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Vector meson decays of nucleons in supersymmetric SU(5) model
Volume 142B, number 5,6
PHYSICS LETTERS
2 August 1984
VECTOR MESON DECAYS OF NUCLEONS IN SUPERSYMMETRIC SU(5) MODEL S. CHADHA and M. DANIEL Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire 0 X l l OQX, UK and A.J. MURPHY 1 Department of TheoreticalPhysics, University of Oxford, 1 Keble Road, OX1 3NP, UK Received 17 March 1984
Using the techniques of chiral dynamics we calculate the branching ratios for the two-body nucleon decay modes involvingvector mesons in the minimal SU(5) supersymmetrymodel with three generations. The widths for these modes are found to be suppressed by about a factor of 100 compared with the correspondingones involvingpseudoscalar mesons.
1. In this note we shall calculate the branching ratios for the two-body nucleon decay modes involving vector mesons in the minimal SU(5) supersymmetric model with three generations using the techniques of chiral dynamics. The inclusion of vector mesons in a phenomenological lagrangian with baryons and pseudoscalar mesons increases the number of possible couplings and hence the number of arbitrary constants in the theory. The vector mesons, however, have to be included, in accordance with the experimentally well-supported idea of vector meson dominance of the vector current matrix elements. A way to implement the above idea is to demand the current to be proportional to the vector meson field. This is achieved by employing the Yang-Mills formalism, which couples the fundamental vector mesons p, ~o and ¢ to the currents of isospin, hyperchange and baryon number [1,2]. In addition the strange vector meson K* is coupled to the partially conserved strangeness changing current. The nonet of vector mesons is thus introduced in the process of gauging the chiral symmetry SU(3)L × SU(3)R X U(1) of strong interactions. It is pointed out in ref. [3] that, in the two-body decay N ~ 12+ V where V is a vector meson, the dom1 Supported by Christ Church CollegeSenior Scholarship. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
inant contribution may come from a pole-type diagram, where there is a coherent emission of a vector meson by three quarks which subsequently annihilate to an antilepton. In this pole approximation method the decay amplitude is determined in terms of the matrix element (0[qqq[N) and the phenomenological coupling of the vector meson V to the baryons. We shall argue in this letter that this approximation is borne out by dimensional analysis within the framework of phenomenological chiral lagrangians. The importance of various terms in a phenomenological lagrangian is assessed through dimensional analysis [4]. It turns out that the decay N ~ + V gets its dominant contribution from tree diagrams with as many non-derivative couplings as possible and, of course, with only one superweak baryon number violating vertex. This implies that pole-type diagrams dominate over non-pole ones. Such a dimensional analysis is reliable whenever E/A is small, E being the energy of the particles involved in the decay and A the chiral symmetry breaking scale. For the problem at hand E/A is not necessarily small, and consequently there is a danger that neglected terms in the phenomenological lagrangian might turn out to be as important as the leading ones. In this letter we shall assume the validity of dimensional analysis and exploit its consequences. For some interesting comments 383
Volume 142B, number 5,6
PHYSICS LETTERS
concerning the strategy for applying current algebra and PCAC to nucleon + antilepton + pseudoscalar meson, despite the long extrapolation from the soft pseudoscalar limit to the physical region, we refer to ref. [5]. 2. The chiral lagrangian for the model consists of two parts L =L 0 +L laBl=l ,
~"~5~B~.~ +, V"Vs~.B~ +,
oUV~B~+F(~), oUUF(-~)~B~+ ,
(2)
=
+g( v. + a . ) l
_+~ + [iOu + g(v u - au) ] ~.
(4)
Here vu and au are the vector and axial * ~ vector gauge fields and
F(.2 ) = ~.(v~ - a~) -- ~ ( v u -- %) - ig[vu - % , v~ -a~]
(5)
%B = a.B - i[ %;B] and elyu and ~u are defined by
To check the transformation properties of the operators in (2) under an (L, R) transformation of SU(3)L X SU(3)R we recall [7,81 that
~-+ L~U+ = U~R +, B-+ UBU*, 9u -+ UguU+'
COuB-+U ~ uBU*,
F(~) -+LF(~)L+'
(6)
where U is a non-linear function of L, R and M. It follows from (6) that all operators in (2) transform like (8 L , 1R). The operators 7u~ ~uB~ +, auU~B~+F(~) and au VF(7)~B~ + can produce non-pole contributions to two-body decays N -+ £ + V. Indeed the operator 7u~ ~uB~ + has been incorporated in the analysis of ref. [7]. Power counting arguments, however, show that the dominant contribution to N -+$ + V arises from pole diagrams and consequently ~Bg+ is the only operator relevant for two-body decays. Using this operator we can transcribe L IzxBl=l(quarks) to its chiral form. This yields LtaBl=l(hadrons) -~ K/5[- memu(1 -,
- ¥- L IUul2)eRP
+
+mumuU11U211aRPL -memuU11U*2 leR+ Y-L + +mumu(1 -- IU21 [2)~-TRZ~ + P6VeLnL +
-vS(p3 - 2memu U21)(VeLNL) • 0 -- #3VeLAL0
+p5v#LnL + lX/~(p 2
mgmuU11
+ ~X/r6(m, m u Url - p 2) (v~--LA O) + p 4(vc--'Ln L) __1
..~
0
1
~
0
~Vr6p lVrLAL + ~x/r2-Pl/.'rL~EL] ,
(7)
where the coefficient r with dimensions [mass] - 2
where [7]
384
(;()
(1)
where L 0 is the lagrangian that describes the strong interaction physics of the vector mesons with the baryons and pseudoscalars and L IzxBl= 1 is the chiral lagrangian for superweak IABI = 1 processes. The supersymmetric SU(5) model implies that L tLXBI= 1 transforms like an (8L, 1R) under the chiral group SU(3)L X SU(3)R. This is because at the quark level LlZXBI= 1 is given as a linear combination of threequark operators all transforming as (8 L, 1R). LlzxBI= 1(quarks) has to be matched, therefore, to LImB I= 1(hadrons) in the effective lagrangian using functions constructed from hadronic fields, which have the same (8 L, 1 R)transformation properties as the quark operators. These functions must also carry the quantum numbers of a baryon. In principle all possible operators with (8L, 1 R) chiral transformation properties have to be included in the effective theory. It is only power counting arguments that can limit their number. The explicit form o f L IABI= l(quarks) in the minimal SU(5) supersymmetric model with three generations is given in ref. [6]. We turn our attention, now, to L IABI= 1(hadrons). There are numerous operators constructed out ofhadronic functions linear in the baryon fields which transform as (8 L, 1R)" Using the special unitary matrix ~ = exp(iM/f~r), where M is the pseudoscalar octet, the baryon octet B, the gauge and chiral covariant derivatives ~ u B and )u' and the field strength F ( S ) one can construct the following operators:
~B~+, ~.~%BU,
2 August 1984
(3)
,1 Actually the physical axail vector mesons are described by a mixture ofav and 0t,M.
Volume 142B, number 5,6
PHYSICS LETTERS
depends on the short distance dynamics of the SU(5) supersymmetric model and the mass spectrum for squarks, sleptons and winos. The strong interaction parameter t3 of dimension [mass] 3 is introduced in the process of transcribing L IAB I= 1(quarks) to its chiral form and it is in principle unmeasurable from conventional low energy hadronic physics. If more operators from the list (2) are included in the process of transcribing L IABI= l(quarks) to its chiral form, then more of the unmeasurable strong interaction parameters will appear in the effective lagrangian with considerable loss of predictability. The chiral symmetric part L 0 of the lagrangian describes the baryon number conserving strong dynamics of the vector mesons with the baryon octet. To lowest order in the expansion ~ = 1 + iM/f,r + ... we obtain the following couplings for the nonet of vector mesons (p, K*, co, 4~):
and (5), it is an easy matter to calculate the various two-body decays involving vector mesons. The values of the vector couplings g, g' in eq. (5)and those of the tensor couplings P1, P2 and P3 are fitted from con. ventional low-energy phenomenology. The numeric values for the various couplings are obtained from the compilation of Nagels et al. [9]. The first observation is that the NNq~coupling is negligibly small. This implies the following requirement among the vector coupling; - g sin 0 ~v/-3 +g' cos 0 = 0 .
e l - 2 P 2 =0,
+ P1 Tr (BFuz, oU~'B) + P2 Tr (BoUVBFu~], (8)
where V is the octet of vector mesons
] v=
p-
-pOl,,/~ +¢slv'ff
K *°
K* -
~-*0
I - X / ~ ¢8J
(9) and Fpv = a p Vp -- a v V#. Gpv = ap dpl .v -- avdPl .,.
~1 is an SUO)L+ R singlet such that the physical states co and ~ are mixtures o f ¢ 1 and ~8 given by (~81 = ( c°s0 ¢1
sin 0
-sin 0 X~°
(11)
The second requirement is that the tensor couplings NN~b and NNw are zero. Consequently
L 0 ~ ~ / 2 g Tr (./~.).u [ Vu, B]) + g ~ l Tr (B~/uB}
+ P3Gu,, Tr {BoU"B) ,
2 August 1984
(10)
cos 0' ' ¢ )"
The "ideal" mixing corresponds to 0 = 54.7 ° giving cos 0 = 1/V~, sin 0 = w / ~ . Experimentally, however, 0 ~--49.7 °. In (8) the vector coupling of the vector meson octet is F-type consistent with Sakurai's universality idea and the singlet ¢1 couples to the baryon number current. 3. With the nucleon decay lagrangian L 0 + L lAB I= 1 expressed totally in terms of hadronic fields eqs. (4)
P3 =0.
(12)
In view of (11) and (12) only two couplings, g and P1 say, are left to be fixed. From ref. [9] we obtain the values g ~ 5.3 and P1 ~ - 3 . 0 g/2~"-2MN . The analytic expressions for the widths as well as their numeric values are collected in table 1. Explicit chiral symmetry breaking has been neglected. This can be taken into account through wave function renormalization of the vector meson fields [2,7]. Some symmetry breaking effects, however, have been taken into account by differentiating between the masses of the various members of the baryon octet. Nucleon two-body decays involving vector mesons in supersymmetric SU(5) model have also been calculated by Salati and Wallet [10] and by Lucha [11] using a different hadronic model. The authors of ref. [10] have neglected the pole (or 3-quark fusion) diagrams and calculated instead only the non-pole ones. The author of ref. [11] on the other hand has adopted a simple pole dominance model. It is now seen that his model is supported by the chiral lagrangian approach. In the work of refs. [10,11], however, the effects of the third generation and those of the phases'¢l and ¢2 (which appear in the expressions for Pi, i = 1 ..... 6) have been neglected. For the numerical computation of the partial widths we have used a Kobayashi-Maskawa matrix whose elements are equal to the mean values of the ranges recently provided by Kleinknecht and Renk [12]. With these values for the matrix elements the terms in Pl ..... P6 which involve the charm quark mass m c (= 1.3 GeV) dominate over the corresponding terms which involve the top quark mass m t (assumed 385
Volume 142B, number 5,6 Table 1 Notation:
*
PHYSICS LETTERS
*
•
,
2 August1984
*
P 1 = m rm u U31 U11 U21 + mrmeel¢~ 1 Ua 1 U12 U22 + mrm t ei$2 U31 U~ 3 U23, * P2=mvmuU111U211 , 2 +mpmceiC~lU2l U*12 U~2+mgmteiO2u21u13u23, ~3 = memuU211U1112 * + memcei¢l U~ ~ U12U22 * * + memt ei~zU1 1 U13U23, * * p4 = m.rmuU31 U~I~ + m.rmceiO1 U31 U12 - - ' 2 +mrmt ei02 U31 U~I2 , as = m#muU21U~ 2 + mpmceiO1u21u~ 2 + m#rntel~bz" U21 U1"23, P6 = -memu U11( l - l Ul l12 ) + memceiO1Ul l U~2 + memteiO2 UIa,,2 ~X= (1/32,rm~)k2#2g 2, Sp = ~x(m~ - m 2 ) , 7%0 = A(m~ - 2 =
X m -mb).
The numerical values of the vector meson widths are given in terms of the same arbitrary unit A = "A/g2f2 which is used for the pseudoscalar case in ref. [61 . Mode
Partial width P
A -1 F (MeV 8)
P--'K*+-d#
1 *2 231+2 2 92 2 AK*(ff(IP2 +ml~muUlll /mx)("4- mN/2mK*+gmK,/m N) + 9(Ip2 - mt, miaU~ 112/m~)(, ~- + rn~/2m~, ~- ] m ~ , / m ~ )
9.68X 1017
+__
p"-~p v# p --* K* +v-r +h
p--*p vr + p'-+p 0 # + p ~ K*÷Ve
+_ p ~ P Ve p ~ K*°e + + p~e p ~ pOe+
2 2 2 2 - gmK,/mNU 9 2 • 2,~ - 3[(iP212 _mpmulUll12)/m~mA](_~+ mN/2mK, Ap{(IPsl2/m~,/)(1 "¢mN/2rn 2 9 2 2 p2 + ~rnp/mN)} 8.43 X 10 ] 7 AK,[~(ip 11+ /mx)(._4_ 2 31 mN/2mK, 2 2 + 9gmK,/m 2 2N) + ~(1011/mzA)(_~ + mN/2mK, 2 2 +gmK,/m 9 2 2N) 8.03 × 1016
2 2 - gmK,/mN) 9 2 2 ] - ~(iPll2/mxmA)(_~ + mN/2mK, 2 2 + 2 2+9 2 2 Ap[(lP41 /mN)(1 mN/2m p gmp/mN) ] Ato [~-(1/eos 20) IrnumuU~l U2112(1/m~)(1 + m~q/2m~)] 1 * 2 2 + 2 2 9 2 2 Ap[~(JmpmuUll U211 /raN)(1 mN/2rn p + ~mp/mN) ] .t * 2 2 31 2 2 9 2 2 AK*{~(IP3 - 2memuU211 /mz)(~- + mN/2mK, + ~mK,/m N) 9 2 2 5 2 2 9 2 2 + ~(IP31 / m A ) ( - ~- + mN/2mK* + ~rnK,/m N) ~ + mN/2mK 2 2 , - gmK,/mN) 9 2 2 ) _ 3[(ip312 _memu(UzlP3 + U21Pa)/mzmA](_ * • A 2 + 2 2 2 p[(IP61 2 /mN)(1 mN/2m p2 + 9 ~mp/mN)] • 2 2 31 2 2 9 2 2 AK*[(ImemuU11U211 /mx:)(-4- + mN/2mK, + gmK,/mN) ] Aw[~(1/cos20)lmemu(1 - IU1112)12(1/m~1)(1 + m~q/2m~)] Ap(~[imemu(l_lUlllZ)12/m~](1 + mN/2m 2 9 2 2 ~2 + ~.mp/mN))
7.20X 1016 2.69 X 10 Is 1.20 × 1014 3.88 X 1014
3 . 1 6 × 1014 6.07 × 109 3.62 X 109 1.58 X 109
Ato [}(1/c os20) IPs 12(1 + m~/2m 2,)1 n ~ p O-v# n ~ K*°~U
n --* t o ~ r n ~p
ovT
n ~ K*°~-r rl --~ p - ~ +
n~p
O--
ve
1
2
2
2
9.42 X 1017
2
~"2
2
2
31 +
2.~
Ap [~(IPsl /m N) (1 + mN/2m p + grnJmN) ] *
2
½(p ~ p+K#) 2
9
2
2
AK*[~(IP2 +m~muU111 /mz)(-~ mN/zmK, +ffmK,/m N) +~(Ip2 - m#muU111 • 2 /mA)(--z; 2 5 2 2 + gmK,/m 9 2 2N) + mN/2mK, 3 2 2 2 +3[(1021 - murnulUlllz)/mxmA](- ~ + mN/2mK , 2 2 _ grnK,/mN) ] 29 2 Ato[3(1/cos20) ip412(1 +mN/2mw)] 2 2
8.05 × 1016 1 +_ ~(p ~ p vr)
1 2 2 + 2 2 9 2 2 Ap[~'(IP41 /mN)(1 mN/2m p + ~mo/mN)] 1
2
3
2
2
31
2
2
-9
2
2
9
2
2
5
2
2
9
2
2
AK*[~(IPl I / m z ) ( ~ + mN/2mK, + gmK,/m N)+~(lpl I / m A ) ( - ~r+mN/2mK, +~mK,/mN) 5.56 × 10 $
2
2
9
2
+ ~(Ip it ImAm×)(- z; + mN/2mK-, - gmK,/mN) ] * 2 2 + 2 2..k9 2 2 Ap[(Im#muUllU211 /mN)(1 mN/2m p ~mp/mN)] Aw[}(1/cos2e)lP612(1 + m~/2m2.)]
2 ( p ~ p ° # +) 3.53 × 1014
1 2 2 2 2 2 A.o[~(!P61 /mN)(1 + mN/2m p2 + ~~mo/mN)] *
.t
*
2
2
~1
2
15
2
~1 ( P 2
9
2
2
n~K*°Ve
AK (~'(IP3 - 2memuU211 /m Z) (-4 + mN/2mK, + ~mK,/m N) 9 2 2 5+ 2 2 9 2 2x +;~(IP31 /mA)(--~ mN/2mK* + gmK*mN) + ~ { [ I p a l z -memu(U21p3 +U~21P~]/mAmx](- ~+ mN/2mK , 2 2
n .-¢-p - e +
2 9 2 2 ~p[(Irnemu(1 - IUI 112)12/m~)( 1 + mN/2m p2 + ~rnJmN)]
386
9.57X 1016
-..-, P + ~ e )
2.75 × 1013
_ ~mK,/mN) 2 2 2(p ~ pOe+)
Volume 142B, number 5,6
PHYSICS LETTERS
equal to 32 GeV). Consequently the modes involving vu dominate over corresponding modes with v r. The latter modes are very sensitive to the numeric value of the matrix element U 31 (=0.0--0.008). This comes about because both Pl and P4 are proportional to U31- The numeric results indicate that the antineutrino (Vu, Vr) modes dominate over modes into charged leptons (p+, e+). Generically F(N -+ V + b-u) and F(N ~ V + p + ) differ by O(10). We also found that F(N ~ V + Ve) is down by O(10 - 3 ) compared with F(N ~ V + ~-u) in contrast with the results of ref. [11]. Finally we would like to compare the nucleon decay modes involving vector mesons with the corresponding ones which involve pseudoscalar mesons. The numeric values of the partial widths given in table 1 are expressed in terms of the same arbitrary unit as the one used in the table of ref. [6] which gives the partial widths into psetidoscalars. A direct comparison of the two tables shows that the modes into vector mesons a are O ( 1 0 - 2 ) below the corresponding ones into pseudo-
2 August 1984
scalars. This is also in agreement with the results of ref. [11], but it disagrees with ref. [10]. References
[1] J.J. Sakurai, Ann. Phys. 11 (1960) 1. [2] S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41 (1969) 531. [3] V.S. Berezinsky, B.L. Ioffe and Ya.l. Kogan, Phys. Lett. 105B (1981) 33. [4] S. Weinberg, Physica 96A (1979) 327; Phys. Rev. D13 (1976) 974. [5] S.J. Brodsky, J. Ellis, J.S. Hagelin and C.T. Sachradja, preprint SLAC-Pub-3141. [6] S. Chadha and M. Daniel, preprint RL-83-099, to be published in Phys. Lett. [7] O. Kaymakcalan, Lo Chong-Huah, K.C. Wali, preprint SU-4217-250 (1983). [8] M. Claudson, K. Hall and M.B. Wise, Nucl. Phys. B195 (1982) 297. [9] M.M. Nagelset al., Nud~ Phys. Bt09(t976) 1. [10] P. Salati and C. Wallet, Nucl. Phys. B209 (1982) 389. [11] W. Lucha, Nucl. Phys. B221 (1983) 300. [12] K. Kleinknecht and B. Renk, Phys. Lett. 130B (1983) 459.
387
PHYSICS LETTERS
2 August 1984
VECTOR MESON DECAYS OF NUCLEONS IN SUPERSYMMETRIC SU(5) MODEL S. CHADHA and M. DANIEL Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire 0 X l l OQX, UK and A.J. MURPHY 1 Department of TheoreticalPhysics, University of Oxford, 1 Keble Road, OX1 3NP, UK Received 17 March 1984
Using the techniques of chiral dynamics we calculate the branching ratios for the two-body nucleon decay modes involvingvector mesons in the minimal SU(5) supersymmetrymodel with three generations. The widths for these modes are found to be suppressed by about a factor of 100 compared with the correspondingones involvingpseudoscalar mesons.
1. In this note we shall calculate the branching ratios for the two-body nucleon decay modes involving vector mesons in the minimal SU(5) supersymmetric model with three generations using the techniques of chiral dynamics. The inclusion of vector mesons in a phenomenological lagrangian with baryons and pseudoscalar mesons increases the number of possible couplings and hence the number of arbitrary constants in the theory. The vector mesons, however, have to be included, in accordance with the experimentally well-supported idea of vector meson dominance of the vector current matrix elements. A way to implement the above idea is to demand the current to be proportional to the vector meson field. This is achieved by employing the Yang-Mills formalism, which couples the fundamental vector mesons p, ~o and ¢ to the currents of isospin, hyperchange and baryon number [1,2]. In addition the strange vector meson K* is coupled to the partially conserved strangeness changing current. The nonet of vector mesons is thus introduced in the process of gauging the chiral symmetry SU(3)L × SU(3)R X U(1) of strong interactions. It is pointed out in ref. [3] that, in the two-body decay N ~ 12+ V where V is a vector meson, the dom1 Supported by Christ Church CollegeSenior Scholarship. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
inant contribution may come from a pole-type diagram, where there is a coherent emission of a vector meson by three quarks which subsequently annihilate to an antilepton. In this pole approximation method the decay amplitude is determined in terms of the matrix element (0[qqq[N) and the phenomenological coupling of the vector meson V to the baryons. We shall argue in this letter that this approximation is borne out by dimensional analysis within the framework of phenomenological chiral lagrangians. The importance of various terms in a phenomenological lagrangian is assessed through dimensional analysis [4]. It turns out that the decay N ~ + V gets its dominant contribution from tree diagrams with as many non-derivative couplings as possible and, of course, with only one superweak baryon number violating vertex. This implies that pole-type diagrams dominate over non-pole ones. Such a dimensional analysis is reliable whenever E/A is small, E being the energy of the particles involved in the decay and A the chiral symmetry breaking scale. For the problem at hand E/A is not necessarily small, and consequently there is a danger that neglected terms in the phenomenological lagrangian might turn out to be as important as the leading ones. In this letter we shall assume the validity of dimensional analysis and exploit its consequences. For some interesting comments 383
Volume 142B, number 5,6
PHYSICS LETTERS
concerning the strategy for applying current algebra and PCAC to nucleon + antilepton + pseudoscalar meson, despite the long extrapolation from the soft pseudoscalar limit to the physical region, we refer to ref. [5]. 2. The chiral lagrangian for the model consists of two parts L =L 0 +L laBl=l ,
~"~5~B~.~ +, V"Vs~.B~ +,
oUV~B~+F(~), oUUF(-~)~B~+ ,
(2)
=
+g( v. + a . ) l
_+~ + [iOu + g(v u - au) ] ~.
(4)
Here vu and au are the vector and axial * ~ vector gauge fields and
F(.2 ) = ~.(v~ - a~) -- ~ ( v u -- %) - ig[vu - % , v~ -a~]
(5)
%B = a.B - i[ %;B] and elyu and ~u are defined by
To check the transformation properties of the operators in (2) under an (L, R) transformation of SU(3)L X SU(3)R we recall [7,81 that
~-+ L~U+ = U~R +, B-+ UBU*, 9u -+ UguU+'
COuB-+U ~ uBU*,
F(~) -+LF(~)L+'
(6)
where U is a non-linear function of L, R and M. It follows from (6) that all operators in (2) transform like (8 L , 1R). The operators 7u~ ~uB~ +, auU~B~+F(~) and au VF(7)~B~ + can produce non-pole contributions to two-body decays N -+ £ + V. Indeed the operator 7u~ ~uB~ + has been incorporated in the analysis of ref. [7]. Power counting arguments, however, show that the dominant contribution to N -+$ + V arises from pole diagrams and consequently ~Bg+ is the only operator relevant for two-body decays. Using this operator we can transcribe L IzxBl=l(quarks) to its chiral form. This yields LtaBl=l(hadrons) -~ K/5[- memu(1 -,
- ¥- L IUul2)eRP
+
+mumuU11U211aRPL -memuU11U*2 leR+ Y-L + +mumu(1 -- IU21 [2)~-TRZ~ + P6VeLnL +
-vS(p3 - 2memu U21)(VeLNL) • 0 -- #3VeLAL0
+p5v#LnL + lX/~(p 2
mgmuU11
+ ~X/r6(m, m u Url - p 2) (v~--LA O) + p 4(vc--'Ln L) __1
..~
0
1
~
0
~Vr6p lVrLAL + ~x/r2-Pl/.'rL~EL] ,
(7)
where the coefficient r with dimensions [mass] - 2
where [7]
384
(;()
(1)
where L 0 is the lagrangian that describes the strong interaction physics of the vector mesons with the baryons and pseudoscalars and L IzxBl= 1 is the chiral lagrangian for superweak IABI = 1 processes. The supersymmetric SU(5) model implies that L tLXBI= 1 transforms like an (8L, 1R) under the chiral group SU(3)L X SU(3)R. This is because at the quark level LlZXBI= 1 is given as a linear combination of threequark operators all transforming as (8 L, 1R). LlzxBI= 1(quarks) has to be matched, therefore, to LImB I= 1(hadrons) in the effective lagrangian using functions constructed from hadronic fields, which have the same (8 L, 1 R)transformation properties as the quark operators. These functions must also carry the quantum numbers of a baryon. In principle all possible operators with (8L, 1 R) chiral transformation properties have to be included in the effective theory. It is only power counting arguments that can limit their number. The explicit form o f L IABI= l(quarks) in the minimal SU(5) supersymmetric model with three generations is given in ref. [6]. We turn our attention, now, to L IABI= 1(hadrons). There are numerous operators constructed out ofhadronic functions linear in the baryon fields which transform as (8 L, 1R)" Using the special unitary matrix ~ = exp(iM/f~r), where M is the pseudoscalar octet, the baryon octet B, the gauge and chiral covariant derivatives ~ u B and )u' and the field strength F ( S ) one can construct the following operators:
~B~+, ~.~%BU,
2 August 1984
(3)
,1 Actually the physical axail vector mesons are described by a mixture ofav and 0t,M.
Volume 142B, number 5,6
PHYSICS LETTERS
depends on the short distance dynamics of the SU(5) supersymmetric model and the mass spectrum for squarks, sleptons and winos. The strong interaction parameter t3 of dimension [mass] 3 is introduced in the process of transcribing L IAB I= 1(quarks) to its chiral form and it is in principle unmeasurable from conventional low energy hadronic physics. If more operators from the list (2) are included in the process of transcribing L IABI= l(quarks) to its chiral form, then more of the unmeasurable strong interaction parameters will appear in the effective lagrangian with considerable loss of predictability. The chiral symmetric part L 0 of the lagrangian describes the baryon number conserving strong dynamics of the vector mesons with the baryon octet. To lowest order in the expansion ~ = 1 + iM/f,r + ... we obtain the following couplings for the nonet of vector mesons (p, K*, co, 4~):
and (5), it is an easy matter to calculate the various two-body decays involving vector mesons. The values of the vector couplings g, g' in eq. (5)and those of the tensor couplings P1, P2 and P3 are fitted from con. ventional low-energy phenomenology. The numeric values for the various couplings are obtained from the compilation of Nagels et al. [9]. The first observation is that the NNq~coupling is negligibly small. This implies the following requirement among the vector coupling; - g sin 0 ~v/-3 +g' cos 0 = 0 .
e l - 2 P 2 =0,
+ P1 Tr (BFuz, oU~'B) + P2 Tr (BoUVBFu~], (8)
where V is the octet of vector mesons
] v=
p-
-pOl,,/~ +¢slv'ff
K *°
K* -
~-*0
I - X / ~ ¢8J
(9) and Fpv = a p Vp -- a v V#. Gpv = ap dpl .v -- avdPl .,.
~1 is an SUO)L+ R singlet such that the physical states co and ~ are mixtures o f ¢ 1 and ~8 given by (~81 = ( c°s0 ¢1
sin 0
-sin 0 X~°
(11)
The second requirement is that the tensor couplings NN~b and NNw are zero. Consequently
L 0 ~ ~ / 2 g Tr (./~.).u [ Vu, B]) + g ~ l Tr (B~/uB}
+ P3Gu,, Tr {BoU"B) ,
2 August 1984
(10)
cos 0' ' ¢ )"
The "ideal" mixing corresponds to 0 = 54.7 ° giving cos 0 = 1/V~, sin 0 = w / ~ . Experimentally, however, 0 ~--49.7 °. In (8) the vector coupling of the vector meson octet is F-type consistent with Sakurai's universality idea and the singlet ¢1 couples to the baryon number current. 3. With the nucleon decay lagrangian L 0 + L lAB I= 1 expressed totally in terms of hadronic fields eqs. (4)
P3 =0.
(12)
In view of (11) and (12) only two couplings, g and P1 say, are left to be fixed. From ref. [9] we obtain the values g ~ 5.3 and P1 ~ - 3 . 0 g/2~"-2MN . The analytic expressions for the widths as well as their numeric values are collected in table 1. Explicit chiral symmetry breaking has been neglected. This can be taken into account through wave function renormalization of the vector meson fields [2,7]. Some symmetry breaking effects, however, have been taken into account by differentiating between the masses of the various members of the baryon octet. Nucleon two-body decays involving vector mesons in supersymmetric SU(5) model have also been calculated by Salati and Wallet [10] and by Lucha [11] using a different hadronic model. The authors of ref. [10] have neglected the pole (or 3-quark fusion) diagrams and calculated instead only the non-pole ones. The author of ref. [11] on the other hand has adopted a simple pole dominance model. It is now seen that his model is supported by the chiral lagrangian approach. In the work of refs. [10,11], however, the effects of the third generation and those of the phases'¢l and ¢2 (which appear in the expressions for Pi, i = 1 ..... 6) have been neglected. For the numerical computation of the partial widths we have used a Kobayashi-Maskawa matrix whose elements are equal to the mean values of the ranges recently provided by Kleinknecht and Renk [12]. With these values for the matrix elements the terms in Pl ..... P6 which involve the charm quark mass m c (= 1.3 GeV) dominate over the corresponding terms which involve the top quark mass m t (assumed 385
Volume 142B, number 5,6 Table 1 Notation:
*
PHYSICS LETTERS
*
•
,
2 August1984
*
P 1 = m rm u U31 U11 U21 + mrmeel¢~ 1 Ua 1 U12 U22 + mrm t ei$2 U31 U~ 3 U23, * P2=mvmuU111U211 , 2 +mpmceiC~lU2l U*12 U~2+mgmteiO2u21u13u23, ~3 = memuU211U1112 * + memcei¢l U~ ~ U12U22 * * + memt ei~zU1 1 U13U23, * * p4 = m.rmuU31 U~I~ + m.rmceiO1 U31 U12 - - ' 2 +mrmt ei02 U31 U~I2 , as = m#muU21U~ 2 + mpmceiO1u21u~ 2 + m#rntel~bz" U21 U1"23, P6 = -memu U11( l - l Ul l12 ) + memceiO1Ul l U~2 + memteiO2 UIa,,2 ~X= (1/32,rm~)k2#2g 2, Sp = ~x(m~ - m 2 ) , 7%0 = A(m~ - 2 =
X m -mb).
The numerical values of the vector meson widths are given in terms of the same arbitrary unit A = "A/g2f2 which is used for the pseudoscalar case in ref. [61 . Mode
Partial width P
A -1 F (MeV 8)
P--'K*+-d#
1 *2 231+2 2 92 2 AK*(ff(IP2 +ml~muUlll /mx)("4- mN/2mK*+gmK,/m N) + 9(Ip2 - mt, miaU~ 112/m~)(, ~- + rn~/2m~, ~- ] m ~ , / m ~ )
9.68X 1017
+__
p"-~p v# p --* K* +v-r +h
p--*p vr + p'-+p 0 # + p ~ K*÷Ve
+_ p ~ P Ve p ~ K*°e + + p~e p ~ pOe+
2 2 2 2 - gmK,/mNU 9 2 • 2,~ - 3[(iP212 _mpmulUll12)/m~mA](_~+ mN/2mK, Ap{(IPsl2/m~,/)(1 "¢mN/2rn 2 9 2 2 p2 + ~rnp/mN)} 8.43 X 10 ] 7 AK,[~(ip 11+ /mx)(._4_ 2 31 mN/2mK, 2 2 + 9gmK,/m 2 2N) + ~(1011/mzA)(_~ + mN/2mK, 2 2 +gmK,/m 9 2 2N) 8.03 × 1016
2 2 - gmK,/mN) 9 2 2 ] - ~(iPll2/mxmA)(_~ + mN/2mK, 2 2 + 2 2+9 2 2 Ap[(lP41 /mN)(1 mN/2m p gmp/mN) ] Ato [~-(1/eos 20) IrnumuU~l U2112(1/m~)(1 + m~q/2m~)] 1 * 2 2 + 2 2 9 2 2 Ap[~(JmpmuUll U211 /raN)(1 mN/2rn p + ~mp/mN) ] .t * 2 2 31 2 2 9 2 2 AK*{~(IP3 - 2memuU211 /mz)(~- + mN/2mK, + ~mK,/m N) 9 2 2 5 2 2 9 2 2 + ~(IP31 / m A ) ( - ~- + mN/2mK* + ~rnK,/m N) ~ + mN/2mK 2 2 , - gmK,/mN) 9 2 2 ) _ 3[(ip312 _memu(UzlP3 + U21Pa)/mzmA](_ * • A 2 + 2 2 2 p[(IP61 2 /mN)(1 mN/2m p2 + 9 ~mp/mN)] • 2 2 31 2 2 9 2 2 AK*[(ImemuU11U211 /mx:)(-4- + mN/2mK, + gmK,/mN) ] Aw[~(1/cos20)lmemu(1 - IU1112)12(1/m~1)(1 + m~q/2m~)] Ap(~[imemu(l_lUlllZ)12/m~](1 + mN/2m 2 9 2 2 ~2 + ~.mp/mN))
7.20X 1016 2.69 X 10 Is 1.20 × 1014 3.88 X 1014
3 . 1 6 × 1014 6.07 × 109 3.62 X 109 1.58 X 109
Ato [}(1/c os20) IPs 12(1 + m~/2m 2,)1 n ~ p O-v# n ~ K*°~U
n --* t o ~ r n ~p
ovT
n ~ K*°~-r rl --~ p - ~ +
n~p
O--
ve
1
2
2
2
9.42 X 1017
2
~"2
2
2
31 +
2.~
Ap [~(IPsl /m N) (1 + mN/2m p + grnJmN) ] *
2
½(p ~ p+K#) 2
9
2
2
AK*[~(IP2 +m~muU111 /mz)(-~ mN/zmK, +ffmK,/m N) +~(Ip2 - m#muU111 • 2 /mA)(--z; 2 5 2 2 + gmK,/m 9 2 2N) + mN/2mK, 3 2 2 2 +3[(1021 - murnulUlllz)/mxmA](- ~ + mN/2mK , 2 2 _ grnK,/mN) ] 29 2 Ato[3(1/cos20) ip412(1 +mN/2mw)] 2 2
8.05 × 1016 1 +_ ~(p ~ p vr)
1 2 2 + 2 2 9 2 2 Ap[~'(IP41 /mN)(1 mN/2m p + ~mo/mN)] 1
2
3
2
2
31
2
2
-9
2
2
9
2
2
5
2
2
9
2
2
AK*[~(IPl I / m z ) ( ~ + mN/2mK, + gmK,/m N)+~(lpl I / m A ) ( - ~r+mN/2mK, +~mK,/mN) 5.56 × 10 $
2
2
9
2
+ ~(Ip it ImAm×)(- z; + mN/2mK-, - gmK,/mN) ] * 2 2 + 2 2..k9 2 2 Ap[(Im#muUllU211 /mN)(1 mN/2m p ~mp/mN)] Aw[}(1/cos2e)lP612(1 + m~/2m2.)]
2 ( p ~ p ° # +) 3.53 × 1014
1 2 2 2 2 2 A.o[~(!P61 /mN)(1 + mN/2m p2 + ~~mo/mN)] *
.t
*
2
2
~1
2
15
2
~1 ( P 2
9
2
2
n~K*°Ve
AK (~'(IP3 - 2memuU211 /m Z) (-4 + mN/2mK, + ~mK,/m N) 9 2 2 5+ 2 2 9 2 2x +;~(IP31 /mA)(--~ mN/2mK* + gmK*mN) + ~ { [ I p a l z -memu(U21p3 +U~21P~]/mAmx](- ~+ mN/2mK , 2 2
n .-¢-p - e +
2 9 2 2 ~p[(Irnemu(1 - IUI 112)12/m~)( 1 + mN/2m p2 + ~rnJmN)]
386
9.57X 1016
-..-, P + ~ e )
2.75 × 1013
_ ~mK,/mN) 2 2 2(p ~ pOe+)
Volume 142B, number 5,6
PHYSICS LETTERS
equal to 32 GeV). Consequently the modes involving vu dominate over corresponding modes with v r. The latter modes are very sensitive to the numeric value of the matrix element U 31 (=0.0--0.008). This comes about because both Pl and P4 are proportional to U31- The numeric results indicate that the antineutrino (Vu, Vr) modes dominate over modes into charged leptons (p+, e+). Generically F(N -+ V + b-u) and F(N ~ V + p + ) differ by O(10). We also found that F(N ~ V + Ve) is down by O(10 - 3 ) compared with F(N ~ V + ~-u) in contrast with the results of ref. [11]. Finally we would like to compare the nucleon decay modes involving vector mesons with the corresponding ones which involve pseudoscalar mesons. The numeric values of the partial widths given in table 1 are expressed in terms of the same arbitrary unit as the one used in the table of ref. [6] which gives the partial widths into psetidoscalars. A direct comparison of the two tables shows that the modes into vector mesons a are O ( 1 0 - 2 ) below the corresponding ones into pseudo-
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scalars. This is also in agreement with the results of ref. [11], but it disagrees with ref. [10]. References
[1] J.J. Sakurai, Ann. Phys. 11 (1960) 1. [2] S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41 (1969) 531. [3] V.S. Berezinsky, B.L. Ioffe and Ya.l. Kogan, Phys. Lett. 105B (1981) 33. [4] S. Weinberg, Physica 96A (1979) 327; Phys. Rev. D13 (1976) 974. [5] S.J. Brodsky, J. Ellis, J.S. Hagelin and C.T. Sachradja, preprint SLAC-Pub-3141. [6] S. Chadha and M. Daniel, preprint RL-83-099, to be published in Phys. Lett. [7] O. Kaymakcalan, Lo Chong-Huah, K.C. Wali, preprint SU-4217-250 (1983). [8] M. Claudson, K. Hall and M.B. Wise, Nucl. Phys. B195 (1982) 297. [9] M.M. Nagelset al., Nud~ Phys. Bt09(t976) 1. [10] P. Salati and C. Wallet, Nucl. Phys. B209 (1982) 389. [11] W. Lucha, Nucl. Phys. B221 (1983) 300. [12] K. Kleinknecht and B. Renk, Phys. Lett. 130B (1983) 459.
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