Hadronic Production and Decays of Charginos and Neutralinos in Split

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Supersymmetry (SUSY) is one of the most elegant solutions, if not the best, to the gauge hierarchy problem. It also provides a dynamical mechanism for electroweak symmetry breaking, as well as a viable candidate for dark matter (DM). Conventional wisdom tells us that SUSY must exist at the TeV scale, otherwise the fine tuning problem returns. Unfortunately these weak scale SUSY models, represented by the minimal supersymmetric standard model (MSSM), also have a number of difficulties such as SUSY flavor and CP problems, and the light Higgs boson. For the past two decades, particle theorists have made every effort to rectify the problems. Recently, Arkani-Hamed and Dimopoulos [1] adopted a rather radical approach to SUSY: they discarded the hierarchy problem and accepted the fine-tuning solution to the Higgs boson mass. They argued that much more serious fine- tuning is required for the observed small cosmological constant. If the cosmological constant problem is to be explained by the anthropic principle [2], an enormous number of metastable vacua, usually called the vast landscape of string theory vacua, are essential. Fine-tuning in the Higgs boson mass is more natural in this anthropic landscape [3]. Once we accept this proposal the finely-tuned Higgs boson mass is not a problem anymore. More concerned issue is to find a consistent set of parameters satisfying the following obser- vations: (i) the refined result of the DM density to be Ω h2 =0.094 0.129 (2σ range) by DM − Wilkenson Microwave Anisotropy Probe (WMAP) [4]; (ii) sub-eV neutrino masses; and (iii) cosmological constant. The last one is accepted as an extremely fine-tuned principle. The second observation requires heavy right-handed neutrinos of a mass scale of 1011−13 GeV; it does not have appreciable effects on electroweak scale physics. The first observation, on the other hand, requires a weakly interacting particle of mass < 1 TeV, in general. It is this ∼ requirement which affects most of the parameter space of the split SUSY scenario [5, 6]. The split SUSY scenario can be summarized as follows:

1. All scalars, except for a CP-even neutral Higgs boson, are super heavy, for which a 9 common mass scalem ˜ is assumed around 10 GeV to MGUT. Dangerous phenomena such as ﬂavor-change neutral currents, CP-violating processes, and large electric dipole moments in the MSSM become safe.1

1 There is another source of CP -violation as pointed out in Ref. [7].

2 2. The gaugino and the Higgsino masses (the µ parameter), which can be much lighter thanm ˜ due to an R and a PQ symmetry, respectively, are assumed near the TeV scale. Light supersymmetric fermions have additional virtue of gauge coupling uniﬁcation as in MSSM, as well as providing a good DM candidate.

3. A light Higgs boson is very similar to the SM Higgs boson, but could be substantially heavier than that of the MSSM [1, 5, 8, 9, 10].

4. The DM density requires suﬃcient mixing in the neutralino sector, so that the lightest neutralino can eﬃciently annihilate [5, 11, 12]. In most cases, the µ parameter is relatively small.

Super heavy masses of SUSY scalar particles practically close oﬀ many neutralino annihilation channels, resulting in smaller annihilation cross section than that in MSSM. Giudice and Romanino [5], Pierce [11], and Masiero et al. [12] identiﬁed three interesting regions of the parameter space where the split SUSY model can accommodate the WMAP data on the DM density.

(i) The lightest neutralino is mostly a Bino, but with a substantial mixing with the Hig-

gsino: µ is comparable to M1. This is because the Bino interacts only with the Higgs and Higgsino when all sfermions are super heavy. Substantial mixing can guarantee eﬃcient annihilation of the Bino into Higgs or Z bosons. The WMAP data allow the gaugino mass parameters as low as the current lower bound.

(ii) The second region is where the LSP is mostly Higgsino with M µ. In this case, 1,2 ≫ the Higgsino LSP annihilates via gaugino-Higgsino-Higgs and Higgsino-Higgsino-gauge couplings. Note that the lightest chargino and the second lightest neutralino are also predominately Higgsino since µ M , and thus degenerate in mass with the LSP. ≪ 1,2 Eﬃcient annihilation and co-annihilation require a rather heavy LSP mass µ 1 1.2 ∼ − TeV for the LSP to be the DM.

(iii) The third region is the wino LSP in anomaly mediation and in this case M2 < M1,µ. Neutral wino can annihilate eﬃciently via an intermediate W ∗, even in the absence of sfermions. Thus, a rather heavy wino with M 2.0 2.5 TeV is required to account 2 ≃ − for the dark matter density.

3 The last two cases require a rather heavy LSP. Therefore, the chance of observing supersymmetric partners at colliders diminishes as the gluino would be even heavier than 2–3 TeV, though it is not impossible. Thus, we concentrate on the phenomenologically more interesting case where M µ M . Note that once we impose the condition of gaugino-mass 1 ∼ ∼ EW unification, M 2M at the weak scale. The current chargino mass bound is almost 104 2 ≃ 1 GeV [13]. If converting to the bounds on M2 and µ, both M2 and µ larger than O(100) GeV should be consistent with the bound. We do not specifically state the bound on M1 and µ, but O(100) GeV should be very safe for M1 and µ with the chargino mass bound. The detailed phenomenological study of the split SUSY scenario, especially focused on high energy colliders, is crucial to clarify the basic structure of the scenario. As pointed out in Ref. [1], the first unique feature is the stable or metastable gluino. It may give rise to stable charged tracks [14, 15, 16] or gluinonium [17] signatures. We focus on the neutralino pair, chargino pair, and neutralino-chargino pair production, and their decays at hadron colliders. These production channels may be the only accessible ones if the gluino mass is more than 3 TeV. There have been some studies of chargino and neutralino production at e−e+ colliders [18] and at hadron colliders [14]. Some variations on split SUSY were also studied, e.g., taking µ to be very large [19], taking gaugino masses to be very large [20], or even taking µ and all gaugino masses to be very heavy [21]. 2 Note that the phenomenology of split SUSY is rather similar to those of focus-point SUSY [22], except for the gluino phenomenology. There is also the PeV scale SUSY scenario [23]. 0 0 0 ± + − At hadron colliders, the production of χi χj , χj χi , and χi χj is mainly via the Drell- Yan-like processes with intermediate W ∗,γ∗, or Z∗ exchanges as in the usual MSSM. The e e e e e e t-channel exchanges viaq, ˜ ℓ˜ are virtually absent. On the other hand, the decays of the 0 neutralino χj (j > 1) and chargino would be quite different from the MSSM case. With 0 0 super-heavy sfermions the decay of the heavier neutralino χj into a lighter neutralino χi e plus ff¯ is mainly via an intermediate Z∗ (the Higgsino component only) instead of a f˜. e e Another channel of importance is χ0 χ±W ∗ χ±ff¯′. We also have χ0 χ0h(∗) j → i → i j → i → 0 ¯ ∗ 0 0 χi +(bb or W W ). Finally, we note that the branching ratio of the radiative decay χj χi γ e e e e e → can be sizable unlike the case with light sfermions. Phenomenologically it leaves clean signal e e e

2 The last scenario was called supersplit supersymmetry which appeared in arXic.org on April fool’s day. It is equivalent to the SM.

4 at the LHC, consisting of a high-energy single-photon plus missing energy. Particularly in the parameter region where µ is comparable to M , the radiative branching ratio would be | | 1 maximal. The chargino decays mainly through χ+ χ0 + W ∗ χ0 + ff¯′. By combining j → i → i the phenomenological studies on neutralinos and charginos, as well as those of gluino, one e e e may be able to distinguish the scenarios of split SUSY, PeV scale SUSY, or focus-point SUSY. Improvement over previous studies [14, 18] can be summarized as follows.

1. We include all decay modes of neutralinos and charginos, in particular, the radiative decay mode. It gives rise to a single-photon plus missing energy signal. Experimentally, it is very clean.

2. We include the CP-violating phases in M1 and µ in order to examine the eﬀect of the CP phases on the decay of neutralinos.

3. We calculate the photon transverse momentum and missing transverse momentum distributions for the single-photon plus missing energy events, which mainly come from the decay of χ0 χ0γ. 3 → 1 The organization of this paper is as follows. We describe the general formalism and e e convention in Sec. II. We calculate the decays and production cross sections for neutralino and chargino pairs in Sec. III and IV, respectively. We conclude in Sec. V.

II. GENERAL FORMALISM OF NEUTRALINOS AND CHARGINOS

The neutralino and chargino sectors are determined by fundamental SUSY parameters:

the U(1) and SU(2) gaugino masses M1 and M2, the Higgsino mass parameter µ, and

tan β = v2/v1 (the ratio between the vacuum expectation values of the two neutral Higgs 3 0 0 ﬁelds). The neutralino mass matrix in the (B, W , H1 , H2 )L basis, where the subscript L denotes the left-handed chirality of neutralinos, is [24] e f e e M 0 m c s m s s 1 − Z β W Z β W   0 M2 mZ cβcW mZ sβcW =  −  , (1) MN    m c s m c c 0 µ   − Z β W Z β W −       mZ sβsW mZ sβcW µ 0   − −    5 where sβ = sin β, cβ = cos β and sW ,cW are the sine and cosine of the electroweak mixing angle θ . Since is symmetric, one unitary matrix N can diagonalize the such that W MN MN = N ∗ N †. The Majorana mass eigenstates are Mdiag MN

0 0 0 0 T 3 0 0 T χ1, χ2, χ3, χ4 L = N B, W , H1 , H2 L. (2) e f e e The mass eigenvalues m 0e(i e=e 1, 2e, 3, 4) in diag can be chosen positive by a suitable χi M deﬁnition of the unitary matrixe N. The chargino mass matrix in the (W −, H−) basis is [25]

fM2 e √2mW cos β = , (3) MC   √2mW sin β µ   † 3 ± ± which is diagonalized by two unitary matrices through UR C UL = diag(mχ , mχ ). The M 1 2 unitary matrices UL and UR can be parameterized in the following way [25]: e e

−iβL iγ1 i(γ1−βR) cos φL e sin φL e cos φR e sin φR UL = , UR = . (4)  eiβL sin φ cos φ   ei(γ2+βR) sin φ eiγ2 cos φ  − L L − R R     In CP-violating theories, the mass parameters are complex. Since M2 can be taken real and positive by rephasing the ﬁelds suitably, the split SUSY scenario allows only two CP- violating phases of M1 and µ:

M = M eiΦ1 and µ = µ eiΦµ (0 Φ , Φ < 2π) . (5) 1 | 1| | | ≤ 1 µ

The universal gaugino mass relation at the GUT scale implies at the weak scale

5 M = tan2 θ M 0.502M . (6) | 1| 3 W 2 ≃ 2

The ﬁve underlying SUSY parameters M , Φ , µ , Φ ;tan β determine the mass spectrum {| 1| 1 | | µ } and couplings of the neutralinos and charginos. The interaction Lagrangian relevant for the production and decay of neutralinos and

3 ∗ UL and UR are related to U and V in Haber-Kane[26] notation by V = UR and U = UL.

6 charginos are expressed in 4-component form as

= e χ−γµχ−A g QccZ χ−γµP χ−Z g QnnZ χ0γµP χ0Z L i i µ − Z αij i α j µ − Z αij i α j µ i α,i,j α,i,j X X X g e Snnhe χ0χ0h0 g Qnnhe χ0P χe0h0 e e − i i i − αij i α j i i

III. DECAYS OF NEUTRALINOS AND CHARGINOS

In general, the production cross section of gaugino-pair in split SUSY is smaller than the case with light sfermions. It brings a challenge to experimental search of gauginos. At the LHC, the gluino-pair production is dominated by the gg-initiated subprocess, which stays the same as in the MSSM. (Though the qq¯-initiated subprocess changes, it is very small compared with gg-initiated one.) On the other hand, the detection is very diﬀerent. Gluinos so produced would be detected as massive stable charged particles [14, 15, 16] or as a gluinonium [17]. Even though the detection of heavy meta-stable gluinos alone can strongly support the split SUSY scenario, it cannot provide any more information on the fundamental SUSY parameters involved in the neutralino and chargino sectors. Even with small production rates, the decays of neutralinos and charginos in split SUSY are to be studied in detail. 0 (∗) In split SUSY, the chargino decay is very simple. It would decay into χ1W giving rise to a single charged lepton or two jets plus missing energy. On the other hand, the heavier e 0 neutralino χj have a few more decay modes.

4 Note that χe− is considered as a particle, contrary to the Haber and Kane notation.

e 7 χ0 χ0Z(∗) χ0(jj/ℓℓ¯), giving rise to a couple of jets or charged leptons plus • j → i → i missing energy. e e e χ0 χ0h(∗) χ0(b¯b/WW ∗). If the Higgs boson is lighter than about 125 GeV, it • j → i → i ∗ decays dominantly into b¯b. With mh 130 GeV the W W mode becomes important, e e e ≥ which is possible because the Higgs mass can be substantially larger in split SUSY [5, 8, 9, 10].

χ0 χ±W ∓(∗) χ±ff¯′. This mode happens when the heavier neutralino is heavier • j → i → i than the lightest chargino, especially, in the region where the µ parameter is smaller e e e than M2.

χ0 χ0γ [27]. This mode goes through the chargino-W loop, as the loops involving • j → i sfermions are highly suppressed. This decay mode gives a single photon and missing e e energy.

The expressions for the decay rates of ﬁrst three decay modes are greatly simpliﬁed in split 0 0 SUSY. We give the formulas for various decays of the neutralino χj into χi . Note that for 0 0 0 decays of χ3,4 they may go through χ2 before they end up in χ1. e e e e e A. Two body decays of χ0 χ0Z, χ0h0, χ±W ∓ j → i i i

For simplicity we introducee somee short-hande e notation of

2 2 m 0(±) χi mX µij = , µXj = , (8) m 0 m 0 eχj ! χj !

0 ± e 0 e where X = Z, h , W . If mχ0 > m 0(±) +mX , χj decays into a lighter neutralino or chargino j χi

associated with the X boson.e The total decay rate is then e e 1/2 0 0(±) λ (1,µij,µXj) 2 Γ χj χi X = , (9) → 16πmχ0 |M| j e e e

8 where λ(a, b, c) = a2 + b2 + c2 2ab 2bc 2ca and 2 is the spin-average amplitude − − − |M| squared. For each decay mode 2 is |M| 2 2 gZmχ0 (1 µ )2 2(χ0 χ0Z) = j ( QnnZ 2 + QnnZ 2) − ij +1+ µ 2µ (10) |M| j → i 2 | Rij | | Lij | µ ij − Zj e Zj nnZ nnZ∗ 12√µij e(Q Q ) , e e − ℜ Rij Lij 2 2 i g mχ0 2 0 0 0 j nnh 2 nnh 2 χj χi h = QRij + QLij (1 + µij µhj) (11) |M| → 2 e | | | | − h nnh nnh ∗ +4√µij e(Q Q ) , e e ℜ Rij Lij 2(χ0 χ−W +) = 2(χ0 χ+W −) i (12) |M| j → i |M| j → i 2 2 g m 0 2 χj cnW 2 cnW 2 (1 µij) e e = e ( Q e + Q ) − +1+ µij 2µWj 2 | L1j | | R1j | µ − e Wj cnW cnW∗ 12√µij e(Q Q ) . − ℜ L1j R1j B. Three body decays of χ0 χ0,±ff¯(′) j → i

e e 0(±) ¯(′) ∗ ∗ ∗ If mχ0 < m 0(±) + mX , the decay will proceed into χi ff via a virtual Z , W or h . j χi

The diﬀerentiale decay width is given by e e dΓ NC mχ0 = j 2 , (13) ¯ 3 dxf dxf 256πe |M|

where NC is the color factor of the fermion f and the kinematic variables are deﬁned in the 0 rest frame of χj by

2Ef 2Ef¯ e xf = , xf¯ = , xi =2 xf xf¯ . (14) m 0 m 0 − − χj χj

e e 2 2 The integration range of xf and xf¯ are, with the deﬁnition of µf(f¯) m ¯ /m 0 , ≡ f(f) χj

e 2√µf xf 1+ µf µ ¯ µij 2√µ ¯µij , (15) ≤ ≤ − f − − f x ¯ x ¯ x ¯ f(−) ≤ f ≤ f(+) where

1 ¯ ¯ xf(±) = (2 xf )(1 + µf + µf µij xf ) (16) 2(1 xf + µf ) − − − − h 2 (x 4µ )λ(1 + µ x , µ ¯, µ ) . ± f − f f − f f ij q i 9 The spin-average amplitude squared for each decay mode is

2 (χ0 χ0Z∗ χ0ff¯) (17) |M| j → i → i 4 f 2 f 2 2 2 2 nnZ 2 nnZ 2 = g ( g + g )dˆ (x + x ¯)(1 µ ) x x ( Q + Q ) e z | Le| | R|e Z { f f − ij − f − f¯} | Lij | | Rij | h nnZ nnZ∗ 4√µij (1 + µij xi) e(Q Q ) , − − ℜ Lij Rij 2 (χ0 χ0h∗ χ0b¯b) i (18) |M| j → i → i g4m2 sin2 α = b dˆ2 (1 + µ x 2µ ) x ( Qnnh 2 + Qnnh 2)+4 µ e(QnnhQnnh∗) , e 2 e 2 he ij i b i Lij Rij √ ij Lij Rij 4mW cos β − − | | | | ℜ 2 (χ0 χ−W ∗ χ−ff¯′) (19) |M| j → i → i 4 2 cnW 2 cnW 2 = g dˆ x (1 µ x ) Q + x ¯(1 µ x ¯) Q e W e f − eij − f | Lij | f − ij − f | Rij | cnW cnW∗ 2√µij (1 + µi xi) e(Q Q ) , − − ℜ Lij Rij ˆ 0 where the propagator factor dX , with X = Z, h , W , is

2 m 0 ˆ χj 1 dX = 2 = . (20) (p + p ¯)2 m 1+ µ x µ f f e − X ij − i − Xj Here the chiral couplings of the fermion f to the Z boson are given by

gf = s2 Q , gf =(T f ) s2 Q , (21) R − W f L 3 L − W f

f where (T3 )L is the third component of the isospin and Qf is the electric charge.

C. Radiative decay

In split SUSY, the radiative decay of neutralino, χ0 χ0γ, may have a substantial j → i 0 0 (∗) 0 0 (∗) branching fraction as the other decay channels are limited to χj χi Z , χj χi h , and e e → → χ0 χ±W ∓(∗). As discussed in Ref. [27], the radiative neutralino decay proceeds through j → i ± ∓ ± ∓ ±e ∓ e e e triangle diagrams mediated by the ff˜, χ W , χ H , and χ G loops in the non-linear Rξ e e ± gauge where the charged Goldstone boson G has the same mass as mW . In split SUSY, e e e only W ±- and G±-mediated diagrams contribute. We expand the results of Ref. [27] into the CP-violating case, and examine whether CP-violating phases can enhance the radiative decay rate of neutralinos. For the decay of χ0(p) χ0(k )+ γ(k ), (22) j → i 1 2 e e 10 the matrix element is, in general, given by

1 ∗ = u¯(k1)(gv + gaγ5)/k2/ǫ u(p) , (23) M m 0 χj

where ǫ denotes the polarization 4-vectore of the photon, and the radiative decay width of 0 χj can be easily calculated as

2 2 3 (m 0 m 0 ) e 0 0 2 2 χj − χi Γ(χj χi γ)=( gv + ga ) 5 . (24) → | | | | 8πm 0 e χj e

e e e 0 In the notation of Ref. [27], the coupling of the incoming neutralino χj to the particles in the loop is denoted by X X X e F = fL PL + fR PR , (25)

0 and the coupling of the outgoing neutralino χi by

X X X G = gL PeL + gR PR , (26)

where X = W ±,G± and igγµ( ig) is omitted for the W ±(G±) loop respectively. Explicitly − − we have

W cnW W cnW ∗ fα = Qαkj , gα =(Qαki ) , (27)

G cnG G cnG ∗ fα = Qαkj , gα =(Qαki¯ ) , whereα ¯ = R(L) for α = L(R). The helicity amplitude in the CP-violating cases is

2 1 eg X X ∗ = 2 u¯(k1)(gV + gA γ5)/k2/ǫ u(p) , (28) M m 0 8π χj XX=W,G e

11 X where gV,A are given by

W W W W W 2 gV = i m(gL fL + gR fR ) m 0 (I2 J K) mχ0 mχ0 (J K) (29) ℑ χj − − − j i − k=1,2 h n o X e e e ± W W W W +2mχ0 mχ m(gL fR + gR fL ) J , j k ℑ W W W W W 2 i e e 0 0 gA = e(gL fL gR fR ) mχ0 (I2 J K)+ mχ mχ (J K) ℜ − j − − j i − k=1,2 h n o X e e e ± W W W W +2mχ0 mχ e(gL fR gR fL ) J , j k ℜ − i i G e G eG G G 2 gV = m(gL fR + gR fL ) m 0 (I2 K)+ mχ0 mχ0 K 4 ℑ χj − j i k=1,2 h n o X e e e ± G G G G + mχ0 mχ m(gL fL + gR fR )I , j k ℑ 1 i G e Ge G G G 2 gA = e(gL fR gR fL ) m 0 (I2 K) mχ0 mχ0 K −4 ℜ − χj − − j i k=1,2 h n o X e e e ± + mχ0 mχ e(gLfL gRfR)I . j k ℜ − i e e The radiative decay width is

2 2 3 3 (mχ0 mχ0 ) α 2 2 j − i Γ= 2 4 ( gV + gA ) 5 , (30) 8π sin θW | | | | m 0 e χj e where g gX . The loop integrals I,I ,J,K are thee same as in Ref. [27]. V,A ≡ X V,A 2 P D. Numerical Results of neutralino decay rates

Focusing on the parameter space of µ compatible with M , we consider µ [160, 230] | | 1 | | ∈ GeV for M = 200 GeV. With the gaugino mass uniﬁcation condition of M 0.502M , 1 | 1| ≃ 2 the mass hierarchy among the neutralinos and charginos is then m 0 < m ± < m 0 < m 0 . χ1 χ1 χ2 χ3 0 0 First, we compute the partial decay widths of χ2 and χ3 ine the CP-conservinge e casee

(Φ1 = Φµ = 0) in Figs. 1 and 2, respectively. Considering the trigger strategy at the e e LHC, we separate the electron and muon signal from other fermion signals. The notation ′ of ff¯( ) includes all appropriate fermions f, f ′ except for e± and µ±. In the decay of 0 0 0 ¯ χ2,P the majority comes from χ2 χ1ff via an on-shell or a virtual Z. It is followed by → 0 ± ¯′ χ2 χ ff via an on-shell or a virtual W . The radiative decay width is down by 3 orders e → e e 0 0 (∗) 0 ¯ of magnitude from χ2 χ1Z χ1ff in the range of µ that we are interested in. Figure e e → → 0 − 2 shows the partial widths of χ3. A substantial mass diﬀerence between mχ0 and mχ for e e e 3 1 e e e 12 1000

0 ¯ 100 χ˜1 P ff 0 + − + − χ˜1(e e + µ µ ) 10 ± ¯′ χ˜1 P ff

1 0 ± ± χ˜1(e νe + µ νµ) decay rate Γ in keV 0 2 ˜ χ 0.1 0 χ˜1γ

0.01 160 170 180 190 200 210 220 230 µ

0 FIG. 1: Partial decay widths of χ2 for M1 = 200 GeV and tan β = 10. All the CP violating phases are set to zero. ff¯(′) includes all possible SM fermion pairs except for e± and µ± involving pairs. e P1000

± ¯′ χ˜1 P ff 100 ± ∓ ± χ˜1 (e νe + µ νµ)

10 0 + − χ˜2 P(qq¯ + l l ) 0 χ˜1 P ff¯ 1 0 + − + − χ˜1(e e + µ µ ) decay rate Γ in keV 0 3

˜ 0 χ 0.1 χ˜1γ 0 χ˜2γ 0.01 160 170 180 190 200 210 220 230 |µ|

0 FIG. 2: Partial decay widths of χ3 for M1 = 200 GeV and tan β = 10. All the CP violating phases are set to zero. e

µ compatible with M leads to the dominant decay mode into χ±ff¯′ via an on-shell or a | | 1 0 0 (∗) 0 ¯ virtual W . It is followed by χ3 χ2Z χ2ff, which drops oﬀ quickly when µ increases. → → e A special dip (around µ 178 GeV with M1 = 200 GeV) shows up in the decay width of | | ≃ e e e χ0 χ0Z(∗) χ0ff¯. This is because of a cancellation in the factor (N N ∗ N N ∗ ) of 3 → 1 → 1 13 33 − 14 34 e e e 13 0.005

0 0 χ3 → χ1γ e e 0.004 ) γ 0 i ˜

χ 0.003 → 0 j χ 0.002 BR(˜

0.001 0 0 χ2 → χ1γ 0 0 e e χ3 → χ2γ 0 e e 160 170 180 190 200 210 220 230 |µ|

0 0 FIG. 3: Radiative decay branching ratios for χ2 and χ3 with M1 = 200 GeV and tan β = 10. All the CP violating phases are set to zero. e e the χ0-χ0-Z coupling. The radiative decay width of χ0 χ0γ is relatively much larger than 3 1 3 → 1 0 0 in the case of χ2. This is the mode that we want to make use of in the collider study of χ3 e e e e 0 0 in split SUSY. Since the masses of χ3 and χ1 cannot be degenerate, the outgoing photon e e is quite energetic. A single-photon plus missing energy event search is possible provided e e that the branching ratio is large enough. We show the radiative decay branching ratios for χ0 χ0γ, χ0 χ0γ, and χ0 χ0γ in Fig. 3. The branching ratio for χ0 χ0γ is much 2 → 1 3 → 1 3 → 2 3 → 2 0 0 smaller than the other two. With increasing µ , BR(χ2 χ1γ) moderately decreases while e e e e e e | | → e e 0 0 BR(χ3 χ1γ) increases rather rapidly. → e e Finally, we examine the dependence of CP-violating phases on the neutralino radiative e e decay. The eﬀect of Φ1 is quite weak, as expected, since M1 is involved only in the neutralino mass matrix. On the other hand, the Φµ dependence is rather strong. The µ parameter 0 ± ∓ aﬀects both the neutralino and chargino sectors, and the χj -χi -W coupling in the loop. 0 0 We compute the Φµ-dependence on the radiative branching ratios of χ2 and χ3 in Fig. 4 e e and Fig. 5, respectively. Three cases of µ = 180, 200, 220 GeV are considered. Figures 4 | | e e and 5 clearly show the CP-phase sensitivity of the neutralino radiative branching ratios. As can be seen in Fig. 5, the BR(χ0 χ0γ) can be enhanced by a factor of about four with 3 → 1 µ = 220 GeV at Φµ = π. In most cases, the maximum of radiative branching ratios occurs | | e e at Φ = π, i.e., negative µ. We also see the tan β dependence by plotting BR(χ0 χ0γ) µ 3 → 1 e e 14 0.002 |µ| = 180 GeV 0.0018 |µ| = 200 GeV |µ| = 220 GeV 0.0016 ) γ

0 1 0.0014 ˜ χ

→ 0.0012 0 2 χ 0.001 BR(˜ 0.0008

0.0006

0.0004 0 π/2 π 3π/2 2π

Φµ

FIG. 4: Radiative decay branching ratio of χ0 χ0γ as a function of Φ (in unit of π) for 2 → 1 µ M = 200 GeV and tan β = 10. Three cases for µ = 180, 200, 220 GeV are shown. 1 e| | e

for tan β = 50 and µ = 220GeV in Fig. 5. At Φ = 0, larger tan β enhances the radiative | | µ branching ratio by a factor of about two. However, at Φµ = π where the radiative BR is maximized, tan β = 50 case has smaller BR, about half of that with tan β = 10. In Fig. 6, we show the variation of the branching ratio versus negative µ. It is now clear that at µ = 220 GeV, a branching ratio about 1.68% can be obtained for the radiative decay − χ0 χ0γ with tan β = 10. 3 → 1 e e

IV. PRODUCTION OF NEUTRALINOS AND CHARGINOS

In split SUSY, gaugino-pair production goes through Drell-Yan-like process via γ,Z, or W s-channel exchange diagrams. Speciﬁcally, we study the production of

q + q χ˜0 +χ ˜0, q + q χ˜− +χ ˜+, q + q′ χ˜± + χ0 . (31) → i j → i j → i j For simplicity we introduce the following notation: e m2 sˆ µ = i ,D = , (32) isˆ sˆ X sˆ m2 + im Γ − X X X 15 tan β = 10, |µ| = 220 GeV 0.02 |µ| = 200 GeV |µ| = 180 GeV tan β = 50 |µ| = 220 GeV )

γ 0.015 0 1 ˜ χ → 0 3

χ 0.01 BR(˜

0.005

0 0 π/2 π 3π/2 2π

Φµ

FIG. 5: Radiative decay branching ratio of χ0 χ0γ as a function of Φ (in unit of π) for 3 → 1 µ M = 200 GeV and tan β = 10. Three cases for µ = 180, 200, 220 GeV are shown. One 1 e | e| additional line (dotted) of tan β = 50 and µ = 220 GeV is also shown. | |

0.02

0.015 χo χo γ 3-> 1

0.01

0.005 Radiative Decay Branching ratio χo χo γ 2-> 1 χo χo γ 3-> 2 0 -220 -210 -200 -190 -180 -170 -160 µ (GeV)

0 0 FIG. 6: Radiative decay branching ratios for χ2 and χ3 with M1 = 200 GeV and tan β = 10. The 0 0 Φµ is set at π, i.e., negative µ. Note that BR(χ χ γ) is very close to 0. e3 → 2e e e 16 0 − where mi denotes the mass of χi or χi and X = Z, W . The helicity amplitude for neutralino-pair production is e e e2 T qq χ0χ0 = (nn)ij [¯v(q)γ P u(q)] u¯(χ0)γµP v(χ0) , (33) → i j sˆ Q αβ µ α i β j α,β=L,R X e e e e where α, β = L, R and (nn) in the superscript of (nn)ij denote neutralino pair production. Q αβ The four generalized bilinear charges (nn)ij are Q αβ

(nn)ij DZ q nnZ αβ = 2 2 gα Qβij . (34) Q sW cW Here the chiral couplings of the quark q with the Z boson are given by

gq = s2 Q , gq =(T q) s2 Q , (35) R − W q L 3 L − W q

q where (T3 )L is the third component of the isospin and Qq is the electric charge of the quark q. The helicity amplitude for chargino pair production is

e2 T qq χ−χ+ = (cc)ij [¯v(q)γ P u(q)] u¯(χ−)γµP v(χ+) , (36) → i j sˆ Q αβ µ α i β j α,β X where the bilinear chargese e are given by e e

(cc)ij q DZ q ccZ αβ = Q δij + 2 2 gαQβij . (37) Q − sW cW Finally, the helicity amplitude for chargino-neutralino associated production is

e2 T du¯ χ−χ0 = (cn)ij [¯v(¯u)γ P u(d)] u¯(χ−)γµP v(χ0) . (38) → i j sˆ Q αβ µ α i β j α,β X where the bilinear chargese e are e e

D (cn)ij = W δ QcnW . (39) αβ 2 αL βij Q √2sW At parton level, the diﬀerential cross sections in the center-of-mass (c.m.) frame for the above three channels have a common expression

dσ 1 1 πα2 = λ1/2 (40) d cos θ∗ N 2ˆs c S 2 2 ∗ ij ij 1/2 ij ∗ 1 (µisˆ µjsˆ) + λ cos θ +4√µisˆµjsˆ +2λ cos θ , × − − Q1 Q2 Q3 h i

17 where θ∗ is the scattering angle in the parton rest frame, λ λ(1,µ ,µ ), N is the color ≡ isˆ jsˆ C factor of q, is the symmetric factor ( = 2 for χ0χ0 production) and θ∗ is the scattering S S i i ij angle in the c.m. frame. The 1,2,3 are combinations of bilinear charges given by Q e e 1 ij = ij 2 + ij 2 + ij 2 + ij 2 , (41) Q1 4 |QRR| |QLL| |QRL| |QLR| 1 ij = e ij ij∗ + ij ij∗ , Q2 2 ℜ QRRQRL QLLQLR 1 ij = ij 2 + ij 2 ij 2 ij 2 , Q3 4 |QRR| |QLL| − |QRL| − |QLR| where the production channel of (nn), (cc) or (cn) in the superscript is omitted. 0 In Fig. 7, we show the production cross sections for χ3 at the LHC versus negative 0 0 0 0 0 + and positive µ with M1 = 200 GeV and tan β = 10, including χ3χ1, χ3χ2, χ3χ1 , and | | e 0 − χ3χ1 . Here we consider only the CP -conserving case (Φµ = π, 0). It is clear that the cross e e e e e e section is in general larger in the region µ M1, because this is region where the Bino and e e | | ∼ Higgsino mix strongly. This is also the favored parameter space for the Bino-Higgsino dark matter [5, 11, 12]. Away from the peak the cross sections in general decrease as the mixing between the Bino and the Higgsino becomes less; in particular, as µ decreases from the | | mixing region (M µ) the χ0 and χ0 become Higgsino-like while χ0 becomes Bino-like, 1 ∼ 1 2 3 0 0 ± and so the production of χ3 with χ1,2 and χ1 all decrease rapidly. e e e 0 The inclusive cross section for χ3 is of the order of O(0.1 0.2) pb for M1 = 200 GeV. We e e e − have also calculated that for M1 = 100 GeV and maintaining the gaugino mass unification e condition, the inclusive cross section for χ0 is as large as O(1 5) pb, while the cross section 3 − goes down to about 0.05 pb for M1 = 300 GeV. Since we are interested in the radiative e 0 decay of χ3, which could have a branching ratio as large as O(1)%, the number of single 3 3 photon plus ET events at the LHC is of the order of O(10 5 10 ), O(100 200), and e 6 − · − O(50) for M1 = 100, 200, 300 GeV, respectively. The next-to-leading order (NLO) SUSY corrections to gaugino pair production have been performed [28]. The K factor is about 1.1 1.4 depending on SUSY parameters. Since these SUSY corrections would not affect − significantly the photon momentum or missing energy distributions, we just note that the event rates can be enhanced by up to about 40% due to NLO corrections. Here we present the differential cross section versus the transverse momentum of the single-photon and versus the missing transverse momentum. We adopt a simple two-body decay of the third neutralino into the lightest neutralino and the photon, without taking into account the spin correlation, which should only be a mild effect. We also assume the decay is

18 0.16 χ+ χo 1 3 χ- χo 0.14 1 3 χo χo 2 3 0.12 χo χo 1 3 β 0.1 LHC M1=200 GeV tan =10

0.08

0.06 Cross Section (pb) 0.04

0.02

0 -250 -230 -210 -190 -170 -150 µ (GeV) 0.16 χ+ χo 1 3 0.14 χ- χo 1 3 χo χo 2 3 0.12 χo χo 1 3 β 0.1 LHC M1=200 GeV tan =10

0.08

0.06 Cross Section (pb) 0.04

0.02

0 150 170 190 210 230 250 µ (GeV)

0 FIG. 7: Production cross sections for χ3 at the LHC versus (a) negative and (b) positive µ with

M1 = 200 GeV and tan β = 10. e

0 prompt, because the decay width of χ3 is of the order of MeV. The contributing production 0 0 0 0 0 ± channels include χ3χ2, χ3χ1, and χ3χ1 , We include all of these production channels to e 0 0 0 account for inclusive χ3 production. We focus on the radiative decay of χ3 χ1γ, which e e e e e e → has a branching ratio of the order of 1% in the region µ M1. The charginos can decay e | | ∼ e e

19 10-4

10-5 LHC

-6 χo χo γ 10 p p -> 3 + X -> 1 + X

) (pb/GeV) -7

γ 10

o 1 χ

-> -8 o 3 10 χ

* B( -9 γ

T 10 /dE

σ -10 d 10

10-11 0 100 200 300 400 500 600 700 800 900 1000

ETγ (GeV)

FIG. 8: Diﬀerential cross section versus the transverse momentum of the photon for the process pp χ0 + X χ0γ + X. We set M = 200 GeV, µ = 220 GeV, and BR= 1.68%. → 3 → 1 1 −

e e ′ into the neutralino and a virtual W boson, which further goes into qq¯ or ℓνℓ. Therefore, the ﬁnal state consists of an isolated single-photon with or without charged leptons or jets, plus missing energies. We are primarily interested in the photon and missing energy distributions, which are shown in Fig. 8 and Fig. 9, respectively. Here we give a brief discussion on possible backgrounds. An irreducible background comes from the SM production of γZ followed by Z νν¯, as well as reducible background → coming from quark fragmentation into a photon. Photon isolation and large p cuts should 6 T be able to suppress the quark fragmentation background. The γZ background, on average, has a smaller missing energy than the signal, because very often in the signal there are two missing neutralino LSP’s. Furthermore, a large portion of the signal will have a charged lepton or jets coming from the associated chargino decay that one can tag so as to remove the γZ background. Therefore, with the above mentioned cuts, tagging, and isolations, the signal of an isolated single-photon plus missing energy signature should be rather clean.

20 10-4

10-5 LHC

-6 χo χo γ 10 p p -> 3 + X -> 1 + X

) (pb/GeV) -7

γ 10

o 1 χ

-> -8

o 3 10 χ

* B( -9 T 10 /dp/ σ d 10-10

10-11 0 100 200 300 400 500 600 700 800 900 1000

p/ T (GeV)

FIG. 9: Diﬀerential cross section versus the missing transverse momentum for the process pp → χ0 + X χ0γ + X. We set M = 200 GeV, µ = 220 GeV, and BR= 1.68%. 3 → 1 1 −

V.e CONCLUSIONSe

We note that the single-photon plus missing energy signal is a possible sign of split SUSY, but not a unique feature. For example, gauge-mediated SUSY breaking (GMSB) models often predict various signals consisting of single- or multi-photon plus multiple leptons and jets, and missing energy. Split SUSY has to be established by gluino production, decay, and detection, and also with neutralino and chargino production and decay. One has to determine the sfermion mass scale and the µ parameter by simultaneously measuring the production and the decay properties of gauginos. One can use the mass spectrum and the photon energy spectrum to diﬀerentiate between split SUSY and GMSB models. We have shown that in general the χ0 χ0γ has a branching ratio of the order of 1%, and the 3 → 1 photon energy spectrum depends on the mass diﬀerence. On the other hand, in the GMSB e e models when the NLSP is a neutralino, the branching ratio into a photon and a gravitino is dominant. Therefore, the frequency of multi-photon plus missing energy events is very high. Also, the photon energy spectrum depends on the mass of the lightest neutralino only. In summary, we have studied the decays and productions of neutralinos and charginos

21 at hadron colliders in the scenario of split SUSY. The decays of neutralinos are particularly interesting because the sfermions are so heavy that only the decays via Z(∗), W (∗), and h(∗) are possible, among which the radiative decay could have a branching ratio as large as O(1)%, unlike the case of MSSM. We have found that χ0 χ0γ has a suﬃciently large 3 → 1 branching ratio when µ M1 such that isolated single-photon plus missing energy events | | ∼ e e may be accessible in the LHC experiments. We argued that such events are rather clean and would be signs of split supersymmetry. Furthermore, counting the event rates will also give a measurement on the radiative decay branching ratio so that it gives information about the parameters of split SUSY.

Acknowledgments

We appreciate the hospitality of KIAS as we completed a part of this paper while we visit KIAS. The work of KC was supported in part by the National Science Council of Taiwan R.O.C. under grant no. NSC 93-2112-M-007-025-. The work of J. Song is supported by KRF under grant no. R04-2004-000-10164-0.

APPENDIX A: COUPLINGS OF NEUTRALINOS AND CHARGINOS

All the couplings of supersymmetric fermions, neutralinos and charginos, to gauge bosons ′ ff X ′ or Higgs bosons are expressed by Qαij . Here f(f ) = c in the superscript denotes the chargino, f(f ′) = n denotes the neutralino, X = Zµ, W µ, h0,H±,G±, and α = L, R in the subscript is the chirality of the fermions. The G± is the Goldstone boson absorbed by W ± boson. In our notation, negatively charged chargino is assigned to be a particle such that

its electric charge Q ± is 1. χ − − + The speciﬁc expressionse for the χi -χj -Z couplings are 3 1 3 1 QccZ = s2 cos2φ , QccZ = s2 cos2φ , L11 W − 4 − 4 e eL R11 W − 4 − 4 R 1 1 QccZ = e−iβL sin 2φ , QccZ = e−i(βR−γ1+γ2) sin 2φ , L12 −4 L R12 −4 R 3 1 3 1 QccZ = s2 + cos2φ , QccZ = s2 + cos2φ . L22 W − 4 4 L R22 W − 4 4 R 0 0 The χi -χj -Z couplings are ∗ ∗ Ni3N Ni4N QnnZ = j3 − j4 , (A1) e e Lij 2

22 where QnnZ = (QnnZ)∗. Neutral couplings of χ0-χ0-h0, where h0 is the light CP -even Higgs Rij − Lij i j boson, are divided into i = j and i = j cases, by using Majorana conditions: 6 e e 1 Snnh = [N t N ][ sin αN cos αN ] , (A2) i 2 i2 − W i1 − i3 − i4 1 Qnnh = (N ∗ t N ∗ )( sin αN ∗ cos αN ∗ )+(i j) , (A3) Lij 2 i2 − W i1 − j3 − j4 ↔

nnh nnh ∗ where QRij =(QLij ) , tW = tan θW and

2 2 2 2 2 mh mA cos β mZ sin β tan α = − 2 2 − . (A4) (mA + mZ ) sin β cos β

− 0 + The χi -χj -W couplings are

cnW ∗ 1 ∗ cnW 1 eQLije =(UL)i1Nj2 + (UL)i2Nj3, QRij =(UR)i1Nj2 (UR)i2Nj4 . (A5) √2 − √2 The charged couplings to H± are

cnH ∗ (UR)k2 ∗ ∗ Q = cos β (UR)k1N + (N + tW N ) , (A6) Lkj j4 √2 j2 j1 (U ) QcnH = sin β (U ) N L k2 (N + t N ) . Rkj L k1 j3 − √ j2 W j1 2 − 0 + Finally we have the χi -χj -G couplings of

cnG cnh e Qαkje = Qαkj (sin β cos β, cos β sin β) . (A7) → − →

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