Partial supersymmetry and a model of unification
K. SHIMA
Laboratory of Physics, Saitama Institute of Technology Okabe-machi, Saitama 369-02, Japan
(ricevuto il 21 Aprile 1997; approvato il 20 Agosto 1997)
Summary. — Based upon 4-dimensional spacetime specified by (xm, um
a(x) ), where um
a(x) is a complex vector-spinor, an attempt to construct a 4-dimensional local gauge field theory consisting of (left-handed) supergravity part and non-supersymmetric left-handed(chiral) spin-1/2 matter part is presented. The invariance of spinor gauge symmetry under the general coordinate and local Lorentz transformations requires chiral symmetry and partial local supersymmetry. A new scheme for the unification of all particles and forces is proposed.
PACS 11.30.Pb – Supersymmetry. PACS 04.65 – Supergravity.
PACS 12.60.Jv – Supersymmetric models.
1. – Introduction
To understand the rationale of all elementary particles including graviton, local supersymmetry (SUSY) [1] and its spontaneous breakdown may be the most important notion. It gives a natural group-theoretical framework to unify graviton with other particles. In this paper, we try to construct a gauge theory consisting of (left-handed) supergravity (SUGRA) [2] part and non-supersymmetric chiral(left-handed) spin-1/2 matter part within 4-dimensional local field theory. The research toward this direction may be worth pursuing from the following phenomenological and theoretical points of view, i.e. SUSY, as a whole exact symmetry for a unified theory including gravity may be so restrictive to realize in a complicated way the observed low-energy chiral symmetry, to break SUSY spontaneously in a severely restricted way and to induce mass hierarchies with fine tunings. And the advocated remarkable success of global SUSY GUT (minimal SUSY standard model, SUSY SU(5), SUSY SO(10), etc.) remains yet to be tested in a unified model including gravity. Further considering the fact that all observed (massive) fermions are presented by complex Dirac spinor fields, it may not be meaningless to regard, at the risk of a priori SUSY invariance, that the spacetime has complex (vector)-spinor quantities instead of (real) Majorana spinors. In this paper, by adopting a four-dimensional spacetime specified by (xm, um
a(x) ) [3], where um
a(x) is a complex vector-spinor, we start surveying a new gauge symmetry by
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considering an invariance of a higher spin gauge symmetry under the general coordinate transformation. The vector-spinor spacetime coordinates um
a(x), which lead to introducing a second-rank tensor-spinor Cmn(x), may be regarded apparently as the vector-spinor world spacetime coordinate analogue of superstring theory(SST) including up to infinitely high spin modes. We attempt to cut out qualitatively the possible contributions of high spin fields to the unified model in the local field theory.
2. – Partial supersymmetry
For the arguments in this paper to be self-contained, we quickly review the way of thinking of partial SUSY starting from the following Lagrangian [4]:
L04 Lc1 Lx1 Lc 2x, (1) where (2) Lc4 c – mn(x)
m
2 i 2g m gbgd¯bcnd(x) 1igm¯ncrr(x) 2 i 2h mn gr¯rcss(x)n
1h.c. , (3) Lx4 x–mn(x)m
2igr¯rxmn(x) 2 i 2g m gbgd¯bxdn(x) 22igr¯mxnr(x) 2 2emngdg5gr¯gxrd(x)n
1h.c. , (4) Lc2x4c–mn(x)]2igngrgs¯sxmr(x)1ihmngr¯sxrs(x)1emlrkg5gn¯lxrk(x)(1h.c. The Lagrangian (1) is invariant under the following gauge transformations which allow the minimal gauge couplings:dcmn(x) 4¯men(x) 1¯nem(x) , dxmn(x) 4¯men(x) 2¯nem(x) , (5)
where Cmn(x) 4cmn(x) 1xmn(x) are complex tensor-spinor fields with symmetric and antisymmetric indices, respectively, and em(x) is an arbitrary complex vector-spinor gauge parameter corresponding to an attempt to generalize SUSY [4]. The Lagrangian (1) is unique and the most general one that is parity-conserving in four-dimensional spacetime [3], if we demand that the gauge transformation (5) hold in the curved spacetime with arbitrary curvatures, i.e. in the absence of uncontracted Riemann tensor terms in the variation of (1) under the covariantized gauge transformations (5). Therefore at the beginning we have demanded only the invariance of the vector-spinor gauge transformation (5) under the general coordinate transformation.
By using the following g-irreducible decompositions: cmn4 Wmn1 ( 1 O6 ) g(mWn)1 ( 1 O4) hmnW , xmn4 jmn1 ( 1 O2 ) g[mjn]1 ( 1 O24 ) g[mgn]j, em4 Lm1 ( 1 O4 ) gmL, gmWmn4
gmW
m4 gmjmn4 gmjm4 gmLm4 0, we find that (1) and (5) can be rewritten straightforwardly as follows [5]: L04 2i j –mn gr ¯rjmn2 i 2 f –m gn ¯nfm1 i 2 f –m ¯mf 2 3 i 16 f – gm ¯mf 1h.c. , (6)
where interestingly fm
4 Wm2 jm and f 4W2j and djmn4 ( 1 O2 ) gr¯rg[mLn]1 ( 1 O6)[gm, gn] ¯rLr, djm4 gr¯rLm2 ( 1 O2 ) gm¯rLr2 ( 1 O2 ) ¯mL 1 (1O8) gmgr¯rL,
dWm4 gr¯rLm2 ( 1 O2 ) gm¯rLr1 ( 3 O2 ) ¯mL 2 (3O8) gmgr¯rL, etc. The absence of the spin-5O2 component Wmn in (6) means that we have reconfirmed by the explicit calculations the well-known difficulty of the gravitational minimal coupling for higher-spin(F5O2) gauge field [6]. Here it is useful for future discussions to notice that
djmninvolves only Lm. Remarkably the first term of (6) (in general the bilinear kinetic term of jmn with the first-order derivative) vanishes identically, which is the direct consequence of the following curious self-duality–like relation of jmn[7]:
jmn4 2
i
2g5emnrsj rs, (7)
which holds in general in 4-dimensional spacetime. Our conventions are ( 1O2)]gm, gn ( 4 4 hmn4 (1 , 2 , 2 , 2), g54 ig0g1g2g3, g254 1, g54 g†5and e01234 2 e01234 1. Now
Lagrangian (6) reduces to the familiar Lagrangian for complex Rarita-Schwinger field
cm defined by the recombinations cmffm1 ( 1 O4 ) gmf and the corresponding spinor gauge symmetry for cm reduces to the familiar gauge symmetry for Rarita-Schwinger field in SUGRA. It seems that there may be nothing new in our arguments. However, as we show in the following, this conclusion is premature.
Note that the above arguments hold when the gauge parameter em(x) is reduced to
gme(x), i.e. Lm4 0, to which from now on throughout the discussions the gauge parameters are restricted. This means that the new gauge symmetry, under which Lagrangian (1) describing spin-3 O2 by means of second-rank tensor-spinors is invariant, is larger than SUSY by the direct product of global SO(3, 1) in the flat space. Furthermore remember that as mentioned above the gauge transformation for jmn vanishes in this reduced case and that all possible bilinear kinetic terms for jmn with the first-order derivative vanish identically due to (7). This indicates that jmn should be regarded as an auxiliary field (or a Lagrange multiplier) in four-dimensional spacetime (1) with em(x) 4gme(x). The existence of such an auxiliary field indicates the existence of certain constraints in the original Lagrangian (1). In the above arguments where we have derived Rarita-Schwinger (supergravity) Lagrangian (6) from (1), these constraints which eliminate automatically some components of cmn and
xmn are ignored. We find that in the flat space the most general Lagrangian that recovers these constraints is the following [8]:
L 4L01 Lj, (8) L04 2 i 2 f –m gn ¯nfm1 i 2 f –m ¯mf 2 3 i 16 f – gm ¯mf 1h.c. , (9) Lj4 i j –mn ¯[mfn]1 iaemnrsj – mn¯[rfs]1h.c. , (10)
where a is an arbitrary constant. Remarkably due to (7) the spin-1 O2 component f is decoupled automatically from (10). We can easily see that by using (7) the Lagrangian (8) disregarding parity invariance is the most general one, which is invariant under the original (flat space) gauge transformations (5) with the reduced gauge parameters. Because dfm and df are now parametrized by only spin-1 O2
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component L, i.e. dfm4 2(¯mL 2 (1O4) gmgr¯rL), df 42gr¯rLand remarkably jmn is invariant as mentioned above under these gauge transformations.
Now we consider the gauge invariance of (8) under the above-mentioned variations in the covariant forms. As mentioned above, L0is invariant up to Ricci tensor terms so
far. However the gauge transformation of (10) produces
dLj4 Rmnabj –mn
(i 22ag5) sabL 1h.c. ,
(11)
where we have used the self-duality–like relation (7) and DmL 4 (¯m1
( 1 O2) vmabsab) L, where vmab is Lorentz spin connection and sab4 ( 1 O4 )[ga, gb]. Euler equation for jmnproduces the following constraints:
(i 22ag5) D[mfn]4 0 . (12)
Remarkably L is gauge invariant, i.e. free from uncontracted Riemann tensor terms provided
a 46 i
2 , L f( 1 6g5) L , fm
f( 1 6g5) fm. (13)
Substituting these into (8), we obtain after rescaling
L 4emnrsc–
Lmg5DngrcLs1 i f
–
LgmDmfL,
(14)
where we have put tentatively a 4iO2 and accordingly cLmf( 1 O2)(11g5) cm4
fLm1 ( 1 O4 ) gmfR, fLmf( 1 O2)(11g5) fm, fRf( 1 O2)(12g5) f, fLf( 1 O2)(11
g5) f. The first term of (14) is a left-handed complex Rarita-Schwinger field of
(SUGRA) Lagrangian coupled to graviton and preserves the reduced original gauge transformation dcLm4 DmLL which guarantees the all-order invariance of the first
term, if we define ordinary SUGRA gauge transformations in part only for the first
term. The all-order invariance, especially the cancellation of the cubic terms of cLm
stemming from the torsions, should be checked explicitly. (The chirality of N 41 SUGRA is already mentioned [9].) If we reinterpret the left-handed complex Dirac spinor (i.e. Weyl spinor) as a four-component (real) Majorana spinor, the all-order invariance is trivial due to SUGRA. However in this case chirality is not manifest.
The second term of (14) is the kinetic term of a chiral (left-handed) spin-1/2 fermion coupled to graviton, which has decoupled from the original gauge transformations and possesses no gauge invariances. Therefore the second term may be regarded as the non-supersymmetric chiral spin-1/2 matter part. In case of a 42i/2 we have the right-handed versions of the above arguments.
Of course, all these arguments can be rewritten straightforwardly by using the original tensor-spinor gauge fields cmn(x) and xmn(x). For example, the Lagrangian (10) for a 4i/2 can be expressed as follows:
(15) Lj4 iemnrs
g
x–mn1 x–kmgkgn2 2 3 x – klsklsmnh
(g52 1 ) Q Qm
Drgt(cts2 xts) 2 1 4Drgs(ct t 2 2 stzxtz)n
1h.c. , and the Lagrangian L0( 1 ) 1Lj( 15 ) gives the total starting Lagrangian.Here we notice a strange uniqueness of the above total Lagrangian (14). If we start with Majorana (real) tensor-spinors cmn(x) and xmn(x) instead of complex ones and perform the similar arguments, we obtain the same final action (14) describing SUGRA and non-SUSY chiral spin-1/2 matter, for the constraint (13) originates from only the second-rank tensor property of jmn. These situations can be easily seen by using Weyl base for g-matrices. Because in Weyl base a four-components Majorana spinor is composed of one Weyl spinor and its conjugate, the choice of the chirality by (13) implies the choice of one of the two Weyl spinors, which makes the theory chiral. 3. – Partial SUSY unification
Next we consider qualitatively a few grand unification (GUT) models by using the above-mentioned scheme. As for the internal symmetry, it is well known that SUGRA sector allows SO(N), N G8, i.e. N(N21)O2 gauge fields are contained as elementary fields. For non-SUSY sector there are no such restrictions on the internal symmetry and we can anticipate ordinary non-SUSY GUT. According to the assignment of particles to SUGRA sector, simple unification models in this scheme may be classified qualitatively into the following three types [10];
i) Some of the observed chiral fermions accompanied by bosonic
supersymmetric partners are assigned to SUGRA sector and other observed chiral fermions are assigned to chiral spin-1/2 non-supersymmetric sector.
ii) All of the observed chiral fermions are assigned to non-SUSY chiral spin-1/2 sector.
iii) All of the observed chiral fermions accompanied by bosonic supersymmetric partners are assigned to SUGRA sector.
As a simple example of type i), it may be natural to take SO( 8 ) for SUGRA sector
and decompose 8 4 313*1111 4 (3, 21/3)1 (3*, 1/3)1 (1, 0)1 (1, 0) under
SU(3) 3 U(1) [11]. It may be interesting and characteristic to assign the observed SU( 2 )-singlet quarks and leptons to 56 of spin-1/2 states of SUGRA, for the
electroweak SU( 2 )-doublet states for quarks and leptons (i.e., the charged weak bosons) are missing in SUGRA sector in this type. Therefore, among all observed quarks and leptons only (some of) the SU( 2 )-singlet right-handed particles have supersymmetric partners. These simple arguments may explain why right-handed quarks and leptons are SU( 2 )-singlets. For chiral spin-1/2 non-SUSY sector, other quarks and leptons should be accommodated. For example, if we take tentatively SU( 8 ) and adopt 56 4 (1, 1)1 (3*, 5)1 (3, 10)1 (1, 10*) under U(1)3SU(3)3SU(5), we can expect three generations of observed quarks and leptons to appear through the spontaneous (super)symmetry breaking, [SO( 8 ) & SU( 3 ) 3U(1)3U(1), SU(8) &
SU( 3 ) 3SU(5)3U(1) ] K [SU(3)3U(1) ], which remains to be studied in detail. However for the unification of all forces in a (semi)simple group it may be interesting to consider for the chiral matter sector a gauge group which has the same dimension as
SO(N) with the same coupling constant, for example, SU( 5 ) 3SU(2)3U(1) for chiral
matter sector and SO( 8 ) for SUGRA sector. Alternatively if we assume SU( 8 ) gauge symmetry instead of SO( 8 ) for SUGRA sector, some missing gauge bosons, quarks and leptons should appear as composite particles [12] in SUGRA sector.
For type ii), the model may be qualitatively similar to ordinary GUT coupled to SUGRA. If we take 56 of SU( 8 ) for non-SUSY chiral spin-1/2 sector, we can assign
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three generations of quarks and leptons as displayed in case i). In this case, the structure of SUGRA sector may not be determined from the present low-energy physics. For example, in the simplest case, SUGRA (N 41) sector may interact with non-SUSY chiral spin-1/2 sector (i.e. ordinary GUT) by means of only gravitational interaction.
For type iii), this is opposite to type ii). The model may be qualitatively similar to
N-extended SUGRA unification apart from the non-SUSY chiral spin-1/2 sector. The
effect of non-SUSY chiral spin-1/2 sector with appropriate internal symmetries should be studied in detail. Otherwise no viable models with elementary fields appear in this case [11].
As easily read off from the arguments, the models given by partial SUSY in the above scenario are not the so called unified models based on a simple group but those attempting to accommodate graviton and non-SUSY chiral (Dirac) fermions. From viewpoints of unity of all forces, it is more interesting to perform the similar arguments in higher spacetime dimensions and to introduce internal symmetries through dimensional reductions to 4-dimensions. These reductions (spacetime compacti-fications) may be different between SUGRA sector and non-SUSY chiral sector, but in 4-dimensional spacetime they should realize SO(N) & SU( 3 ) 3U(1) in SUGRA sector and SU(n) & SU( 3 ) 3SU(2)3U(1) in non-SUSY chiral sector, where SO(N) and
SU(n) should be isomorophic in order to unify all forces in the same bosonic degrees of
freedom of the same gauge action in higher spacetime dimension. Only the couplings with matter fields distinguishes between SO(N) and SU(n) for the unifying gauge force. This new mechanism for the unification (dual unification) of forces is essential for the present scenario. As a simple example, the compactifications from 10 to 4 dimensions may be worthwhile to be considered in detail to see whether partial SUSY model with isometry groups SO( 6 ) 3SO(2) and SU(4)3U(1) can be realized. Therefore, it is interesting to study the compactifications parametrizing the manifolds
SO( 6 ) 3SO(2)/SO(5) and SU(4)3U(1)/SU(3)3U(1)3U(1). As for the spontaneous
(super)symmetry breaking [SO(N) SUGRA, chiral SU(n) ]K[SU(3)3SU(2)3U(1) ]K K [SU( 3 ) 3 U( 1 ) ], it may be useful to point out that the hierarchies of the sym-metries can be different between SUGRA sector and non-SUSY chiral spin-1/2 sector.
4. – Conclusions
The notion of partial SUSY and the derivation mentioned above may be unfamiliar and awkward from the viewpoints of ordinary unified gauge theory. However we can summarize the results as follows: Lagrangian (8)
(
or (1) and (5))
written by complex spinors cmn(x) and xmn(x) is the most general one that is invariant in the above sense under the covariantized gauge transformation (5) with the reduced gauge parametersgme(x) and produces non-SUSY chiral spin-1/2 fermions as gauge particles, provided SUSY is defined in part for left-handed (chiral) spin-3/2 sector. Alternatively we can regard that, in the spacetime specified by
(
xm, uma(x)
)
, Lagrangian (8) is the most general one that is invariant under the general coordinate and local Lorentz transformations, where SUSY(N 41) is necessary in part for left-handed (chiral) spin-3/2 sector.We hope that the new scheme mentioned above gives tractable field-theoretical models which have enough degrees of freedom for the unification of all particles and
forces and suggests a breakthrough toward the unification in the framework of the local field theory. However there still remain many open questions to be studied in detail.
* * *
The author thanks the colleagues of Saitama University for their warm hospitality and Y. TANIIfor useful discussions.
R E F E R E N C E S
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