Orbital angular momentum and the parton model

Volume 192, number 1,2

PHYSICS LETTERSB

25 June 1987

ORBITAL ANGULAR M O M E N T U M A N D THE PARTON MODEL ~ P.G. RATCLIFFE Queen Mary College, London E1 4NS, UK Received 28 February 1987;revised manuscript received 6 April 1987

The role of orbital angular momentum is discussed within the frameworkof the parton model. It is shown that a consistent interpretation of the Altarelli-Parisi equations governingthe Q"-evolutionof helicity-weightedparton distributions necessitates the assumption that partons carry a large orbital angular momentum, contrary to popular belief. In developingthe arguments presented, the Altarelli-Parisi formalism is extended to include orbital angular momentumdependence.

1. Introduction. Many models have been proposed with the aim of incorporating helicity dependence into the parton picture of hadrons, some of which [ 1] compare successfully with data on polarised deepinelastic scattering [2]. Common to most models is the almost unquestioned assumption that the z-component of orbital angular momentum (L=) carried by quarks and gluons is negligible. However, the Altarelli-Parisi (AP) equations [3] for the relevant spin-dependent densities require a solution for which the net gluon polarisation increases with Q2, while that of the quark content is independent of Q2. One model [ 4] does allow for a partonic L=; using the Bjorken sum rule [5] to relate the [3-decay parameters GA/Gv for the neutron and E - to the spinweighted parton distributions of the proton, a prediction is obtained that the partonic Lz should account for about 40% of the proton spin. Unfortunately, this model is based on the naive SU(6) wave-function picture for baryons and fails to agree with data on the x-dependence of the spin-weighted quark distributions. Put simply, the explanation of the growing spin angular momentum component is as follows: gluons are produced by bremsstrahlung (either from quarks or gluons) and, to some extent, "remember" the parent helicity. On the other hand, gluon emission (arising as it does from a vector coupling) cannot flip Research supported by the UK Science and Engineering Research Council. 180

quark spin (in the limit of zero-mass quarks) and the number of valence quarks is constant. Moreover, sea quarks are pair produced with opposite (and thus cancelling) helicities). The apparent contradiction then is that this picture leads to the violation of conservation of angular momentum. In the present paper it is shown that consideration of the orbital angular momentum of the final-state pair, when one parton "splits" into two under the Q2 evolution process, restores angular momentum conservation. (The basic ideas put forward here were first presented by the author in ref [6].) Before continuing, a comment is in order on an alternative approach [7] which has been proposed to this problem: namely to invoke uncertainty in the knowledge of the AP kernels at x----0. It is argued that one should add a delta function at x = 0, whose coefficient is fixed in order to maintain the total parton polarisation independent of Q2. Now the AP prescription for regulating the kernels at x = 1 is vindicated in the diagrammatic approach [ 8 ], where the delta function subtractions are seen to come from wave-function renormalisation (in a physical gauge). However, the spin-dependent kernels are not singular at x = 0 and it is impossible to justify such ad hoc alterations in this context. Moreover, contrary to statements made in ref. [ 7 ], one does not consider k 2 as tending to zero in the calculation of parton splitting functions, but merely as being small compared with Q2. Thus there is cer-

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tainly room for a logarithmically increasing partonic L~ component of the hadron longitudinal polarisation. The layout of this paper is as follows: section 2 deals with the existing AP equations, examining in particular angular momentum conservation in the splitting kernels, from which the necessity of a nonnegligible orbital angular momentum is deduced. The AP formalism is then extended in section 3 to parton densities with an L: dependence; from this picture the consistency of a large partonic orbital angular momentum emerges. Finally, in section 4, the general conclusions are presented and discussed.

2. Helicity dependence in the AP formalism. Let us first briefly recall the formalism developed by AP [ 3 ] and the results they derive; this will illustrate the need to consider orbital angular momentum. Patton densities q + (x) and g-* (x) are defined for quarks and gluons, respectively, where x is the momentum fraction and + refers to the parton helicity relative to the parent hadron (i.e., parallel or antiparallel). From these one can form the spin-averaged and spinweighted densities q + (x) + q - (x) etc., denoted by q(x) and Aq(x), respectively. The AP equations are in fact conveniently diagonal in these quantities. The Q2 evolution of the densities is governed by splitting kernels Pab hh' (X), which describe the probability that a parton of type a (quark or gluon) with momentum fraction x and helicity h is contained within a parton of type b with helicity h'. This is shown diagrammatically in fig. 1 for quark-to-quark splitting. Again one forms spin-averaged and spinweighted quantities P + + + P +-, denoted respectively P and AP; note that parity conservation implies p + + = P - - and P + - = P - + The usual evolution equations (taking the flavour non-singlet sector for simplicity) have the following form: 1

~)Ns(X) = I ? Pqq(X/y)qNS(Y),

(1)

x

where tl=b OqlO In In Q2 with b = ~ C 2 ( G ) _ t T ( R ) , the first coefficient of the QCD beta function. The spin-weighted equations are obtained by substituting in eq. (1) everywhere with the corresponding spinweighted quantities. As is well known, the convo-

25 June 1987

1

) h ~, p

) h, x p

Fig. 1. Graphical representationof the quark-to-quark splitting kernel, P~' (x). lution may be eliminated by projecting onto moments in x; eq. (1) then takes the particularly simple form

~t~"s) = P~q~) q~'~ ,

(2)

where a(")=f~ dx x " - l a ( x ) . In what follows we shall only be interested in quantities averaged over x and thus in the n = 1 moments, which for the spin-weighted kernels are ~qq

= 0,

~]kgqg = 0 ,

Z~q = ~C'2(R), ~u°~ = ~ C 2 ( G ) - I T ( R ) = b ,

(3)

where the index n = 1 has been suppressed and from now on will always be understood (unless the variable x appears explicitly). Physically, the significance of Aq and Ag is rather simple: they are directly related to the total z-component of the quark and gluon spins, respectively; thus (sz)q = ~Aq,

(s~)g =Ag.

(4)

In terms of these quantities and taking into account eqs. (3) the full, singlet, evolution equations are

( L >~= ½zxO=o , (sz)g = Ag---zkP,q Aq+ ZkP~ Ag,

(5)

tO which the solutions are Aq(Q) = Aq(Qo), Aq(Q) =Aq(Qo) -as(Qo) -

as(Q)

+ ? Aq(Q°) k , ~ - - - ~

1 ,

(6) 181

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where as is the usual QCD running coupling constant. Thus the vanishing of both quark (spinweighted) anomalous dimensions at n = 1 implies Aq= 0 and hence the total quark spin is constant, as remarked earlier, while those for the gluon sector being positive and non-vanishing leads to a total gluon spin which grows asymptotically as In Q2. Therefore, taking the QCD parameter A=0.2 GeV and four effective quark flavours, i f , for example, gluons carried one half (quarter) of the proton spin at Q = 2 GeV (with quarks carrying the rest), then at Q = 6 8 (127) GeV their total spin would be approximately twice that of the proton itselfl Clearly then, total spin angular momentum is not conserved. However, one should have expected this; in fig. 1 what is not displayed is the spin of the label (real and collinear) gluon, which is + I. The spin of the quark is preserved and thus the total spin of the system changes by + 1 according to the gluon polarisation. One is therefore led to conclude that the finalstate pair must be produced with a relative L= of -Y-1 to balance this change (note that this argument is independent o f x and so a delta function at x = 0 is not a satisfactory modification).

3. The extension to orbital angular momentum dependence. One is thus obliged to take the unusual step of extending the dependence of parton distributions to include the z-component of orbital angular momentum associated with individual partons (which will be denoted numerically by l:). In order to make the connection with the usual distributions it is necessary to take a backward step to the point before integration over k± has been carried out. The parton state ~Vh(x, k . ) is then specified by its momentum fraction x, transverse momentum k± and helicity h. From the usual k±-dependent wave functions one can obtain wave functions in an l- representation via the simple transformation (up to an unimportant normalisation factor, which will be suppressed) [ 9 ]: 2n

~h(x, k i , l:) = I d0 eiS:°5Vh(x' k . ) ,

(7)

o

where 0 is the polar angle associated with the twovector k j_=k~ (cos 0, sin 0), k± = Ik_ [ a n d L = / ~ + s : (the total angular momentum of the field in ques182

25 J u n e 1987

tion), with s== ½h for quarks and h for gluons. Parton densities for fixed x, k . , l= and h are then given by qh(x, k±,/-) = I Th(x, k±, l:) 12 = f d0 d~ e ij:<~-° ) ~ , ( x , k±, 0) ~h(x, k±, q5) . J

(8) and the total L= carried by the quarks is (suppressing the k~ integral for clarity) q= ~ h

Z l=~(I=),

(9)

+1 l:

the sum over lz is purely symbolic as the question of quantization is not being considered. A similar expression holds for the Lz carried by the gluons. The total z-component of the proton spin is expressed by the following sum rule: sz = ½= ½Aq+Ag+ q+g

,

(10)

where, for convenience and without loss of generality, a proton helicity of + 1 has been chosen. Differentiating this sum rule with respect to In In Q2, noting that the total proton spin is independent of Q2 (it is of course in this case exactly ½) and taking into account eq. (5), one obtains

q+, = - ½Aq- Ag= - A g .

(1 l)

This is just the statement that the total orbital angular momentum inside the proton is not constant (and therefore necessarily non-zero). Note also that eq. (11 ) implies that ( L : ) will become negative for large enough (though not particularly large) values of Q2. Reconsidering the kernel depicted in fig. 1, it now becomes necessary to take into account the possibility of increasing or decreasing the orbital angular momentum of the system (by one unit), according as to whether the gluon emitted carries negative or positive helicity. In fact it turns out that all four of the parton splitting kernels carry with them a net change in orbital angular momentum of precisely one unit. It only remains then to calculate the L: dependence of the kernels, i.e., the way in which the Lz of the final-state pair is distributed between the two particles. Fortunately, at the level of the leading-logarithm

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approximation (LLA), relevant to the ordinary AP equations, the kernels are completely independent of any azimuthal angle and thus the L: dependence is trivial. In order to understand this consider once again the kernel of fig. 1 and let the final (initial) particle have transverse momentum k± ( k i ) . The only possibility for 0 dependence to enter is through terms of the form k±.k% (there are no other transverse vectors in the process). However, the strong ordering of momentum scales associated with the LLA, i.e., k~ >> k~, implies that all terms of order ki should be neglected under the k~ integral (since they will be power suppressed in Q2) and thus no 0dependent terms survive. One might also worry at this point, since the finalstate quark is not moving parallel to the initial quark, that corrections should be made to the way in which its helicity contributes to the total s~. Again, since cos O= 1 + O ( k 2/Q2) for deviations from the parent z-direction, restriction to the LLA regime ensures that such corrections will also be power suppressed. Combining eqs. (8) and (9) one has (for the quark sector)

q=,5.Z/: /--f dOd~eiJ:(°-~}T~(6)T$)

,

(12)

where again (and in what follows) the k~-integral has been suppressed. After differentiating eq. (12) with respect to In In Q2 the AP equations can be applied to the RHS to obtain (again taking the valence-quark sector for simplicity)

s = ; 3 f dOda

f dO'

o', (13)

,=-½A$,

25 June 1987

(15)

Thus eq. (11 ) is indeed satified and one sees that it is possible to include L: dependence in an entirely consistent and natural manner.

4. Conclusions. It has been shown, via detailed inspection of the AP kernels, that total angular momentum conservation in the evolution of parton distributions necessarily implies the presence of a non-negligible z-component of orbital angular momentum in the proton. Furthermore, by extending the existing formalism to include Lz dependence, it has been demonstrated that a large is indeed generated via the Q2 evolution process itself. The question now arises as to how, if at all, this growing might be detected. From eqs. (5) and (15) one can see that the growth is confined entirely to the gluon sector and also to the region of small x. One possible place where these effects might manifest themselves then would be in the angular distribution of pion production at low x. Certainly, when using sum rule (10) in conjunction with experimental data (e.g. the forthcoming data from the present EMC collaboration [ 10 ]) with a view to extracting information on gluon helicity distributions, great care will have to be exercised in respect of the term, whih has not only been shown to be non-negligible, but, but which must also become negative for some (quite moderate) value of Q2. Taking the same parameters as in the example of section 3, but now assuming the proton spin to be divided evenly between q, <&>g and (i.e. 33% each) at Q = 2 GeV, then at Q = 50 GeV, g will represent 125% of the proton spin and minus 58%.

Since, as explained above, the kernels P have no dependence, the 0 and ¢5integrals over e ij:(~-~ effectively reduce to a delta function in j:, which sets / ~ = - ½h. The variables 0' and $ ' can also be integrated over and the sums over h, h' and lz, together with the delta function, then lead to

The author would like to thank Professor E. Leader for initially bringing this problem to his attention, reading the manuscript and again, together with Dr. M. Anselmino, for the many valuable discussions. Thanks are also due to Dr. K. Heller for suggestions on the possible experimental manifestations of the effects discussed in this work.

( L z >NS = -- ½~dOqq AqNs = -- ½AqNs .

References

(14)

The extension to the singlet sector is straightforward and gives

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[2] [3] [4] [5] [ 6] [7]

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