The Thirring interaction in the two-dimensional axial-current-pseudoscalar derivative coupling model

0.23 MB·English

by  L. V. Belvedere

The Thirring interaction in the two-dimensional
axial-current-pseudoscalar derivative coupling model
L. V.Belvedere∗ and A.F. Rodrigues∗∗
Instituto de F´ısica - Universidade Federal Fluminense
Av. Litorˆanea S/N, Boa Viagem, Niter´oi, CEP 24210-340
Rio de Janeiro, Brasil
∗[email protected]ff.br
6
0 ∗∗armfl[email protected]ff.br
0
2 February 1, 2008

n
a
J
Abstract

7
1 Wereexamine the two-dimensional model of massive fermions interacting with a massless pseudoscalar field
via axial-current-pseudoscalar derivative coupling. Performing a canonical field transformation on the Bose

1 field algebra the model is mapped into the Thirring model with an additional vector-current-scalar-derivative
v interaction(Schroer-Thirringmodel). Thecompletebosonizedversionofthemodelispresented. Thebosonized
9 composite operators of the quantum Hamiltonian are obtained as the leading operators in the Wilson short
1
distance expansions.
1
1
0 1 Introduction
6
0
/ The two-dimensional spinor-scalar model with derivative couplings has been the subject of various investigations
h
within different approaches [1, 2, 3, 4, 5, 6, 7, 8, 9]. The model describing a Fermi field interacting via derivative
t
- coupling with two Bose fields, one scalar and the other pseudoscalar, was analyzed in Refs. [6, 7, 8]. For a certain
p
choiceofthecouplingparameterstheequivalencebetweenthefermionicsectorofthederivativecouplingmodeland
e
h the Thirringmodelcanbe establishedinaweakformbetweenthefermionic Green’sfunctions ofthe corresponding
: models. However,this weak equivalence only works under the expense of introducing opposite metric quantization
v
i forthebosonicfields[7,8],orbyconsideringonederivativeinteractiontermwithimaginarycouplingparameter[6].
X
As a matter of fact, this is the only way under which the degrees of freedom in the two models can be artificially
r matched. In Ref. [7] the connection between the two models is analyzed within the operator formulation. In
a
order to establish the isomorphism between the fermionic Green’s functions of the two models, the Bose fields
are considered with oposite metric and a special combination of the original three bosonic degrees of freedom is
introduced to define two new bosonic fields. In this way the operator solution is given in terms of both a spurion
field and the Thirring field operator. Moreover, the transformation performed in Ref. [7] is meaningless in the
modelwithmassivefermionssince itpresupposesthatthe bosonicfieldsarefree andmassless;the spurionfieldhas
no definite parity and spoils the mass operator.
The model describing a massless pseudoscalar field interacting via derivative coupling with fermions (massless
Rothe-Stamatescu model) [2] was considered in Ref. [9] within the smooth bosonization approach. The similarity
between the massless Rothe-Stamatescu model and the Thirring model is suggested. Within this approach, the
claimedsimilaritybetweenthetwomodelsfollowsfromthefactthattheLagrangianoftheThirringmodelmodified
bythe introductionofanauxiliaryvectorfieldis“almost”the Lagrangianofthe masslessRothe-Stamatescumodel
except for the existence of a scalar field with indefinite metric. However, this is a na¨ıve conclusion and does not
impliesneithertheequivalencebetweenthe twomodels,northe presenceoftheThirringinteractioninthemassless
Rothe-Stamatescu model. The structural aspects of the bosonization of the Thirring model with the use of an
auxiliary vector field are discussed in Ref. [10]. The use of an auxiliary vector field to reduce the action of the
Thirring model to a quadratic action in the Fermi field introduces a redundant Bose field algebra which contains
1
more degrees of freedom than those needed for the description of the model. It is shown in Ref. [10] that the only
effect of the redundant decoupled Bose fields is to generate constant contributions to the Wightman functions in
the Hilbertspaceofstates. Incontrasttowhatoccursintwo-dimensionalgaugetheories[11,12],thespuriousBose
field combinationhas no physicalconsequences,since it carriesno chargeselectionrule,andreduces to the identity
in a positive metric Hilbert space of states [10].
From our point of view, the approach followed in Refs. [7, 8] to establish the weak equivalence between the
Thirringmodelandthederivativecoupling(DC)model,doesnotexhibitthetruephysicalpropertiesofthecomplete
Hilbert space of the model. The relation between the Thirring model and the derivative coupling model has never
been very clear because of an incomplete understanding of the actual role played by the fermionic quartic self-
interaction in the derivative coupling model. A clear demonstration at the operator level of the role played by the
Thirringinteractionunderneaththederivativecouplingmodelislackingintheliterature. Thepurposeofthepresent
work is to fulfill this gap. To this end we shall consider the two-dimensionalmodel of a massless pseudoscalar field
interacting via derivative coupling with a massive Fermi field. This model corresponds to the Rothe-Stamatescu
model [2] in the zero mass limit for the boson field and modified to include a mass term for the Fermi field. The
model describes a massless pseudoscalar field interacting with a massive Fermi field via axial-current-pseudoscalar
derivativecoupling. Throughoutthis paperweshallrefertothis modifiedRothe-StamatescumodelasMRSmodel.
The main purpose of the present paper is to analyze the MRS model within the operator formulation in order to
explicitly show that the MRS model is equivalent to the Thirring model with an additional vector-current-scalar
derivative interaction, i. e., the Schroer-Thirring model. The hidden Thirring interaction in the MRS model is
exhibited compactly by performing a canonical transformation on the Bose fields. The operator solution for the
quantum equations of motion of the MRS model is written in terms of the Mandelstam Fermi field operator of
the Thirring model interacting with a scalar field via derivative coupling. The charge sectors of the MRS model
are mapped into the charge sectors of the massive Thirring model. The paper is organized as follows: In section
2 we present the operator formulation to display the Thirring interaction in the MRS model. The equivalence of
the MRS model with the Schroer-Thirring model is established at the operator level. In section 3 the complete
bosonized version of the model is presented. The bosonized composite operators of the quantum Hamiltonian are
computed as the leading operators in the Wilson short distance expansion for the operator products at the same
point. The conclusion and final comments are presented in section 4.
2 The mapping of the MRS model into the Schroer-Thirring model
The two-dimensionalmodeldescribing a massiveFermifield interactingwith a masslesspseudoscalarBosefield via
axial-currentderivative coupling is defined by the classical Lagrangiandensity 1,
1
(x)=ψ¯(x) iγµ∂ m ψ(x) + ∂ φ˜(x)∂µφ˜(x) + g ψ¯(x)γµγ5ψ(x) ∂ φ˜(x). (2.1)
µ o µ µ
L − 2
The Lagrangian(2.1)descr(cid:0)ibes the Rothe(cid:1)-Stamatescumodel[2]inthe zer(cid:16)omasslimitofth(cid:17)e pseudoscalarfieldand
modified to include a mass term for the Fermi field (MRS model). The quantum theory is defined by the following
equations of motion
iγµ∂ m ψ(x) = gγµγ5N[ψ(x)∂ φ˜(x)], (2.2)
µ o µ
−
(cid:0) (cid:1)
. .
2φ˜(x) = g∂ .. ψ¯(x)γµγ5ψ(x) ... (2.3)
µ
−
(cid:16) (cid:17)
The dots in Eq. (2.3) mean that the current is computed as the leading operator in the Wilson short distance
expansion and the normal product in (2.2) is defined by the symmetric limit [2, 15]
1Theconventions usedare:
γ0= 01 10 , γ1= −01 01 , γ5=γ0γ1 , ψ= ψψ12 ,
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
g00=−g11=1,ǫµν =−ǫνµ,ǫ01=ǫ10=1,γµγ5 = ǫµνγν.
ForfreemasslessBosefields,φ(x)=φr(x)+φℓ(x),φ˜(x)=φr(x)−φℓ(x) , φr,ℓ=φ(x∓).
2
N[ψ(x)∂ φ˜(x)]=. lim 1 ∂ φ˜(x+ε)ψ(x)+∂ φ˜(x ε)ψ(x) . (2.4)
µ µ µ
ε 0 2 −
→ n o
As a consequence ofthe axial-current-pseudoscalarderivative interaction,for massivefermions (m =0) the field φ˜
o
6
does not remains free due to the non-conservation of the axial current 2 in Eq. (2.3).
For massless Fermi fields the quantum model described by the Lagrangian (2.1) (massless Rothe-Stamatescu
model) is a scale invariant theory with anomalous scale dimension [2]. As in the standard Thirring model [14], in
orderthatthetheorydescribedbytheLagrangian(2.1)hasthemodelwithamasslessfermionastheshortdistance
fixed point the scale dimension of the mass operator must be
D <2. (2.5)
ψ¯ψ
In what follows the mass term should be understood as a perturbation in the scale invariant model.
The operator solution for the quantum equations of motion is given in terms of Wick-ordered exponentials
[2, 7, 15],
ψ(x) = −12 :eigγ5φ˜(x):ψ(0)(x), (2.6)
Zψ
where is a wave function renormalizationconstant [2, 15] and ψ(0) is the free massive Fermi field,
ψ
Z
iγµ∂ m ψ(0)(x)=0. (2.7)
µ o
−
The bosonized expression for the free Fermi(cid:0)field operato(cid:1)r ψ(0) is given by the Mandelstam field operator [16],
ψ(0)(x)= µ 1/2e−iπ4 γ5:ei√π{γ5ϕ˜(x) + x∞1 ∂0ϕ˜(x0,z1)dz1}:, (2.8)
2π
(cid:16) (cid:17) R
whereµis aninfraredregulatorreminiscentofthe free masslesstheory. For m =0,the field ϕ˜is freeandmassless
o
such that,
ǫ ∂νϕ˜(x)=∂ ϕ(x). (2.9)
µν µ
The meaning of the notation :( ): in the field operators is that the Wick ordering is performed by a point-splitting
•
limit in which the singularities subtracted are those of the free theory. In this way, the Wilson short-distance
expansions are performed using the two-point function of the free massless field
1
[Φ(+)(x), Φ(−)(0)]x 0 = ln µ2(x2 + iǫx0) . (2.10)
≈ −4π {− }
In order to preserve the classical symmetry of the model the vector current is computed with the regularized
point-splitting limit procedure [2, 15],
µ(x)=...ψ¯(x)γµψ(x)...= lim ψ¯(x+ε)γµe−ig xx+ε ǫµν∂νφ˜(z)dzµψ(x) V.E.V. , (2.11)
J ε 0 −
→ n R o
with the wave function renormalization constant given by [2, 15],
ψ
Z
(ǫ) = eg2[φ˜(+)(x+ǫ),φ˜(−)(x)]. (2.12)
ψ
Z
The vector current is given by [2, 15]
g
µ(x)=jµ(x) ǫµν∂ φ˜(x), (2.13)
J f − π ν
where jµ(x) is the free fermion current,
f
jµ(x)=...ψ¯(0)(x)γµψ(0)(x)...= 1 ǫµν∂ ϕ˜(x), (2.14)
f −√π ν
2In the Schroer model [1], which describes a massive Fermi field interacting with a scalar field via vector-current-scalar derivative
coupling,thescalarBosefieldremainsfreeduetotheconservationofthevector currentψ¯γµψ.
3
and the axial current is,
1 g
5(x)=ǫ ν(x)= ∂ ϕ˜(x) + φ˜(x) . (2.15)
Jµ µνJ − µ √π π
(cid:16) (cid:17)
Withtheexpression(2.15)fortheaxialcurrentonecanwritethequantumequationofmotion(2.3)inthebosonized
form
g2 g
1 2φ˜(x) = 2ϕ˜(x). (2.16)
− π √π
(cid:16) (cid:17)
In order to avoid opposite metric quantization for the fields φ˜ and ϕ˜, we shall consider the model defined for g2 in
the range 3
g2
<1. (2.17)
π
We shall ignore the infrared problems of the two-dimendional massless free boson field since the selection rules
carried by the Wick-ordered exponentials ensure the construction of a positive metric Hilbert space [13, 14]. The
bosonized mass operator takes the form
...ψ¯(x)ψ(x)... = µ:cos 2√πϕ˜(x) + 2gφ˜(x) :. (2.18)
−π
(cid:16) (cid:17)
Fromthebosonizedmassoperator(2.18)andfromtheequationofmotion(2.16)weseethatform =0thefieldsϕ˜
o
andφ˜aresine-Gordon-likefields. In this case the fields ϕ˜andφ˜are free masslessfields andthe axia6lcurrent(2.15)
in the equation of motion (2.3) is conserved. In the original Rothe-Stamatescu model (m = 0) the pseudoscalar
o
field φ˜is massive and the axial current has an anomaly. In this case the field φ˜ remains free in the presence of the
axial-vector-pseudoscalarderivative interaction, aside from a finite mass and wavefunction renormalization [2].
In order to have canonical commutation relation for the field φ˜ we perform the field scaling
g2 1
φ˜(x)= 1 −2 φ˜(x). (2.19)
′
− π
(cid:16) (cid:17)
After the field scaling (2.19), the mass operator (2.18), the vector current (2.13) and the equation of motion (2.16)
can be rewritten as
...ψ¯(x)ψ(x)...= µ:cos 2√πϕ˜(x) + 2g φ˜(x) :, (2.20)
′
−π 1 g2
(cid:16) − π (cid:17)
q
1 g
(x) = ǫ ∂ν √πϕ˜(x)+ φ˜(x) , (2.21)
µ µν ′
J −π 1 g2
(cid:16) − π (cid:17)
g q
2 √πφ˜(x) ϕ˜(x) =0. (2.22)
′
− 1 g2
(cid:16) − π (cid:17)
q
The scale dimension of the mass operator is given by
β2
D = , (2.23)
ψ¯ψ 4π
with
. 4π
β2 = . (2.24)
1 g2
− π
3ThemasslessRSmodelatthe criticalpoint g2=π wasconsideredinRef. [9]withinthe Hamiltonianformalisminthecontext of
anenlargedgaugeinvarianttheory.
4
On account of (2.5) and (2.23), for short distances the mass perturbation becomes increasingly negligible for
g2 < π/2. Notice that the scale dimension of the mass operator is the same as that of the massive Thirring
model with coupling parameter g. In terms of the field φ˜ the Fermi field operator can be rewritten as
′
g
ψ(x)=Zψ−1/2:exp i 1 g2 γ5φ˜′(x) :ψo(x). (2.25)
n − π o
q
The combination between the fields ϕ˜ and φ˜ appearing in Eqs. (2.20) and (2.21) corresponds to a sine-Gordon
′
field, whereas the combination appearing in Eq.(2.22) correspondsto a free massless field. This suggest to perform
the following canonical field transformation,
g
δΦ˜(x)=√πϕ˜(x) + φ˜′(x), (2.26)
1 g2
− π
g q
δξ˜(x)= ϕ˜(x) √πφ˜(x). (2.27)
′
1 g2 −
− π
q
The value of the parameter δ is fixed by imposing canonical commutation relations for the fields Φ˜ and ξ˜,
δ2 β2 1
= = . (2.28)
π 4π 1 g2
− π
The fields φ˜ and ϕ˜ can be written in terms of the new fields (Φ˜,ξ˜), as
′
g √π
φ˜′(x)= Φ˜(x) ξ˜(x), (2.29)
√π − δ
√π g
ϕ˜(x)= Φ˜(x) + ξ˜(x). (2.30)
δ √π
The equation of motion (2.22) is now
2ξ˜(x)=0, (2.31)
and the vector current (2.21) of the MRS model is mapped into the current of the Thirring model,
β
(x) Th(x) = ǫ ∂νΦ˜(x). (2.32)
Jµ ≡ Jµ −2π µν
The mass operator (2.20) is identified with the mass operator of the Thirring model
...ψ¯(x)ψ(x)... ...Ψ¯(x)Ψ(x)... = µ:cos βΦ˜(x):. (2.33)
≡ −π
ThehiddenThirringinteractionintheMRSmodelisexhibitedcompactlyinouroperatorapproach. Usingthefact
that ξ˜is a free and massless field,
ε ∂νξ˜(x)=∂ ξ(x), (2.34)
µν µ
the Fermi field operator (2.6) can be rewritten in terms of the Wick-ordered exponential of the scalar field ξ,
ψ(x) = −12 :eigξ(x):Ψ(x), (2.35)
Zψ
where Ψ is the Fermi field operator of the massive Thirring model given by the Mandelstam soliton operator,
Ψ(x)= µ 1/2e−iπ4 γ5:ei{γ5 β2 Φ˜(x) + 2βπ x∞1 ∂0Φ˜(x0,z1)dz1}:. (2.36)
2π
(cid:16) (cid:17) R
The field operator(2.35) correspondsto the Fermifield ofthe Thirringmodelinteracting with the scalarfield ξ via
vector-current-scalar-derivative coupling, i. e., the Schroer-Thirring model. The equation of motion (2.2) for the
Fermi field can be rewritten as,
5
(iγµ∂ m )ψ(x) = g2 N[γµψ(x) Th(x)] + g N[γµψ(x)∂ ξ(x)]. (2.37)
µ − o Jµ µ
From the equations of motion (2.2) and (2.37) we read off the equivalence between the MRS model with a massive
fermionandthemassiveThirringmodelwithavector-current-scalar-derivativeinteraction(Schroer-Thirringmodel)
4. Theequationofmotion(2.37)correspondstoaparticularcaseoftheSchroer-ThirringmodelinwhichtheThirring
couplingparameterandthederivativecouplingparameterarethesame. InthiscasetheThirringinteractioncannot
be turnedoffinordertogivetheSchroermodel[1]. Theequivalence(2.32)impliesthatthe Hilbertsubspaceofthe
MRS model generated by the correlation functions of the vector current is isomorphic to the Hilbert subspace
µ
J
of the Thirring model generated by the correlation functions of the vector current Th. The Wightman functions
Jµ
of the field operator ψ(x) are those of the Fermi field Ψ(x) of the Thirring model clouded by the contributions of
the free massless field ξ,
0 ψ(x ) ψ(x )ψ¯(y ) ψ¯(y ) 0 =
1 n 1 n
h | ··· ··· | i
n n
0 :e igξ(xj): :e−igξ(yk): 0 0 Ψ(x1) Ψ(xn)Ψ¯(y1) Ψ¯(yn) 0 . (2.38)
h | | ih | ··· ··· | i
j=1 k=1
Y Y
In view of Eq. (2.32), the charge and the pseudo-charge 5 carried by the Fermi field ψ of the MRS model
Q Q
are mapped into the charges and 5 carried by the Thirring field Ψ(x)
QTh QTh
[ , ψ(x)]= ψ(x) [ , Ψ(x)]= Ψ(x), (2.39)
Q − ≡ Q −
1 β2
[ 5, ψ(x)]= γ5 ψ(x) [ 5, Ψ(x)] = γ5 Ψ(x). (2.40)
Q − 1 g2 ≡ Q − 4π
(cid:16) − π (cid:17)
The charge sectors of the MRS model are mapped into the charge sectors of the massive Thirring model. The
Hilbert space of the MRS model is a direct product
H
= , (2.41)
H Hφ˜⊗Hψ(0)
with the selection rules
=0 , 5 =0, (2.42)
QHφ˜ Q Hφ˜6
=0 , 5 =0, (2.43)
QHψ(0) 6 Q Hψ(0) 6
and is isomorphic to the Hilbert space
4TheSchroer-ThirringmodelisdefinedbytheclassicalLagrangiandensity,
1
L(x)=ψ¯(x) iγµ∂µ − mo ψ(x) + ∂µξ(x)∂µξ(x)
2
G2 (cid:0) (cid:1)
+ ψ¯(x)γµψ(x) ψ¯(x)γµψ(x) + g ψ¯(x)γµψ(x) ∂µξ(x).
2
Thequantum theoryisdefinedbytheeq(cid:16)uations ofmo(cid:17)ti(cid:16)on (cid:17) (cid:16) (cid:17)
. .
iγµ∂µ − mo ψ(x) = G2..γµψ(x) ψ¯(x)γµψ(x) .. + gγµN[ψ(x)∂µξ(x)],
(cid:0) (cid:1) (cid:16). (cid:17) .
. .
2ξ(x) = −g∂µ. ψ¯(x)γµψ(x) ..
Theoperatorsolutionforthequantum equations ofmotionisgiven(cid:16)byEq. (2.35)(cid:17)with
β2= 4π .
1− G2
π
TheSchroermodel[1]isobtainedwithβ2=4π (G=0).
6
= (2.44)
ξ Ψ
H H ⊗H
with the selection rules,
=0 , 5 =0, (2.45)
ξ ξ
QH Q H
=0 , 5 =0. (2.46)
Ψ Ψ
QH 6 Q H 6
ItshouldberemarkedthattheThirringcurrent(2.32)correspondstothe vectorcurrentoftheSchroer-Thirring
model defined with a regularization prescription for which the contribution of the scalar field ξ is gauged away.
Using the transformation (2.26)-(2.27) into the current definition (2.11) one obtain,
...ψ¯(x)γµψ(x)...RS = lim ψ¯(x+ε)γµe−ig2 2βπ xx+ε ǫµν∂νΦ˜(z)dzµ+ig xx+ε ∂µξ(z)dzµψ(x) V.E.V.
ε 0 −
→ n R R o
= lim Ψ¯(x+ε)γµe−ig2 2βπ xx+ε ǫµν∂νΦ˜(z)dzµ Ψ(x) V.E.V. ...Ψ¯(x)γµΨ(x)...Th. (2.47)
ε 0 − ≡
→ n R o
From(2.47)we readoffthe appropriateregularizationprescriptionfor the computationofthe vectorcurrentofthe
Thirring model. A general prescription for the current definition of the Schroer-Thirringmodel is given by
...ψ¯(x)γµψ(x)...ST = lim ψ¯(x+ε)γµe−iG22βπ xx+ε ǫµν∂νΦ˜(z)dzµ−iag xx+ε γ5ǫµν∂νξ(z)dzµψ(x) V.E.V. , (2.48)
ε 0 −
→ n R R o
where a is an arbitrary parameter and ψ(x) is given by (2.35) with
4π
β2 = . (2.49)
1 G2
− π
q
Taking into account the fact that the model is scale invariant at small distances, the locality of the theory ensures
the path independence of the line integral in the Mandelstam formula, which can be written as a line integral over
a conserved current
jµ(x)=∂ fνµ(x), (2.50)
ν
with
fνµ(x) = ǫνµΦ˜(x), (2.51)
−
and we obtain
β g
µ(x) = ǫµν∂ Φ˜(x) (a+1)∂µξ(x). (2.52)
ST ν
J −2π − 2π
The Thirring current is obtained with a = 1. The current corresponding to the Schroer model is recovered with
−
a=1 and β2 =4π.
7
3 Bosonized quantum Hamiltonian
In this sectionwe obtainthe complete bosonizedLagrangianof the MRS model. As in the case of the free massless
fermiontheory[15]weshallfirstconsiderthe bosonizedquantumHamiltonian. Thebosonizedcomposite operators
of the quantum Hamiltonian are obtained as the leading operators in the Wilson short distance expansion for the
operator products at the same point [17].
From the Lagrangian (2.1), the classical canonical momentum π conjugate to the field φ˜ is formally given by
φ˜
the expression
π (x)=∂0φ˜(x)+gψ¯(x)γ0γ5ψ(x). (3.1)
φ˜
Form =0,thequantumHamiltoniandensityofthescaleinvariantmodelisobtainedfromtheclassicalHamiltonian
o
withtheclassicalfieldsreplacedbytheirquantumoperatorcounterpartsandisgivenintermsofthenormal-ordered
operator products,
(x)= 1: ∂ φ˜(x) 2: + 1: ∂ φ˜(x) 2: i...ψ¯(x)γ1∂ ψ(x)... g: ψ¯(x)γ1γ5ψ(x) ∂ φ˜(x):, (3.2)
0 1 1 1
H 2 2 − −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
with the axial current given by Eq. (2.15). In terms of the spinor components ψ (α = 1,2.) the kinetic term of
α
the Fermi field in the Hamiltonian (3.2) can be written as
. . 2
h(x)= i..ψ¯(x)γ1∂ ψ(x).. = ( 1)α+1h (x), (3.3)
1 α
− −
α=1
X
where,
. .
. .
hα(x) = i.ψα†(x)∂1ψα(x).. (3.4)
We shall compute the composite operatorh (x) as the leading term in the Wilsonshort distance expansionfor the
α
operatorproductatsamepointusingthesameregularizationasthatemployedinthe computationofthe fermionic
current. To begin with let us consider the point-splitting limit
h (x) = lim h (x;ε) + h.c. V.E.V. , (3.5)
α α
ε 0 −
→ n o
where h (x;ε) is defined by the splitted operator product
α
hα(x;ε) = 2i :ψα†(x+ε)e−ig −x∞+ε ǫµν∂νφ˜(z)dzµ: :eig −x∞ ǫµν∂νφ˜(z)dzµ∂1ψα(x): . (3.6)
! !
R R
Withtheoperatorsolution(2.6),theoperatorproduct(3.6)canbewrittenintermsoftheWick-orderedexponentials
of the field φ˜,
hα(x;ε) = Zψ−1(ε)( :e−ig(cid:0)γα5αφ˜(x+ε)+R−x∞+ε ǫµν∂νφ˜(z)dzµ(cid:1)::eig(cid:0)γα5αφ˜(x)+R−x∞ ǫµν∂νφ˜(z)dzµ(cid:1):!h(α0)(x;ε)
− g2γα5α :e−ig(cid:0)γα5αφ˜(x+ε)+R−x∞+εǫµν∂νφ˜(z)dzµ(cid:1)::eig(cid:0)γα5αφ˜(x)+R−x∞ǫµν∂νφ˜(z)dzµ(cid:1)∂1φ˜(x):!ψα(0)†(x+ε)ψα(0)(x)), (3.7)
where h(0)(x;ε) is the contribution of the kinetic term of the free Fermi field,
α
i
h(0)(x;ε)= ψ(0)†(x+ε)∂ ψ(0)(x). (3.8)
α 2 α 1 α
In the computation of (3.7) we shall use that,
8
γ5 εµ∂ + εµǫ ∂ν φ˜(x)= ε ∂ φ˜(x) , α=1,2, (3.9)
αα µ µν ∓ ± ±
(cid:16) (cid:17)
and that if [B,A]= c - number,
e BA = Ae B [B,A]e B, (3.10)
− − −
−
:e iaΦ(x): :eiaΦ(y)∂ Φ(y): =
− 1
(cid:16) (cid:17)(cid:16) (cid:17)
ea2D(+)(x−y) :e−ia[Φ(x)−Φ(y)]∂1Φ(y): ia ∂y1D(+)(x y) :e−ia[Φ(x)−Φ(y)]: , (3.11)
( − − )
(cid:16) (cid:17)
where
D(+)(x)=[Φ(+)(x), Φ( )(0)]. (3.12)
−
Performing the normal ordering of the exponentials of the field φ˜we can decompose (3.7) as follows,
h (x;ε) = h(I)(x;ε) + h(II)(x;ε) + h(III)(x;ε). (3.13)
α α α α
where
h(αI)(x;ε) = :e±igε±∂±φ˜(x):h(α0)(x;ε), (3.14)
hα(II)(x;ε) = −g2 ψα(0)†(x+ε)ψα(0)(x) :e±igε±∂±φ˜(x)∂1φ˜(x):, (3.15)
(cid:16) (cid:17)
g2
hα(III)(x;ε) = i 2 γα5αFα(ε) ψα(0)†(x+ε)ψα(0)(x) :e±igε±∂±φ˜(x):, (3.16)
(cid:16) (cid:17)
where the singular function F (ε) is given by the commutator
α
x+ε 1
Fα(ε)= γα5αφ˜(+)(x+ε)+ ǫµν∂νφ˜(+)(z)dzµ, ∂1φ˜(−)(x) = 2−πε . (3.17)
h Z−∞ i ±
Let us consider the term h(I). In order to compute the free field contribution h(0)(x), given by Eq. (3.8), we
shall make use of the following relations
1
εµ∂µϕℓ,r(x)= ε±∂ ϕ˜(x), (3.18)
∓2 ±
1
∂ ϕ (x)= ∂ ϕ˜(x), (3.19)
1 ℓ,r
−2 ±
corresponding to α = 1,2, respectively. Using (3.10) and normal ordering the exponentials of the field ϕ˜, the free
fermion contribution is given by [15],
i
h(α0)(x;ε)= −4√πε± :e±i√πε±∂±ϕ˜(x)∂±ϕ˜(x):
1
+ ε [ϕ(ℓ+,r)(x+ε), ∂1ϕℓ(−,r)(x)]:e±i√πε±∂±ϕ˜(x):, (3.20)
±
where (cid:0) (cid:1)
1
(+) ( )
[ϕℓ,r (x+ε), ∂1ϕℓ−,r (x)] = 4±πε . (3.21)
±
Expanding the first exponential in (3.20) in powers of ε up to first order and the second exponential up to second
order, we obtain
9
1 2 1
h(0)(x;ε)= : ∂ ϕ˜(x) : + (ε). (3.22)
α ±8 ± ± 4π(ε±)2 O
(cid:16) (cid:17)
Introducing (3.22) into (3.14), in order to compute the leading operator in the ε-expansion of h(I)(x;ε) we need
retain terms up to second order in ε in the exponentials,
h(I)(x)=h(I)(x;ε)+h.c. V.E.V.=
α α −
1 2 1 g2 2
: ∂ ϕ˜(x) : : ∂ φ˜(x) :. (3.23)
± 8 ± ∓ 8 π ±
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Combining the contributions from the two spinor components, the leading operator h(I)(x) is given by
h(I)(x)=h(I)(x) h(I)(x)
1 − 2
1 2 2 1 g2 2 2
= : ∂ ϕ˜(x) : + : ∂ ϕ˜(x) : : ∂ φ˜(x) : + : ∂ φ˜(x) : . (3.24)
0 1 0 1
2 − 2 π
n (cid:16) (cid:17) (cid:16) (cid:17) o (cid:16) (cid:17)n (cid:16) (cid:17) (cid:16) (cid:17) o
The first term in (3.24) corresponds to the bosonized Hamiltonian of the free massless Fermi field [15]. The term
proportional to g2 in (3.24) is the quantum correctionto the free Lagrangianpiece of the field φ˜.
Nowletusconsiderthesecondtermh(II). Tothisendwemustcomputetheoperatorproductofthefreefermion
field appearing in Eq. (3.15). Using (3.18) and normal ordering the exponential, one obtain
1
ψα(0)†(x+ε)ψα(0)(x) = 2iπε :e±i√πε±∂±ϕ˜(x):. (3.25)
±
Introducing the pseudoscalar potential ,
J
1 g
e (x)= ϕ˜(x)+ φ˜(x), (3.26)
J √π π
such that the axial-currentcan be written ase
5(x)= ∂ (x), (3.27)
Jµ − µJ
and using (3.25), we can writte (3.15) as
e
g i
hα(II)(x;ε)= 4π γα5α ε :e±iπε±∂±J(x)∂1φ˜(x):. (3.28)
±
(cid:16) (cid:17)
Expanding the exponential in (3.28) in powers of ε up to first order, onee has
g
h(II)(x)=h(II)(x;ε)+h.c. V.E.V.= : ∂ (x)∂ φ˜(x) : + (ε). (3.29)
α α − ±π ±J 1 O
(cid:16) (cid:17)
The leading operator in the second contribution h(II)(x)=h(II)(x) he(II)(x) is then given by,
1 − 2
h(II)(x)= g:∂1 (x)∂ φ˜(x): = g: ψ¯(x)γ1γ5ψ(x) ∂ φ˜(x):. (3.30)
1 1
− J
(cid:16) (cid:17)
As expected from the operatorsolution (2.6), the contribution (3.30) cancels the correspondingterm in the Hamil-
e
tonian (3.2).
Finally, let us consider the term h(III). Using (3.25), (3.26) and (3.17), we can write (3.16) as follows
g2 1
hα(III)(x;ε) = ± 8π2 (ε )2 :e±iπε±∂±J(x):. (3.31)
±
(cid:16) (cid:17)
Expanding the exponential in powers of ε up to second order, we get e
g2 2 i g2
h(III)(x;ε) = : ∂ (x) : + g2 ∂ (x) + (ε). (3.32)
α ∓16 ±J 8πε± ±J ± 8π2(ε±)2 O
(cid:16) (cid:17) (cid:16) (cid:17)
e e
10