Renormalization of Subleading Dijet Operators in Soft-Collinear Effective Theory
RRenormalization of Subleading Dijet Operators in Soft-CollinearEffective Theory
Simon M. Freedman ∗ and Raymond Goerke † Department of Physics, University of Toronto,60 St. George Street, Toronto, Ontario, Canada M5S 1A7
Abstract
We calculate the anomalous dimensions of the next-to-leading order dijet operators in soft-collinear effective theory (SCET). We use a formulation of SCET where the Lagrangian is multiplecopies of QCD and the interactions between sectors occur through light-like Wilson lines in externalcurrents. We introduce a small gluon mass to regulate the infrared divergences of the individualloop diagrams in order to properly extract the ultraviolet divergences. We discuss this choice ofinfrared regulator and contrast it with the δ -regulator. Our results can be used to increase thetheoretical precision of the thrust distribution. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] O c t . INTRODUCTION Perturbative calculations of jet observables involve multiple scales. In the kinematicregion where all the scales are much greater than Λ
QCD but the ratio of these scales is small,often called the “tail” region, the rate is perturbative in both the strong coupling constant α s and the ratio of the scales involved. However, the rate includes large logarithms of the ratio ofthese scales at each order in perturbation theory. These large logarithms limit the precisionof theoretical predictions. Effective field theory (EFT) techniques provide a framework tosum the terms enhanced by the logarithms using renormalization group equations (RGE).This framework also contains a systematic procedure for including higher order effects inthe small ratio of scales using subleading operators, allowing for logarithms suppressed bythis small ratio to be summed in addition to those at leading order in the ratio. Thesetechniques can be used to improve the precision of the theoretical predictions. In this paperwe renormalize the next-to-leading order dijet operators in soft-collinear effective theory(SCET) with the purpose of using the RGE to sum the logarithms suppressed by the ratioof scales. We will use the SCET operators introduced in the formulation of [1, 2], in whichthe QCD dynamics of jets are described by multiple decoupled copies of QCD, and the EFTexpansion only enters in the external currents. Our results are useful for any observablerequiring dijet operators; however, we will use the concrete example of the thrust rate toillustrate their usefulness.Thrust [3] is a useful jet shape observable for precision studies of high energy collisions,in particular for measuring α s ( M Z ) from LEP data . Thrust is defined as τ = 1 − Q max (cid:126)t (cid:88) i ∈ X (cid:12)(cid:12) (cid:126)t · (cid:126)p i (cid:12)(cid:12) , (1)where X is the final state, Q is the total energy, and (cid:126)t is chosen to maximize the sum. Theintegrated rate of the differential thrust distribution is defined by R ( τ ) = 1 σ (cid:90) τ d τ (cid:48) d σ d τ (cid:48) , (2)where the Born rate is σ . We will call this the thrust rate in the following. A perturbativecalculation of the thrust rate in the tail region where (Λ QCD /Q ) (cid:28) τ (cid:28) Q , the intermediate scale √ τ Q , and the soft scale See [4] and previous works by this collaboration. Q . The rate can be written as an expansion R ( τ ) = R (0) ( τ ) + τ R (1) ( τ ) + O ( τ ) in thisregion, where the superscripts refer to the suppression in τ , with R (0) and R (1) referringto O ( τ ) and O ( τ ) rate respectively. Each of the R ( i ) ( τ ) terms in the thrust rate has anexpansion in α s of the form R (0) ( τ ) = (cid:88) n (cid:88) m ≤ n R (0) nm α ns ln m ( τ ) ,R (1) ( τ ) = (cid:88) n (cid:88) k ≤ n − R (1) nk α ns ln k ( τ ) , (3)where the R ( i ) nm are O (1) constants and the large logarithms ln τ (cid:29) O ( τ ) rate is suppressed by an additionalpower of α s relative to the O ( τ ) rate. When α s ln τ ∼ O (1) the O ( τ ) rate becomes adivergent sum in increasing powers of ln τ , spoiling the expansion in α s ( Q ) (cid:28)
1. Althoughthe O ( τ ) rate has an overall suppression by τ compared to the O ( τ ) rate, the rate is similarlya sum in increasing powers of the logarithm. Therefore, in order to restore a perturbativeexpansion in α s for both the O ( τ ) and O ( τ ) rates, the logarithms must be summed.The O ( τ ) thrust rate has already been calculated to N LL accuracy and included thefixed order O ( τ ) rate at O ( α s ) [5]. In order to increase the theoretical precision in the tailregion, the leading logarithms in the O ( τ ) rate can become more important than furtherincreasing the logarithmic accuracy in the O ( τ ) rate. Therefore, if the precision of the α s ( M Z ) measurement is to be improved, these former contributions to the thrust rate willneed to be calculated.The appropriate EFT for describing thrust is SCET [1, 6–12]. SCET includes collinearand ultrasoft (usoft) fields that reproduce both the highly boosted and low energy degreesof freedom that are relevant in the tail region. The expansion parameter of SCET is usuallydenoted by λ . For thrust λ ∼ √ τ , meaning the O ( τ ) corrections require next-to-next-to-leading order in λ (N LO) corrections to the effective theory . We use a formulation of SCETin which QCD fields are coupled to Wilson lines [1]. Each of the sectors (usoft and collinear)interact amongst themselves via QCD, while the interactions between sectors are describedby Wilson lines in appropriate representations. This picture has been shown explicitly toN LO by doing a tree-level matching from QCD [2]. We contrast this formulation with the Unless otherwise stated, LO, NLO, and N LO refer to the expansion in λ . Q . For the O ( τ ) rate we use theLO dijet operators. The O ( τ ) rate requires the NLO and N LO dijet operators, which arethen run to the intermediate scale √ τ Q using the RGE. At the intermediate scale, the dijetoperators are matched onto soft operators with the help of a factorization theorem. TheWilson coefficients of the soft operators, often called the jet function, are run to the softscale τ Q . The sequence of matching and running sums the large logarithms in the rate.Recently, a factorization theorem has been shown for the O ( τ ) rate [2] that makes thispossible. The appropriate dijet operators and the tree-level matching coefficients were de-rived, as well as the appropriate soft operators. By solving the RGE for the operators in [2]the large logarithms in the O ( τ ) rate can be summed. In this paper we begin this processby calculating the anomalous dimensions of the NLO dijet operators in SCET. Summing allthe logarithms in the O ( τ ) rate of (3) also require the N LO dijet operators, which we leavefor future work.To compute the anomalous dimensions of the subleading effective operators we first com-pute their counterterms. We regulate using the MS scheme and include a separate infrared(IR) regulator to ensure the 1 /(cid:15) poles are ultraviolet (UV) divergences. The decouplingof the collinear and usoft sectors, manifest in the formulation of [1], means the IR cannotbe regulated using a fermion off-shellness because the usoft sector will not be changed bythis regulator. We identify two possible IR regulators that will regulate the formulation of[1]: the δ -regulator and a gluon mass. The δ -regulator [13] is similar to off-shellness butalso modifies the Feynman rules of the usoft Wilson lines. Unfortunately, the regulatorintroduces additional terms that make the calculation unnecessarily complicated. We willdemonstrate this in Section III A. A gluon mass does not introduce any additional terms,meaning fewer calculations are needed. However, this is done at the expense of introducingunregulated divergences in individual diagrams that only cancel if all the diagrams are addedtogether before integrating. Either choice of regulator is equivalent since the countertermsdo not depend on the IR regulator. We chose to use a gluon mass.The rest of the paper is organized as follows: In Section II we briefly summarize the4CET formulation of [1] and write the operators used in this calculation. We note thatit was necessary to generalize the operators of [1] in order to account for the mixing thatoccurs under renormalization. In Section III we discuss our choice of using a gluon mass asan IR regulator over the δ -regulator. We present the anomalous dimensions for the NLOoperators in Section IV and conclude in Section V. II. SCET AND NLO OPERATORS
In the kinematic region where thrust is dominated by collimated jets of light, energeticparticles, SCET is the appropriate description. It is convenient to introduce light-conecoordinates for describing the momentum of the highly boosted particles. In light-conecoordinates the momentum is decomposed into two light-like components described by thevectors n µ and ¯ n µ as p µ = p · n ¯ n µ p · ¯ n n µ p µ ⊥ . (4)The vectors n µ and ¯ n µ satisfy n = 0 = ¯ n and n · ¯ n = 2. A boosted particle with p · ¯ n ∼ Q will be described by n -collinear fields in the effective theory. Similarly, a boosted particlewith p · n ∼ Q will be described by an ¯ n -collinear field. The perpendicular momentum of acollinear particle p µ ⊥ ∼ λQ is suppressed compared to the hard scale. We must also includeusoft fields that have no large components of momentum and whose momentum scales like p µ ∼ λ Q .We follow the approach of [1] in deriving the NLO SCET dijet operators. Since particlesin the same sector have no large momentum transfers, the interactions within each sector aregoverned by QCD. Consequently, the Lagrangian has no expansion in λ and can be writtenas L SCET = L n QCD + L ¯ n QCD + L us QCD , (5)where L i QCD is the QCD Lagrangian involving only i th -sector fields. The interactions ofparticles in different sectors are described by external currents. Since these interactionsinvolve large momentum transfers, the external currents can be organized into an expansionin λ . When computing the thrust rate in the limit τ (cid:28)
1, the relevant external currents are The approach of including decoupled copies of QCD for each sector has also been used to study factor-ization in QCD [14, 15]. ψ ( x )Γ ψ ( x ) = e − iQ ( n +¯ n ) · x/ (cid:34) C (0)2 O (0)2 ( x ) + 1 Q (cid:88) i (cid:90) { dˆ t } C (1 i )2 ( { ˆ t } ) O (1 i )2 ( x, { ˆ t } ) + O ( λ ) (cid:35) (6)for a general Dirac structure Γ. The phase corresponding to the external momentum hasbeen pulled out. The superscripts in the dijet operators refer to the suppression in λ andthe 1 /Q is included because the subleading operators are higher dimensional. We haveintroduced a set of dimensionless shift variables { ˆ t } = { Qt } that were not included in [1]; itwill become apparent below that this shift corresponds to a displacement along a light-likedirection describing the position of a derivative insertion. This generalization is needed inorder to properly describe the mixing of operators under renormalization.The leading order operator in (6) is [1] O (0)2 ( x ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , (0) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) , (7)and its matching coefficient is [16, 17] C (0)2 ( µ ) = 1 + α s C F π (cid:18) − ln µ − Q − µ − Q − π (cid:19) (8)where µ is the renormalization scale. The Dirac structure isΓ (0) = P ¯ n Γ P ¯ n (9)with projectors P n = ( /n/ ¯ n ) / P ¯ n = ( / ¯ n/n ) /
4. The subscripts on the fields denote thesector of the field. Each of the square brackets in (7) are independently gauge invariant andcorresponds to a separate sector. The Wilson lines in the R representation W ( R ) n ( x, y ) = P exp (cid:32) − ig (cid:90) n · ( y − x ) / d s ¯ n · A an ( x + ¯ ns ) T a R e − s(cid:15) (cid:33) Y ( R ) n ( x, y ) = P exp (cid:32) − ig (cid:90) ¯ n · ( y − x ) / d sn · A as ( x + ns ) T a R e − s(cid:15) (cid:33) , (10)represent a light-like colour source corresponding to the total colour of the other sectors (thesymbol P indicates path-ordering). The (cid:15) in the definition above gives the proper i(cid:15) pole6rescription. The W ( R )¯ n and Y ( R )¯ n Wilson lines are defined similarly. The positions in (7) x n = (0 , x · ¯ n, x ⊥ ) x ∞ n = (0 , ∞ , x ⊥ ) x ¯ n = ( x · n, , x ⊥ ) x ∞ ¯ n = ( ∞ , , x ⊥ ) (11) x ∞ n us = ( ∞ , , x ∞ ¯ n us = (0 , ∞ , , come from multipole expanding the total momentum conservation constraint in λ and isneeded to ensure consistent power-counting at each order in λ .The NLO operators are found by including O ( λ ) corrections in the interactions betweenthe sectors [1]. The operators that describe the modification to the n -collinear sector are O (1 a n )2 ( x, t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ¯ n + ¯ nt ) iD µ ⊥ ( x ¯ n + ¯ nt ) W ( ) n ( x ¯ n + ¯ nt, x ∞ ¯ n ) (cid:3) × (cid:104) Y ( ) n ( x ∞ n us , µ (1 a n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 b n )2 ( x, t ) = (cid:2) ¯ ψ ( x ¯ n ) W ( ) n ( x ¯ n , x ¯ n + t ¯ n ) iD µ ⊥ ( x ¯ n + t ¯ n ) W ( ) n ( x ¯ n + t ¯ n, x ∞ ¯ n ) (cid:3) × (cid:104) Y ( ) n ( x ∞ n us , µ (1 b n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 B n )2 ( x ) = (cid:104) ¯ ψ ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) i ←− ∂ µ ⊥ (cid:105) × (cid:104) Y ( ) n ( x ∞ n us , µ (1 b n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 c n )2 ( x, t , t ) = (cid:104) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ¯ n + t ¯ n ) i ←− D µ ⊥ ( x ¯ n + t ¯ n ) W ( ) n ( x ¯ n + t ¯ n, x ∞ ¯ n ) (cid:105) × (cid:104) Y ( ) n ( x ∞ n us , t n ) i ←− D µ ⊥ ( t n ) Y ( ) n ( t n, (1 c n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 d n )2 ( x, t ) = (cid:2) ig ¯ n µ G aµνn ⊥ ( x ¯ n ) W ( ) n ab ( x ¯ n , x ∞ n ) (cid:3) (12) × (cid:104) Y ( ) n bc ( x ∞ n us , tn ) ¯ ψ s ( tn ) T c Y ( ) n ( tn, ν (1 d n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 e n )2 ( x, t ) = (cid:2) ig ¯ n µ G aµνn ⊥ ( x ¯ n ) W ( ) n ab ( x ¯ n , x ∞ n ) (cid:3) (cid:104) Y ( ) n d ˆ d ( x ∞ n us , Y ( )¯ n ˆ dc (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n cb ( x ∞ n , x n + tn ) ¯ ψ ¯ n ( x n + tn ) T b Γ (1 e n ) W ( ) n ( x n + tn, x n ) ψ ¯ n ( x n ) (cid:105) O (1 δ )2 ( x ) = Q (cid:2) ¯ ψ n ( x ¯ n ) x µ ⊥ W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) × (cid:104) Y ( ) n ( x ∞ n us , (1 δ ) ( D µ ⊥ + ←− D µ ⊥ )(0) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) , µ (1 a n ) = P ¯ n Γ γ µ /n µ (1 b n ) = / ¯ n γ µ Γ P ¯ n Γ (1 c n ) = P ¯ n Γ P ¯ n Γ µ (1 d n ) = /n γ µ ⊥ Γ P ¯ n Γ µ (1 e n ) = /n γ µ ⊥ Γ P ¯ n Γ (1 δ ) = P ¯ n Γ P ¯ n . (13)The covariant derivative is defined as D µ ( x ) = ∂ µ − igT a A aµ ( x ) and only couples the gluon tothe corresponding sector on which it acts. The field strength tensor is defined as igG aµν = f abc [ A bµ , A cν ] where f abc are the SU (3) structure constants. The derivative in the (1 B n )operator is strictly a partial derivative and not a covariant derivative because we are workingin a covariant gauge where the gauge transformations at infinity vanish. The Q in front ofthe (1 δ ) operator is required dimensionally. The matching coefficients for the operatorslisted above are [1] C (1 a n )2 (ˆ t ) = − δ (ˆ t ) + O ( α s ) C (1 b n )2 (ˆ t ) = δ (ˆ t ) + O ( α s ) C (1 B n )2 = 1 + O ( α s ) C (1 c n )2 (ˆ t , ˆ t ) = 2 iθ (ˆ t ) δ (ˆ t ) + O ( α s ) (14) C (1 d n )2 (ˆ t ) = − iθ (ˆ t ) + O ( α s ) C (1 e n )2 (ˆ t ) = iθ (ˆ t ) + O ( α s ) C (1 δ )2 (ˆ t ) = 1 + O ( α s ) , which are all dimensionless. The factors of i ensure the convolution in (6) is real.The NLO operators explicitly decouple the sectors, just as in the LO operator. Theseoperators differ from the LO operators by a D ⊥ insertion at an arbitrary point along a Wilsonline (for example the (1 a n ) operator) or by a change in the field content and Wilson linerepresentation (for example the (1 e n ) operator). The operators in (12) are generalizations ofthe NLO operators in [1, 2]. We find the form in (12) is necessary to properly renormalize theoperators, since different values of the parameters can mix under renormalization. We havealso slightly changed the definition of the (1 b n ) operator and included the (1 B n ) operator,which makes the operator basis in (12) diagonal under renormalization.As was done in [1], we can compare the operators in (12) with the subleading operatorsin other formulations of SCET, such as in [18]. In [18] the subleading heavy-to-light currentswere renormalized. While the dijet and heavy-to-light operators obviously differ in the usoftand ¯ n -collinear sectors, the modifications to the n -collinear sector from the vector currentsand subleading Lagrangian insertions in [18] only differ from the corresponding operators in(12) by the appropriate Dirac structure basis. This will serve as a way for us to comparethe anomalous dimensions we calculate in Section IV with the results of [18].8e find it more convenient to work with the Fourier transformed operators ˜ O ( i )2 definedas ˜ O (1 i )2 ( x, u ) = (cid:90) dˆ t (2 π ) e − iu ˆ t O (1 i )2 ( x, ˆ t ) = Q (cid:90) d t (2 π ) e − iQut O (1 i )2 ( x, t )˜ O (1 i )2 ( x, u , u ) = (cid:90) dˆ t (2 π ) dˆ t (2 π ) e − i (ˆ t u +ˆ t u ) O (1 i )2 ( x, ˆ t , ˆ t ) . (15)The matching in (6) is written in terms of these operators as (cid:90) d { ˆ t } C (1 i )2 ( { ˆ t } ) O (1 i )2 ( { ˆ t } ) = (cid:90) d { u } ˜ C (1 i )2 ( { u } ) ˜ O (1 i )2 ( { u } ) (16)where ˜ C (1 i )2 ( u ) = (cid:90) dˆ t e iu ˆ t C (1 i )2 (ˆ t )˜ C (1 i )2 ( u , u ) = (cid:90) dˆ t dˆ t e i ( u ˆ t + u ˆ t ) C (1 i )2 (ˆ t , ˆ t ) . (17)The u ’s are momentum fractions at the vertex of the external current. For collinear mo-mentum 0 ≤ u ≤ ≤ u < ∞ because usoft momentum is not conserved at the vertex. The Fourier transformation of theNLO operators are˜ O (1 a n )2 ( x, u ) = (cid:104) ¯ ψ n ( x ¯ n ) δ ( u − in · ˆ D ) iD µ ⊥ ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:105) × (cid:104) Y ( ) n ( x ∞ n us , µ (1 a n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) ˜ O (1 b n )2 ( x, u ) = (cid:104) ¯ ψ ( x ¯ n ) δ ( u − in · ˆ D ) iD µ ⊥ ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:105) × (cid:104) Y ( ) n ( x ∞ n us , µ (1 b n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) ˜ O (1 c n )2 ( x, u , u ) = (cid:104) ¯ ψ n ( x ¯ n ) δ ( u − in · ˆ D ) i ←− D µ ⊥ ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:105) × (cid:20) Y ( ) n ( x ∞ n us , i ←− D µ ⊥ (0) δ ( u − in · ←− ˆ D )Γ (1 c n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:21) × (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) ˜ O (1 d n )2 ( x, u ) = (cid:2) ig ¯ n µ G aµνn ⊥ ( x ¯ n ) W ( ) n ab ( x ¯ n , x ∞ n ) (cid:3) (18) × (cid:20) Y ( ) n bc ( x ∞ n us ,
0) ¯ ψ s (0) T c δ ( u − in · ←− ˆ D )Γ ν (1 d n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:21) × (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) ˜ O (1 e n )2 ( x, u ) = (cid:2) ig ¯ n µ G aµνn ⊥ ( x ¯ n ) W ( ) n ab ( x ¯ n , x ∞ n ) (cid:3) (cid:104) Y ( ) n d ˆ d ( x ∞ n us , Y ( )¯ n ˆ dc (0 , x ∞ ¯ n us ) (cid:105) × (cid:20) W ( )¯ n cb ( x ∞ n , x n ) ¯ ψ ¯ n ( x n ) T b Γ (1 e n ) δ ( u − in · ←− ˆ D ) ψ ¯ n ( x n ) (cid:21) D µ = D µ /Q is a dimensionless covariant derivative. The tree-level matching coeffi-cients up to O ( α s ) corrections are˜ C (1 a n )2 ( u ) = − C (1 b n )2 ( u ) = 1˜ C (1 c n )2 ( u , u ) = − u ˜ C (1 d n )2 ( u ) = 2 u ˜ C (1 e n )2 ( u ) = − u (19)The (1 B n ) and (1 δ ) are independent of ˆ t so are not transformed. A. Constraining the NLO Operators
We restrict ourselves to the electromagnetic current Γ = γ λ in this paper. This currentis both CP invariant and conserved. We will show how we can exploit these two propertiesto constrain the NLO SCET operators. We will also show how we can use the ambiguity indefining the n µ and ¯ n µ directions to make further constraints.First we use CP invariance to expand the list of operators to include corrections to the¯ n -collinear sector. The action of CP is equivalent to switching n and ¯ n and then taking thecomplex conjugate. Therefore, the NLO corrections to the ¯ n -collinear sector can be obtained10or free from the operators in (12). The operators are O (1 a ¯ n )2 ( x, t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , µ (1 a ¯ n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n ( x ∞ n , x n + tn ) i ←− D µ ⊥ ( x n + nt ) W ( )¯ n ( x n + nt, x n ) ψ ¯ n ( x n ) (cid:105) O (1 b ¯ n )2 ( x, t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , µ (1 b ¯ n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n ( x ∞ n , x n + tn ) i ←− D µ ⊥ ( x n + nt ) W ( )¯ n ( x n + nt, x n ) ψ ¯ n ( x n ) (cid:105) O (1 B ¯ n )2 ( x ) = (cid:2) ¯ ψ ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , µ (1 b ¯ n ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) × (cid:104) i∂ ⊥ µ W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) O (1 c ¯ n )2 ( x, t , t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , (1 c ¯ n ) Y ( )¯ n (0 , t ¯ n ) D µ ⊥ ( t ¯ n ) Y ( )¯ n ( t ¯ n, x ∞ ¯ n us ) (cid:105) × (cid:104) W ( )¯ n ( x ∞ n , x n + t n ) iD µ ⊥ ( x n + t n ) W ( )¯ n ( x n + t n, x n ) ψ ¯ n ( x n ) (cid:105) O (1 d ¯ n )2 ( x, t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) × (cid:104) Y ( ) n ( x ∞ n us , ν (1 d ¯ n ) Y ( )¯ n (0 , t ¯ n ) T c ψ s ( t ¯ n ) Y ( )¯ n bc ( tn, x ∞ ¯ n us ) (cid:105) (20) × (cid:104) ign µ G aµν ¯ n ⊥ ( x n ) W ( )¯ n ab ( x n , x ∞ ¯ n ) (cid:105) O (1 e ¯ n )2 ( x, t ) = (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ¯ n + t ¯ n )Γ (1 e ¯ n ) T b ψ n ( x ¯ n + t ¯ n ) W ( ) n bc ( x ¯ n + t ¯ n, x ∞ ¯ n ) (cid:3) × (cid:104) Y ( ) n c ˆ d ( x ∞ n us , Y ( )¯ n ˆ dd (0 , x ∞ ¯ n us ) (cid:105) (cid:104) ign µ G aµν ¯ n ⊥ ( x n ) W ( )¯ n ad ( x n , x ∞ ¯ n ) (cid:105) with Dirac structuresΓ µ (1 a ¯ n ) = Γ µ (1 b n ) Γ µ (1 b ¯ n ) = Γ µ (1 a n ) Γ (1 c ¯ n ) = Γ (1 c n ) Γ µ (1 d ¯ n ) = P ¯ n γ µ ⊥ Γ ¯ n/ µ (1 e ¯ n ) = P ¯ n γ µ ⊥ Γ ¯ n/ . (21)The (1 δ ) is already CP invariant since the x µ ⊥ can be moved into either collinear sector. CP invariance guarantees the matching coefficients of the (1 i n ) and (1 i ¯ n ) are equal C (1 i n )2 = C (1 i ¯ n )2 (22)for i = { a, b, c, d, e, B } . The Fourier transform of the operators in (20) are similar to thosein (18), and we avoid writing them down for the sake of brevity. In the following we will use CP invariance to avoid talking about the (1 i ¯ n ) operators unless it is necessary.Next, we can exploit the conservation of the electromagnetic current ∂ λ ¯ ψ ( x ) γ λ ψ ( x ) = 0.As was discussed in [19], the EFT dijet operators must also be conserved at each order in11 . The only operators in (12) that are not conserved by themselves are the (1 a n ), (1 b n ),and (1 B ( n, ¯ n ) ) operators. All the other NLO operators are conserved up to O ( λ ). Therefore,conservation of the current requires C (1 a n )2 = − C (1 b n )2 C (1 B n )2 = C (1 B ¯ n )2 (23)to all orders in α s .Finally, we can exploit Reparameterization Invariance (RPI) [20, 21] to constrain thematching coefficients. RPI has been discussed extensively for heavy-to-light currents in thetraditional SCET formulations [18, 22] but has not previously been discussed in the SCETformulation we use. However, insight can be drawn from the traditional SCET formulationsdue to the equivalence of the two approaches.The n -collinear fields represent particles boosted in the n µ direction, where n µ is a vectorwe specify when matching from QCD onto SCET. The ¯ n -collinear particles are describedsimilarily. However, an n -collinear particle does not travel exactly along the n µ direction,and will generically have a momentum perpendicular to n µ of order λ . In fact, we couldhave chosen a slightly different n µ , for example n (cid:48) µ = n µ + (cid:15) µ ⊥ , (24)where (cid:15) ⊥ ∼ O ( λ ). In this case an n -collinear particle also appears to be boosted along the n (cid:48) µ direction and has relative perpendicular momentum of order λ . Therefore, it should notmatter to the physical result whether we include an n -collinear sector or an n (cid:48) µ -collinearsector. We can make use of this equivalence by applying the variation n µ → n µ + (cid:15) µ ⊥ to theoperators in the n -collinear sector and enforcing that they cancel order-by-order in λ . Thisprovides constraints on the matching coefficients that must hold to all orders in α s .Using the equation of motion for a Wilson line n · DW ( R ) n = 0 and a fermion /Dψ = 0,the variation of the LO operator is O (0)2 ( x ) n µ → n µ + (cid:15) µ ⊥ −−−−−−→ O (0)2 ( x )+ (cid:2) ¯ ψ n ( x ¯ n ) W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:3) (cid:104) Y ( ) n ( x ∞ n us , δ (Γ (0) ) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) + (cid:20) ¯ ψ n ( x ¯ n ) (¯ n · x ) (cid:15) ⊥ µ W ( ) n ( x ¯ n , x ∞ ¯ n ) (cid:21) (cid:104) Y ( ) n ( x ∞ n us , (cid:16) D µ + ←− D µ (cid:17) Γ (0) Y ( )¯ n (0 , x ∞ ¯ n us ) (cid:105) (25) × (cid:104) W ( )¯ n ( x ∞ n , x n ) ψ ¯ n ( x n ) (cid:105) + O ( λ ) , We would like to thank Ilya Feige and Ian Moult for this observation δ (Γ (0) ) = / ¯ n /(cid:15) ⊥ P ¯ n . (26)Only the left projector is transformed because the Dirac structure is Γ (0) = P ¯ n Γ P n where n µ and n µ are the light-like directions of the two sectors. However, matching enforces that n ≡ n = ¯ n , so the transformed projector reduces to (26).It is straightforward to show that the (1 δ ) and (1 B n ) operators cancel the variations in(25) if their matching coefficients are constrained to be C (0)2 = C (1 δ )2 = C (1 B n )2 (27)to all orders in α s . We note this is similar to what was found in [18] for heavy-to-lightcurrents.We will check the constraints in (23) and (27) when we renormalize the NLO operators.The anomalous dimensions, being the kernels of a linear integro-differential equation, areexpected to be equal if two operators are constrained to be the same up to a multiplicativeconstant. We will see this in Section IV. III. INFRARED REGULATOR
In order to extract counterterms from loop diagrams we must be able to differentiatebetween UV and IR divergences. Introducing a small mass scale to serve as an IR cut-offallows us to regulate the IR separately from dimensional regularization and ensures that allthe 1 /(cid:15) poles in the loop integrals are UV divergences. A common scheme is to introducea small fermion off-shellness, as was done in [6, 16]. However, in the SCET approach of[1] where the sectors explicitly decouple, a fermion off-shellness leaves the Wilson linesunchanged and will not properly regulate the usoft sector of the LO operator . A regulatorthat produces similar results to a fermion off-shellness is the δ -regulator [13]. The δ -regulatormodifies the Feynman rules of both the usoft and collinear sectors thereby regulating the IRof the SCET approach we use in this paper. However, when there is more then one externalleg in a single sector, the δ -regulator introduces extra terms that complicate the calculation.Using a gluon mass to regulate the IR avoids these additional terms, although the individual In the traditional approach to SCET [6–10] the LO operator does not explicitly decouple until after afield redefinition, which does not affect the counterterms. p = p p ≡ (a) I n = ≡ (b) I us = ≡ (c) I ¯ n FIG. 1: Relevant graphs for the renormalization of the O (0)2 operator. Solid lines anddashed lines are fermions and Wilson lines respectively. We decompose the diagram on theleft into the contribution from each sector in the middle three diagrams. We can alsocompactify the notation by only showing the sector that has the one-loop contribution, asshown on the right.diagrams will contain unregulated divergences, which cancel in the total sum of diagrams.We have chosen to use a gluon mass as our IR regulator, and in this section we will contrastsome of the details of the two approaches.In this and following sections we will use a condensed notation for representing the Feyn-man diagrams considered in our calculations. As an example to illustrate the notation, Fig-ure 1 shows the diagrams for n -collinear quark and ¯ n -collinear anti-quark production usingthe LO dijet operator. This notation becomes especially useful when considering subleadingoperators with a gluon in the final state, as the number of diagrams grows considerably.14 (a) I n (b) I ¯ n (c) I us FIG. 2: One-loop diagrams for O (0)2 with an external n -collinear gluon, using the compactnotation of Figure 1. A. The Delta Regulator
The δ -regulator was introduced when considering SCET with massive gauge particlesto help make the loop integrals of individual diagrams converge [13]. The construction issimilar to using a fermion off-shellness and can be used to regulate the IR of the SCETformulation of [1]. This makes it an obvious choice for regulating the NLO operators in(12). However, the δ -regulator requires extra terms when there is more than one externalleg that do not appear when using a gluon mass to regulate the IR.As an example to show where these extra terms arise, we renormalize the LO dijetoperator with an ¯ n -collinear anti-quark and an n -collinear quark and gluon in the finalstate. The diagrams are shown in Figure 2. The δ -regulator regulates the IR by insertinga small mass term into the Lagrangian for each field. The particles are brought off-shell bymaintaining the massless equations of motion p = 0. The Feynman rules for the Wilson linesare also modified to incorporate this off-shellness. The Feynman rules for the propagatorsand Wilson lines are [13] 1( p i + k ) − ∆ i and ¯ n αi T a R j k · ¯ n i − δ j,n (28)respectively. The momentum of the internal particle is k and ∆ i is the mass term insertedinto the Lagrangian. The Feynman rule for the Wilson line is for a particle in the j th -sector with colour T R j emitting a particle in the i th -sector . The shift in the Wilson line is δ j,n = (2∆ j ) / (( n i · n j )(¯ n j · p i )). The regulator naively breaks the explicit decoupling into We note the colour structure was not in the original δ -regulator definition but is necessary when lookingat O ( g ) processes. ig κ (cid:15) (cid:90) d d k k − ∆ g )(¯ n · k + δ ¯ q, ¯ n ) (cid:18) C F + C A n · k − δ q,n − C A n · k − δ g,n (cid:19) , (29)where κ (cid:15) = ( µ e γ E ) (cid:15) / (2 π ) d . The extra C A terms account for the internal usoft gluon beingemitted by or before the external n -collinear gluon. These extra terms are necessary tocancel all the mixed UV/IR divergences from the n -collinear diagrams. The ¯ n -collineardiagram will also require extra diagrams. However, as expected, the final result reproducesthe expected LO anomalous dimension and is very similar to using a fermion off-shellnessin a theory that does not decouple usoft and collinear fields. B. Gluon mass
Another scheme that can be used to regulate the IR is to introduce a small gluon mass.Unfortunately, massive bosons introduce an obstacle in SCET: the individual diagrams areoften unregulated in dimensional regularization [13]. However, the sum of all the diagramsfrom a particular operator must still be well-regulated by a gluon mass [13]. As an example,we show how a gluon mass can be used to calculate the anomalous dimension of the LOoperator. The necessary diagrams are shown in Figure 1. The n -collinear diagram gives theintegral I n = 2 ig C F κ (cid:15) (cid:90) d d k ¯ n · ( p − k )( k − M )( p − k ) (¯ n · k )= − g C F κ (cid:15) π d/ Γ( (cid:15) ) M − (cid:15) (cid:90) p − d k − k − (cid:18) − k − p − (cid:19) − (cid:15) (30)where M is the gluon mass. The second line above is found by doing the k + integral bycontours and then the k ⊥ integral. The final integral diverges as k − → I us = 2 ig C F κ (cid:15) (cid:90) d d k k − M )( n · k )( − ¯ n · k )= − g C F κ (cid:15) π d/ Γ( (cid:15) ) M − (cid:15) (cid:90) ∞ d k − k − (31)16fter doing the same integrals as the n -collinear diagram. This integral diverges as k − → ∞ . The ¯ n -collinear diagram gives the integral I ¯ n = 2 ig C F κ (cid:15) (cid:90) d d k n · ( p − k )( k − M )( p − k ) (¯ n · k )= 2 g C F κ (cid:15) π d/ Γ( (cid:15) ) (cid:18)(cid:90) ∞ d k − p +2 M + k − p +2 ( − M − (cid:15) + ( − k − p +2 ) − (cid:15) ) + M − (cid:15) − (cid:15) (cid:19) (32)again doing the same integrals as the n -collinear diagram. The first term above diverges as k − → ∞ . As usual, we must also subtract a zero-bin I no / = I us = I ¯ no / for each of the collinearsectors [23]. Therefore, the sum of the diagrams is I n + I ¯ n − I us . (33)Each of the divergences in the above integrals cancel in the sum and we can find the anoma-lous dimension γ = α s C F π (cid:18) ln − Q µ − (cid:19) . (34)This is the well-known result for the anomalous dimension of the LO dijet operator [16].Although the δ -regulator would avoid unregulated divergences in intermediate steps, itrequires keeping track of additional terms. We chose to calculate the counterterms using agluon mass and expect a δ -regulator to give the same results. IV. ANOMALOUS DIMENSIONS
In order to run the NLO Wilson coefficients in (14) from the high scale Q to any otherscale below Q , we must solve the RGE. To do so we must renormalize the NLO operatorsand calculate their anomalous dimensions.The renormalized operators ( R ) and bare operators ( B ) are related by˜ O (1 i )2 ( B ) ( µ ; x, { u } ) = (cid:88) j (cid:90) { d v } Z ij ) ( µ ; { u, v } ) ˜ O (1 i )2 ( R ) ( x, { v } ) (35)where Z ij ) is the counterterm matrix extracted from the UV divergences of the Green’sfunctions of the operator. In general, the continuous set of operators can mix within eachlabel u and with other operators j . The independence of µ of the renormalized operatorsleads to an integro-differential equation for the bare operatorsdd ln µ ˜ O (1 i )2 ( B ) ( µ ; x, { u } ) = − (cid:88) j (cid:90) { d v } γ ij ) ( µ ; { u, v } ) ˜ O (1 j )2 ( B ) ( µ ; x, { v } ) . (36)17he anomalous dimension is calculated from the counterterms γ ij ) ( µ ; { u, v } ) = − (cid:88) k (cid:90) { d w } Z − ik ) ( µ ; { u, w } ) dd ln µ Z kj ) ( µ ; { w, v } ) . (37)The corresponding equation for the Wilson coefficientsdd ln µ ˜ C (1 i )2 ( µ ; { u } ) = (cid:88) j (cid:90) { d v } ˜ C (1 j )2 ( µ ; { v } ) γ ij ) ( µ ; { v, u } ) (38)is the RGE that must be solved.The operators in (18) are written in a diagonal basis in i, j up to O ( α s ) correctionsmeaning Z ij ) = Z i ) if i = j i (cid:54) = j . (39)The counterterms can be written perturbatively as Z i ) ( µ ; { u, v } ) = δ ( { u − v } ) + (cid:16) α s π (cid:17) Z (1)2(1 i ) ( µ ; { u, v } ) + O ( α s ) . (40)The anomalous dimension will also be diagonal in i, j and the lowest order contribution willbe γ i ) ( µ ; { u, v } ) = 2 (cid:18) α s (cid:15) ∂∂α s − ∂∂ ln µ (cid:19) Z (1)2(1 i ) ( µ ; { u, v } ) . (41)The first term comes from the renormalization of the coupling constant g ( R ) = g ( B ) µ − (cid:15) . Wewill suppress the explicit dependence on µ in the anomalous dimension for the sake of moreconcise notation.The diagrams for the calculation of the anomalous dimensions of the NLO operators areshown in Figure 3. We must consider a gluon in the final state for most of the operatorsas these operators have a gluon in the final state at tree-level. The (1 B n ) operator canbe renormalized in a frame where the total perpendicular collinear momentum is non-zeroand it has the same diagrams as the LO operator in Figure 1. We use the backgroundfield method [24] to maintain gauge invariance under renormalization. The backgroundfield method ensures Z g = Z − / A , which properly renormalizes the derivative insertions andthe Wilson lines. Extracting the UV divergences from the diagrams lead to the following18 (a) O (1 a n )2 and O (1 b n )2 (b) O (1 δ )2 (c) O (1 c n )2 (d) O (1 d n )2 (e) O (1 e n )2 FIG. 3: Diagrams for the NLO operators. Each bracket represents the one-loop graphfrom a separate sector. Going from left to right, the diagrams are the n -collinear, usoft,and ¯ n -collinear sectors. The box vertex represents the derivative insertion.19nomalous dimensions γ (1 a n ) ( u, v ) = α s δ ( u − v ) θ (¯ v ) π (cid:18) C F (cid:18) ln − Q µ −
32 + ln ¯ v (cid:19) + C A (cid:19) + α s π (cid:18) C F − C A (cid:19) ¯ u (cid:18) θ (1 − u − v ) uv ¯ u ¯ v + θ (¯ u ) θ (¯ v ) θ ( u + v − uv + u + v − uv (cid:19) − α s C A π ¯ u (cid:18) θ (¯ u ) θ ( u − v ) ¯ v − uvu ¯ v + θ (¯ v ) θ ( v − u ) ¯ u − uvv ¯ u + 1¯ u ¯ v (cid:20) ¯ uθ (¯ u ) θ ( u − v ) u − v + ¯ vθ (¯ v ) θ ( v − u ) v − u (cid:21) + (cid:19) γ (1 b n ) ( u, v ) = γ (1 a n ) ( u, v ) γ (1 B n ) = α s C F π (cid:18) −
32 + ln − Q µ (cid:19) = γ (0) (42) γ (1 c n ) ( u ; u , v ) = α s δ ( u − v ) δ ( u ) π (cid:18) C F (cid:18) −
32 + ln − Q µ (cid:19) + C A v (cid:19) − α s C A δ ( u ) π (cid:18)(cid:20) θ ( v − u ) θ ( u ) v − u + θ ( u − v ) θ ( v ) u − v (cid:21) + − θ ( u − v ) u − θ ( v − u ) v (cid:19) γ (1 d n ) ( u, v ) = α s δ ( u − v ) π (cid:18) − C F C A (cid:18) ln − Q µ + ln( v ) − (cid:19)(cid:19) − α s π (cid:18) C F − C A (cid:19) v (cid:20) vθ ( u − v ) θ ( v ) u − v + uθ ( v − u ) θ ( u ) v − u (cid:21) + γ (1 e n ) ( u, v ) = α s δ ( u − v ) θ (¯ v ) π (cid:18) C F C A (cid:18) ln − Q µ + ln( v ) − (cid:19)(cid:19) − α s π (cid:18) C F − C A (cid:19) v ¯ v (cid:18) θ (¯ v ) θ ( v − u ) u ¯ v + θ (¯ u ) θ ( u − v ) v ¯ u + (cid:20) ¯ uvθ (¯ u ) θ ( u − v ) u − v + u ¯ vθ (¯ v ) θ ( v − u ) v − u (cid:21) + (cid:19) , where ¯ u = 1 − u and ¯ v = 1 − v . We have used a generalized symmetric plus-distribution firstintroduced in [18], which was denoted by square brackets as in [ ] + . The formal definitionof this distribution is[ θ ( u − v ) q ( u, v ) + θ ( v − u ) q ( v, u )] + ≡ − lim β → ddu (cid:20) θ ( u − v − β ) (cid:90) vu dw q ( w, v ) + θ ( v − u − β ) (cid:90) u dw q ( v, w ) (cid:21) , (43)which is the same as the distribution defined in [18] when u, v ≤
1. The above definition isalso valid when u, v >
1, which was not required in [18]. Equation (42) is our main result.20e can compare the results for γ a n ) with [18]. The (1 a n ) operator in (12) is similarto the NLO vector heavy-to-light current in [18]. As expected, the anomalous dimensionfor these two operators are the same for the non-diagonal terms. They only disagree inthe diagonal terms by the difference of the LO dijet and heavy-to-light operator, which isexpected. Also, the anomalous dimensions of the (1 a n ) and (1 b n ) operators are the same,as expected from current conservation in (23). We can also check that γ i ¯ n ) = γ i n ) asexpected from CP invariance. Finally, the (1 B n ), (1 δ ), and (0) operators all have the sameanomalous dimension as expected from RPI.A final check is to compare the anomalous dimension of the (1 e n ) and (1 d n ) operators.From (12) we see the (1 d n ) operator is the limit of the (1 e n ) operator when the quarkbecomes usoft. Therefore, we expect in the limit where u ∼ λ ∼ v in the (1 e n ) anomalousdimension to recover the (1 d n ) anomalous dimension. This is indeed the case as seen in (42) . The NLO operators have a cusp in the usoft light-like Wilson lines at x µ = 0. Therefore,the anomalous dimension depends on at most a single logarithm and can be written in theform γ i ) ( µ ; u, v ) = δ ( u − v )Γ C(1 i ) ( α s ) ln − Q µ + γ NC(1 i ) ( α s ; u, v ) . (44)The coefficient of the logarithm is proportional to the universal cusp anomalous dimensionΓ cusp ( α s ) [25], which means it is possible to perform NLL summation without going to higherloops. This universal form of (44) is confirmed up to O ( α s ) corrections in (42).Obviously, solving the RGE analytically is straightforward for the (1 B n ) and (1 δ ) op-erators because the anomalous dimension is the same as the LO dijet operator. However,solving the RGE analytically for the other operators is more difficult. The non-diagonalterms in the (1 a n ) and (1 b n ) RGE were solved in [18] by exploiting that the non-diagonalterms in the anomalous dimensions can be written as f ( u, v ) S ( u, v ) where S ( u, v ) is a sym-metric function. For example, f ( u, v ) = ¯ u for the (1 a n ) operator and 1 / ( v ¯ v ) for the (1 e n )operator. The authors of [18] were able to expand in an infinite set of Jacobi polynomialswith the appropriate weight functions in order to diagonalize the anomalous dimensions andsolve the RGE. We expect that a similar solution will work for the (1 a n ), (1 b n ) and (1 e n )operators. However, the (1 c n ) and (1 d n ) operators are qualitatively different due to the This limit must be taken carefully, since the u → O ( λ ) limit does not commute with the limit in thedefinition of the plus distribution. V. CONCLUSION
In order to increase the accuracy of the α s ( M Z ) measurement the O ( τ ) corrections arebecoming important. Just like for the O ( τ ) rate, the O ( τ ) rate includes large logarithmsthat must be summed. We describe how this can be done using SCET and the factorizationtheorem in [2]. The required operators in the O ( τ ) factorization theorem must be renormal-ized so they can be run from the hard scale to the usoft scale. The running can be done intwo stages. First the NLO and N LO dijet operators in SCET must be renormalized. Theseoperators are then run from the hard scale to the intermediate scale. In the next step, thesoft operators introduced in [2] will be renormalized and run from intermediate scale to theusoft scale. This sequence of running and matching will sum all the large logarithms in the O ( τ ) rate.In this paper, we have started the first step by renormalizing the NLO dijet operators.Although we have used thrust as a concrete example of an application, our results is ap-plicable to any observable requiring dijet operators. Because we use the SCET formulationof [1] we cannot use fermion off-shellness to regulate the IR. Instead we have used a gluonmass, which leads to individual diagrams being unregulated. However, the sum of all thediagrams from a given operator is well-defined, as expected. The UV divergences are ex-tracted by looking at the 1 /(cid:15) poles allowing us to calculate the anomalous dimensions of theNLO operators. We have checked our results with similar operators for the heavy-to-lightcurrents in [18] and find good agreement.We leave renormalizing the N LO dijets operators and the soft operators to future work.Although we have calculated the anomalous dimensions of the NLO operators, and inves-tigated the possibility of solving the RGE analytically, we believe that it may be morepractical to solve it numerically, which we leave for future work.22
CKNOWLEDGMENTS
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