superfield perturbation theory and renormalization

IL NUOVO CIMENTO VOL. 25A, N. 4 21 Febbraio 1975

Superfield Perturbation Theory and Renormalization (*)C).

R. DELBOURGO

Physics Department, lmperial College - ~ondon

(rieevuto il 14 Agosto 1974)

S u m m a r y . - - The perturbation theory graphs and divergences in super- symmetric Lagrangian models are studied by using superfield techniques. In super ~3-theory very little effort is needed to arrive at the single infinite (wave function) renormalization counter-term, while in ~4-theory the method indicates the counter-Lagrangians needed at the one-loop level and possibly beyond.

1. - P r e l i m i n a r i e s .

I n the i r analysis of the supersymmet r i c ¢a-model , W~ss and ZUMINO (1) discovered the r emarkab l e fact t h a t jus t a wave funct ion fac tor sufficed to renormal ize the theo ry a t the one-loop level, a conclusion which was l a te r conf i rmed to all orders b y IzxoPouL0s and ZUMINO (~) b y using the W a r d iden- t i t ies app rop r i a t e to supergroup t ransformat ions . However elegant thei r work , i t still involved considerable labour s imply because their descript ion br ings in large numbers of componen t fields which have to be tediously wr i t t en out

eve ry t ime. The powerful technique of superfields, invented b y SALA3~ and

STRATHDEE (a), which combines all the componen t fields into a single en t i t y

#(x, 0), avoids these cumbersome features, makes short work of super t rans-

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (**) After completing this work we received a preprint by Dr. D. M. CAI'PE~ which essentially covers the same ground, though his emphasis is on momentum space and restricted mainly to ~a-theory. We thank him for sending us an advance copy of his paper. (1) J. WEss and B. ZUMINO: Phys. Left., 49 B, 52 (1974). (2) J. ILIOPOULOS and B. ZuMINO: N~tcl. Phys., 76B, 301 (1974). (a) A. SALAM and J. STRATHDEE: ~Vuel. Phys., 76 A, 477 (1974). See also S. FERRARA, J. W~ss and B. ZUMINO: CERN preprint TH 1863.

646

SUPERFIELD PERTURBATION TH:EORY AND R~NORMALIZATION 6 ~ 7

format ions , and provides new insights into supergroups. This paper is devoted to a r e -examina t ion of ¢3- theory in the f r amework of superfields, and to

corresponding first look a t ¢4-theory. I n the following we shall borrow heavi ly f rom the recent art icle b y SALAM

and STRATHDEE (4)~ par t i cu la r ly f rom thei r Append ix C where the F e y n m a n

rules for superfields are spelt out. S ta r t ing f rom the free Lagrang ian of a

scalar superfield

{1) ~e o = lV~(~+~_) -- l m V 0 ( ~ + ~_),

t hey obtain the free p ropaga tors

A j:+(xO, x ' O') --~ - - i < T[~+(xO)~v ±(x ' O~) ]> =

- - ~ m ( O - - O ' ) ( O - - O ' ) ± e x p [ ½ i O ~ O ' ] A ~ ( x - - x ' ) ,

(2) A . A x O , x ' O') = - i < T[~v ±(xO)qJ~=(x' O') ]> =

---- exp [l i 0~0 ' -+- ¼ (0 - - 0 ' )~75(0 - - O') ] A ~ ( x - - x ' ) .

Here %@0) refer to complex superfields of par t icular ehiral i ty

(3) ~±(xO) = exp [ =F ~ 0~ 75 0] [A± + 0 ~ + 1 (00±)F±]

with ~± = 1 (1 =[= iys) ~, etc. and V 0 = ~2/~0~ ~ . Of course one m a y easily pass

f rom (2) to the m o m e n t u m - s p a c e expressions

A±±(pO, pO') = -- i m(0 - - 0')(0 - - 0')± exp [10(r.p) O']/(p 2 - m ~ ) , (4)

A.~:(pO, pO') = exp [10(r.p)0' =k ¼ (0 -- 0') i(7.p)rs(O -- O')]/(p 2-:ms) .

The formulae (3) convenient ly summar ize the propaga tors of all the field com- ponents

A± = w~lo-o, V± = ~±/e01 .=o , F ~ = ~Vo~±lo:o,

and one easily deduces (~) t h a t

( T ( A ± F ± ) ) = - - im l (p2 - - m ~) ,

<T(~± ~±)) = ½- ira(1 ::t: i75)1(P 2 - m ~ ) ,

(5) < T ( A ± A T ) > - - i / ( p ~ - u 2 ) ,

<T(~v± vTT)> = ~ i (1 =k i 7 , ) (? ' p ) / (p 2 - m ~ ) ,

<T(F:~ FT)> = ip2 / (p z - m°-) ,

all other propaga tors vanishing.

(4) A. SALAM and J. STRATItDEE: Trieste prcprint 1C/74/42.

6 4 8 R. DELBOURGO

We shall be s tudying theories which are given in the first place b y the interact ion Lagrangian

(6) . ~ - - Vo[v(9+) + v(~_) ] - v o v ( ¢ ) ,

where V(q~) m a y even be a nonpolynomial funct ion of its argument . Then in N- th - o r de r per turba t ion theory the amputa t ed n-point Green's funct ion (m, lines a t tached to ver tex x~O~) is given by the Hori expression

(7)

before we act with the Laplacian V 0 (see Fig. 1).

m 2 rnl

I r%

\ ~ . / /

Fig. 1. - A general supergraph in 2g-th order.

The 0-differentiation can ei ther be applied to the external or internal lines. Thus if all outside legs are A-lines, then the associated Green's funct ion is

V0, ... V0. S~ ...... (x101, ..., x~,O~).

However if all outside legs are pure /~-lines, the diagram is described b y S itself. For in termedia te eases, such as external v-lines, one 0-derivative is t aken internal ly and another external ly. In any case, since ~/~0 ~ has the dimensions of mass, we conclude tha t the most singular diagrams are the

ex te rna l A-line graphs when V 0 is operat ing at each ver tex. One impor tan t point before we part icularize to q}3 and ¢~ theories; namely,

we need not concern ourselves with tadpole graphs. These vanish identically for 0 ---- 0' f rom (2), which can be in terpre ted as cancellation among field com- ponen t loops. Therefore in (7), i and j real ly do refer to different points.

2. - One-loop graphs in ¢3-theory.

Let us specialize now to

(8) v(qs) ~ ' = , g [~+ + ~ ! ] ,

S U P E R F I E L D P E R T U R B A T I O I N T t l E O R Y A N D R E N O R M A L I Z A T I O N ~

where g is a dimensionless coupling constant . Because V 0 annihi lates combina- t ions like 6~750 or ~7~0 we can effectively replace our propaga tors (2) b y the simpler expressions

(9)

in all subsequent computa t ions . To show how simple are the calculations~

consider the basic one-loop graphs, the self-energies 2;. According to Fig. 2 we can read (*) these off:

00) g A+~

i z ± ~ : = g~ ~ , ± = o .

Fig. 2. - Sclf-energics of ~ba-thcory in second order.

I n t e rms of tile componen t fields, the only nonvanish ing pieces are obta ined as

The significant point is t h a t the derivatives appear externally in (10), so the g raph has jus t the usual logar i thmic divergence of ~3-theory ( implying

t h a t the quadrat ic infinities f rom individual field contr ibut ions cancel among themselves) . This infini ty can be r emoved b y a supersymmet r i c counter - te rm

~ ( z - 1)v$(~+w_),

which corresponds to wave funct ion renormal izat ion.

± ¥

_ z

+ m

Fig. 3. - Vertex parts of On-theory in second order.

(*) To understand how formulae (9) follow from (8) note that the argument of A c undergoes the complex displacement ¼i07u(1 =L i75)0' and also that (00~)2 = 0 according to (A.2).

42 - I I Nuovo Cimento A.

650 R. DV~LBOURGO

With the ver tex corrections of Fig. 3 we use the lemma (A.6) proved in the Appendix tha t

[ ( 0 , - 0 ~ ) ( o l - o ~ ) ± ] [ ( 0 ~ - 0 , ) ( o , - o ~ ) ~ ] .. . [(0.-01)(o.-ol)2]= o

to prove t ha t F++~ = 0, while

(11) F ~ = ig 3 A~v(12) Av~(23) A±v(13) =

= ½ ig 3 m[exp [i01~0~T] A~2"exp [-- li03~01+] AI~" (02 -- 03)(02 -- 0a)÷ A~3 ] .

I f one picks out the nonzero-component contr ibut ions

- - " 3 A ~ F ~ A ~ ~mg 23 l[Al~A13],

one again notices the feature tha t all derivetives latch on to the external legs; thus the graph is perfect ly finite and the proper ver tex requires no renormaliza- t ion to order g~.

For o ther one-loop diagrams it is not ve ry difficult to demonst ra te finite- ness by means of dimensional analysis and power counting: i) we need only worry about the most v i rulent diagrams, viz. those with external A-lines; ii) graphs with a surplus of ± compared to ~= vertices necessarily involve a chiral i ty-preserving propagator A++ and contr ibute factors of mass which serve to cut down the degree of singulari ty; iii) thus the worst graphs have equal numbers of + and -- vertices which occur in a l ternat ing fashion, i.e. I"+_+ . . . . . . Evalua t ing the even n-point funct ion F~+~_a+..A - a t zero ex- te rna l momentum, to pick out the possible divergence, we get

FA÷~_...a - = iV0,V0, ...

... vo.fd4k ( k ~ - m~)-" exp [02(?'k)(01- 03)+ + ... + O,(?.k)(O~ -- 0,_1)+] =

-- iV0 V0, . . .Vo._, fd 'k (k~)i"(k ~ -- m~) -"" (01-- 03)(0~ -- 03)... (0,_~-- 01)(0,_1-- 01) = 0

b y (A.6). Hence the graph is bound to contain factors of external momentum, guaranteed to render the resul t finite.

3. - Higher-loop graphs in ~8-theory.

Le t us look at some two-loop graphs contr ibut ing to the self-energy before we make the necessary generalization. Figure 4a) gives zero by simple dimen-

SUPERFIELD PERTURBATION TIIEORY AND RENORMALIZATION 651

sion,d count ing: one encounters products of the t ype (OOmA) 5 with a surplus of 0 - - a f t e r ac t ing with V 4 the surviving 0 2 t e rms mean a zero answer as 0 -+ 0. Figure 4b) contr ibut ion (lis~tppears for ~ different reason:

g4 m2(O, _ 0 ~ ) ( 0 1 - - 0 2 ) [ /~12" (02 - - 03)(02 - - 03)-[- A 23"

• e x p [i02 CO,_ I/I.,4 .exp [i0, ~04-]A 14 .exp [i03 ~ 04_] A3,

leads to

j 2 , / Z ~ ~ "(t4x.,d4x4g4m274[A342~4.J24A~2.JL31 0.

+ +

+ +

+ + a) b)

+ ~ + t~~[[~ +

c) a)

Fig'. 4. L'f i- s~df-('n('rgv, in q~3-theory to second order.

Figure 4c) wmishes again by 0 counting, and d iagram 4d) gives zero for the same reason as d iagram 4b), viz. the der ivat ives migra te to an internal point where they in tegra te out to zero. Thus to order g4 we have obta ined Za+A+ = 0. On the other hand, for Z~_a+, a l though it is t rue t ha t Fig. 5a) and b) give zero, the remain ing Fig. 5c), d) and e) are nonvanishing; thus Fig. 5e) reduces to

g4 m.~f l~J l~ 4 F~j ,,i~ 3 J14 :I~4 J d 4 x 2 ( b x 4 ,

+

a)

Fig. 5. - Z~+ self-energy in ¢'~-theory 1~) second order.

b)

+ + - + -- +

d) e)

6 ~ R. DELBOURGO

which is finite, Fig. 5d) simplifies to

g'm~f A ~, ~[(A~8 As, A 1,) d*xz dtxs,

which is finite once the second-order renormalization of A ~ has been accom- plished, and finally Fig. 5e) reduces to

g' ~*lf A 1, ~(A*~, z]3, A u) d'x8 d ' x s ,

which represents a logarithmic infinity, again connected with wave function renormalization.

The g4 analysis above suggests tha t 2:,~a~ = 0, and tha t the most singular parts (logarithmic) of 2:,~,~ are associated with ehirality-changing self-energy insertions in internal lines. The following general argument substantiates these s ta tements : supergroup invariance of the vacuum (4) entails tha t

i A (zO, z' 0') = <T[O(zO) q~(x' 0 ' ) ] > =

= ( T [ q S ( x - - x ' + ½ i ~ 7 0 ' , 0 - - 0 ' ) q ~ ( 0 ) ] ) = e x p [ l i O ~ O ' ] J ( x - - x ' ; 0 - - 0 ' ) •

By the same token, the self-energies must always be expressible as

(12) iZ(xO, x'O') = exp [½i0~0'] ~-(x - x ' ; 0 - 0 ' ) .

Then if one picks out 0 with particular chiralities

iz~±(x0, ~'0') = exp [ i0~0±] ~ ( ~ - x'; 0 - 0'~) -~ £ ( ~ - ~'; 0 ~ - 0',)

and

(13) i-~.~(xO, x'O') ~ exp [i~:~0'~=] £=~:~(x- x' ; 0 - 0').

I f one expands ~ in powers of (0 -- 0') 2 previous experience shows tha t at least two powers of m are involved, and, since [m(0--0 ')(0--0 ')±] ~= 0, we deduce tha t Z~,~ = 0. On the contrary, for 27,.~ these mass factors do not always arise and all we can say is tha t 2 : ~ . is finite, or at worst diverges logarithm- ically.

Likewise for the vertex function a supertransformation establishes t ha t

['(x,01, xz02, x803) = exp [½i(~1~108 + ~2~208)] -P(Xl-- xs, xz-- xs; 01-- 08, 02-- 08).

Again Fa~A~(123) = P ~ a ~ ( 1 -- 3 ; 2 -- 3) --> 0 by 0-expansion, symmet ry and

S U P ] ~ R F I E L D P E R T I ~ R B A T I O ~ T H E O R Y A ND R~ENORMALIZATION 6 5 3

the use of l e m m a (A.6). On the other hand

l'~ t_(123) ---- exp [.I, i ( 0 , ~ + 02~32)0a_] ~___(1 -- 3; 2 -- 3)

will ei ther vanish or car ry a fac tor m ( 6 ~ - { 5 , ) ( G - G ) + in the 0-expansion, giving a finite answer once internal self-energT renormal izat ions have been performed. By similar a rguments the reader m a y readi ly infer the finiteness of all higher-point irreducible vertices.

4. - One-loop graphs in (/)*-theory.

Now let us begin afresh with the in teract ion

04) v ( ¢ ) = G( G + ~ t ) / 4 !

though, as we shall soon discover, the theory m u s t be funds, menta l ly modified

b y renormal iza t ion te rms . Before get t ing involved with details, note t h a t G has the dimensions of inverse muss, so the effective coupling cons tant is bound to be G°-m s or G '~ 2 in some sense. For reasons a l ready given, tadpole graphs can be disregarded.

Lagrangian (14) provides the self-energies to order G 2

05)

• ~2 3 w =--'t(, A++ 0 ,

G ~ = ia~ A~,~: = it;, exp [iO0 0'~1A~,

corresponding to the componen t fichl self-energies

(16) X,~.~ iG~,~'A3~, w iG~i~ 1 (1 =7 iys)Aa, , 1 ~ = iG~A s, . ~ ' ~ ± Y $ . ~ - - ¥ : L . ~ , •

The /]3 infinity requires two subtract ions; in the ordinary ~ ' - t heo ry the sub- t rac t ions have the significance of mass and wave funct ion renormal izat ion,

bu t in our case one of t h e m cannot be so in te rpre ted (the one corresponding

to fixing the second p2-der ivat ive of the inverse propagator) . I n fact t he

order-G °- counter -Lagrangian

(17) a~f,, = ~ ( z - ~ )v~(q~+~_) + yv i (~+ . ~_)

with Z.-~I q - ( ~ A " and Y , - , G 2 1 n A °- a l ready (lifters radical ly f rom the orig-

inal free Lagrang ian ~aY.o ill the n u m b e r of der ivat ives associated with the t5 te rm. Before worrying abou t the repercussions of (17) let us pursue the

other one-loop corrections due to (14).

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

The G-ver tex corrections (see Fig. 6) are ei ther zero (*) or logar i thmical ly infinite, as in ord inary 9*-theory. Thus

2 2 iF++++ =- G zJ++ = 0

and (18)

2 i~v++__ = a ~÷_ = a~ e x p [Oi~O'_]/1~ ,

Fig. 6. - Reseattering corrections of ~4-theory in second order.

the der iva t ives again migra t ing to the outside lines. Now we need a new counte r -Lagrangian

(19) ~Lf 1 ---- XV~(~_~_)

wi th X , ~ G 2 1 n A 2 to e l iminate the infinities (18) of the scat ter ing graphs. We r e m a r k t h a t ~ f l is inherent ly different f rom the s ta r t ing in terac t ion .L#l

despi te its mani fes t supe r symmet ry .

\

) /

\

/ Fig. 7. - Selected higher-order graphs of O4-theory.

(*): Which does not imply that X++ vanishes identically. Indeed Fig. 4c) gives a finite correction to the ~ mass terms ~) 2:a+l, + , Z'~+~+.

8 U P E R F I E L D PERTURBATION THEORY AND I~ENORMALIZATION 6 ~

We have inves t iga ted a n u m b e r of mul t i loop corrections provided b y (14) - - f o r example those depicted in Fig. 7 - - a n d have found t h a t in each case

the three counter - te rms (17) and (19) (analogous to the three renormal iza t ions of o rd inary ~4) are enough, per one or two loops.

Unfor tuna te ly , t h a t is fa r f rom being the whole s tory. H a v i n g genera ted new bil inear and quadri l inear Lagrangians , these m u s t now be studied in thei r own right. Indeed the s i tuat ion closely resembles the s ta te of affairs

in g rav i ty theory (5) where R,~R "~ and R ~ counter -Lagrangians are genera ted f rom the original Eins te in Lagrangian R. I n par t icular , the modified free

Lagrang ian 5¢o+ ~ o contains quar t ic der iva t ive t e rms which can lead to d a m p e d propagators , a lbei t wi th a possible ghost problem. Since the interac-

t ions have a t mos t two der iva t ives we cannot ven tu re to say t h a t renormal iza-

bi l i ty is lost. Another po in t to be borne in mind is tha t , wi th der iva t ive inter-

act ions present , we m a y expect 64(0) type t e rms as a consequence of canon- ical quant iza t ion which also t end to cancel off the worst effects of (19). I n fac t the s i tuat ion is so m u r k y a t the present t ime t h a t it is ra ther p rema- tu re to announce (6) categorical ly the demise of super ~b4 as a possible re-

normal izable model. Anyhow, wha tever the eventua l ou tcome of a thorough inves t igat ion of this question, we cer ta in ly feel t h a t the superfield methods of SALA1V[ and STRATHDEE offer the mos t a t t r a c t i ve and economical way of

tackl ing the problems, and we hope t ha t the calculat ions outl ined in this paper will have convinced the reader abou t this.

This work owes a great deal to the enl ightening r emarks of Prof. A. SALAM to whom we ~re mos t grateful for discussions.

APPENDIX

Here we shall list a n u m b e r of useful propert ies of Majorana spinors which m a k e for crucial simplifications in the text .

Le t 0 ~ = ½(1 :J=iys)O. Then

(A.1) 0±(0~0:~) = 0 ,

which is obvious in a two-componen t basis. I n par t icular

(A.2) (00~)" = 0 for n>~2.

(5) D.M. ('~hPm.:R, M. J. I)UFV and 1,. HALI'ERN: IC/73/130 (to appear in Phys. Rev.); S. DESEI~ and P. VAN NIEIJWENI1EUZEN: Phys. Rev. Lett., 32, 245 (1974). (6) W. L.XN(~ and J. WEss: Karlsruhc t)rcprint.

656 R. DELBOURGO

W i t h two di f ferent spinors

(A.3) (O(y.t)) 0'~) (0(r. Q) 0~:) : ½ (00~} (0' 0 : ) P . Q

fol lows b y F i e r z t r a n s f o r m a t i o n . H e n c e f r o m (A.1)

P ' O' (A.4) ( 0 ( ~ . ) 0 , ) (O(~.Q) ~) (0(~.R)0'~) = 0 .

T h u s

(A.5) exp [0(7.P)0'~] : 1 q- O(7.P)O'~ Jr 1p2(00~)(0'0 '~).

I n t he t e x t we m e e t t he fol lowing p r o d u c t of sp inor differences:

H o w in v i ew of (A.1)

F . ( 0 1 , . . . , 0 . ) - - - -2(0101~)(0202:~) . . . (O.On~) ~- (--2)"0102:~0203± ... OnOl* o

A Fie rz reshuff le of t he l as t t e r m reorgan izes t he p r o d u c t as

~ . ( 0 ~ , . . . , 0 . ) = 2 (0~0~) [ (0~0~) .. . (0 .0 .~ ) + ( - 2 ) . - ~ 0 : , ~ ... 0 . 0 ~ ] =

= (0~0x~:)F._~(0~, . . . , 0 . ) .

B y success ive i t e ra t ions

(A.6) ~.(01, ..., 0 . ) = (0~o1+)(0~o~:~) . . . (0._~o._~.)~'~(o._~, o . ) = o ,

since F~(o, o ' )= [ ( 0 - 0 ' ) ( o - o').]~ = o .

• R I A S S U N T O (*)

Per mezzo delle tecniche dei supercampi si studiano i grafici delia teoria perturb~tiva e le divergenze nei modelli l~grangiani supersimmetrici. Nella teoria del supercampo ~a con poco sforzo si perviene al singolo controtermine infinite di rinormalizzazione (delle funzioni d'onda), mentre per la teoria ~b4 il metodo indiea il controlagrangiano necessario al livello di un solo cappio e possibilmente oltre.

(*) T~aduzione a cura della Redazione.

T e o p ~ BO3MyUIelUtlH ~IJ-~! cynepnoaefi n nepenopM~Onl~a.

Pe3mMe (*). - - I/Icnon~,3ya TexmIgy cynepnoneii, Hccne~ytoTCa rpaqbmcri TeopaH BO3My- meam~ H pacxo~HMOCTH B Mo~enax cynepcaMMeTprmHI, ix YlarpaHmnarmB. B Teop~a cynepnona ~a Tpe6yeTc~ He3HaqaTenbHOe ycnnHe ~n~ nonyqerma e~i4rlCTBelt~loro 6ec- KOHeqHoro HepeHopMHpOBOqr~oro Ko~rrp-q~IeHa, Tor~a gaK B ~i~ TeOpHH Hpe~no~(eH-HbLYi MeTO~ TpeSyeT KOHTp-.l-larpaHmaaHOB Ha o~i~o-neTenbHOM yponHe H, BO3MO~KHO, IIOMHMO 3TOFO.

(*) HepeaeOeno pec)aKque~.