Formal definition of $d$-dimensional integral for complex dimension $d\in\mathbb{C}$

Sketched existence proof:

1. Ref. 1 assumes Euclidean signature, i.e. that the Wick rotation has already been performed.

2. We would like to define the integral $$I_d[f]~=~\int_{\mathbb{R}^d}\!d^dp~f(p)\tag{i}$$ for complex dimension $d\in\mathbb{C}$. Ref. 1 assumes

   • $\mathbb{C}$-linearity (A1),

   • translation invariance (A2),

   • homogeneity (A3).

3. Using analytic continuation in $d\in\mathbb{C}$, it is enough to define $I_d[f]$ for dimension $d\in ]c_f,\infty[$ in an open interval$^1$ $]c_f,\infty[$ for a sufficiently large number $c_f\geq 0$.

4. Now let us try to extract a rigorous existence proof from the heuristic arguments of Ref. 1. Around eqs. (A14-15) Ref. 1 seems to suggest that if the integrand is of the form $$p~\mapsto~ f(p^2, p\cdot q_1,\ldots, p\cdot q_n)\tag{ii}$$ with $n$ external $d$-momenta$^2$ $q_1,\ldots ,q_n\in \mathbb{R}^d$, where the function $f\in{\cal S}(\mathbb{R}^{n+1})$ belongs to a Schwartz space, we choose the dimension $d>n$, and split $$p~=~\underbrace{p_{\parallel}}_{\in~V}+\underbrace{p_{\perp}}_{\in~V^{\perp}}, \qquad p_{\parallel}~=~\sum_{i=1}^np_i q_i~\in~V~:=~{\rm span}_{\mathbb{R}}(q_1,\ldots ,q_n),\tag{iii}$$ then the integral is factorized as $$I_d[f]~=~\int_{\mathbb{R}^n}\!d^np_{\parallel} \int_{\mathbb{R}^{d-n}}\!d^{d-n}p_{\perp}~f(p_{\perp}^2+p_{\parallel}^2,p_{\parallel}\cdot q_1,\ldots,p_{\parallel}\cdot q_n).\tag{iv}$$

5. With point 4 in mind, in practice, it is therefore enough to define the integral $I_d[f]$ for spherically symmetric integrands $$p~\mapsto~ f(p^2),\tag{v}$$ where the function $f\in{\cal S}(\mathbb{R})$.

   • For positive integer dimension $d\in\mathbb{N}$ we use spherical coordinates $$I_d[f]~=~{\rm Vol}(\mathbb{S}^{d-1})\int_{\mathbb{R}_+} \!p^{d-1}dp ~f(p^2) ~=~\frac{2\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}\int_{\mathbb{R}_+} \!p^{d-1}dp ~f(p^2),\tag{vi}$$ cf. eqs. (A9-10).

   • For positive dimension $d>0$, we naturally generalize (vi) to define
     $$I_d[f]~:=~\frac{2\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}\int_{\mathbb{R}_+} \!p^{d-1}dp ~f(p^2).\tag{vii}$$

6. The integral $I_d[f]$ is essentially uniquely defined up to multiplication with an arbitrary over-all holomorphic normalization function $\mathbb{C}\ni d\mapsto {\cal N}(d)\in \mathbb{C}$ that satisfies $$ {\cal N}(\mathbb{N})~=~1, \tag{viii}$$ whose effect may be compensated by counterterms, cf. the discussion on p. 2924 in Ref. 1.

References:

1. K.G. Wilson, QFT Models in Less Than 4D, Phys. Rev. D7 (1973) 2911; App. A.

$^1$ NB: Be aware that this is different from traditional dimensional regularization, where the integrand is a rational/meromorphic function and the open interval is on the form $]-\infty, c_f[$ for a sufficiently small number $c_f\in\mathbb{R}$. In contrast, Ref. 1 seems to basically focus on integrands that are Schwartz functions.

$^2$ It is unclear what a $d$-momentum $q_i\in\mathbb{R}^d$ exactly means for a non-integer $d$. However, we only use that the $d$-momenta $q_1,\ldots ,q_n$ is a basis for a finite-dimensional $\mathbb{R}$-vector space $V$, and that the integral (iv) effectively only depends on $q_1,\ldots ,q_n$ through the scalar products $q_i\cdot q_j\in\mathbb{R}$.