On a class of supergauge lagrangians

Nuclear Physics B91 (1975) 2 8 9 - 3 0 0 © North-Holland P u b h s h m g C o m p a n y

ON A CLASS OF SUPERGAUGE LAGRANGIANS P. WEST Physics Department, Impertal College, London SW7 Received 15 November 1974 (Revised 13 February 1975)

It xs shown that the Wess-Zummo Lagrangian is the only viable supergauge Lagranglan which can be constructed from scalar superfields (and the covariant derivative) that has an interaction cubic in the scalar superfield.

1. Introduction A year ago Wess and Zumino [ 1 ] proposed a supersymmetric Lagranglan which requires only one infinite renormalization constant. Soon after, Salam and Strathdee developed a compact scheme for setting up superflelds and supergauge Lagrangians. In this scheme the general superfield (a function of 4 commuting x u and 4 anticommuting variables 0a) is composed of 3 irreducible parts under the supergauge transformation.

4)=4)+ +4)- +4)1 " The reader is referred to this paper for details of the construction and reduction of these fields [2]. The rule for writing down super gauge invariant actions is as follows: in the Lagrangian, every 4)± (chiral) type superfield must be operated on by DD and all other types of superfield b y * (DD) 2 . The Wess-Zumino Lagranglan is a particular example

~°w_ z = ~ (DD) 2 (4)+¢_) - ½DD(V(4)+) + V(4)_)), where V(4)±) = i m(4)_+)2 + g(4)+)3 . It is the purpose of this paper to examine the physical viability of the remaining * D a is the covarlant derivanve defined to be

Da

_

i)

~ot

1

2 i(~O)a "

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supergauge Lagranglans which have interaction terms cubic in the superfield and are composed only of ~b±, ~b1 and the covanant derwatwe. Such Lagrangians are 221 = ( D D ) 2 {-~¢ ( b D -

2m)~b +gq~3} ,

22 2 = (DD) 2 {~(~b+ + q5 ) (DD - 2m) (~b+ + ¢ _ ) +g(q~+ + ~_)35 , .2 3 = I(DD)2 (~+q5) + (DD) 2 g(q~+ + qS_)3 , .12 1 having been already suggested by Salam and Strathdee [2]. It will be shown that 221 ~s non-renormalizable and the source of the non-renormallzabllity is q51 . This leads us to 222 which is renormahzable *, but contains ghosts. On the other hand 223 ~s non-renormalizable. In the class o f supergauge Lagranglans considered above the Wess-Zumino Lagranglan is unique that it alone is renormahzable and contains no ghosts.

2. Non-renormalizability of ,L91 The superfield is a function on an eight-dimensional space, 4 space-time coordinates x u and 4 antlcommutmg Majorana parameters, 0 a. The supergauge transformation on this space is O-~O'=O

+e ,

x u ~ x u' = x u + -~i g

7u 0

and a scalar superfield satisfies ~(x, 0) = ~,(x', 0 ' ) . The action is given by

s =f22 (x, o) dx, and is rendered supergauge mvarlant by demanding that 22 is transformed into a 4 divergence under variation. This is implemented by the rule already stated in the introduction. A comparison o f the infinities present an the second-order diagrams and the normal counter terms generated by ./2 1 quickly shows that ./21 is non-renormahzable. This calculation is grossly slmphfied if we work in superfield notation rather than try to treat each component field lndwldually. The superpropagators are calculated using functional methods. To this end we introduce a classical supercurrent ] into the Lagranglan

* This Is proved to all orders m the m = 0 case and to second order for massive fields

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291

.~O'1 = -cO1 -- ( 9 0 ) 2 (21¢). q~can be expanded in a complete set of 0's

¢ = A ( x ) + O-t)(x) + ¼00F(x) + ¼0 7 5 0 G(x) + ¼0iTvvsOAV(x ) 2 D(x).

+ a1 0 0 0 ×(x) +

In order to get the correct source-field combinations we define J to be

2i = JD - ½0ix + lEO iF + ~OvsOiG + lO ,v~TsOJAu + ~O00 (--i¢ ) + ~ 00/a t

When evaluating .6?3 m terms of its component fields we need only pink out the coefficient of the 04 term because

fdx(DD) 2=fdx

"d0 d

'

r

.t21 = 2[-2auAOVF - 2FD - 2 G a l A

+ 2AuaUG + ~ c x - ~ ; ) ×

- f C i ~ x + 0 v--c t) Out) - 2 m ( A D + F 2 + G 2 + A uA u _ -,/c X _ ~ct)) + J- cqj t)] + interaction terms

+].4 A + ] F F +

(1)

The equations of motion f o r / = 0 are

02F = mD ,

byAv = - m G ,

F+mA=O,

a2A = 2 m F + D ,

32t)+iOx=2mx,

3vG = m A u ,

iOt)-x=2mt).

Using the equations of motion F, D, A u and X can be ehminated. The equations of motion can be rewritten in superfield notation: (DD - 2 m ) ¢ = 2],

(2)

from which it follows that ,+ ¢-

lm2 {½DD +~(DD)2/ ] -/--'m. j

32 +

(3)

Using supergauge lnvarlance of the vacuum

i i A ( 1 , Z ) = ~ ( T ¢ ( X l , O1)¢(xz,Oz))=~(Tdp(X, Ol2)¢(O,O)) t We used the fact that (DD) 3 = -4~2DD

6()(x, 012 ) 6]A(O)

(4)

P West/Supergauge

292 where

x = x 1 --X2 +~iglTO 2 ,

X12 =X 1 --X 2 .

012=01 --02 ,

Using (3) and (4) and

-~DDj = ~ OOjA + terms not Involving/A , ½(~D)2j = ~-j 2A - & (0-0)2 02jA + terms not revolving]A it follows that

1 02 + m 2

--JA (0-0)202 0-0 t (0-0)2 ; + terms not involving/A ~ 128 IA +]-gJA ) -6--T~JA

Hence 1

A(1,2)=

1

4m 0 2 + m 2

e~,Ol ~02{e-~mO12012 1 --

1

--

+ 1 (0-12012)2 (02 + m2)}g(x12)

Armed with the propagator we can calculate the Feynman graphs The secondorder diagrams are produced by

-½g2T(f(~D)

24~3(1)dx 1 f(~D) 2 ~b3(2) dx2}.

The second order self-energy diagram with only A external lines is d 2 N A A = ( d d l a d d ) 2 ( d d 2 ¢ / d O ~ ) A ( 1 , 2 ) 2" Now * e½~Oq iJo~ 1 A2(1 , 2 ) - - {e -~m°12°12 +(02 +m2)(OlzO12)A} 2 16m2 The only term to survwe the 0 differentiations is 04 0~, but (012012) 2 0-1~012 = (012012) 2 O12~ 02 = 0 , (012012) (0-1~ 02)3 = 0. Then,

1

* As e~01 $ 02 is just a translation on the 6 function,

P West/Supergauge A(1,2) 2 - (O1 ~J02)4 A2(X12) 4! 24 16m 2

293

terms with less O's

(0-101) 2 (O202) 2 (a2)2 A 2 + terms with less O's.

m 2 82 162 So

~AA

= g__~2(~2)2 A 2 (X12) 4m 2

In m o m e n t u m space this gives rise to a logarithmm divergence (analogous to ~b3 theory) a(q2) 2 In A (q the external momenta). Similarly we can calculate the second order self-energy for other external hnes For example A and F

(d d) (d

~AF=g2 "d01~ d0-~

d02~ d0-~2

0-202A(l'2)2¢xg2t)2A2

(in m o m e n t u m space a q2 In A). The normal counterterms are generated through the substitutions z2 ¢0 = V ~ I q~,

go = - X g ,

m 0 = m + 6m

in the bare Lagrangian. From the expression for ~'1 written out in terms of component fields eq. (1), it is clear that the above substitutions can not produce a counterterm looking anything like F_,AAa(q2)2 in A (the EAF graph, however does have a counterterm from the wave function renormahzation to the 3v F~VA term). Since there exists no counterterm coming from wave function, coupling constant or mass renormalizatlon to account for ~'AA we could only put in an arbitrary counterterm (DD) 2 {(/)D~b) 2} to remove the divergence. This counterterm has no immediate physmal interpretation and we conclude that "/21 is non-renorrealizable in the conventional sense. If we started with the Lagrangian ./21 + (DD) 2 ((/)D¢) 2 ) we could generate the counterterm reqmred to cancel the troublesome divergence. However, this would lead to ghosts in the theory.

3. Origin o f the non-renormalizability of "~1

In order to see which irreducible part of the superfield causes the non-renormalizabihty and to compare Z? 1 with £?w-z, the renormalizatlon of which was carried

P. West/Supergauge

294

out in superfield notation by Delbourgo and Capper [3], we calculate the propagators between the irreducible parts of the superfleld. This is achieved by applying the projectors E+, E 1 to eq. (4): E+ -

1 ~ -~(1 -7-i75)D/)½(1 _+i75)D 432

-

E 1 = 1 + 1 (/~D)2. 432 Then i

i

E+ -~(Tg~(x, 012 ) q~(0, 0)) =~ (TgL+(x,012 ) ~b(0, 0)) = A+ (I, 2) m {1 + 01201232 +- 012")'501232 + 1 - " " 832(32 + m 2) 4m 4m -gO12tTv~'5012t~v

_

32 (0-12012)232 } 6(x).

(5)

1

Similarly

Al(1,2)=
(6)

Using the properties of the projectors and propagators we find that E(1) (x, 012 ) (T(p(x, 012 ) q~(0,0))= (Tq~I(x , 012 ) ~b(0, 0))= (Tgb(0, 0)~b~ (x, 012 )) = (r~b(x2, 02)~b± (x I , 01)). 1

Hence, (T~f (X1,01) ~(X2,02)) = (T~p(x2, 02) ~+(Xl, 01)). 1

1

Apphcatlon of the projectors to the above equation gives the A(~)(~) (1,2) = (T~b(~) (x, 012 ) q~(~)(0, 0)) propagators m terms of E+ acting on the propagators of eqs (5) and (6)*: 1 1

1

A++(1, 2) - 32 32 + m 2

e½,~-i$02 ~112(1 + i~[5)012~(x12) ,

A+~(1 2 ) m e+~o12~VsO126(x12), ' 8 32(32 + m 2) * Use was made of the result (DD)2E+ = -432E_+ -

(8) (9)

P West/Supergauge

~+4 - ~

+4-

'Z+_+_+ _ ~

-*

295

7--+~~ ~-+I +I

z

I

I

II )l

Fig 1.

AI+(1, 2) = 0 , A l l ( l , 2)

(10)

=- ~-~--~[al + ~-~(O12012)~}e~t°-~°28(x12)

A+± and A~+ are very similar in form to those of m A~+. That is for ~w-z [2]: m++

1

--

1

1

1

--

1-

~

", ,

za x12 )

A+.~ = e~°1~°2 z-'4°12#"ls°12A(x

12

except for an extra 1/0 2

- -

_-~101~02 ~ (

__ = --~m012~(1 + i75)012 e -

~w-z

(11)

)

•

(12) (13)

We can now recalculate the self-energy diagrams with only A external hnes (see fig. 1):

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P. West/Supergauge

~++=2~__=E++=E++ =E__=O, ----

+--

+_

as the highest possible number of O's from the propagators is (O12012) 2. E+_+_a (a2) 2 .a2(a 2 + m2) ]

,

which is finite ,

/ 8(X12)/2 Z+__+a (b2)2 [a2(--~ ~m2) j

,

which is finite ;

E++ = ~ _ _ 11

= 0,

11

0t~}2 /

6(X12)

( a 2 ( a 2 + m 2)

There remains

{

/

~I 10~(a2) 2 - ~ 8 {X12)

1 ~(X12) / ~2 J

which is finite.

,

which is the cause of the logarithm divergence. It is clearly ~1 which is responsible for the non-renormalizibllity of the theory. Further this difficulty is inherent m any theory which contains a full self interacting superfield* ¢, as can be seen by the following: ¢1, unlike ~b+_,has the property that ~b2 is a general superfield and so that Lagrangian must be of the form = (DD) 2 {f(¢, De, b)}We consider the case in which f is allowed to be an arbitrary function of ~band D a q~consistent with Z? being at least superficxally a sensible Lagrangian when written in terms of its component fields. The kinetic term can only be constructed from D a ~b, but (DD) 2 has the property that (DD) 2 {g(~)D h (~)} = -(DD) 2 {(Oag(q~)) h (~)), and so can be cast m the form (DD) 2 {q~(DD)q~}. The difficulty stems from the fact that DD q~l = 0 so the ~b1 part of.6? has the form (DD) 2 {m ~12 - 2j 1~1 }" This results in the All propagator of eq. (1 1) and so the divergent graph o~(q2)2 In A. The counterterm must be generated from (DD) 2 {(DDq~)2 } which if added to the original Lagrangian produces ghosts. * i.e. constructed out of ~ and D a.

297

P. West/Supergauge

If the theory were not self-Interacting, but described a gauge interaction between a ~1 superfield and two other chiral superfields the gauge symmetry can make it renormalizable. The (DD) 2 (gD~b) 2 term is still needed to make the Lagranglan renormalizable but this is the usual gauge fixing term. The ghosts introduced from this term decouple from the physical matrix elements as a consequence of gauge Ward identities. We are grateful to the referee for this observation.

4. Renormalizability and ghosts of ~ 2 Motivated by the preceding argument we examine .122. Since ( 9 0 ) 2 {(~b++ tb_)~bl) = 0 = ( 9 0 ) 2 (~blgO(q5 + + ~b_)), the propagators of eqs. (8) and (9) are those for "/~2" Using these propagators it can be shown that to second order only mass-renormahzatlon is required (see appendix 1). By working with the _

(A++ + A__ + A+_ + A_+) m=0 = 76 012012 eU°l

~o 2

A(Xl2),

propagator it can be shown that Z? 2 is renormahzable to all orders in the m = 0 case. The essence of the proof is to find the highest number of a's which can come from the exponentials in the propagators subject to the restrictions that for an n-vertex graph any polynomial of O's greater than 4n vanishes and O12012(01 ~ 02)3 = 0. This number enables us to calculate the superficial degree of divergence of the graphs and leaves only graphs for which vertex + external lines ~< 8 whose finite nature is not accounted for. Of these remaining graphs all are finite except those In fig. 2 which are accounted for by mass-renormalization and adding a harmless (DD) 2 {~) to the Lagrangian respectively (see appendix 2). The theory has in the m = 0 case exactly the same renormalization pattern as ordinary q~3 theory. Examination of the propagator of eq. (9) leads us to suspect that the theory has ghosts. Writing Z?2 out in terms of its component fields confirms this possibility. Z? 2 = - 4 a v A ~ V F - 4avGaVB + 4~v~C~vt ~ - 2m(~vA ~VA + F 2 _ G 2 - 4 m ~ C ( - i ~ b ) + interaction terms,

where the component fields are labelled by (~+ + (~_) =A + O-~k + ~O-OF+ ~0-750iG + ¼0-~Tvv50,~VB

+ ¼000 (--iO~) + ~(0-0) 2 (--O2A) . The equations of motion are

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P. West/Supergauge

Fig. 2. m F = 02A + Interaction terms,

m G = 02B + interaction terms.

After eliminating F and G we obtain terms like OvA b v 02A i.e. ghosts*. Whether the problem concerned with ghosts can be consistently avoided in this theory has not been examined. A possibility for ehmlnating such ghosts is to consider the larger class of Lagranglans which contain terms of the form iOv$ which is a supergauge lnvariant because Sa the supergauge infinitesimal operator commutes with displacements. [Sa, Pu] = 0. To use such terms is in some senses a departure from the superfield techniques.

5. Non-renormalizability of £?3 It is sufficient to consider only second-order diagrams with only A external lines. The propagators for £73 are the same as for Z?w_ z and were given in eqs. (12) and (13). The second-order self-energy graphs for A external lines (see fig. 1) E++ = Z +4-

_ =E+_=E+_=O, ----

++

----

Z+__+=Z+_+_ot(02)2 0 2 + m 2 j

,

as the only term to contribute is (O1 ~ 02)4. In momentum space these divergent graphs are cx(q2) 2 in A. For which there is no normal counterterm generated by £?3" The reader may wonder why the Lagrangian £? = -~(3D) 2 (~+3O~b+ + ~b L)D~_)+½rn2(1)D) ( , 2 + ,2_) + interaction is not considered. This is because when £? is written out in terms of components it contains terms like 0 v $ 0 v $ and so has ghosts. It is nevertheless of interest to note * If we renormalized so that the physical mass was zero then we could no longer eliminate F and G as above. However, as OvA OVF = ~ {Ov(A + F) OV(A + F) - Ov(A - F) OV(A - F)),

where A and F are hermitlan fields, we again have ghosts.

P West/Supergauge

299

that provided the interaction does not couple q~+ to q~_ then in this theory the only non-zero propagators are 1

--

A,+,+(1,2) = O12(i -+ i75)012 e~°1~°2 6(x12) a2 + m2 as a consequence of which only tree graphs are non-zero. The author sincerely appreciates the encouragement and help of Dr. R. Delbourgo and thanks Professor A. Salam for suggesting an examination of the renormalizability of.~o 1 .

Appendix A It was stated that to second order .~°2 required only mass renormalizatlon. The only diagrams at this order which can diverge are of ~,+,+ type. Of these diagrams only those which pick up 04 from external lines will be non-zero because the propagators contribute only a (012012) 2 factor (e.g. A_+ and De external hnes). On calculating the S-matrix element which gwes rise to these divergent diagrams we note that it can be cancelled by the mass renormahzation term of 8 m f ( D D ) 2 q~2 dx. The relevant S-matrix element being: J ( D D ) 2 (1)dx 1 f(~D) 2(2) dx 2 A++ ( 1 , 2 ) A _ _ ( 1 , 2 ) q~(1) ~(2) = f d x I dx 2 ½ A2(x12 ) (A(.)(1)D(_+)(2) - 2 4(,+)(1) 4(_+)(2) - 2x-(C-+)(1) 4(_+)(2)

+ 2F(+_)(1)/7(+)(2) + 2G(+)(1)(7(_+)(2) + 2A(-+)v(1)AV(-+)(2)+ O(+)(1)A(±)(2)}.

Appendix B We noted that for ~o2 in the m = 0 limit all graphs for which the number of external hnes + the number of vertices are greater than 8 are convergent. In this class, with the exception of the two graphs in fig. 2, they all have divergences which are at most logarithmic. Therefore, it suffices to see if they are finate when the momenta on each external line are zero. This proof is considerably slmphfied by the two lemmas. Lemma 1. For subgraphs of the form in fig. 3(a) no derivatives from the exponentials of the propagators can contribute to the dwergence because

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300

/

I

(a)

(c)

(b)

\

Fig. 3.

ff12012023023031031

e {t(°-lt~°2+O-2nJo3+ga~o 1) = 0120120--23023031031e½,(°q2~°23)

= 012012023023031031

as

012012023023031031012~023=0.

Lemma 2. Consider graphs which contain a subgraph o f the form in fig. 3(b). If p is the loop momentum then no more than 3p factors can contribute from the exponentials in the propagators. A ( 1 , 2 ) A(2, 3) A(3, 4) A(4, 1) = = 012012~30230-34034041041

=

e½t(~lt~O2+ff21JO3+g31~04+04~01) ( 1 ) \p2/

4

0--120120-230230-340340-41041e~'(ffa2t~O2a+~a~Oa4+~2nJOa4)(1) 4 \ p2/

Clearly it IS not possible to bring down 4p factors. For example the graph in fig. 3(c) requires 4 internal momenta factors to diverge and so xs finite.

References [1] J. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974); Phys. Letters 49B (1974) 52. [2] A. Salam and J. Strathdee, On superfields and Fermi-Bose symmetry, ICTP preprint ICTP, Trieste, preprint IC/74/42; S. Ferrara, J. Wess and B. Zumino, Phys. Letters 51B (1974) 239. [3] R Delbourgo, Superfield perturbation theory and renormahzation, ICTP preprint ICTP/74/44 D.M. Capper, ICTP, Trieste, preprmt IC/74/66; A. Salam and J. Strathdee, Nucl. Phys. B86 (1975) 142; J. Honerkamp, F Krause, M. Scheumert and M. Schlindwein, Perturbatxon theory in terms of superfield, Freiburg preprmt THEPI; K. Fujlkawa and W. Lang, Karlsruhe preprmt, (September 74).