Spontaneous breaking of chiral symmetry, and eventually of parity, in a σ -model with two Mexican hats
aa r X i v : . [ h e p - ph ] D ec Spontaneous breaking of chiral symmetry, and eventually ofparity, in a σ -model with two Mexican hats Francesco Giacosa
Institute for Theoretical Physics,Johann Wolfgang Goethe University,Max von Laue–Str. 1, 60438 Frankfurt am Main, Germany
December 28, 2018
Abstract A σ -model with two linked Mexican hats is discussed. This scenario could be realized in low-energy QCD when the ground state and the first excited (pseudo)scalar mesons are included, andwhere not only in the subspace of the ground states, but also in that of the first excited states, aMexican hat potential is present. This possibility can change some basic features of a low-energyhadronic theory of QCD. It is also shown that spontaneous breaking of parity can occur in thevacuum for some parameter choice of the model. The ‘Mexican hat’ potential allows for a simple and intuitive description of the phenomenon of spon-taneous symmetry breaking. For this reason it has been widely used -in a variety of versions- in bothcondensed matter and hadron physics, see for instance Ref. [1] and refs. therein.In the context of Quantum Chromodynamics (QCD) nearly massless N f − N f = 2, where N f is the number of light quark flavors) emerge as (quasi) pseudoscalar Goldstonebosons as a consequence of spontaneous breaking of chiral symmetry: U R ( N f ) × U L ( N f ) → SU V ( N f ) . In the context of a linear σ -model this spontaneous breaking is induced by a negative squared massof the scalar and pseudoscalar mesons. This feature is responsible for the typical Mexican hat form ofthe mesonic potential.In this work, beyond the ground state (pseudo)scalar mesons, we also consider the first excited(pseudo)scalar states and we investigate the case in which also in this sector a negative squared massis present. As we shall argue, for some parameter choice this possibility cannot be excluded and leadsto a more complicated scenario, in which ground-state and first-excited scalar and pseudoscalar mesonsmix. Moreover, for some parameter choice it is possible that also one neutral pseudoscalar pionic fieldcondenses, thus realizing a spontaneous symmetry breaking of parity.The paper is organized as it follows: we first briefly review the properties of the Mexican hatpotential and its emergence from an hadronic model of QCD. We then turn to the case of two linkedMexican hats and discuss the consequences of this assumptions. First, the parameter range in whichonly spontaneous breaking of chiral symmetry take place is studied. Then, the parameter range inwhich also spontaneous breaking of parity occurs is investigated. In the end, the conclusions are brieflyoutlined. 1 Mexican hat
In its simplest form the Mexican hat potential is written in terms of two real scalar fields σ and π : V MH = λ (cid:0) σ + π − F (cid:1) = λ (cid:0) ϕ ∗ ϕ − F (cid:1) , (1)where in the last passage the complex scalar field ϕ = σ + iπ has been introduced. The requirement λ ≥ σ represents a scalar field ( σ ≡ σ ( t, x ) → σ ( t, − x ) under parity transformation P ) while π represents apseudoscalar field ( π ≡ π ( t, x ) → − π ( t, − x ) under P ). Note, the quadratic (mass) term of the Mexicanhat potential reads − λ F ϕ ∗ ϕ, i.e. it has a negative coefficient as long as F is a real number, whichcorresponds to an imaginary mass for both the σ and the π fields. For this reason one can immediatelydeduce that the point ϕ = 0 does not correspond to the minimum of the potential. Moreover, anexpansion around this point is instable.The potential V MH is symmetric under SO (2) ∼ U (1) (denoted as chiral) transformation, namely: (cid:18) σπ (cid:19) → (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) σπ (cid:19) or ϕ → e − iθ ϕ. (2)The model does not have a unique minimum: all the points ϕ min = F e iθ for each θ ∈ [0 , π ) areminima. If no other information is given, each one of these minima can be in principle realized.However, we assume that a small perturbation, which breaks chiral symmetry but does not breakparity, V MH → V MH − εσ with ε ∈ + , takes place: as a consequence, the only realized minimumis ϕ min = F. [A change of sign of ε would simply provide the equivalent solution − ϕ min ]. Whenevaluating the fluctuations around the minimum ϕ min = F , one obtains a scalar, massive σ mesonwith M σ = 2 λF and a pseudoscalar, massless Goldstone boson π . The chiral symmetry of the modelis not realized as a degeneracy of the particle spectrum because the minimum (i.e. the vacuum) is notleft invariant by this transformation: spontaneous breaking of chiral symmetry has taken place andthe field π is the corresponding Goldstone boson. For the purpose of this paper we briefly recall how the Mexican hat potential describes the spontaneousbreaking of chiral symmetry which is observed in the context of low-energy QCD. The matrix Φ = S + iP includes N f scalar and N f pseudoscalar fields, S = S a t a and P = P a t a where the matrices t a with a = 1 , ..., N f − SU ( N f ) (with Tr[ t a t b ] = δ ab ) and t = q N f N f . Uponchiral transformation U R ( N f ) × U L ( N f ) the field Φ transforms as Φ → L Φ R † with L ǫ U L ( N f ) and R ǫ U R ( N f ) . The transformation in flavor space SU V ( N f ) is obtained by setting L = R = U V , where U V is a SU ( N f ) matrix. The transformation SU A ( N f ) is obtained by setting L = R † = U A , where U A is a SU ( N f ) matrix. (Note, however, that this set of transformations does not form group for N F > U A (1) axial transformation is obtained by setting L = R † = e − iα N f [2]. (The U V (1) transformationcorresponds to L = R = e iα N f , thus trivially implying the identity transformation Φ → Φ).The effective potential for the field Φ reads [3] V eff [Φ; µ , γ, δ, k, h ] = Tr h µ Φ † Φ + γ (cid:0) Φ † Φ (cid:1) i + δ (cid:0) Tr[Φ † Φ] (cid:1) − k (det Φ † +det Φ) − Tr[ h (Φ † +Φ)] . (3)The first three terms are invariant upon U R ( N f ) × U L ( N f ) transformations. A sufficient condition forthe stability of the potential is that γ > δ > . The term proportional to k is not invariant underthe U A (1) axial transformation and describes the so-called axial anomaly [4]. In the last term thediagonal N f × N f matrix h describes the explicit contribution of nonzero current quark masses. It is2ot invariant under SU A ( N f ) and U A (1) transformations, and if h =const · N f , it is also not invariantunder SU V ( N f ).A first, naive attempt to obtain the Mexican hat of Eq. (1) is to study the case N f = 1 withΦ = q ( σ + iπ ) = q ϕ. In the chiral limit ( h = 0) one can easily identify λ = ( γ + δ ) and µ = − ( γ + δ ) F <
0. The latter is a necessary condition for spontaneous symmetry breaking.However, the anomalous term − k (det Φ † + det Φ) = −√ kσ breaks explicitly chiral symmetry andcannot be regarded as a small perturbation. This is due to the fact that for N f = 1 the chiraltransformation SU A ( N f ) cannot be distinguished from the axial transformation U A (1) . We concludethat, in virtue of the anomaly, the Mexican hat potential cannot be reproduced in the case of onequark flavor only.When N f = 2 the matrix Φ readsΦ = X a =0 φ a t a = ( σ + iη ) t + ( ~a + i~π ) · ~t , (4)where ~t = ~τ / , with the vector of Pauli matrices ~τ , and t = / . In terms of quark degrees of freedom, the scalar isotriplet ~a and the pseudoscalar pion ~π are givenby ud, q ( uu − dd ) , du, while the σ and the η mesons by q ( uu + dd ) . The identification of the piontriplet with the experimentally very well known resonances π ± (139) and π (135) listed in the Particledata Group (PDG) [5] is straightforward. In the pseudoscalar-isoscalar channel, one has in Ref. [5]two resonances η (547) and η (958) , which are a combination of the bare contributions η ≡ q ( uu + dd ) entering in Eq. (4) and the s -quark counterpart ss . The physical field η (547) reads η (547) =cos( ϕ P ) q ( uu + dd ) + sin( ϕ P ) ss where ϕ P ≃ − ◦ [6], while η (958) is the corresponding orthogonalcombination. (One can also ‘unmix’ the two physical fields and obtain that, in an hypothetical N f = 2world without s quark, the η ≡ q ( uu + dd ) would have a mass of about 700 MeV [6]). Theidentification of the fields σ and −→ a is more complicated and addresses the problem of the identificationof scalar mesons in low-energy QCD. Two set of candidates are the resonances { f (600) , a (980) } and { f (1370) , a (1450) } . A detailed description of this issue is not relevant for the scope of this paper, seehowever Ref. [7] and refs. therein.We assume that the charged fields π , π , a , a do not condense. In this case they are not relevantin the study of the minima of the potential and we set their mean value to zero. We are therefore leftwith the diagonal matrix Φ = 12 (cid:18) σ + a + i ( η + π ) 00 σ − a + i ( η − π ) (cid:19) (5)where a and π refer to the neutral a and π mesons.The anomaly term of the potential reads explicitly in the case N f = 2 − k (det Φ † + det Φ) = − k σ + π ) + k a + η ) . (6)For the case k >
0, the absolute minimum is found for a nonzero expectation value of the field σ (or π ) and not for a nonzero value of η (or a ). This is thus the physically interesting case becausea condensation of η (or a ) would imply a parity (or isospin) breaking which is not observed in theprocesses listed in the PDG [5].By further setting a = η = 0 the potential (3) reduces exactly to Eq. (1) by identifying λ = (cid:16) γ δ (cid:17) , µ = − (cid:16) γ δ (cid:17) F + k , ε = 0 . (7)3ote, the choice ε = 0 is denoted as the chiral limit. The explicit inclusion of a breaking termproportional to h = ε = 0 plays the role of the small external perturbation, which induces thecondensation of the σ field and not of π . By further setting the mean value of π to zero, the potentialin terms of the field σ only reads V ( σ ) = 12 (cid:0) µ − k (cid:1) σ + 14 (cid:16) γ δ (cid:17) σ − εσ . (8)The minimum of the latter is realized by a nonzero value σ = φ = 0 if the quantity µ − k is a negativenumber (at zeroth order in ε one has φ = F ). In this case spontaneous breaking of chiral symmetrytakes place and the pions emerge as (quasi) Goldstone bosons.The masses of all the fields, as calculated from Eq. (3) as second derivatives around the minimum σ = φ = 0, read: M π = µ − k + (cid:16) γ δ (cid:17) φ = εφ , M η = µ + k + (cid:16) γ δ (cid:17) φ (9) M σ = µ − k + 3 (cid:16) γ δ (cid:17) φ , M a = µ + k + (cid:18) γ + δ (cid:19) φ (10)It is clear that M η receive a positive contribution form the anomalous term k >
0; this also explainswhile the latter is clearly heavier than the pion fields. It is also renowned that the axial current reads J aA,µ = φ∂ µ π a : the constant φ can then be set equal to the pion decay constant f π = 92 . diag { σ + iπ, σ − iπ } = σt + iπt . Therefore, a SU A (2) transformation Φ → U A Φ U A with U A ǫSU A (2) in the third isospin direction, i.e. U A = e − iαt , is such that Φ → U A Φ U A = U A Φ = e − iαt Φ . The latter reduces exactly to the transformation of Eq. (2), i.e. ϕ = σ + iπ → e − iθ ϕ by identifying α = θ/
2. We thus obtain the simple Mexican hat potential in Eq. (1) as a special case of the general N f = 2 effective potential by identifying σ as the scalar-isoscalar field and π as the pseudoscalarneutral member of the isotriplet field ~π . Let us now turn to the case of interest of this work: two linked Mexican hats. A ‘double Mexican hatpotential’ is introduced in terms of the complex fields ϕ = σ + iπ and ϕ = σ + iπ : V DMH = λ (cid:0) ϕ ∗ ϕ − F (cid:1) + λ (cid:0) ϕ ∗ ϕ − F (cid:1) + c h ( ϕ ∗ ϕ ) + ( ϕ ∗ ϕ ) i . (11)As long as F and F are real numbers, it constitutes of two distinct Mexican hats for ϕ and ϕ , anda c -term, which mixes them [9]. The fields σ and σ are assumed to have positive parity, while thefields π and π negative parity.The model of Eq. (11) is manifestly invariant under the “chiral” U (1) transformation applied to both fields: ϕ → e − iθ ϕ , ϕ → e − iθ ϕ . (12)The condition λ , λ > c must be such that | c | < min { λ + λ , √ λ λ } . We also set, for definiteness, F < F . Note that if c = 0 the model reduces to two decoupled linear sigma models. The symmetry is in thislimit larger: U (1) (1) × U (2) (1) , i.e. it is invariant under ϕ → e − iθ ϕ or ϕ → e − iθ ϕ separately. TwoGoldstone bosons π and π and two massive σ and σ fields with M σ = 2 λ F and M σ = 2 λ F are obtained. 4n terms of the fields ( σ , π ) and ( σ , π ) the potential V DMH takes the form V DMH = λ (cid:0) σ + π − F (cid:1) + λ (cid:0) σ + π − F (cid:1) + c (cid:2)(cid:0) σ − π (cid:1) (cid:0) σ − π (cid:1) + 4 σ π σ π (cid:3) . (13)As usual, the minima of the model must be identified. The sign of the parameter c plays an importantrole: the cases c ≤ c > ϕ, the potential V DMH may ariseas a special case of a more general N f = 2 QCD effective theory in which one starts from two matricesΦ and Φ , each one made of N f scalar and N f pseudoscalar fields as in Eq. (4). The matrix Φ represents the ground state (pseudo)scalar fields, while Φ the first radial excitation. The effectivepotential reads V eff [Φ , Φ ] = V (1)eff [Φ ] + V (2)eff [Φ ] + 2 c Tr h (Φ † Φ ) + (Φ † Φ ) i , (14)where V (1)eff [Φ ] and V (2)eff [Φ ] read as in Eq. (3): V (1)eff [Φ ] = V eff [Φ ; µ , γ , δ , k , h = ε ] , V (2)eff [Φ ] = V eff [Φ ; µ , γ , δ , k , h = ε ] . (15)The U R ( N f ) × U L ( N f ) chiral transformation implies the simultaneous transformation of both fieldsΦ → L Φ R † , Φ → L Φ R † . (16)By performing the same steps as before, we reduce the matrices Φ to their diagonal form Φ = diag { σ + iπ , σ − iπ } . A SU A (2) chiral transformation in the third isospind directionreduces to Eq. (12). The identification of the parameters of Eq. (11) with those of Eq. (14) leads to λ = (cid:16) γ δ (cid:17) , µ − k = − (cid:16) γ δ (cid:17) F , ε = 0 , (17) λ = (cid:16) γ δ (cid:17) , µ − k = − (cid:16) γ δ (cid:17) F , ε = 0 . (18)Two Mexican hats are present as long as F and F are real numbers, i.e. if the quantities µ − k and µ − k are negative real numbers. In this case, one has a Mexican hat for V eff [Φ = diag { σ + iπ , σ − iπ } , Φ = 0] (in the subspace of the ground state fields { σ , π } ) and also for V eff [Φ =0 , Φ = diag { σ + iπ , σ − iπ } ] (in the subspace of { σ , π } ).Note, in Ref. [10] a Lagrangian with (an infinity of) linked Φ k has been introduced, but only oneMexican hat is present: while µ − k < , one has µ p − k p > p = 2 , , ... Similarly, in the N f = 3models of Refs. [11] an additional nonet of scalar and pseudoscalar tetraquark mesons is introduced,but the Mexican hat is present only in the subspace of the ground-state quark-antiquark (pseudo)scalarmesons. In the recent work of Ref. [12] two multiplets Φ and Φ are considered in a general fashion,but the attention is focused on parity breaking at nonzero temperatures/densities.More in general, we also refer to Higgs sector of supersymmetric models (Ref. [13] and refs. therein)and to works on superconductivity (Refs. [14] and refs. therein) where scalar theories, their mixingand spontaneous symmetry breaking are studied. We study the minima of the potential V DMH of Eq. (11) for − c max < c ≤ . One absolute minimum ofthe potential V DMH is given by( π = π = 0 , σ = A , σ = A ) ↔ ( ϕ = A , ϕ = A ) , (19)5 = vuut F − cλ F − c λ λ , A = vuut F − cλ F − c λ λ . (20)Due to the form of the potential this minimum is not unique. All other minima can be obtained byapplying a chiral U (1) transformation to Eq. (19):( ϕ , min , ϕ , min ) = (cid:0) A e iθ , A e iθ (cid:1) with θ ∈ [0 , π ) . (21)The minimum of Eq. (19) is unequivocally realized if we add to the potential the following parity-conserving but chirally breaking terms V DMH → V DMH − ε σ − ε σ with ε , ε ∈ + . (22)Note, the latter shift corresponds to small but nonzero current quark masses, h = ε , h = ε , in Eq. (14).Clearly, the minimum of Eq. (19) is parity-conserving because two scalar fields condense. Being notinvariant under chiral transformation, a spontaneous breaking of this symmetry occurs in the vacuum.The behavior of the condensates as function of the parameter c is reported in Fig. 1, left panel ( c ≤ (cid:18) M σ = 3 λ A − λ F + 2 cA cA A cA A M σ = 3 λ A − λ F + 2 cA (cid:19) ; (23) (cid:18) M π = λ ( A − F ) − cA cA A cA A M π = λ ( A − F ) + 2 cA (cid:19) . (24)The ‘physical masses’ M σ ′ , M σ ′ , M π ′ , M π ′ (the first two states with positive parity, the latter twowith negative parity) are obtained in the standard way as eigenvalues of the latter two matrices. Thespectrum of the system consists of two massive scalar fields, one massive pseudoscalar field and onemassless pseudoscalar Goldstone boson. In fact, one eigenvalue of the pseudoscalar matrix of Eq. (24)vanishes, therefore realizing the Goldstone theorem. In Fig. 1, right panel, the masses are plottedas function of c < M π ′ vanishes for c → − , in agreement with the fact that a second Goldstoneboson exists due to the larger, spontaneously broken symmetry in this limit.Some considerations are in order:(i) If the parameter F -instead of being real- is a purely imaginary number (i.e. if µ − k > σ and π is present. As aconsequence, only the field σ condenses to F (chiral condensate) and π is the Goldstone boson [16].Denoting F = iα one obtains: M σ = 2 λ F , M π = 0 , M σ = λ α + cF and M π = λ α − cF . The mass splitting between σ and π is generated by the chiral condensate σ = F . This is the typicalpicture for low-energy QCD effective theories, in which the fields σ , π are interpreted as the radialexcitations of the ground state σ , π [10]. If, for heavier multiplets Φ k , one has smaller and smaller c ,one recovers the degeneracy of the chiral partners. For a more detailed description of chiral symmetryrestoration see Ref. [16] and refs. therein.(ii) The scenario of two Mexican hats together with c < cannot be excluded as an effectivetheory of QCD. Although a phenomenological study in the framework of a realistic potential shouldbe performed to investigate this possibility, here we simply note that the case with two Mexican hats(with c <
0) is in agreement with all the symmetries and constraints imposed by QCD.(iii) The case c = 0 is interesting. It implies that a larger symmetry group is realized for the effectivetheory than at the fundamental level. In fact, in this limit the effective theory of Eq. (14) is invariant6igure 1: Fixing λ = λ = 1 , F = F / − − c max < c < < c < c max are plotted in the left panel. For c < c > π ′ is the massless Goldstone boson for each c ). For c < c > → L Φ R † and, independently, under Φ → L Φ R † , i.e. under the product of independentchiral transformations U (1) R ( N f ) × U (1) L ( N f ) × U (2) R ( N f ) × U (2) L ( N f ). (The latter transformations reducein the toy model to the already mentioned invariance under U (1) (1) × U (2) (1) , i.e. ϕ → e − iθ ϕ and,independently, ϕ → e − iθ ϕ when axial transformations in the third isospin direction are considered.)If F is a real number, this would imply the presence of two Goldstone bosons, an eventuality which isnot seen in the real world. Indeed, the parameter c should also not be too small, otherwise a second,light pseudoscalar meson would be present in the spectrum, see Fig. 1, right panel, what is excluded byexperimental data (the second pionic excitation has a mass of about 1 . F is imaginaryas described in the point (ii), the condition c = 0 implies the degeneracy M σ = λ α + cF and M π = λ α − cF . In the context of the already mentioned effective restoration of chiral symmetry,where for heavier multiplets a degeneracy is postulated, one indeed would have an approximatelyhigher symmetry, corresponding to a product of U ( k ) R ( N f ) × U ( k ) L ( N f ) for different values of k , where k refers to the k -th excited (pseudo)scalar matrix Φ k . (iv) A generalization to more than 2 Mexican hats can also be easily performed. However, in orderto avoid a proliferation of undesired light pseudoscalar mesons, the mixing among the different Φ k should be large. We regard this possibility as remote for QCD, see next point.(v) QCD in the chiral limit has only dimensional parameter, the Yang-Mills scale Λ QCD . By varyingit, it is -although speculative- conceivable that different phases are realized: a phase in which noMexican hat is present ( F and F both purely imaginary, with no spontaneous chiral symmetrybreaking and no Goldstone boson(s)) obtained for 0 < Λ QCD ≤ Λ [17], a phase in which only for theground state mesons one has a Mexican hat ( F real and F purely imaginary, which is the standardscenario) for Λ ≤ Λ QCD ≤ Λ , a phase in which two Mexican hats are present ( F and F both real)for Λ ≤ Λ QCD ≤ Λ , and so on and so forth. The case Λ ≤ Λ QCD ≤ Λ is the one described by thepotential of Eq. (11) when both F and F are real numbers [18].(vi) In the case of a double Mexican potential ( F and F real), it is not possible to obtain a simplesituation as described by the potential (3), or its reduced form (1). In the scenario of two Mexican hatsit is therefore necessary to take into account both multiplets Φ and Φ . More in general, in the presenceof more Mexican hats, one is obliged to include all of them in a linear hadronic theory of QCD. Needlesto say, a double (or multiple) Mexican hat would correspond to a substantial complication. ‘Life iseasier’ if such a scenario is not realized and if only the ground state σ ≡ σ condenses. Nevertheless,the question why this should be the case is interesting. Is there some yet unknown motivations which7orbids the emergence of a second (or more) Mexican hat(s)? Can it be an accidental fact, whichdepends only on the value of Λ QCD as describe above?
We now study V DMH for 0 < c < c max . One absolute minimum is given by( π = σ = 0 , σ = B , π = B ) ↔ (cid:16) ϕ = B , ϕ = B e iπ/ (cid:17) , (25) B = vuut F + cλ F − c λ λ , B = vuut F + cλ F − c λ λ . (26)The pseudoscalar field π assumes a nonzero vacuum expectation value. This minimum is not unique:the full set of minima is obtained by performing a U (1) rotation of Eq. (25):( ϕ = B e iθ , ϕ = B e i ( θ + π/ ) with 0 ≤ θ < π. (27)Each of these minima breaks parity because π and π never vanish simultaneously. By adding to thesystem the parity conserving but chirally breaking term V DMH → V DMH − ε σ − ε σ , Eq. (25) isthe univocally selected minimum: in fact, this is the point at which σ is maximal for the assumedordering F < F . Note that, although a parity conserving perturbation has been added, still therealized vacuum breaks parity. We conclude that in the proposed model, besides spontaneous breakingof chiral symmetry, also a spontaneous breaking of parity takes place in the vacuum for c > . In Fig.1, left panel, the condensate of Eq. (25) are plotted for c > c < σ and π mix, thus originating twomassive physical states σ ′ and π ′ which are not eigenstates of parity. At the same time also the statesof opposite parity σ and π mix, out of which one massless and one massive bosons π ′ and σ ′ -bothwith undefined parity- are obtained. Numerically, one has a mirror-like picture for c > c > c > cannot be an effective description of QCD even when varying Λ QCD : it is not possible that F and F are real numbers and that at the same time c is negative. However, the validity of the Vafa-Wittentheorem has been questioned in a variety of works (see the discussion in Ref. [20] and refs. therein). Ifit is not valid, it is still conceivable that for a different value of Λ QCD , spontaneous breaking of paritytakes place in the vacuum: in this case the here outlined model -with real F and negative c - wouldcorrespond to its low-energy hadronic (confined) realization.More in general, the original constrain that the charged components of the pion field can also bereleased. All the present treatment is still valid upon replacing π with |−→ π | . We have in this case thecondition |−→ π | = B : as soon as also π = 0 and/or π = 0 not only parity, but also charge conjugationis spontaneously broken. However, it is enough that a further, small perturbation, which originatesfrom other interactions and is invariant under change conjugation, is present: then this additionalperturbation generates a condensation of π ≡ π only, in line with the discussion of the presentpaper. The main interest of this paper has been the possibility that an hadronic, σ -model for QCD is effectivelydescribed by a ‘double’ Mexican hat effective potential. In this scenario not only in the subspace of8he neutral ground state (pseudo)scalar mesons σ ≡ σ and π ≡ π fields, but also in the subspaceof the first excited (pseudo)scalar mesons σ ≡ σ and π ≡ π fields, a typical Mexican hat form ispresent. Mixing among these bare configurations arise: in the case that no spontaneous parity breakingoccurs (here for c <
0) the outlined effective model is in agreement with all the constraints imposed byQCD. In the case that parity symmetry breaking occurs ( c >
0) the described model can provide aneffective description of a underlying QCD-like theory only if the Vafa-Witten theorem does not strictlyhold. More in general, the here presented model can also be conceived as an ‘elementary’ model of(pseudo)scalar fields which generates parity breaking for some choices of the parameters and may playa role in the early Universe.
Acknowledgments: the author thanks T. Brauner and D. H. Rischke for useful discussions.
References [1] A. Zee, “Quantum field theory in a nutshell,”
Princeton, UK: Princeton Univ. Pr. (2003) 518 p [2] In the N f = 2 case one can consider the reduced combination Σ = σt + i −→ π −→ t , which alsotransforms as Σ → R Σ L † under chiral SU R (2) × SU L (2) transformation (but not under U R (2) × SU L (2): the axial transformation, which mixes σ with η and −→ π with −→ a cannot be describedwith Σ only). The contributions of the γ and δ terms are equal in this case and the correspondingpotential can be written as V MH = λ (cid:0) T r (cid:2) Σ † Σ (cid:3) − F (cid:1) = λ (cid:0) σ + −→ π − F (cid:1) . Clearly, themodel of Eq. (1) is recovered by setting π = π = 0 . [3] S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. , 531 (1969).[4] The anomaly term proportional to k is in the present form not renormalizable for N f ≥ . Althoughthis is already in the region of heavy quarks and therefore unimportant for practical purposes,the general issue is if non-renormalizable terms should enter in an effective description of QCD.In principle, being an effective QCD model valid in a restricted low-energy domain, there is noreason to disregard non-renormalizable terms. For instance, Nambu Jona-Lasinio inspired modelsof QCD are non-renormalizable. As another example, one can consider the non-renormalizableFermi Lagrangian of weak interactions, which arises as a low-energy effective term of the morecomplete (and renormalizable) electroweak Lagrangian. On the other hand, what can constrain thedimensionality of terms entering in an hadronic Lagrangian is -rather than the renormalizability-the requirement that dilatation invariance is solely broken by a Yang-Mills scale in the glueballsector, see details in Ref. F. Giacosa, arXiv:0903.4481 [hep-ph] and in Ref.[7].[5] C. Amsler et al. (Particle Data Group), Physics Letters
B667 , 1 (2008)[6] T. Feldmann and P. Kroll, Phys. Scripta
T99 (2002) 13 [arXiv:hep-ph/0201044]. F. Giacosa,arXiv:0712.0186 [hep-ph].[7] S. Gallas, F. Giacosa and D. H. Rischke, arXiv:0907.5084 [hep-ph]. D. Parganlija, F. Giacosa andD. H. Rischke, PoS C
ONFINEMENT8 (2008) 070 [arXiv:0812.2183 [hep-ph]]. D. Parganlija,F. Giacosa and D. H. Rischke, AIP Conf. Proc. (2008) 160 [arXiv:0804.3949 [hep-ph]].[8] V. Koch, arXiv:nucl-th/9512029.[9] Further U (1) symmetric terms could be added to the potential, such as the terms of order 2( ϕ ∗ ϕ + ϕ ∗ ϕ ) and of order 4 ( ϕ ∗ ϕ + ϕ ∗ ϕ ) ϕ ∗ ϕ and ( ϕ ∗ ϕ + ϕ ∗ ϕ ) ϕ ∗ ϕ , and also the U (1) (1) × U (2) (1) symmetric term ( ϕ ∗ ϕ ) ( ϕ ∗ ϕ ) . Here we restrict for definiteness and simplicity to the termof Eq. (11), leaving an extended study for the future.[10] T. D. Cohen and L. Y. Glozman, Mod. Phys. Lett. A (2006) 1939 [arXiv:hep-ph/0512185].911] A. H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D (2005) 034001. M. Napsuciale andS. Rodriguez, Phys. Rev. D (2004) 094043, [arXiv:hep-ph/0407037]. F. Giacosa, Phys. Rev. D (2007) 054007, [arXiv:hep-ph/0611388].For a simple description of the N f = 2 case see: A. Heinz, S. Struber, F. Giacosa and D. H. Rischke,Phys. Rev. D (2009) 037502, arXiv:0805.1134 [hep-ph].[12] A. A. Andrianov, V. A. Andrianov and D. Espriu, Phys. Lett. B (2009) 416 [arXiv:0904.0413[hep-ph]].[13] I. J. R. Aitchison, arXiv:hep-ph/0505105.[14] E. Babaev, arXiv:cond-mat/0302218. E. Di Grezia, S. Esposito and G. Salesi, Physics Letters A (2009) 2385 [arXiv:0807.1414 [cond-mat.supr-con]].[15] In principle a 4 × σ - π , σ - π , ... vanish.[16] L. Y. Glozman, Phys. Rept. (2007) 1 [arXiv:hep-ph/0701081].[17] Note, this possibility is not in agreement with the Casher’s argument, which states that in theconfining mode chiral symmetry is necessarily broken, see: A. Casher, Phys. Lett. B83 , 395(1979). If the Casher’s argument is strictly valid, than we are led to conclude that Λ = 0 , i.e. aconfining but chirally symmetric phase is not possible.[18] Note, it is assumed that confinement holds for each Λ QCD . Obviously, a direct calculations of thevarious Λ i is not doable at present.[19] C. Vafa and E. Witten, Phys. Rev. Lett. (1984) 535.[20] M. B. Einhorn and J. Wudka, Phys. Rev. D67