aa r X i v : . [ h e p - ph ] J a n Chiral Condensate in Two Dimensional Models
V.G. Ksenzov State Scientific Center Institute for Theoreticaland Experimental Physics, Moscow, 117218, Russia
We investigate two different models. In one of them massive fermions interactwith a massive scalar field and in the other the fermion field is in an electrical field(QED2). The chiral condensates are calculated in one-loop approximation. We foundthat the chiral condensate in the case of the Yukawa interaction the fermions andscalar field does not vanish if the mass of the fermion field tends to zero. The chiralcondensate disappears in QED2, if the fermion mass is zero.
I. INTRODUCTION
Dynamical breaking of the symmetry is described by a quantity known as the orderparameter. In models with a broken chiral symmetry the chiral condensate is the orderparameter that disappears, if the chiral symmetry is restored. The investigation of thechiral condensate plays a crucial role in attempts to describe phase transitions related tothe dynamical chiral symmetry breaking.In QCD chiral condensate is investigated via lattice simulations. This approach is usuallyemployed in studies of the chiral condensate in the presence of external factors such astemperature, chemical potential, magnetic fields and magnetic monopoles, etc (see Refs. [1]–[5] and references therein).Apart from numerical studies there are a lot of models employed for the investigation ofthe chiral condensate in various physical systems [6]–[9], [12]–[17]. In the first paper on thissubject Nambu and Jona-Lasinio (NJL) analyzed a specific field model in four dimensions[12]. Later, the chiral condensate was studied by Gross and Neveu (GN) in two dimensionsspacetime in the limit of a large number of fermion flavors N [13]. These two models aresimilar, but in contrast to NJL-model, GN-model is a renormalizable and asymptoticallyfree theory. Due to these properties, GN-model is used for qualitative modeling of QCD.The relative simplicity of both models is a consequence of the quartic fermion interaction.In our previous papers we investigated a system of a self-interacting massive scalar andmassless fermion fields with the Yukawa interaction in a (1 + 1) -dimensional spacetime. Inthe limit of a large mass of the scalar field, the model equivalent to the GN-model. Thechiral condensate was obtained by the path integral using the method of the stationaryphase [14]–[17].In this paper we present a study of two models. One of them is the model with massivefermions and massive scalars with the Yukawa interaction between these two fields. It isworth to note, that the scalars is not a self-interacting fields. The other model is QED intwo-dimensional space-time.The purpose of this paper is to obtain the chiral condensate in QED2-model. To do thiswe must obtain an effective action of the model. Evaluating the one-loop effective potentialis equivalent to the summation of an infinite class of Feynman diagrams, therefore we areunable to calculate a fermionic determinant with an arbitrary vector potential A µ ( x ) . Forthis reason we begin studying of the first model, that we use to investigate the effectivepotential and the chiral condensate, if discrete chiral symmetry was explicitly broken. Thismodel exhibits the essential feature of the techniques, that we use to construct the effectiveaction of QED2. In particular we can see, how the chiral symmetry breaking manifests itselfin the effective potential. As a result we found the chiral condensate in the electrical field. II. FERMIONS IN A SCALAR FIELD
The model, that will be discussed in this section involves N massive fermion fields anda massive scalar field with the Yukawa interaction between those two fields in (1 + 1) -dimensional spacetime.Lagrangian of the model is L = L b + L f = 12 ( ∂ µ φ ) − µ φ ( x ) + i ¯ ψ a ∂ψ a − g (cid:18) φ ( x ) − mg (cid:19) ¯ ψ a ψ a , (1)here φ ( x ) is a real scalar field, ψ a ( x ) is a fermion fields, index a runs from 1 to N ≫ and m is a mass of the fermions.The model with the massless fermions was investigated in our previous papers [14]–[17].In the case, if m = 0 , the Lagrangian is invariant under a discrete symmetry ψ a → γ ψ a , ¯ ψ a → − ¯ ψ a γ , φ → − φ, (2)which is broken by the chiral condensate. If m = 0 , the discrete symmetry (2) disappears,and chiral condensate is not a qualitative criterion.As before in our papers we will determine the chiral condensate by an effective poten-tial. For our purpose we formally define the chiral correlator, using a functional integral inMinkowski space (cid:10) | g ¯ ψ a ψ a | (cid:11) = 1 Z Z DφD ¯ ψ a Dψ a g ¯ ψ a ψ a exp (cid:18) i Z d xL ( x ) (cid:19) , (3)here Z is a normalization constant. The chiral correlator (3) is rewritten as (cid:10) | g ¯ ψ a ψ a | (cid:11) = 1 Z Z Dφ exp (cid:18) i Z d xL b ( x ) (cid:19) i δδφ Z D ¯ ψ a Dψ a exp (cid:18) i Z d xL f ( x ) (cid:19) . (4)The fermionic Lagrangian is quadratic in the field and we can integrate over them, getting (cid:10) | g ¯ ψ a ψ a | (cid:11) = 1 Z Z Dφ g N (cid:16) φ ( x ) − mg (cid:17) π ln g (cid:16) φ ( x ) − mg (cid:17) Λ ×× exp i Z d x
12 ( ∂ µ φ ) − µ φ − N g (cid:16) φ ( x ) − mg (cid:17) π ln g (cid:16) φ ( x ) − mg (cid:17) Λ − , (5)here Λ is the ultraviolet cutoff.We want to obtain the chiral condensate in the framework of one-loop approximation.Therefore we calculate (5) using the method of the stationary phase. A minimum of theeffective action of the system is reached if the effective potential and kinetic energy areminimal on its own: ∂ µ φ = 0 and U eff ( φ ) = min . (6)Let the constant scalar field φ m satisfies the condition (6). The factor in front of the exponentin (5) is fixed at the point φ = φ m and we obtain (cid:10) | g ¯ ψ a ψ a | (cid:11) = N g (cid:16) φ m − mg (cid:17) π ln g (cid:16) φ m − mg (cid:17) Λ . (7)It is worth to note, that the correlator (cid:10) | g ¯ ψ a ψ a | (cid:11) is a chiral condensate only in the mini-mum of the effective potential φ m . The effective potential and the chiral condensate requirerenormalization. We renormalize effective potential following Coleman and Weinberg [20]and Gross and Neveu [13] by demanding that d U eff dφ m (cid:12)(cid:12)(cid:12)(cid:12) φ m = M = µ R . (8) U , MeV eff f R, MeV m f ba Figure 1: a – The value U eff ( φ ) ; b – Chiral condensate R = (cid:10) | m ¯ ψ a ψ a | (cid:11) R at g N = 1 , MeV , µ R = 0 , MeV , m/g = 2 , , M = 100 . Then the renormalized chiral condensate is written as (cid:10) | g ¯ ψ a ψ a | (cid:11) R = N g (cid:16) φ m − mg (cid:17) π ln (cid:16) φ m − mg (cid:17) M . (9) φ m is determined by mean the renormalized effective potential U R eff , which is written as U R eff = 12 µ R φ + g N π (cid:18) φ − mg (cid:19) ln (cid:16) φ − mg (cid:17) M − , (10)We determine the stationary point φ m by numerical solution of equation dU eff dφ (cid:12)(cid:12)(cid:12)(cid:12) φ = φ m = µ R φ m + g N π (cid:18) φ m − mg (cid:19) ln (cid:16) φ m − mg (cid:17) M − = 0 . (11)One can see, that there are two different solutions of (11), and for this reason appear twodifferent vacuums. One of them is a global, and the other is a local minima (see Fig. 1a).Using (9) and (11) we get (cid:10) | m ¯ ψ a ψ a | (cid:11) R = mg Nπ ( g − g cr ) φ m − m Nπ , (12)here g cr = m R πN .It is worth to note, that the chiral condensate does not disappear, if the fermions masstends to zero. In such a case ( m = 0) the effective potential has the minimum at the point a b c m m mm m m ff f Figure 2: The dashed line denotes the fermion loops, solid line denotes the scalar field φ . φ m , which is given in an explicit form φ m = M exp 2 (cid:18) − πµ R g N (cid:19) , (13)and the vacuum energy is E V = − g N π φ m . (14)If g N = πµ R , then φ m = M and the vacuum energy is completely perturbative one. It isknown, that the vacuum energy is determined by the vacuum condensate of the trace of theenergy-momentum tensor θ µµ [10], [11], [18] and [19] as d h | θ µµ | i = E V , (15)here d is the dimension of the spacetime.The diagrams shown in Fig. 2 determine the quantum correction to the trace of theenergy-momentum tensor in the case of the massive fermions. Then the quantum correctionto the trace of the energy-momentum tensor is written as θ µµ = N π ( g φ − mgφ + m ) , (16)here the fist term defines quantum anomaly if mass fermions are zero, the second termdefines the discrete symmetry breaking Fig. 2b, and the term ∝ m defines the vacuumenergy noninteracting fermions Fig. 2c.It will be noted, that although the discrete symmetry is broken in the model with thefree massive fermions, but there is no violation of this symmetry in the effective potentialof the theory. This is due to the fact, that it is impossible to construct a combination of theenergy dimension, that violates this symmetry.The diagrams shown in Fig. 2 determine the quantum correction to the trace of theenergy-momentum tensor in the case of massive fermions. III. FERMIONS IN AN ELECTRICAL FIELD
The model that will be discussed here is QED2. A Lagrangian of the model is well-known L = L b + L f = − F µν + i ¯ ψ a ∂ψ a − m ¯ ψ a ψ a − A µ ¯ ψ a γ µ ψ a , (17)If m = 0 then the Lagrangian is invariant under a discrete chiral simmetry ψ a → γ ψ a , ¯ ψ a → − ¯ ψ a γ . (18)We formally define the chiral correlator as Z d x (cid:10) | m ¯ ψ a ψ a | (cid:11) == 1 Z Z DA µ exp (cid:18) i Z d x L b ( x ) (cid:19) im ddm Z D ¯ ψ a Dψ a exp (cid:18) i Z d xL f ( x ) (cid:19) . (19)It is hard enough to calculate a fermionic determinant with an arbitrary vector potential A µ ( x ) therefore we will use an alternative method of getting an effective potential.It is known that the expression of the effective potential was derived within gluodynamicsby Migdal and Shifman [10] and [11]. In the paper [21] the method was used to obtain thevacuum condensate of the trace of the energy-momentum tensor in massless theories invarious spacetime dimensions. In this paper the method will be used for construction theeffective Lagrangian in QED2. The expression for the effective Lagrangian was obtainedfrom the requirement that the anomaly be reproduced under a scale transformation. Theeffective potential has the form V eff = 1 d σ (cid:16) ln συ − (cid:17) , (20)here σ = θ µµ and υ emerges as a constant of integration in solving the respective differentialequation [10], [11]. We should find θ µµ and υ for our model. a b c m m mm y g y m mA m A n A n aa 5 Figure 3: The dashed line denotes the fermion loops, solid line denotes the vector potential A µ . Above we see that the mass term is fundamentally important for calculating of the chiralcondensate (19) therefore we must construct θ µµ taking into account the mass term. Thediagrams shown at Fig. 3 determine θ µµ in QED2. The chiral symmetry breaking at theexpense of the mass term is given by the diagram Fig. 3b. A result of calculating thediagrams is given θ µµ = − e E π − e m E π + m π , (21)here we introduced a dimensionless coupling constant e = e m and an electrical field E = ε µν ∂ µ A ν .To find υ we assume that the coupling constant e is zero at the time then V eff ( σ ) coincideswith the effective potential for free fermions and we get υ = π M , here M is an arbitrarysubstraction parameter.Now the effective Lagrangian can be expressed as L eff = E σ (cid:18) ln 2 πσM − (cid:19) . (22)A minimum of the effective Lagrangian of the system is reached if dL eff dE = E + 12 dσdE ln 2 πσM = 0 , (23)here dσdE = − e E π − e m π (24)The chiral correlation is Z d x (cid:10) | g ¯ ψ a ψ a | (cid:11) = Z d x m dσdm ln 2 πσM , (25)here m dσdm = − e mE π + m π . (26)Let’s E m is the solution of the equation (23) then using (23) and (25) and accountingthat E m and (cid:10) | g ¯ ψ a ψ a | (cid:11) are constants we get (cid:10) | g ¯ ψ a ψ a | (cid:11) = − mE m dσdm (cid:18) dσdE m (cid:19) − (27)or (cid:10) | g ¯ ψ a ψ a | (cid:11) = (cid:0) m − e mE m (cid:1) E m e m + e E m (28)In generally there are no analytical solutions (23) but we can find a few of them in specialcases if L eff RE E m ba Figure 4: a – The value L eff ( φ ) ; b – Chiral condensate R = (cid:10) | g ¯ ψ a ψ a | (cid:11) R at e = 1 , m = 0 , MeV, M = 100 MeV . m = 0 , then E m = 0 and the chiral condensate disappears,2. e ≪ , then E M ≃ e m π ln m M and (cid:10) | g ¯ ψ a ψ a | (cid:11) = m π ln m M (cid:18) − e m π ln m M (cid:19) , here (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e m π ln m M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It is worth to note that E may be a constant quantity. Really, fixing a gauge A = 0 andusing the fact that the Coulomb potential is a leaner one A = − ax ,we get E = a .In Fig. 4 the effective Lagrangian and the chiral codensate are shown as the function of E . IV. CONCLUSIONS
The models analyzed in this paper, formulated in two dimensional spacetime, are unre-alistic. However, we believe that obtained results are showing the method calculation of thechiral condensate in more realistic models.It was shown that the massive fermions in the massive scalar field have the effectivepotential with two different vacuums. One of them is global the other is local minima. Thechiral condensate obtained in the model does not disappear if the fermion mass tends tozero.The trace of the energy-momentum tensor taking into account the mass of fermionsdemonstrates a clear violation of discrete symmetry. This fact allowed to construct theeffective Lagrangian in QED2. The chiral condensate was obtained in the model. If thefermion mass vanishes, then the chiral condensate disappears.Technically, the central point is the construction of the effective Lagrangian in QED2.Acknowledgment: I am grateful to O.V. Kancheli for useful discussions. [1] P.V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Phys. Lett.
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