Understanding the correlation between $(g-2)_\mu$ and $\mu \rightarrow e \gamma$ in the MSSM

The supersymmetric contributions to the muon anomalous magnetic moment $a_\mu$ and to the decay $\mu\to e\gamma$ are given by very similar Feynman diagrams. Previous works reported correlations in specific scenarios, in particular if $a_\mu$ is dominated by a single diagram. In this work we give an extensive survey of the possible correlations. We discuss examples of single-diagram domination with particularly strong correlations, and provide corresponding benchmark parameter points. We show how the correlations are weakened by significant cancellations between diagrams in large parts of the MSSM parameter space. Nevertheless, the order of magnitude of $\text{BR}(\mu \to e \gamma)$ for a fixed flavor-violating parameter can often be predicted. We summarize the behavior by plotting the correlations as well as resulting bounds on the flavor-violating parameters under various assumptions on the MSSM spectrum.


Introduction
As is well established, the Minimal Supersymmetric Standard Model (MSSM) with fully general soft supersymmetry (SUSY) breaking parameters is strongly constrained by flavorviolating observables. Particularly in the charged lepton sector no sign of flavor violation has been observed yet. In contrast, the muon anomalous magnetic moment a µ = (g µ −2)/2 is a flavor-conserving leptonic observable with a tantalizing deviation ∆a µ between experiment and Standard Model (SM) prediction by more than 3σ, which could be explained beautifully by the contributions from light sleptons, charginos, or neutralinos. On the other hand, the Feynman diagrams for SUSY contributions to a µ and to the branching ratio of the flavor-violating decay µ → eγ, BR(µ → eγ), are essentially identical, except for the flavor transition appearing only in the latter case. For this reason it was put forward to study correlations between the two observables in the MSSM [1,2,3].
Such correlations could prove useful, for example, in constraining scenarios which are in agreement with a µ . Once the SUSY contribution to a µ is fixed, the mass scale of the relevant superparticles is fixed as well. One could then predict the value of BR(µ → eγ) as a function of only the flavor-violating SUSY breaking parameters and thus derive stringent bounds on these parameters, which cannot be evaded by simply raising the overall SUSY mass scale. It has already been noted in [2] that a strong correlation could emerge if the contribution to a µ comes mainly from a single diagram; a strong correlation was also observed in [3] in a parameter scan. The interest of the present work is to identify the parameter space where this happens. In fact, cases with a strong correlation often correspond to certain mass hierarchies among the particles involved. Hence we characterize such hierarchies and establish the corresponding bounds.
On the other hand, in large parts of the MSSM parameter space the SUSY contribution to a µ is large but no single contribution to a µ dominates. It is then also of interest to study to what extent the two observables are correlated and whether we can derive bounds on the flavor-violating parameters even in such cases.
Motivated by the stringent LHC mass limits on colored SUSY particles, which in constrained scenarios imply stringent limits also on uncolored superparticles, we consider the MSSM without assumptions on GUT-scale or SUSY breaking physics. For our study, only electroweak parameters are relevant: the higgsino and gaugino mass parameters µ, M 1 , M 2 , the ratio of the Higgs vacuum expectation values tan β, and the slepton mass and mixing parameters, to be described below.
The work is organized as follows. In Sec. 2 we present the status of the relevant observables. In Sec. 3 we discuss the SUSY contributions to a µ and BR(µ → eγ). In Sec. 4.1, we survey their correlation in the general case where all supersymmetric particles involved in a µ and µ → eγ have no particular hierarchy. For this case the charginos tend to dominate, so in Sec. 4.2 we study the conditions under which correlations between a µ and BR(µ → eγ) could be established for chargino domination. In Sec. 4.3 we study a scenario where the µ parameter is very large and the lightest neutralino is essentially a bino, so the main contribution to a µ is given by diagrams involving binos,μ L , andμ R . Finally, in Sec. 4.4 we consider the case where all left-handed sleptons are very heavy, so that the chargino and most neutralino contributions are suppressed, except for the neutralino contribution withμ R exchange.
Section 5 can be read independently. It provides an extended discussion of the results, and it summarizes the behavior with plots of the correlations and bounds on flavor-violating parameters under various assumptions on the SUSY spectrum.
2 Status of relevant observables 2.1 (g − 2) µ and BR(µ → eγ) The difference between the experimental determination [4] and the SM prediction for the anomalous magnetic moment of the muon is larger than 3σ. Taking the evaluation of hadronic contributions of Ref. [5], including recent updates of the QED [6] and electroweak [7] contributions, and adding theoretical and experimental uncertainties in quadrature, the difference is ∆a µ = a exp µ − a SM µ = (287 ± 80) × 10 −11 .
Alternative theory evaluations [8,9] obtain similar or even larger differences. Further progress can be expected not only from improvements on the theory side, but in particular from Fermilab P989 [10,11] and the new J-PARC approach to the g − 2/EDM measurements [12]. Both aim to improve the experimental uncertainty of a µ by at least a factor of 4. In quantum field theory, a µ can be obtained from the covariant decomposition of the muon-photon three-point function. Written similarly to [2], the relevant term is in the limit q → 0. Here σ αβ = i/2 [γ α , γ β ], ǫ α is the photon polarization vector, k and k + q are on-shell momenta, and finally u µ ,ū µ are spinors that satisfy the Dirac equation.
The current 90% C.L. upper limit on the branching ratio BR(µ → eγ), set by the MEG experiment [13], is BR(µ → eγ) < 5.7 × 10 −13 ≡ BR EXP (µ → eγ), and future upgrades will attempt to explore regions of O(10 −14 ) [14]. We write the amplitude for µ → eγ as where P R,L = (1±γ 5 )/2. The L/R index in a µeγL/R refers to the electron chirality. Thus, L and R are interchanged with respect to the notation in [15,2]. By convention, the photon momentum q is oriented towards the vertex in both Eq. (2) and Eq. (4). The resulting branching ratio is

Superparticle masses
While the negative results from LHC SUSY searches place stringent lower limits on the masses of the colored superparticles, the electroweakly interacting charginos, neutralinos and sleptons are still allowed to be quite light [16,17,18,19,20]. This is very enthralling because a sizable supersymmetric contribution to the muon g − 2 requires masses of O(100) GeV precisely for sleptons and charginos or neutralinos. Consequently, within the general MSSM an appealing possibility is to consider scenarios with the hierarchy mq, mg ≫ ml, mχ±, mχ0.
Collider constraints on the masses relevant for a µ and µ → eγ for certain specific scenarios of this kind were derived in [21,16]. In particular in [21] the recent results on the searches for the non-colored supersymmetric particles were investigated in the parameter region where the muon g − 2 is explained. Under the assumption of the GUT relation M 1 ≈ M 2 /2, lower bounds of around 150-200 GeV were obtained for the wino mass. Further, Refs. [22,23] revealed logarithmically enhanced two-loop corrections from mass hierarchies such as those in Eq. (6). Since we will focus on the parametric dependence of a µ , BR(µ → eγ) and their correlation on the SUSY spectrum, we will choose parameters satisfying Eq. (6) but we will not use more detailed LHC mass limits and restrict ourselves to one-loop accuracy.
3 Contributions to a µ and a µeγ

Chargino-sneutrino contributions
The one-loop contributions to g − 2 in the MSSM have been evaluated in full generality in [24]. If the mixing of sneutrinos of the first two generations can be decoupled from the mixing of the third generation, the contributions to g − 2 from a charginoχ ± k can be nicely decomposed into two terms aχ which correspond to the diagrams mediated by winos or higgsinos, and a combination of higgsino and wino, respectively, and which involve corresponding powers of the gauge and muon Yukawa couplings g 2 and Y µ . Throughout this work we use the conventions of [25] unless specified otherwise. For completeness, the well-known functions F C 1 and F C 2 are defined as the arguments x ki are defined as the mass ratios The sneutrino mass eigenvalues mν i and the mixing angle between the first two generations, θν, are defined via diagonalization of the sneutrino mass matrix, cos θν sin θν − sin θν cos θν and the requirement thatν 1 be the mass eigenstate composed primarily ofν e , which implies that mν 1 is not necessarily smaller than mν 2 . Here m 2 are the flavor-diagonal and off-diagonal soft mass parameters, and D ν L are the D-term contributions to the masses. For the chargino mass matrix Mχ± we use the diagonalization In the following we will neglect terms suppressed by two powers of the muon Yukawa coupling or by the electron Yukawa coupling. Then, as pointed out in [2], the right-handed part of the µ → eγ amplitude vanishes, And for the left-handed part we can write where for each chargino and for each of the two contributions in Eqs. (7) and (14), the ratio of the left-handed µ → eγ amplitude and a µ takes the compact form We can rewrite this ratio as where ). We will use this result to derive analytical approximations later on.

Neutralino-charged slepton contributions
The contributions from the neutralinos and charged sleptons can be decomposed similarly to the chargino-sneutrino contributions, but the mixing structure is more complicated. Even if only the first two generations are allowed to mix, the relevant mixing of the leftand right-handed charged sleptons is described by a 4 × 4 matrix. In [2], this matrix was diagonalized approximately. In the following analysis, we consider the full mixing structure. Then, the neutralino contributions to a µ can be written as (compare Ref. [25] for the flavor-diagonal result) Neglecting Yukawa-suppressed terms like in the chargino case, the µ → eγ amplitudes can be written as In an obvious analogy to the chargino case, one could introduce aχ 0 i µ 1,2 etc., but we will not make use of that. In the previous equations, the abbreviations are defined as the loop functions are and To define the slepton masses and mixing we start from the slepton mass terms for the interaction and flavor eigenstatesl Li ,l Ri with generation index i = 1, 2, 3, with 3 × 3 block matrices m 2 L , m 2R and (m 2 LR ) ij = δ ij m i (A i − µ * tan β). We have omitted the small F -term contribution m 2 i involving the lepton masses. We use a basis where the lepton mass matrix is diagonal. The 6 × 6 mass matrix M 2 leads to slepton mass eigenvalues ml 1 < ml 2 < . . . < ml 6 and the diagonalization matrix K mℓ defined via For the neutralino mass and mixing matrices we use the convention 4 Correlations in different parameter space regions

Similar SUSY masses
In order to obtain a first impression of the strength of the correlation between the SUSY contribution and BR(µ → eγ) for generic SUSY spectra without strong mass hierarchies between sleptons, neutralinos and charginos, we perform a random scan over the parameters M 1 , M 2 , µ, mL 11 , mL 22 , mR 11 , mR 22 , varying them between 300 GeV and 600 GeV while ensuring that a neutralino is the lightest superparticle (LSP). All other superparticle masses are irrelevant for our calculations. We set the trilinear couplings A e and A µ to zero after verifying that they have no impact. We fix tan β = 50 and (δ l 12 ) LL = (δ l 12 ) RR = 2 × 10 −5 for the flavor-violating parameters The results are shown in Fig. 1. Taking into account that the correlated quantities are the amplitudes a µ , a µeγL and a µeγR whereas BR(µ → eγ) involves amplitudes squared, we observe a correlation that is significant but not extremely strong. This indicates that typically several diagrams contribute, either cancelling each other or adding up constructively. In order to investigate this, we encoded in the color of points the importance of the leading contribution, defined as R is a measure for the degree of cancellation: If a single diagram dominates, R ≈ 1, while R > 1 indicates cancellations; R < 1 if diagrams add up constructively. The figure shows that the latter does not occur. In fact, R is larger than about 1.5 for all points. For more than 90% of them we find even R > 2. Consequently, significant cancellations are typical. We have checked that adding the constraints m 2 does not strengthen the correlation significantly. It is also virtually unchanged if we set (δ l 12 ) RR = 0 while keeping (δ l 12 ) LL = 2 × 10 −5 . On the other hand, the correlation becomes very weak if only (δ l 12 ) RR is non-zero. This is to be expected, since only neutralinos contribute to a µeγ for (δ l 12 ) LL = 0, whereas a µ is dominated by charginos for similar SUSY masses. Finally, the correlation is not very sensitive to the value of tan β, since the dominant contributions to all amplitudes are proportional to this parameter, so it does not appear in their ratios, see e.g. Eq. (15). Exceptions can occur for tan β < 10, but in this case a µ is generically too small to be of interest.
In the following we will study scenarios for which the correlation can become stronger. They are characterized by hierarchies within the SUSY spectrum, and Tab. 1 already shows the benchmark parameter choices considered in sections 4.2, 4.3 and 4.4. Comparing these cases with the generic, non-hierarchical case discussed here will further clarify the weak correlation in Fig. 1. Later, in Sec. 5, we will compare bounds on the flavor-violating parameters for all these cases.

Chargino dominance
We first focus on cases in which the chargino contributions are dominant. For large tan β they are essentially given by aχ ± k µ 2 and aχ ± k µeγ 2 ; the corresponding mass-insertion diagrams are shown in Fig. 2   .

Hierarchy
Limiting behavior of

Possible realization
LSP constraint µeγ I /aχ ± k µ I , according to different hierarchies of the parameters x ki , for I = 1, 2. These approximations are valid chargino by chargino, thus one may have to consider different hierarchies for the two charginos. The last column shows the constraints to obtain a neutralino LSP; where no constraint is given, this is not possible. a µeγL from one individual charginoχ ± k and has two immediate implications. The ratio aχ ± k µeγ I /aχ ± k µ I in Eq. (16) depends non-trivially on the mass ratios x k1 and x k2 ; hence (i) it is different for the two charginos and the correlation between the sums over the contributions can be much weaker than the correlations between the individual contributions, and (ii) even for the individual charginos the right-hand side depends on the mass hierarchy between charginos and sneutrinos. Table 2 shows a comprehensive list of hierarchies between chargino and sneutrino masses, following Ref. [2]. For each hierarchy we analytically evaluated the limiting behavior of the ratio aχ ± k µeγ I /aχ ± k µ I . The results are shown in the third and fourth column of the table; they slightly improve similar results of Ref. [2]. Note that the result for I = 2 is the more relevant one since the I = 2 contributions dominate for large tan β. The last column shows the conditions that are necessary for having a neutralino LSP. Note that the lightest neutralino need not be the LSP, if a super-weakly interacting particle such as the gravitino or axino is the LSP, or if R parity is violated. Finally, the table also shows mass patterns which realize the hierarchies and which allow dominance of the contributions of the lightest or both charginos over the neutralino contributions. In the following we will study each mass hierarchy, focussing in particular on the deviations from the approximations in the table and on the impact of possible cancellations between individual contributions.
Let us consider first the situation of very similar masses, i.e. case V in Tab. 2, or in terms of the mass ratios x ki defined in Eq. (10). As a concrete example, we choose the parameters appearing in column II of Tab. 1. The chargino masses are driven by µ and M 2 , respectively, mχ± 1 ≈ 500 GeV and mχ± 2 ∈ [550, 1300] GeV. The first and second plot of Fig. 3 show, respectively, the contributions to a µ and a µeγL from each chargino and demonstrate that the lightest chargino provides the dominant contributions. The larger M 2 , the more pronounced the domination becomes. The third plot of the same figure shows the ratios of the contributions from each chargino individually and the ratio of the sums of all contributions to a µ and a µeγL . The ratios are compared with the prediction for the theoretical limiting behavior given in Tab . Indeed, the ratio for the lightest chargino agrees excellently with this prediction, because the hierarchy condition x 11 ∼ x 12 ∼ 1 is satisfied precisely. For the heaviest chargino, x 21 and x 22 are significantly larger than 1, except at the lower end of the considered interval for M 2 , so for higher M 2 the corresponding ratio deviates more and more from 1 . Due to the dominance of the lightest chargino, the ratio of the sum of all contributions agrees with the prediction within a factor 1.5 for all values of M 2 shown in the plot. The absolute value of this ratio, |a µeγL /a µ |, is larger than both individual ratios |aχ ± k µeγL /aχ ± k µ |, which indicates that the cancellation between the different contributions is stronger for a µ than for a µeγL . As M 1 is the smallest of the soft SUSY breaking parameters, the LSP is a neutralino with dominant bino component.
We will now study systematically all nine cases listed in Tab. 2. We set the mass parameters to definite values according to the hierarchies given in the table, but such that the lightest chargino still represents the leading contribution to a µ , and such that the value of a µ remains inside the region allowed by Eq. (1). We use the following sets of parameters: Here we have used m 2 νe ≡ m 2 + D ν L as more physical inputs. In addition, tan β = 50, m 2 L 12 = (3 GeV) 2 , and the value of M 2 is kept as a variable. For each case, we focus on the following questions: Does the limiting behavior in Tab. 2 provide a reliable prediction for the correlation between a µ and a µeγ for (i) the individual contribution of the lighter chargino and (ii) the sum of all contributions? We consider the prediction reliable if it agrees with the precise numerical result up to a factor of 1.5 or less. In addition, (iii) are the scenarios compatible with a neutralino LSP? We will be briefer in our discussions than for case V discussed above, but we will highlight cases I, • For case I, both µ and M 2 must be considerably bigger than the sneutrino masses to generate the hierarchy x k1 , x k2 ≫ 1, which pushes the masses of both charginos and three neutralinos well above the masses of the sneutrinos. In order to guarantee chargino dominance, we also set M 1 to a high value, which implies that in case I we cannot obtain a neutralino LSP. Fig. 4 shows that the approximation for the ratio aχ ± k µeγ 2 /aχ ± k µ 2 works within 50% accuracy for the lightest chargino. It fails in the low-M 2 region where x 1i becomes smaller than about 8, since there the expansion of the ratio in Eq. (15) is not valid anymore. As it depends on the chargino mass, the approximation for k = 1 plotted in the figure cannot describe the contribution of the heavier chargino. Besides, at µ = M 2 the chargino mixing becomes maximal, and so does the cancellation between the chargino contributions. Hence, approximating one of them well can be meaningless for the sum. The approximation works reliably for the sum only for M 2 1 TeV.
• For case II, the approximation in Tab. 2 contains a factor involving logarithms. This factor can change the result by more than a factor of 2 and therefore should not be neglected. If it is included, the approximation works very well for the contribution of the lighter chargino and satisfactorily for the sum. For the specific example in Eq. (30) and M 2 ≫ µ, the LSP is a higgsino-like neutralino with a mass around 216 GeV.
• For case III the approximation works quite well for the sum and for the contribution of the lighter chargino for all values of M 2 up to about 1 TeV. It is no problem to consider relatively small values of M 1 , so the LSP can be a bino-like neutralino. For the values quoted in Eq. (30), the LSP has a mass around 350 GeV.
• Case IV interchanges the roles ofν e andν µ compared to case III. Due to the higher mν µ the contribution to both observables is suppressed, and the approximation for the ratio a µeγL /a µ depends in a complicated way on the chargino andν µ masses. We find that even the full approximation shown in Tab. 2 does not work reliably. The hierarchy required in case IV forbids very light values of µ and M 2 around 100 GeV, and the requirement of chargino dominance forbids values of M 1 smaller than around 400 GeV. This case thus does not allow a neutralino LSP.
• For case V, see the discussion above.
• For case VI, both µ and M 2 must be kept large to generate the hierarchies x k2 ≫ 1, x k1 ∼ 1. If this is satisfied, the approximation works well for the lighter chargino.
It also provides a reliable prediction for the sum of the contributions with about 40% accuracy for all values M 2 > 800 GeV, see Fig. 5. The bino mass should be heavy, M 1 500 GeV, to avoid neutralino contributions to be dominant, so with the hierarchy of this case it is not possible to obtain a neutralino LSP.
• Case VII is shown in Fig. 6. The plotted range for M 2 is the one where a µ is within the favored 2σ region. The µ parameter is small enough to satisfy the LSP condition mχ0 1 < mν 1 . In this case it is important not to neglect the logarithmic factor in the approximation for aχ ± k µeγ 2 /aχ ± k µ 2 , which then works very well for the lightest chargino. However, interestingly the sum of the contributions is not sufficiently dominated by the lightest chargino, so the approximation fails to predict the ratio a µeγL /a µ correctly.
• Case VIII is similar to case VI in that both µ and M 2 must be kept large to generate the hierarchies, which together with the requirement of subdominant neutralino contributions makes it impossible to obtain a neutralino LSP. The approximation for aχ ± k µeγ 2 /aχ ± k µ 2 works well for the contribution of the lightest chargino but not for the sum, which shows a strong dependence on M 2 .
• For case IX, the approximation of the ninth line in Tab. 2 works quite well for both the lightest chargino contribution and for the sum of the contributions from the two charginos. In this case it is possible to obtain a neutralino LSP. For the example of Eq. (30), it could be higgsino-or wino-like, depending on the value of M 2 in comparison to µ, which remains fixed at 500 GeV. Note that for all these cases the values of mR 11 and mR 22 are not important. These parameters just have to be sufficently large to avoid the appearance of a charged LSP.
In summary, depending on the hierarchy of the sneutrino and chargino masses, the different approximations to aχ ± k µeγ 2 /aχ ± k µ 2 given in Tab. 2 predict the ratio of the lighter chargino's contributions to a µeγL and a µ reliably, i.e. within a factor 1.5 of accuracy, with case IV being the sole exception. In cases I (for M 2 1 TeV), II, III, V, VI, and IX, these approximations can also be used as a reliable substitute for the exact value of a µeγL /a µ stemming from all contributions. However, in case II the approximation depends on three superparticle masses, weakening the link between the two observables. In the remaining cases I, III, V, VI, and IX, there is a strong correlation; we can predict BR(µ → eγ) as a function of a µ and the ratio of the flavor-violating parameter and a single superpartner mass within a factor of roughly 1.5 2 . Figure 7 summarizes the present section and allows further interpretation of the similarmass case of the previous section 4.1. It shows BR(µ → eγ) versus a µ for the benchmark points listed in Eq. (30) and for random SUSY masses. The scatter regions repeat the similar-mass case of Fig. 1, but this time focussing on the effect of different mass intervals instead of the strength of cancellation. The random SUSY masses are generated in the regions [430, 530] GeV (dark blue region), [300, 600] GeV (light blue) and [200, 1000] GeV (yellow). We have fixed (δ l 12 ) LL = (δ l 12 ) RR = 2 × 10 −5 . The benchmark points represent parameter choices with certain extreme mass hierarchies, each of which leads to a different correlation between BR(µ → eγ) and a µ , as we have discussed in this section. Indeed, one finds a wide variety of branching ratios even for almost the same values of a µ , see e.g. dots 2, 5, and 6, which are nearly aligned on a

Large µ limit
If tan β is large, an interesting neutralino contribution to a µ and a µeγ is represented by the mass-insertion diagrams in Fig. 8 with bino exchange. These diagrams grow linearly with µ. Their contribution to a µ is proportional to m 2 µ µ tan β M 1 F (M 1 , mμ R , mμ L ), where F denotes the loop function involved in each diagram. All other contributions involve higgsinos and are therefore suppressed for large µ. Hence, the diagrams of Fig. 8 dominate for sufficiently large µ. We now analyze this parameter region and the behavior of a µ and a µeγ . Very recently, a µ in this scenario has also been studied in detail in Ref. [26]. The most important parameters of the parameter region with large µ are µ, M 1 , m 2 Choosing M 2 significantly larger than M 1 further increases the dominance of the contribution of the lightest neutralino for large µ. Note that for large values of µ and tan β, charge-breaking minima in the scalar potential endanger the stability of the electroweak vacuum. This problem can be alleviated by increasing the stau masses [26], which otherwise have no impact on a µ and a µeγ .
Interplay of contributions to a µ In Fig. 9, we present an example for this scenario and show a µ as a function of µ for the parameter choice of spectrum III in Tab. 1. The thick solid (orange) curve corresponds to the total SUSY contribution to a µ . The solid black line corresponds to the total neutralino contribution and the dot-dashed black line to the contribution of the lightest (bino-like) neutralino. As we can see this lightest neutralino almost fully accounts for the total a µ in the large-µ region, where µ is significantly larger than M 2 . Here the diagrams of Fig. 8 dominate. We have also plotted the other individual neutralino contributions (as blue solid, dotted, and dashed lines) and extended the figure down to smaller values of µ, so that it shows the transition from a regime with chargino dominance at low µ to the bino-dominated regime. At very small values of µ, the chargino contributions are important and the neutralino contributions only account for a fraction of the total a µ . At intermediate values, for µ ≈ M 1 or µ ≈ M 2 , there are two neutralino mass eigenstates whose contributions are strongly enhanced due to large neutralino mixing. However, the enhanced contributions roughly cancel each other, so the sum of all contributions shows no enhancement. This can be understood from the massinsertion diagrams, which have a monotonous behavior. Around µ ≈ 500 GeV, we notice a discontinuity in the plot, which reflects an exchange of identity of the higgsino-like mass eigenstatesχ 0 2 andχ 0 3 that is caused by the mixing between higgsinos and gauginos.  In the large µ limit, the total SUSY contribution can be written as where x 1m is defined in Eq. (21). As long as flavor violation is small, the most important contributions to a µ come from the two mass eigenstates containing mainlyμ L andμ R , which we label as m = b and m = c. Then a µ can be very well approximated by The factors K * m5 K m2 determine the mixing betweenμ L andμ R , which is a necessary ingredient as shown by Fig. 8. As we are dealing with the mixing of only two states, we can introduce a mixing angle θμ with in analogy to Eq. (11), where we have restricted ourselves to real parameters and chosen the states such that ml b < ml c . We can safely neglect the trilinear coupling in the smuon mixing term (m 2 LR ) 22 = m µ (A µ − µ * tan β) and retain only µ tan β. Then we can write a µ as For not too hierarchical smuon masses, the contributions from the two mass eigenstates are of the same order and partially cancel. Note that xF N 2 (x) increases monotonously, so a µ > 0 for µ > 0. The result for this scenario can be summarized as a µ ≈ 2 × 10 −9 µ 4 TeV tan β 50 We could further approximate where the numerical value of the derivative varies slowly around 1 in the region of interest. Thus, increasing M 1 increases a µ if all other parameters are kept fixed.
Interplay of contributions to a µeγ We consider small flavor-violating terms m 2 For large µ, we have then Let us first study the case m 2 R 12 = 0 and m 2 L 12 = 0, which implies that only a µeγL is non-negligible. As we can see from the corresponding diagram in Fig. 8, a µeγL receives contributions from the three mass eigenstates containing mainlyẽ L ,μ L andμ R . 1 Let us denote them byl a ,l b andl c , respectively. For these states we estimate assuming real parameters and that all mixings between selectrons and smuons are small. Plugging these expressions into Eq. (39) yields The ratio a µeγL /a µ is now easily obtained from eqs. (35) and (44). We find that this ratio can be well approximated by the numerical estimate if all slepton masses are below a TeV. For masses between 300 GeV and 600 GeV, it deviates from the exact result by less than 30%, while for the wider mass range [200,900] GeV the difference can be up to a factor of 2. The first plot of Fig. 10 demonstrates that the approximation is even more accurate if we fix m 2 = 470 GeV and vary only the other slepton masses. Note that the points in the figure correspond to mass differences |ml b − ml c | 1 GeV, which is generically expected due to the difference between the Dterms ofμ L andμ R . Fine-tuning the mass eigenvalues could lead to extreme cancellations and thus to points outside the colored regions in Fig. 10. When m 2R

Neutralino-μ R dominance
We consider the contributions from the diagrams of Fig. 11, which involve the right-handed smuonμ R . They dominate if the spectrum satisfies where the hierarchy M 1 < M 2 , |µ| ensures a bino-like lightest neutralino. Note that µ < 0 in order to have a positive a µ [25,27]. The contributions from the diagrams of Figs. 2 and 8 are suppressed due to the large values of the left-handed slepton masses.
Interplay of contributions to a µ From the left diagram of Fig. 11 we can then estimate a µ as 2 where we have labeled the state which is mostlyμ R asl a . Numerically, we can summarize the behavior of a µ for this case as For the set of parameters in column IV of Tab. 1, theχ 0 -μ R contribution dominates and the value of a µ is within the allowed 1σ region if mμ R 90 GeV, within the 2σ region if mμ R 130 GeV and within the 3σ region if mμ R 220 GeV. A benchmark parameter point with this behavior and a value of a µ in the allowed 2σ range was also defined and discussed in Ref. [22].
Interplay of contributions to a µeγ For this case, the relevant flavor-violating amplitude is where sizable contributions come only from the slepton mass eigenstates containing mainlỹ µ R andẽ R , which we label asl a andl b , respectively. The mixing of the two states is given by |K a5 K a4 | ≈ |m 2 )| and K b5 K b4 ≈ −K a5 K a4 for real parameters. Using the same approximation for N 11 N 13 as before, we find Taking into account Eq. (48), |a µeγR /a µ | becomes In Fig. 12 we show the numerical results for |a µeγR /a µ | as a blue band, taking into account the chargino contribution as well. The parameters are set to the values of Tab. 1, column IV; in particular, we varied µ in the range [−550, −650] GeV and M 2 in the range [100, 900] GeV. For µ < −650 GeV, a µ leaves the allowed 3σ region for the considered mass spectrum. Note that as long as the left-handed slepton masses are kept above 2 TeV, there is no change of the behavior presented for this scenario. The horizontal line in Fig. 12 represents the approximation of Eq. (52). The width of the band in the figure and its limited variation with mμ R allow us to conclude that the case of neutralino-μ R domination features a strong correlation between a µ and BR(µ → eγ). 2 We use N 11 N 13 ≈ MZ sin θW sin β µ , which is a good approximation if Eq. (47) holds and in addition |µ| ≫ M Z and tan β 30 [28].

Correlation between a µ and BR(µ → eγ) for similar supersymmetric masses
Our study started with the following basic question: Assuming supersymmetric parameters that reproduce the observed value of a µ , can we predict the amplitude a µeγL (or a µeγR )? Evidently this amplitude depends on the flavor-violating parameter m 2 ), so in the best possible case it would be proportional to a µ times the dimensionless ratio of the flavor-violating parameter and some other SUSY mass m 2 p . I.e. in the best possible case we could write with a constant f . We found several parameter regions in which such correlations hold for a µeγL or a µeγR , but the proportionality constant f and the appropriate mass ratio are specific for each region. In Sec. 4.1 we found that for similar masses of the supersymmetric particles involved in a µ and a µeγ , we can indeed determine the order of magnitude of a µeγL /a µ , employing as mass ratio the commonly used quantity (δ l 12 ) LL defined in Eq. (26). The correlation is rather weak, however, and we also found that significant cancellations among different diagrams contributing to both processes are typical. Nevertheless, using the MEG limit on BR(µ → eγ) [13] we can put bounds on the flavor-violating parameters under the assumptions that a µ is explained by SUSY and all relevant SUSY masses are similar. In the left panel of Fig. 13 we plot bounds on (δ l 12 ) LL as a function of tan β. The bounds are obtained from a random scan as follows. For each value of tan β, the value of the generic parameter M is chosen such that a µ agrees with its central experimental counterpart in Eq. (1) when Afterwards, random SUSY mass spectra are generated by varying the seven mass parameters above within the interval [0.7M, 1.3M], imposing the conditions that (a) the LSP is a neutralino and (b) a µ falls into the 1σ region given in Eq. (1). From the generated spectra we derive three regions and two corresponding bounds on (δ l 12 ) LL . The top-most (red) region in the left panel of Fig. 13 is "totally excluded", i.e. (δ l 12 ) LL is so large that the MEG limit is violated by all generated mass spectra. The lower-most (green) region is "totally allowed", i.e. the MEG limit is never violated. In the yellow region in between some (but not all) spectra satisfy the MEG limit. Thus, the upper, weaker bound delimiting the red region is conservative and must be satisfied (under the above assumptions). The yellow region is allowed by the above assumptions, but with more information on the SUSY masses from either experiment or theoretical models, the bound on (δ l 12 ) LL might go down as low as the lower, stronger bound delimiting the green region.
In order to see how the size of the mass range affects the bounds, we also investigate the smaller interval (1 ± 0.03) × M. The result is a narrower band limited by the dashed boundaries in the plot, which are analogous to the thick blue lines.
In addition to what is shown in the plot, we calculated bounds from a restricted set of spectra, imposing mL 11 = mL 22 and mR 11 = mR 22 . In comparison to the general case, the additional degeneracy conditions result in a slightly relaxed strong bound. One can understand this from the fact that the conditions restrict the spread of BR(µ → eγ) for a fixed mass insertion. By the same token, the weak bound becomes tighter, albeit only by a tiny margin.
In the right plot of Fig. 13, we consider flavor violation in the right-handed slepton sector. The weak bound on (δ l 12 ) RR is not displayed. We omit it since the calculation is based on the approximation BR(µ → eγ) ∝ |(δ l 12 ) RR | 2 . Naively applying this relation would result in bounds of O(1). For so large values of (δ l 12 ) RR , the proportionality approximation is not valid. In any case, the weak bound is expected to lie far outside the plotted vertical range. For such a large mass insertion, it might not be BR(µ → eγ) but the smallest slepton mass eigenvalue that determines the limit on (δ l 12 ) RR . For |(δ l 12 ) RR | ≪ 1, one can always find a set of mass parameters within [0.7M, 1.3M] such that different contributions to µ → eγ cancel [29,30], resulting in a decay rate below the MEG limit. The 3% range, however, is too narrow for a sufficient cancellation. This leads to the upper dashed curve, which depicts the weak bound for this small mass range. After considering the case of similar superparticle masses, we turned to the next step, asking under which conditions a strong correlation arises. The answer is that there should be just one kind of diagram (where only one kind of supersymmetric particle mediates the contribution to both processes) and so we explored cases where either charginos or neutralinos could dominate.

Specific results for the case of chargino dominance
At the end of Sec. 4.2 we have made a comparison between the accuracy of the correlations in cases where the charginos dominate both a µ and a µeγ and the general correlation in the case of Sec. 4.1. We have also pointed out for which cases the correlations become particularly useful. In summary, depending on the hierarchy of the sneutrino and chargino masses, the different approximations given in Tab. 2 predict the ratio of the lighter chargino contributions to a µeγL and a µ , i.e. the ratio aχ ± k µeγ 2 /aχ ± k µ 2 , within a factor 1.5, except for case IV. In cases I (for M 2 1 TeV), II, III, V, VI, and IX, these approximations can even be used as a reliable substitute for the full value of a µeγL /a µ stemming from all contributions.
Again, we can in turn use the obtained correlations to set bounds on the relevant parameter combinations appearing in a µeγL /a µ . In order to highlight the structure of the bounds and the differences in the different parameter regions we do not carry out scans in parameter space but we use the experimental information on a µ from Eq. (1) and require BR(µ → eγ) to satisfy the bound (3). Then we obtain where we have omitted |a µeγR | 2 in the expression for BR(µ → eγ) since it is subdominant. In all cases of Tab. 2 in which the left-hand side can be approximated reliably as a simple mass ratio this bound translates into a lower bound on the corresponding mass mp of the supersymmetric particle driving the correlation between a µ and BR(µ → eγ); for example, for case I this is mχ± 1 . The bounds are functions of the flavor-violating parameter mL 12 . The results are shown in Fig. 14, where we have plotted only those cases of Tab. 2 for which a reliable bound can be extracted. For each case we have restricted mp to the range where the approximation for a µeγL /a µ is valid within a factor 1.5 and where a µ can be made to lie in the favored 2σ region by choosing the superparticle masses appropriately. We can see that these requirements restrict considerably the scale mp. In the figure tan β is fixed to 50; lowering (increasing) it would allow to loosen the lower (upper) bounds on mp. Note that among the cases mentioned here, we have left out case II because the useful approximation involves logarithms and so it is not possible to set a bound on the mass of an individual particle.

Specific results for cases with neutralino dominance
We identified two particular regions of parameter space, also represented by the benchmark points 3 and 4 of Refs. [22,23], where neutralino contributions dominate and a µ and a µeγ are strongly correlated. If µ is very large and the bino mass M 1 sufficiently small, the exchange of the lightest, bino-like neutralino can become the dominant contribution. Contrary to what happens in the case of chargino dominance, for the neutralino case the contributions from a µeγR can be relevant, since the lightest neutralino for this case is bino-like (see Fig. 8). Hence both left-and right-handed slepton contributions to a µeγ and to a µ become important.
For this case, we have found in Sec. 4.3 that indeed a single diagram dominates and there is a strong correlation between a µ and BR(µ → eγ). Hence we analyzed separately the correlations when either left-handed or right-handed charged sleptons dominate a µ (and thus also a µeγ ). The correlations lead to relations of the type wherel a ≈ẽ L andl a ≈ẽ R , respectively. In this sense, just as in the case of chargino dominance, we can obtain reliable lower bounds on the mass of the supersymmetric particle driving the correlation between a µ and BR(µ → eγ). We plot the results in Fig. 14, where we can also compare to the bounds in the case of the chargino dominance. Another case of neutralino dominance for which we have found a strong correlation is the case analyzed in Sec. 4.4, where a µ is dominated by diagrams involving the righthanded smuon and the lightest neutralino. The conditions for this dominance are that (a) left-handed slepton masses be much more bigger than |µ| and (b) M 1 , mμ R , mẽ R < M 2 . We find that a µeγR a µ ∼ |m 2 where now obviouslyl a ≈ẽ R . The lower bound on the mass ofl a , as a function of |m 2 R 12 |, is also visualized in Fig. 14. TeV. The figure confirms the existence of correlations, and it allows to easily look up the expected results for each parameter region. In detail, let us explain why we observe a slightly larger variation of BR(µ → eγ) for fixed a µ than found earlier, for example in Fig. 10. Earlier we fixed the very parameter ratio driving the correlation between a µ and BR(µ → eγ), for example m 2 L 12 /m 2 ℓa . In Fig. 15 we fixed (δ l 12 ) LL ≈ m 2 L 12 /m 2 ℓa × mẽ L /mμ L , i.e. the decisive parameter multiplied by a factor varying between 0.5 and 2 if we vary the slepton masses between 450 GeV and 900 GeV. This illustrates a certain limitation of the correlations: Even if we consider just a single diagram, varying all SUSY masses by a factor of 2 can still change the branching ratio by an order of magnitude, unless we fix the "correct" flavor-violating parameter, which varies from diagram to diagram.

Comparison of different scenarios
Finally, let us compare our results to previous studies in the literature where stronger correlations were found. In particular, Fig. 6 of [3] shows a narrower band than the one for the case of similar supersymmetric masses plotted in our Fig. 15. To clarify, in the left panel of Fig. 16 we reproduce our results for the similar masses scenario together with the results of Fig. 6 of [3] using the same constraints as there, except A U , which we set to zero, and (δ l 12 ) LL , which we set to 2 × 10 −5 . Specifically, in [3] the additional constraints with respect to ours are M 2 ≈ 2M 1 and mL 11 = mL 22 = mR 11 = mR 22 . The resulting band is the black (darkest) one in our Fig. 16. Indeed it agrees with the corresponding one in Ref. [3]. Relaxing the constraint on the gaugino masses produces a wider, dark blue (dark grey) band, and relaxing the constraints on the slepton masses produces the light blue (light grey) band. This shows that the correlation found in [3] is stronger than the one we find (for our similar supersymmetric mass scenario) because of the constraints on the slepton and gaugino masses.
Furthermore, the right panel of Fig. 16 displays an anatomy of the decorrelation effect arising from the variation of each parameter. First, we pick up a representative point, marked by the black dot, out of the black band in the left panel. Then, we vary each of the dimensionful parameters shown in the plot by a factor of 10 around the dot, for instance, 240 GeV ≤ µ ≤ 2400 GeV. The range of tan β is from 10 to 70. One might regard the tan β curve as depicting the principal correlation that guides the black band, as both a µ and a µeγL are approximately proportional to tan β. The other curves are then deviations from this principal correlation. Among them, the mL 11 = mR 11 line should be the easiest to understand: It shows simply that a µ remains fixed while BR(µ → eγ) decreases as the selectron masses increase. One can notice in the left panel that relaxing the equality of the four diagonal slepton masses causes a greater spread than unfixing the gaugino mass ratio. In the right plot, the slepton mass splits are further divided into two classes: the intergeneration split and the left-right split. Comparing the mL 11 = mR 11 and the mR 11 = mR 22 curves, one finds that the left-right split causes a larger deviation from the principal correlation.
In conclusion, the essential goal of the present work was to characterize the sources of the possible correlation between a µ and BR(µ → eγ). We discerned that in the case of similar SUSY masses entering both observables cancellations are typical and hence the correlation is rather weak; we also identified some cases where the correlation is strong. In these cases we derived bounds on the flavor-violating parameters under various assumptions (Fig. 13) and conversely bounds on the masses of supersymmetric particles, as a function of only one flavor-violating parameter (Fig. 14). The ultimate application of this kind of analyses would be attained if BR(µ → eγ) could eventually be measured, because that would give definite indications for the required values of all relevant mass parameters. Finally, Fig. 15 and Fig. 16 can be regarded as a summary, and as an update and generalization of Ref. [3]. The looser the assumptions in particular on slepton masses, the weaker the correlations, but there are interesting further parameter space islands with strong mass hierarchies, where correlations exist.