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---
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title: Generic RGE calculation
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permalink: /Generic_RGE_calculation/
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---
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# Generic RGE calculation
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[Category:Calculations](/Category:Calculations "wikilink")
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Supersymmetric RGEs
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-------------------
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... | ... | @@ -12,7 +9,7 @@ Supersymmetric RGEs |
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SARAH calculates the SUSY RGEs at the one- and two-loop level. In general, the*β*-function of a parameter*c* is parametrized by
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$\\frac{d}{dt} c \\equiv \\beta_c = \\frac{1}{16 \\pi^2} \\beta^{(1)}_c + \\frac{1}{(16 \\pi^2)} \\beta^{(2)}_c$
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$`\frac{d}{dt} c \equiv \beta_c = \frac{1}{16 \pi^2} \beta^{(1)}_c + \frac{1}{(16 \pi^2)} \beta^{(2)}_c`$
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*β*<sub>*c*</sub><sup>(1)</sup>, *β*<sub>*c*</sub><sup>(2)</sup> are the coefficients at one- and two-loop level. The results used by SARAH are mainly based on Ref..
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... | ... | @@ -28,20 +25,16 @@ For the gauge couplings the generic one-loop expression is rather simple and rea |
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The starting point for the calculation of the RGEs for the superpotential terms in SARAH are the anomalous dimensions *γ* for all superfields. These can be also parametrized by
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$\\gamma_{\\hat \\phi_i \\hat \\phi_j} = \\frac{1}{16\\pi^2} \\gamma^{(1)}_{\\hat \\phi_i \\hat \\phi_i} + \\frac{1}{(16 \\pi^2)^2} \\gamma^{(2)}_{\\hat \\phi_i \\hat \\phi_j}$
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$`\gamma_{\hat \phi_i \hat \phi_j} = \frac{1}{16\pi^2} \gamma^{(1)}_{\hat \phi_i \hat \phi_i} + \frac{1}{(16 \pi^2)^2} \gamma^{(2)}_{\hat \phi_i \hat \phi_j}`$
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One has to stress that*i*, *j* are not generation indices but label the different fields. Generic formula for the one- and two-loop coefficients *γ*<sup>(1)</sup>, *γ*<sup>(2)</sup> are given in Ref. as well. SARAH includes the case of an anomalous dimension matrix with off-diagonal entries, i.e. *ϕ̂*<sub>*i*</sub> ≠ *ϕ̂*<sub>*j*</sub>. That’s for instance necessary in models with vector like quarks where the superpotential reads
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$W \\supset Y_u \\hat{u} \\hat{q} \\hat{H}_u + Y_U \\hat{U} \\hat{q} \\hat{H}_u + M_U \\hat{U} \\hat{\\bar{{U}}}$
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$`W \supset Y_u \hat{u} \hat{q} \hat{H}_u + Y_U \hat{U} \hat{q} \hat{H}_u + M_U \hat{U} \hat{\bar{{U}}}`$
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*γ*<sub>*û**Û*</sub> is not vanishing but receives already at one-loop contributions∝*Y*<sub>*u*</sub>*Y*<sub>*U*</sub>.
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From the anomalous dimensions it is straightforward to get the*β*-functions of the superpotential terms: for a generic superpotential ofthe coefficients*β*<sup>(*x*)</sup> are given by
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$\\begin{aligned}
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\\beta^{(x)}_{L^i} \\sim & L^a \\gamma^{(x)}_{a \\hat \\phi_i} \\\\
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\\beta^{(x)}_{M^{ij}} \\sim & M^{ia} \\gamma^{(x)}_{a \\hat \\phi_j} + (j\\leftrightarrow i) \\\\
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\\beta^{(x)}_{Y^{ijk}} \\sim & Y^{ija} \\gamma^{(x)}_{a \\hat \\phi_k} + (k\\leftrightarrow i) + (k\\leftrightarrow j) \\\\
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\\beta^{(x)}_{W^{ijkl}} \\sim & W^{ijka} \\gamma^{(x)}_{a \\hat \\phi_l} + (l\\leftrightarrow i) + (l\\leftrightarrow j) + (l\\leftrightarrow k) \\end{aligned}$
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$` \beta^{(x)}_{L^i} \sim L^a \gamma^{(x)}_{a \hat \phi_i} \\ \beta^{(x)}_{M^{ij}} \sim M^{ia} \gamma^{(x)}_{a \hat \phi_j} + (j\leftrightarrow i) \\ \beta^{(x)}_{Y^{ijk}} \sim Y^{ija} \gamma^{(x)}_{a \hat \phi_k} + (k\leftrightarrow i) + (k\leftrightarrow j) \\ \beta^{(x)}_{W^{ijkl}} \sim W^{ijka} \gamma^{(x)}_{a \hat \phi_l} + (l\leftrightarrow i) + (l\leftrightarrow j) + (l\leftrightarrow k) `$
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up to constant coefficients.
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... | ... | @@ -49,11 +42,11 @@ up to constant coefficients. |
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In the soft-breaking sector SARAH includes also all standard terms of the form
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$- \\mathfrak{L}_{SB} = t^i \\phi_i + \\frac{1}{2} B^{ij} \\phi_i \\phi_j + \\frac{1}{3!} T^{ijk} \\phi_i \\phi_j \\phi_k + \\frac{1}{4!} Q^{ijkl} \\phi_i \\phi_j \\phi_k \\phi_l + \\frac{1}{2} (m^2)^j_i \\phi^{\* i} \\phi_j - \\frac{1}{2} M \\lambda \\lambda$
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$`- \mathfrak{L}_{SB} = t^i \phi_i + \frac{1}{2} B^{ij} \phi_i \phi_j + \frac{1}{3!} T^{ijk} \phi_i \phi_j \phi_k + \frac{1}{4!} Q^{ijkl} \phi_i \phi_j \phi_k \phi_l + \frac{1}{2} (m^2)^j_i \phi^{\* i} \phi_j - \frac{1}{2} M \lambda \lambda`$
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The generic expressions for *B*’s, *T*’s, *m*<sup>2</sup>’s and *M*’s up to two-loop are given again in Ref. which is used by SARAH. The *β*-function for the linear soft-term *t* is calculated using Ref. . For the quartic soft-term *Q* the approach of Ref. is adopted. In this approach *γ̄* is defined by
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$\\bar{\\gamma}^{(x)}_{\\hat \\phi_i \\hat \\phi_j} = \\left(M_A g_A^2 \\frac{\\partial }{\\partial g_A^2} - T^{lmn} \\frac{\\partial}{\\partial Y^{lmn}} \\right) \\gamma^{(x)}_{\\hat \\phi_i \\hat \\phi_j}$
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$`\bar{\gamma}^{(x)}_{\hat \phi_i \hat \phi_j} = \left(M_A g_A^2 \frac{\partial }{\partial g_A^2} - T^{lmn} \frac{\partial}{\partial Y^{lmn}} \right) \gamma^{(x)}_{\hat \phi_i \hat \phi_j}`$
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The *β*-functions for *Q* can then expressed by *γ* and *γ̄*:
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... | ... | @@ -65,13 +58,11 @@ In principle, the same approach can also be used for*B* and*T* terms as long as |
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The running of the Fayet-Iliopoulos*D*-term*ξ* receives two contributions:
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$\\beta_{\\xi_A}^{(x)} = \\frac{\\beta_{g_A}^{(x)}}{g_A} \\xi_A + \\beta^{(x)}_{\\hat \\xi_A}$
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$`\beta_{\xi_A}^{(x)} = \frac{\beta_{g_A}^{(x)}}{g_A} \xi_A + \beta^{(x)}_{\hat \xi_A}`$
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The first part is already fixed by the running of the gauge coupling of the Abelian group, the second part, *β*<sub>*ξ̂*</sub><sup>(*x*)</sup>, is known even to three loops . SARAH has implemented the one- and two-loop results which are rather simple:
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$\\begin{aligned}
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\\beta^{(1)}_{\\hat \\xi_A} =& 2 g_A \\sum_i (Q^A_{\\phi_i} m_{\\phi_i \\phi_i}^2) \\equiv \\sigma_{1,A}\\\\
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\\beta^{(1)}_{\\hat \\xi_A} =& - 4 g_A \\sum_{ij} (Q^A_{\\phi_i}m^2_{\\phi_i \\phi_j} \\gamma^{(1)}_{\\hat \\phi_j \\hat \\phi_i}) \\equiv \\sigma_{3,A} \\end{aligned}$
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$` \beta^{(1)}_{\hat \xi_A} = 2 g_A \sum_i (Q^A_{\phi_i} m_{\phi_i \phi_i}^2) \equiv \sigma_{1,A}\\ \beta^{(1)}_{\hat \xi_A} = - 4 g_A \sum_{ij} (Q^A_{\phi_i}m^2_{\phi_i \phi_j} \gamma^{(1)}_{\hat \phi_j \hat \phi_i}) \equiv \sigma_{3,A} `$
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*σ*<sub>1</sub> and *σ*<sub>3</sub> are traces which are also used to express the *β*-functions of the soft-scalar masses at one- and two-loop, see for instance Ref. .
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... | ... | @@ -79,7 +70,7 @@ $\\begin{aligned} |
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Finally, the*β*-functions for the gaugino mass parameters are
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$\\frac{d}{dt} \\equiv \\beta_M = \\frac{1}{16\\pi^2} \\beta_M^{(1)} + \\frac{1}{(16 \\pi^2)} \\beta_M^{(2)}$
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$`\frac{d}{dt} \equiv \beta_M = \frac{1}{16\pi^2} \beta_M^{(1)} + \frac{1}{(16 \pi^2)} \beta_M^{(2)}`$
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where the expressions for*β*<sub>*M*</sub><sup>(*x*)</sup> are also given in Ref. .*β*<sub>*M*</sub><sup>(1)</sup> has actually a rather simple form similar to the one of the gauge couplings. One finds
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... | ... | @@ -91,7 +82,7 @@ Therefore, the running of the gaugino masses are strongly correlated with the on |
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The changes in the RGEs in the presence of Dirac gaugino mass terms are known today at the two-loop level, see Ref. . SARAH makes use of these results to obtain the*β*-functions for the new mass parameters as well as to include new contribution to the RGEs of tadpole terms in presence of Dirac gauginos. The*β* functions of a Dirac mass terms *m*<sub>*D*</sub><sup>*ϕ̂**λ*</sup>*Ψ**λ*<sub>*i*</sub> are related to the anomalous dimension of the involved chiral superfield*ϕ̂*, whose fermionic component is *Ψ*, and to the running of the corresponding gauge coupling:
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$\\beta_{m^{\\hat \\phi A}_D} = \\gamma_{\\hat \\phi a} m_D^{a A} + \\frac{\\beta_{g_A}}{g_A} m_D^{\\hat \\phi A}$
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$`\beta_{m^{\hat \phi A}_D} = \gamma_{\hat \phi a} m_D^{a A} + \frac{\beta_{g_A}}{g_A} m_D^{\hat \phi A}`$
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The tadpole term receives two new contributions from Fayet-Iliopoulos terms discussed above and terms mimicking*B* insertions
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... | ... | @@ -114,7 +105,7 @@ Non-Supersymmetric RGEs |
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SARAH calculates also the for a general quantum field theory at the one- and two-loop level. The parameterisation is the same as for a SUSY model:
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$\\frac{d}{dt} c \\equiv \\beta_c = \\frac{1}{16 \\pi^2} \\beta^{(1)}_c + \\frac{1}{(16 \\pi^2)} \\beta^{(2)}_c$
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$`\frac{d}{dt} c \equiv \beta_c = \frac{1}{16 \pi^2} \beta^{(1)}_c + \frac{1}{(16 \pi^2)} \beta^{(2)}_c`$
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*β*<sub>*c*</sub><sup>(1)</sup>, *β*<sub>*c*</sub><sup>(2)</sup> are the coefficients at one- and two-loop level. The results used by SARAH are mainly based on Ref..
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... | ... | @@ -135,9 +126,7 @@ The expressions presented in literature do usually not cover all possibilities a |
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In order to include gauge-kinetic mixing in the running of the gauge couplings and gaugino masses one can use the substitutions
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$\\begin{aligned}
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g_A^3 S(R) \\to & G \\sum_{\\hat \\phi} V_{\\hat \\phi} V_{\\hat \\phi}^T \\\\
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g_A^2 M_A S(R) \\to & M \\sum_{\\hat \\phi} V_{\\hat \\phi} V_{\\hat \\phi}^T + \\sum_{\\hat \\phi} V_{\\hat \\phi} V_{\\hat \\phi}^T M\\end{aligned}$
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$` g_A^3 S(R) \to G \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T \\ g_A^2 M_A S(R) \to M \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T + \sum_{\hat \phi} V_{\hat \phi} V_{\hat \phi}^T M`$
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Here, *G* and *M* are matrices carrying the gauge couplings and gaugino masses of all *U*(1) groups, see also sec. \[sec:supported<sub>m</sub>odels\], and I introduced*V*<sub>*ϕ̂*</sub> = *G*<sup>*T*</sup>*Q*<sub>*ϕ̂*</sub>. The sums are running over all chiral superfields *ϕ̂*. Also for all other terms involving gauge couplings and gaugino masses appearing in the *β* functions similar rules are presented in Ref. which are used by SARAH.
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... | ... | |