%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f (with hyperlinking) Copyright 1986, 1994 Radical Eye Software %%Title: hep-ph/9907537 %%CreationDate: Tue Jan 9 16:07:28 2001 %%Pages: 4 %%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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03F00001F00000F0C000F0C000F0C000E0E000E0F001C0FC03C0EFFF0083FC00141C7D9B 1B>I<7FFFFFE07FFFFFE0781F81E0701F80E0601F8060E01F8070C01F8030C01F8030C0 1F8030C01F8030001F8000001F8000001F8000001F8000001F8000001F8000001F800000 1F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800007 FFFE0007FFFE001C1C7E9B21>I86 D89 D<03FC000E0E001C1F003C1F00781F 00780E00F80000F80000F80000F80000F80000F800007800007801803C01801C03000E0E 0003F80011127E9115>99 D<000FF0000FF00001F00001F00001F00001F00001F00001F0 0001F00001F00001F001F9F00F07F01C03F03C01F07801F07801F0F801F0F801F0F801F0 F801F0F801F0F801F07801F07801F03C01F01C03F00F0FFE03F9FE171D7E9C1B>I<03FC 000F07001C03803C01C07801C07801E0F801E0F801E0FFFFE0F80000F80000F800007800 007800603C00601E00C00F038001FC0013127F9116>I<03F0F00E1F383C0F3838073078 07807807807807807807803807003C0F001E1C0033F0003000003000003000003FFF003F FFC01FFFE01FFFF07801F8F00078F00078F000787000707800F01E03C007FF00151B7F91 18>103 D107 DIII<01FC000F07801C01C03C01E07800F07800F0F800F8F800F8F800F8F8 00F8F800F8F800F87800F07800F03C01E01E03C00F078001FC0015127F9118>I<1FD830 786018E018E018F000FF807FE07FF01FF807FC007CC01CC01CE01CE018F830CFC00E127E 9113>115 D<0300030003000300070007000F000F003FFCFFFC1F001F001F001F001F00 1F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F9913>I 119 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: Letter end HPSdict begin /TargetAnchors 21 dict dup (cite.gronberg98) [4 [126 511 126 523] 709] put dup (cite.BrodskyLepage) [4 [126 499 126 511] 697] put dup (cite.Kess+Ong93) [4 [126 367 126 379] 566] put dup (cite.KlKe2) [4 [126 439 126 451] 638] put dup (cite.KlKe4) [4 [126 427 126 439] 626] put dup (cite.CP90) [4 [126 415 126 427] 614] put dup (cite.manohar90) [4 [126 487 126 499] 685] put dup (equation.1) [1 [188 422 188 434] 620] put dup (equation.2) [2 [131 341 131 353] 539] put dup (equation.3) [3 [226 365 226 377] 563] put dup (cite.RobertsDubr) [4 [126 403 126 415] 602] put dup (Doc-Start) [1 [126 623 126 635] 791] put dup (cite.KeBiKl98) [4 [126 451 126 463] 650] put dup (cite.Radyushkin+Rusk3) [4 [126 475 126 487] 674] put dup (cite.KeKl3) [4 [126 525 126 537] 724] 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 (cite.novikov+al84) [4 [126 391 126 403] 590] put dup (cite.TandyDub99) [4 [126 463 126 475] 662] put targetdump-hook def TeXDict begin %%EndSetup %%Page: 1 1 gsave matrix defaultmatrix setmatrix 90 rotate 279 -39 moveto /Times-Roman findfont 20 scalefont setfont 0.3 setgray (arXiv:hep-ph/9907537 28 Jul 1999) show grestore 1 0 bop 241 432 a Fs(THE)17 b(ASYMPTOTIC)f(BEHA)-5 b(VIOR)16 b(OF)g(THE)h Fr(\031)1206 416 y Fq(0)1225 432 y Fr(\015)1248 416 y Fp(?)1268 432 y Fr(\015)h Fs(TRANSITION)699 562 y Fo(DUBRA)l(VK)o(O)12 b(KLABU)1080 552 y(\024)1076 562 y(CAR)250 609 y Fn(Physics)g(Dep)n(artment,)g(F)m(aculty)f(of)j(Scienc) n(e,)d(POB)j(162,)e(University)f(of)i(Zagr)n(eb,)f(Cr)n(o)n(atia)766 696 y Fo(D)o(ALIBOR)h(KEKEZ)418 743 y Fn(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)300 830 y Fm(It)k(is)g(discussed)e(ho)o(w)i(v)n(arious)e Fl(A)o(ns\177)-18 b(atze)15 b Fm(for)g(the)f(dressed)f(quark-photon)e (\()p Fk(q)q(q)q(\015)r Fm(\))k(v)o(ertices)300 869 y(\000)322 858 y Fj(\026)343 869 y Fm(\()p Fk(q)q(;)5 b(q)406 858 y Fi(0)417 869 y Fm(\))12 b(in\015uence)d(the)i(asymptotic)o(s)e(of)i (the)g Fk(\015)967 858 y Fj(?)985 869 y Fk(\015)h Fh(!)e Fk(\031)1081 858 y Fg(0)1111 869 y Fm(transition)f(form)h(factor.)300 1010 y Ff(It)16 b(has)g(b)q(een)h(sho)o(wn)644 995 y Fq(1)p (#cite.KeKl3) [[227 478 231 490] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.KeKl3) [[230 478 230 490] [1 1 1 [3 3]] [0 0 1]] pdfm 678 1010 a Ff(that)f(the)g(Sc)o(h)o(winger-Dyson)g(\(SD\))g(approac)o (h)g(to)f(ph)o(ysics)225 1060 y(of)h(quarks)h(and)f(gluons)g(leads)h (to)f(the)h Fr(\015)884 1045 y Fp(?)904 1060 y Ff(\()p Fr(k)q Ff(\))p Fr(\015)r Ff(\()p Fr(k)1021 1045 y Fe(0)1034 1060 y Ff(\))f Fd(!)g Fr(\031)1149 1045 y Fq(0)1167 1060 y Ff(\()p Fr(p)p Ff(\))h(transition)f(form)f(factor)225 1114 y Fr(T)249 1121 y Fp(\031)269 1113 y Fc(0)288 1114 y Ff(\()p Fr(k)327 1099 y Fq(2)346 1114 y Fr(;)7 b(k)388 1099 y Fe(0)399 1093 y Fq(2)417 1114 y Ff(\))14 b(whic)o(h)g(has)g(the) h(asymptotic)d(momen)o(tum)e(dep)q(endence)485 1230 y Fr(T)509 1237 y Fp(\031)529 1229 y Fc(0)548 1230 y Ff(\()p Fd(\000)p Fr(Q)629 1213 y Fq(2)648 1230 y Fr(;)d Ff(0\))k(=)773 1202 y Fd(K)p 764 1220 52 2 v 764 1258 a Fr(Q)797 1246 y Fq(2)910 1230 y Ff(\()p Fd(K)h(!)f Ff(const)k(as)f Fr(Q)1216 1213 y Fq(2)1246 1230 y Fd(!)d(1)p Ff(\))p Fr(;)213 b Ff(\(1\))225 1349 y(for)16 b(large)h(spacelik)o(e)g Fr(k)595 1334 y Fq(2)630 1349 y Ff(=)g Fd(\000)p Fr(Q)744 1334 y Fq(2)779 1349 y Fr(<)g Ff(0.)26 b(This)17 b(is)g(consisten)o(t)h (with)e(the)i(data)1488 1334 y Fq(2)p (#cite.gronberg98) [[429 397 433 409] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.gronberg98) [[433 397 433 409] [1 1 1 [3 3]] [0 0 1]] pdfm 1524 1349 a Ff(at)e(the)225 1399 y(presen)o(tly)d(largest)g(accessible) g Fr(Q)752 1384 y Fq(2)783 1399 y Ff(and)f(in)g(agreemen)o(t)g(\(up)h (to)f(the)h(precise)g(v)n(alue)f(of)g Fd(K)q Ff(\))225 1449 y(with)i(p)q(erturbativ)o(e)i(QCD)f(where)782 1434 y Fq(3)p (#cite.BrodskyLepage) [[260 373 264 385] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.BrodskyLepage) [[264 373 264 385] [1 1 1 [3 3]] [0 0 1]] pdfm 816 1449 a Fd(K)e Ff(=)g(2)p Fr(f)947 1455 y Fp(\031)969 1449 y Ff(,)i(op)q(erator)g(pro)q(duct)g(expansion)g(\(OPE\))225 1499 y(where)338 1483 y Fq(4)p (#cite.manohar90) [[153 361 157 373] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.manohar90) [[157 361 157 373] [1 1 1 [3 3]] [0 0 1]] pdfm 371 1499 a Fd(K)c Ff(=)h(4)p Fr(f)499 1505 y Fp(\031)522 1499 y Fr(=)p Ff(3,)h(and)g(QCD)h(sum)f(rules)i(where)1078 1483 y Fq(5)p (#cite.Radyushkin+Rusk3) [[331 361 335 373] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.Radyushkin+Rusk3) [[335 361 335 373] [1 1 1 [3 3]] [0 0 1]] pdfm 1110 1499 a Fd(K)d(\031)g Ff(1)p Fr(:)p Ff(6)p Fr(f)1272 1505 y Fp(\031)1294 1499 y Ff(.)300 1550 y(The)g(SD)f(approac)o(h,)g (where)h(the)g(quark)f(propagators)g Fr(S)r Ff(\()p Fr(q)q Ff(\))i(=)f([)p Fr(A)p Ff(\()p Fr(q)1379 1535 y Fq(2)1397 1550 y Ff(\))p Fr(q)-19 b(=)t Fd(\000)t Fr(B)r Ff(\()p Fr(q)1543 1535 y Fq(2)1563 1550 y Ff(\)])1591 1535 y Fe(\000)p Fq(1)225 1600 y Ff(are)12 b(dynamically)d(dressed,)14 b(also)d(requires)i(consisten)o(tly)g(dressed)h(quark-photon)d(\()p Fr(q)q(q)q(\015)r Ff(\))225 1650 y(v)o(ertices)h(\000)398 1635 y Fp(\026)420 1650 y Ff(\()p Fr(q)q(;)7 b(q)495 1635 y Fe(0)506 1650 y Ff(\))k(in)f(order)i(to)e(satisfy)h(the)g(v)o (ector)g(W)m(ard-T)m(ak)n(ahashi)d(iden)o(tit)o(y)j(\(WTI\).)225 1699 y([In)20 b(addition)f(to)h(using)g(for)g(the)h(pseudoscalar)g(v)o (ertex)g(the)g(quark-an)o(tiquark)f(pion)225 1749 y(Bethe-Salp)q(eter)c (\(BS\))f(b)q(ound-state)g(v)o(ertex)h(\000)989 1756 y Fp(\031)1009 1748 y Fc(0)1027 1749 y Ff(\()p Fr(q)q(;)7 b(p)p Ff(\),)14 b(this)g(de\014nes)i Fb(gener)n(alize)n(d)e(im-)225 1799 y(pulse)j(appr)n(oximation)g Ff(\(GIA\).])f(Ev)o(en)h(just)f (appro)o(ximately)e(adequate)j(SD)f(solutions)225 1849 y(for)i(\000)319 1834 y Fp(\026)341 1849 y Ff(\()p Fr(q)q(;)7 b(q)416 1834 y Fe(0)427 1849 y Ff(\))19 b(are)f(not)g(y)o(et)h(a)o(v)n (ailable,)e(and)h(in)f(practice)j(the)f(more)e(or)h(less)h(realistic) 225 1899 y(WTI-satisfying)12 b Fb(A)o(ns\177)-21 b(atze)14 b Ff(still)f(m)o(ust)g(b)q(e)i(used.)300 1950 y(The)h(topic)f(of)g (this)h(writeup)g(is)f(the)h(dep)q(endence)i(of)d(the)h(asymptotic)e (co)q(e\016cien)o(t)225 2000 y Fd(K)e Ff(on)g(the)g(c)o(hoice)g(of)g (the)g Fb(A)o(nsatz)g Ff(for)g(\000)852 1985 y Fp(\026)874 2000 y Ff(\()p Fr(q)q(;)7 b(q)949 1985 y Fe(0)960 2000 y Ff(\).)17 b(This)12 b(topic)g(needs)h(clari\014cation,)e(since)225 2050 y(Ref.)319 2035 y 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g(other)h(fermion)d(leg)h(in)h(the)g(v)o(ertices)i(\000)1176 1783 y Fp(\026)1198 1798 y Ff(\()p Fr(q)1234 1783 y Fe(0)1246 1798 y Fr(;)7 b(q)q Ff(\))k(=)h(\000)1382 1778 y Fp(\026)1382 1810 y(B)q(C)1436 1798 y Ff(\()p Fr(q)1472 1783 y Fe(0)1484 1798 y Fr(;)7 b(q)q Ff(\).)300 1848 y(Ho)o(w)o(ev)o(er,)20 b(this)e(result)h(on)f(the)h(asymptotics)f(of)g Fr(T)1139 1855 y Fp(\031)1159 1847 y Fc(0)1178 1848 y Ff(\()p Fd(\000)p Fr(Q)1259 1833 y Fq(2)1278 1848 y Fr(;)7 b Ff(0\))17 b(when)i(using)f(the)225 1897 y(BC)13 b(v)o(ertex,)h(caused)g(some)f (con)o(tro)o(v)o(ersy)h(since)g(Ref.)1076 1882 y Fq(6)p (#cite.TandyDub99) [[330 265 334 277] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.TandyDub99) [[334 265 334 277] [1 1 1 [3 3]] [0 0 1]] pdfm 1108 1897 a Ff(claimed)d(that)i(ev)o(en)h(for)f(the)h(BC)225 1947 y(v)o(ertex)i Fd(K)e Ff(=)g(4)p Fr(f)485 1953 y Fp(\031)507 1947 y Fr(=)p Ff(3,)h Fb(i.e.)p Ff(,)f(that)h(no)g(mo)q (di\014cation)e(o)q(ccurs)j(for)f(the)h(BC)f(v)o(ertex)h(due)g(to)225 1997 y(one)i(soft)g(quark)g(leg.)30 b(The)19 b(argumen)o(t)e(of)g(Ref.) 1033 1982 y Fq(6)p (#cite.TandyDub99) [[320 241 324 253] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.TandyDub99) [[324 241 324 253] [1 1 1 [3 3]] [0 0 1]] pdfm 1069 1997 a Ff(\(see)j(its)e(Sec.)31 b Fb(4.)p Ff(\))h(is)18 b(that)g(there)225 2047 y(are)e(in)f(fact)g(no)g(soft)g(legs)h(in)f (the)h Fr(q)q(q)q(\015)i Ff(v)o(ertices)f(when)e Fr(Q)1148 2032 y Fq(2)1182 2047 y Ff(b)q(ecomes)h(v)o(ery)f(large.)23 b(The)225 2097 y(on-shell)13 b(condition)f(for)h(the)h(pion)e(and)h (one)h(photon,)f(\()p Fr(p)7 b Fd(\000)i Fr(k)q Ff(\))1215 2082 y Fq(2)1245 2097 y Ff(=)j Fr(M)1334 2082 y Fq(2)1329 2107 y Fp(\031)1360 2097 y Fd(\000)c Ff(2)p Fr(p)f Fd(\001)h Fr(k)h Fd(\000)f Fr(Q)1573 2082 y Fq(2)1603 2097 y Fd(\031)225 2151 y(\000)p Ff(2)p Fr(p)13 b Fd(\001)f Fr(k)i Fd(\000)f Fr(Q)450 2136 y Fq(2)488 2151 y Ff(=)21 b Fr(k)564 2136 y Fe(0)575 2130 y Fq(2)614 2151 y Ff(=)g(0,)f(and)f Fr(k)829 2136 y Fq(2)867 2151 y Ff(=)i Fd(\000)p Fr(Q)985 2136 y Fq(2)1004 2151 y Ff(,)f(are)f(used)h(to)f(argue)h(that)f(the)h(pion) 225 2201 y(momen)o(tum)10 b Fr(p)k Ff(has)g(comp)q(onen)o(ts)g(whic)o (h)g(m)o(ust)f(scale)i(lik)o(e)e Fr(k)i Ff(and)f(th)o(us)h(lik)o(e)e Fr(Q)p Ff(.)19 b(Then,)225 2251 y Fr(A)p Ff(\([)p Fr(q)10 b Fd(\006)g Fr(p=)p Ff(2])430 2235 y Fq(2)447 2251 y Ff(\))j(=)f Fr(A)p Ff(\()p Fr(q)587 2235 y Fq(2)615 2251 y Fd(\006)e Fr(q)g Fd(\001)f Fr(p)h Ff(+)f Fr(M)824 2235 y Fq(2)819 2261 y Fp(\031)843 2251 y Ff(\))14 b(w)o(ould)f(tend)i(to)f (1)g(as)g Fr(Q)1259 2235 y Fq(2)1290 2251 y Fd(!)d(1)j Ff(ev)o(en)h(for)f(v)o(ery)225 2300 y(soft)g(lo)q(op)f(momen)o(ta)e Fr(q)q Ff(,)i(just)h(b)q(ecause)i(of)d Fr(p)f Fd(\030)g Fr(Q)p Ff(,)h(causing)h(\000)1214 2280 y Fp(\026)1214 2313 y(B)q(C)1279 2300 y Fd(!)d Fr(\015)1355 2285 y Fp(\026)1379 2300 y Ff(.)300 2350 y(W)m(e)19 b(will)e(no)o(w)i(demonstrate)f(that)h (this)g(argumen)o(t)f(do)q(es)i(not)f(hold.)32 b(The)20 b(v)o(ery)225 2400 y(fact)c(that)g(the)h(size)g(of)e(the)i Fb(c)n(omp)n(onents)g Ff(is)f(in)o(v)o(ok)o(ed)g(mak)o(es)f(the)h (argumen)o(t)f(susp)q(ect,)920 2490 y(2)p eop %%Page: 3 3 3 2 bop 225 432 a Ff(b)q(ecause)21 b(it)e(is)h(a)f(frame-dep)q(enden)o (t)h(statemen)o(t.)35 b(The)19 b(argumen)o(t)g(of)g(Ref.)1504 416 y Fq(6)p (#cite.TandyDub99) [[433 617 437 629] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.TandyDub99) [[437 617 437 629] [1 1 1 [3 3]] [0 0 1]] pdfm 1542 432 a Ff(relies)225 481 y(on)i(w)o(orking)e(in)i(a)f(Loren)o(tz)i (frame)d(suc)o(h)j(as)f(the)g(one)g(where)h Fr(k)i Ff(=)g(\(0)p Fr(;)7 b Ff(0)p Fr(;)g Ff(0)p Fr(;)1517 445 y Fa(p)p 1556 445 52 2 v 1556 481 a Fr(Q)1589 469 y Fq(2)1607 481 y Ff(\),)225 537 y Fr(k)248 522 y Fe(0)272 537 y Ff(=)14 b(\()p Fr(E)365 543 y Fp(\031)387 537 y Fr(;)7 b Ff(0)p Fr(;)g Ff(0)p Fr(;)g Fd(\000)p Fr(E)549 543 y Fp(\031)569 537 y Ff(\),)15 b Fr(p)e Ff(=)g(\()p Fr(E)738 543 y Fp(\031)760 537 y Fr(;)7 b Ff(0)p Fr(;)g Ff(0)p Fr(;)859 501 y Fa(p)p 898 501 V 898 537 a Fr(Q)931 525 y Fq(2)960 537 y Fd(\000)j Fr(E)1033 543 y Fp(\031)1055 537 y Ff(\),)k(and)h Fr(E)1210 543 y Fp(\031)1245 537 y Ff(=)e(\()p Fr(Q)1339 522 y Fq(2)1367 537 y Ff(+)d Fr(M)1454 522 y Fq(2)1449 547 y Fp(\031)1473 537 y Ff(\))p Fr(=)p Ff(2)1531 501 y Fa(p)p 1572 501 V 1572 537 a Fr(Q)1605 525 y Fq(2)1623 537 y Ff(.)225 587 y(Ev)o(en)21 b(if)f(one)g(stic)o(ks) i(just)e(to)h(that)f(c)o(hoice)h(in)f(one's)h(calculation,)g(one)g(can) f(exp)q(ect)225 637 y(p)q(ersisten)o(t)d(soft)f(con)o(tributions)f(b)q (ecause)j(of)c(those)j(soft)e(lo)q(op)g(momen)o(ta)e Fr(q)j Ff(whic)o(h)g(are)225 686 y(also)f(p)q(erp)q(endicular)i(to)f Fr(p)g Ff(so)g(that)g Fr(p)10 b Fd(\001)g Fr(q)16 b Ff(=)f(0.)24 b(Ho)o(w)o(ev)o(er,)16 b(the)h(shortest)g(and)f(clearest)225 736 y(demonstration)h(that,)h(at)f(least)h(in)f(this)h(application,)f Fr(q)c Fd(\001)e Fr(p)18 b Ff(cannot)f(b)q(e)i(hard)f(if)e Fr(q)j Ff(is)225 786 y(soft,)f(is)f(noting)g(that)g(one)h(can)f(mak)o (e)f(a)h(Loren)o(tz)h(transformation)e(to)h(the)h(pion)f(rest)225 836 y(frame.)f(In)c(this)g(case)g(it)g(is)f(the)h(b)q(o)q(ost)h (transformation)c(along)i(the)h Fr(z)r Ff(-axis)f(and)h(with)f(the)225 886 y(parameter)i Fr(\014)h Ff(=)e(\()p Fr(Q)553 871 y Fq(2)579 886 y Fd(\000)d Fr(M)665 871 y Fq(2)660 896 y Fp(\031)683 886 y Ff(\))p Fr(=)p Ff(\()p Fr(Q)769 871 y Fq(2)795 886 y Ff(+)f Fr(M)880 871 y Fq(2)875 896 y Fp(\031)899 886 y Ff(\).)18 b(In)13 b(that)g(frame,)f Fr(k)g Ff(=)g(\()p Fr(M)1347 892 y Fp(\031)1378 886 y Fd(\000)c Fr(E)1449 892 y Fp(\015)1470 886 y Fr(;)f Ff(0)p Fr(;)g Ff(0)p Fr(;)g(E)1600 892 y Fp(\015)1619 886 y Ff(\))225 935 y(and)i Fr(k)324 920 y Fe(0)347 935 y Ff(=)j(\()p Fr(E)438 941 y Fp(\015)459 935 y Fr(;)7 b Ff(0)p Fr(;)g Ff(0)p Fr(;)g Fd(\000)p Fr(E)621 941 y Fp(\015)640 935 y Ff(\),)j(with)f Fr(E)799 941 y Fp(\015)832 935 y Ff(=)j(\()p Fr(Q)925 920 y Fq(2)944 935 y Ff(+)p Fr(M)1021 920 y Fq(2)1016 946 y Fp(\031)1040 935 y Ff(\))p Fr(=)p Ff(2)p Fr(M)1138 941 y Fp(\031)1160 935 y Ff(,)e(whereas)h Fr(p)g Ff(=)h(\()p Fr(M)1467 941 y Fp(\031)1490 935 y Fr(;)7 b Ff(0)p Fr(;)g Ff(0)p Fr(;)g Ff(0\),)225 985 y(making)12 b(it)j(clear)g(that)g(for)g(the)g(ligh)o(t)f(pion,)g Fr(A)p Ff(\([)p Fr(q)d Fd(\006)f Fr(p=)p Ff(2])1151 970 y Fq(2)1168 985 y Ff(\))15 b(is)g(appro)o(ximated)e(w)o(ell)h(b)o(y)225 1035 y Fr(A)p Ff(\()p Fr(q)292 1020 y Fq(2)311 1035 y Ff(\))g(and)g(not)f(b)o(y)h Fr(A)p Ff(\()p Fd(\006)p Fr(q)d Fd(\001)e Fr(p)p Ff(\))k(whic)o(h)h(allegedly)1017 1020 y Fq(6)p (#cite.TandyDub99) [[316 472 320 484] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.TandyDub99) [[320 472 320 484] [1 1 1 [3 3]] [0 0 1]] pdfm 1050 1035 a Ff(w)o(ould)f(b)q(e)h(1.)300 1093 y(W)m(e)e(w)o(an)o(t)g (to)g(mak)o(e)f(clear)h(that)h(w)o(e)f(of)g(course)h(giv)o(e)f (precedence)j(to)d(the)h(v)n(alue)f Fd(K)g Ff(=)225 1143 y(4)p Fr(f)266 1149 y Fp(\031)289 1143 y Fr(=)p Ff(3)i(for)h(the)h (asymptotic)e(co)q(e\016cien)o(t)i(as)f(the)h(one)g(ha)o(ving)e(the)i (more)e(fundamen)o(tal)225 1192 y(meaning,)e(resulting)j(from)e(the)i Fr(q)q(q)q(\015)i Ff(v)o(ertices)f(suc)o(h)g(as)e(the)i(CP)e(or)h(mCP)e (ones,)i(whic)o(h)225 1242 y(ha)o(v)o(e)c(prop)q(erties)h(closer)f(to)g (the)g(true)h(v)o(ertex)f(solution,)f(suc)o(h)i(as)e(b)q(eing)h (renormalizable.)225 1292 y(Also)19 b(indicativ)o(e)f(is)h(the)h (asymptotics)e(found)g(b)o(y)h(Ref.)1161 1277 y Fq(1)p (#cite.KeKl3) [[351 411 355 423] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.KeKl3) [[355 411 355 423] [1 1 1 [3 3]] [0 0 1]] pdfm 1199 1292 a Ff(for)g(the)g(case)h(when)g(b)q(oth)225 1342 y(photons)14 b(are)g(o\013-shell,)f Fr(k)639 1327 y Fq(2)670 1342 y Ff(=)e Fd(\000)p Fr(Q)778 1327 y Fq(2)809 1342 y Fr(<<)h Ff(0)h(and)h Fr(k)1023 1327 y Fe(0)p Fq(2)1063 1342 y Ff(=)d Fd(\000)p Fr(Q)1171 1327 y Fe(0)p Fq(2)1211 1342 y Fd(\024)h Ff(0:)641 1468 y Fr(T)665 1475 y Fp(\031)685 1467 y Fc(0)704 1468 y Ff(\()p Fd(\000)p Fr(Q)785 1451 y Fq(2)804 1468 y Fr(;)7 b Fd(\000)p Fr(Q)888 1451 y Fe(0)p Fq(2)916 1468 y Ff(\))12 b(=)992 1440 y(4)p 992 1459 21 2 v 992 1497 a(3)1091 1440 y Fr(f)1111 1446 y Fp(\031)p 1030 1459 166 2 v 1030 1500 a Fr(Q)1063 1488 y Fq(2)1091 1500 y Ff(+)d Fr(Q)1165 1488 y Fe(0)1177 1482 y Fq(2)1207 1468 y Fr(:)363 b Ff(\(3\))225 1599 y(This)14 b(is)g(found)g(for)f(the)i Fr(q)q(q)q(\015)i Ff(v)o(ertices)e(whic)o(h)f(reduce)i(to)e(the)h(bare)f Fr(\015)1331 1584 y Fp(\026)1368 1599 y Ff(as)g(so)q(on)h(as)f(just)225 1648 y(one)g(of)f(the)h(quark)f(legs)h(is)g(hard,)f(while)g(the)h (usage)g(of)f(the)h(BC)g(v)o(ertex)h(again)d(mo)q(di\014es)225 1698 y(this)j(result)g(b)o(y)f(the)h(substitution)g Fr(f)805 1704 y Fp(\031)841 1698 y Fd(!)902 1687 y Fa(e)895 1698 y Fr(f)915 1704 y Fp(\031)937 1698 y Ff(.)20 b(Eq.)15 b(\(3)p (#equation.3) [[327 309 332 321] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(agrees)i(with)e(the)h(leading)f(term)225 1748 y(of)h(the)g(OPE)h (result)g(deriv)o(ed)g(b)o(y)f(No)o(vik)o(o)o(v)f Fr(et)g(al)q(:)1038 1733 y Fq(12)p (#cite.novikov+al84) [[321 301 329 313] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.novikov+al84) [[329 301 329 313] [1 1 1 [3 3]] [0 0 1]] pdfm 1088 1748 a Ff(for)h(the)h(sp)q(ecial)f(case)h Fr(Q)1484 1733 y Fq(2)1516 1748 y Ff(=)e Fr(Q)1595 1733 y Fe(0)p Fq(2)1623 1748 y Ff(.)225 1798 y(The)f(distribution-amplitude-dep)q (endence)f(of)g(the)g(pQCD)h(approac)o(h)f(cancels)h(out)f(for)225 1848 y(that)h(symmetric)e(case,)j(so)f(that)f Fr(T)776 1855 y Fp(\031)796 1847 y Fc(0)815 1848 y Ff(\()p Fd(\000)p Fr(Q)896 1833 y Fq(2)915 1848 y Fr(;)7 b Fd(\000)p Fr(Q)999 1833 y Fe(0)p Fq(2)1027 1848 y Ff(\))13 b(in)f(this)h(approac)o(h)g(\() p Fb(e.g.)p Ff(,)f(see)1523 1833 y Fq(13)p (#cite.Kess+Ong93) [[438 277 446 289] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.Kess+Ong93) [[446 277 446 289] [1 1 1 [3 3]] [0 0 1]] pdfm 1560 1848 a Ff(\),)g(in)225 1902 y(the)k(limit)e Fr(Q)433 1887 y Fq(2)466 1902 y Ff(=)h Fr(Q)546 1887 y Fe(0)557 1881 y Fq(2)591 1902 y Fd(!)f(1)p Ff(,)h(exactly)h(agrees)h(with)e(b)q (oth)h(our)g(Eq.)24 b(\(3)p (#equation.3) [[399 261 404 273] [1 1 1 [3 3]] [0 0 1]] pdfm (\))16 b(and)g(Ref.)1588 1887 y Fq(12)p (#cite.novikov+al84) [[453 264 461 276] [1 1 1 [3 3]] [0 0 1]] pdfm (#cite.novikov+al84) [[461 264 461 276] [1 1 1 [3 3]] [0 0 1]] pdfm 1623 1902 a Ff(.)225 1952 y(Therefore,)d(for)f(that)g(symmetric)e (case,)i(w)o(e)h(should)e(ha)o(v)o(e)h(ev)o(en)g(the)h(precise)g (agreemen)o(t)225 2001 y(of)g(the)i(co)q(e\016cien)o(ts)g(irresp)q (ectiv)o(e)h(of)d(the)i(description)f(of)f(the)i(pion)e(in)o(ternal)h (structure)225 2051 y(enco)q(ded)h(in)f(the)h(distribution)e (amplitude.)j(Ob)o(viously)m(,)d(this)h(fa)o(v)o(ors)f(the)i Fr(q)q(q)q(\015)i Ff(v)o(ertices)225 2101 y(whic)o(h)d(reduce)h(to)f (the)g(bare)g Fr(\015)712 2086 y Fp(\026)749 2101 y Ff(as)g(so)q(on)f (as)h(one)g(of)f(the)h(quark)g(legs)g(is)f(hard,)h(o)o(v)o(er)f(the)225 2151 y(BC)h(v)o(ertex,)f(and)g Fd(K)f Ff(=)g(4)p Fr(f)643 2157 y Fp(\031)665 2151 y Fr(=)p Ff(3)h(o)o(v)o(er)g Fd(K)f Ff(=)g(4)924 2140 y Fa(e)917 2151 y Fr(f)937 2157 y Fp(\031)959 2151 y Fr(=)p Ff(3.)18 b(Ho)o(w)o(ev)o(er,)13 b(the)h(BC)f(v)o(ertex,)h(whic)o(h)f(is)225 2201 y(the)g(simplest)f (WTI-preserving)i(v)o(ertex)f(and)g(has)g(b)q(een)h(the)g(one)f(most)e (widely)i(used)g(in)225 2251 y(phenomenological)c(applications,)h(ma)o (y)f(an)o(yw)o(a)o(y)h(b)q(e)i(the)g(one)f(whic)o(h)g(is)g(more)g (accurate)225 2300 y(not)h(only)g(for)g(the)h(presen)o(tly)h (accessible)g Fr(Q)915 2285 y Fq(2)933 2300 y Ff(,)e(but)h(also)f(for)g (m)o(uc)o(h)g(larger)g(v)n(alues)g(b)q(efore)225 2350 y(starting)i(to)g(fail.)k(F)m(or)c(that)g(reason)h(it)f(is)g(imp)q (ortan)o(t)f(to)h(understand)h(the)g(asymptotic)225 2400 y(b)q(eha)o(vior)f(to)f(whic)o(h)h(the)h(BC)f(v)o(ertex)h(leads.)920 2490 y(3)p eop %%Page: 4 4 4 3 bop 225 432 a Fs(Ac)o(kno)o(wledgmen)o(ts)225 508 y Ff(D.K.)16 b(w)o(an)o(ts)h(to)g(thank)g(Da)o(vid)e(Blasc)o(hk)o(e)j (\(who,)f(as)g(the)h(w)o(orkshop)e(co)q(ordinator)h(se-)225 558 y(cured)k(supp)q(ort)g(and)f(made)f(D.K.'s)g(participation)h(p)q (ossible\),)h(F)m(ritjof)e(Karsc)o(h)i(and)225 608 y(Craig)c(Rob)q (erts)h(for)f(organizing)g(In)o(ternational)g(W)m(orkshop)f(on)i (Understanding)g(De-)225 658 y(con\014nemen)o(t)g(in)g(QCD,)g(ECT*)h(T) m(ren)o(to,)g(Italy)m(,)f(Marc)o(h)h(01)f(-)g(13,)h(1999,)f(where)i (this)225 707 y(w)o(ork)13 b(w)o(as)h(initiated,)f(and)g(to)h(P)m(.)f (T)m(andy)g(and)g(P)m(.)g(Maris)h(for)f(the)i(v)n(aluable)d (discussions)225 757 y(in)h(connection)i(with)f(this.)276 849 y(1.)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 Fs(457B)p Ff(,)13 b(359)h(\(1999\).) 276 898 y(2.)20 b(J.)13 b(Gron)o(b)q(erg)h Fb(et)h(al.)j Ff(\(CLEO)d(collab)q(oration\),)d(Ph)o(ys.)18 b(Rev.)g(D)c Fs(57)p Ff(,)f(33)g(\(1998\).)276 948 y(3.)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 Fs(22)p Ff(,)f(2157)g(\(1980\).)276 998 y(4.)20 b(A.)13 b(Manohar,)g(Ph)o(ys.)19 b(Lett.)f Fs(B244)p Ff(,)13 b(101)g(\(1990\).)276 1048 y(5.)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 Fs(481)p Ff(,)f(625)g(\(1996\).)276 1098 y(6.)20 b(P)m(.)13 b(T)m(andy)m(,)f(Fizik)n(a)h(B)h(\(Zagreb\))h Fs(8)p Ff(,)e(295)g(\(1999\).)276 1147 y(7.)20 b(D.)13 b(Kek)o(ez)i Fb(et)g(al.)p Ff(,)e(In)o(t.)18 b(J.)13 b(Mo)q(d.)18 b(Ph)o(ys.)h(A)14 b Fs(14)p Ff(,)f(161)g(\(1999\).)276 1197 y(8.)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 Fs(58)p Ff(,)g(096003)g(\(1998\).) 276 1247 y(9.)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 Fs(8)p Ff(,)e(303)h(\(1999\).)255 1297 y(10.)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 Fs(42)p Ff(,)f(4165)g(\(1990\).)255 1347 y(11.)20 b(C.)13 b(D.)g(Rob)q(erts,)h(Fizik)n(a)f(B)h(\(Zagreb\))h Fs(8)p Ff(,)e(285)g(\(1999\).)255 1396 y(12.)20 b(V.)14 b(A.)f(No)o(vik)o(o)o (v,)g(M.)g(A.)h(Shifman,)e(A.)i(I.)f(V)m(ainsh)o(tein,)h(M.)f(B.)h(V)m (oloshin)f(and)h(V.)329 1446 y(I.)f(Zakharo)o(v,)g(Nucl.)18 b(Ph)o(ys.)h(B)14 b Fs(237)p Ff(,)f(525)g(\(1984\).)255 1496 y(13.)20 b(P)m(.)13 b(Kessler)i(and)f(S.)g(Ong,)f(Ph)o(ys.)19 b(Rev.)e(D)d Fs(48)p Ff(,)f(R2974)f(\(1993\).)920 2490 y(4)p eop %%Trailer end end userdict /end-hook known{end-hook}if %%EOF