Effective field theory approach to lepton number violating τ decays
EEffective field theory approach to lepton number violating τ decays 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
We continue our endeavor to investigate lepton number violating (LNV) processes at low energy in the frameworkof effective field theory (EFT). In this work we study the LNV tau decays τ + → (cid:96) − P + i P + j , where (cid:96) = e , µ and P + i , j are the lowest-lying charged pseudoscalars π + , K + . We analyze the dominant contributions in a series of EFTs fromhigh to low energy scales, namely, the standard model effective field theory (SMEFT), the low-energy effective fieldtheory (LEFT), and the chiral perturbation theory ( χ PT). The decay branching ratios are expressed in terms of theWilson coefficients of dimension-five and -seven operators in SMEFT and hadronic low energy constants. TheseWilson coefficients involve the first and second generations of quarks and all generations of leptons and thus cannot beexplored in low energy processes such as nuclear neutrinoless double decay or LNV kaon decays. Unfortunately, thecurrent experimental upper limits on the branching ratios are too weak to set useful constraints on those coefficients.Or, if we assume the new physics scale is larger than 1 TeV, the branching ratios are well below the current experimentalbounds. We also estimate hadronic uncertainties incurred in applying χ PT to τ decays by computing one-loop chirallogarithms and attempt to improve convergence of chiral perturbation by employing dispersion relations in the short-distance part of the decay amplitudes. [email protected] [email protected] [email protected] a r X i v : . [ h e p - ph ] F e b Introduction
While neutrino oscillation experiments provide definite evidence for the existence of neutrino mass, its origin and thenature of neutrinos remain mysterious. As neutral fermions, neutrinos may well be Majorana particles as it naturallyhappens in conventional seesaw mechanisms of neutrino mass generation thus resulting in lepton number violation. Inthe meantime one searches for new heavy particles presumably involved in Majorana mass generation at high energycolliders through like-sign dilepton production, it is important to explore lepton number violating (LNV) signals inprecision low energy processes. The nuclear neutrinoless double beta decay (0 νβ β ) has so far provided the largestdata sample and set the strongest constraint on lepton number violation in the first generation of leptons and quarks [1,2, 3]. In this circumstance we should keep conscious that new physics might first reveal itself in processes involvingheavier leptons and quarks as the usual wisdom indicates. Indeed, in the past years LNV decays of mesons such as K ± , D ± , D ± s , B ± and the τ lepton have been continuously searched for in many experiments including LHCb [4, 5,6, 7], BaBar [8, 9, 10], Belle [11, 12], CLEO [13] and others [14, 15, 16, 17], and significantly improved constraintson some of the decays are expected in upgraded or proposed experiments [18, 19]. From the theoretical point of viewit is advantageous that we avoid complicated nuclear physics in those decays although we have to cope with hadronicuncertainties in most cases.In the previous publications [20, 21], we investigated the LNV decays K ± → π ∓ (cid:96) ± α (cid:96) ± β (with (cid:96) ± α , β = e ± , µ ± )completely in the framework of effective field theory (EFT) including both short-distance (SD) and long-distance(LD) contributions. Note that K ± are the lightest hadrons whose decays could violate lepton number in the chargedlepton sector. In this work, we want to extend our study to the single charged lepton which can decay hadronicallywhile violating lepton number, i.e., the three-body τ lepton decays, τ ± → (cid:96) ∓ α P ± i P ± j , with P ± i , j = π ± , K ± . The bestupper limits on the branching ratios of those decays are from the Belle experiment [12] B ( τ − → e + π − π − ) < . × − , B ( τ − → µ + π − π − ) < . × − , (1) B ( τ − → e + K − K − ) < . × − , B ( τ − → µ + K − K − ) < . × − , (2) B ( τ − → e + K − π − ) < . × − , B ( τ − → µ + K − π − ) < . × − , (3)which are expected to be improved in the Belle II experiment [19]. While the bounds are weaker by about two ordersof magnitude than those on the LNV K ± decays, they provide the unique information on lepton number violationinvolving the τ lepton and are thus worthwhile to explore. We will continue to work in the EFT framework. The mostsalient feature of the EFT approach is its universality. To study physics below the electroweak scale, we only have toassume whether there are any new and relatively light particles while different high-scale physics is reflected in Wilsoncoefficients in effective field theory at low energy.This paper is organized as follows. Assuming there are no new particles lighter than the electroweak scale Λ EW ,we start in section 2 with the standard model effective field theory (SMEFT) whose dimension-5 (dim-5) and -7operators would give the dominant effective LNV interactions. At the scale Λ EW we do matching calculation betweenthe SMEFT and low energy effective field theory (LEFT) up to dim-9 operators in the latter that are relevant to thedecays under consideration. We then study in section 3 the chiral realization below the chiral symmetry breaking scale Λ χ of the effective interactions in the LEFT, and calculate the decay amplitudes. We estimate in section 4 hadronicuncertainties due to the relatively large mass of the τ lepton by computing one-loop chiral logarithms, and attempt toimprove convergence of chiral perturbation by dispersion relations. Our master formulas for the decay branching ratiosare presented in section 5 together with numerical estimates. We summarize our main results in the last section 6.2 H O LH = ε i j ε mn ( L C , i L m ) H j H n ( H † H ) ψ H O eLLLH = ε i j ε mn ( eL i )( L C , j L m ) H n ψ H D O LeHD = ε i j ε mn ( L C , i γ µ e ) H j ( H m iD µ H n ) O dQLLH = ε i j ε mn ( dQ i )( L C , j L m ) H n ψ H X O LHB = g ε i j ε mn ( L C , i σ µν L m ) H j H n B µν O dQLLH = ε i j ε mn ( d σ µν Q i )( L C , j σ µν L m ) 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 O duLeH = ε i j ( d γ µ u )( L C , i γ µ e ) H j ψ H D O LDH = ε i j ε mn ( L C , i ←→ D µ L j )( H m D µ H n ) O QuLLH = ε i j ( Qu )( L C L i ) H j O LDH = ε im ε jn ( L C , i L j )( D µ H m D µ H n ) ψ D O duLDL = ε i j ( d γ µ u )( L C , i i ←→ D µ L j ) Table 1: Basis of dim-7 lepton number violating but baryon number conserving operators in SMEFT. L , Q are theleft-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. D µ is defined for the gauge symmetries SU ( ) C × SU ( ) L × U ( ) Y , and D µ H n is understood as ( D µ H ) n . In light of the null result in searching for new particles of mass up to the TeV scale, it is plausible to assume thatnew physics appears at a scale Λ NP well above the electroweak scale Λ EW and that there are no new particles with amass of order Λ EW or below. We can then establish an effective field theory, the SMEFT, between the two scales thatis composed of the standard model (SM) fields and respects the SM gauge symmetries SU ( ) C × SU ( ) L × U ( ) Y .Its Lagrangian starts with the SM Lagrangian L SM , and is augmented by an infinite sum of effective interactionsinvolving higher and higher dimensional operators and suppressed by more and more powers of Λ NP : L SMEFT = L SM + L + L + L + · · · . (4)Here L = C αβ LH O αβ contains the unique dim-5 Weinberg operator [22], O αβ = ε i j ε mn ( L C , i α L m β ) H j H n , (5)which induces Majorana neutrino mass when the Higgs doublet field H develops a vacuum expectation value. Here L β refers to the left-handed doublet lepton field of flavor β , and i jmn are the SU ( ) L indices in the fundamentalrepresentation. L collects effective interactions of dim-6 operators [23, 24], and L is a sum of dim-7 operators [25,26]. For the decays under consideration here, LNV L and L would contribute dominantly. The dim-7 operatorswere first systematically studied in [25], and its basis was established in [26] by removing redundancy and furtherrefined in [21] by making flavor symmetries manifest. The dim-7 operators that violate lepton number by two unitsbut conserve baryon number are reproduced in table 1.The SM electroweak symmetries are spontaneously broken to U ( ) EM by the vacuum expectation value of theHiggs field, (cid:104) H (cid:105) = ( , ) T v / √
2, which defines the electroweak scale Λ EW . Upon integrating out the heavy particles inSMEFT, i.e., the Higgs, W ± , and Z bosons and the top quark, we arrive at the low energy effective field theory (LEFT)for remaining SM particles that is another infinite sum of effective interactions involving higher and higher dimensionaloperators and suppressed by more and more powers of Λ EW . We adopt for dim-6 operators in the LEFT the basis in [27]and for dim-7 operators the basis in [28]. The short-distance contribution to the τ decays under consideration comesfrom dim-9 LNV operators involving four quark and two lepton fields, whose basis was determined in Ref [20]. Allthese operators relevant to our discussion here are collected in table 2. They are classified according to the types ofcontributions finally entering the τ decays: Majorana neutrino mass insertion (MM), long-distance (LD), and short-distance (SD), see figure 1. Here long distance refers to the exchange of a light neutrino and short distance meanscontact interactions among the initial and final particles in the τ decays.3ypes and dim operators in LEFT matching LEFT (left) with SMEFT (right) at Λ EW MM: dim-3 L M = − m αβ ν C α ν β m αβ = − v C αβ ∗ LH − v C αβ ∗ LH LD: dim-6 O RL , Spr αβ = ( u pR d rL )( (cid:96) L α ν C β ) C RL , Spr αβ = v √ V wr C wp αβ ∗ ¯ QuLLH O LR , Spr αβ = ( u pL d rR )( (cid:96) L α ν C β ) C LR , Spr αβ = v √ C rp αβ ∗ ¯ dQLLH O LL , Vpr αβ = ( u pL γ µ d rL )( (cid:96) R α γ µ ν C β ) C LL , Vpr αβ = v √ V pr C βα ∗ LeHD O RR , Vpr αβ = ( u pR γ µ d rR )( (cid:96) R α γ µ ν C β ) C RR , Vpr αβ = v √ C rp βα ∗ ¯ duLeH O LR , Tpr αβ = ( u pL σ µν d rR )( (cid:96) L α σ µν ν C β ) C LR , Tpr αβ = v √ C rp αβ ∗ ¯ dQLLH LD: dim-7 O LL , VDpr αβ = ( u pL γ µ d rL )( (cid:96) L α i ←→ D µ ν C β ) C LL , VDpr αβ = − V pr (cid:16) C βα ∗ LHW + C αβ ∗ LDH (cid:17) O RR , VDpr αβ = ( u pR γ µ d rR )( (cid:96) L α i ←→ D µ ν C β ) C RR , VDpr αβ = C rp αβ ∗ ¯ duLDL SD: dim-9 O LLLL , S / Pprst , αβ = ( u pL γ µ d rL )[ u sL γ µ d tL ] j αβ ( ) C LLLL , S / Pprst , αβ = − √ G F V pr V st × (cid:16) C αβ ∗ LHW + C βα ∗ LHW + C αβ ∗ LDH + C αβ ∗ LDH (cid:17) O LRRL , S / Pprst , αβ = ( u pL d rR )[ u sR d tL ] j αβ ( ) C LRRL , S / Pprst , αβ = O LRRL , S / Pprst , αβ = ( u pL d rR ][ u sR d tL ) j αβ ( ) ˜ C LRRL S / Pprst , αβ = − √ G F V pt C rs αβ ∗ ¯ duLDL Table 2: Relevant LEFT operators (middle column) and their Wilson coefficients (right column) obtained from match-ing at scale Λ EW to those of SMEFT dim-5 and dim-7 effective interactions. The Wilson coefficients carry identicalindices as corresponding operators in both SMEFT and LEFT. Here j αβ = ( (cid:96) α (cid:96) C β ) , j αβ = ( (cid:96) α γ (cid:96) C β ) , D µ refers to gaugesymmetries SU ( ) C × U ( ) EM , and ( · · · ) and [ · · · ] indicate two color contractions.The two EFTs, the SMEFT above Λ EW and the LEFT below, are related by matching conditions at the scale Λ EW .We content ourselves here with tree level matching whose results are also shown in table 2. Our convention in theLEFT is that we work with mass eigenstate fields of quarks and charged leptons but with flavor eigenstate fields ofneutrinos since neutrino mass appears only in the form of a matrix in flavor space. Some operators, for instance,dim-7 tensor operators, that generically exist in the LEFT are not induced at this level from the SMEFT. Some otheroperators that are not induced at the matching scale Λ EW are however generated at lower scales from other operatorsby renormalization-group running effects, for instance, the dim-9 operator O LRRL , S / Pprst . Part of dim-7 SMEFT operators( O ¯ QuLLH , O ¯ dQLLH , O ¯ dQLLH , O LeHD , and O ¯ duLeH ) induce LNV dim-6 LEFT operators that involve a charged lepton, aneutrino, and a quark bilinear. These operators supposedly give the leading contributions in figure 1 (b). The other setof operators in SMEFT ( O LDH , O LHW , and O duLDL ) induces dim-7 LEFT operators carrying an additional covariantderivative D µ , which would contribute at the next-to-leading order to figure 1 (b). Finally, among many possible LNVdim-9 LEFT operators involving four quarks and two charged leptons [20], only a few can be induced from dim-7operators in SMEFT ( O LHW , O duLDL , O LDH , and O LDH ). This simplifies life significantly.As we aim to calculate the τ decays at even lower energies, we will match effective operators in LEFT to thosein χ PT at the scale Λ χ = π F π ≈ . C S ( Λ χ ) = . C S ( Λ EW ) , C S ∈ { C RL , Spr αβ , C LR , Spr αβ } , C LR , Tpr αβ ( Λ χ ) = . C LR , Tpr αβ ( Λ EW ) . (6)The other dim-6 and -7 operators involve a quark vector current and thus do not run due to the QCD Ward identity. As4entioned above, renormalization of dim-9 operators induces mixing of operators in addition to running: C LLLL , S / Puiu j ( Λ χ ) = . C LLLL , S / Puiu j ( Λ EW ) , (7)˜ C LRRL , S / Puiu j ( Λ χ ) = .
88 ˜ C LRRL , S / Puiu j ( Λ EW ) , (8) C LRRL , S / Puiu j ( Λ χ ) = .
62 ˜ C LRRL , S / Puiu j ( Λ EW ) . (9)The RG running from higher to lower scales here is a mild suppression with the exception of the dim-6 scalar operatorswhich are enhanced. χ PT At the chiral symmetry breaking scale Λ χ the approximate chiral symmetry G = SU ( ) L × SU ( ) R for the q = u , d , s quarks in the QCD Lagrangian is spontaneously broken to H = SU ( ) V by the quark condensate (cid:104) | ¯ qq | (cid:105) = − BF .This generates an octet of Nambu-Goldstone (NG) bosons living in the coset space G / H . When the small quark massesare taken into account, they become the so-called pseudo-NG bosons and can be identified with the lowest-lying octetof pseudoscalar mesons π ± , π , K ± , K , K , η . Their strong interactions at low energy are best described by chiralperturbation theory ( χ PT) [29, 30]. The framework of χ PT is flexible enough to describe additional interactions of themesons inherited from effective interactions of light quarks in the LEFT. This is exactly what we want to do next forthe nonperturbative matching at the scale Λ χ where the light quark degrees of freedom give way to the mesons. Thebasis of this is the analysis of linear versus nonlinear realizations of the chiral symmetry. For effective interactionsinvolving a single quark bilinear, i.e., the dim-6 and -7 operators in table 2, the products of other fields multiplying thebilinear are treated as external sources coupled to the light quarks in the QCD Lagrangian. For effective interactionswith more quark bilinears one can apply the technique of spurion analysis, which has been elaborated in Refs [31, 20]in the context of the dim-9 operators in table 2.In χ PT the meson fields are parameterized by [32, 33] Σ ( x ) = exp (cid:32) i √ Π ( x ) F (cid:33) , Π = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − (cid:113) η , (10)where F is the decay constant in the chiral limit. The leading-order O ( p ) Lagrangian incorporating scalar andpseudoscalar ( χ ) and vector ( l µ , r µ ) external sources is given by L ( ) χ PT = F (cid:0) D µ Σ ( D µ Σ ) † (cid:1) + F (cid:0) χ Σ † + Σ χ † (cid:1) , (11)where D µ Σ = ∂ µ Σ − il µ Σ + i Σ r µ . The external tensor sources ( t µν l , t µν r ) first appear at O ( p ) [34]: L ( ) χ PT ⊃ i Λ Tr (cid:0) t µν l ( D µ Σ ) † U ( D ν U ) † + t µν r D µ UU † D ν U (cid:1) . (12)Inspecting the dim-6 and dim-7 operators in table 2 which appear as additional terms in the QCD Lagrangian whenmultiplied by their Wilson coefficients, we can read off the external sources relevant to the decays under consideration: ( l µ ) ui = − √ G F V ui ( (cid:96) L α γ µ ν α ) + C LL , Vui αβ ( (cid:96) R α γ µ ν C β ) + C LL , VDui αβ ( (cid:96) L α i ←→ D µ ν C β ) + · · · , (13) ( r µ ) ui = C RR , Vui αβ ( (cid:96) R α γ µ ν C β ) + C RR , VDui αβ ( (cid:96) L α i ←→ D µ ν C β ) + · · · , (14)5Figure 1: Feynman diagrams for decay τ + → (cid:96) − P + i P + j in χ PT. The heavy blob denotes effective LNV interactions,and the arrow on the lepton (meson) line indicates lepton number (positive charge) flow. Crossing diagrams in (a, b)are not shown. ( χ † ) ui = BC RL , Sui αβ ( (cid:96) L α ν C β ) + · · · , (15) ( χ ) ui = BC LR , Sui αβ ( (cid:96) L α ν C β ) + · · · , (16) ( t µν l ) ui = C LR , Tui αβ ( (cid:96) L α σ µν ν C β ) + · · · , (17) ( t µν r ) ui = , (18)where i = d , s (or i = , Σ matrix indices), V is the CKM matrix, and the ellipses denote irrelevantterms. These source terms are the LNV interactions of the mesons with a charged lepton-neutrino pair, and generatethe diagrams in figure 1 (b) when the neutrino field is contracted by a usual interaction vertex which defines the mesondecay constant (see the first term in equation (19)). Since each tensor term includes at least two mesons, this impliesthat it only contributes to the four-body decays of the τ lepton. We reserve this more complicated situation for thefuture study as one necessarily has to include resonances explicitly. Finally, when both neutrinos in the mass term L M are contracted to two usual vertices mentioned above, we obtain the diagram in figure 1 (a).With all of the above details we can now write down the interactions entering the long-distance contribution to the τ decays: L ( ) χ PT ⊃ F G F (cid:0) V ud ∂ µ π − + V us ∂ µ K − (cid:1) (cid:0) (cid:96) L α γ µ ν α (cid:1) + F (cid:104) iB (cid:16) c αβπ π − + c αβ K K − (cid:17) (cid:16) (cid:96) L α ν C β (cid:17) − (cid:16) c αβπ ∂ µ π − + c αβ K ∂ µ K − (cid:17) (cid:16) (cid:96) R α γ µ ν C β (cid:17) − (cid:16) c αβπ ∂ µ π − + c αβ K ∂ µ K − (cid:17) (cid:16) (cid:96) L α i ←→ D µ ν C β (cid:17) (cid:105) , (19)where the first term is the usual one and the others are new LNV interactions. We have introduced the parameters 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) , (20)which are implicitly defined at the scale Λ χ , with P i = π , K for i = d , s . Employing the matching conditions in table 2at Λ EW and the one-loop QCD running effects in equation (6) from Λ EW to Λ χ , we connect the above parameters at Λ χ with 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 , (21)6ecays LEFT operators chiral irrep hadronic operators O LLLL , S / Puiu j , αβ L × R g × F ( Σ i ∂ µ Σ † ) i ( Σ i ∂ µ Σ † ) j j αβ ( ) O LRRL , S / Puiu j , αβ L × R ( a ) g a × F ( Σ † ) i ( Σ ) j j αβ ( ) τ + → (cid:96) − P + i P + j O LRRL , S / Pu jui , αβ L × R ( a ) g a × F ( Σ † ) j ( Σ ) i j αβ ( ) ˜ O LRRL , S / Puiu j , αβ L × R ( b ) g b × F ( Σ † ) i ( Σ ) j j αβ ( ) ˜ O LRRL , S / Pu jui , αβ L × R ( b ) g b × F ( Σ † ) j ( Σ ) i j αβ ( ) O LLLL , S / Pudud L × R g × F ( Σ i ∂ µ Σ † ) ( Σ i ∂ µ Σ † ) ( j / j ) τ + → (cid:96) − π + π + O LRRL , S / Pudud L × R ( a ) g a × F ( Σ † ) ( Σ ) ( j / j ) ˜ O LRRL , S / Pudud L × R ( b ) g b × F ( Σ † ) ( Σ ) ( j / j ) O LLLL , S / Pusus L × R g × F ( Σ i ∂ µ Σ † ) ( Σ i ∂ µ Σ † ) ( j / j ) τ + → (cid:96) − K + K + O LRRL , S / Pusus L × R ( a ) g a × F ( Σ † ) ( Σ ) ( j / j ) ˜ O LRRL , S / Pusus L × R ( b ) g b × F ( Σ † ) ( Σ ) ( j / j ) O LLLL , S / Pudus L × R g × F ( Σ i ∂ µ Σ † ) ( Σ i ∂ µ Σ † ) ( j / j ) O LRRL , S / Pudus L × R ( a ) g a × F ( Σ † ) ( Σ ) ( j / j ) τ + → (cid:96) − K + π + O LRRL , S / Pusud L × R ( a ) g a × F ( Σ † ) ( Σ ) ( j / j ) ˜ O LRRL , S / Pudus L × R ( b ) g b × F ( Σ † ) ( Σ ) ( j / j ) ˜ O LRRL , S / Pusud L × R ( b ) g b × F ( Σ † ) ( Σ ) ( j / j ) Table 3: Chiral realizations (fourth column) of dim-9 LEFT operators (second column) contributing to decays τ + → (cid:96) − P + i P + j with (cid:96) = e , µ , P i = π , K for i = ,
3, and j ( ) = j (cid:96) τ ( ) or j τ (cid:96) ( ) .where Y αβ P i = V wi C w αβ ∗ ¯ QuLLH ( Λ EW ) − C i αβ ∗ ¯ dQLLH ( Λ EW ) , Y αβ P i = V ui C βα ∗ LeHD ( Λ EW ) − C i βα ∗ ¯ duLeH ( Λ EW ) , Y αβ P i = V ui (cid:104) C αβ ∗ LHW ( Λ EW ) + C αβ ∗ LDH ( Λ EW ) (cid:105) − C i αβ ∗ ¯ duLDL ( Λ EW ) . (22)Now we turn to evaluate the SD contribution in figure 1 (c) that arises from matching to the dim-9 LEFT operatorsin table 2. We refer to our previous work [20] for the detail of matching based on spurion analysis, and show theresults in table 3. The matching leaves behind a low energy constant (LEC) multiplying a mesonic operator that canonly be determined by nonperturbative methods. Note that different components in the same irreducible representationof the chiral group share the same LEC in the chiral limit; for instance, all of O LLLL , S / Pudud , O LLLL , S / Pusus and O LLLL , S / Pudus sharethe same LEC g × . Operators belonging to the same irreducible representation but arising from different colorcontractions generally have different LECs, as is the case with the LECs g a × and g b × . Fortunately, these threeparameters are already determined in the literature; here we use the values in Ref [35], which in our notation are, g × = . ± . , g a × = . ± , g b × = . ± .
65 GeV . (23)Expanding the hadronic operators in table 3 to their first terms and attaching their corresponding LEFT Wilson coeffi-cients defined at the scale Λ χ yields the SD interactions for (cid:96) ± α (cid:96) ± β P ∓ i P ∓ j : L SD (cid:96) ± α (cid:96) ± β P ∓ i P ∓ j = F g × ∂ µ P − i ∂ µ P − j (cid:104) C LLLL , Suiu j ( Λ χ ) (cid:96) α (cid:96) C β + C LLLL , Puiu j ( Λ χ ) (cid:96) α γ (cid:96) C β (cid:105) F P − i P − j (cid:104)(cid:16) C LRRL , Suiu j ( Λ χ ) g a × + ˜ C LRRL , Suiu j ( Λ χ ) g b × (cid:17) (cid:96) α (cid:96) C β + (cid:16) C LRRL , Puiu j ( Λ χ ) g a × + ˜ C LRRL , Puiu j ( Λ χ ) g b × (cid:17) (cid:96) α γ (cid:96) C β + ( − δ i j )( i ↔ j ) (cid:105) + h.c. . (24)Utilizing the QCD running effects according to equations (7)-(9) and the LEFT-SMEFT matching results in table 2,we obtain L SD τ + → (cid:96) − P + i P + j = F G F + δ i j (cid:104) c (cid:96) τ , i j P − i P − j + c (cid:96) τ , i j ∂ µ P − i ∂ µ P − j (cid:105) (cid:96) L τ CL , (25)where the parameters c , are defined as c (cid:96) τ , i j = − √ (cid:16) . g a × + . g b × (cid:17) X (cid:96) τ , P i P j , c (cid:96) τ , i j = − √ ( . g × ) V ui V u j X (cid:96) τ . (26)Here X (cid:96) τ , P i P j = V ui C ju (cid:96) τ ∗ ¯ duLDL ( Λ EW ) + V u j C iu (cid:96) τ ∗ ¯ duLDL ( Λ EW ) , X (cid:96) τ = C (cid:96) τ ∗ LHW ( Λ EW ) + C τ (cid:96) ∗ LHW ( Λ EW ) + C (cid:96) τ ∗ LDH ( Λ EW ) + C (cid:96) τ ∗ LDH ( Λ EW ) , (27)are given in terms of the SMEFT Wilson coefficients evaluated at the electroweak scale Λ EW .With all relevant interactions between the mesons and leptons at hand, we can compute the Feynman diagrams infigure 1 to arrive at the complete amplitude for the decay τ + ( p ) → (cid:96) − ( p ) P + i ( q ) P + j ( q ) , M = F G F (cid:2) T SD v τ P R u C (cid:96) + T µν v τ γ µ γ ν P R u C (cid:96) + T µνρ v τ γ µ γ ν γ ρ P R u C (cid:96) + T µνρ v τ γ µ γ ν γ ρ P L u C (cid:96) (cid:3) , (28)where T SD stands for the SD term and the others are the LD ones: T SD = − (cid:16) c τ (cid:96) , i j − c τ (cid:96) , i j ( q · q ) (cid:17) , (29) T µν = G F V ui V u j m τ (cid:96) (cid:0) q µ q ν t − + q µ q ν u − (cid:1) + (cid:104) V ui (cid:16) Bc (cid:96) τ P j − c (cid:96) τ P j ( t − p ) (cid:17) q µ ( p − q ) ν − V u j (cid:16) Bc τ (cid:96) P i − c τ (cid:96) P i ( t − p ) (cid:17) ( p − q ) µ q ν (cid:105) t − + (cid:104) V u j (cid:16) Bc (cid:96) τ P i − c (cid:96) τ P i ( u − p ) (cid:17) q µ ( p − q ) ν − V ui (cid:16) Bc τ (cid:96) P j − c τ (cid:96) P j ( u − p ) (cid:17) ( p − q ) µ q ν (cid:105) u − , (30) T µνρ = V u j c τ (cid:96) P i q µ ( p − q ) ν q ρ t − + V ui c τ (cid:96) P j q µ ( p − q ) ν q ρ u − , (31) T µνρ = V ui c (cid:96) τ P j q µ ( p − q ) ν q ρ t − + V u j c (cid:96) τ P i q µ ( p − q ) ν q ρ u − , (32)with s = ( q + q ) , t = ( p − q ) , and u = ( p − q ) . We have so far been working at the leading order in χ PT. Since it is well-known that chiral perturbation does notconverge fast enough for hadronic τ decays due to its large mass m τ compared to Λ χ , in this section we estimateuncertainties due to ignored higher order corrections and try to improve our leading order results. To estimate un-certainties, we will compute chiral logarithms arising from one-loop diagrams associated with an LNV vertex, whichas in the usual case cannot be cancelled by higher order counterterms. And to improve convergence we will employthe technique of dispersion relation by incorporating experimental data on phase shifts. Inspection of figure 1 showsthat this is nontrivial only for the short-distance part which unfortunately is numerically less important than the long-distance part as we will see in section 5. To put it in short, we are effectively considering one-loop uncertainties or8Figure 2: One-loop diagrams for SD contributions to (cid:104) P + i P + j | O irrep | (cid:105) with an insertion of O irrep in blob.dispersion-relation improvement to the matrix element (cid:104) P + i ( q ) P + j ( q ) | O irrep ( ) | (cid:105) , where O irrep is a dim-9 LEFT op-erator in the chiral representation irrep in table 3 whose lepton bilinear has been stripped off for simplicity and whosechiral realization is also shown in that table.To assess the relative importance of one-loop chiral logarithms to tree level terms, we compute one-loop diagramsin figure 2 at the kinematic point ( q + q ) = K ± decays. The results are, M ππ × = g × F π m π ( + L π ) , (33) M ππ , a / b × = g a / b × F π ( + L π ) , (34) M KK × = g × F K m K (cid:20) + (cid:18) m K + m π m K L π + L K + m K − m π m K L η (cid:19)(cid:21) , (35) M KK , a / b × = g a / b × F K (cid:20) − ( L π − L K + L η ) (cid:21) , (36) M K π × = g × F K ( m K + m π ) (cid:20) − (cid:18) m π − m K ( m K − m π ) L π − m K − m π m K − m π L K + L η (cid:19)(cid:21) , (37) M K π , a / b × = g a / b × F K (cid:20) − (cid:18) m π − m K ( m K − m π ) L π − m K + m π m K − m π L K + L η (cid:19)(cid:21) , (38)where L P = m P / ( π F ) ln ( µ / m P ) with µ being the renormalization scale. The results for the K π channel coincidewith those in Ref [20] while the ones for the KK and ππ channels are newly computed. We have taken into accountboth renormalization of decay constants and wavefunction renormalization which were collected in Ref [20]. To makea rough estimation, the relative corrections at µ = Λ χ ( µ = m τ ) in the ππ , K π , and KK channels (each put in a pair ofsquare brackets) and in the order of the chiral representations L × R , L × R ( a / b ) (separated by a comma withina pair of square brackets) in each channel are, [27%, 17%] ([29%, 18%]), [50%, 28%] ([55%, 27%]), [65%, 50%]([73%, 56%]), respectively. The neglected higher order corrections are thus about 20% − τ decays, see, e.g., Refs [36, 37, 38, 39, 40]. The matrix elements mentioned above are parameterizedas, M P i P j × ( s ) = (cid:104) P + i ( q ) P + j ( q ) | ( u L γ µ d iL )[ u L γ µ d jL ] | (cid:105) = − ( q · q ) F P i P j × ( s )( + δ i j ) , (39) M P i P j , a × ( s ) = (cid:104) P + i ( q ) P + j ( q ) | ( u L d iR )[ u R d jL ] | (cid:105) = F P i P j , a × ( s )( + δ i j ) , (40) M P i P j , b × ( s ) = (cid:104) P + i ( q ) P + j ( q ) | ( u L d iR ][ u R d jL ) | (cid:105) = F P i P j , b × ( s )( + δ i j ) , (41)with s = ( q + q ) . The form factors are normalized to, at leading order in χ PT, F P i P j × ( ) = F g × , F P i P j , a / b × ( ) = F g a / b × . (42)9 .50 0.75 1.00 1.25 1.50 1.75 - - -
100 1.01.52.02.5 s [ GeV ] δ [ ° ] | F i rr e p ππ ( s ) / F i rr e p ( ) | - -
100 1.01.52.0 s [ GeV ] δ [ ° ] | F i rr e p K π ( s ) / F i rr e p ( ) | Figure 3: Phase shifts (blue/solid curve, left vertical axis) and | F P i P j irrep ( s ) / F P i P j irrep ( ) | (red/dashed curve, right verticalaxis) in the ππ (left panel) and K π (right panel) channels shown as a function of √ s .To build the dispersion relation we decompose the elastic meson scattering amplitude into partial wave amplitudes f Il ( s ) with orbital angular momentum l and isospin I . Application of the Cutkosky cutting rules to figure 2 (b) wherethe meson scattering vertex is replaced by a general scattering amplitude yieldsIm F ππ irrep ( s ) = λ / ππ ( s ) s F ππ irrep ( s )[ f ( s )] ∗ θ ( s − s ππ ) , (43)Im F KK irrep ( s ) = λ / KK ( s ) s F KK irrep ( s )[ f ( s )] ∗ θ ( s − s KK ) , (44)Im F K π irrep ( s ) = λ / K π ( s ) s F K π irrep ( s )[ f / ( s )] ∗ θ ( s − s K π ) , (45)where s P i P j = ( m P i + m P j ) , and λ P i P j ( s ) is the basic three-particle kinematic function, λ P i P j ( s ) = m P i + m P j + s − m P i m P j − m P i √ s − m P j √ s . (46)The partial wave amplitude for elastic scattering can be expressed in terms of the phase shift δ Il ( s ) , f Il ( s ) = s λ / ( s ) sin δ Il ( s ) e i δ Il ( s ) . (47)The above dispersion relations imply that the phase of F P i P j irrep ( s ) is equal to the corresponding phase shift, just as theWatson’s final-state theorem for elastic scattering states [41]. Equations (43)-(45) then have a universal solution F P i P j irrep ( s ) = F P i P j irrep ( ) Ω Il ( s ) , (48)where the Omn`es factor Ω Il ( s ) [42] for a once-subtracted dispersion relation is Ω Il ( s ) = exp (cid:34) s π (cid:90) ∞ s PiPj ds (cid:48) δ Il ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) (cid:35) . (49)The phase shifts δ and δ / were measured experimentally long ago [43, 44]. We have taken the fits of δ from [45]and of δ / from [46]. As far as we know there are no available data for the ( I , l ) = ( , ) channel. These phase shiftsand corresponding magnitudes of F P i P j irrep ( s ) / F P i P j irrep ( ) = Ω Il ( s ) are shown in figure 3 in which a cutoff s cut = m τ has beenchosen for the integral. As expected, there appears no hint of resonance in these channels. The above results will beapplied to our evaluation of decay rates in the next section.10 Master formulas for decay rates
We are now in a position to present our master formulas of the decay rates and branching ratios for the decay τ + → (cid:96) − P + i P + j . We will skip kinematic details which are similar to the LNV K + decays [20, 21]. The spin-summed and-averaged decay width is Γ = + δ i j m τ π m τ (cid:90) ds (cid:90) dt ∑ | M | , (50)where the integration domains are s ∈ (cid:2) ( m P i + m P j ) , ( m τ − m (cid:96) ) (cid:3) , (51) t ∈ (cid:104) ( E ∗ + E ∗ ) − (cid:16)(cid:113) E ∗ − m P j + (cid:113) E ∗ − m (cid:96) (cid:17) , ( E ∗ + E ∗ ) − (cid:16)(cid:113) E ∗ − m P j − (cid:113) E ∗ − m (cid:96) (cid:17) (cid:105) , (52)with E ∗ = √ s ( s − m P i + m P j ) , E ∗ = √ s ( m τ − s − m (cid:96) ) . (53)Using the LECs in equation (23) and the SM parameters for the τ lepton width, various particle masses, and the Fermiconstant G F [47], the decay branching ratios become B ( τ + → e − π + π + ) GeV = . × − GeV | m τ e | eV + . | Y e τπ | + . | Y τ e π | + . × − (cid:16) | Y τ e π | + | Y e τπ | (cid:17) + . × − (cid:12)(cid:12) X τ e , ππ (cid:12)(cid:12) + − (cid:16) | X τ e | + | Y τ e π | + . | Y e τπ | (cid:17) + int. , (54) B ( τ + → µ − π + π + ) GeV = . × − GeV | m τµ | eV + . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) + . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) + . × − (cid:16)(cid:12)(cid:12) Y τµπ (cid:12)(cid:12) + (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) (cid:17) + . × − (cid:12)(cid:12)(cid:12) X τµ , ππ (cid:12)(cid:12)(cid:12) + − (cid:16) (cid:12)(cid:12) X τµ (cid:12)(cid:12) + (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) + . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) (cid:17) + int. , (55) B ( τ + → e − K + K + ) GeV = . × − GeV | m τ e | eV + . × − | Y τ eK | + . × − | Y e τ K | + . × − (cid:12)(cid:12) X τ e , KK (cid:12)(cid:12) + . × − (cid:16) | Y τ eK | + | Y e τ K | (cid:17) + − (cid:16) . | Y τ eK | + . | Y e τ K | + . | X τ e | (cid:17) + int. , (56) B ( τ + → µ − K + K + ) GeV = . × − GeV | m τµ | eV + . × − (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + . × − (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) + . × − (cid:12)(cid:12)(cid:12) X τµ , KK (cid:12)(cid:12)(cid:12) + . × − (cid:16)(cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) (cid:17) + − (cid:16) . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) + . (cid:12)(cid:12) X τµ (cid:12)(cid:12) (cid:17) + int. , (57) B ( τ + → e − K + π + ) GeV = . × − GeV | m τ e | eV + . × − | Y τ eK | + . × − | Y e τ K | + . × − | Y e τπ | + − (cid:16) | Y τ e π | + | Y τ eK | + | Y e τ K | + (cid:12)(cid:12) X τ e , K π (cid:12)(cid:12) + . | Y e τπ | + . | Y τ e π | (cid:17) + − (cid:16) | Y τ eK | + | X τ e | + | Y e τ K | + . | Y τ e π | + . | Y e τπ | (cid:17) + int. , (58) B ( τ + → µ − K + π + ) GeV = . × − GeV | m τµ | eV + × − (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + . × − (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) + . × − (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) + − (cid:16) (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) + (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τµ , K π (cid:12)(cid:12)(cid:12) + . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) (cid:17) + − (cid:16) (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) + (cid:12)(cid:12) X τµ (cid:12)(cid:12) + (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) + . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) (cid:17) + . × − (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) + int. , (59)where the Wilson coefficients of the dim-7 SMEFT operators are contained in the X (for the SD part) and Y (for theLD part) parameters defined in equations (27) and (22) whose interference terms (int.) are not explicitly displayed.We have incorporated the dispersion-relation improved hadronic matrix elements in the SD part.The above results show that the contribution from neutrino mass insertion in figure 1 (a) can be totally ignored.If we assume the Wilson coefficients of LNV dim-7 operators in the SMEFT are of a similar size, their relativeimportance is then controlled by the prefactors of the X and Y parameters. The LD contribution from Y dominates,while the ones from Y and Y are suppressed by a factor p and p / Λ EW respectively. The SD contribution of X hasan order of magnitude similar to Y , while the X term is suppressed by p / Λ χ and has a similar size as Y . To obtainconcrete constraints, we have to make a simplifying assumption due to too many Wilson coefficients, i.e., we assumethat only one of the X and Y parameters is nonzero at a time. The experimental upper bounds on the τ decays inequations (1)-(3) translate to the bounds on those parameters as shown in table 4. These bounds are much weaker thanthose from nuclear 0 νβ β decay and LNV K ± decays because of much smaller data samples, and being of order GeVthey should not be taken literally. But they are the first bounds obtained thus far on the LNV Wilson coefficients inthe SMEFT that involve the third generation of leptons. If we parameterize all Wilson coefficients by the same scale C i = Λ − , the branching ratios will be proportional to Λ − as shown in figure 4. For Λ > B ( τ − → e + π − π − ) < . × − , B ( τ − → µ + π − π − ) < . × − , (60) B ( τ − → e + K − K − ) < . × − , B ( τ − → µ + K − K − ) < . × − , (61) B ( τ − → e + K − π − ) < . × − , B ( τ − → µ + K − π − ) < . × − , (62)which are several orders of magnitude smaller than the current experimental upper bounds. We have studied the LNV τ decays τ + → (cid:96) − P + i P + j in the framework of effective field theory. A merit of these decays isthat they could potentially probe LNV interactions in the third generation of leptons which are not accessible in eithernuclear 0 νβ β decay or LNV K ± decays. Assuming absence of new particles of a mass below the electroweak scale,we started from effective interactions of LNV dim-5 and -7 operators in SMEFT, and matched them first to effectiveinteractions in LEFT at the electroweak scale, and then to those in χ PT at the chiral symmetry breaking scale. Wecomputed the decay branching ratios and expressed them in terms of the Wilson coefficients in SMEFT and hadroniclow energy constants. As in the case of LNV K ± decays, the long-distance contribution from exchange of a neutrinogenerically dominates over the short-distance one arising from LNV dim-9 operators in LEFT involving four quarksand two like-charge leptons. We estimated by computing one-loop chiral logarithms theoretical uncertainties due toneglect of higher order terms in chiral perturbation for the hadronic τ decays and found them large. We thereforeattempted to improve convergence in the short-distance part by appealing to dispersion relations. We found the decays τ + → e − π + π + , µ − π + π + have the largest branching ratios among the six channels, but unfortunately they are stillwell below the current experimental bounds for a reasonable choice of new physics scale.12 + → e − π + π + τ + → e − K + K + τ + → e − K + π + names bounds names bounds names bounds names bounds (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τ e , KK (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τ e , K π (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τ e , ππ (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) X τ e (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ eK (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τ K (cid:12)(cid:12) − . (cid:12)(cid:12) X τ e (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . (cid:12)(cid:12) X τ e (cid:12)(cid:12) − . (cid:12)(cid:12) Y τ e π (cid:12)(cid:12) − . (cid:12)(cid:12) Y e τπ (cid:12)(cid:12) − . τ + → µ − π + π + τ + → µ − K + K + τ + → µ − K + π + names bounds names bounds names bounds names bounds (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τµ , KK (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τµ , K π (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12)(cid:12) X τµ , ππ (cid:12)(cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . (cid:12)(cid:12) X τµ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . (cid:12)(cid:12) X τµ (cid:12)(cid:12) − . (cid:12)(cid:12) X τµ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτ K (cid:12)(cid:12) − . (cid:12)(cid:12) Y µτπ (cid:12)(cid:12) − . (cid:12)(cid:12) Y τµπ (cid:12)(cid:12) − . | X i | − / or | Y i | − / parameters of combinations of Wilson coefficients.Note that X αβ i = X βα i . Acknowledgement
This work was supported in part by the Grants No. NSFC-12035008, No. NSFC-11975130, by The National KeyResearch and Development Program of China under Grant No. 2017YFA0402200, by the CAS Center for Excellencein Particle Physics (CCEPP). XDM is supported by the MOST (Grants No. 109-2112-M-002-017-MY3 and 109-2811-M-002-535). We thank Rui Gao and Feng-Kun Guo for electronic communications on the current status of phase shiftdata.
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