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Index: trunk/src/omega/src/color.ml
===================================================================
--- trunk/src/omega/src/color.ml (revision 4008)
+++ trunk/src/omega/src/color.ml (revision 4009)
@@ -1,364 +1,365 @@
(* $Id$
Copyright (C) 1999-2012 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
(* \thocwmodulesection{Quantum Numbers} *)
type t =
| Singlet
| SUN of int
| AdjSUN of int
let conjugate = function
| Singlet -> Singlet
| SUN n -> SUN (-n)
| AdjSUN n -> AdjSUN n
let compare c1 c2 =
match c1, c2 with
| Singlet, Singlet -> 0
| Singlet, _ -> -1
| _, Singlet -> 1
| SUN n, SUN n' -> compare n n'
| SUN _, AdjSUN _ -> -1
| AdjSUN _, SUN _ -> 1
| AdjSUN n, AdjSUN n' -> compare n n'
module type Line =
sig
type t
val conj : t -> t
val equal : t -> t -> bool
val to_string : t -> string
end
module type Cycles =
sig
type line
type t = (line * line) list
(* Contract the graph by connecting lines and return the number of
cycles together with the contracted graph.
\begin{dubious}
The semantics of the contracted graph is not yet 100\%ly fixed.
\end{dubious} *)
val contract : t -> int * t
(* The same as [contract], but returns only the number of cycles
and raises [Open_line] when not all lines are closed. *)
val count : t -> int
exception Open_line
(* Mainly for debugging \ldots *)
val to_string : t -> string
end
module Cycles (L : Line) : Cycles with type line = L.t =
struct
type line = L.t
type t = (line * line) list
exception Open_line
(* NB: The following algorithm for counting the cycles is quadratic since it
performs nested scans of the lists. If this was a serious problem one could
replace the lists of pairs by a [Map] and replace one power by a logarithm. *)
let rec find_fst c_final c1 disc seen = function
| [] -> ((L.conj c_final, c1) :: disc, List.rev seen)
| (c1', c2') as c12' :: rest ->
if L.equal c1 c1' then
find_snd c_final (L.conj c2') disc [] (List.rev_append seen rest)
else
find_fst c_final c1 disc (c12' :: seen) rest
and find_snd c_final c2 disc seen = function
| [] -> ((L.conj c_final, L.conj c2) :: disc, List.rev seen)
| (c1', c2') as c12' :: rest->
if L.equal c2' c2 then begin
if L.equal c1' c_final then
(disc, List.rev_append seen rest)
else
find_fst c_final (L.conj c1') disc [] (List.rev_append seen rest)
end else
find_snd c_final c2 disc (c12' :: seen) rest
let consume = function
| [] -> ([], [])
| (c1, c2) :: rest -> find_snd (L.conj c1) (L.conj c2) [] [] rest
let contract lines =
let rec contract' acc disc = function
| [] -> (acc, List.rev disc)
| rest ->
begin match consume rest with
| [], rest' -> contract' (succ acc) disc rest'
| disc', rest' -> contract' acc (List.rev_append disc' disc) rest'
end in
contract' 0 [] lines
let count lines =
match contract lines with
| n, [] -> n
| n, _ -> raise Open_line
let to_string lines =
String.concat ""
(List.map
(fun (c1, c2) -> "[" ^ L.to_string c1 ^ "," ^ L.to_string c2 ^ "]")
lines)
end
(* \thocwmodulesection{Color Flows} *)
module type Flow =
sig
type color
type t = color list * color list
val rank : t -> int
val of_list : int list -> color
val ghost : unit -> color
val to_lists : t -> int list list
val in_to_lists : t -> int list list
val out_to_lists : t -> int list list
val ghost_flags : t -> bool list
val in_ghost_flags : t -> bool list
val out_ghost_flags : t -> bool list
type power = { num : int; den : int; power : int }
type factor = power list
val factor : t -> t -> factor
val zero : factor
end
module Flow (* [: Flow] *) =
struct
type color =
| Lines of int * int
| Ghost
type t = color list * color list
let rank cflow =
2
(* \thocwmodulesubsection{Constructors} *)
let ghost () =
Ghost
let of_list = function
| [c1; c2] -> Lines (c1, c2)
| _ -> invalid_arg "Color.Flow.of_list: num_lines != 2"
let to_list = function
| Lines (c1, c2) -> [c1; c2]
| Ghost -> [0; 0]
let to_lists (cfin, cfout) =
(List.map to_list cfin) @ (List.map to_list cfout)
let in_to_lists (cfin, _) =
List.map to_list cfin
let out_to_lists (_, cfout) =
List.map to_list cfout
let ghost_flag = function
| Lines _ -> false
| Ghost -> true
let ghost_flags (cfin, cfout) =
(List.map ghost_flag cfin) @ (List.map ghost_flag cfout)
let in_ghost_flags (cfin, _) =
List.map ghost_flag cfin
let out_ghost_flags (_, cfout) =
List.map ghost_flag cfout
(* \thocwmodulesubsection{Evaluation} *)
type power = { num : int; den : int; power : int }
type factor = power list
let zero = []
let count_ghosts1 colors =
List.fold_left
(fun acc -> function Ghost -> succ acc | _ -> acc)
0 colors
let count_ghosts (fin, fout) =
count_ghosts1 fin + count_ghosts1 fout
type 'a square =
| Square of 'a
| Mismatch
let conjugate = function
| Lines (c1, c2) -> Lines (-c2, -c1)
| Ghost -> Ghost
let cross_in (cin, cout) =
cin @ (List.map conjugate cout)
let cross_out (cin, cout) =
(List.map conjugate cin) @ cout
module C = Cycles (struct
type t = int
let conj = (~-)
let equal = (=)
let to_string = string_of_int
end)
let square f1 f2 =
let rec square' acc f1' f2' =
match f1', f2' with
| [], [] -> Square (List.rev acc)
| _, [] | [], _ -> Mismatch
| Ghost :: rest1, Ghost :: rest2 ->
square' acc rest1 rest2
| Lines (0, 0) :: rest1, Lines (0, 0) :: rest2 ->
square' acc rest1 rest2
| Lines (0, c1') :: rest1, Lines (0, c2') :: rest2 ->
square' ((c1', c2') :: acc) rest1 rest2
| Lines (c1, 0) :: rest1, Lines (c2, 0) :: rest2 ->
square' ((c1, c2) :: acc) rest1 rest2
| Lines (0, _) :: _, _ | _ , Lines (0, _) :: _
| Lines (_, 0) :: _, _ | _, Lines (_, 0) :: _ -> Mismatch
| Lines (_, _) :: _, Ghost :: _ | Ghost :: _, Lines (_, _) :: _ -> Mismatch
| Lines (c1, c1') :: rest1, Lines (c2, c2') :: rest2 ->
square' ((c1', c2') :: (c1, c2) :: acc) rest1 rest2 in
square' [] (cross_out f1) (cross_out f2)
(* In addition to counting closed color loops, we also need to count closed
gluon loops. Fortunately, we can use the same algorithm on a different
data type, provided it doesn't require all lines to be closed. *)
module C2 = Cycles (struct
type t = int * int
let conj (c1, c2) = (- c2, - c1)
let equal (c1, c2) (c1', c2') = c1 = c1' && c2 = c2'
let to_string (c1, c2) = "(" ^ string_of_int c1 ^ "," ^ string_of_int c2 ^ ")"
end)
let square2 f1 f2 =
let rec square2' acc f1' f2' =
match f1', f2' with
| [], [] -> Square (List.rev acc)
| _, [] | [], _ -> Mismatch
| Ghost :: rest1, Ghost :: rest2 ->
square2' acc rest1 rest2
| Lines (0, 0) :: rest1, Lines (0, 0) :: rest2 ->
square2' acc rest1 rest2
| Lines (0, _) :: rest1, Lines (0, _) :: rest2
| Lines (_, 0) :: rest1, Lines (_, 0) :: rest2 ->
square2' acc rest1 rest2
| Lines (0, _) :: _, _ | _ , Lines (0, _) :: _
| Lines (_, 0) :: _, _ | _, Lines (_, 0) :: _ -> Mismatch
| Lines (_, _) :: _, Ghost :: _ | Ghost :: _, Lines (_, _) :: _ -> Mismatch
| Lines (c1, c1') :: rest1, Lines (c2, c2') :: rest2 ->
square2' (((c1, c1'), (c2, c2')) :: acc) rest1 rest2 in
square2' [] (cross_out f1) (cross_out f2)
-(* Surprisingly, this is missing from [Pervasives]! *)
+(* $\ocwlowerid{int\_power}: n\, p \to n^p$
+ for integers is missing from [Pervasives]! *)
let int_power n p =
let rec int_power' acc i =
if i < 0 then
invalid_arg "int_power"
else if i = 0 then
acc
else
int_power' (n * acc) (pred i) in
int_power' 1 p
(* Instead of implementing a full fledged algebraic evaluator, let's
simply expand the binomial by hand:
\begin{equation}
\left(\frac{N_C^2-2}{N_C^2}\right)^n =
\sum_{i=0}^n \binom{n}{i} (-2)^i N_C^{-2i}
\end{equation} *)
(* NB: Any result of [square] other than [Mismatch] guarantees
[count_ghosts f1 = count_ghosts f2]. *)
let factor f1 f2 =
match square f1 f2, square2 f1 f2 with
| Mismatch, _ | _, Mismatch -> []
| Square f12, Square f12' ->
let num_cycles = C.count f12
and num_cycles2, disc = C2.contract f12'
and num_ghosts = count_ghosts f1 in
(*i Printf.eprintf "f12 = %s -> #loops = %d\n"
(C.to_string f12) num_cycles;
Printf.eprintf "f12' = %s -> #loops = %d, disc = %s\n"
(C2.to_string f12') num_cycles2 (C2.to_string disc);
flush stderr; i*)
List.map
(fun i ->
let parity = if num_ghosts mod 2 = 0 then 1 else -1
and power = num_cycles - num_ghosts in
let coeff = int_power (-2) i * Combinatorics.binomial num_cycles2 i
and power2 = - 2 * i in
{ num = parity * coeff;
den = 1;
power = power + power2 })
(ThoList.range 0 num_cycles2)
end
(* later: *)
module General_Flow =
struct
type color =
| Lines of int list
| Ghost of int
type t = color list * color list
let rank_default = 2 (* Standard model *)
let rank cflow =
try
begin match List.hd cflow with
| Lines lines -> List.length lines
| Ghost n_lines -> n_lines
end
with
| _ -> rank_default
end
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)
Index: trunk/src/omega/src/color.mli
===================================================================
--- trunk/src/omega/src/color.mli (revision 4008)
+++ trunk/src/omega/src/color.mli (revision 4009)
@@ -1,74 +1,81 @@
(* $Id$
Copyright (C) 1999-2012 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
(* \thocwmodulesection{Quantum Numbers} *)
(* Color is not necessarily the~$\textrm{SU}(3)$ of QCD. Conceptually,
it can be any \emph{unbroken} symmetry (\emph{broken} symmetries correspond
to [Model.flavor]). In order to keep the group theory simple, we confine
ourselves to the fundamental and adjoint representation
of a single~$\textrm{SU}(N_C)$ for the moment. Therefore,
particles are either color singlets or live in the defining
representation of $\textrm{SU}(N_C)$: [SUN]$(|N_C|)$, its conjugate
[SUN]$(-|N_C|)$ or in the adjoint representation of
$\textrm{SU}(N_C)$: [AdjSUN]$(N_C)$. *)
type t = Singlet | SUN of int | AdjSUN of int
val conjugate : t -> t
val compare : t -> t -> int
(* \thocwmodulesection{Color Flows} *)
module type Flow =
sig
type color
type t = color list * color list
val rank : t -> int
val of_list : int list -> color
val ghost : unit -> color
val to_lists : t -> int list list
val in_to_lists : t -> int list list
val out_to_lists : t -> int list list
val ghost_flags : t -> bool list
val in_ghost_flags : t -> bool list
val out_ghost_flags : t -> bool list
+(* A factor is a list of powers
+ \begin{equation}
+ \sum_{i}
+ \left( \frac{\ocwlowerid{num}_i}{\ocwlowerid{den}_i}
+ \right)^{\ocwlowerid{power}_i}
+ \end{equation} *)
type power = { num : int; den : int; power : int }
type factor = power list
+
val factor : t -> t -> factor
val zero : factor
end
module Flow : Flow
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)

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