Heavy QCD Axion in b\to s transition: Enhanced Limits and Projections
Sabyasachi Chakraborty, Manfred Kraus, Vazha Loladze, Takemichi Okui, Kohsaku Tobioka
KKEK-TH-2295
Heavy QCD Axion in b → s transition: Enhanced Limits and Projections Sabyasachi Chakraborty, ∗ Manfred Kraus, † Vazha Loladze, ‡ Takemichi Okui,
1, 2, § and Kohsaku Tobioka
1, 2, ¶ Department of Physics, Florida State University, Tallahassee, FL 32306, USA Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
We study a “heavy” QCD axion whose coupling to the standard model is dominated by aG (cid:101) G but with m a (cid:29) m π f π /f a . This is well motivated theoretically as it can evade the infamous axionquality problem. It also poses interesting challenges for its experimental search due to its suppressedcouplings to photons and leptons. Such axion with mass around a GeV is kinematically inaccessibleor poorly constrained by most experimental probes except B-factories. We study B → Ka transi-tions as a powerful probe of the heavy QCD axion by performing necessary 2-loop calculations forthe first time. We find some of the existing limits are enhanced. For forthcoming data sets of theBelle II experiment, we provide a projection that f a of a few TeV is within its future reach, whichis relevant to the quality problem. I. INTRODUCTION
Null signals of new physics at the TeV scale so farsuggest us to adopt broader perspectives on the priori-ties of theoretical questions and future experimental pro-grams. In particular, the possibility of new physics atscales much lighter than the TeV scale has been gain-ing growing attention. The axion offers a strong motiva-tion for such light new physics, being a compelling solu-tion [1, 2] to the long-standing strong CP problem [3] byutilizing the Peccei-Quinn (PQ) symmetry [4, 5], as wellas being a candidate for cold dark matter [6–8].The original axion model [1, 2], in which QCD isthe sole origin of the axion mass, predicts the relation m a f a (cid:39) m π f π among the axion mass m a , its decay con-stant f a , and the analogous quantities for the pion. If weimagine that the PQ symmetry breaking scale, f a , is re-lated to the origin of the electroweak symmetry breaking,it would be natural to place f a at the TeV scale, as pro-posed in the original axion models by Weinberg [1] andWilczek [2], which then puts m a at the keV scale by therelation above. This possibility, however, is excluded byastrophysical observations [9–12] and beam dump exper-iments [13–15]. Much higher f a around 10 –10 GeV,and hence much lighter m a , can be motivated by an ax-ion as cold dark matter [16]. This part of the parameterspace has also been searched with null results [17–19].The relation, m a f a (cid:39) m π f π , can easily be violated,however, if there are additional contributions to the ax-ion mass [20–36]. This permits us to reconsider the casewhere f a is at or moderately above the TeV scale, butnow with m a much heavier than ∼ m π f π /f a ∼ keV. (Itwould be difficult, if not impossible, to imagine a scenariowhere m a is lighter than ∼ m π f π /f a .) It is particularlyimportant to explore the masses of 10 MeV (cid:46) m a with ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] TeV-scale f a [37]. Such low values of f a can also be mo-tivated theoretically as a solution to the axion qualityproblem. If the violation of global symmetries by quan-tum gravity appears as unsuppressed O (1) coefficientstimes powers of f a /M Pl , the axion solution to the strongCP problem would be ruined [38–41] unless f a is below ∼
10 TeV [32].In this work, therefore, we focus on what we call the heavy QCD axion scenario, where m a is much heavierthan ∼ m π f π /f a and the axion couples to the SM domi-nantly via only the aG µν (cid:101) G µν interaction, where a is theaxion field and G µν the gluon field strength. Our effectiveLagrangian thus has the form L = L SM + α s π af a G aµν (cid:101) G aµν + 12 ( ∂ µ a ) − m a a , (1)where (cid:101) G aµν ≡ (cid:15) µνρσ G aρσ . The additional terms requiredfor renormalization that are phenomenologically relevantwill be discussed shortly. There are many models thatUV-complete this EFT or could do so with minor modi-fications [29, 33, 34, 36].The status of experimental probes into the heavy QCDaxion is the following. For m a (cid:46)
400 MeV, f a at theTeV scale is excluded by the hadronic production anddiphoton decay of the heavy axion, the proton beamdump experiment [15, 42], the kaon experiments [43–50], the precision measurement of pion decay [51–53],the fixed target experiment [42, 54, 55], and the col-lider experiments [42, 56, 57]. For m a (cid:38)
400 MeV, thesearch is difficult because the hadronic decay mode dom-inates, and it is overall poorly explored until m a reaches50 GeV where the CMS dijet search kicks in [37, 58].However, axion production from hadron decays such as φ → γa and η (cid:48) → ππa constrain some parameter space[59] (see also Fig. 4). A part of the experimental loop-hole, m a (cid:38) m a in thefew GeV range and f a at the TeV scale and above hasnot been explored. In this region, B physics should playa crucial role, having the right energy scale as well asgreat experimental precision. Moreover, the experimen- a r X i v : . [ h e p - ph ] F e b tal reach of B physics will be improved further in up-coming years by LHCb (300 fb − ) and Belle II (5 × B -meson pairs). A promising channel is B → Ka with the axion subsequently decaying to hadrons, whichis induced at 2-loop, starting from the tree-level La-grangian (1). The importance of this channel was pointedout in [42, 64], but the required 2-loop calculation has notbeen performed to date.Our goal, therefore, is to perform this calculation andobtain robust and competitive bounds for the heavy QCDaxion. We will also provide a projection for the reach ofBelle II. II. CALCULATION OF b → sa Starting from the Lagrangian (1), the leading contri-bution to b → sa arises at 2-loop as shown in Fig. 1.Cancelling UV divergences in these diagrams requires thefollowing additional interactions to be further included inthe Lagrangian: L = · · · + C qq (cid:88) q ∂ µ af a ¯ qγ µ γ q + C bs ∂ µ af a ¯ s L γ µ γ b L + h.c. , (2)where the ellipses denote the terms in Eq. (1) as well astheose irrelevant for the b → sa phenomenology of ourinterest (see e.g. [65] for those other operators generatedat 1-loop from Eq. (1)). The C qq term is generated at1-loop from the diagram shown in Fig. 2 and necessary tocancel 1-loop sub-divergences in Fig. 1. The C bs term isrequired to remove remaining divergences at 2-loop. Wehave written the same coefficient C qq for all quark flavorsbecause we assume m t / Λ UV (cid:28) ∼ m t / Λ , where Λ UV is the cutoff of our EFT.It is not necessary at the 2-loop level to modify the co-efficient of aG (cid:101) G in Eq. (1) from α s / πf a , provided thatthe α s here is treated as the running coupling α s ( µ ).While this claim is verified by an explicit calculationin Appendix, it may be understood as follows. If wetreat the axion as an external field, the coefficient of( a/f a ) G (cid:101) G is completely fixed by matching the PQ-QCD-QCD anomaly. All corrections from turning a back on asa dynamical field involve the aG (cid:101) G coupling itself at leasttwice and hence negligibly small.Although C qq and C bs are free parameters in the EFT,their sizes must be consistent with the defining featureof our framework that the aG (cid:101) G interaction is the dom-inant coupling of the axion to the SM. As we wouldset C qq and C bs to zero for our scenario if there were If there is an aW ˜ W coupling, b → sa is induced at one-looplevel [62] (see also [63]).
3. For C bs , we further includetwo electroweak gauge couplings and GIM suppression(see Fig. 1), so C bs ∼ C F ( α s / π ) ( α w / π ) (cid:80) k V kb V ∗ ks ξ k ,where V is the CKM matrix and ξ k ≡ m k /M W with k = u, c, t . Therefore, at the cutoff Λ UV of our EFT,where it is matched on to the UV theory, we parametrize C qq and C bs as C qq (Λ UV ) ≡ AC F (cid:18) α s π (cid:19) ,C bs (Λ UV ) ≡ BC F (cid:18) α s π (cid:19) α w π (cid:88) k V ik V ∗ kj ξ k , (3)where A and B are O (1) parameters that depend on theUV model, and all the SM parameters are evaluated atΛ U V . We will show, however, that our bounds on f a arefairly insensitive to the exact values of A and B givenexperimental uncertainties. Then, keeping in mind theserough sizes of C qq and C bs , we find the leading RG run-ning of C qq and C bs (see Appendix for the details of thecalculation): µ d C qq d µ = − C F (cid:18) α s π (cid:19) , (4) µ d C bs d µ = (cid:18) C F (cid:18) α s π (cid:19) + C qq (cid:19) α w π (cid:88) k ξ k V kb V ∗ ks . (5) Λ UV [TeV] C i / h V t b V ∗ t s ‡ α s π · ‡ α w π · i C C C W C FIG. 3. C , C and C refers to the first, second and thirdcontribution in C W respectively, for different UV scales (seeEq. (7)). After running down to µ ∼ M W using Eqs. (4, 5), weswitch to another EFT in which the top quark and W boson are integrated out. In the limit of m b,s /M W → b → sa phenomenology: L bsa = C W ∂ µ af a ¯ s L γ µ γ b L + h.c. , (6)where C W is determined by C qq ( µ w ) and C bs ( µ w ) with µ w ∼ M W and the contributions from integrating out t and W . We find C W = C bs ( µ w ) + α w π C qq ( µ w ) g ( µ w ) + 12 α w π (cid:18) α s π (cid:19) f ( µ w ) , (7)where g and f are 1- and 2-loop matching functions givenrespectively in Eqs. (B7, B6) in Appendix. In the limitof m b,s /M W → C W does not run between M W to m b .This is because in this particular limit, there is no mix-ing between aG ˜ G and flavor changing axial-vector cou-pling. In Fig. 3 we show the 1st, 2nd, and 3rd termsof the right-hand-side of Eq. (7) as well as the net C W ,all as a function of Λ UV , assuming the initial condition A = B = 0 in Eq. (3). We observe that C bs , i.e., the b - s - a operator dominates the overall C W and interferesdestructively with C gg , i.e., a - g - g operator. The dom-inance of C bs can be explained by the operator mixingunder the RGE evolution; C bs acquires leading logarith-mic contributions ∼ ln(Λ /M W ) and ∼ ln (Λ /M W ) m a [GeV] -1 f a [ G e V ] B → Ka ( ηππ ) b → sa (inclusive) B → K a ( K K π ) B → Ka ( φφ ) B → K a ( π ) Λ UV =1 TeV B e l l e I I P r o j e c t i o n ( a → η π π ) LEP
Light MesonBounds
CHARM
E137 -7 -6 -5 -4 -3 -2 C gg Λ [ G e V − ] m a [GeV] -1 f a [ G e V ] B → Ka ( ηππ ) b → sa (inclusive) B → K a ( K K π ) B → Ka ( φφ ) B → K a ( π ) Λ UV =10 TeV B e l l e I I P r o j e c t i o n ( a → η π π ) LEP
Light MesonBounds
CHARM
E137 -7 -6 -5 -4 -3 -2 C gg Λ [ G e V − ] FIG. 4. We portray the constraints from different B -decaymeasurements in the m a - f a plane. Three curves are drawn foreach constraint corresponding to different initial conditions(see Eq.(3)), i.e., the strongest ( A = +3 , B = − A = − , B = +3) and central constraints ( A = B = 0). Wechoose the UV scale Λ UV to be 1 and 10 TeV for the top andthe bottom plot, respectively. The grey shaded regions com-prise bounds from [15, 42, 46, 55–57]. For B → Ka , we use[66] for inclusive analsysis and [67–69] for exclusive channels a → π, ηππ, KKπ, φφ . For the projection at Belle II, 5 × ¯ BB pair is assumed. The right vertical axis is labelled usingthe notation of Ref. [42] for comparison. due to the mixing with a - g - g and a - q - q operators. Sinceln(TeV /M W ) ≈ C bs dic-tates over others.The final step is to evaluate the meson level decay B → aK ( ∗ ) [62, 70]. We findΓ B → Ka = (cid:12)(cid:12) C W (cid:12)(cid:12) m B πf a (cid:18) − m K m B (cid:19) λ Ka (cid:2) f ( m a ) (cid:3) , (8)where λ Ka is given by λ Ka = (cid:20)(cid:18) − ( m K + m a ) m B (cid:19)(cid:18) − ( m K − m a ) m B (cid:19)(cid:21) , (9)while f ( m a ) is the form factor obtained from the light-cone QCD sum rules [71, 72]: f ( m a ) = 0 . − m a / . . (10) III. PHENOMENOLOGY
To derive constraints on the axion decay constant as afunction of the mass we use different B decay measure-ments. • We first derive the constraint on inclusive b → sa decaybased on PDG data BR( B + → ¯ cX ) = 97 ±
4% [66].Thus, we require BR( b → sa ) < − BR( b → c ) (cid:46) b → sa ) (cid:39) (cid:12)(cid:12) C W (cid:12)(cid:12) Γ B f a ( m B − m a ) πm B , (11)where Γ B is the width of B meson. The inclusive b → sa decay rules out the region marked by yel-low in Fig. 4. In fact, this constraint is compara-ble and in some cases more robust than the boundsdrawn for light meson phenomenology [42, 46], e.g., K L → π a ( γγ ), η (cid:48) → ππa (3 π ), φ → γa ( ππγ, ηππ )and γp → pa ( γγ ), displayed in grey in Fig. 4. • Next we use exclusive final states a → π , φφ , KKπ ,and ηππ to perform axion search. We perform a peaksearch except in a → π final state. To calculate cor-responding branching fractions for axion decay we usethe data-driven approach given in Ref. [42] and usebranching fractions given in Fig. 3 of their paper. Theuncertainties in this approach for axion hadronic (par-tial) widths are not estimated and therefore not in-cluded in the following bounds. However, these can beextracted by the same drive-driven method [42].1. The constraints on the a → π channel, shownby the blue region in Fig. 4 is derived based onBelle analysis [67]. This analysis is applicableto 0 .
73 GeV ≤ m a ≤ .
83 GeV. We requireBR (cid:0) B → K a (cid:1) BR (cid:0) a → π + π − π (cid:1) < . × − ,which is from BR (cid:0) B → K ω (cid:1) < . × − [67] andBR (cid:0) ω → π + π − π (cid:1) = 89%.2. We use B → Kφφ data of BaBar [68] to derive a con-straint on the a → φφ channel, which is shown by the orange region in Fig. 4. We assume the axion to beat the center of each bin (see Fig. 5 of Ref. [68]) ofwidth 125 MeV. Despite experimental smearing, thegaussian event distribution coming from the axion de-cay is expected to be completely inside one of thesebins. From the perspective of peak search, we alsorequire the signal from the axion to be less than thecentral value of the measurement augmented with 2 σ uncertainty.3. We analyze B → Ka ( → KKπ ) final state based onBabar measurements [69]. The channel is studied atLHCb using 3fb − data [73], but the sensitivity is cur-rently weaker compared to Babar. The bound is shownby the pink region in Fig. 4. To derive this bound, wefollow a similar strategy mentioned previously withone difference. The bin size for KKπ experimentaldata is only 22.5 MeV (see Fig. 1(e) of Ref. [69]).Hence, instead of assuming the axion mass to be at thecenter of each bin, we assume it to be at the bound-ary of adjacent bins. We then require the number ofevents from the decay of the axion to be less than thesum of central values of those two bins plus 2 σ un-certainty, after subtracting non-resonant backgroundfrom the measurement. The merging of two bins cor-rect for any spilling over effect due to experimentalsmearing. Further, experimental efficiency is calcu-lated based on binned data and final measurement ofthe branching fraction given on Fig. 1 (e) and TA-BLE I of [69] respectively. Finally, the data analysisperformed on KKπ measurement contains mass cut:one of the Kπ pair invariant mass is required to be0 .
85 GeV (cid:46) m Kπ (cid:46) .
95 GeV. To apply this cuton axion decay calculations we use a → KKπ ma-trix element given in Eq. (S59-S61) of [42]. However,the result strongly depends on the experimental inputparameters that have large uncertainties. Because ofthis uncertainties bound from this channel have or-der one error close to the end of the mass spectrum m a ∼ . a → ηππ [69] in the 1 . < m a < . KKπ exceptthe mass cut. For m a < . m a > . m a > . m a < . • Finally we derive Belle II projection for a → ηππ search, shown as the green dashed curve in Fig. 4. Toestimate projection we first extrapolate BaBar’s con-tinuous QCD background given on FIG.1 (f) of [69].Next, we scale it with luminosity, assuming 5 × ¯ BB pair at Belle II. Eventually, based on our result we cal-culate standard deviation and require that signal fromthe axion to be less than 2 times this standard devia-tion. We estimate the experimental width (smearing)of the axion as Γ a ∼ Γ η (cid:48) m a /m η (cid:48) where Γ η (cid:48) ∼ . η (cid:48) fitted from the Fig.1(f) of [69]. IV. CONCLUSION
In this letter, we performed the first 2-loop calculationfor the axion production from B → Ka process startingfrom the minimal interaction of the QCD axion, aG ˜ G (Eq.1). Assuming the UV scale to be at 1 TeV, theconstraints on the m a - f a parameter space (see Fig. 4)turns out to be ∼
10 times stronger than the previousestimate [42]. Increasing the UV scale only increasesthis difference. The reason for this enhancement is twofold. Firstly, in [42] the 2-loop amplitude was approx-imated from a 1-loop matching using an RGE induced att coupling, which does not reproduce the completelogarithmic behaviour. Our improved description pro-vides roughly a factor of five enhancement in the bound.Secondly, we perform a detailed bin by bin analysis in-stead of assuming an overall branching fraction. Thismakes our bound even more robust by roughly a factorof two. Therefore, the bounds on the decay constant isorder of 100 GeV using Belle and BaBar measurementsfor Λ UV = 1 TeV. For the future, although there aremany intensive studies for the heavy QCD axion basedon the (near) future data at kaon factories [46], GlueX[55], LHC with track-trigger [35, 74], DUNE near detec-tor [75], or beam-dump type facilities (summarized inFig.41 of [76]), the B → Ka process is particularly im-portant for GeV mass range of the axion. This is becausethe GeV axion is not produced at light meson precisionexperiments and also because the lifetime is shorter dueto the hadronic decay channels making the beam-dumpexperiments less effective. Belle II will be able to coverthe unique parameter space using B → Ka ( → ηππ ) asshown in Fig. 4, and we expect the other channels andfuture data of LHCb will further improve the sensitivi-ties. Also, B → Ka ( → γγ ) will be another attractivechannel particularly for m a < m π (cid:39)
450 MeV, which isnot yet studied in B-factories.
ACKNOWLEDGMENTS
We thank Mike Williams and Yotam Soreq for corre-spondence regarding Ref. [42]. We also thank Lina Alas-far, Fernando Febres Cordero, Andrei Gritsan, TaeHyunJung and Abner Soffer for discussions. This work is sup-ported by by the US Department of Energy grant DE-SC0010102.
Appendix A: Renormalization Scheme
We start from the EFT Lagrangian: L = L SM + L a + (cid:88) i C i O i + . . . , (A1)where L a denotes the axion kinetic and potential termsand the ellipses represent effective operators irrelevantfor the b → sa phenomenology of our interest, while i ∈{ gg, qq, bs } and O gg = 18 π af a G aµν ˜ G aµν , O qq = (cid:88) q ∂ µ af a ¯ qγ µ γ q , O bs = ∂ µ af a ¯ s L γ µ γ b L + h.c. . (A2)However, we will soon be redefining O qq and O bs below inorder to take into account the subtleties of dealing with γ in dimensional regularization (DR).To simplify our calculations, we will neglect terms oforder m b,s,a /M W or higher. This in particular meansthat we evaluate the diagrams in Fig. 1 at vanishingexternal momenta. The Feynman gauge has been usedthroughout our calculations and thus the inclusion of anunphysical Nambu-Goldstone mode accompanying every W boson is implied in the following discussions. We haveimplemented tensor reduction in FORM [77] and usedKIRA [78] to obtain integration-by-parts relations.We will regulate UV divergences using DR, while wecut off IR divergences explicitly by introducing fictitiousquark masses. Note that all diagrams in Fig. 1 as well asall coefficients in Eq. (7) are O ( α s α w ). At this order, thedependence on the fictitious masses actually cancels outas the IR theory (6) has no IR divergences even in the m b,s → γ and (cid:15) µνρσ . We first redefine O qq and O bs as O qq = i6 ∂ µ af a (cid:15) µνρσ (cid:88) q ¯ qγ ν γ ρ γ σ q , O bs = i6 ∂ µ af a (cid:15) µνρσ ¯ s L γ ν γ ρ γ σ b L + h.c. , (A3)which is equivalent to their original forms in d = 4 butwe use these new forms in d = 4 − (cid:15) because what wedirectly obtain from diagrams in Fig. 1 is actually theproduct of three γ matrices multiplied by the (cid:15) tensorfrom the a - g - g vertex. Therefore, all we need is the totalantisymmetric property of the (cid:15) tensor, which we assumeas part of the definition of our scheme, and the property { γ , γ µ } = 0, which is valid as we have no anomalouschiral fermion loops. We do not use any explicit form ofthe (cid:15) tensor nor any relation between γ and the (cid:15) tensor,until only after all divergences are cancelled and we areback to d = 4.We employ the M S scheme (with one exception men-tioned below) and redefine the Wilson coefficients as C i → (cid:88) j ( e γ E µ / π ) (cid:15)/ C j γ ji , O i → (cid:88) j Z ij O j , (A4)where Z consists of the field-strength renormalizations ofthe SM fields inside O i . For our 2-loop computation de-picted in Fig. 1, it only has three nontrivial components: Z = Z G Z q Z bs , (A5)where Z G and Z q are respectively the gluon and quarkfield-strength renormalizations due to 1-loop QCD cor-rections, while Z bs the renormalization of the b - s kineticmixing induced by a W loop. We use M S to determine Z G and Z q , while we fix Z bs by requiring that the net b - s kinetic mixing should vanish at 1-loop at vanishingquark momentum.All of these are determined completely by the SM andwe find Z G = 1 + α s π (cid:18) N c − N f (cid:19) (cid:15) ,Z q = 1 − α s π C F (cid:15) ,Z bs = − α w π (cid:88) k ξ k V kb V ∗ ks (cid:20) (cid:15) − ln M W µ + 3( ξ k + 1)2( ξ k − − ξ k (2 + ξ k )( ξ k − ln ξ k (cid:21) , (A6)where N c = 3 and N f = 6. Appendix B: Anomalous dimensions andrenormalization group evolutions
To obtain the anomalous dimensions matrix γ inEq. (A4), we calculate 1- and 2-loop diagrams contribut-ing to the a - g - g , a - q - q and a - b - s vertex corrections in M S . We find γ = − α s π β (cid:15) − α s π C F (cid:15) α s π α w π C F S (cid:0) (cid:15) − (cid:15) (cid:1) α w π S (cid:15) , (B1) where β = 11 N c / − N f / S = (cid:80) k ξ k V kb V ∗ ks .The RGEs can then be found by demanding µ d( µ (cid:15) C i γ ij ) / d µ = 0, i.e., µ d C i d µ = − (cid:15)C i − (cid:88) j,k C j µ d γ jk d µ ( γ − ) ki . (B2)In the limit (cid:15) →
0, we get µ d C gg d µ = − β α s π C gg ,µ d C qq d µ = − C F α s π C gg π ,µ d C bs d µ = (cid:18) α s π C gg π C F + C qq (cid:19) α w π (cid:88) k ξ k V kb V ∗ ks . (B3)Here, to see the size of each contribution, recall thatroughly C gg ∼ α s and C qq ∼ ( α s / π ) . We further sim-ply the RGE for C bs by neglecting the u and c quarkmasses. This then allows us to combine α w and ξ t as α w ξ t = y t / π . Therefore, we also incorporate the SMrunning of the top-quark Yukawa coupling: µ d y t d µ (cid:39) y t π (cid:18) y t − g (cid:19) . (B4)We also take into account the running of V ts . The leadingcontribution reads [79] µ d V ts d µ (cid:39) π y t V ts . (B5)Let us first verify our claim in the main text that therunning of C gg is completely accounted for by the SMrunning of α s . This can be trivially seen by solveingEq. (B3) with the initial condition C gg (Λ UV ) = α s (Λ UV ),which leads to C gg ( µ ) = α s ( µ ). Then, setting C gg ( µ ) = α s ( µ ) in the RGEs for C qq and C bs above, we obtain theresults in Eq. (3).After we run from Λ UV down to µ ∼ M W , we integrateout the W and t and match onto the EFT described bythe operator (6) with the coefficient (7), where we find f ( µ ) = 32 C F (cid:88) k V kb V ∗ ks (cid:20) − ξ k M W µ + (cid:26) ξ k (3 ξ k − ξ k + 4)2( ξ k − − ( ξ k − ξ k + 1) ξ k − ξ k − (cid:27) ln ξ k + 3 ξ k − ξ k − ξ k + 42( ξ k − ln ξ k + (cid:26) ξ k ( ξ k + 1)2( ξ k − − ξ k (3 ξ k − ξ k + 8)( ξ k − ln ξ k (cid:27) ln M W µ + π (4 + 11 ξ k − ξ k ) + 3 ξ k (13 ξ k − ξ k −
1) + ( ξ k − ξ k + 1) ξ k − (cid:18) ξ k (cid:19) − ( ξ k + 2)( ξ k + 2 ξ k − ξ k − Li (cid:18) ξ k − ξ k (cid:19)(cid:21) . (B6) g ( µ ) = 14 (cid:88) k V kb V ∗ ks ξ k (cid:20) ξ k + 5 ξ k − ξ k − ξ k + 4)( ξ k − ln ξ k + 2 ln M W µ (cid:21) . (B7)We verified that the difference between Eq. (B7) and the1-loop matching function found in Eq. (71) of Ref. [64]is due to different scheme choices of handling γ and theLevi-Cevita tensor. Appendix C: Input parameters
Parameters Values G F [59] 1 . × − GeV − α s ( M Z ) [59] 0 . ± . V tb [59] 0.9991 V ts [59] 0.0413 M W [59] 80.379 GeV M Z [59] 91.187 GeV m t ( m t ) [80] 163 . m b ( m b ) [59] 4 .
18 GeV m s ( m s ) [59] 92 . m B [59] 5.279 GeV m ± K [59] 493.6 MeV m K [59] 497.6 MeVTABLE I. Input parameters. In Table I we list the input parameters used in ouranalysis. [1] S. Weinberg, “A New Light Boson?,”
Phys. Rev. Lett. (1978) 223–226.[2] F. Wilczek, “Problem of Strong P and T Invariance inthe Presence of Instantons,”
Phys. Rev. Lett. (1978)279–282.[3] G. ’t Hooft, “Symmetry Breaking Through Bell-JackiwAnomalies,” Phys. Rev. Lett. (1976) 8–11.[4] R. Peccei and H. R. Quinn, “CP Conservation in thePresence of Instantons,” Phys. Rev. Lett. (1977)1440–1443.[5] R. Peccei and H. R. Quinn, “Constraints Imposed byCP Conservation in the Presence of Instantons,” Phys.Rev. D (1977) 1791–1797.[6] J. Preskill, M. B. Wise, and F. Wilczek, “Cosmology ofthe Invisible Axion,” Phys. Lett. B (1983) 127–132.[7] M. Dine and W. Fischler, “The Not So Harmless Axion,”
Phys. Lett. B (1983) 137–141.[8] L. F. Abbott and P. Sikivie, “A Cosmological Bound onthe Invisible Axion,”
Phys. Lett. B (1983) 133–136.[9]
CAST
Collaboration, V. Anastassopoulos et al. , “NewCAST Limit on the Axion-Photon Interaction,”
NaturePhys. (2017) 584–590, arXiv:1705.02290 [hep-ex] .[10] G. G. Raffelt, “Astrophysical axion bounds,” Lect.Notes Phys. (2008) 51–71, arXiv:hep-ph/0611350 .[11] G. Raffelt,
Stars as laboratories for fundamentalphysics: The astrophysics of neutrinos, axions, andother weakly interacting particles . 5, 1996.[12] A. Friedland, M. Giannotti, and M. Wise,“Constraining the Axion-Photon Coupling with MassiveStars,”
Phys. Rev. Lett. no. 6, (2013) 061101, arXiv:1210.1271 [hep-ph] .[13] J. Bjorken, S. Ecklund, W. Nelson, A. Abashian,
C. Church, B. Lu, L. Mo, T. Nunamaker, andP. Rassmann, “Search for Neutral MetastablePenetrating Particles Produced in the SLAC BeamDump,”
Phys. Rev. D (1988) 3375.[14] J. Blumlein et al. , “Limits on neutral light scalar andpseudoscalar particles in a proton beam dumpexperiment,” Z. Phys. C (1991) 341–350.[15] CHARM
Collaboration, F. Bergsma et al. , “Search forAxion Like Particle Production in 400- { GeV } Proton -Copper Interactions,”
Phys. Lett. B (1985)458–462.[16] J. E. Kim and G. Carosi, “Axions and the Strong CPProblem,”
Rev. Mod. Phys. (2010) 557–602, arXiv:0807.3125 [hep-ph] . [Erratum: Rev.Mod.Phys.91, 049902 (2019)].[17] CAST
Collaboration, E. Arik et al. , “Probing eV-scaleaxions with CAST,”
JCAP (2009) 008, arXiv:0810.4482 [hep-ex] .[18] ADMX
Collaboration, S. J. Asztalos et al. , “Designand performance of the ADMX SQUID-basedmicrowave receiver,”
Nucl. Instrum. Meth. A (2011) 39–44, arXiv:1105.4203 [physics.ins-det] .[19]
CAST
Collaboration, S. Aune et al. , “CAST search forsub-eV mass solar axions with 3He buffer gas,”
Phys.Rev. Lett. (2011) 261302, arXiv:1106.3919[hep-ex] .[20] S. Dimopoulos, “A Solution of the Strong { CP } Problem in Models With Scalars,”
Phys. Lett. B (1979) 435–439.[21] B. Holdom and M. E. Peskin, “Raising the AxionMass,” Nucl. Phys. B (1982) 397–412.[22] M. Dine and N. Seiberg, “String Theory and the Strong { CP } Problem,”
Nucl. Phys. B (1986) 109–124.[23] J. M. Flynn and L. Randall, “A Computation of theSmall Instanton Contribution to the Axion Potential,”
Nucl. Phys. B (1987) 731–739.[24] K. Choi, C. W. Kim, and W. K. Sze, “MassRenormalization by Instantons and the Strong { CP } Problem,”
Phys. Rev. Lett. (1988) 794.[25] V. A. Rubakov, “Grand unification and heavy axion,” JETP Lett. (1997) 621–624, arXiv:hep-ph/9703409 .[26] K. Choi and H. D. Kim, “Small instanton contributionto the axion potential in supersymmetric models,” Phys.Rev. D (1999) 072001, arXiv:hep-ph/9809286 .[27] Z. Berezhiani, L. Gianfagna, and M. Giannotti, “StrongCP problem and mirror world: The Weinberg-Wilczekaxion revisited,” Phys. Lett. B (2001) 286–296, arXiv:hep-ph/0009290 .[28] A. Hook, “Anomalous solutions to the strong CPproblem,”
Phys. Rev. Lett. no. 14, (2015) 141801, arXiv:1411.3325 [hep-ph] .[29] H. Fukuda, K. Harigaya, M. Ibe, and T. T. Yanagida,“Model of visible QCD axion,”
Phys. Rev. D no. 1,(2015) 015021, arXiv:1504.06084 [hep-ph] .[30] S. Dimopoulos, A. Hook, J. Huang, andG. Marques-Tavares, “A collider observable QCDaxion,” JHEP (2016) 052, arXiv:1606.03097[hep-ph] .[31] P. Agrawal and K. Howe, “A Flavorful Factoring of theStrong CP Problem,” JHEP (2018) 035, arXiv:1712.05803 [hep-ph] .[32] P. Agrawal and K. Howe, “Factoring the Strong CPProblem,” JHEP (2018) 029, arXiv:1710.04213[hep-ph] . [33] P. Agrawal, G. Marques-Tavares, and W. Xue,“Opening up the QCD axion window,” JHEP (2018)049, arXiv:1708.05008 [hep-ph] .[34] M. K. Gaillard, M. B. Gavela, R. Houtz, P. Quilez, andR. Del Rey, “Color unified dynamical axion,” Eur.Phys. J. C no. 11, (2018) 972, arXiv:1805.06465[hep-ph] .[35] A. Hook, S. Kumar, Z. Liu, and R. Sundrum, “HighQuality QCD Axion and the LHC,” Phys. Rev. Lett. no. 22, (2020) 221801, arXiv:1911.12364[hep-ph] .[36] T. Gherghetta, V. V. Khoze, A. Pomarol, andY. Shirman, “The Axion Mass from 5D SmallInstantons,”
JHEP (2020) 063, arXiv:2001.05610[hep-ph] .[37] A. Mariotti, D. Redigolo, F. Sala, and K. Tobioka, “NewLHC bound on low-mass diphoton resonances,” Phys.Lett. B (2018) 13–18, arXiv:1710.01743 [hep-ph] .[38] M. Kamionkowski and J. March-Russell, “Planck scalephysics and the Peccei-Quinn mechanism,”
Phys. Lett.B (1992) 137–141, arXiv:hep-th/9202003 .[39] R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb,R. Watkins, and L. M. Widrow, “Solutions to thestrong CP problem in a world with gravity,”
Phys. Lett.B (1992) 132–136, arXiv:hep-ph/9203206 .[40] S. M. Barr and D. Seckel, “Planck scale corrections toaxion models,”
Phys. Rev. D (1992) 539–549.[41] S. Ghigna, M. Lusignoli, and M. Roncadelli, “Instabilityof the invisible axion,” Phys. Lett. B (1992)278–281.[42] D. Aloni, Y. Soreq, and M. Williams, “CouplingQCD-Scale Axionlike Particles to Gluons,”
Phys. Rev.Lett. no. 3, (2019) 031803, arXiv:1811.03474[hep-ph] .[43] H. Georgi, D. B. Kaplan, and L. Randall, “Manifestingthe Invisible Axion at Low-energies,”
Phys. Lett. B (1986) 73–78.[44] W. A. Bardeen, R. D. Peccei, and T. Yanagida,“CONSTRAINTS ON VARIANT AXION MODELS,”
Nucl. Phys. B (1987) 401–428.[45] D. S. M. Alves and N. Weiner, “A viable QCD axion inthe MeV mass range,”
JHEP (2018) 092, arXiv:1710.03764 [hep-ph] .[46] S. Gori, G. Perez, and K. Tobioka, “KOTO vs. NA62Dark Scalar Searches,” JHEP (2020) 110, arXiv:2005.05170 [hep-ph] .[47] E949
Collaboration, A. V. Artamonov et al. , “Searchfor the decay K+ to pi+ gamma gamma in the pi+momentum region P >
213 MeV/c,”
Phys. Lett. B (2005) 192–199, arXiv:hep-ex/0505069 .[48]
NA62
Collaboration, C. Lazzeroni et al. , “Study of the K ± → π ± γγ decay by the NA62 experiment,” Phys.Lett. B (2014) 65–74, arXiv:1402.4334 [hep-ex] .[49]
KTeV
Collaboration, E. Abouzaid et al. , “FinalResults from the KTeV Experiment on the Decay K L → π γγ ,” Phys. Rev. D (2008) 112004, arXiv:0805.0031 [hep-ex] .[50] KOTO
Collaboration, J. K. Ahn et al. , “Search for the K L → π νν and K L → π X decays at the J-PARCKOTO experiment,” Phys. Rev. Lett. no. 2, (2019)021802, arXiv:1810.09655 [hep-ex] .[51]
PIENU
Collaboration, A. Aguilar-Arevalo et al. ,“Search for heavy neutrinos in π → µν decay,” Phys.Lett. B (2019) 134980, arXiv:1904.03269 [hep-ex] .[52] D. Pocanic et al. , “Precise measurement of the pi+ → pi0 e+ nu branching ratio,” Phys. Rev. Lett. (2004)181803, arXiv:hep-ex/0312030 .[53] W. Altmannshofer, S. Gori, and D. J. Robinson,“Constraining axionlike particles from rare piondecays,” Phys. Rev. D no. 7, (2020) 075002, arXiv:1909.00005 [hep-ph] .[54]
GlueX
Collaboration, H. Al Ghoul et al. ,“Measurement of the beam asymmetry Σ for π and η photoproduction on the proton at E γ = 9 GeV,” Phys.Rev. C no. 4, (2017) 042201, arXiv:1701.08123[nucl-ex] .[55] D. Aloni, C. Fanelli, Y. Soreq, and M. Williams,“Photoproduction of Axionlike Particles,” Phys. Rev.Lett. no. 7, (2019) 071801, arXiv:1903.03586[hep-ph] .[56]
OPAL
Collaboration, G. Abbiendi et al. , “Multiphotonproduction in e+ e- collisions at s**(1/2) = 181-GeV to209-GeV,”
Eur. Phys. J. C (2003) 331–344, arXiv:hep-ex/0210016 .[57] S. Knapen, T. Lin, H. K. Lou, and T. Melia, “Searchingfor Axionlike Particles with Ultraperipheral Heavy-IonCollisions,” Phys. Rev. Lett. no. 17, (2017) 171801, arXiv:1607.06083 [hep-ph] .[58]
CMS
Collaboration, A. M. Sirunyan et al. , “Search forlow mass vector resonances decaying intoquark-antiquark pairs in proton-proton collisions at √ s = 13 TeV,” JHEP (2018) 097, arXiv:1710.00159 [hep-ex] .[59] Particle Data Group
Collaboration, M. Tanabashi et al. , “Review of Particle Physics,”
Phys. Rev. D no. 3, (2018) 030001.[60] X. Cid Vidal, A. Mariotti, D. Redigolo, F. Sala, andK. Tobioka, “New Axion Searches at Flavor Factories,” JHEP (2019) 113, arXiv:1810.09452 [hep-ph] .[Erratum: JHEP 06, 141 (2020)].[61] X. Cid Vidal et al. , “Report from Working Group 3:Beyond the Standard Model physics at the HL-LHCand HE-LHC,” CERN Yellow Rep. Monogr. (2019)585–865, arXiv:1812.07831 [hep-ph] .[62] E. Izaguirre, T. Lin, and B. Shuve, “Searching forAxionlike Particles in Flavor-Changing Neutral CurrentProcesses,” Phys. Rev. Lett. no. 11, (2017) 111802, arXiv:1611.09355 [hep-ph] .[63] M. B. Gavela, R. Houtz, P. Quilez, R. Del Rey, andO. Sumensari, “Flavor constraints on electroweak ALPcouplings,”
Eur. Phys. J. C no. 5, (2019) 369, arXiv:1901.02031 [hep-ph] .[64] M. Bauer, M. Neubert, S. Renner, M. Schnubel, andA. Thamm, “The Low-Energy Effective Theory ofAxions and ALPs,” arXiv:2012.12272 [hep-ph] .[65] M. Bauer, M. Neubert, and A. Thamm, “ColliderProbes of Axion-Like Particles,” JHEP (2017) 044, arXiv:1708.00443 [hep-ph] . [66] Particle Data Group
Collaboration, P. Zyla et al. ,“Review of Particle Physics,”
PTEP no. 8, (2020)083C01.[67]
Belle
Collaboration, V. Chobanova et al. ,“Measurement of branching fractions and CP violationparameters in B → ωK decays with first evidence of CPviolation in B → ωK S ,” Phys. Rev. D no. 1, (2014)012002, arXiv:1311.6666 [hep-ex] .[68] BaBar
Collaboration, J. Lees et al. , “Measurements ofbranching fractions and CP asymmetries and studies ofangular distributions for B → phi phi K decays,” Phys.Rev. D (2011) 012001, arXiv:1105.5159 [hep-ex] .[69] BaBar
Collaboration, B. Aubert et al. , “Study of BMeson Decays with Excited eta and eta-prime Mesons,”
Phys. Rev. Lett. (2008) 091801, arXiv:0804.0411[hep-ex] .[70] B. Batell, M. Pospelov, and A. Ritz, “Multi-leptonSignatures of a Hidden Sector in Rare B Decays,”
Phys.Rev. D (2011) 054005, arXiv:0911.4938 [hep-ph] .[71] P. Ball and R. Zwicky, “New results on B → π, K, η decay formfactors from light-cone sum rules,” Phys.Rev. D (2005) 014015, arXiv:hep-ph/0406232 .[72] P. Ball and R. Zwicky, “ B d,s → ρ, ω, K ∗ , φ decayform-factors from light-cone sum rules revisited,” Phys.Rev. D (2005) 014029, arXiv:hep-ph/0412079 .[73] LHCb
Collaboration, R. Aaij et al. , “Study of B + c decays to the K + K − π + final state and evidence for thedecay B + c → χ c π + ,” Phys. Rev. D no. 9, (2016)091102, arXiv:1607.06134 [hep-ex] .[74] Y. Gershtein, S. Knapen, and D. Redigolo, “Probingnaturally light singlets with a displaced vertex trigger,” arXiv:2012.07864 [hep-ph] .[75] K. J. Kelly, S. Kumar, and Z. Liu, “Heavy AxionOpportunities at the DUNE Near Detector,” arXiv:2011.05995 [hep-ph] .[76] J. Beacham et al. , “Physics Beyond Colliders at CERN:Beyond the Standard Model Working Group Report,” J. Phys. G no. 1, (2020) 010501, arXiv:1901.09966[hep-ex] .[77] J. Vermaseren, “New features of FORM,” arXiv:math-ph/0010025 .[78] P. Maierhofer, J. Usovitsch, and P. Uwer, “Kira—AFeynman integral reduction program,” Comput. Phys.Commun. (2018) 99–112, arXiv:1705.05610[hep-ph] .[79] C. Balzereit, T. Mannel, and B. Plumper, “TheRenormalization group evolution of the CKM matrix,”
Eur. Phys. J. C (1999) 197–211, arXiv:hep-ph/9810350 .[80] P. Marquard, A. V. Smirnov, V. A. Smirnov, andM. Steinhauser, “Quark Mass Relations to Four-LoopOrder in Perturbative QCD,” Phys. Rev. Lett. no. 14, (2015) 142002, arXiv:1502.01030 [hep-ph]arXiv:1502.01030 [hep-ph]