The Angular Momentum and g p 1 Sum Rules for the Proton

Transcription

1 The Angular Momentum an g p Sum ules for the Proton G.M. Shore a a Département e Physique Théorique, Université egenève, 24, quai E. Ansermet, CH-2 Geneva 4, Switzerlan an TH Division, CEN, CH 2 Geneva 23, Switzerlan. The gauge invariant operator formulation of the angular momentum sum rule = Jq + Jg for the proton is 2 presente an contraste with the sum rule for the first moment of the polarise structure function g p. The ecoupling of the axial charge a 0 from the angular momentum sum rule is highlighte an the possible QCD fiel-theoretic basis for an angular momentum sum rule of the form = q + g + Lq + Lg is critically 2 2 iscusse.. Introuction In this talk, base on work in collaboration with. White, I review our recent formulation[] of the angular momentum sum rule for the proton an iscuss its relation to the sum rule for the first moment of the polarise structure function g p.in particular, the role of the axial charge a 0, which is measure in the g p sum rule, is highlighte an it is shown how this ecouples from the angular momentum sum rule. This emphasises the limitations of attempting to ientify a 0 with quark (an gluon spin an an alternative interpretation in terms of topological charge is briefly reviewe. The angular momentum sum rule is shown to take the simple form 2 = J q + J g,wherej q an J g are gauge an Lorentz invariant form factors of the forwar matrix elements of local operators, which may reasonably be interprete as quark an gluon components of the total angular momentum of the proton. Experimentally, they may be measure in, for example, eeply virtual Compton scattering. Their G evolution properties are esearch supporte in part by PPAC grant G/L56374 an EC TM grant FMX-CT Invite talk at QCD00, Montpellier, July CEN-TH/ , SWAT/00-263, UGVA-DPT Permanent aress: Department of Physics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K. erive from a careful analysis of operator mixing. We also iscuss critically whether there is inee a QCD fiel-theoretic basis for a ecomposition of the proton spin into separate quark an gluon spin an orbital angular momentum components, as in the frequently-quote sum rule 2 = 2 q + g + L q + L g. 2. The g p sum rule an topological charge The sum rule for the first moment of g p (see e.g. ref.[2] for reviews of our earlier work an references is x g p (x, Q2 = ( 0 2 CNS a a8 + 9 CS a0 (Q 2 ( where the C are Wilson coefficients an the flavour singlet axial charge a 0 is efine as the form factor in the forwar matrix element of the corresponing axial current, p, s A 0 µ p, s = a0 s µ (2 where s µ is the covariant spin vector. Since A 0 µ is not a conserve current, ue to the U A ( anomaly µ A 0 µ 2n fq 0 (3 where Q = αs 8π ɛµνρσ trf µν F ρσ is the gluon topological charge ensity, the form factor a 0 is scale

2 2 epenent an satisfies the non-trivial G evolution equation a0 (Q 2 =γa 0 (Q 2 (4 where t =lnq 2 /Λ 2 an the anomalous imension α is γ = n s f 2π. 2 Given a 3 an a 8 from low-energy neutron an hyperon ecays, a 0 can be extracte from polarise inclusive DIS processes e(µp e(µx. The simplest preiction, a 0 a 8, is an immeiate consequence of the OZI rule an transcribe into the sum rule for g p gives the Ellis-Jaffe sum rule. However, the flavour singlet pseuovector or pseuoscalar channel is precisely where we woul expect to fin strong OZI violations relate to the U A ( anomaly an inee it is foun experimentally that a 0 a 8. Since we can rewrite a 0 using eq.(3 in terms of the matrix element of the topological charge ensity, a 0 = p, s Q p, s (5 we see immeiately that the observe suppression in a 0 is a manifestation of topological charge screening. This interpretation has been evelope in a series of papers written in collaboration with Veneziano, Narison an De Florian[2]. Our proposal is that this screening is universal, i.e. target-inepenent, being an intrinsic property of the QCD vacuum itself. Specifically, we showe that (in the chiral limit a 0 = [ 2M 2n f χ(0γ Qpp + ] χ (0Γ Φ5pp (6 where the Γ are suitably efine PI vertex functions an χ(k 2 =i 4 xe ik.x 0 TQ(xQ(0 is the topological susceptibility, χ (0 being its slope at k = 0. Since the anomalous chiral War ientity implies χ(0 = 0 in the chiral limit, the sole contribution to a 0 comes from the secon term in eq.(6. Making the motivate assumption that the G invariant vertex Γ Φ5pp obeys the OZI rule to a goo approximation, we conjecture that the principal origin of the suppression in a 0 is an anomalously small value of χ (0 ue to universal topological charge screening by the QCD vacuum. 5 This is anticipate in certain instanton-base moels of the QCD vacuum, has been confirme quantitatively by QCD spectral sum rule calculations, an is currently being investigate in lattice gauge theory. The QCD parton moel gives an alternative interpretation of a 0 through the ientification a 0 (Q 2 α s = q 2n f 4π g(q2 (7 where q = u + + s an g(q 2 are the first moments of the polarise flavour singlet quark an gluon istributions an we have use the A class of renormalisation schemes where q is efine to be Q 2 inepenent. The OZI/Ellis-Jaffe relation a 0 = a 8 follows immeiately from the assumption that in the proton, s =0an g = 0. In this moel, 2 q an g are interprete as the quark an gluon spins, which le to the initial interpretation of the experimental observation a 0 a 8 as inicating that the quarks carry only a small fraction of the spin of the proton the so-calle proton spin crisis. In the rest of this talk, we erive the actual angular momentum sum rule for the proton in terms of gauge invariant operator matrix elements, with particular emphasis on whether an how the axial charge a 0 appears an whether the interpretation of q an g as spin components can be complemente by corresponing efinitions of orbital angular momentum components L q an L g. 3. The angular momentum sum rule The angular momentum sum rule is erive by taking the forwar matrix element of the conserve angular momentum current M µνλ, efine from the energy-momentum tensor as M µνλ = x [ν T λ]µ + ρ X ρµνλ (8 The inclusion of the arbitrary tensor X ρµνλ (antisymmetric uner ρ µ an ν λ just reflects the usual freeom in QFT in efining conserve currents. However, this arbitrariness allows us to 5 Explanations which favour OZI violations ue to a large polarise strange quark component of the proton or the implications of the Skyrme moel of proton structure have been recently reviewe in, for example, ref.[3]

3 3 write ifferent equivalent expressions for M µνλ as a sum of local operators, suggesting corresponing interpretations of the total angular momentum as a sum of components, which we can try to ientify as reasonable efinitions of quark an gluon spin an orbital angular momentum[4]. The best ecomposition is the following: M µνλ = + 2 x ν] + + g µ[λ x ν] L gi + { i {µ cd [λ} c + {µ A [λ} + g µ[λ } L gf x ν] 4 ρ[ x [ν ɛ λ]µρσ ψγσ γ 5 ψ } +EOM+ ρ X ρµνλ (9 The tensor X ρµνλ is chosen to cancel the ivergence an equation of motion (EOM terms, while the forwar matrix element of the operator g µ[λ x ν] L gi vanishes. The term in {...} is the contribution from the covariant gauge-fixing an ghost terms in the Lagrangian, but this turns out to be a S variation so its matrix element between physical states vanishes. The remaining (bare operators are gauge invariant: = 2 ɛµνλσ ψγσ γ 5 ψ 2 ɛµνλσ A 0 σ 2 = i ψγ µ D λ ψ 3 = F µρ F ρ λ (0 We also fin it convenient for later use to efine 4 = 2 O{µλ} 2.Atfirstsight,, which is just the flavour singlet axial current consiere in section 2, looks as if it may be associate with quark spin, with 2 x ν] corresponing to a gauge invariant efinition of quark orbital angular momentum. This leaves to be associate with the gluon total angular momentum. In fact, there is no further ecomposition of the gluon contribution as long as we restrict to gauge invariant operators. We can certainly make the alternative ecomposition: M µνλ = + Õµ[λ 2 x ν] + Õµνλ 3 + Õµ[λ 4 x ν] + g µ[λ x ν] L gi + { A µ [λ + i µ c [λ c + i [λ cd µ c + g µ[λ } L gf x ν] + ρ [ x [ν A λ] F µρ] +EOM+ ρ X ρµνλ ( where = 2 ɛµνλσ ψγσ γ 5 ψ Õ µλ 2 = i ψγ µ λ ψ 3 = F µ[ν A λ] Õ µν 4 = F µρ λ A ρ (2 an try to ientify the first four operators with, respectively, quark spin an orbital an gluon spin an orbital angular momentum. However, even the forwar matrix elements of these operators turn out not to be gauge invariant. Moreover, the gauge-fixing an ghost term in {...} is no longer a S variation so woul contribute a non-vanishing ghost orbital angular momentum. Further iscussion of the problems with such gauge non-invariant ecompositions is given in ref.[], an from now on we restrict attention to the gauge invariant formulation base on eq.(9. The next step is to express the matrix elements of the operators, 2 x ν] an in terms of form factors. There are technical subtleties connecte with efining operators of the form Ox an their renormalisation mixing which are explaine precisely in ref.[]. The prescription is essentially to efine the forwar matrix elements of Ox in terms of the limit of an offforwar matrix element. We therefore write (a little loosely p x ν p = i p p p (3 =p ν where = p p.wethenhave[] p, s p, s = a 0 Mɛ µνλσ s σ p, s 2 x ν] p, s = q (0 2M p ρp {µ ɛ [λ}ν]ρσ s σ + q (0 2M p ρp [µ ɛ [λ]ν]ρσ s σ 2D q (0Mɛ µνλσ s σ p, s p, s = g (0 2M p ρp {µ ɛ [λ}ν]ρσ s σ (4 The crucial observation now follows from the ientity + 2 x ν] = 4 x ν] +ivergence+eom (5

4 4 Since the matrix elements of the ivergence an EOM terms vanish, an recalling that the operator 4 is symmetric in µ, λ, weseethat q (0 = 0 2D q (0 = a 0 (6 Thus a 0 appears not only as the unique form factor in the matrix element O of the axial current, but also as a contribution to O 2 x. Ittherefore cancels from the angular momentum sum rule. Introucing the notation J q = q (0, J g = g (0, then from eq.(4 we may write the sum rule as 2 = J q + J g (7 where J q an J g are gauge an Lorentz invariant quantities which may reasonably be ientifie as total quark an gluon angular momenta respectively. Of course, this is a rather nonrigorous terminology since the corresponing operators O 2 x an O 3 x mix, an inee O 2 x itself contains an explicit gluon fiel component in the covariant erivative, but it is convenient an the closest approximation to a quark-gluon ecomposition that can be given in an interacting QFT such as QCD. Moreover, as we iscuss below, J q an J g are still not G scheme/scale epenent quantities. The point we wish to stress is that provie we restrict to gauge an Lorentz invariant quantities, the true angular momentum sum rule involves only the two form factors J q an J g. The axial charge a 0 is simply not present in the sum rule (7. We return to this point in section 5. Just as the axial charge form factor a 0 can be measure in polarise inclusive DIS, the angular momentum form factors J q an J g can in principle be extracte from measurements of unpolarise off-forwar parton istribution functions in processes such as eeply virtual Compton scattering γ p γp. The require ientifications are ip + µ x xf q(g/p (x, ξ, =0 = J q(g M ɛ+µρσ P ρ s σ (8 where ξ = q. 2q.P an the incoming(outgoing proton momenta are P (+. These form factors may also be calculate nonperturbatively in lattice gauge theory an some initial results for J q in the quenche approximation have recently been obtaine in ref.[5]. 4. Operator mixing an G evolution The operators O, O 2 x an O 3 x in the angular momentum sum rule renormalise an mix in a non-trivial way. The analysis is mae more subtle by the explicit factors of the coorinate x which have to be carefully treate. A etaile iscussion is presente in ref.[] an here we only sketch the main features. First, note that when inserte into forwar matrix elements, operators of the form O a an O i x mix with a block triangular structure: ( Oa O i x ( Z = ab 0 Z ib Z ij ( Ob O j x (9 since gauge-invariant operators with no factors of x only mix with other similar operators. Then since 3 is symmetric, it can only mix with the symmetric operators 3 an 4, which for forwar matrix elements implies that only mixes with itself an + 2 x ν]. Finally, since the full angular momentum current is conserve an therefore not renormalise, the columns of the mixing matrix must all a to one. This implies the following form for the mixing matrix for forwar matrix elements: 2 x ν] = +X 0 0 Z X +Z Y Z Z +Y 2 x ν] an one-loop calculations show Y = 2 3 n f αs 4π ɛ an Z = 8 3 C F αs 4π ɛ. X is ue to the anomaly an is O(α 2 s. For the form factors, this gives: a0 q = g (20

5 5 +X Z Y a0 q 0 Z +Y g (2 We therefore fin the evolution equations for the quark an gluon components of the proton angular momentum: ( Jq J g = α s 4π ( 8 3 C F 2 3 n f 8 3 C F 2 3 n f ( Jq J g together with eq.(4 for a 0. It follows that in the asymptotic limit Q 2, the partitioning of quark:gluon angular momenta is 3n f :6[6]. Interestingly, this is the same result as for the partitioning of momentum obtaine from the first moment of the unpolarise pfs. 5. Partons an orbital angular momentum We have seen how the g p an angular momentum sum rules involve three Lorentz invariant form factors a 0, J q an J g,wherej q(g may be reasonably ientifie as total quark (gluon angular momentum components in the sum rule 2 = J q+j g while the axial charge a 0, which enters the g p sum rule, ecouples an may instea be interprete in terms of topological charge ensity. There is no gauge an Lorentz invariant operator ientification of orbital angular momenta L q an L g. In the parton moel, the axial charge a 0 is interprete as a sum of polarise quark q an gluon g istributions as in eq.(7. In the A scheme, the G evolution equations are q =0 where β 0 g (22 = α s ( 3CF q+β 0 g (23 4π = 2 3 n f, compatible with eq.(4. This evolution can also be irectly obtaine from the splitting functions. Now if as the parton moel suggests, 2 q an g are to be interprete as quark an gluon spins, then to complete the angular momentum sum rule we are force to write 2 = 2 q + g + L q + L g (24 where L q(g are orbital angular momenta. Since there is no intrinsic operator efinition of these partonic quantities, the best we can o is to efine them to be consistent with the sum rule (7, i.e. L q = J q 2 q L g = J g g (25 Of course this means the sum rule (24 is not really preictive, since there is no way to inepenently measure L q(g only the form factors J q(g can be extracte from experiment. Moreover, the ientifications (25 are not very natural, since they involve subtracting quantities belonging to form factors for ifferent Lorentz structures. They are therefore frame-epenent, not surprisingly given that spin an orbital angular momentum are associate with ifferent representations of the Lorentz group. Nevertheless, if we aopt (7,(25,(24 as the best possible gauge-invariant efinition of a quark/gluon spin/orbital angular momentum sum rule, we may etermine the G evolution for the components q, g, L q an L g from eqs. (22,(23. We fin q g L q L g = α s 3C F β π 4 3 C 2 F 3 n f 8 3 C 2 F 3 n f 5 3 C 8 F 3 C F 2 3 n f q g L q L g which is consistent with the non-renormalisation of the full angular momentum current an may, at least in part, also be erive in a splitting function approach [6]. EFEENCES. G.M. Shore an.e. White, hep-ph/ G.M. Shore, Nucl. Phys. (Proc. Suppl. 64 (998 67; Proc. International School of Subnuclear Physics, Erice, J. Ellis, hep-ph/ L. Jaffe an A.V. Manohar, Nucl. Phys. 337 ( N. Mathur et al., hep-ph/ P. Hoobhoy, X. Ji an J. Tang, Phys. ev. Lett. 76 ( (26