526 lines
16 KiB
Plaintext
526 lines
16 KiB
Plaintext
Rio de Janeiro
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December 19, 19Sl
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Dear J'ermio
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:BeiJlc thousands ot miles aw,q I ha.Te onl)" hea.I-db;r amateur radio trom
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trienda 1n the UoSo that :,ou are doing exper1menta in meson acatterinc trom
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protonao I don6t know what :,our theoretical friends are aqing. ao I ahoulcl
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like to molte·some comment■ at the risk ot onl.7 sqing what 1a obvious to every--
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bodT 1n the u:.so
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To begin with lam ot the opinion that Yukawa0a meaon theor:, with
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paeudoacalar me■ona gradient couplill&, 1■ wronge (or at lee.at useless) 1n
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- 1ta present form-because at least perturbation theor:, is NoGoand other-
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wlee divergences cloud the ia11ueo But I think me■one are paeudoacalar 0 and . I think th• amplitude• that a nucleon emits Juat one mq be proportion• al to "
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o0 Q (where o 1a the nuclear apin, Q the meson momentum)tor Q amallo
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(Thia 1• ot course ~aement with the Yukawatheor:,-to a11 ·1puep in
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.
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account, becauae tor low Q one operator in the
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1■ proportiona.l to Q and the othera, involving
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tu. BB aerie■ Hf1+
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E
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_ iEJ.1 + E
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H
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•
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O
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all the virt~ mesons a.re not
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etc
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(the virtual momentaare of orderµ, the meeon maaa) ao tor Q low enough the
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aum will be proportional to Q, and further will be Q0 t1mea the sum with the
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o operator 1n place of one of the H9a-which means Q0 timea a spin 1/2· obJect
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which can onl:, therefore be proportional to o). Let ua •Bl' then the coupling
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1■ 1/µ G (Q)(o0 Q)u for emission of om meson amplitude u. momentumQ, mas■ µ
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where G(Q) 1a a function ot Q (and posaibl7 the nucleon moment■ at higher Qn)
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and l expect G to have the properties of not varying much for Q small, Jur;t la a
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reaaonable function of Q/µo For Q • o. call G(O) a G (If pert theoq were 0
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OXG is Juat·the usual g) • .Further thi■ is most reasonable on nearly 8.ffT
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0
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theor:,-=for the meson bei~ pseudoscalar the coet to emit one (even it
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proton ia a positron+ 18 neutrino■ + 4 neutral mesons) mu.atbe paoscala.rcac,
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/
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wh1Clh,1t it doean8t involve the nucleon momenta (and I can8 t ■ea how it
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ea.ell:, can be galllean invariant..,...but Nature 9a i~1nat1on alwqJ has m:,
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reapect) can hardl:,be other than oaQ. {According to Yuks.wet.heor:,f) stand&rd.
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form, the total aerie■ would give a G(Q) vhicho it g2 were ver:, small and
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*I make all analyse, thinking ot the theor:, non-relativiatic 1n the nucleons.
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2
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intecrala conYerged., would be nearly conetant for all Q equal to c-
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bu.t 1t c2 i■ larger, correction terma ■et in tor Q ot order µ.)o
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I wl■ll to appeal to experiment to tr., to o■ tabli1h, 1t poaalble, whether the abOYe 1■ correct azJd the cou:pltn« 1a like 0°Q tor Jmt me■on ab■orpt1on. You ■ee tho I meanonl.7 to ref er to ~ .IPIW me■og-tor Q.t'JJ. or hiper I ha.Yeno argument■ about what to expect.
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Yet it 1■ 1mpo■■ 1ble to meaeure the ab■orption ot one meaon by nucleon
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41reotl7 tor the con■ erYation of enera demanda that another coupling enter
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to tab out the enero. If we do U with a y-re:r, or a collia1on between llUCleona new unoertaintie1 ariae, but if we do it by mean■ of another me■on (aoattering) the situation would appear to be ae aimple aa poa■ 1ble.
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The •tntermediate atatee• (it they mean a:a37thinc) haYe, mqbe, energy of order 1J.n,; 10 that a■ long a.1 Q remaina amall enouch (non-relo me1ona) the intermediate 1tatea do not depend much on Q. Then, itw a11'\11l9 the coa.pling tor two meaona 1a e1 ■ entiall.7 11k:e the double action of the l■t order coupling, we ■ee that the matrix element tor ■oatterinc ought to be
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proportional bilinearly to 4:i_and~ • It mu■ t therefore haYe the fora
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Xz where ~ • are 1ome tunct1on■ ot Q, inaen11t1Ye to Q tor amall Q.. :But in
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principle knowledge of the oouplbJc ot one meaon doe■ not determine that for
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two, !here could atill be a term with arbitrar,r coefficient 1n the Hamiltonian
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of fol'll ~ ~ which 1■ ■calar. Hence w• might expect
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2 •
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M • Q (~. co■ <>to ■ 1n O X2) + x3
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(0)
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(J'or uample, gradient and direct coupl1Jlc theorie1 acr•• 'on a•Q for one me■on,
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.._
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Becauae, it you like, now 1n therpert ■er1•• one of the B 1■ prop. ~, other
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\o ~ am otberviae nothing 1a 1ena1'1n to the Talue ot ~, ~•
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7
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b\lt for two x3 is ver,- d1fterent being veey amall for grad. and ver:, large tor
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direct-in pert. theo17).
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llaturall.7 auch a form 1a completel.7 general-but what I want to
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verit7 1a that
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(1) x3 1a ve17 a.mall (~b• order µ3/M amaller than ~, x2 )
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(could 1n principle depend on apin-I will aaaume 1t doesn't)
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1i, (2)
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x2 are 1naene1tive to Q2 for Q2 well b.elow µ20
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I am not in poaition to calculate 1j_,~ in terma of -G, nor to get a relation
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between them-tor we have no good theory. (One poes1bilit7 of courae 1•
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that relations ot the let order pert theo17 ma,-be true, but let 111 tin\ t1ncl
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out U (1), (2) are true ·and that being ea tabliahed go on from there.)
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Comment■: (1) 1a a pure gun■~c.r1oua ev1d.ence (sueh as y ualaslon
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coapetlng tavorabl7 with Tr0 em1aa1on in B capturing Tr-) ind1catea ~t 1a 10-
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all th• n14ence whtoh it u■WtJ.17 educed to prefer the grad. to direct o.,upl lng
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X,1a Juat a question ot how big x3 18. I aaaume tor no excellent reaaon that does not depend on apSna.
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(2) could be wrong. U would be Yer,- 1ntereatbco ror 1t probabl.7
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would. mean there exist important •intermediate ■ tatea• at low (rel. toµ)
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ener17-which would be a vital d1acove17~ Bence I urge 7ou to tr,- to aee wbiether the prediction■ of (l), (2) are aatiafiedo
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IDC:14entall7 aince K tor the 1nvene reaction ahould 'be the oomplez
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coDJuca,te I conclude all X8• are real (~t I aa notorioual7 punk at ■uch argu.-
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menta.-..,get a field theor,- or croup theo17 ex.pert) •
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. But, verr interesting ia the relation ot the X9a tor dUterent
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reaction■ (I mean meaona ot dif'terent charges, neutral sc.). It would be
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ve-q 1ntereating U we mulcl ver1f7 that the ■JJllllletric theo17 ia Ta.lid. Let
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... I
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ua look at the prediction• of th11 theor,- for thi1 problem and teat it later ez-
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•
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~
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perimentall7. It u, Tare the Tectora in isotropic apin apace representing the
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H■ona_ in and out, and t' 1a the operator for the nucleon MllUlt be bilinear
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1n u, and v and invariant 1n leotropic spin, or of the torm
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+> I M • Affl
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+ B i ~ c: X t)
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where A,:B are ma.trice• involving spin etc. (Wlall we later write
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A• Ai+ ta~, and we expect nearl.7 to write
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B • :a1 + la :a2
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1n the form
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(4),(5)
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4
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Ai,• Q2~ cos Q + x3• A2 • Q2x2 ain Q
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Bl • Q2Tl coa G + Y:,• B2 • Q2Y2 ain e
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. x3, Y:, tmall,
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X,Y nearl.7 conatant
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amall Q_2oAll real?
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bu,t torm (3) doee not depend on aa1umption1 (:,)(S) ot course, just invarianceo)
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That is, getting down to ca.sea, the matrix element for each proc•••
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ia given in the following table. Proceaaea labeled with the aame •TYPE•letter
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have equal probab111t1e■-aa would oe expected from either reaction • inverse
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or the moat naive use of the charge 17mmetryide&: Tr+ ia to p as Tr-• 11 ~o D
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and n° 11 impartialo
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Nowlet ua look at the X-■ect tor variou■ caaea, In complete generalit7
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Ai A can be written 1n the form A• + ic/~ where Aj,la 1calar /A2 ia 9
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quantities (oomplex)(vector) and B • B1 + 1o•/B2o SWnmingOTer all apin direction• of the nucleon then we obtain that the croae eection is p~oportional.
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in each caae reepectivel7 to~-
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PROCJSS
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ELEMENT .ma
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(a) /'l_+BJ 2+. ··:.·//A2+ /B~2
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+·
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TT ♦
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p
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) Tr++ p
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A+B
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(a)
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',,
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0
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ff
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+
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p
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-fl +p
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) TT+♦ n
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'~ fl 0+ p ) TT0+ D
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-fri (b)
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A
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(c)
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+fiB (b)
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>
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fl
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.... +
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p
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A-B
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(d)
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(b) 2(1:sJ2 + //»22()
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<c>l.&2J+ IIAIi2
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(d) IAi~12 l+ l/A2 - BJ 2
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+
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fl
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+n
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>
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fl
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+ +
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n
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> 0
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fl + p
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.;.fiT A - ll
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(d)
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(b}
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(where/ A2f meana A.fl-A/:t.1s•.f/~Af/A)o
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TTO + 11
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-fl +n
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-> 0 Tr +
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n
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> 1T + p
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> ,r-♦ n
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A
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(c)
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+{"2B (b)
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A + l3 (a)
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Bence the a)'Jllllletric theo17 predict•
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aa+ Ob• 2..0.o+ ob which would be a wonderful thing to verifT for it doea involve the
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idea that neutral meaona have 1/,-;-t1mea
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the coupling ot charged. However, untortunatel.7 o 1a unmeasurable exper1mentall.7o
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C
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.
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( U aomedq we know •1Jll• theo17 1a OX we can uae thia to get o which. aomebocllm' ight O
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want to interpret n° production and aubaequent escape 1n heaV7 nucleio) So ao far
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no teat of subtle parts of a,m. theo17.
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But now let us aubatitute (4},(S). I call x3 zero for aimplicit7
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-7ou can put 1t 1n and aee eftecta. AaaumeX,Y real-I hope it'• true.
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C,
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s
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(a) Ill Q4 (b) • Q4 (d) • Q4
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Renee (A) Cro1s aect1on1 ahorud go as Q4 (up until Q2~µ. 2)
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(B) 1n angl.e should be ot form a+ b coa2e, or say a. coa2Q +• ~ 11n24~o Effect of x3 will be aeen as a small residual con~twi.t x-eecto as a Vlo Q
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11 extrapolated to zeroe---or more 1enaitivel.7 (?) a tarm in cos~ in the
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angular diatribu~ion (le.Gk ot tr?nt-back s,mmetcy Cogo) for low Qo
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(c) S7J11111fltrthlc.eo17 predict,· tor a. i;;,Mfoir ~: or for cr(90o)
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and for o(O) one of the relation,
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(Ta; - I <if •
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rm or rra,r+{'fd}" •
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or
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=-fw •
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(theae are not valid if 'lffT argument XoY real ie f a.ult7)
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which ma,-11erve at1 a teat ot th.J.t theory o Could 7ou tell me to what extant thmsa toredictton■ (A)o·(:a)0 (0)
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are verified b7 experiments? Mq I urge the importance ot lowenergy meson
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e:xper1menta in Gstabliehing be7ond doubt (it the7 agree) some of our basic
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premisea toda7? Righer energy are inter&stina bu.t 1n our ignorance we do not
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know how tc- interpret .them-so 1t ir, well to otudy low enera as lello
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f2 In particular thare 1• hope to check the ot the symmstric theor1 wi'Ul
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low enero data<>
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S1ncerel70 /•/ Dick Fo,nman
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PoSo I ha.To alraaq heard that z...goct riaea rapidly \itith ene1:•a--■ topa
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rieing about Q • µ.. ao I am not entirel;y 1n the dark 1n Brasil.,
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P.P.S. Between ua theoriata (I 1m&&ined you aa a..'l e·xper1menter above--
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1i hence the low remark about seeing a field theo17 expert to aeo it ,x2
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auat be real) I 0d like· to make 1ome rem&rlcao I th:lnk now non-relati-r1at1call7
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about nucleon,, so error■ o~ order (Pnuc/MAaaproton) 2 (c•l) can come in.
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::
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6
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A coupling ot one meaon a0 Q is not Galilean invariant, for at additional velocit7
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V • Q' • Q + OJV where <,J • trequ. ot mes~n. :But nucleon changea mom 'b7 MV hence
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••the ci.ai11ean invariant co~pling must be (error now order r/c 2• not V/c) • 1.eo
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c,)
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•
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.
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(Q - i/P 1a invariant)
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a, (L\u + ~UP)~i"u. + "~icf;.c1-P)'/~
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(6)
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(t)
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where P 1■ the operator nucleon (b) momentum. Mqbe you ehould use this in the z-eect ana.17•1• but it _on:1m7 akes ta.ctore ot l + r,,)/2Mor l + µ/2M to the . accurac7 we expect. ao 11 Just an unknown constant &n7Wq,
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The Pio theol'J grad coupl1ng agree■ , making for the non-relo hamiltoniasl
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BowI arcued abovs tor the term (a) with a g renormalized to G aa ettect tor
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absorption of 1 meaon. Hence the galilean argwaent ahowa the g 1n (b) 1a
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the aameaa that 1n (a)., Bow the (a) looka like the t7Pe ot • 2/A•/A term that
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comae 1n electrodTnamica trom (/P - /A!./A) 2• In that caae renormalising the
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C •
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2
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cha.i"gemust che.r1gethee 1n e(/P 0 /A + /A•/P) and in• /A0 /A b1 the aame amount-
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by a.n argument of gauge 1nYar1ance. Bow1a there aomerea.■on tor a.particular
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aize (c)T Or, ia there aomepr1nc1pl• which shows the renormeJ.iaed.gin (a)
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and. (c) muat be equal? Doea &J17one1n UoS, knCllf about thia? It ia Yer,-
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1ntereat1ng because (o) ot course 1• the origin of the x3 term--eo it x3
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1a known 1n alse it '111&t1e'11 ua •aomethingo
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A.lao when electric potent1a1·1■ present the a 0 Au.get■ another
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- - a0 /Au. So one Wai'G might be got 1• from the croa, sect tor 0
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,r + p-? p + "V capture from n at reat. WeJmowthis competes auoc•••-
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tull.7 with" • +.p--> p + n0 ud the latter a'baolute z◄ect cu be got 'b7 utrac,,
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pol&tlDg ■catter1ng croes ■action■ don. (The latter 1• ■mall either beC&\18•
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lt ua•• the Y3 term. or it th1• 1a aero (aa 1D lat order pert. theol'J) by ao Q ot the out n° followed by the (b) tena tor the in 11jo Call we a:tgu
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- - ... ,..
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. --- ..+·· that they emieaion comes Just from (o•/A)u? I thinlc 7ea. It 7ou imagine
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the pert. aerie• aga~ and t17 to get TT- 1n by 0° Q it cau onl.7 Co by (b)
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° term and hence ii ao ■mall that n could compete. Hence Tr :mustgo 1n via
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G(o•/A)u. Bext ahould the G be G(0)7 I &11 not clear on this. ProbabJ.T not.
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tor 1t the nucleon baa a structure it would depend on they r&3 wave length
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--or otherwiae put, 1n principle we cannot exclude additional terms ot the
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kind (o• (/A z/I))u etc. A-q w,q it mrq be interesting when enoup data ia
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available to put 1n number■ and ••• how comparable a.re the G1 • obtained trom
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x3• trom thi1 reaction, trom an attempt to get~• x2 ••• from pert. theo17.
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etc. Aleo 1ntereating 1a to aee 1t 8:Jl7 electromagnetic properties can be got
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•from the acattering baaed on the principle that MT :tunctlon of momentumtor
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charged particle• goe1 to Q • •/c A.
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P:,So I'm aor17 to have to writ• by bend but ••CT'• here have lazicu,ace trouble,
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8Zl4 are alow, aud are now on X-ma.a vacation, e.nd I've del1qed too lo!lg. 'ft
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aiqthing herelll looks 1ntere■ tinc enoup to tell azq other meson lab• please
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tell them, I am not writin,; thia to arq-one else. (It 1ou melat copiea plea.a•
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send me one.)
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P4s. Leite Lope■ and I tiniahed that teat of the Yuk. theoq potential I aald
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I might t17. 'lhe id.ea vu take lat order in g2 potential from pa. grad. a711o
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meaon theor;r. AaaumeOXfor larger but not for ■mall •• •. Integrate tl'Oll
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ou.ta1d• in, but don•t aaaume r'f goe■ enctl.T to O at origin (becauae 'f there
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la wron,;) o Startlnc with 11n,;let. ■catterin& length and effective l'm1p d.eteraine g2• ~18 he. r'f ~ 0 at -0.1/',J,o lhlt usinc these tor triplet entirely
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too much D ata.te reaulta and no accord 1a cat to experiment no matter what
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phase ia chosen at origin. It 1• 10 'bad that we can wq the potential mwJt be
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wrong by lta own order of magnitude wen aa tar out as at µ.z, • 0.7. The
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next order (c4) potential ot Yuk theo17 make■ chance• ot 2°°" at µ,r • 1.0,
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nen tor g2 u large aa Oo2 (the coefticienta 1n the aerie■ are 10 la.rgd)o 1n
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41rect1one which do J!211eem richt to atra.1&}lten .things out. Bence we ha.Te no
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idea of what the potfential
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ahoulcl be nen it the meaon theo17 wen OX.-per•
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2
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• •
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turbation expana1on■ are inconaiatent tor auch large g • &:ad nan 1t all 11
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clropped arbitrarily but the tirat term the p4otential d.laagre•e1 experiMnt. The terma (b) f.n •(6) produce in 4th order (c )a quite atrong apin orbit tore•
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between nucleons and a oloaecl ■hell aa well. It •••• to be of the r1pt
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.f. •. ·Don't 'believe an:r calcul~tion in meson theo17 which ueea a l'e7DJllalldiacraal
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Ji -• perturbation value, ot X,Y are X, • 0, ~ • +1. X:, • u"J/2M) Simple but •-1, Y2 • O, Y:, • 0 ) falae. Bow talae??
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8
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11gn and order of nui.gnitude (it actual.17 11 too J!H,but ••• ) tor Mlqer
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(order~
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main torce)o I BID writing a.ll p:.a.rUcul1;.ra to :Bethe, 1n detailo
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I\ 1• hard to believe 1n &ZJTth11t1rogm the pa. theo17 bee&uae the pertur'batiom
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~• 1ncon1i1tent. I haTe tried tor 6 montha and 100 cloael.7 written pace, of
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formula.a to vorlc out intermediate coupling problem■• I think I could succeed
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but the grad. meson t~eo17 d1Terge1 ff81'1Vhere ■o I• 411hea.rteneclto pick
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out a:q tal■• mOdol (without 41Tergenc1e■) and puah 11; throup., becau■ e 1t•1
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■o much worko I th1Dk I could do a:q apec1al. problem which didn 1t ha.Te
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dlvergenciea (e.g. a. cut....oft theo1'7) but I don• t want to vaate ,q time.
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So I 8111, with th11 letter to ;youand one to :Betha• g1Y1ilgl1P Yuko
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idea 1934 and BID goillg to the Copacal>ana beach to 1ee 1t I c1m get one ot sr
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own. I get lota of idea.a at the beach.
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