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KAIST-TH 19/96 CBNU-TH 961108 hep-ph/xxxxxxx
Proton Decay with a Light Gravitino or Axino
y y
Kiwo on Choi , Eung Jin Chun , and Jae Sik Lee
Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701,
y
Korea
Department of Physics, Chungbuk National University, Cheongju 360-763, Korea
Abstract
We consider the proton decay in sup ersymmetric mo dels with a gravitino
or axino lighter than the proton. This consideration leads to a stringent
limit on the R parity and B violating Yukawa coupling of the sup erp otential
c c c 00 15 00
op erator U D D as 10 (m =eV ) for a light gravitino, and
3=2
112 112
i j
k
15 10
10 (F =10 GeV ) for a light Dine-Fischler-Srednicki- Zhitnitskii axino. For
a
3
hadronic axino, the constraintisweakened by the factor of 10 .
Typ eset using REVT X
E 1
Proton stability strongly constrains the baryon (B ) and lepton (L)numb er violating cou-
plings. Since all known fermions lighter than the proton carry a nonzero lepton numb er, the
couplings (or the combinations of couplings) relevant for the proton decay should conserve
B L. However if there is a lighter fermion whichdoesnot carry any lepton numb er, proton
decaymay b e induced byaB violating but L conserving interaction alone [1]. There are in
fact very interesting class of mo dels which predict such a light fermion. In sup ersymmetric
mo dels in which sup ersymmetry (SUSY) breaking is mediated by gauge interactions, the
squark and/or gaugino masses, i.e. the soft masses in the sup ersymmetric standard mo del
n
(SSM) sector, are given by m ' ( ) where n is a mo del-dep endent p ositive integer
soft S
and corresp onds to the scale of sp ontaneous SUSY breaking [2]. In such mo dels, in order
S
for m to be of order the weak scale, is assumed to be 10 1000 TeV, leading to
soft S
2
the gravitino mass m = =M 1 keV far b elow the proton mass. If a global U (1)
3=2 P PQ
S
symmetry is intro duced in a gauge-mediated SUSY breaking mo del to solve the strong CP
problem by the axion mechanism [3], SUSY breaking in the axion sector is mediated also
m 2
by some gauge interactions. The axino mass in such mo dels is given by m ' ( = ) =F
a~ a
S
where m is again a mo del-dep endent (but typically bigger than n) p ositive integer and F
a
denotes the scale of sp ontaneous U (1) breaking [4]. Obviously then the axino is lighter
PQ
10
than the proton for a phenomenologically allowed F 10 GeV. In other typ e of mo dels
a
in which SUSY breaking is transmitted by sup ergravity interactions, the gravitino mass is
xed to be of order the weak scale, however there is still a ro om for an axino lighter than
the proton [5]. As was p ointed out in Ref. [5], some sup ergravity-mediated mo dels lead to
1=2
m ' m (m =M ) ' 1 keV for which the axino would be a good warm dark matter
a~ 3=2 3=2 P
candidate [6]. In this pap er, we wish to examine the proton decayinvolving a light gravitino
00 c c c
or axino to derive a constraint on the sup erp otential interaction U D D which violates
ij k i j k
R parity and B , while conserving L.
Let us rst consider the proton decay involving a light gravitino, more precisely the
helicity 1=2 Goldstino comp onent. Our starting p oint is the e ective lagrangian b elow the 2
q
scale = m M but ab ove the weak scale soft mass m :
S 3=2 P soft
L = L + L ; (1)
SSM G
where L denotes the lagrangian density of the SSM elds and the Goldstino lagrangian
SSM
L is given by [7]
G
p
i i
a a
p
L = @ GF +2 2 (1 ) @ GD +h:c (2) G @ G +
G I 5
I
2
4 6m M
p
3=2
where G denotes the four-comp onent Ma jorana Goldstino eld. Here L includes the
SSM
terms asso ciated with the B violating sup erp otential interaction,
00 c c c
W 3 U D D ; (3)
SSM
ij k i j k
a a
and ( ; ) and ( ;F ) stand for the left-handed chiral matter and gauge multiplets in
I I
the SSM sector. Note that the ab ove form of Goldstino lagrangian is enough for the study
of the pro cess involving a single on-shell Goldstino ob eying i @ G = m G.
3=2
Integrating out all elds heavier than the scale of the QCD chiral symmetry breaking, i.e.
' 1 GeV, we are left with an e ective lagrangian of the light quarks, q ( =(u; d; s)),
and gluons together with the light Goldstino (of course also the light leptons and the photon
which are not relevant for our discussion). The op erators resp onsible for the proton decay
in this e ective lagrangian at are induced by the exchange of the SU (2) singlet squarks
L
as
00
2i y
112
c
p
O = (q (1 )q )(@ q (1 )@ G): (4)
e 5 5
2
3m m M
3=2 P
0
2
Here m denotes the squark masses which are assumed to b e (approximately) universal,
0
y = y = y =1; (5)
dsu uds usd
and all other comp onents of y do vanish. Note that the ab ove op erator has B = S = 1,
+
and thus the relevant proton decaymodeisp!G+K . For a generic non-universal squark
+
mass matrix, S =0 op erator can be induced also to give rise to p ! G + , however it is 3
suppressed by a small squark mixing. To arrive at the ab ove interaction op erator, we have
used the equation of motion of the on-shell Goldstino eld and ignored the piece suppressed
by the small m . Also ignored are the renormalization e ects between the weak scale and
3=2
.
The hadronic matrix elements of the ab ove B = S = 1 op erator would b e describ ed by
an e ectivechiral lagrangian including the Goldstino eld. Let us consider a chiral op erator
+
O whichwould induce p ! G + K as a low energy realization of the light quark op erator
O b elow . Obviously it can b e written as O = Z (1 )@ G where Z is a fermionic
e 5
+
B = S = 1 op erator including P and K . If Z do es not include any spacetime derivative,
O is suppressed by the small factor m =m (for on-shell Goldstino) where m denotes the
p p
3=2
proton mass. For Z containing a single spacetime derivative, we have
00
2
112
+
p
O = (P (1 )@ G)@ K ; (6)
5
2
3m m M
3=2 P
0
where again the equations of motion are used together with m m . To estimate the
p
3=2
size of the hadronic co ecient , we use the naive dimensional analysis (NDA) rule of Ref.
[8], yielding
2
j j' 4f ; (7)
where f = 93 MeV is the pion decay constant. In fact, the NDA rule gives =4f and
then the typical energy in the proton decay, i.e. m , is comparable to . This means that,
p
within the NDA rule, chiral op erators with more spacetime derivatives are equally imp ortant
as the op erator of Eq. (6). However for an order of magnitude estimate of the hadronic
matrix element, the consideration of Z with a single derivative would be enough. Then
+ +
applying the exp erimental limit on p ! K + for p ! K + G induced by the interaction
of Eq. (6), we nd the following constraint on the R parity and B violating coupling:
!
2
2
m
4f m
3=2
0
00 16
5 10 ; (8)
112
300 GeV j j 1eV
which is one of the main results of this pap er. 4
Let us now consider the proton decay involving a light axino. Similarly to the case of a
light gravitino, we start from the e ective lagrangian at scales b elow the scale F of U (1)
a PQ
breaking but ab ove m :
soft
L = L + L ; (9)
SSM A
where the axino lagrangian L can b e read o from
A
Z Z
c c
a
y
y 2 a a 2 2 I
(A + A ) + d AW W +h:c ; (10) d d
I
I
2
F 16 F
a a
p
2
where A =(s+ia)+ 2a~ +F is the axion sup er eld containing the axion a, the saxion
A
a
s and the axino a~ , while and W are the chiral sup er elds for the SSM matter and
I
a a
gauge multiplets ( ; ) and ( ;F ), resp ectively. Here c and c are dimensionless real
I I a
I
co ecients. For F de ned as the scale of sp ontaneous U (1) breaking, the co ecients
a PQ
c of the axion coupling to the gauge multiplets are of order unity in general. However
a
as we will discuss later, the size of the co ecients c of the axion coupling to the matter
I
multiplets is somewhat mo del-dep endent. Note that the ab ove lagrangian corresp onds to
the sup ersymmetric generalization of the conventional axion e ective lagrangian [9]:
2c c
a
I a a
~
L = @ a + aF F : (11)
a I 5 I
2
F 32 F
a a
Obviously it is manifestly invariant under the nonlinear U (1) transformation, A ! A + ic
PQ
(c = real constant), up to the PQ anomaly. At any rate, the relevant axino lagrangian is
given by
c 1
I
ia ~ @ a~ (i@ (1 + )~a +h:c) L =
I 5 A
I
2 2F
a
c
a
a a
p
+ ( (1 )~aF +h:c); (12)
5
2
32 2 F
a
wherea ~ denotes the four-comp onent Ma jorana axino eld. Again the exchange of the SU (2)
L
singlet squarks leads to the following B = S = 1 interaction in the e ective lagrangian at
:
00
i y c
112
c
(q (1 )q )@ q (1 + )~a; (13) O =
5 5 e
2
m F
a
0 5
where c ( = u; d; s) denotes the axino coupling to the sup ermultiplet containing the SU (2)
L
singlet right-handed light quark q in Eq. (12) and the squark degeneracy is assumed also.
R
Similarly to the gravitino case, in order to estimate the proton decay rate from the ab ove
e ective interaction, we consider a chiral op erator of the form O = X (1 + )~a where
5
+
X is a B = S = 1 fermionic current made of P and K which are on mass-shell. For
+
X / K P , the chiral op erator O with the smallest number of spacetime derivatives is
given by
00
c
112
+
(P (1 + )~a)K ; (14)
5
2
m F
a
0
where the hadronic co ecients are again determined by the NDA rule as
2 3
j j' 16 f : (15)
This then leads to the exp erimental b ound on the R parity and B violating coupling as
!
2 3
2
m 16 f F
0 a
00 16
7 10 ; (16)
112
10
300 GeV c j j 10 GeV
which is another result of this pap er.
The ab ove constraint from the proton decay involving a light axino dep ends up on the
dimensionless co ecients c describing the axino coupling to the sup ermultiplets of the
SU (2) singlet quarks [see Eq. (12)], as well as the axion scale F . In fact, the size of c
L a
has a certain mo del-dep endence. If the quark sup er elds carry a nonzero U (1) charge,
PQ
which would be the case for the sup ersymmetric extension of the Dine-Fischler-Srednicki-
Zhitnitskii (DFSZ) axion mo del [10], the co ecients c would be of order unity in general.
However in hadronic axion mo dels [11] in which all SSM elds have a vanishing U (1)
PQ
charge, the co ecients c are zero at tree level. However the axino-quark couplings are
radiatively generated through the axino coupling to the gluon multiplet, yielding c '
2 3 2
( = ) ln (F =m ) ' 10 10 [9]. Thus the constraint for hadronic axion mo dels
c a soft
2 3
b ecomes weaker than that for DFSZ mo dels by the factor of 10 10 .
To conclude, we have considered the proton decay involving a gravitino or axino lighter
than the proton. Generic mo dels in which sup ersymmetry breaking is mediated by gauge 6
interactions contain such a light gravitino. Then the R parity and B violating coupling
00
is strongly constrained by the proton stability [see Eq. (8)] to be less than ab out
112
15
10 (m =eV ). Ab out the p ossibility of a light axino, gauge-mediated sup ersymmetry
3=2
breaking mo dels endowed with a global U (1) symmetry generically predict an axino lighter
PQ
than the proton. Also some sup ergravity-mediated mo dels can give rise to a light axino, while
00
the gravitino mass in such mo dels is xed to b e the weak scale. We nd that in mo dels
112
15 10
with a light axino is constrained [see Eq. (16)] to be less than ab out 10 (F =10 GeV )
a
12 10
and 10 (F =10 GeV ) for Dine-Fischler-Srednicki-Zhitnitskii axion mo dels and hadronic
a
axion mo dels resp ectively.
ACKNOWLEDGMENTS
This work is supp orted in part by KOSEF Grant 951-0207-002-2 (KC, JSL), KOSEF
through CTP of Seoul National University (KC), Programs of Ministry of Education BSRI-
96-2434 (KC), and Non Directed ResearchFund of KRF (EJC). EJC is a Brain-Po ol fellow. 7
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