The decoupling theorem in softly broken supersymmetric theories

Volume 221, number 3,4 PHYSICS LETTERS B 4 May 1989

T H E D E C O U P L I N G T H E O R E M IN SOFTLY BROKEN S U P E R S Y M M E T R I C T H E O R I E S ~

J. LEON, J. PI~REZ-MERCADER and M.F. SANCHEZ Instztuto de Estructura de la Materta, CSIC, Serrano 119, E-28006 Madrtd, Spam

Received 3 January 1989

We discuss the Wess-Zumlno model with soft-breaking terms and two mass scales: M>> m. The one loop effective low energy theory is obtained by integration of the heavy degrees of freedom (HDF) using a method originally due to Welsberger We find complete decouphng of the heavy and light sectors m some cases The problem has physical interest because the superpotentml and breaking terms proposed here cames all of the features of the superpotentmls appearing m reahstlc SUSY-GUT models. We check that, under certain conditions, the existence of hght smglets m these models does not destroy the gauge hierarchy

Models for the unification o f all interactions necessarily require the appearance of several mass scales. Phe- nomenological constraints require that these differ by many orders o f magnitude, so that the quantum field theories that one has to deal with are o f the so-called multlscale type. These scales serve to define heavy and light sectors in the models, and one is generally interested in assessing the impact that the physics o f the high mass scales has on the physics of the low energy scales. To do this one appeals to the Appelquist-Carazzone theorem [ 1 ], which, as is well known, asserts that if there is decoupling of the heavy from the light sector, then the effects of the heavy degrees o f freedom ( H D F ) on the physics o f the low mass scales are both calculable and harmless. In fact, the theorem states that "the only observable effects o f the heavy degrees o f freedom come as effectwe operators o f dr menston d > 4 [ hence depressed by powers o f (Large Mass ) 4 - d ] or as logarl thin tc correcttons to the dlmenstonless couphng constants or other effective parameters"; however, it gives no exphclt description for the calculation of the (low mass) effective lagrangian nor the effective parameters.

Central to this distinction between heavy and light sectors, are the following two issues, namely, that (1) ( the contributions due to the propagation of states with virtual mass larger than the light mass scales be truly small, and consistent with perturbation theory as defined at the tree level, and (ii) that the tree level definition of light be stable under perturbation theory. A theory where (i) is not satisfied is said to have a violation of the decoupling theorem and if (ii) is not satisfied either, we then say that the theory has a hierarchy problem. These sicknesses are known to be curable by imposing appropriate symmetries on the models, which thus restrict the radiative corrections to the tree level lagrangian. Familiar examples are supersymmetry and chiral symmetry, with the former playing a fundamental role m grand unified theories and in models derived from the superstring.

For unified theories, supersymmetry, however, must be softly broken and, as is well known, this has the po- tential to destabilize the gauge hierarchy. In principle, this can happen m the case where the hght and heavy sectors are gauge singlets, without any gauge symmetry to protect the low energy scales from the effects of the heavy masses. Also in the case o f unified theories generated from the superstring, the situation is in addition compounded by the presence o f intermediate scales, which are introduced in order to avoid the presence in the low energy spectra o f exotic matter.

Typically, the analysis o f the decoupling of the heavy and light degrees o f freedom, proceeds through a calcu-

¢r Work supported, m part, by Spain's CICyT

324 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Pubhshing Divis ion)


lation of the effective lagrangian, and several methods are available for this [2] ~t. From this effective lagrangian, one obtains the form of the effecttve parameters in terms of the original tree level parameters, and checks on their size and stability under changes of scale. This reformation can then be used for a detailed study of the decoupllng behavior in the model.

In th~s paper we study the problem using the superfield generalization of Weisberger's method presented in ref. [4 ], where it was applied to the two-scale, four-dimension, Wess-Zumino model. Here we will further gen- eralize that method to include soft breakings, which will be included in the tree level action, as is usual [ 5 ], by introducing the appropriate spurions. Let us then, consider the two scale (/t8 and/tL), renormallzable and softly broken Wess-Zummo model in four dimensions, described by a classical action

S(B, t ) = f dax d40( lB]2 + ILI2-O202[I~ IBI2 + p~ ILI2+/~L(B/S+h.c . ) ]}

+ ~ d4x{d20[ W(B, L) -02f(B, L)] +h.c.}, (1)

where W(B, L ) is the most general renormalizable superpotential in four dimensions, andf (B , L ) parametrlzes the soft breaking terms. That is

W( B, L) =pBB + pL L + ½ MB2 + ½mL Z + mBL BL + ~2~ B3+ -~22L3+ ~23BZL+ ½24L 2B, (2)

f( B, L) =b+ hBB + hL L + -~fBB2 + ~fL L 2 + fBL BL + ~g~ B3 + ~gzL 3 + ½g3B2L + ½g4L 2B. (3)

The model described by eqs. ( 1 ) - (3) is the most general, renormalizable Wess-Zumino model that one can construct. We are however, interested in studying models consistent with the N = 1 supergravity scenario ~2 and in particular with the appropriate supersymmetric extensions of the standard model. This translates into restric- tions on the relative sizes of the parameters of the model. In particular, we must impose the followmg require- ments: (i) a tree level mass hierarchy, characterized by

m/M<< 1 and mzBL<<.Mm, (4)

and (ii) a set of constraints on the soft-breaking terms, related to the naturalness of the spontaneous breaking of local supersymmetry. These may be characterized as follows:

2 2 2 2 ltB=m3/2~B, /t2=m2/2~L, /tBL=m3/2CrBL, fB=m3/2MBB, fL=m3/2mBL, fBL=m3/2mBLBBL,

gt = m3/2J.,A,, m3/2 ~ m. (5)

Here, all the a's, B's, A's and 2's are o f O ( 1 ), and z= 1, 2, 3, 4. The gravitino mass m3/2, is assumed to be of the order of the magnitude of the light mass, and characterizes the scale of supersymmetry breaking through a geometric hierarchy [ 7 ] scenario, as implied by the tree level mass hierarchy condition mentioned in (i) above.

The choice of the relative sizes of the parameters given above contains all the essential features of the models with phenomenological applications. One may, nevertheless, view eq. (5) above, as a set of boundary conditions on the tree level parameters, valid at some mass sca le /~=M which fxes their relative values there. For our purposes, the problem Is whether these condttions are respected as we go from the large mass scale to the low mass scales and, if so, under what circumstances. Simultaneously, one can ask about how the soft breakings at large scales influence the low mass scales, and even tf the resulting low energy effective theory is also a softly- broken supersymmetric theory. We will provide answers to these questions.

We are interested in writing the low energy effective action for the model described through eqs. ( 1 ) - ( 3 ) , under the conditions imposed in eqs. (4) and (5). With this aim in mind, we will compute the effecttve, low energy effective action by doing a simultaneous integration of the light and heavy degrees of freedom. This way,

~t See ref. [3 ] for details and a more complete set of references ~2 See ref [ 6 ] for a collection of papers.

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the one-loop corrected low energy effective action takes into account, as it must, the contributions due to the propagation of the heavy degrees of freedom (HDF) in the loops, as well as those due to light degrees of freedom propagating in the loops with virtual momenta comparable to, or higher than, the heavy masses. Performing the calculation in this fashion allows in particular inclusion of mixed loops, where both light and heavy propagators coexist.

In order to implement this program, one proceeds along the following course. First, we compute the quantum correction to the full tree level actton, and then solve the equations of motion in the low momenta kinematic regime, where the external momenta are comparable to the light mass and much smaller than the heavy mass. Then, we take the limit mhglat/Mheavy--~O, finally eliminating the HDF from the action. While doing this, we also carry out the renormalizatlon of the theory by cancelhng infinities in the usual fashion.

The first step is to solve the tree level, heavy superfield equation of motion,

1 - 2 - - 2 2 ~ - 2 - - 2 - zD (B- -m3/2 f fBO 0 B--m3/2ffBLO202ff~) + O W / O B - O 2 O f / O B = O (6)

in the low momenta region. We notice that eq. (6) contains an inhomogeneous soft-breaking term, - m3/2M2rBO 2, which comes from f (B, L). To simplify its solution, it is convenient to do a shift on B so as to ehminate this term; this can be achieved by introducing a new superfield B', related to B, via

B=B' +ao +bo 02. (7)

Plugging this into eq. (6), one obtains the following two equations:

bo + Mao + ½).1 a2 = 0 ,

( M + 2 j ao )bo - m3/2 (M2r~ + MB~ao + m3/2¢rBao + ½,~ iA x ag ) = O,

which we use to,compute both ao and bo. Notice that here we have introduced the following reparametrization of the soft-breaking tadpoles: he = m3/~M2rB and hL = m3/2m 2rL.

Th~s system of equations can, of course, be solved exactly, but for our purposes it is more useful to expand their solution as a power series in terms ofm3/2/M, whose dominant terms are

ao=--m3/2.~B, b o = m 3 / 2 M r B . ( 8 )

Substituting eq. ( 7 ) in eqs. ( 1 ) - ( 3 ), shows that th~s generates contributions to the non-supersymmetric mass term of the light field. These contributions are of the form

-- ½ f dax d20 02 ( m3/2 mBL + m3/2J.4A4ao - 24 bo) L 2. ( 9 )

This can, in prlnc~ple, spoil the original tree level hierarchy. In fact, plugging here the dominant terms for ao and bo, we find a contribution to the tree level, light field, soft-breaking mass, dominated by - m 3 / 2 M 2 4 r B .

Therefore, we see that the non-supersymmetric heavy tadpole ~3 will spoil the hterarchy already at the tree level, unless we require that rB~< m/M.

In addition to the two equatmns for ao and bo, eq. (6) contains the equation for B' itself. We will solve it by first expanding the solution in terms of supersymmetric and Lorentz lnvariants, which provide one with a minimal and complete set of proper functions. A solution which also satisfies the approprtate chirality constraints may be written as

B' =B ' (L) =B~ +B2 +B3 + 02 (b l +b2 + b 3 ) ,

where

~3 Notice that setting .~4=0, does not solve the problem, because there is a term m the supersymmetrlc part of S(B', L) of the form f d4xdZO(mRLao+½23a~)L, which also destroys the hierarchy, independently of the value of ~4 The dominant term here is -m3/zmeLzB, which is well-behaved tf zB <~ m/ M

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B, (L) =al L +a2 ~I~E+ a 3 ~17,RL, 1 - 1 2 B2(L) =a4L 2 +as¼1~E2 +a6~REL +aT ~I~RL 2 +as I~RRLL +a9L~I~RL +a~o¼I)2IS+aI~ zR~D L

+ al2 ~6/~R~I321-.,

b~ (L) =A2L-.I.-A 3 ¼t~ff,, b2(L) =AsL2+A7 ~/~/52 +As 1/~/SL +Al i 102E+A~2 ~R~D2L.

In these equations the functions B3(L) and b3(L) contain contributions that go like M -n, with n>_-2 and n >/1, respectively, and R designates the operator

R = D202 = - 4 + 40I) + 02D 2.

Substituting B' (L) as given above, in the heavy superfield equation of motion, and equating to zero the coefficients of similar functions, leads to a set of coupled algebraic equations from which we determine the coefficients a, and A~.

Of all the coefficients, the only ones that turn out to be relevant are ao, bo and a~, with the latter being deter- mined by an exact equation:

(M+2~ao)a~ +mBL +J, aao =0.

The rest of the coefficients, as well as B 3 (L) and b3 (L) , do not contribute to the low energy limit ofS(B' (L), L ) because their contributions are suppressed by powers of the heavy mass. The contributions from other poten- tially dangerous terms disappear when one uses the equations of motion, as we must.

Substituting then B' (L) into the original tree level action, yields the tree level contribution to the low energy effecUve action.

Upon straightforward calculation one discovers that the parameters in the supersymmetric low energy sector receive contributions from the softly broken sector and viceversa ~a. As a consequence of this, we see that the final, tree level low energy effective parameters in the supersymmetric sector receive contributions which are proportional to the gravltino mass.

Another observation is that there are contributions the low energy action which are not soft breaking, namely, there are terms of the form

f d4OL2f-,, f d4OO202L21D2L, f d4OO20~LI2L, (10)

which show that the low energy theory, although renormalizable, is not soft. However, we can also check that these hard-breaking terms all disappear in the situation where eq. (5) is supplemented by the previously mentioned condition that z8 <~ m / M.

Now we are ready to take the limit m3/2/M~O of S(B(L), L). As is well known [4], in the process one generates a contribution to the kinetic energy term of the light degree of freedom; this contribution is proportional to a ~ and therefore constant, and one can write a minimal action by a simple redefinition of L, according to

L = ( 1 +a~)-t/2L'~,

where L ~ now has a minimal kinetic energy term, and specifies the light, low energy degree of freedom. The low energy tree level action is thus

~4 This 1s, for example, clearly seen by examining the contnbuuons generated by subsUtutmg B for ao+ boo 2 in the full action.

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S*(L~)= lim S(B(L~) ,L~) m3/2/M~O f d4x d40( it~12 _2 jq2ff2,.r* 2 ) J" = --,--3/2v v ~,0 I t~l + d4x{d20[W*(L~)-m3/zO2f*(L'~)]+h.c.}. (11)

In writing eq. ( 11 ), we have regrouped terms in the classical contribution of the low energy effective parameters according to the following definitions:

W . . . . l m * ] ' * 2 . . l . . I ~ * r * 3 f*-b~+m~2~oL~+i,,,o,-,o~o ~g~o~o~o , =poLo + ~, , ,o , - -o T ~m0z~0 , 1 ~ * / ~ , y * 2 ..1.. I ~ * A * /--3

for the low energy superpotential and soft breaking. The parameters appearing here have the following expressions in terms of the parameters in the full original theory:

1 P~ = ( 1 +a 2) i/2 (maLao + ½23a2),

1 m~ - ( 1 + a 2 ~ [m+24ao + (maL +)~3ao)al ],

1 ' ~ - - ( l + a l 2 ) 3/2 [~.2+32.4al +323a2+21a3] ,

1 ~ - ( l+a2~ [ ¢rL +2aleaL +a21(Ta]'

m3/2b~=½b2 +m3/2[b+M2zaao l 2 1 2 1 + ~m3/2craao + ~MBaao + ~2 ~A~ a3],

1 m3/2m'~2~° = ( 1 +a~)1/2 [ - - (reaL +23ao)bo +m3/2(m2"fL + maLBaLao + m3/z~raLao + ½,~3A3ao 2 ) ],

l m3/2 m'~B~ = ( 1 +a 2 ) [m3/2mBL + m3/224A4ao -24b0 + a2(m3/2MBa + m3/221Al ao --21 bo)

+2al (m3/2mBLBBL +m3/223A3ao -23bo) ],

1 2~A~- (1+a2)3/2 (22A2 + 324A4al + 323A3a2 + 2tAla31).

We will denote collectively this set of parameters as O~. (Notice that from these expressions, one sees explicitly that the "hght" tree level parameters are hght when the conditions eq. (5) and r s< m / M are simultaneously satisfied. )

Having studied the structure and stzes of parameters in the tree level part of the lagranglan, we can now compute the one-loop correction to the low energy effective action. As advertised, we will compute the full effective action, i.e. the one-loop effective action, allowing for simultaneous propagation of hght and heavy degrees of freedom, separating the I/E parts from the log M2eavy//t 2 and log mhght/It 2 contributions. The 1/~ parts will be used to renormalize the action by means of e.g. MS, and the contributions proportional to the heavy logs will, eventually, be absorbed in the definition of the one-loop low energy effective [ 8 ] parameters. We will then solve for B = B ( L ) and take the large-mass limit. The O(h) correction to the full a c t i o n , / ~ ) (B, L), is obtained as

( : d4k X' ) F ( ~ ) ( B , L ) = - ½ h T r l o g c~8 (x -x ' )+ (-~n)4X(k,x, ) e x p [ i k . ( x - x ' ) ] , (12)

where the matrix X(k, x, x' ) is a 4 by 4 matrix that may cast as

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Volume 221, number 3,4 PHYSICS LETTERS B

( x l l X12)(~4(0_0,). X(k, x , x ' ) = \X21 X22

The 2 by 2 matrices X v are defined by

X~t = U(x)~[' G(k)[)2D2/16k 2 - [ ~r, j//[, G(k)02D2D2/16k 2 + aUG(k)02~D2~lD2 ],

Xl2 = - U(x)G( k )-~D2 + [ 3 ~' G( k )02~D 2 + a2~ ' G( k )02¼D2~D2D2/16k2],

X21 =X12, X22 ~-Xll ,

and where we have, also, introduced the following matrices:

U(X) =Hi a(x) +A2L(x) - 02m3/2 Gl B(x) - 02m3/2GuL(x).

Al, A2 depend only on the supersymmetric sector and are given by

( ) 21 23 , A2= 23 A I = 23 24 24 22

The rest contain reformation from the soft-breaking sector, and are explicitly

(2lAl 23A3~ (23A3 24A4~ GI =- k,23A 3 24A4/]' G2= k24A 4 22A2J'

v # ' - ( M+2 la o MBL+23ao) =~#+Alao, \msL +23a0 m+24ao

The propagator G(k) is given by

G ( k ) = - ( k Z + j ( ' 2 ) -I.

a2=m2/2( aBaBl. GBL~'~L ,]

( MBB mBLBsL) o~, =m3/2 \mBLBBL mBL ] +m3/zGlao-Albo.

4 May 1989

(13)

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Now we must expand the trace in eq. (12). This is done by well-known methods, and making extensive use of the Grassmann-algebra properties [ 5 ] of the O's. The calculation although lengthy is straightforward and will not be repeated here. The main points are that one only retains in the expansion those terms which are not suppressed by powers of rn'd/M or m3/2/M. In the end one obtains the low energy limit of the one-loop contribution to the effective action,

lim F~I)(B',L), m~/M, m3/2/M~O

and finds that (because terms of order m3/2/M have been neglected) it has the same form as the original tree level action; furthermore all htgher dimensional proper functions containing only the heavy field or combina- tions with the light field, are also suppressed by powers of 1/M, as guaranteed by the decoupling the ~rem of Appelquist and Carrazone.

Out of this expansion one constructs the full one-loop effective action, which will be the sum of the tree level action and the object obtained by expansion of eq. (12), and from these, compute the equation of motion for the heavy field. This equation is then solved in the regime of low mass scales, following techniques similar to the ones used previously for the classical contribution.

Finally, this solution is plugged in the effective action, thus eliminating the HDF. As m the classical contribution, one performs a rescaling of the light field so as to have minimal kinetic energy terms, and finally obtains the following, low energy effective action, corrected to one-loop:

* * ~ * 2 ~ 2 t~ 2 ~--2 ,,-e* * 2) F (Ll.loop) = daxd40( ILl-loop I - - ' "3 /2 wt t t,,l.loop ILl-loop I

f --m3/20 "f l-loop( l-loop) ] -~h.c.} + d4x{d20[ v~:* tL* ~ 2 * L* vv l-loop ~ l-loop I


where we have introduced the low energy superpotential and soft breakings, according to the following:

* * - * L* 1 ~ * Z*2 -t- 1~, L.3 (14) W i_loop (L l - loop) = P l - l o o p l-loop "q- ~ t , , l-loop l-loop r ~r . l-loop l - loop ,

f . I L . x _ • a_~.2 ~* L* . 1 ~ * B* r*2 ± l ~ . A* L .3 (15) l-loop ~. l-loop ] = b l-loop x r t t l-loop t l-loop l-loop / ~ t i t l-loop l-loop -t-" l-loop T ~ l-loop l-loop l-loop •

The effective parameters can be writ ten as

P~-loop =p3[1 -- (h /32n2) ½23 z log(c , / f l 2 ) ],

m l-~oop-m0[1 - (h/32zc2)232 log(c l / / t 2) ],

~. l.loop =~,o [ 1 (h/327~2) 31.2 - - ~ 'o 1og(cl/, t t2)],

[ (2pAp) +30~o2o ] Iog(cl/,U2), q-loop = ~ - (h /32 r~2) . . 2 .2

* 2 * 2 * * * * bY-loop = b 3 - (h/64~z 2) (m3/2O'*oo 2 + 2m3/2m32o~o +m3/zmo Bo + 2po2omoBo) log(cl / /z2),

m.2 ~ _ ~ . . . . . . . 2 . . . 2 1 ~ . 2 ] . 2 , r a ¢ ' ~ l-loop l-loop--m32~ (h/32rc2)(ma/2AoBo2omo+2m3/22"d~m3+poAo2o +B0k0mo + - - ~ ' " t 0 z"0 ~'01

× log(cl//~2),

* ° * ~ * B * - ( h / 3 2 ~ c 2) . .2 • . . .2 (2m020 Bo + 2Aomo2o ) log(c l /#2) , 1~ l-loop Lj l-loop = rr~ 0 0

1 - A * (h/327t2) 9A3,a.33 Iog(cl / / t2) . (16) ~l- loop l - loop=a~A3-

Here cl represents the heavy eigenvalue of the square mass matrix ~ " 2, and/~ is the usual, arbitrary renor- malizat ion scale. The parameters collectiAvely denoted by 03 , are the tree level effective parameters given after eq. ( 11 ), and the "ha t t ed" parameters O3 are related to the 03 , via a relation of the form

O3 = 0 3 + h (function of the original parameters ) log(cl//~ 2) + O(h2) .

These parameters have the proper ty of being fixed under the renormahzat ion group, that is, ( # d / d # ) O 3 = 0, for each of them. In other words, they behave as if they were a class of bare (but finite) parameters with respect to the low energy effective theory. They are obtained as follows: when comput ing the effective action, and after renormahzat ion, one obtains the effective parameters as

OLloop = O3 + h ~ log (Cl/]~2 ) , ( 17 )

where ~ denotes the coefficient o f all the heavy logarithms contributing to O}_loop after taking the low scale hmit. We now add and subtract to the right-hand side of eq. (17) the contr ibution necessary to cancel out the one coming f rom theft-function of O~. In other words if

d . ~ o0 = ~#o~,

we rewrite eq. ( 17 ) in the following way:

O~. l oop = O ~ - ~ h ( ~ 1 -$1- ~ o ~ - - a o ~ ) log(c l / / t2) ,

where 6o, is chosen such that

~l +ao~ = + '/~o~.

The paramete r O3 is defined as

O3 = O3 + h ( ~ +6o~) log(c l / / t2) ,

and, by constructton, is fixed under the renormal izat ion group. This seemmgly arbztrary reparametr izat ion of OT-loop as

330


OT-~oop = O~ - h6o~ log (Cl//t 2 ), ( 18 )

has in addition the interesting, and non-trivial, property that the renormalization group equation satisfied by each of the OT-loop in eq. ( 18 ) is precisely the RGE that one would obtain [ 9 ] from the effective action o f eq. ( 13 ). This can be explicitly checked by applying the scaling operator/ td/d/~ to eq. ( 18 ) and using the RGE for the parameters in the original two-scale action, eqs. ( 1 ) - (3). The parametrization being "closed" under the renormalization group means that, at least to the order of perturbation theory that we are working ~5, it leads to a consistent quantum field theory valid at the low mass scales. In fact, up to terms of O (h2), one obtains the set o f equations

d * = (h/32n)Pl_loopJ, l.loop, ]2 ~ P l - loop 2 * *2

d * - - 2 * *2 ~ - ~ ml-loop -- (h/32z~)2ml-loop21-1oop,

d 2 . 2 /z~--~2T_loop = (h/32n)321.1oop,

d 60~l-loop). l-loop ], /t~--~ ~-Ioop = (h /32n 2) [ 2 (A~qoop,~,~-ioop) 2+ 2 .2

d . ~ ~ .2 B.2 2m3/2 m l-loop O1'1-1oo0 "1- rt~3/2 rtl. l-loop l-loop + 2pl-loop2 l-loop m l-loop B l-loop), l l .~ f lb l_ loop=(h/32rg2)(m3/20. .12oop+ .2 . . . .

d t m . 2 z*, " 2 * * * * 4 ~ ~ * ] * .,.~2 /Z~--~ ~ l-loop l-loop;=(h/32X )(2m3/2Al-loopBl-loopmHoop21-1oop+ '"3/2'"l-loop'~l-loop~'l-loop

On* ~*Z A* + 2 m .2 2* B* _L .2 ~.2 Z'Pl-loop ~" 1 -loop l-loop l-loop 1 -loop l-loop ~ m l-loop ~" l-loop ] ,

d i ~ . B* ~ ( h / 3 2 z c 2 ) t 4 , ~ . ~.2 B* + 4 ~ * 2 *2 A* l / ~ I I t t l-loop l-loop ] ~ I H~ l-loop ~'~ l-loop l-loop ct~ l-loop l-loop l-loop 1,

d * * 2 2 A * /~'3 l t - ~ (21-1oopAl-loop)=(h/3 rC )9 l-loop l-loop, (19)

which are the RGE for a one scale, softly broken, Wess-Zumino model. This result goes to show that for a two-scale softly-broken supersymmetric theory, given by eqs. ( 1 ) - ( 3 ), and

where the tree level parameters are constrained as in eqs. (4), (5), there exists a perturbatively stable, effective low energy theory given by eqs. ( 13 ) - ( 15 ). This theory is parametrized by a set o f effective couphng constants which may be written in terms of the original, full theory, as in eqs. (16); the low energy parameters have the property o f transforming among themselves under the renormalization group, and the low energy theory is therefore stable under changes o f scale.

It ~s important to realize that the soft-breaking character o f the high mass scales is not, in general, preserved when going to the low mass scales. This is so because removal of the heavy degrees of freedom induces terms of the form, eq. (10) which are [ 5 ] hard breaking. However, these terms disappear from the effective lagrangian when eq. (5) and the condition zu< m / M are imposed on the original parameters. This situation must be respected by any models where low energy supersymmetry is a desired feature o f the theory.

The form of the effective parameters computed to one-loop and given m eq. (16) shows several interesting W * t L * features. The supersymmetric parameters, i.e., the ones tn l-loop ~ l-loopS, are all given by the product of its

~5 A proof to higher orders in perturbation theory may be bmlt along the hnes presented in ref [ 10 ]

331


"hatted" counterpart, hmes what essentially amounts to a wave function renormahzat lon. Furthermore, they renormalize among themselves, that is, there is no percolation from the soft-breaking sector ~6. For the soft-

breaking parameters, m f 1'-loop (L T-loop ), we see that they contain the contr ibut ion from the corresponding "hatted" parameter plus a one-loop contrlbut~on which always depends on the soft-breaking sector of the theory.

The radiative structure of the low energy theory is then the one of a general, one-scale, softly-broken theory. The RGEs for the low energy theory are the ones in eq. (19) , and the expressions for the effective parameters

in eq. ( 16),are their solution. This solution satisfies the proper boundary condit ions at the large mass scale, /~ = cl. In par t lcularwe nottce that since as we have seen, each parameter can be written as in eq. (18) , then we are free to set the O3 to any value that we, or the physics of the large mass scales, want, and the only contr ibut ions generated by the changes of scale are given by the O (h) contr ibuUon to the parameters in eq. ( 16 ).

In conclusion, we have shown how the decoupllng theorem works for a general, two-scale softly-broken supersymmetric model in four &mensions. We have used a superfield method previously used by us to study this problem in supersymmetric theories. We have shown explicitly how decouphng takes place, and have provided

the expressions for the low energy parameters in terms of the parameters in the full theory. In particular, we have shown that for the class of models that are relevant from a phenomenological point of view, one can find a parametrlzatxon of the theory, where the low energy parameters may be written m terms of a set of parameters that remain invarmnt under changes of scale, plus a logarithmic scale correction which involves these ("hatted") parameters. Furthermore, we have checked by explicit computat ion, that the resulting low energy theory is radtat~vely self-consistent; in other words, the parameter set is closed under renormalization.

#6 Thls reminds one of a sxmdar property for the renormahzatxon group equatxons of and arbitrary and softly broken theory, and is a consequence of the non-renormahzatlon theorems. See, e.g., ref. [9].

References

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N.P Chang, A Das and J P6rez Mercader, Phys Lett B 93 (1980) 137, Phys. Rev. D 22 (1980) 1414, S Wemberg, Phys Lett. B91 (1980)51, P Bm6truy andT Schucker, Nucl Phys B 178 (1980) 293, W I. Welsberger, Phys. Rev. D 24 ( 1981 ) 481, Y. Kazama and Y P Yao, Phys Rev D 25 (1982) 1605, J B Galv~in, J P6rez Mercader and F. Sfinchez, Phys. Lett B 188 (1987) 347

[ 3 ] R.D.C Mdler and B.H J McJellar, Phys Rep 160 (1984) 169 [4] J. Le6n, J. P6rez Mercader and F Sfinchez, Phys Lett B 208 (1988) 463. [ 5 ] S J Gates Jr, M T Gnsaru, M Rof:ek and W Siegel, Superspace or 1001 lessons m supersymmetry (Benjamin-Cummings, Reading,

MA, 1983) [ 6 ] E g, M Jacob, ed, Supersymmetry and supergravlty (North-Holland/World SoentlfiC, Amsterdam/Singapore, 1986) [7] S Dlmopoulos and S Raby, Nucl. Plays B 219 (1983) 479,

M Dine and W Flschler, Nucl. Phys B 204 (1982) 346; J Polschmsky and L Susskmd, Phys. Rev D 26 (1982) 3661, see also, H P Nllles, M Sredmckl and D Wyler, Phys Lett B 120 (1983) 346

[8] W I Welsberger, Phys Rev D 24 ( 1981 ) 481 [9] B. Gato, J Le6n, J P6rez-Mercader and M. Qmr6s, Nucl Phys. B 253 (1985) 285.

[ 10] N.P Chang, A. Das, D X. L1, D C Xlan and X.J Zhou, Phys Rev. D 25 (1982) 1630, N P Chang and X Z Wu, Phys Rev D 25 (1982) 1425.

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The decoupling theorem in softly broken supersymmetric theories