An approach to solve Slavnov–Taylor identities in nonsupersymmetric non-Abelian gauge theories

21 Pages·2008·0.22 MB·English

USM-TH-122
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An approach to solve Slavnov–Taylor identities
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in nonsupersymmetric non-Abelian gauge theories
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Igor Kondrashuk, Gorazd Cvetiˇc, and Iva´n Schmidt

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Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa,

2 Avenida Espan˜a 1680, Casilla 110-V, Valpara´ıso, Chile
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1 e-mail: igor.kondrashuk@fis.utfsm.cl
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e-mail: gorazd.cvetic@fis.utfsm.cl
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e-mail: ivan.schmidt@fis.utfsm.cl
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h Abstract
:
v We present a way to solve Slavnov–Taylor identities in a general nonsupersym-
i metric theory. The solution can be parametrized by functions bilocal in spacetime
X
coordinates, so that all the effective fields are dressed by these bilocal functions. The
r
a solution restricts the ghost part of the effective action and gives predictions for the
physical part of the effective action.
Keywords: Slavnov–Tayloridentities, Legendre transformation, effective action.
PACS: 11.15.-q,11.15.Tk
1 Introduction
The effective action is an important quantity of the quantum theory. Defined as the
Legendretransformation of thepathintegral, itprovides uswithan instrumentto findthe
true vacuum state of the theory under consideration and to study its behavior taking into
account quantum corrections. Slavnov–Taylor (ST) identities are also an important tool
to prove the renormalizability of gauge theories in four spacetime dimensions [1, 2]. They
generalize Ward–Takahashiidentities ofquantumelectrodynamics tothenon-Abeliancase
and can be derived starting from the property of invariance of the tree-level action with
respect to BRST symmetry [3, 4]. ST identities for the effective action have been derived
in Ref. [5].
Slavnov–Taylor identities areequationsinvolvingvariational derivativesoftheeffective
action. The effective action contains all the information about quantum behaviour of the
theory, and in quantum field theory it is the one particle irreducible diagram generator.
Searching for the solution to Slavnov–Taylor identities can be considered as a complemen-
tary method to the existing nonperturbative methods of quantum field theory such as the
Dyson–Swinger and Bethe–Salpeter equations. The solution to Slavnov–Taylor identities
in the four-dimensionalsupersymmetrictheory has been proposedrecently [6]. In thepro-
ceduretoderivethatsolution, theno-renormalization theorem forsuperpotential[7,8]has
been used extensively. In this paper we will suggest that this point is not crucial and that
arguments similar as those given before[6] can beused in the nonsupersymmetriccase. In
the approach developed below there are no restrictions on the number of dimensions and
renormalizability of the theory. We require only that the theory under consideration can
be regularized in such a way that the Slavnov–Taylor identities are valid and that BRST
symmetry is anomaly free.
We argue that the functional structure of the auxiliary ghost-ghost Lc2 correlator in
nonsupersymmetric gauge theories is fixed by Slavnov–Taylor identities in a unique way.
In this correlator L is a nonpropagating background field and it is coupled at thetree level
to the BRST transformation of the ghost field c. According to our assumption, the vertex
Lc2 is invariant with respect to ST identities and this then gives the following quantum
structure for it:
i
dx′dxdydz G (x′−x) G−1(x′−y) G−1(x′−z) fbcaLa(x)cb(y)cc(z). (1)
c c c 2
Z
Asonecansee,themainfeatureofthisresultisthattheeffective ghostfieldcisdressedby
the unknown bilocal function G−1(x−y). This bilocal dressing contains all the quantum
c
information aboutthis correlator. We can use the structureof this correlator as a starting
point to find the solution for the total effective action.
The solution to the Slavnov–Taylor identities found in the present paper imposes re-
strictions on the ghost part of the effective action. For example, it means that the gluon-
ghost-antighost vertex can be read off from our result for the effective action (58):
G˜ (q2)
A
G (q,p) = iq , (2)
m mG˜ (k2)G˜ (p2)
A c
where G˜ is the Fourier image of the bilocal function that dresses gauge field, while
A
G (q,p) is the gluon-ghost-antighost vertex, q is the momentum of the antighost field b
m
and p is the momentum of the ghost field c, and p+k +q = 0. Another feature of the
result obtained here is that the physical part of the effective action (58) is gauge invariant
in terms of the effective fields dressed by the bilocal functions.
Thepaper is organized in the following way. In Section 2 we review some basic aspects
of BRST symmetry and Slavnov–Taylor identities for the irreduciblevertices. In Section 3
we show how to obtain the functional structure (1) of the Lc2 correlator. In Section 4 we
obtain the correlator linear in another nonpropagating background field K , thus fixing
m
the terms in the effective action which contain ghost and antighost effective fields. In
Section 5 we describe higher correlators in K and L. In Section 6 we make a conjecture
m
about the form of the physical (pure gluonic) part of the effective action and then in
Section 7 we consider renormalization of it to remove infinities. A brief summary is given
at the end. The questions of consistency of this effective action within perturbative QCD
are investigated in a second paper [9]. For simplicity, in the present paper we focus on
pure gauge theories in four spacetime dimensions with SU(N) gauge group. No matter
field is included in the consideration, although their addition does not change our results.
2 Preliminaries
We consider the traditional Yang–Mills Lagrangian of the pure gauge theory
1
S = − dx Tr[F (x)F (x)] (3)
mn mn
2g2
Z
The gauge field is in the adjoint representation of the gauge group. A nonlinear local
(gauge) transformation of the gauge fields exists which keeps theory (3) invariant. This
symmetry must be fixed, Faddeev–Popov ghost fields [10] must be introduced and finally
the BRST symmetry can be established for the theory that in addition to the classical
action (3) contains a Faddeev–Popov ghost action and the gauge-fixing term.
To be specific, we choose the Lorentz gauge fixing condition
∂ A (x) = f(x). (4)
m m
Here f is an arbitrary function in the adjoint representation of the gauge group that is
independent on the gauge field. The normalization of the gauge group generators is
1
Tr TaTb = δab, (Ta)† = Ta, Tb,Tc = ifbcaTa,
2
(cid:16) (cid:17) h i
and we use notation X = XaTa for all the fields in the adjoint representation of the gauge
group, like gauge fields themselves, ghost fields, and their respective sources.
The conventional averaging procedure with respect to f is applied to the path integral
with the weight
1
−i d x Tr f2(x)
e α
Z
and as the result we obtain the path integral
Z[J,η,ρ,K,L] = dA dc db expi S[A,b,c]
(
Z
+2 Tr dx J (x)A (x)+i dx η(x)c(x)+i dx ρ(x)b(x) (5)
m m
(cid:18)Z Z Z (cid:19)
+2 Tr i dx K (x)∇ c(x)+ dx L(x)c2(x) .
m m
(cid:18) Z Z (cid:19))
in which
1 1
S[A,b,c] = d x − Tr[F (x)F (x)]− Tr [∂ A (x)]2
mn mn m m
2g2 α
Z (cid:20) (cid:18) (cid:19)
−2 Tr ( i b(x)∂ ∇ c(x))]. (6)
m m
Here the ghost field c and the antighost field b are Hermitian, b† = b, c† = c in the adjoint
representation of the gauge group. They possess Fermi statistics.
The infinitesimal transformation of the gauge field A is defined by the fact that it is
m
a gauge connection,
A → A −∇ λ,
m m m
whereλ(x)isaninfinitesimalparameterof thegaugetransformation. Thistransformation
comes from the transformation of covariant derivatives,
∇ → eiλ ∇ e−iλ, ∇ = ∂ +iA , φ→ eiλ φ,
m m m m m
where φ is some representation of the gauge group. To obtain the BRST symmetry we
have to substitute i c(x) ε instead of λ. Here ε is Hermitian Grassmannian parameter,
ε† = ε, ε2 = 0. Thus, the BRST transformation of the gauge field is
A → A −i ∇ c ε. (7)
m m m
In order to obtain the BRST transformation of the ghost field c we have to consider two
subsequent BRST transformations
∇ → e−cκe−cε ∇ ecεecκ = e−cε−cκ−(cε)(cκ) ∇ ecε+cκ+(cε)(cκ), (8)
m m m
whereκis aGrassmannian parameter too, κ2 = 0. Thistransformation again is equivalent
to an infinitesimal transformation of the gauge field in covariant derivatives,
A → A −i∇ [cε+cκ+(cε)(cκ)].
m m m
It means that we can consider the inner BRST transformation (with ε) as the substitution
(7) in the outer BRST transformation (with κ). The second term after the covariant
derivative is a transformation of A under the outer BRST transformation while the
m
third term after the covariant derivative is the transformation of i∇ cκ and can it be
m
cancelled by the transformation of the second term cκ
c→ c+c2ε. (9)
Thus, the transformations (7) and (9) together leave the covariant derivative of the ghost
field unchanged. Such a symmetry is very general and always exists if the gauge fixing
procedure has been performed in the path integral for any theory with nonlinear local
symmetry. The noninvariance of the gauge fixing term is cancelled by the corresponding
transformation of the antighost field b.
To collect all things together, the action (6) is invariant with respect to the BRST
symmetry transformation with Grassmannian parameter ε,
A → A −i ∇ c ε,
m m m
c → c+c2ε, (10)
1
b → b− ∂ A ε.
m m
α
The external sources K and L of the BRST transformations of the fields are BRST
invariantbydefinition,sothelasttwo linesintheEq. (5)areBRSTinvariant withrespect
to the transformations (10).
The effective action Γ is related to W = i ln Z by the Legendre transformation
δW δW δW
A ≡ − , ic ≡ − , ib≡ − (11)
m
δJ δη δρ
m
Γ = −W −2 Tr d x J (x) A (x)+ d x iη(x) c(x)+ d x iρ(x) b(x)
m m
(cid:18)Z Z Z (cid:19)
≡ −W −2 Tr(Xϕ), (12)
(Xϕ) ≡ iG(k) Xkϕk ,
X ≡ (J ,η,ρ), ϕ(cid:16)≡ (A(cid:17),c,b).
m m
where G(k) = 0 if ϕk is Bose field and G(k) = 1 if ϕk is Fermi field. We use throughout
the paper notation
δ δ
= Ta
δX δXa
for any field X in the adjoint representation of the gauge group. Iteratively all equations
(11) can be reversed,
X = X[ϕ,K ,L]
m
and the effective action is defined in terms of new variables, Γ = Γ[ϕ,K ,L]. Hence, the
m
following equalities take place
δΓ δΓ δW δΓ δΓ δΓ δW
= −J , = − , = iη, = iρ, = − . (13)
m
δA δK δK δc δb δL δL
m m m
If the change of fields (10) in the path integral (5) is made one obtains the Slavnov–
Taylor identity as the result of invariance of the integral (5) under a change of variables,
δ δ
Tr dxJ (x) − dxiη(x)
m
δK (x) δL(x)
(cid:20)Z m Z (cid:18) (cid:19)
1 δ
+ dxiρ(x) ∂ W = 0, (14)
m
α δJ (x)
Z (cid:18) m (cid:19)(cid:21)
or, taking into account the relations (13), we have [2]
δΓ δΓ δΓ δΓ
Tr d x + d x
δA (x)δK (x) δc(x)δL(x)
(cid:20)Z m m Z
δΓ 1
− d x ∂ A (x) = 0. (15)
m m
δb(x) α
Z (cid:18) (cid:19)(cid:21)
The problem is to find the most general functional Γ of the variables ϕ,K ,L that
m
satisfies the ST identity (15). We present in this paper the solution for Lc2 proper cor-
relator only. That is, we will consider the restrictions that the identity (15) imposes on
that proper correlator. Before doing it, we need in addition to ST identities also the ghost
equation that can be derived by shifting the antighost field b by an arbitrary field ε(x) in
the path integral (5). The consequence of invariance of the path integral with respect to
such a change of variable is (in terms of the variables (11)) [2]
δΓ δΓ
+∂ = 0. (16)
m
δb(x) δK (x)
m
The ghost equation (16) restricts the dependence of Γ on the antighost field b and on the
external source K to an arbitrary dependence on their combination
m
∂ b(x)+K (x). (17)
m m
This equation together with the third term in the ST identities (15) is responsible for
the absence of quantum corrections to the gauge fixing term. Stated otherwise, when
expressing δΓ/δb(x) in the third term in the ST identity (15) as −∂ (δΓ/δK (x)) by
m m
Eq. (16), the sum of the first and the third term in (15) can be rewritten as
δΓ′ δΓ′
Tr dx ,
δA (x)δK (x)
Z m m
where Γ′ ≡ Γ−S(g.f.), and S(g.f.) = −(1/α)Tr dx[∂ A (x)]2 is the gauge–fixing part of
m m
Z
theclassical action (6). Infact, alltheother termsintheSTidentity (15)can berewritten
with Γ′ instead of Γ, yielding
δΓ′ δΓ′ δΓ′ δΓ′
Tr d x + d x = 0. (18)
δA (x)δK (x) δc(x)δL(x)
(cid:20)Z m m Z (cid:21)
This shows explicitly that the gauge–fixing part of Γ remains unaffected by quantum
corrections (Γ = Γ′+Γ(g.f.); Γ(g.f.) = S(g.f.)).
3 Functional structure of Lc2 vertex
One can consider the part of the effective action that depends only on the fields L and c.
We write generally
Γ| = dx dy dy Γ|(a1;b1,b2)(x ;y ,y )La1(x )cb1(y )cb2(y )+...
L,c 1 1 2 L,c 1 1 2 1 1 2
Z
...+ dx ... dx dy ... dy Γ|(a1,...,an;b1,...,b2n)(x ,...,x ;y ,...,y ) La1(x )×
1 n 1 2n L,c 1 n 1 2n 1
Z
×...Lan(x )cb1(y )...cb2n(y )+... (19)
n 1 2n
We assume that the first term is invariant with respect to the second operator in the
identities (15) which is
δΓ δΓ
Tr d x = 0. (20)
δc(x)δL(x)
Z
In such a case, it will be shown below that the only solution for this term is
dxdx dy dy G (x−x ) G−1(x−y ) G−1(x−y ) 2Tr(L(x )c(y )c(y )). (21)
1 1 2 c 1 c 1 c 2 1 1 2
Z
The operator (20) can beconsidered as an infinitesimal substitution in the effective action
δΓ
c(x) → c(x)+ . (22)
δL(x)
In other words, one can consider the result of such a substitution as the difference
δΓ
Γ L,c(x)+ −Γ[L,c(x)],
δL(x)
(cid:20) (cid:21)
δΓ
to linear order in .
δL(x)
To prove (21), we consider the proper correlator
Γ = dx dy dz Γ(x,y,z)TabcLa(x)cb(y)cc(z). (23)
Z
In perturbation theory it can be understood as a correction to the vertex Lc2 (nothing
forbids us to consider the auxiliary field L as a non-propagating background field). Tabc is
somegroupstructure. Equation(23) already suggests thatΓ(a;b,c)(x,y,z) = Γ(x,y,z)Tabc,
where Tabc is a 3-tensor in the adjoint representation of the gauge group, antisymmetric
in its last two indices. This reflects the fact that the global symmetry of the gauge group
must be conserved in the effective action. With respect to that symmetry the auxiliary
fields Ka and La are vectors in the adjoint representation of the gauge group.
δΓ
= dy dz Γ(x,y,z) Tabccb(y)cc(z).
δLa(x)
Z
By substituting this expression in the Slavnov–Taylor identity (15) we have
δΓ δΓ δΓ
dx = dx dy′ dz′ Γ(y′,x,z′)TdabLd(y′) cb(z′)
δca(x)δLa(x) δLa(x)
Z Z
δΓ
− dx dy′ dz′ Γ(y′,z′,x)TdbaLd(y′)cb(z′)
δLa(x)
Z
= dx dy dz dy′ dz′ Γ(y′,x,z′)TdabLd(y′)Γ(x,y,z)Tamncm(y)cn(z)cb(z′)
Z
− dx dy dz dy′ dz′ Γ(y′,z′,x)TdbaLd(y′)cb(z′)Γ(x,y,z)Tamncm(y)cn(z)
Z
= dx dy dz dy′ dz′ Γ(y′,x,z′)Γ(x,y,z)TdabTamnLd(y′)cm(y)cn(z)cb(z′)
Z
− dx dy dz dy′ dz′ Γ(y′,y,x)Γ(x,z,z′)TdmaTanbLd(y′)cm(y)cn(z)cb(z′)
Z
= dx dy dz dy′ dz′ Γ(y′,x,z′)Γ(x,y,z)TdabTamn
Z (cid:16)
−Γ(y′,y,x)Γ(x,z,z′)TdmaTanb Ld(y′)cm(y)cn(z)cb(z′) =0.
(cid:17)
Thus, we come to two conditions
dx Γ(y′,x,z′)Γ(x,y,z) = dx Γ(y′,y,x)Γ(x,z,z′) (24)
Z Z
TdabTamn = TdmaTanb (25)
The second condition must be understood up to symmetric properties of Tabc. As it is
clear from (23) this tensor structure must be antisymmetric in its last two indices. This
condition will be solved at the end of this section. Now we focus on the first condition.
We introduce Fourier transformations1
Γ(x,y,z) = dp dq dk δ(p +q +k )Γ˜(p ,q ,k )exp(ip x+iq y+ik z)
1 1 1 1 1 1 1 1 1 1 1 1
Z
Γ(y′,x,z′) = dp dq dk δ(p +q +k )Γ˜(p ,q ,k )exp(ip y′+iq x+ik z′)
2 2 2 2 2 2 2 2 2 2 2 2
Z
Γ(y′,y,x) = dp dq dk δ(p +q +k )Γ˜(p ,q ,k )exp(ip y′+iq y+ik x)
3 3 3 3 3 3 3 3 3 3 3 3
Z
Γ(x,z,z′) = dp dq dk δ(p +q +k )Γ˜(p ,q ,k )exp(ip x+iq z+ik z′)
4 4 4 4 4 4 4 4 4 4 4 4
Z
The condition (24) in momentum space is
dxdp dq dk dp dq dk δ(p +q +k )δ(p +q +k )Γ˜(p ,q ,k )×
1 1 1 2 2 2 1 1 1 2 2 2 1 1 1
Z
×Γ˜(p ,q ,k )exp(ip x+iq y+ik z+ip y′+iq x+ik z′)
2 2 2 1 1 1 2 2 2
= dxdp dq dk dp dq dk δ(p +q +k )δ(p +q +k )Γ˜(p ,q ,k )×
3 3 3 4 4 4 3 3 3 4 4 4 3 3 3
Z
×Γ˜(p ,q ,k )exp(ip y′+iq y+ik x+ip x+iq z+ik z′).
4 4 4 3 3 3 4 4 4
It can be transformed to
dp dq dk dp dk δ(p +q +k )δ(p −p +k )Γ˜(p ,q ,k )×
1 1 1 2 2 1 1 1 2 1 2 1 1 1
Z
×Γ˜(p ,−p ,k )exp(iq y+ik z+ip y′+ik z′)
2 1 2 1 1 2 2
= dp dq dk dq dk δ(p +q +k )δ(−k +q +k )Γ˜(p ,q ,k )×
3 3 3 4 4 3 3 3 3 4 4 3 3 3
Z
×Γ˜(−k ,q ,k )exp(ip y′+iq y+iq z+ik z′),
3 4 4 3 3 4 4
and then by momentum redefinitions in the second integral one obtains
dp dq dk dp dk δ(p +q +k )δ(p −p +k )Γ˜(p ,q ,k )×
1 1 1 2 2 1 1 1 2 1 2 1 1 1
Z
×Γ˜(p ,−p ,k )exp(iq y+ik z+ip y′+ik z′)
2 1 2 1 1 2 2
= dp dq dk dk dk δ(p +q +k )δ(−k +k +k )Γ˜(p ,q ,k )×
2 1 3 1 2 2 1 3 3 1 2 2 1 3
Z
×Γ˜(−k ,k ,k )exp(iq y+ik z+ip y′+ik z′).
3 1 2 1 1 2 2
1We do not write factors 2π in these Fourier transformations since at the end of the calculations we
will go back to coordinate space in which all the factors 2π will disappear.
By removing one of delta functions in each part one obtains
dq dk dp dk δ(p +k +q +k )Γ˜(p +k ,q ,k )Γ˜(p ,−p −k ,k )×
1 1 2 2 2 2 1 1 2 2 1 1 2 2 2 2
Z
×exp(iq y+ik z+ip y′+ik z′)
1 1 2 2
= dp dq dk dk δ(p +q +k +k )Γ˜(p ,q ,k +k )Γ˜(−k −k ,k ,k )×
2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2
Z
×exp(iq y+ik z+ip y′+ik z′).
1 1 2 2
By making the last simplification one obtains
dk dp dk Γ˜(p +k ,−p −k −k ,k )Γ˜(p ,−p −k ,k )×
1 2 2 2 2 2 2 1 1 2 2 2 2
Z
×exp(i(−p −k −k )y+ik z+ip y′+ik z′)
2 2 1 1 2 2
= dp dk dk Γ˜(p ,−p −k −k ,k +k )Γ˜(−k −k ,k ,k )×
2 1 2 2 2 2 1 1 2 1 2 1 2
Z
×exp(i(−p −k −k )y+ik z+ip y′+ik z′).
2 2 1 1 2 2
Thus, finally the condition (24) takes the form
Γ˜(p +k ,−p −k −k ,k )Γ˜(p ,−p −k ,k ) (26)
2 2 2 2 1 1 2 2 2 2
= Γ˜(p ,−p −k −k ,k +k )Γ˜(−k −k ,k ,k ).
2 2 2 1 1 2 1 2 1 2
This is an equation for a function of three variables, which will be solved below. First we
show that there is simple ansatz which satisfies Eq. (26). Indeed, by choosing ansatz
G˜(q2)G˜(k2)
Γ˜(p,q,k) = , (27)
G˜(p2)
where G˜ is the Fourier image of some bilocal function G−1, we can substitute this expres-
c
sion in Eq. (26):
G˜((p +k )2) G˜(k2) G˜((p +k +k )2) G˜(k2)
2 2 2 × 2 2 1 1 =
G˜(p2) G˜((p +k )2)
2 2 2
G˜((p +k +k )2) G˜((k +k )2) G˜(k2) G˜(k2)
= 2 2 1 2 1 × 1 2 . (28)
G˜(p2) G˜((k +k )2)
2 1 2
Thisisanidentity. Thatis,fortheansatz(27), Eq. (26)isvalid. Nowwewilldemonstrate
that this anzatz is unique solution.
In general, the function Γ˜(p,q,k) is a function of three independentLorentz invariants,
since the moments p, q and k are not independent but related by conservation of the
moments, p+q+k = 0. We can choose p2, q2 and k2 as those independent invariants,
Γ˜(p,q,k) ≡ f(p2,q2,k2).
Therefore, we can rewrite the basic equation (26) as
f (p +k )2,(p +k +k )2,k2 ×f p2,(p +k )2,k2 (29)
2 2 2 2 1 1 2 2 2 2
(cid:16) (cid:17) (cid:16) (cid:17)
= f p2,(p +k +k )2,(k +k )2 ×f (k +k )2,k2,k2 .
2 2 2 1 1 2 1 2 1 2
(cid:16) (cid:17) (cid:16) (cid:17)
Let us introduce into the equation (29) new independent variables,
(p +k )2 = x, (p +k +k )2 = y,
2 2 2 2 1
k2 = z, p2 = u,
1 2
k2 = v, (k +k )2 = w.
2 1 2
The number of the independent variables is six, since in (29) we have only three indepen-
dent Lorentz vectors p ,k ,k . Using these vectors we can construct six Lorentz-invariant
2 2 1
values above. In terms of these new independent variables the basic equation (29) looks
like
f (x,y,z)×f(u,x,v) = f(u,y,w)×f(w,z,v). (30)
We consider equation (30) as an equation for an analytical function of three variables
in R3 space. We observe that the r.h.s. of (30) does not depend on x for any values of
y,z,u,v. There is the unique solution to this - the dependence on x must be factorized in
the following way:
1
f(x,y,z) = F (y,z), f (u,x,v) = ϕ(x)F (u,v), (31)
1 2
ϕ(x)
where ϕ(x) is some function, and F (y,z) and F (u,v) are other functions. The rigorous
1 2
prove of this statement is given below . The two equations in (31) imply
ϕ(y)
f (x,y,z) = ×F(z),
ϕ(x)
where F(z) is some function. By substituting this in Eq. (30) we immediately infer that
F(z) = constant×ϕ(z). Rescaling ϕ(z) by an appropriate constant, we obtain:
ϕ(y)ϕ(z)
f(x,y,z) = . (32)
ϕ(x)
Let us give a rigorous proof that the factorization (31) of the x dependence is the unique
solution to equation (30). Denote h ≡ ln f. Applying logarithm to (30), we have
h(x,y,z) = −h(u,x,v)+ terms independent on x. (33)
dm
Applying , m = 1,2,... to (33), we obtain
dxm
∂mh(x ,y,z) ∂mh(u,x ,v)
1 2
| = − |
∂xm x1=x ∂xm x2=x
1 2
This means that the Taylor expansions in x around the point x= 0 for functionsh(x,y,z)
and h(u,x,v) are
h(x,y,z) = h(0,y,z)−ϕ˜(x,y,z), (34)
h(u,x,v) = h(u,0,v)+ϕ˜(x,y,z), (35)