Ignazio Scimemi Universidad Complutense de Madrid (UCM) In - - PowerPoint PPT Presentation

Ignazio Scimemi

Universidad Complutense de Madrid (UCM) In collaboration with A. Idilbi arXive:1009.xxxx and work in progress with

• M. García Echevarría

 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  Conclusions

 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

4

Light-cone coordinates

Bauer, Fleming, Pirjol, Stewart, „00

,

( ) ( )

ipx n p n p

x e x  





2

~ ~ ~ np Q p Q np Q  



4 4 nn nn                  

Integrated out with EOM

5

Leading order Lagrangian (n-collinear)

Light-cone coordinates

Bauer, Fleming, Pirjol, Stewart, „00

( ) exp ( ) W x P ig ds n A ns x n              

6

The SCET Lagrangian is formed by gauge invariant building blocks. Gauge Transformations:



 n

W

   

 U W W U

n n

 

Is gauge invariant

• 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 f

x  

• “Naïve” Transverse Momentum Dependent PDF (TMDPDF):
• In Full QCD And At Low Transverse Momentum:

Analogous to the W in SCET Ji, Ma,Yuan „04

S Q q / 

This result is true only in “regular” gauges: Here all fields vanish at infinity

) , (



  ) , (



 b 

) , (

  b

 

) , ( 

• 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

) , (



 

) , (



 b 

• In LC Gauge This Gauge Link Is Built From The Transverse Component

Of The Gluon Field:

Ji, Ma, Yuan Ji, Yuan Belitsky, Ji, Yuan Cherednikov, Stefanis

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 at one-loop in Feynman an Gauge e and In LC LC gauge In Feynam amn Gauge ge

† n n

W q 

We calculate at one-loop in Feynman an Gauge e and In LC LC gauge In LC Gauge ge

† n n

W q 

†

1

n n

A W W

 

  

2

( ) k n i D k g k i k k n

        

                

We calculate at one-loop in Feynman an Gauge e and In LC LC gauge In LC Gauge

† n n

W q 

2

( ) k n i D k g k i k k n

        

                

         

 

(Pr ) (Pr ) , , 2

2

es es LC Fey Ax w Fey w Ax

ip n p p p I p I p p        

We calculate at one-loop in Feynman an Gauge e and In LC LC gauge In LC Gauge

† n n

W q 

         

 

(Pr ) (Pr ) , , 2

2

es es LC Fey Ax w Fey w Ax

ip n p p p I p I p p        

  



 

(Pres) 2 , 2

2 2 (2 ) 4

w A F x

d d k p d k p k k I ig C i i k



               

   



 



2 2 ,

2 2 2 (2 )

F n Fey

d d k p d k p k I ig C i i i k k



           



The gauge invariance is ensured when

(Pres) , ,

1 2

w A ey x n F

I I  

Gauge invariance is realized only with one prescription!!

The SCET matrix element is not gauge invariant . Using LC gauge the result of the

• ne-loop correction depends on the used

prescription.

†

0| |

n n

W q   

   

, , i i w Ax w Ax

I p I p

 

 

Gauge invariance is violated with –i0

• prescription. The

same occurs with PV, ML

In order to restore gauge invariance we have to introduce a new Wilson line, T, in SCET matrix elements

†

( , ) exp · ( , ; )

n

T x x P ig d x  

        

        



l A l x

And the new gauge invariant matrix element is

† †

0| |

n n n

T W q   

In covariant gauges , so we recover the SCET results

† 1

T T  

In LC gauge

†

0| |

n n

W q   

†

0| |

n n

T q   

All prescription dependence cancels out and gauge invariance is restored no matter what prescription is used

Covariant Gauges In All Gauges

(Pres) 2 (Pres) (Pres) 2 , 2 2

2 (2 ) ( 0)(( ) 0)

d T Ax F d

C C d k p k I C g k i p k i k i k i i



 

     

             



Prescription C∞ +i0

• i0

1 PV 1/2 ML Θ( )

k 

(Pres) (Pres) , , ,

1 2

n Fey w Ax T Ax

I I I   

† †

0| |

n n n q

W T   

†

0| |

n n

W q   

• TMDPDF

We Can Define A Gauge Invariant TMDPDF In SCET (And Factorize SIDIS)

(2) /

| ( ) ( ) (0) | 2

q P n n n n

n n P y x p P np     

 

           

† †

( ) ( , ) ( ) ( )

n n n n

y T y W y y  

 

 y

• 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

• f factorization. The inclusion of T is so fundamental.

Work in progress in this direction.