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Anatomy and Phenomenology of mu-tau Lepton Flavour Violation in the MSSM

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Andrea Brignole, Anna Rossi

High Energy Physics - Phenomenology · Submitted 23 Apr 2004 · 38 references

Pith's verdict: A solid, useful operator-anatomy paper; the advertised 10^-9 numbers sit at M_S ~ m_Z, where the paper's own leading-order EWSB truncation is least reliable.

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arXiv IDhep-ph/0404211 (version 2)
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CitationAndrea Brignole, Anna Rossi. "Anatomy and Phenomenology of mu-tau Lepton Flavour Violation in the MSSM." arXiv:hep-ph/0404211 (2004). https://pith.science/paper/5Y6WGJS3
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The paper

The paper claims

In a supersymmetric model with strongly mixed muon and tau sleptons, the paper argues that four classes of one-loop operators compete, so tau-to-muon decay rates need not follow the pattern set by the radiative decay alone.

The reading

The paper asks a concrete question: if the muon and the tau are mixed in the slepton mass matrices of a supersymmetric extension of the Standard Model, which of the many tau-to-muon decays actually get a measurable rate, and do their rates track one another? Lepton flavour violation here means a process in which a muon appears where a tau should be, which the Standard Model forbids. The authors first write down every effective operator that can mediate such a decay — a photon or $Z$ dipole, a chirality-conserving contact term, a four-fermion box, and a Higgs-mediated Yukawa coupling — and derive the branching ratios in terms of those coefficients. They then compute the coefficients at one loop in an unconstrained supersymmetric model, keeping the muon-tau entries of the slepton mass matrices large and diagonalising those matrices exactly rather than treating the mixing as a small perturbation. The payoff is a map of which decays are dipole-driven, which are dominated by the other operators, and where the usual correlations between them fail.

What carries the argument

The carrying object is a matched pair: a model-independent operator basis, and an explicit one-loop evaluation of its coefficients. The basis collects the $Z$ monopole coefficients $A^Z$ and $C^Z$, the photon monopole $C^gamma$, the dipoles $D^Z$ and $D^gamma$, the four-fermion coefficients $B^f$, and the Higgs-muon-tau couplings $Delta_{L,R}$ that appear in a dimension-four term $h_tau Delta^*_L H_2^{0*} tau^c mu + h.c.$. Each decay rate is then a specific linear combination of these, so the same coefficient reappears across channels and creates the correlations the paper tests. The coefficients themselves come from triangle and box diagrams with gauginos, higgsinos and sleptons, expressed through the loop functions $I_N$, $J_N$ and $K_N$; the authors diagonalise the left and right slepton mass matrices into eigenstates and mixing angles $theta_L, theta_R$, which lets them follow the coefficients into the large-mixing regime where the mass-insertion expansion would fail. The parametric estimates $A^Z, C^gamma, C^Z, B^f/g_w^2 sim (g_w^2/16pi^2)(1/M_S^2)(tilde m^2_{mu tau}/M_S^2)$ and $D^gamma, D^Z sim (g_w^2/16pi^2)(1/M_S^2)(tilde m^2_{mu tau}/M_S^2$ or $tilde m^2_{mu tau} tanbeta/M_S^2$ or $A_{mu tau} M_S)$ are what make the competition between dipole and non-dipole terms plausible in the first place.

Extended reading notes

Core claim

The central claim is that the operator content of muon-tau lepton flavour violation in the MSSM is richer than the single photon dipole that dominates discussions of $tau to mu gamma$. Concretely, the authors show that the monopole coefficients from $Z$ and photon penguins, the box coefficients, and — at large $tanbeta$ — the dimension-four Higgs couplings $Delta_{L,R}$ can all be of the same order as the dipole coefficients, and that the relative size of these contributions shifts with the region of parameter space. Because each decay samples a different combination, the familiar dipole-dominated relations such as $mathrm{BR}(tau to mu e e)/mathrm{BR}(tau to mu gamma) simeq 10^{-2}$ hold only in the corners where the dipole wins; elsewhere $tau to mu e e$, $tau to 3mu$ and $tau to mu rho$ can sit above or below that scaling by factors of a few, and the semileptonic channels $tau to mu pi$, $tau to mu eta$, $tau to mu eta'$ and $Z to mu tau$ can reach $10^{-9}$. Branching ratios in the range $10^{-9}$ to $10^{-7}$ are attainable, and the lepton-flavour-violating Higgs decays $A, H, h to mu tau$ can reach $10^{-4}$ for at least two of the three neutral Higgs bosons.

Load-bearing premise

The quoted reach depends on choosing maximal muon-tau slepton mixing with widely split eigenvalues and on assuming flavour violation in the electron-muon and electron-tau sectors is negligible, while no flavour model is offered that would generate the large muon-tau entries in the first place, so the numbers are a conditional reach rather than a generic prediction.

Editorial extensions

If this is right

  • If the photon dipole cancels among its own components, then $tau to mu e e$, $tau to 3mu$ and $tau to mu rho$ can be as sensitive to supersymmetric flavour violation as $tau to mu gamma$, so a null result in the radiative channel does not close the muon-tau window.
  • At large $tanbeta$ with heavy superpartners, the Higgs-mediated terms take over: $tau to 3mu$ and $tau to mu eta$ become the discovery channels and follow the lepton-flavour-violating Higgs branching ratio through a fixed relation.
  • The same $Delta$ coefficients that drive $A, H, h to mu tau$ also feed the muon anomalous magnetic moment through terms enhanced by $m_tau/m_mu$, tying two observables to one set of parameters.
  • $Z to mu tau$ and the semileptonic $tau to mu P$ channels are predicted to sit near $10^{-9}$ only where the superpartner masses are light and the dipole is suppressed by cancellation, which makes that corner sharply identifiable.
  • The ratio $mathrm{BR}(tau to mu eta)/mathrm{BR}(tau to 3mu)$ is predicted to be of order a few under Higgs dominance, but is sensitive to the loop-induced corrections $xi_s, xi_b$ that rescale the Higgs couplings to down-type quarks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the non-dipole contributions can add to or subtract from the dipole scaling depending on the interference sign, no single channel can identify the operator mix; only a simultaneous fit of $tau to mu gamma$, $tau to mu e e$, $tau to 3mu$ and $tau to mu rho$ would pin it down.
  • The cancellation windows that keep $tau to mu gamma$ under its bound while leaving the superpartners light require tuning between bino-higgsino and trilinear contributions; a future collider measurement of the spectrum that finds no such tuning would exclude the $10^{-9}$ reach in the semileptonic channels even without better tau bounds.
  • Since $Delta_{L,R}$ depend on $mu$ and $M_1$, the same parameters that fix the neutralino sector, a neutralino mass measurement would convert the $A to mu tau$ prediction from a range into a fairly rigid number.
  • Treating the muon-tau entries as free parameters leaves open whether the flavour symmetry behind the observed neutrino mixing also generates them; if it does, the maximal-mixing, widely-split choice used throughout the numerics may be the natural one rather than a tuning.
Original abstract

We perform a detailed analysis of several mu-tau lepton flavour violating (LFV) processes, namely tau --> mu X (X = gamma, e+ e-, mu+ mu-, rho, pi, eta, eta'), Z --> mu tau and Higgs boson decays into mu tau. First we present a model independent operator analysis relevant to such decays, then we explicitly compute the LFV operator coefficients [and (g_\mu -2)] in a general unconstrained MSSM framework, allowing slepton mass matrices to have large mu-tau entries. We systematically study the role and the interplay of dipole and non-dipole operators, showing how the rates and the mutual correlations of those LFV decays change in different regions of the MSSM parameter space.Values of the LFV branching ratios in the experimentally interesting range 10^{-9}-10^{-7} can be achieved. For at least two MSSM Higgs bosons, the branching ratio of the LFV decay into mu tau can reach values of order 10^{-4}.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We perform a detailed analysis of several mu-tau lepton flavour violating (LFV) processes, namely tau --> mu X (X = gamma, e+ e-, mu+ mu-, rho, pi, eta, eta'), Z --> mu tau and Higgs boson decays into mu tau. First we present a model independent operator analysis relevant to such decays, then we explicitly compute the LFV operator coefficients [and (g_\mu -2)] in a general unconstrained MSSM framework, allowing slepton mass matrices to have large mu-tau entries. We systematically study the role and the interplay of dipole and non-dipole operators, showing how the rates and the mutual correlations of those LFV decays change in different regions of the MSSM parameter space.Values of the LFV branching ratios in the experimentally interesting range 10^{-9}-10^{-7} can be achieved. For at least two MSSM Higgs bosons, the branching ratio of the LFV decay into mu tau can reach values of order 10^{-4}.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We perform a detailed analysis of several mu-tau lepton flavour violating (LFV) processes, namely tau --> mu X (X = gamma, e+ e-, mu+ mu-, rho, pi, eta, eta'), Z --> mu tau and Higgs boson decays into mu tau. contribution 1.00 0.93 0.05 asserted 0.93
s1 First we present a model independent operator analysis relevant to such decays, then we explicitly compute the LFV operator coefficients [and (g_\mu -2)] in a general unconstrained MSSM framework, allowing slepton mass matrices to have large mu-tau entries. method 0.84 0.93 0.07 asserted 0.96
s2 We systematically study the role and the interplay of dipole and non-dipole operators, showing how the rates and the mutual correlations of those LFV decays change in different regions of the MSSM parameter space.Values of the LFV branching ratios in the experimentally interesting range 10^{-9}-10^{-7} can be achieved. result 0.58 0.94 0.30 asserted 0.92
s3 For at least two MSSM Higgs bosons, the branching ratio of the LFV decay into mu tau can reach values of order 10^{-4}. result 1.00 0.90 0.29 hedged possibility 0.76

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from simulation or computation 0.81.

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  • references and citing sentences
  • relations to other papers
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The different contributions to τ → µγ, Z → µτ and τ → µff decays. Z → µτ The effective lagrangian for the Z → µτ decay is easily obtained from eqs. (8-10): L eff Zµτ = gZ h m2 Z  F Z L µσ¯ µ τ + F Z R µ cσ µ τ c  Zµ − 2mτ  D Z L µσ¯ µντ c + D Z R µ cσ µν τ  i∂νZµ i + h.c., (15) where F Z L(R) = A Z L(R) − C Z L(R) . (16) The branching ratio reads as: BR(Z → µ +τ −) = c m4 Z " |F Z L | 2 + |F Z R | 2 + 1 2
Figure 2. Topologies of the MSSM diagrams contributing to the coefficients AZ, DV , C V (V = γ, Z) and Bf . Internal solid lines denote gauginos (λ) or higgsinos (H˜ ). Dashed lines denote sleptons (or other sfermions in the box diagrams). Dotted lines with a cross denote Higgs insertions. also corresponds to take only the d = 6 operators in the SU(2)W × U(1)Y -unbroken phase. We believe that working in this approximation is more transparent and appealing from the theoretical point of view, although this may imply some loss of numerical accuracy when M2 S is very close to m2 Z . A few add
Figure 3. Topologies of the MSSM diagrams contributing to the ∆ coefficients. Dashed lines denote sleptons. where the explicit expressions of the dimensionless coefficients ∆L,R = ∆(b) L,R + ∆(c) L,R are given8 . In the mass-eigenstate basis for both leptons and Higgs bosons, the LFV couplings read as: L eff Higgs µτ = − hτ √ 2cβ (∆∗ L τ cµ + ∆R µ c τ ) [cβ−αh − sβ−αH − iA] + h.c., (55) where A is the physical CP-odd Higgs field, α is the mixing angle in the CP-even Higgs sector [ √ 2Re(H0 1 − hH0 1 i) = cαH −sαh, √ 2Re(H0 2 − hH0 2 i) = sαH + cαh], and a short-hand notation
Figure 4. ∆-contributions to the Higgs boson decays A, H, h → µτ and to the decays τ → 3µ, τ → µπ, τ → µη, τ → µη′ . In the last diagram, curly lines denote gluons. • Higgs→ µτ Concerning the Higgs boson decays, we have BR(A → µ +τ −) = tan2 β (|∆L| 2 + |∆R| 2 )BR(A → τ +τ −), (56) where we have approximated 1/c 2 β ≃ tan2 β since non-negligible effects can only arise in the large tan β limit. If A is replaced with H [or h] in eq. (56), the r.h.s. should also be multiplied by a factor (cβ−α/sα) 2 [or (sβ−α/cα) 2 ]. These LFV decays and the related phenomenology have been extensively…
Figure 5. D γ(b) L isocontours with tan β = 3, ˜mL2 = 1 TeV, θL = π/4 and µ = 140 GeV (1 TeV) in the left (right) panel. The solid and dashed lines refer to M1 = 100 GeV and M1 = −100 GeV, respectively. The coefficient |D γ(a) L | is at most 10−9 GeV−2 , so it fulfils the bound. In contrast, D γ(b) L and D γ(c) L may separately exceed the bound in some regions of the parameter space, especially 18
Figure 6. D γ L contours for tan β = 3, ˜mL2 = 1 TeV, θL = π/4, AL µτ = 0 and five choices of ( ˜mL3 , µ, M2, M1) in GeV: (100, 140, 120, 100) (dotted), (100, 140, 120, −100) (short-dashed), (100, 250, 150, 100) (solid), (100, 250, 150, −100) (dot-dashed), (200, 700, 300, −100) (dashed). For each example the two external lines correspond to |D γ L | = 5 × 10−9 GeV−2 and the middle one to D γ L = 0. tan β = 3, m˜ L2 = 1 TeV, AL µτ = 0 and several choices of ( ˜mL3 , µ, M1, M2). For each case we show three lines: the two external ones delimit the allowed region (|D γ L | ≤ 5 × 10−9 G
Figure 7. Branching ratios of LFV decays for tan β = 3, ˜mL2 = 1 TeV, ˜mL3 = 100 GeV, θL = π/4, AL µτ = 0. In the upper panel, the remaining parameters are: µ = 120 GeV, M2 = 150 GeV, M1 = 100 GeV, m˜ τ˜R = ˜mµ˜R = 100 GeV, ˜me˜ = ˜mq˜ = 1 TeV. In the lower panel: µ = 800 GeV, M2 = 120 GeV, M1 = −100 GeV, Aτ = 0, ˜me˜ = 120 GeV, ˜mq˜ = 1 TeV and ˜mµ˜R = ˜mτ˜R . The solid horizontal line indicates the present bound on BR(τ → µγ). In the upper example: BR(τ → µη) ∼ 6 × 10−10, BR(τ → µη′ ) ∼ 5 × 10−10. In the lower example: BR(τ → µπ) ∼ 2 × 10−12, BR(τ → µη) ∼ BR(τ → µη′ ) ∼ 10−12 . 20
Figure 8. BR(Z → µτ) for tan β = 3, ˜mL2 = 1 TeV, θL = π/4. In the upper panel: ˜mL3 = 100 GeV, m˜ τ˜R = 100 GeV, M2 as shown, M1 = 100 (−100) GeV for the solid (dashed) lines. In the left (right) lower panel: µ = 120 (1000) GeV, M1 = 100 (−100) GeV and ˜mτ˜R = 100 (500) GeV. In
Figure 9. Monopole contribution to BR(τ → µµµ) for tan β = 3, ˜mL2 = 1 TeV, θL = π/4. In the upper panel: ˜mL3 = 100 GeV, m˜ τ˜R = ˜mµ˜R = 100 GeV, M2 as shown, M1 = 100 (−100) GeV for the solid (dashed) lines. In the left (right) lower panel: µ = 120 (1000) GeV, M1 = 100 (−100) GeV and ˜mτ˜R = 100 (500) GeV. 100 200 300 400 500 600 700 800 900 1000 1 0.3 0.1 0.3 1 100 200 300 400 500 600 700 800 900 1000 PSfrag replacements µ [GeV] 10−11 10−10 10−9 10−8 M2 = 120 GeV M2 [GeV] m˜ L3 [GeV] M2 = 160 GeV BR(τ → µµµ)Dγ=0 BR(τ → µµµ)Dγ=0 [10−9 ] µ [GeV] M2 [GeV] m˜ L3 [GeV] m˜ e˜ [GeV] BR(τ →…
Figure 10. Monopole contribution to BR(τ → µee) for tan β = 3, ˜mL2 = 1 TeV, θL = π/4. In the upper panel: ˜mL3 = 100 GeV, m˜ τ˜R = 100 GeV, M2 = 120 GeV, M1 = 100 (−100) GeV for the solid (dashed) lines. In the left (right) lower panel: µ = 120 (1000) GeV, M1 = 100 (−100) GeV, ˜me˜ = 1000 (120) GeV and m˜ τ˜R = 100 (500) GeV. 23
Figure 11. Monopole contribution to BR(τ → µρ) (upper panels) and BR(τ → µπ) (lower panels), for tan β = 3, ˜mL2 = 1 TeV, θL = π/4. In the left panels: ˜mL3 = 100 GeV, m˜ τ˜R = 100 GeV, M2 = 120 GeV, M1 = 100 (−100) GeV for the solid (dashed) lines. In the right panels: µ = 120 GeV, M1 = 100 GeV, m˜ q˜ = 1 TeV and ˜mτ˜R = 100 GeV. In
Figure 12. Branching ratios of τ → µγ (dashed lines), τ → µµµ (dotted lines) and A → µτ (solid lines) for tan β = 40, θL = π/4, ˜mL2 = 3 TeV, ˜mL3 = 1 TeV, |M1| = 1 2 |M2|. In the upper (lower) panel sign(M1M2) is positive (negative) and ˜mτ˜R = ˜mµ˜R = 2.5 TeV (2 TeV). We have also fixed mA = 100 GeV (which affects τ → 3µ) and BR(A → ττ) = 0.1 (which affects A → µτ). 26
Figure 13. Contours of D γ(a) R (left panel) and D γ(b) R (right panel) for ˜mR2 = 1 TeV and θR = π/4. To determine D γ(b) R we have also fixed tan β = 3 and µ = 150 GeV. can see that D γ(a) R and D γ(b) R are comparable to or even larger than the bound in some regions of the parameter space, and can interfere with each other destructively or constructively. Also D γ(c) R may or may not exceed the bound, depending on the range of the extra parameters ˜mτ˜L , Aτ , A R µτ . Anyway, mutual cancellations involving all three contributions can bring the total dipole coefficient D γ R below the…
Figure 14. D γ R contours for tan β = 3, θR = π/4, m˜ R2 = 1 TeV, AR µτ = 0 and three choices of ( ˜mR3 , µ, M1) in GeV: (100,150,50) (dashed), (150,200,100) (solid), (100,600,-50) (dotted). For each example the two external lines correspond to |D γ R| = 5 × 10−9 GeV−2 and the middle one to D γ R = 0. of D γ R (±5 × 10−9 GeV−2 , 0) in the plane (Aτ , m˜ τ˜L ) for tan β = 3, m˜ R2 = 1 TeV, AR µτ = 0 and three choices of ( ˜mR3 , µ, M1). As a general feature the allowed regions are wider than in the (LFV)L examples (see
Figure 15. Branching ratios of LFV decays for tan β = 3, ˜mR2 = 1 TeV, ˜mR3 = 100 GeV, θR = π/4, AR µτ = 0 and ˜me˜ = 100 GeV, m˜ q˜ = 200 GeV. In the upper panel, the remaining parameters are: µ = 150 GeV, M1 = 50 GeV, ˜mτ˜L = ˜mµ˜L = 200 GeV. In the lower panel: µ = 800 GeV, M1 = −50 GeV, Aτ = 0 and m˜ µ˜L = ˜mτ˜L . The solid horizontal line indicates the present bound on BR(τ → µγ). In the upper example, BR(τ → µπ) ∼ 2 × 10−11, BR(τ → µη) ∼ 2 × 10−12, BR(τ → µη′ ) ∼ 4 × 10−11 and BR(Z → µτ) ∼ 10−10 . In the lower example, BR(τ → µπ) ∼ 3 × 10−12, BR(τ → µη) ∼ 4 × 10−12, BR(τ → µη′ ) ∼ 10−11
Figure 16. BR(Z → µτ) for tan β = 3, ˜mR2 = 1 TeV, θR = π/4 and ˜mτ˜L = 300 GeV. In the left panel: m˜ R3 = 100 GeV and |M1| as shown. In the right panel: |µ| = 150 GeV. contribution is varied (through either Aτ or ˜mτ˜L ) and the monopole ones are essentially fixed. The behaviour of BR(τ → µµµ), BR(τ → µee), BR(τ → µρ) reflects the interplay between the dipole and monopole contributions, similarly to the (LFV)L case. Now let us examine in more detail the parameter dependence of monopole contributions, considering one process at a time. In
Figure 17. Monopole contribution to BR(τ → µµµ) for tan β = 3, ˜mR2 = 1 TeV, θR = π/4 and ˜mτ˜L = m˜ µ˜L = 300 GeV. In the left panel: ˜mR3 = 100 GeV and |M1| as shown. In the right panel: |µ| = 150 GeV. 100 200 300 400 500 600 700 800 900 1000 0.3 0.04 0.1 100 200 300 400 500 600 700 800 900 1000 PSfrag replacements |µ| [GeV] |M1| [GeV] m˜ R3 [GeV] m˜ e˜ [GeV] BR(τ → µee)Dγ=0 [10−9 ] 100 150 200 250 300 1 0.1 0.1 0.01 0 50 100 150 200 250 300 PSfrag replacements |µ| [GeV] |M1| [GeV] m˜ R3 [GeV] m˜ e˜ [GeV] BR(τ → µee)Dγ=0 [10−9 ]
Figure 18. Monopole contribution to BR(τ → µee) for tan β = 3, ˜mR2 = 1 TeV, θR = π/4 and ˜mτ˜L = 300 GeV. In the left panel: ˜mR3 = 100 GeV, |M1| = 50 GeV. In the right panel: |µ| = 150 GeV, m˜ e˜ = 100 GeV. 100 200 300 400 500 600 700 800 900 1000 0.3 0.2 0.1 0.03 0.01 0.001 0.001 100 200 300 400 500 600 700 800 900 1000 PSfrag replacements |µ| [GeV] |M1| [GeV] m˜ R3 [GeV] m˜ q˜ [GeV] BR(τ → µρ)Dγ=0 , BR(τ → µπ) [10−9 ] 100 150 200 250 300 0.3 0.1 0.03 0.01 0.05 0.01 0.001 0 50 100 150 200 250 300 PSfrag replacements |µ| [GeV] |M1| [GeV] m˜ R3 [GeV] m˜ q˜ [GeV] BR(τ → µρ)Dγ=0 , BR(τ → µπ)…
Figure 19. Monopole contribution to BR(τ → µρ) (solid lines) and BR(τ → µπ) (dashed lines), for tan β = 3, m˜ R2 = 1 TeV, θR = π/4 and ˜mτ˜L = 300 GeV. In the left panel: ˜mR3 = 100 GeV, |M1| = 50 GeV. In the right panel: |µ| = 150 GeV, m˜ q˜ = 500 GeV. 31
Figure 20. Branching ratios of τ → µγ (dashed lines), τ → µµµ (dotted lines) and A → µτ (solid lines) for tan β = 40, θR = π/4, ˜mR2 = 3 TeV, ˜mR3 = 0.7 TeV and ˜mτ˜L = 0.7 TeV. We have also fixed mA = 100 GeV and BR(A → ττ) = 0.1. and so is Z → µτ . The decays τ → µee and τ → µρ are generically dipole dominated, so the relations (34), (36) [and the bounds (37), (39)] hold. The dominant contributions to BR(τ → µµµ) [eq. (23)] are induced by the dipole term D γ R and the Higgs-mediated term δF µR L , proportional to ∆R [eq. (59)]. Therefore, τ → µµµ is correlated to both τ → µγ and A → µτ [11]
Figure 21. Shorthand notation used in some diagrams. (83) JN (m2 1 , . . . , m2 N ) ≡ i π 2 Z k 2 d 4k (k 2 − m2 1 ). . .(k 2 − m2 N ) = (−1)N Z ∞ 0 s 2 ds (s + m2 1 ). . .(s + m2 N ) (84) KN (m2 1 , . . . , m2 N ) ≡ i π 2 Z k 4 d 4k (k 2 − m2 1 ). . .(k 2 − m2 N ) = (−1)N+1Z ∞ 0 s 3 ds (s + m2 1 ). . .(s + m2 N ) . (85) It is worth recalling that the expression of a diagrammatic computation in terms of these loop integrals is not unique, since such functions are not independent (this fact is also reflected in the arbitrariness in the parametriza
Figure 22. Diagrams that contribute to A Z(a) L (first and second rows) and A Z(a) R (third row). The coefficients AZ L,R receive contributions of three different types: AZ L,R = A Z(a) L,R + A Z(b) L,R + A Z(c) L,R . The diagrams corresponding to A Z(a) L,R are depicted in
Figure 23. Diagrams that contribute to A Z(b) L,R (left side) and A Z(c) L,R (right side). The diagrams corresponding to A Z(b) L,R (
Figure 24. Diagrams that contribute to C γ L,R. A.5 Contributions to C Z L,R The diagrams corresponding to C Z L,R are depicted in
Figure 25. Diagrams that contribute to C Z L,R. A.6 Contributions to D γ L,R The coefficients D γ L,R receive contributions of three different types: D γ L,R = D γ(a) L,R +D γ(b) L,R +D γ(c) L,R . The diagrams corresponding to D γ(a) L,R are depicted in
Figure 26. Diagrams that contribute to D γ(a) L,R . The diagrams corresponding to D γ(b) L,R are depicted in
Figure 27. Diagrams that contribute to D γ(b) L,R . The diagrams corresponding to D γ(c) L,R are depicted in
Figure 28. Diagrams that contribute to D γ(c) L (first row) and D γ(c) R (second row). A.7 Contributions to DZ L,R The leading contributions to DZ L,R are of two types: DZ L,R = D Z(b) L,R + D Z(c) L,R . The diagrams corresponding to D Z(a) L,R are depicted in
Figure 29. Diagrams that contribute to D Z(b) L,R . +c 2 RJ5(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 R3 , m˜ 2 R3 ) i − (L2 → L3) (109) D Z(c) R sRcR = g ′2 16π 2 · 1 4 M1µ tan β h (1 − 2s 2 W )  s 2 LJ5(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 L2 , m˜ 2 R2 ) +c 2 LJ5(M2 1 , M2 1 , m˜ 2 L3 , m˜ 2 L3 , m˜ 2 R2 )  − 2s 2 W  s 2 LJ5(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 R2 , m˜ 2 R2 ) +c 2 LJ5(M2 1 , M2 1 , m˜ 2 L3 , m˜ 2 R2 , m˜ 2 R2 ) i − (R2 → R3) (110) τ c µ B˜0 Z H0 2 ˜ℓRα ˜ℓLβ τ c µ B˜0 H Z 0 2 ˜ℓRα ˜ℓLβ τ µ c B˜0 Z H0 2 ˜ℓLα ˜ℓRβ τ µ c B˜0 H Z 0 2 ˜ℓLα ˜ℓRβ
Figure 30. Diagrams that contribute to D Z(c) L,R . A.8 Contributions to B fL,R L,R Contributions to B fL,R L,R arise from the diagrams in
Figure 31. Diagrams that contribute to B fL,R L (first and second rows) and B fL,R R (third row). In the case of B µL,R L,R , i.e. f = f d = µ, f c = µ c , the following replacements are implied: ˜f u L → ν˜β, ˜fL → ˜ℓLβ , ˜fR → ˜ℓRβ . + g ′4 16π 2 · 1 4 (−Y 2 fL ) h J4(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 ˜fL ) + 2M2 1 I4(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 ˜fL ) i − (L2 → L3) (111) B fR L sLcL = g ′4 16π 2 · 1 4 Y 2 fR h J4(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 ˜fR ) + 2M2 1 I4(M2 1 , M2 1 , m˜ 2 L2 , m˜ 2 ˜fR ) i − (L2 → L3) (112
Figure 32. Diagrams that contribute to ∆(b) L,R (first row) and ∆(c) L,R (second row). can be decomposed into five terms, i.e. a MSSM µ = a (a) µL +a (a) µR +a (b) µL +a (b) µR +a (c) µLR, given as follows: a (a) µL 2m2 µ = g 2 16π 2 · c 2 L 4 h m˜ 2 L2 J5(M2 2 , m˜ 2 L2 , m˜ 2 L2 , m˜ 2 L2 , m˜ 2 L2 ) − 2M2 2 J5(M2 2 , M2 2 , M2 2 , M2 2 , m˜ 2 L2 ) i + g ′2 16π 2 · c 2 L 4 m˜ 2 L2 J5(M2 1 , m˜ 2 L2 , m˜ 2 L2 , m˜ 2 L2 , m˜ 2 L2 ) + (c 2 L → s 2 L , L2 → L3) (123) a (a) µR 2m2 µ = g ′2 16π 2 c 2

Figures reproduced from arXiv:hep-ph/0404211 by the authors.

References
How later papers used it

50 later papers cite it

Evidence: substantial. 50 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 10 of the 90 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Compares with it 1 of 8 papers with this label
How the literature uses it

46 arXiv papers cite this in 83 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 42 sentences without a confident label

Uses its method or tool 2 papers
Uses its data or values 1 paper
Compares with it 7 papers
Cites it as background 23 papers

Discussion

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