Instantons in Field Strength Formulated Yang-Mills Theories

12 Pages·0.13 MB·English

by  H. Reinhardt

November 5, 1992
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a Instantons in field strength formulated Yang-Mills theories1
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1 H. Reinhardt, K. Langfeld, L. v. Smekal

1 Institut fu¨r theoretische Physik, Universit¨at Tu¨bingen
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D–7400 Tu¨bingen, Germany
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: Abstract
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It is shown that the field strength formulated Yang-Mills theory yields the
r same semiclassics as the standard formulation in terms of the gauge poten-
a
tial. This concerns the classical instanton solutions as well as the quantum
fluctuations around the instanton.
1 Supported by DFG under contract Re 856/1 − 1
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1. Introduction
Yang-Mills (YM) theories can be reformulated entirely in terms of field strength
thereby eliminating the gauge potential [1, 2, 3]. The field strength formulation
allows for a non-perturbative treatment of the gluon self-interaction and offers a
simple description of the non-perturbative vacuum where the gauge bosons are pre-
sumable condensed. In the so called FSA an effective action for the field strength
is obtained which contains a term of order h¯ with an explicit energy scale. Hence
in the FSA the anomalous breaking of scale invariance is described already at tree
level [2]. At this level the YM vacuum is determined by homogeneous field configu-
rations with constant hFa Fa i =6 0 and instantons do no longer exist as stationary
µν µν
points of the effective action. This is already dictated by the fact that the effective
action contains an energy scale. This energy scale cannot tolerate instantons which
have a free scale (size) parameter.
In this paper we show that if the FSA is consistently formulated in powers of
h¯ one recovers the same semiclassics as in the standard formulation of YM theory
in terms of the gauge potential. All instantons which extremize the standard YM
action are also stationary points of the action to O(h¯) of the FSA. Furthermore
the leading h¯ corrections originating from the integral over quantum fluctuations
around the instanton are also the same in both approaches. Although one might
have expected this result on general grounds it is completely non-trivial how this
result emerges in the field strength formulation. Furthermore the equivalence proof
will also shed some new light on the FSA.
2. Semiclassical approximation to the Yang-Mills theory
We start from the generating functional of Euclidean YM theory
Z[j] = DA δ(fa(A))DetM exp{−S [A] + d4x jA} , (1)
f YM
Z Z
where
1
S = d4x Fa (A)Fa (A) , Fa (A) = ∂ Aa−∂ Aa+fabcAbAc , (2)
YM 4g2 µν µν µν µ ν ν µ µ ν
Z
is the classical YM action. Furthermore δ(fa(A)) is the gauge fixing constraint and
DetM denotes the Faddeev-Popov determinant.
f
Itiswell known that allfinite actionself-dualor antiself-dual fieldconfigurations
extremize the YM action [5] i.e. solve the classical YM equation of motion
∂ Fa = −fabcAbFc . (3)
µ µν µ µν
2
These classical solutions are referred to as instantons. Expanding the fluctuating
gauge field around the classical instanton solution Ainst up to second order in the
µ
quantum fluctuations and performing the integral in semiclassical approximation
one finds
Z[j = 0] = Q (Det′D−1 (x ,x ))−1/2 e−SYM[Ainst] , (4)
YM 1 2
where
δ2S [A]
D−1 (x ,x )ab = YM | (5)
YM 1 2 µν δAa(x )δAb(x ) Aµ=Aiµnst
µ 1 ν 2
and the prime indicates that the zero modes of D−1 have to be excluded from the
YM
determinant. This canbedoneinthestandardfashionandyields thefactorQin(4).
Furthermore even when the zero modes are excluded the determinant is still singular
andneeds regularization. Aswillbecome clear later, forour purposearegularization
scheme that only depends on eigenvalues is convenient, e.g. Schwinger’s proper time
regularization or ζ-function regularization. The second variation of the YM action
reads
1 δFc (x) δFc (x)
g2D−1 (x ,x )ab = Fˆab(x )δ(x ,x ) + d4x κλ κλ , (6)
YM 1 2 µν µν 1 1 2 2 δAa(x )δAb(x )
Z µ 1 ν 2
where
Fˆab = fabcFc (7)
µν µν
denotes the field strength in the adjoint representation and
δFc (x)
κλ = [Dˆca(x)δ − Dˆca(x)δ ] δ(x−x ) , (8)
δAa(x ) κ µλ λ µκ 1
µ 1
with
Dˆab(x) = ∂ δab − Aˆab(x) (9)
µ µ µ
being the covariant derivative. The functional determinant DetD−1 in an instanton
YM
background has been explicitly evaluated by t’Hooft [6].
3. Field strength formulated Yang-Mills theory
Non-AbelianYMtheory canbeequivalently formulatedin termsof field strength
[1, 2, 3]. Inserting the identity
1 1 i
exp{− d4xF2(A)} = Dχ exp − d4x{ χa χa + χa Fa (A)}
4g2 4 µν µν 2g µν µν !
Z Z Z
(10)
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into (1) we obtain
Z = Dχ DA δ(fa(A)) DetM exp{−S[χ,A] + d4x jA} (11)
f
Z Z
1 1 i
S[χ,A] = d4x χa χa ± χa Fa (A) , (12)
g2 4 µν µν 2 µν µν
Z (cid:18) (cid:19)
If there were no gauge fixing the integral over the gauge field would be Gaussian.
At first sight it seems that with the presence of the gauge fixing constraint the A -
µ
integration can no longer be performed explicitly. However, one can transfer the
gauge fixing from the gauge potential A to the field strengths. For this purpose we
µ
insert the following identity into (12)
1 = DetM (χ) d(θ)δ(ga(χθ)) , (13)
g
Z
where d(θ) is the invariant measure of the functional integration over the group
space, χθ denotes the gauge transformed of χ, and DetM (χ) does not depend
g
on θ. The key observation now is that the action S[χ,A] in (12) is only invariant
under simultaneous gauge transformations of A and χ. Therefore a change of
the integration variable χθ → χ implies also a change in the gauge potential
A → A(−θ) to leave the exponent in (12) invariant. Because of the gauge invariance
µ µ
of the measure and the determinants one then obtains
Z = d(θ) DχDA δ(fa(Aθ−1)) δ(gb(χ)) DetM DetM e−S[χ,A] . (14)
f g
Z Z
Now the integration over the gauge group can be performed again yielding
Z = DχDA δ(gb(χ))DetM exp −S[χ,A]+ d4xjA . (15)
g
Z (cid:26) Z (cid:27)
Fixing the gauge in terms of field strengths leaves a residual invariance with re-
spect to transformations, which leave the field strengths invariant. In the case of
YM theories these transformations belong to the discrete invariant subgroup of the
gauge group. Therefore (in contrast to the Abelian case) this residual invariance is
harmless.
Once the field strength χ is gauge fixed there is no invariance left in the poten-
tials (up to the irrelevant residual invariance mentioned above) and the integration
DA becomes unconstraint. For non-singular χˆab it yields
µν
R
i
Z = Dχ δ(ga(χ)) DetM (Det χˆ)−1/2 exp{−S [χ,j]} (16)
g FS
2g
Z
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S [χ,j] = 1 d4x 1 χa χa + i χa Fa (J)+jaJa −ig2ja(χˆ−1)abjb ,
FS g2 4 µν µν 2 µν µν µ µ 2 µ µν ν!
Z
(17)
where
Ja = χˆ−1 ab ∂ χb (18)
µ ρ ρν
µν
(cid:16) (cid:17)
is an induced gauge potential. For singular χˆ, integration over the gauge potential
Aa(x) yields an expression similar to (16), where the matrix χˆ is, however, replaced
µ
by its projection onto the non-singular subspace. But in addition constraints for the
χ-integration result. These constraints indicate that singular field configurations χˆ
are statistically suppressed. Since χa behaves under gauge transformations as the
µν
field strength Fa (A) the induced potential Ja transforms precisely like the original
µν µ
gauge field Aa. In practice, gauge fixing of the χ can be done by using the familiar
µ
gauges for the induced gauge potential Ja (18).
µ
The presence of the external source ja in the exponent of (16) ensures that
µ
Green’s functions of the original gauge potential Aa are still accessible in the field
µ
strength formulation.
Finally in the field strength formulation a current current interaction is induced
which dominates the fermion dynamics at low energies [7, 8, 9, 10].
4. Instantons in the field strength formulation
We are interested in a semiclassical analysis of the field strength formulated YM
theory. For simplicity we discard the external gluon source (ja = 0). The extrema
µ
of the action S [χ] = S [χ,j = 0] occur for
FS FS
gχa = −iFa (J). (19)
µν µν
It was observed by Halpern [1] that the effective action of the χ field is extremized
by the standard Polyakov t’Hooft instantons. This fact is, however, not only true
for the standard SU(2) instanton but holds for any classical solution extremizing the
Yang-Mills action (2). For a proof we rewrite the classical YM equation of motion
(3) with the definition (18) as
Ja(F(A)) = Aa(x) (20)
µ µ
where we have for simplicity assumed that Fˆab(A) is not singular. Now let Ainst
µν µ
denote an instanton solution to (20). Since Ja(χ = −iF/g) = J(F) it follows from
µ
(20) that the equation of motion of the field strength formulation (19) is solved
indeed for
i
χa = − Fa (Ainst) . (21)
µν g µν µ
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Furthermore, it follows then that also the classical action of the instanton in the
field strength formulation is the same as in the standard approach
i
S [χ = − F(Ainst)] = S [Ainst] . (22)
FS g µ YM µ
The equivalence between both approaches holds, however, not only at the classi-
cal level but also the quantum fluctuations give identical contributions as we will
explicitly prove in the following for the leading order in h¯ corrections.
Inthefield strength formulationquantum fluctuations arounda background field
were considered in [3]. In the semiclassical approximation the background field is
chosen as the instanton (21). If we expand the tensor field χa in terms of the
µν
t’Hooft symbols [6] ηi , η¯j
µν µν
χa = χaZi , Zi = {ηi , η¯j } (23)
µν i µν µν µν µν
the generating functional (16) becomes then
Z[j = 0] = Q (Deti χˆinst)−1/2 (Det′D−1[χinst])−1/2 e−SFS[χinst] (24)
FS g FS
where we have used (22) and [3]
δ2S [A]
DF−S1[χ](x1,x2)aijb = δχa(x F)δSχb(x ) |χinst (25)
i 1 j 2
isthesecondvariationofthefieldstrengthactiontakenatabackgroundfield χinst =
−iF(Ainst)/g. The prime indicates again that zero modes are excluded. Their
contribution is included in the factor Q . For space-time dependent dependent
FS
background fields χa = −iFa (x)/g one finds
µν µν
D−1[F](x ,x )ab = 2δabδ δ(x ,x ) + Kab[F](x ,x ) (26)
FS 1 2 ij ij 1 2 ij 1 2
Kab[F] = −Dˆac(x )Zi (Fˆ−1)cdDˆdb(x )Zj δ(x ,x ) (27)
ij κ 1 κµ µν λ 1 λν 1 2
where Dˆab denotes here the covariant derivative (9) with respect to the induced
µ
gauge potential Ja(χ = −iF) = Ja(F)
µ µ
Dˆab(x) = ∂ δab − Jˆab(x) . (28)
µ µ µ
For a constant background field the above expressions for the fluctuations reduce to
the expressions given in [3]. Comparison of (4) and (24) shows if both approaches
give the same semiclassical result we should have the relation
Q(Det′g2D−1 [A ](x ,x ))−21 = CQ (DetFˆ Det′D−1[F ])−21 , (29)
YM inst 1 2 FS inst FS inst
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where C is an irrelevant (but non-vanishing ) constant, which does not depend on
the instanton solution. We will now explicitly prove this relation.
5. Equivalence proof
In order to establish the validity of (29) we cast the functional matrix (6) of the
standard approach YM into the form of its field strength formulated counterpart
(26) by writing
2g2D−1 [F](x ,x )ab = (Fˆ1/2)ac(x )Mce[F](x ,x )(Fˆ1/2)eb(x ) (30)
YM 1 2 µν µσ 1 σρ 1 2 ρν 2
δFc (x) δFc (x)
Mab[F] = 2δabδ δ(x ,x ) + (Fˆ−1/2)ad d4x κλ κλ (Fˆ−1/2)eb . (31)
µν µν 1 2 µσ δAd(x )δAe(x ) ρν
Z σ 1 ρ 2
We first prove that M[F] has the same eigenvalues (including the zero modes ) as
D−1[F]. Let φa(x) and λ denote the eigenvectors and eigenvalues of M:
FS µ
d4x Mab[F](x ,x )φb(x ) = λφa(x ) (32)
2 µν 1 2 ν 2 µ 1
Z
Defining
δFc (x)
ψc (x) := d4x κλ (Fˆ−1/2)eb(x )φb(x ) (33)
κλ 2 δAe(x ) ρν 2 ν 2
Z ρ 2
the eigenvalue equation becomes
d4x K¯dc [F](y,x)ψc (x) = (λ−2)ψd (y) (34)
ρσ,κλ κλ ρσ
Z
where
δFd (y) δFc (x)
K¯dc [F](y,x) = d4x ρσ [Fˆ−1(x )]ab κλ . (35)
ρσ,κλ 1 δAa(x ) 1 µνδAb(x )
Z µ 1 ν 1
By construction the amplitudes ψa are antisymmetric in (µ,ν) and can hence be
µν
expanded in terms of t’Hooft symbols (c.f. (23))
ψa = ψaZi . (36)
µν i µν
The eigenvalue equation (34) then reads
d4x K¯dc[F](y,x)ψc(x) = (λ−2)ψd(y) (37)
ij j i
Z
1
K¯ab(y,x) = Zi K¯[F](y,x)ab Zj . (38)
ij 4 µν µν,κλ κλ
Inserting the explicit form of δFa (x)/δAb(y) (8) into K¯[F] (35) the integration over
κλ µ
the intermediate coordinate x can be carried out upon using
1
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Dˆab(x)δ(x−y) = −Dˆba(y)δ(x−y). Exploiting also the antisymmetry of the Zi
µ µ µν
¯
the kernel K[F] takes the form
K¯ab(x,y) = −Dˆac(x)Zi (Fˆ−1(x))cdDˆdb(x)Zj δ(x−y) . (39)
ij µ µκ κλ ν νλ
For an instanton background field A we have in view of (20) Ja(x) = Aa(x) and
µ µ µ
the covariant derivatives in (9) and (28) are the same so that the kernels K (27)
¯
and K (39) are identical. We thus proved that all eigenvalues of M (31) are also
eigenvalues of D−1, and the corresponding eigenvectors are related by (33).
FS
On the other hand, not all eigenvalues of D−1 (26) are also eigenvalues of M
FS
(31). This is because the transformation (33) maps the Hilbert space of eigenstates
D
of M of dimension n = D(N2 − 1) onto a subspace of the m = (N2 − 1)
2
!
dimensional space of eigenvectors of K¯. This implies that the m − n additional
eigenvectors of K¯, denoted by ψ(0)c, are zero modes
µν
d4x K¯dc (y,x)ψ(0)c(x) = 0 (40)
ρσ,κλ κλ
Z
satisfying
δFc (x)
d4x κλ ψ(0)c(x ) = 0 . (41)
1 Aa(x ) κλ 1
Z µ 1
These additional zero eigenvalues of K¯ give rise to additional eigenvalues 2 of D−1
FS
in the field strength formulation. The latter contribute only an irrelevant constant
to the functional determinant of D−1, which can be absorbed into the constant C
FS
in (29).
Iftherewerenozeromodestheproofof(29)wouldbecompleted. Thisisbecause
in the absence of zero modes (Q = Q = 1) from (30) would follow
FS
DetD−1 = DetFˆ DetM (42)
YM
and we have shown above that
Det′M[F] = CDet′D−1 (43)
FS
provided the same regularization is used.
In the presence of zero modes some more care is required. Their contribution [4]
to the functional integrals is represented by the preexponential factors in (4) and
(24)
Q = Det1/2hδ A | δ A i and Q = Det1/2hδ χ | δ χ i (44)
i cl k cl FS i cl k cl
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respectively. Here δ A denotes the variation of the classical (instanton) solution
i cl
with respect to its ith symmetry parameter, which is the (unnormalized) zero mode.
Its counterpart δ χ in the field strength formulation is related to δ A by
i cl i cl
i i δFa
δ (χ )a (x) = − δ Fa [A ] = − d4x κλ δ A b(x) . (45)
i cl κλ g i κλ cl g δAb(x) i clν
Z ν
Exploiting the fact D−1 δ A = 0 one readily verifies that
YM i cl
Q2 = Dethδ A | Fˆ | δ A i = Q2 Det(0)Fˆ , (46)
FS i cl k cl
where Det(0)Fˆ is the determinant of the matrix arising from projection Fˆ onto the
space of zero modes of D−1 . Inserting (46) into (29) it remains to be proven that
YM
1 Det(0)Fˆ
= . (47)
Det′D−1 DetFˆ Det′D−1
YM FS
In order to extract Det′D−1 from Det′D−1 it is convenient to introduce the com-
FS YM
plete set of orthonormal eigenvectors ϕ and φ of the symmetric matrices D−1 =
i i YM
Fˆ1/2MFˆ1/2 and M. We denote the normalized zero and non-zero modes of D−1
YM
(M) by ϕ(0), (φ(0)) and ϕ′, (φ′), respectively. Accordingly Det(0)Fˆ and Det′Fˆ de-
i i i i
note the respective subspace determinants of Fˆ. From the defining equation (30)
follows that the vectors {Fˆ1/2ϕ(0)} span the space of zero modes {φ(0)} of M and
k k
conversely, the {Fˆ−1/2φ(0)} span the space of zero modes {ϕ(0)} of Det′D−1 . From
k k YM
the orthogonality of the φ and the ϕ then follows
k k
φ′TFˆ1/2ϕ(0) = 0 and ϕ′TFˆ−1/2φ(0) = 0 (48)
i k i k
respectively, implying that we may write
Fˆ1/2ϕ(0) = U φ(0) and Fˆ−1/2ϕ′ = O φ′ . (49)
l lm m l lm m
For later use we calculate the determinant of the matrices U and O. Due to the
orthonormality of the eigenvectors ϕ we have
l
ϕ(0)Tϕ(0) = U U φ(0)TFˆ−1φ(0) = δ , ϕ′Tϕ′ = O O φ′TFˆφ′ = δ
i k il km l m ik i k il km l m ik
(50)
and thus
1 = DetU Det(0)Fˆ−1DetUT and 1 = DetODet′FˆDetOT , (51)
implying
Det′Fˆ = (DetO)−2 , and Det(0)Fˆ = (DetU)2 . (52)
9
In order to show that
DetFˆ = Det′Fˆ Det(0)Fˆ , (53)
we note that the orthonormal sets ϕ and φ are related by an orthogonal transfor-
i k
mation. Therefore we may write
′
φ
DetFˆ1/2 = Det ϕ′T,ϕ(0)T Fˆ1/2 k . (54)
(cid:18) l l (cid:19) φ(k0) !
In view of (48) the matrix on the right hand side is of triangular block form.
DetFˆ1/2 = Det ϕ′TFˆ1/2φ′ Det ϕ(0)TFˆ1/2φ(0)
l k l k
= Det(cid:16)O φ′ Fˆφ′(cid:17) Det(cid:16) U φ(0)Tφ(0)(cid:17)
lm m k lm m k
= DetO(cid:16) Det′Fˆ Det(cid:17)U (cid:16) (cid:17)
= Det′Fˆ 1/2 Det(0)Fˆ 1/2 , (55)
(cid:16) (cid:17) (cid:16) (cid:17)
An analogous manipulation, again using (48) and (49), shows that the determi-
nant Det′D−1 in (47) factorizes
YM
Det′D−1 = Det ϕ′TFˆ1/2φ′ φ′TMφ′ φ′TFˆ1/2ϕ′
YM k m m n n l
= DetO(cid:16) Det′Fˆ Det′M Det′Fˆ DetOT(cid:17)= Det′Fˆ Det′M . (56)
This completes the proof of (47), which establishes explicitly the equivalence
of the field strength formulation and the standard formulation at the semiclassical
level.
In the so called field strength approach of [2] the (Detχˆ)−1/2 arising from the
integration over the gauge field is included into an effective action
1 µ4 i i
S = d4x {χ2 + trln χˆ/µ2 + χF(J) } (57)
FSA
4 2 g 2g
Z
where the scale µ arises from the regularizations of Trln i χˆ. Due to the appearance
g
of the scale this effective action does no longer tolerate instantons as stationary
points [11] as one might have expected since instantons have a free scale (size)
parameter. This was explicitly shown already in [2] for t’Hooft-Polyakov instantons.
Instead of instantons the effective action (57) has (up to gauge transformations)
constant solutions χ = −iG, which can be interpreted as instanton solids [11]. If
one considers fluctuations around these constant solutions the propagator of the
fluctuations is given by [3]
D−1 = 2(1 + C) + K (58)
FSA
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