Deciphering top flavor violation at the LHC with B factories
arXiv:0704.1482v2  [hep-ph]  7 Sep 2008
UCB–PTH–07/06
YITP–SB–07–11
Deciphering top flavor violation at the LHC with B factories
Patrick J. Fox,1 Zoltan Ligeti,1 Michele Papucci,1, 2 Gilad Perez,3 and Matthew D. Schwartz4
Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
Department of Physics, University of California, Berkeley, CA 94720
C.N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840
Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles St., Baltimore, MD 21218
The LHC will have unprecedented sensitivity to flavor-changing neutral current (FCNC) top quark
decays, whose observation would be a clear sign of physics beyond the standard model. Although
many details of top flavor violation are model dependent, the standard model gauge symmetries
relate top FCNCs to other processes, which are strongly constrained by existing data. We study
these constraints in a model independent way, using a low energy effective theory from which the new
physics is integrated out. We consider the most important operators which contribute to top FCNCs
and analyze the current constraints on them. We find that the data rule out top FCNCs at a level
observable at the LHC due to most of the operators comprising left-handed first or second generation
quark fields, while there remains a substantial window for top decays mediated by operators with
right-handed charm or up quarks. If FCNC top decays are observed at the LHC, such an analysis
may help decipher the underlying physics.
I. INTRODUCTION
The Large Hadron Collider (LHC) will have unprecedented sensitivity to flavor changing neutral currents (FCNCs)
involving the top quark, such as t → cZ. With a tt pair production cross section of about 800 pb and after 100 fb−1
of integrated luminosity, the LHC will explore branching ratios down to the 10−5 level [1, 2]. Flavor changing neutral
currents are highly suppressed in the standard model (SM), but are expected to be enhanced in many models of
new physics (NP). Because top FCNCs are clean signals, they are a good place to explore new physics. There are
important constraints from B physics on what top decays are allowed, and understanding these constraints may help
decipher such an FCNC signal. In this paper, we calculate the dominant constraints on top FCNCs from low energy
physics and relate them to the expected LHC reach using a model-independent effective field theory description.
Flavor physics involving only the first two generations is already highly constrained, but the third generation could
still be significantly affected. Of course, the new flavor physics could be so suppressed that it will not be observable
at all at the LHC. However, since the stabilization of the Higgs mass is expected to involve new physics to cancel the
top loop, it is natural to expect some new flavor structure which may show up in the top quark couplings to other
standard model fields. Thus, one may expect flavor physics to be related to the electroweak scale, and then flavor
changing effects involving the top quark are a natural consequence.
Although there are many models which produce top FCNCs, the low energy constraints are independent of the
details of these models. The new physics can be integrated out, leaving a handful of operators relevant at the weak
scale involving only standard model fields. These operators mediate both FCNC top decays and flavor-changing
transitions involving lighter quarks. Thus, the two can be related without reference to a particular model of new
physics, provided there is no additional NP contributing to the B sector. The low energy constraints can be applied
to any model in which top FCNCs are generated and the constraints on the operators may give information on the
scale at which the physics that generates them should appear.
Analyses of FCNC top decays have been carried out both in the context of specific models [3] and using model
independent approaches [4]. However, in most cases the effective Lagrangian analyzed involved the SM fields after
electroweak symmetry breaking. As we shall see, the scale Λ at which the operators responsible for top FCNC are
generated has to be above the scale v of electroweak symmetry breaking. Thus, integrating out the new physics should
be done before electroweak symmetry breaking, leading to an operator product expansion in v/Λ. The requirement of
SU(2)L invariance provides additional structure on the effective operators [5], which helps constrain the expectations
for top FCNCs. For example, an operator involving the left-handed (t, b) doublet, the SU(2) gauge field, and the right
handed charm quark, can lead to b → sγ at one loop, but also directly to a b → c transition. If we ignored SU(2)L
invariance, we would only have the b → sγ constraint, and the resulting bound would be different. An important
feature of our analysis is that, after electroweak symmetry breaking, the resulting operators can modify even SM
parameters which contribute at tree level to B physics observables, such as |Vcb|.
The organization of this paper is as follows. In Sec. II we introduce the effective Lagrangian relevant for top
FCNCs. We also explain why some operators can be neglected and introduce conventions used throughout the paper.
In Sec. III we calculate how these operators affect top quark decays and integrate out theW and Z bosons and the top
quark to match onto the relevant effective theory at the weak scale. In Sec. IV we relate the experimental constraints
http://arxiv.org/abs/0704.1482v2
to the Wilson coefficients calculated in Section III, focusing mostly on observables related to B physics. This leads
directly to predictions for the top branching ratio. Sec. V contains a summary of the results and our conclusions. We
include an Appendix with details of the calculations.
II. EFFECTIVE LAGRANGIAN FOR TOP FCNC
We consider an effective Lagrangian
Leff =
(CiOi + C
i) . (1)
where the Oi operators involve third and second generation quarks and the O
i involve the third and first generations.
Since we are interested in top quark decays, we define Oi and O
i in the mass basis for the up-type quarks.
A complete set of dimension-six operators which give a tcZ or tcγ vertex are
OuLL = i
H̃†Q2
+ h.c. ,
OhLL = i
+ h.c. ,
OwRL = g2
µνσaH̃
µν + h.c. ,
ObRL = g1
tRBµν + h.c. ,
OwLR = g2
µνσaH̃
µν + h.c. ,
ObLR = g1
cRBµν + h.c. ,
OuRR = i tRγ
+ h.c. . (2)
The brackets mean contraction of SU(2) indices, Q3 and Q2 are the left-handed SU(2) doublets for the third and
second generations, tR and cR are the right-handed SU(2) singlets for the top and charm quarks, H is the SM
Higgs doublet, H̃ = iσ2H
∗, and the index a runs over the SU(2) generators. The first lower L or R index on the
operators denotes the SU(2) representation of the third generation quark field, while the second lower index refers to
the representation of the first or second generation field. In this basis all of the derivatives act on the Higgs fields.
We could also consider operators directly involving gluons, but since the indirect constraints on gluonic currents are
very weak (see, e.g., [6]), we restrict our focus to the electroweak operators in Eq. (2). The form of the operators in
Eq. (2) after electroweak symmetry breaking are given in the Appendix.
Throughout the paper we focus on those new operators that contribute to t → cZ, cγ. In any particular model
there may be additional contributions to Eq. (1) that contribute to ∆F = 1 and ∆F = 2 processes in the down
sector (e.g., four-fermion operators). These operators have suppressed contributions to top FCNCs. When we bound
the coefficients of the operators in Eq. (2) from B physics, we neglect these other contributions. In any particular
model these two sets of operators may have related coefficients. Unless there are cancellations between the different
operators, the bounds will not get significantly weaker.
There are other dimension-six operators that can mediate FCNC top decays (for example tRγ
µDνcRBµν). But
these can always be reduced to a linear combination of the operators included in Eq. (2) plus additional four-fermion
operators and operators involving QLqRHHH fields. For instance, operators involving two quark fields and three
covariant derivatives can be written in terms of operators involving fewer derivatives using the equations of motion.
Operators involving two quark fields and two covariant derivatives (e.g., Q3DµcRD
µH̃) can be written in terms of
operators involving the commutator of derivatives included in Eq. (2) plus operators with one derivative and four-
fermion operators. Finally, operators involving two quark fields and one covariant derivative can be written in a way
that the derivative acts on the H field, as in Eq. (2), plus four-fermion operators.
Of the four-fermion operators which appear after the reduction of the operator basis, some are suppressed by
small Yukawa couplings and can simply be neglected. However, some are not suppressed, and of those, the biggest
concern would be semileptonic four-fermion operators, like (tc)(ℓℓ). These contribute to the same final state as
t → cZ → cℓ+ℓ−. (We emphasize Z → ℓ+ℓ−, because the LHC is expected to have the best sensitivity in this
channel [1, 2].) However, the invariant mass of the ℓ+ℓ− pair coming from a four-fermion operator will have a smooth
distribution and not peak around mZ , so the Z-mediated contribution can be disentangled experimentally. Operators
with (tc)(qq) flavor structure also contribute to t → cℓ+ℓ− or t→ cγ at one loop, but their contributions are suppressed
by α/(4π). Finally, operators with the QLqRHHH structure either renormalize Yukawa couplings, or contribute to
FCNCs involving the Higgs (e.g., t→ ch), but we do not consider such processes, as explained later.
Throughout most of this paper we consider each of the operators one at a time and constrain its coefficient. This
is reasonable as the operators do not mix under renormalization. One exception is that OuLL and O
LL mix with one
another between the scales Λ and v, so it would be unnatural to treat them independently. Their mixing is given by
d lnµ
CuLL(µ)
ChLL(µ)
CuLL(µ)
ChLL(µ)
, (3)
where α2 = α/ sin
2 θW is the SU(2) coupling. (The zero in the anomalous dimension matrix is due to the fact that
custodial SU(2) preserving operator OhLL cannot mix into the custodial SU(2) violating O
LL.) So, we will also carry
out a combined analysis for these two operators.
We have written the operators in Eq. (2) in terms of a single SM Higgs doublet. In principle there may be many
new Higgs scalars, but only those that acquire a vev will contribute to t → cZ and cγ. Since a triplet Higgs vev is
tightly constrained by electroweak precision tests, we concentrate on the possibility of multiple Higgs doublets. With
the introduction of extra Higgs doublets, there are more operators of each particular type (OuLL, O
LL, etc.), one linear
combination of which gives rise to t → cZ and cγ. There are also several physical Higgs states that can contribute
in loops in low energy processes. For each type of operator, a different linear combination of couplings enter in low
energy measurements. However, without cancellations this will only differ from the one Higgs case by a number of
order one. This allows our results to be applied to the general case of multiple Higgs doublets.1 Of course, the Higgs
sector is also relevant to FCNCs involving the Higgs, such as t → ch, but we do not consider such processes as they
are more model dependent.
Once we go beyond models with minimal flavor violation (MFV) [7], the possibility of new CP violating phases in
the NP should be considered. In MFV models, top FCNC is not observable at the LHC. In models such as next-to-
minimal flavor violation (NMFV) [8] top FCNCs could be observable and the Wilson coefficients can be complex. It
is not always the case that the constraints are weaker when the NP Wilson coefficients are real (in the basis where the
up type Yukawa matrix is real and diagonal). Rather, interference patterns realized in some of the observables mean
the constraints are weakest when some of the new phases are different from 0 or π. We shall point out the places
where phases associated with the new operators can play an important role and how we treat them.
In addition to the B physics related constraints we will derive in this paper, one can also use constraints from
electroweak precision observables. However, these bound flavor-diagonal operators strongly, and the flavor non-
diagonal operators in Eq. (2) which contribute to top FCNCs are far less constrained. For instance, the OuLL operator
corrects the W propagator at one loop and so contributes to the T parameter. The loops involve a t or c quark, and
have one insertion of OuLL and one insertion of Vts or Vcb. Thus, the contribution is suppressed by |Vts| ∼ |Vcb| ∼ 0.04
relative to an insertion of the flavor diagonal equivalent of OuLL, Q3H̃D/H̃
†Q3. In contrast, when considering low
energy FCNC processes, OuLL will be more strongly constrained then its flavor diagonal version. That is, flavor
diagonal operators are more tightly constrained by electroweak observables than by low energy FCNCs, while the
off diagonal operators are more tightly constrained by low energy FCNCs. Moreover, the mixing between these two
classes of operators is small. It occurs at one loop proportional to y2b |Vcb|, where the factor of yb, the bottom Yukawa
coupling, is due to a GIM mechanism. Thus, we can think of the flavor diagonal and off diagonal operators as
independent. And so for the purpose of studying top FCNCs, we are justified in neglecting flavor diagonal operators
and the relatively weak constraints from electroweak precision tests.
III. WEAK SCALE MATCHING
In this section we derive how the NP operators modify flavor changing interactions at the electroweak scale and
derive the effective Hamiltonian in which the t,W , and Z are integrated out. For numerical calculations we use besides
the Higgs vev, v = 174.1GeV, and other standard PDG values [9], |Vts| = 41.0× 10−3 [10] and mt = 171GeV [11].
1 One possible exception is if an extended Higgs sector allows Yukawa couplings larger than in the SM, for example, in a two Higgs doublet
model at large tan β. Then a Higgs loop may give additional unsuppressed contributions when we match to the Wilson coefficients at
the electroweak scale.
A. Top quark decays
After electroweak symmetry is broken, the operators in Eq. (2) give rise to t→ cZ and t→ cγ FCNC decays. The
analytic expressions for the partial widths of these decays are given in Eq. (A2) in the Appendix. Numerically, the
t→ cZ branching ratio in terms of the Wilson coefficients is
B(t→ cZ) =
× 10−4 ×
|CbLR|2 + |CbRL|2
− 9.6Re
CbLRC
+ CbRLC
|CwLR|2 + |CwRL|2
− 8.3Re
(ChLL + C
∗ − CbLRCuRR
+ 28Re
ChLL + C
∗ − CwLRCuRR
∣ChLL + C
+ |CuRR|2
. (4)
The tcγ vertex, which has a magnetic dipole structure as required by gauge invariance, is induced only by the left-right
operators. The branching ratio for t→ cγ is
B(t→ cγ) =
× 10−4 × 8.2
∣CbLR + C
∣CbRL + C
. (5)
The analogous expressions for t→ u decays are obtained by replacing Ci by C′i in Eqs. (4) and (5).
The LHC will have unparalleled sensitivity to such decays. With 100 fb−1 data, the LHC will be sensitive (at 95%
CL) to branching ratios of 5.5 × 10−5 in the t → cZ channel and 1.2 × 10−5 in the t → cγ channel [1]. In the SM,
B(t → cZ, cγ) are of order α(Vcbαm2b/m2W )2 ∼ 10−13, so an experimental observation would be a clear sign of new
physics. Equations (4) and (5) will allow one to translate the measurements or upper bounds on these branching
ratios to the scale of the individual operators.
B. B decays
Many of the operators in Eq. (2) modify SM interactions at tree level (this possibility was discussed in [5]). After
electroweak symmetry breaking, OuLL gives rise to a bWc vertex with the same Dirac structure as the SM, so the
measured value of Vcb (which we denote V
cb ) will be the sum of the two. This allows us to absorb the new physics
contribution of CuLL into the known value of V
cb — in processes where Vcb and C
LL enter the same way, the
dependence on CuLL cannot be disentangled. For example, the SM unitarity condition, V
tbVtd + V
cbVcd + V
ubVud = 0,
would be violated if one simply shifted the SM values by the NP contributions. However, the CKM fits have unitarity
built in, so the NP contribution to Vcb causes a shift in the values of Vts and Vtd extracted from the CKM fit, V
and V fittd . Since we cannot measure all CKM elements independently, we have to replace Vts and Vtd by V
ts and V
plus modified NP contributions. (Recall that Vts and Vtd are only constrained from loop processes where they enter
together with new physics contributions.) With these redefinitions we can use V
cb , V
ts and V
td in the CKM fit,
and the NP will only have distinguishable effects in SM loop processes. An analogous procedure applies to the t → u
contribution to V
ub , V
td and V
ts . Some other operators such as C
LR do not generate a bWc vertex with the same
Dirac structure as the SM. Thus, their contributions to observables from which Vcb is extracted may be disentangled
as discussed in the following.
At leading order in the Wolfenstein parameter (Cabibbo angle), λ, these relations are:
Vcb = V
cb + (v
2/Λ2)CuLLVtb ,
Vub = V
ub + (v
2/Λ2)C′uLLVtb ,
V ∗ts = V
ts − (v2/Λ2) (CuLLV ∗cs + C′uLLV ∗us) ,
V ∗td = V
td − (v2/Λ2) (CuLLV ∗cd + C′uLLV ∗ud) . (6)
The OwLR (O
LR) also modifies the bWc (bWu) vertex, but with different Dirac structure from the SM, so its effects
can be separated from the SM contribution. Finally, OhLL (O
LL) gives tree-level FCNC, since it contains a bZs (bZd)
interaction.
At the one-loop level, the operators in Eq. (2) contribute to b→ s transitions. The constraints from B physics are
easiest to analyze by matching these operators onto operators containing only the light SM fields at a scale µ ∼ mW .
We use the standard basis as defined in [12]. Integrating out the top, W , and Z, the most important operators for
B → Xsγ and B → Xsℓ+ℓ− which are affected by NP are
O7γ =
[mbs σ
µν(1 + γ5)b]Fµν ,
O9V = [sγ
µ(1− γ5)b] [ℓγµℓ] ,
O10A = [sγ
µ(1− γ5)b] [ℓγµγ5ℓ] . (7)
For example, the diagram in Fig. 1 gives a contribution from OwRL (denoted by ⊗) to O7γ . The coefficients of the
QCD and electroweak penguin operators, O3,...,10, are also modified, but their effect on the processes we consider are
suppressed.
Summing the relevant diagrams, the contributions of all operators can be expressed in terms of generalized Inami-
Lim functions, presented in the Appendix. Setting Λ = 1TeV, the numerical results are2
C7γ(mW ) = −0.193 +
0.810CuLL + 0.179C
LL + 0.310C
RL − 0.236CbRL + 0.004CwLR − 0.003CbLR
C9V (mW ) =
1.56 +
−0.562CuLL + 44.95ChLL − 0.885CwRL − 1.127CbRL + 0.046CwLR + 0.004CbLR
C10A(mW ) =
−4.41 +
−7.157CuLL − 598ChLL + 3.50CwRL − 0.004CuRR
. (8)
The first term in each expression is the SM contribution. Note that the OhLL contribution is large because it is at
tree level, while ObLR, O
LR, and O
RR are tiny because they are suppressed by mc/mW and so the constraints on
these will be weaker. In the case of b → d transitions the NP contribution has to be rescaled by the O(1/λ) factor,
|V ∗tsVud/V ∗tdVcs| ≈ 5.6, and Ci should be replaced with C′i.
C. ∆F = 2 transitions
The operators OuLL, C
LL, and O
RL also contribute to ∆F = 2 transitions, i.e., neutral meson mixings. Again,
the contribution from OhLL is present at tree level, while the other two contribute starting at one-loop order. The
relevant functions are again listed in the Appendix. The modifications relative to the SM Inami-Lim function can be
parameterized as S0 → S0(1 + hMe2i σM ) for each neutral meson system. Numerically (setting Λ = 1TeV), for B0sB0s
mixing, the effect of the t→ c operators is given by
2iσBs = 800(ChLL)
2 + 0.92ChLLC
LL − 6.84(CuLL)2 + 1.55ChLL − 2.64CuLL − 0.32(CwRL)2 − 1.03CwRL . (9)
The contributions of the O′i operators to B
s mixing is given by replacing Ci with C
i in Eq. (9) and multiplying its
right-hand side by λ.
The contribution of the Oi operators to B
d mixing is obtained by multiplying the right-hand side of Eq. (9) by
eiβ , where β is the CKM phase, β = arg(−VcdV ∗cb/VtdV ∗tb). Whereas the contribution of the O′i operators to B0dB0d
mixing is obtained again by replacing Ci with C
i in Eq. (9) and multiplying its right-hand side by −eiβ/λ.
Finally, the O′i contribution to K
0K0 mixing is the same as that to B0dB
d mixing, up to corrections suppressed by
powers of λ. For the Oi contribution to K
0K0 mixing, one has to replace in Eq. (9) each Wilson coefficient Ci by
Ci + C
iβ (see Eq. (A20) in the Appendix), and add to it the additional contribution
∆(hKe
2iσK ) = 2.26Re(ChLLC
LL) e
iβ − 5.17 |CuLL|2 eiβ − 8.35 |CwRL|2 eiβ . (10)
These expressions are valid up to corrections suppressed by λ2 or more.
b t s
FIG. 1: A one-loop contribution from OwRL (denoted by ⊗) to O7γ .
2 Throughout this paper we will bound Ci(1TeV/Λ)
2 and quote numerical results setting Λ = 1TeV.
IV. EXPERIMENTAL CONSTRAINTS
In this section we use low energy measurements to constrain the Wilson coefficients of the operators in Eq. (2).
Throughout we assume that there are no cancellations between the contributions from different operators.
A. Direct bounds
The best direct bounds on the operators in Eq. (2), as summarized in [9], come at present from searches for FCNCs
at the Tevatron, LEP, and HERA. The strongest direct constraints on t→ cZ and t→ uZ come from an OPAL search
for e+e− → tc in LEP II [13]. The upper limit on the branching ratio B(t→ cZ, uZ) < 0.137 bounds the LL and RR
operators. For neutral currents involving a photon, there is a constraint from ZEUS that looked for e±p→ e±tX [14].
This bounds B(t→ uγ) < 0.0059, and is the strongest constraint on the RL and LR operators with an up quark. The
other bounds come from a CDF search in Tevatron Run I, which bounds B(t → cγ, uγ) < 0.032 [15] and constrains
the LR and RL involving a charm. We translate these branching ratios into bounds on the Wilson coefficients and
list them in the first rows of Tables I and II. The LHC reach with 100 fb−1 data, as estimated in the ATLAS study [1]
is B(t→ cZ, uZ) < 5.5× 10−5 and B(t→ cγ, uγ) < 1.2× 10−5. These will improve the current direct constraints on
the Wilson coefficients by one and a half orders of magnitude, as summarized in the second rows of the tables.
B. B → Xsγ and B → Xsℓ
We first consider the constraints from B → Xsγ. At the scale mb, O7γ gives the leading contribution. Using the
NLO SM formulae from Ref. [16], we obtain
B(B → Xsγ) = 10−4 ×
0.07 +
∣1.807 + 0.081 i+ 1.81∆C7γ(mW )
, (11)
where ∆C7γ(mW ) is the NP contribution to C7γ at the µ = mW matching scale. The current experimental average [17],
B(B → Xsγ) = (3.55± 0.26)× 10−4, implies at 95% CL3 (setting Λ = 1TeV)
− 0.07 < CuLL < 0.04 or 1.2 < CuLL < 1.3 ,
−0.3 < ChLL < 0.16 or 5.3 < ChLL < 5.8 ,
−0.2 < CwRL < 0.1 or 3.1 < CwRL < 3.4 ,
−0.1 < CbRL < 0.24 or −4.5 < CbRL < −4.1 , (12)
The first (left) intervals are consistent with the SM, while the second (right) ones require new physics at the O(1)
level. The non-SM region away from zero is disfavored by b → sℓ+ℓ− discussed below, but we include it here for
completeness. For the operators whose contributions are suppressed by mc, we find
− 14 < CwLR < 7 , −10 < CbLR < 19 , (13)
and no meaningful bound for CuRR. To obtain the results in Eq. (12) and (13), we assumed that the NP contributions
are real relative to the SM, i.e., that there are no new CP violating phases. Had we not made this assumption, the
allowed regions would be annuli in the complex Ci planes.
Next we consider B → Xsℓ+ℓ−. The theoretically cleanest bound at present comes from the inclusive B → Xsℓ+ℓ−
rate measured for 1GeV2 < q2 < 6GeV2 [18]
B(B → Xsℓ+ℓ−)1GeV2<q2<6GeV2 = (1.61± 0.51)× 10−6 . (14)
Due to the unusual power counting in B → Xsℓ+ℓ−, the full set of O(αs) corrections are only included in what
is called NNLL order, achieving an accuracy around 10%. For the SM prediction we use the NNLL calculation as
implemented in Ref. [19]. This calculation does not normalize the rate to the B → Xℓν̄ rate; doing so would not
3 Hereafter all constraints are quoted at 95% CL, unless otherwise specified.
-0.5 0 0.5 1 1.5
-0.03
-0.02
-0.01
FIG. 2: Constraints from B → Xsγ and B → Xsℓ
− in the CuLL – C
LL plane. The red, green, and blue regions denote 68%,
95%, and 99% CL, respectively. The region between the dashed lines is beyond the LHC sensitivity.
improve the prediction significantly and would unnecessarily couple different operators’ contributions. We include the
modifications of C7γ , C9V , and C10A due to the new operators at lowest order. With our input parameters, we obtain
B(B → Xsℓ+ℓ−)1<q2<6GeV2 = 10−6 ×
1.55 + 35100
|∆C9V (mW )|2 + |∆C10A(mW )|2
+ 0.45 |∆C7γ(mW )|2
(180 + 5i)∆C9V (mW )
− 360Re
∆C10A(mW )
(0.17 + 0.04i)∆C7γ(mW )
− 200Re
∆C9V (mW )
∗∆C7γ(mW )
The simplest way to proceed would be to bound C7γ , C9V , and C10A separately at µ = mW , assuming that the others
have their SM values, and use this to constrain new physics. This procedure would not be consistent, since the NP
necessarily affects these Wilson coefficients in a correlated way. Instead, we directly constrain the coefficients of OuLL,
OhLL, O
LR, and O
LR, which also yields stronger constraints. With Λ = 1TeV, we obtain
− 1.1 < CuLL < 0.3 ,
−1.8× 10−2 < ChLL < −1× 10−2 or − 5× 10−3 < ChLL < 3× 10−3 ,
−0.5 < CwRL < 0.7 or 1.7 < CwRL < 3 ,
−2.0 < CbRL < 3.5 . (16)
The combined constraints from B → Xsγ and B → Xsℓ+ℓ− on these four Wilson coefficients are shown in Table I
in the Conclusions. We plot in Fig. 2 the bound on the LL operators in the CuLL – C
LL plane. The SM corresponds
to the point (0, 0). A measurement or a bound on the t→ cZ branching ratio corresponds to a nearly vertical band.
The LHC is sensitive to this whole plane, except for the band between the dashed lines.
The above bounds were derived assuming that the NP contribution is real relative to the SM. It is conceivable that
improved measurements of B → Xsℓ+ℓ− will lead to constraints on the CP violating phases before the LHC is be
able to probe top FCNCs. Thus we postpone a full analysis with complex NP Wilson coefficients until more data
become available.
C. Exclusive and inclusive b → cℓν̄ decays
In this subsection we investigate the constraints on the operators in Eq. (2) due to measurements of semileptonic
b→ c decays. They will allow us to bound the coefficient of the operator OwLR, which contains a right handed charm
field and is weakly constrained otherwise. We focus on three types of constraints coming from the ratio of exclusive
D and D∗ rates, the polarization in the D∗ mode, and moments in inclusive spectra.
We begin with the exclusive case where the B → Dℓν and B → D∗ℓν rates can be calculated in an expansion in
ΛQCD/mb,c using heavy quark effective theory. The form factors at zero recoil, where w = v · v′ = 1 (v and v′ are the
four-velocities of the B and D(∗) mesons, respectively), have been determined from lattice QCD [20]. In the SM the
ratio of rates is independent of Vcb, and therefore it provides a good test for non-SM contributions. The presence of
the new operator, OwLR, affects the two rates differently. The rates are given by [21]
dΓ(B → Dℓν)
r3(w2 − 1)3/2(1 + r)2|Vcb|2(FD)2 ,
dΓ(B → D∗ℓν)
w2 − 1 (1 + w)2
(1 − r∗)2 + 4w
1 + w
(1− 2wr∗ + r∗2)
|Vcb|2(FD∗)2 , (17)
where r = mD/mB and r
∗ = mD∗/mB. The form factors FD and FD∗ can be decomposed in terms of 6 form factors,
h+,−,V,A1,A2,A3 [21]. At leading order in the heavy quark limit F(w) = F∗(w) = h+,V,A1,A3 = ξ(w), where ξ(w) is the
Isgur-Wise function [22], while h−,A2 = 0. Therefore, it is useful to define the following ratios of form factors
R1(w) =
, R2(w) =
hA3 + r
, (18)
which are equal to unity in the heavy quark limit and have been measured experimentally.
Following the analysis of [23], we can absorb the new physics contributions in the form factors. We obtain
∆h+ = k(1 + r)(1 − w)ξ(w) , ∆h− = −k(1− r)(1 + w)ξ(w) ,
∆hA1 = −2k(1− r∗)ξ(w) , ∆hA2 = −2kξ(w) ,
∆hA3 = −2kξ(w) , ∆hV = 2k(1 + r∗)ξ(w) ,
where
2 vmB
CwLRVtb
. (20)
For the new physics contribution we include only the leading term, so we set ξ(1) = 1. Setting Λ = 1TeV, we obtain
FD(1) ≈ FSMD (1)− 1.01× 10−3 × Re
CwLRVtb
FD∗(1) ≈ FSMD∗ (1)− 2.02× 10−3 × Re
CwLRVtb
R1(1) ≈ RSM1 (1) + 6.52× 10−3 × Re
CwLRVtb
R2(1) ≈ RSM2 (1)− 2.48× 10−3 × Re
CwLRVtb
. (21)
Recent lattice QCD calculations [20] give FSMD (1) = 1.074± 0.024 and FSMD∗ (1) = 0.91± 0.04. For RSM1 and RSM2 we
use the results of [24], scanning over the hadronic parameters that enter. The experimental results are |Vcb|FD(1) =
(42.4± 4.4)× 10−3, |Vcb|FD∗(1) = (36.2± 0.6)× 10−3 [17], R1(1) = (1.417± 0.075), and R2(1) = (0.836± 0.043) [25].
We set |Vtb| = 1 and do a combined fit for CwLR and |Vcb|. We find
− 0.2 < Re(V
LRVtb)
|Vcb|
< 1.6 . (22)
We next turn to inclusive B → Xcℓν̄ decays, which is also sensitive to the presence of the additional operators. We
concentrate on the partial branching ratio and moments constructed from the charged lepton energy spectrum (see,
e.g., [26]),
M0(E0) = τB
dEℓ , M1(E0) =
, Mk(E0) =
[Eℓ −M1(E0)]k dΓdEℓ dEℓ
. (23)
-1 -0.5 0 0.5 1 1.5 2
FIG. 3: Constraints on OwLR in the Re(C
LR) – |Vcb| plane from semileptonic B → Xcℓν̄ (solid curves) and B → D
ℓν̄ decays
(dashed curves) and their combination (shaded areas). For each constraint the 68%, 95% and 99% CL regions are shown.
These are well measured and can be reliably calculated. We use the SM prediction including 1/m2b and αs corrections
and compare it in a combined fit with the 20 Babar [27] and a subset [28] of the 45 Belle [29] measurements, including
their correlations. The modification of dΓ/dEℓ due to the C
LR coupling is given by
dΓNP(B → Xcℓν̄)
2 Re(CwLRVcb)
2π3Λ2
ρ y2(3 − y)(1− y − ρ)3
(1− y)3
2G3Fm
4 |CwLR|2
3π3Λ4
y2(3− y)(1− y − ρ)4
(1− y)3
, (24)
where y = 2Eℓ/mb and ρ = m
b . It is known that the data cannot be fitted well with the OPE truncated at
1/m2b. Including the 1/m
b corrections in a more complicated fit would make the agreement with the SM better, and
therefore our bounds stronger.
The combined constraints on CwLR and |Vcb| from exclusive and inclusive decays is shown in Fig. 3. The solid curves
show the constraints from inclusive decays, the dashed curves show the bounds from exclusive semileptonic decays to
D and D∗, and the shaded regions show the combined constraints (the confidence levels are as in Fig. 2).
D. Exclusive and inclusive b → uℓν̄ decays
We now turn to some 3rd → 1st generation transitions. While the experimental constraints are less precise for these
than for 3rd → 2nd generation transitions, the SM also predicts smaller rates, and therefore NP could more effectively
compete with the SM processes. These constraints are particularly important as they bound the O′i contributions
relevant for t→ u decays, which might not be distinguishable at the LHC from t→ c.
As is the case for 3rd → 2nd generation transitions, exclusive and inclusive semileptonic b→ u decays can constrain
the operator O′wLR in t → u transition. Similarly to b → cℓν̄, this operator distorts the lepton energy spectrum, so
information on the lepton energy moments could constrain it. However, such measurements are not yet available for
B → Xuℓν̄. Therefore, to distinguish between the SM Vub contribution and CwLR, we use B → πℓν̄ in addition to the
inclusive data.
For exclusive B → πℓν̄ decay, we use for the SM prediction the parameterization of Ref. [30], which relies on
-2 -1 0 1 2
FIG. 4: Constraints on O′wLR in the Re(C
LR) – |Vub| plane from B → Xuℓν̄ (solid curves) and B → πℓν̄ (dashed curves) and
their combination (shaded areas). For each constraint the 68%, 95% and 99% CL regions are shown.
analyticity constraints and lattice QCD calculations of the form factors at large q2 [31, 32]. The NP contribution is
dΓNP(B → πℓν̄)
G2F |pπ|3
4m2Bv
2|C′wLR|2
(1 + q̂2)f− + (1− q̂2)f+
− 4mBvRe(VtbC
(1− q̂2)f2+ + (1 + q̂2)f−f+
. (25)
where the f± form factors are functions of the dilepton invariant mass, q
2, q̂2 = q2/m2B, and we neglected terms
suppressed by m2π/m
For inclusive B → Xuℓν̄ decay, we focus on the measurement utilizing combined cuts [33] on q2 and the hadronic
invariant mass, mX , and compare it with the Belle and Babar measurements [34]. Using this determination of Vub
is particularly simple for our purposes, because in the large q2 region the mild cut on mX used in the analysis only
modifies the rate at a subleading level. Working to leading order in the NP contribution, we can neglect the effect of
the mX cut on the NP and include the NP contribution to the rate via
dΓNP(B → Xuℓν̄)
192π3
32m2Bv
|C′wLR|2 q̂2(q̂2 − 1)2(q̂2 + 2) . (26)
Since the interference between the SM and NP is suppressed by mu/mb (see the
ρ factor in Eq. (24) in the first
term), there is no dependence on the weak phase of C′wLR in the inclusive decay. Using other determinations of Vub
would be harder to implement and would not change our results significantly.
The combined constraint on C′wLR and |Vub| from inclusive and exclusive decays is shown in Fig. 4. (This uses the
lattice QCD input from Fermilab [31], and the one using the HPQCD calculation [32] would also be similar.)
E. B → ργ and B → µ
The inclusive B → Xdγ decay has not been measured yet, and there is only limited data on B → ργ. Averaging
the measurements [35] using the isospin-inspired4 relation B(B → ργ) = B(B± → ρ±γ) = 2(τB±/τB0)B(B0 → ρ0γ),
4 Isospin is not a symmetry of the electromagnetic interaction. This average relies on the heavy quark limit to argue that the dominant
isospin violation is ΛQCD/mb suppressed. With more precise data, using onlyB
0 decays will be theoretically cleaner, because annihilation
is suppressed in the B0 compared to the B± modes. At present, this would double the experimental error, so we include the B± data.
and the PDG value τB±/τB0 = 1.07, we obtain
B(B → ργ) = (1.26± 0.23)× 10−6 . (27)
To reduce the sensitivity to form factor models, we normalize this rate to B(B → K∗γ) =
B(B± → K∗±γ) + (τB±/τB0)B(B0 → K∗0γ)
/2 = (41.4± 1.7)× 10−6,
B(B → ργ)
B(B → K∗γ)
2 ( m2B −m2ρ
m2B −m2K∗
|C7γ |2
|CSM7γ |2
. (28)
We use ξγ = 1.2 ± 0.15, where this error estimate accounts for the fact that we consider the rates to be determined
by O7γ(mb) alone. The contributions of other operators have larger hadronic uncertainties and are expected to
partially cancel [36]. If first principles lattice QCD calculations of the form factor become available then one can
avoid taking the ratio in Eq. (28), and directly compare the calculation of B(B → ργ) with data. We obtain the
following constraints
− 0.26 < C′uLL < −0.21 or −0.026 < C′uLL < 0.03 ,
−1.2 < C′hLL < −0.9 or −0.11 < C′hLL < 0.13 ,
−0.7 < C′wRL < −0.5 or −0.07 < C′wRL < 0.08 ,
−0.1 < C′bRL < 0.09 or 0.7 < C′bRL < 0.9 . (29)
Note that there are no constraints on O′wLR or O
LR, because of their mu/mW suppression. As for B → Xsγ, the two
solutions in Eq. (29) correspond to the sign ambiguity in interpreting the constraint on |C7γ |2 when we assume that
the NP contributions are real relative to the SM. Had we not made this assumption, the allowed regions would be
annuli in the complex C′i planes.
The NP operators we consider also contribute to the rare decays Bd,s → µ+µ−. This is most interesting for
Bd → µ+µ−, since one expects that the NP contribution is enhanced compared to the SM by [(v2/Λ2)(1/|Vtd|)]2,
which is around 20 for Λ = 1TeV. Moreover, O′hLL contributes at tree level, so its contribution is enhanced by an
additional factor of (4π/α)2. Although this decay mode has not yet been observed and the present upper bound
B(B → µ+µ−) < 3× 10−8 [37] is two orders of magnitude above the SM expectation, it still gives a useful constraint
on O′hLL. In particular, for Λ = 1TeV, we obtain
− 0.023 < C′hLL < 0.026 . (30)
The combined constraints fromB → ργ and B → µ+µ− on O′uLL and O′hLL are shown in Figure 5. The region between
the dashed lines is beyond the LHC reach, but the LHC will be able to exclude (though perhaps not completely) the
non-SM region in Fig. 5. In the case of O′uLL and O
RL the present data are not strong enough to exclude the non-SM
region allowed by B → ργ.
F. ∆F = 2 transitions
In this section we present the results of the analysis of the NP effects in ∆F = 2 processes. Since their contribution
appear at the same time in BdBd, BsBs and K
0K0, we performed a full fit using the CKMFitter code [10], after
having suitably modified it to include the results of Sec. III C. The code simultaneously fits experimental data for the
Wolfenstein parameters ρ and η and for NP (extending earlier studies in ∆F = 2 processes [38, 39]). The observables
used here include the Bd and Bs mass differences, the time dependent CP asymmetries in B → J/ψK, the CP
asymmetries in B → ππ, ρρ, ρπ, the ratio of |Vub| and |Vcb| measured in semileptonic B decays, the CP asymmetries
in B → DK, the width difference in the BsBs system, ∆Γs, the semileptonic CP asymmetry in B decays, ASL, and
the indirect CP violation in K decays, ǫK . We allowed the NP Wilson coefficients to be complex and performed a
scan over their phases. Thus, the results in this section are quoted in terms of the absolute values of the Ci and C
Keeping only one operator at a time, we get
|CuLL| < 0.07 , |ChLL| < 0.014 , |CwRL| < 0.14 ,
|C′uLL| < 0.11 , |C′hLL| < 0.018 , |C′wRL| < 0.26 . (31)
As before, we also performed a combined analysis for the LL operators. This is particularly interesting for O′uLL and
O′hLL, since until B → Xdℓ+ℓ− data becomes available, only ∆F = 2 processes are sensitive to the complex phases. In
-0.4 -0.2 0 0.2 0.4
-0.04
-0.02
FIG. 5: Constraints from B → ργ and B → µ+µ− in the C′uLL – C
LL plane. The red, green, and blue regions denote 68%, 95%,
and 99% CL, respectively. The region between the dashed lines is beyond the LHC sensitivity.
general, allowing for a variation of the phases of C′uLL and C
LL, a cancellation can occur between the two contributions
and the above bounds are relaxed. If their absolute values satisfy |C′hLL| ∼ 0.1 |C′uLL| then arbitrarily large values of
the Wilson coefficients is allowed for some values of the phases. This possibility is ruled out when the B → ργ and
B → µ+µ− constraints are included. Indeed, combining ∆F = 2 with these measurements, we obtain
|C′uLL| < 0.26 , |C′hLL| < 0.026 . (32)
V. COMBINED CONSTRAINTS AND CONCLUSION
In this paper, we studied constraints on flavor-changing neutral current top quark decays, t → cZ, uZ, cγ, uγ.
We used an effective field theory in which beyond the SM physics is integrated out. In the theory with unbroken
electroweak symmetry the leading contributions to such FCNC top decays come from seven dimension-6 operators of
Eq. (2). We assumed that the new physics scale, Λ, is sufficiently above the electroweak scale, v, to expand in v/Λ
and neglect higher dimension operators. We find different and sometimes stronger constraints than starting with an
effective theory which ignores SU(2)L invariance.
The 95% CL constraints on the Wilson coefficients of the operators involving 3rd and 2nd generation fields are
summarized in Table I. We consider one operator at a time, i.e., that there are no cancellations. The top two rows
show the present direct constraints and the expected LHC bounds. The next three rows show the bounds from B
physics. In the B → Xsγ, Xsℓ+ℓ− row the combined bounds from these processes are shown. The two allowed
regions are obtained neglecting the complex phases of the operators (see Fig. 2 and the discussion in Sec. IVB). This
assumption can be relaxed in the future with more detailed data on B → Xsℓ+ℓ−. In the ∆F = 2 row the numbers
refer to upper bounds on the magnitudes of the Wilson coefficients and are derived allowing the phase to vary. The
best bound for each operator is listed and then translated to a lower bound on the scale Λ (in TeV, assuming that the
C’s are unity), and to the maximal t → cZ and t→ cγ branching ratios still allowed by each operator. The last row
indicates whether a positive LHC signal could be explained by each of the operators alone. In this row, the star in
“Closed∗” for CuLL and C
LL refers to the fact that small values of these Wilson coefficients cannot give an observable
top FCNC signal, however, there is an allowed region with cancellations between the SM and the NP, which may still
give a signal. In the same row “Ajar” means that CwRL and C
RL cannot yield an LHC signal in t → cZ but may
manifest themselves in t→ cγ. It is remarkable that the coefficients of several operators are better constrained by B
physics than by FCNC top decays at the LHC.
direct bound 9.0 9.0 6.3 6.3 6.3 6.3 9.0
LHC sensitivity 0.20 0.20 0.15 0.15 0.15 0.15 0.20
B → Xsγ, Xsℓ
− [−0.07, 0.036]
[−0.017, −0.01]
[−0.005, 0.003]
[−0.09, 0.18] [−0.12, 0.24] [−14, 7] [−10, 19] —
∆F = 2 0.07 0.014 0.14 — — — —
semileptonic — — — — [0.3, 1.7] — —
best bound 0.07 0.014 0.15 0.24 1.7 6.3 9.0
Λ for Ci = 1 (min) 3.9 TeV 8.3 TeV 2.6 TeV 2.0 TeV 0.8 TeV 0.4 TeV 0.3 TeV
B(t → cZ) (max) 7.1×10−6 3.5×10−7 3.4×10−5 8.4×10−6 4.5×10−3 5.6×10−3 0.14
B(t → cγ) (max) — — 1.8×10−5 4.8×10−5 2.3×10−3 3.2×10−2 —
LHC Window Closed∗ Closed∗ Ajar Ajar Open Open Open
TABLE I: 95% CL constraints on the Wilson coefficients of the operators involving 3rd and 2nd generation fields for Λ = 1TeV.
The top two rows show the present direct constraints and the expected LHC bounds. The second part shows the bounds from
B physics, which is then translated to a lower bound on the NP scale, Λ, and to the maximal t → cZ and t → cγ branching
ratios each operator could still give rise to (the ATLAS sensitivity with 100 fb−1 is 5.5 × 10−5 and 1.2 × 10−5, respectively).
The last line concludes whether a positive LHC signal could be explained by each of the operators.
direct bound 9.0 9.0 2.7 2.7 2.7 2.7 9.0
LHC sensitivity 0.20 0.20 0.15 0.15 0.15 0.15 0.20
B → ργ, µ+µ−
[−0.26, −0.21]
[−0.026, 0.03]
[−0.023, 0.026]
[−0.7, −0.5]
[−0.07, 0.08]
[−0.1, 0.09]
[0.7, 0.9]
— — —
∆F = 2 0.11 0.02 0.26 — — — —
semileptonic — — — —
[−0.9, 0.1]
[0.8, 1.4]
combined bound 0.10 0.02 0.16 0.9 1.4 2.7 9.0
Λ for Ci = 1 (min) 3.2 TeV 7.2 TeV 2.5 TeV 1.1 TeV 0.8 TeV 0.6 TeV 0.3 TeV
B(t → uZ) (max) 1.6×10−5 6.4×10−7 4.1×10−5 1.2×10−4 3.2×10−3 1.0×10−3 0.14
B(t → uγ) (max) — — 2.1×10−5 6.7×10−4 1.6×10−3 5.9×10−3 —
LHC Window Closed Closed Ajar Open Open Open Open
TABLE II: Constraints on the Wilson coefficients of the operators involving 3rd and 1st generation fields. The entries in the
table are analogous to Table I.
Table II shows the constraints on the operators involving the 3rd and 1st generation quarks. We studied this
because the LHC may not be able to distinguish between t→ c and t→ u FCNC decays, and these processes are also
interesting in their own rights. In this case there are two allowed regions of C′wLR from semileptonic decays, as can be
seen in Fig. 4. The entries in the “combined bound” row show the result of the fit to all the B decay data above it,
as discussed in Sec. IVF. We see from the last row that the LHC window remains open for all of the RR, LR, and
RL operators, except O′wRL.
We conclude from Tables I and II that if the LHC sees FCNC t decays then they must have come from LR or RR
operators, unless there are cancellations. Moreover, if t→ cZ is seen but t→ cγ is not, then only OuRR could account
for the data.
Our analysis used the currently available data and compared it to an estimate of the LHC reach with 100 fb−1.
However, by that time many of the constraints discussed above will improve, and new measurements will become
available. The direct bounds will be improved by measurements from Run II of the Tevatron in the near future. All
the B decay data considered in this paper will improve, and the calculations for many of them may become more
precise. Important ones (in no particular order) are: (i) improved measurements of B → Xsℓ+ℓ− to better constrain
the magnitudes and especially the phases of CuLL, C
LL, C
RL, and C
RL; (ii) measurements of the lepton energy and
the hadronic mass moments in B → Xuℓν̄ to constrain C′wLR; (iii) improvements in B → ργ and measurement of
B → Xdγ to reduce the uncertainties of C′uLL, C′hLL, C′wRL, and C′bRL; (iv) measurement of and B → Xdℓ+ℓ− to reduce
the errors and constrain the weak phases of these last four coefficients. Additional information will also come from
LHCb. For example, the measurement of the CP violating parameter SBs→ψφ, the direct measurements of the CKM
angle γ, and some of the above rare decays will help improve the constraints. With several of these measurements
available, one can try to relax the no-cancellation assumption employed throughout our analysis. Note that not
all NP-sensitive B factory measurements can be connected to FCNC top decays; e.g., the CP asymmetry SK∗γ is
sensitive to right-handed currents in the down sector and cannot receive a sizable enhancement from the operators in
Eq. (2). Thus, there are many ways in which there can be interesting interplays between measurements of or bounds
on FCNC t and b quark decays.
If an FCNC top decay signal is observed at the LHC, the next question will be how to learn more about the
underlying physics responsible for it. With a few tens of events one can start to do an angular analysis or study an
integrated polarization asymmetry [40]. These could discriminate left-handed or right-handed operators (say OuRR or
OuLL). Such interactions could arise in models in which the top sector has a large coupling to a new physics sector,
predominantly through right-handed couplings [41]. However, a full angular analysis that could also distinguish OuRR
from OwLR requires large statistics, which is probably beyond the reach of the LHC.
The observation of FCNC top decays at the LHC would be a clear discovery of new physics, and therefore it would
be extremely exciting. Our analysis shows that an LHC signal requires Λ to be less than a few TeV. This generically
implies the presence of new particles with significant coupling to the top sector. If the new particles are colored,
we expect that they will be discovered at the LHC. It would be gratifying to decipher the underlying structure of
new physics from simultaneous information from top and bottom quark decays and direct observations of new heavy
particles at the LHC.
Acknowledgments
We acknowledge helpful discussions with Chris Arnesen and Iain Stewart on B → πℓν̄ [30], Frank Tackmann on
B → Xsℓ+ℓ− [19], and Phillip Urquijo about the Belle measurement of B → Xcℓν̄ [29]. P.F., M.P., and G.P. thank
the Aspen Center for Physics for hospitality while part of this work was completed. The work of P.F., Z.L., and M.P.
was supported in part by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under contract DE-AC02-05CH11231. M.S. was supported in part by the National Science Foundation under
grant NSF-PHY-0401513.
APPENDIX A: ANALYTIC EXPRESSIONS
We give the form of the operators of Eq. (2) after electroweak symmetry breaking, keeping only trilinear vertices
which do not involve the Higgs:
OuLL =
−sL + bLW/
2mZmW
tLZ/ cL + . . . ,
OhLL =
2mZmW
tLZ/ cL + bLZ/ sL
+ . . . ,
OwRL = mW sLσ
µνtRW
2mW cLσ
µνtR (cwZµν + swAµν) + . . . ,
ObRL =
2mW cLσ
swAµν −
+ . . . ,
OwLR = mW bLσ
µνcRW
2mW tLσ
µνcR (cwZµν + swAµν) + . . . ,
ObLR =
2mW tLσ
swAµν −
+ . . . ,
OuRR =
2mZmW
tRZ/ cR + . . . . (A1)
Here sw = sin θw, cw = cos θw, and the dots denote hermitian conjugate and the neglected vertices involving Higgs
and higher number of fields. Throughout this paper the covariant derivative is defined as Dµ = ∂µ + igA
a + ig′Bµ.
The analytic expressions for the contributions of the operators in Eq. (2) to the top FCNC partial widths are
Γ(t→ cZ) = mt
v2m2t
(1− y)2
|ChLL + CuLL|2 + |CuRR|2
(1 + 2y)
+ 2g22 cos
2 θW (2 + y)
|CbLR tan2 θW − CwLR|2 + |CbRL tan2 θW − CwRL|2
2g2 sin θW tan θW
(CbRL)
∗(ChLL + C
LL)− (CbLR)∗CuRR
2g2 cos θW
(CwRL)
∗(ChLL + C
LL)− (CwLR)∗CuRR
, (A2)
Γ(t→ cγ) = αmt
v2m2t
|CbLR + CwLR|2 + |CbRL + CwRL|2
, (A3)
where y = m2Z/m
t . The analogous expressions for t → u decays are obtained by replacing Ci by C′i above. This
expression makes it straightforward to relate the Wilson coefficients used in this paper with different notation present
in the literature, which defines the couplings in the effective Lagrangian after electroweak symmetry breaking.
Next we present the analytic expression for the Wilson coefficients originating from the operators in Eq. (2). We
use the MS scheme and match at the scale µ = mW . It is easiest to express the results as modifying the Inami-
Lim functions B0, C0, and D0/D
0, coming from box diagrams, Z-penguins, and γ-penguins, respectively. Using the
standard normalization of the effective Hamiltonian
Heff = −
CiOi , (A4)
the Wilson coefficients at the matching scale can be written as
C7γ = −
D′0(x) ,
C9V =
sin2 θW
B0(x) +
sin2 θW
C0(x) −D0(x)
C10A =
sin2 θW
B0(x) − C0(x)
, (A5)
where x = m2t/m
W . In the SM, we have the well-known expressions [12]
B0(x) =
(x− 1)2
C0(x) =
3x+ 2
(x − 1)2
D0(x) = −
lnx− 19x
3 − 25x2
36(x− 1)3
x2(5x2 − 2x− 6)
18(x− 1)4
lnx ,
D′0(x) =
8x3 + 5x2 − 7x
12(x− 1)3
3 − 2x2
2(x− 1)4
lnx . (A6)
The contributions of the OuLL, O
LR, andO
LR operators introduced in Eq. (2) can be included by adding the following
terms to Eq. (A6)
∆B0(x) =
− x ln x
(x− 1)2
, (A7)
∆C0 =
20(x− 1) sin2 θW + 23x+ 7
− 6x(x
2 + x+ 3)
(x− 1)2
− 2πκ
ChLL +
− x lnx
(x− 1)2
− x− 4
, (A8)
∆D0 = −
47x3 − 237x2 + 312x− 104
6(x− 1)3
4 − 30x3 + 54x2 − 32x+ 8
(x− 1)4
49x2 − 89x+ 34
6(x− 1)3
3 − 9x2 + x+ 1
(x− 1)4
2κg CbRL
x lnx
59x− 68
9(x− 1)
3x− 2
(x− 1)2
, (A9)
∆D′0 =
68x3 − 291x2 + 297x− 92
18(x− 1)3 +
x2(3x− 2)
(x− 1)4 lnx
ChLL(sin
2 θW + 3)
3x3 + 33x2 − 25x+ 1
2(x− 1)3
4 − 6x3 + 33x2 − 32x+ 8
(x− 1)4
− 2x(x − 4)
(x − 1)2
CwLR −
CbLR , (A10)
where xc = m
W and
V ∗cs
V ∗ts
. (A11)
Note that the contribution of OhLL to ∆C0 occurs at tree level, as indicated by its 1/α2 enhancement in Eq. (A8), so
OhLL gives tree-level contributions to C9V and C10A. Nevertheless, we shall not include the matrix element of O
LL to
one higher order in α2, in analogy with the conventional approach in which the NNLL calculation of B → Xsℓ+ℓ−
does not include the O(α2s) matrix element of O9V .
Finally we calcuate the ∆F = 2 contributions due to CuLL and C
LR. The shift in the SM contributions read
SSM0 → SSM0 + κi∆Si(x) + κi κj∆S′i,j(x) + κij ∆S′′ij(x) , (A12)
where i = u, h, w labels the contributions from the operators OuLL, O
LL and O
RL, respectively. The expressions for
∆S and ∆S′ are
∆Su = −
x(4x2 − 11x+ 1)
(x− 1)2
2x(x3 − 6x+ 2)
(x− 1)3
lnx , (A13)
∆Sh = −
x[(1 + x) sin2 θW + 2x− 6]
2x[x(x + 2) sin2 θW − 6]
3(x− 1)2
lnx , (A14)
∆Sw = 3 g
x(x+ 1)
(x− 1)2 −
(x− 1)3 lnx
, (A15)
∆S′u,u =
7x3 − 15x2 + 6x− 4
(x − 1)2
− 2x(2x
3 + 3x2 − 12x+ 4)
(x − 1)3
ln x , (A16)
∆S′h,h =
, (A17)
∆S′u,h =
2[(x+ 1)(x+ 2) sin2 θW + 3(x
2 − 9x+ 4)]
3(x− 1)
2x[x(x − 3− 2 sin2 θW ) + 6]
(x− 1)2
lnx , (A18)
∆S′w,w = g
−6x(x+ 1)
(x− 1)2
(x− 1)3
, (A19)
and κi depends on the flavor transition,
Ci Vcs/Vts for t→ c contribution in b→ s ,
Ci Vcd/Vtd for t→ c contribution in b→ d ,
C′i Vus/Vts for t→ u contribution in b→ s ,
C′i Vud/Vtd for t→ u contribution in b→ d ,
(Ci V
tsVcd + C
csVtd)/(V
tsVtd) for t→ c contribution in s→ d ,
(C′i V
tsVud + C
usVtd)/(V
tsVtd) for t→ u contribution in s→ d .
(A20)
The κij are zero except for K
0K0 mixing, where they are given by
κij =
csVcd/(V
tsVtd) for t→ c,
usVud/(V
tsVtd) for t→ u,
(A21)
and ∆S′′ij are given by
∆S′′u,u =
x(29x2 − 84x+ 7)
4(x− 1)2
− x(7x
3 + 9x2 − 64x+ 24)
2(x− 1)3
lnx , (A22)
∆S′′u,h =
2x[2x− 6 + (x+ 1) sin2 θW ]
(x − 1)
− 4x[x(x+ 2) sin
2 θW − 6]
3(x− 1)2
lnx , (A23)
∆S′′w,w = g
−2x(x
2 − 2x− 11)
(x− 1)2
− 12x
2(x2 − 3x+ 4)
(x− 1)3
. (A24)
[1] J. Carvalho et al. [ATLAS Collaboration], Eur. Phys. J. C 52, 999 (2007) [arXiv:0712.1127].
[2] CMS Physics TDR: Volume II, CERN/LHCC 2006-021, http://cmsdoc.cern.ch/cms/cpt/tdr/.
[3] F. Larios, R. Martinez and M. A. Perez, hep-ph/0605003; J. L. Diaz-Cruz, R. Martinez, M. A. Perez and A. Rosado, Phys.
Rev. D 41, 891 (1990); G. Eilam, J. L. Hewett and A. Soni, Phys. Rev. D 44, 1473 (1991) [Erratum-ibid. D 59, 039901
(1999)]; J. A. Aguilar-Saavedra and B. M. Nobre, Phys. Lett. B 553, 251 (2003) [hep-ph/0210360]; J. A. Aguilar-Saavedra
and G. C. Branco, Phys. Lett. B 495, 347 (2000) [hep-ph/0004190]; J. A. Aguilar-Saavedra, Phys. Rev. D 67, 035003
(2003) [Erratum-ibid. D 69, 099901 (2004)] [hep-ph/0210112]; J. L. Diaz-Cruz and G. Lopez Castro, Phys. Lett. B 301
(1993) 405; M. E. Luke and M. J. Savage, Phys. Lett. B 307, 387 (1993) [hep-ph/9303249]; P. J. Fox et al., hep-th/0503249;
B. Grzadkowski, J. F. Gunion and P. Krawczyk, Phys. Lett. B 268, 106 (1991); C. S. Li, R. J. Oakes and J. M. Yang, Phys.
Rev. D 49, 293 (1994) [Erratum-ibid. D 56, 3156 (1997)]; G. M. de Divitiis, R. Petronzio and L. Silvestrini, Nucl. Phys.
B 504, 45 (1997) [hep-ph/9704244]; G. Eilam et al., Phys. Lett. B 510, 227 (2001) [hep-ph/0102037]; X. L. Wang et al.,
Phys. Rev. D 50, 5781 (1994). R. D. Peccei and X. Zhang, Nucl. Phys. B 337, 269 (1990); J. J. Cao et al., hep-ph/0702264;
J. J. Liu, C. S. Li, L. L. Yang and L. G. Jin, Phys. Lett. B 599, 92 (2004) [hep-ph/0406155].
[4] A. Cordero-Cid, M. A. Perez, G. Tavares-Velasco and J. J. Toscano, Phys. Rev. D 70, 074003 (2004) [hep-ph/0407127];
J. A. Aguilar-Saavedra, Acta Phys. Polon. B 35, 2695 (2004) [hep-ph/0409342]; W. Wagner, Rept. Prog. Phys. 68, 2409
(2005) [hep-ph/0507207]; J. M. Yang, Annals Phys. 316, 529 (2005) [hep-ph/0409351]; E. Malkawi and C. P. Yuan,
Phys. Rev. D 52, 472 (1995) [hep-ph/9502225]; F. Larios, E. Malkawi and C. P. Yuan, hep-ph/9704288; O. J. P. Eboli,
M. C. Gonzalez-Garcia and S. F. Novaes, Phys. Lett. B 415, 75 (1997) [hep-ph/9704400]. U. Baur, A. Juste, L. H. Orr and
D. Rainwater, Phys. Rev. D 71, 054013 (2005) [hep-ph/0412021]; F. Larios, R. Martinez and M. A. Perez, Phys. Rev. D
72, 057504 (2005) [hep-ph/0412222]; T. Han, R. D. Peccei and X. Zhang, Nucl. Phys. B 454, 527 (1995) [hep-ph/9506461];
F. del Aguila, M. Perez-Victoria and J. Santiago, Phys. Lett. B 492, 98 (2000) [hep-ph/0007160].
[5] W. Buchmuller and D. Wyler, Nucl. Phys. B 268, 621 (1986).
[6] P. M. Ferreira and R. Santos, PoS TOP2006, 004 (2006) [hep-ph/0603131]; and references therein.
[7] A. Ali and D. London, Eur. Phys. J. C 9, 687 (1999) [hep-ph/9903535]; A. J. Buras, P. Gambino, M. Gorbahn, S. Jager and
L. Silvestrini, Phys. Lett. B 500, 161 (2001) [hep-ph/0007085]; G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia,
Nucl. Phys. B 645, 155 (2002) [hep-ph/0207036].
[8] K. Agashe, M. Papucci, G. Perez and D. Pirjol, hep-ph/0509117.
[9] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
[10] A. Hocker, H. Lacker, S. Laplace and F. Le Diberder, Eur. Phys. J. C 21 (2001) 225 [hep-ph/0104062]; J. Charles et al.,
Eur. Phys. J. C 41 (2005) 1 [hep-ph/0406184]; updates at http://ckmfitter.in2p3.fr/.
[11] A. P. Heinson [CDF and DØ Collaborations], AIP Conf. Proc. 870, 223 (2006) [hep-ex/0609028].
[12] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996) [hep-ph/9512380].
[13] G. Abbiendi et al. [OPAL Collaboration], Phys. Lett. B 521, 181 (2001) [hep-ex/0110009].
[14] S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 559, 153 (2003) [hep-ex/0302010].
[15] F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 80, 2525 (1998).
[16] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002) [hep-ph/0112300];
A. J. Buras, A. Czarnecki, M. Misiak and J. Urban, Nucl. Phys. B 631, 219 (2002) [hep-ph/0203135].
[17] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG)], arXiv:0704.3575, and updates at http://www.slac.stanford.
edu/xorg/hfag/.
[18] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 93, 081802 (2004) [hep-ex/0404006];
M. Iwasaki et al. [Belle Collaboration], Phys. Rev. D 72, 092005 (2005) [hep-ex/0503044].
[19] K. S. M. Lee, Z. Ligeti, I. W. Stewart and F. J. Tackmann, Phys. Rev. D 75, 034016 (2007) [hep-ph/0612156]; and
references therein.
[20] M. Okamoto et al., Nucl. Phys. Proc. Suppl. 140, 461 (2005) [hep-lat/0409116]; S. Hashimoto, A. S. Kronfeld, P. B. Macken-
zie, S. M. Ryan and J. N. Simone, Phys. Rev. D 66, 014503 (2002) [hep-ph/0110253].
[21] A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000).
[22] N. Isgur and M. B. Wise, Phys. Lett. B 232, 113 (1989); Phys. Lett. B 237, 527 (1990).
[23] W. D. Goldberger, hep-ph/9902311; B. M. Dassinger, R. Feger and T. Mannel, Phys. Rev. D 75, 095007 (2007) [hep-
ph/0701054].
[24] B. Grinstein and Z. Ligeti, Phys. Lett. B 526, 345 (2002) [Erratum-ibid. B 601, 236 (2004)] [hep-ph/0111392].
[25] B. Aubert et al. [BABAR Collaboration], hep-ex/0607076.
[26] M. Gremm, A. Kapustin, Z. Ligeti and M. B. Wise, Phys. Rev. Lett. 77, 20 (1996) [hep-ph/9603314];
M. B. Voloshin, Phys. Rev. D 51, 4934 (1995) [hep-ph/9411296].
[27] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 69, 111104 (2004) [hep-ex/0403030].
[28] P. Urquijo, private communications.
[29] P. Urquijo et al. [Belle Collaboration], Phys. Rev. D 75, 032001 (2007) [hep-ex/0610012].
[30] M. C. Arnesen, B. Grinstein, I. Z. Rothstein and I. W. Stewart, Phys. Rev. Lett. 95, 071802 (2005) [hep-ph/0504209].
[31] M. Okamoto et al., Nucl. Phys. Proc. Suppl. 140, 461 (2005) [hep-lat/0409116].
[32] J. Shigemitsu et al., Nucl. Phys. Proc. Suppl. 140, 464 (2005) [hep-lat/0408019].
[33] C. W. Bauer, Z. Ligeti and M. E. Luke, Phys. Rev. D 64, 113004 (2001) [hep-ph/0107074]; Phys. Lett. B 479, 395 (2000)
[hep-ph/0002161].
http://cmsdoc.cern.ch/cms/cpt/tdr/
http://ckmfitter.in2p3.fr/
http://www.slac.stanford.edu/xorg/hfag/
http://www.slac.stanford.edu/xorg/hfag/
[34] H. Kakuno et al. [Belle Collaboration], Phys. Rev. Lett. 92, 101801 (2004) [hep-ex/0311048]; I. Bizjak et al. [Belle Collab-
oration], Phys. Rev. Lett. 95, 241801 (2005) [hep-ex/0505088]; B. Aubert et al. [BABAR Collaboration], hep-ex/0507017.
[35] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 96, 221601 (2006) [hep-ex/0506079]; B. Aubert et al. [BABAR
Collaboration], Phys. Rev. Lett. 98, 151802 (2007) [hep-ex/0612017].
[36] P. Ball, G. W. Jones and R. Zwicky, Phys. Rev. D 75, 054004 (2007) [hep-ph/0612081]; and references therein.
[37] CDF Collaboration, http://www-cdf.fnal.gov/physics/new/bottom/060316.blessed-bsmumu3.
[38] Z. Ligeti, Int. J. Mod. Phys. A 20, 5105 (2005) [hep-ph/0408267]; UTfit Collaboration (M. Bona et al.), JHEP 0603,
080 (2006) [hep-ph/0509219]; F. J. Botella, G. C. Branco, M. Nebot and M. N. Rebelo, Nucl. Phys. B 725, 155 (2005)
[hep-ph/0502133].
[39] Z. Ligeti, M. Papucci and G. Perez, Phys. Rev. Lett. 97, 101801 (2006) [hep-ph/0604112]; Y. Grossman, Y. Nir and
G. Raz, Phys. Rev. Lett. 97, 151801 (2006) [hep-ph/0605028]; M. Blanke, A. J. Buras, D. Guadagnoli and C. Tarantino,
JHEP 0610, 003 (2006) [hep-ph/0604057].
[40] K. Agashe, A. Belyaev, T. Krupovnickas, G. Perez and J. Virzi, Phys. Rev. D 77, 015003 (2008) [hep-ph/0612015].
[41] K. Agashe, G. Perez and A. Soni, Phys. Rev. D 75, 015002 (2007) [hep-ph/0606293];
G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, JHEP 0706, 045 (2007) [hep-ph/0703164].
http://www-cdf.fnal.gov/physics/new/bottom/060316.blessed-bsmumu3
