Weakening of algebra and N = 12 supersymmetry

Volume 166B, number 2

PHYSICS LETTERS

9 January 1986

W E A K E N I N G OF ALGEBRA AND N = 1 / 2 S U P E R S Y M M E T R Y Min Ho J E O N G and Jae Kwan K I M Department of Physics, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryangri, Seoul, Korea Received 11 June 1985; revised manuscript received 30 September 1985

It is shown that weakening of the supersymmetry algebra of Tata, Sudarshan and Schechter can solve the problem of broken Poincar6 invariance in N = 1 / 2 supersymmetry but not in N = 1 supersymmetry.

Supersymmetry (SUSY) is physically and mathematically a subject of considerable interest. Supersymmetric theories have the possibility of being realistic if SUSY is broken either spontaneously or explicitly. On the other hand, in ref. [1] the authors argued that the spontaneous breakdown of global supersymmetry is consistent with unbroken Poincar6 invariance if and only if the supersymmetry algebra " a t = 0" is understood to mean invariance of the dynamical variables ¢ under transformations generated by the algebra, i.e. [at, ~b] = 0 rather than as an operator equation. It is further argued that this "weakening" of the algebra does not alter any of the conclusions about supersymmetry quantum field theories that have been obtained using the original (stronger) form of the algebra. As a result of this weakening the SUSY algebra is changed into a~ = n, where n is a neutral element which commutes with all the generators of the Poincar6 group and all the field operators; hence it is a numerical constant. Applying the process to the algebra (Qa, Q+) = 4p0,

(1)

they obtained the weakened algebra

(Q~, Q~*) = 4P ° + n,

(2)

ot

and explained that the spontaneous breakdown of SUSY is due to the non-vanishing VEV of the neutral element n, while the hamiltonian has vanishing VEV which preserves the Poincar6 symmetry. But in eq. (2), p0 transforms as a four-vector corn-

ponent under the Poincar6 transformation and n transforms as a scalar. So addition o f n t o P 0 is not proper. Applying the weakening process directly to the algebra in the eq. (1) is not appropriate. In this paper, we consider the SUSY algebra =

(3)

I f we take the vacuum expectation value on both sides of eq. (3), the left-hand side does not vanish ff SUSY is spontaneously broken. Then (0lPUl0)7 u 4: 0. So the Poincar~ invariance should be spontaneously broken when SUSY is spontaneously broken. So following the suggestion of ref. [ 1], eq. (3) should be understood to mean the invariance of the dynamical variables under transformations generated by the algebra rather than as an operator equation. Or equivalently,

"[Qa, 0.¢) = (PUTu)a# + n'6~¢.

(4)

Since the transform properties of PUTu and n' under the Poincar~-group transformation are equal, addition of PUTu and n' is proper. According to ref. [2] ,1, SUSY is spontaneously broken if the anticommutator of supersymmetry charge Q with some operator X is non-zero:

(OI(Q,~,X}IO)=(OIQ,~X +XQc, IO):/:O.

(5)

Because ifQa 10) = 0 eq. (5) would have to vanish. In this sense, eq. (2) (for which X = Qa) and eq. (4) (for which X = QO) are breaking SUSY spontaneously. Or, the nonvanishing n' means spontaneous SUSY breaking, since * 1 See especially pp. 516/517 of this reference. 153

Volume 166B, number 2 (0l

9 January 1986

PHYSICS LETTERS

(Qc,, Svo} 10) = n'rluv (~/°Tu)ao 4=0

(6)

means spontaneous supersymmetry breaking, in the above sense. Furthermore, juv (= f d3x (xUT4u - x VT4U)) does not depend on the neutral element n'. But the weakenedN = 1 SUSY algebra in eq. (4) cannot completely solve the problem of broken Poincar6 invariance. Multiplying with ~/0 on both sides of eq. (4) and taking trace, we get

Applying the weakening process of ref. [ 1] to the algebra in eq. (8), we get

~a+~, a+t3} = -PU(h+Fu C)~ + n5 ~ =(H+P+n 0

0) n s0

Furthermore, taking trace of eq. (10), we get

(Q+a, Q+a} =H +P + 2n. {Qa, Q+} = 4?0.

(7)

Ot

As indicated in ref. [1], the algebra in eq. (7) has the problem of broken Poincar6 invariance. So it can be stated that the weakening of algebra cannot completely solve the problem of Poincar6 invariance in N = 1 SUSY. On the other hand, in ref. [3], the author showed that a n N = 112 SUSY algebra exists for 2 (rood 8) dimensions, and constructed an N = 1/2 SUSY lagrangian in 2 dimensions by use of a superfield formulation. According to ref. [3], t h e N = 112 SUSY algebra is

(Q+u, Q+O) = - PU(h+I"u C)a#,

(8)

where h+ = (1 + 1"3)/2 and C is the charge conjugation matrix. Taking the Majorana basis, eq. (8) becomes

(10)

(11)

As we can see, by weakening the algebra, the problem of broken Poincar6 invariance can be solved completely i n N = 112 SUSY. As stated in ref. [3], t h e N = 1/2 SUSY includes N = 1 SUSY as a special case. When special conditions are taken, N = 1/2 SUSY theory h a s N = 1 SUSY. The N = 1[2 SUSY is more basic t h a n N = 1 SUSY. So it is more pleasing to weaken the algebra in the N = 1/2 SUSY t h a n N = I SUSY. However, it is crucial to remark that t h e N = 1 SUSY does not exist for four-dimensional spacetime. This research was supported in part by the KOSEFUS NSF cooperative research grant and by the Korea Research Center for Theoretical Physics.

References {Q+~, Q+o} =

0

0 a¢

Let us return to our previous problem of broken Poincar6 invariance.

154

(9) [1] X.R. Tata, E.C.G. Sudarshan and J.M. Schechter, Phys. Lett. 123B (1983) 308. [2] E. Witten, Nuel. Phys. B188 (1981) 513. [3] M. Sakamoto, Phy~ Lett. 151B (1985) 115.