Volume 121B, number 2,3
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27 January 1983
PROTON DECAY IN MODELS WITH INTERMEDIATE SCALE SUPERSYMMETRY BREAKING N. SAKAI National Laboratory for High Energy Physics, Tsukuba, Ibaraki 305, Japan Received 19 October 1982 A general operator-analysis is performed for proton decay in supersymmetric models with intermediate scale supersymmerry breaking, It is found that induced AB @0 four-scalar interactions can be as important as the dominant supersymmetric operators.
Supersymmetry seems to offer the most promising possibility to solve the gauge hierarchy problem of grand unification [ 1 - 3 ] . In initial attempts to construct realistic models, the mass scale of supersymmetry breaking was assumed to be not very much larger than the weak boson mass M w [2]. Recently, however, several groups have found a class of models with supersymmetry breaking at a mass scale/a intermediate between M w and the grand unification mass M [4,5]. If the supersymmetry breaking is transmitted to low-energy particles (quarks, leptons and Higgs doublets) only through superheavy (~M) particles, the effect is suppressed by inverse powers of M (decoupling). In this case the electroweak mass scale M w can be identified with the effective mass scale of supersymmetry breaking among the low-energy particles
M w ~-tt21M.
(I)
This relation is called geometric hierarchy [5]. On the other hand proton decay is the most spectacular and important experimental prediction of grand unification. Recently a AB 4:0 four-scalar interaction due to the exchange of a superheavy vector multiplet was found and its importance in the case of intermediate scale supersymmetry breaking was stressed [6]. Previously an operator analysis of proton decay in supersymmetric models was performed assuming the mass scale of supersymmetry breaking to be of order M w [7,8]. It was found that the dominant operators are of dimension five (six) if Majorana masses for gauge fermions are significant (negligible) [9]. The purpose 130
of this paper is to perform a systematic operator analysis of proton decay in models with intermediate scale supersymmetry breaking such as that of the geometric hierarchy model. We shall show that new four-scalar AB 4:0 interactions are generally induced and can give contributions to proton decay comparable to the dominant operators up to model-dependent numerical factors. To discuss rare processes such as proton decay, one needs to take two steps: (i) Count all the possible local operators consistent with the symmetries of the low-energy effective theory. (ii) Take into account perturbations in the renormalizable interactions of the low-energy effective theory in order to obtain the relevant amplitude for the process (e.g. four-fermion interactions among quarks and a lepton for proton decay). Let us consider the first step. As low-energy particles in the minimal supersymmetric SU(3) X SU(2) × U(1) effective theory we assume chiral scalar supermultiplets of quarks, leptons and two Higgs doublets as in table 1 besides vector supermultiplets for SU(3), SU(2) and U(I). The possible existence of a righthanded neutrino supermultiplet is relevant for leptonnumber nonconservation, but does not significantly affect our considerations on proton decay. In addition we introduce chiral scalar superfields X± which contain a Nambu-Goldstone fermion and a heavy scalar particle. Supersymmetry is broken by the vacuum expectation value of the F-component of X+ (F x) = U2
(2)
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
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PHYSICS LETTERS
Table 1 Quantum number of supermultiplets in the low-energyeffective theory. Suffix _+indicates chirality. SU(3)
SU(2)
Y
matter parity
q_ u+ d+
3 3 3
2 I 1
-1/6 -2•3 1/3
-
~_ e+ ~_ ~ X+_
1
2
1/2
l
l
1
1 1 l
2 2 1
1/2 1/2 0
27 January 1983
8] except for two points: There is an additional suppression factor p2/M2, and the quarks and the lepton need not have the same chirality. (a) Operators with quark and lepton superfields of mixed chirality. M -3 [d÷ ~ u+ ~q_ i71~_/X+] Dec~,~ei] T -1 ~_]eafi,,/ei], = ( p 2/ g 3)Ad+aAu+f~q_i3,C
+ + +
M
-2
a b [q_i~q_/.#u+Te+X+]De,~#.rei/
aT C-1 ~q_/~Au+q, b = ~ 2'M / 3)~-ia Ae+eaoTeil
at the intermediate mass scale # given in eq. (1) [10]. Hence the X+ have to be singlet under SU(3) X SU(2) × U(I). In order to avoid proton decay through dimension-four supersymmetric operators, we impose invariance under sign change of quark and lepton superfields (matter parity) as in the naive supersymmetric SU(5) model [2]. Under these conditions AB 4= 0 supersymmetric operators with the singlet X+ have dimension six or higher. Taking SU(3) × SU(2) × U(1) quantum numbers into account we obtain three operators of dimension six: M_2 [qa_io,qb/~qC k.~ - n X_ ] Fec~.rei/ekn
a b c = Qa2~'M2)Aq._iaAq_j~Aq_k3tA~_neaf3.rei]ekn, (3)
M -4 [d+ c~u+~1_ iv £- ]X+X_] DeaoTeii = (1.14)1,/4 / )hd+ahu+~Aq_i~;a~_/eoloTei/,
M-2 [t~+au+~d+Te+ b a X+]Fec~,~ b c = 2/ ~4"2 a
M_4 [qa__i c~qb_/ou+7e+X+X_] oe~#Tei/
where suffixes (i,] .... ), (a, fl, ...) and (a,b .... ) correspond to SU(2), SU(3) and generation indices respectively, the scalar component of the q_ superfield is denoted by Aq_, and the F-component projection by [".] F. In terms of component fields we wrote only terms proportional to (F x) = ~t2 in eqs. (3)-(5). Let us note that the operator (3) is symmetric in the generation indices a and b, while the operators (4)'and (5) are antisymmetric. We see that the new AB 4= 0 operators contain interactions among four scalars with the same chirality and are suppressed by p2/M2
~-Mw/M. Dimension-seven operators linear in X are very similar to the dimension-five operators without X [7,
(7)
where [...] D means the projection of the D component and C is the charge conjugation matrix. The operator (7) is symmetric in generation indices a and b. There are also similar operators with X_ instead of X+. (b) Operators with quark and lepton superfields of the same chirality can be obtained from the dimension. five operators without X by adding X of opposite chirality and by taking the D component instead of the F component. Other operators of dimension seven contain heavy particles. So far all operators contain scalar quarks and/or lepton with the same chirality. We can find those with the opposite chirality among dimension-eight operators
M-2[qai~qb_f~qck.r~_nX_]Fe~(~e)ij(.te)kn (4) = t,,2/M2~.~a Ab a c A v-, r )~q_ ia,~ q_/'tF~q_ kT'al2 _ nec,#-t(te)i/(¢e)kn, (5)
(6)
a b = (l,t4M4 / )Aq_oc4q_j~Au+~rAe+ec~o.reij.
(8)
(9)
The operator (9) is symmetric in generation indices a and b. On the other hand dimension-eight operators with scalar quarks and a lepton of the same chirality can be obtained from the dimension-six operators (3)-(5) by adding X of opposite chirality and taking the D instead o f F component. They are identical to the operators (3)-(5) except for an additional suppression factor la2/M 2. There are also three operators of dimension eight which are linear in X, for instance,
M-4[DD(d+au+~)DD(u+~,e+ )X+] FeaO,t T -1 g'u+O~u+'r T C-I ~be+ea#~" = 0a 2/ ~ )¢'d+~C
(10) 131
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Contrary to the dimension-six operators without X [7,8], these four-fermion interactions are among fermions of the same chirality and have an additional suppression factor la2/M 2. There is a dimension-eight AB ~: 0 operator with six matter fields which contributes to neutron--antineutron oscillations [ 11 ] M - a [ d a a~+#~+-~ ,4b ,'lc d d+6 ue+e t~.+~"X +iF" 1
(ll)
- t,.2/M4"~Aa Ab Ac Ad A e ,if e e -v a t .~d+~-'d+#~d+y'ad+5 u+e"~u+~" ~#7 6e~'" Other dimension-eight operators contain either Higgs doublets or other heavy particles. As the second step, we shall now discuss how these AB @ 0 operators contribute to proton decay, assuming that protons do not contain scalar quarks. The four-scalar interactions ( 3 ) - ( 5 ) can be converted into four-fermion interactions through the two-loop diagram of fig. 1. The exchanged fermions can either be those associated with gauge bosons or Higgs doublets. Since the coupling constants of Higgs doublets to quarks and leptons are very small, we shall neglect the Higgs fermion contribution. On the other hand the loop diagram involving gauge fermions vanishes unless the gauge fermion has the Majorana mass which breaks supersymmetry [9]. In general one expects both the Majorana mass and the masses of the scalar quarks and leptons to be of orderM w ~-la2/M, the effective scale of supersymmetry breaking. In that case, by counting only the powers of the mass paramters we find that the two-loop diagrams with the AB 4~ 0 four-scalar interactions ( 3 ) - ( 5 ) give effective four-fermion interactions with the strength
(]a2/M2)/M 2 ~ 1/~u2 .
(12)
This is of the same order as the strength of the effective four-fermion interaction due to the one-loop diagram containing the dimension-five operator with-
71
,>', /
+0
Fig. 1. Two-loop diagram containing ~ ~ 0 four-scalar operator and gauge fermions hw. 132
27 January 1983
out X [ 7 - 9 ] , ( I / M ) ( 1 / M w ) ~ - 1/~ 2. Hence the fourscalar interactions (3)--(5) can contribute to proton decay with a magnitude comparable to that of the dimension-five operator if the Majorana mass for gauge fermions is of orderM w. In practice there are various other model-dependent numerical factors. In particular the loop integration tends to give suppression factors like 1/16rr 2 and disfavors the four-scalar interactions. One should also note that the four-scalar interaction (5) among right-hand particles does not contribute to proton decay since it has to contain at least one charm or top scalar quark which can not be converted to an up quark through the gauge fermion exchange. One can show that there are no other operators which give proton decay of the same order as (12). It has recently been pointed out that the Majorana masses for gauge fermions are absent or negligible in many models [12,6]. In that case the usual dimension-five operator as well as the new four-scalar interactions (3)---(5) may not dominate proton decay. Instead, the dimension-six operators without X may be more important since they contain four-fermion interactions of the strength 1/M 2. Even in such a circumstance, the four-scalar interactions (8) and (9) with mixed chirality can contribute to proton decay since the corresponding two-loop diagrams do not require the Majorana mass for gauge fermions. In fact the effective four-fermion interaction due to the two-loop diagram has a strength comparable to that of the dimension-six operator, i.e. Oa/M)4/M 2 ~- 1/M 2 in the geometric hierarchy picture M w ~- t2~/M. Actually to assess the proton decay rate in models with small or negligible Majorana mass for gauge fermions, more detailed model-dependent analysis is necessary. Namely suppressed contributions from the dimension-five operators without X may be as important as the dimension-six operators without X. Two examples are: loop diagrams with a Higgs fermion, and tree diagrams where virtual scalars are converted into fermions and photinos L t (e.g. qq ~ q~ L~ Lr). In table 2 we schematically summarize operators of various scalar-spinor composition and chirality: coefficients of these operators are listed according to their effective strength of the four-fermion interaction for proton decay. Up to now we have performed an operator analysis of proton decay as model-independently as possible. At this point it is instructive to see how these new
Volume 121B, number 2,3
PHYSICS LETTERS
Table 2 Coefficients of ~tB ~: 0 operators. The leading contributions to proton decay amplitude come from A±A+_qJ+_~k+and A+_A+_A±A±provided the Majorana mass for gauge fermions are of order M w. Operators
Operator coefficient corresponding to the effective strength of four-fermion interactions of order 1 lu 2
¢,+~0+qJ _ 4 _
A±A:t¢,+_~O+_ A±A±A+_A+_ A+A+A A
1 IM2 1/M 2 (refs. [7,8] )
1/M (refs. [7,8]) /~2/M2[eqs. (3), (5) 1
u2/Ma
i.t2/M3 [eqs. (6), (7)1
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,o\ tO/H=
,o\
xI 0
[Q"
xt
~=
"H
I0
/,o.
[b]
,o\. xt. _lx. /,o.
u4/M 4 [eqs. (8), (9)] [¢]
operators in fact emerge in an illustrative model. The most interesting four-scalar interactions ( 3 ) - ( 5 ) are of F-type and have to arise from tree diagrams consisting o f F - t y p e vertices only. Moreover the singlet field X has to couple directly to the baryon-number violating superheavy Higgs field. In the naive supersymmetric SU(5) model [2,3] such a coupling is unacceptable, since it induces supersymmetry breaking of o r d e r / / t o the Higgs doublet besides the superheavy Higgs triplet, both of which are contained in H, the 5 representation. This problem is closely related to the fact that in the naive model fine tuning is still necessary at the tree level to m a k e t h e Higgs doublet light. Therefore, we shall consider models in which the Higgs doublet is naturally massless because of group-theoretical reasons [13]. In this scheme SU(5) is broken to SU(3) X SU(2) X U(1) by the vacuum expectation value of 75 and H acquires mass only through the coupling with 75 and 50 (denoted by 0), namely H .(75). 0. Since 50 has (3, l ) b u t no (1,2) of SU(3) X SU(9), the Higgs doublet is massless whereas the color triplet Higgs becomes superheavy. Suppose that the singlet X field couples to 50 and 50* (denoted by 0 ) and that its F-component develops a vacuum expectation value of order//2 (it is easy to construct an O'Raifeartaigh type model which accomplishes this). Then the superheavy Higgs triplet is affected by the intermediate scale supersymmetry breaking, whereas the Higgs doublet is left untouched. In this model many interesting operators appear already at the tree level. It is easy to see that the four-scalar interactions ( 3 ) - ( 5 ) of the same chirality are contained in the supergraph with chirality-flip superpropagators in fig.
Fig. 2. Supergraphs which reduces to no~enormalizable ~LB¢ 0 operators in the large mass limit of 0 and H. Arrows denote chkality. Cross stands for the vacuum expectation value of the F-component. Quarks and leptons are in 10 and 5* representations. (a) For operators (3)-(5). (b) For operators (6) and (7). (c) For operators (8) and (9). 2a, where an arrow and a cross denote the chirality and the vacuum expectation value o f F x respectively. Operators (6) and (7) with two scalars and two spinors of mixed chirality arise from a similar supergraph with only one chirality flip such as fig. 2b. Another supergraph without chirality-flips gives the four-fermion operator (10). To obtain the four-scalar interactions (8) and (9) of mixed chirality, we should consider a supergraph with X inserted twice as in fig. 2c. The four-scalar interaction found by Derendinger and Savoy [6] is due to the exchange of the superheavy vector multiplet, and can be identified in our classification with one of the dimension-eight operators in eqs. (8) or (9). We acknowledge the hospitality of the CERN Theory Division where most of this work was performed. We would also like to thank Dr. Pietro Rotelli and Dr. Tsuneo Uematsu for useful suggestions improving the manuscript. We are very grateful to Dr. Derendinger and Dr. Savoy for communicating thei(results privately and as a preprint at CERN and for a kind correspondence.
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References [1] E. Witten, Nucl. Phys. B188 (1981) 513. [2] N. Sakai, Z. Phys. C l l (1981) 153; S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150. [3] S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353; M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189 (1981) 575. [4] L. Alvarez-Gaum6, M. Claudson and M.B. Wise, Harvard preprint HUTP-81/AO63 ; C.R. Nappi and B.A. Ovrut, IAS preprint; M. Dine and W. Fischler, IAS preprint ; J. Ellis, L. lb~n'ez and G.G. Ross, Phys. Lett. 113 (1982) 283. [5] M. Dine and W. Fischler, IAS preprint A supersymmetric GUT;
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S. Dimpoulos and S. Raby, LA-UR-82-1282; T. Banks and V. Kaplunovsky, TAUP 1028-82. J.P. Derendinger and C.A. Savoy, Phys. Lett. 118B (1982) 347;and private communications. N. Sakai and T. Yanagida, Nucl. Phys. B197 (1982) 533. S. Weinberg, Phys. Rev. D26 (1982) 287. J. Ellis, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B202 (1982) 43; S. Dimopoulos, S. Raby and F. Wilczek, UMHE 81-64. J. Polchinski and L. Susskind, SLAC-Pub.-2924. G. Costa, F. Feruglio and F. Zwirner, IFPD 15/82; Y. Fujimoto and Z. Zhiyont, IC/82/60. J. Polchinski, SLAC-Pub.-2931-T; A. Masiero et al., Phys. Lett. 115B (1982) 380; B. Grinstein, HUTP-82/AO14; S. Dimopoulos and tl. Georgi, Phys. Lett. 117B (1982) 287; S. Dimopoulos and F. Wilczek, Santa Barbara preprint.