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-189 5413 811 4 v -97 5466 a Ft(a)-62 5490 y Fi(j.o.eeg@fys.uio.no)-100 5548 y Ft(b)-62 5571 y Fi(kkumer@ph)n(y)-6 b(.hr)-93 5627 y Ft(c)-62 5650 y Fi(picek@ph)n(y)g(.hr)1996 2517 y Fw(and)31 b(to)g(infer)f(on)h(the)g(ph)n(ysics)f(b)r(ey)n(ond)h(the)g(standard)f (mo)r(del)1996 2617 y(\(BSM\).)2121 2716 y(By)g(selecting)h(the)g(hea)n (vy-ligh)n(t)e(m)n(uonium)i(system)g Fm(\026)3872 2686 y Fv(+)3927 2716 y Fm(e)3966 2686 y Fk(\000)1996 2816 y Fw(\(where)36 b Fm(m)2350 2828 y Fu(\026)2431 2816 y Fl(\021)h Fm(M)45 b Fl(\035)37 b Fm(m)2852 2828 y Fu(e)2925 2816 y Fl(\021)f Fm(m)p Fw(\),)h(the)f(b)r(ound-state)g(calcu-)1996 2915 y(lation)f(corresp)r(onds)e(to)i(that)g(of)g(the)g(relativistic)f (h)n(ydrogen.)1996 3015 y(Thereb)n(y)i(w)n(e)g(distinguish)g(b)r(et)n (w)n(een)h(the)f(Coulom)n(b)g(\014eld)h(re-)1996 3115 y(sp)r(onsible)25 b(for)e(the)i(binding,)g(and)f(the)h(radiation)e (\014eld)i([6)p (#cite.Sa67) [[526 393 531 405] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])g(par-)1996 3214 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b(no)g(in)n(ten)n(tion)h(to)f(con)n(tribute)g(to)h(it)g(here.)2121 4211 y(This)g(t)n(w)n(o-step)e(metho)r(d)i(is)g(kno)n(wn)f(to)g(w)n (ork)g(w)n(ell)g(for)g(dis-)1996 4310 y(in)n(tegration)43 b(\(annihilation\))h(of)f(the)h(simplest)g(QED)f(atom,)1996 4410 y(p)r(ositronium.)37 b(Generalization)f(of)i(this)f(pro)r(cedure)f (to)i(m)n(uo-)1996 4510 y(nium)d(means)g(that)f(the)h(t)n(w)n(o-photon) e(deca)n(y)h(width)h(of)g(m)n(uo-)1996 4609 y(nium)29 b(is)e(obtained)g(b)n(y)h(using)2419 4838 y(\000)23 b(=)2592 4782 y Fl(j)p Fm( )s Fw(\(0\))p Fl(j)2801 4752 y Fv(2)2857 4782 y Fl(jM)p Fw(\()p Fm(\026)3062 4752 y Fv(+)3117 4782 y Fm(e)3156 4752 y Fk(\000)3235 4782 y Fl(!)g Fm(\015)5 b(\015)g Fw(\))p Fl(j)3492 4752 y Fv(2)p 2592 4819 937 4 v 2912 4895 a Fw(64)p Fm(\031)s(M)k(m)3562 4838 y(;)331 b Fw(\(1\))1996 5053 y(where)21 b Fl(j)p Fm( )s Fw(\(0\))p Fl(j)2439 5022 y Fv(2)2498 5053 y Fw(is)g(the)g(square)f(of)h(the)g(b)r (ound-state)g(w)n(a)n(v)n(e)f(func-)1996 5152 y(tion)28 b(at)g(the)g(origin.)e(After)j(this)f(factorization)e(has)h(b)r(een)h (p)r(er-)1996 5252 y(formed)37 b(the)g(rest)g(of)g(the)g(problem)f (reduces)h(to)f(the)i(ev)-5 b(alua-)1996 5351 y(tion)40 b(of)f(the)h(scattering-annihilation)e(in)n(v)-5 b(arian)n(t)38 b(amplitude)1996 5451 y Fl(M)p Fw(.)e(In)h(the)g(case)e(of)h(p)r (ositronium)g(this)h(expression)d(will)j(in-)1996 5551 y(v)n(olv)n(e)21 b(equal)g(masses)g(\()p Fm(M)9 b Fw(=)p Fm(m)p Fw(\),)22 b(and)g(the)g(in)n(v)-5 b(arian)n(t)21 b(amplitude)1996 5650 y(whic)n(h)g(for)f(the)h(p)r(ositronium)g (annihilation)f(at)h(rest)f(has)g(a)h(text-)1896 5899 y(1)p eop %%Page: 2 2 2 1 bop -189 48 a Fw(b)r(o)r(ok)27 b(form)h([8)p (#cite.Na90) [[100 761 105 773] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])53 289 y Fl(M)23 b Fw(=)300 233 y(i)p Fm(e)362 203 y Fv(2)p 274 270 152 4 v 274 346 a Fw(2)p Fm(m)389 322 y Fv(2)438 289 y Fw(\026)-45 b Fm(v)475 301 y Fu(s)511 289 y Fw(\()p Fm(p)585 301 y Fv(2)622 289 y Fw(\))654 197 y Fh(n)710 289 y Fm(\017)j(=)744 255 y Fk(\003)744 310 y Fv(2)782 289 y Fm(\017)g(=)816 255 y Fk(\003)816 310 y Fv(1)854 289 y Fm(k)j(=)900 301 y Fv(1)955 289 y Fw(+)18 b Fm(\017)-42 b(=)1072 255 y Fk(\003)1072 310 y Fv(1)1110 289 y Fm(\017)g(=)1144 255 y Fk(\003)1144 310 y Fv(2)1182 289 y Fm(k)j(=)1228 301 y Fv(2)1265 197 y Fh(o)1320 289 y Fm(u)1368 301 y Fu(r)1405 289 y Fw(\()p Fm(p)1479 301 y Fv(1)1516 289 y Fw(\))23 b Fm(:)136 b Fw(\(2\))-189 501 y(Only)37 b(the)i(an)n(tisymmetric)e(piece)g(in)h (the)g(decomp)r(osition)g(of)-189 600 y(the)28 b(pro)r(duct)g(of)f (three)h(gamma)e(matrices)h(ab)r(o)n(v)n(e)89 717 y Fh(n)47 b(o)269 809 y Fl(!)23 b Fw(i)p Fm(\017)432 775 y Fu(\026\027)t (\013\014)597 809 y Fm(\015)640 821 y Fv(5)678 809 y Fm(\015)721 821 y Fu(\014)765 809 y Fw(\()p Fm(k)840 821 y Fv(1)897 809 y Fl(\000)18 b Fm(k)1023 821 y Fv(2)1060 809 y Fw(\))1092 821 y Fu(\013)1140 809 y Fw(\()p Fm(\017)1206 775 y Fk(\003)1206 829 y Fv(1)1244 809 y Fw(\))1276 821 y Fu(\026)1321 809 y Fw(\()p Fm(\017)1387 775 y Fk(\003)1387 829 y Fv(2)1425 809 y Fw(\))1457 821 y Fu(\027)1521 809 y Fm(;)186 b Fw(\(3\))-189 1017 y(con)n(tributes)21 b(to)h(the)g(spin)g (singlet)g(parap)r(ositronium)e(t)n(w)n(o-pho-)-189 1117 y(ton)33 b(annihilation.)f(This)h(selects)f(\()p Fg(\017)984 1087 y Fk(\003)984 1138 y Fv(1)1044 1117 y Fl(\002)22 b Fg(\017)1171 1087 y Fk(\003)1171 1138 y Fv(2)1209 1117 y Fw(\),)33 b(a)f(CP-o)r(dd)g(con-)-189 1217 y(\014guration)26 b(of)i(the)g(\014nal)f(t)n(w)n(o-photon)g(state.)-64 1319 y(If)52 b(parap)r(ositronium)e(deca)n(y)i(can)f(serv)n(e)g(as)g (an)h(initial)-189 1419 y(b)r(enc)n(hmark)26 b(in)i(considering)e(QED)g (atom)h(annihilation,)g(then)-189 1518 y(its)49 b(QCD)f(coun)n(terpart) g(w)n(ould)g(b)r(e)h Fm(\031)1103 1488 y Fv(0)1198 1518 y Fl(!)58 b Fm(\015)5 b(\015)g Fw(.)48 b(Ho)n(w)n(ev)n(er,)-189 1618 y(the)41 b(latter)g(pro)r(cess)e(sho)n(ws)h(some)g(subtlet)n(y)-7 b(,)41 b(kno)n(wn)f(as)g(the)-189 1717 y(triangle)26 b(anomaly)-7 b(.)26 b(In)n(terestingly)g(enough,)h(this)g(quark)f(atom) -189 1817 y(double)f(radiativ)n(e)f(deca)n(y)h(can)g(also)f(b)r(e)i (view)n(ed)f(as)g(an)g(o\013-shell)-189 1917 y(e\013ect,)53 b(as)f(explained)h(in)f(some)h(detail)f(in)h([9)p (#cite.EeP94c) [[235 537 240 549] [1 1 1 [3 3]] [0 0 1]] pdfm (].)g (It)g(is)f(the)-189 2016 y(o\013-shellness)31 b(in)h(t)n(w)n(o-photon)e (annihilation)h(of)h(atoms)f(whic)n(h)-189 2116 y(w)n(e)c(further)h (explore)e(in)i(what)g(follo)n(ws.)-64 2218 y(The)38 b(pap)r(er)g(is)g(organized)f(as)h(follo)n(ws:)f(In)i(section)f(2)g(w)n (e)-189 2318 y(w)n(e)31 b(consider)f(the)i(quan)n(tum)f(\014eld)h (treatmen)n(t)f(of)g(the)h(annihi-)-189 2418 y(lation)c(pro)r(cess)f Fm(\026)387 2387 y Fv(+)442 2418 y Fm(e)481 2387 y Fk(\000)561 2418 y Fl(!)d Fm(\015)5 b(\015)32 b Fw(in)c(arbitrary)f(external)g (\014eld\(s\).)-189 2517 y(In)i(section)f(3)g(w)n(e)g(relate)f(the)i (binding)g(forces)e(to)i(the)f(external)-189 2617 y(\014elds)23 b(of)g(section)g(2.)f(In)i(section)e(4)h(w)n(e)f(consider)g(the)i (analogous)-189 2716 y(hea)n(vy-ligh)n(t)33 b(QCD)i(system,)g(and)f(in) h(section)g(5)f(w)n(e)g(giv)n(e)g(our)-189 2816 y(conclusions.)-189 3105 y Fn(2)135 b(Fla)l(v)l(our-c)l(hanging)48 b(op)t(erators)h(for)13 3255 y Ff(\026)83 3211 y Fw(+)152 3255 y Ff(e)206 3211 y Fl(\000)308 3255 y Fe(!)33 b Ff(\015)7 b(\015)-189 3442 y Fw(W)-7 b(e)41 b(treat)f(the)g(lepton)h(\015a)n(v)n(our-c)n (hanging)36 b(pro)r(cess)j(at)i(hand)-189 3541 y(analogously)30 b(to)i(the)h(quark)f(\015a)n(v)n(our)e(c)n(hange,)h(accoun)n(ted)h(for) -189 3641 y(b)n(y)41 b(the)h(electro)n(w)n(eak)d(theory)-7 b(.)41 b(Th)n(us,)g(the)g(double-radiativ)n(e)-189 3741 y(transition)k(is)h(triggered)f(b)n(y)g(t)n(w)n(o)h(classes)e(of)i (one-particle-)-189 3840 y(irreducible)20 b(diagrams)f(\(Fig.)i(1)p (#figure.1) [[163 306 168 318] [1 1 1 [3 3]] [0 0 1]] pdfm (a)e(and)i (Fig.)f(1)p (#figure.1) [[213 306 219 318] [1 1 1 [3 3]] [0 0 1]] pdfm (b\),)h(related)f(b)n(y)g(the)-189 3940 y(W)-7 b(ard)34 b(iden)n(tities.)h(After)g(in)n(tegrating)e(out)h(the)h(hea)n(vy)e (parti-)-189 4039 y(cles)c(in)h(the)g(lo)r(ops,)f(these)h(one-lo)r(op)e (electro)n(w)n(eak)f(transitions)-189 4139 y(can)g(b)r(e)h(com)n(bined) g(in)n(to)f(an)g(e\013ectiv)n(e)h(Lagrangian)d([1)p (#cite.EeP93) [[253 270 258 282] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(],)-116 4361 y Fl(L)p Fw(\()p Fm(e)f Fl(!)f Fm(\026)p Fw(\))224 4373 y Fu(\015)303 4361 y Fw(=)37 b Fm(B)18 b(\017)520 4326 y Fu(\026\027)t(\025\032)675 4361 y Fm(F)728 4373 y Fu(\026\027)824 4361 y Fw(\()868 4340 y(\026)856 4361 y(\011)23 b(i)1013 4287 y Fk($)990 4361 y Fm(D)1059 4373 y Fu(\025)1125 4361 y Fm(\015)1168 4373 y Fu(\032)1207 4361 y Fm(L)14 b( )s Fw(\))32 b(+)18 b(h.c.)23 b Fm(:)73 b Fw(\(4\))-189 4549 y(where)24 b(the)g(m)n(uon)g(and)g(the)h(electron) e(are)g(describ)r(ed)h(b)n(y)g(quan-)-189 4648 y(tum)38 b(\014elds)g(\011)h(=)h Fm( )482 4660 y Fu(\026)564 4648 y Fw(and)e Fm( )k Fw(=)e Fm( )991 4660 y Fu(e)1026 4648 y Fw(.)e(Corresp)r(ondingly)-7 b(,)36 b(for)-169 4727 y(\026)-189 4748 y Fm(B)-122 4718 y Fv(0)-126 4769 y Fu(s)-62 4748 y Fl(!)24 b Fw(2)p Fm(\015)5 b Fw(,)26 b(the)i(in)n(v)n(olv)n(ed)f(\014elds)g(are)g Fm( )1058 4760 y Fu(s)1116 4748 y Fw(=)c Fm(s)28 b Fw(and)f Fm( )1486 4760 y Fu(b)1543 4748 y Fw(=)22 b Fm(b)p Fw(.)-64 4850 y(In)37 b(our)f(case,)g(w)n(e)g(do)h(not)g(need)g(to)g(sp)r(ecify)g (the)g(ph)n(ysics)-189 4950 y(b)r(ehind)25 b(the)g(lepton-\015a)n(v)n (our-violating)c(transition)j(in)h(\(4)p (#equation.4) [[263 173 268 185] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)g(F) -7 b(or)-189 5050 y(instance,)40 b(the)h(strength)f Fm(B)k Fw(migh)n(t)c(con)n(tain)g(some)f(leptonic)-189 5149 y(parameters,)33 b(analogous)f(to)j(the)g(Cabibb)r(o-Koba)n(y)n (ashi-Mas-)-189 5249 y(k)-5 b(aw)n(a)27 b(parameters)e Fm(\025)496 5261 y Fv(CKM)687 5249 y Fw(in)i(the)h(quark)f(sector.)-64 5351 y(Keeping)44 b(in)i(mind)g(that)g(the)g(fermions)f(in)g(the)h(b)r (ound)-189 5451 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D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E131FA2EB0F 80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A414F0A213 03A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A485AA2485A 48C7FC120E5A5A5A5A5A16647BCA26>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A 78891B>46 D97 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc neet 128 4 /Fc 4 5 df[755 609 132 712 747 1 D[838 609 132 712 830 I[755 609 132 712 747 I[838 609 132 712 830 I E %EndDVIPSBitmapFont end TeXDict begin 1 0 bop 324 2544 a Fc(\001)401 2662 y Fb(\011)167 b Fa(W)684 2778 y Fb(.)684 2811 y(.)684 2844 y(.)922 2663 y Fa( )747 2156 y(\015)1372 2544 y Fc(\002)1473 2662 y Fb(\011)1543 2120 y Fa(\015)1734 2662 y(W)1774 2778 y Fb(.)1774 2811 y(.)1774 2844 y(.)2030 2663 y Fa( )1975 2120 y(\015)324 3488 y Fc(\003)401 3606 y Fb(\011)635 3737 y(\(a\))922 3607 y Fa( )478 3351 y(W)241 b(W)747 3100 y(\015)1372 3488 y Fc(\004)1473 3606 y Fb(\011)1543 3064 y Fa(\015)1722 3737 y Fb(\(b\))2030 3607 y Fa( )1975 3376 y(W)-574 b(W)1975 3064 y(\015)1734 3147 y(W)p eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial 1996 1648 a(Figure)e(1:)h(The)g(examples)f(of)g(the)i (one-particle-irreducible)c(di-)1996 1747 y(agrams)26 b(leading)h(to)g(the)h(double-radiativ)n(e)e(\015a)n(v)n(our-c)n (hanging)1996 1847 y(transitions.)k(Only)g(the)h(second-ro)n(w)d (diagrams)h(exist)h(for)g(the)1996 1946 y(leptonic)e(case.)1996 2181 y(\(4)p (#equation.4) [[315 505 320 517] [1 1 1 [3 3]] [0 0 1]] pdfm (\))34 b(obtained)g(within)g(p)r(erturbation)f(theory)g(splits)h(in)n(to)f (the)1996 2281 y(on-shell)27 b(magnetic)g(transition)g(op)r(erator)f Fl(L)3427 2293 y Fu(\033)2086 2426 y Fl(L)2143 2438 y Fu(\033)2188 2426 y Fw(\(1)p Fm(\015)5 b Fw(\))22 b(=)h Fm(B)2515 2438 y Fu(\033)2571 2405 y Fw(\026)2560 2426 y(\011)13 b(\()p Fm(M)c(\033)22 b Fl(\001)c Fm(F)12 b(L)18 b Fw(+)g Fm(m)c(\033)22 b Fl(\001)c Fm(F)12 b(R)q Fw(\))i Fm( )35 b Fw(+)18 b(h.c.)24 b Fm(;)103 b Fw(\(5\))1996 2570 y(and)28 b(an)f(o\013-shell)g(piece)h Fl(L)2845 2582 y Fu(F)2928 2570 y Fw([1)p (#cite.EeP93) [[426 459 431 471] [1 1 1 [3 3]] [0 0 1]] pdfm (])1996 2741 y Fl(L)2053 2753 y Fu(F)2132 2741 y Fw(=)22 b Fm(B)2282 2753 y Fu(F)2363 2720 y Fw(\026)2351 2741 y(\011[\(i)2521 2675 y Fk( )2518 2749 y Fm(D)2542 2741 y(=)12 b Fl(\000)g Fm(M)d Fw(\))14 b Fm(\033)h Fl(\001)d Fm(F)g(L)g Fw(+)g Fm(\033)j Fl(\001)d Fm(F)g(R)q Fw(\(i)q Fm(D)-53 b(=)12 b Fl(\000)g Fm(m)p Fw(\)])p Fm( )28 b Fw(+)12 b(h.c.)23 b Fm(;)3916 2840 y Fw(\(6\))1996 2940 y(where)f Fm(\033)10 b Fl(\001)d Fm(F)35 b Fw(denotes)21 b Fm(\033)2747 2952 y Fu(\026\027)2829 2940 y Fm(F)2894 2910 y Fu(\026\027)2976 2940 y Fw(,)h(and)g Fm(L)h Fw(=)f(\(1)7 b Fl(\000)g Fm(\015)3540 2952 y Fv(5)3577 2940 y Fw(\))p Fm(=)p Fw(2)22 b(and)f Fm(R)j Fw(=)1996 3040 y(\(1)g(+)f Fm(\015)2225 3052 y Fv(5)2263 3040 y Fw(\))p Fm(=)p Fw(2)34 b(denote)i(left-hand)g(and)f (righ)n(t-hand)g(pro)5 b(jectors.)1996 3139 y(T)-7 b(o)31 b(lo)n(w)n(est)g(order)f(in)h(QED)g(\(or)g(QCD\))h Fm(B)3366 3151 y Fu(F)3450 3139 y Fw(=)d Fm(B)3607 3151 y Fu(\033)3682 3139 y Fw(=)g Fm(B)t Fw(,)i(but)1996 3239 y(in)g(general)d(they)j(are)e (di\013eren)n(t)h(due)g(to)g(di\013eren)n(t)g(anomalous)1996 3338 y(dimensions)24 b(of)f(the)h(op)r(erators)e(in)h(\(5)p (#equation.5) [[448 366 453 378] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (and)g(\(6)p (#equation.6) [[483 366 488 378] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)f (\(The)h(o\013-shell)1996 3438 y(part)j Fl(L)2233 3450 y Fu(F)2317 3438 y Fw(has)g(zero)f(anomalous)g(dimension\).)2121 3538 y(By)k(decomp)r(osing)g(the)h(co)n(v)-5 b(arian)n(t)29 b(deriv)-5 b(ativ)n(e,)30 b(i)p Fm(D)-53 b(=)28 b Fw(=)g(i)p Fm(@)-37 b(=)20 b Fl(\000)1996 3637 y Fm(eA)-42 b(=)p Fw(,)38 b(in)g(the)g(o\013-shell)f(op)r(erator)f(\(6)p (#equation.6) [[445 331 450 343] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h(w) n(e)h(separate)e(the)i(one-)1996 3737 y(photon)28 b(piece)1996 3916 y Fl(L)2053 3928 y Fu(F)2109 3916 y Fw(\(1)p Fm(\015)5 b Fw(\))23 b(=)f Fm(B)2436 3928 y Fu(F)2517 3895 y Fw(\026)2505 3916 y(\011[\(i)2671 3837 y Fk( )2680 3916 y Fm(@)-37 b(=)31 b Fl(\000)p Fm(M)9 b Fw(\))14 b Fm(\033)s Fl(\001)p Fm(F)e(L)p Fw(+)p Fm(\033)s Fl(\001)p Fm(F)g(R)q Fw(\(i)p Fm(@)-37 b(=)p Fl(\000)p Fm(m)p Fw(\)])p Fm( )16 b Fw(+h.c.)23 b Fm(;)3916 4016 y Fw(\(7\))1996 4115 y(from)28 b(the)g(t)n(w)n (o-photon)e(piece)2038 4260 y Fl(L)2095 4272 y Fu(F)2150 4260 y Fw(\(2)p Fm(\015)5 b Fw(\))23 b(=)g Fm(B)2478 4272 y Fu(F)2558 4239 y Fw(\026)2547 4260 y(\011)o([)p Fl(\000)p Fm(eA)-42 b(=\033)9 b Fl(\001)d Fm(F)12 b(L)6 b Fw(+)g Fm(\033)j Fl(\001)d Fm(F)12 b(R)q Fw(\()p Fl(\000)p Fm(eA)-42 b(=)p Fw(\)])p Fm( )23 b Fw(+)6 b(h.c.)23 b Fm(:)42 b Fw(\(8\))1996 4404 y(The)37 b(amplitude)g(for)f(the)h(t)n(w)n (o-photon)e(diagram)h(\(Fig.)g(2)p (#figure.2) [[535 239 540 251] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(is) 1996 4504 y(giv)n(en)27 b(b)n(y)2380 4687 y Fm(A)2442 4699 y Fu(a)2505 4687 y Fw(=)c(i)2630 4574 y Fh(Z)2727 4687 y Fw(d)2773 4649 y Fv(4)2810 4687 y Fm(x)14 b Fl(L)2928 4699 y Fu(F)2984 4687 y Fw(\(2)p Fm(\015)5 b Fw(\))23 b(=)f Fm(A)3310 4652 y Fu(L)3310 4707 y(a)3379 4687 y Fw(+)c Fm(A)3524 4652 y Fu(R)3524 4707 y(a)3601 4687 y Fm(;)292 b Fw(\(9\))1996 4869 y(in)35 b(an)f(ob)n(vious)f(notation.)g (The)i(single-photon)e(o\013-shell)h(La-)1996 4969 y(grangian)26 b Fl(L)2395 4981 y Fu(F)2451 4969 y Fw(\(1)p Fm(\015)5 b Fw(\))28 b(leads)f(to)h(the)g(amplitude)h(with)f(the)h(hea)n(vy)1996 5068 y(particle)e(in)h(the)g(propagator)2000 5251 y Fm(A)2062 5263 y Fu(b)2121 5251 y Fw(=)c(i)14 b Fm(B)2310 5263 y Fu(F)2393 5138 y Fh(Z)23 b(Z)2559 5251 y Fw(d)2605 5214 y Fv(4)2642 5251 y Fm(x)14 b Fw(d)2750 5214 y Fv(4)2787 5251 y Fm(y)2856 5230 y Fw(\026)2845 5251 y(\011)o(\()p Fm(y)s Fw(\))3017 5159 y Fh(h)3075 5251 y Fl(\000)k Fw(i)p Fm(eA)-42 b(=)3282 5263 y Fv(2)3319 5251 y Fw(\()p Fm(y)s Fw(\))3427 5159 y Fh(i)3481 5251 y Fw(i)p Fm(S)3560 5208 y Fv(\()p Fu(\026)p Fv(\))3555 5275 y Fu(F)3656 5251 y Fw(\()p 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Fm(@)-37 b(=)1008 1481 y Fu(x)1072 1469 y Fl(\000)p Fm(M)9 b Fw(\))14 b Fm(\033)22 b Fl(\001)c Fm(F)1436 1481 y Fv(1)1474 1469 y Fw(\()p Fm(x)p Fw(\))p Fm(L)h Fw(+)-140 1702 y Fm(\033)j Fl(\001)c Fm(F)23 1714 y Fv(1)61 1702 y Fw(\()p Fm(x)p Fw(\))p Fm(R)q Fw(\(i)q Fm(@)-37 b(=)341 1714 y Fu(x)401 1702 y Fl(\000)18 b Fm(m)p Fw(\))589 1585 y Fh(\025)651 1702 y Fl(\002)32 b Fw(i)p Fm(S)827 1659 y Fv(\()p Fu(e)p Fv(\))822 1726 y Fu(F)914 1702 y Fw(\()p Fm(x)19 b Fl(\000)f Fm(y)s Fw(\))1171 1610 y Fh(h)1229 1702 y Fl(\000)g Fw(i)p Fm(eA)-42 b(=)1436 1714 y Fv(2)1473 1702 y Fw(\()p Fm(y)s Fw(\))1581 1610 y Fh(i)1621 1702 y Fm( )s Fw(\()p Fm(y)s Fw(\))23 b Fm(:)1689 1876 y Fw(\(11\))-189 2057 y(The)h(subscripts)f(1) g(and)g(2)g(distinguish)h(b)r(et)n(w)n(een)f(the)h(t)n(w)n(o)f(pho-) -189 2157 y(tons.)31 b(It)g(is)f(understo)r(o)r(d)h(that)g(a)f(term)h (with)g(the)g(1)d Fl($)g Fw(2)j(sub-)-189 2256 y(script)25 b(in)n(terc)n(hange)e(should)i(b)r(e)h(added)f(in)g(order)f(to)h(mak)n (e)f(our)-189 2356 y(result)j(symmetric)h(in)f(the)h(t)n(w)n(o)f (photons.)-64 2455 y(Within)50 b(the)f(quan)n(tum)g(\014eld)g (formalism,)f(the)h(sum)h(of)-189 2555 y(the)38 b(equations)g(\(9)p (#equation.9) [[117 460 123 472] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (\(10)p (#equation.10) [[138 460 148 472] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(and)h(\(11)p (#equation.11) [[180 460 190 472] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (describ)r(es)f(the)i(pro)r(cess)-189 2655 y Fm(\026)-139 2625 y Fv(+)-84 2655 y Fm(e)-45 2625 y Fk(\000)34 2655 y Fl(!)23 b Fm(\015)5 b(\015)g Fw(,)27 b(or)g Fm(\026)c Fl(!)g Fm(e\015)5 b(\015)g Fw(.)-64 2754 y(Let)35 b(us)h(no)n(w)f(b)r (e)h(v)n(ery)e(general,)g(and)h(assume)g(that)h(b)r(oth)-189 2854 y(particles)j(\()p Fm(e)i Fw(and)f Fm(\026)p Fw(\))g(feel)h(some)e (kind)h(of)h(external)e(\014eld\(s\))-189 2954 y(represen)n(ted)30 b(b)n(y)h Fm(V)423 2969 y Fv(\()p Fu(e)p Fv(\))542 2954 y Fw(and)g Fm(V)755 2969 y Fv(\()p Fu(\026)p Fv(\))852 2954 y Fw(,)g(and)g(ob)r(ey)g(one-b)r(o)r(dy)g(Dirac)-189 3053 y(equations)283 3166 y Fh(\002)317 3234 y Fw(i)p Fm(@)-37 b(=)18 b Fl(\000)h Fm(V)539 3249 y Fv(\()p Fu(i)p Fv(\))618 3234 y Fw(\()p Fm(x)p Fw(\))h Fl(\000)e Fm(m)905 3249 y Fv(\()p Fu(i)p Fv(\))984 3166 y Fh(\003)1019 3234 y Fm( )1073 3249 y Fv(\()p Fu(i)p Fv(\))1176 3234 y Fw(=)k(0)h Fm(;)338 b Fw(\(12\))-189 3414 y(for)21 b Fm(i)h Fw(=)h Fm(e)e Fw(or)f Fm(\026)h Fw(\(in)h(general)e Fm(V)748 3429 y Fv(\()p Fu(i)p Fv(\))851 3414 y Fw(=)i Fm(\015)981 3426 y Fu(\013)1043 3414 y Fm(V)1110 3384 y Fu(\013)1091 3441 y Fv(\()p Fu(i)p Fv(\))1171 3414 y Fw(\))f(,)g(and)g(accordingly) -189 3551 y(the)28 b(particle)f(propagators)e Fm(S)768 3508 y Fv(\()p Fu(i)p Fv(\))763 3575 y Fu(F)874 3551 y Fw(satisfy:)-60 3684 y Fh(\002)-25 3751 y Fw(i)p Fm(@)-37 b(=)18 b Fl(\000)g Fm(V)196 3766 y Fv(\()p Fu(i)p Fv(\))276 3751 y Fw(\()p Fm(x)p Fw(\))h Fl(\000)f Fm(m)562 3763 y Fu(i)590 3684 y Fh(\003)624 3751 y Fm(S)680 3708 y Fv(\()p Fu(i)p Fv(\))675 3775 y Fu(F)760 3751 y Fw(\()p Fm(x)h Fl(\000)f Fm(y)s Fw(\))23 b(=)f Fm(\016)1167 3717 y Fv(\(4\))1257 3751 y Fw(\()p Fm(x)d Fl(\000)f Fm(y)s Fw(\))23 b Fm(:)129 b Fw(\(13\))-189 3932 y(Our)25 b(photon)h(\014elds) g(en)n(ter)g(via)g(p)r(erturbativ)n(e)f(QED,)h(switc)n(hed)-189 4031 y(on)h(b)n(y)g(the)h(replacemen)n(t)e Fm(@)691 4043 y Fu(\026)759 4031 y Fl(!)d Fm(D)934 4043 y Fu(\026)1001 4031 y Fw(=)g Fm(@)1133 4043 y Fu(\026)1196 4031 y Fw(+)17 b Fm(ieA)1408 4043 y Fu(\026)1480 4031 y Fw(in)27 b(\(12)p (#equation.12) [[265 283 275 295] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)g (It)-189 4131 y(should)32 b(b)r(e)h(emphasized)f(that)g Fm(A)891 4143 y Fu(\026)936 4131 y Fw(\()p Fm(x)p Fw(\))i(represen)n (ts)c(the)j(radia-)-189 4230 y(tion)24 b(\014eld)h(and)f(do)r(es)g(not) g(include)g(binding)h(forces,)e(whic)n(h)h(will)-189 4330 y(in)k(the)g(next)g(section)f(b)r(e)h(related)f(to)g(the)h (external)f(\014elds)h Fm(V)1731 4345 y Fv(\()p Fu(i)p Fv(\))1811 4330 y Fw(.)-64 4430 y(No)n(w,)f(using)g(relations)f(\(12)p (#equation.12) [[160 235 170 247] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (and)f(\(13)p (#equation.13) [[201 235 211 247] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(w) n(e)h(obtain)-38 4610 y Fm(A)24 4622 y Fu(b)81 4610 y Fw(=)22 b Fl(\000)p Fm(A)295 4576 y Fu(L)295 4631 y(a)363 4610 y Fw(+)c(\001)p Fm(A)577 4622 y Fu(b)634 4610 y Fm(;)180 b(A)899 4622 y Fu(c)956 4610 y Fw(=)23 b Fl(\000)p Fm(A)1171 4576 y Fu(R)1171 4631 y(a)1244 4610 y Fw(+)18 b(\001)p Fm(A)1458 4622 y Fu(c)1515 4610 y Fm(;)151 b Fw(\(14\))-189 4791 y(resulting)32 b(in)h(a)g(partial)e(cancellation)h (when)h(the)g(amplitudes)-189 4891 y(are)27 b(summed)281 5071 y Fm(A)343 5083 y Fu(a)402 5071 y Fw(+)18 b Fm(A)547 5083 y Fu(b)599 5071 y Fw(+)g Fm(A)744 5083 y Fu(c)801 5071 y Fw(=)23 b(\001)p Fm(A)1020 5083 y Fu(b)1072 5071 y Fw(+)18 b(\001)p Fm(A)1286 5083 y Fu(c)1343 5071 y Fm(:)323 b Fw(\(15\))-189 5252 y(This)39 b(sho)n(ws)f(that)i(the)g(lo)r (cal)e(o\013-diagonal)g(fermion)h(seagull)-189 5351 y(transition)24 b(of)g(Fig.)h(2)p (#figure.2) [[125 125 130 137] [1 1 1 [3 3]] [0 0 1]] pdfm 24 w(cancels,)e(ev)n(en)h(if)h(the)g(external)f(fermions)-189 5451 y(are)31 b(o\013-shell.)h(The)g(left-o)n(v)n(er)f(quan)n(tities)g (\001)p Fm(A)1310 5463 y Fu(b)1376 5451 y Fw(and)h(\001)p Fm(A)1673 5463 y Fu(c)1740 5451 y Fw(in-)-189 5551 y(v)n(olv)n(e)27 b(the)i(in)n(tegrals)e(o)n(v)n(er)f(the)j(Coulom)n(b)f(p)r(oten)n(tial) g(and)g(rep-)-189 5650 y(resen)n(t)f(the)h(net)g(o\013-shell)f (e\013ect.)1959 723 y @beginspecial 100 @llx 370 @lly 460 @urx 500 @ury 2520 @rwi @setspecial %%BeginDocument: MbMc.ps %DVIPSCommandLine: dvips -f fig3.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.03.08:1416 /TeXDict 300 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 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7FA2EB0780EB03C01301EB00E014701438141C140C166476CA26>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E131FA2EB0F 80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A414F0A213 03A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A485AA2485A 48C7FC120E5A5A5A5A5A16647BCA26>I97 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd neet 128 2 /Fd 2 3 df[1003 666 7 756 1245 1 D[1005 653 7 732 1245 I E %EndDVIPSBitmapFont end TeXDict begin 1 0 bop 324 3016 a Fd(\001)884 3265 y Fc(\(a\))329 3109 y(\011)338 2390 y Fb( )1487 3084 y(\015)1538 3099 y Fa(2)1487 2390 y Fb(\015)1538 2405 y Fa(1)919 2365 y Fb(x)921 3108 y(y)1870 3016 y Fd(\002)2428 3265 y Fc(\(b\))1876 3109 y(\011)1885 2390 y Fb( )3034 3084 y(\015)3085 3099 y Fa(2)3034 2390 y Fb(\015)3085 2405 y Fa(1)2467 2366 y Fb(y)2465 3128 y(x)p eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial 1996 906 a(Figure)g(3:)g(The)h(shaded)f(b)r(o)n(xes)f (indicate)i(the)f(com)n(bination)g(of)1996 1005 y(the)39 b(unrotated)f(o\013-shell)h(transition)f(\()p Fl(\030)j Fm(B)3454 1017 y Fu(F)3509 1005 y Fw(\))e(and)f(the)h(on-)1996 1105 y(shell)27 b(magnetic)e(momen)n(t)i(transition)e(\()p Fl(\030)e Fm(B)3418 1117 y Fu(\033)3463 1105 y Fw(\),)k(giving)e(the)i (ef-)1996 1205 y(fectiv)n(e)h(v)n(ertex)f(in)g(Eq.)g(\(26)p (#equation.26) [[407 622 417 634] [1 1 1 [3 3]] [0 0 1]] pdfm (\).) 2121 1469 y(There)20 b(are)g(also)g(amplitudes)i Fm(A)3127 1481 y Fu(d)3187 1469 y Fw(and)e Fm(A)3403 1481 y Fu(e)3460 1469 y Fw(whic)n(h)h(are)f(coun-)1996 1568 y(terparts)31 b(of)i Fm(A)2477 1580 y Fu(b)2542 1568 y Fw(and)f Fm(A)2770 1580 y Fu(c)2837 1568 y Fw(when)g Fl(L)3115 1580 y Fu(F)3171 1568 y Fw(\(1)p Fm(\015)5 b Fw(\))32 b(is)g(replaced)f(b)n(y)h Fl(L)3954 1580 y Fu(\033)3999 1568 y Fw(.)1996 1668 y(The)44 b(total)f(con)n(tribution)g(from)g(our)g(\015a)n(v)n(our-c)n(hanging)d (La-)1996 1768 y(grangian)26 b(\()p Fl(L)2427 1780 y Fu(F)2510 1768 y Fw(and)i Fl(L)2729 1780 y Fu(\033)2801 1768 y Fw(parts\))g(is)f(then)h(giv)n(en)f(b)n(y)1998 1985 y Fm(A)2060 1997 y Fu(d)2118 1985 y Fw(+)18 b(\001)p Fm(A)2332 1997 y Fu(b)2391 1985 y Fw(=)38 b(i)2531 1872 y Fh(Z)23 b(Z)2697 1985 y Fw(d)2743 1948 y Fv(4)2780 1985 y Fm(x)14 b Fw(d)2888 1948 y Fv(4)2925 1985 y Fm(y)2994 1964 y Fw(\026)2982 1985 y(\011\()p Fm(y)s Fw(\))3155 1893 y Fh(h)3213 1985 y Fl(\000)k Fw(i)p Fm(eA)-42 b(=)3420 1997 y Fv(2)3457 1985 y Fw(\()p Fm(y)s Fw(\))3565 1893 y Fh(i)2914 2177 y Fl(\002)18 b Fw(i)p Fm(S)3076 2134 y Fv(\()p Fu(\026)p Fv(\))3071 2201 y Fu(F)3172 2177 y Fw(\()p Fm(y)j Fl(\000)d Fm(x)p Fw(\))c Fm(Q)p Fw(\()p Fm(x)p Fw(\))g Fm( )s Fw(\()p Fm(x)p Fw(\))25 b Fm(;)g Fw(\(16\))1996 2352 y(represen)n(ted)i(b)n(y)g(Fig.)h(3)p (#figure.3) [[399 485 404 497] [1 1 1 [3 3]] [0 0 1]] pdfm (a,)e(and)i (a)f(similar)g(one)1996 2565 y Fm(A)2058 2577 y Fu(e)2113 2565 y Fw(+)18 b(\001)p Fm(A)2327 2577 y Fu(c)2386 2565 y Fw(=)38 b(i)2526 2452 y Fh(Z)23 b(Z)2692 2565 y Fw(d)2738 2528 y Fv(4)2776 2565 y Fm(x)14 b Fw(d)2883 2528 y Fv(4)2920 2565 y Fm(y)2989 2545 y Fw(\026)2978 2565 y(\011)o(\()p Fm(x)p Fw(\))g Fm(Q)p Fw(\()p Fm(x)p Fw(\))g(i)r Fm(S)3439 2522 y Fv(\()p Fu(e)p Fv(\))3434 2590 y Fu(F)3526 2565 y Fw(\()p Fm(x)19 b Fl(\000)f Fm(y)s Fw(\))2909 2766 y Fl(\002)2992 2674 y Fh(h)3050 2766 y Fl(\000)g Fw(i)p Fm(eA)-42 b(=)3257 2778 y Fv(2)3294 2766 y Fw(\()p Fm(y)s Fw(\))3402 2674 y Fh(i)3441 2766 y Fm( )s Fw(\()p Fm(y)s Fw(\))24 b Fm(;)232 b Fw(\(17\))1996 2966 y(corresp)r(onding)37 b(to)i(Fig.)g(3)p (#figure.3) [[412 411 417 423] [1 1 1 [3 3]] [0 0 1]] pdfm (b.)f(The)h (op)r(erator)e Fm(Q)p Fw(\()p Fm(x)p Fw(\))j(in)f(these)1996 3066 y(expressions)26 b(reads)2153 3241 y Fm(Q)p Fw(\()p Fm(x)p Fw(\))e(=)2442 3174 y Fh(\002)2476 3241 y Fm(B)2539 3253 y Fu(\033)2584 3241 y Fm(M)j Fw(+)18 b Fm(B)2838 3253 y Fu(F)2893 3241 y Fm(V)2941 3256 y Fv(\()p Fu(\026)p Fv(\))3038 3241 y Fw(\()p Fm(x)p Fw(\))3149 3174 y Fh(\003)3199 3241 y Fm(\033)j Fl(\001)e Fm(F)3362 3253 y Fv(1)3400 3241 y Fw(\()p Fm(x)p Fw(\))p Fm(L)2637 3374 y Fw(+)f Fm(\033)k Fl(\001)c Fm(F)2883 3386 y Fv(1)2921 3374 y Fw(\()p Fm(x)p Fw(\))p Fm(R)3110 3307 y Fh(\002)3145 3374 y Fm(B)3208 3386 y Fu(\033)3253 3374 y Fm(m)g Fw(+)g Fm(B)3490 3386 y Fu(F)3545 3374 y Fm(V)3593 3389 y Fv(\()p Fu(e)p Fv(\))3681 3374 y Fw(\()p Fm(x)p Fw(\))3792 3307 y Fh(\003)3842 3374 y Fm(:)9 b Fw(\(18\))1996 3550 y(The)36 b(result)g(giv)n(en)f(b)n(y)h(Eqs.)f(\(16)p (#equation.16) [[431 341 441 353] [1 1 1 [3 3]] [0 0 1]] pdfm (\){\(18)p (#equation.18) [[453 341 463 353] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (can)h(also)f(b)r(e)h(under-)1996 3649 y(sto)r(o)r(d)21 b(in)h(terms)f(of)g(the)h(follo)n(wing)e(\014eld)h(rede\014nition.)g (Eq.)g(\(12)p (#equation.12) [[541 329 551 341] [1 1 1 [3 3]] [0 0 1]] pdfm (\))1996 3749 y(can)28 b(b)r(e)g(obtained)f(from)g(the)h(Lagrangian)2014 3924 y Fl(L)2071 3936 y Fu(D)2131 3924 y Fw(\(\011)p Fm(;)14 b( )s Fw(\))24 b(=)2477 3903 y(\026)2465 3924 y(\011)2530 3857 y Fh(\002)2564 3924 y Fw(i)p Fm(D)-53 b(=)18 b Fl(\000)g Fm(V)2794 3939 y Fv(\()p Fu(\026)p Fv(\))2909 3924 y Fl(\000)g Fm(M)3082 3857 y Fh(\003)3117 3924 y Fw(\011)g(+)3300 3902 y(\026)3283 3924 y Fm( )3340 3857 y Fh(\002)3374 3924 y Fw(i)p Fm(D)-53 b(=)18 b Fl(\000)g Fm(V)3604 3939 y Fv(\()p Fu(e)p Fv(\))3710 3924 y Fl(\000)g Fm(m)3866 3857 y Fh(\003)3901 3924 y Fm( )26 b(:)3874 4024 y Fw(\(19\))1996 4124 y(No)n(w,)i(b)n(y)f(de\014ning)g(new)h (\014elds)2038 4299 y(\011)2103 4265 y Fk(0)2149 4299 y Fw(=)22 b(\011)10 b(+)g Fm(B)2449 4311 y Fu(F)2517 4299 y Fm(\033)j Fl(\001)d Fm(F)26 b(L )g(;)180 b( )3086 4265 y Fk(0)3132 4299 y Fw(=)23 b Fm( )13 b Fw(+)d Fm(B)3429 4265 y Fk(\003)3425 4320 y Fu(F)3493 4299 y Fm(\033)j Fl(\001)d Fm(F)i(L)p Fw(\011)23 b Fm(;)55 b Fw(\(20\))1996 4474 y(w)n(e)28 b(obtain)2216 4650 y Fl(L)2273 4662 y Fu(D)2333 4650 y Fw(\()p Fm( )s(;)14 b Fw(\011\))19 b(+)f Fl(L)2715 4662 y Fu(F)2807 4650 y Fw(=)37 b Fl(L)2966 4662 y Fu(D)3026 4650 y Fw(\()p Fm( )3115 4616 y Fk(0)3139 4650 y Fm(;)14 b Fw(\011)3241 4616 y Fk(0)3264 4650 y Fw(\))32 b(+)18 b(\001)p Fl(L)3537 4662 y Fu(B)3618 4650 y Fm(;)233 b Fw(\(21\))1996 4825 y(whic)n(h)26 b(sho)n(ws)e(that)i Fl(L)2703 4837 y Fu(F)2784 4825 y Fw(can)f(b)r(e)h(transformed)e(a)n(w) n(a)n(y)f(from)i(the)1996 4925 y(p)r(erturbativ)n(e)i(terms,)h(but)g(a) f(relic)g(of)h(it,)2038 5100 y(\001)p Fl(L)2164 5112 y Fu(B)2258 5100 y Fw(=)37 b Fm(B)2423 5112 y Fu(F)2503 5079 y Fw(\026)2492 5100 y(\011)2556 5033 y Fh(\002)2591 5100 y Fm(V)2639 5115 y Fv(\()p Fu(\026)p Fv(\))2736 5100 y Fm(\033)16 b Fl(\001)c Fm(F)g(L)26 b Fw(+)f Fm(\033)16 b Fl(\001)c Fm(F)g(R)j(V)3361 5115 y Fv(\()p Fu(e)p Fv(\))3449 5033 y Fh(\003)3483 5100 y Fm( )30 b Fw(+)12 b(h.c.)23 b Fm(;)55 b Fw(\(22\))1996 5276 y(remains)39 b(in)g(the)h(b)r(ound)f (state)g(dynamics)g(.)h(Th)n(us,)f(the)g(o\013-)1996 5375 y(shell)g(e\013ects)f(are)f(non-zero)g(for)g(b)r(ound)i(external)e (fermions.)1996 5475 y(Com)n(bining)27 b(\001)p Fl(L)2544 5487 y Fu(B)2630 5475 y Fw(and)g Fl(L)2848 5487 y Fu(\033)2893 5475 y Fw(,)h(w)n(e)f(obtain)2464 5650 y(\001)p Fl(L)2590 5662 y Fu(B)2679 5650 y Fw(+)32 b Fl(L)2833 5662 y Fu(\033)2924 5650 y Fw(=)3047 5629 y(\026)3035 5650 y(\011)14 b Fm(Q)g( )34 b Fw(+)18 b(h.c.)24 b Fm(;)333 b Fw(\(23\))1896 5899 y(3)p eop %%Page: 4 4 4 3 bop -189 48 a Fw(where)27 b Fm(Q)h Fw(is)g(giv)n(en)e(b)n(y)i(\(18) p (#equation.18) [[143 761 153 773] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)g (This)f(sho)n(ws)g(the)h(equiv)-5 b(alence)28 b(of)-189 148 y(this)i(\014eld)h(rede\014nition)f(pro)r(cedure)f(and)g(the)i (result)f(giv)n(en)f(b)n(y)-189 247 y(Eqs.)e(\(16)p (#equation.16) [[75 737 85 749] [1 1 1 [3 3]] [0 0 1]] pdfm (\){\(18)p (#equation.18) [[98 737 108 749] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)-64 347 y(This)20 b(is)g(ho)n(w)f(far)h(w)n(e)f(can)h(push)g(the)h(problem) e(within)i(quan-)-189 446 y(tum)32 b(\014eld)f(theory)-7 b(.)30 b(Up)h(to)g(no)n(w)f(w)n(e)h(ha)n(v)n(e)e(made)i(no)f(appro)n (xi-)-189 546 y(mations)c(except)h(for)f(standard)g(p)r(erturbation)g (theory)-7 b(.)26 b(In)h(the)-189 646 y(next)k(section)g(w)n(e)f(will)h (adapt)g(the)g(results)f(of)h(this)g(section)g(to)-189 745 y(the)d(relev)-5 b(an)n(t)27 b(b)r(ound)h(state)f(e\013ects.)-189 1019 y Fn(3)135 b(O\013-shellness)82 b(in)g(the)f(m)l(uonium)13 1168 y(annihilation)46 b(amplitude)-189 1350 y Fw(As)35 b(announced)g(in)h(the)f(In)n(tro)r(duction,)g(w)n(e)g(c)n(ho)r(ose)f (the)i(sim-)-189 1449 y(plest)49 b(hea)n(vy-ligh)n(t)f(QED)h(atom,)f(m) n(uonium.)i(Naiv)n(ely)-7 b(,)48 b(the)-189 1549 y(pro)r(duct)137 1528 y(\026)126 1549 y(\011)13 b Fm( )34 b Fw(corresp)r(onds)29 b(to)i(the)h(b)r(ound)f(state)g(of)g Fm(\026)1617 1519 y Fv(+)1703 1549 y Fw(and)-189 1649 y Fm(e)-150 1619 y Fk(\000)-67 1649 y Fw(\(whic)n(h)c(migh)n(t)g(b)r(e)g(true)g(only)g (in)g(the)h(asymptotically)e(free)-189 1748 y(case\).)i(Ho)n(w)n(ev)n (er,)e(relativistic)i(b)r(ound)h(state)f(ph)n(ysics)g(is)g(a)g(dif-) -189 1848 y(\014cult)44 b(sub)5 b(ject,)43 b(out)g(of)g(our)f(scop)r (e.)g(W)-7 b(e)44 b(will)f(stic)n(k)f(to)h(the)-189 1948 y(t)n(w)n(o-step)27 b(pro)r(cedure)g([7)p (#cite.ItZ80) [[138 533 143 545] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])h(as)f(explained)h(in)g(the)g(In)n(tro)r(duction.)-189 2047 y(W)-7 b(e)28 b(p)r(erform)f(the)h(calculations)e(in)i(the)f(m)n (uonium)h(rest)f(frame)-189 2147 y(\(CM)38 b(frame)g(of)f Fm(\026)415 2117 y Fv(+)508 2147 y Fw(and)h Fm(e)719 2117 y Fk(\000)775 2147 y Fw(\))g(where)f(w)n(e)h(put)g(the)g(external) -189 2246 y(\014eld\(s\))c(equal)g(to)f(a)h(m)n(utual)f(Coulom)n(b)g (\014eld,)h Fm(V)1408 2261 y Fv(\()p Fu(i)p Fv(\))1522 2246 y Fl(!)f Fm(\015)1681 2258 y Fv(0)1732 2246 y Fm(V)1780 2258 y Fu(C)-189 2346 y Fw(\(where)f Fm(V)136 2358 y Fu(C)222 2346 y Fw(=)e Fl(\000)p Fm(e)421 2316 y Fv(2)458 2346 y Fm(=)p Fw(4)p Fm(\031)s(r)r Fw(\).)i(In)g(calculating)f(the)i Fm(\026)1448 2316 y Fv(+)1503 2346 y Fm(e)1542 2316 y Fk(\000)1628 2346 y Fl(!)d Fm(\015)5 b(\015)-189 2446 y Fw(amplitude)33 b(in)f(momen)n(tum)g(space,)g(w)n(e)g(tak)n(e)f(for)h Fm(V)1490 2458 y Fu(C)1578 2446 y Fw(the)h(a)n(v-)-189 2545 y(erage)c(o)n(v)n(er)g(solutions)h(in)h(the)g(Coulom)n(b)e(p)r (oten)n(tial,)i(whic)n(h)g(is)-189 2645 y Fl(h)p Fm(V)-109 2657 y Fu(C)-52 2645 y Fl(i)h Fw(=)f Fl(\000)p Fw(\()p Fm(m\013)331 2615 y Fv(2)368 2645 y Fm(=)p Fw(2\).)h(In)h(this)g(w)n(a) n(y)f(the)h(m)n(uonium-deca)n(y)f(in-)-189 2745 y(v)-5 b(arian)n(t)26 b(amplitude)h(acquires)f(the)h(form)f(whic)n(h)h(is)g(a) f(straigh)n(t-)-189 2844 y(forw)n(ard)31 b(generalization)g(of)i(the)g (p)r(ositronium-deca)n(y)f(in)n(v)-5 b(ari-)-189 2944 y(an)n(t)27 b(amplitude)h(\(2)p (#equation.2) [[118 414 123 426] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(in) g(momen)n(tum)g(space.)-64 3043 y(The)36 b(amplitudes)h Fm(A)611 3055 y Fu(d)674 3043 y Fw(+)24 b(\001)p Fm(A)894 3055 y Fu(b)964 3043 y Fw(from)36 b(Eq.)g(\(16)p (#equation.16) [[235 402 245 414] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (together)-189 3143 y(with)22 b Fm(A)56 3155 y Fu(e)98 3143 y Fw(+)6 b(\001)p Fm(A)300 3155 y Fu(c)356 3143 y Fw(from)21 b(\(17)p (#equation.17) [[141 390 151 402] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (transformed)f(to)h(the)h(momen)n(tum)-189 3243 y(space)27 b(tak)n(e)g(the)h(form)-189 3455 y Fl(M)d Fw(=)35 3399 y(2)p Fm(eB)179 3411 y Fu(\033)p 35 3436 188 4 v 93 3512 a Fm(m)236 3455 y Fw(\026)-45 b Fm(v)273 3467 y Fu(\026)318 3455 y Fw(\()p Fm(p)392 3467 y Fv(2)429 3455 y Fw(\))461 3363 y Fh(n)535 3399 y Fm(m)p 527 3436 90 4 v 527 3512 a(M)626 3455 y(k)-39 b(=)672 3467 y Fv(2)709 3455 y Fm(\017)d(=)743 3421 y Fk(\003)743 3476 y Fv(2)781 3455 y Fw(P)18 b Fl(\000)g Fw(P)p Fm(\017)-42 b(=)1030 3421 y Fk(\003)1030 3476 y Fv(2)1067 3455 y Fm(k)j(=)1113 3467 y Fv(2)1169 3455 y Fw(+)18 b(\(1)23 b Fl($)g Fw(2\))1529 3363 y Fh(o)1584 3455 y Fm(u)1632 3467 y Fu(e)1667 3455 y Fw(\()p Fm(p)1741 3467 y Fv(1)1779 3455 y Fw(\))p Fm(;)1700 3608 y Fw(\(24\))-189 3783 y(where)h Fm(v)88 3795 y Fu(\026)157 3783 y Fw(and)g Fm(u)363 3795 y Fu(e)423 3783 y Fw(are)f(m)n(uon)h(and)h(electron)e (spinors,)g(and)i Fm(\017)1747 3753 y Fk(\003)1747 3804 y Fv(1)p Fu(;)p Fv(2)-189 3883 y Fw(are)j(photon)g(p)r(olarization)f(v) n(ectors.)g(The)i(factor,)e(incorp)r(orat-)-189 3982 y(ing)f(the)g(binding)g(in)f(the)i(form)e(of)h(a)f(four-v)n(ector)f Fm(U)1464 3952 y Fu(\013)1534 3982 y Fw(=)e(\()p Fm(\032;)14 b Fd(0)p Fw(\),)119 4157 y(P)23 b Fl(\021)g Fw(\(1)18 b Fl(\000)g Fm(xU)-46 b(=)p Fw(\))p Fm(k)-39 b(=)640 4169 y Fv(1)677 4157 y Fm(\017)d(=)711 4123 y Fk(\003)711 4178 y Fv(1)749 4157 y Fm(L)17 b Fw(+)i Fm(xk)-39 b(=)1000 4169 y Fv(1)1037 4157 y Fm(\017)d(=)1071 4123 y Fk(\003)1071 4178 y Fv(1)1109 4157 y Fm(R)q Fw(\(1)18 b Fl(\000)g Fm(U)-46 b(=)p Fw(\))22 b Fm(;)211 b Fw(\(25\))-189 4332 y(accoun)n(ts)26 b(for)h(the)h(aforemen)n(tioned)f(factorization)f(of)h (a)g(bind-)-189 4432 y(ing)d(and)g(a)g(deca)n(y)-7 b(,)23 b(and)h(is)g(represen)n(ted)f(b)n(y)h(the)g(shaded)g(b)r(o)n(x)g(of) -189 4532 y(Fig.)k(3)p (#figure.3) [[70 223 75 235] [1 1 1 [3 3]] [0 0 1]] pdfm (:)-81 4635 y Fh(")-32 4777 y Fm(M)9 b Fw(\(1)18 b Fl(\000)g Fm(x\032\015)371 4742 y Fv(0)408 4777 y Fw(\))p Fm(\033)k Fl(\001)d Fm(F)604 4789 y Fv(1)641 4777 y Fm(L)f Fw(+)g Fm(m)c(\033)22 b Fl(\001)c Fm(F)1049 4789 y Fv(1)1087 4777 y Fm(R)q Fw(\(1)g Fl(\000)g Fm(\032\015)1417 4742 y Fv(0)1454 4777 y Fw(\))1486 4635 y Fh(#)1557 4777 y Fm(:)109 b Fw(\(26\))-189 5022 y(Here)23 b(w)n(e)f(in)n(tro)r(duced)h (abbreviations)f(for)g(t)n(w)n(o)h(small)f(constan)n(t)-189 5122 y(parameters,)277 5340 y Fm(x)i Fl(\021)454 5284 y Fm(m)p 446 5321 V 446 5397 a(M)568 5340 y(;)180 b(\032)23 b Fl(\021)g(\000)1013 5284 y Fm(B)1076 5296 y Fu(F)1145 5284 y Fl(h)p Fm(V)1225 5296 y Fu(C)1282 5284 y Fl(i)p 1013 5321 301 4 v 1073 5397 a Fm(mB)1209 5409 y Fu(\033)1347 5340 y Fm(;)319 b Fw(\(27\))-189 5551 y(in)33 b(terms)g(of)f(whic)n(h)h (the)g(sough)n(t)f(o\013-shell)h(e\013ect)g(will)g(b)r(e)g(ex-)-189 5650 y(pressed.)f(Note)g(that)h(in)f(the)h(e\013ectiv)n(e)g(in)n (teraction)e(\(26)p (#equation.26) [[258 89 268 101] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (the)1996 48 y(left-handed)25 b(part)e(corresp)r(onding)f(to)i Fm(V)3281 63 y Fv(\()p Fu(\026)p Fv(\))3402 48 y Fw(has)g(gotten)g(an)f (ex-)1996 148 y(tra)36 b(suppression)f(factor)g Fm(x)j Fw(=)f Fm(m=)-5 b(M)45 b Fw(in)36 b(fron)n(t)g(of)g(the)h(bind-)1996 247 y(ing)26 b(factor)f Fm(\032)p Fw(,)h(in)g(agreemen)n(t)f(with)h (the)h(exp)r(ectation)e(that)h(the)1996 347 y(hea)n(vy)i(particle)g(\() p Fm(\026)2618 317 y Fv(+)2674 347 y Fw(\))h(is)g(appro)n(ximately)e (free,)h(and)h(the)g(ligh)n(t)1996 446 y(particle)36 b(\()p Fm(e)2376 416 y Fk(\000)2432 446 y Fw(\))i(is)e(appro)n (ximately)f(the)i(reduced)f(particle,)h(in)1996 546 y(analogy)26 b(with)i(the)g(H-atom.)2121 651 y(The)35 b(annihilation)f(amplitude)h (\(24)p (#equation.24) [[456 689 466 701] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(can)g(no)n(w)g(b)r(e)h(ev)-5 b(alu-)1996 751 y(ated)33 b(explicitly)-7 b(.)33 b(A)g(tedious)g(calculation,)f(p)r(erformed)g (in)h(the)1996 851 y(m)n(uonium)e(rest)f(frame)f(with)i(photons)f (emitted)h(along)e(the)i Fm(z)t Fw(-)1996 950 y(axis,)c(giv)n(es)2134 1208 y Fl(M)d Fw(=)h Fl(\000)p Fw(2)p Fm(eB)2557 1220 y Fu(\033)2600 1208 y Fm(M)2690 1173 y Fv(2)2727 1076 y Fh(r)p 2810 1076 152 4 v 2820 1152 a Fw(2)p Fm(M)p 2820 1189 132 4 v 2849 1265 a(m)2961 1116 y Fh(h)3000 1208 y Fw(\(1)18 b(+)h Fm(x\032)p Fw(\))p Fg(\017)3338 1173 y Fk(\003)3338 1228 y Fv(2)3395 1208 y Fl(\001)f Fg(\017)3477 1173 y Fk(\003)3477 1228 y Fv(1)2283 1400 y Fw(+)g(i\(1)g(+)h(2)p Fm(x)f Fw(+)g Fm(x\032)p Fw(\)\()p Fg(\017)2950 1366 y Fk(\003)2950 1421 y Fv(2)3007 1400 y Fl(\002)g Fg(\017)3130 1366 y Fk(\003)3130 1421 y Fv(1)3168 1400 y Fw(\))h Fl(\001)3265 1378 y Fw(^)3260 1400 y Fg(k)3312 1412 y Fv(1)3368 1400 y Fw(+)f Fl(O)r Fw(\()p Fm(\032)3594 1366 y Fv(2)3631 1400 y Fm(;)c(x)3715 1366 y Fv(2)3753 1400 y Fw(\))3785 1308 y Fh(i)3848 1400 y Fm(;)s Fw(\(28\))1996 1615 y(where)32 b(w)n(e)f(ha)n(v)n(e)f(k)n(ept)i(only)f(the)h(leading)g (terms)f(in)h Fm(\032)g Fw(and)f Fm(x)p Fw(.)1996 1714 y(In)k(comparison)d(to)i(the)h(expressions)e(\(2)p (#equation.2) [[466 561 471 573] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (and)g(\(3)p (#equation.3) [[503 561 508 573] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (for)e(para-)1996 1814 y(p)r(ositronium,)27 b(w)n(e)g(notice)g(that)h (in)f(addition)g(to)g Fg(\017)3593 1784 y Fk(\003)3593 1834 y Fv(2)3649 1814 y Fl(\002)17 b Fg(\017)3772 1784 y Fk(\003)3772 1834 y Fv(1)3837 1814 y Fw(there)1996 1913 y(app)r(ears)k(also)g Fg(\017)2499 1883 y Fk(\003)2499 1934 y Fv(2)2545 1913 y Fl(\001)8 b Fg(\017)2616 1883 y Fk(\003)2616 1934 y Fv(1)2654 1913 y Fw(,)22 b(a)g(CP-ev)n(en)f(t)n (w)n(o-photon)f(con\014guration.)2121 2019 y(The)39 b(explicit)h (expression)e(for)g Fm(\032)i Fw(dep)r(ends)f(on)g(some)g(as-)1996 2118 y(sumptions.)51 b(As)f(explained)h(previously)-7 b(,)49 b(w)n(e)h(use)g Fl(h)p Fm(V)3807 2130 y Fu(C)3864 2118 y Fl(i)61 b Fw(=)1996 2218 y Fl(\000)p Fm(m\013)2187 2188 y Fv(2)2224 2218 y Fm(=)p Fw(2)25 b(whic)n(h)h(giv)n(es)e Fm(\032)f Fw(=)g Fm(\013)2977 2188 y Fv(2)3015 2218 y Fm(=)p Fw(2)i(for)g Fm(B)3312 2230 y Fu(\033)3379 2218 y Fw(=)e Fm(B)3530 2230 y Fu(F)3608 2218 y Fw(=)g Fm(B)t Fw(,)j(whic)n(h)1996 2318 y(is)i(a)f(go)r(o)r(d)g(appro)n(ximation)f (in)i(the)g(leptonic)f(case.)1996 2423 y(Eq.\(1)p (#equation.1) [[330 476 335 488] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (\014nally)h(giv)n(es)2382 2670 y(\000)22 b(=)2554 2614 y(2)p Fm(\013M)2739 2584 y Fv(4)p 2554 2651 222 4 v 2610 2727 a Fm(m)2683 2703 y Fv(2)2786 2670 y Fl(j)p Fm( )s Fw(\(0\))p Fl(j)2995 2636 y Fv(2)3032 2670 y Fl(j)p Fm(B)3118 2682 y Fu(\033)3163 2670 y Fl(j)3186 2636 y Fv(2)3237 2670 y Fw(\(1)c(+)g(2)p Fm(x\032)p Fw(\))38 b Fm(:)237 b Fw(\(29\))1996 2887 y(Th)n(us,)32 b(for)g(m)n(uonium,)g(the)g(sough)n (t)f(o\013-shell)h(con)n(tribution)f(is)1996 2987 y(only)22 b(a)f(tin)n(y)h(correction,)f(2)p Fm(x\032)i Fw(=)f Fm(\013)3102 2957 y Fv(2)3140 2987 y Fm(m=)-5 b(M)31 b Fl(')23 b Fw(2)p Fm(:)p Fw(6)7 b Fl(\001)g Fw(10)3678 2957 y Fk(\000)p Fv(7)3766 2987 y Fw(,)22 b(to)f(the)1996 3087 y(magnetic)39 b(momen)n(t)g(dominated)g(rate)3261 3056 y Fv(1)p (#Hfootnote.1) [[463 397 468 409] [1 1 1 [3 3]] [0 0 1]] pdfm 3297 3087 a Fw(.)h(Ho)n(w)n(ev)n(er,)d(the)i(cor-)1996 3186 y(resp)r(onding)34 b(o\013-shellness)g(in)i(a)e(strongly)g(b)r (ound)h(QCD)g(sys-)1996 3286 y(tem)28 b(should)f(b)r(e)h(signi\014can)n (tly)f(larger.)f(W)-7 b(e)27 b(also)g(tak)n(e)f(in)n(to)i(ac-)1996 3385 y(coun)n(t)35 b(the)h Fm(B)2446 3397 y Fu(F)2501 3385 y Fm(=B)2606 3397 y Fu(\033)2685 3385 y Fw(correction)e(in)h(\(29) p (#equation.29) [[458 361 468 373] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (when)g(considering)1996 3485 y(the)2159 3464 y(\026)2139 3485 y Fm(B)2206 3455 y Fv(0)2202 3506 y Fu(s)2267 3485 y Fl(!)23 b Fm(\015)5 b(\015)32 b Fw(deca)n(y)26 b(b)r(elo)n(w.)2121 3590 y(Before)36 b(ending)h(this)g(section,)g(w)n(e)g(should)g(also)f (men)n(tion)1996 3690 y(that)28 b(the)g(Lagrangian)c(giv)n(en)j(b)n(y)f (\(18)p (#equation.18) [[446 324 456 336] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(and)g(\(23)p (#equation.23) [[486 324 496 336] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (can)g(b)r(e)h(used)1996 3790 y(to)21 b(calculate)g(the)g(amplitude)g (for)g(m)n(uonic)g(h)n(ydrogen)e(deca)n(ying)1996 3889 y(to)28 b(a)g(photon)f(and)h(ordinary)e(h)n(ydrogen,)h(that)h(is,)g (the)g(pro)r(cess)1996 3989 y Fm(\026)2046 3959 y Fk(\000)2129 3989 y Fl(!)f Fm(e)2278 3959 y Fk(\000)2354 3989 y Fw(+)20 b Fm(\015)34 b Fw(for)c(b)r(oth)g(leptons)g(b)r(ound)g(to)g(a)g (proton.)f(This)1996 4089 y(is)39 b(a)g(leptonic)g(v)n(ersion)e(of)i (the)g(celebrated)g Fm(B)t Fw(-meson)f(deca)n(y)1996 4188 y Fm(B)2059 4200 y Fu(d)2121 4188 y Fl(!)23 b Fm(K)2304 4158 y Fk(\003)2342 4188 y Fm(\015)5 b Fw(.)2121 4293 y(As)38 b(a)f(to)n(y)g(mo)r(del,)h(one)f(migh)n(t)h(consider)f(a)g(pro) r(cess)g(\\)p Fm(\026)p Fw(")1996 4393 y Fl(!)j Fw(\\)p Fm(e)p Fw(")e Fm(\015)44 b Fw(in)c(an)f(external)f(Coulom)n(b)h (\014eld,)h(with)g(\\)p Fm(\026)p Fw(")e(and)1996 4493 y(\\)p Fm(e)p Fw(")e(rather)g(close)g(in)h(mass)f(suc)n(h)h(that)g(the) g(non-relativistic)1996 4592 y(descriptions)19 b(of)g(the)h(\\leptons") e(migh)n(t)h(b)r(e)h(used.)f(The)h(e\013ectiv)n(e)1996 4692 y(\\)p Fm(\026)p Fw(")32 b Fl(!)g Fw(\\)p Fm(e)p Fw(")f Fm(\015)37 b Fw(in)n(teraction)32 b(is)g(giv)n(en)f(in)i(\(23)p (#equation.23) [[477 204 487 216] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)f (If)h(w)n(e)f(assume)1996 4792 y(that)c(\()p Fm(M)e Fl(\000)16 b Fm(m)p Fw(\))27 b(is)g(of)g(order)f Fm(\013)14 b(m)p Fw(,)27 b(w)n(e)f(obtain)h(o\013-shell)g(e\013ects)1996 4891 y(of)d(order)e Fm(\013)2353 4861 y Fv(2)2414 4891 y Fw(due)i(to)g Fl(L)2722 4903 y Fu(F)2777 4891 y Fw(,)g(relativ)n(e)e (to)i(the)g(standard)e(magnetic)1996 4991 y(momen)n(t)36 b(term)g Fl(L)2589 5003 y Fu(\033)2634 4991 y Fw(.)g(Bigger)f(mass)g (di\013erences)g(giv)n(es)g(bigger)1996 5090 y(e\013ects,)k(un)n(til)g (the)g(non-relativistic)e(appro)n(ximation)f(breaks)1996 5190 y(do)n(wn.)p 1996 5415 811 4 v 2089 5469 a Ft(1)2123 5493 y Fi(Note)h(that)g(it)e(is)g(not)i(necessary)g(to)f(kno)n(w)g(the) h(precise)f(v)l(alue)g(of)1996 5571 y Fc(j)p Fb( )r Fi(\(0\))p Fc(j)2173 5548 y Ft(2)2243 5571 y Fc(\030)e Fi(\()p Fb(m\013)p Fi(\))2493 5548 y Ft(3)2530 5571 y Fb(=\031)r Fi(,)e(in)g(order)g(to)h (kno)n(w)g(the)h(relativ)n(e)e(o\013-shell)g(con-)1996 5650 y(tribution.)1896 5899 y Fw(4)p eop %%Page: 5 5 5 4 bop -189 63 a Fn(4)135 b(O\013-shellness)46 b(in)973 32 y Fa(\026)945 63 y Ff(B)1040 19 y Fw(0)1034 92 y Fm(s)1119 63 y Fe(!)32 b Ff(\015)7 b(\015)-189 244 y Fw(By)31 b(the)g(replacemen) n(ts)f Fm(\026)f Fl(!)f Fm(s)j Fw(and)g Fm(e)d Fl(!)h Fm(b)p Fw(,)i(the)g(expressions)-189 344 y(\(4)p (#equation.4) [[53 726 58 738] [1 1 1 [3 3]] [0 0 1]] pdfm (\))37 b(to)g(\(8)p (#equation.8) [[84 726 89 738] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (apply)g(to)g Fm(b)h Fl(!)h Fm(s\015)5 b(\015)41 b Fw(induced)1262 323 y(\026)1242 344 y Fm(B)1309 314 y Fv(0)1305 365 y Fu(s)1386 344 y Fl(!)d Fw(2)p Fm(\015)k Fw(deca)n(y)-189 444 y(amplitude.)26 b(Then)f(one)g(has)g(to)g(scale)f(the)i(op)r (erators)d Fl(L)1592 456 y Fu(F)r(;\033)1726 444 y Fw(de-)-189 543 y(\014ned)k(at)g(the)h Fm(M)338 555 y Fu(W)440 543 y Fw(scale,)f(do)n(wn)f(to)h(the)h Fm(B)t Fw(-meson)e(scale.)g(The)-189 643 y(co)r(e\016cien)n(ts)d Fm(B)290 655 y Fu(F)369 643 y Fw(of)h Fl(L)517 655 y Fu(F)572 643 y Fw(,)g(and)f Fm(B)839 655 y Fu(\033)908 643 y Fw(of)g Fl(L)1055 655 y Fu(\033)1100 643 y Fw(,)h(in)g(Eqs.)f(\(6)p (#equation.6) [[246 690 251 702] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (and)f(\(5)p (#equation.5) [[281 690 286 702] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)-189 743 y(b)r(oth)i(b)r(eing)f(equal)g(to)g Fm(B)k Fw(at)c(the)h Fm(W)36 b Fw(scale,)24 b(ma)n(y)f(ev)n(olv)n(e)g(di\013er-)-189 842 y(en)n(tly)j(do)n(wn)f(to)g(the)h Fm(\026)d Fw(=)g Fm(m)708 854 y Fu(b)767 842 y Fw(scale.)i(This)g(di\013erence)h(b)r(et) n(w)n(een)-189 942 y Fm(B)-126 954 y Fu(F)-42 942 y Fw(and)j Fm(B)184 954 y Fu(\033)257 942 y Fw(is)g(due)g(to)f(di\013eren)n(t)h (anomalous)e(dimensions)i(of)-189 1041 y(the)f(resp)r(ectiv)n(e)f(op)r (erators.)e(Within)k(the)f(SM)g(one)f(can)g(write)257 1258 y Fm(B)320 1270 y Fu(\033)o(;F)455 1258 y Fw(=)553 1202 y(4)p Fm(G)660 1214 y Fu(F)p 553 1239 162 4 v 578 1256 a Fl(p)p 647 1256 42 4 v 68 x Fw(2)738 1258 y Fm(\025)786 1270 y Fv(CKM)1039 1202 y Fm(e)p 973 1239 171 4 v 973 1315 a Fw(16)p Fm(\031)1107 1291 y Fv(2)1167 1258 y Fm(C)1232 1218 y Fu(\033)o(;F)1226 1280 y Fv(7)1367 1258 y Fm(:)299 b Fw(\(30\))-189 1480 y(The)30 b(co)r(e\016cien)n(t)h Fm(C)440 1450 y Fu(\033)434 1501 y Fv(7)515 1480 y Fw(has)f(b)r(een)h (studied)f(b)n(y)h(v)-5 b(arious)29 b(authors)-189 1580 y([10)p (#cite.GrSW90) [[52 577 62 589] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)c(11)p (#cite.BuMMP94) [[68 577 78 589] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)f(12)p (#cite.GrOSN88) [[83 577 93 589] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)h(13)p (#cite.CeCRV90) [[99 577 109 589] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)g(14)p (#cite.CeCRV94) [[115 577 125 589] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)f(The)h(co)r(e\016cien)n(t)f Fm(C)1127 1550 y Fu(F)1121 1601 y Fv(7)1207 1580 y Fw(w)n(as)g(considered)f(in)-189 1680 y([3)p (#cite.EeP94b) [[52 565 57 577] [1 1 1 [3 3]] [0 0 1]] pdfm (],)k (where)g(at)h(the)g Fm(b)p Fw(-quark)e(scale)g(w)n(e)i(obtained)374 1858 y Fm(C)439 1828 y Fu(F)433 1878 y Fv(7)p 374 1895 121 4 v 379 1971 a Fm(C)444 1942 y Fu(\033)438 1993 y Fv(7)528 1914 y Fl(')22 b Fw(4)p Fm(=)p Fw(3)165 b(\()p Fm(\026)23 b Fw(=)g Fm(m)1172 1926 y Fu(b)1205 1914 y Fw(\))g Fm(:)406 b Fw(\(31\))-189 2138 y(Although)32 b(the)g(o\013-shell)f(e\013ect)h(for)f Fm(B)j Fl(!)c Fw(2)p Fm(\015)36 b Fw(is)c(exp)r(ected)g(to)-189 2238 y(b)r(e)23 b(suppressed)f(b)n(y)h(the)g(ratio)f(\(binding)h(energy\)/)p Fm(m)1487 2250 y Fu(b)1519 2238 y Fw(,)g(it)g(could)-189 2337 y(still)28 b(b)r(e)g(n)n(umerically)f(in)n(teresting.)-189 2569 y FB(4.1)112 b(Coulom)m(b-t)m(yp)s(e)37 b(QCD)h(mo)s(del)-189 2722 y Fw(The)d(con)n(v)n(en)n(tional)d(pro)r(cedure)i(when)g(ev)-5 b(aluating)34 b(the)h(pseu-)-189 2822 y(doscalar)30 b(meson)i(deca)n(y) f(amplitudes)h(is)g(to)h(express)e(them)h(in)-189 2922 y(terms)24 b(of)g(the)h(meson)e(deca)n(y)h(constan)n(ts,)f(b)n(y)h (using)f(the)i(PCA)n(C)-189 3021 y(relations)281 3200 y Fl(h)p Fw(0)p Fl(j)s Fw(\026)-45 b Fm(s\015)460 3212 y Fu(\026)504 3200 y Fm(\015)547 3212 y Fv(5)584 3200 y Fm(b)p Fl(j)663 3179 y Fw(\026)643 3200 y Fm(B)710 3166 y Fv(0)706 3221 y Fu(s)747 3200 y Fw(\()p Fm(P)12 b Fw(\))p Fl(i)26 b Fw(=)e Fl(\000)p Fw(i)p Fm(f)1152 3212 y Fu(B)1209 3200 y Fm(P)1262 3212 y Fu(\026)1330 3200 y Fm(;)336 b Fw(\(32\))368 3325 y Fl(h)p Fw(0)p Fl(j)s Fw(\026)-45 b Fm(s\015)547 3337 y Fv(5)584 3325 y Fm(b)p Fl(j)663 3304 y Fw(\026)643 3325 y Fm(B)710 3290 y Fv(0)706 3345 y Fu(s)747 3325 y Fw(\()p Fm(P)12 b Fw(\))p Fl(i)26 b Fw(=)e(i)q Fm(f)1088 3337 y Fu(B)1144 3325 y Fm(M)1225 3337 y Fu(B)1305 3325 y Fm(:)361 b Fw(\(33\))-189 3503 y(These)27 b(relations)g(will)h(b)r(e)g(useful)g(after)f(reducing) g(our)g(general)-189 3603 y(expression)36 b(\(24)p (#equation.24) [[102 335 112 347] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (con)n(taining)f(the)h(terms)g(with)g(pro)r(ducts)g(of)-189 3703 y(up)e(to)g(\014v)n(e)f(gamma)g(matrices.)g(After)i(some)e (calculation)g(w)n(e)-189 3802 y(arriv)n(e)26 b(at)i(the)h(expression)d (for)i(the)g Fm(B)1027 3814 y Fu(s)1091 3802 y Fw(meson)g(deca)n(y)f (at)h(rest,)-189 3902 y(whic)n(h)f(is)f(analogous)e(to,)j(and)f(in)h (fact)g(con\014rms)f(our)g(previous)-189 4002 y(relation)h(\(28)p (#equation.28) [[90 287 100 299] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (obtained)g(in)h(di\013eren)n(t)g(w)n(a)n(y)-7 b(,)-67 4234 y Fl(M)33 4199 y Fu(B)115 4234 y Fw(=)24 b Fl(\000)p Fw(i)303 4177 y Fm(e)p 302 4215 42 4 v 302 4291 a Fw(3)353 4234 y Fm(B)416 4246 y Fu(\033)461 4234 y Fm(f)502 4246 y Fu(B)559 4234 y Fm(M)649 4199 y Fv(2)696 4177 y Fw(\(1)18 b(+)g Fm(x)p Fw(\))950 4147 y Fv(2)p 696 4215 293 4 v 818 4291 a Fm(x)998 4141 y Fh(h)1037 4234 y Fw(\(1)g(+)g Fm(x\034)9 b Fw(\))14 b Fg(\017)1392 4199 y Fk(\003)1392 4254 y Fv(2)1448 4234 y Fl(\001)19 b Fg(\017)1530 4199 y Fk(\003)1530 4254 y Fv(1)1587 4234 y Fw(+)115 4419 y(+)47 b(i\(1)19 b(+)f(2)p Fm(x)g Fw(+)g Fm(x\034)9 b Fw(\)\()p Fg(\017)814 4385 y Fk(\003)814 4440 y Fv(2)871 4419 y Fl(\002)18 b Fg(\017)994 4385 y Fk(\003)994 4440 y Fv(1)1032 4419 y Fw(\))h Fl(\001)1129 4397 y Fw(^)1124 4419 y Fg(k)1176 4431 y Fv(1)1232 4419 y Fw(+)f Fl(O)r Fw(\()p Fm(\034)1460 4385 y Fv(2)1498 4419 y Fm(;)c(x)1582 4385 y Fv(2)1620 4419 y Fw(\))1652 4327 y Fh(i)1691 4419 y Fm(:)-25 b Fw(\(34\))-189 4618 y(Here,)18 b(the)i(parameter)d Fm(\034)28 b Fw(represen)n(ts)18 b(the)h(o\013-shell)f(e\013ect)h(in)g (the)-189 4718 y(QCD)31 b(problem)f(at)g(hand,)h(and)f(will)h(b)r(e)g (more)f(mo)r(del)g(dep)r(en-)-189 4818 y(den)n(t)k(than)g(its)h(QED)e (coun)n(terpart)g Fm(\032)p Fw(.)h(With)h(the)f(amplitude)-189 4917 y(\(34)p (#equation.34) [[53 177 63 189] [1 1 1 [3 3]] [0 0 1]] pdfm (\))27 b(w)n(e)h(arriv)n(e)d(at)j(the)g(total)f(deca)n(y)g(width)283 5149 y(\000)c(=)462 5093 y Fm(\013M)605 5063 y Fv(5)p 456 5130 194 4 v 456 5206 a Fw(18)p Fm(m)613 5182 y Fv(2)659 5149 y Fm(f)709 5115 y Fv(2)700 5170 y Fu(B)757 5149 y Fl(j)p Fm(B)843 5161 y Fu(\033)888 5149 y Fl(j)911 5115 y Fv(2)962 5149 y Fw(\(1)18 b(+)g(2)p Fm(x\034)9 b Fw(\))38 b Fm(;)325 b Fw(\(35\))-189 5351 y(where)26 b(b)n(y)g(switc)n(hing)g(o\013)h Fm(\034)36 b Fw(w)n(e)26 b(repro)r(duce)g(the)g(result)h(of)f(Ref.)-189 5451 y([15)p (#cite.DeDL96) [[52 113 62 125] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)-64 5551 y(In)i(order)f(to)h(estimate)g(the)h(v)-5 b(alue)28 b(of)g(the)h(o\013-shell)f(con)n(tri-)-189 5650 y(bution)d Fm(\034)9 b Fw(,)26 b(in)f(this)g(subsection)f(w)n(e)h (assume)f(a)g(QED-lik)n(e)f(QCD)1996 48 y(mo)r(del)h(with)g(the)f (Coulom)n(bic)g(w)n(a)n(v)n(e)f(function)i([16)p (#cite.Go97) [[494 761 504 773] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)f(17)p (#cite.TrM88) [[509 761 519 773] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])h Fm( )s Fw(\()p Fm(r)r Fw(\))g Fl(\030)1996 148 y Fw(exp\()p Fl(\000)p Fm(mr)r(\013)2385 160 y Fv(e\013)2464 148 y Fw(\).)30 b(Th)n(us)g(w)n(e)g(rely)g(again)f(on)h(an)g(exact)f (solution)1996 247 y(corresp)r(onding)23 b(to)i(e\013ectiv)n(e)h(p)r (oten)n(tial)f Fm(V)18 b Fw(\()p Fm(r)r Fw(\))25 b(=)d Fl(\000)p Fw(4)p Fm(\013)3734 259 y Fv(e\013)3811 247 y Fm(=)p Fw(\(3)p Fm(r)r Fw(\),)1996 347 y(with)46 b(e\013ectiv)n(e)g (coupling)f Fm(\013)2945 359 y Fv(e\013)3022 347 y Fw(\()p Fm(r)r Fw(\))c(=)d Fl(\000)p Fw(\(4)p Fm(\031)s(b)3494 359 y Fv(0)3544 347 y Fw(ln)q(\()p Fm(r)r Fw(\003)3743 359 y Fv(p)r(ot)3845 347 y Fw(\)\))3909 317 y Fk(\000)p Fv(1)3999 347 y Fw(.)1996 446 y(Here)31 b Fm(b)2232 458 y Fv(0)2299 446 y Fw(=)e(\(1)p Fm(=)p Fw(8)p Fm(\031)2601 416 y Fv(2)2637 446 y Fw(\)\(11)21 b Fl(\000)g Fw(\(2)p Fm(=)p Fw(3\))p Fm(N)3149 458 y Fu(f)3191 446 y Fw(\).)31 b(The)h(mass)f(scale)f(\003)3920 458 y Fv(p)r(ot)1996 546 y Fw(appropriate)d(to)h(the)g(hea)n(vy-ligh)n(t)f(quark)3375 525 y(\026)3356 546 y Fm(Q)o(q)32 b Fw(p)r(oten)n(tial)c(is)g(re-)1996 646 y(lated)40 b(to)g(the)h(more)e(familiar)g(QCD)h(scale)g(parameter,) e(e.g.)1996 745 y(\003)2054 757 y Fv(p)r(ot)2179 745 y Fw(=)22 b(2)p Fm(:)p Fw(23)14 b(\003)p 2487 723 101 4 v 28 x Fv(MS)2613 745 y Fw(\(for)27 b Fm(N)2839 757 y Fu(f)2882 745 y Fw(=3\).)g(Within)h(this)f(mo)r(del,)h(w)n(e)e(ob-) 1996 845 y(tain)2741 979 y Fm(\034)33 b Fw(=)2908 923 y(2)p 2908 960 42 4 v 2908 1036 a(3)2959 979 y Fm(\013)3012 945 y Fv(2)3012 1000 y(e\013)3100 923 y Fm(C)3165 893 y Fu(F)3159 944 y Fv(7)p 3100 960 121 4 v 3105 1036 a Fm(C)3170 1008 y Fu(\033)3164 1058 y Fv(7)3254 979 y Fm(:)597 b Fw(\(36\))1996 1162 y(By)24 b(matc)n(hing)f(the)h(meson)f (deca)n(y)g(constan)n(t)g Fm(f)3469 1174 y Fu(B)3550 1162 y Fw(and)g(the)h(w)n(a)n(v)n(e)1996 1261 y(function)29 b(at)e(the)h(origin)2061 1485 y Fm(N)2128 1497 y Fu(c)2171 1429 y Fl(j)p Fm( )2248 1441 y Fu(B)2306 1429 y Fw(\(0\))p Fl(j)2435 1399 y Fv(2)p 2171 1466 301 4 v 2277 1542 a Fm(M)2505 1485 y Fw(=)2593 1368 y Fh(\022)2664 1429 y Fm(f)2705 1441 y Fu(B)p 2664 1466 98 4 v 2692 1542 a Fw(2)2772 1368 y Fh(\023)2833 1386 y Fv(2)2907 1485 y Fw(;)97 b Fl(j)p Fm( )3104 1497 y Fu(B)3161 1485 y Fw(\(0\))p Fl(j)3290 1451 y Fv(2)3350 1485 y Fw(=)3448 1429 y(\()p Fm(m\013)3606 1441 y Fv(e\013)3684 1429 y Fw(\))3716 1399 y Fv(3)p 3448 1466 306 4 v 3576 1542 a Fm(\031)3787 1485 y(;)64 b Fw(\(37\))1996 1693 y(w)n(e)25 b(obtain)g(the)h(v)-5 b(alue)26 b(for)e(the)i(strong)e(in)n(teraction)h(\014ne)h(struc-)1996 1792 y(ture)f(strength)f Fm(\013)2547 1804 y Fv(e\013)2650 1792 y Fl(\030)p Fw(1.)g(Then,)g(including)h(\(31)p (#equation.31) [[483 552 494 564] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (for)f(the)h(QCD)1996 1892 y(case,)i(the)h(correction)e(factor)2810 2054 y Fm(x\034)33 b Fl(')k Fw(0)p Fm(:)p Fw(1)f Fm(;)680 b Fw(\(38\))1996 2216 y(is)30 b(m)n(uc)n(h)f(larger)e(than)j Fm(x\032)g Fw(in)f(the)h(corresp)r(onding)e(QED)h(case.)1996 2316 y(Corresp)r(ondingly)-7 b(,)31 b(one)h(exp)r(ects)g(ev)n(en)g (more)g(signi\014can)n(t)f(o\013-)1996 2415 y(shell)i(e\013ects)g(in)g (ligh)n(t)g(quark)f(systems,)g(in)h(compliance)f(with)1996 2515 y(our)27 b(previous)g(results)g([1)p (#cite.EeP93) [[403 465 408 477] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)h(2)p (#cite.EeP94) [[414 465 420 477] [1 1 1 [3 3]] [0 0 1]] pdfm (,)f(4)p (#cite.KeKKP95) [[426 465 431 477] [1 1 1 [3 3]] [0 0 1]] pdfm (].) 1996 2744 y FB(4.2)113 b(A)37 b(constituen)m(t)f(quark)i(calculation) 1996 2897 y Fw(No)n(w)32 b(w)n(e)g(adopt)f(a)h(v)-5 b(arian)n(t)31 b(of)h(the)h(approac)n(h)d(in)i(Refs.)g([3)p (#cite.EeP94b) [[535 419 540 431] [1 1 1 [3 3]] [0 0 1]] pdfm (,)g(5)p (#cite.VaE98) [[547 419 552 431] [1 1 1 [3 3]] [0 0 1]] pdfm (])1996 2997 y(as)g(an)g(alternativ)n(e)e(to)i(the)h(Coulom)n(b-t)n(yp)r(e)e (QCD)h(mo)r(del)g(de-)1996 3096 y(scrib)r(ed)27 b(ab)r(o)n(v)n(e.)e (One)i(migh)n(t)f(use)h(the)g(PCA)n(C)f(relations)g(\(32)p (#equation.32) [[536 395 546 407] [1 1 1 [3 3]] [0 0 1]] pdfm (\){) 1996 3196 y(\(33)p (#equation.32) [[315 383 325 395] [1 1 1 [3 3]] [0 0 1]] pdfm (\))33 b(together)f(with)i(a)e(kinematical)h(assumption)f(for)h(the)j(\026)-45 b Fm(s)p Fw(-)1996 3295 y(quark)27 b(momen)n(tum,)h(similar)e(to)i (those)f(in)h(Refs.)g([15)p (#cite.DeDL96) [[505 372 515 384] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)f(18)p (#cite.HeK92) [[521 372 531 384] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)h(As-)1996 3395 y(suming)k(the)g(b)r(ound)k(\026)-46 b Fm(s)32 b Fw(and)g Fm(b)g Fw(quarks)e(in)3392 3374 y(\026)3372 3395 y Fm(B)3439 3365 y Fv(0)3435 3416 y Fu(s)3508 3395 y Fw(to)i(b)r(e)g(on)g(their)1996 3495 y(resp)r(ectiv)n(e)21 b Fj(e\013e)l(ctive)29 b Fw(mass-shells)21 b(\(e\013ectiv)n(e)h(mass)f(b)r(eing)h(cur-)1996 3594 y(ren)n(t)32 b(mass)g(plus)g(a)g(constituen)n(t)g(mass)g Fm(m)3355 3606 y Fv(0)3424 3594 y Fw(of)g(order)f(200-300)1996 3694 y(MeV\),)i(the)f(structure)f(of)h(the)g(amplitude)g(comes)f(out)h (essen-)1996 3794 y(tially)c(as)f(in)h(\(34)p (#equation.34) [[365 312 375 324] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f (with)h(a)f(relativ)n(e)g(o\013-shell)g(con)n(tribution)2655 3995 y Fm(x)19 b Fw(~)-47 b Fm(\034)47 b Fw(=)2911 3938 y(2)p Fm(m)3026 3950 y Fv(0)p 2911 3976 152 4 v 2933 4052 a Fm(m)3006 4064 y Fu(b)3109 3995 y Fl(\030)23 b Fw(0)p Fm(:)p Fw(1)f Fm(;)525 b Fw(\(39\))1996 4197 y(of)33 b(the)g(same)f(order)f(as)h(in)h(\(38)p (#equation.38) [[422 263 432 275] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)f (Ho)n(w)n(ev)n(er,)f(unlik)n(e)h(\(34)p (#equation.34) [[520 263 530 275] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (the)1996 4297 y(o\013-shell)25 b(e\013ect)h(is)f(no)n(w)f(only)h(in)h (the)f(CP-o)r(dd)g(term)g(\()p Fg(\017)3758 4266 y Fk(\003)3758 4317 y Fv(1)3810 4297 y Fl(\002)14 b Fg(\017)3928 4266 y Fk(\003)3928 4317 y Fv(2)3966 4297 y Fw(\),)1996 4396 y(the)28 b(square)f(brac)n(k)n(et)f(in)i(\(34)p (#equation.34) [[411 239 421 251] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(b) r(eing)h(replaced)e(b)n(y)2369 4491 y Fh(h)2409 4583 y Fg(\017)2449 4549 y Fk(\003)2449 4604 y Fv(2)2505 4583 y Fl(\001)19 b Fg(\017)2587 4549 y Fk(\003)2587 4604 y Fv(1)2644 4583 y Fw(+)f(i\(1)g(+)g(2)p Fm(x)g Fw(+)g Fm(x)t Fw(~)-46 b Fm(\034)10 b Fw(\)\()p Fg(\017)3313 4549 y Fk(\003)3313 4604 y Fv(2)3370 4583 y Fl(\002)18 b Fg(\017)3493 4549 y Fk(\003)3493 4604 y Fv(1)3531 4583 y Fw(\))3563 4491 y Fh(i)3626 4583 y Fm(:)225 b Fw(\(40\))1996 4770 y(This)34 b(ma)n(y)f(b)r(e)g(di\013eren)n(t)h(in)g(other)f (approac)n(hes)e([2)p (#cite.EeP94) [[505 195 510 207] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(],)j(sho)n(wing)1996 4870 y(the)20 b(mo)r(del)g(dep)r(endence)g (of)g(the)g(o\013-shell)f(e\013ect.)h(F)-7 b(or)19 b(instance,)1996 4969 y(p)r(oten)n(tial-QCD)27 b(mo)r(dels)g(in)g(general,)e(b)r(esides) i(a)g(v)n(ector)e(Cou-)1996 5069 y(lom)n(b)j(p)r(oten)n(tial,)f(also)g (con)n(tain)g(a)g(scalar)f(p)r(oten)n(tial.)1996 5298 y FB(4.3)113 b(A)37 b(b)s(ound)h(state)g(quark)g(mo)s(del)1996 5451 y Fw(F)-7 b(or)2165 5430 y(\026)2145 5451 y Fm(B)2212 5421 y Fv(0)2208 5472 y Fu(s)2272 5451 y Fl(!)23 b Fw(2)p Fm(\015)5 b Fw(,)27 b(w)n(e)g(ha)n(v)n(e)f(previously)g([3)p (#cite.EeP94b) [[462 113 467 125] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)i(5)p (#cite.VaE98) [[473 113 478 125] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])f(applied)h(a)e(b)r(ound)1996 5551 y(state)f(mo)r(del,)f(where)g (the)h(p)r(oten)n(tials)f Fm(V)3268 5563 y Fu(i)3321 5551 y Fw(in)g(\(12)p (#equation.12) [[486 101 496 113] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (are)g(replaced)1996 5650 y(b)n(y)k(a)f(quark-meson)e(in)n(teraction)i (Lagrangian)1896 5899 y(5)p eop %%Page: 6 6 6 5 bop 278 48 a Fl(L)335 60 y Fv(\010)386 48 y Fw(\()p Fm(s;)14 b(b)p Fw(\))37 b(=)g Fm(G)766 60 y Fu(B)820 26 y Fw(\026)823 48 y Fm(b)14 b(\015)916 60 y Fv(5)967 48 y Fm(s)g Fw(\010)k(+)g(h)p Fm(:)p Fw(c)p Fm(:)23 b(;)333 b Fw(\(41\))-189 248 y(where)24 b(\010)h(is)g(the)g(B-meson)f(\014eld.) h(Then,)g(the)g(term)f Fl(L)1521 260 y Fu(F)1602 248 y Fw(can)g(b)r(e)-189 348 y(transformed)i(a)n(w)n(a)n(y)g(b)n(y)h (means)g(of)h(the)g(\014eld)g(rede\014nitions:)-118 548 y Fm(s)-79 514 y Fk(0)-33 548 y Fw(=)23 b Fm(s)18 b Fw(+)g Fm(B)258 560 y Fu(F)327 548 y Fm(\033)k Fl(\001)d Fm(F)25 b(L)14 b(b)22 b(;)180 b(b)884 514 y Fk(0)930 548 y Fw(=)23 b Fm(b)18 b Fw(+)g Fm(B)1222 514 y Fk(\003)1218 569 y Fu(F)1287 548 y Fm(\033)k Fl(\001)d Fm(F)12 b(L)i(s)22 b(:)71 b Fw(\(42\))-189 749 y(Ho)n(w)n(ev)n(er,)21 b(its)i(e\013ect)h (reapp)r(ears)d(in)j(a)e(new)h(b)r(ound-state)g(in)n(ter-)-189 848 y(action)k(\001)p Fl(L)186 860 y Fv(\010)238 848 y Fw(,)163 1049 y Fl(L)220 1061 y Fv(\010)272 1049 y Fw(\()p Fm(s;)14 b(b)p Fw(\))k(+)g Fl(L)606 1061 y Fu(F)698 1049 y Fw(=)37 b Fl(L)857 1061 y Fv(\010)909 1049 y Fw(\()p Fm(s)980 1014 y Fk(0)1003 1049 y Fm(;)14 b(b)1076 1014 y Fk(0)1099 1049 y Fw(\))33 b(+)18 b(\001)p Fl(L)1373 1061 y Fv(\010)1448 1049 y Fm(;)218 b Fw(\(43\))-189 1249 y(where,)27 b(after)g(using)h Fm(R)q(\015)595 1261 y Fv(5)655 1249 y Fw(=)22 b Fm(R)29 b Fw(and)e Fm(L\015)1095 1261 y Fv(5)1155 1249 y Fw(=)22 b Fl(\000)p Fm(L)p Fw(,)-147 1450 y(\001)p Fl(L)-21 1462 y Fv(\010)67 1450 y Fw(=)37 b Fm(B)232 1462 y Fu(F)301 1450 y Fm(G)366 1462 y Fu(B)437 1382 y Fh(\002)469 1428 y Fw(\026)472 1450 y Fm(b)508 1415 y Fk(0)531 1450 y Fm(\033)10 b Fl(\001)d Fm(F)12 b(L)i(b)790 1415 y Fk(0)833 1450 y Fl(\000)23 b Fw(\026)-45 b Fm(s)957 1415 y Fk(0)994 1450 y Fm(\033)10 b Fl(\001)d Fm(F)12 b(R)i(s)1262 1415 y Fk(0)1286 1382 y Fh(\003)1320 1450 y Fw(\010)21 b(+)7 b(h)p Fm(:)p Fw(c)p Fm(:)22 b(:)42 b Fw(\(44\))-189 1650 y(Also)f(in)g(these)g(cases)f([3)p (#cite.EeP94b) [[143 569 148 581] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)h(5)p (#cite.VaE98) [[156 569 161 581] [1 1 1 [3 3]] [0 0 1]] pdfm (],)g (the)g(net)h(o\013-shell)e(e\013ects)i(are)-189 1750 y(found.)f(F)-7 b(urther)40 b(calculations)g(of)g Fm(B)49 b Fl(!)c Fw(2)p Fm(\015)g Fw(within)c(b)r(ound)-189 1849 y(state)32 b(mo)r(dels)g(of)h(the)f(t)n(yp)r(e)h(in)f(\(41)p (#equation.41) [[177 545 187 557] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (will)h(b)r(e)f(presen)n(ted)g(else-)-189 1949 y(where.)-189 2270 y Fn(5)135 b(Conclusions)-189 2469 y Fw(W)-7 b(e)38 b(ha)n(v)n(e)f(demonstrated)h(the)g(app)r(earance)f(of)h(the)g (o\013-shell)-189 2568 y(e\013ects)23 b(in)g(the)g(\015a)n(v)n(our-c)n (hanging)c(t)n(w)n(o-photon)i(deca)n(y)h(of)g(m)n(uo-)-189 2668 y(nium)g(and)e(its)h(hadronic)635 2647 y(\026)615 2668 y Fm(B)682 2638 y Fv(0)678 2689 y Fu(s)743 2668 y Fl(!)i Fm(\015)5 b(\015)25 b Fw(coun)n(terpart.)20 b(It)h(is)g(a)f(quite)-189 2768 y(signi\014can)n(t)34 b(10)f(p)r(ercen)n(t)h(e\013ect)h(in)g(the)f(latter)g(case,)g(whereas) -189 2867 y(in)28 b(the)h(leptonic)f(case)g(it)g(is)g(v)n(ery)f(small)h (\(of)g(order)f(10)1540 2837 y Fk(\000)p Fv(7)1628 2867 y Fw(\),)i(but)-189 2967 y(clearly)d(iden)n(ti\014able.)-64 3075 y(The)c(presen)n(t)h(\\atomic")e(approac)n(h)g(enables)h(us)h(to)f (see)h(in)g(a)-189 3175 y(new)h(ligh)n(t)g(the)g(e\013ect)g(studied)g (\014rst)g(for)f(the)h Fm(K)1308 3187 y Fu(L)1381 3175 y Fl(!)f Fm(\015)5 b(\015)28 b Fw(ampli-)-189 3275 y(tude)j(in)g(the)h (c)n(hiral)d(quark)h(mo)r(del)h([1)p (#cite.EeP93) [[191 374 196 386] [1 1 1 [3 3]] [0 0 1]] pdfm (,)g(2)p (#cite.EeP94) [[203 374 208 386] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(],)g(and)g(subsequen)n(tly)-189 3374 y(in)h(the)g(b)r(ound-state)f (mo)r(del)h([4)p (#cite.KeKKP95) [[168 362 173 374] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)g(The)g(observ)-5 b(ation)30 b(that)i(o\013-)-189 3474 y(shell)f(e\013ects)g(can)g(b)r(e)h(clearly)e(isolated)g(from)h (the)g(rest)g(in)g(the)-189 3574 y(hea)n(vy-ligh)n(t)25 b(quark)h(atoms)g([3)p (#cite.EeP94b) [[161 338 166 350] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(])h(w)n(as)f(still)h(plagued)f(b)n(y)g(the)i(un-)-189 3673 y(certain)n(t)n(y)36 b(in)h(the)g(QCD)g(binding)h(calculation)e ([5)p (#cite.VaE98) [[242 326 247 338] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)i(Here,)e(in)-189 3773 y(the)c(Coulom)n(b-t)n(yp)r(e)f(QCD)h (mo)r(del)f(w)n(e)h(are)f(able)g(to)g(subsume)-189 3872 y(the)i(e\013ect)h(in)n(to)e(an)h(univ)n(ersal)e(binding)j(factor,)e (in)h(the)g(same)-189 3972 y(w)n(a)n(y)27 b(as)h(for)f(the)i(t)n(w)n (o-photon)e(deca)n(y)g(of)h(m)n(uonium)h(in)f(the)h(ex-)-189 4072 y(actly)19 b(solv)-5 b(able)18 b(QED)g(framew)n(ork.)f(As)j(a)e (result,)h(w)n(e)f(obtain)h(the)-189 4171 y(explicit)27 b(expressions)e(describing)h(ho)n(w)g(the)h(\015a)n(v)n(our-c)n (hanging)-189 4271 y(op)r(erators)e(that)i(v)-5 b(anish)26 b(on-shell)g(mo)r(dify)h(b)r(oth)g(the)g(CP-ev)n(en)-189 4371 y(\()p Fg(\017)-117 4340 y Fk(\003)-117 4391 y Fv(1)-63 4371 y Fl(\001)15 b Fg(\017)15 4340 y Fk(\003)15 4391 y Fv(2)54 4371 y Fw(\),)26 b(and)g(CP-o)r(dd)f(\()p Fg(\017)674 4340 y Fk(\003)674 4391 y Fv(1)727 4371 y Fl(\002)15 b Fg(\017)847 4340 y Fk(\003)847 4391 y Fv(2)885 4371 y Fw(\))27 b(con\014guration)d(of)i(the)h(\014nal)-189 4470 y(photons.)21 b(In)h(a)f(constituen)n(t)h(quark)e(calculation)h(w) n(e)g(get)h(a)f(sim-)-189 4570 y(ilar)28 b(result)g(for)g(the)h (o\013-shell)f(e\013ect.)h(As)g(a)f(di\013erence,)g(in)h(this)-189 4669 y(case)k(the)i(o\013-shellness)e(resides)h(solely)f(in)i(the)f (CP-o)r(dd)g(part)-189 4769 y(of)28 b(the)g(amplitude.)-64 4878 y(The)19 b(main)f(result)h(of)g(the)g(presen)n(t)f(pap)r(er)h(is)g (a)f(clear)g(demon-)-189 4977 y(stration)28 b(of)i(the)g(parallelism)e (of)h(the)h(strict)f(nonzero)f(o\013-shell)-189 5077 y(e\013ects)41 b(in)g(the)g(leptonic)f(and)h(quark)e(hea)n(vy-ligh)n(t) g(systems.)-189 5176 y(Th)n(us,)f(w)n(e)g(ha)n(v)n(e)f(established)h(a) g(solid)g(ground)f(for)h(estimat-)-189 5276 y(ing)45 b(the)g(o\013-shell)g(b)r(ound-state)f(e\013ects)h(in)h(the)f(imp)r (ortan)n(t)-189 5376 y Fm(B)-126 5388 y Fu(d)-64 5376 y Fl(!)23 b Fm(K)119 5346 y Fk(\003)156 5376 y Fm(\015)k Fw(deca)n(y)-7 b(,)20 b(whic)n(h)h(will)h(b)r(e)g(presen)n(ted)f (elsewhere)f([19)p (#cite.EKPprep) [[277 122 287 134] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(].)-189 5559 y Fs(Ac)n(kno)n(wledgemen)n(t.)25 b FE(One)h(of)i(us)f(\(J.O.E.\))h(thanks)e(H.)h(Pilkuhn)-189 5650 y(for)g(discussions)f(related)h(to)f(this)f(pap)r(er.)1996 48 y Fn(References)2035 213 y FE([1])42 b(J.)27 b(O.)e(Eeg,)i(I.)f (Picek,)g(Ph)n(ys.)g(Lett.)f Fs(B301)p FE(,)i(423)g(\(1993\))2035 338 y([2])42 b(J.)27 b(O.)g(Eeg,)g(I.)g(Picek,)g(Ph)n(ys.)f(Lett.)h Fs(B323)p FE(,)g(193)h(\(1994\),)g(hep-)p [[537 726 555 738] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9312320) pdfm 2157 429 a(ph/9312320)p [[312 716 378 728] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9312320) pdfm 2035 554 a([3])42 b(J.)27 b(O.)e(Eeg,)i(I.)f(Picek,)g(Ph)n(ys.)g(Lett.) f Fs(B336)p FE(,)i(549)g(\(1994\))2035 678 y([4])42 b(D.)c(Kek)n(ez,)f (et)g(al.,)i(Ph)n(ys.)e(Lett.)h Fs(B348)p FE(,)g(239)h(\(1995\),)g (hep-)p [[537 686 555 698] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9501350) pdfm 2157 769 a(ph/9501350)p [[312 675 378 687] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9501350) pdfm 2035 894 a([5])j(E.)27 b(V)-6 b(anem,)24 b(J.)i(O.)g(Eeg,)h(Ph)n(ys.)e (Rev.)g Fs(D58)p FE(,)h(114010)i(\(1998\))2035 1019 y([6])42 b(J.)25 b(J.)g(Sakurai,)f(Adv)l(anced)f(Quan)n(tum)f(Mec)n(hanics)j (\(Addison-)2157 1110 y(W)-6 b(esley)g(,)26 b(Reading,)g(1967\))2035 1234 y([7])42 b(C.)31 b(Itzykson,)f(J.-B.)h(Zub)r(er,)f(Quan)n(tum)f (Field)h(Theory)h(\(Mc-)2157 1326 y(Gra)n(w-Hill,)c(New)f(Y)-6 b(ork,)25 b(1980\),)j(p.)d(233)2035 1450 y([8])42 b(O.)31 b(Nac)n(h)n(tmann,)e(Elemen)n(tary)h(P)n(article)i(Ph)n(ysics:)f (Concepts)2157 1542 y(and)26 b(Phenomena)f(\(Springer,)g(Berlin,)i (1990\))2035 1666 y([9])42 b(J.)27 b(O.)e(Eeg,)i(I.)f(Picek,)g(Fizik)l (a)g Fs(B3)p FE(,)g(135)h(\(1994\))1996 1791 y([10])43 b(B.)j(Grinstein,)g(R.)e(Springer,)h(M.)h(B.)f(Wise,)h(Nucl.)f(Ph)n (ys.)2157 1882 y Fs(B339)p FE(,)27 b(269)f(\(1990\))1996 2006 y([11])43 b(A.)27 b(J.)h(Buras,)f(et)g(al.,)h(Nucl.)f(Ph)n(ys.)g Fs(B424)p FE(,)g(374)h(\(1994\),)h(hep-)p [[537 526 555 538] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9311345) pdfm 2157 2098 a(ph/9311345)p [[312 515 378 527] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9311345) pdfm 1996 2222 a([12])43 b(R.)26 b(Grig)r(janis,)i(et)e(al.,)h(Ph)n(ys.)e (Lett.)h Fs(B213)p FE(,)g(355)h(\(1988\))1996 2347 y([13])43 b(G.)27 b(Cella,)g(et)f(al.,)h(Ph)n(ys.)e(Lett.)h Fs(B248)p FE(,)g(181)h(\(1990\))1996 2471 y([14])43 b(G.)e(Cella,)i(et)d(al.,)i (Ph)n(ys.)f(Lett.)g Fs(B325)p FE(,)g(227)h(\(1994\),)g(hep-)p [[537 470 555 482] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9401254) pdfm 2157 2563 a(ph/9401254)p [[312 459 378 471] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9401254) pdfm 1996 2687 a([15])h(G.)22 b(G.)g(Devidze,)f(G.)h(R.)f(Dzhibuti,)g(A.)h (G.)g(Liparteliani,)h(Nucl.)2157 2779 y(Ph)n(ys.)j Fs(B468)p FE(,)g(241)h(\(1996\))1996 2903 y([16])43 b(T.)i(Goldman,)f(presen)n (ted)f(at)h(2nd)g(In)n(ternational)g(Confer-)2157 2994 y(ence)26 b(on)g(Quark)e(Con\014nemen)n(t)g(and)i(the)f(Hadron)g(Sp)r (ectrum,)2157 3086 y(Como,)h(Italy)-6 b(,)26 b(26-29)h(Jun)e(1996,)j (hep-ph/9703222)p [[451 397 515 409] [1 1 1 [3 3]] [0 0 1]] (http://xxx.uni-augsburg.de/abs/hep-ph/9703222) pdfm 1996 3210 a([17])43 b(H.)30 b(D.)h(T)-6 b(rottier,)31 b(R.)f(R.)g(Mendel,)h(in)f(Pro)r(c.)i(of)f(10th)g(Ann)n(ual)2157 3302 y(Mon)n(treal-Ro)r(c)n(hester-Syracuse-T)-6 b(oron)n(to)20 b(Mtg.)g(on)e(High)h(En-)2157 3393 y(ergy)j(Theory)-6 b(,)21 b(T)-6 b(oron)n(to,)22 b(Canada,)h(Ma)n(y)e(9-10,)i(1988,)g(pp.) e(134-)2157 3484 y(155)1996 3609 y([18])43 b(S.)19 b(Herrlic)n(h,)h(J.) g(Kalino)n(wski,)h(Nucl.)e(Ph)n(ys.)g Fs(B381)p FE(,)h(501)g(\(1992\)) 1996 3733 y([19])43 b(J.O.)27 b(Eeg,)g(K.)e(Kumeri)n(\024)-36 b(cki,)25 b(I.)h(Picek,)g(in)g(preparation)1896 5899 y Fw(6)p eop %%Trailer end end userdict /end-hook known{end-hook}if %%EOF