Six-fermion (B−L)-violating operators of arbitrary generational structure

Nuclear Physics B232 (1984) 143-179 t~) North-Holland Publishing Company

SIX-FERMION (B-L) -VIOLATING OPERATORS OF ARBITRARY GENERATIONAL STRUCTURE

Sumathi RAO and Robert E. SHROCK

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA

Received 3 February 1983

We enumerate and construct the complete set of six-fermion SU(5)-invariant operators of arbitrary generational structure and prove that they, and, more generally, all six-fermion SU(3)cXSU(2)L X U(1)y-invariant operators, violate (B-L) and, in the standard theory with no va's, do so by +2 units. The process for which the six-fermion SU(5)-invariant operators are most important is the n,--~fi transition. Accordingly, we enumerate and construct the complete set of multigenerational six-quark B - 13 operators which are invariant under SU(3)c × SU(2) L × U(1) and SU(3)cxU(1)e.m. and determine which of these contribute to n~--~fi transitions. An interesting result of this analysis is that the contributions of certain six-fermion operators containing higher- generation fields can dominate over those of entirely first-generation operators, despite mixing- angle suppression, because of much larger coefficient functions. Finally, we comment on the absolute n ~ fi transition rate and derive a new lower bound on the masses of color sextet ( B - L)-violating Higgs bosons.

1. Introduction

The violat ion of ba ryon and (total) l ep ton numbers , B and L, is a na tu ra l fea ture

of g rand unified theories. In addi t ion to the predic t ion of p ro ton and b o u n d n e u t r o n

decay, there is the in t r iguing and equal ly testable possibility of n e u t r o n - a n t i n e u t r o n

t rans i t ions [1, 2, 4, 5, 7, 8, 9]*. In an earlier work [10] we analyzed the single-

genera t ion opera tors which are responsible for such t ransi t ions at the SU(5) ,

SU(3)color × SU(2) elect . . . . . k x U(1) y , and SU(3)c x U(1) . . . . levels, and computed the

matr ix e lements of the resu l tan t s ix-quark opera tors in the M I T bag model

[11, 12]**. In this paper , we shall ex tend this work to e nume r a t e and construct the

full set of SU(5) - inva r i an t s ix-fermion ( B - L ) - v i o l a t i n g opera tors of arbi t rary

genera t iona l s t ructure and, correspondingly , the full set of mul t igenera t iona l

SU(3)c×SU(2)L × U(1) y and SU(3)c x U(1) . . . . - invar ian t s ix-quark opera tors con-

t r ibu t ing to B,~ ,B transit ions. The subset of these opera tors that cont r ibutes to

n ~--~fi t ransi t ions is exhibited. We shall show that all s ix-fermion opera tors which

* n-~ fi oscillations have also been considered in a partially unified theory by Mohapatra and Marshak (ref. [3]); also the phenomenological possibility of n~-+fi oscillations was considered previously by Kuzmin (ref. [6]) (for recent reviews of n-~+fi oscillations, see refs. [7-9]).

** For other work on bag models of hadron structure, see ref. [11].

143

144 S. Rao, R.E. Shrock / Six-fermion operators

are invariant under the group SU(3)c x SU(2)L x U(1) y violate ( B - L) and, in the standard theory with no VR'S*, they do so by +2 units. The same statement applies, a fortiori, to six-fermion operators which are invariant under SU(5). Thus, in the terminology used throughout the paper, "SU(5)-invariant six-fermion ( B - L)-violating (or Iza(B-L)I=2) operators", although all of the properties are included, it should be understood that they are not all independent. One of our interesting new results is that the contributions of certain multigeneration operators can dominate over those of many single-(in practice, first-)-generation operators, despite mixing-angle suppression, because they have much larger coefficient functions than the latter, as a result of different Higgs mass dependences.

To make our discussion self-contained, it is useful to recall some salient facts about grand unification and ( B - L) symmetry. In the simplest realistic grand unified theory, based on the group SU(5) [13] with a 5Rg = ~/~g and a 10Lg = ~b~g of each generation, g, together with a 5 and 24, or [14] 5, 24, and 45 of Higgs, the lagrangian is invariant under a global "X" symmetry [15, 16] with the assignments

X(5R) = 3 , (1.1)

X(10L) = 1, (1.2)

X(5, ) = - 2 , (1.3)

X ( 4 5 . ) = - 2 , (1.4)

where the subscript "H" denotes Higgs field, and as usual, X ( 5 ) R = 3 means that the field ¢J~ destroys 3 units of X. (X(24)= 0, since the 24 of Higgs in SU(5) is self-adjoint.) This X-symmetry is spontaneously broken by the non-zero vacuum expectation values of the Higgs 5 (and 45, if present). The local gauge symmetry associated with U(1)y, is, of course, also broken by these vacuum expectation values. Remarkably, however, a linear combination of these broken global and local symmetries remains invariant, namely

B - L = ½ ( X + 2 Y ) , (1.5)

which .also explains why the spontaneous symmetry breaking of the global X symmetry does not lead to the appearance of a Goldstone boson. This exact B - L symmetry forbids processes such as n,~fi, nn-~ If's, n--> e-zr +, and p--> w "+. The remaining Higgs fields that can couple directly to fermions are the 10, 15, and 50.

* If one extended the standard SU(2)L x U ( 1 ) Y theory to include r ight-handed neutrinos, VR, then these would be SU(5) singlets and would have dynamics determined by the terms (with generation dependence suppressed in the notation)

[ 6~ i i¢~'R] -- m~[ ~"~ C~R] -- mD[ 6 ~ ~R]( H *s )~ +h.c.

in the SU(5) lagrangian. The presence of such a massive v R would render the lagrangian X- and ( B - L ) - n o n - i n v a r i a n t even if the theory did not contain any 10 or 15 of Higgs. Unless otherwise specified, we shall assume the usual standard electroweak theory with no z,R's and corresponding SU(5) singlet ~/R fields.

S. Rao, R.E. Shrock / Six-fermion operators 145

It is easy to check that one can add a 50 of Higgs, with X(50H)=-2 , and maintain an X-invariant lagrangian and (B-L) - invar ian t physical theory. However, if one adds a 10 or 15 of Higgs, then there is no set of X-assignments for which the various fields will render the lagrangian X-invariant [1, 2, 13, 14]. The terms which break the X symmetry are characterized by the selection rule IA (B - L)[ -- 2. This is evident from an explicit consideration of the relevant terms:

~C~,xv T,~ ~ * = h5gl,5g2;lO[~lRgxCl[IRg2](nlo )o~

+hs.,.s.2.1s[~bT~l ~bRg2](H15)<,~

+ 5,,52.T6 (/-/5)

+ (Hs) (Us)

+h.c . , (1.6)

where (Hs) ~, (H10) ~ = -(/-/10) ~ , and (H15) ~ = (His) ~<' denote respectively, the 5, 10, and 15 of Higgs in SU(5); a sum on SU(5) indices and, in the case of the Yukawa terms, also generation indices, is understood; the h's and d/ 's denote Yukawa and triple Higgs coupling constants; and the 5-5-10 triple Higgs term vanishes unless the theory contains at least two Higgs 5's labelled 51 and 52. If one chooses X(10H)=X(15H)= 6 SO as to make the Yukawa couplings X-invariant, then the triple Higgs terms violate X by / iX = - 1 0 . On the other hand, if one chooses X(10H)=X(15H)= 4 in order to make the triple Higgs terms invariant, then the Yukawa terms have zaX = 10. Thus, in such an SU(5) theory with a 10 or 15 of Higgs, the lagrangian violates ( B - L ) and is characterized by the selection rule IAXI = 10, or equivalently, I (B-L)I =2. As is proved in appendix A, any 2nooo-fermion SU(3)×SU(2)xU(1)- invar iant operator, and thus, afor t ior i , any 2nodo-fermion SU(5)-invariant operator (where hood denotes a (positive) odd integer), automatically violates ( B - L) and hence X. In particular, in the standard theory with no VR'S, an SU(3)×SU(2)×U(1)-invariant , and hence also SU(5)- invariant six-fermion operator 6 has I(B-L)~I = 2 or equivalently, Ix l = 10. This property will form the basis for our analysis in sect. 2.

In order to analyze ( B - L)-violating processes and n o f i transitions in particular, one uses a technique based on calculating an effective lagrangian involving only products of light-mass fermions fields, multiplied by c-number coefficients which contain all dependence on the heavy particles that mediate the respective process. The most important application of six-fermion SU(5)-invariant (B-L)-v io la t ing operators is n-~--~fi oscillations and the consequent matter instability. For this process,

6f~n(n<--~ f i )=E Cp~+h.c . , (1.7) i

where the ~7~ are generic six-fermion operators, which thus have dimension do = 9 in mass units; and the C~ are generic coefficient functions, with dc = - 5 . Since U(1) .. . . and, presumably SU(3)~ are exact symmetries, the 6~ must be invariant under these groups. Further, if the mass scale which characterizes the violation of

1 4 6 S. Rao, R.E. Shrock / Six-fermion operators

( B - L ) (e.g. the minimum mass of (B-L) -v io l a t i ng Higgs) is much greater than row,z, then, to a very good approximation, the G are fully invariant under the group SU(3)¢ × SU(2)L x U(1)y. Accordingly, in sect. 3 we shall construct all multigeneration six-quark operators satisfying these two types of invariances. Parenthetically, it should be remarked that one might consider an electroweak gauge theory based on a group G'.w. different than the standard SU(2)L X U(1)y [17], and carry out an analogous classification of operators that are invariant under SU(3)c x G" .... The main alternate group at the present time is SU(2)LXSU(2)RXU(1)B_L [18]. However, it is immediately clear that there are no n ~ fi operators invariant under SU(3)¢ x SU(2)L x SU(2)R x U(1)B-L, since the n <--> fi transition violates B - L.

It is worthwhile to examine explicitly how our SU(3)~x U(1) .. . . -invariant and SU(3)c x SU(2)L X U(1)y-invariant operators arise in the context of an SU(5) grand unified theory, and for this purpose, in sect. 2 we construct the complete set of SU(5)-invariant six-fermion (B-L) -v io l a t i ng operators. Such operators also give rise to other processes like n-->e-~ +. However, in contrast to n<-->fi oscillations, these other processes can be mediated by operators of dimension lower than 9, and hence, by standard arguments, the contributions of the six-fermion operators to these processes are negligible. Thus, for example, the decays n o e-~ -÷ or p--> Verr ÷ can be effected by d = 7 operators of the generic form [OTCO][@Tc@]4~, via non-zero vacuum expectation value of the Higgs field ~b, v~. The contributions of these operators would be larger than those of six-fermion operators by factors of order (v6m~B-L)V/ 3 m l i g h t ) , where m(n-L)V is the minimal mass scale characterizing ( B - L)-violation, and m l i g h t ~ a few hundred MeV is the low mass scale characterizing the matrix element between physical particle states. Some explicit examples of such operators are

T a f t y8 TA P" H* e,.,/3~,8,, [@L~, CCLg2]E~Rs3C~Rg,]( 5 ) . (1.8) - - /3 T a y I" [0L~/3g, ORg2][@Rs3CORg,](H5 )3", (1.9)

- - 3' T / 3 6 t a [ ~,bL,/3g z t~Rg z ][¢Rg3C~bRg4 ] ( H 4 5 ) 3"8 , ( 1 . 1 0 )

all of which contribute at the first-generation level. In contrast, the lowest-dimension operator that can mediate n*->fi transitions is a six-fermion one. Of course, not all six-fermion n<--~fi transition operators are SU(5)-invariant. For example, one class of non-invariant six-fermion operators arises from SU(5)-invariant d = 10 operators of the generic form [~ITc~[I][~ITcI~][~ITc~I]~, through the vacuum expectation value of the Higgs field, &. Some explicit examples are

T a 13 T3" ~ -- - - T A [@Rg l CORg 2 ] [0Rg 3 CORg4 ][ t/JL a 3"g 5 CI///3Ag 6 ] ( (H24) 8)0, (1 .1 1)

T a /3 - 8 - " H 3" [@RglC~Rg2][~]-JL3"Sg3~Rg4][~L/3I.Lg5@Rg6]((24) or)0 , (1.12) T a /3 -- K T y 8 h.u. p

E3"8XtzpEl[lRg, Cl[lRg2]EI]lLa,<g3~[YRg4]EllJLg s Ctl/Lg6]((H24)/3)o, ( 1 . 1 3 )

Ta /3 3"8 TAp, v'0 T o K p e,~/33"sp ex,~,,no. [OLg, C@Lg2][l~tLg3 CI]iLg4][~JRgsCI~Rg6]((H24)K)o, ( 1 . 1 4 )


where

I /)24 , O/ = /3 = 1, 2, or 3

((H24)Z)0 = t - - 3 t ) 2 4 ( 1 q - 8 ) , O~ =f l = 4 (1.15)

[ - 3 v 2 , ( 1 - 8) , a =,8 = 5 ,

with [81 - 10 -12. It is clear that the Higgs content of an SU(5) theory plays a crucial role in

determining the presence or absence, and, in the former case, the size, of ( B - L)-violation. In sect. 4 we work out correspondences between tree-level SU(5) diagrams and six-fermion (B-L)-violat ing operators on the one hand, and on the other, SU(3)xSU(2)×U(1) diagrams and the corresponding six-quark operators. Some comments are also made concerning the implications for ( B - L) violation of embedding SU(5) in SO(10). Using these results and lower bounds on the masses of various Higgs fields, we then derive some expected inequalities between coefficient functions. Finally, incorporating our earlier calculation of the relevant matrix elements in the MIT bag model, we comment on the absolute n ~ f i transition rate. Appendix A has already been referred to; appendix B contains some further results on the quark mass dependence of the bag model matrix elements.

2. The general set of SU(5)-invariant, six-fermion, (B-L)-violating operators

In this section we shall exhibit the general set of (Lorentz-invariant) SU(5)- invariant, six-fermion ( B - L)-violating operators which have arbitrary generational structure. As demonstrated in sect. 1, these operators necessarily violate ( B - L )

and do so by two units. The analysis of these operators is obviously necessary to describe ( B - L)-violating processes involving particles of more than a single generation. But this is not all; the analysis of multigeneration operators is also necessary for (B-L)-violat ing processes involving only particles of a single generation (in practice, the first), since such operators can contribute to these processes through mixing and can dominate over first-generation operators because of larger coefficient functions.

The problem of enumerating and constructing the arbitrary-generation operators can and will be solved using only SU(5) group theory* and considerations of Lorentz and Dirac structure, without recourse to any Feynman diagrams or analysis of Higgs particles. Subsequently, we shall exhibit the specific (tree-level) diagrams and Higgs which yield various operators.

The analysis begins, as in the one-generation case, with the observation that the operators change Xaccording to laxl-- 10, and thus IA ( B - L) I = 2. There are four ways to assign Xi to each of the six fermion fields in order to obtain laxl--- IZ,61 x,I = 10. Here, Xi = 3 and 1 correspond to the SU(5) fields ff~ and ~b~ ~, respectively.

* A helpful compendium of SU(5) tables can be found in ref. [19].


Next, one must construct Lorentz-invariant products of the six fields having ]AX[ = 10. We recall here the fact that operators involving vector, axial-vector, or tensor Dirac structure can be Fierz-transformed into operators involving only scalar and pseudoscalar Dirac structure. There are three types of bilinear products of fermion fields that one can form; these are (suppressing generation indices)

,ITOL 1,~,1,~ a R R ~ mR ~'FWR, w i t h X b = 6 , (2.1)

,h TOtB t.'~1, y8 aLL---- WL '-'~'L , with Xb = 2 , (2.2)

aLR--= #L.~¢'~ +

= aRL, with Xb = 2 , (2.3)

where Xb=Xbi , . . . . and C = i yzy ° is the Dirac charge conjugation matrix with C*= C T= C - 1 = - C . There are five independent types of Lorentz-invariant six- fermion operator products which can be formed from these three types of bilinears. We denote these as classes 1 through 5. For each of the sets {Xi}, = 1, 4, and 5, there is a unique Lorentz-invariant grouping of the Xi into pairs, viz.

{Xi} 1 ={3, 3, 3, 3, 1, - 3 } 4 { ( 3 , 3), (3, 3), (1 , -3 )}

~'*{Xb), = {6, 6, --2}, (2.4)

{X/} 4 ={3, 3, 3, --1, 1, 1}~ {(3, 3), (3, --1), (1, 1)}

<"') {Xbj}4 ={6, 2, 2}, (2.5)

{X~}5 ={3, 3, 1, 1, 1, 1}+{(3, 3), (1, 1), (1, 1)}

~ { X b ) 5 = {6, 2, 2}. (2.6)

In contrast, for the set {Xi} ={3, 3, 3, 3, - 1 , -1} there are two such groupings,

{(3, 3), (3, 3), ( - 1 , - 1 ) } ~ { X b ) 2 ={6, 6,--2}, (2.7)

{(3, 3), (3, --1), (3, --1)}~'{Xb,}3 ={6, 2, 2}. (2.8)

Viewing the correspondence in reverse, it is interesting that {Xb)l = {Xbfl2 although {Xi}l ~ {Xi}2 and {Xb)3 = {Xb)4 ={Xb)s although none of the corresponding {Xi} sets is equal to any other. Note also that all of the six-fermion operators involve at least one a R R factor, since there is no way to obtain lY3=1 Xb,] = 10 with any combination of +2 taken three times.

The operator products corresponding to the Clebsch-Gordan decomposition of the bilinears are

. T e e f l

5 ) ( 5 ~ 10a" [ ~ g l C ~ R g 2 ] a = . T /3 (2.9, 2.10) t15s. ['/'R~, C~'Rg2]s,


F,i, Ta/3 7~ 5 s : Ecq3ySA k W L g 1 C ~ L g 2 ] ( s ) (2.11)

- - r,I, Ta/3 r , l , 7 ' s 1 1 0 X 1 0 = 4 5 a : ea/3.yAp. LVLg 1 ~.---tFLg2Ja (2.12)

5 - 0 s : [-'l'T°t~ ('~'l" Y8 "[ L V L g l t ' V L g 2 I s , (2.13)

_ _ f 5: /3 [qJL~/3g, ~Oag2 ] (2.14) 10x5 =

. 3' 45-[qJL~/3g, qJRg2 ]d, (2.15)

where we have used the convenient abbreviations T a /3 __ 1 T a /3 Tf l o~ ~{[ORga CORg~ ] + (2.16) [ ~Rg, Cl~tRg 2 ]s = [ ~Rg, C f f I R g 2 ] }

l / F , / T o t /3 T a /3 ~-~ ~ I L W R g 1 C ~ t R g 2 ] 4- [ ~ t R g 2 C ~ I R g 1 ] } , ( 2 . 1 7 )

[',t ToqS l.'~ l, y~5 "] similarly for tVLg, "-'V'Lg2J, and

-- 3, - - -- 3, [g.,~/3~ g ' ~ h = [ 0,_o/3~, #'.~ ] 1 3, - x .a t_ 3' - g

- - ~ ( (~ ~ [ ~ t A / 3 g a ~ R g 2 ] ~/3 [ ~ t ~ A g l ~ R g 2 ] ) , ( 2 . 1 8 )

in which the subscripts "s" , "a" , and "d" stand for "symmetric", "antisymmetric", and "detraced", respectively. As indicated, the (anti) symmetrizations in eqs. (2.16) and (2.17) can equivalently be viewed as properties of the SU(5) or generation indices, respectively. Note that in the single-generation case, gl = g2, it was not necessary to carry out these (anti)symmetrizations in order that the various bilinear products have well-defined transformation properties. For example, the term

T a /3 [~ORglC~ORg, ] by itself is symmetric in the SU(5) indices a and/3 (because of Fermi statistics and the fact that C a = - C ) and hence transforms as a 15 of SU(5). However,

T a /3 [q'RglCqJRg~ ] with gl ~ g2 has no well-defined transformation property under SU(5). Finally, we will use the subscripts "(s)" and "(a)" rather than "s" and "a" if the (anti)symmetry of the given operator is an automatic consequence of its contractions with other operator(s) or SU(5)~ tensor(s).

We proceed to list the (Lorentz-invariant), SU(5)-invariant, six-fermion z I ( B - L) = - 2 operators that we have constructed in each of the five classes. The order of the factors in the operator products is chosen to be the same as in our previous paper. Subsequently, we shall derive a relation between certain of these operators; thus the list given below is complete but not minimal.

For operators of the form aZRaRL, denoted as class 1, we find

(~1-1 Tot fl T3" 6 -- oh = e~/33,~[Oag, Cq~ag~](a)[~,ORg~Cq'Rg4](a)[OR,g~qJLg6] , (2.19)

- - A A T a /3 [~1 -2 - - ( Eotfl~Tzv ~ 6 - - E ySap. v ~ fl ) [ l ~ R g l C l / / R g 2 ] a

T3" 6 -- p.v × [qJRg~C4'Rg,]a[~ORag~q'Lg6]O, (2.20)

(~1-3 ~ T a /3 T3, 6 - /xv eafl3,1~v[ff lRg, C f f lRg2] (a ) [~ tRg3Cf f JRg4]s[ f f lR ,Sgs f f lLg6]O. ( 2 . 2 1 )

As an example, the SU(5) representation structure of the contractions involved in these operator products is shown in fig. 1.


5 5

I t I0 a , I0 a , 15 s

5

I

15 s

I 50 s , 70 s , 105 a

5s , 450 ,50s I_._~

5 5 ~0

I [ I

5 , 45

0 H

11-2

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

~ 3 - 1 - - -- ~ ]d[~t 'YSg3~-JRg4]d[~Rg5C~Rg6]s, - I ~ L ~ / 3 ~ , ~ : - " T/3 +

~ 3 - 2 - - -- /3 -- 8 Tt~ 3' -- [ ~Lot/3gl ~Rg2 ][ ~ L ~g3 ~Rg4][ ~lIRgs C ~ R g 6 ]s ,

/3 -- ~ Ta C T 6+-~ = [~Lo/3,, ¢ ' ~ ][¢'.~+M'~ ][ ~ , ~ SRg+]a,

- v - '~ T/3 c ~ , C 3 - 4 = [ ffJLct/3gl ffJRg2 ] d [ ~tL'ySg3 ff/Rg4 ]d [ f f /Rg5 ffJRg6 ]a

~ 3 - 5 = -- /3 -- ct T) ' ~, [ I//Lct/3gl t l /Rg 2 ] [ I#LySg3 ~ / r g , , ] d [ I] /Rg 5 CfflRg6 ] ( a ) "

45,1 105 01_3

Fig. 1. SU(5) contractions for multigeneration six-fermion A ( B - L) = - 2 SU(5)-invariant operators of class 1 (a2Rt~LR).

- 2 For opera tors of the form aLLaRR , denoted as class 2, we find

(~2-1 = [tffLot/3gl --T T a )' TB 8 Cl#L v6g2 ](s)[ l~tRg3 Cl#Rg4 ]sr ~Rgs C~IRg6 ]s , ( 2 . 2 2 )

-- - T T a /3 T y ¢72-2 = [¢L~/3glCCLvSg2 ]s[¢R~3CCRg+](a)[Oag, CCRg6 ](a), (2.23)

T/~ v T p o" x [ q'Rg~ Cq'Rg+ ]~a~[ ~b ag~ C~bRg+ L ) ' (2.24)

(~2-4 - - a/3TAlx [ R8 8 -- --T -- e \eAl~trKrl~p--eAl~rlpo.(~K)[l~lL~t~gl Cl[ILT~g2]a

TK "r t Tp <r X [ ffJRg 3 Cff.lRg 4 ] a [ ffJRg 5 CffJRg 6 ]a, (2.25)

__ or/37A/.~ r 7 ,'~ "/-T 1 (~2-5 - - E EvValxpL~lLct~gllJ~llycrg2Ja

Tv "0 TO o" × [¢Rg+C$Rg4]~a>[¢Rg+C$p.g+]s. (2.26)

There are five opera tors of the form a2LRaRR, denoted as class 3; they are


Class 4, of the form aLLaLRaRR, consists of seven operators:

T a r 3"8 -- v TA g (74-1 ~-~ Eotf3,SA [ [ / /Lg , C~Lg2](s)Fi[lLgvg3ffJRga][l[lRgsC~Rg6]s ' ( 2 . 3 2 )

T a r 3",~; - g TA z, (74_2 = e,~f,~,~[g'Lg, CqJLg2]a[g'La~gflJag~]d[g/Rg, CORg~]s, (2.33)

T a r 3"8 -- v TA g (74-3 = e~f3"sA[qJLg~ C~OLg~](~)[@Lg~g~6Rg~][~ORg~CORg6]a, (2.34)

T a r 3,6 -- h T/x 'q 0 4 _ 4 = Ec~f3,Xg rff..tLg, Cff.lLg2]a[l~L,3rtg3@Rge,]d[l]lRg5CffJRg6]a, ( 2 . 3 5 )

r ~Tctf 3,6 -- v TA g (74-5 = eotf~AgtqJLg! C~,OLg~]~[OL3,SgflJRg4]d[q'Rg, CORg~]¢,), (2.36)

(74-6 ~ Tc t f 3,$ - A T g v e~fvax [0Lg, C~OLg~]~s)[OLg~g~ORg~]d[tPRg~CORgj(a), (2.37)

Tctfl 3,6 -- v TA g (74-7 = Eot f3 ,Ag[ I ] /Lg l Cff-lLg2]a[~lLSug3ffIRg4][~tRgsCfflRg6](a) • ( 2 . 3 8 )

Finally, there are seven operators of the form a~LaRR, denoted as class 5:

Ta/~ 3,8 T g ~ per TA "q C5-I = Ectf3,6A Eguptrrl [l~/Lg l C I ] / L g 2 ] (s)[ l ] /Lg3 C~[]Lg4](s)[ff-lRg5C~IRg6 I s , (2.39)

Tocf 3"~ T g v po" TA 7/ (75-2 = Ec~fgvAEy,3po'~7[l]lLgl C~Lg2qs[~Lg3 C~Lg4qs[l~lRg5CffJRg6 I s ' ( 2 . 4 0 )

Totf y6 Tgv po- TA ~q ~ S - 3 = Eaf3"SXe~vo~n[@Lg, C6Lg2](s)[OLg3 COLg4](s)[ORgsCt~Rg6]a ' ( 2 . 4 1 )

- - Toq3 3"~5 T v r t po" TA g ( 7 5 - 4 - EotfvAgE6vnpo'[~/ILgl CI] /Lg2]a [ f f /Lg3 CfflLg4](s)[~tRgsCl]tRg6](a)' ( 2 . 4 2 )

Taf t 3,8 Tpo" ~'/A T g tc (75-5 = e,~t33,xgeo,~na,,[~bLg~ C ~ L g z ] a [ I / / L g 3 CffJLg4]aFffJRgsCi ] lRg6] s ' (2.43)

Taf t 3,~ Tpo- "0A Yg K (75-6 = eotf3,Agep,~,a, [qJLg, COLg~]a[qJLg~ C~OLg,]a[0Rg~C~bRgs]~, (2.44)

T a r 3'6 TAg po- Trt K COLg2]a[ffJLg 3 COLg4]s[ffJRgsC~Rg6](a) ( 2 . 4 5 ) (75-7 = e o t f ~ x g e s p , ~ , ~ [ ~ t L g l

From general symmetry considerations, one can determine the generation structure of the various six-fermion operators. Fermi statistics and the fact that C T = - C prevent the antisymmetric coupling of two identical fields, either two 5R'S with the same generation index, or two 10L'S with the same property. Explicitly, for the former case,

T a ,13 T f a = [ I~Rg2 CI [ IRg I ( 2 . 4 6 ) [~R~, Cq~R~2] ]. T o O • Hence, if gl = g2 as mentioned before, [q'Rg, Cq'Rg2 ] is symmetric in the SU(5) indices

a and/3 implying that any non-,6anishing coupling must also be symmetric in these Tot F ,I,T°~f f ' , l , Y 6 1 indices. It follows that in terms of the form [6RglC~g2]a and tWLg, "-'WLg~Ja, gl

cannot be equal to g2- It is useful to classify the operators according to the number, Nhg, of SU(5) fermion fields which must be of higher generation than g = 1. This classification is given in table 1. If there were no fermion mixing, so that mass and gauge group eigenstates were identical, then first generation processes such as n <--> fi would receive contributions only from operators with Nhg = 0. In the right-most column of table 1 we also list the total number of independent SU(5)-invariant six-fermion operators of each type, N(n) , where n denotes the number of fermion generations.


TABLE 1

Number of higher generation fields Nhg, and total number of operators, N(n), for each type of SU(5)-invariant six-fermion A ( B - L) = - 2 operator (n = total number of ferrnion generations)

Operator Nhg N(n)

(7>1 2 ln3 (n -1 ) [n (n -1 )+2] ~71_ 2 2 ~ n 3 ( n - 1 ) [ n ( n - 1 ) - 2 ] 61. ~ 1 ¼n4(n-1)(n+l)

{~2-1 0 ~ne(n+l)Z[n(n+l)+2] ~2-2 2 l n 2 ( n + l ) ( n - 1 ) [ n ( n - 1 ) + 2 ] ~2-3 2 l n 2 ( n + l ) ( n - 1 ) [ n ( n - 1 ) + 2 ] ~2-4 3 aA-~n2(n-1)Z[n(n-1)-2] 62_ 5 2 lna(n--1)2(n+l)

~3-1 0 ~n3(n+l)(n2+l) ~3-2 0 ln3(n+l) (n2+l) ~3-3 1 l n3 (n -1 )2 (n+l ) {~3-4 1 a4n3(n-1)2(n+l) ~3-5 1 ½nS(n - 1)

~7,i_ 1 0 ¼n4(n + 1) 2 ~4-2 1 ¼n4(n-1)(n+l) ~4-3 1 ¼n'*(n-1)(n+l) ~74-4 2 ¼ n 4 ( n - 1 ) 2 ~4-5 1 ¼na(n-1)(n+l) ~7,,_ 6 1 ~na(n-1) (n+l ) ~74_ 7 2 ¼n4(n - 1) 2

(75_ 1 0 ~n2(n+l)2[n(n+l)+2] ~5-2 0 1Azn2(n+l)2[n(n+l)+2] ~75_ 3 1 1%n2(n-1) (n+l)[n(n+l) -2] ~75.4 2 l n 3 ( n - - 1 ) 2 ( n + l )

~-5 2 l n 2 ( n + l ) ( n - 1 ) [ n ( n - 1 ) + 2 ] ~5-6 3 l nZ (n - 1)2[n(n -- 1) --2] ~5-7 2 ~n3(n-1)2(n+l)

B y m e a n s o f F i e r z t r a n s f o r m a t i o n s , w e h a v e f o u n d t h a t

- - 2 e , , m , aa e,,.,,~,7 [q'agl Cq 'ag ,][¢ tg , C~bLg=][0Rg, CCRg6]s ( 1~5-1 ) gl g2g3g4gsg6 - - Ta,8 po" TI.zv 3'8 TA "0

1 F.I, TOt.OCo r ,I,'Y~ ][-,I, TI-~ 'Co.K¢,I , po" "l - -4Eotf ly~SA~p.vpo"qlt fYLgl ~" K~tf~Lg2JLWLg3 ~ W'LggJ

TA x [0RgsCORg6]s. (2.47)

If gl = g2 or g3 = g4, the term involving Lorentz tensor Dirac matrices vanishes. If, in addition, g~ = g4 and g2 = g3, then we have

( {~5 .1) gag, g~glgsg 6 = - - 2 ( 6 5 _ 2) g191glglg5g 6 . ( 2 . 4 8 )

Thus, to obtain the total number of independent SU(5)-invariant operators of types 65_~ and {25_2, one cannot simply add the respective entries in table 1 ; rather, the

S. Rao, R.E. Shrock / Six-fermion operators

TABLE 2

Nhg and total N(n) for the classes of SU(3) x U(1)-invariant six-fermion operators contributing to B ° ~ B ° transitions

Type of operator Nhg Ntot~l(n)

~Ta, s 0 lnZ(n+ l)2[n(n+ l)+ l] Oa,~2 2 n3(n + 1)(n- 1) 2 (?l,a2, (saa) 2 ½n2(n+l) (n-1)[n(n-1) +1 ] ~1,~3 3 ½n2(n-1)2[n(n-1) - l ] ~2,s 0 na(n + 1)(2n 2 + 1) (~2,a2 0 n3(n + 1)(2n2 + 1) (~2,a2' 1 4nS(n - 1) ~2,a3 2 na(n-1)(2n2-1)

153

number is given by

N ( n ) ~5.1+c5_~ = ~n2( n + 1)2In( n + 1) + 2]-½ne(n + 1). (2.49)

If one takes all the gi equal for i = 1 . . . . . 6, then the general set of operators

reduces to five single-generation (sg) operators as follows:

~2-1 "~ ~ l(sg) , (2.50)

(~3-1 + ( ~ 2 - - 2 ~ 3 ) ( s g ) , ( 2 . 5 1 )

63-2"~ ~3(sg) , . (2.52)

(74-1 + ~4(sg), (2.53)

~s-1 + ~5(sg), (2.54)

~5-2 "+ ~6(sg) , (2.55)

with

~i~5(sg ) = --2~6(sg ) . (2.56)

The additional term in eq. (2.51) results from the fact that in our present analysis, in order to elucidate the group-theoretic structure of the operators, we have removed the trace from the bilinear [ ~ L ~ ] (cf. eq . (2.18)), so that it transforms purely as a 4--5, whereas in the previous work [2, 10], this trace was not removed.

3. The general set o| n~-+fi transition operators invariant under SU(3)× U(1) and SU(3) × SU(2) × U(1)

Since U(1) . . . . and presumably also SU(3)color a r e exact symmetries, the effective local six-fermion operators which mediate n,~+fi transitions must be SU(3 )x U(1) singlets (we shall drop the subscripts on these groups where they are obvious). It is thus useful to construct all possible SU(3)xU(1)- invar iant six-fermion n,~+fi


transition operators. Among these, it is worthwhile to determine the subset that is also invariant under the full standard SU(3)cxSU(2)LXU(1)v low-energy symmetry group [ 17]. For the special case of a single generation of fermions (in practice, the first), these problems have been studied [4, 10, 20]. Here, we shall extend these studies to the case of an arbitrary number of fermion generations. We shall actually solve the more general problem of constructing all SU(3)xU(1)- invar iant and SU(3) x SU(2) x U(1)-invariant six-fermion (neutral) baryon-antibaryon (B ° - 13 °) transition operators. This is useful since, as was noted in sect. 1, some of these operators consisting of one or more fields of higher than first generation can still contribute to purely first generation processes, such as n a i l transitions, through fermion mixing. Indeed, their corresponding coefficient functions can be sufficiently large that they actually dominate over certain operators consisting of only first- generation fields.

The reasoning for considering fully SU(3)×SU(2)xU(1) - invar ian t operators proceeds along lines similar to those well known from the case of proton decay. If the lowest mass scale characterizing the particles which mediate n<--> fi transitions is much larger than the mass scale of - 3 0 0 GeV characterizing the spontaneous breaking of SU(2)L X U(1)y, where Y denotes weak hypercharge, then the resultant effective local six-fermion operators are also, to a very good approximation, SU(3)c × SU(2)LXU(1)y singlets. (Given the invariance under SU(2)L, the condition of U(1)y invariance is equivalent to that of U(1) .... .) In order that n<--~fi transitions be observable in forseeable experiments, the mass scale characterizing the Higgs which mediate these transitions probably must be no larger than ~103-104 GeV (see sect. 4), substantially lower than the figure of ~105 GeV often quoted in the past. Thus, it may be, as noted before in ref. [10], that SU(3)c×U(1) .... -invariant operators which are invariant under SU(3)cXU(1) .... but not SU(3)c×SU(2)L× U(1) v ........ can make contributions which are not negligible relative to those of the operators which are fully invariant under the latter group.

In order to proceed with the analysis, we observe first that t h e constraint of U(1) .... invariance is automatically satisfied for any n ~ fi transition operator. Next we note that there are three ways of forming an SU(3) singlet from a product of six 3 representations.

The corresponding color SU(3) tensors are

(T~) ~ap~ = e~pe~a~ + e ~ e ~ p + e~pe~ap + e ~ e ~ s p , (3.1)

( Z a 2 ( a a s ) ) ~ v a o ( r ------ ( Z a 2 ) a/3v,Sp(7

= e~,,~pe r~o- + e,~o-ev~p , (3.2)

( Za3),~/3vapo" = e ~13p E ~,&~ - e al3(r e V,~ p • (3.3)

The subscripts "s" and "a" denote "symmetric" and "antisymmetric", respectively, and the label (aas) in Tae indicates which pairs of SU(3) indices have which symmetries. For the case of non-identical 3 representations of SU(3), which will be

S. Rai, R.E. Shrock / Six-fermion operators 155

relevant in our analysis, there are two other types of Ta2 contraction tensors:

(Za2'(saa) ) ,~/3vap(~ = (Ta2) vap(~,~13 , (3.4)

(Ta2'(asa) ) ,~/3vap,~ -- (Za2) ,~/3p,~Y,~ • (3 .5 )

Define the symbols (ab) and [ab] to mean symmetry and antisymmetry under the exchange of a and b, respectively, where these can be single indices or sets of indices. It will be useful to record the various symmetries that the SU(3) tensors have:

(Ts) ~ p ( , : (aft), (y6), (p(r), (a/~, y6), (y& p~r), (oq3, p(r), (3.6)

(Ta2(aas) ) a/3.~aOa.: [O~j~], ['y($], (pO' ) , (O~j~, 'y(~) , (3 .7 )

and analogously for the Ta2, tensors, and

(Ta3)~erap~: [o~fl],[y6],[po'],[o~)8, ya],[y&po'],[off3, po']. (3.8)

In the single-generation case only T~ and Ta2 occur; thus, Ta2 was denoted simply as Ta in ref. [10]. When the symbol T~2, is used, it will mean that the tensors Ta2'(~a~) and T~2'(~a) yield equivalent SU(3) contractions.

The quark fields which naturally occur in the operators are gauge-group eigenstates. For the SU(3)xU(1)-invariant operators, we label these as U~g and dxg, where u is the SU(3) color index, X = L , R denotes the chirality, and g is the

ot ~ ~ a generation index. Thus, ux~ --- u~, ux2 = c x, etc. where u x is a gauge group eigenstate, and similarly for dxg. For the SU(3)×SU(2)×U(1)- invariant operators, the quark fields that occur are the SU(2) singlets U~g, d~g, and the SU(2) doublets

ict qLg, where

qLg \dL,,ig (3.9)

Since in practice these fields arise as components of representations of a grand unified group, their mixing must be analyzed in the latter context. For example, in the SU(5) theory of interest here, if one has a 5, 24 and 45 of Higgs, then the mixing of mass eigenstates to form gauge group eigenstates is specified by

t - / L ¢'~ R ( m ) I [(e)* c I//5Rg e~ = ~ ' L eR(m) ,

\pcR/g ,~(e)* t - ' L P e R ( m ) g

and (in abbreviated notation)

c

I~ lOLg -~- U L

uL} e~

dL g

U(U)* c R U L ( m )

Ukd) dL( m )

u(e)*~,c 1 R ¢ ' L ( m ) l ,

(3.10)

(3.11)


where eg=l = e, eg=2 = IX, etc. the subscripts " ( m ) " denote mass eigenstates, and we absorb all phases into the definitions of the unitary transformations U(~ ° for each

fermion fx. Since the neutrinos are massless and hence degenerate, the U(L ~) can be chosen freely; the choice in eq. (3.10) is standard. One can further choose the mass eigenbasis so as to render U(L ~) = 1 and U(L d) = 1; we avoid doing this in order to retain a symmetric description of the mixing. Further simplification of the fermion mixing would take place if the theory contained only a 5 and 24 Higgs, but to avoid the untenable mass relation rod~ m~ = m~/rn~, it is probably necessary to include a 45 of Higgs [14]. The presence of a 10 or 50 of Higgs does not alter eq. (3.10) or eq. (3.11), since these fields both have zero vacuum expectation values. Since we

are studying ( B - L ) - v i o l a t i n g operators, we are obviously interested in theories which contain one or more 10 and /or 15 Higgs representations. If, in particular, the theory has any 15's of Higgs, then U(~ ~ will in general be independent of U(L e),

- _ y . uL >* . so that the lowest member of the fermion 5 will be ~ 5 _ k=l ( )gk l lRek(m) ,

i.e. there will be (relative) lepton mixing.

We find the following complete set of S U(3 )x U(1)-invariant six-fermion B ~ 1] transition operators of arbitrary generational structure:

-- uT~ /3 T~' (~l,z)XaXt, X~ -- [ x .g~Clgxbg2][dxbg3Cdxbg4]

Tp o- x [ dxcg ~ Cdxcg ~ ](Tz) ,~t3~,~f,o-

for z = s, a2, a2'(saa), and a3 ; (3.12)

( 62,Z ) XaXbXC -- Ta /3 T')" ~5 -- [ U~agl Cdxag21[ Uxbg3 Cdxbg41

X Tp o -

for z = s, a2, a2' , and a3 , (3.13)

where each X = L or R and we suppress the generational dependence on the left-hand sides of these equations. In the single-generation (sg) case, certain of these operators reduce to those given in ref. [10]:

( C2,a2) Xa,lfbX c "~ ( ~3 ) XaXbXe(sg) ,

(3.14)

(3.15)

(3.16)

while the others vanish. In the present context, the operators (6m)x,x~xc, m = 1, 2, 3, will be considered to be multigenerational unless specified as (sg), so that

(Cm,s)x~,XbXc=((~,,,)XoXbXc, m = 1, 2, and ( [ ~ 2 , a 2 ) . ~ a ) ( b X c = ( ~ 3 ) X a X b g c . The operators have generational and chiral symmetry properties which follow from the symmetries of their SU(3) tensors*. These include the extension to arbitrary generations of the

* For lack of space, we shall not list the results here. They can be found in ref. [34].


symmetry relations

( ~ I ) x ~ L R = ( ~ I ) X a R L '

given before [10] for the single-generation case.

m = 2 , 3 ,

(3.17)

(3.18)

(T3)a~'yapo" = ~ pO:Se O.~,y -It-eo.a~pp,y •

In terms of these tensors,

We find that

( ~ 1 ,s) ,VaXaXc ;gl g2g3g4gsg6

T~ = T2+ T3, (3.21)

T~2 = T 2 - T3. (3.22)

T a /3 T3' 8 Tp o" = 2[Uxag~Cdx.g4][Uxag2Cdx~g3][dx.gsCdxcg6](T3- Ta2),~,ap,~

1 T a /3 Ty ~¢~ 6 Tp o- -~[UxaglCo',,¢Ux.g2][dx.g3Co" dxag4][dxcgsCdx~g6](rs),~.r~p,y. (3.23)

If gl -- g2 or g3 = g4, then the term involving Lorentz tensor Dirac matrices vanishes, and we obtain the relation, for gx = g2

(~2,S)XaXaXc;glg4glg3gsg6-- ((~l,s)XaXaXc;glglg3g4gsg6 = 3(~2,a2)XaXaXc;glg4glg3gsg6 ' (3.24)

and, for g3 = g4,

( ~2,S) Xa)(a)(c;glg3g2g3gsg6--( ~l,s)/ya)(aX_c;glg2g3g3gSg6 = 3( ~2,a2) x,,x.x~;g,g3g2g3gsg6 " (3.25)

It is of interest to determine the generational structure of a given type of SU(3)xU(1)- invar ian t operator. For example, in the case of the operator (~l,a2)XaXbXc, it is necessary that gl # g2 and g3 # g4 in order that the operator should not vanish. Consequently, there must be at least two quark fields of generation higher than the first, i.e. Nhg = 2. The multigenerational operators ~,s, ~72,s, and ~l,a2 with one or more g~ indices not equal to one still can contribute to the first generation process n o f f through fermion mixing, albeit with some mixing-angle suppression. None of the other operators in eqs. (3.12) and (3.13) contributes to n o f f transitions*. Previous studies of quark mixing have demonstrated that it is consistent with being hierarchical [21], that is, [(VcKM),[> [(VcKM)m,[ iff [i--j[ < [rn -- n I, where VCKM is the Cabibbo-Kobayashi-Maskawa quark mixing matrix [22].

* Similar results have been obtained by L. Wolfenstein and J. Basecq (L. Wolfenstein, private communication).

(3.19)

(3.20)

Recently, a Fierz relation between the single-generation operators (~Ti)x~c, i = 1, 2, 3 was derived [20]. By similar methods, we have derived the full multigenerational relation. Define the SU(3) color tensors


More direct evidence for this result has been obtained through measurements of B meson decays at CESR [21]. Assuming that not only the CKM matrix, but also the individual mixing matrices UCx q) (q = u, d; 2( =L , R) which occur in grand unified theories, are close to the identity, we would expect that, other factors (such as coefficient functions; see below) being equal, among operators with Nhg>0, the main contributions to n-,--~fi oscillations would arise from those with Nhg = 1 and the relevant gk = 2.

It is also of interest to calculate the number of independent SU(3) x U( 1)-invariant operators of each of the types (3.12z) through (3.13z). We define N((~,,,z)xox~xo)

to be the number of independent operators of type m, z for each independent choice

of the chirality indices Xa, Xb, and 2(c. Similarly, N((61.z)xoLR) denotes the number of independent operators of type 1, z for each of the two possibilities Xa = L and X, = R. Thus, for example,

N((I~I,s) XoXbXb ) = N((GI,s)LLL) = N((~I,s)LRR)

= N((~'I,s)RLL) = N((~I,s)RRR)

= l n 2 ( n + 1)2[n(n + 1) + 2] , (3.25)

N((~,,s)xaLR) = N((~71,s)LLR)

= N((ffl.s)RLR)

= ~n3(n + 1) 3 , (3.26)

where, as before, n is the number of fermion generations. The set of (~7~,s)xoRL operators is not independent of the set of (~71,s)xoLR operators, and hence is not included. Similar analyses can be carried out for the other SU(3)x U(1)-invariant operators (see footnote on p. 157). Adding the numbers of independent operators of the different chirality types to obtain the total number of independent operators of a given type, m, z, one finds the results given in table 2. As with the SU(5) operators, because of the Fierz relation (3.23) (or (3.24)), one cannot in general simply add the values of N ( n ) for each operator type to obtain the total number of independent operators. In particular, in the single-generation case, eq. (3.24) yields four relations, so that there are only fourteen independent sg SU(3 )x U(1)-invariant n ~ fi transition operators.

We next proceed to list the complete set that we have constructed of six-quark, SU(3)cxSU(2)L xU(1)y- invar iant operators of arbitrary generational structure which contribute to B ° ~ B ° transitions. The possible SU(3) contractions are the same as those for the SU(3)cxU(1) .... -invariant operators. The possible SU(2) contractions will be obvious from the actual operators, which are listed below:

f m = l , z = s , a2, a2'(saa) and a3 ~m,~ = (~Tm,z)RRR, for ~rn 2, z = s, a2, a2', a3 , (3.27)


~ 4 , z - - r Tit~ j/3 T3, 8 Tp tr - eijtqLsl CqLg2 ][uRg3CdRg,][dRs~CdRs6 ](T~) ~/3vap~,

for z = s, a2, a2'(saa), a2'(asa) and a3 ; (3.28)

- - E E r Tict..'~ j/3 lr T k y ~ 1~ "~ ~ 5 . z - - q kleqLgl(-'qLg2JUqLg3 L~qLg4J

Tp o" x [dag~Cda~6](T~) ~/3vsp~,

for z -- s, a2, a2', and a3 ; (3.29)

, E Tia j/3 Tk3, 18 ~ 6 z = ( EikEjl -[- ilEjk )[ qLgl aqLg2 ][ qLg3 C q L g 4]

X Tp o" [ ~ , ca~6 ](Tz) ~/3,~,

for z = s, a2, a2', and a3. (3.30)

As before with the SU(3)xU(1)- invar ian t operators, we define the natural multigenerational extensions of the single-generation S U ( 3 ) x S U ( 2 ) x U ( 1 ) - invariant operators of refs. [4, 10] as follows:

~m,s--- ~,~, m = 1, 2 (3.31)

~2,a2 ~ ~ 3 (3.32)

P4,a2 ------ ~4 (3.33)

~5,a2 ~ ~ 5 (3.34)

~ 6 , s ~ ~6" (3.35)

In the single-generation case, these operators will be explicitly denoted as (sg). The generational symmetries of the SU(3 )xSU(2)xU(1 ) - inva r i an t operators can be determined in a straightforward way. Using these, we have computed the minimal number of higher-generation fields, Nhg, and the number of independent operators N ( n ) , for each operator type (see footnote on p. 157). The single-generation Fierz

relations [20] ~ 2 ( s g ) - - ~ l ( s g ) = 3~3(sg) and ~/~6(sg)=--3~5(sg) reduce the number of independent operators to four. The extensions of these relations to the multigenerational case have the form of eq. (3.23) (after the SU(2) contractions are carried out) with X~ = X~ = R and X~ = L, X~ = R, respectively.

Of these multigeneration SU(3) x SU(2) × U(1)-invariant operators, obviously all of the ones given in eqs. (3.31)-(3.35) contribute to n ~--~ fi transitions through quark mixing. In addition, ~4,~, ~5,~, and ~6.~2 also contribute to these transitions. For e x a m p l e , (~4,s)121111 (where the subscripts outside the parenthesis are generation indices) yields the n,~-~fi operator

{ ( U ( U ) ) 1 1 ( u ( d ) ) 2 1 - ( U ( L U ) ) 2 1 ( u ( d ) ) l l } ( U ( ~ ) U~I d) )11{( R )11} 3

Ta /3 T3, 8 Tp o" X[UL(, , )CdL(, , )][UR(m)CdR(, , )][dR(, , )CdR(m)](T~)~/3v8 ~ , (3.36)

where, as before, (m) denotes mass eigenstate.


This completes our construction of six-quark SU(3) x U(1)-invariant and SU(3) × SU(2) x U(1)-invariant multigenerational B ° - 13 ° operators and our determination of the subset of these which contributes to n~-~fi transitions. It should be stressed that this analysis is independent of assumptions regarding details of grand unification and only relies on the established empirical success of the standard theory of strong and electroweak interactions.

4. Operator-graph correspondences, implications of various Higgs sectors for six-fermion ( B - L)-violating operators, and comments on the toal n~--~h

transition rate

As was emphasized earlier, the enumeration and construction of the various six-fermion (B-L) -v io l a t i ng operators can be performed purely by means of group-theoretical techniques, without any reference to the Higgs sector of the particle theory. However, having done this, it is of considerable interest to determine which Higgs fields are necessary in order that a given operator will result from some tree-level graph (and thus have the possibility of a large coefficient) or, equivalently, to specify which six-fermion, ( /3-L)-violat ing operators will occur in a theory with a given Higgs content, and what the corresponding coefficient functions will be. As in sect. 3, we shall concentrate on the case of the particular ( B - L)-violating process, n~--~fi. In order to carry out this program, we shall work out the correspondence between the multigenerational SU(5)-invariant, six-fermion, A ( B - L ) = - 2 operators constructed in sect. 2 and specific tree-level Feynman diagrams for B ° o 13 ° transitions. In a related analysis, the correspondence between the multigenerational SU(3)×SU(2)×U(1) - invar ian t operators constructed in sect. 3 and tree-level graphs will be given. Using these results, we shall discuss the relative sizes of the coefficient functions for the various operators. Finally, incorporating our earlier calculation of n o f f transition matrix elements in the MIT bag model, we shall comment on the actual n a i l transition rate as a function of the relevant Higgs masses and couplings.

The correspondence between the SU(5)-invariant multigeneration A (B - L) = - 2 six-fermion operators and tree-level SU(5) graphs can be inferred directly from the operators in eqs. (2.19)-(2.45). As an example, the tree-level graphs for the operators of class 2 are given in fig. 2. The others can be inferred in a similar way (see footnote on p. 157). The generation indices on the fermion lines are suppressed in the notation. Observe that all of the graphs involve triple-Higgs vertices. Several of these vertices consist of antisymmetric couplings of two Higgs fields transforming in the same way under SU(5), in order for such vertices to be non-zero, it is necessary that there be at least two different such Higgs (which also couple to fermions). This is indicated by the subscripts "1" and "2". Since the momenta carried by the virtual Higgs lines are negligibly small relative to the Higgs masses, a standard approximation replaces the Higgs propagators by the appropriate (mass) -2 values, thereby


5 ) i,[lOH < 5 5 > ,101H < 5 ~e

; i 5 :> f ~IOH I02H < 5 5 > ~ < 5

~SH ~45H

I0 ) ~ < T I0 > ~ < "5" 0 1-1 0 i -a

161

5 ) < 5 iWlOH

~e

5 > ~ ,l, isl't < 5

'~45 H

I0 > " ( 5 0 I-3

Fig. 2. Tree-level graphs corresponding to SU(5)-invariant multigeneration six-fermion zl ( B - L) = - 2 operators of class 1. Solid and dashed lines represent fermion and Higgs bosons, respectively. Numbers labelling the lines refer to SU(5) representations. Generation indices on the fermion fields are suppressed

in the notation.

yielding effective local operators, multipled by coefficient functions depending on the Higgs masses, Yukawa couplings and triple-Higgs vertices. On the basis of this correspondence, we can answer the question regarding the operators which occur in theories with given Higgs sector. Our results are presented in table 3. The left-hand column of this table is a list of the SU(5) operators of interest here, while the uppermost horizontal categories refer to various Higgs sectors, including the totality of possible Higgs fields that can couple to fermions (viz. the 5, 10, 15, 45, and 50), together with the 24 used to break SU(5) to SU(3)xSU(2)xU(1). The inclusion of additional Higgs such as the 40, 75, etc. would not directly affect our analysis. The entry "1" ("0") means that the operator does (does not) occur directly in a theory with the indicated Higgs sector*. The entry I(R) signifies that the operator occurs if and only if there are at least two R-dimensional Higgs representations in the theory. Some of the salient conclusions which can be derived from this table can be summarized as follows. First, note that for a given (B-L)-violating Higgs subsector, the presence or absence of a particular six-fermion, A(B-L)=-2 operator depends also on the nature of the (B-L)-conserving Higgs subsector. Evidently, if a theory contains a 5, 24, and 10 of Higgs, there are no mixing-angle favored, or equivalently, single-generation, six-fermion zl(B-L)=-2 operators that result. This is also true of a theory with the Higgs sector (5, 24, 45, 10). In contrast, if a theory contains a 5, 24, and 15 of Higgs, then the general multigenerational operators ~3, ~4, and ~5 are present. As will be discussed further below, Higgs propagator suppression can be far more important than mixing-angle suppression in determining the overall contribution of an SU(5)-invariant operator.

* Although an operator does not occur directly, one may be able to obtain it, for special values of generation indices, by a Fierz transformation such as eq. (2.18), from an operator that does occur directly. The table applies for arbitrary values of the generation indices.


TABLE 3

Correspondences between various SU(5) Higgs sectors and SU(5)-invariant six-fermion A ( B - L ) = - 2 operators

5,24 , 5 ,24 , 5 ,24 , 5 ,24 , 5 ,24 ,

5 ,24 , 5 ,24, 5 ,24 , 5 ,24 , 5 ,24 , 45 ,10 , 5 ,24 , 5 ,24 , 50 ,10 , 45 ,50 , 45 ,50 , 45 ,50 ,

10 15 10 ,15 4 5 , 1 0 4 5 ,1 5 15 50 ,10 50 ,15 15 10 15 10 ,15

~7t_ 1 1 0 1 1 0 1 1 0 1 1 0 1

01_ 2 0 0 0 1(10) 0 1(10) 0 0 0 1110) 0 1(10)

0'1 3 0 0 0 0 0 1 0 0 0 0 0 1

02_ 1 ~ ~ 0 0 0 0 0 0 0 1 1 0 1 1

02_ 2 0 0 0 0 0 0 1 0 1 1 0 1

02_ 3 1 0 1 1 0 1 1 0 1 1 0 1

02_ 4 0 0 0 1(10) 0 1(10) 0 0 0 1(10) 0 1(10)

£Y2-5 0 0 0 0 0 1 0 0 0 0 0 1

1~3_1 ~ ~2 , 3 0 0 0 0 1 1 0 0 0 0 1 1

~3_2 ~ ~ 3 0 1 1 0 1 1 0 1 1 0 1 1

~3-3 1(5) 0 1(5) 1(5) 0 1(5) 1(5) 0 1(5) 1(5) 0 1(5)

03_ 4 0 0 0 1(45) 0 1(45) 0 0 0 1145) 0 1145)

03_ 5 0 0 0 1 0 1 0 0 0 1 0 1

04-1 = ~ 4 0 1 1 0 1 1 0 1 1 0 1 1

0 4 2 0 0 0 0 1 1 0 0 0 0 1 1

• -3 1(5) 0 1(5) 1(5) 0 1(5) 1(5) 0 1(5) 1(5) 0 1(5)

~4-4 0 0 0 1(45) 0 1(45) 0 0 0 1(45) 0 1(45)

C4_ 5 0 0 0 0 0 0 0 0 0 1 0 l

0,*_ 6 0 0 0 1 0 1 0 0 0 1 0 1

0 4 7 0 0 0 1 0 1 0 0 0 1 0 1

C5_1 ~ ~5 0 1 1 0 1 1 0 1 1 0 1 1

05_2 ~ ~ 6 0 0 0 0 0 0 0 1 1 0 1 1

c5_ 3 1(5) 0 1(5) 115) 0 1(5) 1(5) 0 115) 1(5) 0 115)

Os 4 0 0 0 1 0 1 0 0 0 1 0 1

~s s 0 0 0 0 1 1 0 0 0 0 1 1

05_ 6 0 0 0 1(45) 0 1(45) 0 0 0 1(45) 0 1145)

~75_ 7 0 0 0 0 0 0 0 0 0 l 0 1

The entries 0, 1, and 1 (R) mean that for the given Higgs sector, the operator is absent, present, and present lit there exist at least two

R-dimensional Higgs representations, respectively, for arbitrary values of the generation indices.

In this connection, we note that the operators ~74_2 and ¢75-5 can make relatively large contributions to n ~ f i oscillations even though they are mixing-angle suppressed. Neither of these operators results from the Higgs sectors (5, 24, 10), (5, 24, 45, 10) or (5, 24, 15), but both are present in a theory with the Higgs sector (5, 24, 45,15) . The last of the above-mentioned sectors also yields the mixing-angle favored operator 1~3-1 ~ ~ 2 ' Finally, if one has the Higgs sector (5, 24, 50, 15), then one obtains the two remaining mixing-angle favored operators ¢Y2-1 =- ~1 and 65_2 -= ~6 (with arbitrary gi indices).

It is perhaps appropriate to comment on non-minimal Higgs sectors of the type of interest here. A possible view is that it is preferable to use as small a Higgs sector


as would be allowed by the requirement of producing phenomenologically acceptable fermion masses. However, one is led to a generally different viewpoint if one considers a given grand unified theory, such as SU(5), as embedded in a larger one, such as an SO(10) theory [23]. The status of ( B - L ) as a symmetry is, of course, fundamentally different in SO(10) than in SU(5), since it is a generator of the former group, whereas it is not a generator of the latter group, but rather a linear combination of local and global symmetries (cf. eq. (1.5)). Thus, B - L may or may not be broken at the lagrangian level in SU(5), depending on whether a 10 or 15 of Higgs is absent or present, whereas it is necessarily exact at this level in SO(10). Moreover, B - L is spontaneously broken in realistic SO(10) theories, either (a) at the highest mass level, as is the case if one uses the 16 or 126 of SO(10) Higgs to break S O( 10 )~ SU(5); or (b) in a possibly distinct stage from the initial breaking, as is the case if one uses an SO(10) 45 of Higgs to break SO(10) ~ SU(5) x U(1)B-L, and then breaks the U(1)B--L at a lower mass scale; or (c) via the symmetry breaking scheme relying on an SO(10) 54 of Higgs to yield S O ( 1 0 ) ~ S O ( 6 ) x S O ( 4 ) ~- S U ( 4 ) × S U ( 2 ) × S U ( 2 ) followed by further breaking, for example, to SU(3)× SU(2) x S U ( 2 ) x U ( 1 ) B _ L × U ( 1 ) ' . In contrast in an SU(5) theory without a 10 or 15 of Higgs, B - L is an exact symmetry after all spontaneous symmetry breaking has occurred. The Higgs fields that play a role in the spontaneous breakdown of SO(10) all introduce non-minimal SU(5) Higgs components (i.e. Higgs fields other

than the 5, 24, and probably also 45). Thus, the 16so~10~ contains a 10su~5~; the 45so(1o) contains 10su(5) and 1-0sv~5); the 54so(~0) brings in a 15su(5) and 15su(5); and the 126so(10) contains a 10su(5), 15su(5) and 5--0su(5). Similarly, the generation of fermion masses at the SO(10) level can introduce non-minimal SU(5) Higgs. The SO(10) Higgs that couple to fermions are the 10, 120, and 126. The first of these cannot be used alone to give the fermions masses, because this would lead to the untenable mass relation m~,, = m% (as well as the probably unacceptable relation m% = mdg). Hence, one must also rely on the 120 or 126; in the former case one thereby introduces a 10 and 1--0 of SU(5) Higgs, and in the latter case the additional Higgs fields mentioned above. It is certainly true that in SO(10) one can arrange the symmetry breaking in such a way as to preserve B - L as an approximate (local) symmetry down to mass scales far below 1014 GeV. Our point is simply that when one views SU(5) as embedded in SO(10), it is quite reasonable to consider non- minimal Higgs sectors such as those listed in table 3.

The correspondence between multigeneration SU(3) xSU(2) ×U(1)-invariant n ~ fi transition operators and tree-level Feynman diagrams can also be determined. The results are presented in fig. 3. As before, the quark fields are group eigenstates, with generational indices suppressed. The representations of the Higgs are indicated in the parentheses adjacent to each Higgs line, where the numbers refer, respectively, to the SU(3), SU(2), and U(1)y representations.

The SU(3)xSU(2)×U(1) - inva r i an t B°,~--~l] ° transition operators which result from the SU(5)-invariant operators are listed in table 4. The operators of class 1


( d R d R • • u R dR > nV(6,1, - 4 / 3 ) ~ ( 6 , 1 , 2 / 3 )

dR > i~ (6 ,1, -4 /3)• dR dR • ~/ ,~ (6 ,1 ,2 /3) • UR

\ ~ \ ( 6 , I 8/3) \ , I~x ( 6 , I , - 4 / 3 )

U R • ~ < u R d R ) ~ • d R

PI , s = P I P2 ,s -= P2

< UR qL • • qL dR • ~ (3 ,1 ,2 /3 ) ~(3, I ,2 /3)

/ / ( 3 , 1 , 2 / 3 ) i / / ( ~ , I ,2 /3 ) d R > ~ • U R d R • ~ • u R

\ \

~x (6, l , - 4 / 3 ) ~\ (6 , I , - 4 / 3 )

d R > ~ < d R d R • ~ < d R P 2 , 0 2 - P5

P 4 , o 2 - P 4

< qL qL ) < qL qL • ~ .~(3 '1 ,2 /3) 0~(6,1,2/3) f ~¢~,1,2/3) ," ;(6,1,2/3)

qL • k < qL qL ) I x ~ < qL ( 6 , l , - 4 / 3 ) x \ ~\ ( 6 , 1 , - 4 / 3 )

d R > ~ ,t d R d R > ~ < dR

P 5 , 0 2 - P5 P5 ,s

qL ' TI6,3,2/ )qL qL ) qL ~(6 3,213) f - ~ ( ~ , 3 213)

qL • t , I ' x ( qL qL > ~ < qL I~ ( 6 , 1 , - 4 / 3 ) I~ ( 6 , 1 , - 4 / 3 )

k x

d R > \ • d R dR • \ • d R

P6 ,s = P6 P 6 , o 2

Fig. 3. Tree-level graphs corresponding to multigeneration six-fermion SU(3) ×SU(2) xU(1)-invariant operators contributing to n - f i transitions. Generation indices on the fermion fields are suppressed in the notation. The dashed lines refer to Higgs bosons; these transform according to representations indicated by (Rsu(3), Rsu(2), Ru(1)v). The order of the fermion lines is chosen to be the same as that in the SU(5) operator which yielded the given SU(3) × SU(2) x U(1)-invariant operator, but the graphs are symmetric, up to sign and redefinition of generational indices, under interchange of initial and final fermions on a particular line. Thus, for example, in the graph for ~3, either or both of the upper two

lines could have been drawn in the reverse order, (ui~dR), rather than (dRUR).

and , in add i t ion , (~2-3 and ~74_5 do n o t c o n t r i b u t e to t he se t r ans i t i ons at all. A g iven

S U ( 5 ) o p e r a t o r y ie lds o n e o r a l i nea r c o m b i n a t i o n of t w o S U ( 3 ) x S U ( 2 ) x U ( 1 )

o p e r a t o r ( s ) . A s d i scussed b e f o r e , o n l y a subse t of t h e s e B ° o ~o o p e r a t o r s c o n t r i b u t e

to n~-~fi t rans i t ions . I n t ab l e 5 we g ive t h e coef f ic ien t f u n c t i o n s wh ich m u l t i p l y

t h e S U ( 3 ) x S U ( 2 ) × U ( 1 ) - i n v a r i a n t n~-~fi t r an s i t i on o p e r a t o r s t ha t ar ise f r o m


TABLE 4

SU(3) × SU(2) xU(1)- invariant B°,--~B ° transition operators resulting from the six-fermion

/t (B - L) = - 2 SU(5)-invariant operators

SU(3) x SU(2) x U(1) SU(5) operator B%--~B ° operator

02 - 1 ~ 1 ,s ~2-2 ~1 ,a2' 02-4 ~1,a3 02-5 ~1,a2 03.1 ~2,~ 03-2 302,a2 03-3 ~°2,a3

~2 , a2 ' 03-4 ',~2,a3

i~2,a2, 03"5 ~2.a3 04-1 ~4,a2 04-2 ~4,S

04-4 ~4,a2'(saa) 04.6 ~f~4,a2'(asa)

k~4,a3 04-7 ~4,.2'(~a) 05-1 ~5,.2 05- 2 ~b6, s 05-3 ~5,a3 05.4 ~5,a2'

05-5 L~6,~2 05.6 ~'6,~3

05,a2, 05"7 k~6,a2 ,

165

six-fermion SU(5)-invariant 9perators in the context of an SU(5) grand unified theory. In order to express the mixing-angle dependence, we actually list the coefficient functions for the SU(3) × SU(2) × U(1) operators comprised of u and d quark mass eigenstates rather than gauge group eigenstates as in eqs. (3.12) and (3.13). In accordance with the assumption of hierarchical quark mixing, we include only the lowest-generation contributions. Thus, for an SU(5) operator ~7 which can occur at the single-generation level, we only give the contribution to n~--~fi transitions from (~7) 111111, where the subscripts are generation indices. Similarly, for an operator which makes a non-vanishing contribution only if Nhg = 1, we present only the term from the

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opera to r in which the relevant gk = 2 ra ther than a higher value. The Yukawa couplings

~J~R2CI~xR1 where the R ' s are deno ted as hRI,R2;RH for the vertex of the form x , t~RH, are SU(5) representat ions. The triple Higgs couplings are denoted d /R. RH R.3 for a vertex of the form HR~HR:HR~, i.e. all of the Higgs lines are taken to point into the vertex. The

Higgs p ropaga tor masses are labelled as in fig. 4 by (Rsu(3), Rsu(a ), Ru(1)y)Rsucs). A concise character izat ion of the mixing angle and Higgs p ropaga to r mass dependence of

the coefficient functions is given in the last column of table 5. A generic constant , e, is used to refer to off-diagonal quark mixing-matrix e l -ments , with [( U(x q) )ijl - e I~-Jl. The

masses of various Higgs fields t ransforming as different representat ions of

SU(3) × SU(2) × U(1) are g rouped into three sets, labelled I1,12, and I3, which reflect the lower bounds on their masses (see below), such that (mlj)min < (mlk)min iff j < k.

These coefficient functions are naturally defined on mass scales greater than mN or

Ac ~ 0.1 5 G e V and will undergo some renormal izat ion when considered at the latter mass scales relevant for n*--~fi transit ion matrix elements. Since these effects are

logari thmic and hence relatively small compared with the o ther uncertaint ies which

a l ready exist, such as the precise values of the Higgs masses and couplings, it is not of

immedia te interest to include these renormalizat ions. Moreover , for experimental ly impor tan t operators , the dominan t mass scales characterizing the diagrams

responsible probably canncrt be much larger than about 103-10 4 G e V (see below), so

that any perturbat ively calculable renormal izat ion effects would be ra ther small.

In order to determine the relative and absolute sizes of the coefficient functions,

one must examine the various factors that contr ibute to them. As noted above,

singly off-diagonal mixing-matr ix elements are assumed to be of o rder e; we take

e ~ 0.1 f rom the known Cabibbo mixing ]( VCKM) I2 ] --1( VCKM)21 ] ~ Isin 0Cabibbo] 0.23. Regard ing the H iggs -Yukawa couplings, some of these are constra ined by

SU(3) × SU(2) × U(1) SU(5) Higgs relevant Mass quantum numbers to n-fi oscillation

10 TM GeV mGU

~>101° GeV mi3

~> 10 7 GeV miz

~>10 3-4 GeV mH

--10 2 GeV row, z

(3, 1,-~-) (3, 1,-2) 5 1, _8) (3, 1,-38)~

((33: 3,-~) (3, 3, --2)45 {(3, 1,4) (3, 1, ~)~6

I (6, 1,-4) (6, 1,--4)15 (6, 1, 2) (6, 1, 32)~

l (6, 1, 8) (6, 1,8)so (6, 3,2) (6, 3, ~)~o

Fig. 4. Approximate lower bounds on masses of Higgs bosons relevant for n*-~fi transitions. The SU(3) ×SU(2)×U(1)y representations of the Higgs fields are labelled as in fig. 3. Subscripts on the Higgs in the right-most column refer to the SU(5) Higgs representation containing the indicated

SU(3) × SU(2)× U(1)v Higgs field. See text for further details concerning these bounds.

168 s. Rao, R.E. Shrock / Six-fermion operators

fermion mass generation. Thus, the diagonal couplings are

hl%,10g,;~ = mu~/(8vs) ,

hssclO,,;~ = ( Amdg i, Ame~,)/ V5 ,

and

(4.1)

(4.2)

hs~ 1% .4~ = ( / t rna , , -~Am% )//(2/)45 ) (4.3)

where, as before, Ug=l = u, Ug=2 = c, ug=3 = t, etc.; (/45) ~ and (H45)~ ~ are, respectively, the 5 and 45 of Higgs in SU(5), with vacuum expectation values

((H5)5)0-- vs, (4.4)

((Has)iS)o 1 45 = - ~ ( ( H 4 ~ ) 4 )o

/')45, for i = 1, 2, or 3 , (4.5)

and Amf refers to the contribution to the mass of the fermion f, from the given Higgs coupling. The empirically observed hierarchial fermion mixing implies that the off-diagonal Yukawa couplings, involving fermions of generations gi and gj ~ gi are smaller than the diagonal ones by factors of order e Ig`-gjl. Next,

hsg,.sgi;15 = m~) /v15 , (4.6)

where m (M).~, denotes the (Majorana) mass of the neutrino v% ; (H15)=o(= (His) ~ ) is the SU(5) 15 of Higgs; and

((H15)55)0 --- v15<< Vs, v45. (4.7)

The latter inequality is required, first, in order to avoid upsetting the phenomenologically successful relation mw = mz cos 0w in the standard model (since this is violated by the presence of SU(2) triplet Higgs) and, secondly, to avoid violating experimentally established upper limits on neutrino masses. Since the 10 and 50 of Higgs do not contribute to fermion masses at the lagrangian level, the couplings hs~. ,.5,/1°. and h1%.1o~S6 are not constrained in the way that the previously discussed Yukawa couplings are. It is plausible, nevertheless, to assume that these latter couplings are comparable to those involved in fermion mass generation. Thus, finally, we shall take for the relevant first generation Yukawa couplings, hR,=,.R~=,;H g( m~.d/ row) ~ 1 0 - 4 - 10 -5.

We next consider the question of the allowed masses for the Higgs bosons relevant to multigeneration n o f i transition operators. In fig. 4 we show a schematic level ordering for the minimum allowed masses of various Higgs fields. The mass scales of ~ 102 GeV and ~ 1 0 1 4 GeV characterizing, respectively, the breaking of SU(2)x U(1) and SU(5) are indicated on the figure. The associated severe gauge hierarchy problem is well-known. For our present purposes, we recall that it is standard to take the mass of the physical electroweak Higgs boson in the (1, 2, 1) of the SU(5)


5 of Higgs (in the same notation as in Fig. 4) to be of order v s - 102 GeV, which (a) introduces an extremely large mass splitting between the masses of different Higgs fields in the theory, in addition to the already existing huge splitting between vacuum expectation values and gauge boson masses, and (b), even more drastically, introduces the same great mass splitting between different SU(3)xSU(2)xU(1) components of a single SU(5) Higgs representation, here the 5. Given this fact, it is no less unnatural to consider possible values of Higgs masses intermediate between mGu and mw.z and, furthermore, to allow large splittings between different components of a given SU(5) Higgs representation. There have been a number of studies of allowed Higgs mass hierarchies in grand unified theories [2, 5, 15, 24, 25]. A general requirement is that any component of Higgs fields that can mediate proton or bound neutron decay at tree level must be sufficiently heavy not to contradict the current lower limit on the proton lifetime rp i> 3 x 103o y. Since the actual limit

2 2 applies to the product of Yukawa couplings and the Higgs masses hv/mn, any resultant bound on Higgs masses obviously entails some assumption about the relevant hr. If one takes h v - 10 -4"5, then one obtains mH~ 101° GeV. The Higgs that can mediate proton decay via tree-level graphs which yield effective four- fermion operators which thus have minimal dimension (in mass units) d - -6 , have been determined by Weinberg [15]; they are (3, 1, _2), (3, 1, -~), and (3, 3, _2), in the same notation used before in fig. 4. These must therefore have masses greater than or equal to the lower bound cited above, which we label as the intermediate mass m13. A Higgs field transforming as a (3, 1, 4) representation of SU(3) x SU(2) × U(1) y cannot cause proton decay, but can lead to bound neutron decay if the theory also contains (3, 2,1) or (3, 2, 7) Higgs fields, via graphs involving two Higgs propagators and a triple Higgs vertex [5]. The resultant effective local operators have dimension d = 7 and are of the form

h2~/ _. 4~m2r3 2 l~e'~t3v[dTgxCd~g2][ILg3dRg'*]e6~bIY=-l' (4.8)

m : ( ~ , l , - ~ j ~ , ,g~

h2 ~ ~c~.~.[dTag, Cd~gz][ l-~_g3U~g4]Eqf~Jy=l , (4.9) m2(3, 1, -4)m2(3, 2, 7)

where

/Lg=(veL) , (4.10) x e L / g

and the Higgs fields transform like (1, 2, q: 1) and have vacuum expectation values (~b ~=±1 )0 = 6i,~1v±~. The SU(2) contractions of the relevant parts of these operators thus reduce to V_l[~Lg3d~g4] and V+I[gLg3U~g4], respectively. The two operators yield the respective decays n--> e-(~r +, p+, K +" • ") and n--> u(Tr °, pO, wo, K o" . . ) . Tak- ing h y - 10 -4.5 and the illustrative value d/--, 10 3 GeV (see comments below), we obtain the lower bounds [m(3,1,4)m(3,2,1)]1/2>m12 and [m(3,1, 4)


m(3, 2,7)]1/2> m12,- 106_107 GeV. To facilitate graphical presentation, we take

m(3, 1,4) _ m(3, 2, ~) or - m ( 3 , 2, 7), which yields the lower bound shown in fig. 4.

As was emphasized in ref. [15], even if a certain set of Higgs fields does not contribute to zaB ~ 0 baryon decay at tree level, it is still necessary to check that diagrams with one or more loops do not give rise to too rapid a rate for such decay. We have performed this check for all of the intermediate mass Higgs relevant to our analysis, shown in fig. 4, and have found that loop effects do not imply more severe constraints than those which one already has from other sources. Intermedi- ate-mass Higgs have two further effects. First, they alter the conventional scenario for the generation of a baryon-number asymmetry in the universe [28]. However, various studies have found consistent schemes for baryon-number generation, with SU(3) sextet Higgs having very low masses << 107 G e V [24]. Secondly, Higgs with

sufficiently low masses ~<103GeV can affect flavor-changing processes such as K ° - I ~ ° and D ° - I ) ° mixing, but this does not provide us with a more severe constraint than the totality of limits from other sources [5]*. Finally, of course, low-mass Higgs mediate the n <--> fi oscillations and consequent mat ter instability of interest here; as will be discussed below, the current limit on mat ter instability, in conjunction with our earlier calculation of n<-->fi transition matrix elements and an analysis of fi annihilation (see below) yields the lower bound for color sextet Higgs rnll ~> 103-4 GeV. This set of lower bounds is summarized in fig. 4.

Regarding the size of the triple Higgs couplings in the coefficient functions of table 4, it is natural to take Id~H1.H2,H31- mH if all of the three Higgs fields have masses of the same o r d e r - mn. If the masses of the Higgs are very different, then, in order to avoid rendering perturbation theory unreliable we shall assume that

[ddH,,H2,H3[-- min (mH,, mH2, mH3)**. Given these inputs and the assumption that the various Higgs in the sets I1 and

I2 do indeed have respective masses of order mi1 and ml2 (some Higgs in set I1 must have a mass - rn~ l if n ~ f i oscillations are to be experimentally observable), we derive the coefficient function inequalities (suppressing common Yukawa couplings)

12-1Cl,sl, ]B_lC2,s[, 15_2C6,s1- 1/m~, > 14_2C4,s[

> 15-5C5,s1 misl

-- E2/m51 )) ACB, (4.11)

for other values of A and B. These results demonstrate the importance of considering multigeneration ( B - L)-violating operators; the SU(5) operators G-2 and ¢75_5, and

* Intermediate-mass Higgs also modify the values of sin 2 0w and mot:; studies of these effects (see ref. [25]) help to set the mass bounds given in fig. 4.

** Maintaining this condition in higher orders of perturbation theory can require fine tuning. We view this as one more manifestation of the mass hierarchy problem in grand unified theories.

s. Rao, R.E. Shrock / Six-fermion operators 171

the resulting S U ( 3 ) x S U ( 2 ) x U ( 1 ) operators ~4,s and ~s,s would vanish in the single-generation limit, or equivalently, in the limit of no fermion mixing, but, given the reasonable assumption made about this mixing and the possible Higgs mass hierarchy discussed above, they dominate by many orders of magnitude over other operators contributing to n ~ fi transitions, because of different Higgs mass dependences of their coefficient functions, in particular, the presence of only Higgs SU(3)

sextets rather than any Higgs SU(3) triplets. We proceed to consider the actual n*+fi transition rate due to operators of the

type analyzed above. Let us first define some notation. In the presence of n*+fi transitions, the mass matrix describing the coupled n, fi system is

[(nlnln) (nlnlfi)~ m = \ ( N H i n ) < IHIfi>]" (4.12)

In the idealized case of field-free vacuum, this takes the form

=(mn-½iAn 8rn ) (4.13) m \ 8m mn-½iAn '

where

8m -= (filHIn)

= - ( i l l / ~ e f r l n > , (4.14)

and An = 1/rn is the rate for the regular weak decay of the neutron. The resulting mass eigenstates are easily determined to be

In±) =,/~[In) ± Ifi)], (4.15)

with masses

m± = rand: 8m. (4.16)

Thus, if one starts with a pure beam of neutrons at proper time r = 0 , after propagation for a time r, there,is a non-zero probability to find a c.omponent of

antineutrons in the beam given by

p(n( r ) = fi) ~ I( ln(r))l 2 = sin 2 [(Sin)r] e -a.~

=- sin 2 ('/'//'/'nil) e -*"~. (4.17)

(This definition " / ' n i l = 1/8rn conforms with a common usage; however, the reader is cautioned that some authors use definitions of rn~ which differ from eq. (4.17) by factors of 2 or 2~r. The n*+fi oscillation time enters in a similar way for realistic propagation experiments, but the mass matrix has a more complicated form because of the presence of residual terrestrial magnetic fields.)


The n o fi transition gives rise to matter instability. In turn, a lower bound on rn~ can be derived from the existing lower bound [26, 27] "gmatter(instability) ~ " 3 x 1030 y. To obtain this limit, one begins with the mass matrix for bound neutrons, which is

= ((mn)et r 8m

m \ 6rn (m~)efr] ' (4.18)

where

(rn(,))eff = ran+ V(~, (4.19)

and V(~) is the effective nuclear potential to which an (fi) is subject in the nucleus. In particular, V, has an imaginary part, which represents the decay of matter via fi annihilation. Carrying out the diagonalization of the mass matrix (4.18), calculating the eigenstates and eigenvalues for the nuclear potentials, and finally computing the decay rate/'matter due to n o fi transitions, one finds that the present lower limit cited above on the stability of matter implies the lower limit rna~>fewX 107 s, or equivalently, the upper limit on the n o f i transition amplitude [8, 9, 29-31] 16ml <~

10-31-10 -32 GeV. Planned future fine-grained nucleon decay detectors will be able to improve the limit on matter instability and hence, a for t io r i , on this limit on n o fi transitions (or, of course, observe an effect). With an optimized detector, the signature for matter decay due to such transitions is definitely distinguishable from the baryon-number violating decay of an individual nucleon, since it yields a final state consisting primarily of pions, with average multiplicity of 4-5, total charge zero, total energy - 2mN, and total momentum zero, to within Fermi motion correc- tions. An alternative technique is to search for n o f f transitions in a beam of neutrons. There are currently several such propagation experiments, either running or proposed [30-33]. One such experiment, has now been completed by a collabor- ation at the Grenoble reactor and has obtained the lower limit [32] rn~> 105s (90% CL), which is a weaker limit than the one derived from the stability of matter but has the advantage of not being subject to uncertainties due to nuclear physics effects, as the latter, to some extent, is. Future propagation experiments may be able to probe for n o f i oscillations up to the level Tnfi ~ 10 8 s.

In order to use a limit on rn~ to gain information about (B-L) -v io l a t i ng parameters in the underlying particle theory, one must know the matrix elements of the n o f i transition operators. These were calculated, within the framework of the successful MIT bag model, in our earlier work. Some further results concerning the (gentle) dependence of the matrix elements on the quark masses are given in appendix B. The main feature of this work that is relevant here is that the sizes of the bag-model matrix elements are set primarily by the overall dimensional factor N6p-3/(47r)2, o r more transparently, ( / ~ ) 6 p 6 / ( 4 7 r ) 2 (see ref. [10] or appendix B for definitions). This factor is equal to ~(0.5-0.6) x 10 -5 GeV 6, and the resultant matrix elements, which also involve (three types of) dimensionless integrals of sixth-order polynomials of spherical Bessel functions, are in the range o f ~ f e w x 10 -5 GeV,

s. Rao, R.E. Shrock / Six-fermion operators 173

much smaller than the most naive estimate based on dimensional considerations alone, viz. m 6.

The SU(5) operators that have the potentially largest coefficient functions are

~2_1~1, ~3_1~-~3, and ~5_2~-~ ~6 (with gl . . . . . g 6 = l ) and thus the SU(3 )x SU(2) x U(1)-invariant operators ~1, ~2, and ~6. Using these in conjunction with the results discussed above, we obtain the lower limit

rail ~ 103 G e V . (4.20)

Our result can be viewed in the other direction as a statement that Higgs bosons in the set I1 with masses in this range (for the values of Yukawa and triple Higgs couplings assumed) can give rise to n*-~fi oscillations that may be experimentally observable soon. Clearly, given the strong dependence 8 m o c m l ~ , unless the necessary Higgs masses lie in a rather narrow interval above this lower limit, such oscillations will not be likely to be observed in feasible experiments. In any case, it is of great interest to pursue further the theoretical study of, and experimental search for (B-L) -v io l a t i on , and in particular, n,~-~fi transitions.

We would like to thank L. Wolfenstein for valuable discussions and G. Senjanovi6 for informing us of the work in ref. [20]. This research was supported in part by NSF grant no. PHY 81-09110.

Appendix A

( B - L ) SELECTION RULES FOR FERMION OPERATORS

In this appendix we shall analyze the (B - L) selection rules for (Lorentz-invariant) fermion operators which are invariant under SU(3)c × SU(2)L x U(1) y and SU(3)c × U(1) ..... Our results will apply, a fortiori, to SU(5)-invariant fermion operators. For the first analysis we begin with:

Theorem 1. All SU(3)c×SU(2)r- invariant operators composed of nf=2nodd fermion fields, where nodo is a (positive) odd integer, and having dimension d = 3nf, violate ( B - L).

Proof. Let I and q denote left-handed SU(2)-doublet leptons and quarks, respectively, and u, d, and e, the right-handed SU(2) singlets in the standard SU(3)x SU(2) x U(1) theory [17]. We also allow for the possibility of right-handed neutrino singlets, VR (see the footnote on p. 2). Further, let l -denote either l~-= (I¢)R or TL~ (ICT0)R; e denote either e~_ o r ( e C y o ) L , and so forth for the other fields. An SU(3) × SU(2)-invariant fermion operator ~7 has the generic form

t7 = l", rnrqnqq°"u'"fi~"dn~d na ene~"~v "~ ~ ° . (A. 1)


The condition that (7 be Lorentz-invariant implies that

nl"~- nq '{- n i l+ nd-~ n~-~ n~, = 0 mod 2

-~ 2p, (A.2)

n i + n q + n u + rid+ h e + n~ = 0 rood 2

---2q, (A.3)

where p and q are (positive semidefinite) integers. The condition that 6 be SU(3)¢- invariant requires that its total triality vanish, i.e.

n q - n~l -{- n u - n o + n d - na = 0 mod 3

--- 3r. (A.4)

Finally, the condition that ~7 be SU(2)-invariant implies that its total 2-ality vanishes, i.e.

nq-- net+ ne-- n g = 0 mod 2

-= 2s, (A.5)

where r and s are integers. Now

( B - L ) e = ~ ( n q - n q + n , - n Q + n o - n a ) - ( n e - n ~ + n e - n e + n ~ - n ~ ) , (A.6)

so that ( B - L ) c = 0 iff

( n e - ne+ n e - ne + n ~ - n~) = r. (A.7)

Hence, the total number of fermion fields comprising 6, nf, can be written as

nf= 3r + 2nq + 2nQ+ 2na+ ne + n g+ ne+ n~ + n~ + n~

= 4 r + 2 ( n ~ + n ~ + n a + n ~ + n e + n v )

= 4(r + p - s ) , (A.8)

which contradicts the premise that nf= 2nodal QED. Note that this theorem does not require that (7 be U(1 )y invariant. Two corollaries are:

Corollary 1. All SU(3)c × SU(2)L× U(1)y-invariant operators composed of nf = 2nodd fermion fields, where no0d is a (positive) odd integer, and having d =3nf, violate (B - L).

Corollary 2. In a ( B - L)-conserving theory, SU(3)c × SU(2)L-invariant operator products with 2nodd fermions are not generated to any order in perturbation theory.

Selection rules for SU(3 )x SU(2)× U(1)-invariant operators are specified by the next result.

Theorem 2. Let ~ be an SU(3)xSU(2)×U(1) - inva r i an t operator consisting of n~ fermion fields, with dimension d =3nf . The maximum value of I(B-L)cl in the


standard theory with no VR'S is given by

[ (B-L)c[max=nf -4-4 ~ O ( n r l O k ) , (A.9) k = l

where 0 ( x ) = 0 for x~<0 and 1 for x > 0 . The possible values of [(B-L)cl are given by

I(B-z) l--[(B-Z) lmax-4l, (m.10)

where

1 = 0, 1 . . . . [~[(B- L)eI] I (A.11)

and [v]i denotes the integral part of v. All values of ( B - L ) are reached for single-generation operators and hence, a fortiori, for multi-generation operators.

Proof. These results are proved by maximizing [ ( B - L ) d , subject to the constraints (A.3)-(A.5) and, in addition, the constraint of U(1)y invariance. If one extends the theory to include VR fields, then [(B-L)c[max = nf, as realized by the operator [ pT CPR](n~/2).

As one would expect, operators which are only invariant under the smaller group SU(3)c x U(1) .... obey less restrictive ( B - L) selection rules. By methods similar to those used above, it is easy to prove the following theorem.

Theorem 3. Let t7 be an SU(3)c×U(1) .. . . -invariant operator comprised of nf fermion fields. Then in the standard theory the possible values of [ (B-L)~[ are given by

I(B - L)c[ = I(B - L)olmax- 21, (A.12)

where I(B-L)olm.x is listed in eq. (A.9), and l is given in eq. (A.11). Again, all values of ( B - L ) are realized for single-generation operators.

For the main text, we only need the special cases of eqs. (A.9)-(A. 12) for six-quark operators, viz. that [ ( B - L ) ~ l = 2. For our analysis of SU(5)-invariant six-fermion operators in sect. 2, we use the fact that, given the selection rule of theorem 2 and eq. (1.5), such operators have IXc[ = 10.

We have actually explicitly constructed all possible types of six-fermion SU(3) x SU(2) x U(1)-invariant operators, but since the ones of primary physical interest are those dealt with in the text, namely six-quark operators, we shall not present the results of this construction here.

Appendix B

FURTHER RESULTS ON THE QUARK MASS DEPENDENCE OF THE MATRIX ELEMENTS OF n~-*fi TRANSITION OPERATORS IN THE MIT BAG MODEL

In our earlier calculation of the matrix elements of the full set of n*-~fi transition operators in the MIT bag model [10], we used the effective quark masses

176 s. Rao, R.E. Shrock / Six-fermion operators

mu = md =- mq and neutron bag radius, R, in the standard MIT " A " fit and, for

comparison, also the values of these quantities in the MIT "B" fit. These are, respectively, mq = 0, R = 5.00 GeW -1 and mq = 0.108 GeV, R = 5.59 GeV -1. It was

found that the three types of dimensionless integrals, Ja.b.c, and the actual matrix

elements, did not differ markedly for these two parameter sets. It is also of interest to

examine how the matrix elements vary as continuous functions of quark masses intermediate between the two discrete values of the MIT A and B fits. It is obviously

reasonable to retain the property m, = rnd for such intermediate values. To proceed,

one must construct an approximate interpolating formula for the bag radius as a

function of rnq. Since R changes only by about 10% between the two MIT fits, it is justified to use a linear formula

R mq 5.00+5.46 (B.1)

1 GeV -1 1 GeV"

The resulting MIT bag model matrix elements for the different (first-generation) SU(3) x U(1)-invariant operators, including a subset proportional to the SU(3)x

SU(2) x SU(1)-invariant operators, are shown in fig. 5 as functions of mq and mqR.

As is evident from fig. 5, the matrix elements are slowly varying functions of mq.

In all cases except ( ~ I ~ R L L and ( (~2 )RLL, the magnitudes of the matrix elements

4

3

2

I (D > 0

2 - 1 i 0 -2

A -3

~ - 4 V -5

-6

-7

-8 0

m q ( M e V )

25 50 75 I00

~ f<(~2>LLR

< 0 2 > R L L

125 150 i i

<01> 7 - - - - - - - - ~ ~ R L L

<03>LRR

< 0, ~,R

< (~I >RRR I I I

0. I 0.2 0.3 I I

0.4 0.5

mqR

I I I 0.6 0.7 0.8 0.9

Fig. 5. Matrix elements of n,~,fi transition operators in the MIT bag model, as functions of u, d quark masses, mq, and neutron bag radius, R.


decrease with increasing mq and hence increasing R. As was observed in ref. [10], this is easily understood as being primarily due to a change in the common dimensional factor, N6p-3/(4"n')2=(ffl)6p6/(47r) 2, w h e r e p is the bound state momentum of the u or d quark in the neutron bag and/V =- p - 3 / 2 N is the dimensionless quark wave-function normalization factor, given by

1 , [ # J J [ _ ~ + : sin 2 , + c o s 2 , ~ ] , /~_2 =.~ [{so_ ~ sin 2~} + E q - mq 1 1 (B.2) Eq + mq

where s r is the smallest positive solution to the equation

tan ¢ = 1 - m q R - (s ~2 + (mqR)2) 1/2' (B.3)

and p - - ¢ / R . T h e above-mentioned dimensional factor decreases monotonically from 0.650 × 10 -5 GeV 6 to 0.529 × 10 -5 GeV 6 as mq increases from 0 to 0.108 GeV. (This is mainly due to the decrease in N from 0.777 to 0.752, since the increase in R is largely offset by an increase in ¢ from 2.043 to 2.281, so that p remains nearly constant, varying only from 0.409 to 0.408 GeV.) In the two cases where matrix elements cross through zero, the behavior is controlled by cancellations between terms from the J integrals.

These dimensionless integrals, Ja, Jb, and Jo defined in ref. [10], change slowly and monotonically between their values in the MIT A and B fits. Since plots of these functions are essentially just straight lines, they can be inferred from our earlier results, and we omit a graph of them.

Finally, for reference, we note that the phase conventions used in ref. [10] and here in the definition of the n and fi states are indicated by

"t~ tv In)=4~se~(/~*d~.b~b*~--/~*d~bd .bu )10), (B.4)

= (OI4~e,.~(du.d~ do ~ "~ ' '~ , (ill X "~ * "A-.dudddd) (B.5)

where 6~ * is the creation operator for a quark with flavor f, color A, and spin up (as indicated by dot), and d~ * is the analogous creation operator for an antiquark.

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Six-fermion (B−L)-violating operators of arbitrary generational structure