THE DYSON-SCHWINGER EQUATION FOR A MODEL WITH INSTANTONS - THE SCHWINGER MODEL

11 Pages·0.12 MB·English

by  C. Adam

UWThPh-1994-39
15 Dec 1994
5
9 The Dyson-Schwinger equation for a model with instantons –
9
the Schwinger model*
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n
a
J
C. Adam

1
Institut fu¨r Theoretische Physik
1
Universita¨t Wien

1
v
3
7
2
1
0
5
9
/
h
p
-
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e Abstract:
h
:
v Using the exact path integral solution of the Schwinger model – a model where instantons are
i
X present – the Dyson-Schwinger equation is shown to hold by explicit computation. It turns out
r that the Dyson-Schwinger equation separately holds for every instanton sector. This is due to
a
Θ invariance of the Schwinger model.
PACS–No 11.10 Mn ...Field theory, Schwinger source theory
PACS–No 11.15 Tk ...Gauge field theories, other nonperturbative techniques
*)Supportedby“FondszurF¨orderungderwissenschaftlichenForschunginO¨sterreich”,Projekt
Nr. P8444-TEC
1
Introduction
The Schwinger model (SM), which is QED with one massless fermion, is known to be exactly
2
soluble ([1]), and explicit solutions within the operator formalism ([2]) and the path integral
(PI) approach ([3], [4], [5], [6], [7], [8]) are wellknown. Further instantons and an instanton
vacuum have to be present in the SM in order to obtain a consistent quantization ([7], [8]). So
thequestionofwhetherthereisamodificationoftheDyson-Schwinger (DS)equation([10],[11],
[12]) when an instanton vacuum exists may be answered by explicit calculation in the SM. On
one hand we will compute that for the trivial sector (instanton number k = 0) the DS equation
holds as it must be. On the other hand it will turn out that the DS equation even holds for
the nontrivial sector k = 0. More precisely it holds separately for every instanton sector k.
6
So, at least for the model at hand, the DS equation may be extended to the case where the
true vacuum is not the perturbative one but an instanton or Θ vacuum. The reason why every
instanton sector separately fulfills the DS equation is related to the Θ invariance of the SM, as
we will discuss below.
Our computations are done in flat Euclidean space E2, and we use conventions like in [7],
[8], [9]:
g = δ , γ = iγ0γ1 =: iγ0γ1 ,
µν µν 5 − E E −
γ γ = ǫ γν , ǫ ǫνλ = gλ , ǫ = i. (1)
µ 5 µν µν µ 01 −
The abelian gauge field A is parametrized like in [7], [8],
µ
1
A (x) = (∂ α(x)+ǫ ∂νβ(x)), (2)
µ µ µν
e
where e is the dimensionfull charge. The gauge fixing α = 0 is chosen in the sequel. β, which is
the dynamical part of A , may carry an integer instanton number (Pontryagin index)
µ
i ie
ν = dx2β(x) = dxF˜(x) = k Z. (3)
−2π −2π ∈
Z Z
As representative for a ”k-instanton” (wellbehaving at all spacetime but not minimizing the
action) we choose
xν ik x2+λ2
eA (x) = ikǫ = ǫ ∂ν ln =:ǫ ∂νβ (x). (4)
µ µνx2+λ2 µν 2 λ2 µν k
The vacuum functional has to be summed over all k instanton configurations,
∞
Z[J ,η,η¯] = Z [J ,η,η¯], (5)
µ k µ
k=−∞
X
where
Zk[Jµ,η,η¯]= N (DAµ)kDΨ¯DΨe dx[Ψ¯(i∂/−eA/)Ψ−14FµνFµν+AµJµ+Ψ¯η+η¯Ψ]. (6)
Z R
When one separates the zero modes of the Dirac operator and integrates out the fermions, the
resulting expression for the vacuum functional leading to the correct quantum theory of the SM
is (see [7], [8] for details)
k−1
Z [J ,η,η¯]= N Dβ (η¯Ψβ¯)(Ψ¯β¯η)
k µ i0 i0 ·
Z iY0=0
2
ei η¯(x)G(x,y)η(y)dxdye (12β¯Dβ¯+1eβ¯λ)dx, (7)
·
where β¯= β +β , β is the ”insRtanton” (4) and βRhas zero instanton number; further
k k
A Jµ = βλ,
µ
so
δ δ
= ǫµν∂ ; (8)
ν
δJ (x) δλ(x)
µ
G(x,y) is the exact fermion propagator,
G(x,y) = ei(β¯(x)−β¯(y))γ5G (x y), (9)
0
−
and G is the free fermion propagator. The brackets mean
0
(η¯Ψ) dzη¯α(z)Ψα(z). (10)
≡
Z
D is the operator of the effective photon action after the integration of the fermions (see [9],
[8])
2 e2
D= (2 µ2) , µ2 := , (11)
e2 − π
which has the Green’s function ([7], [8])
G(x y) =π(D(µ,x y) D(0,x y)),
− − − −
D G(x y)= δ(x y), (12)
x
− −
D(µ,x) and D(0,x) being the massive and massless scalar field propagators in two dimensions.
Ψβ¯ in (7) is a zero mode of the Dirac operator with respect to the gauge field β¯. It may be
i0
related to a zero mode with respect to the ”instanton” β of (4) by
k
Ψβ¯(x) = eσiβ(x)Ψβk(x), (13)
i0 i0
where σ = 1 is the eigenvalue of γ , γ Ψβk = Ψβk for k>0. The zero modes Ψβk may be
± 5 5 i0 ± i0 < i0
computed ([7], [8]),
1 x2+λ2 1
Ψβi0k = √2π(x−)i0( λ2 )−k2 0! , k > 0 , i0 = 0...k−1,
1 x2+λ2 0
Ψβi0k = √2π(x+)i0( λ2 )k2 1! , k < 0 , i0 = 0...|k|−1,
x+ = x +ix , x− = x ix . (14)
1 0 1 0
−
3
The Dyson-Schwinger equations
The general derivation of the Dyson-Schwinger (DS) equations for QED starts with the obser-
vation that a field derivative of the vacuum functional vanishes ([10]):
δZ[J ,η,η¯] δS δ δ δ
µ
= ( [ , , ]+η (x))Z[J ,η,η¯]= 0 (15)
δΨ¯ (x) δΨ¯ (x) δJ δη δη¯ α µ
α α µ
and
δZ[J ,η,η¯]
µ
= 0. (16)
δAν(x)
Inserting the Dirac equation in (15) or the Maxwell equation in (16), performing derivatives
with respect to the sources and setting the sources equal to zero at the end results in equations
for the exact n-point functions of the theory - the DS equations.
Now the only nontrivial vacuum expectation values (VEVs) of the SM involving instan-
ton contributions are VEVs of products of (pseudo)scalar currents ([3], [7], [8]). The Maxwell
equations however generate only gauge field and vector current correlators, so they are not in-
teresting for the problem which we want to discuss - the DS equations in an instanton vacuum.
Therefore it suffices to work with equation (15), which is the quantum analog of the Dirac
equation.
The simplest nontrivial DS equation is obtained from (15) by performing three fermion
source derivatives, resulting in an equation between fermionic two- and four-point functions. In
the explicit calculation of this DS equation we will observe a cancellation mechanism between
different Green’s functions that may be easily generalized to higher n-point functions. So it is
enough to deal with this DS equation (another reason for doing the computation for the four-
point function is its importance for a Bethe-Salpeter analysis of the SM, which will be done in
a forthcoming paper). It reads
δ δ δ δ δ
(ηα(x )+γαβ(i∂µ e ) )Z = 0, (17)
δη (y )δη (x )δη¯ (y ) 1 µ x1 − δJ (x ) δη¯ (x ) |0
ǫ 2 δ 2 γ 1 µ 1 β 1
where the vertical line indicates that the sources have to be set equal to zero after the differen-
tiation. We rewrite (17) in three terms that we will compute separately, and they have to sum
up to zero:
δ δ δ δ
[ δαδ(x x ) +δαδ(y x ) ]Z A (18)
− δ 2 − 1 δη (y )δη¯ (y ) ǫ 2 − 1 δη (x )δη¯ (y ) |0 ≡
ǫ 2 γ 1 δ 2 γ 1
δ δ δ δ
iγαβ∂µ Z B (19)
µ x1δη (y )δη (x )δη¯ (y )δη¯ (x ) |0 ≡
ǫ 2 δ 2 γ 1 β 1
δ δ δ δ δ
eγαβ Z C. (20)
− µ δJ (x )δη (y )δη (x )δη¯ (y )δη¯ (x ) |0 ≡
µ 1 ǫ 2 δ 2 γ 1 β 1
Next we must remember that for a VEV of n fermions and n antifermions instanton sectors
up to k = n contribute ([3], [7], [8], see formula (7)). So for the term A k = 0, 1 will be
± ±
important, and for B,C k = 0, 1, 2.
± ±
Further the general spinor structure is important in the DS equation only for the indices
α,β,δ belonging to the x variables, on which the DS equation acts. We therefore may focus on
i
4
thescalar component γ = ǫ for they variables withoutloss of information (remember that only
i
scalar components are important for instanton effects). This will simplify the computations and
is fixed in the sequel.
Now we are ready to start the computations. We will do them separately for k = 0,+1,+2
( 1 and 2 are analogous to +1, +2) and find that the DS equation (17) separately holds for
− −
every instanton sector.
Computation for k = 0
Using (7) and (8) we get for the term C0 (20) (the zero index indicates zero instanton sector)
δ δ δ δ δ
eγαβ Z =
− µ δJ (x )δη (y )δη (x )δη¯ (y )δη¯ (x ) |0
µ 1 γ 2 δ 2 γ 1 β 1
= γαβǫµνN Dβ(∂x1β(x ))[Gγδ(y ,x )Gβγ(x ,y )
µ ν 1 1 2 1 2 −
Z
Gβδ(x1,x2)Gγγ(y1,y2)]e12 dzβDβ, (21)
−
where the rules of Grassmann differentiation have been uRsed. With Gγγ = trG = 0 we further
obtain (see (9))
ǫµνγαβN Dβ(∂x1β(x ))[eiγ5(β(x1)+β(x2)−β(y1)−β(y2))]βγ
µ ν 1 ·
Z
·Gγ0ǫ(x1−y2)Gǫ0δ(y1−x2)e21 dzβDβ (22)
′ R
and, introducing the real field β = iβ (we omit the prime) and using the short notation
Dµ[β]:= N Dβe−21 dzβDβ (23)
Z R
we get
iǫµνγαβ Dµ[β](∂x1β(x )) [1cosh(β(x )+β(x ) β(y ) β(y ))+
− µ ν 1 { 1 2 − 1 − 2
Z
+γ sinh(β(x )+β(x ) β(y ) β(y ))]G (x y )G (y x ) βδ = (24)
5 1 2 1 2 0 1 2 0 1 2
− − − − }
= iγαβ[γ G (x y )G (y x )]βδǫµν∂x1 Dµ[β]
− µ 5 0 1− 2 0 1− 2 ν ·
Z
cosh(β(x )+β(x ) β(y ) β(y )), (25)
1 2 1 2
· − −
where we used the Leibnitz rule for the cosh and the fact that in the Gaussian PI only even
powers of the variable β contribute. With the identity γ γ ǫµν = γν (see (1)) we finally arrive
µ 5
−
at
C0 = iγαβGβγ(x y )Gγδ(y x )∂µ Dµ[β]
µ 0 1− 2 0 1− 2 x1 ·
Z
cosh(β(x )+β(x ) β(y ) β(y )). (26)
1 2 1 2
· − −
Thecomputation of B0 is nearly identical upto formula(22), justinstead of ǫµν(∂x1β(x ))we
− ν 1
have to insert i∂µ acting on the whole four-point function (see (19), (20)). Introducing again a
x1
real β like in (24), this time only the 1cosh part of the PI contributes and we finally get
B0 = iγαβ∂µ Gβγ(x y )Gγδ(y x ) Dµ[β]
− µ x1 0 1− 2 0 1− 2 ·
Z
5
cosh(β(x )+β(x ) β(y ) β(y )) = (27)
1 2 1 2
· − −
= C0 i[∂/ G (x y )]αγGγδ(y x ) Dµ[β]
− − x1 0 1− 2 0 1− 2 ·
Z
cosh(β(x )+β(x ) β(y ) β(y )) = (28)
1 2 1 2
· − −
= C0 iδ(x y )Gαδ(y x ) Dµ[β]cosh(β(x ) β(y )). (29)
− − 1 − 2 0 1− 2 2 − 1
Z
So the first term in B0 just cancels C0, whereas the second one results in a two-point function
that will cancel A0 of (18) as we now shortly compute.
Keeping β real from the very beginning we find (because we fixed γ = ǫ again the first term
of (18) is zero!)
δ δ
A0 = δ(y x ) Z =
2 1 0
− δη (x )δη¯ (y ) |
δ 2 α 1
= iδ(y x ) Dµ[β]Gαδ(y x ) =
2 1 1 2
− −
Z
= iδ(y x )Gαδ(y x ) Dµ[β]cosh(β(y ) β(x )), (30)
2 − 1 0 1− 2 1 − 2
Z
which indeed cancels the rest of B0 in (29).
So the DS equation is separately fulfilled for k = 0, which however is the perturbative
vacuum. Therefore this result is not too surprizing.
Computation for k = 1
We compute the case k = +1, k = 1 is nearly identical. In formula (7) now one zero mode is
−
present and we obtain for C1 (20)
δ δ δ δ
C1 = ǫµνγαβ N Dβ(∂x1β¯(x ))
− µ δη (y )δη (x )δη¯ (y )δη¯ (x ) ν 1 ·
γ 2 δ 2 γ 1 β 1 Z
(η¯Ψ0)(Ψ¯0η)ei dzdz′η¯(z)G(z,z′)η(z′)e12 dzβ¯Dβ¯ =
·
R R
= iǫµνγαβN Dβ(∂x1β¯(x ))[ Ψβ(x )Ψ¯δ(x )Gγγ(y ,y )+Ψβ(x )Ψ¯γ(y )Gγδ(y ,x )+
− µ ν 1 − 0 1 0 2 1 2 0 1 0 2 1 2
Z
+Ψγ0(y1)Ψ¯δ0(x2)Gβγ(x1,y2)−Ψγ0(y1)Ψ¯γ0(y2)Gβδ(x1,x2)]e12 dzβ¯Dβ¯, (31)
where again Gγγ = 0 in the first term is zero. Inserting for the G formulaR(9), transforming the
zero modes like in (13) and introducing the zero mode projectors (see (14))
1
Sαβ(x,y) := (Ψ(β1)(x))α(Ψ¯(β1)(y))β = Pαβei(β1(x)+β1(y)) (32)
0 0 2π +
we get
C1 = iǫµνγαβN Dβ(∂x1β¯(x ))[eiβ(x1)+iβ(x2)Sβγ(x ,y )
− µ ν 1 1 2 ·
Z
(eiγ5(β¯(y1)−β¯(x2)))γǫGǫδ(y x )+eiβ(y1)+iβ(x2)Sγδ(y ,x )
· 0 1− 2 1 2 ·
(eiγ5(β¯(x1)−β¯(y2)))βǫGǫγ(x y ) eiβ(y1)+iβ(y2)Sγγ(y ,y )
· 0 1− 2 − 1 2 ·
6
·(eiγ5(β¯(x1)−β¯(x2)))βǫGǫ0δ(x1−x2)]e12 dzβ¯Dβ¯, (33)
R
and, turning to real β and using γ P = P γ = P , we further get
5 + + 5 +
C1 = −21πǫµνγµαβN Dβ(∂νx1β¯(x1))e−12 dzβ¯Dβ¯·
Z R
[sinh(β¯(x ) β¯(x )+β¯(y )+β¯(y ))PβγGγδ(y x )+
· 1 − 2 1 2 + 0 1− 2
+sinh( β¯(x )+β¯(x )+β¯(y )+β¯(y ))PγδGβγ(x y )
− 1 2 1 2 + 0 1− 2 −
eβ¯(y1)+β¯(y2)sinh(β¯(x ) β¯(x ))γβǫGǫδ(x x )
1 − 2 5 0 1− 2 −
eβ¯(y1)+β¯(y2)cosh(β¯(x ) β¯(x ))δβǫGǫδ(x x )]. (34)
− 1 − 2 0 1− 2
Here the first two terms in (34) may be treated in a way identical to that for the k = 0 case
(leading from formula (24) to (26)). The third and fourth term, after applying Leibnitz rule for
the derivative ∂x1 and keeping track of even β powers only, look like cosh(...) cosh(...) and
ν ·
sinh(...) sinh(...) respectively. However by using the formulae
·
1
coshacoshb = (cosh(a+b)+cosh(a b)),
2 −
1
sinhasinhb = (cosh(a+b) cosh(a b)), (35)
2 − −
they may be brought into standard form. Besides, these two terms, being proportional to 1 and
γ5, combine into P+, P− via formulae (35), and after some gamma matrix algebra we find the
final result
1
γαβGβγ(y x )Pγδ∂ν Dµ[β]cosh(β(x ) β(x )+β(y )+β(y ))+
2π{ ν 0 1− 2 − x1 1 − 2 1 2
Z
+γαβGβγ(x y )Pγδ∂ν Dµ[β]cosh( β(x )+β(x )+β(y )+β(y ))
ν 0 1− 2 + x1 − 1 2 1 2 −
Z
−γναβGβ0γ(x1−x2)P−γδ∂xν1 Dµ[β]cosh(β(x1)−β(x2)+β(y1)+β(y2))−
Z
γαβGβγ(x x )Pγδ∂ν Dµ[β]cosh( β(x )+β(x )+β(y )+β(y )) . (36)
− ν 0 1− 2 + x1 − 1 2 1 2 }
Z
For the computation of B1 we may again use an intermediate result for C1, (33), substitute
iǫµν(∂x1β¯(x ))by ∂x1, and find
ν 1 µ
1
B1 = γαβ∂µ PβγGγδ(y x ) Dµ[β]cosh(β(x ) β(x )+β(y )+β(y ))
2π µ x1{− + 0 1− 2 1 − 2 1 2 −
Z
βγ γδ
G (x y )P Dµ[β]cosh( β(x )+β(x )+β(y )+β(y ))+
− 0 1− 2 + − 1 2 1 2
Z
βδ
+G (x x ) Dµ[β]cosh(β(x ) β(x ))cosh(β(y )+β(y ))+
0 1− 2 1 − 2 1 2
Z
βγ γδ
+γ G (x x ) Dµ[β]sinh(β(x ) β(x ))sinh(β(y )+β(y )) = (37)
5 0 1− 2 1 − 2 1 2 }
Z
7
1
= C1 (∂/ G (x y ))αγPγδ Dµ[β]cosh( β(x )+β(x )+β(y )+β(y ))+
− − 2π x1 0 1− 2 + − 1 2 1 2
Z
+(∂/ G (x x ))αδ Dµ[β]cosh(β(x ) β(x ))cosh(β(y )+β(y ))
x1 0 1− 2 1 − 2 1 2 −
Z
(∂/ G (x x ))αγγγδ Dµ[β]sinh(β(x ) β(x ))sinh(β(y )+β(y )) = (38)
− x1 0 1− 2 5 1 − 2 1 2 }
Z
1
= C1+ δ(x y )Pαδ Dµ[β]cosh(β(x )+β(y ))+
− 2π{− 1 − 2 + 2 1
Z
+δ(x x )δαδ Dµ[β]cosh(β(y )+β(y )) , (39)
1 2 1 2
− }
Z
where the remaining two-point functions are again cancelled by A1:
δ δ δ δ
A1 = δαδδ(x x ) Z +δαγδ(y x ) Z =
2 1 0 2 1 0
− − δη (y )δη¯ (y ) | − δη (x )δη¯ (y ) |
γ 2 γ 1 δ 2 γ 1
= −δαδδ(x2 −x1) DβΨγ0(y1)Ψ¯γ0(y2)e12 dzβ¯Dβ¯+
Z R
+δαγδ(y2−x1) DβΨγ0(y1)Ψ¯δ0(x2)e12 dzβ¯Dβ¯ =
Z R
1
= δαδδ(x x ) Dµ[β]cosh(β(y )+β(y ))+
2 1 1 2
2π{− −
Z
+Pαδδ(y x ) Dµ[β]cosh(β(x )+β(y )) , (40)
+ 2 − 1 2 1 }
Z
which we wanted to prove.
The computation for k = 1 is analogous.
−
Computation for k=2
For k = +2 we find for C2
δ δ δ δ
C2 = γαβǫµν N Dβ(∂x1β¯(x ))
− µ δη (y )δη (x )δη¯ (y )δη¯ (x ) ν 1 ·
γ 2 δ 2 γ 1 β 1 Z
(η¯Ψ0)(Ψ¯0η)(η¯Ψ1)(Ψ¯1η)e12 dzβ¯Dβ¯ =
·
R
= −γµαβǫµν Dβ(∂νx1β¯(x1))e12 dzβ¯Dβ¯·
Z R
[Ψγ(y )Ψ¯δ(x )Ψβ(x )Ψ¯γ(y )+Ψβ(x )Ψ¯γ(y )Ψγ(y )Ψ¯δ(x )
· 0 1 0 2 1 1 1 2 0 1 0 2 1 1 1 2 −
Ψγ(y )Ψ¯γ(y )Ψβ(x )Ψ¯δ(x ) Ψβ(x )Ψ¯δ(x )Ψγ(y )Ψ¯γ(y )] = (41)
− 0 1 0 2 1 1 1 2 − 0 1 0 2 1 1 1 2
= −4π12γµαβǫµνN Dβ(∂νx1β¯(x1))e12 dzβ¯Dβ¯·
Z R
ei(β¯(x1)+β¯(x2)+β¯(y1)+β¯(y2))[PγδPβγx−y++
· + + 1 2
+PβγPγδy−x+ Pβδx−x+ Pβδy−y+], (42)
+ + 1 2 − + 1 2 − + 1 2
8
where we used the zero modes (14) and the transformation (13). Introducing real β we get
(ǫµνγ P = γνP )
µ + +
−
i
C2 = γαβPβδ(x− y−)(y+ x+)
−4π2 ν + 1 − 1 2 − 2 ·
∂ν Dµ[β]cosh(β(x )+β(x )+β(y )+β(y )). (43)
· x1 1 2 1 2
Z
For the B2 part we may, as usual, starting from (42), read off
i
B2 = γαβ∂µ Pβδ(x− y−)(y+ x+)
4π2 µ x1 + 1 − 1 2 − 2 ·
Dµ[β]cosh(β(x )+β(x )+β(y )+β(y )). (44)
1 2 1 2
·
Z
Further there is no contribution from k = 2 to the two-point function in A, (18), so (43) and
(44) must cancel completely, or the derivative
γαβ∂µ Pβδx− (45)
µ x1 + 1
has to vanish. This is indeed true and is in fact used to construct the zero modes (14) (see [7],
[8] for details):
γµ∂xP x− = 0 ∂x− 1 0 x− = 0 0 x− = 0. (46)
µ + ∂ 0 0 0 ∂ 0
x+ ! ! x+ !
So the DS equation holds for the sector k = 2, too.
Using
x−y+ = x yµ+ǫ xµyν (47)
µ µν
we may obtain the final result (which we display for k = 2)
±
i
B±2 = 4π2γµαβ∂xµ1P±βδ[(x1 −y1)(y2−x2)±ǫµν(x1−y1)µ(y2−x2)ν)]·
Dµ[β]cosh(β(x )+β(x )+β(y )+β(y )). (48)
1 2 1 2
·
Z
General discussion
We showed the DS equation to hold for the simplest nontrivial case, the fermion four-point
function. For the model at hand it is however easy to generalize the result to higher n-point
functions.
To study the DS equation in an instanton vacuum it remains enough to consider scalar
(or pseudoscalar) bilocals of the fermion fields, with the exception of the two fields with vari-
ables x ,x on which the DS equation acts. Therefore the spinor structure reduces to simple
1 2
gamma matrix products. Further the (∂ν β(x )) term enters the computation of the C term in
x1 1
a way completely analogous to our computation, and the PI remains Gaussian. So the partial
cancellation between B and C persists to hold.
For a n-pointfunctionin the cosh( ) thereare nβ(x ). Whenever two of them stem froma
i
···
zero modeprojector,bothβ have thesame sign;wheneverthey stem froma fermionpropagator
9
G(x ,x )theirsignsareopposite.SowhenafreepropagatorG (x ,x )isreducedtoaδ-function
i j 0 i j
by aderivative thecorrespondingβ(x ),β(x )cancel in thecosh( )andlead to then 2-point
i j
··· −
function that cancels with the term A. So indeed it remains true for higher n-point functions
that the DS equations separately hold for every instanton sector k.
The reason for this separate cancellation is easy to understand. The Schwinger model (SM)
is wellknown tobeindependentofthevacuumangle Θ([2],[7],[13],[14],[15]). Letusremember
how this feature may be seen quickly within the PI formalism. There the Θ vacuum may be
taken into account by an additional term in the action
Θ
S S +ikΘ =S + dxF˜(x) (49)
→ 2π
Z
and by integrating over all instanton configurations (summing over all k) in the vacuum func-
tional. But the additional term on the r.h.s. of (49) is nothing else than the index density. An
axial transformation of the fermion fields creates a similar term in the action via the chiral
anomaly:
1
= dxβF˜. (50)
A π
Z
Usually formula (50) is derived for an infinitesimal axial transformation β, for a constant β
formula (50) remains true for finite β. Therefore by choosing
Θ
β = (51)
−2
the Θ term may be absorbed by a simple change of the fermionic integration variables in the
PI via an axial transformation.
Our instanton vacuum corresponds to the choice Θ = 0. For general Θ a VEV from the
sector k acquires an additional phase eikΘ. Now when the SM is invariant with respect to Θ the
DS equations have to be invariant, too. When the DS equations enforced cancellations between
different instanton sectors, those cancellation conditions would acquire a Θ dependence. On
the other hand, when the DS equations are fulfilled separately for every instanton sector, they
automatically are Θ invariant.
Of course, the interesting question remains to be answered if these simple features of the
DS equations in an instanton vacuum may be generalized to more complicated theories.
References
[1] J. Schwinger, Phys. Rev. 128 (1962) 2425
[2] J. Lowenstein, J. Swieca, Ann. Phys. 68 (1971) 172
[3] C. Jayewardena, Helv. Phys. Acta 61 (1988) 636
[4] I. Sachs, A. Wipf, Helv. Phys. Acta 65 (1992) 653
[5] A. Dettki, I. Sachs, A. Wipf, preprint ETH-TH/93-14
[6] A. V. Smilga, Phys. Rev. D46 (1992) 5598
[7] C. Adam, Z. Phys. C63 (1994) 169