Electroweak two-loop contribution to the mass splitting within a new heavy SU(2)$_L$ fermion multiplet

New heavy particles in an SU(2)_L multiplet, sometimes introduced in extensions of the standard model, have highly degenerate tree-level mass M if their couplings to the Higgs bosons are very small or forbidden. However, loop corrections may generate the gauge-symmetry-breaking mass splitting within the multiplet, which does not vanish in the large M limit due to the threshold singularity. We calculate the electroweak contribution to the mass splitting for a heavy fermion multiplet, to the two-loop order. Numerically, two-loop electroweak contributions are typically O(MeV).


Introduction
In some extensions of the standard model, there are new heavy particles which belong to an SU(2) L ×U(1) multiplet F and have no, or very small, mixing with other particles. The masses of these particles are almost degenerate to a value M by the gauge symmetry. Although the spontaneous breaking of the SU(2) L ×U(1) symmetry by Higgs bosons may generate mass splitting δM among them, the tree-level mass splitting generally behaves as δM ∼ m 2 W /M and becomes very small for M ≫ m W . This is especially the case for a very heavy fermion multiplet where tree-level renormalizable couplings to the Higgs bosonsF F H are forbidden by symmetry. Some of the examples are the almost pure winos or higgsinos [1,2,3,4,5,6,7,8] in special parameter regions of the minimal supersymmetric standard model [9], SU(2) L triplet fermions in Type III seesaw model for neutrino masses [10,11], and also models [12,13] where vector-like heavy fermion multiplets are added to the standard model by hand.
In such cases, it has been known [2,4,6,8,11,12,13,14] that the dominant part of the gauge-symmetry-breaking mass splitting within the multiplet F comes from the radiative correction. Although the form of the mass correction strongly depends on models, the contributions involving electroweak gauge bosons V = (γ, Z, W ), shown in Fig. 1(a) for the one-loop, are common in a wide class of extended models. Since gauge symmetry breaking in this diagram comes from the squared masses (m 2 W , m 2 Z ) in the loops, one naively expect the O(α 2 m 2 W /M) contribution to the mass splitting. However, due to the singularity of the diagram near the threshold, at p 2 = M 2 ∼ (M + m V ) 2 , O(α 2 m W ) contribution to the mass splitting appears, which does not vanish in the M ≫ m W limit: Roughly speaking, it is "nondecoupling". This mass splitting is phenomenologically interesting, especially in the case where the neutral component f 0 of F , either fermion or boson, is stable or has very long lifetime, and may be a candidate for the cosmological dark matter. In such a case, the loop-generated mass splitting between charged components f Q (Q = 0) of F and f 0 is crucial for estimating the rates of the f Q → f 0 + · · · decays expected at colliders, and also for possible resonant annihilation [8,13].
To evaluate the mass splitting within F to the next-to-leading order, we need twoloop calculation of the mass correction for the members of F . In this paper, we perform such calculation for the loop corrections by the standard model particles, generated by the electroweak gauge interactions of F . For simplicity, we concentrate on the SU(2) Lbreaking and "nondecoupling" part of the mass correction, which should be relevant for the mass splitting in the M ≫ m W case.

One-loop mass correction
Since the electroweak contributions to the mass correction should be determined by the SU(2) L ×U(1) representation of F , we work in the framework of the Minimal Dark Matter model [13], which has been proposed as a minimalist approach to the dark matter problem, for the fermion case. In this case, Dirac or Majorana fermions in an SU(2) L multiplet F with SU(2) L isospin I and U(1) hypercharge Y (and having no SU(3) color) are added to the standard model. The lagrangian is where c = 1(1/2) for Dirac(Majorana) fermions, respectively. Note that the mass corrections presented in this paper are common to both types of fermions. D µ denotes SU(2) L ×U(1) gauge covariant derivative for F . Since F has no direct couplings to the Higgs boson, the members of F , f Q (with charge Q = I 3 + Y , I 3 = −I, −I + 1, . . . , I) have a common mass M at the tree-level. We assume that M is sufficiently larger than the masses of standard model particles (W , Z, top quark t, Higgs boson h), typically M = O(TeV) which is cosmologically favored in the Minimal Dark Matter model [13].
We also use approximation that all other particles in the standard model are massless. The pole mass M p of f Q at the two-loop order is given in terms of the self energy of as Here Σ K,M and Σ (2) K,M are the one-loop and two-loop parts, respectively. The dot in Eq. (3) denotes the derivative with respect to the external momentum squared. The absorptive part of the self energy is O(g 6 ) and need not be considered here. Loop integrals are regularized by the dimensional regularization (D = 4 − 2ǫ) with the MS subtraction scheme.
The form of the one-loop mass correction δM (1) is well known [2,6,8,11,12,13,14,15]. Abbreviating the factor α 2 /(4π), it is expressed as where We use the Passarino-Veltman one-loop functions [16] defined as and The O(m V ) term of Eq. (5) gives the nondecoupling mass splitting within the multiplet. For example, for Y = 0, the one-loop mass splitting between f Q and the neutral component f 0 of F is written as [13], independent of I, where, in the M ≫ m W limit, The numerical value in Eq. (9) is obtained by using the pole masses m W = (80.398±0.025) GeV, m Z = 91.1876 GeV, , and the QED running coupling in the MS scheme α(m Z ) = (127.93 ± 0.03) −1 , cited from Ref. [17], as input parameters. Note that the value (9) should change by ∼ 1 MeV depending on choices of the renormalization scheme for the input parameters.

Two-loop mass correction
We now calculate the two-loop mass correction δM (2) coming from diagrams shown in Fig. 1(b-e). We use Feynman gauge fixing for simplicity, although the final result should not depend on the gauge fixing method.
The contribution of the diagram Fig. 1(b) with the insertion of the one-loop self energy of the electroweak gauge boson, , is written as where Here we list the analytic forms of Π V 1 V 2 (k 2 ) in the standard model for completeness [18]. The contributions from the (t, b) quark loops are, up to the overall factor N c α 2 /(4π) (N c = 3 is the color number of quarks), where The contributions of other quarks and leptons are obtained by appropriate changes of m t , Q q , and N c . For the gauge and Higgs boson loops, we have, abbreviating the overall factor α 2 /(4π), In addition, there are also the contributions of F to Π V 1 V 2 . However, it is shown that the resulting O(m W ) contributions to δM (2) are completely cancelled by the renormalization of the parameters in δM (1) . We may calculate the integrals (11) by extending the general formulas for the twoloop mass corrections [19], by including finite masses for (W, Z). However, since we are interested in the SU(2) L -breaking and nondecoupling part of Eq. (11), it is prefered to expand the integrals (11) in m W (∼ m Z , m t , m h ) and then separate the O(m W ) terms from the dominant and gauge-symmetric O(M) terms, before numerical evaluation. This is achieved by applying the asymptotic expansion of the Feynman integrals near the threshold p 2 = M 2 , as described in Ref. [20]. The O(m W ) part of the integral (11) is then obtained as In the following, we show only the O(m W ) part (21) of the corrections ∆Σ V 1 V 2 . By substituting the self energies (12)(13)(14)(15)(16)(17)(18)(19)(20), the integrals (21) are expressed in terms of the two-loop functions (a = 1, 2) and products of the one-loop functions. Note that the functions in Eq. (22) are independent of M and have no overall divergences. We calculate these functions by numerical integration of the Feynman parameter integrals shown below, where r 1,2 ≡ m 2 1,2 /m 2 V . µ is the MS renormalization scale. Here we show the explicit forms of the integrals (21), after subtracting O(1/ǫ) divergences from Π V 1 V 2 by the MS scheme, and separating the mass corrections to the gauge bosons δm 2 . The (t, b) contributions coming from Eqs. (12)(13)(14)(15)(16) are, up to the overall factor N c α 2 /(4π), with Here we used the notations and substituted analytic forms of the two-loop integrals (22) and, for leptons, changing (Q q , N c ). Similarly, the gauge and Higgs boson contributions coming from Eqs. (17)(18)(19)(20) are, up to the factor α 2 /(4π), Other diagrams shown in Fig. 1(c-e) are also evaluated by using the threshold expansion [20], keeping only the O(m V ) parts. Their sum, with subtracting subdivergences by the MS scheme and after the (one-loop)×(one-loop) term in Eq. (3) is added, is given as Here is the two-loop function appearing in Fig. 1(c,d) with both W and Z bosons. We then need to add the counterterms coming from the renormalization of the parameters in the one-loop contributions (4,5); (m W , m Z ) in X (1) W,Z and (α 2 , c 2 W , . . . ) in the coupling constants. We adopt the scheme where the pole masses (m W , m Z ) and the MS running coupling of QED α(m Z ), which are used in Eq. (9), are chosen as the input parameters. In this scheme, the renormalization is achieved by removing the last O(δm 2 V ) terms from ∆Σ W W (27, 38) and ∆Σ ZZ (30, 41), and adding the counterterms for (α 2 , c 2 W , . . . ) expressed as tree-level functions of (m Z , m W , α(m Z )). It is checked that the final form of the two-loop O(m W ) mass correction to f Q is finite and independent of the MS renormalization scale µ.
Here we comment on the mass splitting of a new heavy scalar SU(2) L multiplet S. In contrast to the case of the fermion multiplet, direct couplings of S to the Higgs bosons, such as S * SH † H, should always exist [13,21]. Nevertheless, assuming that the effect of these direct couplings is negligible, we have verified that the nondecoupling O(m W ) parts of the one-loop [13,22] and two-loop mass corrections δM are identical to those for the fermions in the same gauge representation. This result is quite natural in the view that the O(m W ) mass correction could be understood as the energy of the electroweak gauge fields around a static point source, and should be insensitive to the spin of the source particle [13].
The contribution ∆M (2,ql) of the quark-lepton subloop diagrams (including corresponding counterterms) is shown in Fig. 2   The remaining contribution ∆M (2,V h) from diagrams without quarks or leptons (again including corresponding counterterms) is shown in Fig. 3 as a function of m h . At m h = 140 GeV, the shift is −0.9 MeV, smaller than the quark-lepton loops.
These two-loop contributions are much smaller than the O(m W ) part of the leading one-loop contribution (4), as expected. However, for Y = 0, it may compete with the M-dependence of the one-loop contribution (8)

Conclusion
We have calculated the two-loop electroweak contribution to the O(m W ) correction to the masses of new heavy fermions in an SU(2) L multiplet F , which causes gauge-symmetrybreaking and "nondecoupling" mass splitting within F . Analytic formula of the O(m W ) mass corrections have been presented for F in general SU(2) L ×U(1) representation. The two-loop contribution has turned out to be typically O(MeV), which is of similar order to the M dependence of the one-loop contribution for the Y = 0 case.