Effective lagrangian for spontaneously broken N = 2 superconformal symmetry

Nuclear Physics B309 (1988) 669-679 North-Holland, Amsterdam

EFFECTIVE LAGRANGIAN FOR SPONTANEOUSLY BROKEN N = 2 SUPERCONFORMAL SYMMETRY Ken-ichiro Department

KOBAYASHI

LEE*

of Physics, Faculty of Science, University of Tokyo, Tokyo I1 3, Jupan

Tsuneo Deportment

and Kyung-Hwa

UEMATSU

of Physics, College of Liberal Arts and Sciences, Qoto

University, Kyoto 606, Japan

Received 28 July 1987 (Revised 4 May 1988)

In a general framework of non-linear realizations for space-time symmetries, we investigate the effective lagrangian for N = 2 superconformal symmetry which is spontaneously broken down to N = 2 super-PoincarC symmetry. For the case in which the dilaton multiplet is a massless N = 2 gauge multiplet, we derive a low-energy effective lagrangian which describes the interaction of Nambu-Goldstone particles.

1. Introduction Non-linear realization is known to be a powerful method for studying the low-energy effective theory when a certain symmetry, characterized by a group G, is spontaneously broken down to its subsymmetry given by a subgroup H [l-5]. In such a case there appear Nambu-Goldstone fields which transform under G as the coordinates

of the coset space G/H.

Within

tions, we can construct effective lagrangians massless Nambu-Goldstone particles. A general

framework

for non-linear

the framework describing

realizations

of non-linear

the interaction

of internal

symmetries

realiza-

among

the

was given

by Coleman et al. [l]. However, in order to investigate spontaneously broken space-time symmetries, we have to extend their method to a more general framework. The two main features characterizing non-linear realizations of space-time symmetries, which we shall consider in this paper, are as follows. (I) The coset space to be considered is parametrized not only by broken generators but also by the translational operator P, (as well as by the super-charge Q, for *On leave Korea.

of absence

from

Department

of Physics,

0550-3213/88/$03.500Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

Kyungpook

National

University,

Taegu

635.

670

K.-i. Kobayashi et al. / Effective lagrangian

supersymmetric theories if supersymmetry is preserved) though P. (Q~) is unbroken. (II) Not all the broken generators correspond to true Nambu-Goldstone degree of freedom. For illustration, let us consider the case of conformal symmetry which is spontaneously broken down to Poincar~ symmetry. The conformal group is generated by special conformal operator K~, dilatation operator D and the generators of Poincar6 group: P. (translations) M.. (Lorentz rotations). Although the broken generators for this example are K~ and D, the relevant coset representative is given by these broken generators as well as by the unbroken translational operator P. L ( x ) = exp( ix"P~)exp(

ie~K~)exp( ioD ).

(1.1)

This situation was noticed some time ago by a number of authors who studied the non-linear realization of the conformal group [5-12]. While, for the second feature, we note that there appears a true NambuGoldstone particle called the dilaton, o, corresponding to the broken-scale transformation operator D, whereas the ~. associated with broken conformal boost operator K . is not a true Nambu-Goldstone degree of freedom but can be expressed as the derivative of o i.e. q,~- 3~o. Now, we are interested in the supersymmetric version of the non-linearly realized conformal symmetry. The non-linear realization of extended superconformal symmetry was first studied some years ago [13]. In that paper the authors have given the Cartan forms, the non-linear transformation laws, and the effective action in the case where the N-extended superconformal symmetry is broken down to Poincar6 symmetry. Hence, they considered the effective lagrangian where the supersymmetry is broken. Our interest here is to investigate the effective lagrangian for the case in which superconformal symmetry is broken down to super-Poincar~ symmetry. Thus, in our case, supersymmetry is preserved and hence the Nambu-Goldstone fields form a supermultiplet. In the previous paper [14], we investigated the non-linear realization of N = 1 superconformal symmetry spontaneously broken down to super-Poincar6 symmetry. We applied the general method to that case and derived non-linear transformation laws and a low-energy effective lagrangian describing the interaction among Nambu-Goldstone particles. The lagrangian we obtained may be regarded as a superconformal extension of the Volkov-Akulov lagrangian [28]. It is not merely of academic interest, but it has been applied to the chaotic inflaton scenario where the dilaton plays a role of the inflaton [29]. Here, in the present paper, we shall extend the N = 1 case to an extended or N = 2 case, where the dilaton belongs to a massless N = 2 gauge multiplet. The motivations for studying the N = 2 case are as follows. First of all, we can incorporate a non-trivial internal symmetry, which is SU(2) in the N = 2 case, into our problem only when we go to the extended superconformal symmetry. Secondly,

671

K.-i. Kobayashi et al. / Effective lagrangian

there is new aspect which is absent in the N = 1 case. Namely, the N = 2 supermultiplet of the dilaton contains a massless particle which is not a N a m b u - G o l d s t o n e field but a gauge field of the central charge. This gauge field participates in the low-energy non-linear interaction. This work is a first step toward the more interesting N = 4 superconformal case, realized by N = 4 supersymmetric Yang-Mills theory, which is a finite theory and is free of conformal anomalies [15-17]*. This paper is organized as follows. In sect. 2 we present the N = 2 superconformal algebra and by parametrizing the relevant coset space we write down Cartan differential forms and covariant derivatives in the harmonic superspace formalism. In sect. 3 we obtain a simple effective lagrangian corresponding to a super-determinant of the chiral subspace of N = 2 harmonic superspace. This describes the interaction among the particles contained in the dilaton multiplet which is a N = 2 gauge multiplet. Sect. 4 is devoted to discussion and conclusion.

2. N - - 2 superconformal algebra and N = 2 dilaton muitiplets Although the Caftan differential forms for N-extended superconformal symmetry were given in ref. [13], in order to make our paper self-contained, we present our result based on harmonic superspace formalism which is necessary for deriving the effective action in sect. 3. We study non-linear realization of N = 2 superconformal symmetry spontaneously broken down to N = 2 super-Poincar4 symmetry**. By studying the group structure, we set up the coset representative and compute the covariant differential which is the necessary procedure to construct effective lagrangians. In order to linearly realize unbroken N = 2 supersymmetry, we have to introduce a N = 2 superspace formalism which we shall take to be the harmonic superspace of Galperin et al. [20]. Let us consider the N = 2 superconformal group usually denoted by SU(2, 2 / 2 ) [21-23], the algebra of which is given in the appendix. We shall adopt the 2-component Weyl-spinor notation to make SU(2) internal symmetry more explicit [22,23]. The generators of SU(2,2/2) are those of conformal group: P~, M~, K~ and D together with Q-supersymmetry generators: Q~i, Qa (i = 1,2), S-supersymmerry generators: S ~i, S~ai (i = 1, 2), SU(2) generators: T/j (i = 1, 2) and the U(1)R charge A. In the present paper, we shall consider the case in which the dilaton associated with the broken dilatation operator D is contained in the massless N = 2 gauge multiplet. - -

i

* Some comments on conformal anomalies will be given in sect. 4. ** Non-linear realization of extended super-Poincar6 symmetry was studied in refs. [26,27].

K.-i. Kobayashiet al. / Effectivelagrangian

672

As it was discussed by Fayet [24], the internal symmetry breaking for this case is* SU(2) × U(1)R -~ SU(2). The broken generators are therefore D, K~, S% S&i and A. We associate these generators with the fields o, ~ , ~b~i, ~ i and 0, respectively. The relevant left-invariant coset representative is given by

L ( x , O, O) = TFU,

(2.1)

where T = exp(ix- P + iOiQi + iOiQi), F = exp(iO. K + i~iSi + i~i~)exp(ioD + ioA), U = exp( iv _ T+ + iv + T_ + 2iv3T3).

(2.2)

Here one should note that the broken generators are contained in the middle factor F. On the other hand, the first factor T is characteristic to the non-linear realization of space-time symmetries and is necessary for super-Poincard symmetry to be linearly realized. The last factor U is introduced so as to realize SU(2) internal symmetry non-linearly in accord with the harmonic superspace formalism. Now let us calculate the Cartan differential form in the following form

L-ldL=iDx.

P+iDO~Qi+iD~-Q'+iDo

T++iDv+T_+2iDv3T

+/D 0 K+iD~piSi+iD~+iDoD+iDoA+uA%~M •

1

•

3 l~v,

(2.3)

where the covariant differentials relevant to our problem turn out to be

Dx ~= e x p ( - o ) [ d x ~ + i d O i ,,~0,. + i d ~ 6~0 '] = exp( - o ) a t ~ , DOi = e x p ( - ½o + ½ip )[dO j - i dP' ~:6~,] Uj i, D 0 i = e x p ( - ½o - ½ip)[d~ - i dU q:,%](U 1)i',

D o = do - 2[dP* ep,- dOi ~, - ~' dOi] , Do=do

-

2[dP*~'6~,~i+

i deiq:, - i~ idOi].

* There is another possibility [24] SU(2) × U(1)R ~ U(1)R, where SU(2) symmetryis not left over. We shall leave this possibility for future investigation.

(2.4)

K.-i. Kobayashi et al. / Effective lagrangian

673

Our results for the Cartan forms and covariant differentials (2.3) and (2.4) are essentially consistent with those of ref. [13]. Before going to the construction of effective lagrangians, we now examine the structure of the N = 2 gauge multiplet in which the dilaton o is contained. As was discussed by Fayet [24], the relevant N = 2 multiplet consists of the dilaton o (scalar), the axion O (pseudoscalar), the dilatinos q,~, ~ (i = 1,2)(spinors), and the vector A~ which can be interpreted as the gauge field corresponding to the central charge. In the harmonic superspace formalism, the N = 2 gauge multiplet can be expressed as an analytic superfield V++(fA) where fA = (XA, 0+, 0+, U +) with X A ~ = X ~ - - i ( O + o t ~ O + 0 0~0+), 0 +-= 0 i ui-, + 0+_= Oiu i + and ui-+ being harmonic coordinates. In the Wess-Zumino gauge V++(fA) reads [20,251 V++(fA)

=

lq 0 1 qO. ( 0 + ) 2 ~22 + ( 0 + ) 2 ~-2 +iO+°"O+A,

+ (0+)20+. 2q/ui-+ (0+)20 +. 2 ~ i u i - + (O+)2(O+)2~(i/)ui u/-

(2.5)

where cp = f~7(o + io) is a complex scalar and N(~J) is the auxiliary field. Now, in order to eliminate the unphysical degrees of freedom we shall impose constraints on the covariant derivatives as follows

D o / D x ~ = 0,

(2.6a)

D o / D 0 i = 0,

D o / D 0 i = 0,

(2.6b)

D o ~ D r +,3 = O,

D o ~ D r ±,3 = 0.

(2.6c)

These constraints lead to

,~. =

10.o,

i

D~=0,

(2.7a) ~ = o - i0,

D++~=D

~=D3~=0,

where

D~ i= O/a0 i ~ D ++=

u+iO/OU

_]_

i( o~Oi)~a~ ,

-i ,

D - - = u-iO/Ou +i ,

D 3= 1[u+'0/Ou+'- u '0/0u 'l, are spinor and harmonic derivatives, respectively [20, 25].

(2.7b) (2.7c)

K.-i. Kobayashi et al. / Ef]ective lagrangian

674

It can be easily shown that the chiral subspace is invariant under the transformations in the non-linear realization. Here we also note that the analytic subspace, another subspace of the N = 2 harmonic superspace, is not invariant under the transformations in the non-linear realization.

3. Effective lagrangian N o w the simplest possibility for an invariant action turns out to be the invariant phase volume

l=fDxDO+h.c.=fd4xad2OsdetW+h.c.,

(3.1)

where the covariant differentials read

2iOio~dOi ),

D x " = exp ( - o )(dxR" -

DOi=exp(-~o-½io)[d~-(dxr~+ZOko"dOk)~jo~](U-1)/,

(3.2)

and h.c. stands for the hermitian conjugate. We have introduced a supervierbein W~tN which connects the covariant differentials with the ordinary differentials as follows

DOai] = ( WMN ) d. dOBj

:

dO~j t '

C

(3.3)

where the elements of the supervierbein are obtained from eq. (3.2) A = W"~ = e x p ( - o ) 8 " ~ ,

B = W "/jj= -2iexp(-o)(8~OJ) l~, C= Wai~= - i e x p ( - ½ o -

~ip)(t)jo~)a(U 1)iJ ,

D=exp(-½o-½io)[(U-1)/6a~-Z(~ko,)s(U

')ik(6"0J)/~],

N o w we can evaluate the superdeterminant of the supervierbein the formula sdet

(WMN)

= det A det I ( D -

CA-1B).

WMN

(3.4) by using

(3.5)

675

K.-i. Kobayashi et aL / Effective lagrangian

F r o m eqs. (3.4) and (3.5) we get

sdet(WMN) = exp (-- 20 + Zip).

(3.6)

T h e r e f o r e the invariant action (3.1) is found to be

I=

f d"x.d2t~exp { -

2 , ( x R, 0)} + h.c.,

(3.7)

where we have introduced a normalization factor ~4T h e chiral superfield , ( x R, 0) can be d e c o m p o s e d into the c o m p o n e n t s of the N = 2 gauge multiplet V + +

*(XR, 0) = ~'//'lchiralbasis,

(3.8)

where :oK is the field strength derived from V ++ as

"~1 analyticbasis =

-- I ( D + ) 2 D - - u ,

(3.9)

where D+

=

D--=

0/00 -~, u-'O/Ou +i- 2iO-oUO- O/OXA" + O-~O/O0+~ + 0 aO/Ot~ +a ,

are the spinor and harmonic derivatives in the analytic basis. The v is so-called p r e p o t e n t i a l and can be expressed for the abelian case as u = - ( 1 / D + + ) V ++ .

(3.10)

By substituting the expression of V ++ in the W e s s - Z u m i n o gauge (2.5) into eq. (3.10) and then the resulting v into eq. (3.9) together with changing the basis we get

,(x.,0) =

0++ / x R ) +~[(O+)2~(--)(XR)--2O+O

~+ )(XRI+(0-)ZN¢++)(XR)]

+ ½~ a"o"Ur.,(xR) - i(0+)20 a~'o.+ (x.)

--i(0 )20+OP'Op,@+(XR) -- ~(0-)2(0+)2E]q0*(XR).

(3.11)

676

K.-i. Kobayashi et al. / Effective htgrangian

Note that the above expression actually does not depend on u;-+. This can be seen from the relation +;Xi = q~+X - +-X +, ++-= +iui+-. Eqs. (3.7) and (3.11) lead to ;=

f d4XR exp(-- V~-~o)[--~[]9~* + i~

o 0~# 6~0~#+- ;~ "-+-~

1 --+ 2

, + 1 ~7(+ -+

2

) 2(d7)2] + h . c . .

(3.12)

Thus, the effective lagrangian is given by ~(~eff =

• --i--/~

__

!~;"

F~v

exp(-v~-qn)[-rpDqp* + t ¢ o O,~b; 4-,~--

+ 4,~,~

~ ( ~7V7J) ( ~7,~7,)] + b.c.

(3.13)

This is our main result in this paper. The lagrangian describes the non-linear interaction of the component fields of the massless N = 2 gauge multiplet

(qP;~i,~i(i=l,2);Aj,), where ~ = ~-~!(o + ip) is a complex scalar corresponding to a complex combination of the dilaton o and the axion p, whereas ~', ~ ( i = 1,2) stand for dilatinos associated with broken S;, ~ (i = 1, 2) and A, is not a Nambu-Goldstone field but a gauge field of the spontaneously generated central charge. One should note that the action can be obtained from the free action for the N = 2 gauge multiplet I=

fd4xad20 + d20 - du :CUR ,

(3.14)

after replacing Y¢/" by its exponential e - ~ , where YCf=qS. This gives us another example of an exponential prescription for deriving the non-linear effective lagrangian.

4. Discussion and conclusion

In the present paper, we have studied the non-linear realization of N = 2 superconformal symmetry which is spontaneously broken down to super-Poincar6 symmetry.

K.-i. Kobayashiet al. / Effectivelagrangian

677

For the case in which the dilaton multiplet is a massless N = 2 gauge multiplet, we have written down a simple effective lagrangian describing a non-linear interaction between N a m b u - G o l d s t o n e particles and a massless vector which is interpreted as the gauge field of the central charge. We have seen that an exponentiation prescription for a non-linear lagrangian works also in this case. Here, some comments on conformal anomalies are in order. In our effective lagrangian, we have not taken into account the contribution from conformal anomalies. They are proportional to/3-functions. Our lagrangian provides a good description when we are sitting near the zero of the B-function. The exactly vanishing/9-function is realized in the N = 4 Super-Yang-Mills theory. Even in the N = 2 case, it is known that the suitable choice of matter multiplets leads to the vanishing B-function and hence to the absence of conformal anomalies [30,31]. Thus, we are not necessarily far from the real situation. On the other hand, if one could construct the W e s s - Z u m i n o term for conformal anomalies, then it would describe the effective action including the conformal anomaly. However, it has been unknown, so far, even for the non-supersymmetric case. Or it may well be that it does not exist. This is one of the future problems to be pursued. Another future problem to be studied is to derive an effective lagrangian for the dilaton multiplet which is a matter multiplet of 0(2) or U(1) N = 2 supersymmetry corresponding to the breaking: S U ( 2 ) × U(1)R--+ U(1)R. This would complete the non-linear realization of N = 2 supersymmetry which might provide an important step for applying the present method to the most interesting case, i.e. N = 4 superconformal case. We thank Professor H. Miyazawa for useful discussion and encouragement. One of us (K.H.L.) would like to thank the High-Energy Theory Group at Department of Physics, University of Tokyo for the kind hospitality.

Appendix N= 2 SUPERCONFORMAL ALGEBRA SU(2,2/2)AND CONVENTIONS

[¢, =iP~,

[K~, D] = -iK~,,

[P. K.] = 2 i ( ~ D - M ~ , ~ ) ,

[M~,~,D] =0,

K.-i. Kobayashi et al. / Effective lagrangian

678

[~,~, ~,~1: [K~, K~] : [A, M~] = [A, ~'~]

[A,/~,] =[A,D]=0,

(eo,,o;):

26/(0.') ~ P , ,

i, j = 1,2,

(eoi, S"J) = 6/[(o""),fM~,.( Q°i, QBJ } =

{~,g)

2iD6~ ~] - 4 6 . f B , ' ,

= ( e o , , g ) =o,

1 B QB,' [Qo~, M.~] : 7(o0,),

[soi, M~J = - vJ , o ~ . t0..~)a, o ,

[~,/~1 =0, [Qo~,K.I [Q,~,,D] =~iQ,~,,

[~,i, D ]

[Q~,,A] = I Q ~ , ,

[S~,,A] = _ ±2~~ o i

=

__

~lSil.-& ,

'

[eo~, ~ ~] =SikQoj _ 7*,~kt9 j ~oi, = 8,~

- ~

~,

= 6/Tk j - 8fiT, t,

where we use the following conventions 7]00 =

-- ~ii = 1,

i = 1,2,3, %= (1,-o),

0." = (1, 0.),

with o = (0 "1, 0"2, 0 3) being Pauli matrices, and ,

0..~

½i(6.0.~

6ft.).

All spinors are 2-component Weyl spinors. The U(2) generators B i j are decomposed

679

K.-i. Kobayashi et al. / Effective lagrangian

into SU(2) generators T/and

U(1)R generator A 1

Another

J

b a s i s of the S U ( 2 ) g e n e r a t o r s :

T + = T21, T =

~

i

T + = T 1_+ iT2, T 3 are r e l a t e d to

~ J as

T12 a n d T 3 = ½(T11 - T22). F i e r z r e a r r a n g e m e n t s r e a d

References [1] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247 [2] C.J. Isham, Nuovo Cim. A59 (1969) 356 [3] A. Salam and J. Strathdee, Phys. Rev. 184 (1969) 1750 [4] J. Wess, Springer tracts in modem physics, vol. 50 (1969) 132 [5] D.V. Volkov, Fiz. Elem. Chast. Atom. Yad. 4 (1973) 3 [6] G. Mack and A. Salam, Ann. of Phys. (N.Y.) 53 (1969) 174 [7] A. Salam and J. Strathdee, Phys. Rev. 184 (1969) 1760 [8] B. Zumino, in Proc. 1970 Brandeis Univ. Summer institute in theoretical physics, vol. 2, eds. S. Deser, M. Grisaru, M. Pendleton (MIT Press, Boston, MA, 1970) p. 437 [9] C.J. Isham, A. Salam and J. Strathdee, Ann. of Phys. (N.Y.) 62 (1971) 98 [10] A.B. Borisov and V.I. Ogievetskii, Theor. Mat. Fiz. 21 (1974) 329 [11] V.I. Ogievetsky, in Proc. 10th Winter school of theoretical physics, Karpacz, vol. 1 (Wroclaw, 1974), p. 117 [12] E.A. Ivanov and V.I. Ogievetskii, Thoer. Mat. Fiz. 25 (1975) 164 [13] V.P. Akulov, I.V. Bandos and V.G. Zima, Theor. Mat. Fiz. 56 (1983) 3 [14] K. Kobayashi and T. Uematsu, Nucl. Phys. B263 (1986) 309 [15] S. Mandelstam, Nucl. Phys. B213 (1983) 149 [16] P.S. Howe, K.S. Stelle and P.K. Townsend, Nucl. Phys. B236 (1984) 125 [17] M.F. Sohnius and P.C. West, Phys. Lett. B100 (1981) 245 [18] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39 [19] S. Ferrara, Nucl. Phys. B77 (1974) 73 [20] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 1 (1984) 469 [21] P.H. Dondi and M.F. Sohnius, Nucl. Phys. B81 (1974) 317 [22] R. Haag, J.T. Lopuszanski and M. Sohnius, Nucl. Phys. B88 (1975) 257 [23] M.F. Sohnius, Nucl. Phys. B138 (1978) 109; in Quantum theory of particles and fields, eds. B. Jancewicz, J. Lukierski (World Scientific, Singapore, 1983) p. 161: in Supersymmetry and supergravity 1983, ed. B. Milewski (World Scientific, Singapore, 1983), p. 520 [24] P. Fayet, Nucl. Phys. B149 (1979) 137 [25] N. Ohta, Phys. Rev. 32 (1985) 1467 [26] S. Ferrara, L. Maiani and P.C. West, Z. Phys. C19 (1983) 267; S. Ferrara, in Quantum theory of particles and fields, eds. B. Jancewicz, J. Lukierski (World Scientific, Singapore, 1983) p. 49 [27] J.A. de Azcarraga and J. Lukierski, Class. Quantum Grav. 2 (1985) 683 [28] D.V. Volkov and V.P. Akulov, Phys. Lett. B46 (1973) 109 [29] E. Papantonopoulos, T. Uematsu and T. Yanagida, Phys. Lett. B183 (1987) 282 [30] P.S. Howe, K.S. Stelle, P.C. West Phys. Lett. B124 (1983) 55 [31] P.C. West, in Supersymmetry and supergravity 1983, ed. B. Milewski (World Scientific, Singapore, 1983) p. 341