Principle of minimal singularity for Green’s functions

Recently, two approaches were proposed to resolve the indeterminacy of the nonperturbative Dyson-Schwinger equations in D -dimensional spacetime. One approach utilizes the asymptotic behavior of the Green’s functions G n = ⟨ ϕ n ⟩ at large n , while the other one makes use of the null state condition. In this work, we point out that these two seemingly different approaches can be unified by a novel principle: Singularities in the complex plane should be minimal. For D = 0 , the exact Green’s functions of the general gϕ m theory can be determined by minimizing the complexity of the essential singularities at n = ∞ . For D = 1 , we revisit the quartic theory and discover the merging of different branches of Green’s functions at exact solutions. Then we solve the one-dimensional Hermitian quartic and non-Hermitian cubic theories using the principle of minimal singularity.


I. INTRODUCTION
In quantum field theory, the Green's functions describe the correlation of quantum fields and encode information about the particle spectrum and vacuum structure.As quantum equations of motion, the Dyson-Schwinger (DS) equations [1][2][3] imply that the Green's functions are related to each other, providing a self-consistent way to determine the Green's functions.However, the nonperturbative DS equations usually form an underdetermined system [4].Additional assumptions or constraints are needed to close the system.A simple scheme is to set the higher connected Green's functions to zero.Even though one may find convergent results, their limiting values deviate from the exact values, as emphasized recently in [5,6].To resolve this issue, Bender et al. replaced the naive vanishing constraints with more sophisticated approximations from the large n asymptotics [5,6].Alternatively, one can resolve the indeterminacy unbiasedly using the null state condition [7]. 1 For unitary solutions, one can also use positivity to constrain the solution space [9][10][11][12]. 2   The use of large n asymptotic behaviors in [5,6] was based on an implicit assumption.To study different Green's functions at the same time, the full set of G n = ⟨ϕ n ⟩ should exhibit good analytic behavior in n, but n is usually thought of as an integer parameter.This is reminiscent of the quantum angular momentum, which is usually known as a quantized quantity that takes discrete values.In 1959, Regge analytically continued the angular momentum to complex values, leading to deep insights into the asymptotic behavior of scattering amplitudes [14].This original idea had significantly influenced the developments of the bootstrap program [15,16].As the DS equations furnish a self-consistent method for determining Green's functions, the analytic continuation in n should also provide useful insights.
There is another motivation for complexifying n.Inspired by the properties of Yang-Lee edge singularities [17][18][19][20][21], Bessis and Zinn-Justin studied the quantum mechanical version of the iϕ 3 theory and noticed the reality of the energy spectrum despite non-Hermiticity.Later, Bender and Boettcher proposed a novel family of non-Hermitian PT -invariant theories with real and bounded spectra [22], in which the power of the interaction term can take non-integer values.To bootstrap the theories with non-integer power, it is inevitable to take into account ⟨ϕ n ⟩ with non-integer n [23].
Along these lines, we revisit the issue of the underdetermined DS equations from the complexified-n perspective.It turns out that there exists a novel principle that can resolve the problem: Singularities in the complex plane should be minimal. 3The null state approach [7] mentioned earlier can be viewed as an unbiased way to minimize the complexity of singularities.Below, we will determine the Green's functions using the principle of minimal singularity.The examples include Hermitian solutions, non-Hermitian PT symmetric solutions [5,[24][25][26][27], as well as fully complex solutions.

A. Cubic theory
For the D = 0 cubic theory, the DS equation from the Lagrangian L = gϕ 3 reads with G 2 = 0 and G 0 = 1.The general solutions are given by where G 1 is a free parameter, p is a non-negative integer, and (a) b = Γ(a + b)/Γ(a) is the Pochhammer symbol.To minimize the complexity of the singularity structure, we construct an ansatz for the general n expression which is manifestly compatible with the normalization condition G n=0 = 1.Note that α n and β n are related to two possible kinds of essential singularities at n = ∞.The general solutions in (2) imply They should be satisfied for generic p, so we have the periodicity conditions where α ̸ = 1.The two choices of α are not independent and we choose α = e 2πi 1 3 .Then we can derive G 1 from (3): where k = 0, 1, 2. We obtain the three exact solutions For g = i 3 , the PT -symmetric solution corresponds to k = 2.We can also derive the exact solutions using the large-n expansion.As the factor (1 − α n+1 )/(1 − α) is responsible for the vanishing Green's functions at n = 3p + 2, we will focus on the remaining part of (8).Note that the DS equation (1) gives a recursion relation between G n and G n+3 .We assume the existence of a stronger relation: where F n should be minimally singular in n.Then the DS equation (1) implies which can be solved asymptotically at large n.The leading terms in 1/n are where k = 0, 1, 2. In general, the resummation of the n −1 series is not unique.To minimize the complexity of singularities, we use the gamma function to construct the "cube root" of the DS equation ( 1): n + 1 3 In this way, we also obtain the exact solutions in (8).

B. Quartic theory
For the D = 0 quartic theory with L = gϕ 4 , the DS equation is where G 3 = 0 and G 0 = 1.The general solutions read where G 1 and G 2 are free parameters.As in the cubic case, we consider a minimally singular ansatz where α n and β n are associated with two possible types of essential singularities at infinity.The consistency with ( 14) implies the periodicity conditions We can use (15) to determine G 1 and G 2 : where we have used the fact that α ̸ = 1.The independent choices of α are In this way, we find all the exact solutions: • In the first case α = i, there are four different solutions: where k = 0, 1, 2, 3.For g = − 1 4 , the PT -symmetric solution is associated with k = 3. • In the second case α = −1, all the parity-odd Green's functions vanish.We obtain two solutions: For g > 0, the standard Hermitian solution corresponds to the positive case with (+1).
Let us also deduce them from the large n expansion.We focus on the α = −1 case for simplicity.Since 1+(−1) n 2 is implied by the vanishing case of n = 2p + 1, we concentrate on the remaining part.To minimize the complexity of the singularity structure, we impose where F n should be minimally singular in n.The DS equation ( 13) implies It is straightforward to solve this equation at large n The resummation of this 1/n series also contains ambiguities.To minimize the complexity of singularities, we again use the gamma function to construct the complete expression.As a result, the "square root" of the DS equation ( 13) is given by which gives the two exact solutions in (22).Since both α 1 and α 2 are consistent with the periodicity condition, we could consider a non-minimal ansatz Then ( 14) implies 1 + c 1 α 3 1 + c 2 α 3 2 = 0 .After introducing one more type of singular behavior to the ansatz, the final solutions are parametrized by a free parameter and interpolate between the minimally singular solutions.This is reminiscent of the two-cut solutions in matrix models.
where G m−1 = 0 and G 0 = 1.The general solutions are where p is an integer.If m ≥ 3 is an integer, {G 1 , . . ., G m−2 } are free parameters.The minimally singular ansatz reads The α part is related to G m−1 = 0.The β part is associated with the D = 0 version of the Symanzik/Sibuya rotation [28] that relates different Stokes sectors.Then (29) implies the periodicity conditions under the constraint α ̸ = 1.Therefore, the exact solutions are labelled by two integers.We choose the independent solutions based on the condition Im(α) ≥ 0. The number of different types of essential singularities increases with m.In analogy to the multi-cut solutions in matrix models (see [10,11]  5 , e 2πi 3 5 ) , (e 2πi 2 5 , 1) .We can also consider a rational power m = (1/m 1 )m 2 , where (m 1 , m 2 ) are non-negative and mutually prime integers.If we introduce α ′ m1 = α 1/m1 and β ′ m1 = β 1/m1 , then the imaginary G 1 solutions from (α ′ m1 ) m2 = (β ′ m1 ) m2 = 1 lie precisely on the interpolating curves in Fig. 1. , where the curves are labelled by k with α = e 2πi k m .We do not plot the full interpolating curves because they oscillate more and more rapidly as m decreases.In addition, infinitely many of them would intersect at the black dots due to α k+m = α k .

III. ONE-DIMENSIONAL QUARTIC THEORY
For D = 1, we will use the Hamiltonian formalism for simplicity.The Hamiltonian of the massive quartic theory is [29] The position operator x and the momentum operator p satisfy the canonical commutation relation [x, p] = iℏ.For an eigenstate with energy E, the expectation values satisfy some consistency relations [30] where the expectation values are associated with the standard Hermitian inner product.One can derive from (33) a recursion relation: where G n = G n (t 1 , t 2 , . . ., t n )| ti→t1 = ⟨x n ⟩ E is the equal time limit of the n-point Green's function. 4The normalization is set by G 0 = 1.We assume that the solutions respect the parity symmetry, so the odd-n Green's functions vanish and we can focus on those with even n.Instead of imposing positivity constraints as in [30], we will study (34) by analytically continuation in n [23].
According to (34), we can express G n in terms of E and G 2 .In Fig. 2, we consider some cases of (E, G 2 ) around the exact values, where G n has been divided by the leading asymptotic behavior 3 n/3 n 1/6 Γ n 6 2 .At large n, the solutions for G n exhibit three oscillatory curves alternatively, corresponding to G 6n+2k with k = 0, 1, 2. When (E, G 2 ) are close to the exact values, the three branches of G n merge into a single curve at relatively small n.Around the exact values, the three branches of solutions merge into one decay curve at relatively small n.At the same precision, the merging of the excited-state results (orange) extends to larger n than the ground-state case (blue).
To show the existence of three branches of solutions, let us carry out the large-n asymptotic analysis.Assuming that the non-vanishing Green's functions grow faster than n, the leading asymptotic behavior is determined by where the second and third terms in (34) have been omitted.This is a third-order difference equation for the even-n Green's functions.At large n, the leading behavior is given by Below we will set g = 1.The three independent constants c 0 , c 1 , c 2 lead to three independent branches of solutions at integer n. 5 In the merging limit c 1 , c 2 → 0, two types of essential singularities are removed.Then the third term in (34) implies a subleading factor e −(n/2) 1/3 , which is consistent with the decay behavior in Fig. 2. The merging phenomena indicate that the low-lying information is strongly constrained the principle of minimal singularity.In terms of F n = G n+2 /G n , the recursion relation (34) becomes where F n should be minimally singular in n.If F n grows with n, then the leading behavior of F n is encoded in Note that the large n expansion is different from the small ℏ expansion in the semiclassical WKB method.Below we will further set ℏ = 1.Taking the "cube root" of (38), we find three minimal solutions and two of them are complex.The properties of the Hermitian inner product suggest that we should choose the real one, corresponding to (36) with c 1 = c 2 = 0.The systematic large-n expansion of the real solution gives It is not clear how to express (39) in terms of well-known special functions.Nevertheless, as the counterpart of the (+1) solution in (22), we expect that the main implications of minimal singularity have been captured.Although nonperturbative corrections are missing, a high-order truncation of the n −1 series (39) provides an accurate approximation for F n with sufficiently large n.
FIG. 3: The absolute error in the ground-state energy E 0 of the D = 1 quartic theory (32) from the matching conditions (40), where M is the matching order.The 1/n series of n −2/3 F n is truncated to order n −N .The estimates converge rapidly to the exact value.
To determine E and G 2 , we will impose some matching conditions on G n at relatively large n.Above, the n −1 series (39) encodes the constraints from the principle of minimal singularity.To connect with the observables in the nonperturbative regime with small n, we solve the recursion relation (34) exactly to relatively large n.The analytic solutions for G n are given by high-degree polynomials in E and G 2 .To determine (E, G 2 ), we impose two matching conditions: where M denotes the matching order and N indicates the truncation order of the n −1 series.As a result, we have two polynomial equations.The real roots provide accurate approximations for the low-lying data.Fig. 3 shows that the estimate of the groundstate energy E 0 improves rapidly with the matching order M and the truncation order N .Since the results for the higher states merge into one curve more easily, the principle of minimal singularity leads to more accurate results for the low-lying states.For example, the energy spectrum from the matching conditions (40) with (M, N ) = (100, 10) is (1.39235164153029206, 4.6488127042152, 8.6550499661, 13.15680465, 18.05715, 22.76, 28.69, 39.2, . . .), where the last two digits of the approximate energies deviate from the exact values.The results for G 2 exhibit similar features.
We also revisit the iϕ 3 theory at D = 1.In this example, the principle of minimal singularity is less manifest because there is no merging phenomena.As in the generic situation, the exact PT symmetric solutions also exhibit five branches of curves.Nevertheless, it turns out that they have only two types of essential singularities, instead of all the five possible types in a general case.Therefore, we can again use the principle of minimal singularity to determine the PT symmetric solutions.More details are presented in the Appendix.

IV. DISCUSSION
The analytic continuation of n in the Green's functions G n leads to the novel principle that the complexity of the singularity structure should be minimized.We show how to resolve the indeterminacy of the self-consistency equations in this new approach.We obtain the exact solutions (30), (31) of the general gϕ m theory at D = 0 and rapidly convergent results at D = 1.Besides good analytic properties, the minimality of a self-consistent solution is closely related to the simplicity of an asymptotic behavior at large n [5,6] and the irreducibility of an operator-algebra representation [23].
In the recursion relation for consecutive Green's functions, an exact solution is related to a minimally singular coefficient function F n .The null state approach [7,23] can be viewed as a rational approximation for the exact, minimally singular recursion relation. 6Without a priori knowledge of F n , a truncated null state condition leads to an approximate recursion relation for multiple Green's functions, in which the coefficients are constant and unbiasedly determined by the consistency equations themselves.
If we consider the DS equations at D > 0, there will be infinitely many free parameters.For the emergence of a proper inner product, they need to exhibit better analytic behavior than an arbitrary set of numbers.The null state approach is a particularly useful way to derive the minimally singular solutions.To further extract physically more meaningful solutions, we can impose additional constraints, such as positive semidefiniteness from unitarity [9][10][11][12] or spectral boundedness from stability [7].
where n = M, M + 1, M + 2, M + 3. The nonperturbative expressions of G (NP) n are derived from the exact recursion relation (A2) at n = −3, −2, . . ., M − 2. They are high-degree polynomials in E, but at most linear in G 1 .The perturbative expressions of G (P) n are deduced from the large n series to order n −N , where the first four leading terms are given explicitly in (A5).The results again converge rapidly to the exact values with the matching order M and the truncation order N .For (M, N ) = (100, 10), the stable solutions with positive real energies are (E, G 1 ) = (1.1562670719881112,−0.59007253309070011i), (4.10942, −0.982086i), (A7) which correspond to the PT symmetric ground state and first-excited state.Their last two digits are different from the exact values.Compared with the quartic case, we obtain less energy levels because some solutions in the non-Hermitian cubic case are spurious and unstable.Although there are only two types of essential singularities at n = ∞, the Green's functions still exhibit five branches of solutions.In fact, if we take into account the odd n Green's functions, the Hermitian quartic case also has two types of essential singularities at infinity.In the end, it seems that (c 2 , c 3 ) are not independent, as c 3 /c 2 = tan(−3π/10) is satisfied to high precision near the exact solutions.
C. General gϕ m theoryAfter discussing two basic examples at D = 0, we are in the position to consider the general L = gϕ m theory.The DS equation is FIG. 1: The purely imaginary solutions for G 1 in the non-Hermitian theories L = −(i m /m) ϕ m from the general formulae (30) and (31).Many of them are PT symmetric.The exact solutions at integer m are denoted by black dots.The colored curves interpolate the integer-m solutions with the same k, according to α = e 2πi k m .All the non-zero solutions with integer m are located at the intersection points.

FIG. 2 :
FIG. 2: The solutions of G n from the recursion relation (34) with ℏ = g = 1.The subfigures are labelled by the input (E, G 2 ).Around the exact values, the three branches of solutions merge into one decay curve at relatively small n.At the same precision, the merging of the excited-state results (orange) extends to larger n than the ground-state case (blue).