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
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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
Bauer, Fleming, Pirjol, Stewart, ‘00
U-soft
Light-cone coordinates
 ( x)   eipx n, p ( x)
np ~ Q
p ~ Q
 nn
  nn
 
        
4 
 4
np ~  2Q
n
p
Integrated out with EOM
Soft modes ( ,  ,  )
do not interact with
(anti) collinear or u-soft
In covariant gauge
4
Bauer, Fleming, Pirjol, Stewart, ‘00
Light-cone coordinates
Leading order Lagrangian (n-collinear)



Wn ( x)  P exp  ig  ds n An (ns  x) 


0


iD  i   gAus
inD  Y inYn
†
n
n
(0)



Y ( x)  P exp  ig  ds n A (ns  x) 
n
us


0


 YnW 
†
n
The new fields do not
interact anymore with
u-soft fields
5
6
The SCET Lagrangian is formed by gauge invariant building blocks.
Gauge Transformations:
  U
Wn  WnU 
Wn
Is gauge invariant
•PDF In Full QCD
•Factorization In SCET
•PDF In SCET:
[Neubert et.al,
Manohar]
[Stewart et.al]
 ( x,  2f ) is gauge invariant because each
building block is gauge Invariant
• In Full QCD And At Low Transverse Momentum:
Ji, Ma,Yuan ‘04
• “Naïve” Transverse Momentum Dependent PDF (TMDPDF):
q Q/S
Analogous to the W in SCET
This result is true only in “regular” gauges:
Here all fields vanish at infinity

( , b )


(0,0)

(, b )

(,0 )
Ji, Ma, Yuan
Ji, Yuan
Belitsky, Ji, Yuan
Cherednikov, Stefanis
• For gauges not vanishing at infinity [Singular Gauges] like
the Light-Cone gauge (LC) one needs to introduce an additional


Gauge Link which connects (,0 ) with (, b ) to make it
Gauge Invariant
• In LC Gauge This Gauge Link Is Built From The Transverse Component
Of The Gluon Field:
Are TMDPDF fundamental matrix elements in SCET?
Are SCET matrix elements gauge invariant?
Where are transverse gauge link in SCET?
LC gauge
W † 

We calculate 0 Wn† n q at one-loop in Feynman Gauge and In LC
gauge
In Feynamn Gauge
We calculate 0 Wn† n q at one-loop in Feynman Gauge and In LC
gauge
In LC Gauge
[Bassetto, Lazzizzera, Soldati]
Canonical quantization
imposes ML prescription
A  0  Wn  Wn †  1
n k  n k
i 
 g  
D (k )  2
k  i0 
 k  





We calculate 0 Wn† n q at one-loop in Feynman Gauge and In LC
gauge
In LC Gauge
n k  n k
i 
 g  
D (k )  2
k  i0 
 k  





2
ip
n
(Pr es )
(Pr es )
 LC  p    Fey  p    Ax  p    I w, Fey  p   I w, Ax  p   
p 2
We calculate 0 Wn† n q at one-loop in Feynman Gauge and In LC
gauge
In LC Gauge
)
 LC  p    Fey  p   (AxML )  p    I w, Fey  p   I w( ML
, Ax  p  
I
(ML)
w, Ax
ddk
 4ig CF  
(2 )d
2
2

p  k 
k 2  i 0  p  k 2  i 0  k  
 
1
 (k  )
 (k  )
 
 


[k ] k  ip  k  ip 


ip n
p 2
2
The gauge
invariance is
ensured when
1 ( ML )
 I w, Ax  I n , Fey
2
The result of this is independent of
 and has got only a single pole. Zero-bin
subtraction is nul in ML.
The SCET matrix element 0| Wn†n | q is not gauge
invariant . Using LC gauge the result of the
one-loop correction depends on the used
prescription.
In order to restore gauge invariance we have
to introduce a new Wilson line, T, in SCET
matrix elements


T ( x , x )  P exp ig  d l  ·A  (  , x  ; l   x  ) 
 0

†
n

And the new gauge invariant matrix element is
0| Tn†Wn†n | q
In covariant gauges T
SCET results 0| Wn†n | q
In LC gauge
0 | Tn†n | q
 T  1 , so we recover the
†
)
IT( ML
, Ax
( ML )
( ML )
d




C
C
d
k
p

k


 2CF g 2  2 i 

(2 ) d (k 2  i 0)(( p  k ) 2  i 0)  k   i 0 k   i 0 
C( ML )   (k  )
I n , Fey
1 (Pres)

I w, Ax  IT(Pres)
, Ax
2
All prescription dependence cancels out and gauge
invariance is restored no matter what prescription is used
0 | W  | q
†
n n
Covariant Gauges
0 | Tn†Wn†n | q
In All Gauges
Let us consider the pole part of the interesting integral with1 / [k ]  1/ (k  i )
or the PV prescription. The result is


I
 i
n
 2  1
s
 2
CF  2 
4
 p  

(1  z )1 z 
g2
1
0 dz z i   4 2 CF  1  ln i   finite
1
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 Prescription C∞
+i0
0
-i0
1
PV
OR with zero-bin
Subtraction!!
1/2
The values of this constant depend
also on the convention for inner/outer moments
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.
(nA, nA, A ) ~ Q(1,  2 ,  )
nA  0
In the canonical quantization of of the gauge field (Bassetto et al.)
A (k )  T (k ) (k )  n
a
a
2
 (nk )
k2
 (nk , k  ) 
a
ik 
k2
 (nk )U a (nk , k  )
n  Ta (k )  0; k Ta (k )  0;
We define
()
def

A ( x , x )  A( x  ,   , x )


def
A( x , x , x )  A( x  , x  , x )  A(  ) ( x  , x )
And we can show
iD
iD


def

 iD


 gA ( )


 Ti D T †


T  P exp ig  d l ·A(  ) ( x  , x  l ) 


0
†
In SCET-I only collinear and u-soft fields. The first step to obtain the
SCET Lagrangian is integrating out energetic part of spinors
1

n
L  n  inD  iD
iD  n
inD

2
xn ~ 1/ Q(1,1/  2 ,1/  )
And then applying multipole expansion,
xus ~ 1/ Q(1/  2 ,1/  2 ,1/  2 )
1
n
T†
LI   n (inDn  gnAus ( x )  iDn W
Wn iDn  )  n
in 
2

Where
T
n
WnT  TnWn
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!!
Now the degrees of freedom are just collinear and soft
(nAn , nAn , An ) ~ Q(1, 2 , ); (npn , npn , pn  ) ~ Q(1, 2 , );
(nAs , nAs , As  ) ~ Q( , , );
(nps , nps , ps  ) ~ Q( , , );
No interaction is possible for on-shell states
LI I   n (inDn  iDn Wn
1
n
Wn†iDn  )  n
in 
2
Is this true in every gauge?
(nAn , nAn , An ) ~ Q(1, 2 , );
(npn , npn , pn  ) ~ Q(1, 2 , );
(nAs , nAs  0, As  ) ~ Q( , , );
(nps , nps , ps  ) ~ Q( , , );
The gauge ghost however acts only on some momentum components
i



i



(
x
)
A
(
x
,
x
)


(
x
)
A
(0,
x
)
 n s
 n s
i
i
Thus the covariant derivative is
iD  i   gAn ( x)  gAs()  (0 , x )
The decoupling of soft fields requires
An(0)  ( x)  Tsn ( x ) An ( x)Tsn† ( x )


Tsn  P exp ig  d l ·As ( ) (0 , x  l ) 
 0

The new SCET-II Lagrangian is
LI I  
(0) 
n
A
(0)
n
(inD
(0)
n

 iD W
(0)
n
T (0)
n
1
n (0)
T (0)†
(0)
Wn iDn  )  n
in 
2
( x)  Tsn ( x ) An ( x)T ( x )
†
sn
Dn(0)   i   gAn(0) 
 n(0)  Tsn ( x ) n ( x)


Tsn  P exp ig  d l ·As ( ) (0 , x  l ) 
 0

TMDPDF
Drell-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
…
]
•PDF In Full QCD
•Factorization In SCET
•PDF In SCET:
[Neubert et.al]
[Stewart et.al]
 ( x,  2f ) is gauge invariant because each
building block is gauge Invariant
• In Full QCD And At Low Transverse Momentum:
F ( xB , zh , Ph  , Q) 

q u , d , s ,..
Ji, Ma,Yuan ‘04
eq2  d 2 k d 2 p d 2l q ( xB , k ,  2 , xB ,  ) qˆT ( zh , p ,  2 , ˆ / zh ,  )
 S (l ,  2 ,  ) H (Q 2 ,  2 ,  )  2 ( zh k  p  l  Ph )
 Is renorrmalization scale;  is a rapidity cut-off
• “Naïve” Transverse Momentum Dependent PDF (TMDPDF):
q Q/S
Analogous to the W in SCET
This result is true only in “regular” gauges:
Here all fields vanish at infinity

( , b )


(0,0)

(, b )

(,0 )
Ji, Ma, Yuan
Ji, Yuan
Belitsky, Ji, Yuan
Cherednikov, Stefanis
• For gauges not vanishing at infinity [Singular Gauges] like
the Light-Cone gauge (LC) one needs to introduce an additional


Gauge Link which connects (,0 ) with (, b ) to make it
Gauge Invariant
• In LC Gauge This Gauge Link Is Built From The Transverse Component
Of The Gluon Field:
n ( y)  T ( y , y )W ( y)n ( y)
†
n
q / P

†
n

nP  (2)
n
  Pn | n ( y)  x 
 n (0) | Pn 
  ( p  P )
np 
2

We Can Define A Gauge Invariant TMDPDF In SCET (And
Factorize SIDIS)
Introduce Gauge Invariant Quark Jet:
The TMDPDF Is Indeed Gauge Invariant.
Notice That The Cross-Section Is Independent Of
The Renormalization Scale (RG Invariance).
Also (Up To Three Loop Calculation!)
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.
In 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 Regulation
Then Classical Invariance Is Lost However The Q-Dependence
Is Obtained
The TMDPDF Is Ill-Defined And We need To Introduce New Set
Of NP Matrix Elements
Re-factorization Into The Standard PDF
The Analysis Of Becher-Neubert Ignores Two Notions: Transverse Gauge Links
And Soft-Gluon Subtraction Needed To Avoid Double Counting!
(Currently Investigated.)
•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.
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.
Conclusions
It is definetely possible to understand the origin of TWilson 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!