On the universal structure of Higgs amplitudes mediated by heavy particles

Transcription

1 On the universa structure of Higgs ampitudes mediated by heavy partices a,b, Féix Driencourt-Mangin b and Germán Rodrigo b a Dipartimento di Fisica, Università di Miano and INFN Sezione di Miano, I- Mian, Itay. b Instituto de Física Corpuscuar, Universitat de Vaència Consejo Superior de Investigaciones Científicas, Parc Científic, E-4698 Paterna, Vaencia, Spain. E-mai: german.sborini@unimi.it,feix.dm@ific.uv.es,german.rodrigo@csic.es We describe the cacuation of the one-oop corrections to H γγ and gg H within the fourdimensiona unsubtraction/oop-tree duaity FDU/LTD approach. The fact that these corrections are both IR and UV finite is not enough to perform the computation without a proper reguarization of the intermediate steps. We show how the FDU/LTD method unambiguousy eads to the proper resut with a pure four-dimensiona representation of the oop ampitude. Moreover, this method aows us to obtain very compact expressions which share the same functiona structure independenty of the partices circuating the oop. Besides this, asymptotic expansions for the ow and high mass regime naturay arise as a consequence of the smoothness of the integrands obtained to describe the ampitudes. EPS-HEP 7, European Physica Society conference on High Energy Physics 5- Juy 7 Venice, Itay Speaker. c Copyright owned by the authors under the terms of the Creative Commons Attribution-NonCommercia-NoDerivatives 4. Internationa License CC BY-NC-ND 4..

2 . Introduction The one-oop ampitudes for Higgs to massess gauge bosons are finite because there is no a tree-eve contribution in the Standard Mode. However, the expressions invoved are individuay i-defined and the four-dimensiona imit has to be considered carefuy. In other terms, a finite resut does not guarantee that we can eave aside the reguarization framework. In fact, a nonconsistent four-dimensiona impementation woud produce miseading resuts. Since the Higgs ampitudes exhibit this pathoogica behaviour, we used them as a case of study for our method: the four-dimensiona unsubtraction FDU framework. We center the discussion in two processes: Higgs production through guon fusion and Higgs decay into a photon pair. Whist in the first case ony massive cooured fermions can be incuded in the oop, the diphoton decay aso invoves W bosons. We show in Ref. [] that the structure of a these contributions is universa, i.e. independent of the nature of the partice inside the oop. However, the specific UV behaviour of the different contributions is sighty different: this eads to potentia discrepancies when naivey considering the computation in four dimensions []. This probem can be easiy soved within the FDU framework because we achieve a fuy oca reguarization.. Four-dimensiona Unsubtraction FDU and Loop-Tree Duaity LTD The FDU framework is based on three key ideas: the oop-tree duaity LTD theorem, a momentum mapping between the virtua and rea kinematics, and the introduction of oca counterterms. The LTD estabishes that oop ampitudes can be expressed as the sum of dua ampitudes, which are buit starting from tree-eve ike objects and repacing the oop measure with a phasespace one. Expicity, the master formua is given by N i= G F q i N i= δq i G D q i ;q j,. j i for any scaar one-oop Feynman integra, with G F q i = /q i m i ı and the dua propagators G D q i ;q j = /q j m j ıη k ji []. Notice that the prescription inside the dua propagators invoves an arbitrary space-ike vector η and depends on both the cut ine and the momentum fow associated to the origina propagator. Moreover, the theorem is vaid beyond one-oop and, aso, when mutipe powers of the propagators are present [4, 5]. The ast statement is crucia for the definition of oca renormaization counter-terms, as we wi briefy mention in the foowing section. Aside the purey theoretica deveopments, the LTD theorem has been succesfuy appied to compute numericay many tensoria muti-eg oop ampitudes [6]. The impementation is very efficient since the expressions to be integrated do not contain any Gram determinant which are usuay responsibe of many numerica instabiities and there are oca canceation among dua contributions that render the whoe integrand smooth. On the other hand, the method reies on the possibiity of reguarizing the expressions at integrand eve and in four space-time dimensions. To dea with UV singuarities, we need integrandeve formuae for the renormaization factors, whie the rea-emission is directy used as a oca counter-term in the IR region through the appication of a proper momentum mapping [7, 8, 9].

3 This mapping is motivated by the universa factorization of IR singuarities in the scattering ampitudes [, ].. Resuts and discussion In order to appy the LTD/FDU to the Higgs ampitudes, we introduce a tensor basis and then define two projectors, i.e. P µν = η µν pν pµ d pµ pν, P µν = pµ pν,. d s s s which aow us to extract the reevant scaar coefficients. The momenta of the photons/guons is p and p with s = p p. Whist the first projector is associated to the ony non-trivia contribution to the physica ampitude, the second one vanishes, because of Ward s identities, after integration. So, we define A, f i = P µν i A µν, f as the one-oop projected scaar ampitudes. Then, we appy the LTD decomposition into dua contributions. In this case, there are four different propagators, so we obtain four dua terms associated to the interna momenta q = p, q = p, q = and q 4 = p. Since the interna on-she momenta can be reated by unifying the coordinate system, we add the four contributions and obtain A, f = g f [ δ q, p s M f q s 4, p ı p p c f c f s ] s p ı c f,. c f = g f δ q, q,. 4, A, f with f = {φ,t,w} denoting the favour of the partice circuating the oop. It is important to appreciate that the functiona structure of the ampitudes is independent of the partice inside the oop; a the information is contained in the coefficients c f i []. We can further simpify A, f by expoiting the fact that A, f vanishes after integration. Thus, we get A, f = g f s [ δ q, s s p ı c f q 4, p M f s p ı p p c f ],.4 which depends ony on c f and c f = c f c f. Notice that the first ine in Eq..4 is integrabe in four-dimensions, but the second one is UV divergent. However, this UV singuarity is mutipied, which is proportiona to d 4; a non-rigorous impementation of the four-dimensiona imit might ead to discrepancies in the finite parts of the ampitude. In order to remove the ambiguities reated to the four-dimensiona imit, we wi introduce a oca renormaization counter-term. Even if the fu Higgs ampitude is UV finite, Eq..4 shows that the integrand is ocay singuar in the high-energy region. The oca UV counter-term is by c f

4 obtained by expanding the dua integrand around the UV propagator, as expained in Refs. [, 8]. Expicity, we write A, f,r = d=4 A, f,uv A, f A, f,uv = g f s 4q UV,,.5 d=4 UV c f d 4,.6 µ q UV, where the sub-eading term in the UV counter-term is defined to mimic the conventiona DREG resut i.e. A, f,uv =.. Asymptotic expansions Another important feature of the LTD approach is reated with the possibiity of performing expansions directy at integrand eve. This is due to two properties of the expressions that we obtain. On one side, the integration measure is defined in an Eucidean space namey, the spatia components of the oop-momentum which aows to define a norm for the integration variabe. On the other, we are aways deaing with ocay reguarized expressions, that are integrabe. In consequence, we guarantee the commutativity between asymptotic expansions and integration. In Ref. [], we used the Higgs ampitudes at one-oop as a benchmark exampe to show the feasibiity of the asymptotic expansions in the LTD/FDU framework. To have an insight about how the method works, et s consider the imit M f s and use the expansion q = 4, p p / = n= Γn Γ n p p n,.7 with q, = p M f and = M f. These expressions are vaid because > M f for any vaue of the oop three-momentum. Using these formuae in each term of Eq..5, we manage to obtain the ow and high mass imits at integrand eve in Ref. []. Of course, after integration, we recovered the we-known resuts avaiabe in the iterature []. In particuar, we succeeded in recovering the ogarithmic factors that appear when M f, without any additiona handing of the expansions.. Concusions and outook We appied the LTD/FDU framework to tacke the computation of one-oop scattering ampitudes for Higgs bosons. The compact expressions obtained for the dua contributions aowed to achieve a universa integrand-eve representation of the compete virtua ampitude. The information reated with the nature of the virtua partices inside the oop is encoded in constant coefficients but the functiona form is shared. This universaity might be present even at higher-orders due to the properties of the LTD representation. Aso, we managed to remove the scheme ambiguities by impementing a oca renormaization inspired by the FDU method []. A the previousy known resuts were recovered within a pure four-dimensiona framework. Another important resut of our

5 work is the possibiity of an straightforward impementation of asymptotic expansions at integrandeve. This is due to the fact that the dua integration is performed in an Eucidean space, namey the oop three-momentum space. Finay, we expect further simpifications of higher-order computations and asymptotic expansions by expoiting the four-dimensiona nature of the LTD/FDU approach. Acknowedgments This work is partiay supported by the Spanish Government, EU ERDF funds grants FPA4-56-C--P and SEV-4-98, by Generaitat Vaenciana GRISOLIA/5/5 and Fondazione Caripo under the grant number References [] F. Driencourt-Mangin, G. Rodrigo and G. F. R. Sborini, arxiv:7.758 [hep-ph]. [] S. Weinzier, Mod. Phys. Lett. A 9 4 no.5, 45. [] S. Catani, T. Geisberg, F. Krauss, G. Rodrigo and J. C. Winter, JHEP [4] I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, JHEP 7. [5] I. Bierenbaum, S. Buchta, P. Draggiotis, I. Maamos and G. Rodrigo, JHEP 5. [6] S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, Eur. Phys. J. C 77 7 no.5, 74. [7] R. J. Hernández-Pinto, G. F. R. Sborini and G. Rodrigo, JHEP [8] G. F. R. Sborini, F. Driencourt-Mangin, R. J. Hernández-Pinto and G. Rodrigo, JHEP [9] G. F. R. Sborini, F. Driencourt-Mangin and G. Rodrigo, JHEP [] S. Buchta, G. Chachamis, P. Draggiotis, I. Maamos and G. Rodrigo, JHEP [] G. Sborini, F. Driencourt-Mangin, R. Hernández-Pinto and G. Rodrigo, PoS EPS-HEP TIF-UNIMI-7-7. [] U. Agietti, R. Bonciani, G. Degrassi and A. Vicini, JHEP 7 7. [] C. Gnendiger et a., Eur. Phys. J. C 77 7 no.7, 47. 4