Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.

💡 Research Summary

The paper introduces a differential‑reduction algorithm that systematically rewrites generalized hypergeometric functions (pF{p-1}) with arbitrary complex parameters into linear combinations of hypergeometric functions whose parameters differ from the original ones by integer amounts. The authors motivate the method by the observation that many multiloop Feynman integrals can be expressed in terms of a single variable (z) (e.g. a ratio of external momentum squared to a mass squared) and that the resulting integrals are naturally represented by hypergeometric functions of the type ({L+1}F{L}), where (L) is the loop order.

The core of the technique relies on two families of identities satisfied by hypergeometric functions: (i) contiguous relations, which connect functions whose parameters differ by ±1, and (ii) differential relations that follow from the defining differential equation of (pF{p-1}). By combining these, the authors construct shift operators (D_{a_i}^{\pm}) and (D_{b_j}^{\pm}) that raise or lower the upper and lower parameters by one unit. Applying a sequence of such operators reduces any set of parameters to a “standard” set where all differences are integers. In the language of Feynman diagram reduction, a diagram is called “reducible” if the sequence of shifts terminates in a function with only integer‑shifted parameters; otherwise it is “irreducible”.

The paper focuses on the one‑variable case because most practical multiloop integrals depend on a single kinematic invariant after appropriate parametrisation. The authors demonstrate the algorithm on two representative families of diagrams. The first example is a two‑point, two‑loop self‑energy diagram whose Mellin‑Barnes representation leads to a (_2F_1) function. After applying the shift operators, the function collapses to a (_1F_0) (essentially a rational function) multiplied by a polynomial coefficient, showing that only one master integral is needed—exactly the same conclusion reached by traditional integration‑by‑parts (IBP) reduction. The second example is a three‑point, three‑loop vertex diagram that yields a (_3F_2) function. Differential reduction brings it to a linear combination of two (_2F_1) functions, confirming that the IBP analysis would produce three master integrals but only two are linearly independent.

A key result is the identification of a precise correspondence between the number of master integrals obtained by differential reduction and the number obtained by IBP. The authors prove that the integer‑shift criterion for hypergeometric reducibility is equivalent to the existence of linear relations among the IBP‑generated integrals. Consequently, the differential‑reduction method provides a mathematically transparent way to predict the minimal set of masters without solving large systems of linear equations.

The discussion also addresses the algorithm’s limitations and possible extensions. Currently the method is implemented for single‑variable hypergeometric functions; however, the authors outline how the same shift‑operator formalism can be generalized to multivariate hypergeometric functions (e.g. Appell or Lauricella functions) that appear in more intricate topologies. They suggest that an automated implementation—integrating the shift operators into symbolic algebra systems—could dramatically speed up the reduction of high‑loop, multi‑mass diagrams.

In conclusion, the paper establishes differential reduction as a powerful complementary tool to IBP. By translating the problem of Feynman integral reduction into the language of hypergeometric function theory, it offers a clear criterion for reducibility, reduces the computational overhead associated with large IBP systems, and opens the door to systematic treatment of even more complex multiloop integrals.