Measuring Two Angularities on a Single Jet

40 Pages·2014·2.52 MB·English

Measuring Two Angularities on a Single Jet
Andrew Larkoski, Ian Moult, Duff Neill
QCDEvolution2014
May 13, 2014
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Motivation
Angular and energy resolutions of modern detectors
unprecedented.
With high luminosity one can probe increasingly detailed
questions about the structure of QCD radiation.
Such detailed questions often require take the form of
multi-differential cross-sections, often on a single “sector.”
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan: (cid:126)q and Beam Thrust
T
Process: pp l+l−+X and measure:
→
Lepton-Pair Transverse Momentum (cid:126)q .
T
Lepton-Pair Invariant Mass Q2.
Beam thrust: τ = (cid:80)i|p(cid:126)Ti|e−|yi|
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
l− dΣ
dΤdqT2dQ^2
qT,Τ(cid:60)(cid:60)Q^2
Τ
P H P¯
l+ qT
(cid:200) (cid:200)
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan: (cid:126)q and Beam Thrust
T
Factorization in region (cid:126)q2 Qτ Q2:
T ∼ (cid:28)
(cid:90)
dσ
= σ dYdt dt d2k d2k H(Q2)δ(2)((cid:126)q (cid:126)k (cid:126)k )
dτd(cid:126)q2dQ2 0 a b Ta Tb T − Ta− Tb
T
(cid:16) e−Yt +eYt (cid:17)
B (x ,(cid:126)k ,t )B (x ,(cid:126)k ,t )S τ a b
n a Ta a n¯ b Tb b
− Q
Q
x = e±Y
a,b
E
cm
1110.0839: Jain, Procura, Waalewijn
See also:
0708.2833 J. Collins, T. Rogers, and A. Stasto,
0807.2430 T. C. Rogers.
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan and Fully Unintegrated PDFs
Operators and phase space of region (cid:126)q2 Qτ Q2:
T ∼ (cid:28)
Bn(x,(cid:126)kT,t)=θ(p−)(cid:104)p(p−)|χ¯n(0)δ(xp−−P−)δ(2)((cid:126)kT −P(cid:126)T)δ(t−xp−P+)χn(0)|p(p−)(cid:105)
1
S(τ)= Nctr(cid:104)0|T{Sn(0)Sn†¯(0)}δ(τ−τˆ)T¯{Sn†(0)Sn¯(0)}|0(cid:105)
(cid:16) (cid:17)
Resumsαnlnm τ termsinperturbationtheory.
s Q
Bn(x,(cid:126)kT,t)correctlycomputeslowerboundaryonly.
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
dΣ
dΤdqT2dQ^2
qT2(cid:126)QΤ(cid:60)(cid:60)Q^2
Τ
FU(cid:45)PDFPhaseSpace
qT
(cid:200) (cid:200)
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan and TMD-PDFs
Now factorize in region (cid:126)q τ Q
T
| | ∼ (cid:28)
(cid:90)
dσ
= σ dYd2k d2k d2k H(Q2)δ(2)((cid:126)q (cid:126)k (cid:126)k )
dτd(cid:126)q2dQ2 0 Ta Tb Ts T − Ta− Tb
T
(cid:16) (cid:17)
B (x ,(cid:126)k )B (x ,(cid:126)k )S τ,(cid:126)k
n a Ta n¯ b Tb Ts
Q
x = e±Y
a,b
E
cm
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan and TMD-PDFs
Operators and phase space of region (cid:126)q τ Q:
T
| | ∼ (cid:28)
Bn(x,(cid:126)kT)=θ(p−)(cid:104)p(p−)|χ¯n(0)δ(xp−−P−)δ(2)((cid:126)kT −P(cid:126)T)χn(0)|p(p−)(cid:105)
1
S(τ,(cid:126)kT)= Nctr(cid:104)0|T{Sn(0)Sn†¯(0)}δ(τ−τˆ)δ(2)((cid:126)kT −P(cid:126)T)T¯{Sn†(0)Sn¯(0)}|0(cid:105)
(cid:16) (cid:17)
Resumsαnlnm |q(cid:126)T| termsinperturbationtheory.
s Q
S(τ,(cid:126)kT)correctlycomputesupperboundary.
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
dΣ
dΤdqT2dQ^2
qT(cid:126)Τ(cid:60)(cid:60)Q^2
Τ Space
Phase
MD(cid:45)PDF
T
qT
(cid:200) (cid:200)
ThisisanSCETII factorization.
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan: Beam thrust and (cid:126)q phase space
T
How do we interface the different factorization descriptions?
(cid:40)
dσ Bn(xa,q(cid:126)T,τ) τ Bn(xb,q(cid:126)T,τ) τ S(τ)
= H ⊗ ⊗
dτd2(cid:126)qT Bn(xa,(cid:126)qT)⊗q(cid:126)T B(xb,(cid:126)qT)⊗q(cid:126)T S(τ,(cid:126)qT)
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
dΣ
dΤdqT2dQ^2
qT,Τ(cid:60)(cid:60)Q^2
Τ
DFs
(cid:45)P
D
M
T F U(cid:45) P D Fs
q
T
(cid:200) (cid:200)
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan: Beam thrust and (cid:126)q phase space
T
τ (cid:126)q phase space interpolates between EFT power countings:
T
−
SCET SCET
I II
p Q(1,λ2,λ) p Q(1,λ2,λ)
n n
∼ ∼
p Q(λ2,λ2,λ2) p Q(λ,λ,λ)
us s
∼ ∼
Wrong soft or collinear function in a region wrong phase
→
space boundaries:
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
dΣ dΣ
dΤdqT2dQ^2 dΤdqT2dQ^2
qT2(cid:126)QΤ(cid:60)(cid:60)Q^2 qT(cid:126)Τ(cid:60)(cid:60)Q^2
Τ FU(cid:45)PDFPhaseSpace Τ TMD(cid:45)PDFPhaseSpace
qT qT
(cid:200) (cid:200) (cid:200) (cid:200)
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet
Drell-Yan: Beam thrust and (cid:126)q phase space
T
τ (cid:126)q phase space interpolates between EFT power countings:
T
−
SCET SCET
I II
p Q(1,λ2,λ) p Q(1,λ2,λ)
n n
∼ ∼
p Q(λ2,λ2,λ2) p Q(λ,λ,λ)
us s
∼ ∼
SCET has only UV renormalization group.
I
SCET also has rapidity resummation, e.g., CS eqn.
II
1202.0814,Chiu,DN,Jain,Rothstein
pp(cid:174)l(cid:43)l(cid:45)(cid:43)X
dΣ
dΤdqT2dQ^2
qT,Τ(cid:60)(cid:60)Q^2
Τ
TMD(cid:45)PDFs FU(cid:45)PDFs
qT
(cid:200) (cid:200)
AndrewLarkoski,IanMoult,DuffNeill MeasuringTwoAngularitiesonaSingleJet