QED radiative corrections to the decay pi^0 to e^+e^-

0.19 MB·English

by  A. E. Dorokhov

QED radiative corrections to the decay π0 e+e
−
→
A. E. Dorokhov, E. A. Kuraev, Yu. M. Bystritskiy and M. Seˇcansky´
JINR-BLTP, 141980 Dubna, Moscow region, Russian Federation
(Dated: February 2, 2008)
Abstract
8 We reconsider QED radiative corrections (RC) to the π0 e+e− decay width. One kind of RC
0 →
0
investigated earlier has a renormalization group origin and can be associated with the final state
2

n interaction of electron and positron. It determines the distribution of lepton pair invariant masses
a
J
in the whole kinematic region. The other type of RC has a double-logarithmic character and is

4
1 related to almost on-mass-shell behavior of the lepton form factors. The total effect of RC for the

h] π0 e+e− decay is estimated to be 3.2% and for the decay η e+e− is 4.3%.
→ →
p
-
p
PACS numbers:
e
h
[

1
v
8
2
0
2
.
1
0
8
0
:
v
i
X
r
a
1
I. INTRODUCTION
Rare decays of mesons serve as the low-energy test of the Standard Model. Accuracy of
experiments has increased significantly in recent years. Theoretically, the main limitation
comesfromthelargedistancecontributionsofthestrongsectoroftheStandardModelwhere
the perturbative theory does not work. However, in some important cases the result can be
essentially improved by relating these poorly known contributions to other experimentally
known processes. The famous example is the Standard Model calculation of the anomalous
magnetic moment of muon (g 2) where the data of the processes e+e− hadrons and
− µ →
τ hadrons are essential to reduce the uncertainty. It turns out that this is also the case
→
for the rareneutral piondecay into an electron-positronpair measured recently by the KTeV
collaboration [1] and reconsidered theoretically in [2].
The measured branching is [1]
KTeV 0 + − −8
B π e e ,x > 0.95 = (6.44 0.25 0.22) 10 , (1)
D
→ ± ± ·
where the kinematic c(cid:0)ut over the Dalitz vari(cid:1)able xD (p+ +p−)2/M2, ν2 4m2/M2
≡ ≡ ≤
x 1, was used in order to suppress the Dalitz decay events π0 e+e−γ. Then, the
D
≤ →
important step in extraction of the branching consists in correct treating the radiative cor-
rections (RC) to the process which has been considered earlier in [3] and [11]. Extrapolating
the full radiative tail beyond x > 0.95 and scaling the result back up by the overall RC
D
leads to the final result [1]
KTeV + − −8
B (π e e ) = (7.49 0.29 0.25) 10 , (2)
0 0
→ ± ± ·
where the leading order radiative corrections have been taken into account [3]. It is the
motivation of our paper to revise the calculation of QED RC to the π e+e− decay width.
0
→
In the lowest order of QED perturbation theory (PT), the photonless decay of the neutral
pion,
− + 2 2 2 2
π0(q) e (p−)+e (p+), q = M , p± = m ,
→
(M meson mass, m lepton mass) is described by the one-loop Feynman amplitude (Fig. 1a)
corresponding to the conversion of the pion through two virtual photons into an electron-
positron pair. The normalized branching ratio is given by [4, 5, 6]
B (π0 e+e−) αm 2
+ − 0 2 2
R (π e e ) = → = 2β M , (3)
0 0 → B(π0 γγ) πM |A |
→ (cid:16) (cid:17)
(cid:0) (cid:1)
2
ee−−((pp ))
q1 −−
ππ00((qq))
= +
q2
a) ee++((pp++)) b)
+ + + +...
c) d) e)
FIG. 1: Set of the lowest order QED RC to π0 e+e− process: virtual corrections.
→
where β = √1 ν2, B(π0 γγ) = 0.988 and the reduced amplitude is
− →
2 d4k (qk)2 q2k2
2 2 2
q = − F ( k , (q k) ), (4)
A M2 iπ2 (k2 +iǫ)[(q k)2 +iǫ][(p− k)2 m2 +iǫ] π − − −
Z − − −
(cid:0) (cid:1)
with the pion transition form factor F ( k2, q2) being normalized as F (0,0) = 1. The
π π
− −
imaginary part of (q2) can be found in a model independent way [5]
A
π 1+β
2
Im (M ) = ln , (5)
A −2β 1 β
(cid:18) − (cid:19)
while the real part is reconstructed by using the dispersion approach up to a subtraction
constant
1 π2 1 1+β
2 2
Re (M ) = (0)+ + ln . (6)
A A β 12 4 1 β
(cid:20) (cid:18) − (cid:19)(cid:21)
Usually this constant, containing the nontrivial dynamics of the process, is calculated
within different models describing the form factor F (k2,q2) [2, 6, 7, 8]. However, it has
π
recentlybeenshownin[2]thatthisconstantmaybeexpressedintermsoftheinversemoment
of the pion transition form factor given in symmetric kinematics of spacelike photons
2 me 3 µ2 Fπγ∗γ∗ (t,t) 1 ∞ Fπγ∗γ∗ (t,t) 5
q = 0 = 3ln dt − + dt . (7)
A µ − 2 t t − 4
(cid:18) (cid:19) "Z0 Zµ2 #
(cid:0) (cid:1)
Here, µisanarbitrary(factorization)scale. Onehastonotethatthelogarithmicdependence
of thefirst termonµ is compensated by the scale dependence of theintegrals in thebrackets.
The accuracy of these calculations are determined by omitted small power corrections of the
order O(m2) and O(m2L) in the r.h.s. (6), where Λ . M is the characteristic scale of the
Λ2 M2 ρ
form factor Fπγ∗γ∗ (t,t) and L is the large logarithm parameter
M2 1+β
L = ln ln .
m2 ≈ 1 β
(cid:18) (cid:19) (cid:18) − (cid:19)
3
For the decay π0 e+e− one has L 11.2.
→ ≈
By using the representation (7), and the CELLO [9] and CLEO [10] data on the pion
transition form factor FCLEO(t,0) given in asymmetric kinematics the lower bound on
πγ∗γ∗
the decay branching ratio was found in [2]. This lower bound follows from the property:
Fπγ∗γ∗ (t,t) < Fπγ∗γ∗ (t,0) for t > 0. It considerably improves the so-called unitary bound
2 2
obtained from the property (Im ) . Further restrictions follow from QCD and allow
|A| ≥ A
one to make a model independent prediction for the branching [2]
Theor 0 + − −8
B π e e = (6.2 0.1) 10 , (8)
0
→ ± ·
(cid:0) (cid:1)
which is 3.3σ below the KTeV result (2). The main source of the error in (8) is defined
by indefiniteness in the knowledge of the pion form factor Fπγ∗γ∗ (t,t) [2]. The discrepancy
between (8) and (2) requires further attention from experiment and theory to this process
because there are not many places where experiment is in conflict with the Standard Model.
Considering the higher orders of QED PT, there are two sources of RC to the width of
the π0 e+e− decay (Fig. 1). One of them has a renormalization group origin and can be
→
associatedwiththefinalstateinteractionofelectronandpositron. Therelevantcontribution
corresponds to taking into account the charged particle interaction at large distances. The
other is ofdouble-logarithmiccharacter andis relatedto short distance contributions. Let us
note that the branching ratio (3) is proportional to the electron mass squared and thus the
Kinoshita–Lee-Nauenberg theorem of cancellation of mass singularities in the limit m 0
→
is not violated.
ee−−((pp ))
kk −−
ππ00((qq))
= + +...
a) ee++((pp++)) b)
FIG. 2: RC due to soft photon emission.
The first kind of corrections was considered in [3]. Later, the effect of the higher order
RC was estimated in [11] by using the exponentiation of soft photon contributions, which
is essentially equivalent to the Yennie, Frautchi-Suura factorization procedure. Considering
only the two-virtual-photon conversion to a lepton pair, it is originated from the box type
Feynman amplitude (Fig. 1e) as well as the contribution from the emission of real photons
4
by leptons (Fig. 2) and produces the single-logarithmic enhanced terms ( L) which are
∼
described by the lepton nonsinglet structure function method [12]. It was shown in [11] that
the soft photon emission can drastically change the results obtained at the Born level when
the invariant mass of leptons is close to the pion mass.
Furthermore, we find an additional source of RC which is of the so-called ”double-
logarithmic” (DL) nature (αL2/π 1). This kind of asymptotics was intensively inves-
∼
tigated in the 70s in a series of QED processes [13, 14]. The DL type contribution to the
decay width was not considered earlier for the π0 e+e− decay.
→
II. LARGE LOGARITHM REGIME AND DOUBLE-LOGARITHMIC CORREC-
TION
First, it is instructive to reproduce the results discussed above in a simple and physical
way. For this aim we note that the main contribution to the real part of (M2) comes from
A
the kinematic region of loop momenta corresponding to the intermediate virtual electron (or
positron) close to the mass shell (Fig. 1a). Really, by changing the integration variable in
(1) as q1 = k = p− κ, q2 = q k = p+ +κ and omitting terms of order O(κ2/M2) we
− −
can rewrite the amplitude in the Born approximation (M2) as
0
A
1 d4κ M2
2
M .
A0 ≈ 2 iπ2 (κ2 m2 +iǫ)((κ p−)2 +iǫ)((κ+p+)2 +iǫ)
Z − −
(cid:0) (cid:1)
Let us find the real part of the amplitude within the leading logarithmic accuracy that
corresponds to the restriction of the kinematic region by conditions
2 2 2 2 2
m κ q , q M . (9)
1 2
≈ | | ≪ | | | | ≪
(cid:0) (cid:1)
To this end one performs the substitutions
d4κ iπ κ2d κ dO
−→ −→ κ
− | | [Θ(κ )+Θ( κ )] , (10)
∆+iǫ → 2ω 0 − 0 |κ20=ω2
2 2
q = 2Mκ u, q = 2Mκ (u+β cosθ),
1 0 2 0 ω
−
m2 1 β cosθ
ω = −→κ2 +m2, ∆ = κ2 −m2, βω = 1− ω2, u = − 2ω
r
p
where θ is the angle between the directions of electron momentum (the rest frame of the
initial pion implied) and the 3-momentum of the virtual electron. Let us note that by
5
kinematical reasons in the region of maximal contribution the signs of q2 and q2 must be
1 2
opposite. Performing the angular integration we obtain the leading term of (6)
M2 d3κ M2 β dω ω2 1
2 ω 2
Re M = = ln L . (11)
A0 − π ωq2q2 ω m2 ≈ 4
Z 1 2 Zm
(cid:0) (cid:1)
Then, let us consider the vertex type RC (Fig. 1 c,d). The lowest order evaluation arising
from the diagrams of Figs. 1b) and 1c) leads to the correction Γ(q1,κ,p−) = 1−2απIV(q12,κ2)
with
q2 q2 1 q2 3 q2 π2 1 m 3π2
2 2 1 1 2 1 1 2
I (q ,κ ) = ln | | ln | | + ln | | ln | | + + ln Θ q ,
V 1 m2 κ2 2 κ2 − 2 m2 3 2 − λ − − 1 2
| | | |
(cid:0) (cid:1)
which is consistent with the result of similar calculations in [15]. The last term arises from
a renormalization procedure. A similar contribution I (q2,κ2) comes from the diagrams of
V 2
Figs. 1b) and 1d).
Let us consider now the box-type diagram (see Fig. 1e) and demonstrate the calculations
in more detail. The corresponding contribution to the amplitude has the form
∆ d4k N
1
Re ,
2 iπ2 (k12 λ2 +iǫ)((p− +k1)2 m2 +iǫ)((p+ k1)2 m2 +iǫ)((κ+k1)2 m2 +iǫ)
Z − − − − −
(12)
where
N = u¯(p−)γµ(p− +k1 +m)γλ(κ+k1 +m)γγ( p+ +k1 +m)γµv(p+).
−
In the leading kinematic region, where k and κ are small, we can reduce the numerator to
1
2
N 2mM u¯(p−)γλγγv(p+).
≈ −
The calculation of the scalar 4-denominator integral is standard: by using the Feynman
parametrization and performing loop momentum integration we arrive at
1 1 1 z2dz
2
2mM ∆ dx ydy ,
− (Az2 +Bz +C)2
0 0 0
Z Z Z
with
2 2 2
A = (yp y¯κ) ; B = y¯∆ λ ; C = λ , (13)
x
− − −
2 2 2
px = xp+ −x¯p−, px = m −M xx¯, x¯ = 1−x, y¯= 1−y.
6
First, we perform the integration in z
1 z2dz 2C +B 2C 2C +B +√R
= ln , (14)
0 (Az2 +Bz +C)2 (A+B +C)R − R32 2C +B √R
Z −
2
R = B 4AC > 0.
−
Calculating the y-integral of (14) results in
1 1 ∆2 1 p2 p2
ln + ln x ln x .
−p2 2 λ2m2 2 m2 − q2x¯+q2x
x (cid:20) (cid:12) 1 2 (cid:12)(cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
Integration in x by using
(cid:12) (cid:12)
1 dx 2 1 dx p2 +iǫ 1 4
Re = L, Re ln x = (L2 π2),
p2 +iǫ −M2 p2 +iǫ m2 −M2 − 3
Z0 x Z0 x
leads to the correction α I with
2π B
1 m 1 1
2 2 2 2 2
I q ,q = L 2(L 1)ln L(L +L ) (L L ) + π , (15)
B 1 2 1 2 1 2
−2 − − λ − − 2 − 2
(cid:0) (cid:1)
where
q2
L = ln | 1,2|.
1,2 m2
Finally, oneneedstointegrateover photonmomentaq andq . Again,thelogarithmically
1 2
enhanced contribution comes from the kinematic regions m ω M/2, cosθ 1 (see
≤ ≤ → ±
definitionsin(10)). TheBornamplitude(one-loop)andthelowest-order radiativecorrection
to it can be written as (we take into account the equal contributions of regions q2 q2
1 2
| | ≪ | |
and q2 q2 )
2 1
| | ≪ | |
M 1
2 dω du α
2 2 2 2 2 2
β 1+ (I q ,m +I q ,m +I q ,q ) (16)
ω V 1 V 2 B 1 2
∼ Zm ω Z4mω22 u h 2π i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
with (we put here ∆ m2)
≈
1 1 3 m 3 3
2 2 2 2 2 2 2 2
I q ,m +I q ,m +I q ,q = L Ll l 2(L 1)ln + L+ l
V 1 V 2 B 1 2
−4 − 2 − 4 − − λ 2 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 1 3 π2
2
l L+ (l 1) l + +1, (17)
− u − 2 2 − u 6
(cid:18) (cid:19)
where we use substitutions (10) and introduce the notation
4ω2 1 β cosθ
ω
l = ln , l = ln − . (18)
m2 u 2
(cid:18) (cid:19) (cid:18) (cid:19)
7
Integration of (16) leads to (we keep only terms of order L2,L and L0)
R
virt
= 1+δ ,
virt
R
0
α 13 m 3 π2
2
δ = L 2(L 1)ln + L+ +2 . (19)
virt
π −24 − − λ 4 6
(cid:20) (cid:21)
Consider now the real photon emission corrections. One can distinguish two mechanisms
of the radiative decay π0 e+e−γ. One of them, the so-called Dalitz process, corresponds
→
to decay mode of the pion to real and virtual photons with a subsequent decay of the virtual
photon to the e+e− pair. The corresponding contribution to the width is not suppressed by
lepton mass and provides an important background to the π e+e− process [3, 16, 17].
0
→
However, the Dalitz matrix element squared and its interference with the double virtual
3 2
photon amplitude are suppressed by (1 x ) and (1 x ) as x 1 [3] (Fig. 3).
D D D
∼ − ∼ − →
This results ina negligible (of order 0.02%) interference contribution integrated inthe region
of interest for this measurement, 0.9 < x < 1 [19].
D
Another mechanism consists in creation of a lepton pair by two virtual photons with
emission of real photon by a pair components. For emission of a soft photon (with energy
ω not exceeding ∆ǫ << M in the pion rest frame) the standard calculations [13] give
2
α 2∆ε m 1 π2
2
δ = 2(L 1)ln +2(L 1)ln + L . (20)
soft
π − M − λ 2 − 3
(cid:20) (cid:21)
Emission of a hard photon was investigated in [3] with the result
α 2∆ε 3 π2 7
δ = 2(L 1)ln (L 1) + . (21)
hard
π − − M − 2 − − 3 4
(cid:20) (cid:21)
In order to find the distribution over a lepton pair invariant mass, the adequate way is to
use the method of structure functions [12] based on the application of the renormalization
group approach to QED. Here, the nonsinglet structure function is associated with a final
state fermion line. In partonic language it describes the probability for a fermion to stay a
fermion. Omitting theevents withcreationofmorethanonelepton, thenonsinglet structure
function of electron relevant to the process valid at all orders in perturbation theory is1
3 1
b−1 2
F(x ) = b(1 x ) 1+ b b(1+x )+O b , (22)
D D D
− 4 − 2
(cid:18) (cid:19)
(cid:0) (cid:1)
1 Note thatthe valueofthe K-factorin(22), K =1+3α,differs fromthatobtainedin[12]forthe process
4π
of hadron production in single-photon e+e− annihilation channel.
8
where b = 2α(L 1). To the lowest order in b and in the region x >> ν2 = 4m2/M2, the
π − D
above expression is in agreement with the leading order expression [3]
1 dRbrem(x ) α 1 1+β
LO D 2 x
= 1+x ln 2x β , (23)
R dx π 1 x D 1 β − D x
0 D − D (cid:26) (cid:18) − x(cid:19) (cid:27)
(cid:0) (cid:1)
where
ν2
β = 1 .
x
s − xD
Thus we arrive at the differential rate
1 dRRC(x )
π D = JF(x ), (24)
D
R dx
0 D
where the normalization factor J takes into account to total RC. The distribution (24),
shown in Fig. 3, in contrast to (23) is free of nonintegrable singularity at x = 1. So, we
D
see the importance of taking into account the higher orders of perturbation theory.
Adding the virtual and real photon emission contributions we finally obtain
RRC 1
π = J F(x ,L)dx = 1+α +α +α
D D virt soft hard
R
0 0
Z
α 1 3 π2 21
2
= 1 L + L + 0.968. (25)
− π 24 4 − 2 4 ≈
(cid:20) (cid:21)
The total radiative correction contains the large logarithm term L because the γ
5
∼
current is not conserved. Compared with [3] we provide a more detailed analysis of the
effective π0 e+e− vertex revealing its DL structure2. However, numerically the ratio of
→
the total RC corrections to the lowest level rate estimated in (25) is very close to 3.4%
−
found in [3] and used by the KTeV Collaboration in their analysis.
The branching ratio of the pion decay into an electron-positron pair has been measured
by the KTeV collaboration in the restricted kinematic region in order to avoid a large
backgroundfromtheDalitzprocessdominatingatlower valuesx . Byusingthedistribution
D
(22) we can estimate the factor of extrapolation of the full radiative tail beyond x > 0.95
D
as f = 1.114. With this factor and scaling the result back up by the overall radiative
0.95
correction (25) we confirm the result (2) obtained by the KTeV Collaboration.
2 It is naturally to expect the existence of DL-type RC in higher orders of perturbative theory. We do not
touch this problem here.
9
3
10
Dalitz
RC
2
10 Int
1
10
)
x
d
/
B 0
d 10
(
)
B
/
1 -1
( 10
-2
10
-3
10
0,80 0,85 0,90 0,95 1,00
2 2
x=q /M
FIG. 3: Distribution of different contributions over x = Me2e: solid line - Dalitz mechanism con-
M2
tribution, dashed line - inner bremsstrahlung contribution (24), dotted line - interference of these
two mechanisms.
III. CONCLUSIONS
In this work, we reconsidered the contribution of QED radiative corrections to the π0
→
e+e− decay which must be taken into account when comparing the theoretical prediction (8)
with experimental result (1). Comparing with earlier calculations [3], the main progress is
in detailed consideration of the γ∗γ∗ e+e− subprocess and revealing of dynamics of large
→
and small distances. The large distance subprocess associated with final state interaction
produces the terms linear in the large logarithm parameter L. The double logarithmic
contributions ( L2) correspond to configurations when the particles in the loop are highly
∼
virtual. The total result is in reduction of the normalization factor by 1 J 0.032.
− ≈
10