Leading two-loop Yukawa corrections to the pole masses of SUSY fermions in the MSSM

We have calculated the leading Yukawa corrections to the chargino, neutralino and gluino pole masses in the DR-bar scheme in the Minimal Supersymmetric Standard Model (MSSM) with the full set of complex parameters. We have performed a numerical analysis for a particular point in the parameter space and found typical corrections of a few tenths of a percent thus exceeding the experimental resolution as expected at the ILC. We provide a computer program which calculates two-loop pole masses for SUSY fermions with complex parameters up to the respective order in pertubation theory.


Introduction
In the Minimal Supersymmetric Standard Model (MSSM), each Standard Model particle is part of a SUSY multiplet containing its superpartners of opposite statistics and a spin differing by 1/2 unit. Moreover, due to anomaly cancellation the SUSY Higgs-sector consists of two chiral multiplets whose fermionic degrees of freedom are called higgsinos. On the other hand, the vector supermultiplets contain the standard model gauge bosons and their spin-1/2 superpartners called gauginos.
Organizing these particles in mass eigenstates one finds four mixing neutral Majorana fermions composed of the superpartners of the photon, the Z 0 -boson, and the neutral Higgs bosons H 0 1,2 . Then there are two charginosχ ± 1 andχ ± 2 , which are the fermion mass eigenstates of the partners of the W ± and the charged Higgs bosons H ± 1,2 . Finally, the gluino is the superpartner of the SU(3) C vector-boson, the gluon.
The next generation of future high-energy physics experiments at Tevatron, LHC and a future e + e − linear collider (ILC) will hopefully discover some of these particles if supersymmetry (SUSY) is realized at low energies. The physics interplay of these experiments has been studied in [1] and it turned out that particularly a linear collider [1,2,3] will allow to test the underlying SUSY model with great accuracy. In fact, the accuracies of the masses of the lighter SUSY fermions are expected to be in the permille region which makes the inclusion of higher order loop-corrections indispensible.
Within the real MSSM important results on quark self-energies were obtained in [4]- [6]. In [7]- [10] the gluino pole mass was calculated to two-loop order O(α 2 S ). Moreover, the scalar MSSM Higgs-sector has been studied in detail, also in the full complex model [11]- [15] and even some three-loop results have been included recently [16].
In previous works [17,18] we studied the SUSYQCD two-loop corrections to chargino and neutralino pole masses and found that these effects have to be taken into account when matching DR -input with experimental data. However, the reference scenario used for the numerical analysis was the benchmark point SPS1a' [19], a scenario where the lighter mixing partners have a dominating gaugino component. But once the higgsino component becomes more important one can expect that also leading two-loop Yukawa corrections are of the same size.
In this paper we therefore study these leading two-loop Yukawa corrections to the pole masses of charginos, neutralinos and the gluino. Since the one-loop correction to the gluino pole mass is of order O(α S ) the leading two-loop Yukawa correction is of order O(α S Y 2 ). On the contrary, leading Yukawa corrections for charginos and neutralinos are of order O(Y 4 ). We neglect the Yukawa coupling to leptons. The reference point for the numerical study is chosen such that the lighter mixing partners have a dominant higgsino component and their masses are roughly of the size of their SPS1a'-values so that we can compare the magnitude of the corrections to the expected experimental accuracy at this point.
We conclude that the leading two-loop Yukawa corrections typically exceed experimental resolution for the case of neutralinos. For charginos we find smaller corrections comparable to their two-loop SUSYQCD corrections [18] which should be taken into account when extracting DR parameters from experiment. The leading two-loop Yukawa corrections to the gluino are comparably small but extending the known two-loop SUSYQCD corrections [9,10] to include the gluino phase for the numerical evaluation gives important results.
For the calculation of the self-energy amplitudes we employ semi-automatic Mathematica tools [20,21] and cross-check the results with the generic analytic formulae for two-loop corrections to fermion pole masses in a general renormalizable field-theory derived in [9].
In principle, some of the complex phases of the full complex MSSM can be rotated away by rigid field transformations, e.g. the phase of the gluino mass parameter. However, we do not pose restrictions on these phases for convenience. Instead we generalize the explicit results on the SUSYQCD corrections to SUSY fermions derived in [9] to the case of a complex gluino mass parameter. For the above reason this is not a generalization of the physics.
All of these new results are implemented in the C-program Polxino 1.1 [22]. It currently is the only publicly available program for the calculation of even the one-loop chargino and neutralino pole masses with complex parameters and described in [18].
We list all the analytic results in the appendix.

Diagrammatics and calculation
The masses of a set of mixing fermions are defined through the complex poles of the full renormalized propagator of the mixing system as gauge-invariant and renormalization scale invariant quantities [23]- [31]. The tree-level values are the eigenvalues of the treelevel DR mass-matrix m. This complex pole masses can be inferred from the zeros of the determinant of the renormalized effective action Γ (2) (p 2 ) in terms of the renormalized self-energiesΣ Here, we use the usual definition for the Lorentz-generators σ µ = (1, σ i ) andσ µ = (1, −σ i ). The diagonalized tree-level mass-matrix m could be chosen real and semipositive definite by a rigid reparametrization of the fields. However, in the case of a complex gluino mass parameter we find it more natural to keep its complex phase in the bilinear Lagrangian and therefore we keep the complex conjugates in Eq. (1). The pole mass condition in the Weyl representation (1) becomes particularly simple in the rest frame where kinematics is trivial Note that the objectsΣ L,R m,p are finite matrix valued functions of the external momentum invariant s = p 2 . The iterative solution to Eq. (2) is [9] All self-energies have to be evaluated at an external momentum invariant s = |m i | 2 with an infinitesimal positive imaginary part [32].
According to the DR ′ -scheme the UV-divergencies of the Feynman-integrals are regulated dimensionally by d = 4 − ǫ, and the unphysical scalar mass parameter m ǫ for the evanescent fields is absorbed according to [33]. The resulting two-loop integrals have a rather complex tensor structure which has already been worked out in detail in [9], implemented in a C-program [10] and tested thoroughly [17,18]. In this work we therefore use the notation of [9] for the basis integrals.

One-loop level
In Fig. 1-3 we show the one-loop diagrams for the gluino, the neutralinos and the charginos, respectively. Note, that diagrams with charge conjugated inner particles are not shown explicitly. We checked our analytic one-loop calculation against previous work [34,35] in the on-shell scheme and found agreement.
In the case of the gluino the one-loop correction is proportional to g 2 s , thus any squared one-loop self-energies cannot contribute to leading two-loop Yukawa corrections O(g 2 s Y 2 ). The result is which slightly generalizes Eq.(5.2) of [9] for a complex gluino mass parameter.
The O(α) one-loop mass shifts for charginos and neutralinos are given in Eq.(5.15) and Eq.(5.18) of [9] and are not repeated here. However, for a fixed order calculation O(α, Y 4 ) we need the O(Y 2 ) one-loop self-energies because their squares appear in Eq. (3).
The results arê where summation over the squark index s and q = (u, d, c, s, t, b) is understood implicitly. Q denotes the iso-spin partner of q and is not summed over independently. The Yukawa couplings are given by where we employ the conventions of [36] for the mixing parameters. Furthermore, In Eq. (6) and Eq. (7) the symbols u and d denote the up-type and down-type quark, respectively. The generation index is suppressed.

Two-loop results
In Fig. 4 and Fig. 5 we show the leading two-loop self-energy diagrams contributing O(Y 4 ) to the pole mass. For simplicity we put the external neutralinos and the gluino in Fig. 4 into a generic Majorana field η. Without loss of generality neutralino and chargino masses are taken to be real, the gluino mass however is allowed to have a complex phase thus avoiding the introduction of a one-dimensional rotation matrix for this particle.
The renormalization scheme adopted is the familiar DR -scheme which regulates UVdivergencies dimensionally but introduces an unphysical scalar field for any gauge field in the theory in order to restore the counting of degrees of freedom in supersymmetry. These unphysical mass parameters can be absorbed in the sfermion mass parameters [33], the resulting scheme is called DR ′ .
The diagrams and the amplitudes were generated in FeynArts 3.2 [20] and simplified analytically using FeynCalc 4.0.2 [21]. The resulting tensor integrals were translated into the conventions of [9]. We then independently derived the same results from the generic formulae given in Eq. (4.4, 4.5) of [9].
The arguments of these self-energy tensor-functions are the squared masses of the inner particles and we abbreviate them by their respective symbol. The external momentum invariant is the squared tree-level DR -input mass-parameter s = |m ii | 2 with an infinitesimal positive imaginary part.
For the numerical evaluation of the tensor integrals in our program Polxino [22] we use the implementation of [10] and the two-loop self-energy library [32].
We split the rather lengthy analytic expression into five parts according to which particles occur in the loops, neutral (scalar and pseudoscalar) Higgs bosons, charged Higgs bosons, neutralinos, charginos or only quarks and squarks where χ in Eq.

Numerical results
Our reference scenario for the numerical analysis is a higgsino scenario defined through M 1 = 200 GeV, M 2 = 400 GeV and µ = 130 GeV at Q = 1 TeV. All other parameters are taken from SPS1a', in fact tan β = 10, A t = −532.38 GeV, A b = −938.91 GeV, MQ 3 = 470.91 GeV, MŨ 3 = 385.32 GeV and MD 3 = 501.37 GeV, for further details see [19]. Evaluating the DR -spectrum, the one-loop corrections and our two-loop results Eq. (8,9) as well as the respective two-loop SUSYQCD corrections [9] we find the following result for the real parts of the one-and two-loop masses : In this scenario the lighter mixing partners of charginos and neutralinos (χ 0 1 ,χ 0 2 ,χ − 1 ) have a dominating higgsino component but their tree-level masses are roughly of the size of the SPS1a' values. Therefore, we can compare the magnitude of the two-loop corrections to the expected experimental accuracy at SPS1a'.
In Fig. 6 we show the neutralino pole masses mχ0 1 and mχ0 2 and the two-loop mass shifts on the right as a function of tan β. The dotted line contains the two-loop SUSYQCD corrections only whereas the solid line is the full O(αg 2 s , Y 4 ) result. Despite the obvious cancellation between SUSYQCD-and Yukawa corrections forχ 0 1 the corrections exceed the expected experimental accuracy of about ±0.05 GeV [1].
Forχ − 1 in Fig. 7 we find corrections of a few hundred MeV for rather high values of tan β which should be included when matching DR-parameters to on-shell masses. However, corrections toχ − 1 tend to stay below expected experimental accuracy. From the dependence of δ (2) mχ0 1 and δ (2) mχ0 2 on A t in Fig. 8 we learn that A t severely effects these two-loop corrections and typical results exceed experimental resolution by an order of magnitude. On the contrary, δ (2) mχ− 1 in Fig. 9 as a function of A t is again smaller than the expected experimental accuracy. In Fig. 10 we show the pole masses and the corrections as a function of φ At for the two light neutralinos. The two-loop corrections substantially depend on this phase.
Note that forχ 0 1 andχ − 1 the Yukawa corrections tend to balance the SUSYQCD corrections.
In Fig. 11 and Fig. 12 we investigate the dependence on φ M 3 of the lightest masseigenstates and the gluino. For the gluino it turns out that only the variation of the oneloop correction is numerically significant. Note also, that the two-loop Yukawa corrections to the gluino pole mass are very small. On the other hand, δ (2) mχ0 1 and δ (2) mχ− 1 have a comparably strong dependence on φ M 3 coming from the SUSYQCD corrections alone.

Conclusions
We have calculated leading two-loop Yukawa corrections to the pole masses of charginos, neutralinos and the gluino. The magnitude of these corrections typically exceeds experimental resolution for neutralinos. For charginos we find smaller corrections which still need to be taken into account when analyzing precision experiments. The leading twoloop Yukawa corrections to the gluino pole mass are rather small but we extended existing results on the SUSYQCD corrections to the case of a complex gluino phase giving rise to numerically relevant one-and two-loop effects.
Finally, we included all results in the publicly available program Polxino 1.1 [22] which has an convenient interface to the commonly used SUSY Les Houches accord [37] for further numerical studies.

A Analytic results
In the following q is summed over (u, d, c, s, t, b) and Q denotes the iso-spin partner of q. The chargino index c, the neutralino indicex n as well as the squark-indices s, t, u are summed over independently. The generation index g appearing in Eq. (14) is summed over (1,2,3). Finally, φ 0 is summed over (h 0 , H 0 , A 0 , G 0 ) and φ − over (H − , G − ). |φ 0 | is 0 for the scalar (h 0 , H 0 ) and +1 for the pseudo-scalar Higgs (A 0 ,G 0 ). The tensor-integral functions are listed in [9], their arguments denote the squared DR -masses of the particles.

A.1 Gluino
For the gluino we get with

A.2 Neutralinos
Similarly we find for the neutralinos where The four squark couplings are and finally the Higgs couple to squarks according to In Eq. (29-32) u and d denote the up-type resp. down-type quark, the generation index is suppressed.