SLIDE 1 On the numerical evaluation of 3-loop self-energy integrals
University of Pittsburgh
- 1. Introduction
- 2. Review: General 3-loop vacuum integrals
- 3. Planar-type 3-loop self-energy integrals
- 4. Public program TVID 2
SLIDE 2
Introduction
1/21
Need for 3-loop corrections: Electroweak precision tests: Current exp. Current theory∗ CEPC FCC-ee MW [MeV] 15 4 1 0.5–1 ΓZ [MeV] 2.3 0.4 0.5 0.1 Rb = Γb
Z/Γhad Z
[10−5] 66 10 4.3 6 sin2 θℓ
eff [10−5]
16 4.5 <1 0.5
∗ Full 2-loop and leading 3-/4-loop corrections
Higgs mass calculation in SUSY
Harlander, Kant, Mihaila, Steinhauser ’08,10 Reyes, Fazio ’19
Mixed EW-QCD corrections to Higgs prod. at LHC
Bonetti, Melnikov, Tancredi ’17 Anastasiou et al. ’18
...
SLIDE 3
Two-loop electroweak corrections
2/21
Analytical evaluation of master integrals with diff. eq. or Mellin-Barnes rep.
Kotikov ’91; Remiddi ’97; Smirnov ’00,01; Henn ’13; ...
→ Result in terms of Goncharov polylogs / multiple polylogs
Goncharov ’98 Gehrmann, Remiddi ’00,01
→ Some problems need iterated elliptic integrals / elliptic multiple polylogs
Levin, Racinet ’07; Bloch, Vanhove ’07 Adams, Bogner, Weinzierl ’14; ...
→ Full set of functions for all 2-loop diagrams not known
SLIDE 4 Two-loop electroweak corrections
3/21
Problem has multiple scales: MZ, MW, MH, mt (mf → 0, f = t) Numerical techniques needed Self-energies (incl. from renormlization) and vertices with sub-loop bubbles using dispersion relation technique
Awramik, Czakon, Freitas ’06
Non-trivial vertex diagrams:
Dubovyk, Freitas, Gluza, Riemann, Usovtisch ’16,18
- Sector decomposition
- Mellin-Barnes representations (MB / AMBRE 3 / MBnumerics)
- No tensor reduction (besides trivial cancellations)
→ > 1000 different two-loop vertex integrals
s MZ MZ s 1 2 3 5 4 6 s 1 2 4 3 6 5 s 1 2 3 5 4 6
SLIDE 5
Direct numerical integration
4/21
Two general (automizable) approches: Sector decomposition:
Binoth, Heinrich ’00,03
Advantageous for diagrams with many massive propagators Public programs: SecDec
Carter, Heinrich ’10; Borowka et al. ’12,15,17
FIESTA
Smirnov, Tentyukov ’08; Smirnov ’13,15
Mellin-Barnes representations:
Smirnov ’99; Tausk ’99 Czakon ’06; Anastasiou, Daleo ’06
... with fewer independent parameters Public programs: MB/MBresolve
Czakon ’06; Smirnov, Smirnov ’09
AMBRE/MBnumerics
Gluza, Kajda, Riemann ’07 Dubovyk, Gluza, Riemann ’15 Usovitsch, Dubovyk, Riemann ’18
Can be applied to any number of scales and loop order Automated extraction of UV and IR divergencies Requires sizeable computing resources Diagrams with internal thresholds can cause numerical instabilities
SLIDE 6 General 3-loop vacuum integrals
5/21
Relevant for low-energy precision observables (p2 ≪ MZ) Coefficients of low-momentum expansions Building block for more general 3-loop calculations Master integrals:
M(ν1, ν2, ν3, ν4, ν5, ν6; m2
1, m2 2, m2 3, m2 4, m2 5, m2 6)
= i e3γEǫ π3D/2
1 − m2 1]−ν1[(q1 − q2)2 − m2 2]−ν2
× [(q2 − q3)2 − m2
3]−ν3[q2 3 − m2 4]−ν4[q2 2 − m2 5]−ν5[(q1 − q3)2 − m2 6]−ν6
b b b
1 1 2 3 4
b b b
1 2 4 3 5
b b b b
1 2 4 3 5 6
U4 U5 U6
= M(2, 1, 1, 1, 0, 0) = M(1, 1, 1, 1, 1, 0) = M(1, 1, 1, 1, 1, 1)
SLIDE 7 Sub-loop dispersion relations
6/21
Topologies with self-energy sub-loop can easily be integrated by using dispsersion relation for B0 function:
B0(p2, m2
1, m2 2) = −
∞
(m1+m2)2 ds ∆B0(s, m2 1, m2 2)
s − p2 with ∆B0(s, m2
1, m2 2) = (4πµ2)4−DΓ(D/2 − 1)
Γ(D − 2) λ(D−3)/2(s, m2
1, m2 2)
sD/2−1 , λ(a, b, c) = (a − b − c)2 − 4bc
the k
tegrations and
the s-in tegrations an b e p erformed and yield T 1::: N +2 (p i ; m 2 1 ; : : : ; m 2 N +1 ; m 2 N +2 ) = B (m 2 N +2 ; m 2 N ; m 2 N +1 )T (1) (p i ; m 2 1 ; : : : ; m 2 N 1 ; m 2 N +2 )
2 i Z 1 s ds B (s; m 2 N ; m 2 N +1 ) s
2 N +2 T (1) (p i ; m 2 1 ; : : : ; m 2 N 1 ; s) : (68) T (1) denotes a
N-p
t fun tion in whi h s en ters in the remaining
in tegration as a mass v ariable. A diagram with t w
erti es leads to a result whi h is similar to the remaining in tegration in (68), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 1 p 2 p N 1 p N m 1 m N 1 m N m N +1 . . . . . . t t t t T 1::: N +1 (p i ; m 2 i ) = 1 2 i 1 Z s ds B (s; m 2 N ; m 2 N +1 )
k 2
1 (k + p 1 ) 2
2 1 : : : 1 (k + p 1 + : : : + p N 1 ) 2
2 N 1
2 i 1 Z s ds B (s; m 2 N ; m 2 N +1 )T (1) (p i ; m 2 1 ; : : : ; m 2 N 1 ; s) : (69) 4.2 Examples An appli ation
(69) to the London transp
diagram leads to T 123 (p 2 ; m 2 1 ; m 2 2 ; m 2 3 ) =
2 i 1 Z (m 2 +m 3 ) 2 ds B (s; m 2 2 ; m 2 3 ) B (p 2 ; s; m 2 1 ) ; (70) a result whi h w
also follo w from (5). In that ase a suitable subtra tion [6 ℄ is T 123N (p 2 ; m 2 1 ; m 2 2 ; m 2 3 ) = T 123 (p 2 ; m 2 1 ; m 2 2 ; m 2 3 )
123 (p 2 ; m 2 1 ; 0; m 2 3 ) (71) T 123 (p 2 ; 0; m 2 2 ; m 2 3 ) + T 123 (p 2 ; 0; 0; m 2 3 ) : F
T 1234
from (68) T 1234 (p 2 ; m 2 1 ; m 2 2 ; m 2 3 ; m 2 4 ) 17
TN+1(pi; m2
i ) = −
∞
s0
ds ∆B0(s, m2
N, m2 N+1)
×
1 q2−s 1 (q+p1)2−m2
1
· · ·
1 (q+p1+···+pN−1)2−m2
N−1
SLIDE 8 U4
7/21
b b b b
1 1 2 3 4
p → = B0,m1(p2, m2
1, m2 2)B0(p2, m2 3, m2 4)
=
∞
ds ∆Idb(s) s − p2 − iε ∆Idb(s, m2
1, m2 2, m2 3, m2 4) = ∆B0,m1(s, m2 1, m2 2) B0(s, m2 3, m2 4)
+ B0,m1(s, m2
1, m2 2) ∆B0(s, m2 3, m2 4),
∆B0(s, m2
a, m2 b ) = 1
sλ(s, m2
a, m2 b ) Θ
∆B0,m1(s, m2
a, m2 b ) = m2 a − m2 b − s
s λ(s, m2
a, m2 b ) Θ
U4(m2
1, m2 2, m2 3, m2 4) = − eγEǫ
iπD/2
∞
ds ∆Idb(s) q2
3 − s + iε
b b b
1 1 2 3 4
= −
∞
ds A0(s) ∆Idb(s)
SLIDE 9 U4
8/21
Problem: U4 is divergent
b b b
1 1 2 3 4
Solution: U4(m2
1, m2 2, m2 3, m2 4) = U4(m2 1, m2 2, 0, 0) + U4(m2 1, 0, m2 3, 0)
+ U4(m2
1, 0, 0, m2 4) − 2 U4(m2 1, 0, 0, 0) + U4,sub(m2 1, m2 2, m2 3, m2 4)
→ U4(m2
X, m2 Y , 0, 0) can be computed analytically
→ U4,sub is finite
U4,sub(m2
1, m2 2, m2 3, m2 4) = −
∞
ds A0,fin(s) ∆Idb,sub(s) Idb,sub(s, m2
1, m2 2, m2 3, m2 4) =
∆B0,m1(s, m2
1, m2 2) Re
3, m2 4) − B0(s, 0, 0)
1, 0) Re
3) + B0(s, 0, m2 4) − 2B0(s, 0, 0)
1, m2 2)
∆B0(s, m2
3, m2 4) − ∆B0(s, 0, 0)
1, 0)
∆B0(s, 0, m2
3) + ∆B0(s, 0, m2 4) − 2 ∆B0(s, 0, 0)
SLIDE 10 U5, U6
9/21
b b b
1 2 4 3 5
= − eγEǫ iπD/2
∞
ds
[q2
3 − s][q2 3 − m2 5] × Disc
b b
1 2 3 4
=
∞
ds B0(0, s, m2
5) Disc[...]s
b b b b
1 2 4 3 5 6
= − eγEǫ iπD/2
∞
ds
[q2
3 − s][q2 3 − m2 5] × Disc
b b b
1 2 4 3 6
=
∞
ds B0(0, s, m2
5) Disc[...]s
2-loop self-energy known in terms of 1-dimensional numerical integral
Bauberger, B¨
U5, U6 made UV-finite by subtracting terms that can be computed analytically
SLIDE 11
Results
10/21
U4, U5 given in terms of one-dimensional numerical integrals U6 given in terms of two-dimensional numerical integral Special cases (e.g. m1 = 0) can also be handled Public code: TVID 1
Freitas ’16; Bauberger, Freitas ’17
Algebraic part (Mathematica) performs subtraction of UV-divergencies Numerical part (C++) performs numerical integrals Timing (single core Xeon 3.7 GHz):
0.1 s for U4, U5 30 s for U6
At least ten digit agreement with literature (for one/two-scale cases)
Broadhurst ’98; Chetyrkin, Steinhauser ’99 Grigo, Hoff, Marquard, Steinhauser ’12
Available at www.pitt.edu/˜afreitas/
SLIDE 12
Alternative approach
11/21
Differential equations: (with respect to mass parameters)
Martin, Robertson ’16 [slide from S. Martin]
Public code: 3VIL (timing below 1 sec. and similar precision as TVID)
SLIDE 13 Planar-type 3-loop self-energy integrals
12/21
3-loop self-energy diagrams with ladder-type topology Useful for:
- On-shell renormalization in full SM
- Higgs mass corrections in SUSY
- ...
Find set of master integrals:
- Generate diagrams with FEYNARTS 3 [Hahn ’01]
- Reduce using IBP relations with FIRE 5 [Smirnov ’14]
Set of masters with only denominators, no numerators (may not be optimal) All masters can be evaluated in terms of 2-dim. numerical integrals
→ Fast and high-precision results
SLIDE 14 Master integrals
13/21
p →
b b
8 6 3 1
b b b
1 3 7 6 5
b b b
4 3 7 6 2
b b b
1 3 7 6 2
U4a U5a U5b U5c
b b b b
1 3 7 6 5 4
b b b b
5 4 3 8 6 1
b b b b
1 4 3 6 7 2
b b b b
1 2 3 8 6 4
U6a U6b U6c U6m
b b b b
1 2 3 6 7 8
b b b b b
1 2 3 5 8 6 4
b b b b b
1 2 3 4 6 7 8
b b b b b b
1 2 3 4 5 6 7 8
U6n U7m U7a U8a
SLIDE 15 Master integrals with doubled propagators
14/21
b b b
1 1 3 6 8
b b b b
1 1 3 3 6 8
b b b b
1 1 1 3 6 8
b b b b
3 1 7 6 5 5
b b b b
3 1 1 7 6 5
U4a1 U4a2 U4a3 U5a1 U5a2
b b b b
4 4 3 7 6 2
b b b b
4 3 7 6 2 2
b b b b
1 3 7 7 6 2
b b b b b
1 2 3 3 8 6 4
b b b b b
1 2 3 8 6 6 4
U5b1 U5b2 U5c1 U6m1 U6m2
b b b b b
1 2 2 3 8 6 4
b b b b b
1 2 3 3 6 7 8
b b b b b
1 2 2 3 6 7 8
b b b b b b
1 2 3 3 4 6 7 8
b b b b b b
1 2 2 3 4 6 7 8
U6m3 U6n1 U6n2 U7a1 U7a2
SLIDE 16 Double-bubble integrals
15/21
Example: U5b
b b b
4 3 7 6 2
∞
ds1
∞
ds2 ∆B0(s1, m2
6, m2 7) ∆B0(s2, m2 1, m2 3)
× eγEǫ iπD/2
1 [q2
2 − m2 4][q2 2 − s1][(q2 + p)2 − s2]
- ne-loop integral, known analytically
Subtraction of divergencies:
∞
ds1
∞
ds2 ∆B0(s1, m2
6, m2 7) ∆B0(s2, m2 1, m2 3)
s1 − m2
4 − iε
B0(p2, s1, s2) =
∞
ds1
∞
ds2 ∆B0(s1, m2
6, m2 7) ∆B0(s2, m2 1, m2 3)
s1 − m2
4 − iε
×
- B0(0, s1, s2)
- vacuum integrals
+ p2B′
0(0, s1, s2)
- + [B0(p2, s1, s2) − B0(0, s1, s2) − p2B′
0(0, s1, s2)]
SLIDE 17 Planar master topology: U8a
16/21
b b b b b b
1 2 3 4 5 6 7 8
d4q
iπ2 C0(p2, y, x, m2
1, m2 2, m2 3)C0(x, p2, y, m2 6, m2 7, m2 8)
[x − m2
4 + iε][y − m2 5 + iε]
x = q2, y = (q + p)2 In cms frame, p = (p0, 0): x = q2
0 − |
q|2, y = q2
0 − |
q|2 + p2 + 2q0p0 Integrate angles of q; variable transformation (q0, | q|) → (x, y): U8a = 1 2iπp2
∞
−∞ dx
∞
−∞ dy
- λ(x, y, p2) Θ(λ(x, y, p2))
Ghinculov ’96
× C0(p2, y, x, m2
1, m2 2, m2 3)C0(x, p2, y, m2 6, m2 7, m2 8)
[x − m2
4 + iε][y − m2 5 + iε]
.
λ(x, y, z) = x2 + y2 + z2 − 2(xy + yz + xz)
Integration over poles:
∞
−∞ dx
f(s) x − ξ ± iε = ∓iπf(ξ) +
∞
dx′ f(ξ + x′) − f(ξ − x′) x′ .
SLIDE 18 Planar master topologies
17/21
Similar approach for
b b b b b b
1 2 3 4 5 6 7 8
b b b b b
1 2 3 4 6 7 8
b b b b b b
1 2 3 3 4 6 7 8
b b b b b b
1 2 2 3 4 6 7 8
U8a U7a U7a1 U7a2
→ UV-finite, no subtractions necessary → Two-dimensional numerical integrals, suitable for high-precision evaluation
Implementation in TVID 2 Adaptive Gauss-Kronrod integration [QUADPACK Piessens, de Doncker-Kapenga, ¨
Uberhuber, Kahanger ’83],
quad precision floating points Note: Evaluation of C0 in quad precision is rather slow
→ Use double precision C0 from FF/LOOPTOOLS
Hahn, Perez-Victoria ’99
→ Final precision reduced to 6–8 digits
SLIDE 19
Implementation notes
18/21
Currently no treatment for IR divergencies or threshold singularities Numerical instabilites in tail of
∞
s0 ds...
→ Use asymptotic formulas for integrand above s > scut
Some double-propagator integrals U6m1(...) =
∂ ∂m2
3
U6m(...) U6m3(...) =
∂ ∂m2
2
U6m(...) U6n2(...) =
∂ ∂m2
2
U6n(...) produce big rational expressions in integrand, leading to 0/0 instabilities in integration region
→ Use numerical differentiation of final function, using 5-point stencil
U6m1(...) ≈ −U6m(m2
3 + 2δ) + 8U6m(m2 3 + δ) − 8U6m(m2 3 − δ) + U6m(m2 3 − 2δ)
12δ
SLIDE 20 Checks and performance
19/21 p2 = 1.0, m2
1 = 1.1, m2 2 = 1.2, m2 3 = 1.3, m2 4 = 1.4, m2 5 = 1.5, m2 6 = 1.6, m2 7 = 1.7, m2 8 = 1.8
TVID 2.0 FIESTA 4.1
[Smirnov ’16]
Result Time∗ [s] Result Time∗ [s] U4a 38.7964435845(4) 6.6 38.80(1) 283
9.828362321(2) 0.5 9.830(2) 283
1.196967810(2) 0.5 1.1970(1) 315
−9.64795183(6) 160 −9.6480(1) 336 U8a 0.1224166(1) 502 0.122418(1) 542 U4a1 −1.4651121210(1) 1.5 1.465(3) 163 U7a2 0.2200785(2) 559 0.220080(3) 269
∗ only numerical integration time, for O(ǫ0) parts
- U5a,
- U6a,
- U6m are linear combinations that avoid O(ǫ) terms of 2-loop
self-energy integrals
CurrentIntegratorSettings = {{"epsrel","1e-05"},{"maxeval","5e6"}}; ComplexMode = False;
SLIDE 21 Checks and performance
20/21 p2 = 40, m2
1 = 1.1, m2 2 = 1.2, m2 3 = 1.3, m2 4 = 1.4, m2 5 = 1.5, m2 6 = 1.6, m2 7 = 1.7, m2 8 = 1.8
TVID 2.0 FIESTA 4.1
[Smirnov ’16]
Result Time∗ [s] Result Time∗ [s] U4a −149.6944621(5) 17.6 −149.7(1) 3052 +9.6099138(5) i +9.6(1) i
53.705925142(1) 0.5 53.71(8) 2865 −20.874552008(1) i −20.88(8) i
−11.094545131(6) 989 −11.094(7) 5585 +4.390391111(6) i +4.391(7) i U8a 0.01238717(2) 253 0.012353(3) 11407 −0.16344185(2) i −0.016361(3) i
∗ only numerical integration time, for O(ǫ0) parts
- U5a,
- U6a,
- U6m are linear combinations that avoid O(ǫ) terms of 2-loop
self-energy integrals
CurrentIntegratorSettings = {{"epsrel","1e-05"},{"maxeval","5e6"}}; ComplexMode = True;
SLIDE 22
Summary
21/21
Public code: TVID 2
Bauberger, Freitas, Wiegand ’19
Algebraic part (Mathematica) performs subtraction of UV-divergencies
→ Symbolic expressions can get large!
Numerical part (C++) performs numerical integrals
→ Uses LOOPTOOLS for some integrals
precision [digits] timing [single core] 3-loop vacuum integrals ≥ 10 0.1s–30s 3-loop self-energy integrals 9–10 some cases 6–8 0.5s–16m Tested on Linux systems Soon available at www.pitt.edu/˜afreitas/
