# piKaon.wxm 

 π(+, m, p1) + K(+, M, p2) --> π(+, m, p3) + K(+, M, p4)

Pi Kaon Elastic QED Scattering

piKaon.wxm,  Dec. 19, 2018
 Edwin (Ted) Woollett,   Maxima by Example Ch. 12,
Dirac Algebra and Quantum Electrodynamics, ver. 3, "Dirac3"
http://web.csulb.edu/~woollett/

 We use Heaviside-Lorentz electromagnetic units, e^2 = 4πα
In general, we follow the conventions in Peskin & Schroeder: Quantum Field Theory.

We ignore the finite size and quark content (and potential strong 
interaction effects in small impact parameter collisions) in this 
example of scalar QED, and only take into account the first order
electromagnetic interactions of these spinless bosons.

First load the dirac3 package by loading dirac3.mac:

load(dirac3);

Note from the last two printouts, that invar_flag = true and stu_flag = false (our defaults).

The fact that invar_flag = true means that the symbolic trace function tr
will automatically make use of the list invarR (produced by the function set_invarR) which defines
the replacements for D(pa,pb) in terms of the Mandelstam variables s, t, u.
(After the package loads, invarR has the default value [].)

The fact that stu_flag = false means that the function noncov will not
automatically replace the Mandelstam variables s, t, u, by s_th, t_th, and u_th.

Dirac3 package functions used here are set_invarR, ev_Ds, pullfac,  comp_def,
noncov, noncov_ratio, VP, sub_stu, and div_tb.

## Center of Momentum Cross Section Without Using Mandelstam Variables

Dirac3 package functions used in this section are comp_def, noncov, noncov_ratio, and div_tb.

The fastest approach is to express the reduced amplitude in terms of D(pa,pb)'s, 
define 4-momentum components using comp_def, and then use noncov_ratio on the result.

D(pa, pb) is a symbolic representation of the 4-vector dot product of
4-momenta pa and pb.  D(pb, pa) = D(pa, pb) since we have declared 
D symmetric in our package.

We define Mr as the "reduced amplitude" for the scattering process,
  where Mfi is the amplitude:
            Mfi =  -e^2 Mr

Mr : D(p1+p3,p2+p4)/D(p1 - p3, p1 - p3);

 
We next define 4-momentum components in the CM frame, using our function
 comp_def, with p representing the magnitude of 3-momentum, Epi the 
 pion particle energy, EK the kaon particle energy, with the initial 
 particles approaching along the z-axis, the final particles confined
 to the z-x plane, and th representing the cms scattering angle th of the final pion relative to the
direction of the incident pion (the positive z-axis).

comp_def ( pa(Ea, pax, pay, paz), ....) is the syntax used here.

assume ( p > 0, th >= 0, th <= %pi )$
comp_def ( p1( Epi, 0, 0, p),
             p2( EK, 0, 0, -p),
             p3 (Epi, p*sin(th), 0, p*cos(th)),
             p4 (EK, -p*sin(th), 0, -p*cos(th)) )$

noncov_ratio (Mr);

We use here the package function div_tb(expr, fac),  divide top and bottom
 of expr  by fac.
 

Mr_th :  div_tb(%, 2*p^2);

The above is the "reduced amplitude" Mr for arbitrary energies.

For high energy collisions in CM frame, neglect masses, and particle energies
are the same as the 3-momentum magnitude p --> E, and Mr becomes:

Mr_th_HE :   Mr_th, Epi = E, EK = E, p = E;

Now for the differential scattering cross section dsigdo in the CMS.

We have absorbed e^4 into the constant A, with e^2 = 4πα.  

In general 2 --> 2 scattering, A = α^2 * (pf/pi) /(4*s) in cms
  where s = E(cms)^2.

Let p be  the magnitude of the 3-momentum of the initial electron and
initial Kaon and pf be the magnitude of the final electron and final Kaon in CMS.

Conservation of energy implies
   sqrt(m^2 + p^2) + sqrt(M^2 + p^2) = sqrt(m^2 + pf^2) + sqrt(M^2 + pf^2)

This implies pf = p, so pf = pi and A = α^2/( 4 s).

A : alpha^2/(4*s);

s is the Mandelstam variable (p1 + p2)^2  = E(cms)^2

s_th : noncov (D(p1 + p2, p1 + p2));

s_th_HE : s_th, Epi = E, EK = E;

For arbitrary energies (note Mr_th is real):

dsigdo : A * Mr_th^2;

and in the high energy limit:

dsigdo_HE : A * Mr_th_HE^2;

## Cross Section Using Mandelstam Variables

### Use set_invarR to construct list invarR; Use ev_Ds( expr ) to replace
4-momentum dot products with Mandelstam variables s, t, and u.

Use our  set_invarR function to express all invariant dot products of 4-momenta 
in terms of the Mandelstam variables s, t,  and u, as well as the π mass m 
and the kaon mass M. The use of the function set_invarR in this way 
populates a list called invarR, which is later used by other package 
functions to transform expressions.

Recall conservation of 4-momentum: p1^μ + p2^μ = p3^μ + p4^μ
 and our definitions of the Mandelstam variables for this process:

    s = (p1 + p2)^2 = (p3 + p4)^2, 
    t = (p1 - p3)^2   = (p2 - p4)^2,
    u = (p1- p4)^2   = (p2 - p3)^2

  s, t, and u are not all independent since the sum s + t + u = sum of mass-square,
  or here:
   s + t + u = 2 m^2 + 2 M^2.

The following dot product replacement rules come from the definitions
of s, t, and u, as well as our chosen symbols for particle masses.

set_invarR (D(p1,p1) = m^2,
         D(p1,p2) =  (s - m^2 - M^2)/2,
         D(p1,p3) = (2*m^2 - t)/2,
         D(p1,p4) = (m^2 + M^2 - u)/2,
         D(p2,p2) = M^2,
         D(p2,p3) = (M^2 + m^2 - u)/2,
         D(p2,p4) = (2*M^2 - t)/2,
         D(p3,p3) = m^2,
         D(p3,p4) = (s - m^2 - M^2)/2,
         D(p4,p4) = M^2)$

invarR;

Use of ev_Ds (expr) function to replace D(pa,pb) symbols with expressions
in terms of the Mandelstam variables s, t, and u, once the list invarR has
been populated using the function set_invarR. Also use of pullfac(expr,fac) to
pull a factor (fac) out of expression expr.

We define Mr as the "reduced amplitude" for the scattering process,
  where Mfi is the amplitude:
            Mfi =  -e^2 Mr

In lowest order there is one photon exchange diagram contributing. 
We could immediately replace D (p1 - p3, p1 - p3) with t, but let's leave
the denominator unevaluated to show how ev_Ds( expr) uses the list
invarR to replace all 4-momentum dot products in terms of s, t, and u.

Mr : D(p1+p3,p2+p4)/D(p1 - p3, p1 - p3);

Mr : ev_Ds (Mr);

Mr : pullfac(Mr,1/t);

If we had used ratsimp(Mr) instead of pullfac(Mr,1/t), we would have 
ended up with -(u - s)/t. 

We now want to evaluate the (reduced) amplitude in terms of CM frame parameters.
Our package has a function sub_stu(expr) which replaces, in expr, the
  symbol s by s_th, replaces the symbol t by t_th, and replaces the symbol
 u by u_th. We can use fundef to see the function definition:

fundef(sub_stu);

Note: trigsimp(%%) in this context means evaluate trigsimp of the previous
 expression in this function definition.

If you have previously bound values to the three symbols s_th, t_th, 
and u_th, the expression produced by sub_stu will reflect those bindings.

We have already used   comp_def ("components-define") to define
arrays which hold the components of the 4-momentum vectors in the CM frame
above.  We can then use either noncov or VP  to evaluate
the Mandelstam variables in terms of the CM frame parameters p,
Epi, EK, and the scattering angle th, as we show below.
   

### Define s_th, t_th, and u_th using VP or noncov

In order to use sub_stu(expr) to substitute values of s_th, t_th, and
 u_th  (Mandelstam variables in terms of frame dependent parameters) for s,
 t, and u, it is easiest to use our function VP (pa,pb)  (Vector Product)
 for evaluation of the vector products of 4-momenta.

Returning to the task of giving values to the symbols s_th, t_th, and u_th:
From the definitions of the Mandelstam varibles:

We defined s_th above using noncov:

s_th;

 VP (p1+p2,p1+p2), factor;

t_th : VP (p1-p3,p1-p3), factor;

u_th :  VP (p1-p4,p1-p4);

### Use sub_stu (expr) to replace s, t, u by s_th, t_th, u_th

Mr;

Mr_th : sub_stu(Mr), ratsimp;

ratsimp was used to define a common denominator and achieve
a rational expression.

As we did above we  use here the package function div_tb(expr,fac),(divide top and bottom
by the same thing) which returns the result of dividing both the numerator 
and denominator (top and bottom) of expr by fac. 

You might think this ratio would work here:

(num(Mr_th/p^2) / denom(Mr_th/p^2));

The function div_tb (expr, fac) is more complicated on purpose:

fundef (div_tb);

so here goes:

  Mr_th :   div_tb (Mr_th, p^2);

The above is the "reduced amplitude" Mr for arbitrary energies, and is the same
expression we derived earlier without using Mandelstam variables.

For high energy collisions in CM frame, neglect masses, and particle energies
are the same as the 3-momentum magnitude p, and Mr becomes:

subst([Epi=p,EK=p],Mr_th);

We can also use the interactive substitution method:

Mr_th_HE :   Mr_th, Epi = p, EK = p;

Now for the differential scattering cross section dsigdo in the CMS.
We have already defined the prefactor A:

A;

We can replace s by s_th,   E(cms)^2: 

s_th;

For arbitrary energies (note Mr_th is real):

dsigdo : A * Mr_th^2;

 and in the high energy limit:

dsigdo_HE : A * Mr_th_HE^2;