Reparametrization Invariance, the controversial extraction of $\alpha$
  from $B\to\pi\pi$ and New Physics
IFIC/07-17
FTUV-07-0402
Reparametrization Invariance, the
controversial extraction of α from B → ππ
and New Physics
Francisco J. Botella a, Miguel Nebot b
a Departament de F́ısica Teòrica and IFIC,
Universitat de València-CSIC,
E-46100, Burjassot, Spain
b Centro de F́ısica Teórica de Part́ıculas (CFTP),
Instituto Superior Técnico,
P-1049-001, Lisboa, Portugal
Abstract
The extraction of the weak phase α from B → ππ decays has been
controversial from a statistical point of view, as the frequentist vs.
bayesian confrontation shows. We analyse several relevant questions
which have not deserved full attention and pervade the extraction of α.
Reparametrization Invariance proves appropriate to understand those
issues. We show that some Standard Model inspired parametriza-
tions can be senseless or inadequate if they go beyond the minimal
Gronau and London assumptions: the single weak phase α just in the
∆I = 3/2 amplitudes, the isospin relations and experimental data.
Beside those analyses, we extract α through the use of several ade-
quate parametrizations, showing that there is no relevant discrepancy
between frequentist and bayesian results. The most relevant informa-
tion, in terms of α, is the exclusion of values around α ∼ π/4; this
result is valid in the presence of arbitrary New Physics contributions
to the ∆I = 1/2 piece.
http://arxiv.org/abs/0704.0174v1
1 Introduction
The extraction of the CP violating phase α [1] has lead to some recent con-
troversy confronting the results and statistical methods of two different col-
laborations: the frequentist approach advocated in references [2, 4] and the
bayesian approach employed in reference [3]. In reference [2] J. Charles et al.
presented an important criticism to the bayesian methods used by the UTfit
collaboration in order to extract the angle α of the unitarity triangle b–d from
ππ and ρρ data. The criticism relies heavily on the statistical treatment of
data: frequentist vs. bayesian. The answer of the UTfit collaboration [3]
rises some interesting points, both on the interpretation of the results and
on the importance of the physical assumptions on the hadronic amplitudes.
The authors of [2] have recently answered to this UTfit reply in [4]. The
aim of the present work is to clarify several issues central to an adequate
understanding of the physics at stake. We also want to call the attention
on the importance of reparametrization invariance (RpI) in the sense intro-
duced by F.J.B. and J. Silva in reference [5] to do so. We will not enter
the polemic arena of statistical confrontation. With regard to this, we will
instead illustrate the compatibility of results obtained in both approaches as
long as things are done properly; notwithstanding, we will not ignore some
“obscure” aspects of both approaches that are somehow swept under the
rug as the statistical confrontation rages on, they illustrate that rather than
sticking to one approach and deprecating the other it may be wiser to learn
lessons from both.
This work is organized as follows. We start section 2 with a short re-
minder on reparametrization invariance and its implications, then we use the
exclusion or inclusion of B → π0π0 data together with RpI to clarify the ori-
gin of our knowledge on α. In section 3 we study critically Standard Model
inspired parametrizations. We devote section 4 to a detailed analysis of the
impact on the results of allowed ranges for some parameters. The lessons
from previous sections set up the stage for an adequate extraction of α, to
which section 5 is dedicated, especially in the presence of New Physics (NP)
in loops. Several appendices deal with aspects left out of the main flow of
the discussion.
2 Reparametrization invariance and B → ππ
2.1 Weak Phases
We start this section with a short reminder of the findings presented in
reference [5] concerning the parametrization of decay amplitudes and the
election of weak phases. A generic parametrization of the decay amplitude
of a B meson to a given final state and the CP-conjugate amplitude is the
following1:
A = M1 e
+iφ1 eiδ1 +M2 e
+iφ2 eiδ2 ,
Ā = M1 e
−iφ1 eiδ1 +M2 e
−iφ2 eiδ2 , (1)
where φj are CP-odd weak phases, δj are CP-even strong phases and Mj
the magnitudes of the different contributions. The first property to consider
is the full generality, as long as φ1 − φ2 6= 0 mod [π], of Eq. (1), i.e. any
additional contribution M3e
±iφ3eiδ3 can be recast into the previous form as
e±iφ3 =
sin(φ3 − φ2)
sin(φ1 − φ2)
e±iφ1 +
sin(φ3 − φ1)
sin(φ2 − φ1)
e±iφ2 , (2)
and thus
A′ = A +M3e
+iφ3eiδ3 = M ′1e
+iφ1eiδ
1 +M ′2e
+iφ2eiδ
Ā′ = Ā +M3e
−iφ3eiδ3 = M ′1e
−iφ1eiδ
1 +M ′2e
−iφ2eiδ
2 , (3)
M ′1e
1 = M1e
iδ1 +M3e
sin(φ3 − φ2)
sin(φ1 − φ2)
M ′2e
2 = M2e
iδ2 +M3e
sin(φ3 − φ1)
sin(φ2 − φ1)
. (4)
We can also use Eq. (2) to change our basic set {φ1, φ2} of weak phases to any
other arbitrary set of weak phases {ϕ1, ϕ2}, as long as ϕ1−ϕ2 6= 0 mod [π]:
A = M1 e+iϕ1 ei∆1 +M2 e+iϕ2 ei∆2 ,
Ā = M1 e−iϕ1 ei∆1 +M2 e−iϕ2 ei∆2 , (5)
where
M1ei∆1 = M1eiδ1
sin(φ1 − ϕ2)
sin(ϕ1 − ϕ2)
sin(φ2 − ϕ2)
sin(ϕ1 − ϕ2)
M2ei∆2 = M1eiδ1
sin(φ1 − ϕ1)
sin(ϕ2 − ϕ1)
sin(φ2 − ϕ1)
sin(ϕ2 − ϕ1)
. (6)
1If the final state is ±1 CP eigenstate, Ā should include an additional ±1 factor.
This change in the basic set of chosen weak phases should have no physical
implications, hence the name reparametrization invariance. We remind two
main consequences of RpI in the absence of hadronic inputs. For an extensive
discussion see [5]:
1. Consider two basic sets of weak phases {φ1, φ2} and {φ1, ϕ2} with φ2 6=
ϕ2; if an algorithm allows us to write φ2 as a function of physical
observables then, owing to the functional similarity of equation (1) and
(5), we would extract ϕ2 with exactly the same function, leading to
φ2 = ϕ2, in contradiction with the assumptions; then, a priori, the
weak phases in the parametrization of the decay amplitudes have no
physical meaning, or cannot be extracted without hadronic input.
2. If, experimentally, the direct CP asymmetry C = (|A|2 − |Ā|2)/(|A|2+
|Ā|2) is C = 0, then the decay amplitudes can be expressed in terms of a
single weak phase, which could be sensibly extracted, up to discrete am-
bigüities, through the indirect CP asymmetry S = 2 Im(ĀA∗)/(|A|2 +
|Ā|2). Additionally, if the theoretical description of the decay ampli-
tudes only involves a single weak phase from a basic Lagrangian, then
it can be identified with the phase measured through S.
As we will see, this two results apply respectively to the π+π− and π+π0
channels. Essentially, the first one will be operative in the ∆I = 1/2 piece
and the second one in the ∆I = 3/2.
2.2 Removing π0π0 information
To make our point transparent we will start by studying the extraction – in
fact the non-extraction – of α from ππ data when B → π0π0 experimental
information is removed. Let us start with a widely used [2, 3], Standard
Model inspired, parametrization of the decay amplitudes:
A+− ≡ A(B0d → π+π−) = e−iαT+− + P ,√
2A+0 ≡
2A(B+ → π+π0) = e−iα(T+− + T 00) ,
2A00 ≡
2A(B0d → π0π0) ≡
2A+0 −A+− = e−iαT 00 − P ,
Ā+− ≡ A(B̄0d → π+π−) = e+iαT+− + P ,√
2Ā+0 ≡
2A(B− → π−π0) = e+iα(T+− + T 00) ,
2Ā00 ≡
2A(B̄0d → π0π0) ≡
2Ā+0 − Ā+− = e+iαT 00 − P . (7)
When π0π0 experimental information is removed we have two decoupled de-
cays:
1. π+π0 data, i.e. the average branching ratio B+0 and the direct CP
asymmetry C+0, provide, respectively, |T+− + T 00| and a consistency
check C+0 = 0; α is irrelevant there.
2. π+π− data, i.e. B+−, C+− and the mixing induced CP asymmetry
S+−, give information on α decoupled from π+π0, on |T+−|, |P | and
the relative (strong) phase δPT+− between T
+− and P .
With three observables and four parameters everybody knows or suspects
that one cannot really extract α: we have C+− 6= 0, as reminded in section
2.1, α cannot be extracted from B → π+π− in this limited case. One can try,
nevertheless, to obtain a probability distribution function (PDF) for α as in
reference [2]. This PDF, obtained in an analysis with three observables and
four unknowns, has obviously a strong dependence in the priors, as in figure
2 of [2]. Even worse, reparametrization invariance [5] tells us that A+−, Ā+−
can also be written as
A+− = e
−iα′T ′+− + P ′, Ā+− = e
+iα′T ′+− + P ′ , (8)
where α′ is any weak phase – known or unknown, α′ 6= 0, π –. In this scenario
the conclusion is clear: any information one would get for α would also be
valid for any α′ and thus it cannot be assigned to α. This solves the puzzle
raised in the MA and RI parametrizations within figure 4 of reference [2]:
those PDFs cannot be attributable to α. Just with that data alone we cannot
extract α′ – whatever it is –, as we have emphasized in 2.1. To illustrate this
issue we compute the PDFs of figure 1 in the following parametrization:
A+− ≡ A(B0d → π+π−) = e−iα
T+− + P ,
2A+0 ≡
2 A(B+ → π+π0) = e−iα(T+− + T 00) ,
2A00 ≡
2 A(B0d → π0π0) ≡
2A+0 − A+− ,
Ā+− ≡ A(B̄0d → π+π−) = e+iα
T+− + P ,
2Ā+0 ≡
2 A(B− → π−π0) = e+iα(T+− + T 00) ,
2Ā00 ≡
2 A(B̄0d → π0π0) ≡
2Ā+0 − Ā+− .
Notice that just with α = α′, Eq. (9) recovers the parametrization in Eq. (7).
The phase of T+− is set to zero (i.e. all strong phases are relative to arg(T+−))
and flat priors are used for all the parameters2, that is, moduli |T+−|, |T 00|,
2The allowed ranges for the different moduli and the sensitivity to them in this and
other cases will be addressed later, for instance, for this example, they are all limited to
lie in the range [0; 10]× 10−3 ps−1/2.
|P | and phases δP = arg(P ), δ0 = arg(T 00), α and α′. Results in other
parametrizations, being equally illustrative, are relegated to appendix C.
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
0 25 50 75 100 125 150 175
α = α′
(c) Joint (α′, α) PDF
25 50 75 100 125 150 175
(d) α = α′ PDF
Figure 1: PDFs of α and α′ from B → ππ without π0π0 data.
The lesson of this example is rather obvious: the set of observables being
insensitive to α, its PDF is uninformative (just the flat prior in this case);
the PDF in figure 1(d), erroneously identified with α, is nothing else than α′
itself, whatever it could be.
2.3 Including back π0π0 information
When we incorporateB → π0π0 data to the isospin construction, |A00| (|Ā00|)
gives the angle among A+0 (Ā+0) and A+− (Ā+−); using then the known
phase difference between A+− and Ā+−, the angle among A+0 and Ā+0 is
obtained. This is just the isospin analysis giving α. Knowing α, i.e. with α
fixed, A+− = e
−iαT+− + P would have full meaning and {B+−, C+−, S+−}
would fix the three hadronic parameters. Unfortunately the isospin analysis
as explained above yields allowed values for α spanning a wide range. The
degeneracy of solutions together with the experimental errors do not fix α,
just exclude some region. In this situation {B+−, C+−, S+−} do not really fix
the hadronic parameters and, consequently, they tend to generate a spurious
PDF for α as we have seen. The final “α” is thus a sort of convolution of the
α obtained from the isospin analysis and the spurious one “extracted” purely
from π+π− data. This is illustrated with the PDFs of figure 2, making use
of the parametrization in Eq. (9).
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
0 25 50 75 100 125 150 175
α = α′
(c) Joint (α′, α) PDF
25 50 75 100 125 150 175
(d) α = α′ PDF
Figure 2: PDFs of α and α′ from B → ππ.
To stress the importance of this issue we repeat the previous example
while arbitrarily reducing all experimental uncertainties by a common factor
of 5. The PDFs corresponding to this fake scenario are displayed in figure 3.
The results shown in figures 1, 2 and 3 deserve some comment:
1. Figures 1(b) and 2(b) are almost identical; in the former we were not
using B → π0π0 information while in the later we were doing so. This
similarity is a dramatic illustration of the spurious nature of the “ex-
tracted” α′.
2. Figure 2(d) is the cut of the joint PDF in figure 2(c) along the line α =
α′. Therefore the so called MA extraction of α is a sort of convolution
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
25 50 75 100 125 150 175
(c) α = α′ PDF
Figure 3: PDFs of α and α′ from B → ππ with experimental uncertainties
reduced by a factor of 5.
of the Gronau-London α – figure 2(a) – and the spurious one.
3. This α′ PDF basically allows any value of α′ except the neighborhoods
of 0 and π, which are a priori forbidden by S+−, C+− 6= 0: obviously
there is no way to produce CP violation in the π+π− channel without
two weak phases in the amplitude that controls it. The exclusion of
α′ = 0, π is the only physical information one can extract in the SM
from the PDF of α′.
4. The deep in the α distributions around α ∼ π/4, which is transmitted
to the α = α′ PDF, is senseful. The exclusion of α ∼ 0, π is also
physical inside the SM. Nevertheless, how strongly these 0, π regions
are excluded is highly sensitive to the allowed ranges for |T+−|, |T 00|
and |P | – see section 4 –. As we move away form the α = 0, π points, the
final PDF of α would be more influenced by the spurious α′ distribution.
One can see that in the shape of the α distribution for α < 25◦ or
α > 75◦.
5. As uncertainties are reduced, even with α ≡ α′, the valid ranges for
the “real” α emerge, despite the α′ distribution. That is, as experimen-
tal uncertainties are reduced, the α′ “pollution” of α through α ≡ α′
becomes increasingly ineffective, as it should, and just transmits the
physical exclusion of α = 0, π inside the SM.
The main lesson from the previous example is: α is obtained from purely
∆I = 3/2 amplitudes, without additional hadronic input. Including it in
∆I = 1/2 pieces, as reparametrization invariance shows, pollutes the legiti-
mate extraction with information that one cannot claim is concerning α.
3 Standard Model inspired parametrizations
As stated above, following the consequences of reparametrization invariance,
the really legitimate sources of our knowledge on α are A+0, Ā+0. We have
referred to the parametrization in Eq. (7) as a “SM inspired parametrization”
of the amplitudes and we have discussed how the inclusion of α in A+−, Ā+−
is dangerous with present uncertainties. Nevertheless, it is clear that the
exclusion of α ∼ 0, π inside the SM is a valid physical consequence that comes
from having α in A+− and Ā+−. To further illustrate the importance and the
subtlety of this issue let us consider in detail what can be interpreted as a
“SM inspired parametrization”. Once we take into account reparametrization
invariance, we only need3 to focus on A+− and Ā+−:
1. RpI allows us to write {A+−, Ā+−} in terms of any pair of weak phases
{φ1, φ2} (as long as φ1 − φ2 6= 0 mod [π]), nothing enforces the use of
{0, α}.
2. SM compliance of any parametrization only requires that the vanishing
of all the SM phases leads to no CP violation, once again nothing singles
out or requires the use of {0, α}.
Consequently, as we have at our disposal other SM phases that we can choose
to parametrize A+−, Ā+−, namely
4 γ, β, χ, χ′, instead of A+− = e
−iαT+−+P
and Ā+− = e
iαT+− + P , we can for example write, on equal footing,
A+− = M1e
iδ1e−iχ +M2e
iδ2e−iβ, Ā+− = M1e
iδ1e+iχ +M2e
iδ2e+iβ , (10)
A+− = e
−iχT+− + P, Ā+− = e
+iχT+− + P . (11)
Within the SM χ ∼ O(λ2), had we used this last parametrization (Eq. (11)),
we would have found extreme compatibility problems5 that would be absent
with another SM inspired parametrization: this is a dramatic illustration
of the consequences of RpI mentioned in section 2.1. In other words, pre-
tending that one obtains information on SM “theoretical” phases just by
parametrizing A+− and Ā+− with them is in general senseless. In this case
we would have obtained that figure 2(b) is the PDF of the phase χ, the one
that appears in Bs–B̄s mixing [8, 9, 10, 11, 12, 13].
3A+0 and Ā+0 can be parametrized with a single weak phase, identifiable with α, A00
and Ā00 will follow from the isospin relations.
4γ = arg(−VudVcbV ∗ubV ∗cd), β = arg(−VcdVtbV ∗cbV ∗td), χ = arg(−VcbVtsV ∗csV ∗tb) and χ′ =
arg(−VusVcdV ∗udV ∗cs) [6].
5Just look, for example, to the O(λ2) ∼ 2 − 3◦ region of the different α′ PDFs in the
plots of previous sections [7].
4 Physics and parametrical problems
In section 2 we mentioned that the exclusion of the “dangerous” α′ near 0 and
π depended on the allowed ranges for the parameters |T ij| and |P |. Figure 4
shows the PDFs of α, α′ and α = α′ for four different sets of allowed ranges of
|T ij| and |P |. On the one hand, the PDFs of α in figures 4(a), 4(d), 4(g) and
4(j) are quite similar. On the other hand, the PDFs of α′ in figures 4(b), 4(e),
4(h) and 4(k) are completely different: the “dangerous” α′, especially in the
regions close to 0,π, is sensitive to the applied bounds. This is automatically
transmitted to the α = α′ PDF and it is in this way that the region with
“α” close to 0,π is suppressed (even wipped out as in figures 4(c) and 4(i))
through the cuts on the spurious α′, induced by the cuts on |T ij| and |P |.
One could think that this is particular to the bayesian statistical approach,
figure 5 shows the frequentist confidence level curves for α computed under
the same parametric restrictions. As we use the parametrization of Eq. (7),
they correspond to the α = α′ plots in Figure 4. It is rather clear that
without regard to the statistical approach, limiting the values of |T ij| and
|P | has observable effects in the extraction of α. Note that figure 5(c) differs
from figure 5(a) not by a cut but by a change in the shape, even if it is not
a dramatic change.
The authors of reference [2] pointed out that there is some peculiar limit
with α → 0 together with P/T+−, T 00/T+− → −1, |T+−| → ∞ – using the
parametrization of Eq. (7) – that keeps all the observables “in place”: it is
in fact a question of having α′ → 0 rather than α → 0. This peculiar limit
is useful to understand the α ∼ 0, π exclusion above mentioned. To obtain
parameter configurations with high likelihood when α(′) approaches 0 or π,
the required values of |T ij| and |P | are increasingly large. Imposing bounds
on |T ij| and |P | automatically limits how close to 0, π one can push the weak
phase while producing likely branching ratios and asymmetries. The use of
the parametrization in Eq. (9) shows how this works for the dangerous α′
and is then transmitted to α.
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
25 50 75 100 125 150 175
(c) α = α′ PDF
Allowed ranges: |T ij| ∈ [0; 10]× 10−3 ps−1/2, |P | ∈ [0; 2.5]× 10−3 ps−1/2
25 50 75 100 125 150 175
(d) α PDF
25 50 75 100 125 150 175
(e) α′ PDF
25 50 75 100 125 150 175
(f) α = α′ PDF
Allowed ranges: |T ij| ∈ [0; 10]× 10−3 ps−1/2, |P | ∈ [0; 10]× 10−3 ps−1/2
25 50 75 100 125 150 175
(g) α PDF
25 50 75 100 125 150 175
(h) α′ PDF
25 50 75 100 125 150 175
(i) α = α′ PDF
Allowed ranges: |T ij| ∈ [0; 5]× 10−3 ps−1/2, |P | ∈ [0; 1.25]× 10−3 ps−1/2
25 50 75 100 125 150 175
(j) α PDF
25 50 75 100 125 150 175
(k) α′ PDF
25 50 75 100 125 150 175
(l) α = α′ PDF
Allowed ranges: |T ij| ∈ [0; 25]× 10−3 ps−1/2, |P | ∈ [0; 25]× 10−3 ps−1/2
Figure 4: PDFs obtained using the parametrization in Eq. (9) and different
allowed ranges for |T ij| and |P |.
25 50 75 100 125 150 175
(a) |T ij| < 10, |P | < 2.5
25 50 75 100 125 150 175
(b) |T ij | < 10, |P | < 10
25 50 75 100 125 150 175
(c) |T ij | < 5, |P | < 1.25
25 50 75 100 125 150 175
(d) |T ij | < 25, |P | < 25
Figure 5: α CL; as usual |T ij| and |P | in units of 10−3 ps−1/2.
5 The extraction of α from B → ππ and New
Physics
Recently the UTfit collaboration has proposed to add information on the
moduli of the amplitudes in order to extract α inside the SM. In particular, to
add reasonable QCD based cuts on the moduli of T ij and P . Even if we agree
with this procedure, we must stress that the resulting PDF of α – see figures
4(c) or 4(i) – in the non zero region mixes ∆I = 3/2 information with spurious
∆I = 1/2 information. In this case it does not seem dramatic, but it can be so
in the B → ρρ case – see [2] –. In addition, if one is trying to make a general
fit of the SM it is more natural to use the ∆I = 3/2 piece of B → ππ to
get reliable bounds on α and once α is fixed by the general unitarity triangle
analysis, use the ∆I = 1/2 piece of B → ππ to obtain better information
on the hadronic parameters. In fact, the UTfit collaboration presents results
along this line in [3]. This implies our recommendation of using α in the A+0
amplitude and another phase in A+− or in the ∆I = 1/2 piece.
After confronting the SM à la CKM with data, the most important ob-
jective in overconstraining the unitarity triangle is in fact to look for New
Physics (NP) [14, 15, 16, 8, 9, 10, 11, 12, 13]. When there is NP – just in the
mixings or also in the ∆I = 1/2 decay amplitudes6 – it is not appropriate
to use a SM inspired parametrization. In the limit where all SM phases
go to zero, C+− and S+− can still be reproduced by NP loops. So, if we
want to interpret the α PDF as7 ᾱ we have to use a different CP-violating
phase in the ∆I = 1/2 piece or in A+−. Parametrizations that fulfill these
requirements are the so-called PLD, ES, the ’τ ’ parametrization in [2] and
even our SM-like parametrization with α′ in Eq. (9) despite having one more
parameter. A similar one, which additionally factorizes an overall scale of
the amplitudes, is the following, that we call ’1i’:
A+− ≡ e−iαT3/2(T + iP ),
2A00 ≡ e−iαT3/2(1− T − iP ),√
2A+0 ≡ e−iαT3/2,
2Ā+0 ≡ e+iαT3/2,
Ā+− ≡ e+iαT3/2(T − iP ),
2Ā00 ≡ e+iαT3/2(1− T + iP ).
Notice that a global weak phase in A+− is irrelevant in C
+− and amounts to
a global shift of arg(Ā+−A
In this section we will “extract” α in a bayesian approach making use of
different parametrizations; we will show the consistency of all those results
6With great accuracy – up to small electroweak penguins – this case corresponds to
having NP everywhere except in tree level amplitudes.
7Where ᾱ = π − β̄ − γ, β̄ = β − φd and the NP phase in B0d–B̄0d mixing is defined by
= r2de
−i2φd [Md
]SM .
and then compare to frequentist results. From a fundamental point of view,
as stressed in previous sections, we are not willing to use information beside
assuming the triangular isospin relations, the single “tree level” weak phase of
the ∆I = 3/2 piece and experimental results themselves. Reparametrization
invariance and the presence of a single weak phase, α, in the ∆I = 3/2
amplitudes A+0 and Ā+0 imply that all the results to be presented in this
section will be valid in the presence of New Physics in loops.
Figure 6 shows the PDF of α in three different cases: the ’PLD’ [17] and
’1i’ (Eq. (12)) parametrizations, and the explicit extraction (as in [17] or [5]).
Corresponding 68%, 90% and 95% probability regions are displayed in table
1, together with the frequentist 68%, 90% and 95% CL regions (in the fol-
lowing, frequentist calculations are carried with the ’PLD’ parametrization).
These regions are represented in figure 7. Despite some small differences in
the 68% regions, somehow expectable as they are more sensitive to details,
the results are consistent, they coincide rather well. B → ππ data are still
too uncertain to really provide important constraints on α, the only relevant
feature being the exclusion of the α ∼ π/4 region, which could be understood
(see section C.1 in appendix C) in terms of the smallness of B00.
25 50 75 100 125 150 175
(a) PLD parametrization
25 50 75 100 125 150 175
(b) 1i parametrization
25 50 75 100 125 150 175
(c) Explicit extraction
Figure 6: α PDFs.
68% 90% 95%
PLD [0; 5]◦ ∪ [85; 101]◦∪ [0; 8]◦ ∪ [82; 107]◦∪ [0; 9]◦ ∪ [82; 110]◦
[121; 150]◦ ∪ [168; 180]◦ [114; 157]◦ ∪ [162; 180]◦ ∪[113; 180]◦
1i [95; 174]◦ [0; 1]◦ ∪ [89; 180]◦ [0; 5]◦ ∪ [85; 180]◦
[2; 8]◦ ∪ [82; 88]◦∪ [0; 9]◦ ∪ [81; 91]◦
Explicit [100; 120]◦ ∪ [125; 145]◦∪ [95; 175]◦ ∪ [179; 180]◦ [0; 10]◦ ∪ [80; 180]◦
[150; 170]◦
CL [0; 7]◦ ∪ [83; 104]◦ [0; 12]◦ ∪ [78; 180]◦ [0; 14]◦ ∪ [76; 180]◦
[115; 154]◦ ∪ [166; 180]◦
Table 1: α regions within [0; 180◦].
25 50 75 100 125 150 175
Figure 7: α regions (the ordering, top to bottom, is in each case: ’PLD’ pa-
rametrization, ’1i’ parametrization, Explicit extraction and frequentist anal-
ysis).
Conclusions
To our knowledge the discrepancies between frequentist and bayesian ap-
proaches using the so-called MA and RI parametrizations with Eq. (7) have
not been previously understood. We explain that with present experimen-
tal uncertainties it is extremely unsecure to introduce the phase α in the
∆I = 1/2 piece. To a great extent a spurious PDF of α tends to be gener-
ated. The Gronau and London analysis is critically based on the appearance
of one weak phase in the ∆I = 3/2 piece (C+− = 0). Introducing α in the
∆I = 1/2 piece – or A+− – (C
+− 6= 0) brings this “second” α to the category
of ’not observable’ even if one is using a Standard Model inspired parametri-
zation. This difficulty is operative in the so-called MA and RI parametriza-
tions. The introduction of α in the ∆I = 1/2 piece and some QCD-based
bounds on the amplitudes allows – as done by the UTfit collaboration – to
eliminate the solutions around α ∼ 0, π inside the SM. The PDF can still be
partially contaminated with the spurious α distribution. In B → ππ it is not
dramatic but it could be so in other channels. This last procedure cannot be
applied to an analysis with NP in loops. Therefore, we strongly recommend
to use parametrizations where α is just included in the ∆I = 3/2 piece. We
partially agree with the UTfit collaboration that, in spite of the differences
among the frequentist and bayesian methods, both approaches give similar
results if one uses parametrizations with a clear physical meaning. In this
sense the most relevant result is the exclusion of the region ᾱ ∼ 25◦ − 75◦.
Acknowledgments
This research has been supported by European FEDER, Spanish MEC under
grant FPA 2005-01678, Generalitat Valenciana under GVACOMP 2007-172,
by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the
projects PDCT/FP/63912/2005, PDCT/FP/63914/2005, CFTP-FCT UNIT
777, and by the Marie Curie RTN MRTN-CT-2006-035505. M.N. acknowl-
edges financial support from FCT. The authors thank J. Bernabéu and P.
Paradisi for reading the manuscript and useful comments.
A Inputs and numerical methods
Along this work we use the set of experimental measurements [18, 19, 20, 21,
22,23,24,25,26,27,28], combined by the Heavy Flavour Averaging Group [29],
in table 2.
B+−ππ B
5.2± 0.2 1.31± 0.21 5.7± 0.4
C+−ππ S
−0.39± 0.07 −0.59± 0.09 −0.37± 0.32
Table 2: Experimental results, branching ratios are multiplied by 10−6.
In terms of B → ππ amplitudes,
Bij = τBi+j
|Aij|2 + |Āij |2
, C ij =
|Aij|2 − |Āij|2
|Aij|2 + |Āij|2
, Sij =
2 Im(ĀijAij
|Aij|2 + |Āij |2
All frequentist CL computations are performed by: (1) minimizing χ2 with
respect to all parameters except the one of interest which is fixed (in this case
α), (2) computing the corresponding CL through an incomplete Γ function.
All bayesian PDFs are computed using especially adapted Markov Chain
MonteCarlo techniques.
B Experimental results and isospin relations
The isospin relations
A+− +
2A00 =
2A+0 ,
Ā+− +
2Ā00 =
2Ā+0 , (14)
define two triangles in the complex plane whose relative orientation fixes α.
The sizes of the different sides follow from Eq. (14).
|A+−|
2|A00|
2|A+0| |Ā+−|
2|Ā00|
2|Ā+0|
1.441 1.040 2.634 2.176 1.533 2.634
Table 3: Numerical values of the sides of the isospin triangles computed with
experimental central values, to be multiplied by 10−3 ps−1/2.
This allows the reconstruction, up to a number of discrete ambigüities -
namely up to eight -, of both triangles. Central values of present measure-
ments yield the values of the sides in table 3. One straightforward question
is mandatory: do those would-be triangles “close”? The answer is in the
negative because
|A+−|+
2|A00| = 2.481 ≯ 2.634 =
2|A+0| ,
|Ā+−|+
2|Ā00| = 3.709 > 2.634 =
2|Ā+0| .
In fact, for those central values, the first triangle is not a triangle [30]. In
terms of likelihood, the closest configuration to that situation, the most likely
one, is having the first triangle flat, a feature which naturally explains the
reduced – by a factor of two, from eight to four – degeneracy of α “solutions”.
That is, while for old data the almost flatness of this same isospin triangle
yielded eight different solutions distributed in four almost-degenerate pairs,
those pairs are now degenerate and rather than exact solutions for the central
values of the observables they produce best-fitting points.
Consequently, the use of explicit solution constructions requires the rejec-
tion of the joint regions of experimental input incompatible with the isospin
relations Eqs. (14). For old data, this meant rejecting some 48.2% of allowed
experimental input (weighting each observable with a gaussian with mean
and standard deviation given by the corresponding central value and uncer-
tainty), for the new data set this rejection rate is 70.9%. In the bayesian
and frequentist treatments the isospin relations are assumed valid and all
the subsequent analyses are “normalized” to that assumption.
C Removing B → π0π0 information
C.1 Explicit extraction of α
This appendix is devoted to some complementary results extending what is
presented in section 2.2. The first issue we will address is the explicit extrac-
tion8 of α when B → π0π0 information is removed, that is, no knowledge of
B00 and C00. The explicit extraction of α assumes the isospin relations in
Eqs. (14) so to start with, the ignorance on B00 is not ”just plain ignorance”
(whatever this could stand for) as it will operatively mean that for any ex-
perimental set of results {B+−, B+0, C+−, S+−}, B00 and C00 should be such
8Beside the explicit formula for α in terms of the available observables presented in
reference [17] we also make use of the extraction of α explained in [5]; the results are com-
pletely equivalent, however the later does not make any use of a particular parametrization
of the amplitudes and is easily interpreted in terms of the isospin construction.
that both would-be isospin triangles are in fact isospin triangles. C00 is obvi-
ously restricted to be in the range [−1; 1]; what about B00? One could argue
that if there is no information on B → π0π0 it should be smaller than a given
bound or one can just let it be as large as allowed by other data and isospin
constraints. This rather trivial fact is apparently at the origin of the discrep-
ancy in the results presented in references [2,3] for the explicit extraction of
α “without” B → π0π0 information: figure 8 shows two PDFs of α. They
are obtained by generating known experimental sets {B+−, B+0, C+−, S+−}
according to gaussian distributions with central values and standard devia-
tions given by the quoted measurements and uncertainties (C+− and S+− are
also restricted to be within [−1; 1]), then C00 and B00 are generated through
flat distributions, C00 in the range [−1; 1] and B00 in a range [0;B00Max]. Sets
{B+−, B+0, C+−, S+−, B00, C00} which fulfill the isospin relations Eqs. (14)
are retained and used to extract α. The PDFs of α represented in figure 8
only differ in the value of B00Max, Fig. 8(a) was obtained with B
Max equal to
two times the present measurement while Fig. 8(b) was obtained with B00Max
equal to twenty times the present measurement. On the one hand, the PDF
in figure 8(a) coincides with the one presented in figure 4 ’ES’ of reference [2];
on the other hand the PDF in figure 8(b) agrees, more or less, with figure
4 of reference [3]. It is now clear that the difference among both may be
just due to the numerical procedure. Figure 8(b) shows that the removal of
B → π0π0 information leads to a loss of knowledge on α. Ironically, there is
a lesson in this example: numerics apart, the smallness of B00 is responsible
for the exclusion of values α ∼ π/4.
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α PDF
Figure 8: Explicit extraction without B → π0π0; lighter curves correspond
to the different individual contributions related by the discrete ambigüities.
C.2 Parametrizations
To complete the picture we now proceed to repeat the extraction of α when
B → π0π0 information is removed in several parametrizations. We will make
use of the ’PLD’ parametrization [17], of the ’1i’ parametrization with fixed
weak phases in {A+−, Ā+−} (Eq. (12)) and, finally, of the parametrization in
Eq. (9) but in this case, apart from α and α′, instead of moduli and phases we
will use real and imaginary parts of T+−, P and T 00 (the RI parametrization
in reference [2]). The PDFs of α obtained for the first two parametrizations
are shown in figure 9, they are eloquent: no knowledge on α.
25 50 75 100 125 150 175
(a) α PDF, PLD parametrization
25 50 75 100 125 150 175
(b) α PDF, 1i parametrization
Figure 9: Extraction without B → π0π0.
For the RI parametrization we show the PDFs of α, α′ and the one
obtained by setting α = α′ in figure 10. Once again it is clear that there
is no information on α and that inappropriately insisting on including it in
{A+−, Ā+−} produces the senseless result of figure 10(c).
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
25 50 75 100 125 150 175
(c) α = α′ PDF
Figure 10: Extraction without B → π0π0, RI parametrization.
The conclusion of this appendix is straightforward: just dealing with
a reduced scenario in which B → π0π0 information is removed, a proper
understanding of the subtleties involved in the parametrization of B → ππ
amplitudes avoids peculiar results as for instance the ’MA’ and ’RI’ ones
included in figure 4 of reference [2]. We have shown here that starting with a
flat prior for α consistently gives highly non-informative posteriors in several
sensible parametrizations.
D Using the RI parametrization
In section 2.3 we used the parametrization in Eq. (9) to obtain figure 2 with
flat |T+−|, |P |, |T 00|, arg(P ), arg(T 00), α and α′ priors. For completness
we also show – figure 11 – the PDFs of α, α′ and α = α′ in case one uses
flat |T+−|, Re [P ], Im [P ], Re [T 00], Im [T 00], α and α′ priors. Beside the
effect of the spurious α′ in the PDF of α = α′, we can also appreciate the
influence of the change in the priors: the integration domain is the same as in
figure 2 but the integration measure is now different. The main effect is the
relative enhancement of the contributions from regions with large parameters,
including the contributions from the α′ → 0 driven region.
25 50 75 100 125 150 175
(a) α PDF
25 50 75 100 125 150 175
(b) α′ PDF
25 50 75 100 125 150 175
(c) α = α′ PDF
Figure 11: α extraction, RI parametrization.
E One short statistical comment
Leaving completely aside philosophical aspects of probability, both frequen-
tist and bayesian approaches start with a common likelihood function. Each
approach reduces the information provided by the likelihood function in a
different manner. Consequently, they do not yield strictly coincident results:
• Bayesian posteriors obviously depend on the priors, for example the
allowed ranges or the shape. As we have seen, we obtain different
posteriors with different priors. However, as long as one is using sen-
sible parametrizations and reasonable priors, we end up finding rather
compatible results.
• Frequentist CL curves do depend on the parametrization, to be precise,
they depend on the allowed ranges for the parameters; once sensible
parametrizations and adequate ranges are used, CL curves obtained
with them are identical. The α → 0 limit in the SM inspired parame-
trization of Eq. (7) illustrates this issue.
Beside those well known issues, we may find troublesome that:
1. Most probable values in the bayesian PDFs do not coincide with the
analytical solutions for α.
2. Intimately related to this aspect, bayesian PDFs seem unable to dis-
tinguish among degenerate solutions.
We remind that these statements concern one dimensional PDFs of α. Fre-
quentist one dimensional CL curves distinguish α solutions because they are
obtained through best fitting points for fixed α. Bayesian PDFs do not dis-
tinguish them as the uncertainties produce distributions for the degenerate
solutions which overlap and add up in the complete PDF. One can still have
a hint of the proximity of different solutions from this kind of overlap, but
this is not the point here. For reduced experimental uncertainties, bayesian
PDFs would not overlap and would distinguish among those different solu-
tions. This could be sufficient to think that, per se, there is no discriminating
advantage in using one or the other approach. With present uncertainties,
bayesian analyses seem incapable of pinning down the right location of the
solutions in α and telling us something about their degeneracy. It is not a
fundamental problem of bayesian methods as reduced uncertainties would
overcome these “difficulties”. If it is not a fundamental problem, could we
somehow overcome these “difficulties” with present uncertainties? The an-
swer is in the positive as the problem only arises because we are insisting
in the reduction of the available experimental information to obtain one-
dimensional PDFs of α; let us take a look to the joint PDFs in figure 12.
These are the joint PDFs of (δ, α) and (αeff , α) obtained with the ’PLD’
parametrization. They are quite illustrative, one can see the different solu-
tions in α concentrated around the values of α dictated by the analytical
expectations. The pretended fundamental drawbacks of bayesian methods
to adequately place and distinguish the solutions are just a consequence of
pushing too far, for the present level of experimental uncertainty in the re-
sults, the statistical “reduction of information process”. A simultaneous look
to both frequentist and bayesian results will not put an end to the statistical
discrepancies, notwithstanding it will be very helpful to understand the phys-
ical results we are interested in. Both approaches are “information reduction
processes” and strictly sticking to one and deprecating the other may not be
the wiser strategy.
0 25 50 75 100 125 150 175
(a) Joint (δ, α) PDF
0 25 50 75 100 125 150 175
(b) Joint (αeff , α) PDF
Figure 12: Joint PDFs obtained with the ’PLD’ parametrization.
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	Introduction
	Reparametrization invariance and bold0mu mumu BBBBBB
	Weak Phases
	Removing bold0mu mumu 000000000000 information
	Including back bold0mu mumu 000000000000 information
	Standard Model inspired parametrizations
	Physics and parametrical problems
	The extraction of bold0mu mumu  from bold0mu mumu BBBBBB and New Physics
	Inputs and numerical methods
	Experimental results and isospin relations
	Removing bold0mu mumu B00B00B00B00B00B00 information
	Explicit extraction of bold0mu mumu 
	Parametrizations
	Using the RI parametrization
	One short statistical comment
