Solving the muon g-2 anomaly in deflected AMSB with messenger-matter interactions

TL;DR

This paper proposes a modification to the deflected anomaly mediated SUSY breaking scenario by adding messenger-matter interactions, successfully explaining the muon g-2 anomaly within current experimental constraints and predicting accessible gluino masses for future collider searches.

Contribution

It introduces messenger-matter interactions in deflected AMSB, demonstrating their effectiveness in resolving the muon g-2 anomaly and analyzing the implications for gluino masses and dark matter constraints.

Findings

Muon g-2 anomaly can be explained in both complete and incomplete GUT multiplet scenarios.

Gluino mass upper bounds are around 2.5-3.0 TeV, accessible at future LHC runs.

Dark matter constraints favor the incomplete GUT multiplet scenario B, with much of its parameter space testable by future experiments.

Abstract

We proposed to introduce general messenger-matter interactions in the deflected anomaly mediated SUSY breaking scenario to explain the $g_\mu-2$ anomaly. Scenarios with complete or incomplete GUT multiplet messengers are discussed, respectively. The introduction of incomplete GUT mulitiplets can be advantageous in various aspects. We found that the $g_\mu-2$ anomaly can be solved in both scenarios under current constraints including the gluino mass bounds, while the scenarios with incomplete GUT representation messengers are more favored by the $g_\mu-2$ data. We also found that the gluino is upper bounded by about 2.5 TeV (2.0 TeV) in Scenario A and 3.0 TeV (2.7 TeV) in Scenario B if the generalized deflected AMSB scenarios are used to fully account for the $g_\mu-2$ anomaly at $3\sigma$ ($2\sigma$) level. Such a gluino should be accessible in the future LHC searches. Dark matter constraints, including DM relic density and direct detection bounds, favor the scenario B with incomplete GUT multiplets. Much of the allowed parameter space for the scenario B could be covered by the future DM direct detection experiments.

Tables

Table 1. Table 1: The range of d 𝑑 d within which the wino will be lighter than bino for various messenger species N 𝑁 N in Scenario A.

	N=1	N=2	N=3	N=4
$d$

Table 2. Table 2: The range of d 𝑑 d within which the wino will be lighter than bino for various messenger species M 𝑀 M in Scenario B.

		M=1	M=2	M=3
Scenario B1	d	$-0.92\lesssim d\lesssim 1.23$	$-0.51\lesssim d\lesssim 0.78$	$-0.35\lesssim d\lesssim 0.95$
Scenario B2	d	$-1.21\lesssim d\lesssim 1.05$	$-0.69\lesssim d\lesssim 0.64$	$-0.49\lesssim d\lesssim 0.46$

Equations

$$\mathcal{Z}(\tilde{\mu };\tilde{X},{\tilde{X}}^{†})=Z(\mu ;X,X^{†})+\left[{\theta }^{2}F\frac{\partial }{\partial X}+{\bar{\theta }}^2{F}^{†}\frac{\partial }{\partial X^{†}}+({\theta }^2\tilde{F}+{\bar{\theta }}^2{\tilde{F}}^{†})\frac{\partial }{\partial \mu }\right]Z(\mu ;X,X^{†})$$

$$+{\theta }^2{\bar{\theta }}^2\left(F^{†}F\frac{{\partial }^2}{\partial X\partial X^{†}}+F^{†}\tilde{F}\frac{{\partial }^2}{\partial X^{†}\partial \mu }+{\tilde{F}}^{†}F\frac{{\partial }^2}{\partial X\partial \mu }+{\tilde{F}}^{†}\tilde{F}\frac{{\partial }^2}{\partial {\mu }^2}\right)Z(\mu ;X,X^{†}).$$

$$Q^{\prime }\equiv Z^{1/2}\left[1+{\theta }^2\left(\frac{F}{M}\frac{\partial }{\partial \ln X}+\frac{\tilde{F}}{\mu }\frac{\partial }{\partial \ln \mu }\right)\ln Z(\mu ;X,X^{†})\right],$$

$$m^2$$

$$\mathcal{L}$$

$$\frac{A_{abc}}{y_{abc}}$$

$$\frac{F}{M}\to d{F}_{\varphi },\frac{\tilde{F}}{\mu }\to -F_{\varphi }/2.$$

$${\overline{Q}}_{\varphi }^I(1,2{)}_{1/2},{\tilde{Q}}_{\varphi }^I(1,\bar{2}{)}_{-1/2},{\overline{T}}_{\varphi }^I(\bar{3},1{)}_{1/3},T_{\varphi }^I(3,1{)}_{-1/3},(I=1,\cdots ,N)$$

$$W$$

$${\Sigma }_O^I(8,1{)}_0,{\sigma }_T^I(1,3),Z^J(1,1{)}_1,{\bar{Z}}^J(1,1{)}_{-1},I,J=(1,\cdots ,M);$$

$${\overline{Q}}_{\varphi }^A(1,2{)}_{1/2},{\tilde{Q}}_{\varphi }^A(1,\bar{2}{)}_{-1/2},{\overline{T}}_{\varphi }^A(\bar{3},1{)}_{1/3},T_{\varphi }^A(3,1{)}_{-1/3}.$$

$$W$$

$$Q_{\varphi }^A$$

$$W$$

$${\tilde{y}}_{ij}^E\frac{{\lambda }_X}{\sqrt{{\lambda }_A^2+{\lambda }_X^2}}=y_{ij}^E,-{\tilde{y}}_{ij}^D\frac{{\lambda }_A}{\sqrt{{\lambda }_A^2+{\lambda }_X^2}}=y_{ij}^D.$$

$${\tilde{y}}_{ij}^E\frac{{\lambda }_A}{\sqrt{{\lambda }_A^2+{\lambda }_X^2}}\equiv -{\lambda }_{ij}^E,-{\tilde{y}}_{ij}^D\frac{{\lambda }_X}{\sqrt{{\lambda }_A^2+{\lambda }_X^2}}\equiv {\lambda }_{ij}^D,\sqrt{{\lambda }_A^2+{\lambda }_X^2}\equiv {\lambda }_S.$$

$$W$$

$$\Delta {\beta }_{g_i}=\frac{1}{16{\pi }^2}{g}_i^3\Delta b_{g_i},$$

$$\Delta (b_3,b_2,b_1)=(N,N,N),$$

$$\Delta (b_3,b_2,b_1)=(3M+1,2M+1,1)\mathrm{Scenario}B1$$

$$\Delta (b_3,b_2,b_1)=(3M+1,2M+1,\frac{6M}{5}+1)\mathrm{Scenario}B2$$

$${\delta }^I$$

$$G^i\equiv \frac{d{Z}_{ij}}{d\ln \mu }\equiv -\frac{1}{8{\pi }^2}\left(\frac{1}{2}{d}_{kl}^i{\lambda }_{ikl}^{\ast }{\lambda }_{jmn}{Z}_{km}^{-1\ast }{Z}_{ln}^{-1\ast }-2c_r^i{Z}_{ij}{g}_r^2\right),$$

$$\Delta G_{y_t}$$

$$\Delta G_{y_b}$$

$$\Delta G_{y_{\tau }}$$

$$2{\delta }_{Q_{L,i}}^I$$

$$2{\delta }_{U_{L,i}^c}^I$$

$$2{\delta }_{D_{L,i}^c}^I$$

$$2{\delta }_{L_{L,i}}^I$$

$$2{\delta }_{E_{L,i}^c}^I$$

$$2{\delta }_{H_u}^I$$

$$2{\delta }_{H_d}^I$$

$${\delta }^G=\frac{{\partial }^2}{\partial \ln X\ln X^{†}}\ln Z^D=\frac{{\partial }^2}{\partial \ln X\ln X^{†}}{Z}^D-\frac{\partial Z^D}{\partial \ln X}\frac{\partial Z^D}{\partial \ln X^{†}}.$$

$$\Delta G_{Q_i{Q}_i}$$

Full text

1 Introduction

Low energy supersymmetry (SUSY) is strongly motivated and regarded as one of the most appealing candidates for TeV-scale new physics beyond the Standard Model(SM). SUSY can not only solve the gauge hierarchy problem of the SM, but also elegantly explain the cosmic dark matter puzzle. Besides, the gauge coupling unification, which can not be achieved in the SM, can be successfully realized in the framework of SUSY. Especially, the 125 GeV Higgs boson discovered by the LHC ATLAS:higgs ; CMS:higgs lies miraculously in the narrow range of $115-135$ GeV predicted by the Minimal Supersymmetric Standard Model (MSSM).

Although SUSY is an appealing extension of the SM, it seems to have some tensions with the current LHC data. In particular, no evidences of SUSY partners (sparticles) have been observed at the LHC. Actually, the LHC data has already set stringent constraints on sparticle masses CMSSM1 ; CMSSM2 in simplified SUSY models, e.g., the gluino mass $m_{\tilde{g}}\gtrsim 1.9$ TeV for a massless lightest sparticle (LSP), the lightest stop mass $m_{\tilde{t}_{1}}\gtrsim 850$ GeV and even stronger bounds on the first two generations of squarks. In fact, the LHC data agrees quite well with the SM predictions and no significant deviations have been observed in flavor physics or electroweak precision measurements. So far the only sizable deviation comes from the so-called anomalous magnetic moment of the muon $a_{\mu}=(g_{\mu}-2)/2$ measured by the E821 experiment at the Brookhaven National Laboratory muong-2:0208067 , which shows a 3.2$\sigma$ discrepancy from the SM prediction. The SUSY explanation of this anomaly requires relatively light sleptons and electroweak gauginos. If SUSY is indeed the new physics to explain all these experimental results, its spectrum must display an intricate structure. Therefore, the origin of SUSY breaking and its mediation mechanism, which determines the low energy SUSY spectrum, is a crucial issue.

There are many popular ways to mediate the SUSY breaking effects from the hidden sector to the visible MSSM sector, such as the gravity mediation SUGRA , the gauge mediation GMSB and the anomaly mediation AMSB SUSY breaking(AMSB) mechanisms. Spectrum from the AMSB is insensitive to the ultraviolet(UV) theory deflect:RGE-invariance and automatically solves the SUSY flavor problem. Unfortunately, the AMSB scenario predicts tachyonic sleptons so that the minimal theory must be extended. There are several ways to tackle the tachyonic slepton problem tachyonslepton . A very elegant solution is the deflected AMSB deflect scenario, in which additional messenger sectors are introduced to deflect the Renormalization Group Equation (RGE) trajectory and give new contributions to soft SUSY breaking termsokada ; Hsieh:2006ig ; Luty:2001zv ; Nelson:2002sa ; Everett:2008qy . On the other hand, a relatively large number of messenger species are needed to give positive slepton masses with small negative deflection parameters. It is known that too many messenger fields may lead to strong gauge couplings below GUT scale or Landau pole below Planck scale. So it is preferred to introduce less messenger species to deflect the RGE trajectory and at the same time give positive slepton masses. In our previous work Fei:1508.01299 , we proposed to solve this problem by introducing general messenger-matter interactions in the deflected AMSB which has advantages in several aspects.

Note that in order to preserve gauge coupling unification, the messenger species are generally fitted into complete representations of the GUT group. However, sometimes it is economic and well motivated to introduce incomplete representations of GUT group, such as the $SU(3)_{c}$ and $SU(2)_{L}$ adjoint messengers in GMSBhan ; ilia ; tianjun . The introduction of incomplete representations of messengers, which seems to spoil successful gauge coupling unification, can be natural in AMSB. This is due to the ${}^{\prime}decoupling~{}theorem^{\prime}$ in ordinary anomaly mediation scenario which states that the simple messenger threshold (by pure mass term) will not deflect the AMSB trajectory. By assigning different origin for messenger thresholds (determined by moduli VEV or pure mass term), even a complete GUT group representation at high energy may seem as ${}^{\prime}incomplete^{\prime}$ in AMSB at low energy. Therefore, the messengers in incomplete GUT representations should also be considered in the study of deflected AMSB.

In this work, we propose to study a generalized deflected AMSB scenario involving messenger-matter interactions with incomplete GUT multiplets. As noted before, the introduction of incomplete GUT mulitiplets in anomaly-type mediation scenarios can be advantageous in various aspects. Besides, virtues of ordinary deflected AMSB are kept while the undesirable Landau-pole type problems can be evaded. Such scenarios can be preferable in solving the muon $g-2$ anomaly. It is known that a SUSY spectrum with heavy colored sparticles and light non-colored sparticles is needed in order to solve the muon $g-2$ anomaly and at the same time be compatible with the LHC data. We try to realize such a spectrum in the deflected AMSB scenario with general messenger-matter interactions, where the messengers can form complete or incomplete GUT representations. In our scenario, the slepton sector can receive additional contributions from both the messenger-matter interactions and ordinary deflected anomaly mediation to avoid tachyonic slepton masses, while the colored sparticles can be heavy to evade various collider constraints.

This paper is organized as follows. In Sec 2, we study the soft parameters in the deflected AMSB scenarios with different messenger-matter interactions. The explanation of the muon $g-2$ in our scenarios and the relevant numerical results are presented in Sec 3. Sec 4 contains our conclusions.

2 General matter-messenger interactions in deflected AMSB

It is well known that the ordinary AMSB is bothered with the tachyonic slepton problem. Deflected AMSB scenario, which can change the RGE trajectory below the messenger thresholds, can elegantly solve such a problem. However, possible strong couplings at the GUT scale or the Landau pole problem may arise with a small negative deflection parameter. Positively deflected AMSB, which may need specific forms of moduli superpotential okada or strong couplings Fei:1505.02785 , could be favored in certain circumstances. However, our previous study indicated that the Landau pole problem may still persist with a small positive deflection parameter in order to solve the $g_{\mu}-2$ anomaly.

In Fei:1508.01299 , we proposed to introduce general messenger-matter interactions in the messenger sector which can have several advantages. In this work, the scenarios with complete or incomplete GUT representation messengers accompanied by messenger-matter interactions will be studied. Note that, the introduction of both adjoint messengers in 3 and 8 representations of $SU(2)_{L}$ and $SU(3)_{c}$, respectively, will not spoil the gauge coupling unificationhan .

Besides, even if the low energy messenger sector seems to spoil the gauge coupling unification, the UV theory can still be consistent with the GUT requirement. As noted previously, the decoupling theorem in anomaly mediation ensures that the vector-like thresholds with pure mass terms $M_{T}>M_{mess}$ will not affect the AMSB trajectory upon messenger scales. So each low energy (deflected) AMSB theory with incomplete GUT multiplet messengers below messenger scale $M_{mess}$ could be UV completed to a high energy theory with completed GUT multiplets at certain scale upon $M_{mess}$. Incomplete GUT multiplet messengers can also origin from orbifold GUT models by proper boundary conditions.

The formulas in deflected AMSB with messenger-matter interactions can be obtained from the wavefunction renormalization approachgiudice with superfield wavefunction

[TABLE]

After canonically normalize the field

[TABLE]

we can obtain the sfermion masses for the most general forms of deflected AMSB

[TABLE]

From the canonicalized normalized superpotential

[TABLE]

we can obtain the trilinear soft terms

[TABLE]

In our scenario, we have the following replacement

[TABLE]

Details on general messenger-matter interactions in deflected AMSB can be found in our previous work Fei:1508.01299 .

2.1 Two scenarios with messenger-matter interactions

• •

Scenario A: deflected AMSB with complete SU(5) GUT representations messengers.

We introduce the following ${}^{\prime}N^{\prime}$ family of new messengers which are fitted into ${\bf 5}$ and ${\bf\overline{5}}$ representation of SU(5) GUT group to deflect the AMSB trajectory

[TABLE]

We introduce the following superpotential that involves messenger-MSSM-MSSM interaction, typically the slepton-slepton-messenger interaction:

[TABLE]

with certain form of superpotential $W(S)$ for pseduo-moduli field $S$ to determine the deflection parameter $d$. From the form of the interaction, we can see that the slepton soft SUSY breaking parameters will be different from the ordinary deflected AMSB results.

• •

Scenario B: deflected AMSB with incomplete SU(5) GUT representations messengers.

Motivated by the GMSB with adjoint messenger scenario, we introduce the following incomplete SU(5) GUT representation messengers to deflect the AMSB trajectory

[TABLE]

We note that additional singlet messengers $Z^{I}$ with non-trivial $U(1)_{Y}$ quantum number can be introduced to deflected the $\tilde{E}_{L}^{c}$ slepton RGE trajectory. As in the previous scenario, the superpotential also involves messenger-MSSM-MSSM interaction, typically the slepton-slepton-messenger interaction:

[TABLE]

We can see that there will be mixing between the messenger $\tilde{Q}^{A}_{\phi}$ and $\tilde{H}_{d}$ (as well as $Q^{B}_{\phi}$ and $H_{u}$). We will define the new states

[TABLE]

After the substitution of the new states, the superpotential changes to

[TABLE]

We have the following relation

[TABLE]

We define

[TABLE]

So the superpotential can be rewritten as

[TABLE]

For simplicity, we chose $\lambda_{ij}^{E}=\lambda_{E}\delta_{ij},\lambda^{D}_{ij}=\lambda_{D}\delta_{ij}$ to be diagonal. Below the messenger threshold determined by the VEV of pseudo-moduli $S$, we can integrate out the heavy fields $F_{H},Q_{\phi}^{A}$ and obtain the low energy MSSM.

2.2 The soft SUSY spectrum in two scenarios

From the superpotential, the soft SUSY breaking parameters can be calculated. In the calculation, the wavefunction renormalizatin approach wavefunction:hep-ph/9706540 is used in which messenger threshold $M_{mess}^{2}$ is replaced by spurious chiral fields $X$ with $M_{mess}^{2}=X^{\dagger}X$. The most general type of expressions in AMSB can be found in our previous work Fei:1508.01299 .

We can calculate the change of the gauge beta-function

[TABLE]

with

[TABLE]

for Scenario A. For Scenario B we consider two cases. One is

[TABLE]

in which ${}^{\prime}I=M,J=0^{\prime}$ is adopted to guarantee apparently gauge coupling unification. The other is

[TABLE]

with ${}^{\prime}I=J=M^{\prime}$ in which apparently the gauge coupling unification is spoiled. However, as we discussed previously, successful GUT may still be possible if certain additional incomplete messengers upon ${}^{\prime}X^{\prime}$ threshold determined by pure mass terms are introduced in the UV completed theory.

From the general expressions in Eq.(2), we can see that there are three types of contributions to the soft SUSY breaking parameters:

• •

The interference contribution part given by

[TABLE]

In our convention, the anomalous dimensions are expressed in the holomorphic basis shih ; chacko

[TABLE]

We define $(\Delta G\equiv G^{+}-G^{-})$, the discontinuity across the integrated heavy field threshold with $G^{+}(G^{-})$ denoting the value upon (below) such threshold, respectively.

The discontinuities of the relevant couplings are given as

[TABLE]

We take into account the terms involving $y_{t},y_{b},y_{\tau},g_{i},\lambda$, and the subleading terms are neglected in the calculation. The new interference contributions from the messenger-matter interactions are given as

[TABLE]

[TABLE]

with $\delta_{i,j}$ the Kronecker delta. Terms involving the gauge parts are absorbed in the deflected AMSB contributions involving $G_{i}$.

• •

The pure gauge mediation part given by

[TABLE]

Note that

[TABLE]

and

[TABLE]

and also the anomalous dimension above the messenger threshold

[TABLE]

so we have

[TABLE]

• •

The pure deflected AMSB contributions without messenger-matter interactions given by

[TABLE]

The expressions are given by

[TABLE]

[TABLE]

with

[TABLE]

So we obtain the final results of soft SUSY breaking parameters for sfermions

[TABLE]

with ${}^{\prime}d^{\prime}$ being the deflection parameter.

The trilinear coupling will also receive new contributions which are given by

[TABLE]

[TABLE]

The gaugino masses are determined by

[TABLE]

So we have

[TABLE]

Therefore, the gaugino masses at the messenger scale are given as

[TABLE]

It is well known in AMSB that naively adding a supersymmetric $\mu$ term to the Lagrangian will lead to unrealistic large $B\mu=\mu F_{\phi}$. So the generations of $\mu$ and $B_{\mu}$ in AMSB may have a different origin and are model dependent. In fact, there are already many proposals to generate realistic $\mu$ and $B\mu$, for example, by promoting to NMSSM cao or introducing a new singlet 1008.2024 . We will treat them as free parameters in this scenario.

3 Solving the muon g-2 anomaly in our scenario

The E821 experimental result of the muon anomalous magnetic moment at the Brookhaven AGS ex:g-2

[TABLE]

is larger than the SM predictionsm:g-2

[TABLE]

The deviation is about $3.2\sigma$

[TABLE]

SUSY can yield sizable contributions to the muon $g-2$ which dominantly come from the chargino-sneutrino and the neutralino-smuon loop diagrams. The muon $g-2$ anomaly, which is order $10^{-9}$, can be explained for $m_{\rm SUSY}={\cal O}(100)$ GeV and $\tan\beta={\cal O}(10)$. In our scenario, slepton masses as well as $M_{1},M_{2}$ can be relatively light. On the other hand, the colored sparticles can be heavy to evade possible constraints from the LHC, the SUSY flavor and CP problems. Some recent discussions can be seen in muon:g-2 .

The soft terms are characterized by the following free parameters at the messenger scale

[TABLE]

All the inputs should be seen as the boundary conditions at the messenger scale, which after RGE running to the EW scale, could give the low energy spectrum. About these parameters, we have the following comments:

• •

The value of $F_{\phi}$ is chosen to lie in the range $1{\rm TeV}<F_{\phi}<500{\rm TeV}$. We know that the value of $F_{\phi}$ determines the whole spectrum. On the one hand, $F_{\phi}$ cannot be very low due to the constraints from the gaugino masses. A very heavy $F_{\phi}$ will spoil the EWSB requirement and give a Higgs mass heavier than the LHC results.

• •

The messenger scale $M_{mess}$ can be chosen to be less than the GUT scale and at the same time heavier than the sparticle spectrum. So we choose $1{\rm TeV}\leq M_{mess}\leq 10^{15}{\rm GeV}$.

• •

We choose the deflection parameter in the range $-5\leq d\leq 5$ and $\tan\beta$ in the range $2\leq\tan\beta\leq 50$.

• •

The parameters $\lambda_{D},\lambda_{E},\cdots$ can be chosen in the range $0<|\lambda|<\sqrt{4\pi}$ which ensure positive contributions to slepton masses regardless of the (sign of) deflection parameter $d$. This is the advantage of our scenario which needs less messenger species with a given $d$.

We also take into account the following collider and dark matter constraints:

• (1)

The mass range for the Higgs boson $123{\rm GeV}<M_{h}<127{\rm GeV}$ from ATLAS and CMS ATLAS:higgs ; CMS:higgs .

• (2)

The lower bounds on neutralino and charginos masses, including the invisible decay bounds for $Z$-boson EW-precision .

• (3)

The dark matter relic density from the Planck result $\Omega_{DM}=0.1199\pm 0.0027$ planck (in combination with the WMAP data wmap ) and the limits of the LUX-2016Akerib:2016vxi ,the PandaXPANDAX spin-independent dark matter scattering cross section .

• (4)

Flavor constraints from the rare decays of B-mesons

• –

Constraints from $Br(B_{s}\rightarrow\mu^{+}\mu^{-})$bsmu

[TABLE]

• –

Constraints from $Br(B_{S}\rightarrow X_{s}\gamma)$ etcbsg

[TABLE]

• (5)

The electroweak precision obsearvables pdg , such as

[TABLE]

• (6)

Current LHC constraints on sparticle masses SUSYmass :

• –

Gluino mass $m_{\tilde{g}}\gtrsim 1.5\sim 1.9$ TeV;

• –

Light stop mass $m_{\tilde{t}_{1}}\gtrsim 0.85$ TeV;

• –

Light sbottom mass $m_{\tilde{b}_{1}}\gtrsim 0.84$ TeV;

• –

First two generation squarks $m_{\tilde{q}}\gtrsim 1.0\sim 1.4$ TeV.

From the numerical results, we have the following observations:

• •

Scenario A: Fig.1 shows the scan results of Scenario A in which the $\Delta a_{\mu}$ versus $m_{\tilde{g}}$ plots with complete GUT multiplets are given. The blue (cyan) dashed line indicate the $2\sigma$ ($3\sigma$) range of $g_{\mu}-2$ data. All survived points satisfy the constraints (1-6) except the bounds from the dark matter relic density and the gluino mass. The most stringent constraints come from the LHC bounds on gluino mass, which excluded a great majority of the survived points that solve the $g_{\mu}-2$ anomaly at $2\sigma$ level. As the messenger species number $N$ gets larger, more and more points can survive the gluino mass bound.

The gluino is upper bounded by about 2.5 TeV (2.0 TeV) if the $g_{\mu}-2$ anomaly is solved at $3\sigma$ ($2\sigma$) level. We know that the $g_{\mu}-2$ anomaly can be solved if the relevant sparticles $\tilde{\mu},\tilde{\nu}_{\mu},\tilde{B},\tilde{W}$ are lighter than $600\sim 700$ GeV muong-2:0208067 (the region with a smaller $\tan\beta$ needs even lighter sparticles). In AMSB, the whole low energy spectrum is determined by the value of $F_{\phi}$. So, in order to solve the $g_{\mu}-2$ anomaly, the mass scale of $\tilde{\mu},\tilde{\nu}_{\mu},\tilde{B},\tilde{W}$ determines the upper bound of $F_{\phi}$, which, on the other hand, sets a bound on gluino mass. The allowed range of $F_{\phi}$ versus the messenger scale $M_{mess}$ in Scenario A is shown in the left panel of Fig 2. It is obvious from the plots that the scale of $F_{\phi}$ is indeed upper bounded to account for the $g_{\mu}-2$ anomaly. We should note that the deflection of the RGE trajectory and the messenger-matter interactions can loosen the bound of $F_{\phi}$ in comparison with the ordinary AMSB.

The deflection parameter $d$ versus the messenger-matter couplings $\lambda_{E}\equiv\lambda$ is plotted in the right panel of Fig.2. We see that additional messenger-matter interactions are welcome to explain the $g_{\mu}-2$ anomaly. Only a small range of $d$ is allowed without leptonic messenger-matter interactions ($\lambda_{E}=0$). However, the allowed range for $d$ enlarges with non-trivial messenger-matter interactions.

Our numerical results indicate that the majority part of the allowed parameter space can not satisfy the the upper bound of dark matter relic density. This result can be understood from the hierarchies among the gauginos at the EW scale. From Eq.(64), the gaugino mass ratios at the weak scale are given by

[TABLE]

Knowing the range of the deflection parameter $d$, the lightest gaugino can be identified.

It can be seen in case $N=1$ that the deflection parameter $d$ is lower bounded to $d\gtrsim 1.5$ for a positive $d$ while $d\lesssim-4.5$ for a negative $d$. From Eq.(75) we can see that for $-4.6<d<2.8$ the lightest gaugino will be the wino, otherwise the lightest gaugino will be the bino. It is well known that the relic density constraints for bino-like dark matter is very stringent and possible co-annihilation with sleptons or resonance are needed to obtain the correct DM relic density. So in a majority of the parameter space allowed by $g_{\mu}-2$ and gluino mass bound, the LSP will be bino-like and can hardly give the right DM relic density. On the other hand, a small portion of the allowed parameter space will predict a wino-like LSP which will lead to insufficient dark matter abundance for a wino mass below 3 TeV unless other DM components (for example, axion) will be present. Heavy wino-like LSP of order 3 TeV will always lead to heavy bino and sleptons which otherwise can not explain the $g_{\mu}-2$ anomaly. Given the upper bounds on $F_{\phi}$ from $g_{\mu}-2$ and gluino mass, the wino will always be much lighter than 3 TeV. We give in Table 1 the range of $d$, within which the wino will be lighter than bino for various messenger species $N$. We can see that only a small portion of parameter space with a positive $d$ can satisfy the dark matter relic density upper bound. The vast parameter space with a bino-like LSP will be stringently constrained by dark matter relic density upper bound. We checked that a very small region can satisfy such relic density constraints. So generalized deflected AMSB scenarios with complete GUT representation of messengers are not favored in solving the $g_{\mu}-2$ discrepancy.

We should note that the constraints from the gluino can be alleviated if we introduce pure colored messenger particles (without $SU(2)_{L}$ and $U(1)_{Y}$ quantum numbers). We can see from the expressions for the soft SUSY parameters that the value of $\Delta b_{3}$ can essentially control the gluino mass. More pure colored messenger particles always mean a heavy gluino for a positive deflection parameter which, on the other hand, may spoil the gauge coupling unification. As noted in the previous section, the complete representation messengers may seem ${}^{\prime}incomplete^{\prime}$ at the low energy $X$ threshold. However, the perturbative gauge coupling unification may be spoiled with more additional messenger species. We will discuss the detailed consequence of general messenger sectors versus gauge coupling unification in our subsequent studies.

• •

Scenario B:

The scatter plots of the survived samples showing $a_{\mu}$ versus $m_{\tilde{g}}$ in Scenario B are shown in Fig.3, in which the upper panel is for Scenario B1 and the lower pannel is for Scenario B2. We can see that a lot of points which can fully account for the $g_{\mu}-2$ anomaly can survive the LHC gluino mass bound, especially, for a larger $M$. So scenarios with the incomplete GUT representation of messengers are more favored by the $g_{\mu}-2$ data.

Similar to Scenario A, the upper bound of gluino mass can be understood from the upper bound of $F_{\phi}$, which is obvious in Fig.4 for both cases. The upper mass bound of gluino is around 3 TeV (2.7 TeV) in both scenarios if the muon $g-2$ is explained at $3\sigma$ ($2\sigma$) level. Such a light gluino will be accessible at future LHC experiments.

The deflection parameter $d$ versus the messenger-matter couplings $\lambda_{E}\equiv\lambda$ in Scenario B is plotted in Fig.5 with all points satisfying both the upper and lower bound of DM relic density. Again, additional non-trivial messenger-matter interactions are obviously advantageous in solving the $g_{\mu}-2$ anomaly with which the allowed range for $d$ enlarges. Besides, the non-vanishing messenger-matter interactions $\lambda\neq 0$ can be used to solved the $g_{\mu}-2$ anomaly for a relatively small deflection parameter $d$, especially for the Scenario B1. We can see from Fig.5 that in Scenario B1 the maximum negative $d$ is $-3.5$ with $\lambda=0$. However, the maximum negative $d$ changes to almost $-2$ with non-vanishing messenger-matter interactions. A small deflection parameter $|d|$ is relatively easy for model buildings. In Scenario B2, it is not possible to solve the $g_{\mu}-2$ anomaly with $\lambda=0$ for a positive $d$. With messenger-matter interactions, a positive deflection parameter also works.

In Fig.5 the survived points which satisfy both the upper and lower bounds of dark matter relic density are shown as green ${}^{\prime}\Box^{\prime}$. The numerical calculation indicates that the number of points which satisfy the dark matter relic density decreases with $M$ in Scenario B1, but increases with $M$ in Scenario B2. This can be understood from the mass ratio between the bino and the gluino with (the most favorite) large negative deflection parameter $d\sim-4$. For a gluino mass between 1.5 TeV and 3 TeV, the mass ratio should be adjusted to a proper value at $M_{3}:M_{1}\sim{\cal O}(10)$ to fully account for the dark matter relic density by decreasing (Scenario B1) or increasing (Scenario B2) the value of $M$. Bino dominated neutralino often leads to over-abundance of DM, unless (co)annihilation processes reduce the relic density to levels compatible with Planck.

We should note that some portion of the parameter space with insufficient DM relic abundance is not displayed in Fig.4 and Fig.5. Following the discussions in Scenario A, we obtain Table 2 from Eq.(75), showing the range of the deflection parameter $d$ within which the wino is lighter than bino. Constrained by $F_{\phi}$, a light wino-like DM will always lead to insufficient relic abundance.

The DM Spin-Independent(SI) direct detection constraints from LUX and PandaX are shown in Fig.6. It can be seen that a large portion of points that satisfy the DM relic density can survive the SI direct detection constraints. We know that interactions between bino DM and the nucleons are primarily mediated by t-channel scalar Higgses ($h_{0}$ and $H_{0}$), or by s-channel squarks (with t-channel Z-boson exchange process highly suppressed). As the squarks are not found at the LHC, their masses should be significantly larger than the Higgs masses. So the SI cross section is dominated by Higgs-mediated process, despite the associated suppression by Yukawa couplings and the small Higgsino fraction. In scenario B, the type of the neutralino which can give the right DM relic abundance is almost bino-like with small Higgsino component, thus suppress the SI direct detection cross sections.

4 Conclusions

We proposed to introduce general messenger-matter interactions in the deflected anomaly mediated SUSY breaking scenario to explain the $g_{\mu}-2$ anomaly. Scenarios with complete or incomplete GUT multiplet messengers are discussed, respectively. The introduction of incomplete GUT mulitiplets can be advantageous in various aspects. We found that the $g_{\mu}-2$ anomaly can be solved in both scenarios under current constraints including the gluino mass bounds, while the scenarios with incomplete GUT representation messengers are more favored by the $g_{\mu}-2$ data. We also found that the gluino is upper bounded by about 2.5 TeV (2.0 TeV) in Scenario A and 3.0 TeV (2.7 TeV) in Scenario B if the generalized deflected AMSB scenarios are used to fully account for the $g_{\mu}-2$ anomaly at $3\sigma$ ($2\sigma$) level. Such a gluino should be accessible in the future LHC searches. Dark matter constraints, including DM relic density and direct detection bounds, favor the scenario B with incomplete GUT multiplets. Much of the allowed parameter space for the scenario B could be covered by the future DM direct detection experiments.

Acknowledgement

This work was supported by the Natural Science Foundation of China under grant numbers 11375001, 11675147, 11675242, by the Open Project Program of State Key Laboratory of Theoretical Physics, ITP, CAS (No.Y5KF121CJ1), by the Innovation Talent project of Henan Province under grant number 15HASTIT017 and the Young-Talent Foundation of Zhengzhou University, by the CAS Center for Excellence in Particle Physics (CCEPP), and by the CAS Key Research Program of Frontier Sciences.