PHYSICAL REVIEW D VOLUME 36, NUMBER 11 1 DECEMBER 1987

Fourth generation and nucleon decay in supersymmetric theories

R. ArnowittCenter for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843

Pran NathLyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138

and Department of Physics, Northeastern University, Boston, Massachusetts 02115(Received 12 June 1987)

Analysis of nucleon decay in N =1 supergravity unified models including the effect of a fourthgeneration of matter is given. Experimental constraints from nucleon-lifetime limits on theKobayashi-Maskawa (KM j matrix that enters nucleon decay are obtained. The decays

KL ~p+p and K + ~~+ vv are analyzed under these constraints, since the combination of theKM matrix that enters nucleon decay also enters these rare decays. The branching ratioK ~~+vv in four generations is shown to be considerably larger than for the three-generationcase except for certain narrow domains of the KM matrix for two of the four branches of solu-

tions. Bounds on V» and V„~ are also obtained.

I. INTRODUCTION

M =(1.0+0.6) X 10' GeV, (1.2)

and thus Eq. (1.1) puts a significant upper bound on theproton lifetime, which, when combined with the experi-mental lower bounds, ' eliminates certain supergravitymodels. ' '

Since supergravity nucleon decay proceeds throughHiggsino interactions, the decay amplitudes depend ex-plicitly on Kobayashi-Maskawa (KM) matrix elementsV; . Thus the decay rates are sensitive to the values ofV,- and the number of generations. Most significant isthe fact that the same combinations of KM matrix ele-ments also appear in the rare K-meson decay modesKL ~pp and K+~~+vv. Thus in supergravity modelsit is possible to correlate the proton lifetime with therare K decay rates, and in this fashion probe for the ex-

Proton decay provides a strong experimental test forany grand unified theory (GUT). Thus the current ex-perimental bounds' on the decay p~e+w clearly ruleout the minimal SU(5) GUT model. A great deal ofwork exists in the literature on nucleon decay in super-gravity unified models. ' In supergravity models, "proton decay proceeds through the exchange of the su-perheavy Higgsino triplet. Since the Higgsino mass Mzis governed by physics at the GUT scale, it is not deter-mined theoretically. The absolute decay rates thus arenot predicted, though branching ratios into the variousmodes are. However, as pointed out by Enqvist,Masiero, and Nanopoulos, GUT models which preservethe gauge hierarchy generally require

M~ SMG

(and often MH is considerably less than the GUT massMG). For the standard SU(5) supergravity model withtwo Higgs doublets one has

istence of a fourth generation of quarks and leptons. Wewill see below that for the standard SU(5) supergravitymodels, the existing data are consistent with the ex-istence of a fourth generation, but that strong restric-tions can be placed on various KM matrix elements,which in fact eliminate some conjectured four-generationKM matrices. Furthermore, the decay rate forK+~~+vv is predicted to be generally larger for fourgenerations than for three generations, a result that isexperimentally testable. ' Thus when proton decay iscombined with the rare K decays, supergravity modelsmake experimental predictions which allow one to dis-tinguish the number of generations which have lightneutrinos.

In Sec. II we review the supergravity proton-decay re-sults for two and three generations. Section III then ex-tends this analysis to four generations and examines thecorrelations with the rare K decay modes. Section IVdiscusses the constraints proton decay imposes on theKM matrix elements V„b and V„&. Section V gives asummary of the results and conclusions. The Appendixlists the main proton-decay formulas.

II. PROTON-DECAY AMPLITUDES

The effective dimension-six nucleon-decay amplitudesin supergravity GUT s arise from Higgsino-triplet ex-change followed by gaugino "dressing. " The charac-teristic diagrams for 8'-ino dressing are shown in Fig. 1

where KM factors arise at each vertex. The full resultinvolves, in addition, gluino and Z-ino dressing as wellas RRRR dimension-five operators ' and is quite com-plicated. It is given, generalized for an arbitrary numberof generations, in the Appendix for the decay modesX~vK and N ~v~. We restrict our discussion in thispaper to the supergravity models with large D terms[e.g. , renormalization-group (RG) models] where theHiggs mixing angle a& is small, i e., oH = 10'—25 '.

36 3423 1987 The American Physical Society

3424 R. ARNOWITT AND PRAN NATH 36

Xo(N~v, K) =(a~ /MH )(m; V;„)(2Mii sin2aH )

X g P, A,'m.,"F,, (aL+P~)+ b, ,

1 =2

(2.1)

where V," are the KM matrix elements (with phases sothat V,d ——real),

A;=VdV;,', (2.2)

F, is the form factor resulting from the 8'-ino triangleloop integral of Fig. 1, m,-" and m," are the u- and s-quark masses, P, is the additional (PC-violating) phasesof nucleon decay, a, and P, are the LLLL four-fieldquark-lepton interactions, and 6; are additional, gen-erally small, contributions from other gaugino clothings.F,-, a, , and 6, are given in the Appendix.

The KM factors governing the N~v, K mode are V„,and the combination A, of Eq. (2.2). The A, also enterinto the rare K decay modes. Thus for the branching ra-tio of K+~vr+vv one has'

2B(K+~~+vv) =1.5N, X 10 g A;D(x; ) (2.3)

where N, is the number of generations (i.e., light neutri-nos) x; =(m;"/Mii, ) and

(Analyses may also be carried out for the case aH -45'. )

In this case, if the gravitino mass m3/2 is not too small,i.e., m3/2 150 GeV, the 8'-ino dressing dominates, andthe eA'ective dimension-six Lagrangian for N ~v, K(where v; is the ith-generation neutrino) reduces to

CreV. Hence models where the second generation dom-inates are excluded for squarks which are in the massrange where they would be detectable at the FermilabTevatron and/or the Superconducting Super Collider.

When three generations are considered, the possibilityof suppressing the v, K modes arises via an approximatecancellation between the second and third generations. '

From Eq. (2.1) this can occur if

32m, PzF,, + 33m, P3F,, =0 . (2.8)

The form factors F, are approximately independent ofthe generation index i and so the suppression occursuniversally for all modes v, K. As discussed in Refs. 7and 8, Eq. (2.8) can be satisfied for m, ~ 50 GeV andprovided the PC-violating phase 5 is =180', P3/P2 isapproximately real and Az, the Polonyi constant, is notsmall, i.e., Az = 1 (Ref. 17). Simultaneously, the v, vr and

v;p modes are enhanced, making them comparable to orlarger than the v, K modes. To satisfy Eq. (1.1), the can-cellation in Eq. (2.8) need not be precise, i.e. ,

~A2m, P2F,, +A3m, P3F,, ~

&0.2~

Azm, PzF, , (2.9)

Thus one finds Eq. (1.1) is obeyed for a wide range of pa-rameters. ' The three-generation supergravity models,then, are consistent with existing proton-decay data, andmake interesting predictions which could be tested bythe Kamiokande Collaboration with their planned"Super Kamiokande" detector.

As discussed in Ref. 18, existing data plus three-generation unitarity of the KM matrix puts an upperbound on 3 3. We find

1 3 x 3 xC(x) = —x —— +—4 4 x —1 4 x —1

lnx .

In addition, the 2, obey the unitarity relation

QA, =O.

D(x)= —x+ — +— lnx .1 3 x 3 (x —2)x4 4x —1 4 (x —1)2

Also, from the bound on KL ~pp, one has' ''

Q(KL )= Re+A;C(x, ) ~2X10

where2

(2.4)

(2.5)

(2.6)

(2.7)

(2.10)

Since the dominant contribution to Q(KI ) comes fromthe third generation, the K~pP constraint (2.5) alsoputs a bound on A3. This bound dominates Eq. (2. 10)when m, ~ 130 GeV. Bounds on 3 3 then produce upperbounds on the K+~vr+vv rate from Eq. (2.3). As canbe seen in Table I, the three-generation B(K+~rr+ vv)reaches a maximum of about 8 )& 10 ' for m, = 140GeV, and is less at higher and lower values of m, . Theseresults will help distinguish three- and four-generationmodels.

From Eqs. (2.1), (2.3), and (2.5), one sees it is possible torelate proton decay and the rare K decay modes.

The proton-decay rates depend sensitively on the num-ber of generations. Thus if one considers only the firsttwo generations, one finds that N ~vK is the dominantmode. ' From the experimental upper bound on thesemodes' and Eq. (2.1), one can obtain a lower bound onMH (Refs. 7 and 8). One finds, for a squark massm =180 GeV, MH ~(10—70)X10' seriously violatingthe theoretical constraint of Eq. (1.1). (A similar resultoccurs for no-scale models. } One may, of course,reduce the decay rate by increasing the squark mass,which enters in the triangle loop of Fig. 1. However, theinconsistency remains for squarks with mass m 5350

q

m,

5060

100120130140150200

0.001 510,001 510.001 510.001 510.001 510.001 410.001 270.000 81

B(K+~~+vv), „1.67 x 102.18x 104.89 x 10-"6.66x 10-"7.66x 107.81x 10-"7.43 x 106.» x10-"

(m Ipi )-.199199199199199213237371

TABLE I. Bounds on A„B(K+~m+vv), and p, in three-generation models, as a function of the t-quark mass m, . (Allenergies are in GeV. )

36 FOURTH GENERATION AND NUCLEON DECAY IN. . . 3425

A ] + A 2:0.0246+0.03 1 5 (3.5)

MH- MGuT

In analyzing this case we will make use of the UA1 ex-perimental lower bounds on the t- and t'-quark massesand the ~'-lepton mass

m„mr + 40 GeV,

m, ~41 GeV .

(3.6a)

(3.6b)

U

W

Furthermore, in the renorrnalization-group analysis ofthe supergravity models, the requirement thatSU(2) XU(1) breaking correctly occurs at the W massscale gives upper bounds on the t', 6', and ~' masses:

U

FIG. 1. Diagrams leading to proton decay with 8'-ino (W)dressing. u and d are the u and d squarks.

m, &140 GeV, m& &135 GeV,

m, &70 GeV .

(3.7a)

(3.7b)

P3 F,,P P F3 2 jc

(2.11)

Since~

A~~

=0.20, Eq. (2.11) yields a lower bound on

~ p3 ~, which is also shown in Table I. Of course in su-pergravity models, the value of p3 is determined dynami-cally by the loop integrals of Fig. 1, and in general thelower bounds of Table I can be satisfied. Note that onlythe last column for m, p3 of Table I depends on theSUSY model, and the other columns hold equally wellfor the three-generation standard model.

III. FOUR-GENERATION MODEL

For four generations, the situation is more complicat-ed as less is known about the four-generation KM ma-trix. The condition that the v, K proton-decay modes besuppressed so that Eq. (1.1) remains valid, therefore, is auseful constraint. From Eq. (2.1) we write this in theform

A3m, p3+ A4m, .p4- —m, A2,

where

P3 F,, P4F, ,P F ' P F2 Ic 2 jc

(3.1)

(3.2)

and m, . is the t'-quark mass. Equation (3.1) and the uni-tarity condition (2.7) allow one to solve for A 3 and A 4..

A3 ——[m, (A&+A2)p4 —m, Az]/(m, p3 —m, p4) (3.3)

A& ——[ —m, (A &+ Az)p3+m, A2]/(m, p3 —m, p4) . (3.4)

The results depend sensiti vely on the combinationA

& + A 2 which unfortunately is not well determined ex-perimentally. Using four-channel unitarity and experi-ment, one may derive V„=0.9171+0.1085 which isslightly better than V,', "'=0.95+0. 14. This yieldsA 2 ———0. 1898+0.0314 and

One may write the supersymmetry (SUSY) proton-decay constraint (2.8) as

In general, SU(2) &&U(1) breaking in RG models willnot occur at 8 mass scale unless at least one quark massis large (i.e. , ~M~) while if the quark masses are tooheavy, the breaking will occur at too high a mass scale.Since m, & m, we will assume in the following thatm, 5 M~ and m, ~M~ as well as the constraints (3.6)and (3.7). We will also assume P3/Pz are relatively realso that Eq. (3.1) can be approximately satisfied.

In three generations, the condition (2.8) whichsuppresses the v, K modes, required that p3 ~

=2 —4 ascan be seen from Table I. For four generations, the cor-responding condition (3.1) does not require p3 and p~ tobe large, as the two terms on the left-hand side can addcoherently. Thus one may consider two possible cases.

(i) Models wi th no L Rmixing -Here . we assumeA p 0 and hence there is no L -R mixing in the squarkmass matrices (such a situation is realized in certain sec-tors of the superstring-inspired models ). For this casethe squarks in diAerent generations are approximatelydegenerate so that F;, =F;, =F;, . Then p3-+1 andp4-+1 where the signs of p, and p4 are determined bythe phases P, /P2 and P4/P2.

(ii) Models with large L Rmixing He-re on.e charac-teristically expects A~=1 so that there is large L-Rmixing in the squark mass matrices for the heavy-quarkgenerations. Then one expects

~ p3 &~ p4 and both

can be large, particularly when m, and m, are large. Inthe following analysis, we will consider only case (i).

The solutions for A3 and A„of Eqs. (3.3) and (3.4)can be used to calculate B(K+ «~+vv) and —Q(KL ) ofEqs. (2.3) and (2.5). However, these quantities will de-pend on A, + A& which, as seen in Eq. (3.5), is poorlydetermined experimentally. It is therefore better tothink of A &+ A z as a free parameter [within the allowedranges of Eq. (3.5)] and see what constraints can be puton it.

We first ask what is the allowed ranges of A]+ A2which do not violate the K~pp constraint Eq. (2.5)when m, and m, are in the above ranges 40GeV m, ~M~, and M~ ~ m, '5140 GeV. For the fourcases of p3=+1 and p~=+I (no L-R mixing) results aregiven in Table II. We see that even when all four casesare taken together, the range allowed for A &+ A2 by

3426 R. ARNOWITT AND FRAN NATH 36

p3

+1—1

+1—1

+ 1

—1

—1

+1

(A, +22)—0.0257

0.0047—0.0098—0.0032

(Hi+82)—0.0049

0.02230.00320.0095

TABLE II. Minimum and maximum values for 3 l+ 3& al-

lowed by the K~pP constraint of Eq. (2.5).IV. CONSTRAINTS ON Vub AND Vub

Condition (3.1) is sufficient to suppress the decayN ~v„K. From Eq. (2. 1) one sees that it will alsosuppress the N v+ and N~v, K [and hence guaran-tee that Eq. (1.1) is obeyed] provided the frontcoefficients mb V b and I& V &. are the same size orsmaller than the second-generation factor m, V„,. Thusone has approximately

Eq. (2.5) is much narrower than even the direct experi-mental 2cr bounds from Eq. (3.5):

V„b~

&m,~

V„, i/mb-6&&10

fV„b.

[m, .

[ V„, //mt, m,/ V„, //(2m, ) 4&&10

(4.1}

(4.2)

—0.0384& 3 i+ A2 &0.0876 . (3.8)

We next ask under what circumstances does the four-generation E +~~+vv rate fall below the three-generation rate of Table I while not violating the K ~ppconstraint of Eq. (2.5). We find that for the entire massranges of m, and I, this can never happen when

p3——p4

——+1. For p3———p4

——+1 and fixed m, and m,there is only a narrow band in the values of 3 &+ Azabout 0.002 wide where this occurs around 2, + A2close to zero. For any I, and m, the four-generationK+ ~m+vv rate lies below the three-generation one onlywhen

Vud —C1, Vus S1C3 V„b —S )S3C57 V~b' —S iS3$5

(4.3)

where c, = cos0&, s, = sin0, , etc. , one can convert Eqs.(4. 1) and (4.2) to constraints on 0~ and Os. One finds

where we have used the experimental bound Eq. (3.6b)and the theoretical estimate' m& -2m, Equation (4.1)is not much stronger than the experimental bound

~V„b &0.009 [based I (bu )/I (b c) &0.008 and a

B lifetime rb ——1. 1 ps]. Equation (4.2), however, is quitelimiting.

Parametrizing the KM matrix elements in terms ofthe usual four-generation KM angles 0, 06,

—0.006 5 3 ] + A ~ 5 0.002 . (3.9)~

sinH&~

&3&(10,~

sings~

&6&&10 (4.4)

TABLE III. Examples of Q(KL ) and B(K+~vr+vv) forvarious t and t ' masses (in Ge V) for the case p &

———1 =p4.Large values of A, + A2 [still consistent with the experimentalbounds of Eq. (3.8)) cause Q(KI ) to exceed the experimentallimit of 2&(10

m

40408050

1004040

mt

14010014010014014060

0.0100.0100.0100.015

—0.0180.0560.056

Q(K1 )

0.8 ~ 10—'0.9 X 10—'0.9 ~ 10-'0.9~ 10—'

16.6~ 10—'14.4&&10 '8.0~ 10—'

B(K+ )

15.7)& 1015.8X10-"

15X10-"11.2~ 10

67' 10—"4. 1Z10-"3.7&& 10—"

Thus almost always the K+~~+vv rate for supergravi-ty models with four generations is larger than that ex-pected from the standard model with three generations,and so this decay is a good indicator of new physics.

When 2, + A2 lies in the ranges of Table II allowedby the K~pp constraint, the B (K+~~+vv) is usuallyconsiderably larger than the three-generation limits ofTable I [except in the small domain of Eq. (3.9) wherethere is an accidental cancellation between the third andfourth generations]. Some examples are given in TableIII where it is also shown that when the limits ofA t + A ~ of Table II are exceeded, the bound Eq. (2.5) onQ(KL ) can be badly violated.

In deducing Eq. (4.2), we have, of course, assumedthat the fourth-generation neutrino v, . is massless so thatthe decay N ~v, K is energetically possible. Asufficiently heavy v ~ could prevent this decay fromoccurring, eliminating the constraint (4.2)

V. CONCLUSIONS

The above discussion has shown that the fourth gen-eration can eA'ect nucleon decay in supergravity modelsin a significant way both through additional contribu-tions to the dressing loop integrals of Fig. 1, as well asthrough the new decay modes involving the fourthsequential neutrino. The condition Eq. (1.1) thatMH 5 MzU T plays a role similar to the requirementM& ——M&UT in nonsupersymmetric SU(5} GUT models.This, combined with the fact that the same combinationof KM matrix elements which enter in proton decay,i.e., A, of Eq. (2.2), also enter into the rare K decays,KL ~pp and K+~m+vv, allows one to determine infor-mation both on the values of KM matrix elements andthe number of generations. Thus, aside from a smallrange Eq. (3.8), the four-generation K+ ~~+vv rate willexceed the three-generation rate, and the branching ratiofor the latter is bounded by 8&&10 ' for any t-quarkmass. Hence a precision measurement of this decay'will shed light on the existence of a fourth generation,and the determination of I, will further greatly restrictsupergravity models and the allowed range of A ] + 3 2.

36 FOURTH GENERATION AND NUCLEON DECAY IN. . . 3427

APPENDIX: NUCLEON-DECAY AMPLITUDES

The nucleon-decay amplitudes for three generations are given in Ref. 8. We generalize these results here for an ar-bitrary number of generations. In order to make contact with the notation of Ref. 8, we rewrite Eq. (2.1) factoringout the second-generation 8'-ino dressing contribution. For N~v, E dimension-six Lagrangian we have

%6(N~v, K)=[(az) (2MHM~ sin2aH ) 'Pzm, m, V, , Vz, Vzz][F(c;d;; W)+F(c;e, ; W)]

1+ yyf+(yg+y, )&;z+&; a, + 1+ yy;, —(yg —y, )&;z p;+(yi, &; +yzp;") y&„J J J

(A 1)

where j) 3. In Eq. (Al) one has, for the t, t', . . . squark contributions to the W-ino dressing diagrams,

P m"ViVz F(u;d, ;W)+F(u;e, ;W)mg Vzi Vzz F(c;d;; W)+F(c;e, ; W)

(A2)

where the triangle loop form factors F are defined in Eq. (3.11) and the Appendix of Ref. 8. The gluino and Z-inodressing contributions, y and y, are defined in Eqs. (5.2) and (5.3) of Ref. 8. The contributions from the RRRRdimension-five operator are given by

3'ii =p 2

gmdm "V„VzV,,Q(e, ;u, ; W)J

m, m; Vzi Vzz V;, [F(c",d;; W)+F(c;e, ; W)](A3)

Jzi= p 2

gm, m,"V,z V, , V, g(e, ;u, ; W)J

m, m, . Vzi Vzz V, i [F(c;d, ; W)+F(c;e, ; W)](A4)

The dimension-six quark-lepton operators are

'4b ( LP iibL )(s L3 viL ) (A5)

a, is a; with (dL, uL ~dz, u~ ), and p, " is a, ' with d~s. The quantity 5, is the generalization of b, , and is gen-

erally small.The dimension-six N~v, ~ effective Lagrangian may be written as (j=3,4, . . . )

C6(N y, ~)=[(az) (2MHMw sin2ctH ) 'Pzm, m;"(Vzi ) V;i ]

Here

X[F(c,d;)+F(c, e, )] 1+ gy; +y," +$, ),. +y,& g Q, . y.J J

(A6)

P m "( V i) F(u;d, ; W)+F(u e, ; W)

m, ( Vz, ) F(c;d, ; W)+F(c;e, ; W)(A7)

The 5; are generalizations of b, , of Ref. 8 and are generally small. y," is defined in Eq. (5.11) of Ref. 8. The y,z arise

from the RRRR dimension-five couplings, and may be written as

V„md gm, "V, V;, Q(e, u, ; W)P] J

' m, m, ( V„)'V, , F(c;d;; W)+F(c;e, ; W)](A8)

where the dimension-six quark-lepton operators y, are defined in Eq. (5.9) of Ref. 8. The condition whichsuppresses the N ~v,-K modes

(A9)

generally tends to enhance the N ~v, ~ modes, as the corresponding structure in Eq. (A6), i.e., 1+gy,", does not can-cel.

3428 R. ARNOWITT AND PRAN NATH 36

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