Orbital Momentum Effects in Heavy Ion and Hadron Collisions Sergey Troshin, IHEP, Protvino.
Hadron collisions in SCET · • Factorization of any process in hadron-hadron collisions needs...
Transcript of Hadron collisions in SCET · • Factorization of any process in hadron-hadron collisions needs...
Hadron collisions in SCET
Grigory Ovanesyan
UC Berkeley/LBL
Santa Fe 2010 Summer Workshop "LHC: From Here to Where?"
July 6, 2010
In collaboration with B.Lange and C.Bauer
earlier collaboration with C.Lee and Z.Ligeti
Grigory Ovanesyan
UC Berkeley/LBL
Santa Fe 2010 Summer Workshop "LHC: From Here to Where?"
July 6, 2010
In collaboration with B.Lange and C.Bauer
earlier collaboration with C.Lee and Z.Ligeti
On Exclusive Drell-Yan in SCET
Outline
• Introduction
• Drell-Yan factorization: a Review
• Breakdown of SCET in exclusive Drell-Yan
• Conclusions
Introduction
Introduction• Most of the interesting BSM signals on
Tevatron and LHC suffer from large QCD backgrounds
• Factorization is the main tool to understanding QCD backgrounds
Introduction• Most of the interesting BSM signals on
Tevatron and LHC suffer from large QCD backgrounds
• Factorization is the main tool to understanding QCD backgrounds
~ part f1 f2
Collins, Soper, Sterman, 82-85
Effective theories
• In recent years Effective Field Theories (EFT) have been developed to derive factorization
• Method of EFT relies on the hierarchy of physical scales, a small parameter
• Natural language to derive factorization theorems
• Allows clean resummation of large logarithms (Chris Lee’s talk)
• A systematic field theoretic tool that is straightforwardly expanded to higher orders and higher twist
• Clear separation of scales between hard emission, collinear splittings and soft radiation
• In SCET the small parameter describes how close to the jet axis the collinear emissions occur
• Power counting of SCET requires couplings between collinear quarks, gluons, and soft gluons
Soft Collinear Effective Theory
e+ e-
, AQ
E
Mj
QCD
n1, n2, An1, An2
Bauer, Fleming, Luke, Pirjol, Stewart, (00-01)
In Bauer, Cata, GO, (08), the SCET Lagrangian was derived for the first time on the functional integral levelAs
1/Q
-1QCD
1/MJ
IntroductionFactorization theorems in SCET
lepton collisions:
hadron collisions:
Bauer, Manohar,Wise, 02Bauer, Lee, Manohar,Wise, 03Lee, Sterman, 07Becher, Schwartz, 08 / Stewart et al (10)Bauer, Fleming, Lee, Sterman, 08Fleming, Hoang, Mantry, Stewart, 07 (I,II)Hornig, Lee, GO 09Ellis, Hornig, Lee, Vermilion, Walsh, 09,10
Becher Neubert, 07Ahrens, Becher, Neubert, Yang 08Becher, Schwartz, 09Bauer, Lange 09Stewart, Tackmann, Waalewijn 09,10Mantry, Petriello 09Ahrens, Ferroglia, Neubert, Pecjak, Yang 10
DIS Manohar 03Becher, Neubert, Pecjak 06
IntroductionFactorization theorems in SCET
lepton collisions:
hadron collisions:
Bauer, Manohar,Wise, 02Bauer, Lee, Manohar,Wise, 03Lee, Sterman, 07Becher, Schwartz, 08 / Stewart et al (10)Bauer, Fleming, Lee, Sterman, 08Fleming, Hoang, Mantry, Stewart, 07 (I,II)Hornig, Lee, GO 09Ellis, Hornig, Lee, Vermilion, Walsh, 09,10
Becher Neubert, 07Ahrens, Becher, Neubert, Yang 08Becher, Schwartz, 09Bauer, Lange 09Stewart, Tackmann, Waalewijn 09,10Mantry, Petriello 09Ahrens, Ferroglia, Neubert, Pecjak, Yang 10
DIS Manohar 03Becher, Neubert, Pecjak 06
this talk: Exclusive Drell-Yan in SCETA new mode in the SCET
Introduction
• In going from lepton to hadron collisions there are many important obstacles
• We are trying to resolve one of them: are we sure we have all the the important degrees of freedom in the theory?
• In this talk we will show that we need a new mode for one observable: exclusive Drell-Yan cross-section
• We will perform an explicit one loop matching calculation between QCD and SCET
A new mode in the SCET?
Drell-Yan factorization: Review
Tree level Drell-Yan-p
p
q-q
_
_
n
n_
M1
M2
p, p (1, 2, )
q, q ( 2,1, )
_
_PM1=p+p
_
PM2=q+q_
Tree level Drell-Yan-p
p
q-q
_
_
n
n_
M1
M2
p, p (1, 2, )
q, q ( 2,1, )
_
_PM1=p+p
_
PM2=q+q_
Glauber gluon: k ( 2, 2, )
Tree level Drell-Yan-p
p
q-q
_
_
n
n_
M1
M2
p, p (1, 2, )
q, q ( 2,1, )
_
_PM1=p+p
_
PM2=q+q_
Glauber gluon: k ( 2, 2, )
~ -4
Tree level Drell-Yan-p
p
q-q
_
_
n
n_
M1
M2
p, p (1, 2, )
q, q ( 2,1, )
_
_PM1=p+p
_
PM2=q+q_
Glauber gluon: k ( 2, 2, )
~ -4
Off-shellness as an infrared regulator
Pinch analysis of loop integrals
a
bx1(z)
x2(z)
a
bx1(z)x2(z)
I(z) =∫
C
dx f(x, z)
I(z → z0) =?
Pinch analysis of loop integrals
a
bx1(z)
x2(z)
a
bx1(z)x2(z)
I(z) =∫
C
dx f(x, z)
I(z → z0) =?
Pinch singularity, leads to a true pole
Not a true singularity, can be avoided by deforming the contour
• The pinched singularities appear only in the collinear n, collinear n and soft regions
• Glauber region is not pinched
Active-Active Interactions
-pp
q-q_
n
n_
M1
M2
_
Collins, Soper, Sterman, `82-`85_
• The leading pinched singularities appear only in the collinear n and soft regions
• Glauber region is not pinched
-pp
q-q_
n
n_
M1
M2
_
Spectator-Active InteractionsCollins, Soper, Sterman, `82-`85
• The pinched singularities appear in the Soft and Glauber regions
• This mode break the simple factorization of Drell-Yan exclusive cross-section
-pp
q-q_
n
n_
M1
M2
_
Spectator-Spectator InteractionsCollins, Soper, Sterman, `82-`85
• Glauber contribution is shown to be purely imaginary
• Thus it cancels in the inclusive cross-section
• These one-loop results are generalized to all orders
Cancellation of Glaubers
M1
M2
+ = 0
Our goal: should we include Glauber modes into Effective Theory?
Collins, Soper, Sterman, `82-`85
Drell-Yan factorizationCollins, Soper, Sterman, `82-`85
DY(Q2)~ part f1(x1) f2(x2)
DY(Q2, QT )~ part f1(x1,p1 ) f2(x2,p2 )
(Glauber gluons cancel)
(Glauber gluons contribute)
inclusive:
exclusive:
Pinches and Effective theory modes
A
BC
D
Full Theory pinches:
A,B,C,D
Pinches and Effective theory modes
A
BC
D
Full Theory pinches:
Need Effective Theory modes A, B, C, D:
A
BC
DA,B,C,D
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
inclusive
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
inclusivethreshold
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
inclusivethresholdisolated
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
inclusivethresholdisolatedexclusive (soft, collinear)
ps2~pc2
Drell-Yan factorization in SCET
Bauer, Fleming, Pirjol, Rothstein, Stewart, 02Becher, Neubert, Xu, 07
Stewart, Tackmann, Waalewijn, 09,10Mantry, Petriello, 09
inclusivethresholdisolatedexclusive (soft, collinear)
ps2~pc2
we will address the SCETI for exclusive Drell-Yan (ultrasoft, collinear pus2<<pc2)
Breakdown of SCET in exclusive Drell-Yan
• Factorization of the Drell-Yan process
• Loop diagrams contain a Glauber region which gives a leading order IR divergent contribution (on top of Soft and Collinear regions)
• “Glaubers” break the traditional factorization of the exclusive Drell-Yan cross-section
• In the inclusive cross-section this contribution cancels: G+G*=0 and factorization is restored
Why do we expect SCET to break down?
Do we need Glauber modes in the Effective Theory?
Collins, Soper, Sterman, `82,`85Bodwin, Brodsky, Lepage, `81
Bodwin, `85
• Glauber interactions happen between initial state spectator partons and they break the simple factorization in the exclusive cross-section
• Factorization is the key ingredient to make predictions for high energy QCD cross-sections
• Factorization of any process in hadron-hadron collisions needs analysis of Glauber modes
• Conceptual issue: do we have all the necessary low energy modes included into SCET?
• “Glaubers” play an important role for jet propagating in dense QCD media
Why is the presence of Glauber modes important?
• Idilbi, Majumder `08,
• D`Eramo, Liu, Rajagopal, 10
A Matching calculation for Drell-Yan
Idea of the calculation
• We want to set up a matching calculation which involves “Drell-Yan”-like one loop diagrams
• Should be a matching between QCD and EFT1, and EFT2, where
EFT1=SCET
EFT2=SCET+Glaubers
• By comparing the two matching calculations we should be able to find out which effective theory consistently describes the Drell-Yan amplitude
(collinear, ultrasoft)
Operator O2
Operator O2 arises from matching the QCD current onto a 2-jet Effective Theory:
where O2 is defined as:
C2 is Wilson coefficient which is well known to higher orders
J = q̄ Γq
J = C2O2
O2 = χn Γχn̄
Operator O2
< |J|qq>=C2< |O2|qq>
The Idea
EFT1
EFT2
<qq|J|0>=C2<qq|O2|0>_ _
Simplest final states to calculate C2 are <0| and |qq>:_
For our purpose we will chose instead states: < | and |qq>_
_ _ C2=?
C2=?γγ
g
g
Operator O2
< |J|qq>=C2< |O2|qq>
The Idea
EFT1
EFT2
<qq|J|0>=C2<qq|O2|0>_ _
Simplest final states to calculate C2 are <0| and |qq>:_
For our purpose we will chose instead states: < | and |qq>_
_ _ C2=?
C2=?
We know the answer for C2
γγ
g
g
〈γ∗γ∗|O2 |q̄q〉FT =1
p2q̄2I3 +
1q̄2
I(nn̄)4 +
1p2
I(n̄n)4 + I5
〈γ∗γ∗|O2 |q̄q〉EFT1=
1p2q̄2
(Ic3 + I c̄
3 + Is3) +
1q̄2
(I(nn̄)c4 + I
(nn̄)s4 ) +
1p2
(I(n̄n)c̄4 + I
(n̄n)s4 ) + Is
5
γγ
g
Outline of the matching
EFT1
EFT1I
Full Theory
〈γ∗γ∗|O2 |q̄q〉EFT2=
1p2q̄2
(Ic′3 + I c̄′
3 + Ig3 + Is
3) +1q̄2
(I(nn̄)c′
4 + I(nn̄)g4 + I
(nn̄)s4 )
+1p2
(I(n̄n)c̄′
4 + I(n̄n)g4 + I
(n̄n)s4 ) + Ig
5 + Is5 .
〈γ∗γ∗|O2 |q̄q〉FT =1
p2q̄2I3 +
1q̄2
I(nn̄)4 +
1p2
I(n̄n)4 + I5
〈γ∗γ∗|O2 |q̄q〉EFT1=
1p2q̄2
(Ic3 + I c̄
3 + Is3) +
1q̄2
(I(nn̄)c4 + I
(nn̄)s4 ) +
1p2
(I(n̄n)c̄4 + I
(n̄n)s4 ) + Is
5
γγ
g
Outline of the matching
EFT1
EFT1I
Full Theory
〈γ∗γ∗|O2 |q̄q〉EFT2=
1p2q̄2
(Ic′3 + I c̄′
3 + Ig3 + Is
3) +1q̄2
(I(nn̄)c′
4 + I(nn̄)g4 + I
(nn̄)s4 )
+1p2
(I(n̄n)c̄′
4 + I(n̄n)g4 + I
(n̄n)s4 ) + Ig
5 + Is5 .
〈γ∗γ∗|O2 |q̄q〉FT =1
p2q̄2I3 +
1q̄2
I(nn̄)4 +
1p2
I(n̄n)4 + I5
〈γ∗γ∗|O2 |q̄q〉EFT1=
1p2q̄2
(Ic3 + I c̄
3 + Is3) +
1q̄2
(I(nn̄)c4 + I
(nn̄)s4 ) +
1p2
(I(n̄n)c̄4 + I
(n̄n)s4 ) + Is
5
γγ
g
Outline of the matching
EFT1
EFT1I
Full Theory
〈γ∗γ∗|O2 |q̄q〉EFT2=
1p2q̄2
(Ic′3 + I c̄′
3 + Ig3 + Is
3) +1q̄2
(I(nn̄)c′
4 + I(nn̄)g4 + I
(nn̄)s4 )
+1p2
(I(n̄n)c̄′
4 + I(n̄n)g4 + I
(n̄n)s4 ) + Ig
5 + Is5 .
Zero-bin subtractions in
k
k+
1
2
2
Collinear
GlauberSoft
EFT2Manohar, Stewart (`06)
Zero-bin subtractions in
k
k+
1
2
2
Collinear
GlauberSoft
k+
1
2
2
GlauberSoft
k
EFT2Manohar, Stewart (`06)
Zero-bin subtractions in
k
k+
1
2
2
Collinear
GlauberSoft
k+
1
2
2
GlauberSoft
k
C`n=C-(Cg-Cgs+Cs)
EFT2Manohar, Stewart (`06)
Zero-bin subtractions in
k
k+
1
2
2
Collinear
GlauberSoft
k+
1
2
2
GlauberSoft
k
C`n=C-(Cg-Cgs+Cs)
k
k+
2
GlauberSoft
EFT2Manohar, Stewart (`06)
Zero-bin subtractions in
k
k+
1
2
2
Collinear
GlauberSoft
k+
1
2
2
GlauberSoft
k
C`n=C-(Cg-Cgs+Cs)
Gn=G-Gs
k
k+
2
GlauberSoft
EFT2Manohar, Stewart (`06)
Active-Active topology
I3c∫
dDl
(2π)D
1l2(l + p)2[−q̄−l+]=
∫dDl
(2π)D
1l2[p+l−](l − q̄)2=
∫dDl
(2π)D
1l2(l + p)2(l − q̄)2
∫dDl
(2π)D
1l2[p+l− + p2][−q̄−l+ + q̄2]I3s =
QCD=I3=
n-collinear (1, 2, ):
n-collinear ( 2,1, ):_
Soft ( 2, 2, 2):
-4
-4
-4
∫dDl
(2π)D
1[−l2⊥][p+(l− + p−) − (l⊥ + p⊥)2][−q̄−(l+ − q̄+) − (l⊥ − q̄⊥)2]I3g =
Glauber ( 2, 2, ):-4
EFT1
EFT2
I3c_
Active-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
EFT1=I3c+ +I3sI3c_
EFT2=I3c`+ `+I3g +I3sI3c_
Active-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
EFT1=I3c+ +I3sI3c_
EFT2=I3c`+ `+I3g +I3sI3c_
Active-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
I3-EFTI=C2EFT1=I3c+ +I3sI3c
_
EFT2=I3c`+ `+I3g +I3sI3c_
Active-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
I3-EFTI=C2EFT1=I3c+ +I3sI3c
_
EFT2=I3c`+ `+I3g +I3sI3c_
All zero bin integrals and Glauber integral scaleless!
Active-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
I3-EFTI=C2EFT1=I3c+ +I3sI3c
_
EFT2=I3c`+ `+I3g +I3sI3c_
All zero bin integrals and Glauber integral scaleless!
= EFT1
Active-Active topologyContribution to the matching
So, for active-active graph we find: EFTI EFT2
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
I3-EFTI=C2EFT1=I3c+ +I3sI3c
_
EFT2=I3c`+ `+I3g +I3sI3c_
All zero bin integrals and Glauber integral scaleless!
= EFT1
γγ
g
< |J|qq>=C2< |O2|qq>EFT1
EFT2
_ _ C2=C2+?+?
C2=C2+?+?
gives C2
Status of the calculation
Spectator-Active topology
QCD=I4=∫
dDl
(2π)D
1[l2][(l − p̄)2][(l + p)2][(l − q̄)2]
∫dDl
(2π)D
1[l2][(l − p̄)2][(l + p)2][−l+q̄−]∫
dDl
(2π)D
1[l2][−p̄+l−][p+l− + p2][(l − q̄)2]
∫dDl
(2π)D
1[−l2⊥][−p̄+(l− − p̄−) − (l⊥ − p̄⊥)2][p+(l− + p−) − (l⊥ + p⊥)2][−q̄−(l+ − q̄+) − (l⊥ − q̄⊥)2]
∫dDl
(2π)D
1[l2][−p̄+l− + p̄2][p+l− + p2][−q̄−l+ + q̄2]
n-collinear (1, 2, ):
n-collinear ( 2,1, ):_
Soft ( 2, 2, 2):
Glauber ( 2, 2, ):
-4
-2
-4
-4
=
=
I4s=
I4g=
EFT1 EFT2
I4c
I4c_
Spectator-Active topologyContribution to the matching
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
EFT1=I4c+I4s
EFT2=I4c`+I4g +I4s
The only non-zero zero bin is: (I3c`)0g= I4g
Spectator-Active topologyContribution to the matching
So, for spectator-active graph we find: EFTI EFT2
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
EFT1=I4c+I4s
EFT2=I4c`+I4g +I4s
The only non-zero zero bin is: (I3c`)0g= I4g
Spectator-Active topologyContribution to the matching
So, for spectator-active graph we find: EFTI EFT2
In the First effective theory all modes including overlaps equal:
In the Second effective theory all modes including overlaps equal:
C2=(I4-EFTI,II)/Tree=?
EFT1=I4c+I4s
EFT2=I4c`+I4g +I4s
The only non-zero zero bin is: (I3c`)0g= I4g
Spectator-Active topologyContribution to the matching
I4=
I(nn̄)s4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(−
ln p2p̄+
p̄2p+ + iπ
ε+
12
lnp̄2p+
p2p̄+ln
μ4p+p̄+(q̄−)2
p2p̄2(q̄2)2− iπ ln
μ2p2(p̄+)2q̄−
q̄2(p̄2)2p++
32π2
)
lnp̄2p+
p2p̄+
(ln
(p̄2p+ + p2p̄+)2
(p + p̄)2p+p̄+p̄2− 1
2ln
μ4p+
p2p̄2p̄+
) )I(nn̄)c4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(ln p2p̄+
p̄2p+ + iπ
ε− 7π2
6− 2 Li2
(−p2p̄+
p̄2p+
)+ iπ ln
μ2p2(p + p̄)2(p̄+)2
(p̄2p+ + p2p̄+)2p̄2+
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(π2
3− 2 Li2
(−p2p̄+
p̄2p+
)+
(ln
(p̄2p+
p2p̄+
)− i π
)ln
(q̄−
(p+p̄2 + p̄+p2
)2
q̄2(p + p̄)2p+p̄2
) )+ O (
ε, λ0).
Spectator-Active topologyContribution to the matching
I4=
C2=(I4-EFTI)/Tree=0
I(nn̄)s4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(−
ln p2p̄+
p̄2p+ + iπ
ε+
12
lnp̄2p+
p2p̄+ln
μ4p+p̄+(q̄−)2
p2p̄2(q̄2)2− iπ ln
μ2p2(p̄+)2q̄−
q̄2(p̄2)2p++
32π2
)
lnp̄2p+
p2p̄+
(ln
(p̄2p+ + p2p̄+)2
(p + p̄)2p+p̄+p̄2− 1
2ln
μ4p+
p2p̄2p̄+
) )I(nn̄)c4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(ln p2p̄+
p̄2p+ + iπ
ε− 7π2
6− 2 Li2
(−p2p̄+
p̄2p+
)+ iπ ln
μ2p2(p + p̄)2(p̄+)2
(p̄2p+ + p2p̄+)2p̄2+
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(π2
3− 2 Li2
(−p2p̄+
p̄2p+
)+
(ln
(p̄2p+
p2p̄+
)− i π
)ln
(q̄−
(p+p̄2 + p̄+p2
)2
q̄2(p + p̄)2p+p̄2
) )+ O (
ε, λ0).
Spectator-Active topologyContribution to the matching
I4=
C2=(I4-EFTI)/Tree=0In both effective theories contribution
to the matching coefficient is zero
I(nn̄)s4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(−
ln p2p̄+
p̄2p+ + iπ
ε+
12
lnp̄2p+
p2p̄+ln
μ4p+p̄+(q̄−)2
p2p̄2(q̄2)2− iπ ln
μ2p2(p̄+)2q̄−
q̄2(p̄2)2p++
32π2
)
lnp̄2p+
p2p̄+
(ln
(p̄2p+ + p2p̄+)2
(p + p̄)2p+p̄+p̄2− 1
2ln
μ4p+
p2p̄2p̄+
) )I(nn̄)c4 =
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(ln p2p̄+
p̄2p+ + iπ
ε− 7π2
6− 2 Li2
(−p2p̄+
p̄2p+
)+ iπ ln
μ2p2(p + p̄)2(p̄+)2
(p̄2p+ + p2p̄+)2p̄2+
i
16π2
1q̄−
1p̄2p+ + p2p̄+
(π2
3− 2 Li2
(−p2p̄+
p̄2p+
)+
(ln
(p̄2p+
p2p̄+
)− i π
)ln
(q̄−
(p+p̄2 + p̄+p2
)2
q̄2(p + p̄)2p+p̄2
) )+ O (
ε, λ0).
γγ
g
< |J|qq>=C2< |O2|qq>EFT1
EFT2
_ _ C2=C2+0+?
C2=C2+0+?
gives C2
Status of the calculation
gives 0(both theories) (both theories)
Spectator-Spectator topologyQCD=I5=
∫dDl
(2π)D
1l2(l − p̄)2(l + p)2(l − q̄)2(l + q)2
∫dDl
(2π)D
1l2[−l−p̄+][l−p+](l − q̄)2(l + q)2
∫dDl
(2π)D
1l2(l − p̄)2(l + p)2[−l+q̄−][l+q−]
n-collinear (1, 2, ):
n-collinear ( 2,1, ):_
Soft ( 2, 2, 2):
Glauber ( 2, 2, ):
∫dDl
(2π)D
1l2[−l−p̄+ + p̄2][l−p+ + p2][−l+q̄− + q̄2][l+q− + q2]
-2
-2
-4
∫dDl
(2π)D
1[−l2⊥][−p̄+(l− − p̄−) − (l⊥ − p̄⊥)2][p+(l− + p−) − (l⊥ + p⊥)2][−q̄−(l+ − q̄+) − (l⊥ − q̄⊥)2]
1[q−(l+ + q+) − (l⊥ + q⊥)2]
-4
Spectator-Spectator topology
EFT1= I5s =(1/ UV+1/ IR+finite)
EFT2=I5g+I5s
Contribution to the matching
In the First effective theory we have only Soft mode present:
In the Second effective theory all modes including overlaps equal:
Spectator-Spectator topology
EFT1= I5s =(1/ UV+1/ IR+finite)
EFT2=I5g+I5s
Contribution to the matching
In the First effective theory we have only Soft mode present:
In the Second effective theory all modes including overlaps equal:
• So, for spectator-active graph EFTI and EFT2 are Not equivalent
Spectator-Spectator topology
EFT1= I5s =(1/ UV+1/ IR+finite)
EFT2=I5g+I5s
Contribution to the matching
In the First effective theory we have only Soft mode present:
In the Second effective theory all modes including overlaps equal:
• So, for spectator-active graph EFTI and EFT2 are Not equivalent
• In the matching I5-EFT1 there is an extra UV divergence which will change the anomalous dimension of C2
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
(I5g)0s=0zero-bin integral is scaleless:
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
UV divergence
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
(I5g)0s=0zero-bin integral is scaleless:
Matching contribution in EFT2
Spectator-Spectator topology
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
(I5g)0s=0zero-bin integral is scaleless:
IR divergence
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
Matching contribution in EFT2
Spectator-Spectator topology
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
(I5g)0s=0zero-bin integral is scaleless:
IR divergence
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
Matching contribution in EFT2
Spectator-Spectator topology
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
(I5g)0s=0zero-bin integral is scaleless:
IR divergence
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
Finite term
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
(I5g)0s=0zero-bin integral is scaleless:
Matching contribution in EFT2
Spectator-Spectator topology
i
16π2
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠
Pentagon integral is reduced to sum of 5 box integrals:
I5= +
+ O(
ε,1λ2
)
Finite term
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
(ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−Q4
)+ π2
)]
i
16π2
[Q2
p+p̄+(p + p̄)2q−q̄−(q + q)2
(2πi
ε− 2π2 + 2π i ln
(μ2
Q2
))
2πi Q2
p+p̄+(p + p̄)2 − q−q̄−(q + q̄)2
⎛⎝ ln
(Q4
q−q̄−(q+q̄)2
)q−q̄−(q + q̄)2
−ln
(Q4
p+p̄+(p+p̄)2
)p+p̄+(p + p̄)2
⎞⎠ ]
I5g = +
+
I5s = i
16π2
Q2
p+p̄+(p + p̄)2q−q̄−(q + q̄−)2
[−2 iπ
ε+ ln
(p̄+p2
p+p̄2
)ln
(q−q̄2
q̄−q2
)+ iπ ln
(p̄2p2q̄2q2
p̄+p+q̄−q−μ4
)+ 3π2
]
C2=(I5-I5s-I5g)/Tree= 0
Spectator-Spectator topology
EFT1 EFT2
EFT1
EFT2
Spectator-Spectator topology
EFT1 EFT2
C2=(I5-I5s)/Tree=1/ UV+1/ IREFT1
EFT2
Spectator-Spectator topology
EFT1 EFT2
C2=(I5-I5s)/Tree=1/ UV+1/ IR
C2=(I5-I5s-I5g)/Tree= 0
EFT1
EFT2
γγ
g
< |J|qq>=C2< |O2|qq>EFT1
EFT2
_ _C2=C2+0+1/ UV+1/ IR
C2=C2+0+0
gives C2
Status of the calculation
gives 0(both theories) (both theories)
gives 0 in EFT2
but not in EFT1
γγ
g
< |J|qq>=C2< |O2|qq>EFT1
EFT2
_ _C2=C2+0+1/ UV+1/ IR
C2=C2+0+0
gives C2
Status of the calculation
gives 0(both theories) (both theories)
gives 0 in EFT2
but not in EFT1
EFT2 is an Effective Theory with Glauber modes, and it is the Right one!
γγ
g
< |J|qq>=C2< |O2|qq>EFT1
EFT2
_ _C2=C2+0+1/ UV+1/ IR
C2=C2+0+0
gives C2
Status of the calculation
gives 0(both theories) (both theories)
gives 0 in EFT2
but not in EFT1
EFT2 is an Effective Theory with Glauber modes, and it is the Right one!
Matching calculationSummary of the matching calculation
Matching calculationSummary of the matching calculation
• In active-active and spectator-active topologies putting the Glauber mode or not into SCET doesn’t make any difference
• In spectator-spectator topology, the presence of Glauber mode makes a non-trivial contribution to the Drell-Yan amplitude and only including Glaubers into effective theory we get the right answer for the matching coefficient C2
• Our results are in no conflict with Collins, Soper, Sterman’s “pinch” analysis of Drell-Yan loop integrals
• Taking into account the zero-bins, or the overlaps between the different modes was crucial for our analysis
• We completed a one-loop matching calculation for the operator O2 with special partonic final states
• SCET needs to be expanded for with Glauber mode
• Understanding the cancellation of Glauber gluons in the Drell-Yan cross-section using Effective Theory hasn’t been achieved(yet)
Conclusions