arXiv:2108.00902v1 [hep-th] 2 Aug 2021 hs ihripiil repiil,ipisteintrodu the implies explicitly, Λ or the implicitly require either that This, fluctuations exampl short-distance for see, many divergences: anomalies a of of (for context types 4] other the 3, and [2, in th chiral approaches of of derived different review one been following is by has This and anomaly models, emphasized. chiral be the hardly why can asp (QFT) phenomenological Theory and Field theoretical diverse axia to the of relevance divergence rese anomalous intense the of of derivation objects oneering been have anomalies other and Chiral osrainlw ilto hc uvvstermvlof removal viol the the survives Λ in which itself violation manifests a breaking law, conservation This scale. such any of hrlaoay nue urn,advacuum and current, induced anomaly, Chiral hc eut ntebekn fasmer htdpnso t on depends that symmetry a of breaking the in results which , ∞ → npyia em,aoaishv hi rgni h existe the in origin their have anomalies terms, physical In oaiaintno o ia edi the in field Dirac a for tensor polarization eeti h aea hnn eet r rsn,bt in both present, are defects no when as same the is b defect bag imposes limit special a in which conditions. defect, of kind ticular 1 + 1 1 + 3 in field Dirac . eas hwta h hrlaoayi h rsneo uha such of presence the in anomaly chiral the that show also We eeaut h aumplrzto esr(P)framassl a for (VPT) tensor polarization vacuum the evaluate We iesosdet h oetodrVTi exact. is VPT order lowest the to due dimensions iesos hsipista h nue aumcretin current vacuum induced the that implies This dimensions. etoAóioBrlceadIsiuoBalseiro Instituto and Bariloche Atómico Centro 80APS .d aioh,Argentina. Bariloche, de C. S. R8402AGP oiinNcoa eEegaAtómica Energía de Nacional Comisión 1 + 1 rsneo defect a of presence .D oc n .Silva A. and Fosco D. C. and uut3 2021 3, August 1 + 3 Abstract iesos ntepeec fapar- a of presence the in dimensions, 1 to fa of ction s faregulator. a of use urn 1.Their [1]. current l to faclassical a of ation h euao,i.e., regulator, the rh ic h pi- the since arch, cso Quantum of ects [5]). e 1 + 1 comprehensive oundary c fUV of nce eabsence he mass reasons e and ess diverse scale On the other hand, introducing nontrivial boundary conditions in QFT models is a rather explicit breaking of (at least) translation symmetry. This leads to many interesting effects, a noteworthy example of which being the Casimir effect [6, 7], as well as many other related phenomena [8]. In the case of fermionic fields, non-trivial boundary conditions, and the resulting Casimir effect, has been studied extensively within the bag model of QCD [9]. Besides, interesting applications of the fermionic Casimir effect to carbon nanotube models have been presented [10, 11]. It is the aim of this letter to study the interplay between the two phenomena, anomalies and non trivial boundary conditions, in a concrete system: fermions in the presence of a potential, which may be used to impose bag boundary conditions. As shown in [12], in order to impose that kind of condition, the potential has to be a singular, space-dependent mass term. We analyze the results of that interplay on two objects: the chiral anomaly and the vacuum polarization tensor (VPT), a correlator between current fluctuations, which one should expect to exhibit a strong dependence with the distance to the boundary. The structure of this work is as follows: in Sect. 1 we introduce the model; then, in Sect. 2 we evaluate the chiral anomaly for that model, and in Sect. 3 the induced vacuum current and the VPT. Finally, in Sect. 4 we present our conclusions.

1 The model

We consider a quantum fermionic field (ψ, ψ¯) in D = d + 1 dimensions (d =1, 3), endowed with an Euclidean action S(ψ,¯ ψ; A), with A an exter- nal Abelian gauge field. We impose non-trivial boundary conditions on the fermionic field at xd =0, by coupling it to a singular scalar potential:

d+1 S(ψ,¯ ψ; A) = d x ψ¯(x) D6 + gδ(xd) ψ(x) , (1) Z   with D6 =6∂ +ie A6 (x), and g a dimensionless constant. It is worth noting that in [13] a 1+1 dimensional model has been studied, where a singular potential is part of the gauge potential; namely, Aµ(x) = Bµ(x)+ sµ(x0)δ(x1), with Bµ a bulk contribution of the gauge potential, and sµ a singular term. Recalling the approach of [14, 15], one sees that g =2 imposes bag bound- ary conditions [14], namely, the normal component of the vector current Jµ due to the Dirac field, vanishes on the interface xd = 0. We assume that the fermions are confined to the region which, in our choice of coordinates, corresponds to xd > 0. For g = 2, it becomes disconnected (independent)

2 from its complement. For g =6 2, however, that is no longer true, as part of the current may cross the interface between them. In our conventions, both ~ and the speed of light are equal to 1, spacetime coordinates are denoted by xµ, µ = 0, 1, ...,d, and the metric tensor is gµν ≡ diag(1, 1,..., 1). Regarding Dirac’s γ-matrices, for d = 1 they are chosen in the representation:

0 1 0 −i γ0 ≡ σ = , γ1 ≡ σ = , (2) 1  1 0  2  i 0  and 1 0 γ5 ≡ γ ≡ −iγ0γ1 = σ = , (3) 5 3  0 −1  with σi (i =1, 2, 3) representing the standard Pauli’s matrices. On the other hand, for d =3:

† I 0 σµ 2×2 0 γµ = , γ5 ≡ γ0γ1γ2γ3 = , (4)  σµ 0   0 −I2×2  where σ0 ≡ iI2×2. The boundary conditions that this system imposes on the fields are as follows: the singular potential in the Dirac equation implies a discontinuity in ψ which, following [16], may be replaced by the average of the two lateral limits: g γ ψ(xq, ǫ) − ψ(xq, −ǫ) + ψ(xq, ǫ)+ ψ(xq, −ǫ) = 0 , (5) d 2

with xq ≡ (x0, x1,...,xd−1). ± 1±γd Setting g =2, and introducing the (orthogonal) projectors: P ≡ 2 ,

+ − P ψ(xq, ǫ) = − P ψ(xq, −ǫ) . (6)

Thus, the orthogonality of these projectors leads to:

+ − P ψ(xq, ǫ)= 0 , P ψ(xq, −ǫ) =0 . (7)

Each one of these conditions implies the vanishing of ψ¯(x)γdψ(x), the nor- mal component of the vacuum current (see next Section below) for g = 2, approaching the border, or ‘wall’ either from xd > 0 or from xd < 0.

3 2 Chiral anomaly in the presence of a defect

5 5 Let jµ(x) ≡ hJµ(x)i the vacuum expectation value of the axial current 5 ¯ Jµ(x) ≡ ieψ(x)γµγ5ψ(x), where the average symbol h...i is defined as follows:

Dψ¯Dψ...e−S(ψ,ψ¯ ;A) h ... i ≡ , (8) R Dψ¯Dψ e−S(ψ,ψ¯ ;A)

R 5 with S as in (1). A naive evaluation of the divergence of jµ(x), using the equations of motion satisfied by the Dirac field, yields the wrong (classical) 5 ¯ result: ∂µjµ(x)=2iegδ(xd)hψ(x)γ5ψ(x)i. This is wrong because of the ill- defined nature of the fermion bilinear. A possible way to tackle this in a gauge invariant manner is to use a (single), bosonic, Pauli-Villars regulator field φ, with a mass Λ, having a Dirac action with the same couplings as the Dirac field. This produces for the divergence 1: 5 ¯ ¯ ¯ ∂µjµ(x) = lim 2iegδ(xd) hψ(x)γ5ψ(x)i + hφ(x)γ5φ(x)i + Λhφ(x)γ5φ(x)i . Λ→∞    (9) Therefore 5 ¯ ∂µjµ(x)= A(x)+2iegδ(xd) hψ(x)γ5ψ(x)i (10) where

−1 A(x)= 2ie lim (Λ + gδ(xd))tr γ5hx| D6 + gδ(xd)+Λ |xi (11) Λ→∞ h
i where Dirac’s bra-ket notation has been used to denote matrix elements of the inverse of the Dirac operator (which includes the singular potential and the gauge field). At this point we note that, for g = 0 (i.e., no defect), one can show that: D6 2 A|g=0(x) ≡ A0(x) =2ie lim tr hx|f(− 2 )|xi (12) Λ→∞ h Λ i 1 where f(x) ≡ 1+x . Since f is a function which satisfies f(0) = 1, and (k) limx→∞ f (x)=0 for all k, this reproduces the proper result for the anomaly, namely,

D 1+ 2ie 2 A (x)= D ǫ F (x)...F (x), (13) 0 D µ1ν1...µ D ν D µ1ν1 µ D ν D 2 2 2 2 2 (4π) ( 2 )! where ǫµ1ν1...µ D ν D is the Levi-Civita Symbol in D = d +1 dimensions. 2 2 1The average symbol for the regulator field is defined in an entirely analogous fashion as for the original field, except for its mass and opposite statistics.

4 The anomaly A can also be obtained in terms of the anomalous Jacobian ieφ(x)γ5 Jφ due to the (infinitesimal version of the) transformation ψ(x) → e ψ(x), ψ¯(x) → ψ¯(x)eieφ(x)γ5 , as follows: δ log[J ] A(x)=2ie φ , (14) δφ(x) where ∞ k−D 5 2 log[Jφ] ≡ lim t 2 ak(φγ , D/ ), (15) t→0+ Xk=0 where ak are functions of matrix-valued arguments. Let us consider now the form of the anomaly for two cases: bag boundary conditions and the general (g not necessarily equal to 2):

2.1 Bag boundary conditions For the case of bag boundary conditions (in our set-up: g =2), and more gen- eral geometries, the calculation of the coefficients in (15) has been presented in [17]. For a planar wall, the first 5 coefficients reduce to:

−D 5 2 D a0(φγ , D/ )=(4π) 2 ( d x tr(φ(x)γ5)), (16) Zxd>0 −d 2 1 d a1(φγ5, D/ )= (4π) 2 ( d xq tr(χφ(xq, 0)γ5)), (17) 4 Z −D 2 1 D Fµν (x) a2(φγ5, D/ )= (4π) 2 ( d x tr(6φ(x)γ5[γν,γµ] ) 6 Zxd>0 4

d + d xqtr(3χ∂dφ(xq, 0)γ5)), (18) Z − D− 2 1 ( 1) 5 Fµν (xq, 0) a3(φγ5, D/ )= (4π) 2 ( dxqtr(φ(xq, 0)γ (96χ[γν,γµ] ) 384 Z 4 5 + 24χ∂d∂dφ(xq, 0)γ )), (19)

−D 5 2 1 D 5 Fµν (x) a4(φγ , D/ )= (4π) 2 ( d x tr(φ(x)γ (60∂i∂i[γν,γµ] 360 Zxd>0 4 2 2 + 180E (X) − 30Fµν))

Fµν(x) + dxq tr(φ(x)γ5((240 ⊓+ −120⊓−)∂d[γν,γµ] ) Z 4 F (x) + ∂ φ(x)γ (180χ[γ ,γ ] µν )+30∂ ∂ ∂ φ(x)γ χ)), (20) d 5 ν µ 4 i i d 5 1 5 1 5 where ⊓+ ≡ 2 (1 + iγ γd), ⊓− ≡ 2 (1 − iγ γd), χ ≡ ⊓+ − ⊓− and ∂i denotes derivation with respect to all coordinates except xd. Higher-order coefficients

5 are multiplied by a positive power of t in the expansion 15, so they do not contribute. In D =2, just the coefficients 16, 17 and 18 are relevant. Given that the trace of an odd number of γ matrices vanishes [18], and taking into account the Dirac algebra, one sees that the contribution from a1 vanishes. 5 2 The coefficient a2(φγ , D/ ) contains both boundary (x1 = 0) and bulk (x1 > 0) terms. Only the latter is non-vanishing, and it produces the anomaly in 2 dimensions ie2 A(x)= ǫ F (x) (21) 2π µν µν where ǫµν is the 2-dimensional Levi-Civita tensor. Thus, there is no quantum correction to the anomaly in 2 dimensions when confining fields to the half- line. In 4 dimensions, a0 and a1 are zero, as well as the boundary term in a2. 5 The bulk term is proportional to tr(γ γµγν), which is a trace of 6 γ matrices. This trace is evaluated in more detail in [19], and is equal to zero. Similarly to the precedents coefficients, a3 has two terms and they are proportional to 5 5 5 tr(γ χγµγν) and tr(γ χγ ) respectively. Both of them are zero because they are traces of 11 and 13 gamma matrices. We are left with the contributions of the coefficient a4. The boundary contribution has 4 terms. These are all proportional to the trace of an odd 5 number of γ matrices or proportional to tr(γ γµγν), so there is no quantum boundary contribution. The bulk term yields the usual result ie3 A(x)= ǫ F (x)F (x) (22) 16π2 µναβ µν αβ where ǫµναβ is the Levi-Civita tensor in 4 dimensions. Thus, so far we have proved that axial anomaly has no boundary contri- butions when bag boundary conditions are imposed at xd =0.

2.2 General case In this case, we come back to the general expression (11) for A, and note that the inverse operator appearing there may be rendered as follows:

−1 −1 hx| D6 + gδ(xd)+Λ |yi = hx| D6 +Λ |yi

d d ′ −1
′ ′
−1 −g d zqd zq hx| D6 +Λ |zq, 0iM(zq, zq)hzq, 0| D6 +Λ |yi (23) Z

with ′ −1 −1 ′ M(zq, zq) = hzq, 0| 1+ g(D6 + Λ) |zq, 0i . (24)   6 Now, we see that the first term in (23) reproduces the previous, bag model anomaly. The second term, whenever one considers points in the bulk xd = ε > 0 will produce a vanishing contribution when Λ → ∞. The reason is that that contribution is UV finite for ε> 0, since one needs to consider that term for y = x =(xq,ε), and there are no coincident points in any operator. Therefore, the whole contribution is finite, and, since it has an extra negative power of Λ, it vanishes in the limit.

3 Induced vacuum current and vacuum polar- ization tensor

3.1 Induced vacuum current

Let jµ(x) ≡ hJµ(x)i the vacuum expectation value of the (vector) current, which is given by

−1 jµ(x) = − e tr γµhx| 6∂ + ie A6 (x)+ gδ(xd) |yi . (25) h
i In order to express jµ in terms of the VPT, we expand that inverse in powers of A:

−1 d+1 hx| 6∂ +ie A6 +gδ |yi = SF (x, y) − d z SF (x, z) ie A6 (z) SF (z,y)+ ... Z
(26) where SF (x, y) is the exact propagator in the presence of the defect, and no A. Therefore, to the lowest non-trivial order, we obtain:

2 d+1 jµ(x) = i e d y tr γµ SF (x, y)γν SF (y, x) Aν (y) . (27) Z h i Namely, to this order, the response to the external gauge field is linear, the proportionality being given by the VPT, Πµν , defined by:

2 Πµν(x, y) = − e tr [γµ SF (x, y)γν SF (y, x)] , (28) such that, d+1 jµ(x) = −i d y Πµν(x, y) Aν(y) . (29) Z

7 3.2 Vacuum polarization tensor in the presence of the defect, in 1+1 dimensions

Since the defect is static, SF , and therefore also Πµν , will depend on the time arguments only through their difference. Using a mixed Fourier representa- tion whereby we just transform the time coordinate,

dp0 ip0(x0−y0) SF (x0, x1; y0,y1) = SF (x0 − y0; x1,y1) = e SF (p0; x1,y1) , Z 2π e (30) we see that:

+∞ 2 dp0 Πµν (k0; x1,y1) = − e tr γµ SF (p0 + k0; x1,y1)γν SF (p0; y1, x1) . Z−∞ 2π h i e e e (31) By an analogous procedure to the one applied in the previous Section, the form of SF may be obtained exactly, for example, by expanding in powers of g, taking into account the form of the singular term, and afterwards summing e the resulting series. The outcome of this procedure may be put in terms of the (0) free-space fermion propagator, SF (x1,y1) (we omit, for the sake of clarity, writing the p0 argument: it is the same in all the instances where it appears here) as follows:

(0) (0) g (0) SF (x1,y1) = SF (x1,y1) − SF (x1, 0) (0) SF (0,y1) . (32) 1+ g SF (0, 0) e e e e (0) e On the other hand, since SF (x1,y1) may be shown to be given by: 1 S(0)(p ; x ,y )= [−iγ σ(p ) + γ σ(x − y )]e−|p0||x1−y1| , (33) F 0 1 1 2 0 0 1 1 1 e where σ denotes the sign function, we can render (32) into a form which will be more convenient in our study of Πµν :

1 g2(1 − σ(x )σ(y )) S (p ; x ,y )= −iγ σ(p ) e−|p0||x1−y1|− 1 1 e−|p0|(|x1|+|y1|) F 0 1 1 2 0 0 4+ g2 n   e g2 + γ σ(x − y ) e−|p0||x1−y1| − (σ(x ) − σ(y ))e−|p0|(|x1|+|y1|) 1 1 1 4+ g2 1 1  o g −|p0|(|x1|+|y1|) + 2 1+ σ(x1)σ(y1)+ γ5σ(p0)(σ(x1)+ σ(y1)) e . (34) 4+ g h i Introducing this expression for the propagator into (31), one can obtain

Πµν . Note that because of the lack of explicit Lorentz invariance (time and e 8 space coordinates are treated differently), one should expect the calculation to miss a ‘seagull term’ [20]. Indeed, Πµν being the correlator between current operators: Πµν (x, y) = −hJµ(x) Jν(y)i , (35) a seagull term τµν (x, y) should be concentrated on x = y. Or, for the partially Fourier transformed version, concentrated on x1 = y1 and a local polynomial in k0. Coming back to the result for Πµν , due to its dependence on the sign of x1 and y1, we have found it convenient to present the result according to the value of those signs:

• x1 > 0 and y1 > 0

In this case, we obtain for Πµν (k0; x1,y1) a result that may be written explicitly: e 2 2 e −|k0||x1−y1| g −|k0||x1+y1| Πµν (k0; x1,y1) = − |k0| e + e δµ0δν0 2π (1 + g2 )2 n 4  e 2 −|k0||x1−y1| g −|k0||x1+y1| − e − 2 e δµ1δν1 (1 + g )2  4  2 −|k0||x1−y1| g −|k0||x1+y1| + iσ(k0) σ(x1 − y1) e − 2 e δµ0δν1 (1 + g )2  4  2 −|k0||x1−y1| g −|k0||x1+y1| + iσ(k0) σ(x1 − y1) e + 2 e δµ1δν0 . (1 + g )2  4  o (36)

• x1 > 0 and y1 < 0

e2 1 − ( g )2 Π (k ; x ,y ) = − 2 e−|k0||x1−y1| |k | δ δ − δ δ µν 0 1 1 2π 1+( g )2 0 µ0 ν0 µ1 ν1 2 h
e +ik0 δµ0δν1 + δµ1δν0 . i
(37)

Note that, for the particular choice g = 2, Πµν vanishes, exhibiting the decoupling from the current in the x > 0 region from the gauge field at 1 e x1 < 0. In other words, in this situation the induced current is insensitive to the existence of a gauge field in the x1 < 0 region.

9 The form of the vacuum polarization function in this case is identical, albeit suppressed by a g-dependent factor, to the one for a fermionic field in free spacetime. Besides, there is a covariantizing seagull term missing, due to the lack of explicit Lorentz covariance in our calculation. Also, note that that term should indeed be missing from the result obtained for x1 > 0 and y1 < 0, which excludes x1 = y1. (0) Denoting by Πµν the VPT in the absence of the wall, we recall the result for the vacuum polarization tensor corresponding to the well-known result for e the Schwinger model in the absence of borders, with both arguments Fourier transformed, e2 k k Π(0) = δ − µ ν , (38) µν π µν k2
we find that: e e2 Π(0)(k ; x ,y ) = − e−|k0||x1−y1| |k | δ δ − δ δ µν 0 1 1 2π 0 µ0 ν0 µ1 ν1 h
e + ik0 σ(x1 − y1) δµ0δν1 + δµ1δν0 i e2
+ δ(x − y )δ δ , (39) π 1 1 µ0 ν0

(0) where the last term is the seagull. Πµν (k0; x1,y1) does of course satisfy the Ward identity: e (0) (0) ik0Π0ν (k0; x1,y1) + ∂x1 Π1ν (k0; x1,y1)=0 . (40) e e(0) Indeed, it is just another version of kµΠµν = 0, which (38) clearly satisfies. When g =6 0, it is straightforward to verify that also the part of the VPT e which depends on the presence of the wall, satisfies the Ward identity by itself. Therefore,

ik0Π0ν (k0; x1,y1) + ∂x1 Π1ν(k0; x1,y1)=0 . (41) e e 3.3 Behaviour of the normal-current correlator

We study here the behaviour of Πµν (k0; x1,y1) when µ = 1 and one ap- proaches the boundary with x : x → 0 (the situation would be identical if 1 1e one considered ν = 1 and y1 → 0, instead). This is the correlator between the normal component of the current on the boundary, and both components of the current Jν. We find, for y1 > 0:

2 2 e g −|k0|y1 Π1ν(k0;0,y1) = 1 − e (ik0δν0 + |k0|δν1) , (42) 2π g2 2 h (1 + 4 ) i e 10 and

2 2 e g |k0|y1 Π1ν (k0;0,y1) = 1 − 2 e (−ik0δν0 + |k0|δν1) , (43) 2π (1 + g )2 h 4  e for y1 < 0. In both cases, the result vanishes only if g =2. Finally, we note that Πµν may be conveniently written in an equivalent way: e 2 (0) g ν (0) Πµν (k0; x1,y1) = Π (k0; x1,y1) + (−1) Π (k0; x1, −y1) , (44) µν g2 2 µν (1 + 4 ) e e e (0) (no sum over ν) in terms of Πµν . The second term proceeds from the corre- lator between J (x , x ) and J ′ (y ,y )=(−1)νJ (y , −y ) (no sum over ν), µ 0 1 e ν 0 1 ν 0 1 the current reflected on the wall.

3.4 Vacuum polarization tensor in the presence of the defect, in 3+1 dimensions Most of the previous results generalize in a rather straightforward way. Here, we use a Fourier representation where xq ≡ (x0, x1, x2) are transformed, since there is translation invariance in that hyperplane:

3 d pq ipq·(xq−yq) SF (x; y) = e SF (pq; x3,y3) , (45) Z (2π)3 e and we see that:

3 2 d pq q q q q Πµν(k ; x3,y3) = − e 3 tr γµ SF (p + k ; x3,y3)γν SF (p ; y3, x3) . Z (2π) h i e e e (46) In this case, we will just present the most relevant result for Πµν(kq; x3,y3), (0) namely, for the case x3,y3 > 0. Taking into account that SF (xe3,y3) is given by: (0) 1 −|pq||x −y | S (pq; x ,y )= [−iγq · pˆq + γ σ(x − y )]e 3 3 , (47) F 3 3 2 3 3 3 pα q e where pˆ ≡ ( |pq| ), α =0, 1, 2, we find that (for x3,y3 > 0)

(0) g/2 (0) q q q SF (p ; x3,y3) = SF (p ; x3,y3) + g 2 SF (p ; x3, −y3)γ3 . (48) 1+( 2 ) e e e

11 Again, as it happened in 1+1 dimensions, Πµν may be conveniently written is terms of the known, free-space result for the VPT: e 2 (0) g iπδν3 (0) Πµν(kq; x3,y3) = Π (kq; x3,y3) + e Π (kq; x3, −y3) , (49) µν g2 2 µν (1 + 4 ) e e e (no sum over ν) which may also be thought of as the combination of a direct term and a suppressed image contribution.

4 Conclusions and discussion

We have obtained the anomaly in the presence of a planar defect of a kind that can impose bag boundary conditions in a special limit, showing that the anomaly is independent of the value of the coupling constant g. Since the chiral anomaly is the same, away from the defect, as the one for the free, no defect case (g =0). This has important consequences in the 1+1 dimensional case. To that end, we recall here some well-known relations, which depend on that property, and apply them to the case at hand: The 5 ¯ axial current, Jµ ≡ ieψγµγ5ψ is, in 1+1 dimensions, completely determined by the vector current. Indeed, in terms of expectation values,

5 jµ = ǫµν jν . (50)

Now, since the vector current is conserved and, because of the previous rela- tion, its curl is the anomaly, we have:

ie2 ∂ j = 0 , ǫ ∂ j = ǫ ∂ A . (51) µ µ µν µ ν π µν µ ν Using the decomposition:

jµ = ∂µϕ + iǫµν ∂ν χ , (52) one finds: e2 ∂2 χ = − ǫ ∂ A . (53) π µν µ ν The general solution to the previous equation for χ may be written as follows:

e2 1 χ = χ − ∂ A (54) 0 π ∂2 µ ν 1 R2 where, as usual, ∂2 denotes the inverse of the Laplacian in , with null con- ditions at infinity, and χ0 is a harmonic function that enforces the boundary

12 conditions. When one considers the g = 0 case, χ0 = 0, and inserting (54) (f) into (52) one recovers the result for Πµν . In the g = 2 case, on the other hand, one considers the x1 > 0 region, and a non-trivial χ0 function is re- quired in order to make j1 = 0 on the boundary. The result for Πµν in this case may be interpreted as the one for g = 0 plus a contribution which can be understood as due to an image source, outside of the region. Note that the combination of two effects: invariance of the anomaly and boundary conditions, completely determine the form of the VPT.

Acknowledgements

The authors thank CONICET, ANPCyT, and UNCuyo for financial support.

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