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Proton instability and the SU5 ⊗ U1 subgroup of SO10
Volume 127B. number 6
PROTON INSTABILITY
PHYSICS LETTERS
11
August 1983
AND THE SU5 cs U1 SUBGROUP OF SOlo
W. ALLES Istituto di Fisica dell’Universit4 Bologna, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy
Received 14 January 1983
It is shown, for the case of three generations of fermions, that for gauge interactions of the SUs 0 U1 subgroup of Solo, the nucleon can be made stable to zeroth order in the weak interactions with no restriction on the Kobayashi and Maskawa mixing matrix.
Grand unification of weak, electromagnetic and strong interactions based on the Georgi and Glashow SUj group [l] implies baryon number non-conservation. The proton lifetime would be unambiguously determined, but for computational difficulties, in the case of one generation of fermions. For more than one generation of fermions, mixing between generations can in principle lead to dependency of the proton lifetime on generalized Cabibbo-type mixing angles. One may then raise the question of making the proton stable to lowest order in the baryon non-conserving interaction arising from gauge meson exchange and to zeroth order in the weak interaction [2]. For the case of only one Higgs multiplet responsible for SUY breaking (a 5 of SUs), it has been shown that proton decay cannot be rotated away [3]. Relaxing this restriction on the Higgs sector, one can show that proton decay can be rotated away to lowest order with no limitation on the Kobayashi and Maskawa [4] mixing matrix if there are four or more generations of fermions [5]. For the three-generation case, the requirement of proton stability implies that the generalized Cabibbo angle 03 = 0 [5]. The Kobayashi and Maskawa matrix therefore depends only on two mixing angles, 191and 02. The phase S can be made equal to zero for an appropriate choice of the field phases. No CP violation results for weak interactions from Kobayashi and Maskawa mixing and the beautiful reason for requiring six quark flavors would be incom0 03 1-9163/83/0000-0000/S
03.00 0 1983 North-Holland
patible with proton stability *l. Stability of the proton in this scheme holds to zeroth order in the weak interactions. The baryon number non-conserving interaction is constructed so as to lead only to channels, for example p + 7+ + TO, that are forbidden by energy conservation. An extra weak interaction could lead to energetically allowed channels [6]. A good feature of these schemes is the fact that the leading logarithmic diagrams for proton decay are zero in the limit of zero mass fermions. A calculation of these box diagrams leads to an amplitude for proton decay that is 0((cu2/2n)(m2/&) X, log (M,/m)) times the amplitude for the one generation case (m is a typical quark or lepton mass). Terms of order (cu2/27r) log(MIM,), where M is the unification level, cancel. Classification of one generation of fermions in a reducible set of representations 5 + 10 of SU5 is unaesthetic. Consequently, SOlo has been proposed as grand unifying group [7]. One generation of lefthanded fermion fields would correspond to a 16-com*’ On the contrary, it is possible to forbid proton decay into a charged lepton (p j+ e+ . . . . p % c1+ . ..) without any restria tion on the Kobayashi and Maskawa matrix. Effective current-current interactions responsible for these decays are either of the form (u -+ II), (d(s) + e+b+)) or (d(s) -+ a), (u + ef(wf)). Both are eliminated by the assignment: 10: @(KD)’ u (<(KD)’ c e+b+))i, cc+(e+))L where D is the column Cd s b) and 7 contains only f and C and no ii. The assignment of fields to 3 is left completely open.
;+)L,
(E’(KDj3t
423
Volume 127B, number 6
PHYSICS LETTERS
ponent spinor of SO10 that decomposes under SU 5 into 1 + 5-+ 10. The extra SU 5 singlet corresponds to a right-handed neutrino that gets a superheaw Majorana mass (~1014 GeV) in the SU 5 symmetry limit, if SO10 is broken down to SU 5 by an SU 5 singlet Higgs meson transforming like 126 of SO10. Thus generating the PR Majorana mass at the tree level. As a consequence, a small mass for the low mass neutrino results, my ~ m2/M ~" 1 0 - t 1 - 1 0 . 5 eV. Several ways of breaking SO10 down to SU~' e U 1 ® SU 3 are possible [8]. A maximal subgroup of SO10 is SU 5 ® U 1 and the Georgi and Glashow choice corresponds to U1 carrying no component of the electric charge. An alternative choice consisting in the assignment of both an SU 5 and a U 1 component to the charge operator has been discussed by Barr [9]. For this SU 5 ® U 1 subgroup of SO10, the 16-component spinor decomposes, in the one-generation case, into multiplets with content: 1: (e+)L,
5-: ( g i v e e ) L ,
10: (clidiui~e) L.
This decomposition is obtained from the Georgi and Glashow assignment by interchanging gL ~ dL and e [ <+~eL. The transition SO10-+ SU 5 ® U 1 can be realized by a Higgs meson transforming like the SU 5 singlet of 45 of SO10. Transition SU 5 ® U 1 -+ SU2 ® UI® SU 3 could be realized either by a Higgs meson transforming like the SU 2 and SU 3 singlet contained in 16 or 126 of SO10. In the latter case, the right-handed neutrino gets a Majorana mass, not in the SU 5 limit as for the Georgi and Gtashow choice, but only in the SU 2 ® U l ® SU 3 limit. One expects for it, therefore, a Majorana mass ~Mw. As a consequence the light neutrino could have mass "~m2/Mw ~ 1 - 1 0 6 MeV2/ 100 GeV, 10 eV <~rn v <~ 10 MeV, a range easily accessible to experiment. In the Georgi and Glashow SO10 -+ SU5 case, vR still gets a very large mass at the two-loop level if a 16 Higgs is responsible for the SO10 -+ SU5 transition [10]. On the other hand, in the present case, if the Higgs responsible for breaking SO10 down to SU 5 ® U 1 were the neutral SU2 and SU 3 singlet of 16, the/)R Majorana mass resulting from two-loop graphs would be much too small, both for the nucleon stabilization discussion of this paper and for explaining the smallness of the neutrino mass. The remaining element of the chain 424
11 August 1983
SO10-+SU 5 ® U I - + S U 2 ® U 1® SU 3-+U~ m ® SU 3, SU 2 ® U 1® SU 3-+U~~n ® SU 3, can be obtained as in the standard SO10 scheme by Higgs doublets contained in 10, 120 or 126 of SO10. The right-handed neutrino has too large a mass in this scheme to be present in proton decay channels and only one up-type quark (u) is lighter than the proton. These two facts make up all the difference between the Georgi and Glashow SU5 and the Barr SU 5 ® U 1 cases when forbidding proton decay ~ la Jarlskog [2]. For the three-generation case proton decay is forbidden to lowest order by assigning to:
5: (ui Vr r ) L ,
(c-~vu U)L,
q i Ve e)L
(c~"and t-~ are mixtures of ci and i-i fields only), and 10: (bi di (K+Ui) 1 V')L,
(si si (K+Ui) 2 v")L,
(di bi (K+Ui) 3 v")L and 1: e~,,
/at,
r[,
i is the SU 3 colour index, K + is the hermitean conjugate of the Kobayashi and Maskawa matrix [4] and Ui is the column (ui ci ri). Inspection shows that no restriction on the matrix K is necessary to forbid proton decay. As for the Georgi-Glashow SU s case, proton decay is thus forbidden only to zeroth order in the weak interactions. Calculation of the box diagrams giving leading logarithmic corrections shows that to first order in the weak interactions, the proton decay amplitude is again the onegeneration amplitude times a factor O((a2/2rr)(m2/M2w)log(Mw/m))~ 10 -6. As for the Georgi and Glashow case [6] a level of validity of SU 5 ® U 1 of 1014 GeV would lead to a proton lifetime of 1042 yr or, alternatively, a lifetime of 1030 yr would correspond to a symmetry level of 10 lj GeV. Proton decay would still be possible at the SO10 level. If one wants to forbid proton decay at this level, it must be forbidden both for SU5 and SU 5 ® U 1 interactions. Therefore, the restriction 03 = 0 for the matrix K must be imposed. One is lead to assign to representation 16 of SO10 the contents:
Volume 127B, number 6
PHYSICS LETTERS
16: (di (K+Ui) 1 bi c-~Vev (ell) ~' £~)L , + 2 .... g,, (si (1(Ui) si t i Veu (e//)' J~)L , (bi (K+Ui) 3 di ui Vr r ~" ~ ) L , with 03 = 0. g' and T', Veu and Veu, (e/J) and (e/~)' are respectively arbitrary mixtures o f e- and t, Ve and v u and e and #, and 2+ is any positive lepton. A p p a r e n t l y an effective SU5 (Barr) current X SU 5 ( G - G ) current interaction results, (SL -+ (K+U) 2) X (e+(/u+) -+ UL), b u t this violates SU~v so it is suppressed b y a factor of order (Mw/M).
References [ 1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
11 August 1983
[2] C. Jarlskog, Phys. Lett. 82B (1979) 401. [3] R.N. Mohapatra, Phys. Rev. Lett. 43 (1979) 893; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Phys. Lett. 88B (1979) 320. [4] M. Kobayashi and M. Maskawa, Progr. Theor. Phys. 49 (1973) 652. [5] S. Nandi, A. Stern and E. Sudarshan, Phys. Lett. 113B (1982) 165. [6] D. Altsch(iler, P. Eckert and T. Schficker, Phys. Lett. l19B (1982) 351. [7] H. Georgi, in: Particles and fields 1974, ed. C.E. Carlson (AIP, New York, 1975) p. 575; H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193. [8] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B159 (1979) 16. [9] S.M. Barr, A New Symmetry Breaking Pattern for SO(10) and Proton Decay, Univ. of Washington, preprint 40048-82-PT2 (nov. 1981). [10] E. Witten, Phys. Lett. 91B (1980) 81; R. Barbieri, D.V. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. 90B (1980) 91.
425
PROTON INSTABILITY
PHYSICS LETTERS
11
August 1983
AND THE SU5 cs U1 SUBGROUP OF SOlo
W. ALLES Istituto di Fisica dell’Universit4 Bologna, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy
Received 14 January 1983
It is shown, for the case of three generations of fermions, that for gauge interactions of the SUs 0 U1 subgroup of Solo, the nucleon can be made stable to zeroth order in the weak interactions with no restriction on the Kobayashi and Maskawa mixing matrix.
Grand unification of weak, electromagnetic and strong interactions based on the Georgi and Glashow SUj group [l] implies baryon number non-conservation. The proton lifetime would be unambiguously determined, but for computational difficulties, in the case of one generation of fermions. For more than one generation of fermions, mixing between generations can in principle lead to dependency of the proton lifetime on generalized Cabibbo-type mixing angles. One may then raise the question of making the proton stable to lowest order in the baryon non-conserving interaction arising from gauge meson exchange and to zeroth order in the weak interaction [2]. For the case of only one Higgs multiplet responsible for SUY breaking (a 5 of SUs), it has been shown that proton decay cannot be rotated away [3]. Relaxing this restriction on the Higgs sector, one can show that proton decay can be rotated away to lowest order with no limitation on the Kobayashi and Maskawa [4] mixing matrix if there are four or more generations of fermions [5]. For the three-generation case, the requirement of proton stability implies that the generalized Cabibbo angle 03 = 0 [5]. The Kobayashi and Maskawa matrix therefore depends only on two mixing angles, 191and 02. The phase S can be made equal to zero for an appropriate choice of the field phases. No CP violation results for weak interactions from Kobayashi and Maskawa mixing and the beautiful reason for requiring six quark flavors would be incom0 03 1-9163/83/0000-0000/S
03.00 0 1983 North-Holland
patible with proton stability *l. Stability of the proton in this scheme holds to zeroth order in the weak interactions. The baryon number non-conserving interaction is constructed so as to lead only to channels, for example p + 7+ + TO, that are forbidden by energy conservation. An extra weak interaction could lead to energetically allowed channels [6]. A good feature of these schemes is the fact that the leading logarithmic diagrams for proton decay are zero in the limit of zero mass fermions. A calculation of these box diagrams leads to an amplitude for proton decay that is 0((cu2/2n)(m2/&) X, log (M,/m)) times the amplitude for the one generation case (m is a typical quark or lepton mass). Terms of order (cu2/27r) log(MIM,), where M is the unification level, cancel. Classification of one generation of fermions in a reducible set of representations 5 + 10 of SU5 is unaesthetic. Consequently, SOlo has been proposed as grand unifying group [7]. One generation of lefthanded fermion fields would correspond to a 16-com*’ On the contrary, it is possible to forbid proton decay into a charged lepton (p j+ e+ . . . . p % c1+ . ..) without any restria tion on the Kobayashi and Maskawa matrix. Effective current-current interactions responsible for these decays are either of the form (u -+ II), (d(s) + e+b+)) or (d(s) -+ a), (u + ef(wf)). Both are eliminated by the assignment: 10: @(KD)’ u (<(KD)’ c e+b+))i, cc+(e+))L where D is the column Cd s b) and 7 contains only f and C and no ii. The assignment of fields to 3 is left completely open.
;+)L,
(E’(KDj3t
423
Volume 127B, number 6
PHYSICS LETTERS
ponent spinor of SO10 that decomposes under SU 5 into 1 + 5-+ 10. The extra SU 5 singlet corresponds to a right-handed neutrino that gets a superheaw Majorana mass (~1014 GeV) in the SU 5 symmetry limit, if SO10 is broken down to SU 5 by an SU 5 singlet Higgs meson transforming like 126 of SO10. Thus generating the PR Majorana mass at the tree level. As a consequence, a small mass for the low mass neutrino results, my ~ m2/M ~" 1 0 - t 1 - 1 0 . 5 eV. Several ways of breaking SO10 down to SU~' e U 1 ® SU 3 are possible [8]. A maximal subgroup of SO10 is SU 5 ® U 1 and the Georgi and Glashow choice corresponds to U1 carrying no component of the electric charge. An alternative choice consisting in the assignment of both an SU 5 and a U 1 component to the charge operator has been discussed by Barr [9]. For this SU 5 ® U 1 subgroup of SO10, the 16-component spinor decomposes, in the one-generation case, into multiplets with content: 1: (e+)L,
5-: ( g i v e e ) L ,
10: (clidiui~e) L.
This decomposition is obtained from the Georgi and Glashow assignment by interchanging gL ~ dL and e [ <+~eL. The transition SO10-+ SU 5 ® U 1 can be realized by a Higgs meson transforming like the SU 5 singlet of 45 of SO10. Transition SU 5 ® U 1 -+ SU2 ® UI® SU 3 could be realized either by a Higgs meson transforming like the SU 2 and SU 3 singlet contained in 16 or 126 of SO10. In the latter case, the right-handed neutrino gets a Majorana mass, not in the SU 5 limit as for the Georgi and Gtashow choice, but only in the SU 2 ® U l ® SU 3 limit. One expects for it, therefore, a Majorana mass ~Mw. As a consequence the light neutrino could have mass "~m2/Mw ~ 1 - 1 0 6 MeV2/ 100 GeV, 10 eV <~rn v <~ 10 MeV, a range easily accessible to experiment. In the Georgi and Glashow SO10 -+ SU5 case, vR still gets a very large mass at the two-loop level if a 16 Higgs is responsible for the SO10 -+ SU5 transition [10]. On the other hand, in the present case, if the Higgs responsible for breaking SO10 down to SU 5 ® U 1 were the neutral SU2 and SU 3 singlet of 16, the/)R Majorana mass resulting from two-loop graphs would be much too small, both for the nucleon stabilization discussion of this paper and for explaining the smallness of the neutrino mass. The remaining element of the chain 424
11 August 1983
SO10-+SU 5 ® U I - + S U 2 ® U 1® SU 3-+U~ m ® SU 3, SU 2 ® U 1® SU 3-+U~~n ® SU 3, can be obtained as in the standard SO10 scheme by Higgs doublets contained in 10, 120 or 126 of SO10. The right-handed neutrino has too large a mass in this scheme to be present in proton decay channels and only one up-type quark (u) is lighter than the proton. These two facts make up all the difference between the Georgi and Glashow SU5 and the Barr SU 5 ® U 1 cases when forbidding proton decay ~ la Jarlskog [2]. For the three-generation case proton decay is forbidden to lowest order by assigning to:
5: (ui Vr r ) L ,
(c-~vu U)L,
q i Ve e)L
(c~"and t-~ are mixtures of ci and i-i fields only), and 10: (bi di (K+Ui) 1 V')L,
(si si (K+Ui) 2 v")L,
(di bi (K+Ui) 3 v")L and 1: e~,,
/at,
r[,
i is the SU 3 colour index, K + is the hermitean conjugate of the Kobayashi and Maskawa matrix [4] and Ui is the column (ui ci ri). Inspection shows that no restriction on the matrix K is necessary to forbid proton decay. As for the Georgi-Glashow SU s case, proton decay is thus forbidden only to zeroth order in the weak interactions. Calculation of the box diagrams giving leading logarithmic corrections shows that to first order in the weak interactions, the proton decay amplitude is again the onegeneration amplitude times a factor O((a2/2rr)(m2/M2w)log(Mw/m))~ 10 -6. As for the Georgi and Glashow case [6] a level of validity of SU 5 ® U 1 of 1014 GeV would lead to a proton lifetime of 1042 yr or, alternatively, a lifetime of 1030 yr would correspond to a symmetry level of 10 lj GeV. Proton decay would still be possible at the SO10 level. If one wants to forbid proton decay at this level, it must be forbidden both for SU5 and SU 5 ® U 1 interactions. Therefore, the restriction 03 = 0 for the matrix K must be imposed. One is lead to assign to representation 16 of SO10 the contents:
Volume 127B, number 6
PHYSICS LETTERS
16: (di (K+Ui) 1 bi c-~Vev (ell) ~' £~)L , + 2 .... g,, (si (1(Ui) si t i Veu (e//)' J~)L , (bi (K+Ui) 3 di ui Vr r ~" ~ ) L , with 03 = 0. g' and T', Veu and Veu, (e/J) and (e/~)' are respectively arbitrary mixtures o f e- and t, Ve and v u and e and #, and 2+ is any positive lepton. A p p a r e n t l y an effective SU5 (Barr) current X SU 5 ( G - G ) current interaction results, (SL -+ (K+U) 2) X (e+(/u+) -+ UL), b u t this violates SU~v so it is suppressed b y a factor of order (Mw/M).
References [ 1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
11 August 1983
[2] C. Jarlskog, Phys. Lett. 82B (1979) 401. [3] R.N. Mohapatra, Phys. Rev. Lett. 43 (1979) 893; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Phys. Lett. 88B (1979) 320. [4] M. Kobayashi and M. Maskawa, Progr. Theor. Phys. 49 (1973) 652. [5] S. Nandi, A. Stern and E. Sudarshan, Phys. Lett. 113B (1982) 165. [6] D. Altsch(iler, P. Eckert and T. Schficker, Phys. Lett. l19B (1982) 351. [7] H. Georgi, in: Particles and fields 1974, ed. C.E. Carlson (AIP, New York, 1975) p. 575; H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193. [8] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B159 (1979) 16. [9] S.M. Barr, A New Symmetry Breaking Pattern for SO(10) and Proton Decay, Univ. of Washington, preprint 40048-82-PT2 (nov. 1981). [10] E. Witten, Phys. Lett. 91B (1980) 81; R. Barbieri, D.V. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. 90B (1980) 91.
425