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Update N3LO (2512.10783) #484

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3ca56fc
update splittings python
giacomomagni Dec 18, 2025
f9516dc
minor fix
giacomomagni Dec 18, 2025
0d74f83
update rust
giacomomagni Dec 18, 2025
3ab54ee
extend tests and fix minor signs
giacomomagni Dec 18, 2025
baf38bd
lint
giacomomagni Dec 18, 2025
b200b26
Update tests/ekore/anomalous_dimensions/unpolarized/space_like/test_a…
giacomomagni Dec 19, 2025
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2 changes: 1 addition & 1 deletion crates/ekore/src/anomalous_dimensions/unpolarized/spacelike/as4.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -44,7 +44,7 @@ mod tests {
let NF = 5;
let mut c = Cache::new(cmplx!(2., 0.));
// Numbers are coming from the python implementation.
let quark_refs: [f64; 3] = [0.053441, 0.225674, -0.118792];
let quark_refs: [f64; 3] = [0.040173, 0.260616, -0.18027];
let gluon_refs: [f64; 3] = [-0.0300842, 0.283004, -0.343172];
for imod in [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2]] {
let g_singlet = gamma_singlet(&mut c, NF, imod);
Expand Down
36 changes: 31 additions & 5 deletions crates/ekore/src/anomalous_dimensions/unpolarized/spacelike/as4/ggg.rs
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Expand Up @@ -10,9 +10,10 @@ use crate::harmonics::log_functions::{

/// Compute the singlet gluon-to-gluon anomalous dimension.
///
/// The routine is taken from [\[Falcioni:2024qpd\]][crate::bib::Falcioni2024qpd].
/// The routine is taken from [\[Falcioni:2024qpd\]][crate::bib::Falcioni2024qpd],
/// with the update for :math:`n_f=6` from [\[Falcioni:2025hfz\]][crate::bib::Falcioni:2025hfz].
///
/// These are approximations for fixed `nf` = 3, 4 and 5 based on the
/// These are approximations for fixed `nf` = 3, 4, 5 and 6 based on the
/// first 10 even moments together with small-x/large-x constraints.
/// The two sets providing the error estimate are called via `variation = 1`
/// and `variation = 2`. Any other value of `variation` invokes their average.
Expand Down Expand Up @@ -70,7 +71,7 @@ pub fn gamma_gg(c: &mut Cache, nf: u8, variation: u8) -> Complex<f64> {
+ x1L4cff * lm14m1(c)
+ x1L3cff * lm13m1(c);

// The selected approximations for nf = 3, 4, 5
// The selected approximations for nf = 3, 4, 5, 6
let P3ggApp1: Complex<f64>;
let P3ggApp2: Complex<f64>;
if nf == 3 {
Expand Down Expand Up @@ -147,8 +148,33 @@ pub fn gamma_gg(c: &mut Cache, nf: u8, variation: u8) -> Complex<f64> {
+ 2285.0 * lm11m1(c)
+ 19295.0 * lm12m1(c)
- 13719.0 * lm12m2(c);
} else if nf == 6 {
P3ggApp1 = P3gg01
- 476018.0 * (-(1. / (-1. + n).powu(2)) + 1. / n.powu(2))
- 469289.0 * 1. / ((-1. + n) * n)
+ 2049351.0 * (1. / (n + n.powu(2)))
- 1589000.0 * (1. / (2. + 3. * n + n.powu(2)))
+ 3185549.0 * (-(1. / n.powu(2)) + 1. / (1. + n).powu(2))
+ 1994521.0 * 2. / n.powu(3)
+ 527723.0 * (-(6. / n.powu(4)))
- 340674.0 * lm11m1(c)
+ 22460.0 * lm12m1(c)
- 394556.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3ggApp2 = P3gg01
- 709863.0 * (-(1. / (-1. + n).powu(2)) + 1. / n.powu(2))
- 2134347.0 * 1. / ((-1. + n) * n)
+ 1605315.0 * (1. / (2. + 3. * n + n.powu(2)))
+ 360743.0
* ((12. + 9. * n + n.powu(2))
/ (6. * n + 11. * n.powu(2) + 6. * n.powu(3) + n.powu(4)))
- 2426250.0 * (-(1. / n.powu(2)) + 1. / (1. + n).powu(2))
+ 230631.0 * 2. / n.powu(3)
- 185804.0 * (-(6. / n.powu(4)))
- 7992.9 * lm11m1(c)
+ 15918.0 * lm12m1(c)
- 32771.0 * lm11m2(c);
} else {
panic!("nf=6 is not available at N3LO");
panic!("Select nf=3,..,6 for N3LO evolution");
}

// We return (for now) one of the two error-band representatives
Expand Down Expand Up @@ -196,7 +222,7 @@ mod tests {
mom_list[(N - 2) / 2]
}
for variation in [0, 1, 2] {
for NF in [3, 4, 5] {
for NF in [3, 4, 5, 6] {
for N in [2.0, 4.0, 6.0, 8.0, 10., 12., 14., 16., 18., 20.] {
let mut c = Cache::new(cmplx!(N, 0.));
let test_value = gamma_gg(&mut c, NF, variation);
Expand Down
169 changes: 97 additions & 72 deletions crates/ekore/src/anomalous_dimensions/unpolarized/spacelike/as4/ggq.rs
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Expand Up @@ -8,9 +8,11 @@ use crate::harmonics::log_functions::{lm11, lm12, lm12m1, lm13, lm14, lm14m1, lm

/// Compute the singlet quark-to-gluon anomalous dimension.
///
/// The routine is taken from [\[Falcioni:2024xyt\]][crate::bib::Falcioni2024xyt].
/// The routine is taken from [\[Falcioni:2025hfz\]][crate::bib::Falcioni2025hfz].
/// A previous version was given in [\[Falcioni:2024xyt\]][crate::bib::Falcioni2024xyt],
/// while a version based only on the lowest 10 moments was given in [\[Moch:2023tdj\]][crate::bib::Moch2023tdj].
///
/// These are approximations for fixed `nf` = 3, 4 and 5 based on the
/// These are approximations for fixed `nf` = 3, 4, 5 and 6 based on the
/// first 10 even moments together with small-x/large-x constraints.
/// The two sets providing the error estimate are called via `variation = 1`
/// and `variation = 2`. Any other value of `variation` invokes their average.
Expand Down Expand Up @@ -48,80 +50,103 @@ pub fn gamma_gq(c: &mut Cache, nf: u8, variation: u8) -> Complex<f64> {
+ y1L4cff * lm14m1(c)
+ y1L5cff * lm15m1(c);

// The selected approximations for nf = 3, 4, 5
// The selected approximations for nf = 3, 4, 5, 6
let P3gqApp1: Complex<f64>;
let P3gqApp2: Complex<f64>;
if nf == 3 {
P3gqApp1 = P3GQ01 + 6.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 744384.0 * 1. / ((-1. + n) * n)
+ 2453640.0 * 1. / n
- 1540404.0 * (2. / (1. + n) + 1. / (2. + n))
+ 1933026.0 * -1. / n.powu(2)
+ 1142069.0 * 2. / n.powu(3)
+ 162196.0 * -6. / n.powu(4)
- 2172.1 * lm13(c)
- 93264.1 * lm12(c)
- 786973.0 * lm11(c)
+ 875383.0 * lm12m1(c);
P3gqApp2 =
P3GQ01 + 3.0 * bfkl1 * (-(1. / (-1. + n).powu(2))) + 142414.0 * 1. / ((-1. + n) * n)
- 326525.0 * 1. / n
+ 2159787.0 * ((3. + n) / (2. + 3. * n + n.powu(2)))
- 289064.0 * -1. / n.powu(2)
- 176358.0 * 2. / n.powu(3)
+ 156541.0 * -6. / n.powu(4)
+ 9016.5 * lm13(c)
+ 136063.0 * lm12(c)
+ 829482.0 * lm11(c)
- 2359050.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp1 = P3GQ01 + 3.5 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 27891.0 * 1. / ((-1. + n) * n)
- 309124.0 * 1. / n
+ 1056866.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 124735.0 * -1. / n.powu(2)
- 16246.0 * 2. / n.powu(3)
+ 131175.0 * -6. / n.powu(4)
+ 4970.1 * lm13(c)
+ 60041.0 * lm12(c)
+ 343181.0 * lm11(c)
- 958330.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp2 = P3GQ01 + 7.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 1139334.0 * 1. / ((-1. + n) * n)
+ 143008.0 * 1. / n
- 290390.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 659492.0 * -1. / n.powu(2)
+ 303685.0 * 2. / n.powu(3)
- 81867.0 * -6. / n.powu(4)
+ 1811.8 * lm13(c)
- 465.9 * lm12(c)
- 51206.0 * lm11(c)
+ 274249.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
} else if nf == 4 {
P3gqApp1 = P3GQ01 + 6.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 743535.0 * 1. / ((-1. + n) * n)
+ 2125286.0 * 1. / n
- 1332472.0 * (2. / (1. + n) + 1. / (2. + n))
+ 1631173.0 * -1. / n.powu(2)
+ 1015255.0 * 2. / n.powu(3)
+ 142612.0 * -6. / n.powu(4)
- 1910.4 * lm13(c)
- 80851.0 * lm12(c)
- 680219.0 * lm11(c)
+ 752733.0 * lm12m1(c);
P3gqApp2 =
P3GQ01 + 3.0 * bfkl1 * (-(1. / (-1. + n).powu(2))) + 160568.0 * 1. / ((-1. + n) * n)
- 361207.0 * 1. / n
+ 2048948.0 * ((3. + n) / (2. + 3. * n + n.powu(2)))
- 245963.0 * -1. / n.powu(2)
- 171312.0 * 2. / n.powu(3)
+ 163099.0 * -6. / n.powu(4)
+ 8132.2 * lm13(c)
+ 124425.0 * lm12(c)
+ 762435.0 * lm11(c)
- 2193335.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp1 = P3GQ01 + 3.5 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 8302.8 * 1. / ((-1. + n) * n)
- 347706.0 * 1. / n
+ 1105306.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 127650.0 * -1. / n.powu(2)
- 29728.0 * 2. / n.powu(3)
+ 137537.0 * -6. / n.powu(4)
+ 4658.1 * lm13(c)
+ 59205.0 * lm12(c)
+ 345513.0 * lm11(c)
- 995120.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp2 = P3GQ01 + 7.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 1129822.0 * 1. / ((-1. + n) * n)
+ 108527.0 * 1. / n
- 254166.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 667254.0 * -1. / n.powu(2)
+ 293099.0 * 2. / n.powu(3)
- 77437.0 * -6. / n.powu(4)
+ 1471.3 * lm13(c)
- 1850.3 * lm12(c)
- 52451.0 * lm11(c)
+ 248634.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
} else if nf == 5 {
P3gqApp1 = P3GQ01 + 6.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 785864.0 * 1. / ((-1. + n) * n)
+ 285034.0 * 1. / n
- 131648.0 * (2. / (1. + n) + 1. / (2. + n))
- 162840.0 * -1. / n.powu(2)
+ 321220.0 * 2. / n.powu(3)
+ 12688.0 * -6. / n.powu(4)
+ 1423.4 * lm13(c)
+ 1278.9 * lm12(c)
- 30919.9 * lm11(c)
+ 47588.0 * lm12m1(c);
P3gqApp2 =
P3GQ01 + 3.0 * bfkl1 * (-(1. / (-1. + n).powu(2))) + 177094.0 * 1. / ((-1. + n) * n)
- 470694.0 * 1. / n
+ 1348823.0 * ((3. + n) / (2. + 3. * n + n.powu(2)))
- 52985.0 * -1. / n.powu(2)
- 87354.0 * 2. / n.powu(3)
+ 176885.0 * -6. / n.powu(4)
+ 4748.8 * lm13(c)
+ 65811.9 * lm12(c)
+ 396390.0 * lm11(c)
- 1190212.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp1 =
P3GQ01 + 3.5 * bfkl1 * (-(1. / (-1. + n).powu(2))) + 14035.0 * 1. / ((-1. + n) * n)
- 384003.0 * 1. / n
+ 1152711.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 126346.0 * -1. / n.powu(2)
- 42967.0 * 2. / n.powu(3)
+ 144270.0 * -6. / n.powu(4)
+ 4385.5 * lm13(c)
+ 58688.0 * lm12(c)
+ 348988.0 * lm11(c)
- 1031165.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp2 = P3GQ01 + 7.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 1117561.0 * 1. / ((-1. + n) * n)
+ 76329.0 * 1. / n
- 218973.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 670799.0 * -1. / n.powu(2)
+ 282763.0 * 2. / n.powu(3)
- 72633.0 * -6. / n.powu(4)
+ 1170.0 * lm13(c)
- 2915.5 * lm12(c)
- 52548.0 * lm11(c)
+ 223771.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
} else if nf == 6 {
P3gqApp1 =
P3GQ01 + 3.5 * bfkl1 * (-(1. / (-1. + n).powu(2))) + 39203.0 * 1. / ((-1. + n) * n)
- 417914.0 * 1. / n
+ 1199042.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 120750.0 * -1. / n.powu(2)
- 55941.0 * 2. / n.powu(3)
+ 151383.0 * -6. / n.powu(4)
+ 4149.2 * lm13(c)
+ 58466.0 * lm12(c)
+ 353589.0 * lm11(c)
- 1066510.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
P3gqApp2 = P3GQ01 + 7.0 * bfkl1 * (-(1. / (-1. + n).powu(2)))
- 1102470.0 * 1. / ((-1. + n) * n)
+ 46517.0 * 1. / n
- 184858.0 * (3. + n) / (2. + 3. * n + n.powu(2))
- 670056.0 * -1. / n.powu(2)
+ 272689.0 * 2. / n.powu(3)
- 67453.0 * -6. / n.powu(4)
+ 905.0 * lm13(c)
- 3686.2 * lm12(c)
- 51523.0 * lm11(c)
+ 199594.0 * (S1 - n * (ZETA2 - S2)) / n.powu(2);
} else {
panic!("nf=6 is not available at N3LO");
panic!("Select nf = 3..6 for N3LO evolution");
}

// We return (for now) one of the two error-band boundaries
Expand Down Expand Up @@ -172,15 +197,15 @@ mod tests {
mom_list[(N - 2) / 2]
}
for variation in [0, 1, 2] {
for NF in [3, 4, 5] {
for NF in [3, 4, 5, 6] {
for N in [2.0, 4.0, 6.0, 8.0, 10., 12., 14., 16., 18., 20.] {
let mut c = Cache::new(cmplx!(N, 0.));
let test_value = gamma_gq(&mut c, NF, variation);
assert_approx_eq_cmplx!(
f64,
test_value,
cmplx!(gq3_moment(N as usize, NF as f64), 0.),
rel = 4e-4
rel = if NF != 6 { 4e-4 } else { 2e-3 }
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@felixhekhorn felixhekhorn Dec 19, 2025

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both gamma_gq and gq3_moment are coming from Falcioni et al., right? should we tell them that they got worse, i.e. they don't follow their own boundary condition? I know we are talking 1e4 here, so it is stupid - it may still indicate a possible typo ...

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@giacomomagni giacomomagni Dec 19, 2025

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eh yes, I saw your email, but it could well be that the typo is from my side!! I'll try to double check the changes nex week.

);
}
}
Expand Down
32 changes: 26 additions & 6 deletions crates/ekore/src/anomalous_dimensions/unpolarized/spacelike/as4/gps.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -7,9 +7,10 @@ use crate::harmonics::log_functions::{lm11m1, lm12m1, lm12m2, lm13m1, lm13m2, lm

/// Compute the pure-singlet quark-to-quark anomalous dimension.
///
/// The routine is taken from [\[Falcioni:2023luc\]][crate::bib::Falcioni2023luc].
/// The routine is taken from [\[Falcioni:2023luc\]][crate::bib::Falcioni2023luc],
/// with the update for :math:`n_f=6` from [\[Falcioni:2025hfz\]][crate::bib::Falcioni:2025hfz].
///
/// These are approximations for fixed `nf` = 3, 4 and 5 based on the
/// These are approximations for fixed `nf` = 3, 4, 5 and 6 based on the
/// first ten even moments together with small-x/large-x constraints.
/// The two sets spanning the error estimate are called via `variation = 1`
/// and `variation = 2`. Any other value of `variation` invokes their average.
Expand Down Expand Up @@ -43,7 +44,7 @@ pub fn gamma_ps(c: &mut Cache, nf: u8, variation: u8) -> Complex<f64> {
+ y1L3cff * lm13m2(c)
+ y1L4cff * lm14m2(c);

// The selected approximations for nf = 3, 4, 5
// The selected approximations for nf = 3, 4, 5, 6
let P3psApp1: Complex<f64>;
let P3psApp2: Complex<f64>;
if nf == 3 {
Expand Down Expand Up @@ -103,8 +104,27 @@ pub fn gamma_ps(c: &mut Cache, nf: u8, variation: u8) -> Complex<f64> {
+ 804.5 * lm11m1(c)
- 1760.8 * lm12m1(c)
- 10295.0 * lm12m2(c);
} else if nf == 6 {
P3psApp1 = P3ps01 + 134701.0 * xm1lm1 + 518318.0 * 1. / ((-1. + n) * n)
- 195241.0 * (1. / n - n / (2. + 3. * n + n.powu(2)))
+ 66517.0 * 1. / (6. + 5. * n + n.powu(2))
+ 658832.0 * (-(1. / n.powu(2)) + 1. / (1. + n).powu(2))
+ 19605.0 * 2. / n.powu(3)
+ 76125.0 * -6. / n.powu(4)
- 4734.5 * lm11m1(c)
- 2035.2 * lm12m1(c)
+ 1633.1 * lm12m2(c);
P3psApp2 = P3ps01 + 110032.0 * xm1lm1 + 341158.0 * 1. / ((-1. + n) * n)
- 365676.0 * 1. / (n + n.powu(2))
+ 25934.0 * 2. / (3. + 4. * n + n.powu(2))
+ 3614.4 * (-(1. / n.powu(2)) + 1. / (1. + n).powu(2))
- 194868.0 * 2. / n.powu(3)
- 4172.2 * -6. / n.powu(4)
+ 3924.3 * lm11m1(c)
- 1324.9 * lm12m1(c)
- 12520.0 * lm12m2(c);
} else {
panic!("nf=6 is not available at N3LO");
panic!("Select nf=3,..,6 for N3LO evolution");
}
// We return (for now) one of the two error-band boundaries
// or the present best estimate, their average
Expand Down Expand Up @@ -162,15 +182,15 @@ mod tests {
mom_list[(N - 2) / 2]
}
for variation in [0, 1, 2] {
for NF in [3, 4, 5] {
for NF in [3, 4, 5, 6] {
for N in [2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0, 20.0] {
let mut c = Cache::new(cmplx!(N, 0.));
let test_value = gamma_ps(&mut c, NF, variation);
assert_approx_eq_cmplx!(
f64,
test_value,
cmplx!(qq3ps_moment(N as usize, NF as f64), 0.),
rel = 4e-4
rel = if NF != 6 { 4e-4 } else { 2e-3 }
);
}
}
Expand Down
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