10/96/217

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSICS

SINGLE MESON PHOTOPRODUCTIONAND IR RENORMALONS

Shah in S. Agaev1

International Centre for Theoretical Physics. Trieste, Italy.

ABSTRACT

Single pseudoscalar and vector mesons inclusive pliotoproduction 7/}. —V MXvia higher twist mechanism is calculated using the Q(.T) running coupling constantmethod. It is proved that in the context of this method a higher twist contributionto the pliotoproduction cross section cannot be normalized in terms of the mesonelectromagnetic form Parlor. The st.riid.nre of infrared renormalon singularities oft.Tic: higher t.vvist. snbprocesH cross sect.ion and t.he resinnmed expression (the RorelKinn) for it are found. Comparisons are iviridc wil.h carli<;r rcKiill.s, a.K vvdl as wil.hleading l.vvisl. cross Hed.ion. PlKmoTncTiologicril effecl.s of KJ.ndied contributions for7T. K. -TVKJSOTI photoprodnd.ioTi MV. dis<:nsK<;d.

MTRAMARE TRIESTE

October 1996

1 Perm alien I. addr<;sK: High ErKjrgy PhysicK Labor al.ory. Baku Sl.al.e University,Z.Khalilov si.. 23. 370148 Baku, Azerbaijan. E-mail: azhep'Silan.ab.a/,

1 INTRODUCTION

OTIC of llie finidamenlal achievements of'QCT) is the; prediction ofasymptolic scalinglaws for large-angle exclusive; processes and their calculalion in I.IK; franiework ofperturbative QCD (pQCD) [1-3]. In the context of the factorized QC1) an expressionfor an amplitude of an exclusive process can be written as integral over x. y ofhadron wave functions (w.f.)2 cJ>,;(x. 0 2 ) (an initial hadron), <&J('x, Q2) (a finalhadron) and amplitude !/'w(x. y; as{Q2)- Q2) of the hard-scattering subprocess [2].The hard-scattering amplitude TH('x,y; o:s{Q2), Q2) depends on a process and canbe obtained in 1.1K; framework of pQCD, whereas llic w.f. $ (x . Q1) describes all1.1K; TioTi-p<;rlTirbalivc and process-independent cffccls of liadroriic binding. TIK;liadrori w.f. giv<;s llie amplitudes for finding par Ions (quarks, gluons) carrying 1.1K;longiludinal fraclional rnoTncTila x = (J.'I. x2, ....'(•„) and virtnalness up lo Ql witliin1.1K; liadrori and, in general, includes all Fock slates willi quaTilurri numbers of I.IK;liadron. "Bui only I.IK; lowest Fock stale {q\q2 - ft"" mesons, uud - for prolon. elc.)contributes to the leading scaling behavior, other Fock states' contributions aresuppressed by powers of I / O 2 . In our work we shall restrict ourselves by consideringthe lowest Fock state for a meson. Then, x = x \. x2 and x \ + x2 = 1.

This approach can be applied for investigation, not only exclusive processesbut also for the calculation of higher twist (ITT) corrections to some inclusive pro-cesses, sncli as large-/??1 dilepl.on production [4], 1.wo-jel-|-iTiesoTi production in I.IK;elecl.ron-posil.ron annihilation [5], elc. TIK; HT correclions to a singles meson inclu-sive; plioloproduclion and jet. pliotoprodnction cross seclions were sludied by variousaulliors [6.7]. ^T\ these early papers for calculation of integrals over x = xi, .T-2, like;

.Q2)l''U,as(Q2). Q2)S(1 - xt - x2)dxl(ix2 (1)

whie:h appear in an expression e>f the; ampliluele;, llie [rt)z.t:n e:e)upling e;enislanl. ap-pre)xiTnalie)ii was use;el. Semie; e:t"nTnenls ares in ortk:r e:e)ne;esrniTig this pe)int. It iswe;ll known [8], that in pQCD e:ale:ulalie)ns llie argurnesnl e>f llie rnnning e:onplinge:e)nsl.aTil (e>r llie remornialization an el fae;le)rizalie)n se:ale;) Ql slieinlel be lakesn e;e ualto lilts se.|iiares e>f llie Tnemiemtmn t.ranster of a. harel gluon in a e;e)rresspe)neling Fe;yn-man diagram. But defined in this way, as(Q2) suffers from infrared singularities,indeed, in a meson form factor calculations , for example, O2 equals to X\y\Q2 orz-ilJ-iQ2, —Q2 being the four momentum square of the virtual photon. In the singlemeson photoproduction -fh —V MX, this scale has to be chosen equal to —x\it orx2s. where it,s are the subprocess's Mandelstam invariants [6]. Therefore, in the

3Strie:l.ly spe;aking. ^^(x.Q2) is a lia.elreni distribution amplitude a.nel it. differsfre)TTi a. lia.elreni wa.ves fiine;lie)n; the; lbrme;r e:an be obtaine;d by inle;gra.ling the; e:e)rre;-sponding wave; fnne:lie>n ovesr pa.rl.ons7 l.ransvesrses momenla up le> llie [atM.tse:ale Q2. "But. in this pape;r we; use; l.he;scs I wo lerms on lilts same; fooling.

soft regions x\ —V 0, ij\ —V 0; x2 —Y 0, y2 —> 0 ori ' i —V 0, £2 — 0 integrals (1) divergeand for their calculation some regularization methods of as(Q2) in these regionsare needed. In the frozen coupling approximation these difficulties were avoidedsimply by equating Q2 to some fixed quantity characterizing the process. In formfactor calculations this is O2 = Q2jA [9]. in the single meson photoproduction -g 2 = .?/2,-«/2[6].

Recently, in our pa.p<;i'K [10,11] ele;vol.e;d t.o t.lie iriveKl.igat.ioTi of the; light, mesonseled.roma.gTiet.ic form [Victors, for t.heir calculation we; applied the; niTining couplingcoiiKtant. method, where; t.heKe Kirigiilarit.ies lia<] been regnla.riz.ed by means of theprincipal value; prescription [12]. Tn our recent work we consider t.Tie inclusive pho-toproduction of single pseudoscalar and vector mesons 7/1 —> MX using the sameapproach.

2 CALCULATION OF THE HIGHER TWISTDIAGRAMS

The two 11T subprocesses, namely 7 1 —> Mq2 and ^q2 —V Mqx contribute to thepliol.oprodiid.ion of I.IK; single; meKon M in I.IK; r<;a.c:l.ion jh —> MX . T h e Feynman

diagrams for t.he firsl. snbproc:<;sH are; slienvn in Fig A. We; do ne)l. pre>viele; t.he se;l. of

d iagrams c:e)rre;sponding j.o t.he He;c:e)nel snbproc:e;sH -f7j2 —> Mqt: l.he;y can be e)bl.a.ineel

from Fig A by e;xclia.nging t.he e.|iia.rk a.nel an t.i quark lines. T h e Tnemieml.aand c}ia.rgeK

e>f t.he pa.rt.icle;s in queKJ.ion are indie;at.e;d in Fig A (a). Tn our inve;st.iga.l.ion the;

meson mass is neglected. As is seen from I'igA. in the 11T subprocess the mesonM is coupled directly to the photon and the hadron quark and its suppression incomparison with leading twist subprocesses is caused by a hard gluon exchange inthe higher twist diagrams.

The amplitude for the subprocess 7Q1 —> Mq2 can be found by means of theRrexlsky-Lepa.ge; Tiie;l.he)el [2],

M =

Tn (2). TJJ is the; smvi of graphs cont.ribiit.iiig t.o t.he hard-scat.l.eriiig pa.rt. of the;snbproce;sH, which for the; Kiibprocens under coiiKideration is 7 + <]i —'r {q\q2) + <]2->w}i<;re a. quark a.nel ant.iquark froin t.he meson forin a. color Kinglet, st.aie (qiq2) •

The iinporLaril. ingrcxlie;nl. of our KJ.ndy is t.he choice of t.he meKon model w.f. $M-Tn this work we calculate t.he phot.oproduct.ioii of t.he pK<;u<]oscalar (pion, kaon)arid vect.or (^-TVKJSOTI) TVKJSOTIS. For t.heK<; TVKJSOTIS in t.he lit.e;ra.t.ure; [3],[ 13] variousw.f. were proposed. Here for the pion and kaon we use the phenomenologicalw.f. obtained in [3] by applying the QC1) sum rules method, for calculation ofthe p-meson photoproduction we utilize both w.f. found in [3] and derived in [13].The reason is that, in accordance with results of Ii,ef.[l3], the wave functions of

3

longitudinally and transversely polarized p-mesons are similar (coincide in shape),whereas in [3] a significant difference between them was predicted. In [13] theauthors suggested also that the change in shape of the transverse p-meson w.f. mayincrease the rate of the production of transversely polarized p-mesons by a factor2. We think that the single meson photoproduction is a suitable arena for checkingthis conclusion.

The piori and ^-niesoTi wave Functions have the form

(3)

Tor the model w.f. the coefficients a,b take the following values:Chernyak-Zhitnitsky w.f. [3];

a = 0. b = 5. for the pion,

a = 0.7. h = 1.5. for ihc longitudinally polarized pi — rncson. (4)

a = 1.25. b = —1.25. for the transversely polarized />/• — meson.

Ball-Braun w.f. [13];a = 0.7, b = 1.5,

for both longitudinally and transversely polarized p-meson. Here we have denotedby x = X\ the longitudinal fractional momentum carrying by the quark within themeson. Then, x2 = 1 — x and :ri — x2 = 2x — 1.

The pion and p-meson w.f. are symmetric under replacement x \ — x2 O x-2—X\.But the kaon w.f. is non-symmetric; Q>K(X\ — x2) ^ ^Ki'^2 — £\) [3]. indeed, thekaon w.f. inclndcK a. tcrin proportioTia.1 l.o odd power of (2;r — 1),

a = 0,1. 6 = 3 , c=1 .25 ,

and may be written as the sum of the symmetric $.,(£,//„) and antisymmetricQa{x,f.tl) parts,

Tn (3).(5).(6) ^,it,/(-'') is ''lie aHynipt.otic w.f.

(7)

where fM is the meson decay constant; / I = 0.093 Ge\'\fK = 0.112 CeV. In thecase of the p-meson we take fj; = fj = 0.2 CeV for the CZ w.f., and fj; = 0.2 6VI/,fj = 0.16 6Vi'"for BB w.f.

The TioTTnalizatioTi of $ M ( - ' ^ / ' U ) <-- /•'•o = 0.5 Gf.-:V IK given by l.lie condil.ion

-i f,.,(8)

The factor \/2 appearing in the normalization of a vector meson is included in thep-meson decay constant.

The formalism for calculation of the 11T subprocess cross section is well knownand described in [6.11]. We omit details of our calculations and write down thefinal expression for daHT jdi. We find:

for the pKeudoscalar and longitudinally pola.riz.cd vector mesons,

dl

(9)

sui

for the trariKversely polarized vector meson,

c/(TH7"(ci , Co) 647r26'rO:r: -/* r . ,

\S + hii)u] \ .

lh-wK2? (10)

Tn (9),(10), ct-E — 1/137 IK t}i<; fine structure constant, Cjr = 4/3 is the colorfactor. The Mand<;lstam invariants for the subprocess are defined as

s = (zp + qf = zs,

T = {q-P)2 = L

u = (zp-Pf = zu

(11

where ,H,!,,U ar<; 1.1K; Ma.nd<;lsta.m invariants for the process -fh —'r MX , z is thelongitudinal fractional momentum of the <.|iiark q\ out of the hadron h.

The main probl<;m in our investigation is the calculation of quantities /i,2 , ft'1,2?

-xt -x-i)as(Q2

1)$M{x]..x-i;Qt)(12)

and

k\ =

" d:r1

" d:H {•!•!, •'•2; W-->_)

•'•I

idxldx.i5(].-xl-x.i)as(Q'2i)4>M(xi..x.i-iQl)

•f'2

(13)

(14)

(15)

where for 1\. 12 the renormalization and factorization scale is Q\ = x2s. for A'I . K2

it is given by O\ = — xyu.Let us first consider the frozen coupling constant approximation. In this ap-

proximation we put Q2/2 equal to their mean values 5/2, —0/2 and remove o.s{Q2 2)

as the constant factor in (12-15). After such manipulation, the integrals (12-15) aretrivial and can easily be found. For t.Tic: mesons wil.li symmetric w.f. we gel

w h e r e s u p e r s c r i p t " 0 " i n d i c a t e s tlm.t t.Tic: q m i n t i l i e s T,K a.re found in t h e [VOZ.CT

c o u p l i n g a p p r o x i r n a l i o n . H<;rc t.Tic: [inid.ioTi TMIQ2) IK

- : > •

in this approximation, using the last expressions and (9).(10).(12-15) one can easilyreproduce results of [6] for the subprocess cross section3.

In the case of the kaon we find

(16)- X

2) -7V2yJo 1-x

it is evident that

The same is also true for A'" and A". This means that in the case of a pseu-doscalar meson with the non-symmetric w.f. the result of [6] is not valid and in thecalculations our expressions (9).(12-15) have to be applied.

The important problem in the single meson inclusive photoproduction is thepossibility of normalization of the iiT subprocess cross section (9),(10) in terms ofl.lie d e d . r o n m g n c l i c form [Victor FM(Q2) <>'' I l ie corr<;spoTidiTig rncKon.

T h e dc<:l.roTnagTi<;ti<: form fax:lor FM(Q2) <>f ^1K^ m e s o n M is g iven by t.Tic: e x -

p r e s s i o n

FM{Q2)= f:'The difference between our expressions and corresponding formulas in Ref.[6] is

c:.riiiK<;cl by o u r de f in i l ion of j.}i<; ar i l ic^i iark 's enlarge:, i .e . in o u r e x p r e s s i o n s j.}i<; charge;

of 1.1K; rinl. iqmirk [rorn M is — c2- w}i<;rc:a.K in Ref.[6] it is <]<;nol<;d b y f. .

6

Here

;.)7T(_,-j"

Q'1

For t.Tic: meson with symmelric vv.f. using the frozen coupling approximation (Q2

Q2/4) we- got

(18)

it is not difficult to conclude that for such mesons the quantities (I0)2 and (A'0)2

in the cross section can be expressed in terms of l')\-;

(19)

2,

F o r T i i o s o T i s w i l . l i T i o T i - K y i n T V K J J . r i c . v v . f . f r o m ( 1 7 ) w e f i n d

FM(Q2) =

(20)

it is now clear that (I0)2. (A'0)2 (16) are not proportional to I'M (20). This meansthat even in the context of the frozen coupling approximation the 11T subprocesscroKS K<;d.ion may bo normalized in terms of the meson form fax:lor only if thepliotoprodiid.ion of the meson with symmetric vv.f. is considered.

3 THE RUNNING COUPLING CONSTANT METHODAND IR RENORMALONS

Tn this section we Hlia.ll calculate the integrals (12-15) using t.Tic: running couplingconHta.nl. method and also discuss the problem of norm a.liza.l. ion of t.Tic: higher l.vvisf,process cross section in terms of t.Tic: meson electromagnetic form [Victor obtained inthe context, of t.Tic: same approach.

As is seen from (12-15). in general, one: lias to take int.o account, not only thedependence of a(Qfj) on the scale Q[-2, but also an evolution of ^M(X, Q[t2) withQ\ 2. TTic: meson vv.f. evolves in accordance with a. Bet he-Sa.l peter type equation,but its dependence on O2 is mild and may be neglected by replacing <$>M (X, Q'l2) —V$:V;(A\//O)' Such approximation does not change considerably numerical results,but phenomenon considering in this article (effect of infrared renormalons) becomestransparent.

Let us clarify our method by calculating the integral (12); the quantities 12, K\,2can be: worked out in the same way. For the mesons wit.Ti symmetric vv.f. Eq.(12)in t.Tic: framework of t.Tic: running coupling approach l.a.kes the form

1 — x

The Q-.sffl — x)s) has the infrared singularity at x —V 1 and as a result integral (21)diverges (the pole associated with the denominator of the integrand is fictitious,because <&,v; ~ (1 — x). and therefore, the singularity of the integrand at :r = 1 iscaused only by cxs((l — x)s)). ibr the regularization of the integral let us relate therunning coupling at scaling variable a.s(fl —$)s) with the aid of the renormalizationgroup equation in terms of the fixed one as(£). T'}i<; r<;normalizatioTi group equationfor t.Tic: running coupling a(,s) = fts(,?)/j

fiffW^, (22)has the solution [12]

In (22).(23). j.}i<; onc:-loop QCD coupling conHta.nl. as(/./2) is definc:<] as

asiji ] = A J ^ Mtio = 11 — 2n,f/'S being t.Tic: QCD beta-function first coefficient.

Having inserted (23) into (21) we get

where /. =The integral (24) is. of course. still divergent, but, now il. is recast.ed iiij.o a forin,which is suil.able for calculation. Using the method described in det.ails in our work[10] il. may be found us a pertnrbative series in a.s'(?)

( ^ ) 6'«, 6' CW r1- (25)• f i . = l

The coefficients Cr,. of this series demonstrate factorial growth Cr,. — (n — 1)!, whichmight indicate an infrared renormalon nature of divergences in the integral (21) andcorresponding series (25). The procedure for dealing with such ill-defined series iswell known; one Im.s l.o perform t.be Borel transform of j.}i<; seri<;s [15]

v i . = l

then invert ii[Y|](i() to obtain the res ummed expression (the Borel sum) for l\(s).This Tnel.bod is slraighlforwi-ird but, tedious. Therefore, il. is convenient l.o applythe second Tnel.bod proposed also in our work [11], which allows us to bypass allthese int.ermediate st.eps and find directly the resummed expression for /i(£). Forthese purposes lei. us introduce t.be inverse Laplace transform of '1/(1, -\- z)

\xp[-(t + ,z)u]du. (27)

Then l\ (s) may be readily carried out by the change of the variable :r to z = ln(l — x)and using (27)

86

3 — u

sr16

1 -

exp

)H i

4rru

. «s(5)ftij(a + b

n.

a

2

+ rob- u

(28)

Eq.(28) is nothing more than the Borel sum of the perturbative series (25) and thecorresponding Borel transform is

The series (25) can be recovered by means of the following formula.

• n - 1

d

u = 0

The Rorel l.ra.nsform B[Fi](u) 1ms poles GTI 1.1K: real •?/ a.xis al. v/ = 1:2: 3: 4, whichconfirms our conclusion concerning l.lie infrared renorma.lon nalun; of divergencesin (25). To remove: l.hem from Eq.(2S) some: regnUi.riza.l.ion mel.hods Imve l.o beapplied. In l.his ri.rt.icle we: adopt, t.lic: principal value; prescription [12]. We obtain

,<i- + b) —— - (a +

(30)

w h e r e Li(\) is 1.1K; loga.r i t . lnnic iTit<;gral [16]. for A > 1 de f ined in i t s p r i n c i p a l v a l u e

Li(X) = P.V. r ^ - . A = ?/A2. (31)Jo In x

F o r o t h e r i n t e g r a l s f r o m ( 1 3 - 1 5 ) w e f ind

fh(a + b) - Ab + 4b-

and

[R\ t - S ) ] " " = [T, (-S)]™ , [A'2 (-2)]™ =

(32)

(33)

From (30).(32).(33). we conclude that in the framework of the running couplingapproximation even for mesons wit.h Hynmi<;1 ri<; w.f. we: }iri.ve

Therefore only our results for the subprocess cross section (9),(10) are correct.Another question is, as we have discussed in Sect.2, the normalization of the

meson pliot.oproduct.ion cross K<;d.ioTi in t e rms of l.lie TVK;SOTI elm form fa.ct.or. Tlie

pkni rind kaoTi [OTTTI fa.ct.ors have Tx;<;n c;-i.lciila.t.(;d by Tnea.Tis of t.he riniTiing conplnig

i-i.pproa.ch in our previous papers [10, 11]. Fei. us write down t.he pion form [kclor

obt.i-i.ined using t.he pion 's simplesl. w.f.. t.ha.t. is. 1.1K; i-i.syTnpJ.ot.ic one (a = 1.6 = 0 in

(4))

a sy

3_ ^ + ( l T , A - 2 ) ^ - ^ + (lnA + 2)-

2 A(34)

10

From (30),(34) it follows that the relations (19) do no longer hold. The same is alsotrue for the pion's other w.f.. as well as for p\ — and />/•—mesons, in other words., inthe running coupling approach the IFF subprocess cross section (9).(10) cannot benormalized in terms of the meson form factor neither for mesons with symmetricw.f. nor for non-symmetric ones.

Let, UK, for oornplotenosK, write down T(.H). K( — u) calculated for TIOTI- Kymmotriow.f. (5)

4 \/37T/,.;•;, . ; • ; Li(\) , , .- (a + 56 + 7c)

Li{\2)

A

(35)

°A1 (36)

The expressions for [A'I ('—0)]'™ and [K2 { — u)Y'"'s may be obtained from (35).(36)by c —> —c.X = S/A2 —> —it/A2 replacements, respectively. With these explicitexpressions and the results of [11] at hand one can check our statements concerningthe normalization of the subprocess cross section for kaons.

Some comments are in order concerning these results. First of all, it is instructiveto compare; sources of the infrared ronornmloTiK in our case and in oilier QCDprocesses considered in [17]. Tn those arliclos t.Tie running coupling eonsUnit rnelliodwas used in one-loop order calculations for resuniTnalion of any innnTx;r of form ionbubble; insertions in l.lie gluon propagalor. This U;clini(.|ii<; corresponds to parlialreKinriTnalioTi of the; perturbative K<;rieK for a quantily under eonsid<;ral,ion. Tnd<;<;d,for such quantities, the coefficients S.n in (25) have the following expansion in powers

of A*?

Therefore, by defining t.Tie Ror<;l transform as in (26) and inv<;rl.iiig it one obtainsa partially resummed expreKsion for a physical quanlity. ^T\ our calculations ofthe; HT cross section wo use; the; leaeling order term for TJJ: t.Tiere; are no gluonloops in the corresponding Feynman diagrams in Fig.i. It is not accidental that5,,. (25) in our case has exactly ~ $',''"' dependence and, hence, the expressions(30),(32),(35),(36) are exact sums of the corresponding perturbative series. In [17]it has been demonstrated that the only source of terms of order ~ cx^S^1 in (25)is the running coupling constant. The source of these terms in the IFF calculationsis alse> the; running coupling r.v;s(.S(1 — x)) (or tv^(— u("I — .'(•))), vvliicli runs elue (,o

the integration in (1) over the meson quark's (antiquaries) longitudinal fractionalmomentum, but not because of a loop integration.

Another question commonly discussed in papers involving Hi, renormalons is anambiguity produced by the principal value prescription used for the regularizationof divergent integrals (28). The ambiguity introduced by our treatment of (28) isa. higher t.wist. and behaves as A 2 /Q J

(I.IK; first renorTnalon pole is u = 1). "But. I.IK;

snbprocess under consideration il.self'is already l.he higher t.wist. one. Therefore, wecan safely ignore; such " TTT-I.o-HT" corrections.

At. l.he end of this sect.ion let us write; down the; HT ce>rre;d.ioTi t.e> l.he; single;iviejsoTi phe)l.oproelue;l.ion exoss se;e;l.ion by taking into aexounl. both HT subproe;e;sse;s;•7(71 —> Mq-i and -fq2 —> Mq~\. it is not difficult to prove that the second subprocesscross section can be obtained from (9).(10) by e\ O e2 replacement. Then the HTcorrection to the single meson photoproduction cross section is given by

HT ((T

where

\fs - prey

He;re; the; sum runs over l.he; hadron's e.|iia.rk r/i a.nel an t.i quark q2 flavors. Tn (37)qfi {'**,—t), ?2(*-K:~0 a r e the clLiai'k a n ^ antiquark distribution functions, respec-tively. All r.h.s. quantities are expressed in terms of the process cm. energy y's,the meson transverse momentum p-r and rapidity y using the following expressions

u = - (38)/ // •• * -t v

sJ.H — pT(:JI

Eq.(37) is the final result which will be used later in our numerical calculations.

4 PHOTOPRODUCTION OF MESONS AT THELEADING TWIST LEVEL

Tn our sl.nely of l.he; single; TVKJSOTI phe>l.oprodnd.ion a crncia.1 pe>inl. is l.he; e;ennpa.rise)ne>f einr re;snll.s with leading twist (LT) enie;s. This will enable us l.e> fine! sue;h domainsin the phase space in which the higher twist photoproduction mechanism is actuallyobservable.

The LT subprocesses. which contribute to a meson photoproduction are:a photon-quark (antiquark) scattering

- q-i.(p\) -> qAp-i) + g{px),

12

and photon-gluon fusion reactions

Tn this article we consider the inclusive cross section difference in t.Tie photon-protoncollision, namely

d<r , ,,_,_ , r da\ i".'-ri —* M Y\ = Y.**M — ^

pfdy, , - . (41)

Th<; TT1 snbproc<;sK vvliicli dominates in this difference is 71:/ —> <yc/ with r/ —> Af. Ttscross section at the tree level is well known,

— — = r^ as(s)—ho's -< - • (-'12)

wh<;re l.lie Mand<;lsl.am invariants of l.lie subprocess are

In (12), the running coupling constant as is evaluated at momentum scales s and/ |. wliicli ar<; <;(]iial l.o off-Kliell Tvi()Tn<;nJ.a. Cri.rri<;d by l.lie virLiuil c^nark propagators

in l.lie corresponding Feynnian dirigrarriK of OK; leading twist snbproc<;sK "fq —> qq.

Many ol.}i<;r KiibproceKseK con t r ibu te to j.}i<; TVKJSOTI p l io toprodnct ion. among them

i) ~j<] —'r gq vvil.li g —> M, ii) jq —> (jq wil.li q —> M, iii) --"<r/ —> (jq vvil.li g —> M,

iv)7,c/ —> qq vvil.li q —> M or q —f M. Considering l.lie cross section difference AM

we not only reduce the number of subprocesses contributing to AM, because thesubprocesses involving gluons or antiquaries contribute equally to ^J+ and >j££_and cancel in A^f, but also solve two other important problems. The first oneis the next-to-leading order correction to the meson photoproduction cross sectioncalculated in [18]. In this paper the authors have investigated the ratio

r r + _ T - = dano(7p -+ x+xydprdy - dano(7p -> --X)jdPrdy = K _ i

daHnrn(jp -^ -+X)/dprdy - daHnrn(jp -^ --X)/dprdy i '

for l.lie cross K<;d.ion differ<;nce as a finiction of p-j at s/s = 14.1 GcV and y = 0.5.Th<;y hav<; found tliat this ratio is negative and almost constant vvil.li p-j (J>T =2 — 6 GcV/c). Tins means tliat l.lie K-facl.or for the cross section difference isless than 1. Tn other words, using the LT cross section (42) for calculations ofR-M =| &M' /^W I ^ l 'nc saTnc- Qr slightly different kinematic regimes, we onlyunderestimate the ratio R^j and related quantities and give lower bounds for them.The second problem solved by our choice of AM is a contribution to the photopro-duction cross section originating from the photon's quark and gluon content. It iswell known that the photoproduction process -fp —> h + A' may proceed via twodistinct mechanisms;photon can interact either directly with the hadron's partons

13

(direct photoproduction), or via its quark and gluon content (resolved photoproduc-tion). As was demonstrated in [18]. the contribution from the resolved photopro-duction almost completely cancel in the TT+ — TT~ difference. These results obtainedin [18] for pions at certain kinematic domain seemingly are valid also for other lightmesons at the same or slightly different kinematic conditions.

Then the leading twist conlribulion l.o t.Tic: single; meson pholoproduclion in"fp —> MX is given by l.lie expression,

da'--r

dprdy ^ J-rmin 2 At

where

pre.-y preM pre~v

in (13), qp(x,—i) and DM/q{'*,—i) are a quark q distribution and fragmentationfiiTicl.ioTiK, respectively. The snbprocesH invririanls x. I,, v. in (43) are [inid.ioTis of

xpTyfseyi PrVse A xpTyfses = ;r.s, t = . u = . (-'15)

z z

Eq.(-13) together with (37) for the 11T contributions will be applied in the nextsection for numerical calculations.

5 NUMERICAL RESULTSTn this seclion we compute the jp —> M+X rnul -fp —'r M~X inchiKive cross sectionsXy+, ^>M~, as well as the difference AM = ^ M + ~ ^M- by taking into account thedominant LT faq —> gq with q —> M), and 11T (jq —> Mq) contributions to theinclusive photoproduction. Only the 11T cross section of K~ photoproduction iscalculated using the proton s and u quarks induced subprocesses, which contributeat the same order. Our calculations are performed for M = IT . K. p at ^fs =14.1 GcV, 25 GcV.

Tn l.liIK work, for c^iark diKtribntion [inictions. we borrow th<; leading orderparri.m<;trizri.tion of Owens [19]. This pa.ra.Trielriz.alion is Kuilable for our pnrpoK<;s,T)<;cmiK<; llie HT mechanism probe ihe <.|iiai'k dislribulion [inictions alz* = pT<;xp (—;(/)/y/s — PT <'*]> (:(/), which Tor chosen proc<;sK's parrnn<;ters IK alwayKmore ihan 0.01. T'}i<; Kri.m<; is also true Tor ;?:,„;„ in the LT CTOKS seclion (43). Thatis. kinematical conditions allow us to avoid the region of small x < 0.01, whereOwen's parameterization may give incorrect results. The same reason will enable usto compute X^/ ignoring a contribution from the leading twist subprocess jg —V qq.:which otherwise may be considerable.

14

The quark fragmentation functions are taken from [20]. Recently, in [21], a newset of fragmentation functions for charged pions and kaons, both at leading andnext-to-leading order, have been presented. These functions give JJ^I++M (x,Q2),bill. not. Dy1 (x.Q2). Therefore;, we cannot, apply t.heni in our calculations.

The other problem is a choice of t.he QCD scale parameter A and number ofactive; quark flavors nj. Tli<; HT snbprocesses probe t.he meson w.f. over a largerange; of Ql. Q2 being e<.|nal t.o ,s or —u. It. is <;asy t.o find t.hat. —umin > 4.04 G'cV*2,while .s„,,,;,,. > 16 (!eVi. For momentum scales s, —t used in (A2) as arguments of asin the LT cross section we get

-imin > 6 CeV\ smin > 16 CeY2.

In other kinematic domains these scales take essentially larger values. Tak-ing into account these facts we find it reasonable to assign A = 0.1 C/eK •»•/ = 5throughout in this section.

Results of our numerical calculations are plotted in Figs.2 — 8. First of all,it is interesting to compare the resummed 11T cross sections with the ones ob-tained in the framework of the frozen coupling approximation. In Fi<y.2, t.he ratiorM = (S^7")rR7(S^r)'"' for negatively charged particles (TT". K~) is shown. In thecomputing of ( 5 ^ / )° we have neglected t.he meson's w.f. dependence; on the scaleQl. Let. us emphasize that for the kaon we have used t.he frozen coupling version ofour expression (9). but. not. t.he Bagger-Gnnion formula from [6], which is incorrectin that case.

As is seen from Fig.2(a), ?v- — 1 almost for all -pr, whereas r^- falls fromrK- ^ 2.75 at pr = 2 CtVjc, y = 0 until rK- ^ 2.13 at pr = 11 CeVjc. y = 0 andfrom rK- ^ 2.71 at pr = 2 CeVjc. y = 0.5 till rK- ~ 1.5 at pr = 11 CeVjc. y =0.5. Tor K~ this ratio demonstrates also a sharp dependence on y at fixed y^s, -pr

1T\ all of t.he following figures we have used t.he resummed expression for the HTcross section. In Fig.3 the ratio R^j = | A^1 /A'^ | is depicted. For all part.icl<;s

'•'11.5the; LT cross section difference; is positive; A^j > 0, sine;e £'{/+ ~J up{x.while; £({/- ^ C/P(;J:. — !•)<;?;• The smaller quark e:harge; c£j and the; smaller <list.vibiit.ionfunction dv both suppress X({/_ [6]. The 11T cross section difference may change signat small -pr and become negative A ^ r < 0. For example. A^J < 0 at 2 (.ieV/c <pr < 11 CeV/c for yfl = 25 CeV.' y = 0 and at 2 GeYfc"< pr < 9 CeV/c fory 7 ! = 25 (.t'eV, y = 0.5. Only at the phase-space boundary pr > 11 (JeV/c in thefirst case or at pv > 9 GeV/c in the second one >j{ _r > ^JI. Therefore, we plot theabsolute value of 11M. The similar picture has been also found for other mesons.

As is seen from Fi(.js.'i{a).{b) for pion an el kae>n t.he HT e:ont.ribiif.ie>n is coin-paraT)le with the; LT OTIC e>nly at pj < 3 GcV/c. We <\o not. find a considerableanel stabler grenvth e>f HT e;ontributie)ns at large vahie;s e>f pj for all t.he e;re>ss se;ctiondiffere;ne:eK A £ / (M = TT. A'), as we;ll as, for all E^7 ' . Tims. S^-l ' /Sj;^ is small at

high -pr for different sfs (Fig.A(a)). At the same time >j^+/X£-+ is a rising functionof pr for pr > A CeV/c (y/s = H.I CeV) and pr > 7 CeY/c (y/s = 25 CeV).

The cross section differences A^f and A'^ = A^f + A ^ r as functions of -pr areshown in Fig.o. For the pion the total cross section difference A{:'1 in the region ofsmall pr is smaller than A^T due to A^ / r < 0 in this region (Fig.5(a)). But forkaon Af

t"f > A1^1 in t.Tic: same kinematic domain. For bol.li mesons 1.1K: difference

between A*{}* and A',;/ cross sections is small.The rapidil.y dependence; of R^j at \ / ^ = ' ^ GcV,p-j = 3 GaV/c plotled in

Fig.'i{c) illustrates nol. only l.lie tendency of I.IK; HT conlribulions l.o be enhancedin I.IK; region of negative ra.pidiI.y, but. also reveals an inleresting feature of I.IK;HT terms; as is seen from Fig.?>{c) the ratio 11M is an oscillating function of therapidity. This property of the 11T terms may have important phenomenologicalconsequences, in fact, in Fig.fi we have depicted Al$ and A£f versus rapidity,in both cases, owing to observed property of A^J

;r(y), in certain domains of the

rapidity interval — 2 < y < 2.105 the total cross section difference is more than A1^i-ind in some ones less l.lian A',;/ . in l.lie <;a.K<; of I.IK; kaon pliot.oprodnct.ion

A1^1 > Ajf. far - 2 < y < 0.3 and 1.8 < y < 2.105,

A^f < A{;7': for 0.3 < y < 1.8.

Tlie properl.ieH of l.lie HT I.ernis foinid in I.IK; pion and kaon pliol.oprodncl.ion pro-cesses persist also in I.IK; /j-nieHon pliol.oprodncl.ion. "Bill, now I.IK; HT cont.rilml.ionHcliange I.IK; whole pict.nre of I.IK; process arising from l.lie ordinary IT1 ealenlal.ions.Tims. a.K in I.IK; case of I.IK; pion phol.oproduel.ion, I.IK; HT I.ernis ar<; enhanced rela-tive l.o the leading ones and A ^ ' < 0 almost for all pj. Rut now | A^' | tal«;s suchlarge values that it even changes the sign of the total cross section difference. Thatis. if in accordance with the LT estimations Xjj+ > Xjj^ must be valid for all pr., forpr < Px we find >j')+

i < >J^-'. The value of -pj- depends on the process parameters,as well as on the p-meson w.f. used in calculations. At pf ^ pT we have Xjj+ ^ >J^'.

Our results are shown in Fig.7. Tor the parameters indicated in the figure a.cril.ical value o [ > r is: pri ~ 5.05 GcV/c for CZ w.f., and pT2 2 6.25 GcV/c for "BRw.f. In all kineTnal.ic doTiiairiH I.IK; HT conl.ribiil.ionK found iiKing RB w.f. exceedI.IK; OTK;S obtained by applying CZ w.T., l.lial, IH, | A^r(BB) \>\ A%r(CZ) |. Forexample, l.he ral.io | A^7"( BB)/A^r(CZ) \ equals l.o 2.39 at ^ = 25 GcV, pT =5 GaV/c, y = 0, or l.o 2.63 al, ^ = 25 GcV, pT = 3 GaV/c, y = - 1 . Ourresults confirm the conclusion made by the authors in [13] concerning a possibilityof increasing the rate of the production of transversely polarized p-meson. Similarpictures persist in Fig.S. where A^T and A1^'1 are depicted as functions of therapidity y. in Fig.S(a), for the process parameters A/S = 25 (.ieV.pr = 3 (JeV/Owe have: in domain 1' ( — 1.71 < y < 1.3) the total cross section difference for CZw.f. is negative, in 1 ( -1 .5 < y < 1.5) - A'f{BB) < 0. in Fig.S(b) the sameis shown for ^fx = 25 GcV, p-j = 5 GcV/c. 1T\ two oilier r<;gioriH lying oulside of

1{1') the ^ exceeds ^ (for - 2 < y < - 1 . 5 . ^l(BB) and for - 2 < y < -1 .74 ,11l

fj'l(CZ) are negligible and are not shown).

It is worth noticing that in [6] the authors considered the p-meson photoproduc-tion at the same process's parameters and predicted X^)£ < 0 at pr < 3 (JeV/c, butcould not find similar effects for >j'.°' in dependence on the rapidity. Our investiga-tions prove; that Yf

p"f < 0 at. p-j < Pr <"nKl Pr w<!^ iTlli° deep pe;rlurbalive ele)main.

We; have also ele;me)TiKtrateel llial llie same; plienemiemon exiKJ.K for yf < y < y -

6 CONCLUDING REMARKS

Tn illIK work we; have; calculated the; single; meson inclusive; phe)toprodiiction vialiighe;r t.vvisl me;chaniKm and obtaine;d the; e;xpre;sKie)iiK for the; snbproce;sK ~j<] —> MqCTOKS see:lie)ii for me;sons with bolli symmelrie: and noTi-symmelrie: wave; ['unctions.Fe>r llie calculalie)n e>f the; e:re>SK see:lie>n we; have; applieel llie running cempling e;on-stant method and revealed Hi, renormalon poles in the cross section expression. Hi,renormalon induced divergences have been regularized by means of the principalvalue prescription and the resummed expression (the Borel sum) for the highertwist cross section has been found. Phenomenological effects of the obtained resultshave been discussed.

Summing up we can slate llial:i) for meKe>TiK willi TK)n-KyiTHTie;tric w.f. in the; frame;we)rk of llie fre>ze;n e;onplingapproximation llie liiglier twist Kubpre>e:eKS CTOKS see:lie)n canne)!. be; normalize^] interms e>f a me;son e;le;ctre)Tnagnelic form fae;le)r;ii) in llie e;onle;xt of the; running coupling e;onslant me;the)el llie HT Kubpre>e:eKS e:re>SKsee:lie)ii e;annol be; normalize;d in le;rmK ofmeKenVK e;lm form factor ne;ithe;r for me;sonswith symmetric w.f. nor for non-symmetric ones;III> the resummed 11T cross section differs from that found using the frozen couplingapproximation, in some cases, considerably;iv) HT contributions to the single meson photoproduction cross section have im-portant phenomenological consequences, specially in the case of p-meson photopro-dnction. Tn this proce;sK llie HT e:oTilribulie>nK wash llie LT re;snll.K off. changing the;TT1 pre;die;lie)iiK.

ACKNOWLEDGMENTS

The; anthe)r we>ulel like; le> thank the; Tnle;rnatioTial (.•e;ntre; for T}ie;e)relical PhyKics,Trie;sle;, for lioKpilality and Pre>f. S. Raneljbar-Dae;mi for IHK interest, in this work.

17

REFERENCES

1. S.J.Brodsky. B.Blankenbeder and J.F.Gunion: Phys.Bev.D6. 2651 (1972);S.J.Brodsky and C.B.Farrar: Phys.Hev.Lett.31, 1153 (1973).

2. C.P.Lepage and S.J.Brodsky: Phys.Pev.D22. 2157 (1980).

3. V.L.Chernyak and A.R.Zhitnitsky: Phys.Rep.112, 173 (1980).

4. S.S.Agaev: Phys.Lett. 13283, 125(1992); Z.Phys.C-Particles and Lields57, 403

E.L.Berger and S.J.Brodsky: Phys.Pev.Lett. 42, 940 (1979);E.L.Berger: Z.Phys. C-Particles and Fields 1. 289 (1980).

5. V.N.Baierand A.G.Grozin: Phys.Lett.B96, 181 (1980);S.Gupta: Phys.Pev.D21. 1169 (1981).

6. J.A.Bagger and J.F.Gunion: Phys.Pev.D25. 2287 (1982).

7. J.A.Hassan and J.K.Storrow: Z.Phys.C-Particles and Fieldsll. 65 (1982).

8. S.J.Brodsky. C.P.Lepage and P.B.Mackenzie: Phys.Pev.D28. 228 (1983).

9. R.D.FIdd, R.GnpU, S.Olto and L.Clmng: Nnd.Phyn.B1S6, 429 (1981).

10.S.S.Agaev: Phys.Lell .B360, 117 (1995); E. Phys.LeU.B369, 379 (1996);S.S.Agaev: Mod.Phys.Lell .A10, 2009 (1995).

11.S.S.Agaev: Mod.Phys.Lell .Al 1, 957 (1996): TCTP preprint TC/95/291, Septem-ber 1995, hep-ph/9611215.

12.A.H.Mueller: Nuel.Phys.B250, 327 (1985);ii.Contopanagos and G.Sterman: Nud.Phys. B419, 77 (1994).

13.P.Ball and V.M.Rrmni: Phys.Rev.D54, 2182 (1996).

14.S.S.Agaev: TTil.J.Mod.Pliys. AS: 2605 (1993); TTII .J.Mod.Phys.A9, 5077 (1994).

15.G.'1. Hoort: TTI: The WIIVH of Subnuelear PliysicH, Proc.TnJ..School, Ericc, 1977,od. A.Zicliiclii, Pl< iiiTn, N<w York, 1978;V.T.Zaklmrov: Nnd.Phyn.B3So, 452 (1982).

16.A.Er<]<;lyi: Higher iraTisceTideTiUil linielioTiH, v.2, McGrow-Hill Book Company,New York, 1953.

17.M.Neubert: Phys.Rrv.D51, 5924 (1995);P.Ball,M.B(:Ti<:kcaTi<] V.M.Brann: Nuel.Phys.B452, 563 (1995);P.Ball, M.Beneke and V.M.Braun: Phys.llev.D52, 3929 (1995);M.Benekeand V.M.Braun: Phys.Lett.B318. 513 (1995);C.N.Lovett-Turner and C.J.Maxwell: Nud.Phys.B452, 188 (1995).

18

18.P.Aurenche. R.Baier. A.Douiri, M.ibntannaz and D.Schiff: Nucl.Phys.B286, 553

(1987).

19.J.E.Owens: Phys.Lett.B266, 126(1991).

20.J.E.Owens: Phys.Rev.D19, 3279 (1979);J.E.Owens. E.Reya and M.Gliick: Phys.Hev.D18. 1501 (1978).21.J.Binnewies, C.Kramer and B.A.Kniehl: DESY preprint DESY-95-0'18, March1995.

19

FIGURE CAPTIONS

Fig.l Teynman diagrams contributing to the higher twist subprocess -/q —> Mq.Here p and P are the hadron h and meson M four momenta, respectively.

Fig.2 Ratio rM = (>>&T)r*»/'(l^f')", where (X&T) r" and (^MT)° are IIT con-

tributions to the photoproduction cross section calculated using the running andfrozen coupling approximations, respectively. The ratio is depicted as a function of•pr a), and of the rapidity b).

Fig.3 Ratio RM =1 ^M'/^M' I [ o r l n c P'OT1 a)> ' n u l [o r l n c 1<aoT1 ^) a l fixc<1

rapidity y=0. in c) 11M is plotted as a function of y for the pion (dashed curve)and for the kaon (solid curve).

Fig.4 Tlic dcpcTidcTicx; of t}i<; ratio S^'/Sj.;-' for K+ and K~ on p-j a) and on

Fig.5 The cross sccl.ion difference; AM IK sliovvn al. fixed rapidity for pious a),and for kaonn b). For the curves 1 the process cm. energy is s/s = 14.1 GaV, fort}i<; curves 2 - yG = 25 GcV.

Fig.6 The cross section difference A,.;/ as a function of the rapidity for pions a);for kaons b).

Fig.7 Ap for p-meson. The solid curve describes A^T, whereas the dashed curvescorrespond to A'p°

l. The long-dashed curve has been obtained using the CZ w.f.,the short-dashed one- 1313 w.f. in the domains 1(BB w.f.) and l'(CZ w.f.) theabsolute value of | X^)£ | or Il^'l — Xjj+ is plotted.

Fig.8 Ap dependence on t.Tie rapidity at y s = 25 GaV.pT = 3 GcV/c for a);at i/s = 25 GeV,-pr = 5 (leV/c for b). The solid curve corresponds to A£T, thelong-dashed and short-dashed curves describe A1^'1 obtained using CZ and 1313 w.f.,respectively. In regions 1(BB w.f.) and l'(CZ w.f.) the cross section differenceyUU _ vtof Js s n o w n i

20

00

n

21

HS3

i i r [ i ~n n i n i | i n i i n i i | i n i i i i i i | n ! i n r~r

s-25GeVsolid — y~0.dashed—y~0.5

7T

(a)

°2i i i i i i i i i i I i i i i i i i i i I i i i i i i t i i I i i i i i i i i i I i i i i i I i

4 6 8 10 12pT(GeV/c)

Fig-2

22

3-

o

i i rn i i | i i i i i i f [ f | i \ i i T i i

1 :

0

K

7T

(b)

— 1

Vs=25GeV

1 I ; I r I I I I I ) I I 1 I I I I I I I I I I I I

0y

.2

23

1 _ i i i f • i • " i i

10

< 10 - 2

10 - 3

10

I 1 | -T"r~i i i r n i j i i ri n Tr\

solid- Vs=25GeVdashed-Vs=14.1GeV

\\

\

\\

\ /

• ( a )

—4 M i i i i i i i I I n I i i i i I i [ i I I I i I i I I I I I M I I i I

4 6 8 10 12pT(GeV/c)

ig.3

24

\ LI i i i i i I T i i i i r

10 "^

-2<

IT

10

i i i i i i i i i [ r i i i i i i i i [ i f i i i i i i i

y=o.solid- Vs = 25GeVdashed-Vs-14.1GeV

\\

\

10 - 4

(b)I 1 I I I 1 I I I I I L i I I I I I I I I I I I I 1 I I ! I I [ 1 I I I I I I I I I I I I i I 1

2 4 6 8 10 12pT(GeV/c)

Fig. 3

25

10 I r~r t i i | i n i r i 1 r i | I I I ! ! l r "i [ "\ I i i i i i i ] i | i i f r

=25GeVpT=3GeV/c

•s- 1

<

10 - i

(c)

10 i i i i i i i i I i • i t i i i ; i I i i L.j i t I i i i i

-2 - 1 0

Fig.3

26

1 E

10

10 - 2

10 - 3

10 - 4

i i i n i n i i i n i i i i t i i i i rn i n i n i i i i i i i n n r

solid- Vs^SSGeVdashed-Vs=14.1GeV

\\

K

- (a)i i i I i i i i i i i i i 1 i i i i i i i i i I i i i t i i i i i I i i i i i i i i i

6 8pT(GeV/c)

10

Fig. 4

27

10 P l i i [ i i i l i i i i i | i i i i i i i i r~| i i i i i i i

10 -^

s=25GeVpT=3GeV/c

K

Fig.4

28

~ n i i 1 i | r r i T r i T i i [~r t i r r I t r i } \ l r i T^

10 "^

w> 10o

- 3"MO ^

10- 4

10 - 5

: (a)

0

y=o.solid - LTdashed-LT+HT

4 6pT(GeV/c)

8 10

Fig.5

29

10

10

CS2

<3

10

10

10

T T T T T T I ' 1 I I I I I I I I | . T 1 - | | | | ] | i ] iTTHr111) '1 I I I

y=o.solid - LTdashed-LT+HT

4 6 8pT(GeV/c)

Fig. 5

30

100

80 :

N> 60hO

Jr 40

0

i i i i i T i ri [ n

Vs=25GeVpT=3GeV/csolid - LTdashed-LT+HT

i i | i i i i i i i i i | i i i r

: (a)I i i I I i I I I I \ I I I I Mill

-2 -1 0

Fig.6

31

100

80

M> 6 0CDo

<? 40

T T T T T I I I | I I I I I I T I"11 I "7

Vs=25GeVpT=3GeV/csolid — LTdashed-LT+HT

\ r i ^ [ \ \ \ i I I

y

Fig.6

\

32

10

10 - i

10 - 2

10 - 3

10

i [ 1 I I I 1 I 1 I

4 6 8pT(GeV/c)

10

Fig.7

33

OP

00

Ap(pb/GeV2)

oo

tooo

COoo

E T T I I I I I I T H ' H ' H I I I f I I I I I I I \~ \ \ "\" \ I I I I I I T " T " 1 I V~Y~\ f f ! I I I T

i

I I I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

12

8CM

>a

I

i i i r i i r

I

"T i r

Fig.8(b)

35