John K. Elwood and Mark B. Wise. California Institute of Technology, Pasadena, CA and. Martin J. Savage

Transcription

1 π e e * John K. Elwood and Mark B. Wise California Institute of Technology, Pasadena, CA 9115 arxiv:hep-ph/95488v1 11 Apr 1995 and Martin J. Savage Departent of Physics, Carnegie Mellon University, Pittsburgh PA 1513 Abstract We calculate all of the for factors for the one-photon, π γ π e e contribution to the π e e decay aplitude at leading order in chiral perturbation theory. These for factors depend on one unknown constant that is a linear cobination of coefficients of local O(p 4 ) operators in the chiral lagrangian for weak radiative kaon decay. We deterine the differential rate for π e e and also the agnitude of two CP violating observables. CALT CMU-HEP 95- DOE-ER/ DE-FG3-9-ER471 April 1995 * Work supported in part by the Departent of Energy under contracts DE FG 91ER468 (CMU) and DE-FG3-9-ER471 (Caltech). 1

2 1. Introduction At the present tie about twenty π e e events have been observed and a detailed experiental study of this decay ode will be possible in future experients [1]. The π e e weak decay aplitude is doinated by the process, π γ π e e, where a single virtual photon creates the e e pair. This one photon contribution to the decay aplitude has the for M (1γ) = s 1G F α 4πfq [ igε µλρσ p λ p ρ q σ F p µ F p µ ] u(k )γ µ v(k ), (1.1) where G F is Feri s constant, α is the electroagnetic fine structure constant, s 1. is the sine of the Cabibbo angle and f 13 MeV is the pion decay constant. The π and four-oenta are denoted by p and p and the e and e four-oenta are denoted by k and k. The su of the electron and positron four-oenta is q = k k. The Lorentz scalar for factors G, F ± depend on the scalar products of the four-oenta q, p and p. Neglecting CP nonconservation, under interchange of the pion four-oenta p p and p p (1.) the for factors becoe G G, F F, F F. (1.3) In this paper we copute the CP conserving contribution to the for factors G, F ± using chiral perturbation theory at one-loop order (the O(p ) aplitude vanishes). The coefficients of soe of the local operators appearing at the sae order in the chiral expansion (i.e., order p 4 counter ters, where p is a typical oentu) are deterined by the experiental value of the pion charge radius and the easured K π e e and π γ decay rates and spectra. We also copute (in chiral perturbation theory) an iportant tree level contribution to the for factors F ± that arises fro the sall CP even coponent of the state. This contribution to the F ± for factors fro indirect CP nonconservation has the opposite syetry property under interchange of pion oenta when copared with the CP conserving contribution to F ± (see eqs. (1.) and (1.3)). If p p and p p (1.4)

3 then the CP violating one-photon for factors becoe F F, F F. (1.5) The decay aplitude that follows fro squaring the invariant atrix eleent in eq. (1.1) and suing over e and e spins is syetric under interchange of e and e oenta, k k. Physical variables that are antisyetric under interchange of the e and e oenta arise fro the interference of the short distance contributions (Zpenguin and W-box diagras) and the two photon piece with the one photon aplitude given in eq. (1.1). for In the inial standard odel the coupling of the quarks to the W-bosons has the L int = g u j L γ µv jk d k L W µ h.c.. (1.6) Here repeated generation indices j, k are sued over 1,,3 and g is the weak SU() gauge coupling. V is a 3 3 unitary atrix (the Cabibbo Kobayashi Maskawa atrix) that arises fro diagonalization of the quark ass atrices. By redefining phases of the quark fields it is possible to write V in ters of four angles θ 1, θ, θ 3 and δ. The θ j are analogous to the Euler angles and δ is a phase that, in the inial standard odel, is responsible for the observed CP violation. Explicitly V = c 1 s 1 c s 1 c 3 c 1 c c 3 s s 3 e iδ s 1 s 3 c 1 c s 3 s c 3 e iδ s 1 s c 1 s c 3 c s 3 e iδ c 1 s s 3 c c 3 e iδ, (1.7) where c i cos θ i and s i sin θ i. It is possible to choose the θ j to lie in the first quadrant. Then the quadrant of δ has physical significance and cannot be chosen by a phase convention for the quark fields. A value of δ not equal to or π gives rise to CP violation. The short distance W-box and Z-penguin Feynan diagras depend on the V ts eleent of the Cabibbo Kobayashi Maskawa atrix. It is very iportant to be able to deterine this coupling experientally. In this paper we calculate the contribution to the π e e decay aplitude arising fro the Z-penguin and W-box diagras which can be deterined using chiral perturbation theory since the left handed current sγ µ (1 γ 5 )d is related to a generator of chiral syetry. At the present tie all observed CP nonconservation has its origin in K K ass ixing. A CP violating variable can be constructed in the decay π e e that gets an iportant contribution fro 3

4 CP nonconservation in the Z-penguin and W-box diagras, that is, direct CP violation. The variable (in the rest frae) A CP =< ( p p ) ( k k ) ( p p ) ( k k ) >, (1.8) is even under charge conjugation and odd under parity. It is also odd under interchange of k and k. The real and iaginary parts of V ts are coparable, and hence the CP conserving and CP violating parts of the Z-penguin and W-box diagras are of roughly equal iportance. A CP gets a significant contribution fro this direct source of CP nonconservation. In this paper we calculate A CP in the inial standard odel but unfortunately we find that it is quite sall; A CP 1 4. The decay π e e has been studied previously by Sehgal and Wanninger [] and by Heiliger and Sehgal [3]. These authors adopted a phenoenological approach, relating the π e e decay aplitude to the easured π γ decay aplitude. In the systeatic expansion of chiral perturbation theory we find iportant additional contributions to the π e e decay aplitude for q = (k k ) >> 4 e that were not included in this previous work. It was pointed out in refs. [] and [3] that indirect CP nonconservation fro K K ixing gives an iportant contribution to the π e e decay rate and consequently there is a CP violating observable, B CP, that is quite large. We reexaine B CP using the for factors deterined in this paper.. The One-Photon Aplitude Chiral perturbation theory provides a systeatic approach to understanding the onephoton part of the π e e decay aplitude. It uses an effective field theory that incorporates the SU(3) L SU(3) R chiral syetry of QCD and an expansion in powers of oentu to reduce the nuber of operators that occur. In the chiral Lagrangian the π s, K s and η are incorporated into a 3 3 special unitary atrix ( ) im Σ = exp, (.1) f where M = π / η/ 6 π K π / η/ 6 K K K η/ 6. (.) 4

5 At leading order in chiral perturbation theory f 13 MeV is the pion decay constant. Under SU(3) L SU(3) R transforations the Σ field transfors as where L ǫ SU(3) L and R ǫ SU(3) R. Σ LΣR, (.3) At leading order in chiral perturbation theory (i.e., order p, where p is a typical fouroentu) the strong and electroagnetic interactions of the pseudo Goldstone bosons are described by the chiral Lagrange density L (1) S = f 8 Tr(D µσd µ Σ ) vtr( q Σ q Σ ), (.4) where v is a paraeter with diensions of ass to the third power and q is the quark ass atrix q = u d s. (.5) In this paper we neglect isospin violation in the quark ass atrix and set u = d. In this approxiation the K and K have equal asses which we denote by K, and the Gell-Mann Okubo ass relation holds. 3 η 4 K π =, (.6) The effective Lagrangian for S = 1 weak nonleptonic decays transfors as (8 L, 1 R ) (7 L, 1 R ) under SU(3) L SU(3) R. The (8 L, 1 R ) aplitudes are uch larger than the (7 L, 1 R ) aplitudes and so we will neglect the (7 L, 1 R ) part of the effective Lagrangian. The effective Lagrangian for weak radiative kaon decay is obtained by gauging the effective Lagrangian for weak nonleptonic decays with respect to the U(1) Q of electroagnetis. At leading order in chiral perturbation theory the S = 1 transitions are described by W = g 8G F s 1 f 4 4 Tr [ D µ ΣD µ Σ T ] h.c.. (.7) L (1) The atrix T in (.7) projects out the correct flavour structure of the octet T =, (.8) 1 5

6 and g 8 is a constant deterined by the easured K S π decay rate; g In (.4) and (.7) D µ represents a covariant derivative: where D µ Σ = µ Σ iea µ [Q, Σ], (.9) Q = /3 1/3, (.1) 1/3 is the electroagnetic charge atrix for the three lightest quarks, u, d and s. The state is ostly the CP odd state > K > ǫ K 1 >, (.11) K >= 1 ( K > K >), (.1) with sall adixture of the CP even state K 1 >= 1 ( K > K >). (.13) The paraeter ǫ characterizes CP nonconservation in K K ixing. At leading order in chiral perturbation theory the π γ π e e decay aplitude arises though the CP even coponent of. Writing the for factors contributing to π γ as a power series in the chiral expansion F ± = F (1) ± F () ±..., G = G(1) G ()..., (.14) where the superscript denotes the order of chiral perturbation theory, we find that the Feynan diagras in fig. 1 give G (1) = F (1) = 3g 8f ( K π )π ǫ [q q p ] F (1) = 3g 8f ( K π)π ǫ [q q p ]. (.15) Despite the fact that ǫ.3 e i44o (in a phase convention where the K ππ(i = ) decay aplitude is real), is sall it is iportant to keep this part of the decay aplitude. 6

7 Other contributions not proportional to ǫ don t occur until higher order in chiral perturbation theory. We neglect direct sources of CP nonconservation in the one-photon part of the decay aplitude. Experiental inforation on ǫ suggests that they are sall. At the next order in the chiral expansion the for factors G (), F () ± arise fro O(p 4 ) local operators and fro one-loop Feynan diagras involving vertices fro the leading Lagrange densities in (.4) and (.7). However, the for factor G () arises solely fro local operators as the one loop Feynan diagras and tree graphs involving the Wess Zuino ter [4] [5] do not contribute. The contribution of the O(p 4 ) local operators to G () is fixed by the easured π γ decay rate [6] [7] to be G () 4. (.16) The experientally observed π γ Dalitz plot suggests that the for factor G has significant oentu dependence. This indicates that G (3) is not negligible, and that our extraction of G () fro the rate is not copletely justified [8]. The for-factors F () ± get contributions both fro local operators of O(p4 ) [9] and fro one-loop diagras involving vertices fro the leading Lagrange densities in (.4) and (.7). For π e e the local operators that contribute are and L () S = ieλ cr(µ) 16π F µν Tr [ Q(D µ ΣD ν Σ D µ Σ D ν Σ) ], (.17) L () W = i G Fs 1 ef g 8 [ a1 (µ)f µν Tr[QT(ΣD µ Σ )(ΣD ν Σ )] 16π a (µ)f µν Tr[Q(ΣD µ Σ )T(ΣD ν Σ )]. a 3 (µ)f µν Tr[T[Q, Σ]D µ Σ ΣD ν Σ TD µ ΣD ν Σ Σ[Σ, Q]] a 4 (µ)f µν Tr[TΣD µ Σ [Q, Σ]D ν Σ ] ] h.c. (.18) The coefficients λ cr, a 1, a, a 3 and a 4 depend on the renoralization procedure used and we eploy diensional regularization with M S subtraction. The dependence of the coefficients λ cr, a 1,,3,4 on the subtraction point µ cancels that coing fro the one-loop diagras. Note that the basis of operators in eq. (.18) is slightly different than that used in [9]. With this basis of operators the cobination of counterters w L = a 3 a 4, (.19) 7

8 is independent of the subtraction point µ at one loop. The value of λ cr is fixed by the easured π charge radius; < r π >=.44 ±. f. The one-loop diagras in fig. give (using MS) ( ) π λ cr (µ) = f < rπ 3 > 1 [ ln( 4 π /µ ) ln( K /µ ) ], (.) which iplies that (at the subtraction point µ = 1GeV) λ cr (1GeV ) =.91 ±.6. (.1) A linear cobination of a 1 and a is fixed by the easured K π e e decay aplitude. Fortunately it is the sae cobination of a 1 and a that enters into the π e e decay aplitude. The one-photon part of the K π e e decay aplitude can be written in ters of a single for factor f(q ) M (1γ) (K π e e ) = s 1G F α 4π f(q ) p µ π u(k )γ µ v(k ). (.) The one-loop diagras in fig. 3 and the operators in (.17) and (.18) give [1] f(q ) = g 8 (φ K (q ) φ π (q ) 1 6 ln( K /µ ) 1 6 ln( π /µ ) 3 (a 1(µ) a (µ)) 4λ cr (µ) 1 ) 3 ( = g 8 φk (q ) φ π (q ) ) w, (.3) where φ i (q ) = 1 ( ) ) dx i x(1 x) ln (1 q q x(1 x) i. (.4) This relation defines the µ independent constant w [1] which has been experientally deterined to be [11] Using the central values of λ cr (1GeV ) and w we find that w = (.5) a 1 (1GeV ) a (1GeV ) = 6., (.6) with an associated error around 1% (which is correlated with the uncertainty in λ cr (1GeV )). Throughout the reainder of this work we will use the central values of λ cr (1GeV ) and a 1 (1GeV ) a (1GeV ) and suppress the associated uncertainties. Note 8

9 that the contributions λ cr (1GeV ) and (a 1 a )(1GeV ) to f(q ) are separately quite large but they alost cancel against each other. At O(p 4 ) the for factors F () ± for π e e decay follow fro the Feynan diagras in fig. 4 and tree level atrix eleents of the operators in (.17) and (.18). We find, using MS subtraction, that ( F () = g 8 3 q [a 1 (µ) a (µ) 6a 3 (µ) 6a 4 (µ)] 4q λ cr (µ) 3 q φ Kη φ Kπ where 1 4 dx ( q [ ( q x(1 x)ln q ) x(1 x) K π q q (p p ) µ ( πln 1 q x(1 x) ) ( φ K (q ) φ π (q ) 1 6 ln ( π K π )) ) ) ]. (.7) φ Kη = 9 ( K π ) ( 1 with 1 (q q p ) 1 3 ( K π ) ( 1 3 dy 1 dyy y 1 y 1 y dyy dy dxln 1 µ 1 [ dxln 1 y ( 1 dx 1 µ 1 x(1 x)(q )) q p ) K (1 x) η x πx(1 x) ( (q x(1 x) q p xy) dxln 1 ( 1 K (1 y) η y πy(1 y) ) (q x(1 x) q p xy) K (1 y) ηy πy(1 y) (q(1 x) p y) (4q 6p 4p ) K p (q p ) ( dx 1 x (x 1)(3 K π ) q p q ) ( ln 1 x(1 x)(q )) q p ) K (1 x) ηx πx(1 x) (.8) ) ] µ 1 = K (1 y) η y π y(1 y) q x(1 x) q p xy. (.9) 9

10 The other function is φ Kπ = ( K π ) ( 1 with y dy dy 1 y dyy ( dx 1 y ( dxln 1 (q x(1 x) q p xy) K y π(1 y) ( dxln 1 q x(1 x) q p xy K y π (1 y) dx (p p q) (q(1 x) yp p ) 1 x K (x 1) q q p µ ) ) ) ( ln 1 x(1 x)(q q p ) π (1 x) K x )) (.3) µ = Ky π(1 y) q x(1 x) q p xy. (.31) The Gell-Mann Okubo ass forula (.6) has been used to siplify soe of the dependence on the pseudoscalar asses in (.8) and (.3). F () is obtained fro (.7) by taking p p and p p. Notice that the cobination of coefficients (a 1 a ) and λ cr that appear in the expression for F () ± has a relative sign difference copared to the cobination that appears in the expression for f(s) given in eq.(.3). The uncertainty in λ cr (1GeV ) and w gives rise to about a 1% uncertainty in the cobination of counterters that appears in (.7). The one photon part of the π e e decay aplitude is the largest and doinates the rate. In the next section we use the for factors calculated here to obtain dγ( π e e )/dq. One (scale independent) linear cobination of counterters w L = a 3 a 4 is not deterined by the present experiental data and consequently we cannot predict the rate for π e e. However, this is the only undeterined constant and the entire function dγ( π e e )/dq is experientally accessible. 3. The Differential Decay Rate The π e e decay rate is obtained by squaring the invariant atrix eleent (1.1), suing over the e and e spins, and integrating over the phase space. Since the e and e four oenta only occur in the lepton trace, Tr [ /k γ ν /k γ µ ], the phase space integrations over k and k produce a factor d 3 k d 3 k (π) 3 k (π) 3 k (π) 4 δ 4 (q k k )Tr [ ] /k γ ν /k γ µ = 1 6π (q µq ν q q µν ) 1. (3.1)

11 The reaining phase space integrations can be taken to be over q and the su and difference of the pion energies in the rest frae, E S = p p, E D = p p. The contribution of the for factors F ± and G to dγ/dq do not interfere. Therefore, we can write where dγ dq ( π e e ) = dγ G dq dγ F dq, (3.) dγ G dq = G F α s 1 de K f 6 (π) 7 3q S de D G ] [ 4πq π(p q) π(p q) (p p )(q p )(q p ) q (p p ) dγ F dq = G F α s 1 K f 6 (π) 7 3q 4 de S de D [ F q p F q p q ( F π F π Re(F F )p p ) ]. (3.3) In eq. (3.) and eq. (3.3) the difference of pion energies is integrated over the region E (ax) D < E D < E (ax) D E (ax) D = where K E S q K 4 π K E S q (K E S ) q, (3.4) K and the su of pion energies is integrated over the region E (in) S the boundaries are E (ax) S = K q E (in) S < E S < E (ax) S where = K q 4 π K. (3.5) The scalar products appearing in the expression for the rates are easily expressed in ters of E S, E D and q : p p = 1 (q K π KE S ) q p = 1 ( KE S K E D q K). (3.6) q p = 1 ( KE S K E D q K) 11

12 The for factors F (1) ± and F () ± have the opposite property under interchange of pion oenta and consequently they do not interfere in dγ/dq. Neglecting ters in chiral expansion of O(p 6 ) and higher the differential decay rate given in (3.) becoes dγ dq ( π e e ) = dγ G () dq dγ F (1) dq dγ F () dq. (3.7) In fig. 5 we have graphed for each of the three ters on the right hand side of (3.7) 1 Γ KL dγ dy = y ( K π ) 1 Γ KL dγ dq, (3.8) where y = q /( K π ), Γ KL is the total width of the and we have set w L =. Integrating the three ters on the rhs of (3.7) over the invariant ass interval q > (3MeV ) (corresponding to y >.14) we find that for w L = 1 8 Br( π e e ; q > (3MeV ) ) = = 8.. (3.9) The branching fraction over this range of e e invariant ass is doinated by the region of low q and for typical values of w L it receives coparable contributions fro the for factors G and F (). However, in the region of high q the branching fraction is likely to be doinated by the F () ± for factor. For q > (8MeV ) (corresponding to y >.37) and w L = the three ters on the rhs of (3.7) contribute 1 8 Br( π e e ; q > (8MeV ) ) = =.6. (3.1) A suary of our results for the rate can be found in Table 1. We have displayed the contribution to the branching ratio (in units of 1 8 ) fro the three for factors G, F (1) and F () for different values of the iniu lepton pair invariant ass qin. Since the loop contribution to the for factor F () ± is sall, it will be difficult to extract a unique value for w L fro dγ/dq data alone; a two-fold abiguity in the value of w L will persist. 1

13 Lower cut qin Br(1 8 ) G Br(1 8 ) F (1) Br(1 8 ) F () (1MeV) w L.8wL (MeV) w L.8wL (3MeV) w L.8wL (4MeV) w L.7wL (6MeV) w L.6wL (8MeV) w L.4wL (1MeV) w L.3wL (1MeV) w L.16wL (18MeV) w L.6wL Table 1: Contributions to the Branching Ratio (1 8 ) for a range of q in 4. The Z-penguin and W-box Aplitude The short distance W-box and Z-penguin diagras give the effective Lagrange density L SD = ξ s 1G F α sγ µ (1 γ 5 )d eγ µ γ 5 e h.c.. (4.1) Here we only keep the part that contains the lepton axial current (the vector current is neglected). It is only the axial current that gives rise to observables that are antisyetric under interchange of e and e four oenta, k k. In (4.1) the quantity ξ receives significant contributions fro both the top quark and char quark loops and is given by ξ = ξ c ( ) V ts V td Vus V ξ t, (4.) ud where ξ q = ξ (Z) q ξ (W) q, (4.3) is the su of the contributions of the Z-penguin and W-box diagras. It is convenient to express the cobination of eleents of the Cabibbo Kobayashi Maskawa atrix that enters in ξ in ters of V cb and the standard coordinates ρ iη of the unitarity triangle V ts V td/v us V ud = (ρ 1 iη) V cb. (4.4) 13

14 A value of V cb.4 is obtained fro inclusive B X c eν e decay and fro exclusive B D eν e decay. Although the values of ρ and η are not deterined by present data, they are expected to be of order unity. The quantities ξ c and ξ t have been calculated including perturbative QCD corrections at the next to leading logarithic level [1] [13]. There is soe sensitivity to the values of Λ QCD, c and t but ξ c is of order 1 4 and ξ t is of order unity. The quark-level Lagrange density in eq. (4.3) can be converted into a Lagrange density involving the π, K and η hadrons using the Noether procedure. Equating the QCD chiral currents with those obtained fro chiral variations of the effective lagrangian in eq. (.4) leads to L SD = ξ ig Fαs 1 f Tr( µ ΣΣ T) eγ µ γ 5 e h.c.. (4.5) Expanding out Σ in ters of the eson fields M we find that the Lagrange density (4.5) iplies that the short distance contribution to the π e e decay aplitude fro the W-box and Z-penguin diagras is M (SD) = G Fs 1 α f ( ξ p µ ξ p µ ) u(k )γ µ γ 5 v(k ). (4.6) 5. The Asyetry A CP It is the interference of M (SD) in (4.6) with M (1γ) in (1.1) that produces the asyetry A CP defined in (1.8). For calculation of A CP it is convenient to use the phase space variables used by Pais and Treian [14] for K l4 decay (rather than those used for the total rate in Section 3). They are: q = (k k ) ; s = (p p ) ; θ π the angle fored by the π three oentu and the three-oentu in the π rest frae; θ l, the angle between the e three oentu and the three oentu in the e e rest frae; φ, the angle between the norals to the planes defined in the rest frae by the π pair and the e e pair. In ters of these variables and the asyetry is 1 A CP = 7 (π) 6 3 K Γ ( p p ) ( k k ) ( p p ) ( k k ) ( π ) dφ sign(sin φ) 14 = sign(sinφ), (5.1) ) dc π dc e ds dq β X Re (M (SD) M (1γ)) (5.),

15 where c π = cos θ π, c e = cos θ e. The other kineatic functions appearing in this expression are β = [1 4 π /s]1/ [ ( X = K s q ) ] 1/. (5.3) sq In order to evaluate the contributing for factors the following scalar products of four vectors are required: q p = 1 4 ( K s q ) 1 βx cos θ π q p = 1 4 ( K s q ) 1 βx cos θ π p p = 1 (s π). (5.4) ε αβσρ p α p β k σ k ρ = 1 4 βx sq sin θ e sin θ π sin φ If the variables s and q are not integrated over the coplete phase space then it is understood that the sae is to be done for the width Γ KL eq. (5.). in the denoinator of The for factor G does not enter into Re ( M (SD) M (1γ)) (a su over e and e spins is understood). Integrating our cosθ e and φ we find that G F A CP = s 1 α 8 (π) 6 f 3 K Γ dc π ds dq sin θ π [ ]. s β X I(ξ) (Re(F q ) Re(F )) Re(ξ) (I(F ) I(F )) (5.5) The integration over cos θ π iplies that at leading non-trivial order of chiral perturbation theory I(F ) I(F ) I(F (1) ) I(F (1) ) reflecting indirect CP violation fro ǫ and Re(F ) Re(F ) Re(F () () ) Re(F ) in eq.(5.5). Using (4.) and (4.4) we can write the CP violating asyetry in ters of the real and iaginary parts of the CKM eleents A CP = A 1 ( (ρ 1) V cb ξt ξ c ) A η V cb ξt, (5.6) where A 1 arises fro indirect CP nonconservation (i.e. K K ixing ) and A arises fro direct CP nonconservation. We are only able to predict A CP since the sign of g 8 is not known. Our expressions for F (1) () ± and F ± with w L = give (up to an overall sign) A 1 =.7 1, A = 3.9 1, (5.7) 15

16 for q (3MeV ) and A 1 =.4 1, A = 8.4 1, (5.8) for q (8MeV ). In Table we present A 1 and A for a range of values of the iniu lepton pair invariant ass, qin, noralized to the branching ratios given in Table 1 assuing w L =. Lower cut qin A 1 A (1MeV) (MeV) (3MeV) (4MeV) (6MeV) (8MeV) (1MeV) (1MeV) (18MeV) Table : The CP violating quantities A 1, A with w L = for different values of q in We find that direct and indirect sources of CP nonconservation give coparable contributions to A CP. In our coputation we have neglected final state ππ interactions which are forally higher order in chiral perturbation theory. With the values of A 1 and A given in Table, A CP is only of order 1 4 and further refineents of our calculation do not see warranted. 6. The Asyetry B CP Using the kineatic variables introduced in the previous section the CP violating observable B CP is defined as B CP =< sign(sin φ cos φ) >. (6.1) 16

17 At leading order in chiral perturbation theory it arises fro the interference of F (1) ± with G (). The CP violating for factors F (1) ± are not sall because they occur at a lower order in chiral perturbation theory than the other for factors, F () ± and G (). Consequently, as was noted in refs. [] and [3], B CP is quite large. Neglecting M (SD) we find after integrating over φ and cosθ e that G F B CP = s 1 α 3 7 (π) 8 f 3 K Γ dc π ds dq sin θ π β 3 X s q I[ G (F F ) ]. (6.) If the variables s and q are not integrated over the entire phase space then it is understood that the sae is to be done to the width Γ KL in the denoinator of (6.). The for factor G is real at leading order in chiral perturbation theory and the iaginary part arises fro the phase in F F induced by K K ixing. The integration over cosθ π iplies that F F F (1) F (1) (1) in eq.(6.). Using our expressions for F ± and the value of G () we find that with w L = B CP 6.3% for q > (3 MeV ) and B CP.4% for q > (8 MeV ). The asyetry for a range of values of q in are shown in Table 3. Lower cut qin B CP Br(1 8 ) (%) (1MeV) 134 (MeV) 78 (3MeV) 5 (4MeV) 33 (6MeV) 14 (8MeV) 6.3 (1MeV).5 (1MeV).9 (18MeV).86 Table 3: The CP violating observable B CP Br(1 8 ) for a range of values of qin Note that in Table 3 Br(1 8 ) denotes the π e e branching ratio in units of 1 8 with the sae cut on q iposed. We have neglected final state ππ interactions because they arise at higher order in chiral perturbation theory. Our prediction for B CP has considerable uncertainty because of the neglect of final state ππ interactions and because neglected O(p 6 ) contributions to G see to be iportant. 17

18 7. Conclusions In this paper we have calculated the one-photon contribution to the π e e decay rate. We used chiral perturbation theory to deterine the for factors and for e e pairs with high invariant ass ( q >> 4 e) found iportant new contributions that were not included in previous work [][3]. The aplitude for π e e depends on the undeterined (renoralization scale independent) cobination of counterters w L. We found that for q = (k k ) > (3 MeV ) the branching ratio for π e e is approxiately (8. 3.w L.8wL ) 1 8 and for q > (8 MeV ) the branching ratio is approxiately (.6 1.8w L.4wL ) 1 8. One interesting aspect of this decay ode is that the CP even coponent of the state contributes at a lower order in chiral perturbation theory than the CP odd coponent. This enhances CP violating effects in π e e decay. For exaple, the CP violating observable [] [3] B CP =< sign(sin φ cos φ) >, where φ is the angle between the norals to the π and e e planes, is about 6% for q > (3MeV ) if w L =. The CP violating observable A CP =< sign(sinφ) > arises fro the interference of W-box and Z-penguin aplitudes with the one-photon part of the decay aplitude. Unfortunately, we find that A CP is of order 1 4 and hence ost likely uneasurable. Chiral Perturbation theory has been extensively applied to nonleptonic, seileptonic and radiative kaon decays. The study of π e e offers an opportunity to deterine the linear cobination of coefficients in the O(p 4 ) chiral lagrangian that we call w L and to test the applicability of O(p 4 ) chiral perturbation theory for kaon decay. Soe iproveents in our calculations are possible. While a full coputation of the O(p 6 ) contribution to F ± and G arising fro two-loop diagras and new local operators does not see feasible it should be possible to calculate the leading contribution to the absorptive parts of G and F F. Note that the absorptive parts coe fro both ππ ππ rescattering and because of CP nonconservation fro ππ ππγ. We hope to present results for this in a future publication. Acknowledgeents MJS would like to thank the High Energy Physics group at Caltech for kind hospitality during part of this work. 18

19 References [1] Y.W. Wah, in Proceedings of the XXVI International Conference on High Energy Physics, Dallas, Texas, AIP Conf. Proc. No. 7, edited by Jaes R. Sanford (AIP, New York), 199. [] L.M. Sehgal and M. Wanninger, Phys. Rev. D46 (199) 135. [3] P. Heiliger and L.M. Sehgal, Phys. Rev. D48 (1993) [4] G. Ecker, H. Neufeld and A. Pich, Phys. Lett. 78B (199) 337. [5] J. Bijnens, G. Ecker and A. Pich, Phys. Lett. 86B (199) 341. [6] G. Donaldson et al, Phys. Rev. Lett. 33 (1974) 554; Phys. Rev. D14 (1976) 839. [7] E.J. Raberg et al, Phys. Rev. Lett. 7 (1993) 55. [8] G. Ecker, H. Neufeld and A. Pich, Nucl. Phys. B413 (1994) 31. [9] G. Ecker, J. Kabor and D. Wyler, Nucl. Phys. B394 (1993) 11. [1] G. Ecker, A. Pich and E. derafael, Nucl. Phys. B91 (1987) 69. [11] C. Alliegro etal, Phys. Rev. Lett. 68 (199) 78. [1] G. Buchalla and A.J. Buras, Nucl. Phys. B398 (1993) 85. [13] G. Buchalla and A.J. Buras, Nucl. Phys. B4 (1993) 5. [14] A. Pais and S. Triean, Phys. Rev. 168 (1968)

20 Figure Captions Fig. 1. Feynan diagras contributing to F (1) ±. Fig.. Feynan diagras contributing to the π ± charge radius, < rπ order in chiral perturbation theory. >, at leading Fig. 3. The Feynan diagras contributing to the aplitude for K π γ at leading order in chiral perturbation theory. The solid square denotes a vertex fro the gauged weak lagrangian in (.7), the solid circle denotes a vertex fro the gauged strong lagrangian in (.4). Figures 3(a) involve only weak and electroagnetic vertices while Figures 3(b) also has a strong vertex. Figures 3(c) are the contributions fro the kaon and pion charge radii (including both loop graphs and the tree-level counterter). Figure 3(d) is the contribution of the weak counterter as given by (.18). We have not shown the wavefunction renoralization of the tree graphs for the process as the su of these graphs vanish. Fig. 4. Feynan diagras contributing to the CP conserving aplitude for π γ at leading order in chiral perturbation theory. The notation is the sae as in fig. 3 and we have not shown the wavefunction renoralization of the tree graphs for the process as the su of these graphs vanish. Fig. 5. The differential decay spectru as a function of y the invariant ass of the lepton pair noralized to K π. The dot-dashed curve is the contribution fro F (1) () ±, the dotted curve is the contribution fro F ± with w L = and the dashed curve is the contribution fro G (). The total differential decay rate for w L = is given by the solid curve.

21 π π π Figure 1 1

22 π π K π, K π, π π π π π π K, π K, π π π Figure

23 K π K π, K K, π π K π K π K π, Figure 3(a) K π, 3

24 K π K π K π, K π, K, π K π K π K π, K π K π K π, K π, Figure 3(b) 4

25 K π K π Figure 3(c) K π Figure 3(d) 5

26 K π, π K π, π K, K, K π, π K π, π K, K, K π, π K π, π K, K, Figure 4(a) 6

27 π π π π π (, ) = ( K, π ), ( K, η ) ( π, K ), ( π, K ) Figure 4(b) 7

28 π π π π π (, ) = ( K, π ), ( K, η ) ( π, K ), (, K ) Figure 4(c) 8

29 π π π K, π K, π π π K, π K, Figure 4(d) 9

30 π π π Figure 4(e) 3

31 π Figure 4(f) 35 dγ Γ 1 dy 3 5 ( ) G() ( 1 ) F(1) ( ) F() Total Figure 5 y 31