Proceedings to the 12th Workshop 'What Comes Beyond the Standard Models', Bled, July 14. - 24., 2009, Slovenia

TL;DR

This collection of workshop papers covers diverse topics in beyond Standard Model physics, including supersymmetry, dark matter, particle symmetries, and theories unifying spins and charges, presenting recent theoretical and phenomenological advances.

Contribution

The workshop introduces new models, analyses, and ideas in beyond Standard Model physics, with specific focus on supersymmetry, dark matter, and unification theories.

Findings

Likelihood analysis of supergravity models

Characterization of phases via the Multiple Point Principle

Proposals for dark matter composed of stable quark families

Abstract

Contents: 1. Likelihood Analysis of the Next-to-minimal Supergravity Motivated Model (C. Balazs and D. Carter) 2. The Multiple Point Principle: Characterization of the Possible Phases for the SMG (D.L. Bennett) 3. Does Dark Matter Consist of Baryons of New Stable Family Quarks? (G. Bregar and N.S. Mankoc Borstnik) 4. P, C and T for Truly Neutral Particles (V.V. Dvoeglazov) 5. Relativistic Equations for Spin Particles: What Can We Learn From Noncommutativity? (V.V. Dvoeglazov) 6. Radiative Charged Fermion Masses and Quark Mixing (VCKM)4x4 in a SU(3) Gauged Flavor Symmetry Model (A. Hernandez-Galeana) 7. Low Energy Binding of Composite Dark Matter with Nuclei as a Solution for the Puzzles of Dark Matter Searches (M.Yu. Khlopov, A.G. Mayorov and E.Yu. Soldatov) 8. On the Possibilities and Impossibilities of Random Dynamics (A. Kleppe) 9. Spin Connection Makes Massless Spinor Chirally Coupled to Kaluza-Klein Gauge Field After Compactification of $M^{1+5}$ to $M^{1+3}$ x Infinite Disc Curved on $S^2$ (D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen) 10. Offering the Mechanism for Generating Families - the Approach Unifying Spins and Charges Predicts New Families (N.S. Mankoc Borstnik) 11. Confusions That are so Unnecessary (R. Mirman) 12. - 17. Discussion Sections 18. Presentation of VIA and Bled 2009 Workshop Videoconferences (M.Yu. Khlopov)

Figures

Tables

Table 1. Table 1: The properties of a cluster of the fifth family quarks within the extended Bohr-like (hydrogen-like) model from Appendix. m q 5 subscript 𝑚 subscript 𝑞 5 m_{q_{5}} in TeV/c 2 is the assumed fifth family quark mass, α c subscript 𝛼 𝑐 \alpha_{c} is the coupling constant of the colour interaction at E ≈ ( − E c 5 / 3 ) 𝐸 subscript 𝐸 subscript 𝑐 5 3 E\approx(-E_{c_{5}}/3)\; (Eq. 15 ) which is the kinetic energy of quarks in the baryon, r c 5 subscript 𝑟 subscript 𝑐 5 r_{c_{5}} is the corresponding average radius. Then σ c 5 = π ​ r c 5 2 subscript 𝜎 subscript 𝑐 5 𝜋 superscript subscript 𝑟 subscript 𝑐 5 2 \sigma_{c_{5}}=\pi r_{c_{5}}^{2} is the corresponding scattering cross section.

$\frac{m_{q_5}{c}^2}{\mathrm{TeV}}$	${\alpha }_c$	$\frac{E_{c_5}}{m_{q_5}{c}^2}$	$\frac{r_{c_5}}{{10}^{-6}\mathrm{fm}}$	$\frac{\Delta m_{ud}{c}^2}{\mathrm{GeV}}$
$1$	0.16	-0.016	$3.2\cdot {10}^3$	0.05
$10$	0.12	-0.009	$4.2\cdot {10}^2$	0.5
${10}^2$	0.10	-0.006	$52$	5
${10}^3$	0.08	-0.004	$6.0$	50
${10}^4$	0.07	-0.003	$0.7$	$5\cdot {10}^2$
${10}^5$	0.06	-0.003	$0.08$	$5\cdot {10}^3$

Table 2. Table 2: The fifth family quark mass is presented (Eq.( 20 )), calculated for different choices of η c 5 subscript 𝜂 subscript 𝑐 5 \eta_{c_{5}} (which takes care of the inaccuracy with which a colourless cluster of three quarks (anti-quarks) cross section was estimated and of η ( q ​ q ¯ ) b subscript 𝜂 subscript 𝑞 ¯ 𝑞 𝑏 \eta_{(q\bar{q})_{b}} (which takes care of the inaccuracy with which the cross section for the annihilation of a bound state of quark—anti-quark was taken into account) from Eqs. ( 20 , 18 , 17 ). * denotes non stable calculations.

$\frac{m_{q_5}{c}^2}{\mathrm{TeV}}$	${\eta }_{{\left(q\overline{q}\right)}_b}=\frac{1}{10}$	${\eta }_{{\left(q\overline{q}\right)}_b}=\frac{1}{3}$	${\eta }_{{\left(q\overline{q}\right)}_b}=1$	${\eta }_{{\left(q\overline{q}\right)}_b}=3$	${\eta }_{{\left(q\overline{q}\right)}_b}=10$
${\eta }_{c_5}=\frac{1}{50}$	21	36	71	159	417
${\eta }_{c_5}=\frac{1}{10}$	12	20	39	84	215
${\eta }_{c_5}=\frac{1}{3}$	9	14	25	54	134
${\eta }_{c_5}=1$	8	11	19	37	88
${\eta }_{c_5}=3$	7	10	15	27	60
${\eta }_{c_5}=10$	7*	8*	13	22	43

Table 3. Table 3: The 8-plet of quarks - the members of S ​ O ​ ( 1 , 7 ) 𝑆 𝑂 1 7 SO(1,7) subgroup, belonging to one Weyl left handed ( Γ ( 1 , 13 ) = − 1 = Γ ( 1 , 7 ) × Γ ( 6 ) superscript Γ 1 13 1 superscript Γ 1 7 superscript Γ 6 \Gamma^{(1,13)}=-1=\Gamma^{(1,7)}\times\Gamma^{(6)} ) spinor representation of S ​ O ​ ( 1 , 13 ) 𝑆 𝑂 1 13 SO(1,13) . It contains the left handed weak charged quarks and the right handed weak chargeless quarks of a particular colour ( ( 1 / 2 , 1 / ( 2 ​ 3 ) ) 1 2 1 2 3 (1/2,1/(2\sqrt{3})) ). Here Γ ( 1 , 3 ) superscript Γ 1 3 \Gamma^{(1,3)} defines the handedness in ( 1 + 3 ) 1 3 (1+3) space, S 12 superscript 𝑆 12 S^{12} the ordinary spin (which can also be read directly from the basic vector), τ 23 superscript 𝜏 23 \tau^{23} the weak charge and Y 𝑌 Y defines the hyper charge. Let the reader notice (by taking into account the relations γ a ( k ) a ​ b = η a ​ a [ − k ] a ​ b , ( − k ) a ​ b ( k ) a ​ b = η a ​ a [ − k ] a ​ b \gamma^{a}\stackrel{{\scriptstyle ab}}{{(k)}}=\eta^{aa}\stackrel{{\scriptstyle ab}}{{[-k]}},\,\stackrel{{\scriptstyle ab}}{{(-k)}}\stackrel{{\scriptstyle ab}}{{(k)}}=\eta^{aa}\stackrel{{\scriptstyle ab}}{{[-k]}} ) that γ 0 ( − ) 78 superscript 78 superscript 𝛾 0 absent \gamma^{0}\stackrel{{\scriptstyle 78}}{{(-)}} (appearing in − ℒ Y = ψ † γ 0 { ( + ) 78 p 0 + + ( − ) 78 p 0 − } ψ -{\mathcal{L}}_{Y}=\psi^{\dagger}\;\gamma^{0}\{\stackrel{{\scriptstyle 78}}{{(+)}}\;p_{0+}\;+\;\stackrel{{\scriptstyle 78}}{{(-)}}\;p_{0-}\}\psi ) transforms u R c ​ 1 superscript subscript 𝑢 𝑅 𝑐 1 u_{R}^{c1} of the 1 s ​ t superscript 1 𝑠 𝑡 1^{st} row into u L c ​ 1 superscript subscript 𝑢 𝐿 𝑐 1 u_{L}^{c1} of the 7 t ​ h superscript 7 𝑡 ℎ 7^{th} row, while γ 0 ( + ) 78 superscript 78 superscript 𝛾 0 absent \gamma^{0}\stackrel{{\scriptstyle 78}}{{(+)}} transforms d R c ​ 1 superscript subscript 𝑑 𝑅 𝑐 1 d_{R}^{c1} of the 3 r ​ d superscript 3 𝑟 𝑑 3^{rd} row into d L c ​ 1 superscript subscript 𝑑 𝐿 𝑐 1 d_{L}^{c1} of the 5 t ​ h superscript 5 𝑡 ℎ 5^{th} row, doing what the Higgs and γ 0 superscript 𝛾 0 \gamma^{0} do in the standard model.

i		${|}^a{\psi }_i>$	${\Gamma }^{(1,3)}$	$S^{12}$	${\tau }^{23}$	$Y$
		$\mathrm{Octet}\mathrm{of}\mathrm{quarks}$
1	$u_R^{c1}$	$\stackrel{03}{\left(+i\right)}\stackrel{12}{(+)}\left|\stackrel{56}{(+)}\stackrel{78}{(+)}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	1	$\frac{1}{2}$	0	$\frac{2}{3}$
2	$u_R^{c1}$	$\stackrel{03}{\left[-i\right]}\stackrel{12}{[-]}\left|\stackrel{56}{(+)}\stackrel{78}{(+)}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	1	$-\frac{1}{2}$	0	$\frac{2}{3}$
3	$d_R^{c1}$	$\stackrel{03}{\left(+i\right)}\stackrel{12}{(+)}\left|\stackrel{56}{[-]}\stackrel{78}{[-]}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	1	$\frac{1}{2}$	0	$-\frac{1}{3}$
4	$d_R^{c1}$	$\stackrel{03}{\left[-i\right]}\stackrel{12}{[-]}\left|\stackrel{56}{[-]}\stackrel{78}{[-]}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	1	$-\frac{1}{2}$	0	$-\frac{1}{3}$
5	$d_L^{c1}$	$\stackrel{03}{\left[-i\right]}\stackrel{12}{(+)}\left|\stackrel{56}{[-]}\stackrel{78}{(+)}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	-1	$\frac{1}{2}$	$-\frac{1}{2}$	$\frac{1}{6}$
6	$d_L^{c1}$	$\stackrel{03}{\left(+i\right)}\stackrel{12}{[-]}\left|\stackrel{56}{[-]}\stackrel{78}{(+)}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	-1	$-\frac{1}{2}$	$-\frac{1}{2}$	$\frac{1}{6}$
7	$u_L^{c1}$	$\stackrel{03}{\left[-i\right]}\stackrel{12}{(+)}\left|\stackrel{56}{(+)}\stackrel{78}{[-]}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	-1	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{6}$
8	$u_L^{c1}$	$\stackrel{03}{\left(+i\right)}\stackrel{12}{[-]}\left|\stackrel{56}{(+)}\stackrel{78}{[-]}\right||\stackrel{\mathrm{9\hspace{0.33em}10}}{(+)}\stackrel{\mathrm{11\hspace{0.33em}12}}{(-)}\stackrel{\mathrm{13\hspace{0.33em}14}}{(-)}$	-1	$-\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{6}$

Table 4. Table 4: The Yukawa couplings for u − limit-from 𝑢 u- quarks after the break of S ​ O ​ ( 1 , 7 ) × U ​ ( 1 ) 𝑆 𝑂 1 7 𝑈 1 SO(1,7)\times U(1) into S ​ O ​ ( 1 , 3 ) × S ​ U ​ ( 2 ) × U ​ ( 1 ) 𝑆 𝑂 1 3 𝑆 𝑈 2 𝑈 1 SO(1,3)\times SU(2)\times U(1) .

	$I$	$II$	$III$	$IV$	$V$	$VI$	$VII$	$VIII$
$I$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$II$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$III$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$IV$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$V$	$0$	$0$	$0$	$0$	$-\frac{\tilde{g}}{c}{\tilde{A}}_{-}^{23}$	$\frac{\tilde{g}}{\sqrt{2}c}{\tilde{A}}_{-}^{2-}$	$0$	$0$
$VI$	$0$	$0$	$0$	$0$	$-\frac{\tilde{g}}{\sqrt{2}c}{\tilde{A}}_{-}^{2+}$	$-\frac{\tilde{g}}{c}{\tilde{A}}_{-}^{23}$	$0$	$0$
$VII$	$0$	$0$	$0$	$0$	$0$	$0$	$\frac{\tilde{g}}{c}{\tilde{A}}_{-}^{23}$	$\frac{\tilde{g}}{\sqrt{2}c}{\tilde{A}}_{-}^{2-}$
$VIII$	$0$	$0$	$0$	$0$	$0$	$0$	$-\frac{\tilde{g}}{\sqrt{2}c}{\tilde{A}}_{-}^{2+}$	$\frac{\tilde{g}}{c}{\tilde{A}}_{-}^{23}$

Equations

$$W_{NMSSM}=W_{MSSM}{|}_{\mu =0}+\lambda \hat{S}{\hat{H}}_u\cdot {\hat{H}}_d+\frac{\kappa }{3}{\hat{S}}^3,$$

$$P=\{M_0,M_{1/2},A_0,\tan \beta ,\lambda ,\mathrm{sign}(\mu )\}.$$

$$\mathcal{P}(P|D;H)=\mathcal{P}(D|P;H)\frac{\mathcal{P}(P|H)}{\mathcal{P}(D|H)}.$$

$${\mathcal{L}}_i(D,P;H)=\frac{1}{\sqrt{2\pi }{\sigma }_i}\exp ({\chi }_i^2(D,P;H)/2),$$

$$\frac{{\int }_{\mathrm{within}\mathrm{LHC}\mathrm{reach}}\mathcal{P}(p_i|D;H)d{p}_i}{{\int }_{\mathrm{outside}\mathrm{LHC}\mathrm{reach}}\mathcal{P}(p_i|D;H)d{p}_i}=\mathrm{0.57.}$$

$$gA^{\mu}(x)\rightarrow\Lambda^{-1}(x)gA^{\mu}(x)\Lambda(x)-i\Lambda^{-1}(x)\partial^{\mu}\Lambda(x)\;\;\;(\mbox{$g=$ coupling constant})$$

$$\varphi (x)\to \Lambda (x)\varphi (x)$$

$$\displaystyle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rightarrow$$

$$\displaystyle\approx\Lambda^{-1}(x)U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})(\Lambda(x)+\partial^{\rho}\Lambda(x)a\delta^{\rho}_{\nu})$$

$$\displaystyle=\Lambda^{-1}(x)U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\Lambda(x)(1+\partial^{\rho}(\log\Lambda(x))a\delta^{\rho}_{\nu})\approx$$

$$\displaystyle\approx\Lambda^{-1}(x)U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x^{\mu}\;\;y^{\mu}})\Lambda(x)\exp(\partial^{\rho}(\log\Lambda(x))a\delta^{\rho}_{\nu})$$

$${\Lambda }^{-1}(x)(1+ig{A}^{\rho }(x)a{\delta }_{\rho }^{\nu })\Lambda (x)(1+{\partial }^{\rho }(\log \Lambda (x))a{\delta }_{\rho }^{\nu })\approx$$

$$\approx 1+{\Lambda }^{-1}(x)(ig{A}^{\rho }(x)a{\delta }_{\rho }^{\nu })\Lambda (x)+{\partial }^{\rho }(\log \Lambda (x))a{\delta }_{\rho }^{\nu }=$$

$$=1+{\Lambda }^{-1}(x)(ig{A}^{\rho }(x)a{\delta }_{\rho }^{\nu })\Lambda (x)+{\Lambda }^{-1}(x)\Lambda (x)\frac{1}{\Lambda (x)}{\partial }^{\rho }(\Lambda (x))a{\delta }_{\rho }^{\nu }$$

$$=1+i[{\Lambda }^{-1}(x)g{A}^{\rho }(x)\Lambda (x)-i{\Lambda }^{-1}(x){\partial }^{\rho }(\Lambda (x))]a{\delta }_{\rho }^{\nu }$$

$${\Lambda }_{Const}\in \{\Lambda :\text{mbox}{\mathbf{R}}^4\to G|\exists \alpha [\forall x\in \text{mbox}{\mathbf{R}}^4[\Lambda (x)=e^{i\alpha }]]\}$$

$${\Lambda }_{Linear}\in \{\Lambda :\text{mbox}{\mathbf{R}}^4\to G|\exists {\alpha }_{\mu }[\forall x\in \text{mbox}{\mathbf{R}}^4[\Lambda (x)=e^{i{\alpha }_{\mu }{x}^{\mu }}]]\}.$$

$$h_2\stackrel{g\in SO(3)}{⟶}{h}_1(h_1,h_2\in SO(3)/SO(2);$$

$$\{g\in SO(3)|g{h}_2=h_1\}$$

$$\kappa dist^{2}(\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x}),U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!\!\!y}))$$

$$dist^{2}(\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x}),U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!\!\!y}))\stackrel{{\scriptstyle def}}{{=}}$$

$$inf\{dist^{2}(g_{x}\cdot SO(2),U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})g_{y}\cdot SO(2)|$$

$$\mbox{ $g_{x}$ \& $g_{y}$ \footnotesize are reps. of respectively the cosets $\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x})$ \& $\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!\!\!y})$}\}$$

$$U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rightarrow\Lambda^{-1}(x)U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\Lambda(x)\cdot\underbrace{\exp(\partial^{\rho}(\log\Lambda(x))\cdot a\delta^{\nu}_{\rho}}_{\mbox{\footnotesize gradient part of transf.}},$$

$$U=U(g,k,h)\text{mbox}withg\in G/K,k\in K/H,h\in H.$$

$$E_{c_5}\approx -3\frac{1}{2}{\left(\frac{2}{3}{\alpha }_c\right)}^2\frac{m_{q_5}}{2}{c}^2,r_{c_5}\approx \frac{\hslash c}{\frac{2}{3}{\alpha }_c\frac{m_{q_5}}{2}{c}^2}.$$

$$<\sigma v{>}_{q\bar{q}}$$

$$<\sigma v{>}_{gg}$$

$$<\sigma v{>}_{c_5}$$

$$<\sigma v{>}_{(q\bar{q}{)}_b}$$

$${\sigma }_T$$

$$a^{-3}\frac{d(a^3{n}_{q_5})}{dt}$$

$$a^{-3}\frac{d(a^3{n}_{c_5})}{dt}$$

$$a^{-3}\frac{d(a^3{n}_{q_5})}{dt}$$

$${\rho }_{dm}$$

Full text

1. Preface
2. DISCUSSION SECTION
3. PRESENTATION OF VIRTUAL INSTITUTE OF ASTROPARTICLE PHYSICS AND BLED 2009 WORKSHOP VIDEOCONFERENCES

Preface

C. Balázs and D. Carter csaba.balazs@sci.monash.edu.audaniel.carter@sci.monash.edu.auBrookes Institute for Advanced Studies,

Bøgevej 6, 2900 Hellerup, DenmarkDepartment of Physics, FMF, University of Ljubljana,

Jadranska 19, 1000 LjubljanaUniversidad de Zacatecas

Ap. Postal 636, Suc. 3 Cruces, C. P. 98064

Zacatecas, Zac., MéxicoUniversidad de Zacatecas, Apartado Postal 636, Suc. 3 Cruces

Zacatecas 98064, Zac., México

URL: http://planck.reduaz.mx/ṽaleri/

e-mail address: valeri@fisica.uaz.edu.mxDepartamento de Física, Escuela Superior de Física y Matemáticas, I.P.N.,

U. P. ”Adolfo López Mateos”. C. P. 07738, México, D.F., México1Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

2 Centre for Cosmoparticle Physics ”Cosmion” 115409 Moscow, Russia

3 APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, FranceSCAT, Oslo, Norway1 Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000

2 Department of Physics, Niels Bohr Institute,

Blegdamsvej 17, Copenhagen, DK-2100Department of Physics, FMF, University of Ljubljana,

Jadranska 19, 1000 Ljubljana, Slovenia

http://www.fmf.uni-lj.si14U

155 E 34 Street

New York, NY 10016 (http://viavca.in2p3.fr/bled_09.html)1 Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000

2 Department of Physics, Niels Bohr Institute,Blegdamsvej 17, Copenhagen, DK-2100Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000Centre for Cosmoparticle Physics Cosmion, 115409 Moscow, Russia;

Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, France1Departamento de Física, Escuela Superior de Física y Matemáticas, I.P.N.,

U. P. ”Adolfo López Mateos”. C. P. 07738, México, D.F., México

2Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 100014U

155 E 34 Street

New York, NY 10016 1Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

2 Centre for Cosmoparticle Physics ”Cosmion” 125047 Moscow, Russia

3 APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, France

The series of workshops on ”What Comes Beyond the Standard Model?” started in 1998 with the idea of organizing a real workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. The picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks, was chosen to stimulate the discussions.

The idea was successful and has developed into an annual workshop, which is taking place every year since 1998. This year the twelfth workshop took place. Very open-minded and fruitful discussions have become the trade-mark of our workshop, producing several published works. It takes place in the house of Plemelj, which belongs to the Society of Mathematicians, Physicists and Astronomers of Slovenia.

In this twelfth workshop, which took place from 14th to 24th of July 2009, we were discussing several topics, most of them presented in this Proceedings mainly as talks and partly in the discussion section. The main topic was this time the ”approach unifying spin and charges”, proposed by Norma, as the new way beyond the ”standard model of the electroweak and colour interactions”, accompanied by the critical discussions about the chance which this theory has to answer the open questions which the ”standard model” leaves unanswered. Proposing the mechanism for generating families, this ”approach” is predicting the fourth family to be possibly seen at LHC and the stable fifth family which have a chance to form the dark matter. The discussions of the questions: Is the ”approach unifying spin and charges” the right way beyond the standard model? Are the clusters of the fifth family members alone what constitute the dark matter? Can the fifth family baryons explain the observed properties of the dark matter with the direct measurements included? What if such a scenario is not confirmed by the direct measurements? What are next steps in evaluating properties of the predicted Yukawa couplings? Can we find the way out (besides by a choice of appropriate boundary conditions) of the ”no go theorem” of Witten, saying that there is a little chance for these kind of theories (to which also the ”approach unifying spins and charges” belong), since the masses of the fermions, predicted by these theories should be too high?

Talks and discussions in our workshop are not at all talks in the usual way. Each talk or discussions lasted several hours, divided in two hours blocks, with a lot of questions, explanations, trials to agree or disagree from the audience or a speaker side. Most of talks are ”unusual” in the sense that they are trying to find out new ways of understanding and describing the observed phenomena. Although we always hope that the discussions will in the very year proceedings manifest in the progress published in the corresponding proceedings, it happens many a time that the topics appear in the next or after the next year proceedings. This happened also in this year. Therefore neither the discussion section nor the talks published in this proceedings, manifest all the discussions and the work done in this workshop.

Several videoconferences were taking place during the Workshop on various topics. It was organized by the Virtual Institute for Astrophysics (www.cosmovia.org) of Maxim Khlopov with able support by Didier Rouable. We managed to have ample discussions. The transparent and very systematic overview of what does the LHC, which is in these days starting again, expect to measure in the near future, was presented by John Ellis, who stands behind the theoretical understanding of the LHC. The talks and discussions can be found online at

http://viavca.in2p3.fr/bled_09.html

The organizers thank all the participants for fruitful discussions and talks.

Let us present the starting point of our discussions: What science has learned up to now are several effective theories which, after making several starting assumptions, lead to theories (proven or not to be consistent in a way that they do not run into obvious contradictions), and which, some of them, are within the accuracy of calculations and experimental data, (still) in agreement with the observations, the others might be tested in future, and might answer at least some of the open questions, left open by the scientific community accepted effective theories. It is a hope that the law of Nature is ”simple” and ”elegant”, on one or another way, manifesting symmetries or complete randomness, whatever the ”elegance” and ”simplicity” might mean (as few assumptions as possible?, very simple starting action?), while the observed states are usually not, suggesting that the ”effective theories, laws, models” are usually very complex.

Let us write in this workshop discussed open questions which the two standard models (the electroweak and the cosmological) leave unanswered:

• •

Why has Nature made a choice of four (noticeable) dimensions while all the others, if existing, are hidden? And what are the properties of space-time in the hidden dimensions?

• •

How could ”Nature make the decision” about breaking of symmetries down to the noticeable ones, if coming from some higher dimension d?

• •

Why is the metric of space-time Minkowskian and how is the choice of metric connected with the evolution of our universe(s)?

• •

Why do massless fields exist at the low energy regime at all? Where does the weak scale come from?

• •

Why do only left-handed fermions carry the weak charge? Why does the weak charge break parity?

• •

Where do families come from?

• •

What is the origin of Higgs fields? Where does the Higgs mass come from?

• •

Can all known elementary particles be understood as different states of only one particle, with a unique internal space of spins and charges?

• •

Can one find a loop hole through the Witten’s ”no-go theorem” and give them back a chance to the Kaluza-Klein-like theories to be the right way beyond the ”standard model of the electroweak and colour interaction”?

• •

How can all gauge fields (including gravity) be unified (and quantized)?

• •

What is our universe made out of besides the (mostly) first family baryonic matter?

• •

What is the role of symmetries in Nature?

We have discussed these and other questions for ten days. The reader can see our progress in some of these questions in this proceedings. Some of the ideas are treated in a very preliminary way. Some ideas still wait to be discussed (maybe in the next workshop) and understood better before appearing in the next proceedings of the Bled workshops.

The organizers are grateful to all the participants for the lively discussions and the good working atmosphere.

*Norma Susana Mankoč Borštnik, Holger Bech Nielsen,

Maxim Yu. Khlopov, Dragan Lukman* Ljubljana, December 2009

11institutetext: School of Physics, Monash University,

Melbourne Victoria 3800, Australia

Likelihood Analysis of the Next-to-minimal Supergravity Motivated Model

C. Balázs and D. Carter csaba.balazs@sci.monash.edu.audaniel.carter@sci.monash.edu.auBrookes Institute for Advanced Studies,

Bøgevej 6, 2900 Hellerup, DenmarkDepartment of Physics, FMF, University of Ljubljana,

Jadranska 19, 1000 LjubljanaUniversidad de Zacatecas

Ap. Postal 636, Suc. 3 Cruces, C. P. 98064

Zacatecas, Zac., MéxicoUniversidad de Zacatecas, Apartado Postal 636, Suc. 3 Cruces

Zacatecas 98064, Zac., México

URL: http://planck.reduaz.mx/ṽaleri/

e-mail address: valeri@fisica.uaz.edu.mxDepartamento de Física, Escuela Superior de Física y Matemáticas, I.P.N.,

U. P. ”Adolfo López Mateos”. C. P. 07738, México, D.F., México1Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

2 Centre for Cosmoparticle Physics ”Cosmion” 115409 Moscow, Russia

3 APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, FranceSCAT, Oslo, Norway1 Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000

2 Department of Physics, Niels Bohr Institute,

Blegdamsvej 17, Copenhagen, DK-2100Department of Physics, FMF, University of Ljubljana,

Jadranska 19, 1000 Ljubljana, Slovenia

http://www.fmf.uni-lj.si14U

155 E 34 Street

New York, NY 10016 (http://viavca.in2p3.fr/bled_09.html)1 Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000

2 Department of Physics, Niels Bohr Institute,Blegdamsvej 17, Copenhagen, DK-2100Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 1000Centre for Cosmoparticle Physics Cosmion, 115409 Moscow, Russia;

Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, France1Departamento de Física, Escuela Superior de Física y Matemáticas, I.P.N.,

U. P. ”Adolfo López Mateos”. C. P. 07738, México, D.F., México

2Department of Physics, FMF, University of Ljubljana,

Jadranska 19,Ljubljana, 100014U

155 E 34 Street

New York, NY 10016 1Moscow Engineering Physics Institute (National Nuclear Research University), 115409 Moscow, Russia

2 Centre for Cosmoparticle Physics ”Cosmion” 125047 Moscow, Russia

3 APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, France

Abstract

In anticipation of data from the Large Hadron Collider (LHC) and the potential discovery of supersymmetry, we calculate the odds of the next-to-minimal version of the popular supergravity motivated model (NmSuGra) being discovered at the LHC to be 4:3 (57 %). We also demonstrate that viable regions of the NmSuGra parameter space outside the LHC reach can be covered by upgraded versions of dark matter direct detection experiments, such as super-CDMS, at 99 % confidence level. Due to the similarities of the models, we expect very similar results for the constrained minimal supersymmetric standard model (CMSSM).

Abstract

We investigate the possibility that the dark matter consists of clusters of the heavy family quarks and leptons with zero Yukawa couplings to the lower families. Such a family is predicted by the approach unifying spin and charges as the fifth family. We make a rough estimation of properties of baryons of this new family members, of their behaviour during the evolution of the universe and when scattering on the ordinary matter and study possible limitations on the family properties due to the cosmological and direct experimental evidences. This paper will be published in October 2009 in Phys. Rev D. We add it here since in the discussion sections the derivations and conclusions of this paper are commented.

Abstract

We present a realization of a quantum field theory, envisaged many years ago by Gelfand, Tsetlin, Sokolik and Bilenky. Considering the special case of the $(1/2,0)\oplus(0,1/2)$ field and developing the Majorana construct for neutrino we show that a fermion and its antifermion can have the same properties with respect to the intrinsic parity ($P$) operation. The transformation laws for $C$ and $T$ operations have also been given. The construct can be applied to explanation of the present situation in neutrino physics. The case of the $(1,0)\oplus(0,1)$ field is also considered.

Abstract

We derive relativistic equations for charged and neutral spin particles. The approach for higher-spin particles is based on generalizations of the Bargmann-Wigner formalism. Next, we study , what new physical information can the introduction of non-commutativity give us. Additional non-commutative parameters can provide a suitable basis for explanation of the origin of mass.

Abstract

We report the analysis on charged fermion masses and quark mixing, within the context of a non-supersymmetric $SU(3)$ gauged family symmetry model with hierarchical one loop radiative mass generation mechanism for light fermions, mediated by the massive bosons associated to the $SU(3)$ family symmetry that is spontaneously broken, meanwhile the top and bottom quarks as well as the tau lepton are generated at tree level by the implementation of Dirac See-saw mechanisms through the introduction of new vector like fermions. A quantitative analysis shows that this model is successful to accommodate a realistic spectrum of masses and mixing in the quark sector as well as the charged lepton masses. Furthermore, the above scenario enable us to suppress within current experimental bounds the tree level $\Delta F=2$ processes for $K^{o}-\bar{K^{o}}$ and $D^{o}-\bar{D^{o}}$ meson mixing mediated by these extra horizontal gauge bosons.

Abstract

Positive results of dark matter searches in experiments DAMA/NaI and DAMA/ LIBRA taken together with negative results of other groups can imply nontrivial particle physics solutions for cosmological dark matter. Stable particles with charge -2 bind with primordial helium in O-helium ”atoms” (OHe), representing a specific Warmer than Cold nuclear-interacting form of dark matter. Slowed down in the terrestrial matter, OHe is elusive for direct methods of underground Dark matter detection like those used in CDMS experiment, but its low energy binding with nuclei can lead to annual variations of energy release in the interval of energy 2-6 keV in DAMA/NaI and DAMA/LIBRA experiments. Schrodinger equation for system of nucleus and OHe is considered and reduced to an equation of relative motion in a spherically symmetrical potential, formed by the Yukawa tail of nuclear scalar isoscalar attraction potential, acting on He beyond the nucleus, and dipole Coulomb repulsion between the nucleus and OHe at distances from the nuclear surface, smaller than the size of OHe. The values of coupling strength and mass of meson, mediating scalar isoscalar nuclear potential, are rather uncertain. Within these uncertainties and in the approximation of rectangular potential wells we find a range of these parameters, at which the sodium and/or iodine nuclei have a few keV binding energy with OHe. At nuclear parameters, reproducing DAMA results, the energy release predicted for detectors with chemical content other than NaI differ in the most cases from the one in DAMA detector. In particular, it is shown that in the case of CDMS germanium state has binding energy with OHe beyond the range of 2-6 keV and its formation should not lead to ionization in the energy range of DAMA signal. Due to dipole Coulomb barrier, transitions to more energetic levels of Na(I)+OHe system with much higher energy release are suppressed in the correspondence with the results of DAMA experiments. The proposed explanation inevitably leads to prediction of abundance of anomalous Na and I, corresponding to the signal, observed by DAMA.

Abstract

Random Dynamics is an anti-grand unification project, based on the assumption that at a fundamental scale Nature is not necessarily ”simple”, but probably enormously complicated and is most simply described in terms of randomness. The ambition is to ”derive” all the known physical laws as an almost unavoidable consequence of a random fundamental ”world machinery”, which is a very general, random mathematical structure, which contains non-identical elements and some set-theoretical notions.

But how can one extract anything from something very general and random, which is not even well described in detail?

Abstract

One step towards realistic Kaluza-Klein-like theories and a loop hole through the Witten’s ”no-go theorem” is presented for cases which we call ”an effective two dimensionality” cases: We present the case of a spinor in $d=(1+5)$ compactified on an (formally) infinite disc with the zweibein which makes a disc curved on $S^{2}$ and with the spin connection field which allows on such a sphere only one massless spinor state of a particular charge, which couples the spinor chirally to the corresponding Kaluza-Klein gauge field. In refs. dhn1hnkk06 ; dhn1hn07 we achieved masslessness of spinors with the appropriate choice of a boundary on a finite disc, in this paper the masslessness is achieved with the choice of a spin connection field on a curved infinite disc. In $d=2$, namely, the equations of motion following from the action with the linear curvature leave spin connection and zweibein undetermined dhn1dhnproc04 .

Abstract

The ”approach unifying spin and charges” sn2snmb:n92 ; sn2snmb:pn06 ; sn2snmb:gmdn07 ; sn2snmb:n07 offers the explanation for all the internal degrees of freedom—the spin, all the charges and the family quantum number—by introducing two kinds of the spin, the Dirac kind and the second kind anticommuting with the Dirac one. It offers a new way of understanding the properties of quarks and leptons: their charges and their connection to the corresponding gauge fields and their appearance in families and their Yukawa couplings. In this talk I present the way from a simple starting Lagrange density for a spinor—carrying in $d=1+13$ only two kinds of the spin, no charges, and interacting with the vielbeins and the two kinds of the spin connection fields—to the effective Lagrangean, postulated by the ”standard model of the electroweak and colour interactions”. The way of breaking the starting symmetries determines the observed properties of the families of spinors and of the gauge fields, predicting that there are four families at low energies and that a much heavier fifth family with zero Yukawa couplings to the lower four families, might, by forming baryons in the evolution of the universe, contribute a major part to the dark matter. I comment on properties of the Yukawa couplings following from the simple starting Lagrangean, as well as on the possibility that the starting Lagrangean for spin connection and vielbeins fields linear in the curvature might lead to the observable properties of the gauge fields and their couplings to almost massless observed fermions.

Abstract

There are too many aspects of science, particularly quantum mechanics, that should be obvious but are quite unclear to too many people (especially physicists and journalists who seem to enjoy flaunting their confusions). We summarize and analyze these here; detailed discussion and proofs are well known.

Abstract

A short report on talks and discussions taken place at Bled through the video conferences, organized by the Virtual Institute for Astrophysics (www.cosmovia.org) is presented. Talks and discussions can be found on http://viavca.in2p3.fr/bled_09.html. The list of open questions proposed for wide discussions with the use of VIA facility is added.

Abstract

The ”approach unifying spin and charges”, proposed by Norma Susana Mankoč Borštnik gn2n09 ; gn2pn06 ; gn2gmdn08 , predicts four families, which are connected with the (non zero) Yukawa couplings. The masses of the fourth family quarks lie above a few $100$ GeV/$c^{2}$, the masses of the fourth family leptons are at around $100$ GeV/$c^{2}$ or above. The masses of quarks might be low enough to be possibly measured at the LHC gn2gmdn08 . The approach predicts also the stable fifth family (with no Yukawa couplings to the lower four families), which is the candidate to form the main part of the dark matter. The work done by Gregor Bregar and Norma Susana Mankoč Borštnik gn2gn assumes that the neutron is the lightest fifth family baryon and the neutrino the lightest fifth family lepton. Following the evolution of the fifth family members in the expanding universe, and analysing carefully the interaction of the fifth family neutrons and neutrinos with the ordinary matter in the direct measurements of the DAMA and the CDMS experiments and in other published measurements which could concern our fifth family members as the dark matter constituents and accordingly their properties, the authors of the paper Phys. Rev. D 80, 083534 (2009) predict that the fifth family quarks with the masses of a few $100$ TeV/$c^{2}$ and the fifth family neutrinos with the mass of a few TeV/$c^{2}$ are the candidates for forming the dark matter. This is true also for not too large interval of matter-antimatter asymmetry of the fifth family baryons (which could contradict the measured dark matterdensity) Possible weak points pf the evaluations in the work gn2gn are discussed bellow by Gregor and Norma.

Abstract

The ”approach unifying spin and charges”, proposed by Norma Susana Mankoč Borštnik mkd2n09 ; mkd2pn06 ; mkd2gmdn08 ; mkd2gn predicts the stable fifth family (with no Yukawa couplings to the lower four families). The conclusion on stability of this family is strongly motivated in this approach and the extensive study of possible candidates for the dark matter is challenging.

In view of the uncertainty of fifth family masses all the possible variants for the lightest stable particle can be considered following the methods, developed in mkd2N ; mkd2Q ; mkd2FK ; mkd2FKS . The possibility of stable charged leptons and quarks is generally in serious trouble, related with inevitable presence of stable positively charged species that behave as anomalous isotopes of hydrogen. However there is one exception. It is the solution of composite dark matter, which assumed an excess of -2 charged species, bound in atom like systems with He nuclei that formed in primordial nucleosynthesis. This O-helium (OHe) nuclear interacting form of dark matter was shown to avoid any direct contradiction with experimental constraints mkd2I ; mkd2Q ; mkd2FKS ; mkd2KK . It provides Warmer than Cold Dark Matter scenario, can explain the excess of positron annihilation line observed by INTEGRAL and can resolve the puzzles of direct and indirect dark matter searches. It was shown that electroweak $SU(2)_{ew}$ sphaleron transitions in very early Universe can provide relationship between the observed baryon asymmetry and excess of -2 charged species over their antiparticles, if these species have nontrivial $SU(2)_{ew}$ charges. If sphaleron transitions are possible for the fifth family members, predicted by the ”approach unifying spin and charges” of N.S.M.B. and having nontrivial $SU(2)_{ew}$ charges, and if their masses assure that $\bar{u}_{5}\bar{u}_{5}\bar{u}_{5}$ is the lightest stable fifth family antibaryon, the excess of $\bar{u}_{5}$ over $u_{5}$ can be generated in the early Universe and OHe composite dark matter scenario with $\bar{u}_{5}\bar{u}_{5}\bar{u}_{5}$ constituent can be realized. For highly improbable masses of the fifth family quarks at around $300$ GeV/$c^{2}$, such scenario can reproduce all the features of composite dark matter scenario. For case of quarks with the masses of a few $100$ TeV/$c^{2}$ that are assumed more realistic for the ”approach” some of these features still hold true, with the lack of explanation for the excess of positron annihilation line and of anomalies in spectra of cosmic high energy electrons and positrons. These astrophysical data may not, however, require dark matter solution and can be explained by natural astrophysical sources.

The problems of composite dark matter solution for the puzzles of direct dark matter searches and of realization of this scenario with the use of stable fifth family are discussed.

Abstract

The ”approach unifying spin and charges”, proposed by Norma Susana Mankoč Borštnik, predicts two kinds of the Yukawa couplings. One kind distinguishes on the tree level only among the members of one family (among the u-quark, d-quark, neutrino and electron), while the other kind distinguishes only among the families. Long discussions at the present workshop between Norma and Albino lead to the first step of collaboration presented in this contribution: to a toy model with evaluated contributions bellow the tree level, done by Albino.

Abstract

Virtual Institute of Astroparticle Physics (VIA) has evolved in a unique multi-functional complex, combining various forms of collaborative scientific work with programs of education on distance. The activity on VIA website includes regular videoconferences with systematic basic courses and lectures on various issues of astroparticle physics, participation at distance in various scientific meetings and conferences, library of their records and presentations, a multilingual forum. VIA virtual rooms are open for meetings of scientific groups and for individual work of supervisors with their students. The format of a VIA videoconferences was effectively used in the program of Bled Workshop to discuss the open questions of physics beyond the standard model.

0.1 Introduction

Supersymmetry is one of the most robust theories that can solve outstanding problems of the standard model (SM) of elementary particles. The theory naturally explains the dynamics of electroweak symmetry breaking while preserving the hierarchy of fundamental energy scales. It also readily accommodates dark matter, the asymmetry between baryons and anti-baryons, the unification of gauge forces, gravity, and more. But if supersymmetry is the solution to the problems of the standard model, then its natural scale is the electroweak scale, and it is expected to be observed in upcoming experiments, most notably the CERN Large Hadron Collider (LHC). In this work, we will attempt to determine, quantitatively, what the chances are that this may occur for the simplified case of a constrained supersymmetric model.

The minimal supersymmetric extension of the standard model (MSSM) faces several significant issues, such as the little hierarchy problem Giudice:2008bi and the so-called $\mu$ problem Kim:1983dt . However extensions of the MSSM by gauge singlet superfields not only resolve the $\mu$ problem, but can also ameliorate the little hierarchy problem Dermisek:2005ar ; BasteroGil:2000bw ; Gunion:2008kp . In the next-to-minimal MSSM (NMSSM), the $\mu$ term is dynamically generated and no dimensionful parameters are introduced in the superpotential (other than the vacuum expectation values that are all naturally weak scale), making the NMSSM a truly natural model (see Balazs:2008ph for references).

For the sake of simplicity and elegance, we choose to impose minimal supergravity-motivated (mSuGra) boundary conditions; specifically, universality of sparticle masses, gaugino masses, and tri-linear couplings at the grand unification theory (GUT) scale. Thus we define the next-to-minimal supergravity-motivated (NmSuGra) model.

Using a Bayesian likelihood analysis, we identify the regions in the parameter space of the NmSuGra model that are preferred by the present experimental limits from various collider, astrophysical, and low-energy measurements. Thus we show that, given current experimental constraints, the favored parameter space can be detected by a combination of the LHC and an upgraded CDMS at the 95 % confidence level.

In the next section we define the next-to-minimal version of the supergravity motivated model (NmSuGra). Then, in Section 0.3, we summarize the main concepts of Bayesian inference that we use in this work. Section 0.4 contains the numerical results of our likelihood analysis, and Section 0.5 gives the outlook for the experimental detection of NmSuGra.

0.2 The next-to-minimal supergravity motivated model

The next-to-minimal supersymmetric model (NMSSM) is defined by the superpotential

[TABLE]

where $W_{MSSM}|_{\mu=0}$ is the MSSM superpotential containing only Yukawa terms and having $\mu$ set to zero Ellwanger:2005dv , and $\hat{S}$ is a standard gauge singlet with dimensionless couplings $\lambda$ and $\kappa$. The couplings $\lambda$, $\kappa$, and $y_{i}$ are dimensionless, and $\hat{X}\cdot\hat{Y}=\epsilon_{\alpha\beta}\hat{X}^{\alpha}\hat{Y}^{\beta}$ with the fully antisymmetric tensor normalized as $\epsilon_{11}=1$.

We use supergravity motivated boundary conditions to parametrize the soft masses and tri-linear couplings. Defining a constrained version of the NMSSM, we assume unification of the gaugino masses to $M_{1/2}$, the sfermion and Higgs masses to $M_{0}$, and the tri-linear couplings to $A_{0}$ at the grand unified theory (GUT) scale where the three standard gauge couplings meet $g_{1}=g_{2}=g_{3}=g_{GUT}$. After electroweak symmetry breaking, our constrained NMSSM model has only five free parameters and a sign. Defining $\tan\beta=\langle H_{u}\rangle/\langle H_{d}\rangle$, the parameters of the next-to-minimal supergravity motivated model (NmSuGra) are

[TABLE]

Furthermore, from Eq.1 we see that when the singlet acquires a vev, the MSSM $\mu$ term is dynamically generated as $\mu=\lambda\langle S\rangle$, and thus the NMSSM naturally solves the $\mu$ problem.

Different constrained versions of the NMSSM have been studied in the recent literature Djouadi:2008yj ; Hugonie:2007vd ; Belanger:2005kh ; Cerdeno:2007sn ; Djouadi:2008uw . In the spirit of the CMSSM/mSuGra, we adhere to universality and use only $\lambda$ to parametrize the singlet sector. This way, we keep all the attractive features of the CMSSM/mSuGra while the minimal extension alleviates problems rooted in the MSSM, making the NMSSM a more natural model.

As we have shown in our previous work Balazs:2008ph , NmSuGra phenomenology bears a high similarity to the minimal supergravity motivated model. The most significant departures from a typical mSuGra model are the possibility of a singlino-dominated neutralino and the extended Higgs sector, which may provide new resonance annihilation channels and Higgs decay channels, potentially weakening the mass limit from LEP.

0.3 Bayesian inference

Since several excellent papers have appeared on this subject recently Feroz:2008wr ; Trotta:2008bp ; AbdusSalam:2009qd , in this section, we summarize the concepts of Bayesian inference that we use in our analysis in a compact fashion. Our starting hypothesis $H$ is the validity of the NmSuGra model. The conditional probability $\mathcal{P}(P|D;H)$ quantifies the validity of our hypothesis by giving the chance that the NmSuGra model reproduces the available experimental data $D$ with its parameters set to values $P$. When this probability density is integrated over a region of the parameter space it yields the posterior probability that the parameter values fall into the given region.

Bayes’ theorem provides us with a simple way to calculate the posterior probability distribution as

[TABLE]

Here $\mathcal{P}(D|P;H)$ is the likelihood that the data is predicted by NmSuGra with a specified set of parameters. The a-priori distribution of the parameters within the theory $\mathcal{P}(P|H)$ is fixed by purely theoretical considerations independently from the data. The evidence $\mathcal{P}(D|H)$ gives the probability of the hypothesis in terms of the data alone, equivalent to integrating out the parameter dependence.

For statistically independent data the likelihood is the product of the likelihoods for each observable. For normally-distributed measurements the likelihood is given by:

[TABLE]

where the exponents $\chi_{i}^{2}(D,P;H)/2=(d_{i}-t_{i}(P;H))^{2}/2\sigma_{i}^{2}$ are defined in terms of the experimental data $D=\{d_{i}\pm\sigma_{i,e}\}$ and theoretical predictions $T=\{t_{i}\pm\sigma_{i,t}\}$ for these measurables. Independent experimental and theoretical uncertainties combine into $\sigma_{i}^{2}=\sigma_{i,e}^{2}+\sigma_{i,t}^{2}$. In cases when the experimental data only specify a lower (or upper) limit, we replace the Gaussian likelihood with a likelihood based on the error function. Often, the profile if the likelihood distribution is used for statistical inference, however this disregards information about the structure of the parameter space itself. In Bayesian statistics we use the so-called marginalized probability, given by the integral of the posterior probability density over all parameter space except the quantity of interest.

0.4 Likelihood analysis of NmSuGra

Our main aim is to calculate the posterior probability distributions for the five continuous parameters of NmSuGra and check the consistency of the model against available experimental data. To this end, we use the publicly available computer code NMSPEC Ellwanger:2006rn to calculate the spectrum of the superpartner masses and their physical couplings from the model parameters given in Eq. (2). Then, we use NMSSMTools 2.1.0 and micrOMEGAs 2.2 Belanger:2006is to calculate the abundance of neutralinos ($\Omega h^{2}$) Komatsu:2008hk , the spin-independent neutralino-proton elastic scattering cross section ($\sigma_{SI}$) Ahmed:2008eu , the NmSuGra contribution to the anomalous magnetic moment of the muon ($\Delta a_{\mu}$) Jegerlehner:2009ry , and various b-physics related quantities Barberio:2008fa ; Artuso:2009jw . We also impose limits from negative searches for the sparticle masses AbdusSalam:2009qd , applying a lower lightest Higgs mass limit where appropriate, as shown in Barger:2006dh . Among the standard input parameters, $m_{b}(m_{b})=4.214$ GeV and $m_{t}^{pole}=171.4$ GeV are used.

Using the above specified tools, we generate theoretical predictions for NmSuGra in the following part of its parameter space: $0<M_{0}<5~{}{\rm TeV},0<M_{1/2}<2~{}{\rm TeV},-3~{}{\rm TeV}<A_{0}<5~{}{\rm TeV},0<\tan\beta<60,10^{-5}<\lambda<0.6,{\rm sign}(\mu)>0$. In this work, we only consider the positive sign of $\mu$ because, similarly to mSuGra Feroz:2008wr , the likelihood function is suppressed by $\Delta a_{\mu}$ and $B(b\to s\gamma)$ in the negative $\mu$ region. We calculate posterior probabilities using two methods: a uniform random scan, and Markov Chain Monte Carlo (implementing the metropolis algorithm) as described in Baltz:2006fm , which is significantly more efficient but marginally less consistent. In general, the two methods are in good agreement.

0.4.1 Posterior probabilities

We now turn to our numerical results in Figure 1, which shows the posterior probability marginalized to different pairs of NmSuGra input parameters. In the left frame we show the posterior probability marginalized to the plane of the common scalar and gaugino masses, $M_{0}$ vs. $M_{1/2}$. The slepton co-annihilation region combined with Higgs resonance corridors, at low $M_{0}$ and low to moderate $M_{1/2}$ supports most of the probability. This region is clearly separated from the focus point at high $M_{0}$ and moderate to high $M_{1/2}$, a large part of which falls in the 68 % confidence level. While the contribution from $\Delta a_{\mu}$ strongly suppresses the likelihood at higher values of $M_{0}$ and $M_{1/2}$, the volume of the focus point region is quite large, contrasted with the highly-sensitive sfermion coannihilation region. This shifts the expectation for $M_{0}$ much higher than its likelihood distribution might suggest, and implies that it would probably not be reasonable to confine $M_{0}$ to low values. Most of the focus point happens at high $\tan\beta(\sim 50)$ where the traditional focus point region merges with multiple Higgs resonance corridors creating very wide regions consistent with WMAP.

In $M_{1/2}$, there appears a narrow region close to 150 GeV that corresponds to neutralinos resonantly self-annihilating via the lightest scalar Higgs boson in the s-channel. This ’sweet spot’ emerges as a combined high-likelihood and volume effect. Part of this region is allowed in NmSuGra due to the somewhat relaxed mass limit by LEP on the lightest Higgs. The narrowness of this strip correlates with the smallness of the lightest Higgs width.

The top right frame of Figure 1 shows the distribution of the posterior probability in the $M_{0}$ vs. $\tan\beta$ frame. This makes it clear that most of the probable points are carried by Higgs resonant corridors toward higher $\tan\beta$, and the sfermion co-annihilation, due to its narrowness in $M_{0}$, falls only in the 95 % confidence, but is outside the 68 % region. The exception is a minute corner of the parameter space at very low $M_{0}$, $M_{1/2}$, and $\tan\beta\sim 10$ where all theoretical results conspire to match experiment, raising the sfermion co-annihilation region into the 68 % confidence region. At the opposite, high $M_{0}$ and $\tan\beta$ corner multiple Higgs resonances combined with neutralino-chargino co-annihilation in the focus point lead to substantial contribution to the total probability. A similar plot shows that positive values of $A_{0}$ are preferred over negative ones, because Higgs resonance annihilation occurs overwhelmingly at low to moderately positive values of $A_{0}$, and that $\lambda$ has little impact on the posterior.

0.5 Experimental detection of NmSuGra

We examine prospects of NmSuGra being detected at the LHC by plotting the posterior probability marginalized to the masses of relevant sparticles in Figure 2. Here we see that part of the NmSuGra parameter space, specifically the focus point, is out of the reach of the LHC, as shown by the posterior probability distribution of the gluino mass. In the mSuGra model the LHC is able to reach about 3 TeV gluinos with 100 fb*-1* luminosity, provided the model has low $M_{0}$ Baer:2003wx . In the focus point this reach is reduced to about 1.75 TeV.

In the lower left frame of Figure 2 shows that the lighter stop is also expected to be heavier than the likelihood alone would suggest. Even the sharp peak at low values in the stop likelihood function is overwhelmed due to the minute volume of the parameter space it occupies.

While the LHC will not be able to cover the full viable NmSuGra parameter space, fortunately a large part of the remaining region will be accessible to direct detection, measuring the spin-independent neutralino-nucleon elastic recoil cross section, $\sigma_{SI}$. From several of these experiments, we single out CDMS as the most illustrative example. Figure 3 shows the posterior probability density marginalized to the plane of $\sigma_{SI}$ and the lightest neutralino mass.

This plot clearly shows that direct detection experiments can play a pivotal role in discovering or ruling out simple constrained supersymmetric scenarios. Even a 25 kg CDMS will reach a substantial part of the focus point region, complementing the LHC.

In the possession of the above results, we can quantify the chances for the discovery of NmSuGra at the LHC by calculating the ratio of posterior probabilities inside and outside the reach of the LHC:

[TABLE]

According to this the odds of finding NmSuGra at the LHC are 4:3 (assuming, of course, that the model is chosen by Nature). If we then include the reach of a ton equivalent of CDMS (CDMS1T), the NmSuGra model lies within the combined reach of the LHC and CDMS1T at 99 percent confidence level. This result strongly underlines the complementarity of collider and direct dark matter searches.

0.6 Conclusions

The next-to-minimal supergravity motivated model is one of the more compelling models for physics beyond the standard model due to its naturalness and simplicity. In this work we applied a thorough statistical analysis to NmSuGra based on numerical comparisons with present experimental data. Using Bayesian inference we find that the LHC and future CDMS limits cover the viable NmSuGra parameter region at 99 % confidence level, underlining the complementarity of these approaches to discovering new physics at the TeV scale. Thanks to the similarity between our model and the CMSSM, we expect these conclusions to be broadly valid in that model as well. However, this poses a challenge to the LHC experimentalists to disentangle these models.

Acknowledgements

This research was funded by the Australian Research Council under Project ID DP0877916.

0.7 Introduction

The Multiple Point Principle (MPP) abel ; nonabel states that Nature takes on intensive parameter values (coupling constant values) that correspond to a maximally degenerate vacuum where these degenerate vacua all have essentially vanishing cosmological constants. The MPP was originally applied in the context of lattice gauge theory for the purpose of predicting the values of the three gauge coupling constants for the Standard Model Group (SMG). This pursuit entailed among other things a way in which to characterize the possible phases of of a non-simple gauge group such as the SMG. Having such a phase classification scheme, it was subsequently necessary to parameterize the action in such a way that these various phases could be provoked. In such an action parameter space our claim is that Nature takes on parameter values corresponding to the the point (or surface) — the multiple point — at which a maximum number phases come together.

The presentation at the 12th International Workshop “What Comes Beyond the Standard Model” in Bled (2009) was an attempt at a somewhat comprehensive review of the original implementation of the MPP. I was very happy that my talks were interrupted by so many questions. Many of these were about the formal way that different possible phases of the SMG are distinguished. So rather than a review of MPP I shall in this proceedings contribution address the questions posed. These were centered around the way in which the various possible phases for a non-simple gauge group such as the SMG are characterized in terms of subgroups $K\subseteq SMG$ and invariant subgroups $H\triangleleft K$. It will be seen that the subgroups $K$ and $H$ are defined according to the way that they transform under gauge transformations $\Lambda_{Const}$ and $\Lambda_{Linear}$ having respectively constant and linear gauge functions. The quantum fluctuation patterns characteristic of a given phase are defined in terms of $K\subseteq SMG$ and $H\triangleleft K$. We are working with a lattice formulation of a gauge theory. The different phases in such a theory are generally regarded as lattice artefacts. However we assume that a lattice is just one implementation of a fundamental really existing Planck scale regulator. In light of this assumption “lattice artefact” phases become ontological. That transitions between such phases are most often first order plays an important role in the finetunning mechanism inherent to MPP.

0.8 Distinguishing the possible phases of a non-simple

group

Using a lattice formulation of a gauge theory with gauge group 111The symbol $G$ denotes a generic gauge group where we should have the $SMG$ or at least a non-simple gauge group in mind unless the context indicates otherwise. $G$, let the dynamics of the system be described by a Lagrangian ${\cal L}(A^{\mu},\phi)$ that is invariant under (local) gauge transformations $\Lambda$ of the gauge potential $A^{\mu}(x)$ and the (complex) scalar field $\phi(x)$. In the continuum, the fields $A^{\mu}(x)$ and $\phi(x)$ transform under gauge transformations as

[TABLE]

[TABLE]

In the lattice formulation, each of the four components of the $A^{\mu}$ field corresponds to a group-valued variable U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}}) defined on links of the lattice. The index $\nu$ specifies the direction of the link connecting sites with coordinates $x^{\rho}$ and $x^{\rho}+a\delta^{\rho}_{\nu}$; often such coordinates are written more briefly as $x$ and $x+a\delta_{\nu}$. Under a local gauge transformation, the U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}}) transform as

[TABLE]

That this corresponds to the transformation (6) for the continuum fields $A^{\rho}$ is readily verified: write U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})=\exp(igA^{\rho}(x)a\delta_{\rho}^{\nu})\approx 1+igA^{\rho}(x)a\delta_{\rho}^{\nu} in which case the gauge transformation above is

[TABLE]

which corresponds to the transformation rule (6). On the lattice, the group-valued field $\phi$ is defined on lattice sites; the transformation rule is as in (7) above.

0.8.1 “Phase” classification according to symmetry properties

of vacuum

We are interested in the case in which the gauge field U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}}) takes values in a non-simple gauge group such as $G=SMG$. The gauge field for the SMG has 12 degrees of freedom: if we allow a slight simplification one can say that 8 of these are associated with $SU(3)$ degrees of freedom, 3 with $SU(2)$ degrees of freedom   and one with the $U(1)$ degree of freedom.  It is possible for these degrees of freedom to take values in various structures all of which are determined for each choice $(K,H)$ such that $K\subseteq SMG$ and $H\triangleleft K$. The various structures are the subgroup $K$, the invariant subgroup $H$, the homogeneous space $SMG/K$ and the factor group $K/H$. For gauge field degrees of freedom there is a correspondence between distributions that characterize qualitatively different physical behaviors (e.g., quantum fluctuation patterns) and which structures the gauge field degrees of freedom take values in (e.g., elements of $K\subseteq G$ and $H\triangleleft K$ and cosets of $G/K$ and $K/H$. As already hinted, there is a one-to-one correspondence between the possible phases for the gauge field theory and the possible combinations $(K,H)$ with $K\subseteq G$ and $H\triangleleft K$. Discrete subgroups must be included among the possible subgroups. The choice of the pair $(K,H)$ specifies which degrees of freedom are in a Higgsed phase and also whether the un-Higgsed degrees of freedom are in a confined or Coulomb-like phase. Now I need to reveal how $K\subseteq G$ and $H\triangleleft K$ are defined.

The subgroups $K\subseteq G$ and $H\triangleleft K$ are defined by the transformation properties of the vacuum according to whether or not there is spontaneous breakdown of gauge symmetry under gauge transformations corresponding to the sets of gauge functions $\Lambda_{Const}$ and $\Lambda_{Linear}$ that are respectively constant and linear in the spacetime coordinatesfrogniel ; nonabel :

[TABLE]

and

[TABLE]

Here $\alpha=\alpha^{a}t^{a}$ and $\alpha_{\mu}=\alpha^{a}_{\mu}t^{a}$ where $a$ labels the Lie algebra generators in the case of non-Abelian subgroups. The $t^{a}$ denote a basis of the Lie algebra satisfying the commutation relations $[t^{a},t^{b}]=c^{ab}_{c}t^{c}$ where the $c^{ab}_{c}$ are the structure constants.

Spontaneous symmetry breakdown is manifested as non-vanishing values for gauge variant quantities. However, according to Elitzur’s theorem, such quantities cannot survive under the full gauge symmetry. Hence a partial fixing of the gauge is necessary before it makes sense to talk about the spontaneous breaking of symmetry. We choose the Lorentz gauge for the reason that this still allows the freedom of making gauge transformations of the types $\Lambda_{Const}$ and $\Lambda_{Linear}$ to be used in classifying the lattice artifact “phases” of the vacuum. On the lattice, the choice of the Lorentz gauge amounts to the condition \prod_{\;\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x^{\mu}\;\;}\;\tiny\mbox{emanating from }\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}}U(\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} })=1 for all sites \mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x}.

By definition the degrees of freedom belonging to the subgroup $K$ exhaust the un-Higgsed degrees of freedom if, after fixing the gauge in accord with say the Lorentz condition, $K\subseteq G$ is the maximal subgroup of gauge transformations belonging to the set $\Lambda_{Const}$ that leaves the vacuum invariant For the vacuum of field variables defined on sites (denoted by \langle\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}})\rangle), invariance under transformations $\Lambda_{Const.}$ is possible only if \langle\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}})\rangle=0. For the vacuum of field variables defined on links (denoted by \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rangle, invariance under transformations $\Lambda_{Const}$ requires that \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rangle takes values in the centre of the subgroup $K$. Conventionally, the idea of Higgsed degrees of freedom pertains to field variables defined on sites. With the above criterion using $\Lambda_{Const}$, the notion of Higgsed degrees of freedom is generalised to also include link variables.

If $K\subseteq G$ is the maximal subgroup for which the transformations $\Lambda_{Const}$ leave the vacuum invariant, the gauge field variables taking values in the homogeneous space $G/K$ (see for example dubrovin ; nakahara ) are by definition Higgsed in the vacuum. For these degrees of freedom, gauge symmetry is spontaneously broken in the vacuum under gauge transformations $\Lambda_{Const}$ (i.e., global gauge transformations).

In the vacuum, the un-Higgsed degrees of freedom - taking values in the subgroup $K$ - can be in a confining phase or a Coulomb-like phase according to the way these degrees of freedom transform under gauge transformations $\Lambda_{Linear}$ having linear gauge functions.

Degrees of freedom taking values in the invariant subgroup $H\triangleleft K$ are by definition confined in the vacuum if $H$ is the maximal invariant subgroup of gauge transformations $\Lambda_{Linear}$ that leaves the vacuum invariant; i.e., $h$ consists of the set of elements $h=\exp\{i\alpha^{1}_{a}t_{a}\}$ such that the gauge transformations with linear gauge function $\Lambda_{Linear}$ exemplified by222In the quantity $x^{1}/a$, $a$ denotes the lattice constant; modulo lattice artifacts, rotational invariance allows the (arbitrary) choice of $x^{1}$ as the axis $x^{\mu}$ that we use.$\Lambda_{Linear}\stackrel{{\scriptstyle def.}}{{=}}h^{x^{1}/a}$ leave the vacuum invariant.

If $H\triangleleft K$ is the maximal invariant subgroup of degrees of freedom that are confined in the vacuum, degrees of freedom taking as values the cosets belonging to the factor group $K/H$ are by definition in a Coulomb phase (again, in the Lorentz gauge). For degrees of freedom corresponding to this set of cosets, there is invariance of the vacuum expectation value under coset representatives of the type $\Lambda_{Const}$ while gauge symmetry is spontaneously broken in the vacuum under coset representatives of the type $\Lambda_{Linear}$.

Having now formal criteria for distinguishing the different phases of the vacuum, it would be useful to elaborate a bit further on what is meant by having a phase associated with a subgroup - invariant subgroup pair $(K_{i}\subseteq G,H_{j}\lhd K_{i})$. A phase is a characteristic region of action parameter space. Where does an action parameter space come from and what makes a region of it characteristic of a given phase $(K_{i}\subseteq G,H_{j}\lhd K_{i})$? An action parameter space comes about by choosing a functional form of the plaquette action. This will normally be a sum of terms each of which is a product of an action parameter (action parameters are related to coupling constants) multiplied by a sum over lattice plaquettes each term of which is the trace of group-valued plaquette variable in one of the desired representations (e.g., the fundamental representation, the adjoint representation, etc.).

Having an action allows the calculation of the partition function and subsequently the free energy. As each phase $(K_{i}\subseteq G,H_{j}\lhd K_{i})$ corresponds to different micro physical patterns of fluctuations along various group structures and homogeneous spaces as described above, the partition function and hence the free energy is a different function of the plaquette action parameters for each phase $(K_{i}\subseteq G,H_{j}\lhd K_{i})$. Have in mind that transitions between these “lattice artefact” phase are first order. A region of plaquette action parameter space is in a given phase $(K_{i}\subseteq G,H_{j}\lhd K_{i})$ if the free energy $-\log Z_{K_{i}\subseteq G,H_{j}\triangleleft K_{i}}$ associated with this phase has the largest value of all free energy functions. One should imagine that at any given point in the action parameter space all free energy funcions (one for each possible phase $(K,H)$) are defined (and have values). The realized phase at the point in question is determined by which of these free energy functions is the largest.

In seeking the multiple point, we seek the point or surface in parameter space where “all” (or a maximum number of) phases $(K_{i}\subseteq G,H_{j}\lhd K_{i})$ “touch” one another. MPP claims that action parameter values (which are simply related to coupling constants) at the multiple point are those realized in Nature.

0.9 The Higgsed phase

On a lattice consisting of sites (\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}) and site-connecting links (\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x^{\mu}\;\;y^{\mu}}) denote by \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}) a scaler field variable defined on lattice sites. We want to describe the conditions to be fulfilled if the field variable \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}) is a Higgsed degree of freedom. The appropriate mathematical structure is that of a homogeneous space. If $K\subseteq SMG$ ($K$ not an invariant subgroup of SMG) is the subgroup of not-Higgsed gauge degrees of freedom, the Higgsed degrees of freedom \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}) take values in the homogeneous space $SMG/K$

It might be be useful with a reminder about the mathematical structure of a homogeneous space. For the purpose of exposition it is expedient to use the example of the group $G=SO(3)$ instead of the $G=SMG$ and the subgroup $SO(2)\subset SO(3)$ (instead of the unspecified $K\subseteq SMG$). So we consider the homogeneous space $SO(3)/SO(2)$. In this case the cosets (i.e. elements) of $SO(3)/SO(2)$ are in one-to-one correspondence with the points on a $S_{2}$ sphere: for an arbitrary coset $h\in SO(3)/SO(2)$, the orbit of the action of $SO(3)$ on $h$ is just $S_{2}$. The homogeneous space $SO(3)/SO(2)$ is mapped onto itself under the action of $SO(3)$:

[TABLE]

Note that there is no multiplication (i.e., composition) rule for the cosets (i.e., elements) of a homogeneous space. For example, $h_{1}\cdot h_{2}$ for $h_{1},h_{2}\in SO(3)/SO(2)$ is not meaningful. It can be shown that the action the group $G$ on the homogeneous space $G/K$ is transitive which means that for any two cosets $h_{1},h_{2}\in G/K$ there exists at least one element $g\in G$ such that $h_{1}=gh_{2}$. In the example with $G=SO(3)$ and $G/K=SO(3)/SO(2)$ this means that for any two points $h_{1}$ and $h_{2}$ on $S_{2}\cong SO(3)/SO(2)$ there is at least one element $g\in SO(3)$ such that $h_{1}=gh_{2}$. The set of such elements $g$:

[TABLE]

is the coset of $SO(3)/SO(2)$ associated with $h_{1}$ (here $h_{2}$ can be thought of as a (arbitrarily chosen) basis coset from which all other cosets of $SO(3)/SO(2)$ can be obtained by the appropriate action of $SO(3)$).

Any element $g^{\prime}$ belonging to the coset $\{g\in SO(3)|gh_{2}=h_{1}\}$ is a representative of this coset associated with $h_{1}$. Other representatives of this same coset are obtained by letting $g^{\prime}$ act on the $SO(2)\subset SO(3)$ that leaves the basis coset $h_{2}$ invariant (denote the latter by $SO(2)_{h_{2}\,inv}$). In fact all of the representatives of the coset $\{g\in SO(3)|gh_{2}=h_{1}\}$ are given by $g^{\prime}\cdot SO(2)_{h_{2}\,inv}$. So when $g^{\prime}$ is a representative of the coset associated with $h_{1}$, so is $g^{\prime}\cdot k$ when $k\in SO(2)_{h_{2}\,inv}$. It goes without saying that a representative of a coset always belongs to the coset that it represents.

To get a feeling for it means to have a Higgsed phase, think of having an $S_{2}$ situated at each site \mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}} of the (space-time) lattice. In this picture, the variable \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}}) at each site \mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x^{\mu}} corresponds to a point on the $S_{2}$ at this site. A priori there is no special point in this homogeneous space $SO(3)/SO(2)\stackrel{{\scriptstyle-}}{{\simeq}}S_{2}$ which implies \langle\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} })\rangle=0. The Higgs mechanism comes into play when, for all sites on the lattice, the vacuum value of \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }) - modulo parallel transport between sites by link variables - is (in a classical approximation) the same coset of $SO(3)/SO(2)$ or in other words the same point on all the (site situated) $S_{2}$’s (modulo parallel transport) inasmuch as $SO(3)/SO(2)\cong S_{2}$. With one point of $S_{2}$ singled out globally - call it $h_{2}$ - it is obvious that \langle\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} })\rangle\neq 0. The symmetry of the homogeneous space $SO(3)/SO(2)$ is broken globally down to the $SO(2)\subset SO(3)$ that leaves the point $h_{2}\in S_{2}$ invariant. This is just the isotropy group of $h_{2}\in S_{2}$ which we can - using a notation defined above - denote as $SO(2)_{h_{2}\>inv.}$. After a Higgsning corresponding to singling out $h_{2}\in S_{2}$ we can think of $h_{2}$ as the axis about which the symmetry remaining after this Higgsning are just the rotations $SO(2)_{h_{2}\>inv.}$.

In a quantum field theoretic description of a Higgsed phase corresponding to $h_{2}\in S_{2}$ where we allow for quantum fluctuations, we expect a clustering of the values of \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }) about the coset $h_{2}\in S_{2}$ for all sites of the lattice (modulo parallel transport). This brings us to a technical problemhbnpriv : the average value of such quantum fluctuations is expected to be $h_{2}$: \langle\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} })\rangle=h_{2}. But the average value of for example two cosets of in the neighborhood of the coset corresponding $h_{2}$ does not lie in $S_{2}\cong SO(3)/SO(2)$ but rather in the interior of $S_{2}$ (the convex closure). In order to have such average values in our target space we need the convex closure 333 If we want for example to include averages of the cosets of the homogeneous space $SO(3)/SO(2)$ (which we know is metrically equivalent to an $S_{2}$ sphere), it would generally be necessary to construct the convex closure (e.g., in a vector space). In this case, one could obtain the complex closure as a ball in the linear embedding space $\mbox{\bf{R}}^{3}$. Alternatively, we can imagine supplementing the $SO(3)/SO(2)$ manifold with the necessary (strictly speaking non-existent) points needed in order to render averages on the $S_{2}$ meaningful. Either procedure eliminates the problem that an average taken on a non-convex envelope is generally unstable in the following way: e.g., think of the “north pole” of an $S_{2}$ about which quantum fluctuations are initially clustered (the Higgsed situation); if the fluctuations become so large that that they are concentrated near the equator, the average on an $S_{2}$ will jump discontinuously back and forth between the north and south poles depending respectively on whether the fluctuations are concentrated just north of or just south of the equator). It is interesting to note that by including the points in the ball enclosed by an $S_{2}$, it is possible for $\langle\phi\rangle$ to have a value lying in the symmetric point (i.e., center) when quantum fluctuations are large enough. This point, corresponding to $\langle\phi\rangle=0$, is of course unique in not leading to spontaneous breakdown under rotations of the $S_{2}$. This scenario describes an inverse Higgs mechanismhbnpriv . of $S_{2}$.

The Higgs mechanism outlined above can be provoked if there is a term in the action of the form

[TABLE]

where $\kappa$ is a parameter and dist^{2}(\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x}),U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})\phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!\!\!y})) is the suitably defined squared distance on the $S_{2}$ at the site \mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x} between the point \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x}) and the point \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!\!\!y}) after the latter is “parallel transported” to \mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }^{\!\!\!x} using the link variable U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})\in G. This is the so-called Manton actionmanton .

In terms of of elements $g\in SO(3)$,

[TABLE]

[TABLE]

[TABLE]

In order to provoke the Higgs mechanism, not only must the parameter $\kappa$ be sufficiently large to ensure that it doesn’t pay not to have clustered values of the variables $\phi(\cdot)$. It is also necessary that “parallel transport” be well defined so that it makes sense to talk about the values of \phi(\mbox{\begin{picture}(4.15,4.0)\put(2.0,2.0){\circle*{2.0}} \end{picture} }) being organised (i.e., clustered) at some coset of $SO(3)/SO(2)$. This would obviously not be the case if the theory were confined. In confinement, \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;y})\rangle=0 and parallel transport is meaningless. In the continuum theory, this would correspond to having large curvature (i.e., large $F_{\mu\nu}$) which in turn would make parallel transport very path dependent

0.10 The un-Higgsed Phases

The un-Higgsed gauge field degrees of freedom (i.e., link variables) take values that correspond to the Lie algebra of the subgroup $K\subseteq G$. The confined degrees of freedom take as values the elements of the invariant subgroup $H\triangleleft K$. The Coulomb-like degrees of freedom take as values the cosets of the factor group $K/H$.

0.10.1 Confined degrees of freedom

The confined phase is characterized by large quantum fluctuations in the group-valued link variables so that at least crudely speaking the whole confined subgroup $H$ is accessed. So roughly speaking all elements $h\in H$ are visited with nearly the same probability. In other words the distribution of quantum fluctuations for confined link variables is not strongly clustered in a small part of the group space (e.g. at the group identity or in the center of the group). Since the distribution of confined degrees of freedom is essentially flat (i.e., without much characteristic structure) the effect of gauge transformations is not noticeable. The subgroup $H$ is therefore essentially invariant under all classes of gauge transformations including the for us interesting types of gauge transformation $\Lambda_{Const}$ and $\Lambda_{Linear}$.

0.10.2 The Coulomb-like phase

The claim above is that Coulomb-like link variable degrees of freedom take as values the cosets of the factor group $K/H$. Recall that by definition of a factor group all of the elements of $H$ are identified (i.e., not distinguishable from one another) and the (invariant) subgroup $H$ becomes the identity element in the coset space. That elements of $H$ are not distinguished from one another is consistent with the intuitive properties of having confinement along the subgroup $H$ as sketched above: a consequence of having large quantum fluctuations along $H$ is that all elements of $H$ enter into the fluctuation pattern (which is a manifestation of the underlying physics) with essentially the same weight (as opposed to e.g., a Coulomb-like phase in which the fluctuation pattern is more or less tightly clustered around the group identity).

The transformation properties of the vacuum that are appropriate for having a Coulomb-like phase are suggested by examining the requirementsfrogniel for getting a massless gauge particle as the Nambu-Goldstone boson accompanying the spontaneous breakdown of gauge symmetry. To this end we need to examine the Goldstone Theorem

As already pointed out, a gauge choice must be made in order that spontaneous breakdown of gauge symmetry is at all possible. Otherwise Elitzur’s Theorem insures that all gauge variant quantities vanish identically. Once a gauge choice is made - the Lorenz gauge is strongly suggested inasmuch as we want, in order to classify phases, to retain the freedom to make gauge transformations of the types $\Lambda_{Const}$ and $\Lambda_{Linear}$ - the symmetry under the remaining gauge symmetry must somehow be broken in order to get a Nambu-Goldstone boson that, according to the Nambu-Goldstone Theorem, is present for each generator of a spontaneously broken continuous gauge symmetry.

Recalling from (8) that a link variable U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x^{\mu}\;\;y^{\mu}}) transforms under gauge transformations as

[TABLE]

it is seen that, for the special case of an Abelian gauge group, a gauge function that is linear in the coordinates (or higher order in the coordinates) is required for spontaneous breakdown because the only possibility for spontaneously breaking the symmetry comes from the “gradient” part of the transformation (13). So the needed spontaneous breakdown of gauge symmetry is garanteed if gauge symmetry for gauge transformations of the type $\Lambda_{Linear}$ is spontaneously broken (i.e., the vacuum is not invariant under this class of gauge transformations). Let $Q_{\nu}$ denote the generator of such gauge transformations.

However the proof of the Nambu-Goldstone Theorem also requires the assumption of translational invariance. This amounts to the requirement that the vacuum be invariant under gauge transformations generated by the commutator of the momentum operator with the generator of the spontaneously broken symmetry which is just $Q_{\nu}$ as defined above. Then the requirement of translational symmetry is equivalent to requiring that the vacuum \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rangle is annihilated by the commutator $[P_{\mu},Q_{\nu}]=ig_{\mu\nu}Q$ where $Q$ denotes the generator of gauge transformations with constant gauge functions. So the condition for having translational invariance translates into the requirement that the vacuum \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rangle be invariant under gauge transformations with constant gauge functions. An examination of (13) verifies that this is always true for Abelian gauge groups and also for non-Abelian groups if the vacuum expectation value \langle U(\;\mbox{\begin{picture}(4.15,10.0)\put(0.0,3.0){\circle*{2.0}} \put(0.0,2.75){\line(1,0){7.0}} \put(7.3,3.0){\circle*{2.0}}\mbox{ } \end{picture} }^{\!\!\!\!\!\!\!x\;\;x+a\delta_{\nu}})\rangle lies in the centre of the group (which just means that the vacuum is not “Higgsed”).

0.11 Summary and Concluding Remarks

We have presented a formalism that can be used to define the various possible phases for a non-simple gauge group in the context of lattice gauge theory (LGT). Specifically we are interested in the non-simple $SMG$. These phases are normally said to be artefacts of the unphysical lattice regulator. As we assume that a lattice is one way to implement what we take to be a fundamental ontological (roughly Planck scale) regulator, the “artefact” phases take on a physical meaning.

The various phases are realized by adjusting intensive parameters (which are closely related to the couplings) in the action. These span the so-called action parameter space which is the space in which the boundaries separating the various possible phases can be constructed in a way analogous to the way that temperature and pressure span the space in which the boundaries separating the solid, liquid and gaseous phases of $H_{2}O$ can be drawn. In LGT a typical term in the action is the product of such an intensive parameter with the trace in some representation of a gauge group element defined on a lattice plaquette. In each action term these traces are summed over the plaquettes of the lattice

In this contribution we have developed the formalism for distinguishing the possible phases of a non-simple gauge group $G$ each of which corresponds to a pair of subgroups $(K,H)$ such that $K\subseteq G$ and $H\triangleleft K$.

For each phase $(K,H)$ the free energy $-\log Z_{K_{i}\subseteq G,H_{j}\triangleleft K_{i}}$ is defined for the entire action parameter space. At any point in this space, the phase realized is that for the free enery function has the largest value.

The point in the action parameter space at which the maximum number of different phases come together - the multiple point - corresponds according to the MPP to the parameter values (couplings) realized in Nature. At this point the free energy functions for all the phases that come together at the multiple point are of course all equal.

The degrees of freedom belonging to the subgroup $K$ are the un-Higgsed degrees of freedom if, after fixing the gauge in accord with say the Lorentz condition, $K\subseteq G$ is the maximal subgroup of gauge transformations belonging to the set $\Lambda_{Const}$ that leaves the vacuum invariant. The field variables taking values in the homogeneous space $G/K$ are by definition Higgsed in the vacuum. For these degrees of freedom, gauge symmetry is spontaneously broken in the vacuum under global gauge transformations $\Lambda_{Const}$.

Degrees of freedom taking values in the invariant subgroup $H\triangleleft K$ are by definition confined in the vacuum if $H$ is the maximal invariant subgroup of gauge transformations $\Lambda_{Linear}$ that leaves the vacuum invariant.

The degrees of freedom in a Coulomb-like phase take as values the cosets of the factor group $K/H$. The symmetry properties of the vacuum for a Coulomb-like phase are dictated by the requirements of the Goldstone Theorem. The conditions to be fulfilled in order that the Nambu-Goldstone boson accompanying a spontaneous breakdown of gauge symmetry can be identified with a massless gauge particle (the existence of which is the characteristic feature of a Coulomb-like phase) suggests that the Coulomb phase vacuum is invariant under gauge transformations having a constant gauge function but spontaneously broken under gauge transformations having linear gauge functions.

Summarizing one can say that each phase corresponds to a partitioning of the degrees of freedom (these latter can be labelled by a Lie algebra basis) - some that are Higgsed, others that are un-Higgsed; of the latter, some degrees of freedom can be confining, others Coulomb-like. It is useful to think of a group element $U$ of the gauge group as being parameterized in terms of three sets of coordinates corresponding to three different structures that are appropriate to the symmetry properties used to define a given phase $(K,H)$ of the vacuum. These three sets of coordinates, which are definable in terms of the gauge group $G$, the subgroup $K$, and the invariant subgroup $H\triangleleft K$, are the homogeneous space $G/K$, the factor group $K/H$, and $H$ itself:

[TABLE]

The coordinates $g\in G/K$ will be seen to correspond to Higgsed degrees of freedom, the coordinates $k\in K/H$ to un-Higgsed, Coulomb-like degrees of freedom and the coordinates $h\in H$ to un-Higgsed, confined degrees of freedom.