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GAUGE SUPERSYMMETRY
Pran Nath, R. Arnowitt
To cite this version:
Pran Nath, R. Arnowitt. GAUGE SUPERSYMMETRY. Journal de Physique Colloques, 1976, 37 (C2), pp.C2-85-C2-89. �10.1051/jphyscol:1976210�. �jpa-00216483�
J O U R N A L DE PHYSIQUE Colloque C2, supplément au n° 2, Tome 37, février 1976, page C2-85
GAUGE SUPERSYMMETRY (*)
PRAN NATH and R. ARNOWITT
Department of Physics, Northeastern University, Boston, Massachusetts 02115, U.S.A.
Résumé. — Supersymétrie est extendue à devenir une gauge invariance locale. Le groupe de supergauge unifie la gauge électromagnétique (ou Yang Mills) avec les invariances des coordonnées spatio-temporelles. La géométrie dans « superspace » et les équations du mouvement des super- champs sont discutées.
Abstract. — The extension of supersymmetry to be a local gauge invariance is given. The super- gauge group unifies electromagnetic (or Yang-Mills) gauge with the space-time coordinate inva- riances. The geometry in superspace and the superfield equations of motion are discussed.
1. Introduction. — During the past two years there has been much interest in the subject of supersym- metry [1, 2]. As is well-known, the supersymmetry transformations may be viewed as linear transfor- mations in an 8-dimensional superspace characterized by coordinates zA = { x11, 0" } where x11 are the usual four spacetime coordinates and 0* constitute a set of four additional anticommuting Fermi coordinates [3].
Supersymmetry transformations, however, are global transformations. Recently, we have proposed [4] that one extend the supersymmetry transformations to a local gauge group of arbitrary coordinate transfor- mations in superspace.
The above extension provides one with a new framework for a possible unification of elementary particle interactions. Modern approaches to the unification of two or more elementary particle interactions involve the use of gauge mesons to mediate the interactions [5-7]. The importance of gauge mesons resides in that they lead to renormali- zable interactions which may also be asymptotically free. However, whereas in a non-abelian gauge theory the self-interactions among the vector mesons is uniquely determined by the gauge invariance itself, the sources of the gauge fields, e.g., the fermions are left arbitrary by the constraints of the gauge invariance itself. This arbitrariness could be removed if one incorporated the matter fields other than the vector mesons also as gauge fields in the theory. In refe- rence [4], the proposed framework based on the extension of supersymmetry to a local gauge inva- riance treats all fields on an equal footing as gauge fields since they arise from a common supergauge multiplet gAB(x, 6). The interactions among the mem- bers of the multiplet are then determined through an underlying non-abelian gauge invariance.
In the gauge theories unifying the weak and the electromagnetic interactions such as the SU(2) x U(l) model of Weinberg [5] and Salam [6], the observed (*) Research supported in part by the National Science Founda- tion.
interactions appear as the spontaneously broken solutions of the original gauge symmetry. In this fashion the relation GF « e2/M^ arises between the Fermi constant and the electric charge. Thus the couplings for the observed interactions may be vastly different after the breaking has occurred. Similar arguments also apply when one includes the strong interactions in the unification and attempts in this direction have been made by several authors [7]. The inclusion of the gravitational interactions, however, would introduce new complications. First the current- ly prevalent unified gauge theories all involve gauge vector mesons whereas gravity is a tensor gauge theory. In addition, gravity theory has an infinite order of non-linearities (compared to finite order non-linearities of vector gauge theories) and the coupled quantum gravity is also non-renorma- lizable [8].
The generalization of supersymmetry to a local gauge invariance includes remarkably both the elec- tromagnetic (or Yang-Mills) and the gravitational gauge groups. Further, the field equations obeyed by the superfields give rise to a set of coupled Maxwell, Dirac and Einstein field equations provided a spon- taneous (or dynamical) breaking does occur. Since the superfields are polynomials in 0 a set of field-current identities result. Thus comparison with experiment is possible only after symmetry breaks and some of the fields become massive. Then the Maxwell stress tensor appears correctly as the source of the gravitational field equation with the Einstein constant satisfying GE x e2/M2 in a manner similar to the unification of the weak and the electromagnetic interactions. Such a relation holds even though the gravitational and the electromagnetic forces are both long range.
In section 2 we discuss the supergauge transfor- mations and exhibit the electromagnetic, Yang-Mills and gravitational gauge groups to be the subgroups of the supergauge group. In section 3 we discuss the geometry in curved superspace and also discuss the field equations for the coupled scalar-gravity, Maxwell-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1976210
C2-86 PRAN NATH A N D R. ARNOWITT
gravity and Maxwell-Dirac subsectors. Further com- ments and discussions are given in section 4.
2. Gauge transformations. - As already noted in section 1, the conventional supersymmetry trans- formations may be viewed as linear transformations on
To incorporate local gauge invariance we consider arbitrary coordinate transformations zA' = zA'(z) which leave the line element ds2 = dzA gAB(z) dzB invariant. Thus the superfields gAB(z) appear as the metric tensor for the superspace. They satisfy the symmetry property.
In eq. (2. l), the number a is associated with the index A and assumes the values (0, 1) depending on whether A is Bose or Fermi. The superfields gAB(z) transform according to the rule [9]
where L,B, and R:, stand for the left and the right derivatives of zB with respect to zA'. The transforma- tions of eq. (2.2) preserve the symmetry eq. (2.1) in the new frame.
Under the infinitesimal coordinate transformations
eq. (2.2) takes the form
where eq. (2.3) involves only the right derivatives. The supergauge transformations given by eq. (2.3) of course include as a subcase the Einstein gauge trans- formations. This can be seen by setting xS ' = xS'(x), 8" = 0" and transforming g, (z). Remarkably howe- ver, eq. (2.3) also includes, in addition, the electroma- gnetic (or Yang-Mills) gauge transformations when the fermion coordinates are chosen to be the Dirac spinors (or an n-tuplet of Dirac spinors). We illustrate in the following discussion, the case of the Yang-Mills gauge group as a subgroup of the supergauge group.
Dirac spinor coordinates may conveniently be intro- duced by adding a charge index q to the Majorana spinor which gives the doublet P q , q = 1, 2. The combination 0"' - iOZ2 then describes a Dirac spinor.
Similarly, one may consider an n-tuplet of spinor coordinates 8"qa, a = 1, 2, ..., n. We will see that this automatically produces a Yang-Mills invariance. The infinitesimal Yang-Mills transformations are in fact induced by the superspace transformations
zA' = ZA + tA
with
t A = ( t , = 0 , Y = AA(x)(TAt))"). (2.4)
AA(x) are the gauge functions and TA are the real antisymmetric SU(n) (or U(n)) matrices in the Majo- rana spinor representation [lo] where A = 1 ... n2 - 1 (or A = 1 ... n2).
The transformation properties of various elements of the superfield gAB may be deduced from eqs. (2.3) and (2.4). Expanding gAB(z) in a few of the lowest terms in powers of 8, we have [I 11
g,,(z) = giV(x) + (FSV 8) + (&TA 8) P$(x) + ...
g,(z) = G a ( x ) + (BT,,). B," F + (BT,)& 8 +
= qua F(x) + @TA)U$ - @TA)@
X:
+ " ' .(2.5) As a consequence of eqs. (2.3) and (2.4), the scalar, vector and tensor fields in eq. (2.5) obey the trans- formation laws
We note that the scalar field F(x) and the gravity field g,Ov(x) are Yang-Mills invariant whereas the fields B,A transform like the regular representation of SU,. The transformation properties of P,A, are such that
where P; transforms like the regular representation.
The spinor field transformations that follow from eqs. (2.3) and (2.4) are
Thus $; transforms like the fundamental represen- tation whereas the f transform like the fundamental x the regular representation (e.g., 3 x 8 for SU(3)).
The structure of $,, however, determined by the transformation above is
where Y,A transforms like the fundamental x the regular representation. Similarly II/, has the form
where Y,, transforms as the fundamental represen- tation.
One may note that it is the non-linear terms of eq. (2.3), which are the superspace analogues of the non-linear terms of the Einstein transformations of
GAUGE SUPERSYMMETRY C2- 87
general relativity, that supply the precise non-linear terms in the Yang-Mills transformations. We have then in the coordinate transformations of superspace, a scheme for unification of the space-time coordinate invariances and the gauge invariances of elementary particle physics.
3. Superspace geometry and applications. - The discussion of the gauge group shows that a single super- field gAB(z) contains a variety of fields of different spins and statistics, i-e., gravitational, Maxwell (or Yang-Mills), Dirac, etc. A similar unification also appears at the level of field equations when we consi- der geometry of the curved superspace. A single superfield equation then describes the Einstein, Max- well (or Yang-Mills), Dirac, etc., field equations.
The geometry in superspace is a straightforward generalization of the ordinary Riemanian geometry.
The invariant action in superspace has the form [12]
where R = ( - 1)" gAB RBA is the curvature scalar in superspace and RAB is the contracted curvature tensor.
As shown in reference [4], RAB has the form [13]
where the affinity T,CB is determined to be [14, 151
and obeys the symmetry property
The form of T:, is similar to the usual Christoffel symbol except for the additional signs factors. These are the signs that allow one to have electromagnetic and gravitational phenomena in a common formalism.
Finally the metric density in eq. (3.1) is
,& [- d e t ~ ~ ~ ] " ' and may be explicitly evaluated out to be [15]
6
= [- (det g,,) (det ga8)]112 . (3.5) The action of eq. (3.1) yields the field eqations [12]postulated in reference [4]
Applications : To recover the full content of the superfield eq. (3.6), a complete expansion of gAB(z) in powers of 8 is necessary. This expansion when inserted in eq. (3.6) would produce a closed set of coupled field equations. The above procedure, howe-
ver, is rather involved and we examine here some general properties and a few special cases.
First, without regard to the detailed dynamical considerations a theory constructed from superfield geometry explains automatically the great diversity in the nature of gauge interactions. For instance, whereas gravitational theory involves infinite order of non- linearities, the vector gauge theories involve only cubic or quartic interactions. This phenomenon is easily understood as due to the property (0")' = 0 of the Fermi coordinates in eq. (3.6). Thus the fields which appear in the metric as coefficients of 0 (such as the Maxwell or Yang-Mills fields) have a finite order of non-linearity whereas the gravitational field which appears without any coefficient in 8 in g,, exhibits the full non-linearities of eq. (3.6).
All field equations resulting from eq. (3.6) are massless and a spontaneous or dynamical symmetry breaking would have to occur to grow masses for some of these fields. Since the theory is self-contained, the allowed forms of the symmetry breakdown must be supplied by the field equatiops themselves without any additional ad hoc assumptions. Next, we discuss some specific applications of eq. (3.6).
A. - Coupled scalar gravity : As a simple example we retain in the superfield expansion of eq. (2.5), only the gravity field gz,(x) and the scalar field F(x).
Using eqs. (3.2), (3.3) and (3.6) we obtain
where gz) = F - g$) and F F exp(acp(x)). In eq. (3 .7)
and T,, is precisely the stress tensor of the scalar field cp(x). The gravitational constant GE = a2/8 x is also automatically positive. The above example illustrates the self-sourced nature of the theory.
B. - Coupled Maxwell-gravity : For a discussion of the coupled Maxwell-gravity sector we consider the spinor coordinates to be doublets of Majoranas Pq, q =. 1, 2 and retain in eq. (2.5) only a minimum of the relevant fields. Eq. (2.5) then acquires the form [16]
g,O,(x) and f(x) are then electromagnetic gauge invariant whereas for p,, one finds analogous to eq. (2.6), p,, = P,, + A , A, where P,, is gauge invariant under the electromagnetic gauge transfor- mations.
The curvature superfield R,,(z) has the expansion
C2-88 PRAN NATH AND R. ARNOWITT
For the metric under consideration it is sufficient to limit the expansion of R,,(z) to the two terms exhibited in eq. (3.8). Using eq. (3.6) we obtain then a set of two coupled field equations :
K,,(P,,) + iJpv = ) e2 F,, F: (3 .lo) where K,,(P,J is the Klein-Gordon for the tensor field PAP [17], FPA = a,A, - a,A, are the electromagnetic field strengths. Equations of motion for the field f,
which appears in eq. (3.10) arise from the section Rij(z) = 0. The coefficient of @E), @ E ) ~ in the expan- sion of the equations of motion then gives
First we note that eqs. (3.9)-(3.11) give us a set of field-current identities. Further, a symmetry break- down must occur and the tensor field P, must develop a mass in order t h d G,,(gO) has as its source the electromagnetic stress tensor. For the moment we do not delve into the question of how the symmetry breakdown is to be induced but discuss the solutions if it does. Thus if the left hand side of eq. (3.10) acquires a mass M, one may solve for P,, [18]. Assu- ming the symmetry breaking occurs, one may insert this value of P,, along with eq. (3.11) into eq. (3.9).
This gives [19]
where GE = (2 n)-l e2/M2, M being the symmetry breaking mass. Thus if the symmetry breaking occurs there would be a unification of electromagnetism and gravity similar to the unification of weak and electro- magnetic interactions [5,6]. The size of the symmetry breaking mass M is superheavy of x 1017 GeV.
(A similar size superheavy mass also appears in the breakdown of SU(5) [20].)
C. - Coupled Yang-Mills and matter : Next we consider the field equations that arise when we consider an n-tuplet of spinor coordinates P", a = 1, 2, ..., n. We have seen already that the n2 - 1 (or n2)-plet of vector fields B,A defined in eq. (2.5) transform like the regular representation of SU(n) (or U(n)). For an adequate discussion of the field equations in this sector we exhibit explicitly some additional terms in the superfield expansion of g, not shown in eq. (2.5). The relevant part of g,,(z) then takes the form [16]
Using eqs. (2.5) and (3.13) in gq. (3.6) and examining the coefficient of the term (BT,), in the superfield expansion of the equations of motion one has
where F,A, is the gaugecovariant curl of B,A, i.e.,
and C,A depends linearly on C,Ay y with gauge trans- formation properties appropriate to eq. (3.14). The presence of the term C,A in eq. (3.14) reminds us once again of the field-current identity nature of field equations. The equation of motion obeyed by C,A itself is obtained by examining the coefficient of (GT,), BP OY in the superfield equations of motion.
One has
where j,, contains among other things a contribution from the field $$. First, for the mechanics of field- current identity to work in a normal fashion C,A must develop a mass analogous to the generation of mass for P,, in the coupled Maxwell-gravity case. Second,
$, may appear in j,, only in the form cpFy, TA $ [21]
and cp must therefore develop a vacuumexpectation value for a conventional Dirac current $y, TA $ to appear as source of the Yang-Mills field equations.
We find then, that the symmetry breakdown is inextricably tied in making contact with nature.
4. Conclusions. - Recent theoretical schemes unifying two or more fundamental interactions involve the use of gauge fields. This technique is generally powerful; the presence of gauge fields limits the allowed interactions and may also lead to renorma- lizability. However, fermions appear as non-gauge particles in such schemes and their interactions are left arbitrary. It is worthwhile to contemplate a theoretical framework where all fields including the fermionic fields are treated as gauge fields. The theoretical scheme discussed here proposes this approach through an extension of supersymmetry to be a local gauge invariance. First, we have shown that the supergauge group includes the Maxwell (or Yang-Mills) and Einstein gauge groups. A superspace geometry is developed with four Bose coordinates of space-time and N Fermi coordinates. The superfield equations of motion are shown to contain field equations for particles of various spins and statistics.
An application of the general framework of geometric superspace to discuss dynamics in the conventional supersymmetry has recently been made by Woo [22].
Related to the ideas discussed here is a recent work by Cho and Freund [23] who also develop field theories in higher dimensions. However, their forma- lism is limited only to Bose coordinates.
GAUGE SUPERSYMMETRY
References and Footnotes
[I] WESS, J. and ZUMINO, B., NucI. Phys. B 70 (1974), 39 ;
VOLKOV, D. V. and AKULOV, V. P., Phys. Lert. 46 (1973) 109 ; ZUMINO, B., Proc. 17th Int. Conf. on High Energy Physics,
London, 1974.
[2] SALAM, A. and STRATHDEE, J., NucI. Phys R76 (1974). Also see the Proceedings of this meeting.
[3] We use p, v, I., ... for tensor indices and a, p, 7, ... for spinor indices. For the m-oment we assume 0's to be Majorana spinors satisfying 8, = Bg 3#. where 3 = - C -' and C in the charge conjugation matrix.
[4] NATH, P. and ARNowrTT, R., Phy.~. Lell. 56B (1975) 177.
[5] WEINBERG, S., Phys. Rev. Lert. 18 (1967) 507.
[6] SALAM, A., Elementary Particle Physics, N . Svartholm, ed.
(Almguist and Wiksells, Stockholm, 1968).
(71 PATI, J. C. and SALAM, A., Phys. Rev. D 8 (1973) 1240;
GEORGI, H. and GLASHOW, S. L., Phys. Rev. Lerr. 32 (1974) 438 ;
WEINBERG, S., Phys. Rev. D 8 (1974) 4482.
[8] DEER, S. and V A N NIEWENHUIZEN, P., Phys. Rev. D l 0 (1974) 401 and 41 1.
[9] For an arbitrary transformation zA' = zA'(z) one has .
where Ri' and L;' are related by.
R i ' = ( - l)b+ob L,A' and
R i ' RF8 = 6: = LF L j ,
[lo] TA are { i iAt', $EI$) f where iYJ are the n x n (symmetric, antisymmetric) SU(n) (or U(n)) matrices and cqq. is the antisymmetric charge matrix with E* = - 1. TA satisfy TA TB = - 4 f A B C TC + : t;dABC Tc.
[I 1 ) We suppress the charge and the SU(n) (or U(n)) indices when no possibility of an ambiguity exists.
[I21 ARNOWITT, R,, NATH, P. and ZUMINO, B., Phys. Letr. 56B (1975) 81.
[I31 Unless otherwise stated, all our derivatives are understood to be right derivatives.
[I41 There is a typographical error in the expression for the affinity given by eq. (3) of ref. [4].
[IS] The determinant of III,, In suprspacc ib defined by det M = exp Tr In M where Tr M = 1 (- I)" MA,.
[16] Here we have made a supergauge transformation of the form 5" = (ox) - 1) 0" to set the coefficjent of qM in eq. (2.5) to unity. Similarly the term independent of 0 in g,,(z) may be eliminated by a supergauge transformation of the form 5' = goLV IC/: 6.
[I71 All indices in eqs. (3.9) and (3.10) are raised and lowered by g: and ordinary derivatives are replaced by tovariant derivatives (denoted by semi-colon) with respect to this metric. Thus K, is given by
[I81 A full analysis would involve other tensor fields which appear as coefficients of (oO)2, etc., and would involve a diago- nalization of the mass matrix.
[I91 The linear part of the last three terms in eq. (3.12) are a set of superpotential terms and have an identically vanishing divergence.
[20] GEORGI, H., QUINN, H. R. and WEINBERG, S., P/I.I,S. Rev. Lert.
33 (1 974) 45 1.
[21] We write J/i = TA 7, $ for this discussion.
[22] Woo, G., MIT Preprint : Center for Theoretical Physics Publication, 465.
[23] CHO, Y. M. and FREUND, P. G. O., University of Chicago Preprint EFT 75-1 5.
