Limits on W' mass and mixing angle from a (Subscript[P, μ]) ξ measurement

Andrei Gaponenko, February 2003

The definitions

1 - (Subscript[P, μ]) ξ  =  
      4  ((Subscript[ζ, g])^2)  +  2 (1  +   ((Subscript[C, R])^2) ) (t^2)  +  
        4  (Subscript[C, R]) (Subscript[ζ, g]) 
          t Cos[α + ω]
1 - (Subscript[P, μ]) ξδ/ρ  =  
      2  ((Subscript[ζ, g])^2)  +  2 (1  +   ((Subscript[C, R])^2) ) (t^2)  +  
        4  (Subscript[C, R]) (Subscript[ζ, g]) 
          t Cos[α + ω]
where  (Subscript[C, R])  =   cos ((Subscript[θ, R]))/
            cos ((Subscript[θ, L]))  ~ =  
        cos ((Subscript[θ, R]))/0.97 ... ,    (Subscript[ζ, g]) = 
      ((Subscript[g, R])/(Subscript[g, L])) ζ,  
    t  =  (((Subscript[g, R])^2)/((Subscript[g, L])^2)) * 
        (((Subscript[M, 1])^2)/((Subscript[M, 2])^2))  ,  
α is a CP - 
      viloating phase in the right - 
      handed CKM matrix, 
ω is a CP - 
      violating phase in  W  mixing . 
(P .  
              Herczeg,  Phys .  Rev .  D 34,  
            pp3449 - 
              3456 (1986))

\(\(\(Exlusion\ plots\ are\ obtained\ from\ experimental\ limits\ \ \ 1 - \(P\_μ\) ξ\)\(≤\)\(\ \)\(a\^2\ \ \ or\ \ \ 1 - \(P\_μ\) ξδ/ρ\)\(≤\)\(\ \)\(a\^2\)\(\ \)\)\)

To incorporate both  (1 - (Subscript[P, μ]) ξ) and (1 - 
            (Subscript[P, μ]) ξδ/ρ) cases I introduce 
k= (({{4, for  PmuXi}, {2, for PmuXi delta/rho}}))

dPmuXi  =  
      k zetag^2  +  2 (1 + cr^2) t^2  +  
        4  cr  zetag t Cos[alphaomega]

sols  =  Solve[dPmuXi == a^2,  zetag];

zetag1[cr_,  
        alphaomega_]  =  sols[[1]][[1]][[2]]
zetag2[cr_,  
        alphaomega_]  =  sols[[2]][[1]][[2]]

Examples of exclusion plots for some specific values of (Subscript[C, R]) and Cos[α+ω]

experimentalA  =  {a -> Sqrt[0.0040]};
funcsmin  =  {
zetag1[1. ,  
              0. ] //. {k -> 4}, 
zetag1[1. ,  
              0. ] //. {k -> 
                2}
}//.experimentalA

funcsmax =  {
zetag2[1. ,  
              0. ] //. {k -> 4}, 
zetag2[1. ,  
              0. ] //. {k -> 2}
} //. 
        experimentalA

Plot[
        Evaluate[Flatten[{funcsmin,  funcsmax}]],  {t,  0. ,  
          0.1},  {PlotStyle -> {Hue[0.1],  Hue[0.3]}}];

\(Obtaining\ limits\ without\ assumptions\ about\ \ Cos[α + ω]\ and\ C\_R\)

1) Eliminating CP viloating phases

Maximization of zetag2[] with respect to alphaomega gives (see the logbook):  Cos[alphaomega]==-1,

zetag2m1 = (1/(2  k))(\(TraditionalForm\`\@\(4  ka\^2 - 8 \(\( kt\^2\)(1 + C\_R\^2)\) + \((4  t\ C\_R)\)\^2\)\)+4 t (Subscript[C, R]))

Verify:

zetag2m1[t_]  =  
      1/(2  k) (Sqrt[
              4  k a^2  - 
                8  k t^2 (1 + cr^2) + (4  t cr)^2]  +  
            4 t cr)

testassumptions  =  
        Flatten[{cr -> 1.03,  k -> 4,  
            experimentalA}];
bfmaxaw  =  
        Table[zetag2[cr, alphaomega], {alphaomega,  0. ,  2.  Pi,  
            Pi/20. }];
bfmaxawplot  =  
        Plot[Evaluate[bfmaxaw //. testassumptions],  {t,  0. ,  
            0.1},  {DisplayFunction -> 
              Identity}];
calcmaxawplot  = 
        Plot[zetag2m1[t] //. testassumptions, {t, 0. , 
            0.1},  {PlotStyle -> {Hue[0.9]}, 
            DisplayFunction -> Identity}];
Show[bfmaxawplot, calcmaxawplot, 
        DisplayFunction -> $DisplayFunction];

testassumptions = 
        Flatten[{cr -> 1.03, k -> 2, 
            experimentalA}];
bfmaxaw = 
        Table[{zetag2[cr, alphaomega], zetag1[cr, alphaomega]}, {alphaomega, 
            0. , 2.  Pi, Pi/20. }];
bfmaxawplot = 
        Plot[Evaluate[bfmaxaw //. testassumptions], {t, 0. , 
            0.1}, {DisplayFunction -> 
              Identity}];
calcmaxawplot = 
        Plot[zetag2m1[t] //. testassumptions, {t, 0. , 
            0.1}, {PlotStyle -> {Hue[0.9]}, 
            DisplayFunction -> Identity}];
Show[bfmaxawplot, calcmaxawplot, 
        DisplayFunction -> $DisplayFunction];

2) Eliminating the right-handed CKM matrix

The next step is to maximize zetag2m1[] w.r.t.  (Subscript[C, R]).   The cases k==2 and k==4 are different:
\(TraditionalForm\`ζ\_max = \ \@\(a\^2/2\  - \ t\^2\)\  + \ \(C\_max\) t\)  for k==2 
(Subscript[ζ, max]) =  (Sqrt[(a^2)/2  -  (t^2)])  for k==4 and (t^2) >  ((a^2)/(2 (1 + ((Subscript[C, R])^2))))
(Subscript[ζ, max]) =  (1/2) ((Subscript[C, max]) 
                t  +  (Sqrt[(a^2) - (2 + ((Subscript[C, max])^2)) (t^2)]) )  for k==4 and (t^2) ≤  
        ((a^2)/(2 (1 + ((Subscript[C, R])^2))))

(Subscript[ζ, min]) is simply (-(Subscript[ζ, max])).

Verify:

\(zetagmax[t_]\  = \ 
      If[\((k == 
                2)\) || \((\((k == 4)\) && \((t^2 > 
                    a^2/\((2 \((1 + cmax^2)\))\))\))\), \ 
          Sqrt[a^2/2 - t^2], \ 0]
	 + \ 
        If[k == 2, \ cmax\ t, \ 0]
	 + \ 
        If[\((k == 4)\) && \((t^2 <= 
                a^2/\((2 \((1 + cmax^2)\))\))\), \ \((cmax\ t\ \  + \ 
                Sqrt[a^2 - \((2 + cmax^2)\) t^2])\)/2, \ 0]\)

testassumptions = 
        Flatten[{cmax -> 1.03, k -> 2, 
            experimentalA}];
bfmaxawcr = 
        Table[zetag2[cr, alphaomega], {alphaomega, 0. , 2.  Pi, Pi/20. },  
          Evaluate[{cr, 0, cmax,  cmax/10. } //. 
              testassumptions]];
bfmaxawcrplot = 
        Plot[Evaluate[bfmaxawcr //. testassumptions], {t, 0. , 
            0.1}, {PlotStyle -> {Hue[0.3]}, 
            DisplayFunction -> Identity}];
calcmaxcrplot = 
        Plot[zetagmax[t] //. testassumptions, {t, 0. , 
            0.1}, {PlotStyle -> {Hue[0.9]}, 
            DisplayFunction -> Identity}];
Show[bfmaxawcrplot, calcmaxcrplot, 
        DisplayFunction -> $DisplayFunction];

(* Convert from t to W' mass. Assume gR==gL *)

tsubst  =  {t -> ML^2/M^2,  
          ML -> 80.423 } ;

zgm[M_]  =  zetagmax[t] //. tsubst

zgtest2  =  
        Flatten[{k -> 2,  cmax -> 1.03,  
            experimentalA}];
zgtest2plot  =  
        Plot[Evaluate[{zgm[m],  -zgm[m]} //. zgtest2],  {m, 0, 
            1200. },  {PlotStyle -> {Hue[0.3]}}];

zgtest4  =  
        Flatten[{k -> 4,  cmax -> 1.03,  
            experimentalA}];
zgtest4plot  =  
        Plot[Evaluate[{zgm[m],  -zgm[m]} //. zgtest4],  {m, 0, 
            1200. }, {PlotStyle -> {Hue[0.9]}}];

Show[zgtest2plot, 
      zgtest4plot]

Prepare curves for TWIST and other muon decay experiments

<< Graphics`FilledPlot`
zetaplotmin := -0.1;  
    zetaplotmax = 0.1;
zg4fplot[x_, style_] := 
        FilledPlot[
          Evaluate[{zgm[m], -zgm[m]} //. {k -> 4, cmax -> 1.03, 
                a -> Sqrt[x]}], {m, 0, 1200. }, 
          Flatten[{style, DisplayFunction -> Identity}, 
            1]];
zg2fplot[x_, style_] := 
        FilledPlot[
          Evaluate[{zgm[m], -zgm[m]} //. {k -> 2, cmax -> 1.03, 
                a -> Sqrt[x]}], {m, 0, 1200. }, 
          Flatten[{style, DisplayFunction -> Identity}, 
            1]];

TWIST anticipated:  (Subscript[P, μ]) ξ ≥  
        0.00014*1.6,  90%CL

twistPmuXiLim  :=  0.00014*1.6;
twistRhoLim  := 0.00010*1.6;

(*Limits on mixin angle from rho*) 
 (* 
      Rho in LRS models is always  <  
          3/4.   Can get a greater value in experiment 
 = > strict limit .  
                Or can get 1 sigma below 3/
                4  = > allow 2.6 sigma for 90  % CL *) 

twistZetaLim  :=  Sqrt[1.6*twistRhoLim/2]

twistRhoColor := 
        CMYKColor[0. , 0. , 0.9, 1.05];
\(twistRhoPlot := 
      Graphics[{
	
          CMYKColor[0.4, 0.4, 0. , 
            0. ], 
	{Thickness[ .007], 
            Dashing[{0.01, 0.015}], 
            Line[{{0, twistZetaLim}, {1200, 
                  twistZetaLim}}]}, 
	{Thickness[ .007], 
            Dashing[{0.01, 0.015}], 
            Line[{{0, \(-twistZetaLim\)}, {1200, \(-twistZetaLim\)}}]}, 
	
          Text["TWIST rho", {310. , \ 0.02}, \ {1. , 
              0. }]
}]\)

twistPmuXiColor := 
        CMYKColor[0. , 0.9, 0. , 0. ];
twistPmuXiPlot := 
        zg4fplot[
          twistPmuXiLim, {Fills -> twistPmuXiColor, 
            Curves -> None}];
twistPmuXiLabel := 
        Graphics[{twistPmuXiColor, 
            Text["TWIST PmuXi", {1200. , 0.26*zetaplotmax}, {1. , 
                0. }]}];

twistManifestColor  :=  
        CMYKColor[0. ,  0. ,  0. ,  1. ];
twistManifestPlot  :=  
        Plot[Evaluate[{zetag1[1,  0],  zetag2[1, 0]} //. {k -> 4, 
                  cmax -> 1.03, a -> Sqrt[twistPmuXiLim]} //. 
              tsubst],  {M, 0, 
            1200. }, {PlotStyle -> {{Thickness[0.006], 
                  twistManifestColor}},  
            DisplayFunction -> Identity}];
twistManifestLabel  :=  
        Graphics[{twistManifestColor, 
            Text["TWIST PmuXi, MLRS", {1200. , -0.025}, {1. , 
                0. }]}];

Jodidio et al.combined with Stocker et al.(PDG value): (Subscript[P, μ]) ξδ/ρ ≥  
        0.99682, CL=90%

pdgPmuXiDeltaRhoColor := CMYKColor[0., 0., 0.9, 0.05]; 
pdgPmuXiDeltaRhoPlot :=  zg2fplot[1. - 0.99682, {Fills -> pdgPmuXiDeltaRhoColor, Curves -> None}];
pdgPmuXiDeltaRhoLabel := Graphics[{pdgPmuXiDeltaRhoColor, Text["PmuXiDeltaRho", {100., -0.076}, {-1., 0.}]}];

\(Beltrami\ et\ al . \((PDG\ value)\) : 
\(P\_μ\) ξ = \(1.0027 ± 0.0079\) ± 
        0.0030\)
(Subscript[σ, total]) = (Sqrt[(0.0079^2) + (0.0030^2)]) = 
        0.00845
1.0027 - 1.6 (Subscript[σ, total]) = 0.989179

pdgPmuXiColor  :=  
        CMYKColor[0.9, 0. , 0. , 0. ];
pdgPmuXiPlot := 
        zg4fplot[
          1 - 0.989179, {Fills -> pdgPmuXiColor, 
            Curves -> None}];
\(\(pdgPmuXiLabel\  := \ 
        Graphics[{
	
            pdgPmuXiColor,  (*\(CMYKColor[0, 0, 0, 
                1]\)\(,\)*) 
	
            Text["PmuXi", \ {320. , \ 0.05}, \ {1. , 
                0. }]
}];\)\)

Data from collider experiments

D0 Collaboration, Phys. Rev. Letters 76, pp.3271-3276 (1996).
Using limts from the peak search ((Subscript[N, R ])can only be light in μ-decay). 
Using (Subscript[g, R])/(Subscript[g, L]) = 1

d0yr1996MWbest  :=  720.    ;
d0yr1996MWworst  :=  510.

\(d0Plot\  := \ 
      Graphics[{
	
          CMYKColor[1. , \ 0. , \ 1. , \ 
            0.1], 
	{Thickness[ .01], 
            Line[{{d0yr1996MWworst, \ \ zetaplotmin}, {d0yr1996MWworst, \ 
                  zetaplotmax}}]}, 
	{Thickness[ .005], \ 
            Dashing[{0.03, 0.02}]\ , \ 
            Line[{{d0yr1996MWbest, \ zetaplotmin}, {d0yr1996MWbest, \ 
                  zetaplotmax}}]}, 
	
          Text["D0", \ {d0yr1996MWbest, \ 0.82*zetaplotmax}, \ {1. , 
              0. }]
}]\)

CDF Collaboration, Phys. Rev. Letters 87, p.231803 (2001). 
Using (Subscript[g, R])/(Subscript[g, L]) = 1
Combined result:  (Subscript[M, W]) ≥  786, CL=95%

Need to convert everything to the same CL

cdf2001MWbest  :=  786

\(cdfPlot\  := \ 
      Graphics[{
	
          CMYKColor[0. , \ 1. , \ 1. , \ 
            0.1], 
	{Thickness[ .005], \ 
            Dashing[{0.03, 0.02}]\ , \ 
            Line[{{cdf2001MWbest, \ zetaplotmin}, {cdf2001MWbest, \ 
                  zetaplotmax}}]}, 
	
          Text["CDF", \ {cdf2001MWbest, \ 0.88*zetaplotmax}, \ {\(-1. \), 
              0. }]
}]\)

All together

\(\(finalPlot\  = \ 
        Show[
	pdgPmuXiPlot, \ 
          pdgPmuXiLabel, 
	pdgPmuXiDeltaRhoPlot, 
          pdgPmuXiDeltaRhoLabel, 
	twistPmuXiPlot, 
          twistPmuXiLabel, 
	d0Plot, 
	
          cdfPlot, 
	twistRhoPlot, 
	
          twistManifestPlot, twistManifestLabel, 
	
          DisplayFunction -> $DisplayFunction, 
	
          PlotRange -> {{0. , 1200. }, {zetaplotmin, \ 
                zetaplotmax}}
	 (*\ \(,\)\(AxesOrigin -> {0. , \(-0.1\)}\)\(,\)\(\ \)\(AxesLabel -> {\*""(Subscript[M, W])"", \ "ζ"}\)*) 
, 
          Frame -> True
, 
          FrameLabel -> {StyleForm[\*""(Subscript[M, (Subscript[W, R])]), GeV"", 
                FontSize -> 10], 
	
              StyleForm["mixing angle", 
                FontSize -> 
                  10]}
 (*\(,\)\(RotateLabel -> 
              False\)*) 
];\)\)
Display["lrs_limits.eps", finalPlot, "EPS"];