QCD Evolutions of Twist-3 Chirality-Odd Operators
aa r X i v : . [ h e p - ph ] O c t QCD Evolutions of Twist-3 Chirality-Odd Operators
J.P. Ma , , Q. Wang and G.P. Zhang Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100190, China Center for High-Energy Physics, Peking University, Beijing 100871, China Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu210097, P.R.China School of Physics, Peking University, Beijing 100871, China
Abstract
We study the scale dependence of twist-3 distributions defined with chirality-odd quark-gluonoperators. To derive the scale dependence we explicitly calculate these distributions of multi-partonstates instead of a hadron. Taking one-loop corrections into account we obtain the leading evolutionkernel in the most general case. In some special cases the evolutions are simplified. We observe that theobtained kernel in general does not get simplified in the large- N c limit in contrast to the case of thosetwist-3 distributions defined only with chirality-odd quark operators. In the later, the simplificationis significant. Predictions can be made from QCD with the concept of factorizations for processes with large momen-tum transfers. A typical example is DIS. For unpolarized DIS, the differential cross section at the leadingpower of the momentum transfer Q is predicted as a convolution of perturbative coefficient functionswith parton distribution functions(PDF’s). PDF’s are defined with twist-2 QCD operators and describenonperturbative effects of hadrons. Although PDF’s can not be predicted with perturbative QCD, butthey scale dependence, hence the Q -dependence of the differential cross section, can be determined byperturbation theory. The dependence is governed by the famous DGLAP equation. In the past, thepredicted scale-dependence or DGLAP equation has played an indispensable role for testing QCD as thecorrect theory of strong interaction.In general, factorized differential cross sections also contain contributions involving hadronic matrixelements of higher-twist operators. Although these contributions are suppressed by inverse powers of Q ,they contain more informations about inner structure of hadrons. Among them, the most interestingare those involving twist-3 operators. These contributions are responsible for certain asymmetries indifferential cross sections. These asymmetries can be measured in experiment and hence provide infor-mation about hadronic matrix elements of twist-3 operators. A well-known example is the study of Singletransverse-Spin Asymmetries(SSA). The asymmetries can be factorized with the ETQS matrix elementsdefined with chirality-even quark-gluon operators at twist-3[1, 2]. The scale dependence of these twist-3 operators have been studied in [3, 4, 5, 6, 7, 8]. Besides them, there are chirality-odd quark-gluonoperators at twist-3. In this work, we study the scale dependence of these operators.We consider a spin-1/2 hadron moving in the z -direction with the momentum P µ = ( P + , P − , , a µ is expressed as a µ = ( a + , a − , ~a ⊥ ) =(( a + a ) / √ , ( a − a ) / √ , a , a ) and a ⊥ = ( a ) + ( a ) . In this coordinate system we introduce twolight-cone vector: n µ = (0 , , ,
0) and l µ = (1 , , , T F ( x , x ) = g s Z dy dy π e − iy x P + + iy x P + h P | ¯ ψ ( y n ) (cid:0) iγ ⊥ µ γ + (cid:1) G + µ (0) ψ ( y n ) | P i ,λ ˜ T ∆ ( x , x ) = g s Z dy dy π e − iy x P + + iy x P + h P, λ | ¯ ψ ( y n ) (cid:0) iγ ⊥ µ γ + γ (cid:1) G + µ (0) ψ ( y n ) | P, λ i , (1)where the matrix element in the first line is spin-averaged and that in the second line is of a longitudinallypolarized hadron with the helicity λ = ± / x , are momentum fractions. The definitions are given inthe light-cone gauge n · G = 0. In other gauges gauge links along the direction n should be supplementedto make the definitions gauge invariant. From general principle one can show that the two distributionsare real and: ˜ T F ( x , x ) = ˜ T F ( x , x ) , ˜ T ∆ ( x , x ) = − ˜ T ∆ ( x , x ) . (2)Replacing the gluon field strength tensor G + µ (0) in Eq.(1) with the covariant derivative D µ = ∂ µ + ig s G µ (0), one can obtain another two twist-3 distributions. With equation of motion one can relate thosetwo distributions to the two distributions defined in Eq.(1), respectively (see, e.g., [9, 10, 14]).The operators in Eq.(1) contains the operator of the gluon field strength tensor. There are twochirality-odd twist-3 operators which only consist of quark field operators. Correspondingly, one canalso define two distributions. One is e ( x ) for unpolarized hadrons, another is h L ( x ) for longitudinallypolarized hadrons. Again, these two distributions are not independent. One can use operators identitiesin [11] or equation of motion to show that these two are related with the above two and plus somecontributions with local operators. The evolution of e ( x ) and h L ( x ) have been studied in [12, 13, 14]and the evolution equations have been solved in moment space. The evolution of twist-3 quark-gluonoperators has been studied in [15] with the emphasis on the solutions of evolution equations. In this workwe derive the evolution kernel for the two twist-3 distributions defined in Eq.(1) in momentum fractionspace with a different method.Under renormalization there is no mixing between the two operators in Eq.(1). The evolution kernelscan be conveniently expressed with one function by introducing the combinations:˜ T ± ( x , x , µ ) = h P, ±|O ( x , x ) | P, ±i = ˜ T F ( x , x , µ ) ± ˜ T ∆ ( x , x , µ ) , O ( x , x ) = g s Z dy dy π e − ix y P + + ix y P + ψ ( y n ) iγ ⊥ µ γ + (1 + γ ) G + µ (0) ψ ( y n ) . (3)The functions T ± ( x , x ) are nonzero in the region | x , | ≤ | x − x | ≤
1. The scale-dependencecan be written in the form: ∂∂ ln µ ˜ T ± ( x , x , µ ) = α s π Z dξ dξ F ± ( x , x , ξ , ξ ) ˜ T ± ( ξ , ξ , µ ) . (4)The integration region of ξ , is fixed by the support of ˜ T ± . It is easy to find that F + is related to F − .Here we will take F − to give our results.The distributions are defined for a given hadron, but the kernel does depend on the hadron. It iscompletely determined by the dynamics of QCD. For large µ it can be calculated with perturbativeQCD. Because of this one can use various parton states to calculate the distribution ˜ T − to find its µ -dependence, hence the kernel F − . For the case with operators of twist-2 one can use single-parton statefor the purpose. But for the two distributions defined here, one simply finds null results with single-partonstates because the chirality is flipped by the operators. Therefore, one has to use multi-parton statesto calculate the distribution. By using multi-parton states factorizations of SSA with twist-3 operators2ave been studied in [16, 17, 18]. In [7] such multi-parton states have been employed to study the scaledependence of twist-3 operators relevant for SSA.We introduce the following state | n ( λ ) i as a superposition of single- and multi-parton states: | n ( λ ) i = | q i + c | qG i + c | qGG i + c | q ¯ qq i + · · · . (5)The state | n i is with the momentum ( P + , , , z -direction and the sum of their momenta is that of | n i . The sum of helicity of partons is λ . In the abovewe have suppressed quantum numbers of partons, which will be specified later. In principle one canintroduce wave functions depending on momenta of partons. For simplicity, we take these wave-functionsas δ -functions and hence c i ( i = 1 , , · · · ) are real constants. If we calculate ˜ T − of the state | n ( λ ) i instead of a hadron, we obtain nonzero contributions from interference between different partonic states.At tree-level, the contributions can be schematically written as:˜ T − ( x , x ) = C h q ( − ) |O ( x , x ) | q (+) g ( − ) i + C h q ( − ) g (+) |O ( x , x ) | q (+) i + C h q ( − )¯ q ( − ) |O ( x , x ) | g ( − ) i + C h g (+) |O ( x , x ) | q (+)¯ q (+) i , (6)where ± in brackets indicate the helicity of partons. It should be noted that there are possible spectators.E.g., in the first term, the spectators must carry the total helicity λ s = 0 which can be a quark pairor gluon pair. In the second term the spectators must have λ s = − λ = − / λ s = ± /
2, respectively. The contributions from spectators give overallfactors as products of δ -functions for each term. These overall factors are contained in the constants C i which also depend on c i in Eq.(5). Because of existence of spectator-partons, the parton states in theabove are not necessarily with the total momentum P .Beyond tree-level, the four matrix elements of partons in Eq.(6) receive corrections. These correctionsmake ˜ T − µ -dependent. We define the four matrix elements as: T − qg ( x , x , y , z ) = h q ( p, − ) |O ( x , x ) | q ( p , +) g ( p , − ) i ,T + qg ( x , x , y , z ) = h q ( p , − ) g ( p , +) |O ( x , x ) | q ( p, +) i ,T − q ¯ q ( x , x , y , z ) = h ¯ q ( p , − ) q ( p , − ) |O ( x , x ) | g ( p, − ) i ,T + q ¯ q ( x , x , y , z ) = h g ( p, +) |O ( x , x ) | q ( p , +)¯ q ( p , +) i , (7)with the momenta: p +1 = y P + , p +2 = ( z − y ) P + , p + = z P + . (8)The color of the state | qg i is the same as the single-quark state. Details can be found in [16, 18]. Thefour matrix elements are not independent. They are pair-wise related: T − qg ( x , x , y , z ) = T + qg ( x , x , y , z ) , T − q ¯ q ( x , x , y , z ) = T + q ¯ q ( x , x , y , z ) . (9)As we will see, the µ -dependence of the four matrix elements determine the kernel F − in different regionsof ξ , . Because of this, the determined kernel will not depend on the states, i.e., the coefficients c i and C i . To determine the kernel in the full region of ξ , , one also needs to calculate ˜ T − of the state | ¯ n i –The charge-conjugated state of | n i . However, ˜ T − of | ¯ n i can be obtained from ˜ T − of | n i through chargeconjugation. 3 a ) ( b ) ( c ) ( d )( f ) ( g ) ( h ) ( i ) ( e )( j )( k ) ( l ) ( m ) ( n ) ( o ) Figure 1: A set of diagrams of one-loop corrections to the defined twist-3 matrix elements. This set onlycontains the self-energy corrections and corrections from the gluon emission from a gauge link representedby a double line.We take T − qg as an example to show how the corresponding contribution to F − is determined. Attree-level, we have: T (0) − qg ( x , x , y , z ) = 2 πg s p z y ( N c − z − y ) δ ( z − x ) δ ( x − y ) . (10)It should be noted z > y because of p +2 >
0. If we have the one-loop result T (1) − qg , we can substitute theresults of T (0 , − qg into Eq.(4) through Eq.(6). Then we can find for ξ > g s α s F − ( x , x , ξ , ξ ) θ ( ξ − ξ ) = 12( N c − ξ − ξ ) √ ξ ξ ∂∂ ln µ T (1) − qg ( x , x , ξ , ξ ) . (11)The θ -function θ ( ξ − ξ ) appears because of z − y >
0. Therefore, F − in the region of ξ > ξ and ξ > T (1) − qg . Similarly, one can find that F − in the region of ξ > ξ and ξ > T (1)+ qg . Combining the two contributions one has the kernel in the region of positive ξ , .The one-loop correction to T − qg are from the diagrams given in Fig.1 and Fig.2. The calculationof these diagrams is rather standard. Therefore we give the result directly. We introduce the followingfunction which is just the kernel in the region of ξ , > F ( x , x , ξ , ξ ) = θ ( x ) (cid:20) δ ( ξ − x ) f ( ξ , x , x ) + δ ( ξ − ξ − x + x ) f ( ξ , x , x )+ δ ( ξ − x ) f ( ξ , x , x ) (cid:21) + ( x ↔ x , ξ ↔ ξ ) , (12)4 a ) ( b ) ( c ) ( d ) ( e )( f ) ( g ) ( h ) ( i ) Figure 2: Another set of diagrams for one-loop corrections of the defined twist-3 matrix elements.with f ( ξ, x , x ) = 12 N c x θ ( ξ − x ) ξ ( x − ξ ) + ,f ( ξ, x , x ) = N c (cid:18) θ ( x − x ) θ ( x − ξ ) − θ ( x − x ) θ ( ξ − x ) (cid:19)(cid:18) x − ξ ) + − x − x ( x − ξ ) − x − ξ (cid:19) + N c θ ( ξ − x ) x ξ (cid:18) ξ − x ) + − ξ − x (cid:19) + 12 δ ( ξ − x ) (cid:20) N c − N c − N c − N c ln x − N c ln | x − x | (cid:21) ,f ( ξ, x , x ) = − N c ξ − x ) θ ( x ) (cid:20) θ ( x − ξ ) θ ( x − x ) x − x x ( ξ − x ) (3 x − x − ξ )+2 θ ( ξ − x ) θ ( x − x ) x ξx ( x − x − ξ ) − θ ( ξ − x ) θ ( x − x ) 2 x x − ξx − x + x ξ + 2 ξ ξ ( ξ − x ) (cid:21) + N c − N c θ ( x − x ) θ ( x ) x ( x − x ) x ( ξ − x )+ θ ( x − x )2 N c ( ξ − x ) ( x − x − ξ ) (cid:20) − ξ − x + x ξ ( ξ − x ) θ ( − x ) θ ( ξ − x + x ) − θ ( x ) (cid:18) − x θ ( x − ξ − x ) x ( ξ − x ) + x − x ξx θ ( ξ − x + x ) (cid:19)(cid:21) . (13)The +-distributions here are defined as: Z x dz f ( z )( x − z ) + = Z x dz f ( z ) − f ( x ) x − z + f ( x ) ln x, Z x dz f ( z )( z − x ) + = Z x dz f ( z ) − f ( x ) z − x + f ( x ) ln(1 − x ) . (14)It should be noted that our +-distribution is not the standard +-distribution defined later. The function f and f are determined by the contributions from Fig.1. In general for each diagram with a gluonemitted from the gauge link there is a light-cone singularity. But, the sum is free from the singularity.5ow we turn to the contributions from T − q ¯ q . At tree-level we have T (0) − q ¯ q ( x , x , y , z ) = − πg s ( N c − q y ( z − y ) z δ ( x − y ) δ ( x + z − y ) . (15)With the one-loop result of T − q ¯ q one can determine the kernel g s α s F − ( x , x , ξ , ξ ) θ ( ξ ) θ ( − ξ ) = − N c − ξ − ξ ) √− ξ ξ ∂∂ ln µ T (1) − q ¯ q ( x , x , ξ , ξ − ξ ) , (16)The kernel determined from T − q ¯ q is in the region of ξ > ξ <
0. From T + q ¯ q the kernel in theregion ξ < ξ > T − q ¯ q is still given by the diagramsin Fig.1 and Fig.2, where the quark line in the left side should be taken as an outgoing anti-quark. Thecalculation of these diagrams are again straightforward. The result can be summarized by the function F ( x , x , ξ , ξ ) = δ ( ξ − x ) θ ( x ) (cid:18) h ( − ξ , x , x ) + h ( − ξ , − x , − x ) (cid:19) + δ ( ξ − x ) θ ( − x ) (cid:18) h ( ξ , − x , − x ) + h ( ξ , x , x ) (cid:19) + δ ( ξ − ξ − x + x ) (cid:18) θ ( − x ) h ( − ξ , x , x ) + θ ( x ) h ( ξ , − x , − x ) (cid:19) , (17)with the functions h ( ξ, x , x ) = 12 N c θ ( x + ξ ) x ξ ( ξ + x ) + ,h ( ξ, x , x ) = N c (cid:20) θ ( − x ) θ ( x + ξ ) (cid:18) ξ + 2 x − x ( ξ + x ) − x ξ ( ξ + x ) (cid:19) x − x ( ξ + x ) + + θ ( x − x ) θ ( x ) ξ + 2 x − x ( ξ + x ) x − x ξ + x (cid:21) + 12 θ ( − x ) δ ( x + ξ ) (cid:20) N c − N c − N c − N c ln | x | − N c ln | x − x | (cid:21) ,h ( ξ, x , x ) = − N c ξ − x ) (cid:20) θ ( x ) θ ( ξ − x ) − ξ − x ξ + 2 x + 3 ξx − x x ξ + θ ( − x ) θ ( x − x ) x − x x (2 ξ + 2 x − x ) (cid:21) + 12 N c ξ + x − x ( ξ − x ) θ ( − x ) θ ( ξ − x + x ) (cid:20) θ ( x − x ) x x + θ ( x − x ) ξ − x + x ξ (cid:21) + N c − N c θ ( − x ) θ ( x − x ) x ( x − x ) x ( ξ − x ) . (18)The function F is just the kernel in the region of ξ > ξ <
0. The kernel in the region of ξ < ξ > T + q ¯ q through the relation in Eq.(9).Finally, we have the kernel in the full region of ξ , as: F − ( x , x , ξ , ξ ) = θ ( ξ ) θ ( − ξ ) F ( x , x , ξ , ξ ) + θ ( ξ ) θ ( − ξ ) F ( x , x , ξ , ξ )+ θ ( ξ ) θ ( ξ ) F ( x , x , ξ , ξ ) + θ ( − ξ ) θ ( − ξ ) F ( − x , − x , − ξ , − ξ ) . (19)6he kernel in the region with ξ , < ξ , > T F and ˜ T ∆ are not mixed under renormalization,the correct kernel should satifiy: F − ( x , x , ξ , ξ ) = F − ( x , x , ξ , ξ ) . (20)Our result has the property. Therefore, taking the part of F − symmetric between x and x , one canobtain the evolution of ˜ T F as a convolution with ˜ T F . The anti-symmetric part of F − gives the kernel ofthe evolution of ˜ T ∆ .The evolution kernel for arbitrary x , is rather lengthy. But the convolution in Eq.(4) is a one-dimensional integral at one-loop. In some special cases, the kernel is simplified. E.g., for x = x = x > ∂∂ ln µ ˜ T F ( x, x, µ ) = α s π (cid:26) − N c + 34 N c ˜ T F ( x, x, µ ) + Z x dzz − z ) + (cid:18) N c ˜ T F ( x, ξ ) − zN c ˜ T F ( ξ, ξ ) (cid:19) + 1 N c Z − x dξ ξ ( ξ + x ) ˜ T F ( x, − ξ ) (cid:27) ,∂∂ ln µ ˜ T ∆ ( x, x, µ ) = 0 (21)with z = x/ξ . Here the +-distribution is the standard one defined as: Z dz θ ( z − x )(1 − z ) + t ( z ) = Z x dz t ( z ) − t (1)1 − z + t (1) ln(1 − x ) . (22)The result in Eq.(21) agrees with that in [8]. There are also cases in which a quark carries zero momentumfraction entering a hard scattering. In these cases, one has either x = 0 or x = 0. The evolutions inthese cases are: ∂∂ ln µ ˜ T − (0 , x, µ ) = α s π (cid:26) Z x dzz (cid:20) (cid:18) − N c z + z ) ˜ T − (0 , ξ ) + 12 N c ˜ T − ( ξ, x ) + 12 N c ˜ T − ( ξ − x, ξ ) (cid:19) + 1(1 − z ) + (cid:18) N c ˜ T − (0 , ξ ) + N c T − ( x − ξ, x ) − N c ˜ T − ( ξ − x, ξ ) (cid:19) + (1 − z ) N c ˜ T − (0 , − ξ ) + 3( N c − N c ˜ T − (0 , x ) (cid:27) . (23)Changing every ˜ T − ( x , x ) to ˜ T − ( x , x ) in the above equation, we obtain the evolution of ˜ T − ( x, e ( x ) and h L ( x ). But the evolution of the distribution e ( x ) or h L ( x ) does mix with ˜ T F and ˜ T ∆ [12, 13, 14]. One may derive the evolution of e ( x ) and h L ( x ) throughtheir relations to ˜ T F and ˜ T ∆ , respectively. But the derivation is very tedious. It has been shown in [12]that in the large- N c limit the evolution of e ( x ) or h L ( x ) is simplified and obeys DGLAP-type equations.The result for this in momentum fraction space can be found in [14]. These evolutions in the large- N c limit can also be derived with the approach used here. We take e ( x ) as an example to give our result. e ( x ) is defined in the light-cone gauge as: e ( x ) = P + Z dy π e − iyxP + h P | ¯ ψ ( yn ) ψ (0) | P i . (24)7o derive the evolution of the two-parton twist-3 distribution in the large- N c limit one can use the sameapproach for deriving the evolution of three-parton twist-3 distributions. The derivation is straightfor-ward. We have for x > ∂e ( x ) ∂ ln µ = α s N c π (cid:20) e ( x ) + Z x dzz e ( ξ ) (cid:18) − z ) + + 1 (cid:19)(cid:21) + O ( N − c ) , (25)with z = x/ξ . This result agrees with that in [14]. Although the evolution of e ( x ) and h L ( x ) getssimplified in the large- N c limit, the evolutions of ˜ T F, ∆ are not much simplified in the limit. This can beobserved from our results. This supports the observation made in [19] that simplifications of evolutionsof higher-twist distributions can be accidental.To summarize: We have calculated the twist-3 distributions of multi-parton states. The distributionsare defined with chirality-odd quark-gluon operators. The evolution kernel of the distributions has beenobtained by the calculation at one-loop. The evolution at one-loop is an one-dimensional convolution.In some special cases the evolution takes a short form. We have also derived the evolution of e ( x ) in thelarge- N c limit and found an agreement with existing results. Acknowledgments
The work of J.P. Ma is supported by National Nature Science Foundation of P.R. China(No. 10975169,11021092, 11275244), and the work of Q. Wang is supported by National Nature Science Foundation ofP.R. China(No.10805028).
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