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

Bauer, Fleming, Pirjol, Stewart, „00 Light-cone coordinates   ~ np Q    ipx ( ) ( ) x e x , n p  ~ p Q n p   2 ~       np Q nn nn             4 4 Integrated out with EOM 4

Bauer, Fleming, Pirjol, Stewart, „00 Light-cone coordinates Leading order Lagrangian (n-collinear)           ( ) exp ( ) W x P ig ds n A ns x   n   0 5

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The SCET Lagrangian is formed by gauge invariant building blocks. Gauge Transformations:      U W Is gauge invariant   n   W W U n n

• PDF In Full QCD • Factorization In SCET [Neubert et.al] • PDF In SCET: [Stewart et.al]  x  is gauge invariant because each 2 ( , ) f 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

  Ji, Ma, Yuan  b  b  ( , ) ( , )  Ji, Yuan   Belitsky, Ji, Yuan   Cherednikov, Stefanis ( , 0 ) ( 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   b ( , 0 ) ( , )   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? W     † LC gauge

We calculate at one-loop in Feynman  an Gauge e and In LC LC † 0 W q n n gauge In Feynam amn Gauge ge

We calculate at one-loop in Feynman  an Gauge e and In LC LC † 0 W q n n gauge In LC Gauge ge      † 0 1 A W W n n     n k n k i         ( ) D k g           2 0 k i k    

We calculate at one-loop in Feynman  an Gauge e and In LC LC † 0 W q n n gauge In LC Gauge     n k n k i         ( ) D k g           2 0 k i k     2    ip n                  (Pr ) (Pr ) es es p p p I p I p p  , , LC Fey Ax w Fey w Ax 2

We calculate at one-loop in Feynman  an Gauge e and In LC LC † 0 W q n n gauge In LC Gauge The gauge 2    ip n                  invariance is (Pr es ) (Pr es ) p p p I p I p p  LC Fey Ax w Fey , w Ax , 2 ensured when   d  d k p k   1   (Pres) 2 2      4 I ig C   (Pres)  ,   I I w A x F d k 2  2     (2 ) 0 0 i p k i k , , w A x n F ey   2     d  d k p k      2 2        2 I ig C  , n Fey F d k  2 2      (2 ) 0 0 0 i p k i k i Gauge invariance is realized only with one prescription!!

The SCET matrix element is not gauge    † 0| | W q n n invariant . Using LC gauge the result of the one-loop correction depends on the used prescription.         0 0 i i Gauge invariance is I p I p , , w Ax w Ax 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 · ( , ; ) T x x P ig d l A x l x          n 0    † † And the new gauge invariant matrix element is 0| | T W q n n n

† 1   In covariant gauges , so we recover the T T SCET results    † 0| | W q n n In LC gauge    † 0| | T q n n

     (Pres) (Pres) d d k p k C C        (Pres) 2 2 2   I C g i         , T Ax F 2 2 d (2 ) ( 0)(( ) 0)  0 0  k i p k i k i k i Prescription C∞ +i0 0 -i0 1 PV 1/2  1   (Pres) (Pres) k  ML Θ ( ) I I I , , , n Fey w Ax T Ax 2 All prescription dependence cancels out and gauge invariance is restored no matter what prescription is used    † †    † 0| | T W n q 0| | W q n n n n Covariant Gauges In All Gauges

• TMDPDF     † † ( ) ( , ) ( ) ( ) y y T y W y y  n n n n    n n           (2) | ( )   ( ) (0) | P y x p P   q P / n n n n   np 2 We Can Define A Gauge Invariant TMDPDF In SCET (And Factorize SIDIS)

• 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.