PROJECT X AND THE LHC
JOSEPH LYKKEN
FERMI NATIONAL ACCELERATOR LABORATORY
PROJECT X PHYSICS WORKSHOP, FERMILAB, 16-17 NOVEMBER 2007
SCENARIOS FOR PRECISION PHYSICS WITH
PROJECT X IN THE “LHC ERA”
➡
SCENARIOS WITH SUPERSYMMETRY AND CHARGED
LEPTON FLAVOR VIOLATION
➡
KAON PHYSICS AND MINIMAL FLAVOR VIOLATION
➡
ADD YOUR OWN EXAMPLES HERE!
SCENARIOS WITH SUPERSYMMETRY AND
CHARGED LEPTON FLAVOR VIOLATION
➡
A neutrino see-saw implies larger, perhaps order 1, neutrino
Yukawa couplings up at some high energy scale
➡
If there is also supersymmetry, this implies that the slepton soft
mass matrix will acquire CLFV entries from the renormalization
group running down from the high scale
➡
These in turn will induce CLFV decays at low energies via loop
effects
FIG. 1: The diagrams contributing to µ → e, γ decays
Irrespective of the source, LFV at the weak scale can be parametrised in a model-independent
l
SCENARIOS WITH SUPERSYMMETRY AND
CHARGED LEPTON FLAVOR VIOLATION
➡
If supersymmetry is discovered by CMS and ATLAS, one of the
most compelling challenges will be to connect this discovery to
neutrino physics via charged lepton flavor violation
➡
This will (most probably) be our best chance to explore unification
scale physics in the next decade (i.e. in the pre- “ILC Era”)
SCENARIOS WITH SUPERSYMMETRY AND
CHARGED LEPTON FLAVOR VIOLATION
➡
One immediate difficulty is that we don’t know the neutrino
Yukawas matrices at the high scale
➡
One can look at two extremes: where the mixing seen by the
sleptons is small, CKM-like
➡
Or the mixing is large, PMNS-like
➡
Then one can scan over SUSY models, computing the rates
for µ → e γ
τ → µγ
µ→e
Super B reach is interesting but not comprehensive
τ → µ γ at tan β = 10
10
se
00
PMNS-case
CKM-case
1
Now
Now
SuperB
BR(τ → µ γ) · 107
0.1
SuperF
0.01
0.001
1e-04
1e-05
1e-06
1600
1e-07
0
200
400
600
800
1000
1200
1400
1600
M1/2 (GeV)
100
se
τ → µ γ at tan β = 40
L. Calibbi, A. Faccia, A. Masiero, S. Vempati, hep-ph/0605139
PMNS-case
CKM-case
10
µ → e in Ti at tan β = 10
1000
PMNS
CKM
100
10
CR(µ → e) · 1012 in Ti
n SUSY–
amplitude
esponsible
a strong
→ e conth respect
1
0.1
0.01
ome from
0.001
ve bounds
tance, the
1e-04
−12
) is al1e-05
n µ → eγ
1e-06
parameter
1e-07
EG sensi0
200
scenarios
a µ → e
10000
d a sensi-Project X reach
1000
intensity
100
that proe emitted
10
he µ → e
1
PARC ex-
012 in Ti
Now
PRIME
400
600
800
1000
1200
1400
1600
M1/2 (GeV)
µ → e in Ti at tan β = 40
PMNS
CKM
L. Calibbi, A. Faccia, A. Masiero, S. Vempati, hep-ph/0605139
Now
CONNECTING CLFV TO SUSY
➡
Suppose that you observe CLFV in the MEG experiment, in a
Project X experiment, and at a Super Flavor factory
➡
What can we learn about supersymmetry from these
measurements?
➡
Can we make a direct connection between CLFV and its SUSY
origins?
amplitude depends on several supersymmetric masses. On the other hand,
in the mass insertion approximation is easily seen that all four coefficients
ore, it is necessary
to observe to
at the
least
two of theseoff-diagonal
processes and
to and consequently
are proportional
corresponding
element
or possible correlations
them.
T ∝ (m2L )among
ij .
cause it provides The
the strongest
constraint
µ-e flavor
violation,
we from
branching
ratio of theondecay
µ →CLFV
eγ
is obtained
(3),
CONNECTING
TO
SUSY
se the decay µ → eγ as the reference process to correlate with. Other
#$ $
$ R3e,
$2 %
48π 3τα →
processes that might be related BR(µ
to µ →
eγ
include
µγ,
L $2 µ →
$
$
→ eγ) =
A2 + A2 $ .
(4)
2
G
-e conversiontherefore,
in nuclei. it is necessary to observe
F
at least two of these processes and to
∗
µ →fore #γj →
→ µgives
γ come
and
from independent
e off-shell amplitude
#i γ τ, that
a significant
contribution parts Rof theL
It
depends
only
on
the
A
coefficients,
which
fulfill the relation A2 $ A2 .
2
look for
possible
correlations
among
them.
ide class of lepton
flavor
violating
processes,
may
be
written
asThey have the same loop
slepton
mass
matrix
τ
→
µγ
is
a
decay
analogous
to
µ
→
eγ.
it provides the strongest constraint
on µ-e flavor violation, we
! 2 Because
"
α∗
L but are
R induced by different
β
L off-diagonal
R
structure
elements
in the left-handed
e$ ūi(p − q) will
q γαuse
(A1 P
A1 PR )µ+→
mj iσ
q (A
uj (p) to
L +decay
αβas
L + A2 PR ) process
2 Preference
the
eγ
the
correlate with. Other
slepton mass matrix, (m2L )21 for µ → eγ and (m2L )32 for
τ
→
µγ.
Since these
(3)
LFV
processes
that
might
be
related
to
µ
→
eγ
include
τ → µγ, µ → 3e,
two quantities
in principle
correlations
q is the momentum
of the are
photon,
e is theindependent,
electric charge,
$∗ the between τ → µγ
and
µ-e
inwave
nuclei.
and
µ →conversion
canunot
be
used
to get information
aboutand
the msugra parameter
n polarization
vector,
ueγ
functions
of the initial
i and
j the
∗
The
off-shell
amplitude
for
#jrelated
→
#i γ in
,expressions
that
gives
a significant
contribution
we
must
for
other
processes
induced
by
the same
µ →To
e γthisand
µ→
e #search
eptons, p is thespace.
momentum
ofend,
the
particle
, and
explicit
the underlying
SUSY
model
jare
off-diagonal
element.
That each
is, other
µ-e transitions.
a be
wide
class
flavor
violating
processes,
e coefficients to
can
found
in of
[7].lepton
Since
coefficient
receives
con- may be written as
Being constant over the whole parameter space, this ratio cannot be used to
The decay µ → !3e
receives
contributionsloops,
from penguin-type
"
ions from both chargino-sneutrino
and
neutralino-slepton
the β L diagrams
α∗
2
L constrain it.
R
R
=several
e$ ūbox-type
q γα (A
+On
A1out,
PR )other
+nuclei
mhand,
q (A
PL + A2 from
PisR )penguin
uj (p)and box
and
from
ItLµ-e
turns
however,
that
the2contributions
amplitude
i (p − q) diagrams.
j iσ
αβ
1P
conversion
in
also
receives
tude depends T
on
supersymmetric
masses.
the
diagrams,
the
γ-penguin the
diagram
dominates
over a large portion
by a penguin-type
involving
same
combination
(3)of the
mass insertiondominated
approximation
is easily seencontribution
that alland
four
coefficients
L
R
parameter
space.
In that region, the rate of µ-e conversion in nuclei
is given
∗
of Acorresponding
enters into element
µ
→the
eγ.and
Indeed,
oportional towhere
the
off-diagonal
consequently
q isAthe
momentum
of
photon,
e
is
the
electric
charge,
$
the
2 and
2 that
by
Thus measuring both probes/constrains
&
'the SUSY model
m2L )ij .
photon polarization
vector,α ui16and umj µthe 14
wave functions
of the initial and
BR(µ → 3e)
Γ(µ →
e)7 × 10−3 .
≈
log
−
=
(5)
e branching final
ratio of
the decay
→
eγmomentum
is obtained
from
(3),=particle
e)
leptons,
p isµ →
the
of→
the
#
,
and
explicit
expressions (6)
BR(µ
eγ)
8π 3 R(µ
2m
9
j
e
Γcapt
4
2 5
$ L $be
$ R $2 % in [7]. Since
"
4α5 Zef
mµ ! L
for the coefficients
found
each
coefficient
con48π 3 α #can
2
f Z|F (q)|
Rreceives
2
R
L 2
$
$
$
$
"
|A
−
A
|
+
|A
−
A
|
(7)
BR(µ → eγ) =
A2 + A2
.
(4)
1
2
1
2
3
2
Γ
capt
tributions fromG both chargino-sneutrino and neutralino-slepton loops, the
➡
➡
➡
F
depends
on several
supersymmetric
ef f On the other hand,
R
Lmasses.
ends only onamplitude
the A2 coefficients,
which
fulfill the
the1s
relation
A
2 2 $ A2 .
state, F (q ) is the nuclear form factor, and Γcapt is the total capture
in theanalogous
mass insertion
islosseasily
seen we
that
all four
coefficients
→ µγ is a decay
to µ → approximation
eγ. They
have
the
same
loop
rate. Without
of generality,
will limit
our following
discussion on
48
µ-e conversion
rates
to the nucleus
[7, 8],
Zef f = 17.6, F (q 2 "
are proportional
to the corresponding
off-diagonal
element
and
consequently
ure but are induced
by different off-diagonal
elements
in the
left-handed
22 T i. Then
2
−m
"→
0.54,µγ.
andSince
Γcapt =these
2.59 × 106 s−1 .
2 2
2
µ) τ
n mass matrix,
(m(m
)
for
µ
→
eγ
and
(m
)
for
T ∝
)
.
21
32
L L ij
L
Owing to the additional dependence of R(µ T i → e T i) on the AL,R
1
uantities are in principle
independent,
between
τ →
The branching
ratio correlations
of the decay
µ → eγ
is µγ
obtained from (3),
where Z denote the proton number, Z
is the effective charge of the muon in
4
2 5
"
4α5 Zef
Z|F
(q)|
mµ ! L
f
R 2
R
L 2
"
|A1 − A2 | + |A1 − A2 |
Γcapt
denote the proton number, Zef f is the effective charge of the muo
tate, F (q 2) is the nuclear form factor, and Γcapt is the total cap
Without loss of generality, we will limit our following discussio
version rates to the nucleus 48
22 T i. Then [7, 8], Zef f = 17.6, F (
" 0.54, and Γcapt = 2.59 × 106 s−1 .
➡ Thus suppose we can measure (with pretty good precision) the
ng to the ratio
additional dependence of R(µ T i → e T i) on the
des, the ratio
BR(µ → eγ)
≡C
R(µ T i → e T i)
xpected to be constant. Its value might contain useful informa
upersymmetric parameters. In fact, since both BR(µ → eγ)
→ e T i) are proportional to |(m2L )21 |2 , C does not depend o
is determined exclusively by msugra parameters.
have thus identified µ-e conversion in a nucleus as a process th
220
µ>0
µ<0
200
C
180
160
140
120
200
300
400
500
600
700
m0 (GeV)
800
900
1000
Figure 1: Scatter plot of C as a function of m0 for µ > 0 and µ < 0.
AR
2
AL1
C. Yaguna, hep-ph/0502014
in the relative sign between
and
when µ changes sign. As expected
from these considerations, C(µ < 0) is always larger than C(µ > 0), and the
scan
over
mSUGRA
models
gap between them
is small
because
one of the two
-|AR
2 |- is much larger than
the other.
note C is always between ~ 120 and 220
Since practically any given C is compatible with all possible values of
m0 and M1/2 , C cannot constrain them. C does determine the sign of µ,
however. For instance, C ∼ 180 implies µ < 0 whereas C ∼ 150 implies
µ > 0.
More interesting is the correlation of C with tan β (see figure 3). The
220
µ>0
µ<0
200
C
180
160
140
120
5
10
15
20
25
30
tan !
35
40
45
50
Figure 3: Scatter plot of C as a function of tan β for µ > 0 and µ < 0.
C. Yaguna, hep-ph/0502014
4
Conclusion
sensitivity to both tan β and the sign of µ
We have shown that in msugra models the values of BR(µ → eγ) and the
µ-e conversion rate in a nucleus determine the sign of µ and constrain tan β
practically in a model independent way. In fact, this result holds as long
as the dominant source of lepton flavor violation resides in the left-handed
slepton mass matrix. In particular, it is valid, independently of the value of
the off-diagonal elements, in all models with seesaw induced neutrino masses.
10-8 %1 = -1.7
BR (µ ! e ")10
mN = (1010, 1011, 1014 ) GeV
A0 = 0, tan # = 50
10-12
CR (µ - e, Ti)
10-13
10
-12
10-13
10-10
$13 = 10°, $i = 0
%1 = -1.8
250 GeV < MSUSY < 1000 GeV
%1 = -1.7
%1 =-9 -1.6
10-8
%110= 0.
-11
10
10-15
250 GeV < MSUSY < 1000 GeV
10-13 -11
10
10-7
10-10
10-8
%1 = -1.8
%1 = -1.7
%1 = -1.6
%1 = 0.
10-15
10
mN = (10-16
, 1011, 1014 ) GeV
10
A0 = 0, tan # = 50
10-14
$13 = 1°, -17
$i = 0
A0 = 0, tan # = 50
250 GeV < MSUSY < 1000 GeV $13 = 0.2°, $i = 0
-11
BR (µ ! e ")
10-10
10-11
10-16
10-17
A0 = 0, tan # = 50
10
250 GeV < MSUSY < 1000 GeV
250 GeV < MSUSY < 1000 GeV
10-1210
%1 = -1.8
%1 = -1.7
%1 = -1.6
%1 = 0.
mN = (1010, 1011, 1014 ) GeV
mN = (1010, 1011, 1014 ) GeV
$13 = 1°, $i = 0
-15
-12
1010
10-9
BR (µ ! e ")
-13
%1 = -1.8
%1 = -1.7
%1 = -1.6
%1 = 0.
10-11
mN = (1010, 1011, 1014 ) GeV
A0 = 0, tan # = 50
$13 = 5°, $i = 0
10-12
BR (µ ! e ")
10-14
10-15
10-7
%1 = -1.6
%1 = 0.
CR (µ - e, Ti)
-12
CR (µ - e, Ti)
CR (µ - e, Ti)
10-9
10-11
10
CR (µ - e, Ti)
10-10
CR (µ - e, Ti)
%1 = -1.8
%1 = -1.7
%1 = -1.6
%1 = 0.
10-10
10-18
10-9
10-15
10-10
10-14
10
10-9
-13
10-18
10-12
BR (µ ! e ")
10-15
BR (µ ! e ")
Figure 10: CR(µ − e, Ti) versus BR(µ → eγ) for 250 GeV ≤ MSUSY ≤ 1000 GeV, and δ1 =
−1.8, −1.7, −1.6, 0 (crosses, triangles, asterisks, dots, respectively). We set δ2 = 0, and take mNi =
(1010 , 1011 , 1014 ) GeV, A0 = 0, tan β = 50 and R = 1 (θi = 0). From left to right and top to bottom, the
panels are associated with θ13 = 10◦ , 5◦ , 1◦ and 0.2◦ . In each case, the horizontal and vertical dashed (dotted) lines denote the present experimental bounds (future sensitivities) for CR(µ − e, Ti) and BR(µ → eγ),
respectively.
10
11
14
Figure 10: CR(µ − e, Ti) versus BR(µ → eγ) for 250 GeV ≤ M
−1.8, −1.7, −1.6, 0 (crosses, triangles, asterisks, dots, respectively). W
(10 , 10 , 10 ) GeV, E.
A0Arganda,
= 0, tan
= 50 A.
and
R = arXiv:0707.2955
1 (θi = 0). From lef
M-JβHerrero,
Teixeira,
panels are associated with θ13 = 10◦ , 5◦ , 1◦ and 0.2◦ . In each case, the ho
Considering
larger
values of MSUSY and
identical
intervals known
for δ1,2 leadsfrom
to somewhat
if basic
SUSY
parameters
are
already
ted)
lines
denote
the
present
experimental
bounds
(future
sensitivities) fo
similar results: one finds the same pattern of clusters departing from the constant value of
LHC,
can constrain
other
high
scale
parameters
respectively.
the universal
case, but the maximum
value of
CR(µ−e,
Ti)/BR(µ
→ eγ) is in general smaller
than the 0.04 obtained in the scan of Fig. 11. The reason why this ratio is not improved at
larger values of MSUSY than 1 TeV is because the acceptable solutions producing the proper
SU(2) × U(1) breaking do not lead to sufficiently light Higgs bosons.
more dedicated and refined study.
CR (µ - e, Nuclei)
10-10
Sb
Sr
Ti
Au
Pb
Al
10-11
CRexp(Ti)
10-12
10-13
exp
CR
(Au)
mN = (1010, 1011, 1014 ) GeV
A0 = 0, tan ! = 50
"13 = 5°, "i = 0
#1 = -1.8, #2 = 0
300
400
500
600
700
800
900
M0 = M1/2 (GeV)
Figure 9: µ − e conversion rates for various nuclei as a function of M0 = M1/2 in the NUHM-seesaw. We
E. Arganda, M-J Herrero, A. Teixeira, arXiv:0707.2955
display the theoretical predictions for Sb, Sr, Ti, Au, Pb and Al nuclei (diamonds, triangles, dots, asterisks,
times and crosses, respectively). We have taken mNi = (1010 , 1011 , 1014 ) GeV, A0 = 0, tan β = 50, θ13 = 5◦
and R = 1 (θi = 0). The non-universality parameters are set to δ1 = −1.8 and δ2 = 0. From top to bottom,
the horizontal dashed lines denote the present experimental bounds for CR(µ − e, Ti) and CR(µ − e, Au).
using other nuclei gives added handles
26
(SLIDE FROM MY PREVIOUS TALK)
IMPLICATIONS OF AN ALMOST-MFV WORLD
➡
Measuring small deviations from MFV will be of great importance,
since these can tell us about things like (i) the SUSY breaking
scale, (ii) flavor symmetries related to unification, (iii)
compositeness, extra dimensions, etc.
➡
Such measurements will be directly complementary to the central
physics program of the LHC
➡
Future quark flavor experiments had better focus on modes with
small theoretical uncertainties, and had better achieve
experimental precision comparable to those small theory errors
➡
More $ for lattice gauge theory!
tainties which affect non-leptonic process both in B and K decays. This implies
that there is still much room in the new-physics parameter space which can only
be explored by means of KL → π 0 ν ν̄.
• One of the most popular (and well motivated) scenarios about the flavor structure
of physics beyond the SM is the so-called Minimal Flavor Violation (MFV) hypothesis [29, 30]. Within this framework (which can be regarded as the most pessimistic
case for new-physics effects in rare decays), flavor- and CP-violating interactions
are induced only by terms proportional to the SM Yukawa couplings. This implies
that deviations from the SM in FCNC amplitudes rarely exceed the O(20%) level,
or the level of irreducible theoretical errors in most of the presently available observables, although model independently effects of order 50% cannot be excluded
at present [31]. Moreover, theoretically clean quantities such as aCP (B → J/ΨKS )
and ∆MBd /∆MBs , which measure only ratios of FCNC amplitudes, turn out to be
insensitive to new-physics effects. Within this framework, the need for additional
clean and precise information on FCNC transitions is therefore even more important. A precise measurement of B(KL → π 0 ν ν̄) would offer a unique opportunity
in this respect.
3.2
General parameterization and phenomenological considerD. Bryman, A. Buras, G. Isidori, L. Littenberg, hep-ph/0505171
ations
An important consequence of the first item in the above list, is the fact that in most
SM extensions the new physics contributions in K + → π + ν ν̄ and KL → π 0 ν ν̄ can be
parameterized in a model-independent manner by just two parameters, the magnitude
2
and the phase of the Wilson coefficient of the operator Qνν
sd in Eq. (2.1). More explicitly,
we can encode all the new-physics effects around and above the electroweak scale into
KAON PHYSICS AND
MINIMAL FLAVOR VIOLATION
➡
Look at the general MFV analysis of Buras et al
➡
Note that their definition of MFV is very strict, thus some effects
called MFV SUSY by others are not MFV for them
➡
Thus their analysis is the most conservative scenario
Moreover, there exist several relations between various branching ratios that allow straightforward tests of these models. A review has been given in [8].
This formulation of MFV agrees with the one of [9, 10] except for the case of models with
two Higgs doublets at large tan β, where also additional operators, strongly suppressed in the
SM, can contribute significantly and the relations in question are not necessarily satisfied. In
the present paper, MFV will be defined as in [5, 8].
As reviewed in [8], this class of models can be formulated to a very good approximation
in terms of eleven parameters: four parameters of the CKM matrix and seven values of the
universal master functions Fi (v) that parametrize the short distance contributions to rare
decays with v denoting symbolically the parameters of a given MFV model. However, as
argued in [8], the new physics contributions to the functions
S(v),
C(v),
D! (v),
(1.1)
representing respectively ∆F = 2 box diagrams, Z 0 -penguin diagrams and the magnetic photon
penguin diagrams, are the most relevant ones for phenomenology, with the remaining functions
producing only minor deviations from the SM in low-energy processes. Several explicit calculations within models with MFV confirm this conjecture. We have checked the impact of these
additional functions on our analysis, and we will comment on it in Section 3.
Now, the existence of a UUT implies that the four CKM parameters can be determined
independently of the values of the functions in (1.1). Moreover, only C(v) and D ! (v) enter the
branching ratios for radiative and rare decays so that constraining their values by (at least)
two specific branching ratios allows to obtain straightforwardly the ranges for all branching
ratios within the class of MFV models. Analyses of that type can be found in [8, 9, 11].1
The unique decay to determine the function D! (v) is B → Xs γ, whereas a number of decays
such as K + → π + ν ν̄, KL → π 0 ν ν̄, Bs,d → µ+ µ− , B → Xs,d ν ν̄ and KL → π 0 l+ l− can be used
1
+ −
2
−6
(B → Xs l l , 0.04 < q (GeV) < 1) = 1.16 · 10
!
1 + 0.38 (∆B ll̄ )2 + 0.46 ∆C7eff ∆B ll̄
to determine C(v). The decays B → Xs,d l+ l− depend on both C(v) and D" (v) and can be
ll̄
eff
ll̄
eff 2
− 3.47 ∆C
+ → π∆C∆B
+ ν ν̄ to determine
7 + 0.56 ∆B + 4.31(∆C7 )
used together with B → Xs γ and K+0.41
C(v).
"
0
Eventually the decays K + → π + ν ν̄ and KL →
cleanest ones
2 π ν ν̄, being the theoretically
eff
+0.19 (∆C) + 0.38∆C − 0.11 ∆C7 ∆D ,
[15, 16], will be used to determine C(v). However, so far only three events of K + → π + ν ν̄
!
have
been
of
K
→1.33
π 0 ν ν̄,
with
+ − observed
2 [17, 18, 19] and no event −6
ll̄ the
ll̄
Br(B → Xs l l , 1 < q (GeV) < 6) = 1.61 · 10
1L +
(∆B
)2 +same
1.26 comment
∆C7eff ∆Bapplying
to Bs,d → µ+ µ− , B → Xs,d ν ν̄ and KL → π 0 l+ l− . On the other hand the branching ratio
ll̄the branching ratio
ll̄ for B" → X
ll̄ s l+ l− has
for B → Xs γ has been known for +1.43
some time
and
!
∆C∆B
−
0.31
∆D∆B
+
2.08
∆B
+ 1.42(∆C7eff )2
2
−6
ν ν̄
→by
XBelle
= 1.50
· 10 [21]1 collaborations.
+ 0.65 (∆C +The
∆B
) combined
,
d ν ν̄) [20]
been recently Br(B
measured
and BaBar
latter,
with"
2
eff
(∆C)
+the
1.36∆C
− 0.18∆D
"2MFV
K + → π + ν ν̄, provide presently the +0.67
best estimate
range −
for0.29
C(v)∆C
within
models. ,
7
−5 !of
ν ν̄ ∆D
Br(B → Xs ν ν̄) = 3.67 · 10
1 + 0.65 (∆C + ∆B ) ,
!
The main goals of the present paper are
eff
ll̄
eff
B → Xs l+ l− , 14.4 < q 2 (GeV)
<+25) = 3.70 · 10−7−111#+ 1.18 (∆B ll̄ )2 + 0.70ν∆C
+
ν̄ 2 7 ∆B + 0.60 ∆C7ν ν̄
Br(Kvarious
→ πbranching
ν ν̄) = 8.30
10functions
1 +of0.20(∆C
) + 0.89(∆C + ∆B
! · as
"2+ ∆B
• to calculate
ratios
C(v)ν ν̄within
MFV models,
−6
Br(B → Xd ν ν̄) = 1.50 · 10
1 + 0.65 (∆C
+ ∆B ) ,ll̄
ll̄
ll̄
eff 2
+1.27
∆C∆B
−
0.27
∆D∆B + 2.18
0
−11
ν ν̄ 2∆B + 0.21(∆C7 )
• to determine
allowed
for
available
data,)) ,
Br(Kthe
π ν ν̄)range
= −5
3.10
· 10 from(1presently
+ 0.65(∆C
+ ∆B
! C(v)
L →
ν ν̄ "2
"
Br(B → Xs ν ν̄) = 3.67 · 10
1 + 0.652(∆C + ∆B ) ,
eff
+ 1.24∆C
0.24∆D
+ ν∆C
0 ν ν̄,
+ − for +0.60
−10 ratios
l7l̄ L ∆D
2
# ·(∆C)
• to findBr(K
the upper
bounds
the
branching
of K +−→0.16
π
ν̄, K
→. π−
Bs,d $→ ,
→
µ
µ
)
=
8.58
10
(1
+
0.82(∆C
+
∆B
))
(
L
+
+
−11
ν ν̄ 2
ν ν̄
+
−
0
+
−
Br(K
→
π
ν
ν̄)
=
8.30
·
10
1
+
0.20(∆C
+
∆B
)
+
0.89(∆C
+
∆B
)
,
! MFV models as "
µ µ , B → Xs,d ν ν̄ and KL → π l l within
defined
here,
2
+ −
−10
ll̄
Br(Bd → µ µ ) = 1.08 · 10
1 + ∆B + ∆C ,
0
−11
ν ν̄ 2
π νimpact
ν̄) = 3.10
· 10 measurements
(1 + 0.65(∆C
+ ∆B
)) ,
•Br(K
to assess
of future
on MFV
models.
L →the
Numerical Analysis+
!
"2
ll̄
Br(B
µ 8.58
µ
) ·=103.76
1 +be∆B
+ll̄∆C
−10Section
Our paper
organized
as follows.
2 can
considered
Br(K
µ+
µs −→
)=
(1· 10
+ 0.82(∆C
+
∆B
))2 as
. a ,guide to the literature,
(2.5)
L → is
+ −
numerical
consists
of with
three
steps:
where
the discarded
formulae
for
the
branching
ratios
in question
canin be
found.
Notice
that
weanalysis
have
terms
coefficients
smaller
than 0.1
Br(B
→ XIn
l ).Section we also
s l this
give the list of the input parameters. In Section 3 we present our numerical analysis of various
merical
Analysis
Extracting
CKM
parameters
using
theand
UUT analysis;
branching
ratios
as functions
of C(v)
5 their expectation values and upper bounds. A brief
summary of our results is given in Section 4.
−
−9
ical
analysis consists
of threerange
steps:for ∆C and ∆C7eff from presently available data;
Determining
the allowed
a)
a)
b)
b)
1: Leading
relevant
toν̄Kdecays
→ πν(a);
ν̄ decays
CKM un
Figure 1:Figure
Leading
FeynmanFeynman
diagramsdiagrams
relevant to
K → πν
CKM (a);
unitarity
from
→ πν ν̄ (b).
triangle triangle
from K →
πν ν̄K(b).
Vts∗ VVtdij, being
with Vthe
the elements
of the
CKM
matrix
γ) the
ij being
where λtwhere
= Vts∗ Vλtdt ,=with
elements
of the CKM
matrix
and
(β, γ)and
the (β,
angles
of the unitarity
triangle
Fig. 1).
the measurements
of the branching
of the unitarity
triangle (see
Fig. (see
1). With
the With
measurements
of the branching
ratios at ra
±5%
level
these estimates
the ±5%the
level
these
estimates
change tochange to
◦
◦
2β) = ±0.03,
σ(Imλ
±3%,
±4%,
.
σ(sin 2β)σ(sin
= ±0.03,
σ(Imλt ) =
±3%,t ) =σ(|V
±4%,
σ(γ)
= ±6σ(γ)
. = ±6
(2.3)
td |) =
td |) =σ(|V
Further details
be found
in found
[9]. in [9].
Furthercan
details
can be
It is worth
stressing
that the determination
of CKM parameters
via K → πν
decays
It is
worth stressing
that the determination
of CKM parameters
viaν̄ K
→ πν ν̄
is mainlyisan
efficient
to compare
measured
value of these
FCNC
transitions
mainly
an way
efficient
way to the
compare
the measured
valueclean
of these
clean
FCNC tran
Rare K- (vs.) B-decays
Ulrich Haisch
3%
4%
!t , EW
!+
!L
scales
12%
15%
CKM
"Pc,u
mt
12%
33%
16%
CKM
mt, !s
69%
11%
mc
25%
Figure 3: Error budget of the SM prediction of BR(KL → π 0 ν ν̄) (left) and BR(K + → π + ν ν̄(γ)) (right).
See text for details.
finds, by adding errors quadratically, the following
SM predictions
for the two K → πν ν̄ rates
U. Haisch,
arXiv:0707.3098
BR(KL → π 0 ν ν̄)SM = (2.54 ± 0.35) × 10−11 ,
+
+
(2.3)
BR(K
→π
ν ν̄(γ))
0.86) fully
× 10 exploit
.
(2.4)
SM = (7.96
lots of work
to be
done
before
we±can
the
precision
of Project
kaon
The error budgets of
the SM predictions
of both X
K→
πν ν̄ experiments
decays are illustrated by the pie charts
−11
in Fig. 3. As the breakdown of the residual uncertainties shows, both decay modes are at present
subject mainly to parametric errors (81%, 69%) stemming from the CKM parameters, the quark
masses mc and mt , and αs (MZ ). The non-parametric errors in KL → π 0 ν ν̄ are dominated by the
+
+
Probability density
"10-3
0.8
0.6
0.4
0.2
!C
0
0.5
1
1
1.5
!Ceff
7
0
-4
-2
0
2
!C
C. Bobeth, M. Bona, A. Buras, T. Ewerth,
M. Pierini, L. Silvestrini, A. Weiler, hep-ph/0505110
0.5
MFV new physics parameter ∆C not well-constrained
0
-0.5
MFV SUSY effects on K → πνν
Figure 1: Regions in the mt̃ – mχ̃ plane (lightest stop and chargino masses) allowing enhancements of B(K + → π + ν ν̄) of more than 11% (yellow/light gray), 8.5% (red/medium
gray) and 6% (blue/dark gray) in the MFV scenario, for tan β = 2 and MH + > 1 TeV [the
corresponding enhancements for B(KL → π 0 ν ν̄) are 15%, 12.5% and 10%, respectively,
see Eq. (21)].
G. Isidori, F. Mescia, P. Paradisi, C. Smith, S. Trine, hep-ph/0604074
R(K → f ) = B(K → f )/B(K → f )SM ,
(22)
in the region of maximal enhancements (i.e. 10% to 16% for the neutral mode). In
principle, the relation (21) would allow the best test of the MFV hypothesis. However, the
experimental challenges of the KL → π 0 ν ν̄ mode make the correlation between B(K + →
π + ν ν̄), mt̃ and mχ̃ outlined in Figure 1 a more useful test for the near future.
In Figure 2, we present a more detailed analysis of the parameter-space region with
• Assuming that future more precise measurements of the K → πν ν̄ branching ratios will
be consistent with the MFV upper bounds presented here, the determination of C through
these decays will imply much sharper predictions for various branching ratios that could
confirm or rule out the MFV scenario. In thisC.
context
the
between
various
Bobeth,
M.correlations
Bona, A. Buras,
T. Ewerth,
+ →
M.
Pierini,
Silvestrini,
A. Weiler,
hep-ph/0505110
Figure ratios
8: P.d.f.discussed
for the branching
ratiosplay
of thethe
rarecrucial
decays
Br(K
π 0 ν ν̄) vs Br(K
branching
in [8] will
role.L.
L →
π + ν ν̄). Dark (light) areas correspond to the 68% (95%) probability region. Very light areas
correspond
to the range obtained
without
the experimental information.
of
such correlations
predicts
thatusing
the measurement
of sin 2β and of Br(K +
• One
→ π + ν ν̄)
implies only two values of Br(KL → π 0 ν ν̄) in the full class of MFV models that correspondAstoin two
signs of
thethe
function
X corresponds
[42]. Figure
8 demonstrates
that the solution with
the previous
cases,
HI solution
to a much
lower upper bound.
us now considerto
B decays:
X < 0, Let
corresponding
the values in the left lower corner, is practically ruled out so
that a unique
Br(K
π 01.56]
ν ν̄)∪can
the
be×obtained.
L →
Br(Bprediction
→ Xs ν ν̄) = for
[0, 5.17]
(LOW
: [0,
[1.59,in
5.4],
HIfuture
: [0, 3.22])
10−5 ,
−6
Br(B → Xd ν ν̄) = [0, 2.17] (LOW : [0, 2.26], HI : [0, 1.34]) × 10 ,
• A strong violation
of any of the 95% probability upper bounds on the branching ratios
Br(Bsby
→ µµ̄)
= [0,measurements
7.42] (LOW : [0, 7.91],
: [0, 3.94])
× 10−9 ,of MFV as defined in [5],
considered here
future
will HI
imply
a failure
−10
Br(Bd →
µµ̄) =
[0, 2.20] (LOW
: [0,
2.37], HI
[0, 1.15])
× 10
.
unless an explicit
MFV
scenario
can be
found
in :which
the
contributions
of(3.9)
box diagrams
are significantly
larger than assumed here. Dimensional arguments [43] and explicit
The reader may wonder whether other observables could help improving the constraints on
!C
7
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
!Ceff
7
Figure 7: P.d.f.’s for ∆C7eff (top-left), ∆C (top-right), ∆C vs. ∆C7eff (bottom-left) and
Br(K + → π + ν ν̄) vs ∆C (bottom-right) obtained without using Br(K + → π + ν ν̄) as a constraint. Dark (light) areas correspond to the 68% (95%)
probability
region.
C. Bobeth,
M. Bona,
A. Buras, T. Ewerth,
0
+ →
Pierini,
Silvestrini,
A. Weiler,
hep-ph/0505110
Figure 8: P.d.f. for the branching ratios of the rare M.
decays
Br(KL.
π ν ν̄) vs Br(K
L →
π + ν ν̄). Dark (light) areas correspond to the 68% (95%) probability region. Very light areas
correspond to the range obtained without using the experimental information.
(see Figures 2, 5 and 6). In Figure 8 we see explicitly the correlation between the charged
and neutral Kaon decay modes. We observe a very strong correlation, a peculiarity of models
0 bound.
As [42].
in the In
previous
cases, thea HI
solution
corresponds to
much lower
upper
with MFV
particular,
large
enhancement
ofa Br(K
L → π ν ν̄) characteristic of models
Let us now consider B decays:
with new complex phases is not possible [43]. An observation of Br(KL → π 0 ν ν̄) larger than
Xs ν ν̄)signal
= [0, 5.17]
: [0, 1.56] ∪phases
[1.59, 5.4],
: [0,flavour
3.22]) × changing
10−5 ,
6 · 10−11 wouldBr(B
be a→clear
of (LOW
new complex
or HI
new
contributions
that violate the
correlations
B and
K decays.
Br(B
→ Xd ν ν̄) = between
[0, 2.17] (LOW
: [0, 2.26],
HI : [0, 1.34]) × 10−6 ,
The 95% probability
ranges
for (LOW
Br(K: L[0,→7.91],
µ+ µHI− ): SD
are × 10−9 ,
Br(Bs → µµ̄)
= [0, 7.42]
[0, 3.94])
Br(B → µµ̄) = [0, 2.20] (LOW : [0, 2.37], HI : [0, 1.15]) × 10−10 .
(3.9)
d
Br(KL →
µ+ µ− )SD = [0, 1.36] (LOW : [0, 1.44], HI : [0, 0.74]) × 10−9 .
The reader may wonder whether other observables could help improving the constraints on
(3.8)
KAON PHYSICS AND
MINIMAL FLAVOR VIOLATION
➡
If supersymmetry is discovered at the LHC, it will be of great
importance to understand to what extent it is minimally flavor
violating
➡
Project X kaon experiments could play a decisive role here.