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JHEP05(2018)142 Springer May 14, 2018 May 23, 2018 : : January 2, 2018 : on function at one- Accepted Published Received r renormalization, and the lization of gluon operators. istribution functions contain mology, ity, osmology, t the gluon distribution in large ivergences in the cut-oﬀ scheme. to the newly introduced operators. Published for SISSA by https://doi.org/10.1007/JHEP05(2018)142 [email protected] , . 3 1712.09247 The Authors. c Perturbative QCD, Lattice QCD

Recent perturbative calculation of quasi gluon distributi , [email protected] INPAC, Shanghai Key Laboratory forMOE Particle Key Physics Laboratory for and Particle Cos School of Physics, Physics Astrophysics and and800 C Astronomy, Dongchuan Shanghai RD, Jiao Shanghai, Tong Univers 200240, China E-mail: Open Access Article funded by SCOAP Abstract: Wei Wang and Shuai Zhao On the power divergencefunction in quasi gluon distribution Keywords: ArXiv ePrint: linear divergences in quasi distribution functionAfter can including be the in mixing, we findonly the logarithmic improved divergences, quasi and gluon thus can d momentum be effective used theory. to extrac loop level shows the existenceWe employ of the extra auxiliary linear field ultravioletThe approach, d non-local and study gluon the operator renorma can mix with new operators unde JHEP05(2018)142 1 2 4 9 17 18 20 es ] and the “lattice aMET), has been ]. In this approach, 46 ]. These divergences ]. Another popular 7 istribution functions – 8 4 , – 6 40 1 ff scheme show that, the red due to the technical er, since PDFs are defined nt, and can be calculated n not be calculated in QCD shed [ rgences [ ing the experimental data for e lattice, defined by equal-time ution function is evaluated on rvables share the same infrared ions. This approach has been tion has been examined at one- lculation [ ]. Therefore, if the matching coefficient 9 , 8 ], it is a formidable task to calculate the PDFs – 1 – 5 ] are also developed. 48 , 47 ], and some encouraging progresses on LQCD calculation are , 9 30 – 10 ]. Some other frameworks, e.g., the pseudo-PDFs [ 39 – 31 A novel approach, named as large momentum effective theory (L The one-loop calculations of matching coefficient in the cuto proposed to extract light-cone observables from lattice ca studied extensively [ reported [ cross section” approach [ quasi quark PDF contains linear power ultraviolet (UV) dive correlation matrix elements. The(IR) quasi and structures. light-cone obse Theperturbatively. difference between Factorization of them theloop is quasi IR level quark and independe distribu provedis to calculated hold in to perturbationthe all theory, lattice, orders and one [ the can quasi distrib extract light-cone distribution funct on the lattice.difficulties. Instead only the lowest few moments are explo one can calculate the corresponding quasi observables on th non-perturbative method is to useas lattice light-cone QCD correlation (LQCD). matrix Howev elements [ (PDFs). Because of theperturbation non-perturbative nature, theory. the PDFs Traditionally, ca the PDFs are processes extracted in us which the factorization theorem is establi 1 Introduction The high energy behaviour of a hadron is encoded into parton d 6 Summary and outlook A Cancelation of the linear divergences in no-Wilson line diagrams 3 One-loop corrections to quasi4 gluon distribution Auxiliary field method and5 renormalization of linear Matching divergenc coefficient for improved gluon distribution Contents 1 Introduction 2 Definition of quasi gluon distribution JHEP05(2018)142 . ]. 5 54 ], as (2.2) (2.1) , 51 ), with 53 ⊥ . ,~a i . − i P | P , a | + ve some detailed (0) a (0) , the renormaliza- + i 4 iz G G = ( ) quantity especially for ) + z a n n , we introduce a slightly rk distribution, the quasi L 2 L ]. gluon distribution. We will inear divergences can not be that linear divergences exist 0; hing coefficient in section 0; cal matrix element which is 52 , , ergences can be renormalized o all orders recently [ , we present the perturbative ld method. Then we propose gluon quasi PDF is much more z In section ved. 0) is a light-cone vector. Sim- , 3 + ing, as well as the mass counter erturbative coefficient for quasi , energy diagram. The renormal- n lso crucial for understanding the nvariant momentum-subtraction ve been made to solve the prob- 0 F, one can perform nonperturba- 37 zn − , ( , ion is defined by a non-local matrix ξ 1 ( , W 35 ) , W z ) 24 + = (0 zn is expressed as ( n i + z − a n ξ G ], a mass counter term of Wilson line [ ( | i + 2. P 50 – 2 – h G , | √ z ]. All these methods are based on the fact that / ], in which the results are presented in both the cut- P 49 ) 52 h z 56 izxP + a e z xP − − 0 iξ a dz ] − πxP e 2 56 = ( + , Z − − 48 a dξ , ] , in which the UV divergence can be removed to all orders by a πxP 6 ) = 2 z 55 Z 2 and x, P ( √ ) = / ) z g/H ˜ a f x, µ ( is the summary and outlook of this work. In the appendix, we gi + 6 0 g/H a f The rest of this paper is organized as follows. In section Among various PDFs, the gluon distribution function is a key = ( + a element of two light-cone separated gluon strength tensor calculations of the real diagrams with no Wilson lines invol 2 Definition of quasiThe gluon light-cone distribution (or conventional) gluon distribution funct modified definition of quasicalculation gluon of distribution. quasi In gluontion section of distribution linear at divergence one-loopan will improved level. be quasi analyzed gluonSection with distribution auxiliary and fie calculate the matc involved. In this workshow we focus that, on in power the divergencesby formalism considering in of the quasi auxiliary contribution from field,term possible the of operator linear Wilson mix div line. gluon distribution is lessgluon explored. distribution is A presented first in ref. calculationoff [ of and p dimensional regularization (DR)even in schemes. the It diagrams is without shown absorbed any into Wilson the line, renormalization and of thus Wilson these line. l Hence the tive renormalization of quasi-PDF(RI/MOM) in scheme the [ regularization-i renormalization factor determined on the lattice [ the processes like the productionspin of structure Higgs particles. of It hadron. is a However, compared to the quasi qua well as the auxiliarythe field linear formalism power divergences [ izability arise of from Wilson quasi lines’ quarkBased self- distribution on the has multiplicative renormalizability also of been quasi-PD proved t lem, e.g., the non-dipolar Wilson line [ arise from Wilson-line’s self-energy. Some suggestions ha In the light-cone coordinate, a four-vector ilarly, the quasi gluonequal-time distribution separated [ can be defined by non-lo JHEP05(2018)142 . . n n L 1 ) can 2 z with (2.4) (2.8) (2.3) (2.5) (2.6) (2.7) (2.9) ) n (2.10) z L P ent, i.e. ( / 2 (1) = th a scalar distribution , x 2 M i ) 1) is an unite 2 ( z , P oint on 0 | O M P , and ( 0 (0) , 1+ n , 2 µz O 2 z ) G z ) = (0 + )) z λ n field strength tensor is i P µ z n , ( P n )( L x | ( µ (0) = ( 0; . ) denotes a Wilson line along αβ A x µν . , a · , g C z G ! , ) 2 ; ) βα = 2 . λ µ λ ) zn αβ 2 2 ( G can be expressed as ( M ( z ) = 0. On the other hand, if we take 2 a ˙ − G x, y x 1 4 i + M αβ P ( W 1 ( zz M = P ) P G | − ++ , g 4 1 z dλ W i g )( 2 4 1 ν ≡ µν 1 ) P + µ 0 g zn | z P ( 2 ( + ¯ Z λ 4 1 2 QCD µ ) a µ P z z ++ ( Λ P zz ig − = P T G T

( | − | ν i | – 3 – ) λ ) P P O P P µ h h G | . The Wilson line can be parametrized as µ h ( z 2 ( 2 n a , the gluon field is in adjoint representation and , where ) µλ exp a ) y sums over the transverse directions. In this paper, ) + ++ n + z 1 1 G = 1 µ P T = i izxP ( | P P i e ( ( a = λz P ], z 2 and P h | ) = = + µν 56 x n M disappears because µν ) = ) = dz T , n zz πxP 2 T ,L 2 | 2 g , µ between a hadron state 2 48 z x, µ M 1 4 P ¯ λz , ( h , z Z µν 6 1 ) is a proper definition of quasi gluon distribution, we begin − z 2 T x, P ( ) g/H ( 2.4 z ) = ) = is the velocity vector of the contour. W λ P ( ( ) are related to the first moment of light-cone and quasi gluon g/H ˜ x x, µ f ) dx xf ( /dλ 2.8 µ direction. In the above definitions, ) ( Z is the hadron mass. This tensor is symmetric and traceless wi λ z a ) characterizing the magnitude. If we pick up the “++” compon g/H ( dx x 2. ˜ f µ , µ with two endpoints = ( 1 , then M denotes that the operator exponential is path ordered. Any p Z i a , z dx ) and ( C P | P = = +, we have ≡ 2.7 = 0 zz ) ν ν According to refs. [ To see that eq. ( T µ λ | ( = = µ P ˙ h x be expressed as vector along for µ contour is a straight line with direction vector Eqs. ( where The averaged value of the energy-momentum tensor of gluon field in which The term proportional to summed over all the directions except “3”, i.e., we adopt a modified definition, where the Lorentz index of gluo In quasi PDF we still adopt the Cartesian coordinates, where µ function respectively, JHEP05(2018)142 . . ), 2 /a 2.5 (0)] ) will (2.11) − µ 2.4 G ). This re- − , µ 2 arises from z (0) / + ) and ( µ ]. The quadratic x, P G ( 57 inuum theory with 2.2 )[ z mponents of internal n g/H eory but are essential ˜ s. ( transverse momentum. f f the external gluon. It L rization diagram, which zation scales (or the UV 0; the gauge invariance. For ht two are from the non- , ion is the lattice perturba- easure of the path integral. components: is the gluon momentum. To z dx x tion, the non-local operator . The factor 1 rguments will no longer hold. s not necessary to include the R ently, to the distribution func- arated the non-Abelian terms. P -gluon case. The hadron state ± fer to ref. [ zn by additional tadpole-like dia- e the hadron state with a single ( ions, and will not contribute to n the gluon self energy diagram W ) = )] z x, µ . We will perform the perturbative , where ( g zn i ( ) being the lattice spacing. The collinear ) is a reasonable definition of quasi gluon m µ P − g/H a ( 2.4 G g . The left two sub-diagrams are from the | 1 − ) z dx xf – 4 – R zn ( µ + , (a) has the continuum analog and diverge like 1 2 is only summed over transverse directions in eq. ( G [ ), we will re-calculate the one-loop correction to quasi 1 2 λ in LQCD, with 2.4 /a component is suppressed and thus the operator for quasi-PDF (0) = (0) can be decomposed into the µz − µz G G ) ) z z n n limit, one can obtain L L 0; 0; , , limit, the z z → ∞ z themselves. It indicates that eq. ( zn zn z ˜ P ( f ( divergence corresponds to the quadratic divergence in cont P W W 2 ) ) z and z /a f zn zn ( The Feynman rules are listed in figure We firstly make some remarks on the self-energy corrections o Actually, one can easily validate that both definitions in eq can be replaced by an on-shell gluon state ( µ µ i z z P G calculation in Feynman gauge. the conversion of Euclidean and light-cone coordinates. 3 One-loop corrections toBased quasi on gluon the distribution definition in eq. ( divergence is regulated by a small gluon mass lation can be generalized to hightions order moments, and consequ regulate the UV divergence,This we cut-off introduce corresponds an to UV 1 cut-off Λ on the a detailed review ondivergence lattice in perturbation lattice theory, one perturbationgrams, can theory as re well will as be theThese canceled counter diagrams term are which shown originates in from figure the m is known for amomentum) long leads time to a that quadraticbreaks the divergence the UV in Ward cut-off QCD identity vacuum scheme andtion pola the (on theory gauge all that invariance. provides the an An co UV except cut-off scheme which preserve Abelian part ofAbelian the part. gluon In field these Feynman strength rules, we tensor, have explicitly while sep the rig gluon distribution. To perform theparton matching, one state. can replac In this| paper we are interested in the gluon-in give the same resultsG for the matching at tree-level. In addi distribution. As a comparison,the if tensor will not be Lorentz covariant, and then the above a In the large This 1 Under the recovers the same Lorentz structure with the light-cone PDF a naive UV cut-off.to restore the The gauge (b),cut-offs) symmetry. (c) for One quasi and can and (d)contributes set light-cone a only the gluon same same exist distributions, factor renormali in the tothe matching light-cone lattice coefficient. and th quasi Based distribut ongluon’s the self above energy discussions, at it present. i JHEP05(2018)142 0 1 1 (3.1) ) < x < d 0 ρ n ρ ′ . x < ν n ′ ν µ z , x > g µ c Λ ′ g z P c bb Λ ′ P , aa 3+ z gf Λ tor. The left two sub- P − gf − x +3+ + 2 lson line. The Feynman = = 5 x 2 2 tion theory. (a) denote the 2 g 5 ) + z b, ν 2 m +2) ′ − ) x P x ρ, c 2 2 2 , µ 5 ′ x )( x ′ a − x ole diagrams involving a quark or )( − + , ν c ′ tensor, while the right two are from 1 2 +2 − b x us limit. (b) is a counter-term which x ) x 1 ρ, c − − (1 2 x x − x x a, µ . (1 x 4 +10 ln + 3 x ln ln x 2 6 − 2 2 − + + ′ − x − − 4 bb ′ 2(1 x x δ x aa ) (a), the non-local vertex is from the Abelian +2 ′ δ 3 ν 2 ) – 5 – +2 +2 ′ n x µ 2 2 1)(4 ν µ 3 x x k − 3 3 − x 3 k g − − − · x ′ (2 3 3 )( 2 b x x n νν + 2 2 − − k g ′ . In figure · µ                          3 n n A µ ( i π k C ( 2 s i − α = = = ) b, ν + a ( ′ 3 , µ ′ ′ a Fig. , ν

′ k b Λ) , z k a, µ )( a . The Feynman rules for non-local vertex from gluonium opera . One-loop self energy diagrams of gluon in lattice perturba ( x,P ( (1) g/g We start with the calculation of one-loop diagrams with no Wi ˜ f x term of field strength tensor. A direct calculation gives Figure 2 Figure 1 diagrams are shown in figure diagrams which have correspondingarises diagrams in from the the continuo measureghost of loop, which path only integral. exist in (c) lattice and perturbation (d) theory are tadp diagrams are from the Abelianthe part non-Abelian of part. the gluon field strength JHEP05(2018)142 1 1 0 (3.5) (3.4) (3.3) (3.2) < x < ) d 0 ( , z -Abelian term Λ P 1 1 0 0 1 1 − ) g ( < x < 1)+1 < x < 0 − 0 any Wilson line. The cross x , x < ( z , x > x e. These diagrams give Λ sult to the quasi distribution. P z these two parts are separated. ,, x > . x < . x < Λ P +2 − (e), is zero for light-cone distri- z z z z f definition. ) 3 2 c Λ Λ Λ Λ 2 g ( , , x > P P P − ) P z +1 z z 1 m Λ Λ + ) x − P P P − 2 2 1 + 1 + x x 2 )( x x − − − x − − − + 1 − 2 2 x x +2 ], the non-Abelian term is combined with − x 1 x x , x − 1              − ) (1 56              x x f − (b) x ( A 3 (1 A x 4 π C π – 6 – C 2 s 2 s Fig. α α

+1)ln x = = Λ) +1)ln +1)ln ( , x x x z ( ( (e) ) (d) 3 − b x x 3 (                x,P ( Fig. Fig. A

π (1) C g/g ˜ Λ) 4 s f Λ) , α , x z z = = ) x, P e (c) x, P (b) ( ( 3 ( 3 (1) g/g (1) g/g ˜ f ˜ Fig. Fig. f

x x Λ) (b), (c) and (d) correspond to the contributions from the non Λ) , , 3 z z ) . One-loop corrections to quasi gluon distribution without a ( x,P x,P ( ( The four-gluon coupling diagram, depicted in figure Figure (1) (1) g/g g/g ” denotes the non-local vertex from the operator structure o ˜ ˜ f f × x x and This contribution is linearly divergent: bution. However this diagram contributes with a non-zero re in the field strength tensor.the Wilson Note line. that In in theThis ref. present is [ work, more the convenient contributions to from find the source of linear divergenc Figure 3 “ JHEP05(2018)142 . 5 1 1 0 0 1 1 (3.8) (3.7) (3.6) (3.9) 1 1 0 (3.10) and < x < 4 < y < 1 1 0 0 . A direct 0 5 < x < , 0 + < y < lson line, while ion. , x < z figures 0 3 , P , x > ) 2 g 2 utions are propor- − x ) , Λ z m x 2 − 2 g +3 ) ) , P 2 (1 x z 2 2 g m 2 ) +2 y 2 )( ) , y < P + z 2 , x < 2 m y − − 1 + ) x y )( P , x > x 2 2 , y > 2 . y < +1 y − y y − x + )( on line in figure x x 1 z x 1 − , y > 2 − y x y +8 P − − (1 + y z y − ) 1 1 − − 2 1 +2 y − − rgences cancel with each other (1 x x P 2 Λ y x y x ) 4 (1 y − − ) 1 y x − x − (1 (1 Λ 2 2 4 y x x 15 − (1 x − x − (1 4 x − (1 + + 2 ) + 1) ln 3 ) 2 x x +1)ln z x + y 2)ln x x 2 ( + 1) ln + 1) ln y P x − − + x − +1)ln +1)ln − ( 2)ln 2)ln 2 1 )( x 1 y y − y y − x . x − ( ( − − − ( ( 1 − +21 1 − − 2(1 (g) together, we get the total contribution x x 4 (f)                  (1 1                  +3 3 + (1 2 g x , ∼ x − 2 x 1 m +3 +3 dy 4 x dy x 12 (a) x – 7 – − 2 2 (a) 4 2 Fig. x ln ln x x

− 3 Z Z − 2 2 5 ln 3 A A Fig. Λ) x x x − −

x +1) +1) , 8 π π C C − − 3 3 ): x z x x 4 s 2 s 1 1 ( ( +1) Λ) (2 x x x + − , x x α α x 1 ( z − ) ) (2 (2 − x − − x, P x x (                          (1 − x,P −                    A δ ( (1) g/g (1 π ˜ A C f (1 δ 2 s (1) π C g/g δ x ˜ 4 s f α + α x = = = = = 3 (f) (g) 3 3 (a) (b) 4 4 Fig.

Fig. Fig.

Fig.

Fig. (e). In the appendix, we give more details for this cancellat

Λ)

, Λ) ∼ Λ) : z (f), (g) are the virtual correction diagrams. Their contrib Λ) Λ) , , 3 , , 3 z z z z , only one end of the internal gluon line is connected to the Wi (a) 4 3 x,P ( x,P x, P x,P x, P ( ( ( ( (1) g/g ˜ We come to the diagrams which involve one Wilson line shown in Figure f (1) (1) g/g (1) g/g (1) g/g g/g ˜ ˜ x ˜ f ˜ f f f x x x x from figure This result is freein of figure linear divergence, and the linear dive Summing the contributions from figure both the ends ofcalculation internal shows gluon that line are connected to the Wils In figure tional to the tree-level result JHEP05(2018)142 is 4 (3.11) (3.12) (3.15) (3.13) (3.14) 0 1 1 0 1 1 < x < < x < 0 0 , ion from figure , x > . + , , . x < , z + z P (1)] (1)] ) P z z nvolve one Wilson line and (1)] ) x T Λ T x P P Λ T − ) ) − − x x − . x < Λ Λ (1 − ) (1 ) − − , x > ) x − x (1 (1 − ( f definition. x ( 2 z 1 e line is the Wilson line. The cross ) ( ) x x T ntroduced to regulate the singular − − P T b d 2 z ) ( ( − T − 1 1 )[ x P − Λ 1 )[ ) 1 )[ x − x x Λ (                    x ( (1 + − ( A 2 2 g (1 ) − π C z 2 s m x dx f ) − dx f α P 1 2 − dx f x )( x 1 1 ∞ 0 1 = x − + 0 1 −∞ 1 + − x Z Z Z – 8 – 1 (d) − − (1 4 x − x 1 (1 x 4 x − ) = ) = ) = Fig. x x x x ln ln

( ( ( ) ) ln x x Λ) T T T x x ) , + x − − z ) x ) 1 1 a c (1+ (1+ )] )] )] ( ( − x x x x x 1 (1+ ( ( ( x,P x − − ( f f f [ [ [                    (1) g/g ˜ f A dx dx dx x π C 1 2 s ∞ 0 = 0 α 1 −∞ Z Z Z (c) = 4 4 Fig. Fig.

Λ) Λ) , ]: , z z ) is an arbitrary smooth function. Then, the total contribut . One-loop corrections to quasi gluon distribution, which i 24 x ( x,P x,P ( T ( (1) g/g (1) g/g ˜ ” denotes the non-local vertex from the operator structure o f ˜ f x × x given as integral [ In the above expressions, three generalized functions are i One can find that the linear divergences do not vanish. Figure 4 Here “ connects only one endpoint of internal gluon line. The doubl and JHEP05(2018)142 or perty (3.16) to local ], because all 1 1 0 gences 52 , f definition. 51 , < x < 0 18 or Wilson line lead to d diagrams, which have tor with the method of th ordered exponential of nvolves one Wilson line and gy of Wilson line, which is rections. unctional of gluon field, the ,, x > . x < defined by the gauge invari- me approach. to retaining the gauge invari- + he Wilson line’s self energy. In re cured [ z z z P P P on line. The double line is the Wilson 2 2 2 ) ) ) ) is the length of Wilson line. Based x x x Λ Λ Λ c ( − − − that Wilson line’s self energy is not the L 3 (1 (1 (1 + + + 1 1 x – 9 – 1 1 1 − − − x 1 x                    ) has the similar structure with the linear divergence in A π C 2 s α 3.16 = is the UV cut-off and 5 ]. The power divergences can be summed to an exponential fact ]. These contributions can be easily derived by using the pro /a 60 56 Fig. –

. The result is 5 58 Λ) ]. In this method, the non-local operator can be converted in , ” denotes the non-local vertex from the operator structure o z 61 × ). )) where 1 x x, P ( c ˜ ( ( f − (1) . One-loop correction to quasi gluon distributions, which i g/g ˜ f x π/aL ) = − It is convenient to discuss gauge invariant non-local opera At last we note that there are also contributions from crosse At last we calculate the one-loop correction to the self ener x − ( ˜ on this, the linear divergence in quasi quark distribution a gluon quasi PDF, itonly has source been of shown linear ingluon section divergence. operator and Since Wilson the line Wilson might line be entangled is in aauxiliary loop f field cor [ exp( the power divergences in quasi quark distribution are from t quasi quark distribution, thus may be renormalized in thebeen sa discussed in ref. [ Figure 5 f ance. It haspower divergences been [ known for a long time that loop corrections f 4 Auxiliary field method andThe renormalization light-cone of and linear quasiant diver parton quark bi-linear distribution or functionsa gluonium are line operator. integral over A gluon Wilson field line, along a a contour, pa is essential The linear divergence in eq. ( both the endpoints of internalline. gluon The line cross connect “ to the Wils presented by figure JHEP05(2018)142 (4.4) (4.9) (4.3) (4.6) (4.8) (4.7) (4.5) (4.1) (4.2) (4.11) (4.10) (4.12) . Z i - field is inte- ed by the two ear operator is ) Z 2 , , z Z ( i (0) b ρσ )) Z 2 G z ) ( 2 (0) ψ s well as the dimensional λ ¯ ) ( Z 2 i b , ¯ λ ) , Z . ( been converted into the prob- + a µ a c e theory is given by . , ¯ xing operators are constrained ′ ) vided into two local operators, ¯ c Z A Z Z b Z ν Z ) i dλ b ˙ 1 , ) ))( x A D , 2 1 λ Z µ A · ( ˙ λ µ λ − 2 )( · x ˙ a ( ˙ ( x ˙ 2 x a µ x α ˙ a ν a x ¯ ˙ Z ˙ x α Z only. Therefore, when Z t x can only mix with two operators under A ) A ˙ ) ) x a µ ν abc 1 µ 1 1 ˙ abc z Z x (1) να A z x λ (1) µν ( (1) να gf ( ( ˙ ( µ Ω gf ¯ 2 − ˙ ¯ x Ω ψ a µν − ( − a ν − hZ | − h ig G a − ˙ A x ν ¯ a | Z ˙ h µ – 10 – ν + line is just the Wilson line. 2 x ] , a Z ˙ λ ˙ ) = 2 x x ) ′ a , x α ) = λ ( ˙ ˙ ∂ ˙ x α 1 64 Z x C d 2 Z ∂ ˙ D ←− a µν z – ) = x − ; dλ − ( 2 ¯ 2 | a µν G (1) µα Z j = z ˙ 61 = x a (1) µα ( ψ , z | G − a ¯ ¯ ) ¯ a 1 Ω Z Z ) b ρσ ) z = = = Ω C λ ( Z , which is a gauge invariant field strength tensor, may mix = = = ( G ; 0 D a ←− ) 1 field [ Z D ¯ W (3) µν (1) µν (2) µν Z ( Z i C (2) µν (1) µν (3) µν , z ( Ω Ω Ω ; 2 a µν ¯ ¯ ¯ Ω Ω Ω 2 + z Z− ( G , z ij 1 ] = z QCD ( W ), all the internal 65 ] ) S (1) µν ab 2 , z 64 4.1 = ( W – 64 field technique, the gauge invariant non-local quark bi-lin i ) ¯ S 1 ψ 61 z Z ( is the action of QCD. Obviously the Wilson line can be express a µν G QCD denotes the expectation value respect to S Z The operator Ω With the by the gauge symmetry, orconstraint. more It strictly, has the been BRST demonstratedrenormalization symmetry, that [ a the Ω with other operators under renormalization. The allowed mi Now the problem of renormalizing thelem non-local of operator renormalizing has the localwhich operators. can be The considered operator individually. is di grated out in eq. ( and the gauge invariant non-local gluonium operator is The Hermitian adjoint of these operators are with with h· · · i where point Green’s function of operators by introducing an auxiliary field. The action of th then given by [ JHEP05(2018)142 . ) z 2) r. , )( i 1 ( µν δm , (4.14) (4.17) (4.13) (4.15) (4.16) . Since = 0 (3) zµ µ , ) denotes that a r. Z ′ z · . One can easily find should be ip a e ) m operator with full ¯ ) , we have Z p Z z e the tree level matrix component ( ( D µ ∗ µ , ( ǫ , . are not distinguishable. ome remarks on the renor- ) and | λzn a µ r. z z | A Z (2) µν = 0 = 0 z 2 ) | (3)( µν ) = b z a a z ( or λ on, the operator should contract Ω | ¯ δm ( Z 3 denotes the counter-term from matrix |Z | c x − z z = and Ω | | field gets a renormalized mass + N z z = a ) ) a (1) µν Z r. direction, the ) Z δm δm z Z D (2)( µν + − can be replaced by the operators Ω ( D c c Ω 2 )) is used. ¯ | 2 )( Z Z z z c (1) µν b b a z | ′ A A A – 11 – + 4.16 z · · · µ ) ˙ ip x ˙ ˙ r. x x e ) − p (eq. ( (1)( µν abc abc ( a µ Ω ∗ µ A Z 1 ǫ gf gf ) can be expressed as z c and external gluon state gives z − x n − + . Note that the operator we are interested in is Ω ) ( ˙ r. → 2 a a ) ,L (3) µν ¯ a − Z Z ) = ( | 0 ′ λ λ (3)( zµ ˙ . (1) µν x z ) | ∂ ∂ z, ( Ω r. ( ) = r. 3) are the mixing parameters, and the superscript ( W (3)( zµ , ¯ Ω (3) zµ (3)( zµ 2 , Ω Ω | or ) field. Under renormalization, the ) . Therefore, in the present work, Ω p = 1 r. ( Z i g (2) zµ ( h (3)( zµ i c = Ω 3) as . The tree-level counter-term contribution, where , , which is the unit tangent vector of the Wilson line contour. 2 z (1) zµ field in the operators are renormalized. , n is zero. To study the renormalization of operator we should calculat Before calculating the tree-level matrix elements we make s However, the operator we are interested in is not the gluoniu After renormalization, the operator Ω Z x = 1 i element from the operator Ω Lorentz indices. In thewith definition of quasi gluon distributi an arbitrary point on that Ω Because the path in the Wilson line is along the the of ˙ where the equation of motion for Figure 6 ( where the Therefore, after renormalization, the equation of motion f elements of Ω malization of Then, the contraction of Ω JHEP05(2018)142 Z i (4.18) (4.19) (4.20) (0) ′ (a), with ) a 6 z ¯ z Z ik − (1) e a , ) − hZ | C z 1 1 0 (1 ; z ) | 2 − 2 ) . For figure ) ǫ z 6 , z z z + 1 2 z ik ) < x < ( z δm − k ( e 0 z (( W nt. In the above expression, z α ) − , x, > . x < (0) k − iP i ) e nce has the same structure with + − (1 ) 2 z e 2 p z iπ k ǫ ( ) x x x , λ 2 α z g z 1 z 4 . For a Wilson line in fundamental | 2 3 + 1 1 1 iP k z − − − λ 7 − Z z dk ( 2 k 2 i z 1 1 1 ) − πα C z ∞ αǫ z k k √ (0) + ixP − z ) and also the fact that the matrix element ( ) 2 −∞ P                    e r. z Z z | ze ] z 1) z 4.2 π δm | dk ǫ ), its contribution to quasi gluon distribution 60 (3)( zµ π dz − dα dα − πP δm − – – 12 – ¯ Ω 2 e 2 ∞ x ) ∞ ∞ y | 58 + z 0 0 z 4.17 (( z z −∞ 1 Z = | Z Z − δ k exp ( Z ) zn 0 0 0 6 1 1 0 1 1 0 p − )( at tree level are shown in figure − ( → → → lim lim lim ǫ ǫ ǫ → ) µ dy ) lim ǫ Z Z z Fig. ∗ i z z z r.

µ z ǫ z z z P z z ) Z = P P P p P Λ) ) < x < G ) (3)( zµ 1) ( 1) 1) 1) , ( x r. 1) by recalling eq. ( 0 z − − − ( |h − − µ,i ) x x x ′ ) − ( ( ( ǫ x i i i x , . x < , x > ( p C aa i e e e ( ; δ x, P (1 i e (( + g z z z 2 ( δ (a) δ z X h 6 ′ z z , z x x x z − dz dz dz g/g πP πP πP 1 π dz ˜ (b) gives the same contribution. Thus we have aa πP f z 2 2 2 1 1 1 dk − − − x δ 6 2 Fig. ( x ixP

1 1 1 1 e − ∞ 1) Z Z Z W Z z Λ) + z z z 1 z , − −∞                    z 1 Z dz 2 c πP δm δm δm z z z z z 2 δm z z z N is taken to be z x, P ). Figure δm δm πP πP 2( iP Z P δm iP iP ( Z i ′ ======a g/g 3.15 ˜ The contributions from Ω Now we calculate the contribution from figure ¯ f Z x a Here the superscript (0)hZ denotes the tree level matrix eleme representation, it has been shown that [ is at tree level.eq. It ( is interesting to notice that the diverge the contraction rule presentedreads in eq. ( JHEP05(2018)142 5 . The (4.21) (4.23) (4.22) (1) zµ . Figure 5 In the language and . ) 4 z ··· (0) i P ) ) p x ) + corrections to Ω ( z 2 g | − ) n y L (1 1 (0) − 0; , − µz denotes the mass counter-term of ), one can determine that z gence in this diagram should be z ) (1 z G z 2 k ) ) n izP N zn ( z dy z 4.22 ( . 1 1 0 − L δ k e ( zn Z 2 (1) 0; − Λ) ( ) ) , | µ ) π z z is a renormalization constant. Similarly, x z W ( z z | P G z A P Z ( ) and ( zn − < x < + | | z ) ( π ) | C . z 0 P x 2 s p | z (1 | ( C z W 1) δ α | 3.16 z z g − , x, > . x < z − h | − – 13 – x + ) z + ( | ], the leading order contribution from the counter δm i ixP 2 ((1 z = e e | ) δm 2 2 2 δ 51 − z z ) ) ) z π x − z dz 2 x x x e 1 π − dz 1 1 1 δm πP dk δm − − − = 1 = 2 Z − (1 ) ( z Z (1 (1 (1 r. z Z − ( . From eqs. ( z P z ) ) ) 7 z P ) in the adjoint representation, we have z z z z                    n z z ixP n e z z L δm δm z δm πP δm ,L 0; − − δm πP , 0 − − dz z πP 2 z, = ( = = ( = = ( zn ( Z W W is the one-loop correction to the gluon-Wilson line vertex. 4 denotes the length of the contour k . The tree-level counter-term contribution, where C k field formalism, it can also been interpreted as the one-loop Figure Now we turn back to the one-loop corrections, shown by figures Z Therefore, with the same trick in ref. [ for a Wilson line of term reads canceled by the one in figure denotes the self-energy of Wilson line, and the linear diver Figure 7 the Wilson line. where JHEP05(2018)142 ] ) ... 3.15 -loop , with (4.24) n ) compo- r. z , racket [ (3)( zµ ), we have k j N D 3.15 , the 4 nal regularization ··· ··· . From eqs. ( entically the same. 2 2 -components of the 6 z si gluon distribution field strength tensor. N D es. 1 1 off scheme, which may gate the cancelation of N D n gration over the “0” and ntribution from a ces. l gauge invariant operators, elies on the lattice pertur- ) and eq. ( 3 ce from another viewpoint. ate the d ntities for a general diagram sent work, the linearly diver- and section t is likely that our results are . r a future study, and here we ) 4.23 ns can be generated by adding 3 ··· alized by operator Ω een the cut-off and DR schemes terms are the same. It is widely r. ! 2 l function, or left unintegrated. It z 3 (3)( zµ P δ d ) 1 l x 3 d − Z (1 − ! z i z l i – 14 – P c ) x n =1 i denote the numerator and denominator respectively, X − i

(1 D δ k j − undetermined. From eq. ( N D z i l and 3 i should be canceled by the one in ﬁgure c i c ··· ··· 4 2 2 N n =1 i N D X , and the matrix elements of Ω 1 1 z

N D δ n z n δm l . Here 4 dl ⊥ d i and leave them unintegrated. Then the loop integrals in the b l propagators can be generally written as 2 i ··· l d ··· k z 2 0 i 2 l dl dl 4 z 1 d ], we have calculated the matching coefficient in the dimensio ) one can find that the structures of linear divergences are id ≡ 2. Then, at one-loop level, all the linear divergences in qua 1 dl : the number of three-gluon vertices and gluon-ghost vertic l : the number of vertices corresponding to the Abelian part of / i g 56 4 : the number of internal gluons and ghosts. : the number of internal fermions. l are some constants. In the discussions of section : the number of loops. 1 3 G d Z 3 L F G In addition, no external line should connects to these verti V V i 4.18 − d c To further validate the above proposal, one needs to investi The above calculation has been performed in the “naive” cut- = Z • • • • • = 3 are all three-dimensional (3D)as integrals. follows: We define a few qua linear divergences for alloutline loops. the proof A at completeinternal two-loop proof lines level. is step Higher left by order fo step correctio from lower order diagrams. The co diagram with loop momenta believed that gauge invariance isgent preserved terms in have DR. been In absorbedleaving the into the pre the finite matrix terms elements consistentgauge of with invariant. the the DR. Therefore, i scheme. It is interesting toonly notice reside that in the the differences linearly betw divergent terms, while the finite and ( It indicates that all these linear divergences can bec renorm and nent of loop momentum is either constrained by the has been shown thattransverse the components of linear loop divergences momenta. come Therefore, from we the separ inte only the mixing parameter linear divergence in figure with can be absorbed by break the gauge symmetrybation at theory. first sight. But A itIn complete is ref. analysis viable [ r to examine the gauge invarian JHEP05(2018)142 . G 8 V + 1, . For ) G r. ditional (a), (b), + 1, and 8 → (3)( zµ g for the 3D 3 ) G c V ( ounter terms. ure D → will be decreased g ) vertices. In this 3 rgences, which are Z D V rgence for this case ne, which are shown n line. This is shown ed at the beginning of tions from Ω ce, so there is no need al degree of divergence. h are shown in figure luon- + 2, he lower order diagram. opagator as well, but es and three gluon propa- rom the one-loop diagram. G f divergence n, in which the added internal → (c) contains gluon’s self energy, G of the Wilson line, as well as the (b), there will be two more gluon 8 ine. 9 z will be increased by 1. This case +1. In this case L D δm → L ) b ( . The propagator of Wilson line only involves will be increased by 1. The power divergence G – 15 – V D + +2 and g g 3 3 V V + → g G (a), there will be two more gluon propagators, one three- 3 2 9 V − F will not be modified; for figure +3, − G D L → is increased by 1, the power divergence must be canceled by ad = 3 G . In this case there will be one more gluon propagator, i.e., . For figure D 9 D 10 . 3 (a) just one counter term is needed, while for (b) we need two c + 1, then ) 9 a ( L . Typical two-loops corrections to quasi gluon distributio → in figure case, the power divergence can be removed by adding contribu diagrams due to thesection gauge invariance, which has been discuss in this diagram canfrom be the one-loop viewed corrections as of the the product gluon-Wilson of line (or two g linear dive figure gluon vertex, and one Wilson line propagator, i.e., propagators, one three-gluon vertex, and one Wilson line pr although This is the sameadding one with gluon ordinary line Feynman willgators, introduce i.e., diagrams two in three-gluon QCD. vertic For fig L will be modified from 1 to 2, then Both the endpoints of the added gluon line touch the Wilson li to further subtract the linear divergence. Figure None of the added internal line touches the Wilson line, whic by 1. A linear divergence will reduce to a logarithm divergen in figure Only one endpoint of the added gluon line is attached to Wilso can be explained asshould be Wilson cured line’s by self adding mass energy. counter term The power dive and two more Wilson line propagators, then corresponding counter term for the linear divergence from t -component of momentum, hence will not affect the 3D superfici There are three generic ways to generate two-loop diagrams f z • • • integrals, with the With these quantities, we introduce the superficial degree o line (denoted by dotted line) is not connected to the Wilson l Figure 8 JHEP05(2018)142 oop (4.25) (4.26) remarkable ], i.e., 65 n the logarithm diver- , om other components of red to hold to all orders, 64 . ) n, in which both the endpoints ) ) tracted with the tangents of iscussed in DR scheme for a 2 the renormalization of gauge ion of Wilson line. The quasi b b n, in which only one endpoint of can be defined by the matrix z zed [ ( σ ˙ x ρσ ) . G r. ( ) ) C ; 2 µνρσ , z G 1 ( z ( µ x W ) – 16 – = ˙ 1 z σ ( ˙ x µν G µνρσ = G µ )( )( ˙ a a x ( ( µνρσ G one can find that there is no logarithm divergence in the one-l 3 ]. Note that in DR the linear divergence is not regularized. A 66 – 64 . Typical two-loops corrections to quasi gluon distributio . Typical two-loops corrections to quasi gluon distributio After discussing the linear divergence we add some remarks o It is also shown that the quasi distributions can be defined fr The above discussions on two-loop corrections are conjectu the correlators. For example, the quasi quark distribution though we are lack of a complete all-order proof. the contour, in this case, the operator will not be renormali real corrections. By employinginvariant quark the bi-linear auxiliary and fieldlong gluonium approach time operators [ has beensimplification d is achieved when the gluonium operator is con with This can be traced backgluon to distribution the belongs property to of this functional case. derivat Figure 10 of the added gluon line are connected to the Wilson line. Figure 9 the added internal gluon line is connected to the Wilson line gence. From section JHEP05(2018)142 . ) , (5.3) (5.1) (5.2) (5.4) P 4.25

2 Z − x 2 7 (0) . Eq. ( in the matrix z components of + ν 1)+ µz = ) ν 2 g − r. ν x m and ( ) 2 g . (3)( 2 = and 2 µ ¯ x Ω m x 2 µ µ Λ) 3 ) c 2 Λ + , +2 y, y y in LQCD. According to x 2 + ( ) as already been calculated Λ 2 + − ion vector corresponding to ur of the Wilson line (e.g., in µz z x y ) i/H Λ (1 P r. + − gence will be greatly simpliﬁed ecifying x yf x ln ξ, 3 (1 (1)( 1 − ¯ Ω z ln ) +1 − Λ n P y y ( 2(1 x x gg ) , z − Z − y x 1+ 1 n + 2 g zn ( 2 g dy m s π gi ) 1 m α 2 2 µ ) 0 Z z 2 ) x Z 2 y x r. ) 2 dy Λ + – 17 – x =0 Λ + ∞ x 1 i ]), it is more convenient to modify the components X (3)( 0 ) x − − 2 Z Ω 49 = − (1 3 x (1 c δ (1 + (0), in present work we choose x + − z ν x 3 ρ µ , we propose the subtracted quasi gluon distribution as Λ Λ) = 2 g z . P − 2)ln ) 4 , can be expanded as G z m 1)ln r. ) − ξ, ) z gg 2 x/y − 2(1 x n (1)( x Z x 2 x, P − ≡ ) is the mixing parameter determined by perturbation theory Ω gg ( ,L Λ + +2 s 2 ]. For gluon, one can pick up any indices of 0 ξ ) Z 2 +4 x α 8 − r. 2 x ( ( z, − ), g/H 3 x x

˜ ( ξ f O 2 2 7 (1 − P x (0) [ . W − 3 − ) ψ ln 3 x z z 2 + 0 (1 x ) ( 1)+ total / (2 γ δ x

) 1 (2 − µρ x Λ) z ixzP x − , ( Λ) e G − ( z ¯ ) = z (1+ ψ A A 2 1 x, ξ 2 = ], with result x π π ( ( C C x 2 2 s s dz 3 πP x, P 56 (0) α α c gg (1) 2 ( g/g − +2 ) Z and the matrix elements can be evaluated non-perturbativel = = The one-loop correction to light-cone gluon distribution h r. xf Z ( g/H z ˜ f = x LaMET, one can expect a factorization formula as δm in ref. [ with where elements of In perturbation theory the of gluon strength tensor correspondingly. 5 Matching coefficient forWith improved the gluon discussions distribution in section indicates that, if one defines quasi gluon distribution by sp the non-dipolar Wilson line approach [ gluonium operator, the renormalizationwhen of the logarithm tangents diver ofthese the components; on contour the are other collinear hand, if to one the modifies the project conto element of JHEP05(2018)142 l, 1 0 1 1 0 1 (5.6) (5.5) x < x > < x < < ξ < 0 0 + 1 2 . ξ < ) x − 3 1 z , ξ > x − x P − ξ x )( +3 2 ln ξ ξ Λ 1 +2 2 2 ln 2 g − + , ) 2 -off Λ. It is free of IR +1 − gluon distribution. The 1 − z ξ 2 m (1 2 2 x x ) 1 ξ +1 − 2 P ξ − 2 contracted with each other. 4 − x x x − )( x x 2 − olds for gluon distribution at tion to sum all components of x +2 x ln + efinition, we have re-calculated 1 1 + +8 − 1 + x x 2 − 1 − +1 ξ 2 − 2 g p level, in which the Abelian and 1 − (1 x ) ξ ξ x ) ξ x − ξ z x − 2 m (1 4 x 15 ) ξ x ln P 0. On the other hand, the one-loop + 2 − 1 2 )( ln + x 1)ln 3 ) x − +1 x 1 z + x − 1)ln 1)ln ξ ξ − x < +1 − − P x x − − ξ ξ )( 2 − (1 . , − x x ξ +21 Λ x 3 +4 2(1 1 4 (1 + − 4 2 ξ +3 − x +4 +4 − 1 x – 18 – (1 2 2 ξ 1 and , x x ln 2 ξ − ξ 12 x x 2 ξ 4 1 − 2 2 +1 − +2 − 3 − +1 ξ 5 2 2 − − 1)ln x x > x x x 3 3 x x + − 1)ln 1)ln 8 2 (2 x x 3 ξ ξ − − + − − +2 + (2 (2 ξ ξ +4 +2                                                  2 +4 +4 ξ + A 2 2 2 ), one can determine the matching coefficient at one-loop leve ξ ξ π C 1 − 2 2 2 2 s 3 ξ − 5.5 − − α ξ 2 ξ 3 3 ξ (2 ξ = . + − (2 (2                              ) and ( total A

C 5.4 Λ) = , 1. The result is 0 for z z Λ P x,P ( ξ, < x < )(1) From eqs. ( (1) r. gg ( g/g Z ˜ f x gluon PDF is parametrized byFrom two the gluon viewpoint field of strength covariance, we tensors Lorentz proposed indices in instead the of contrac onlythe transverse gluon-in-gluon ones. quasi With distribution thisnon-Abelian function d contributions at are one-loo separated. 6 Summary and outlook In this work we have discussed power divergences in the quasi This coefficient hasdivergence, only which the indicates that logarithmicone-loop the dependent level. LaMET on factorization the h cut for 0 correction to gluon-in-gluon quasi distribution reads JHEP05(2018)142 ), z n ,L ∞ z, ( W continuum to the ) = m QCD, the non- he light-cone PDF z ion in lattice gauge ]. The renormalized ( ps under fundamen- . However, applying uark PDF, one is the ilson line, and not all luon PDF, the reason n, thus one can match Z 52 th the gauge invariant , moment, a perturbative ngth tensor and Wilson hermore, to arrive at a lson loops under funda- uon operators should be r divergence. Therefore, luon PDF on the lattice, n PDF on the lattice and 37 , attice is also a crucial step. seful in practical extraction e needs to define the gauge ctation value of the product or non-local gluon operator, -loop level. This cancelation tching coefficient at one-loop 35 lson line. We then have used vergence can be absorbed into , on distributions from LaMET ion of Wilson line. Employing 34 , 24 ], another one is the nonperturbative renor- field as a Wilson line MS renormalized light-cone PDF, and work 51 Z , – 19 – 38 , 28 , 25 invariant non-local gluon operatorsmental by representation. the The renormalized mixing Wi considered of as the well. gauge invariant gl local gluon operators arelines constructed are by under the adjoint gluonone representation. field should stre To define simulate quasi quasitheory, PDF g in and which perform the thetal gauge matching representation. field calculat is For a expressed practical by lattice the evaluation, Wilson on loo In this work, we only consider the quasi gluon PDF in continuu is defined in continuum,quasi one PDF has to calculated match withreliable the result, the light-cone renormalizing lattice the PDF matrix regularization. in Two elements approaches on have Furt the been l employedlattice to perturbation renormalize theory the [ quasi q Since the quasi PDF should be evaluated on the lattice while t during to the operator mixing discussedrenormalization in with this lattice paper. perturbation theory Atof the will gluon be PDF. u malization approach, e.g., the RI/MOM schemematrix [ element in RI/MOM is independentthe of lattice UV evaluated regularizatio quasiout PDF the to matching the coefficientRI/MOM in to DR, gluon instead operators ofis is the not that naive straightforward the cut-off for multiplicative quasi renormalizability g is not clear f We have to note that it is still a challenge of evaluating gluo We note that since one can identify the We found that power divergences cancel in diagrams without W • • the present work leaves substantial room for improvement. and LQCD calculation. this new operator isan a improved product quasi of gluon covariant distribution, derivativeslevel. and of determined This Wi the matching ma coefficientthe is present IR work finite provides and the free possibility of of UV extracting powe glu of power divergence is conjectured to hold to all orders. linear divergences can beauxiliary attributed field into method, we the expressof renormalizat the two Wilson auxiliary line fields. asgluon the field We strength expe found tensor, and that pointedthe a out matrix that new elements the operator power of di can this newly mix introduced wi operator at one These issues will be addressed in the forthcoming work. JHEP05(2018)142 z P (A.1) (A.6) (A.3) (A.4) (A.2) (A.5) ≫ component z agrams ), we calculate en the diagrams function, the UV 2.4 , ) δ . oratory for Particle 0 nce the i 0). To compare with ) , , (Λ y ) P e Physics, Astrophysics , or valuable suggestions. . , ( 0 ) O nspiring discussions, and ) ) , k undation of China under g 0 0 0 | x + (Λ by the (Λ (Λ (Λ undation of Shanghai under y when this work is finalized. , k n line has been eliminated. O 2 (0) ⊥ 0 3 O O O k ) k + νz ν 2 + + + ˜ k G ˜ k 2 ) µ ( µν 2 2 2 ⊥ ν = ( ν ν ⊥ ˜ k z as well as the external momentum is the loop momentum and Λ ) ) ) ˜ k g ˜ k ˜ k k 2 2 2 ˜ k 4 4 z µ k µ µ ⊥ ˜ ˜ ˜ k k k zn ) ) ˜ ˜ k k ˜ k k ˜ ( ( k ( k ( 3 3 π π 4 4 4 d d zµ ) ) ) ˜ ˜ ˜ k k k (2 (2 3 3 3 π π π G | , we have d d d ) Z Z (2 (2 (2 3 ), where P z z 2 2 ( z Z Z Z n P P g n – 20 – h z z z 2 2 2 2 A ,P z n n n A ] and the new definition in eq. ( P P P C Λ is replaced by 2 , C A A A 56 2 k ixzP Λ C C C ig , ). In the following discussion, we will show that how e , 2 2 2 ig 2 3 z ), for figure 48 ig ig ig (Λ z , dz πP 6 ∼ ∼ − ∼ ∼ − ∼ ∼ 2 ,P k Λ (c) (e) (a) (b) (d) , 3 3 Z 3 3 3 Λ = , Fig. Fig. Fig. Fig. Fig.

(Λ µν µν µν µν µν µν ∼ M M M k M M M , we have concluded that the linear divergences cancel betwe 3 The UV divergence appears when loop momenta go to infinity. Si In the region . Then, under this approximation, of loop momentum in one-loop real correction is constrained the original definition in refs. [ divergence is from the region the correlation matrix element the cancelation works. which with no Wilson line (figure Since we are interested in the no-Wilson diagrams, the Wilso A Cancelation of the linear divergencesIn in section no-Wilson line di Grant No.11575110, 11655002, 11735010,Grant Natural No. Science 15DZ2272100 Fo Physics and and No. Cosmology, 15ZR1423100, andand by Shanghai Cosmology. MOE Key Key Lab Laboratory for Particl We are grateful to J.W.T. Chen, Ishikawa, Y. X.D. Jia, Ji, J.P. Ma, H.N.W.W. and Li, thanks R.L. Cai-Dian J. Zhu Lüand Qiang Xu, for Zhao Y.B. i This for Yang their work and hospitalit is J.H. supported Zhang f in part by National Natural Science Fo Acknowledgments is the largest scale.P In this region, one can neglect JHEP05(2018)142 (A.8) (A.9) (A.7) (A.15) (A.16) (A.17) (A.18) (A.11) (A.19) (A.14) (A.10) (A.12) (A.13) , . ) ) 0 0 , (Λ (Λ ) , 1 ) , O O 0 ) , , 0 (Λ ) ) , ∼ ∼ (Λ , , . 0 0 ) O ) ) ) , (Λ . 0 O 0 0 ) 0 ) (e) (c) O (Λ (Λ 0 ∼ 0 3 3 (Λ + (Λ (Λ (Λ ) O O + (Λ (Λ 0 O 2 O O O ) 2 Fig. Fig. + + 3 O 2 O ⊥ 2

) + ) (Λ + + + k 2 2 ˜ lson line diagrams under the k 2 2 ⊥ + ) ) 2 + ( O Λ) Λ) ˜ 2 2 2 k 2 2 ⊥ ⊥ 1 k 2 2 ˜ k ) ) ) ( , , 2 4 ( 2 k k etween diagrams with no Wilson ˜ ˜ 2 2 2 k k ⊥ ⊥ ⊥ 1 2 2 2 1 z z ) + ˜ ˜ 4 k k ˜ k ( ( 4 k k k ˜ ˜ ˜ ) k k k 3 π ˜ k ) 2 ˜ 4 0 k 4 4 4 ( ( ( 3 d π ) 3 π ˜ ) ) ) k k ˜ ˜ ˜ k k k (2 4 4 4 d , we return to the deﬁnition proposed 3 2 d π 0 x, P x, P 3 3 3 π π π ) ) ) (2 ˜ ˜ ˜ k k k (2 ( ( d k 3 d d d 3 3 3 π π π ˜ ˜ µν (2 ) Z (2 (2 (2 f f d d d 2 ⊥ Z − x x Z (2 (2 (2 z 2 ), we have g ˜ k Z z Z Z Z z 2 ( ), 2 z n ⊥ 2 P + + z z n k Z Z Z 2 z z z P n 2 2 2 P A 2 0 ,P n z z z P n n n A 2 2 2 P P P (d) (b) A k C ,P Λ 3 3 n n n 2 A P P P C A A A 4 C , Λ 2 – 21 – ) 2 ˜ k C , A A A C C C ig Λ 3 2 π 2 2 2 ig 2 , Λ Fig. Fig. C C C ig d

, 2 2 2 2 ig (2 ig ig ig (Λ ig ig ig Λ) Λ) (Λ ∼ ∼ − ∼ ∼ ∼ − , , Z ∼ we arrive at the deﬁnition of quasi gluon PDF adopted z z ∼ ∼ − ∼ ∼ ∼ − ∼ z 2 k (c) (e) (a) (b) (d) n P 3 3 k 3 3 3 µν (e) (c) (a) (b) (d) is contracted with g 3 3 3 x, P x, P A 3 3 ( ( C ˜ ˜ Fig. Fig. Fig. Fig. Fig. f f µν 2

Fig. Fig. x x Fig. Fig. Fig.

ig with M Λ) Λ) Λ) Λ) Λ) + , , , µν µν , , µν µν µν ∼ z z z z z µν 3 (a) M M M M M 3 M µν µν x, P x, P µν x, P µν µν x, P x, P ( ( Fig. ( ( ( ⊥ ⊥ ⊥ ⊥ ⊥

˜ ˜ ˜ ]. In the region Fig. ˜ ˜ f f f g g f f g g g

x x x x x µν 56 Λ) , , M z 48 µν , ⊥ 6 g x, P ( ˜ f x By contracting On the other hand, if Totally, One can find that in the region which indicates that the linear divergenceline will under not the cancel original b definition. in this work. Then, from the above results, we have definition proposed in this work. Therefore, the linear divergences cancel between the no-Wi by refs. [ JHEP05(2018)142 , ]. SPIRE ommons ]. IN ][ EQ-TEA ]. ]. SPIRE ]. ]. IN ]. ][ SPIRE SPIRE SPIRE SPIRE ]. IN IN IN , SPIRE IN ]. , ][ ][ (2013) 262002 , IN , ][ ][ ][ SPIRE (2015) 074009 110 arXiv:1412.3401 ]. ]. redited. SPIRE [ IN IN ]. ][ ][ D 91 SPIRE SPIRE ]. arXiv:1511.05242 IN IN [ SPIRE ][ ][ IN (2015) 112 arXiv:1310.7471 ][ SPIRE [ arXiv:1404.6680 arXiv:1609.07968 [ Quasi-parton distribution functions: a study [ IN arXiv:1506.00248 [ Phys. Rev. Lett. arXiv:1509.08016 ][ One-Loop Matching for Generalized Parton Camb. Monogr. Part. Phys. Nucl. Phys. Phys. Rev. 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