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$Q^2$ dependence of chiral-odd twist-3 distribution $e(x,Q^2)$

$Q^2$ dependence of chiral-odd twist-3 distribution $e(x,Q^2)$
$Q^2$ dependence of chiral-odd twist-3 distribution $e(x,Q^2)$

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Q 2dependence of chiral-odd twist-3distribution e (x,Q 2)Y.Koike and N.Nishiyama Graduate School of Science and Technology,Niigata University,Niigata 950-21,Japan ABSTRACT We discuss the Q 2dependence of the chiral-odd twist-3distribution e (x,Q 2).The anomalous dimension matrix for the corresponding twist-3operators is calculated in the one-loop level.This study completes the calculation of the anomalous dimension matrices for all the twist-3distributions together with the known results for the other twist-3distributions g 2(x,Q 2)and h L (x,Q 2).We also have con?rmed that in the large N c limit the Q 2-evolution of e (x,Q 2)is wholely governed by the lowest eigenvalue of the anomalous dimension matrix which takes a very simple analytic form as in the case of g 2and h L .1.Introduction The nucleon has three twist-3distributions g 2,h L and e [1].g 2is chiral-even and the other two are chiral-odd.e is spin-independent and the other two are https://www.sodocs.net/doc/6910696535.html,pared to e ,g 2and h L have more chance to be measured ex-perimentally since they become leading contribution to proper asymmetries in the polarized deep inelastic scattering (DIS)and Drell-Yan process,respectively.Their Q 2evolution has been studied in [2,3]for g 2and in [4,5]for h L .Similarly to the distribution functions,there are three twist-3fragmentation functions which describes hadronization processes of partons in semi-inclusive pro-cesses [6],?g 2,?h L and ?e .(Their naming is parallel to the corresponding distribution functions.)In the inclusive pion production in the transversely polarized DIS,the chiral-odd fragmentation function ?e of the pion appears as a leading contribution

together with h 1(twist-2)of the nucleon.Although the Q 2evolution of the twist-2fragmentation functions is known to be obtained from that of the corresponding distributions in the one-loop level (Gribov-Lipatov reciprocity [7]),no such relation is known for the higher twist fragmentation functions.

In our recent work [8],we have investigated the Q 2evolution of e (x,Q 2).Theo-retically,this completed the calculation of the anomalous dimension matrices of all the twist-3distributions.Phenomenologically,we expect it will shed light on the Q 2evolution of ?e (z,Q 2),anticipating the day when their relation is clari?ed.This talk is a brief summary of [8].

2.Q 2evolution of e (x,Q 2)

The chiral-odd twist-3distribution function e (x,Q 2)is de?ned by the relation [1]

where|P is the nucleon(mass M)state with momentum P.Two light-like vectors p and n de?ned by the relation,P=p+M2

2

S nˉψσαμ1iDμ2···gGμlα···iDμnψ?traces,(5) Eμ1···μn=

1

2 +1 .(7)

By this combination,R n,l has a de?nite charge conjugation(?1)n.Note the simi-larity between the present{U l,R n,l}and{θl,R n,l}which appeared in h L[1,4].In fact the presence ofγ5inθl is the mere di?erence from U l.R n,l of h L is de?ned as θn?l+2?θl and has a charge conjugation(?1)n+1which is opposite to the above R n,l in(7).We also recall that e(x)does not mix with the gluon-distribution owing to the chiral-odd nature.

For the renormalization of e(x,Q2),we choose R n,l(l=2,..., n

2 +1 .(8)

This Z ij gives the anomalous dimension matrix for{R n,l,E}.

https://www.sodocs.net/doc/6910696535.html,rge-N c limit

In[3,5],it has been proved that all the(nonsinglet)twist-3distributions g2, h L and e obey a simple GLAP equation similarly to the twist-2distributions in the N c→∞limit.In this limit,Q2evolution of these distributions is completely determined by the lowest eigenvalue of the anomalous dimension matrix which has a simple analytic form.For e,this can be also checked using the obtained result for the mixing matrix Z ij.At N c→∞,i.e.,C F→N c/2,the lowest eigenvalue of the anomalous dimension matrix for R n,l is given by(ignoring the factor g2/8π2)

γe n=2N c S n?12(n+1) .(9) Furthermore the Q2evolution of e becomes

M n e(Q2) =Lγe n/b0M n e(μ2) .(10) As an example we compare the results for n=4.The exact Q2evolution is

M4 e(Q2) =(0.983a4,2(μ)+0.520a4,3(μ)) α(Q2)

α(μ2) 15.3/b0

,(11)

and the leading N c evolution is

M4 e(Q2) = a4,2(μ)+1α(μ2) 10.4/b0,(12) where a n,l is de?ned by

P|Rμ1···μn

n,l

(μ2)|P =2a n,l(μ2)M S n(Pμ1···Pμn?traces),(13)

and b0=11

3N f.One sees clearly that the coe?cients in the second term of

(11)are small(i.e.,1/N2c suppressed)and the anomalous dimension in(12)is close to the smaller one in(11).

References

[1]R.L.Ja?e and X.Ji,Nucl.Phys.B375,527(1992).

[2]J.Kodaira,K.Tanaka,T.Uematsu and Y.Yasui,hep-ph9603377,and references therein.

[3]A.Ali,V.M.Braun and G.Hiller,Phys.Lett.B266,117(1991).

[4]Y.Koike and K.Tanaka,Phys.Rev.D51,6125(1995).

[5]I.I.Balitsky,V.M.Braun,Y.Koike and K.Tanaka,hep-ph/9605439,Phys.Rev.Lett. in press.

[6]R.L.Ja?e and X.Ji,Phys.Rev.Lett.71,2457(1993).

[7]V.N.Gribov and L.N.Lipatov,Sov.J.Nucl.Phys.15,675(1972).

[8]Y.Koike and N.Nishiyama,hep-ph/9609207.

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