Finiteness at integer space-time dimension of resummed perturbatively nonrenormalizable amplitudes: completing the Parisi 1973 ladder construction

Can perturbatively nonrenormalizable interactions be given a finite nonperturbative definition at integer space-time dimension, in which the order-by-order ultraviolet divergences appear as artifacts of Taylor-expanding amplitudes that are nonanalytic in the coupling? Parisi's 1973 letter established a controlled instance of this at noninteger dimension: for the massless cubic scalar theory with $D>6$ held noninteger, the planar-ladder series for the four-point amplitude -- every term of which is ultraviolet divergent by power counting -- sums to a finite closed form that is nonanalytic in the coupling $g$; but Euler Gamma-function factors of the summed expression develop poles at integer values of $D$ (the prefactors $\Gamma(-2/\varepsilon)$ and $\Gamma(-1/\varepsilon)$, with $\varepsilon=(D-6)/2$, for instance, at $D=7,8,10$), and the computation covers only the ladder diagram subclass, massless internal lines, and the special kinematics with two external momenta zero. The problem is to determine whether this integer-dimension gap can be removed -- whether the resummation extends to a finite nonperturbative definition at integer dimension, beyond the ladder subclass and with controlled treatment of the omitted diagrams and kinematics. Later literature gives model-dependent partial answers: all-loop resummations of maximally supersymmetric Yang-Mills theory in $D=8,10$ find surviving infinite periodic poles and no limit as the regulator is removed; large-$N$ vector models in four and five dimensions can be nonperturbatively renormalizable; scalar $\phi^4$ theory in $d>4$ is rigorously trivial. The core question is open.

Background

Perturbative renormalizability. In relativistic quantum field theory (QFT), scattering amplitudes are computed as formal series in the coupling constants (perturbation theory), each term a sum of Feynman diagrams -- graphical objects encoding integrals over internal loop momenta. A theory is called (power-counting) renormalizable if all ultraviolet (UV) divergences, i.e. divergences of loop integrals at large internal momenta, can be absorbed into finitely many parameters of the Lagrangian; otherwise an infinite tower of counterterms with increasingly many derivatives is required and the perturbative theory loses predictive power at high energy. For the massless cubic scalar theory with interaction $g\phi^3$ in $D$ Euclidean space-time dimensions, the engineering dimension of the coupling is $(6-D)/2$, negative for $D>6$, so the theory is power-counting nonrenormalizable there.

Nonanalyticity in the coupling. Dyson's 1949 analysis of the structure of the perturbation series, and the 1958 conjecture of Redmond and Uretsky on nonrenormalizable field theories, raised the question of whether a divergent or meaningless perturbation series reflects a genuine inconsistency of the underlying theory or only of its Taylor expansion in the coupling. The second possibility is real: several exactly known physical quantities are nonanalytic at $g=0$ (the superconducting gap behaves like $e^{-1/g}$ at weak coupling; mass gaps in certain two-dimensional models are likewise nonanalytic), so their Taylor series in integer powers of $g$ either miss the true answer or spuriously diverge. If the exact Green's functions of a nonrenormalizable theory were themselves nonanalytic in $g$ (noninteger powers, logarithms, essential singularities), then the termwise UV divergences of the perturbative expansion could be artifacts of expanding a nonanalytic object. Wilson's renormalization-group program supplied the standard modern criterion: a perturbatively nonrenormalizable theory can still exist as a continuum limit if its couplings flow to a nontrivial ultraviolet fixed point, i.e. a scale-invariant point of the flow other than the free theory.

Ladder diagrams. A ladder diagram (in this context) is the planar Feynman graph in which two external lines exchange a chain of massless propagators (the rungs), adjacent rungs joined by two cubic vertices, so that the graph resembles a ladder with $N$ rungs. Restricting the four-point function to this subclass is the planar or ladder approximation; ladders form a geometrically simple subclass whose sum can sometimes be evaluated in closed form.

Parisi's construction. In a 1973 letter, Parisi treated the massless $g\phi^3$ theory with $D>6$ and $D$ regarded as a noninteger parameter, so that every loop integral is first defined by analytic continuation in the space-time dimension $D$ -- the mechanism underlying dimensional regularization, the standard loop-integral regulator of QFT. For the four-point amplitude with two external momenta set to zero, $A(0,0,p,-p)=\sum_N g^{2(N+1)}A_N$, the ladder coefficients obey the recursion $A_{N+1}(p^2)=\int d^Dk\, A_N(k^2)/\big[(k^2)^2\,(k+p)^2\big]$ (Eq. (3) of the letter); each $A_N$ is separately UV divergent for $D>6$. Parisi resummed the series in closed form, as an Euler Gamma-function expression in $N$ (Eq. (6)) and equivalently as an iterated one-dimensional contour integral (Eq. (8)), with $\varepsilon=(D-6)/2$: the summed amplitude is finite for noninteger $D$ and nonanalytic in $g$ (of the type of effective monomials such as $g^{8-D}$), supporting the interpretation that the perturbative UV divergences arise from Taylor-expanding a nonanalytic function.

The gap. The construction is defined only for noninteger $D$: the Gamma-function factors of the summed expression develop poles at integer space-time dimensions -- the prefactors $\Gamma(-2/\varepsilon)$ and $\Gamma(-1/\varepsilon)$ with $\varepsilon=(D-6)/2$, for instance, are singular exactly at $D=7$, $8$ and $10$ among the integers $D>6$. Moreover it sums only the planar-ladder subclass, only for massless internal lines, and only at the special kinematics with two external momenta zero. Whether the mechanism can be promoted to a genuine nonperturbative definition at the physical integer dimensions, beyond the ladder subclass, and in realistic theories, is the subject of this problem.

Problem Statement

Parisi's 1973 construction shows, for the massless cubic scalar theory at noninteger space-time dimension $D>6$, that the planar-ladder series for the four-point amplitude $A(0,0,p,-p)$ -- every term of which is ultraviolet divergent by power counting -- can be summed to a finite closed form that is nonanalytic in the coupling $g$, so that the order-by-order ultraviolet divergences can be interpreted as artifacts of Taylor-expanding a nonanalytic function. The construction, however, exists only for noninteger $D$ (the Euler Gamma-function factors of the summed expression develop poles at integer $D$ -- the prefactors $\Gamma(-2/\varepsilon)$ and $\Gamma(-1/\varepsilon)$ with $\varepsilon=(D-6)/2$, for instance, at $D=7,8,10$), and covers only one diagram subclass, massless internal lines, and the special kinematics with two external momenta zero.

Determine whether this mechanism can be turned into a finite nonperturbative definition of perturbatively nonrenormalizable interactions at integer space-time dimension: whether a resummation prescription or nonperturbative definition of the theory's amplitudes can remain finite as the analytic continuation in $D$ approaches integer values by removing or taming the pole singularities that the continuation develops there, so that the tamed prescription has a well-defined limit as $D\to$ integer. The target is every integer $D>6$: the named prefactors are singular at $D=7,8,10$, which are the highlighted singular cases rather than the whole target, and the complete set of integer dimensions at which the closed forms of Eqs. (6) and (8) remain finite is part of the question. Lifting the restrictions of the original construction is part of the question: what becomes of the pole structure once the planar-ladder subclass, the massless internal lines, and the special kinematics with two external momenta zero are left behind -- for crossed ladders (ladders whose rungs are exchanged, so the graph is no longer planar), vertex and self-energy insertions (corrections to, respectively, the basic interaction vertex and the propagator), nonplanar topologies, and generic external kinematics and/or nonzero masses. These extensions are not independent sub-problems to be settled one at a time: settling any one of them requires controlling the omitted contributions as well.

References

1. G. Parisi, 'On renormalizability of not renormalizable theories', Lettere al Nuovo Cimento (Series 2) 6, 450-452 (1973), DOI 10.1007/BF02759056, https://doi.org/10.1007/BF02759056.
2. F. J. Dyson, 'The Radiation Theories of Tomonaga, Schwinger, and Feynman', Physical Review 75, 486-502 (1949), DOI 10.1103/PhysRev.75.486, https://doi.org/10.1103/PhysRev.75.486.
3. P. J. Redmond and J. L. Uretsky, 'Conjecture Concerning the Properties of Nonrenormalizable Field Theories', Physical Review Letters 1, 147-148 (1958), DOI 10.1103/PhysRevLett.1.147, https://doi.org/10.1103/PhysRevLett.1.147.
4. K. G. Wilson, 'Feynman-Graph Expansion for Critical Exponents', Physical Review Letters 28, 548-551 (1972), DOI 10.1103/PhysRevLett.28.548, https://doi.org/10.1103/PhysRevLett.28.548.
5. G. Parisi, 'The theory of non-renormalizable interactions. 1. The large N expansion', Nuclear Physics B 100, 368-388 (1975), DOI 10.1016/0550-3213(75)90624-0, https://doi.org/10.1016/0550-3213(75)90624-0.
6. G. Parisi, 'On Non-Renormalizable Interactions', in 'New Developments in Quantum Field Theory and Statistical Mechanics' (Cargese 1976), Plenum Press, pp. 281-305 (1977), DOI 10.1007/978-1-4615-8918-1_12, https://doi.org/10.1007/978-1-4615-8918-1_12.
7. M. Aizenman, 'Proof of the Triviality of $\phi_d^4$ Field Theory and Some Mean-Field Features of Ising Models for $d>4$', Physical Review Letters 47, 1-4 (1981), DOI 10.1103/PhysRevLett.47.1, https://doi.org/10.1103/PhysRevLett.47.1.
8. H. W. Hamber and R. M. Williams, 'Renormalization group running of Newton's constant $G$: The static isotropic case', Physical Review D 75, 084014 (2007), DOI 10.1103/PhysRevD.75.084014, arXiv:hep-th/0607228, https://arxiv.org/abs/hep-th/0607228.
9. H. W. Hamber, 'Vacuum Condensate Picture of Quantum Gravity', Symmetry 11(1), 87 (2019), DOI 10.3390/sym11010087, arXiv:1707.08188, https://arxiv.org/abs/1707.08188.
10. H. W. Hamber and L. H. S. Yu, 'Dyson's Equations for Quantum Gravity in the Hartree-Fock Approximation', Symmetry 13(1), 120 (2021), DOI 10.3390/sym13010120, arXiv:2010.10500, https://arxiv.org/abs/2010.10500.
11. D. I. Kazakov and D. E. Vlasenko, 'Leading and subleading UV divergences in scattering amplitudes for $D=8$, $N=1$ SYM theory in all loops', Physical Review D 95, 045006 (2017), DOI 10.1103/PhysRevD.95.045006, arXiv:1603.05501, https://arxiv.org/abs/1603.05501.
12. D. I. Kazakov, L. V. Bork, A. T. Borlakov, D. M. Tolkachev and D. E. Vlasenko, 'High Energy Behaviour in Maximally Supersymmetric Gauge Theories in Various Dimensions', arXiv:1812.11084 (2018), https://arxiv.org/abs/1812.11084.
13. P. Romatschke, 'Quantum Field Theory in Large N Wonderland: Three Lectures', Acta Physica Polonica B 55, 4-A2 (2024), DOI 10.5506/APhysPolB.55.4-A2, arXiv:2310.00048, https://arxiv.org/abs/2310.00048.