invariance, Goldstone bosons, and gauge-fermion masses

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<ul><li><p>PHYSICAL REVIEW D VOLUME 27, NUMBER 11 1 JUNE 1983 </p><p>Supersymmetry at ordinary energies. 11. R invariance, Goldstone bosons, and gauge-fermion masses </p><p>Glennys R. Farrar Department of Physics and Astronomy, Rutgers University, New Brunswick, New Jersey 08903 </p><p>Steven Weinberg Department of Physics, University of Texas, Austin, Texas 78712 </p><p>(Received 18 October 1982) </p><p>We explore the observable consequences of supersymmetry, under the assumption that it is broken spontaneously at energies of order 300 GeV. Theories of this sort tend automati- cally to obey a global R symmetry, which presents us with a choice among phenomenologi- cally unacceptable alternatives. If the R symmetry is broken by scalar vacuum expectation values of order 300 GeV, there will be a semiweakly coupled light Goldstone boson, similar to an axion. If it is not broken by such vacuum expectation values but is broken by quantum-chromodynamic (QCD) anomalies, then there will be a light ninth pseudoscalar meson. If it is not broken by QCD anomalies, then the asymptotic freedom of QCD is lost at high energies, killing the hope of an eventual meeting of the electroweak and strong cou- plings within the regime of validity of perturbation theory. We also confront the problem of an uncomfortably light gluino. A general analysis of gaugino masses shows that the gluino mass is at most of order 1 GeV, and in many cases much less. </p><p>I. INTRODUCTION </p><p>This paper will continue the study of supersym- metry at ordinary energies that was begun in Ref. 1. </p><p>Our theoretical framework is as follows. We as- sume that supersymmetry survives down to energies of order 300 GeV, where, along with the electroweak gauge symmetry, it is spontaneously broken by vacu- um expectation values (VEV's) of weakly coupled scalar fields. Where relevant, we assume that these VEV's are also responsible for quark and lepton masses. In order to avoid light scalars2 and fast pro- ton decay,' we pursue the suggestion of Fayet that the gauge group at low energies should contain in addition to the usual SU(3)XSU(2) XU(1) at least an additional U(1) factor, called here 0(1) , with genera- tor F. However, most of our discussion would also apply in the alternative case3 where the gauge group is just SU(3) x SU(2) x U(1) with light scalars avoid- ed by having all quark, lepton, and associated scalar masses arise from radiative corrections, and we shall occasionally refer to SU(3) x SU(2) x U(1) theories. We try here to avoid basing our considerations on any specific menu of superfields, but we have in mind a model including the following left chiral superfields: </p><p>plus additional chiral superfields like the SU(3) x SU(2) x U( 1)-neutral X with F= + 4 of Ref. 2, which was introduced to allow a spontaneous violation of supersymmetry. </p><p>One severe problem with this class of theories is that they are beset with triangle anomalies in gauge currents.' (For instance, with just the ab_ove super- fields, there is a QCD anomaly in the Y current.) Furthermore, if enough new chiral scalar superfields </p><p>2732 @ 1983 The American Physical Society </p></li><li><p>27 - SUPERSYMMETRY AT ORDINARY ENERGIES. 11. . . . 2733 </p><p>are introduced to cancel these anomalies, one tends to find that the scalar VEV's either do not break su- persymmetry or do break charge or color gauge in- variance^.^ We will not deal with this problem here, but will simply assume that some set of chiral scalar superfields has been found which allow a realistic </p><p>of VEV's while at the same time canceling all anomalies in gauge currents. The considerations resented here will not be sensitive to the details of how this is done. </p><p>Our chief concern in this paper is with another problem: the phenomenological implications of a global symmetry known as R invariance. This sym- metry is not put into these theories by hand, but is automatic in a wide class of supersymmetric models, including all _those containing a gauge quantum number like Y whose values for left chiral super- fields are restricted to 1 (mod 3). We do not know if this symmetry is broken by scalar VEV's of order 300 GeV, or by QCD anomalies, or by QCD con- densates, or by some of these mechanisms, or by none of them, so we explore all of these alternatives. Our conclusion, summarized in the last section, is that each one of these alternatives leads to a severe conflict with experiment or with current theoretical ideas. </p><p>R invariance is introduced in its several forms in Sec. 11. The possible mechanisms for breaking R in- variance are outlined in Sec. 111. Then in Sec. IV the properties of the Goldstone bosons associated with each of these mechanisms are considered. Sec- tion V deals with the masses of the gauge fermions, in the light of the previous analysis of R invariance. </p><p>11. R INVARIANCE </p><p>A large class of renormalizable supersymmetric theories automatically have a global symmetry of the type called R invariance. By an R invariance is meant any global U(1) symmetry which acts non- trivially on the superfield coordinate 8, and there- fore acts differently on the spinor and the scalar or vector components of a superfield. R symmetries were introduced by Salam and strathdee5 and by Fayet6 as a means of imposing lepton conservation in supersymmetric models, and also in order to con- strain the Lagrangian to rule out the possibility that the scalar fields could have vacuum expectation values which would leave supersymmetry unbroken. The discussion here will differ in that R symmetry is not imposed on the theory, but is found to arise in the theories that interest us whether we like it or not. </p><p>As one example of a large class of theories which automatically have an R symmetry, consider those renormalizable supersymmetric theories which are </p><p>prohibited by gauge symmetries from including any sort of super-renormalizable linear or bilinear F terms. [For instance, this is the case if there is a U(1) gauge symmetry like that discussed by Fayet2 and in I, for which the left chiral scalar superfields carry only the quantum numbers 1, -2,4, - 5,7,. . . .I The Lagrangian of any such theory will contain only the kinetic terms and gauge couplings of the chiral superfields S (x, 0) [contained in the D terms ( S * ~ " S ) ~ ] , plus the Yang-Mills F terms ( WWIF ( W is defined below in terms of V 1, and possible ~a~et-11iopoulos7 D terms ( V )D involv- ing the gauge vector superfields V(x,8) alone, plus trilinear F terms ( s 3 ) F and (s3);. Any Lagrangian constructed from such ingredients will automatically be invariant under a global U(1) transformation whose g5neratyr Ro has the values + 1 ( - 1) for OL (OR), + for all left (right) chiral superfields S (S*), and 0 for all gauge vector superfields V. </p><p>To see this, note that the D term and F term of any function of superfields are the coefficients of t3L28R and OL 2, respectively, so if we arbitrarily as- sign the value Ro= + 1 to OL (and hence Ro = - 1 to OR cc 82), then the D terms and F terms of any func- tion have Ro values equal to those of the function and the function minus 2, respectively. The func- tions S*e 'S and V obviously have Ro =0, so their D terms conserve Ro. The left chiral spinor superfield W which contains the Yang-Mills curl is given schematically by </p><p>so it has Ro=+l; w2 has Ro=+2; and so its F term con;erves Ro. Finally the function S' has Ro = 3 X 7 = 2, so again its F term conserves R 0 . </p><p>An R symmetry sometimes arises also in theories that do contain super-renormalizable F terms. For instance, if there are just two kinds of left chiral sca- lar superfields S+ which carry values k 1 for some U(1) gauge quantum number, then the only allowed renormalizable term [ f (S)IF is (S+S- IF, and the Lagrangian is then invariant under an R symmetry for which S+ both carry the R values R = 1. Where not otherwise indicated the discussion here will be restricted to theories without super- renormalizable couplings, i; which all left chiral scalar superfields have Ro = T , but much of this dis- cussion would also apply in more general cir- cumstances. </p><p>The scalar and spinor component fields Y and SL of a left chiral scalar superfield S are the coefficients of 1 and OL in the expansion of S ( x , 8 ) , while the spinor and vector component fields hL and V p of a </p></li><li><p>2734 GLENNYS R. FARRAR AND STEVEN WEINBERG </p><p>real gauge superfield V are the coefficients of and in the expansion of V(x,O). Hence these component fields have the Ro values: </p><p>left chiral scalars Y: R, = f -o= f , 2 1 </p><p>left chiral spinors st : R, = 7 - 1 = - - 3 ' (1) </p><p>left gauge spinors hL : R = 0 + 1 = 1 , vector gauge fields V p : Ro = O + O = O </p><p>Any such R symmetry is surely bryken by the vacuum expectation values of the Ro = 7 Higgs sca- lars which break SU(2) XU( 1) and give masses to the quarks and leptons. However, it is sometimes possi- ble to combine this broken global R symmetry with a suitable broken gauge symmetry to obtain an un- broken global R symmetry. (We do not consider the possibility of combining Ro with a broken global symmetry to obtain an unbroken R symmetry, be- cause this would lead to consequences similar to those of breaking R-specifically, a semiweakly coupled Goldstone boson.) The neutral Higgs sca- lars whose vaccum expectation values give masses to the quarks of charge f and - (and charged lep- tons) belong to left chiral superfields with opposite values for electroweak hypercharge and zero values for charge and color, so there is no way that the Ro symmetry defined above could be combined with SU(3) X SU(2) XU( 1) generators to yield an unbroken symmetry. On the other hand, suppose there is an_ additional U(1) gauge symmetry whose generator Y has equal values for the Higgs superfields (as in the models of ~ a ~ e t ~ and Secs. IV-VI of I). To keep the same notation as in I, let us take this value as F= -2. Then we can define a new global symme- try </p><p>which has the value zero for the Higgs scalars, and is therefore not broken by their vacuum expectation values. Even so it is still an open question whether k conservation is broken by other vacuum expecta- tion values, or by Adler-Bell-Jackiw (ABJ) anomalies, or by dynamical effects of the strong in- teractions, or by suppressed nonrenormalizable terms in an effective Lagrangian resulting from an R-noninvariant theory at a higher energy scale. All these possibilities will be considered in following sections. </p><p>Using the values for Ro given above and taking k = 0 and -2 for gauge and Higgs superfields, the </p><p>values of the component fields are as follows: </p><p>Higgs scalars: k = 0 , </p><p>left-handed Higgs spinors: k = - 1 , left-handed gauge spinors: E = + 1 , (3) - gauge vector bosons: R =O . </p><p>If we suppose for simplicity that all left chiral quark and lepton superfields have equal values, then in order for them to couple in pairs to the Higgs super- fields they would have to have Y = +_I, and their component fields would thus have the R values </p><p>quarks and leptons: k = 0 , (4) </p><p>left chiral quark and lepton scalars: R = + 1 </p><p>Finally, in order to find a suitable supersymmetry- breaking solution it has generally been found neces- sary to introduce left chiral X superfields which can couple to pairs of Higgs superfields and hence have ?= + 4 (see Ref. 2 and App5ndix B of I). Their component fields would have R values </p><p>left chiral X scalars: = + 2 , ( 5 ) left chiral X spinors: k= + 1 . </p><p>It is convenient that the known particles of low mass, including- quarks, lepLons, gluons, and photons, all have R =O. Hence R invariance if unbroken would rule out interactions in which exot- ic particles with Rf 0 such as quark or lepton sca- lars or Higgs or gauge spinors or X scalars or spi- nors (including the Goldstone fermion) are produced singly in collis~ons of known low-mass particles.8 An unbroken R invariance would also severely re- strict the mass matrices of these exotic particles, and prohibit their mixing with known low-mass parti- cles. We will return to these masses in Sec. V but first it is nec_essary to study the mechanisms that might break R invariance. </p><p>111. MECHANISMS FOR a BREAKING We can distinguish five different mechanisms </p><p>which can either individually or jointly break R in- variance in supersymmetric theories. </p><p>A. Intrinsic R o breaking </p><p>As explained in the previous section, Ro invari- ance can only be broken in a renormalizable super- symmetric Lagrangian by super-renormalizable P terms of the form (S1SZlF or (S3IF, where Si are generic left chiral scalar superfields. These are not allowed if there is a U(1) symmetry whose generator </p></li><li><p>27 - SUPERSYMMETRY AT ORDINARY ENERGIES. 11. . . . 2735 </p><p>F has values equal to 1 (mod 3) (e.g., + 1, - 2, +4) for all left chiral superfields, as in the models of ~ a ~ e t ~ and Secs. IV-VI of I. Bilinear F terms would be allowed if there were also left chiral super- fields which belong to the complex conjugates of some of the representations of SU(3) x SU(2) x U( 1) XU( 1) furnished by quark, lepton, Higgs, etc., superfields; in particular these would have F= - 1 (mod 3). This would have the advantage of making it easy to cancel ABJ anomalies, and the disadvan- tage of making it easy to find sets of scalar vacuum expectation values which leave supersymmetry un- broken. </p><p>Alternatively it is possible to add left chiral super- fields which belong _to real representations of SU(3) x SU(2) x U( 1) x U(1), and therefore cannot have renormalizable F-term interactions with "known" superfields, but which can have both bilin- ear and trilinear F-term interactions with each other. One possible addition of this sort is a superfield So that is neutral under all gauge groups. This could have (so3)F, (sO2)F, and (So), interactions, which would break Ro, but since other superfields would have no interaction whatever with So, their own R o would still be conserved. A more interesting possi- bility is to add superfields which have no F-term in- teractions with known superfields, but belong to nontrivial real representations of gauge groups. R violation in the new F-term sector could then induce transitions of left gauge fermions (Ro= + 1) into their antiparticles, breaking R o for all particles that feel these gauge forces. For instance, a color octet SU(2) x U( 1) x g( 1)-neutral chiral superfield E could have ( E ~ ) ~ and ( E ~ ) ~ interactions, leading through radiative corrections to a Majorana mass term for the gluino,9 which violates R o. </p><p>Of course, if the gauge group at ordinary energies were just SU(3) x SU(2) XU( 1) then it would be easy to break Ro by including an interaction of the form (HH')F, which generates a Majorana Higgs-fermion mass. However if the only left chiral superfields in additon to H and H ' were quark superfields Q and lepton superfields L, with F-term interactions schematically of the form (HH1)F, (QQH (or and (LLHIF, then the Lagrangian would automati- cally be invariant under an R symmetry for which H and H ' carry the ~a lues R = 1 while Q and L carry...</p></li></ul>

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