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1944 y Fm(W)p 1868 1981 299 4 v 1868 2057 a Fn(2)p Fm(K)1981 2069 y Fk(eq)2049 2057 y Fm(m)2122 2069 y Fk(n)2177 2000 y Fm(R)2240 2012 y Fk(T)g(T)2340 2000 y Fm(:)275 2197 y Fn(F)-7 b(or)31 b(pion)f(pro)r(duction)h(from)f (the)i(proton,)g(the)g(ab)r(o)n(v)n(e)e(prescription)f(is)i(unam)n (biguous.)46 b(Giv)n(en)30 b(an)i(ev)n(en)n(t,)150 2296 y(one)k(simply)e(needs)i(to)h(calculate)d(\000)1339 2308 y Fk(v)1415 2296 y Fn(in)i(the)h(lab,)g(b)r(o)r(ost)g(the)g(pion)e(to)i (the)g(cen)n(ter)f(of)g(mass)f(and)i(calculate)150 2396 y Fm(d\033)s(=d)p Fn(\012)388 2366 y Fl(\003)388 2417 y Fk(\031)433 2396 y Fn(.)k(Then)29 b(one)g(m)n(ust)f(transform)f(the)i (virtual)d(photon)j(cross)e(section)h(bac)n(k)g(to)h(the)g(lab)f(via)f (a)i(Jacobian)150 2496 y(\(see)f(App)r(endix)f(A\).)150 2728 y Fg(3.2)112 b(A)p Fe(>)p Fg(1)150 2881 y Fn(When)31 b(A)f(is)g(greater)e(than)j(one,)f(w)n(e)g(tak)n(e)g(the)g(approac)n(h) f(that)h(the)h(mo)r(del)d(cross)h(section)g(is)g(just)i(the)g(proton) 150 2981 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(e,)f(ho)n(w)n(ev)n(er,)g(w)n(e)h(m)n(ust)g(accoun)n(t)f(for)h(the)150 3579 y(relativ)n(e)37 b(v)n(elo)r(cit)n(y)g(of)j(the)h(target)e (particle.)72 b(Th)n(us)40 b(w)n(e)g(m)n(ust)f(correct)g(the)i(inciden) n(t)d(\015ux,)44 b(v,)f(b)n(y)d(a)g(factor)150 3678 y([1)19 b Fj(\000)g Fm(~)-42 b(p)360 3690 y Fk(f)7 b(er)r(mi)568 3678 y Fj(\001)20 b Fm(~)-43 b(p)652 3690 y Fk(e)688 3678 y Fm(=)p Fn(\()p Fm(E)823 3690 y Fk(tar)r(g)r(et)1012 3678 y Fm(p)1054 3690 y Fk(e)1089 3678 y Fn(\)].)41 b(F)-7 b(urthermore,)27 b(if)h(one)g(tak)n(es)g Fm(K)2229 3690 y Fk(eq)2326 3678 y Fn(to)g(b)r(e)h(the)g(real)e(photon)i(energy)e (needed)i(to)150 3778 y(excite)e(a)g(mass)f(W)i(from)e(the)i(target)f (n)n(ucleon)f(w)n(e)h(then)h(migh)n(t)d(naiv)n(ely)g(calculate,)1345 4020 y Fm(K)1416 4032 y Fk(eq)1507 4020 y Fn(=)1625 3953 y Fm(W)1715 3923 y Fh(2)1771 3953 y Fj(\000)18 b Fn(\()p Fm(E)1952 3923 y Fh(2)1947 3974 y Fk(tar)r(g)r(et)2155 3953 y Fj(\000)g Fm(p)2280 3923 y Fh(2)2280 3977 y Fk(f)7 b(er)r(mi)2469 3953 y Fn(\))p 1605 4001 917 4 v 1605 4077 a(2\()p Fm(E)1740 4089 y Fk(tar)r(g)r(et)1947 4077 y Fj(\000)19 b Fm(~)-43 b(p)2072 4089 y Fk(f)7 b(er)r(mi)2280 4077 y Fj(\001)18 b Fm(~)-41 b(q)s(=)p Fj(j)p Fm(~)f(q)r Fj(j)p Fn(\))2531 4020 y Fm(:)150 4250 y(K)221 4262 y Fk(eq)315 4250 y Fn(in)26 b(the)g(virtual)e(photon)i(\015ux)g(cancels)f (when)h(com)n(bined)e(with)i(the)g Fm(K)2529 4262 y Fk(eq)2623 4250 y Fn(in)g(the)h(kinematic)c(factors)i(for)h(the)150 4350 y(cen)n(ter)f(of)h(mass)e(photon)i(cross)f(section.)35 b(Ho)n(w)n(ev)n(er,)24 b(since)h(it)g(is)g(used)h(in)f(the)h(Multip)r (ole)e(mo)r(del)g(lo)r(ok-up)g(table,)150 4449 y(the)k(c)n(hoice)e(of)h (con)n(v)n(en)n(tion)f(is)g(still)f(imp)r(ortan)n(t.)35 b(It)28 b(turns)f(out)h(that)g(when)g(the)g(ab)r(o)n(v)n(e)e(form)h(of) g Fm(K)3312 4461 y Fk(eq)3408 4449 y Fn(is)f(used)i(in)150 4549 y(the)h(Multip)r(ole)e(mo)r(del,)g(the)i(cen)n(ter)f(of)h(mass)e (cross)g(section)h(do)r(es)g(not)h(ha)n(v)n(e)f(the)h(correct)e(dep)r (endence)i(on)g Fm(W)12 b Fn(.)150 4648 y(F)-7 b(or)32 b(that)h(reason,)g(the)g(same)e(form)g(for)i Fm(K)1544 4660 y Fk(eq)1644 4648 y Fn(as)f(w)n(as)g(used)h(in)f(the)h(stationary) d(n)n(ucleon)h(case)h(is)g(used)h(in)f(the)150 4748 y(mo)n(ving)c(n)n (ucleon)i(case)g(\(this)h(giv)n(es)d(the)k(correct)e Fm(W)43 b Fn(dep)r(endence\).)48 b(Note)31 b(that,)h(regardless)c(of)j (mo)r(del,)f Fm(W)43 b Fn(is)150 4848 y(calculated,)1277 4947 y Fm(W)1367 4913 y Fh(2)1428 4947 y Fn(=)22 b(\()p Fm(E)1608 4959 y Fk(tar)r(g)r(et)1816 4947 y Fn(+)c Fm(!)s Fn(\))1986 4913 y Fh(2)2042 4947 y Fj(\000)g Fn(\()q Fm(~)-43 b(p)2199 4959 y Fk(f)7 b(er)r(mi)2407 4947 y Fn(+)18 b Fm(~)-42 b(q)s Fn(\))2562 4913 y Fh(2)2600 4947 y Fm(:)275 5097 y Fn(A)29 b(more)f(subtle)g(issue)g(is)g(ho)n(w)h (to)g(treat)g(the)h(\\o\013-shellness")25 b(of)k(the)h(target)e(n)n (ucleon.)41 b(Recall)27 b(that)i(in)g(the)150 5196 y(generation)e(of)i (an)f(ev)n(en)n(t,)h(the)h(sp)r(ectator)e(system)g(is)g(on)g(the)i (mass)d(shell)g(while)g(the)j(target)e(is)g(not.)41 b(Ho)n(w)n(ev)n (er,)150 5296 y(the)30 b(mo)r(dels)e(w)n(e)i(use)g(generally)c(assume)j (the)h(target)f(is)g(on)h(mass)e(shell.)42 b(W)-7 b(e)30 b(are)f(free)h(to)g(rede\014ne)g(the)g(target)150 5396 y(n)n(ucleon)d(energy)g(suc)n(h)h(that)h(the)f(particle)e(is)h(on)h (shell,)f(but)i(then)g(this)e(energy)h(will)d(b)r(e)k(inconsisten)n(t)d (with)i(the)1561 5656 y(3)p eop %%Page: 4 4 4 3 bop 150 390 a Fn(energy)26 b(calculated)f(for)i(the)h(pion.)36 b(More)27 b(sp)r(eci\014cally)-7 b(,)24 b(when)k(the)g(pion)e(is)h(b)r (o)r(osted)g(to)g(the)h(cen)n(ter)f(of)h(mass,)e(it)150 490 y(will)e(not)i(actually)e(b)r(e)i(b)r(o)r(osted)h(to)f(a)g(frame)f (where)h(the)g(sum)g(of)g(all)e(momen)n(ta)g(is)h(zero.)36 b(Curren)n(tly)-7 b(,)24 b(w)n(e)i(c)n(ho)r(ose)150 589 y(to)e(treat)f(the)h(struc)n(k)f(n)n(ucleon)f(the)j(same)d(in)h(b)r (oth)h(ev)n(en)n(t)f(generation)f(and)h(in)g(the)h(ph)n(ysics)f(mo)r (dels,)f(i.e.)35 b(assume)150 689 y(the)28 b(strcuk)f(n)n(ucleon)f(is)h (o\013-shell.)275 789 y(Once)22 b(w)n(e)g(c)n(ho)r(ose)f(a)h(con)n(v)n (en)n(tion)f(for)h(the)h(target)e(n)n(ucleon)g(energy)-7 b(,)23 b(it)f(is)f(not)i(alw)n(a)n(ys)d(p)r(ossible)g(to)i(implemen)n (t)150 888 y(it)37 b(in)g(a)g(consisten)n(t)f(manner.)64 b(Both)38 b(MAID)g(and)f(Multip)r(ole)e(require)h(that)h(the)h (fundamen)n(tal)e(amplitudes)150 988 y(\(squared\))27 b(b)r(e)i(m)n(ultiplied)24 b(b)n(y)j(v)-5 b(arious)26 b(kinematic)f(factors.)37 b(Since)27 b(these)h(are)f(calculated)f (outside)g(the)j(mo)r(del,)150 1088 y(w)n(e)23 b(are)f(free)i(to)f(c)n (ho)r(ose)f(whatev)n(er)g(con)n(v)n(en)n(tion)f(w)n(e)i(lik)n(e.)34 b(The)23 b(amplitudes)e(themselv)n(es,)g(ho)n(w)n(ev)n(er,)i(are)f (another)150 1187 y(story)-7 b(.)53 b(F)-7 b(or)32 b(the)i(Multip)r (ole)c(mo)r(del,)j(once)g(one)f(has)h(c)n(hosen)f(a)h(prescription)d (for)j(calculating)c(the)k(\\o\013-shell")150 1287 y Fm(K)221 1299 y Fk(eq)289 1287 y Fn(,)h(there)f(are)f(no)g(am)n (biguities.)49 b(The)33 b(MAID)g(mo)r(del)f(is)f(a)i(bit)f(more)g (complicated,)f(though.)52 b(Recall)31 b(that)150 1386 y(the)40 b(MAID)g(mo)r(del)d(tak)n(es)h Fm(W)12 b Fn(,)42 b Fm(Q)1279 1356 y Fh(2)1316 1386 y Fn(,)g(and)d(cos)13 b Fm(\022)1720 1356 y Fl(\003)1798 1386 y Fn(as)39 b(its)f(inputs.)71 b(It)40 b(is)e(unkno)n(wn)h(ho)n(w)f(it)h(determines)e(the)150 1486 y(fundamen)n(tal)27 b(amplitudes,)h(but)i(from)e(the)i(pap)r(er)e (describing)f(this)i(mo)r(del)e(it)i(almost)e(certainly)g(requires)g (the)150 1586 y(cen)n(ter)22 b(of)g(mass)e(virtual)g(photon)i(energy)f (and)h(momen)n(tum.)33 b(The)22 b(problem)e(arises)g(from)h(the)h(fact) h(that)f(w)n(e)g(ha)n(v)n(e)150 1685 y(calculated)h Fm(W)38 b Fn(and)26 b(cos)12 b Fm(\022)981 1655 y Fl(\003)1046 1685 y Fn(assuming)23 b(some)h(o\013-shell)f(prescription.)33 b(When)27 b(MAID)f(determines)e Fm(!)3377 1655 y Fl(\003)3440 1685 y Fn(from)h(the)150 1785 y(input)33 b(quan)n(tities,)e(it)i (assumes)d(the)j(target)f(n)n(ucleon)g(is)f(on)i(shell)d(so)i(that)i (the)f(virtual)d(photon)i(energy)g(\(and)150 1885 y(momen)n(tum\),)22 b(and)g(therefore)h(the)g(amplitudes)d(it)j(returns)f(ma)n(y)f(not)i(b) r(e)h(consisten)n(t)d(with)i(the)g(kinematics)d(that)150 1984 y(w)n(e)32 b(generated)f(for)h(this)f(ev)n(en)n(t.)50 b(This)31 b(is)g(not)i(a)e(problem)f(if)i(w)n(e)g(c)n(hange)f(the)h(ev) n(en)n(t)g(generation)e(sc)n(heme)g(and)150 2084 y(put)d(the)f(target)f (n)n(ucleon)f(on)i(shell)e(and)h(put)i(the)f(sp)r(ectator)f(o\013)h (shell,)e(but)j(then)f(at)g(some)e(kinematic)f(settings,)150 2183 y(the)36 b(agreemen)n(t)d(b)r(et)n(w)n(een)i(data)g(and)g(SIMC)h (su\013ers)e(in)h(regard)f(to)h(distribution)d(o\013sets.)60 b(In)35 b(addition,)h(the)150 2283 y(sp)r(ectator)e(n)n(ucleon)g(is)g (no)n(w)g(o\013)h(shell)e(in)h(the)i(\014nal)d(state.)59 b(In)36 b(the)f(end,)i(w)n(e)e(ha)n(v)n(e)f(decided)g(to)h(tolerate)e (the)150 2383 y(p)r(ossible)26 b(inconsistency)h(in)h(MAID's)h 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y(1)p 2237 2019 207 4 v 2237 2095 a(2)p Fm(p)2321 2071 y Fl(\003)2321 2115 y Fk(\031)2365 2095 y Fm(q)2405 2071 y Fl(\003)2487 1982 y Fm(d\033)p 2463 2019 142 4 v 2463 2095 a(d)p Fn(\012)2566 2071 y Fl(\003)2615 2038 y Fm(:)1006 b Fn(\(3\))150 2227 y(The)28 b(full)e(transformation)e(is)i(then,)1206 2365 y Fm(d\033)p 1179 2402 149 4 v 1179 2478 a(d)p Fn(\012)1282 2490 y Fk(\031)1360 2421 y Fn(=)d Fm(J)8 b Fn(\()p Fm(t;)14 b(\036)1650 2387 y Fl(\003)1650 2442 y Fk(q)r(\031)1751 2421 y Fj(!)23 b Fn(cos)13 b Fm(\022)2021 2433 y Fk(q)r(\031)2099 2421 y Fm(;)h(\036)2185 2433 y Fk(q)r(\031)2263 2421 y Fn(\))2387 2365 y(1)p 2305 2402 207 4 v 2305 2478 a(2)p Fm(p)2389 2454 y Fl(\003)2389 2499 y Fk(\031)2433 2478 y Fm(q)2473 2454 y Fl(\003)2559 2365 y Fm(d\033)p 2531 2402 149 4 v 2531 2478 a(d)p Fn(\012)2634 2454 y Fl(\003)2634 2499 y Fk(\031)2708 2421 y Fm(;)913 b Fn(\(4\))150 2621 y(where)27 b(the)h(Jacobian,)d Fm(J)8 b Fn(,)28 b(is)e(giv)n(en)g(b)n (y)i(the)g(determinan)n(t,)1156 2848 y 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y(to)j(the)g(cen)n(ter)g(of)g(mass)e(is)h (along)f Fc(q)p Fn(,)i(so)g(that)g Fm(\036)1671 900 y Fk(q)r(\031)1776 888 y Fn(is)f(the)h(same)f(in)g(the)h(lab)f(and)h(the) g(cen)n(ter)g(of)g(mass)e(since)h(the)150 988 y(b)r(o)r(ost)h(do)r(es)g (not)g(a\013ect)g(transv)n(erse)e(comp)r(onen)n(ts)g(of)i(the)h(pion)e (momen)n(tum.)34 b(When)27 b(the)h(target)e(is)g(allo)n(w)n(ed)d(to)150 1088 y(mo)n(v)n(e,)j(ho)n(w)n(ev)n(er,)f(the)j(b)r(o)r(ost)g(is)e(not)i (alw)n(a)n(ys)c(along)i Fc(q)p Fn(,)i(so)f Fm(\036)2069 1057 y Fl(\003)2069 1108 y Fk(q)r(\031)2174 1088 y Fn(m)n(ust)g(b)r(e)h (recalculated.)275 1187 y(T)-7 b(o)32 b(calculate)f Fm(\036)803 1157 y Fl(\003)803 1208 y Fk(q)r(\031)881 1187 y Fn(,)j(the)f(\\q")f (system)g(is)g(de\014ned)h(as)g(ab)r(o)n(v)n(e)e(in)i(terms)e(of)i Fc(q)h Fn(and)f Fc(P)3010 1199 y Fb(b)r(eam)3191 1187 y Fn(,)i(but)e(ev)-5 b(aluated)150 1287 y(in)30 b(the)h(cen)n(ter)f(of) h(mass.)45 b(F)-7 b(or)30 b(clarit)n(y)-7 b(,)29 b(the)i 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b(Jans)g Fa(et)i(al.)p Fn(,)g(Nucl.)e(Ph)n(ys.)g Fc(A475)p Fn(,)g(687)f(\(1987\).)192 3584 y([3])41 b(P)-7 b(.)34 b(Brauel)f Fa(et)j(al.)p Fn(,)i(\\Electropro)r(duction)31 b(Of)k(Pi+)e(N,)i(Pi-)e(P)h(And)i(K+)e(Lam)n(b)r(da,)321 3683 y(K+)d(Sigma0)e(Final)h(States)i(Ab)r(o)n(v)n(e)f(The)h(Resonance) f(Region,")e(Zeit.)j(Ph)n(ys.)f Fc(C3)p Fn(,)321 3783 y(101)26 b(\(1979\).)192 3949 y([4])41 b(D.)29 b(Drec)n(hsel,)f(O.)g (Hanstein,)h(S.S.)g(Kamalo)n(v)c(and)j(L.)h(Tiator,)f(\\A)g(Unitary)g (isobar)321 4049 y(mo)r(del)22 b(for)h(pion)g(photopro)r(duction)f(and) h(electropro)r(duction)e(on)i(the)h(proton)f(up)h(to)321 4148 y(1-GeV,")i(Nucl.)h(Ph)n(ys.)g Fc(A645)p Fn(,)g(145)f(\(1999\))h (n)n(ucl-th/9807001.)192 4314 y([5])41 b(G.)28 b(v)n(on)e(Gehlen,)h (Nucl.)g(Ph)n(ys.)g Fc(B20)p Fn(,)g(102)f(\(1970\).)192 4480 y([6])41 b(X.)28 b(Li,)e(L.)i(E.)f(W)-7 b(righ)n(t,)27 b(and)g(C.)h(Benhold,)e(Ph)n(ys.)h(Rev.)g Fc(C45)p Fn(,)g(2011)f (\(1992\).)1561 5656 y(7)p eop %%Trailer end userdict 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