Efficient interpolation and evolution of parton distribution functions. Riccardo Nagar Deutsches Elektronen-Synchrotron (DESY) XXVII International Workshop on Deep Inelastic Scattering 8–12 April 2019, Turin, Italy work in collaboration with Markus Diehl and Frank Tackmann (DESY) 1 / 15
Motivation Why do we need efficient and precise PDF interpolation and evolution? Precision predictions ◮ current interpolation precision becomes insufficient at N 3 LO [see Dulat et al. 1710.03016] ◮ fast on-the-fly Mellin convolutions with “complicated” kernels required for analytical higher-order resummation ◮ PDF derivatives appear in subleading power calculations Double parton scattering ◮ cross-section formula for DPS includes double parton distributions (DPDs) F a 1 a 2 ( x 1 , x 2 , ② , µ 1 , µ 2 ) 5D grids? optimistic est. is 1 TB! → ◮ DPDs needed for DPS phenomenology (e.g. W +jets) at different factorization scales ( µ 1 , µ 2 ) and differential in transverse separation ② ◮ already implemented solutions [Gaunt, Stirling ’11] [Elias, Golec-Biernat, Sta´ sto ’18] ◮ our goal: “fast” on-the-fly evolution and interpolation for DPDs 2 / 15
PDF interpolation methods The usual interpolation [LHAPDF, APFEL, HOPPET, QCDNUM, ...] ◮ one or more grids in log x in interval [ x min , 1] (usually O (100) points) ◮ one or more grids in log α s ( µ ) or log( µ/ GeV) ◮ splines or polynomial interpolation on equispaced grids (not always on LHAPDF!) 10 - 2 ◮ LHAPDF : log cubic splines, 10 - 4 continuous 1st derivative 10 - 6 ◮ Mathematica : log cubic splines, 10 - 8 continuous 2nd derivative 10 - 10 10 - 6 10 - 5 10 - 4 0.001 0.010 0.100 1 3 / 15
PDF interpolation methods The usual interpolation [LHAPDF, APFEL, HOPPET, QCDNUM, ...] ◮ one or more grids in log x in interval [ x min , 1] (usually O (100) points) ◮ one or more grids in log α s ( µ ) or log( µ/ GeV) ◮ splines or polynomial interpolation on equispaced grids (not always on LHAPDF!) 10 - 2 ◮ LHAPDF : log cubic splines, 10 - 4 continuous 1st derivative 10 - 6 ◮ Mathematica : log cubic splines, 10 - 8 continuous 2nd derivative 10 - 10 10 - 6 10 - 5 10 - 4 0.001 0.010 0.100 1 3 / 15
Chebyshev interpolation Our interpolation ◮ one or more grids in u = log x ◮ interpolating N -th order polynomial in the Chebyshev points u k = cos k π ◮ Chebyshev ( N + 1)-grid on [ − 1 , 1]: ˜ → shifted grids in u N ◮ barycentric formula is simple, fast and numerically stable: ( − 1) j c j � u − u j f ( u ) ≃ f ( u j ) b j ( u ) , with b j ( u ) = , ( c 0 , N = 1 2 , c i =1 ) ( − 1) i c i � j i u − u i � �� � barycentric basis function Some advantages ◮ higher accuracy with considerably smaller number of points ◮ built-in Clenshaw-Curtis integration on full grid ◮ high-precision differentiation 4 / 15
Comparing splines vs barycentric MMHT grid : 64 points 10 - 3 ◮ LHAPDF: 64 pts ◮ Mathematica: 64 10 - 7 pts ◮ Chebyshev: 63 pts 10 - 11 10 - 15 10 - 6 10 - 5 10 - 4 0.001 0.010 0.100 1 Relatively small amount of points can reach remarkable accuracy. 5 / 15
Comparing splines vs barycentric HERAPDF grid : 199 points 10 - 3 ◮ LHAPDF: 199 pts ◮ Mathematica: 199 10 - 7 pts ◮ Chebyshev: 71 pts 10 - 11 10 - 15 10 - 6 10 - 5 10 - 4 0.001 0.010 0.100 1 Zero-crossings do not degrade the accuracy too much. 5 / 15
Comparing splines vs barycentric Comparing: ◮ single grid [10 − 6 , 1] with 64 points ◮ double grid [10 − 6 , 0 . 2] and [0 . 2 , 1] with 32 points each. 10 - 3 10 - 7 10 - 11 10 - 15 10 - 6 10 - 5 10 - 4 0.001 0.010 0.100 1 Generally, better accuracy with (few) subintervals. 5 / 15
Mellin moments: a measure of accuracy ◮ truncated Mellin moment of a PDF defined as � 1 dz z j − 1 f ( z ) M j ( f , x 0 ) = x 0 tends to the full Mellin moment as x 0 → 0 ◮ global measure of interpolation accuracy 10 - 3 10 - 3 10 - 6 10 - 6 10 - 9 10 - 9 10 - 12 10 - 12 10 - 15 10 - 15 40 60 80 100 120 40 60 80 100 120 6 / 15
Mellin convolution ◮ Mellin convolution with a PDF � 1 � x � dz ( K ⊗ f )( x ) = z K ( z ) f , z x where K ( z ) is a kernel and f ( x ) the parton distribution ◮ use barycentric formula f ( x ) = � n f n b n ( x ), � 1 � x m � dz ( K ⊗ f )( x m ) = K mn f n , with K mn = z K ( z ) b n z x m ◮ obtain ( K ⊗ f ) at any x via interpolation Benefits 1. precision: pre-compute kernel matrix K ij once at desired precision 2. efficiency: apply same matrix to any discretized PDF-like distribution 7 / 15
Mellin convolution accuracy Comparing Chebyshev (63 pts.) vs LHAPDF (MMHT grid, 64 pts.) 10 10 0.01 0.01 10 - 5 10 - 5 10 - 8 10 - 8 10 - 11 10 - 11 10 - 14 10 - 14 10 - 5 10 - 4 10 - 5 10 - 4 0.001 0.010 0.100 1 0.001 0.010 0.100 1 10 10 0.01 0.01 10 - 5 10 - 5 10 - 8 10 - 8 10 - 11 10 - 11 10 - 14 10 - 14 10 - 5 10 - 4 10 - 5 10 - 4 0.001 0.010 0.100 1 0.001 0.010 0.100 1 8 / 15
Discretized DGLAP equations ◮ DGLAP evolution equations d d log µ f a ( x , µ ) = ( P ( α s ) ⊗ f a ) ( x , µ ) ◮ define ˜ f a ( x , µ ) = x f a ( x , µ ) � 1 d ˜ dz P ( z , α s ( µ )) ˜ f ( x , µ ) = f ( x / z , µ ) d log µ x ◮ discretize splitting kernel � 1 d f m = P mn ˜ ˜ f n with P mn = dz P ( z ) b n ( x m / z ) d log µ x m ◮ solve the linear system of differential equations 9 / 15
■ ◆ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ● ● ◆ ◆ ◆ ■ ● ◆ ◆ ◆ ◆ ◆ ● ● ● ● ● ● ● ● ● ● ● ◆ Runge–Kutta methods We solve the system of homogeneous differential equations using an explicit Runge–Kutta routine. Various Runge–Kutta implementations Accuracy in orders p of the step-size h : O ( h p ) ◮ classic RK4: “Fiat”, 4 function calls, 4th order ◮ Cash–Karp: “Alfa Romeo”, 6 function calls, 5th order ◮ Dormand–Prince: “Lamborghini”, 8 function calls, 6th order 10 - 6 10 - 8 10 - 10 10 - 12 10 - 14 10 - 16 10 4 10 100 1000 10 / 15
Comparison with benchmark [hep-ph/0204316, hep-ph/0511119] We agree with all the digits shown in the benchmark tables. ◮ benchmark evolution using HOPPET (G. Salam) and PEGASUS (A. Vogt) ◮ benchmark values given with 5 significant digits (some points with 4) √ ◮ evolution from µ 0 = 2 GeV to µ = 100 GeV in variable flavour number scheme from N f = 3 to N f = 5, with matching at µ = m c and µ = m b ◮ HOPPET result obtained with a total of 1,170 pts in x ∈ [10 − 8 , 1] and 220 pts in µ 2 ∈ [2 , 10 6 ] GeV 2 Absolute difference of our NNLO evolution vs. the benchmark results 2 x L + x x u v x d v x L − x s + x c + x s − x g 10 − 7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10 − 6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.0 0.0 4e-13 1e-12 3e-13 1e-13 1e-14 0.0 u ± ¯ our settings: N = 70 pts in x and h = 0 . 02 with L ± = ¯ d , q ± = q ± ¯ q 11 / 15
PDF evolution accuracy Settings ◮ NNLO VFN evolution ◮ Runge–Kutta relative discrepancy between h = 0 . 02 and h = 0 . 004 ◮ discretization error estimated comparing N = 70 and N = 106 grids ◮ Evolution to µ = 1 . 01 m b (just after matching from n F = 3 to 5) ◮ Inaccuracy grows in valence distributions at low x 10 - 7 10 - 7 10 - 10 10 - 10 10 - 13 10 - 13 10 - 5 10 - 5 0.001 0.100 0.001 0.100 12 / 15
PDF evolution accuracy Settings ◮ NNLO VFN evolution ◮ Runge–Kutta relative discrepancy between h = 0 . 02 and h = 0 . 004 ◮ discretization error estimated comparing N = 70 and N = 106 grids ◮ Evolution to µ = 1 . 01 m b (just after matching from n F = 3 to 5) ◮ Inaccuracy grows in valence distributions at low x 10 - 8 10 - 8 10 - 11 10 - 11 10 - 14 10 - 14 10 - 5 0.001 0.100 0.2 0.4 0.6 0.8 1.0 12 / 15
PDF evolution accuracy Settings ◮ NNLO VFN evolution ◮ Runge–Kutta relative discrepancy between h = 0 . 02 and h = 0 . 004 ◮ discretization error estimated comparing N = 70 and N = 106 grids ◮ Evolution to µ = 10 TeV ◮ Numerical errors still under control; Runge-Kutta error negligible 10 - 7 10 - 7 10 - 10 10 - 10 10 - 13 10 - 13 10 - 5 10 - 5 0.001 0.100 0.001 0.100 12 / 15
PDF evolution accuracy Settings ◮ NNLO VFN evolution ◮ Runge–Kutta relative discrepancy between h = 0 . 02 and h = 0 . 004 ◮ discretization error estimated comparing N = 70 and N = 106 grids ◮ Evolution to µ = 10 TeV ◮ Numerical errors still under control; Runge-Kutta error negligible 10 - 8 10 - 8 10 - 11 10 - 11 10 - 14 10 - 14 10 - 5 0.001 0.100 0.2 0.4 0.6 0.8 1.0 12 / 15
Extension to DPDs H 1 p p ◮ DPD cross section is factorized as H 2 d σ DPS ∝ d σ (1) a 1 b 1 ( µ 1 ) ⊗ d σ (2) a 2 b 2 ( µ 2 ) � d 2 ② F a 1 a 2 ( ② , µ 1 , µ 2 ) ⊗ F b 1 b 2 ( ② , µ 1 , µ 2 ) ⊗ ◮ DGLAP equations extended to DPDs are separate evolution equations w.r.t. µ 1 or µ 2 [Diehl, Ostermeier, Sch¨ afer ’11] � � d P ( i ) ( α s ) ⊗ d log µ i F a 1 a 2 ( x 1 , x 2 , y , µ 1 , µ 2 ) = ( x 1 , x 2 , y , µ 1 , µ 2 ) i F a 1 a 2 EXPERT NOTE: homogeneous equations only valid for y -dependent DPDs! ◮ defining F a 1 a 2 ( x 1 , x 2 ) = 0 for x 1 + x 2 > 1, obtain µ 1 -evolution independent from x 2 and viceversa ◮ same discretization as done for PDFs → matrices P mn are the same 13 / 15
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