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

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

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

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

( , , )

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

6

SCET

SCET building blocks

The SCET Lagrangian is formed by gauge invariant building blocks.

Gauge Transformations:

nW

UWW

U

nn

Is gauge invariant

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

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

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

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

Gauge invariance of SCET building blocksWe calculate at one-loop in Feynman Gauge and In LC gauge

In Feynamn Gauge

†0 n nW q

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

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

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.

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

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

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

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

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

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.

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

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

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?

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

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

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 ]…

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

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

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

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

Drell-Yan At Low PT

Introduce Gauge Invariant Quark Jet:

The TMDPDF Is Indeed Gauge Invariant.

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!)

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

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

•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 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!