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Page 1: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

Order p6 Chiral Couplings from the Scalar K Form Factor

José A. OllerJosé A. OllerUniv. Murcia, SpainUniv. Murcia, Spain

• Introduction

• Strangeness Changing Scalar K π , K η´ Form Factors

• O(p6) Counterterms, F+(0), FK/ Fπ

• Conclusions

Vienna, February 13th 2004

In Collaboration with:

M. Jamin and A. Pich

hep-ph/0401080 JHEP

Page 2: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

1. Introduction

We focus our attention on the Scalar Kπ Form Factors:

P= π, 1, 8 CKP Clebsch-Gordan coefficients such

K π =MK2- M π 2 that FKP(t)=1 at lowest order in U(3) CHPT

We respect isospin symmetry

<P|μ ūus γμu |K> = i( )ms mu <P|ūus u|K> = iΔKπ

2CKPFKP( )t

Due to the absence of scalar probes the scalar form factors seem to be rather theoretical observables....

Nevertheless, they have important implications for the knowledge of input parameters for the Standard Model like ms or Vus

Page 3: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

The previous scalar form factors were calculated by solving the corresponding coupled channel (Kπ and K´) integral equations in Jamin, Pich, J.A.O., NPB622(´02)279, NPB587(´00)331 for the first, and still unique, time.

• In Jamin, Pich, J.A.O. EPJC24(´02)237 they were applied to calculate mms s

ms(2 GeV)=(99 16) MeV in MS

Making use of Scalar Sum Rules, by studying the correlator

ψ( )p2 =idxe ipx <Ω|T[J(x) J(0)]| Ω > , J( )x = μ( )ūus γμq ( )x = i( )ms âam (ūusq)(x)

Page 4: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

• In Jamin, Pich, J.A.O. EPJC24(´02)237 they were applied to calculate mms s

ms(2 GeV)=(99 16) MeV in MS

Making use of Scalar Sum Rules, by studying the correlator

ψ( )p2 =idxe ipx <Ω|T[J(x) J(0)]| Ω > , J( )x = μ( )ūus γμq ( )x = i( )ms âam (ūusq)(x)

• Now we are interested in calculating FKπ(0), necessary to determine Vus from Kl3 decays

The previous scalar form factors were calculated by solving the corresponding coupled channel (Kπ and K´) integral equations in Jamin, Pich, J.A.O., NPB622(´02)279, NPB587(´00)331 for the first, and still unique, time.

Page 5: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

Charge Changing currents in the SMCharge Changing currents in the SM

LCC=g

2 2Wμ

+ ŪUU i γμ( )1 γ5 Vij D j ūuν kγμ( )1 γ5 lk + h.c. +

e

l

b

s

d

D

t

c

u

Ue

utustd

utuscd

utusud

VVV

VVV

VVV

V VV†= V†V= I

*VVCKM

matrix

Unitariy Matrix

Page 6: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

The most accurate test of CKM unitarityThe most accurate test of CKM unitarity

|Vud|=0.9739 0.0005 (Superallowed FT, np e n-decay)

|Vus|=0.2196 0.0026 (Ke3)

|Vub|pdg=0.0036 0.0010

|Vud|2 + |Vus|2 + |Vub|2 =1 |Vus|=0.2270 0.0021

• There is around a 2.5 deviation among both values of Vus

(this disappears with the high-stadistics K+e3 E865 Collaboration hep-

ex/0305042 experiment)

• Vub is negligible (1% of the uncertainty from Vus or Vud )

• There is a problem between unitarity of CKM and presently accepted values of Vij

• Close relationship between Vus F+(0) from Ke3

Page 7: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

M=GF

2Vus

* Lμ CK [ ]F+( )t ( )pK pπ + μ

F_( )t ( )p k pπ μ +

F0( )t = F+( )tt

MK2 Mπ

2F_( )t +

Kl3 Decays: K l+ ν

t=(pK - pπ )2 ; Lμ = ūuu( )p ν γμ ( )1 γ5 ν( )p l

CK=1

21, for ( )K ,K0 +

F+(t) is the vector K π form factor

F0(t) is the scalar K π form factor

< π( )pπ |ūus γμu |K( )pK >

F+(0)= F0(0)

|F0(t)|2 is always multiplied by ml2/MK

2 and can be disregarded in the calculation of the width for Ke3.

K+

Page 8: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

•From ΓK one can obtain |Vus| , once F+(0) is known Leutwyler, Roos

Z.Phys C25(´84)91

•New high precission experiments on Ke3 are ongoing or prepared (CMD2,

NA48, KTEV, KLOE) to improve ΓK

Decay Rate

F0( )t = F+( )0 1 λ0

t

Mπ2 + c0

t2

Mπ4 +

F+( )t = F+( )0 1 λ+

t

Mπ2 + c+

t2

Mπ4 + m l

2 t ( )MK Mπ2

IK results from integrating over

phase space the form factor dependence on t

ΓK = CK2

GF2 MK

5

192π3S ew |Vus |

2 |F+(0)|2 IK

Page 9: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

These facts, together with the possible unitarity violation of the CKM matrix, has triggered important efforts within (G)CHPT

•O(p4) Gasser, Letwyler NPB250(´85)517

• + estimated O(p6) analytical contribution Leutwyler, Roos Z.Phys C25(´84)91

•O(p4) within GCHPT Fuchs, Knecht, Stern PRD62(´00)033003

•O(p4,(md-mu)p2,e2p2) Cirigliano, Knecht, Neufeld, Rupertsberger and Talavera EPJC23(´02)121

•O(p6) isospin limit Post, Schilcher, EPJC25(´02)427;

Bijnens, Talavera NPB699(´03)341 (BT)

O(p4)~O(p6) ; O(p4) suppressed by large Nc

•From ΓK one can obtain |Vus| , once F+(0) is known Leutwyler, Roos

Z.Phys C25(´84)91

•New high precission experiments on Ke3 are ongoing or prepared (CMD2,

NA48, KTEV, KLOE) to improve ΓK

Page 10: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

•From BT, F+(0) only depends on two new order p6 chiral counterterms: Cr

12, Cr34

•The same countererms are also the only ones that appear in the slope and the curvature of F0(t), and hence by a knowledge of F0(t) one can determine those counterterms.

This is our aimSo that an order p6 calculation of F+(0) becames feassible

Page 11: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

2. Strangeness Changing Scalar K π , K η´ Form Factors

<P|μ ūus γμu |K> = i( )ms mu <P|ūus u|K> = iΔKπ

2CKPFKP( )t

•FKP(t) are analytical functions in t, except for a cut from the lightest threshold, (MK+M)2 to (the unitarity cut).

•Its discontinuity through the cut is fixed by unitarity:

P= π, η´, η

F0(t)FKπ(t)

TKP,KQ is the I=1/2 S-wave T-matrix coupling toghether Kπ, K , K ´ and calculated in Jamin, Pich, J.A.O. NPB587(´00)331 (JOP00)

ρKQ( )t =pKQ

8π sθ t ( )MK MQ + 2ImFKP( )t =

Q

FKQ ρKQ TKP,KQ*

Page 12: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

The K turns out to be negligible, both experimentally as well as theoretically, and we neglect it in the following.

The form factors satisfy the dispersion

relations:

F1( )s =1

πs th

ds´ρ1( )s´ F1( )s´ T11

* ( )s´

s´ s iε1

πs th

ds´ρ2( )s´ F2( )s´ T12

* ( )s´

s´ s iε +

F2( )s =1

πs th

ds´ρ1( )s´ F1( )s´ T12

* ( )s´

s´ s iε1

πs th

ds´ρ2( )s´ F2( )s´ T22

* ( )s´

s´ s iε +

This is the so called Muskhelishvili-Omnès problem in coupled channels solved in Jamin, Pich, J.A.O. NPB622(´02)279 (JOP02)

For only one channel (elastic case), namely K, one has the Omnès solution:

F1( )s = F1( )0 exps

πs th

ds´δ1( )s´

s´( )s´ s iε

Page 13: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

s G i( )s 1

sχ isuch that χ1 χ2 + = m

According to Muskhelishvili, Singular Integral Equations (Dover,1992) ; Babelon, Basdevant, Mennessier NPB113(´76)445

The solutions of the Muskhelishvili-Omnès problem for two coupled channels can be expressed in terms of two linearly indepedent form factors

G1(s){G11(s) , G12(s)} and G2(s){G21(s) , G22(s))

F(s)=1 G1(s) + 2 G2(s) with F(s)={F1(s),F2(s)} {FK(s), FK´(s)}

1 and 2 are, in general, polynomials.

The behaviour at inifinity of the basic solutions Gij(s) is determined by the times the argument of the determinant of the S-matrix (L=0, I=1/2) rounds to 2, m

For the T-matrices of Jamin, Pich, J.A.O. NPB587(´00)331 m= 2 or 1

then for vanishing form factors at inifinite 1= 1 (1) , 2= 1 (0)

Page 14: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

m=2 1, 2 constants ; m=1 1 0 constant , 2 =0

In JOP00 we presented several fits: m=2 m=1Fits:6.10K2, 6.10K36.11K2, 6.11K3

Fits: 6.10K1, 6.11K1

All of them are equally acceptable, tiny differences above 2 GeV

degr

ees

a 0

Data: Sol. A of Aston et al. NPB296(´88)493 (We showed in JOP00 that Sol. B of Aston et al. is unphysical)

Page 15: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

m=1 and m=2 fits

F0(0)=0.981

FK/F=1.22 JOP02

m=2 fits

F0(0)=0.981

FK/F=1.220.01 JOP02

Page 16: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

3. O(p6) Counterterms ( Cri ), F+(0) , FK / Fπ

F+(t) , F0(t) calculated recently at order p6 in CHPT .

Post,Silcher EPJC25(´02)427, Bijnens, Talavera NPB669(´03)341 (BT03)

Some differences were found in both calculations for t0.

We follow BT03 that employs the O(p6) CHPT Lagrangian of Bijnens,Colangelo,Ecker NPB508(´97)263

F+( )0 = 1 Δ( )0 + 8

Fπ4

C12r C34

r + ΔKπ2

?

Lir O(p4);

No Cri

Lir from

Amoros,Bijnens,Talavera NPB602(01)87

Δ( )0 = 0.00800.0057 [ ]loops 0.0028 L ir +

= 0.0080 0.0064

Page 17: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

C12r C34

r + = F0´( )0 ŪUΔ 1

FK Fπ 1

ΔKπ

8ΣKπ

Fπ4

C12r

Fπ4

8ΣKπ

C12r = [ ]2ŪUΔ 2 F0´´( )0

Fπ4

16

F0( )t = F+( )0FK Fπ 1

ΔKπ

t + 8

Fπ4

2C12r C34

r + ΣKπ t + 8

Fπ4C12

r t2 ŪUΔ ( )t +

Momentum dependence ΔKπ = MK2 Mπ

2 ΣKπ = MK2 Mπ

2 +

ŪUΔ ( )t = ŪUΔ 1( )t t ŪUΔ 2( )t t2 + ŪUΔ 3( )t t3 + O( )t4 +

= 0.259( )9 t 0.840( )31 t2 + 1.291( )170 t2 + Dependence on Lr

i not on Cr

i

rr CC 3412 and Can be fixed from the t-dependence, slope and curvature, of the scalar form factor F0(t)=FK(t)

3

Page 18: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

F+( )0 = 1 Δ( )0 + 8

Fπ4

C12r C34

r + ΔKπ2

C12r C34

r + = F0( )0 G11(0)´ ŪUΔ 1 ΣKπŪUΔ 2

FK Fπ 1

ΔKπ

ΣKπF0( )0 G11(0)´´ 2 + Fπ

4

8ΣKπ

m=1: Fits 6.10K1, 6.11K1

Only one vanishing independent solution G1(s)={G11(s), G12(s)}

Normalized: G11(0) =1 F0(s) = F0(0) G11(s) , F0(s)´=F0(0) G11(s)´ , F0(s)´´=F0(0) G11(s)´´

C12r = [ ]2ŪUΔ 2 F0( )0 G11(0)´´

Fπ4

16

But F0(0) appears as well as a necessary input... CONSISTENCY

We solve for F+(0)=F0(0)

Page 19: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

F+(0)|C =8

Fπ4

C12r C34

r + ΔKπ2

G11(0)´ = 0.803 , G11(0)´´= 1.661

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ

ŪUΔ 1 ΣKπŪUΔ 2 + FK Fπ 1

ΔKπ

+ +

GeV-2 GeV-4

F0(0) = 0.979 0.009

Cr12=(2.8 2.9) 10-7

Cr12+Cr

34=(2.4 1.6) 10-6

F+(0)|C= - 0.0130.009 The presently used estimated value is the one from Leutwyler&Ross ZPC25(´84)91 F+(0)|CLR= - 0.0160.008

Model depedent calculation based on the overlap between the Kaon and Pion quark wave functions in the light cone. Only analytic piece, no loops.

=M , in agreement with SMD estimates after running to 1.2-1.4 GeV

Page 20: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ

ŪUΔ 1 ΣKπŪUΔ 2 + FK Fπ 1

ΔKπ

+ +

To be improved...

We take FK/F = 1.22 0.01 from Leutwyler, Roos ZPC25(´84)91 (PDG source) F0(0) = 0.979 0.009.

LR84 already employed F+(0) (O(p4) CHPT + F+(0)|CLR )

Our results for F+(0)|C is biassed by the previous F+(0)|CLR

FK/F = 1.20 F+(0)|C= 0.0280.009 F0(0) = 0.965 0.009

FK/F = 1.24 F+(0)|C= + 0.0010.009 F0(0) = 0.993 0.009One needs an independent determination of FK/F

F0(0) = 0.979 0.009

F0(0)=0.976 0.010 BT03 employing F+(0)|CLR

Page 21: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

As pointed out in BT03, the loops corrections cancel partially the negative contribution from loops of O(p4). Thus:

F+(0)|C F+(0) tends to be larger |Vus| smaller

Unitarity of CKM then worses?

Depends on the taken FK/F :

For FK/F 1.19 the unitarity problem disappears

Within GCHPT this points was also stressed in Fuchs,Knecht,Stern PRD62(´00)033003 in a O(p4) GCHPT

calculation

Page 22: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

m=2: Fits 6.10K2, 6.10K3, 6.11K2, 6.11K3

Two vanishing independent solutions G1(s)={G11(s), G12(s)}, G2(s)={G21(s), G22(s)}

Normalized: G11(0) =1 G21(0) =0

G21(K) =0 G21( K ) =0

F0(s) = F0(0) G11(s)+ F0( K) G21(s)

Dashen-Weinstein relation: F0(K)= FK/F + CT

CT=O(msmu, msmd) =-3 10-3 in Gasser,Leutwyler

NPB250(´85)517The extra dependence in this case on F0(K) introduces more uncertainty (4.5errorbar) in our results for F0(0)|C following the scheme developed for m=1

Consistency Check: We take F0(0) = 0.979 0.009 (0.976 0.010) BT03

FK/F = 1.22 0.01

Input(m=1)/Output(m=2) compatible ?

Page 23: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

Input: F0(0) = 0.979 0.009 FK/F = 1.22 0.01

Output: F0(0) = 0.977 0.010

Cr12=(1 5) 10-7

Cr12+ Cr

34=(2.7 1.4) 10-6

F+(0)|C= - 0.0150.008 F+(0)|CLR= - 0.0160.008

F+(0)|C= - 0.0130.009 m=1

F0(0)´=0.804 0.048 GeV-2

Resulting Slope:

CHPT O( )p4 : < rKπ2 > = 0.200.05 fm2 ; λ0 = 0.0170.004

r2K=(0.192 0.012) fm2 ; 0 = 0.0160.001

PDG02: K+3 0 = 0.0300.005

K03 0 = 0.0130.005

Page 24: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

4. Conclusions

Present value for FK/F =1.22 0.01

Cr12=(2.8 2.9) 10-7

Cr12+Cr

34=(2.4 1.6) 10-6

F+(0)|C= - 0.0140.009 (F+(0)|CLR= - 0.016 0.008)

F+(0)=1-0.0227[p4]+0.0113 [p6 loops]+0.0033[p6-Li]– 0.014 [p6 –Ci]

0.0057[loops] 0.0028[Li] 0.0028[Ci]

= 0.978 0.009

It seems that Unitarity problem of CKM gets worse

FK/F 1.19 (without considering E865 K+e3)

Page 25: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ

ŪUΔ 1 ΣKπŪUΔ 2 + FK Fπ 1

ΔKπ

+ +

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ2 ŪUΔ 1 ΣKπŪUΔ 2 +

FK Fπ 1

ΔKπ

+ FK

G21(0)´ G21(0)´´ΣKπ

2 + +

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

1. m=1

m=2

By equating both results we can solve for FK/F in terms of (0), i(0), Gi1(0)´ and Gi1(0)´´

FK/F = 1.19 0.06

F+(0) = 0.96 0.05

The large errors are mainly due to (0)= -0.0080 0.0064 BT03

For a 5% of error for (0), similar size

as those of i(0) , 0.06 0.02

0.05 0.016

Page 26: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

Within this method one needs to improve definitely the knowledge of (0) ..... We must face then the problem of diposing of more precise Li

r

Page 27: José A. Oller Univ. Murcia, Spain Order p 6 Chiral Couplings from the Scalar K  Form Factor José A. Oller Univ. Murcia, Spain Introduction Strangeness.

Ynduráin PLB578 ( )´04 99: < rKπ2 > = 0.310.06 fm2

2 higher, incompatibleY04 claims that this is due to (800) resonance... We also have a (800) in our amplitudes. BWs, employed by Y04, are not appropriate in the scalar sector to parameterize phase shifts

Had we followed the strategy of m=1 fits (solving for F0(0)) then:

F0(0) = 0.96 0.04

F+(0)|C= - 0.030.03