The role of the transverse gauge links in soft collinear effective theory
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Transcript of The role of the transverse gauge links in soft collinear effective theory
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The role of the transverse gauge links in soft
collinear effective theory Ignazio Scimemi
Universidad Complutense de Madrid (UCM)In collaboration with A. Idilbi , M. García Echevarría
A.I., I.S. Phys. Lett. B695 (2011) 463, M.G.E., A.I., I.S. arXive:1104.0686[hep-ph] and work in progress
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SCET and its building blocks Gauge invariance for covariant gauges Gauge invariance for singular gauges (Light-cone gauge) A new Wilson line in SCET: T The origin of T-Wilson lines in SCET
Lagrangian: gauge conditions for different sectors
Phenomenology Conclusions
Outline
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SCET (soft collinear effective theory) is an effective theory of QCD
SCET describes interactions between low energy ,”soft” partonic fields and collinear fields (very energetic in one light-cone direction)
SCET and QCD have the same infrared structure: matching is possible
SCET helps in the proof of factorization theorems and identification of relevant scales
SCET, an effective theory of QCD
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4
SCET: Kinematics
Light-cone coordinates
Bauer, Fleming, Pirjol, Stewart, ‘00
,( ) ( )ipxn p
n p
x e x 2
~~
~
np Qp Q
np Q
4 4nn nn
Integrated out with EOM
U-soft
Soft modesdo not interact with (anti) collinear or u-softIn covariant gauge
( , , )
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5
SCET
Leading order Lagrangian (n-collinear)
Light-cone coordinates
Bauer, Fleming, Pirjol, Stewart, ‘00
( ) exp ( )0
n nW x P ig ds n A ns x
( ) exp ( )0
Y x P ig ds n A ns xn us
†
us
n n
iD
i
i gA
nD Y in Y
(0) †
n n nY W The new fields do not interact anymore with u-soft fields
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6
SCET
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SCET building blocks
The SCET Lagrangian is formed by gauge invariant building blocks.
Gauge Transformations:
nW
UWW
U
nn
Is gauge invariant
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Factorization Theorem For DIS
•PDF In Full QCD
•PDF In SCET:
is gauge invariant because each building block is gauge Invariant
•Factorization In SCET[Neubert et.al, Manohar]
[Stewart et.al]),( 2
fx
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Factorization Theorem For SIDIS in QCD: Covariant gauge
• “Naïve” Transverse Momentum Dependent PDF (TMDPDF):
• In Full QCD And At Low Transverse Momentum:
Analogous to the W in SCET
Ji, Ma,Yuan ‘04
SQq /
This result is true only in “regular” gauges:Here all fields vanish at infinity
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Transverse Gauge Link in QCD
)0,( ),( b
),(
b
)0,0(
• For gauges not vanishing at infinity [Singular Gauges] like the Light-Cone gauge (LC) one needs to introduce an additional Gauge Link which connects with to make it Gauge Invariant
)0,(
),( b
• In LC Gauge This Gauge Link Is Built From The Transverse Component Of The Gluon Field:
Ji, Ma, YuanJi, YuanBelitsky, Ji, YuanCherednikov, Stefanis
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Gauge Invariant TMDPDF In SCET?Are TMDPDF fundamental matrix elements in SCET?
Are SCET matrix elements gauge invariant?
Where are transverse gauge link in SCET?
† LC gaugeW
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Gauge invariance of SCET building blocksWe calculate at one-loop in Feynman Gauge and In LC gauge
In Feynamn Gauge
†0 n nW q
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Gauge invariance of SCET building blocksWe calculate at one-loop in Feynman Gauge and In LC gauge
In LC Gauge
†0 n nW q
†0 1n nA W W
2( )0
kniD k gk i
kk
n
[Bassetto, Lazzizzera, Soldati] Canonical quantization imposes ML prescription
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Gauge invariance of SCET building blocksWe calculate at one-loop in Feynman Gauge and In LC gauge
In LC Gauge
†0 n nW q
2( )0
kniD k gk i
kk
n
(Pr ) (Pr ), ,
2
2es es
LC Fey Ax w Fey w Axip np p p I p I pp
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Gauge invariance of SCET building blocksWe calculate at one-loop in Feynman Gauge and In LC gauge
In LC Gauge
†0 n nW q
( ) ( )2
, , 2ML ML
LC Fey Ax w Fey w Axip np p p I p I pp
(ML 2 2), 22(2 )
1 ( ) ( )
40 0
[ ]
w Ax FkI ig C
i i
dd k pd k p k k
k kk k ip k ip
The gauge invariance is ensured when
( ), ,
12
MLw yA Fex nI I
The result of this is independent of and has got only a single pole. Zero-bin subtraction is nul in ML.
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The SCET matrix element is not gauge invariant . Using LC gauge the result of the one-loop correction depends on the used prescription.
Gauge invariance in SCET†0 | |n nW q
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In order to restore gauge invariance we haveto introduce a new Wilson line, T, in SCET
matrix elements
Gauge invariance in SCET
†
0( , ) exp · ( , ; )nT x x P ig d x
l A l x
And the new gauge invariant matrix element is † †0 | |n n nT W q
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The T-Wilson Line
In covariant gauges , so we recover the SCET results
† 1T T
In LC gauge
†0 | |n nW q
†0 | |n nT q
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The T-Wilson Line
All prescription dependence cancels out and gauge invariance is restored no matter what prescription is used
Covariant Gauges In All Gauges
( ) ( )( ) 2, 2 2
2
( )
2(2 ) ( 0)(( ) 0) 0 0
( )
ML MLdML
T Ax F d
ML
C Cd k p kI C gk i p k i k i k i
C
i
k
(Pres) (Pres), , ,
12n Fey w Ax T AxI I I
† †0 | |n n n qWT †0 | |n nW q
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Prescription
C∞
+i0 0-i0 1PV 1/2
The T-Wilson Line in other prescriptionsLet us consider the pole part of the interesting integral with or the PV prescription. The result is
2 1 21
2 20
1 (1 ) 12 1 ln4 4
i sn F F
z z gI C dz C i finitep z i
And in PV the result does not have any imaginary part. The gauge invariance is restored either with the T with a prescription dependent factor
OR with zero-bin Subtraction!!
The values of this constant depend also on the convention for inner/outer moments
] 1/ (1 [ )/ kk i
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Where does the T-Wilson Line come from?
Is there a way to understand the T-Wilson lines from the SCET Lagrangian?
An example, the quark form factor: from QCD to SCET in LCG
The T-Wilson line is born naturally in One loop matching.
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Where does the T-Wilson Line come from?
In the canonical quantization of of the gauge field (Bassetto et al.) 2
2 2
( )( ) ( ) ( ) ( , ) ( ) ( , )
( ) 0; ( ) 0;
a a a a
a a
iknkA k T k k n nk k nk U nk kk k
n T k k T k
2, , ) ~ (1( , , )0
nA nA A QnA
We define ( )
( )
( , ) ( , , )
( , , ) ( , , ) ( , )
def
def
A x x A x x
A x x x A x x x A x x
And we can show
( )
†
D D
D D
def
i i gA
i Ti T
† ( )
0exp · ( , )T P ig d l A x x l
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The T-Wilson Lines in SCET-I In SCET-I only collinear and u-soft fields. The first step to obtain theSCET Lagrangian is integrating out energetic part of spinors
And then applying multipole expansion,
Where Tn n nW T W
U-soft field do not give rise to any transverse gauge link!!There are no transverse u-soft fields and they cannot depend on transverse coordinates!!
†1( )2
( )T Tn n us n n n n n
ninD gnA x iD W W iDin
IL
12n nninD iD iD
inD
L2
2 2 2
~1/ (1,1/ ,1/ )
~ 1/ (1/ ,1/ ,1/ )n
us
Q
x Q
x
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The T-Wilson Lines in SCET-II
†12
( )n n n n n n nninD iD W W iD
in
I IL
Now the degrees of freedom are just collinear and soft2 2, , ) ~ (1, , ); , , ) ~ (1, , );(
, , )(
~ ( , , ); , , ) ~ ( , ,( ( );n n n n n n
ss s s s s
nA nA A Q np np p QnA nA A Q np np p Q
No interaction is possible for on-shell states
Is this true in every gauge?
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The T-Wilson Lines in SCET-II
The gauge ghost however acts only on some momentum components
2 2( ((
, , ) ~ (1, , ); , , ) ~ (1, , );, , ) ~ ( , , ); , , ) ~ ( , , ;0 ( )
n n n n n n
ss s s s s
nA nA A Q np np p QnA nA A Q np np p Q
( ) ( , ) ( ) (0, )i in s n s
i i
x A x x x A x
Thus the covariant derivative is ( )( ) (0 , )n siD i gA x gA x
(0) †( ) ( ) ( ) ( )n sn n snA x T x A x T x
The decoupling of soft fields requires
( )
0exp · (0 , )sn sT P ig d l A x l
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The T-Wilson Lines in SCET-II
(0) †
(0) (0)
(0)
( ) ( ) ( ) ( )
( ) ( )
n sn n sn
n n
n sn n
A x T x A x T x
D i gA
T x x
The new SCET-II Lagrangian is
(0) (0) (0) (0) (0)† (0) (0)12
( )T Tn n n n n n n
ninD iD W W iDin
I IL
( )
0exp · (0 , )sn sT P ig d l A x l
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Applications
TMDPDFDrell-Yan at low Pt [Becher,Neubert]Higgs production at low Pt [Mantry,Petriello]Beam functions [Jouttenus,Stewart, Tackmann,Waalewijn] Heavy Ion physics – Jet Broadening [Ovanesyan,Vitev ]…
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Factorization Theorem For DIS
•PDF In Full QCD
•PDF In SCET:
is gauge invariant because each building block is gauge Invariant
•Factorization In SCET[Neubert et.al]
[Stewart et.al]),( 2
fx
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Factorization Theorem For SIDIS in QCD: Covariant gauge
• “Naïve” Transverse Momentum Dependent PDF (TMDPDF):
• In Full QCD And At Low Transverse Momentum:
Analogous to the W in SCET
Ji, Ma,Yuan ‘04
SQq /
This result is true only in “regular” gauges:Here all fields vanish at infinity
2
, , ,.
2 2 2 2 2
2 2 2 2
.
ˆ, , , , , , ) , , , / , )
( , )
ˆ( , ) ( (
, ( , ), )(
B h h q B B T h h
h h
q u d s
F x Q e d d p d l q x q z
H p l
z P k k x p z
S l Q z k P
Is renorrmalization scale; is a rapidity cut-off
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Transverse Gauge Link in QCD
)0,( ),( b
),(
b
)0,0(
• For gauges not vanishing at infinity [Singular Gauges] like the Light-Cone gauge (LC) one needs to introduce an additional Gauge Link which connects with to make it Gauge Invariant
)0,(
),( b
• In LC Gauge This Gauge Link Is Built From The Transverse Component Of The Gluon Field:
Ji, Ma, YuanJi, YuanBelitsky, Ji, YuanCherednikov, Stefanis
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TMDPDF
We Can Define A Gauge Invariant TMDPDF In SCET (And Factorize SIDIS)
(2)/ | ( ) ( ) (0) |
2q P n n n nn nP y x p Pnp
P P
† †( ) ( , ) ( ) ( )n n n ny T y W y y y
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Drell-Yan At Low PT
Introduce Gauge Invariant Quark Jet:
The TMDPDF Is Indeed Gauge Invariant.
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Collinear Anomaly
Notice That The Cross-Section Is Independent Of The Renormalization Scale (RG Invariance).
For Vanishing Soft Function, The Product Of Two TMDPDFs Has to Be Logarithmically Dependent On The Renormalization Scale. This is Impossible Unless There Is Anomaly.
Also (Up To Three Loop Calculation!)
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Origin Of Collinear AnomalyIn The Absence Of Soft Interactions Different Collinear
Sectors Do Not Interact So There Is No Way To Generate The Q-Dependence
Classically Each Collinear Lagrangian Is Invariant Under Rescaling of Collinear Momentum.
For TMDPDF Quantum Loop Effects Needs RegulationThen Classical Invariance Is Lost However The Q-Dependence Is Obtained
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The TMDPDF Is Ill-Defined And We need To Introduce New Set Of NP Matrix Elements
The Analysis Of Becher-Neubert Ignores Two Notions: Transverse Gauge Links And Soft-Gluon Subtraction Needed To Avoid Double Counting! (Currently Investigated.)
Re-factorization Into The Standard PDF
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•Application To Heavy-Ion Physics
D´Eramo, Liu, Rajagopal
In LC Gauge The Above Quantity Is Meaningless. If We Add To It The T-Wilson line Then We Get A Gauge Invariant Physical Entity.
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Conclusions
The usual SCET building blocks have to be modified introducing a New Gauge Link, the T-Wilson line.Using the new formalism we get gauge invariant definitions of non-perturbative matrix elements. In particular the T is compulsory for matrix elements of fields separated in the transverse direction. These matrix elements are relevant in semi-inclusive cross sections or transverse momentum dependent ones.It is possible that the use of LC gauge helps in
the proofs of factorization. The inclusion of T is so fundamental. Work in progress in this direction.
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Conclusions
It is definetely possible to understand the origin of T-Wilson lines in a Lagrangian framework for EFT.
Every sector of the SCET can be appropriately written in LCG.
The LCG has peculiar property for loop calculation and can avoid the introduction of new ad-hoc regulators
There is a rich phenomenology to be studied… so a lot of work in progress!! THANKS!