Radiative corrections in Kl3

decays

Andrea MarroccoPh.D. student, University “Roma Tre”

[Ph.D. thesis supervisor: G. Isidori]

Thesis Decay rate calculation including

radiative corrections

0

0

K

K

Independent determination of |Vus|Improve the theoretical indetermination of this parameter

The rates ratio of the electronic and the muonic channel is |Vus| independent

It is possible to test the accuracy required in the CHPT expansion

Dimensional regularization for both UV & IR divergences

The decay amplitude

Real photon emissionγ,q

K+,p4 π0,p1

νµ,p3

µ+,p2

γ,qK+,p4

π0,p1

νµ,p3µ+,p2

K+,p4 π0,p1

µ+,p2

γ,q

νµ,p3

2

Virtual photon

exchange

K+,p4

µ+,p2

π0,p1

νµ,p3 kq

hνµ,p3

π0,p1K+,p4

µ+,p2

+

π0,p1K+,p4

µ+,p2

νµ,p3

+

q

h

K+,p4 π0,p1

νµ,p3µ+,p2

+

K+,p4

µ+,p2

π0,p1

νµ,p3

+

K+,p4 π0,p1

νµ,p3

µ+,p2

+

K+,p4

µ+,p2

π0,p1

νµ,p3+

K+,p4 π0,p1

νµ,p3

µ+,p2

+

K+,p4π0,p1

νµ,p3µ+,p2+

K+,p4

µ+,p2

π0,p1

νµ,p3

2

1414

25

3*

,,

12

ppvtfppvtf

pvpuVG

A

KK

usF

Self energy

Vertex modification

51 AZ

Wave function renormalization

512

1 AZ

+ 512

1 AZ

A,p A,p

pA

22

2

Amp

AA p

pZ

“A” particle self energy

1

02)1(

1

bzaz

dz

ab

Standard Feynman parameterization

1

2

22

2

2

22

11

224

1

2

12

4

11

2

dn

dn

E

Ed

Ed

dn

dn

E

dE

d

n

dn

d

l

lld

n

dn

l

ld

K+,p4

h

q

K+,p4

K+,p4 K+,p4+K+,p4

h

q

K+,p4

K+,p4 K+,p4+

2

6522

212

2222

2422

9

20

3

8

2

2

2kk

kkk

kd

dd

kK pkkepkkeimppqqiq

qpqdiep

Ki = coefficients of the O(e2p2) mesonic Lagrangian [Urech, ‘95]

652

212

2

22

22

2

22

9

20

3

8,1

2,1,

21

2

2,2

2,3,

22

4

8

,12,2,

22

2

8,2,1,

22

2

2

4

22

kkekkenn

Fn

nnF

nn

nnF

nn

nnF

nm

n

me kn

n

kk

2

2

k

k

m

p

1

0

11 11,,, dxxxxbcb

ccbaF abcb

Hypergeometric functions

65

221

22

22

2

22

12 9

20

3

8

3212

1

4

22

1kkekke

nnn

n

me

mZ n

n

k

K

kK

μ+,p2 μ+,p2+

h

q

μ+,p2μ+,p2

pXe

iqiqpq

mpqqdiep n

nd

62

2242

22

6

22

22

2

22

13

1

4

22

1Xe

n

nn

me

mZ n

n

6222

2

22

2

22 ,1

2,2,

22

4,2,1,

22

2

2

4

22

Xemnn

Fn

nnF

n

nm

n

me n

n

m

p

Xi = coeff. of the O(e2p2) leptonic Lagrangian [Neufeld & Rupertsberger, ’95-96]

vpunn

Fnn

nnF

n

n

m

VGiekn

n

k

usF52

22

2

2*2

1,12,2,

22

2

4,2,1,

22

2

2

4

22

2

vppuvppuvpu kkk 555 12

11

2

11

K+,p4 π0,p1

νµ,p3

µ+,p2

222254

*2

2

12

22 kkk

kn

nnusF

mpqpqiq

vqpuqdVGe

2

2

k

k

m

p

K+,p4π0,p1

νµ,p3µ+,p2

2222

54*2

2

1

22 kn

nnusF

mpqpqiq

vmpquqdVGe

vum

n

nn

m

VGien

n

usF52

22

2

2*2

13

1

4

22

2

vppuvum k 55 11 Contributes only to the f-

function

µ+,p2νµ,p3

π0,p1K+,p4

k

q

h

2425413

42

222

42

21

22

1

22

pvqpmqpqpppu

iqpqiqpqiq

qdVGe

rs

n

nnusF

22153

2

12

pvIIpuVGe rsusF

1

03

1

0 11

12

1

cyxxybxadyxdx

abcFeynman parameterization

251432

251432

2

1,1,2

pvpppuppmHpvpppuppmGVGie rs

Kjrs

KjusF

Full agreement with Cirigliano et al.

24

4log2log

22

mm

M

mm KK

Real photon emission process

pvpuqVieG rusF

5*

*

12

pvqpppu

pq

pVieGk

k

kr

usF5

**

12

pvmqppppu

qp

qVieG

k

r

usF

5

**

1

22

0K

γ,q

K+,p4 π0,p1

νµ,p3

µ+,p2

γ,qK+,p4

π0,p1

νµ,p3µ+,p2

K+,p4 π0,p1

µ+,p2

γ,q

νµ,p3

2

Phase space separation 21

324 2,,;,;,,,; dlqplpdppldqppppd nkk

nlG

ln

mlppld

nnn

nn ,

21

2

,;2 2

1222

342

322

2

This formula is valid in n dimensions and the

result is based on Lorentz-covariance

considerations

In K+ rest frame

qppppdMm

mmd k

n

k

,,,;2 4

2

m

j jn

jnm

iimm

E

pdpPppPd

11

1

41

22,....,;

Decay rate

2

22

1

0

1

022

2

32

3

2

322

32

2

322

22

2

1222

21

1

21

1

,

,

22

21

2

12

222

2

2

M

xEC

mE

xddx

xEC

mE

xx

ECllEA

lEAmEdEdl

nnm

mm

nn

n

nnm

lmm

m

mm

m

nnk

k

k

k

22

22

2222 2,

mEEmEC

mEEmEC

lEmmmlEA

k

k

kk

The infrared divergence is hidden in this

factor

It has no divergent terms and can be

analytically expressed using hypergeometric

functions

522

2

2222

,1

0,2

n

k

k

lEAE

M

lEAm

lmmE

21cos

21

2

, 222

xEC

mE

EC

lEAE

Strategy to isolate the divergences

25222

,,,,,

222

lEnflEAlEnkdEn

m

lmm

m

k

k

result of the integration

on the photon

variables

All other factors are in this function

Coordinate transformation

22

2,

lmm

lEAz

k

1

0 0

221

1

0

21 ,,0,,,, dz

lzjlzjzdzlzjz

0,0,0,0,0,,0,0, 21

0

222

lj

dzz

ljzljlj

• Full analytical agreement with Cirigliano et al. in the virtual corrections

• Differential decay rate calculation with real emission completed -> explicit check of the cancellation of infrared divergences

• The IR-safe observable differential rate depends on z and l2. For each bin of l2 we are numerically calculating the O(α) corrections to the decay rate (numerical results in progress..).

• Calculation performed for both K+ and K0 decays

• For each channel we expect to reach the same accuracy of Cirigliano et al. (counterterms error ~ few x 0.001)

• For the ratio between the decay rates of electronic channel and muonic channel we expect a better accuracy because many counterterms cancel (~ 0.001)

Conclusions & Outlook