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Mass Splitting between Charged and Neutral Winos at Two-Loop Level

The recent result of the higgs search at the LHC experiment has lead to more attention to the supersymmetric standard models with heavy sfermions. Among them, the models with the almost pure wino being the lightest supersymmetric particle (LSP) have been widely discussed due to their success in providing a consistent dark matter candidate. The notable phenomenological feature of the wino LSP is the degeneracy with its charged SU(2)_L partner (the charged wino) in mass. The tiny mass splitting makes the charged wino long-lived, which allows us to detect the wino production at the LHC experiment by searching for the disappearing charged tracks inside the detectors. Since the reach of the experiment is sensitive to the mass splitting, it is mandatory to estimate it very precisely. We therefore perform a full calculation of the mass splitting at two-loop level, and find that the splitting is reduced by a few MeV compared to the one-loop calculation. This reduction leads to about a 30 % longer lifetime of the charged wino, with which the current constraint on the wino mass by the ATLAS experiment is improved by about 10 %.


Introduction
The supersymmetric standard model (SSM) is one of the most attractive candidates for physics beyond the standard model (SM). Both the discovery of the Higgs boson [1,2] and the null-observation of supersymmetry (SUSY) signals at the LHC experiment have given us some hints for SUSY model buildings. In the minimal SSM (the MSSM), for example, the Higgs boson mass is predicted to be smaller than the Z boson mass at the tree-level. The observed mass of the Higgs boson at around 126 GeV, therefore, indicates that huge radiative corrections to the Higgs self-coupling from the SUSY breaking effects are required [3]- [6].
One of the simplest scenarios leading to such huge corrections is putting the masses of the sfermions (especially of the squarks) at the scale of O(10-100) TeV [3]- [7]. It should be noted that although the squarks are far beyond the accessible range of the LHC experiment, in such cases, this class of the scenarios does not necessarily mean that all the SUSY particles are as heavy as O(10-100) TeV. For example, if we suppose that the SUSY breaking field is charged under some (gauge) symmetries, gauginos cannot acquire their masses through the linear term of the SUSY breaking field in the gauge kinetic functions of the MSSM. In this case, the leading contributions to the gaugino masses come from the anomaly mediated contribution [8,9], which are one-loop suppressed compared to the squark masses. The gaugino masses are therefore predicted to be O(0.1-1) TeV which are accessible at the LHC experiment. This class of the high-scale SUSY scenarios has recently attracted more attention, and phenomenological and cosmological aspects of the scenarios have been studied extensively [10]- [23]. 1 One of the most prominent features of the anomaly mediated gaugino spectrum is that the lightest supersymmetric particle (LSP) is predicted to be the almost pure neutral wino, which is highly degenerate with the charged wino in mass. This is because the mass splitting between these two particles is forbidden at the tree-level due to the approximate custodial symmetry. The mass splitting is dominated by the radiatively generated contributions which are estimated to be 160-170 MeV at the one-loop level [25]- [27]. 1 In this Letter, we base our discussion on the pure gravity mediation scenarios [10], where the Higgsinos and the heavier Higgs bosons in the MSSM are predicted to be as heavy as the sfermions.
Our formulas obtained in the following analysis are applicable to any heavy sfermion scenarios as long as the Higgsinos are much heavier than the gauginos as in the scenarios such as PeV-Scale Supersymmetry [24] and Spread Supersymmetry [11].
Due to the degeneracy between the charged and the neutral wino masses, the charged wino decays mainly into the neutral wino and a soft pion with the decay length of cτ = O(1-10) cm, which allows the wino production to be detected by looking for a disappearing charged track inside the detectors at the LHC experiment [28]. This signal is characteristic for the high-scale SUSY scenarios with the anomaly-mediated gaugino mass spectrum. The expected number of the charged track is proportional to exp[−L/cτ ] with L being the distance between a detector and a collision point, and hence, the reach of the experiment is very sensitive to cτ .
Therefore, a precise calculation of the decay length is mandatory.
In this article, we calculate the mass splitting between the charged and the neutral winos at the two-loop level. In Ref. [29], the splitting of the winos has been evaluated at the two-loop level in the heavy wino limit by calculating non-decoupling contributions. The result, however, cannot be directly applied to the wino mass in the range of O(100) GeV where the LHC experiment is searching for the winos. We therefore perform a full two-loop calculation of the splitting including the non-decoupling effects. In the next section (section 2), we calculate the mass splitting at the twoloop level. We will see that the contributions from the SM particles are dominant, while those from the SUSY particles are negligible. In section 3, we discuss the decay length of the charged wino and compared with the recent experimental results by the ATLAS collaboration [28]. As a result, we find that the decay length of the charged wino becomes 10-30 % longer than that obtained at the one-loop calculation. This result makes the current constraint on the wino mass by the ATLAS experiment severer than the LEP2 constraints [30]- [33]. Section 4 is devoted to summary of our discussion.

The mass splitting
As mentioned above, the neutral wino (χ 0 ) and its charged SU(2) L partner (the charged wino,χ ± ) are almost degenerated in mass at the tree-level due to the approximate custodial symmetry. The dominant mass splitting, δm = mχ± − mχ0, is generated by radiative corrections, which pick up the breaking of the custodial symmetry as pointed out in Ref. [27]. In this section, we calculate the radiative corrections at the two-loop level.

SM contributions
When the sfermions, Higgsinos, and the heavier Higgs bosons are in the range of O(10-100) TeV and decouple from the low energy physics below the TeV scale, the neutral and the charged winos only couple to the SM particles through the SU(2) L gauge interaction. In such cases, the radiative correction to the mass splitting from the SM sector can be calculated by using the effective Lagrangian, where L SM is the SM Lagrangian and M 2 is the invariant mass of the winos. The notation for the SM gauge fields is understood, and SU(2) L gauge coupling is denoted by g, while c W (s W ) = cos θ W (sin θ W ) with θ W being the weak mixing angle.
The mass splitting between the charged and the neutral winos is caused by the custodial symmetry breaking by U(1) Y gauge and Yukawa interactions. It should be noted that the breaking of the custodial symmetry is highly suppressed at the treelevel in the wino-SM system. In fact, at the tree-level, the breaking of the custodial symmetry is mediated through the Higgsino mixing. As a result, the tree-level mass splitting is highly suppressed by the Higgsino mass, µ, which is given by Here, m W denotes the mass of the W -boson, β the Higgs mixing angle of the MSSM, and M 1 the mass of the bino. 2 As we will see below, the above tree-level mass splitting is sub-dominant compared to the radiatively generated mass splitting. 3

The pole mass
The pole mass of a spin half particle can be extracted from the 1PI effective two-point function, with p being the four momentum of the particle and M 0 the tree-level mass. Thus, for given self-energy functions, Σ K and Σ M , the pole mass is iteratively given by In a perturbative analysis, we expand the above pole mass as a power series of coupling constants. At the two-loop level, the above iterative expression of the pole mass is reduced to Here, Σ K,M and Σ

Renormalization scheme and input parameters
We take the input parameters to the above effective Lagrangian: where the hatted variables denote the MS variables, and Q is the renormalization scale. All the quark and lepton masses except for the top quark mass are neglected in our analysis.
To relate the above listed input parameters (the MS variables) to the experimental observables, we have to take finite renormalization effects into account. In the following analysis, we extract the input parameters by using the renormalized relations at the one-loop level, where all the self-energies (Π xx and Σ Once we obtain the input parameters,α,m W , andm Z from Eqs. (6)-(8), we can calculateĝ,ĝ ′ using tree-level relations. In deriving the one-loop relations in Eqs. (6)-(9), we also obtain the counter-terms to subtract ultra-violet (UV) divergences. These counter-terms play important roles to calculate Σ K,M , as will be discussed later.

The mass splitting at one-loop level
The one-loop result of the mass splitting between neural and charged winos is well known [25]- [27] and used in the earlier literature. The loop diagrams of the winos and gauge bosons shown in Fig. 1 lead to the functions Σ K,M given in the appendix B, the mass splitting δm = mχ± − mχ0 at the one-loop level is given by where the function f (z) is defined as f (z) = In the heavy wino limit,M 2 ≫m Z,W , the mass splitting is reduced to which is about 160-170 MeV.

The mass splitting at two-loop level (strategy)
The two-loop self-energies, Σ K,M (M 2 2 ), are obtained from the two-loop diagrams (Fig. 2) and from the diagrams including counter-terms which cancel the one-loop UV divergences (Fig. 3). In our actual analysis, we first calculated the two-loop 1PI amplitudes using FeynArts [35] and FeynCalc [36], which were reduced to a set of basis integrals by TARCER [37]. We finally evaluated the integrals numerically using TSIL [38]. For the diagrams including counter-terms, we used the ones given in Appendix B.3. As a nontrivial cross check, we have confirmed that all the UV divergences are properly canceled.
We also have to care about infra-red (IR) singularities. For the charged wino, the amplitude in and hence, they are IR divergent. In addition, the derivatives,Σ K,M | p 2 =M 2 2 , are also IR divergent due to the diagram including a photon propagator. We have checked that all the IR divergences are canceled with each other when we evaluate the pole mass in Eq. (5). See the appendix C for more discussions on the cancellation of the IR divergences.     Here, we takem t (m t ) =163.3 GeV and m h =125.5 GeV.
In Fig. 4, we show the renormalization scale dependence of δm, which is the dominant source of the theoretical uncertainty of the mass splitting at the two-loop level. In the figure, the two-loop result is numerically evaluated by Eq. (5) in the MS scheme, while the one-loop result is evaluated by Eq. (10) in terms of the on-shell W and Z boson masses, i.e., where we definedc 2 In addition to the above uncertainty, there are expected to be other uncertainties from the higher-loop corrections which are not encapsulated in the choice of the renormalization scale. At the three-loop level, for example, the dominant nondecoupling contribution to the mass splitting is expected to be proportional to m t and the QCD coupling. Although the numerical factors of those corrections cannot be determined unless explicitly calculated, we give naive estimations to those higher loop corrections by where α 2 = g 2 /4π. Here, we have multiplied a factor of π which is expected to  The experimental errors of the input parameters also lead to uncertainties of the mass splitting. As we summarize in Tab which can be understood as non-decoupling effect. We can see that numerical value of mass splitting at two-loop level is consistent with the result of Ref. [29]. For mχ ≃ 100 GeV, two-loop contribution is about −3.5 MeV. Then, we can see decoupling effect also diminishes wino mass splitting if wino mass is small, although this effect is smaller than non-decoupling effect. We also show the theoretical and experimental uncertainties as green/red bands. As a result, we find that the uncertainties are 6 We have confirmed that the naive estimation of the two-loop contribution, gives a fair estimation of our two-loop numerical results.
significantly reduced by the two-loop analysis. By numerical calculation, we have also confirmed that the limit m W,Z ≪ mχ, our result reproduces the one in Ref. [29] at this level of precision in the heavy wino limit, M 2 ≫ m Z .
for the central values of the SM input parameters. Deviation of the above fitting function from our two-loop result is smaller than 0.02 % for the wino mass being 100-4000 GeV.

SUSY contributions
Before closing this section, let us evaluate the contributions to the mass splitting from the diagrams including the heavy SUSY particles in the tens to hundreds TeV range.
Since the winos couple to the other gauginos (bino and gluinos) only through the exchange of those heavy particles, all the SUSY contributions to the wino masses can be expressed by the higher-dimensional operators suppressed by the heavy masses.
At a first glance, a five-dimensional operator,   The next lowest-dimensional operator which contributes to the mass splitting is the dimension-seven operator where M denote the insertion of the gaugino mass. 7 For Λ = O(10-100) TeV, the contribution from this operator to the mass splitting is again negligibly small.

The charged wino decay
As we have seen in the previous section, the charged and the neutral winos are highly degenerated in mass. Therefore, the decay width of the charged wino is highly suppressed by the phase space integral, and hence, the charged wino is long-lived and has the decay length about cτ = O(1-10) cm. With such a rather long decay length, it is possible to detect the charged wino production at the LHC experiment by looking for disappearing tracks. In this section, we estimate the lifetime of the charged wino and compare with the constraint from the disappearing track search by the ATLAS collaboration [28].
With the small mass splitting δm ∼ 160 MeV, the charged wino dominantly decays into a neutral wino and a soft charged pion. At the leading order, the decay width of the charged wino can be expressed in terms of the decay width of the charged pion, where m π and m µ denote the masses of the charged pion and the muon, respectively. 8 The decay width of the sub-leading leptonic decay mode into a pair of the electron and the neutrino [44] is given by We consider the above two decay modes. 7 This operator can be obtained from, for instance, a dimension-eight operator (q L χ a H) † (q L χ a H)/Λ 4 which is generated by integrating out the squarks (especially stops) at the tree-level. By integrating the quark-loop and inserting the gaugino mass, we obtain the dimensionseven operator in Eq. (20). 8 At the next-to-leading order, Eq. (21) receives radiative corrections from the QED and the electroweak interactions which are expected to be around (α/π) log(mχ/m π ) ≃ 2 %. In this Letter, we neglect these corrections to the total decay width and leave the detailed analysis of the decay width for future study [43].

Summary
We have calculated the mass splitting of the charged and the neutral wino in the almost pure wino LSP scenario at the two-loop level. Such a scenario attracts more attention after the discovery of the Higgs-like boson at the LHC experiment. As a result, we found that the lifetime of the charged wino becomes about 10-30 % longer due to the two-loop contributions. Furthermore, we also found that the precise determination of the mass splitting improves the constraint on the mass of the wino obtained by the disappearing track search at the ATLAS experiment by about 10%.
subsection, we present the contributions to the amplitude Π(p 2 ) from both the SM particles and the winos, which are divided into three parts; where V 1 V 2 = γγ, γZ, ZZ, and W W . The first term in the right-hand side is the contributions from the quarks and the leptons, the second term is those from the gauge-Higgs sector of the SM, and the third term is from the neutral and charged winos. The fourth and fifth terms show the counter-terms given in appendix B.3.

B.2 Wino self-energies
With the use of the 1PI amplitudes Σ K (p 2 ) and Σ M (p 2 ), the full propagators (2point functions) of the winos are given by i/[{1 + Σ K (p 2 )} / p −M 2 + Σ M (p 2 )]. In this subsection, we explicitly present the amplitudes for both neutral and charged winos at the one-loop level. For the neutral wino, the amplitudes are given by On the other hand, the two amplitudes for the charged wino are given by where explicit forms of the counter-terms, δ Zχ and δ Mχ , are given in Appendix B.3.

B.3 Counter-terms
Finally, we give the counter-terms in the MS scheme in the framework of the SM plus the winos. These are used in the calculations of the self-energies mentioned above and of the mass splitting at the two-loop level as shown in Fig. 3.

B.3.1 Gauge boson self-energies
where N g is the number of the generation, namely N g = 3 for the SM.
where we have neglected the masses of all the SM fermions except the top quark.

B.3.3 Gauge interaction of the wino
The neutral and charged winos have the SU(2) L gauge interaction which is described by the term, L int = iǫ abc (ĝ + δχχ W )χ a † / W bχ c , and the counter-term is given by where we have introduced a photon mass m γ to control the IR divergences. The ellipses stand for the contributions from the loop diagrams of the W and Z bosons, which are nothing to do with the IR divergences. The derivative of the one-loop amplitude F 1L (q 2 ) with respect to p 2 gives the IR-divergent contribution, where the ellipses represent the terms which do not cause the IR divergences, namely the IR-safe terms. The second contribution, F 2L ≡ (−1)(Σ As a result, the IR-divergent part of the two-loop contribution can be reduced to, Therefore, we find that the IR-divergences cancel with each other, and hence, the pole mass is an IR-safe quantity.