# Bnl-Het-04/6 Ic/hep/04-2 A critique of the angular momentum sum rules and a new angular momentum sum rule

###### Abstract

We present a study of the tensorial structure of the hadronic matrix elements of the angular momentum operators . Well known results in the literature are shown to be incorrect, and we have taken pains to derive the correct expressions in three different ways, two involving explicit physical wave packets and the third, totally independent, based upon the rotational properties of the state vectors. Surprisingly it turns out that the results are very sensitive to the type of relativistic spin state used to describe the motion of the particle i.e. whether a canonical (i.e. boost) state or a helicity state is utilized. We present results for the matrix elements of the angular momentum operators, valid in an arbitrary Lorentz frame, both for helicity states and canonical states.

These results are relevant for the construction of angular momentum sum rules, relating the angular momentum of a nucleon to the spin and orbital angular momentum of its constituents. It turns out that it is necessary to distinguish carefully whether the motion of the partons is characterized via canonical or helicity spin states. Fortunately, for the simple parton model interpretation, when the proton moves along , our results for the sum rule based upon the matrix elements of agree with the often used sum rule found in the literature. But for the components the results are different and lead to a new and very intuitive sum rule for transverse polarization.

Submitted to Physical Review D

## 1 Introduction and summary of results

Sum rules, relating the total angular momentum of a nucleon to the spin and orbital angular momentum carried by its constituents, are interesting and important in understanding the internal structure of the nucleon. Indeed it is arguable that the main stimulus for the tremendous present day experimental activity in the field of spin-dependent structure functions was the surprising result of the European Muon Collaborations polarized DIS experiment in 1988 [1], which, via such sum rules, led to what was called a “spin crisis in the parton model” [2], namely the discovery that the spins of its quarks provide a very small contribution to the angular momentum of the proton. A key element in deriving such sum rules is a precise knowledge of the tensorial structure of the expectation values of the angular momentum operators in a state of the nucleon, labeled by its momentum , and with some kind of specification of its spin state, denoted here non-commitally by .

In a much cited paper [3], Jaffe and Manohar stressed the subtleties involved in deriving general angular momentum sum rules. As they point out, too naive an approach leads immediately to highly ambiguous divergent integrals, and a careful limiting procedure has to be introduced in order to obtain physically meaningful results. In this it is essential to work with non-diagonal matrix elements and, as discussed below, this can have some unexpected consequences. Jaffe and Manohar comment that to justify rigorously the steps in such a procedure requires the use of normalizable wave packets, though they do not do this explicitly in their paper.

In a later paper [4], Shore and White utilized the approach of Ref. [3], including an explicit treatment with wave-packets, to derive some far reaching conclusions about the rôle of the axial anomaly in these sum rules.

We shall argue that despite all the care and attention to subtleties, there are flaws in the analysis in [3] and the results presented there are not entirely general. Indeed there are cases where the results of [3] are incorrect. This, in turn, throws doubt upon some of the conclusions reached in [4], which we will examine.

The bulk of our analysis is based on a straightforward wave-packet approach. However, as we explain, this is rather subtle for particles with non-zero spin. The key points underlying our results are:

(1) Our wave packets are strictly physical, i.e. a superposition
of physical plane-wave states. This requirement turns out to be
incompatible with some of the steps in [3].

(2) We give a careful treatment of the Lorentz covariance properties of the
matrix elements involved in the subsidiary steps of the anaylsis.
This leads to tensorial structures which differ in some cases from
those in [3].

(3) Because our results differ from [3] we have looked
for and found a totally independent way to check our results. This
does not use wave packets and is based upon the transformation
properties of momentum states under rotations. This very direct
approach holds for arbitrary spin, whereas in the wave packet
treatment we are only able to deal with spin
particles. It also brings to light some peculiar and unintuitive
properties of helicity states, and this must be taken into account
when deriving spin sum rules. This is important since
we have to deal with gluons in our sum rules.

Our results for the matrix elements of are as follows:

For a massive particle of spin with 4-momentum in a canonical spin state (i.e. in a ‘boost’ state of the kind generally used in textbooks on Field Theory e.g. in Bjorken and Drell [5], or in Peskin and Schroeder [6] ) , where is the spin eigenvector in the rest frame , we show that, for the forward matrix elements,

(1) |

The states are normalized conventionally to

(2) |

and we note that in the rest frame

(3) |

where has spin projection along the -direction in the rest frame and rotates a unit vector in the -direction into by first a rotation about and then a rotation about .

For the purpose of deriving sum rules our result for the matrix elements non-diagonal in the spin label is actually more useful, namely, for a spin particle

(4) |

The generalization of these results for arbitrary spin is given in Eq. (150).

Helicity states are more suitable for massless particles such as gluons. Using the Jacob-Wick conventions for helicity states [7] we find a surprisingly different result, namely,

(5) |

where

(6) |

and are the polar angles of .

The first term in Eq. (1) differs from the results of Jaffe and Manohar [3]. If we rewrite their expression (see Eq. (59)) in terms of the independent vectors and , we find, for the expectation value

(7) |

to be compared to

(8) |

arising from the first term in Eq. (1). In general these are different. However, one may easily check that if the Jaffe-Manohar value agrees with Eq.(8), while if they are not the same.

The agreement for is consistent with the much used and intuitive sum rule

(9) |

In the case that we find a new sum rule. For a proton with transverse spin vector we find

(10) |

where is the component of along . The structure functions are known as the quark transversity or transverse spin distributions in the nucleon. Note that no such parton model sum rule is possible with the Jaffe-Manohar formula because, as , Eq. (7) for diverges.

The organization of our paper is as follows: In Section 2 we explain why the calculation of the angular momentum matrix elements is so problematical. Because of the unexpected sensitivity of the matrix elements of to the type of spin state used, and because we are forced to use helicity states for gluons, Section 3 presents a resumé of the difference between Jacob-Wick helicity states and canonical ( i.e as in Bjorken-Drell) spin states. Further, given that our results disagree with one of the classic papers in the literature, we have felt it incumbent to summarize, in Section 4, the treatments of Jaffe-Manohar and Shore-White, pointing out the incorrect steps in these derivations.

In Section 5 we present a detailed wave-packet derivation of the structure of the matrix elements of for spin , first for a relativistic Dirac particle, then in a field theoretic treatment. We comment here on the claims made in Shore-White on the role of the axial anomaly in the structure of the matrix elements. Sections 4 and 5 are heavy going, and the reader only interested in a quick and direct derivation of the key results should skip these and read Sections 6, 6.1, 6.3 and 7.

In Section 6 we confirm the results of Section 5 in a completely independent approach, which is valid for arbitrary spin, based on the rotational properties of canonical and helicity spin states. We also prove that our results are in conformity with the demands of Lorentz invariance.

In Section 7 we derive the most general form of an angular momentum sum rule for a nucleon and show that it reduces to the standard, intuitive, sum rule for when the nucleon is moving along . We also derive a new sum rule for a transversely polarized nucleon.

## 2 The origin of the problem

In the standard approach one relates the matrix elements of the angular momentum operators to those of the energy-momentum tensor.

Let be the total energy-momentum density which is conserved

(11) |

Later we shall distinguish between the conserved canonical energy-momentum tensor , which emerges from Noether’s theorem, and which is, generally, not symmetric under , and the symmetrised Belinfante tensor , which for QCD is given by

(12) |

and which is also conserved. For the moment, however, this distinction is irrelevant.

Being a local operator, transforms under translations as follows

(13) |

where the are the total momentum operators of the theory.

By contrast the various angular momentum density operators which are of interest, the orbital angular momentum densities

(14) |

or the version constructed using the symmetrised stress-energy tensor,

(15) |

are not local operators (we shall call them compound) and do not transform according to Eq. (13).

Note that, strictly speaking, the operators relevant to the angular momentum are the components where are spatial indices. However, for reasons of simplicity in utilising the Lorentz invariance of theory, the authors of Ref. [3] prefer to deal covariantly with the entire tensor . We shall loosely refer to them also as angular momentum densities.

The total angular momentum density is

(16) |

where the structure of depends on the type of fields involved. From Noether’s theorem is a set of conserved densities, i.e.,

(17) |

As a consequence of the densities being conserved, it follows that the total momentum operators

(18) |

and the total angular momentum operators ,

(19) |

with

(20) |

are conserved quantities, independent of time.

The relationship between the , constructed using the symmetrical energy-momentum density and the constructed from the canonical energy-momentum tensor is extremely interesting and will be commented on later in Section 4.2. One can show (see e.g. Ref. [4]) that

(21) |

The [E of M terms] vanish if it is permissible to use the equations of motion of the theory. The [divergence terms] are of the form .

As mentioned we shall be primarily interested in the expectation values of the physical operators, i.e. in their forward matrix elements. If were a local operator, it would follow directly that the forward, momentum-space, matrix elements of the divergence terms in Eq. (2.11) vanish. But it is not a local operator. Nonetheless, a careful treatment using wave packets [4] demonstrates that the forward matrix elements do indeed vanish. See Section 4 below.

Dropping, as is customary, the [E of M terms], we shall thus assume the validity of

(22) |

We shall return to this question in Sections 5.1 and 5.2.

Of primary interest are the matrix elements of the angular momentum operators or, equivalently, the . Consider the forward matrix element, at ,

(24) |

The integral in Eq. (24) is totally ambiguous, being either infinite or, by symmetry, zero.

The essential problem is to obtain a sensible physical expression, in terms of and , for the above matrix element. The fundamental idea is to work with a non-forward matrix element and then to try to approach the forward limit. This is similar to what is usually done when dealing with non-normalizable plane wave states and it requires the use of wave packets for a rigorous justification.

It will turn out that the results are sensitive to the type of
relativistic spin state employed, so in Section 3 we present a
brief resumé of the properties of relativistic spin states. We
then proceed to discuss the approaches of [3] and
[4] in Section 4, where we shall comment on the dubious
steps in these treatments. The most crucial error in these
treatments is the mishandling of the matrix elements of a
covariant tensor operator. If transforms as a
second-rank tensor its *non-forward* matrix elements do
*not* transform covariantly. This was the motivation, decades
ago, for Stapp to introduce -functions [11]. Namely,
the covariance is spoilt, for canonical spin states by the Wigner
rotation, and, for helicity states by the analogous Wick helicity
rotation [10]. Only by first factoring out the
wave-functions (in our case Dirac spinors) i.e. by writing

(25) |

does the remaining -function, in this case , transform covariantly. For local operators the transformations of the spinors and cancel between themselves for forward matrix elements and so the result does have the naively expected tensor expansion. This is not true in general for compound operators, in particular the angular momentum and boost operators.

## 3 Relativistic Spin States

The definition of a spin state for a particle in motion, in a relativistic theory, is non-trivial, and is convention dependent. Namely, starting with the states of a particle at rest, which we shall denote by , where is the spin projection in the -direction, one defines states for a particle with four-momentum by acting on the rest frame states with various boosts and rotations, and the choice of these is convention-dependent. The states are on-shell so

There are three conventions in general use [8]

a) Canonical or boost states as used e.g. in Bjorken and Drell [5] or Peskin and Schroeder [6]

(26) |

where is a pure boost along , and denotes the three-vector part of

b) Jacob-Wick helicity states [7]

(27) |

where is a boost along , and the later introduced, somewhat simpler

c) Wick helicity states [10]

(28) |

From the canonical states of spin- one can construct the states

(29) |

which, in the rest frame, are eigenstates of spin with spin eigenvector along the unit vector . (The rotation was explained after Eq. (3)).

The canonical states, with their reference to a rest frame, are clearly not suitable for massless particles like gluons. Helicity states, on the other hand, can be used for both massive and massless particles. However, it turns out that the results for the canonical states are much more intuitive, so we will generally use them for .

The reason we are emphasizing this distinction between canonical and helicity states is that the matrix elements of the angular momentum operators between helicity states are quite bizarre! Since, for arbitrary , helicity states are just linear superpositions of canonical states, one may wonder why this is so. It results from the facts i) that the coefficients in the linear superposition are -dependent, i.e. depend upon the polar angles of and ii) that the matrix elements of the angular momentum operators contain derivatives of -functions, and these, as usual, must be interpreted in the sense of partial integration, i.e.

(30) |

The technical details are explained in Section 6.

In almost all studies of hard processes, where a mixture of perturbative and non-perturbative QCD occurs, nucleons are taken to be in helicity states moving with high energy along the -axis, and typically one is utilizing matrix elements of local products of quark or gluon field operators between these states. For these operators there is no problem in dealing with diagonal matrix elements. But when it comes to an angular momentum sum rule for the nucleon , care must be taken to decide whether one is dealing with helicity states , where or with canonical states , where . The point is that even though the initial states are the same,

(31) |

the singular nature of forces one to deal with non-diagonal matrix elements i.e. to utilize where is not along the -axis, and for these

(32) |

In this paper we show that it is possible to give a rigorous derivation of the structure of the expectation values for canonical states

(33) |

where is a unit vector along the rest frame spin eigenvector, and for helicity states

(34) |

In general, for the arbitrary component of , for spin

(35) |

even though lies along the direction of in both cases, and even if is along OZ where Eq.(31) holds. Only for the specific component of along do the matrix elements agree, i.e. for arbitrary ,

(36) |

In using the sum rules based on Eq.(33) or Eq.(34) for arbitrary to test any model of the nucleon in terms of its constituents, it is essential to construct wave-functions appropriate to the type of spin state being used for the nucleon. The equations Eq.(34) and Eq.(33) contain delta functions and derivatives of delta-functions and this is the reason for the special care required. Throughout this paper, with the exception of the discussion in Section 6 and in Section 7, we will use canonical spin states. In these latter sections we shall utilize Jacob-Wick helicity states for the massless gluons.

## 4 The Jaffe-Manohar and Shore-White Approaches

These authors employ the standard approach of trying to relate the matrix elements of the angular momentum operators to those of the energy-momentum density operator, and utilize the version Eq. (15) based on the symmetrized energy-momentum tensor.

### 4-1 The Jaffe-Manohar treatment

In order to make efficient use of the Lorentz invariance the authors of [3] prefer to label their states using the covariant spin 4-vector and to consider the entire tensor and to integrate the tensor densities over four-dimensional space time, i.e. they consider

(37) |

and eventually take the limit . In Ref. [3] the LHS is written in the abbreviated form , but we shall use the above notation for clarity. Note that Jaffe and Manohar use the notation to mean the covariant spin which we denote by . It is times the expectation value of the Pauli-Lubanski operator; see e.g. [8]:

(38) |

and

(39) |

In terms of components

(40) |

Also we take while Ref.[4] takes the covariant normalization to be .

The Lorentz invariant normalization of the states is conventional

(41) |

The extra integration in Eq. (37) , is argued to be harmless, leading to an infinite which cancels out when calculating a genuine expectation value.

Comment 1 It will be seen in Section 5.3 that the choice as done in [3] is not consistent in a proper wave-packet treatment.

Analogously to the steps leading to Eq. (24) we have

(42) | |||||

Comment 2 The last step in Eq. (42) can only be justified if in Eq. (37) is considered an independent variable. We may not take . Once this is recognized it is no longer so evident that Eq. (37) followed by the limit provides a natural definition of the ambiguous forward matrix element.

Continuing from Eq. (42) one has

(43) | |||||

In Ref. [3] the limit is given as

(44) | |||||

This form is a little puzzling, given that in Eq. (37), as discussed in Comment 2. We thus prefer to write Eq. (44) in the form

(45) | |||||

The highly singular first term in Eq. (45) can only be understood in a wave-packet analysis. It corresponds to the angular momentum about the origin arising from the motion of the center of mass of the wave-packet, and has nothing to do with the internal structure of the nucleon. Thus we will take as the definition of the ambiguous forward matrix element in the Jaffe-Manohar (JM) approach

(46) | |||||

where we have used the fact that for the on mass-shell momenta forces .

The last part of the analysis concerns the structure of the matrix element of . For Eq. (46) we require an expansion in of up to terms linear in . It is at this point that the choice in Eq. (37) of becomes significant: it greatly simplifies the tensorial structure of the expansion.

Following Ref. [3] one writes, with ,

(47) |

where and are scalar functions of . The terms in Eq. (47) are chosen to respect the relation

(48) |

(To see that one should recall that the nucleon states are physical states with .)

For the forward matrix element

(49) |

Comment 3 While Eq.(48) *is*
correct the crucial expansion (47) is not. The reason is
the following.The RHS of Eq. (47) has been constructed to
transform under Lorentz transformations as a genuine second-rank
tensor on the grounds that transforms as a
second-rank tensor. But the *non-forward* matrix elements of
a tensor operator do *not* transform covariantly, as was
explained in the discussion of Eq. (25). We shall
see the consequences of this in Section 5.1.

Continuing with the derivation in Ref. [3], we have from Eqs. (18) and (41)

(50) | |||||

where we have used the fact that is the total energy or Hamiltonian operator. But the LHS, taking , equals

(51) |

from Eq. (49).

Comparing Eqs. (50) and (51) yields

(52) |

Now one uses Eq. (47) to calculate the derivative needed in Eq. (46). The result in Ref. [3] is

(53) |

Lastly the value of is found by choosing a nucleon state at rest and spin along This is an eigenstate of

(54) |

Then with

(55) | |||||

But from Eq. (53) the LHS is just equal to
^{2}^{2}2There is a factor of
missing in the value of given in [3].

(56) |

so

(57) |

Finally, then, in the Jaffe-Manohar treatment the interpretation given to the forward matrix element of the angular momentum operator is

Equation (LABEL:eq.36) is meant to provide a general basis for angular momentum sum rules. So, for example, if we have a theory of the nucleon in terms of quark and gluon fields and we construct the operator from these fields, then the requirement that our satisfy Eq. (LABEL:eq.36) for an arbitrary state of the nucleon yields a set of conditions on some of the elements of the theory.

As indicated in the comments, there are flaws in the derivation and Eq. (LABEL:eq.36) is incorrect. We shall present the correct result, in a wave-packet treatment in Sections 5.1 and 5.2. There our states or wave-functions will be normalized to so that we calculate actual expectation values. Moreover, as will be explained in Section 4.2, the wave-packet approach seems only able to deal with the physically relevant operators . For these conserved densities the in Eq. (37) is simply equivalent to the factor in Eq. (53). Thus, dividing the expression (LABEL:eq.36) by this factor and by the normalization given in Eq. (41) we have, for the expectation values in the JM-treatment

(59) |

which can be compared directly with the result we shall obtain in Section 5.1 and Section 5.2. In Section 6 we shall give a completely independent corroboration of these results.

We turn now to the treatment of Shore and White, which basically follows the pattern of Ref. [3], but attempts to put the argument on a rigorous footing via the use of wave-packets.

### 4-2 The Shore-White treatment

As already mentioned, the authors of Ref. [3] remark that a wave-packet approach is needed to justify some of the manipulations involved, more precisely, to get rid of the unwelcome derivatives of -functions. This is done in Ref. [4], but, as we shall see, the treatment also suffers from some of the incorrect elements commented on in Section 4.1. The notation in Ref. [4] differs somewhat from that of Ref. [3], so we will rephrase the notation in Ref. [4] to match as closely as possible the development in Ref. [3].

The authors of Ref. [4] try to give a sensible definition to the forward matrix elements defined in Eq. (24), i.e.

(60) |

by utilizing a wave packet. Thus they define

(61) |

where drops rapidly to zero as , and they interpret Eq. (60), modulo normalization, as

(62) |

It turns out to be sufficient to use just one wave packet, either for the inital or the final state.

Note that this differs from in two respects. Firstly, consideration is given here only to the spatial elements of the tensor and secondly, the integral is over three-dimensional space. The latter difference is not significant given that the operators in Eq. (60) are supposed to be time-independent.

The derivation then runs as follows:

(63) | |||||

Exchanging the order of integration, then integrating by parts w.r.t. , and discarding the surface terms at ,

all antisymmetrized under .

Now

(65) |

so in the Shore-White (SW) approach

(66) |

which is identical to in Eq. (46), aside from
normalization, which is irrelevant, since, at the end, it cancels
out when computing actual expectation values.

Comment 4 It is actually not possible in Eq. (66) to take the same in both initial and final states, as in eq. (37). The reason is the following. A wave packet is, by definition, a superposition of physical states. But for a spin-1/2 particle in a physical state

(67) |

and we cannot integrate independently over each component of . That is one reason why we have chosen to define the states using the rest-frame spin vector instead. The correct procedure is then to take the same rest frame for the initial and final states, not the same . Thus the wave packet Eq. (61) should be modified to

(68) |

and all three components of can then be integrated over independently.

The use of the Lorentz tensor form, as in Eq. (LABEL:eq.36), is central to some of the most interesting arguments given by Shore and White—for example their conclusion that the axial charge does not contribute to the angular momentum sum rules. But, as pointed out in Comment 3, this form is based on the incorrect expansion Eq. (47), and so one must be skeptical about their results. We shall see, however, in Section 5.2, that the conclusion regarding the role of the axial charge in the sum rules is correct in spite of the erroneous argument.

To summarize: the Shore-White wave-packet analysis provides a justification for the manipulations in the Jaffe-Manohar treatment for the spatial components of the angular momentum tensor. But the analysis is not correct in general because it utilizes the impermissible simplification and also makes an incorrect use of Lorentz covariance.

Given these criticisms, it is interesting to consider one important practical case of a sum rule that can be derived from Eq. (LABEL:eq.36). Namely, for a longitudinally polarized proton moving along i.e. in a state of definite helicity with along one has the sum rule

(69) |

where , are the first moments of the polarized quark flavour-singlet and the polarized gluon densities, respectively, and , are contributions from the quark and gluon internal orbital momenta. This sum rule is so manifestly intuitively right that it is almost impossible to contemplate it being incorrect [12]. And indeed, it is correct. This is the one unique case where the flaws in the deduction are irrelevant, as will be seen later.

In Section 5 we will present what we believe to be a correct evaluation of the wave-packet definition of the recalcitrant forward matrix element.

## 5 A detailed wave-packet treatment

Given our critical stance vis-à-vis the treatments of Refs. [3] and [4] it is incumbent upon us to proceed with caution. We shall therefore present a detailed wave-packet treatment for two separate situations, for a relativistic quantum-mechanical Dirac particle and for the field-theoretical case. It will emerge that there is complete agreement between the results for the two cases and that they differ, in general, from Eq. (59).

We shall also address the question as to the validity of Eq. (22), which was assumed in the derivations in Refs. [3] and [4]. In order to do this we shall need to distinguish between constructed out of the canonical energy-momentum density as in Eq. (14) and used in Section 4.1 and Section 4.2 based upon the symmetrized , as in Eq. (15).

### 5-1 Relativistic quantum-mechanical Dirac particle

We construct the wave-function corresponding to a superposition of physical states centered around momentum , all of which have rest-frame spin vector :

(70) |

where , and

(71) |

with

(72) |

The constant , whose value is irrelevant for the moment, is adjusted so that is normalized to 1, i.e.

(73) |

Since our aim is to provide a sensible prescription for a plane-wave state of definite momentum , we shall, at the end, take the limit . In this limit the values of that contribute in the integral Eq. (70) are forced towards , so we make a Taylor expansion of about the point and keep only those terms that survive ultimately in the limit .

Note that we are unable to carry out the analysis if can be arbitrarily large. The point of using the wave-packet is to produce a cut-off in the divergent spatial integrals in Eq. (24), but this does not produce a cutoff in . Thus the wave-packet approach can not be used if the operator densities are integrated over four-dimensional space-time. Since we are only interested in conserved densities, we will take advantage of the time-independence to choose in Eq. (70).

It turns out to be sufficient to keep just the first two terms of the Taylor expansion, which yield:

(74) | |||||

with .

The term is removed from under the integral in Eq. (70) via

(75) |

The remaining integral is just the Fourier transform of a Gaussian, yielding a factor on the RHS of Eq. (75) then produces a term . The result is , and the

(76) |

where

(77) |

and

(78) | |||||

The structure of Eq. (78) is extremely instructive in understanding both the difference between matrix elements of local operators like and compound ones like , and the question as to how many terms of the Taylor expansion are necessary.

On the one hand the Gaussian implies that the effective values of satisfy . On the other, the term in the Taylor expansion provides a term of order . For a local operator there are no other factors of present, so even the first-order term of a Taylor expansion can be ignored in the limit