Challenges for a QCD Axion at the 10 MeV Scale
EEFI-21-2
Challenges for a QCD Axion at the 10 MeV Scale
Jia Liu,
1, 2, a
Navin McGinnis,
3, 4, b
Carlos E.M. Wagner,
4, 5, 6, c and Xiao-Ping Wang
7, 8, d1
School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China TRIUMF, 4004 Westbrook Mall, Vancouver, BC, Canada, V6T 2A3 High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Physics Department and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA School of Physics, Beihang University, Beijing 100083, China Beijing Key Laboratory of Advanced Nuclear Materials and Physics,Beihang University, Beijing 100191, China
Abstract
We report on an interesting realization of the QCD axion, with mass in the range O (10) MeV.It has previously been shown that although this scenario is stringently constrained from multiplesources, the model remains viable for a range of parameters that leads to an explanation of theAtomki experiment anomaly. In this article we study in more detail the additional constraintsproceeding from recent low energy experiments and study the compatibility of the allowed param-eter space with the one leading to consistency of the most recent measurements of the electronanomalous magnetic moment and the fine structure constant. We further provide an ultravioletcompletion of this axion variant and show the conditions under which it may lead to the observedquark masses and CKM mixing angles, and remain consistent with experimental constraints onthe extended scalar sector appearing in this Standard Model extension. In particular, the decayof the Standard Model-like Higgs boson into two light axions may be relevant and leads to a novelHiggs boson signature that may be searched for at the LHC in the near future. a [email protected] b [email protected] c [email protected] d [email protected] a r X i v : . [ h e p - ph ] F e b ONTENTS
I. Introduction 2II. Effective Low Energy Model 5III. ( g − e I. INTRODUCTION
The Standard Model (SM) provides a consistent description of particle interactions, basedon a local and renormalizable gauge theory [1]. The gauge interactions of the theory havebeen tested with great precision, while the Yukawa interactions of quarks and leptons withthe recently discovered Higgs boson are being currently tested at the Large Hadron collider(LHC). In the SM, these Yukawa interactions lead to a rich flavor structure based on thediversity of quark and lepton masses and the CKM and PMNS matrices in the quark andlepton sector, respectively. These matrices include CP violating phases that are sufficientto explain all CP-violating phenomena observed in B and K meson, as well as neutrinoexperiments.Although the gluon interactions with quarks preserve the CP symmetry, there is a CPviolating interaction in the SM QCD sector that is related to the interactions induced bythe θ QCD parameter. Even in the absence of a tree-level value, a non-vanishing value of θ QCD would be generated via the chiral anomaly, by the chiral phase redefinition associated with2aking the quark masses real. In general, hence, the physical value of θ QCD comes from theaddition of the tree-level value and the phase of the determinant of the quark masses, andis naturally of order one. In the presence of an unsuppressed value of θ QCD , however, theneutron electric dipole moment (EDM) would acquire an unobserved sizable value [2]. Theexperiment measuring neutron EDM has found that such CP phase should be smaller than10 − [3], which leads to a severe fine tuning problem in the SM [1].One elegant way to solve the neutron EDM problem is to introduce a chiral symmetry,the so-called PQ symmetry, which is spontaneously broken, leading to a Goldstone boson, a , which has been called the axion [4–7]. The axion interactions lead to a field dependent θ QCD angle, and the strong interactions induce a non-trivial potential which is minimized atvanishing values of θ QCD , therefore leading to the disappearance of the CP interactions inthe strong sector in the physical vacuum and the solution of the strong CP problem.Being associated with a Goldstone boson, the axion field has an approximate shift symme-try, thus it can be naturally light. Its couplings to other particles may be rendered small dueto the suppression from large decay constant, f a , proportional to the vacuum expectationvalue of the field that breaks the chiral PQ symmetry. Since the f a scale is not necessarilyrelated to the weak or QCD scales, the axion mass and couplings to fermions and gaugebosons, which are proportional to 1 /f a , can span many orders of magnitude. Thus, thereis a rich phenomenology to explore this wide region of parameter space with different ex-perimental setups [1, 8–10]. The original visible axion scenario [6, 7], where f a is related tothe weak scale, has been severely constrained by laboratory experiments [1, 10]. There arealso invisible axion scenarios, with f a much larger than electroweak scale, where the axionremains hidden due to its ultralight mass and ultra small couplings. A realization of theseinvisible axions are given in the so-called KSVZ and DFSZ axion scenarios [11–14].Regarding the visible axion, there are efforts [15–17] to save it from experimental con-straints by making it pion-phobic . In this scenario, the mixing with the neutral pion issuppressed by choosing appropriate PQ charges and an accidental cancellation arising fromthe low energy quark mass and coupling parameters. This visible axion scenario has beencombined with the recent experimental anomalies from Atomki collaboration [18, 19]. Theexperiment looks for high excitation states of nuclei from Be and He, which de-excite intoground states and emit a pair of electron and positron. A peak was observed in the angularcorrelation of e − e + , which can be interpreted as an intermediate pseudo-scalar particle with3ass around 17 MeV produced in the de-excitation and later decay into e − e + [16, 17, 20–22].Recently, it has been suggested that higher order SM effects may lead to an effect similar tothe one observed by the Atomki experiment [23]. However, further theoretical and experi-mental analyses are needed to determine whether or not the observed peak may indeed beaccounted by SM effects. In this work we shall assume that this is not the case and identifythe associated pseudo-scalar particle with a 17 MeV QCD axion. We will, however, commenton how the constraints on the proposed QCD axion at the 10 MeV scale would be modifiedif its resonant signal were absent. In general, since m a (cid:39) m π f π /f a , an axion mass in the10 MeV range implies f a (cid:39) f a [16, 17].The recent accurate measurement of the fine structure constant [24] revealed that thereis a mild discrepancy with the electron magnetic dipole moment measurements [25, 26],compared to the SM theoretical calculation [27, 28]. The discrepancy, ∆ a e , associated withthis measurement used to be negative and had a significance of about 2 . σ . On the otherhand, a very recent, more accurate measurement of the fine structure constant [29] suggests asmaller discrepancy for ∆ a e and with positive sign. Since the axion has to decay to electron-positron pairs promptly in the Atomki experiment, it has a sizable coupling to electron. Asa pseudo-scalar, it can naturally give a negative contribution to electron ( g − e at the one-loop level [30]. Although positive correction would be therefore in tension with the one-loopcorrections induced by the 17 MeV axion particle, there are relevant two loop correctionsthat, depending on the axion- η meson mixing, may lead to a positive value of ∆ a e .In addition to experimental constraints from precision measurement of mesons, this sce-nario also is constrained by low energy electron collider and beam dump experiments. There-fore, it is interesting to ask if this 17 MeV visible axion can evade all current experimentalconstraints and also be consistent with the current measurements of ( g − e and the finestructure constant. It is also important to find a field theoretical realization of this axion.The most natural realizations are associated with new physics at the weak scale and aretherefore subject to further constraints beyond those associated with the effective low en-ergy axion model. In this article, we will present an extension of the SM which leads to a10 MeV axion and study the associated constraints on this model.This article is organized as follows. In section II, we review the properties of this visible4xion model and how the most relevant experimental constraints may be avoided under the pion-phobic limit. In section III, we consider the electron and photon couplings to the axionand show the region of parameters which can lead to a consistency between the measuredvalues of ( g − e and α , what leads to a relation between the axion electron and photoncouplings. In section IV, we further show how it can survive the constraints coming fromlow energy electron colliders and beam dump experiments, while at the same time leadto an explanation of the Atomki experimental constraints. Furthermore, in section V, weconstruct a concrete UV model and show how this scenario can couple to only the firstgeneration fermions while successfully generating the CKM matrix. We also discuss theadditional constraints implied by the new physics associated with this scenario. We reservesection VI for our conclusions. II. EFFECTIVE LOW ENERGY MODEL
We consider the effective theory for a variant of the QCD axion which couples mostly tothe first generation fermions in the SM. While the low-energy structure of this setup hasappeared some time ago [15, 31], it has recently been discussed how this scenario remainsrobust to a variety of constraints [16, 17]. From the ultraviolet (UV) perspective, for theaxion to couple only to u and d quarks in the infrared (IR), the PQ breaking mechanismmust allow for the generation of the appropriate quark and lepton masses as well as theCKM matrix. We discuss how this is achieved in a particular UV model later in section V.From the IR perspective, the relevant low-energy interactions are simply written as L int = (cid:88) f = e,u,d, m f e iQ f a/f a ¯ f L f R + h.c., (1)where Q f denotes the PQ charge of the fermion f , and f a is the axion decay constant,which, as emphasized in the introduction must be f a (cid:39) (cid:88) f g fa ia ¯ f γ f, g fa = Q f m f f a . (2)At energies below the QCD scale, the couplings of the axion to u and d quarks will induce5ixing of the axion in particular to the pion as well as other pseudo-scalar mesons. In thisregime, the physical axion state and meson mixing angles can be determined by standardchiral perturbation theory ( χ PT) techniques. For detailed explanations and results of thisprocedure we refer to Refs. [16, 17]. The same mixing angles can be used to parameterizethe effective operator controlling the axion coupling to photons g γγa = α πf π (cid:32) θ aπ + 53 θ aη ud + √ θ aη s (cid:33) , (3)where θ aπ is the mixing angle between the axion and neutral pion, and θ aη ud,s are the mix-ing angles with the eta-mesons in the light-heavy basis. The relationships between theseparameters and other bases for the meson octet in leading order of χ PT are given also inRef. [16]. For the purpose of this work, it is important to note that for Q u , Q d (cid:39) O (1), thenatural ranges for θ aη ud and θ aη s are in the range of O (10 − − − ) [16].Searches for axions in the decay products of pions presents the most stringent constraintsfor the setup we consider. In particular, the bound on the charged pion decay BR ( π + → e + ν e ( a → e − e + )) < . × − is quite severe [32]. For m a = O (10) MeV, this translates toa bound on the mixing angle as | θ aπ | (cid:46) × − (cid:112) BR( a → e + e − ) . (4)As we will see the range of couplings needed for ( g − e will require the axion width intophotons to be subdominant with respect to the one into electrons. Hence, if decays only toSM particles are considered, one expects BR( a → e + e − ) (cid:39)
1. Thus, this variant of the axionis pion-phobic [16, 17], in the sense that the mixing of the axion with the pion is below itsnatural values. Let us emphasize that, in leading order of perturbation theory m a (cid:39) ( Q u + Q d ) m u m d ( m u + m d ) m π f π f a (5)and θ aπ (cid:39) ( Q d m d − Q u m u )( m u + m d ) f π f a . (6)Since the ratio m u /m d (cid:39) . ± .
06 [1], a cancellation of θ aπ (cid:46) − demands Q u (cid:39) Q d .Even assuming this relation between the quark PQ charges, the value of the mixing θ aπ becomes naturally of the order of a few times 10 − and hence an accidental cancellation6etween the leading order contribution associated with the precise value of the up and downquark masses and the higher order contributions must occur for this scenario to be viable.One could in principle alleviate this aspect of the model assuming that the axion widthis dominated by invisible decays, allowing the reduction of the axion decay branching ratiointo electrons. This possibility, however, is strongly constrained by charged Kaon decaysconstraints, in particular the decay K + → ( π + + Missing Energy). Indeed, considering aneutral pion state with a relevant mixing with an axion with dominant invisible decays leadsto a bound on the axion-pion mixing θ aπ (cid:46) − , and therefore this scenario does not leadto a relaxation of the pion axion mixing bound.One interesting aspect of a mixing angle θ aπ below but close to 10 − is that it leads to apossible resolution of the so-called KTeV anomaly, related to a decay branching ratio of theneutral pion into pairs of electrons larger than the one expected in the SM [16, 17]. For thisreason, for the rest of this article, we will fix the pion mixing to values close to the upperbound, θ aπ = 0 . × − . Apart from the KTeV anomaly, however, the phenomenologicalproperties of this model do not change by setting this mixing angle to any other valueconsistent with the Kaon decay bounds. III. ( g − e ( a ) ( b ) FIG. 1. 1- and 2-loop contributions to ( g − e from a pseudo-scalar with couplings to the firstgeneration fermions. The effective vertex is generated at low energies through the meson mixingangles, Eq. (3). A pseudo-scalar particle with a mass m a (cid:39)
17 MeV and a coupling to electrons of order m e /f a (cid:39) × − for f a (cid:39) a e = ( g − e /
2. The measured deviation of the electron magnetic dipole7oment [25, 26] from the SM prediction [27, 28] extracted from the fine structure constantmeasured in recent cesium recoil experiments [24] is∆ a e ≡ a exp e − a SM e = ( − ± × − . (7)However, more recently the deviation inferred from a new, independent measurement ofthe fine-structure constant has been reported as [29]∆ a (cid:48) e = (48 ± × − . (8)In the present case, the largest contribution of the model to ∆ a e comes from the tree-levelcoupling of the axion to electrons, induced by the one-loop diagram in Fig. 1∆ a e = − π ( g ea ) (cid:90) dx (1 − x ) (1 − x ) + x ( m a /m e ) . (9)At the 2-loop level, however, the magnetic moment receives an additional contributionfrom a Barr-Zee type diagram proportional to the product g ea g γγa , where the effective couplingto photons is defined by Eq. (3). This contribution is given by∆ a e = g ea g γγa m e π f P S (cid:20) πf π m a (cid:21) , (10)where f P S [ z ] = (cid:90) dx z / x (1 − x ) − z log (cid:18) x (1 − x ) z (cid:19) , (11)and we have taken an effective cutoff at the scale 4 πf π as in [16].It is clear that when g ea and g γγa have the same sign, the 2-loop contribution can partiallycancel the 1-loop result. Thus, both couplings will have a relevant impact to the predictionof ∆ a e . In Fig. 2, we show the total contribution to ∆ a e for m a = 17 MeV and f a = 1 GeVwith respect to the electron and photon effective couplings. The cyan region shows therange of couplings predicting ∆ a e at the 95% CL according to the values reported in Eq. (7),whereas the blue shaded region shows the region of parameters which can explain the newmeasurement of the deviation of the electron anomalous magnetic moment measurement,Eq. (8).Relevant constraints on the values of the electron coupling are extracted from the decaysof the axion to electron-positron pairs. As these constraints are placed on the couplings to8 . × - × - × - × - × - × - × - × - × - g a γγ [ MeV - ] g a e m a =
17 MeV
KLOE NA64
FIG. 2. Parameter space for the 17 MeV axion which can accommodate ( g − e . The purpleand red shaded regions are excluded by KLOE and NA64 respectively [30, 33]. The cyan region isconsistent at the 95% C.L. with the deviation ∆ a e obtained from Eq (7). The blue region showsthe 95% C.L. region consistent with the value inferred from the recent measurement of the finestructure constant, reported in [29]. electrons, the constraints will remain the same regardless of which measurement of ∆ a e isassumed. Thus, in Fig. 2 we show an upper limit on g e from the KLOE experiment in thepurple shaded region [30]. A lower limit can be derived from various beam dump experi-ments. We show the strongest available bound, from NA64, in the red shaded region [33]. Itis important to note that both bounds have been derived assuming BR( a → e + e − ) = 1. Wesee that without the 2-loop contribution (or g γγa →
0) the values of ∆ a e would be negativeand therefore the new determination of the fine structure constant demands sizable twoloop corrections for the axion contributions to be consistent with ∆ a e . Upper bounds onthe couplings of the axion to vector bosons from LEP have also been explored in various re-alizations of axions and ALPS [34–36]. However, those bounds are based on the assumptionof an axion which mostly decay into photons, contrary to our model in which the dominant9xion decay is into electron pairs. Although searches for an axion decaying to electron pairsare difficult at LEP due to large backgrounds, this may be feasible at future runs of theLHC [37].Let us stress that the values of the axion coupling to electron leads to a constraint onthe ratio of the PQ charges of the first generation quark and leptons. For instance, since inthis scenario, Q u ∼ Q d , from the expression of the axion mass one obtains that f a Q d ∼ . (12)Therefore, the coupling of the axion to the electrons, Eq. (2), is given by g ea = Q e Q d m e GeV . (13)Taking into account the range of values consistent with ( g − e , which from Fig. 2 arebetween 1 . × − and 1 × − , one obtains Q e Q d = 0 . − . (14)In particular, if the above ratio of PQ charges takes the values 1/3, 1/2 or 1, the axionis free of experimental constraints from the KLOE and NA64 experiments. This range ofvalues of the ratio of PQ charges will play an important role in the UV model described insection V. IV. ATOMKI ANOMALY
Recently, it has been shown that the variant of the QCD axion analyzed in this work, withcouplings only to the first generation of quarks and leptons, can lead to an explanation of theanomalies reported by the Atomki experiment [17]. The relevant axion-nuclear transitionrates were also considered much earlier in [38–41] with further studies aimed at genericpseudo-scalar couplings to nucleons more recently in [20, 42]. In terms of the mixing anglesof the axion to mesons, the predicted axion emission rate for Be ∗ (18 .
15) nuclear transitionscan be parameterized as − ( θ aη ud (∆ u + ∆ d ) + √ θ aη s ∆ s ) (cid:12)(cid:12)(cid:12) Be ∗ (18 . (cid:39) (1 . − . × − , (15)10here ∆ q parametrizes the proton spin contribution from a quark, q , while for He ∗ (21 . − ( θ aη ud (∆ u + ∆ d ) + √ θ aη s ∆ s ) (cid:12)(cid:12)(cid:12) He ∗ (21 . (cid:39) (0 . − . × − , (16)where the R.H.S of each equation gives the range of possible values explaining the Atomkianomalies. For the aim of the computations, we have chosen ∆ u + ∆ d = 0 .
52 and ∆ s = − . θ aη ud and θ aη s , which are determined by the Atomkianomaly, effectively fix the coupling of the axion to photons. We see in Fig. 2 that for a givenvalue of g γγa , the combined one- and two-loop contributions to ( g − e sets a value for g e which leads to the correct prediction for ∆ a e . Thus, combining the regions of mixing angleswhich favor the Atomki anomalies with the requirement to satisfy ( g − e parametricallyconnects the PQ charges of the u and d quarks to that of the electron, giving complementaryinformation on the viable region of the scenario.If SM effects are confirmed to be dominant feature of the Atomki anomaly, as in [23], thiscould imply further constraints on the mixing angles θ aη ud and θ aη s or more specific relationsbetween them. For instance, in order to suppress the resonant signal, one possibility wouldbe that those mixing angles are much smaller than their natural values, leading to a smallphoton axion coupling. Beyond the fact that such small mixing values are difficult to realize,a small photon axion coupling would be inconsistent with the values of ∆ a e associated withthe new fine structure constant determination, Eq. (8), but would remain consistent withthe old one, Eq. (7). Alternatively, there could be an approximate cancellation between the θ aη ud and θ aη s contributions to Eqs. (15) and (16), what would imply a mixing θ aη s about anorder of magnitude larger than θ aη ud .In Fig. 3 we show the regions of parameters given in [17] favored by the nuclear transitionanomalies. We include the correlation between the predicted value of g γγa in this plane withthe values of g e which are consistent with ∆ a (cid:48) e , Eq. (8), at the 95% CL. Thus, the red andpurple shaded regions are excluded by the NA64 and KLOE experiments, respectively, asshown in Fig. 2. Given the tension between the previous measurements, we also show thebound inferred from the central value of ∆ a e , Eq. (7), by the red dashed lines. The lightgray region is excluded by bounds on the K + → π + ( a → e + e − ) branching ratio in the non11 . × - × - × - × - × - × - | θ a η s | | θ a η ud | sgn ( θ a η ud ) = sgn ( θ a η s ) , m a =
17 MeV
NA64KLOE K + →π + ( a → e + e - )( not octet - enhanced ) × - × - × - × - × - × - | θ a η s | | θ a η ud | θ a η ud < θ a η s > m a =
17 MeV
NA64 K + →π + ( a → e + e - )( not octet - enhanced ) FIG. 3. Mutual parameter space for the ( g − e and Atomki anomalies. Left:
The light anddark yellow bands indicate the parameter space for the 17 MeV axion which can accommodate the( g − e and Atomki anomalies for Be and He, respectively, when sgn( θ aη ud ) = sgn( θ aη s ). Weshow exclusion regions from NA64 and KLOE in red and purple respectively, as in Fig. 2, assumingthe correlation of electron couplings consistent with ∆ a (cid:48) e at the 95% C.L.. We also show the boundinferred from the central value of ∆ a e with the red dashed line. Light gray regions are excludedfrom kaon decays in the non octet-enhanced enhanced regime. Gray lines indicate the region thatwould be excluded in the octet-enhanced regime. Right:
Parameter space for the 17 MeV axionwhich can accommodate the ( g − e and Atomki anomalies when θ aη ud < θ aη s >
0. All colorcoding follows that from the left panel. In both panels, the light (dark) blue shaded contours showregions where g γγa is consistent with ∆ a (cid:48) e at the 95% C.L. for Q e /Q d = 1 / / Q e /Q d = 1 / octet-enhanced regime, see [17]. The regions that would be excluded in the octet enhancedregime, which are therefore not firm bounds, are indicated by gray lines. In the left panel,we show the predictions and bounds for the case when sgn( θ aη ud ) = sgn( θ aη s ), while in theright panel we show the corresponding results when θ aη ud < θ aη s > It would be interesting to compare the allowed values of the axion- η mixing parameters When θ aη ud > θ aη s < g − e and Atomkianomalies. − and a few 10 − , and hence, they are consistent with most of the allowedparameter space shown in Fig. 3. Between the yellow regions the contributions from θ aη ud and θ aη s cancel in Eqs. (15) and (16) leading to a suppression of the axion signal in the Be ∗ (18 .
15) and He ∗ (21 .
01) transitions.The light and dark blue bands in Fig. 3 also show the regions of parameter space consistentwith a ratio of PQ charges Q e /Q d = 1 / Q e /Q d = 1 / θ aη s that is much larger than its natural values. V. UV MODELA. The generation of CKM matrix and the couplings to first generation fermions
In this section, we present a possible UV completion of the effective model presentedin Eq. (1), where the axion couples exclusively to the first generation fermions in the SM.Generating the effective interactions to the up- and down-quarks is non-trivial as the PQbreaking mechanism must be carefully intertwined with electroweak symmetry breaking ina way that reproduces the correct flavor structure in the CKM matrix. To this end, weconsider an extension of the SM by three additional Higgs doublets and three singlets Higgsfields. The PQ symmetry is realized by assigning charges to the additional Higgs bosons aswell as the right-handed SM fermions u R , d R , and e R . The particle content and their PQcharges of the model is summarized in Table I.The relevant Yukawa interactions allowed by the PQ symmetry of the Higgs doublets, H f , and those of the SM Higgs, H , are given by L YukPQ ⊃ − (cid:88) i =1 , , (cid:0) ¯ Q i Y i u H u u R + ¯ Q i Y i d H d d R + ¯ L i Y i e H e e R (cid:1) + h.c. (17) L YukSM ⊃ − (cid:88) i =1 , , (cid:88) j =2 , (cid:16) ¯ Q i Y iju ˜ Hu jR + ¯ Q i Y ijd Hd jR + ¯ L i Y ije He jR (cid:17) + h.c., (18)where L YukPQ gives the couplings of H u,d,e to the first generation fermions, while L YukSM gives thecouplings of the second and third generation fermions to the SM Higgs. Below the scales of13 articles
H H u H d,e u R d R e R φ f SU (2) L U (1) Y −
12 12 23 − − U (1) PQ − Q u − Q d,e Q u Q d Q e − Q f TABLE I. SU (2) L × U (1) Y × U (1) PQ charges of the SM Higgs, additional Higgs doublets H f , right-handed fermions, and Higgs singlets φ f , where U (1) PQ is a global Peccei-Quinn-like symmetry and f = u, d, e . The PQ charges are Q u = 2, Q d = 1 and Q e = 1 /n , with n = 2 or 3. All other particlesare considered to be PQ singlets. PQ and electroweak symmetry breaking the up-quark mass matrix is then given by M u ≡ √ Y u v u Y u v Y u vY u v u Y u v Y u vY u v u Y u v Y u v = V u L m u m c
00 0 m t V † u R (19)where v and v u are the vacuum expectation values of H and H u respectively. The V uL and V uR matrices given the rotation between the flavor- and mass-eigenstate basis (denoted withsuperscript m ), (cid:126)u L = V uL (cid:126)u mL , (cid:126)u R = V uR (cid:126)u mR , (20)where (cid:126)u L/R = ( u L/R , c
L/R , t
L/R ) T . For simplicity, we work in a basis where V u R is the identitymatrix. The diagonalization of the down-type quarks follows similarly.After diagonalization of both up- and down-type quark sectors, the CKM matrix is givenby V CKM = V † u L V d L . (21)Since H u is charged under the PQ symmetry while H is not, we expect the axion field to becontained in H u but not H . Therefore, to connect to the effective model, this requires that H u couples only to the first generation up-quark in the mass eigenstate basis,¯ Q i Y i u H u u R = Q mi (cid:0) V † u L (cid:1) ij Y j u H u u ,mR = √ m u v u Q m H u u ,mR , (22)which is equivalent to (cid:0) V † u L (cid:1) ij Y j u = √ m u v u (1 , , T . (23)14aking Y j = ( Y u , Y u , Y u ) T , the unitary matrix V u L can be decomposed in terms of thenormalized vector ( Y j ) (cid:107) , and two perpendicular normalized vectors ( Y j ) ⊥ and ( Y j ) ⊥ V † u L = c x s x − s x c x ( Y j u ) T (cid:107) ( Y j u ) T ⊥ ( Y j u ) T ⊥ , (24)where T represents the transposed column vector, and x is an auxiliary rotation angle whichdoes not affect the equality Eq. (23). Similarly for V d L we obtain V † d L = c y s y − s y c y ( Y j d ) T (cid:107) ( Y j d ) T ⊥ ( Y j d ) T ⊥ . . (25)We choose the auxiliary angles explicitly as x = θ , y = 0 , (26)and for the Yukawas we take ( Y j u ) T (cid:107) ( Y j u ) T ⊥ ( Y j u ) T ⊥ = c s − s c , ( Y j d ) T (cid:107) ( Y j d ) T ⊥ ( Y j d ) T ⊥ = c − s s c
00 0 1 , (27)where θ , θ , θ angles are the CKM mixing angles in the SM. Then, the explicit forms ofthe vectors Y j u and Y j d are given by, Y j u = √ m u v u c s , Y j d = √ m d v d c − s . (28)The rest of the Yukawa matrix Y u and Y d can be reconstructed according to Eq. (19). Thiscompletes the realization of the CKM matrix in the UV model. We omit details of theprocedure of the mixing in the lepton sector. However, this may be accomplished in ananalogous way.Besides the Yukawa interactions, the scalar potential needs to be studied to single outthe physical axion field in the model as well as its mixing with other pseudo-scalars. The15ff-diagonal scalar potential, V PQ , and diagonal scalar potential, V dia , allowed by symmetriesare V PQ = (cid:0) A u φ ∗ u H · H u + A d φ ∗ d H † H d + A e φ ∗ e H † H e + A φ φ ∗ u φ d + B φ φ ∗ d φ ne (cid:1) + h.c., (29) V dia = (cid:88) Φ − µ Φ † Φ + λ Φ (cid:0) Φ † Φ (cid:1) , (30)where Φ = H, H u , H d , H e , φ u , φ d , φ e . We assume that all the coefficients are real. In Eq. (29),the first three terms generate the couplings of axion to first generation fermions throughmixing with H u,d,e . The last two terms reflect the PQ charge assignments of the scalars φ u , φ d and φ e . V dia provides all the diagonal terms which do not provide mixing, but providemasses to CP-even scalars. We omit other operators found by other combinations of thescalar fields allowed by the imposed symmetries. The above potential will be sufficient for acomplete description of the physical axion and the interactions relevant for our discussion.We require that the potential has a minimum, which determines the values of the µ Φ ’s.After applying this condition, we solve the mixing between the seven pseudo-scalars, (cid:126) Φ I ≡ ( h I , − h Iu , h Id , h Ie , φ Iu , φ Id , φ Ie ) T , of which five will be massive, where we have chosen − h Iu sinceit is associated with the imaginary part of the neutral component of ˜ H u , carrying the samehypercharge as H d and H e . The two massless pseudoscalars correspond to the Goldstonebosons after breaking of SU (2) L × U (1) Y ( (cid:126)G SM eigenvector) and the global U (1) P Q ( (cid:126)G PQ eigenvector) symmetries (cid:126)G SM = 1 (cid:112) v + v u + v d + v e (cid:16) v, v u , v d , v e , , (cid:17) , (31) (cid:126)G PQ ≈ (cid:113)(cid:80) f Q f ( v f + v φ f ) × (cid:16) − (cid:80) f ( − f Q f v f v , − Q u v u , Q d v d , Q e v e , Q u v φ u , Q d v φ d , Q e v φ e (cid:17) , (32)where f = u, d, e , Q f is the associated PQ charge, and we define ( − f ≡ − f = u and+1 for f = d, e . We have expressed (cid:126)G PQ at leading order in v (cid:29) v f , v φ f .Note that (cid:126)G PQ gives the mixing between the physical axion states among the sevenpseudo-scalars, e.g. (cid:126) Φ I ⊃ (cid:126)G PQ a . For example, if we further assume v φ f (cid:29) v f , it is clear thatthe axion is dominantly composed of φ If . In the doublet Higgs H f , the axion is containedwith the suppressed factor v f /v φ f , while for the SM Higgs H , it is contained with a doublesuppressed factor v f / ( vv φ f ). 16fter rotating to the physical basis, the couplings of axion to SM fermions are given by (cid:88) f = u,d,e m f f a Q f ia ¯ f γ f − (cid:80) f ( − f Q f v f v (cid:88) F =2nd , m F f a ia ¯ F γ F, (33)where f a ≡ (cid:113)(cid:80) f Q f ( v f + v φ f ) and the second term gives the axion couplings to the 2nd and3rd generation fermions, generated by mixing with the SM Higgs. Note that this generatesan effective PQ charge to heavy fermions, namely Q PQ F, eff ≡ − (cid:80) f ( − f Q f v f v . (34)The couplings of the QCD axion to the 2nd and 3rd generation fermions are constrainedfrom the exotic decays of heavy mesons. The relevant constraints are J/ Ψ → γa leading to | Q PQ c | (cid:46) . f a / GeV) and Υ → γa leading to | Q PQ b | (cid:46) . × − ( f a / GeV) [16]. If we takethe hierarchy v f ≈
100 MeV (cid:28) v and f a ≈ Q PQ F, eff ≈ × − which is quitesafe from the above constraints. B. Additional Massive Scalar and Pseudo-scalar States
In addition to the two Goldstone bosons, there are five massive pseudo-scalars. For n = 3,e.g. Q PQ e = 1 /
3, one can only decouple four heavy massive pseudo-scalars leaving one lightpseudo-scalar. More specifically, we assume the following parameters A φ (cid:29) A u,d,e ≡ A f ≈
20 GeV , v u,d,e ≡ v f ≈
20 MeV , v φ u,d,e ≡ v φ f ≈ , (35)where the last term determines the axion decay constant f a ≈ O (1)GeV. In this case, B φ isa dimensionless parameter. There are then three massive pseudo-scalars, composed mostlyof linear combinations of the h If states, with masses given by A f vv φ f /v f , (36)where f = u, d, e . Additionally, there is one mass of the form A φ v φ f , (37)which dominantly comes from linear combination of φ Iu,d states and may have a mass closeto 100 GeV, and one pseudo-scalar of mass B φ v φ . (38)17he light pseudo-scalar with mass B φ v φ ∼ GeV , which we denote as φ I light , mainly comesfrom φ Ie . The lightness of this state can be understood in the limit B φ = 0. In this case, thereis a global U (1) e symmetry and associated charged scalars φ e and H e . As a result, there isone additional exact goldstone boson. With B φ (cid:54) = 0, the U (1) e is explicitly broken in thescalar potential. Therefore, φ I light becomes a pseudo-goldstone boson with mass proportionalto B φ v φ f and not related to any other dimensionful trilinear couplings. The associatedeigenvector can be easily represented with the simplificiation A φ = A u = 0. In this case, weobtain (cid:126)ω ( φ I light ) ≈ (cid:8) , , v d ( v e + v φ e ) , − v e ( v d + v φ d ) , , v φ d ( v e + v φ e ) , − v φ e ( v d + v φ d ) (cid:9) , ≈ (cid:8) , , v f , − v f , , v φ f , − v φ f (cid:9) , (39)where we have omitted the normalization factor. Thus, φ I light is dominantly composed of φ Ie since v f (cid:28) v φ f . Moreover, it has a mixing of v f /v φ f from H Ie and an even smallermixing from H Id . Therefore, this state mainly couples to the electron with a coupling of ∼ m e /v φ ∼ × − . This coupling can be constrained using the Dark photon search atBaBar, e + e − → γA (cid:48) [45]. Refs. [46, 47] have converted this constraint to scalars, whichrequires a coupling smaller than ∼ × − for scalar masses 1–10 GeV, and thereforethis light pseudo-scalar coupling would be close to the current experimental bounds. Thecorresponding 1-loop contribution to ∆ a e is − . × − (taking m φ I light = 1 GeV andcoupling 5 × − ) making its contribution to ∆ a e much smaller than the uncertainty of theexperiment.For n = 2, B φ is dimensionful. In this case, the mass of φ I light is a linear combination of A φ v φ and B φ v φ . One can choose A φ and B φ large enough such that φ I light avoids existingconstraints. Moreover, one can also add the term A φ e H † d H e , which can also further raiseits mass. As a result, we conclude that for n = 3 the light pseudo-scalar is still viable but issubject to strong constraints, while for n = 2, the five massive pseudo-scalars can be heavyand completely decoupled from the low energy phenomenology.Having dealt with the pseudo-scalar sector, we briefly discuss the seven massive CP-even scalars. For n = 3, there are seven massive CP-even scalars. Under the assumption A f (cid:29) v φ f (cid:29) v f , we find that there is one SM-like Higgs with mass squared 2 λ H v . In this18imit, there are three CP-even masses squared given by A f v v φ f v f , (40)where f = u, d, e , and are all on the order of ∼ (500 GeV) . These states are mainlycomposed of the neutral CP-even components of doublet Higgs H f . There are two massivescalars with masses squared ∼ A φ v φ f (cid:39) (100 GeV) , which are mostly composed of theCP-even components of singlet Higgs bosons φ f . Similar to the pseudo-scalar sector, thereis still one light scalar with mass squared ∼ O ( v φ f ) , whose eigenvector is given by (cid:126)ω ( φ R light ) ≈ A e v e v φ e m h, SM (cid:113) v e + v φ e , , , v e (cid:113) v e + v φ e , , , v φ e (cid:113) v e + v φ e . (41)Just as φ I light , φ R light couples dominantly to the electron with coupling ∼ m e /v φ e and suffersfrom similar constraints. For n = 2, the light scalar masses become A φ v φ or B φ v φ . Thus,in this case the masses for CP-even scalars are also heavy and can be decoupled from lowenergy phenomenology.Finally, we come to the phenomenology related to the SM Higgs decay. Working againin the limit A f (cid:29) v φ f (cid:29) v f the mass eigenstate of the SM Higgs is almost exactly given bythe neutral CP-even component of H . The interactions which determine the decay of theSM Higgs to two axions are given by L int = h (cid:32) λvh I − √ (cid:88) f = u,d,e ( − f A f h If φ If (cid:33) (42) ⊃ − √ (cid:80) f = u,d,e ( − f Q f A f v f v φ f (cid:80) f = u,d,e Q f ( v f + v φ f ) ha ≈ − √ A f v f v φ f ha , (43)where f = u, d, e , and ( − f ≡ − f = u and +1 for f = d, e , as defined above. Wealso assume v f (cid:28) v φ f . The SM Higgs decay to aa has a width ofΓ( h → aa ) = A f v f πm h v φ f (cid:115) − m a m h ≈ .
025 MeV , (44) Since v φ f = 1 GeV, this requires values for A φ which may induce a second minimum in the scalar potentialaway from the EW one, particularly in the direction where the doublet fields vanish. In this case, the EWvacuum must be in a metastable configuration just as in the SM. However, as mentioned previously wehave omitted additional terms in Eq.(29) which are irrelevant to the main issues in this work and a fullevaluation of the potential and its vacuum structure is beyond the present discussion. .
6% exotic Higgsbranching ratio). Assuming the boost factor for the axion is m h / (2 m a ) and width of a → e + e − as Γ( a → e + e − ) ≈ π Q e m e f a m a , (45)we obtain the lifetime of axion in the lab frame γcτ a ≈ .
65 cm for Q e = 1 / m a =17 MeV.Since the axion a is pretty light, the positron and electron pair (we will refer to them aselectrons collectively) in the a → e + e − decay will be highly collimated with a separation of θ ∼ m a /E a ≈ × − radian at the decay vertex. With the 3.8 Tesla magnetic field inthe tracker, the two tracks will separate with the curvature given by R = | p T / ( q × B ) | =( p T / GeV) × .
88 meter. The separation angle between two tracks at the EM calorimetercells will be θ sep ≈ r/ (2 R )), where r is the radius for the EM calorimeter cellslocated in the transverse plane. Assuming the electrons have p T ∼
10 GeV, this angle willbe as large as 0 .
11 radian and the two tracks can in principle be recognized as electronand positron with the correct charge assignment. However, the separation is still smallerthan the cone size of the typical EM cluster ∆ R = 0 . a can be reconstructed, which isdisplaced to the primary vertex by O (cm). This is similar to the signature of a convertedphoton in SM events, where the photon passing through the material in the detector hasbeen converted into an electron-positron pair. In that case, one can sum the momentum of e + and e − and obtain the momentum for the converted photon. Extrapolating the directionof the momentum from the vertex, it will go back to the primary vertex . Therefore, theconverted photon has zero impact parameter. In the case of the axion, since it will be highlyboosted, we can assume the electrons in its decay are parallel. The electron tracks can havenon-zero impact parameters, d = (cid:112) R + d a − R ≈ d a R , (46)where d a is the distance between the displaced vertex (where a decay) and primary vertex inthe transverse plane. Since d a ∼ O (cm), we see d ∼ − cm which is too close to measure. This is different from the case analyzed in Ref. [49], where the converted photon has a large impactparameter. a decaying into e + e − indeed looks similar to the convertedphoton in SM event. But there is one critical difference in that the conversion vertex forthe signal depends only on the lifetime of a , while in SM events it has to be in the detectormaterial.Refs. [50–52] have considered the case of a light and highly boosted A (cid:48) , which decaysinto electron pairs, A (cid:48) → e + e − . Such A (cid:48) can be reconstructed as converted photon, as wepreviously discussed for light a → e + e − decay. Ref. [52] further pointed out that whether itis reconstructed as a converted photon highly depends on the A (cid:48) decay location. If it decaysbefore the first layer of the pixel tracker (about 3.4 cm away from the central axis of thedetector in the barrel region), the electrons (including positrons) will leave hits in the firstlayer of pixel detector. In this case, since the conversion occurs outside the material theevent cannot be reconstructed as a converted photon. In the flowchart for reconstruction ofelectrons or photons [53], such event is also classified as “ambiguous”. Thus, as the electronscannot pass the isolation criteria, this will be identified as a lepton-jet signature. If A (cid:48) decaysafter the first layer of the pixel detector but before the 3rd-to-the -last layer of silicon-stripdetectors (SCT), it can be identified as a converted photon.In our scenario, the lifetime of a → e + e − in the lab is about ∼ .
65 cm for the benchmark.Therefore, it has the probability 1 − e − . / (1 . ≈
87% to decay before the first layerof the pixel detector. As a result, it will dominantly look like a lepton-jet event. Thereis a ∼
13% probability to decay after the first layer of the pixel detector. From Ref. [53],the two Si tracks (SCT tracks) have an efficiency of 0.35 to be reconstructed as convertedphoton for E true T ∼
20 GeV. Therefore, with two a both reconstructed as converted photons,the probability is as small as 0.002. As a result, we conclude that the signal h → aa with a → e + e − will be negligible in the contribution to the h → γγ branching ratio. It also doesnot fit to the h → e + e − searches because the electrons in the signal is not isolated.The lepton-jet signature has been generally discussed and also specifically for two lepton-jets from the SM Higgs decay in Refs. [54–57]. The ATLAS and CMS collaboration havesearched for lepton jets from a light boson [58–62], where the lepton jets come from thedecay topology h → χ χ → χ χ A (cid:48) A (cid:48) , where A (cid:48) is the dark photon, and χ is an excitedfermion DM, which decay promptly to χ , and χ is the DM, recognized as missing energyat the collider. The A (cid:48) decay is displaced but Refs. [58–62] focused on the di-muon channelonly. The prompt electron-jet signature has been studied in the ATLAS search [63] focusing21n the topology h → XX, X → A (cid:48) A (cid:48) with A (cid:48) → e + e − . There, a constraint is set on theelectron-jets decay BR( h → e − jets) (cid:46) . m A (cid:48) ≥
100 MeV. However, this search doesnot cover lower masses O (10) MeV and the topology is different from our model. Therefore,we conclude that h → aa for an axion with mass O (10) MeV is still viable and provides aninteresting electron-jet signature for future searches. VI. CONCLUSIONS
The strong CP problem is one of the most intriguing aspects of the Standard Model ofparticle physics. It can be naturally solved by the introduction of an axion field, implyingthe presence of a new pseudo-scalar particle in the spectrum. Generically, in order to avoidexperimental constraints, the axion decay constant is assumed to be much larger than theweak scale, implying a very light axion particle. In this article, we have investigated thesuggestion that in spite of these constraints, the axion may be as heavy as 17 MeV, openingthe possibility that this axion particle may lead to an explanation of the di-lepton resonanceobserved in nuclear transitions at the Atomki experiment. For this to happen, the axionshould couple only to the first generation quark and leptons, and should have a small mixingwith the pion, something that may be achieved for values of the ratio of the PQ charges ofup and down quarks Q u /Q d = 2.One of the most relevant constraints on this scenario is imposed by the modifications tothe anomalous magnetic moment of the electron. In this work, we demonstrated that theexistence of relevant contributions at the two-loop level may lead to a compatibility of theelectron ( g − e with the recent determinations of the fine structure constraint. This impliesa correlation between the electron coupling to the axion, and hence its charge under PQ,and the coupling to photons at low energies, induced mostly by the mixing with the lightmesons, π and η . Moreover, the corresponding range of couplings leads to PQ charges oforder one and that take values between Q e /Q d = 0 . SU (2) L doublets and singlets, associated with the up and down quarksand the electron sectors, respectively. These fields acquire vacuum expectation values partly22nduced by interactions with the Standard Higgs doublet, with the doublet vevs being muchsmaller than the singlet ones. The axion field is then mostly associated with a combinationof the singlet pseudo-scalar components, and their vevs should be of the order of 1 GeV togenerate the axion decay constant f a = O (1GeV). This scenario leads naturally to smallfirst generation quark and lepton masses. The axion couples to first generation fermionsdominantly with suppressed effective PQ charge to 2nd and 3rd generations through themixing with SM Higgs. Thus, searches for the axion in the exotic decays of heavy mesonsare safely evaded. In order to avoid a large coupling of the SM Higgs to two axions, theHiggs trilinear couplings with the new singlets and doublets cannot be too large, leadingnaturally to a relatively light spectrum of doublet and singlets with masses of the order ofseveral hundreds or tens of GeV, respectively.The model we constructed generates the right quark and charged lepton masses and isconsistent with the observed CKM mixing. It also avoids all experimental constraints andleads naturally to a decay mode of the standard model Higgs into two axions, that due tothe large boost and its corresponding lifetime, may be searched for in the electron channelin ways that are not yet studied by experiments. VII. ACKNOWLEDGMENTS
We would like to thank Wolfgang Altmannshofer, Daniele Alves, Stefania Gori, DavidMorrissey, and Tim Tait for useful discussions and comments. CW and NM have beenpartially supported by the U.S. Department of Energy under contracts No. DEAC02-06CH11357 at Argonne National Laboratory. The work of CW at the University of Chicagohas been also supported by the DOE grant DE-SC0013642. TRIUMF receives federal fund-ing via a contribution agreement with the National Research Council of Canada. The workof JL is supported by National Science Foundation of China under Grant No. 12075005 andby Peking University under startup Grant No. 7101502458. The work of XPW is supportedby National Science Foundation of China under Grant No. 12005009. [1]
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