Intermediate Field Representation Research Articles

The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries

AbstractWe consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane $$\mathbb {C}_{\pi } =\mathbb {C}{\setminus } \mathbb {R}_-$$ C π = C \ R - . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any $$g=|g| e^{\imath \varphi }$$ g = | g | e ı φ on the sector of the Riemann surface with $$|\varphi |<3\pi /2$$ | φ | < 3 π / 2 , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, $$Z_n(g)$$ Z n ( g ) and $$W_n(g)$$ W n ( g ) , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of $$Z_n(g)$$ Z n ( g ) is readily accessible: much like Z(g, N), for any n, $$Z_n(g)$$ Z n ( g ) has a zero- and a one-instanton contribution. The transseries of $$W_n(g)$$ W n ( g ) is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of $$W_n(g)$$ W n ( g ) and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, $$W_n(g)$$ W n ( g ) exhibits contributions of only up to n-instanton sectors.

Intermediate Field Representation for Positive Matrix and Tensor Interactions

In this paper we introduce an intermediate field representation for random matrices and random tensors with positive (stable) interactions of degree higher than 4. This representation respects the symmetry axis responsible for positivity. It is non-perturbative and allows to prove that such models are Borel–Le Roy summable of the appropriate order in their coupling constant. However, we have not been able yet to associate a convergent loop vertex expansion with this representation; hence, our Borel summability result is not of the optimal expected form when the size N of the matrix or of the tensor tends to infinity.

Note on the Intermediate Field Representation of ϕ 2 k $\phi ^{2k}$ Theory in Zero Dimension

This note is a sequel to Rivasseau and Wang (J. Math. Phys. 51, 092304, 2010). We correct the intermediate field representation for the stable \(\phi ^{2k}\) field theory in zero dimension introduced there and extend it to the case of complex conjugate fields. For \(k = 3\) in the complex case we also provide an improved representation which relies on ordinary convergent Gaussian integrals rather than oscillatory integrals.

Tensorial Gross-Neveu models

We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.

Loop vertex expansion for higher-order interactions

This note provides an extension of the constructive loop vertex expansion to stable interactions of arbitrarily high order, opening the way to many applications. We treat in detail the example of the $(\bar \phi \phi)^p$ field theory in zero dimension. We find that the important feature to extend the loop vertex expansion is not to use an intermediate field representation, but rather to force integration of exactly one particular field per vertex of the initial action.

Phase transition in tensor models

Generalizing matrix models, tensor models generate dynamical triangulations in any dimension and support a $1/N$ expansion. Using the intermediate field representation we explicitly rewrite a quartic tensor model as a field theory for a fluctuation field around a vacuum state corresponding to the resummation of the entire leading order in $1/N$ (a resummation of the melonic family). We then prove that the critical regime in which the continuum limit in the sense of dynamical triangulations is reached is precisely a phase transition in the field theory sense for the fluctuation field.

An analysis of the intermediate field theory of T4 tensor model

In this paper we analyze the multi-matrix model arising from the intermediate field representation of the tensor model with all quartic melonic interactions. We derive the saddle point equation and the Schwinger-Dyson constraints. We then use them to describe the leading and next-to-leading eigenvalues distribution of the matrices.

How to Resum Feynman Graphs

In this paper, we reformulate the combinatorial core of constructive quantum field theory. We define universal rational combinatorial weights for pairs made of a graph and any of its spanning trees. These weights are simply the percentage of Hepp’s sectors of the graph in which the tree is leading, in the sense of Kruskal’s greedy algorithm. Our main new mathematical result is an integral representation of these weights in terms of the positive matrix appearing in the symmetric “BKAR” Taylor forest formula. Then, we explain how the new constructive technique called Loop Vertex Expansion reshuffles according to these weights the divergent series of the intermediate field representation into a convergent series which is the Borel sum of the ordinary perturbative Feynman’s series.