Bennett, JF

Gracey, John

English

University of Liverpool Repository

arXiv:hep-ph/9808380v1  21 Aug 1998Large Nf calculations in deep inelastic scatteringJ.F. Bennett∗& J.A. Gracey,Theoretical Physics Division,Department of Mathematical Sciences,University of Liverpool,Peach Street,Liverpool,L69 7ZF,United Kingdom.Abstract. We describe the evaluation of the anomalous dimensions of twist-2 deep in-elastic light cone operators to O(1/Nf) as a check on future perturbative calculations. Inparticular we present recent results for the singlet gluonic operator dimension in polarisedand unpolarised scattering and give three loop predictions for the O(1/Nf) gluonic eigen-operator. The section of the 3-loop O(1/Nf) DGLAP splitting function proportional tothe adjoint quadratic Casimir is also calculated for the singlet gluonic operators.LTH-432∗Talk presented at Quarks ’98, Suzdal, Russia.11 Introduction.Our modern view of hadronic matter is based upon the rich quantum field theory ofQuantum Chromodynamics (QCD) (for a review see [1]) in which we picture hadrons asbound states of valence quarks in a sea of vacuum excitation quark/antiquark pairs andgluons. Because QCD has the remarkable property of asymptotic freedom [2] we can usehigh momentum transfer reactions such as deep inelastic scattering (lepton + nucleon−→ lepton + hadrons) to compare experiment with the predictions of perturbative QCD(pQCD) and so refine our ideas on the structure of strongly interacting particles.The formal QCD approach to deep inelastic scattering (DIS) allows us to predict struc-ture function behaviour using moment sum rules. Key to this method is the formalismof the light cone expansion (LCE) in which the non-local time ordered product of elec-tromagnetic quark currents appearing in the general DIS cross-section is expanded in aseries of local, spin-n operators together with c-number Wilson co-efficients. The LCEis easily seen to be dominated by operators of lowest twist (τ = operator dimension −operator spin). For QCD the lowest twist operators available are twist-2. It can be shownby using the anomalous dimensions associated with the renormalisation of insertions ofthese operators in 2-point Green functions, that we can determine the evolution of themoment sum rules with the large momentum transfer scale.The unpolarised twist-2 operators are [3],ONSiµ1...µn= in−1Sψ¯γµ1Dµ2 . . .Dµn12λiψOqµ1...µn = in−1Sψ¯γµ1Dµ2 . . .DµnψOgµ1...µn =12in−2S Gµ1αDµ2 . . .Dµn−1Gαµn(1.1)and the polarised twist-2 operators are [4],RNSiσµ1 ...µn−1 = in−1Sψ¯γ5γσDµ1 . . .Dµn−112λiψRqσµ1 ...µn−1= in−1Sψ¯γ5γσDµ1 . . .Dµn−1ψRgσµ1 ...µn−1 =12in−2S ǫσαβγGβγDµ1 . . .Dµn−2Gαµn−1(1.2)Here, ψ is the quark field, Gµν is the gluon field strength tensor, Dµ is the QCD covariantderivative, λi are the generators of SU(Nf ), S stands for symmetrisation over Lorentzindices and NS denotes the non-singlet operator under SU(Nf ) as opposed to the othersinglet operators.An alternative, less abstract means of providing theoretical predictions for DIS isgiven by the parton model. Here the structure functions may be expressed in terms ofparton distribution functions (PDFs). These give the probability that the struck nucleonconstituent (parton) carries a particular fraction of the total nucleon momentum. Theevolution of the PDFs are then governed by the DGLAP equation [5]. This containsthe DGLAP ‘splitting functions’ which express the probabilities for the struck partonundergoing certain collinear decays as the energy scale changes. The DGLAP splittingfunctions can be explicitly obtained from the previously mentioned operator anomalousdimensions through an inverse Mellin transform. Solutions of the full DGLAP equationprovide us with scale dependent PDFs which may than be used as inputs for other hard2scattering processes where the formal QCD approach is not necessarily applicable as wellas giving us a deeper insight into hadron structure in the asymptotic regime.With facilities such as HERA currently opening up new kinematical regimes for unpo-larised and (soon) polarised DIS, there presently exists a need to perform a NNLO twist-2analysis to ensure a high precision examination of pQCD. This is especially importantwhere the new order’s effects are expected to be seen experimentally (as in low x, highQ2 scattering at HERA). This programme requires the 2-loop Wilson co-efficients and3-loop anomalous dimensions of all the DIS twist-2 light cone operators with full spin de-pendence. Although the complete 2-loop operator dimensions have been known for manyyears now [3, 6, 7, 8], the full 3-loop computations have only become viable in recentyears. The current state of play is that the 3-loop unpolarised singlet and non-singletoperator dimensions have been calculated for particular spins of the operator (n = 2, 4, 6,8 and n = 2, 4, 6, 8, 10 respectively) [9, 10]. In addition, the 2-loop finite parts requiredfor the full 3-loop results have recently been calculated for all twist-2 unpolarised andpolarised operators [11, 12]. As one might expect, these are particularly difficult calcu-lations involving thousands of Feynman diagrams. It is clear that a check on the finalfull 3-loop results would be useful. One way of approaching this would be to calculatethe dimensions using an alternative expansion parameter to the QCD coupling and checkthat there is agreement where overlap exists. We can do this by use of the 1/Nf expansionand a critical point approach.2 The Method.By applying the critical large Nf method developed in the series of papers [13] it is possibleto obtain expressions for the twist-2 operator dimensions to O(1/Nf). The method entailsthe analysis of operator insertions in QCD Green functions at a d-dimensional non-trivialrenormalisation group fixed point. This fixed point may be found as a stable zero of thed-dimensional four-loop QCD β-function [14] and is located atac =3ǫ4T (R)Nf+[3316C2(G)ǫ −(2716C2(R) +4516C2(G))ǫ2+(9964C2(R) +237128C2(G))ǫ3+(7764C2(R) +53128C2(G))ǫ4 + O(ǫ5)]1T 2(R)N2f+ O(1N3f)(2.1)where ac is the strong coupling constant at criticality, C2(R) and C2(G) are the funda-mental and adjoint quadratic Casimirs respectively, tr(T aT b) = 12δab for T a the generatorsof SU(Nc) and d = 4 − 2ǫ.Several good things come out of using such an approach;1. Propagators take on a simple, dressed, scaling form due to the scaling properties ofGreen functions at a renormalisation group fixed point.3For example, the quark and gluon propagators are respectively [15],ψ(k) ∼Ak/(k2)µ−α, Aµν(k) ∼B(k2)µ−β[ηµν − (1− b)kµkνk2](2.2)where b is the covariant gauge parameter, d = 2µ, A and B are momentum indepen-dent amplitudes and the exponents α and β are defined using simple dimensionalanalysis of the massless QCD action asα = µ − 1 + 12η , β = 1 − η − χ (2.3)Here, the critical exponent η is the quark anomalous dimension associated with thequark wave function renormalisation at the critical coupling and similarly χ is theanomalous dimension of the quark-gluon vertex.2. Using the critical renormalisation group it can be shown that the form of a particularO(1/Nf) operator anomalous dimension at ac may be calculated by evaluating theresidues of first order poles (we use an analytic regularisation of the gluon dimension)in appropriate O(1/Nf) two-point Green functions at ac. These Green functionsconsist of an insertion of the operator into a QCD two-point function. (We alsoneed to take the field renormalisations into account).3. Since, in the language of statistical mechanics, we are working at a critical point, weknow that we have an essentially massless theory which enables us to use integrationtricks such as uniqueness and conformal transformations to evaluate diagrams.4. In d-dimensions and to O(1/Nf) there exists a universality equivalence betweenQCD and the non-abelian Thirring model (NATM) at ac [16]. This may be seenthrough the reproduction of the QCD triple and quartic gluon vertices by fermionloop integration in three and four point NATM Green functions at ac using the abovepropagators. The upshot of this is that we can use the simpler NATM interactionsin our calculations and dispense with the tricky three and four-point gluon vertices.5. The fact that we work with a fixed d-dimensional spacetime with an analytic ratherthan a dimensional regularisation seems to alleviate some of the problems causedwhen treating γ5 in arbitrary dimensions [17]. This obviously becomes importantin the calculation of the dimensions of the polarised operators.3 Comparing singlet operator dimensions at O(1/Nf )with perturbation theoryA slight complication in this calculation arises due to mixing of the singlet operators inboth polarised and unpolarised scattering. Since the singlet light cone operators sharethe same quantum numbers and have equal canonical dimension in strictly four dimen-sions, they mix under renormalisation. This means that we have to consider a matrix ofrenormalisation constants for these operatorsOiren = ZijOjbare (3.1)4Here the indices i, j = q, g and we are refering to the operators (1.1) and (1.2). Theanomalous dimensions, γij(a), are defined byγij(a) =(γqq(a) γgq(a)γqg(a) γgg(a))(3.2)where γij(a) = β(a)(∂/∂a) lnZij and β(a) is the renormalization group function governingthe running of the QCD coupling constant a. The entries in γij(a) depend on the colourgroup parameters, Nf and n and since it is the 1/Nf corrections that we are interested in,we define the explicit form of the entries asγqq(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)(3.3)where N˜f = T (R)Nf and the coefficients aij , bij , cij and dij depend on n and the colourgroup Casimirs.We note that this matrix of anomalous dimensions has eigenvaluesλ±(a) =12(γqq + γgg) ±12[(γqq − γgg)2 + 4γqgγgq]12 (3.4)Expanding in powers of a and retaining the same orders in 1/Nf with the definitions (3.3)we find,λ−(a) =(a1 −b1c1d11)a+(a21 −b21c1d11)Nfa2 +(a31 −b31c1d11)N2f a3 +O(N3f a4)λ+(a) =(d11Nf + d12 +b1c1d11)a+(d21 +b21c1d11)Nfa2+(d31 +b31c1d11)N2f a3 +O(N3f a4) (3.5)Here we can see that λ+(a), λ−(a) are dominated by contributions from the gluonic andfermionic operators respectively.When in our approach we consider the singlet sector operators in d-dimensions at ac,we find that they no longer mix. (It is easy to see that the difference is O(ǫ).) This meansthat by calculating in d-dimensions, we are accessing the above mixing matrix eigenvaluesin perturbation theory. This is borne out by explicit calculations. By evaluating thegraphs required for γgg(ac), γqq(ac) we actually obtain λ+(ac) and λ−(ac) respectively,with the universality equivalence between QCD and NATM at O(1/Nf) accounting forthe contributions from the off-diagonal dimensions.4 ResultsThe unpolarised non-singlet operator dimension at O(1/Nf) was published in [18]. Thepolarised non-singlet operator result was published in [19] together with the expressionsfor λ−(ac) to O(1/Nf) and λ+(ac) to O(1) for polarised and unpolarised scattering.5We now give new results [20, 21] for λ+(ac) to O(1/Nf) for the unpolarised and po-larised gluonic operator together with the 3-loop n-dependent eigenvalue predictions forcomparison with perturbation theory (3.5). The relevant graphs are given in Fig. 1 witha different operator insertion Feynman rule [7, 22] used for the separate unpolarised andpolarised cases.Fig. 1. Leading order diagrams for λ+(ac).For the unpolarised gluonic singlet operator we have,λ+,1(ac) = − [8µ3n2 + 8µ3n+ 8µ3 + 2µ2n4 + 4µ2n3 − 22µ2n2 − 24µ2n− 28µ2 − 6µn4 − 12µn3 + 14µn2 + 20µn+ 32µ+ 5n4+ 10n3 + n2 − 4n− 12]Γ(n+ 2− µ)Γ(µ− 1)µC2(R)ηo1/[(µ− 2)2(n+ 2)(n+ 1)(n− 1)Γ(2− µ)Γ(µ+ n)nT (R)]+2µ(µ− 1)S1(n)C2(G)ηo1(2µ− 1)(µ− 2)T (R)− [32µ5n2 + 32µ5n + 32µ5 − 144µ4n2 − 144µ4n− 160µ4 − 4µ3n4− 8µ3n3 + 240µ3n2 + 244µ3n+ 316µ3 + 16µ2n4 + 32µ2n3− 180µ2n2 − 196µ2n− 306µ2 − 20µn4 − 40µn3 + 59µn2+ 79µn+ 146µ+ 8n4 + 16n3 − 6n2 − 14n− 28]µC2(G)ηo1/[8(2µ− 1)(µ− 1)3(µ− 2)(n+ 2)(n+ 1)(n− 1)nT (R)]+ [32µ5n2 + 32µ5n+ 32µ5 + 8µ4n4 + 16µ4n3 − 120µ4n2 − 128µ4n− 160µ4 − 32µ3n4 − 64µ3n3 + 160µ3n2 + 192µ3n+ 316µ3 + 48µ2n4+ 96µ2n3 − 78µ2n2 − 126µ2n− 306µ2 − 31µn4 − 62µn3 + 31µn+ 146µ+ 7n4 + 14n3 + 7n2 − 28]Γ(n+ 2− µ)Γ(µ− 1)µC2(G)ηo1/[8(2µ− 1)(µ− 1)2(µ− 2)(n+ 2)(n+ 1)(n− 1)Γ(2− µ)Γ(µ+ n)nT (R)](4.1)6implying,d31 +b31c1d11=64(n2 + n + 2)2(S1(n))2C2(R)3(n+ 2)(n+ 1)2(n− 1)n2T (R)−64(10n6 + 30n5 + 109n4 + 168n3 + 155n2 + 76n+ 12)S1(n)C2(R)9(n+ 2)(n+ 1)3(n− 1)n3T (R)− 4[33n10+ 165n9 − 32n8 − 1118n7 − 5807n6− 12815n5 − 16762n4 − 13800n3 − 7112n2− 2112n− 288]C2(R)/[27(n+ 2)(n+ 1)4(n− 1)n4T (R)]−8(8n6 + 24n5 − 19n4 − 78n3 − 253n2 − 210n− 96)S1(n)C2(G)27(n+ 2)(n+ 1)2(n− 1)n2T (R)− 2[87n8 + 348n7 + 848n6 + 1326n5 + 2609n4 + 3414n3 + 2632n2+ 1088n+ 192]C2(G)/[27(n+ 2)(n+ 1)3(n− 1)n3T (R)] (4.2)For the polarised gluonic singlet operator we have,λ+,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)] (4.3)giving,d31 +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(4.4)Throughout these calculations we used reduce [23] and form [24] to handle tediousamounts of algebra. The quantity ηo1 is defined byηo1 =(2µ− 1)(µ− 2)Γ(2µ)4Γ2(µ)Γ(µ+ 1)Γ(2− µ)(4.5)7and we have set λ+(ac) =∑∞i=0 λ+,i(ac)/Nif . The finite sum S1(n) is given by S1(n) =∑ni=1 1/i.These expressions agree exactly with all known perturbative results. This can be seenby putting µ = 2 − ǫ in (4.2) and (4.4) and expanding in powers of ǫ/Nf . An interestingfeature of the mixing matrix (3.2) is that the O(1/Nf) contribution to γgq(a) depends onlyon the Casimir C2(R). This is evident from the one and two loop results for all n andthe three loop results for n ≤ 8. Assuming this to be true for all n at three loops we cansee from (4.2) and (4.4) that we can deduce the exact form of the coefficients d31 in theC2(G) sector. This means that by using an inverse Mellin transform we can calculate theO(1/Nf) part of the three loop DGLAP function Pgg for both unpolarised and polarisedcases. For the unpolarised splitting function we obtain,Pgg(d31, C2(G)) =14C2(G)(6427[11− x]+−6427−589δ(1− x) +1289(x+ 1)Li2(x)+827(x− 1)(52x2 + 19x+ 52)xln(1− x) −1289ψ′(1)(x+ 1)−827(52x2 + 43x+ 76) ln(x) +329(x+ 1) ln2(x)+881(x− 1)(236x2 + 47x+ 236)x)(4.6)Similarly for the polarised splitting function,Pgg(d31, C2(G)) =14C2(G)(6427[11− x]+−6427−589δ(1− x) +1289(x+ 1)Li2(x)−1289ψ′(1)(x+ 1) +329(x+ 1) ln2(x) −827(67x− 56) ln(x)−3289(1− x) ln(1− x) −92027(x− 1))(4.7)These results complete the programme to calculate the O(1/Nf) corrections for theanomalous dimensions of the twist-2 light cone operators. At present it seems that acontinuation of this programme to include contributions of O(1/N2f ) may be viable. Weconclude by noting that the results may also be useful in estimating the full three loopcorrections to the operator dimensions by using asymptotic Pade´ approximant techniques[25].Acknowledgements. The author would like to thank the hosts of QUARKS ’98 fortheir extremely kind hospitality and is also grateful to many members of the workshop forhelpful and interesting discussions. This work was carried out with the support ofPPARCthrough an Advanced Fellowship (JAG) and a Postgraduate Studentship (JFB).References.[1] A.J. Buras, Rev. Mod. Phys. 52 (1980), 199.8[2] D.J. Gross & F.J. Wilczek, Phys. Rev. Lett. 30 (1973), 1343; H.D. Politzer, Phys.Rev. Lett. 30 (1973), 1346.[3] H. Georgi & H.D. Politzer, Phys. Rev. D8 (1974), 416; D.J. Gross & F.J. Wilczek,Phys. Rev. D8 (1973) 3633.[4] M.A. Ahmed & G.G. Ross, Nucl. Phys. B111 (1976), 441.[5] G. Altarelli & G. Parisi, Nucl. Phys. B126 (1977), 298; Yu.L. Dokshitzer, Sov. Phys.JETP 46 (1977), 641; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1975), 95; V.N. Gribov& L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972), 438.[6] E.G. Floratos, D.A. Ross & C.T. Sachrajda, Nucl. Phys. B129 (1977), 66; B139(1978), 545(E); A. Gonza´lez-Arroyo, C. Lo´pez & F.J. Yndura´in, Nucl. Phys. B153(1979), 161; G. Curci, W. Furmanski & R. Petronzio, Nucl. Phys. B175 (1980), 27.[7] E.G. Floratos, D.A. Ross & C.T. Sachrajda, Nucl. Phys. B152 (1979), 493.[8] A. Gonza´lez-Arroyo & C. Lo´pez, Nucl. Phys. B166 (1980), 429; C. Lo´pez & F.J.Yndura´in, Nucl. Phys. B183 (1981), 157; E.G. Floratos, C. Kounnas & R. Lacaze,Phys. Lett. B98 (1981), 89, 285; Nucl. Phys. B192 (1981), 417.[9] S.A. Larin, T. van Ritbergen and J.A.M. Vermaseren, Nucl. Phys. B427 (1994), 41.[10] S.A. Larin, P. Nogueira, T. van Ritbergen & J.A.M. Vermaseren, Nucl. Phys. B492(1997), 338.[11] Y. Matiounine, J. Smith & W.L. van Neerven, Phys. Rev. D57 (1998), 6701.[12] Y. Matiounine, J. Smith & W.L. van Neerven, hep-ph/9803439.[13] A.N. Vasil’ev, Yu.M. Pis’mak & J.R. Honkonen, Theor. Math. Phys. 46 (1981), 157;47 (1981), 291; A.N. Vasil’ev & M.Yu. Nalimov, Theor. Math. Phys. 55 (1982), 163;56 (1983), 15.[14] T. van Ritbergen, J.A.M. Vermaseren & S.A. Larin, Phys. Lett. B400 (1997), 379.[15] J.A. Gracey, Phys. Lett. B373 (1996), 178.[16] A. Hasenfratz & P. Hasenfratz, Phys. Lett. B297 (1992), 166.[17] G. ’t Hooft & M. Veltman, Nucl. Phys. B44 (1972), 189; S.A. Larin, Phys. Lett.B303 (1993), 113; T.L. Trueman, Phys. Lett. B88 (1979), 331.[18] J.A. Gracey, Phys. Lett. B322 (1994), 141.[19] J.A. Gracey, Nucl. Phys. B480 (1996), 73.[20] J.F. Bennett & J.A. Gracey, Nucl. Phys. B517 (1998), 241.[21] J.F. Bennett & J.A. Gracey, Phys. Lett. B432 (1998), 209.[22] R. Mertig & W.L. van Neerven, Z. Phys. C70 (1996), 637.9[23] A.C. Hearn, “Reduce Users Manual” version 3.4, Rand publication CP78, (1991).[24] J.A.M. Vermaseren, “Form” version 2.2c, CAN publication, (1992).[25] M. Karliner, hep-ph/9804381.10

Large N_f calculations in deep inelastic scattering

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Large N_f calculations in deep inelastic scattering

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