Apparently Noninvariant Terms of U(N) × U(N) Nonlinear Sigma Model in the OneLoop Approximation
Koji Harada
2
3
4
) Hirofumi Kubo
1
2
4
Yuki Yamamoto
0
2
4
0
Department of Community Service and Science, Tohoku University of Community Services and Science
, Sakata 9988580,
Japan
1
Synchrotron Light Application Center, Saga University
, Saga 8408502,
Japan
2
Subject Index: 131, 132
3
Department of Physics, Kyushu University
, Fukuoka 8108560,
Japan
4
Apparently Noninvariant Terms of U (N )
We show how the Apparently Noninvariant Terms (ANTs), which emerge in perturbation theory of nonlinear sigma models, are consistent with the nonlinearly realized symmetry by employing the WardTakahashi identity (in the form of an inhomogeneous ZinnJustin equation). In the literature the discussions on ANTs are confined to the SU(2) case. We generalize them to the U(N ) case and demonstrate explicitly at the oneloop level that despite the presence of divergent ANTs in the effective action of the pions, the symmetry is preserved.

to employ the WardTakahashi (WT) identity for the effective action. The invariance
under the transformation of the fields, collectively denoted as (x), (x) (x) +
(x), can be expressed as
= 0,
where, [] is the effective action for the classical field a(x) = a(x) .
A crucial observation is that in general a(x) is not equal to a(x) for a
nonlinearly realized symmetry. Thus at the quantum level, the symmetry
transformation for is different from the defining transformation law for the quantum field
.
From this observation, it is obvious that the appearance of ANTs is not a
phenomenon particular to NLSMs, but is a general feature of the theories with
nonlinearly realized symmetry.
To treat nonlinear transformations, we introduce an external field that couples
to , as ZinnJustin10) did for the BRST symmetry of nonabelian gauge theory. The
difference between the present case and the BRST case is that the BRST symmetry
is nilpotent while the present symmetry is not. Thus, the resultant WT identity has
an inhomogeneous term in the present case.
Furthermore, in order to express the WT identity in a compact form, we need
to make a special choice for the pion fields. In the case of SU (2) SU (2) NLSM,
a useful parameterization of the field U (x) SU (2) is
U (x) =
Under the infinitesimal SU (2) SU (2) transformation, these fields transform as
follows,
where and are parameters for the vector and the axialvector transformations.
The point is that a finite number of fields ( and in the present case) form a closed
set under the transformation (i.e., a multiplet), in the sense that they transform
mutually among them. This property is absent for a generic parameterization. For
example, in the case of the exponential parameterization, U (x) = exp(i (x)/F ),
the axialvector transformation law for the sodefined pion field introduces another
operator ab tr aU 1 bU + a b ,
If one transforms ab, the result cannot be expressed as a linear combination of
and ab, and introduces another operator. No finite sequence of this procedure does
form a closed set under the axialvector transformation.
A special parameterization such as Eq. (1.2) for the SU (2) SU (2) case is
desired also for a general group, but it is not obvious to find one for a group other
than SO(N ). (Note that SU (2) SU (2) SO(4).)
In this paper, we consider U (N ) U (N ) NLSMs and formulate the WT identity
in the form of an inhomogeneous ZinnJustin equation.) By using it, we investigate
how the WT identity is satisfied despite the presence of ANTs in the oneloop
approximation. We determine the form of the divergent part of the ANTs to all order
in . We also show that ANTs of the sixpoint proper vertices (the 1PI parts of the
amputated connected Green function) do not vanish on shell.
Several comments on the literature are in order.
(i) Tataru5) calculates the effective action for the SO(N ) NLSM and shows that
the ANTs are proportional to S/ at the oneloop level, where S is a classical
action. It implies that there is a field redefinition that eliminates ANTs. It thus
implies that the ANTs do not contribute to the Smatrix elements in the oneloop
approximation.
(ii) Appelquist and Bernard9) emphasize that the ANTs can be eliminated by a
field redefinition and thus the symmetry is not lost. They however do not examine
the WT identity. Their statement may be translated as follows in terms of the
effective action: let us assume that the field (x) can be expressed as a function of a
new field and its derivatives,) = ( , , ). Their statement is that, with
a suitable choice of , the effective action in term of ,
[ ] = []=( , , ) ,
= 0.
Note that, unlike the field , the new field does not have a direct connection to
the quantum field , and thus the relation between (1.1) and (1.7) is obscure. The
definition of the new field in terms of the old one may be found only after calculating
the ANTs.
(iii) Brezin, ZinnJustin, and Le Guillou11) consider the SU (2) SU (2) NLSM
in 2 + dimensions. They examine the symmetry with the help of the
inhomogeneous ZinnJustin equation. In this respect, their work is very close to our present
work. As they noted the use of the special parameterization (1.2) is essential for the
formulation of the ZinnJustin equation. In two dimensions, the NLSM is
renormalizable if the Lagrangian does not contain the higher derivatives of the fields than
two. In such a case, the field redefinition that eliminates ANTs does not depend
on the derivatives and is nothing but a usual wavefunction renormalization. They
can also obtain the most general form of the ANTs in this case. But it is a very
special situation. If one includes more derivatives in the Lagrangian, the theory is
not renormalizable, and the field redefinition inevitably involves derivatives. See also
Ref. 12).
(iv) Recently Ferrari and his collaborators13)16) have discussed the SU (2)
SU (2) NLSM in perturbation theory based on a local functional equation, which
is nothing but the SchwingerDyson equation in the functional form. Although our
work looks similar to theirs in the respect that both utilize functional identities,
there are important differences between them. First of all, they try to construct
a theory with only two constants, while we regard the NLSM as an effective field
theory, that is, a nonrenormalizable theory with infinitely many higher dimensional
operators. Secondly, their functional identity is a local one, while our analysis is
based on the usual WT identity for rigid transformations. Thirdly (and perhaps
most importantly), they subtract ANTs as well as symmetric divergences, while we
only subtract divergences occurring in apparently invariant terms keeping the ANTs
intact.
We consider the loopwise expansion in order to examine how the WT identity
is satisfied. At the tree level, the effective action is nothing but a classical action,
and the invariance is trivial, irrespective of the number of derivatives it contains.
ANTs emerge only in loop corrections.
For an effective theory with infinitely many terms with an increasing number
of derivatives, it is impossible to calculate all the oneloop corrections. Fortunately,
however, because of the chiral symmetry, the NLSMs admit a derivative
expansion.17), 18) To O(p4), to which order the oneloop corrections start to contribute,
there are only a finite number of independent operators.
Although the main result of the present paper is the clarification of the
consistency of ANTs with the underlying symmetry by the extensive use of the
inhomogeneous ZinnJustin equation in the oneloop approximation, we also clarify several
confusing points in the literature; (i) The counterterms for ANTs break the
symmetry therefore should not be added. The resulting effective action contains apparently
noninvariant divergences which however do not contribute to the Smatrix. (ii) No
field redefinition is needed to make the theory invariant. (iii) ANTs for the proper
vertices in general do not vanish even on shell.
The investigation of ANTs would have crucial importance in the Wilsonian
renormalization group (RG) analysis of theories with nonlinearly realized symmetries. RG
transformations generate all the terms consistent with the symmetry, thus ANTs as
well. One needs to understand ANTs well to obtain correct RG equations. An
interesting example is a Wilsonian RG analysis of the chiral perturbation theory with
one or more nucleons, which would require proper treatment of ANTs. In the usual
treatment, a power counting scheme, which allows one to exploit the perturbation
theory, is assumed, but one may consider an alternative power counting that implies
the resummation of the certain contributions by examining the Wilsonian RG flows.
See Refs. 19) and 20) for such an example. We expect that a Wilsonian RG
analysis gives dynamical explanations for nonperturbative aspects of the theory such as
resonances.
The structure of the paper is the following: In 2, we introduce the special
parameterization of the U (N ) U (N ) NLSM according to Ref. 21), which allows us
to formulate the WT identity in the form of an inhomogeneous ZinnJustin equation.
In 3, an explicit oneloop calculation of O(3) is shown, which illustrates how the
WT identity is satisfied despite the presence of ANTs up to including O(p4). In 4
the exact form of the divergent part of the ANTs of the oneloop effective action
is determined to all order in by using the invariance argument. In 5, we show
that the ANTs of the sixpoint proper vertices do not vanish on shell, while the
sixpoint amputated connected Green functions do vanish on shell. Finally in 6,
we summarize the results. Appendix A collects some formulae for U (N ) which are
useful in simplifying the results. We present the results for the SU (2) SU (2) case
in Appendix B to compare them with those in the literature.
WT identity for U (N ) U (N ) NLSMs
U (x) =
where (x) and (x) are Hermitian N N matrices, and can be expressed by using
the generators of U (N ), Ti, (i = 0, 1, , N 2 1):
The generators Ti are normalized as
and satisfy the following relations,
[Ti, Tj ] = ifijkTk,
{Ti, Tj } = dijkTk,
where fijk is a completely antisymmetric structure constant and dijk is a completely
symmetric tensor. Note in particular that T0 is proportional to the identity matrix,
T0 =
f0jk = 0,
d0jk =
The U (N ) U (N ) transformation law for U is given by
Here and hereafter we ignore the irrelevant normalization factor. The constraints we
impose are
U U =
2 + i [, ] + 2 = I, [, ] = 0.
Note that these constrains are invariant under U (N )U (N ) transformation Eq. (2.7).
They are to be inserted as delta functions:
where gL and gR are elements of U (N )L and U (N )R respectively. Under the
infinitesimal transformation, and transform as
DD(2 + 2 F 2I) ([, ]) .
The first delta function may be written as
F 2 2 is a matrix, and is defined as a power series of 2,
F 2 2 = F 21F 2 8F1 3 4 + .
where we have dropped (0) from the second constraint as an irrelevant constant.
Since we are going to work with the dimensional regularization, the determinant
factor does not give nontrivial contributions, and we ignore it hereafter. Our experience
with the lattice calculation1) leads us to believe that this factor has nothing to do
with ANTs.
Note that the field U introduced in Eq. (2.1) originally has 2N 2 real components
but now it has N 2 real components: N 2 1 for the NambuGoldstone (NG) bosons
associated with the broken SU (N )A symmetry and one for the NG boson associated
with the broken U (1)A. If one wishes to incorporate the axial anomaly and the
masses, one needs to introduce explicit symmetry breaking terms, as we do in the
effective theory of QCD. In this paper, however, we do not consider such breaking
terms for simplicity.
The generating functional in Euclidean space is now given by
eW [J,K] =
where i(x) is actually given as in Eq. (2.14). The Lagrangian is now for a general
U (N ) U (N ) invariant NLSM,
where the ellipsis denotes the terms with more than two derivatives, and U (x) is
written in terms of (x) as in Eq. (2.1) with Eq. (2.14). The explicit forms of the
terms of O(p4) will be shown in Eqs. (3.22)(3.27) in 3.
As usual, we introduce the effective action,
under the infinitesimal axialvector transformation leads to the following WT
identity,
d4x [ Ai(x) J,KKi(x) Ai(x) J,KJi(x)] = 0.
where (x) is the expectation value of (x) in the presence of the external fields
J(x) and K(x),
Ji =
into the WT identity, we arrived at the following inhomogeneous ZinnJustin
equation,
[, K] [, K]
Kj(x) k(x)
+ j(x)Kk(x)
= 0.
Note that this is not a local equation. If we consider a local axialvector
transformation function j(x), we would arrive at a local equation which however contains
the expectation value of the divergence of the axialvector current as an additional
term. The additional term cannot be written in terms of the derivatives of the
effective action with respect to and K. Furthermore, since we are considering
an effective theory with infinitely many terms, the axialvector current is not just
tr(U U U U ) but depends on infinitely many terms with derivatives. We
find that a local version is not useful.
The effective action [, K] may have a loopwise expansion:
= 0 ,
j
which just implies the invariance of the classical action, S[] d4xL() under the
axialvector transformation, Ai = dijkjk(). Note that in deriving Eq. (2.29),
we have used
At the first order (at the oneloop level), Eq. (2.25) gives
= 0.
where the zeroth order term is the classical action,
where () is a function of the classical field, , with the definition Eq. (2.14).
At the zeroth order (at the tree level), Eq. (2.25) gives
S[, K] S[, K]
Kj(x) k(x)
+ j(x)Kk(x)
= 0.
With the definition Eq. (2.27), it is
This can be expanded in powers of the external field K. At the zeroth order of this
expansion, we have
= 0.
This is an important equation. The first term is the variation of the oneloop
contribution to the effective action (1)[] (1)[, K] K=0 under the naive axialvector
transformation. This equation tells us that the first term does not need to vanish.
The symmetry requires that the sum of these two terms should vanish, but not
necessarily individually. If the first term does not vanish, we see there is an ANT.
Another important point is that the apparent noninvariance of the oneloop
contribution to the effective action (1)[] is proportional to S/. With the use of
the equations of motion, (1)[] is invariant under the naive transformation. Note
that we have reached this conclusion without doing any explicit calculations.
In the next section, we demonstrate the explicit calculation of (1)[, K], and
show how Eq. (2.32) is satisfied for the first few terms in the expansion in powers of
.
3. An explicit oneloop calculation
In this section, we present a detailed oneloop calculation for the effective action
of the U (N ) U (N ) NLSM. As we explained in Introduction, we also do a
lowmomentum expansion and calculate the effective action up to O(p4). Only the
vertices of O(p2) (obtained by expanding the terms explicitly shown in Eq. (2.17) in
powers of ) can enter oneloop diagrams at this order.)
3.1. Preliminary: expansion in powers of
(1)[, K] = I(T1)s[] + A(1N)Ts[] + (1)[, K] ,
= 0 ,
and A(1N)Ts[] is the apparently noninvariant part. All the Kdependent terms are
contained in (1)[, K], which vanishes when we set K = 0.
There is a potential ambiguity in the separation into the apparently
invariant and noninvariant parts. Even if one subtracts an invariant piece from the
) This refers to the case with dimensional regularization. In the cutoff scheme such as lattice
regularization, the loopwise expansion does not match the expansion in powers of momenta.
invariant part and adds it to the noninvariant one, they are still invariant
and noninvariant parts respectively. This ambiguity does not affect the following
discussion in any significant way.
Let us define the momentum space quantities in d dimensions as follows:
j (p) (2)d S[, K]
Kj (p) K=0
where i(p) and Ki(p) are defined as
i(x) =
Ki(x) =
In terms of them, Eq. (2.32) is now written as
Let us expand the each quantity in powers of as follows:
j (p) = (0)(p) + (2)(p) + ,
j j
Bk(p) = B(3)(p) + B(5)(p) + ,
k k
Cj (p) = C(2)(p) + C(4)(p) + ,
j j
Dk(p) = D(1)(p) + D(3)(p) + .
k k
Note that B(1)(p) = 0, because, with the dimensional regularization, the oneloop
k
contributions to the twopoint function vanish when K = 0.
3.2. Cubic terms in
The first nontrivial order in the expansion of the lefthand side of Eq. (3.9) is
cubic in ,
(0)(p)Bk(3)(p) + C(2)(p)Dk(1)(p) .
j j
We immediately get
Apparently Noninvariant Terms of U (N ) U (N ) Nonlinear Sigma Model 485
ln Det[DF1] =
Tr Ln[DF1]
ddk2 1 1
(2)d k2 Gij(k1, k2) k2 Gji(k2, k1)
1 2
where DF and Gij are defined as
ddy[DF1(x, y)]ijeiq1xiq2y = (2)d
= q12(2)dd(q1 + q2)ij Gij(q1, q2). (3.18)
Note that Gij is the field dependent part of the inverse propagator.
The classical action S[, K] (2.27) consists of two parts; the part independent
of K and the part linear in K. We therefore divide Gij accordingly,
Gij(q1, q2) G0,ij (q1, q2) + G1,ij (q1, q2).
Note that the first term of Eq. (3.17) vanishes because of the property of the
dimensional regularization.
At the zeroth order in K, we have
ddk2 1 1
(2)d k12 G0,ij(k1, k2) k22 G0,ji(k2, k1) + O(6) ,
where G0,ij starts with the terms quadratic in 2. See Fig. 1.
A straightforward calculation gives the following expression for the first term,
dkij dkl1l2 dgjidgl3l4 (p21 + p22)(p23 + p24)
+ 2dkij dkl1l2 (dgjidgl3l4dgjl3 dgil4 )(p21 + p22)(p3 p4)
+ 2dgjidgl3l4 (dkij dkl1l2dkil1 dkjl2 )(p23 + p24)(p1 p2)
Fig. 1. The graphical representation of the term explicitly shown in Eq. (3.20), the contribution to
the effective action to the zeroth order in K, starting with 4. The blob stands for the vertex
G0,ij, which contains an even number of s.
Fig. 2. The graphical representation of the second term explicitly shown in Eq. (3.31), the
contribution to the effective action to the first order in K, starting with 2. The blob stands for the
vertex G0,ij, which contains an even number of , and the square for the vertex G1,ji, which
contains a single K as well as zero or an even number of s.
4
3 dkil1 dkjl2 dgjl3 dgil4 [(p1 p4)(p2 p3) (p1 p3)(p2 p4)]
+(finite terms) ,
where 1 1 + ln 4.
The next task is to separate the contributions which can be cancelled by
symmetric counterterms from those which cannot be. The symmetric terms are already
contained in the Lagrangian L[] in Eq. (2.17), so that the addition of the
counterterms is absorbed by the change of the coefficients of these operators. We list all of
those terms up to including O(p4) that are invariant under axialvector
transformations. There are six independent ones;
L1 = c1 tr
L2 = c2 tr U U U U ,
L3 = c3 tr U U U U ,
L4 = c4 tr U U tr U U ,
Apparently Noninvariant Terms of U (N ) U (N ) Nonlinear Sigma Model 487
L5 = c5 tr U U tr U U ,
L6 = c6 tr( U U ) tr(U U ) + tr(U U ) tr(U U ) .
Expanding them in powers of , and comparing the contributions to the effective
action with those in Eq. (3.21) with the help of the formulae given in Appendix A,
we see that we can cancel some of the divergences in Eq. (3.21) with the following
values of cis,
c1 = 0,
for the case of M S renormalization scheme. But, importantly, there are some
divergences which cannot be canceled by the symmetric counterterms with any choice of
the cis. They are ANTs, which contribute to B(3)(p).
k
After adding the counterterms, we may define the apparently invariant (and
finite) part of the effective action I(T1)s[] of O(p4) coming from oneloop diagrams
as
2 ln
The rest gives the apparently noninvariant (and infinite) part,
The O(4) terms contribute to B(3).
k
+2dkijdkl1l2(dgjidgl3l4 dgjl3dgil4)(p21 + p22)(p3 p4)
ddk2 1 1
(2)d k12 G0,ij (k1,k2) k22 G1,ji(k2,k1) + O(4K) .
See Fig. 2.
We obtain for the second term
1 + 2 ln(p2/2) dgijdgl1l2 p2 2dgil1 dgjl2(p1 p2) dkij ,
which contributes to C(2).
j
Now we are ready to calculate Eq. (3.14). Noting that we only need B(3)(p) at
k
p = 0 (because of the delta function in Eq. (3.15)) and that only the terms with j = 0
survive (because of the 0j) and di0k is proportional to ik simplifies the calculation.
We finally obtain for the first term of Eq. (3.14)
dhnmdhl3i dgmndgl1l2 p3 dgml1 dgnl2 (p1 p2) p23 .
2
The calculation of the second term of Eq. (3.14) is simpler and one easily finds that
it is the same as Eq. (3.33) but with the opposite sign. Thus we have explicitly
demonstrated that Eq. (3.9) is satisfied in O(3).
We have shown that the oneloop effective action (1)[] is not invariant under
the naive axialvector transformation. But it does satisfy the WT identity so that
the symmetry is preserved.
siKj = dijkHk ,
siHj = dijkKk ,
Invariants are most easily formed in terms of U = ( + i)/F . In the same way, we
introduce
1 2
L = (K + iH) = (Ki + iHi) Ti , (4.8)
F F
and form invariants in terms of U and L. Note that L has the same property under
parity as that of U .
We remark that the divergence of the effective action occurs in the terms at
most quadratic in K in the oneloop approximation. To represent them, therefore,
we only consider the invariants at most quadratic in L. We also remark that the
vertices containing K do not involve derivatives, thus the divergent part of the
effective action quadratic in L does not involves derivatives, either.
There are five independent invariants contributing to (1)
ANTs[] + (1)[, K] up
to including O(p4) and O(4) when expanded and consisting of U and L,
by introducing a differential operator si,
where Hk is defined as
si dijk
ddx j (x) k(x) Hk(x) Kj(x)
si A(1N)Ts[] + (1)[, K] = 0 .
In the following, we discuss only the divergent parts, that is, the local functionals
of and K.
An important observation is that K and H transform in the same way as and
do:
tr( U L) + tr(L U ) ,
besides the ones consisting only of U listed in Eqs. (3.22)(3.27). There are also
other (linearly independent) invariants, such as
tr(LU LU ) + tr(U LU L),
but it turns out that they do not contribute. Incidentally, the term (4.14) is shown
to be independent of K and can be written solely in terms of U :
tr(U L) tr(LU ) = 2 tr U U + U U ,
up to terms with higher derivatives.
One might think that there should be more invariants which start with 6 when
expanded, but because of the constraint U U = I, such an invariant requires at least
six derivatives; there is no other invariant than Oi (i = 1, , 5) to this order. Thus
(1)
ANTs[] + (1)[, K] can be expressed as a linear combination of the integrals of
these invariants.
By comparing the O(4) and O(2K) terms with Eqs. (3.30) and (3.31), we
can determine the coefficients. We can determine the divergent part of A(1N)Ts[] +
(1)[, K] to all order in as
ddx [N (O1 O3) + O2 + O4 2O5] .
We therefore see that the divergent part of A(1N)Ts[] is proportional to S/.
It is well known22) that the part of the oneloop effective action that is
proportional to S/ does not contribute to the Smatrix elements, due to the equivalence
theorem.23) That is, ANTs of the amputated connected Green function (ACGF)
vanish on shell.
5. Sixpoint functions
There seems to be a little confusion about the use of the equations of motion,
S/ = 0, and the onshell condition, pi2 = 0. Some authors use them
interchangeably. But they are essentially different. ANTs of the oneloop effective action
vanish by the use of the equations of motion.
One might think that it implies that the ANTs contained in the npoint proper
vertices generated by the effective action also vanish on shell. But this is not the
case.
Apparently Noninvariant Terms of U (N ) U (N ) Nonlinear Sigma Model 491
Fig. 3. The graphical representation of the righthand side of Eq. (5.6). A solid circle stands an
amputated external line and the solid line for the full propagator, W, abbreviated as W2. The
gray blobs stand for the proper vertices. 6 is the abbreviated notation for 16 , and 4 for
i1 i2 i3 and i4 i5 i6 . The sum is taken over the permutation of the external lines.
It is not the npoint proper vertices, but the ACGF whose ANTs vanish on shell,
because of the equivalence theorem.
For the fourpoint functions, however, the difference is not seen; the proper
vertex coincides with the ACGF in the oneloop approximation. Thus ANTs of the
proper vertex happen to vanish on shell. See Eq. (3.30).
It is therefore interesting to examine the sixpoint functions because the
difference between the ACGF and the proper vertex first appears in the sixpoint functions
at the oneloop order. Since we have obtained the divergent part of the oneloop
effective action to all order in , we are ready to examine the npoint functions with
n 6.
We first show that the ANTs of the sixpoint proper vertex do not vanish on
shell. Because we are interested in the proper vertices in the absence of K, we set
K = 0. By expanding Oi (i = 1, , 5) in terms of , we see that the contributions
from O2, O3 vanish on shell. The O(6) part of O1 is given by
ik (pk) di1i2,i3i4,i5i6 (p1 p2)(p3 p4)
up to the terms proportional to i, which vanish on shell. Similarly, from O4 and
O5, we have the contributions that do not vanish on shell,
respectively, where di,jk,lm stands for dighdgjkdhlm. Substituting these contributions
in Eq. (4.17), we see that the divergent part of A(1N)Ts[] contains
that gives rise to the nonvanishing contribution to the sixpoint proper vertex. Here
we have introduced another notation,
+(di5,i3i4,i2i6 + di6,i3i4,i2i5 )i10 + (di5,i3i4,i1i6 + di6,i3i4,i1i5 )i20 .
(5.5)
Thus, we have established that the divergent part of ANTs in the sixpoint proper
vertex does not vanish on shell.
We will show that ANTs contained in the sixpoint ACGF do vanish on shell.
In an abbreviated notation, the sixpoint ACGF is written as
12n 1 n =0
Note that because the invariant terms are made finite by the addition of the invariant
counterterms, the divergent parts of the both sides of Eq. (5.6) come only from ANTs.
The assertion is that, despite the nonvanishing divergent part of ANTs in
oneloop contributions to the first term in Eq. (5.6) on shell, the sum does not contain
such ANTs. Note that the tree contributions of course do not contain ANTs.
In the oneloop approximation, the propagator W is the free one thanks to a
property of dimensional regularization. In the product 123456 one of the
two s must be the oneloop contribution while the other must be the tree one.
Thus the noninvariant divergent contributions to the 123456W in the
oneloop approximation contain the following divergent part,
4S[]
i4 (q4)i5 (q5)i6 (q6)k(k) =0
Apparently Noninvariant Terms of U (N ) U (N ) Nonlinear Sigma Model 493
di1i2i3,i4i5i6 (q1 q2)(q4 q5) + di1i2i3,i4i6i5 (q1 q2)(q4 q6) + di1i2i3,i5i6i4 (q1 q2)(q5 q6)
+di1i3i2,i4i5i6 (q1 q3)(q4 q5) + di1i3i2,i4i6i5 (q1 q3)(q4 q6) + di1i3i2,i5i6i4 (q1 q3)(q5 q6)
+di2i3i1,i4i5i6 (q2 q3)(q4 q5) + di2i3i1,i4i6i5 (q2 q3)(q4 q6) + di2i3i1,i5i6i4 (q2 q3)(q5 q6)
+(terms those do not contribute on shell),
where we introduce
Note that, with the help of the formulae given in Appendix B, one can show that
di1i2,i3i4,i5i6 = di1i2i5,i3i4i6 + di1i2i6,i3i4i5 ,
which is useful in the following demonstration.
In a similar way, we can calculate all the divergent terms contained in
at oneloop that do not vanish on shell. There are 90 similar terms to
On the other hand, the differentiation of Eq. (5.4) with respect to six times
produces 6! = 720 terms but every 16 (= 24) terms are actually identical. (Note the
symmetry under the interchange of the indices, i1 i2, i3 i4, i5 i6, and
(i1i2) (i3i4) of the representative term di1i2,i3i4,i5i6 (p1 p2)(p3 p4).) Thus we have
45 (= 720/16) similar terms to
Now substituting Eq. (5.11), we see that the divergent terms in the first term of the
righthand side of Eq. (5.6) is exactly equal to those in the second term with the
opposite sign. Thus we have shown that the divergent terms cancel in the ACGF on
shell.
6. Summary
In this paper, we have explained why ANTs emerge in theories with nonlinearly
realized symmetry. The naive variation of the classical field under the symmetry
is in general not equal to the expectation value of the variation of the quantum field
, so that the effective action [] is not invariant under the naive variation of .
In this respect, the emergence of ANTs is not limited to NLSMs.
We have formulated the WT identity as an inhomogeneous ZinnJustin equation
for the U (N ) U (N ) NLSMs to investigate the nature of ANTs. We have shown
that ANTs are consistent with the symmetry.
An explicit oneloop calculation is presented for the fourpoint functions. We
then determine the divergent part of the ANTs in the oneloop effective action,
A(1N)Ts[], to all order in .
It is shown that the divergent part of the ANTs in the sixpoint proper vertex
does not vanish when the external momenta are all set to be on shell, in contrast to
the fourpoint proper vertex, whose divergent part of the ANTs vanishes on shell.
We have shown that, despite its presence in the proper vertices, ANTs contained in
the sixpoint ACGF do vanish on shell. We believe that higherpoint ACGF also
have this property.
Note that we do not subtract divergences in (1)
ANTs[]; the introduction of
noninvariant counterterms cause the violation of the original symmetry. Even though they
are divergent, ANTs do not contribute to the Smatrix elements after all. It provides
an example where the effective action is divergent while the Smatrix elements are
finite.
We expect that similar properties hold even in higher order in the loopwise
expansion.
Acknowledgements
Appendix A
Some Useful Formulae for U (N )
[[Ti, Tj ], Tk] + [[Tj , Tk], Ti] + [[Tk, Ti], Tj] = 0,
[{Ti, Tj }, Tk] + [{Tj , Tk}, Ti] + [{Tk, Ti}, Tj ] = 0,
[Tk, [Ti, Tj ]] = {Tj , {Tk, Ti}} {Ti, {Tj , Tk}},
we find the following relations between fijks and dijks,
From them one can derive useful relations.
Let us introduce N 2 N 2 matrices, (Dj)ik,
filmfmjk + fjlmfimk + fklmfijm = 0,
filmdmjk + fjlmdimk + fklmdijm = 0,
fijmfklm dikmdjlm + djkmdilm = 0.
(Dj)ik dijk,
Apparently Noninvariant Terms of U (N ) U (N ) Nonlinear Sigma Model 495
tr(DiDjDk) =
tr(DiDj DkDl) =
N 1
4 (dijmdmkl fijmfmkl) + 2 (ij kl + ikjl + iljk)
2N
+ 4 (i0djkl + j0dikl + k0dijl + l0dijk).
They are used in reducing the result of Eq. (3.21) in a compact form for the
comparison with the local terms listed in Eqs. (3.22) through (3.27).
Appendix B
SU (2) SU (2) Case
In this appendix, we present the results for the SU (2) SU (2) case. Because
of the extra U (1) factors, the SU (2) SU (2) results cannot be obtained directly
from those for U (N ) U (N ). The purpose of this Appendix is to provide a useful
reference for those who are familiar with the literature devoted to the SU (2) SU (2)
case.
eW [J,K] =
and define the effective action as usual. The inhomogeneous ZinnJustin equation is
given by
[, K] [, K]
K(x) a(x)
+ a(x)K(x)
= 0.
The oneloop calculation for the effective action, (1)[, K], can be done in a
similar way, but the results are much simpler. The Kindependent part contains
pi a(p1)a(p2)b(p3)b(p4)
3(p21 + p22)(p23 + p24) + 4(p21 + p22)(p3 p4) + 4(p23 + p24)(p1 p2) +
8 8
3 (p1 p4)(p2 p3) + 3 (p1 p3)(p2 p4) + (finite terms),
instead of Eq. (3.21). There are only three independent symmetric counterterms of
this order because of the relations,
due to the tracelessness of the generators. Thus there are only three independent
invariants, L1, L4, and L5. By expanding Li (i = 1, 4, 5) in and comparing the
momentum dependence of them with the terms in Eq. (B.3), we see that some of the
divergences can be removed by these counterterms with the coefficients,
which agree with the result Eq. (34) of Ref. 9). (Note that there is a factor 2
difference coming from the difference between their = 4 d and our = (4 d)/2,
and the difference of signs coming from that we are working in Euclidean space.)
Similarly, for the linear part in K, we have
(1 + 2 ln(p2/2)) 3p2 2(p1 p2) ,
which hold for the SU (2) case, and
1
tr(U U U U ) = [tr(U U )]2 2 [tr(U U )]2,
2
O1 = F 2
4
O2 = F 4
L =
is independent of K. In fact, they are invariant under the axialvector transformation
SU (2)A as we are going to show. Note that the infinitesimal generator now takes
the form
sa
ddx (x) a(x) Ha(x) K(x)
Applications of sa to Sb and (bSb) give
saSb =
sa(bSb) =
= 0.
It means that, modulo invariant terms consisting only of U , O1 is equivalent to O2/2.
Similarly we can rewrite O3 and O4 as
O3 =
O4 =
aSa ,
thus the combination
2
O3 + O4 = F 4 F 2Sa a + ()2 + (a)2 bSb
is K independent and is invariant under SU (2)A,
sa (O3 + O4) = 0.
bination of the independent ones,
It means that, modulo invariant terms consisting only of U , O3 is equivalent to
O4. Thus we have shown that there are only two independent ones, O2 and O4,
for example.
The divergent terms of (1)
ANTs[] + (1)[, K] can be expressed as a linear
com