Systematic approximation of multi-scale Feynman integrals - - PowerPoint PPT Presentation
Systematic approximation of multi-scale Feynman integrals arXiv:1804.06824 Daniel Hulme In collaboration with: Thomas Gehrmann and Sophia Borowka Amplitudes in the LHC era, Galileo Galilei Institute 29.10.2018 Daniel Hulme arXiv:1804.06824
arXiv:1804.06824 Daniel Hulme In collaboration with: Thomas Gehrmann and Sophia Borowka
Amplitudes in the LHC era, Galileo Galilei Institute
29.10.2018
Daniel Hulme arXiv:1804.06824 29.10.2018 1
Feynman integrals are often the bottleneck for multi-scale multi-loop calculations, especially with massive propagators! Analytic evaluation of very complicated Feynman integrals not generally understood, although progress is being made! Numerical evaluation often the only option. Producing accurate numerical results over full phase space in an automated way is difficult - divergent nature of loop integrals. An algorithm to analytically approximate Feynman integrals - TayInt!
Daniel Hulme arXiv:1804.06824 29.10.2018 2
Three aims: to produce an algebraic integral library with full phase space validity for any kind of integral The idea - the integrand has to be Taylor expanded in the Feynman parameters - otherwise you can’t integrate it - the kinematics are not to be touched. The algorithm brings an integral into a form optimised for an accurate Taylor expansion with validity in all kinematic regions. Divide and rule - take all the nastiness in the Feynman integral - distribute it so that it does not hinder a TAYLOR EXPANSION IN THE FEYNMAN PARAMETERS.
Daniel Hulme arXiv:1804.06824 29.10.2018 3
A generic Feynman loop integral G in an arbitrary number of dimensions D at loop level L with N propagators, wherein the propagators Pj with mass mj can be raised to arbitrary powers νj, G µ1...µR
α1...αR ({p}, {m}) =
L
α1 · · · kµR αR
N
j=1 Pνj j ({k}, {p}, m2 j )
dDκα =µ4−D iπ
D 2
dDkα , Pj({k}, {p}, m2
j ) = q2 j − m2 j + iδ ,
The qj are linear combinations of external momenta pi and loop momenta kα. Henceforth scalar integrals considered.
Daniel Hulme arXiv:1804.06824 29.10.2018 4
Rewrite scalar integrals in terms of Feynman parameters tj, j = 1...N. Integrate the loop momenta to give, G = (−1)Nν N
j=1 Γ(νj) N
∞ dtj tνj−1
j
δ(1 −
N
tl) UNν−(L+1)D/2 FNν−LD/2 , U and F are the first and second Symanzik polynomials.
Daniel Hulme arXiv:1804.06824 29.10.2018 5
U1: reduce the Feynman Integral to a quasi-finite basis U2: perform an iterated sector decomposition below threshold above threshold BT1: tj → yj OT1: tj → θj, generate K BT2: Taylor expand OT2: find Θo(0),...o(J−1) and integrate OT3: perform one fold integrations OT4: find θ∗
j and the
OT5: determine Pj OT6: Taylor expand and integrate
Daniel Hulme arXiv:1804.06824 29.10.2018 6
m m p p
(a) S1401220
p1 p2 m m m p1 + p2
(b) T41
m2 p1 m1 m1 m1 p4 p2 + p3
(c) I10
p1 p4 m1 m1 m1 m1 m2 p2 + p3
(d) I21
p4 p1 m1 m1 m1 m1 m2 p2 + p3
(e) I246
m2 m1 p1 p2 p3 m1 m1 m1 p4
(f) I39
The finite sunrise S1401220 and the triangle T41 are used to illustrate
calculation - are computed with TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 7
Universal step 1 (U1) - Feynman integral G expressed as a superposition of finite Feynman integrals G F multiplying poles in ǫ Quasi-finite basis: (von Manteuffel, Panzer, Schabinger, arXiv:1411.7392),(Panzer, arXiv:1401.4361) these integrals contain no divergences from integration of Feynman parameters - all divergent parts restricted to prefactors. The G F have a shifted number of dimensions or dotted propagators or both. Automated shell script used to direct all required Reduze (von Manteuffel, Panzer, Schabinger, arXiv:1411.7392),(von Manteuffel, Studerus, arXiv:1201.4330) jobs towards generating the quasi-finite basis.
Daniel Hulme arXiv:1804.06824 29.10.2018 8
The divergent sunrise S1401110 in terms of the finite integrals S1401220, S1401320 and the tadpole S630300, S1401110 = 8m2 (p2 − 4 m2)(p2 + 2 m2) (−3 + D)(−8 + 3D)(−10 + 3D) · S1401320 + ((4 − D)p4 + (−5 + D)8 m4 + (18 − 5D)4 p2m2) (−3 + D)(−8 + 3D)(−10 + 3D) · S1401220 − 16m4 ((−4 + D) p2 + 2 (−24 + 7D) m2) (−3 + D)(−4 + D)2(−8 + 3D)(−10 + 3D) · S530300 , Poles in ǫ: (−4 + D)−1 terms.
Daniel Hulme arXiv:1804.06824 29.10.2018 9
Universal step 2 (U2) - decompose integrals G F to iterated sectors using version 3 of SecDec (Borowka, Heinrich et al., arXiv:1703.09692) Iterated sectors: G F
l = N
1 dtj tAl−Bljǫ
j
UNν−(L+1)D/2
l
l
, where l = 1, . . . , r , and r is the number of iterated sector integrals. Ak and Bk are numbers independent of the regulator ǫ. Remap so that j runs from 0 to J − 1. Iterated sector integrals G F
l - building blocks of TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 10
O(ǫ0) coefficient of an I10 iterated sector,
I101 =
2
1 dtj 1 (1 + t0 + t1 + t2 + t1t2) · [t0(−u − m2t1) + m2
1(1 + t2 0 + t2 + t2 1(1 + t2)+
t1(2 + 2t2) + t0(2 + t2 + t1(2 + t2)))]−1,
Three Feynman parameters after iterated decomposition, three kinematic scales, m1, m2 and u.
Daniel Hulme arXiv:1804.06824 29.10.2018 11
Below Threshold step 1 (BT1) - maximise distance to nearest point
The iterated sectors G F
l - still have non-analytic points outside the
integration region. To move these as far away as possible, import G F
l into Mathematica
(Wolfram), apply conformal mappings, tj = ayj + b cyj + d . Thus far: optimum mapping found.
Daniel Hulme arXiv:1804.06824 29.10.2018 12
0.002
1 2 3 4 I1011(t0,t1=0,t2=0) t0
0.0001 0.0002 0.0003
1 2 3 4 I1011(y0) y0 8.0 · 10-6 1.6 · 10-5 2.4 · 10-5 3.2 · 10-5 0.5 1 8.0 · 10-6 1.6 · 10-5 2.4 · 10-5 3.2 · 10-5
Plot of a one dimensional integrand, I101(t0, t1 = 0, t2 = 0) = 1/((1 + t0)(−m2t0 + m2
1(1 + 2t0 + t2 0))),
before and after a conformal mapping y0 = −1−t0
t0
.
Daniel Hulme arXiv:1804.06824 29.10.2018 13
Below Threshold steps 2 and 3 (BT2-3) - Taylor expand and integrate in the Feynman parameters BT2 - Taylor expand the integrand in the re-mapped Feynman parameters yj. BT3 - integrate over the yj. Done in FORM (Kuipers, Ueda, Vermaseren, arXiv:1310.7007).
Daniel Hulme arXiv:1804.06824 29.10.2018 14
1 1.002 1.004 1.006 1.008 1.01 1.012 30000 60000 90000
SecDec/Taylor u (Gev2)
w/o conformal map with conformal map
Ratio of SecDec and TayInt calculation of the ǫ0 coefficient of I10.
Daniel Hulme arXiv:1804.06824 29.10.2018 15
Above lowest threshold of an integral - discontinuities on the real
TayInt returns to the result of U2, the iterated sector integrands G F
l (tj). The Feynman +iδ prescription is implemented in
Mathematica. TayInt determines the contour configuration in the complex plane to avoid the discontinuities. Over threshold part of TayInt is fully automated in Mathematica.
Daniel Hulme arXiv:1804.06824 29.10.2018 16
Over Threshold step 1 (OT1) - generate all possible contour configurations for each iterated sector integrand The first Over Threshold step (OT1) - transform the Feynman parameters of the J − 1 iterated sectors, tj → 1
2 + 1 2 exp (iθj).
Generate representative sample of the kinematic region. A nested list of values K = {{s1, . . . , sβ}1, . . . , {s1, . . . , sβ}γ} = {K1, . . . , Kγ} for a β scale integral, sample size of γ points.
Daniel Hulme arXiv:1804.06824 29.10.2018 17
Over Threshold step 2 (OT2) - select contour configuration
OT2 - calculate the mean absolute value of the θj derivatives (MAD)
l (θj).
Kinematic scales first set to the mean of the sample, MAD calculated at the edges. Kinematic scales then set to each sample value, MAD calculated over bulk. MAD calculated for all possible contour configurations, Θo(1),...o(J−1)
Contour configuration which minimises the MAD selected.
Daniel Hulme arXiv:1804.06824 29.10.2018 18
Slice of I102 without a complex mapping...
Daniel Hulme arXiv:1804.06824 29.10.2018 19
with a complex mapping, contour orientation {o(1), ...o(J − 1)} determined via TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 20
Over Threshold steps 3 and 4 (OT3-4) - generate all possible post-integration contour configurations for each iterated sector integrand - select optimum one. OT3 - perform all possible one-fold integrations in the θj exactly. In OT4 the resultant J − 2 variable contour configuration with the lowest MAD is selected - same process as in OT2. If MAD is lower than without integration, this contour configuration is used. The θj integrated to yield the selected contour configuration is θ∗
j .
Daniel Hulme arXiv:1804.06824 29.10.2018 21
I102: pre-integration contour chosen by TayInt, arbitrary post-integration contour.
Daniel Hulme arXiv:1804.06824 29.10.2018 22
I102: pre-integration contour arbitrary, post-integration contour chosen by TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 23
I102: full TayInt algorithm.
Daniel Hulme arXiv:1804.06824 29.10.2018 24
Over Threshold steps 5 and 6 (OT5-6) - maximise the convergence
OT5 - determine the optimal partitioning, Pj= {(l, h)1, ..., (l, h)N}j,
±π dθj =
N
hk,j
lk,j
dθj , hN, j = ±π and l1, j = 0. New integrands expanded and integrated within each partition in OT6
Results are functions of the kinematic scales valid everywhere above threshold!
Daniel Hulme arXiv:1804.06824 29.10.2018 25
200000 600000 1000000 Re(integral) u (Gev2) Taylor Partitioned Taylor SecDec
0.0 · 10+0 200000 600000 1000000 Im(integral) u (Gev2) Taylor Partitioned Taylor SecDec
I10 calculated to ǫ0 over the 4m2
1 threshold with and without
partitioning.
Daniel Hulme arXiv:1804.06824 29.10.2018 26
Slice of I2461 at ǫ0 in the first over threshold region before and after applying TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 27
p4 p1 m1 m1 m1 m1 m2 p2 + p3
Slice of I2461 at ǫ0 in the second over threshold region before and after applying TayInt.
Daniel Hulme arXiv:1804.06824 29.10.2018 28
p4 p1 m1 m1 m1 m1 m2 p2 + p3
I10: O
, u > 4m2
1, m2 2 = 0.5m2 1, m1 = 173GeV
Re(integral) TayInt SecDec
Im(integral) TayInt SecDec 0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec error 0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 29
m2 p1 m1 m1 m1 p4 p2 + p3
Re(integral) TayInt SecDec
Im(integral) TayInt SecDec 0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 500000 700000 900000 Errors u (Gev2)
Truncation Error 8 SecDec error Truncation Error 4
0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 500000 700000 900000 Errors u (Gev2)
Truncation Error 8 SecDec Error Truncation Error 4
Daniel Hulme arXiv:1804.06824 29.10.2018 30
m2 p1 m1 m1 m1 p4 p2 + p3
Feynman integral Quasi-finite integrals Iterated Sectors
Once-integrated complex iterated sectors
Partitioned complex-valued iterated sectors
1 2+1 8 64 →8
Sum each integrated Taylor series Sum all iterated sectors Sum all finite integrals
Remove and calculate easy integrals
TayInt Result
64
Complex iterated sectors
I10 24 →8
C h
i n g A l g
i t h m
F(q1,q2,q3,…)
Daniel Hulme arXiv:1804.06824 29.10.2018 31
Daniel Hulme arXiv:1804.06824 29.10.2018 32
New method of contour deformation, removed dependence on Reduze U1 removed - divergent integrals can be computed. OT3-4 removed - exact integration no longer performed as precision can be more reliably controlled with partitions. OT2 changed - MAD computed at a sample of kinematic points - select the contour with the most invariant MAD over the sample. Check - is the chosen contour the optimal one at the majority of points in the kinematic sample?
Daniel Hulme arXiv:1804.06824 29.10.2018 33
If not - use partial complex contours - one Feynman parameter not mapped - and perform the same contour selection. This continues until a contour is optimum for the majority of points in the kinematic sample. The new method of contour selection is more reliable as it cannot be distorted by small regions of extreme change. It is based on our observation that the smoothest surface changes the least when you change the kinematics. This invariance is a more general indicator of a surfaces suitability for a Taylor expansion than the mean MAD over a kinematic sample.
Daniel Hulme arXiv:1804.06824 29.10.2018 34
I59: O
, s > 4m2
1, u = −2m2 1, m2 2 = 0.5m2 1, m1 = 173GeV
0.0 · 10+0 2.0 · 10-10 4.0 · 10-10 6.0 · 10-10 8.0 · 10-10 1.0 · 10-9 Re(integral) TayInt SecDec 1.0 · 10-10 3.0 · 10-10 5.0 · 10-10 7.0 · 10-10 Im(integral) TayInt SecDec 0.99 1.01 1.03 120000 160000 200000 240000 280000 Errors s (Gev2)
Truncation Error SecDec Error
0.96 1 1.04 1.08 120000 160000 200000 240000 280000 Errors s (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 35
p1 p2 m1 m1 m1 m1 p3 p4 m2
I503: O
, 0 < s > −m2
µ, u = −0.5m2 µ, u = −0.5m2 µ, mµ = 106MeV
1.5 · 10-11 2.5 · 10-11 3.5 · 10-11
Integral
TayInt Analytic 0.995 1 1.005 1.01
Errors s (MeV2)
Truncation Error SecDec/TayInt Daniel Hulme arXiv:1804.06824 29.10.2018 36 µ e e µ
Feynman integral Iterated Sectors Partitioned complex-valued iterated sectors
1 22 2816 →19
Sum each integrated Taylor series Sum all iterated sectors Sum all finite integrals TayInt Result
22
Complex iterated sectors
I59 Invariant-based choosing Algorithm F(q1,q2,q3,…)
Partially complex iterated sectors
48 →3
Daniel Hulme arXiv:1804.06824 29.10.2018 37
Three aims: to produce an algebraic integral library with full phase space validity for any kind of integral TayInt is flexible. Results generated for different: numbers of propagators numbers of external scales ǫ orders kinematic regions, above and below thresholdS diagrams relevant for Higgs+jet production at two loop.
Daniel Hulme arXiv:1804.06824 29.10.2018 38
Future goals: Do we need the Sector decomposition? Automate fully, Apply to phenomenological processes, Apply to five-point integrals.
Daniel Hulme arXiv:1804.06824 29.10.2018 39
Daniel Hulme arXiv:1804.06824 29.10.2018 40
1, m2 2 = 0.5m2 1,m1 = 173GeV
0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 Re(integral) TayInt SecDec 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 Im(integral) TayInt SecDec 0.998 1 1.002 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec error 0.998 1 1.002 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 41
m2 p1 m1 m1 m1 p4 p2 + p3
1, m2 2 = 0.5m2 1,m1 = 173GeV
Re(integral) TayInt SecDec
Im(integral) TayInt SecDec 0.99992 0.99996 1 1.00004 1.00008 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec error 0.99975 0.99985 0.99995 1.00005 1.00015 1.00025 200000 600000 1000000 Errors u (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 42
m2 p1 m1 m1 m1 p4 p2 + p3
1, m2 2 = 0.5m2 1,m1 = 173GeV
Re(integral) TayInt SecDec
Im(integral) TayInt SecDec 0.9996 0.9998 1 1.0002 1.0004 200000 300000 400000 Errors s (Gev2)
Truncation Error SecDec Error
0.996 0.998 1 1.002 1.004 200000 300000 400000 Errors s (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 43
m2 m1 p1 p2 p3 m1 m1 m1 p4
1, m2 2 = 0.5m2 1,m1 = 173GeV
0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 1.0004 150000 200000 Re(integral) s (Gev2)
Truncation Error 16 Truncation Error 8 SecDec Error
Daniel Hulme arXiv:1804.06824 29.10.2018 44
m2 m1 p1 p2 p3 m1 m1 m1 p4
1, m2 2 = 0.5m2 1,m1 = 173GeV
0.99 0.995 1 1.005 1.01 150000 200000 Im(integral) s (Gev2)
Truncation Error 16 Truncation Error 8 SecDec Error
0.99992 0.99996 1 1.00004 1.00008 170000 220000
Daniel Hulme arXiv:1804.06824 29.10.2018 45
m2 m1 p1 p2 p3 m1 m1 m1 p4
1, m2 2 = 0.5m2 1,m1 = 173GeV
8.7 · 10-4 8.9 · 10-4 9.1 · 10-4 9.3 · 10-4 Re(integral) TayInt SecDec 5.0 · 10-5 1.5 · 10-4 2.5 · 10-4 Im(integral) TayInt SecDec 0.996 0.998 1 1.002 1.004 200000 300000 400000 Errors s (Gev2)
Truncation Error SecDec Error
0.96 0.98 1 1.02 1.04 200000 300000 400000 Errors s (Gev2) Truncation Error SecDec Error Daniel Hulme arXiv:1804.06824 29.10.2018 46
m2 m1 p1 p2 p3 m1 m1 m1 p4
1, m2 2 = 0.5m2 1,m1 = 173GeV
Re(integral) TayInt SecDec
Im(integral) TayInt SecDec 0.999 0.9994 0.9998 1.0002 1.0006 1.001 200000 300000 400000 Errors s (Gev2)
Truncation Error SecDec Error
0.992 0.996 1 1.004 1.008 200000 300000 400000 Errors s (Gev2)
Truncation Error SecDec Error
Daniel Hulme arXiv:1804.06824 29.10.2018 47
m2 m1 p1 p2 p3 m1 m1 m1 p4
Input Feynman parameters, propagators, kinematics Generate complex transformations Generate finite iterated sectors (U1,U2) Find the complex sectors and kinematic sample (OT1) Create lists needed for finding the Θ Find the contour configuration (Θ) using the MAD (OT2) On that Θ, perform all possible one fold exact integrations (OT3) Use the MAD to choose the best n-1 variable surface, Θ (OT4) Compare the MAD of the n variable and n-1 variable integrands. Check the behaviour
lowest MAD along their edges and at their endpoints and initial points to determine how to discretise the Taylor series (OT5) Create lists needed for performing the discretisation Can we find the series to high
Yes No Increase discretisation Make sure we do not pick an iD with sporadic singularities
the bulk Compute result (OT6)
Daniel Hulme arXiv:1804.06824 29.10.2018 48
Mean TayInt Error Number of Partitions 4 8 Re Im Re Im Order 0.530165 0.623989 0.0812167 0.242449 2 0.0221554 0.0242271 0.000642405 0.00237282 4 0.00278254 0.00242541 0.000163342 0.000079292 6 0.000284179 0.000281809 0.0000239721 0.000038864
Daniel Hulme arXiv:1804.06824 29.10.2018 49
Graph Re (∆) Im (∆) I10 0.000658179 0.000270775 I21 0.00126601 0.000277579 I39 0.0000763027 0.0000668706 The mean difference ∆ between TayInt, using a sixth order expansion, and SecDec, normalised to the SecDec result.
Daniel Hulme arXiv:1804.06824 29.10.2018 50
Daniel Hulme arXiv:1804.06824 29.10.2018 51
Daniel Hulme arXiv:1804.06824 29.10.2018 52
Daniel Hulme arXiv:1804.06824 29.10.2018 53
Integral
literature result 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 10 20 30 40 50 60 70 80 90 100
Errors x
truncation error
0.1 0.2 0.3
(k)
Integral
literature result 0.97 0.98 0.99 1 1.01 1.02 1.03 10 20 30 40 50 60 70 80 90 100
Errors x
truncation error
0.1 0.2 0.3
(l)
x = √ s + 4 m2 − √s √ s + 4 m2 + √s
Daniel Hulme arXiv:1804.06824 29.10.2018 54
1, m2 2 = 0.5m2 1,m1 = 173GeV
Integral
TayInt SecDec 0.998 1 1.002 50000 100000
Errors u (GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 55
1, m2 2 = 0.5m2 1,m1 = 173GeV
Integral
TayInt SecDec 0.999 1.001 50000 100000
Errors u (GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 56
1, m2 2 = 0.5m2 1,m1 = 173GeV
Integral
TayInt SecDec 0.999 1.001 50000 100000
Errors u (GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 57
1, m2 2 = 0.5m2 1,m1 = 173GeV
Integral
TayInt SecDec 0.9985 0.999 0.9995 1 1.0005 1.001 1.0015 50000 100000
Errors s (GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 58
1, m2 2 = 0.5m2 1,m1 = 173GeV
0.00075 0.00076 0.00077 0.00078 0.00079 0.0008 0.00081
Integral
TayInt SecDec
0.998 0.999 1 1.001 1.002 50000 100000
Errors s(GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 59
1, m2 2 = 0.5m2 1,m1 = 173GeV
Integral
TayInt SecDec
0.998 0.999 1 1.001 1.002 50000 100000
Errors s(GeV2)
Truncation Error SecDec Error
Figure:
Daniel Hulme arXiv:1804.06824 29.10.2018 60