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Radiatively-induced flavor-changing neutral Higgs-Boson couplings
Volume
103B, number
PHYSICS
2
RADIATIVELY-INDUCED
FLAVOR-CHANGING
16 July 1981
LETTERS
NEUTRAL HIGGS-BOSON COUPLINGS
Mark B. WISE ’ Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138,
USA
Received 22 December 1980 Revised manuscript received 27 April 198 1
In the standard model of weak and electromagnetic interactions with several Higgs doublets there are several physical neutral Higgs particles. A discrete symmetry can be imposed which forbids flavor-changing neutral Higgs-boson couplings at tree level. In this case it is shown that radiative corrections induce significant flavor-changing neutral Higgs-boson couplings.
In the standard model of weak and electromagnetic interactions [l] with one Higgs doublet the tree-level couplings of the physical Higgs scalar to quarks are flavor diagonal. Radiative corrections induce finite flavor-changing Higgs couplings; however, they have been shown to be small [2] . In addition the Higgs mass must be greater than 7 GeV so the production of a real Higgs scalar in kaon or B-meson decays is forbidden [3]. If several Higgs doublets are present there will be more than one physical Higgs particle in the theory. A discrete symmetry can be used to forbid tree-level flavor-changing couplings of the neutral Higgs particles. The purpose of this note is to show that in this case radiative corrections induce significant flavor-changing couplings for some of the neutral Higgs bosons. For simplicity consider a model with two Higgs doublets
(1)
and
The discrete symmetry, G2 + -$2, is imposed to forbid Yukawa couplings of @I to the quarks *l. In the
absence of CP violation the vacuum is characterized by two real vacuum expectation values 0
(@l)O=
0
Science and the
($2)o =
rl
0
.
t
Rotating the Higgs doublets $1 = cos cQ1 t sin “Q2,
(34
qb2= -sin a@1 + cos c~$~ ,
WI
where sin cy= $/(E2 + q2)l I2 and_cos (Y= q/(g2 +&l/2 it IS evident that only $Q has a vacuum expectation’ value [5] . The physical neutral spin-zero particles are two “Higgs scalars” that are linear combinations of Re p and Re @ and one “Higgs pseudoscalar” I-0 _There are also physical charged Higgs P” =fi Im $2 particles with fields $5. The Higgs potential can be arranged so that P” is light compared with the scale of symmetry breaking. It is the flavor-changing couplings of PO, which are caused by radiative corrections, that will be studied in this paper. The tree-level couplings of P” to the quarks are diagonal in flavor and follow from the term “cy = (ig/mw)
’ Research supported in part by the National Foundation under Grant No. PHY77-22864, Harvqd Society of Fellows. ” This model has been discussed in ref. [4]
0
and
(l/Q)P’ (o+Ygu
- DMDySD)
(4)
in the lagrangian density. Here and
D= ! 0b
(5)
121
Volume
103B, number
2
PHYSICS
s(p)
--T@T \ ‘1
‘1 P”(k) Fig. 1. Feynman diagram contributing to leading order in large masses.
to the s + d P” transition
m,, m,)
and
MD = diag(md , m,, mb),
E(p, k) = i(E/s) (g3/32n2)(ms/Mw)
+ s1s2(c1s2c3
tron pairs *3 this would give a branching ratio for K+ + rr+e+e- that is well above the experimental value of 2.6 X low7 *4 , unless (E/v)~ is very small *5. If P” is lighter than the b-quark then a diagram like fig. 1 induces the decay b + s PO. Treating s1 and s3 as small quantities and neglecting the masses of the strange quark and P” the rate for b + s P” is
X (s2 + ~3s~)~ [ln(M$/mt)]
(6) are the quark-mass matrices, Mw is the W-boson mass, and g is the weak SU(2) gauge coupling. One-loop radiative corrections induce an s + d P” transition. In Feynman gauge, to leading order in the W-boson and large quark masses, only fig. 1 contributes. Assuming MG; = M,, other diagrams do not produce a large logarithm in the W-boson mass. Fig. 1 gives rise to the effective s -+ dP” vertex * 2
X ]slc2(c1~2~3
16 July 1981
I’(b + sP”) = (g/q)2(G~.m~m~/&
are fields for the quark-mass eigenstates with charge ti and charge -i, respectively Mu = diag(m,,
LETTERS
- s2s3)(m,/Mw)21n(M~/m2c) +c~s3)(m~/Mw)~
ln(M$/mf)l
x J(P - kg (1 + Yg)S(P>PO(k>
(7)
in momentum space. If P” is lighter than the kaonpion mass difference, mK - m,, this vertex causes kaon decays of the form K + .P”. Setting s2 = s3 = 0 and neglecting the pion and P” masses the rate for K” + n+P” is
27 n5)
2 ci .
(9)
The total width for b-quark decay is P,, x (5G;/192m3)mi(s2
+ c2s3)2F(mc/mb)
)
(10)
when the contribution from Cabibbo suppressed modes and final states with more than one heavy particle are neglected. The function F(x) is defined by F(x)=l-8x2+8x6-x8-24x41nx.
(11)
Using m, = 1.5 GeV, mb = 4.5 GeV and mt = 20 GeV, the branching ratio for b -+ s P” is about 0.02 (,$/Q)~. Since the t-quark mass is not small in comparison with the W-boson mass this calculation only serves as a crude order of magntude estimate, It is not inconceivable that such a “pseudoscalar” be detected in B-meson decays. For example, a P” with mass around 2 GeV could give rise to decays like B -+ KrrP” --f Krrp+pwith branching ratios of order of a fraction of a percentf6. In a multi-Higgs-model tree-level flavor-changing couplings of the neutral Higgs bosons can be forbidden by a discrete symmetry. However, in this case radiative corrections induce finite flavor-changing couplings.In this paper the radiatively-induced flavor-changing couplings of a neutral “pseudoscalar” Higgs boson were *3 This possibility
X [W4$/m~)12f?m~.
(8)
Ineq. (8) GF is the Fermi constant and f+is the form factor in K, 3 decay. The branching ratio for K’ -+ n+P” is then about 10-4(g/~)2. It is amusing to note that if P” decayed primarily to electron-posi*zHeres~-sint?randc~=cos0rfori=1,2,3.TheangJesBr, 02 and 83 are the weak mixing angles in the KobayashMaskawa sixquark model [ 61. CP violation is being neglected so the phase 6 has been set to zero. The down quark mass has also been neglected.
122
periments.
appears to be ruled out by beam dump See ref. [7].
*’ The K+ + n+e+e- experiment
ex-
has a cut on the effective mass of an e’e- pair of 140 MeV [ 8 ] Thus a Pi decaying to e’ewith a mass less than 140 MeV might have escaped detection in kaon decays even if (E/q)2 is not small. *’ An upper bound on (g/v)2 was derived from charged Higgs boson contributions to K”-Ro mixing in ref. [9]. For a very light P” the vertex in eq. (7) gives a small contribution to the KL-KS mass difference of order 10-l 6 ([/n)2 MeV when the approximation sa = sa = 0 is made.
*6 A possible background for this is the decay B --) J/$ Kn + p’~-Kn. (See for example, ref. [ lo]). However, the invariant mass of the p’p- will be different in this case.
Volume
103B, number
2
PHYSICS
estimated in a two-Higgs-doublet model and found to be significant. Strong-interaction corrections were neglected in this computation. Some of the effects of strong interactions could be taken into account using renormalization group techniques; however, these corrections are not expected to change the character of the free quark model result. Finally I note that the results of this paper can be relevant if the SU(2) X U(1) symmetry of weak and electromagnetic interactions is broken dynamically. In this case there may be light pseudo-Goldstone bosons (PGBs) [l 1 ] . At low energies * 7 the interactions of the PGBs with quarks and leptons are described by an effective field theory with apparently fundamental Higgs multiplets. In the absence of spontaneous CP violation the jlavor-diagonal tree-level couplings of the neutral PGBs are parity conserving if the PGBs are assigned a negative intrinsic parity. If the effective low-energy lagrangian has a discrete symmetry which prevents tree-level flavor-changing neutral PGB couplings +a radiative corrections will induce flavor-changing couplings that are significant *9. ‘7 That is at energies low compared
with the typical scale of the new strong interactions which cause the condensate that breaks SU(2) X U(1) to form. *8 For some thoughts on how this may occur, see ref. [ 121.
‘9 Of course these flavor-changing
couplings are only relevant if the PGB is very light. A light PGB might also be detectable in r decays through its tree-level flavor-diagonal couplings. For example, in the two-Higgs-doublet model that was considered in this paper, the branching ratio for r -+ Pay is about 2 X 10m4 (t/n)’ when the mass of P” is neglected. (See ref. [ 131).
16 July 1981
LETTERS
I am grateful to H. Georgi for many useful discussions. I have also benefited from conversations with E. Eichten and J. Preskill. I thank J. Ellis and the Theory Group at CERN for their hospitality during a visit when work on this project began. References [l] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264;
[2] [3] [4] [5] [6]
A. Salam, in: Elementary particle physics: relativistic groups and analyticity, Nobel Symp. No. 8, ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) p. 367. J. Ellis et al., Nucl. Phys. B109 (1976) 213. S. Weinberg, Phys. Rev. Lett. 36 (1976) 294; A.D. Linde, Pis’ma Zh. Eksp. Teor. Fiz. 23 (1976) 64. H. Haber et al., Nucl. Phys. B161 (1979) 493. H. Georgi and D. Nanopoulos, Phys. Lett. 82B (1979) 95. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
[7] H. Faissner et al.. Phys. Lett. 96B (1980) 210; and references therein. [8] P. Bloch et al., Phys. Lett. 56B (1975) 201. [9] L.F. Abbott et al., Phys. Rev. D21 (1980) 1393. [lo] M.B. Wise, Phys. Lett. 89B (1980) 229. [Ill E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125; S. Dimopoulos, Nucl. Phys. B168 (1980) 93; M. Beg et al., Phys. Rev. Lett. 43 (1979) 1701. [ 121 J. Ellis et al.,CERN 2938 (1980), unpublished. [13] F. WiJczek, Phys. Rev. Lett. 39 (1977) 1304.
123
103B, number
PHYSICS
2
RADIATIVELY-INDUCED
FLAVOR-CHANGING
16 July 1981
LETTERS
NEUTRAL HIGGS-BOSON COUPLINGS
Mark B. WISE ’ Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138,
USA
Received 22 December 1980 Revised manuscript received 27 April 198 1
In the standard model of weak and electromagnetic interactions with several Higgs doublets there are several physical neutral Higgs particles. A discrete symmetry can be imposed which forbids flavor-changing neutral Higgs-boson couplings at tree level. In this case it is shown that radiative corrections induce significant flavor-changing neutral Higgs-boson couplings.
In the standard model of weak and electromagnetic interactions [l] with one Higgs doublet the tree-level couplings of the physical Higgs scalar to quarks are flavor diagonal. Radiative corrections induce finite flavor-changing Higgs couplings; however, they have been shown to be small [2] . In addition the Higgs mass must be greater than 7 GeV so the production of a real Higgs scalar in kaon or B-meson decays is forbidden [3]. If several Higgs doublets are present there will be more than one physical Higgs particle in the theory. A discrete symmetry can be used to forbid tree-level flavor-changing couplings of the neutral Higgs particles. The purpose of this note is to show that in this case radiative corrections induce significant flavor-changing couplings for some of the neutral Higgs bosons. For simplicity consider a model with two Higgs doublets
(1)
and
The discrete symmetry, G2 + -$2, is imposed to forbid Yukawa couplings of @I to the quarks *l. In the
absence of CP violation the vacuum is characterized by two real vacuum expectation values 0
(@l)O=
0
Science and the
($2)o =
rl
0
.
t
Rotating the Higgs doublets $1 = cos cQ1 t sin “Q2,
(34
qb2= -sin a@1 + cos c~$~ ,
WI
where sin cy= $/(E2 + q2)l I2 and_cos (Y= q/(g2 +&l/2 it IS evident that only $Q has a vacuum expectation’ value [5] . The physical neutral spin-zero particles are two “Higgs scalars” that are linear combinations of Re p and Re @ and one “Higgs pseudoscalar” I-0 _There are also physical charged Higgs P” =fi Im $2 particles with fields $5. The Higgs potential can be arranged so that P” is light compared with the scale of symmetry breaking. It is the flavor-changing couplings of PO, which are caused by radiative corrections, that will be studied in this paper. The tree-level couplings of P” to the quarks are diagonal in flavor and follow from the term “cy = (ig/mw)
’ Research supported in part by the National Foundation under Grant No. PHY77-22864, Harvqd Society of Fellows. ” This model has been discussed in ref. [4]
0
and
(l/Q)P’ (o+Ygu
- DMDySD)
(4)
in the lagrangian density. Here and
D= ! 0b
(5)
121
Volume
103B, number
2
PHYSICS
s(p)
--T@T \ ‘1
‘1 P”(k) Fig. 1. Feynman diagram contributing to leading order in large masses.
to the s + d P” transition
m,, m,)
and
MD = diag(md , m,, mb),
E(p, k) = i(E/s) (g3/32n2)(ms/Mw)
+ s1s2(c1s2c3
tron pairs *3 this would give a branching ratio for K+ + rr+e+e- that is well above the experimental value of 2.6 X low7 *4 , unless (E/v)~ is very small *5. If P” is lighter than the b-quark then a diagram like fig. 1 induces the decay b + s PO. Treating s1 and s3 as small quantities and neglecting the masses of the strange quark and P” the rate for b + s P” is
X (s2 + ~3s~)~ [ln(M$/mt)]
(6) are the quark-mass matrices, Mw is the W-boson mass, and g is the weak SU(2) gauge coupling. One-loop radiative corrections induce an s + d P” transition. In Feynman gauge, to leading order in the W-boson and large quark masses, only fig. 1 contributes. Assuming MG; = M,, other diagrams do not produce a large logarithm in the W-boson mass. Fig. 1 gives rise to the effective s -+ dP” vertex * 2
X ]slc2(c1~2~3
16 July 1981
I’(b + sP”) = (g/q)2(G~.m~m~/&
are fields for the quark-mass eigenstates with charge ti and charge -i, respectively Mu = diag(m,,
LETTERS
- s2s3)(m,/Mw)21n(M~/m2c) +c~s3)(m~/Mw)~
ln(M$/mf)l
x J(P - kg (1 + Yg)S(P>PO(k>
(7)
in momentum space. If P” is lighter than the kaonpion mass difference, mK - m,, this vertex causes kaon decays of the form K + .P”. Setting s2 = s3 = 0 and neglecting the pion and P” masses the rate for K” + n+P” is
27 n5)
2 ci .
(9)
The total width for b-quark decay is P,, x (5G;/192m3)mi(s2
+ c2s3)2F(mc/mb)
)
(10)
when the contribution from Cabibbo suppressed modes and final states with more than one heavy particle are neglected. The function F(x) is defined by F(x)=l-8x2+8x6-x8-24x41nx.
(11)
Using m, = 1.5 GeV, mb = 4.5 GeV and mt = 20 GeV, the branching ratio for b -+ s P” is about 0.02 (,$/Q)~. Since the t-quark mass is not small in comparison with the W-boson mass this calculation only serves as a crude order of magntude estimate, It is not inconceivable that such a “pseudoscalar” be detected in B-meson decays. For example, a P” with mass around 2 GeV could give rise to decays like B -+ KrrP” --f Krrp+pwith branching ratios of order of a fraction of a percentf6. In a multi-Higgs-model tree-level flavor-changing couplings of the neutral Higgs bosons can be forbidden by a discrete symmetry. However, in this case radiative corrections induce finite flavor-changing couplings.In this paper the radiatively-induced flavor-changing couplings of a neutral “pseudoscalar” Higgs boson were *3 This possibility
X [W4$/m~)12f?m~.
(8)
Ineq. (8) GF is the Fermi constant and f+is the form factor in K, 3 decay. The branching ratio for K’ -+ n+P” is then about 10-4(g/~)2. It is amusing to note that if P” decayed primarily to electron-posi*zHeres~-sint?randc~=cos0rfori=1,2,3.TheangJesBr, 02 and 83 are the weak mixing angles in the KobayashMaskawa sixquark model [ 61. CP violation is being neglected so the phase 6 has been set to zero. The down quark mass has also been neglected.
122
periments.
appears to be ruled out by beam dump See ref. [7].
*’ The K+ + n+e+e- experiment
ex-
has a cut on the effective mass of an e’e- pair of 140 MeV [ 8 ] Thus a Pi decaying to e’ewith a mass less than 140 MeV might have escaped detection in kaon decays even if (E/q)2 is not small. *’ An upper bound on (g/v)2 was derived from charged Higgs boson contributions to K”-Ro mixing in ref. [9]. For a very light P” the vertex in eq. (7) gives a small contribution to the KL-KS mass difference of order 10-l 6 ([/n)2 MeV when the approximation sa = sa = 0 is made.
*6 A possible background for this is the decay B --) J/$ Kn + p’~-Kn. (See for example, ref. [ lo]). However, the invariant mass of the p’p- will be different in this case.
Volume
103B, number
2
PHYSICS
estimated in a two-Higgs-doublet model and found to be significant. Strong-interaction corrections were neglected in this computation. Some of the effects of strong interactions could be taken into account using renormalization group techniques; however, these corrections are not expected to change the character of the free quark model result. Finally I note that the results of this paper can be relevant if the SU(2) X U(1) symmetry of weak and electromagnetic interactions is broken dynamically. In this case there may be light pseudo-Goldstone bosons (PGBs) [l 1 ] . At low energies * 7 the interactions of the PGBs with quarks and leptons are described by an effective field theory with apparently fundamental Higgs multiplets. In the absence of spontaneous CP violation the jlavor-diagonal tree-level couplings of the neutral PGBs are parity conserving if the PGBs are assigned a negative intrinsic parity. If the effective low-energy lagrangian has a discrete symmetry which prevents tree-level flavor-changing neutral PGB couplings +a radiative corrections will induce flavor-changing couplings that are significant *9. ‘7 That is at energies low compared
with the typical scale of the new strong interactions which cause the condensate that breaks SU(2) X U(1) to form. *8 For some thoughts on how this may occur, see ref. [ 121.
‘9 Of course these flavor-changing
couplings are only relevant if the PGB is very light. A light PGB might also be detectable in r decays through its tree-level flavor-diagonal couplings. For example, in the two-Higgs-doublet model that was considered in this paper, the branching ratio for r -+ Pay is about 2 X 10m4 (t/n)’ when the mass of P” is neglected. (See ref. [ 131).
16 July 1981
LETTERS
I am grateful to H. Georgi for many useful discussions. I have also benefited from conversations with E. Eichten and J. Preskill. I thank J. Ellis and the Theory Group at CERN for their hospitality during a visit when work on this project began. References [l] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264;
[2] [3] [4] [5] [6]
A. Salam, in: Elementary particle physics: relativistic groups and analyticity, Nobel Symp. No. 8, ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) p. 367. J. Ellis et al., Nucl. Phys. B109 (1976) 213. S. Weinberg, Phys. Rev. Lett. 36 (1976) 294; A.D. Linde, Pis’ma Zh. Eksp. Teor. Fiz. 23 (1976) 64. H. Haber et al., Nucl. Phys. B161 (1979) 493. H. Georgi and D. Nanopoulos, Phys. Lett. 82B (1979) 95. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
[7] H. Faissner et al.. Phys. Lett. 96B (1980) 210; and references therein. [8] P. Bloch et al., Phys. Lett. 56B (1975) 201. [9] L.F. Abbott et al., Phys. Rev. D21 (1980) 1393. [lo] M.B. Wise, Phys. Lett. 89B (1980) 229. [Ill E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125; S. Dimopoulos, Nucl. Phys. B168 (1980) 93; M. Beg et al., Phys. Rev. Lett. 43 (1979) 1701. [ 121 J. Ellis et al.,CERN 2938 (1980), unpublished. [13] F. WiJczek, Phys. Rev. Lett. 39 (1977) 1304.
123