Parameterisation of energy loss and total cross-section

Instead of using the explicit formula () and (), we have chosen to parameterise directly $E$_loss^pair(Z,T, E_c ) and $\sigma (Z,T,E$_c) as:

$\sigma (Z,T,E$c)	$=$	$Z\; [Z+\; \xi$σ(1+ γlnZ)][ln{EmaxE}]αFσ(Z,X,Y)
$E$losspair(Z,T,Ec)	$=$	$Z\; [Z+\; \xi$l(1+ δlnZ)]E[ {Ec- EcminE}]βFl(Z,X,Y)

where $\xi$_σ,l , $\alpha$ , $\beta$ , $\gamma$ and $\delta$

are parameters obtained by a fitting procedure and $E$_c^min= E&sp;v_min= 4 m_e and $E$_max= E&sp;v_max

The functions $F$_i (Z,X,Y) ($i=\sigma ,\hspace{1em}l$ ) have the form: $F$_i(Z,X,Y) = F_i0(X,Y)+ZF_i1(X,Y)

where $F$_ij(X,Y) denotes a function constructed from two polynomials $F$_ij(X,Y) =

$P$ijneg(X,Y)	$if$	$Y\; \le 0$
$P$ijpos(X,Y)	$if$	$Y\; >\; 0$

$.$

and $P$_ij^neg , $P$_ij^pos fulfil the conditions:

$P$_ij^neg(X,Y)_Y=0= P_ij^pos(X,Y)_Y=0&sp;. {P_ij^negY }|_Y=0=. {P_ij^posY }|_Y=0

The detailed form of the $P$_ij polynomials is:

$P$i0neg(X,Y)	$=$	$(C$1+C2X+...+C6X5)+(C7+C8X+...+C12X5)Y
	$+$	$(C$13+C14X+...+C18X5)Y2+...+(C31+...+C36X5)Y5
$P$i0pos(X,Y)	$=$	$(C$1+C2X+...+C6X5)+(C7+C8X+...+C12X5)Y
$$	$+$	$(C$37+C38X+...+C42X5)Y2+...+(C55+...+C60X5)Y5
$P$i1neg(X,Y)	$=$	$(C$61+C62X+...+C65X4)+(C66+C67X+...+C70X4)Y
	$+$	$(C$71+C72X+...+C75X4)Y2+...+(C81+...+C85X4)Y4
$P$i1pos(X,Y)	$=$	$(C$61+C62X+...+C65X4)+(C66+C67X+...+C70X4)Y
	$+$	$(C$86+C87X+...+C90X4)Y2+...+(C96+...+C100X4)Y4

with

$X\; =\; ln\{Em$_μ}&sp;Y = ln{E_cv_jE}&sp;j = σ, l

where $v$_σ,l are additional parameters of the fit. By a numerical (twofold) integration of the formulae () and () above, we have calculated 1800 data points in the range Z = 1, 6, 13, 26, 50, 82, 92 and the energy ranges 1GeV $\le T\; \le$ 10 TeV and $4m\; \le E$_c≤T

and performed a least-squares fit to determine the parameters. The fitted values of the parameters are in the DATA statements in the functions GPRSGM and GPRELM, which compute the formulae () and () respectively. The accuracy of the fit is:

The function GPRELM contains a second formula to calculate the total energy lost by the muon due to direct -production used when $E$_c≥E_max= E-0.75 e m_μZ^1/3 . This formula describes the total energy loss with an error less than 1%:

$E$losspair(Z,T)	$=$	$E$losspair(Z,T,Ec=Emax)
	$=$	$Z(Z+1)\; E$max[d1+(d2X+d3Y)+(d4X2+d5XY+d6Y2)+
	$+$	$...+(d$22X6+d23X5Y+...+d28Y6)]

where $X=ln(E/m$_μ) and $Y=Z1/3$ . The fitted parameters $d$_i can be found in the DATA statement in the function GPRELM.

PHYS451