The $q$-extension of iterated integrals and nested sums in quantum field theory

Technical Summary: The q-extension of iterated integrals and nested sums in quantum field theory

Problem Statement
Perturbative calculations in quantum field theory (QFT), particularly for zero- and single-scale quantities, yield results expressed in terms of special functions and numbers. These include polylogarithms, Nielsen integrals, harmonic polylogarithms, and their generalizations (cyclotomic, quadratic form-induced, and square-root valued). The series expansions of these functions around $x=0$ (or their Mellin transforms) correspond to classes of nested sums, such as harmonic sums, generalized harmonic sums, and sums involving central binomial coefficients.

While previous work established the "$\mu$-extension" of these function spaces, the authors identify a need to construct the q-extension (or q-analogue) of these specific function classes. This extension is motivated by physical applications involving q-deformed commutation relations in quantum mechanics and quantum field theory, specifically where creation and annihilation operators satisfy modified relations of the form $a_\lambda(k)a^\dagger_{\lambda'}(k') - q a^\dagger_{\lambda'}(k')a_\lambda(k) = g_{\lambda\lambda'}\delta^{(3)}(k-k')$. The paper notes that q-extensions define new function spaces of higher transcendental functions that are structurally distinct from $\mu$-extended functions.

Methodology
The authors employ a systematic algorithmic approach to construct q-extensions, relying on the following core principles:

1. Series Expansion Basis: The starting point is the known closed-form solution $f[n]$ for the expansion coefficients of a function $F(x) = \sum_{n=0}^\infty f[n]x^n$. The q-extension $F(x; q)$ is defined by the series $\sum_{n=0}^\infty f[n; q]x^n$, where the coefficients $f[n]$ are replaced by their q-analogues $f[n; q]$.
2. q-Deformation Rules: The construction utilizes standard asymmetric q-definitions:
   • q-integers: $\{n\}_q = \frac{1-q^n}{1-q}$.
   • q-derivatives (Jackson derivative): $D_q^x f[x] = \frac{f(x) - f(qx)}{(1-q)x}$.
   • q-factorials and q-binomials.
   • Crucially, powers $c^n$ ($c \in \mathbb{C}$) are not q-extended, while sum-product structures are.
3. Algorithmic Derivation:
   • Nested Sums: The authors derive recurrence relations for the q-extended nested sums. These relations are finite-order linear difference equations with polynomial coefficients in $q^n$. They utilize computer algebra packages (specifically qFunctions, QREGuess, Sigma, and QRE2DE) to guess recurrences from expansion coefficients and to verify identities.
   • Iterated Integrals: The q-extensions of iterated integrals are constructed by replacing the sum-product structures in the expansion coefficients with their q-extensions. The authors derive the defining q-differential equations and q-shift relations (difference equations in the argument $x$) for these functions.
   • Method Name: The authors term this series expansion approach around $x=0$ the "q-Frobenius method," drawing an analogy to the Frobenius method used in standard QFT calculations.
4. Algebraic Structure: The paper investigates the algebraic properties of these extensions, specifically the shuffle algebra for iterated integrals and the quasi-shuffle algebra for nested sums, verifying that these structures are preserved under q-deformation.

Key Contributions and Results
The paper provides explicit constructions and defining equations for q-extensions across several classes of functions:

• Nested Sums:

  • Derivation of q-extensions for harmonic sums, generalized harmonic sums, and sums containing central binomials.
  • Presentation of specific recurrence relations for single and multiple sums (e.g., $S_{2,1}(n; q)$, $S_{3,-2}(n; q)$).
  • Demonstration that these sums satisfy finite-order q-difference equations.

• Iterated Integrals:

  • Polylogarithms: Closed-form q-extensions are derived, expressed as q-hypergeometric functions ($_{k+1}\phi_k$). The paper defines q-zeta values $\zeta_k(q)$ as limits $x \to 1$.
  • Nielsen Integrals: q-extensions are constructed, and their associated q-differential equations and q-shift relations are presented for weights up to $w=5$.
  • Harmonic Polylogarithms (HPLs): The authors extend HPLs over the alphabet $\{1/x, 1/(1-x), 1/(1+x)\}$. They provide recurrence relations and differential equations for specific cases like $H_{-1,0,1}(x; q)$.
  • Generalized and Cyclotomic HPLs: Algorithmic steps are detailed for generalized harmonic polylogarithms (Kummer-Poincaré integrals) and cyclotomic harmonic polylogarithms.
  • Quadratic Forms and Square Roots: The paper addresses the most complex cases, including iterated integrals implied by quadratic forms and those containing square-root valued letters (which involve central binomial coefficients). Despite the complexity, the authors show these also obey first-order factorizing q-differential and q-difference equations.

• Algebraic Relations:

  • The paper verifies that the shuffle product for iterated integrals and the quasi-shuffle product for nested sums hold for the q-extended quantities.
  • Explicit examples of quasi-shuffle relations (e.g., for $S_{1,-1}$ and $S_{1,-1,-2}$) are provided and verified using the Sigma package.

Significance and Claims
The paper claims to provide a comprehensive framework for the q-extension of the specific special functions arising in perturbative QFT calculations. Its significance lies in:

1. Completeness: It covers all currently known classes of iterated integrals and nested sums that obey first-order factorizing differential/difference equations in single-scale QFT, extending them to the q-deformed case.
2. Structural Distinction: It highlights that q-extensions create new function spaces of higher transcendental functions, distinct from the $\mu$-extensions studied previously.
3. Algorithmic Utility: For complex alphabets where closed forms are not immediately obvious, the paper provides the algorithmic steps (recurrence generation, differential equation derivation) to obtain the q-extension for individual cases.
4. Physical Relevance: The constructed functions are posited to play a role in perturbative calculations involving q-deformed commutation relations, such as those found in deformed oscillator algebras, non-commutative spacetime approaches, and quantum group symmetries.

The authors conclude that while the q-extended functions are more complex (higher transcendental), they remain uniquely defined by their q-recurrence relations, q-differential equations, and q-shift relations, allowing for their systematic study and potential application in deformed quantum field theories.