On the Radiative Pion Decay

0.24 MB·

by  Rene Unterdorfer

PSI-PR-07-05
December 2007
On the Radiative Pion Decay
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Rene Unterdorfer1
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Hannes Pichl2
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1) Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

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2) Helvetia Insurance, Dufourstrasse 40, CH-9001 St. Gallen, Switzerland
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Abstract
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1
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A reanalysis of the radiative pion decay together with the calculation of the radiative
8
0 corrections within chiral perturbation theory (CHPT) is performed. The amplitude of
: this decay contains an inner Bremsstrahlung contribution and a structure-dependent
v
i part that are both accessible in experiments. In order to obtain a reliable estimate of
X
the hadronic contributions we combine the CHPT result with a large-N expansion and
C
r
a experimental data on other decays, which makes it possible to determine the occurring
coupling constants.
1 Introduction
Raredecays area useful source of informationonparticle interactions. Searches for new physics
effects can take place at the high-energy or the high-precision frontier. At low energies new
heavy particles can appear in the quantum loops. The advantage of high-precision physics is
that one does not need to know the particle content of possible new physics in order to detect
discrepancies between experimental results and theoretical predictions. One works with known
external particles.Ontheotherhand,it isofcoursenotpossible todetect newparticles directly
at low energies.
The radiative pion decay π+ e+ν γ is interesting as it is not dominated by inner
e
→
Bremsstrahlung and therefore sensitive to the so-called ”structure-dependent” contributions
that are generated by QCD effects. These are described with the help of two form factors,
the vector and the axial-vector form factor. Via the conserved-vector-current (CVC) hypoth-
esis the vector form factor can be related to the decays π0 γγ and π0 γe+e , where
−
→ →
already precise data exist [1]. Therefore it is possible to measure the axial-vector form factor
in experiments on the radiative pion decay directly. If one extracts both form factors experi-
mentally with high precision, deviations from the CVC hypothesis can by investigated. Isospin
breaking effects are of interest as they are not completely understood in the case of two-pion
electroproduction and the corresponding τ decay. There has been a discussion on a possible
tensor interaction that could be detected in experiments on the radiative pion decay [2,3]. The
induced tensor form factor due to radiative corrections is expected to be very small, but an
explicit calculation seems to be useful.
The typical energy scale of pion decays lies far below the region were perturbative standard
model calculations are possible. At low energies chiral perturbation theory (CHPT) [4–7] is
used as the effective field theory of the standard model in this energy region. All high-energy
effects are included in the coupling constants of the effective Lagrangian. If it is not possible
to determine all these coupling constants by use of experimental data from other processes,
large-N QCD [8] is used to estimate them.
C
Theoretical calculations of the structure-dependent contributions to the radiative pion de-
cay havebeen presented in[9]and[10]attwo-looporder.The lowest-order radiativecorrections
that are relevant for the inner Bremsstrahlung part are given in [11]. We complete the analysis
by calculating the radiative corrections to the structure-dependent part within the framework
of CHPT to lowest order in the large-N expansion.
C
On the experimental side an investigation has been performed at PSI [12], where the form
factors have been measured with errors of only a few percent. Older experimental data [13–15]
serve as an additional check. Future experiments on the radiative kaon decay will be helpful
as the ratio of the decay widths of the pion and the kaon mode can be predicted theoretically
with higher precision than the decay widths individually.
The paper is organized as follows. In Sec. 2 we explain the basic facts of CHPT. The
kinematics and the structure of the amplitude and the decay width of the radiative pion decay
are presented in Sec. 3. The strong interaction contributions are explained in Sec. 4. Values of
2
the NNLO coupling constants are given. In Sec. 5 we report how the structure of the amplitude
is modified by radiative corrections. Apart from that, the treatment of soft photon radiation
and the application of the Low theorem is explained and the large-N form factors used to
C
calculate the radiative corrections to the structure-dependent contributions are introduced.
In Sec. 6 our results for the form factors, the radiative corrections and the decay width are
presented. Our conclusions are summarized in Sec. 7.
2 Low-energy expansion
TheasymptoticstatesinCHPTarenotquarksandgluonsbutthemembersofthelightestoctet
of pseudoscalar mesons1, the photon and the light leptons. CHPT is a low-energy expansion in
the external momenta and masses that should be small compared to the natural scale of chiral
symmetry breaking, which is expected to have a value of about 1.2 GeV. The order n of this
expansion is indicated by pn. A squared momentum of a pseudoscalar meson is of (p2). The
O
lowest-order effective Lagrangian is of the form
F2 1
= u uµ +χ +e2F4Z UQemU Qem F Fµν
Leff 4 h µ +i h L † R i− 4 µν
¯
+ [ℓ(i∂ +eA m )ℓ+ν i∂ν ] (2.1)
ℓ ℓL ℓL
6 6 − 6
ℓ
X
with
u = i[u (∂ ir )u u(∂ il )u ] ,
µ † µ µ µ µ †
− − −
χ = u χu uχ u. (2.2)
† † †
± ±
The pseudoscalar mesons are collected in a matrix u:
π0 + 1 η π+ K+
iΦ √2 √6 8
u = exp √2F! , Φ =  Kπ− −√π02K+¯0√16η8 K20η  , U = u2. (2.3)
 − −√6 8 
 
The symbol stands for the trace in three-dimensional flavour space. In order to introduce
h i
masses the external field χ is set equal to an expression proportional to the quark mass matrix
from now on:
m 0 0
u
χ = 2B 0 m 0 . (2.4)
0 d 
0 0 m
s
 
 
˜
By adding terms determined via gauge symmetry to the external fields l and r˜ of the purely
µ µ
mesonic case the coupling of the photon A and the leptons ℓ,ν to the pseudoscalar mesons
µ ℓ
1Other hadrons like the baryons can also be incorporated.
3
is fixed.
lµ = ˜lµ −eQeLmAµ + (ℓ¯γµνℓLQwL +νℓLγµℓQwL†),
ℓ
X
r = r˜ eQemA . (2.5)
µ µ − R µ
As the electroweak interactions break chiral symmetry, the spurion matrices Qem , Qw can be
L,R L
equated with the following expressions:
2/3 0 0 0 V V
ud us
Qem = 0 1/3 0 , Qw = 2√2 G 0 0 0 . (2.6)
L,R  −  L − F 
0 0 1/3 0 0 0
 −   
   
Every term in the Lagrangian (except kinetic terms) is multiplied with a coupling constant.
The constant F in Eq. (2.1) is identified with the pion decay constant in the chiral limit
without electroweak interactions. In the same limit the constant B can be related to the
0
quark condensate. Z dominates the pion electromagnetic mass difference. We use the SU(3)
formalism in this work because information from processes involving strange quarks is needed
to determine some of the coupling constants.
The lowest-order Lagrangian is not enough to make connection with experiment. Higher
orders have to be included. At every order the Lagrangian contains all terms that respect
the symmetries. In [6] the SU(3) Lagrangian to (p4) was presented considering also the
O
Wess-Zumino-Witten functional [16]. Here and in the following only the terms relevant for our
calculation are shown:
= L u uµ 2 +L u u uµuν +L u uµu uν
p4 1 µ 2 µ ν 3 µ ν
L h i h ih i h i
L
iL fµνu u + 10 f fµν f fµν
− 9 h + µ νi 4 h +µν + − −µν − i
i
εµναβ ΣLU ∂ r Ul ΣRU∂ l U r +ΣLl ∂ l +ΣL∂ l l
−16π2 h µ † ν α β − µ ν α † β µ ν α β µ ν α β
3i
iΣLΣLΣLl +iΣRΣRΣRl + ΣL(U r U +l ) [v ,v ] +... (2.7)
− µ ν α β µ ν α β 2 µ † ν ν h α β ii
where
fµν = uFµνu u Fµνu, Fµν = ∂µlν ∂νlµ i[lµ,lν],
± L † ± † R L − −
Fµν = ∂µrν ∂νrµ i[rµ,rν], ΣL = U ∂ U, ΣR = U∂ U . (2.8)
R − − µ † µ µ µ †
To (p6) one has [18–20]:
O
= C χ h hµν +C χ h hµν +C χ f fµν
p6 12 + µν 13 + µν 61 + +µν +
L h i h ih i h i
+C χ f fµν +iC f χ ,uµuν +iC χ f uµuν
62 + +µν + 63 +µν + 64 + +µν
h ih i h { }i h ih i
+iC f uµχ uν +iC f fνρ,hµ +C χ f fµν
65h +µν + i 78 +µν − ρ 80h + −µν − i
+C χ f fµν +C fD [fhµν,χ ] i+E C f ρfµν
81 + µν 82 +µν 87 ρ µν
h ih − − i h − − i h∇ − ∇ − i
+iC f [hµρ,uν] +iCWεµναβ χ f f
88h∇ρ +µν i 7 h − +µν +αβi
+iCWεµναβ χ [f ,f ] +CWεµναβ u f ,f + ... (2.9)
11 h − +µν −αβ i 22 h µ{∇γ +γν +αβ}i
4
with
h = u + u , X = ∂ X +[Γ ,X] ,
µν µ ν ν µ µ µ µ
∇ ∇ ∇
1 1 1
Γ = [u ,∂ u] iu r u iul u . (2.10)
µ † µ † µ µ †
2 − 2 − 2
The Lagrangian of (e2p2) can be found in [21,22]. We will not present an (e2p4) Lagrangian
O O
because at this order we use the expression that is of lowest order in the large-N expansion.
C
As CHPT is a quantum field theory loops have to be taken into account. The primitive
degree of divergence of a loop amplitude [17] is equivalent to the chiral dimension. A one-loop
Feynman graph including only lowest-order vertices is of (p4) in the purely mesonic case and
O
of (e2p2) if there is one internal photon propagator. The counterterms used to compensate
O
the ultraviolet divergences of the loop integrals have to be of the same order in the external
momenta as the loops. By renormalizing the appropriate coupling constants (e.g. L , C , ...)
9 12
that appear in the Lagrangian an UV finite amplitude is achieved.
The scale dependent2 finite parts of all coupling constants are determined experimentally
or estimated by performing resonance exchange calculations. In the large-N limit the values
C
of the coupling constants are given by exchange of infinitely narrow resonances. It turns out
that at (p4) the values one gets in this approximation agree quite well with the experimental
O
values at a renormalization scale equal to the mass of the ρ particle [23]. This agreement is
obtained by using only the lowest-lying vector, axial-vector, scalar and pseudoscalar octets.
In [23] also the constant Z of Eq. (2.1) has been determined. In this case and whenever
one wants to calculate a coupling constant of an (e2pn) Lagrangian resonance propagators
O
appear in the loop and the correct momentum dependence of the involved form factors also
for high energies [24] is needed. We summarize how this can be achieved in the case of the
electromagnetic pion form factor F . From Eqs. (2.1) and (2.7) one gets3:
e
2Lr
F (q2) = 1+ 9q2 +A + (q4). (2.11)
e F2 loop O
At leading order in the 1/N expansion including the lowest-lying vector resonance with mass
C
M we have
V
k q2
F (q2) = 1+ V . (2.12)
e F2M2 q2
V −
We will identify M with the mass of the ρ meson. The chiral loops are of higher order and
V
introduce the width of the ρ [25]. Imposing that the form factor should vanish at infinite
momentum transfer due to the Brodsky-Lepage behavior [26], the constant k becomes equal
V
to F2 and
M2
F (q2) = V + (1/N ). (2.13)
e M2 q2 O C
V −
2The whole amplitude does not depend on the renormalizationscale.
3The renormalized coupling constants are labeled with an r.
5
Therefore, one concludes
F2
Lr(M2) = . (2.14)
9 V 2M2
V
The resonance Lagrangian that leads to a pion form factor with the correct low- and high-
energy behavior to the order indicated in Eqs. (2.11) and (2.13) is of the form [24]
1 M2 F iG
= λV Vνµ V V Vµν + V V fµν + V V uµuν (2.15)
Lres −2h∇ λµ∇ν − 2 µν i 2√2 h µν + i √2 h µν i
withthehigh-energyconditionF G = F2.Thevector mesons aredescribed byantisymmetric
V V
tensor fields V = 1 8 λ Vi . To (p4) this formalism is equivalent to the more familiar
µν √2 i=1 i µν O
notation with vector fields if one introduces explicit local terms [24].
P
3 General structure of amplitude and decay width
The amplitude of π+(p) e+(p )ν(p )γ(k) has the following structure [27]:
e ν
→
M = ieG V ǫ F Lµ Hµνl (3.1)
0 − F u∗d ∗µ{ π − ν}
with
pµ 2pµ+ kγµ
Lµ = m u¯(p )(1+γ )( e 6 )v(p ),
e ν 5 e
p k − 2p k
e
· ·
i
Hµν = (F (p2)ǫµναβk p F (p2)(k p gµν pµkν)),
−√2m V w α β − A w · −
π+
lµ = u¯(p )γµ(1 γ )v(p ), p = p +p (3.2)
ν 5 e w e ν
−
where F is the physical pion decay constant. One distinguishes between the inner
π
Bremsstrahlung (IB) contribution and the structure-dependent (SD) part. The first is given by
the term with L and corresponds to the radiation of a pointlike pion and positron. The latter
µ
contains the two structure functions F (p2) and F (p2) including the hadronic contributions.
V w A w
In the process π eνγ the IB part is helicity suppressed, allowing the detection of the
→
structure-dependent terms. The IB contribution diverges if the photon energy goes to zero.
This divergence is canceled in the total rate by loop corrections to the decay π eν implying
→
virtual photons. In experiments usually an energy cut is applied. Only photons above a certain
energy are detected.
Whereas the IB part is completely determined by the Low theorem [28] the structure-
dependent part reflects the influence of QCD on this decay. The form factors F and F to
V A
(p4) in the chiral expansion are given by
O
m 1
π+
F = = 0.027 0.003, (3.3)
V
F 4√2π2 ±
π
6
4√2(Lr +Lr )
F = m 9 10 = 0.010 0.004. (3.4)
A π+
F ±
π
They include a mass m that is of no physical meaning and drops out in the amplitude
π+
(see Eq. (3.2)). At higher orders and by including radiative corrections, the form factors get a
momentum dependence.
The importance of the different contributions can be seen from the differential rate (here
normalized to the non-radiative mode):
dΓ α F m2 2
eγν Γ = IB(x,y)+ V π [(1+γ)2SD+(x,y)+(1 γ)2SD (x,y)]
eν −
dxdy 2π 2√2Fπme! × −
.(cid:16) (cid:17)
F m
+ V π (1+γ)S+ (x,y)+(1 γ)S (x,y) (3.5)
√2Fπ! int − i−nt
h i
with
γ = F /F . (3.6)
A V
IB,SD andS arefunctionsofthetwokinematicvariablesx = 2p k/m2 andy = 2p p /m2.
± i±nt · π · e π
For m /m = 0 one has:
e π
(1 y)((1+(1 x)2)
IB(x,y) = − − ,
x2(x+y 1)
−
SD+(x,y) = (1 x)(x+y 1)2,
− −
SD (x,y) = (1 x)(1 y)2. (3.7)
−
− −
In Eq. (3.5) the terms including SD dominate over those with S because of the additional
± i±nt
factor m2/m2. When x + y goes to 1 the function IB(x,y) diverges. SD+(x,y) reaches its
π e
maximum at x = 2/3, y = 1 and SD (x,y) at x = 2/3, y = 1/3 (i.e. x + y = 1). One can
−
define an angle between the positron and photon momenta:
θ x+y 1
sin2 eγ = − . (3.8)
2 xy
For θ = 0 the function IB(x,y) goes to infinity (SD (x,y) has its maximum), whereas
eγ −
SD+(x,y) has its maximum for θ = π. Therefore an experiment performed in the region
eγ
near θ = π is sensitive to (1+ γ)2. It is difficult to distinguish experimentally between the
eγ
terms proportional to IB and SD .
−
In the standard model weak transitions are described by V A interactions. New physics
−
could lead to tensor interactions of the form
eG V
T = i F u∗dǫ k F u¯(p )σµν(1+γ5)v(p ). (3.9)
√2 ∗µ ν T ν e
Radiative corrections generate an induced tensorial form factor as described in Sec. 5.
7
4 Contributions due to the strong interaction
The form factors F and F have been calculated up to (p6) for the chiral group SU(2) in [9]
V A
O
and for SU(3) in [10]. The momentum dependence of the form factors starts at (p6). We will
O
use the SU(3) result which is in the isospin limit of the following form:
m 256 64 1 10
F (p2) = π+ 1 π2m2CWr + π2p2CWr + p2
V w 4√2π2Fπ ( − 3 π 7 3 w 22 32π2Fπ2 (cid:20) 9 w
1 m2 4
p2 ln π + G(p2 /m2,m2) (4.1)
−3 w Mρ2 3 w π π #)
with
z z 4 √z 4+√z
G(z,m2) = m2 1 − ln − 2m2. (4.2)
− 4 s z √z 4 √z −
(cid:18) (cid:19)
− −
and
4√2m m 1 m2
F (p2) = π+ (Lr +Lr )+ π+ ( 2Lr +Lr)m2 ln π
A w Fπ 9 10 Fπ3 (√2π2 " − 1 2 π Mρ2!
1 m2 m2
( Lr +Lr +Lr ) m2 ln K +2m2 ln π
− 2 3 9 10 " K Mρ2! π Mρ2!##
m2 p2 m2
+ π I w − π 4√2 4m2 (6Cr 2Cr +Cr +2Cr )
6(2π)8 2 2m2π !− K 13 − 62 64 81
h
+2m2(6Cr +6Cr 2Cr 2Cr +2Cr +Cr +Cr Cr +2Cr
π 12 13 − 61 − 62 63 64 65 − 78 80
1
+2Cr 2Cr +Cr ) (p2 m2)(2Cr 4Cr +Cr ) . (4.3)
81 − 82 87 − 2 w − π 78 − 87 88
(cid:21)(cid:27)
All coupling constants are taken at a scale equal to the ρ mass. For the two-loop integral I in
2
Eq. (4.3) the numerical approximation given in [10] is used:
I (z) = 44.5z 10304.2. (4.4)
2
−
Therenormalizedlow-energyconstantsCWr,Cr havetobedeterminedbyuseoflarge-N QCD
i i C
or experimental data. Values for CW and CW can be obtained via the conserved vector current
7 22
hypothesis4 with the help of experimental data [1] on the decays π0 γγ and π0 γe+e .
−
→ →
1 2Γ(π0 γγ)
|FVπ0e→xγpγ(0)| = αv πm→ = 0.0262±0.0005. (4.5)
u π0
u
t
π0 γγ
The slope parameter of FV e→xp is given by
exp
a = 0.032 0.004. (4.6)
π ±
4The relation that leads to Eqs. (4.5) and (4.6) is reproduced within CHPT if one neglects the kaon loops
in case of the decay π0 γe+e−.
→
8
Cr(M ) Value [10 5] Source
i ρ −
Cr 0.6 0.3 scalar resonance exchange
12 − ±
Cr 0 0.2 resonance exchange
13 ±
Cr 1.0 0.3 τ decays, < VV > correlator
61 ±
Cr 0 0.2 resonance exchange
62 ±
2Cr Cr 1.8 0.7 K charge radius
63 − 65 ± 0
Cr 0 0.2 resonance exchange
64 ±
Cr 10.0 3.0 resonance exchange
78 ±
Cr 1.8 0.4 a , K differences
80 ± 1 1
Cr 0 0.2 resonance exchange
81 ±
Cr 3.5 1.0 resonance exchange
82 − ±
Cr 3.6 1.0 resonance exchange
87 ±
Cr 3.5 1.0 resonance exchange
88 − ±
Table 1: Values of the coupling constants appearing in Eq. (4.3) and the source of information
used to fix them.
One gets the following values for the low-energy constants in Eq. (4.1)
CWr(M ) = (0.1 1.2) 10 9MeV 2,
7 ρ ± × − −
CWr(M ) = (5.4 0.8) 10 9MeV 2. (4.7)
22 ρ ± × − −
In[29]theconstantC hasbeenfixedbytaking intoaccount theexchange ofscalarresonances.
12
The constant C can be determined with the help of experiments on τ-decays by considering
61
the correlator of two vector-currents and using finite-energy sum rule techniques [30,31]. The
combination 2C C also appears in the expression for the electromagnetic K charge
63 65 0
−
radius [32,33] that has been measured [34,35]. In [36] a large-N expression for the correlator
C
of vector, axial-vector and pseudoscalar currents with the correct high-energy behavior fixed by
the operator product expansion is used to determine amongst others the low-energy constants
C , C , C and C . The contribution to C with three propagating resonances is not
78 82 87 88 82
constrained by the high-energy behavior and will be neglected. The constant C is fixed with
80
the help of mass and decay constant differences of the a and K particles following an idea
1 1
presented in [31] for vector mesons (see App. C). The situation in the case of axial-vector
mesons is more complicated as the states with the quantum numbers JPC = 1++ and 1+
−
mix. The other constants C , C , C , and C are set to zero as resonance exchange does
13 62 64 81
not contribute in this case [37].
In Table 1 the values of the constants C at the ρ mass and the information needed for their
i
determination are shown.
The best procedure to get precise values for F (0) and for the slope of this form factor is to
V
take the values obtained in the isospin limit via the conserved vector current hypothesis (see
Eqs. (4.5) and (4.6)) and to add/subtract the theoretical predictions for the isospin breaking
9
(ISB) contributions. The latter are proportional to m m , e2 and m2 m2 .
u − d π+ − π0
FV = FVπ0e→xγpγ −FVπI0S→Bγγ +FVπI+S→Be+νγ. (4.8)
In [38] the ISB contribution for π0 γγ has been calculated with the result
→
FVπI0S→Bγγ(0) = 0.00066±0.0001 (2.5 % of FVπ0e→xγpγ(0)). (4.9)
What has to be considered concerning FVπI+S→Be+νγ are the radiative corrections discussed in
Sec. 5 and the following contribution proportional to an additional constant CW
11
m m m
FVπm+d→−em+uνγ = 4√2ππ2Fπ 256π2m2π mdd −+muu C1W1 . (4.10)
From experimental data [39] on K+ ℓ+νγ one gets 5 CWr(M ) = (0.68 0.21) 10 9MeV 2
→ 11 ρ ± × − −
which leads to FVπm+d→−em+uνγ = 0.00025±0.00009 (0.9 % of FVπ0e→xγpγ(0)). Numerical results for the
form factors FVπ,+A→e+νγ can be found in Sec. 6.
5 Radiative corrections
The amplitude including radiative corrections contains additional terms compared to Eq. (3.2)
and is of the form
M = iG eV ǫ F LµF (x,y) Hµνl +T(x,y),
− F u∗d ∗µ{ π IB − ν}
i
Hµν = (ǫµναβV (x,y) F (x,y)(k pgµν pµkν)
αβ A
−√2m − · −
π
Fˆ (x,y)(k p gµν pµkν)), (5.1)
− A · l − l
where V (x,y) has a tensor structure more complicated then F (q2)k p and one can dis-
αβ V α β
tinguish between two different axial-vector form factors. As mentioned above the radiative
corrections generate in addition an induced tensorial form factor T(x,y), that is very small,
i.e. 0.5 - 1.5 % of the IB part of the differential branching ratio depending on x and y. To
(e2p2) there is no contribution to V (x,y) and the contributions to F (x,y) and Fˆ (x,y)
αβ A A
O
turn out to be proportional to m2/m2 and can be neglected. Therefore the lowest-order radia-
e π
tivecorrections(includingalsotheinducedtensorialformfactor)canberegardedascorrections
to the IB part.
The squared amplitude M 2 receives radiative corrections due to virtual loop photons ∆
0 V
| |
and additional soft real photons ∆ :
S
α
M 2 = M 2 1+ (∆ +∆ ) . (5.2)
0 V S
| | | | π
spin spin (cid:18) (cid:19)
X X
5We have assumed that the (p6) contribution is smaller then the (p4) part.
O O
10