Non-Standard Interactions
&
the study of
Ï„
→
ηπν
Ï„
Riazuddin
1
Centre for Advanced Mathematics and Physics
National University of Science and Technology
and
High Energy Theory Group
National Centre for Physics
December 29, 2009
1
In Collaboration with Nello Paver (ICTP, Trieste, Italy)
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
1 / 25
Motivation
The decays of
Ï„
−
leptons
provide a good laboratory for the analysis of
various aspects of Particle Physics.
In particular,
Ï„
−
decays
into hadrons allow us to study hadronization
of vector and axial vector currents and thus can be used to study to
determine intrinsic properties of hadronic resonances that, together
with chiral symmetry, governs the dynamics of these processes.
Consider a general decay
Ï„
(
k
)
→
X
(
p
x
) +
ν
Ï„
(
k
0
)
X
is any number of hadrons allowed by energy conservation.
The
T
matrix is
T
=
−
G
0
√
2
h
X
|
J
µ
|
0
i
u
(
k
0
)
γ
µ
(
1
−
γ
5
)
u
(
k
)
J
µ
=
V
µ
−
A
µ
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December 29-31, 2009
2 / 25
Example (1):
X
=
Ï€
−
Ï€
0
Ï€
+
[Ref: D. Gomez Dumm et al. arXiv:0911.4436]
In the iso-spin limit
h
Ï€
−
(
p
1
)
Ï€
0
(
p
2
)
Ï€
+
(
p
3
)
|
A
µ
|
0
i
=
V
µ
1
F
1
+
V
µ
2
F
2
+
θ
µ
F
p
V
µ
1
= (
g
µν
−
θ
µ
θ
ν
θ
2
)(
p
1
−
p
3
)
ν
V
µ
2
= (
g
µν
−
θ
µ
θ
ν
θ
2
)(
p
2
−
p
3
)
ν
θ
µ
=
p
µ
1
+
p
µ
2
+
p
µ
3
F
1
and
F
2
derive a
J
p
=
1
+
transition while
F
p
accounts for
J
p
=
0
−
transitions: very much suppressed,
O
(
m
Ï€
2
)
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
3 / 25
Example (2):
X
=
Ï€
+
Ï€
0
,
J
µ
=
V
µ
h
Ï€
+
(
k
)
Ï€
0
(
p
)
|
V
µ
|
0
i
=
−
√
2
G
F
V
ud
[
f
+
(
t
)(
p
−
k
)
µ
−
f
−
(
t
)(
p
+
k
)
µ
]
t
= (
p
+
k
)
2
Using iso-spin limit and CVC
f
+
(
t
) =
F
Ï€
(
t
)
,
f
−
(
t
) =
0
F
Ï€
(
t
)
is the pion electromagnetic form factor
T
µ
=
h
Ï€
+
(
k
)
Ï€
−
(
p
)
|
V
µ
em
|
0
i
= (
p
−
k
)
µ
F
Ï€
(
t
)
F
Ï€
(
0
) =
1, normalization condition for the pion electric charge
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
4 / 25
Theoretical Frame work
At low energies
E
<
M
p
(770
Mev
), Chiral symmetry is an approximate
symmetry of QCD which then derives the interaction of light
pseudoscalar mesons. However, in general this approximation cannot
be excited to the intermediate energy range, in which the dynamics of
resonant states play a major role.
Example I:
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5 / 25
Example II: For
t
≥
4
m
2
Ï€
,
t
= (
p
+
4
)
2
is time-like
F
Ï€
(
t
)
being analytically continued to the space-like region,
corresponds to
h
Ï€
+
(
k
)
|
V
µ
em
|
Ï€
+
(
−
p
)
i
related to
h
Ï€
+
(
k
)
Ï€
−
(
p
)
|
V
µ
em
|
0
i
, by assuming crossing symmetry.
Perturbative QCD gives
lim
t
→−∞
F
Ï€
(
t
)
∼
ds
(
t
)
t
An important role is played by the dispersion relation.
F
Ï€
(
t
) =
1
Ï€
Z
∞
4
m
Ï€
2
dt
=
F
Ï€
(
t
0
)
t
0
−
t
−
ic
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
6 / 25
The above asymptotic behaviour allows an unsubtracted dispersion
relation. Now
Abs
[
T
µ
] =
1
2
X
h
Z
d
Ï„
h
h
Ï€
+
(
k
)
Ï€
−
(
p
)
|
h
ih
h
|
V
µ
em
|
0
i
The states which contribute
J
pc
(
I
G
) =
1
−−
(
1
+
)
i.e.
Ï
,
Ï
0
, ...
f
(
t
)
≡
F
Ï€
(
t
) =
R
Ï
M
2
Ï
M
2
Ï
−
t
+
R
Ï
0
M
2
Ï
0
M
2
Ï
0
−
t
+
...
R
Ï
M
2
Ï
=
F
p
g
Ïππ
,
h
Ï
0
|
V
µ
em
|
0
i
=
f
Ï
µ
h
Ï€
+
(
k
)
Ï€
−
(
p
)
|
Ï
0
i
= (
p
−
k
)
ν
ν
g
Ïππ
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Contributions of Excited States
One possibility is to adopt the "dual resonance model" inspired by
Veneziano.
(see for Example:
Frampton, Phys. Rev. D1, 3141 (1969);
Urrutia, Phys. Rev. D9, 3213 (1974);
C. A. Dominguez, Phys. Lett. B512, 331 (2001);
Bruch, Eur. Phys. J. C39, 41 (2005)
)
Sum of poles corresponding to radial excitations n (with
n
=
0
≡
Ï
,
n
=
1
≡
Ï
0
,
etc
.
):
F
Ï€
(
t
) =
X
n
g
n
f
n
1
M
2
n
−
t
with, in our case,
n
=
0 for
Ï
and
n
=
1 for
Ï
0
.
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December 29-31, 2009
8 / 25
Ingredients:
Ï
-meson “universal†linear Regge trajectory
α
Ï
(
t
) =
1
+
α
0
t
−
M
2
Ï
,
α
0
'
1
2
M
2
Ï
giving the spectrum
M
2
n
=
M
2
0
·
(
1
+
2
n
)
This model would give for the
Ï
0
(
1450
)
:
M
Ï
0
=
1
.
33
GeV
, a
reasonable approximation to the measured mass (10%). The couplings
are given by
g
n
f
n
=
(
−
1
)
n
Γ(
β
−
1
/
2
)
α
0
√
Ï€
Γ(
n
+
1
)Γ(
β
−
n
−
1
)
with
β
related to the asymptotic behavior
F
(
t
)
→
1
/
t
β
−
1
as
t
→ −∞
.
F
Ï€
(
t
) =
g
Ïππ
f
Ï
M
2
Ï
−
t
+
g
Ï
0
ππ
f
Ï
0
M
2
Ï
0
−
t
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
9 / 25
g
Ï
0
ππ
f
Ï
0
g
Ïππ
f
Ï
=
−
Γ(
β
−
1
)
Γ(
β
−
2
)
=
−
(
β
−
2
)
For
β
'
2
.
2-2.3 as used by Bruch to fit
F
Ï€
(
t
)
in the timelike region,
one finds numerically:
g
Ïππ
f
Ï
M
2
Ï
'
1
.
17
;
g
Ï
0
ππ
f
Ï
0
g
Ïππ
f
Ï
'
0
.
3
,
F
Ï€
(
0
)
'
1
.
04
Therefore:
Ï
VDM plus corrections, normalization at
t
=
0 missed by
only 4% taking into account finite width of resonances.
Final ansatz for
F
Ï€
(
t
)
is:
F
Ï€
(
t
) =
g
Ïππ
f
Ï
M
2
Ï
"
M
2
Ï
M
2
Ï
−
t
−
iM
Ï
Γ
Ï
(
t
)
−
(
β
−
2
)
M
2
Ï
M
2
Ï
0
−
t
−
iM
Ï
0
Γ
Ï
0
(
t
)
#
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
10 / 25
We could add as many resonances as we want, the masses and
couplings are explicitly determined by the model.
For the widths, in principle we should take them
t
-dependent, in order
not to miss the normalization at
t
→
0, i.e.:
Γ
n
(
t
) =
M
2
n
t
q
(
t
)
q
(
M
2
n
)
3
Γ
n
,
where
q
is the C.M. momentum.
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December 29-31, 2009
11 / 25
The Decay
Ï„
+
→
ηπ
+
ν
Ï„
Why it is interesting?
We may assign to parts of hadronic weak current,
J
λ
, quantum
numbers, that the strong interactions conserve: Charge conjugation
parity, hypercharge, isospin, charge symmetry
U
=
e
i
Ï€
·
I
2
and
G
-parity.
Such an assignment is very important and is guided by selection rules
which holds for weak interaction.
Classification w.r.t
G
.parity
G
=
Ce
i
Ï€
·
I
2
=
CU
gives
GJ
0
λ
G
−
1
=
η
J
0
λ
η
=
±
according as
J
is
V
or
A
, superscript 0 indicates that it is
∆
Y
=
0 current.
The current is called first or second class, accordingly as
= +
1 or
−
1
under
G
-parity.
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December 29-31, 2009
12 / 25
In the Standard model currents are of first class. Thus for
I
=
1 vector
current, which is relevant for the above process is
V
λ
i
,
i
=
1
,
2
,
3
G
=
1
.
On the other hand
G
(
ηπ
+
) =
−
1
Thus within the Standard Model, the decay is iso-spin and
G
−
parity
violating and as such is suppressed by small value of
(
m
d
−
m
u
)
Λ
QCD
or
α
EM
.
Various estimates, indicate, its Branching ratio
B
expt
'
10
−
5
far below
B
expt
<
1
.
4
×
10
−
4
.
Thus the detection of the decay
Ï„
→
ηπν
Ï„
which is expected in near
future, might provide a unique signature for the "
second class
currents"
. It is worthwhile to point out interesting consequences of
various
B
values.
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December 29-31, 2009
13 / 25
We focus on the rate as expected within standard model due to isospin
violation.
T
µ
=
h
Ï€
+
(
k
)
η
(
p
)
|
V
µ
1
+
i
2
|
0
i
=
−
√
2
[
f
+
(
t
)(
p
−
k
)
µ
+
f
−
(
t
)(
p
+
k
)
µ
]
=
−
√
2
{
f
1
(
t
)[(
p
−
k
)
µ
−
M
2
η
−
M
2
Ï€
t
(
p
+
k
)
µ
]
+
f
0
(
k
)
M
2
η
−
M
2
Ï€
t
q
µ
}
q
=
p
+
k
,
q
2
=
t
h
Ï€
+
(
k
)
η
(
p
)
|
i
∂
µ
V
µ
1
+
i
2
|
0
i
=
−
(
−
√
2
)[(
M
2
η
−
M
2
Ï€
)
f
+
(
t
) +
tf
−
(
t
)]
=
√
2
f
0
(
t
)[
M
2
η
−
M
2
Ï€
]
f
0
(
t
) =
f
+
(
t
) +
t
M
2
η
−
M
2
Ï€
f
−
(
t
)
f
1
(
t
) =
f
+
(
t
)
f
1
(
0
) =
f
+
(
0
) =
f
0
(
0
)
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December 29-31, 2009
14 / 25
Chiral Symmetry Constraints
i
∂
µ
V
µ
1
+
i
2
= (
m
d
−
m
u
)¯
ud
= (
m
d
−
m
u
)
S
1
+
i
2
= (
m
d
−
m
u
)
Ï€
+
η
|
S
1
+
i
2
|
0
√
2
F
Ï€
lim
k
→
0
Ï€
+
η
|
S
1
+
i
2
|
0
=
i
η
F
5
1
−
i
2
,
S
1
+
i
2
0
Commutator
F
5
1
−
i
2
,
S
1
+
i
2
=
id
1
−
i
2
,
1
+
i
2
,
k
P
k
=
i
(
2
P
)
where (
P
j
= ¯
qi
γ
5
(
λ
j
/
2
)
q
, λ
0
=
p
2
/
3
I
)
2
P
=
2
√
3
(
P
8
+
√
2
P
0
)
=
2
1
2
¯
ui
γ
5
u
+ ¯
di
γ
5
d
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December 29-31, 2009
15 / 25
Finally, the soft-pion low-energy theorem for
k
→
0 [
M
Ï€
→
0]reads
f
0
(
M
2
η
) =
−
1
2
F
Ï€
m
d
−
m
u
M
2
η
−
mM
2
Ï€
h
η
(
p
)
|
2
P
|
0
i
.
where in the Chiral limit
h
η
(
0
)
|
2
P
|
0
i
= +
1
F
8
2
√
3
ν
−
ν
=
h
0
|
¯
uu
|
0
i
=
h
0
|
¯
dd
|
0
i
=
h
0
|
¯
ss
|
0
i
F
Ï€
=
F
8
In this limit
M
2
η
=
2
¯
m
+
4
m
s
3
F
2
Ï€
ν
Thus
f
0
(
M
2
η
) = ˜
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December 29-31, 2009
16 / 25
where
˜
=
√
3
(
m
d
−
m
u
)
4
(
m
s
+
¯
m
2
)
'
Limiting to just
Ï
and
Ï
0
√
2
f
1
(
t
) =
g
Ïηπ
f
Ï
+
M
2
Ï
M
2
Ï
M
2
Ï
−
t
−
iM
Ï
Γ
Ï
(
t
)
−
(
β
−
2
)
Γ
0
2
Ï
M
2
Ï
0
M
2
Ï
0
−
t
−
iM
Ï
0
Γ
Ï
0
(
t
)
f
Ï
+
=
−
√
2
f
Ï
g
Ïηπ
is defined as
Ï€
+
(
p
)
η
0
(
k
)
|
Ï
+
=
g
Ïηπ
(
p
−
k
)
ν
ν
Now
|
η
i
=
−
sin
|
Ï€
3
i
+
cos
|
Ï€
8
i
|
Ï€
0
i
=
cos
|
Ï€
3
i
+
sin
|
Ï€
i
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December 29-31, 2009
17 / 25
Thus
Ï€
+
(
p
)
η
0
(
k
)
|
Ï
+
=
−
sin
Ï€
+
(
p
)
Ï€
0
(
k
)
|
Ï
+
giving
g
Ïηπ
'
g
Ïππ
where
g
Ïππ
appears in the pion form factor. Thus
f
+
(
t
) =
F
Ï€
(
t
)
=
g
Ïππ
f
Ï
M
2
Ï
"
M
2
Ï
M
2
Ï
−
t
−
iM
Ï
Γ
Ï
(
t
)
−
(
β
−
2
)
Γ
0
2
Ï
M
2
Ï
0
M
2
Ï
0
−
t
−
iM
Ï
0
Γ
Ï
0
(
t
)
#
where
g
Ïππ
f
Ï
M
2
Ï
'
1
.
17
,
≈
10
−
2
q
(
t
) =
1
2
√
t
q
t
−
(
M
η
−
M
Ï€
)
2
q
t
−
(
M
η
+
M
Ï€
)
2
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
18 / 25
Scalar Form Factor
f
0
(
t
)
Dominated by states with quantum number
J
PC
(
I
G
) =
0
++
(
1
+
)
.
Restricting to the two resonances
a
0
(980 MeV) and
a
0
0
(1450 MeV),
call them
a
0
and
a
1
.
f
0
(
t
) =
g
0
M
2
0
−
t
−
iM
0
Γ
0
(
t
)
+
g
1
M
2
1
−
t
−
iM
1
Γ
1
(
t
)
Γ
n
(
t
) =
M
2
n
t
q
(
t
)
q
(
M
2
n
)
Γ
n
and
g
0
=
F
a
0
g
a
0
ηπ
;
g
1
=
F
a
0
0
g
a
0
0
ηπ
where
F
a
0
is defined by
a
0
(
q
)
V
µ
1
+
i
2
0
=
√
2
F
a
0
q
µ
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2nd National Winter Meeting on Particles & fields
December 29-31, 2009
19 / 25
Using the constant unit
f
0
M
2
η
=
retaining only
a
0
f
0
(
t
) =
f
0
M
2
η
M
2
0
−
M
2
η
−
iM
0
Γ
0
M
2
η
M
2
0
−
t
−
iM
0
Γ
0
(
t
)
Retaining
a
0
and
a
0
0
,
f
0
(
t
) =
f
0
M
2
η
M
2
1
−
M
2
η
−
iM
1
Γ
1
M
2
η
M
2
1
−
t
−
iM
1
Γ
1
(
t
)
+
g
0
M
2
0
−
M
2
η
−
iM
0
Γ
0
M
2
η
×


t
−
M
2
η
−
iM
0
(
Γ
0
(
M
2
η
)
−
Γ
0
(
t
)
)
M
2
0
−
t
−
iM
0
Γ
0
(
t
)
−
t
−
M
2
η
−
iM
1
(
Γ
1
(
M
2
η
)
−
Γ
1
(
t
)
)
M
2
1
−
t
−
iM
1
Γ
1
(
t
)


Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
20 / 25
Fix
g
0
=
F
a
0
g
a
0
ηπ
from exprimental width of
a
0
and take
[
from QCD Sum Rules
]
F
a
0
=
1
.
28
MeV
=
128
MeV
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
21 / 25
Decay rate corresponding to
a
0
verses
q
2
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
22 / 25
Decay rate corresponding to
f
0
with two
resonances verses
q
2
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
23 / 25
Decay rate corresponding to
f
1
with two
resonances verses
q
2
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
24 / 25
Total Decay Rate verses
q
2
:
Branching Ratio
=
5
.
45
×
10
−
6
Riazuddin (CAMP NUST and NCP)
2nd National Winter Meeting on Particles & fields
December 29-31, 2009
25 / 25