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B-L nonconservation and neutron oscillation
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
B - L NONCONSERVATION AND NEUTRON OSCILLATION Lay-Nam CHANG 1 Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA and Ngee-Pong CHANG 2 Physics Department, City College of the City University of New York, New York, NY10031, USA Received 29 January 1980
We present a general discussion of B - L violation in an SU(5) invariant theory, for a single generation of light fermions. We realize the simplest of such mechanism through the 15 Higgs. Its consequences for neutron oscillation are discussed.
Recently Weinberg [1 ] and independently Wilczek and Zee [2] have reported on a theorem concerning the effective four-fermion operator for proton decay. The theorem says that a local, nonderivative effective four-fermion operator for proton decay that is invariant under SU(3) × SU(2) X U(1) automatically conserves B - L. If the theory has no other scale apart from MX, the grand unification scale, then six-fermion and higher fermion operators are naturally suppressed so that B - L conservation must remain true in the nucleon sector even when B - L is not strictly a good quantum number. In this paper * 1 we shall take the point of view that B and L are not good quantum numbers. We shall discuss (i) in the context o f SU(5) how it is possible to have B - L nonconservation without breaking SU(5) and all this to take place within each single generation o f light fermions; and (ii) given the B - L nonconservation, the implications it has for neutron oscillation , 2 .
1 Supported in part by NSF-PHY 78-21975. 2 SuppoIted in part by NSF-PHY 77-01350 and by City University Research Foundation PSC-BHE. ,1 The results of this paper were presented at the 1980 Conference on Theoretical Particle physics, Canton, People's Republic of China (Jan. 5-10, 1980).
As Wilczek and Zee [4] have already pointed out, the usual 5 R and 10 L assignments for the light fermions under SU(5) carry an extra global hypercharge (named X) quantum number. Let X(5R) = 3 ,
X(10L) = 1,
(1)
then under the combined U(1) gauge and X phase change diR:
Yw-
+ c eR, VR:
Yw=l,
2 a,
X=3,
B-L=1,
X=3,
B-L=I,
(2)
where 2 Yw + ~ X = B - L and Yw is the U(1) hypercharge. X is a global quantum number which as we shall see plays a role analogous to triality and may be conserved by SU(5) only modulo, in these units, 10. In that case, the B - L global quantum number would be conserved, according to eq. (2), modulo 2. That is to say, B - L violation can occur even when SU(5) is good. The B - L violation may occur through the SU(5) ,2 Mohapatra and Marshak have recently discussed the same phenomenon within the context of left-right symmetric models [3]. 103
Volume 92B, number 1,2
PHYSICS LETTERS
invariant ec~uvx coupling. Since fermion gauge couplings do not involve ea~uvx, B -- L violation can occur only through the Higgs couplings. For the minimal Higgs system, 5 and 24, it is easy to see that the assignment X(5H) = --2,
X(24H) = O,
+b.c.
(Ilia)
-
a ~ ~ -1
~
~ ~,~C-1 ~a~vh~arp ~ ,~'L
vcr~X -1 r
,i,uv ~L
x
X ~UL ~
~
~R C
0R.
(8)
Corollary, For the minimal set of _5, 24 and I_55, the only tree diagrams are type (Ia), (IIa) and (IIIa).
(4)
M H ~ H a H ~ + h.c.,
(5)
The assignment X(15H) = 6 is respected by the Yukawa but not by trilinear Higgs coupling. To sharpen our discussion we present our results in a series of theorems.
Theorem 1. The four-fermion operator which violates B - L but conserves SU(5) must be of the forms ~c~¢~ -1 x~ -1 (6) e
(lib)
(3)
is conserved by both the allowed, renormalizable Yukawa couplings and Higgs quartic couplings. To achieve B - L violation, it is simplest to introduce a 15, with the couplings R
5 May 1980
,"g,o~c-l~ ,t,x ,"ff,~C-1
Proof of theorem 2. Nota that the B - L violating six-fermion operators must have the X quantum number add up to 10, with basic building blocks -+3 and -+1. The only partitions are 3+3+3+3
-3+1
= 10,
3+3+3+3-1-1=10, 3+3+3+1
+ 1 - 1 = 10,
3+3+1+1
+1 + 1 = 1 0 .
(9)
These partitions specify uniquely the fermion operators allowed. Putting in SU(5) as well as Lorentz indices, the theorem follows. Q.E.D.
plus hermitean conjugate.
Corollary. The lowest order diagram which contributes to the four-fermion operator is at the two-loop level and is proportional to (M X >>Mlt)
(M/M 3) [h 5/(47r 2) 2 l ,
(7)
where h refers generically to the Yukawa couplings and M H is the heavier Higgs mass of the pair 5_and 15.
Theorem 2. The only lowest order Lorentz and SU(5) invariant six-fermion operators of a single generation which violate B - L are (Ia)
-
~-
o~
(Ib)
-t~Lafl~Rt~Lar~R~C ~r a~ -1 ~rR,
([C)
~ L a f l ~ ~Lor~rR"~C -1 ~J~,
~Laf3~R~Lor~R~
C-1
~R'
(Id) (IIa) 104
o ~a[3C -ld,uv,'T,,r h ea~v?t~Lor~R~L ~'L ~'R c - l a ,~"R,
Note that in theorem 2, type (I), (IIa) and (Ilia) operators contribute only to n ~ e [ n +, ULTrOwhile type (IIb), (IIIb) contribute to both n -~ e~n +, uLnO and n -+ eRrr+. On the other hand, all the operators listed in theorem 2 contribute to neutron oscillation. For the minimal set of 5, 24 and 1_55,only types (Ia), (lla) and (Ilia) are tree diagrams. For the neutron B - L violating decay processes the matrix elements for p ~ 1 GeV are proportional to generically
Mh3/M22M23M25,
(10)
while for the neutron oscillations, the matrix elements are proportional to
Mh3/M43M25 .
(11)
The difference between eqs. (10) and (11) comes from the fact that at 1 GeV, the effective mass of H i (i = I, 2,3)( -= MHa) is much greater than the effective mass of H a (a = 4, 5)(=MH2). From t h e B - L conserving proton decay lifetime constraint, one can set a lower limit on MH3 of around 1010 GeV.
Volume 92B, number 1,2
PHYSICS LETTERS
Since H a does not mediate proton decay at the four-fermion level, there is no similar constraint on MH2. The same is true of the 1_55.It appears tempting to assume the effective mass of H a at the 1 GeV scale to be of order TeV. In that case, a viable range for M15 may be obtained from the requirement that the two-loop contribution, eq. (7), be smaller than the tree diagram contribution, eq. (10), both evaluated at the p ~ 1 GeV scale. This requirement translates into (in GeV units) M2 ./(41r2) 2 MH3 - (4rr2) 2 M 2 MH3
1s-
g2
or
M15 < 10 9 GeV.
(13)
A lower limit on M15 will come upon examination of neutron oscillation. Before passing on to that topic, we remark on an alternative way to accommodate B - L nonconservation. This is by spontaneous symmetry breakdown of X carrying Higgs fields, (qS) -~ u. We assume here that the lagrangian otherwise strictly conserves X. Again, the set of Higgs field must be enlarged in order to generate B - L nonconservation. Minimal generalization requires a 4_55.The result may be summarized in a theorem.
Theorem 3. Let X be st'rictly conserved by all interaction terms in the lagrangian, and let the vacuum soak up g3 unit of X through spontaneous symmetry breakdown. Then the B - L nonconserving piece of the four-fermion operator must involve the vacuum transition at least twice and this minimum occurs only for the operator (~RdL)(d~tdl).
Proof. By charge constraint, the only B - L nonconserving four-fermion operator involving e - production is the pairing (g-d)(dCd). By enumerating helicities and counting the X quantum numbers, it is easy to find the least number of vacuum transitions needed, with each transition involving a loss of ~ unit of X. Upon repeating the process for v production, the theorem follows. As a result of this theorem, the B - L nonconserving amplitude is at least of order (v2/M2) × (17 - L)
5 May 1980
conserving amplitude. All the operators of eq. (8) can induce not only neutronB L violating decays but also induce n -+ fi transitions. In this section we study briefly the phenomenology of this neutron oscillation. The effective lagrangian for this, induced by theorem 2 operators, in the n - f i sector, reads
M ~ + eM~C-I~ + e M ~ C - I ~ + Ln_decay. Note that the n-fi transition operator is maximally parity violating. The property under CP is not, however, unique. The possibility exists that the neutron oscillation is intimately related to CP violation, but we shall not discuss it further in this paper. For an estimate of the size of this effect, we assume M to be of some order as M15. Neutron oscillation time we shall take to be ~ 1 0 30 years, since it would otherwise contradict the stability of matter as determined by experiment [5]. This is because when n has changed into an fi, the 1~ can annihilate with other nucleons in matter into multipions resulting in an effective baryon number nonconserving decay. Since oscillation time is (x 1/eMp, we find (M = M15 ) in GeV units,
e 0c h3/M43M15 ~ 10 - 6 0 , or
M15 >~ 10 5 GeV. With this lower limit, we deduce that the direct B - L violating decay is extremely small, being [from eq. (10)] of order 10 -46 GeV - 2 compared with the p r e s ent limit 10 -30 GeV -2. The surprising conclusion of this analysis therefore is that in the same nucleon decay experiment, B - L violation may occur through neutron oscillation rather than through decay. The signal for this neutron oscillation may thus be a dramatic spontaneous decay of nuclear matter into pions, losing in the process two units of baryon number, e.g. (A, Z) -+ (A 2, Z) + 7r+lr- . This mechanism of B L violation is different from the one proposed by Wilczek and Zee [4]. Our mechanism does not involve inter-generational transitions. Also, in their case,B L violation is manifested through decay rather than through oscillation; or, in other words, the oscillation occurs in their case as a second-order effect. 105
Volume 92B, number 1,2
PHYSICS LETTERS
While completing this work, we became aware of the work by Mohapatra and Marshak [3]. One of us (LNC) acknowledges discussions on the subject with both of them at VPI.
Note added in proof. The estimates on the mass of the 15-plet given above is in error. From the observed limit of nuclear stability of 10 30 years, the expected masses of the 15 and 4.~5mediating AB ¢ AL reactions are of the order of 104 GeV/c 2.
106
5 May 1980
References [1] [2] [3] [4] [5]
S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566. F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571. R.N. Mohapatra and R.E. Marshak, to be published. F. Wilczek and A. Zee, UPR 0135 T. F. Reines, J. Learned and A. Soni, Phys. Rev. Lett. 43 (1979) 907.
PHYSICS LETTERS
5 May 1980
B - L NONCONSERVATION AND NEUTRON OSCILLATION Lay-Nam CHANG 1 Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA and Ngee-Pong CHANG 2 Physics Department, City College of the City University of New York, New York, NY10031, USA Received 29 January 1980
We present a general discussion of B - L violation in an SU(5) invariant theory, for a single generation of light fermions. We realize the simplest of such mechanism through the 15 Higgs. Its consequences for neutron oscillation are discussed.
Recently Weinberg [1 ] and independently Wilczek and Zee [2] have reported on a theorem concerning the effective four-fermion operator for proton decay. The theorem says that a local, nonderivative effective four-fermion operator for proton decay that is invariant under SU(3) × SU(2) X U(1) automatically conserves B - L. If the theory has no other scale apart from MX, the grand unification scale, then six-fermion and higher fermion operators are naturally suppressed so that B - L conservation must remain true in the nucleon sector even when B - L is not strictly a good quantum number. In this paper * 1 we shall take the point of view that B and L are not good quantum numbers. We shall discuss (i) in the context o f SU(5) how it is possible to have B - L nonconservation without breaking SU(5) and all this to take place within each single generation o f light fermions; and (ii) given the B - L nonconservation, the implications it has for neutron oscillation , 2 .
1 Supported in part by NSF-PHY 78-21975. 2 SuppoIted in part by NSF-PHY 77-01350 and by City University Research Foundation PSC-BHE. ,1 The results of this paper were presented at the 1980 Conference on Theoretical Particle physics, Canton, People's Republic of China (Jan. 5-10, 1980).
As Wilczek and Zee [4] have already pointed out, the usual 5 R and 10 L assignments for the light fermions under SU(5) carry an extra global hypercharge (named X) quantum number. Let X(5R) = 3 ,
X(10L) = 1,
(1)
then under the combined U(1) gauge and X phase change diR:
Yw-
+ c eR, VR:
Yw=l,
2 a,
X=3,
B-L=1,
X=3,
B-L=I,
(2)
where 2 Yw + ~ X = B - L and Yw is the U(1) hypercharge. X is a global quantum number which as we shall see plays a role analogous to triality and may be conserved by SU(5) only modulo, in these units, 10. In that case, the B - L global quantum number would be conserved, according to eq. (2), modulo 2. That is to say, B - L violation can occur even when SU(5) is good. The B - L violation may occur through the SU(5) ,2 Mohapatra and Marshak have recently discussed the same phenomenon within the context of left-right symmetric models [3]. 103
Volume 92B, number 1,2
PHYSICS LETTERS
invariant ec~uvx coupling. Since fermion gauge couplings do not involve ea~uvx, B -- L violation can occur only through the Higgs couplings. For the minimal Higgs system, 5 and 24, it is easy to see that the assignment X(5H) = --2,
X(24H) = O,
+b.c.
(Ilia)
-
a ~ ~ -1
~
~ ~,~C-1 ~a~vh~arp ~ ,~'L
vcr~X -1 r
,i,uv ~L
x
X ~UL ~
~
~R C
0R.
(8)
Corollary, For the minimal set of _5, 24 and I_55, the only tree diagrams are type (Ia), (IIa) and (IIIa).
(4)
M H ~ H a H ~ + h.c.,
(5)
The assignment X(15H) = 6 is respected by the Yukawa but not by trilinear Higgs coupling. To sharpen our discussion we present our results in a series of theorems.
Theorem 1. The four-fermion operator which violates B - L but conserves SU(5) must be of the forms ~c~¢~ -1 x~ -1 (6) e
(lib)
(3)
is conserved by both the allowed, renormalizable Yukawa couplings and Higgs quartic couplings. To achieve B - L violation, it is simplest to introduce a 15, with the couplings R
5 May 1980
,"g,o~c-l~ ,t,x ,"ff,~C-1
Proof of theorem 2. Nota that the B - L violating six-fermion operators must have the X quantum number add up to 10, with basic building blocks -+3 and -+1. The only partitions are 3+3+3+3
-3+1
= 10,
3+3+3+3-1-1=10, 3+3+3+1
+ 1 - 1 = 10,
3+3+1+1
+1 + 1 = 1 0 .
(9)
These partitions specify uniquely the fermion operators allowed. Putting in SU(5) as well as Lorentz indices, the theorem follows. Q.E.D.
plus hermitean conjugate.
Corollary. The lowest order diagram which contributes to the four-fermion operator is at the two-loop level and is proportional to (M X >>Mlt)
(M/M 3) [h 5/(47r 2) 2 l ,
(7)
where h refers generically to the Yukawa couplings and M H is the heavier Higgs mass of the pair 5_and 15.
Theorem 2. The only lowest order Lorentz and SU(5) invariant six-fermion operators of a single generation which violate B - L are (Ia)
-
~-
o~
(Ib)
-t~Lafl~Rt~Lar~R~C ~r a~ -1 ~rR,
([C)
~ L a f l ~ ~Lor~rR"~C -1 ~J~,
~Laf3~R~Lor~R~
C-1
~R'
(Id) (IIa) 104
o ~a[3C -ld,uv,'T,,r h ea~v?t~Lor~R~L ~'L ~'R c - l a ,~"R,
Note that in theorem 2, type (I), (IIa) and (Ilia) operators contribute only to n ~ e [ n +, ULTrOwhile type (IIb), (IIIb) contribute to both n -~ e~n +, uLnO and n -+ eRrr+. On the other hand, all the operators listed in theorem 2 contribute to neutron oscillation. For the minimal set of 5, 24 and 1_55,only types (Ia), (lla) and (Ilia) are tree diagrams. For the neutron B - L violating decay processes the matrix elements for p ~ 1 GeV are proportional to generically
Mh3/M22M23M25,
(10)
while for the neutron oscillations, the matrix elements are proportional to
Mh3/M43M25 .
(11)
The difference between eqs. (10) and (11) comes from the fact that at 1 GeV, the effective mass of H i (i = I, 2,3)( -= MHa) is much greater than the effective mass of H a (a = 4, 5)(=MH2). From t h e B - L conserving proton decay lifetime constraint, one can set a lower limit on MH3 of around 1010 GeV.
Volume 92B, number 1,2
PHYSICS LETTERS
Since H a does not mediate proton decay at the four-fermion level, there is no similar constraint on MH2. The same is true of the 1_55.It appears tempting to assume the effective mass of H a at the 1 GeV scale to be of order TeV. In that case, a viable range for M15 may be obtained from the requirement that the two-loop contribution, eq. (7), be smaller than the tree diagram contribution, eq. (10), both evaluated at the p ~ 1 GeV scale. This requirement translates into (in GeV units) M2 ./(41r2) 2 MH3 - (4rr2) 2 M 2 MH3
1s-
g2
or
M15 < 10 9 GeV.
(13)
A lower limit on M15 will come upon examination of neutron oscillation. Before passing on to that topic, we remark on an alternative way to accommodate B - L nonconservation. This is by spontaneous symmetry breakdown of X carrying Higgs fields, (qS) -~ u. We assume here that the lagrangian otherwise strictly conserves X. Again, the set of Higgs field must be enlarged in order to generate B - L nonconservation. Minimal generalization requires a 4_55.The result may be summarized in a theorem.
Theorem 3. Let X be st'rictly conserved by all interaction terms in the lagrangian, and let the vacuum soak up g3 unit of X through spontaneous symmetry breakdown. Then the B - L nonconserving piece of the four-fermion operator must involve the vacuum transition at least twice and this minimum occurs only for the operator (~RdL)(d~tdl).
Proof. By charge constraint, the only B - L nonconserving four-fermion operator involving e - production is the pairing (g-d)(dCd). By enumerating helicities and counting the X quantum numbers, it is easy to find the least number of vacuum transitions needed, with each transition involving a loss of ~ unit of X. Upon repeating the process for v production, the theorem follows. As a result of this theorem, the B - L nonconserving amplitude is at least of order (v2/M2) × (17 - L)
5 May 1980
conserving amplitude. All the operators of eq. (8) can induce not only neutronB L violating decays but also induce n -+ fi transitions. In this section we study briefly the phenomenology of this neutron oscillation. The effective lagrangian for this, induced by theorem 2 operators, in the n - f i sector, reads
M ~ + eM~C-I~ + e M ~ C - I ~ + Ln_decay. Note that the n-fi transition operator is maximally parity violating. The property under CP is not, however, unique. The possibility exists that the neutron oscillation is intimately related to CP violation, but we shall not discuss it further in this paper. For an estimate of the size of this effect, we assume M to be of some order as M15. Neutron oscillation time we shall take to be ~ 1 0 30 years, since it would otherwise contradict the stability of matter as determined by experiment [5]. This is because when n has changed into an fi, the 1~ can annihilate with other nucleons in matter into multipions resulting in an effective baryon number nonconserving decay. Since oscillation time is (x 1/eMp, we find (M = M15 ) in GeV units,
e 0c h3/M43M15 ~ 10 - 6 0 , or
M15 >~ 10 5 GeV. With this lower limit, we deduce that the direct B - L violating decay is extremely small, being [from eq. (10)] of order 10 -46 GeV - 2 compared with the p r e s ent limit 10 -30 GeV -2. The surprising conclusion of this analysis therefore is that in the same nucleon decay experiment, B - L violation may occur through neutron oscillation rather than through decay. The signal for this neutron oscillation may thus be a dramatic spontaneous decay of nuclear matter into pions, losing in the process two units of baryon number, e.g. (A, Z) -+ (A 2, Z) + 7r+lr- . This mechanism of B L violation is different from the one proposed by Wilczek and Zee [4]. Our mechanism does not involve inter-generational transitions. Also, in their case,B L violation is manifested through decay rather than through oscillation; or, in other words, the oscillation occurs in their case as a second-order effect. 105
Volume 92B, number 1,2
PHYSICS LETTERS
While completing this work, we became aware of the work by Mohapatra and Marshak [3]. One of us (LNC) acknowledges discussions on the subject with both of them at VPI.
Note added in proof. The estimates on the mass of the 15-plet given above is in error. From the observed limit of nuclear stability of 10 30 years, the expected masses of the 15 and 4.~5mediating AB ¢ AL reactions are of the order of 104 GeV/c 2.
106
5 May 1980
References [1] [2] [3] [4] [5]
S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566. F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571. R.N. Mohapatra and R.E. Marshak, to be published. F. Wilczek and A. Zee, UPR 0135 T. F. Reines, J. Learned and A. Soni, Phys. Rev. Lett. 43 (1979) 907.