BNL–HET01/7 CERN–TH/2001–056 July 10, 2021 hepph/0102317
Precision Observables in the MSSM:
[0.5em] Leading Electroweak Twoloop Corrections
S. Heinemeyer and G. Weiglein
HET, Brookhaven Natl. Lab., Upton NY 11973, USA
CERN, TH Division, CH1211 Geneva 23, Switzerland
The leading electroweak MSSM twoloop corrections to the parameter are calculated. They are obtained by evaluating the twoloop selfenergies of the and the boson at in the limit of heavy scalar quarks. A very compact expression is derived, depending on the ratio of the odd Higgs boson mass, , and the top quark mass, . Expressions for the limiting cases and are also given. The decoupling of the nonSM contribution in the limit is verified at the twoloop level. The numerical effect of the leading electroweak MSSM twoloop corrections is analyzed in comparison with the leading corrections of in the SM and with the corrections in the MSSM.
Presented by S. Heinemeyer at the
5th International Symposium on Radiative Corrections
(RADCOR–2000)
[4pt] Carmel CA, USA, 11–15 September, 2000
1 Introduction
Theories based on Supersymmetry (SUSY) [1] are widely considered as the theoretically most appealing extension of the Standard Model (SM). They predict the existence of scalar partners to each SM chiral fermion, and spin–1/2 partners to the gauge bosons and to the scalar Higgs bosons. So far, the direct search for SUSY particles has not been successful. One can only set lower bounds of GeV on their masses [2]. Contrary to the SM, two Higgs doublets are required in the Minimal Supersymmetric Standard Model (MSSM) resulting in five physical Higgs bosons [3]. The direct search resulted in lower limits of about for the neutral Higgs bosons [4].
An alternative way to probe SUSY is to search for the virtual effects of the additional particles via precision observables. The most prominent role in this respect plays the parameter [5]. The leading radiative corrections to the parameter, , constitute the leading processindependent corrections to many electroweak precision observables, such as the boson mass, , and the effective leptonic weak mixing angle, . Within the MSSM the full oneloop corrections to and have been calculated already several years ago [6, 7]. More recently also the leading twoloop corrections of to the quark and scalar quark loops for and have been obtained [8, 9]. Contrary to the SM case, these twoloop corrections turned out to increase the oneloop contributions, leading to an enhancement of the latter of up to 35% [8].
We summarize here the result for the leading twoloop corrections to at [10]. For a large SUSY scale, , the SUSY contributions decouple from physical observables. This has been verified with existing results at the oneloop [11] and at the twoloop level [8, 10]. Therefore, in the case of large the leading electroweak twoloop corrections in the MSSM are obtained in the limit where besides the SM particles only the two Higgs doublets needed in the MSSM are active. We derive the result for the [10] corrections in this case and provide a compact analytical formula for it, depending on the odd Higgs boson mass, , and the top quark mass, . Furthermore, we present formulas for the limiting cases (i.e. the SM limit) and . The numerical effect of the corrections is compared with the corresponding SM result [12] and the gluonexchange correction of in the MSSM.
2 Calculation of the corrections
2.1 and the Higgs sector
The quantity ,
(1) 
parameterizes the leading universal corrections to the electroweak precision observables induced by the mass splitting between fields in an isospin doublet [5]. denote the transverse parts of the unrenormalized and boson selfenergies at zero momentum transfer, respectively. The shifts induced by in the prediction for the boson mass, , and the effective leptonic weak mixing angle, , are approximately given by
(2) 
Contrary to the SM, in the MSSM two Higgs doublets are required [3]. At the treelevel, the Higgs sector can be described in terms of two independent parameters (besides and ): the ratio of the two vacuum expectation values, , and , the mass of the odd boson. The diagonalization of the bilinear part of the Higgs potential, i.e. the Higgs mass matrices, is performed via orthogonal transformations with the angle for the even part and with the angle for the odd and the charged part. The mixing angle is determined at lowest order through
(3) 
One gets the following Higgs spectrum:
2 charged bosons  
3 unphysical scalars  (4) 
The treelevel masses, expressed through and , are given by
(5) 
where the last two relations, which assign mass parameters to the unphysical scalars and , are to be understood in the Feynman gauge.
2.2 Evaluation of the contributions
In order to calculate the corrections to in the approximation that all superpartners are heavy so that their contribution decouples, the Feynman diagrams generically depicted in Fig. 1 have to be evaluated for the boson () and the boson () selfenergy. We have taken into account all possible combinations of the doublet and the full Higgs sector of the MSSM, see Sect. 2.1.
The twoloop diagrams shown in Fig. 1 have to be supplemented with the corresponding oneloop diagrams with subloop renormalization, depicted generically in Fig. 2. The corresponding insertions for the fermion and Higgs mass counter terms are shown in Fig. 3.
The amplitudes of all Feynman diagrams, shown in Figs. 1–3, have been created with the program FeynArts2.2 [13], making use of a recently completed model file for the MSSM^{1}^{1}1 Only the nonSM like counter terms had to be added. . The algebraic evaluation and reduction to scalar integrals has been performed with the program TwoCalc [14]. (Further details about the evaluations with FeynArts2.2 and TwoCalc can be found in Ref. [15].) As a result we obtained the analytical expression for depending on the oneloop functions and [16] and on the twoloop function [14, 17]. For the further evaluation the analytical expressions for , and have been inserted. In order to derive the leading contributions of we extracted a prefactor . Its coefficient can be evaluated in the limit where and (and also ) are set to zero. Furthermore we made use of the mass relations in the MSSM Higgs sector, see eq. (5). In the limit they reduce to
(6) 
In the limit the relation between the angles and , see eq. (3), becomes very simple, , i.e. term thus depends only on the top quark mass, , the odd Higgs boson mass, , and (or ). . The coefficient of the leading ,
We explicitly verified the UVfiniteness of our result. As a further consistency check of our method we also recalculated the SM result for the corrections with arbitrary values of the Higgs boson mass, as given in Ref. [18], and found perfect agreement.
3 Analytical results
3.1 The full result
The analytical result obtained as described in Sect. 2.2 can conveniently be expressed in terms of
(7) 
The corresponding twoloop contribution to then reads:
(8)  
with
(9) 
In the limit of large (i.e. ) one obtains
(10) 
Thus for large the SM limit with [12] is reached.
3.2 The expansion for large
The result for in eq. (8) can be expanded for small values of , i.e. for large values of :
(11)  
In the limit one obtains
(12) 
i.e. exactly the SM limit for is reached. This constitutes an important consistency check: in the limit the heavy Higgs bosons decouple from the theory. Thus only the lightest even Higgs boson remains, which has in the approximation the mass , see eq. (6). This decoupling of the nonSM contributions in the limit where the new scale (i.e. in the present case ) is made large is explicitly seen here at the twoloop level.
3.3 The expansion for small
The result for in eq. (8) can also be expanded for large values of , i.e. for small values of (with ):
In the limit or one obtains
(14) 
4 Numerical analysis
4.1 The expansion formula
We first analyze the validity of the two expansion formulas, eqs. (11) and (3.3). In Fig. 4 we show the result for , defined by
(15) 
as a function of for . The expansion for is sufficiently accurate nearly up to . The other expansion gives accurate results for . For larger the expansion becomes better, enlarging the validity region for the large expansion up to .
4.2 Effects on precision observables
In this section we analyze the numerical effect on the precision observables and , see eq. (2), induced by the additional contribution to . In Fig. 5 the size of the leading MSSM corrections, eq. (8), is compared for with the leading contribution in the SM for [12], with the leading MSSM corrections arising from the sector at [7], and with the corresponding gluonexchange contributions of [8] (the gluinoexchange contributions [8], which go to zero for large , have been omitted here). For illustration, the left plot () is shown as a function of , which affects only the MSSM contributions, while the right plot () is given as a function of the common SUSY mass scale in the scalar quark sector, , which affects only the and MSSM contributions. We have furthermore chosen the case of “maximal mixing” in the scalar top sector, which is realized by setting the offdiagonal term in the mass matrix, , to and yields the maximal value for for a given (see Ref. [19] for details). In the right plot the case of no mixing, , is also shown. The mixing in the scalar bottom sector has been determined by using a bottom quark mass of , and by setting the trilinear couplings to and the Higgs mixing parameter to . The contributions in the SM and the MSSM are negative and are for comparison shown with reversed sign.
While for small values of the gluonexchange contribution in the MSSM is much larger than the contribution from eq. (8) (note that in this region of parameter space the approximation of neglecting the scalarquark contributions in the result is no longer valid), they are of approximately equal magnitude for (this refers to both the nomixing and the maximalmixing case) and compensate each other as they enter with different sign. In this region the twoloop contributions are about one order of magnitude smaller than the MSSM contribution. For the leading MSSM contribution is about three times bigger than the gluonexchange contribution in the MSSM.
For small (left plot of Fig. 5) and moderate () the new MSSM corrections are about two times larger than the leading contributions in the SM for . For large the decoupling of the extra contributions in the MSSM takes place and the MSSM correction approaches the value of the leading contributions in the SM for , as indicated in eqs. (11), (12). For large (right plot of Fig. 5) the MSSM correction and the contribution in the SM for are indistinguishable in the plot, in accordance with eq. (10).
It is well known that the SM result with underestimates the result with realistic values of by about one order of magnitude [18]. One can expect a similar effect in the MSSM once higher order corrections to the Higgs boson sector are properly taken into account, which can enhance up to [19], see Ref. [10].
In Fig. 6 the approximation formulas given in eq. (2) have been employed for determining the shift induced in and by the new correction to . In Fig. 6 the effect for both precision observables is shown as a fuction of for . The effect on varies between and for small and is almost constant, , for . As above, the constant behavior can be explained by the analytical decoupling of when , see eq. (10). The induced shift in lies at or below and shows the same qualitative dependence as .
5 Conclusions
We have calculated the leading corrections to in the MSSM in the limit of heavy squarks. Short analytical formulas have been obtained for the full result as well as for the cases and . As a consistency check we verified that from the MSSM result the corresponding SM result can be obtained in the decoupling limit (i.e. ).
Numerically we compared the effect of the new contribution with the leading SM contribution with and with the leading MSSM corrections originating from the sector of and . The numerical effect of the new contribution exceeds the one of the leading QCD correction of in the scalar quark sector for . It is always larger than the leading SM contribution with , reaching approximately twice its value for small and moderate .
The numerical effect of the new contribution on the precision observables and is reletively small, up to for and for . It should be noted, however, that the SM result with , to which the new result corresponds, underestimates the result with realistic values of by about one order of magnitude. A similar behavior can also be expected for the MSSM corrections. An extension of our present result to the case of nonzero values of the lightest even Higgs boson mass will be undertaken in a forthcoming publication.
Acknowledgments
S.H. thanks the organizers of “RADCOR2000” for the invitation and the inspiring and constructive atmosphere at the workshop.
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