1

62. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays

62. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion(π`2γ ) and Kaon (K`2γ ) Decays
Revised August 2023 by M.A. Bychkov (Virginia U.) and G. D’Ambrosio (INFN, Napoli).
±±
62.1 π`2γ
and K`2γ
Form Factors

The radiative decays, π ± → l± νγ and K ± → l± νγ, with l standing for an e or a µ, and γ fora real or virtual photon (e+ e− pair), provide a powerful tool to investigate the hadronic structureof pions and kaons. The structure-dependent part SDi of the amplitude describes the emission ofphotons from virtual hadronic states, and is parametrized in terms of form factors V, A, (vector,
axial vector), in the standard description [1–4]. Note that in the Listings and some literature,equivalent nomenclature FV and FA for the vector and axial form factors is often used. Exotic,
non-standard contributions like i = T, S (tensor, scalar) have also been considered. Apart fromthe SD terms, there is also the Inner Bremsstrahlung amplitude, IB, corresponding to photon
radiation from external charged particles and described by Low theorem in terms of the physicaldecay π ± (K ± ) → l± ν. Experiments try to optimize their kinematics so as to minimize the IB partof the amplitude.
The SD amplitude in its standard form is given as−eGF Uqq0 µ ν P√
 l V µνστ k σ q τ
2mP−ieGF Uqq0 µ ν P
M (SDA ) = √ l {A [(qk − k 2 )gµν − qµ kν ]2mP
+ RP k 2 gµν } ,M (SDV ) =(62.1)(62.2)

which contains an additional axial form factor RP which only can be accessed if the photon remains
virtual. Uqq0 is the Cabibbo-Kobayashi-Maskawa mixing-matrix element; µ is the polarization
vector of the photon (or the effective vertex, µ = (e/k 2 )u(p− )γ µ v(p+ ), of the e+ e− pair); `ν =u(pν )γ ν (1−γ5 )v(p` ) is the lepton-neutrino current; q and k are the meson and photon four-momenta(k = p+ + p− for virtual photons); and P stands for π or K.
For decay processes where the photon is real, the partial decay width can be written in analyticalform as a sum of IB, SD, and IB/SD interference terms INT [1, 4]:d2 ΓP →`νγd2 (ΓIB + ΓSD + ΓINT )
=
dxdy
dxdy
(

α
1
=
ΓP →`ν
IB(x, y)2π
(1 − r)2
+

1
r



mP
2fP

2 h

i

(V + A)2 SD+ (x, y) + (V − A)2 SD− (x, y)
)

i
mP h
−
+ P
(V + A)S+.
INT (x, y) + (V − A)SINT (x, y)fP
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:18am(62.3)

2

62. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays

Here
"

1−y+r
IB(x, y) = 2
x (x + y − 1 − r)"#

2xr(1 − r)
x + 2(1 − x)(1 − r) −x+y−1−r#2h

SD+ (x, y) = (x + y − 1 − r) (x + y − 1)(1 − x) − rhSD− (x, y) = (1 − y + r) (1 − x)(1 − y) + r"

S+
INT (x, y) ="S−
INT (x, y) =1−y+rx(x + y − 1 − r)

#"

1−y+r
x(x + y − 1 − r)

#"

i

i
#

(1 − x)(1 − x − y) + r#
2x − (1 − x)(1 − x − y) − r(62.4)

0.12

0.8

-

1

0.14
0.12

0.8

0.1

0.1
0.6

0.08
0.06

0.4

0.6

0.08
0.06

0.4

0.04

0.04
0.2

0.2
0.02

0.02
0
0

SD

0.14

y

+

1

SD

y

where x = 2Eγ /mP , y = 2E` /mP , r = (m` /mP )2 , fP is the meson decay constant, and P is +1
for pions and -1 for kaons. The structure dependent terms SD+ and SD− are shown in Fig. 1.
The SD− term is maximized in the same kinematic region where overwhelming IB term dominates(along x + y = 1 diagonal). Thus experimental yields with less background are dominated by SD+
contribution and proportional to AP + V P making simultaneous precise determination of the form
factors difficult.

0.2 0.4 0.6 0.8

1
x

0

0
0

0.2 0.4 0.6 0.8

1x0

Figure 62.1: Components of the structure dependent terms of the decay width. Left: SD+ , right:
SD−
Recently, formulas 62.3 and 62.4 have been extended to describe polarized distributions inradiative meson and muon decays [5].
The “helicity” factor r is responsible for the enhancement of the SD over the IB amplitude in
the decays π ± → e± νγ, while π ± → µ± νγ is dominated by IB. Interference terms are important forthe decay K ± → µ± νγ [6], but contribute only a few percent correction to pion decays. However,they provide the basis for determining the signs of V and A. Radiative corrections to the decay
31st May, 2024

3

62. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays

π + → e+ νγ have to be taken into account in the analysis of the precision experiments. They makeup to 4% corrections in the total decay rate [7]. In π ± → e± νe+ e− and K ± → `± νe+ e− decays,
all three form factors, V P , AP , and RP , can be determined [8, 9].
Theoretically, the first non-trivial χP T contributions to AP and V P appear at O(p4 ) [4], respectively from Gasser-Leutwyler coefficients, Li ’s, and the anomalous lagrangian:√√
4 2MP r
2MP
P
r
P
A =
(L9 + L10 ), V =.
(62.5)
Fπ
8π 2 FπIn case of the kaon AK = 0.042 and V K = 0.096. O(p6 ) contributions to AK can be predictedaccurately: they are flat in the momentum dependence and shift the O(p4 ) value to 0.034. O(p6 )contributions to V K are model dependent and can be approximated by a form factor linearly
dependent on momentum. For example, when looking at the spread of results obtained within two
different models, the constant piece of this linear form factor is shifted to 0.078 ± 0.005 [1, 2, 4].
We give the experimental π ± form factors V π , Aπ , and Rπ in the Listings. In the K ± Listings, wegive the extracted sum AK +V K and difference AK −V K , as well as V K , AK and RK . In particular
KLOE has measured for the constant piece of the form factor AK + V K = 0.125 ± 0.007 ± 0.001 [10]while ISTRA+, V K − AK = 0.21 ± 0.04 ± 0.04 [11].
0The pion vector form factor, V π , is related via CVC (Conserved
q Vector Current) to the π → γγπdecay width. The constant term is given by |V (0)| = (1/α) 2Γπ0 →γγ /πmπ0 [3]. The resulting
value, V π (0) = 0.0259(9), has been confirmed by calculations based on chiral perturbation theory(χP T ) [4], and by two experiments given in the Listings.
A recent experiment by the PIBETA collaboration [12] obtained a V π (0) that is in excellent
agreement with the CVC hypothesis. It also measured the slope parameter a in V π (s) = V π (0)(1 +a·s), where s = (1−2Eγ /mπ ), and Eγ is the gamma energy in the pion rest frame: a = 0.095±0.058.
A functional dependence on s is expected for all form factors. It becomes non-negligible in the caseof V π (s) when a wide range of photon momenta is recorded; proper treatment in the analysis of K
decays is mandatory.
The form factor, RP , can be related to the electromagnetic radius, rP , of the meson [2]: RP =1
2
3 mP fP hrP i using PCAC (Partial Conserved Axial vector Current).
In lowest order χP T , the ratio Aπ /V π is related to the pion electric polarizability αE =[α/(8π 2 mπ fπ2 )] × Aπ /V π [13]. Direct experimental and theoretical status of pion polarizability
studies currently is not settled. Most recent theoretical predictions from χP T at O(p6 ) [14] andexperimental results from COMPASS collaboration [15] favor a small value of pion polarizability απ ∼ (2 ÷ 3) × 10−4 fm3 . Dispersive analysis of γγ → π + π − crossection [16] and experimental
results from MAMI collaboration [17] report a much larger value of απ ∼ 6×10−4 fm3 . Precise measurement of the pion form factors by PIBETA collaboration favors smaller values of polarizability−4 fm3 .
απ = 2.7+0.6−0.5 × 10
Several searches for the exotic form factors FTπ , FTK (tensor), and FSK (scalar) have been pursued
in the past. In particular, FTπ has been brought into focus by experimental as well as theoreticalwork [18]. New high-statistics data from the PIBETA collaboration have been re-analyzed together
with an additional data set optimized for low backgrounds in the radiative pion decay. In particular,
lower beam rates have been used in order to reduce the accidental background, thereby making thetreatment of systematic uncertainties easier and more reliable. The PIBETA analysis now restricts
FTπ to the range −5.2 × 10−4 < FTπ < 4.0 × 10−4 at a 90% confidence limit [12]. This result is inexcellent agreement with the most recent theoretical work [4].
Precision measurements of radiative pion and kaon decays are effective tools to study QCD in
the non-perturbative region and are of interest beyond the scope of radiative decays. Meanwhile
31st May, 2024

462. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays

other processes such as π + → e+ ν that seem to be better suited to search for new physics at the
precision frontier are currently studied. The advantages of such process are the very accurate andreliable theoretical predictions and the more straightforward experimental analysis.
±0Form Factors
62.2 K`3
and K`3

Assuming that only the vector current contributes to K → π`ν decays, we write the matrix
element ash

M ∝ f+ (t) (PK + Pπ )µ `γµ (1 + γ5 )ν
hii+ f− (t) m` `(1 + γ5 )ν ,(62.6)

where PK and Pπ are the four-momenta of the K and π mesons, m` is the lepton mass, and f+
and f− are dimensionless form factors which can depend only on t = (PK − Pπ )2 , the square of the
four-momentum transfer to the leptons. If time-reversal invariance holds, f+ and f− are relatively
real. Kµ3 experiments, discussed immediately below, measure f+ and f− , while Ke3 experiments,
discussed further below, are sensitive only to f+ because the small electron mass makes the f−term negligible.
62.2.1 Kµ3 Decays
Analyses of Kµ3 data frequently assume a linear dependence of f+ and f− on t, i.e.,ihf± (t) = f± (0) 1 + λ± (t/m2π+ ) .

(62.7)

Most Kµ3 data are adequately described by formula 62.7 for f+ and a constant f− (i.e., λ− = 0).
There are two equivalent parametrizations commonly used in these analyses: λ+ , ξ(0) parametrization and λ+ , λ0 parametrization.
Older analyses of Kµ3 data often introduce the ratio of the two form factors
ξ(t) = f− (t)/f+ (t) .

(62.8)The Kµ3 decay distribution is then described by the two parameters λ+ and ξ(0) (assuming time
reversal invariance and λ− = 0).
More recent Kµ3 analyses have parametrized in terms of the form factors f+ and f0 , which areassociated with vector and scalar exchange, respectively, to the lepton pair. f0 is related to f+ and
f− by
h

i

f0 (t) = f+ (t) + t/(m2K − m2π ) f− (t) .

(62.9)

Here f0 (0) must equal f+ (0). The earlier assumption that f+ is linear in t and f− is constant leads
to f0 linear in t:hif0 (t) = f0 (0) 1 + λ0 (t/m2π+ ) .

(62.10)With the assumption that f0 (0) = f+ (0), the two parametrizations, (λ+ , ξ(0)) and (λ+ , λ0 ) areequivalent as long as correlation information is retained. (λ+ , λ0 ) correlations tend to be lessstrong than (λ+ , ξ(0)) correlations.
Since the 2006 edition of the Review [19], we no longer quote results in the (λ+ , ξ(0)) parametrization. We have removed many older low statistics results from the Listings. See the 2004 version of0
this note [20] for these older results, and the 1982 version [21] for additional discussion of the Kµ3
parameters, correlations, and conversion between parametrizations.
31st May, 2024

562. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays

More recent high-statistics experiments have included a quadratic term in the expansion off+ (t),"

f+ (t) = f+ (0)00λ
1 + λ+ (t/m2π+ ) + + (t/m2π+ )2
0

#

.

2

(62.11)

If there is a non-vanishing quadratic term, then λ+ of formula 62.7 represents the average slope,0which is then different from λ+ . Our convention is to include the factor 12 in the quadratic term,++
and to use mπ+ even for Ke3
and Kµ3
decays. We have converted other’s parametrizations to±
0 Form Factors" sections ofmatch our conventions, as noted in the beginning of the “K`3
and K`3
the Listings.
There are two alternatives to the Taylor parametrization: The Pole Parametrization and Dispersive Parametrization.
The pole model describes the t-dependence of f+ (t) and f0 (t) in terms of the exchange of thelightest vector and scalar K ∗ mesons with masses MV and MS , respectively:"

MV2
f+ (t) = f+ (0)MV2 − t#"MS2
f0 (t) = f0 (0)MS2 − t,#.

(62.12)

The Dispersive Parametrization approach, valid in a much wider kinematic range and able todescribe at the same time τ -decay data, [22] uses dispersive techniques and the known low-energy
K-π phases to parametrize the vector and scalar form factors:f+ (t) = f+ (0) expt(Λ+ + H(t)) ;
m2π


"

(62.13)#t
f0 (t) = f+ (0) exp(ln[C] − G(t)) ,2(mK − m2π )(62.14)

where Λ+ is the slope of the vector form factor, and ln C = ln [f0 (m2K − m2π )] is the logarithm of
the scalar form factor at the Callan-Treiman point. The functions H(t) and G(t) are dispersive
integrals.
62.2.2 Ke3 Decays
Analysis of Ke3 data is simpler than that of Kµ3 because the second term of the matrix element
assuming a pure vector current [formula 62.6 above] can be neglected. Here f+ can be assumed tobe linear in t, in which case the linear coefficient λ+ of formula 62.7 is determined, or quadratic, in
000which case the linear coefficient λ+ and quadratic coefficient λ+ of formula 62.11 are determined.
If we remove the assumption of a pure vector current, then the matrix element for the decay,in addition to the terms in formula 62.6, would contain+2mK fS `(1 + γ5 )ν
+(2fT /mK )(PK )λ (Pπ )µ ` σλµ (1 + γ5 )ν ,(62.15)where fS is the scalar form factor, and fT is the tensor form factor. In the case of the Ke3 decays
where the f− term can be neglected, experiments have yielded limits on |fS /f+ | and |fT /f+ |.
For Ke3 data, we determine best values for the three parametrizations: linear (λ+ ), quadratic
000(λ+ , λ+ ) and pole (MV ). For Kµ3 data, we determine best values for the three parametrizations:
31st May, 2024

662. Form Factors for Semileptonic Kaon (K`3 ), Radiative Pion (π`2γ ) and Kaon
(K`2γ ) Decays
000linear (λ+ , λ0 ), quadratic (λ+ , λ+ , λ0 ) and pole (MV , MS ). We then assume µ − e universality sothat we can combine Ke3 and Kµ3 data, and again determine best values for the three parametriza0
00
tions: linear (λ+ , λ0 ), quadratic (λ+ , λ+ , λ0 ), and pole (MV , MS ). When there is more than one
parameter, fits are done including input correlations. Simple averages suffice in the two Ke3 cases
where there is only one parameter: linear (λ+ ) and pole (MV ).
A comprehensive global analysis of the semileptonic kaon decay data and its effect on the CKM
unitarity debate can be found in [23,24]. An update on experimental data including NA48/2 newest
results can be found in [25].
Both KTeV and KLOE see an improvement in the quality of their fits relative to linear fits whena quadratic term is introduced, as well as when the pole parametrization is used. The quadratic
00
parametrization has the disadvantage that the quadratic parameter λ+ is highly correlated with0
the linear parameter λ+ , in the neighborhood of 95%, and that neither parameter is very well determined. The pole fit has the same number of parameters as the linear fit, but yields slightly better fit
probabilities, so that it would be advisable for all experiments to include the pole parametrization
as one of their choices.
The “Kaon Particle Listings" show the results with and without assuming µ-e universality. The“Meson Summary Tables" show all of the results assuming µ-e universality, but most results notassuming µ-e universality are given only in the Listings.
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