Table Of ContentLU TP 04-40
hep-ph/0501163
December 2004
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J Isospin Breaking in K 3π Decays III:
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1 Bremsstrahlung and Fit to Experiment
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Johan Bijnens and Fredrik Borg
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Department of Theoretical Physics, Lund University
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h S¨olvegatan 14A, S 22362 Lund, Sweden
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Abstract
We complete here our work on isospin violation in the K 3π system. We first calculate K 2π to the same
order as we did K 3π in papers I and II of this series. Thi→s adds effects of order G p2(m →m ) and G p2e2
27 u d 27
→ −
to earlier work. We calculate also the lowest order Bremsstrahlung contributions, K 2πγ,3πγ. With these and our
→
earlier results we perform a full fit to all available CP conserving data in the K 2π,3π system including isospin
→
violationeffects.Weperformthesefitsundervariousinputassumptionsaswellastestthefactorizationandthevector
dominance model for the weak NLO low energy constants.
PACS: 13.20.Eb, 12.39.Fe, 14.40.Aq, 11.30.Rd
EPJ manuscript No.
(will be inserted by the editor)
K 3π
Isospin Breaking in Decays III: Bremsstrahlung and Fit
⋆ →
to Experiment
Johan Bijnens and Fredrik Borg
Department of Theoretical Physics, Lund University
S¨olvegatan 14A, S 22362 Lund,Sweden
the dateof receipt and acceptance should beinserted later
Abstract. We complete here our work on isospin violation in the K → 3π system. We first calculate
K → 2π to the same order as we did K → 3π in papers I and II of this series. This adds effects of
order G27p2(mu−md) and G27p2e2 to earlier work. We calculate also the lowest order Bremsstrahlung
contributions, K →2πγ,3πγ. With these and our earlier results we perform a full fit to all available CP
conserving data in theK →2π,3π system including isospin violation effects. Weperform thesefitsunder
various input assumptions as well as test the factorization and the vector dominance model for the weak
NLO low energy constants.
PACS. 1 3.20.Eb, 12.39.Fe, 14.40.Aq, 11.30.Rd
1 Introduction inclusion of those led to changes of a few percent in the
amplitudes.Thelocalelectromagneticpartwasalsocalcu-
lated in [10], in full agreement with our result after sort-
Low-energy QCD is non-perturbative, which calls for al-
ing out some misprints in [10], corrected in [12]. In [13]
ternativemethods ofcalculatingprocessesincluding com-
the radiative corrections were added, which means that
posite particles such as mesons and baryons. A method
the full effects of isospin breaking were studied. This led
used to describe the interactions ofthe lightpseudoscalar
to changes in the amplitudes of order 5–10 percent. Note
mesons (K,π,η) is Chiral Perturbation Theory (ChPT).
that the results in [13] disagree numerically with the re-
It was first presentedby Weinberg, Gasserand Leutwyler
sults for K+ π0π0π+ of [14].
[1–3] and it has been very successful. Pedagogical intro-
→
ductions to ChPT can be found in [4]. The theory can be
extended to also cover the weak interactions of the pseu-
To answer the question whether isospin breaking re-
doscalars,first done in [5].
moves the problem of fitting the quadratic slopes, a new
The first calculation of a kaon decaying into pions full fit has to be done. That is the main result in this pa-
(K 2π,3π) was presented in [6], and reviews of other per, and in this new fit we also include new experimental
→
applicationsofChPTtononleptonicweakinteractionscan data [15,16]. We also present recalculations of the ampli-
be found in [7]. tudes K 2π, K 2πγ and K 3πγ, all calculated
The details from [6] were lost, but a recalculation in to next-to→-leading o→rder and includ→ing first order isospin
the isospin limit of K 2π to next-to-leading order was breaking,i.e. we include contributions proportionalto p2,
→
made in [8,9] and of K 3π in [9,10]. In [9] a full fit m2, e2, m m (leading order),and p4,p2m2,m4,p2e2,
to all experimental data →existing at the time was made, m2e2, p2(um− dm ) and m2(m m ) (next-to-leading
u d u d
and it was found that the decay rates and linear slopes order). The co−rrections needed to−be added to determine
agreed well. However, a small discrepancy was found in ππ scattering lengths from K 3π [17] are beyond the
the quadratic slopes, and this can have several different order calculated in this paper. →
origins. It could be an experimental problem or it could
haveatheoreticalorigin.Inthelattercasethecorrections
to the amplitude calculated in [9] are threefold: strong Theoutlineofthispaperisasfollows.Thenextsection
isospin breaking, electromagnetic (EM) isospin breaking describes isospin breaking in more detail. In section3 the
or higher order corrections. basisofChPT,theChiralLagrangians,arediscussed.Sec-
tion4specifiesthedecaysanddescribestherelevantkine-
In [11] the strong isospin and local electromagnetic
matics. The divergences appearing when including pho-
corrections were investigated and it was found that the
tons are discussedin section 5. In section 6 the analytical
⋆ Supported in part by the European Union TMR network, results are discussed, section 7 contains the numerical re-
Contract No. HPRN-CT-2002-00311 (EURIDICE). sults and the last section contains the summary.
Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment 1
2 Isospin Breaking wherethespecialunitarymatrixU containstheGoldstone
boson fields
Isospin symmetry is the SU(2) symmetry under the ex-
i√2
change of up- and down-quarks. This symmetry is only U = exp M ,
F
exact in the approximation that m = m and electro- 0 !
u d
magnetism is neglected, i.e. in the isospin limit. Calcula- 1 π + 1 η π+ K+
tions are sometimes performed in the isospin limit since √2 3 √6 8
this is simpler and gives a good first estimate of the re- M = π− √−12π3+ √16η8 K0 . (4)
sult. However, to get a more accurate result one should K− K0 √−26η8
include the effects from mu = md and electromagnetism,
6
i.e. isospin breaking. Weusetheformalismoftheexternalfieldmethod[2],and
The two different sources of isospin breaking give rise to include photons we set
to different effects. Strong isospin breaking, coming from
mu = md, include mixing between π0 and η. This mix- mu
ing l6eads to changes in the formulas for both the physical χ=2B0 md (5)
masses of π0 and η as well as the amplitude for any pro- ms
cess involving either of the two. For a detailed discussion
see [18]. and
D U =∂ U ieQA U ieUQA , (6)
The other source, electromagnetic isospin breaking, µ µ µ µ
− −
comingfromthefactthattheup-andthedown-quarksare where A is the photon field and
µ
charged, implies interactions with photons. This means
both the addition of new Lagrangians at each order, as 2/3
wellastheintroductionofnewdiagramsincludingexplicit Q= 1/3 . (7)
photons. − 1/3
−
The quadraticterms in (2) are diagonalizedby a rotation
3 The ChPT Lagrangians
π0 = π cosǫ+η sinǫ
3 8
η = π sinǫ+η cosǫ, (8)
3 8
The basis of our ChPT calculation is the various Chiral −
Lagrangians of relevant orders. We work to leading order where the lowest order mixing angle ǫ satisfies
in m m and e2 but next-to-leading order in p2 and
u d
m2.Fo−rsimplicity wecallinthe remaindertermsoforder tan(2ǫ)=√3 md−mu . (9)
pp22,mm22,,me24,, mp2ue−2,mmd2ele2a,dpi2n(gmorderman)datnedrmms2o(fmordermp4), 2ms−mu−md
u d u d
− − The weak, ∆S = 1, part of the Lagrangian has the
next-to-leading order.
form [19]
3.1 Leading Order LW2 =CF04"G8h∆32uµuµi+G′8h∆32χ+i
The lowest order Chiral Lagrangian is divided in three
+G tij,kl ∆ u ∆ uµ + h.c..
parts 27 h ij µih kl i#
= + + , (1)
2 S2 W2 E2
L L L L
The tensor tij,kl has as nonzero components
where refers to the strong ∆S = 0 part, the
S2 W2
L L
weak ∆S = 1 part, and the strong-electromagnetic 1 1
± LE2 t21,13 =t13,21 = , t22,23 =t23,22 = ,
andweak-electromagneticparts combined.For the strong 3 −6
part we have [2] 1 1
t23,33 =t33,23 = , t23,11 =t11,23 = , (10)
−6 3
F2
LS2 = 40 huµuµ+χ+i. (2) and the matrix ∆ij is defined as
HerehAistandsfortheflavourtraceofthematrixA,and ∆ij ≡uλiju†, (λij)ab ≡δiaδjb. (11)
F isthepiondecayconstantinthechirallimit.Wedefine
0 The coefficient C is defined such that in the chiral and
the matrices u , u and χ as
µ
± large Nc limits G8 =G27 =1,
uµ =iu†DµUu† =u†µ, u2 =U, χ± =u†χu†±uχ†(u3,) C =−53 G√F2VudVu∗s =−1.06·10−6 GeV−2. (12)
2 Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment
Finally, the remaining electromagnetic part, relevant for 4 Kinematics
this calculation, looks like (see e.g. [20])
4.1 K 2π and K 2πγ
=e2F4Z +e2F4 Υ , (13) → →
LE2 0 hQLQRi 0h QRi
In the limit of CP-conservation, there are three different
where the weak-electromagnetic term is multiplied by a
decays of the type K 2π (K decays are not treated
constant GE (gewkG8 in [20]), separatelysincethey a→recounter−partstothe K+ decays):
Υ =G F2∆ +h.c. (14)
E 0 32 K (k) π0(p )π0(p ), [AS ],
S → 1 2 00
and K (k) π+(p )π (p ), [AS ],
L =uQu†, R =u†Qu. (15) S → 1 − 2 +−
Q Q K+(k) π+(p )π0(p ), [A ], (19)
1 2 +0
→
3.2 Next-to-leading Order where we have indicated the four-momentum defined for
eachparticleandthesymbolusedfortheamplitude.With
Chiral Perturbation Theory is a non-renormalizable the- an external photon it changes to:
ory.This means that new terms have to be addedat each
K (k) π0(p )π0(p )γ(q), [AS ],
ordertocompensateforthedivergencescomingfromloop- S → 1 2 00γ
diagrams.Thus the Lagrangiansincrease in size for every K (k) π+(p )π (p )γ(q), [AS ],
neworderandthenumberoffreeparametersrisesaswell. S → 1 − 2 +−γ
K+(k) π+(p )π0(p )γ(q), [A ]. (20)
At next-to-leading order the Lagrangian is split in four → 1 2 +0γ
parts which, in obvious notation, are
The kinematics for K 2πγ is treated using
→
= + + + (G ). (16)
4 S4 W4 S2E2 W2E2 8 r k q, r p q, r p q, (21)
L L L L L 0 1 1 2 2
≡− · ≡ · ≡ ·
The notation (G ) indicates that here only the dominant
8 where
G -partisincludedintheLagrangianandthereforeinthe
8 r +r +r =0. (22)
calculation. 0 1 2
TheLagrangiansofnext-to-leadingorderarequitelarge
and we will not write them explicitly here since they can 4.2 K 3π and K 3πγ
be found in many places [2,5,20–24]. For a list of all the → →
piecesrelevantforthisspecificcalculationsee[11,13].Note For the corresponding process K 3π, there are five
however that one contributing term was forgotten when different decays: →
writing in [11], namely
S2E2
L K (k) π0(p )π0(p )π0(p ), [AL ],
−ie−2∇F0µ2QKR12QhR(∇b+µQQRLQ∇LµQ−RQ)uLµ∇biµ,QL (17) KKLLS((kk)) →→→ππ++((pp111))ππ−−((pp222))ππ00((pp333)),, [[AA00S+L+0−−00]],,
where K+(k) π0(p )π0(p )π+(p ), [A ],
b b → 1 2 3 00+
i K+(k) π+(p )π+(p )π (p ), [A ], (23)
µ L = µ L+ [uµ, L]=uDµQLu†, → 1 2 − 3 ++−
∇ Q ∇ Q 2 Q
and here the variables are
i
∇bµQR = ∇µQR− 2[uµ,QR]=u†DµQRu. (18) s (k p )2 , s (k p )2 , s (k p )2 ,
1 1 2 2 3 3
≡ − ≡ − ≡ −
It contrbibutes to the calculation of the decay constants, (24)
F and F . It only contributes to the amplitudes of where
π+ K+
K 2π and K 3π via the rewriting of the lowest s1+s2+s3 =k2+p21+p22+p23. (25)
→ →
order in terms of Fπ+ and FK+ rather than F0. The amplitudes are expanded in terms of the Dalitz plot
variables x and y defined as
3.2.1 Ultraviolet Divergences s s s s 1
3 0 2 1
y = − , x= − , s = (s +s +s ) .
m2 m2 0 3 1 2 3
The processesK 2π andK 3π receives higher-order π+ π+
→ → (26)
contributionsfromdiagramsthatcontainloops.Thestudy
With an external photon the decays are:
of these diagrams is complicated by the fact that they
needtobepreciselydefined.Theloop-diagramsinvolvean K (k) π0(p )π0(p )π0(p )γ(q), [AL ],
integrationoverthe loop-momentum Q,and the integrals L → 1 2 3 000γ
K (k) π+(p )π (p )π0(p )γ(q), [AL ],
are divergent in the ultraviolet region, i.e. when Q→∞. L → 1 − 2 3 +−0γ
Theseultravioletdivergencesarecanceledbyreplacingthe K (k) π+(p )π (p )π0(p )γ(q), [AS ],
coefficients, X , in the next-to-leading order Lagrangians S → 1 − 2 3 +−0γ
by the renormialized coefficients, Xir, and a subtraction K+(k) →π0(p1)π0(p2)π+(p3)γ(q), [A00+γ],
part. See [9,11] and references therein. K+(k) π+(p )π+(p )π (p )γ(q), [A ],(27)
1 2 − 3 ++ γ
→ −
Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment 3
where the kinematics is treated using
s (k p )2, s (k p )2, s (k p )2, (28)
1γ 1 2γ 2 3γ 3
≡ − ≡ − ≡ −
t k q, t p q, t p q, t p q, (29)
0 1 1 2 2 3 3
≡− · ≡ · ≡ · ≡ ·
where
t +t +t +t =0 (30)
0 1 2 3
and
s +s +s =k2+p2+p2+p2 2t . (31)
1γ 2γ 3γ 1 2 3− 0
Fig. 1. The diagrams for K → 2π. An open square is a ver-
5 Infrared Divergences
tex from LW4 or LW2E2, a filled square a vertex from LW2
Beside the ultraviolet divergences which are removed by or LE2(∆S = 1) and a filled circle a vertex from LS2 or
renormalization of the higher order coefficients, diagrams LE2(∆S = 0). A straight line is a pseudoscalar meson and
a wiggly line a photon.
includingphotonsintheloopscontaininfrared(IR)diver-
gences. These infinities come from the Q 0 end of the
→
loop-momentumintegrals.They arehandled by including
6.1.1 Lowest Order
also the Bremsstrahlung process, where a real photon is
radiatedoff one of the chargedmesons.It is only the sum
ThereisonlyonediagramcontributingtothedecayK
ofthevirtualloopcorrectionsandtherealBremsstrahlung
→
2π atlowestorder,seetopleftinFig.1,andthe resulting
which is physically significant and thus needs to be well
amplitudes are also quite simple. To first order in isospin
defined.
they can be written
We regulate the IR divergences in both the virtual
photon loops and the real emission with a photon mass
mγ and keep only the singular terms plus those that do AS00 = iF(G8−G27) (36)
not vanish in the limit m 0. We include the real
γ sinǫ
Btrreeamtsitstirnahthluensgoffotrpphhoototonnaepnp→errogxiiemsautpiotno.a cut-off ω and 4 √3 (m2π −m2K)−2m2π0 +2m2K0!, (37)
TheexactformoftheamplitudesquaredfortheBrems-
strahlungdepends onwhichspecific amplitude thatis be-
ing calculated. For a detailed presentation of the calcula- AS = iFG ( 2m2 +2m2 )
tion and resulting expressions for K 3π see [13]. The +− 8 − π+ K0
4 4
corresponding amplitudes for K →2π→are +iFG27 −3m2π+ + 3m2K+ −2iF3e2GE,
(cid:18) (cid:19)
|AS00|2BS =0, (32) (38)
e2 ω2
AS 2 = AS 2 log I (m2,m2,m2 )
| +−|BS −| +−|LO 4π2 (cid:20) m2γ − IR π π K (cid:21) sinǫ sinǫ
(33) A =iFG 2 m2 +2 m2 m2 +m2
e2 ω2 +0 8 (cid:18)− √3 π √3 K − π+ π0(cid:19)
A 2 = A 2 log I (m2,m2 ,m2)
| +0|BS −| +0|LO 4π2 m2 − IR π K π sinǫ sinǫ
(cid:20) γ (cid:21) +iFG 3 m2 +3 m2 7/3m2
(34) 27 − √3 π √3 K − π+
where
IIR(m21,m22,m23)≡−4xπs2mm23m−m(121−xm)22 logxslogmω22. +2/3m2π0+5/3m2K+!−iF3e2GE. (39)
1 2 − s γ
(35)
See section 7.1.1 for a discussion of the masses used.
When using these expressions, the divergences from the
photon loops cancel exactly.
A similar problem shows up in the definition of the
decayconstants,since we normalizethe lowestorderwith 6.1.2 Next-to-leading Order
F and F . See [13] for details.
π+ K+
There are seven more diagrams contributing to next-to-
leading order, see Fig. 1. The resulting amplitudes are
6 Analytical Results
long, and we decided to not include them here but in-
6.1 K 2π stead make them available for download [27]. Note that
→ we have also included contributions proportional to G ,
27
The most complete work on isospin violation in K 2π not included in [25]. These are included for consistency
→
is in [25], earlier work can be found in [26]. between the K 2π(γ) and K 3π(γ) calculations.
→ →
4 Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment
Fig.2.ThediagramsforK →2πγ.Asquareisaweakvertex,
a circle a strong vertex, a straight line a pseudoscalar meson
and a wiggly line a photon.
6.2 K 2πγ
→
TheamplitudesfortheprocessesK 2πγ havebeencal-
→
culatedbefore.Hereweonly needthe lowestordercontri-
butiontobeconsistentwiththeK 2πcalculation.This
→
werecalculatedandthestartingpointisthetwodiagrams
that contribute to the process, shown in Fig. 2.
Sincephotonsatthisorderonlycoupletochargedpar-
ticles, there are only two different amplitudes and the re-
sults are Fig.3.ThediagramsforK →3πγ.Asquareisaweakvertex,
a circle a strong vertex, a straight line a pseudoscalar meson
2
AS = eF G + G (m2 m2) and a wiggly line a photon.
+−γ 8 3 27 K − π
(cid:18) (cid:19)
1 1 1 1 G G 3m2 +8m2
k.ε +(p2.ε p1.ε) + , 8 27 − K π
(cid:20)− (cid:18)r1 − r2(cid:19) − (cid:18)r1 r2(cid:19)(cid:21) (cid:18)− 3 − 6 m2π−m2K (cid:19)×
(40) ( 3(s 2t 2t )+m2 +3m2)
− 3γ − 1− 2 K π ×
and p1 ε p2 ε
· ·
t − t
5 1 1 (cid:18) 1 2 (cid:19)
A+0γ =eF 6G27(m2K −m2π) (cid:20)−k.ε (cid:18)r0 + r1(cid:19) −6 −G38 − G627 −3mm22K +m82m2π ×
1 1 1 1 (cid:18) π − K (cid:19)
+p .ε +p .ε + (.41)
1 (cid:18)−r0 − r1(cid:19) 2 (cid:18)−r0 r1(cid:19)(cid:21) (t +t ) p1·ε p2·ε , (43)
2 1
These amplitudes canbe decomposedinto anelectricand (cid:18) t1 − t2 (cid:19)#
a magnetic part:
5 G
A(K 2πγ)=eεµ(k)(E +ε Mνρσ), (42) AS =ie 27 (2m2 3m2 )
→ µ µνρσ +−0γ 6 (cid:18)−m2π−m2K(cid:19) π− K ×
butatlowestorderthemagneticamplitudeMνρσ vanishes p ε p ε
1 2
(s s ) · ·
since there is no ε tensor in the corresponding lowest 1γ 2γ
µνρσ − t − t
order Lagrangian. (cid:20) (cid:18) 1 2 (cid:19)
p ε k ε
The electric amplitude, on the other hand, is com- +2t 1· + ·
0
pletelydeterminedbythecorrespondingnon-radiativeam- (cid:18) t1 t0 (cid:19)
plitude via Low’s theorem [28].
p ε k ε
2
+2t · + · , (44)
0
t t
(cid:18) 2 0 (cid:19)#
6.3 K 3πγ
→
G G
ThedecayK 3πγ isdiscussedindetailin[29].Weonly A =ie 8 (m2 +m2) 27 (5/3m2m2
→ 00+γ − 2 K π − m2 m2 π K
need the lowest order amplitude for consistency with the (cid:20) π− K
calculation of K 3π. We have calculated the four dif- G G
→ 13/6m4 +1/2m4 ) 8 + 27
ferent amplitudes using Chiral Perturbation Theory, and − π K − 2 m2 m2 ×
checked that they agree with Low’s theorem. The calcu- (cid:18) π − K
lation is based on seven diagrams, see Fig. 3. The four (7/6m2π−17/6m2K) ×
amplitudes are (2(s 2t 2t )(cid:17) m2 3m2)
3γ − 1− 2 − K − π
G G G
AL =ie 8− 27 m2 + 27 5/6(2m2 3m2 )
+−0γ (cid:20) 3 K −m2π−m2K π− K ×
Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment 5
GE −0.4 Lr1 0.38·10−3
(−(s3γ −2t1−2t2)+m2K +3m2π)#× siZnǫ 1.19·01.08−052 LLr2r −21..9519··1100−−33
3
k ε p3 ε Fπ 0.0924 GeV Lr4 0
t· + t· (45) FK 0.113 GeV Lr5 1.46·10−3
(cid:18) 0 3 (cid:19) N14 −10.4·10−3 Lr6 0
N15 5.95·10−3 Lr7 −0.49·10−3
D13 0 Lr8 1.0·10−3
A++−γ = ie" −G8(m2π+m2K)+G27(m2K +13/3m2π) D15 0 Lr9 7.0·10−3
Table 1. The various input values used, LECs given at µ =
+(G8 13/3G27)(s3γ 2t1 2t2) 0.77 GeV.
− − − !×
k ε p ε p ε p ε
1 2 3
· + · + · · 7.1.1 Strong and Electromagnetic Input
t t t − t
(cid:18) 0 1 2 3 (cid:19)
p ε p ε There aredifferent ways to treatthe masses,especially in
1 2
+(2G8 26/3G27)(t2 t1) · · . the isospin limit case. In [9] the masses used in the phase
− − (cid:18) t1 − t2 (cid:19)# spacewereobtainedfromthephysicalmassesoccurringin
(46) the decays.However,in the amplitudes the physicalmass
ofthe kaon involvedin the process wasused and the pion
Once again, these amplitudes can be decomposed into an mass was given by m2 = 1 m2 with i = 1,2,3
electric and a magnetic part: π 3 i=1,3 πi
being the three pions participating in the reaction. This
P
A(K 3πγ)=eεµ(k)(E +ε Mνρσ), (47) allowedfor the correctkinematicalrelations1+s2+s3 =
→ µ µνρσ m2 +3m2 tobesatisfiedwhilehavingthe isospinlimitin
K π
andthemagneticamplitudeMνρσ vanishesforthereasons theamplitudebutthephysicalmassesinthephasespace.
given above. The results in [9] were obtained with the physical mass
Theelectricamplitudeatthisorderisagaincompletely for the eta. Results with the Gell-Mann-Okubo (GMO)
determinedbythecorrespondingnon–radiativeamplitude relation for the eta mass in the loops gave small changes
A(s,ν) via Low’s theorem [28,29]: within the general errors given in [9].
In the decays here, we work to first order in isospin
Eµ = A(s,ν)Σµ breaking. We have rewritten explicit factors of m m
u d
−
∂A(s,ν) ∂A(s,ν) in terms of sinǫ according to
+2 Λµ + (Λµ Λµ )
∂s 12 ∂ν 14− 24 1
m m = (2m m m ) sinǫ. (50)
+ (k) (48) u− d −√3 s− u− d
O
with(the mesonchargesinunits ofe aredenotedq ,with In general we use the physical masses of pions and kaons
i
4 q =0) intheloopsbutassoonasafactorofsinǫore2 ispresent
i=1 i
we use a common kaon and a common pion mass. This
P s =(p +p )2 simplifies the analytical formulae enormously. The kaon
1 2
mass chosen is the mass from the kaon in the decay and
ν =k (p p )
· 1− 2 the pion mass used is 3m2 = m2 with i = 1,2,3 the
4 q pµ three pions in the final sπtate, i.ie. πthie mass we used in
Σµ = it i the isospin limit case. For thePeta mass we use in general
i
Xi=1 the physical mass in the loop integrals.We have used the
Λµij =Λµji =(qitj −qjti)Diµj GMO mass relation with isospin violation included,
Dµ = Dµ = pµi pµj . (49) m2 = 2(m2 +m2 m2 )+ 1m2 , (51)
ij − ji ti − tj η 3 K+ K0 − π+ 3 π0
to simplify the amplitudes, except in the loops as stated
Since there are no terms of (k) at lowest order in the
O above. The possible lowest order contributions from the
chiral expansion, the leading–order electric amplitude is
etamasshavebeenremovedfromtheamplitudesusingthe
completely determined by the explicit terms in (48).
corresponding next-to-leading order relation as described
in [11].
The strongLECs,Lr to Lr, as wellas sinǫ come from
7 Numerical Results 1 8
the one-loopfit in [18], Lr from [30]and the G estimate
9 E
is from [31].
7.1 Experimental Data and Input
The constant Z from we estimate via
E2
L
For the numerical studies we use the input given in Ta- 1
Z = (m2 m2 ), (52)
ble 1. 2F2e2 π+ − π0
π
6 Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment
which corresponds to the value in Table 1. The higher ω(GeV) IR photon Extraphoton Sum(GeV)
ordercoefficients of ,Kr...Kr ,areratherunknown. (GeV) (GeV)
SomeroughestimateLsEe4xist1butwe12putthemtozerohere, KS →π+π−
at the relevant scale. 0.01 4.34·10−17 1.59·10−17 5.92·10−17
The IR divergences are canceled by adding the soft- 0.001 2.17·10−17 3.69·10−17 5.86·10−17
photon Bremsstrahlung. We have used a 1 MeV cut-off 0.0005 1.52·10−17 4.34·10−17 5.85·10−17
in energy for this and used the same cut-off in the defini- 0.0001 0 5.85·10−17 5.85·10−17
tion of F and F . We also use m = 1 MeV, which K+ →π+π0
effectivelyπ+removesKth+e infrared part. γ 0.01 4.30·10−20 1.54·10−20 5.84·10−20
Thesubtractionscaleµischosentobe0.77GeVunless 0.001 2.15·10−20 3.61·10−20 5.77·10−20
stated otherwise. 0.0005 1.50·10−20 4.26·10−20 5.76·10−20
0.0001 0 5.77·10−20 5.77·10−20
KL→π+π−π0
0.01 2.39·10−21 3.39·10−22 2.73·10−21
7.1.2 Input Relevant for the Photon Reducible Diagrams 0.001 1.19·10−21 1.39·10−21 2.58·10−21
0.0005 8.34·10−22 1.74·10−21 2.58·10−21
The two constants D and D are set to zero since no 0.0001 0 2.56·10−21 2.56·10−21
knowledge exist of th1e3ir value1s5. One can determine the KS →π+π−π0
constantsN andN fromK πl+l decays.Forade- 0.01 5.88·10−24 8.93·10−25 6.77·10−24
tailed analys1i4s, see [1135]. The res→ulting−values are given in 0.001 2.94·10−24 3.43·10−24 6.37·10−21
Table1.Notehowever,thatasdescribedin[13],D ,D , 0.0005 2.05·10−24 4.28·10−24 6.34·10−21
N and N only contributes via the photon re1d3ucib15le 0.0001 0 6.29·10−24 6.29·10−24
di1a4grams. T1h5ese diagrams are negligible numerically, un- K+ →π0π0π+
less the constants are ordersofmagnitude largerthan ex- 0.01 3.77·10−22 4.03·10−23 4.18·10−22
pected. 0.001 1.89·10−22 1.98·10−22 3.86·10−22
0.0005 1.32·10−22 2.53·10−22 3.85·10−22
0.0001 0 3.83·10−22 3.83·10−22
K+ →π+π+π−
7.2 Bremsstrahlung and Dependence on mγ and ω 0.01 5.05·10−21 5.86·10−22 5.63·10−21
0.001 2.52·10−21 2.73·10−21 5.25·10−21
TheisospinbreakingamplitudesforK 2πandK 3π 0.0005 1.76·10−21 3.46·10−21 5.23·10−21
both depend on m , introduced to regu→larizethe inf→rared 0.0001 0 5.19·10−21 5.19·10−21
γ
divergences coming from loops containing photons. This
m -dependence iscanceledby addingthe bremsstrahlung Table2. K →2,3π decayratescalculatedfordifferentvalues
γ
amplitudes, where a real soft photon is radiated off one on ω. Here we use mγ = 0.1 MeV and the same value is used
of the mesons. This cancellation was checked for K 3π as a cut-off in the decay constants, see theappendix in [13].
→
in [13], and we have now also checked it for K 2π.
→
However,after the addition of bremsstrahlung the de-
cay rates depend instead on ω, the cut-off energy of the 7.3 Fit to K 2π
radiated real photon. This is a parameter that should be →
settoavalue dependingonthe experimentthatonecom-
pares to. The process K 2π in the presence of isospin breaking
→
Another possibility, which we use in this paper, is to has been discussed in detail in [25]. We have reproduced
add the full amplitudes with an extra radiated photon, that calculationbut added in addition also all the isospin
K 2πγ and K 3πγ. When doing that the decay breakingcontributionsfromthe 27amplitudes,exceptfor
→ →
rates should be independent of ω. We have checked this the parts from the weak-electromagnetic27 Lagrangian.
numerically and the results are presented in Table 2. For Theisospinbreakingcorrectionstothedecayratesare
thiscomparisonwehavechosenmγ =0.1MeVandvaried rather small, but they have an impact on the phase shift
omega over a large range. One can see that up to photon between the I = 2 and I = 0 amplitudes, δ δ , as
2 0
−
energiesof1MeV,thesumisconstantwithintheexpected describedindetailin[25].Ourresultsarecompatiblewith
uncertainties. The different sum when including energies the ones presented there. The phase shift we use here is
upto10MeVisanindicationthatthesoftphotonapprox- defined via
imation, used in calculating the infrared contribution, is
breaking down. √2 2
AS = A A ,
The way we treated the Bremsstrahlung contribution 00 √3 0− √3 2
in the fits is as follows. We assume that the measuredde-
√2 1
cay widths are including all photons. To compare to our AS = A + A ,
amplitudes (calculated without hard photons), we there- +− √3 0 √3 2
foresubtractnumericallythe calculatedhardphotoncon- √3
tributions from the experimental numbers. A+0 = 2 A+2 ,
Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment 7
K →2π This means that the effect of Bremsstrahlung is in-
Order µ [GeV] G8 G27 δ2−δ0 cluded fully in the decay rates, but only via the soft pho-
LO 10.4 0.55 −60.3◦ ton approximation with a 1 MeV cut-off for the Dalitz
NLO 0.6 6.43 0.44 −58.9◦ plot distributions. We have not included the preliminary
NLO 0.77 5.39 0.36 −58.0◦ data from KTeV, NA48 and KLOE.
NLO 1.0 4.60 0.30 −57.4◦ Thenumberoffreeinputparametersonthetheoryside
is very large. Since it turns out that the isospin breaking
Table 3. Quantities fitted from K → 2π only. The scale µ is effectsareverysmall,weputthoseextraNLOparameters
the scale at which the unknown low energy constants are put equal to zero at the scale µ indicated in the tables.
to zero. Let us repeat here the definitions of the various ex-
tra NLO parameters. The Lr are taken from the stan-
i
dardfit done at one loop to be compatible with the order
A2 = A2 ei(δ2−δ0). (53) of this calculation. The Kir are the extra parameters at
A0 (cid:12)A0(cid:12) NLO in the p2e2 sector. Those are always put to zero at
(cid:12) (cid:12) the scale indicated. In the isospin limit 11 combinations
Inthefitswehaveleftδ(cid:12) δ(cid:12)asanadditionalfreeparam-
2(cid:12)− 0(cid:12) of the weak NLO low-energy coefficients show up, as dis-
eter,asitis knownthatthis phaseisbadly reproducedat cussed in [9]. These are K˜ , i=1,...,11. In the presence
i
one-looporderin ChPT.Note that because of the isospin
of isospin breaking many more combinations of these, as
breaking the A+ amplitude is different from A .
2 2 wellasfromtheweakoctetordere2p2Lagrangian,emerge
We have performed a lowest order and a NLO fit to
and they were classified in [11]. The 27-part of the weak
tohnelyetxhtreaKpa→ram2πetearms,pKl˜itud=esK. Irn=thZerNL=O0fiattwtehesestcaallel Lagrangianofordere2p2hasnotbeenworkedoutandwill
i i i leadto somemorefree parameters.We havenotusedany
µ indicated. The Bremsstrahlung contribution has been
estimates of these extra parameters but set all of them
subtracted as discussed above. to zero at the scale indicated, except for K˜ , i=1,...,7.
i
As can be seen in Table 3, there is a sizable vari-
Thereasonforthischoiceisthattheyaretheleadingcon-
ation depending on the input scale used. There is very tributions.K˜ areoctetenhancedandcomemultiplied
1,2,3
little change in the absolute values of G and G com-
8 27 withfactorsoforderm4 andK˜ are27-pletsbutalso
paredto theisospinlimitfitof[9],whereonlythe fitwith K 4,5,6,7
come multiplied with factors of order m4 . The neglected
µ = 0.77 GeV was done. The angle is similar to the fit K
ones are thus suppressed by either isospin breaking, fac-
there, but this is a combinationof two different effects. It
tors of m2/m2 or by the ∆I = 1/2 rule, i.e. an extra
was lowered because the new KLOE data have now been π K
factor of G /G .
included in the PDG averaging, but the isospin break- 27 8
ing effects induced a positive correctionas wasalso found
in [25].
7.4.1 General Fits
The values of G and G are determined by fitting
8 27
CF4G and CF4G and then setting F = F numeri-
0 8 0 27 0 π
Hereweperformthefitswithsimilarassumptionsasused
cally to provide the numbers in the tables.
in the isospin limit fit, as well as a few additional ones.
FirstG andG areextremelycorrelatedwiththevalues
8 27
7.4 Fit to K 2π and K 3π ofK˜1andK˜4respectively.Theyareverydifficulttoobtain
→ → separately without additional assumptions. The main fit
Thequantitieswefitarethemeasureddecayratesandthe is therefore the one with
variousparametersoftheDalitzplotdistributionsdefined
K˜ =K˜ =K˜ =K˜ =0, (56)
via 1 4 8 9
2
A(s ,s ,s )
1 2 3 =1+gy+hy2+kx2. (54) ata scale µ=0.77GeV. The results aregivenin Table 6.
A(s ,s ,s )
(cid:12) 0 0 0 (cid:12) ThistableisverysimilartoTable6in[9].Thelargevalues
(cid:12) (cid:12)
For the de(cid:12)cay K 3π(cid:12)0, k = h/3 and g = 0. The decay of K˜ and the resulting large value of K˜ have the same
K π+π(cid:12) π0 isLin→clud(cid:12)ed via origin6 asinthatreference.Inordertofitγ7 well,K˜ isput
s − S 6
→
large because it gets multiplied there with a small factor
AS+−0 =γSx−ξxy. (55) avanldueiss tfohreKo˜nlytoocnoemcponentrsiabtuetiinngo.tThehrispilnacteusr.n leads large
7
The decay rates are included in the fit as follows. We
The fit with
subtract from the decay rates the Bremsstrahlung contri- K˜ =0 (57)
butions as described above as a function of G and G . 6
8 27
Wethenconvertthedecayrateusingthecentralvaluesof in addition has only a slightly larger χ2 and a smaller
the measured Dalitz plot distribution into a value for the K˜ . The χ2 is larger than in [9] because the experimental
7
amplitude squared at the center of the Dalitz plot. These errorsonseveralquantitieshavedecreasedsincethen.The
squared amplitudes together with the parameters g,h,k overallfitis slightlybetter thantheoneof[9]becausethe
and γ are used as the 18 experimental parameters to be newer measurements of the Dalitz distribution in K+
S
fitted. π0π0π+ agree better with the chiral expressions. →
8 Johan Bijnens, Fredrik Borg: Isospin Breaking in K →3π Decays III:Bremsstrahlung and Fit to Experiment
Decay Width [GeV] ChPT [GeV] Fact. [GeV]
K+ →π+π0 (1.1231±0.0078)·10−17 1.123·10−17 1.127·10−17
KS →π0π0 (2.2828±0.0104)·10−15 2.282·10−15 2.283·10−15
KS →π+π− (5.0691±0.0108)·10−15 5.069·10−15 5.069·10−15
KL→π0π0π0 (2.6748±0.0358)·10−18 2.618·10−18 2.698·10−18
KL →π+π−π0 (1.5998±0.0271)·10−18 1.658·10−18 1.711·10−18
K+ →π0π0π+ (9.195±0.0213)·10−19 8.934·10−19 8.816·10−19
K+ →π+π+π− (2.9737±0.0174)·10−18 2.971·10−18 2.933·10−18
Table 4. The various decay widths from thePDG tables [32], and ourresults from themain fit and thebest factorization fit.
Decay Quantity Experiment ChPT Fact.
KL →π0π0π0 h −0.0050±0.0014 -0.0062 -0.0025
KL→π+π−π0 g 0.678±0.008 0.678 0.654
h 0.076±0.006 0.088 0.083
k 0.0099±0.0015 0.0057 0.0068
KS →π+π−π0 γS (3.3±0.5)·10−8 3.0·10−8 2.9·10−8
K± →π0π0π± g 0.638±0.020 0.636 0.648
h 0.051±0.013 0.077 0.080
k 0.004±0.007 0.0047 0.0069
K+ →π+π+π− g −0.2154±0.0035 −0.215 −0.226
h 0.012±0.008 0.012 0.019
k −0.0101±0.0034 −0.0034 −0.0033
K− →π−π−π+ g −0.217±0.007
h 0.010±0.006
k −0.0084±0.0019
Table 5. Experimental values and the main fit and best factorization fit of the Dalitz plot distribution parameters. The data
are from thePDG tables [32] except γS from [33].
Constraint Eq. (56) Eq. (56) Eq. (56) Eq. (56,57)
µ 0.77 GeV 1.0 GeV 0.6 GeV 0.77 GeV
G8 5.39(1) 4.60(1) 6.43(1) 5.39(1)
G27 0.359(2) 0.301(1) 0.438(2) 0.359(2)
δ2−δ0 −57.(1.5)o −57.3(1.4)o −58.9(1.4)o −57.9(1.4)o
103K˜1/G8 ≡0 ≡0 ≡0 ≡0
103K˜2/G8 48.5(2.4) 56.5(2.4) 41.2(1.9) 46.6(1.6)
103K˜3/G8 2.6(1.2) −1.7(1.1) 6.7(1.0) 3.5(0.8)
103K˜4/G27 ≡0 ≡0 ≡0 ≡0
103K˜5/G27 −41.2(16.9) −52.0(17.7) −31.1(12.0) −27.0(8.3)
103K˜6/G27 −102(105) −114(105) −93(76) ≡0
103K˜7/G27 78.6(33) 78.0(33.5) 79.6(22.7) 50.0(13.0)
103K˜8/G8 ≡0 ≡0 ≡0 ≡0
103K˜9/G8 ≡0 ≡0 ≡0 ≡0
χ2/DOF 29.3/10 27.2/10 33.0/10 30.5/11
Table 6. TheresultsforG8 andG27 andtheK˜i forthevariousconstraintsdescribedinthetext.InbracketsaretheMINUIT
errors. The K˜i are quoted at thescale µ mentioned.
We get fits of roughly similar quality for all values of variation with µ from µ=0.77 GeV to µ=0.6 GeV and
µ where the other parametershave been put to zero.The µ=1.0 GeV for K˜ , i=1,...,11 in Table 7
i
fits tend to be slightly better for the larger values of µ.
The fitted values for the K˜ are µ-dependent, albeit not Inordertocomparewiththefactorizationmodelofthe
i
extremely strongly. weaklowenergyconstants,wealsoperformafitwhereall
next-to-leadingorderLECsproportionaltoG aresetto
27
TheK˜ themselveshaveaµ-dependencewhichisgiven zero, but we keep in addition the sub-leading octet ones.
i
by the cancellation of divergences, and this can be calcu- This fit is shown for µ = 0.77 GeV in Table 7. The fit
lated from the known subtractions. We have shown the is somewhat worse than those of Tab. 6 but not much. A
