Colour modification of factorisation in single-spin asymmetries

arXiv:0911.4306v1 [hep-ph] 22 Nov 2009

IN SINGLE-SPIN ASYMMETRIES

Philip G. Rat lie

1, 2 †

and Oleg V. Teryaev

3

(1)Dip.to di Fisi a e Matemati a, Università degliStudi dell'Insubria,Como

(2) IstitutoNazionale di Fisi a Nu leare, Sezione di MilanoBi o a, Milano

(3)Bogoliubov Lab. of Theoreti al Physi s, Joint Inst. for Nu lear Resear h, Dubna

†

^E-mail: philip.rat lieunisubria.it

Abstra t

Wedis ussthewayinwhi hfa torisationispartiallymaintainedbutnevertheless

modied by pro ess-dependent olour fa tors in hadroni single-spin asymmetries.

WealsoexamineQCDevolutionofthetwist-three gluoni -polestrengthdeningan

ee tive T-odd Sivers fun tion in the large-

x

^{limit, where evolution of the T-even}

transverse-spinDIS stru ture fun tion

g ₂

^{isknown to be} multipli ative.

1 Preamble

1.1 Motivation

Single-spin asymmetries (SSA's) have long been something of an enigma in high-energy

hadroni physi s. Prior to the rst experimental studies, hadroni SSA's were predi ted

to be very small for a variety of reasons. Experimentally, however, they turn out to be

large(uptotheorderof50%andmore)inmanyhadroni pro esses. Itwasalsolongheld

that su h asymmetries should eventuallyvanish with growing energy and/or

p _T

^{. Again,}

however, the SSA's sofar observed show nosigns of high-energy suppression.

1.2 SSA Basi s

Typi ally, SSA's ree tspinmomenta orrelations ofthe form

s · (p

∧

k)

^{, where}

s

^issome

parti lepolarisationve tor,while

p

^and

k

^areinitial/nalparti le/jetmomenta. Asimple example might be:

p

^{the beam dire tion,}

s

^{the target} polarisation(transverse therefore withrespe t to

p

^)and

k

^{the nal-stateparti ledire tion}(ne essarilythen outof the

p

^

s

plane). Polarisations involved in SSA's must usually thus be transverse (although there

are ertain spe ial ex eptions).

It is more onvenient touse an heli itybasis via the transformation

|↑ / ↓i = ^{√ 1} ₂  |+i ± i |−i .

⁽¹⁾

A transverse-spin asymmetry then takes onthe (s hemati )form

A N ∼ h↑ | ↑i − h↓ | ↓i

h↑ | ↑i + h↓ | ↓i ∼ 2 Im h+|−i

h+|+i + h−|−i .

⁽²⁾

Theappearan eofboth

|+i

^and

|−i

^{inthenumeratorsignalsthepresen eofa}heli ity-ip amplitude. The pre ise form of the numerator implies interferen e between two dierent

heli ityamplitudes: oneheli ity-ipandonenon-ip,witharelativephasedieren e(the

imaginary phaseimplying naïve T-odd pro esses).

Early on Kane et al. [1℄ realised that in the massless (or high-energy) limit and the

Bornapproximationagaugetheorysu hasQCD annotfurnisheitherrequirement: fora

masslessfermion,heli ityis onserved and tree-diagramamplitudesare alwaysreal. This

ledtothe nowinfamousstatement[1℄: ... observationof signi antpolarizationsin the

above rea tions would ontradi t either QCD or its appli ability.

It therefore aused mu h surprise and interest when large asymmetries were found;

QCDnevertheless survived! Efremov andTeryaev[2℄soondis overed oneway out within

the ontext of perturbative QCD. Consideration of the three-parton orrelators involved

in,e.g.

g ₂

^{,leads to the following ru ial} observations: the relevantmass s ale is not that of the urrent quark, but of the hadronand the pseudo-two-loop natureof the diagrams

an generate animaginarypart in ertainregions of partoni phase spa e [3℄.

It took some time, however, before real progress was made and the ri hness of the

newly availablestru tures wasfullyexploitedsee[4℄. Indeed,itturns outthat thereare

a variety of me hanisms that an generate SSA's:

•

Transversity: this orrelates hadronheli ity ip toquark ip. Chirality onservation, however, requiresanother T-odd (distribution orfragmentation)fun tion.

•

^{Internal quarkmotion: the transverse} polarisationof a quarkmay be orrelated with itsowntransversemomentum. This orrespondstotheSiversfun tion[5℄andrequires

orbital angularmomentum together with soft-gluonex hange.

•

^Twist-3 transverse-spin dependent three-parton orrelators ( f.

g 2

^{): here the pseudo}

two-loopnatureprovidesee tivespinip(viatheextraparton)andalsotherequired

imaginary part (viapoleterms).

The se ond and third me hanisms turn out to be related.

2 Single-Spin Asymmetries

2.1 Single-Hadron Produ tion

As a onsequen e of the multipli ity of underlying me hanisms, there are various types

of distribution and fragmentation fun tionsthat an bea tive ingenerating SSA's (even

ompetingin the same pro ess):

•

higher-twist distributionand fragmentationfun tions,

• k _T

^-dependent distribution and fragmentation fun tions,

•

interferen e fragmentationfun tions,

•

higher-spin fun tions, e.g.ve tor-meson fragmentation fun tions.

Consider then hadron produ tionwith one initial-state, transversely polarised hadron:

A ^↑ (P A ) + B(P B ) → h(P h ) + X,

⁽³⁾

where hadron

A

^is transversely polarised while

B

^{is not. The} unpolarised (or spinless) hadron

h

^{is produ edat large transverse momentum}

P _hT

^{and PQCD is thus}appli able.

Typi ally,

A

^and

B

^{are protonswhile}

h

^{may be apion or kaon et .}

A

^↑

(P

_A

)

B (P

_B

)

X

X a

b d

c

X h (P

_h

)

Figure 1. Fa torisation insingle-hadronprodu tion witha transversely polarised hadron.

The following SSAmay then be measured:

A ^h _T = dσ(S T ) − dσ(−S T )

dσ(S T ) + dσ(−S T ) .

⁽⁴⁾

Assuming standard fa torisationto hold, the dierential ross-se tion for su h a pro ess

may be writtenformally as( f. Fig.1)

dσ = X

abc

X

αα

^′

γγ

^′

ρ ^a _α

^′

_α f a (x a ) ⊗ f b (x b ) ⊗ dˆ σ αα

^′

γγ

^′

⊗ D ^γ _h/c

^′

^γ (z),

⁽⁵⁾

where

f a

⁽

f b

^{) is the density of parton type}

a

⁽

b

^{) inside hadron}

A

⁽

B

^),

ρ ^a _αα

^{′ is the spin}

density matrix for parton

a

^,

D ^γγ _h/c

^{′ is the} fragmentation matrix for parton

c

^{into the nal}

hadron

h

^and

dˆ σ αα

^′

γγ

^{′ is the partoni} ross-se tion:

 dˆ σ dˆt



αα

^′

γγ

^′

= 1

16πˆ s ² 1 2

X

βδ

M _αβγδ M ^∗ _α

^′

_βγ

^′

_δ ,

⁽⁶⁾

where

M _αβγδ

^{is the amplitude for the hard partoni pro ess, see Fig. 2.}

The o-diagonal elements of

D ^γγ _h/c

^{′ vanish for an} unpolarised produ ed hadron; i.e.,

D ^γγ _h/c

^′

∝ δ γγ

^{′. Heli ity} onservationthenimplies

α = α ^′

^{,sothatthere anbenodependen e}

on the spin of hadron

A

^{and all SSA's must vanish. To avoid su h a} on lusion, either intrinsi quark transverse motion,or higher-twist ee ts must beinvoked.

M αβγδ

⁼

α, α

^′

γ, γ

^′

β δ

k

b

k

a

k

c

k

d

Figure 2. The hard partoni amplitude,

αβγδ

^{areDira indi es.}

Quark intrinsi transverse motion an generate SSA's in three essentially dierent ways

(allne essarily

T

^{-odd ee ts):}

1.

k _T

^inhadron

A

^requires

f a (x a )

^{toberepla ed by}

P a (x a , k T )

^{, whi hmay thendepend}

on the spin of

A

(distribution level);

2.

κ _T

^{in hadron}

h

^allows

D _h/c ^γγ

^{′ tobe}non-diagonal (fragmentation level);

3.

k ^′ _T

^{in hadron}

B

^requires

f b (x b )

^{to be repla ed by}

P b (x b , k ^′ _T )

^{the spin of}

b

^{in the}

unpolarised

B

^{may then oupleto the spin of}

a

(distribution level).

The three orresponding me hanisms are: 1.the Sivers ee t [5℄; 2. the Collins ee t [6℄;

3. an ee t studied by Boer [7℄ in Drell-Yan pro esses. Note that all su h intrinsi -

k _T

^,

-

κ _T

^{or -}

k ^′ _T

^{ee ts are}

T

^{-odd; i.e., they require ISI orFSI. Notetoothat whentransverse}

parton motionis in luded, the QCD fa torisationtheorem isnot ompletelyproven, but

see [8℄.

Assuming fa torisationto bevalid, the ross-se tion is

E h

d ³ σ

d ³ P _h = X

abc

X

αα

^′

ββ

^′

γγ

^′

Z

dx a dx b d ² k _T d ² k ^′ _T d ² κ _T πz

× P a (x a , k T ) ρ ^a _α

^′

_α P b (x b , k ^′ _T ) ρ ^b _β

^′

_β  dˆ σ dˆt



αα

^′

ββ

^′

γγ

^′

D ^γ _h/c

^′

^γ (z, κ T ),

⁽⁷⁾

where again

 dˆ σ dˆt



αα

^′

ββ

^′

γγ

^′

= 1

16πˆ s ² X

βδ

M αβγδ M ^∗ _α

^′

_βγ

^′

_δ .

⁽⁸⁾

TheSivers ee trelieson

T

^-odd

k _T

^-dependentdistributionfun tionsandpredi ts anSSA of the form

E h

d ³ σ(S T ) d ³ P _h − E h

d ³ σ(−S T ) d ³ P _h

= |S T | X

abc

Z

dx a dx b

d ² k _T

πz ∆ ^T ₀ f a (x a , k ² _T ) f b (x b ) dˆ σ(x a , x b , k T )

dˆt D h/c (z),

⁽⁹⁾

where

∆ ^T ₀ f

^(relatedto

f _1T ^⊥

^{) is a}

T

^-odd distribution.

2.3 Higher Twist

EfremovandTeryaev[2℄showedthatinQCDnon-vanishingSSA's analsobeobtainedby

invokinghighertwistandtheso- alledgluoni polesindiagramsinvolving

qqg

orrelators.

Su h asymmetries were later evaluated in the ontext of QCD fa torisation by Qiu and

Sterman, whostudiedboth dire t-photonprodu tion[4℄and hadronprodu tion[9℄. This

programhasbeenextendedbyKanazawaandKoike[10℄tothe hirally-odd ontributions.

The various possibilities are:

dσ = X

abc

n G ^a _F (x a , y a ) ⊗ f b (x b ) ⊗ dˆ σ ⊗ D _h/c (z)

+ ∆ T f a (x a ) ⊗ E _F ^b (x b , y b ) ⊗ dˆ σ ^′ ⊗ D h/c (z)

+ ∆ T f a (x a ) ⊗ f b (x b ) ⊗ dˆ σ ^′′ ⊗ D ⁽³⁾ (z) o

.

in [2℄ and studied in [4,9℄; the se ond ontains transversity and is the hirally-odd on-

tributionanalysed in [10℄; and the third also ontains transversity but requires a twist-3

fragmentationfun tion

D _h/c ⁽³⁾

^.

2.4 Phenomenology

Anselmino et al. [11℄ have ompared data with various models inspired by the previous

possible (

k T

-dependent) me hanisms and nd good des riptions although they were not able to dierentiate between ontributions. The al ulations by Qiu and Sterman [4℄

(based on three-parton orrelators) also ompare well but are rather omplex. However,

the twist-3 orrelators (as in

g 2

^)obey onstraining relationswith

k T

^{-dependent densities}

and alsoexhibit a novelfa torisationproperty,to whi hwe nowturn.

2.5 Pole Fa torisation

Efremov and Teryaev [2℄ noti edthat the twist-3 diagrams involving three-parton orre-

lators an supply the ne essary imaginary part via apoleterm; spin-ip is impli it(and

Figure 3. Example of

a propagator pole in a

three-parton diagram.

due tothe gluon). The standard propagatorpres ription(

−•−

ⁱⁿ

Fig. 3,with momentum

k

^),

1

k ² ± i ε = IP 1

k ² ∓ i πδ(k ² ).

⁽¹¹⁾

leads to an imaginary ontribution for

k ² → 0

^{. A gluon with}

momentum

x _g p

^{inserted into an (initial or nal) external line}

p ^′

sets

k = p ^′ − x g p

^{and thus as}

x g → 0

^{we have that}

k ² → 0

^.

The gluon vertex may then be fa tored out together with the

quark propagatorand pole, see Fig. 4. Su h fa torisation an be

performed systemati ally for all poles (gluon and fermion): on

all external legs with all insertions [12℄. The stru tures are still

omplex: for a given orrelator there are many insertions, leading to dierent signs and

momentum dependen e.

The olour stru tures of the various diagrams (with the dierent types of soft inser-

tions) are also dierent (we shall examine this question shortly). In all ases (exam-

ined) it turns out that just one diagramdominates in the large-

N c

^{limit,see Fig. 5. All}

other insertions into external (on-shell) legs are relatively suppressed by

1/N c 2

. This

Pole Part

p

^′

x

_g

= −iπ p

^′

.ξ p

^′

.p ×

Figure4. Anexampleofpolefa torisation:

p

^{isthein omingprotonmomentum,}

p ^′

^theoutgoing

hadronand

ξ

^{is the gluon}polarisation ve tor (lyinginthetransverseplane).

Figure 5. Example of

a dominant propagator

polediagram.

sis [13℄): the leading diagrams provide a good approximation.

The analysis has yettoberepeated forallthe othertwist-3 on-

tributions (e.g. alsoinfragmentation).

A questionimmediatelyarises: ouldtherebeanydire trela-

tionship between the twist-3 and

k T

^-dependent me hanisms? It might be hoped that, via the equations of motion et ., unique

predi tions for single-spin azimuthal asymmetries ould be ob-

tained by linking the (e.g. Sivers- or Collins-like)

k T

^-dependent

me hanismsto the (EfremovTeryaev)higher-twist three-parton

me hanisms. An earlyattempt was made by Ma and Wang [14℄

for the Drell-Yan pro ess, but the predi tions were found not to

be unique. Ji et al. [15℄ have sin e also examined the relation-

ship between the

k T

^{-dependent and} higher-twist me hanismsby mat hing the two in an intermediate

k T

^{region of ommonvalidity.}

3 More on Multiparton Correlators

3.1 Colour Modi ation

In[16℄weprovidedanaposterioriproofoftherelationbetweentwist-3and

k _T

-dependen e.

The starting pointis a fa torisedformulaforthe Sivers fun tion:

d∆σ ∼ Z

d ² k _T dx f

S

(x, k _T ) ǫ ^{ρsP k}

^T

Trγ ρ H(xP, k _T ) .

⁽¹²⁾

Expanding the subpro ess oe ient fun tion

H

^{in powers of}

k _T

^{and keeping the rst}

non-vanishingterm leads to

∼ Z

d ² k _T dx f

S

(x, k _T ) k ^α _T ǫ ^{ρsP k}

^T

Tr



γ _ρ ∂H(xP, k _T )

∂k _T ^α



k

_T

=0

.

⁽¹³⁾

By exploitingvarious identities and the fa t that there are other momentainvolved, this

an be rearranged into the following form:

d∆σ ∼ M Z

dx f

S

⁽¹⁾ (x) ǫ ^{αsP n} Tr



P / ∂H(xP, k _T )

∂k ^α _T



k

_T

=0

,

⁽¹⁴⁾

where

f

S

⁽¹⁾ (x) = Z

d ² k _T f

S

(x, k _T ) k ² _T

2M ² .

⁽¹⁵⁾

Thenalexpression oin ideswiththemasterformulaofKoikeandTanaka[17℄fortwist-3

gluoni poles in high-

p _T

^{pro esses. The Sivers fun tion an thus be identied with the}

gluoni -polestrength

T (x, x)

^{multipliedby a} pro ess-dependent olour fa tor.

The sign of the Sivers fun tiondepends on whi h of ISI or FSIis relevant:

f ⁽¹⁾ (x) = X C i

1 T (x, x),

where

C i

^{is arelative olour fa tor dened with respe t toan Abelian} subpro ess. Emis- sion of an extra hard gluon is needed to generate high

p _T

^{and, a ording to the pro ess}

under onsideration, FSI may o ur before or after this emission, leading to dierent

olour fa tors. In this sense, fa torisation is broken in SIDIS, albeit in a simple and

a ountable manner. Figure 6depi ts the appli ation of this relationto high-

p _T

^SIDIS.

π

π

Figure 6. Twist-3 SIDIS

π

^{produ tion viaquark andgluon} fragmentation.

3.2 Asymptoti Behaviour

The relation between gluoni poles (e.g. the Sivers fun tion, and T-even transverse-spin

ee ts, e.g.

g ₂

^{[1822℄) still remains un lear. Although there are} model-based estimates and approximate sumrules, the ompatibilityofgeneraltwist-3evolution withdedi ated

studies of gluoni -poleevolution ([23, 24℄and at NLO[25℄)is stillunproven.

In the large-

x

^{limitthe evolution equations for}

g ₂

diagonalise in the double-moment arguments[26℄. FortheSiversfun tionandgluoni poles,thisistheimportantkinemati al

region [4℄. The gluoni -polestrength

T (x)

^, orresponds to a spe i matrix element [4℄.

It is alsothe residue of ageneral

qqg

^{ve tor orrelator}

b _V (x ₁ , x ₂ )

^[27℄:

b V (x 1 , x 2 ) = T ( ^x

¹

^+x ₂

²

)

x ₁ − x ₂ +

^regularpart

,

⁽¹⁷⁾

whi his dened as

b V (x 1 , x 2 ) = i M

Z dλ 1 dλ 2

2π e ^iλ

¹

^(x

¹

^−x

²

^)+iλ

²

^x

²

ǫ ^µsp

¹

ⁿ hp 1 , s| ¯ ψ(0) / n D µ (λ 1 ) ψ(λ 2 )|p 1 , si .

⁽¹⁸⁾

There isthough one other orrelator, proje ted onto anaxial Dira matrix:

b A (x 1 , x 2 ) = 1 M

Z dλ 1 dλ 2

π e ^iλ

¹

^(x

¹

^−x

²

^)+iλ

²

^x

²

hp 1 , s| ¯ ψ(0) / n γ ⁵ s·D(λ 1 ) ψ(λ 2 )|p 1 , si .

⁽¹⁹⁾

This lastis required to omplete the des ription of transverse-spin ee ts, inboth SSA's

and

g 2

^{. The two} orrelatorshave opposite symmetry properties for

x 1 ↔ x 2

^:

b A (x 1 , x 2 ) = b A (x 2 , x 1 ), b V (x 1 , x 2 ) = −b V (x 2 , x 1 ),

⁽²⁰⁾

determined by

T

invarian e. In both DIS and SSA's justone ombinationappears [19℄:

b ₋ (x ₁ , x ₂ ) = b _A (x ₂ , x ₁ ) − b _V (x ₁ , x ₂ ).

⁽²¹⁾

whi his determined by matrix elements of the gluon eld strength:

Y (x 1 , x 2 ) = (x 1 − x 2 ) b ₋ (x 1 , x 2 ).

⁽²²⁾

It should be safeto assume that

b ₋ (x ₁ , x ₂ )

^{has no doublepoleand thus}

T (x) = Y (x, x).

⁽²³⁾

Evolution is easier studied in Mellin-moment form, implying double moments for

Y (x, y)

^:

Y ^mn = Z

dx dy x ^m y ⁿ Y (x, y),

⁽²⁴⁾

wherethe allowed regionsare

|x|

^,

|y|

^and

|x − y| < 1

^{(re allthat negative values indi ate}

antiquark distributions). We wish to examine the behaviour for

x

^and

y

^{both lose to}

unity and therefore lose to ea h other. The gluoni pole thus provides the dominant

ontribution:

x,y→1 lim Y (x, y) = T ( ^x+y ₂ ) + O(x − y).

⁽²⁵⁾

In this approximation(now large

m, n

^{) the} leading-order evolutionequations simplify:

d

ds Y ⁿⁿ = 4



C F + C A

2



ln n Y ⁿⁿ ,

⁽²⁶⁾

where the evolution variable is

s = β ₀ ⁻¹ ln ln Q ²

^{. In terms of}

T (x)

^{this is}

T (x) = 4 ˙



C F + C A

2

 Z 1 x

dz (1 − z) (1 − x)

1 (z − x) +

T (z),

⁽²⁷⁾

whi hissimilarto theunpolarised ase, but diers by a olourfa tor

(C F + C A /2)

^anda

softening fa tor

(1 − z)/(1 − x)

^.

The extra pie e in the olour fa tor (

C A /2

^{) vis-a-vis the} unpolarised ase (just

C F

⁾

ree ts the presen e of a third a tive partonthe gluon. That is, the pole stru ture of

three-partonkernelsisidenti al,buttheee tive olour hargeoftheextragluonis

C A /2

^.

The softening fa tor isinessential to the asymptoti solution,it merely implies standard

evolution for the fun tion

f (x) = (1 − x)T (x)

^{. For an initial}

f (x, Q ² ₀ ) = (1 − x) ^a

^{, the}

asymptoti solution [28℄ is the same but modied by

a → a(s)

^{, with}

a(s) = a + 4



C F + C A

2



s.

⁽²⁸⁾

For

T (x)

^,

a

^{shifts to}

a − 1

^{; the evolution} modi ation is identi al; the spin-averaged asymptoti solutions are thus also valid for

T (x)

^{. This large-}

x

^{limit of the evolution}

agrees, barringthe olourfa tor itself,with studiesof gluoni -poleevolution [2325℄. We

should note here that Braun et al. [29℄ have found errors in [23℄ and [25℄; our results for

the logarithmi term are, however, unae ted.

ViewingtheSivers fun tionasanee tivetwist-3gluoni -pole ontribution[16℄,itisseen

tobe pro ess dependent: besides asign (ISIvs.FSI), thereis apro ess-dependent olour

fa tor. This fa tor is determined by the olour harge of the initial and nal partons.

It generates the sign dieren e between SIDIS and Drell-Yanat low

p _T

^{, but in hadroni}

rea tions at high

p _T

^{it is more} ompli ated. Su h a pi ture is omplementary to the mat hing in the region of ommon validity. The mat hing between various

p _T

^regions

now takes the form of a

p _T

^{-dependent olour fa tor. It also lends some} justi ation to the possibility of globalSivers fun tion ts [30℄.

We have shown that generi twist-3 evolution is appli ableto the Sivers fun tion. Its

ee tive nature allows us to relate the evolution of T-odd (Sivers fun tion) and T-even

(gluoni pole) quantities. A vital ingredient here is the large-

x

approximation, where gluoni -polesdominateandthe evolutionsimplies. TheSivers-fun tionevolutionisthen

multipli ativeanddes ribedby a olour-fa tormodiedtwist-2spin-averagedkernel[31℄.

A knowledgments

I wish to thank the organisers for invitingmeto this delightful and stimulatingmeeting:

despitetheirrepeatedinvitations,thisistherst timeIhavebeenable toparti ipateina

DSPIN workshopand tovisitDubna. The studiespresented herehavebeen performedin

ollaborationwith Oleg Teryaev, see [16℄ and work in progress [31℄. The support for the

visitsof Teryaev toComo was provided by the Landau Network (Como)and also by the

re ently ompleted (and hopefully to be renewed) Italian ministry-funded PRIN2006 on

Transversity. Someof the ideas have already been presented atother workshops [3234℄.

Referen es

[1℄ G.L. Kane,J. Pumplin and W.Repko, Phys. Rev. Lett. 41 (1978) 1689.

[2℄ A.V. Efremov and O.V. Teryaev, Phys. Lett. B150 (1985)383.

[3℄ A.V. Efremov and O.V. Teryaev, Sov. J.Nu l. Phys. 36 (1982)140.

[4℄ J.Qiuand G.Sterman,Phys.Rev.Lett.67 (1991)2264; Nu l. Phys.B378(1992)52.

[5℄ D. Sivers, Phys. Rev. D41(1990) 83.

[6℄ J.C. Collins,Nu l. Phys. B396(1993) 161.

[7℄ D. Boer, Phys. Rev.D60(1999) 014012.

[8℄ X. Ji, J.-P. Ma and F. Yuan, Phys. Rev.D71(2005) 034005.

[9℄ J. Qiu and G. Sterman, Phys. Rev.D59 (1998)014004.

[10℄ Y. Kanazawa and Y. Koike,Phys. Lett. B478 (2000)121; ibid. B490 (2000) 99.

[11℄ M. Anselmino,U. D'Alesio and F. Murgia, Phys. Rev.D67 (2003)074010.