vakint

Library for the computation of (single-scale) vacuum integrals in High Energy Physics.

The code will perform the matching of user input onto known topologies, and then perform their computation either:

• analytically, if the topology is known, using tensor reduction and parametric integration by parts. To this end, it uses a combination of Symbolica and FORM scripts, and FMFT for the four-loop case.

• numerically, if the topology is not known, using the sector decomposition algorithm implemented in pySecDec.

Python usage

Vakint is also part of the symbolica-community Python module where vaccuum graphs can be evaluated directly using the Python API exposed in that module.

Rust usage

Checkout the examples directory for a few examples of how to use this library. For example:

use vakint::{vakint_parse, Vakint, VakintExpression, VakintSettings};

fn main() {
    let vakint = Vakint::new(Some(VakintSettings {
        allow_unknown_integrals: false,
        ..VakintSettings::default()
    }))
    .unwrap();

    //println!("Supported topologies:\n{}", vakint.topologies);

    let input = vakint_parse!(
        "(k(11,2)*k(11,2)+k(11,77)*k(22,77)+k(22,33)*p(42,33))*topo(\
                    prop(9,edge(7,10),k(11),mUVsqA,1)*\
                    prop(33,edge(7,10),k(22),mUVsqA,2)*\
                    prop(55,edge(7,10),k(11)+k(22),mUVsqA,1)\
                )+\
                (2*k(33,2)*k(33,2)+17*k(44,33)*p(17,33))*topo(\
                    prop(7,edge(9,21),k(33),mUVsqB,1)*\
                    prop(13,edge(9,21),k(44),mUVsqB,2)*\
                    prop(17,edge(9,21),k(33)+k(44),mUVsqB,1)\
                )"
    )
    .unwrap();

    let output = vakint.to_canonical(input.as_view(), true).unwrap();

    println!(
        "\nInput:\n\n{}\n\nhas been matched to\n\n{}\n",
        VakintExpression::try_from(input).unwrap(),
        VakintExpression::try_from(output).unwrap()
    );
}

yields:

When carrying a complete evaluation, the workflow is as follows:

use ahash::HashMap;
use vakint::{vakint_parse, Vakint, VakintExpression, VakintSettings};

fn main() {
    let vakint = Vakint::new(Some(VakintSettings {
        allow_unknown_integrals: false,
        use_dot_product_notation: true,
        integral_normalization_factor: vakint::LoopNormalizationFactor::Custom("1".into()),
        run_time_decimal_precision: 16,
        ..VakintSettings::default()
    }))
    .unwrap();

    let mut integral = vakint_parse!(
        "(k(3,11)*k(3,22)+k(3,77)*p(8,77))*topo(\
        prop(9,edge(66,66),k(3),MUVsq,1)\
    )"
    )
    .unwrap();
    println!(
        "\nInput integral:\n{}\n",
        VakintExpression::try_from(integral.clone()).unwrap()
    );

    integral = vakint.to_canonical(integral.as_view(), true).unwrap();
    println!(
        "Matched integral:\n{}\n",
        VakintExpression::try_from(integral.clone()).unwrap()
    );

    integral = vakint.tensor_reduce(integral.as_view()).unwrap();
    println!(
        "Tensor reduced integral:\n{}\n",
        VakintExpression::try_from(integral.clone()).unwrap()
    );

    integral = vakint.evaluate_integral(integral.as_view()).unwrap();
    println!("Evaluated integral:\n{}\n", integral);

    let mut params = HashMap::default();
    params.insert("MUVsq".into(), vakint.settings.real_to_prec("1.0"));
    params.insert("mursq".into(), vakint.settings.real_to_prec("1.0"));

    let numerical_partial_eval =
        Vakint::partial_numerical_evaluation(&vakint.settings, integral.as_view(), &params, None);
    println!("Partial eval:\n{}\n", numerical_partial_eval);

    params.insert("g(11,22)".into(), vakint.settings.real_to_prec("1.0"));
    let numerical_full_eval = Vakint::full_numerical_evaluation_without_error(
        &vakint.settings,
        integral.as_view(),
        &params,
        None,
    )
    .unwrap();
    println!(
        "Full eval (metric substituted with 1):\n{}\n",
        numerical_full_eval
    );
}

yielding:

Disclaimer about Symbolica namespace

In the current version of Vakint, the interface to FORM is such that the namespace of all Symbolica symbols from your input expression will be overwritten to be the Vakint one, i.e. vk::<symbol>. For now, the user sensitive to namespaces is responsible for renaming the symbols in their input expression to avoid conflicts, and possibly remap them afterwards to their original namespaces. For simple usage, it is recommended to parse the input expression with the macro vakint_parse! which will automatically assign vk as the namespace of all your symbols with unspecified namespaces.