TRANSVERSE POLARISATION IN QCD AND RELATED PROBLEMS

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TRANSVERSE POLARISATION IN QCD AND RELATED PROBLEMS

P. Ratcliffe

To cite this version:

P. Ratcliffe. TRANSVERSE POLARISATION IN QCD AND RELATED PROBLEMS. Journal de

Physique Colloques, 1985, 46 (C2), pp.C2-31-C2-43. �10.1051/jphyscol:1985204�. �jpa-00224514�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-31

TRANSVERSE POLARISATION IN QCD AND RELATED PROBLEMS

P.G. R a t c X i f f e

Cavendish Laboratory, Madingley Road, Cambridge, U.K.

Résumé - On examine le problème suivant : comment expliquer dans le ca- dre de la QCD perturbative les grandes asymétries à une polarisation transverse observées, par exemple, dans la production des A . En par- ticulier un travail récent a montre que l'échelle de masse appropriée n'est pas celle du quark frappé,mais du hadnan polarisé et que la pha- se imaginaire nécessaire peut provenir naturellement au niveau du terme de Born,

On discute aussi le traitement des violations d'invariance d'échelle.

Abstract - The problem of explaining, within the framework of perturb- ative QCD, the large observed single transverse-spin asymmetries in, for example, A production is reviewed. In particular, recent work has shown that the relevant mass-scale is not that of the struck quark but of the polarised hadron and that the necessary imaginary phase may arise naturally at the Born level. The treatment of scaling violations is also discussed.

For some years now the experimentally determined large single transverse-spin effects in hadronic interactions [l] have been widely held to contradict QCD and the parton model [2]. In the na'ive parton picture transverse-spin effects are proportional to the quark mass [3] and therefore heavily suppressed. Moreover a single spin

asymmetry, arising from interference between spin-flip and non-spin-flip amplitudes, requires an imaginary phase; standard Born amplitudes are purely real and thus such effects are also proportional to ct

. However, recent work [4] has shown, via a study of the hadron spin-density matrix, that the correct mass-scale, indeed for all spin effects, is that of the polarised hadron. In addition, the same authors have demon- strated that Born diagrams with three partonic legs, arising naturally in the description of transverse spin [5], can produce the necessary imaginary phase without any a

suppression [6]. Looking beyond this leading order analysis, transverse spin also provides an excellent opportunity for the study of logarithmic corrections

(giving rise to scaling violations) to the higher twist effects relevant to this case.

The non-perturbative nature of transverse-spin effects, not surprisingly leads to subtle complications in respect of renormalisation; indeed much of the existing work in this regard is in error and only recently has the role of twist-three operators been fully understood [7,8].

Experimentally the polarisation of inclusively produced hyperons at large P T is well documented [l]; the most striking feature is the magnitude of the .polarisation which increases with p^ £.1 GeV reaching values of about 50%, see fig.l. Similarly there are large left-right asymmetries in inclusive meson production with either beam [9]

or target [lo] polarised normally to the production plane.

QCD via the hard scattering of partons permits the prediction of many of the qualit- ative features of hyperon polarisation. In particular it explains the independence from energy and target type and the weak dependence on xp, it can also explain the large pion asymmetry [ll].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985204

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Polarisation of inclusively produced A particles plotted against p

I'

However an analysis of the subprocess within the framework of QCD leads to a very small polarisation, proportional to both the quark mass and as. To see this consider the scattering of a free quark, in accordance with the na'ive parton picture, forwhich the spin density matrix is

where m is the current quark mass and the spinor normalisation is taken to be

= 2m.

The covariant spin is thus defined as

and is dimensionless. The transverse-spin dependence comes from two terms in (1) : my5 $ depending explicitly on the quark mass and fiy5 $ which, in order to have the required even number of gamma matrices in a trace, must pick out a factor of m from another spin projector or quark propagator. However, longitudinal-spin effects are clearly not suppressed since in the limit m

0, my5$ + y 5 ~ . Furthermore the ys implicitly contains a factor i; thus, since parton-parton Born amplitudes are real, the polarisation requires a loop diagram and is therefore proportional to as.

This talk is divided into three parts: the first deals with the problem of the mass parameter for spin asymnetries, the second discusses the possibility of a Born level contribution to the imaginary phase in single transverse-spin asymnetries and finally the third part goes on to explore the problem of logarithmic corrections within the framework of perturbative QCD.

§l The Mass Parameter

As a point of departure let us adopt the following representation of the cross-section for the process AB + CX, in the large p~ region, with one particle (B) polarised (fig.

2) :

ig/ A.dzl d o = /d~drl<p~,sl&(~)e o

<P~I?.-.Y~-..$IP~><P~~?...Y "... ^{$\pc> ; ! E (5,rl), ( 3 )}

where all singularities at 6-rl

0 are contained in the function E describing the

short-distance partonic subprocess.

Fig.2. The diagram corresponding to cross-section (3). Particle B is described by the covariant polarisation four-vector

The proof of the factorised picture is essentially the same as for unpolarised particles [12]. Applying the operator product expansion:

--

(using the notation <X>

<p,s(~lp,s>) and the Fierz transformation:

the spin-dependent matrix element (for particle B) can be written in terms of the following operator matrix elements (retaining only those of minimal twist):

where factors i-n have been suppressed. The term in bd only contributes in the case of longitudinal polarisation while the pseudotensor.contribution is only present for transverse polarisation, the vector and scalar operators relate to the spin-averaged part. Although the quark mass has been used here, it is not necessarily the true scale parameter of the matrix elements. This fact is related to the requirement of gauge invariance of the separate parts of the above decomposition (necessary for infra-red stability). For the matrix elements to be gauge invariant the external partons must be on mass-shell [13]. Thus we can apply the equations of motion to the matrix elements (6a-d) and obtain relationships between the various projections, allowing their expression in terms of general distribution functions:

d B f (B) gn+l ,

'n,~

a n + l , ~

j0 7

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Thus the matrix element for hadron B in cross-section (3) can be represented by a density matrix which, for transverse polarisations, takes the form

where now the hard-scattering part, E, is evaluated for a parton with momentum Bp (i.e. fraction 8 of the parent hadron momentum).

Similarly, for longitudinal polarisations, defining

and substituting

, , h(p'/~) we have

- 11

It is interesting to note that using the spin representation, rather than pure helicity, raises the question of mass parameter even in the case of longitudinal polarisation.

The introduction of the quark mass into the distribution functions is merely an auxilliary method of isolating the subprocess in a gauge invariant way. In order to find the true parameter we appeal to the conservation of angular momentum and the experimentally known value of the axial operator of neutron 6-decay.

The first of these equations, valid if the gluon contribution is neglected, leads to the sum rule:

while the second gives the well-known Bjorken sum rule [l41 which, written in terms of the functions takes the form

Thus the true scale of the axial current is not the mass of the quark but rather the mass of the hadron.

Physically this result has a simple explanation on dimensional grounds; massless particles cannot be polarised transversely, therefore there must be another scale which can only be that of the polarised hadron itself, To understand this within

the context of the parton model recall that whereas the na-ive parton picture is constructed from free on-shell states, the hadron is actually composed of bound and therefore off-shell quarks. The off-shellness is related to the hadronic mass-scale and leads to an effective (constituent) mass for the quarks. The off-shellness is also related to the transverse momentum of the partons and indeed a non-zero trans- verse momentum itself implies that they are not moving parallel to the parent hadron and thus partonic helicity can actually contribute to transverse polarisation at the hadronic level. Thus in terms of parton fragmentation, degradation of a parton momentum by gluon emission (ql # 0) implies that the q2-evolution of the transverse

spin distribution function is related directly to that of the longitudinal polar- isation.

It is natural then to introduce f

(m/~) $ and f

(m/M) $//, note that since

boA

0 the functions f L and satisfy the kame s k l rule:

implying that transverse spin effects must be of the same order of magnitude as those expected in the case of longitudinal polarisations.

However calculation of the quark-quark scattering process, at the one-loop level, contributing to the pionasymmetrygives, in the central region (y

0), assuming f. a f:

where m2

: p + : m and M is the mass of the polarised hadron. For p

1 GeV/c this gives an a s y e t r y of roughly 3%. Other effects could lead to an enfiancement of this

asynunetry: quark-gluon scattering is known to make a significant contribution to the cross-section in this kinematical region and an important r61e is played by the smearing over parton tzansverse momentum. It is also necessary to take into account operators of the form $ F ' " $ and it is to this that we now turn our attention.

52 The imaginary phase

These new operators represent the effects of diagrams containing three partonic legs, i.e. two quarks and one gluon (fig.3). Infact, as will be discussed in the next section, they are indispensable to a correct analysis of the Q~-evolution of the structure functions.

Fig.3. The two- and three-parton diagrams relevant to photon scattering off

- a transversely polarised hadronic target. E is the short-distance scattering process and r the long-distance distribution function describing a hadron of momentum p and polarisation

Including their effects in the case of photon scattering on a polarised hadronic target, as depicted i r i fig.3, we write schematically:

W

Id4k r ( k ) E(k) + jd4k d k P (k ,k

' E (kl, 1 2 1 1

k2) .

The hadron-parton amplitudes are given by d4z ikz

raB(k)

l - ^e <P,s~;~(o) $B(~)l~,s>

V*)

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The factorisation procedure is most conveniently carried out in an axial gauge,

17.A

0, performing an expansion of the struck quark propagator in powers of four-

momentum in the vicinity of its longitudinal component [l51 and discarding all terms of order p2 (leading to power corrections). Making the standard Sudakov expansion:

k

'

an'' + + ky ,

with

k . p = k n = n 2 = p 2 = 0 ,

and normalisation 17-P

1, we have

W

[: E(xp)

- aE (xp) (k-xp)']I'(x) akp

where

using the notation A(A)

A(Arl) and <A>

<p,s(~lp,s>. The subprocess coefficients are related by the Ward identity:

- aE (xp)

E,,(xp,xp). (22)

a k '

The Fiertz identity then allows the spin-dependent part of W to be rewritten in terms of vector and axial vector projections [5] :

ldxl dx2 B v (x1,xz)K Eu(xi ,x2) v p n cP'sp, where [ . . .

^{f Tr (.} . ^{.) and}

^{sapB and}

dX ,iAx

C(x)

J ^{T;; <} ?(o) S us $(A>> ,

A d A 1 dX2 .i[~l ( x ~ - x ~ ) + A ~ x ~ ~ B (~19x2)

1 (2,)2

T-invariance leads to the following symmetry properties for the two-argument functions (correlators) :

A A V V

B (xl,x2)

B (x2,xi) ; B (xl,x2)

-B (x2,xl) , (25) which also obey the following sum rule, derived from the equations of motion:

A V

dx2 [B

,XZ) - ^{i B (xl,x211}

^C(x1)

This sum rule may be viewed as allowing the transformation of the diagram of fig.4b to the two parton form of fig. 4a; the overall effect then being to give a mass (of the hadron scale) to the quark propagator, fig. 4c.

Fig.4. Diagrams giving rise to an effective mass (on the hadronic scale) for the quark propagator.

Some manipulation then leads to the parton formulae [5]:

~ [ g ~

g2

l

C(X)

where gl and g2 are the standard structure functions of deep-inelastic scattering.

Although, as discussed above, the standard parton model picture leads to zero

imaginary phase at the Born level, the situation is changed when contributions of the correlators are taken into account. An imaginary part (discontinuity in s

2p.q) can appear in the diagram of fig. 3b, i.e. when an "external" gluon enters the Born subprocess. This can be thought of as a box diagram with the quark-gluon vertex being included in the long-distance parton distribution. The integral over x2,inthe vicinity of x2

xl, of the gluon propagator [(xl-x2)s+ie]-' gives rise to an

imaginary part i2v6(x1-x2) and thus to the appearance of an imaginary part of the hadron-parton amplitude, proportional to B(x,x). The asymmetry in this case is not proportional to as; more,correctly, as is absorbed into the function B which contains

<qgAq>,as can be confirmed by analysing the simplest diagrams of perturbation theory.

The situation here is similar to that of vacuum condensate corrections to QCD sum rules [16].

It is interesting to note that since the imaginary part comes from the region xl

xp and the gluon therefore carries zero momentum, one can view this as the interaction with the static field of the polarised hadron providing an effective mass to the quark and an imaginary phase to its wave function.

The authors of [6] consider the implications for the process

+ yX (i.e. inclusive Compton scattering off a polarised target), for which they obtain the following expression for the asymmetry

where the sum runs over flavour, $ is the spin-averaged distribution, the dimension- less distributionsb are defined by

M bV(x,y)

2vi ~'(x,y) ,

M bA(x,y)

B A (x,y) , (29)

m2

ut/s, XI..= -t/(u+s) and x2

-u/s =

. Thus the left-right asymmetry does not

depend on energy (just as in the parton mozel), but does depend strongly on x ~ . 1.

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83 Leading-logarithm summation

The inclusion of diagrams with an extra (gluonic) leg has profound implications in the evaluation of logarithmic corrections [7,17] ; indeed all previous papers dealing with scale-breaking effects in g2 are entirely erroneous, failin as they do to take

into account these extra twist-three effects, as pointed out in f7,17].

To have some indication of their importance let us consider the calculation of the logarithmic corrections to one-loop level using a simple model for the hadronic bound state [17], i.e. an effective proton-quark-diquark interaction, which respects gauge invariance, represented by the interaction Lagrangian:

t

'int ^{= h ( $} $ ~ + h . c . ) , (30)

where I),$ and

are effective elementary quark (fermionic),diquark (scalar) and hadronic (fermionic) fields respectively.

P Ay- ^{\ Y}

h.c. + self energies

Fig.5. Model calculation of the logarithmic corrections to the deep- inelastic structure functions. The dashed line represents a scalar diquark propagator.

The lowest order contribution comes from the box diagram in fig. 5a. As far as corrections to g1 are concerned the factorising diagrams of fig.5b give the usual anomalous dimension and the non-parton picture diagrams of fig.5~ give zero contri- butions independently of the gauge used. However, in the case of a transversely polarised hadron (for which the combination g1 + g2 is relevant) the non-parton-like contributions are not zero and must be included to obtain a gauge invariant answer.

Examining the diagrams in more detail one finds that the transverse-spincontribution, even at lowest order, is non-zero solely by virtue of the off-shellness (or equiv- ently non-zero kl) of the quark legs. Consideration of pure quark-photon scattering requires on-shell quarks for gauge invariance but this requirement leads to zero transverse-spin. On the other hand off-shellness requires full account of the bound state, including extra gluonic legs.

For simplicity let us consider the deep-inelastic scattering of polarised electrons

and protons; the hadronic part of such a process is described by the Fourier trans-

form of the commutator of two electromagnetic currents sandwiched between hadronic

states. At short distances use may be made of the Wilson operator product expansion

for the product of two currents:

The coefficient functions Cn obey the renormalisation group equations

where yn are the anomalous dimensions of the operators 0 (related to its renorm- alisation) and are in general matrices reflecting the mi"xng, under renormalisation, of operators of the same type (twist and dimension). The solution of equations (32) gives the Q~-evolution of the coefficient functions and hence that of the structure functions themselves.

The leading operators relevant to g1 are the set of twist-two operators:

where symmetrisation over the indices upl, ...,pn is understood (as is subtraction of traces). The calculation of the renormalisation of this operator is essentially the same as that for the spin-averaged equivalent and has identical anomalous dimensions.

There is also a twist-three operator contributing to transverse polarisations at leading order:

7 0 p 1 p 2

~ " = i ~ $ y y D D ... ^D

2

5

where now

indicates antisymmetry on these two indices. Another operator, describing genuine quark mass effects and which may mix with R2, arises:

.n-l V v2 un

= 1

m G y 5 y o y D ... ^{D J I ,}

3

where m is the bare quark mass. Since such quark mass effects are inessential complications and vanish smoothly as m/M, for clarity in what follows they are neglected.

Although this exhausts the operators contributing to leading order in the coupling constant g, there are two families of operators which can mix with R2 under renorm- alistion 173:

=

in-3 - V D

...g Pp' ^{... D 'n-I n Y : y S}

4 - R

where g is the bare charge,

-i[Do,DP] is the stress tensor of the gluon field and *FOP

Jcappv F its dual; note that the number of operators increases with n.

Since the FUv may tize any one of n-l positions there are this many of each operator for a given spin.

The coefficient functions for the operators are defined in terms of their matrix elements, which writing

are

C2-40 JOURNAL DE PHYSIQUE

where the factors of n+l are combinatorical arising from the (anti)synunetry over (owl) wl,. . . ,un. The equations of motion for the field 9 then lead to

1 n-l

An

- n+l COn

1 (n-R) yn1. ^R R=1

The quantities 0 and En satisfy the same evolution equation:

n

where 5

b-l RnRn ( Q ~ / A ~ ) with b

7 11 ^CA - 4 C p f the lowest order B-function coefficient, the SU(N) gauge group Casimirs are CF : ( ~ ~ - 1 ) / 2 ~ and CA

N, nf is the number of quark flavours. The lowest order coefficients of the anomalous dimensions,

are given by

The Yn satisfy a coupled set of evolution equations which take the form a

The f. are complicated but rational functions given in full in [18]. Unlike the twist- two cise there is now a dependence on the Casimir of the adjoint representation, CA.

In order to obtain a more physical picture one can consider the technique of ladder diagram summation in an axial gauge, see fig.6a. For twist-two the use of such a gauge in the leading-logarithm approximation eliminates graphs of non-planar topology (fig.6b) and also those containing the triple gluon vertex (fig.6~). For higher twist however, one expects a more complex structure. A preliminary analysis reveals that, in the case of transverse spin, account must be taken of diagrams in which one, but only one, gluon rung crosses any number of other rungs. Moreover it appears that diagrams with triple gluon vertices, as in fig.6c, also contribute.

Fig.6. (a) Ladder type diagram contributing in the leading-logarithm

approximation, (b) and (c) new diagrams of different topology

contributing in the case of transverse polarisation.

Nalve consideration of the hadron spin-density matrix, i.e. (1) with the quark mass replaced by that of the hadron, suggests a projector My5kl for a transversely polar- ised hadronic blob. However, convoluting this with a simple gluon-rung kernel produces a new structure:

M ~5 X 2kl.sl/k:

M YS [Bl

8$2kl.sl/k; 1 , ⁽⁴²⁾

where is defined by the Sudakov decomposition (17) of the vector k. The second term on the right-hand side of eq. (42) cannot be neglected since the one odd power of kl necessary to generate the large logarithm can come from the next rung;symmetric integration over the two-vector kL then leads to the following equivalence:

/d2kL f (k2) ky 2kl. sl/kl

dkf f (kf) sy . ⁽⁴³⁾

This is another way of seeing that eq. (8a) is not the complete picture. The new projector, it turns out, iterates through the ladder structure with the same kernel as the helicity projector hy5$, and does not contribute to non-ladder graphs. It corresponds, in fact, to the quantity On in eq. (38).

The simple projector however, also contributes in diagrams of the type shown in fig.6(b) and (c). Examining

these crossed-rung diagrams shows that this projector then generates the following two projectors for the three parton matrix elements:

P:

M y5 $ sy ,

These structures correspond directly to the axial-vector and vector projections of eq. (24). In practice, though, it is more convenient to use the following combi- nations which are diagonal for all the relevant kernels:

These structures then result in the Bethe-Salpeter equations shown in fig.7 for the corresponding distributions functions; the relevant two- and three-particle

irreducible kernels are shown in fig. 8. Recall that these kernels are in general singular and are regularised by the use of a planar gauge (i.e. q2 # 0) or the introduction of a small gluon mass for example. The singularities are thencancelled against similar singularities, similarly regularised, in the self-energy insertions of the quark, and now the gluon, propagators. The evaluation of these kernels is in progress and the results will appear shortly.

Fig.7. The Bethe-Salpeter equations governing the Q~-evolution of (a)

- the two- and (b) the three-parton amplitudes.

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Fig.8. The two- and three-particle irreducible kernels (the lower legs are amputated as they are already accounted for in the hadronic blob) determining the evolution e,quations of fig.7.

What again emerges from this analysis is that the description one might na'ively give to leading order in the coupling constant is not sufficient and that there are contributions which, although superficially proportional to as, have to be considered at the Born level. In the above physical approach these extra contributions were seen to arise naturally from the diagrams themselves, even if initially ignored.

In the previous section the explanation was given that the power of as (and associated colour factor) was absorbed into the two-argument distributions. Physically the Bethe-Salpeter equations shown above indicate that this power of as also has an associated large logarithm and thus in terms of a perturbative expansion one actually has a power series in a , & n ~ ~ starting at order one, not zero as is usual. Thus summing the series gives a contribution that is really of order zero in as.

Conclusions

While much progress has been made in reconciling perturbative QCD with the experi- mentally measured large single transverse-spin asyrmnetries via a modified parton model and it is now true to say that QCD need not be in conflict with experimental data (indeed it may even explain them), the explanations discussed above have yet to be tested. In particular it is highly desirable to measure both g1 and

independ- ently so as to obtain the necessary information on the two-argument structure functions corresponding to the three-parton hadronic matrix elements.

As for logarithmic corrections, the operator product expansion approach has now been

correctly applied and a complete set of evolution equations derived. The interesting

development here is the appearance of operators proportional to the coupling constant

g which, while therefore not present at lowest order, nevertheless via mixing under

renormalisation influence the Q*-evolution of the structure functions. In the

physical ladder diagram picture this is seen to arise from the fact that now all but

one of the gluons is constrained to be emitted and absorbed at the same momentum

scale, this odd one gluon while unable to contribute in leading order to the hard

scattering process still carries a large logarithm and infact represents the summation

of a power series in a,iln~~ with only the first term missing. It should be pointed out that standard arguments relating higher twist to more partonic legs [l91 already implied the importance of these contributions at twist three.

Transverse-spin then within the context of hadronic interactions, owing to its inherently non-perturbative origin, has presented many difficulties for treatment within the framework of perturbative QCD, the application of which requires a great deal of care. In recompense there are many lessons and information to be learnt from this elementary property of particles.

References

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