gg! hh in the high energy limit

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1 gg! hh in the high energy limit Go Mishima Karlsruhe Institute of Technology (KIT), TTP in collaboration with Matthias Steinhauser, Joshua Davies, David Wellmann work in progress

2 gg! hh : previous works exact analytic@lo [Eboli, Marques, Novaes, Natale, 87, Glover, van der Bij 88] Born-improved HEFT@NLO [Dawson, Dittmaier Spira, 98] FTapprox, FT approx [Maltoni, Vryonidou, Zaro, 14] HEFT@NNLO with 1/mt corr. [Grigo, Hoff, Melnikov, Steinhauser, 13, Grigo, Melnikov, Steinhauser, 14, Grigo, Hoff, Steinhauser, 15, Degrassi, Giardino, Gröber, 16] exact numerical@nlo (14TeV, 100TeV) [Borowka, Greiner, Heinrich, Jones, Kerner, Schlenk, Zicke, 16] Padé approximation using the large top-mass and the threshold expansion@nlo [Gröber, Maier Rauh, 17] We supplement these works by providing the high energy expansion@nlo. 2 /19

3 gg! hh : our aim is to obtain the high energy expansion@nlo. = S/2, ( = /2, p T = p S/2) dσ/dθ [fb/rad] exact O((m 2 t ) 0 ) O((m 2 t ) 1 ) O((m 2 t ) 2 ) O((m 2 t ) 4 ) S [GeV] 3 /19

4 Outline (1) Introduction (2) high energy expansion of Feynman integral (3) calculation of the two-loop gg->hh amplitude (reduction) (4) analytic result of the two-loop massive double box diagram in the high energy limit (preliminary) (5) summary and outlook 4 /19

5 asymptotic expansion of Feynman integral [Smirnov `90, Beneke, Smirnov 97, Smirnov `02, Jantzen `11] is useful when (i) the integral is hard to solve due to multi-scale complexity (ii) certain hierarchy in dimensionful parameters makes sense In our case, we assume m 2 h <m 2 t S T U Then, (1) Expansion in is the Taylor expansion. m h I(m 2 h)=i(0) + m 2 hi 0 (0) + m t m t m t m h (2) Expansion in m t is not the Taylor expansion. m t m h I(m t )= X n (m t ) n f n (S, T, log m t ) We use asy.m to perform the asymptotic expansion. [Pak, Smirnov 10, Jantzen, Smirnov, Smirnov 12] 5 /19

6 Expansion in m h (Taylor expansion) = +m 2 h( O(m 4 h) The massive-higgs diagram can be expressed as an infinite sum of the massless-higgs diagrams. 6 /19

7 Expansion in m t = Z Dk k 2 1 m 2 t 1 (k + p 1 ) 2 m 2 t 1 (k + p 1 + p 2 ) 2 m 2 t 1 (k + p 3 ) 2 m 2 t = 1X (m 2 t ) n f n (S, T, log m t ) n=0 Naive expansion of the integrand like k 2 1 m 2 t = 1 k 2 + m2 t (k 2 ) 2 + gives wrong result. massive massless is finite. = 1 st ( 4 2 log st ε2 ε + 2 log s log t 4π2 3 ) +O( ) 7 /19

8 Expansion by region blue: hard-scaling propagator red: soft-scaling propagators [Beneke, Smirnov 97, Smirnov `02, Jantzen `11] = + + ``all-hard region = massless integral + + the scaling of propagators in terms of alpha-parameter representation 0 ( 4 n=1 dα n ) ) α d/ e m2 α 1234 (sα 1 α 3 +tα 2 α 4 )/α 1234 α 1234 = α 1 + α 2 + α 3 + α 4 8 /19

9 Expansion by region blue: hard-scaling propagator red: soft-scaling propagators [Beneke, Smirnov 97, Smirnov `02, Jantzen `11] = the scaling of propagators in terms of alpha-parameter representation 0 ( 4 n=1 dα n ) ) α d/ e m2 α 1234 (sα 1 α 3 +tα 2 α 4 )/α 1234 α 1234 = α 1 + α 2 + α 3 + α 4 9 /19

10 ( Expansion by region: ``all-hard region In our case, the expansion in this region corresponds to the naive Taylor expansion. The right hand side consists of massless diagrams with dots. ( = + m 2 t ( + (m 2 t ) 2 ( + We can apply the integration by parts (IBP) reduction. 10/19

11 Expansion by region: soft regions = Z 1 0 ( 4 n=1 dα n ) ( ) α d/2 12 e m2 α 12 (sα 1 α 3 +tα 2 α 4 )/α 12 α d/ (α 3 + α 4 )((d/2)α 12 + m 2 (α 12 ) 2 sα 1 α 3 tα 2 α 4 ) e m2 α 12 (sα 1 α 3 +tα 2 α 4 )/α 12 + Usual momentum representation is not always possible The integrals are ill-defined, so we have to introduce analytic regularization of the exponent of propagators. f (2) 0 = (m2 ) ε st f (3) 0 = (m2 ) ε st f (4) 0 = (m2 ) ε st f (5) 0 = (m2 ) ε st [ 1 ε [ 1 ε ε ( 1 1 )] + log st δ 3 δ 4 [ 1 ε ε [ 2 ε ε ( 1 1 ) ] + log t/m 2 + π2 δ 3 δ 4 12 ) ] ( 1δ3 + 1δ4 + log s/m 2 + π2 12 ( ) ] 2 log m 2 + π2 δ 3 δ 4 6 Cancellation of auxiliary parameters between soft regions occurs. 11/19

12 Expansion by region: total = f (1) 0 = 1 st f (2) 0 = (m2 ) ε st f (3) 0 = (m2 ) ε st f (4) 0 = (m2 ) ε st f (5) 0 = (m2 ) ε st ( 4 2 log st ε2 ε [ 1 ) + 2 log s log t 4π2 3 )] ( 1 δ 3 1 δ 4 + log st ε [ 1 ε ε [ 1 ε ε [ 2 ε ε ( 1 1 ) ] + log t/m 2 + π2 δ 3 δ 4 12 ) ] ( 1δ3 + 1δ4 + log s/m 2 + π2 12 ( ) ] 2 log m 2 + π2 δ 3 δ 4 6 I = (m 2 ) n f n n=0 f 0 = 1 st (2 log s m 2 log t m 2 π2 ) [ ( ) Cancellation of auxiliary parameters between soft regions occurs. 12/19

13 Expansion 2 t ) m t : using differential equation 2(d 2) 2m 2 (s + t)+st m 4 (4m 2 + s)(4m 2 + t)(4m 2 (s + t)+st) [Kotikov 91] We used LiteRed [Lee 13] for obtaining the diff.-eq. 2(d 3)t m 2 (4m 2 + t)(4m 2 (s + t)+st) 2(d 3)s m 2 (4m 2 + s)(4m 2 (s + t)+st) (d 4)s 4m 4 (s + t)+m 2 st (d 4)t 4m 4 (s + t)+m 2 st 2(d 5)(s + t) 4m 2 (s + t)+st Substituting the form, = X we obtain recursive relations of C_n s. n 1,n 2 c n1,n 2 (m 2 t ) n1 (log m t ) n2 See also [Melnikov, Tancredi, Wever 16] =(m 2 t ) 0 f 0 +(m 2 t ) 1 f 1 +(m 2 t ) 2 f /19

14 setup to calculate the two-loop amplitude qgraf [Nogueira, 93] : generate amplitudes q2e/exp [Harlander, Seidensticker, Steinhauser, 98, Seidensticker, 99] : rewrite output to FORM notation FIRE [Smirnov, 14] (with LiteRed rules [Lee, 13]) tsort [Smirnov, Pak] : reduction to master integrals : minimization of master integrals Up to this point, we retain the full top mass dependence. 14/19

15 Master integrals at 2 loop 167 =135(planar&crossing)+32(nonplanar&crossing) =29+( ) +crossing +17+(9+3) +crossing +1+(2) +crossing +11+(6+6) +crossing +9 15/19

16 high energy expansion of massive double box There are 13 regions. (all-hard region + 12 soft regions) α[2] Region2 α[5] α[2] Region5 α[5] α[2] Region8 α[5] α[2] Region11 α[5] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[1] α[4] α[1] α[4] α[1] α[4] α[1] α[4] α[2] Region3 α[5] α[2] Region6 α[5] α[2] Region9 α[5] α[2] Region12 α[5] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[1] α[4] α[1] α[4] α[1] α[4] α[1] α[4] α[2] Region4 α[5] α[2] Region7 α[5] α[2] Region10 α[5] α[2] Region13 α[5] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[3] α[7] α[6] α[1] α[4] α[1] α[4] α[1] α[4] α[1] α[4] 16/19

17 Analytic result of massive double box diagram = = T/S, l t = log, l m = log( m 2 t /S) We can evaluate this expression with the package HPL.m [Maitre 05]. +O(m t, ep 2 ) 17/19

18 Analytic result of massive double box diagram leading order in m = S/2, ( = /2, p T = p S/2) Comparison between our result and the numerical result from pysecdec. [Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zicke, 16] finite part epsilon-linear part analytic (real) analytic (imaginary) pysecdec (real) pysecdec (imaginary) analytic (real) analytic (imaginary) pysecdec (real) pysecdec (imaginary) S [GeV] S [GeV] We multiply the integral by m 6 t to make it dimensionless. 18 /19

19 Summary We are calculating the two-loop gg->hh amplitude in the high energy approximation. Reduction to the master integrals is done. Some of the most complicated integrals are evaluated. To Do Complete the evaluation of the planar diagrams. Including higher order of Non-planar diagrams m h 19/19

20 Application to physical process dσ/dθ S = 2000 GeV O(m t 0 ) O(m t 4 ) O(m t 8 ) O(m t 16 ) exact θ/π 20/19

21 higgs-top coupling 0.87 ± 0.15 [ ] see also ATLAS-CONF ! 7% (prospect) [ ] 21/19

22 definition of Vfin 22/19

23 Master integrals at 2 loop 167 =135(planar&crossing)+32(nonplanar&crossing) =29+( ) +17+(9+3) +[s $ t] +[t! u]+[s $ t&t! u] +[s! u]+[s $ t&s! u] +[t! u]+[s! u] +1+(2) +11+(6+6) +[t! u]+[s! u] +[s $ t]+[s $ t&s! u] +9 23/19

24 High pt makes the previous approximation worse Padé approximation using the large top-mass and the threshold [Gröber, Maier Rauh, 17] V fin [GeV 2 ] 10 4 M HH [GeV] p T [GeV] HEFT [n/m] [n/n ± 0, 2] full ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 2: Numbers for the virtual corrections for some representative phase space points for the HEFT result reweighted with the full Born cross section (as in Ref. [78]), the Padé-approximated ones and the full calculation [85]. 24/19