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April 17, 2008

Gauge Mediation

I was remiss (read: lazy, overworked, or whatever) in not writing, earlier, about Meade, Seiberg and Shih’s paper on gauge mediation. But Patrick Meade was visting this week, so perhaps I can make amends.

There is a huge literature on models of gauge mediated supersymmetry-breaking. And there are a variety of characteristic predictions that emerge from particular classes of models. What these guys do is provide a model-independent characterization of gauge-mediation and try to isolate what features are generic to all models versus those which are special to particular subclasses of models of gauge mediation.

Their characterization is very simple: in gauge mediation, there the MSSM sector and a hidden sector, SS, in which supersymetry is dynamically broken. When the MSSM gauge couplings, α r,r=1,2,3\alpha_r,\, r=1,2,3, are set to zero, the two sectors decouple and the MSSM sector is supersymmetric.

The coupling between the two sectors is described by gauging an SU(3)×SU(2)×U(1)SU(3)\times SU(2)\times U(1) subgroup of the global symmetry group of SS. The corresponding conserved current(s), j μj_\mu, is part of supermultiplet, a real linear linear multiplet, D 2𝒥=D¯ 2𝒥=0 𝒥=J+iθj−iθ¯ȷ¯−θσ μθ¯j μ+12θθθ¯σ¯ μ∂ μj−12θ¯θ¯θσ μ∂ μȷ¯−14θθθ¯θ¯□J \begin{gathered} D^2\mathcal{J}= \overline{D}^2\mathcal{J}=0\\ \mathcal{J} = J + i\theta j - i\overline{\theta}\overline{\jmath} - \theta \sigma^\mu\overline{\theta} j_\mu +\tfrac{1}{2} \theta\theta\overline{\theta}\overline{\sigma}^\mu\partial_\mu j -\tfrac{1}{2} \overline{\theta}\overline{\theta}\theta\sigma^\mu\partial_\mu\overline{\jmath} -\tfrac{1}{4} \theta\theta\overline{\theta}\overline{\theta} \Box J \end{gathered}

The physics of gauge mediation is governed by the two-point functions

(1)⟨J(p)J(p′)⟩ = (2π) 4δ (4)(p+p′) C 0(p 2/M 2,Λ 2/M 2) ⟨j α(p)j¯ α˙(p′)⟩ = −(2π) 4δ (4)(p+p′) σ αα˙ μp μC 1/2(p 2/M 2,Λ 2/M 2) ⟨j μ(p)j ν(p′)⟩ = −(2π) 4δ (4)(p+p′) (p μp ν−η μνp 2)C 1(p 2/M 2,Λ 2/M 2) ⟨j α(p)j β(p′)⟩ = (2π) 4δ (4)(p+p′) ϵ αβMB(p 2/M 2)\begin{aligned} \langle J(p)J(p')\rangle &=& {(2\pi)}^4 \delta^{(4)}(p+p')& C_0(p^2/M^2,\Lambda^2/M^2)\\ \langle j_\alpha (p)\overline{j}_{\dot{\alpha}}(p')\rangle &=& -{(2\pi)}^4 \delta^{(4)}(p+p')& \sigma^\mu_{\alpha\dot{\alpha}}p_\mu C_{1/2}(p^2/M^2,\Lambda^2/M^2)\\ \langle j_\mu(p)j_\nu(p')\rangle &=& -{(2\pi)}^4 \delta^{(4)}(p+p')& (p_\mu p_\nu -\eta_{\mu\nu}p^2)C_{1}(p^2/M^2,\Lambda^2/M^2)\\ \langle j_\alpha (p)j_\beta(p')\rangle &=& {(2\pi)}^4 \delta^{(4)}(p+p')& \epsilon_{\alpha\beta} M B(p^2/M^2) \end{aligned}

Here, MM is a mass scale characterizing physics in the hidden sector, and Λ\Lambda is a UV cutoff, to regulate the short-distance singularity in the 2-point function.

For the case of an abelian symmetry, there’s also the possibility of a 1-point function, ⟨J(p)⟩=(2π) 4δ (4)(p)ζ \langle J(p)\rangle = {(2\pi)}^4 \delta^{(4)}(p) \zeta But, for various phenomenological reasons, it’s best to assume that the hidden sector has a 𝒥→−𝒥\mathcal{J}\to-\mathcal{J}, which enforces ζ=0\zeta=0.

In the supersymmetric limit,

(2)C 0=C 1/2=C 1 B=0\begin{gathered} C_0 = C_{1/2} = C_1\\ B = 0 \end{gathered}

Since supersymmetry is restored in the large-pp limit, this means that

(3)C s=clog(Λ 2/p 2)+finiteC_s = c\, \log(\Lambda^2/p^2) +\text{finite}

for some constant, cc, independent of the spin, and that BB is UV-finite. The divergent part of C 1C_1 gives the contribution of the hidden sector to the β\beta-function of the gauge coupling. Specifically, the shift in the 1-loop β\beta-function coefficient from integrating out the hidden sector is b high=b low−16π 2c b_{\text{high}} = b_{\text{low}} - 16\pi^2 c In a model which preserves coupling unification, this means that all of the c (r),r=1,2,3c^{(r)},\, r=1,2,3 are equal.

The contribution to the gaugino masses comes from a tree-level insertion of the ⟨j αj β⟩\langle j_\alpha j_\beta\rangle two point function, represented by the magenta blob: Tree Graph, with insertion of B ⇒m (r)=g r 2MB (r)(0) \array{\arrayopts{\rowalign{top}} \begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="150" height="30"> <desc>Tree Graph, with insertion of B</desc> <defs> <marker id="arrowheadA" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="6" orient="auto"> <path d="M 0 0 L 10 5 L 0 10 z" /> </marker> </defs> <g fill="none" stroke="black" stroke-width="1"> <path d="M 5 15 c 10 10 10 -10 20 0 s 10 -10 20 0 s 10 -10 20 0 s 10 -10 20 0 s 10 -10 20 0 s 10 -10 20 0 s 10 -10 20 0" /> <path d="M 5 15 l 30 0 l 30 0" marker-mid="url(#arrowheadA)" /> <path d="M 85 15 l 30 0 l 30 0" marker-mid="url(#arrowheadA)" /> </g> <circle fill="magenta" cx="75" cy="15" r="10" /> </svg> \end{svg}&\Rightarrow\quad m^{(r)} = g_r^2 M B^{(r)}(0) }

The contributions to the sfermion masses come from one-loop diagrams with an insertion of the two-point functions (1), C s (r)C^{(r)}_s, represented by the red (s=0s=0), yellow (s=1/2s=1/2) and blue (s=1s=1) blobs, respectively: Loop Graph, with insertion of C_0 + Loop Graph, with insertion of C_{1/2} + Loop Graph, with insertion of C_1 + Another Loop Graph, with insertion of C_1 ⇒ m f 2=∑ r=1 3g r 4c 2(f;r)A (r) \begin{gathered} \begin{matrix} \begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="150" height="50"> <desc>Loop Graph, with insertion of C_0</desc> <g fill="none"> <path stroke="#666" stroke-width="2" stroke-dasharray="1" d="M 45 45 a 29 29 0 0 1 58 0" /> <path stroke="black" stroke-width="1" stroke-dasharray="2" d="M 5 45 l 138 0 " /> </g> <circle fill="red" cx="75" cy="15" r="10" /> </svg> \end{svg}&+& \begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="150" height="50"> <desc>Loop Graph, with insertion of C_{1/2}</desc> <defs> <marker id="arrowheadB" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="6" orient="auto"> <path d="M 0 0 L 10 5 L 0 10 z" /> </marker> </defs> <g fill="none" stroke="black" stroke-width="1"> <path d="M 0 0 c -15 -15 -19 -31 4 -38 s 27 -8 15 -35 s 10 -36 23 -31 s 28 6 30 -22 s 19 -27 35 -15 s 20 17 38 -4 s 27 -20 38 6 s 11 23 34 14 s 37 -3 30 23 s 0 28 23 28 s 28 20 14 36 s -10 27 5 37" transform="translate(45 45) scale(0.2)" stroke-width="5" /> <path d="M 45 45 a 29 29 0 0 1 8.49 -20.51 a 29 29 0 0 1 20.51 -8.49 a 29 29 0 0 1 20.51 8.49 a 29 29 0 0 1 8.49 20.51" marker-mid="url(#arrowheadB)" /> <path d="M 103 45 l -29 0 l -29 0" marker-mid="url(#arrowheadB)" /> <path stroke-dasharray="2" d="M 5 45 l 40 0 m 58 0 l 40 0" /> </g> <circle fill="yellow" cx="75" cy="15" r="10" /> </svg> \end{svg}&+&\\ \begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="150" height="50"> <desc>Loop Graph, with insertion of C_1</desc> <g fill="none" stroke="black" stroke-width="1"> <path d="M 0 0 c -15 -15 -19 -31 4 -38 s 27 -8 15 -35 s 10 -36 23 -31 s 28 6 30 -22 s 19 -27 35 -15 s 20 17 38 -4 s 27 -20 38 6 s 11 23 34 14 s 37 -3 30 23 s 0 28 23 28 s 28 20 14 36 s -10 27 5 37" transform="translate(45 45) scale(0.2)" stroke-width="5" /> <path stroke-dasharray="2" d="M 5 45 l 138 0 " /> </g> <circle fill="blue" cx="75" cy="15" r="10" /> </svg> \end{svg}&+& \begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="150" height="50"> <desc>Another Loop Graph, with insertion of C_1</desc> <g fill="none" stroke="black" stroke-width="1"> <path d="M0,0c21,0,15,23,31,27s23-16,40-7s4,28,15,39 s28,-4,37,13s-12,23-8,40s28,12,28,32s-24,14-29,31s19,28,10,41s-27-1-40,12 s4,33-13,41s-27-12-43-9s-14,29-31,29s-16-25-31-29s-24,19-41,10s-1-29-13-40 s-32,1-41-14s13-23,9-41s-24,-12,-24,-32s22-17,26-33s-18,-23,-7,-39s27-3,39-15 s-1-30,14-39s23,13,39,8s14,-25,33,-25" transform="translate(75 10) scale(0.12)" stroke-width="8" /> <path stroke-dasharray="2" d="M 5 45 l 138 0 " /> </g> <circle fill="blue" cx="75" cy="15" r="10" /> </svg> \end{svg}&&\Rightarrow \end{matrix}\\ m_f^2 = \sum_{r=1}^3 g_r^4 c_2(f;r)A^{(r)} \end{gathered} where c 2(f;r)c_2(f;r) is the quadratic Casimir in representation ff and

(4)A (r) =−∫d 4p(2π) 41p 2(3C 1 (r)(p 2/M 2)−4C 1/2 (r)(p 2/M 2)+C 0 (r)(p 2/M 2)) =−M 216π 2∫dy(3C 1 (r)(y)−4C 1/2 (r)(y)+C 0 (r)(y))\begin{aligned} A^{(r)} &= -{\int \frac{d^4 p}{{(2\pi)}^4}\frac{1}{p^2}\left(3C_1^{(r)}(p^2/M^2)-4C_{1/2}^{(r)}(p^2/M^2)+C_0^{(r)}(p^2/M^2)\right)}\\ &= -\frac{M^2}{16\pi^2}{\int dy \left(3C_1^{(r)}(y)-4C_{1/2}^{(r)}(y)+C_0^{(r)}(y)\right)} \end{aligned}

Note that the integrand in (4) is UV-finite, and vanishes in the supersymmetric limit, as a consequence of (2),(3). The sfermion masses obey two sum rules TrYm 2=0 Tr(B−L)m 2=0 \begin{gathered} Tr Y m^2 = 0\\ Tr (B-L) m^2 = 0 \end{gathered} or, more concretely, m Q 2−2m U 2+m D 2−m L 2+m E 2=0 2m Q 2−m U 2−m D 2−2m L 2+m E 2=0 \begin{gathered} m_Q^2-2m_U^2+m_D^2-m_L^2+m_E^2=0\\ 2m_Q^2-m_U^2-m_D^2-2m_L^2+m_E^2=0 \end{gathered} but are otherwise arbitrary.

And, since the B (r)B^{(r)} have no a-priori relation to the C s (r)C^{(r)}_s, there is, in this formalism, no prediction for gaugino mass unification.

Almost all concrete models, including those that come from String Theory, have more structure. In particular, supersymmetry is often broken at a scale much lower than MM. That is, there’s a small parameter, F/M 2F/M^2. And, at least at leading order in F/M 2F/M^2, the supersymmetry-violating bits of the C sC_s and BB seem to be related. This is what gives things like gaugino mass unification, and the prediction that the NLSP is a Bino or a stau.

They are, according to Patrick, hard at work trying to incorporate this feature into their analysis.

Posted by distler at April 17, 2008 2:33 AM

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