Precision Physics at the LHC

Transcription

1 Precision Physics at the LHC V. Ravindran The Institute of Mathematical Sciences, Chennai, India IMHEP-2019, IOP, Bhubaneswar, Jan 2019

2 Precision Physics

3 Plan Why Precision Calculation (PC) Impact of PCs on discoveries Part-1 Methods for Multi-leg processes Part-2 Methods for Multi-loops processes Part-3 Infrared physics

4 Large Hadron Collider Excellent Discovery Reach Higgs Supersymmetry Extra-Dimensions Anything else

5 Large Hadron Collider Large amount of events W! e : 10 8 events Z! e + e : 10 7 events tt production 10 7 events Higgs production 10 5 events

6 Standard Model Theories Quantum Chromodynamics Electroweak Theory (SM) Theory of Gravity

7 Large Hadron Collider Large background Large number of,l ±,Z,W ± Jets Large number of tt, bb

8 Theoretical Issues Issues to be tackled Kinematics Normalisation Renormalisation and Factorisation Scales Parton distribution functions Phase space boundary e ects, resummation

9 Why Precision?

10 The truth is in the Details

11 Precision Physics Experiment Theory

12 Tests of Quantum Chromodynamics QCD RGE prediction for DIS QCD Jets at LEP

13 Hits from LEP for Top and Higgs EW Radiative Corrections M Z,M H,m t, s (M Z ), (M Z ) m t = ± 3.9 GeV m H = GeV

14 Stability of our Vacuum NNLO Electroweak Correction : Fate of the universe depends on the mass of top

15 Leading order is often Crude in QCD LHC Leading Order

16 Exclusion Plot for Higgs mass

17 LO is often a crude estimate 2S H (x, m H )= Z 1 x dz z (0) gg (z,µ )2ŝˆ(0) F gg x z,m2 H, µ R + 2ŝˆ(0) gg x z,m2 H, µ R = 2 s(µ R )G F F (m t,m H ) LO prediction is unreliable due to % scale uncertainty

18 True Result for Higgs pb

19 QCD + EW for Higgs 13 Tev LHC

20 Theory Vs Experiment Agreement with SM f Higgs Boson

21 Top production Large theory uncertainty s (µ 2 R ) f g(x, µ 2 F )

22 Theory Prediction

23 Theory Vs Experiment

24 Parton Model in QCD Hadronic Cross section: Partonic Flux: Partonic cross section: Precision Measurements Precise theory Discover/Test Physics

25 Inputs that can affect UV Renormalisation Scale, Strong coupling s (µ R ) Factorisation Scale and Parton Distribution Functions Missing Higher Order corrections f a (x, µ F ) Stability of the perturbation theory Resummation Methods Hadronisation models

26 LO (Tree level)

27 No. of diagrams For Jet / background to BSM n g + g! ng no. of diagrams

28 Polarisation Vectors Gauge Choice Polarisation Sum

29 SU(N) color algebra f abc f abc

30 QCD improved Parton Model Partial Amplitude Partial Amplitudes - No color information - Gauge Invariant

31 Cyclic Ordered Feynman Rules No Color Factors!!!

32 Berends-Giele Recursion - Off-Shell currents - No Feynman diagrams

33 BCFW relation

34 Park-Taylor Amplitude MHV n - gluon Amplitudes

35 Twister space Momenta in bi-spinor p µ! a ȧ Scaling a!z a ȧ! 1 z ȧ Transform ȧ! i ȧ ȧ Fourier Transform f( ȧ) = Z d 2 p (2 ) 2 exp(i ȧ ȧ) f( ȧ)

36 Weinzierl s comparison

37 NLO

38 Beyond LO Loop integral I = YN k i=1 Z d n k i N({k l }, {p m }) (2 ) n D a 1 1 Da 2 2 Da N d N d Phase space integrals d N = NY i=1 Z d n p i (2 ) n (p2 i m 2 i ) (2 ) n n (q X i p i )

39 Loop Integrals Numerator: Reducible Irreducible Reducible if (p i k j ) a k (k i k j ) b k } are expressible in terms of Denominators

40 Loop Integrals k + p! p! p! Z d n k (k p) a (2 ) n D 1 D 2 D 1 = k 2 + i k p D 1 = 1 2 k p D 2 = 1 2 k D2 D p 2 D 1 D 1 p 2 D 2 D 2 D 2 =(k + p) 2 + i, p 2 < 0 } Reducible Z d n k (2 ) n k p D 1 D 2 = 1 2 [ - - p 2 ]

41 Tensorial Reduction D 2 D 1 I µ = Z d n k (2 ) n k µ D 1 D 2 D 3 D 3 I µ = A 1 p 1µ + A 2 p 2µ I 1 = p 1 I = A 2 p 1 p 2 I 2 = p 2 I = A 1 p 2 p 1 A 1 = 1 2[ 2p 1 p - + s ]

42 Ossola-Papadopoulos-Pittau (OPP method) Integrand : N(k, p i ) D 1 D 2 D 3 D 4 D i =(k + X j c j p j ) 2 m 2 i N(k, p i )= mx i 1 <i 2 <i 3 <i 4 mx mx i 1 <i 2 h d(i 1 i 2 i 3 i 4 )+ d(k, i i 1 i 2 i 3 i 4 ) i 1 <i 2 <i 3 [c(i 1 i 2 i 3 )+ c(k, i 1 i 2 i 3 )] mx i 1 h i b(i 1 i 2 )+ b(k, i 1 i 2 ) my D i i6=i 1 i 2 i 3 Y m i6=i 1 i 2 D i h i Y m b(i 1 i 2 )+ b(k, i 1 ) D i + P (q) i6=i 1 m Y i6=i 1 i 2 i 3 i 4 D i my i=1 Best suited for Numerical Methods D i

43 NLO QCD - Tool Kits Analytical Tools Faster generation of Feynman diagram QGRAF Symbolic Manupulation: FORM,Mathematica On-Shell Methods BCFW Recursion techniques BG Merging NLO with Showers MC@NLO POWEG SHERPA VINCIA GENeVa amc@nlo KRKMC Semi-Numerical Methods Helac-NLO CutTools BlackHat Rocket SAMURAI MADLoop GoSam Ngluon

44 NLO revolution Golem, HELAC BlackHat

45 2- loop

46 Integration By Parts (IBP) N k Y i=1 Z d n k i 1 (2 ) n D a 1 1 Da 2 D i = q 2 i + i 2 Da N d N d q i = X j k l + X l p j n-dimensional Gauss theorem Z YN k i=1 d n k i (2 j,µ k l,µ D a 1 1 Da 2 2 Da Ne N d! =0 j, l =1, N k Z YN k i=1 d n k i (2 j,µ p l,µ D a 1 1 Da 2 2 Da Ne N d! =0 l =1, N e 1 Integration By Parts (IBP) identities

47 Integration By Parts (IBP) Number of IBP identities = N k (N k + N e 1) IBP identities: X C i (s ij,n) I i ({p j },n,a 1, a )=0 Ne i Integrals: i =1, N I s ij =(p i + p j ) 2 Solving IBP identities -> Master Integrals: I i ({p j },a 1, a Ne ) i =1, N MI Good News N MI << N I

48 Integration By Parts (IBP) I(a 1,a 2 )= Z Z d n k (2 ) n d n k 1 (2 ) n (k 2 ) a 1 ((k + p)2 ) µ v µ (k 2 ) a 1 ((k + p)2 ) a 2 =0 v = k, p I(a 1,a 2 )= a 1 + a 2 n 1 p 2 (a 2 1) I(a 1,a 2 1) + 1 p 2 I(a 1 1,a 2 ) a 2 MI I(1, 1) a 1

49 Lorentz Invariant Identities Integrals are Lorentz scalars! µ =! µ p µ i! pµ i + pµ i = pµ i +!µ p, I(p i + p i )=I(p i )+! µ X j p I(p i )=I(p i µ j Anti-symmetry! µ =! µ X i ] i I(p i )=0 Anti-symmetry of p [µ j p ] k p [µ j p ] k X i ] i I(p i )=0

50 Solving Master Integrals Consider a Master Integral I(s ij,n)= N k Y i=1 Z d n k i 1 (2 ) n D 1 D 2 D Nf Define s 12 = s D i = q 2 i + i q i = X j k l + X l p j s =(p 1 + p 2 ) 2 Differential w.r.t I(s ij,n)= N k Y i=1 Z d n k i (2 ) 1 D 1 D 2 D Nf

51 Master Integrals Generalization with set of MIs {I i (~x)} ~I = I 1,I 2,,,,I N depend on Scaling variables x ~x = 1,x 2,,,,x M x i = f i sij Q 2 Differential equation: 2 d ~ I = i I N 3 MX i=1 2 4 A i dx i ~ I A 11 A 1N A N1 A NN I 1 I N 3 5 i =1, 2,,, M

52 Canonical/Henn's Basis Consider Diff equation: d ~ I(~x, n) = X i A i (~x, n)dx i ~ I(~x, n) Choose U Transformation such that U 1 A(~x, n)u U 1 du =(n 4) A(~x) Diff equation contains n independent A d ~ I(~x, n) =(n 4) X i A i (~x)dx i ~ I(~x, n) Solution ~ I(~x, n) = ~ I(~x0,n) P exp (n 4) Z d A( ) P - Path Ordered exponential

53 Canonical/Henn's Basis Start with Henn s Diff I(s, n) =(n 4)A(s) I(s, n) If A contains poses at s i A(s) = X i à i (s i ) s s i I(s, n) =I (0) (s 0 )+(n 4) X i à i (s i ) log s s 0 si s i +(n 4) 2 X i Ã(s i )L i (s 0,s i )+ Polylogarithms - Uniform transcendental terms

54 NNLO

55 NNLO corrections: Pure Virtual ( ) Virtual - Real ( ) Real-Real ( )

56 Reverse Unitarity Real-Real Reverse Unitarity Loop Integrals

57 Higgs production to N 3 LO in QCD at the LHC Anastasiou,Duhr,Dulat,Furlan,Gehrmann,Herzog,Mistlberger Integration By Parts Lorentz Invariance Integrals Master Integrals

58 NNLO n-jettiness: T N < T cut N T N NNLO - analytical { T cut N H- pure virtual B - Initial state beam fns. S - Soft distribution fns. J - Final state jet fns. T N T cut N NLO with two jets finite- numerically

59 NNLO n-jettiness: T N < TN cut and T N TN cut

60 qt subtraction at N3LO: Note that qt resummation gives

61 qt subtraction at N3LO:

62 Resummation

63 Small qt Resummation for DY

64 Small qt Resummation for DY

65 Small qt Resummation for Higgs

66 Small qt Resummation for Higgs

67 Rapidity Distribution Rapidity Distribution of any colorless particle: d I dy =ˆIB X ab=q, q,g Z 1 x 0 1 dz 1 z 1 Z 1 x 0 2 dz 2 z 2 Ĥ I ab x 0 1, x0 2,µ 2 z 1 z 2 ˆ I d,ab z 1,z 2,q 2,µ 2 DY production of lepton pairs Higgs through gluon (bottom anti-bottom), I = d q,q 2,y dq 2. I = g(b),q 2,y. Rapidity: y = 1 2 ln p2 q p 1 q =ln x 0 1 x 0 2, = x 0 1x 0 2 Partonic Scaling variables: z 1 = x0 1 x 1, z 1 = x0 2 x 2

68 Soft and Virtual terms ( I d = (1 z 1 ) (1 z 2 )+a s c (1) 1 (1 z 1 ) (1 z 2 ) +c (1) 2 ln(1 z1 ) 1 z R (1) (z 1,z 2 )+z 1 $ z 2 ) + O(a 2 s) I d(z 1,z 2 )= I,SV (z d 1.z 2 )+ I,hard (z d 1,z 2 ) Virtual, Soft (1 z i ) ln(1 zi ) (1 z i ) + I,SV d (!) = Z 1 0 dz 1 z N Z 1 0 dz 2 z N I,SV d (z 1,z 2 )

69 Resummation to NNLL Logarithms that are resummed in g I d O(a s ) ln 2 ( N 1 N2 ) ln( N 1 N2 ) O(a 2 s) ln 3 ( N 1 N2 ) ln 2 ( N 1 N2 ) ln( N 1 N2 ) O(a 3 s) ln 4 ( N 1 N2 ) ln 3 ( N 1 N2 ) ln 2 ( N 1 N2 ) Resummed terms: LL NLL NNLL a m s ln m+1 ( N 1 N2 ) a m s ln m ( N 1 N2 ) a m+1 s ln m ( N 1 N2 ) Function that resums : g I d,1 ln( N 1 N2 ) g I d,2 a s g I d,3

70 Soft Gluon Resummation Double Mellin Transformation: I,SV d (!) = Z 1 0 dz 1 z N Z 1 0 dz 2 z N I,SV d (z 1,z 2 ) Resumed Rapidity distribution: Ni dependent! = a s 0 ln N 1 N 2 Ni independent N i = e E N i

71 Rapidity of Higgs at NNLO + NNLL 14 dσ/dy [pb] LO NLO NNLO LO+LL NLO+NLL NNLO+NNLL M H = 125 GeV µ = [M /2,2M H ] H MMHT2014 LHC K-factor 2 NLO/LO NNLO/LO Fixed order CS M H /2 apple µ R,F apple 2M H Resummed CS =) 2 NLO+NLL/LO+LL NNLO+NNLL/LO+LL 1 y At NNLO+NNLL result stabilises convergence of perturbation series!

72 Rapidity of DY at NNLO + NNLL

73 Resummation at N3LL for Higgs

74 Pseudoscalar Higgs at N3LO(A) + N3LL

75 Conclusion The truth is in the Details

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