Renormalization of the Orientable Non-commutative Gross-Neveu Model

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Renormalization of the Orientable Non-commutative

Gross-Neveu Model

Fabien Vignes-Tourneret

To cite this version:

Fabien Vignes-Tourneret. Renormalization of the Orientable Non-commutative Gross-Neveu Model.

Annales Henri Poincaré, Springer Verlag, 2007, 8 (3), pp. 427-474. �10.1007/s00023-006-0312-6�.

�hal-00083455�

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arXiv:math-ph/0606069v1 29 Jun 2006

Non- ommutative Gross-Neveu Model

Fabien Vignes-Tourneret

Laboratoire de Physique Théorique a

, Bât. 210, CNRS UMR 8627

Université Paris XI, F-91405 Orsay Cedex,Fran e

e-mail: Fabien.Vignesth.u-psud.fr

Abstra t

Weprove thatthenon- ommutative Gross-Neveumodelonthetwo-dimensional

Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing,

renormalizability an be a hieved. However, in the massive ase, this for es us to

introdu e an additional ounterterm of the form

¯

ψ ıγ

0 γ

1 ψ

. The massless ase is renormalizable withoutsu h anaddition.

1 Introdu tion

Fromtherebirthofnon- ommutativequantumeldtheories[1,2,3℄,peoplewere fa edto

a major di ulty. A new(with respe t tothe usual ommutativetheories) kindof

diver-gen esappearedinnon- ommutativeeldtheory[4,5℄. ThisUV/IRmixingin itedpeople

tode laresu htheoriesnon-renormalizable. Nevertheless H.Grosse andR.Wulkenhhaar

found re ently the way toover omesu haproblemby modifyingthe propagator. Su h a

modi ationwill be now alled vul anization. They proved the perturbative

renormal-izability,to allorders, of the non- ommutative

Φ

4

theory onthe four-dimensional Moyal

spa e [6, 7℄. Their proof is written in the matrix basis. This is a basis for the S hwartz

lassfun tionswheretheMoyalprodu tbe omesasimplematrixprodu t[8,9℄. AMoyal

based intera tion has a non-lo al os illating kernel. The main advantage of the matrix

basis is that the intera tion is then of the type

Tr Φ

4

. This form is mu h easier to use

to get useful bounds. The main drawba k is the very ompli ated propagator (see [10℄

for a omplete study of the Gross-Neveu propagator in the matrix basis). This is one

of the reasons whi h lead us to re over in a simplied manner the renormalizability of

the non- ommutative

Φ

4

theory in

x

-spa e [11℄. Thedire t spa ehas several advantages. First of all, the propagator may be omputed exa tly (and used). It has a Mehler-like

form in the

Φ

4

, LSZ and Gross-Neveu theories [10, 11, 12℄. The

x

-spa e allows to om-pare the behaviour of ommutative and non- ommutative theories. It seems to allow

a simpler handling of symmetries like parity of integrals. This point is very useful for

the renormalizationof the Gross-Neveu model. We alsoplan to extend renormalizability

proofs into the non-perturbative domain thanks to onstru tive te hniques developed in

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wewouldliketohavesomeexperien ewithourphysi alspa e. Of ourse

x

-spa ehasalso drawba ks. It for es to deal with non absolutely onvergent integrals. We have to take

are ofos illations. Untilnowitis mu hmoredi ulttoget the exa t topologi alpower

ounting of the known non- ommutativeeld theories indire t spa e than inthe matrix

basis. The non- ommutative parametri representation would ertainly provide an other

way toget the full power ounting [13℄.

Apartfromthe

Φ

4

,themodiedBosoni LSZmodel[11℄andsupersymmetri theories, we now know several renormalizable non- ommutative eld theories. Nevertheless they

eitheraresuper-renormalizable(

Φ

4

2

[9℄)or(and)studiedataspe ialpointintheparameter spa ewheretheyaresolvable(

Φ

3

2 , Φ

3

4

[14,15℄,theLSZmodels[16,17,18℄). Althoughonly logarithmi allydivergent for parity reasons, the non- ommutative Gross-Neveu model is

a just renormalizablequantum eld theory as

Φ

4

. One ofits main interesting features is that it an be interpreted as a non-lo al Fermioni eld theory in a onstant magneti

ba kground. Then apart from strengthening the vul anization pro edure to get

renor-malizablenon- ommutative eld theories, the Gross-Neveu model may alsobe useful for

the study of the quantum Hall ee t. It is alsoa good rst andidate for a onstru tive

study [19℄ of a non- ommutative eld theory as Fermioni models are usually easier to

onstru t. Moreover its ommutative ounterpartbeing asymptoti allyfree and

exhibit-ingdynami almass generation[20, 21,22℄, astudy of thephysi s ofthis modelwould be

interesting.

In this paper, we prove the renormalizability of the non- ommutative Gross-Neveu

model to all orders. For only te hni al reasons, we restri t ourselves to the orientable

ase. An interesting feature of the model is a kind of remaining UV/IR mixing. Some

(logarithmi ally)divergentgraphsenteringthe four-pointfun tionarenotrenormalizable

by a lo al ounterterm b

. Nevertheless these riti al omponents only appear as

sub-divergen es of two-point graphs. It turns out that the renormalization of the two-point

fun tion make the (four-point) riti algraphsnite. In the massive ase, we have toadd

to the Lagrangian a ounterterm of the form

δm ¯

ψıγ

0 _γ

1 _ψ

. The massless model is also

renormalizablewithout su h a ounterterm.

In se tion 2, we present the model and x the notations. We state our main result.

Se tion 3 is devoted to the main te hni al di ulty of the proof. Here is explained how

to exploit properly the vertex os illationsin order to get the power ounting. In se tion

4,we ompute this power ountingwith amultis aleanalysis. In se tion5, weprovethat

allthedivergentsubgraphs an berenormalizedby ounterterms oftheformofthe initial

Lagrangian. Finally,appendi esfollowabout te hni al details and additionalproperties.

A knowledgement I amvery grateful toJ. Magnenfor onstantdis ussions and

rit-i al omments. In parti ular he found howto use properly the vertex os illations. I also

thank V. Rivasseau and R. Gurau for enlightening dis ussions at various stages of this

work and J.-C. Wallet for areful reading.

b

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The non- ommutativeGross-Neveu model (GN

2 Θ

) onsists in a Fermioni quarti ally in-tera ting eld theory on the (two-dimensionnal) Moyal plane

R

2 Θ

. The algebra

A

Θ

of fun tions on

R

2 Θ

may be dened as

S(R

2 ₎

(it may also be extended to an algebra of

tempered distributions, see [23, 24, 8, 25℄ for rigorous des riptions) endowed with the

asso iative non- ommutative Moyalprodu t:

(f ⋆

Θ

g) (x) =(2π)

−2

Z

R

2 Z

R

2 dydk f (x +

1 ₂

Θk)g(x + y)e

ık·y

(2.1)

The skew-symmetri matrix

Θ

is

Θ =

0 −θ

θ

0

(2.2)

where

θ

is a real parameter of dimension length

2

. The a tion of the non- ommutative

Gross-Neveu model is

S[ ¯

ψ, ψ] =

Z

dx ¯

ψ

_−ı/

_{∂ + Ω/e}

x + m + ıδm θγΘ

−1

γ

ψ + V

o

( ¯

ψ, ψ) + V

no

( ¯

ψ, ψ)

(x)

(2.3) where

e

x = 2Θ

−1

_x

and

V = V

o

+ V

no

is the intera tion part given later. The term

in

δm

will be treated perturbatively as a ounterterm. It appears from the two-loop order (see se tion 5.2.2). Throughout this paper we use the Eu lidean metri and the

Feynman onvention

/a = γ

µ

_a

µ

. The matri es

γ

0

and

γ

1

onstitute a two-dimensionnal

representationoftheCliordalgebra

{γ

µ

_{, γ}

ν

_{} = −2δ}

µν

. Notethatwithsu ha onvention

the

γ

µ

's are skew-Hermitian:

γ

µ†

₌

_−γ

µ

.

Propagator The propagator of the theory is given by the followinglemma:

Lemma 2.1 (Propagator 1 [10 ℄) The propagator of the Gross-Neveu model is

C(x, y) =

_−ı/

_{∂ + Ω/e}

x + m

−1

(x, y)

(2.4)

=

Z

∞

0 dt C(t; x, y),

C(t; x, y) =

₋

Ω

θπ

e

−tm

2 sinh(2e

Ωt)

e

−

Ω

e

2 coth(2e

Ωt)(x−y)

2 +ıΩx∧y

(2.5)

×

n

ıe

Ω coth(2e

Ωt)(/

x

− /y) + Ω(/ex − /e

y)

− m

o

e

−2ıΩtγΘ

−1

γ

with

Ω =

e

2Ω

θ

and

x

∧ y = 2xΘ

−1

_y

. We also have

e

−2ıΩtγΘ

−1

γ

_{= cosh(2e}

_Ωt)

1

2 − ı

θ

2 sinh(2e

Ωt)γΘ

−1

γ

.

Thepropagatormayalsobe onsideredasdiagonalinsome olor spa eindi esif wewant

to study

N

opies of spin

1

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Intera tions Con erningthe intera tion part

V

, rst remindthat

∀f

1 , f

2 , f

3 , f

4 ∈ A

Θ

,

Z

dx (f

1 ⋆ f

2 ⋆ f

3 ⋆ f

4 ) (x) =

1 π

2 _{det Θ}

Z

_Y

4 j=1

dx

j

f

j

(x

j

) δ(x

1 − x

2 + x

3 − x

4 )e

−ıϕ

,

(2.6)

ϕ =

4 X

i<j=1

(

−1)

i+j+1

_x

i

∧ x

j

.

(2.7)

This produ tisnon-lo aland only y li allyinvariant. Then,in ontrasttothe

ommut-ativeGross-Neveutheory forwhi hthereisonlyonepossible(lo al)intera tion,theGN

2 Θ

modelexhibits, at least, six dierent ones: the orientable intera tions

V

o

=

λ

1

4 X

a,b

Z

dx ¯

ψ

a

⋆ ψ

a

⋆ ¯

ψ

b

⋆ ψ

b

(x)

(2.8a)

+

λ

2

4 X

a,b

Z

dx ψ

a

⋆ ¯

ψ

a

⋆ ψ

b

⋆ ¯

ψ

b

(x)

(2.8b)

+

λ

3

4 X

a,b

Z

dx ¯

ψ

a

⋆ ψ

b

⋆ ¯

ψ

a

⋆ ψ

b

(x),

(2.8 )

where

ψ

's alternate with

ψ

¯

'sand the non-orientable intera tions

V

no

=

λ

4

4 X

a,b

Z

dx ¯

ψ

a

⋆ ¯

ψ

b

⋆ ψ

a

⋆ ψ

b

(x)

(2.9a)

+

λ

5

4 X

a,b

Z

dx ¯

ψ

a

⋆ ¯

ψ

b

⋆ ψ

b

⋆ ψ

a

(x)

(2.9b)

+

λ

6

4 X

a,b

Z

dx ¯

ψ

a

⋆ ¯

ψ

a

⋆ ψ

b

⋆ ψ

b

(x).

(2.9 )

All these intera tions have the same

x

-kernel thanks to (2.6). The indi es

a, b

are spin indi estakingvaluein

{0, 1}

(or

{↑, ↓}

). Theymaybeadditionnally olorindi esbetween

1

and

N

. For only te hni al reasons, we willrestri t ourselves to orientableintera tions. Su h a quali ationwill be ome lear in the next se tion. This paper is mainlydevoted

to the proof of

Theorem 2.2 (BPHZ Theorem for GN

2 Θ

) The quantum eld theory dened by the a tion (2.3) with

V = V

o

is renormalizable to all orders of perturbation theory.

Multi-s ale analysis In the followingwe usea multi-s aleanalysis[19℄. The rststep

onsists in sli ingthe propagator as

C

l

=

∞

X

i=0

C

_l

i

, C

_l

i

=











Z

M

−2(i−1)

M

−2i

dt C

l

(t; )

if

i > 1

Z

∞

1 dt C

l

(t; )

if

i = 0.

(2.10)

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A

G

=

X

µ

A

µ

_G

(2.11)

where

µ =

{i

l

}

runs over all possible attributions of a positive integer

i

l

for ea h line

l

in

G

. This index represents the s ale of the line

l

. The usual ultraviolet divergen es of eld theory be omes, in the multi-s ale framework, the divergen e of the sum over

attributions

µ

ofindi es. Toworkwith well-denedquantities, weput anultraviolet ut-o

ρ

:

i

∈ {0, . . . , ρ}

. In ea hsli e, the followinglemmagivesa bound onthe propagator.

Lemma 2.3 For all

i

∈ N

, there exists

K, k

∈ R

+

su h that

C

i

_{(x, y)}

_6KM

i

_e

−kM

i

_|x−y|

.

(2.12)

This bound also holds in the ase

m = 0

.

To any assignment

µ

and s ale

i

are asso iated the standard onne ted omponents

G

i

k

, k

∈ {1, . . . , k(i)}

of the subgraph

G

i

made of all lines with s ales

j > i

. These tree omponentsare partially ordered a ording totheir in lusionrelations and the

(ab-stra t)treedes ribingthese in lusionrelationsis alledthe Gallavotti-Ni olòtree[26℄; its

nodes are the

G

i

k

'sand itsrootis the omplete graph

G

.

More pre isely for anarbitrary subgraph

g

one denes:

i

g

(µ) = inf

l∈g

i

l

(µ),

e

g

(µ) =

_{l external line of g}

sup

i

l

(µ).

(2.13)

The subgraph

g

isa

G

i

k

foragiven

µ

ifand onlyif

i

g

(µ) > i > e

g

(µ)

. Asis wellknown in the ommutativeeldtheory ase, thekeytooptimizetheboundoverspatialintegrations

is to hoose the real tree

T

ompatible with the abstra t Gallavotti-Ni olò tree, whi h means that the restri tion

T

i

k

of

T

toany

G

i

k

must stillspan

G

i

k

. This isalwayspossible (by asimple indu tion fromleaves toroot).

Let us dene

i

ν

(µ)

as the index of the line of highest s ale hooked to the vertex

ν

. Then any (amputed)

N

-point fun tion

S

has anee tive expansion:

S

N

(x

1 , . . . , x

N

; ρ) =

X

N

-pointgraphs

G

X

µ(G)

Y

ν∈G

λ

i

ν

A

µ

G

(x

1 , . . . , x

N

; ρ).

(2.14)

Stri tly speaking, we prove here that allthe orders of the ee tive series are nite asthe

ut-o goestoinnity and that there exists a onstant

K

∈ R

su hthat:

lim

ρ→∞

Z

R

2N

N

Y

i=1

dx

i

f

i

(x

i

)

|A

G

µ

(x

1 , . . . , x

N

; ρ)

| 6 K

n(G)

(2.15)

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The delta fun tion in (2.6) implies that the vertex is parallelogram shaped. To simplify

the graphs,we willnevertheless drawit eitheras a lozenge(Fig. 1)or asa square.

Weasso iate a sign,

+

ou

−

, to ea h of the four positionsat avertex. This sign hanges from aposition toitsneighbouringone and ree ts the signs enteringthe delta fun tion.

For example, the delta fun tion asso iated tothe vertex of gure 1 has to be thought to

be

δ(x

1 −x

2 +x

3 −x

4 )

andnot

δ(

−x

1 +x

2 −x

3 +x

4 )

. Thevertexbeing y li allyinvariant, we an freely hoose the sign of one amongthe four positions. The three other signs are

then xed. Letus allorientablealinejoiningapoint

+

toa point

−

. On the ontrary if it joins two

+

(or

−

),we allit lashing. By denition, a graphis orientableif allits lines are orientable. We willdraw orientablelines with anarrowfromits

−

to its

+

end. The

−

positions are then dened asout oming a vertex and the

+

ones asin oming.

Letagraph

G

. Werst hoose a(optimal)spanning rootedtree

T

. The

x

1 x

2 x

3 x

4 −

−

+

Figure 1: A vertex omplete orientation of the graph, whi h orresponds to the hoi e of the

signs at ea h vertex, is xed by the orientation of the tree. For the root

vertex, we hoose an arbitrary position to whi h we give a

+

sign. If the graph is not a va uum graph, it is onvenient to hoose an exernal eld

for this referen e position. We orient then all the lines of the tree and all

the remaining half-loop lines or loop elds, following the y li ity of the

verti es. This means that starting from an arbitrary referen e orientation

at the root and indu tively limbing into the tree, at ea h vertex we follow the y li

order to alternate in oming and out oming lines as in Figure 2a (where the verti es are

pi tured aspoints). Letusremarkthat withsu hapro edure, atreeisalwaysorientable

(and oriented). The looplines may now be orientableor not.

Denition 2.1 (Sets of lines). We dene

T

=

_{

tree lines

} ,

L

=

{

looplines

} = L

0 ∪ L

+

∪ L

−

with

L

0 =

{

looplines

(+,

−)

or

(

−, +)} ,

L

+

=

{

looplines

(+, +)

} ,

L

−

=

{

looplines

(

−, −)} .

Itis onvenient toequip ea hgraph withatotal orderingamongthe vertex variables.

Westartfromthe rootandturn aroundthe treeinthe trigonometri alsense. Wenumber

all the vertex positions in the order they are met. See Figure 2b. Then it is possible to

order the lines and externalpositions.

Denition 2.2 (Order relations). Let

i < j

and

p < q

. For all lines

l = (i, j), l

′

₌

(p, q)

_{∈ T ∪ L}

,for all externalposition

x

k

, wedene

l

_{≺ l}

′

if

i < j < p < q

l

_{≺ k}

i < j < k

l

⊂ l

′

_{p < i < j < q}

k

_{⊂ l}

i < k < j

:

l

ontra ts above

x

k

l ⋉ l

′

_{i < p < j < q.}

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(a)Orientationofatree

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

2 (x

2 )

1 (x

1 )

₃

4 5 (y)

6

7 8 (z)

9 (x

3 )

10

11

12

13

14 15 (x

4 )

16 ℓ

4 ℓ

1 l

6 l

3 l

2

4

3

2

1

(b)Totalordering

Figure 2: Orientability and ordering

We extend these denitions to the sets of lines dened in 2.1. For example, we write

L

0 ⋉

L

+

instead of

{(ℓ, ℓ

′

₎

_{∈ L}

0 × L

+

, ℓ ⋉ ℓ

′

}

. We also dene the following set. Let

S

1

and

S

2

twosets of lines,

S

1 ⋉

⋊

S

2 =

{(l, l

′

)

∈ S

1 × S

2 , l ⋉ l

′

or

l ⋊ l

′

_{} .}

(2.16)

For example, in Figure 2b,

ℓ

1 ≺ ℓ

4

,

l

2 ⊂ ℓ

1

,

l

3 ≻ x

1

. Note also that with su h sign onventions, orientable lines always join an even (

−

) to an odd (

+

) numbered position. It is now onvenient to dene new variables. These are relative tothe lines of the graph

whereas thevariablesused untilnowwere vertexvariables. Ea horientableline

l

joinsan out omingposition

x

l−

toanin omingone

x

l+

. Wedene

u

l

= x

l+

−x

l−

asthedieren e between the in oming and the out oming position. For the lashing lines,

u

l

is also the dieren ebetween itstwoendsbut thesign isarbitraryand hosenindenition2.3. The

u

l

are the short variables. The long ones are dened as the sum of the two ends of the lines. Wewrite them

v

l

= x

l+

+ x

l−

fortree lines and

w

ℓ

= x

ℓ+

+ x

ℓ−

for the loops. Denition 2.3 (Short and long variables). Let

i < j

. Forallline

l = (i, j)

∈ T ∪ L

,

u

l

=











(

₋₁₎

i+1

_s

i

+ (

−1)

j+1

s

j

∀l ∈ T ∪ L

0 ,

s

i

− s

j

∀l ∈ L

+

,

s

j

− s

i

∀l ∈ L

−

.

(2.17)

v

l

=s

i

+ s

j

∀l ∈ T

(2.18)

w

l

=s

i

+ s

j

∀l ∈ L.

(2.19)

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orient-eld only ontra ts toits omplex onjugate. Forthe Gross-Neveu model, aline an also

beoriented fromits

ψ

end toits

ψ

¯

end. Then weare leadtodene twodierentsigns for a same line.

Denition 2.4 (Signs of a line). Let

i < j

. Forall line

l = (i, j)

∈ T ∪ L

,

ε(l) = +1

_{∀l ∈ T ∪ L}

0

if

i

even

= +1

L

−

=

₋₁

_{T ∪ L}

0

if

i

odd

=

₋₁

_L

+

ǫ(l) = +1

if

ψ(x

i

) ¯

ψ(x

j

)

=

₋₁

if

ψ(x

¯

i

)ψ(x

j

).

Corollary 2.4 (Propagator 2) From the denitions 2.3 and 2.4, the propagator

or-responding to a line

l

may be written as

C

l

(u

l

, v

l

) =

Z

∞

0 dt

l

C(t

l

; u

l

, v

l

)

(2.20)

C(t

l

; u

l

, v

l

) =

Ω

θπ

e

−t

l

m

2 sinh(2e

Ωt

l

)

e

−

Ω

e

2 coth(2e

Ωt

l

)u

2 l

−ı

Ω

2 ǫ(l)ε(l)u

l

∧v

l

(2.21)

×

n

ıe

Ω coth(2e

Ωt

l

)ǫ(l)ε(l)/

u

_l

+ Ωǫ(l)ε(l)/e

u

l

+ m

o

e

−2ıΩt

l

γΘ

−1

γ

with

Ω =

e

2Ω

θ

and where

v

l

will be repla ed by

w

ℓ

if the propagator orresponds to a loop line.

2.2 Position routing

We give here a rule to solve in an optimal way the vertex delta fun tions. In parti ular

this will allow us to fa torize the global delta fun tion (see (2.6)) for ea h four-point

subgraph. There is no anoni al way to do it but we an reje t the arbitrariness of the

pro ess intothe hoi eof atree. Then itis onvenienttointrodu eabran h system. To

ea h tree line

l

we asso iate a bran h

b(l)

ontaining the verti es above

l

. Let us dene above. Atea h vertex

ν

, there exists aunique tree linegoing down towards the root. We denote it by

l

ν

. A ontrario, to ea h tree line

l

orresponds a unique vertex

ν

su h that

l

ν

= l

. We alsodene

P

ν

as the unique set of tree lines joining

ν

to the root. Then the bran h

b(l)

is the set of verti es dened by

b(l) =

_{{ν ∈ G| l ∈ P}

ν

} .

(2.22)

On Figure 2b, the bran h

b(l

2 ) =

{2, 3, 4}

. We an now repla e the set of vertex delta fun tions by a new set asso iated to the bran hes. Let a graph

G

with

n

verti es. A tree is made of

n

− 1

lines whi h give raise to

n

− 1

bran hes. At ea h vertex

ν

, we repla e

δ

ν

(

P

4 i=1

(

−1)

i+1

x

ν

i

)

by

δ

l

ν

(

P

ν

′

_∈b(l

ν

)

P

4 i=1

(

−1)

i+1

x

ν

i

′

)

. To omplete this new

system of delta fun tions, we add to these

n

− 1

rst ones the root delta given by

δ

G

(

P

ν

′

_∈G

P

4 i=1

(

−1)

i+1

x

ν

′

(10)

variables. To this aim, we dene the set

b

(l)

of lines ontra ting inside a given bran h

b(l)

:

b

_{(l) =}

_{l

′

_{= (x}

_ν

_{, x}

_ν

′

)

∈ G|ν, ν

′

∈ b(l)} .

(2.23) Therealsoexists lines

l = (x

ν

, x

ν

′

)

with

ν

∈ b(l)

and

ν

′

_{∈ b(l)}

_/

. Moreover

b(l)

may ontain external positions. We denote by

X

(l)

the set made of the external positions in the bran h

b(l)

and of the ends (in

b(l)

) of lines joining

b(l)

to an other bran h. From the denition 2.3 of short and long variables,for xed

ν

, we have

X

ν

′

_∈b(l

ν

)

4 X

i=1

(

−1)

i+1

x

ν

′

i

=

X

l∈(T ∪L

0 )∩b(l

ν

)

u

l

+

X

ℓ∈L

+

∩b(l

ν

)

w

ℓ

−

X

ℓ∈L

−

∩b(l

ν

)

w

ℓ

+

X

e∈X (l

ν

)

η(e)x

e

(2.24)

where

η(e) = 1

iftheposition

i

isin omingand

−1

ifnot. Forexample,thedeltafun tion asso iated tothe bran h

b(l

2 )

inthe Figure2bis

δ(y

_{− z + x}

3 + x

4 + u

l

3 + u

ℓ

5 + u

l

6 − w

ℓ

4 ).

(2.25) In the same manner, the delta fun tion of the omplete bran h is

δ(x

1 − x

2 + x

3 + x

4 + u

ℓ

1 + u

l

2 + u

l

3 + u

ℓ

5 + u

l

6 − w

ℓ

4 ).

(2.26) Letus emphasize the parti ular ase of

δ

G

δ

G

X

l∈T ∪L

0 u

l

+

X

ℓ∈L

+

w

ℓ

−

X

ℓ∈L

−

w

ℓ

+

X

e∈E(G)

η(e)x

e

(2.27)

where

E

(G)

is the set of external points in

G

. Remark that for an orientable graph

G

(

L

+

=

L

−

=

∅

), the root delta fun tion (2.27) only ontains the external points and the sum of all the

u

l

variablesin

G

.

Remark. In the

Φ

4

model[11℄,these delta fun tionswere used to solveall thelong tree

variables

v

l

, l

∈ T

. This isthe optimal hoi e. Integrations overthe long variables

v

l

(or

w

l

) ost

M

2i

l

. Moreover the tree being hosen optimal, the

v

l

are the most expensive long variables. From (2.24), wehave

δ

b(l)

_X

l

′

_{∈(T ∪L}

0 )∩b(l)

u

l

′

+

X

ℓ∈L

+

∩b(l)

w

ℓ

−

X

ℓ∈L

−

∩b(l)

w

ℓ

+

X

e∈X (l)

η(e)x

e

.

(2.28)

There exists

e

l

∈ X (l)

su h that

x

e

l

=

1

2 (η(e

l

)u

l

+ v

l

)

(see denition 2.3). This external pointis anend of the line

l

. Thus

δ

b(l)

gives

v

l

=

−η(e

l

)u

l

− 2η(e

l

)

_X

l

′

_{∈(T ∪L}

0 )∩b(l)

u

l

′

+

X

ℓ∈L

+

∩b(l)

w

ℓ

−

X

ℓ∈L

−

∩b(l)

w

ℓ

+

X

e∈X (l)\{e

l

}

η(e)x

e

.

We have then used

n

− 1

delta fun tions (one per tree line). The last one is kept. It is the equivalent of the global momentum onservation in usualeld theories.

(11)

integrals. In the orientable ase, wehave

δ

b(l)

X

l

′

_∈b(l)

u

l

′

+

X

e∈X (l)

η(e)x

e

=

Z

d

2 _p

l

(2π)

2 e

ıp

l

·(

P

_l′∈b(l)

u

l′

+

P

e∈X(l)

η(e)x

e

)

_.

(2.29)

After some manipulations on these os illations (see se tion 3.2), we will get de reasing

fun tions for the

v

l

's and

p

l

's. For ea h tree line

l

, we will integrate over

v

l

and

p

l

, the nal result being bounded by

O(1)

.

3 From os illations to de reasing fun tions

Inthe pre edingse tion,wede idedtoexpressallthe vertex deltafun tionsasos illating

integrals. Then we have

2

independant variables per internal propagator. One is integ-rated over with the exponential de rease of the propagator (see 2.4). The other uses the

propagator and verti es os illations. Then it is useful to pre isethe os illationsin terms

of the

u

's and

v (w)

's variables. This is done in se tion 3.1. We will see how to use the os illationsto get enoughde reasing fun tions in se tion3.2.

3.1 The rosette fa tor

We have seen in the pre eding se tion that the os illationsare expressed in terms of the

vertex variableswhereas the propagatorsarenaturallyexpressed with shortand longline

variables. It is not very onvenient to deal with two equivalent sets of variables. We are

then going to express the vertex os illationswith the linevariables.

In the following we all rosette fa tor the set of all the vertex os illationsplus the

rootdelta fun tion. Wealso distinguish tree lines

l

and looplines

ℓ

. The rst step to a

omplete rewritingofthe vertex os illationsisatree redu tion. It onsistsinexpressing

alltree variablesintermesof

u

and

v

variables. Letagraph

G

of order

n

. Ithas

2(n

− 1)

tree positions. Theremaining

2n + 2

loopandexternalvariablesare subsequently written

s

j

. Byusing the y li symmetry of the verti es and the delta fun tions,we get (see [11℄ for a proof):

Lemma 3.1 (Tree redu tion) The rosette fa tor after the srt Filk moveis [27, 11℄:

δ(s

1 − s

2 +

· · · − s

2n+2

+

X

l∈T

u

l

) exp ıϕ

(3.1) where

ϕ =

2n+2

_X

i<j=0

(

−1)

i+j+1

_s

i

∧ s

j

+

1

2 X

l∈T

ε(l)v

l

∧ u

l

+

X

T ≺T

u

l

′

∧ u

_l

+

X

{l∈T , i≺l}

u

l

∧ (−1)

i+1

s

i

+

X

{l∈T , i≻l}

(

−1)

i+1

s

i

∧ u

l

.

(12)

Thenextstepistoexpressalltheloopvariableswiththe orresponding

u

and

w

variables. In [11℄, we omputed the result for planarregular graph (

g = 0

and

B = 1

,see appendix A for graphologi denitions and also [6, 28℄). Here we need the general ase

d

. We

now denote the (true) external variables by

s

j

k

, k

∈ J1, NK

def

= [1, N]

∩ N

. We write

∁L

0

def

=

_L

+

∪ L

−

.

Lemma 3.2 The rosette fa tor of a general graph is:

δ

N

X

k=1

(

₋₁₎

j

k

+1

_s

j

k

+

X

l∈T ∪L

0 u

l

+

X

ℓ∈L

+

w

ℓ

−

X

ℓ∈L

−

w

ℓ

exp ıϕ

(3.2) with

ϕ = ϕ

E

+ ϕ

X

+ ϕ

U

+ ϕ

W

,

ϕ

E

=

N

X

k<l=1

(

₋₁₎

j

k

+j

l

+1

_s

j

k

∧ s

j

l

,

ϕ

X

=

N

X

k=1

X

((T ∪L

0 )≺j

k

)

∪(∁L

0 ⊃j

k

)

(

−1)

j

k

+1

_s

j

k

∧ u

l

+

X

(T ∪L

0 )≻j

k

u

l

∧ (−1)

j

k

+1

s

j

k

,

ϕ

U

=

1

2 X

T

ε(l)v

l

∧ u

l

+

1

2 X

L

ε(ℓ)w

ℓ

∧ u

ℓ

+

1

2 X

L

0 ⋉

L

0 ε(ℓ)w

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

L

0 ⋉

∁L

0 ε(ℓ)w

ℓ

∧ u

ℓ

′

− ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

L

0 ⋊

∁L

0 −ε(ℓ)w

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

(L

+

⋉

⋊

L

−

)

∪(L

+

⋉

L

+

)∪(L

−

⋉

L

−

)

u

ℓ

∧ ε(ℓ

′

)w

ℓ

′

+ u

_ℓ

′

∧ ε(ℓ)w

_ℓ

+

X

((T ∪L

0 )⊂L

0 )

∪((T ∪L

0 )≻∁L

0 )

ε(ℓ

′

)w

ℓ

′

∧ u

_l

+

X

(∁L

0 ⊂∁L

0 )

∪((T ∪L

0 )≺∁L

0 )

u

l

∧ ε(ℓ

′

)w

ℓ

′

+

X

(T ∪L

0 )≺(T ∪L

0 )

u

l

′

∧ u

_l

+

X

(T ∪L

0 )⊂∁L

0 u

l

∧ u

ℓ

′

+

1

2 X

(L

0 ⋉

L

0 )

∪(L

+

⋉

L

+

)∪(L

−

⋉

L

−

)

u

ℓ

′

∧ u

_ℓ

+

1

2 X

(L

0 ⋉

⋊

∁L

0 )

∪(L

+

⋊

L

−

)∪(L

−

⋊

L

+

)

u

ℓ

∧ u

ℓ

′

,

d

Stri tlyspeaking, weonly need, in thispaper,the orientable ase. Neverthelessthenon-orientable

(13)

ϕ

W

=

X

(∁L

0 ≺j

k

)

∪(L

0 ⊃j

k

)

ε(ℓ)w

ℓ

∧ (−1)

j

k

+1

s

j

k

+

X

∁L

0 ≻j

k

(

₋₁₎

j

k

+1

_s

j

k

∧ ε(ℓ)w

ℓ

+

1

2 X

(L

0 ⋉

L

0 )

∪(∁L

0 ⋉

∁L

0 )∪(L

0 ⋉

⋊

∁L

0 )

ε(ℓ

′

)w

ℓ

′

∧ ε(ℓ)w

_ℓ

+

X

(L

0 ⊃∁L

0 )

∪(∁L

0 ≺∁L

0 )

ε(ℓ

′

)w

ℓ

′

∧ ε(ℓ)w

_ℓ

,

where

l

(

ℓ

) belongsto the set on the left-hand-side.

Proof. As explained inse tion 2.2, the root

δ

fun tion isgiven by

δ

N

X

k=1

(

−1)

j

k

+1

_s

j

k

+

X

l∈T ∪L

0 u

l

+

X

ℓ∈L

+

w

ℓ

−

X

ℓ∈L

−

w

ℓ

.

(3.3)

We express all the loop eld variables with the

u

and

w

variables. Then the quadrati term in the external variablesis

N

X

k<l=1

(

−1)

j

k

+j

l

+1

_s

j

k

∧ s

j

l

.

(3.4) Letanexternal variable

s

j

k

. The linear termswith respe t to

s

j

k

are

ϕ

j

k

=

X

i<j

K

(

₋₁₎

i+1

s

i

∧ (−1)

j

k

s

j

k

+

X

i>j

k

(

₋₁₎

j

k

_s

j

k

∧ (−1)

i+1

_s

i

+

X

T ≻j

k

(

₋₁₎

j

k

_s

j

k

∧ u

l

+

X

T ≺j

k

u

l

∧ (−1)

j

k

s

j

k

(3.5)

where the

s

i

'sare allloopvariables. Leta loopline

ℓ = (i, j)

≺ j

k

. Its ontributionto

ϕ

j

k

is:

(

₋₁₎

i+1

s

i

+ (

−1)

j+1

s

j

∧ (−1)

j

k

_s

j

k

.

(3.6)

The result in terms of the

u

ℓ

and

w

ℓ

variables depends on the orientability of the loop line. From denitions 2.3 and 2.4, we have

(

₋₁₎

i+1

s

i

+ (

−1)

j+1

s

j

∧ (−1)

j

k

_s

j

k

(3.7)

= u

ℓ

∧ (−1)

j

k

s

j

k

if

ℓ

∈ L

0 =

− ε(l)w

ℓ

∧ (−1)

j

k

s

j

k

if

ℓ

∈ L

+

∪ L

−

.

In the same way, if a loop line ontra ts above an external variable

s

j

k

, its ontribution to

ϕ

j

k

is:

(

−1)

i+1

_s

i

+ (

−1)

j

s

j

∧ (−1)

j

k

_s

j

k

(3.8)

=

_{− ε(l)w}

ℓ

∧ (−1)

j

k

s

j

k

if

ℓ

∈ L

0 = u

ℓ

∧ (−1)

j

k

s

j

k

if

ℓ

∈ L

+

∪ L

−

.

(14)

Finally the linear term for

s

j

k

is

ϕ

j

k

=

X

((T ∪L

0 )≺j

k

)

∪(∁L

0 ⊃j

k

)

u

l

∧ (−1)

j

k

s

j

k

+

X

(T ∪L

0 )≻j

k

(

−1)

j

k

_s

j

k

∧ u

l

(3.9)

+

X

(∁L

0 ≺j

k

)

∪(L

0 ⊃j

k

)

(

₋₁₎

j

k

_s

j

k

∧ ε(ℓ)w

ℓ

+

X

∁L

0 ≻j

k

ε(ℓ)w

ℓ

∧ (−1)

j

k

s

j

k

.

Let us now onsider a loop line

ℓ = (p, q)

. Its ontribution to the rosette fa tor de omposes into a loop-loop term and a tree-loop term. We will detail the rst one,

the se ondone being obtained with the same method. The loop-loopterm is:

ϕ

ll

=

X

i<p

(

₋₁₎

i+1

s

i

∧ (−1)

p

s

p

+

X

p<i

i6=q

(

₋₁₎

p

s

p

∧ (−1)

i+1

s

i

+ (

−1)

p+q+1

s

p

∧ s

q

+

X

i<q

i6=p

(

₋₁₎

i+1

s

i

∧ (−1)

q

s

q

+

X

q<i

(

₋₁₎

q

s

q

∧ (−1)

i+1

s

i

=

X

i<p

(

₋₁₎

i+1

s

i

∧ [(−1)

p

s

p

+ (

−1)

q

s

q

] +

X

q<i

[(

₋₁₎

p

s

p

+ (

−1)

q

s

q

]

∧ (−1)

i+1

s

i

+

X

p<i<q

(

−1)

i+1

_s

i

∧ [(−1)

p+1

s

p

+ (

−1)

q

s

q

] + (

−1)

p+q+1

s

p

∧ s

q

.

(3.10) An otherloopline

ℓ

′

_{= (i, j)}

has now sixpossibilities. It mayfolloworpre ede

ℓ

, ontain or be ontained in

ℓ

, ross it by the left orthe right. Moreover the lines

ℓ

and

ℓ

′

may be

orientable or not. I will not exhibit allthese dierent ontributions but will explain our

methodthanks to two examples.

Let

(ℓ, ℓ

′

₎

_{∈ L}

2

0

su h that

ℓ

′

_⋉

_ℓ

. The line

ℓ

′

rosses

ℓ

by the left as dened in 2.2. The orresponding term is:

(

₋₁₎

i+1

s

i

∧ [(−1)

p

s

p

+ (

−1)

q

s

q

] + (

−1)

j+1

s

j

∧ [(−1)

p+1

s

p

+ (

−1)

q

s

q

]

= (

₋₁₎

i+1

s

i

∧ (−u

ℓ

) + (

−1)

j+1

s

j

∧ (−ε(ℓ)w

ℓ

)

=

1

2 (u

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

_)w

ℓ

′

∧ u

_ℓ

+ ε(ℓ)w

_ℓ

∧ u

_ℓ

′

+ ε(ℓ)w

_ℓ

∧ ε(ℓ

′

)w

_ℓ

′

) .

(3.11) In the same way,if

ℓ

∈ L

0

,

ℓ

′

_{∈ L}

+

su h that

ℓ

⊂ ℓ

′

,wehave:

(

−1)

i+1

_s

i

∧ [(−1)

p

s

p

+ (

−1)

q

s

q

] + [(

−1)

p

s

p

+ (

−1)

q

s

q

]

∧ (−1)

j+1

s

j

= (

₋₁₎

i+1

_s

i

∧ (−u

ℓ

) + (

−u

ℓ

)

∧ (−1)

j+1

s

j

= u

ℓ

∧ u

ℓ

′

(3.12)

(15)

ϕ

ll

=

1

2 X

L

ε(ℓ)w

ℓ

∧ u

ℓ

(3.13)

+

X

(L

0 ⊂L

0 )

∪(L

0 ≻∁L

0 )

ε(ℓ

′

_)w

ℓ

′

∧ u

_ℓ

+

X

(L

0 ≺∁L

0 )∪(∁L

0 ⊂∁L

0 )

u

ℓ

∧ ε(ℓ

′

)w

ℓ

′

+

1

2 X

L

0 ⋉

L

0 ε(ℓ)w

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

L

0 ⋉

∁L

0 ε(ℓ)w

ℓ

∧ u

ℓ

′

− ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

L

0 ⋊

∁L

0 −ε(ℓ)w

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

1

2 X

(L

+

⋉

⋊

L

−

)

∪(L

+

⋉

L

+

)∪(L

−

⋉

L

−

)

u

ℓ

∧ ε(ℓ

′

)w

ℓ

′

+ u

_ℓ

′

∧ ε(ℓ)w

_ℓ

+

1

2 X

(L

0 ⋉

L

0 )∪(∁L

0 ⋉

∁L

0 )

∪(L

0 ⋉

⋊

∁L

0 )

ε(ℓ

′

)w

ℓ

′

∧ ε(ℓ)w

_ℓ

+

X

(L

0 ⊃∁L

0 )

∪(∁L

0 ≺∁L

0 )

ε(ℓ

′

)w

ℓ

′

∧ ε(ℓ)w

_ℓ

+

X

L

0 ≺L

0 u

ℓ

′

∧ u

_ℓ

+

X

L

0 ⊂∁L

0 u

ℓ

∧ u

ℓ

′

+

1

2 X

(L

0 ⋉

L

0 )

∪(L

+

⋉

L

+

)∪(L

−

⋉

L

−

)

u

ℓ

′

∧ u

_ℓ

+

1

2 X

(L

0 ⋉

⋊

∁L

0 )

∪(L

+

⋊

L

−

)∪(L

−

⋊

L

+

)

u

ℓ

∧ u

ℓ

′

The tree-loop term is:

ϕ

tl

=

X

{l

′

_{∈T , l}

′

_≺p}

u

l

′

∧ (−1)

p

s

_p

+

X

{l

′

_{∈T , l}

′

_≻p}

(

₋₁₎

p

_s

p

∧ u

l

′

(3.14)

+

X

{l

′

_{∈T , l}

′

_≺q}

u

l

′

∧ (−1)

q

s

_q

+

X

{l

′

_{∈T , l}

′

_≻q}

(

−1)

q

_s

q

∧ u

l

′

=

X

{l

′

_{∈T , l}

′

_≺p}

u

l

′

∧ [(−1)

p

s

_p

+ (

−1)

q

s

_q

] +

X

{l

′

_{∈T , l}

′

_≻q}

[(

₋₁₎

p

s

p

+ (

−1)

q

s

q

]

∧ u

l

′

+

X

{l

′

_{∈T , p≺l}

′

_≺q}

u

l

′

∧

(

−1)

p+1

s

_p

+ (

−1)

q

s

_q

=

X

L

0 ≻T

u

ℓ

∧ u

l

′

+

X

(L

0 ≺T )

∪(∁L

0 ⊃T )

u

l

′

∧ u

_ℓ

+

X

(L

0 ⊃T )

∪(∁L

0 ≺T )

ε(ℓ)w

ℓ

∧ u

l

′

+

X

∁L

0 ≻T

u

l

′

∧ ε(ℓ)w

_ℓ

.

(16)

δ

N

X

k=1

(

₋₁₎

j

k

+1

_s

j

k

+

X

l∈T ∪L

u

l

exp ıϕ

(3.15) with

ϕ = ϕ

E

+ ϕ

X

+ ϕ

U

+ ϕ

W

,

ϕ

E

=

N

X

k<l=1

(

₋₁₎

j

k

+j

l

+1

_s

j

k

∧ s

j

l

,

ϕ

X

=

N

X

k=1

X

(T ∪L)≺j

k

(

₋₁₎

j

k

+1

_s

j

k

∧ u

l

+

X

(T ∪L)≻j

k

u

l

∧ (−1)

j

k

+1

s

j

k

,

ϕ

U

=

1

2 X

T

ε(l)v

l

∧ u

l

+

1

2 X

L

ε(ℓ)w

ℓ

∧ u

ℓ

+

1

2 X

L⋉L

ε(ℓ)w

ℓ

∧ u

ℓ

′

+ ε(ℓ

′

)w

_ℓ

′

∧ u

_ℓ

+

X

(T ∪L)⊂L

ε(ℓ

′

_)w

ℓ

′

∧ u

_l

+

X

(T ∪L)≺(T ∪L)

u

l

′

∧ u

_l

+

1

2 X

L⋉L

u

ℓ

′

∧ u

_ℓ

,

ϕ

W

=

X

L⊃j

k

(

−1)

j

k

_s

j

k

∧ ε(ℓ)w

ℓ

+

1

2 X

L⋉L

ε(ℓ

′

)w

ℓ

′

∧ ε(ℓ)w

_ℓ

.

Proof. It is enoughto set

L

+

=

L

−

=

∅

in the general expression of lemma 3.2.

Corollary 3.4 Let a planar regular graph (

g = 0

and

B = 1

). Its rosette fa tor is [11℄

δ

N

X

k=1

(

₋₁₎

k+1

x

k

+

X

l∈T ∪L

u

l

exp ıϕ

(3.16) ave

ϕ = ϕ

E

+ ϕ

X

+ ϕ

U

,

ϕ

E

=

N

X

i<j=1

(

₋₁₎

i+j+1

x

i

∧ x

j

,

ϕ

X

=

N

X

k=1

X

(T ∪L)≺k

(

₋₁₎

k+1

x

k

∧ u

l

+

X

(T ∪L)≻k

u

l

∧ (−1)

k+1

x

k

,

(17)

ϕ

U

=

1

2 X

T

ε(l)v

l

∧ u

l

+

1

2 X

L

ε(ℓ)w

ℓ

∧ u

ℓ

+

X

(T ∪L)⊂L

ε(ℓ

′

)w

ℓ

′

∧ u

_l

+

X

(T ∪L)≺(T ∪L)

u

l

′

∧ u

_l

.

Proof. As the graph has only one broken fa e, there is always an even number of elds

between two external variables. In this ase,

j

k

and

k

have the same parity. Thus by swit hing

s

j

k

into

x

k

, the quadrati term inthe external variables is:

N

X

i<j=1

(

−1)

i+j+1

_x

i

∧ x

j

.

(3.17)

Moreover the onstraints

g = 0

and

B = 1

imply that the graph is orientable (

L = L

0

). Indeed, letus onsider a lashingloopline

ℓ

joining

s

i

to

s

i+2p

. These twopositionshave same parity. Between the two ends of

ℓ

are an odd number of positions. Then either

ℓ

ontra tsaboveanexternalvariableand

B > 2

,oranotherloopline rossesitand

g > 1

. Finallyby skippingfromthe resultoflemma3.2theterms on erning rossinglines,lines

ontra ting aboveexternal variables and non-orientablelines, we get (3.16).

3.2 The masslets

Contrary to the

Φ

4

ase, the Gross-Neveu propagator

C

i

(2.21) does not ontain any

term of the form

exp

−M

−2i

_w

2

(we allthemmasslets) [11℄. This term is repla ed by an

os illation of the type

u

∧ w

. Whereas masslets are not in the propagator, they appear after integration over the

u

variables:

Z

d

2 u e

−M

2i

u

2 +ıu∧w

= KM

−2i

e

−M

−2i

w

2 .

(3.18)

Let

G

a onne ted graph. Itsamplitude is

A

G

=

Z

_Y

N

i=1

dx

i

f

i

(x

i

)δ

G

Y

l∈T

du

l

dv

l

δ

b(l)

C

l

(u

l

, v

l

)

Y

ℓ∈L

du

ℓ

dw

ℓ

C

ℓ

(u

ℓ

, w

ℓ

)e

iϕ

.

(3.19)

The points

x

i

, i

∈ J1, NK

are the external positions. For the delta fun tions, we use the notations of se tion 2.2. The total vertex os illation

ϕ

is given by the lemma 3.2. It is onvenient to split the propagator into two parts. We dene, for all line

l

∈ G

,

¯

C

_l

(u

l

)

by

C

l

(u

l

, v

l

) = ¯

C

l

(u

l

) e

−ı

Ω

2 ǫ(l)ε(l)u

l

∧v

l

. On e more we repla e

v

by

w

for loop lines. This splitting allows to gather the propagators os illations with the vertex ones. The

total os illation

ϕ

Ω

is simply dedu ed from

ϕ

by repla ing the terms

1

2 ε(l)v

l

∧ u

l

by

1

2 (1 + ǫ(l)Ω)ε(l)v

l

∧ u

l

. The graph amplitude be omes

A

G

=

Z

_Y

N

i=1

dx

i

f

i

(x

i

)δ

G

Y

l∈T

du

l

dv

l

δ

b(l)

C

¯

l

(u

l

)

Y

ℓ∈L

du

ℓ

dw

ℓ

C

¯

ℓ

(u

ℓ

)e

iϕ

Ω

.

(3.20)

(18)

In ontrast with the

Φ

4

theory [11℄, we won't solve the bran h delta fun tions. Instead

we keep

δ

G

but express the

n

− 1

other delta fun tionsas os illatingintegrals:

δ

b(l)

X

l

′

_∈b(l)

u

l

′

+

X

e∈X (l)

η(e)x

e

=

Z

d

2 _p

l

(2π)

2 e

ıp

l

·(

P

l′∈b(l)

u

l

+

P

e∈X(l)

η(e)x

e

)

.

(3.21)

As already explainedinse tion 2.2, thereexists

e

l

∈ X (l)

su hthat

x

e

l

=

1

2 (η(e

l

)u

l

+ v

l

)

. Remark that

η(e

l

) = ε(l)

. Then

X

l

′

_∈b(l)

u

l

′

+

X

e∈X (l)

η(e)x

e

=

1

2 (u

l

+ ε(l)v

l

) +

X

l

′

_∈b(l)

u

l

′

+

X

e∈X (l)\{e

l

}

η(e)x

e

.

(3.22)

In the following wewilluse anadditionnal notation. Forallline

l

∈ T

,letusdene

ν

l

as the unique vertex su h that

l = l

ν

where

l

ν

isdened in se tion2.2.

ν

l

is the vertex just above

l

inthe tree. Wewrite

ϕ

′

Ω

forthetotalos illationwhereweaddthenewos illations resulting fromthe delta fun tions

e

. The graph amplitude isnow

A

G

=

Z

_Y

N

i=1

dx

i

f

i

(x

i

)δ

G

Y

l∈T

du

l

dv

l

dp

l

C

¯

l

(u

l

)

Y

ℓ∈L

du

ℓ

dw

ℓ

C

¯

ℓ

(u

ℓ

)e

iϕ

′

Ω

.

(3.23)

Remarkthatwehaveomittedthefa tors

2π

aswehavedoneuntilnowandwillgoondoing withthe

−λ

4π

2 _{det Θ}

vertexfa tors. Togetthe masslets,we ould,forexample,integrateover the variables

u

l

. This exa t omputationwould be the equivalentof equation (3.18). We shouldintegrate

2n

− N/2

oupledGaussianfun tions. Wewould get Gaussianfun tions insomevariables

W

l

whi hwouldbelinear ombinationsof

w

ℓ

′

. Apartfromthedi ulty of this omputation, we should then prove that the obtained de reasing fun tions are

independant. Forgeneralgraphs, itis somewhat di ult. Then insteadof omputing an

exa t result, we get round the di ulty by exploiting the os illations before integrating

over the

u

's,

v

'sand

w

's. The rest of this se tionis devoted to the proof of

Lemma 3.5 Let

G

an orientable graph with

n

verti es and

µ

a s ale attribution. For all

Ω

_{∈ [0, 1)}

, there exists

K

∈ R

su h that the amplitude (3.23), amputed, integrated over test fun tions, with the