IL NUOVO CIMENTO VOL. l l A , N. 4 10 t tobre 1972

Renormalization of a Weak-Interaction Model.

]~. DELBOURGO

Imperial College - London

(ricevuto il 3 Marzo 1972)

S u m m a r y . - - We consider the prototype weak-interaction coupling ~-g@W'y(1--iys)~f of a boson W (mass /~) to a spinor field (mass m), gauge-transformed to a nonpolynomial form, and set up the detailed renormalization programme for it. The effective cut-off for the longi- tudinal W interactions is providcd by the (~ Fermi constant ~) G : g2/t~ and the remaining (renormalizable) (~ infinities ~> due to the transverse W interactions can be rendered finite by invoking the furthcr gravitational interactions of the fields. The finite radiative corrections are given as an ~synlptotic series in g and m//~, assumed small, and we illustrate the procedure by treating spinor-spinor scattering.

1 . - I n t r o d u c t i o n .

Even since the accept~nce of V - - A theory the infinities of the weak inter-

actions have proved u major source of difficulty, par t icular ly so because the

radiat ive corrections of the lowest-order couplings are expected to give non-

negligible and observable physie~l effects. There have been many a t tempts

with varying degrees of success to show how the weak-interaction corrections

have to be summed and/or renormalizcd (~), but by and large the problem is

still an outs tanding one today. Of late, a significant advance has however

been made by SgAF[ (3); based on recent ly developed nonpolynomial Lagrungian

methods, he has sketched how the pseudovector coupling theory when gauge-

t ransformed to the form (p(igA "Vvs - -m exp [2gvsB//~])W produces infinities no

(1) T .D. LEn and C. N. YANG" Phys. Rev., 126, 2239 (1962); T. D. LEE: Phys. Bey., 128, 899 (1962); G. FEI~BERG and A. PAIS: Phys. Rev., 131, 2724 (1963); 133, B 477 (1964); B. ARBUZOV and A. FILIPPOV: Nuovo Cimento, 38, 796 (1965). (3) Q. SHAH: On the renormalizability o/ massive spin-one theory with pseudovector interaction, ICTP/69/22, Phys. Rev., to be published.

943

944 R. DELBO~GO

worse than those of QED and is therefore renormalizable; essentially this is thanks to the exponent i :d in terac t ion by i tself which introduces a cut-off,

whose scale is effectively de te rmined by the Fermi constant G = g2/#2, tha t renders all ma t r ix elements finite.

In this paper we shall apply these techniques in some detai l to the more

realistic par i ty-viola t ing model coupling ½gCpW.y(1--i75)~ proposed by LEE (3) since i t typifies the weak interactions. We shall develop the ent i re renormal- izat ion scheme and implement i t by explici t ly computing the lowest-order radia t ive corrections to spinor-spinor scattering, the analogue of ~ decay. I f desired, the infinite renorm:,~lizations which persist , corresponding to mass and coupling constant of the t ransverse W-field, can be made finite by intro- ducing a fu r the r gravi ta t ional cut-off (4), on a parqllel wi th recent work on

QED. We quickly review in Sect. 2 the difficulties of the model f rom the tradi-

t ional point of view. In Sect. 3 we tu rn to the Stfiekelberg-gauge var ian t of the theory whereby the in terac t ion is t ransformed to ~L~ + JL~, where

and discuss the renormalizations needed when m = 0; tha t is, we t r ea t J5~ as ~ per turb~t ion of a chiral invar ian t theory. As we have in mind a t rue weak-interact ion theory whereby g and m/# are small parameters , we show in

Sect. 4 tha t ma t r ix elements corrected by JL~ are character ized b y an effec- t ive coupling paramete r g(m/#) and an effective cut-off tt'~/g2~ 1/G so tha t no fur ther infinities a r i se- - for clear appreciat ion of the circumst~mce i t is useful not to separate off the mass t e rm m ~ for absorpt ion into the free Lagrangian. Some pe r t inen t diagrams ~re calculated. Section 5 is devoted to the struc- ture of the renormalizations wi th special a t t en t ion paid to the self-energy and ve r t ex par ts involving A and thei r effect in spinor-spinor scattering; the lowest- order radia t ive corrections to the Fermi constant are tiros eva.luated. ]~eing a simplified version of leptonic physics this work is a necessary prel iminary for the unders tanding of the na tura l e.m. and weak interactions.

2. - A model of weak interactions.

A synthesis of the leptonic weak and e.m. interact ions (5) in N~tm'e points

a t tile ve ry least to the existence of an SU~ × U~ gauge group in physics, genera ted by the e.m. and charged weak currents together with another (parity-

(3) T . D . LEE: Nuovo Cime'~to, Z9A, 579 (1969). (4) C . J . ISHA~I, A. SALA~t and J. STICATnDEE: Phys. l{ev., 3, 1805 (1971). (~) A. SALAM and J. C. WARD: Phys. Lett., 13, 168 (1964). See also the subsequent work of S. WEINBERG: Phys. l~ev. Lett., 19, 1264 (1967).

E, ENORMALIZATION OF A WEAK-INTERACTION MODEL ~

violating) neu t ra l current . The in terac t ion of these currents wi th the photon

and the mass ive vec tor -meson t r ip le t is well known to produce unrenormal izable

infinities when "~ pe r tu rba t ion expansion in the weak-coupl ing constant is made,

excep t when the gauge s y m m e t r y is spontaneous ly broken. Analysis of the

p rob lem is considerably compl ica ted b y the mul t ip l ic i ty of bosons and

leptons, so we have chosen to s imula te this s t a te of '~ffairs b y examining

a weak pa r i ty -v io la t ing in te rac t ion model , first s tudied b y LEE (a), whereby

a single mass ive neu t ra l vector boson W is coupled to le f t -handed spinors

W. Thus we adop t %s the model Lagr 'mg ian

( 1 )

t- ½/~-w ~ + ½ g f W . e ( ~ - ~ ) , e .

I n se t t ing up the usual pe r tu rba t ion expansion in powers of g wi th the free

propaga tors

(W~W~} = A~z(p) -- (--g~a + p,,p~/#~)(p~--tt~) -~ ,

(% ~} - S(p) -- ( p . y - - m ) -1,

one can expect the overal l infinities of tile F e y n a m n d iagrams to become pro- gress ively worse as the order of g is increased because of the longi tudinal W-components .

One m a y highlight this difficulty by tak ing the Sti ickelberg va r i an t (e) of the above theory :

(2) L(V , A, B) == (o(i°c . y - - m) ~ - - ½ ( (~zA , ) (~A , ) - -#2A2) +

+ ½ ( (~B)~- -# :B ~-) + ½g~(A + ~B/#) . )~(1-- i ts) ~o.

I n this vers ion (*) i t is the der iva t ive B in terac t ion which is responsible for the virulence of the infinities when the pe r tu rba t i on graphs are ca lcula ted in the t rad i t iona l way. For instance, if one evaluates the L e h m a n n spectral func- t ions a of the renornl~dized spinor propag'~tor

p "7 + m ([ma4(s) -~_p '7a2(s) + ip .r?J,~as(s)] ds (3) S'(p) = p~ _ mT + 3 p ~ - - s . . . . .

(6) E . STUCKELBERG: Helv. Phys. Acta, 11, 225 (1938); A. SALAM: Phys. Rev., 127, 331 (1962). The rcnorm~lizability of vector-meson theory with a eonserved current was first shown by P. T. MATTHEWS: Phys. Rev., 76, 1254 (1949), and has been rediscovered many times since. (*) By construction this equivalent to theory (1) because ~.A--I~B satisfies a free-field equation. Therefore, the subsidiary condition on physical state vectors 19} (~'A(+)-B(~)) Iq~)- 0 (:an be imposed, just like the gauge condition in QED, to ensure that, with no extcrnM B-particles, no longitudinM A-states ever materialize. Thus only transverse external A-lines need be taken to describe processes involving W-mesons.

60 - II Nuovo Cimento A.

946 R. DELBOURGO

to second order in g one finds

~ ( s ) _ a~(8) _ ~ ( s )

s --}- m ~ 2 s s - - m 2

_ g2A(3(s Jr m2--# 2) + A~/# ~) 647~2 s2(s - - m2)~

A 2 ~ 0 ( s - - ( m + # )2 ) ( s - - (m + tt)~}(s-- (m- -# )~} .

Hence the convent ional definitions of 3m, Z and the fu r the r renormalizat ion constant Zs:

Z -1 ~m = mf(o'2(8 ) --0'4(8)) 48 ,

Z -1 = 1 ~- (0~2(8) d8 (4)

Z; ~ =fa~(s) ds

appear to yield quadratic infinities. The s i tuat ion deteriorates with higher orders in g and the theory is then deemed to be unrenormal izab le - -apar t f rom the exceptional circumstance tha t m = # = 0 when these infinities are rendered harmless by being absorbed in the gauge degrees of f reedom of A.

Actual ly these s ta tements about nonrenormaliz~bil i ty must be viewed cau- t iously for physical ma t r ix elements, because Ward identi t ies can operate to produce cancellations among separately divergent corrections which contr ibute to a given order in g, e.g. to g3 the leading quadrat ic infinities of the (~v/A)

ve r t ex cancel in part because of the asymptot ic Ward iden t i ty

F~(p, p) ,,~ ~X*/~p~.

We can look forward to these cancellations, p,~rticularly when m = 0, to sim- pl i fy m an y of the renormalizat ions.

3 . - N o n p o l y n o m i a l v a r i a n t o f t h e m o d e l .

Le t us redefine the fermion field by making the lef t -handed gauge trans-

format ion

~0 ->exp [1i(1 --iy~)Bg/#]

in order to cast the Lagrangian into (~ exponent ia l ~ form. We can hope tha t

the nonpolynomi,M form of the Lagrangian

(5) L ~ ( o i ~ ' y ~ - - I ( ~ A ~ A ~ - - # ~A2) +½((~B)~--# 2B°') ÷ L ~ v , ÷ L ~ , ,

where

(6) LA~ = ½g~A .r(1 --iys)~o , L ~ ------ --m(f exp [rsgB/#] v2 ,

R E N O R M A L I Z A T I O S ~ O F A W E A K - I N T E R A C T I O N M O D E L 947

give an equivalent theory because the exponent ia l t r ans fo rmat ion is locali-

zable (7)--this is cer ta in ly t rue to any fixed order in g. The newin t e ruc t i on

Lagrangian (6) consists of a polynoInial coupling of the t ransverse W-field A

and a nonpolynomia l coupling of the longi tudinal W-field B. When m ~ 0

we are assured t ha t the theory is renorinal izable (~) as i t is s imply a pa r i ty -

v iola t ing general izat ion of QED as we describe in this Section. The whole

purpose of this pape r is to show tha t even for m e 0 the theory remains

renormalizable , if we take a d w m t a g e of the desirable proper t ies possessed

by the exponent ia l in terac t ion , and this i~ done in the nex t Section.

For the p resen t we set m ~ 0 and discttss the possible infinities. On a

simple power count b~sis t imre would seem to be five p r imi t ive infinities (neglecting v a c u u m graphs). I n ac tua l fact the cur ren t conservut ion of the A field in this l imit , "malogously to QED and s y m m e t r y proper t ies , implies

t ha t there arc ac tual ly three p r imi t ive infinities (*) corresponding to the self-

energies of A and ~ and the v v A ver t ex , ;dl of which are logari thmic. The

Appendix contains a complete discussion of the renormal iza t ion needed in

a pa r i ty -v io la t ing theory (s) gnd we refer the reader to i t for details. Le t us

sketch the i r appl icat ion to our model in i ts chiral l imit .

i) ~ sell-energy. F r o m the charac ter of J5~ we see tha t only the left- handed componen t ½(1 - - i~5 )? is affected by in teract ion. ]~y adding ~ left- handed counter t e r m

(7) 8L = ½ (Z~-- 1) ~?i~-y(1 - - iy~)~

to the Lagrangian we sub t rac t aw'~y the logar i thmic infini ty of the self-energy and obta in the complete renornIal ized p ropaga to r

(s) 1 S'-,(p) = y .p + ~ (Z~--])y.p(1--i7~) + X*(p) =

1 . , ~ ? ~ ( s ) d s l

(7) A. JAFFE: Anm Phys., 32, 127 (1965); Phys. Rev., 158, 1454 (1967). That the two theories are equivalent is implied in the work of F. J. DYsoN: Phys. Rev., 73, 929 (1948). ~*) The box diagram (A 4) vanishes with each of the external A momenta and requires no subtractions. The more dangerous looking triangle graph (A 3) being the difference of two loops is described by th(~ typical integral TrSy~lS(k3)?~2S(kl)?~3S(k2)i?sdk i which inust assume the typical forln e~2~a~k~F(/c), where k connotes the external momenta. I t is not possible for F(0) ¢ 0 since there is no nonzero symmetric Lorentz tensor having that form. Thus F vanishes with each momentum meaning that the triangles arc certainly finite, if not zero. This result is not to be confused with the anomalous Adler graphs which are intrinsically nonsymmetrie in their external lines. (8) K. SEKINE: ~TUO'~IO Cimento, l l , 87 (1959); R. D~'LBOURGO: J~UO~'O Cimento, 32, 1380 (1964); R. A. CARHART: Phys. Rez., 132, 2337 (1963).

9 4 8 R. DELBOURGO

where

(9)

and

(lO)

1 1 f ~(s) ds 2 * ( p ) = -~ ~ , . p O - i~,~) . z ~ ( p ~ ) ~- :j ~, p O - i~,~) s - p ~

Z r ----- 1 - - Z '~(0) = 1 -/?~(s)/" ds d s

defines the lef t -handed wave funct ion renormalizat ion constant . We have, of course, Z a = 1 for the r ight-handed renormalizat ion and 3m m = 0 at this

stage. The bare and physical fields are connected by (see Appendix)

(11) % = exp [-- ½ ~(1 - - ivy)] ~o with exp [-- $] = Zzt ,

so tha t upon addi t ion of the counter t e rm the free pa r t is correct ly in te rpre ted

as the bare quan t i t y v~o@'~%. For instance to second order in g one has

(12) 1 f Z*(P)---~Y " p ( 1 - i y s ) - i g 2 d4k 2 2 p . ( p ~ k ) 2

(2~), p (p - k V ( ~ - t~ ) ' _ g2 s2__#4

~(s) 16~ 2 s 2 + 0(9 4)

and

g2 ( l n A 2 _ l ) + o ( g 4 ) , z ~ l - i - C ~ \ t, ~

where A is an ul t raviole t cut-off.

ii) A sell-energy. The contr ibut ions here would appear to be quadrat ical ly infinite; however, for cer ta in reasons (see Appendix) t hey are in real i ty only logari thmic and only so in the t ransverse component--- the longitudinal par t is essential ly decoupled by A current conservat ion when m = 0. We add counter

te rms

(13) ~L = - - 1 (Za - - 1)((O.Az)(O.A~) - - # 2 A 2 - - (~B) ~ + #2B2) - -

- - ½ Z, ~ 2 ( A ' - - B-')

in order tha t Ltre.+ L has a ready in te rpre ta t ion as the free bare Lagrangian

2 2 2 2 2 --½ ((~,,Aoa)(~,,Aox)--#oAo --(c~Bo) +/~oBo)

if we connect the bare fields to the physical fields in the usual way:

(14) Ao/A = Bo/B = Zta .

RENOI~MALIZATION OF A ~VEAK-INTERACTION MODEL 949

The complete renormalized propagator of A becomes (*)

(15) g~(k) = - - Z~(k2 - -# ~ + ~tt 2) + II~(k)* . =

k~ k~ + k~- ( - G ( ~ - ~ + ~ ) ) '

where, in this l imit ,

(16) I I~(k) = _ ( g ~ _ k~k~ ~c"- ] Hdk~)

and

,~, ~" ~3(s)ds

d s ( s - - k ~)

(17) f ra(s) ds Z A = ] - - ( s - -y~)~ '

F T3(8) ds - - ZA ~#~ # q

= J

define the renorm~fiization constants. The fact tha t the longitudinal A prop-

agator has mass #0 and carries an infinite Z-factor is i r re levant if we remember

that , like B, ~A propagates f reely and permi ts us to apply subsidiary condi- t ions

on our physical s ta te vectors even when m = O, i.e. both B and ~A can be disreg~,rded for the interact ions of the vector mesons.

For example the g2 correct ion is described b y

(is)

k ~ j 3 d(2z) • p ~ ( p - - k ) ~ '

g2s + o(g 4) , T3(s) = ' z 4 ~

ZA ~ 24;~- _

However , note the finite combin,~tion

I_ d ( s - -# - ) j

g2 = 1 - - ~ - 2 + ° ( g 4) •

iii) y~AyJ vertex. Last ly there are the obvious logari thmic infinities of the proper ve r t ex graphs which clearly only affect the lef t-handed spinors

(*) And will correspond to an unstable particle having a width governed by g2.

950 R. D:ELBOURGO

and accordingly give the overal l v e r t e x factor

F~(p', p) = IF~(p' , p)(1 - i 7 5 ) ,

where Y~ is a vec tor m a t r i x of the type 74, (P + P')~Y'(P ~- P~), etc., mul t ip l ied into scalar funct ions E ( p '~, p2, (p p,)2). I n the l imi t p'---~p, CP invar iance

entai ls t h a t

(20) Y~(p, p) = I (7~ F + P~7 .pF')(1 --i7~ )

and t aken be tween f ree-par t ic le spinors of zero mass gives

(21) ~F~u = I ~y~(1 - - i75) uF(0, 0, 0) .

An a l t e rna t ive descr ipt ion first places ex te rna l spinors on mass shell:

(21') ~'~r~ u = 1 ~ % ( 1 - - i75) uG(0, 0, (p - -p ' )~)

and then takes the l imi t p-~p ' . Of course bo th E and G are logar i thmical ly

infinite and the renorm,~lization cons tant of the ve r t ex is defined us Z = = F ( 0 , 0~ 0) ~ G(O, 0,0) . I n our case i t is i m p o r t a n t to recognize (~) the

exis tence of a Takahushi i den t i ty

(22) (p ' - -p) l~(p', p) = ( S'-~(p ') - - S'-~(p)) ½- (1 - - i75),

or i ts differential Ward form

1 1 (22 ~) F~(p, p) = -~ 7~( - - i75) -~ ~p~ X*(p) ,

which serves to equate Z~ wi th the wave funct ion renormal iza t ion constant Z~ wi thout the necess i ty of explici t calculation. Thus to e l iminate the ve r t ex

infinit ies we add the counter t e r m

(g~-- 1) Ig~y~(1 - - i75)y~A~,

bear ing in mind tha t Z ~ - Z z. I n this way, for example , the lowest-order rad ia t ive correct ion to the

v e r t e x reads

(9) J. C. WARD: Phys. Rev., 78, 1824 (1950); Y. TAKAHASItI: NUOVO Cimento, 6, 370 (1957).

RENORMALIZATION OF A WEAK-INTERACTION MODEL ~ 5 1

giving

~(p') ~i(p', p)u(p) ~(p,) ~ ~ (~ - ~ ) ~ ( p ) ( Z o + r ( ( p - p ' ) ~ ) ) ,

where

• ~ d~k ~p';(p--t , .)p.(p'o--k)/p.{

c o

- ~ - z , + t fd t ' ~(r) ( t '~ t ) t '

with

t

g~ f s~ds g2__ t (23) T(t) - - 8z,,t ~ t ~ - # ~ - - s ~ 8 7 I '2 I n - - , #2

0

fo rm fac tor ~ 1 g2 2 t ... + ~ l n t- 5 .

Summariz ing , the model is a chirM general izat ion of QED which corresponds

to a bare Lagra, ngi~n (m = O)

(5') L = Cfoiy.~/,o--½(~Ao~?~Ao~--/~2oA~o) ÷ ½go~oAo.~(1--iys)V,o

expressible in t e rms of renormal ized fields and masses viu

] A o = Z~A, (24) /

% e x p [ - - ½ $ ( l - - i y s ) ] ~ o with e x p [ - - 2 ~ ] = Z L ,

go gZ ZZ ~ Z~ ~ -~ = = Z ~ g ,

whereby only the t ransverse A field and the le f t -handed W field are coupled. All calculated ma t r i x e lements ~re then au tomat ica l ly finite a f te r these renor- malizutions and given us a pe r tu rba t ion series in g.

To i l lust ra te take spinor-spinor sca t te r ing to order g4 which includes the

three p r imi t ive corrections ~bove as depicted in Fig. 1. The renormMized

3- + + . . . ÷ + . . . + + - c r o s s e d

Fig. 1. - A-field contributions to V~-~ scattering.

graphs

952

pole diagrams combine to give

R. DELBOURGO

~ya(1 - - ivy) u~ay~(1 - - iy~)u~B(t) - - (3 +-~ 4) ,

~O

B(t) fdt' (t') f dt-- 1 - - 1 + 2t ( t - - #~) t - - # ~ t ' ( t ' - - t ) ( t ' - - t ) ( t ' - - # 2 ) ~ ] '

o

where v and ~3 are the spectral funct ion of the ve r t ex par t and the A self-energy given previously in (18) and (23). The box diagrams on the other hand give

~ u4y,,(1 - - iys) u2u3y~(t - - iys) u~K~x(stu ) - - (3 +~ 4) ,

1;,,~(stu) =

4ig4~ d4k 1 • r g~ k~ J k~ k~ 1

In the final analysis we know tha t the ampl i tude must reduce to the form ½~47a(1 --i75) u~ 1~37~(1 - - iys) UlM(s tu ) , where M is obta ined f rom B ~nd K. I t suffices for our purpose to ew~luate M in the soft l imit as s, t, u -~ 0 as i t essentially provides the radiat ive corrections to the Fermi constant . In tha t

l imit the integrals become ve ry simple since

(25)

B - - ~ t ( t_#~)2 j = ~ + ,

K-~ 3ig 4 ( d4k __ 3g 4

(~2~), I k~(k~-- ~)~ 1 6 ~ # ~'

... ~ - + 2 ~ 1 + 2 4 ~ ~ j = 2G ~ 4 s ~ J "

I f we ident i fy g wi th the e.m. coupling constant e, the radiat ive correction to

the Fermi constant is given by the factor 1 - -7~ /12 . Tha t is the main con- clusion of this Section, apar t f rom the v iab i l i ty of the renorm~lization prog-

ramme.

4. - T h e m a s s Lagrang ian .

We now switch on the mass in teract ion L ~ which induces transit ions between right- and lef t -handed spinors. The first thing to simplify calculations is to nor-

mal-order ~ and this amounts to summing all the (~ tadpole graphs ~. We note

I ~ E N O R M A L I Z A T I O N O F A W E A K - I N T E R A C T I O N M O D E L ~ 5 3

t ha t ~v(x)y~(x)=:f(x)~f(x): because 8 ( 0 ) = 0 for massless fermions, so we need

only sum the B tadpole graphs (*) of Fig. 2. Thus

(26) exp [gysB(x)/it] = exl)[ g%l(O)/it ~] :exp [gy~B(x)/it]:,

Fig. 2. - The tadpole graphs.

/ = \ \ !

- \ /+-\ / / x

- - + \ + i + + . . .

which corresponds to a mul t ip l ica t ive muss r eno rma l i za t i on - - i t is precisely

in this way tha t the most t roublesome infinities of the theory are absorbed (~.3),

since an expansion in powers of g2 brings out powers of the quadrat ic infinity

~(o)-=fdp(p~--/) -~. We m a y re in te rpre t our fermion mass by

m exp [-- G~I(0)] ~ m

therefore and work with the exponent ia l in terac t ion

L~v . . . . m: ~ exp [gy~B/#] y~ : ,

which we t r ea t as a pe r tu rba t ion of the A theory. This is a useful procedure

for two reasons:

i) Ls~ gives complete ly finite ma t r ix elements,

ii) the effective expansion p a r a m e t e r (for dimensional reasons) is m/it which we deem to be small, e.g. m ~ I GeV and tt ~ 30 GeV, as typi-

cally taken , g ive m/it ~0.03.

The main burden of this Section is to demons t r a t e i) and ii) wi th some per- t inen t examples .

One 's first incl inat ion would be to separa te off the free mass t e r m mv~y~

for absorp t ion into the free Lagrangian. This is not a course to be recommended

because the chi ra l i ty of L~ is des t royed and the remain ing in terac t ion can

produce infinite graphs which mus t be summed to all orders before they can be

seen to cancel off (**). For ins tance the sequence of d iagrams in Fig. 3 would

(*) In general, if U(~) stands for an interaction Lagrangian, then thenormal-ordering operation U(~0) = : V(~)" is accomplished through V(~0) -- exp [½ A(0)(d2/d~)] U(~). (**) Thus the interaction m p ( e x p [ , ~ ] - - l ) ~ has a nonpolynomial piece which can combine with the interaction in higher orders apparently giving infinite radiative corrections.

9 5 4 R. D E L B O U R G O

contain quadrat ic and logar i thmic infinities, ye t adds up to zero! On the other

hand when t rea t ing (8) as an en t i t y we have to keep an eye open for first order

mass renormal iza t ion and be c~reful to p roper ly t r e a t the connected F e y n m a n

graphs in comput ing S-mat r ix elements .

/ / , \ \ I I I ~ \ \ / / I \ \ / I "\ ,/ I \

Fig. 3. - Vanishing sums of graphs having massive-fermion propagators.

+ . . .

An extens ive s tudy of the exponeut ia l in te rac t ion exp [2~0] of a massless

field ~ is a l ready to be found in the l i t e ra ture (~o). Of grea tes t significance to

us is the fact t h a t the d is t r ibut ion

cxp [2~D(x)] = exp [ - - ,~'~/4z"-x ~]

is so defined t h a t

(27) l i m D " exp [,~2D] ~ 0 , $ --->o

a resul t which is obvious when the approach x--->0 is mude along the t ime- l ike direct ion so t ha t D ~ - - c ~ , bu t is not so obvious for other directions where an appropr ia te cont inuat ion has to be made in 2: (~o). The resul t (27) renders (ultraviolet) finite all m a t r i x e lements of the in teract ion L , ~ ~ exp [~q~]. I n our case (8) one has to cope wi th a t least one mass ive field. Never theless , since the mos t singular behav iour of its p ropaga t ion is mass independent , one can still asser t t h a t to each order in m the ma t r i x e lements should be ul t ra- violet finite; fur thermore , t hey do not obviously give rise to infra-red infinities

e i ther because of the mass carr ied by B. To see how finiteness is achieved

and the effective cut-off on integrals consider the selection of graphs below.

1) _Fermion sel/-energy and associated graphs. To second order in LB~ ,

and neglect ing LA~ for the momen t , one gets (see Fig. 4)

S'-l(p) = y "p - - m - - im ~.

• f d ~ x exp [ipx] ((exp [ g y j ~ ( x ) / f f ] - 1), iy . ~D(x ) . (exp [ g x ~ B ( O ) / f f ] - 1)} ==

= y .p - - m - - i m 2 f d 4 x exp [ipx] i X • ~D(x ) . (exp [ g ' A ( x ) / # ~] - - 1) .

(lo) S. OxuBo: Progr. Theor. Phys., 11, 80 (1954); M. VOLKOV: Comm. Math. Phys, . 7, 289 (1968); H. L~HMA~N and K. POI~LM~YER: Comm. Math. Phys., 20, 101 (1971).

l Z E N O R M A L I Z A T I 0 1 q O F A W E A K - I N T E R A C T I O N M O D E L 955

The self-energy pa r t can be reduced to a 1-dimensional integral :

(2s) x;<v> = -++,+ +,<x>/,++,+- + + ( + ) :

= '2 7 .p - f p~l#~ o

(1 - - exp [g2 K,(~)/4z~2 ~]) .

blIIIILIIIIIIIIIIIIIIIIIIILII[I 1

+ -,- + . . . + - - ~ + , tt +

~ IIII illlll[llllilliLlllll LIIIt +...+ ~- + /I ~ llI~ +...

Fig. 4. - Fermion self-energy.

Analogously to the proof of (27), we shall ,~SSulne gO+< 0 to begin with, so

theft the in tegral is well defined (*) and then cont inue our results to other g2(> 0) values. Clearly (28) is finite (**) because as

+ --> 0 , (1 - e~l) [g+/(,(o)/~+++]) J+(+q) -~ :I e+q ~ ,

e -+-+ ~ , (1 - - e x p [ g ~ I q ( e ) / 4 , ~ o ] ) J ~ ( e q ) -~ _ g O (xI) [ - - ~o] cos oq/4~°-e ~ .

Indeed, in (28) we see the existence of two cut-offs: thus the oscillations in J2 correspond to ~2 values cut off at --#~/p~, while the damping factor in the

exponent ia l corresponds to a e~ cut off in the v ic in i ty of _ g 2 ; we conclude t h a t for p ~ < #~/g2 = G-~ i t is the exponent ia l in terac t ion cut-off which oper- ates first, while for p ~ > tt2/g 2 i t is the Compton cut-off which pr imar i ly gov- erns the integral . In this way one c~n a r r ive a.t separa te ~symptot ic exp~msions for the integral :

wri te a) For p 2 < G -1 use the series representa t ion of the Besset function J to

~ * ( P ) = - - 4 ~ ' P ~ n ! ( n + 2 ) ! . , v J

(*) In much the same way that the gamma-function representation n ! a - " - l = =f t " exp [--at] dt starts life with a > 0 and is then analytically continued. (**) The behaviour of the massive superpropagator will be the subject of a separate publication with Dr. B. [][ARTLEY. Extensive computer work is required for non- asymptotic p2 values.

9 5 6 R . D ~ L B O U R G 0

with

(29)

co

fde (e2) (1 - exp [g2K~(o)/4~'~] ) . o

For small g2/4~ one obtains the lowest few coefficients (*) as

(30)

Io(g 2) m - 4 ~ 2 0 . 7 - - 1 n ~ + O ( g ~lng~),

g 2 I,(g ~) ~ ~ + O(g 4 In g2),

8g ~ I~(g ~) ~ - - ~ f + O(g 4 In g2), etc.

b) For p 2 > G-~ we a l t e rna t ive ly expand the K-func t ion and obta in the well-known leading behav iour (lo)

[c oy .l Z;(p) - - , \16z~] exp 3 \~67e~ ] j y . p m "

Similar ly for higher orders in ZB~. I f we are in te res ted in values of p~ </~2,

i t is the a sympto t i c behuviour a) which is appropri,~te, in which event we learn t h a t the effective expansion parameter is (gin I#) 2 -- m~G, tip to In gO fac- tors, and moreover t ha t the Fermi constant G sets the scale/or the momentum (11). I n par t icu lar for the quest ion of mass and wave funct ion renormal iza t ion , one

finds to order L ~

(31) 8 ' - I ( P ) = r ' P 1 - - ~ -~ Io(g 2) - - m 1 - - ~ Io(g 2) + 0 \ / ~ !

Hence the renormal ized mass is finite and equals

1 (32) m 1 + 8 Io(g2)+. . . m m 1 - - ~ \ 4 z ~ ]

(*) The emergence of In g2 factors should cause no distress. They have been previously noted in ref. (1) in the ~-limiting formalism and in other ways of ~( summing the most infinite graphs ~>. I t is quite conceivable too that we could encounter in (m//~) contribu- tions in higher orders when the fermion propagators acquire mass and integrals of the type Sd~ ~ exp [--Qm/#] exp [g~/o ~] arise as generalizations of (29). (11) A SALAM: Computation o] renormalization constants, in Coral Gables Con]erenee (New York, 1971).

RENORMALIZATION O F A WEAK-INTERACTION MODET,

while the wave function renormalization is

We can go further and evaluate diagrams where soft B lines are attached to the vertices of the self-energy graphs. This is important because the mass renormnliz:%tion will affect the entire nonpo1ynomi:il B inter:wtion. In fact

if we consider the associated graphs of Fig. 5 we notice that they are essentially obtainable from Z* by forming combinations, for example like

in second order, etc., and will have (finite) renormalizations related to the self-energy.

2 ) Meson, self-energy and associated graphs. Let us consider the full AB propagator matrix. One notices first of a11 that to order m the (AB) transi- tion vanishes. Thus all B corrections are intrinsically nonpolynomial and finite, and therefore a t least of order ( g m l , ~ ) ~ . Secondly we observe that the R self-energy DBB(k) vanishes as k 2 + 0 because of c:rncelhtions with (( tadpol,% like o graphs. These results are to be expected, since the equations of motion following from (7) include the pair

and

which imply that the elements of self-energy matrix (see Pig. 6)

958

are connected generMly b y

R. D E L B O U R G O

• A A (35) ~k, , l l~a = t toFI~ B , i k ~ [ l ~ ~ = I I ~B .

In fact , as we shall see~ a pa i r of renormMizat ion constants suffices to t r ea t

the ent i re m a t r i x (34).

/

. . . . . . . . ÷ . . . .

Fig. 6. Meson self-energy.

The associated graphs are ob ta ined b y a t t ach ing B lines to the meson self-

energy ver t ices . We shall not discuss t h e m a t all except for ment ion ing tha t the A - - B ~ t rans i t ions van ish to order m, demons t ra t ing the intrinsic non-

po lynomia l i ty of the remain ing contr ibut ions (see Fig. 7). Note also t ha t

they also van i sh a t zero A m o m e n t u m .

Fig. 7. - B-associated meson self-energy graphs.

+ . . .

3) M e s o n - / e r m i o n v e r t e x . Take the correct ions to the ~0~A ve r t ex shown

in Fig. 8 which~ in the i r B-modifications~ are finite through the act ion of the

exponent ia l Lagrangian . In the l imi t of soft A emission we can evaluate

%"'%ll,,H,f Fig. 8• - A meson-fermion vertex correction•

RENORMALIZATION OF A WEAK-INTERACTION MODEL 9~9

these in the usual Ward way (s), viz.

82:;~: 1 a_r]: 1 (:l + i s ' y ~ , ) = 8 2 * l ~ p a , (36) ]).(p,p)--lya(1--iTa)= ~1,~ + ~ ~ ( l + , i s ) ~ ) . cq,,~ ~ ' ' '~ '

whose le f t -handed piece we shall p resen t ly renormalize. The impor t an t thing

to notice front (36) is theft even the B-line corrections to the ve r t ex give equal i ty

of lef t -handed wave funct ion and ve r t ex renormal izat ions .

~, '~ / 1 \ \ ~ , / / \ \ / / ""+ I v I +"'+ < ~ ' / , +

IIl[q I[i I]]11 IIII II IIII1~ Fig. 9. - Mcson-fermion vertices.

Final ly, there are the associa.ted B-emission graphs of Fig. 9 in which the

A-line corrections to the basic exp [B] in6en~etion vanish identically. Thus

B-emission corrections are again intr insical ly nonpolynomial and finite.

4) Other graphs. Le t us focus on those radk~tive corrections, besides 1), 2)

and 3), which leave t ha t p r in t on spinor-spinor scat ter ing. Broadly these can be divided into pure B-line corrections plus mixed A-B corrections as shown

',,rllllll IIIIII1~ I + , ' ~ ' =- + .-~

:Fig. 10. - Some basic contributions to fcrmion-fcrmion scattering.

in Fig. 10. The first of these d iagrams is the mos t impor t an t and gives the M-funct ion

(37) im2 f ddx exp [i(pl- Pa) .x].

-(ex I) [gy~ B(x)//~] - - 1, exp [g7~ B(o)/#] - - 1 ) - - (crossed term) =

= i m 2 f d 4 x o x p [i(Pi - - Pa) "x ] ( l @ 1 {cosh g~ ~d (x)/}t 2 - - ] } @

e

r , _ . . . .

960 R. DELBOURGO

For s, t, u ~ # ~ the in tegra l (37) can be ewdua ted like (28) to yield the con-

s t an t ampl i tude

dme~ (3s) o .

These induced pseudoscalar t e rms equal the B-pole contr ibut ion a t zero mo- m e n t u m tr :msfer . The other graphs of in te res t because they involve exchange

of ut least 2 meson lines ,~re ,~t least of order gd(m/#)" as g2(m/lu)" and can be safely disregarded for the small masses and couplings we have in mind.

5. - R e n o r m a l i z a t i o n s and radiative corrections.

Taking stock we see the necess i ty of three logar i thmical ly infinite renormal-

izations Z~, ZL, Z~ which ar ise in the ehiral l imi t plus a finite number of other

renormal iza t ions . We wan t to do these b y ~dding counter t e rms such t h a t

the resul t ing Lagrangian exac t ly corresponds to the bare one. The resul t ing renormal iza t ion constants will be g iven as an a sympto t i c series in g2 and (m/#) 2 up to in g2, as described earlier. We can t r y to es t imate the (~ infinite contr ibu-

t ions )) b y int roducing g r a v i t y as the universa l cut-off (~) whereby A is inter-

p re t ed as K~ ~ ~ 2 . 1 0 ~s GeV/c. Thus the logar i thmic t e r m In (A2/# 2) ~ 80,

mak ing of infinite constants ,

(39)

5g 2 Z~ = Z~ ~ 1 - - - ~ + 0(gd) ,

10g ~ Z~ ~ l - - ~ + O(g 4).

F r o m this poin t of v iew there are never any infinities a t all in the theory , the

only significant po in t be ing t h a t the F e r m i cons tan t will p rov ide the cut-off

before g r av i t y whenever the longi tudinal vector meson in terac ts (11).

The re levan t renormal izat ions pe r t a in to the fermion and meson propag-

ators and the fermion-meson v e r t e x ( together wi th some associated graphs),

and we fix the constants in the t ime-honoured way b y ar ranging for poles

wi th un i t residues a t physical masses and a le f t -handed coupling cons tant g

of soft vector mesons to spinors. The essentials of the procedure are as follows.

i) ~ self-energy. Express ing the to ta l cont r ibut ion as

(40) 2*(p) = Z,.m + + .p + (Zs -i ZL)ir'py5

invi tes definit ion of th ree renormul izat ion constants ~m, Z and Z5 (or Z~, Z t as

RENORMALIZATIOS O F A \VEAK-INTERACTIOK NODEL

described in the Appendix) which we associate with counter terms

(41) (2- 1) piy .ay - Z$5y .2y5y -+ (2 Bm- (2-1) m) $5 exp [gy,B/p] y .

These are the familiar terms for a parity-violating theory when B = 0, and ensure correct propagation of the renormalized field. To order ( m / ~ ) ~ one gets

so if we set m/,u m 0.03 a>nd identify g2/4n with the fine-structure constant a, then (gm/Z,~)~ln (g2/4n2) m proving that mass perturbations give quite small corrections t o the lowest-order chirarl renormalization 2, in (39).

The role of the exponential counter term needs some clarification. Recall that every fermion self-energy diagram has associated soft B-emission diagrams with essentially the same structure, occurring in the form {y,, Z*) or 2"- - y,Z* y,, and thereby involving just m Z4. Now on the mass shell mZ4(m2) =

= - Z6m + (I - 2) m-2mz(Zi(m2) + ~ i ( m " ) , so, to a large extent, these as- sociated graphs are subtracted away leaving small finite corrections (*) to

q exp [y,Bl,ul y of order (gmlpI4.

ii) A self-elzergy. Ideally one must consider the A-B matrix propagator, but, if we remember that AB transitions are a t least of order ( g m / , ~ ) ~ , the mixing effect is practically negligible, and the renormalization is essentially provided by 2, of the chiral limit. Still as a matter of principle we would like to show that despite all complications the mass Bp2 and wave function 2, renormaliza- tion of A suffice to treat all the meson ( A and B) self-energies. The proof rests on relations (35) between A and B line corrections, for when we add the counter terms (13) the complete inverse propagator matrix reads

( ) The point being that a t least one B-line will be connected to other graphs when evaluating radiative corrections, so the associated remainder 2 7 n 2 ( ~ i ,'f 2:) will be multiplied by the factor ( g r n / ~ ) ~ .

61 - 11 Nuoao Cimento A.

~ 6 2 R . D E L B O U R G O

Therefore one finds correct ly t h a t in the new field basis

(44) ~ ' -~ -~

W~ = A~ 4- ikz Bilbo, C = ikz Azt[~o - - B ,

- g ~ + k~ ] (z~(k~-#~) +JT~(k~))+

+ -~ - ~o - ~ (z~°~-ul(k~)) _1)t / there is a decoupling, wi th the t ransverse W-components carrying all the

physical informat ion . Thus the counter t e rms are as before (13) except t ha t

Z A --~ Z 3 = Z , + B-correct ions etc.

iii) ~ A vertex. Once again we t ake a d v a n t a g e of the soft l imi t and

go to the mass shell when

(45) ~(p)(Yz(p , p) - - ½yz(1 - - i~,~)) u(p) = ~(p)(SX*(p)/Sp~) u(p) =

= ~(p)(~(1 - z ) - i ~ r ~ z ~ ) u(p) .

The <(infinity ~> is only carr ied b y Z - - Z ~ = ZI and is rcnormal ized away b y

int roducing the counter t e r m (Z~ = Z~)

g ( Z l - 1) ½ ~A .~(1 -- ~o) ~,

to make the soft le f t -handed coupling equal g. There does survive , however ,

a finite r igh t -handed coupling a f t e r the v e r t e x renormal iza t ion , viz. on mass

shell

(46) /'z -+ ~ y~(1 - - iy~) ÷ 74(1 + ~75)(1 - - Z?) =

The only other counter t e r m we need to add corresponds to B-emission and arises

f rom the bare quant i t ies in the exponent of L ~ . We shall not s tudy i t fu r the r (*)

since this renorm~lizat ion is finite

2 _I (47) go Bo//'o = gB {/~o Z3} °

(*) ID connection with the study of associated graphs thc role of this exponent rcnor- realization is perhaps the one loose end in our argument. These points are highly non- trivial and are being investigated further. One conjecture is that the exponent renor- realization cancels soft B-emission graphs associatcd with A mass corrections.

RENORMALIZATION OF A ],VEAK-INTERACTION MODEL ~63

even in the chiral limit and corresponds to radiative corrections of order

(gm/ff)'. Finally, then with all counter terms included one has interpreted every field

and mass in (7) as bare (subscript 0). The connexion between bare and phys- ical quantities being given by

(48)

v'0 = exp [½ i¢7~] z ~ ,

(Ao, Bo) -Z~(A, ~) ,

mo - - ( m - - 8m) eosh ~ ,

r, -1 -~ = = Z 3 - g , go gZ~ Z-~ Z~- -~

Z~ eosh ~ = Z , Z~ sinh ~ = Z5,

it rem,~ins to apply the formalism to some physical matrix element to understand the workings of the renormalizt~tion programme, and especially to determine the remaining radiative corrections. Spinor-spinor scuttering is un ideal vehicle for this because of its analogy to physical leptonie and semi-leptonic decays. Now

in addition to the pure A-field diagrams of Fig. 1 we must consider the B-field

corrections of Fig. 10, plus B-line insertions into the external ~v-lines and the A~v~v vertex as depicted in Fig. 4 and 8 (which obviously correspond to renor- malizutions (48)). To first order in m2/~: or g2 one is left with the radiative

A-correction (25) plus the radiative B-correction {38); the finite right-handed correction (46) and other renormaliz~tions are essentially negligible. In toto at near vanishing momentum transfer

(49) - ( M-~lig*(1--iy~)uuy~(1-- ' i75)uG 4 1 48~2] +

oo ] - - crossed graphs.

One sees the exact correspondence in zeroth order with tile Born term of the original W-meson theory (1). In this connection it is important to appreciate that double graphs are included as part of the infinite set of B plus A-exchange illustrated in Fig. 11. The uni tar i ty of the theory could be verified by ex-

w w w w w c ~ + ,',,,1111111111111,,,,

WWW WW C ~ + WWWWW

F i g . 11. - I n c l u s i o n of W g r a p h s in t h e :I-B g r a p h s .

t~;,,tlllfllllllllt~ t

+ _= ,-~,-_- ÷

~:~,',11 II III11111t,i"~,~1

+ .,,lllllllllllllll,,.i

964 R. DELBOURGO

panding to any given order in g and summing the A + B discontinui t ies to

reproduce the W discontinuit ies . We do not expect a n y surprises here a l though we have not carr ied out a deta i led check.

***

I would like to t h a n k Prof. A. SALA~ and Dr. I . HALLIDAY for m a n y

helpful discussions.

APPENDIX

Renormal izat ion constants in CP-eonserving theories.

Let us assume t h a t we are dealing with a par i ty-v io la t ing renormal izable t heo ry of vec to r -meson- fe rmion interact ions whereby infinities are never worse t h a n quadrat ic and only entai l mass and wave funct ion renormal iza t ions apa r t f rom coupling constant renormal izat ion. This being so we m a y pass f rom the Lagrangian wr i t t en in t e rms of bare quant i t ies

1 2 2 Lo = VTo(i ~ - ~ --too) ~fo-- ¼ (~ Uo~-- ~ Uo~)(a,, Uo~ - - ~ Uo,,) + ~po Uo

to the renormal ized fo rm

-Lo = Z2 y5 [exp [ - - ½ i~75 ] i v • ~ exp [½ i~ys] - - (m - - 8m) cosh ~] V ÷

+ Z~ [-- ¼ (~, U~ - - ~ U, ) (~ u~- - ~ u~) + 1 ( / ~ _ ~ ) u s]

if we define the physical fields and masses th rough (9)

mo = (m - - 3m) cosh $ ,

U0~ = Z] U~.

Observe t h a t there are two Z-factors for the fermion because of pa r i t y violat ion and an a l te rna t ive way of wri t ing the fermionie piece of Zo indeed is

~[i 7 • ~(Z ~- iy~Zs) - - Z ( m - - ~m)] V,

where

Z~ cosh ~ = Z and

Par i ty -conserv ing theories would have

Z~ sinh ~ = Z5 •

~ = Z s = 0 .

Zo can be construed as the addi t ion to the physical free Lagrangian (i.e.

R E N O R M A L I Z A T I O N OF A W E A K - I N T E R A C T I O N ~ O D E L

writ ten in te rms of physical ~p, U, m and #) of the counter terms

~L = (Z~-- ] )[-- { (~ tT~ - - ~ U~)( ~ Y~ - - ~ U~) + ~ ~ a ] - - ~ Z~ ~ V'- +

Let us now see how the renormalizat ion constants relate to the proper self- energies.

a) Fermion sell-energy. The complete renormalized propagator is given by

s~-,(p) = v .p(Z + i ~ z ~ ) - - z ( m - - ~ m ) + 2*(p) ,

where by CP conservation

Z*(p) = X27 "P + Z 4 m + Z s i y ' p 5

and formally the

are logari thmically infinite.

f ~:i(s) ds

965

We arrange our subtractions on the X* such tha t S' describes a propagator with a pole at y . p = m having uni t residue, and this fixes

- - Z ~m = m[Zd(m 2) + 22(m2)] = m f ~4(s) + z2(s) ds 3 s _ m 2

Z = 1 - - Z2(m 2) - - 2m 2 [X~(m 2) + X~(m2)] = I - - f 2mZ r,(s) + (s + m z) v~(s) ds (s__m~)2

Z~ = -- Z~(m~) = - - ( T#)ds . / 8 _ m 2~

whence

S'-~(p) = ( y . p - - m) .

• [ 1 - - f {[(s+m~)~4+2sT~]m-+-[2m27:~+(s.+m2)':~]~p+i'5"'pT~(s-m~)}( ' 'p-m)ds](,~, _ m2)2(,~ - - p~)

is adequate ly renormalized. I t will sometimes prove convenient to define right- and lef t-handed renor-

malization constants through

z ~ = z + z ~ , z t = z - z o ,

which are obta ined from spectral functions z? and 31 re la ted to v~ and v2 in an obvious way. The connection with the ehiral angle ~ and the net renormal-

9 6 6 R. D~LBOVRGO

ization Z~ is t hen

exp [2¢] = Zt/Z¢ , z~ = z~z~.

b) Vector-meson self-energy. - The complete renormalized propagator is

= Z k * . Zl:5'(k) (ff2--~f2--k2)Z3g, n + 3k,, ~+ll;~(k)

We can wri te

II*a(k) = -- (g~, - - k~ k~/k ~) //a(k ~) + k. ka//l(k2)/k 2

a n d / / I and Ha can formally be given a spectral representat ion. The F e y n m a n integrals desc r ib ing / /* 4o not pe rmi t a singulari ty us k -> 0 and consequent ly we must impose the condition

Ha(0) + / / , ( 0 ) = o ,

whereas in fact a direct calculation of Ha and //1 as infinite integrMs often does not satisfy this constraint. We are therefore led to define the t ransverse and longitudinal functions by

f T , ( s ) ds / /~(k2)= s - - k ~ '

ds

The renormMization constants enter via the formulae

v,(s) + ,u 2 v~(s)] ds

s(s - - ff~)

1 f ~3(s) ds = 1 = - J

so t ha t finally

A'~I(k) = (g~a---~C] - - ( k 2 - - # ~)+ (k2- -# 2)2 ( s_k2) ( s_#2)2 +

f a(s)ds ,0 ~ Vl(8)ds]

gives a requisi te pole in the t ransverse pa r t at k 2 = if2 with uni t residue. Note in par t icular t ha t the formula for Z 3 and the re la ted formula

f v3(s)ds f ~l(s) ds Z~ff~ ---- #~-- # s(s -- #2)~ s

RENORlgALIZATION OF A WEAK-INTERACTION MOD]]L 9 6 7

accord fully with canonical commutat ion sum rules (12). The significant remark, however, is that , owing to tile definition f o r / / 3 above, the integrals are loga- rithmically infinite at worst.

There is the special case of conserved currents where vl ~ 0 and

f r3(s)ds

agrees with our preconceived notions. Moreover, in this l imit because

Z ~ o = / ~ 1 - - # J s ( s - -#~)~J

we see tha t the particle remains massless af ter interact ion if it is massless to begin w i t h - - t h e photon analogue of the neutrino.

c) Vector-meson-/ermion vertex. - We are presuming tha t infinities are always logarithmic. By CP conservation, if we place the fermions on their mass shells, the proper ver tex function reduces to the expression

u(p') G~(p', p) u(p) =

where the infinities are carried by G~v and G~ on dimensional grounds. We therefore define vector and axial vector coupling renormalizations to write the in teract ion ~y. U[Z~vgv-~ Zl~g~iys]~. To relate Z~v and Z~ to the wave func- t ion renormalizations we look for Ward identit ies. Let us therefore presume a PCAC equat ion ~j~ = mj5 , where J5 acts like a pseudoscalar current. By considering ~(T(~j~v~)} one obtains the Takahashi ident i ty

(p~-- p) G~(p', p) -- S~-~(p')(gv + ig~ y~) -- (gv-- ig~7~) ~'-~(P) ÷ mG~(p', p) ,

where G refers to the ver tex funct ion for ]~. Indeed, one has

mQ(p, p) = g~(i75 , S'-l(p)}.

5Ioreover, to lowest order in p ' - - p one establishes the Ward iden t i ty

G~(p, p) = ~ S~-l(p)(g~-~ ig, y~) + m ~G~p~

and taking this between ~(p), u(p) one deduces tha t

Z~g~÷ Z,~g~iT~ = ( Z ÷ i~Z~) (g~ ÷ ig~n) ,

(12) G. ~T]~NTZ]~L: Quantum Theory o] Fields (New York, 1949); K. JOttNSON: Nuel. Phys., 25, 435 (1961); A. SALi~: Phys. Rev., 130, 1287 (1963).

968

w h e n c e

R. DELBOURGO

2 2 Z = Z..g.2 - - Z~.~ ga , Z~ ~ (Z~A --~ Z.,)~ ga gv

gv - - ga gv - - g~

T h e l e f t - h a n d e d ch i r~ l l i m i t ga = - - g v , Z~ = - Z is ~ p ~ r t i c u l ~ r l y t r i v i u l e~se s i n c e Z a v - - - - - Zia = Z t h e n .

@ R I A S S U N T O (*)

Si s tudia il proto t ipo di aceoppiamento di interazionc debole ½gf~W.7(1-- i75) ~p di un bosone W (di massa /~) ad un campo spinoriale (di m a s s a m ) , t r a s f o r m a t o di gauge in una fo~ma non polinomiale, e si costruiscc un de t tag l ia to p rogramma per r inormaliz- zarlo. I1 taglio effet t ivo per le in terazioni di W longi tudinal i ~ fornito dalla (( cos tante di F e r m i ) ) G = g2/l~z e le r imanen t i (rinormalizzabili)(~infinith,> dovutc alle in tera- zioni di W trasvcrsal i possono esscrc rcsc finite invoeando le ul ter ior i in tcrazioni gravi- tazional i dei campi. Si danno le corrczioni rad ia t ive finite come una scric as intot ica in g e m/~, supposto piccolo, e si i l lustra il procedimento con la t r a t t az ione dell() scat- ter ing spinore-spinore.

(*) Traduzione a cura della Redazione.

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