We compute the O(1/N_f) correction to the predominantly gluonic flavour
singlet twist-2 anomalous dimension used in polarized deep inelastic
scattering. It is consistent with known two loop perturbation theory and we
determine the three loop contribution at O(1/N_f). The treatment of the
epsilon-tensor in the large N_f d-dimensional critical point formalism is also
discussed.Comment: 8 Latex pages, 1 postscript figur

Ahmed

Altarelli

Broadhurst

Gracey

Hasenfratz

Hooft

J.A. Gracey

J.F. Bennett

Kaplan

Kodaira

Larin

Mertig

Trueman

van Ritbergen

Physics Letters B

English

arXiv

Crossref

Anomalous dimension of gluonic operator in polarized deep inelastic scattering at O(1/Nf)

We compute the O(1/N_f) correction to the predominantly gluonic flavour
singlet twist-2 anomalous dimension used in polarized deep inelastic
scattering. It is consistent with known two loop perturbation theory and we
determine the three loop contribution at O(1/N_f). The treatment of the
epsilon-tensor in the large N_f d-dimensional critical point formalism is also
discussed

Bennett, JF

Gracey, JA

University of Liverpool Repository

arXiv:hep-ph/9803446v1  25 Mar 1998Anomalous dimension of gluonic operator in polarized deepinelastic scattering at O(1/Nf )J.F. Bennett & J.A. Gracey,Theoretical Physics Division,Department of Mathematical Sciences,University of Liverpool,Peach Street,Liverpool,L69 7ZF,United Kingdom.Abstract. We compute the O(1/Nf ) correction to the predominantly gluonic flavour singlettwist-2 anomalous dimension used in polarized deep inelastic scattering. It is consistent withknown two loop perturbation theory and we determine the three loop contribution at O(1/Nf ).The treatment of the ǫ-tensor in the large Nf d-dimensional critical point formalism is alsodiscussed.LTH-4241There has been intense experimental activity in recent years into understanding the originof the spin of the nucleons. This is primarily because of the EMC effect, [1], which appearedto suggest that the proton spin was due to a sea quark effect with no significant contributionfrom the valence quarks. Consequently there has been theoretical interest in the problem. Forexample, one avenue of research has been to improve theoretical predictions from perturbativeQCD by going to higher loop orders, [2]. A problem related to this concerns the constructionof the perturbative sector of the operator product expansion used in polarized deep inelasticscattering which is important, for example, for current and future HERA experiments. Unlikethe unpolarized sector, polarized calculations of twist-2 operator dimensions have only recentlybecome available at two loops, [3, 4], as a function of the operator moment, n, or equivalentlyas a function of x, the conjugate variable of the Mellin transform in the DGLAP formalismof Dokshitzer, Gribov, Lipatov, Altarelli and Parisi, [5, 6]. Although the two loop results of[3, 4] represent progress for polarized scattering, building on the one loop results of [7], there isclearly a need to push calculations as far as is possible. Indeed for unpolarized scattering therelevant twist-2 operator dimensions are available exactly at three loops for the even moments,n ≤ 8, [8] (and additionally n = 10 for the flavour non-singlet case, [9]). Although the explicitn-dependence of the three loop result is the main goal, since it is needed to carry out the full twoloop renormalization group evolution, some progress has been made in that direction already.Recently, the finite parts of the two loop graphs have been determined in [10] as a function of x.These represent the first stage of a full three loop calculation since they will contribute to thefinal value when multiplied by one loop counterterms. Further, some information is availablefrom the large Nf expansion where the 1/Nf leading order non-singlet anomalous dimensionshave been deduced to all orders in the coupling, explicitly as a function of n, [11], with Nfthe number of quark flavours. (The method used is based on a critical point argument whereone considers the theory in the neighbourhood of a d-dimensional non-trivial zero of the β-function and computes critical exponents. The initial development was for the O(N) σ modelin [12]. Through the critical renormalization group equation these exponents are related to theperturbative renormalization group functions.) The restriction of [11] to compare with the exactthree loop non-singlet results of [9] gave agreement where there is overlap. In anticipation of afull polarized calculation at three loops the large Nf unpolarized calculations have been extendedin [13] to the polarized case. However, in the calculation of [13] only the dimension of the twist-2flavour singlet operator which was predominantly fermionic in nature, was considered. The aimof this article is to complete the gap in the twist-2 singlet polarized sector by computing the firstnon-trivial large Nf contribution to the flavour singlet operator which is gluonic in character.In providing this result we will not only be able to check the recent two loop results but alsobe able to provide the first insight into the n-dependence of the three loop dimension, albeit atO(1/Nf ). As the calculation of the unpolarized singlet gluonic operators was treated at length in[14], we will be brief concerning the technical details of the integration and focus on the salientdifferences which arise in the polarized case.We recall that the basic twist-2 flavour singlet polarized operators are, [7],Oµ1...µnq = in−1Sψ¯γ5γµ1Dµ2 . . . Dµnψ − trace termsOµ1...µng =12in−2Sǫµ1αβγ trGaβγDµ2 . . . Dµn−1Ga µnα − trace terms (1)where ψ is the quark field, Gaµν = ∂µAaν − ∂νAaµ + fabcAbµAcν , Aaµ is the gluon field, fabc are thecolour group structure functions, S denotes symmetrization over the Lorentz indices and ǫµνσρis a totally antisymmetric four dimensional pseudotensor. As has been observed in [7], since Oqand Og have the same dimension, they will mix under renormalization in perturbation theory.This leads to a mixing matrix of renormalization constants and hence anomalous dimensions2which we denote byγij(a) =(γqq(a) γgq(a)γqg(a) γgg(a))(2)where we use the same definition for the coupling constant, a, as [8]. To three loops the explicitNf dependence of each of the entries will beγqq(a) = a1a+ (a21N˜f + a22)a2 + (a31N˜2f + a32N˜f + a33)a3 +O(a4)γgq(a) = b1a+ (b21N˜f + b22)a2 + (b31N˜2f + b32N˜f + b33)a3 +O(a4)γqg(a) = c1N˜fa+ c2N˜fa2 + (c31N˜2f + c32N˜f + c33)a3 +O(a4)γgg(a) = (d11N˜f + d12)a+ (d21N˜f + d22)a2 + (d31N˜2f + d32N˜f + d33)a3 +O(a4) (3)where N˜f = T (R)Nf and the coefficients aij , bij, cij and dij will depend on n and the colour groupCasimirs. This form will be important when we apply the critical point formalism to deducethe O(1/Nf ) anomalous dimensions. Moreover, for concreteness and to clarify the conventionswe use, we recall the one loop polarized results of Ahmed and Ross are, [7],a1 = C2(R)[4S1(n) − 3 −2n(n+ 1)], b1 = −2(n+ 2)C2(R)n(n+ 1)c1 = −4(n− 1)T (R)n(n+ 1), d11 =43T (R) (4)where the finite sums Sl(n) are defined by Sl(n) =∑nr=1 1/rl. Although we have indicated theform of the mixing matrix in (2), one could in principle consider a basis where it is diagonal.Indeed it turns out that in the 1/Nf formalism of [13] that that method produces the anomalousdimensions of the twist-2 eigenoperators of (2). This is primarily due to the fact that thecanonical dimensions of the operators of QCD in perturbation theory are the same, whereascomputing at the fixed point of QCD in d-dimensional spacetime, where the large Nf expansionis constructed, the canonical dimensions are different. Indeed it is elementary to deduce thisdifference is O(ǫ) where d = 4 − 2ǫ. Further, it was shown in [13] that the critical exponent ofthe operator Of corresponded to the eigen-anomalous dimension of (1) with a predominantlyfermionic content evaluated at the non-trivial zero of the β-function. Likewise in [14], it wasdemonstrated that the remaining operator corresponded to the predominantly gluonic operator.More concretely, diagonalizing (2) produces the eigen-anomalous dimensionsλ±(a) =12(γqq + γgg) ±12[(γqq − γgg)2 + 4γqgγgq]12 (5)The critical value of the QCD coupling has been given before in [14] and isac =3ǫ4T (R)Nf+[3316C2(G)ǫ −(2716C2(R) +4516C2(G))ǫ2+(9964C2(R) +237128C2(G))ǫ3 +(7764C2(R) +53128C2(G))ǫ4]1T 2(R)N2f(6)where we have included the contribution from the recently computed four loop β-function, [15].Therefore substituting (6) in (5) and expanding in powers of 1/Nf , we find that the gluonicdimension to third order in ǫ is given byλ+(ac) =3d11ǫ4+1T (R)Nf[34(d12 +b1c1d11+114C2(G)d11)ǫ+916(d21 +b21c1d11− (3C2(R) + 5C2(G)) d11)ǫ2+2764(d31 +b31c1d11+(113C2(R) +7918C2(G))d11)ǫ3]+ O(ǫ4N2f)(7)3To reproduce this at O(ǫ2) together with the subsequent terms at O(1/Nf ) one applies thecritical point formalism to the operator Og which exploits the fixed point equivalence of QCDwith the non-abelian Thirring model, [16], at leading order in 1/Nf . As the method has beenwell documented, [14], we will comment only on those technical details of the computation whichare peculiar to Og. Indeed as the calculation virtually parallels that of the unpolarized case werefer the interested reader to [14]. There details of the integration procedure for the Feynmandiagrams have been recorded, including the construction and solution of the recurrence relationsfor deducing the contributions from subintegrals in the graphs. These have been implemented inthe symbolic manipulation language form, [17], which was used to perform the tedious algebrain the integration. The four contributing Feynman graphs for the O(1/Nf ) part of (7) areillustrated in fig. 1, where the operator insertion is indicated on an internal gluon line. We haveused the same Feynman rule for the insertion and conventions as [3].One technical issue deserves detailed comment. As has been noted elsewhere, [18], onehas to deal with the problem of treating an object, such as ǫµνσρ, which is defined only infour dimensions, in a formalism which is manifestly d-dimensional. Such a situation arisesnot only in large Nf calculations but also in explicit perturbative multiloop computations. Itsresolution in the latter case is to treat products of two ǫ-tensors, where they arise, as thedeterminant of a 4 × 4 matrix of ηµν tensors, [19]. As the latter can have a d-dimensionalinterpretation one can perform a calculation without reference to the original ǫ-tensors. Tocompensate for the obvious discrepancy and to restore the Ward identity, one introduces anadditional (finite) renormalization constant, [20]. This approach was developed for the treatmentof the renormalization of the singlet axial vector current which has an anomaly, [20, 21]. Asimilar approach is adopted for 1/Nf calculations, [22], which has already been applied to thecalculation of the exponent corresponding to the anomalous dimension of the axial vector current,[13]. For each graph of the present calculation, we project out the contribution to the exponentby multiplying by ǫµνσρ∆σpρ, where ∆µ is a null 4-vector used to define the Feynman rules onthe light cone and pµ is the external momentum flowing through each diagram. Therefore eachintegral contains the product of two ǫ-tensors and like [20] we replace these by a determinant ofηµν tensors and compute in d-dimensions. However, we also need to introduce a compensatingfactor since not only is ǫµνσρ four dimensional but the vectors ∆µ and pµ are strictly physicalobjects and cannot be extended to d-dimensions, [18]. In the split γ-algebra notation of [18], ∆µ≡ ∆µ¯ and pµ ≡ pµ¯ where the bar denotes the four dimensional subspace of the d-dimensionalextended spacetime, which corresponds to the physical spacetime in the limit as d → 4. Tosummarize we have replaced the correct ǫ-tensor combination given on the right side ofǫµνσρǫαβγδ∆σpρ∆γpδ = f(µ)ǫµνσ¯ρ¯ǫαβγ¯δ¯∆σ¯pρ¯∆γ¯pδ¯ (8)by the object on the left side which is formally d-dimensional and involves the η-determinant.The discrepancy in this procedure is represented by a compensating function of the spacetimedimension. We have computed this proportionality function, f(µ), by considering contractionsof the d-dimensional object and comparing it with similar contractions of the object restrictedto 4-dimensional spacetime. We foundf(µ) = (2µ− 3)(µ − 1) (9)To check that this is a correct procedure we computed the first graph of fig. 1 by using thedeterminant substitution approach and then compared the result with that obtained by directcalculation of the integral. Such a computation is possible in this case since the graph is ineffect a series of chain integrals in the language of [12]. It turns out that by inspecting each ofthe other graphs a combination which is effectively the same as (8) arises and therefore (9) isused for each of the graphs. Moreover, this argument was also used to justify the inclusion of a4similar factor in the large Nf calculation of the anomalous dimension of the singlet axial vectorcurrent, [13]. The final expression for that case was consistent with three loop perturbationtheory, [21, 19].In light of these remarks, we can now quote the final value we obtained for λ+,1(ac) whereλ+(ac) =∑∞i=0 λ+,i(ac)/Nif . After evaluating the contribution from each graph we findλ+,1(ac) = −(n+ 2)(n − 1)Γ(n + 2− µ)Γ(µ+ 1)C2(R)ηo1(µ− 2)2(n+ 1)Γ(2 − µ)Γ(µ+ n)nT (R)+2µ(µ− 1)S1(n)C2(G)ηo1(2µ− 1)(µ− 2)T (R)− [4µ3n2 + 4µ3n− 8µ3 − 8µ2n2 − 8µ2n+ 16µ2 + 5µn2+ 5µn− 9µ− n2 − n+ 2]Γ(n + 2− µ)Γ(µ)µC2(G)ηo1/[8(2µ − 1)(µ− 1)3(n+ 1)Γ(3 − µ)Γ(µ+ n)nT (R)]− [32µ4 − 4µ3n2 − 4µ3n− 120µ3 + 16µ2n2 + 16µ2n+ 160µ2 − 20µn2 − 20µn− 89µ+ 8n2 + 8n+ 18]µC2(G)ηo1/[8(2µ − 1)(µ− 1)3(µ− 2)(n + 1)nT (R)] (10)where we have used reduce, [23], to handle tedious amounts of algebra. The quantity ηo1 isdefined byηo1 =(2µ− 1)(µ− 2)Γ(2µ)4Γ2(µ)Γ(µ+ 1)Γ(2 − µ)(11)There are various checks on the validity of (10). First, we find that expanding (10) in powersof ǫ it agrees with two loop perturbation theory, [7, 3, 4]. From one point of view this justifiesthe use of (8), since (9) is ǫ-dependent and therefore will affect the form of the O(ǫ2) termsof (10) when it is expanded. Second, if one substitutes n = 1 in (10), we find that it vanishesidentically. This is consistent with the observations of [24, 25], where it was shown that the axialanomaly forces the n = 1 gluonic operator to have vanishing anomalous dimension. Thereforewe are confident that (10) is correct and can make an observation on the form of the 3-loopcontribution to the gluonic eigen-anomalous dimension. By expanding (10) to O(ǫ3) and using(7), we find thatd31 +b31c1d11= −64(7n2 + 7n+ 3)(n + 2)(n − 1)S1(n)C2(R)9(n+ 1)3n3+64(n + 2)(n − 1)S21(n)C2(R)3(n + 1)2n2− 4[33n8 + 132n7 + 142n6 − 36n5 − 263n4 − 312n3+ 280n2 + 408n + 144]C2(R)/[27(n + 1)4n4]−8(8n4 + 16n3 − 19n2 − 27n+ 48)S1(n)C2(G)27(n + 1)2n2−2(87n6 + 261n5 + 249n4 + 63n3 − 76n2 − 64n − 96)C2(G)27(n + 1)3n3(12)Evaluating this at n = 1, one obtains a value which, from (7), is correctly proportional to thethree loop coefficient of ac. As an aid to future explicit perturbative calculations similar to[8], we have evaluated (12) for odd n, n ≤ 23, and listed the results in table 1. Unlike in the5unpolarized calculations, here it is the odd moments which are important since the originaloperator does not exist when n is even as can be seen, for example, from the Feynman rules, [3].A further remark on (12) concerns the status of its C2(G) sector. In the unpolarized case, [14],it was noted that at O(1/Nf ) b31 will only involve contributions proportional to C2(R) and notC2(G). Assuming the situation is similar here, this implies that the terms involving C2(G) in(12) are in fact the correct n-dependent part of the coefficient set d31 of the original perturbativeoperator. Finally, several general remarks on the overall structure are in order. First, the fullexpression is much simpler in form than that of the unpolarized case, [14]. Indeed this was alsoobserved in the fermionic singlet sector as well. Second, in the evaluation of each integral anextra n-dependent structure was present which was proportional to Γ(µ−1)Γ(n)/Γ(µ+n). Thisis similar to the unpolarized situation, [14], and like in that calculation, it also cancels in thefinal summation.To conclude with we note that all the 1/Nf leading order terms are now available for theanomalous dimensions of the twist-2 operators used in polarized deep inelastic scattering. Thenext stage in this 1/Nf programme would be to consider the unpolarized flavour non-singletsector and compute the O(1/N2f ) correction to the dimension of the twist-2 operator as a functionof n.Acknowledgements. This work was carried out with the support of PPARC through aPostgraduate Studentship (JFB) and an Advanced Fellowship (JAG).6References.[1] J. Ashman et al (EMC), Phys. Lett. B206 (1988), 364; Nucl. Phys. B328 (1989), 1; V.W.Hughes et al, Phys. Lett. B212 (1988), 511; M.J. Alguard et al (SLAC), Phys. Rev. Lett.37 (1978), 1262; Phys. Rev. Lett. 41 (1978), 70.[2] S.A. Larin, T. van Ritbergen & J.A.M. Vermaseren, Phys. Lett. B404 (1997), 153.[3] R. Mertig & W.L. van Neerven, Z. Phys. C70 (1996), 637.[4] W. Vogelsang, Phys. Rev. D54 (1996), 2023; Nucl. Phys. B475 (1996), 47.[5] V.N. Gribov & L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972), 438; L.N. Lipatov, Sov. J.Nucl. Phys. 20 (1975), 95; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977), 641.[6] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977), 298;[7] M.A. Ahmed & G.G. Ross, Nucl. Phys. B111 (1976), 441.[8] S.A. Larin, P. Nogueira, T. van Ritbergen & J.A.M. Vermaseren, Nucl. Phys. B492 (1997),338.[9] S.A. Larin, T. van Ritbergen and J.A.M. Vermaseren, Nucl. Phys. B427 (1994), 41.[10] Y. Matiounine, J. Smith & W.L. van Neerven, hep-ph/9801224; hep-ph/9803439.[11] J.A. Gracey, Phys. Lett. B322 (1994), 141.[12] A.N. Vasil’ev, Yu.M. Pis’mak & J.R. Honkonen, Theor. Math. Phys. 46 (1981), 157; A.N.Vasil’ev, Yu.M. Pis’mak & J.R. Honkonen, Theor. Math. Phys. 47 (1981), 291; A.N. Vasil’ev& M.Yu. Nalimov, Theor. Math. Phys. 55 (1982), 163; 56 (1983), 15.[13] J.A. Gracey, Nucl. Phys. B480 (1996), 73.[14] J.F. Bennett & J.A. Gracey, hep-ph/9710364 to appear in Nucl. Phys. B.[15] T. van Ritbergen, J.A.M. Vermaseren & S.A. Larin, Phys. Lett. B400 (1997), 379.[16] A. Hasenfratz & P. Hasenfratz, Phys. Lett. B297 (1992), 166.[17] J.A.M. Vermaseren, “Form” version 2.3, CAN publication, (1992).[18] G. ’t Hooft & M. Veltman, Nucl. Phys. B44 (1972), 189.[19] S.A. Larin, Phys. Lett. B303 (1993), 113.[20] T.L. Trueman, Phys. Lett. B88 (1979), 331.[21] J. Kodaira, Nucl. Phys. B165 (1980), 129.[22] D.J. Broadhurst & A.L. Kataev, Phys. Lett. B315 (1993), 179.[23] A.C. Hearn, “Reduce Users Manual” version 3.4, Rand publication CP78, (1991).[24] D.B. Kaplan & A. Manohar, Nucl. Phys. B310 (1988), 527.[25] G. Altarelli & G.G. Ross, Phys. Lett. B212 (1988), 391.7n C2(R) coefficient C2(G) coefficient1 − 449− 158273 − 5527972− 23422435 − 716824151875− 341683303757 − 15575276334574400− 45247033704409 − 2698556278602791875− 29607251229635011 − 28232856616762758939320− 182233283313584186013 − 50772702510711120171987935− 2787601430992014295283015 − 915045986064491200354394240000− 16575124393116756640017 − 18938338446769024941180834365746075− 936055607761764532854386019 − 4489933300070894563970409113734060000− 1074675403337372725291100021 − 934060610086114519200816623282959684− 310974856859920703406323623 − 12883853884052147853727573179435993423232− 777007271915095098296739536Table 1. Coefficients of [d31 + b31c1/d11] as a function of moment.8Fig. 1. Leading order diagrams for λ+(ac).9

Anomalous dimension of gluonic operator in polarized deep inelastic scattering at O(1/N<sub>f</sub>)

Anomalous dimension of gluonic operator in polarized deep inelastic
  scattering at O(1/N_f)

Anomalous dimension of gluonic operator in polarized deep inelastic scattering at O(1/N_f)

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