aa r X i v : . [ h e p - ph ] F e b Exact accidental U ( ) symmetries for the axion Luc Darm´e ∗ and Enrico Nardi † INFN, Laboratori Nazionali di Frascati, C.P. 13, 100044 Frascati, Italy
We study a class of gauge groups that can automatically yield a perturbatively exact Peccei-Quinn symmetry, and we outline a model in which the axion quality problem is solved at all operatordimensions. Gauge groups belonging to this class can also enforce and protect accidental symmetriesof the clockwork type, and we present a toy model where an ‘invisible’ axion arises from a singlebreaking of the gauge and global symmetries.
Introduction . The non-trivial structure of the vac-uum of Yang-Mills theories  implies that CP viola-tion is a built-in feature in QCD [2, 3]. Strong CP vi-olation is parametrized in terms of an angular variable θ ∈ [0 , π ] whose value is not determined by the the-ory, but is experimentally bounded to lie surprisinglyclose to zero | θ | < ∼ − . It is hard to believe thatthis could occur simply as whim of nature, especiallybecause any value θ . − would leave our Universebasically unaﬀected [4–6], precluding an anthropic ex-planation. A convincing rationale for θ ≈ U (1) PQ - SU (3) C anomaly and bro-ken spontaneously. This unavoidably implies a quasi-massless spin zero boson, the axion [9, 10], whose centralrole is to relax dynamically θ to 0. Remarkably, the axionalso provides a novel solution to the apparently unrelatedpuzzle of the origin of dark matter [11–13], as well as aplethora of other implications for astrophysics and cos-mology (for a recent review see ). However, it alsoraises various new issues. Among the deepest new ques-tions stands the very origin of the axion or, more pre-cisely, which is ‘the origin of the PQ symmetry’ ? Thereare in fact good reasons to believe that global symme-tries cannot be fundamental, and this is especially truefor a symmetry that, being anomalous, does not surviveat the quantum level. A satisfactory explanation wouldarise if, in some suitable extension of the Standard Model(SM), the PQ symmetry occurs accidentally, in the sensethat all renormalizable Lagrangian terms respecting ﬁrstprinciples (Lorentz and local gauge invariance) preserveautomatically also a global U (1) with the required prop-erties. A second problem emerges because to comply withthe bound | θ | < − , U (1) PQ must be respected by alleﬀective operators acquiring a vacuum expectation value(VEV) up to dimension D > ∼
11. This is at odd with thewell founded belief that all global symmetries are even-tually violated by operators of all types and dimensionsinduced by quantum gravity [15–25]. This is known as ‘the PQ symmetry quality problem’ . A third issue is re-lated with ‘the axion scale’ . The axion is a periodic ﬁeldthat, to comply with phenomenological constraints, musttake values over a compact space of rather large radius v a ∼ ± GeV. In benchmark models this is generally engineered by identifying v a with the PQ spontaneoussymmetry breaking (SSB) scale v PQ . This, however,brings in the usual problem of stabilising the electroweakscale against O ( v PQ ) corrections. Various strategies havebeen put forth to explain the origin of the PQ symmetryand protect it up to a suitable operator dimension D :discrete gauge symmetries Z D [26–32], multiple scalarswith values of U (1) gauge charges of order D , non-Abelian gauge symmetries, which generally have degreenot less than D [33, 34], often assisted by supersymme-try [35–37] or by higher dimensional constructions [38–41]. However, an unsatisfactory aspect of all these solu-tions is that if the scale of PQ-breaking (PQ ✟✟ ) eﬀects liesbelow m P , if PQ SSB occurs at a scale v PQ ≫ GeV,or if future experimental limits will hint to θ ≪ − ,the value of D will have to be accordingly increased. Asregards the axion scale problem, certain solutions havebeen attempted exploiting the so-called clockwork mech-anism [42–50]. Clockwork PQ symmetries allow to boostselectively some axion couplings [51–54], and to expo-nentially enhance  or suppress  the ratio v a /v PQ .Clearly, also these symmetries call for an explanation oftheir origin and required high quality . However, devisingways to generate and protect clockwork symmetries em-ploying ﬁrst principles is an even more challenging task.In this Letter we show that a so far unchartedtype of ﬂavor gauge symmetries of the form G MN = SU ( M ) × SU ( N ) with M = N , that we henceforthdenote as ‘rectangular’ symmetries, allow to solveat the root the PQ origin and quality problems, byenforcing automatically global U (1) symmetries thatare either perturbatively exact at the Lagrangian level,or that become exact on the vacuum . We outline asimple example where axion protection is enforced by SU (4) × SU (2). Finally, we speculate how rectangularsymmetries might prove useful to solve also the axionscale problem. To illustrate this we construct a toy modelwherein a clockwork PQ symmetry arises automatically, In this work ‘ﬂavor’ refers to a replication of exotic quarks. This term refers to global symmetries that are broken explicitlysolely by operators whose VEV vanishes. Vacua having moreglobal symmetries than the Lagrangian yield additional masslessscalars besides the usual Nambu-Goldstone-Bosons (NGB) . and a large axion scale v a results from a gauge/globalsymmetry spontaneously broken by VEVs v ≪ v a . Rectangular gauge groups and accidental U ( ) PQ .Consider a scalar multiplet Y transforming in the bi-fundamental representation ( M, N ) of the gauge group G MN = SU ( M ) × SU ( N ) with M > N ≥
2. Let us denotea generic component as Y αi where Greek indices span SU ( M ) and Latin indices SU ( N ). Each group factorhas a pair of Kronecker and Levi-Civita invariant tensors( δ M , ǫ M ), ( δ N , ǫ N ) which can be used to construct invari-ants by contracting the indices of ﬁeld components. Therenormalizable Lagrangian always contains the two in-variants T ≡ Tr ( Y † Y ) and T ≡ Tr ( Y † Y ) constructedfrom δ M and δ N . Being Hermitian, they are manifestlyinvariant under a global U (1) ξ Y phase redeﬁnition Y → e iξ Y . Let us denote the trace of the matrix of the minorsof order k of Y † Y as C k = Tr [Mnr ( Y † Y, k )]. We have T = C and we replace T with A = ( T − T ) = C .The C k ’s up to C N = det [ Y † Y ] form a fundamental set of G MN × U (1) ξ Y invariants: it can be proven  that anyhigher order invariant T k = Tr ( Y † Y ) k can be expressedin terms of this set. The accidental U (1) ξ Y can only bebroken by non-Hermitian invariants, that are monomi-als with an unequal number of Y and Y † components,which must then involve the ǫ tensors. However, all in-variants involving ǫ M and a single scalar multiplet vanishsymmetrically. Consider in fact the SU ( M ) singlet ǫ α ...α M Y α ,i . . . Y α M ,i M ≡ ( ǫ M Y M ) i ...i M , (1)where the right hand side (r.h.s) deﬁnes a shorthand no-tation for the contraction of SU ( M ) indices with ǫ M .Since M > N at least two components have the same SU ( N ) index, so that the string vanishes symmetrically. Thus the Lagrangian for a scalar multiplet Y transform-ing under a rectangular gauge symmetry automaticallyenjoys a global U (1) ξ Y which is perturbatively exact .To promote U (1) ξ Y to a PQ symmetry, it must beendowed with a QCD anomaly. This requires assigning U (1) ξ Y charges to fermions that carry color and coupleto Y . Let us introduce two sets of chiral exotic quarksin the fundamental of SU (3) C , singlets under the elec-troweak gauge group, and transforming under G MN as Q L ∼ ( M,
1) and Q R ∼ (1 , N ) so that the Yukawa op-erator Q L Y Q R is gauge invariant. To prevent a colorgauge anomaly we add P = M − N quarks q R , and a newscalar multiplet Z acquiring a VEV so that all the quarkscan be massive. This step can be arranged in diﬀerentways, the two extreme possibilities are: Considering the M × M matrix Y Y † one has Tr [Mnr ( Y Y † , k )] = C k for k ≤ N and 0 for k > N which yields the same result. Only for ‘square’ symmetries with M = N are U (1) ξ Y -breakingoperators like ǫ M ǫ N Y M ∝ det Y allowed [33, 58, 60]. (I) Add a set of G MN -singlets q Ra ( a = 1 , . . . , P ) whichcouple to a scalar multiplet Z ∼ ( M,
1) via P Yukawa operators P Pa =1 Q L Z q Ra .(II) Assign the q R ’s to the fundamental representationof a new gauge factor SU ( P ), and Z to the bi-fundamental ( M, P ) of G MP , so that there is a sin-gle Yukawa operator Q L Z q R .Note that for M = N + 1 the two cases coincide, hencewe restrict case (II) to P ≥ G MN ( P ) gauge anomaliescan be canceled by adding three copies of M, N, ( P )-pletsof colorless ‘leptons’ of chirality opposite to that of thequarks, which can acquire mass from the VEVs of thesame multiplets Y and Z , e.g. P r =1 L rR Y ℓ rL etc.Scalar terms involving only Z also enjoy an exact ac-cidental symmetry U (1) ξ Z , i.e. V ( Z ) = V ( Z † Z ). How-ever, by contracting the SU ( M ) indices of Y and Z it ispossible to construct certain mixed non-Hermitian oper-ators that break U (1) ξ Y × U (1) ξ Z to a single U (1), thatis deﬁned by some speciﬁc condition between the U (1)charges X Y and X Z . As it will become clear below, de-pending if SU ( M ) index contraction is performed with δ αβ or with ǫ α ...α M the two possibilities are δ M : U (1) ξ Y × U (1) ξ Z → U (1) ξ , X Y − X Z = 0 (2) ǫ M : U (1) ξ Y × U (1) ξ Z → U (1) ξ ′ , N X Y + P X Z = 0 . (3)The charge relation in Eq. (3) implies that U (1) ξ ′ has noQCD anomaly. Hence the symmetry preserved by the op-erators constructed with ǫ M cannot be promoted to a PQsymmetry. To see this let us consider a chiral transfor-mation with generic quark charges X Q L , X Q R , X q R . The U (1)-QCD anomaly coeﬃcient is precisely | N | = M X Q L − N X Q R − P X q R = N X Y + P X Z , (4)where the relation with the charges of the scalars followsfrom requiring U (1) invariance of the Yukawa terms. • U (1) -breaking operators . Eqs. (2) and (3) show thatoperators involving δ M break U (1) ξ ′ , while ǫ M -type ofoperators break U (1) ξ , so that in the presence of bothno U (1) would survive. To see which operators can arise,let us start with case (I) where the multiplets have com-ponents Y α i , Z α . Let us deﬁne a set of SU ( N ) vectors( X n ) i = ( Z † ( Y Y † ) n − Y ) i , n = 1 , . . . , N . The operator O I ( X n ) = ǫ N Π Nn =1 X n (5)does not vanish symmetrically, is non-renormalizable( D = N ( N + 1) ≥ N ≥
2) and preserves U (1) ξ .Since for M − N ≥ ǫ M contractions must in-volve at least two Z α , they vanish symmetrically, andthus U (1) ξ survives as a perturbatively exact accidentalsymmetry, broken only by the anomaly with coeﬃcient | N | = ( N + P ) X Y . For M − N = 1 instead we can write O ′ I ( Y, Z ) = ( N !) − ǫ α ...α N α M (cid:0) ǫ N Y N (cid:1) α ...α N Z α M , (6) Case D ( O ′ ) D ( O )(I) M − N = 1 M N ( N + 1) M − N > − N ( N + 1)(II) N = P M MN > P M D ( L ) TABLE I. Dimension of the operators O ′ and O of lowest or-der that break respectively U (1) ξ and U (1) ′ ξ . The expressionfor D ( L ) is given in the text. that has dimension D = M (and hence is renormalizablefor G and G ). Then, in this particular case U (1) ξ ′ getsbroken at D = M · N and no protected U (1) survives.In case (II) the multiplets components are Y αi , Z αa where a, b, . . . span SU ( P ). Let us take N ≥ P ≥ N ≤ P amounts to interchange Y ↔ Z ) and let us con-sider the SU ( P ) and SU ( N ) singlets ( ǫ P Z P ) α ...α P and( ǫ N Y N ) β ...β N . Since M = P + N the SU ( M ) indices oftheir product can be exactly saturated with ǫ M , yieldingthe G MNP invariant operator of dimension D = M O ′ II ( Y, Z ) = ( P ! N !) − ǫ M (cid:0) ǫ P Z P (cid:1) (cid:0) ǫ N Y N (cid:1) , (7)that preserves U (1) ξ ′ (and is renormalizable for G ). δ M -type of operators can be constructed starting from( ǫ P Z † P ) α ...α P and by contracting the SU ( M ) indiceswith P components of Y . Deﬁning ( X ) ai = ( Z † Y ) ai thisyields ( ǫ P X P ) i ,...i P . The SU ( N ) indices can be con-tracted with ǫ N only if N = P = M/
2, that is when X is a N × N square matrix. The D = M operator O II ( X ) = ( P !) − ǫ N ǫ P X N = det X , (8)is also renormalizable only for G , and is invariant un-der U (1) ξ . For P < N ≤ P , adding N − P new ob-jects ( X ) ai = ( Z † Y Y † Y ) ai allows for the contraction[ ǫ N ( ǫ P X P ) X N − P ] a ...a N − P . However, unless N = 2 P this cannot be contracted into a P -singlet. We thus needto consider the least common multiplier L ≡ lcm( P, N ),in terms of which the structure of these operators is O II ( X n ) ∼ ( ǫ N ) LN ( ǫ P ) LP (cid:0) X P . . . X P F X N −F P F +1 (cid:1) LN (9)where F ≡ ﬂoor(
N/P ) denotes the greatest integer lessor equal to
N/P . Operators of this type preserve thesymmetry deﬁned by X ( X n ) = 0, that is U (1) ξ ofEq. (2), while they break U (1) ξ ′ . However, the dimen-sion D ( L ) = ( L/N )( F + 1)(2 N − F P ) grows rapidly with L (for N = 4 and P = 3, D = 30 !) so that in most cases U (1) ξ ′ breaking remains an academic issue. The dimen-sion of the eﬀective operators of lowest order that breakrespectively U (1) ξ and U (1) ξ ′ are given in Table I. Diﬀerent ﬁeld combinations X = X are needed because oth-erwise ( ǫ N X N ) would vanish symmetrically since pairs of X would necessary have the same SU ( P ) index. Thus, for N = mP , m diﬀerent objects up to X m = Z † ( Y Y † ) m − Y are needed. Vacuum structure of the operators . The PQ so-lution is endangered when the minimum of the axionpotential is shifted away from the one selected by thenon-perturbative QCD eﬀects. Therefore, operators thatbreak explicitly U (1) PQ in the Lagrangian but have van-ishing VEVs are harmless, since they do not contributeto determine the minimum. Thus we need to study thebehaviour of hOi , hO ′ i at the potential minimum. Let usconsider the renormalizable potential for Y . It reads V ( Y ) = κ (cid:0) T − µ Y (cid:1) + λ A , (10)where T and A are the two invariants introduced above,we require κ > λ > − NN − κ to ensure a potentialbounded from below, and µ Y > Y ( x ) in its singular value decomposition (SVD): √ v Y Y = U ˆ Y V † = U ˆ Y e iϕ Y V † −→ ˆ Y e iϕ Y , (11)where v Y = p h T i , U and V are U ( M ) and U ( N ) uni-tary matrices, U and V are the corresponding special uni-tary (det ( U, V ) = +1), ϕ Y ≡ a Y ( x ) /v Y = M arg det U − N arg det V is the NGB of the global U (1) ξ Y , and ˆ Y is thematrix of real non-negative singular values, which can betaken to lie in the diagonal upper N × N block, while allother entries vanish. We will henceforth denote as Y | N ↑ the N × N upper left block of a matrix Y . The last formin Eq. (11) is obtained by gauging away U ( x ) and V ( x ).In this gauge the two invariants read: T ( ˆ Y ) = N X i =1 y i , A ( ˆ Y ) = X i
1) (with a = 1 , ﬂavor relevant scalarterms and the quark Yukawa operators are V f = − λA ( Y ) + η O ZY + h η I O (6) I + h.c. i , (14) V q = κ Q Q L Y Q R + X a =1 , κ a Q L Zq aR + h.c. (15)with λ, η > A ( Y ) drives ˆ Y → ˆ Y c | ↑ ∼ diag (1 ,
1) atthe minimum, while O ZY misalignes h Z i ∼ (0 , , z , z ) T and h Y i . G → SU (2) V and all the quarks are massive.As regards the global symmetries U (1) ξ Y × U (1) ξ Z = U (1) ξ × U (1) ξ ′ , the D = 6 operator O (6) I preserves U (1) ξ (see Eq. (5)) and breaks U (1) ξ ′ . However, VEVs mis-alignment implies hO (6) I i = 0 which yields two NGB: a = 1 v a ( v Y a Y + v Z a Z ) , a ′ = 1 v a ( v Y a Y − v Z a Z ) , (16)where v a = v Y + v Z and, given that all the ﬁelds havethe same periodicity, we have set X Y = X Z = 1. a ( x )gets a mass m a ∼ m π f π /f a from the QCD anomaly,with f a = v a / | N | and | N | = 2( X Y + X Z ) = 4. Thereare, however, only two domain walls because under the Z center of SU (2) V h a i → h a i + π . At this stage a ′ ( x )remains massless. However, considering that breaking U (1) ξ ′ does not imply breaking the gauge symmetry,it might acquire a mass `a la Coleman-Weinberg once all the eﬀects, including those of the fermions, areincluded in the eﬀective potential. A gauge symmetry for a clockwork axion . We nowdiscuss a construction based on rectangular gauge sym-metries that enforces a mechanism for a highly protected‘clockwork’ e U (1) PQ . Although we use suggestive namesfor some group factors, this should be regarded as a toymodel not intended to describe real phenomenology.Consider the gauge group U (1) Y × [ SU (2) × SU (3)] n +1 .We call U (1) Y hyperchage , and the ﬁrst SU (2) × SU (3) isospin and ﬂavor . We introduce three sets of quarksin the fundamental of color transforming under thesefactors as Q L ∼ (2 , , u aR ∼ (1 , , d aR ∼ (1 , − ( a = 1 , ,
3) (we leave understood that gauge anomaliesare compensated by suitable sets of ‘leptons’) and twoscalar multiplets Y d,u ∼ (3 , ± which acquire VEVs h T ( Y d,u ) i = v d,u /
2. The Yukawa Lagrangian reads: L q = − X a =1 (cid:0) κ au Q L Y u u aR + κ ad Q L Y d d aR (cid:1) + h.c, (17)where κ au,d are coupling constants. Note that a coupling( ǫ Y u Y d ) αβ is forbidden because of unsaturated ﬂavor in-dices, so that the potential involving the two scalars hasthe form V ( Y † u Y u , Y † d Y d ) and carry an accidental globalsymmetry U (1) ξ u × U (1) ξ d = U (1) Y × U (1) ξ . Orthogonal-ity with hypercharge Y u X u v u + Y d X d v d = 0 ﬁxes the ratioof the U (1) ξ charges of the scalars as X u / X d = v d /v u , andwe normalize their sum to X u + X d = 2. We now addtwo sets of hyperchargeless ﬁelds Σ p , Y p ( p = 1 , . . . , n )which transform under the additional gauge factors. For SU (3) × SU (2) × SU (3) we add Σ α i α ∼ (3 , , )and Y α i ∼ (1 , , ), and for the successive factorsΣ p ∼ (3 p − , p , p ) and Y p ∼ (1 , p , p ) with p >
1. Thisallows to write a chain of n renormalizable operators( ǫ ǫ Y u Y d Σ ) α i Y α i + n X p =2 ( ǫ ǫ Y p − Σ p ) α p i p Y pα p i p . (18)For each ﬁeld Σ p there is an operator ǫ ( ǫ ǫ ′ Σ p )( ǫ ǫ ′ Σ p )of dimension D = 6 which, together with the opera-tors in Eq. (18), breaks the global symmetry U (1) ξ × [ U (1) Σ × U (1) Y ] n to e U (1) PQ , under which ˜ X Y p = ( − p and ˜ X Σ = 0. Let us now assume that all dimensionalparameters in the scalar potential have values of order v u,d so that there is no large scale in the model. Theoperators in Eq. (18) have multifold eﬀects. First, nonvanishing VEVs would lower the potential by an amount ∼ |h Y Y Σ ′ Y ′ i| so that the VEVs of the ﬁelds tend to alignin speciﬁc directions. The combination ǫ Y u Y d in theﬁrst operator Eq. (18) misaligns h Y u i and h Y d i in isospinspace, in such a way that after U (1) Y × SU (2) breakinga U (1) gauge factor is preserved in the usual way. At thesame time ǫ Y u Y d Σ rotates h Σ i in the direction in ﬂavorspace orthogonal to the plane h Y u i - h Y d i while, because of δ and δ index-contraction, h Σ i and h Y i tend to getaligned in SU (3) × SU (2) space. Isospin breaking pro-vides a negative mixed-term − v u v d Σ Y and thus thereare regions in parameter space where these two ﬁelds ac-quire a VEV proportional to v u,d even if their squaredmasses are non-negative. Hence, regions exist in whichall the VEVs of the chain vanish if isospin is unbrokenand v u,d →
0. Let us verify if e U (1) PQ remains preservedby higher order operators. For each pair (Σ p +1 , Y p +1 ) letus deﬁne ( X n ) γ p = [Σ(Σ † Σ) n − Y ] γ p with n = 1 , , . . . .It is indeed possible to write e U (1) PQ breaking operatorslike ǫ p X X X etc. However, since h Σ p +1 i is orthogo-nal in SU (3) p space to the plane h Y p i α - h Y p i β , it has onlyone non-zero γ p component, and thus all these operatorsvanish on the vacuum. Thus the accidental e U (1) PQ isperturbatively exact, and is broken by the QCD anomalywith | N | = 3( X u + X d ) = 6. The corresponding NGB is˜ a ( x ) = 1 v a ( v u a u + v d a d + n X p =1 v p a p ) , (19)where v u,d,p and a u,d,p are the VEVs and orbital modesand of Y u,d,p and v a = X u v u + X d v d + P p X p v p ≈ v n +1 ,where the approximation holds if all v u,d,p ≈ v . If we nowtake the VEVs that break isospin and PQ symmetries at v ∼
100 GeV, then for n ∼
20 the radius of the axioncompact space is boosted to v a > ∼ GeV without theneed of introducing any large fundamental parameter.
Conclusions . The ‘origin’ and ‘quality’ problems of thePQ symmetry can be solved by assigning the scalar multi-plets hosting the axion to representations of semi-simplegauge groups with a ‘rectangular’ structure. No groupfactors of large degree are required, which renders thesolution particularly elegant. It should have not goneunnoticed that such constructions require that (exotic)quarks must replicate, with some ‘generations’ obtaininga mass from diﬀerent VEVs than others. Admittedly,the embedding into the SM of rectangular symmetries toplay the role of ﬂavor symmetries appears to be a chal-lenging undertaking, but hopefully not insurmountable.Succeeding in this venture might uncover unexpected im-plications for the SM ﬂavor problem.
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