Radiative kaon decays
2 processes:
Internal Bremsstrahlung (IB)Radiation from external charged particles
Completely determined by non-radiative amplitudes

Observables:

Direct-emission (DE) or Structure-Dependent (SD)Radiation from intermediate hadronic states
Theoretical approaches:•! Current algebra: Fearing et al. [FFS] ’70•! Increasingly precise ChPT calculations

RX!" (Emin, #min) =!
BRX!" (E"*>Emin, #!!*>#min)
BRX!(")
SD/(IB + SD) for E"*>Emin, #!!*>#min

Amplitudes are IR divergent at any
fixed order in !
Physical amplitudes strongly peaked
for E"* ! 0 (or for #!!* ! 0 if m ! 0)
Separate by analysis of E"* spectrum

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 20092

Ke2 and Ke2!: Current status
RK !

"(K # e$ (% IB ))"(K # µ$ (% IB ))Cirigiliano, Rosell ’07

Definition of RK inclusive of IB only

RKSM = 2.477(1) ! 10"5Helicity suppression: IB ~ SD

MSSM with R parityu~lR

H+s,dlR

Masiero,Petronzio,Paradisi ’06

Experimental data is from ’70s:#"(SD)/"(SD) " 15%~B~#L#$~1% effect on RK for:!13 ~ 5!10!4 tan " ~ 40 MH+ ~ 500 GeV

IB + SD
IBE!* [MeV]KLOE ’09 final:
NA62 ’09 prelim:

RK = 2.493(31) ! 10"5RK = 2.500(16) ! 10"5SD contribution to KLOE error
reduced from 0.5% ! 0.2%

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 20093

Ke2! amplitudes !
d!( K " e#$ )dx dynegligible= %IB ( x, y ) + % SD ( x, y ) + %INT ( x, y )22Fx = 2E!* mKy = 2Ee* mK

G Vus "

22(*5%'mV+A
fx,y+
V$A
fx,y()
()
()
()
K&SD+
SD$)
64# 2
V, A: effective vector and axial couplings

pe* [MeV]

pe* [MeV]! SD ( x, y ) =SD+
V+A

SD!!
V"Ae+

#

endpoint of Ke3 decays

e+

#!

!
E!* [MeV]!

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009

E!* [MeV]!
4

1.! ChPT at O(p4)No dependence on ! energy
V ! 0.0945
A ! 0.0425
Bijnens, Ecker, Gasser ’93

106 dBR(Ke3!)/dx

Ke2!: Theoretical predictions for SD

2.! ChPT at O(p6)Linear energy dependence for V
V ! 0.082[1 + "(1 – x)] with " ! 0.4
A ! 0.034

ChPT O(p4)
ChPT O(p6)
LFQM

IBAmetller, Bijnens, Bramon, Cornet ’93
Geng, Ho, Wu ’04
Chen, Geng, Lih ’08

3.! Light Front Quark Model (LFQM)
Non-trivial x dependence
V = A = 0 at x = 0 or t = tmax = mK2

SDx = 2E!*/mKChen, Geng, Lih ’08
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 20095Ke2! event selection
Starting sample of Ke2(!) events same as for analysis of RK
1-prong selection with tight track-quality cuts
Make hard cut on neural-network output to reject Kµ2! with accidental photon
Neural network uses EmC shower profile information, TOF
Require detection of ! with Ecal > 20 MeV to further reject accidentals/fragments
Cluster times for photon and electron must be compatible:!t!e = (t! – r!/c) – (te – Le/c) < 2"Ke2"!

K!2!t!e/"!t!e/"! from #0$(#+) ! 0.8, not 1

Kµ2!t!e/"Accidentals flat in time

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 20096

Ke2! sample definition
Dominant backgrounds:
Kµ2 for M2! < 20000 MeV2
Ke3 for M2! > 20000 MeV2
No sensitivity for Ke2! withpe < 200 MeV (SD! amplitude)

dataKe3
Kµ2

We measure Ke2! with:
E!* > 10 MeV
cos "e!* < 0.9
pe* > 200 MeV
This selection has:
Acceptance for SD+ events ~90%
SD! events ~2%
Residual IB events ~1%

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009

Ke2!
E! > 10 MeV
pe > 200 MeV

Ke2!
E! > 10 MeV
pe > 200 MeV

M2" [MeV2]7

Ke2! photon association
E!lab can be evaluated from Ke2!
kinematics, using measurements of:•! track momenta pK, pe•! photon direction n! from cluster
and vertex positionsElab!

mK2 + me2 " 2E K Ee + 2p K # pe=2( E K " Ee " p K # n ! + pe # n ! )#Elab ! 12 MeV#Ecal ! 30 MeV

Ke2!
Ke3!E! = E!lab – E!cal useful for signal/
background separation
Perform 2-dimensional binned
likelihood fit in M"2, !E!/# in 5 bins of E!*
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009!E!/"8p"* [MeV]MC kinematics for samples in Ke2! fit

K ! e#!
E!*(MC) > 10 MeV
and cos "!#*(MC) < 0.9
SD enhanced

K ! e#!
E!*(MC) < 10 MeV
or cos "!#* (MC) > 0.9
Essentially all IB

K ! µ#(!)
including accidental !s

Other background
Mainly Ke3
E!* [MeV]

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 20099

Ke2! fit results
Projections on M#2 axis for 2 most populated E!* bins

100 < E! < 150 MeV+! data!! fit!! Ke2!
!! Kµ2!! Ke3

Ne2! = 463 ± 32
"2/ndf = 87/106
M#2 [MeV2]150 < E! < 200 MeV
Ne2! = 494 ± 38
"2/ndf = 100/106
M#2 [MeV2]
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200910

Ke2! fit results
Projections on !E!/" axis for all 5 E!* bins, with cuts on M#2Signal
enhancedsignaldominated

Backgroundenhanced
data
fit
Kµ2
M!2 [MeV2]
Ke3bkgdominated

data
fit
Kµ2
M!2 [MeV2]
Ke3+! data!! fit!! Kµ2!! Ke3!E"/#$!E!/"
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009!E!/"11We measure
d!SD+ (K e2 " )1!(K µ 2( " ) )dE"
where “SD+” means:
E!* > 10 MeV
cos "e!* < 0.9
pe* > 200 MeV

d!SD+(Ke2!)/!(Kµ2(!)) ! 105

Ke2! spectrum vs O(p4) ChPT•! data!! O(p4) ChPT!! IB!2 = 5.4/5

Summed over all bins in E!*:

NSD+(Ke2!) = 1378 ± 63
"SD+(Ke2!)"5=1.484(66)(16)!
10stsy"(Kµ2(!))

E!* [MeV]!

in agreement with ChPT O(p4) prediction, 1.447 ! 10"5 [Bijnens, Ecker, Gasser ’93]
KLOE MC implements O(p4) ChPT for SD – used in analysis of RK
Validated to within 4.6% - systematic error on RK from SD = 0.2%
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200912

Fit data to extract V + A for SD+

Get slope of vector form factor
V = V0 [1 + "(1 – x)]
Not sensitive to SD" amplitude
Acceptance ~2%Fix V " A to O(p6) prediction
Obtain:V0 + A = 0.125 ± 0.007!
! = 0.38 ± 0.21

d!SD+(Ke2!)/!(Kµ2(!)) ! 105

Ke2! spectrum: fit to O(p6) ChPT•! data!! O(p6) ChPT fit!! IB

Compare to ChPT O(p6):
V0 + A ! 0.116! ! 0.4
Chen, Geng, Lih ’08

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009

!2 = 1.9/3

E!* [MeV]!

13

Light Front Quark Modelwith parameters as inChen, Geng, Lih, ’08
Excluded by our data
!2 = 127/5

d!SD+(Ke2!)/!(Kµ2(!)) ! 105

Ke2! spectrum vs LFQM•! data!! LFQM!! IB!2 = 127/5

E!* [MeV]!

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200914

Preexisting data on KL ! !e"# (Ke3#)
Most common observable: Re3" (E"$ > 30 MeV, %e"$ > 20°)
KTeV
PRD 64 (2001)

Re3" = 0.908(8)(+13#12)%
15575 evts over 1% bkg
Attempt to study d!/dE#*

NA31 ’96

Isolation of SD complicatedby use of cumbersome
theoretical frameworkKTeV
PRD 71 (2005)

NA48
PLB 605 (2005)

KTeV !01
KTeV ’05

Re3" = 0.916(17)%4309 evts over 1% bkg
Reanalysis of previous data
Tighter cuts, 30% subsample
No attempt to isolate SD
Re3" = 0.964(8)(+11#9)%
18977 evts over 7% bkg
No attempt to isolate SD

NA48 ’05
GKPV ’05
0.96(1)%

Re3! [%]

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009

IB only15

Ke3! theory statusO(p6) decomposition of photon spectrum:d!Vid!Aid! d!IB=+ # Vi+ Ai***
dE" dE" i=1,4 dE"dE"*
V1, V2, A1 amplitudes dominant,have similar dependence on E!*d! d!IB#=+$X%f(E" )##dE" dE"

Contribution to !(Ke3!) [a.u.]

Gasser, Kubis, Paver, Verbeni [GKPV] ’05! X " # !V1 " + 0.4!V2 " + 0.4! A1 "IB/200
V1
V2
A1
A2E!* [MeV]O(p6) estimates (for V) and full calculations (for A) give ! X " = #1.2 ± 0.4
Error from estimates of O(p6) LECs
Also have Re3! = (0.963 + 0.006" X # ± 0.010)%
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200916

Inclusive sample (Ke3(!))•! 328 pb!1 of ’01 + ’02 data (same sample as for Vus)•! KL decay tagged by KS ! "+"!
•! Pos. and neg. tracks with vertex in fiducial volume•! Kinematic rejection for Kµ3 and "+"!"0 decays (e.g. Mmiss)•! PID/charge assignment by TOF: "tx = tmeas ! tpred(mx)~ 3!106 Ke3(!) events, 0.7% background"t("+) ! "t(e!) [ns]Ke3! at KLOE: Event selection
"+e!

"!e+

Radiative sample (Ke3!)•! Look for photon cluster with timing compatible with"t("!) ! "t(e+) [ns]distance from track vertex•! Predict its energy from kinematics with # " 1 MeV•! Cut on measured energy•! Refine using neural networks to reject Kµ3 and "+"!"0decays to ~1% level•! Use "+"!"0 decays to check photon reconstruction:KL e3

p2!2 = 0 = (pK – p" – p" – p!1)2
p2$ = 0 = (pK – p" – pe – p!)2

KS " "+"!

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009!

17

Ke3!: Fit to (E!*, "#!*) distribution"e!* [deg]Monte Carlo distributions in 9!9 bins in "e!* vs. E!* for:
IBSD

3" + Kµ3(!)

Ke3(!)
true E!*< 20 MeV

Fit results: E!* distributions for in 9 bins in "e!*"NE!* [MeV]NIBNSD

8981 ± 206
9083 ± 213!102 ± 59

Ne3(!)

6726 ± 194

e3!

N3$ + µ3(!) fixed%2/ndf = 60/69 (P = 77%)
E!* [MeV]

Destructive interference
between IB and SD

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200918

Ke3!: Results for Re3! and !X"Source

Re3! (E!" > 30 MeV, #e!" > 20°) = 0.924(23)st(16)sy %!X" = !2.3 ± 1.3st ± 1.4sy!Re3! " 105 !!X" " 105

Tag & trigger

4.0

0.7

Tracking

1.5

0.8

Clustering

5.5

0.1

TOF cuts

1.3

0.5

p calibration

3.50.2p resolution

7.2

0.4Fiducial volume

3.0

0.5

Accidentals

5.20.4! association

2.90.3Background

9.0

0.1

Total

15.5

1.4

ChPT constraint on KLOE result [GKPV ’05]

Re3! = (0.963 + 0.006!X" ± 0.010)%
Re3! [%]KLOE
EPJC 55 (2008)

Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 2009

!X"

19

Summary
Ke2!:!SD+(Ke2!)"5=1.484(66)(16)!
10stsy!(Kµ2(!))
Agreement with O(p4) ChPT used in KLOE MC gives systematic
error on RK from SD = 0.2%
Fits performed to determine vector form-factor parameters forcomparison with ChPT at O(p6)
LFQM ruled out

Ke3!:
Re3! (E!" > 30 MeV, #e!" > 20°) = 0.924(23)st(16)sy %!X" = "2.3 ± 1.3st ± 1.4sy
First complete study of d!/dE!* for KL ! "e#!
Measurement (esp. for !X") limited by statistics –~7! more data available
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200920The KLOE experiment
Be beam pipe (0.5 mm thick)Instrumented permanent magnet
quadrupoles (32 PMTs)
Drift chamber (4 m ! " 3.3 m)
90%He+10% IsoB, composite frame
12582 stereo sense wires
Electromagnetic calorimeter
Lead/scintillating fibers
4880 PMTs
Superconducting coil (5 m bore)
B = 0.52 T ( # B dl = 2 T!m)Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200922

KLOE generators for kaon decays
include radiation with no cutoff
energy•! Full O(!) amplitudes (real and virtualcontributions) summed to all orders
in ! by exponentiation (soft-photonapproximation)•! Carefully checked against allavailable data and calculations

dN/dE!*Generators for radiative kaon decays

Re3! (E! > 30 MeV, "e! > 20°)KLOE generator

0.93%

Bijnens, Ecker, Gasser ’93
Gasser, Kubis, Paver, Verbeni ’05

0.94%
0.96(1)%

b%(d!
O(#)
E
( ) $' " *d!
=dE"dE"& mK )

Ke3! generator:
2.4% E"* > 30 MeV
E!* [MeV]
C. Gatti, EPJC 40 (2005)

Accuracy of KLOE generator more than sufficient for inclusion of radiation in BR mmts.
Separate generator [GPKV ‘05] used to estimate !X" in study of Ke3"
Recent KLOE results on radiative kaon decays M. Moulson (Frascati) - KAON '09 - Tsukuba, Japan - 10 June 200923