dimensional regularization for weak interactions

IL NUOVO CIMENTO VoL. 23 A, N. 2 21 Settembre 1974

Dimensional Regularization for Weak Interactions (').

1~. DELBOURGO and V. B. PRASAD

Physics Department, Imperial College - J~ondon

(ricevuto il 18 Aprile 1974)

Summary. - - In generalizing the current-current form of the weak Lagrangian to arbitrary dimensions we encounter two kinds of polar vector and two kinds of axial vector among the possible set of currents. One of these weak polar vector currents is not conserved except in four dimensions and undergoes a finite renormalization from quantum loops.

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

Except for certain gauge field model calculations (1), users of dimensional regularization (2) have steered clear of the weak interactions. There are good

reasons for the evasion, not the least of which is the problem of finding the proper generalization of V - - A theory to 21 dimensions before the final descent to 1----2. Those computat ions which have appeared in the l i terature (1) all

assume tha t the only weak currents are of vector-pseudovector type, viz.

O) v~F~(1- - iF_ l )~ , where F _ I = FoF~IP~ ... F2z_~,

the presumption being tha t there is a weak vector current, identical to the

electromagnetic current, plus a (2 / - -1 ) - index an t i symmetr ic pseudovec tor weak current brought in by the par i ty-viola t ing lef t-handed neutrinos. On the

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) For instance, W. A. BAI~DEEN, ~R. GASTMANS and B. LAU~RC~: Nuel. Phys., 46 B, 319 (1972); T. API'~LQU~ST, J. C~,RRAZO:SE, T. GOLDMAN and H. QUINN: Phys. Rev., 8, 1747 (1973); W. J. MARCIANO and A. SIRLIlU: Phys. Rev., 8, 3612 (1973). (2) J. ASI~Y[OICE: Lett. JVuovo Cimento, 4, 289 (1972); C. BOLLINI and J. GIAMBIAGI: .LYUOVO Cimento, 12 B, 20 (1972); G. "THooFT and M. VELTMAN: Nuel. Phys., 44B, 189 (1972).

257

2 5 ~ R. DELBOURGO and v . B. PRASAD

o the r hand, f r o m our work on anomalous PCAC identit ies (3), we have learnt t h a t t he hadronic axial cur rent consists ins tead of a three- index an t i symmet r i c t ensor (the pseudoscalar mesons being associated by four- index tensors), a conclusion which seems to be at var iance wi th the fo rm (1).

To reconcile these two viewpoints we shall r e tu rn to first principles and t r y to m a k e a respec tab le guess a t the s t ruc ture of the four -Fermi weak La- g rang ian for a r b i t r a r y integer l, withou t categorical ly ty ing ourselves solely to the currents (1). There are th ree useful guides for the appropr ia te choice of leptonic Lag rang ian :

i) in terac t ions should involve (( le f t -handed >> neutrinos,

ii) t h e y mus t reduce to V - - A fo rm when l = 2 and

iii) the weak Lagrang ian ough t to be Fierz-reshuffiing invar ian t in 21

dimensions , a p r o p e r t y which we know to be t rue for 1 = 2.

This last cri terion is pe rhaps not to ta l ly compell ing and we shall la ter discuss the consequences of re laxing it.

Now the classification of massless- and mass ive-par t ic le s ta tes given in the Append ix shows t h a t neutr inos of the le f t -handed va r i e ty have associated the p ro jec to r 1 ( 1 - iF_l), the direct genera l iza t ion of ½(1- - iys ) in four dimensions. So far, so obvious. The nex t s tep is to exploi t the propert ies of the Fierz- t r a n s f o r m a t i o n m a t r i x (4) in order to find the crossing-invariant general izat ion of V - - A theory . This we do in Sect. 2 and p rove t h a t

(2)

where

(3)

oLZwoC K_(1) - - K_(3) -~- K_(5) - - ... -~- (-- 1)z-~K_(2/-- 1 ) ,

K_(r) =_ ¼ ySFc~,...~rl(1 - - i1~1 ) ~flV~/~'"'M'](1 - - iF_l)~fl.

This resul t is pe rhaps not so obvious and it leads to some unexpec ted consequences , t he mos t i m p o r t a n t of which is t he emergence of a new current ~FE~z~jF_I~v which is not conserved bu t which ye t reduces to a vector current in four d imensions! Na tu ra l l y a t the classical t ree level this current behaves innocuous ly as a vec tor current bu t a t the nex t level of comput ing q u a n t u m loops we m e e t some curious renormal iza t ion effects. The leptonic and nonleptonic effect ive Lagrangians for hadronie weak interact ions are l is ted in Sect. 3, and in the following Section we e labora te on the renormal izat ions of the weak currents b y eva lua t ing the one-loop corrections of the axial currents and the e x t r a o r d i n a r y new vec tor current , in a par t icu lar model of s t rong in- te rac t ions . We do not know if this new p h e n o m e n o n of weak vec tor current r eno rma l i za t ion is a desirable result or needs to be cancelled out as one does

(3) D. AKYEAMPONG and R. DELBOURGO: ~VUOVO Cimento, 17A, 578 (1973); l t A , 94 (1974).

DIMF~NBIONAL REC-ULARIZATIOI~ FOR W E A K I N T E R A C T I O N S 9 ~ 9

with axial anomalies. Certa inly it is one of the oddes t fea tures of d imensional regular izat ion which unt i l now has been blessed wi th a spec tacu la r list of successes.

2. - The weak leptonie Lagrangian.

The two basic proper t ies enjoyed b y the usual four -Fermi cu r ren t -cu r ren t in terac t ion

(4)

f~ re

~ w ac ~1~(1 - - i~5) ~2 ~3Y~( 1 - - i75)YJ4

i) lef t -handedness of the 2 -componen t lepton spinors,

ii) F ie rz - t r ans fo rmat ion invar iance of the Lagrangian , i.e.

(4') ~ c c G 7.( ] - i7~) w~ G y " ( 1 - i r ~ ) w , •

Since we have no prior knowledge abou t the s t ruc tu re of weak in terac t ions in a rb i t r a ry dimensions (2/) let us for the present assume t h a t character is t ics i) and ii) are re ta ined for all 1. P r o p e r t y (1) means (see Appendix) t h a t the lepton fields have 2 ~-I components and are represented b y the spinors ½ ( 1 - i F 1)F, where, according to eq. (]), F_I is the na tu ra l genera l iza t ion of y~ to 21 dimensions. To inves t igate r equ i remen t ii) we have to know someth ing a b o u t Fierz reshuffling.

I n an earlier pape r (~) we spelt out the proper t ies of the Fierz crossing mat r ix . I f K(r) FLj,,M, J (~I ~'M'1 s tand for the 21q -1 pa r i ty -conserv ing k inema t i c covar iants per ta in ing to one channel, the covar ian ts in the crossed channel /~(r) are given b y

(5) ~(s) : ~ C(s, r)K(r) , $

where the F ie rz - t r ans fo rmat ion e lement C(s, r) is defined b y

(6) V(s, r)G~,.....~ = 2-' ~: r r r,-,...-.~ s

and numerical ly given as

(7") C ( s , r ) : 2 - z ( - - 1 ) ~ q ( - - l )q (21- -q~: ) .

(4) K.M. CASE: Phys. Rev., 97, 810 (1955); R. DELBOURCxO and V. B. PRASAD: .IYUOVO Cimento, 21A, 32 (1974).

2 ~ 0 R . D E L B O U R G O and v . B . P R A S A D

I n the case of weak in teract ions there is the minor modificat ion t h a t we should be dealing wi th the pa r i ty -v io la t ing le f t -handed covar iants K_(r) as defined in (3). B y crossing, for s odd, we obta in

(8) R_(s) = 2 -~ ~ ¼/~(.,(I - - i F ~)/ ' ( ,F("(1 - - iF_~) ® F(" = r

: Z 1C(s, r)F(,~(1 - - iF_t) @/ '( ' ) = Z C(s, r)K_(r). r ~*odd

Thus Fierz reshuffling for these par t icu la r covar iants involves jus t the odd-odd entr ies in C (up to r, s ~ 1 - 1 b y reflection symmet ry ) .

We are now in a posi t ion to p rove t h a t the sum (2) is crossing invar iant . We m a k e use of the represen ta t ion

(9) 2~(--1)~'sIC(s'r)= dzz {(1--z)~( l+z)"~- '}]~=°"

Then, f rom (8) and (9),

~ ( - - 1 ) ½ c " - l ) / ~ _ ( s ) = i 2 - ~ ~ ~ i ~ {(1--z) ' (1-t-z)2~-~}l~=oK_(r)= sodd s , r o d d

: i2 -~-1 ~ (exp [i d/dz] - - exp [ - - i d/dz]) {(1 - - z)~(1 + z)2~-~}l~=oK_(r ) =

}*odd

: (--1)½~ ~ (--1)½(~-l)K_(r), ~*odd

showing t h a t the sum (2) is a crossing e igenvector wi th eigenvalue ( - -1 ) ½z. Fina l ly t ak ing note of the a n t i c o m m u t a t i v i t y of spinor fields we conclude t h a t the a p p r o p r i a t e weak four -Fermi in te rac t ion in 21 dimensions is

t--1

~Lf w oc ~ ~IF(,)(1 - - i F _ l ) ~ ~ / ~ ' ) ( 1 - - iF_l) F4 = t o d d

1--1

= (-- 1) ½~+~ ~ v~lF(~)(1 - - iF_l) ~, v/3/~"(1 - - i I '_ , ) ~f2. f" odd

I n pa r t i cu la r the leptonic Lagrangian , wi th correct normalizat ion, wr i t t en in cur ren t -cur ren t form, is

2 / ~ I

(10) ~ w = V 2 G , z., ~ ~(,) Tt ='T(') t odd

with

1 - (11) J(,) ~- ~ V, FE~,~...~,~(1 - iF_l ) ~ .

At t h e t r ee g raph level, which m a y b e is t he only sensible way to look upon the fou r -Fe rmi ~fw, we observe t h a t in the l imit 1 = 2 there occur two cur-

D I M E N S I O N A L R E G U L A R I Z A L T I O N F O R W E A K I N T E R A C T I O N S 261

rents which are replicas of one another, namely

J,---- 1~7,(1 -- i75)~ and J,,~ ~ ½ i~7~,,7~7~,~(1 - - i75)~ .

(The same c~rrents arise in

(12) ~Lf~, ---- e ~ (J+, W ( " + h.e.) , r odd

the weak-intermediate-boson var iant of weak interactions.) Such repeti t ions

of V - - A in J(~) and J(3) for 1 ~ 2 are uninterest ing in themselves if we stick to the Born diagrams. However as soon as we calculate higher-order q u a n t u m loops differences begin to show up. Indeed for 1 :~ 2 the currents

a true vec tor ,

a t rue p s e u d o v e c t o r ,

an (~ ax ia l ~> vector and

the (, p s e u d o a x i a l ~> vector, s a y ,

are all dist inct f rom one another. I n fact J r is the only t ru ly conserved current ; the pseudovector current is conserved in the zero-mass fermion limit when chiral t ransformations ~v-~exp[0F_l]y~ become an exact s y m m e t r y of the

theory ; bu t the axial and pseudoaxial currents are not at all conserved as we know from our work on anomalies (5). We shall r e tu rn to the consequences of this ext raordinary fact shortly.

3. - Ef fect ive hadron ic w e a k L a g r a n g i a n .

I t is a good idea to list first of all the 2/-dimensional forms of the Lagran- gians responsible for the weak decays before we come to the question of quan-

t um loops. The baryons are represented by 2 z component spinors and we have to bear in mind tha t pseudoscalar mesons are 4-index tensors. Beginning with

the semi-leptonic ba ryon decays B-~B ' lO, the s traight generalization of (10) and (12) is indicated, and we should include in (11) the hadronic-current contr ibution

j(~o~ ~ 1 ~F(r)( 1 __ i F 1 ) B

before renormalization, lqaturally one expects radia t ive corrections due to current nonconservat ion to be significant, so one should modify the effective

(5) D. fl~KYEAMPONG and R. DELBOURGO: NUOVO Cimento, 19A, 219 (1974).

262 R, DELBOURGO and v. m ~RASA~

hadronic currents to

J(1) --> ½BF~(g~ - - ig~t_lF_l ) B ,

1 - - - - - - , (13) J(3) -~ ~BFEE~j(g~ -- ig~ 3F ~) B

J (~) --> ½ B F ~ m ( g ~ -- ig~_~F_~) B , etc. ,

where the g~ are coupling cons tant renormalizations. The vector current is of course conserved so g~ ~ 1; one would also be inclined to suppose tha t g~-a

is unrenormal ized at un i ty since the associated pseudoaxial current looks vectorial in four dimensions and is conserved in tha t limit. That however is con t ra ry to the rules of dimensional regularization which stipulate tha t all

pe r tu rba t ion calculations have to be performed be]ore going to the four- dimensional limit. I n fact we shall prove later tha t g2~_~ differs f rom 1 by a

finite amount , calculable in any given model of the strong interactions. Thus we have the bizarre fact t ha t the weak vector current is renormalized and

( ]4) g~/gv = l im (g3 ~- g2~--l)/( 1 -~ g21--]) "

~--~2

W'e shall enlarge upon this curiosity in the next Section, but for the present let us car ry on writing effective Lagrangian for purely leptonic and semi- leptonic decays, P --> lY and P --> P'~V respectively• The effective weak cur-

rents J(,) which couple the mesons to the lepton currents tu rn out to be

(25)

q+-P.;~,~, ~,,~ ,

• + K I ~ M

/ ~J P.rEL~,m

vec to r ,

pseudoveetor,

axial vec to r ,

the pseudoaxial not coupling with mesons. ~on lep ton ic hadron decays have a similar flavour.

principle described b y a host of terms

Thus B - ~ B ' P is in

(]6) ~fsw = BFExzu~ ( b + a F z) B P K z ~ ~-

+ Z B ~ J ~ J " ... ~ " Ft,r,j....j,xL~j(d , + c , F _ , ) B P ~Lrz~ , r

though d and c disappear at the classical level. Similarly the nonleptonic

decays K ° --> P P ' include

~ A B C D ~ ~ J z . . . v J l Z - 6 ~ K L M N - - l

among others (pari ty violation being signalled by the presence of the s tensor). W h a t we have learnt f rom axial anomalies is of some help in cut t ing down some

D I M E N S I O N A L R E G U L A R I Z A T I O N F O R W E A K I N T E I ~ A C T I O N S 263

of the arbi t rar iness of possible couplings: recall ing t h a t PCAC takes t he fo rm

tsAx~M1 = Pt,rXZMl -~- P' [ J K . L M ] '

where P ~ L ~ couples in the normal way v ia y~FE~L~j~p while / ) ~ N couples v ia the anomalous current ~ + F ~ + ~ ~, we gua ran tee the (( c o r r e c t , ~ o - + 2 y ra te b y the anom a l y by dissociat ing pions f rom P ' currents . This suggests t h a t we should discard all dr and c~ t e rms in (16) which anyhow seem to lead to nonrenormMizabi l i ty .

I t is general ly believed t h a t nonleptonic Lagrang ians do not hold the s ame privi leged s ta tus as semi-leptonic cur ren t -cur ren t Lagrangians , possibi ly me- diated b y weak bosons. I f t h a t is t rue, the re is l i t t le poin t in compu t ing higher-order weak corrections ensuing f rom (16) unt i l one unders t ands b e t t e r the origin of the nonleptonic in terac t ions ; the l a t t e r half of this Sect ion can then be jus t regarded as an academic exercise in tensor analysis.

4. - One- loop r e n o r m a l i z a t i o n s .

Let us suppose t h a t s t rong in teract ions are renormal izable in four dimensions.

The ¥ u k a w a meson-ba ryon coupling

(17) ~ e t ~ ~ = g~F~,~ ...~,.~ B~ E~ '~ ' '

can serve as a suitable model for the purposes of the following discussion.

Using the p ropaga tors

~...~,~ a~N~..~,~ 1~2 _ #~)-1 ( B ( p ) B ( p ) ) : i ( F . p - - m ) -~ , ( ~ , . .~(k)~ ( - k)) = =E~, ~,~,.= ,

we can enquire abou t the na tu re of one-loop renormMizat ions of the weak currents. Before we plunge into an analysis of the ve r t ex pa r t s let us t r e a t t he

wave funct ion renormMizations. The fermion loop cont r ibut ion to the meson serf-energy par t s (see Fig. 1)

.l .l[~'-.-N.lb ~/-712v'--2v,l ~ [ M 1 . . . M r I ~ / ~ ~ ~ [Mx,..Mr ]

Fig. 1. - Meson wave function renormalization.

26~ R. DELBOURO0 and v. B. PRASAD

was e v a l u a t e d in a p r e v i o u s p a p e r (~). T h e r e s u l t was

(~8) i l i a , . ..~.]@~ e~F(1 -- l)

1

• ~ f U[M,. . .Mr] -1-- MS.. .Mrl] J ( - - k ~ ~(1 - - ~)) ~-~ { ( - 1) ' (21 - - r - - ] ~ -~[;~"'~'] " 2 (l - - 1) k[~, k [; ' d ; ' ' ' ' ; ' ] / k ~

0

fo r mass l e s s f e r m i o n s (*). To d e t e r m i n e t h e a s s o c i a t e d w a v e f u n c t i o n r e n o r m a l -

i z a t i o n c o n s t a n t s Z~(r) we n o t e t h a t in t h e v e c t o r sec to r (r = 1)

(19) / /~~(k ) --

1

2e {e-1} l ~ 3(-k~(1-~))~-~ ~ 1'

0

w h e r e a s in t h e p s e u d o v e c t o r s e c t o r ( r = 21--1) we m e e t t h e d u a l

~ N N I . . . N r ~ [ M , . . .Mr] ~ " ' / ! ~ - - ~ •

b e c a u s e F _ I c h i r a l i t y is a g o o d s y m m e t r y . O n t h e o t h e r h a n d t h e a x i a l v e c t o r

s e c t o r (r = 3) has

(20) n~?(k) = 2e 2 r (1 - ~ ) (2~)~ 1

f ax • / [ i ,,"El l ." (_k2~( 1_¢1)~_, {( / --2)6[~] ] + ( / - - l ' k k[~O~;]'k ~

0

u n d t h e d u a l p s e u d o a x i a l s e l f - e n e r g y (r = 21--3) is i t s i m a g e :

--1[1J~3 " , " 1 ~ ~IJ 'K~ ' I . , ,RI I - - " ' ~ a . . . ~ s l , , ' - I * ~ [ I J ~ ] U ' ! "

T h u s fo r m = 0, Z ~ ( 1 ) = Z ( 2 1 - - 1 ) a n d Z ¢ ( 3 ) = Z¢(21--3). H o w e v e r

(21) 1

2e2(l--2)F(1--1) f d~ --e 2 Z~(21 - - 3 / - - Z~(1) l i r a (2~7-k2 j ( _ k2~(1 _ ~)) , -z - - 12~2

0

is f i n i t e a n d n o n v a n i s h i n g ! T h e e x p l a n a t i o n for t h e d i f fe rence b e t w e e n v e c t o r

a n d p s e u d o a x i a l r e n o r m a l i z a t i o n s is t h e s a m e one t h a t is of fered w h e n one

m e e t s a n o m a l o u s W a r d i d e n t i t i e s , viz. b e c a u s e t h e p s e u d o a x i a l c u r r e n t is on ly

(*) Fc rmion masses affect longi tudinal pa r t s of H and are not d i rect ly relevant to the de te rmina t ion of Z except for removing infra-red divergences.

D I M E N S I O N A L R E G U L A . F ~ I Z A T I O ~ F O R W E A K I N T E R A C T I O : N S 2 ~ 5

conserved in four dimensions we have the p roduc t of a k inema t i c f ac to r which vanishes as 1--> 2 and a singular fac tor ( 1 - 2) -~ due to the d ivergent q u a n t u m loop. The same d iscrepancy (~) be tween axiul and pseudovcc to r renormal izat ions was observed some t ime ago, bu t is not so s t r ik ing because the axial current will not in general be conserved.

Consider nex t the fermion self-energy due to the in te rac t ion (17), as de- p ic ted in Fig. 2. Again, the fe rmion wave funct ion renormal iza t ion cons tan t Zv can for s implici ty be de te rmined b y se t t ing the fe rmion mass equal to zero. Thus

i 2 ~F(~)F.(p- - k)F(~) d2*k z ( p ) = - g j (-~__ k V i ~ Z ~ ) (2~)~,

l / \ f . s

Fig. 2. - Fermion wave function rcnormalization.

B y s tandard procedures this can be reduced to the pa r ame t r i c integral

(22) X ( p ) = F . p g 2 ~ t l ) 1

f tl~ - - F ( 2 - - 1 ) {(#2 p2c~)( 1 _ ~ ) } 2 - ~ , 0

where the Fierz m a t r i x e lement C(s, 1) can be worked out f rom eq. (7). ]3Tear four dimensions, Z ~ - - 1 ~ g2C(s, 1)/16s2(l - - 2).

F inal ly we can tu rn to the ve r t ex pa r t of Fig. 3 which we shall eva lua te a t zero m o m e n t u m t ransfer

(23) A<,I(p) = ig~f d~k F. ,F. (p -- k) F,.m" (p-- k ) F (s) (2~) 2~ (p - - k)4(k2-- / t~)

11 F "\ F s i*

Fig. 3. - Proper vertex renormalization.

18 - I I N u o v o C i m e n t o A .

266 R. DELBOURGO and v. B. eRASeD

The A(,) are l oga r i t hm i c a l l y infinite a t l = 2 and have e~ch to be r enormal - ized b y a ve r t ex fac tor , s a y Z~(r). Clearly, s ince t he vec to r ve r t ex satisfies

W a r d ' s i d e n t i t y A~(p)----~Z(p)]~p ~, t h e v e c t o r r eno rma l i za t i on c o n s t a n t equa ls t h e f e r m i o n w~ve f u n c t i o n r e no rm u l i za t i on

Z ( 1 ) = Z ~ ,

b u t t he re is no reason in genera l to expec t s imilar equali t ies for t he r ema in ing

c o n s t a n t s ; in our mode l h o w e v e r it h a p p e n s t h a t

Zg(21 -- 1) ---- (-- 1)*+~Z~.

Of all t h e o the r v e r t e x p a r t s t he p seudoax ia l is t he m o s t in t r igu ing and it can

be d e d u c e d f r o m t h e p a r a m e t r i c in teg ra l r e p r e s e n t a t i o n of eq. (23):

A ( , ) ( p ) = (4~) ' [ {(it - - P ~)( - -~ )} - 0

T h u s

v(1, r)C(s, r)F(,)(2 - - / ) ) ]

(24) Z,(21-- 3) - - 1 = - - l i m C(1, 21 - - 3) C(s, 21 - - 3)-

1

g2F(2--1) f ~do~ (4~), d { ( ~ - - p ~ ) ( 1 - ~ ) } ~ - ~ "

0

T h e a n o m a l o u s finite difference b e t w e e n v e c t o r and pseudoax ia l r enormal iza -

t i ons is

(25) Z~(1) - - Zo(21-- 3) = l im 92{C(1' 1) C(s, 1) - - C(1, 21 - - 3) C(s, 21 -- 3)} I--+2 3 2 : ~ ( 1 - - 2 )

a n d for va lues up to s = 4 equals

(25') 32:~2 [ 4

A s imilar difference arises be tween p s e u d o v e c t o r and axial ve r t ex re-

n o r m a l i z a t i o n s :

(25") Z ~ ( 3 ) - - Z ~ ( 2 1 - - 1 ) = l i m g 2 { C ( l ' 3 ) C ( s ' 3 ) - c ( l ' 2 1 - 1 ) c ( s ' 2 l - 1 ) } = ~-~2 3 2 ~ ( I - - 2 )

= z , ( 2 t - - 3 ) - z , (~ ) .

A l t o g e t h e r t h e one- loop cu r r en t coup l ing renormal i za t ions in (13) fol low f r o m t h e f o r m u l a

g, = z~(r) z: l (r) z ~,

D I M E N S I O N A L R E G U L A R I Z A T I O N F O R W E A K I N T E R A C T I O N S 267

since all bare couplings are normal ized to uni ty . I f the renormal ized couplings g, t ru ly refer to weak interact ions we can disregard differences be tween Z~(r) as being of order e ~ and therefore small ( r emember however t h a t Z¢ is not r - independent) and we need only concern ourselves wi th the s t rong correct ions to Z~ and Z v. Besides t he expec ted resul t g~ ~-- 1 connot ing absence of vec to r current renormal izat ion, we have the unexpec ted resul t

g2 [ 1 1 1 1 ]

as the one-loop renormal iza t ion of the o ther weak vec tor current . The axia l and pseudovec to r renormal iza t ions are infinite if s is even because (17) v io la tes F_~ chiral i ty, bu t finite if s is odd. Fo r ins tance a vec tor g iven model (s = 1) gives

a s

g3 -~ g2~-a = 1 + 3 2 ~ 2 ,

while an axial vector s t rong in terac t ion (s--~ 3) gives

ga = g2t-a ~ 1 - - - - - g~

9 6 ~ "

I n each of these cases

g~/gv = (gl Jr g2~-a)/(g2~-, -~ g3) = 1

because of F_~ chiral s y m m e t r y . I n more general c i rcumstances when eve ry current (with the except ion of the pure vector) is nonconserved we an t i c ipa te t h a t g2~-a :/: 1 and g~-i :/: g3. H o w e v e r the previous discussion shows how and why g ~ - 3 - 1 and g 3 - g2z_~ are finite and calculable to a n y order in pe r tu rba - t ion t h e o r y - - w h e n the s t rong interact ions are renormal izable the differences depend only on the magn i tude g and charac te r of the s t rong coupling.

Apa r t f rom cer ta in conformal anomalies (6) in q u a n t u m grav i ty , the renor- real izat ion of this weak vec tor cur rent is the first unconven t iona l resul t to come out of dimensional regular iza t ion and one m a y wonder how it can be avoided. One possibil i ty is to contr ive a cancel lat ion of g2z_3- ] wi th a counter -Lagrangian , bu t this can surely be dismissed as be ing too artificial and ra the r ugly. A second possibil i ty is to re ject the Fierz invar iance require- men t and to re tu rn to the vector and pseudovec tor currents , v ~ F ~ and i~ / '~F_ l~ ; this too is unpala table , for the simple reason t h a t hadronic contri- but ions to axial currents mus t sat isfy anomalous PCAC identi t ies (even if the anomaly cancels in toto); the pseudovec tor iv~F~F_ly~ is po ten t ia l ly incapable of yielding a n y anomaly (since F_l - invar iance is an exac t s y m m e t r y of zero-

(6) D. CAFPEg and M. DUFF: Imperial College, London, prcprint ICTP/73/12.

268 n. DELBOURGO and V. B. PRASAD

mass fermions) in contrast to the axial i~FEKz, ~ ~p and its par tner i~lPE~zm F ~ ~f; therefore even if we abandon the not ion of Fierz symmet ry of Lf~ we are obliged to consider at least these four kinds of vector current and their a t t endan t renormalizations. Another possibility is to cancel off the anomalous renormal-

izations altogether (by increasing the number of fermion fields, including V + A interactions, etc.) in the accustomed manner which ensures the renormalizubil i ty

of gauge theories. One last possibility, which we favour, is to accept (*)the pseudouxial renormaliz~tion g~,_~ bu t to m~ke it independent of the source; being universal it would then be essentially unobservable and cancel out

against g~-- g~_~.

A P P E N D I X

We present here ,~ resume of the classification of p~rticle states in 21 dimensions with p~rt icular :~ttention paid to the massless limit. We begin with the general izat ion of the Poincar6 group 0 ( 2 / - - 1 , 1 ) A T(2 / )genera ted by the linear- m o m e n t u m oper~tors PL and angu la r -momentum operators J ~ which satisfy the usmfi commuta t ion rules. Because of the changed dimensionali ty we can define l - - 1 sets of Paul i -Lubanski ~, spin ,> operators W(~)

W(3)

W(5)

~/V(2I-1)

or W t z ~ ~ ~- ~O~L J~2~ l ,

or Wt+~,L~,v] ~-- Pt.,, JxL J ~ l , ... ,

all of which are t rans la t ional ly invar ian t

[P~, WI,)]----- 0 , r---- 3, 5, .. . , 2 1 - - 1 ,

and whose squares W¢~)W ~r) are C~simir operators of the inhomogeneous group like p2.

To unders t and b~t ter the significance of the W, suppose first t h a t we are deal ing wi th P~ > 0 vectors. I n t h a t case we c~n induce all momen ta from a f rame where p is at rest : j 6 = (m, 0). The li t t le group is 0 ( 2 l - - 1 ) and is generx ted by all the spat ial Jk,; the W(,) have one index equal to zero for rest s ta tes and are given by direct products of the spa t i i l J . In fact since an 0 ( 2 / - - 1 ) representxt ion is described b y 1 - -1 Casimirs, these invar iants are precise ly re la ted to the W~,). The remaining ½/ ( l - - l ) labels needed to specify the s ta te vector ful ly correspond "to picking out par t icular W components and sub-Casimirs.

(*) One should not entertain any hopes that this could explain the origin of the Cabibbo angle; for although it is easy to obtain a small ratio for (AS= 1)/(AS= 0) vector couplings one would be at a loss to account for the near equality of G~ and G~.

DIMENSIONAL I~EGULAI:CIZATION FOR W E A K INT:EI~ACTIONS 2 6 ~

Next t ake p 2 = 0. Here we only p e r m i t t e d to induce our vectors f rom a f rame in which the m o m e n t u m has 0 and 3 components : /~ = (E, 0, 0, E, 0, ..., 0) say. Now the li t t le group is the Euc l idean group in 2 / - - 2 dimensions, 0(21--2)AT(21--2); the ro ta t ion generators ~re J~t, k, l : 1, 2, 4, ..., 21--1, and the t ransla t ion gene?ators are Jko--J~3. Again these Eucl idean opera- ~ors are just what the WraLm reduce to on such a m o m e n t u m eigenvector , the higher W(~) being direct products of P and the J . I n pract ice one is only in teres ted in the finite-dimensionM bases where in the t rans la t ions are t r iv ia l ly represented; in these circumstances ¢h'e l i t t le group is effectively 0(2l--2) with its 1 - -1 Casimirs and fu r the r 1 ( / - - 2 ) ( l + 1 ) labels for des ignat ing the weights. I n par t icular one has

Wiz~...M,z_z] ~ ~M~...~2z -pM2:

where A is the 0(21--2) Casimir if degree 1- -1 . I n four dimensions, of course, A has the significance of helicity.

The finite-dimensional spinor representa t ion of 0 ( 2 / - - 1 , 1) holds special in teres t because it is the direct general izat ion of the four-dimensional Dirac spinor. For massive part icles the ((Dirac equation~> ( F . p - - m ) u = O serves to cut down the number of degrees of f reedom from 2 ~ to 2 ~-~ and these correspond to the spinor representa t ion of the li t t le group 0(21--1). Howeve r for massless particles there is F_~-invariance of the equat ion which breaks up u instead into left- and r igh t -handed 2 z-~ component spinors:

u_---- ½ (1 - - iF_~)u and u + - - ½ (1 + iF_~)u

corresponding here ~o the li t t le group 0 ( 2 / - - 2 ) . I n fact when m - - 0 the H~mil tonian u~For.pu can be re-expressed as lplAu+F-lu, where 2 is the 0 ( 2 / - - 2 ) Casimir, repr -sented spinorially now by

1 ®1 ®... ®6 '~ @1

with eigenvalues ± 1 . The other commut ing W(r) operators which serve to remove the degeneracy in the classification of neut r ino states have a similar s t ructure, viz.

~.~ @1 ®... ®1, i ®a./~ @... ®1, etc.

• R I A S S U N T O (*)

Nel generalizzare la forma corrente-corrente del lagrangiano debole a dimensioni arbi- trarie si incontrano nell'insieme dells possibili eorrenti due tipi di vettori polari e due tipi di vettori assiali. Una di quests eorrenti deboli vettoriali polari non B conservata eccetto che nel caso quadridimensionale e subisce una rinormalizzazione finita per anse quantiche.

(*) T r a d u z i o n e a cura del la R e d a z i o n e .

270 R. DELBOURGO a n d V. B. PRASAD

P e r y ~ p H 3 a U n f l pa3MepH0CTH ~.~l[ C.rla~blX B3aHM0~e~tCTBHfi.

P e 3 m M e (*). - - I I p n o 6 o 6 m e H ~ r t TOK-TOKOBO~ ~)OpMbI c a a 6 o r o J I a r p a r D r n a n a Ha c n y q a ~

IIpOI43BOJabaoro n n c a a HaMepeHrn~ Mbt CTaaKrIBaeMC~I C KByM~t TrIllaMI~ n o a a p u b l X BeKTOpOB

H nayM~I THHaMH aKcHanbm,IX BeKTOpOB c p e ~ n BO3MO)KHOfi CHGTeMI:,I TOKOB. O,//HH H3 3TtlX cJ/a6blX iIOJIapn'blX BeKTOpHFIX TOKOB He coxpanfleTCfl, 3a HCK.nIottel-meM cJIyqa~t

t teTblpex r i3MepeHi~ . ~TOT TOK no£ tBepraeTc~ KoHeqHOH nepeHopMI, ipoBKe, I.tCXO;i~ 143

KBaHTOBbIX IIeTeIIb.

(*) llepeeec)eno pec)ar#ue~.

dimensional regularization for weak interactions

Documents