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arXiv:0810.3633v1 [hep-ph] 20 Oct 2008

34thInternational Conference on High Energy Physics, Philadelphia, 2008

Charged-Higgs effects in B → (D)τν decays

St´ ephanie Trine

Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D-76128 Karlsruhe, Germany

We update and compare the capabilities of the purely leptonic mode B → τν and the semileptonic mode B → Dτν

in the search for a charged Higgs boson.

1. INTRODUCTION

Supersymmetric extensions of the standard model (SM) – or more generally extensions that require the existence

of at least one additional Higgs doublet – generate new flavour-changing interactions already at tree-level via the

exchange of a charged Higgs boson. The coupling of H+to fermions grows with the fermion mass. It is thus natural

to look at (semi)leptonic B decays with a τ in the final state to try to uncover this type of effects. In a two-Higgs-

doublet-model (2HDM) of type II, where up-type quarks get their mass from one of the two Higgs doublets and

down-type quarks from the other one, H+effects are entirely parametrized by the H+mass, MH, and the ratio

of the two Higgs vacuum expectation values, tanβ = vu/vd. They can compete with the exchange of a W+boson

for large values of tanβ [1]. In the minimal supersymmetric extension of the SM (MSSM), the tree-level type-II

structure is spoilt by radiative corrections involving supersymmetry-breaking terms. The effective scalar coupling gS

then exhibits an additional dependence on sparticle mass parameters when tanβ is large (q = u,c) [2, 3]:

HH+

eff= −2√2GFVqbmbmτ

M2

B

gS[qLbR][τRνL] + h.c.,gS=M2

Btan2β

M2

H

1

(1 + ε0tanβ)(1 + ετtanβ), (1)

where ε0,τ denote sparticle loop factors. The correction induced can be of order one. However, the access to the

Higgs sector remains exceptionally clean. In Eq.(1), gShas been normalized such that it gives the fraction of effects

in the B → τν amplitude, which is very sensitive to H+exchange: B(B → τν)/B(B → τν)SM= |1 − gS|2. The

B → Dτν channel is less sensitive (though better in this respect than other modes such as B → D∗τν) but, as we will

see, exhibits a number of features that make it, too, play an important part in the hunt for the charged Higgs boson.

2. B(B → Dτν) VERSUS B(B → τν)

The current capabilities of B(B → Dτν) and B(B → τν) to constrain H+effects are compared in Fig.1 for

gS ≥ 0 (as is typically the case in the MSSM or the 2HDM-II). The lower sensitivity of the B → Dτν mode

comes from the different momentum dependence of the Higgs contribution with respect to the longitudinal W+one1:

(dΓ(B → Dτν)/dq2)W+

prediction for B(B → τν) suffers from large parametric uncertainties from the CKM matrix element Vub and the

B decay constant fB. In contrast, Vcbis known with better than 2% accuracy from inclusive B → Xcℓν (ℓ = e,µ)

decays, |Vcb| = (41.6 ± 0.6) × 10−3[4], and the form factors f+(q2) and f0(q2) describing the B → D transition are

very well under control, as we now discuss in more detail.

To this end, we introduce the following conformal transformation:

?+H+

∝ |1 − gS(q2/M2

B)/(1 − mc/mb)|2with q ≡ pB− pD. On the other hand, the theory

q2→ z(q2,t0) ≡

?(MB+ MD)2− q2−?(MB+ MD)2− t0

?(MB+ MD)2− q2+?(MB+ MD)2− t0

,(2)

1Note that the latter, though helicity-suppressed, is still (slightly) larger than the transverse W+contribution for all q2values.

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34thInternational Conference on High Energy Physics, Philadelphia, 2008

0 0.51 1.52 2.53 3.5

gS

0

0.5

1

1.5

2

2.5

3

R

B

BØ n

t

0

1-4

0 0.51 1.52 2.53 3.5

gS

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

R

Figure 1: Left: B(B → τν) as a function of gS. Light gray band: Bexp= (1.51 ± 0.33) × 10−4[5]. Gray (blue) band: Bthfor

|Vub| = (3.95±0.35) ×10−3[4] and fB = 216±38MeV [6] (in this last case, we add errors linearly to stay on the conservative

side). Right: R ≡ B(B → Dτν)/B(B → Dℓν) as a function of gS. Light gray band: Rexp= (41.6 ± 11.7 ± 5.2)% [7]. Dark

gray (dark blue) band: Rthwith the HFAG vector form factor before ICHEP08 [8]. Gray (blue) band: Rthwith the HFAG

vector form factor after ICHEP08 [5]. The dashed lines indicate the 2 and 3-sigma limits. The ratio mc/mb in the MS scheme

has recently been determined with very high accuracy: mc/mb = 0.2211 ± 0.0044 [9]. We inflated the error on this number

and set mc/mb= 0.22 ± 0.01 to reduce the discrepancy with the HFAG estimation [5].

which maps the complex q2plane, cut along q2≥ (MB+ MD)2, onto the disk |z| < 1. The form factors f+and f0

are analytic in z in this domain, up to a few subthreshold poles, and can thus be written as a power series in z after

these poles are factored out (i = +,0) [10]:

fi(q2) =

1

Pi(q2)φi(q2,t0)

?ai

0(t0) + ai

1(t0)z(q2,t0) + ...?,(3)

where the function Pigathers the pole singularities and an arbitrary analytic function φican be factored out as well.

This parametrization has been used in Ref.[11] with the choice t0= q2

spin symmetry inputs, to derive the following ansatz for the vector form factor:

max= (MB−MD)2, together with heavy-quark

f+(q2) ≡MB+ MD

2√MDMB

V1(q2),V1(q2) = G(1)?1 − 8ρ2z(q2,t0) + (51ρ2− 10)z(q2,t0)2− (252ρ2− 84)z(q2,t0)3?, (4)

where V1is defined such that it reduces to the Isgur-Wise function in the heavy-quark limit and G(1) ≡ V1(q2

parameters |Vcb|G(1) and ρ2can be determined from B → Dℓν experimental data. Before this summer, the HFAG

averages [8] based on BELLE, CLEO, and ALEPH data read: |Vcb|G(1) = (42.3 ± 4.5) × 10−3and ρ2= 1.17 ± 0.18

(with a |Vcb|G(1)-ρ2correlation of 0.93). The recent BABAR results [12] and [13] have now been included, leading

to a substantial improvement [5]: |Vcb|G(1) = (42.4±0.7±1.4)×10−3and ρ2= 1.19±0.04±0.04 (with |Vcb|G(1)-ρ2

correlation 0.88). The old and new vector form factors are compared in Fig.2 (left), where we have defined as usual

w = (M2

For the scalar form factor, we adopt the ansatz of Ref.[14]:

f0(q2) ≡(w + 1)√MDMB

MB+ MD

z(q2,M2

max). The

B+ M2

D− q2)/(2MBMD).

S1(q2) =

1

1)z(q2,M2

2)φ0(q2,t0)

?a0

0(t0) + a0

1(t0)z(q2,t0)?,(5)

where t0= (MB+MD)2?

M2= 7.108GeV [15] are the subthreshold poles, and φ0is obtained from Eq.(10) of Ref.[14] setting Q2= 0 and η = 2:

1 −?1 − (MB− MD)2/(MB+ MD)2?

such that |z|maxis minimized, M1= 6.700GeV and

φ0(q2,t0) =

?

2(M2

B− M2

16π

D)2

?(MB+ MD)2− q2

((MB+ MD)2− t0)1/4

z(q2,0)2

(q2)2

?z(q2,t0)

t0− q2

?−1/2?z(q2,(MB− MD)2)

(MB− MD)2− q2

?−1/4

.(6)

Following [16], we truncate the series (3) after the first two terms. This is motivated by the fact that |z|max =

0.032 and that a similar parametrization for f+, when fitted to experimental data, produces the same result as

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11 1.11.1 1.2 1.21.3

ww

1.41.41.5 1.51.6 1.6

0.0250.025

0.030.03

0.0350.035

0.04 0.04

0.0450.045

Vb

Vb

c V1

1.3

c V1

11 1.11.1 1.21.21.3

ww

1.41.4 1.51.5 1.61.6

0.025 0.025

0.030.03

0.035 0.035

0.040.04

0.045 0.045

Vb

Vb

c S1

1.3

c S1

Figure 2: Vector (left) and scalar (right) form factors corresponding to the |Vcb|G(1) and ρ2determinations of HFAG before

(dark gray/dark blue)[8] and after (gray/blue)[5] ICHEP08. With the new determination, the errors on |Vcb|V1 and |Vcb|S1

are smaller than 4% and 7%, respectively.

100 100200 200 300300400 400500500600 600

MH

MH

10 10

2020

3030

4040

5050

6060

n

a t

n

a t

bb

gS=0.1

gS=0.2

100100200200300 300400400500500600600

MH

MH

10 10

2020

3030

40 40

50 50

6060

n

a t

n

a t

bb

gS=0.1

gS=0.2

Figure 3: 95% C.L. exclusion zones in the (MH,tanβ) plane from B(B → τν) (dark gray/dark blue) and R (gray/blue) in a

2HDM (left) and in the MSSM with ε0 = 0.01 and ετ ≃ 0 (right). The exclusion limits are directly read from the gray (blue)

bands in Fig.1. They differ from those usually found in the literature in that experimental and theory errors are not simply

added in quadrature and the dependence of the errors on the H+contribution is taken into account.

Eq.(4) in very good approximation [16]. Then, |Vcb|a0

|Vcb|S1(0) = |Vcb|V1(0) and (ii) |Vcb|S1(q2

B → Xcℓν [4] and S1(q2

old and new |Vcb|V1are not very different, as one can see on Fig.2 (right).

The recent progress on |Vcb|V1allows to reduce the errors on the SM predictions for the two B → Dτν branching

fractions: B(B−→ D0τ−¯ ν)SM=?0.70+0.06

τB0 ?= τB+). The errors from |Vcb|V1, however, already cancel to a large extent in the ratio R ≡ B(B → Dτν)/B(B →

Dℓν), which is why the nice improvement in Fig.2 has little impact on Fig.1 (right), already dominated by the error

on S1(q2

RSM

lead to a similar error on R. Still, an interesting 95% C.L. bound on gScan already be obtained from R: gS< 1.79,

complementary to the bounds from B(B → τν): gS< 0.36 ∪ 1.64 < gS< 2.73. The corresponding exclusion zones

in the (MH,tanβ) plane are depicted in Fig.3. The error assigned to S1(q2

constraints are robust. At the three-sigma level, it is not possible to extract any interesting bound from R yet, but

its experimental knowledge is expected to improve in the near future. Its role to constrain H+effects will then of

course depend on the new central value. For the moment, a 15% measurement with the same central value would

exclude gS> 0.29 at the 95% C.L..

0(t0) and |Vcb|a0

1(t0) are determined imposing the conditions (i)

max) = (4.24 ± 0.27)% (corresponding to |Vcb| = (41.6 ± 0.6) × 10−3from

max) = 1.02 ± 0.05 from HQET [16]). The scalar form factors obtained in this way from the

−0.05

?%, B(¯B0→ D+τ−¯ ν)SM=?0.65+0.06

−0.05

?% (differing essentially due to

max): RSM= 0.31 ± 0.02. This estimation is compatible with the one obtained from lattice methods:

latt= 0.28± 0.02 [17]. Note that replacing condition (ii) by a constraint on S1(q2

max)/V1(q2

max) from HQET would

max) is quite conservative, so the above

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34thInternational Conference on High Energy Physics, Philadelphia, 2008

111.11.1 1.21.2 1.31.31.41.4

ww

00

2020

4040

6060

80 80

100100

dG w

d

dG w

00

1- 6

1

1- 6

V

e

G

e

G

d

1

V

111.051.051.1 1.11.151.151.2

ww

1.251.251.31.31.351.35

0.50.5

11

1.51.5

22

d3G

d3G

0

1

0

1

- 6

11

1.2

- 6

Figure 4: dΓ(B → Dτν)/dw (left) and dΓ(B → Dντ[→ πν])/dEπdcosθDπdw with Eπ = 1.8GeV and cosθDπ = −1 (right)

for gS = 0 (gray/red), gS = 0.35 (light gray/light blue), and gS = 1.75 (dark gray/dark blue). These values are still allowed

by B(B → Dτν) and B(B → τν) at the 95% C.L.. The various curves have been obtained using the more recent HFAG vector

form factor [5]. The lighter bands take all errors into account, while the darker bands only take into account the error on

S1(q2

max). One could of course also normalize the above differential distributions to dΓ(B → Dℓν)/dw to reduce the impact of

the errors on f+.

3. B → Dτν DIFFERENTIAL DISTRIBUTIONS

If a hint for a charged Higgs boson is seen at the branching fraction level, B → Dτν has a great advantage over

B → τν: it allows to analyze the same data points on a differential basis, better suited to discriminate between

effective scalar-type interactions and other effects. The dΓ(B → Dτν)/dq2distribution, in particular, has already

been studied in great detail [18]. The polarization of the τ is also known as a H+analyzer [19], yet it requires the

knowledge of the τ momentum, which cannot be accessed at B factories as the τ does not travel far enough for a

displaced vertex and decays into at least one more neutrino.

A straightforward way to nevertheless exploit the sensitivity of the τ polarization to H+effects and at the same

time retain the information from the q2spectrum is to look at the subsequent decay of the τ into a pion and a

neutrino [16]. The direction of the pion is indeed directly correlated with the polarization of the τ. Integrating over

the neutrino momenta, we end up with a triple differential decay distribution dΓ(B → Dντ[→ πν])/dq2dEπdcosθDπ.

An explicit formula is given in Eqs.(9-11) of Ref.[16] (with FV ≡ f+and FS≡ f0). Its sensitivity to gSis illustrated in

Fig.4 for Eπ= 1.8GeV and cosθDπ= −1. For comparison, we also display the q2spectra corresponding to the same

gSvalues. Of course, in practice, one should not fix Eπor θDπ, but rather perform a (unbinned) maximum likelihood

fit of the triple differential decay distribution to the available data points. The information from the q2spectrum in

the dominant τ → ℓν¯ ν decay channel should also be included in the fit to make the most out of experimental data.

4. CONCLUSION

The form factors f+(q2) and f0(q2) in the B → Dτν transition are under good control. As a result, the ratio

R ≡ B(B → Dτν)/B(B → Dℓν) can be predicted with 7% accuracy in the SM: RSM= 0.31 ± 0.02, where the

5% uncertainty on the scalar form factor at zero recoil S1(q2

useful constraints on the effective H+coupling gS. Together with the constraints from B(B → τν), we obtain:

gS< 0.36 ∪ 1.64 < gS< 1.79, i.e., the window around gS= 2 left over by B(B → τν) is now nearly completely

excluded by R alone. These bounds should be strengthened soon thanks to the current considerable experimental

efforts on both modes. In this respect, one should pay particular attention to the B → Dτν differential distributions

as these are especially well-suited to discriminate between effective scalar interactions and other types of effects and,

if the former are seen, to extract the coupling gSwith good precision.

max) is the main error source. This allows to derive

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34thInternational Conference on High Energy Physics, Philadelphia, 2008

Acknowledgments

It’s a pleasure to thank my collaborators Ulrich Nierste and Susanne Westhoff. Discussions with Matthias Stein-

hauser about the ratio mc/mband with Christoph Schwanda and Laurenz Widhalm about experimental issues are

also warmly acknowledged. Work supported by the DFG grant No. NI 1105/1-1, by the DFG-SFB/TR9, and by the

EU contract No. MRTN-CT-2006-035482 (FLAVIAnet).

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