Non-local non-linear sigma models

TL;DR

This paper investigates non-local non-linear sigma models across dimensions, revealing unique renormalization properties and potential links to M2-brane dynamics, with implications for both Archimedean and ultrametric spaces.

Contribution

It introduces a scale-invariant non-local sigma model framework, analyzing one-loop divergences and renormalization without metric renormalization, and discusses possible applications to M2-branes.

Findings

One-loop divergences are canceled by bi-local terms.

Metric renormalization is absent in the non-local case.

Potential connections to M2-brane dynamics are proposed.

Abstract

We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes.

Equations

$$S=\frac{{\mu }^{n-s}}{2\gamma }{\int }_{V\times V}\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}d(\varphi (x),\varphi (y){)}^2,$$

$$S=\frac{1}{2}{\int }_V{d}^{n}k\hat{\varphi }(-k)|k{|}^s\hat{\varphi }(k)+{\int }_V{d}^{n}xU(\varphi (x)),$$

$$\varphi (x)={\int }_V{d}^{n}k{e}^{2\pi ik\cdot x}\hat{\varphi }(k)\hat{\varphi }(k)={\int }_V{d}^{n}x{e}^{-2\pi ik\cdot x}\varphi (x).$$

$${\int }_V{d}^{n}k{e}^{2\pi ik\cdot x}|k{|}^s=\frac{{\Gamma }_V(n+s)}{|x{|}^{n+s}}+\text{(contact terms)}.$$

$${\Gamma }_V(s)=\frac{{\zeta }_v(s)}{{\zeta }_v(n-s)}$$

$${\zeta }_{\infty }(s)\equiv {\pi }^{-s/2}{\Gamma }_{\text{Euler}}(s/2){\zeta }_p(s)\equiv \frac{1}{1-p^{-s}}.$$

$$\int d^{n}k{e}^{2\pi ik\cdot x}={\delta }^n(x).$$

$$\int d^{n}k{e}^{2\pi ik\cdot x}|k{|}^s\hat{\varphi }(k)=D^s\varphi (x),$$

$$D^s\varphi (x)={\Gamma }_V(n+s)\int d^{n}y\frac{\varphi (y)-\varphi (x)}{|x-y{|}^{n+s}}.$$

$$D^s\varphi (x)=\frac{1}{(2\pi {)}^s}(-\square {)}^{s/2}\varphi (x)\text{for positive even s},$$

$$D^s\varphi (x)={\Gamma }_V(n+s){\int }^{\prime }{d}^{n}y\frac{\varphi (y)-\varphi (x)}{|x-y{|}^{n+s}},$$

$${\int }^{\prime }{d}^{n}y\frac{G(x,y)}{|x-y{|}^{n+s}}$$

$${\int }^{\prime }\frac{d^{n}y}{|y{|}^{n+s}}\left[\varphi (y)-\varphi (0)\right]=\int \frac{d^{n}y}{|y{|}^{n+s}}\left[\varphi (y)-\varphi (0)-y_i{\partial }_i\varphi (0)-\frac{1}{2}{y}_{i_1}{y}_{i_2}{\partial }_{i_1}{\partial }_{i_2}\varphi (0)\right].$$

$$D^s\varphi (0)=\int \frac{d^{n}y}{|y{|}^{n+s}}\left[\varphi (y)-\varphi (0)-\frac{y^2}{2n}\square \varphi (0)\right].$$

$$D^s\varphi (0)={\Gamma }_V(n+s)\int \frac{d^{n}y}{|y{|}^{n+s}}\left[\varphi (y)-\sum _{r=0}^{\lfloor s/2\rfloor }{y}^{2r}{b}_r{\square }^r\varphi (0)\right]$$

$$b_r=\frac{{\Gamma }_{\text{Euler}}(\frac{n}{2})}{2^{2r}{\Gamma }_{\text{Euler}}(r+\frac{n}{2}){\Gamma }_{\text{Euler}}(r+1)}.$$

$$\int d^{n}x{e}^{-2\pi ik\cdot x}{D}^s\varphi (x)$$

$${\int }_{V\times V}\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}G(x,y)$$

$${\int }^{\prime }\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}G(x,y)\equiv \int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}\left[G(x,y)-G(x,x)\right].$$

$${\int }^{\prime }\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}G(x,y)$$

$$\int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}$$

$$\int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}\left[\varphi (y{)}^2-\varphi (x{)}^2\right]=\frac{1}{{\Gamma }_V(n+s)}\int d^{n}x{D}^{s}f(x)=0.$$

$${\int }^{\prime }\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}{\left[\varphi (x)-\varphi (y)\right]}^2=\int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}\left({\left[\varphi (x)-\varphi (y)\right]}^2-\frac{(y-x{)}^2}{n}(\partial \varphi (x){)}^2\right)$$

$$-\frac{2}{{\Gamma }_V(n+s)}\int d^{n}k$$

$${\int }^{\prime }\frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}{\left[\varphi (x)-\varphi (y)\right]}^2=-\frac{2}{{\Gamma }_V(n+s)}\int d^{n}k$$

$$\int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}\left[f(y)-f(x)-\frac{(y-x{)}^2}{2n}\square f(x)\right]=\frac{1}{{\Gamma }_V(n+s)}\int d^{n}x{D}^{s}f(x)=0.$$

$$\int \frac{d^{n}x{d}^{n}y}{|x-y{|}^{n+s}}$$

$$\square_{y}^{m}\left(\phi(x)-\phi(y)\right)^{2}\Big{|}_{x=y}=-2\phi(x)\square^{m}\phi(x)+\square^{m}\phi(x)^{2}\,.$$

$$R_{ab}{}^c{}_d={\partial }_a{\Gamma }_{bd}^c-{\partial }_b{\Gamma }_{ad}^c+{\Gamma }_{ae}^c{\Gamma }_{bd}^e-{\Gamma }_{be}^c{\Gamma }_{ad}^e{R}_{ac}=g^{bd}{R}_{abcd}.$$

Full text

1. 1 Introduction

2. 2 Fourier transforms

3. 3 Bi-local integrals

4. 4 The bi-local non-linear sigma model

5. 5 Loop divergences in momentum space

6. 6 The propagator through one loop

7. 7 Curvature and arc length calculations

8. 8 Three-point vertices

9. 9 Renormalization through cubic order in the fields

10. 10 Quartic counterterms

11. 11 Renormalization at quartic order

12. 12 Recovering the local non-linear sigma model

13. 13 Outlook

1 Introduction

Scalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the $p$-adic numbers were considered in [3] as a continuum description of Dyson’s hierarchical model [4]. A unifying point of view on the bi-local $O(N)$ vector model was provided in [5], showing that the standard large $N$ development can be framed in terms that are largely independent of whether the theory is formulated over the reals or the $p$-adics. The present work extends the study of bi-local theories to bi-local non-linear sigma models, starting with the action

[TABLE]

where $|x-y|$ is the distance function on the $n$-dimensional base space $V$ and $d(\phi(x),\phi(y))$ is the distance function on the target manifold. In the limit $s\to n$, where the theory (1) becomes classically scale invariant, we find logarithmic divergences in one-loop diagrams which can be canceled by counterterms that can be expressed in terms of the target space laplacian of the square of the distance function, together with field redefinitions.111An exception, as we will see, is when $s$ is an even integer and the base space $V=\mathbb{R}^{n}$. Through a procedure we will outline in section 12, one recovers in this case a local non-linear sigma model, and at least for $s=2$ we can use our results to check the standard analysis [6] of the one-loop beta function.

Ricci flatness suppresses the one-loop divergences that we encounter, so in a sense (and with significant caveats) we may claim that we are deriving the vacuum Einstein equations from conformal invariance, as in [6]. Our work was partly motivated by the more recent results of [7], which were derived for a nearest neighbor arc length model on the Bruhat-Tits tree—in other words, on the other side of the $p$-adic AdS/CFT duality [8, 9] from our results for field theories over the $p$-adic numbers. However, the particular structure of counterterm we find suggests that renormalization of our theories have less to do with renormalization of the local metric as normally understood (i.e. Ricci flow) than with an augmentation of the action (1) to include the target space laplacian of $d(\phi(x),\phi(y))^{2}$.

A conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable. Theories with purely quadratic bi-local kinetic terms, as studied in [1, 3] (as well as many subsequent works) avoid such problems through a non-renormalization theorem: If we write

[TABLE]

then the claim is that the quadratic bi-local term is never renormalized (at least perturbatively), though the purely local term $U(\phi(x))$ certainly is—and depending on details, derivative terms might be radiatively generated. Non-local interaction terms vitiate this non-renormalization theorem, and one’s suspicions could be renewed that there is no sensible theory. We will not be able in this work entirely to allay such concerns, because we do not give a demonstration parallel to the one in [6] that Ward identities based on diffeomorphism invariance guarantee that loop divergences can only modify the original form of the action. Indeed, the counterterms we generate at one loop do modify the bi-local action in an unexpected way, but one which appears to be controlled in a derivative expansion, so that higher derivative terms can be radiatively generated at each new order without spoiling results from lower orders. We will revisit the question of renormalizability in section 13.

The organization of the rest of this paper is as follows. In section 2 we present the main results in Fourier analysis that we need, both over the reals and the $p$-adics. In section 3 we explain how double integrals such as the one in (1) can be regulated if divergences arise as $|x-y|\to 0$. In section 4 we introduce the classical action for the bi-local non-linear sigma model. In section 5 we discuss loop divergences in general terms, including an introductory account of the non-renormalization property of the kinetic term in (2). In sections 6-10 we investigate the simplest one-loop divergences of the bi-local non-linear sigma model, and then in section 11 we argue that all these divergences can be canceled by a laplacian counterterm in place of renormalization of the local metric, together with field redefinitions. As a byproduct of our analysis, we recover in section 12 the usual beta function for the two-derivative theory in two dimensions. We conclude in section 13 with a summary of possible future directions.

2 Fourier transforms

In loop calculations we will often need to go back and forth between momentum space expressions similar to the ones presented in (2) and their real space counterparts, using the Fourier transforms

[TABLE]

The relevant results are fairly similar between real and $p$-adic cases, so we present them together. When $V=\mathbb{R}^{n}$, the definitions (3) are entirely standard, and $k\cdot x$ can be understood as the ordinary dot product. Likewise, in this case, $|x|$ is understood as the standard $L^{2}$ norm on $\mathbb{R}^{n}$. We refer to the real case as Archimedean because the norm $|\cdot|$ has the property that if $0<|x|<|y|$, then there is some $n\in\mathbb{Z}$ such that $|y|<|nx|$.

The simplest $n$-dimensional $p$-adic construction is based on letting $V=\mathbb{Q}_{p^{n}}$ be the (unique) unramified $n$-dimensional extension of $\mathbb{Q}_{p}$. Let $\operatorname{N}$ and $\operatorname{Tr}$ be the field norm and field trace with respect to the extension $\mathbb{Q}_{p^{n}}/\mathbb{Q}_{p}$. Then we define $|x|=|\operatorname{N}(x)|_{p}^{1/n}$ where $|\cdot|_{p}$ is the usual $p$-adic norm. We will refer to the $p$-adic case as ultrametric because the norm $|\cdot|$ just defined has the property $|x+y|\leq\max\{|x|,|y|\}$. Next we define $k\cdot x={1\over n}\operatorname{Tr}(kx)$. Note that $k\cdot x\in\mathbb{Q}_{p}$, so to give meaning to $e^{2\pi ik\cdot x}$ we now only need to define $e^{2\pi i\xi}$ for $\xi\in\mathbb{Q}_{p}$. To this end we find the unique $p$-adic integer $\lfloor\xi\rfloor$ such that $\xi-\lfloor\xi\rfloor\in[0,1)\cap\mathbb{Q}$, and we understand that by $e^{2\pi i\xi}$ we really mean $e^{2\pi i(\xi-\lfloor\xi\rfloor)}$.

We are particularly interested in the Fourier transform of powers of $|k|$:

[TABLE]

Here $\Gamma_{V}(s)$ is a meromorphic function of $s$ which can be evaluated as

[TABLE]

where we set $v=\infty$ in the Archimedean case and $v=p$ in the ultrametric case, with

[TABLE]

Intuitively, $\Gamma_{V}$ is a variant of the Euler gamma, specific to the choice of $V$, and constructed so as to be the coefficient of the $1/|x|^{n+s}$ term in (4). In the remainder of our discussion, integrals are over $V$ unless otherwise indicated.

The contact terms in (4) are somewhat delicate and dependent on detail. When $-n<s<0$, the integral in (4) is convergent, and no contact terms are needed. One can easily check that $\Gamma_{V}(n+s)\to 0$ as $s\to 0$, so when $s=0$ the power law term goes away and we recover the obvious result

[TABLE]

For $s>0$, the integral in (4) diverges, and we need a more careful approach. A good first step is to understand (4) in terms of its action on a test function $\phi\colon V\to\mathbb{R}$:

[TABLE]

where $D^{s}$ is some linear map on functions $\phi(x)$. A suitable class of test functions are so-called Schwartz-Bruhat functions. When $\phi\colon\mathbb{Q}_{p^{n}}\to\mathbb{R}$, we require that $\phi$ is locally constant with compact support. For example, the characteristic function of the $p$-adic integers is a Schwartz-Bruhat function on $\mathbb{Q}_{p}$. When $\phi\colon\mathbb{R}^{n}\to\mathbb{R}$, the test functions are more appropriately called Schwartz functions, and their defining property is that they go to [math] faster than any power of $|x|$, as do all their derivatives. An example is a Gaussian. Both in the real and ultrametric cases, the Fourier transform $\hat{\phi}(k)$ of a Schwartz-Bruhat function is again a Schwartz-Bruhat function.

With (8) taken as the definition of $D^{s}$, our task is to find a representation of $D^{s}$ entirely in position space. In the ultrametric case for arbitrarily positive $s$, one finds

[TABLE]

This is the Vladimirov derivative. In the Archimedean case, the same expression (9) is valid for $0<s<2$. There is one more easy case to dispose of: even positive integer $s$ for Archimedean $V$. Then $\Gamma_{V}(n+s)=0$, which makes sense in (4) because the right hand should be purely distributional, on account of $|k|^{s}=(k^{2})^{s/2}$ being analytic in $k^{2}$. Explicitly,

[TABLE]

where $\square=\sum_{i=1}^{n}\partial_{x^{i}}^{2}$.

We are left with the task of defining $D^{s}$ for Archimedean $V$ and for $s>2$ but not an even integer. Heuristically, the contact terms in (4) are a sum of terms of the form $\square^{r}\delta^{n}(x)$, where $0\leq r<s/2$, with divergent coefficients. To state this more precisely, we write

[TABLE]

where a regulated integral

[TABLE]

is rendered finite (if possible) by allowing the subtraction from $G(x,y)$ of a finite sum of smooth functions of either of the following types:

1. I.

Pure powers: more precisely, any function whose $y$ dependence comes solely through a factor $|x-y|^{\alpha}$ where $\alpha$ is a real number. This is meant to include, through the case $\alpha=0$, functions which have no $y$ dependence. 2. II.

Higher partial waves: more precisely, any function of the form $Y(\widehat{y-x})g(|x-y|)$ where $Y(\hat{z})$ is a spherical harmonic on $S^{n-1}$ other than the $s$-wave.

Type I functions are never integrable, whereas type II functions may or may not be; so at best there is a unique choice of type I functions that will work, whereas many choices of type II functions are possible. An alternative approach, generalizing the principle value prescription, is to eschew modifications of the integrand and instead carry out $y$ integration in polar coordinates centered around $x$, as follows. One first performs the angular integrals. Then the radial integral is restricted to run from $l$ to $L$. One next allows the subtraction of an arbitrary finite sum of negative powers of $l$ and/or positive powers of $L$, chosen (if possible) so that the limits $l\to 0$ and $L\to\infty$, taken independently, lead to a finite result. Doing the angular integration first obviates the need for type II functions, while the ultraviolet and infrared cutoffs, $l$ and $L$, obviate the need for type I.222The alert reader may notice that the alternative approach using cutoffs is not quite equivalent to adjusting $G(x,y)$ by pure powers of $|x-y|$: For example, if $s$ is a positive even integer and $G(x,y)=|x-y|^{s}$, then we get a logarithmic divergence that would obviously be canceled using an appropriate type I function but cannot be cured using powers of $l$ and/or $L$ after a cutoff integration. Because we avoid even integer $s$ as well as functions $G(x,y)$ which grow as positive powers of large separation $|x-y|$, we do not need to specify a resolution to this inequivalence.

While the subtractions described can in principle cure either ultraviolet (UV) or infrared (IR) divergences, we will be interested only in applications where UV divergences matter: that is, divergences arising when $|x-y|\to 0$ (with $x$ held fixed). Type II subtractions are relatively innocuous because they follow automatically from performing angular integrations first; therefore we will use the notation $\int d^{n}y\dots$ to indicate a $y$ integration with type II subtractions which we usually omit to write explicitly.

Although we have stated our integration prescriptions in the abstract, it is easy to see how to apply them to (11) when $\phi$ is a Schwartz function. Consider the case $2<s<3$, and set $x=0$ for simplicity. Then (11) becomes

[TABLE]

The extra terms in square brackets on the right hand side of (13) evidently render the integral convergent near $y=0$ for $2<s<3$. The term linear in $y$ is clearly a type II function, and the term quadratic in $y$ is a sum of a type II function proportional to $y_{i_{1}}y_{i_{2}}-{y^{2}\over n}\delta_{i_{1}i_{2}}$ (a $d$-wave term) and a type I function proportional to $y^{2}$. If $3\leq s<4$, then we would need one additional term in the Taylor series expansion of $\phi$ around $y=0$, and this additional term is a type II function. In summary, for $2<s<4$, and omitting type II subtractions,

[TABLE]

Evidently, if $0<s<2$, a simpler subtraction scheme would work, resulting in (14) with the laplacian term omitted, in agreement with (9).

For general $s>0$ (other than positive even integers) and Archimedean $V$,

[TABLE]

where

[TABLE]

In principle, one may derive (15) by subtracting an appropriate number of terms in the Taylor series expansion of $\phi(y)$ and then finding appropriate type II subtractions to bring the result into the form (15).

A more efficient way to determine the coefficients $b_{r}$ is to start from (15) and Fourier transform:

[TABLE]

In the second equality of (17), we have partially carried out the $y$ integral in polar coordinates around the point $y=x$, introducing a radial variable $\tilde{y}=|x-y|$. In the third equality, we have carried out the angular $\theta$ integral and introduced a new radial variable, $\rho=2\pi|k|\tilde{y}$. The $\rho$ integral in the last line of (17) converges, provided $s$ is positive but not an even integer, and provided the coefficients $a_{r}$ are coefficients in the Taylor series expansion of the Bessel function around $\rho=0$. These coefficients $a_{r}$ are well known, and from them one can recover the expression (16) for the $b_{r}$.

3 Bi-local integrals

We are particularly interested in double integrals of the form

[TABLE]

where $s>0$ and $G(x,y)$ is piecewise constant if $V$ is ultrametric and smooth if $V$ is Archimedean. Unless otherwise noted, all double integrals over $x$ and $y$ will by taken over all of $V\times V$. In the ultrametric case, for any $s>0$, following (9) we define

[TABLE]

In the Archimedean case, we define

[TABLE]

by performing the $y$ integration first and allowing the subtraction of type I and type II functions to $G(x,y)$ in order to achieve a finite result (if possible). As in the previous section, type II subtractions are deemed relatively inconsequential, so even unprimed integration over $x$ and $y$ means to perform the $y$ integration first, allowing the subtraction of type II functions in order to achieve a finite result (if possible). Explicitly, for $s$ not a positive even integer,

[TABLE]

where the coefficients $b_{r}$ are as given in (16). We avoid positive even integer $s$ when $V$ is Archimedean because in this case we expect that our constructions will lead instead to purely local theories; also, precisely in this case, the subtleties pointed out in footnote 2 regarding logarithmic divergences come into play.

Our computational strategy will turn on converting bi-local position space integrals into Fourier space integrals. Let’s start with the simplest example of that calculation, valid for ultrametric $V$ and any $s>0$, and also for Archimedean $V$ and $0<s<2$. Let $\phi\colon V\to\mathbb{R}$ be a Schwartz-Bruhat function. Then

[TABLE]

The first step is actually the trickiest, because it is not clear from the rules of integration set forth following (20) that we are allowed to add a function like $\phi(x)^{2}-\phi(y)^{2}$ to the integrand. To justify this step, we denote $f(x)=\phi(x)^{2}$, and we argue that

[TABLE]

The second integral in (23) is the $k=0$ component of the Fourier transform of $D^{s}f(x)$. But this Fourier transform is $|k|^{s}\hat{f}(k)$, and since $s>0$ the $k=0$ component indeed vanishes.

Let’s now pursue the same computation for the Archimedean case with $2<s<4$. On one hand, using (21),

[TABLE]

On the other hand, using (14),

[TABLE]

In order to conclude

[TABLE]

we must therefore argue that the final integrals in (24) and (25) agree. Subtracting (25) from (24) and simplifying slightly with the definition $f(x)=\phi(x)^{2}$, we arrive at

[TABLE]

The first equality in (27) follows from (14), and the second is by the same argument used following (23). To summarize, for Archimedean $V$ and for $2<s<4$,

[TABLE]

$\Gamma_{V}(n+s)>0$ for $2<s<4$, and so without the $(\partial\phi)^{2}$ on the left hand side of (28) we would have a sign problem. The equality (26) can be checked in a similar manner for $s>4$. A key relation is

[TABLE]

Two take-away lessons are:

• •

When we write simple $|k|^{s}$ kinetic terms in momentum space, in position space we are combining non-local position space terms and local terms involving derivatives in a precisely tuned ratio.

• •

There is some freedom in the precise structure of the position space form, as exemplified by the equality of the last integrals in (24) and (25) due to a manipulation which is the non-local version of integration by parts.

4 The bi-local non-linear sigma model

We are now in a position to present the action for the bi-local non-linear sigma model. Let $M$ be a smooth $D$-dimensional manifold with a Riemannian metric $g_{ab}$, whose Riemann and Ricci tensors are

[TABLE]

Given any two points $X$ and $Y$ on $M$, let

[TABLE]

be the square of the shortest distance between $X$ and $Y$. Clearly, $Q(X,Y)$ is a smooth function of $X$ and $Y$, provided $X$ and $Y$ are not too far apart. For smooth functions $\phi\colon V\to M$ whose range is sufficiently localized, we consider the action functional

[TABLE]

where $\int^{\prime}$ indicates a regulated double integral of the type discussed around (19)-(21).333One may wonder whether the primed integral, as defined following (20), spoils coordinate invariance of the integrand. For instance, if $s$ is sufficiently large we may, in light of (21), be required to include a $\square_{y}Q$ term to the integrand, which if written only in terms of partial derivatives does not appear to be coordinate invariant. In fact, it is easy to convince oneself that, e.g., $\square_{y}Q$ can be constructed from covariant quantities: $\partial_{y^{a}}\partial_{y^{b}}Q=\frac{\partial^{2}Q}{\partial\phi^{i}\partial\phi^{j}}\frac{\partial\phi^{i}}{\partial y^{a}}\frac{\partial\phi^{j}}{\partial y^{b}}+\frac{\partial Q}{\partial\phi^{i}}\frac{\partial^{2}\phi^{i}}{\partial y^{a}\partial y^{b}}=\left(\nabla_{\phi^{i}}\frac{\partial Q}{\partial\phi^{j}}\right)\frac{\partial\phi^{i}}{\partial y^{a}}\frac{\partial\phi^{j}}{\partial y^{b}}+\frac{\partial Q}{\partial\phi^{i}}\frac{\partial\phi^{j}}{\partial y^{a}}\nabla_{\phi^{j}}\frac{\partial\phi^{i}}{\partial y^{b}}$. Note that this discussion requires us to avoid positive even integer $s$ when $x$ and $y$ are valued in $\mathbb{R}^{n}$ (as opposed to $\mathbb{Q}_{p^{n}}$). In the Archimedean setting, when $s>2$, there are derivative terms like $(\partial\phi)^{2}$ implicitly built into (32), with coefficients tuned so as to ensure convergence of the integral. The parameter $\mu$ has dimensions of energy so that we can regard $\phi$ and $Q(\phi(x),\phi(y))$ as dimensionless. The factor $\gamma$ is a loop-counting parameter: Classical effects are $O(\gamma^{-1})$, one-loop amplitudes are $O(\gamma^{0})$, two loop amplitudes are $O(\gamma)$, and so forth. In other words, $\gamma$ plays the role of $\hbar$.

A close cousin of the action (32) was considered in [7]:

[TABLE]

where now $x$ and $y$ are vertices of a graph and the sum is over undirected edges. The formula actually appears earlier in [6], though it was intended there to be considered on a square lattice, as a regulator for the local non-linear sigma model, rather than on the Bruhat-Tits tree as in [7].

We require the range of the maps $\phi$ to be sufficiently localized in order to ensure that we do not encounter any failures of smoothness in $Q(X,Y)$, and in order to ensure that we can use a single system of Riemann normal coordinates for $\phi$ throughout. One can now solve the geodesic equation perturbatively in the curvature and use that to evaluate $Q(X,Y)$ and expand the action (32),

[TABLE]

where

[TABLE]

and

[TABLE]

Here $g_{ab}$ and $R_{abcd}$ are evaluated at the origin of the Riemann normal coordinates, which we assume is at $X=0$. Often, the definition of Riemann normal coordinates includes the requirement $g_{ab}=\delta_{ab}$, but this is not necessary for our calculations, and we find it more convenient to retain explicit factors of $g_{ab}$ and the inverse metric $g^{ab}$. Put differently, we are choosing a coordinate system so that all geodesics passing through the origin are linear in the affine parameter:

[TABLE]

The ellipsis in (34) indicates higher order interactions, involving five or more powers of $\phi(x)$ and/or $\phi(y)$, as well as derivatives and powers of the curvature. We will consider up to six point interactions in section 10.

5 Loop divergences in momentum space

Our aim in this section is to introduce the main concepts related to divergent loop diagrams that we will need in our analysis of the non-local non-linear sigma model. As a warmup, we first exhibit the simplest manifestation of the non-renormalization theorem of the non-local quadratic kinetic term in the action (2), where $V=\mathbb{R}^{n}$ or $\mathbb{Q}_{p^{n}}$, with

[TABLE]

The purely cubic theory is unstable, but it serves our purpose because we are only interested in analyzing the behavior of the one-loop correction to the propagator. Using the diagram shown in figure 1a, we obtain the one-loop contribution to the quadratic part of the one-particle irreducible (1PI) effective action:

[TABLE]

We continue the convention of integrating over all of $V$ except as otherwise indicated. Let’s assume $n>2s$, so $I$ is UV divergent (and IR convergent). To regulate the divergence, we introduce a hard cutoff: $|\ell|\leq\Lambda$. If $V=\mathbb{R}^{n}$, then $\Lambda$ can be any positive real number. If $V=\mathbb{Q}_{p^{n}}$, then we will require that $\Lambda$ is an integer power of $p$.

The ultrametric case is easy to analyze, because when $|\ell|>|k|$ we have $|\ell|=|k-\ell|$ exactly. So, except in the compact region where $|\ell|\leq|k|$, the integrand has no $k$ dependence at all. Therefore, any UV divergences are entirely independent of $k$, and to evaluate them we can set $k=0$:

[TABLE]

The last equality comes from splitting the integration region into shells with fixed $|\ell|$; then the integral becomes a geometric sum. Because the divergent part of $I(\Lambda)$ has no $k$-dependence, the counterterm required to cancel it is proportional to $\int d^{n}k\,\hat{\phi}(-k)\hat{\phi}(k)=\int d^{n}x\,\phi(x)^{2}$. In other words, it is a mass term. This argument is easy to generalize to the statement that only purely local terms (powers of $\phi(x)$) can be radiatively generated starting from the action (2) over $V=\mathbb{Q}_{p^{n}}$. An essentially equivalent argument was made in a Wilsonian picture in [3].

The Archimedean case is more subtle because of the possibility of subleading divergences. A straightforward approach is to expand

[TABLE]

in powers of $k$. Terms with an odd number of powers of $k$ vanish by parity, leaving only terms analytic in $k^{2}$. Of these, only terms proportional to $k^{2r}$ with $r\leq{n\over 2}-s$ are UV divergent. In short, the divergent part of $I(\Lambda)$ is a polynomial in $k^{2}$ whose order is $\left\lfloor{n\over 2}-s\right\rfloor$. A divergent term proportional to $k^{2r}$ requires a counterterm proportional to $\int d^{n}x\,\phi(x)\square^{r}\phi(x)$. These are the radiatively generated derivative terms mentioned in section 1.

We should note a troublesome feature of the hard momentum cutoff for Archimedean theories: The coefficients one finds for sub-leading divergences depend on how one implements the cutoff. For example, it is easy to check that the coefficient of the $k^{2}$ term in $I(\Lambda)$ changes if instead of requiring $|\ell|\leq\Lambda$ we impose the more democratic condition $\left|\ell-{k\over 2}\right|\leq\Lambda$. However, the feature that we care about, namely the fact that the divergent terms have only polynomial dependence on $k^{2}$, doesn’t depend on the details of the cutoff. It is perhaps instructive to consider one other alternative, namely dimensional regularization, in which one first computes

[TABLE]

by continuing to a domain of $n$ in which the integral is convergent. (In the current example, $s<n<2s$ is such a domain.) The only divergences one then tracks are poles of the right hand side of (42) as a function of $n$. These occur precisely when ${n-2s\over 2}$ is a non-negative integer. It is characteristic of dimensional regularization that there is (at most) one divergent term for a given $n$, corresponding to a logarithmic divergence in the original integral.

The loop diagrams we will need to consider in our analysis of the non-local non-linear sigma model are simpler than (39) in one regard: The loop is a single propagator starting and ending at the same vertex. This matters because there is then only one internal momentum $\ell$, and imposing the hard cutoff $|\ell|\leq\Lambda$ is a privileged choice because it corresponds to integrating $\ell$ over an $O(n)$-invariant region. An example is the diagram shown in figure 1b, which is proportional to

[TABLE]

assuming that whatever vertex factor is needed to fully evaluate the diagram doesn’t depend on $\ell$. We also assume $n>s$ so that $I_{0}$ is UV divergent but IR convergent. We straightforwardly find

[TABLE]

There are obviously no subleading divergences in $I_{0}$.

For convenience we introduce

[TABLE]

We are interested in divergences proportional to $\log\Lambda$ that arise when $\epsilon=n-s=0$. As a technical trick to isolate these divergences, we make $\epsilon$ small and positive, and we look for divergences of the form $\Lambda^{\epsilon}/\epsilon$, which in the $\epsilon\to 0^{+}$ limit give rise to $\log\Lambda$ terms. To characterize this limit precisely, given $n_{0}>0$ and $\lambda\in\mathbb{R}$, we set $n=n_{0}+\lambda\epsilon$ and $s=n_{0}+(\lambda-1)\epsilon$ and then take the $\epsilon\to 0^{+}$ limit with $n_{0}$ and $\lambda$ held fixed. (Clearly then we are allowing non-integer $n$, in the spirit of [6].) For the most part, our final results are independent of $\lambda$. When $\epsilon$ is sufficiently small, we may replace (44) with

[TABLE]

If we lift the requirement that $\Lambda$ is an integer power of $p$ when $V=\mathbb{Q}_{p^{n}}$, then (46) is unaltered, because $I_{0}={\zeta_{p}(\epsilon)\over\zeta_{p}(n)}p^{\epsilon\lfloor\log_{p}\Lambda\rfloor}$, and $p^{\epsilon\lfloor\log_{p}\Lambda\rfloor}$ differs from $\Lambda$ at most by a factor of $p^{\epsilon}=1+O(\epsilon)$. The important point is that in the limit $\epsilon\to 0^{+}$, $I_{0}$ includes a logarithmic term $i_{0}\log\Lambda$, and isolating this term is our stated objective.

We will encounter one other loop integral:

[TABLE]

It comes from graphs similar to the one in (43), but with a vertex prefactor $|k-\ell|^{s}$. Using the same reasoning that led to (40), we see that when a hard cutoff $|\ell|\leq\Lambda$ is imposed, one obtains in the ultrametric case

[TABLE]

If $V=\mathbb{R}^{n}$ and $n$ is positive but not an even integer, then

[TABLE]

for some coefficients $c_{r}$. If we choose $n_{0}$ positive but not an even integer and fix any finite value of $\lambda$, then for sufficiently small $\epsilon$, (49) applies, and the least singular power of $\Lambda$ appearing in it is $\Lambda^{n-2\lfloor n/2\rfloor}$. As $\epsilon\to 0^{+}$, this power remains positive and finite, tending to $\Lambda^{n_{0}-2\lfloor n_{0}/2\rfloor}$. So there is no $\log\Lambda$ behavior, even in the $\epsilon\to 0^{+}$ limit. If instead we make $n_{0}$ a positive even integer, then by choosing the very particular value $\lambda=1$, so that $s=n_{0}$ exactly, we find (for sufficiently small $\epsilon>0$) that the least singular term in (49) is $c_{\lfloor n/2\rfloor}|k|^{s}{\Lambda^{\epsilon}\over\epsilon}$, which does contribute a $c_{\lfloor n/2\rfloor}|k|^{s}\log\Lambda$ divergence in the $\epsilon\to 0^{+}$ limit; moreover, in this case, by calculation, $c_{\lfloor n/2\rfloor}=i_{0}+O(\epsilon)$.444Although we have argued that the hard cutoff prescription $|\ell|\leq\Lambda$ is the natural one to use, it is interesting to note that if instead we impose $|k-\ell|\leq\Lambda$, then still $c_{\lfloor n/2\rfloor}=i_{0}+O(\epsilon)$ when $s=2\lfloor n/2\rfloor$ and $\epsilon$ is sufficiently small.

We can summarize the situation, both for the Archimedean and ultrametric cases, by stating that for $\epsilon=n-s$ positive but sufficiently small, then subject to the restriction that $n$ cannot be a positive even integer when $V=\mathbb{R}^{n}$,

[TABLE]

where

[TABLE]

The higher powers of $\Lambda$ in (50) are accompanied by non-negative integer powers of $k^{2}$, and they correspond to operators which remain relevant in the $\epsilon\to 0^{+}$ limit. The only $\log\Lambda$ behavior arising from $I_{2}(k)$, in the $\epsilon\to 0^{+}$ limits described above, is the $i_{2}|k|^{s}\log\Lambda$ term coming from the $i_{2}|k|^{s}{\Lambda^{\epsilon}\over\epsilon}$ term shown in (50). We are not concerned about $O(\epsilon)$ terms to $i_{0}$ and $i_{2}$ because they drop out of the $\log\Lambda$ behavior in the $\epsilon\to 0^{+}$ limit. In the following sections, therefore, we will drop $O(\epsilon)$ terms from (46) and (51), and we will evaluate $i_{0}$ and $i_{2}$ in terms of $n$ rather than $n_{0}$.

6 The propagator through one loop

To derive the tree-level propagator, we use an obvious generalization of (26) to multi-component scalar fields to rewrite the free action in momentum space:

[TABLE]

where we recall that $\epsilon=n-s$, and for notational convenience we have introduced555A point worthy of remark is that while $\hat{\gamma}$ and $\gamma$ have the same sign in the ultrametric case, and also in the Archimedean case for $0<s<2$, for $2<s<4$ they have the opposite sign. The integral in (52) is well-defined and positive, so to make our theory sensible we should always choose $\hat{\gamma}>0$. This means that $\gamma<0$ for $2<s<4$. As explained in (28) for a single real scalar, the regulated position space integral used to define the action (32) includes a $(\partial\phi)^{2}$ term that enters with the opposite sign of the non-local $\left[\phi(x)-\phi(y)\right]^{2}$ term, so positivity conditions are difficult to judge in position space.

[TABLE]

We immediately extract from (52) the propagator

[TABLE]

We are primarily interested in $\epsilon$ small, so that $G^{ab}(x)$ is nearly logarithmic.

Informally, we can understand the one-loop correction to the propagator as a contribution to the 1PI effective action coming from all possible Wick contractions of the two of the four factors of $\phi$ in $S_{\text{int}}$. The calculation is done most straightforwardly in momentum space, where we can express

[TABLE]

where $d^{4n}k=\prod_{i=1}^{4}d^{n}k_{i}$. As usual, derivative terms are implied in $\int^{\prime}$ in the Archimedean case when $s>2$. Symbolically, the Wick-contracted quartic action is

[TABLE]