K → π transition form factors
and the transverse spin density
Hyeon-Dong Son
Inha University, Korea@ Progress of J-PARC Hadron Physics,
Nov. 30. - Dec. 02. 2014.

in collaboration withSeung-il Nam
Hyun-Chul Kim

Contents•

Motivation•

Kaon Tensor Transition Form Factors
from the Nonlocal Chiral Quark Model•

Generalised Parton Distributions &Transverse Densities for the Kaon transition•

Numerical Results•

Summary

Motivation•

Semileptonic decays (Kl3): CKM matrix elements |Vus|•Tensor transition provides non-standard typeinteraction[I. Baum et al, Phys. Rev. D 84, 074503 (❨2011)❩]•Transverse charge & spin densities of thetransition[C. Carlson and M. Vanderhaeghen,
Phys. Rev. Lett. 100, 032004 (❨2008)❩]

Kaon l3 decay
The decay amplitude

Weak leptonic element

Hadronic matrix element

수식, Dirac structure, flavor 더 명확하게 적어서 improve 할 것.
su가 되려면 K0 \pi- 여야 함.
예를 들어 마지막 식에 F -> \GammaHadronic Matrix Elements✤ Vector transition✤ Tensor transition✤ Scalar transition

Nonlocal Chiral Quark Model✤ The chiral effective action
derived from the instanton vacuum✤ No free parameter- Average Instanton size & separation✤ Nonlocality- Momentum-dependent dynamical quark mass
✤ Nicely reproduces pion properties: Fpi, EMFF
✤ Explicit SU(❨3)❩ symmetry breaking[D. Diakonov, Instantons at work, arXiv:hep-ph/0212026v4]

Nonlocal Chiral Quark Modelmodbess ﬀ 써서 수치계산 하는데 무슨 문제가 있었지?

✤Momentum-dependent dynamical quark masspp
M (i@) = M0 f (m)F 2 (i@)
kkk
F (k) =I0 ( )K1 ( )⇤2⇤
2⇤
FN (k) =✤✓2N ⇤2N ⇤2 + k 22⇤kk
I1 ( )K1 ( )k2⇤
2⇤◆NCurrent quark mass correction f(❨m)❩
f (m) =✤2kk
I1 ( )K0 ( )2⇤2⇤rm21+ 2dm, d ⇡ 198M eVdZero-momentum dynamical quark mass M0[M. Musakhanov Eur.Phys.J.C9,235(❨1999)❩]

Pion Decay ConstantCalculationa)b)
σµν
k2

K0

σµν
k3

k1

k2

π−K0

Experimental Values

k3π−

QCD RG Evolution✤ At the leading order[Glück et al. Zeits.Für.Phys. C.67(❨3)❩,
Barone et al. Phys.Repts., 359(❨1-2)❩, 1–168.]

Numerical Results

Display the form factor only in thespace like region.
Discuss the slope parametrisation andthe values.

1.0µ = 0.6 [GeV]

BT (t)

0.8µ = 2 [GeV]

0.6
0.4
0.2
0.0
1.00.80.60.42t [GeV ]0.20.0arX

ctor at zero-momentum transfer is fTKπ (0) = twisted-mass
0.417(14stat )(5fermionssystematic
error
syst ), where[3, 4]theandpreliminary
results haveot include the eﬀect of quenching the strangebeen
and presented
charm quarks.
Our
resultdiﬀers signifialreadyinRef.
[5].
Summaryandconclusions
from the old quenched6.resultfTKπ (0)= 0.78(6)obtained by the SPQcd R Collaboration withThe EM operator involved in the weak s → d transiinvestigated:
asses
above 500 MeV. We investigate the source of this diﬀerence andµν conclude that it istion
isgivenbys̄
F
σd, where Fµν is the EM fieldµν−
related to the chiral extrapolation.
We hasalsobeen
study
the tensor
of dethe pion and obtainstudiedusingcharge
in-flight
The Ke3 decay
tensor.
Therefore
itsmatrix
elements between kaon and−ue fTππ (0) = 0.195(8stat)(6)ingood
agreement
with,
but
more
accuratethan the result
syst
bythe“ISTRA+”
magcays of
25 GeV K , detectedpion states
involvetheonesof the weak tensor current,=0.216(34)obtainedbytheQCDSF
Collaboration
usinghigher
pion
masses.
0.003; !fS =netic spectrometer. Due tothe high
adequatewhich
can statistics,
be written
in terms of a single form factor,
2
cs is very small
resolutionof the detector,fTKπand(qgoodoverassensitivityI.), Baumet al,μ all= 2 GeV
Presentwork
umbers: 11.15.Ha,12.38.Gc, 13.20.Eb

Lattice Results

method (see ex-the Dalitz plot space, the errors in the fitted parameters√ Kπ 2are significantly reduced as comparedwiththe µprevi2fT (q )0µν0νν µ⟨π |s̄σ d|K ⟩ = (pπ pK − pπ pK ), (1)NTRODUCTION ous measurements.
tensor form factor at zero-momentum
transfer
MK + Mπ isbility cut:The λ+ parameter of the vector form-factor isfTKπ(0)(p=0.417) (5syst ) = 0.417(15)
0.0144; !fS =measured to bewhereq=pπ ) is(14thestat4-momentum
transferand. the (2K−
rements of hadron weak decays canfactor (MK + Mπ )−1 is conventionally inserted in order
meters(SM)0.0286 ±0.0008(stat)± the0.0006(syst).
λ+ = Model
where
systematic
error
does
not include any estimat
to makethetensor
form
factordimensionless.
ility
cut: of the Standard
Extrapolated to Physical meson masses
ndson
New
Physics(NP)
models.
Inof theeﬀectof quenching
thestrange
and charm
quark
Ournonlinearity
simulations
coverpion
as light
as 270 MeV±0.008; !fS =The limit on the quadratic
forf+ (t)is masses
decays of the kaon obtained
are ideally suited
Our
diﬀersfrom fromthe oldquenchedand finding
three valuesof significantly
the lattice spacing≃ 0.07up to rpossibly large CP-violating eﬀects in
sult
fTKπ0.78(6)obtainedin Ref.
theKSPQ≃ 0.1
fm.(0)
At=thephysicalpoint ourresult[6]forbythe→ πcd+0.0011′′I. V. Ajinenko
et −0.0015
al. Physics
Letterswith
B 574,(❨2003)❩.
(stat).
λ+ = −0.00042
tor(see Ref.
[1]).
Collaboration
pion 14massesabove ∼ 500 MeV. Th
0.0081;!fS =present a lattice studyTheof thematrix
reason tensor
is mainly
to couthe non-analytic behavior of thlimits
on the possible
and duescalare electromagnetic (EM)betensor form factor fTKπ (0) in terms of the quark masse
plingsoperator
are derived:
onstates,
may be relevant in+0.064introduced by the factor (MK +Mπ )−1 in the paramete
0.0104;!fSwhich=/f+semileptonic(0) = 0.021−0.075ization(stat) ±(1).
0.026(syst),
art of the K → πℓf+Tℓ−Such a behavior was not taken into accoungauge+0.020in
Ref. [6] (see later on).
ut:has been performedfSusing/f+ (0)the= 0.002−0.022(stat) ± 0.003(syst).
erated [2]In the case of the degenerate π → π transition, u±0.010;!fby
S =the European Twistedn (ETMC) with Nf = 2 maximallying the predictions of the Chiral Perturbation Theor

Generalised Parton Distributions[D. Brömmel, Pion Structure Frome the Lattice, Regensburg Univ., Thesis]
2 Generalised Parton Distributions
bybyby∼ Q12b⊥xpbxxρ(b⊥ )0a | Form factor!

b⊥b⊥

dx

xpbx

x
R∼ Q12∆→0q(x, b⊥ )b⊥xb | GPD, ξ = 0bxx"q(x)

0x
c | Parton density

Formfactors
GPD,
ξ=0
Partondistribution
Figure 2.1 | A simplified sketch of the diﬀerent phenomenological observables and theirinterpretation in the infinite momentum frame: a | The form factor as a charge density inDVCS & GPDs

How about the corresponding
processes to the weak transitions?
are they measurable from anexperiment?

e(k’)

e(k’)

γe(k)

γe(k)

γ*

γ*pQCDX+ξDVCS
φ(pi)

φ(pf)

φ(pi)

GPDs

X-ξφ(pf)

Factorization : [Collins, J., & Freund, A. (1999).Phys. Rev. D, 59(7), 074009.]

Light cone frame

Generalised Parton Distributions

lightcone gauge: n \cdot A =0

Generalised parton distributions (GPDs)

Kaon Transition GPDs for K0 →π-Expanding the nonlocal operators → Generalised Form Factors

Generalised Form Factors
Generalised form factors

VectorTensor

Polynomiality & Transverse charge density
Polynomiality: Mellin moments of the GPDs
⇒ Finite polynomial of ξ➔ Vector FF➔ EMT FFs
Transverse densities:
2D Fourier transformation into the impact parameter b⊥ at ξ=0

Probability distribution of the partons inside the hadrons
in the transverse impact parameter plane

GFFs & Transverse Densities for the Kaon TransitionK^0 스테이트에 ( p_i)가 빠져있다 ㅠㅠㅠㅠ
ㅠGeneralised form factors for kaon transitions: n =1

Quarks with definite transverse polarisation sTransverse transition density, ξ=0

[2], in LQCD, fe& !0" # 0:960 + 0:009 [12], and 0:952 +fe& !t" and fe0 !t" are almost linearly increasing func0:006 [11].
of t, whereas fe$ !t" decreases. At t # 0, our results
In the right panel of Fig. 2 we draw the ratio of fe& !
onstrate that fe& !0" # fe0 !0" # 0:947 and fe$ !0" #and fe& !0" with respect to the CPLEAR experimental da
37. In the chiral limit, fe& !0" and fe$ !0" should be[39], and linear (dashed) and quadratic (dotted) fits for th
and zero, respectively, which is related to theratio using the PDG data [8]: &e& # !2:960 + 0:05" *mollo-Gatto theorem in the case of pseudoHow the quark spinis, distributed
10$2&0e& # !2:485 + 0:163" * 10$2 , and &00e& #stone bosons [3–5]:+ 0:062"* 10$3 . Inprocess
the present calculation, w
inlocal!a"the transverse plane during!1:920
the K-πtransition
lim F!
’ 2p! & O!mq ":(29)obtain &e& # 3:028 * 10$2 for the linear fit, which
q!0very close to the experimental one, 2:960 * 10$2 . Sinc
our result for fe& is almost linear as shown in Fig. 1, we g
Ademollo-Gatto theorem in Eq. (29) can be easily
almost a negligible value for the slope parameter &00 whe
d in the nonlocal %QM. Considering q ! 0 and ignorhe terms being proportional to k ' p, we can rewrite thethe quadratic fit is taken into account. Being compare
ng contribution of Eq. (26) to order O!mq " as follows:with other model calculations, the present results are com

Transverse charge & spin density

1.20

1.50
1.25

1.15

fe (t)

0.75

f e+ (t)/ fe+ (0)1.00
f e+ (t)f e− (t)f e0(t)0.50
0.25

CPLEAR experimentLinear fit (PDG)
Quadratic fit (PDG)
Our result

1.101.05

0.00− 0.25− 0.50
0.00

0.02

0.04

0.06
0.08
t [GeV2]

0.10

0.12

1.00
0.00

0.02

0.04

0.06
0.08
t [GeV2]

0.10

0.12

2. Ke3 form factors, fe& !t" (solid), fe$ !t" (dotted), and fe0 !t" (dashed) are shown in the left panel, while in the right panel th(solid).
We
alsodraw Kim,
the CPLEAR
data[39], and linear (dashed) and quadratof fe& !t" and fe& !0" is given[S.-i.
NamandH.-Ch.
Phys. Rev.experimental
D 75, 094011(❨2007)❩.]ed) fits using the PDG data [8].

Vector form factor f+(t)1.4

The local & nonlocal things might bemisleading. It’s better to show only thetotal result.

f+

1.2

f+(t)

1.0
0.8
0.6
0.4
0.2
0.0
0.2
1.0

0.8

0.6

0.4
2
t[GeV ]

0.2

0.0

P-pole parametrisation
p-pole parametrisationis this slide necessary?

p1.31

2.2

M [GeV]

0.85

0.78

t=0

0.95

0.70

Transverse Charge Density

0.40
0.8
0.36
0.6
0.32
0.4
0.28

0.40

0.2

0.240.2

0.24

0.0

by [fm][fm 2 ]

0.200.0

0.200.2

0.160.2

0.16

0.12
0.4
0.08
0.6
0.04
0.8
0.00 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8
bx [fm]

0.12

0.8
0.6

by [fm]

0.4

0.4
0.6
0.8

0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8
bx [fm]

0.36
0.32
0.28

0.08
0.04
0.00

[fm 2 ]

Transverse Spin Density그림 업데이트할 것.
이 슬라이드랑 이 다음 슬라이드를 합칠 수 있을 것 같다.

Transverse Spin Density
3.0µ = 0.6 [GeV]⇢K 0⇡(by , sx = +1)

2.5µ = 2 [GeV]

2.0
1.5
1.00.5
0.0
0.8

0.6

0.4

0.2 0.0 0.2
by [fm]

0.4

0.6

0.8

Summary & Outlook• K to pi tensor transition form factor• In good agreement with the lattice result• Distorted

spin structure during the transition
process when the quark spin is polarized• Transition GPDs of the pseudo scalar mesons

Thank you very much!