Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

MASSIVE SUPERSPACE - A SUPERSPACE WITH MASSIVE SCALARS AND GAUGINOS

K.T. MAHANTHAPPA and G.M. STAEBLER Department of Physics, Box 390. University of Colorado, Boulder, CO 80309 USA

Received 11 April 1983

We introduce a superspace whose global metric is modified in the fermiomc sector and hence does not have the canon- ical supersymmetry. This introduces masses for physical scalar particles and gauginos with their superpartners remaining massless. Cancellation of quadratic divergences among fermion and boson loops is exhibited at one-loop level.

The suggestion thal N = 1 global supersymmetry (SUSY) [1 ] may help solve [21 the gauge hierarchy problem has generated a lot of activity +1 to con- struct a realistic model of SUSY grand unified theo- ries (GUTs). Necessarily SUSY has to be broken. It has been done in many different ways: explicit soft breaking, spontaneous breaking and more recently by coupling to N = 1 supergravity. There are models [4] in which SUSY is broken at energies of O(mw) and models in which it is broken at energies of 1012 GeV or higher [5] . Recently, on the basis of cos- mology and the stability of gravitino, the region 106-1011 GeV has been excluded for SUSY break- ing [6] . In this letter we introduce a superspace whose global metric in the fermionic sector is modi- fied. This has the effect of giving mass to physical scalars. We call this space the massive superspace (MASSP). This has the desired property of cancella- tion o f quadratic divergences among fermion and boson loops at the one-loop level. A gauge superfield is introduced and the existence of a massive gaugino is exhibited. In the following paper [7] a grand uni- fied model based on SU(5) in MASSP is given.

Massive superspace. As pointed out by Arnowitt and Nath [8] the most general superspace matrix invariant under global SUSY can be written as

g = 6 t ~ 6 , (1)

I I:or an exhaustive list of recent references see ref. [3].

where r/is the OSP (3, 1/4) invariant tangent space metric = Lorentz metric X charge conjugation matrix and 6 is the superspace vielbein. The vielbein (in the appropriate gauge) is the map

~ " x m -+ x + iOo m O

0 ~ 0c~, 0& ~ 0 a. (2)

Acting on superfields (without external indices) this map has the representation

= exp (iOoUOO). (3)

The action for the free chiral superfield is

dO dO 6 t S t (x,O) ~ S(x,O), (4)

where ~2 S(x,O) =A + v ~ O +FO0. This is obvious- ly just tire Hilbert superspace innerproduct [10] of chiral superfields with the global metric g. Note the flat metric r /would produce a nonpropagating lag- rangian field theory.

The modification we propose is simply to replace the vielbein by

6M = exp (iO°UOOu + c ~ ) , (5)

where c-~ = 1M(O0 ~ ~/~0 + 000 ~/~0) is real with pa- rarneter M having dimension of mass. This is a super- field representation of the map

~2 We use the conventions of ref. [91.

0.031-9163/83/0000 0000/$ 03.00 © 1983 North-Holland 43

Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

~M : Xm --*Xm + iOOmO'

0 -+0 +~MO60 ,

O& -+ O~ + ~MOOO&. (6)

The new vacuum metric i sg M = (5~r/... ~M and it does not preserve the canonical supersymmetry. We call the space described by this metric the "massive super- space" since the inner product of chiral superfields with this metric adds to the usual lagrangian a term

M(AF + FA). (7)

This term is a mass term for the scalar field A when the auxiliary field F is eliminated. There is no mass degeneracy between the scalar and spinor fields with this choice of the massive superspace metric. It is re- markable that the geometry of the fermionic sector of superspace both causes particles to propagate and damps out certain modes by making them massive.

Massive superfields are easily introduced in anal- ogy with usual superfields. In the usual theory the vielbein maps between what we call the base space and the tangent space. If U(x,O,O) is a general func- tion of the base space coordinates then CU will be in the tangent space and hence will transform under the full OSP (3, 1/4) group. We define a real vector superfield as

V(x,O,O) = Cu + Ct u+. (8)

This is just the conventional vector superfield if we make the obvious field redefinitions. On the base space we may use the chirality constraint

aSlaO a = 0. (9)

We shall call the tangent space image of a chiral su- perfield S a scalar superfield ~ -= C S. Now the con- straint on the scalar superfield reads

¢ ( a / a O a ) ¢ t ¢ S = D a + = O . (10)

The covariant derivatives D and D commute with su- persymmetry transformations because the metric preserves supersymmetry. Therefore the chirality constraint is supersymmetric and leads to a reduction of a supermultiplet.

For the massive superspace we need only replace the vielbein with C M . SUSY transformation in the tangent space will remain the same, but we have new vector (V) and scalar (~ ) superfields.

V = EMU+~?MU? , ~ - - ~M S. (11)

In the base space of MASSP the chirality condition on S has the same form as eq. (9). In the tangent space this reads

~M(O/()0 a) ~M 1 (~M S = DM& cI) = O. (12)

The metric is not supersymmetry invariant and so the covariant derivative D M does not commute with su- persymmetry transformations in the tangent space.

Loop eancellations. We indicate ,3 how one-loop self-energy divergences for a S 3 theory may be ab- sorbed into a single wavefunction renormalization in much the same way as in the usual SUSY. We rely heavily on the superfield formulation of Fujikawa and Lang [12]. The free massive lagrangian is

£o(x,O,O) = -2 exp (-ioomO~m + c~ )s+' (x,O)

X exp(ioomO~m + QI{)S(x,O). (13)

The propagators for this case are

- ( ~ p - 1 ~. 20~O /p 2 D12(P,O ,0) = ~i 2 + M 2)

M[8(0) + 6(0)1 +p28(O)8(O)/(p2 + M 2) + ~ + M ~ - ] '

D120P,0,0) = D21(-P,O,O),

(14)

(15)

where 6(0) = 00 and p2 = _(/90)2 +p.p. The lack of mass degeneracy is clearly evident in the propagator. In the usual SUSY the cubic interaction term [12] is

£I = (}~/3!)[$3(x'0)~(0) -- S?3(x'O)~(O)]" (16)

The corresponding action in the usual superspace is just

f d x dO dO £1" (17)

In the massive superspace the field ~ - eC~S(x,O) is in the base space of the usual space since C l ~M = e c~ . The D-component of q53 is just M times the F-component of S 3. We may thus use the same inter- action lagrangian and action as in (16) and (17). The

3 Details are contained in ref. [ 11 ].

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Volume 129, number 1,2 PHYSICS LETTERS

generating functional and Feynman rules were ob- tained in ref. [12]. The loop integral for the one-loop diagram for the propagator of momentum p with in- ternal propagators of momenta k and p - k and ver- tices containing 0 and 0 is

~-J.~--~aD21(k,O,O)D21(P -- k,O,O). (18) tzn)

Note that since we do not have a propagator between 0 ~ 0 and 0 -+ 0, there are no tadpole diagrams. For large momenta (>>M) the propagator (14) approaches - i exp (-20~bO)/2p 2 and therefore the integral (18) has the asymptotic form

?2 - 2d_~) 1 . (19) 8 exp( 20/b0)f-( 4 ( p _ k ) 2 ( k 2)

This indicates that there will still be cancellations of quadratic divergences allowing the one-loop integral to be expressed as a single logarithmic divergence. This is borne out by a detailed calculation +3. Thus the divergence can be absorbed into the wavefunc- tion renormalization.

Gauge fields. A Iocalinternal symmetry may be introduced exactly as in the usual superspace. For example, ifS(x,O) is a rank-one lensor under a ma- trix representation of SU(n), then we introduce the SU(n) metric e V where Visa vector superfield [9] , as previously defined in eq. (1 1), belonging to the adjoint representation of SU(n). The SU(n) invariant lagrangian is

t" * £ = 6 M S eV6MS. (20)

There are no new gauge field couplings in the lag- rangian in the Wess-Zumino gauge. Let us first con- sider the gauge group U(1). An infinitesimal gauge transformation is

V ~ V+ i ( C M S - 6?MS+,). (21)

This has the same effect on the components of V as in the usual case. In fact since e-Q'l(does not affect the transverse components of V we have

15 September 1983

e-C/~V ~ e-C/7~ V + i (ES - ~ t St) . (22)

Hence e-C/'~V has the same infinitesimal gauge trans- formation as the usual gauge superfield. The chirality condition on the scalar field q5 = gMS is

DMa ~ = 0 with BM~ = e of~ D~ e- Qg. (23)

We may use the operator DM~ to construct the gauge invariant field strength +a

[4/a = 1 DM& D~M DMa V, (24)

which is chiral. In fact Wc~ is exactly the usual field strength tensor. The base space form of 1,9 a is

W ( 0 ) = ---~(~2/c302) exp (--2ioomOOm)

X (O/00 '~)exp (ioomOOm - cl~ )V. (25)

The lagrangian in the usual base space is (W a W a + IP a Wa)/4. In the massive superspace the base space is shifted by an additional factor of e-C//~as com- pared to the usual base space. We therefore need to shift back with e cf/~ in the inner product between the field strengths (on both the external index and the internal coordinates). We can deduce the matrix rep- resentation U t~ for e c7/'(- from o~

e-QrQa/a0 a) eQT~

= (1 + ½MOO - ½M20000)OlOO ~ +MOOO/o0. (26)

The action of U p acting on the external index and & internal coordinates of Wc~ yields

W'c~(O ') = (1 + ½MOO - ½M2OOOO)Wa(O ')

+ MO c~ W(O '), (27)

where 0'c~ = (eC)~0)c~. The appropriate MASSP lag- rangian is the inner product

~CGF = ( l /8M) (W'W' + V/W')

= (1/SM) {(1 + MOO)WW + 2MWOOW

- ~M2OOOOW~ / + (1 +MOO)WW + 2MWOOW

- 3M20000 WW}. (28)

#4 Note that D M and D M obey the same ant icommutat ion relations as D and D of the canonical SUSY and hence W a is gauge invariant.

The D-component of this lagrangian is

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Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

£GFIooo 0 - ¼FuF'~zu + ½D 2

- + ½D 2 + +

(29)

The new terms are the last two. Thus we have D 2 in-

stead of D2/2 and a mass for the gaugino. Once again

it is the metric of the massive superspace which gives

mass to the superpartner of the massless vector gauge

field. These results are easily extended to non-abelian

gauge groups in precise analogy to the usual super-

space. The only new terms are identical to the ones

above (in the Wess-Zumino gauge).

We have seen that the metric of the massive super-

space has generated masses M and 3M/16 for the scalar

particle and gaugino respectively keeping their super-

partners massless. In terms of cancellation of quadratic

divergences, the mass produced is similar to the case

of soft breaking of SUSY, but here tile mass is related

to the geometry of superspace. One can see the ad-

vantages of the massive superspace in the construc-

tion of GUTs in this space which is done in ref. [7].

As to the origin of the metric we have used, it may

possibly occur as the vacuum metric in a local gauge

theory relating M to gravity.

This work was supported in part by the US Depart-

ment of Energy Grant No. DE-AC02-81ER 400025.

Note added in proof. The statement concerning the

renormalization needs modification. There is a loga-

rithmic divergence which requires a renormalization

of M. Also, there exists an induced logarithmic diver-

gent term (00) (00) SS at one-loop level.

References

[ 1 ] Y.A. Gol'fand and E.P. Likhlman, Pis'ma Zh. Eksp. Teor. Fiz. 13 (1971) 323; D. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109; J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39; A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 353.

[2] L. Maiani, in: Proc. Summer School of Gif-sur-Yvelte (1979) p. 3; E. Witten, Nucl. Phys. B188 (1981) 513; R.K. Kaul, Phys. Lett. 109B (1982) 19.

[3] J. Ellis, L.E. lbanez and G.G. Ross, CERN preprint Tt1- 3382 (1982).

[4] For example, S. Dimopolous and H. Georgi, Nucl. Phys. B193 (1981) 150; N. Sakai, Z. Phys. Cl l (1982) 153; P. l:ayet, Phys. Lett. 69B (1977) 489; 70B (1977) 461.

[5] S. Dimopoulos and S. Raby, Los Alamos preprint LA- UR-1282 (1982); J. Polchinski and L. Susskind, SLAC preprint SLAC- PUB-2924 (1982).

[6] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1303. [7] K.T. Mahanthappa and G.M. Staebler, preprint COLO-

HEP-61. [8] R. Arnowitt and P. Nath, Proc. Orbis Scientiae (Univer-

sity of Miami, Coral Gables, t:L, 1979). [9] J. Wess and J. Bagger, Supersymmetry and supergravi-

ty (Princeton U.P., Princeton, NY), to be published. [10] W. Siegel and S.J. Gates Jr., Nucl. Phys. B147 (1979)

77. [11] K.T. Mahanthappa and G.M. Staebler, in preparation. [12] K. Fujikawa andW. Lang, Nuel. Phys. B88 (1975) 61.

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