LU TP 04-37 hep-ph/0410333 revised November 2004 5 0 0 2 n Isospin Breaking in K 3π Decays II: a → J Radiative Corrections 0 1 2 v 3 3 3 0 Johan Bijnens and Fredrik Borg 1 4 0 / h Department of Theoretical Physics, Lund University p S¨olvegatan 14A, S 22362 Lund, Sweden - p e h : v i X r a Abstract ThefivedifferentCPconservingamplitudesforthedecaysK 3π arecalculated → using Chiral Perturbation Theory. The calculation is made to next-to-leading order and includes full isospin breaking. The squared amplitudes are compared with the corresponding ones in the isospin limit to estimate the size of the isospin breaking effects. Inthispaperweaddtheradiativecorrectionstotheearliercalculatedm m u d − and local electromagnetic effects. We find corrections of order 5-10 percent. PACS numbers: 13.20.Eb; 12.39.Fe; 14.40.Aq; 11.30.Rd 1 Introduction Thenon-perturbativenatureoflow-energyQCDcallsforalternativemethodsofcalculating processes including composite particles such as mesons and baryons. A method describing the interactions of the light pseudoscalar mesons (K,π,η) is Chiral Perturbation Theory (ChPT). It was introduced by Weinberg, Gasser and Leutwyler [1, 2, 3] and it has been very successful. Pedagogical introductions to ChPT can be found in [4]. The theory was later extended to also cover the weak interactions of the pseudoscalars [5], and the first calculation of a kaon decaying into pions (K 2π,3π) appeared shortly thereafter [6]. → Reviews of other applications of ChPT to nonleptonic weak interactions can be found in [7]. A recalculation in the isospin limit of K 2π to next-to-leading order was made in → [8, 9] and of K 3π in [9, 10]. In [9] also a full fit to all experimental data was made → and it was found that the decay rates and linear slopes agreed well. However, a small discrepancy was found in the quadratic slopes and that is part of the motivation for this further investigation of the decay K 3π in ChPT. → The discrepancies found can have several different origins. It could be an experimental problem or it could have a theoretical origin. In the latter case the corrections to the amplitude calculated in [9] are threefold: strong isospin breaking, electromagnetic (EM) isospin breaking or higher order corrections. These effects have been studied in many papers for the K 2π decays, references can be traced back from [11]. For K 3π → → less work has been done. In [12] the strong isospin and local electromagnetic corrections were investigated and it was found that the inclusion of those led to changes of a few percent in the amplitudes. The local electromagnetic part was also calculated in [10], in full agreement with our result after corrections of some misprints in [10]. In this paper we add also the radiative corrections, i.e. the nonlocal electromagnetic isospin breaking. The full (first order) isospin breaking amplitude to next-to-leading chiral order is thus calculated, and we will try to estimate the effect of this in the amplitudes. A new full fit, including also new experimental data [13, 14], has to be done to answer the question whether isospin breaking removes the problem of fitting the quadratic slopes. This, together with a study of models for the higher order coefficients, we plan to do in an upcoming paper. Other recent results on K 3π decays can be found in [15, 16]. In [15] Nicola → Cabibbo discusses the possibility of determining the a a pion scattering length from 0 2 − the threshold effects of K+ π0π0π+. He gives an approximate theoretical result with → very few unknown parameters. We have a possibly better theoretical description of these effects but it includes more unknown parameters. In [16] an attempt was made to calculate the virtual photon corrections to the K+ π0π0π+ decay. Our result disagrees with the → result presented there. The outline of this paper is as follows. The next section describes isospin breaking in more detail. In section 3 the basis of ChPT, the Chiral Lagrangians, are discussed. Section 4 specifies the decays and describes the relevant kinematics. The divergences appearing when including photons are discussed in section 5. In section 6 the analytical results 1 are discussed, section 7 contains the numerical results and the last section contains the conclusions. 2 Isospin Breaking Isospin symmetry is the symmetry under exchange of up- and down-quarks. Obviously this symmetry is only true in the approximation that m = m and electromagnetism is u d neglected, i.e. in the isospin limit. Calculations are often performed in the isospin limit since this is simpler and gives a good first estimate of the result. However, to get a precise result one has to include isospin breaking, i.e. the effects from m = m and electromagnetism. Effects coming from m = m we refer to as strong u d u d 6 6 isospin breaking and include mixing between π0 and η. This mixing leads to changes in the formulas for both the physical masses of π0 and η as well as the amplitude for any process involving either of the two. For a detailed discussion see [17]. The other source is electromagnetic isospin breaking, coming from the fact that the up- and the down-quarks are charged, which implies different interactions with photons. This part can be further divided in local electromagnetic isospin breaking and explicit photon contributions (radiative corrections). The former are described by adding new Lagrangians at each order and the latter by introducing new diagrams including photons. Our first calculation of K 3π [9] was done in the isospin limit. In the next paper, → [12], we included strong and local isospin breaking (there collectively referred to as strong isospin breaking) and we now present the calculation including all isospin breaking effects. 3 The ChPT Lagrangians The basis of our ChPT calculation is the various Chiral Lagrangians. They can be divided in different orders. The order parameters in the perturbation series are p and m, the momenta and mass of the pseudoscalars. Including isospin breaking also e, the electron charge, and the mass difference, m m , are used as order parameters. All of these u d − are independent expansion parameters. We work to leading order in m m and e2 but u d − next-to-leading order in p2 and m2. For simplicity we call in the remainder terms of order p2, m2, e2 and m m leading order, and terms of order p4 ,p2m2, m4, p2e2, m2e2, u d − p2(m m ) and m2(m m ) next-to-leading order. u d u d − − 3.1 Leading Order The leading order Chiral Lagrangian is usually divided in three parts = + + , (1) 2 S2 W2 E2 L L L L where refers to the strong ∆S = 0 part, the weak ∆S = 1 part, and the S2 W2 E2 L L ± L strong-electromagnetic and weak-electromagnetic parts combined. For the strong part we 2 have [2] F2 = 0 u uµ +χ (2) S2 µ + L 4 h i Here A stands for the flavour trace of the matrix A, and F is the pion decay constant 0 h i in the chiral limit. We define the matrices u , u and χ as µ ± u = iu D U u = u , u2 = U , χ = u χu uχ u, (3) µ † µ † †µ ± † † ± † where the special unitary matrix U contains the Goldstone boson fields 1 π + 1 η π+ K+ i√2 √2 3 √6 8 U = exp F0 M! , M =  Kπ− √−12π3K+0√16η8 K2η0  . (4)  − √−6 8    The formalism we use is the external field method of [2], and to include photons we set m u χ = 2B m and D U = ∂ U ieQA U ieUQA , (5) 0 d  µ µ µ µ − − m s     where A is the photon field and µ 2/3 Q = 1/3 . (6)   − 1/3  −    We diagonalize the quadratic terms in (2) by a rotation π0 = π cosǫ+η sinǫ 3 8 η = π sinǫ+η cosǫ, (7) 3 8 − where the lowest order mixing angle ǫ satisfies m m tan(2ǫ) = √3 d − u . (8) 2m m m s u d − − The weak part of the Lagrangian has the form [18] = CF4 G ∆ u uµ +G ∆ χ +G tij,kl ∆ u ∆ uµ + h.c.. (9) LW2 0 " 8h 32 µ i ′8h 32 +i 27 h ij µih kl i# The tensor tij,kl has as nonzero components 1 1 t21,13 = t13,21 = ; t22,23 = t23,22 = ; 3 −6 1 1 t23,33 = t33,23 = ; t23,11 = t11,23 = , (10) −6 3 3 and the matrix ∆ is defined as ij ∆ uλ u , (λ ) δ δ . (11) ij ≡ ij † ij ab ≡ ia jb The coefficient C is defined such that in the chiral and large N limits G = G = 1, c 8 27 3 G C = FV V = 1.06 10 6 GeV 2. (12) −5 √2 ud u∗s − · − − Finally, the remaining electromagnetic part, relevant for this calculation, looks like (see e.g. [19]) = e2F4Z +e2F4 Υ (13) LE2 0 hQLQRi 0h QRi where the weak-electromagnetic term is characterized by a constant G (g G in [19]), E ewk 8 Υ = G F2∆ +h.c. (14) E 0 32 and = uQu , = u Qu. (15) L † R † Q Q 3.2 Next-to-leading Order The factthat ChPT isanon-renormalizable theorymeans thatnewterms have tobeadded at each order to compensate for the loop-divergences. This means that the Lagrangians increase in size for every new order and the number of free parameters rises as well. At next-to-leading order the Lagrangian is split in four parts which, in obvious notation, are = + + + (G ). (16) 4 S4 W4 S2E2 W2E2 8 L L L L L Here the notation (G ) indicates that here only the dominant G -part is included in the 8 8 Lagrangian and therefore in the calculation. These Lagrangians are quite large and we choose not to write them explicitly here since they can be found in many places [2, 20, 5, 21, 22, 19, 23]. For a list of all the pieces relevant for this specific calculation see [12]. Note however that four terms producing photon interactions should be added to in [12]. The two new terms in the octet part W4 L are N i ∆ fµν,u u +N i ∆ u fµνu (17) 14 32 + µ ν 15 32 µ + ν h { }i h i and in the 27 part D itij,kl ∆ u ∆ [u ,fµν] +D itij,kl ∆ u u ∆ fµν , (18) 13 ij µ kl ν + 15 ij µ ν kl + h ih i h ih i where fµν = uFµνu +u Fµνu, Fµν = eQ(∂µAν ∂νAµ). (19) + † † − 4 3.2.1 Ultraviolet Divergences The process K 3π receives higher-order contributions from diagrams that contain loops. → The study of these diagrams is complicated by the fact that they need to be defined pre- cisely. The loop-diagrams involve an integration over the undetermined loop-momentum q, and the integrals are divergent in the q ultraviolet region. These ultraviolet diver- → ∞ gences are canceled by replacing the coefficients in the next-to-leading order Lagrangians by the renormalized coefficients and a subtraction part, see [9, 12] and references therein. The divergences can be used as a check on the calculation and all our infinities (except the ones left since the G -part in is not known) cancel as they should. 27 W2E2 L 3.2.2 Loop Integrals The prescription we use for the loop integrals can be found in many places, e.g. [24]. The only one needed in addition to the ones given there is the one-loop three point function 1 ddp 1 C(m2,m2,m3,p2,p2,p2) = , (20) 1 2 3 1 2 3 i (2π)d(p2 m2)((p p )2 m2)((p p )2 m2) Z − 1 − 1 − 2 − 3 − 3 where p = p +p . For its numerical evaluation we use the program FF [25]. This program 3 1 2 also deals with possible infrared divergences consistently. 4 Kinematics There are five different CP-conserving decays of the type K 3π (K decays are not − → treated separately since they are counterparts to the K+ decays): K (k) π0(p )π0(p )π0(p ), [AL ], L → 1 2 3 000 K (k) π+(p )π (p )π0(p ), [AL ], L → 1 − 2 3 +−0 K (k) π+(p )π (p )π0(p ), [AS ], S → 1 − 2 3 + 0 − K+(k) π0(p )π0(p )π+(p ), [A ], 1 2 3 00+ → K+(k) π+(p )π+(p )π (p ), [A ], (21) 1 2 − 3 ++ → − where we have indicated the four-momentum defined for each particle and the symbol used for the amplitude. The kinematics is treated using s = (k p )2 , s = (k p )2 , s = (k p )2 . (22) 1 1 2 2 3 3 − − − The amplitudes are expanded in terms of the Dalitz plot variables x and y defined as s s s s 1 3 0 2 1 y = − , x = − , s = (s +s +s ) . (23) m2 m2 0 3 1 2 3 π+ π+ 5 Figure 1: Bremsstrahlung, the emission of an extra final-state photon. The amplitude for K π0π0π0 is symmetric under the interchange of all three final state L → particles and the one for K π+π π0 is antisymmetric under the interchange of π+ and S − → π because of CP. The amplitudes for K π+π π0,K+ π+π+π and K+ π0π0π+ − L − − → → → aresymmetric undertheinterchangeofthefirsttwopionsbecauseofCPorBose-symmetry. 5 Infrared Divergences In addition to the ultraviolet divergences which are removed by renormalization, diagrams includingphotonsintheloopscontaininfrared(IR)divergences. Theseinfinitiescomefrom the q 0 end of the loop-momentum integrals. They are canceled by including also the → Bremsstrahlung diagram, where a real photonis radiated off oneof the charged mesons, see Fig. 1. It is only the sum of the virtual loop corrections and the real Bremsstrahlung which is physically significant and thus needs to be well defined. We regulate the IR divergence in both the virtual photon loops and the real emission with a photon mass m and keep γ only the singular terms plus those that do not vanish in the limit m 0. We include the γ → real Bremsstrahlung for photon energies up to a cut-off ω and treat it in the soft photon approximation. The exact form of the amplitude squared for the bremsstrahlung diagram depends on which specific amplitude that is being calculated. For K+(k) π0(p )π0(p )π+(p ) it can 1 2 3 → be written in the soft photon limit (see e.g. [26]) d3q 1 k ǫ(λ) p ǫ(λ) 2 A 2 = A 2 e2 · 3 · , (24) | |BS | |LO Z (2π)3 2q λX=0,1" q·k − q ·p3 # where A is the lowest order isospin limit amplitude. The number of terms inside the LO | | parentheses is the number of charged particles in the process and the sign of those terms depends both on the charge of the radiating particle and on whether it is incoming or outgoing. Writing out the square and using ǫ(λ)ǫ(λ) = g , you get λ=0,1 µ ν − µν P d3q 1 k2 p2 2p k A 2 = A 2 e2 + 3 3 · (25) | |BS −| |LO Z (2π)3 2q "(q·k)2 (q ·p3)2 − (q ·k)(q ·p3)# 6 To solve the first integral term, place the vector k along the z-axis, i.e. k = (k0,0,0,kz) and (k q)2 = (k0q0 kzqz)2. (26) · − Changing to polar coordinates that part of the integral now looks like m2 q A 2 e2 K dqd(cosθ) , (27) −| |LO 8π2 (k0E kzqcosθ)2 Z γ − where k2 = m2 , q = E and q = qcosθ have been used. Solving the d(cosθ) part is now K 0 γ z straightforward and leads to m2 1 1 1 A 2 e2 K dq , (28) −| |LO 8π2 Z kz k0Eγ −kzq − k0Eγ +kzq! Putting the two terms on a common denominator and changing variable to E leads to γ m2 ω E A 2 e2 K dE γ , (29) −| |LO 4π2 Zmγ γEγ2(k0)2 −(Eγ2 −m2γ)(kz)2 whereω isthephotonenergyabovewhichthedetectoridentifiesitasarealexternalphoton. We are only interested in the result in the limit m 0, so it’s enough to consider γ → m2 ω 1 A 2 e2 K dE , (30) −| |LO 4π2 Zmγ γm2KEγ which gives the result e2 ω2 A 2 log . (31) −| |LO 8π2 m2 γ In a similar way one gets the result for the mixed term d3q 1 p k x s m2 m2 ω2 2 3 · = s 3 − K − π logx log I (m2 ,m2,s ), Z (2π)3 2q " (q·k)(q ·p3)# −4π2mKmπ(1−x2s) s m2γ ≡ IR K π 3 (32) where 1 4m m /(s¯ (m m )2) 1 K π 3 K π x = − − − − . (33) s q 1 4m m /(s¯ (m m )2)+1 K π 3 K π − − − q In order to obtain the correct imaginary part we use the iε-prescription, which means s¯ = s +iε. 3 3 For the other amplitudes the calculations are similar and the resulting bremsstrahlung amplitudes are AL 2 = 0, (34) | 000|BS e2 ω2 AL 2 = AL 2 log I (m2,m2,s ) , (35) | +−0|BS −| +−0|LO 4π2 " m2γ − IR π π 3 # 7 Figure 2: The tree level diagrams for K 3π. A filled square is a weak vertex, a filled → circle a strong vertex, a straight line a pseudoscalar meson and a wiggly line a photon. e2 ω2 AS 2 = AS 2 log I (m2,m2,s ) , (36) | +−0|BS −| +−0|LO 4π2 " m2γ − IR π π 3 # e2 ω2 A 2 = A 2 log I (m2,m2 ,s ) , (37) | 00+|BS −| 00+|LO 4π2 " m2γ − IR π K 3 # e2 ω2 A 2 = A 2 2log I (m2,m2 ,s ) I (m2,m2 ,s ) | ++−|BS −| ++−|LO 4π2 " m2γ − IR π K 1 − IR π K 2 +I (m2,m2 ,s ) I (m2,m2,s ) I (m2,m2,s ) IR π K 3 − IR π π 1 − IR π π 2 +I (m2,m2,s ) . (38) IR π π 3 i When using the above, the divergences from the explicit photon loops cancel exactly. A similar problem shows up in the definition of the decay constants since we normalize the lowest order with F and F . Our prescription for the decay constants is described π+ K+ in App. A. 6 Analytical Results 6.1 Lowest order The four diagrams that could contribute to lowest order can be seen in Fig. 2. However, the two diagrams including photons turn out to give zero. This is obviously so for K+ π+π0π0 and K π0π0π0 since the γπ0π0 vertex vanishes as a consequence L → → of charge conjugation. Thereasonwhyitvanishesfortheotherdecaysissomewhat moresubtleandisthesame as why the lowest order result for K πℓ+ℓ vanishes [27]. When doing a simultaneous − → diagonalizationofthecovariant kineticandmasstermsquadraticinthepseudoscalar fields, including those of the weak lagrangian , p2-terms of the form ∂ K∂µπ are absent W2 µ L and all weak vertices involve at least three pseudoscalar fields. This result should not change as compared to our calculation where the weak Lagrangian was not included in the 8 Figure 3: Examples of diagrams of next-to-leading order with no photons. An open square is a vertex from or , an open circle a vertex from or , a filled square a W4 W2E2 S4 S2E2 L L L L vertex from or (∆S = 1) and a filled circle a vertex from or (∆S = 0). W2 E2 S2 E2 L L L L diagonalization. Thus in our case, the two diagrams on the right in Fig. 2 will together give zero contribution. This means that the lowest order result in the full isospin case is the same as when just including strong and local EM isospin breaking. This result we published before, the full expressions can be found in [12]. 6.2 Next-to-leading order There are 51 additional diagrams contributing to next-to-leading order. They can be divided in three different classes and examples will be shown of each class. It should be noted that the argument in the previous subsection is not valid at this order. There now exist Kπγ vertices. The reason for this is that one can not diagonalize simultaneously all terms with two pseudoscalar fields when going to next-to-leading order. The first class of diagrams are the 13 which do not include explicit photons. They are the ones used in our earlier papers [9, 12] and a complete list of them can be found there. Some examples are shown in Fig. 3. The second class of diagrams are the ones with a photon running in a loop. There are 18 of these and some examples can be found in Fig. 4. Their evaluation is the main new result of this paper. They are also responsible for the infrared divergences discussed in Sect. 5. The first diagram is an example where the photon is the only particle in the loop, a photon tadpole diagram. These vanish in dimensional regularization when only singular and nonzero terms in the limit m 0 are kept. γ → 9