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Splitting Strong and Electromagnetic Interactions in K Decays ℓ4 ∗ 4 0 0 A. Nehme 2 n a February 1, 2008 J 1 3 1 v 27 rue du Four de la Terre 7 F-84000 Avignon, France 0 0 2 [email protected] 0 4 Abstract 0 / h We recently considered K decays in the framework of chiral per- ℓ4 p turbation theory based on the effective Lagrangian including mesons, - p photons, and leptons. There, we published analytic one-loop-level ex- e pressionsfor formfactors f andg correspondingto themixed process, h : K0 π0π−ℓ+ν . We propose here a possible splitting between strong v → ℓ i and electromagnetic parts allowing analytic (and numerical) evalua- X tion of Isospin breaking corrections. The latter are sensitive to the r a infrared divergence subtraction scheme and are sizeable near the ππ production threshold. Our results should be used for the extraction of the P-wave iso-vector ππ phase shift from the outgoing data of the currently running KTeV experiment at FNAL. keywords: ElectromagneticCorrections,KaonSemileptonicDecay,Form Factors, Pion Pion Phase Shifts, Chiral Perturbation Theory. ∗ This work is dedicated to my son 1 Contents 1 Introduction 2 2 Kinematical variables 4 2.1 The photonic contribution . . . . . . . . . . . . . . . . . . . . 6 2.2 The non photonic contribution . . . . . . . . . . . . . . . . . . 7 2.3 Splitting strong and electromagnetic interactions . . . . . . . . 9 3 The photonic contribution 10 4 The non photonic contribution 13 4.1 One-point functions . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Two-point functions . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Isospin limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 The ǫ-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.5 The ∆ -terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 π 4.6 The ∆ -terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 K 5 Results 26 5.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 The f form factor . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3 The g form factor . . . . . . . . . . . . . . . . . . . . . . . . . 31 6 Conclusion 33 A Loop Integrals 34 A.1 B-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.2 τ-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.3 C-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1 Introduction Every time that a kaon decays into a couple of pions and a lepton-neutrino pair, a ππ scattering occursinthe finalstate. Whenever apionscatters onits twin, it offers to us an additional opportunity to scrutinize the fundamental state of strong interaction (see [1] for references). Let δI be the phase of a l 2 two-pion state of angular momentum l and Isospin I and consider the K ℓ4 decay process, K(p) π(p )π(p )ℓ+(p )ν (p ), (1) 1 2 ℓ ℓ ν −→ where the lepton, ℓ, is either a muon, µ, or an electron, e, and ν stands for the corresponding neutrino. In the Isospin limit, the decay amplitude, , for A process (1) can be parameterized in terms of three vectorial (F, G, and R) and one anomalous (H) form factors, . G = i F V∗u(p )γ (1 γ5)v(p ) A √2 us ν µ − ℓ × i [(p +p )µF +(p p )µG+(p +p )µR] 1 2 1 2 ℓ ν (cid:26)MK± − 1 ǫµνρσ(p +p ) (p +p ) (p p ) H , (2) −M3 ℓ ν ν 1 2 ρ 1 − 2 σ K± (cid:27) where V denotes the Cabibbo-Kobayashi-Maskawa flavor-mixing matrix el- us ement and G is the so-called Fermi coupling constant. Note that form F factors are made dimensionless by inserting the normalizations, M−1 and K± M−3. The fact that we have used the charged kaon mass is a purely con- K± ventional matter and corresponds to the choice of defining the Isospin limit in terms of charged masses. In the following, we will be interested only in two form factors, F and G, and denote by (F,G)+− and (F,G)0− the ones corresponding to the physical processes, K+(p) π+(p )π−(p )ℓ+(p )ν (p ), (3) 1 2 ℓ ℓ ν −→ and K0(p) π0(p )π−(p )ℓ+(p )ν (p ), (4) 1 2 ℓ ℓ ν −→ respectively. Form factors are analytic functions of three independent Lorentz invariants, . . s = (p +p )2, s = (p +p )2, (5) π 1 2 ℓ ℓ ν and the angle θ formed by p , in the dipion rest frame, and the line of flight π 1 of the dipion as defined in the kaon rest frame [2, 3]. It has been shown in [4] that, in the experimentally relevant region, the partial wave expansion, F+− = (f (s )+f s )eiδ00(sπ) +f˜ XY cosθ eiδ11(sπ), (6) S π ℓ ℓ P π 3 G+− = (g +g′ s +g s )eiδ11(sπ) +g˜ XY cosθ eiδ20(sπ), (7) P P π ℓ ℓ D π is proving sufficient to parameterize form factors. In the preceding, . 1 . 1 X = λ1/2(s ,s ,M2 ), Y = λ1/2(s ,M2 ,M2 ), (8) 2 π ℓ K± s π π± π± π with, . λ(x,y,z) = x2 +y2+z2 2xy 2xz 2yz, (9) − − − the usual Källén function. Note the linear dependence of the first term in the partial wave expansion of form factors on s . Isospin symmetry, Bose ℓ symmetry, and the ∆I = 1/2 rule lead to, F0− = √2f˜ XY cosθ eiδ11(sπ), (10) P π G0− = √2(g +g′ s +g s )eiδ11(sπ). (11) P P π ℓ ℓ It follows that K decay of the neutral kaon is dominated by P waves. ℓ4 Therefore, a precise measurement of form factors for the decay in question would allow an accurate determination of the P-wave iso-vector ππ phase shift. The currently running KTeV experiment [5] aims at measuring form factors for K decay of the neutral kaon with an accuracy 3 times better than ℓ4 the one offered by previous measurement [6, 7]. The outgoing data on form factors contain, besides strong interaction contribution, a contribution com- ing from electroweak interaction. The latter breaks Isospin symmetry and is expected to be sizeable near ππ production threshold [8]. In order to extract ππ scattering parameters fromthe KTeV measurement, Isospin breaking cor- rection to form factors should therefore be under control. In this direction, we recently published analytic expressions for F0− and G0− form factors cal- culatedatone-looplevel intheframeworkofchiralperturbationtheorybased on the effective Lagrangian including mesons, photons, and leptons [1]. In the present work, we will split analytically Isospin limit and Isospin breaking part in form factors allowing a first evaluation of Isospin breaking effects in K decays. ℓ4 2 Kinematical variables In the following, we shall consider process (4) and use, unless mentioned, notations of reference [1]. In the presence of Isospin breaking, the decay 4 amplitude for process (4) can be written as follows by Lorentz covariance, . G V∗ 0− = F us u(p )(1+γ5) A √2 ν × 1 (p +p )µf0− +(p p )µg0− +(p +p )µr0− γ 1 2 1 2 ℓ ν µ (cid:26)MK± − i (cid:2) (cid:3) + ǫµνρσ(p +p ) (p +p ) (p p ) h0− M3 ℓ ν ν 1 2 ρ 1 − 2 σ K± 1 + [γ ,γ ]pµpν T v(p ). 2M2 µ ν 1 2 l K± (cid:27) The quantities f, g, r, and h, will be called the corrected K form factors ℓ4 since their Isospin limits are nothing else than the K form factors, F, G, R, ℓ4 and H, respectively. The tensorial form factor T is purely Isospin breaking and does not contribute to the mixed process at leading chiral order. The corrected form factors as well as the tensorial one are analytic functions of five independent Lorentz invariants, s , s , θ , θ , and φ. θ is the angle π ℓ π ℓ ℓ formed by p , in the dilepton rest frame, and the line of flight of the dilepton ℓ as defined in the kaon rest frame. φ is the angle between the normals to the planes defined in the kaon rest frame by the pion pair and the lepton pair, respectively. Let us denote by δF and δG the next-to-leading order corrections to the F0− and G0− form factors, respectively, f0− = MK± 0+δF , F 0 (cid:18) (cid:19) M g0− = K± 1+δG . F 0 (cid:18) (cid:19) The analytic expressions for δF and δG were given in [1]. We shall dis- tinguish between photonic and non photonic contributions to δF and δG. The photonic contribution comes from those Feynman diagrams with a vir- tual photon exchanged between two meson legs or one meson leg and a pure strong vertex. Obviously, this contribution is proportional to e2, where e is the electric charge, and depends in general on the five independent kinematical variables, s , s , θ , θ , and φ through Lorentz invariants like (p +p )2, π ℓ π ℓ 2 ℓ say. The non photonic contribution comes from diagrams having similar topology as the ones in the pure strong theory with Isospin breaking allowed in propagators and vertices. This contribution generates Isospin breaking 5 terms proportional to the rate of SU(2) to SU(3) breaking, . √3 m m . 1 d u ǫ = − , mˆ = (m +m ), (12) u d 4 m mˆ 2 s − and to mass square difference between charged and neutral mesons, . ∆ = M2 M2 = 2Z e2F2 + (p4), (13) π . π± − π0 0 0 O ∆ = M2 M2 = 2Z e2F2 B (m m )+ (p4), (14) K K± − K0 0 0 − 0 d − u O or equivalently, (m m )/(m mˆ), Z e2, and m m . The kinematical d u s 0 d u − − − dependence is on three Lorentz invariants, (p +p )2, (p p )2, and (p p )2 1 2 1 2 − − which represent respectively the dipion mass square, the exchange energy between the kaon and the neutral pion, and that between the kaon and the charged pion. In terms of independent kinematical variables, the preceding scalars are functions of s , s , and cosθ . π ℓ π 2.1 The photonic contribution A generic term in the photonic contribution can be, photonic contribution = e2 ξ (p +p )2,... , (15) i 2 ℓ i X (cid:0) (cid:1) where ξ is an arbitrary loop integral function of (p +p )2. To the order we i 2 ℓ are working, that is, to leading order in Isospin breaking, the power counting scheme we use dictates the following on-shell conditions to be used in the argument of ξ , i . . p2 = M2 = B (m +mˆ), p2 = p2 = M2 = 2B mˆ . (16) K 0 s 1 2 π 0 Therefore, (p +p )2 in(15)shouldbereplacedbythefollowing expression [1], 2 ℓ M2 +m2 π ℓ 1 m2 + 1+ ℓ (M2 s s ) 4 s K − ℓ − π (cid:18) ℓ (cid:19) 1 m2 4M2 1/2 1+ ℓ 1 π λ1/2(s ,s ,M2)cosθ − 4 s − s π ℓ K π (cid:18) ℓ (cid:19)(cid:18) π (cid:19) 1 m2 + 1 ℓ λ1/2(s ,s ,M2)cosθ 4 − s π ℓ K ℓ (cid:18) ℓ (cid:19) 6 1 m2 4M2 1/2 1 ℓ 1 π (M2 s s )cosθ cosθ − 4 − s − s K − ℓ − π π ℓ (cid:18) ℓ (cid:19)(cid:18) π (cid:19) 1 m2 4M2 1/2 + 1 ℓ 1 π (s s )1/2sinθ sinθ cosφ. π ℓ π ℓ 2 − s − s (cid:18) ℓ (cid:19)(cid:18) π (cid:19) Fromtheforegoing,itisclear thatfors = m2 thephotoniccontributiondoes ℓ ℓ not depend neither on θ nor on φ. In order to reduce the complexity of the ℓ study and allow the treatment of photonic and non photonic contributions to 0− on an equal footing, we will assume that, A s = m2, (17) ℓ ℓ and use for (p +p )2 in (15) the following expression, 2 ℓ 1 (p +p )2 = (M2 +2M2 +m2 s ) 2 ℓ 2 K π ℓ − π 1 4M2 1/2 1 π λ1/2(s ,m2,M2)cosθ . (18) − 2 − s π ℓ K π (cid:18) π (cid:19) It follows that (15) can be written as, photonic contribution = e2ς(s )+e2ϑ(s )cosθ , (19) π π π where ς and ϑ are analytic functions of s . π 2.2 The non photonic contribution In order to split strong and electromagnetic terms in the non photonic contribution, one has to expand the exchange energies, (p p )2 and (p p )2, 1 2 − − in powers of the fine structure constant, α, and m m . To this end, we d u − shall first express these scalars in terms of s and cosθ for s = m2 and in π π ℓ ℓ the presence of Isospin breaking. From [1], (p p )2 = M2 +M2 − 1 K0 π0 1 (M2 m2 +s )(s +M2 M2 ) − 2s K0 − ℓ π π π0 − π± π + λ1/2((cid:2)s ,m2,M2 )λ1/2(s ,M2 ,M2 )cosθ , (20) π ℓ K0 π π0 π± π (p p )2 = M2 +M2 − 2 K0 π± (cid:3) 7 1 (M2 m2 +s )(s M2 +M2 ) − 2s K0 − ℓ π π − π0 π± π λ1/2((cid:2)s ,m2,M2 )λ1/2(s ,M2 ,M2 )cosθ . (21) − π ℓ K0 π π0 π± π (cid:3) Let us denote by t andu the Isospin limits of thepreceding Lorentz scalars, π π 1 t = (M2 +2M2 +m2 s ) π 2 K± π± ℓ − π 1 4M2 1/2 1 π± λ1/2(s ,m2,M2 )cosθ , (22) − 2 − s π ℓ K± π (cid:18) π (cid:19) 1 u = (M2 +2M2 +m2 s ) π 2 K± π± ℓ − π 1 4M2 1/2 + 1 π± λ1/2(s ,m2,M2 )cosθ . (23) 2 − s π ℓ K± π (cid:18) π (cid:19) For completeness, it is convenient to note the following proposition, 1 cosθ = 0 = t = u = (M2 +2M2 +m2 s ). (24) π ⇒ π π 2 K± π± ℓ − π Using the replacements, M2 M2 ∆ , M2 M2 ∆ , (25) π0 −→ π± − π K0 −→ K± − K and expanding (20) and (21) to first order in ∆ and ∆ , we obtain, π K 1 (p p )2 = (M2 +2M2 +m2 s ) − 1 2 K± π± ℓ − π 1 1 + (M2 m2 s )∆ ∆ 2s K − ℓ − π π − 2 K π 1 4M2 1/2 + 1 π± λ1/2(s ,m2,M2 ) −2 − s π ℓ K± " (cid:18) π (cid:19) 1 4M2 −1/2 1 π λ1/2(s ,m2,M2)∆ −2s − s π ℓ K π π (cid:18) π (cid:19) 1 4M2 1/2 + 1 π 2 − s × (cid:18) π (cid:19) (M2 m2 s )λ−1/2(s ,m2,M2)∆ cosθ , (26) K − ℓ − π π ℓ K K π (cid:21) 8 1 (p p )2 = (M2 +2M2 +m2 s ) − 2 2 K± π± ℓ − π 1 1 (M2 m2 +s )∆ ∆ − 2s K − ℓ π π − 2 K π 1 4M2 1/2 + 1 π± λ1/2(s ,m2,M2 ) 2 − s π ℓ K± " (cid:18) π (cid:19) 1 4M2 −1/2 + 1 π λ1/2(s ,m2,M2)∆ 2s − s π ℓ K π π (cid:18) π (cid:19) 1 4M2 1/2 1 π −2 − s × (cid:18) π (cid:19) (M2 m2 s )λ−1/2(s ,m2,M2)∆ cosθ . (27) K − ℓ − π π ℓ K K π (cid:21) Note that terms of order (∆ ∆ ) are forbidden by our power counting π K O scheme since they are first order in Isospin breaking. Although equations (26) and (27) are simple to derive, their utility is of great importance to the present study. In fact, the involved expansion could be generalized to any K observable as we will see below. ℓ4 2.3 Splitting strong and electromagnetic interactions The first step in our program consists on injecting equations (26) and (27) in the non photonic contribution to the decay amplitude 0−. Then, we A expand once more to first order in ∆ and ∆ dropping out terms of order π K (∆ ∆ ). As a result, form factors for K decay of the neutral kaon can π K ℓ4 O be written in the following compact form which shows explicitly the splitting between strong and electromagnetic interactions, x0− s ,(p p )2,(p p )2,(p +p )2,... = π 1 2 2 ℓ − − M (cid:0) K± [δ +Ux(s )+Vx(s )cosθ ] ,(cid:1) x = f , g, (28) xg π π π F 0 where, Wx = Wx +Wx∆ +Wx∆ s π π K K ǫ +Wxe2 +Wx , W = U , V , (29) e2 ǫ √3 9 are analytic functions of s . If one makes the following substitutions, π ∆ 2Z e2F2, (30) π −→ 0 0 4ǫ ∆ 2Z e2F2 (M2 M2), (31) K −→ 0 0 − √3 K − π then, equations (28) and (29) read, ǫ Wx = Wx +Wxe2 +Wx , (32) s α md−mu √3 Wx = Wx +2Z F2(Wx +Wx), (33) α e2 0 0 π K Wx = Wx 4(M2 M2)Wx . (34) md−mu ǫ − K − π K The aim of the present work is to determine the U functions corresponding to f and g form factors for K decay of the neutral kaon. ℓ4 3 The photonic contribution From now on, we will work under proposition (24) keeping in mind that, in the Isospin breaking contribution, the power counting dictates the following, 1 Isospinbreaking t = u = (M2 +2M2 +m2 s ). (35) −→ π π 2 K π ℓ − π Taking the photonic contribution from [1], applying assumption (17), and performing the preceding expansion, it is easy at a first sight to derive Ue2. The problemis that, inpractice, oneencounters loopintegrals with vanishing Gramm determinant when reducing vector and tensor integrals to scalar ones [9]. After a long and tedious calculation one obtains, 1 Uf = ( 6K +3K +2K +2K 6X ) e2 3 − 3 4 5 6 − 1 2 M2 π (6K 3K 2K 2K +2K +2K ) 3 4 5 6 9 10 − 3 M2 M2 − − − π − η + B(M2,0,M2) 1 π π (cid:26) 1 (M2 m2 s )2 4m2M2 λ−1(t ,m2,M2) − 4 K − ℓ − π − ℓ π π ℓ π 3 (cid:2) (cid:3) + m2M2(s 4M2)(M2 m2 s )λ−2(t ,m2,M2) 4 ℓ π π − π K − ℓ − π π ℓ π (cid:27) 10