Effective field theory approach to lepton number violating decays K ± → π ∓ l ± α l ± β : long-distance contribution
EEffective field theory approach to lepton number violating decays K ± → π ∓ l ± α l ± β : long-distance contribution Yi Liao a , c , Xiao-Dong Ma b , Hao-Lin Wang a a School of Physics, Nankai University, Tianjin 300071, China b Department of Physics, National Taiwan University, Taipei 10617, Taiwan c Center for High Energy Physics, Peking University, Beijing 100871, China
Abstract
This is a sequel to our recent work [1] in which we calculated the lepton number violating (LNV) K ± decaysdue to contact dimension-9 (dim-9) quark-lepton effective interactions that are induced at a high energy scale. Inthis work we investigate the long-distance contribution to the decays arising from the exchange of a neutrino. Thesedecays can probe LNV interactions involving the second generation of fermions that are not reachable in nuclearneutrinoless double- β decays. Our study is completely formulated in the framework of effective field theories (EFTs),from the standard model effective field theory (SMEFT) through the low energy effective field theory (LEFT) to chiralperturbation theory ( χ PT). We work to the first nontrivial orders in each effective field theory, collect along the waythe matching conditions and renormalization group effects, and express the decay branching ratios in terms of theWilson coefficients associated with the dim-5 and dim-7 operators in SMEFT. Our result is general in that it does notdepend on dynamical details of physics at a high scale that induce the effective interactions in SMEFT and in thatit does not appeal to any hadronic models. We find that the long-distance contribution overwhelmingly dominatesover the contact or short-distance one. Assuming the new physics scale to be around a TeV, the branching ratios arepredicted to be below the current experimental upper bounds by several orders of magnitude. [email protected] [email protected] [email protected] a r X i v : . [ h e p - ph ] J a n Introduction
The origin of neutrino mass and the nature of neutrinos remain a challenging issue in physics beyond the standardmodel. If neutrinos are Majorana fermions, the lepton number is violated by two units. In that case it is desirable toexplore lepton number violating (LNV) signals beyond the Majorana neutrino masses. At a high energy collider suchas the LHC, the LNV signals usually manifest themselves as like-sign multileptons that supposedly originate fromthe decays of new heavy particles engaged in neutrino mass generation [2]. The null search result then sets a lowerbound on the masses of new particles under some simplifying assumptions. Complementary to these direct searchesare high-precision experiments at low energy that seek the imprints of new physics in rare or forbidden processes. Themost extensively studied so far are the so-called nuclear neutrinoless double β (0 νβ β ) decays, X → X (cid:48) e ∓ e ∓ , in whicha parent nucleus X decays into a daughter nucleus X (cid:48) with the release of a pair of like-sign electrons or positrons [3, 4].The null result in current experiments can be then used to set a strong bound on the relevant LNV physics [5, 6].Other nuclear processes proposed to search for include, for instance, the muon to positron or antimuon conversion µ − X → e + ( µ + ) X (cid:48) in the upcoming Mu2e experiment [7]. On the other hand, there is a plethora of flavor physicsexperiments in recent years that search for LNV decays in flavored and charged mesons such as K ± , D ± , D ± s , B ± andthe τ lepton [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], and the bounds on some of the decays are expected tobe considerably improved in future experiments [22, 23].From the theoretical point of view the LNV decays of the flavored mesons and the τ lepton are sensitive to theeffective interactions of the fermions beyond the first generation that cannot be probed in the nuclear 0 νβ β decays dueto kinematical limitations, and can thus provide complementary information on underlying new physics. These decayscan be best investigated with the aid of various effective theories while avoiding theoretical uncertainties associatedwith nuclear physics. In a recent publication [1] we started the endeavor with the decays K ± → π ∓ l ± l ± arising fromeffective contact interactions among light quarks and charged leptons l = e , µ . In this work we make a comprehensiveanalysis on the decays by incorporating the long-distance contribution due to the exchange of neutrinos. Before divinginto technical details we describe briefly the strategy of our study in the framework of effective field theory (EFT).In the low energy region defined by the kaon and pion masses, the relevant dynamical degrees of freedom are theoctet of the pseudo-Nambu-Goldstone bosons ( π , K , η ), charged leptons, neutrinos, and the photon, if we assumethere are no new very light particles. The low-energy manifestations of lepton number violation from any high-scalenew physics are reflected in the effective interactions of those light particles, which can be systematically organizedin chiral perturbation theory ( χ PT) formulated in terms of external sources [24, 25, 26]. In particular, working to theleading order in χ PT and in LNV effects, lepton number violation could manifest itself through the neutrino mass(shown in figure 1(a)), effective interactions of a single meson with a charged lepton and neutrino pair (figure 1(b))or of the two mesons with a pair of likely-charged leptons (figure 1(c)). The short-distance (SD) contribution infigure 1(c) was thoroughly studied in [1], and this work will focus on the long-distance (LD) terms in figure 1(a, b)due to the exchange of neutrinos. It is easy to parameterize the above effective interactions, but our aim is to workthem out systematically in the EFT approach by matching sequentially to EFTs closer to new physics at a high scale.In this manner we are able to express the decay branching ratios in terms of the Wilson coefficients in the EFT definedat the electroweak scale or an even higher scale when necessary.The paper is organized as follows. We start in section 2 with the low energy effective field theory (LEFT) de-fined between the electroweak scale Λ EW and chiral symmetry breaking scale Λ χ [27, 28]. We collect the relevantdimension-3 (dim-3), -6 and -7 LNV effective operators and discuss their one-loop QCD running effects. (The dim-3,24, and -5 operators are the mass, kinetic, and electromagnetic (transition) moment terms.) We then match them toother EFTs along the ladder of scales. In section 3, the dim-6 and -7 operators in LEFT are matched at the scale Λ χ to χ PT, thus determining the vertices in figure 1 in terms of the Wilson coefficients in LEFT and low energy constants(LECs) of QCD strong dynamics. Then in section 4 we match LEFT upwards the scale to the standard model effectivefield theory (SMEFT), assuming that there are no new particles with a mass of order Λ EW or lower. The relevantleading LNV operators in SMEFT are the dim-5 and -7 ones known in the literature [29, 30, 31, 32] and reproducedin appendix A. We express the branching ratios in terms of the Wilson coefficients in SMEFT and the LECs of strongdynamics through the above two-step matching, whose complete expressions can be found in appendix B, and makesome numerical estimates. Our main findings are finally recapitulated in section 5.Figure 1: Generic Feynman diagrams for the decay K − → π + l − α l − β in χ PT, where the heavy blob stands for effectiveLNV interactions. Diagrams with the two charged leptons crossed are not shown in (a, b).
The low energy effective field theory is an EFT defined between the electroweak scale Λ EW ∼ GeV and chiralsymmetry breaking scale Λ χ = π F π ∼ u , d , s , c , b ),all charged leptons ( e , µ , τ ) and neutrinos ( ν e , ν µ , ν τ ), the photon and gluons, and enjoy QED and QCD gaugesymmetries. LEFT has been fruitfully applied particularly in flavor physics [27]. As a low energy effective theory,it is an organized and infinite series of effective interactions whose importance is relatively measured by canonicaldimensions of effective operators with similar symmetry properties. For our purpose here we focus on the effectiveoperators that violate the lepton number L by two units and potentially contribute to the decays under consideration atthe leading orders.We work without losing generality in the convention that quarks and charged leptons have been diagonalized whileneutrinos are in their flavor states. Different conventions amount to different ways to incorporate quark and leptonmixing matrices in generally unknown Wilson coefficients. We first recall that the SD contribution in figure 1(c) arisesat leading order from dim-9 operators involving two like-sign leptons and four quarks which have been thoroughlyanalyzed in Ref. [1]. In the following we investigate systematically the effective operators that could finally dominatethe LD contribution in figure 1(a,b). The dim-3 operator is unique, i.e., the Majorana neutrino mass term in the3ffective Lagrangian: − m αβ ν C α ν β , (1)where m αβ is the neutrino mass matrix in the flavor basis of the neutrinos ν α = ν e , ν µ , ν τ and the superscript C refersto charge conjugation. For the LNV interactions in figure 1(b), the relevant operators in LEFT involve one chargedlepton, one neutrino, and a pair of quarks. These operators first appear at dimension six and have been classified inRef. [28]. Following our notations in [1], we denote them as follows: O RL , Spr αβ = ( u pR d rL )( l L α ν C β ) , O LR , Spr αβ = ( u pL d rR )( l L α ν C β ) , (2) O LL , Vpr αβ = ( u pL γ µ d rL )( l R α γ µ ν C β ) , O RR , Vpr αβ = ( u pR γ µ d rR )( l R α γ µ ν C β ) , (3) O LR , Tpr αβ = ( u pL σ µν d rR )( l L α σ µν ν C β ) . (4)Here the Latin letters p , r indicate the flavors of the up- and down-type quarks u p , d r whose chiralities ( L , R ) areshown by their subscripts and by the first two superscripts of the operators O . Since ν C β is right-handed, the chiralityof the charged lepton field in a lepton bilinear is automatically determined by the type of the operators, S , V , T . Inaddition we also require the SM effective operators due to charged-current interactions between leptons and quarks: O (SM) pr αβ = V pr ( u pL γ µ d rL )( l L α γ µ ν β ) δ αβ , (5)where the V pr is the Cabibbo-Kobayashi-Maskawa (CKM) matrix.At the next order LNV operators carry a covariant derivative D µ to become dimension 7. Considering the restric-tions and reductions due to gauge symmetry, equations of motion, integration by parts, and Fierz identities, we obtainthe following LNV operators relevant to our purpose here [33]: O LL , VDpr αβ = ( u pL γ µ d rL )( l L α i ←→ D µ ν C β ) , O RR , VDpr αβ = ( u pR γ µ d rR )( l L α i ←→ D µ ν C β ) , (6) O LR , TDpr αβ = ( u pL σ µν d rR )( l R α γ [ µ ←→ D ν ] ν C β ) , O RL , TDpr αβ = ( u pR σ µν d rL )( l R α γ [ µ ←→ D ν ] ν C β ) , (7)where A ←→ D µ B = A ( D µ B ) − A ←− D µ B and γ [ µ D ν ] = γ µ D ν − γ ν D µ . The operators in equations (2)-(7), as well as thedim-3 Majorana mass term (1), make up the main body for the LD contribution. These operators will be matched inthe next section to those in χ PT where the lepton bilinears act as external sources.Since we will match the effective interactions in LEFT to those in SMEFT at the scale Λ EW and to those in χ PT atthe scale Λ χ , it is necessary to sum the large logarithms between the two scales using renormalization group equations.In our case the leading effect arises from the 1-loop QCD renormalization. While the vector-type operators are free ofrenormalization, the scalar- and tensor-type operators are indeed renormalized, whose Wilson coefficients satisfy therenormalization group equations: µ dd µ C S = − α s π C F C S , C S ∈ (cid:110) C RL , Spr αβ , C LR , Spr αβ (cid:111) , (8) µ dd µ C T = α s π C F C T , C T ∈ (cid:110) C LR , Tpr αβ , C LR , TDpr αβ , C RL , TDpr αβ (cid:111) , (9)where C F = ( N − ) / ( N ) = / N = µ and µ are C S ( µ ) = (cid:18) α s ( µ ) α s ( µ ) (cid:19) C F / b C S ( µ ) , C T ( µ ) = (cid:18) α s ( µ ) α s ( µ ) (cid:19) − C F / b C T ( µ ) , (10)4here b = − + n f / n f being the number of active quark flavors. Incorporating quark threshold effects, weobtain the numerical results between the scales Λ χ and Λ EW : C S ( Λ χ ) = . C S ( Λ EW ) , C T ( Λ χ ) = . C T ( Λ EW ) . (11)Thus the scalar-type interactions are enhanced while the tensor-type ones are suppressed when evolving down fromthe high scale Λ EW to the low scale Λ χ . χ PT While the charged leptons and neutrinos retain their identities at low energy, the quark and gluon degrees of freedomwill condense into hadrons due to strong dynamics. Since the process K − → π + l − α l − β in question involves only thelight quarks q = u , d , s , its transition matrix element due to effective interactions in LEFT can be beautifully evaluatedby matching to chiral perturbation theory, which is the low energy effective field theory of QCD. χ PT is based on thefact that the QCD Lagrangian has the approximate chiral symmetry SU ( ) L × SU ( ) R for the three light quarks whichis spontaneously broken by the quark condensate (cid:104) ¯ qq (cid:105) = − BF to the diagonal SU ( ) V . The symmetry breakdownbrings about eight pseudo-Nambu-Goldstone bosons (pNGBs), which are identified with the octet of the lowest-lyingpseudoscalars π ± , π , K ± , K , K , η . In the χ PT formalism they are represented by the element in the coset space SU ( ) L × SU ( ) R / SU ( ) V and take the matrix form, U ( x ) = exp (cid:32) i √ Π ( x ) F (cid:33) , Π = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − (cid:113) η , (12)where F is the decay constant in the chiral limit. Corresponding to chiral transformations of quarks q L → Lq L and q R → Rq R , U transforms as U → LUR † with L ∈ SU ( ) L and R ∈ SU ( ) R .The interactions of pNGBs with leptons due to dim-6 and dim-7 operators in equations (2)-(7) can be realizedthrough the external source method in which the global chiral symmetry is promoted to a local one [24, 25, 26]. Atthe quark-gluon level, the QCD Lagrangian with all possible external sources is parameterized as follows, L = L QCD + q L l µ γ µ q L + q R r µ γ µ q R + (cid:2) q L ( s − ip ) q R + q L t µν l σ µν q R + h . c . (cid:3) , (13)where L QCD is the QCD Lagrangian for massless u , d , s quarks. The external sources, l µ = l † µ , r µ = r † µ , s = s † , p = p † , t µν r = t µν † l , are 3 × l µ → Ll µ L † + iL ∂ µ L † , r µ → Rr µ R † + iR ∂ µ R † , χ → L χ R † , t µν l → Lt µν l R † , where χ = B ( s − ip ) . By comparing the external sources inequation (13) with the effective interactions in LEFT formed with the operators in equations (2-7) multiplied by theirWilson coefficients, one singles out the terms in external sources specific to the K − → π + transition: ( l µ ) ui = − √ G F V ui ( l L α γ µ ν α )+ C LL , Vui αβ ( l R α γ µ ν C β ) + C LL , VDui αβ ( l L α i ←→ D µ ν C β ) + · · · , (14) ( r µ ) ui = C RR , Vui αβ ( l R α γ µ ν C β ) + C RR , VDui αβ ( l L α i ←→ D µ ν C β ) + · · · , (15) ( χ † ) ui = C RL , Sui αβ ( l L α ν C β ) + · · · , (16) ( χ ) ui = C LR , Sui αβ ( l L α ν C β ) + · · · , (17)5 t µν l ) ui = C LR , Tui αβ ( l L α σ µν ν C β ) + C LR , TDui αβ ( l R α γ [ µ ←→ D ν ] ν C β ) + · · · , (18) ( t µν r ) ui = C RL , TDui αβ ( l R α γ [ µ ←→ D ν ] ν C β ) + · · · , (19)where i can be either d or s quark, and the ellipsis denotes terms not relevant to the transition. In χ PT the vector andscalar sources already appear at order O ( p ) [24, 25] L ( ) χ PT = F (cid:0) D µ U ( D µ U ) † (cid:1) + F (cid:0) χ U † + U χ † (cid:1) , (20)where D µ U = ∂ µ U − il µ U + iUr µ , (21)while the tensor sources first appear at O ( p ) [26] L ( ) χ PT ⊃ i Λ Tr (cid:0) t µν l ( D µ U ) † U ( D ν U ) † + t µν r D µ UU † D ν U (cid:1) , (22)where Λ is a low energy constant (LEC). Since the tensor structure in equation (22) involves at least two pNGBs, onecharged and one neutral, it cannot contribute at tree level to the process under consideration and will be ignored below.The expansion of equation (20) yields the following terms relevant to the LD contribution to the decay K − → π + l − α l − β : L ( ) χ PT ⊃ F (cid:104) G F (cid:0) V ud ∂ µ π − + V us ∂ µ K − (cid:1) (cid:0) l L α γ µ ν α (cid:1) + iB (cid:16) c αβπ π − + c αβ K K − (cid:17) (cid:16) l L α ν C β (cid:17) − (cid:16) c αβπ ∂ µ π − + c αβ K ∂ µ K − (cid:17) (cid:16) l R α γ µ ν C β (cid:17) − (cid:16) c αβπ ∂ µ π − + c αβ K ∂ µ K − (cid:17) (cid:16) l L α i ←→ D µ ν C β (cid:17) (cid:105) , (23)where the parameters defined at the scale Λ χ are c αβ P i = √ (cid:16) C RL , Sui αβ − C LR , Sui αβ (cid:17) , c αβ P i = √ (cid:16) C LL , Vui αβ − C RR , Vui αβ (cid:17) , c αβ P i = √ (cid:16) C LL , VDui αβ − C RR , VDui αβ (cid:17) , (24)with P i = π , K for i = d , s (and sometimes i = , O ( p / B ) relative to their scalar-type counterparts while dim-7 vector-type operators are furthersuppressed by O ( p / Λ EW ) . To put it in short, among the dim-6 and -7 LNV operators in LEFT, the scalar-type dim-6operators generically dominate the LD contribution. Now we make connections between the effective interactions in SMEFT and LEFT, so that we can parameterize thedecay branching ratios as a function of the SMEFT Wilson coefficients. The dim-5 and -7 LNV operators in SMEFTare reproduced in appendix A where we slightly improve the basis of dim-7 operators over Ref. [31]. At the scale Λ EW ( W , Z , h , t ) toinduce effective interactions in LEFT. The matching results at Λ EW for the Wilson coefficients of the relevant dim-3,-6, and -7 operators in LEFT are, in terms of those of the dim-5 and -7 operators in SMEFT, m αβ = − v C αβ ∗ LH − v C αβ ∗ LH , C LR , Tpr αβ = v √ C rp αβ ∗ ¯ dQLLH , (25) C RL , Spr αβ = v √ V wr C wp αβ ∗ ¯ QuLLH , C LR , Spr αβ = v √ C rp αβ ∗ ¯ dQLLH , (26) C LL , Vpr αβ = v √ V pr C βα ∗ LeHD , C RR , Vpr αβ = v √ C rp βα ∗ ¯ duLeH , (27) C LL , VDpr αβ = V pr (cid:16) C αβ ∗ LHW + C αβ ∗ LDH (cid:17) , C RR , VDpr αβ = C rp αβ ∗ ¯ duLDL , (28)where v ≈
246 GeV is the vacuum expectation value of the Higgs field and we have neglected contributions suppressedby small Yukawa couplings. Incorporating the 1-loop QCD running effect in equation (11), the c i parameters inequation (23) defined at Λ χ are expressed in terms of the SMEFT Wilson coefficients defined at Λ EW : c αβ P i = v ( . ) Y αβ P i , c αβ P i = v Y αβ P i , c αβ P i = √ Y αβ P i , (29)where Y αβ P i = V wi C w αβ ∗ ¯ QuLLH − C i αβ ∗ ¯ dQLLH , Y αβ P i = V ui C βα ∗ LeHD − C i βα ∗ ¯ duLeH , Y αβ P i = V ui (cid:16) C αβ ∗ LHW + C αβ ∗ LDH (cid:17) − C i αβ ∗ ¯ duLDL . (30)We are now in a position to employ equation (23) to calculate the LD contribution to the K − decay shown infigure 1(a,b). To make our answer complete, we include the SD contribution in figure 1(c) which takes the form [1] , L SD F G F = c αβ K − π − l L α l CL β + c αβ ∂ µ K − ∂ µ π − l L α l CL β , (31)where c αβ = − √ (cid:16) . g a × + . g b × (cid:17) X αβ , c αβ = − √ ( . g × ) V ud V us X αβ . (32)The X parameters are sums of the Wilson coefficients of the dim-7 operators in SMEFT defined at Λ EW , X αβ = (cid:16) V us C αβ ∗ ¯ duLDL + V ud C αβ ∗ ¯ duLDL (cid:17) , X αβ = C αβ ∗ LHW + C βα ∗ LHW + C αβ ∗ LDH + C αβ ∗ LDH , (33)while the QCD LECs determined in [34] are, in our notation [1], g × = . ± . g a × = ( . ± ) GeV , and g b × = ( . ± . ) GeV . The complete amplitude for the decay K − ( k ) → π + ( p ) l − α ( p ) l − β ( p ) is, M F G F = T SD u α P R u C β + T µν u α γ µ γ ν P R u C β + T µνρ u α γ µ γ ν γ ρ P R u C β + T µνρ u α γ µ γ ν γ ρ P L u C β , (34)where T SD stands for the SD term and the others are the LD ones: T SD = c αβ + c αβ k · p , (35) The convention of c i differs from that in Ref. [1] by a factor 2 F G F . µν = G F V ud V us m αβ (cid:0) k µ p ν t − + p µ k ν u − (cid:1) + t − (cid:104) V ud (cid:16) Bc αβ K − c αβ K ( t − p ) (cid:17) ( k − p ) µ p ν + V us (cid:16) Bc βαπ − c βαπ ( t − p ) (cid:17) k µ ( k − p ) ν (cid:105) + u − (cid:104) V ud (cid:16) Bc βα K − c βα K ( u − p ) (cid:17) p µ ( k − p ) ν + V us (cid:16) Bc αβπ − c αβπ ( u − p ) (cid:17) ( k − p ) µ k ν (cid:105) , (36) T µνρ = V ud c αβ K k µ ( k − p ) ν p ρ t − − V us c αβπ p µ ( k − p ) ν k ρ u − , (37) T µνρ = V us c βαπ k µ ( k − p ) ν p ρ t − − V ud c βα K p µ ( k − p ) ν k ρ u − , (38)with s = ( p + p ) , t = ( k − p ) , and u = ( k − p ) . The amplitude has the correct antisymmetry under interchangeof the two leptons upon using the relations for bilinear spinor wavefunctions u α P R u C β = − u β P R u C α , u α γ µ γ ν P R u C β = − u β γ ν γ µ P R u C α , u α γ µ γ ν γ ρ P R u C β = u β γ ρ γ ν γ µ P L u C α , (39)and obvious relations for T SD and T i tensors. The decay width is calculated as Γ = + δ αβ m K π m K (cid:90) ds (cid:90) dt | M | , (40)where the first factor removes double counting in phase space integration for identical particles, and the integrationdomains are s ∈ (cid:2) ( m α + m β ) , ( m K − m π ) (cid:3) , (41) t ∈ (cid:20) ( E ∗ + E ∗ ) − (cid:18)(cid:113) E ∗ − m β + (cid:113) E ∗ − m π (cid:19) , ( E ∗ + E ∗ ) − (cid:18)(cid:113) E ∗ − m β − (cid:113) E ∗ − m π (cid:19) (cid:21) , (42)with m K , π , α , β being the masses of the K − , π + , l α , and l β respectively and E ∗ = √ s ( s − m α + m β ) , E ∗ = √ s ( m K − s − m π ) . (43)Now we make some numerical analysis. The values of the SM parameters are taken from the Particle DataGroup [35]: Γ exp K = . × − MeV , G F = . × − GeV − , V ud = . , V us = . , m K = .
677 MeV , m π = .
570 MeV , m e = .
511 MeV , m µ = .
658 MeV , (44)together with the χ PT parameters F =
87 MeV [36] and B = . K − → π + l − α l − β are B ( e − e − ) GeV = . × − GeV | m ee | eV + | Y eeK | + . | Y ee π | − → π + e − e − K − → π + µ − µ − K − → π + e − µ − names bounds names bounds names bounds names bounds | Y eeK | − . (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y ee π (cid:12)(cid:12) − . (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) − . | X ee | − . (cid:12)(cid:12) X µµ (cid:12)(cid:12) − . (cid:12)(cid:12) X e µ (cid:12)(cid:12) − . | Y eeK | − . (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y ee π (cid:12)(cid:12) − . (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) − (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) − . | X ee | − . (cid:12)(cid:12) X µµ (cid:12)(cid:12) − . (cid:12)(cid:12) X e µ (cid:12)(cid:12) − . (cid:12)(cid:12) Y eeK (cid:12)(cid:12) − . (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y ee π (cid:12)(cid:12) − (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) − . | X i | − / or | Y i | − / ) of combinations ofWilson coefficients for dim-7 operators in SMEFT. Note that X αβ i = X βα i . + − × (cid:16) | X ee | + | Y eeK | + . | Y ee π | (cid:17) + − × (cid:16) | Y eeK | + | X ee | + . | Y ee π | (cid:17) + int. , (45) B ( µ − µ − ) GeV = . × − GeV | m µµ | eV + (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + . (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) + − × (cid:16) (cid:12)(cid:12) X µµ (cid:12)(cid:12) + (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) (cid:17) + − × (cid:16) (cid:12)(cid:12) X µµ (cid:12)(cid:12) + (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + . (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) (cid:17) + int. , (46) B ( e − µ − ) GeV = . × − GeV | m e µ | eV + (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) + (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) + (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) + . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) + − × (cid:16) (cid:12)(cid:12) X e µ (cid:12)(cid:12) + (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) + (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) + . (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) + . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) (cid:17) + − × (cid:16) (cid:12)(cid:12) X e µ (cid:12)(cid:12) + (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) + (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) + . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) + . (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) (cid:17) + int. , (47)where int. stands for interference terms between any pair of Wilson coefficients, whose complete forms are displayedin appendix B. We can see a few features from the above results. First of all, since the neutrino mass scale is at most O ( eV ) [39, 40], the contribution from the neutrino mass matrix is negligible for any measurable branching ratios in acollider-type experiment. Second, if we assume the Wilson coefficients associated with the dim-7 operators in SMEFTare similar in size, all of X and Y parameters will be of a similar order of magnitude. Their relative importance ismeasured by their prefactors, which have the rough ratios: Y : X : Y : X : Y ∼ : 10 − : 10 − ∼− : 10 − : 10 − ∼− . (48)Generically speaking, a long-distance contribution ( Y P i , j term) due to a neutrino exchange in figure 1(b) dominatesover its similar short-distance one ( X j term) in figure 1(c), which in turn is similar to a Y P i j + term.The current experimental upper bounds on the above decays are B exp ( e − e − ) < . × − [ ] , B exp ( µ − µ − ) < . × − [ ] , B exp ( e − µ − ) < × − [ ] . (49)To get some numerical feel on what these bounds would imply and considering the limited number of experimentalbounds compared to that of Wilson coefficients, we assume only one of the X i or Y i is nonzero. The above upper9ounds on branching ratios then translate into the lower bounds on their inverse cubic roots as displayed in table 1.The bounds are rather weak, especially when compared with those from nuclear 0 νβ β decays [38, 32]. This relativeweakness originates from much smaller data samples accumulated in kaon experiments than the number of nucleiavailable in a ton-level experiment of 0 νβ β decays as we estimated roughly in Ref. [1]. Thus the weak boundsshould not be interpreted as if the SMEFT approach would be valid for a new particle with a mass as low as tens oreven a few GeV; on the contrary, if there are such particles, they must be incorporated explicitly into the expandedversion of SMEFT and even LEFT. Nevertheless, we stress that however weak the bounds are, they are the first onesworked out in a systematic effective field theory approach that involve the second generation of fermions and are thuscomplementary to those obtained from nuclear 0 νβ β decays. Conversely, if we assume the dim-7 Wilson coefficientsare all of order Λ − where Λ is the new physics scale, the branching ratios are dominated by the terms with the largestcoefficients, i.e., the long-distance terms of Y αβ P i . In figure 2 we plot our theoretical predictions as a function of Λ ,together with the current experimental bounds; also shown are the contributions from the neutrino mass matrix aloneassuming m ν ≈ . Λ > νβ β decays imply, we have for SMEFT: B ( e − e − ) < . × − , B ( µ − µ − ) < . × − , B ( e − µ − ) < . × − . (50)These branching ratios are several orders of magnitude smaller than the current experimental upper bounds. We have accomplished a comprehensive analysis on the lepton number violating decays K ± → π ∓ l ± α l ± β in the effectivefield theory approach. We focused in this work on the long-distance contribution due to an exchange of neutrinos, andincorporated the short-distance contribution obtained in our previous work [1]. It turns out that the long-distancecontribution overwhelmingly dominates over the short-distance one by about three orders of magnitude in the decaybranching ratios. Assuming there are no new particles with a mass of the order of or below the electroweak scale,we related the decay branching ratios to the leading LNV effective interactions in SMEFT due to dim-5 and dim-7operators. Our results are general in that subject to the above loose assumption they are independent of dynamicaldetails at an even higher new physics scale; instead, different dynamics are hidden in the Wilson coefficients inSMEFT. Our results do not employ any hadronic models, but are completely based on the well-established symmetriesand effective field theories from SMEFT through LEFT to χ PT. While the hadronic LECs entering in the short-distance contribution were fixed previously by experimental measurements and lattice calculations, the long-distancecontribution involves no parameters other than the pion decay constant and quark condensate making our results veryrobust. Unfortunately, the current experimental upper bounds on the decay branching ratios are too weak to set a usefulbound on the scale of new physics that is responsible for lepton number violation.
Acknowledgement
We would like to thank Chang-Yuan Yao for discussions on numerical calculations. This work was supported in part bythe Grants No. NSFC-11975130, No. NSFC-11575089, by The National Key Research and Development Program ofChina under Grant No. 2017YFA0402200, by the CAS Center for Excellence in Particle Physics (CCEPP). Xiao-DongMa is supported by Grant No. MOST 106-2112-M-002-003-MY3.10 - →π + e - e - K - →π + μ - μ - K - →π + e - μ - Λ EW - - - - - - Λ [ TeV ] B r an c h i ng r a t i o Excluded by the current experimental search m ν ≈ m ν ≈ Figure 2: Branching ratios for K − → π + l − α l − β are shown as a function of the new physics scale Λ in the SMEFTframework. Also shown are the current experimental bounds (upper horizontal line) and the neutrino mass contributionalone (lower horizontal lines). Appendix A Baryon number conserving dim-7 operators in SMEFT ψ H + h.c. ψ H D + h.c. O LH ε i j ε mn ( L C , i L m ) H j H n ( H † H ) O LeHD ε i j ε mn ( L C , i γ µ e ) H j ( H m iD µ H n ) ψ H D + h.c. ψ H X + h.c. O LDH ( ∗ ) ε i j ε mn ( L C , i ←→ D µ L j )( H m D µ H n ) O LHB g ε i j ε mn ( L C , i σ µν L m ) H j H n B µν O LDH ( ∗ ) ε im ε jn ( L C , i L j )( D µ H m D µ H n ) O LHW g ε i j ( ετ I ) mn ( L C , i σ µν L m ) H j H n W I µν ψ D + h.c. ψ H + h.c. O duLDL ( ∗ ) ε i j ( d γ µ u )( L C , i i ←→ D µ L j ) O eLLLH ε i j ε mn ( eL i )( L C , j L m ) H n O dQLLH ( ∗ ) ε i j ε mn ( dQ i )( L C , j L m ) H n O dQLLH ( ∗ ) ε i j ε mn ( d σ µν Q i )( L C , j σ µν L m ) H n O duLeH ( ∗ ) ε i j ( d γ µ u )( L C , i γ µ e ) H j O QuLLH ε i j ( Qu )( L C L i ) H j Table 2: Basis of dim-7 lepton number violating but baryon number conserving operators in SMEFT. L , Q are the SMleft-handed lepton and quark doublet fields, u , d , e are the right-handed up-type quark, down-type quark and chargedlepton singlet fields, and H denotes the Higgs doublet, respectively. D µ H n is understood as ( D µ H ) n .11he dim-5 operator in SMEFT is well-known [29]: O = ε i j ε mn ( L C , i L m ) H j H n . (A.1)The dim-7 operators were first systematically studied in Ref. [30], and corrected by Ref. [31]. In this appendix weimprove further over the basis of operators in Ref. [31] so that flavor symmetries are apparently realized as advocatedin Ref. [32]. This only concerns the subset of operators that violate lepton number but conserve baryon number. Intable 2 the newly chosen basis operators are indicated by ( ∗ ) , which replace the following old basis operators [31]: O prLHD = ε i j ε mn ( L C , ip D µ L jr )( H m D µ H n ) , (A.2) O prLHD = ε im ε jn ( L C , ip D µ L jr )( H m D µ H n ) , (A.3) O prst ¯ duLLD = ε i j ( d p γ µ u r )( L C , is iD µ L jt ) , (A.4) O prst ¯ dLQLH = ε i j ε mn ( d p L ir )( Q C , js L mt ) H n , (A.5) O prst ¯ dLueH = ε i j ( d p L ir )( u Cs e t ) H j , (A.6) O prst ¯ dLQLH = ε im ε jn ( d p L ir )( Q C , js L mt ) H n . (A.7)The relations between the new (left) and old (right) operators are as follows, O prLDH = O prLHD + O rpLHD , (A.8) O prLDH = − (cid:0) O prLHD + O rpLHD (cid:1) + (cid:0) O prLHD + O rpLHD (cid:1) + EoM , (A.9) O prst ¯ duLDL = O prst ¯ duLLD + O prts ¯ duLLD , (A.10) O prst ¯ dQLLH = O psrt ¯ dLQLH + O ptrs ¯ dLQLH − O ptrs ¯ dLQLH , (A.11) O prst ¯ duLeH = O psrt ¯ dLueH , (A.12) O prst ¯ dQLLH = − (cid:16) O psrt ¯ dLQLH − O ptrs ¯ dLQLH + O ptrs ¯ dLQLH (cid:17) , (A.13)where EoM refers to equations of motion terms. Appendix B Complete results for branching ratios in SMEFT
In this Appendix we show the complete results for the branching ratios in terms of the Wilson coefficients in SMEFT: B ( e − e − ) GeV (B.1) = . × v − | m ee | + . × v − ℜ ( m ee Y ee ∗ K ) + . × v − ℜ ( m ee Y ee ∗ π )+ × | Y eeK | − . × v − ℜ ( m ee X ee ∗ ) + . × ℜ ( Y eeK Y ee ∗ π ) + . | Y ee π | − . ℜ ( Y eeK X ee ∗ ) − × − ℜ ( Y ee π X ee ∗ ) + . × − v − ℜ ( m ee Y ee ∗ K ) − . × − v − ℜ ( m ee Y ee ∗ K ) − . × − v − ℜ ( m ee X ee ∗ ) − . × − v − ℜ ( m ee Y ee ∗ π )+ . × − | X ee | + . × − | Y eeK | − . × − v − ℜ ( m ee Y ee ∗ π ) − . × − ℜ ( Y eeK Y ee ∗ π ) + . × − ℜ ( Y eeK Y ee ∗ K ) − . × − ℜ ( Y eeK Y ee ∗ K ) − . × − ℜ ( Y eeK X ee ∗ ) + . × − ℜ ( Y ee π Y ee ∗ K ) − . × − ℜ ( Y eeK Y ee ∗ π )+ . × − | Y ee π | − . × − ℜ ( Y ee π Y ee ∗ K ) − . × − ℜ ( Y eeK Y ee ∗ π ) . × − ℜ ( Y ee π X ee ∗ ) − . × − ℜ ( Y ee π Y ee ∗ π ) − × − ℜ ( Y eeK X ee ∗ ) − . × − ℜ ( Y ee π Y ee ∗ π ) + . × − ℜ ( X ee Y ee ∗ K ) + . × − ℜ ( X ee X ee ∗ )+ × − ℜ ( Y ee π X ee ∗ ) + . × − ℜ ( X ee Y ee ∗ π ) − . × − ℜ ( Y eeK X ee ∗ ) − . × − ℜ ( Y eeK Y ee ∗ K ) + . × − ℜ ( Y eeK X ee ∗ ) − . × − ℜ ( Y eeK Y ee ∗ π )+ . × − | Y eeK | + . × − | X ee | + . × − ℜ ( X ee Y ee ∗ π )+ . × − ℜ ( Y ee π Y ee ∗ K ) + . × − ℜ ( Y eeK Y ee ∗ π ) + . × − ℜ ( X ee Y ee ∗ π )+ . × − ℜ ( Y ee π Y ee ∗ π ) + . × − | Y ee π | B ( µ − µ − ) GeV (B.2) = . × v − (cid:12)(cid:12) m µµ (cid:12)(cid:12) + . × v − ℜ (cid:0) m µµ Y µµ ∗ K (cid:1) + . × v − ℜ (cid:0) m µµ Y µµ ∗ π (cid:1) − × v − ℜ (cid:0) m µµ X µµ ∗ (cid:1) + . × v − ℜ (cid:0) m µµ Y µµ ∗ K (cid:1) + . × (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + . × ℜ (cid:0) Y µµ K Y µµ ∗ π (cid:1) − . v − ℜ (cid:0) m µµ Y µµ ∗ π (cid:1) + . (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) − ℜ (cid:0) Y µµ K X µµ ∗ (cid:1) + . × − ℜ (cid:0) Y µµ K Y µµ ∗ K (cid:1) + . × − ℜ (cid:0) Y µµπ Y µµ ∗ K (cid:1) − . × − ℜ (cid:0) Y µµπ X µµ ∗ (cid:1) − . × − ℜ (cid:0) Y µµ K Y µµ ∗ π (cid:1) − . × − ℜ (cid:0) Y µµπ Y µµ ∗ π (cid:1) − × − v − ℜ (cid:0) m µµ X µµ ∗ (cid:1) − . × − v − ℜ (cid:0) m µµ Y µµ ∗ K (cid:1) − . × − ℜ (cid:0) X µµ Y µµ ∗ K (cid:1) + . × − (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + . × − (cid:12)(cid:12) X µµ (cid:12)(cid:12) − . × − v − ℜ (cid:0) m µµ Y µµ ∗ π (cid:1) − . × − ℜ (cid:0) Y µµ ∗ π Y µµ K (cid:1) + . × − ℜ (cid:0) Y µµπ X µµ ∗ (cid:1) − . × − ℜ (cid:0) Y µµ K X µµ ∗ (cid:1) − . × − ℜ (cid:0) Y µµ K Y µµ ∗ K (cid:1) + × − (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) − . × − ℜ (cid:0) Y µµπ X µµ ∗ (cid:1) − . × − ℜ (cid:0) Y µµπ Y µµ ∗ K (cid:1) − × − ℜ (cid:0) Y µµ K Y µµ ∗ π (cid:1) − . × − ℜ (cid:0) Y µµπ Y µµ ∗ π (cid:1) + . × − ℜ (cid:0) X µµ X µµ ∗ (cid:1) − . × − ℜ (cid:0) Y µµ K X µµ ∗ (cid:1) + . × − ℜ (cid:0) X µµ Y µµ ∗ K (cid:1) − . × − ℜ (cid:0) Y µµ K Y µµ ∗ K (cid:1) − . × − ℜ (cid:0) Y µµ K Y µµ ∗ π (cid:1) + × − ℜ (cid:0) X µµ Y µµ ∗ π (cid:1) + . × − ℜ (cid:0) Y µµπ X µµ ∗ (cid:1) + . × − ℜ (cid:0) Y µµπ Y µµ ∗ K (cid:1) + . × − ℜ (cid:0) Y µµπ Y µµ ∗ π (cid:1) + . × − ℜ (cid:0) X µµ Y µµ ∗ K (cid:1) + . × − (cid:12)(cid:12) X µµ (cid:12)(cid:12) + . × − (cid:12)(cid:12) Y µµ K (cid:12)(cid:12) + . × − ℜ (cid:0) X µµ Y µµ ∗ π (cid:1) + . × − ℜ (cid:0) Y µµ K Y µµ ∗ π (cid:1) + . × − (cid:12)(cid:12) Y µµπ (cid:12)(cid:12) , B ( e − µ − ) GeV (B.3) = . × v − (cid:12)(cid:12) m e µ (cid:12)(cid:12) + . × v − ℜ (cid:0) m e µ Y µ e ∗ K (cid:1) + . × v − ℜ (cid:0) m e µ Y e µ ∗ K (cid:1) + . × v − ℜ (cid:0) m e µ Y e µ ∗ π (cid:1) + . × v − ℜ (cid:0) m e µ Y µ e ∗ π (cid:1) − . × v − ℜ (cid:0) m e µ X e µ ∗ (cid:1) + . × v − ℜ (cid:0) m e µ Y µ e ∗ K (cid:1) + × ℜ (cid:0) Y µ eK Y e µ ∗ K (cid:1) + . × (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) + . × (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) + . × ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) + . × ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) + . ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) + . ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) − . v − ℜ (cid:0) m e µ Y µ e ∗ π (cid:1) + . ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) − . ℜ (cid:0) Y µ eK X e µ ∗ (cid:1) + (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) − . ℜ (cid:0) Y e µ K X e µ ∗ (cid:1) + . (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) + . ℜ (cid:0) Y µ eK Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) Y e µ K Y µ e ∗ K (cid:1) − . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) X e µ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y e µπ Y µ e ∗ K (cid:1) + × − ℜ (cid:0) Y µ e π Y µ e ∗ K (cid:1) + . × − v − ℜ (cid:0) m e µ Y e µ ∗ K (cid:1) − . × − v − ℜ (cid:0) m e µ X e µ ∗ (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) − . × − v − ℜ (cid:0) m e µ Y e µ ∗ K (cid:1) − × − v − ℜ (cid:0) m e µ Y µ e ∗ K (cid:1) − . × − ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) X e µ Y µ e ∗ K (cid:1) . × − (cid:12)(cid:12) X e µ (cid:12)(cid:12) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) − . × − v − ℜ (cid:0) m e µ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ e π Y µ e ∗ π (cid:1) + . × − (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) − . × − v − ℜ (cid:0) m e µ Y µ e ∗ π (cid:1) + . × − (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) − . × − v − ℜ (cid:0) m e µ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) X e µ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y e µ K Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) X e µ Y µ e ∗ K (cid:1) + × − ℜ (cid:0) Y µ eK Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) X e µ Y e µ ∗ K (cid:1) − × − ℜ (cid:0) Y µ eK Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) Y µ e π Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ K (cid:1) + × − ℜ (cid:0) Y e µπ Y e µ ∗ K (cid:1) + . × − (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ K (cid:1) − . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) + . × − (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ e π X e µ ∗ (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) Y e µπ Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ e π Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) Y e µπ Y µ e ∗ K (cid:1) − . × − ℜ (cid:0) Y µ e π Y µ e ∗ K (cid:1) − × − ℜ (cid:0) Y µ e π Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y e µπ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) X e µ Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) X e µ X e µ ∗ (cid:1) − . × − ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) − × − ℜ (cid:0) Y e µπ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ e π Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) Y µ e π Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y µ eK X e µ ∗ (cid:1) + . × − ℜ (cid:0) X e µ Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) Y e µ K Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ K (cid:1) − × − ℜ (cid:0) Y µ eK Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) X e µ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) − . × − ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y µ e π X e µ ∗ (cid:1) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y µ e π Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y µ e π Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) Y µ e π Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) Y µ e π Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) X e µ Y e µ ∗ K (cid:1) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ K (cid:1) + . × − (cid:12)(cid:12) X e µ (cid:12)(cid:12) + . × − ℜ (cid:0) X e µ Y µ e ∗ K (cid:1) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) − . × − ℜ (cid:0) Y e µ K Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) Y e µπ Y e µ ∗ K (cid:1) + . × − ℜ (cid:0) Y e µ K Y µ e ∗ K (cid:1) + . × − (cid:12)(cid:12) Y e µ K (cid:12)(cid:12) − . × − ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) X e µ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) X e µ Y e µ ∗ π (cid:1) + . × − (cid:12)(cid:12) Y µ eK (cid:12)(cid:12) + . × − ℜ (cid:0) Y e µ K Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y µ eK Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) Y e µ K Y e µ ∗ π (cid:1) + × − ℜ (cid:0) Y µ eK Y µ e ∗ π (cid:1) + . × − ℜ (cid:0) Y e µπ Y µ e ∗ K (cid:1) + . × − ℜ (cid:0) Y e µπ Y e µ ∗ π (cid:1) + . × − ℜ (cid:0) Y e µπ Y µ e ∗ π (cid:1) + . × − (cid:12)(cid:12) Y µ e π (cid:12)(cid:12) + . × − (cid:12)(cid:12) Y e µπ (cid:12)(cid:12) . Some interference terms have a smaller coefficient than their separate terms because of phase space integration.
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