View
216
Download
0
Category
Preview:
Citation preview
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
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
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)
• 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.
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
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
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+
•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
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
•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
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*
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
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)
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)
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
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
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
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)
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
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
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
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 ?
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
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)
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
Fπ
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
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
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
Recommended