Three-loop corrections to the mass of the light Higgs boson in the MSSM
aa r X i v : . [ h e p - ph ] J u l SFB/CPP-10-70
Three-loop Corrections to the Mass of the LightHiggs Boson in the MSSM
Philipp KANT ∗ TTP KarlsruheE-mail: [email protected]
The Minimal Supersymmetric extension of the Standard Model (
MSSM ) predicts the existence ofa light neutral Higgs boson. Once found at the
LHC , its mass will immediately become a precisionobservable. The theoretical value of the Higgs mass M h is subject to large radiative corrections.Due to the large top Yukawa coupling, loops of top quarks and their superpartners provide thedominant contribution to the radiative corrections.We present a calculation of the SUSY - QCD corrections to these diagrams, up to the three-loop or-der. We find that our three-loop results can be in the range of one GeV, and are thus relevant whencompared with the expected experimental accuracy at the LHC. We also find a significantly re-duced dependency on the renormalisation prescription, thus decreasing the theoretical uncertaintyof the prediction of M h . European Physical Society Europhysics Conference on High Energy PhysicsJuly 16-22, 2009Krakow, Poland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ hree-loop Corrections to the Mass of the Light Higgs Boson in the MSSM
Philipp KANT
1. Introduction
An important feature of the minimal supersymmetric extension of the Standard Model (
MSSM )is the existence of a light neutral Higgs boson. The
MSSM
Higgs sector is a Two-Higgs DoubletModel, the parameters of which are related to the gauge couplings through Supersymmetry. Theserelations reduce the number of new (in comparison with the Standard Model) parameters to two,which are usually chosen to be the ratio v / v = tan b of the vacuum expectation values of the twoHiggs doublets and the mass M A of the pseudoscalar Higgs. Once these are fixed, the mass M h ofthe light neutral Higgs boson is not a free parameter, but a calculable prediction of the theory. Atthe tree level, there is an upper limit M h ≤ M Z . At higher orders, this bound gets shifted up byradiative corrections to the Higgs self-energy, which depend on the spectrum of the superpartnermasses.The one- and two-loop corrections to M h have been extensively studied in the literature (forreviews, see e.g. Refs. [1,2]). The studies show that, due to the large top-Yukawa coupling, the mostsizeable corrections stem from loops of top quarks and their superpartners in the Higgs propagator.For a light Higgs boson, the Large Hadron Collider ( LHC ) will be able to measure its mass with anexpected experimental accuracy of 100 −
200 MeV [3]. To take full advantage of the experimentaldata, the theoretical prediction for M h has to match this precision. However, based mostly on therenormalization scale and scheme dependence, the theoretical uncertainty on the prediction of M h has been estimated to 3 − M h ,there are only two calculations available that go to the third order of perturbation theory. In [5],Renormalisation Group methods have been used to calculate the leading- and next-to leading termin ln ( M SUSY / M t ) , where M SUSY is the typical scale of
SUSY particle masses. Motivated by the factthat the corrections from top and stop loops dominate the overall corrections, the three-loop
SUSY - QCD corrections to these diagrams have been computed in [6]. Because current methods are notsufficient to solve three-loop multiscale integrals exactly, the calculation in [6] assumed a strong hi-erarchy among the superparticle masses and performed nested asymptotic expansions [7] to reducethe problem to single-scale integrals. While [6] considered two rather simple mass hierarchies, it isthe aim of this talk to report on recent progress on computing more involved scenarios and discusswhich are important for phenomenological studies.
2. Outline of the Calculation
We calculate the corrections to M h by evaluating virtual corrections to the Higgs propagator.As in [6], we restrict ourselves to diagrams where the Higgs couples to top quarks or their super-partners, including SUSY - QCD corrections up to Order O ( a s ) . This leaves us with the followingvirtual particles: top quarks t and their superpartners, the stops ˜ t / , the gluons g and gluinos ˜ g , aswell as the light quarks q and squarks ˜ q , which enter at the three-loop level.One can perform the calculation assuming different hierarchies amongst the superpartnermasses. To estimate the error introduced by expanding around these hierarchies, we systemati-cally compare, at the two-loop level, with the exact result which is given in [8] in very compact2 hree-loop Corrections to the Mass of the Light Higgs Boson in the MSSM Philipp KANT
PSfrag replacements O ( a t ) O ( a t a s ) O ( a t a s ) all masses on-shell D M ( n ) h [ G e V ]
010 15202530354045505 M SUSY [GeV]0 1000 1200 1400 160018002000200 400 600 800010 152025303540455050120 . . . . − . − . − . − . − . − . − . − . . . . . . − − . − . − . − . − . − . − . − . PSfrag replacements O ( a t ) O ( a t a s ) O ( a t a s ) D M ( n ) h [ G e V ]
010 15202530354045505 M SUSY [GeV] m t DR , M SUSY on-shell0 1000 1200 1400 160018002000200 400 600 800010 152025303540455050120 . . . . − . − . − . − . − . − . − . − . . . . . . − − . − . − . − . − . − . − . − . Figure 1:
Renormalisation scheme dependence of the corrections D M h = M h − M treeh . This figure assumesa degenerate mass spectrum of the SUSY particles. In the left panel, all masses are renormalised on-shell,while in the right panel the DR scheme is used for the top mass. The masses of the SUSY particles arerenormalised on-shell in both cases, to ensure that D M h is plotted over the same parameter. Choosing the DR scheme also for the superpartner masses has a small effect on the Higgs mass. form. A detailed study of this comparison, which shows that the relative error can be brought below5% for the SPS benchmark scenarios from [9], will be presented in a forthcoming publication [10].
3. Renormalisation Prescription Dependence
We use Dimensional Reduction [11] in order not to spoil Supersymmetry through the regulari-sation. This leaves us with the choice between using on-shell renormalisation conditions and usingminimal subtraction, i.e. the DR scheme. In order to make a justified choice for one scheme overthe other, we analysed the magnitude of the corrections of different loop orders in both schemes(see Fig. 1).The figure demonstrates three points: i) , the large one-loop corrections to M h are an artefactof using the on-shell top mass. The corrections of higher orders are significantly smaller in the DR scheme, indicating a better behaviour of the perturbative expansion, ii) , while there is a sig-nificant deviation between the two schemes at the one-and two-loop level, at the third loop orderthe discrepancy all but vanishes, indicating a stabilisation of the theoretical prediction, and iii) , thethree-loop corrections amount to some hundred MeV. After this analysis, we resolved to use the DR scheme for the renormalisation of our parameters. An exception to this is the mass of the e -scalarthat appears in Dimensional Reduction, which we renormalise on-shell and set it to zero.
4. Results
As an example for the numerical impact of our calculation we show results for the
SPS bench-mark line
SPS
2. Fig 2 shows M h including corrections of one-, two- and three-loop order in black,blue and red, respectively. To get the best prediction possible, we extract all available one- andtwo-loop corrections from the program F EYN H IGGS [4, 12–14]. In particular, we use the exacttwo-loop result from [8] for the O ( a t a s ) corrections. It is notable that the three-loop correctionsamount to about one GeV for large m / . 3 hree-loop Corrections to the Mass of the Light Higgs Boson in the MSSM Philipp KANT
PSfrag replacements O ( a t ) O ( a t a s ) , exact O ( a t a s ) , expansion O ( a t a s ) m / [GeV] M h [ G e V ]
100 200 300 400 500 600700112 114116118120122011 . . . . . . . . . . . . . . − − . − . − . Figure 2:
Higgs mass for the benchmark line
SPS
2. The red line includes our three-loop corrections, as wellas all the corrections of one- and two-loop order that are implemented in F
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