Transverse spin and higher twist in QCD

Nuclear Physics B264 (1986) 493-512 © North-Holland Publishing Company

TRANSVERSE SPIN AND HIGHER TWIST IN QCD* P.G. RATCLIFFE

Caoendish Laboratory, Universityof Cambridge, Cambridge, UK Received 25 March 1985 (Revised 18 September 1985)

A complete, but non-standard, parton model description is made for transverse-spin effects in hadronic interactions. The Q2 dependence of the various structure functions involved (corresponding to operators of both twist two and three) is evaluated to leading-logarithmic order via the ladder-diagram summation technique.

1. Introduction

In recent years polarisation effects in hard processes have stimulated much interest both from experimental and theoretical points of view. On the one hand there is a growing body of data which indicate that transverse-spin effects can be very significant [1, 2], in direct conflict with early perturbative calculations. On the other hand the higher-twist contributions (which enter at the same level as the leading-twist effects) render this aspect of the theory interesting with regard to renormalisation and evaluation of the resultant In Q2 scale-violating effects [3]. Although there have as yet been no direct measurements of the structure functions governing transverse-spin effects in deep-inelastic scattering (DIS), there is a large amount of very striking data from hadron-hadron interactions. In particular the polarisation of lambda particles inclusively produced in proton-proton collisions is very strongly correlated with the normal to the production plane [2], see fig. 1; the polarisation reaching values of up to about 50%. Early calculations of such effects [4], using the framework of perturbative QCD, predicted small polarisations (of the order of 1%). However, more recent work [5, 6] has shown that the relevant mass parameter is that of the polarised hadron and not, as previously assumed, that of the struck parton. Moreover the same authors make the claim that the imaginary phase necessary for such single-spin asymmetries could be generated at the Born level by diagrams with three partonic legs [7]. Such diagrams are necessary to take into account fully the twist-three contributions. Thus the theory may well be able to explain all known phenomena without any real modifications. Unfortunately * Research supported by the UK Science and Engineering Research Council. 493

494

P.G. Ratcliffe / Transversespin

pcA° O.f

I

-b---'

0

!

|

|

I I

-0.! -0.2

-O.q

-

-0.6

0

!

I

o.q

I

o. 8

1.2

P.L , GeVlc Fig. 1. Polarisation of inclusively produced A hyperons as a function of PT (BIS-2 collaboration).

numerical estimates are not possible without the input of, as yet, unknown structure functions. The study of leading-logarithmic corrections to the structure functions governing transverse polarisation is made particularly interesting by the fact that operators of both twist-two and three contribute equally. The reason for this surprising fact can be understood quite simply by noting that while the longitudinal component of the spin vector receives the same boost as longitudinal momentum in the infinitemomentum frame, the transverse component, by definition, is unaffected; thus terms proportional to s x are suppressed relative to those of s L by a factor Q = ~ - q 2 . Here too, the early work on twist-three anomalous dimensions was erroneous [8] owing to the neglect of certain twist-three operators [9] which led not only to discrepancies but also to a complete loss of gauge invariance [10]. However more recent work, adopting the operator-product expansion approach and correctly taking into account all twist-three operators, has produced unambiguous results [11-13]. The purpose of the present paper is to complete the analysis of [6], which through use of an explicitly gauge-invariant basis avoided an important structure in the hadronic polarisation matrix, and then using this parton model basis (via the technique of ladder-diagram summation [14]) to evaluate the Q2 dependence of the relevant structure functions in the leading-logarithm approximation (LLA). The advantages of the approach are three-fold: firstly this parton-like formulation provides a physically intuitive picture of processes involving transverse polarisations, it also forms a basis from which a correct implementation of the ladder summation procedure automatically generates the complete set of twist-three structures necessary for gauge invariance and finally it is immediately applicable to all processes whether space-like or time-like.

P.G. Ratcliffe / Transverse spin

495

The lay-out of this paper is as follows: sect. 2 deals with the general definitions relevant to a discussion of polarisation in hard processes and the relation of higher twist to transverse spin, sect. 3 displays the necessary modifications and extensions to the standard parton model to include a description of transverse polarisations, sect. 4 contains the details and results of the calculation of the Q2 evolution of the parton densities and finally sect. 5 contains some concluding remarks. Appendix A contains some calculational details while appendix B discusses the rble of self-energy contributions in the axial gauge.

2. Polarisation structure functions and higher twist Since it will be useful later on to make reference to some physical process let us first recall the definitions of structure functions etc. governing the DIS of polarised electrons off a polarised proton target. This process can be described in terms of the imaginary part W ~ of the forward Compton scattering amplitude. The part of this tensor symmetric in the indices/,v is related to the spin-averaged cross section and will be omitted here. The contribution which depends on polarisation comes from the antisymmetric part: Ve'~A =

[

2Mrr. p o , e ~ , o q S gl + p.q

s °-

s. p.

qpo)]

g2 ,

(1)

where p and q are respectively the hadron and virtual photon momenta, M is the hadron mass and the hadron polarisation four-vector s ° obeys s°=~(p,s)y°Ysu(p,s)/2M,

s 2= - 1 ,

(2)

p.s=O.

The two structure functions gl and g2 depend on the energy scale Q 2 = _ q2 and the Bjorken scaling variable x a = Q E / 2 p . q. It is useful to make a Sudakov decomposition of the polarisation vector: (3)

M s ° = an ° + bp ° + Ms~-,

where n ° = q,O/p, q,,

q'°=q°

+ xBp~ ,

n.sr=O=p.s

T.

(4)

Inasmuch as contributions suppressed by powers of M 2 / Q 2 are to be neglected, the vectors n and p may be considered light-like. The constraints (2) then show the n component of s to be power suppressed and that b - O(1) - I s ~ l .

(5)

Thus, since M s . n - O(Q2), the longitudinal component (Ms~. = bp °) can be consid-

496

P.G. Ratcliffe / Transversespin

-~

P

Fig. 2. Schematic representation of the quark-helicity contribution to the transverse polarisation of the parent hadron.

ered as being of order Q while the perpendicular component is explicitly of order M and hence there are twist-three contributions to transverse-polarisation dependence. Note, however, that there is also a twist-two contribution to the transverse-spin structure function; this can be understood intuitively by noting that even a quark carrying pure helicity, owing to a non-zero kT, will contribute to the transverse polarisation of the parent hadron (see fig. 2). The decomposition (3) allows (1) to be written in the following simplified form (suppressing the index A):

W~v=

2M~r p'q ie~s,qo[S[gI + S ~ 2 ] ,

(6)

£l~vqo = e..ooq°.

(7)

where ~2 = gl q- g2 ,

It should also be pointed out that the structure function gl is the exact axial counterpart of the spin-averaged structure function, i.e. in the language of Feynman diagrams and light-cone operators the only difference is the additional presence of a 75 which factors out of all calculations; hence the well-known result that the anomalous dimensions governing the Q 2 evolution of gl are identical to those for F 1 and F 2. Physically, gl can be interpreted as measuring the fraction of longitudinal polarisation carried by the partons as a function of their fractional momenta; as we shall see later there is no such simple interpretation for g2-

3. Patton model and parton densities

As a starting point let us adopt the following representation for a short-distance process involving a hadron of definite momentum and polarisation interacting via some hard scattering mechanism (denoted generically below as E), see fig. 3:

W= f d4kM(k)E(k)+ f d4kd4k'N"(k,k')E,,(k,k'),

(8)

P.G. Ratcliffe / Transversespin

{a}

497

{b}

Fig. 3. (a) Two-parton and (b) three-patton contributions to the hard scattering of a polarised hadron via the short-distance partonic subprocess E.

where M and N are the hadron-parton amplitudes: M~(k)

= f d4z exp( ik.z )(

N~a(k, k) = f d4z d4z'exp[i(k

p, s [ ~ ( 0 ) + a ( z )IP, s ) ,

- k').z + ik'-z']

( p, sl~(O)gA°(z)~B(z')lp, s), (9) with d4z = d4z/(2~r) 4. And terms which make no contribution to the leading terms of twist three in the axial gauge, n. A = 0, are omitted. The second term of (8), representing the diagram of fig. 3b, contains only higher-twist contributions and thus only contributes to transverse spin. Using the Fiertz identity with respect to the indices a and fl and keeping only the relevant structures, (8) can be rewritten in terms of its vector and axial-vector projections:

W= f d4k {[ E( k )'CIMV( k ) + I E( k )'Y5"~p]MA(k ) }

+ fd4kd'k'{[E°(k,k')v~]lMV(k,k')+l[E"(k,k')vsvP]MA(k,k')}, 0o) where [ ...1 = ¼Tr[...] and quark-mass contributions have been neglected for simplicity (these will be reinstated at a later stage). Consider first of all the simpler case of purely longitudinal polarisation: the only projector contributing to (10) is then

MA(k)'y£y"

= AL(k) ~,5&.

(11)

P.G. Ratcliffe / Transversespin

498

Comparing this with the spin-density matrix for a fermion, 51(~b+ m)(1 + Ts~), one sees that the parton density A L (having dimensions of mass) measures the fractional longitudinal polarisation (or helicity) of the parton. Now applying a Sudakov decomposition of k: k ° = y n ° + xp ° + k ~ ,

(12)

the integrations over y and the transverse degrees of freedom may be performed, since in the LLA powers of kv may be neglected, to give

w= f dx [ E ( x ) ' Y 5 ~ L ] A L ( X ) ,

(13)

fdyd2kTAL(k).

(14)

where AL(X ) =

The range of x, the momentum fraction, is - 1 < x < 1. Thus, using (9) and (1I), one can write A L

(X)H'S

=

f dX exp(iXx)< ~ (0) y5~ ~ ( h ) ) ,

(15)

where ( . . . ) = ( p , s]... ]p, s). The inverse Fourier transform is then defined by ( t~(0)Ts~ ~b(X)) = A L ( X ) n ' s .

(16)

For deep-inelastic scattering one has

/x +q)2

Vp_-q?

(17)

and thus

w?,----it~,q~c[Ai.(x ) +AL(--X)], p'q or

2 M g l ( x ) = AL(X ) + A c ( - x ) .

(18)

The ratio gl/F1 is known to be of order unity for large x [15] and thus (18) demonstrates that the correct mass scale is indeed that of the polarised hadron. If one considers the limit of zero quark mass this result follows immediately since the hadron mass is then the only scale relevant to polarisation. There are, as mentioned above, effects due to the (current) quark mass, but these are suppressed by a factor mq/m.

P.G. Ratcliffe / Transverse spin

(a)

499

{b)

Fig. 4. (a) Two-particle irreducible and (b) three-particle irreducible ladder diagrams contributing in the LLA to transverse spin.

Turning now to the case of transverse polarisations one finds a much richer structure of projectors and operators. The naive extension to (11) is MA(k)'/,7 p ....

+ AA(k)"t,4tT .

(19)

However, as will be shown explicitly in the next section, under renormalisation (i.e. considering the effect of hard gluon emission) this projector, from a simple one-rung diagram (see fig. 4a), generates a further structure: M ~ ( k ) 757 p . . . .

+BA(k)751~2kT • ST/k~-.

(20)

In the twist-two case no such less-than-logarithmically divergent term occurs, however it cannot be neglected since it is possible to pick up a further power of k T from the next rung (in [6] such a term was only included implicitly). Furthermore for both of these structures the ordering of the virtuality in the ladder rungs is weakened and thus diagrams of the type shown in fig. 4b may also contribute. The intermediate three-parton state (~tgq) then generates the following vector and axial-vector projectors

M~°(k,

k')757 p =

MV°( k,

k')7' =

DA(k,k')751Js}, DV( k, k') t~iep"°s.

(21)

In turn these lead to vector contributions to the two-parton density: M V ( k ) g p = AV ( k )iePnSPTp+ BV( k )iepnsk~/k~.

(22)

This then forms a complete set of projector under renormalisation, indeed it is not difficult to verify that these are the only projectors one can construct. As is indicated by the explicit appearance of the gauge-fixing vector n (and implicitly where the vectors s T or s L appear) in these expressions, the above

500

P.G. Ratcliffe / Transverse spin

hadron-parton amplitudes are not separately gauge invariant. Moreover, inserting these projectors into the expressions for W in DIS, one sees that individually they violate conservation of the hadronic current. One expects then some constraints relating the various densities. Let us first of all rewrite (19)-(22) in an analogous way to (16): ( ~ (0)~5~¢ ( X ) ) = A~(X )s~, ( ~ (0)'/-~¢ (~.)) = AVT( X )ie pns" , (~/(0) vs~ia-~¢ ( X ) ) = BA(X)s~., ( ~ ( 0 ) ~ i 0 ~ (X)) = BV(A)ie p"s",

(t~(0)Vs~gA~(Jk) tp (~k')) = DA(~k, ~k')S~-, (~/(0) ¢~gA~-(X)~b(X'))= DV(X, X')ie p's°.

(23)

The first set of constraints is a consequence of the equations of motion: O ( z ) q ~ ( z ) = 0 = ~(O)D(O),

(24)

where the covariant derivative is defined as D r ( z ) = ion+ g A ' ( z ) = - i'O" + gA"( z ) .

(25)

As discussed in [16], naive application of the equations of motion is permissible, since any corrections due to quantum effects or renormalisation vanish when evaluated between physical partical states. To obtain the constraints let us apply the Chisholm identity to the relations

o 0 = ( ~ (0)D (0)~y°y5 q~(X)),

(26)

the first leads to O=(~(O)[ys"y°n.D(Xl-YsOD°(X)+ie°"D°'yo]+(X)).

(27)

The Fourier transform of the transverse component (in the index o) of this equation, using (23), gives 0= x[A~(x)-AV(x)]-

- f dx'

[BA(x) - BV(x)]

x')- z)V(x, x')],

(28)

P.G. Ratcliffe / Transverse spin

501

where use has been made of the relation

0~(x) n.D(~')+(X)=i

O)t

'

(29)

which, under the integral, leads to the factor x multiplying the first term on the RHS of (28). Similarly the second equation of (26) leads to

o= x [A~(x) + A~(x)] -IBm(x) + sV(x)]

- f d x ' [D~(x,, x) + DV(x,, ~)].

(30)

These two constraints are sufficient to guarantee conservation of the hadronic current. Note that x, x ' and (x - x') are all restricted to lie in the range [ - 1,1]. The second set of constraints is obtained from the requirement that the final physical expression should be gauge invariant, ie that it should not depend explicitly on the gauge-fixing vector n. Consider first the term involving longitudinal spin (13) (this only contains axial projections), using (3) to write the projection of s ° onto the vector pO one has

wA= f dxaA(x)AL(X),

(31)

where the gauge-dependent quantity a A is defined by

aA(x) = [ E(x)Ysl~]s. n.

(32)

Now let us turn to the axial terms involving transverse spin:

we = f a x ~E(x)~5~T~A~(x)+ fd"* [e(*)~,~IB~(*)2*T" ~T/*~

+fdxdx'[Eo(x,x')vsklDA(x,x')s~.

(33)

The A term can be rewritten as

f dx[E(x)vs#La~(x)=f dx [aA(x)-a~(x)]A~(x),

(34)

a A ( x ) = [E(x)~'5~]] •

(35)

where

P.G. Ratcliffe / Transoersespin

502

The B term is a little more complicated. The function B itself only depends on x and the extra power of kT must come from E ( k ) . Expanding under the integral, one has

E(k)2kT.ST/k~

e(xp) +-~(xp)(k-xp) ~'2kT'ST/kZT OE Ok~(xp)s~,

(36)

where, for the last step, symmetric integration over the transverse variables has been performed by means of the relation (valid in four dimensions).

f d2kT2k~k~;/k 2 =

f d2kTg~ :"

(37)

Use of the Ward identity, OE Ok" (xp) = e , ( x p , x p ) ,

(38)

then allows the B contribution to be written as

f dx [E~(x, X ) y s k ] B A ( x ) s ~ - . The collinear Ward identities [17]:

pOEo(xp,x'p) =

[E(xp)

-E(~,p)]/(x - ~'),

(39)

lead to OE p°Eo(x, x) = ffX-X( x ) , Substituting s T arrives at °

= S

o

--

(40)

p°s. n and applying (40), after integration by parts one

A OB f d x b A ( x , x)B A (x)+ f dxaL(X)Tx(X), for the B term, where

hA(x, X') = lEo(X, x')Vsk]s °.

(41)

P.G.Ratcliffe/ Transversespin

503

Finally the D term may also be similarly rewritten to give 0B A wA=fdxaA(x){ AL(X)--AA(x) +--ff~-x (X)

-fdx'[D"(x,x')+DA(x',x)]/(x-x')} + f dx [aA(x)A~(~)+ bA(~, x)B'(x)] + f dxdx' b~(x, x')DA(~, ~'). (42) The requirement that the gauge-dependent terms vanish for arbitrary a~ then implies the relation 0B A

AL(x)=A~(x)---~x (X)+fdx' [D~(x'x')+DA(x"x)]

(x-x,)

(43)

A similar procedure applied to the vector projections gives

oBv

O=AV(x)----~-x (X )

+f

dx'

[•V(x,x,)+DV(x, x)] (x-x')

(44)

Comparing (28), (30), (43) and (44) one sees that the set of equations is diagonalised when expressed in terms of the combinations X + = X A _+ X v (for X--- A, B, D), the constraints then reduce to

o = xAT(x)- B+(x)-fdx'D+(x, x'), oB ) + f dx' AL(x)=A.~(x)--~x(X

(45)

plus a similar pair of equations for the "minus" components (with the x and x' arguments of D interchanged in the first). There is one further set of constraints which it is now convenient to apply; time-reversal invariance imposes the following set of relations, obtained by considering the complex conjugates of the Fourier transforms of the densities in (23):

X+(x)=X-(x) D+(x, x') = D-(x', x).

forX=A,B, (46)

P.G. Ratcliffe / Transverse spin

504

This means that (44) is simply the identity and one need only consider the set of equations for the axial projections of A and B and the plus projection of D. Substituting for B from the first equation of (45) into the second one obtains,

aA

+f

dx' [ aY (x-x')

ax'aY( , ] x' x) ,

(47)

where the superscripts A and + have been suppressed and Y is defined as

Y(x, x') = ( x - x')D(x, x ' ) ,

(48)

which corresponds to eq. (43) of ref. [6]. For DIS, in terms of the above densities, the structure function governing transverse polarisations is 2M~2(x) =

AT(x) +AT(--x),

(49)

and thus from (47) one sees that the gauge-invariant quantities are the functions

A(x) and Y(x, x'). Furthermore in the lowest order one still has the naive picture of a simple effective spin-density matrix for the proton, the various terms being multiplied by parton distribution functions. However, as shown in the next section, in higher orders of perturbation theory it is necessary to include the effects of the nonstandard parton densities introduced above. Before moving on to an analysis of the leading-logarithmic behaviour let us examine the effects of a non-zero quark mass. In the expansion (10) one structure was omitted:

W ....

fd4k~E(k)vsvOqMoeJ(k),

(50)

where "/P° = ¼[yPy° - y°y°]. This pseudotensor structure, having an even number of gamma matrices, requires a mass term from E (since E is the hard partonic process this will in fact be the current quark mass). The corresponding projector is then 1

M~7( k )757 p° = -~C( k ),/5~Tj~ ,

(51)

the explicit factor of 1/M is included in recognition of the fact that, although the density C is dimensionless, the appropriate mass-scale is still that of the polarised hadron. The new density is already in a gauge-invariant form and so does not contribute to the constraint equations related to gauge invariance. The equations of motion however, now have a contribution coming from the quark mass. The following extra term is then introduced into (27): m

m( ~(O)ys~y-~+( 2~) )= - ~ C ( ~ )s~,

(52)

P.G. Ratcliffe / Transverse spin

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and (47) becomes

A(x)=

- - X

OAT

m OC +f (Xl+-d-y-;(x)

dx'

(x--x')

JOY

-

OY -

X t ,x)

]

. (47')

4. Evolution equations for twist two and three

The method most suited to the evaluation of the Q2 dependence of the parton densities defined in the previous section is the well-established technique of ladderdiagram summation [14]. The novel feature, mentioned above, encountered with the twist-three structures is that the ordering of virtuality or Ik21 now no longer forces gluon rungs to be attached to points of the same scale of virtuality. And thus one has to consider diagrams with intermediate three-parton states, arising from non-planar diagrams and diagrams containing the triple-gluon vertex. As in the previous section, it turns out that the structures which diagonalise the evolution equations are the + projections. These correspond to the following projectors (for the plus components only) representing the hadron-parton amplitudes of fig. 3.

f d'kM(k)=fdx [A(x)½3'5/~T + B(X)2tSt~T#T/k2]. f d4k d4k ' N°(k, k')= f d x d x ' D(x,x')ysl~-cyg ;,

(53)

where, although the integrals are to be considered included in the hadronic blobs M and N, the B term has been left with explicit factors of k x to absorb one extra power from the next gluon rung. Let us first consider the effect of one gluon rung (fig. 4a) on the A part of the amplitude; suppressing a trivial colour factor and the coupling constant, the integral to be evaluated is

~' 1 ~' DP°(k- k') f d4k' M(k ) K ( k ' ) = f d'k'-k~Vp[~vsb~iT]VO k, 2 --(-~--k---,)-5 ,

(54)

where the gluon projector in the light-like axial gauge (n- A = 0) is

Dp"( k ) = gpO - ( kPn ,, + kPn~)/k, n .

(55)

The integral is performed by closing the y' contour to pick up the residue from the

506

P.G. Ratcliffe/ Transversespin

Fig. 5. Non-standardgluonrung. (k - k ' ) 2 pole, this restricts the x ' integration to the region 0 < x ' < x. Discarding all more-than-logarithmically divergent terms (since these will be power suppressed), the remaining integral is

fdln

d x ' [1y5~T + x"ysJ~;T~'T/k'T2] ~x '. kCfox -"~X

Thus the structure of the B term has emerged naturally, since again one is forced to retain this less-than-logarithmically divergent term. It is perhaps of interest to note that the contents of the brackets in the above expression, in the LLA under the integral, can be written (adding to it the "minus" component) as ,{5~'k'. ST/k ~. Consider now the A term and a non-standard gluon rung, fig. 5; the integral to be evaluated is Doo(

-

f d4k'M(k)K°(k')=f d4k'[~1),,b~v]V,,k,2 ~_~,~ ,

(56)

notice that there is one less small denominator than usual; in the twist-two case this leads to the vanishing of this diagram in the LLA. Using the method of integration as for (54) one obtains the following integral ,x dx'

x'

f din k ,2TjO/--x --'"YS~T' x Y'~ Appendix A contains a detailed calculation of the one-rung diagrams for the B term, where care is required in order not to discard terms which can provide the necessary extra power of k r. Having demonstrated the way in which the various structures arise let us now construct the full Bethe-Salpeter equations, fig. 6, since the full expressions are very

P.G. Ratcliffe / Transversespin

507

(a)

(b)

+

h.c.

+

+

h.c.

+

h.c.

(c) +

h.c.

Fig. 6. (a) Bethe-Salpeter equations for the densities A, B, C and D, with (b) 2PI kernels and (c) 3PI kernels.

I

i

t~

I

+

i

i

I-,i

+

-t-

t,,i

+

I'-i

'-11

b~

I

I

L

r

bt

I

bl

-t-

I

i

I,,i

Ixl

I

t,i

I

I

i

i

I

b,i

t-i

bt

+

-t-

I

I

t-i

Jr

+ I

f

,

t-,i

4~1~ 4-

I

I

bt

I i

i

v

i

bl

I

-t-

i

-t-

bl

I

J

I

4-

I

i

I

I

÷

i

i

I

I

i

i

I

I

I

I

II

~ °

J

°°

o

o

P.G. Ratcliffe / Transversespin

509

Note that, as they stand, the evolution equations all contain divergences. The divergences which arise from the use of an axial gauge cancel between ladder graphs and self-energy contributions, for a more detailed discussion see appendix B. The remaining divergences cancel between the various contributions on taking into account the equations of motion (45) and considering the combination corresponding to the RHS of (47). However there still remains a problem in the equations for D(x, y), since they lead to integrals of the type

f y-x d2 o y-z

,

which gives In(x/y) and therefore do not correspond to moments of the parton density D(x, y). However, as discussed earlier, the correct density to consider is actually Y(x, y), see (48). The complete set of equations written in moment form is then A[ = CF(3 -- 2S n - 2S,,+2)A [ , 4

d n = --4CFS,,C n ,

m

""r/,'* = k ( k + 1)(k + 2) CF--MC"

+ YJ.'*( CF(3 -- 2 S j . 2Sk) + (2C v - CA)

2(-1)*

k ( k + 1 ) ( k + 2) 1

+CA

,_1

{

+ y ' yd-,,,k+-, (2CF--CA)(--1) m

1

k+l

Cn-1 k+m (n + m)

j+l

n

] s j - s, / J

( - 1 ) k cmj-1 ]

?lm

+ Ca

k_l

j+ 1 1

k+2

C2-1

m=l

( - 1 ) j + _1

( j - m ) ( j - m + 1) k ( k + l)m

/

{

+ ~_, YJn+m'k-m ( 2 C F - C A ) ( - 1 ) m m=l

[/ t"J+" ""7 ~ (~ + m) X [ CJ-1 I'lm

2(_1)i

( k - m + 1 ) ( k - m + 2)

c~ k ( k + 1 ) ( k + 2)

(58)

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P.G. Ratcliffe / Transverse spin

where C~ are the binomial coefficients, S, =

A"= f dxx"A(x),

E~l/j; the moments are defined by

c'= f dxx"-lc(x) n=j+k,

(59)

A[=(n+ l)A~r+n--C'- ~_, ( n - k ) Y k''-k M k=l

(60)

U "k= fdxdyxJ-lyk-~r(x,y), and relation (47) takes the form m

n-1

These results are in complete agreement with those obtained in [12] using the operator-product expansion approach (note that the expression given in [12] corresponds to the minus component and thus to the above with the arguments of Y interchanged). Note that the function C(x) acts as an inhomogeneous driving term in the equation for the evolution of D(x, y) and, in principle, the functions A, B and D could also appear in the equation for C. Evaluation of the relevant graphs reveals zero contribution, which is as required for self-consistency in the limit of zero quark mass. 5. Conclusions While it is clear from the preceding discussion that there is no simple interpretation of 4l, relation (49) indicates it to be a naive measure of the fractional transverse spin carried by the parton. However, it has also been shown that current conservation requires the introduction of at least the density B(x) (even if the gluon field is neglected) and that gauge invariance (necessary for evaluation of higher-order corrections) forces one to consider the clgq density. In the case of DIS at least, it has been found that to leading order the only density contributing to g2 is A r. Thus in the leading QCD order the naive patton picture carries over to transverse spin; although already at the level of the LLA, where off-shellness and non-zero k T become relevant, the picture is clouded by the presence of the gluon field of the hadron. It should be pointed out that the apparent pole in 1 / ( x - x'), eg. in eq. (43), coming from the gluon propagator, is only an artifact of this way of writing the expression and disappears on rewriting in moment form. Thus the suggestion that three-parton diagrams could provide a Born-level imaginary phase [7] seems improbable. However the predictions made in [6] use only plausible mean values for the distributions, whereas it has been noted [18] that g2 is much larger than gl for small x and could thus reproduce the large experimentally measured spin asymmetries.

P.G. Ratcliffe / Transverse spin

511

On this last point, it would be very useful to calculate double-spin asymmetries adopting the patton-like basis set up here. Such a calculation would be relevant for example to A production off a polarised target as will be measured by the UA6 collaboration [19].

Appendix A The gluon rung of fig. 4a for the B part of the hadron-parton amplitude gives the following integral

[¢'

(A.I)

~¢' DP°(k - k')

fd4k'-~'/p[Vsl~T[CT/kElv*k,2

( k _ k,)2

Closing the y ' contour to pick up the ( k - k ' ) 2 pole gives

(A.2)

(1 - x ' / x ) k '2 = ( k'T2 - 2k'T" kTX'/X) • Thus the k T in the denominators may be removed by making the shift to X !

kT=k~

--r

m

- - k r.

(A.3)

x

The usual symmetric integrations in k T followed by k~r may then be made, taking care to keep all terms of the type ~:~r°,/k'2. The resulting integral is then

f dln

fXo

(1

+

X I// X + 2 x ' 2 / x 2 )

x(1 - x ' / x )

'

(A.4)

where the shift in k~ is no longer important.

Appendix B The cancellation of divergences in the three-parton evolution equations is somewhat more complicated than in the well-discussed twist-two case. The usual cancellation of the singularity in the fermion wave-function renormalisation takes place, however one also has to include the gluon self-energy correction (for the gluon leg associated with the density D) which can be written in the following form

1 CAf dz -1-- z

1 + z + b'

(B.1)

where b is the first coefficient of the B-function and cancels an identical contribu-

512

P.G. Ratcliffe / Transverse spin

t i o n c o m i n g f r o m t h e r e n o r m a l i s a t i o n o f the c o u p l i n g c o n s t a n t i m p l i c i t in t h e d e n s i t y D . T h e d i v e r g e n t p i e c e s t h e n c a n c e l s i m i l a r d i v e r g e n c e s in the 3PI k e r n e l s c o m i n g f r o m g r a p h s c o n t a i n i n g the t r i p l e - g l u o n vertex.

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