Primitive asymptotics in vector theory

Related material

Preprint	hep-th/2412.08617
Program	Mathematica file for 0-dimensional asymptotics
Data set	Leading primitive graphs 4 to 19 loops (2.1MB bzip2)
Data set	Leading primitive graphs 20 to 22 loops (37MB bzip2)
Data set	Subleading primitive graphs 4 to 15 loops (52MB bzip2)
Related project	Statistics of periods in $\phi^4$ theory
Previous data set	DOI 10.5683/SP3/NLEDGH

Vector theory

This project is about the 4-regular vector theory in 4 spacetime dimensions. This theory is a version of $\phi^4$ theory, where the field variable $\phi$ is not just a scalar, but a $N$ -component vector $\vec \phi$ . The square of this field is the usual dot product $\vec \phi^2=\vec \phi \cdot \vec \phi = \phi_1^2 + \phi_2^2 + \ldots + \phi^2_N$ . For example, for $N=2$ the field is by itself a -dimensional vector, but this is an “inner” symmetry, the dimension of this vector is unrelated to the ambient spacetime being 4-dimensional. An example of such internal symmetry are small magnets confined to a fixed axis of rotation. Indeed, for different integer values of $N$ , this theory is the “continuum limit” of various interesting physical systems. We had computed a lot of numerical data for this theory, but without emphasis on the $O(N)$ symmetry, in the project statistics of Feynman periods.

The treatment of this symmetry in perturbative quantum field theory is relatively straightforward: Each Feynman graph $G$ behaves as always, but it additionally obtains a new symmetry factor $T(G,N)$ which can be constructed combinatorially. For a finite graph $G$ , this $T(G,N)$ is a polynomial in $N$ . In the original definition of the theory, $N$ must be a positive integer to be interpreted as the number of components of a vector, but since $T(G,N)$ is a polynomial, one may evaluate it also for non-integer $N$ .

The presence of the parameter $N$ implies that one can construct a series expansion also in this parameter, instead of the usual series expansion in the coupling constant. In fact, a series expansion in orders of $N$ is relatively similar to the original small-coupling expansion, but one obtains a strikingly different result by expanding in orders of $\frac 1 N$ . This is the large- $N$ -expansion. It has been known for decades that the large- $N$ expansion is dominated by a particular class of Feynman graphs, the “bubble” graph, which are chains of 1-loop multiedges. The total number of Feynman graphs grows factorially with the loop order (compare the statistics project), but there are only very few such bubble graphs. Indeed, it turns out that the large- $N$ expansion is convergent (while the series expansion in small coupling, or loop number, is factorially divergent).

Zero-dimensional asymptotics

The term “0-dimensional quantum field theory” is somewhat curious. It means that one considers a path integral without any spacetime integration. This amounts to a theory, where the Feynman rules for all propagators and vertices are just constants, and not functions of masses or momenta. There are no “Feynman integrals” in the ordinary sense, instead, the value of a Feynman graph is just the product of the constants of the propagators and vertices contained in it. In particular, one can choose these constant such that the value of a Feynman graph is unity. Then, the coefficient of loop order $L$ in the perturbation series is the sum of all Feynman integrals, weighted by their symmetry factors, and the 0-dimensional path integral is a generating function for these sums. Thereby, 0-dimensional quantum field theory amounts to counting the number of Feynman graphs, where each graph is weighted by its symmetry factor.

The zero-dimensional path integral

$\begin{align*} Z(\hbar, \eta) &:= \int_{-\infty}^\infty \frac{\d^N \vec \phi}{(2\pi \hbar)^{\frac N2}} e^{\frac{1}{\hbar}\left( -\frac{\vec \phi^2}{2}+ \frac{(\vec \phi^2)^2}{4!} +\frac{\eta}{\hbar} \vec \phi^2 \right) } \end{align*}$

is a generating function of all Feynman graphs. The source term $\eta$ generates a pair of external legs, including a sum over their vector indices. Evaluating at $\eta=0$ produces vacuum graphs, the first derivative at $\eta=0$ are graphs with two external legs, and so on. It turns out that this integral can be solved analytically in terms of a formal power series, thereby one explicitly knows the sum of graphs with arbitrarily many external legs. The series coefficients depend on $N$ , one thereby obtains graphs weighted by their $O(N)$ symmetry factors $T(G,N)$ .

We are more interested in certain special classes of graphs. Their generating functions can be computed by algebraic transformations of $Z(\hbar, \eta)$ : Taking the logarithm yields the generating function of connected graphs, a Legendre transform gives 1PI graphs, and a series inversion of the 1PI 4-point function is (almost) the vertex counterterm, which at the same time enumerates subdivergence-free graphs. All these operations are perfectly concrete and explicit, they are implemented in the Mathematica notebook above.

The fact that one knows the power series $Z(\hbar, \eta)$ explicitly in terms of gamma functions implies that one can also compute its exact asymptotics. All these power series are factorially divergent, the asymptotics of the $n$ -th term $f_n$ always has the form

$\begin{align*} f_n \sim a^{n} \Gamma (n + c_s) \left( c_0 + \frac{c_1}{n+c_s-1} + \frac{c_2}{(n+c_s-1)(n+c_s-2)} +\ldots \right). \end{align*}$

This equation means that $f_n$ grows according to this form in the limit $n \rightarrow \infty$ , where the index $n$ can be the loop number of the graph or any other expansion parameter. It does not mean that for finite $n$ , the value of $f_n$ is given by this formula. In particular, one can not determine the coefficients $a,c_s, c_0, c_1, \ldots$ by computing a few values $f_n$ and set them equal to the asymptotic form.

For the generating function $Z(\hbar, \eta)$ , all (infinitely many) coefficients in the asymptotic expansion are known. Then, through a method pioneered by Michael Borinsky, one essentially repeats the algebraic transformations above, and obtains the exact asymptotics for connected graphs, 1PI graphs, primitive graphs, counterterms and so on. The results have no closed-form expression, but it is reasonably easy to compute at least the first 10 subleading correction coefficients (which is way beyond anything that can be “measured” by examining the growth rate for $n\approx 100$ , say). These computations, too, are implemented in the Mathematica notebook.

The crucial question for applications in QFT is whether the asymptotic expansion gives reliable estimates not only for loop order $L \rightarrow \infty$ , but also at finite $L$ . We are particularly interested in primitive (=subdivergence-free) graphs, because they are not affected by choices of renormalization scheme, and therefore it makes sense to examine them in 0-dimensional QFT. Let $p_L(N)$ be the sum of primitive graphs at $L$ loops, weighted by symmetry factor, and including $N$ -dependence of the vector theory. When we fix $N=1$ and plot $p_L$ against its asymptotics (green line), we get something like this:

At first, this looks as if the asymptotics captures the data points quite well, but pay attention to the logarithmic $y$ -axis: A small shift in $y$ -direction indicates a quite substantial factor of difference. It is more instructive to divide the true value by the leading asymptotics, that is, to scale the plot such that the green line is the constant unity function.

In this plot, we see that even at 20 loops, the difference between the leading asymptotics and the true value is larger than 20%. The colored curves in this plot are subleading corrections, they improve the situation, but they, too, fail below 10 loops.

In practice, it is hard to work with quantities that grow factorially, simply because the numerical values are so huge that one needs ridiculous scaling to even visualize them in plots. It is much easier to consider the growth ratio

$\begin{align*} r_L := \frac{p_{L+1}} {L \cdot p_L}. \end{align*}$

$r_L$ is defined in such a way that when $p_L$ grows factorially, $r_L$ has a finite non-zero limit as $L \rightarrow \infty$ . One can then plot these data points, and the corresponding asymptotics, as function of $\frac 1 L$ . The subleading correction of the asymptotics amounts to a polynomial in $\frac 1 L$ . For our case of primitive graphs, we obtain a plot like this:

The limit of $r_L$ as $L \rightarrow \infty$ , or $\frac 1 L \rightarrow 0$ , is exactly $\frac 2 3$ . However, if we only consider the data points between 10 and 18 loops (red dots), they appear to be converging to a wrong limit. Notice, in particular, that for distinct values of $N$ , the red lines converge towards the same point. This is deceptive because it gives the impression that the red lines represent the asymptotics (which must have a unique limit independent of $N$ ). The true asymptotics, indicated by green lines, can only be seen from the data above 25 loops. This is one of the main findings of the paper: Below 25 loops, numerical data for the beta function of $\phi^4$ theory does not represent the correct asymptotic growth rate.

Everything so far was for the 0-dimensional theory. Of course, the physically interesting case is the 4-dimensional theory. Unfortunately, we only have numerical data up to 18 loops, which allows us to compute the growth rate $r_L$ up to 17 loops. Nevertheless, the observed pattern in the 4-dimensional theory is strikingly similar to the 0-dimensional one:

Again, the green lines indicate the expected asymptotics, and they intersect the observed data around 25 loops. In view of this, we conjecture that the growth rate indicated by the red lines, which results from the data points up to 18 loops, is probably not the correct growth rate at $L\rightarrow \infty$ . We expect to need at least 25 loops to see the true asymptotic growth rate.

Our asymptotic analysis has also yielded several other results. For example, we compute the exact asymptotics of Martin invariants, and we observe that the primitive beta function has zeros at all negative integer $N\leq -4$ . The latter is combinatorially interesting, it can be interpreted to say that asymptotically, almost all orientations of subgraphs of high valence are primitive.

We also obtain a very nice relation to the large- $N$ expansion: Why is it that the large- $N$ expansion is convergent, but the large loop order expansion is factorially divergent? To understand this, we first observe that at each finite loop order $L$ , the sum of primitive graphs $p_L(N)$ is a polynomial in $N$ of degree $\frac 2 3 L$ . Hence, the degree grows linearly with $L$ . The coefficients of the polynomial are positive. Evaluating the polynomial at a fixed value of $N$ is essentially a weighted sum of these polynomial coefficients. However, the numerical value of the coefficients is strongly dependent on their order. That is, for a fixed loop order $L$ , the coefficient of $N^0$ or $N^1$ is much larger than the coefficient of $N^{\frac 23 L}$ . To “measure” this, we define the average order

$\begin{align*} \langle k \rangle_L := \sum_j j \cdot [N^j] p_L(N) = \frac{\frac{\partial}{\partial N} p_L(N)}{p_L(N)} \big|_{N=1} .\end{align*}$

With our asymptotic analysis, we can compute the asymptotics of $\langle k \rangle_L$ for $L \rightarrow \infty$ , and we find that it grows only logarithmically, not linearly, with $L$ . This means that at large $L$ , the polynomial $p_L(N)$ is “infinitely much” dominated by terms of low order in $N$ , while in comparison, the high-order coefficients in $N$ are vanishingly small. This is the reason why the large- $N$ expansion is not divergent: It is a sum of only these leading terms, which grow much slower than the sum of all terms. Conversely, it also means that the large- $N$ expansion misses “almost all” contributions that are included in the perturbation series.

The plot shows how strong this effect is: Even at $L=25$ loops, the mean order $\langle k \rangle_{25}$ is smaller than unity, while the leading term already is $\propto N^{18}$ . The fact that $/langle k \rangle$ is smaller than unity means that the coefficient of $N^0$ is larger than all other coefficients together.

Paul-Hermann Balduf

Research

Primitive asymptotics in vector theory

Vector theory

Zero-dimensional asymptotics