The running coupling method with next-to-leading order accuracy and pion, kaon elm form fac
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THE RUNNING COUPLING METHOD WITH NEXT-TO-LEADING ORDER ACCURACY AND PION,KAON ELM FORM F ACTORS Shahin S.Agaev ?High Energy Physics Lab.,Baku State University,Z.Khalilov st.23,1370148Baku,Azerbaijan Abstract The pion and kaon electromagnetic form factors F M (Q 2)are calculated at the leading order of pQCD using the running coupling constant method.In calculations the leading and next-to-leading order terms in αS ((1?x )(1?y )Q 2)expansion in terms of αS (Q 2)are taken into account.The resummed expression for F M (Q 2)is found.Results of numerical calculations for the pion (asymptotic distribution amplitude)are presented.
During last years a considerable progress were made in understanding of infrared (ir)renormalon e?ects in various inclusive and exclusive processes [1-4].It is well known that all-order resummation of ir renormalons corresponds to the calculation of the one-loop Feynman diagrams with the running coupling constant αS (?k 2)at the vertices,or to calculation of the same diagrams with non-zero gluon mass.Both these approaches are generalization of the Brodsky,Lepage and Mackenzie (BLM)scale-setting method [5].Studies of ir renormalon e?ects have also opened interesting prospects for evalution of higher twist corrections to processes’di?erent characterictics [6].
As was proved in our works [2-4],exclusive processes have additional source of ir renormalon corrections.Indeed,integration over longitudinal fractional momenta of hadron constituents in the expression,for example,of the electromagnetic (elm)form factor generates ir renormalon e?ects.
In this letter we extend our previous consideration of the pion and kaon elm form factors in the context of the running coupling constant method.
A meson M form factor in the framework of pQCD has the following form [7]F M Q 2 = 10 10dxdyφ?M y,Q 2 T H x,y ;Q 2,αS ?Q 2 φM x,Q 2 .(1)
Here Q 2=?q 2is the square of the virtual photon’s four-momentum,φM (x,Q 2)is the meson distribution amplitude.In Eq.(1)T H x,y ;Q 2,αS ?Q
2 is the hard-scattering amplitude of the subprocess q q ′,which at the leading order of pQCD is [7]
T H x,y ;Q 2,αS ?Q
2 =16πC F 3αS ((1?x )(1?y )Q 2)3αS (xyQ 2)
3is the color factor,?Q 2is taken as the square of the momen-tum transfer of the exchanged hard gluon in corresponding Feynman diagrams for F M (Q 2).Unlike the frozen coupling approximation,where for calculation of F M (Q 2)one ?xes the argument of αS ((1?x )(1?y )Q 2)→αS (Q 2/4),in the run-ning coupling method we use Eq.(2)in Eq.(1)and,as a result,encounter with new troubles.Indeed,it is evident that αS ?Q 2 in Eq.(2)su?ers from ir singular-ities associated with the behaviour of the αS ?Q
2 in the regions x →0,y →0;x →1,y →1.Therefore,F M (Q 2)can be found only after proper regularization of αS ?Q
2 in these end-point regions.For solving of this problem let us relate the run-ning coupling αS (λQ 2)in terms of αS (Q 2)by means of the renormalization group equation.The renormalization group equation for the running coupling α≡αS /πhas the form
?α(λQ 2)
where
b2=1
48
(306?38n f).
The solution of Eq.(3)with initial conditionα(λ)|λ=1=α≡αS(Q2)/πis[8]
α(λ)
b2 lnα(λ)b2/b3+α ?1.(4)
This transcendental equation can be solved iteratively by keeping the leadingαk ln kλand next-to-leadingαk ln k?1λpowers.Forλ=(1?x)(1?y)these terms are given by
α (1?x)(1?y)Q2 ?α(Q2)
b2[1+α(Q2)b2ln(1?x)(1?y)]2
.(5)
The?rst term in Eq.(5)is the solution of the renormalization group equation(3) with leading power accuracy,whereas the whole expression(5)is the solution of Eq.(3)with next-to-leading power accuracy.
In our previous papers[3,4]for calculation of the pion and kaon elm form factors we used only the?rst term from Eq.(5).In this work for evaluation of F M(Q2)we use both of them.Let us clarify our approach by considering the pion form factor and pion’s simplest distribution amplitudeφasy(x)
φasy(x)=
√
b2 10 10xydxdy
b2 2b
3
[t+ln(1?x)+ln(1?y)]2
,(7)
where t=1/αb2.
The resummed expression for Fπ(Q2)can be found using an approach advocated in Ref.[3].For these purposes it is instructive to change variables from x,y in Eq.(7) to z=ln(1?x),w=ln(1?y)and apply the inverse Laplace transforms[9] 1
Γ(ν) ∞0exp[?u(t+z+w)]uν?1du,Reν>0,(8)
and
ln(t+z+w)
b2 ∞0exp(?tu)B Q2Fπ (u)du
? 8πfπb2 ∞0exp(?tu)u[1?C?ln t?ln u]B Q2Fπ (u)du,(10) where B[Q2Fπ](u)is given by expression
B Q2Fπ (u)=1(2?u)2?22?u,(11)
and is the Borel transform of the pion elm form factor obtained in our work[3]. It is worth noting that Eq.(10)is the general expression valid for both the pion and kaon;one needs only to replace in Eq.(10)the Borel transform B[Q2Fπ](u) and fπwith corresponding ones B[Q2F K](u)and f K from Ref.[4].The?rst term in Eq.(10)is the pion form factor found in Ref.[3]at the leading order of pQCD in the framework of the running coupling method using only the?rst term from Eq.(5).Of course,the second term in Eq.(10)is the contribution to the form factor coming from the second term of Eq.(5).But interpretation of the integrand in this term(without exp(?tu))as a”traditional”Borel transform of corresponding perturbative series is problematical,because some regularization prescription at u=0for recovering of perturbative series is needed.Here we bypass this problem and concentrate on calculation of the resummed expression for F M(Q2).As we shall see later,integration in Eq.(10)using the principal value prescription[8,10]removes ir renormalon poles at u=1,2...and gives correct results for[Q2F M(Q2)]res.
For the pion we get
Q2Fπ(Q2) res= Q2Fπ(Q2) res1? 8πfπb2{(C+ln t)[f1(0)+f2(0)] +(1?C?ln t)[(t?2)f1(1)+(t+2)f2(1)]+g1(0)+g2(0)
?(t?2)g1(1)?(t+2)g2(1)}.(12)
Here[Q2Fπ(Q2)]res
1is the resummed elm form factor of the pion obtained in Ref.[3]
by means of the leading term of Eq.(5),
Q2Fπ(Q2) res1=(8πfπ)22+(t?2)f1(0)+(t+2)f2(0) .(13) In Eqs.(12,13)we introduce following notations,
f k(n)=P.V. ∞0exp(?tu)u n du
k?u
.(14) For the pion’s Chernyak-Zhitnitsky(CZ)distribution amplitude[11]in the form
φCZ(x)=5φasy(x)(2x?1)2,
we get
Q2Fπ(Q2) res= Q2Fπ(Q2) res1? 40πfπb2 (1?C?ln t) (t?14
3
)f4(1) +(C+ln t)[f1(0)+25f2(0) +64f3(0)+16f4(0)]+g1(0)+25g2(0)+64g3(0)+16g4(0)?(t?
14
3
)g4(1) ,
with
Q2Fπ(Q2) res1=(40πfπ)26+(t?14
3
)f4(0) .
Expressions for the resummed pion form factor obtained using more general than the asymptotic and CZ distribution amplitudes[12]are rather lengthy and will be published elsewhere.
For the kaon we?nd
Q2F K(Q2) res= Q2F K(Q2) res1
? 40πfπb25 k=1{(1?C?ln t)(t m k+n k)f k(1)
+(C+ln t)m k f k(0)?(t m k+n k)g k(1)+m k g k(0)}.
The values of(m k,n k)as well as[Q2F K(Q2)]res
1can be found in Ref.[4].
As an example,results of numerical calculations for[Q2Fπ(Q2)]res carried out using the pion’s asymptotic distribution amplitude are shown in Fig.1.Here the
ratio R=[Q2Fπ(Q2)]res/[Q2Fπ(Q2)]0,where[Q2Fπ(Q2)]0is the pion elm form factor calculated in the frozen coupling approximation using the same distribution
amplitude,is depicted.As is seen,the next-to-leading order correction changes the
shape of the curve in the region of small Q2(2GeV2≤Q2<6GeV2),where the whole result is smaller than the leading order one.At Q2=2GeV2the correction
amounts to～15%of the leading order result decreasing toward Q2=6GeV2.
At the end of the considered values of Q2the next-to-leading order correction is positive and equals to3?6%of the leading order result.The similar calculations can be also ful?lled for the pion using its alternative distribution amplitudes[12] and for the kaon.
As was pointed out in Refs.[3-5],the ir renormalon corrections can be hidden
into the scale ofαS(Q2)at the leading order elm form factor[Q2F M(Q2)]0.In the studied case of the pion(asymptotic distribution amplitude)we?nd
αS(Q2)→αS(e f(Q2)Q2),
f(Q2)??6.71+16.67αS(Q2),(15) where in numerical?tting we have used Eq.(12).
It is known[3,4]that ir renormalon e?ects enhance the perturbative predictions for the pion,kaon elm form factors approximately two times.Our recent stud-ies con?rm that next-to-leading order term in Eq.(5)does not change the picture considerably.This feature of ir renormalons may help one in solution of a contradic-tion between theoretical interpretations of experimental data for the photon-to-pion transition form factor Fγπ(Q2)[13]from one side and for the pion elm form fac-tor Fπ(Q2)[14]from another side.Thus,in works[15,16]the authors noted that the scaling and normalization of the photon-to-pion transition form factor tends to favor of the pion asymptotic-like distribution amplitude.But then prediction for Fπ(Q2)obtained using the same distribution amplitude is lower than the data by approximately a factor of2.We think that in this discussion a crucial point is a chosen method of integration in Eq.(1).Indeed,unlike Fπ(Q2)the expression
for Fγπ(Q2)at the leading order of pQCD does not contain an integration over αS(xQ2).In other words,the running coupling method being applied to Eq.(1)and to the expression for Fγπ(see,Refs.[15,16])enhances the perturbative result for the pion elm form factor and,at the same time,does not change Fγπ1.This allows us to suppose that in the context of pQCD the same pion distribution amplitude may explain experimental data for both Fγπ(Q2)and Fπ(Q2).This problem is a subject of separate publication.
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FIGURE CAPTION
Fig.1The ratio R=[Q2Fπ(Q2)]res/[Q2Fπ(Q2)]0as a function of Q2.In calculation the pion asymptotic distribution amplitudeφasy(x)is used.The curve1is R found using the whole result for the resummed form factor Eq.(12),whereas the dashed curve is the ratio R=[Q2Fπ(Q2)]res1/[Q2Fπ(Q2)]0.The curve2corresponds to R with[Q2Fπ(Q2)]0in the frozen coupling approximation and after scale-setting procedure(15).In calculations the QCD scale parameterΛhas been taken equal to0.1GeV.