Weighted power counting and Lorentz violating gauge theories. I: General properties

0.35 MB·English

by  Damiano Anselmi

IFUP-TH 2008/17
Weighted Power Counting
And Lorentz Violating Gauge Theories.
9
0
I: General Properties
0
2

n
a
J
8 Damiano Anselmi
2
Dipartimento di Fisi
a (cid:16)Enri
o Fermi(cid:17), Università di Pisa,

] Largo Ponte
orvo 3, I-56127 Pisa, Italy,
h
t and INFN, Sezione di Pisa, Pisa, Italy
-
p
damiano.anselmidf.unipi.it
e
h
[

2
v
0
7 Abstra
t
4
3
.
8
We
onstru
t lo
al, unitary gauge theories that violate Lorentz symmetry expli
itly at high energies
0
8 andarerenormalizablebyweightedpower
ounting. They
ontainhigherspa
ederivatives,whi
himprove
0
: thebehaviorofpropagatorsatlargemomenta,butnohighertimederivatives. Weshowthattheregularity
v
i of the gauge-(cid:28)eldpropagatorprivilegesa parti
ularspa
etimebreaking, the oneinto into spa
eand time.
X
We then
on
entrate on the simplest
lass of models, study four dimensional examples and dis
uss a
r
a
number of issues that arise in our approa
h, su
h as the low-energyre
overy of Lorentz invarian
e.
1
1 Introdu
tion
Lorentz symmetry has been veri(cid:28)ed in many experiments with great pre
ision [1℄. However,
di(cid:27)erent types of arguments have lead some authors to suggest that it
ould be violated at very
high energies [2, 3, 4℄. This possibility has raised a
onsiderable interest, be
ause, if true, it would
substantially a(cid:27)e
t ourunderstanding ofNature. TheLorentz violating extension oftheStandard
Model[3℄
ontainsalargeamountofnewparameters. Boundsonmanyofthem,parti
ularlythose
belonging to the power-
ounting renormalizable subse
tor, are available. Their updated values
are reported in ref. [5℄.
In quantum (cid:28)eld theory, the
lassi(cid:28)
ation of lo
al, unitarity, polynomial and renormalizable
models
hanges dramati
ally if we do not assume that Lorentz invarian
e is exa
t at arbitrarily
high energies [6, 7℄. In that
ase, higher spa
e derivatives are allowed and
an improve the
behavior of propagators at large momenta. A number of theories that are not renormalizable by
ordinary power
ounting be
ome renormalizable in the framework of a (cid:16)weighted power
ounting(cid:17)
[6℄, whi
h assigns di(cid:27)erent weights to spa
e and time, and ensures that no term
ontaining higher
time derivatives is generated by renormalization, in agreement with unitarity. Having studied
s
alar and fermion theories in ref.s [6, 7℄, here we begin the study of gauge theories, fo
using on
the simplest
lass of models. The investigation is
ompleted in a se
ond paper [8℄, to whi
h we
refer as paper II, whi
h
ontains the
lassi(cid:28)
ation of renormalizable gauge theories.
The theories we are interested in must be lo
al and polynomial, free of infrared divergen
es in
Feynman diagrams at non-ex
eptional external momenta, and renormalizable by weighted power

ounting. We (cid:28)nd that in the presen
e of gauge intera
tions the set of renormalizable theories
is more restri
ted than in the s
alar-fermion framework. Due to the parti
ular stru
ture of the
gauge-(cid:28)eld propagator, Feynman diagrams are plagued with
ertain spurious subdivergen
es. We
are able to prove that they
an
el out when spa
etime is broken into spa
e and time, and
ertain
other restri
tions are ful(cid:28)lled.
A more deli
ate physi
al issue is the low-energy re
overy of Lorentz symmetry. On
e Lorentz
symmetryisviolatedathighenergies,itslow-energyre
overyisnotguaranteed, be
auserenormal-
ization makes the low-energy parameters run independently. One possibility is that the Lorentz
invariant surfa
e is RG stable [9℄, otherwise a suitable (cid:28)ne-tuning must be advo
ated.
Inotherdomainsofphysi
s,su
hasthetheoryof
riti
alphenomena, whereLorentzsymmetry
is not a fundamental requirement,
ertain s
alar models of the types
lassi(cid:28)ed in ref. [6℄ have
already been studied [10℄ and have physi
al appli
ations.
The paper is organized as follows. In se
tion 2 we review the weighted power
ounting for
s
alar-fermiontheories. Inse
tion3weextendittoLorentzviolatinggaugetheoriesandde(cid:28)nethe

lass of models we fo
us on in this paper. We study the
onditions for renormalizability, absen
e
of infrared divergen
es in Feynman diagrams and regularity of the propagator. In se
tion 4 we
2
provethatthetheoriesarerenormalizabletoallorders,usingtheBatalin-Vilkoviskyformalism. In
se
tion 5 we study four dimensional examples and the low-energy re
overy of Lorentz invarian
e.
In se
tion 6 we dis
uss stri
tly renormalizable and weighted s
ale invariant theories. In se
tion
7 we study the Pro
a Lorentz violating theories, and prove that they are not renormalizable.
Se
tion 8
ontains our
on
lusions. In appendix A we
lassify the quadrati
terms of the gauge-
(cid:28)eld lagrangian and in appendix B we derive su(cid:30)
ient
onditions for the absen
e of spurious
subdivergen
es.
2 Weighted power
ounting
In thisse
tion we brie(cid:29)y review the weighted power
ounting
riterion of refs. [6, 7℄. The simplest
d
framework to study Lorentz violations is to assume that the -dimensional spa
etime manifold
M = Rd Mˆ ×M¯ dˆ Mˆ =
is split into the produ
t of two submanifolds, a -dimensional submanifold
Rdˆ d¯
,
ontaining time and possibly some spa
e
oordinates, and a -dimensional spa
e submanifold
M¯ = Rd¯ d O(1,d − 1)
, so that the -dimensional Lorentz group is broken to a residual Lorentz
O(1,dˆ−1)×O(d¯)
group . In this paper we study renormalization in this simpli(cid:28)ed framework.
The generalization to the most general breaking is done in paper II.
∂ (∂ˆ,∂¯) ∂ˆ ∂¯ Mˆ
The partial derivative is de
omposed as , where and a
t on the subspa
es and
M¯
,respe
tively. Coordinates,momentaandspa
etimeindi
esarede
omposedsimilarly. Consider
a free s
alar theory with (Eu
lidean) lagrangian
1 1
L = (∂ˆϕ)2+ (∂¯nϕ)2,
2 2Λ2n−2 (2.1)
free L
Λ n > 1
L
where is an energy s
ale and is an integer . Up to total derivatives it is not ne
essary to
∂¯ (∂¯nϕ)2
spe
ify how the 's are
ontra
ted among themselves. The
oe(cid:30)
ient of must be positive
to have a positive energy in the Minkowskian framework. The theory (2.1) is invariant under the
weighted res
aling
xˆ → xˆ e−Ω, x¯ → x¯ e−Ω/n, ϕ → ϕ eΩ( /2−1),
ž
(2.2)
= dˆ+d¯/n Λ
L
where ž is the (cid:16)weighted dimension(cid:17). Note that is not res
aled.
The intera
ting theory is de(cid:28)ned as a perturbative expansion around the free theory (2.1).
For the purposes of renormalization, the masses and the other quadrati
terms
an be treated
perturbatively, sin
e the
ounterterms depend polynomially on them. Denote the (cid:16)weight(cid:17) of an
O [O]
obje
t by and assign weights to
oordinates, momenta and (cid:28)elds as follows:
1 1
[xˆ]= −1, [x¯]= − , [∂ˆ] = 1, [∂¯] = , [ϕ] = ž −1,
n n 2 (2.3)
Λ ϕ
L
while is weightless. Polynomiality demands that the weight of be stri
tly positive, so we
> 2
assume ž .
3
P (pˆ,p¯) pˆ p¯ k k
k,n
We say that is a weighted polynomial in and , of degree , where is a multiple
1/n P (ξnpˆ,ξp¯) kn ξ G L V
k,n
of , if is a polynomial of degree in . A diagram with loops, verti
es
I
and internal legs gives an integral of the form
ddLp I V
I (k) = P (p,k) V (p,k),
G (2π)d i j
Z i=1 j=1
Y Y
p k P (p,k)
i
where are the loop momenta, are the external momenta, are the propagators and
V (p,k) ddp
j
are the verti
es. The momentum integration measure has weight ž. The propagator
1 2
is equal to divided by a weighted polynomial of degree . We
an assume that, as far as their
δ
j
momentum dependen
e is
on
erned, the verti
es are weighted monomials of
ertain degrees .
k (kˆ,k¯) → (λkˆ,λ1/nk¯) I (k)
G
Res
aling and as , the integral res
ales with a fa
tor equal to its total
ω(G) I (k) ω(G)
G
weight . By lo
ality, the divergent part of is a weighted polynomial of degree .
Assume that the lagrangian
ontains all verti
es that have weights not greater than ž and only
those. This bound ex
ludes terms with higher time derivatives. Then we (cid:28)nd
−2
ω(G) ≤ −E ž ,
s
ž 2 (2.4)
E > 2
s
where is the number of external s
alar legs. Formula (2.4) and ž ensure that every

ounterterm has a weight not larger than ž, therefore it
an be subtra
ted renormalizing the
(cid:28)elds and
ouplings of the lagrangian, and no new vertex needs to be introdu
ed.
The lagrangian terms of weight ž are stri
tly renormalizable, those of weights smaller than
ž super-renormalizable and those of weights greater than ž non-renormalizable. The weighted
power
ounting
riterion amounts to demand that the theory
ontains no parameter of negative
weight.
ϕ4 = 4
Simple examples of renormalizable theories are the , ž models
L = 1(∂ϕ)2 + 1 (∂nϕ)2 + λ ϕ4
ž=4 2 2Λ2L(n−1) 4!ΛdL−4 (2.5)
b
ϕ6 = 3
and the , ž models
L = 1(∂ϕ)2+ 1 (∂nϕ)2 + 1 λ ∂nϕ4 + λ6 ϕ6.
ž=3 2 2Λ2L(n−1) 4!Λ2L(n−1) Xα αh iα 6!Λ2L(n−1) (2.6)
b
∂nϕ4 n ∂
where α denotes a basis of inequivalent terms
onstru
ted with derivatives a
ting
hϕ i
1
on four 's . Only the stri
tly-renormalizable terms have been listed in (2.5) and (2.6). It is
straightforward to
omplete the a
tions adding the super-renormalizable terms, whi
h are those
ϕ
that
ontain fewer derivatives and/or -powers.
1 O(d) n
Be
ause of -invarian
e, theseexist no su
h terms if is odd.
4
The
onsiderations just re
alled are easily generalized to fermions. The weight of a fermion
−1)/2 d > 1
(cid:28)eld is (ž , so polynomiality is ensured, be
ause ž is ne
essarily greater than 1 (if ).
Formula (2.4) be
omes −2 −1
ω(G) ≤ −E ž −E ž ,
s f
ž 2 2
E
f
where is the number of external fermioni
legs.
n ≥ 2
Our investigation fo
uses on the theories that do
ontain higher spa
e derivatives ( ).
n = 1
Indeed, the theories with , whi
h
an be either Lorentz invariant or Lorentz violating, obey
the usual rules of power
ounting.
3 Lorentz violating gauge theories
∂ = (∂ˆ,∂¯)
Having de
omposed the partial derivative operator as , the gauge (cid:28)eld has to be
A′ = (Aˆ′,A¯′)≡ gA = g(Aˆ,A¯) g
de
omposed similarly. We write , where is the gauge
oupling and
A =AaTa Ta
µ µ
, with anti-Hermitian. The
ovariant derivative is de
omposed as
D =(Dˆ,D¯)= (∂ˆ+Aˆ′,∂¯+A¯′).
(3.1)
With the weight assignments
1
[Aˆ′]= [Dˆ] = 1, [A¯′] = [D¯]= ,
n
the de
omposition (3.1) is
ompatible with the weighted res
aling. The (cid:28)eld strength is split into
three sets of
omponents, namely
Fˆ ≡ F , F˜ ≡ F , F¯ ≡ F .
µν µˆνˆ µν µˆν¯ µν µ¯ν¯
(3.2)
(∂ˆAˆ)2 Aˆ /2−1 [g] = 2− /2
Sin
e the kineti
lagrangian must
ontain , the weight of is ž , hen
e ž .
[A¯] [F˜] = [∂¯]+[Aˆ] = [∂ˆ]+[A¯]
We
an read from . In summary,
1 1 2
[Aˆ] = ž−1, [A¯]= ž−2+ , [Fˆ] = ž, [F˜]= ž−1+ , [F¯]= ž−2+ .
2 2 n 2 2 n 2 n (3.3)
g
Sin
e the weight of
annot be negative, we must have
≤ 4.
ž (3.4)
1/α
In this paper we fo
us on the (cid:16) theories(cid:17), namely those that have a lagrangian of the form
1
L = L (gA,gϕ,gψ,gC¯,gC,λ).
r
α (3.5)
C C¯ ϕ ψ
Here and are the ghosts and antighosts, are the s
alar (cid:28)elds and are the fermions.
L g λ
r
Moreover, the redu
ed lagrangian depends polynomially on and the other parameters , and
[λ] ≥ 0
. The renormalizability of the stru
ture (3.5) is easy to prove (see (3.24)).
5
= 4 g
When ž the gauge
oupling is weightless and the theory
an always be written in the
1/α < 4 1/α
form with a suitable rede(cid:28)nition of parameters. Instead, when ž the theories are a
d = 4 L ∂ˆ gAˆ gϕ gC¯
r
small subset of the allowed theories. For example, in the weights of , , , , ,
gC gψ ∂¯ gA¯
and
oin
ide with their dimensions in units of mass, and only the weights of and ,
1/n
whi
h are equal to , di(cid:27)er from their dimensions. Then the lagrangian
ontains just the usual
D¯ F¯ F˜
power-
ounting renormalizable terms, plus the terms that
an be
onstru
ted with , and .
d 6= 4
The form of the lagrangian does not
hange in . Polynomiality is always ensured.
1/α
Even if the theories are not parti
ularly interesting from the physi
al point of view, it is

onvenient to start from them, be
ause the simpli(cid:28)ed stru
ture (3.5) allows us to illustrate the
basi
properties of Lorentz violating gauge theories without unne
essary
ompli
a
ies. The most
general
ase is studied in paper II.
Observe that the theories (3.5)
annot
ontain higher time derivatives, as desired. Indeed, by
O(1,dˆ−1) ∂ˆ L ∂ˆ gAˆ
r
-invarian
e a term with three 's in must
ontain at least another , or a , or a
g2ψ¯γˆψ ∂ˆ4 g∂ˆ3Aˆ
fermion bilinear su
h as . However, the weights of and are already equal to four,
g2∂ˆ3ψ¯γˆψ
so no other leg
an be atta
hed to su
h obje
ts, and the weight of is equal to six.
It is
onvenient to write the a
tion
S = ddx(L +L ) ≡ S +S ,
0 Q I Q I
(3.6)
Z
S
Q
as the sum of two gauge-invariant
ontributions, the quadrati
terms plus the vertex terms
S
I
. By (cid:16)quadrati
terms(cid:17) we mean the terms
onstru
ted with two (cid:28)eld strengths and possibly

ovariant derivatives. By (cid:16)vertex terms(cid:17) we mean the terms
onstru
ted with at least three (cid:28)eld
strengths, and possibly
ovariant derivatives.
L
Q
In Appendix A we prove that, up to total derivatives, the quadrati
part of the lagrangian
reads (in the Eu
lidean framework)
1 1
L = F2 +2F η(Υ¯)F +F τ(Υ¯)F + (D F )ξ(Υ¯)(D F ) .
Q 4 µˆνˆ µˆν¯ µˆν¯ µ¯ν¯ µ¯ν¯ Λ2 ρˆ µ¯ν¯ ρˆ µ¯ν¯ (3.7)
(cid:26) L (cid:27)
Υ¯ ≡ −D¯2/Λ2 η τ ξ n−1 2n−2 n−2
Here L and , and are polynomials of degrees , and , respe
tively.
We have expansions n−1
2j
η(Υ¯) = ηn−1−iΥ¯i, [ηj]= ,
n (3.8)
i=0
X
η
i
and similar, where are dimensionless
onstants of non-negative weights.
In momentum spa
e we see that the free a
tion is positive de(cid:28)nite if and only if
k¯2
η > 0, η˜≡ η+ ξ > 0, τ >0,
Λ2 (3.9)
L
η τ ξ k¯2/Λ2
where now , and are fun
tions of L.
6
Λ
L
Intheparametrization (3.7)thes
ale isaredundant parameter. Itismainly usedtomat
h
η
j
the dimensions in units of mass, so that the other parameters (e.g. the 's)
an be assumed to
Λ Λ
L L
be dimensionless, but possibly weightful. The -redundan
y implies that is RG invariant,
so its beta fun
tion vanishes by de(cid:28)nition.
BRST symmetry and gauge (cid:28)xing The usual BRST symmetry [12℄
g
sAa =DabCb = ∂ Ca+gfabcAbCc, sCa = − fabcCbCc,
µ µ µ µ 2
sC¯a=Ba, sBa = 0, sψi = −gTaCaψj,
ij
Ba
et
., where are Lagrange multipliers for the gauge-(cid:28)xing, isautomati
ally
ompatible with the
C¯∂ˆ2C B2
weighted power
ounting. The quadrati
terms of the ghost Lagrangian
ontain and ,
and have weight ž, so we have the weight assignments
[C]= [C¯]= ž −1, [s] = 1, [B] = ž.
2 2 (3.10)
Ga
The most
onvenient gauge-(cid:28)xing is linear in the gauge potential and gives
λ
L = sΨ, Ψ = C¯a − Ba+Ga , Ga ≡ ∂ˆ·Aˆa+ζ(υ¯)∂¯·A¯a,
gf 2 (3.11)
(cid:18) (cid:19)
λ υ¯ ≡ −∂¯2/Λ2 ζ
where is a dimensionless, weightless
onstant, L and is a polynomial of degree
n−1
. We demand
ζ >0,
(3.12)
∂¯·A¯a
to in
lude the (cid:16)Coulomb gauge-(cid:28)xing(cid:17) .
The total gauge-(cid:28)xed a
tion is (cid:28)nally
S = ddx(L +L +L ) ≡ S +S .
Q I 0
gf gf (3.13)
Z
Propagator The (Eu
lidean) gauge-(cid:28)eld propagator
an be worked out from the free subse
tor
Ba
of (3.13), after integrating out, whi
h amounts to add
1
(Ga)2
2λ (3.14)
L
Q 2
to . The result is
hAˆAˆihAˆA¯i uδˆ+skˆkˆ rkˆk¯
hA(k) A(−k)i = = ,
hA¯AˆihA¯A¯i! rk¯kˆ vδ¯+tk¯k¯! (3.15)
2
A similar propagator, in a di(cid:27)erent
ontext,has already appeared in ref. [11℄.
7
with
1 λ −kˆ2+ζ ηζ −2 k¯2 λ− ηζ
u= , s = + , r = ,
D(1,η) D2(1,ζ) D(1,η)(cid:16)D2(1,ζ(cid:17)) D2(1,ζ)
τ˜ −2ζ kˆ2−ζ2k¯2
1 λ η
v= , t = + ,
D(η˜,τ) D2(1,ζ) (cid:16)D(η˜,τ(cid:17))D2(1,ζ)
where k¯2 kˆ2
D(x,y) ≡ xkˆ2+yk¯2, η˜= η+ ξ, τ˜= τ + ξ,
Λ2 Λ2
L L
η τ ξ ζ x y k¯2/Λ2
and now , , and , as well as and , are meant as fun
tions of L. The ghost propagator
is 1
.
D(1,ζ) (3.16)
A simple gauge
hoi
e ((cid:16)Feynman gauge(cid:17)) is
λ = 1, ζ =η.
(3.17)
hAˆA¯i s
Then, both and vanish, so
1 1 τ˜−η2
u = , s= r = 0, v = , t = .
D(1,η) D(η˜,τ) ηD(η˜,τ)D(1,η) (3.18)
Physi
al degrees of freedom and dispersion relations To study the physi
al degrees of
freedom we
hoose the Coulomb gauge-(cid:28)xing
Ga = ∂¯·A¯a.
C
λ → ∞ ζ → ∞
It
an be rea
hed from the more general gauge-(cid:28)xing (3.11) taking the limit ,
ς ≡ λ/ζ2
in (3.11), (3.14) and (3.15), with (cid:28)xed, and res
aling the antighosts and the Lagrange
C¯a → C¯a/ζ Ba → Ba/ζ L +(∂¯·A¯a)2/(2ς)
Q
multiplier as , . The quadrati
lagrangian gives the
propagators
1 kˆkˆ ςkˆkˆ ςkˆk¯
hAˆ(k) Aˆ(−k)i= δˆ+ + , hAˆ(k) A¯(−k)i = ,
D(1,η) k¯2η (k¯2)2 (k¯2)2
!
1 k¯k¯ ςk¯k¯
hA¯(k) A¯(−k)i= δ¯− + .
D(η˜,τ) k¯2 (k¯2)2
(cid:18) (cid:19)
kˆ = (iE,kˆ) A¯ d¯−1
Writing and studying the poles, we see that the -se
tor propagates degrees of
freedom with energies
E = kˆ2+k¯2τ(k¯2/Λ2L),
s η˜(k¯2/Λ2)
L
8
Aˆ dˆ−1
while the -se
tor propagates degrees of freedom with energies
E = kˆ2+k¯2η(k¯2/Λ2).
L
q
δˆ+ kˆkˆ/(k¯2η)
Indeed, the matrix has one null eigenve
tor on the pole, sin
e its determinant is
D(1,η)/(k¯2η)
equal to . STOh(e1,rdeˆs−idu1e)s are positive in tkˆhe=M0inkowk¯sk6=ia0n frameSwOo(rdk¯). This
an be
immediately seen using invarian
e to set (at ) and invarian
e to
k¯
set all -
omponents but one to zero.
1/k¯2
Finally, the ghost propagator be
omes , whi
h has no pole. In total, the physi
al degrees
d−2
of freedom are , as expe
ted.
Regularity of the propagator A propagator is regular if it is the ratio
P (kˆ,k¯)
r
P′ (kˆ,k¯) (3.19)
2s
r 2s r s
of two weighted polynomials of degrees and , where and are integers, su
h that the
P′ (kˆ,k)
denominator 2s is non-negative (in the Eu
lidean framework), non-vanishing when either
kˆ 6= 0 k 6= 0
or , and has the form
P′(kˆ,k¯) = ωˆ(kˆ2)s +ω¯(k¯2)ns+··· ,
s
(3.20)
ωˆ > 0 ω¯ > 0 (kˆ2)j−m(k¯2)mn j < s 0 ≤ m ≤ j
with , , where the dots
olle
t the terms with , , and
j = s 0 < m < s
, .
a kˆ
The regularity
onditions just stated ensure that: ) the derivatives with respe
t to improve
k¯ ω¯ 6= 0 kˆ b
the large- behavior (be
ause ), besides the large- and overall ones; and ) the derivatives
k¯ kˆ ωˆ 6= 0 k¯
with respe
t to improve the large- behavior (be
ause ), besides the large- and overall
kˆ k¯
ones. The overall divergen
es of the -subintegrals are lo
al in and the overall divergen
es of
k¯ kˆ
the -subintegrals are lo
al in (on
e subdiagrams have been indu
tively subtra
ted).
Inthispaperweusethedimensional-regularization te
hnique. Were
all[6℄thatitisne
essary
dˆ d¯ dˆ−ε d¯−ε
1 2
to
ontinue both and to
omplex values, say and , respe
tively. In the framework
kˆ k¯
of the dimensional regularization the absen
e of - and -subdivergen
es is immediate to prove:
kˆ k¯
being lo
al, the -subdivergen
es are killed by the (dimensionally
ontinued) -subintegrals and
k¯ kˆ
the -subdivergen
esarekilledbythe -subintegrals. Moreexpli
itly,atoneloopwehaveintegrals
of the form ddˆ−ε1kˆ dd¯−ε2k¯ V(kˆ,k¯;pˆ,p¯)
,
Z (2π)dˆ "Z (2π)d¯ Ii=1P2′s(kˆ,k¯;pˆi,p¯i)#
I pi Q
where denotes the number of propagators, are linear
ombinations of the external momenta
P
r
and the numerator
olle
ts both the verti
es and the polynomials of (3.19). Consider (cid:28)rst the
kˆ
integral
ontained in the square braket. Here
an be treated as an external momentum. The
9
kˆ pˆ
i
regularity of the propagator ensures that di(cid:27)erentiating the integrand with respe
t to (or , or
p¯ k¯
i
) a su(cid:30)
ient number of times the -integral be
omes
onvergent. Thus, the divergent part of
k¯ Q kˆ pˆ p¯
i i
the -integral is a polynomial in (and , ). However,
ddˆ−ε1kˆ
Q(kˆ;pˆ,p¯) = 0
(2π)dˆ
Z
in dimensional regularization, be
auseitistheintegral ofapolynomial. Thus the(sub)divergen
e
k¯ kˆ
of the -integral is killed by the -integral. An analogous
on
lusion holds ex
hanging the roles
kˆ k¯
of and . The arguments
an be generalized to higher loops after in
luding the
ounterterms

orresponding to the proper subdiagrams.
kˆ k¯
In a more general regularization setting the absen
e of - and -subdivergen
es is proved as
kˆ k
follows. Theoverall divergen
esofthe - -integrals aresubtra
ted, forexample, bythe(cid:28)rstterms
of the (cid:16)weighted Taylor expansion(cid:17) around vanishing external momenta [6℄. When the regularity
kˆ

onditionsstatedaboveareful(cid:28)lled, those
ountertermsautomati
ally
urealsothe -subintegrals
k
and the -subintegrals. Indeed, the subintegrals
annot behave worse than the full integrals over
kˆ k
and , be
ause (3.20) ensures that the propagators tend to zero with maximal velo
ity also in
the subintegrals, the loop-integration measures grow less rapidly and the verti
es grow not more
kˆ k
rapidly than in the - -integrals.
The s
alar and fermion propagators are
learly regular. On the other hand, a propagator of
the form Λn−1
L
|kˆ||k|n
kˆ k
is not, and
ould generate (cid:16)spurious subdivergen
es(cid:17) when tends to in(cid:28)nity at (cid:28)xed, or
N
vi
eversa. The problem appears in
ertain large fermion models [7℄, and be
omes
ru
ial
whenever gauge (cid:28)elds are present, as we now dis
uss.
The propagators (3.15) and (3.16) are regular at non-vanishing momenta, be
ause the
ondi-
tions (3.9) and (3.12) ensure that the denominators are positive-de(cid:28)nite in the Eu
lidean frame-
work. To have the best ultraviolet behaviors we must strengthen those
onditions requiring also
η > 0, τ > 0, η˜ = η +ξ > 0, ζ > 0,
0 0 0 0 0 0
(3.21)
whi
h we assume from now on. However, attention must be paid to the behaviors of propagators
kˆ k k kˆ
when is sent to in(cid:28)nity at (cid:28)xed , and when is sent to in(cid:28)nity at (cid:28)xed .
The
onditions (3.21) ensure that all gauge and ghost propagators are regular in the Feynman
hA¯A¯i ω¯ 6= 0 ωˆ = 0 k
gauge (3.17)-(3.18), ex
ept , whi
h has , but , so it is regular when tends
kˆ kˆ k
to in(cid:28)nity at (cid:28)xed, but not when tends to in(cid:28)nity at (cid:28)xed: in that region of momentum
hA¯A¯i ∼ 1/kˆ2
spa
e behaves like . To ensure that no spurious subdivergen
es are generated by
kˆ
the -subintegrals, we have to perform a more
areful analysis, whi
h is done in appendix B and
10