field theory techniques
• Jets are defined by two cones
around the thrust axis
• Out-of-jet energy is cutoff by Λ to
veto additional jets
• Different jet algorithms differ by
some nonsingular terms
Y.-T. Chien
Jet cross sections using SCET
7 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Soft-Collinear Effective Theory (SCET)
• Effective field theory techniques are most useful when
there is clear scale separation
• SCET separates physical degrees of freedom in QCD by a
QCD
systematic expansion in power counting
n
• Match SCET with QCD at the hard scale by
integrating out the hard modes
• Integrating out the off-shell modes gives collinear
Wilson lines which describe the collinear radiation
• The soft sector is described by soft Wilson lines along
the jet directions
n̄
SCET
Soft cross talk
• At leading power, soft-collinear decoupling holds in the Lagrangian and it leads to
the factorization of cross sections
Y.-T. Chien
Jet cross sections using SCET
8 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Power counting in SCET
• The scaling of modes in lightcone coordinates:
ph : Q(1, 1, 1), pc : Q(1, λ2 , λ) and ps : Q(λ, λ, λ)
p−
• Q is the hard scale which is the energy of the jet
√
• λ is the power counting parameter (λ ≈ mJ /Q ≈ τ )
• Qλ is the jet scale which is significantly lower than Q
• Different jet substructure observables may be
hard
soft
sensitive to soft, ultrasoft modes or others
• QCD = O(λ0 ) + O(λ1 ) + · · · in SCET
collinear
ultrasoft
p+
• For exclusive 2-jet cross sections without measuring jet substructures, we expect
that at least the following modes are relevant
ph : Q(1, 1, 1),
Y.-T. Chien
pc : Q(1, R2 , R), ps : (Λ, Λ, Λ)
Jet cross sections using SCET
9 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Factorization of 2-jet cross section
• The cross section factorizes as
• At one loop,
• H is the hard matching coefficient
alg.
• Jun
are unmeasured jet functions
c
• Sveto
is a multi-scale soft function
alg.
• Jun
(µ) =
P
Xc h0|χ̄(0)|Xc ihXc |χ(0)|0i
and χ is the collinear jet field
• Xc is constrained within jets by the jet algorithm with no extra measurements
• The factorization theorem has a product form instead of a convolution
• We will further examine the factorization structure in the soft sector and the
relation to the differential jet thrust distribution
Z τ max
dσ
dτ ′ ′
σ=
dτ
0
Y.-T. Chien
Jet cross sections using SCET
10 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Phase space and EFT modes
• At one loop, it is clear that the soft function
can be decomposed into pieces each
sensitive to only a single scale 2Λ or 2ΛR
(1)
S(1) = Ss
(1)
+ 2Ssc
• 2ΛR is the soft-collinear scale and the
region suggests the relevance of the
soft-collinear mode
• Yn are Wilson lines of soft gluons and
Xs are soft final states
Y.-T. Chien
Jet cross sections using SCET
psc ≈ Λ(1, R2 , R)
11 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
All-order generalization
• This suggests the all-order generalization of global soft
function and soft-collinear functions
• At one-loop, we have
(1)
S(1)
=
Ss
S(1)
=
Γ0S ln R ln
(1)
=
Γ0S ln2
(1)
=
Ss
Ssc
(1)
+ 2Ssc
µ2
+ c1S
4Λ2 R
µ
+ c1ss
2Λ
µ
Γ0sc ln2
+ c1sc
2ΛR
• Γ0sc = − 21 Γ0S = −Γ0cusp
• The global soft gluons can not resolve the small angle
R, and the soft-collinear gluons in the n direction can
only resolve angles with respect to n
Y.-T. Chien
Jet cross sections using SCET
12 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Jet thrust and EFT modes
• With the measurement of jet
thrust, the collinear radiation
inside jets is further restricted
√
pc ≈ Q(1, τ, τ ), τ < R2
• The angular constraint of jets
introduces the collinear-soft
(csoft) mode
pcs ≈
Qτ
1
1
(1, R2 , R) = Qτ ( 2 , 1, )
R2
R
R
• The small component of the csoft
mode is fixed by
(pc + pcs )2 = Q2 τ
• A similar csoft mode was
introduced in the studies of dijet
invariant masses in the ninja
region (Bauer et al) – SCET+
Y.-T. Chien
• With the addition of the soft-collinear mode, we call this
hierarchy of hierarchies SCET++
Jet cross sections using SCET
13 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Jet thrust factorization and soft refactorization
• The jet thrust factorizes as follows
• This suggests the all-order refactorization
as a convolution and can be conveniently
studied in Laplace space
• J alg. is related to inclusive jet functions
• At one loop, the soft function decomposes
into
Y.-T. Chien
Jet cross sections using SCET
14 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Jet thrust factorization and soft refactorization
• The components satisfy multiplicative
RG equations with anomalous
dimensions linear in ln µ
• Each piece has the following form
• The coefficients are determined by the RG
equation with known lower-order Γcusp , γ, β
and unknown two-loop constants
Y.-T. Chien
Jet cross sections using SCET
15 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Anomalous dimension and consistency relation
e+
+
e
e−
−
−
e
e+e−
e
e−
Drell − Yan
• By direct comparison to known two-loop
jet thrust soft function, we have
DIS
• In fact, e+ e− , Drell-Yan and DIS are
closely related processes
• For R = 1, we have
• This leads to the all-order relation
Y.-T. Chien
Jet cross sections using SCET
16 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
From jet thrust to jet cross section
• The two-loop jet thrust can be organized
using the refactorized expression (detail
omitted here)
• We can construct the jet cross section by
• It satisfies a multiplicative RG equation
integrating over jet thrust
• We can derive an all-order relation for its
anomalous dimension
• This gives a relation for the unmeasured
jet function
Y.-T. Chien
Jet cross sections using SCET
17 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Two-loop jet cross section and EERAD3
• We check the two-loop prediction with numerical calculations by EERAD3
Y.-T. Chien
Jet cross sections using SCET
18 / 20
Conclusions
Jet cross section
SCET and refactorization
Two-loop checks
Resummed 2-jet cross section
• With the refactorized expression we
can resum the jet cross section at
partial next-to-next-to leading
logarithmic (NLL) accuracy using
renomalization group techniques
• Resummation does not have a large
effect for quark jets, but for gluon jets
it can be significant
Y.-T. Chien
Jet cross sections using SCET
19 / 20
Conclusions
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Conclusions
• The factorization of jet thrust and jet cross section is analyzed using SCET
• A new soft-collinear mode is introduced to resum log R
• Predictions at two-loop are checked against numerical calculations with excellent
agreement
• The precise calculations of cross sections will remain a key theme in high energy
physics
• Effective field Theory techniques are very useful and effective in exclusive cross
section calculations
Thank you
Y.-T. Chien
Jet cross sections using SCET
20 / 20
Jet cross section
SCET and refactorization
Two-loop checks
Conclusions
Conclusions
• The factorization of jet thrust and jet cross section is analyzed using SCET
• A new soft-collinear mode is introduced to resum log R
• Predictions at two-loop are checked against numerical calculations with excellent
agreement
• The precise calculations of cross sections will remain a key theme in high energy
physics
• Effective field Theory techniques are very useful and effective in exclusive cross
section calculations
Thank you
Y.-T. Chien
Jet cross sections using SCET
20 / 20