Progress in the solving nonperturbative renormalization ... · tensorial group eld theory. These...

Plan

Progress in the solving nonperturbativerenormalization group for tensorial group field

theory

Vincent Lahoche and Dine Ousmane Samary

CEA-LIST (France) and CIPMA-CHAIR-UNESCO (Benin)International online talk Heidelberg

arXiv:1809.06081, arXiv:1809.00247

22 novembre 2018

1 / 41

Plan

Outline

1 Introduction : motivations2 Truncation method and flow equations3 Effective vertex expansion4 Conclusion

2 / 41

Introduction : motivationsTruncation method and flow equations

Effective vertex expansionConclusion

AbstractThis presentation aims at giving our new advance on the functionalrenormalization group applied to tensorial group field theory. It is based on a serieof our three papers [arXiv :1803.09902], [arXiv :1809.00247] and[arXiv :1809.06081]. We consider the polynomial Abelian U(1)d models withoutclosure constraint, especially we discuss the case of quartic melonic interaction. Byusing the effective vertex expansion method we studied the Wetterich flowequation and the possible existence or not of the phase transition in the leadingorder melonic sector. We provide the so called structure equations compactiblewith the Ward identities. These equations allow a new constraint on the flow i.e.on to the β-functions and the anomalous dimension η. We also show that, byadding in the functional renormalization group program other leading ordercontribution in the interaction, the physical conclusions such as the non-existenceof fixed points for quartic model are not modified.

3 / 41



Introduction

Motivations1 In the search of unify theory of modern physics, i.e. a well defined theory of

quantum gravity, a lot of efforts have been made. Despite the fact that today,none of them may give entirely the complete definition on this issue, severalmajor advances are observed to tackle this very important problem. Amongwhich we can identify a very recent directions such as loop quantum gravity,dynamical triangulation, noncommutative geometry, group field theories(GFTs) and tensors models (TMs).

2 These approachs are considered as new background independent approachesaccording to several theoreticians. GFTs are quantum field theories over thegroup manifolds and are considered as the second quantization version ofloop quantum gravity. These theories are caracterized by the specific form ofnon-locality in their interactions. TM, especially colored ones, allow one todefine probability measures on simplicial pseudo-manifolds such that thetensor of rank d represents a (d − 1)-simplex.

4 / 41



Introduction

Motivations1 TMs admit the large N-limit dominated by the graphs called melons thanks

to the Gurau breakthrough. This limit behaviour is a powerful tool whichallows us to understand the continuous limit of these models through, forinstance, the study of critical exponents and phase transitions.

2 TM and GFT are merged to give birth to a new class of field theories calledtensorial group field theory. These class of field models enjoy renormalizationand asymptotic freedom. Using the functional renormalization group (FRG)method, it is also possible to identify the equivalent of Wilson-Fisher fixedpoint for some particular cases of models and to prove the asymptoticallyfreedom and safety.

5 / 41



Introduction

Wilson and Polchinski1 There are several way to introduce the FRG in field theories. The first

approach is the one pioneered by Wilson, which follows from simple andintuitive and therefore yieds a powerful way to think about quantum fieldtheories. This method allows to interpolate smoothly between the knownmicroscopic laws IR-regime and the complicated macroscopic phenomena inphysical systems UV-regime and is constructed with the incompleteintegration as cutoff procedure.

2 Well after a new approach to address the same question inspired from theWilsonian method, called Wilson-Polchinski FRG equation is given. This verypracticable method, may be integrated with an arbitrary cutoff function andexpanded up to the next to leading order of the derivative expansion. Despitethe fact that all these approachs seem to be nonperturbatively, in practice,the perturbative solution has appeared more attractive.

6 / 41



Introduction

Wetterich equation1 After Wilson-Polchinski, the so called Wetterich flow equation is proposed to

study the nonperturbative FRG and whose study requires approximations ortruncations and numerically analysis due to the nonlinearity form.

2 The FRG equation allows to determine the fixed points and probably thephase transition. These phase transitions in the case of TGFT models mayhelp to identify the emergence of general relativity and quantum mechanicsthrough the geometrogenesis scenario.

3 In our recent and modest contributions the effective vertex expansion methodis used in the context of the FRG. This leads to the definition of new class ofequations called structure equations that help to solve the Wetterich flowequation. Taking into account the leading order contribution in the symmetricphase, the non-perturbative regime without truncation can be studied.

7 / 41



Introduction

Our contribution and main goal of this presentation1 The Ward-Takahashi (WT) identities is derived and become a constraint

along the flow. These identities are universal for all field theories having asymmetry, and are not specific to TGFT. Therefore all the fixed points mustbelong inside to the domain of this constraint line, before being considered asan acceptable fixed points.

2 In the case of quartic melonic TGFT models we have showed that the fixedpoint occurring from the solution of Wetterich equation violate thisconstraint for any choices of regulator functions. This violation is alsoindependent of the method used to find this fixed point, whether it is thetruncation, or the EVE method.

3 We will discuss all this point in the following presentation

8 / 41



The model

In the context of TGFT, we consider the pair of complex fields φ and φ whichtakes values of d-copies of arbitrary group G :

φ, φ : G d → C (1)

The particular case is G = U(1) the Abelian compact Lie group. For the rest weconsider only the Fourier transform of the fields φ and φ denoted respectively byT~p and T~p, ~p ∈ Zd written as (for ~g ∈ U(1)d , gj = e iθj ) :

φ(~θ ) =∑~p∈Zd

T~p ei∑d

j=1 θjpj , φ(~θ ) =∑~p∈Zd

T~p e−i

∑dj=1 θjpj . (2)

The description of the statistical field theory is given by the partition function

Z[J, J] =

∫dµC e−Sint+〈J,T〉+〈T ,J〉, (3)

9 / 41



The model

C is the covariance taking to be

C (~p ) =1

~p 2 + m2=

∫dµC T~p T~p (4)

In order to prevent the UV divergences and supress the high momentacontributions, the propagator (4) has to be regularized. Schwinger regularization :

CΛ(~p ) =e−(~p 2+m2)/Λ2

~p 2 + m2. (5)

In general case, Let ϑ(t) such that |1− ϑ(t)| ≤ Ce−κt for C , κ > 0 and t → +∞,a Laplace transform yields :

CΛ(~p) =

∫ +∞

0

dt ϑ(tΛ2) e−t(~p 2+m2), [ϑ(t) = Θ(t − 1)⇒ Schwinger] (6)

10 / 41



The model

We introduce tensorial unitary invariants. An invariant is a polynomial P(T , T ) inthe tensor entries T~p and T~p which is invariant under the following action ofU(N)⊗d as follows :

T~p →∑~q

U(1)p1q1· · ·U(d)

pdqdT~q, T~p →∑~q

U(1)p1q1· · · U(d)

pdqd T~q. (7)

The algebra of invariant polynomials is generated by a set of polynomials labeledby bubbles :

Figure – The 4-vertex bubble from which the dots indicate multiple edges.

11 / 41



The model

We consider the quartic melonic T 45 model which is proved to be renormalizable in

all orders in the perturbative theory. The interaction of this model taking intoaccount the leading order contributions : (melon and pseudo-melon) is writtengraphically as :

S4int = λ41

5∑i=1

i

i

T~p1 T~p2

T~p3T~p4

+ λ42

5∑i,j=1

T~p1 T~p2

T~p3T~p4

i

j + · · · (8)

S4kin = T~p T~p (9)

12 / 41



Wetterich flow equation

The Wetterich equation is a functional integro-differential equation for theeffective action Γ, now taking into account the quantum fluctuations caracterizedby the parameter s, and called average effective action denoted by Γs ,−∞ < s < +∞. It is the Legendre transformation of the standard free energyWs = lnZs :

Γs [M, M] = 〈J,M〉+ 〈M, J〉 −Ws [J, J]− Rs [M, M] (10)

where Rs [M, M] := Tr(MrsM) and rs is called the IR regulator. The appearance ofthis regulator rs is introduced as new parameter function, which controls the scalefluctuation from IR to UV such that

lims→−∞

rs = 0, lims→+∞

rs =∞. (11)

This definition ensures that Γs satisfies the boundary conditionsΓs=ln Λ = S , Γs=−∞ = Γ, where Λ is the UV cutoff.

13 / 41




The fields M and M are the mean values of T and T respectively and are given by

M =∂W∂J

, M =∂W∂J

(12)

where W :=Ws=−∞. In general the regulator rs is chosen to be

rs = Z (s)k2f(~p 2

k2

), k = es , and such that the conditions (11) is well satified. Let

Γ(2)s is the second order partial derivative of Γs with respect to the mean fields M

and M, the Wetterich equation is then given by

∂sΓs = Tr ∂s rs(Γ(2)s + rs)−1 (13)

The average effective action is chosen to be of the form

Γs = Z (s)∑~p∈Zd

T~p(~p 2 + e2sm2(s))T~p +∑n

Z (s)n2 λnVn(T , T ) (14)

14 / 41




In the case of quartic melonic interaction and by taking the standard modifiedLitim’s regulator :

rs(~p ) = Z (s)(e2s − ~p 2)Θ(e2s − ~p 2) (15)

the Wetterich equation can be solved analytically and the phase diagram may begiven. The flow equations are

m2 = −2dλI2(0)

Z (s) = −2λI ′2(q = 0)

λ41 = 4λ241I3(0)

In(q) =∑

~p∈Z(d−1)

rs(Z (s)~p 2 + Zq2 + m2 + rs)n

. (16)

with the renormalization condition

m2(s) = Γ(2)s (~p = ~0), λ41(s) =

1

4Γ(4)s (~0,~0,~0,~0). (17)

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Explicitly using the integral representation of the above sum and with d = 5,η = Z/Z we get

In(0) =π2e6s−2ns

6Z (s)n−1(m2 + 1)n(η + 6), I ′n(0) = − π2e4s−2ns

2Z (s)n−1(m2 + 1)n(η + 4). (18)

In term of dimensionless parameter λ41 = Z 2λ41, m2 = e2sZm2 the system (16)becomes

βm = −(2 + η)m2 − 2d λ π2

(1+m2)2

(1 + η

6

),

β41 = −2ηλ+ 4λ2 π2

(1+m2)3

(1 + η

6

),

(19)

where βm := ˙m2, β41 := ˙λ and :

η :=4λπ2

(1 + m2)2 − λπ2. (20)

16 / 41




The solutions of the system (19) is given analytically :

p± =(m2± = −23∓

√34

33, λ41,± =

328∓ 8√

34

11979π2

). (21)

Numerically

p+ = (−0.52, 0.0028), p− = (−0.87, 0.0036). (22)

Apart from the fact that we have a singularity line around the point m2 = −1 inthe flow equation (16), another second singularity arise from the anomalousdimension denominator, and corresponds to a line of singularity, with equation :

Ω(m, λ) := (m2 + 1)2 − π2λ41 = 0 (23)

This line of singularity splits the two dimensional phase space of the truncatedtheory into two connected regions characterized by the sign of the function Ω.

17 / 41




The region I , connected to the Gaussian fixed point for Ω > 0 and the region IIfor Ω < 0. For Ω = 0, the flow becomes ill defined. The existence of thissingularity is a common feature for expansions around vanishing means field, andthe region I may be viewed as the domain of validity of the expansion in thesymmetric phase. Note that to ensure the positivity of the effective action, themelonic coupling must be positive as well. Therefore, we expect that the physicalregion of the reduced phase space correspond to the region λ41 ≥ 0. Fromdefinition of the connected region I and because of the explicit expression ofanomalous dimension, we deduce that :

η ≥ 0 , In the symmetric phase . (24)

Then, only the fixed point p+ much be taken into account.

18 / 41



Phase diagram

The phase diagram is given :

0.000 0.002 0.004 0.006 0.008

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

λs

ms

19 / 41



Ward-identities

Let U = (U1,U2, · · · ,Ud), where the Ui ∈ U∞ are infinite size unitary martices inmomentum representation. We define the transformation :

U [T ]~p =∑~q

U1 ,p1q1U2 ,p2q2 · · ·Ud ,pdqdT~q , (25)

such that the interaction term is invariant i.e. U [Sint ] = Sint . Then consider aninfinitesimal transformation :

U = I + ~ε, ~ε =∑i

I⊗(i−1) ⊗ εi ⊗ I⊗(d−i) , (26)

where I is the identity on U∞, I = I⊗d the identity on U ⊗d∞ , and εi denotes

skew-symmetric hermitian matrix such that εi = −ε†i and~εi [T ]~p = εi piqiTp1,··· ,qi ,··· ,pd .

20 / 41



Ward-identities

The invariance of the path integral (3) means ~ε [Zs [J, J]] = 0, i.e. :

~ε [Zs [J, J]] =

∫dTdT

[~ε [Skin] + ~ε [Sint ] + ~ε [Ssource ]

]e−Ss [T ,T ]+〈J,T〉+〈T ,J〉 = 0.

(27)Computing each term separately, we get successively using linearity of theoperator ~ε : ~ε [Sint ] = 0, ~ε [Ssource ] = −

∑di=1

∑~p,~q

∏j 6=i δpjqj [J~p T~q − T~pJ~q ]εi piqi ,

~ε [Skin] =∑d

i=1

∑~p,~q

∏j 6=i δpjqj T~p

[Cs(~p 2)− Cs(~q 2)

]T~q εipiqi . where∏

j 6=i δpjqj := δ~p⊥i~q⊥i

, p⊥i := ~p \ pi, C−1s = C−1

−∞ + rs and

C−1−∞ = Z−∞~p

2 + m2−∞. Z−∞ is the renormalized wave function usualy denoted

by Z .

21 / 41



Ward-identities

The ward identity gives relation between two and four point functions as :∑~r⊥i

,~s⊥i

δ~r⊥i~s⊥i

(C−1s (~r)− C−1

s (~s))〈T~r T~sT~pT~q〉 = −δ~p⊥i~q⊥i

(Gs(p)− Gs(q))δri si , (28)

where, defined by Γ(4)s , the 1PI four point function, we get

〈T~r T~sT~pT~q〉 = Γ(4)s,~r~s;~p~q

(Gs(~p)Gs(~q) + δ~r~pδ~s~q

)Gs(~r)Gs(~s) (29)

22 / 41



Ward-identities

The formal invariance of the path integral implies that the variations of theseterms have to be compensate by a non trivial variation of the source terms.

d∑i=1

∑~p⊥i

,~q⊥i

δ~p⊥i~q⊥i

[∂

∂J~p

[Cs(~p 2)− Cs(~q 2)

] ∂

∂J~q− J~p

∂

∂J~q+ J~q

∂

∂J~p

]eWs [J,J] = 0 ,

(30)

Then after few simplification we come to∑~r⊥1

G 2s (~r )

dC−1s

dr21

(~r )Γ(4)s,~r ,~r ,~p,~p =

d

dp21

(C−1∞ (~p )− Γ(2)

s (~p )). (31)

or

Z−∞Ls :=∑~p∈Zd

(Z−∞ +

∂rs∂p2

1

(~p )

)G 2s (~p )δp10 . (32)

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Structure equations

The Structure equations is the relations between correlation function and allowsto establish a constraint between β-functions for mass, interactions couplings andwave function renormalization. These relations are obtained in the deep UV limit(i.e. in the domain 1 es Λ) without any assumption about the β-functionsand without any truncation of the effective action Γs . The only assumptionconcern the choice of the initial conditions, ensuring the perturbative consistencyof the full partition function. The first structure equation concern the self energy(or 1PI 2-point functions). It takes place as the closed equation for self energy. 1

Let us summarize in the following proposition

1. The rank of the tensors is fixed to 5, and we denote it by d to clarify the proof(s).24 / 41



Structure equations

In the melonic sector, the self energy Σs(~p ) is given by the closed equation whichtakes into account the effective coupling λ41(s) as :

− Σs(~p ) = 2λr41Zλ∑~q

(d∑

i=1

δpiqi

)Gs(~q ) . (33)

In the same way, in the melonic sector, the perturbative zero-momenta 1PI

four-point contribution Γ(4),i

s,~0~0;~0~0is given by :

Γ(4),i

s,~0~0;~0~0= 2π00 =

4Zλλr41

1 + 2λr41ZλAs, (34)

where As is defined as :

As =∑~p⊥

[Gs(~p⊥)]2 , ~p⊥ := (0, p1, · · · , pd) , (35)

Gs(~p) being the effective propagator : G−1s (~p ) = Z−∞~p

2 + m2 + rs(~p )−Σs(~p ) .25 / 41



Structure equations

In other words, we have an explicit expression for the effective coupling

λ41(s) := 14 Γ

(4),i

s,~0~0;~0~0,

λ41(s) =λr41

1 + 2λr41As

, (36)

from which we get

∂sλ41(s) = − 2(λr41)2As

(1 + 2λr41∆As)2= −2λ2

41(s)As . (37)

In the above relation we introduce the dot notation As = ∂sAs

As =∑~p⊥

1

[Γ(2)s (~p⊥) + rs(~p⊥)]2

, As = −2∑~p⊥

Γ(2)s (~p⊥) + rs(~p⊥)

[Γ(2)s (~p⊥) + rs(~p⊥)]3

. (38)

26 / 41



Constraint equation

The constraint providing from the Ward identity, which relies the β-functions andthe anomalous dimension is given by :

β41 = −ηλ41

(1− λ41π

2

(1 + m2)2

)+

2λ241π

2

(1 + m2)3βm (39)

This relation need to be taking into account in the Wetterich flow equation andtherefore in the search of fixed point. To prove this relation, let us consider thederivative of Z with respect to s using the structure equation

Z = (Z−∞ − 2λ41Z−∞Ls)λ41

λ41− 2Z−∞∆s λ41. (40)

In the above relation we have used the decomposition of Ls = As + ∆s . Remarkthat the Ward identity can be written as 2λ41Ls = 1− Z where Z = Z/Z−∞.

27 / 41



Constraint equation

Then (40) becomes :

Z

Z=λ41

λ41− 2

Z−∞Z

∆sλ41. (41)

We now use the dimensionless quantities m, λ41, Bs such that ∆s = ZZ 2 Bs and

reexpressing (41) as :

β41 = −ηλ41 + 2λ41(−ηBs + ˙Bs) (42)

where Bs and ˙Bs much be simply compute using the integral representation of thesum. We come to :

Bs = − π2

2(1 + m2)2, ˙Bs =

π2βm(1 + m2)3

, (43)

28 / 41



Constraint equation

Let p is a arbitrary fixed point of the theory. We get βm(p) = 0 = β41(p) = 0.Then the constraint (39) implies that a the point p

ηλ41

(1− λ41π

2

(1 + m2)2

)(p) = 0. (44)

The particular solution λ41 = 0 correspond to the Gaussian fixed point. Forλ41 6= 0 we have only

η = 0, orλ41π

2

(1 + m2)2= 1 (45)

It is clear that the fixed point p+ = (−0.55, 0.0025), η ≈ 0.7 violate theseconstraints.

29 / 41



Constraint equation

Remark that other way to prove the violation of the Ward identity, is to extractthe coupling λ41 in the constraint (39) and solve the flow of mass and couplingtaking into account the assumption that the coupling is determine by theconstraint equation. Now we get

λ341 = 0 or λ41 =

11(1 + m2)2

5π2. (46)

By replacing this solution λ341 = 0 in the flow equations of mass and coupling (16)

we get

βm = −2m2, β41 = 0. (47)

Now setting βm = 0 = β41, only the Gaussian fixed point (m∗ = 0, λ∗41 = 0)survives. Also the last solution leads to

βm =4

9(12m + 11), β41 =

484(m + 1)(15m + 13)

225π2. (48)

Now setting βm = 0 = β41, no solution for the mass exist in this case.30 / 41



Flow equations using the EVE

Let us consider the flow equation for Γ(2), obtained from (13) deriving withrespect to M and M :

Γ(2)(~p ) = −∑~q

Γ(4)~p,~p,~q,~q G

2s (~q )rs(~q ) , (49)

where we discard all the odd contributions, vanishing in the symmetric phase.Deriving on both sides with respect to p2

1 , and setting ~p = ~0, we get :

Z = −∑~q

Γ(4) ′~0,~0,~q,~q

G 2s (~q )rs(~q )− Γ

(4)~0,~0,~q,~q

G 2s (~q )rs(~q ) , (50)

where the ”prime” designates the partial derivative with respect to p21 . In the deep

UV (k 1) the argument used in the T 4-truncation to discard non-meloniccontributions holds, and we keep only the melonic diagrams as well. Moreover, to

capture the momentum dependence of the effective melonic vertex Γ(4)melo and

compute the derivative Γ(4) ′melo ,~0,~0,~q,~q

, the knowledge of πpp is required. It can be

deduced from the same strategy as for the derivation of the structure equation, upto the replacement :

31 / 41




As → As(p) :=∑~p∈Zd

G 2s (~p )δp1p , (51)

from which we get :

πpp =2λr

1 + 2λr As(p), As(p) := As(p)−A−∞(0) . (52)

The derivative with respect to p21 may be easily performed, and from the

renormalization condition (17), we obtain :

π′00 = −4λ2(s)A′s , (53)

and the leading order flow equation for Z becomes :

Z = 4λ2A′s(0) I2(0)− 2λI ′2(0) . (54)

32 / 41




As announced, a new term appears with respect to the truncated version (16),which contains a dependence on η and then move the critical line. The flowequation for mass may be obtained from (49) setting ~p = ~0 on both sides. Finally,the flow equation for the marginal coupling λ may be obtained from the equation(13) deriving it twice with respect to each means fields M and M. As explained

before, it involves Γ(6)melo at leading order, and to close the hierarchy, we use the

marginal coupling as a driving parameter, and express it in terms of Γ(4)melo and Γ

(2)melo

only. One again, Γ(6)melo have to be split into d monocolored components Γ

(6) ,imelo :

Γ(6)melo =

d∑i=1

Γ(6) ,imelo . (55)

The structure equation for Γ(6) ,imelo may be deduced following the same strategy as

for Γ(4) ,imelo

33 / 41




Starting from a vacuum diagram, a leading order 4-point graph may be obtainedopening successively two internal tadpole edges, both on the boundary of acommon internal face. This internal face corresponds, for the resulting 4-pointdiagram to the two external faces of the same colors running through the interiorof the diagram. In the same way, a leading order 6-point graph may be obtainedcutting another tadpole edge on this resulting graph, once again on the boundaryof one of these two external faces. From this construction, it is not hard to seethat the zero-momenta Γ

(6) ,imelo vertex function must have the following structure :

Γ(6) ,imelo = (3!)2

G

G

G

π π

π

ii

i

, (56)

34 / 41




the combinatorial factor (3!)2 coming from permutation of external edges.Translating the diagram into equation, and taking into account symmetry factors,we get :

Γ(6) ,imelo = 24Z 3(s)λ3(s)e−2sA2s , (57)

with :A2s := Z−3e2s

∑~p∈Zd−1

G 3s (~p ) . (58)

Note that this structure equation may be deduced directly from Ward identities.The equation closing the hierarchy is then compatible with the constraint comingfrom unitary invariance. The flow equations involve now some new contributionsdepending on two sums, A2s and A′s , defined without regulation function rs .

35 / 41




However, they are both power-counting convergent in the UV, and therenormalizability theorem ensures their finitness for all orders in the perturbationtheory. For this reason, they becomes independent from the initial conditions atscale Λ for Λ→∞ ; we get, using the Litim’s regulator :

A2s =1

2

π2

1 + m2

[1

(1 + m2)2+

(1 +

1

1 + m2

)], (59)

and

A′s =1

2π2 1

1 + m2

(1 +

1

1 + m2

). (60)

The complete flow equation for zero-momenta 4-point coupling write explicitly as :

Γ(4) = −∑~p

rs(~p )G 2s (~p )

[Γ

(6)

~p,~0,~0,~p,~0,~0− 2

∑~p ′

Γ(4)

~p,~0,~p ′,~0Gs(~p ′)Γ

(4)

~p ′,~0,~p,~0

+2Gs(~p )[Γ(4)

~p,~0,~p,~0]2]. (61)

36 / 41




Keeping only the melonic contributions, we get finally the following autonomoussystem by using the Litim’s regulation : βm = −(2 + η)m2 − 2d λ π2

(1+m2)2

(1 + η

6

),

βλ = −2ηλ+ 4λ2 π2

(1+m2)3

(1 + η

6

) [1− 1

2π2λ(

1(1+m2)2 +

(1 + 1

1+m2

)) ].

(62)

where the anomalous dimension is then given by :

η = 4λπ2 (1 + m2)2 − 12 λπ

2(2 + m2)

(1 + m2)2Ω(λ, m2) + (2+m2)3 λ2π4

. (63)

37 / 41




The new anomalous dimension has two properties which distinguish him from itstruncation version. First of all, as announced, the singularity line Ω = 0 movestoward the λ axis, extending the symmetric phase domain. In fact, theimprovement is maximal, the critical line being deported under the singularity linem2 = −1. In standard interpretations, the presence of the region II is generallyassumed to come from a bad expansion of the effective average action aroundvanishing means field, becoming a spurious vacuum in this region. However, theEVE method show that this singularity line is completely discarded taking intoaccount the momentum dependence of the effective vertex. The secondimprovement come from the fact that the anomalous dimension may be negative,and vanish on the line of equation L(λ, m2) = 0, with :

L(λ, m2) := (1 + m2)2 − 1

2λπ2(2 + m2) . (64)

Interestingly, there are now two lines in the maximally extended region I ′ wherephysical fixed points are expected. However, numerical integrations, show that theimproved flow equations admit a non-Gaussian fixed point, numerically very closefrom the fixed point p+ obtained in the truncation method, and then unphysical.

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Violation of Ward-identity

0.000 0.005 0.010 0.015 0.020 0.025 0.030

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

λ

m2

Figure – The relevant lines over the maximally extended region I ′, bounded at thebottom with the singularity line m2 = −1 (in green). The blue and red curves correspondrespectively to the equations L = 0 and Ω = 0. Moreover, the black point correspond tothe numerical non-Gaussian fixed point, so far from the two previous physical curves.

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To sum up, in this work

• In this presentation we show that the IR fixed point obtained in the FRGapplications for TGFT lack an important constraint coming from Wardidentities. This constraint reduces the physical region of the phase space to aone-dimensional subspace without fixed point, suggesting that the phasetransition scenario abundantly cited in the TGFT literature may be anartifact of an incomplete method.

• This suggestion is improved with a more sophisticated method, taking intoaccount the momentum dependence of the effective vertex, and providing amaximal extension of the symmetric region. Despite with this improvement,the resulting numerical fixed point does not cross any of the physical linesprovided from the Ward constraint. In the literature, the quartic truncationhas been largely investigated, for various group manifold and dimensions.

• We expect from our analysis that none of these models modify ourconclusions, except possibly for TGFT including closure constraint as aGauge symmetry.

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44

Thank you for your attention

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