Understanding SCET: Effective Theory of QCD
SCET, a soft collinear effective theory, describes interactions between low energy, soft partonic fields, and collinear fields in QCD. It helps prove factorization theorems and identifies relevant scales. The SCET Lagrangian is formed by gauge invariant building blocks, enabling gauge transformations and providing insights into PDF factorization in full QCD. Collaborative work by researchers from Universidad Complutense de Madrid sheds light on the origin and applications of SCET in modern particle physics.
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Ignazio Scimemi Universidad Complutense de Madrid (UCM) In collaboration with A. Idilbi , M. Garc a Echevarr a 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
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 ~ ~ np p Q = ipx ( ) x ( ) x e , n p Q n p ( , , ) Soft modes do not interact with (anti) collinear or u-soft In covariant gauge 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) = ds n A ns + ( ) exp ( ) W x P ig x = + ( ) x exp ( ) Y P ig ds n A ns x n n n us 0 0 = + iD i gA The new fields do not interact anymore with u-soft fields us = (0) YW = i nD Y in Y n n n n n 5
The SCET Lagrangian is formed by gauge invariant building blocks. Gauge Transformations: U + W Is gauge invariant n + n + + W W U n
PDF In Full QCD Factorization In SCET [Neubert et.al, Manohar] [Stewart et.al] PDF In SCET: x 2 f is gauge invariant because each building block is gauge Invariant ( , )
In Full QCD And At Low Transverse Momentum: Ji, Ma,Yuan 04 Na ve Transverse Momentum Dependent PDF (TMDPDF): S Q / q Analogous to the W in SCET This result is true only in regular gauges: Here all fields vanish at infinity
Ji, Ma, Yuan Ji, Yuan Belitsky, Ji, Yuan Cherednikov, Stefanis b ( , ) b ( , ) ( 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 Gauge Invariant In LC Gauge This Gauge Link Is Built From The Transverse Component Of The Gluon Field: b ( 0 , ) ( , )
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 Gauge gauge Feynman Gauge and In LC LC 0 W q n n In Feynamn Feynamn Gauge Gauge
We calculate at one-loop in Feynman Gauge gauge Feynman Gauge and In LC LC 0 W q n n In LC Gauge LC Gauge [Bassetto, Lazzizzera, Soldati] Canonical quantization imposes ML prescription += = = 0 1 A W W n n + n k n k + i = ( ) k D g 2 + 0 k i k
We calculate at one-loop in Feynman Gauge gauge In LC Gauge LC Gauge k n i D k g k i k + Feynman Gauge and In LC LC 0 W q n n + n k = ( ) 2 + 0 2 ip p+ n ( ) ( ) p ( ) p ( ) p ( ) p ( ) p = + = + (Pr Ax ) (Pr , w Ax ) es es I I , LC Fey w Fey 2
We calculate at one-loop in Feynman Gauge gauge In LC Gauge LC Gauge ( ) ( ) ( ) ( , LC Fey Ax w Fey p p p I = + = Feynman Gauge and In LC LC 0 W q n n The gauge invariance is ensured when 1 2 2 ip p+ n ) ( ) p ( ) p + ( ) ( w Ax ) ML ML I , 2 + + d + d k p k = (ML , w Ax ) 2 2 4 I ig C ( ) ( )( = ( w ) ML A I I F d + 2 2 ) (2 ) + p k + + 0 0 k i i k , , x n Fe y + + 1 k ( + ) + ( ) k k ip = + + + [ ] k ip k 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 invariant . Using LC gauge the result of the one-loop correction depends on the used prescription. is not gauge 0| | W q n n
In order to restore gauge invariance we have to introduce a new Wilson line, T, in SCET matrix elements = And the new gauge invariant matrix element is n + + + l A l x ( , ) exp ( , ; ) nT x x P ig d x 0 0| | T W q n n
T T = q = 1 In covariant gauges , so we recover the SCET results 0| | n n W In LC gauge 0| | T q n n
+ + + ( ) ( ) ML ML d C k C k d k p k + = ( ) 2 2 ML T Ax 2 I C g i , F + + (2 ) ( + + + 2 2 d 0)(( ) 0) 0 0 k i p k i i i = ( ) ML ( ) C k 1 = + (Pres) , w Ax (Pres) , T Ax I I I , n Fey 2 All prescription dependence cancels out and gauge invariance is restored no matter what prescription is used nq 0| | T W 0| | W q n n n n Covariant Gauges In All Gauges
+ + ] 1/( = i 1 [ / ) k k Let us consider the pole part of the interesting integral with or the PV prescription. The result is 2 1 2 1 (1 ) 1 z z g 1 i = = 1 ln + i + 2 s I C dz C finite n F F i 2 2 4 4 p z 0 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 Prescription C C OR with zero-bin Subtraction!! +i0 -i0 PV 0 1 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.
2 ( nA , = , ) ~ (1 , , ) nA nA A Q 0 In the canonical quantization of of the gauge field (Bassetto et al.) ik k ( k ) nk = + + 2 a a a a ( ) ( ) ( k ) ( , ) ( ) ( , ) A k T k n nk k nk U nk k 2 2 = = a a ( ) k 0; ( ) k 0; n T k T def = ( ) + + We define ( , ) ( , , ) A x x A x x def = + + ( ) + ( , , ) ( , , ) ( , ) A x x x A x x x A x x And we can show def = ( ) + D D i i gA = D D i Ti T ( ) + = d l A exp ( , x x ) T P ig 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 inD iD iD n n 2 x inD 2 ~1/ (1,1/ ,1/ ) Q And then applying multipole expansion, n 2 2 2 ~1/ (1/ ,1/ ,1/ ) x Q us 1 n ( ) + = + + T T I L ( ) inD gnA x iD W W iD n n us n n n n n 2 in = T W TW Where n n n 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 , , ) ~ (1, ( , , ) ~ ( , , ); ( s s s nA nA A Q and soft ) ~ ) ~ Q 2 2 , ); ( ( , , , , (1, ( , , , ); ); nA nA A Q np np p np np p Q n n n n n n s s s No interaction is possible for on-shell states 1 n ( ) = + L inD iD W W iD I I n n n n n n n 2 in Is this true in every gauge?
2 2 ( ( , , , ) ~ A (1, , ); ( , ( ( , , , , ) ~ ) ~ (1, ( , , , ); nA nA A nA nA Q np np p np np p Q Q n n n n n n = , ); 0 , ) ~ ) ; Q s s s s s s The gauge ghost however acts only on some momentum components i i ( ) x A ( , ) ( ) x A (0, ) x x x n s n s i i Thus the covariant derivative is = + + ( ) s ( ) x (0 , ) iD i gA gA x n The decoupling of soft fields requires = (0) n ( ) x ( ) ( ) x T ( ) A T x A x sn n sn ( ) = d l A exp (0 , ) T P ig x l sn s 0
The new SCET-II Lagrangian is 1 n ( ) = + T T (0) n (0) n (0) n (0) (0) (0) n (0) n L inD iD W W iD I I n n 2 in = (0) n ( ) x = + = ( ) ( ) x T ( ) A T x A x sn n sn (0) n (0) n x D i gA (0) n ( ) ( ) T x sn n ( ) = d l A exp (0 , ) T P ig x l sn s 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 [Neubert et.al] PDF In SCET: [Stewart et.al] x 2 f is gauge invariant because each building block is gauge Invariant ( , )
In Full QCD And At Low Transverse Momentum: Ji, Ma,Yuan 04 = , ) 2 q 2 2 2 2 2 q z ( , , , ) Q ( , , , , ) ( , , , / F x z P e d k d p d l q x k x p z B h h B B T h h q u d s = , , ,. . + + 2 2 2 2 ( , , ) ( , , ) ( ) S l H Q z k p l P h h Is renorrmalization scale; is a rapidity cut-off Na ve Transverse Momentum Dependent PDF (TMDPDF): S Q / q Analogous to the W in SCET This result is true only in regular gauges: Here all fields vanish at infinity
Ji, Ma, Yuan Ji, Yuan Belitsky, Ji, Yuan Cherednikov, Stefanis b ( , ) b ( , ) ( 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 Gauge Invariant In LC Gauge This Gauge Link Is Built From The Transverse Component Of The Gluon Field: b ( 0 , ) ( , )
+ y ( ) y ( , ) ( ) y ( ) y T y W n n n n P n np n = (2) P | ( ) y ( ) (0)| P x p P / q P n n n n 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 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!
