%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f (with hyperlinking) Copyright 1986, 1994 Radical Eye Software %%Title: hep-ph/9909273 %%CreationDate: Wed Jan 10 05:26:06 2001 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSParameters: dpi=300, comments removed %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N 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*Resolution 300dpi TeXDict begin %%PaperSize: Letter end HPSdict begin /TargetAnchors 39 dict dup (cite.gronberg98) [8 [126 475 126 487] 674] put dup (cite.BrodskyLepage) [8 [126 353 126 365] 551] put dup (cite.Kess+Ong93) [8 [126 233 126 245] 432] put dup (cite.KlKe2) [8 [126 365 126 377] 563] put dup (cite.KlKe4) [8 [126 210 126 222] 408] put dup (cite.Roberts) [8 [126 413 126 425] 611] put dup (cite.CP90) [8 [126 269 126 281] 468] put dup (cite.manohar90) [8 [126 341 126 353] 540] put dup (cite.BC) [8 [126 281 126 293] 480] put dup (cite.gagasCEBAF) [8 [126 198 126 210] 396] put dup (equation.1) [1 [254 213 254 225] 412] put dup (equation.2) [2 [175 468 175 480] 666] put dup (equation.3) [3 [126 204 126 216] 403] put dup (equation.4) [4 [164 514 164 526] 712] put dup (equation.5) [4 [175 259 175 271] 458] put dup (equation.6) [5 [226 241 226 253] 440] put dup (cite.bando94) [8 [126 425 126 437] 623] put dup (cite.RobertsDubr) [8 [126 257 126 269] 456] put dup (Doc-Start) [1 [126 623 126 635] 791] put dup (cite.KeBiKl98) [8 [126 437 126 449] 635] put dup (cite.jain91+munczek92) [8 [126 329 126 341] 528] put dup (cite.MarisRoberts97PRC56) [8 [126 293 126 305] 492] put dup (figure.1) [3 [181 465 181 477] 663] put dup (cite.jain93b) [8 [126 305 126 317] 504] put dup (cite.Radyushkin+Rusk3) [8 [126 461 126 473] 659] put dup (cite.behrend91) [8 [126 221 126 233] 420] put dup (cite.M+Rady97) [8 [126 389 126 401] 587] put dup (cite.Frank+al) [8 [126 401 126 413] 599] put dup (cite.KeKl1) [8 [126 377 126 389] 575] put dup (cite.KeKl3) [8 [126 449 126 461] 647] put dup (page.1) [1 [126 638 126 650] 791] put dup (page.2) [2 [126 638 126 650] 791] put dup (page.3) [3 [126 638 126 650] 791] put dup (page.4) [4 [126 638 126 650] 791] put dup (page.5) [5 [126 638 126 650] 791] put dup (page.6) [6 [126 638 126 650] 791] put dup (page.7) [7 [126 638 126 650] 791] put dup (page.8) [8 [126 638 126 650] 791] put dup (cite.novikov+al84) [8 [126 245 126 257] 444] put targetdump-hook def TeXDict begin %%EndSetup %%Page: 1 1 gsave matrix defaultmatrix setmatrix 90 rotate 283 -39 moveto /Times-Roman findfont 20 scalefont setfont 0.3 setgray (arXiv:hep-ph/9909273 7 Sep 1999) show grestore 1 0 bop 385 432 a Fu(The)15 b Ft(\015)505 416 y Fs(?)525 432 y Ft(\015)f Fr(!)d Ft(\031)638 416 y Fq(0)673 432 y Fu(transition)h(in)j(a)h(b)q(ound-state)d(approac)o(h)1455 416 y Fs(a)699 564 y Fp(DUBRA)l(VK)o(O)f(KLABU)1080 555 y(\024)1076 564 y(CAR)409 614 y Fo(Physics)g(Dep)n(artment,)f(F)m (aculty)h(of)h(Scienc)n(e,)e(University)g(of)i(Zagr)n(eb,)660 660 y(Bijeni)o(\024)-19 b(cka)11 b(c.)17 b(32,)c(Zagr)n(eb,)e(Cr)n(o)n (atia)766 749 y Fp(D)o(ALIBOR)i(KEKEZ)418 799 y Fo(R)o(udjer)g(Bo)n (\024)-18 b(skovi)o(\023)f(c)11 b(Institute,)g(POB)j(1016,)e(10001)g (Zagr)n(eb,)f(Cr)n(o)n(atia)300 895 y Fn(W)m(e)16 b(presen)o(t)e(a)i (recen)o(t)e(treatmen)o(t)f(of)j(the)f Fm(\031)934 883 y Fl(0)952 895 y Fm(\015)972 883 y Fk(?)990 895 y Fm(\015)j Fn(transition)c(form)g(factor)h(in)g(the)g(cou-)300 934 y(pled)g(Sc)o(h)o(winger-Dyso)o(n)e(and)h(Bethe-Salp)q(eter)e(approac)o (h)h(to)i(stress)g(the)g(desirabilit)o(y)e(of)300 973 y(measuring)c(it)i(at)g(CEBAF.)300 1118 y Fj(The)i(form)e(factor)h Ft(T)622 1125 y Fs(\031)642 1117 y Fi(0)661 1118 y Fj(\()p Fr(\000)p Ft(Q)742 1103 y Fq(2)761 1118 y Ft(;)7 b Fj(0\))12 b(for)g(the)i(transition)e Ft(\015)1173 1103 y Fs(?)1193 1118 y Fj(\()p Ft(k)q Fj(\))p Ft(\015)r Fj(\()p Ft(k)1310 1103 y Fh(0)1322 1118 y Fj(\))g Fr(!)f Ft(\031)1428 1103 y Fq(0)1447 1118 y Fj(\()p Ft(p)p Fj(\))i(\(where)225 1168 y Ft(k)248 1152 y Fq(2)287 1168 y Fj(=)22 b Fr(\000)p Ft(Q)406 1152 y Fq(2)445 1168 y Fr(6)p Fj(=)f(0)f(is)f(the)h(momen)o (tum)o(-squared)d(of)i(the)h(spacelik)o(e)g(o\013-shell)f(photon)225 1217 y Ft(\015)248 1202 y Fs(?)268 1217 y Fj(\),)g(is)f(usually)g (though)o(t)g(to)g(b)q(e)h(almost)d(completely)i(understo)q(o)q(d)h(on) f(the)h(basis)g(of)225 1267 y(the)14 b(p)q(erturbativ)o(e)h(QCD)f (\(pQCD\))g(o)o(v)o(er)f(the)h(whole)g(range)g(of)f(the)h(momen)o (ta-squared)225 1332 y(co)o(v)o(ered)k(b)o(y)g(the)g(new)g(CLEO)g(data) f([1)p (#cite.gronberg98) [[275 397 280 409] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (\(1.5)g(GeV)1064 1316 y Fq(2)1102 1319 y Ft(<)1102 1346 y Fr(\030)1153 1332 y Ft(Q)1186 1316 y Fq(2)1223 1319 y Ft(<)1223 1346 y Fr(\030)1274 1332 y Fj(9)h(GeV)1394 1316 y Fq(2)1413 1332 y Fj(\).)29 b(Ho)o(w)o(ev)o(er,)225 1381 y(Radyushkin)17 b(and)g(Rusk)o(o)o(v's)g([2)p (#cite.Radyushkin+Rusk3) [[248 386 253 398] [1 1 1 [3 3]] [0 0 1]] pdfm (])f(QCD)h(sum)g(rule)h(analysis)e(indicates)i(the)g(presence)225 1431 y(of)d(large)h(nonp)q(erturbativ)o(e)h(con)o(tributions)f(ev)o(en) h(for)e(the)i(largest)f(of)f(these)j(presen)o(tly)225 1481 y(accessible)h(momen)o(ta.)25 b(This)17 b(motiv)n(ated)e(us)j(to)f (address)h(in)f(Ref.)g([3)p (#cite.KeKl3) [[393 362 398 374] [1 1 1 [3 3]] [0 0 1]] pdfm (])f(the) h(transition)225 1531 y(form)f(factor)i(in)f(the)h(coupled)g(Sc)o(h)o (winger{Dyson)g(\(SD\))g(and)f(Bethe{Salp)q(eter)j(\(BS\))225 1581 y(approac)o(h)15 b(in)g(all)f(spacelik)o(e-momen)o(tum)e(regimes,) j(from)e(v)o(ery)j(large)f Ft(Q)1399 1566 y Fq(2)1433 1581 y Fj(all)f(the)i(w)o(a)o(y)225 1630 y(to)k Ft(Q)315 1615 y Fq(2)355 1630 y Fj(=)h(0.)36 b(The)20 b Ft(Q)601 1615 y Fq(2)641 1630 y Fj(=)i(0)e(limit)d(corresp)q(onds)k(to)f(the)h Ft(\031)1234 1615 y Fq(0)1272 1630 y Fj(deca)o(y)f(in)o(to)g(t)o(w)o(o) f(real)225 1680 y(photons)13 b(\()p Ft(k)420 1665 y Fq(2)451 1680 y Fj(=)e Ft(k)517 1665 y Fh(0)p Fq(2)557 1680 y Fj(=)h(0\))h(explained)g(b)o(y)g(the)h(Ab)q(elian)f(axial)e Fg(alias)i Fj(Adler-Bell-Jac)o(kiw)225 1730 y(\(ABJ\))19 b(anomaly)m(,)d(whic)o(h)j(is)f(usually)f(di\016cult)h(to)g(incorp)q (orate)h(in)o(to)f(a)g(b)q(ound-state)225 1780 y(approac)o(h)13 b([4)p (#cite.KeBiKl98) [[172 290 177 302] [1 1 1 [3 3]] [0 0 1]] pdfm (].)j(F)m(ortunately)m(,)c(as)h(sho)o(wn)g(already)f(in)g(Refs.)h([5)p (#cite.bando94) [[358 290 363 302] [1 1 1 [3 3]] [0 0 1]] pdfm (,)t(6) p (#cite.Roberts) [[367 290 372 302] [1 1 1 [3 3]] [0 0 1]] pdfm (])e (and)i(used)g(in)g(the)g(\014rst)225 1830 y(analysis)i(of)g Ft(T)456 1837 y Fs(\031)476 1829 y Fi(0)495 1830 y Fj(\()p Fr(\000)p Ft(Q)576 1815 y Fq(2)595 1830 y Ft(;)7 b Fj(0\))15 b(in)g(a)h(SD)f(approac)o(h)h([7)p (#cite.Frank+al) [[316 278 321 290] [1 1 1 [3 3]] [0 0 1]] pdfm (],)f (suc)o(h)h(approac)o(hes)h(whic)o(h)e(rely)h(on)225 1879 y(the)j(c)o(hirally)e(w)o(ell{b)q(eha)o(v)o(ed)h(SD)g(\(and)h(BS\))g (equations)f(for)g(the)h(ligh)o(t)e(pseudoscalar)225 1929 y(mesons)e(\()p Ft(\031)q(;)7 b(K)q(;)g(\021)q(;)g(:::)p Fj(\))13 b(while)i(resp)q(ecting)j(the)e(W)m(ard-T)m(ak)n(ahashi)d (iden)o(tities)j(\(WTI\))g(of)225 1979 y(QED,)d(repro)q(duce)i(\(in)e (the)h(c)o(hiral)f(and)g(soft)g(limit\))e(the)i(fundamen)o(tal)f (anomaly)e(result)759 2099 y Ft(T)783 2106 y Fs(\031)803 2098 y Fi(0)822 2099 y Fj(\(0)p Ft(;)d Fj(0\))k(=)1018 2070 y(1)p 975 2089 108 2 v 975 2127 a(4)p Ft(\031)1021 2115 y Fq(2)1039 2127 y Ft(f)1059 2133 y Fs(\031)1582 2099 y Fj(\(1\))225 2218 y(naturally)k(and)h(without)f(an)o(y)g(\014ne) i(tuning.)23 b(This)16 b(is)g(an)f(adv)n(an)o(tage)g(with)h(resp)q(ect) i(to)225 2268 y(pQCD)c(approac)o(hes,)g(whic)o(h)g(ha)o(v)o(e)g (problems)e(at)i(lo)o(w)f Ft(Q)1132 2253 y Fq(2)1151 2268 y Fj(.)18 b(\()p Fg(E.g.)p Fj(,)13 b(see)i(Ref.)e([8)p (#cite.M+Rady97) [[423 173 428 185] [1 1 1 [3 3]] [0 0 1]] pdfm (].\)) p 225 2322 564 2 v 225 2349 a Fk(a)244 2361 y Fn(The)h(talk)g(giv)o(en) f(b)o(y)h(D.)h(Klabu)o(\024)-17 b(car)13 b(at)h(Semi-Exclusiv)n(e)e(W)m (orkshop,)h(Je\013erson)g(Lab,)i(Ma)o(y)f(19{22,)225 2400 y(1999.)g(T)m(o)e(app)q(ear)e(in)h(the)g(pro)q(ceeding)o(s)e(of)i (the)g(W)m(orkshop.)p eop %%Page: 2 2 2 1 bop 300 432 a Fj(In)12 b(the)g(presen)o(t)h(coupled)f(SD-BS)g (approac)o(h)g([4)p (#cite.KeBiKl98) [[320 613 325 625] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s (9)p (#cite.KeKl1) [[329 613 334 625] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s (10)p (#cite.KlKe2) [[337 613 347 625] [1 1 1 [3 3]] [0 0 1]] pdfm (],)d(the)k(anomalous)c(amplitude)225 481 y(\(1)p (#equation.1) [[130 602 135 614] [1 1 1 [3 3]] [0 0 1]] pdfm (\))17 b(is)f(also)g(obtained)h(mo)q(del-indep)q(enden)o(tly)f(and)g(without)h (imp)q(osing)d(an)o(y)i(require-)225 531 y(men)o(ts)d(on)g(the)g (solutions)g(for)g(the)h Ft(\031)805 516 y Fq(0)837 531 y Fj(w)o(a)o(v)o(e)f(function)g(or)g(the)h(quark)f(propagator)f(solu-) 225 581 y(tions)h(\()p Fg(e.g.)p Fj(,)g(see)i([5)p (#cite.bando94) [[194 578 199 590] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(6)p (#cite.Roberts) [[204 578 209 590] [1 1 1 [3 3]] [0 0 1]] pdfm (]\).)16 b(This)e(is)f(exactly)h(as)f(it)g(should)h(b)q(e,)g(since)g (the)g(anomaly)d({)i(and)225 631 y(its)k(result)g(\(1)p (#equation.1) [[173 566 178 578] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g({)f (m)o(ust)f(not)i(dep)q(end)h(on)e(the)h(in)o(ternal)g(structure)h(of)e (the)h(pion.)26 b(Sub-)225 681 y(sequen)o(tly)m(,)15 b(in)f(Ref.)g([3)p (#cite.KeKl3) [[208 554 213 566] [1 1 1 [3 3]] [0 0 1]] pdfm (],)f(w)o (e)i(sho)o(w)o(ed)g(that)g(the)g(SD-BS)g(approac)o(h)g(to)f(mo)q (deling)f(QCD)225 730 y(pro)o(vides)k(a)f(go)q(o)q(d)g(description)h (for)f Fg(b)n(oth)h Fj(lo)o(w)e(and)i(high)e(v)n(alues)i(of)f Ft(Q)1369 715 y Fq(2)1387 730 y Fj(.)26 b(W)m(e)16 b(thereb)o(y)225 780 y(obtained)j(the)g(b)q(eha)o(vior)g(similar)e(to)i(the)g(Bro)q (dsky{Lepage)h(in)o(terp)q(olation)f(form)o(ula)225 830 y Ft(T)249 837 y Fs(\031)269 829 y Fi(0)288 830 y Fj(\()p Fr(\000)p Ft(Q)369 815 y Fq(2)388 830 y Ft(;)7 b Fj(0\))k(=)h(\(1)p Ft(=)p Fj(4)p Ft(\031)603 815 y Fq(2)621 830 y Ft(f)641 836 y Fs(\031)664 830 y Fj(\))7 b Ft(=)g Fj(\(1)g(+)g Ft(Q)831 815 y Fq(2)849 830 y Ft(=)p Fj(8)p Ft(\031)916 815 y Fq(2)934 830 y Ft(f)958 815 y Fq(2)954 840 y Fs(\031)977 830 y Fj(\))13 b(prop)q(osed)h([11)p (#cite.BrodskyLepage) [[359 518 369 530] [1 1 1 [3 3]] [0 0 1]] pdfm (])d(as)h(a)h(desirable)g(b)q(eha)o(v-)225 880 y(ior)e(for)h Ft(T)371 887 y Fs(\031)391 879 y Fi(0)410 880 y Fj(\()p Fr(\000)p Ft(Q)491 865 y Fq(2)510 880 y Ft(;)7 b Fj(0\))k(as)h(it)f (reduces)j(to)e(the)g(ABJ)h(anomaly)c(amplitude)h(\(1)p (#equation.1) [[408 506 413 518] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i(at) g Ft(Q)1529 865 y Fq(2)1559 880 y Fj(=)g(0,)225 930 y(while)h(agreeing) h(with)g(the)g(follo)o(wing)e(t)o(yp)q(e)i(of)f(leading)g(b)q(eha)o (vior)h(for)f(large)h Ft(Q)1490 915 y Fq(2)1508 930 y Fj(:)430 1038 y Ft(T)454 1045 y Fs(\031)474 1037 y Fi(0)493 1038 y Fj(\()p Fr(\000)p Ft(Q)574 1021 y Fq(2)593 1038 y Ft(;)7 b Fj(0\))k(=)h Fr(J)756 1010 y Ft(f)776 1016 y Fs(\031)p 752 1028 52 2 v 752 1066 a Ft(Q)785 1054 y Fq(2)898 1038 y Fj(\()p Fr(J)19 b Fj(=)12 b Ft(constant)i Fj(for)g(large)g Ft(Q)1377 1021 y Fq(2)1395 1038 y Fj(\))p Ft(;)159 b Fj(\(2\))225 1149 y(whic)o(h)14 b(is)g(fa)o(v)o(ored)f(b)q (oth)h(exp)q(erimen)o(tally)e([1)p (#cite.gronberg98) [[293 441 298 453] [1 1 1 [3 3]] [0 0 1]] pdfm (])h (and)h(theoretically)g(\()p Fg(e.g.)p Fj(,)f([2)p (#cite.Radyushkin+Rusk3) [[409 441 414 453] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(8)p (#cite.M+Rady97) [[418 441 423 453] [1 1 1 [3 3]] [0 0 1]] pdfm (,)g (11)p (#cite.BrodskyLepage) [[427 441 438 453] [1 1 1 [3 3]] [0 0 1]] pdfm (,)g(12)p (#cite.manohar90) [[442 441 452 453] [1 1 1 [3 3]] [0 0 1]] pdfm (]\).)300 1199 y(In)20 b(the)h(coupled)g(SD-BS)f(approac)o(h,)h(the)g (BS)g(equation)f(for)g(the)g(pion)g(b)q(ound-)225 1249 y(state)e Ft(q)5 b Fj(\026)-25 b Ft(q)19 b Fj(v)o(ertex)f(\000)543 1256 y Fs(\031)563 1248 y Fi(0)582 1249 y Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))17 b(emplo)o(ys)f(the)i(dynamically)d(dressed)k(quark)e (propagator)225 1298 y Ft(S)r Fj(\()p Ft(k)q Fj(\))c(=)e([)p Ft(A)p Fj(\()p Ft(k)445 1283 y Fq(2)464 1298 y Fj(\))p Ft(k)-22 b(=)7 b Fr(\000)h Ft(B)r Fj(\()p Ft(k)620 1283 y Fq(2)639 1298 y Fj(\)])667 1283 y Fh(\000)p Fq(1)711 1298 y Fj(,)13 b(obtained)f(b)o(y)h(solving)f(its)h(SD)f(equation.)18 b(Then,)13 b(in)f(the)225 1348 y(case)19 b(of)e(ligh)o(t)g (pseudoscalars,)j(the)e Ft(q)5 b Fj(\026)-25 b Ft(q)19 b Fj(b)q(ound)f(states)h(are)f(sim)o(ultaneously)e(also)h(the)225 1398 y(\(pseudo-\)Goldstone)d(b)q(osons)h(of)e(dynamical)f(c)o(hiral)h (symmetry)f(breaking)i(\(D)p Ft(\037)p Fj(SB\).)300 1448 y(F)m(ollo)o(wing)9 b(Jain)i(and)h(Munczek)h([13)p (#cite.jain91+munczek92) [[273 369 283 381] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s(14)p (#cite.jain93b) [[286 369 296 381] [1 1 1 [3 3]] [0 0 1]] pdfm (],)c (w)o(e)j(adopt)f(the)i(ladder-t)o(yp)q(e)f(appro)o(xima-)225 1498 y(tion)g(emplo)o(ying)f(bare)j(quark{gluon{quark)d(v)o(ertices)j (but)g(dressed)h(propagators.)i(F)m(or)225 1547 y(the)f(gluon)e (propagator)h(w)o(e)h(use)g(an)f(e\013ectiv)o(e,)h(\()p Fg(p)n(artial)r(ly)p Fj(\))f(mo)q(deled)f(one)i(in)f(Landau-)225 1597 y(gauge)i([13)p (#cite.jain91+munczek92) [[158 334 168 346] [1 1 1 [3 3]] [0 0 1]] pdfm (,)6 b(14)p (#cite.jain93b) [[172 334 182 346] [1 1 1 [3 3]] [0 0 1]] pdfm (],)17 b(giv)o(en)g(b)o(y)g Ft(G)p Fj(\()p Fr(\000)p Ft(l)768 1582 y Fq(2)786 1597 y Fj(\)\()p Ft(g)839 1582 y Fs(\026\027)892 1597 y Fr(\000)12 b Ft(l)949 1582 y Fs(\026)972 1597 y Ft(l)985 1582 y Fs(\027)1005 1597 y Ft(=l)1039 1582 y Fq(2)1058 1597 y Fj(\))17 b Ft(:)g Fj(\(This)h(Ansatz)g(is)f(often)g (called)225 1647 y(the)c(\\Ab)q(elian)f(appro)o(ximation")e([15)p (#cite.MarisRoberts97PRC56) [[259 322 269 334] [1 1 1 [3 3]] [0 0 1]] pdfm (].\))16 b(The)d(e\013ectiv)o(e)h(propagator)e(function)h Ft(G)f Fj(is)g(the)225 1697 y(sum)h(of)h(the)h(p)q(erturbativ)o(e)g (con)o(tribution)f Ft(G)942 1703 y Fs(U)s(V)1011 1697 y Fj(and)g(the)h(nonp)q(erturbativ)o(e)g(con)o(tribu-)225 1747 y(tion)h Ft(G)346 1753 y Fs(I)r(R)390 1747 y Fj(:)24 b Ft(G)p Fj(\()p Ft(Q)508 1732 y Fq(2)526 1747 y Fj(\))17 b(=)g Ft(G)641 1753 y Fs(U)s(V)695 1747 y Fj(\()p Ft(Q)744 1732 y Fq(2)763 1747 y Fj(\))11 b(+)h Ft(G)867 1753 y Fs(I)r(R)911 1747 y Fj(\()p Ft(Q)960 1732 y Fq(2)978 1747 y Fj(\))17 b Ft(;)40 b Fj(\()p Ft(Q)1112 1732 y Fq(2)1147 1747 y Fj(=)17 b Fr(\000)p Ft(l)1241 1732 y Fq(2)1260 1747 y Fj(\))g Ft(:)g Fj(The)g(p)q(erturbativ)o(e)225 1797 y(part)f Ft(G)350 1803 y Fs(U)s(V)420 1797 y Fj(is)f(required)i (to)f(repro)q(duce)h(correctly)g(the)f(ultra)o(violet)f(\(UV\))h (asymptotic)225 1846 y(b)q(eha)o(vior)j(that)h(unam)o(biguously)d (follo)o(ws)h(from)g(QCD)i(in)f(its)h(high{energy)f(regime.)225 1896 y(Therefore,)j Ft(G)465 1902 y Fs(U)s(V)539 1896 y Fj(m)o(ust)c(b)q(e)i(giv)o(en)f(b)o(y)h Ft(\013)914 1902 y Fs(s)931 1896 y Fj(\()p Ft(Q)980 1881 y Fq(2)999 1896 y Fj(\))p Ft(=Q)1069 1881 y Fq(2)1087 1896 y Fj(,)h(where)f(the)g (running)g(coupling)225 1946 y(constan)o(t)c Ft(\013)421 1952 y Fs(s)438 1946 y Fj(\()p Ft(Q)487 1931 y Fq(2)506 1946 y Fj(\))f(is)g(w)o(ell-kno)o(wn)f(from)f(pQCD,)i(so)g(that)h Ft(G)1217 1952 y Fs(U)s(V)1286 1946 y Fj(is)f Fg(not)h Fj(mo)q(deled.)k(As)225 1996 y(in)13 b(Refs.)h([4)p (#cite.KeBiKl98) [[165 238 170 250] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(9)p (#cite.KeKl1) [[174 238 180 250] [1 1 1 [3 3]] [0 0 1]] pdfm (,) g(10)p (#cite.KlKe2) [[184 238 194 250] [1 1 1 [3 3]] [0 0 1]] pdfm (],)11 b(w)o(e)j(follo)o(w)e(Refs.)i([13)p (#cite.jain91+munczek92) [[273 238 283 250] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(14)p (#cite.jain93b) [[288 238 298 250] [1 1 1 [3 3]] [0 0 1]] pdfm (])12 b(and)h(emplo)o(y)f(the)j(t)o(w)o(o{lo)q(op)d(asymptotic)225 2046 y(expression)20 b(for)f Ft(\013)526 2052 y Fs(s)543 2046 y Fj(\()p Ft(Q)592 2031 y Fq(2)611 2046 y Fj(\).)34 b(F)m(or)19 b(the)h(mo)q(deled,)f(IR)f(part,)i(w)o(e)g(adopt)f(from)e (Ref.)h([14)p (#cite.jain93b) [[452 226 462 238] [1 1 1 [3 3]] [0 0 1]] pdfm (])225 2101 y Ft(G)258 2107 y Fs(I)r(R)302 2101 y Fj(\()p Ft(Q)351 2086 y Fq(2)369 2101 y Fj(\))12 b(=)g(\(16)p Ft(\031)524 2086 y Fq(2)542 2101 y Ft(=)p Fj(3\))7 b Ft(a)g(Q)669 2086 y Fq(2)687 2101 y Ft(e)706 2086 y Fh(\000)p Fs(\026Q)778 2074 y Fi(2)811 2101 y Fj(and)14 b Ft(a)d Fj(=)h(\(0)p Ft(:)p Fj(387)7 b(GeV)o(\))1185 2086 y Fh(\000)p Fq(4)1229 2101 y Fj(,)14 b Ft(\026)d Fj(=)h(\(0)p Ft(:)p Fj(510)7 b(GeV)o(\))1551 2086 y Fh(\000)p Fq(2)1596 2101 y Fj(.)300 2151 y(Our)14 b(calculational)e(pro)q(cedures)k(are)e(detailed)f(in)g (our)h(Refs.)f([4)p (#cite.KeBiKl98) [[384 201 389 213] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(9)p (#cite.KeKl1) [[393 201 398 213] [1 1 1 [3 3]] [0 0 1]] pdfm (,) g(10)p (#cite.KlKe2) [[402 201 412 213] [1 1 1 [3 3]] [0 0 1]] pdfm (].)15 b(W)m(e)e(repro-)225 2201 y(duce)21 b(the)f(solutions)f(of)g (Ref.)g([14)p (#cite.jain93b) [[250 189 260 201] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (for)h(the)i(dressed)g(propagator)e Ft(S)r Fj(\()p Ft(q)q Fj(\),)j Fg(i.e.)p Fj(,)e Ft(A)p Fj(\()p Ft(q)1600 2186 y Fq(2)1619 2201 y Fj(\))225 2251 y(and)i Ft(B)r Fj(\()p Ft(q)383 2235 y Fq(2)402 2251 y Fj(\),)i(as)e(w)o(ell)f(as)i(the)f (solutions)g(for)f(the)i(four)f(functions)g(comprising)e(the)225 2300 y(pion)d(b)q(ound-state)i(v)o(ertex)f(\000)710 2307 y Fs(\031)730 2299 y Fi(0)749 2300 y Fj(.)29 b(Actually)m(,)17 b(Ref.)g([14)p (#cite.jain93b) [[330 165 340 177] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (emplo)o(ys)g(the)j(BS)f(amplitude)225 2350 y Ft(\037)251 2357 y Fs(\031)271 2349 y Fi(0)290 2350 y Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))k Fr(\021)h Ft(S)r Fj(\()p Ft(q)f Fj(+)e Ft(p=)p Fj(2\)\000)656 2357 y Fs(\031)676 2349 y Fi(0)695 2350 y Fj(\()p Ft(q)q(;)e(p)p Fj(\))p Ft(S)r Fj(\()p Ft(q)k Fr(\000)e Ft(p=)p Fj(2\),)k(whic)o(h)h(is)f(completely)g(equiv)n (alen)o(t.)300 2400 y(W)m(e)j(assume)h(that)f(the)i Ft(\031)713 2385 y Fq(0)731 2400 y Ft(\015)754 2385 y Fs(?)774 2400 y Ft(\015)i Fj(transition)c(\(and)h(other)g(pseudoscalar)h(meson)d Fr(!)920 2490 y Fj(2)p eop %%Page: 3 3 3 2 bop 225 961 a @beginspecial 0 @llx 0 @lly 306 @urx 162 @ury 2551 @rwi @setspecial %%BeginDocument: 2photon.fig1.eps 80 dict begin /s {stroke} def /l {lineto} def /m {moveto} def /t {translate} def /sw {stringwidth} def /r {rotate} def /rl {roll} def /R {repeat} def /d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def /c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def /cl {closepath} 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> } image } if PT 12 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < E3 E3 DD DD BE BE BE BE BE BE DD DD E3 E3 FF FF 3E 3E DD DD EB EB EB EB EB EB DD DD 3E 3E FF FF > } image } if PT 13 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FE FE 7D 7D BB BB D7 D7 EF EF D7 D7 BB BB 7D 7D FE FE 7D 7D BB BB D7 D7 EF EF D7 D7 BB BB 7D 7D > } image } if PT 14 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 00 00 EE EF EE EF EE EF 0E E0 EE EE EE EE EE EE 00 EE FE EE FE EE FE EE 00 00 FE EF FE EF FE EF > } image } if PT 15 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < DD DD AA AA DD DD FF FF 77 77 AA AA 77 77 FF FF DD DD AA AA DD DD FF FF 77 77 AA AA 77 77 FF FF > } image } if PT 16 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF > } image } if PT 17 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF > } image } if PT 18 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF > } image } if PT 19 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 1F FC 67 F3 7B EF BD DE BD DE DE BD E6 B3 F8 0F E6 B3 DE BD BD DE BD DE 7B EF 67 F3 1F FC 7F FF > } image } if PT 20 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < DD DD EE EE EE EE EE EE DD DD BB BB BB BB BB BB DD DD EE EE EE EE EE EE DD DD BB BB BB BB BB BB > } image } if PT 21 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 0E 0E EF EF EF EF EF EF E0 E0 FE FE FE FE FE FE 0E 0E EF EF EF EF EF EF E0 E0 FE FE FE FE FE FE > } image } if PT 22 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 70 70 F7 F7 F7 F7 F7 F7 07 07 7F 7F 7F 7F 7F 7F 70 70 F7 F7 F7 F7 F7 F7 07 07 7F 7F 7F 7F 7F 7F > } image } if PT 23 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < AA AA 55 55 A9 A9 D1 D1 E1 E1 D1 D1 A9 A9 55 55 AA AA 55 55 A9 A9 D1 D1 E1 E1 D1 D1 A9 A9 55 55 > } image } if PT 24 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FF FE FF FC EA A8 D5 54 EA A8 D5 54 E8 28 D4 54 E8 E8 D4 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/Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute 212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex 217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex 221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex 228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236 /Ucircumflex 237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis 244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def /Times-Roman /Times-Roman accvec ReEncode /Times-Italic /Times-Italic accvec ReEncode /Times-Bold /Times-Bold accvec ReEncode /Times-BoldItalic /Times-BoldItalic accvec ReEncode /Helvetica /Helvetica accvec ReEncode /Helvetica-Oblique /Helvetica-Oblique accvec ReEncode /Helvetica-Bold /Helvetica-Bold accvec ReEncode /Helvetica-BoldOblique /Helvetica-BoldOblique accvec ReEncode /Courier /Courier accvec ReEncode /Courier-Oblique /Courier-Oblique accvec ReEncode /Courier-Bold /Courier-Bold accvec ReEncode /Courier-BoldOblique /Courier-BoldOblique accvec ReEncode /oshow {gsave [] 0 sd true charpath stroke gr} def /stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf text sw pop xs add /xs exch def} def /stwb { /fs exch def /fn exch def /nbas exch def /textf exch def textf length /tlen exch def nbas tlen gt {/nbas tlen def} if fn findfont fs sf textf dup length nbas sub nbas getinterval sw pop neg xs add /xs exch def} def /accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal 69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral 74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright 79 /greaterequal 80 /braceleft 81 /braceright 82 /radical 83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal 88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus 50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup 55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110 /notelement 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def /Symbol /Special accspe ReEncode gsave .25 .25 scale gsave 0 0 t black [] 0 sd 1 lw 1224 646 0 0 C /xs 0 def ( p) /Times-Italic 49 stwn gsave 205 347 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m ( p) show gr /xs 0 def (G) /Symbol 49 stwn (P) /Times-Italic 34 stwn gsave 313 374 t 0 r xs 2 div neg 0 t 0 0 m /Symbol findfont 49 sf 0 0 m (G) show currentpoint pop 0 t /Times-Italic findfont 34 sf 0 -16 m (P) show gr /xs 0 def (q+p/2) /Times-Italic 49 stwn gsave 452 481 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (q+p/2) show gr /xs 0 def (G) /Symbol 49 stwn (n) /Symbol 34 stwn gsave 645 561 t 0 r xs 2 div neg 0 t 0 0 m /Symbol findfont 49 sf 0 0 m (G) show currentpoint pop 0 t /Symbol findfont 34 sf 0 25 m (n) show gr /xs 0 def (k+q) /Times-Italic 49 stwn (-) /Symbol 49 stwn (p/2) /Times-Italic 49 stwn gsave 721 307 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (k+q) show currentpoint pop 0 t /Symbol findfont 49 sf 0 0 m (-) show currentpoint pop 0 t /Times-Italic findfont 49 sf 0 0 m (p/2) show gr /xs 0 def (+ \(k) /Times-Italic 49 stwn (\253) /Symbol 49 stwn (k\302,\040) /Times-Italic 49 stwn (m\253n) /Symbol 49 stwn (\)) /Times-Italic 49 stwn gsave 1060 307 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (+ \(k) show currentpoint pop 0 t /Symbol findfont 49 sf 0 0 m (\253) show currentpoint pop 0 t /Times-Italic findfont 49 sf 0 0 m (k\302,\040) show currentpoint pop 0 t /Symbol findfont 49 sf 0 0 m (m\253n) show currentpoint pop 0 t /Times-Italic findfont 49 sf 0 0 m (\)) show gr /xs 0 def (q) /Times-Italic 49 stwn (-) /Symbol 49 stwn (p/2) /Times-Italic 49 stwn gsave 452 132 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (q) show currentpoint pop 0 t /Symbol findfont 49 sf 0 0 m (-) show currentpoint pop 0 t /Times-Italic findfont 49 sf 0 0 m (p/2) show gr /xs 0 def (G) /Symbol 49 stwn (m) /Symbol 34 stwn gsave 645 52 t 0 r xs 2 div neg 0 t 0 0 m /Symbol findfont 49 sf 0 0 m (G) show currentpoint pop 0 t /Symbol findfont 34 sf 0 25 m (m) show gr /xs 0 def (k\302) /Times-Italic 49 stwn gsave 748 453 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (k\302) show gr /xs 0 def (k) /Times-Italic 49 stwn gsave 748 161 t 0 r xs 2 div neg 0 t 0 0 m /Times-Italic findfont 49 sf 0 0 m (k) show gr 1222 644 1 1 C 3 lw /w 44 def /w2 {w 2 div} def /w3 {w 3 div} def 341 323 m20 [12 12] 0 sd 341 323 m -272 X s 1 lw [] 0 sd 197 323 m -5 10 d 32 -10 d -32 -10 d f 3 lw /w 33 def /w2 {w 2 div} def /w3 {w 3 div} def 612 526 m20 612 526 m -271 -203 d s 1 lw 470 420 m -10 4 d 32 12 d -20 -27 d f 3 lw 612 120 m20 612 120 m 406 Y s 1 lw 612 331 m 10 5 d -10 -32 d -10 32 d f 3 lw 341 323 m 271 -203 d s 1 lw 483 217 m -2 -11 d -20 27 d 32 -11 d f 3 lw 612 526 m 10 X 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 3 -9 d 2 -5 d 3 -2 d 2 2 d 3 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 3 -9 d 2 -5 d 3 -2 d 2 2 d 3 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 37 X s 1 lw 856 526 m -5 10 d 32 -10 d -32 -10 d f 3 lw 612 120 m 10 X 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 3 9 d 2 9 d 3 5 d 2 2 d 3 -2 d 2 -5 d 3 -9 d 2 -9 d 3 -5 d 2 -2 d 3 2 d 2 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 3 -9 d 2 -5 d 3 -2 d 2 2 d 3 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 3 -9 d 2 -5 d 3 -2 d 2 2 d 3 5 d 2 9 d 3 9 d 2 5 d 3 2 d 2 -2 d 3 -5 d 2 -9 d 37 X s 1 lw 856 120 m -5 10 d 32 -10 d -32 -10 d f gr gr showpage end %%EndDocument @endspecial 299 1052 a Fn(Figure)10 b(1:)31 b(The)11 b(GIA)h(diagram)d(for)i Fm(P)838 1041 y Fl(0)865 1052 y Ff(!)g Fm(\015)931 1041 y Fl(\()p Fk(?)p Fl(\))973 1052 y Fm(\015)993 1041 y Fl(\()p Fk(?)p Fl(\))1047 1052 y Fn(pro)q(cesses)e(\()p Fm(P)1240 1041 y Fl(0)1267 1052 y Fn(=)h Fm(\031)1325 1041 y Fl(0)1343 1052 y Fm(;)c(\021)q(;)f(\021) 1412 1041 y Fe(0)1424 1052 y Fm(;)g(\021)1457 1056 y Fk(c)1473 1052 y Fm(;)h(\021)1507 1058 y Fk(b)1522 1052 y Fn(\).)225 1231 y Fj(t)o(w)o(o)k(photon)h(transitions\))f(pro)q (ceeds)j(through)d(the)i(pseudoscalar-v)o(ector-v)o(ector)g(trian-)225 1280 y(gle)h(graph)f(\(Fig.)g(1\),)h(and)f(that)h(w)o(e)h(calculate)f (the)g(p)q(ertinen)o(t)h(amplitude)d Ft(T)1442 1260 y Fs(\026\027)1436 1294 y(\031)1456 1286 y Fi(0)1483 1280 y Fj(\()p Ft(k)q(;)c(k)1564 1265 y Fh(0)1575 1280 y Fj(\))12 b(=)225 1330 y Ft(")244 1315 y Fs(\013\014)q(\026\027)327 1330 y Ft(k)349 1336 y Fs(\013)372 1330 y Ft(k)395 1315 y Fh(0)394 1342 y Fs(\014)416 1330 y Ft(T)440 1337 y Fs(\031)460 1329 y Fi(0)479 1330 y Fj(\()p Ft(k)518 1315 y Fq(2)537 1330 y Ft(;)7 b(k)579 1315 y Fh(0)p Fq(2)607 1330 y Fj(\))37 b(as)18 b(in)g(Refs.)h([4)p (#cite.KeBiKl98) [[285 398 290 410] [1 1 1 [3 3]] [0 0 1]] pdfm (,)7 b(9)p (#cite.KeKl1) [[295 398 300 410] [1 1 1 [3 3]] [0 0 1]] pdfm (,) g(10)p (#cite.KlKe2) [[305 398 315 410] [1 1 1 [3 3]] [0 0 1]] pdfm (],)18 b(using)h(the)g(framew)o(ork)e(adv)o(o)q(cated)225 1380 y(b)o(y)m(,)12 b Fg(e.g.)p Fj(,)g(Refs.)g([5)p (#cite.bando94) [[190 386 195 398] [1 1 1 [3 3]] [0 0 1]] pdfm ({)o(7) p (#cite.Frank+al) [[200 386 205 398] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (in)h(the)g(con)o(text)h(of)e(electromagnetic)g(in)o(teractions)h(of)g (BS)g(b)q(ound)225 1430 y(states,)22 b(and)d(often)h(called)f(\()p Fg(e.g.)p Fj(,)h(b)o(y)f(Ref.)g([7)p (#cite.Frank+al) [[303 374 308 386] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))g (the)h(generalized)g(impulse)e(appro)o(xi-)225 1480 y(mation)d (\(GIA\).)i(W)m(e)g(therefore)i(use)g(the)f Fg(dr)n(esse)n(d)f Fj(quark)g(propagator)g Ft(S)r Fj(\()p Ft(q)q Fj(\))i(and)e(the)225 1529 y(pseudoscalar)g(BS)g(b)q(ound{state)g(v)o(ertex)h(\000)933 1535 y Fs(P)960 1529 y Fj(\()p Ft(q)q(;)7 b(p)p Fj(\),)16 b(as)h(w)o(ell)e(as)i(consisten)o(tly)g(dressed)225 1579 y Fg(ele)n(ctr)n(omagnetic)j Fj(v)o(ertex)h(\000)682 1564 y Fs(\026)704 1579 y Fj(\()p Ft(q)740 1564 y Fh(0)752 1579 y Ft(;)7 b(q)q Fj(\),)21 b(whic)o(h)f(satis\014es)h(the)g(v)o (ector)g(W)m(ard{T)m(ak)n(ahashi)225 1629 y(iden)o(tit)o(y)13 b(\(WTI\))g(\()p Ft(q)547 1614 y Fh(0)567 1629 y Fr(\000)8 b Ft(q)q Fj(\))643 1635 y Fs(\026)666 1629 y Fj(\000)692 1614 y Fs(\026)714 1629 y Fj(\()p Ft(q)750 1614 y Fh(0)762 1629 y Ft(;)f(q)q Fj(\))k(=)h Ft(S)899 1614 y Fh(\000)p Fq(1)944 1629 y Fj(\()p Ft(q)980 1614 y Fh(0)992 1629 y Fj(\))c Fr(\000)h Ft(S)1084 1614 y Fh(\000)p Fq(1)1129 1629 y Fj(\()p Ft(q)q Fj(\))k Ft(:)g Fj(Namely)m(,)e(assuming)h(that) 225 1679 y(photons)e(couple)f(to)h(quarks)f(through)h(the)g(bare)g(v)o (ertex)g Ft(\015)1130 1664 y Fs(\026)1162 1679 y Fj(w)o(ould)f(b)q(e)h (inconsisten)o(t)g(with)225 1729 y(our)j(dressed)j(quark)d(propagator)g Ft(S)r Fj(\()p Ft(q)q Fj(\),)h(whic)o(h)g(con)o(tains)f(the)h(momen)o (tum)o(-dep)q(enden)o(t)225 1778 y(functions)h Ft(A)p Fj(\()p Ft(q)472 1763 y Fq(2)491 1778 y Fj(\))h(and)f Ft(B)r Fj(\()p Ft(q)674 1763 y Fq(2)693 1778 y Fj(\).)23 b(The)16 b(bare)g(v)o(ertex)g Ft(\015)1075 1763 y Fs(\026)1113 1778 y Fj(ob)o(viously)e(violates)h(the)h(v)o(ector)225 1828 y(WTI,)d(implying)d(the)k(nonconserv)n(ation)f(of)g(the)h (electromagnetic)f(v)o(ector)h(curren)o(t)h(and)225 1878 y(c)o(harge.)i(Solving)9 b(the)i(p)q(ertinen)o(t)h(SD)e(equation)g(for) g(the)h(dressed)h(quark{photon{quark)225 1928 y(\()p Ft(q)q(q)q(\015)r Fj(\))17 b(v)o(ertex)g(\000)491 1913 y Fs(\026)530 1928 y Fj(is)f(still)f(an)h(unsolv)o(ed)g(problem,)f(and) h(using)g(the)h(realistic)f(Ans\177)-21 b(atze)225 1978 y(for)16 b(\000)317 1963 y Fs(\026)356 1978 y Fj(still)f(remains)g(the) j(only)d(practical)h(w)o(a)o(y)g(to)g(satisfy)h(the)g(WTI.)e(The)i (simplest)225 2028 y(particular)d(solution)f(of)g(the)h(v)o(ector)h (WTI)f(is)f(the)i(Ball{Chiu)d(\(BC\))29 b([16)p (#cite.BC) [[398 230 408 242] [1 1 1 [3 3]] [0 0 1]] pdfm (])12 b(v)o(ertex)225 2137 y(\000)251 2117 y Fs(\026)251 2149 y(B)q(C)305 2137 y Fj(\()p Ft(q)341 2120 y Fh(0)353 2137 y Ft(;)7 b(q)q Fj(\))k(=)h Ft(A)494 2143 y Fq(+)522 2137 y Fj(\()p Ft(q)558 2120 y Fh(0)p Fq(2)586 2137 y Ft(;)7 b(q)625 2120 y Fq(2)643 2137 y Fj(\))664 2109 y Ft(\015)687 2094 y Fs(\026)p 664 2128 47 2 v 677 2166 a Fj(2)725 2137 y(+)778 2109 y(\()p Ft(q)814 2094 y Fh(0)835 2109 y Fj(+)i Ft(q)q Fj(\))912 2094 y Fs(\026)p 771 2128 170 2 v 771 2166 a Fj(\()p Ft(q)807 2151 y Fh(0)p Fq(2)845 2166 y Fr(\000)g Ft(q)906 2151 y Fq(2)925 2166 y Fj(\))946 2137 y Fr(f)p Ft(A)998 2143 y Fh(\000)1026 2137 y Fj(\()p Ft(q)1062 2120 y Fh(0)p Fq(2)1090 2137 y Ft(;)e(q)1129 2120 y Fq(2)1147 2137 y Fj(\))1168 2109 y(\()p Ft(q)-19 b(=)1205 2094 y Fh(0)1226 2109 y Fj(+)10 b Ft(q)-19 b(=)p Fj(\))p 1168 2128 137 2 v 1226 2166 a(2)1319 2137 y Fr(\000)9 b Ft(B)1391 2143 y Fh(\000)1420 2137 y Fj(\()p Ft(q)1456 2120 y Fh(0)p Fq(2)1484 2137 y Ft(;)e(q)1523 2120 y Fq(2)1541 2137 y Fj(\))p Fr(g)14 b Ft(;)37 b Fj(\(3\))225 2251 y(where)16 b Ft(H)381 2257 y Fh(\006)408 2251 y Fj(\()p Ft(q)444 2235 y Fh(0)p Fq(2)472 2251 y Ft(;)7 b(q)511 2235 y Fq(2)530 2251 y Fj(\))12 b Fr(\021)h Fj([)p Ft(H)s Fj(\()p Ft(q)689 2235 y Fh(0)p Fq(2)717 2251 y Fj(\))c Fr(\006)h Ft(H)s Fj(\()p Ft(q)858 2235 y Fq(2)877 2251 y Fj(\)],)k(for)g Ft(H)h Fj(=)e Ft(A)h Fj(or)h Ft(B)r Fj(.)20 b(It)14 b(do)q(es)i Fg(not)e Fj(in)o(tro)q(duce)225 2300 y(an)o(y)e(new)h(parameters)f(as)h(it)f(is)g(completely)f (determined)h(b)o(y)g(the)h(dressed)i(quark)d(prop-)225 2350 y(agator)h Ft(S)r Fj(\()p Ft(q)q Fj(\).)20 b(In)13 b(the)i(SD-BS)f(approac)o(h,)f(this)h(minima)o(l)d(WTI-satisfying)h (Ansatz)j(\(3)p (#equation.3) [[456 153 461 165] [1 1 1 [3 3]] [0 0 1]] pdfm (\))225 2400 y(is)c(still)g(the)h(most)e(widely)h(used)h Ft(q)q(q)q(\015)j Fj(v)o(ertex)d(\()p Fg(e.g.)p Fj(,)f(Refs.)g([6)p (#cite.Roberts) [[347 141 352 153] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s(7) p (#cite.Frank+al) [[356 141 361 153] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s (9)p (#cite.KeKl1) [[364 141 369 153] [1 1 1 [3 3]] [0 0 1]] pdfm (,)s (10)p (#cite.KlKe2) [[373 141 383 153] [1 1 1 [3 3]] [0 0 1]] pdfm (]\).)j(A)e(general)g(WTI-)920 2490 y(3)p eop %%Page: 4 4 4 3 bop 225 432 a Fj(satisfying)13 b(v)o(ertex)j(can)e(b)q(e)h(written) g([16)p (#cite.BC) [[271 613 281 625] [1 1 1 [3 3]] [0 0 1]] pdfm (])e (as)i(\000)973 416 y Fs(\026)1007 432 y Fj(=)e(\000)1078 412 y Fs(\026)1078 444 y(B)q(C)1142 432 y Fj(+)d(\001\000)1245 416 y Fs(\026)1267 432 y Fj(,)k(where)h(the)g(addition)225 481 y(\001\000)286 466 y Fs(\026)318 481 y Fj(do)q(esn't)c(con)o (tribute)h(to)e(the)i(WTI,)d(since)j(it)e(is)g(transv)o(erse,)j(\()p Ft(q)1298 466 y Fh(0)1313 481 y Fr(\000)s Ft(q)q Fj(\))1384 487 y Fs(\026)1406 481 y Fj(\001\000)1467 466 y Fs(\026)1489 481 y Fj(\()p Ft(q)1525 466 y Fh(0)1537 481 y Ft(;)7 b(q)q Fj(\))k(=)225 531 y(0.)20 b(That)14 b(is,)g(\001\000)497 516 y Fs(\026)519 531 y Fj(\()p Ft(q)555 516 y Fh(0)566 531 y Ft(;)7 b(q)q Fj(\))15 b(lies)f(in)g(the)h(h)o(yp)q(erplane)g (spanned)g(b)o(y)f(the)i(v)o(ectors)f Fr(T)1510 511 y Fs(\026)1500 543 y(i)1533 531 y Fj(\()p Ft(q)1569 516 y Fh(0)1580 531 y Ft(;)7 b(q)q Fj(\))225 581 y(\()p Ft(i)12 b Fj(=)g(1)p Ft(;)7 b(:::;)g Fj(8\))k(transv)o(erse)16 b(to)e(the)h(photon)f(momen)o(tum)c Ft(k)i Fj(=)g Ft(q)1231 566 y Fh(0)1252 581 y Fr(\000)e Ft(q)q Fj(.)18 b(Curtis)d(and)f(P)o (en-)225 631 y(nington)j(\(CP\))i([17)p (#cite.CP90) [[192 566 203 578] [1 1 1 [3 3]] [0 0 1]] pdfm (])d(adv)o (o)q(cated)j(an)f(Ansatz)h(for)e(\001\000)1108 616 y Fs(\026)1130 631 y Fj(\()p Ft(q)1166 616 y Fh(0)1178 631 y Ft(;)7 b(q)q Fj(\))18 b(exclusiv)o(ely)f(along)g(the)225 681 y Ft(i)h Fj(=)h(6)e(basis)h(v)o(ector:)26 b(\001\000)661 666 y Fs(\026)683 681 y Fj(\()p Ft(q)719 666 y Fh(0)731 681 y Ft(;)7 b(q)q Fj(\))18 b(=)g Fr(T)887 661 y Fs(\026)877 692 y Fq(6)909 681 y Fj(\()p Ft(q)945 666 y Fh(0)957 681 y Ft(;)7 b(q)q Fj(\)[)p Ft(A)1055 687 y Fh(\000)1082 681 y Fj(\()p Ft(q)1118 666 y Fh(0)p Fq(2)1147 681 y Ft(;)g(q)1186 666 y Fq(2)1204 681 y Fj(\))p Ft(=)p Fj(2)p Ft(d)p Fj(\()p Ft(q)1320 666 y Fh(0)1331 681 y Ft(;)g(q)q Fj(\)].)29 b(The)18 b(co)q(e\016-)225 730 y(cien)o(t)c(of)g Fr(T)406 710 y Fs(\026)396 741 y Fq(6)428 730 y Fj(\()p Ft(q)464 715 y Fh(0)476 730 y Ft(;)7 b(q)q Fj(\))13 b(can)h(b)q(e)h(c)o (hosen)g(to)e(ensure)j(renormalizabilit)o(y)11 b([17)p (#cite.CP90) [[390 542 400 554] [1 1 1 [3 3]] [0 0 1]] pdfm (],)h Fg(e.g.)p Fj(,)384 846 y Ft(d)406 852 y Fh(\006)434 846 y Fj(\()p Ft(q)470 829 y Fh(0)482 846 y Ft(;)7 b(q)q Fj(\))k(=)655 818 y(1)p 597 836 138 2 v 597 874 a Ft(q)617 862 y Fh(0)p Fq(2)654 874 y Fj(+)f Ft(q)716 862 y Fq(2)746 787 y Fd(\032)777 846 y Fj(\()p Ft(q)813 829 y Fh(0)p Fq(2)851 846 y Fr(\006)f Ft(q)912 829 y Fq(2)931 846 y Fj(\))947 829 y Fq(2)975 846 y Fj(+)1016 800 y Fd(h)1036 846 y Ft(M)1081 829 y Fq(2)1100 846 y Fj(\()p Ft(q)1136 829 y Fh(0)1147 823 y Fq(2)1166 846 y Fj(\))g(+)h Ft(M)1278 829 y Fq(2)1296 846 y Fj(\()p Ft(q)1332 829 y Fq(2)1351 846 y Fj(\))1367 800 y Fd(i)1387 808 y Fq(2)1405 787 y Fd(\033)1457 846 y Ft(;)113 b Fj(\(4\))225 964 y(where)14 b Ft(M)5 b Fj(\()p Ft(q)425 948 y Fq(2)443 964 y Fj(\))12 b Fr(\021)g Ft(B)r Fj(\()p Ft(q)584 948 y Fq(2)603 964 y Fj(\))p Ft(=)n(A)p Fj(\()p Ft(q)705 948 y Fq(2)724 964 y Fj(\))h(is)f(the)i(D)p Ft(\037)p Fj(SB-generated)g(dynamical)c (mass)i(function,)225 1013 y(whic)o(h)i(in)f(our)h(case)h(has)f(the)g (large-)p Ft(q)821 998 y Fq(2)854 1013 y Fj(dep)q(endence)i([4)p (#cite.KeBiKl98) [[333 474 338 486] [1 1 1 [3 3]] [0 0 1]] pdfm (])d (in)g(agreemen)o(t)h(with)f(pQCD.)300 1063 y(The)18 b(c)o(hoice)f Ft(d)g Fj(=)g Ft(d)625 1069 y Fh(\000)670 1063 y Fj(corresp)q(onds)i (to)e(the)g(CP)h(v)o(ertex)g(Ansatz)g(\000)1403 1043 y Fs(\026)1403 1075 y(C)r(P)1473 1063 y Fj([17)p (#cite.CP90) [[428 462 438 474] [1 1 1 [3 3]] [0 0 1]] pdfm (].)26 b(W)m(e)225 1113 y(use)17 b(it)f(in)g(analytic)g(calculations)g(of)g Ft(T)858 1120 y Fs(\031)878 1112 y Fi(0)897 1113 y Fj(\()p Fr(\000)p Ft(Q)978 1098 y Fq(2)997 1113 y Ft(;)7 b Fj(0\),)16 b(whic)o(h)g(are)h(p)q(ossible)g(for)f Ft(Q)1532 1098 y Fq(2)1566 1113 y Fj(=)g(0)225 1163 y(and)d Ft(Q)338 1148 y Fq(2)368 1163 y Fr(!)e(1)p Fj(.)17 b(Ho)o(w)o(ev)o(er,)c(in)g (the)h(n)o(umerical)d(calculations,)i(whic)o(h)g(are)g(necessary)i(for) 225 1213 y(\014nite)i(v)n(alues)f(of)f Ft(Q)542 1198 y Fq(2)577 1213 y Fr(6)p Fj(=)h(0,)g(w)o(e)h(prefer)g(the)g Fg(mo)n(di\014e)n(d)g Fj(CP)f(\(mCP\))g(v)o(ertex,)h(\000)1480 1193 y Fs(\026)1480 1225 y(mC)r(P)1580 1213 y Fj([3)p (#cite.KeKl3) [[454 426 459 438] [1 1 1 [3 3]] [0 0 1]] pdfm (],)225 1262 y(resulting)g(from)e(the)i(c)o(hoice)g Ft(d)f Fj(=)g Ft(d)808 1268 y Fq(+)835 1262 y Fj(,)h(since)h(it)e(is)g(easier)i(to)e (deal)h(with)f(n)o(umerically)m(.)225 1312 y(Unlik)o(e)g(the)g(BC)h (one,)f(the)h(mCP)e(v)o(ertex)i(is)f(consisten)o(t)h(with)f (renormalizabilit)o(y)d(just)225 1362 y(lik)o(e)j(the)g(CP)h(one.)25 b(In)17 b(the)g(presen)o(t)h(con)o(text,)f(the)g(imp)q(ortan)o(t)d Fg(qualitative)i Fj(di\013erence)225 1412 y(b)q(et)o(w)o(een)i(the)e (BC)h(v)o(ertex,)g(and)f(the)g(CP)h(as)f(w)o(ell)f(as)i(the)f(mo)q (di\014ed,)f(mCP)g(v)o(ertex,)i(is)225 1462 y(that)e(\000)342 1442 y Fs(\026)342 1474 y(B)q(C)396 1462 y Fj(\()p Ft(q)432 1447 y Fh(0)444 1462 y Ft(;)7 b(q)q Fj(\))13 b Fr(!)g Ft(\015)590 1447 y Fs(\026)628 1462 y Fj(when)i Fg(b)n(oth)g Ft(q)845 1447 y Fh(0)p Fq(2)873 1462 y Ft(;)7 b(q)912 1447 y Fq(2)944 1462 y Fr(!)13 b(\0061)p Fj(,)h(whereas)i(\000)1283 1442 y Fs(\026)1283 1474 y(C)r(P)1337 1462 y Fj(\()p Ft(q)1373 1447 y Fh(0)1385 1462 y Ft(;)7 b(q)q Fj(\))12 b Fr(!)h Ft(\015)1530 1447 y Fs(\026)1568 1462 y Fj(and)225 1511 y(\000)251 1492 y Fs(\026)251 1524 y(mC)r(P)334 1511 y Fj(\()p Ft(q)370 1496 y Fh(0)382 1511 y Ft(;)7 b(q)q Fj(\))k Fr(!)g Ft(\015)524 1496 y Fs(\026)561 1511 y Fj(as)j(so)q(on)g(as)g Fg(one)g Fj(of)f(the)i(squared)g(momen)o(ta)c (tends)k(to)e(in\014nit)o(y)m(.)300 1561 y(Assuming)i(the)h(isospin)f (symmetry)m(,)e Ft(\037)i Fr(\021)g Ft(\037)1022 1567 y Fs(u)r Fq(\026)-19 b Fs(u)1078 1561 y Fj(=)14 b Ft(\037)1150 1571 y Fs(d)1173 1564 y Fq(\026)1167 1571 y Fs(d)1203 1561 y Fj(is)h(the)h(BS)g(amplitude)e(for)225 1611 y(b)q(oth)20 b Ft(u)s Fj(\026)-24 b Ft(u)19 b Fj(and)g Ft(d)511 1600 y Fj(\026)504 1611 y Ft(d)g Fj(pseudoscalar)h(b)q(ound)g(states,)h(and) f Ft(\037)1176 1618 y Fs(\031)1196 1610 y Fi(0)1214 1611 y Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))21 b Fr(\021)g Ft(\037)p Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))g Ft(\025)1529 1596 y Fq(3)1547 1611 y Ft(=)1568 1577 y Fr(p)p 1603 1577 21 2 v 34 x Fj(2)o(.)225 1661 y(Then,)14 b(for)f Ft(\031)433 1646 y Fq(0)452 1661 y Fj(,)h(GIA)f(yields)h(\(see)h(Fig.)e(1\))h(the)g (tensor)h(amplitude)312 1777 y Ft(T)342 1758 y Fs(\026\027)336 1791 y(\031)356 1783 y Fi(0)383 1777 y Fj(\()p Ft(k)q(;)7 b(k)464 1760 y Fh(0)475 1777 y Fj(\))k(=)h Fr(\000)p Ft(N)611 1783 y Fs(c)668 1749 y Fj(1)p 641 1768 77 2 v 641 1811 a(3)662 1776 y Fr(p)p 696 1776 21 2 v 696 1811 a Fj(2)729 1721 y Fd(Z)800 1749 y Ft(d)822 1734 y Fq(4)841 1749 y Ft(q)p 782 1768 97 2 v 782 1806 a Fj(\(2)p Ft(\031)q Fj(\))860 1794 y Fq(4)884 1777 y Fj(tr)p Fr(f)p Fj(\000)963 1760 y Fs(\026)985 1777 y Fj(\()p Ft(q)f Fr(\000)1077 1749 y Ft(p)p 1077 1768 21 2 v 1077 1806 a Fj(2)1103 1777 y Ft(;)c(k)j Fj(+)f Ft(q)h Fr(\000)1271 1749 y Ft(p)p 1271 1768 V 1271 1806 a Fj(2)1297 1777 y(\))p Ft(S)r Fj(\()p Ft(k)h Fj(+)e Ft(q)i Fr(\000)1506 1749 y Ft(p)p 1506 1768 V 1506 1806 a Fj(2)1532 1777 y(\))522 1907 y Fr(\002)e Fj(\000)589 1889 y Fs(\027)610 1907 y Fj(\()p Ft(k)h Fj(+)g Ft(q)g Fr(\000)775 1879 y Ft(p)p 775 1897 V 775 1935 a Fj(2)801 1907 y Ft(;)d(q)j Fj(+)895 1879 y Ft(p)p 895 1897 V 895 1935 a Fj(2)921 1907 y(\))p Ft(\037)p Fj(\()p Ft(q)q(;)d(p)p Fj(\))p Fr(g)i Fj(+)h(\()p Ft(k)i Fr($)f Ft(k)1253 1889 y Fh(0)1265 1907 y Ft(;)c(\026)k Fr($)g Ft(\027)s Fj(\))p Ft(;)157 b Fj(\(5\))225 1997 y(leading)13 b(to)h(the)h(analytical)e(repro)q (duction)i(of)f(the)h Ft(\031)1073 1982 y Fq(0)1103 1997 y Fr(!)d Ft(\015)r(\015)18 b Fj(amplitude)12 b(\(1)p (#equation.1) [[415 238 420 250] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (with)g(the)225 2047 y(CP)g(and)g(mCP)e(v)o(ertices,)j(in)f(the)g(same) f(w)o(a)o(y)g(as)h(with)g(the)g(BC)h(v)o(ertices.)300 2097 y(Let)d(us)g(stress)h(that)f(in)f(the)h(regime)e(of)h (asymptotically)e(large)i Ft(Q)1322 2082 y Fq(2)1341 2097 y Fj(,)g(w)o(e)h(obtained)f([3)p (#cite.KeKl3) [[457 214 462 226] [1 1 1 [3 3]] [0 0 1]] pdfm (])225 2147 y(analytical)i(results)j(for)f(the)g(transition)g(form)e(factor)i (in)f(agreemen)o(t)h(with)f(the)i(b)q(eha)o(v-)225 2196 y(ior)h(\(2)p (#equation.2) [[146 190 151 202] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)30 b(Since)19 b(the)f(pion)f(is)h(ligh)o(t,)f Ft(k)c Fr(\001)e Ft(k)909 2181 y Fh(0)939 2196 y Fr(\031)19 b Ft(Q)1023 2181 y Fq(2)1041 2196 y Ft(=)p Fj(2)e(for)h(large)f(negativ)o(e)h Ft(k)1464 2181 y Fq(2)1501 2196 y Fj(=)g Fr(\000)p Ft(Q)1616 2181 y Fq(2)225 2251 y Fj(and)k Ft(k)337 2235 y Fh(0)349 2230 y Fq(2)393 2251 y Fj(=)k(0.)44 b(T)m(aking)20 b(in)o(to)i(accoun)o (t)h(the)g(b)q(eha)o(vior)f(of)g(the)h(propagator)f(func-)225 2300 y(tions)e Ft(A)p Fj(\()p Ft(q)400 2285 y Fq(2)419 2300 y Fj(\))g(and)g Ft(B)r Fj(\()p Ft(q)611 2285 y Fq(2)630 2300 y Fj(\))g(\()p Fg(e.g.)p Fj(,)h(see)g([4)p (#cite.KeBiKl98) [[279 165 284 177] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))e (w)o(e)i(can)f(then)h(in)e(Eq.)37 b(\(5)p (#equation.5) [[396 165 402 177] [1 1 1 [3 3]] [0 0 1]] pdfm (\))20 b(appro)o(ximate)225 2350 y(those)f(quark)e(propagators)h(that)g(dep)q (end)h(on)f(the)g(photon)g(momen)o(ta)d Ft(k)k Fj(and)e Ft(k)1549 2335 y Fh(0)1561 2350 y Fj(,)h(b)o(y)225 2400 y(their)h(asymptotic)e(forms:)25 b Ft(S)r Fj(\()p Ft(q)14 b Fr(\000)f Ft(p=)p Fj(2)e(+)i Ft(k)q Fj(\))19 b Fr(\031)g Ft(S)r Fj(\()p Ft(k)14 b Fr(\000)e Ft(p=)p Fj(2\))19 b Fr(\031)g(\000)p Fj(\(2)p Ft(=Q)1430 2385 y Fq(2)1448 2400 y Fj(\)\()p Ft(k)-22 b(=)12 b Fr(\000)h(6)o Ft(p)o(=)p Fj(2\))920 2490 y(4)p eop %%Page: 5 5 5 4 bop 225 432 a Fj(and)16 b(analogously)e(for)i(the)h(propagator)e (where)j Ft(k)f Fj(is)f(replaced)h(b)o(y)f Ft(k)1344 416 y Fh(0)1355 432 y Fj(.)25 b(Although)15 b(the)225 481 y(relativ)o(e)j(lo)q(op)f(momen)o(tum)d Ft(q)19 b Fj(can)f(b)q(e)h(large)f(in)f(the)i(course)g(of)f(in)o(tegration,)g (its)g(ne-)225 531 y(glecting)c(is)g(justi\014ed)g(b)q(ecause)i(the)f (BS)g(amplitude)d Ft(\037)p Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))14 b(deca)o(ys)h(quic)o(kly)e(and)i(th)o(us)225 581 y(strongly)20 b(damps)f(the)h(in)o(tegrand)g(for)g(the)h(large)f Ft(q)1075 566 y Fh(0)1086 581 y Fj(s.)37 b(Ref.)20 b([3)p (#cite.KeKl3) [[374 578 379 590] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (then)h(sho)o(w)o(ed)h(that)225 631 y Ft(T)255 611 y Fs(\026\027)249 645 y(\031)269 636 y Fi(0)296 631 y Fj(\()p Ft(k)q(;)7 b(k)377 616 y Fh(0)388 631 y Fj(\))15 b(=)h Fr(\000)p Fj(\(2)p Ft(=)n(N)588 637 y Fs(c)612 631 y Ft(Q)645 616 y Fq(2)663 631 y Fj(\))7 b Ft(")705 616 y Fs(\026\025\027)r(\033)786 631 y Fj(\()p Ft(k)12 b Fr(\000)f Ft(k)902 616 y Fh(0)913 631 y Fj(\))929 637 y Fs(\025)951 631 y Ft(p)972 637 y Fs(\033)1002 631 y Ft(f)1022 637 y Fs(\031)1044 631 y Fj(,)17 b(\014nally)d(leading)i(to)f (Eq.)25 b(\(2)p (#equation.2) [[432 566 437 578] [1 1 1 [3 3]] [0 0 1]] pdfm (\))16 b(with)225 681 y(the)f(co)q(e\016cien)o(t)h Fr(J)k Fj(=)13 b(4)p Ft(=)p Fj(3.)20 b(W)m(e)14 b(th)o(us)h(found)f Fg(mo)n(del)i(indep)n(endently)g Fj(that)e(the)h(asymp-)225 730 y(totic)f(b)q(eha)o(vior)f(in)g(the)h(presen)o(t)h(approac)o(h)f (\(with)f Ft(q)q(q)q(\015)j Fj(v)o(ertices)f(suc)o(h)g(as)e(the)h(bare) g Ft(\015)1600 715 y Fs(\026)1623 730 y Fj(,)225 780 y(mCP)f(and)g(CP\),)h(agrees)h(exactly)f(with)g(the)g(leading)f(term)g (predicted)j(b)o(y)d(OPE)i([12)p (#cite.manohar90) [[442 530 452 542] [1 1 1 [3 3]] [0 0 1]] pdfm (].) 300 830 y(The)c(asymptotic)f(b)q(eha)o(vior)g Fr(/)i Fj(1)p Ft(=Q)878 815 y Fq(2)907 830 y Fj(obtained)e(in)h(the)g(presen)o (t)h(approac)o(h)f(is)g(esp)q(e-)225 880 y(cially)h(satisfying)g(when)i (compared)f(with)g(the)h(one)f(kno)o(wn)g(to)g(result)h(from)e(the)h (simple)225 930 y(constituen)o(t)h(quark)g(mo)q(del)e(\(with)h(the)h (constan)o(t)g(ligh)o(t{quark)e(mass)h(parameter)g Ft(m)1585 936 y Fs(u)1607 930 y Fj(\),)225 979 y(where)i Ft(T)369 986 y Fs(\031)389 979 y Fi(0)5 b Fj(\()p Fr(\000)p Ft(Q)489 964 y Fq(2)508 979 y Ft(;)i Fj(0\))k Fr(/)i Fj(\()p Ft(m)672 964 y Fq(2)672 990 y Fs(u)694 979 y Ft(=Q)748 964 y Fq(2)766 979 y Fj(\))7 b(ln)824 961 y Fq(2)842 979 y Fj(\()p Ft(Q)891 964 y Fq(2)910 979 y Ft(=m)967 964 y Fq(2)967 990 y Fs(u)989 979 y Fj(\))14 b(as)g Ft(Q)1103 964 y Fq(2)1134 979 y Fr(!)d(1)p Fj(,)j(whic)o(h)g(o)o(v)o(ersho)q(ots)g(the)225 1029 y(data)g(considerably)g(b)q(ecause)h(of)e(the)i(additional)d(ln) 1062 1011 y Fq(2)1080 1029 y Fj(\()p Ft(Q)1129 1014 y Fq(2)1148 1029 y Fj(\)-dep)q(endence.)300 1079 y(It)21 b(is)h(imp)q(ortan)o(t)d(to)i(note)h(that)g(this)f(asymptotic)f(b)q (eha)o(vior,)j Ft(T)1385 1086 y Fs(\031)1405 1078 y Fi(0)1424 1079 y Fj(\()p Fr(\000)p Ft(Q)1505 1064 y Fq(2)1523 1079 y Ft(;)7 b Fj(0\))24 b(=)225 1129 y(\(4)p Ft(=)p Fj(3\))p Ft(f)340 1135 y Fs(\031)362 1129 y Ft(=Q)416 1114 y Fq(2)435 1129 y Fj(,)15 b(w)o(as)h(obtained)f(for)h(the)g Ft(q)q(q)q(\015)j Fj(v)o(ertices)e(whic)o(h)f(reduce)h(to)f(the)g(bare)g(one,)225 1179 y Ft(\015)248 1164 y Fs(\026)271 1179 y Fj(,)c(as)g(so)q(on)g(as)g Fg(one)h Fj(of)e(the)i(squared)f(momen)o(ta)e(tends)j(to)e(in\014nit)o (y)m(,)g(suc)o(h)i(as)f(the)g(renor-)225 1229 y(malizable)17 b(CP)j(and)g(mCP)f(v)o(ertices.)37 b(Subsequen)o(tly)m(,)21 b(Ref.)e([18)p (#cite.RobertsDubr) [[372 422 382 434] [1 1 1 [3 3]] [0 0 1]] pdfm (]) g(sho)o(w)o(ed)h(that)g(this)225 1278 y(asymptotic)13 b(b)q(eha)o(vior)g(m)o(ust)g(in)g(fact)h(hold)f(for)h Fg(any)g Fj(renormalizable)f Ft(q)q(q)q(\015)j Fj(v)o(ertex.)j(F)m(or) 225 1328 y(the)13 b(simplest)e(dressed)k(WTI-preserving)e(v)o(ertex,)g (the)g(BC)g(one,)f(whic)o(h)h(reduces)h(to)e(the)225 1378 y(bare)j(one)g(only)f(when)h(mom)o(en)o(ta)d(are)j(large)f(in)g(b) q(oth)h(of)f(its)h(fermion)d(legs,)j(and)f(whic)o(h)225 1428 y(is)f(not)g(consisten)o(t)h(with)f(m)o(ultiplicativ)n(e)e (renormalizabilit)o(y)f([17)p (#cite.CP90) [[364 374 374 386] [1 1 1 [3 3]] [0 0 1]] pdfm (],)h(the) j(asymptotic)d(b)q(e-)225 1483 y(ha)o(vior)g(is)g(\(4)p Ft(=)p Fj(3\))491 1472 y Fd(e)484 1483 y Ft(f)504 1489 y Fs(\031)526 1483 y Ft(=Q)580 1468 y Fq(2)599 1483 y Fj(,)g(where)i(the)f(quan)o(tit)o(y)980 1472 y Fd(e)973 1483 y Ft(f)993 1489 y Fs(\031)1027 1483 y Fj(is)f(giv)o(en)g(b)o(y)h (the)g(same)e(Mandelstam-)225 1533 y(formalism)g(expression)15 b(as)f Ft(f)689 1539 y Fs(\031)726 1533 y Fj(except)h(that)f(its)g(in)o (tegrand)g(is)f(mo)q(di\014ed)g(b)o(y)g(the)i(factor)225 1583 y([1)6 b(+)g Ft(A)p Fj(\()p Ft(q)369 1568 y Fq(2)388 1583 y Fj(\)])416 1568 y Fq(2)434 1583 y Ft(=)p Fj(4)12 b([3)p (#cite.KeKl3) [[192 337 197 349] [1 1 1 [3 3]] [0 0 1]] pdfm (].)k(F)m(or)c(our)g(mo)q(del)f(solutions)h([4)p (#cite.KeBiKl98) [[315 337 320 349] [1 1 1 [3 3]] [0 0 1]] pdfm (],) 1076 1572 y Fd(e)1068 1583 y Ft(f)1088 1589 y Fs(\031)1122 1583 y Fr(\031)g Fj(1)p Ft(:)p Fj(334)p Ft(f)1282 1589 y Fs(\031)1304 1583 y Fj(,)g(so)g(that)h(the)g(usage)225 1633 y(of)d(the)i(BC)f(v)o(ertices)i(causes)f(the)g(increase)g(of)f Fr(J)18 b Fj(in)11 b(Eq.)17 b(2)p (#equation.2) [[341 325 346 337] [1 1 1 [3 3]] [0 0 1]] pdfm 11 w(from)9 b Fr(J)20 b Fj(=)11 b(4)p Ft(=)p Fj(3)g(to)g Fr(J)19 b(\031)12 b Fj(1)p Ft(:)p Fj(78.)300 1682 y(Of)i(course,)i(the) f(mo)q(del-indep)q(enden)o(t)f(asymptotic)f(co)q(e\016cien)o(t)j(4)p Ft(f)1380 1688 y Fs(\031)1402 1682 y Ft(=)p Fj(3)e(is)h(the)g(one)225 1732 y(ha)o(ving)e(the)h(more)f(fundamen)o(tal)f(meaning,)f(resulting)j (from)e(the)j(renormalizable)d Ft(q)q(q)q(\015)225 1782 y Fj(v)o(ertices)21 b(suc)o(h)f(as)g(the)g(CP)g(or)g(mCP)e(ones,)j (whic)o(h)f(ha)o(v)o(e)f(prop)q(erties)i(closer)g(to)e(the)225 1832 y(true)c(v)o(ertex)g(solution.)j(Also)13 b(indicativ)o(e)h(is)g (the)g(asymptotics)f(found)h([3)p (#cite.KeKl3) [[400 277 405 289] [1 1 1 [3 3]] [0 0 1]] pdfm (])f(for) g(suc)o(h)i Ft(q)q(q)q(\015)225 1882 y Fj(v)o(ertices)f(when)f(b)q(oth) g(photons)g(are)g(o\013-shell,)f Ft(k)989 1867 y Fq(2)1019 1882 y Fj(=)g Fr(\000)p Ft(Q)1128 1867 y Fq(2)1159 1882 y Ft(<<)g Fj(0)g(and)h Ft(k)1371 1867 y Fh(0)p Fq(2)1410 1882 y Fj(=)f Fr(\000)p Ft(Q)1519 1867 y Fh(0)p Fq(2)1559 1882 y Fr(\024)g Fj(0:)641 1983 y Ft(T)665 1990 y Fs(\031)685 1982 y Fi(0)704 1983 y Fj(\()p Fr(\000)p Ft(Q)785 1966 y Fq(2)804 1983 y Ft(;)7 b Fr(\000)p Ft(Q)888 1966 y Fh(0)p Fq(2)916 1983 y Fj(\))12 b(=)992 1955 y(4)p 992 1974 21 2 v 992 2012 a(3)1091 1955 y Ft(f)1111 1961 y Fs(\031)p 1030 1974 166 2 v 1030 2015 a Ft(Q)1063 2003 y Fq(2)1091 2015 y Fj(+)d Ft(Q)1165 2003 y Fh(0)1177 1997 y Fq(2)1207 1983 y Ft(:)363 b Fj(\(6\))225 2091 y(The)18 b(usage)g(of)f(the)i(BC)f(v)o(ertex)g(\(3)p (#equation.3) [[259 215 264 227] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(w)o (ould)f(again)f(mo)q(dify)g(this)h(result)i(b)o(y)e(the)i(sub-)225 2147 y(stitution)h Ft(f)420 2153 y Fs(\031)464 2147 y Fr(!)535 2136 y Fd(e)528 2147 y Ft(f)548 2153 y Fs(\031)570 2147 y Fj(.)37 b(Ho)o(w)o(ev)o(er,)21 b(Eq.)f(\(6)p (#equation.6) [[289 202 294 214] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (agrees)h(with)f(the)g(leading)g(term)f(of)g(the)225 2196 y(OPE)g(result)g(of)f(No)o(vik)o(o)o(v)e Fg(et)j(al.)f Fj([19)p (#cite.novikov+al84) [[263 190 273 202] [1 1 1 [3 3]] [0 0 1]] pdfm (])f(for)h(the)g(sp)q(ecial)h(case)g Ft(Q)1274 2181 y Fq(2)1311 2196 y Fj(=)h Ft(Q)1396 2181 y Fh(0)p Fq(2)1424 2196 y Fj(.)31 b(Also,)18 b(the)225 2246 y(distribution-amplitude-dep)q (endence)12 b(of)f(the)h(pQCD)g(approac)o(h)f(cancels)i(out)f(for)f (that)225 2296 y(symmetric)16 b(case,)j(so)e(that)h Ft(T)707 2303 y Fs(\031)727 2295 y Fi(0)746 2296 y Fj(\()p Fr(\000)p Ft(Q)827 2281 y Fq(2)845 2296 y Ft(;)7 b Fr(\000)p Ft(Q)929 2281 y Fh(0)p Fq(2)957 2296 y Fj(\))18 b(in)f(this)g(approac)o(h)h(\()p Fg(e.g.)p Fj(,)f(see)i([20)p (#cite.Kess+Ong93) [[432 166 443 178] [1 1 1 [3 3]] [0 0 1]] pdfm (]\),)d(in)225 2350 y(the)f(limit)d Ft(Q)430 2335 y Fq(2)461 2350 y Fj(=)h Ft(Q)539 2335 y Fh(0)550 2329 y Fq(2)581 2350 y Fr(!)f(1)p Fj(,)i(exactly)h(agrees)g(with)f(b)q(oth)h(our)f(Eq.) 20 b(\(6)p (#equation.6) [[393 153 398 165] [1 1 1 [3 3]] [0 0 1]] pdfm (\))15 b(and)f(Ref.)g([19)p (#cite.novikov+al84) [[449 153 459 165] [1 1 1 [3 3]] [0 0 1]] pdfm (].)225 2400 y(F)m(or)i(that)g(symmetric)f(case,)i(w)o(e)f(should)g(th) o(us)h(ha)o(v)o(e)f(ev)o(en)h(the)g(precise)h(agreemen)o(t)e(of)920 2490 y(5)p eop %%Page: 6 6 6 5 bop 225 432 a Fj(the)13 b(co)q(e\016cien)o(ts)g(irresp)q(ectiv)o(e) g(of)f(the)g(description)h(of)e(the)i(pion)e(in)o(ternal)h(structure)i (en-)225 481 y(co)q(ded)g(in)e(the)i(distribution)e(amplitude.)k(Ob)o (viously)m(,)11 b(this)i(fa)o(v)o(ors)f(the)i(renormalizable)225 531 y Ft(q)q(q)q(\015)20 b Fj(v)o(ertices,)f(suc)o(h)e(as)h(CP)f(and)g (mCP)f(ones,)i(o)o(v)o(er)f(the)h(BC)f(v)o(ertex.)28 b(Ho)o(w)o(ev)o(er,)18 b(the)225 581 y(BC)g(v)o(ertex)g(ma)o(y)e(an)o (yw)o(a)o(y)g(b)q(e)j(the)f(one)g(whic)o(h)f(is)g(more)g(accurate)i (not)e(only)g(for)g(the)225 631 y(presen)o(tly)e(accessible)g Ft(Q)623 616 y Fq(2)642 631 y Fj(,)e(but)h(also)f(for)h(larger)g(v)n (alues)f(b)q(efore)i(starting)f(to)g(fail.)339 1834 y @beginspecial 0 @llx 0 @lly 510 @urx 510 @ury 2834 @rwi @setspecial %%BeginDocument: FormF.fig2.eps 80 dict begin /s {stroke} def /l {lineto} def /m {moveto} def /t {translate} def /sw {stringwidth} def /r {rotate} def /rl {roll} def /R {repeat} def /d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def /c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def /cl {closepath} def /sf {scalefont setfont} def /black {0 setgray} def /box {m dup 0 exch d exch 0 d 0 exch neg d cl} def /NC{systemdict begin initclip end}def/C{NC box clip newpath}def /bl {box s} def /bf {box f} def /Y { 0 exch d} def /X { 0 d} def /mp {newpath /y exch def /x exch def} def /side {[w .77 mul w .23 mul] .385 w mul sd w 0 l currentpoint t -144 r} def /mr {mp x y w2 0 360 arc} def /m24 {mr s} def /m20 {mr f} def /mb {mp x y w2 add m w2 neg 0 d 0 w neg d w 0 d 0 w d cl} def /mt {mp x y 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16 ] { < BB BB EE EE BB BB EE EE BB BB EE EE BB BB EE EE BB BB EE EE BB BB EE EE BB BB EE EE BB BB EE EE > } image } if PT 3 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FF FF BB BB FF FF EE EE FF FF BB BB FF FF EE EE FF FF BB BB FF FF EE EE FF FF BB BB FF FF EE EE > } image } if PT 4 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < DF DF BF BF 7F 7F FE FE FD FD FB FB F7 F7 EF EF DF DF BF BF 7F 7F FE FE FD FD FB FB F7 F7 EF EF > } image } if PT 5 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 7F 7F BF B F DF DF EF EF F7 F7 FB FB FD FD FE FE 7F 7F BF BF DF DF EF EF F7 F7 FB FB FD FD FE FE > } image } if PT 6 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB > } image } if PT 7 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FF FF FF FF FF FF 00 00 FF FF FF FF FF FF 00 00 FF FF FF FF FF FF 00 00 FF FF FF FF FF FF 00 00 > } image } if PT 8 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < EE EE 47 47 83 83 C5 C5 EE EE 5C 5C 38 38 74 74 EE EE 47 47 83 83 C5 C5 EE EE 5C 5C 38 38 74 74 > } image } if PT 9 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < EF EF EF EF D7 D7 38 38 FE FE FE FE 7D 7D 83 83 EF EF EF EF D7 D7 38 38 FE FE FE FE 7D 7D 83 83 > } image } if PT 10 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < EF EF EF EF EF EF 00 00 FE FE FE FE FE FE 00 00 EF EF EF EF EF EF 00 00 FE FE FE FE FE FE 00 00 > } image } if PT 11 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < F7 F7 B6 B6 D5 D5 E3 E3 D5 D5 B6 B6 F7 F7 FF FF 7F 7F 6B 6B 5D 5D 3E 3E 5D 5D 6B 6B 7F 7F FF FF > } image } if PT 12 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < E3 E3 DD DD BE BE BE BE BE BE DD DD E3 E3 FF FF 3E 3E DD DD EB EB EB EB EB EB DD DD 3E 3E FF FF > } image } if PT 13 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FE FE 7D 7D BB BB D7 D7 EF EF D7 D7 BB BB 7D 7D FE FE 7D 7D BB BB D7 D7 EF EF D7 D7 BB BB 7D 7D > } image } if PT 14 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 00 00 EE EF EE EF EE EF 0E E0 EE EE EE EE EE EE 00 EE FE EE FE EE FE EE 00 00 FE EF FE EF FE EF > } image } if PT 15 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < DD DD AA AA DD DD FF FF 77 77 AA AA 77 77 FF FF DD DD AA AA DD DD FF FF 77 77 AA AA 77 77 FF FF > } image } if PT 16 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF F1 F1 EE EE 1F 1F FF FF > } image } if PT 17 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF EE EE DD DD BB BB FF FF > } image } if PT 18 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF BB BB DD DD EE EE FF FF > } image } if PT 19 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 1F FC 67 F3 7B EF BD DE BD DE DE BD E6 B3 F8 0F E6 B3 DE BD BD DE BD DE 7B EF 67 F3 1F FC 7F FF > } image } if PT 20 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < DD DD EE EE EE EE EE EE DD DD BB BB BB BB BB BB DD DD EE EE EE EE EE EE DD DD BB BB BB BB BB BB > } image } if PT 21 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 0E 0E EF EF EF EF EF EF E0 E0 FE FE FE FE FE FE 0E 0E EF EF EF EF EF EF E0 E0 FE FE FE FE FE FE > } image } if PT 22 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < 70 70 F7 F7 F7 F7 F7 F7 07 07 7F 7F 7F 7F 7F 7F 70 70 F7 F7 F7 F7 F7 F7 07 07 7F 7F 7F 7F 7F 7F > } image } if PT 23 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < AA AA 55 55 A9 A9 D1 D1 E1 E1 D1 D1 A9 A9 55 55 AA AA 55 55 A9 A9 D1 D1 E1 E1 D1 D1 A9 A9 55 55 > } image } if PT 24 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FF FE FF FC EA A8 D5 54 EA A8 D5 54 E8 28 D4 54 E8 E8 D4 D4 E8 EA 54 D5 A8 EA 54 D5 00 C0 00 80 > } image } if PT 25 eq { 16 16 1 [ 16 0 0 16 neg 0 16 ] { < FF FE FF FC FF F8 FF F0 F0 00 F0 00 F0 20 F0 60 F0 E0 F1 E0 F3 E0 F0 00 E0 00 C0 00 80 00 00 00 > } image } if gr } def /FA { /PT exch def gsave clip 0 0 translate 1 1 54 { 1 sub 47.2 mul /Xcurr exch def 1 1 74 { 1 sub 47.2 mul /Ycurr exch def gsave Xcurr Ycurr translate PT DP gr } for } for gr newpath } def /reencdict 24 dict def /ReEncode {reencdict begin /nco&na exch def /nfnam exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict {exch dup /FID ne {dup /Encoding eq {exch dup length array copy newfont 3 1 roll put} {exch newfont 3 1 roll put} ifelse} {pop pop} ifelse } forall newfont /FontName nfnam put nco&na aload pop nco&na length 2 idiv {newfont /Encoding get 3 1 roll put} repeat nfnam newfont definefont pop end } def /accvec [ 176 /agrave 181 /Agrave 190 /acircumflex 192 /Acircumflex 201 /adieresis 204 /Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute 212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex 217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex 221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex 228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236 /Ucircumflex 237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis 244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def /Times-Roman /Times-Roman accvec ReEncode /Times-Italic /Times-Italic accvec ReEncode /Times-Bold /Times-Bold accvec ReEncode /Times-BoldItalic /Times-BoldItalic accvec ReEncode /Helvetica /Helvetica accvec ReEncode /Helvetica-Oblique /Helvetica-Oblique accvec ReEncode /Helvetica-Bold /Helvetica-Bold accvec ReEncode /Helvetica-BoldOblique /Helvetica-BoldOblique accvec ReEncode /Courier /Courier accvec ReEncode /Courier-Oblique /Courier-Oblique accvec ReEncode /Courier-Bold /Courier-Bold accvec ReEncode /Courier-BoldOblique /Courier-BoldOblique accvec ReEncode /oshow {gsave [] 0 sd true charpath stroke gr} def /stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf text sw pop xs add /xs exch def} def /stwb { /fs exch def /fn exch def /nbas exch def /textf exch def textf length /tlen exch def nbas tlen gt {/nbas tlen def} if fn findfont fs sf textf dup length nbas sub nbas getinterval sw pop neg xs add /xs exch def} def /accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal 69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral 74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright 79 /greaterequal 80 /braceleft 81 /braceright 82 /radical 83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal 88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus 50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup 55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110 /notelement 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def /Symbol /Special accspe ReEncode gsave .25 .25 scale gsave 0 0 t black [] 0 sd 1 lw 2040 2040 0 0 C /xs 0 def (Q) /Times-Italic 69 stwn (2) /Times-Italic 48 stwn (\040) /Times-Italic 69 stwn (M) /Special 69 stwn (GeV) /Times-Italic 69 stwn (2) /Times-Italic 48 stwn (N) /Special 69 stwn gsave 1927 85 t 0 r xs neg 0 t 0 0 m /Times-Italic findfont 69 sf 0 0 m (Q) show currentpoint pop 0 t /Times-Italic findfont 48 sf 0 35 m (2) show currentpoint pop 0 t /Times-Italic findfont 69 sf 0 0 m (\040) show currentpoint pop 0 t /Special findfont 69 sf 0 0 m (M) show currentpoint pop 0 t /Times-Italic findfont 69 sf 0 0 m (GeV) show currentpoint pop 0 t /Times-Italic findfont 48 sf 0 35 m (2) show currentpoint pop 0 t /Special findfont 69 sf 0 0 m (N) show gr /xs 0 def (Q) /Times-Italic 69 stwn (2) /Times-Italic 48 stwn (T\(-Q) /Times-Italic 69 stwn (2) /Times-Italic 48 stwn (,0\)\040) /Times-Italic 69 stwn (M) /Special 69 stwn (GeV) /Times-Italic 69 stwn (N) /Special 69 stwn gsave 102 1927 t 90 r xs neg 0 t 0 0 m /Times-Italic findfont 69 sf 0 0 m (Q) show currentpoint pop 0 t /Times-Italic findfont 48 sf 0 35 m (2) show currentpoint pop 0 t /Times-Italic findfont 69 sf 0 0 m (T\(-Q) show currentpoint pop 0 t /Times-Italic findfont 48 sf 0 35 m (2) show currentpoint pop 0 t /Times-Italic findfont 69 sf 0 0 m (,0\)\040) show currentpoint pop 0 t /Special findfont 69 sf 0 0 m (M) show currentpoint pop 0 t /Times-Italic findfont 69 sf 0 0 m (GeV) show currentpoint pop 0 t /Special findfont 69 sf 0 0 m (N) show gr 5 lw /w 32 def /w2 {w 2 div} def /w3 {w 3 div} def 1262 652 m24 gsave 1326 632 t 0 r 0 0 m /Times-Roman findfont 60 sf 0 0 m (CELLO) show gr 1262 567 m26 gsave 1326 547 t 0 r 0 0 m /Times-Roman findfont 60 sf 0 0 m (CLEO) show gr 1632 1700 295 227 C NC 1632 1700 295 227 bl 1632 1700 295 227 C NC 1 lw 295 227 m 1700 Y s 5 lw 335 227 m -40 X s 315 269 m -20 X s 315 312 m -20 X s 315 354 m -20 X s 335 397 m -40 X s 315 439 m -20 X s 315 482 m -20 X s 315 524 m -20 X s 335 567 m -40 X s 315 609 m -20 X s 315 652 m -20 X s 315 694 m -20 X s 335 737 m -40 X s 315 779 m -20 X s 315 822 m -20 X s 315 864 m -20 X s 335 907 m -40 X s 315 949 m -20 X s 315 992 m -20 X s 315 1034 m -20 X s 335 1077 m -40 X s 315 1119 m -20 X s 315 1162 m -20 X s 315 1204 m -20 X s 335 1247 m -40 X s 315 1289 m -20 X s 315 1332 m -20 X s 315 1374 m -20 X s 335 1417 m -40 X s 315 1459 m -20 X s 315 1502 m -20 X s 315 1544 m -20 X s 335 1587 m -40 X s 315 1629 m -20 X s 315 1672 m -20 X s 315 1714 m -20 X s 335 1757 m -40 X s 315 1799 m -20 X s 315 1842 m -20 X s 315 1884 m -20 X s 335 1927 m -40 X s /xs 0 def (0) /Times-Roman 68 stwn gsave 249 204 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0) show gr /xs 0 def (0.02) /Times-Roman 68 stwn gsave 249 374 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.02) show gr /xs 0 def (0.04) /Times-Roman 68 stwn gsave 249 544 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.04) show gr /xs 0 def (0.06) /Times-Roman 68 stwn gsave 249 714 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.06) show gr /xs 0 def (0.08) /Times-Roman 68 stwn gsave 249 884 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.08) show gr /xs 0 def (0.1) /Times-Roman 68 stwn gsave 249 1054 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.1) show gr /xs 0 def (0.12) /Times-Roman 68 stwn gsave 249 1224 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.12) show gr /xs 0 def (0.14) /Times-Roman 68 stwn gsave 249 1394 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.14) show gr /xs 0 def (0.16) /Times-Roman 68 stwn gsave 249 1564 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.16) show gr /xs 0 def (0.18) /Times-Roman 68 stwn gsave 249 1734 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.18) show gr /xs 0 def (0.2) /Times-Roman 68 stwn gsave 249 1904 t 0 r xs neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0.2) show gr 1 lw 295 227 m 1632 X s 5 lw 314 267 m -40 Y s 352 247 m -20 Y s 390 247 m -20 Y s 428 247 m -20 Y s 465 247 m -20 Y s 503 267 m -40 Y s 541 247 m -20 Y s 579 247 m -20 Y s 617 247 m -20 Y s 655 247 m -20 Y s 693 267 m -40 Y s 731 247 m -20 Y s 769 247 m -20 Y s 807 247 m -20 Y s 845 247 m -20 Y s 883 267 m -40 Y s 921 247 m -20 Y s 959 247 m -20 Y s 997 247 m -20 Y s 1035 247 m -20 Y s 1073 267 m -40 Y s 1111 247 m -20 Y s 1149 247 m -20 Y s 1187 247 m -20 Y s 1225 247 m -20 Y s 1262 267 m -40 Y s 1300 247 m -20 Y s 1338 247 m -20 Y s 1376 247 m -20 Y s 1414 247 m -20 Y s 1452 267 m -40 Y s 1490 247 m -20 Y s 1528 247 m -20 Y s 1566 247 m -20 Y s 1604 247 m -20 Y s 1642 267 m -40 Y s 1680 247 m -20 Y s 1718 247 m -20 Y s 1756 247 m -20 Y s 1794 247 m -20 Y s 1832 267 m -40 Y s 314 267 m -40 Y s 1832 267 m -40 Y s 1870 247 m -20 Y s 1908 247 m -20 Y s /xs 0 def (0) /Times-Roman 68 stwn gsave 314 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (0) show gr /xs 0 def (1) /Times-Roman 68 stwn gsave 503 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (1) show gr /xs 0 def (2) /Times-Roman 68 stwn gsave 693 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (2) show gr /xs 0 def (3) /Times-Roman 68 stwn gsave 883 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (3) show gr /xs 0 def (4) /Times-Roman 68 stwn gsave 1073 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (4) show gr /xs 0 def (5) /Times-Roman 68 stwn gsave 1262 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (5) show gr /xs 0 def (6) /Times-Roman 68 stwn gsave 1452 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (6) show gr /xs 0 def (7) /Times-Roman 68 stwn gsave 1642 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (7) show gr /xs 0 def (8) /Times-Roman 68 stwn gsave 1832 159 t 0 r xs 2 div neg 0 t 0 0 m /Times-Roman findfont 68 sf 0 0 m (8) show gr 1632 1700 295 227 C 443 879 m 44 Y s 443 954 m 44 Y s 409 939 m 18 X s 459 939 m 6 X s 443 939 m24 492 973 m 53 Y s 492 1058 m 52 Y s 465 1042 m 11 X s 508 1042 m 14 X s 492 1042 m24 553 948 m 74 Y s 553 1054 m 73 Y s 522 1038 m 15 X s 569 1038 m 29 X s 553 1038 m24 636 1126 m 112 Y s 636 1270 m 113 Y s 598 1254 m 22 X s 652 1254 m 60 X s 636 1254 m24 725 1351 m 257 Y s 725 1640 m 256 Y s 741 1624 m 85 X s 725 1624 m24 625 1183 m 56 Y s 625 1271 m 57 Y s 598 1255 m 11 X s 641 1255 m 14 X s 625 1255 m26 674 1156 m 49 Y s 674 1237 m 49 Y s 655 1221 m 3 X s 690 1221 m 3 X s 674 1221 m26 712 1327 m 57 Y s 712 1416 m 56 Y s 693 1400 m 3 X s 728 1400 m 3 X s 712 1400 m26 750 1226 m 64 Y s 750 1322 m 65 Y s 731 1306 m 3 X s 766 1306 m 3 X s 750 1306 m26 788 1285 m 73 Y s 788 1390 m 73 Y s 769 1374 m 3 X s 804 1374 m 3 X s 788 1374 m26 826 1411 m 83 Y s 826 1526 m 84 Y s 807 1510 m 3 X s 842 1510 m 3 X s 826 1510 m26 872 1286 m 89 Y s 872 1407 m 89 Y s 845 1391 m 11 X s 887 1391 m 15 X s 872 1391 m26 938 1352 m 91 Y s 938 1475 m 92 Y s 902 1459 m 20 X s 954 1459 m 24 X s 938 1459 m26 1023 1227 m 106 Y s 1023 1365 m 105 Y s 978 1349 m 30 X s 1039 1349 m 34 X s 1023 1349 m26 1118 1236 m 114 Y s 1118 1382 m 114 Y s 1073 1366 m 29 X s 1134 1366 m 34 X s 1118 1366 m26 1213 1387 m 133 Y s 1213 1552 m 132 Y s 1168 1536 m 29 X s 1229 1536 m 33 X s 1213 1536 m26 1308 1302 m 141 Y s 1308 1475 m 141 Y s 1262 1459 m 30 X s 1324 1459 m 33 X s 1308 1459 m26 1403 1354 m 174 Y s 1403 1560 m 174 Y s 1357 1544 m 30 X s 1419 1544 m 33 X s 1403 1544 m26 1541 1311 m 158 Y s 1541 1501 m 157 Y s 1452 1485 m 74 X s 1557 1485 m 85 X s 1541 1485 m26 1813 1431 m 199 Y s 1813 1662 m 199 Y s 1642 1646 m 155 X s 1829 1646 m 98 X s 1813 1646 m26 314 227 m 19 201 d 19 172 d 19 146 d 19 123 d 19 103 d 19 87 d 18 73 d 19 63 d 19 54 d 19 46 d 19 39 d 19 34 d 19 30 d 19 26 d 19 22 d 19 20 d 19 18 d 19 15 d 19 13 d 19 12 d 19 10 d 19 9 d 19 9 d 19 7 d 38 12 d 38 9 d 38 8 d 38 6 d 38 5 d 38 3 d 38 3 d 38 3 d 47 2 d 48 2 d 47 1 d 47 1 d 48 X 47 1 d 95 X 48 1 d 284 X s 0 0 1 c /w 2 def /w2 {w 2 div} def /w3 {w 3 div} def 314 227 333 428 352 600 371 746 390 869 409 972 428 1059 446 1132 465 1195 484 1249 503 1295 522 1334 541 1368 560 1398 579 1424 598 1446 617 1466 636 1484 655 1499 674 1512 693 1524 712 1534 731 1543 750 1552 769 1559 807 1571 845 1580 883 1588 921 1594 959 1599 997 1602 1035 1605 1073 1608 1120 1610 1168 1612 1215 1613 1262 1614 1310 1614 1357 1615 1405 1615 1452 1615 1500 1616 1547 1616 1595 1616 1642 1616 1689 1616 1737 1616 1784 1616 48 { m20} R [12 12] 0 sd black 333 493 m 19 147 d 19 139 d 19 120 d 19 102 d 19 87 d 18 74 d 19 63 d 19 54 d 19 46 d 19 40 d 19 34 d 19 29 d 19 25 d 19 21 d 19 19 d 19 16 d 19 13 d 19 11 d 19 10 d 19 8 d 19 7 d 19 6 d 19 4 d 38 7 d 38 4 d 38 2 d 38 X 38 -2 d 38 -2 d 38 -3 d 38 -3 d 47 -5 d 48 -5 d 47 -6 d 47 -6 d 48 -5 d 47 -6 d 48 -6 d 47 -5 d 48 -5 d 47 -5 d 48 -5 d 47 -5 d 47 -5 d 48 -4 d 47 -5 d 48 -4 d 379 -28 d s [] 0 sd /w 18 def /w2 {w 2 div} def /w3 {w 3 div} def 333 428 352 527 371 629 390 721 409 803 428 875 446 937 465 992 484 1040 503 1081 522 1117 541 1148 560 1175 579 1199 598 1220 617 1238 636 1253 655 1267 674 1278 693 1288 712 1297 731 1305 750 1311 769 1317 807 1326 845 1332 883 1337 921 1340 959 1342 997 1343 1035 1343 1073 1343 1120 1342 1168 1341 1215 1340 1262 1338 1310 1337 1357 1335 1405 1333 1452 1331 1500 1330 1547 1328 1595 1326 1642 1325 1689 1323 1737 1322 1784 1320 1832 1319 2211 1310 49 { m2} R [12 12] 0 sd 314 1274 m 1613 X s gr gr showpage end %%EndDocument @endspecial 300 1894 a Fp(Figure)g(2:)j(Our)c(n)o(umerically)j (obtained)f Fc(Q)942 1878 y Fl(2)960 1894 y Fc(T)982 1902 y Fk(\031)1001 1895 y Fi(0)1019 1894 y Fp(\()p Fb(\000)p Fc(Q)1094 1878 y Fl(2)1112 1894 y Fc(;)6 b Fp(0\))13 b(and)g(the)g(data.)300 2001 y Fj(F)m(or)j(\014nite)g Ft(Q)518 1986 y Fq(2)537 2001 y Fj(,)g(the)h(transition)e(form)g (factor)h(is)g(ev)n(aluated)g(n)o(umerically)m(.)22 b(Fig.)15 b(2)225 2051 y(compares)10 b(so)g(obtained)g Ft(Q)651 2036 y Fq(2)670 2051 y Ft(T)694 2058 y Fs(\031)714 2050 y Fi(0)733 2051 y Fj(\()p Fr(\000)p Ft(Q)814 2036 y Fq(2)833 2051 y Ft(;)d Fj(0\))i(with)h(the)h(CLEO)g([1)p (#cite.gronberg98) [[359 225 364 237] [1 1 1 [3 3]] [0 0 1]] pdfm (])e (and)h(CELLO)h([21)p (#cite.behrend91) [[427 225 437 237] [1 1 1 [3 3]] [0 0 1]] pdfm (])e (data.)225 2101 y(The)16 b(solid)f(curv)o(e)h(is)f(obtained)h(with)f (the)h(BC)g(v)o(ertices,)g(and)g(the)g(dashed)g(curv)o(e)g(with)225 2151 y(the)11 b(mCP)f(ones.)17 b(The)11 b(horizon)o(tal)f(dashed)h (line)f(denotes)i(4)p Ft(f)1176 2157 y Fs(\031)1199 2151 y Ft(=)p Fj(3,)e(the)h(asymptotic)f(v)n(alue)225 2201 y(for)17 b(renormalizable)f Ft(q)q(q)q(\015)k Fj(v)o(ertices)e(suc)o(h) g(as)f(the)h(mCP)e(one.)29 b(It)17 b(barely)g(touc)o(hes)h(the)225 2251 y(exp)q(erimen)o(tal)12 b(error)i(bars)f(from)e(b)q(elo)o(w.)17 b(Note)d(that)f(also)f(Manohar)g([12)p (#cite.manohar90) [[396 177 406 189] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (w)o(arned)h(that)225 2300 y(his)d(OPE)h(approac)o(h)f(\(also)g (yielding)f(4)p Ft(f)849 2306 y Fs(\031)872 2300 y Ft(=)p Fj(3\))h(indicates)g(the)h(p)q(ossibilit)o(y)e(of)h(large)g(correc-)225 2350 y(tions)h(to)h(his)g(leading)e(term,)h(but)h(p)q(ossibly)f(also)g (to)h(the)g(pQCD)g(asymptotic)e(co)q(e\016cien)o(t)225 2400 y(2)p Ft(f)266 2406 y Fs(\031)289 2400 y Fj(,)g(whic)o(h)h(is)f (ho)o(w)o(ev)o(er)h(exact)g(in)f(the)h Fg(strict)e Fj(ln\()p Ft(Q)1028 2385 y Fq(2)1047 2400 y Fj(\))i Fr(!)g(1)f Fj(limit)e([11)p (#cite.BrodskyLepage) [[381 141 391 153] [1 1 1 [3 3]] [0 0 1]] pdfm (].)15 b(W)m(e)10 b(get)h(it)f(lo)o(w)o(er)920 2490 y(6)p eop %%Page: 7 7 7 6 bop 225 432 a Fj(b)o(y)16 b(the)h(factor)g(2/3)e(b)q(ecause)k(our)d (presen)o(t)i(approac)o(h)e(do)q(es)i(not)e(incorp)q(orate)h(the)g(ef-) 225 481 y(fects)12 b(of)f(the)g(pQCD)g(ev)o(olution.)17 b(Ho)o(w)o(ev)o(er,)11 b(as)g(Ref.)g([2)p (#cite.Radyushkin+Rusk3) [[330 602 335 614] [1 1 1 [3 3]] [0 0 1]] pdfm (])e(seems)j(to)f(indicate)g(that)g(e\013ects)225 531 y(other)i(than)f(the)h(pQCD)g(ev)o(olution)e(\(including)h(the)h (presen)o(tly)g(in)o(teresting)g(dynamical)225 581 y(dressing)i(of)f (quarks\))g(ma)o(y)e(still)h(pla)o(y)h(imp)q(ortan)o(t)e(\(and)i(ma)o (yb)q(e)f(ev)o(en)i(dominan)o(t\))d(role)225 631 y(ev)o(en)f(at)g Ft(Q)397 616 y Fq(2)415 631 y Fj(-v)n(alues)f(larger)h(than)g(the)g (presen)o(tly)g(accessible)h(ones,)g(one)f(should)f(consider)225 681 y(seriously)17 b(also)f(nonp)q(erturbativ)o(e)i(approac)o(hes)g (suc)o(h)f(as)g(ours.)27 b(On)18 b(the)f(other)h(hand,)225 730 y(high-precision)c(measuremen)o(ts)g(of)g Ft(T)833 737 y Fs(\031)853 729 y Fi(0)872 730 y Fj(\()p Fr(\000)p Ft(Q)953 715 y Fq(2)971 730 y Ft(;)7 b Fj(0\))14 b(can)h(test)g(\(and)f (giv)o(e)g(a)g(hin)o(t)g(on)g(ho)o(w)225 780 y(to)g(impro)o(v)o(e\))e (the)i(SD)g(and)g(BS)g(mo)q(del)e(solutions)h(whic)o(h)h(ha)o(v)o(e)g (so)g(far)f(b)q(een)i(successful)225 830 y(in)g(\014tting)g(the)h(lo)o (w-energy)f(hadron)h(prop)q(erties)h(suc)o(h)f(as)f(the)h(meson)f(sp)q (ectrum.)23 b(F)m(or)225 880 y(example,)14 b(our)h(mo)q(del)f(c)o (hoice)h([14)p (#cite.jain93b) [[249 506 259 518] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (somewhat)g(o)o(v)o(ersho)q(ots)i(\(for)f(b)q(oth)h(BC)f(and)g(mCP)225 944 y(v)o(ertices\))g(the)g(presen)o(t)g Ft(T)631 951 y Fs(\031)651 943 y Fi(0)670 944 y Fj(\()p Fr(\000)p Ft(Q)751 929 y Fq(2)770 944 y Ft(;)7 b Fj(0\))13 b(data)h([1)p (#cite.gronberg98) [[299 490 304 502] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(21)p (#cite.behrend91) [[308 490 318 502] [1 1 1 [3 3]] [0 0 1]] pdfm (])12 b(in)h(the)i(region)f Ft(Q)1328 929 y Fq(2)1365 932 y Ft(<)1365 959 y Fr(\030)1416 944 y Fj(4)f(GeV)1532 929 y Fq(2)1551 944 y Fj(.)300 1017 y(Once)k(one)f(has)g(a)g(solution)f (for)h Ft(\037)p Fj(\()p Ft(q)q(;)7 b(p)p Fj(\))15 b(that)h(leads)g(to) g(the)g(correct)i(v)n(alue)d(of)h Ft(f)1601 1023 y Fs(\031)1623 1017 y Fj(,)225 1067 y(the)e(transition)f(form)e(factor)i(is)g(most)f (sensitiv)o(e)i(on)f Ft(A)p Fj(\()p Ft(q)1132 1052 y Fq(2)1151 1067 y Fj(\),)g(or,)f(more)h(precisely)m(,)g(on)g(its)225 1117 y(v)n(alues)d(at)g(small)e(and)i(in)o(termediate)f(momen)o(ta)f Fr(\000)p Ft(q)1041 1102 y Fq(2)1060 1117 y Fj(,)i(where)i Ft(A)p Fj(\()p Ft(q)1266 1102 y Fq(2)1284 1117 y Fj(\))f(is)f(still)f (appreciably)225 1167 y(di\013eren)o(t)17 b(from)d(1)i([3)p (#cite.KeKl3) [[202 437 207 449] [1 1 1 [3 3]] [0 0 1]] pdfm (,)7 b(22)p (#cite.KlKe4) [[211 437 221 449] [1 1 1 [3 3]] [0 0 1]] pdfm (].)21 b(T)m(o)15 b(illustrate)h(what)g(happ)q(ens)g(when)h(the)f Ft(A)p Fj(\()p Ft(q)1429 1152 y Fq(2)1448 1167 y Fj(\)-pro\014le)g(is) 225 1217 y(decreased,)i(let)e(us)g(enforce)h(b)o(y)e(hand)h(the)g (extreme,)g(arti\014cial)e(case)j Ft(A)p Fj(\()p Ft(q)1419 1201 y Fq(2)1438 1217 y Fj(\))e Fr(\021)f Fj(1.)24 b(\(T)m(o)225 1266 y(a)o(v)o(oid)18 b(confusion,)i(w)o(e)g(stress)h(it)e(is)h(for)f (illustrativ)o(e)f(purp)q(oses)j(only)m(,)f(as)f(w)o(e)h(cannot)225 1316 y(ha)o(v)o(e)14 b(suc)o(h)g(a)f(SD-solution)g(in)g(the)h(adopted)g (approac)o(h)g(of)f(Refs.)g([13)p (#cite.jain91+munczek92) [[382 401 392 413] [1 1 1 [3 3]] [0 0 1]] pdfm (,)5 b(14)p (#cite.jain93b) [[396 401 407 413] [1 1 1 [3 3]] [0 0 1]] pdfm (].\)) 16 b(This)e(leads)225 1366 y(to)20 b(the)h(curv)o(e)g(traced)g(on)f (Fig.)f(2)h(b)o(y)g(small)e(crosses,)23 b(p)q(ertaining)d(to)g(the)h (usage)g(of)225 1416 y(b)q(oth)16 b(BC)g(and)f(mCP)g(v)o(ertices)i (\(as)f(w)o(ell)f(as)h(the)g(CP)g(ones\),)h(since)f Ft(A)p Fj(\()p Ft(q)1391 1401 y Fq(2)1410 1416 y Fj(\))f Fr(\021)g Fj(1)g(mak)o(es)225 1466 y(\000)251 1446 y Fs(\026)251 1478 y(mC)r(P)353 1466 y Fr(!)k Fj(\000)440 1446 y Fs(\026)440 1478 y(B)q(C)495 1466 y Fj(.)32 b(This)18 b(curv)o(e)i(rev)o(eals)f(ho) o(w)f(the)h(heigh)o(ts)g(of)f(the)h(curv)o(es)h(depicting)225 1515 y Ft(Q)258 1500 y Fq(2)276 1515 y Ft(T)300 1522 y Fs(\031)320 1514 y Fi(0)339 1515 y Fj(\()p Fr(\000)p Ft(Q)420 1500 y Fq(2)439 1515 y Ft(;)7 b Fj(0\))i(dep)q(end)i(on)e(ho)o (w)h(m)o(uc)o(h)e(the)i Ft(A)p Fj(\()p Ft(q)1020 1500 y Fq(2)1039 1515 y Fj(\)-pro\014le)g(exceeds)h(1.)17 b(Ob)o(viously)m(,)8 b(for)225 1565 y(b)q(oth)j(the)h(solid)e(curv)o(e) i(and)f(the)g(dashed)h(one,)f(the)h(agreemen)o(t)e(with)h(exp)q(erimen) o(t)g(w)o(ould)225 1615 y(b)q(e)18 b(impro)o(v)o(ed)e(b)o(y)h(lo)o(w)o (ering)f(them)h(somewhat)f(\(at)h(least)h(in)f(the)h(momen)o(tum)13 b(region)225 1679 y Ft(Q)258 1664 y Fq(2)295 1667 y Ft(<)295 1694 y Fr(\030)346 1679 y Fj(4)e(GeV\),)g(whic)o(h)g(could)h(b)q(e)g (ac)o(hiev)o(ed)f(b)o(y)g(mo)q(difying)d(the)k(mo)q(del)e([14)p (#cite.jain93b) [[406 314 416 326] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (and/or)h(its)225 1729 y(parameters)h(so)g(that)g(suc)o(h)h(a)f(new)g (solution)f(for)h Ft(A)p Fj(\()p Ft(q)1067 1714 y Fq(2)1086 1729 y Fj(\))g(is)g(somewhat)f(lo)o(w)o(ered)h(to)o(w)o(ards)225 1779 y(its)k(asymptotic)e(v)n(alue)h Ft(A)p Fj(\()p Ft(q)676 1764 y Fq(2)710 1779 y Fr(!)f(1)p Fj(\))g Fr(!)g Fj(1.)23 b(\(Of)16 b(course,)h(in)f(order)g(to)g(b)q(e)g(signi\014can)o(t,)225 1829 y(this)11 b(m)o(ust)e(not)i(b)q(e)g(a)f(sp)q(ecialized)i (re-\014tting)f(aimed)e Fg(only)h Fj(at)h Ft(A)p Fj(\()p Ft(q)1250 1814 y Fq(2)1269 1829 y Fj(\).)17 b(Lo)o(w)o(ering)10 b(of)g Ft(A)p Fj(\()p Ft(q)1600 1814 y Fq(2)1619 1829 y Fj(\))225 1879 y(should)j(b)q(e)h(a)f(result)h(of)f(a)g(broad)g (\014t)g(to)g(man)o(y)f(meson)g(prop)q(erties,)i(comparable)e(to)h(the) 225 1928 y(original)f(\014t)i([14)p (#cite.jain93b) [[178 254 188 266] [1 1 1 [3 3]] [0 0 1]] pdfm (].)j (This,)c(ho)o(w)o(ev)o(er,)h(is)f(b)q(ey)o(ond)i(the)f(scop)q(e)h(of)e (this)h(pap)q(er.\))300 2001 y(By)k(the)g(same)e(tok)o(en,)i(high)f (precision)h(measuremen)o(ts)f(of)g Ft(T)1301 2008 y Fs(\031)1321 2001 y Fi(0)5 b Fj(\()p Fr(\000)p Ft(Q)1421 1986 y Fq(2)1440 2001 y Ft(;)i Fj(0\))17 b(can)g(b)q(e)225 2051 y(esp)q(ecially)11 b(helpful)e(in)h(obtaining)f(information)f(on)i (whic)o(h)h(solutions)f(for)g Ft(A)p Fj(\()p Ft(q)1457 2036 y Fq(2)1476 2051 y Fj(\))g(are)h(em-)225 2101 y(pirically)d (acceptable)i(and)f(whic)o(h)g(are)h(not.)16 b(Of)10 b(course,)h(measuremen)o(ts)d(of)h Ft(T)1443 2108 y Fs(\031)1463 2100 y Fi(0)1482 2101 y Fj(\()p Fr(\000)p Ft(Q)1563 2086 y Fq(2)1582 2101 y Ft(;)e Fj(0\))225 2151 y(giv)o(e)14 b(information)e(on)i(the)i(in)o(tegrated)f(strength)g(of)g Ft(A)p Fj(\()p Ft(q)1146 2136 y Fq(2)1164 2151 y Fj(\))g(rather)h(than) e(on)h Ft(A)p Fj(\()p Ft(q)1544 2136 y Fq(2)1563 2151 y Fj(\))f(it-)225 2201 y(self.)j(Ho)o(w)o(ev)o(er,)12 b(since)g(it)f(is)g(kno)o(wn)f(that)h(the)h(form)e(of)g(that)h (function)g(m)o(ust)f(b)q(e)i(a)f(rather)225 2251 y(smo)q(oth)h (transition)i(\()p Fg(e.g.)p Fj(,)f(see)h([4)p (#cite.KeBiKl98) [[251 177 256 189] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))f (from)f Ft(A)p Fj(\()p Ft(q)971 2235 y Fq(2)990 2251 y Fj(\))f Ft(>)h Fj(1)h(for)h Ft(q)1179 2235 y Fq(2)1211 2251 y Fj(near)g(0,)f(to)g Ft(A)p Fj(\()p Ft(q)1466 2235 y Fq(2)1485 2251 y Fj(\))f Fr(!)f Fj(1)i(in)225 2300 y(the)g Ft(q)315 2285 y Fq(2)334 2300 y Fj(-domain)c(where)14 b(QCD)e(is)g(p)q(erturbativ)o(e,)h(suc)o(h)h(measuremen)o(ts)d([23)p (#cite.gagasCEBAF) [[403 165 413 177] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (w)o(ould)h(giv)o(e)225 2350 y(a)i(useful)g(hin)o(t)f(ev)o(en)i(ab)q (out)e Ft(A)p Fj(\()p Ft(q)745 2335 y Fq(2)764 2350 y Fj(\))h(itself)g({)f(namely)f(ab)q(out)i(what)g(solutions)f(for)h Ft(A)p Fj(\()p Ft(q)1600 2335 y Fq(2)1619 2350 y Fj(\))225 2400 y(one)f(ma)o(y)f(ha)o(v)o(e)h(in)f(sensible)i(descriptions)g(of)f (dynamically)d(dressed)15 b(quarks)f(and)f(their)920 2490 y(7)p eop %%Page: 8 8 8 7 bop 225 440 a Fj(b)q(ound)17 b(states.)28 b(Therefore,)18 b(the)g(in)o(termediate-mom)o(en)o(tum)13 b(\()p Ft(Q)1288 425 y Fq(2)1325 427 y Ft(<)1325 455 y Fr(\030)1376 440 y Fj(4)k(GeV)1496 425 y Fq(2)1514 440 y Fj(\))g(mea-)225 490 y(suremen)o(ts)12 b(of)f Ft(T)487 497 y Fs(\031)507 489 y Fi(0)526 490 y Fj(\()p Fr(\000)p Ft(Q)607 475 y Fq(2)626 490 y Ft(;)c Fj(0\))k(at)h(Je\013erson)i(Lab,)d(suc)o(h)i(as)e (those)i(prop)q(osed)g(in)e(Ref.)g([23)p (#cite.gagasCEBAF) [[449 599 459 611] [1 1 1 [3 3]] [0 0 1]] pdfm (],) 225 539 y(w)o(ould)k(b)q(e)i(v)o(ery)f(desirable.)25 b(Hop)q(efully)m(,)15 b(the)i(study)f(of)g(virtual)f(Compton)f (scattering)225 589 y(at)g(CEBAF)g(can)h(pro)q(duce)g(the)f(data)g(on)f Ft(T)917 596 y Fs(\031)937 588 y Fi(0)956 589 y Fj(\()p Fr(\000)p Ft(Q)1037 574 y Fq(2)1056 589 y Ft(;)7 b Fj(0\))13 b(as)h(en)o(visioned)g(in)g(Ref.)f([23)p (#cite.gagasCEBAF) [[438 576 448 588] [1 1 1 [3 3]] [0 0 1]] pdfm (].) 225 697 y Fu(Ac)o(kno)o(wledgmen)o(ts)225 774 y Fj(D.)j(Klabu)o(\024) -20 b(car)18 b(thanks)f(the)h(organizers)g(C.)f(Carlson)g(and)g(A.)g (V.)g(Radyuskin)f(for)h(the)225 824 y(w)o(onderful)g(w)o(orkshop.)29 b(The)18 b(authors)g(ac)o(kno)o(wledge)f(the)h(supp)q(ort)h(of)e(the)h (Croatian)225 873 y(Ministry)c(of)f(Science)i(and)f(T)m(ec)o(hnology)m (.)225 981 y Fu(References)276 1058 y Fj(1.)20 b(J.)13 b(Gron)o(b)q(erg)h Fg(et)h(al.)j Fj(\(CLEO)d(collab)q(oration\),)d(Ph)o (ys.)18 b(Rev.)g(D)c Fu(57)p Fj(,)f(33)g(\(1998\).)276 1108 y(2.)20 b(A.)13 b(V.)h(Radyushkin)f(and)h(R.)f(T.)g(Rusk)o(o)o(v,) g(Nucl.)18 b(Ph)o(ys.)g(B)c Fu(481)p Fj(,)f(625)g(\(1996\).)276 1158 y(3.)20 b(D.)13 b(Kek)o(ez)i(and)f(D.)f(Klabu)o(\024)-20 b(car,)13 b(Ph)o(ys.)19 b(Lett.)f Fu(457B)p Fj(,)13 b(359)h(\(1999\).) 276 1207 y(4.)20 b(D.)13 b(Kek)o(ez)i Fg(et)g(al.)p Fj(,)e(In)o(t.)18 b(J.)13 b(Mo)q(d.)18 b(Ph)o(ys.)h(A)14 b Fu(14)p Fj(,)f(161)g (\(1999\).)276 1257 y(5.)20 b(M.)10 b(Bando,)h(M.)f(Harada,)h(and)g(T.) f(Kugo,)h(Prog.)17 b(Theor.)g(Ph)o(ys.)g Fu(91)p Fj(,)11 b(927)f(\(1994\).)276 1307 y(6.)20 b(C.)13 b(D.)g(Rob)q(erts,)h(Nucl.)k (Ph)o(ys.)h(A)14 b Fu(605)p Fj(,)f(475)g(\(1996\).)276 1357 y(7.)20 b(M.R.)12 b(F)m(rank)i Fg(et)g(al.)p Fj(,)f(Ph)o(ys.)19 b(Lett.)f(B)d Fu(359)p Fj(,)e(17)g(\(1995\).)276 1407 y(8.)20 b(I.)13 b(V.)h(Musato)o(v)f(and)h(A.)g(V.)f(Radyushkin,)g(Ph)o (ys.)18 b(Rev.)g(D)c Fu(56)p Fj(,)f(2713)g(\(1997\).)276 1456 y(9.)20 b(D.)13 b(Kek)o(ez)i(and)f(D.)f(Klabu)o(\024)-20 b(car,)13 b(Ph)o(ys.)19 b(Lett.)f(B)d Fu(387)p Fj(,)e(14)g(\(1996\).) 255 1506 y(10.)20 b(D.)13 b(Klabu)o(\024)-20 b(car)14 b(and)g(D.)f(Kek)o(ez,)h(Ph)o(ys.)19 b(Rev.)f(D)13 b Fu(58)p Fj(,)g(096003)g(\(1998\).)255 1556 y(11.)20 b(G.)13 b(P)m(.)g(Lepage)h(and)g(S.)f(J.)h(Bro)q(dsky)m(,)g(Ph)o(ys.)k(Rev.)g (D)c Fu(22)p Fj(,)f(2157)g(\(1980\).)255 1606 y(12.)20 b(A.)13 b(Manohar,)g(Ph)o(ys.)19 b(Lett.)f Fu(B244)p Fj(,)13 b(101)g(\(1990\).)255 1656 y(13.)20 b(P)m(.)14 b(Jain)g(and)h(H.)f(J.)h(Munczek,)h(Ph)o(ys.)21 b(Rev.)g(D)14 b Fu(44)p Fj(,)h(1873)e(\(1991\);)h(Ph)o(ys.)22 b(Rev.)329 1705 y(D)13 b Fu(46)p Fj(,)g(438)h(\(1992\).)255 1755 y(14.)20 b(P)m(.)13 b(Jain)g(and)h(H.)g(J.)f(Munczek,)i(Ph)o(ys.)k (Rev.)e(D)d Fu(48)p Fj(,)f(5403)g(\(1993\).)255 1805 y(15.)20 b(P)m(.)13 b(Maris)h(and)f(C.)h(D.)f(Rob)q(erts,)h(Ph)o(ys.)k (Rev.)g(C)c Fu(56)p Fj(,)f(3369)g(\(1997\).)255 1855 y(16.)20 b(J.)13 b(S.)h(Ball)f(and)h(T.-W.)e(Chiu,)h(Ph)o(ys.)18 b(Rev.)g(D)c Fu(22)p Fj(,)f(2542)g(\(1980\).)255 1905 y(17.)20 b(D.)13 b(C.)g(Curtis)h(and)g(M.)g(R.)e(P)o(ennington,)i(Ph)o (ys.)k(Rev.)g(D)c Fu(42)p Fj(,)f(4165)g(\(1990\).)255 1955 y(18.)20 b(C.)13 b(D.)g(Rob)q(erts,)h(Fizik)n(a)f(B)h(\(Zagreb\))h Fu(8)p Fj(,)e(285)g(\(1999\).)255 2004 y(19.)20 b(V.)13 b(A.)h(No)o(vik)o(o)o(v)e Fg(et)j(al.)p Fj(,)e(Nucl.)18 b(Ph)o(ys.)g(B)d Fu(237)p Fj(,)e(525)g(\(1984\).)255 2054 y(20.)20 b(P)m(.)13 b(Kessler)i(and)f(S.)g(Ong,)f(Ph)o(ys.)19 b(Rev.)e(D)d Fu(48)p Fj(,)f(R2974)f(\(1993\).)255 2104 y(21.)20 b(H.-J.)10 b(Behrend)j Fg(et)g(al.)k Fj(\(CELLO)11 b(Collab)q(oration\),)f(Z.)h(Ph)o(ys.)18 b(C)11 b Fu(49)p Fj(,)g(401)f(\(1991\).)255 2154 y(22.)20 b(D.)13 b(Klabu)o(\024)-20 b(car)14 b(and)g(D.)f(Kek)o(ez,)h(Fizik)n(a)f(B)h(\(Zagreb\))h Fu(8)p Fj(,)e(303)h(\(1999\).)255 2204 y(23.)20 b(CEBAF)c(Letter)h(of)e (In)o(ten)o(t)h(#)g(LOI{94/005.)21 b(Co{Sp)q(ok)o(esmen:)g(A.)15 b(Afanasev,)329 2253 y(J.)e(Gomez,)g(and)g(S.)h(Nanda;)f(A.)g (Afanasev,)h Fa(hep-ph/9608305)p [[323 176 397 188] [1 1 1 [3 3]] [0 0 1]] (http://xxx.lanl.gov/abs/hep-ph/9608305) pdfm Fj(.)920 2490 y(8)p eop %%Trailer end end userdict /end-hook known{end-hook}if %%EOF