Hadron collisions in SCET · • Factorization of any process in hadron-hadron collisions needs analysis of Glauber modes • Conceptual issue: do we have all the necessary low energy

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_


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, )


p

q-q

_

_

n

n_

M1

M2

p, p (1, 2, )

q, q ( 2,1, )

_

_PM1=p+p

_

PM2=q+q_


~ -4


p

q-q

_

_

n

n_

M1

M2

p, p (1, 2, )

q, q ( 2,1, )

_

_PM1=p+p

_

PM2=q+q_


~ -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




inclusive




inclusivethreshold




inclusivethresholdisolated




inclusivethresholdisolatedexclusive (soft, collinear)

ps2~pc2




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


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


1p2q̄2

(Ic3 + I c̄

3 + Is3) +

1q̄2

(I(nn̄)c4 + I

(nn̄)s4 ) +

1p2

(I(n̄n)c̄4 + I

(n̄n)s4 ) + Is

5

γγ

g


EFT1

EFT1I

Full Theory


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


1p2q̄2

(Ic3 + I c̄

3 + Is3) +

1q̄2

(I(nn̄)c4 + I

(nn̄)s4 ) +

1p2

(I(n̄n)c̄4 + I

(n̄n)s4 ) + Is

5

γγ

g


EFT1

EFT1I

Full Theory


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)


k

k+

1

2

2

Collinear

GlauberSoft

k+

1

2

2

GlauberSoft

k



k

k+

1

2

2

Collinear

GlauberSoft

k+

1

2

2

GlauberSoft

k

C`n=C-(Cg-Cgs+Cs)



k

k+

1

2

2

Collinear

GlauberSoft

k+

1

2

2

GlauberSoft

k

C`n=C-(Cg-Cgs+Cs)

k

k+

2

GlauberSoft



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


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_




EFT1=I3c+ +I3sI3c_





I3-EFTI=C2EFT1=I3c+ +I3sI3c

_






_


All zero bin integrals and Glauber integral scaleless!





_



= EFT1


So, for active-active graph we find: EFTI EFT2




_



= 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]



Soft ( 2, 2, 2):

Glauber ( 2, 2, ):

-4

-2

-4

-4

=

=

I4s=

I4g=

EFT1 EFT2

I4c

I4c_

Spectator-Active topologyContribution to the matching



EFT1=I4c+I4s

EFT2=I4c`+I4g +I4s

The only non-zero zero bin is: (I3c`)0g= I4g


So, for spectator-active graph we find: EFTI EFT2



EFT1=I4c+I4s

EFT2=I4c`+I4g +I4s



So, for spectator-active graph we find: EFTI EFT2



C2=(I4-EFTI,II)/Tree=?

EFT1=I4c+I4s

EFT2=I4c`+I4g +I4s



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).


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).


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


EFT2

_ _ C2=C2+0+?

C2=C2+0+?

gives C2


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−]



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:




EFT2=I5g+I5s




• So, for spectator-active graph EFTI and EFT2 are Not equivalent



EFT2=I5g+I5s




• 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


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

)]



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

⎞⎠


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

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

⎞⎠


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 = +

+



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

⎞⎠


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:



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

⎞⎠


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 = +

+





I5= +

+ O(

ε,1λ2

)


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

]




I5= +

+ O(

ε,1λ2

)


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

]




I5= +

+ O(

ε,1λ2

)


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

]



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

⎞⎠


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

]




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

⎞⎠


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


EFT1 EFT2

EFT1

EFT2


EFT1 EFT2

C2=(I5-I5s)/Tree=1/ UV+1/ IREFT1

EFT2


EFT1 EFT2

C2=(I5-I5s)/Tree=1/ UV+1/ IR

C2=(I5-I5s-I5g)/Tree= 0

EFT1

EFT2

γγ

g


EFT2

_ _C2=C2+0+1/ UV+1/ IR

C2=C2+0+0

gives C2



gives 0 in EFT2

but not in EFT1

γγ

g


EFT2

_ _C2=C2+0+1/ UV+1/ IR

C2=C2+0+0

gives C2



gives 0 in EFT2

but not in EFT1

EFT2 is an Effective Theory with Glauber modes, and it is the Right one!

γγ

g


EFT2

_ _C2=C2+0+1/ UV+1/ IR

C2=C2+0+0

gives C2



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

Documents

Hadron collisions in SCET · • Factorization of any process in hadron-hadron collisions needs analysis of Glauber modes • Conceptual issue: do we have all the necessary low energy