Truncation errors in Legendre analysis of N -> Δ quadrupole ratios

James J. Kelly

Most previous analyses of EMR = Re[(Subscript[E, 1+])/(Subscript[M, 1+])]  and SMR  =  
        Re[(Subscript[S, 1+])/(Subscript[M, 1+])] fit Legendre expansions to unpolarized cross sections, usually without Rosenbluth separation, and extract EMR and SMR from those coefficients assuming (Subscript[M, 1+]) dominance and truncation to (Subscript[ℓ, π]) ≤ 1.  We derive more complete multipole expansions and test the accuracy of those approximations using multipole amplitudes from a realistic phenomenological model.

Created: July 31, 1998
Last revised: July 5, 2005

There has been long-standing interest in deformed components of the N and Δ wave functions.  The dominant amplitude for pion electroproduction of the Δ resonance is the (Subscript[M, 1+]) multipole, but smaller (Subscript[S, 1+]) and (Subscript[E, 1+]) multipole amplitudes arise from configuration mixing within the quark core, meson or gluon exchange currents between quarks, or coupling to the pion cloud outside the quark core.  Those mechanisms are not entirely independent, but recent models attempt to distinguish between the intrinsic deformation of the wave functions and dynamical effects.  It is important to deduce accurate quadrupole ratios, EMR = Re[(Subscript[E, 1+])/(Subscript[M, 1+])]  and SMR  =  
        Re[(Subscript[S, 1+])/(Subscript[M, 1+])], from pion electroproduction data with as little model dependence as possible.  Most previous analyses relied upon the assumptions of (Subscript[M, 1+]) dominance and sp truncation to relate the quadrupole amplitudes to Legendre coefficients fitted to cross section angular distributions.  More recently, a recoil polarization experiment at Jefferson Laboratory measured angular distributions for 16 independent response functions and extracted complex multipole amplitudes in a very nearly model-independent analysis. Therefore, the quadrupole amplitudes for the p (π^0) channel at (Q^2) = 1 (((GeV/c))^2) were obtained without need of either assumption of the traditional Legendre analysis, both of which proved unreliable.  In this report we examine in some detail the accuracy of the traditional Legendre analysis.

The unpolarized differential cross section in the π N center of mass can be expressed in the form

\(σ\  = \ \(ν\_0\) \((\ ϵ\_S\ R\_L\  + \ 
            R\_T\  + \ \(\@\(2 \( ϵ\_S\) \((1 + ϵ)\)\)\) \(R\_LT\) Sin[θ] 
              Cos[ϕ]\  + \ ϵ\ \(R\_TT\) \(Sin[θ]\^2\) 
              Cos[2  ϕ]\ )\)\)

where (Subscript[ν, 0]) is a phase-space factor, ϵ is the transverse polarization of the virtual photon, (Subscript[ϵ, S]) = ϵ (Q^2)/(q^2), and (θ, ϕ) are polar and azimuthal pion angles relative to the (Overscript[q,  ⇀])  vector and scattering plane.  The response functions can be expanded in Legendre series as

(Subscript[R, L]) =  (∑_(n = 0)^∞) (Subscript[A, n]^L) 
          (Subscript[P, n])[x]
(Subscript[R, T]) =  (∑_(n = 0)^∞) (Subscript[A, n]^T) 
          (Subscript[P, n])[x]
(Subscript[R, LT]) =  (∑_(n = 0)^∞) (Subscript[A, n]^LT) 
          (Subscript[P, n])[x]
(Subscript[R, TT]) =  (∑_(n = 0)^∞) (Subscript[A, n]^TT) 
          (Subscript[P, n])[x]

where x = Cos[θ] and where the expansion coefficients (Subscript[A, n]^λ) are functions of (W, (Q^2)) that can be fit to cross section data.  Each of the Legendre coefficients can in turn be expressed in terms of multipole amplitudes of the form \(TraditionalForm\`Re[\(B\_\(\(ℓ\)\(±\)\)\) \(C\_\(\(ℓ\)\(±\)\)\^*\)]\) where B, C ∈ {M, E, S} are magnetic, electric, or scalar multipole amplitudes for specified ℓ and j = ℓ ± (1/2).  The traditional Legendre analysis truncates those expansions by requiring ℓ ≤ 1, described as sp truncation, and drops any contribution that does not contain (Subscript[M, 1+]), described as (Subscript[M, 1+]) dominance.  Thus, one obtains the traditional formulas

\(\(R\&~\)\_EM\  = \(\(\ \)\(3  A\_2\%\(L + T\) - 2  A\_0\%TT\)\)\/\(12  A\_0\%\(L + T\)\)\)
\(\(R\&~\)\_SM\  = \(\(\ \)\(A\_1\%LT\)\)\/\(3  A\_0\%\(L + T\)\)\)

where the tildes identify these as approximations and where (Subscript[A, n]^(L + T)) = 
        (Subscript[A, n]^T) + (Subscript[ϵ, S]) (Subscript[A, n]^L) is the linear combination used when Rosenbluth separation is not available.  Note that (Subscript[R, L]) would vanish if sp truncation and (Subscript[M, 1+]) dominance were exact.

Below we derive symbolic expressions for those expansions and study the effects of various truncation schemes.  Much of the code for performing the expansions and manipulating the expressions is borrowed from a more comprehensive analysis of recoil polarization response functions developed in the closely related notebook RecoilPolarization.nb.  Therefore, we will not explain the details of Mathematica code here.  Most readers will probably want to skip to the numerical analysis section without attempting to parse the somewhat intimidating intervening code.

Initialization

Defaults and packages

ClearAll["Global`*"];
Off[General::spell,  General::spell1];
$TextStyle = {FontFamily -> "Times", 
          FontSize -> 12};

Needs["Utilities`Notation`"]

Symbolize[\(M\_+\)];Symbolize[\(M\_-\)];
Symbolize[\(E\_+\)];Symbolize[\(E\_-\)];
Symbolize[\(S\_+\)];Symbolize[\(S\_-\)];

Manipulation of complex quantities

\(\(\(conjugate::usage\  = 
	"A simple method for computing the conjugate of an object which is explicitly complex.";\)\(
\)
    \)\)
conjugateRule  =  
        Complex[re_, im_] :> Complex[re, -im];
conjugate[exp__]  :=  exp /. conjugateRule;

\(\(\(ContractAmplitudeProducts = 
	{
	
          c_.  \(A_\_a_\) \(\((A_\_a_)\)\^*\) :> 
            c\ Abs[A\_a]\^2, 
	
          d_.  + c_.  \(A_\_a_\) \(\((B_\_b_)\)\^*\) :> 
            d + c\ \((Re[\(A\_a\) \(\((B\_b)\)\^*\)] + 
                      I\ Im[\(A\_a\) \(\((B\_b)\)\^*\)])\) /; b > a, 
	
          d_.  + c_. \ \(A_\_a_\) \(\((B_\_b_)\)\^*\) :> 
            d + c\ \((Re[\(B\_b\) \(\((A\_a)\)\^*\)] - 
                      I\ Im[\(B\_b\) \(\((A\_a)\)\^*\)])\) /; b < a, 
	
          d_. \ Re[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ 
              e_.  Re[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] :> 
            d\ Re[\((A\_a + e\/d\ C\_c)\) \(\((B\_b)\)\^*\)], 
	
          d_. \ Im[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ 
              e_.  Im[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] :> 
            d\ Im[\((A\_a + 
                      e\/d\ C\_c)\) \(\((B\_b)\)\^*\)]
	};\)\(	\)\)\)

\(MyRules = {\(0\^*\) -> 0}; 
    MyAssumptions = {α ∈ Reals, β ∈ 
          Reals, γ ∈ Reals, δ ∈ 
          Reals, θ ∈ Reals, ϕ ∈ 
          Reals, ω > 0, q > ω, Q > 0, 
        0 < x < 1};\)
MySimplify = 
        Simplify[# /. MyRules, MyAssumptions] &;
MyFullSimplify = 
        FullSimplify[# /. MyRules, MyAssumptions] &;

Multipole expansion of CGLN amplitudes

CGLN1[ℓ_] = {
	(Subscript[ℱ, 1])->Sum[\((ℓ\ \(M\_+\)[ℓ] + \(E\_+\)[ℓ])\) (Subscript[P, ℓ + 1]^′)[x] + \((\((ℓ + 
                                1)\) \(M\_-\)[ℓ] + \(E\_-\)[ℓ])\)(Subscript[P, ℓ - 1]^′)[x],{ℓ, 0, ℓmax}],
	(Subscript[ℱ, 2])->Sum[\((\((ℓ + 
                              1)\) \(M\_+\)[ℓ]\  + \ ℓ\ \(M\_-\)[ℓ])\)(Subscript[P, ℓ]^′)[x],{ℓ, 0, ℓmax}],
	(Subscript[ℱ, 3])->Sum[\((\(E\_+\)[ℓ]\  - \ \(M\_+\)[ℓ])\) (Subscript[P, ℓ + 1]^″)[x] + \((\(E\_-\)[ℓ] + \(M\_-\)[ℓ])\)(Subscript[P, ℓ - 1]^″)[x],{ℓ, 0, ℓmax}],
	(Subscript[ℱ, 4])->Sum[\((\(M\_+\)[ℓ] - \(E\_+\)[ℓ] - \(M\_-\)[ℓ] - \(E\_-\)[ℓ])\)(Subscript[P, ℓ]^″)[x],{ℓ, 0, ℓmax}],
	(Subscript[ℱ, 5])->(ω/q)Sum[((ℓ + 
                              1)\(\(S\_+\)[ℓ]\)(Subscript[P, ℓ + 1]^′)[x] - ℓ \(\(S\_-\)[ℓ]\)(Subscript[P, ℓ - 1]^′)[x]),{ℓ, 0, ℓmax}],
	(Subscript[ℱ, 6])->(ω/q)Sum[(\((ℓ\ \(S\_-\)[ℓ] - \ \((ℓ + 1)\) \(S\_+\)[ℓ])\)(Subscript[P, ℓ]^′)[x]),{ℓ, 0, ℓmax}]
		};

\(\(rule[
          mp] = {\(M\_-\)[0] -> 0, \(E\_-\)[0] -> 
            0, \(S\_-\)[0] -> 0, \(M\_+\)[0] -> 
            0, \(E\_-\)[1] -> 0, \(0\^*\) -> 0};\)\)

\(\(ExpandAmplitudeProducts = {
Abs[ℱ\_a_]\^2 -> \(ℱ\_a\) \(\((ℱ\_a)\)\^*\), 
Re[\((A_.  ℱ\_a_ + 
                    B_. \ ℱ\_b_)\)\ \(\((C_. \ ℱ\_c_\  + \ D_.  ℱ\_d_)\)\^*\)] -> 
            A\ C\ Re[\(ℱ\_a\) \(\((ℱ\_c)\)\^*\)] + A\ D\ Re[\(ℱ\_a\) \(\((ℱ\_d)\)\^*\)] + 
              B\ C\ Re[\(ℱ\_b\) \(\((ℱ\_c)\)\^*\)] + 
              B\ D\ Re[\(ℱ\_b\) \(\((ℱ\_d)\)\^*\)], 
          Im[\((A_.  ℱ\_a_ + 
                    B_. \ ℱ\_b_)\)\ \(\((C_. \ ℱ\_c_\  + \ D_.  ℱ\_d_)\)\^*\)] -> 
            A\ C\ Im[\(ℱ\_a\) \(\((ℱ\_c)\)\^*\)] + A\ D\ Im[\(ℱ\_a\) \(\((ℱ\_d)\)\^*\)] + 
              B\ C\ Im[\(ℱ\_b\) \(\((ℱ\_c)\)\^*\)] + 
              B\ D\ Im[\(ℱ\_b\) \(\((ℱ\_d)\)\^*\)], 
          Re[\((A_.  ℱ\_a_ + 
                    B_. \ ℱ\_b_)\)\ \(\((C_. \ ℱ\_c_)\)\^*\)\ ] -> 
            A\ C\ Re[\(ℱ\_a\) \(\((ℱ\_c)\)\^*\)] + B\ C\ Re[\(ℱ\_b\) \(\((ℱ\_c)\)\^*\)], 
          Im[\((A_.  ℱ\_a_ + 
                    B_. \ ℱ\_b_)\)\ \(\((C_. \ ℱ\_c_)\)\^*\)\ ] -> 
            A\ C\ Im[\(ℱ\_a\) \(\((ℱ\_c)\)\^*\)] + B\ C\ Im[\(ℱ\_b\) \(\((ℱ\_c)\)\^*\)]};\)\)

\(\(ExpandCGLN\  = \ 
        Join[{Sum[A_, {ℓ1_, ℓ1min_, ℓ1max_}] 
                Sum[B_, {ℓ2_, ℓ2min_, ℓ2max_}] :> 
              MySum[Expand[
                  A\ B], {ℓ1, ℓ1min, ℓ1max}, {ℓ2, ℓ2min, ℓ2max}]\ }, CGLN1[ℓ1], 
          CGLN1[ℓ2] /. {ℱ\_a_ -> \(\((ℱ\_a)\)\^*\), \(M\_+\)[
                  a_] -> \(\((\(M\_+\)[a])\)\^*\), \(E\_+\)[
                  a_] -> \(\((\(E\_+\)[a])\)\^*\), \(S\_+\)[
                  a_] -> \(\((\(S\_+\)[a])\)\^*\), \(M\_-\)[
                  a_] -> \(\((\(M\_-\)[a])\)\^*\), \(E\_-\)[
                  a_] -> \(\((\(E\_-\)[a])\)\^*\), \(S\_-\)[
                  a_] -> \(\((\(S\_-\)[a])\)\^*\)}];\)\)

\(AnyMP = \(M\_-\) | \(M\_+\) | \(E\_-\) | \(E\_+\) | \(S\_-\) | \(S\_+\); \(AnyMP\_-\) = \(M\_-\) | \(E\_-\) | \(S\_-\); \(AnyMP\_+\) = \(M\_+\) | \(E\_+\) | \(S\_+\);\)

\(\(ExpandMultipoleProducts = {
\(\((a_ \((x : AnyMP)\)[
                      m_])\)\^*\) -> 
            a\ \(\((x[m])\)\^*\), 
Re[
              a_. \ \ x_\ \(\((x_)\)\^*\)] -> a\ Abs[x]\^2, \ 
          Im[a_. \ x_\ \(\((x_)\)\^*\)] -> 0, \ 
          Abs[f_[θ]]\^2 -> f[θ]\^2, 
          Abs[a_. \ \((b_\  + \ c_)\)]\^2 -> \(Abs[
                    a]\^2\) \((Abs[b]\^2 + Abs[c]\^2 + 2  Re[b\ \(c\^*\)])\), 
          Abs[a_. \ \((x : \((AnyMP)\))\)[m_]]\^2 -> \(a\^2\) 
              Abs[x[m]]\^2, 
\((f : \((Re | Im)\))\)[
              a_. \ \((b_\  + \ c_)\)] -> 
            f[a\ b] + f[a\ c], \((f : \((Re | Im)\))\)[
              a_. \ \(\((\((b_\  + \ c_)\) d_. )\)\^*\)] -> 
            f[a\ \(\((b\ d)\)\^*\)] + 
              f[a\ \(\((c\ d)\)\^*\)], \((f : \((Re | Im)\))\)[
              a_\ \((x : AnyMP)\)[
                  m_] \(\((\((y : AnyMP)\)[n_])\)\^*\)] -> 
            a\ f[\(\((y[n])\)\^*\) x[m]], 
          Re[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[
                        m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) | \(S\_+\))\))\)[n_]] -> Re[\(\((y[n])\)\^*\) x[m]], 
          Im[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[
                        m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) | \(S\_+\))\))\)[n_]] -> \(-Im[\(\((y[n])\)\^*\) x[m]]\), 
          Re[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[
                        m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[
                  n_]] -> Re[\(\((y[n])\)\^*\) x[m]], 
          Im[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[
                        m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[
                  n_]] -> \(-Im[\(\((y[n])\)\^*\) x[m]]\), \ 
          Re[\(\((x : \(M\_+\)[m_])\)\^*\) \(y : \(M\_-\)[n_]\)] -> 
            Re[\(y\^*\) 
                x], 
Im[\(\((x : \(M\_+\)[
                          m_])\)\^*\) \(y : \(M\_-\)[
                    n_]\)] -> \(-Im[\(y\^*\) 
                  x]\), 
Re[\(\((x : \(E\_+\)[
                          m_])\)\^*\) \(y : \(E\_-\)[n_]\)] -> 
            Re[\(y\^*\) 
                x], 
Im[\(\((x : \(E\_+\)[
                          m_])\)\^*\) \(y : \(E\_-\)[
                    n_]\)] -> \(-Im[\(y\^*\) 
                  x]\), 
Re[\(\((x : \(S\_+\)[
                          m_])\)\^*\) \(y : \(S\_-\)[n_]\)] -> 
            Re[\(y\^*\) 
                x], 
Im[\(\((x : \(S\_+\)[
                          m_])\)\^*\) \(y : \(S\_-\)[
                    n_]\)] -> \(-Im[\(y\^*\) x]\), 
          Re[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < 
                  m)\) -> Re[\(x[m]\^*\) x[n]], 
          Im[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < 
                  m)\) -> \(-Im[\(x[m]\^*\) x[n]]\)
};\)\)

recombine  =  
        a_.  Re[b_] +  c_.  Re[d_]  -> 
          Re[a b  + c d];

Expansion of response functions

ExpandLegendre = {(Subscript[P, λ])_[x] -> LegendreP[λ, x],(Subscript[P, λ_]^′)[x]->D[LegendreP[λ, x], x],(Subscript[P, λ_]^″)[x]->D[LegendreP[λ, x], x, x]};

\(ExpandR[R_, lmax_Integer /; \((lmax ≥ 0)\)] := \ 
      Module[{ℛ}, 
		ℛ = \(TrigExpand[\(\((\(R //. ExpandCGLN\) /. {ℓmax -> lmax, 
                          MySum -> Sum})\) //. 
                    ExpandMultipoleProducts\) //. 
                  rule[mp]] /. {Cos[θ] -> x, 
                Sin[θ] -> \@\(1 - x\^2\)}\) /. 
            ExpandLegendre; 
Collect[
          MySimplify[ℛ], x, MySimplify]]\)

ToLegendreP[x_, m_?EvenQ] :=  
        Sum[(((2^(2  n)) (4  n + 
                      1) m! ((m/2) + n)!)/((m + 2  n + 
                        1)! ((m/2) - n)!)) (Subscript[P, 2  n])[x], {n, 
            0, (m/2)}];
ToLegendreP[x_, m_?OddQ] :=  
        Sum[(((2^(2  n + 1)) (4  n + 
                      3) m! (((m + 1)/2) + n)!)/((m + 
                        2  n + 2)! (((m - 1)/2) - n)!)) 
            (Subscript[P, 2  n + 1])[x], {n, 0, (m/2) + 1}];

xExpand  :=  (x^n)_.  :> ToLegendreP[x, n];

RtoLegendre[R_, lmax_Integer] := 
      Collect[MySimplify[ExpandR[R, lmax] /. xExpand /. 
            (Subscript[P, 0])[x] -> 1], (Subscript[P, λ])_[x], MySimplify]

Unpolarized response functions

multipole expansions

First we quote the formulas relating unpolarized response functions to CGLN amplitudes

\(\(R\_L = \(q\^2\/ω\^2\) \((Abs[ℱ\_5]\^2 + 
              Abs[ℱ\_6]\^2 + 
              2\ Cos[θ]\ Re[ℱ\_5\ \(\((ℱ\_6)\)\^*\)])\);\)\)
\(\(R\_T = 
        Abs[ℱ\_1]\^2 + Abs[ℱ\_2]\^2 - 
          2\ Cos[θ]\ Re[ℱ\_1\ \(\((ℱ\_2)\)\^*\)] + 
          1\/2\ \((Abs[ℱ\_3]\^2 + 
                Abs[ℱ\_4]\^2 + 
                2\ \((Re[ℱ\_2\ \(\((ℱ\_3)\)\^*\)] + 
                      Re[ℱ\_1\ \(\((ℱ\_4)\)\^*\)] + 
                      Cos[θ]\ Re[ℱ\_3\ \(\((ℱ\_4)\)\^*\)])\))\)\ Sin[θ]\^2;\)\)
\(\(R\_LT = \(-\(q\/ω\)\)\ \((Re[ℱ\_2\ \(\((ℱ\_5)\)\^*\)] + 
              Re[ℱ\_3\ \(\((ℱ\_5)\)\^*\)] + 
              Re[ℱ\_1\ \(\((ℱ\_6)\)\^*\)] + 
              Cos[θ]\ \((Re[ℱ\_4\ \(\((ℱ\_5)\)\^*\)] + 
                    Re[ℱ\_3\ \(\((ℱ\_6)\)\^*\)])\) + 
              Re[ℱ\_4\ \(\((ℱ\_6)\)\^*\)])\)\ ;\)\)
\(\(R\_TT = 
        1\/2\ \((Abs[ℱ\_3]\^2 + 
              Abs[ℱ\_4]\^2 + 
              2\ \((Re[ℱ\_2\ \(\((ℱ\_3)\)\^*\)] + 
                    Re[ℱ\_1\ \(\((ℱ\_4)\)\^*\)] + 
                    Cos[θ]\ Re[ℱ\_3\ \(\((ℱ\_4)\)\^*\)])\))\)\ ;\)\)

and then perform multipole expansions for ℓ ≤ 6, which is more than sufficient to study their convergence.

Lexp = RtoLegendre[(Subscript[R, L]), 6]

Texp = RtoLegendre[(Subscript[R, T]), 6]

LTexp = RtoLegendre[(Subscript[R, LT]), 6]

TTexp = RtoLegendre[(Subscript[R, TT]), 6]

Legendre coefficients

Next we extract the Legendre coefficients used by the traditional analysis of quadrupole ratios.

(Subscript[A, T0]) =  Texp /. {(Subscript[P, n])_[x] -> 0};
(Subscript[A, L0]) =  Lexp /. {(Subscript[P, n])_[x] -> 0};
(Subscript[A, TT0]) = TTexp /. {(Subscript[P, n])_[x] -> 0};
(Subscript[A, LT1]) = 
        Coefficient[Expand[LTexp], (Subscript[P, 1])[x]];
(Subscript[A, T2]) = 
        Coefficient[Expand[Texp], (Subscript[P, 2])[x]];
(Subscript[A, L2]) = Coefficient[Expand[Lexp], (Subscript[P, 2])[x]];

It will be useful to define truncation and recombination functions.

\(\(M1truncation[A_] := 
        Select[Expand[A], 
          Not[\((FreeQ[#, \(M\_+\)[1]])\)] &];\)\)
\(\(M1truncation[A_\/B_] := 
        M1truncation[A]\/M1truncation[B];\)\)
\(truncate[
        A_, ℓ_]\  := \ \((A /. {_[n_?\((# > ℓ &)\)] -> 
                0})\) /. {\(0\^*\) -> 0}\)

The number of terms in each expansion grows rapidly with the maximum ℓ

\(TableForm[
      Table[Length /@ \(\((truncate[#, ℓ] &)\) /@ {A\_L0, A\_T0, 
              A\_LT1, A\_TT0, A\_L2, A\_T2}\), {ℓ, 1, 6}], 
      TableHeadings -> {Range[
            6], {\*""(Subscript[A, 0]^L)"", \*""(Subscript[A, 0]^T)"", \*""(Subscript[A, 1]^LT)"", \*""(Subscript[A, 0]^TT)"", \*""(Subscript[A, 2]^L)"", \*""(Subscript[A, 2]^T)""}}]\)

but is severely limited by the assumption of (Subscript[M, 1+]) dominance.

\(TableForm[
      Table[Length /@ \(\((truncate[#, ℓ] &)\) /@ \(M1truncation /@ {A\_L0, A\_T0, A\_LT1, A\_TT0, A\_L2, A\_T2}\)\), {ℓ, 1, 6}], 
      TableHeadings -> {Range[
            6], {\*""(Subscript[A, 0]^L)"", \*""(Subscript[A, 0]^T)"", \*""(Subscript[A, 1]^LT)"", \*""(Subscript[A, 0]^TT)"", \*""(Subscript[A, 2]^L)"", \*""(Subscript[A, 2]^T)""}}]\)

Notice that (Subscript[M, 1+]) truncation by itself is sufficient to eliminate even multipoles.

M1truncation /@ {(Subscript[A, L0]), (Subscript[A, T0]), (Subscript[A, LT1]), (Subscript[A, TT0]), (Subscript[A, L2]), (Subscript[A, T2])} // 
      TableForm

Thus, we verify that the traditional formulas are recovered when sp truncation and (Subscript[M, 1+]) dominance are applied.

\(R\&~\_EM = 
      truncate[M1truncation[\(3 \((A\_T2 + ϵ\ A\_L2)\) - 2  A\_TT0\)\/\(12 \((A\_T0 + ϵ\ A\_L0)\)\)], 1] // MySimplify\)

\(R\&~\_SM = 
      truncate[M1truncation[A\_LT1\/\(3 \((A\_T0 + ϵ\ A\_L0)\)\)], 
          1] // MySimplify\)

However, relaxation of either assumption permits additional terms to contribution to the traditional formulas such that it is no longer possible to isolate Re[(Subscript[E, 1+])/(Subscript[M, 1+])] or Re[(Subscript[S, 1+])/(Subscript[M, 1+])] if those terms are numerically significant.  For example, the traditional combinations of Legendre coefficients take the following forms when (Subscript[M, 1+]) dominance is applied without sp truncation.

M1truncation[((3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]))/(12 ((Subscript[A, T0]) + ϵ (Subscript[A, L0]))))] //. recombine // MySimplify

M1truncation[((Subscript[A, LT1])/(3 ((Subscript[A, T0]) + ϵ (Subscript[A, L0]))))] //. 
        recombine // MySimplify

Similarly, using sp truncation without (Subscript[M, 1+]) dominance yields the following.

truncate[((3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]))/(12 ((Subscript[A, T0]) + ϵ (Subscript[A, L0])))), 1] //. recombine // MySimplify

truncate[((Subscript[A, LT1])/(3 ((Subscript[A, T0]) + ϵ (Subscript[A, L0])))), 1] //. 
        recombine // MySimplify

Notice that the formulas without (Subscript[M, 1+]) dominant become dependent upon ϵ.  Therefore, elimination of (Subscript[R, L]) by Rosenbluth separation would help.

Traditional quadrupole formulas

The functions below form the combinations of Legendre coefficients traditionally used to extract quadrupole ratios from cross section data.  To study the convergence of these formulas, the maximum ℓ for contributing multipoles is given as an argument.  In addition, there are versions which truncate according to (Subscript[M, 1+]) dominance.  However, experimentally one cannot perform such truncations and the question is how well \(TraditionalForm\`\(R\&~\)\_EM\) and \(TraditionalForm\`\(R\&~\)\_SM\) in the limit ℓ -> ∞ approximate (Subscript[R, EM]) and (Subscript[R, SM]).  Although most attempts to extract quadrupole amplitudes have not used Rosenbluth separation, one would expect better accuracy if the longitudinal response function were eliminated because it vanishes under the assumption of (Subscript[M, 1+]) dominance.  Rosenbluth separation can be simulated here by choosing ϵ -> 0.

M1 dominance

EMR[M1, ℓ_Integer?NonNegative] := 
        truncate[
          M1truncation[((3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]))/(12 ((Subscript[A, T0]) + ϵ (Subscript[A, L0]))))], ℓ];
SMR[M1, ℓ_Integer?NonNegative] := 
      truncate[M1truncation[
          ((Subscript[A, LT1])/(3 ((Subscript[A, T0]) + ϵ (Subscript[A, L0]))))], ℓ]

\(\(EMR[M1, ℓ_, eps_, 
          values : {__Rule}] := \(\(EMR[M1, ℓ] /. 
              values\) /. ϵ -> eps\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
\(SMR[M1, ℓ_, eps_, 
        values : {__Rule}] := \(\(SMR[M1, ℓ] /. 
            values\) /. ϵ -> eps\) /. \(x_\^*\) -> 
          Conjugate[x]\)

truncation without M1 dominance

EMR[ℓ_Integer?NonNegative] := 
        truncate[((3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]))/(12 ((Subscript[A, T0]) + ϵ (Subscript[A, L0])))), ℓ];
SMR[ℓ_Integer?NonNegative] := 
      truncate[((Subscript[A, LT1])/(3 ((Subscript[A, T0]) + ϵ (Subscript[A, L0])))), ℓ]

\(\(EMR[ℓ_, eps_, 
          values : {__Rule}] := \(\(EMR[ℓ] /. 
              values\) /. ϵ -> eps\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
\(SMR[ℓ_, eps_, 
        values : {__Rule}] := \(\(SMR[ℓ] /. 
            values\) /. ϵ -> eps\) /. \(x_\^*\) -> 
          Conjugate[x]\)

Ranking of omitted terms

Additional insight into the behavior of the traditional Legendre analysis can be obtained by ordering the contributions to the numerators and the denominator by magnitude.  The following function performs that operation.

\(arrange[coeff_, mpamps_, eps_] := 
      Module[{list}, list = List @@ Expand[coeff]; 
        list[[Ordering[\((\(\(list /. 
                    mpamps\) /. ϵ -> eps\) /. \(x_\^*\) -> 
                  Conjugate[x])\), All, 
            Abs[#1] > Abs[#2] &]]]]\)

Numerical analysis

To analyze the quantitative accuracy of the traditional Legendre analysis, we employ the MAID2003 model.  This is a realistic phenomenological model fitted to previous pion electro- and photo-production data.  Although the details are clearly model dependent, other recent models give qualitatively similar results for the convergence and accuracy of Legendre coefficients and the quadrupole ratios extracted from them using the traditional method.  The following p (π^0) multipole amplitudes were obtained for (W, (Q^2)) = (1.23, 1.0).

Ideally the quadrupole amplitudes are obtained for isospin-3/2 amplitudes for W = (Subscript[M, Δ]).  Although (Subscript[M, Δ]) ≈ 1.232 GeV, this parameter is model dependent.   Furthermore, experimental analyses average over finite bins of W which are not necessarily centered upon the nominal (Subscript[M, Δ]); the closest bin for JLab experiment  e91011 was actually 1.23 GeV.  Finally, experimental separation of isospin amplitudes is not performed often and would entail additional uncertainties if it were done.  Therefore, it is useful to test the traditional Legendre analysis for both p (π^0) and isospin-3/2 channels and for both 1.232 and 1.230  GeV.  This becomes a little tedious, but if any of these variations were omitted, someone would question the conclusions.

p(π^0), W=1.232

\(\(maidvalues[pπ0, 
          1.232] = 
Join[{\(E\_+\)[0] -> 
              3.894×10\^\(-1\) + I\ 1.117×10\^\(-1\), \(S\_+\)[
                0] -> \(-3.516\)×10\^\(-1\) + I\ 9.778×10\^\(-2\), 
\(M\_+\)[
                1] -> \(-4.855\)×10\^\(-3\) + 
                1.895  I, \(E\_+\)[1] -> 
              4.735×10\^\(-2\) - I\ 3.088×10\^\(-2\), \(S\_+\)[1] -> 
              1.932×10\^\(-2\) - I\ 1.261×10\^\(-1\), \(M\_-\)[
                1] -> \(-4.837\)×10\^\(-1\) - I\ 8.949×10\^\(-3\), \(S\_-\)[1] -> 
              2.638×10\^\(-1\) + \ I\ 2.989×10\^\(-2\), 
\(M\_+\)[
                2] -> \(-2.964\)×10\^\(-2\) + I\ 3.000×10\^\(-4\), \(E\_+\)[2] -> 
              2.258×10\^\(-3\) + I\ 9.037×10\^\(-5\), \(S\_+\)[
                2] -> \(-2.768\)×10\^\(-3\) + I\ 6.564×10\^\(-5\), 
\(M\_-\)[2] -> 
              1.854×10\^\(-2\) + I\ 1.686×10\^\(-3\), \(E\_-\)[
                2] -> \(-8.363\)×10\^\(-2\) - I\ 1.176×10\^\(-3\), \(S\_-\)[2] -> 
              2.135×10\^\(-2\) + I\ 5.058×10\^\(-5\), 
\(M\_+\)[3] -> 
              9.687×10\^\(-3\) + I\ 3.213×10\^\(-5\), \(E\_+\)[
                3] -> \(-3.276\)×10\^\(-4\) - I\ 4.398×10\^\(-6\), \(S\_+\)[3] -> 
              2.939×10\^\(-4\) - I\ 1.771×10\^\(-6\), 
\(M\_-\)[3] -> 
              3.727×10\^\(-4\) + I\ 3.154×10\^\(-5\), \(E\_-\)[3] -> 
              1.394×10\^\(-2\) + I\ 1.823×10\^\(-5\), \(S\_-\)[
                3] -> \(-4.856\)×10\^\(-3\) - I\ 6.440×10\^\(-5\), 
\(M\_+\)[
                4] -> \(-1.999\)×10\^\(-3\), \(E\_+\)[4] -> 
              1.027×10\^\(-4\), \(S\_+\)[
                4] -> \(-1.137\)×10\^\(-4\), 
\(M\_-\)[4] -> 
              1.807×10\^\(-4\), \(E\_-\)[
                4] -> \(-3.194\)×10\^\(-3\), \(S\_-\)[4] -> 
              1.036×10\^\(-3\), \(M\_+\)[5] -> 
              5.752×10\^\(-4\), \(E\_+\)[
                5] -> \(-2.540\)×10\^\(-5\), \(S\_+\)[5] -> 
              2.648×10\^\(-5\), 
\(M\_-\)[
                5] -> \(-8.473\)×10\^\(-8\), \(E\_-\)[5] -> 
              8.124×10\^\(-4\), \(S\_-\)[
                5] -> \(-2.738\)×10\^\(-4\)}, 
          Flatten[Table[{\(M\_+\)[ℓ] -> 
                  0, \(E\_+\)[ℓ] -> 
                  0, \(S\_+\)[ℓ] -> 
                  0, \(M\_-\)[ℓ] -> 
                  0, \(E\_-\)[ℓ] -> 
                  0, \(S\_-\)[ℓ] -> 0}, {ℓ, 6, 
                6}]]];\)\)

values = maidvalues[pπ0, 1.232];

The correct quadrupole ratios are

\({R\_EM, 
        R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], 
          Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)

and would be reproduced by the traditional analysis

\(\({R\&~\_EM, R\&~\_SM} /. values\) /. \(x_\^*\) -> 
        Conjugate[x]\)

if its assumptions were accurate.  Consider first the convergence with respect to ℓ without assuming (Subscript[M, 1+]) dominance.

Table[{EMR[ℓ, 0.95, values], 
          SMR[ℓ, 0.95, values]}, {ℓ, 1, 5}] // 
      TableForm

Table[{EMR[ℓ, 0.95, values], 
            SMR[ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

Although these values do appear to converge, their limits are incorrect.  The relative errors are about 18  % for SMR and a whopping 30  % for EMR.  Evidently, multipoles with ℓ > 1 and terms with ℓ ≤ 1 that do not involve (Subscript[M, 1+]) spoil the traditional Legendre analysis.  Although the multipoles do tend to decrease with ℓ, the coefficients in the multipole expansions of Legendre coefficients tend to increase with ℓ.  Multipole expansions for each of the required Legendre coefficients are given below.

(Subscript[A, L0])

(Subscript[A, T0])

(Subscript[A, LT1]) //. recombine

(Subscript[A, TT0]) //. recombine

(Subscript[A, L2]) //. recombine

(Subscript[A, T2]) //. recombine

Therefore, it becomes a detailed quantitative issue whether or not higher multipoles decline rapidly enough to obtain rapid convergence.  This is especially important for (Subscript[A, 0]^T) and (Subscript[A, 0]^L) because all contributions are positive and there is no possible benefit from fortuitous cancellations.  The table below evaluates the convergence of each term in \(TraditionalForm\`R\&~\_EM\) and \(TraditionalForm\`R\&~\_SM\) separately.  Notice that terms with ℓ = 2 produce a 6  % change in (Subscript[A, 1]^LT) and hence in \(TraditionalForm\`R\&~\_SM\).  Even terms with ℓ = 3 make a perceptible contribution.  Similarly, the longitudinal contribution to the denominator of \(TraditionalForm\`R\&~\_SM\), namely (Subscript[A, 0]^L), results in another 4  % reduction when ϵ ∼ 1 even at the level of sp truncation due to errors in (Subscript[M, 1+]) dominance and increases with ℓ.  Consequently, the relative accuracy of \(TraditionalForm\`R\&~\_SM\) is no better than 10  %.  The accuracy of \(TraditionalForm\`R\&~\_EM\) is even worse because the cancellation between (Subscript[A, 2]^T) and (Subscript[A, 0]^TT) amplifies numerical errors.

\(TableForm[
      Table[\(\((\(truncate[#, ℓ] &\) /@ {A\_L0, A\_T0, A\_LT1, 
                  A\_TT0, A\_L2, A\_T2})\) /. values\) /. \(x_\^*\) -> 
            Conjugate[x], {ℓ, 1, 5}], 
      TableHeadings -> {Range[
            5], {\*""(Subscript[A, 0]^L)"", \*""(Subscript[A, 0]^T)"", \*""(Subscript[A, 1]^LT)"", \*""(Subscript[A, 0]^TT)"", \*""(Subscript[A, 2]^L)"", \*""(Subscript[A, 2]^T)""}}]\)

Notice that elimination of the longitudinal contribution by means of Rosenbluth separation, such that

\(R\&~\_EM\ ⟶\ \(\(\(\ \)\(3  A\_2\%T - 
            2  A\_0\%TT\)\)\/\(12  A\_0\%T\)\)\)
\(R\&~\_SM\ ⟶\ \(\(\(\ \)\(A\_1\%LT\)\)\/\(3 
             A\_0\%T\)\)\)

can be simulated in the present analysis by taking ϵ -> 0.  This improves the accuracy of the extracted quadrupole ratios, as shown in the table below,

Table[{EMR[ℓ, 0. , values], 
            SMR[ℓ, 0. , values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

but results with approximately 15  % relative errors still remain unsatisfactory.  If (Subscript[M, 1+]) dominance were valid, one would obtain much faster convergence

Table[{EMR[M1, ℓ, 0. , values], 
            SMR[M1, ℓ, 0. , values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

and more accurate results, but in none of the models considered does (Subscript[M, 1+]) dominance apply at better than the 10  % level.

It is also instructive to examine the relative sizes of the various contributions to both the numerators and the common denominator of these formulas. Thus, we find that the most important neglected term is \(TraditionalForm\`Re[\(M\_\(\(1\)\(-\)\)\) \(E\_\(\(1\)\(+\)\)\^*\)]\).  That term alone would explain an error of approximately 30  %.  However, the numerical values of neglected terms do not decrease especially rapidly and there are very important cancellations.  For example, the third term involves (Subscript[E, 2-]) but is cancelled almost complete by \(TraditionalForm\`\(\(|\)\(S\_\(\(1\)\(+\)\)\)\( | \^2\)\)\) when ϵ -> 1.  Thus, final value depends upon ϵ, contrary to the original assumptions, and depends upon delicate cancellations.  Therefore, one cannot expect good accuracy from this method.

EMRterms = 
        arrange[3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]), values, 
          0.95];
Take[EMRterms, 10]

\(Take[\(\(EMRterms /. values\) /. ϵ -> 
            0.95\) /. \(x_\^*\) -> Conjugate[x], 10]\)

Similarly, the most important neglected term in the numerator of \(TraditionalForm\`R\&~\_SM\) is \(TraditionalForm\`Re[\(E\_\(\(2\)\(-\)\)\) \(S\_\(\(0\)\(+\)\)\^*\)]\), but its magnitude relative to the leading term is smaller than found for \(TraditionalForm\`R\&~\_EM\).  However, convergence is far from monotonic because there are many other contributions of similar size and variable sign.  Nevertheless, one expects the convergence of \(TraditionalForm\`R\&~\_SM\) to be superior to that of \(TraditionalForm\`R\&~\_EM\).

ALT1terms = 
        arrange[(Subscript[A, LT1]), values, 0.95];
Take[ALT1terms, 10]

\(\(ALT1values = \(\(ALT1terms /. values\) /. ϵ -> 
              0.95\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
Take[ALT1values, 10]

FoldList[Plus, 0, ALT1values] // Rest

Finally, the most important neglected term in the common denominator is \(TraditionalForm\`|M\_\(\(1\)\(-\)\)|\^2\), but its convergence is faster and more monotonic because all terms are positive.

DenomTerms = 
        arrange[(Subscript[A, T0]) + ϵ (Subscript[A, L0]), values, 
          0.95];
Take[DenomTerms, 10]

\(\(DenomValues = \(\(DenomTerms /. values\) /. ϵ -> 
              0.95\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
Take[DenomValues, 10]

p(π^0), W=1.230

\(\(maidvalues[pπ0, 
          1.230] = 
Join[{\(E\_+\)[0] -> 
              3.906×10\^\(-1\) + I\ 1.111×10\^\(-1\), \(S\_+\)[
                0] -> \(-3.516\)×10\^\(-1\) + I\ 9.654×10\^\(-2\), 
\(M\_+\)[1] -> 
              6.332×10\^\(-2\) + 1.923  I, \(E\_+\)[
                1] -> 
              4.596×10\^\(-2\) - I\ 3.327×10\^\(-2\), \(S\_+\)[1] -> 
              1.461×10\^\(-2\) - I\ 1.300×10\^\(-1\), \(M\_-\)[
                1] -> \(-4.847\)×10\^\(-1\) - I\ 7.778×10\^\(-3\), \(S\_-\)[1] -> 
              2.619×10\^\(-1\) + \ I\ 2.822×10\^\(-2\), 
\(M\_+\)[
                2] -> \(-2.965\)×10\^\(-2\) + I\ 2.918×10\^\(-4\), \(E\_+\)[2] -> 
              2.260×10\^\(-3\) + I\ 8.675×10\^\(-5\), \(S\_+\)[
                2] -> \(-2.765\)×10\^\(-3\) + I\ 6.300×10\^\(-5\), 
\(M\_-\)[2] -> 
              1.825×10\^\(-2\) + I\ 1.613×10\^\(-3\), \(E\_-\)[
                2] -> \(-8.314\)×10\^\(-2\) - I\ 1.138×10\^\(-3\), \(S\_-\)[2] -> 
              2.138×10\^\(-2\) + I\ 5.008×10\^\(-5\), 
\(M\_+\)[3] -> 
              9.608×10\^\(-3\) + I\ 3.070×10\^\(-5\), \(E\_+\)[
                3] -> \(-3.301\)×10\^\(-4\) - I\ 4.180×10\^\(-6\), \(S\_+\)[3] -> 
              2.974×10\^\(-4\) - I\ 1.648×10\^\(-6\), 
\(M\_-\)[3] -> 
              3.544×10\^\(-4\) + I\ 2.999×10\^\(-5\), \(E\_-\)[3] -> 
              1.384×10\^\(-2\) + I\ 1.730×10\^\(-5\), \(S\_-\)[
                3] -> \(-4.830\)×10\^\(-3\) - I\ 6.157×10\^\(-5\), 
\(M\_+\)[
                4] -> \(-1.984\)×10\^\(-3\), \(E\_+\)[4] -> 
              1.028×10\^\(-4\), \(S\_+\)[
                4] -> \(-1.135\)×10\^\(-4\), 
\(M\_-\)[4] -> 
              1.759×10\^\(-4\), \(E\_-\)[
                4] -> \(-3.163\)×10\^\(-3\), \(S\_-\)[4] -> 
              1.031×10\^\(-3\), \(M\_+\)[5] -> 
              5.685×10\^\(-4\), \(E\_+\)[
                5] -> \(-2.543\)×10\^\(-5\), \(S\_+\)[5] -> 
              2.651×10\^\(-5\), 
\(M\_-\)[
                5] -> \(-3.043\)×10\^\(-9\), \(E\_-\)[5] -> 
              8.028×10\^\(-4\), \(S\_-\)[
                5] -> \(-2.718\)×10\^\(-4\)}, 
          Flatten[Table[{\(M\_+\)[ℓ] -> 
                  0, \(E\_+\)[ℓ] -> 
                  0, \(S\_+\)[ℓ] -> 
                  0, \(M\_-\)[ℓ] -> 
                  0, \(E\_-\)[ℓ] -> 
                  0, \(S\_-\)[ℓ] -> 0}, {ℓ, 6, 
                6}]]];\)\)

values = maidvalues[pπ0, 1.230];

The correct quadrupole ratios are

\({R\_EM, 
        R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], 
          Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)

while the traditional estimators give the following convergence patterns.   These results are very similar, of course, to those for the nominal (Subscript[M, Δ]).

Table[{EMR[ℓ, 0.95, values], 
          SMR[ℓ, 0.95, values]}, {ℓ, 1, 5}] // 
      TableForm

Table[{EMR[ℓ, 0.95, values], 
            SMR[ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

EMRterms = 
        arrange[3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]), values, 
          0.95];
Take[EMRterms, 10]

ALT1terms = 
        arrange[(Subscript[A, LT1]), values, 0.95];
Take[ALT1terms, 10]

Isospin-3/2,  W=1.232

Values below are for W = 1.232.

\(\(maidvalues[1.232] = 
        Join[{\(E\_+\)[
                0] -> \(-1.389\)×10\^\(-1\) + I\ 3.689×10\^\(-2\), \(S\_+\)[
                0] -> \(-5.420\)×10\^\(-1\) + I\ 1.440×10\^\(-1\), 
\(M\_+\)[1] -> 
              2.675×10\^\(-2\) + 2.840  I, \(E\_+\)[
                1] -> \(-4.286\)×10\^\(-4\) - I\ 4.314×10\^\(-2\), \(S\_+\)[
                1] -> \(-1.762\)×10\^\(-3\) - I\ 1.876×10\^\(-1\), \(M\_-\)[
                1] -> \(-1.042\)×10\^\(-1\) + I\ 1.275×10\^\(-2\), \(S\_-\)[
                1] -> \(-2.074\)×10\^\(-1\) + \ I\ 1.772×10\^\(-2\), 
\(M\_+\)[
                2] -> \(-5.386\)×10\^\(-2\) + I\ 3.182×10\^\(-4\), \(E\_+\)[
                2] -> \(-4.579\)×10\^\(-3\) + I\ 2.705×10\^\(-5\), \(S\_+\)[
                2] -> \(-7.989\)×10\^\(-3\) + I\ 4.720×10\^\(-5\), 
\(M\_-\)[2] -> 
              1.217×10\^\(-2\) + I\ 1.225×10\^\(-4\), \(E\_-\)[
                2] -> \(-1.338\)×10\^\(-1\) - I\ 1.878×10\^\(-3\), \(S\_-\)[2] -> 
              6.870×10\^\(-3\) + I\ 2.739×10\^\(-5\), 
\(M\_+\)[3] -> 
              1.620×10\^\(-2\) + I\ 4.758×10\^\(-5\), \(E\_+\)[
                3] -> \(-2.076\)×10\^\(-3\) - I\ 6.017×10\^\(-6\), \(S\_+\)[
                3] -> \(-7.631\)×10\^\(-4\) - I\ 2.216×10\^\(-6\), 
\(M\_-\)[
                3] -> \(-1.249\)×10\^\(-3\) + I\ 1.252×10\^\(-6\), \(E\_-\)[3] -> 
              1.969×10\^\(-2\) - I\ 1.823×10\^\(-6\), \(S\_-\)[
                3] -> \(-3.956\)×10\^\(-4\) - I\ 1.107×10\^\(-7\), 
\(M\_+\)[
                4] -> \(-3.388\)×10\^\(-3\), \(E\_+\)[
                4] -> \(-7.337\)×10\^\(-5\), \(S\_+\)[
                4] -> \(-3.104\)×10\^\(-4\), 
\(M\_-\)[4] -> 
              1.498×10\^\(-4\), \(E\_-\)[
                4] -> \(-4.892\)×10\^\(-3\), \(S\_-\)[4] -> 
              2.098×10\^\(-4\), \(M\_+\)[5] -> 
              9.758×10\^\(-4\), \(E\_+\)[
                5] -> \(-1.024\)×10\^\(-5\), \(S\_+\)[
                5] -> \(-1.822\)×10\^\(-5\), 
\(M\_-\)[5] -> \(-2.805\)×10\^\(-5\), \(E\_-\)[
                5] -> 
              1.216×10\^\(-3\), \(S\_-\)[
                5] -> \(-3.496\)×10\^\(-5\)}, 
          Flatten[Table[{\(M\_+\)[ℓ] -> 
                  0, \(E\_+\)[ℓ] -> 
                  0, \(S\_+\)[ℓ] -> 
                  0, \(M\_-\)[ℓ] -> 
                  0, \(E\_-\)[ℓ] -> 
                  0, \(S\_-\)[ℓ] -> 0}, {ℓ, 6, 
                6}]]];\)\)

values = maidvalues[1.232];

\({R\_EM, 
        R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], 
          Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)

Table[{EMR[ℓ, 0.95, values], 
          SMR[ℓ, 0.95, values]}, {ℓ, 1, 5}] // 
      TableForm

Table[{EMR[ℓ, 0.95, values], 
            SMR[ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

Table[{EMR[M1, ℓ, 0.95, values], 
            SMR[M1, ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 
          1, 5}] // TableForm

Interestingly, the relative importance of various neglected terms is different for isospin-3/2 and p (π^0) channels.  For \(TraditionalForm\`\(R\&~\)\_EM\) we now find that longitudinal contamination is most import and that the \(TraditionalForm\`Re[\(E\_\(\(0\)\(+\)\)\) \(E\_\(\(2\)\(-\)\)\^*\)]\) contribution becomes the most important transverse term.  Therefore, significant improvement in \(TraditionalForm\`\(R\&~\)\_EM\) can be made, in principle, by performing Rosenbluth separation, but the results would still not be satisfactory.

EMRterms = 
        arrange[3 ((Subscript[A, T2]) + ϵ (Subscript[A, L2])) - 2  (Subscript[A, TT0]), values, 
          0.95];
Take[EMRterms, 10]

\(Take[\(\(EMRterms /. values\) /. ϵ -> 
            0.95\) /. \(x_\^*\) -> Conjugate[x], 10]\)

ALT1terms = 
        arrange[(Subscript[A, LT1]), values, 0.95];
Take[ALT1terms, 10]

\(\(ALT1values = \(\(ALT1terms /. values\) /. ϵ -> 
              0.95\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
Take[ALT1values, 10]

Table[{EMR[ℓ, 0. , values], 
            SMR[ℓ, 0. , values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

Isospin-3/2,  W=1.230

Values below are for W = 1.230.

\(maidvalues[1.230] = 
      Join[{\(E\_+\)[
              0] -> \(-1.380\)×10\^\(-1\) + I\ 3.623×10\^\(-2\), \(S\_+\)[
              0] -> \(-5.416\)×10\^\(-1\) + I\ 1.423×10\^\(-1\), 
\(M\_+\)[1] -> 
            1.291×10\^\(-1\) + 2.883  I, \(E\_+\)[
              1] -> \(-2.110\)×10\^\(-3\) - I\ 4.678×10\^\(-2\), \(S\_+\)[
              1] -> \(-8.658\)×10\^\(-3\) - I\ 1.934×10\^\(-1\), \(M\_-\)[
              1] -> \(-1.047\)×10\^\(-1\) + I\ 1.257×10\^\(-2\), \(S\_-\)[
              1] -> \(-2.077\)×10\^\(-1\) + \ I\ 1.743×10\^\(-2\), 
\(M\_+\)[
              2] -> \(-5.382\)×10\^\(-2\) + I\ 3.100×10\^\(-4\), \(E\_+\)[
              2] -> \(-4.462\)×10\^\(-3\) + I\ 2.570×10\^\(-5\), \(S\_+\)[
              2] -> \(-7.902\)×10\^\(-3\) + I\ 4.552×10\^\(-5\), 
\(M\_-\)[2] -> 
            1.207×10\^\(-2\) + I\ 1.179×10\^\(-4\), \(E\_-\)[2] -> \(-1.330\)×10\^\(-1\) - I\ 1.816×10\^\(-3\), \(S\_-\)[2] -> 
            6.870×10\^\(-3\) + I\ 2.739×10\^\(-5\), 
\(M\_+\)[3] -> 
            1.607×10\^\(-2\) + I\ 4.547×10\^\(-5\), \(E\_+\)[3] -> \(-2.051\)×10\^\(-3\) - I\ 5.727×10\^\(-6\), \(S\_+\)[
              3] -> \(-7.358\)×10\^\(-4\) - I\ 2.059×10\^\(-6\), 
\(M\_-\)[
              3] -> \(-1.232\)×10\^\(-3\) + I\ 1.199×10\^\(-6\), \(E\_-\)[3] -> 
            1.957×10\^\(-2\) - I\ 1.915×10\^\(-6\), \(S\_-\)[3] -> \(-4.245\)×10\^\(-4\) - I\ 9.674×10\^\(-8\), 
\(M\_+\)[
              4] -> \(-3.362\)×10\^\(-3\), \(E\_+\)[
              4] -> \(-6.589\)×10\^\(-5\), \(S\_+\)[
              4] -> \(-3.040\)×10\^\(-4\), 
\(M\_-\)[4] -> 
            1.466×10\^\(-4\), \(E\_-\)[
              4] -> \(-4.845\)×10\^\(-3\), \(S\_-\)[4] -> 
            2.130×10\^\(-4\), \(M\_+\)[5] -> 
            9.644×10\^\(-4\), \(E\_+\)[
              5] -> \(-1.000\)×10\^\(-5\), \(S\_+\)[
              5] -> \(-1.653\)×10\^\(-5\), 
\(M\_-\)[5] -> \(-2.697\)×10\^\(-5\), \(E\_-\)[5] -> 
            1.202×10\^\(-3\), \(S\_-\)[
              5] -> \(-3.635\)×10\^\(-5\)}, 
        Flatten[Table[{\(M\_+\)[ℓ] -> 
                0, \(E\_+\)[ℓ] -> 
                0, \(S\_+\)[ℓ] -> 
                0, \(M\_-\)[ℓ] -> 
                0, \(E\_-\)[ℓ] -> 
                0, \(S\_-\)[ℓ] -> 0}, {ℓ, 6, 
              6}]]]\)

values = maidvalues[1.230];

\({R\_EM, 
        R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], 
          Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)

Table[{EMR[ℓ, 0.95, values], 
          SMR[ℓ, 0.95, values]}, {ℓ, 1, 5}] // 
      TableForm

Table[{EMR[ℓ, 0.95, values], 
            SMR[ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

Table[{EMR[M1, ℓ, 0.95, values], 
            SMR[M1, ℓ, 0.95, values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 
          1, 5}] // TableForm

Table[{EMR[ℓ, 0. , values], 
            SMR[ℓ, 0. , values]}/{(Subscript[R, EM]), (Subscript[R, SM])}, {ℓ, 1, 
          5}] // TableForm

Other traditional formulas

One sometimes sees attempts to extract other multipole products from truncated Legendre expansions, but extreme care must be exercised.  For example, the following truncations appear to give simple estimators.

(Subscript[A, LT0]) =  
        LTexp /. {(Subscript[P, n])_[x] -> 0};
(Subscript[A, T1]) = Coefficient[Expand[Texp], (Subscript[P, 1])[x]];

M1truncation[truncate[(Subscript[A, T1]), 1]] // MySimplify

M1truncation[truncate[(Subscript[A, LT0]), 1]] // MySimplify

M1truncation[
        truncate[-(((Subscript[A, T2]) + 2  (Subscript[A, T0]) + 2  (Subscript[A, TT0]))/8), 1]] // 
      MySimplify

Thus, it is useful to define the following accuracy parameters.

\(\(f\_E0p[lmax_, 
          values_] := \(truncate[A\_T1, lmax]\/\(2  Re[\(M\_+\)[1] \(\(E\_+\)[0]\^*\)]\) /. values\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
\(\(f\_S0p[lmax_, 
          values_] := \(truncate[A\_LT0, lmax]\/Re[\(M\_+\)[1] \(\(S\_+\)[0]\^*\)] /. values\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)
\(\(f\_M1m[lmax_, 
          values_] := \(truncate[\(-\((A\_T2 + 2  A\_T0 + 2  A\_TT0)\)\)/8, lmax]\/Re[\(M\_+\)[1] \(\(M\_-\)[1]\^*\)] /. values\) /. \(x_\^*\) -> 
            Conjugate[x];\)\)

\(TableForm[
      Table[{f\_E0p[ℓ, maidvalues[pπ0, 1.232]], 
          f\_S0p[ℓ, maidvalues[pπ0, 1.232]], 
          f\_M1m[ℓ, maidvalues[pπ0, 1.232]]}, {ℓ, 1, 
          5}], TableHeadings -> {Range[
            5], {\*""(Subscript[f, E0+])"", \*""(Subscript[f, S0+])"", \*""(Subscript[f, M1-])""}}]\)

However, these estimators are hopelessly inaccurate at (Q^2) = 1.  The estimator for \(TraditionalForm\`Re\ \(M\_\(\(1\)\(-\)\)\) \(M\_\(\(1\)\(+\)\)\^*\)\) isn't even the numerically leading term; for MAID2003 it is only fifth!  Therefore, these formulas are worthless under these conditions.

Take[arrange[(Subscript[A, T1]), maidvalues[pπ0, 1.232], 0. ], 10]

\(\(% /. maidvalues[pπ0, 1.232]\) /. \(x_\^*\) -> 
        Conjugate[x]\)

Take[arrange[(Subscript[A, LT0]), maidvalues[pπ0, 1.232], 0. ], 10]

\(\(% /. maidvalues[pπ0, 1.232]\) /. \(x_\^*\) -> 
        Conjugate[x]\)

Take[
      arrange[-(((Subscript[A, T2]) + 2  (Subscript[A, T0]) + 2  (Subscript[A, TT0]))/8), 
        maidvalues[pπ0, 1.232], 0. ], 10]

\(\(% /. maidvalues[pπ0, 1.232]\) /. \(x_\^*\) -> 
        Conjugate[x]\)

Conclusions

Despite the intuitive appeal of the traditional Legendre analysis for quadrupole amplitudes, neither of its truncation hypotheses are sufficiently accurate at the level of experimental precision that is now possible.  Even if truncation to ℓ ≤ 1 were accurate, contributions to the Legendre coefficients from terms that do not involve (Subscript[M, 1+]) are not negligible.  Rosenbluth separation would help for \(TraditionalForm\`\(R\&~\)\_EM\), but errors would remain about 15  % for both p (π^0) quadrupole ratios.  The  accuracy expected for \(TraditionalForm\`\(R\&~\)\_SM\%\((p π\^0)\)\) and \(TraditionalForm\`\(R\&~\)\_SM\%\((3/2)\)\) are similar but, surprisingly, the accuracy is even worse for \(TraditionalForm\`\(R\&~\)\_EM\%\((3/2)\)\) than for \(TraditionalForm\`\(R\&~\)\_EM\%\((p π\^0)\)\): errors of about 25  % are expected in  \(\(R\&~\)\_EM\%\((3/2)\)\) with and almost 60  % without Rosenbluth separation.  The cancellation between (Subscript[A, 2]^T) and (Subscript[A, 0]^TT) amplifies the relative error.  The present estimates were made using the MAID2003 model.  Qualitatively similar results have been obtained with other models also, but the details are model dependent because these estimators involve rather delicate cancellations among many nonnegligible terms.  Therefore, the traditional Legendre analysis does not offer reliable estimates of N -> Δquadrupole ratios, with truncation errors that are especially severe for EMR.  These errors probably increase with (Q^2) as (Subscript[M, 1+]) dominance  deteriorates, but for that analysis I will return to numerical evaluation methods.  The behavior of truncated Legendre estimates for (Subscript[E, 0+]), (Subscript[S, 0+]), and (Subscript[M, 1-]) is even worse.