Chiral lagrangian calculation of nucleon decay modes induced by d=5 supersymmetric operators

Nuclear Physics B229 (1983) 105-114 © North-Holland Publishing Company

CHIRAL L A G R A N G I A N CALCULATION OF NUCLEON D E C A Y MODES INDUCED BY d = 5 SUPERSYMMETRIC OPERATORS S. CHADHA and M. DANIEL

Rutherford Appleton Laboratory, Chilton, Didcot, England Received 17 June 1983 We consider the general baryon-number violating operators which are produced by dressing the supersymmetricdimension-five operators by gaugino exchanges. We then use chiral dynamics to calculate the widths for the decay modes p~K+lTa,K°e~, and n~K°~d. The resulting branching ratios are rather sensitive to the precise admixture of the variousbaryon-numberviolatingoperators involved. In particular the decay mode p-~K+Oais generally dominant in qualitative agreement with previous quark model results.

1. Introduction The subject of proton decay has been of intense recent interest as possibly the most spectacular prediction of grand unified theories [1]. As is well known, in supersymmetric theories there are certain supersymmetric operators of dimension 5 which can give rise to this phenomenon [2, 3]. These operators, and the induced dimension-6 operators derived from them, have been studied both within the context of the minimal SU(5) model as well as more generally [4-7]. The resulting nucleon branching ratios have been calculated using either the quark model [5, 7] or the technique of Q C D sum rules [8]. In this note we use yet another technique, that of phenomenological lagrangians [9], to estimate these branching ratios. We first write down the general baryon-number violating operators which are produced by dressing the supersymmetric dimension-five operators by gaugino exchanges, and then use chiral dynamics to obtain an effective lagrangian for nucleon decay written directly in terms of the phenomenological hadronic fields [ 10]. Part of our motivation for this calculation is to compare the results with those obtained from the quark model, particularly as there has been some recent controversy concerning the relative importance of the " n o n - p o l e " and the "pole" (i.e. three-quark fusion) diagrams in the quark model calculations [11].

2. Operators for supersymmetric nucleon decay We assume the standard minimal set of low-energy chiral matter superfields: QL~ia, LLia, Ua,~a, OR,~a, ERa, HLi, H[i, where a is a colour index, i a flavour 105

106

S. Chadha and M. Daniel / Nucleon decay modes

index, and a is a generation label. In addition there are the usual gauge vector superfields associated with the low-energy gauge group S U ( 3 ) × S U ( 2 ) x U ( 1 ) . Elementary dimensional analysis then yields the following supersymmetric operators of low dimension which are SU(3) x SU(2) × U(1) invariant and which violate baryon a n d / o r lepton number conservation: (a) dimension 3: (LLHL)F; (b) dimension 4: (E*HLHL)F; (C) dimension 5: ( L L H L ~ ) D , (LLLLE~R)F, (LLOLD*)F, (DRDRUR)F, * * * (OLU*RL*L)D, ( U R*D R E R*) D , (QLQLQLLL)F, (QLQLQLHL)F, ( Q L U R*E R*H L ) F , * I-'I* IS;* I r ¢ ¢ "I -T'*Rf VTR ' - " R ~ ) F , (LLLLHLHL)F, (LLHLHLHL)F. It is customary to impose a certain reflection symmetry on the low-energy fields under which all the quark and lepton superfields change sign while the Higgs superfields do not [12]. This serves to eliminate the potentially dangerous renormalizable operator (DRDRUR)F, * * * • in addition it also removes (QLOLQLHL)F. The remaining operators of lowest dimensionality (d = 5) are then (OLQLOLLL)F and ( ~r T* n * r:* n ' ~r T* R'~'R~R)F. Of these the latter, more explicitly written as (

* * * * U R a a UR13bDa,,/cgadeaB.v)F

,

(1)

are irrelevant for proton decay [4]. This is simply because these operators are antisymmetric in a and b, and must therefore contain either a charm ( C * ) or a top (T*R) superfield. These fields, however, are SU(2) blind and cannot change flavour; hence the induced dimension-six operators must contain either a charm (c) or a top (t) quark which makes them irrelevant. This then leaves us only with the supersymmetric dimension-5 operators of the type Mabcd ----( QLaia QLI3jb QL vkcLLId ea~veqe'kl)V

= (e~q3~,U,~,D~bU~,cEa)r-(e,,~,D,~aU~bU~cEa)F + (e,,~,D,,,U~bD~,cNa)F- (e,,~,U,,aD~bD~,~Na)F,

(2)

which are baryon-number violating but ( B - L ) conserving. From eq. (2) it is easily verified that these operators, which are composed of left-handed matter fields only, are symmetric in the generation indices a and b and satisfy the relations Maaad = 0 ,

(3)

Mabcd + Mcbad + Macbd = 0 .

(4)

We must now combine the operators M~b~d with the other renormalizable gaugino couplings in the manner indicated in fig. 1 to obtain the four-fermion operators relevant to nucleon decay. These latter couplings have the form*

ix/~A*hO+h.c.,

h = T"A~,

(5)

where T a are the generators of the gauge group S U ( 3 ) × S U ( 2 ) × U(1), ha are the corresponding gaugino fields, and A and 0 generically denote the scalar and • Here, and subsequently, we use two-component spinor notation.

107

S. Chadha and M. Daniel / Nucleon decay modes

,%

/,

_

(g.w,b) L

\\

il X

I \X

I \

I

"x

11 \

/

Fig. 1. A typical diagram giving rise to baryon non-conserving operators in supersymmetric theories. fL denotes a left-handed quark or lepton, (g, w, b)L a left-handed gluino, wino or bino, and ~ a squark or slepton. The cross represents a Majorana mass term for the gaugino. fermionic components of the chiral matter superfields. The cross in fig. 1 represents a source which either absorbs two right-handed antiparticles or emits two left-handed particles. Possible such sources are the supersymmetry violating Majorana mass terms for the gauginos. It is convenient to express the resulting four-fermion operators as appropriate linear combinations of 6abca = ( qL,~iaqL~jb )( qL~&c ILtd ) e ,~#.: ejkSil ,

(6)

which are the only SU(3) x SU(2) × U(1) invariant operators comprising left-handed fields only, but which are not all independent [13, 14]: Oabcd +

Ob,,~a = O~bd + O:bad •

(7)

To obtain the requisite linear combinations it is not necessary to calculate explicitly the various Feynman diagrams, but only to determine their group theoretical weights. When this is done the following expressions are obtained for the induced dimensionsix operators produced by the gluino (g), wino (w), and bino (b) exchanges:

-(fF' o(b)

=

~/'¢(b)

(w) )(2Obc~a +f~,a + 6cb~a)], (b)

(9) X JOc

--

(b)

3fba

) O"~ b a

[ ~t'(b)

- , :b~ - 3f<~)) d~b,a]

(10) In deriving these expressions we have assumed, consistent with SU(2) invariance, that squarks and sleptons within the same generation a have equal masses ma ; this assumption enables us to write the O~bcd in terms of the S U ( 3 ) x S U ( 2 ) x U ( 1 ) invariant operators O~bcd- Had we assumed instead the complete degeneracy of all squark and slepton masses the operators vg"~(g)abcd and ,Jfl(b)abcdwould have vanished

108

S, Chadha and M. Daniel / Nucleon decay modes

identically. The functions lab =--f(m,, rob) result from the loop integrations in fig. 1, and are proportional to the Majorana mass terms of the gauginos; their specific forms are given by

trna gem b [ m2 a m2 fab - 32 ~r27--7-- m 2 ) t.m2_ rn 2 ln~ma

m2_m

ln~--~b

(11)

where m is the appropriate Majorana mass and g the appropriate coupling. Finally, we should note that the operators listed in eqs. (8)-(10) above are symmetric in a and b, vanish identically if all three quark generations a, b, and c are the same and satisfy Oabc d ,..I- Ocba d .~ Oacb d = 0 "~

(12)

these properties simply reflect the analogous properties of the supersymmetric dimension-5 operators M~bcd given in eqs. (3) and (4). In the simplified case where we neglect intergenerational mixings the third generation of quarks cannot enter the lagrangian for nucleon decay. This is so because the third generation always involves the t- or the b-quark which is not kinematically allowed. From just the first two generations we may construct the o p e r a t o r s 0112d , O121d , 0221d and O122d ; however, not all these operators are independent since eq. (12) yields the following relations amongst them: Ol12d = --20121d ,

O221a = - - 2 0 1 2 2 d .

(13)

In fact the operators 022 ld are also irrelevant, since they contain either two strange quarks, or strange and charm quarks, or two charm quarks, and an initial strange or charm quark is not available inside a nucleon. The only relevant remaining operators are then O,~2a, and the general nucleon decay lagrangian (neglecting mixings) must involve a linear combination of these.

3. Chiral lagrangian for supersymmetric nucleon decay Let us now consider the general chiral lagrangian associated with nucleon decay in supersymmetric theories ignoring intergenerational mixings. We will then use it to determine some of the nucleon branching ratios. As mentioned above the relevant operators are the 0112d , o r alternatively the two independent operators (3112d and O121d. In terms of these, or rather the combinations

6112a = - - % ~ ( dL~UL~ )( SL~ULa) ,

(14)

0112d + O121d = e,~(SL~UL~)(UL~eLd -- dLrl'Le) ,

(15)

a model-independent nucleon decay lagrangian may be written as

+ C2(d)' e,~v(SL~UL¢)(ULveLd-- dLv~'Ca)+ h . c . ,

(16)

109

S. Chadha and M. Daniel / Nucleon decay modes

where convert

C’p”

and

C$“” are arbitrary

LZ, into an equivalent

of chiral dynamics.

complex

constants.

phenomenological

The procedure

The strategy

lagrangian

now is to

using the techniques

for doing this is well known

[9]. The operators

in eqs. (14) and (15) both transform as (8,l) under the strong chiral group SU,(3) X SU,(3). We therefore need certain functions constructed directly from the hadronic fields, i.e., from the baryon and the pseudoscalar meson octets J&F’+

J&i0

E:-

B=

(17)

E-

(18) which have the same chiral transformation properties and (15). Such functions are given by the combination

fw= 139

(19)

MeV is the usual pion decay constant.

(16) may be transcribed

in eqs. (14)

iM

5=w,,

and

as the operators &B&+, where

Now using these functions

eq.

into the chiral form [lo]

_YB= C(ld)vLd tr (S”tJB&‘) +Cid)[eLd

tr (SgBJ+)-

vLd tr (S’[BL5’)1+h

Here, again, C’p’ and Cid’ are arbitrary complex C(ld)’ and Cy” respectively through an incalculable C(ld) = pC’,d”, Cp’ = PCY”. The matrices

S=(-;

I

%),

s$

-;

.c .

(20)

constants which are related to strong-interaction parameter:

i),

s$

w

;),

(21)

project out those specific components of the hadronic functions [BLt’ which are required by eq. (16). The lagrangian of eq. (20) then contains all the necessary baryon-number violating vertices. These must, however, be combined with other baryon-number conserving vertices coming from the usual strong-interaction chiral lagrangian L&. Those terms of interest to us here are given by (for details see ref.

[loI)* J%+%o - F) tr C~r“r~B{(~,A~‘- (~,c?bt~l -iW + F) tr [&“-Ys(@,~~- 5’~,tWl +ib, l

tr[Br5(5+m5’-5m5)B]+ibz

We have now reverted

to the usual four-component

tr[&sB(~‘m[t-&5)]. spinor

notation.

(22)

S. Chadha and M. Daniel/ Nucleon decay modes

110

The first two terms in eq. (22) are SUE(3) X SUR(3) invariant, while the remaining two terms, which are proportional to the quark mass matrix

m

=

(iu ° °0) m d

0

(23)

,

m~

transform like (3, 3) + (3, 3) under the chiral group and break it down to SUv(3). From a fit to the semileptonic baryon decays F = 0.44 and D = 0.81, [15] but the symmetry breaking parameters b~ and b2 are not well known since they do not enter into a determination of the baryon masses. We shall now apply the chiral lagrangian to calculate the amplitudes for nucleon decay into a pseudoscalar meson and an antilepton. Expanding ~x + 5¢B in terms of the meson and baryon fields, and retaining only the terms which involve either a proton or a neutron, we finally obtain the explicit form of the nucleon decay lagrangian as (y5 = y0y~ y2 3 )

i

.,-,(a),:-,-"U

.

~_~

i

::(d)

)/~O(~Ldne )

+ ~ C~2d)l~°( e~,~pc) - C~d) ( e~ e,Y~ ) -'* ~,-"~
--

2C~d))(~-Ldao)+ i ( D -F) --.~-6 ~ , ~-~--O~,K (2. "y YsP)

i (D+_3F)O~K_(--ff6y~,ysp ) + (m~ + m~)(202 K_(~-6ysp ) 46f~

f~

- (rnu+ m~) ~/~(2b~ - b2) K_(~---675p) + i ~ F )

o~,i~o(-~-~yUysp)

L +(md+m~)

2~ - - (D-F) o -2K°(~+ysp)-i , - ~ O , K (X°y"ysn) 42/~

• ( D + 3F) o,,Ro(A--~3,%,sn) _ (rn~ + m~) 42b: Ro(~-~3,sn)

- fred+ m,) 4~(2b' -- b2) i~o(--£675n).

g

(24)

It is now an easy matter to calculate the widths for the decays p-->K + 9d, n-->K°0d and p-->K°e~ from this lagrangian. The relevant Feynman diagrams are displayed in fig. 2, and are divided into two classes which we shall refer to as the " n o n - p o l e "

S. Chadha and M. Daniel / Nucleon decay modes

K* /

K* / /

¢/

//~¢

//

m

~

Vd

o

Ko

Ko

/ /

/

111

/

#

//

/

4

I/

Ko

Ko /

/ /4

/

//

4

//

/

p (a)

(b)

Fig. 2. Feynman diagrams contributing to the decays p+K+tTd, n-*K°~d and p+K°e~. Fig. 2a are the "non-pole" diagrams and fig. 2b the "pole" diagrams.

(fig. 2a) and the "pole" diagrams (fig. 2b). The widths are F(P--> K+ ~a)

1

2 3 32,rrf=mN(m~

m2) 2

× c]d)[l+~

(D+3F)+2(m"+rns)(2bl-bi)]3ma

-- c~d) [ ½DmN(-~+31mA) + ½FmN(~ --~) +(mu+ms){3~a+(-~ F(n--> K°~a) =

31a)b2}]l 2,

(25)

1

32"rrf=mN2 (m~ - m 2)2 3

--c(2d)[l-k½OmN(31a-~)-l-½FmN(l I.Oma \m~

'

(26)

S. Chadha and M. D a n i e l / Nucleon decay modes

112

and

r(p~K°e~) = 3 2 rrf2 m~ [m 2 - ( rnK + rnen)e]l/Z[m~ - ( i n K - men)Z]1/2 (27)

× [AZ(p°n + men) +B2(P°n - m~n)], where the quantities A and B are given by A=I

m2

1 _

2 [ ( D - F ) { - m e a ( m z + m N ) + mea2+m.~mN} men (28)

+ 2be(ran + ms)(-men + rnz)], B=I

1

m~_mZd[(D-F)(men(mz.+mN)+mZed +m~mN} + 2bz(md + m~)(men + rn~)].

(29)

As they stand these expressions for the widths are clearly rather unwieldy. They may, however, be considerably simplified if we employ the following approximations: m~-rna-~ mB, rn,, ma<
3rob (30)

r(n-~/¢ ,'d)-.,.,

,'2

3

(rn~-m~) 2

1+

(D+3F)

~ z "n'f ~.m N

(31)

r(p~K°e~)-~ 3

2 3 (m~-m2) 2 1Jw

D-F

.

(32)

N

4. Discussion

The partial width expressions given in eqs. (30)-(32) constitute the main results of this paper. The coefficients of the constants C] d) and C~2d) are completely determined in terms of the known low-energy strong-interaction parameters. Inserting the numerical values of these parameters we see that the widths F(p~K+~Td) and F(n~K%Ta) vanish for p = 0.28, 0 = 0 and p = 0.72, 0 = 0 respectively, where f ~ ( d ) / f-~ (d) p and 0 denote the modulus and phase of the complex number '--1 /,--2 • We might

S. Chadha and M. Daniel/Nucleon decay modes

113

also mention that, using a lower limit for the decay of a proton into the K°e~ m o d e of 10 31 years [16], we find the bound IC~2d) ]<~l.3x 10 -31GeV. Finally, let us consider the branching ratios F(P--* K + ~a) rl = F(p_~KOe~),

F ( p ~ K + ~a)

F(n_~KO~d).

r2 -

(33)

F r o m eqs. (30)-(32) they are given by rl-~4.96p 2 - 2.73p cos 0 + 0.38,

(34)

2.46p 2 - 1.35p cos 0 + 0 . 1 9 r 2 = 2 . 4 6 p Z - 3.56p cos 0 + 1.29 "

(35)

To obtain more specific information about these ratios we must make further assumptions. Thus, if we assume, e.g., that the up, down and strange squark masses

11l r2

~ r2

7

I

! //rl

S / 4

~

0

. . . . . . . . . . . . . . . . . . .

1

2

r2

3

? Fig. 3. T h e branching ratios rI and r2 (eq. (33)) plotted as a function of p for cos 8 = + 1 (solid curve)

and cos 0 = - 1 (dashed curve).

114

s. Chadha and M. Daniel/ Nucleon decay modes

a r e e q u a l [17], t h e r e is a definite r e l a t i o n b e t w e e n t h e constants C~ a) a n d C~d), viz. C~1d) = 2C~2d). This implies t h a t rl--- 1 5 ,

r2=1.8.

(36)

T h e s e ratios a r e in q u a l i t a t i v e a g r e e m e n t with t h e results r e c e n t l y o b t a i n e d by Ellis et al. [7]; in p a r t i c u l a r , t h e y confirm the d o m i n a n c e of t h e p-->K+~Td d e c a y m o d e . If we w e r e to neglect t h e " p o l e " d i a g r a m s of fig. 2b in the c a l c u l a t i o n of the p a r t i a l widths we w o u l d o b t a i n r 1 = 4 a n d r 2 - 4 in c o m p l e t e q u a n t i t a t i v e a g r e e m e n t with ref. [7]. W e e m p h a s i z e , h o w e v e r , that the n u m b e r s in eq. (36) a r e o n l y illustrative; i n d e e d t h e expressions for rl a n d r2 in eqs. (34) a n d (35) show a s t r o n g v a r i a t i o n with p. This is e x h i b i t e d in fig. 3 w h e r e rl a n d r2 a r e p l o t t e d for two different values of 0: cos 0 = +1 (solid line) a n d cos 0 = - 1 ( d a s h e d line). W e g r a t e f u l l y a c k n o w l e d g e s e v e r a l useful c o n v e r s a t i o n s with G r a h a m Ross.

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