Singularity Cancellation in Fermion Loops through Ward Identities

11 Pages·0.15 MB·English

by  C. Kopper

Singularity Cancellation in Fermion Loops through Ward
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C. Kopper and J. Magnen
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t Centre de Physique Th´eorique, CNRS UPR 14
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t. Ecole Polytechnique
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m 91128 Palaiseau Cedex, FRANCE
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Recently Neumayr and Metzner [1] have shown that the connected N-point density-correlation

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functions of the two-dimensional and the one-dimensional Fermi gas at one-loop order generi-
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4 momentum/small energy-momentum limits. Their result is based on an explicit analysis in the
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sequel of the results of Feldman et al. [2]. In this note we use Ward identities to give a proof of
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The infrared properties of the connected N-point density-correlation function of the inter-
acting Fermi gas at one-loop order, to be called N-loop for shortness, are important for the
understanding of interacting Fermi systems, in particular in the low energy regime. The N-
loops appear as Feynman (sub)diagrams or as kernels in effective actions. In two dimensions
e.g., their properties are relevant for the analysis of the electron gas in relation with questions
such as the breakdown of Fermi liquid theory and high temperature superconductivity. We refer
to the literature in this respect, see [3,4] and references given there. Whereas the contribution
of a single loop-diagram to the N-point function for N ≥ 3 generally diverges in the small
energy-momentum limit, these singularities have been known to cancel each other in various
situations [3,4,5] in the symmetrized contribution, i.e. when summing over all possible order-
ings of the external momenta, a phenomenon called loop-cancellation. The two-loop has been
known explicitly in one, two and three dimensions for quite some time [1], the calculation in
two dimensions goes back to Stern [6]. We introduce the following notations adapted to those
of [1] :
Π (q ,...,q ) denotes the Fermionic N-loop for N ≥ 3, see (2) below, as a function of the
N 1 N
(outgoing) external energy-momentum variables q , q ...,q and q = −(q +...+q ).
1 2 N−1 N 1 N−1
Here the (d+1)-vector q stands for (q , q ,...,q )= (q ,~q). We also introduce the variables
0 1 d 0
p = q + q + ...+ q , p = 0, 1 ≤ i≤ N . (1)
i 1 2 i−1 1
By definition we then have
dk ddk N
0
Π (q ,...,q ) = I (k; q ,...,q ) with I (k; q ,...,q ) = G (k−p ) (2)
N 1 N 2π (2π)d N 1 N N 1 N 0 j
Z j=1
Y
1 ~k2
and G (k) = , ε = , µ being the Fermi energy.
0 ik −(ε −µ) ~k 2m
0 ~k
To have absolutely convergent integrals for N ≥ 3, we restrict the subsequent considerations
to the physically interesting cases d ≤ 3. At the end of the paper we indicate how the same
results can be obtained for d ≥ 4. We also assume that the variables q have been chosen such
j
that the integrand is not singular (see below (8)). In the following we will choose units such
that µ = 1, 2m = 1. By convention the vertex of q will be viewed as the first vertex.
1
Symmetrization with respect to the external momenta (q ,...,q ) diminishes the degree of
1 N
singularity of the Fermion loops. To prove this fact we have to introduce some notation on
permutations. We denote by σ any permutation of thesequence (2,...,N). By Πσ (q ,...,q )
N 1 N
we then denote Π (q ,q ,...,q ). For the completely symmetrized N-loop we write1
N 1 σ−1(2) σ−1(N)
ΠS (q ,...,q ) = Πσ (q ,...,q ) . (3)
N 1 N N 1 N
σ
X
We will also have to consider subsets of permutations : For n≤ N −2 and 2 ≤ j < j < ... <
1 2
j ≤ N we denote by σ(i1,...,in) the permutation mapping j → i =σ(j )∈ {2,...,N}, which
n (j1,...,jn) ν ν ν
1Wedonotdividebythenumberofpermutations,here (N−1)!,toshortensomeofthesubsequentformulae.
2
preserves the order of the remaining sequence (2,...,N)−(j ,...,j ) , i.e. σ(ν) < σ(µ) for
1 n
ν < µ, if ν, µ 6∈ {j ,...,j }. When the targ(cid:16)et positions (i ,...,i )(cid:17)are summed over (see
1 n 1 n
e.g. (4) below), we will write shortly σ(j ,...,j ), or also σ (j ,...,j ), σn , if we want to
1 n N 1 n N
indicate the number N . Note that n = N −2 is already the most general case, since fixing
the positions of N −2 variables (apart from q ) fixes automatically that of the last. For the
1
(i)
permutation σ , which maps j onto the i-th position in the sequence (2,...,N) (preserving
(j)
the order of the other variables), we use the shorthands σi or σ . We then also introduce the
j j
N-loop, symmetrized with respect to the previously introduced subsets of permutations, i.e.2
ΠSn(j1,...,jn)(q ,...,q )= ΠσN(j1,...,jn)(q ,...,q ), in particular ΠS = ΠSN−2 . (4)
N 1 N N 1 N N N
σN(jX1,...,jn)
The notations corresponding to (3 - 4) will be applied in the same sense also to I .
N
The recent result [1] of Neumayr and Metzner, based on the exact expression for the N-loop
from [2], which however is nontrivial to analyse, shows that for N > 2 and d = 1, 2 one has
generically :
ΠS(λq ,...,λq ) = O(1) for λ → 0 , (5)
N 1 N
ΠS (q , λ~q ,..., q , λ~q ) = O(λ2N−2) for λ → 0, (6)
N 10 1 N0 N
ΠS (q ,...,q ) = O(|~q |) for ~q → 0 . (7)
N 1 N j j
We are not completely sure about the authors’ definition of ’generically’. In any case their
restrictions on the energy-momentum variables include the following one:
The energy momentum set {q ,...,q } is nonexceptional, if for all J ⊂ {1,...,N} we have
1 N 6=
| q |≥ η > 0 . (8)
i0
i∈J
X
Our bounds given in the subsequent proposition are based on this condition.3. Though we
cannot exclude (and did not really try) that linear relations among the momentum variables
{~q ,...,~q } could even improve those bounds, it seems quite clear that they are saturated
1 N
apart from subsets of momentum configurations of measure zero, cf. also the numerical results
mentioned in [1]. Furthermore they deteriorate with the parameter η−1 (cf. the remarks in the
end of the paper).
Proposition : For nonexceptional energy-momentum configurations {q ,...,q } (as defined
1 N
through (8) with η fixed) and for N ≥ 3 and n ≤ N −2 the following bounds hold:
A) In the small λ limit q → λq , ~q → λq~ , λ→ 0 (9)
i0 i0 i i
A1) |Π (λq ,...,λq )| ≤ O(λ−(N−2)), (10)
N 1 N
2Again we do not multiply by (N−n−1)!.
(N−1)!
3Asfor thespatial components ~qi we only suppose that they lie in some fixed compact region {|~qj|≤K}.
3
A2) |ΠSn(λq ,...,λq )| ≤ O(λ−(N−2−n)), |ΠS (λq ,...,λq )| ≤ O(1) . (11)
N 1 N N 1 N
B) In the dynamical limit ~q → λ~q , λ → 0 (12)
i i
B1) |Π (q , λ~q ,..., q , λ~q )| ≤ O(λ2) , (13)
N 10 1 N0 N
B2) |ΠSn(q , λ~q ,...)| ≤ O(λ2n+2), |ΠS (q , λ~q ,...)| ≤ O(λ2N−2) . (14)
N 10 1 N 10 1
The functions λN−2Π (λq ,...,λq ), ΠS (λq ,...,λq ), Π (q , λ~q ,..., q , λ~q ) and
N 1 N N 1 N N 10 1 N0 N
ΠS(q , λ~q ,..., q , λ~q ) are analytic functions of λ in a neighbourhood of λ = 0 (depend-
N 10 1 N0 N
ing on the momentum configuration, in particular on η).
Proof : To prove A1) we perform the k -integration using residue calculus, so that (2) takes the
0
form (cf. [2,1])
N ddk N −1
Π (q ,...,q ) = f (~k) , where (15)
N 1 N Xi=1Z|~k−p~i|<1 (2π)d (cid:16)j=Y1,j6=i ij (cid:17)
f (~k) = ε(~k−~p )−ε(~k−p~ )+i(p −p ) = 2~k·(p~ −p~ )+(p~2−p~2)+i(p −p ) .
ij i j i0 j0 j i i j i0 j0
This implies that
N ddk N −1
Π (λq ,...,λq ) = λ−(N−1) fλ(~k) , where (16)
N 1 N Xi=1Z|~k−λp~i|<1 (2π)d (cid:16)j=Y1,j6=i ij (cid:17)
fλ(~k) = 2~k·(p~ −p~ )+λ(p~2−p~2)+i(p −p ) .
ij j i i j i0 j0
By Lemma 1 below we find
N ddk N −1
fλ(~k) = 0 (17)
Xi=1Z|~k|<1 (2π)d (cid:16)j=Y1,j6=i ij (cid:17)
so that we obtain
N ddk N −1
Π (λq ,...,λq ) = λ−(N−1) { − } fλ(~k) . (18)
N 1 N Xi=1 Z|~k−λp~i|<1 Z|~k|<1 (2π)d (cid:16)j=Y1,j6=i ij (cid:17)
Thus each entry in the sum in (16) has to be integrated only over a domain of measure O(λ).
Since the integrands are bounded in modulus by O(1) due to the nonexceptionality of the
momenta, this leads to the statement (10).
To prove B1), (13) we use again (15)
N ddk N −1
Π (q , λ~q ,..., q , λ~q ) = { − } f (λ; ~k) (19)
N 10 1 N0 N Xi=1 Z|~k−λp~i|<1 Z|~k|<1 (2π)d (cid:16)j=Y1,j6=i ij (cid:17)
with f (λ; ~k) = 2λ~k·(p~ −~p )+λ2(p~2−p~2)+i(p −p ) .
ij j i i j i0 j0
Onperformingthe change of variables ~k˜ = ~k−λp~ in theintegral over |~k−λp~ |< 1 one realizes
i i
that the difference between the two integrals is of order λ2. Or one may convince oneself that
4
both integrals are even functions of λ. In any case this proves B1).
The previous considerations also imply that λN−2 Π (λq ,...,λq ) and Π (q , λ~q ,...)
N 1 N N 10 1
are analytic around λ = 0: The imaginary parts of the denominators in (15, 16) stay bounded
away from zero, and a convergent Taylor expansion for λN−2×(18) is easily obtained on per-
forming the change of variables ~k˜ = ~k−λp~ in the first integrals.
i
For the proof of the proposition we will need also a slight generalization of the bounds on (15),
which we have just obtained. We have to regard integrals of the type
dk ddk
Π (ϕ; q ,...,q ) := 0 I (k; q ,...,q ) ϕ(~k; q ,...,q ) , (20)
N 1 N 2π (2π)d N 1 N 1 N
Z
where we demand that the functions ϕ(~k; q ,...,q ) be continuous and uniformly bounded in
1 N
the domain specified by (8) : |ϕ(~k; q ,...,q )| ≤ C for some suitable C > 0, and furthermore
1 N
|ϕ(~k; q ,...,q ) − ϕ(~k+~q; q ,...,q )| ≤ sup |~q·~q | C uniformly in ~q ∈ IRd . (21)
1 N 1 N J
J
Here we set
~q = ~q for J ⊂ {1,...,N} . (22)
J j
j∈J
X
The scaling and dynamical limits A1) and B1) can also be studied for Π (ϕ; q ,...,q ) on
N 1 N
introducing
ϕ (~k; q ,...,q ) = ϕ(~k; λq ,...,λq ), ϕ (~k; q ,...,q ) = ϕ(~k; q ,λ~q ,..., q ,λ~q ). (23)
s 1 N 1 N d 1 N 10 1 N0 N
To perform the k -integration as before, it is important to note that ϕ does not depend on k .
0 0
The bounds
|Π (ϕ ; λq ,...,λq )| ≤ O(λ−(N−2)), |Π (ϕ ; q ,λ~q ,..., q ,λ~q )| ≤ O(λ2) (24)
N s 1 N N d 10 1 N0 N
are then proven as previously. In particular to prove (13) we perform the same change of
variables as after (19) and use (after telescoping the integrand)
|ϕ (~k˜+λp~ ; q ,...,q ) − ϕ (~k˜; q ,...,q )| ≤ O(λ2)
d i 1 N d 1 N
for |p~ | ≤K, to obtain the factor of λ2 required in B1). We finally note that (24) also holds for
i
N ≥ 2, by same method of proof, if the function ϕ assures the integrability of the integrand.
On multiplying any admissible ϕ by the function ∆ or from (34), this is assured, and ∆ϕ has
the properties required for ϕ above. Therefore we also find for N ≥ 2
|Π ((∆ϕ) ; λq ,...,λq )| ≤ O(λ−(N−2)), |Π ((∆ϕ) ; q ,λ~q ,..., q ,λ~q )| ≤ O(λ2),
N s 1 N N d 10 1 N0 N
(25)
and the same bound for Π (1 ϕ), using 1 from (32) (in case ~q does not vanish).
N A A j
It remains to prove
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Lemma 1 : For any n ≥ 2 and pairwise distinct complex numbers a , i ∈ {1,...,n}, we set
i
a = a −a . Then we have
i,j i j
n n
1
= 0 . (26)
a −a
i j
i=1j6=i,j=1
X Y
Proof : By isolating the term i = 1 in the sum
n n n n n
1 1 1 1
= − ( ) (27)
a −a a −a a −a a −a
i=1j6=i,j=1 i j j=2 1 j i=2 j6=i,j=2 i j 1 i
X Y Y X Y
we obtain a presentation of (26) in terms of a difference of two rational functions of the complex
variable a . They both have simple poles at a , ... ,a with identical residues. So the left
1 2 n
hand side is an entire function of a which vanishes for |a | → ∞ and thus equals zero. (This
1 1
implies that thesecond term on theright hand sideis thepartial fraction expansion of thefirst.)
We now want to show how to obtain the statements A2) and B2) from A1) and B1) using the
Ward identity,4 in form of the simple propagator identity
(iq −2~q·~k+~q2)G (k−q)G (k) = G (k−q) − G (k) . (28)
0 0 0 0 0
When applying this identity to the productof the two subsequent propagators in I (2), which
N
differ by the momentum q , and then summing over all possible positions of the momentum q
j j
in the loop, the sum telescopes, and we are left with the very first and very last contributions,
the last being obtained from the first on shifting the variable k by q . Thus we obtain for
j
j ∈ {2,...,N}
{iq −2~q ·(~k − p~σj)+~q2}Iσj(k; q ,q ,...,q ) = (29)
j,0 j j N 1 2 N
Xσj
I (k; q +q ,q ,...,6q ,...,q ) − I (k+q ; q +q ,q ,...,6q ,...,q ) .
N−1 1 j 2 j N N−1 j 1 j 2 j N
Note that the term on the r.h.s. vanishes on integration over k. The momentum pσj , short for
pσji , is defined to be the momentum arriving at the vertex of qj for the permutation σji, i.e.
q +...+q +q +...+q , for i> j
pσji = q1+ qσ−1(2)+...+qσ−1(i−1) = 1 j−1 j+1 i (30)
( q1+...+qi−1, for i< j. )
We can rewrite (29) as
A(~k,q )IS1(j)(k; q ,...,q ) = − 2~q ·p~σjIσj(k; q ,...,q ) + (31)
j N 1 N j N 1 N
Xσj
I (k; q +q ,...,6q ,...,q ) − I (k+q ; q +q ,...,6q ,...,q )
N−1 1 j j N N−1 j 1 j j N
4Whenregardingmoregeneralsituations,amoregeneralformofthisidentitycanbederivedfromthefunctional
integral defining the interacting fermion theory, in a way analogous to the famous Ward identity of QED. This
identitybetween N-and N−1-pointfunctionsisrelated tofermion numberconservation. Inthepresentcase we
avoid introducing functional integrals and restrict to thesimple propagator identity(28).
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with the definition A(~k,q ) = iq −2~q ·~k + ~q2 . (32)
j j0 j j
On dividing by A(~k,q ), which is bounded away from 0 due to (8), we obtain (in shortened
j
notation)
2
IS1(j)(k; q ,...)= − ~q ·~pσjIσj(k; q ,...) + ∆(~k,q )I (k+q ; q +q ,...) (33)
N 1 A(~k,q ) j N 1 j N−1 j 1 j
Xσj j
1 1
+ I (k; q +q ,...)− I (k+q ; q +q ,...) .
A(~k,q ) N−1 1 j A(~k+~q ,q ) N−1 j 1 j
j j j
Here we used the definition
1 1 2~q2
∆(~k,q ) = − = j . (34)
j A(~k+~q ,q ) A(~k,q ) A(~k+~q ,q )A(~k,q )
j j j j j j
Regarding (33) we realize that the last two terms give a vanishing contribution on integration
over ~k, by a shift of k. And the prefactors of the first two terms scale as λ2 in the dynamical
limit (see below (45, 46)), whereas there appears a gain of a factor of λ in the scaling limit
when taking into account the change N → N −1 in the second term and using an inductive
argument based on A2) (11). To make work this inductive argument, we have to generalize (33)
σn
to symmetrization w.r.t. more than one variable. The Ward identity for I N with n ≥ 2 is
N
obtained from (28) in the same way as (29) :
{iq −2~q ·(~k−p~σNn)+~q2}IσNn(k; q ,...,q ) = (35)
jν,0 jν jν jν N 1 N
σN(jX1,...,jn)
σn−1 σn−1
I N−1(k; q +q ,...,q ) − I N−1(k+q ; q +q ,...,q ) .
N−1 1 jν N N−1 jν 1 jν N
σN−1(j1X,...,6jν,...,jn)(cid:18) (cid:19)
σn
Here the momentum p~ N is the one arriving at the vertex of q , j ∈ {j ,...,j }, for the
jν jν ν 1 n
permutation σn . For each permutation σn appearing on the l.h.s. we sum on the r.h.s. over
N N
a permutation [σn−1(σn,j )](j ,...,6j ,...,j ) which is defined as
N−1 N ν 1 ν n
σn(j), if σn(j) < σn(j )
[σn−1(σn,j )](j) = N N N ν (36)
N−1 N ν (σNn(j)−1, if σNn(j) > σNn(jν) )
(so that σn−1(σn,j ) is indeed a map onto {2,...,N − 1}). To proceed to an identity in
N−1 N ν
terms of ISn we have to analyse and eliminate (as far as possible) the dependence of the term
N
∼ ~q ·p~σNn on the permutations σn . We use the following
jν jν N
Lemma 2 :
n n
a) ~q ·p~σ(j1,...,jn) = ~q ·~q + ~q ·~pˆσ(j1,...,jn) . (37)
jν jν i j jν jν
ν=1 i,j,i<j ν=1
X X X
i,j∈{j1,...,jn}
Here pˆ is obtained from p by setting to zero the momenta q ,...,q , i.e.
j1 jn
pˆσ(j1,...,jn)(q ,...,q ) = pσ(j1,...,jn)(qˆ ,...,qˆ ), (38)
jν 1 N jν 1 N
7
where qˆ = q , if i6∈ {j ,...,j } and qˆ = 0, if i∈ {j ,...,j }.
i i 1 n i 1 n
b) pˆjσν(j1,...,jn)(q1,...,qN)= pˆσˆjν := qσˆj−ν1(k) . (39)
k∈({2,...,N}−X{j1,...,j6ν,...,jn}),
k<σˆj−ν1(jν)
Here σˆ is a permutation of the type σ1 , and it is defined as the permutation of the
jν N−(n−1)
sequence (2,...,N)−(j ,...,j6 ,...,j ) which transfers j to the same position relative to
1 ν n ν
(2,...,N(cid:16))−(j ,...,j ) as σ(j ,...,j (cid:17)) does.
1 n 1 n
(cid:16) (cid:17)
c) Iσ(j1,...,jn) = (ISn−1(j1,...,6jν,...,jn))σˆjν . (40)
N N
σ(j1X,...,jn) Xσˆjν
Proof : a) To extract all terms ∼ ~q · ~q , i, j ∈ {j ,...,j } from n ~q · p~σ(j1,...,jn), we
i j 1 n ν=1 jν jν
go through the sum over j according to the order in which q appears in the N-loop for the
ν jν P
permutationσ(j ,...,j ), startingfromthelastmomentum. Werealize thatwepick upexactly
1 n
once each pair ~q ·~q , i, j ∈ {j ,...,j }. Once these terms have been extracted the remainder
i j 1 n
obviously takes the form from (37).
b) Since pˆσ(j1,...,jn)(q ,...,q ) equals the sum of the momenta arriving at the vertex of q in
jν 1 N jν
the permutation σ(j ,...,j ), with all q , j ∈ {j ,...,j } set to zero, it is equal to the sum
1 n j 1 n
over those momenta, which lie in the complementary set and arrive at the vertex of q . So it
jν
equals pˆσˆjν .
c) On the r.h.s. of (40), ISn−1, which has been symmetrized w.r.t. (j ,...,j6 ,...,j ), depends
N 1 ν n
only on the sequence (2,...,N) − (j ,...,6j ,...,j ) , which is acted upon by σˆ . The
1 ν n jν
statement (40) then fol(cid:16)lows from the observation that s(cid:17)umming over all possible orderings of
(j ,...,j ) within(2,...,N),keepingtheorderoftheremainingvariablesfixed,canbeachieved
1 n
bysumming,forfixedorderingof (2,...,N)−(j ,...,j6 ,...,j ) ,overallpossibleorderingsof
1 ν n
(j ,...,j6 ,...,j ) within (2,...,N(cid:16)), and then over the position o(cid:17)f j relative to (2,...,N)−
1 ν n ν
(j ,...,j ) . (cid:16)
1 n
(cid:17)
Using Lemma 2 we come back to the analysis of (35). On summing over ν we obtain:
n
{iq −2~q ·~k+~q2} + ~q ·~q ISn(j1,...,jn)(k; q ,...,q ) = (41)
jν,0 jν jν i j N 1 N
(cid:20)ν=1 i,j,i<j (cid:21)
X X
i,j∈{j1,...,jn}
n
−2 ~q ·~pˆσˆjν (ISn−1(j1,...,6jν,...,jn))σˆjν(k; q ,...,q )
jν jν N 1 N
νX=1Xσˆjν
n
σn−1 σn−1
+ I N−1(k; q +q ,...,6q ,...,q ) − I N−1(k+q ; q +q ,...,6q ,...,q ) .
N−1 1 jν jν N N−1 jν 1 jν jν N
νX=1σXNn−−11(cid:18) (cid:19)
For the prefactor on the l.h.s. of (41) we write
n
A(~k; q ,...,q ):= {iq −2~q ·~k+~q2} + ~q ·~q , (42)
j1 jn jν,0 jν jν i j
ν=1 i,j,i<j
X X
i,j∈{j1,...,jn}
8
and we also introduce
∆(~k,~q ; q ,...,q ):= 1 − 1 = 2 nµ=1~qjµ ·~qjν . (43)
jν j1 jn A(~k+~qjν; qj1,...) A(~k; qj1,...) A(~k; .P..)A(~k+~qjν;...)
We divide by A(~k; q ,...,q ) (similarly as in (33) above) and obtain
j1 jn
n n
ISn = −2 ~q ·~pˆσˆjν (ISn−1)σˆjν + ∆(~k,~q ;...)ISn−1(k+q ;q +q ,...)
N A(~k; q ,...) jν jν N jν N−1 jν 1 jν
j1 νX=1Xσˆjν νX=1
n
1 1
+ ISn−1(k;q +q ,...) − ISn−1(k+q ;q +q ,...) . (44)
A(~k;...) N−1 1 jν A(~k+~q ;...) N−1 jν 1 jν
νX=1(cid:18) jν (cid:19)
In the first line on the r.h.s. there appear the terms ISn−1 and ISn−1. Regarding their prefac-
N N−1
tors, ~qj ·~pσˆjν scales as λ2 in the small λ and dynamical limits, and, by (8),
|A(~k; q ,...,q )| > η, |A(~k; λq ,...,λq )| > λη, |A(~k; q , λ~q ,...)| > η , (45)
j1 jn j1 jn j1,0 j1
K K
|∆(~k,λ~q ;λq ,...,λq )| ≤ 1 , ∆(~k,λ~q ;q ,λ~q ,...,q ,λ~q )| ≤ λ2 2 , (46)
jν j1 jν η2 jν j1,0 j1 jν,0 jν η2
(where K , K depend on the (compact) sets of momenta considered).
1 2
Our inductive proof of A2) B2) is based on (44) together with (24, 25). We use an inductive
scheme proceeding upwards in N ≥ 3, and for fixed N upwards in n for 0 ≤ n ≤ N −2. The
induction hypotheses are
|ΠSn(ϕ ; λq ,...,λq )| ≤ O(λ−(N−2−n)), |ΠSn(ϕ ; q ,λ~q ,..., q ,λ~q )| ≤ O(λ2+2n). (47)
N s 1 N N d 10 1 N0 N
For any N and n = 0 (the unsymmetrized case) the claim follows from (24). For n > 0 we
regard the scaling resp. dynamical limit for (44), multiplied by ϕ and integrated over k. We
can apply the induction hypothesis to the r.h.s. of (44) noting again that the functions ∆ϕ and
Aϕ have the properties reqired for ϕ. We also use (25) if N = 3. For each entry in the sum in
the second line of (44) we perform in the second term the change of variables k˜ = k+λq resp.
j
k˜ = k+(q ,λ~q ) and then use (21). With the aid of (45, 46) and the induction hypothesis one
j0 j
then shows
dk ddk
| 0 ISn(k; λq ,...,λq )ϕ(~k; λq ,...,λq )| ≤ O(λ−(N−2−n)) , (48)
2π (2π)d N 1 N 1 N
Z
dk ddk
| 0 ISn(k; q , λ~q ,...,q , λ~q )ϕ(~k; q , λ~q ,...,q , λ~q )| ≤ O(λ2+2n) . (49)
2π (2π)d N 1,0 1 N,0 N 1,0 1 N,0 N
Z
On specializing to ϕ ≡ 1, this ends the proof of the proposition.
9
We join a few comments on various extensions of the results obtained.
a) For dimensions d ≥ 4 the N-loop integrals are absolutely convergent for 2N > d+1 and
canbeobtainedaslimits Λ → ∞oftheirregularizedversions, whicharedefinedonintroducing
0
a regulating function ρ(~k2) in the propagators
Λ2
0
1
G (k) → G (Λ ,k) = .
0 0 0 ik − (ρ−1(~k2)~k2 − 1)
0 Λ2
0
We suppose ρ to be smooth, monotonic, positive, of fast decrease and such that ρ(x) ≡ 1 for
x ≤ 1. The regulator then appears in the A-factors when using the Ward identity, e.g. (32)
changes into
(~k−~q )2 ~k2
iq + ρ−1( j )(~k−~q )2 − ρ−1( )~k2 .
j,0 Λ2 j Λ2
0 0
But since these factors are still independentof k , the regulator disappears without leaving any
0
trace after performing the k -integration, if Λ ≥ |~q |+1. So we still obtain the same results
0 0 j
for d ≥ 4, if 2N > d+1, and we obtain them without this last restriction in case we define the
P
integrals as Λ → ∞- limits of their regulated versions from the beginning.
0
b) Neumayr and Metzner [1] also prove |ΠS (q ,...,q )| ≤ O(|~q |) for ~q → 0, keeping the
N 1 N j j
other variables fixed. In our framework this result is obtained immediately from (33), and we
realize that it holds already on symmetrization with respect to ~q , full symmetrization is not
j
required. This result can be generalized to several vanishing external momenta ~q ,... ,~q , in
j1 jn
the same way as we did for the proof of A2) and B2) in the proposition. Using (44) we obtain
on induction
n
|ΠSn(j1,...,jn)(q ,...,q )| ≤O( |~q |) (50)
N 1 N jν
ν=1
Y
and of course the same bound on ΠS .
N
c) From the proof one can straightforwardly read off a bound w.r.t. the dependence on the
parameter η from (8). This bound is in terms of η−(N+n), stemming from the contributions
with a maximal number of factors of ∆. It is of course rather crude, since it does not take into
account the effects of the nonvanishing spatial variables and can be improved, depending on the
hypotheses made on those.
In conclusion we have recovered previous results on the infrared behaviour of the connected
N-pointdensity-correlationfunctions,inshortN-loops,bysimple,butrigorousargumentsbased
on the Ward identity. We obtain bounds for the fully symmetrized N-loop, in showing, how
successive symmetrization improves the infrared behaviour.5 The bounds hold in any spatial
dimension (taking into account the remarks from a) above). Since the Ward identities are
explicit and easy to handle, they permit generalizations such as (50).
5 We recently learned from W. Metzner, that they were also aware of the fact that partial symmetrization
improves theinfrared behaviour, but did not mention it in [1].
10