Volume 164B, number 1,2,3 PHYSICS LETTERS 5 December 1985

FINITENESS CONDITIONS FOR A GENERAL SOFTLY BROKEN GAUGE THEORY

J. LEbN and J. PEREZ-MERCADER

Institute de Estructura de la Materia, Serrano I1 9, 28006 Mudrrd, Spain

Received 19 December 1984

A finite theory may be defined as one with vanishing beta functions. By using renormalization-group methods and taktng advantage of the non-renormalization theorems we find the conditions on the physical parameters required by imposing

finiteness on a general softly broken gauge theory. The finiteness of the supersymmetric parameters is automatic once the Yukawas are; the soft-breaking parameters require for their finiteness that they be related to the supersymmetric ones through

the gaugino mass. The latter is finite only if the gauge coupling is made finite.

The idea that unification of all forces including gravity may take place at dimensions higher than four and that the observed gauge theory structure at low energies is the result of the compactification of the extra dimensions, has received a lot of attention lately through the Kaluza-Klein approach [l] and super- string theories [2].

The non-preonic Kaluza-Klein approach encoun- tered difficulties to incorporate the observed chiral low-energy fermion spectrum with the correct SU(3) X SU(2) X U( 1) quantum numbers. On the other hand, the ten-dimensional superstring reincarnation of spontaneous compactification [3] seems to be an extremely promising candidate for a scenario of com- plete unification, and has started attracting the inter- est of the community to this type of theories and

their phenomenological consequences [4] . In fact for type I superstring theories the 2 are in-

dications that after compactification of the extra di- mensions and decoupling of the heavy degrees of free- dom has taken place, the Yang-Mills part of the re- sulting theory is UV finite [4] . When compactification on a six-torus is used to break supersymmetry sponta- neously the resulting effective theory contains soft supersymmetry breaking terms that preserve finiteness. Work along these lines [5] has led Hamidi and

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Schwarz [4,6] to propose a three chiral families SU(5) model that is finite.

In view of these developments it seems appropriate to study the restrictions imposed by finiteness on the parameters of a theory with softly broken supersym- metry. We do this here by a direct study and as an ap- plication of the renormalization group equations (RGE) for an arbitrary gauge theory with softly bro- ken SUSY that were previously obtained by the pres-

ent authors [7] . At this point we would also like to notice that the results we present have also been ob- tained by other authors [8] using different methods.

Also, another context where these results are of in- terest is in N = 2 supersymmetric theories, which, as is well known [9] enjoy the remarkable property that if finite to one-loop they are fmite to all orders of per- turbation theory.

Our results are valid for an arbitrary gauge theory with softly broken N = 1 SUSY, and they are one-loop results. Extending them to N > 1 may be done by just imposing the necessary restrictions on the parameters and superfield content of the N = 1 theory.

A quantum field theory is said to be finite when no infinite renormalization of its physical parameters is required to any order in perturbation theory. It means in particular that the bare parameters of the lagrangian are finite. From the point of view of the renormaliza- tion group this translates in that the effective parame- ters are constant, i.e., once they have been assigned a

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Volume 164B, number 1,2,3 PHYSICS LETTERS 5 December 1985

value at a given scale, they wilt maintain it as we change scales; said another way: the RGE for each physical parameter has a fixed point at values of the parameters which are not necessarily trivial. This is ob- vious, since we compute the p-functions ‘r by taking the derivative of the renormalized coupling constant with respect to the logarithm of the scale. Finiteness then means that the set of P-functions vanishes and may be achieved by either an appropriate choice of the field content of the theory and (in some cases) by se- lecting its parameters so that certain relationships be satisfied among them. An example of a theory where this situation attains is the N = 4 supersymmetric Yang-Mills theory, known to be finite to all orders of perturbation theory, and where the one-loop /3-func- tion vanishes due to the field multiplet contents.

For an arbitrary gauge theory with softly brokenN = 1 supersymmetry [7] , the scalar potential may be written as

v. = c f,f” + m&&zazb + m3,2(B + B*) + ;cDz, a cx

(14 where the index a lumps together all the internal quan- tum numbers, .za is the lowest component of a chiral

superfield @‘,f, = af/aza I e + and f is the superpoten- tial restricted by gauge invariance and renormalizability to

(lb)

B describes the d = 3 soft-breaking which may be written as

B = raza +Mabzazb t (l/3!)GabCzazbzc. (Ic)

The D-term is D, = g(z+T,z), and the part of the lagrangian quadratic in the fermionic fields is

-fi~(~,,, =;$afab~b+id%haDaatia

(Id)

with X, the gauginos and .$ a parameter controlling their tree level mass.

The soft-breaking terms are the ones proportional to the gravitino mass, rn3,*. By definition they induce

*’ By p-functions we mean collectively the right-hand sides of the RGEs controlling the evolution of the physical parame- ters such as coupling constants or masses.

divergencies in the effective action which are at most logarithmic; they have the property of not contribut- ing to the RGE for the supersymmetric parameters.

This, then, implies that there is a natural classification of the parameters into supersymmetric and soft-break. ing, with the latter receiving contributions in their RGE from the former but not the converse.

The one-loop RGE [7] for the supersymmetric pa- rameters are

32a2ba = -4g2C2(Ra)pa t Xl!ipi, (2a)

32n2r!rab = -4g2 W2(Ra) + C#b)]mab

t maiXL + m .Xj bJ a’ (2b)

32n2fuua = 2 -4g [C,(-RJ + C2(R,) + C2(Ra)lfu,,

+ fuuix: +f,iaxt + fiUaxi 9 (2c)

where the gauge coupling constant satisfies

& = -i (

3c2(5) - C T(R~) 1

(Ye, ( 2d) 0

with the usual definitions for C2(R) and T(R) *‘.

In addition, the effective gaugino mass term

cmgaugino = m 3/ 2r) has a scale evolution controlled by

.& = &/CL (2e)

The quantity Xj is defined through

x; = f i*yfjxy . (3)

Eqs. (2a)-(2c) are solved by

PW =Z;WPW), (4a)

mXY(p) = Z~(p)Z/(p)mv(M), (4b)

fXYZ(/J) = ZX(p)ZY(jd)Zg_t)f i,‘Q!q i i 3 (4c)

where Zj(,u) is a solution to

32a2i; = [-4g2C2(Rx)S; + X;]Z;, (5)

with boundary condition Z;(M) = 6;.

From (4a)-(4c) and (5) we see that the supersym-

*’ Also the covariant derivative was taken as a,1 + igu(ti) [ T (00 ] , where 1 is the identity in the irrep of the gaugtgroup G, and T(OL) are the representation of the gen- erators in the irrep.

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Volume 164B, number 1,2,3 PHYSICS LETTERS 5 December 1985

metric parameters are finite if the cubic couplings ffjk are chosen so that the condition

Xj@) = 4&&C,@ $; (6)

ismet*‘. In addition, and as is well known, making the gauge

sector finite [cf. (2d)] requires that the chiral multi- plets be in irreps of G satisfying t4

3C,(G) = c T($,). (7) 0

We now turn to the soft-breaking sector of the

theory. It turns out [7] that this sector may be con- veniently

% arametrized in terms of quantities Auva,

Bab and ua. The last one we encountered in (l);A,,, and Bob are defined through

G uva =A uvafuwy %b = m,bBaby

where no-summation over repeated u, u, (I indices is meant. These parameters satisfy RGEs

32n2AUva = -8g24. tC&) + C&,) + C2(Ra)l

+ 2(%xy lfuxy12+~vxylfv~y12+~axylfaxy~2~~ (84

32n2m,vi,v = muv {-8g=t [C2(R,) + C2(RJ1

+2A xzulfxru12+2AyrvlSypv12}

+2fXyzfyuvmxZBxZ1 WI

32,r2b~ = 4d.f ..G’iy x j ibx fibs + o?‘X’ +X?oi t 2G

I x 1 x Xl?

- 8g2t26y [C (R ) + C (R )] x2x 2y’ (8~)

For the case where there are no singlets participat- ing in cubic interactions (cf. footnote 3 and comment in section 4 of ref. [7]), eqs. (8a) and (8b) admit of a solution

Auva(p) = Z:‘)(p) + ZsB’(p) + Z;B,gl),

and

*’ Notice that (6) forbids the presence of couplings involving singlets: an iriteresting property of finite theories. It also requires that X! be diagonal.

*’ It has been rec&tly shown that (6) and (7) are also suffi- cient for two-loop finiteness [IO] .

B, v(,u) = Zf ‘(p) + ZzB)(p) + iA0 - 1,

where ZF)@) itself is a solution to

32&y) = -8g2C2($,).$ t 2A uxy KUX,12~ (9)

with boundary condition that Z:‘)(M) = $Ao. Here A, is the value at the boundary of A,,,, , which in N = 1 softly broken supersymmetry theories from N = 1

supergravity contains the “memory” of the hidden

sector. We immediately ;ee that A,,, and B,, will remain

constant as long as ZL” = 0 is satisfied. From (9) and (6) we find that it is sufficient to have

G uxy = Efuxy, (10)

for simultaneous finiteness in the soft-breaking param- eters Guxy and Mxy *’ .

Applying the same arguments to (8~) leads one to the condition that

o(R), 34‘ yy x=1 28X

(11)

which relates the soft-breaking scalar mass term a(R)

to the gaugino mass terms. This completes the set of one-loop finiteness condi-

tions on the parameters of the most general, renormal- izable lagrangian with softly broken N = 1 supersym- metry.

Aside from the obvious appeal that a theory with a small number of parameters has, finite grand unified models offer a framework to fulfill the hope of pro- viding us with a theory of fermion masses, as specu- lated by Jones and Raby in ref. [4] . However to make contact with the low energy phenomenology [4] one now needs to study models that are finite only up to a given scale.

The finiteness of the theory needs to be spoiled at some S&k so that high energy predictions such as mb = m7 may renormalize to their observed values. A scenario of this type might be one where the theory quits being finite as a consequence of the combined ef fects of gauge symmetry breakdown and the decou- pling [ 1 l] *6 theorem [cf. eq. (2d)] .

*’ The gaugino mass term is also constant because of (2e), (2d) and (7).

*6 Decoupling in finite theories has been studied by McKeon and Rajpoot [ 121.

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Volume 164B, number 1,2,3 PHYSICS LETTERS 5 December 1985

References

I l] See for example M. Duff, in: Proc. XVth GIFT Intern. Seminar on Theoretical physics (San Feliu de Guixols, Spain, 1984), eds. F. de1 Aguila, J.A. Azc&raga and L. Iba;iez (World Scientific, Singapore, 1984).

[2] For reviews cf. J. Schwarz, in: Proc. Trieste Spring

(31

[41

SchooI on Supersymmetry and supergravity (Trieste, Italy, 1984), eds. B. de Wit, P. Fayet and P. van Nieuwenhuizen (World Scientific, Singapore, 1984); see also J. Schwarz, Phys. Rep. 89 (1982) 223; M. Green, Surveys in High Energy Physics 3 (1983) 127; E. Witten, in: Proc. Fourth Workshop on Grand unifica- tion (Philadelphia, USA 1983), eds. H. Weldon, P. Langacker and P. Steinhardt (Birkh%user, Boston 1983). F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253; L. Brink, J. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77; D. Freedman, G. Gibbons and P. West, Phys. Lett. 1243 (1983) 491. D.R.T. Jones and S. Raby, Los Alamos Preprint LA-UR- 84-2692; S. Hamidi and J. Schwarz, Caltech Preprint CALT-68- 1159 (1984), and references therein.

[S] S. Thomas and P. West, King’s College preprint. [6] S. Hamidi, J. Patera and J. Schwarz, Phys. Lett. 141B

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