zotero/storage/GDCJFLN8/.zotero-ft-cache

Rio de Janeiro

December 19, 19Sl

Dear J'ermio

:BeiJlc thousands ot miles aw,q I ha.Te onl)" hea.I-db;r amateur radio trom

trienda 1n the UoSo that :,ou are doing exper1menta in meson acatterinc trom

protonao I don6t know what :,our theoretical friends are aqing. ao I ahoulcl

like to molte·some comment■ at the risk ot onl.7 sqing what 1a obvious to every--

bodT 1n the u:.so

To begin with lam ot the opinion that Yukawa0a meaon theor:, with

paeudoacalar me■ona gradient couplill&, 1■ wronge (or at lee.at useless) 1n
- 1ta present form-because at least perturbation theor:, is NoGoand other-
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 "

o0 Q (where o 1a the nuclear apin, Q the meson momentum)tor Q amallo

(Thia 1• ot course ~aement with the Yukawatheor:,-to a11 ·1puep in

.

account, becauae tor low Q one operator in the
1■ proportiona.l to Q and the othera, involving

tu. BB aerie■ Hf1+

E

_ iEJ.1 + E

H
•

O

all the virt~ mesons a.re not

etc

(the virtual momentaare of orderµ, the meeon maaa) ao tor Q low enough the

aum will be proportional to Q, and further will be Q0 t1mea the sum with the

o operator 1n place of one of the H9a-which means Q0 timea a spin 1/2· obJect

which can onl:, therefore be proportional to o). Let ua •Bl' then the coupling

1■ 1/µ G (Q)(o0 Q)u for emission of om meson amplitude u. momentumQ, mas■ µ

where G(Q) 1a a function ot Q (and posaibl7 the nucleon moment■ at higher Qn)

and l expect G to have the properties of not varying much for Q small, Jur;t la a

reaaonable function of Q/µo For Q • o. call G(O) a G (If pert theoq were 0
OXG is Juat·the usual g) • .Further thi■ is most reasonable on nearly 8.ffT
0
theor:,-=for the meson bei~ pseudoscalar the coet to emit one (even it

proton ia a positron+ 18 neutrino■ + 4 neutral mesons) mu.atbe paoscala.rcac,

/

wh1Clh,1t it doean8t involve the nucleon momenta (and I can8 t ■ea how it

ea.ell:, can be galllean invariant..,...but Nature 9a i~1nat1on alwqJ has m:,
reapect) can hardl:,be other than oaQ. {According to Yuks.wet.heor:,f) stand&rd.
form, the total aerie■ would give a G(Q) vhicho it g2 were ver:, small and

*I make all analyse, thinking ot the theor:, non-relativiatic 1n the nucleons.

2
intecrala conYerged., would be nearly conetant for all Q equal to c-
bu.t 1t c2 i■ larger, correction terma ■et in tor Q ot order µ.)o
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.
Yet it 1■ 1mpo■■ 1ble to meaeure the ab■orption ot one meaon by nucleon
41reotl7 tor the con■ erYation of enera demanda that another coupling enter
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.
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
proportional bilinearly to 4:i_and~ • It mu■ t therefore haYe the fora

Xz where ~ • are 1ome tunct1on■ ot Q, inaen11t1Ye to Q tor amall Q.. :But in

principle knowledge of the oouplbJc ot one meaon doe■ not determine that for

two, !here could atill be a term with arbitrar,r coefficient 1n the Hamiltonian

of fol'll ~ ~ which 1■ ■calar. Hence w• might expect

2 •

M • Q (~. co■ <>to ■ 1n O X2) + x3

(0)

(J'or uample, gradient and direct coupl1Jlc theorie1 acr•• 'on a•Q for one me■on,
.._
Becauae, it you like, now 1n therpert ■er1•• one of the B 1■ prop. ~, other
\o ~ am otberviae nothing 1a 1ena1'1n to the Talue ot ~, ~•

7

b\lt for two x3 is ver,- d1fterent being veey amall for grad. and ver:, large tor
direct-in pert. theo17).

llaturall.7 auch a form 1a completel.7 general-but what I want to

verit7 1a that

(1) x3 1a ve17 a.mall (~b• order µ3/M amaller than ~, x2 )

(could 1n principle depend on apin-I will aaaume 1t doesn't)

1i, (2)

x2 are 1naene1tive to Q2 for Q2 well b.elow µ20

I am not in poaition to calculate 1j_,~ in terma of -G, nor to get a relation

between them-tor we have no good theory. (One poes1bilit7 of courae 1•

that relations ot the let order pert theo17 ma,-be true, but let 111 tin\ t1ncl

out U (1), (2) are true ·and that being ea tabliahed go on from there.)

Comment■: (1) 1a a pure gun■~c.r1oua ev1d.ence (sueh as y ualaslon
coapetlng tavorabl7 with Tr0 em1aa1on in B capturing Tr-) ind1catea ~t 1a 10-

all th• n14ence whtoh it u■WtJ.17 educed to prefer the grad. to direct o.,upl lng
X,1a Juat a question ot how big x3 18. I aaaume tor no excellent reaaon that does not depend on apSna.

(2) could be wrong. U would be Yer,- 1ntereatbco ror 1t probabl.7

would. mean there exist important •intermediate ■ tatea• at low (rel. toµ)

ener17-which would be a vital d1acove17~ Bence I urge 7ou to tr,- to aee wbiether the prediction■ of (l), (2) are aatiafiedo

IDC:14entall7 aince K tor the 1nvene reaction ahould 'be the oomplez

coDJuca,te I conclude all X8• are real (~t I aa notorioual7 punk at ■uch argu.-

menta.-..,get a field theor,- or croup theo17 ex.pert) •

. But, verr interesting ia the relation ot the X9a tor dUterent

reaction■ (I mean meaona ot dif'terent charges, neutral sc.). It would be

ve-q 1ntereating U we mulcl ver1f7 that the ■JJllllletric theo17 ia Ta.lid. Let

... I
ua look at the prediction• of th11 theor,- for thi1 problem and teat it later ez-

•

~

perimentall7. It u, Tare the Tectora in isotropic apin apace representing the

H■ona_ in and out, and t' 1a the operator for the nucleon MllUlt be bilinear

1n u, and v and invariant 1n leotropic spin, or of the torm

+> I M • Affl

+ B i ~ c: X t)

where A,:B are ma.trice• involving spin etc. (Wlall we later write

A• Ai+ ta~, and we expect nearl.7 to write

B • :a1 + la :a2

1n the form

(4),(5)

4

Ai,• Q2~ cos Q + x3• A2 • Q2x2 ain Q

Bl • Q2Tl coa G + Y:,• B2 • Q2Y2 ain e

. x3, Y:, tmall,
X,Y nearl.7 conatant
amall Q_2oAll real?

bu,t torm (3) doee not depend on aa1umption1 (:,)(S) ot course, just invarianceo)

That is, getting down to ca.sea, the matrix element for each proc•••

ia given in the following table. Proceaaea labeled with the aame •TYPE•letter

have equal probab111t1e■-aa would oe expected from either reaction • inverse
or the moat naive use of the charge 17mmetryide&: Tr+ ia to p as Tr-• 11 ~o D

and n° 11 impartialo

Nowlet ua look at the X-■ect tor variou■ caaea, In complete generalit7
Ai A can be written 1n the form A• + ic/~ where Aj,la 1calar /A2 ia 9
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.

in each caae reepectivel7 to~-

PROCJSS

ELEMENT .ma

(a) /'l_+BJ 2+. ··:.·//A2+ /B~2

+·
TT ♦

p

) Tr++ p

A+B

(a)

',,

0
ff

+

p

-fl +p

) TT+♦ n
'~ fl 0+ p ) TT0+ D

-fri (b)

A

(c)

+fiB (b)

>

fl

.... +

p

A-B

(d)

(b) 2(1:sJ2 + //»22()
<c>l.&2J+ IIAIi2
(d) IAi~12 l+ l/A2 - BJ 2

+
fl

+n

>

fl

+ +

n

> 0
fl + p

.;.fiT A - ll

(d)

(b}

(where/ A2f meana A.fl-A/:t.1s•.f/~Af/A)o

TTO + 11
-fl +n

-> 0 Tr +

n

> 1T + p

> ,r-♦ n

A

(c)

+{"2B (b)

A + l3 (a)

Bence the a)'Jllllletric theo17 predict•
aa+ Ob• 2..0.o+ ob which would be a wonderful thing to verifT for it doea involve the

idea that neutral meaona have 1/,-;-t1mea

the coupling ot charged. However, untortunatel.7 o 1a unmeasurable exper1mentall.7o

C

.

( U aomedq we know •1Jll• theo17 1a OX we can uae thia to get o which. aomebocllm' ight O

want to interpret n° production and aubaequent escape 1n heaV7 nucleio) So ao far

no teat of subtle parts of a,m. theo17.
But now let us aubatitute (4},(S). I call x3 zero for aimplicit7
-7ou can put 1t 1n and aee eftecta. AaaumeX,Y real-I hope it'• true.

C,
s

(a) Ill Q4 (b) • Q4 (d) • Q4

Renee (A) Cro1s aect1on1 ahorud go as Q4 (up until Q2~µ. 2)
(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
11 extrapolated to zeroe---or more 1enaitivel.7 (?) a tarm in cos~ in the
angular diatribu~ion (le.Gk ot tr?nt-back s,mmetcy Cogo) for low Qo
(c) S7J11111fltrthlc.eo17 predict,· tor a. i;;,Mfoir ~: or for cr(90o)

and for o(O) one of the relation,

(Ta; - I <if •

rm or rra,r+{'fd}" •

or

=-fw •

(theae are not valid if 'lffT argument XoY real ie f a.ult7)

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)
are verified b7 experiments? Mq I urge the importance ot lowenergy meson

e:xper1menta in Gstabliehing be7ond doubt (it the7 agree) some of our basic

premisea toda7? Righer energy are inter&stina bu.t 1n our ignorance we do not

know how tc- interpret .them-so 1t ir, well to otudy low enera as lello
f2 In particular thare 1• hope to check the ot the symmstric theor1 wi'Ul

low enero data<>

S1ncerel70 /•/ Dick Fo,nman

PoSo I ha.To alraaq heard that z...goct riaea rapidly \itith ene1:•a--■ topa
rieing about Q • µ.. ao I am not entirel;y 1n the dark 1n Brasil.,
P.P.S. Between ua theoriata (I 1m&&ined you aa a..'l e·xper1menter above--
1i hence the low remark about seeing a field theo17 expert to aeo it ,x2
auat be real) I 0d like· to make 1ome rem&rlcao I th:lnk now non-relati-r1at1call7
about nucleon,, so error■ o~ order (Pnuc/MAaaproton) 2 (c•l) can come in.

::

6

A coupling ot one meaon a0 Q is not Galilean invariant, for at additional velocit7

V • Q' • Q + OJV where <,J • trequ. ot mes~n. :But nucleon changea mom 'b7 MV hence

••the ci.ai11ean invariant co~pling must be (error now order r/c 2• not V/c) • 1.eo

c,)

•

.

(Q - i/P 1a invariant)

a, (L\u + ~UP)~i"u. + "~icf;.c1-P)'/~

(6)

(t)

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,
The Pio theol'J grad coupl1ng agree■ , making for the non-relo hamiltoniasl

BowI arcued abovs tor the term (a) with a g renormalized to G aa ettect tor

absorption of 1 meaon. Hence the galilean argwaent ahowa the g 1n (b) 1a

the aameaa that 1n (a)., Bow the (a) looka like the t7Pe ot • 2/A•/A term that

comae 1n electrodTnamica trom (/P - /A!./A) 2• In that caae renormalising the

C •

2

cha.i"gemust che.r1gethee 1n e(/P 0 /A + /A•/P) and in• /A0 /A b1 the aame amount-

by a.n argument of gauge 1nYar1ance. Bow1a there aomerea.■on tor a.particular

aize (c)T Or, ia there aomepr1nc1pl• which shows the renormeJ.iaed.gin (a)

and. (c) muat be equal? Doea &J17one1n UoS, knCllf about thia? It ia Yer,-
1ntereat1ng because (o) ot course 1• the origin of the x3 term--eo it x3
1a known 1n alse it '111&t1e'11 ua •aomethingo

A.lao when electric potent1a1·1■ present the a 0 Au.get■ another

- - a0 /Au. So one Wai'G might be got 1• from the croa, sect tor 0
,r + p-? p + "V capture from n at reat. WeJmowthis competes auoc•••-
tull.7 with" • +.p--> p + n0 ud the latter a'baolute z◄ect cu be got 'b7 utrac,,

pol&tlDg ■catter1ng croes ■action■ don. (The latter 1• ■mall either beC&\18•

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

- - ... ,..

. --- ..+·· that they emieaion comes Just from (o•/A)u? I thinlc 7ea. It 7ou imagine

the pert. aerie• aga~ and t17 to get TT- 1n by 0° Q it cau onl.7 Co by (b)
° term and hence ii ao ■mall that n could compete. Hence Tr :mustgo 1n via

G(o•/A)u. Bext ahould the G be G(0)7 I &11 not clear on this. ProbabJ.T not.

tor 1t the nucleon baa a structure it would depend on they r&3 wave length

--or otherwiae put, 1n principle we cannot exclude additional terms ot the

kind (o• (/A z/I))u etc. A-q w,q it mrq be interesting when enoup data ia

available to put 1n number■ and ••• how comparable a.re the G1 • obtained trom
x3• trom thi1 reaction, trom an attempt to get~• x2 ••• from pert. theo17.
etc. Aleo 1ntereating 1a to aee 1t 8:Jl7 electromagnetic properties can be got

•from the acattering baaed on the principle that MT :tunctlon of momentumtor

charged particle• goe1 to Q • •/c A.

P:,So I'm aor17 to have to writ• by bend but ••CT'• here have lazicu,ace trouble,

8Zl4 are alow, aud are now on X-ma.a vacation, e.nd I've del1qed too lo!lg. 'ft

aiqthing herelll looks 1ntere■ tinc enoup to tell azq other meson lab• please

tell them, I am not writin,; thia to arq-one else. (It 1ou melat copiea plea.a•

send me one.)
P4s. Leite Lope■ and I tiniahed that teat of the Yuk. theoq potential I aald
I might t17. 'lhe id.ea vu take lat order in g2 potential from pa. grad. a711o

meaon theor;r. AaaumeOXfor larger but not for ■mall •• •. Integrate tl'Oll

ou.ta1d• in, but don•t aaaume r'f goe■ enctl.T to O at origin (becauae 'f there

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

too much D ata.te reaulta and no accord 1a cat to experiment no matter what

phase ia chosen at origin. It 1• 10 'bad that we can wq the potential mwJt be

wrong by lta own order of magnitude wen aa tar out as at µ.z, • 0.7. The
next order (c4) potential ot Yuk theo17 make■ chance• ot 2°°" at µ,r • 1.0,
nen tor g2 u large aa Oo2 (the coefticienta 1n the aerie■ are 10 la.rgd)o 1n

41rect1one which do J!211eem richt to atra.1&}lten .things out. Bence we ha.Te no

idea of what the potfential

ahoulcl be nen it the meaon theo17 wen OX.-per•

2

• •

turbation expana1on■ are inconaiatent tor auch large g • &:ad nan 1t all 11

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•

between nucleons and a oloaecl ■hell aa well. It •••• to be of the r1pt

.f. •. ·Don't 'believe an:r calcul~tion in meson theo17 which ueea a l'e7DJllalldiacraal
Ji -• perturbation value, ot X,Y are X, • 0, ~ • +1. X:, • u"J/2M) Simple but •-1, Y2 • O, Y:, • 0 ) falae. Bow talae??

8

11gn and order of nui.gnitude (it actual.17 11 too J!H,but ••• ) tor Mlqer

(order~

main torce)o I BID writing a.ll p:.a.rUcul1;.ra to :Bethe, 1n detailo

I\ 1• hard to believe 1n &ZJTth11t1rogm the pa. theo17 bee&uae the pertur'batiom

~• 1ncon1i1tent. I haTe tried tor 6 montha and 100 cloael.7 written pace, of

formula.a to vorlc out intermediate coupling problem■• I think I could succeed

but the grad. meson t~eo17 d1Terge1 ff81'1Vhere ■o I• 411hea.rteneclto pick

out a:q tal■• mOdol (without 41Tergenc1e■) and puah 11; throup., becau■ e 1t•1

■o much worko I th1Dk I could do a:q apec1al. problem which didn 1t ha.Te

dlvergenciea (e.g. a. cut....oft theo1'7) but I don• t want to vaate ,q time.

So I 8111, with th11 letter to ;youand one to :Betha• g1Y1ilgl1P Yuko

idea 1934 and BID goillg to the Copacal>ana beach to 1ee 1t I c1m get one ot sr

own. I get lota of idea.a at the beach.