Solution of the self-dual $\Phi^4$ QFT-model on four-dimensional Moyal space

Previously the exact solution of the planar sector of the self-dual $\Phi^4$-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant $\lambda>-\frac{1}{\pi}$, the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension $4-2\frac{\arcsin(\lambda\pi)}{\pi}$ for $|\lambda|<\frac{1}{\pi}$. It is this dimension drop which for $\lambda>0$ avoids the triviality problem of the matricial $\Phi^4_4$-model. We also establish the power series approximation of the Fredholm solution to all orders in $\lambda$. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters $0$ and $-1$. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.


Introduction
Many quantum field theory models have been solved or constructed in two dimensions, see e.g. [Thi58,Sch62,GN74]. Until now there is nothing comparable in four dimensions. The perturbative renormalisation of the Φ 4 -model on four-dimensional Moyal space with harmonic propagation [GW05] and the proof that the β-function vanishes [DGMR07] at a self-duality point provided some hope to construct this particular four-dimensional model.
At a special self-duality point [LS02], the model reduces to a dynamical matrix model with action for self-adjoint N ×N -matrices Φ, where E has eigenvalues E n = n √ V + µ 2 bare 2 which arise with multiplicity n. The parameter V ∈ R is the deformation parameter of the Moyal space, λ ∈ R is the coupling constant and µ 2 bare the unrenormalised mass square. The action S[Φ] is employed to define correlation functions Integration by parts produces many relations between these correlation functions. Further relations result from a Ward-Takahashi identity discovered in [DGMR07]. It was shown in [GW14] that these relations can be organised into a closed nonlinear equation for the planar two-point function and a hierarchy of Dyson-Schwinger equations for all other functions. The latter are linear in the function of interest with an inhomogeneity that only depends on finitely many functions known by induction. As characteristic to matrix models, the two-point function has a formal genus expansion Its planar part G (0) ab can be isolated in a limit V → ∞. Particularly transparent is a combined limit where also the size N of the matrices is sent to ∞, with the ratio N √ V = Λ 2 fixed. The previously discrete eigenvalues E n become in this limit functions E x = x + ab converges to a non-linear integral equation [GW14] µ 2 bare + x + y + λ Λ 2 0 dt t ZG(x, t) ZG(x, y) = 1 + λ (2) It is understood that µ 2 bare and Z depend on the cut-off Λ. According to the renormalisation philosophy, the task is to determine the precise dependence µ 2 bare (Λ), Z(Λ) so that the solution G(x, y) of (2) has a limit Λ → ∞.
In our recent work [GHW19b] we succeeded in solving the analogue of (2) for general eigenvalues E a and without requiring the special limit N , V → ∞, up to the determination of an implicitly defined measure function. In case of (2) this solution specifies to: where J is the solution of a Fredholm integral equation of second kind: . (3) Here µ > 0 is a free renormalisation parameter, and G(0, 0) = 1 is already implemented.
As main result of this paper we prove that (3) is solved by a hypergeometric function, Moreover, we show that the particular choice µ 2 = α λ (1−α λ ) λ provides the same normalisation as the expansion into renormalised ribbon graphs.
The following sections present several methods which we employed to find the solution (4) of (3). In sec. 2 we show that a rescaling of J satisfies a hypergeometric differential equation from which we deduce (4). Some steps rely on Appendix A where the spectrum of an integral operator is determined. In subsection 2.1 we determine the spectral dimension. The treatment via a differential equations is probably the most elegant one. We first obtained this solution via a perturbative expansion described in sec. 3. We understand the pattern of the power series solution of (3) to O(λ 10 ) and resum it to (4). The advantage of this approach is that it identifies the renormalisation parameter µ 2 for which our solution matches the usual perturbative renormalisation prescription. Finally, in sec. 4 we directly verify (4) via integrals for Meijer-G functions.
We compute derivatives and integrate by parts, taking the boundary values at 0 and ∞ into account: Also the product with 1 + x simplifies by integration by parts: We differentiate once more: We multiply by x and integrate by parts: We subtract twice (7) and add four times (8): Finally, we add 2c λ +λ andÂ µ is the integral operator with kernelÂ µ (t, u) = 1 u+t+µ 2 . The arguments given in appendix A show thatÂ µ has spectrum [0, π] for any µ ≥ 0. Therefore, equation (10) has for λ > − 1 π only the trivial solution g(x) = 0, which is a standard hypergeometric differential equation. The normalisation φ(0) = 1 uniquely fixes the solution to It remains to satisfy the boundary condition c λ = 1 + λ ∞ 0 dt φ(t) 1+t given after (5). The integral can be evaluated via the Euler integral [GR07, §9.111], The branch is uniquely selected by the requirement lim λ→0 c λ = 1. For λ < − 1 π there is no solution for which c λ and φ are real. Transforming back toρ λ and J gives the result announced in (4), which provides the two-point function G(x, y) via Thm. 1.

Spectral dimension
Let ̺ 0 (x)dx be the spectral measure of the operator E in the initial action (1).
The change of spectral dimension is important. If instead of (3) the function J was given byJ( , then for ̺ 0 (x) = x this functioñ J is bounded above. Hence,J −1 needed in higher topological sectors could not exist globally on R + , which would render the model inconsistent for any λ > 0. The dimension drop down to D = 4 − 2 arcsin(λπ) π avoids this (triviality) problem.

Perturbative expansion
In this section we study two different perturbative expansions of an angle function which is behind the solution of G(x, y). In Sec. 3.2 we directly expand (14) order by order in λ, whereas in Sec. 3.3 we expand (3) and compare with the other result via Corollary 3.1. For a special choice of µ 2 which we determine, both expansions coincide order by order in λ (we played the game up to the 10 th order with a computer algebra system).

Recalling earlier results
Equation (2) is a nonlinear singular integral equation of Carleman type. The solution theory for linear integral equations is known (see e.g. [Tri85]) and suggests the ansatz where a+ε dp f (p) p−a denotes the finite Hilbert transform. Inserting (13) into (2) gives with identities established in [PW19] the consistency relation Renormalisation by Taylor subtraction at 0 suggests to choose the bare mass according to We will later see that another form of (15) is for the exact solution more efficient.
The key step in [GHW19b] to solve (14) (actually in larger generality) was to define a λ-deformation ̺ λ (x) of a spectral measure function ̺ 0 . This deformed measure then gives rise to a function J(x) which in four dimensions reads The system of functions (̺ 0 , ̺ λ , J) is closed by the final condition ̺ 0 (J(x)) = ̺ λ (x). In general this is a complicated system of equations. Here, the integral equation (2) encodes the spectral measure ̺ 0 (x) = x so that J(x) = ̺ λ (x) and (16) is reduced to (3). We now have the following corollary of [GHW19b, Thm. 2.7]: Corollary 3.1. Adjusting the bare mass to then the consistency relation (14) is solved by Re(a + I(p + iε)), where Note that (17) fixes the renormalisation different than (15). It is actually a family of renormalisations which depend on a free parameter µ 2 (λ). Setting G(0, 0) = 1 does not mean µ 2 = 1, nevertheless both approaches coincide in the limit Λ 2 → ∞.

Direct expansion
Expanding equation (14) with renormalisation (15) and finite cut-off gives pλπ cot(τ a (p)) = 1 + a + p + λp log The first order is read out directly pλπ cot(τ a (p)) = 1 + a + p + O(λ 1 ) ⇒ τ a (p) = pλπ 1 + a + p + O(λ 2 ), which gives after inserting back at the next order pλπ cot(τ a (p)) =1 + a + p + λ (1 + p) log(1 + p) − p log(p) The limit Λ 2 → ∞ gives finite results for cot(τ a (p)) as well as for τ a (p) order by order, however the limit has to be taken with caution. Integral and limit do not commute. Namely, for and expansion τ a (p) = ∞ n=1 λ n τ (n) As an example we will look at the next order of both integrals. They give respectively, where Li n (x) is the n th polylogarithm and ζ n ≡ ζ(n) is the Riemann zeta value at integer n. The last term makes the difference. Taking the "wrong" second result and plugging it back into (19) would lead to divergences at the next order. Consequently, we have to treat the perturbative expansion of (19) with a finite cut-off Λ 2 at all orders, where each order has a finite limit.
The integration theory of the appearing integrals is completely understood in form of iterated integrals [Bro09]. They form a shuffle algebra, which is symbolically implemented in the Maple package HyperInt [Pan15].

Multiplying this equation by p and subtracting it from (19) again leads to
where the limit Λ 2 → ∞ is now safe from the beginning and commutes with the integral. We divide (22) by −p 2 and integrate it for all orders higher than λ 1 over p from −1 (here (21) is assumed) up to some q to get lim Λ 2 →∞ λπ cot(τ a (q)) on the lhs. On the rhs the order of integrals q −1 dp ∞ 0 dt can be exchanged. The integral over p is assuming Hölder continuity of τ p (t) so that the integral splits after taking principal values. The last term is computed for small ǫ and all O(λ >1 )-contributions via integration by parts The first term in (23) cancels. The second term in (23) integrates to a boundary term +2 τ 0 (t) ǫ , which is also canceled by the last term of (24). Multiplying by q and including the special O(λ)-contribution we arrive in the limit Λ 2 → ∞ where (21) is (conjecturally) available at qλπ cot(τ a (q)) = 1 + a + q − λq log(q) This equation is much more appropriate for the perturbation theory because the number of terms is reduced tremendously order by order. Obviously, the first six order coincide with the earlier but much harder perturbative expansion of (19). Using (25) the perturbative expansion is increased up to λ 9 with HyperInt. As consistency check of assumption (21) we inserted the next orders τ (n) a (p) into (13) to get the expansion G(a, b) = ∞ n=0 λ n G (n) (a, b). This confirmed the symmetry G (n) (a, b) = G (n) (b, a) which would easily be lost by wrong assumptions. We are thus convinced to have the correct expressions for τ (n) a (p) for 6 < n < 10.

Expansion of the Fredhom equation
To access the angle function τ a (p) through Corollary 3.1 we first have to determine the expansion of the deformed measure ̺ λ (x) = J(x) through the Fredholm equation (3). The constant µ 2 (λ) is not yet fixed and needs a further expansion First orders of the deformed measure are given iteratively through (3) Recall that the inverse of J(x) = ̺ λ (x) = p exists for all p ∈ R + in case λ < . If ̺ λ (x) had the same asymptotics as ̺ 0 (x) = x then J −1 could not be defined globally for λ > 0. We proved in sec. 2.1 that the asymptotics of ̺ λ (x) is altered in such a way that J −1 is defined. Anyway, in each order of perturbative expansion the inverse J −1 is globally defined on R + . At this point it suffices to assume that J −1 (p) is a formal power series in λ, which is achieved by (16) .
Both series are convergent for |λ| < 1 π with the result (up to order λ 10 ) This result suggests that arcsin(λπ) π is a better expansion parameter than λ itself. The factors π 2n are produced by ζ 2n in the iterated integrals. We thus reorganise the perturbative solution of (6) into a series in arcsin(λπ) π . The power of arcsin(λπ) λπ depends on the number of letters of the hyperpolylogarithm, which alternate between −1 and 0. The expansion which holds up to order λ 10 is given by Hlog ( where the underbrace with n means that we have n times the letters 0 and −1 in an alternating way.
In the limit x → 0 only the terms with n = 0 in both sums survive, This value was found in sec. 2 by another method. We also remark that c λ = 1 µ 2 for the special renormalisation.

Stieltjes transform of the measure function
We find it interesting to directly check that the hypergeometric function̺ λ (x) = 1 µ 2 φ( x µ 2 ), see (11), solves the integral equation (5). The hypergeometric function can be expressed through the more general Meijer-G function. A Meijer-G function is defined by with m, n, p, q ∈ N, with m ≤ q and n ≤ p, and poles of Γ(b j −s) different from poles of Γ(1 − a j + s). The infinite contour L separates between the poles of Γ(b j − s) and Γ(1 − a j + s), and its behavior to infinity depends on m, n, p, q (see [ where primed sum and the ⋆ means that the term with j = k is omitted. We need another identity which is derived directly from the definition where the contour is changed L → L ′ such that it is moved through s = 0 and picked up the residue. The contour L ′ fulfills the definition (31) for G 2,2 2,2 z 0,1 b 1 ,b 2 . From (34) one can establish 1,1 t x+µ 2 0 0 . The convolution theorem (33) of Meijer-G functions thus allows to evaluate the integral We have used the expansion of a Meijer-G function into hypergeometric functions and applied in the last step [GR07, §9.132.1]. The result is precisely (5) provided that c λ = λ α λ (1−α λ ) (see (11)) and sin(α λ π) = λπ (see (12)).

Outlook
With the identification of J we have completed the solution of the planar 2-point function of the Φ 4 -QFT model on four-dimensional Moyal space at the self-duality point. From the 2-point function one directly builds all planar correlation functions [GW14,dJHW19]. In our earlier work [GHW19a] on the much simpler cubic Kontsevich-like model we gave an algorithm to compute also all non-planar correlation functions from the planar sector. It remains to be seen whether a similar endeavour can succeed for the Φ 4 -model, too. We expect that non-planar functions are expressed in terms of the inverse function J −1 . Inverses of hypergeometric functions do not seem to be studied. There is now strong motivation to try it. Of course one can approximate J −1 perturbatively via the expansion of J into hyperlogarithms which we established. A nonperturbative characterisation of J −1 could provide useful identities between these number-theoretic functions.
Note that A µ is symmetric and positive. The equation can thus be solved for ψ if λ > λ c = − A µ −1 . By scaling, the spectrum of A µ is independent of µ for µ > 0. We claim that In particular, λ c = −1/π. Since A µ has a positive kernel which is monotone in µ, one readily obtains A µ ≤ A 0 . On the other hand, A 0 is the weak limit of A µ as µ → 0, hence A 0 ≤ lim inf µ→0 A µ , which proves that A µ = A 0 . Now A 0 (t, u) = (u + t) −1 . Introducing logarithmic coordinates, we have which can be diagonalised via Fourier transforms. Since R 1 2 cosh(v/2) dv = π, this shows that the spectrum of A 0 equals [0, π], and indeed A 0 = A µ = π.