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Natural flavor conservation and the absence of radiatively induced cabibbo angles
Volume 86B, number 3,4
PHYSICS LETTERS
8 October 1979
NATURAL FLAVOR CONSERVATION AND THE ABSENCE OF RADIATIVELY INDUCED CABIBBO ANGLES Gino SEGRI~ 1 and H. Arthur WELDON
2
Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA Received 29 June 1979
Non-trivial flavor conservation of neutral Higgs couplings, enforced by discrete symmetries, is known to lead to the vanishing of generalized mixing angles. We prove that this result, valid in the tree approximation, is not altered by radiative corrections.
It has long been hoped that some of the fermion masses and/or mixing angles that occur in the weak interactions might somehow be predicted by the theory. The standard SU(2)L × U(1) model [1] with one Higgs doublet makes no such predictions. It is now known that to determine the mixing angles in terms of quark masses one must add more Higgs fields and impose some additional horizontal symmetry ,1 This leads, however, to a second problem, that of flavor changing, Higgs mediated, neutral currents. If we have more than one Higgs doublet coupling to righthanded quarks of a given charge, diagonalizing the mass matrix will not generally diagonalize the Higgs couplings and we are left with a Higgs mediated A S = 2 effective interaction [3]. The limits on the K L K S mass [4] o f ( M L - M s ) / M K < 0 . 7 × 10 -14 lead to the requirement that Higgs masses be > 1 TeV* 2,
i Supported by the U.S. Department of Energy under Contract No. EY-76-C-02-3071. 2 Supported by a grant from the National Science Foundation. ,1 In fact, Barbieri et al. [2] show that the additional Higgs fields cannot just be one-dimensional representations of the horizontal symmetry. ¢2 The effective AS = 2 interaction is the form (dLSR)2 X f2/M~ where fandM H are characteristic Higgs couplings • and masses, respectively. Using f ~ GF112m, with m a typical quark mass, we estimate (ML - MS)/M K ~ 10GFMK × m2/M~ so that even for m 2 = mdms, we must have MH ~ 1 TeV.
in possible conflict with the conventional structure of the theory ,3 [5]. The obvious way out of this dilemma is "to arrange the horizontal symmetries such that the neutral Higgs couplings are flavor diagonal regardless of the spontaneous symmetry breaking. Surprisingly, this requirement turns out to be so strong that it forces all mixing angles to be either zero or 7r/2,4,5. The question that naturally arises is whether these mixing angles could be zero at the tree level, so as to have flavor diagonal Higgs couplings, but be non-zero after radiative corrections are included, and hence be calculable. This possibility has been raised by several authors recently [9]. The purpose of this note is to show that, contrary to these hopes, the mixing angles remain trivial (i.e. 0 or n/2) even after radiative corrections are included. The proof of the result is simple. We merely note ,3 A possible loophole, however, is suggested by Georgi and Nanopoulos [6]. ¢4 We also show in ref. [7] that when quarks are in reducible representations of the horizontal symmetry not all of the mixing angles need be trivial (i.e. 0 or n/2) but unfortunately the non-trivial angles are still not determined by the quark masses. CS Gatto et al. [8] show that when the quarks are in an irreducible representation of the horizontal symmetry then all the non-zero entries in the Cabibbo matrix must have the same magnitude. Our result, ref. [7], is that each row of the Cabibbo matrix contains only one non-zero entry (of magnitude one). 291
Volume 86B, number 3,4
PHYSICS LETTERS
8 October 1979
that after the spontaneous breaking of SU(2)L × U(1) a discrete symmetry remains unbroken that forbids mixings in higher order. For example, with six quarks but no mixing angles the charged current interaction is
coupling of Z 0 and 3' to quarks. The coupling ofW + to quarks in (1) is clearly invariant. All that remains to be checked are the Yukawa couplings. Here the condition that the couplings of the neutral Higgs naturally conserve quark flavor is essential, for it means that
W~(~LTVdL +-CL7ta SL +t-LTVbL ),
•/2yuk = fl(UL 4 +1 +dL 41o ) dR +f2 e L 4~ +-SL
(1)
where u,c,t,d,s,b refer to the mass eigenstate fields obtained in tree approximation. To obtain a Cabibbo angle the radiative corrections must generate some non-diagonal mass term of the form A ~ = eldS + e2~c + h.c.
(2)
Such terms would yield a Cabibbo angle when the fields u,d,c,s in the charged current (1) are expressed in terms of the corrected mass eigenstates. We now show that the transformation R = B × Q with
B:(W -+,z °,-r) +
Q: (d,s,b) (u,c,t)
0
~
(_W ±,z0,7),
-'
(-4a,+ # ) ,
~
( - d,s,b),
-~
( u , - c , - t),
is an exact symmetry of the spontaneously broken theory. (Note that both d L and d R change sign under Q; a labels the different Higgs doublets.) This symmetry obviously prevents any mass terms of the form (2) from occurring in higher order. . It is probably worth emphasizing that d,s,b,u,c,t are the mass eigenstates calculated at the tree level. They are related to the original fields in the unbroken lagrangian by a unitary transformation. (This means that the discrete symmetry Q will generally not take such a simple form in the original lagrangian.) Since, as we show, radiative corrections induce no further mixing then d,s,b,u,c,t turn out to be the exact mass eigenstates. To show that R is a valid symmetry we note that since the only bosons that change sign under B are the charged ones, charge conservation guarantees B invariance (and hence R invariance) for all interactions involving just scalars and vectors, but no quarks. The remaining interactions are J~quark = -qi(l~ + m)ij
qj + (I~a)ij ~ti4c~q].
(3)
The kinetic energy of the quarks involves only diagonal terms ~tiqi and is B X Q invariant; similarly for the 292
O)s R
+f3(-fL4; +bL~0)bR+f4(O-L 4 0 - d L 4 ~ ) u R
(4)
+f5 (CL 40 --SL 45-) CR +f6 (tL 460 -- bL 46) tR +h.c., where 41, 42 .... 46 may be arbitrary combinations (not necessarily distinct) of the original Higgs fields. It is easy to check that (4) is invariant under B × Q. This shows that R remains unbroken. Hence no radiative corrections can yield mass terms like (2) and there. fore the Cabibbo angle remains zero. One may extend this argument in several directions: (I) Obviously a term of the form e3db + e4~t is already forbidden by R, but e5~-b+ e6~t is not. However, the latter is forbidden by another symmetry B X Q'where Q' changes only the signs of u,s,t. Thus the charged current interaction is of the form (1) to all orders. (II) Another possibility is to begin with some mixing angle n/2 in tree approximation. Note that the labels d,s,b refer to the fact that the masses produced by (4) satisfy rn d < m s < m b. To start with a mixing angle of n/2 means, for example, to increase f2 and decrease f3 so much that the particle labelled s in (4) is heavier than the one labelled b. To make sense (i.e. recover m d < m s < rob) we therefore interchange the labels s and b everywhere they appear in (1) and (4). It is then trivial to find symmetries analogous to B × Q and B X Q' that forbid any radiative corrections to the mixing angles. (III)As yet we have considered only SU(2)L × U(1). In SU(2)L × SU(2)R X U(1) imposing symmetries to enforce flavor conservation of neutral Higgs exchanges also constrains the generalized Cabibbo angles to be 0 or n/2 [10]. By a slight extension of the above arguments one can show that no radiative corrections can modify these angles. Is there any way our result could be evaded? There is a theorem by Georgi and Pais [11] which shows that a discrete symmetry which holds at the tree level can only be broken by radiative corrections if the Higgs potential at the tree level has zero mass, nonGoldstone boson, states. There is therefore the loophole that tree level massless scalars could invalidate our result and yet become pseudo-Goldstone bosons
Volume 86B, number 3,4
PHYSICS LETTERS
via radiative corrections. This seems to be a remote possibility since for tree level massless scalars to be pseudo-Goldstone bosons the discrete symmetry must force the Higgs potential to have a larger "accidental symmetry" than the gauge symmetry G. Since in our case R for bosons is equivalent to charge conservation, we do not expect this to be the case ,6 It appears that we have two choices: Accept the minimal model in which there is one Higgs doublet coupling to each quark charge sector, and undetermined mixing angles; or allow more Higgs doublets and suitable restrictions so that mixing angles are nontrivially fixed, in which case we have flavor changing neutral Higgs couplings. In this latter situation the mixing angles probably need to be introduced at the tree level, since calculable mixing angles induced by radiative corrections seem to be much too small to agree with experiment [12].
¢6 There may, however, be pseudo-Goldstone bosons because of the discrete symmetry which enforces flavor conservation for the Higgs couplings.
8 October 1979
References [1] S. Weinberg, Phys. Rev. Lett. (1967) 1264; A. Salam, in: Elementary particle physics, Nobel Syrup. No. 8, ed. N. Svartholm (Stockholm, 1968) p, 367. [2] R. Barbieri, R. Gatto and F. Strocchi, Phys. Lett. 74B (1978) 344. [3] S. Weinberg and S. Glashow, Phys. Rev. D15 (1977) 1958; E. Paschos, Phys. Rev. D15 (1977) 1966. [4] Particle Data Group, Phys. Lett. 75B (1978) 1. [5] B.W. Lee, C. Quigg and H.B. Thacker, Phys. Rev. Lett. 38 (1977) 883; Phys. Rev. D16 (1977) 1519. M. Veltman, Acta Phys. Polon. B8 (1977) 475. [6] H. Georgi and D.V. Nanopoulos, Phys. Lett. 82B (1979) 95. [7] G. Segr?zand tt.A. Weldon, Univ. of Penna. preprint UPR-0125T, to be published in Ann. Phys. [8] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 83B (1979) 348. [9] R. Gatto, Proc. Jerusalem Einstein Symp. (1979); D. Grosser, Phys. Lett. 83B (1979) 355 and Tubingen preprint (1979). [10] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 80B (1979) 265. [11] H. Georgi and A. Pais, Phys. Rev. D10 (1974) 1246. [12] For examples, see A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 258; G. Segr~ and H.A. Weldon, Hadronic J. 1 (1978) 424.
293
PHYSICS LETTERS
8 October 1979
NATURAL FLAVOR CONSERVATION AND THE ABSENCE OF RADIATIVELY INDUCED CABIBBO ANGLES Gino SEGRI~ 1 and H. Arthur WELDON
2
Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA Received 29 June 1979
Non-trivial flavor conservation of neutral Higgs couplings, enforced by discrete symmetries, is known to lead to the vanishing of generalized mixing angles. We prove that this result, valid in the tree approximation, is not altered by radiative corrections.
It has long been hoped that some of the fermion masses and/or mixing angles that occur in the weak interactions might somehow be predicted by the theory. The standard SU(2)L × U(1) model [1] with one Higgs doublet makes no such predictions. It is now known that to determine the mixing angles in terms of quark masses one must add more Higgs fields and impose some additional horizontal symmetry ,1 This leads, however, to a second problem, that of flavor changing, Higgs mediated, neutral currents. If we have more than one Higgs doublet coupling to righthanded quarks of a given charge, diagonalizing the mass matrix will not generally diagonalize the Higgs couplings and we are left with a Higgs mediated A S = 2 effective interaction [3]. The limits on the K L K S mass [4] o f ( M L - M s ) / M K < 0 . 7 × 10 -14 lead to the requirement that Higgs masses be > 1 TeV* 2,
i Supported by the U.S. Department of Energy under Contract No. EY-76-C-02-3071. 2 Supported by a grant from the National Science Foundation. ,1 In fact, Barbieri et al. [2] show that the additional Higgs fields cannot just be one-dimensional representations of the horizontal symmetry. ¢2 The effective AS = 2 interaction is the form (dLSR)2 X f2/M~ where fandM H are characteristic Higgs couplings • and masses, respectively. Using f ~ GF112m, with m a typical quark mass, we estimate (ML - MS)/M K ~ 10GFMK × m2/M~ so that even for m 2 = mdms, we must have MH ~ 1 TeV.
in possible conflict with the conventional structure of the theory ,3 [5]. The obvious way out of this dilemma is "to arrange the horizontal symmetries such that the neutral Higgs couplings are flavor diagonal regardless of the spontaneous symmetry breaking. Surprisingly, this requirement turns out to be so strong that it forces all mixing angles to be either zero or 7r/2,4,5. The question that naturally arises is whether these mixing angles could be zero at the tree level, so as to have flavor diagonal Higgs couplings, but be non-zero after radiative corrections are included, and hence be calculable. This possibility has been raised by several authors recently [9]. The purpose of this note is to show that, contrary to these hopes, the mixing angles remain trivial (i.e. 0 or n/2) even after radiative corrections are included. The proof of the result is simple. We merely note ,3 A possible loophole, however, is suggested by Georgi and Nanopoulos [6]. ¢4 We also show in ref. [7] that when quarks are in reducible representations of the horizontal symmetry not all of the mixing angles need be trivial (i.e. 0 or n/2) but unfortunately the non-trivial angles are still not determined by the quark masses. CS Gatto et al. [8] show that when the quarks are in an irreducible representation of the horizontal symmetry then all the non-zero entries in the Cabibbo matrix must have the same magnitude. Our result, ref. [7], is that each row of the Cabibbo matrix contains only one non-zero entry (of magnitude one). 291
Volume 86B, number 3,4
PHYSICS LETTERS
8 October 1979
that after the spontaneous breaking of SU(2)L × U(1) a discrete symmetry remains unbroken that forbids mixings in higher order. For example, with six quarks but no mixing angles the charged current interaction is
coupling of Z 0 and 3' to quarks. The coupling ofW + to quarks in (1) is clearly invariant. All that remains to be checked are the Yukawa couplings. Here the condition that the couplings of the neutral Higgs naturally conserve quark flavor is essential, for it means that
W~(~LTVdL +-CL7ta SL +t-LTVbL ),
•/2yuk = fl(UL 4 +1 +dL 41o ) dR +f2 e L 4~ +-SL
(1)
where u,c,t,d,s,b refer to the mass eigenstate fields obtained in tree approximation. To obtain a Cabibbo angle the radiative corrections must generate some non-diagonal mass term of the form A ~ = eldS + e2~c + h.c.
(2)
Such terms would yield a Cabibbo angle when the fields u,d,c,s in the charged current (1) are expressed in terms of the corrected mass eigenstates. We now show that the transformation R = B × Q with
B:(W -+,z °,-r) +
Q: (d,s,b) (u,c,t)
0
~
(_W ±,z0,7),
-'
(-4a,+ # ) ,
~
( - d,s,b),
-~
( u , - c , - t),
is an exact symmetry of the spontaneously broken theory. (Note that both d L and d R change sign under Q; a labels the different Higgs doublets.) This symmetry obviously prevents any mass terms of the form (2) from occurring in higher order. . It is probably worth emphasizing that d,s,b,u,c,t are the mass eigenstates calculated at the tree level. They are related to the original fields in the unbroken lagrangian by a unitary transformation. (This means that the discrete symmetry Q will generally not take such a simple form in the original lagrangian.) Since, as we show, radiative corrections induce no further mixing then d,s,b,u,c,t turn out to be the exact mass eigenstates. To show that R is a valid symmetry we note that since the only bosons that change sign under B are the charged ones, charge conservation guarantees B invariance (and hence R invariance) for all interactions involving just scalars and vectors, but no quarks. The remaining interactions are J~quark = -qi(l~ + m)ij
qj + (I~a)ij ~ti4c~q].
(3)
The kinetic energy of the quarks involves only diagonal terms ~tiqi and is B X Q invariant; similarly for the 292
O)s R
+f3(-fL4; +bL~0)bR+f4(O-L 4 0 - d L 4 ~ ) u R
(4)
+f5 (CL 40 --SL 45-) CR +f6 (tL 460 -- bL 46) tR +h.c., where 41, 42 .... 46 may be arbitrary combinations (not necessarily distinct) of the original Higgs fields. It is easy to check that (4) is invariant under B × Q. This shows that R remains unbroken. Hence no radiative corrections can yield mass terms like (2) and there. fore the Cabibbo angle remains zero. One may extend this argument in several directions: (I) Obviously a term of the form e3db + e4~t is already forbidden by R, but e5~-b+ e6~t is not. However, the latter is forbidden by another symmetry B X Q'where Q' changes only the signs of u,s,t. Thus the charged current interaction is of the form (1) to all orders. (II) Another possibility is to begin with some mixing angle n/2 in tree approximation. Note that the labels d,s,b refer to the fact that the masses produced by (4) satisfy rn d < m s < m b. To start with a mixing angle of n/2 means, for example, to increase f2 and decrease f3 so much that the particle labelled s in (4) is heavier than the one labelled b. To make sense (i.e. recover m d < m s < rob) we therefore interchange the labels s and b everywhere they appear in (1) and (4). It is then trivial to find symmetries analogous to B × Q and B X Q' that forbid any radiative corrections to the mixing angles. (III)As yet we have considered only SU(2)L × U(1). In SU(2)L × SU(2)R X U(1) imposing symmetries to enforce flavor conservation of neutral Higgs exchanges also constrains the generalized Cabibbo angles to be 0 or n/2 [10]. By a slight extension of the above arguments one can show that no radiative corrections can modify these angles. Is there any way our result could be evaded? There is a theorem by Georgi and Pais [11] which shows that a discrete symmetry which holds at the tree level can only be broken by radiative corrections if the Higgs potential at the tree level has zero mass, nonGoldstone boson, states. There is therefore the loophole that tree level massless scalars could invalidate our result and yet become pseudo-Goldstone bosons
Volume 86B, number 3,4
PHYSICS LETTERS
via radiative corrections. This seems to be a remote possibility since for tree level massless scalars to be pseudo-Goldstone bosons the discrete symmetry must force the Higgs potential to have a larger "accidental symmetry" than the gauge symmetry G. Since in our case R for bosons is equivalent to charge conservation, we do not expect this to be the case ,6 It appears that we have two choices: Accept the minimal model in which there is one Higgs doublet coupling to each quark charge sector, and undetermined mixing angles; or allow more Higgs doublets and suitable restrictions so that mixing angles are nontrivially fixed, in which case we have flavor changing neutral Higgs couplings. In this latter situation the mixing angles probably need to be introduced at the tree level, since calculable mixing angles induced by radiative corrections seem to be much too small to agree with experiment [12].
¢6 There may, however, be pseudo-Goldstone bosons because of the discrete symmetry which enforces flavor conservation for the Higgs couplings.
8 October 1979
References [1] S. Weinberg, Phys. Rev. Lett. (1967) 1264; A. Salam, in: Elementary particle physics, Nobel Syrup. No. 8, ed. N. Svartholm (Stockholm, 1968) p, 367. [2] R. Barbieri, R. Gatto and F. Strocchi, Phys. Lett. 74B (1978) 344. [3] S. Weinberg and S. Glashow, Phys. Rev. D15 (1977) 1958; E. Paschos, Phys. Rev. D15 (1977) 1966. [4] Particle Data Group, Phys. Lett. 75B (1978) 1. [5] B.W. Lee, C. Quigg and H.B. Thacker, Phys. Rev. Lett. 38 (1977) 883; Phys. Rev. D16 (1977) 1519. M. Veltman, Acta Phys. Polon. B8 (1977) 475. [6] H. Georgi and D.V. Nanopoulos, Phys. Lett. 82B (1979) 95. [7] G. Segr?zand tt.A. Weldon, Univ. of Penna. preprint UPR-0125T, to be published in Ann. Phys. [8] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 83B (1979) 348. [9] R. Gatto, Proc. Jerusalem Einstein Symp. (1979); D. Grosser, Phys. Lett. 83B (1979) 355 and Tubingen preprint (1979). [10] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 80B (1979) 265. [11] H. Georgi and A. Pais, Phys. Rev. D10 (1974) 1246. [12] For examples, see A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 258; G. Segr~ and H.A. Weldon, Hadronic J. 1 (1978) 424.
293