Abstract
We discuss properties of the є-expansion in d = 4 − є dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in є in terms of the beta function coefficients. The є-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the є-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in є in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar O (1) symmetric model to O(є7).
Highlights
We use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar O(1) symmetric model to O( 7)
Using certain properties of Lagrange inversion we argue that the -expansion of the Wilson-Fisher fixed point coupling well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra
In this work we have used Lagrange inversion to derive the exact form of the Wilson-Fisher fixed point coupling and critical exponents in the -expansion in terms of the coefficients of the appropriate renormalization group functions
Summary
2i−2 i−1 is known as the (i − 1)’th Catalan number and is the leftmost column in the number triangle. To each coefficient gi−1 of the Wilson-Fisher fixed point coupling we associate the associahedron Ki. We will present the general result for any i and motivate it by looking at a number of examples. The sum is over all faces of the associahedron In this way the facial structure of the associahedra controls the -expansion of the coupling at the Wilson-Fisher fixed point. What we have arrived at is a simple and stunningly beautiful closed form expression for the coupling at the Wilson-fisher fixed point to all orders in. It is dictated by the geometry of the associahedra and at each order in is uniquely related to its facial structure i g∗( ) = gi−1 i=1.
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