Anomalous couplings of the third generation in rare B decays
Gustavo Burdman*Department of Physics, University of Wisconsin, Madison, Wisconsin 53706
M. C. Gonzalez-Garcia
Instituto de Fı´sica Corpuscular – IFIC/CSIC, Departament de Fı´sica Teo`rica, Universitat de Vale`ncia, 46100 Burjassot, Vale`ncia, Spain and Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, Brazil
S. F. Novaes
Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, Brazil
共Received 15 June 1999; published 9 May 2000兲
We study the potential effects of anomalous couplings of the third generation quarks to gauge bosons in rare
B decays. We focus on the constraints from flavor changing neutral current processes such as b→s␥ and b
→sl⫹l⫺. We consider both dimension-four and dimension-five operators and show that the latter can give
large deviations from the standard model in the still unobserved dilepton modes, even after the bounds from
b→s␥ and precision electroweak observables are taken into account.
PACS number共s兲: 13.20.He, 14.65.Fy, 14.65.Ha
I. INTRODUCTION
The continuing experimental success of the standard model 共SM兲 suggests the possibility that additional particles and/or non-standard interactions may only be found at scales much larger than MW. On the other hand, several questions remain unanswered within the SM framework that may re-quire new dynamics in order to be addressed. Chief among these questions are the origin of electroweak symmetry breaking and of fermion masses. In principle, it could be argued that the energy scales of the new dynamics related to these questions may be so large as to be irrelevant to observ-ables at the electroweak scale. However, it is known that the physics behind the Higgs sector, responsible for the breaking of the electroweak symmetry, cannot reside at scales much higher than few TeV. Furthermore, it is possible that the origin of the top quark mass might be related to electroweak symmetry breaking. Thus, at least in some cases, the dynam-ics associated with new physdynam-ics may not reside at arbitrarily high energies and there might be some observable effects at lower energies.
The effects of integrating out the physics residing at some high energy scale⌳ⰇMW, can be organized in an effective field theory for the remaining degrees of freedom. Such a theory for the electroweak symmetry breaking共EWSB兲 sec-tor of the SM involves the electroweak gauge bosons as well as the Nambu-Goldstone bosons 共NGB兲 associated with the spontaneous breaking of SU(2)L⫻U(1)Y down to U(1)EM
关1,2兴. The effective theory must be studied up to
next-to-leading order for the possible departures from the SM to appear. This program resembles that of chiral perturbation theory for pions in low energy QCD where, for instance, the presence of the resonance results in deviations from the low energy theorems. For the case of the electroweak
inter-actions, a variety of electroweak precision measurements and flavor changing neutral current processes provide testing ground for possible deviations originating in the EWSB sec-tor of the SM. The next-to-leading order terms in the effec-tive theory will generally contribute to oblique corrections, triple and quartic anomalous gauge boson couplings, and corrections to the NGB propagators that result in four-fermion interactions 关3兴.
In addition to the low energy description of the interac-tions of the EWSB sector共i.e. gauge bosons plus NGBs兲 one may consider the possibility that the new physics above ⌳ may also modify the effective interactions of the SM fermi-ons to the electroweak gauge bosfermi-ons. In principle, this also has a parallel in low energy QCD, as it is pointed out in Ref.
关4兴, where symmetry alone is not enough to determine the
axial coupling of nucleons to pions. In fact, the departure of this coupling from unity is a non-universal effect, only de-termined by the full theory of QCD. Thus, in Ref. 关4兴 it is suggested that in addition to the effects in the EWSB sector of the theory, it is possible that the interactions of fermions with the NGBs are affected by the new dynamics above ⌳, resulting in anomalous interactions with the electroweak gauge bosons. This is particularly interesting if fermion masses are dynamically generated, as is the case with the nucleon mass. Interestingly, the proximity of the top quark mass to the electroweak scalev⫽246 GeV, hints the
possi-bility the top quark mass might be a dynamically generated ‘‘constituent’’ mass. Thus, it is of particular interest to study the couplings of third generation quarks to electroweak gauge bosons.
Processes involving FCNC transitions in B and K decays are a crucial complement to precision electroweak observ-ables, when constraining the physics of the EWSB sector. The effects of anomalous triple gauge boson couplings 关5兴, as well as of the corrections to NGB propagators关6兴 give in each case a distinct pattern of deviations from the SM expec-tations in rare B and K decays. On the other hand, the anomalous couplings of third generation quarks to the W and *Current address: Department of Physics, Boston University, 590
the Z can come from dimension-four and dimension-five op-erators. The indirect effects of the dimension-four operators have been considered in relation to electroweak observables in Refs.关4,7兴, as well as the b→s␥ transitions关8兴. The con-straints on dimension-five operators from electroweak phys-ics have been studied in Ref.关9兴. In this paper, we consider the effects of all possible dimension-five operators in B fla-vor changing neutral current 共FCNC兲 transitions such as b
→s␥ and b→sl⫹l⫺. For completeness, we also present the analysis of the dimension-four operators. We discuss the stringent limits found for one dimension-four parameter in Ref.关8兴 for b→s␥ in light of underlying symmetries. More-over, we will see that the effects of dimension-five operators are comparable and may even dominate over the supposedly leading lower dimension contributions.
In Sec. II we present a brief introduction to the effective theory approach and set our notation. We present the con-straints from rare B decays on the coefficients of dimension-four operators in Sec. III. In Sec. IV we discuss the possible effects in rare B decays from all possible dimension-five op-erators involving the third generation quarks. Finally, we dis-cuss the results and conclude in Sec. V.
II. THE EFFECTIVE THEORY
If the Higgs boson, responsible for the electroweak sym-metry breaking, is very heavy, it can be effectively removed from the physical low-energy spectrum. In this case and for dynamical symmetry breaking scenarios relying on new strong interactions, one is led to consider the most general effective Lagrangian which employs a nonlinear representa-tion of the spontaneously broken SU(2)L⫻U(1)Y gauge symmetry 关10兴. The resulting chiral Lagrangian is a non-renormalizable nonlinear model coupled in a gauge-invariant way to the Yang-Mills theory. This model-independent approach incorporates by construction the low-energy theorems 关11兴 that predict the general behavior of Goldstone boson amplitudes, irrespective of the details of the symmetry breaking mechanism. Unitarity requires that this low-energy effective theory should be valid up to some en-ergy scale smaller than 4v⯝3 TeV, where new physics
would come into play.
In order to specify the effective Lagrangian for the Gold-stone bosons, we assume that the symmetry breaking pattern is G⫽SU(2)L⫻U(1)Y→H⫽U(1)em, leading to just three Goldstone bosons a (a⫽1,2,3). With this choice, the building block of the chiral Lagrangian is the dimensionless unimodular matrix field⌺,
⌺⫽exp
冉
i aav
冊
, 共1兲where a (a⫽1,2,3) are the Pauli matrices. We implement the SU(2)C custodial symmetry by imposing a unique di-mensionful parameter,v, for charged and neutral fields.
Un-der the action of G the transformation of⌺ is
⌺→⌺
⬘
⫽L⌺R†,where L⫽exp(i␣aa/2) and R⫽exp(iy3/2), with ␣a and y being the parameters of the transformation.
The gauge fields are represented by the matrices Wˆ
⫽aW
a/(2i), Bˆ
⫽3B/(2i), while the associated field
strengths are given by
Wˆ⫽Wˆ⫺Wˆ⫺g关Wˆ,Wˆ兴, Bˆ⫽Bˆ⫺Bˆ.
In the nonlinear representation of the gauge group SU(2)L
⫻U(1)Y, the mass term for the vector bosons is given by the lowest order operator involving the matrix⌺. Therefore, the kinetic Lagrangian for the gauge bosons reads
LB⫽ 1 2Tr共WˆWˆ ⫹Bˆ Bˆ兲⫹ v2 4Tr共D⌺ †D⌺兲, 共2兲
where the covariant derivative of the field⌺ is D⌺⫽⌺
⫺gWˆ⌺⫹g
⬘
⌺Bˆ.The effects of new dynamics on the couplings of fermions with the SM gauge bosons can be, in principle, also studied in an effective Lagrangian approach. For instance, if in anal-ogy with the situation in QCD, fermion masses are dynami-cally generated in association with EWSB, residual interac-tions of fermions with Goldstone bosons could be important
关4兴 if the mf⯝ f⯝v. Thus residual, non-universal interac-tions of the third generation quarks with gauge bosons could carry interesting information about both the origin of the top quark mass and EWSB.
In order to include fermions in this framework, we must define their transformation under G. Following Ref. 关4兴, we postulate that matter fields feel directly only the electromag-netic interaction f→ f
⬘
⫽exp(iyQf)f, where Qf stands for the electric charge of fermion f. The usual left-handed fermion doublets are then defined with the following transformation under G: ⌿L⫽⌺冉
f1 f2冊
L →⌿L⬘
⫽L exp共iyY/2兲⌿L, 共3兲 where Qf1⫺Qf2⫽1 and Y⫽2Qf1⫺1. Right-handed
fermi-ons are just the singlets fR. This definition is useful since it permits the construction of linearly realized left-handed dou-blet fields in the same way that, when studying the breaking of SU(2)R⫻SU(2)L→SU(2)R⫹L in QCD, one introduces auxiliary fields for the nucleons which transform linearly un-der the broken axial group. In this framework, the lowest-order interactions between fermions and vector bosons that can be built are of dimension four, leading to anomalous vector and axial-vector couplings, which were analyzed in detail in Ref. 关7兴.
In order to construct the most general Lagrangian describ-ing these interactions, it is convenient to define the vector and tensor fields
⌺a⫽⫺ i 2Tr共 aV R兲⫽⫺i 2Tr共 a⌺†D ⌺兲, ⌺a ⫽⫺i Tr†a⌺†关D ,D兴⌺‡. 共4兲
Under G transformations⌺3 and⌺3 are invariant while
⌺()⫾ →⌺
⬘
()⫾ ⫽exp共⫾iy兲⌺()⫾ ,where⌺()⫾ ⫽(⌺()1 ⫿i⌺()2 )/
冑
2.The basic fermionic elements for the construction of neutral- and charged-current effective interactions are
⌬X共q,q
⬘
兲⫽q¯PXq⬘
,⌬X共q,q
⬘
兲⫽q¯␥PXq⬘
,⌬˜X共q,q
⬘
兲⫽q¯PXD˜q⬘
,⌬X共q,q
⬘
兲⫽q¯PXq⬘
, 共5兲 where PX (X⫽0, 5, L, and R) stands for I,␥5, PL, and PR respectively, with I being the identity matrix and PL(R) the left共right兲 chiral projector. The fermionic field q (q⬘
) repre-sents any quark flavor, and D˜stands for the electromagnetic covariant derivative.The most general dimension-four Lagrangian invariant under nonlinear transformations under G is
L4⫽dL NC⌬ L 共t,t兲⌺ 3⫹d R NC⌬ R 共t,t兲⌺ 3⫹d L CC⌬ L 共t,b兲⌺ ⫹ ⫹dL CC†⌬ L 共b,t兲⌺ ⫺⫹d R CC⌬ R 共t,b兲⌺ ⫹ ⫹dR CC†⌬ R 共b,t兲⌺ ⫺. 共6兲
In principle it is also possible to construct neutral current operators involving only the bottom quark. We will assume however, that these vertices are not modified by the dynam-ics of the symmetry breaking or, at most, that these modifi-cations are suppressed as compared to those of the top quark. In a very general parametrization, the dimension-four anomalous couplings of third generation quarks can be writ-ten in terms of the usual physical fields as
L4⫽⫺ g
冑
2关CL共 t¯L␥bL兲⫹CR共 t¯R␥bR兲兴W ⫹ ⫺2cg W关NL t共 t¯ L␥tL兲⫹NR t共 t¯ R␥tR兲兴Z⫹H.c., 共7兲where sW(cW) is the sine共cosine兲 of the weak mixing angle, W. The parameters CL,R, NL,R
t
can be written in terms of the constants dL,RNC,CC of Eq. 共6兲 and contain the residual, non-universal effects associated with the new dynamics, per-haps responsible for the large top quark mass. Then, if we assume that the new couplings are C P conserving关12兴, there are four new parameters. They are constrained at low ener-gies by a variety of experimental information, mostly from electroweak precision measurements and the rate of b→s␥. In the case of dimension-five operators, the most general neutral-current interactions, which are invariant under non-linear transformations under G, are关13兴
L5 NC⫽a 1 NC⌬ 0共t,t兲⌺⫹⌺⫺⫹a2 NC⌬ 0共t,t兲⌺ 3⌺3⫹ia 3 NC⌬ 5共t,t兲⌺ 3⫹ib 1 NC⌬ 0 共t,t兲Tr关TWˆ 兴⫹b2 NC⌬ 0 共t,t兲B ⫹ib3 NC⌬ 0 共t,t兲共⌺ ⫹⌺ ⫺⫺⌺ ⫹⌺ ⫺兲⫹ic 1 NC关⌬˜ 0 共t,t兲⫺⌬˜ 0 共t,t兲兴⌺3, 共8兲
and the charged-current interactions are
L5 CC⫽a 1L CC⌬ L共t,b兲⌺⫹⌺ 3⫹a 1R CC⌬ R共t,b兲⌺⫹⌺ 3⫹ia 2L CC⌬ L共t,b兲D˜⌺⫹⫹ia2R CC⌬ R共t,b兲D˜⌺⫹⫹b1L CC⌬ L 共t,b兲⌺ ⫹ ⫹ b1R CC⌬ R 共t,b兲⌺ ⫹⫹b 2L CC⌬ L 共t,b兲共⌺ ⫹⌺ 3⫺⌺ ⫹⌺ 3兲⫹ b 2R CC⌬ R 共t,b兲共⌺ ⫹⌺ 3⫺⌺ ⫹⌺ 3兲⫹ic 1L CC⌬˜ L 共t,b兲⌺ ⫹ ⫹ic1R CC⌬˜ R 共t,b兲⌺ ⫹⫹H.c. 共9兲
In general, since chiral Lagrangians are related to strongly interacting theories, it is hard to make firm statements about the expected magnitude of the couplings. Requiring the loop corrections to the effective operators to be of the same order of the operators themselves suggests that these coefficients are of O(1) 关14兴. However, this is not necessarily the case when the operators result from small symmetry breaking effects, as we will see below.
In the unitary gauge, we can rewrite the interactions共8兲 and 共9兲 as a scalar, a vector, and a tensorial Lagrangian involving the physical fields. For the Lagrangian involving scalar currents we have
LS⫽ g2 2⌳
冋
t¯t冉
␣1 NCW ⫹W⫺⫹␣2 NC 2cW2 Z Z 冊册
⫹i g 2cW⌳␣3 NCt¯␥5tZ ⫹ g2 2冑
2⌳cW 兵t¯关␣1LCC共1⫺␥5兲⫹␣ 1R CC共1⫹␥5兲兴bW ⫹Z ⫹b¯关␣1L CC共1⫹␥5兲⫹␣ 1R CC共1⫺␥5兲兴tW ⫺Z其⫹i g 2冑
2⌳兵t¯关␣2L CC共1⫺␥5兲⫹␣ 2R CC共1⫹␥5兲兴b共W ⫹⫹ieAW ⫹兲 ⫺b¯关␣2L CC共1⫹␥5兲⫹␣ 2R CC共1⫺␥5兲兴t共W ⫺⫺ieAW ⫺兲其. 共10兲The Lagrangian containing vectorial current is given by LV⫽ 1 2⌳
再
i g cW␥ NCt¯共D˜ t兲Z⫺i g cW␥ NC共D˜ t兲tZ⫹i g冑
2t¯关␥L CC共1⫺␥5兲⫹␥ R CC共1⫹␥5兲兴共D˜ b兲W⫹⫺i g 2cW共D ˜b兲 ⫻关␥L CC 共1⫹␥5兲⫹␥ R CC 共1⫺␥5兲兴tW⫺冎
. 共11兲Finally, the piece involving a tensorial structure is
LT⫽ 1 4⌳
冋
t¯ t冉
 1 NC eF⫹2NC g cWZ⫹4ig 2 3 NC W⫹W⫺冊册
⫹ g 2冑
2⌳再
t¯ 关 1L CC 共1⫺␥5兲⫹ 1R CC 共1⫹␥5兲兴b关W ⫹⫹ie ⫻共AW⫹⫺AW⫹兲兴⫹b¯关1L CC共1⫹␥5兲⫹ 1R CC共1⫺␥5兲兴t关W ⫺⫺ie共A W⫺⫺AW⫺兲兴⫹i g cW t¯关2LCC共1⫺␥5兲 ⫹2R CC 共1⫹␥5兲兴b共Z W⫹⫺ZW⫹兲⫺i g cW¯b 关 2L CC 共1⫹␥5兲⫹ 2R CC 共1⫺␥5兲兴t共Z W⫺⫺ZW⫺兲冎
. 共12兲The couplings constants ␣’s,’s and␥’s are linear combi-nations of the a’s, b’s and c’s in Eqs.共8兲 and 共9兲. In writing the interactions 共10兲 and 共12兲, the coupling constants were defined in such a way that we have a factor g/(2cW) per Z boson, g/
冑
2 per W⫾, and a factor e per photon. Similar interactions were obtained in Ref. 关13兴 and for a linearly realized symmetry group, in Ref.关15兴.As an example of the above anomalous couplings, we show their couplings for the SM with a heavy Higgs boson integrated out. In this case, we can perform the matching between the full theory and the effective Lagrangian 关16兴. For instance, if we concentrate on the non-decoupling ef-fects, the leading contributions come at one-loop order关17兴. Setting mb⫽0 and keeping only the leading terms of the order mtlog(MH2), we find that only the first two effective operators of Eq. 共10兲 are generated with coefficients,
␣1 NC⫽␣ 2 NC⫽ g 2m t⌳ 162MW2 log MH2 mt2 .
III. RESULTS FOR THE b\s␥ AND b\sl¿lÀ
TRANSITIONS
For the b→s␥ and b→sl⫹l⫺ transitions it is useful to cast the contributions of the four and dimension-five anomalous couplings as shifts in the matching condi-tions at MWfor the Wilson coefficient functions in the weak effective Hamiltonian Heff⫽⫺ 4GF
冑
2 Vtb*Vtsi兺
⫽1 10 Ci共兲Oi共兲, 共13兲with the operator basis defined in Ref.关18兴. Of interest in our analysis are the electromagnetic penguin operator
O7⫽ e
162mb共s¯LbR兲F
, 共14兲
and the four-fermion operators corresponding to the vector and axial-vector couplings to leptons,
O9⫽ e2 162共s¯L␥bL兲共 l¯␥ l兲, 共15兲 and O10⫽ e2 162共s¯L␥bL兲共 l¯␥ ␥ 5l兲. 共16兲
The operators above are already present in the SM. In addi-tion, the dimension-five anomalous couplings generate the operators O11⫽ e2 162 mb MZ2关s¯L共iQ 兲b R兴共 l¯␥␥5l兲, 共17兲 O12⫽ e2 162 mb MZ2关s¯L共iQ 兲b R兴共 l¯␥l兲. 共18兲
However, these new operators will not lead to important ef-fects as will see below, due to the fact that they are further suppressed by the weak scale.
The anomalous couplings of Eq. 共7兲, 共10兲, 共11兲 and 共12兲 will induce shifts in the Wilson coefficient functions Ci() at the matching scale, which we take to be ⫽MW. We make use of the next-to-leading order calculation of the Wil-son coefficients as described in Ref.关19兴.
A. Effects of the dimension-four operators
The dimension-four operators defined in Eq. 共7兲 induce new contributions to the b→s␥ and b→sZ loops as well as the box diagram. They appear in the effective Hamiltonian formulation as shifts of the Wilson coefficients C7( MW), C9( MW) and C10( MW). The contributions from the b→s␥ loops to C7( MW) are ␦C7⫽⫺mt mb CR
再
1 12共x⫺1兲2共5x 2⫺31x⫹20兲 ⫹ 1 2共x⫺1兲3x共3x⫺2兲log共x兲冎
⫹CL再
1 4共x⫺1兲4x 2 ⫻共3x⫺2兲log共x兲⫺ 1 24共x⫺1兲3x共8x 2⫹5x⫺7兲冎
, 共19兲where we have defined the dimensionless quantity x
⫽mt 2
/ MW2 . We should notice that the above result is finite, i.e. independent of⌳, and agrees with the previous result in the literature 关8兴. On the other hand, the result for all other operators is not finite and in this case we have kept only the leading non-analytic, i.e. logarithmic, dependence on the new physics scale⌳. In this way, for C9( MW) we have
␦C9␥⫽⫺ 1
12CL共3x⫺16兲log
冉
⌳2
MW2
冊
. 共20兲 The corrections arising from b→sZ loops to C9( MW) and C10( MW) are ␦C10Z⫽ ⫺1 1⫺4sW2 ␦C9 Z⫽ 1 16sW2 共4NL t⫺N R t⫹C L兲xlog冉
⌳2 MW2冊
, 共21兲while box loops contributions can be written as
␦C9box⫽⫺␦C10box⫽ 1
16sW2 CL共x⫺16兲log
冉
⌳2
MW2
冊
. 共22兲 The measured b→s␥ branching ratio imposes a stringent bound on CR as its contribution to Eq. 共19兲 is enhanced by the factor mt/mb. This has been discussed in the literature关8兴, where the obtained bounds on CR: ⫺0.05⬍CR⬍0.01. Although this is very small, it is possible to generate such value for CR in a large variety of generic strongly coupled theories. For instance, the pseudo Nambu-Goldstone bosons
共PNGBs兲 of extended technicolor 共ETC兲 that result from the
breaking of the various fermion chiral symmetries关20兴, gen-erate at one loop a small CR proportional to mb:
CR⯝ 1 4 mbmt f2 log
冉
m2 mt2冊
. 共23兲This is well within experimental bounds in all ETC incarna-tions, and it is even smaller in modern ETC theories such as top-color-assisted technicolor关21兴, where the top quark mass entering in Eq. 共23兲 is only a few GeV 关22兴. Thus, CR is a consequence of the explicit ETC-breaking of chiral symme-try responsible for mb. Another hint of this, is the fact that in general CR contributes to the renormalization of the b-quark line with a term which does not vanish with mb:
⌺共mb兲⫽ g2
322CRmt共x⫺4兲 log
冉
⌳2
MW2
冊
. 共24兲 Thus if we take into account the potential role of chiral sym-metry in suppressing CR and we rescale this coefficient by defining CˆR asCR⫽ mb
冑
2vCˆR 共25兲
共where v⫽246 GeV兲, the rescaled bounds on CˆR areO(1), which is a more natural value of this coefficient once the underlying symmetry is taken into account.
In Fig. 1 we plot the b→s␥ branching fraction as a func-tion of CˆR. We also include the effect of CL, which is now comparable for similar values of the coefficients. The hori-zontal band corresponds to the latest CLEO result 关23兴 Br(b→s␥)⫽(3.15⫾0.35⫾0.32⫾0.26)⫻10⫺4, where we take a 1 interval after adding the statistical, systematic and model-dependent errors in quadrature.
On the other hand, the effect in b→sl⫹l⫺ is dominated by the coefficients CL, NL
t and N R
t in Eqs. 共20兲, 共21兲, and
FIG. 1. The b→s␥ branching ratio vs CˆR 共solid兲 and CL
共dashed兲. The horizontal band corresponds to the 1 interval from
共22兲. In principle, these coefficients are constrained by
elec-troweak precision measurements, most notably ⑀1⫽⌬
⫽␣T and⑀b 关7兴: ⑀1⫽ GF 2
冑
223mt 2共⫺N L t⫹N R t⫹C L兲log冉
⌳2 MW2冊
⑀b⫽ GF 2冑
223mt 2冉
⫺1 4NR t⫹N L t冊
log冉
⌳ 2 MW2冊
. 共26兲In general, the bounds obtained on a particular coupling from electroweak observables strongly depend on assumptions about the other couplings. For instance, enforcing custodial isospin symmetry in order to avoid the strong constraints from T will imply that NLt⫽CL and NRt⫽0. On the other hand if CL⫽0, then the combination (NL
t⫺N R t
) is strongly constrained since it breaks custodial isospin symmetry and contributes to T. Imposing CL⫽NL
t
, then NRt⬍0.02 关4,24兴 since it is the only linear source of isospin breaking.
We study here three cases in which the stringent con-straints from electroweak observables can be evaded.
共i兲 CL⬇NL
t . In this case the contributions of C
Land NL t to the T parameter cancel, leaving NRt as the only seriously constrained quantity. However, Rb still gives the bound
⫺0.03⬍NL
t⬍0.15 共for ⌳⫽1 TeV兲. In Fig. 2 we plot the b
→sl⫹l⫺branching ratio, normalized to the SM expectation,
as a function of CL⫽NLt 共solid line兲. From this plot it can be seen that, when incorporating the Rbconstraint, the effect in b→sl⫹l⫺is bound to be smaller than roughly a 10% devia-tion.
共ii兲 NL t⬇N
R
t . In this scenario the measurement of the T parameter greatly constrains CL, which prompts us to take this coefficient as equal to zero in this portion of the analysis. The dashed line in Fig. 2 corresponds to the effect of NL t
⫽NR
t in the dilepton branching fraction. As we can see, the effect in this decay is rather small.
共iii兲 Finally and for completeness, we consider the case
NLt⬇NRt/4. With this approximate relation these two coeffi-cients cancel in Rb leaving no sizeable effect from the dimension-four Lagrangian 共7兲 in this quantity. However, this relation leads to a potentially large contribution to the T parameter proportional to (CL⫹3NR
t
/4). When this bound is incorporated, the effect in b→sl⫹l⫺branching ratio is con-strained to be below 15%.
In sum, we have seen that the leading effects of the dimension–four operators in rare B decays are given by CˆR in b→s␥, and the effects in b→sl⫹l⫺ due to CL, NL
t and NRt are below 15% deviations once the constraints from elec-troweak precision measurements are considered. This dis-tinction comes from the fact that Z-pole quantities are not significantly sensitive to CˆR. The effects of CˆR in b
→sl⫹l⫺can be significant, but b→s␥ is considerably more
sensitive to this parameter.
B. Effects of the dimension-five operators
Although in principle dimension-five operators are con-sidered sub-leading with respect to the operators in Eq. 共7兲 due to the additional suppression by the high energy scale⌳, they can still induce large deviations in both electroweak observables and FCNC processes. In Ref.关9兴 bounds on the coefficients of dimension-five operators were derived from data at the Z pole. Here we consider the effect of these op-erators in b→s␥ and b→sl⫹l⫺. They induce new contribu-tions to the b→s␥ and b→sZ loops as well as the box diagram. They appear in the effective Hamiltonian formula-tion as shifts of the Wilson coefficients C7( MW), C9( MW) and C10( MW), C11( MW), and C12( MW).
The contribution from the b→s␥ loops to these coeffi-cients are ␦C7⫽⫺ 1 12 MW2 mb⌳ 关 4␣2RCCx⫺41RCC共3x⫺7兲 ⫹␥R CC共x⫹2兲兴log
冉
⌳ 2 MW2冊
, 共27兲 and ␦C12␥⫽1 24 MZ2 mb⌳兵关3共2␣2R CC⫺␥ R CC兲x ⫹4共3␣2R CC⫺6 1R CC⫹␥ R CC兲兴其log冉
⌳ 2 MW2冊
. 共28兲 The corrections arising from b→sZ loops to the different coefficients areFIG. 2. The b→sl⫹l⫺branching ratio共normalized to the SM兲 vs CL⫽NL t 共solid兲 and N L t⫽N R t 共dashed兲.
␦C10 Z⫽ ⫺1 1⫺4sW2 ␦C9 Z⫽⫺ 1 96sW2 mt ⌳ 关3共2␣1L CC⫹12␣ 2L CC⫹12 2L CC⫺␥ L CC⫹8␥NC兲x⫺18c W 2共␣ 2L CC⫹2 1L CC兲x ⫹2sW 2共␥ L CC⫺12␣ 2L CC兲x⫺9共2␣ 1L CC⫺4 1L CC⫹12 2L CC⫺␥ L CC⫺4␥NC兲⫺18c W 2共␣ 2L CC⫺2 1L CC兲 ⫺6sW 2共␥ L CC⫹12 1L CC兲兴log
冉
⌳ 2 MW2冊
, 共29兲 and ␦C11Z⫽ ⫺1 1⫺4sW2 ␦C12 Z⫽⫺ 1 48sW2 MZ2 mb⌳ 关3共␣1R CC⫹␣ 2R CC⫺2 2R CC兲x⫹3c W 2 共␣2R CC⫺6 1R CC⫹␥ R CC兲x⫺2s W 2 共4␣2R CC⫺␥ R CC兲x ⫹3共2␣1R CC⫹8 1R CC⫺4 2R CC⫺␥ R CC兲⫹6c W 2共␣ 2R CC⫹6 1R CC⫹␥ R CC兲⫺4s W 2共8 1R CC⫺␥ R CC兲兴log冉
⌳ 2 MW2冊
. 共30兲The box loops contributions can be written as
␦C9box⫽⫺␦C10box⫽⫺ 1 96sW2 mt ⌳ 关4␣2L CC共2x⫺5兲⫺36 1L CC⫺␥ L CC共x⫺1兲兴log
冉
⌳ 2 MW2冊
, 共31兲 and ␦C12box⫽⫺␦C11box⫽⫺ 1 16sW2 MZ2 mb⌳ 关 2␣2RCCx⫺41RCC⫺␥RCC共x⫺2兲兴log冉
⌳ 2 MW2冊
. 共32兲In order to quantify the effect of the operators of Eqs.
共10兲, 共11兲 and 共12兲 we classify them into two groups: the
right-handed and left-handed couplings. The RH couplings that are of interest due to their potential effects are ␣2RCC,
1R CC
and␥RCC. Just as for the dimension-four coefficient CR, the effects of these coefficients can be very large in operators such as O7 generating important deviations in b→s␥, as seen in Eq. 共27兲. This is particularly so since their contribu-tions appear to be unsuppressed by mb. As we noted for the dimension-four coefficient CR, the coefficients␣2R
CC
and1RCC are chirally unsuppressed in Eq. 共10兲, which results in an renormalization of the b-quark mass. Although, unlike CR, the dimension-five coefficients are not tightly bound by cur-rent experiments, we could also build strongly coupled theo-ries where these are small due to chiral symmetry. If this suppression is present, it would make their effect on the op-erator O7 negligible. On the other hand, the coefficient␥R
CC
has a chiral suppression already present in the accompanying operator in Eq. 共11兲. Because of this its contribution to the renormalization of the b-quark line vanishes with mb:
⌺共mb兲⫽ ⫺3g2 1282⌳␥R CC mb2共x⫹1兲 log
冉
⌳ 2 MW2冊
. 共33兲 Thus it is possible that some of the RH coefficients are natu-rally suppressed while others may not be. We will illustrate the effects of the RH couplings using ␥RCC as an example. The effects of␣2RCC
and1R CC
would be of similar size if they
are not chirally suppressed. The contribution of ␥RCC to the penguin operator gives rise to a deviation of the b→s␥ branching ratio from its SM expectation. In Fig. 3 we plot this branching fraction as a function of this coefficient. This measurement is the most constraining bound on these type of operators. It can be seen that even for rather small values of
␥R CC
there could be considerable deviations from the SM expectations. On the other hand, the effect is less dramatic in b→sl⫹l⫺, as shown in Fig. 4, where an observable
devia-FIG. 3. The b→s␥ branching ratio vs ␥R CC
tion from the SM will result only if ␥RCC is large enough to dominate the b→s␥ branching ratio.
The effects of the new operators O11 andO12 are negli-gible. Although the presence of mb in the denominators in Eqs. 共28兲, 共30兲 and 共32兲 suggests the possibility of an en-hancement, this is not enough. The effect is suppressed by an effective scale given by mb⌳/MZ⯝55 GeV, which should be compared with the typical momentum transfers in B decays. The remaining group of coefficients we dubbed left-handed includes ␣1LCC, ␣2LCC, CC1L , 2LCC, ␥LCC plus ␥NC which actually is the coefficient of a vector operator, but since is not chirally suppressed is included with the LH in this part of the analysis. These operators affect mainly the b→sl⫹l⫺ rates. Then we could imagine that the underlying new physics preserves chiral symmetry in such a way that does not generate sizeable contributions to the RH operators. Thus it is possible a scenario where no deviations in b
→s␥ occur; but where the new interactions give rise to large effects in the dilepton modes. Although these have not been observed yet, the current experimental sensitivity is very close to the SM predictions and it will reach them in the near future. The leading effects in b→sl⫹l⫺come from the coefficients␣2LCC,1LCCand␥NC. For simplicity we only con-sider these and plot in Fig. 5 the branching ratio, normalized to the SM one, for two cases:␣2LCC⫽1LCC⫽␥NC 共solid line兲 and␣2L
CC⫽␥NC⫽0 共dashed line兲. From Fig. 5 it is apparent that cancellations occur when the three coefficients are simi-lar. The effect of considering only 1LCC shows than even larger effects are possible. In any case, sizeable deviations in b→sl⫹l⫺ are possible even in the absence of effects in b
→s␥.
IV. DISCUSSION
Processes involving FCNC transitions in B and K decays are a crucial complement to precision electroweak observ-ables, when constraining the physics of the EWSB sector. In this paper, we have considered the effects of anomalous cou-plings of third generation quarks to the W and Z gauge
bosons. We computed the effects of all possible dimension-five operators in B FCNC transitions such as b→s␥ and b
→sl⫹l⫺. For completeness, we have also also presented the
analysis of the dimension-four operators.
We have shown that with the natural assumption of chiral symmetry, in fact enforcing vanishing fermion mass renor-malization in the chiral limit, the effects of the dimension-four operators with coefficient CR for b→s␥, are not as dramatic as found in Ref. 关8兴, and somehow smaller than those of CL, which can produce important deviations in the branching ratio that could be resolved in the next round of experiments at B factories.
The effects in b→sl⫹l⫹due to CL, NL t
and NRt are below 15% deviations once the constraints from electroweak preci-sion measurements are taken into account.
On the other hand, we have found that the dimension-five operators with RH coefficients can give rise to observable deviations of the b→s␥ branching ratio from its SM expec-tation. We illustrated these effects for the coefficient ␥RCC, for which no additional chiral suppression is expected. Left handed operators, on the other hand, affect mainly the b
→sl⫹l⫺ rates and we have illustrated that in several
sce-narios sizeable deviations in b→sl⫹l⫺are possible even in the absence of effects in b→s␥.
The dimension-five operators, just as in the case of the more studied dimension-four operators, can be generated at high energies scales by the presence of new particles and/or interactions. For instance, as a simple example, a heavy sca-lar sector with both charged and neutral states, would give contributions to many of the coefficients of the Lagrangian in Eqs. 共10兲. Richer dynamics at the TeV scale might generate also some of the vector and/or tensor couplings of Eqs. 共11兲 and共12兲.
The e⫹e⫺ B factories at Cornell, KEK and SLAC are expected to reach better measurements of the b→s␥ branch-ing ratio, which will largely constrain the dimension-five co-efficient␥RCCand to a lesser extent the dimension-four coef-ficients CLand CˆR. Furthermore, these experiments as well
FIG. 4. The b→sl⫹l⫺branching ratio共normalized to the SM prediction兲 vs␥R
CC .
FIG. 5. The b→sl⫹l⫺branching ratio共normalized to the SM prediction兲 vs 1L
CC
, for ␣2L
CC⫽
1L
CC⫽␥NC 共solid line兲 and ␣
2L
CC
as those at the Fermilab Tevatron, will reach the SM sensi-tivity for the b→sl⫹l⫺ branching fraction. The present analysis, together with previous ones addressing the effects of other anomalous higher dimensional operators in these decay modes, will enable us to interpret a possible pattern of deviations from the SM and perhaps point to its dynamical origin.
ACKNOWLEDGMENTS
We thank O. J. P. E´ boli for a critical reading of the manu-script. M. C. Gonzalez-Garcia is grateful to the Instituto de Fı´sica Teo´rica from Universidade Estadual Paulista for its kind hospitality. G. Burdman acknowledges the hospitality
of the Enrico Fermi Institute at the University of Chicago, where part of this work was completed. This work was sup-ported by the U.S. Department of Energy under Grant No. DE-FG02-95ER40896 and the University of Wisconsin Re-search Committee with funds granted by the Wisconsin Alumni Research Foundation, by Conselho Nacional de De-senvolvimento Cientı´fico e Tecnolo´gico 共CNPq兲, by Fun-dac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo
共FAPESP兲, by Programa de Apoio a Nu´cleos de Exceleˆncia 共PRONEX兲, by CICYT 共Spain兲 under grant AEN96-1718, by
DGICYT 共Spain兲 under grants PB95-1077 and PB97-1261, and by the EEC under the TMR contract ERBFMRX-CT96-0090.
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