On the numerical evaluation of 3-loop self-energy integrals A. - PowerPoint PPT Presentation

On the numerical evaluation of 3-loop self-energy integrals A. Freitas 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

Introduction 1/21 Need for 3-loop corrections: Electroweak precision tests: Current theory ∗ Current exp. CEPC FCC-ee M W [MeV] 15 4 1 0.5–1 Γ Z [MeV] 2.3 0.4 0.5 0.1 R b = Γ b Z / Γ had [ 10 − 5 ] 66 10 4.3 6 Z sin 2 θ ℓ 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 ...

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

Two-loop electroweak corrections 3/21 Problem has multiple scales: M Z , M W , M H , m t ( m f → 0 , f � = t ) Numerical techniques needed Self-energies (incl. from renormlization) and vertices with sub-loop bubbles using dispersion relation technique S. Bauberger et al. ’95 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 0 0 0 0 1 1 1 0 2 5 2 2 5 6 s s s s M Z 5 6 M Z 3 0 6 3 4 3 4 4 0 0 0 0 0

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

b b b b b b b b b b General 3-loop vacuum integrals 5/21 Relevant for low-energy precision observables ( p 2 ≪ M Z ) Coefficients of low-momentum expansions Building block for more general 3-loop calculations Master integrals: M ( ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , ν 6 ; m 2 1 , m 2 2 , m 2 3 , m 2 4 , m 2 5 , m 2 6 ) � = i e 3 γ E ǫ 1 ] − ν 1 [( q 1 − q 2 ) 2 − m 2 d D q 1 d D q 2 d D q 3 [ q 2 1 − m 2 2 ] − ν 2 π 3 D/ 2 × [( q 2 − q 3 ) 2 − m 2 5 ] − ν 5 [( q 1 − q 3 ) 2 − m 2 3 ] − ν 3 [ q 2 3 − m 2 4 ] − ν 4 [ q 2 2 − m 2 6 ] − ν 6 5 5 1 2 3 4 2 3 2 3 1 4 1 4 6 1 U 4 U 5 U 6 = M (2 , 1 , 1 , 1 , 0 , 0) = M (1 , 1 , 1 , 1 , 1 , 0) = M (1 , 1 , 1 , 1 , 1 , 1)

the k -in tegrations and one of the s -in tegrations an b e p erformed and yield 2 2 2 T ( p ; m ; : : : ; m ; m ) 1 ::: N +2 i 1 N +1 N +2 2 2 2 (1) 2 2 2 = B ( m ; m ; m ) T ( p ; m ; : : : ; m ; m ) 0 i N +2 N N +1 1 N � 1 N +2 Z 1 2 2 1 � B ( s; m ; m ) 0 N N +1 (1) 2 2 � ds T ( p ; m ; : : : ; m ; s ) : (68) i 1 N � 1 2 2 � i s � m s 0 N +2 (1) T denotes a one-lo op N-p oin t fun tion in whi h s en ters in the remaining one-dimensional in tegration as a mass v ariable. A diagram with t w o four-v erti es leads to a result whi h is similar to the remaining in tegration in (68), m N � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sub-loop dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p . . . . . . . p . . . . . . . 6/21 N . . . . . . . . N � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m . . . m . . . . . N +1 . . . N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t t Topologies with self-energy sub-loop can easily be integrated by using . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m . . . . . . . . . . p . . . . 1 . . . . p . . . . . . . . . 1 . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dispsersion relation for B 0 function: S. Bauberger et al. ’95 1 Z 1 2 2 2 � ∞ ( m 1 + m 2 ) 2 d s ∆ B 0 ( s, m 2 T ( p ; m ) = 1 , m 2 ds � B ( s ; m ; m ) 1 ::: N +1 i 0 i N N +1 2 ) 2 � i B 0 ( p 2 , m 2 1 , m 2 2 ) = − s 0 � � s − p 2 � 1 1 1 � : : : 2 2 2 2 2 k � s ( k + p ) � m ( k + p + : : : + p ) � m 1 1 N � 1 1 N � 1 1 Z λ ( D − 3) / 2 ( s, m 2 1 , m 2 2 ) 2 ) = (4 πµ 2 ) 4 − D Γ( D/ 2 − 1) 1 2 2 (1) 2 2 ∆ B 0 ( s, m 2 1 , m 2 = � ds � B ( s ; m ; m ) T ( p ; m ; : : : ; m ; s ) : (69) with 0 , i N N +1 1 N � 1 2 � i s D/ 2 − 1 Γ( D − 2) s 0 λ ( a, b, c ) = ( a − b − c ) 2 − 4 bc 4.2 Examples An appli ation of (69) to the London transp ort diagram leads to 1 Z 1 2 2 2 2 2 2 2 2 � ∞ T ( p ; m ; m ; m ) = � ds � B ( s ; m ; m ) B ( p ; s; m ) ; (70) 123 0 0 1 2 3 2 3 1 2 � i T N +1 ( p i ; m 2 d s ∆ B 0 ( s, m 2 N , m 2 i ) = − N +1 ) 2 ( m + m ) 2 3 s 0 a result whi h w ould also follo w from (5). In that ase a suitable subtra tion [6 ℄ is � 1 1 1 d 4 q · · · 2 2 2 2 2 2 2 2 2 2 2 × T ( p ; m ; m ; m ) = T ( p ; m ; m ; m ) � T ( p ; m ; 0 ; m ) (71) 123 N 123 123 1 2 3 1 2 3 1 3 q 2 − s ( q + p 1 ) 2 − m 2 ( q + p 1 + ··· + p N − 1 ) 2 − m 2 2 2 2 2 2 1 � T ( p ; 0 ; m ; m ) + T ( p ; 0 ; 0 ; m ) : N − 1 123 123 2 3 3 F or T one obtains from (68) 1234 2 2 2 2 2 T ( p ; m ; m ; m ; m ) 1234 1 2 3 4 17

b b b b b b b U 4 7/21 2 4 p → = B 0 ,m 1 ( p 2 , m 2 1 , m 2 2 ) B 0 ( p 2 , m 2 3 , m 2 4 ) � ∞ ds ∆ I db ( s ) = 1 1 3 s − p 2 − iε 0 ∆ I db ( s, m 2 1 , m 2 2 , m 2 3 , m 2 4 ) = ∆ B 0 ,m 1 ( s, m 2 1 , m 2 2 ) B 0 ( s, m 2 3 , m 2 4 ) + B 0 ,m 1 ( s, m 2 1 , m 2 2 ) ∆ B 0 ( s, m 2 3 , m 2 4 ) , � s − ( m a + m b ) 2 � b ) = 1 ∆ B 0 ( s, m 2 a , m 2 sλ ( s, m 2 a , m 2 b ) Θ b ) = m 2 a − m 2 � s − ( m a + m b ) 2 � b − s ∆ B 0 ,m 1 ( s, m 2 a , m 2 b ) Θ a , m 2 s λ ( s, m 2 � � ∞ 4 ) = − e γ E ǫ ds ∆ I db ( s ) 1 U 4 ( m 2 1 , m 2 2 , m 2 3 , m 2 d D q 3 2 3 4 q 2 iπ D/ 2 3 − s + iε 0 � ∞ 1 = − ds A 0 ( s ) ∆ I db ( s ) 0

b b b U 4 8/21 Problem: U 4 is divergent 1 Solution: 2 3 4 1 U 4 ( m 2 1 , m 2 2 , m 2 3 , m 2 4 ) = U 4 ( m 2 1 , m 2 2 , 0 , 0) + U 4 ( m 2 1 , 0 , m 2 3 , 0) + U 4 ( m 2 1 , 0 , 0 , m 2 4 ) − 2 U 4 ( m 2 1 , 0 , 0 , 0) + U 4 , sub ( m 2 1 , m 2 2 , m 2 3 , m 2 4 ) → U 4 ( m 2 X , m 2 Y , 0 , 0) can be computed analytically → U 4 , sub is finite � ∞ U 4 , sub ( m 2 1 , m 2 2 , m 2 3 , m 2 4 ) = − ds A 0 , fin ( s ) ∆ I db , sub ( s ) 0 I db , sub ( s, m 2 1 , m 2 2 , m 2 3 , m 2 4 ) = � � ∆ B 0 ,m 1 ( s, m 2 1 , m 2 B 0 ( s, m 2 3 , m 2 2 ) Re 4 ) − B 0 ( s, 0 , 0) � � − ∆ B 0 ,m 1 ( s, m 2 B 0 ( s, 0 , m 2 3 ) + B 0 ( s, 0 , m 2 1 , 0) Re 4 ) − 2 B 0 ( s, 0 , 0) � � � � B 0 ,m 1 ( s, m 2 1 , m 2 ∆ B 0 ( s, m 2 3 , m 2 + Re 2 ) 4 ) − ∆ B 0 ( s, 0 , 0) � � � � B 0 ,m 1 ( s, m 2 ∆ B 0 ( s, 0 , m 2 3 ) + ∆ B 0 ( s, 0 , m 2 1 , 0) 4 ) − 2 ∆ B 0 ( s, 0 , 0) − Re