Mass-corrections to double-Higgs production & TopoID
aa r X i v : . [ h e p - ph ] J u l Mass-corrections to double-Higgs production &TopoID
Jonathan Grigo and Jens Hoff ∗ † Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT)E-mail: [email protected] , [email protected] We consider power corrections due to a finite top-quark mass M t to the production of a Higgs bo-son pair within the Standard Model at next-to-leading order (NLO) in QCD. Previous calculationsfor this process and at this precision were done in the limit of an inifinitely heavy top quark. Ourresults for the inclusive production cross section at NLO include terms up to O (cid:0) / M t (cid:1) .We present the Mathematica package
TopoID which for arbitrary processes aims to performthe necessary steps from Feynman diagrams to unrenormalized results expressed in terms of mas-ter integrals. We employ it for advancing in this process towards next-to-next-to-leading order(NNLO) where further automatization is needed.
Loops and Legs in Quantum Field Theory27 April 2014 - 02 May 2014Weimar, Germany ∗ Speaker. † This work was supported by the DFG through the SFB/TR9 “Computational Particle Physics” and the KarlsruheSchool of Elementary Particle and Astroparticle Physics (KSETA). We would like to thank Kirill Melnikov and MatthiasSteinhauser for the productive collaboration and also for numerous cross-checks and useful suggestions concerning
TopoID . c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ass-corrections to double-Higgs production & TopoID
Jens Hoff
1. Introduction
Its is still an open question whether the scalar particle discovered by ATLAS and CMS [1,2] at CERN is indeed the Higgs boson of the Standard Model (SM). In forthcoming years itscouplings to the various gauge bosons and fermions will be measured with improved precisionto verify their compatibility with the values dictated within the SM. But to gain insight into themechanism of electroweak symmetry breaking the particles self-interactions need to be probed,too. The process granting this possibility is production of a Higgs boson pair via gluon fusionwhich has two contributions: One where both Higgs bosons couple to top quarks, the other oneinvolves the cubic coupling l of the SM Higgs potential (see fig. 3) V ( H ) = m H H + l vH + l H , (1.1)with the Higgs mass m H , vacuum expectation value v , and l = m H / v ≈ .
13 for the SM. Notethat the influence of the second contribution is strongly suppressed compared to the first one, but be-comes noticeable through its large destructive interference. The process has a relatively small crosssection and suffers from large backgrounds, making the extraction of the Higgs self-interaction atthe LHC a challenge. However, a number of studies suggest the prospect of measuring l [4, 3, 5, 6],some within an accuracy of about 30% with at least 3000 fb − accumulated luminosity [5, 6].The leading order (LO) result with exact dependence on the top quark mass M t has been knownsince long [7, 8]. Further terms in the perturbation series have been computed in the approxima-tion of an infinitely heavy top quark M t → ¥ at NLO [9] and just recently at NNLO [10]. It isimportant to remark that doing so, the exact LO result has been factored off in the NLO and NNLOcontributions.
2. Results
It is known that the 1 / M t expansion works extremely well for the case of a single Higgs boson[11, 12, 13] employing the aforesaid factorization procedure. For that reason we computed fordouble-Higgs production at NLO power corrections due to a finite top quark mass to the total crosssection in the following way: s NLOexpanded → s LOexact s NLOexpanded s LOexpanded , (2.1)where numerator and denominator are expanded to the same order in 1 / M t . In [14] we presentedresults expanded up to O (cid:0) / M t (cid:1) and in [15] to O (cid:0) / M t (cid:1) , here they are available to O (cid:0) / M t (cid:1) .The discussion of results has not changed by including the new terms. Therefore we only want toshow updated plots for the hadronic cross section, see fig. 1 and fig. 2 and summarize our findings. • The common enhancement by gluon luminosity of low- ˆ s contributions, for which we observegood convergence, enlarges the validity range of the expansion. • Including 1 / M t corrections is necessary to detect deviations in l of O ( ) .2 ass-corrections to double-Higgs production & TopoID Jens Hoff ds N L O (f b ) √ s cut (GeV) r r r r r r r Figure 1:
NLO contribution (without LO) to the hadronic cross section. The color coding indicates higherexpansion orders in r = m H / M t . √ s cut , a cut on the partonic ˆ s , can be seen as an approximation for theinvariant mass of the Higgs pair. s N L O (f b ) no r m a li ze d √ s cut (GeV) r ± D r r ± D r l ± r r ± r r ± r l ± r Figure 2:
The straight black line shows the hadronic NLO cross section up to O (cid:0) r (cid:1) , the dashed black linesindicate the variation from changing the SM value of l within ± O (cid:0) r (cid:1) and to the O (cid:0) r (cid:1) expansion, respectively. ass-corrections to double-Higgs production & TopoID Jens Hoff s tot. ( gg → HH ) ∼ Disc. ( M ( gg → gg )) Z d PS (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∼ + + Figure 3:
For Higgs boson pair production we only need to consider cuts (denoted by short dashed verticallines) through two Higgs bosons (long dashed black lines) and additional partons (beginning at NLO). Thiscorrespondence is depicted here for the LO order contributions. Curly red lines represent gluons, straightblue lines massive top quarks. • Compared to the prediction in the M t → ¥ limit we obtain for the LHC at 14 TeV s NLO ( pp → HH ) = . LO + . NLO, M t → ¥ fb → s NLO ( pp → HH ) = . LO + ( . ± . ) NLO, 1 / M t fb , (2.2)where no cut on the partonic center-of-mass energy ˆ s was applied and equal factorization andrenormalization scale m = m H was chosen. • This can be either seen as an improvement of current precision with corrections of about20% or at least as reliable error estimate for a NLO computation of this process.
3. Techniques
Being interested mainly in the total cross section gg → HH , we can make use of the op-tical theorem (see, e.g., [16]) and compute imaginary parts or discontinuities of the amplitude M ( gg → gg ) related to a Higgs pair instead of squaring M ( gg → HH ) and performing the phasespace integration (see fig. 3). On the one hand this method simplifies the calculation, namely:forward scattering kinematics, common treatment of contributions related to different phase spaceintegrations and computation of the latter only in the very end at master integral level. On the otherhand, one has to compute a larger number of diagrams with more loops.The second ingredient making this calculation feasible is the asymptotic expansion at diagram-matic level (see, e.g., [17]) in the hierarchy M t ≫ ˆ s , m H which corresponds to a series expansion ofan analytic result in the parameter r = m H / M t . This procedure effectively reduces the number ofloops and scales in the integrals to be evaluated (see fig. 4), thus diminishing some of the drawbacksconnected to use of the optical theorem.Within this framework our toolchain for the various steps of the calculation looks as follows:1. generation of Feynman diagrams with QGRAF [18],2. selection of diagrams which have the correct cuts [19],3. asymptotic expansion with q2e and exp [20, 21],4 ass-corrections to double-Higgs production & TopoID
Jens Hoff ( M t , m H , s ) −→ ( M t ) × ( m H , s ) × ( M t ) Figure 4:
Applying the rules for asymptotic expansion to a single Feynman diagram one obtains in general asum of contributions (there is only one in this example). Each contribution in turn is a product of subgraphs(containing the hard scale; M t in our case) and co-subgraphs (containing the soft scales; m H , s ). The notationis as in fig. 3.
4. reduction to scalar integrals in
FORM [22, 23] and/or
TFORM [24],5. reduction to master integrals by rows [19] and/or
FIRE [25, 26],6. minimization of the set of master integrals [19].Step 2 is necessary since one cannot steer
QGRAF in such a way that only diagrams with a spe-cific cut structure are generated. Because of that we filter the diagrams provided by
QGRAF forthose which exhibit an appropriate cut in the s-channel corresponding to an interference term fromsquaring the amplitude for gg → HH . (Usually only about 10-30% of the initial diagrams pass thefilter.) At NLO step 4 turned out to be the bottleneck of the calculation for going to higher ordersin the expansion parameter r .
4. TopoID
Up to now the input for steps 3-6 in the above list was usually provided manually. For goingbeyond NLO we use
TopoID to provide all that information in an automatic fashion. More pre-cisely: all the graphs corresponding to a topology as “mapping patterns” for step 3,
FORM codeprocessing aforementioned topologies in step 4 and definitions of topologies suitable for reductionwith the programs listed for step 5.When performing a multi-loop calculation one often works with a set of topologies and withineach topology integrals are reduced to a finite set of master integrals. The same master integral maythus be represented in different ways by single integrals of various topologies.
TopoID is capableof providing such an identification, as are recent versions of
FIRE [26]. Moreover, there exist alsonon-trivial linear relations involving multiple “master integrals” which can be found with the helpof this package (step 6).A diagram class or family T , usually referred to as topology, is a set of N scalar propagators { d i } with arbitrary powers { a i } , usually referred to as indices, composed of masses { m i } and linemomenta { q i } . The line momenta { q i } are linear combinations of E external momenta { p i } and I ass-corrections to double-Higgs production & TopoID Jens Hoff
12 p1p2p1p2 1 23 p1p2p1p2 3 1 p12 4 p2p1p2 3 1 p22 4 p1p1p2
Figure 5:
Sample one-loop topologies appearing after asymptotic expansion at LO, NLO and NNLO (thelast two). The first graph is an example of a linearly independent, but incomplete topology. The secondtopology is a linearly independent and complete. The last two topologies, one planar, one non-planar, arelinearly dependent and complete. Plain black lines are massless, the double lines carry the Higgs mass. internal momenta { k i } with integers c i j , d i j , T ( a , . . . , a N ) = ( I (cid:213) i = Z dk Di ) N (cid:213) j = h m j + q j i a j , (4.1) q i = E (cid:229) j = c i j p j + I (cid:229) j = d i j k j . (4.2)For particular kinematics, i.e. given external and internal momenta, supplemented by possibleconstraints, e.g. putting particles on-shell, one can form all occuring scalar products x p i p j = p i · p j , s p i p j = p i · p k , s k i k j = k i · k j . (4.3)If the denominators of a topology { d i } allow for expressing each of the internal scalar products s i j the topology is complete, otherwise incomplete. In the latter case affected scalar products arecalled irreducible scalar products and appear only as numerators (fig. 5 shows some examples forHiggs pair production).Diagram topologies, i.e. mapping patterns for Feynman diagrams, in general are incompleteand also linearly dependent, viz. linear relations among the { d i } exist. In contrast, reductiontopologies need to be linearly independent and complete. This is exemplified in fig. 6 with the two-loop topologies emerging after asymptotic expansion of the purely virtual five-loop diagrams atNNLO . The mapping between these two types of topologies can in general become quite intricatefor larger sets but is handled easily by TopoID .The foundation of this automatization is the a -representation of Feynman integrals T ( a , . . . , a N ) = c ( N (cid:213) i = Z ¥ d a i ) d (cid:0) − S Ni = a i (cid:1) ( N (cid:213) j = a a j j ) U a W b , (4.4)where c , a and b depend on I , D and the { a i } only. The polynomials U and W are homogeneousin the { a i } and encode the complete information on the topology (for further details see, e.g.,ref. [27]). This representation is unique up to renaming of the a -parameters, but this ambiguity In this case two massive tadpole diagrams containing the top quarks (one with one loop and one with two loops)and a two-loop box diagram with Higgs mass remain after asymptotic expansion. ass-corrections to double-Higgs production & TopoID Jens Hoff
6, 7 < 8
6, 7 <8 < 8 <8 < Figure 6:
The left hand side shows the set of diagram topologies, the right hand side the set of reductiontopologies. Their order is chosen by
TopoID in a fixed way, numbers in braces denote the presence ofirreducible numerators. The last topology in both sets is an example of a factorizing topology. Note themodification of self-energy insertions from the left to the right side, propagators carrying the same momen-tum are identified. Furthermore, there is a non-trivial mapping from the second and fourth diagram topologyto the fourth reduction topology which cannot be deduced from the graphs alone. can be eliminated by applying the procedure described in [28] to derive a canonical form of the a -representation, making it a suitable identifier. TopoID is a generic, process independent tool and bridges the gap between Feynman dia-grams and unrenormalized results expressed in terms of actual master integrals, i.e. including thenon-trivial relations, in a completely automatic way. It is written as a package for
Mathematica which offers a high-level programming environment and the demanded algebraic capabilities.However, for the actual calculation
FORM code is generated to process the diagrams in an effectiveway. Let us briefly summarize features the package has to offer: • topology identification and construction of a minimal set of topologies, • classification of distinct and scaleless subtopologies, • access to properties such as completeness, linear dependence, etc., • construction of partial fractioning relations, • revealing symmetries (completely within all levels of subtopologies), • graph manipulation, treatment of unitarity cuts, factorizing topologies, • FORM code generation (diagram mapping, topology processing, Laporta reduction), • master integral identification (arbitrary base changes, non-trivial relations).As one cross-check we repeated the NLO calculation within this automatized setup and foundagreement with our previous calculation. 7 ass-corrections to double-Higgs production & TopoID Jens Hoff
References [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B (2012) 1 [arXiv:1207.7214 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B (2012) 30 [arXiv:1207.7235 [hep-ex]].[3] U. Baur, T. Plehn and D. L. Rainwater, Phys. Rev. D (2004) 053004 [hep-ph/0310056].[4] J. Baglio, A. Djouadi, R. Gröber, M. M. Mühlleitner, J. Quevillon and M. Spira, JHEP (2013)151 [arXiv:1212.5581 [hep-ph]].[5] F. Goertz, A. Papaefstathiou, L. L. Yang and J. Zurita, JHEP (2013) 016 [arXiv:1301.3492[hep-ph]].[6] V. Barger, L. L. Everett, C. B. Jackson and G. Shaughnessy, Phys. Lett. B (2014) 433[arXiv:1311.2931 [hep-ph]].[7] E. W. N. Glover and J. J. van der Bij, Nucl. Phys. B (1988) 282.[8] T. Plehn, M. Spira and P. M. Zerwas, Nucl. Phys. B (1996) 46 [Erratum-ibid. B (1998) 655][hep-ph/9603205].[9] S. Dawson, S. Dittmaier and M. Spira, Phys. Rev. D (1998) 115012 [hep-ph/9805244].[10] D. de Florian and J. Mazzitelli, Phys. Rev. Lett. (2013) 201801 [arXiv:1309.6594 [hep-ph]].[11] A. Pak, M. Rogal and M. Steinhauser, JHEP (2010) 025 [arXiv:0911.4662 [hep-ph]].[12] S. Marzani, R. D. Ball, V. Del Duca, S. Forte and A. Vicini, Nucl. Phys. B (2008) 127[arXiv:0801.2544 [hep-ph]].[13] R. V. Harlander and K. J. Ozeren, JHEP (2009) 088 [arXiv:0909.3420 [hep-ph]].[14] J. Grigo, J. Hoff, K. Melnikov and M. Steinhauser, Nucl. Phys. B (2013) 1 [arXiv:1305.7340[hep-ph]].[15] J. Grigo, J. Hoff, K. Melnikov and M. Steinhauser, PoS RADCOR (2014) 006 [arXiv:1311.7425[hep-ph]].[16] C. Anastasiou and K. Melnikov, Nucl. Phys. B (2002) 220 [hep-ph/0207004].[17] V. A. Smirnov, Springer Tracts Mod. Phys. (2002) 1.[18] P. Nogueira, J. Comput. Phys. (1993) 279.[19] J. Hoff, A. Pak, (unpublished).[20] R. Harlander, T. Seidensticker and M. Steinhauser, Phys. Lett. B (1998) 125 [hep-ph/9712228].[21] T. Seidensticker, hep-ph/9905298.[22] J. A. M. Vermaseren, math-ph/0010025.[23] J. Kuipers, T. Ueda, J. A. M. Vermaseren and J. Vollinga, Comput. Phys. Commun. (2013) 1453[arXiv:1203.6543 [cs.SC]].[24] M. Tentyukov and J. A. M. Vermaseren, Comput. Phys. Commun. (2010) 1419 [hep-ph/0702279[HEP-PH]].[25] A. V. Smirnov, JHEP (2008) 107 [arXiv:0807.3243 [hep-ph]].[26] A. V. Smirnov and V. A. Smirnov, Comput. Phys. Commun. (2013) 2820 [arXiv:1302.5885[hep-ph]].[27] V. A. Smirnov, Springer Tracts Mod. Phys. (2012) 1.[28] A. Pak, J. Phys. Conf. Ser. (2012) 012049 [arXiv:1111.0868 [hep-ph]].(2012) 012049 [arXiv:1111.0868 [hep-ph]].