Consistent treatment of axions in the weak chiral Lagrangian
Martin Bauer, Matthias Neubert, Sophie Renner, Marvin Schnubel, Andrea Thamm
IIPPP/20-82, MITP/21-007, ZU-TH 01/21
Consistent treatment of axions in the weak chiral Lagrangian
Martin Bauer a , Matthias Neubert b,c,d , Sophie Renner e , Marvin Schnubel b , and Andrea Thamm f a Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham, DH1 3LE, UK b PRISMA + Cluster of Excellence & MITP, Johannes Gutenberg University, 55099 Mainz, Germany c Department of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A. d Department of Physics, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland e SISSA International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy f School of Physics, The University of Melbourne, Victoria 3010, Australia
We present a consistent implementation of weak decays involving an axion or axion-like particle inthe context of an effective chiral Lagrangian. We argue that previous treatments of such processeshave used an incorrect representation of the flavor-changing quark currents in the chiral theory. Asan application, we derive model-independent results for the decays K − → π − a and π − → e − ¯ ν e a atleading order in the chiral expansion and for arbitrary axion couplings and mass. In particular, wefind that the K − → π − a branching ratio is almost 40 times larger than previously estimated. Axions and axion-like particles (collectively referred toas ALPs in this work) are new types of elementary par-ticles, which arise in a large class of extensions of theStandard Model (SM) and are well motivated theoreti-cally. They can provide an elegant solution to the strongCP problem based on the Peccei–Quinn mechanism [1–8]. More generally, ALPs can arise as pseudo Nambu–Goldstone bosons in models with explicit global symme-try breaking. Low-energy weak-interaction processes im-ply some of the most stringent bounds on the couplingsof ALPs to gluons and other SM particles [9–12].In a seminal paper [13], Georgi, Kaplan and Randallhave derived the effective chiral Lagrangian accountingfor the interactions of a light ALP (with mass below thescale of chiral symmetry breaking, µ χ = 4 πf π ) with thelight pseudoscalar mesons, opening the door to a model-independent description which does not rely on the de-tails of Peccei–Quinn symmetry breaking. In this Let-ter, we reanalyze this problem and point out a small butimportant omission in the representation of the weak-interaction quark currents in the effective theory, whichhas far-reaching consequences. Despite the 35-year his-tory of the subject, we find that even recent papers onweak decays such as K − → π − a and π − → e − ¯ ν e a omitthe contributions of relevant Feynman diagrams and thusemploy incomplete expressions for the decay amplitudes(see e.g. [14–16]). In many phenomenological studies,the amplitudes are derived by starting from an ampli-tude for a decay process involving a π or η meson andaccounting for the (kinetic) mixing of the ALP with theseneutral mesons by means of mixing angles θ πa and θ ηa .Below we recall the well-known fact that in the approachof [13] the mixing angles are unphysical, because theydepend on the parameters of the chiral rotation used toeliminate the ALP–gluon coupling in the effective La-grangian. It is customary to adopt a “default choice”for these parameters, which eliminates the mass mixingin the effective Lagrangian. However, there always ex-ist other contributions to the decay amplitude, in whichthe ALP participates in the relevant interaction vertices. Neglecting these “direct” contributions leads to incorrectpredictions. In fact, they are essential to ensure that theauxiliary parameters of the chiral rotation cancel out inpredictions for physical quantities. (Only a very specialclass of models, in which the ALP couples to SM fieldsonly through phases in the quark mass matrices, with noderivative interactions and no couplings to gluons at thelow scale µ χ , is an exception to this rule, see e.g. [11, 17].)The starting point of our study is the effective ALPLagrangian at a scale of order µ χ ≈ . L eff = L QCD + 12 ( ∂ µ a )( ∂ µ a ) − m a, a + c GG α s π af G aµν ˜ G µν,a + c γγ α π af F µν ˜ F µν + ∂ µ af (cid:16) ¯ q L k Q γ µ q L + ¯ q R k q γ µ q R + . . . (cid:17) . (1)Here q is a 3-component vector in generation space con-taining the three light quark flavors u, d, s . The ALPdecay constant f is related to the scale of global (Peccei–Quinn) symmetry breaking by Λ = 4 πf and is assumedto lie above the scale of electroweak symmetry breaking.It governs the overall magnitude of the ALP interactionswith SM particles, the leading of which are mediated bydimension-5 operators. (In the literature on QCD ax-ions, one often defines the axion decay constant f a interms of the strength of the axion–gluon coupling, suchthat 1 /f a = − c GG /f .) The parameters c GG and c γγ de-termine the strengths of the ALP interactions with glu-ons and photons, while the hermitian matrices k Q and k q contain the ALP couplings to left-handed and right-handed quarks. The off-diagonal entries of these matricesaccount for the possibility of flavor-changing s → d tran-sitions. The dots represent analogous couplings to lep-tons. The ALP couplings are scale-dependent quantities.Their evolution from the new-physics scale Λ down to thescale µ χ has recently been studied in detail [18, 19]. Themass parameter m a, provides an explicit soft breakingof the shift symmetry a → a + c , which is a (classical) a r X i v : . [ h e p - ph ] F e b symmetry of the effective Lagrangian (1). In QCD axionmodels m a, vanishes and the axion mass is generated bynon-perturbative QCD dynamics [6, 20]. In more generalALP models a non-zero bare mass can be generated bymeans of non-abelian extensions of the SM.To study the low-energy interactions of a light ALPwith the pseudoscalar mesons ( π, K, η ), the Lagrangian(1) is matched onto a chiral effective Lagrangian, in which Σ ( x ) = exp (cid:2) i √ f π λ a π a ( x ) (cid:3) contains the pseudoscalar me-son fields ( λ a are the Gell-Mann matrices). In orderto find the bosonized form of the ALP–gluon interac-tion, one eliminates the aG ˜ G term in favor of ALP cou-plings to quark bilinears, whose chiral representation iswell known. This is accomplished with a chiral rotation[12, 13, 22] q ( x ) → exp (cid:20) − i ( δ q + κ q γ ) c GG a ( x ) f (cid:21) q ( x ) , (2)where δ q and κ q are hermitian matrices, which we chooseto be diagonal in the quark mass basis. Under this fieldredefinition the measure of the path integral is not in-variant [23], and this generates extra contributions tothe ALP couplings to gluons and photons. Imposing thecondition Tr κ q = κ u + κ d + κ s = 1 (3)ensures that the ALP–gluon interaction is eliminatedfrom the Lagrangian at the expense of modifying theALP–photon and ALP–fermion couplings as well as thequark mass matrix. Denoting the modified couplingswith a hat, one finds (with N c = 3 the number of colors)ˆ c γγ = c γγ − N c c GG Tr Q κ q , ˆ k Q ( a ) = e i φ − q a/f (cid:0) k Q + φ − q (cid:1) e − i φ − q a/f , ˆ k q ( a ) = e i φ + q a/f (cid:0) k q + φ + q (cid:1) e − i φ + q a/f , (4)where φ ± q = c GG ( δ q ± κ q ), and Q = diag( Q u , Q d , Q s )contains the electric charges of the quarks in units of e .The phase factors in the last two relations cancel for alldiagonal elements of the matrices ˆ k Q and ˆ k q . As long asthe condition (3) is satisfied, any choice of the matrices δ q and κ q describes the same physics. The derivativecouplings of the ALP to the left- and right-handed quarkcurrents are implemented by including the ALP field inthe definition of the covariant derivative [24], such that i D µ Σ = i∂ µ Σ + eA µ [ Q , Σ ] + ∂ µ af (cid:16) ˆ k Q Σ − Σ ˆ k q (cid:17) , (5)where A µ is the photon field. This definition implies( D µ Σ ) Σ † + Σ ( D µ Σ ) † = ∂ µ (cid:0) Σ Σ † (cid:1) = 0 . (6)The leading-order chiral Lagrangian can then be ex- pressed in the form L χ eff = f π (cid:2) D µ Σ ( D µ Σ ) † (cid:3) + f π B Tr (cid:2) ˆ m q ( a ) Σ † + h.c. (cid:3) + 12 ∂ µ a ∂ µ a − m a, a + ˆ c γγ α π af F µν ˜ F µν , (7)where the parameter B ≈ m π / ( m u + m d ) is proportionalto the chiral condensate. Throughout this Letter we workconsistently at lowest order in the chiral expansion andneglect the effects of π – η – η (cid:48) mixing. With our choiceof diagonal matrices δ q and κ q , the modified quark massmatrix takes the formˆ m q ( a ) = exp (cid:18) − i κ q c GG af (cid:19) m q , (8)where m q = diag( m u , m d , m s ).The effective chiral Lagrangian (7) has been the basisfor numerous studies of low-energy phenomena involv-ing axions or light ALPs. Expanding the Lagrangian toquadratic order in fields, one finds that the ALP acquiresthe mass term m a = c GG f π m π f m u m d ( m u + m d ) + m a, (cid:20) O (cid:18) f π f (cid:19)(cid:21) , (9)up to higher-order corrections in the chiral expansion[6, 20]. Higher-order terms generate a periodic poten-tial for the ALP field a , which breaks the continuous shiftsymmetry of the classical Lagrangian to the discrete shiftsymmetry a → a + nπf /c GG . One also finds that thereare mass-mixing and kinetic-mixing contributions involv-ing the ALP and the neutral mesons π and η , whose ex-plicit form depends on the parameters κ q . For instance,at first order in 1 /f one obtains π = π + θ πa a phys with the mixing angle θ πa = f π √ f (cid:20) m a (ˆ c uu − ˆ c dd ) m π − m a − m π ∆ κ m π − m a (cid:21) , (10)where ˆ c qq = c qq + 2 κ q c GG with c qq = ( k q − k Q ) , ∆ κ = 4 c GG m u κ u − m d κ d m d + m u . (11)Via the quantities ˆ c qq and ∆ κ the mixing angle dependson the auxiliary parameters κ q in (2). The special choice κ q = m − q / Tr( m − q ) eliminates the mass-mixing contri-bution ∆ κ , leaving a contribution from kinetic mixingthat is proportional to m a and hence is negligible for aQCD axion with m a ∼ f π /f . This “default choice”defines a scheme, which is frequently adopted in the lit-erature. It is important to realize, however, that θ πa isnot a physical quantity. For instance, one can find valuesof κ u , κ d and κ s such that θ πa = 0 and θ ηa = 0 [19]. Inour discussion below we treat the quantities δ q and κ q in the field redefinition (2) as free parameters, subjectonly to condition (3). We study in detail how the depen-dence on these auxiliary variables cancels in predictionsfor physical observables. For flavor-conserving processessuch as a → γγ and a → πππ , an analogous study wasperformed in [19].In (7) the ALP enters in the quark mass matrix ˆ m q ( a )and through the covariant derivative defined in (5). Forthe very special situation in whichTr (cid:2) k Q ( µ χ ) − k q ( µ χ ) (cid:3) = 2 c GG , (12)it is possible to choose the matrices κ q and δ q in sucha way that ˆ k q and ˆ k Q both vanish. In this case, theALP only enters the Lagrangian through the quark massmatrix (8), see e.g. [17]. However, condition (12) is notinvariant under renormalization-group evolution, and itwould need a fine tuning to realize this condition at thelow scale µ χ .The effective chiral Lagrangian (7) can also be usedto study flavor-changing processes such as K − → π − a and π − → e − ¯ ν e a , which in the SM are mediated by theweak interactions and at low energies are described by4-fermion operators built out of products of left-handedcurrents. Under a left-handed, flavor off-diagonal rota-tion q L → U L q L of the quark fields, the meson fieldstransform non-linearly as Σ → U L Σ . The effective La-grangian is invariant under this transformation if we treatthe quark mass matrix and the left-handed ALP cou-plings as spurions transforming as ˆ m q ( a ) → U L ˆ m q ( a )and ˆ k Q → U L ˆ k Q U † L . Applying the Noether procedureto the Lagrangians in the quark and meson pictures, andaccounting for an additional phase factor arising from thechiral rotation of the fields, we find that the left-handedquark currents ¯ q iL γ µ q jL must be represented in the chiraltheory by L jiµ = − if π e i ( φ − qi − φ − qj ) a/f (cid:2) Σ ( D µ Σ ) † (cid:3) ji (cid:51) − if π (cid:20) i ( δ q i − δ q j − κ q i + κ q j ) c GG af (cid:21) (cid:2) Σ ∂ µ Σ † (cid:3) ji + f π ∂ µ af (cid:2) ˆ k Q − Σ ˆ k q Σ † (cid:3) ji . (13)This generates both non-derivative and derivative cou-plings of the ALP to the weak-interaction vertices. Withthe special choice δ q = κ q one can eliminate the non-derivative couplings; however, the derivative couplingsremain. Astoundingly, it appears that the contributioninvolving the derivative of the ALP field has been omittedin the literature. It has neither been taken into accountin the original paper [13] nor in later work based on it.The chiral representation of the effective weak La-grangian mediating the decays K − → π − π , K S → π + π − and K S → π π at leading order in the chiral ex-pansion involves an operator transforming as an SU (3)octet and two transforming as 27-plets [25–27]. (A second ⇡ ⇡ ⇡ g K K aK ⇡ ⇡ ⇡ aK a⇡ K ⇡ aK K g g ag a g ⌘ FIG. 1. Feynman graphs contributing to the K − → π − a de-cay amplitude at leading order in the chiral expansion. Weak-interaction vertices are indicated by a crossed circle, whiledots refer to vertices from the Lagrangian (7). octet operator can be transformed into the first one usingthe equations of motion.) The octet operator receives ahuge dynamical enhancement known as the ∆ I = se-lection rule [28]. The corresponding Lagrangian reads L weak = − G F √ V ∗ ud V us g [ L µ L µ ] , (14)where | g | ≈ . s L → d L transition. We have calculated the K − → π − a decay amplitude from the Lagrangians (7) and (14), eval-uating the Feynman graphs shown in Figure 1. The firsttwo diagrams account for the ALP–meson mixing contri-butions mentioned above, while the third graph containsthe ALP interactions at the weak vertex derived from(13). The following two graphs describe ALP emission ofan initial- or final-state meson. They give nonzero con-tributions if the ALP has non-universal vector-currentinteractions with different quark flavors. The last di-agram contains possible flavor-changing ALP–fermioncouplings, as parameterized by the off-diagonal elementsof the matrices k Q and k q in (1). To simplify the analy-sis we set m u = m d ≡ ¯ m in order to eliminate the π – η mass mixing. (More general expressions, including alsothe contribution from the 27-plet operators, will be pre-sented elsewhere.) The meson masses are then given by m π = 2 B ¯ m , m K = B ( m s + ¯ m ), and 3 m η = 4 m K − m π .Before considering the resulting decay amplitude, it is in-structive to see how the scheme-dependent contributionsinvolving the δ q and κ q parameters cancel between thevarious diagrams. In units of N = − G F √ V ∗ ud V us g f π ,with | N | ≈ . · − , we find for these contributions D (cid:51) N f c GG ( κ u − κ d )( m π − m a ) ,D (cid:51) − N f c GG (2 m K + m π − m a ) ( κ u + κ d − κ s ) ,D (cid:51) N f c GG (cid:104) − ( δ d − δ s − κ d + κ s )( m K + m π − m a )+ ( δ u − δ d + κ u + κ s )( m K − m π + m a )+ ( δ u − δ s + κ u + κ d )( m K − m π − m a ) (cid:105) , D (cid:51) − N f c GG m K ( δ u − δ d ) ,D (cid:51) N f c GG m π ( δ u − δ s ) , (15)while the last diagram is scheme independent. Via themixing angles θ πa and θ ηa the results for D and D de-pend on the κ q parameters, see (10). The expressions for D and D , on the other hand, depend only on the δ q parameters. Only the third diagram, in which the ALPis emitted from the weak-interaction vertex, depends onboth sets of parameters. In the sum of all contribu-tions the dependence on the auxiliary parameters cancels(apart from an unambiguous contribution proportional to κ u + κ d + κ s = 1). But this cancellation only works ifthe derivative ALP interactions in (13) are included.Adding up all contributions, we obtain for the decayamplitude (for m u = m d ) i A K − → π − a = N f (cid:20) c GG ( m K − m π )( m K − m a )4 m K − m π − m a + 6( c uu + c dd − c ss ) m a m K − m a m K − m π − m a + (2 c uu + c dd + c ss ) ( m K − m π − m a ) + 4 c ss m a + ( k d + k D − k s − k S ) ( m K + m π − m a ) (cid:21) − m K − m π f [ k q + k Q ] . (16)Note that the transition K − → π − a proceeds via thedynamically enhanced octet operator, whereas the corre-sponding decay K − → π − π receives contributions fromthe 27-plet operator with isospin change ∆ I = only.This effect is well known and is referred to as “octet en-hancement” [9, 10]. Attempts to estimate the K − → π − a decay rate as θ πa times the K − → π − π rate miss thisimportant effect. Another interesting feature of the re-sult (16) is its dependence on the flavor-conserving ALPvector couplings ( k d + k D ) and ( k s + k S ) to down andstrange quarks. In the presence of the weak interactionsthe currents ¯ dγ µ d and ¯ sγ µ s are not individually con-served (unlike in QCD), and hence these couplings canhave observable effects.In order to compare our result (16) with some previouscalculations, we work to leading order in the ratio ¯ m/m s ,consider the limit where m a (cid:28) m K and assume the caseof a minimal flavor-violating ALP, for which c ss = c dd and k d + k D = k s + k S [19]. We then obtain the simpleresult (still with m u = m d , neglecting the small 27-pletcontributions, and setting 1 /f a = − c GG /f ) A K − → π − a ≈ im K f a (cid:34) N (cid:18) c uu + c dd c GG (cid:19) − [ k q + k Q ] c GG (cid:35) . (17)Barring cancellations, the contribution proportional to N dominates as long as | [ k q + k Q ] /c GG | (cid:28) · − , which we assume from now on. Eliminating the parame-ter N via the K S → π + π − decay amplitude, we obtainBr( K − → π − a )Br( K S → π + π − ) ≈ τ K − τ K S f π f a (cid:20) c uu + c dd c GG (cid:21) . (18)For a long-lived ALP with mass m a (cid:28) m π , the upperlimit Br( K − → π − X ) < . · − (90% CL) reportedby NA62 [30] from a search for a feebly interacting newparticle X implies1 f a (cid:12)(cid:12)(cid:12)(cid:12) c uu + c dd c GG (cid:12)(cid:12)(cid:12)(cid:12) < . . (19)Estimating the weak-interaction contribution to the de-cay amplitude from kinetic ALP–meson mixing (see e.g.[14–16]) corresponds to retaining only the first two dia-grams in Figure 1, evaluated with the default choice of κ q parameters. Under the approximations described abovethis leads to A K − → π − a ≈ iN m a f a (cid:18) − c uu − c dd c GG (cid:19) , (20)which underestimates the amplitude by a factor m a / (4 m K ) and predicts the wrong sign for the contri-bution proportional to c uu . If mass mixing with the η (cid:48) isincluded, one finds an additional small contribution pro-portional to sin θ ηη (cid:48) m π /m K [15, 16] relative to the lead-ing term in our result. The authors of [13] performed amore careful evaluation of the K − → π − a decay rate forthe case of a QCD axion ( m a ≈
0) without couplings tomatter ( c qq = 0). In this case diagrams D and D van-ish when one adopts the default choice of κ q parameters,and the graphs D and D vanish if one chooses δ q = 0.In the evaluation of the third diagram the authors omit-ted the derivative couplings of the axion shown by thelast term in (13). They obtained (this formula was notexplicitly shown in the paper, but we have derived it fromtheir arguments and the presented numerical result) A K − → π − a ≈ iN m K f a m u m u + m d . (21)This contribution to the amplitude is smaller than thecorresponding term in (17) by a factor m u m u + m d ) ≈ . δ q = κ q , in which the ALP is removed from theweak-interaction vertex. With their omission, we can-not reproduce that the two treatments lead to the sameexpression.)We have also applied our matching prescription (13) toderive the π − → e − ¯ ν e a decay amplitude, finding again aresult that is independent of the choice of the δ q and κ q parameters. It reads A π − → e − ¯ ν e a = − G F √ V ud f π f ¯ u e (/ p π + / p a )(1 − γ ) v ¯ ν e × (cid:20) c GG m d − m u m d + m u + k u − k d + m a m π − m a ∆ c ud (cid:21) , (22)where k q are the ALP couplings to right-handed quarkcurrents in (1). We omit a contribution with (/ p π − / p a )inside the spinor product, which is proportional to theelectron mass. For the default choice of the κ q parame-ters, the term involving ∆ c ud ≡ c uu − c dd + 2 c GG m d − m u m d + m u in the second line is due to ALP–pion mixing. For theQCD axion or a light ALP with m a (cid:28) m π this contribu-tion is negligible. In “pion-phobic axion models” [11] onetunes the couplings c GG , k u and k d in such a way thatthe amplitude (22) vanishes. This tuning is unnatural,because the couplings k q change under scale evolutionwhereas c GG is scale invariant [19].Our model-independent predictions in (16) and (22)can be compared with results obtained in the context ofspecific axion models. In the “variant-axion models” thecoupling parameters in the effective Lagrangian (1) areobtained as c GG = − N ( x + x ), k u = z , k d = k s = x and k U = k D = k S = 0, where z and x are the Peccei–Quinncharges of the right-handed up and down quarks, and N is the number of up-type quarks with the same chargeas u R . With these identifications, our result (22) agreeswith eq. (4.1) in [12], and our result (16) agrees with eq.(4.66) upon setting m a = 0, apart from some subleadingcorrections of O ( m π /m K ). For the “short-lived axionmodel” the relevant couplings are k u = − c GG = 1 and k d = 0, and with these values our result (22) agrees witha corresponding relation obtained in [11].In summary, we have present a consistent implemen-tation of weak decay processes involving an axion oraxion-like particle in the context of the chiral Lagrangian.We have pointed out that previous calculations have ne-glected to include important weak-interaction vertices in-volving derivative couplings of the ALP, which as shownin (13) arise when the relevant chiral quark currents arederived from the Noether procedure. Other phenomeno-logical treatments based on the notion of ALP–mesonmixing have omitted several relevant contributions. Inparticular, we find that that K − → π − a branching ra-tio is about a factor 37 larger than the prediction ob-tained in [13], which has important phenomenologicalconsequences. We have derived the model-independentexpressions for the K − → π − a and π − → e − ¯ ν e a decayamplitudes, including all relevant ALP couplings and theeffects of the ALP mass. The methods we have developedcan be applied to a variety of other low-energy observ-ables of phenomenological interest. Acknowledgements:
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