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Text File  |  1991-06-14  |  73KB  |  3,468 lines
  1. C Calculation of the one loop longitudinal WW sattering amplitude.
  2.   M. Veltman and F. Yndurain.
  3.  
  4.  U-particle with mass M or m depending on _Sw0=1 or 0.
  5.  Can be set on command line: S=1
  6.  
  7.   Running times quoted are for a 68020 system. For 68000 multiply by 2.
  8.  
  9.   External file used: WWb.e
  10.  
  11. C WW-scattering 1.  Polarization vectors.
  12. C WW-scattering 2.  Tree amplitude, general expression.
  13.             Produces file TreeWW, containing TreeWW.
  14. C WW-scattering 3.  Tree amplitude, case of longitudinal W's.
  15.             Uses output of 2.
  16. C WW-scattering 4.  Tree amplitude, Fi-Fi scattering.
  17. C WW-scattering 5.  One loop diagrams, two external lines.
  18. C WW-scattering 6.  One loop diagrams, three external lines.
  19. C WW-scattering 7.  One loop Fi-Fi scattering, part 1. 33 sec.
  20.             Produces BoxFF1, containing BoxFF.
  21. C WW-scattering 8.  One loop Fi-Fi scattering, part 2. 228 sec.
  22.             Uses output from 7.
  23.             Produces BoxFF2, containing BoxFF.
  24. C WW-scattering 9.  One loop Fi-Fi scattering, part 3. 125 sec.
  25.             Uses output from 8.
  26.             Produces BoxFF_comm, containing Ftot.
  27. C WW-scattering 10. Fi-Fi Renormalization. Result Fi-Fi amplitude.
  28.             Uses output from 8.
  29. C WW-scattering 11. One loop W-W scattering, part 1. 681 sec.
  30.             Produces BoxWW1, containing BoxWW.
  31. C WW-scattering 12. One loop W-W scattering, part 2. 605 sec.
  32.             Uses output from 11.
  33.             Produces BoxWW2, containing BoxWW.
  34. C WW-scattering 13. One loop W-W scattering, part 3. 447 sec.
  35.             Uses output from 12.
  36.             Produces BoxWW_comm, containing Wtot.
  37. C WW-scattering 14. W-W Renormalization. Result W-W amplitude.
  38.             Uses output from 13.
  39. C WW-scattering 15. Verification of part of WW scattering calculation.
  40.             Uses output from 11.
  41. C WW-scattering 16. Infinities and Log's of irreducible 4-point W function.
  42.  
  43. *end
  44.  
  45. C WW-scattering 1.  Polarization vectors.
  46.   Dot-products in restframe.
  47.   Verification of equations used below.
  48.  
  49. C A k0,kl,sin,cos,M
  50.  
  51. D Xk(n) = 0,    0,    kl,        i*k0
  52. D Xp(n) = 0,    0,    -kl,        i*k0
  53. D Xpp(n) = -kl*sin,    0,    -kl*cos,    i*k0
  54. D Xkp(n) = kl*sin,    0,    kl*cos,        i*k0
  55.  
  56. D Xek(n) = 0,    0,    k0/M,        i*kl/M
  57. D Xep(n) = 0,    0,    -k0/M,        i*kl/M
  58. D Xfp(n)=-k0*sin/M,    0,    -k0*cos/M,    i*kl/M
  59. D Xfk(n)=k0*sin/M,    0,    k0*cos/M,    i*kl/M
  60.  
  61. X Dot(Xk,Xp) = DS(j,1,4,(Xk(j)*Xp(j)))
  62.  
  63. Z ekDp  = Dot(Xek,Xp)
  64. Z ekDpp = Dot(Xek,Xpp)
  65. Z ekDep = Dot(Xek,Xep)
  66. Z ekDfk = Dot(Xek,Xfk)
  67. Z ekDfp = Dot(Xek,Xfp)
  68. Z epDk  = Dot(Xep,Xk)
  69. Z epDpp = Dot(Xep,Xpp)
  70. Z epDfk = Dot(Xep,Xfk)
  71. Z epDfp = Dot(Xep,Xfp)
  72. Z fkDk  = Dot(Xfk,Xk)
  73. Z fkDp  = Dot(Xfk,Xp)
  74. Z fkDfp = Dot(Xfk,Xfp)
  75. Z fpDk  = Dot(Xfp,Xk)
  76. Z fpDp  = Dot(Xfp,Xp)
  77. Id,sin^2=1-cos^2
  78. *end
  79.  
  80. C WW-scattering 2.  Tree amplitude, general expression.
  81.   Result is written to file TreeWW. Used in part 3.
  82.  
  83.   The term V4 is maintained to show how the pure 4-vertex behaves.
  84.   Renamed Fourv in part 3.
  85.  
  86. P ninput
  87.  
  88. Read WWb.e
  89. VERT{}
  90. *fix
  91.  
  92. Common TreeWW
  93.  
  94. P stats
  95. I mu,nu
  96. I a=3,b=3,c=3,d=3,e=3,f=3,g=3,h=3,j=3
  97.  
  98. A kl,k0,sin,cos
  99.  
  100. Z TreeWW(al,be,ga,de) = Tree("W,a,al,k,"W,b,be,p,"W,c,ga,pp,"W,d,de,kp)
  101.  
  102. Id,Tree(I1~,a~,al~,k~,I2~,b~,be~,p~,I3~,c~,ga~,pp~,I4~,d~,de~,kp~) =
  103.  
  104.    DS(I1;I4;-J,(TreeT(I1,I2,I3,I4,J,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  105.  + DS(I1;I2;-K,(TreeS(I1,I2,I3,I4,K,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  106.  + DS(I1;I3;-L,(TreeU(I1,I2,I3,I4,L,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  107.  
  108.  + (1+V4)*VE4(I1,I2,I3,I4,*,a,al,k,*,b,be,p,*,c,ga,pp,*,d,de,kp)
  109.  
  110. Id,TreeS(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  111.   VE3(I1,I2,-K1,*,a,al,k,*,b,be,p,*,l1,ka,-qs)*
  112.   VE3(I3,I4,K1,*,c,ga,-pp,*,d,de,-kp,*,l2,kap,qs)*
  113.   PROP(K1,-K1,*,l1,ka,qs,*,l2,kap,-qs)
  114. Al,TreeT(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  115.   VE3(I1,I4,-K1,*,a,al,k,*,d,de,-kp,*,l1,ka,-qt)*
  116.   VE3(I2,I3,K1,*,b,be,p,*,c,ga,-pp,*,l2,kap,qt)*
  117.   PROP(K1,-K1,*,l1,ka,qt,*,l2,kap,-qt)
  118. Al,TreeU(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  119.   VE3(I1,I3,-K1,*,a,al,k,*,c,ga,-pp,*,l1,ka,-qu)*
  120.   VE3(I2,I4,K1,*,b,be,p,*,d,de,-kp,*,l2,kap,qu)*
  121.   PROP(K1,-K1,*,l1,ka,qu,*,l2,kap,-qu)
  122.  
  123. Id,Anti,TAP
  124.  
  125. Id,Compo,<X>,VE4,VE3,PROP
  126. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  127.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  128. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  129.        FF(l1,al,k,l2,be,q,l3,ga,p)
  130. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  131.  
  132. Id,kp(al~)=p(al)+k(al)-pp(al)
  133. Al,Dotpr,kp(al~)=p(al)+k(al)-pp(al)
  134. *yep
  135. Id,qt(al~)=-p(al)+pp(al)
  136. Al,Dotpr,qt(al~)=-p(al)+pp(al)
  137. Id,qs(al~)=p(al)+k(al)
  138. Al,Dotpr,qs(al~)=p(al)+k(al)
  139. Id,qu(al~)=k(al)-pp(al)
  140. Al,Dotpr,qu(al~)=k(al)-pp(al)
  141. Id,k(al)=0
  142. Al,p(be)=0
  143. Al,pp(ga)=0
  144. Al,pp(de)=p(de)+k(de)
  145. Id,pDp=-M^2
  146. Al,kDk=-M^2
  147. Al,ppDpp=-M^2
  148. Id,Epfred
  149. *yep
  150. C s = - (p+k)^2 = -pDp - 2*pDk - kDk
  151.   t = - (p-pp)^2 = - pDp + 2*pDpp - ppDpp
  152.   u = - (k-pp)^2 = - kDk + 2*kDpp - ppDpp
  153.   s + t + u = 4*M^2
  154.  
  155. Id,pDk =-0.5*s-0.5*pDp-0.5*kDk
  156. Al,pDpp= 0.5*t+0.5*pDp+0.5*ppDpp
  157. Al,kDpp= 0.5*u+0.5*kDk+0.5*ppDpp
  158. Id,pDp=-M^2
  159. Al,kDk=-M^2
  160. Al,ppDpp=-M^2
  161. *yep
  162.  
  163. Id,NOM(-qs,M)=-1/s*(1+M^2/s+M^4/s^2)
  164. Al,NOM(-qt,M)=-1/t*(1+M^2/t+M^4/t^2)
  165. Al,NOM(-qu,M)=-1/u*(1+M^2/u+M^4/u^2)
  166. Id,Count,-3,s,2,t,2,u,2,k,1,p,1,pp,1,V4,10
  167. *begin
  168.  
  169. Write TreeWW
  170. *end
  171.  
  172. C WW-scattering 3.  Tree amplitude, case of longitudinal W's.
  173.             Uses output of 2.
  174.  
  175.   Filling in the transversal polarization vectors.
  176.  
  177. Enter TreeWW
  178. *fix
  179.  
  180. P ninput
  181.  
  182. V ek,ep,fp,fk
  183. A cos,sin
  184.  
  185. Names TreeWW
  186.  
  187. Z Ampl= ek(al)*ep(be)*fp(ga)*fk(de) * TreeWW(al,be,ga,de)
  188.  
  189. C In restframe:
  190.   k  = 0    0    kl        i*k0
  191.   p  = 0    0    -kl        i*k0
  192.   pp = -kl*sin    0    -kl*cos        i*k0
  193.   kp = kl*sin    0    kl*cos        i*k0
  194.  
  195.   ek = 0    0    k0/M        i*kl/M
  196.   ep = 0    0    -k0/M        i*kl/M
  197.   fp=-k0*sin/M    0    -k0*cos/M    i*kl/M
  198.   fk=k0*sin/M    0    k0*cos/M    i*kl/M
  199.  
  200. Id,ekDp  = -2*k0*kl/M
  201. Al,ekDpp = -kl*k0*cos/M - kl*k0/M
  202. Al,ekDep = -k0^2/M^2 - kl^2/M^2
  203. Al,ekDfk= k0^2*cos/M^2 - kl^2/M^2
  204. Al,ekDfp=-k0^2*cos/M^2 - kl^2/M^2
  205. Id,epDk  = -2*k0*kl/M
  206. Al,epDpp = kl*k0*cos/M - kl*k0/M
  207. Al,epDfk=-k0^2*cos/M^2 - kl^2/M^2
  208. Al,epDfp= k0^2*cos/M^2 - kl^2/M^2
  209. Id,fkDk = k0*kl*cos/M - k0*kl/M
  210. Al,fkDp =-k0*kl*cos/M - k0*kl/M
  211. Al,fkDfp= -k0^2/M^2 - kl^2/M^2
  212. Id,fpDk = -k0*kl*cos/M - k0*kl/M
  213. Al,fpDp = k0*kl*cos/M - k0*kl/M
  214. *yep
  215.  
  216. C Remember:    pDk  =-0.5*s + M^2
  217.         pDpp = 0.5*t - M^2
  218.         kDpp = 0.5*u - M^2
  219. Id,cos=1+0.5*t/kl^2
  220. Id,Multi,kl^2 = 0.25*s - M^2
  221. Al,Multi,k0^2=0.25*s
  222. Id,Count,2,s,2,t,2,u,2,NOM,2,kl,1
  223. *yep
  224.  
  225. Id,Multi,kl^-2= 4/s*(1 + 4*M^2/s + 16*M^4/s^2)
  226. Id,Count,2,s,2,t,2,u,2,NOM,2,kl,1
  227. *yep
  228.  
  229. IF s
  230. Id,3,s^n~*u^-2 = s^(n-1)*(- t - u + 4*M^2)/u^2
  231. ENDIF
  232. Id,Count,2,s,2,t,2,u,2,NOM,2,kl,1
  233. *yep
  234.  
  235. IF s
  236. Id,3,s^n~*u^-1 = s^(n-1)*(- t - u + 4*M^2)/u
  237. ENDIF
  238. Id,Count,2,s,2,t,2,u,2,NOM,-2
  239. *yep
  240.  
  241. Id,t^3*u^-1=t^2*(- s - u + 4*M^2)/u
  242. Id,Count,2,s,2,t,2,u,2,NOM,-2
  243. *yep
  244.  
  245. C Id,u*s^-1 = ( - s - t + 4*M^2)/s
  246. Id,u=- s -t + 4*M^2
  247. Id,Count,2,s,2,t,2,u,2,NOM,-2,V4,10
  248. IF D(a,c)
  249. Id,t=-s-u
  250. ENDIF
  251.  
  252. F Fourv
  253. Id,V4=Fourv
  254. *end
  255.  
  256. C WW-scattering 4.  Tree amplitude, Fi-Fi scattering.
  257.  
  258.   Tree diagrams with four FI lines.
  259.   Verifying the equivalence theorem.
  260.  
  261. P ninput
  262. Read WWb.e
  263. VERT{}
  264. *fix
  265.  
  266. P stats
  267. I mu,nu
  268. I a=3,b=3,c=3,d=3,e=3,f=3,g=3,h=3,j=3
  269.  
  270. A kl,k0,sin,cos
  271.  
  272. Z TreeFF = Tree("F,a,al,k,"F,b,be,p,"F,c,ga,pp,"F,d,de,kp)
  273.  
  274. Id,Tree(I1~,a~,al~,k~,I2~,b~,be~,p~,I3~,c~,ga~,pp~,I4~,d~,de~,kp~) =
  275.  
  276.    DS(I1;I4;-J,(TreeT(I1,I2,I3,I4,J,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  277.  + DS(I1;I2;-K,(TreeS(I1,I2,I3,I4,K,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  278.  + DS(I1;I3;-L,(TreeU(I1,I2,I3,I4,L,a,al,k,b,be,p,c,ga,pp,d,de,kp)))
  279.  
  280.  + VE4(I1,I2,I3,I4,*,a,al,k,*,b,be,p,*,c,ga,pp,*,d,de,kp)
  281.  
  282. Id,TreeS(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  283.   VE3(I1,I2,-K1,*,a,al,k,*,b,be,p,*,l1,ka,-qs)*
  284.   VE3(I3,I4,K1,*,c,ga,-pp,*,d,de,-kp,*,l2,kap,qs)*
  285.   PROP(K1,-K1,*,l1,ka,qs,*,l2,kap,-qs)
  286. Al,TreeT(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  287.   VE3(I1,I4,-K1,*,a,al,k,*,d,de,-kp,*,l1,ka,-qt)*
  288.   VE3(I2,I3,K1,*,b,be,p,*,c,ga,-pp,*,l2,kap,qt)*
  289.   PROP(K1,-K1,*,l1,ka,qt,*,l2,kap,-qt)
  290. Al,TreeU(I1~,I2~,I3~,I4~,K1~,a~,al~,k~,b~,be~,p~,c~,ga~,pp~,d~,de~,kp~)=
  291.   VE3(I1,I3,-K1,*,a,al,k,*,c,ga,-pp,*,l1,ka,-qu)*
  292.   VE3(I2,I4,K1,*,b,be,p,*,d,de,-kp,*,l2,kap,qu)*
  293.   PROP(K1,-K1,*,l1,ka,qu,*,l2,kap,-qu)
  294.  
  295. Id,Anti,TAP
  296.  
  297. Id,Compo,<X>,VE4,VE3,PROP
  298. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  299.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  300. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  301.        FF(l1,al,k,l2,be,q,l3,ga,p)
  302. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  303.  
  304. Id,kp(al~)=p(al)+k(al)-pp(al)
  305. Al,Dotpr,kp(al~)=p(al)+k(al)-pp(al)
  306. *yep
  307. Id,qt(al~)=-p(al)+pp(al)
  308. Al,Dotpr,qt(al~)=-p(al)+pp(al)
  309. Id,qs(al~)=p(al)+k(al)
  310. Al,Dotpr,qs(al~)=p(al)+k(al)
  311. Id,qu(al~)=k(al)-pp(al)
  312. Al,Dotpr,qu(al~)=k(al)-pp(al)
  313. C Id,pDp=-M^2
  314. C Al,kDk=-M^2
  315. C Al,ppDpp=-M^2
  316. Id,Epfred
  317. *yep
  318. C s = - (p+k)^2 = -pDp - 2*pDk - kDk
  319.   t = - (p-pp)^2 = - pDp + 2*pDpp - ppDpp
  320.   u = - (k-pp)^2 = - kDk + 2*kDpp - ppDpp
  321.   s + t + u = - kDk - pDp - ppDpp - kpDkp = 4*M^2
  322.  
  323. Id,pDk =-0.5*s-0.5*pDp-0.5*kDk
  324. Al,pDpp= 0.5*t+0.5*pDp+0.5*ppDpp
  325. Al,kDpp= 0.5*u+0.5*kDk+0.5*ppDpp
  326. C Id,pDp=-M^2
  327. C Al,kDk=-M^2
  328. C Al,ppDpp=-M^2
  329. *yep
  330. Id,NOM(-qs,m)= 1/m^2 + s/m^4 + s^2/m^4*NOM(-qs,m)
  331. Al,NOM(-qt,m)= 1/m^2 + t/m^4 + t^2/m^4*NOM(-qt,m)
  332. Al,NOM(-qu,m)= 1/m^2 + u/m^4 + u^2/m^4*NOM(-qu,m)
  333. *yep
  334. Id,u = - s - t - kDk - pDp - ppDpp - kpDkp
  335. IF D(a,c)
  336. Id,t = - s - u - kDk - pDp - ppDpp - kpDkp
  337. ENDIF
  338. Id,NOM(-qs,M)=-1/s*(1+M^2/s+M^4/s^2)
  339. Al,NOM(-qt,M)=-1/t*(1+M^2/t+M^4/t^2)
  340. Al,NOM(-qu,M)=-1/u*(1+M^2/u+M^4/u^2)
  341. *yep
  342. IF s
  343. Id,3,s^n~*u^-2 = s^(n-1)*(- t - u + 4*M^2)/u^2
  344. ENDIF
  345. Id,Count,2,s,2,t,2,u,2,NOM,2,kl,1,m,20
  346. *yep
  347. IF s
  348. Id,3,s^n~*u^-1 = s^(n-1)*(- t - u + 4*M^2)/u
  349. ENDIF
  350. Id,Count,2,s,2,t,2,u,2,NOM,-2,m,20
  351. *yep
  352. Id,t^3*u^-1=t^2*(- s - u + 4*M^2)/u
  353. Id,Count,2,s,2,t,2,u,2,NOM,-2,m,20
  354. *yep
  355. C Id,u*s^-1 = ( - s - t + 4*M^2)/s
  356. Id,u= - s -t + 4*M^2
  357. Id,Count,2,s,2,t,2,u,2,NOM,-2,m,20
  358. *yep
  359. Id,NOM(-qs,m)= 1/m^2*( 1 + s/m^2 + s^2/m^4)
  360. Al,NOM(-qt,m)= 1/m^2*( 1 + t/m^2 + t^2/m^4)
  361. Al,NOM(-qu,m)= 1/m^2*( 1 + u/m^2 + u^2/m^4)
  362. Id,u= - s -t + 4*M^2
  363. Id,Count,2,s,2,t,2,u,2,NOM,-2,m,20
  364. *end
  365.  
  366. C WW-scattering 5.  One loop diagrams, two external lines.
  367.  
  368. P ninput
  369.  
  370. A N,N_,M,M2,m,m2,n,n1,n2,n3,n4,Fact,Nom,Nohm,Shi,LogM2,Logm2
  371. F Log,Fq,Tad,Fxx,Two
  372.  
  373. Read WWb.e
  374.  
  375. VERT{}
  376.  
  377. ETE1{}
  378.  
  379. C q1 = q+p
  380.   q2 = q+p+pp
  381.   q3 = q-k
  382.   q4 = q-k-pp
  383.   q5 = q-k-p
  384.   q6 = q+pp
  385.   qu = k+pp
  386.   qs = q-k-p
  387.   qt =
  388.  
  389. V q,q1,q2,q3,q4,q5,q6,qs,qu,qt
  390.  
  391. I al=N,be=N,la=N,de=N,ga=N,la=N
  392.  
  393. I a=3,b=3,c=3,d=3
  394.  
  395. X dede(al,be,ga,de)=D(al,be)*D(ga,de)+D(al,ga)*D(be,de)+D(al,de)*D(be,ga)
  396.  
  397. C n1: -2 for every factor 1/(q^2+m^2)
  398.   n2: number of factors m
  399.   n3: degree of divergence with respect to integration variable q not
  400.       counting n1 types. Integral is convergent if n3+4 < 0.
  401.  
  402. X Fdiv(n1,n2,n3)= DT(-n3-4)*DT(n1+n2) + DT(n3+4-1)*DT(n1+n2+4+n3)
  403.  
  404. C Series expansion for { Nohm/(1-x*Nohm) }^n4
  405. C
  406. X Exp(n1,n2,n3,x,n4) =
  407.    DT(-n3-4)*Nohm^n4*DS(J,0,n1+n2,(DB(n4+J-1,J)*x^J*Nohm^J))
  408.  + DT(n3+4-1)*Nohm^n4*DS(K,0,n1+n2+4+n3,(DB(n4+K-1,K)*x^K*Nohm^K))
  409.  
  410. BLOCK MASS{}
  411. Id,pDp=-M^2
  412. Al,kDk=-M^2
  413. Al,ppDpp=-M^2
  414. Al,pDk=0.5*M^2
  415. Al,kDpp=0.5*M^2
  416. Al,pDpp=0.5*M^2
  417. ENDBLOCK
  418.  
  419. BDELETE COUNT
  420. BDELETE HCOUNT
  421. BDELETE SHIFT
  422. BDELETE STINT
  423.  
  424. BLOCK COUNT{}
  425. Al,NOM(q~,m)=Fact*NOM(q,m)
  426. Id,Count,Fxx,Nohm,-2,Fact,-2 : m,1,[m2-M2],2,m2,2
  427.   : q,1,Fact,2,NOM,-2,Nom,-2,Two,-4,Three,-6
  428.   : Nohm,1
  429. Al,Fact=1
  430. ENDBLOCK
  431.  
  432. BLOCK HCOUNT{}
  433. C Count behaviour with respect to m for large m.
  434.   Eliminate if zero in that limit.
  435. IF Nohm
  436. AND NOT Ztag
  437. COUNT{}
  438. Id,Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)
  439. ELSE
  440. Id,Count,0,m,1,[m2-M2],2,Three,10,DLP,10,Ztag,10
  441. ENDIF
  442. ENDBLOCK
  443.  
  444. BLOCK SHIFT{}
  445. IF Shi^1
  446. Al,qDq=qDq-2*qDp+pDp
  447. Al,q(al~)=q(al)-p(al)
  448. Al,Dotpr,q(al~)=q(al)-p(al)
  449. ENDIF
  450. IF Shi^3
  451. Al,qDq=qDq+2*qDk+kDk
  452. Al,q(al~)=q(al)+k(al)
  453. Al,Dotpr,q(al~)=q(al)+k(al)
  454. ENDIF
  455. IF Shi^6
  456. Al,qDq=qDq-2*qDpp+ppDpp
  457. Al,q(al~)=q(al)-pp(al)
  458. Al,Dotpr,q(al~)=q(al)-pp(al)
  459. ENDIF
  460.  
  461. IF NOT Nohm
  462. Id,Shi=1
  463. ENDIF
  464.  
  465. *yep
  466.  
  467. C Working out of shifted 1/(q^2+m^2)^n
  468.  
  469. IF Nohm^n~*Shi^l~
  470. COUNT{}
  471. Al,Nohm=1
  472.  
  473. Id,Shi^1*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(2*qDp-pDp),n4)
  474. Al,Shi^3*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDk-kDk),n4)
  475. Al,Shi^6*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(2*qDpp-ppDpp),n4)
  476. ENDIF
  477.  
  478. ENDBLOCK
  479.  
  480. BLOCK STINT{}
  481. C Standard integrals.
  482.  
  483. C Type Fn = 1/(q^2+M^2)^n
  484.   Gn = q(mu)*q(nu)*Fn (exclusive D(mu,nu))
  485.   Hn = q(mu)*q(nu)*q(al)*q(be)*Fn (exclusive D*D etc part).
  486.  
  487. Id,F(1,m2~) = 2*i*Pi^2*m2/N_ + i*Pi^2*m2*(-1+Log(m2))
  488. Al,F(2,m2~) = - 2*i*Pi^2/N_ - i*Pi^2*Log(m2)
  489. Al,F(3,m2~) = 0.5*i*Pi^2/m2
  490. Al,F(4,m2~) = i*Pi^2/6/m2^2
  491. Al,F(5,m2~) = 1/12*i*Pi^2*m2^-3
  492. Al,F(6,m2~) = 1/20*i*Pi^2*m2^-4
  493. Al,F(7,m2~) = 1/30*i*Pi^2*m2^-5
  494.  
  495. Id,G(1,m2~) = - 0.5*i*Pi^2*m2^2/N_ + 3/8*i*Pi^2*m2^2
  496.   - 0.25*i*Pi^2*m2^2*Log(m2)
  497.  
  498. Al,G(2,m2~) = i*Pi^2 * ( - 1/2*m2 + m2*N_^-1 )
  499.  + 0.5*m2*Log(m2)*i*Pi^2
  500.  
  501. Al,G(3,m2~) = i*Pi^2 * ( - 1/2*N_^-1 )
  502.  - 1/4*Log(m2)*i*Pi^2
  503.  
  504. Al,G(4,m2~) = 1/12*i*Pi^2*m2^-1
  505.  
  506. Al,G(5,m2~) = 1/48*i*Pi^2*m2^-2
  507.  
  508. Al,G(6,m2~) = 1/120*i*Pi^2*m2^-3
  509.  
  510. Al,G(7,m2~) = 1/240*i*Pi^2*m2^-4
  511.  
  512. Id,H(1,m2~) = 1/12*i*Pi^2*m2^3/N_ - 11/144*i*Pi^2*m2^3
  513.  + 1/24*i*Pi^2*m2^3*Log(m2)
  514.  
  515. Al,H(2,m2~) = i*Pi^2 * ( 3/16*m2^2 - 1/4*m2^2*N_^-1 )
  516.  - 1/8*Log(m2)*i*Pi^2*m2^2
  517.  
  518. Al,H(3,m2~) = i*Pi^2 * ( - 1/8*m2 + 1/4*m2*N_^-1 )
  519.  + 1/8*Log(m2)*i*Pi^2  *m2
  520.  
  521. Al,H(4,m2~) = - 1/12*i*Pi^2*N_^-1
  522.  - 1/24*Log(m2)*i*Pi^2
  523.  
  524. Al,H(5,m2~) = i*Pi^2/96/m2
  525.  
  526. Al,H(6,m2~) = 1/480*i*Pi^2*m2^-2
  527.  
  528. Al,H(7,m2~) = 1/1200*i*Pi^2*m2^-3
  529.  
  530. ENDBLOCK
  531.  
  532. *fix
  533. I mu,nu
  534. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  535. V k,p,pp,kp,q0
  536.  
  537. BLOCK WORK{}
  538. Id,Self(I1~,I2~)=
  539.    DS(I1;J2;-J1;Sym;J2;-J1;TAP,(DIB(I1,J1,J2,I2)
  540.   *DC("F,TFE,-1,J1,J2) ))
  541.  
  542.  + DS(I1;J3;-J3;I2;Sym;J3,-J3;TAP,(DIC(I1,J3,I2) ))
  543.  
  544.  + CONTR(I1,I2)*DLP
  545.  
  546. C + DS(I1;I2;-J4;TAP,{ DS(J4;J5;-J5;Sym;J5;-J5,TAP,{DIT(I1,J4,J5,I2)
  547.   *DC("F,TFE,-1,J5) } ) } )
  548.  
  549. Id,Anti,TAP
  550.  
  551. Id,DIB(I1~,K1~,K2~,I2~)=
  552.    VE3(I1,K2,-K1,*,a,al,p,*,l4,m4,q,*,l1,m1,-q1)*
  553.    VE3(K1,I2,-K2,*,l2,m0,q1,*,b,be,-p,*,l3,m3,-q)*
  554.    PROP(K1,-K1,*,l1,m1,q1,*,l2,m0,-q1)*
  555.    PROP(K2,-K2,*,l3,m3,q,*,l4,m4,-q)
  556. Al,DIC(I1~,K1~,I2~)=
  557.    VE4(I1,K1,-K1,I2,*,a,al,p,*,l1,m1,-q,*,l2,m0,q,*,b,be,-p)*
  558.    PROP(K1,-K1,*,l1,m1,q,*,l2,m0,-q)
  559.  
  560. Al,DIT(I1~,K1~,K2~,I2~) = Tad*
  561.    VE3(I1,I2,-K1,*,a,al,p,*,b,be,-p,*,l1,m1,-q0)*
  562.    PROP(K1,-K1,*,l1,m1,q0,*,l2,m0,-q0)*
  563.    VE3(K1,K2,-K2,*,l2,m0,q0,*,l3,m3,-q,*,l4,m4,q)*
  564.    PROP(K2,-K2,*,l3,m3,q,*,l4,m4,-q)
  565.  
  566. Al,CONTR(I1~,I2~)=CONT(I1,I2,"N,*,a,al,p,*,b,be,-p,*,c,ga,q0)
  567.  
  568. Id,Compo,<X>,VE4,VE3,PROP,CONT
  569.  
  570. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  571.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  572. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  573.        FF(l1,al,k,l2,be,q,l3,ga,p)
  574. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  575.  
  576. Id,Even,NOM,1
  577. Al,Commu,NOM
  578.  
  579. *yep
  580. Id,q0(al~)=0
  581. Al,Dotpr,q0(al~)=0
  582. Al,NOM(q0,M~) = 1/M^2
  583. Id,q1Dq1=qDq+2*pDq+pDp
  584. Al,q1(al~)=q(al)+p(al)
  585. Al,Dotpr,q1(al~)=q(al)+p(al)
  586. Id,qDq*NOM(q,M~)=1-M^2*NOM(q,M)
  587. Id,Adiso,qDp^n~*NOM(q,M~)*NOM(q1,m~)=-0.5*qDp^(n-1)*
  588.  {NOM(q1,m) - NOM(q,M) + (pDp-M^2+m^2)*NOM(q,M)*NOM(q1,m)}
  589. Id,Commu,NOM
  590. Id,Epfred
  591. B Nohm,Nom,i,Pi,Ztag,Xetid
  592. *yep
  593.  
  594. Id,NOM(q,m)=Nohm
  595.  
  596. IF NOM(q~,m)
  597. AND NOT Ztag
  598. COUNT{}
  599. Id,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q1,m)=Fdiv(n1,n2,n3)
  600.  *Exp(n1,n2,n3,(-2*qDp-pDp),1)
  601. ENDIF
  602.  
  603. HCOUNT{}
  604. *yep
  605.  
  606. IF NOM(q,M)=Nom
  607. AND Nohm
  608. Id,Nohm^n~=a1^-n
  609. Al,Nom^n~=a2^-n
  610. Id,Ratio,a2,a1,[m2-M2]
  611. Id,a1^n~=Nohm^-n
  612. Al,a2^n~=Nom^-n
  613. ENDIF
  614.  
  615. Id,Nom*NOM(q1~,M)= Two(q,q1,M)
  616. Al,Nohm*NOM(q1~,m)= Two(q,q1,m)
  617. Id,NOM(q1,M)=Nom*Shi
  618. SHIFT{}
  619.  
  620. IF Nohm^n~*Nom^l~
  621. Id,Nohm^n~=a1^-n
  622. Al,Nom^n~=a2^-n
  623. Id,Ratio,a2,a1,[m2-M2]
  624. Id,a1^n~=Nohm^-n
  625. Al,a2^n~=Nom^-n
  626. ENDIF
  627. *yep
  628.  
  629. IF Nohm
  630. OR Nom
  631. Id,All,q,N,Fq
  632. ENDIF
  633.  
  634. Id,Fq(al~)=0
  635. Al,Fq(al~,be~,ga~)=0
  636. Al,Fq(al~,be~,ga~,de~,la~)=0
  637. Al,Fq(al~,be~,ga~,de~,la~,a~,b~)=0
  638.  
  639. *yep
  640.  
  641. Id,Fq(al~,be~,ga~,de~)*Nom^l~ = dede(al,be,ga,de)*H(l,M2)
  642. Al,Fq(al~,be~)*Nom^l~ = D(al,be)*G(l,M2)
  643. Al,Nom^n~ = F(n,M2)
  644. Id,Fq(al~,be~,ga~,de~)*Nohm^l~ = dede(al,be,ga,de)*H(l,m2)
  645. Al,Fq(al~,be~)*Nohm^l~ = D(al,be)*G(l,m2)
  646. Al,Nohm^l~ = F(l,m2)
  647.  
  648. STINT{}
  649.  
  650. Id,Multi,M2^n~=M^(2*n)
  651. Al,Multi,m2^n~=m^(2*n)
  652.  
  653. *yep
  654.  
  655. IF NOT Two(q~,q1~,M~)
  656. Id,Count,0,m,1,m2,2,[m2-M2],2,DLP,10
  657. Id,Count,x,m2,2,m,1,DLP,10
  658. ..IF x
  659. ..Id,x^n1~*[m2-M2]^n~=m2^n*DS(K,0,n+(n1+1)/2,(DB(-n+K-1,K)*M2^K*m2^-K))
  660. ..Id,m2^n~=m^(2*n)
  661. ..Al,M2^n~=M^(2*n)
  662. ..ELSE
  663. ..Id,[m2-M2]^n~=m^(2*n)
  664. ..ENDIF
  665. Id,Count,0,m2,2,m,1,DLP,10,Ztag,10
  666. Al,x=1
  667. ENDIF
  668.  
  669. IF Two(q~,q1~,M~)
  670. Id,All,q,N,Fq,"F_
  671. Id,Adiso,Two(q,q1~,M~)*Fq(al~,be~)=B21(pDp,M,M)*Fxx(q1,al,be)
  672.  +B22(pDp,M,M)*D(al,be)
  673. Al,Adiso,Two(q,q1~,M~)*Fq(al~)=B1(pDp,M,M)*Fxx(q1,al)
  674. Al,Two(q,q1~,M~)=B0(pDp,M,M)
  675. Id,Fxx(q1,al~)=p(al)
  676. Id,Fxx(q1,al~,be~)=p(al)*p(be)
  677. ENDIF
  678.  
  679. Id,B22(u~,M~,m~)=(-0.5*F1(m)+M**2*B0(u,M,m)
  680.       -0.5*(u+m^2-M**2)*B1(u,M,m))/[1-N]
  681. Id,B21(u~,M~,m~)=-((0.5*N-1)*F1(m)
  682.        -0.5*N*(u+m^2-M**2)*B1(u,M,m)
  683.        +M**2*B0(u,M,m) )/u/[1-N]
  684. Id,B1(u~,M~,m~)= (0.5*F1(M)-0.5*F1(m)
  685.               -0.5*(u+m^2-M**2)*B0(u,M,m) )/u
  686.  
  687. Al,F1(M~) = 2*i*Pi^2*M^2/N_ + i*Pi^2*M^2*(-1+LogM2)
  688. Id,N=N_+4
  689. Al,[1-N]^-1=-1/3 + N_/9
  690. Id,N_=0
  691. Id,N=4
  692.  
  693. IF Ztag
  694. Id,B0(u~,M,M)=i*Pi^2*(Logm2-LogM2-2)*(u+m^2)/m^2
  695.    -2*i*Pi^2/N_-i*Pi^2*(Logm2-2)
  696. Al,B0(u~,m,m)=-i*Pi^2*[Pi/Sqrt(3)-2]*(u+m^2)/m^2
  697.    -2*i*Pi^2/N_- i*Pi^2*(Logm2+[Pi/Sqrt(3)-2])
  698. ELSE
  699. Id,B0(u~,M,M)= -2*i*Pi^2/N_ - i*Pi^2*LogM2 + u*BB0F(u,M,M)/M^2
  700. ENDIF
  701.  
  702. Id,Log(m2)=Logm2
  703. Al,Log(M2)=LogM2
  704.  
  705. Al,CONT(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  706.        FF(l1,al,k,l2,be,q,l3,ga,p)
  707.  
  708. Id,Count,1,m,1,N_,-1,Logm2,1,DLP,10,Ztag,10,Xetid,10
  709. *yep
  710. ETE1{}
  711. P output
  712. *yep
  713. Id,DLP=-1
  714. Id,Count,1,m,1,N_,-10,Logm2,1,DLP,10,Ztag,10,Xetid,10
  715. Id,Ztag=1
  716. ENDBLOCK
  717.  
  718. Z TADP=DS("Z;J1;-J1;Sym;J1;-J1,(DIT("Z,J1)*DC("F,TFE,-1,J1) ))
  719.   + DLP*NNZ(b,be,p,c,ga,q,a,al,q0)
  720.    + Xetid*Et
  721.  
  722. Id,DIT(K1~,K2~)=
  723.    VE3(K1,K2,-K2,*,a,al,q0,*,l3,m3,-q,*,l4,m4,q)*
  724.    PROP(K2,-K2,*,l3,m3,q,*,l4,m4,-q)
  725. Id,Compo,<X>,VE4,VE3,PROP
  726.  
  727. Id,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  728.        FF(l1,al,k,l2,be,q,l3,ga,p)
  729. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  730.  
  731. Id,Even,NOM,1
  732. Al,Commu,NOM
  733.  
  734. *yep
  735. Id,q0(al~)=0
  736. Al,Dotpr,q0(al~)=0
  737. Id,Commu,NOM
  738. Id,Epfred
  739.  
  740. B Nohm,Nom,i,Pi,DEL,Xetid
  741.  
  742. Id,NOM(q~,M)=F(1,M2)
  743. Al,NOM(q~,m)=F(1,m2)
  744. Id,F(1,m2~) = 2*i*Pi^2*m2/N_ + i*Pi^2*m2*(-1+Log(m2))
  745.  
  746. Id,Multi,M2^n~=M^(2*n)
  747. Al,Multi,m2^n~=m^(2*n)
  748.  
  749. Id,N=N_+4
  750. Al,[1-N]^-1=-1/3 + N_/9
  751. Id,N_=0
  752. Id,N=4
  753.  
  754. Id,Log(m2)=Logm2
  755. Al,Log(M2)=LogM2
  756. P output
  757. *yep
  758.  
  759. ETE1{}
  760. Id,DLP=-1
  761. *next
  762.  
  763. Z SelfWW = Self("W,"W)
  764. C - {2*M^2*Ew + 2*M^2*E1 + 2*pDp*Ew}*D(a,b)*D(al,be)*DLP
  765.  + 2*p(al)*p(be)*D(a,b)*Ew*DLP
  766. WORK{}
  767. *next
  768.  
  769. Z SelfFF = Self("F,"F)
  770. C - 2*pDp*Eh*D(a,b)*DLP
  771.  - 1/2*m^2*Et*D(a,b)*DLP
  772. WORK{}
  773. *next
  774.  
  775. Z SelfWF = Self("W,"F)
  776. C + M*{Ew+Eh+E1}*D(a,b)*(-i*p(al))*DLP
  777. WORK{}
  778. *next
  779. Z SelfZZ = Self("Z,"Z)*Ztag
  780. C + m^2*{-2*Eh-1/2*Et+2*E2-2*E1}*DLP
  781.  - 2*pDp*Eh*DLP
  782. WORK{}
  783. P output
  784. *yep
  785. Id,pDp=-m^2
  786. Id,Count,1,m,1,N_,-10,Logm2,1
  787. *end
  788.  
  789. C WW-scattering 6.  One loop diagrams, three external lines.
  790.  
  791. P ninput
  792.  
  793. A N,N_,M,M2,m,m2,n,n1,n2,n3,n4,Fact,Nom,Nohm,Shi,LogM2,Logm2
  794. F Fxx,Two,Three,Fq
  795.  
  796. Read WWb.e
  797.  
  798. VERT{}
  799.  
  800. C q1 = q+p
  801.   q2 = q+p+pp
  802.   q3 = q-k
  803.   q4 = q-k-pp
  804.   q5 = q-k-p
  805.   q6 = q+pp
  806.   q7 = q+kp
  807.   qu = k+pp
  808.   qs = q-k-p
  809.   qt =
  810.  
  811. V q,q1,q2,q3,q4,q5,q6,qs,qu,qt
  812.  
  813. I al=N,be=N,la=N,de=N,ga=N,la=N
  814.  
  815. I a=3,b=3,c=3,d=3
  816.  
  817. X dede(al,be,ga,de)=D(al,be)*D(ga,de)+D(al,ga)*D(be,de)+D(al,de)*D(be,ga)
  818.  
  819. C n1: -2 for every factor 1/(q^2+m^2)
  820.   n2: number of factors m
  821.   n3: degree of divergence with respect to integration variable q not
  822.       counting n1 types. Integral is convergent if n3+4 < 0.
  823.  
  824. X Fdiv(n1,n2,n3)= DT(-n3-4)*DT(n1+n2) + DT(n3+4-1)*DT(n1+n2+4+n3)
  825.  
  826. C Series expansion for { Nohm/(1-x*Nohm) }^n4
  827. C
  828. X Exp(n1,n2,n3,x,n4) =
  829.    DT(-n3-4)*Nohm^n4*DS(J,0,n1+n2,(DB(n4+J-1,J)*x^J*Nohm^J))
  830.  + DT(n3+4-1)*Nohm^n4*DS(K,0,n1+n2+4+n3,(DB(n4+K-1,K)*x^K*Nohm^K))
  831.  
  832. BLOCK MASS{}
  833. Id,pDp=-M^2
  834. Al,kDk=-M^2
  835. Al,ppDpp=-M^2
  836. Al,pDk=0.5*M^2
  837. Al,kDpp=0.5*M^2
  838. Al,pDpp=0.5*M^2
  839. ENDBLOCK
  840.  
  841. BDELETE COUNT
  842. BDELETE HCOUNT
  843. BDELETE SHIFT
  844. BDELETE STINT
  845.  
  846. BLOCK COUNT{}
  847. Al,NOM(q~,m)=Fact*NOM(q,m)
  848. Id,Count,Fxx,Nohm,-2,Fact,-2 : m,1,[m2-M2],2,m2,2
  849.   : q,1,Fact,2,NOM,-2,Nom,-2,Two,-4,Three,-6
  850.   : Nohm,1
  851. Al,Fact=1
  852. ENDBLOCK
  853.  
  854. BLOCK HCOUNT{}
  855. C Count behaviour with respect to m for large m.
  856.   Eliminate if zero in that limit.
  857. IF Nohm
  858. COUNT{}
  859. Id,Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)
  860. ELSE
  861. Id,Count,0,m,1,[m2-M2],2,Three,10
  862. ENDIF
  863. ENDBLOCK
  864.  
  865. BLOCK SHIFT{}
  866. IF Shi^1
  867. Al,qDq=qDq-2*qDp+pDp
  868. Al,q(al~)=q(al)-p(al)
  869. Al,Dotpr,q(al~)=q(al)-p(al)
  870. ENDIF
  871. IF Shi^3
  872. Al,qDq=qDq+2*qDk+kDk
  873. Al,q(al~)=q(al)+k(al)
  874. Al,Dotpr,q(al~)=q(al)+k(al)
  875. ENDIF
  876. IF Shi^6
  877. Al,qDq=qDq-2*qDpp+ppDpp
  878. Al,q(al~)=q(al)-pp(al)
  879. Al,Dotpr,q(al~)=q(al)-pp(al)
  880. ENDIF
  881.  
  882. IF NOT Nohm
  883. Id,Shi=1
  884. ENDIF
  885.  
  886. *yep
  887.  
  888. C Working out of shifted 1/(q^2+m^2)^n
  889.  
  890. IF Nohm^n~*Shi^l~
  891. COUNT{}
  892. Al,Nohm=1
  893.  
  894. Id,Shi^1*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(2*qDp-pDp),n4)
  895. Al,Shi^3*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDk-kDk),n4)
  896. Al,Shi^6*Fxx(n1~,n2~,n3~,n4~)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(2*qDpp-ppDpp),n4)
  897. ENDIF
  898.  
  899. ENDBLOCK
  900.  
  901. BLOCK STINT{}
  902. C Standard integrals.
  903.  
  904. C Type Fn = 1/(q^2+M^2)^n
  905.  
  906. Id,F(1,m2~) = 2*i*Pi^2*m2/N_ + i*Pi^2*m2*(-1+Log(m2))
  907. Al,F(2,m2~) = - 2*i*Pi^2/N_ - i*Pi^2*Log(m2)
  908. Al,F(3,m2~) = 0.5*i*Pi^2/m2
  909. Al,F(4,m2~) = i*Pi^2/6/m2^2
  910. Al,F(5,m2~) = 1/12*i*Pi^2*m2^-3
  911. Al,F(6,m2~) = 1/20*i*Pi^2*m2^-4
  912. Al,F(7,m2~) = 1/30*i*Pi^2*m2^-5
  913.  
  914. Id,G(1,m2~) = - 0.5*i*Pi^2*m2^2/N_ + 3/8*i*Pi^2*m2^2
  915.   - 0.25*i*Pi^2*m2^2*Log(m2)
  916.  
  917. Al,G(2,m2~) = i*Pi^2 * ( - 1/2*m2 + m2*N_^-1 )
  918.  + 0.5*m2*Log(m2)*i*Pi^2
  919.  
  920. Al,G(3,m2~) = i*Pi^2 * ( - 1/2*N_^-1 )
  921.  - 1/4*Log(m2)*i*Pi^2
  922.  
  923. Al,G(4,m2~) = 1/12*i*Pi^2*m2^-1
  924.  
  925. Al,G(5,m2~) = 1/48*i*Pi^2*m2^-2
  926.  
  927. Al,G(6,m2~) = 1/120*i*Pi^2*m2^-3
  928.  
  929. Al,G(7,m2~) = 1/240*i*Pi^2*m2^-4
  930.  
  931. Id,H(1,m2~) = 1/12*i*Pi^2*m2^3/N_ - 11/144*i*Pi^2*m2^3
  932.  + 1/24*i*Pi^2*m2^3*Log(m2)
  933.  
  934. Al,H(2,m2~) = i*Pi^2 * ( 3/16*m2^2 - 1/4*m2^2*N_^-1 )
  935.  - 1/8*Log(m2)*i*Pi^2*m2^2
  936.  
  937. Al,H(3,m2~) = i*Pi^2 * ( - 1/8*m2 + 1/4*m2*N_^-1 )
  938.  + 1/8*Log(m2)*i*Pi^2  *m2
  939.  
  940. Al,H(4,m2~) = - 1/12*i*Pi^2*N_^-1
  941.  - 1/24*Log(m2)*i*Pi^2
  942.  
  943. Al,H(5,m2~) = i*Pi^2/96/m2
  944.  
  945. Al,H(6,m2~) = 1/480*i*Pi^2*m2^-2
  946.  
  947. Al,H(7,m2~) = 1/1200*i*Pi^2*m2^-3
  948.  
  949. ENDBLOCK
  950.  
  951. BLOCK COEF{}
  952. C Generated with program BCij.e
  953.  
  954. Id,BB0=i*Pi^2*( - LogM2 - 2*N_^-1 - [Pi/Sqrt(3)-2] )
  955. Al,BB1=i*Pi^2*( 1/2*LogM2 + N_^-1 + 1/2*[Pi/Sqrt(3)-2] )
  956. Al,BB21=i*Pi^2*( 1/18 - 1/3*LogM2 - 2/3*N_^-1 )
  957. Al,BB22=i*Pi^2*M^2*( - 4/9 + 5/12*LogM2 + 5/6*N_^-1 + 1/4*[Pi/Sqrt(3)-2] )
  958.  
  959. Id,C11= - 2/3*C0
  960. Al,C12= - 1/3*C0
  961. Al,C21= 1/3*i*M^-2*Pi^2
  962. Al,C22= 1/3*i*M^-2*Pi^2 - 1/3*C0
  963. Al,C23= 1/6*i*M^-2*Pi^2
  964. Al,C24= - 1/2*i*N_^-1*Pi^2 + 1/4*i*Pi^2 - 1/4*i*Pi^2*LogM2 - 1/4*i*Pi^2
  965.   *[Pi/Sqrt(3)-2] - 1/3*M^2*C0
  966. Al,C31= - 19/27*i*M^-2*Pi^2 - 2/9*i*M^-2*Pi^2*[Pi/Sqrt(3)-2]
  967.   + 16/27*C0
  968. Al,C32= - 8/27*i*M^-2*Pi^2 + 2/9*i*M^-2*Pi^2*[Pi/Sqrt(3)-2]
  969.   + 11/27*C0
  970. Al,C33= - 19/54*i*M^-2*Pi^2 - 1/9*i*M^-2*Pi^2*[Pi/Sqrt(3)-2]
  971.   + 8/27*C0
  972. Al,C34= - 17/54*i*M^-2*Pi^2 + 1/9*i*M^-2*Pi^2*[Pi/Sqrt(3)-2]
  973.   + 10/27*C0
  974. Al,C35= 1/3*i*N_^-1*Pi^2 - 1/6*i*Pi^2 + 1/6*i*Pi^2*LogM2 + 1/6*i*Pi^2
  975.   *[Pi/Sqrt(3)-2] + 2/9*M^2*C0
  976. Al,C36= 1/6*i*N_^-1*Pi^2 - 1/12*i*Pi^2 + 1/12*i*Pi^2*LogM2
  977.   + 1/12*i*Pi^2*[Pi/Sqrt(3)-2] + 1/9*M^2*C0
  978. Id,C0=i*Pi^2*CC0
  979. ENDBLOCK
  980.  
  981. *fix
  982.  
  983. I mu,nu
  984. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N,m9=N
  985. V k,p,pp
  986.  
  987. BLOCK WORK{TADP}
  988. Id,VERT(K1~,K2~,K3~)=
  989.   DS(K1;J3;-J1;TAP,(DS(K2;J1;-J2;TAP,(
  990.   DIB(K1,K2,K3,J1,J2,J3)*DC("F,TFE,-1,J1,J2,J3) ))))
  991.  
  992.  +DS(K1;K2;-J4;J5;Sym;-J4;J5;TAP,(VIR1(K1,K2,K3,J4,J5) ))
  993.  
  994.  +DS(K2;K3;J6;-J7;Sym;J6;-J7;TAP,(VIR2(K1,K2,K3,J6,J7) ))
  995.  
  996.  +DS(K1;K3;-J8;J9;Sym;-J8;J9;TAP,(VIR3(K1,K2,K3,J8,J9) ))
  997.  
  998.  
  999. Id,DIB(K1~,K2~,K3~,J1~,J2~,J3~)=
  1000.    VE3(K1,-J1,J3,*,a,al,k,*,l1,m1,-q,*,l6,m6,q3)*
  1001.    VE3(K2,J1,-J2,*,b,be,p,*,l2,m0,q,*,l3,m3,-q1)*
  1002.    VE3(K3,J2,-J3,*,c,ga,pp,*,l4,m4,q1,*,l5,m5,-q3)*
  1003.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  1004.    PROP(J2,-J2,*,l3,m3,q1,*,l4,m4,-q1)*
  1005.    PROP(J3,-J3,*,l5,m5,q3,*,l6,m6,-q3)
  1006.  
  1007. Al,VIR1(K1~,K2~,K3~,J1~,J2~)=
  1008.    VE4(K1,K2,-J1,J2,*,a,al,k,*,b,be,p,*,l1,m1,-q,*,l4,m4,q6)*
  1009.    VE3(K3,J1,-J2,*,c,ga,pp,*,l2,m0,q,*,l3,m3,-q6)*
  1010.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  1011.    PROP(J2,-J2,*,l3,m3,q6,*,l4,m4,-q6)
  1012.  
  1013. Al,VIR2(K1~,K2~,K3~,J1~,J2~)=
  1014.    VE4(K2,K3,J1,-J2,*,b,be,p,*,c,ga,pp,*,l2,m0,q,*,l3,m3,-q3)*
  1015.    VE3(K1,-J1,J2,*,a,al,k,*,l1,m1,-q,*,l4,m4,q3)*
  1016.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  1017.    PROP(J2,-J2,*,l3,m3,q3,*,l4,m4,-q3)
  1018.  
  1019. Al,VIR3(K1~,K2~,K3~,J1~,J2~)=
  1020.    VE4(K1,K3,-J1,J2,*,a,al,k,*,c,ga,pp,*,l1,m1,-q,*,l4,m4,q1)*
  1021.    VE3(K2,J1,-J2,*,b,be,p,*,l2,m0,q,*,l3,m3,-q1)*
  1022.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  1023.    PROP(J2,-J2,*,l3,m3,q1,*,l4,m4,-q1)
  1024.  
  1025. Id,Anti,TAP
  1026.  
  1027. Id,Compo,<X>,VE4,VE3,PROP
  1028. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  1029.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  1030. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  1031.        FF(l1,al,k,l2,be,q,l3,ga,p)
  1032. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  1033.  
  1034. Id,Even,NOM,1
  1035. Id,Commu,NOM
  1036.  
  1037. C q1 = q+p
  1038.   q3 = q+p+pp = q-k
  1039.   q6 = q+pp
  1040.  
  1041. Id,q1Dq1=qDq+pDp+2*qDp
  1042. Al,q3Dq3=qDq+kDk-2*qDk
  1043. Al,q6Dq6=qDq+ppDpp+2*qDpp
  1044. Id,q1(al~)=q(al)+p(al)
  1045. Al,Dotpr,q1(al~)=q(al)+p(al)
  1046. Id,q3(al~)=q(al)-k(al)
  1047. Al,Dotpr,q3(al~)=q(al)-k(al)
  1048. Id,q6(al~)=q(al)+pp(al)
  1049. Al,Dotpr,q6(al~)=q(al)+pp(al)
  1050.  
  1051. Id,qDq*NOM(q,M~)=1-M^2*NOM(q,M)
  1052. Id,Adiso,qDp^n~*NOM(q,M~)*NOM(q1,m~)=-0.5*qDp^(n-1)*
  1053.  {NOM(q1,m) - NOM(q,M) + (pDp-M^2+m^2)*NOM(q,M)*NOM(q1,m)}
  1054. Id,Adiso,qDk^n~*NOM(q,M~)*NOM(q3,m~)=0.5*qDk^(n-1)*
  1055.  {NOM(q3,m) - NOM(q,M) + (kDk-M^2+m^2)*NOM(q,M)*NOM(q3,m)}
  1056. Id,Adiso,qDpp^n~*NOM(q,M~)*NOM(q6,m~)=-0.5*qDpp^(n-1)*
  1057.  {NOM(q6,m) - NOM(q,M) + (ppDpp-M^2+m^2)*NOM(q,M)*NOM(q6,m)}
  1058.  
  1059. Id,Commu,NOM
  1060. Id,Epfred
  1061. Id,ppDpp=kDk+pDp+2*kDp
  1062. Id,pp(al~)=-k(al)-p(al)
  1063. Al,Dotpr,pp(al~)=-k(al)-p(al)
  1064.  
  1065. Id,NOM(q1~,M)*NOM(q3~,M)*NOM(q6~,M)= Three(M,q1,q3,q6)
  1066.  
  1067. B Nohm,Nom,i,Pi,DEL
  1068.  
  1069. *yep
  1070.  
  1071. Id,NOM(q,m)=Nohm
  1072.  
  1073. IF NOM(q~,m)
  1074. COUNT{}
  1075. Id,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q1,m)=Fdiv(n1,n2,n3)
  1076.  *Exp(n1,n2,n3,(-2*qDp-pDp),1)
  1077. Al,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  1078.  *Exp(n1,n2,n3,(2*qDk-kDk),1)
  1079. Al,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q6,m)=Fdiv(n1,n2,n3)
  1080.  *Exp(n1,n2,n3,(-2*qDpp-ppDpp),1)
  1081. ENDIF
  1082.  
  1083. HCOUNT{}
  1084. MASS{}
  1085. *yep
  1086.  
  1087. IF NOM(q~,m)
  1088. COUNT{}
  1089. Id,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q1,m)=Fdiv(n1,n2,n3)
  1090.  *Exp(n1,n2,n3,(-2*qDp-pDp),1)
  1091. Al,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  1092.  *Exp(n1,n2,n3,(2*qDk-kDk),1)
  1093. Al,Adiso,Fxx(n1~,n2~,n3~,n4~)*NOM(q6,m)=Fdiv(n1,n2,n3)
  1094.  *Exp(n1,n2,n3,(-2*qDpp-ppDpp),1)
  1095. ENDIF
  1096.  
  1097. HCOUNT{}
  1098. MASS{}
  1099.  
  1100. IF NOM(q,M)=Nom
  1101. AND Nohm
  1102. Id,Nohm^n~=a1^-n
  1103. Al,Nom^n~=a2^-n
  1104. Id,Ratio,a2,a1,[m2-M2]
  1105. Id,a1^n~=Nohm^-n
  1106. Al,a2^n~=Nom^-n
  1107. ENDIF
  1108.  
  1109. Id,Nom*NOM(q1~,M)= Two(q,q1)
  1110.  
  1111. *yep
  1112.  
  1113. IF NOT NOM(q1,M)=Nom*Shi
  1114. AND NOT NOM(q3,M)=Nom*Shi^3
  1115. Al,NOM(q6,M)=Nom*Shi^6
  1116. ENDIF
  1117.  
  1118. *yep
  1119.  
  1120. Id,Shi^1*NOM(q3,M)=NOM(q6,M)*Shi
  1121.  
  1122. SHIFT{}
  1123.  
  1124. *yep
  1125.  
  1126. HCOUNT{}
  1127. MASS{}
  1128.  
  1129. IF Nohm^n~*Nom^l~
  1130. Id,Nohm^n~=a1^-n
  1131. Al,Nom^n~=a2^-n
  1132. Id,Ratio,a2,a1,[m2-M2]
  1133. Id,a1^n~=Nohm^-n
  1134. Al,a2^n~=Nom^-n
  1135. ENDIF
  1136.  
  1137. Id,Nom*NOM(q1~,M)=Two(q,q1)
  1138.  
  1139. *yep
  1140.  
  1141. Id,NOM(q6,M)=Nom*Shi^6
  1142.  
  1143. SHIFT{}
  1144. HCOUNT{}
  1145. MASS{}
  1146.  
  1147. IF Nohm^n~*Nom^l~
  1148. Id,Nohm^n~=a1^-n
  1149. Al,Nom^n~=a2^-n
  1150. Id,Ratio,a2,a1,[m2-M2]
  1151. Id,a1^n~=Nohm^-n
  1152. Al,a2^n~=Nom^-n
  1153. ENDIF
  1154.  
  1155. IF NOT Nohm
  1156. AND NOT Three(m,q1~,q3~,q6~)
  1157. Id,Count,0,m,1,[m2-M2],2,DLP,10
  1158. ENDIF
  1159.  
  1160. *yep
  1161.  
  1162. IF Nohm
  1163. OR Nom
  1164. Id,All,q,N,Fq
  1165. ENDIF
  1166.  
  1167. Id,Fq(al~)=0
  1168. Al,Fq(al~,be~,ga~)=0
  1169. Al,Fq(al~,be~,ga~,de~,la~)=0
  1170. Al,Fq(al~,be~,ga~,de~,la~,a~,b~)=0
  1171.  
  1172. *yep
  1173.  
  1174. Id,Fq(al~,be~,ga~,de~)*Nom^l~ = dede(al,be,ga,de)*H(l,M2)
  1175. Al,Fq(al~,be~)*Nom^l~ = D(al,be)*G(l,M2)
  1176. Al,Nom^n~ = F(n,M2)
  1177. Id,Fq(al~,be~,ga~,de~)*Nohm^l~ = dede(al,be,ga,de)*H(l,m2)
  1178. Al,Fq(al~,be~)*Nohm^l~ = D(al,be)*G(l,m2)
  1179. Al,Nohm^l~ = F(l,m2)
  1180.  
  1181. STINT{}
  1182. MASS{}
  1183.  
  1184. Id,Multi,M2^n~=M^(2*n)
  1185. Al,Multi,m2^n~=m^(2*n)
  1186.  
  1187. IF NOT Two(q~,M~)
  1188. AND NOT Three(M~,q~,q1~,q2~)
  1189. Id,Count,0,m,1,m2,2,[m2-M2],2
  1190. Id,Count,x,m2,2,m,1
  1191. ..IF x
  1192. ..Id,x^n1~*[m2-M2]^n~=m2^n*DS(K,0,n+(n1+1)/2,(DB(-n+K-1,K)*M2^K*m2^-K))
  1193. ..Id,m2^n~=m^(2*n)
  1194. ..Al,M2^n~=M^(2*n)
  1195. ..ELSE
  1196. ..Id,[m2-M2]^n~=m^(2*n)
  1197. ..ENDIF
  1198. Id,Count,0,m2,2,m,1,DLP,10
  1199. Al,x=1
  1200. ENDIF
  1201.  
  1202. Id,ppDpp=kDk+pDp+2*kDp
  1203. Al,pp(al~)=-k(al)-p(al)
  1204. Al,Dotpr,pp(al~)=-k(al)-p(al)
  1205.  
  1206. Id,Log(m2)=Logm2
  1207. Al,Log(M2)=LogM2
  1208.  
  1209. IF Three(M~,q,q1,q3)=Fxx(M)
  1210. Al,All,q,N,Fq,"F_
  1211. Id,Adiso,Fxx(m~)*Fq(al~,be~,ga~)=
  1212.     p(al)*p(be)*p(ga)*C31
  1213.   + pp(al)*pp(be)*pp(ga)*C32
  1214.   + (pp(al)*p(be)*p(ga)+p(al)*pp(be)*p(ga)+p(al)*p(be)*pp(ga))*C33
  1215.   + (p(al)*pp(be)*pp(ga)+pp(al)*p(be)*pp(ga)+pp(al)*pp(be)*p(ga))*C34
  1216.   + (p(al)*D(be,ga)+p(be)*D(al,ga)+p(ga)*D(al,be))*C35
  1217.   + (pp(al)*D(be,ga)+pp(be)*D(al,ga)+pp(ga)*D(al,be))*C36
  1218. Al,Adiso,Fxx(m~)*Fq(al~,be~)=
  1219.     p(al)*p(be)*C21
  1220.   + pp(al)*pp(be)*C22
  1221.   + (p(al)*pp(be)+pp(al)*p(be))*C23
  1222.   + D(al,be)*C24
  1223. Al,Adiso,Fxx(m~)*Fq(al~)= p(al)*C11 + pp(al)*C12
  1224. Al,Fxx(m~)=C0
  1225. ENDIF
  1226.  
  1227. IF Two(q~,q1~)
  1228. Id,All,q,N,Fq,"F_
  1229. Id,Adiso,Two(q,q1~)*Fq(al~,be~)=BB21*Fxx(q1,al,be)+BB22*D(al,be)
  1230. Al,Adiso,Two(q,q1~)*Fq(al~)=BB1*Fxx(q1,al)
  1231. Al,Two(q,q1~)=BB0
  1232. Id,Fxx(q1,al~)=p(al)
  1233. Al,Fxx(q3,al~)=-k(al)
  1234. Al,Fxx(q6,al~)=-p(al)-k(al)
  1235. Id,Fxx(q1,al~,be~)=p(al)*p(be)
  1236. Al,Fxx(q3,al~,be~)=k(al)*k(be)
  1237. Al,Fxx(q6,al~,be~)=(p(al)+k(al))*(p(be)+k(be))
  1238. ENDIF
  1239.  
  1240. *yep
  1241.  
  1242. COEF{}
  1243.  
  1244. Id,pp(al~)=-k(al)-p(al)
  1245. MASS{}
  1246. Id,N=N_+4
  1247. Id,N_=0
  1248. C Id,[Pi/Sqrt(3)-2]= - BB0F - LogM2
  1249. *yep
  1250. Id,Count,0,m,1,m2,2,[m2-M2],2,DLP,10
  1251. Id,Count,x,m2,2,m,1
  1252. ..IF x
  1253. ..Id,x^n1~*[m2-M2]^n~=m2^n*DS(K,0,n+(n1+1)/2,(DB(-n+K-1,K)*M2^K*m2^-K))
  1254. ..Id,m2^n~=m^(2*n)
  1255. ..Al,M2^n~=M^(2*n)
  1256. ..ELSE
  1257. ..Id,[m2-M2]^n~=m^(2*n)
  1258. ..ENDIF
  1259. Id,Count,0,m2,2,m,1,DLP,10
  1260. Al,x=1
  1261.  
  1262. Id,Count,1,m,1,Logm2,1,N_,-1,DLP,10
  1263.  
  1264. ETE1{}
  1265. P output
  1266. *yep
  1267. Id,DLP=-1
  1268.  
  1269. ENDBLOCK
  1270.  
  1271. Z IWWW = VERT("W,"W,"W)
  1272.   + (Eg+3*Ew)*WWW(a,al,k,b,be,p,c,ga,pp)*DLP
  1273. WORK{IWWW}
  1274. *next
  1275. Z IFFW = VERT("F,"F,"W)
  1276.   + (Eg+Ew+2*Eh)*FFW(a,al,k,b,be,p,c,ga,pp)*DLP
  1277. WORK{IFFW}
  1278. *next
  1279. Z IFWZ = VERT("F,"W,"Z)
  1280.   + (Eg+Ew+2*Eh)*FWZ(a,al,k,b,be,p,c,ga,pp)*DLP
  1281. WORK{IFWZ}
  1282. *next
  1283. Z IWWZ = VERT("W,"W,"Z)
  1284.   + (Eg+2*Ew+Eh+E1)*WWZ(a,al,k,b,be,p,c,ga,pp)*DLP
  1285. WORK{IWWZ}
  1286. *next
  1287. Z IFFZ = VERT("F,"F,"Z)
  1288.  + (Eg+3*Eh-2*E2+E1)*FFZ(a,al,k,b,be,p,c,ga,pp)*DLP
  1289. WORK{IFFZ}
  1290. *next
  1291. Z IZZZ = VERT("Z,"Z,"Z)
  1292.  + (Eg+3*Eh-2*E2+E1)*ZZZ(a,al,k,b,be,p,c,ga,pp)*DLP
  1293. WORK{IZZZ}
  1294. *end
  1295.  
  1296. C WW-scattering 7.  One loop Fi-Fi scattering, part 1. 33 sec.
  1297.   Result to file BoxFF1.
  1298.  
  1299. C One loop diagrams.
  1300.   Four point function. FF scattering.
  1301.   Evaluated in the limit m^2 >> s,t,u >> M^2, where m = Higgs boson
  1302.   mass and M = W boson mass.
  1303.  
  1304. C Terms in the output are labelled by A0, A1, A2, A3, and R, T for the
  1305.   reducible and tadpole types.
  1306.   The connection is:
  1307.   Rx: reducible diagrams (in u-channel, as A3).
  1308.   R1 type: One propagator.
  1309.   R5 type: Two propagators: selfenergy insertion.
  1310.   Diagrams marked with R1Z and R5Z are Z-exchange diagrams.
  1311.   Tx: tadpole types.
  1312.   A0: Box diagram, a,al,k and c,ga,pp in opposite corners.
  1313.   A1: Inverted Triangle diagram, a,al,k and c,ga,pp on 4-vertex.
  1314.   A2: Triangle diagram, a,al,k and c,ga, pp on triangle basis.
  1315.   A3: Bubble diagram, a,al,k and c,ga,pp on one end.
  1316.  
  1317. P ninput
  1318.  
  1319. C Work done:
  1320.   - Generate the diagrams.
  1321.   - Reduce as much as possible q ocurrences in the numerator.
  1322.   - Eliminate Higgs mass in terms containing at least one Higgs
  1323.     and one non-Higgs propagator. Cost: each m^2 gives one q.
  1324.  
  1325.   As it happens, of the four-propagator terms only some are left,
  1326.   with numerator pDq^4. That one is zero, because there are two
  1327.   Higgs propagators, and a non-zero result for large m obtains
  1328.   only for the most divergent part, i.e. when qqqq = D(,,,)
  1329.   Then the result is proportional to M^4, where behaviour as
  1330.   s^2 (or u^2, s*t etc.) is to be computed.
  1331.  
  1332.   These terms are put to zero, and Error wil be attached if there is
  1333.   any other four propagator term.
  1334.  
  1335.   It is assumed that there are no more than two non-Higgs propagators.
  1336.   If there are Error will be attached.
  1337.  
  1338. P ninput
  1339.  
  1340. C This order is of importance when ordering NOM.
  1341.  
  1342. A M,M2,m,m2,x,qq2,qqM,q2M
  1343. V q,q1,q3,q2,q4,q0
  1344.  
  1345. Read WWb.e
  1346. VERT{}
  1347. *fix
  1348.  
  1349. BLOCK REDUC{}
  1350. Id,Count,x,NOM,1
  1351. IF x^4
  1352. Id,Adiso,qDk^n~*NOM(q,M~)*NOM(q3,m~)=0.5*qDk^(n-1)*
  1353.  {NOM(q3,m) - NOM(q,M) + (kDk-M^2+m^2)*NOM(q,M)*NOM(q3,m)}
  1354. Id,Adiso,qDp^n~*NOM(q1,M~)*NOM(q,m~)=0.5*qDp^(n-1)*
  1355.  {NOM(q,m) - NOM(q1,M) + NOM(q,m)*NOM(q1,M)*(-pDp+m^2-M^2)}
  1356. Id,Adiso,qDpp^n~*NOM(q2,M~)*NOM(q1,m~)=0.5*qDpp^(n-1)*
  1357.  {NOM(q1,m) - NOM(q2,M) + NOM(q1,m)*NOM(q2,M)*(- 2*pDpp -ppDpp+m^2-M^2)}
  1358. Id,Adiso,qDkp^n~*NOM(q2,M~)*NOM(q3,m~)=0.5*qDkp^(n-1)*
  1359.  {NOM(q2,M) - NOM(q3,m) + NOM(q2,M)*NOM(q3,m)*(
  1360.  - kpDkp - 2*pDkp - 2*ppDkp + M^2 - m^2)}
  1361. Id,pDp=-M^2
  1362. Al,kDk=-M^2
  1363. Al,ppDpp=-M^2
  1364. Al,kpDkp=-M^2
  1365. Id,NOM(q~,M~)=NOM(M,q)
  1366. Id,Commu,NOM
  1367. Id,NOM(M~,q~)=NOM(q,M)
  1368. ENDIF
  1369. Id,x=1
  1370. ENDBLOCK
  1371.  
  1372. BLOCK Q2RED{X}
  1373. Id,NOM(q~,M~)=NOM(M,q)
  1374. Id,Commu,NOM
  1375. Id,NOM(M~,q~)=NOM(q,M)
  1376. Id,qDq=qq2
  1377.  
  1378. C Do only for X.
  1379.  
  1380. Id,qq2^n~*NOM(q~,'X')=qq2^n/qqM*Fxx(q,'X')
  1381.  
  1382. C This works for M and m.
  1383.  
  1384. Id,Ratio,qq2,qqM,q2M
  1385. Id,q2M^n~*Fxx(q,m~)=m^(2*n)*Fxx(q,m)
  1386. Al,q2M^n~*Fxx(q1,m~)={2*qDp-M^2+m^2}^n*Fxx(q1,m)
  1387. Al,q2M^n~*Fxx(q2,m~)={2*qDp+2*qDpp+2*pDpp-2*M^2+m^2}^n*Fxx(q2,m)
  1388. Al,q2M^n~*Fxx(q3,m~)={-2*qDk-M^2+m^2}^n*Fxx(q3,m)
  1389. Al,q2M^n~*Fxx(q4,m~)={-2*qDk-2*qDpp+2*kDpp-2*M^2+m^2}^n*Fxx(q4,m)
  1390. Id,qqM^-1*Fxx(q~,m~)=NOM(q,m)
  1391. Al,Fxx(q~,m~)=1
  1392. Al,qq2=qDq
  1393. ENDBLOCK
  1394.  
  1395. P stats
  1396. P input
  1397.  
  1398. Common BoxFF
  1399.  
  1400. V q1,q2,q3,q4
  1401. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  1402. I a=3,b=3,c=3,d=3
  1403.  
  1404. F F4q,F3,F2,F1
  1405.  
  1406. C Momenta: all taken to be ingoing.
  1407.   k,p in, pp,kp out. k + p = - pp - kp.
  1408.  
  1409. Z BoxFF(a,al,k,b,be,p,c,ga,pp,d,de,kp) =
  1410.     VIER("F,a,al,k,"F,b,be,p,"F,c,ga,pp,"F,d,de,kp)
  1411.  
  1412. C  For information: this is to be added to get the full result:
  1413.    + VIER("F,a,al,k,"F,c,ga,pp,"F,b,be,p,"F,d,de,kp)
  1414.    + VIER("F,a,al,k,"F,b,be,p,"F,d,de,kp,"F,c,ga,pp)
  1415.  
  1416. FOUR{}
  1417.  
  1418. Id,q1(al~)=q(al)+p(al)
  1419. Al,q1Dq1=qDq+pDp+2*qDp
  1420. Al,Dotpr,q1(al~)=q(al)+p(al)
  1421. Al,q0(al~)=0
  1422. Al,Dotpr,q0(al~)=0
  1423. Id,q2(al~)=q(al)+p(al)+pp(al)
  1424. Al,q2Dq2=qDq+pDp+ppDpp+2*qDp+2*qDpp+2*pDpp
  1425. Al,Dotpr,q2(al~)=q(al)+p(al)+pp(al)
  1426. Id,q3(al~)=q(al)-k(al)
  1427. Al,q3Dq3=qDq-2*qDk+kDk
  1428. Al,Dotpr,q3(al~)=q(al)-k(al)
  1429. Id,q4(al~)=q(al)-k(al)-pp(al)
  1430. Al,q4Dq4=qDq+kDk+ppDpp-2*qDk-2*qDpp+2*kDpp
  1431. Al,Dotpr,q4(al~)=q(al)-k(al)-pp(al)
  1432. Id,qu(al~)=k(al)+pp(al)
  1433. Al,quDqu=kDk+ppDpp+2*kDpp
  1434. Al,Dotpr,qu(al~)=k(al)+pp(al)
  1435.  
  1436. Al,Even,NOM,1
  1437.  
  1438. C By definition:
  1439.   Extm = 1/{1 + (2*kDpp + kDk + ppDpp)/m^2}
  1440.        = 1/{1 + 2*(kDpp - M^2)/m^2}
  1441.   ExtM = - 2*kDpp/(u-M^2)
  1442.   This makes their principal behaviour explicit.
  1443.  
  1444. Id,NOM(qu,m)= Extm/m^2
  1445. Al,NOM(qu,M)= ExtM/kDpp/2
  1446. Al,NOM(q0,M~)= 1/M^2
  1447.  
  1448. Id,pDp=-M^2
  1449. Al,kDk=-M^2
  1450. Al,ppDpp=-M^2
  1451. Al,kpDkp=-M^2
  1452. Id,Count,4,M,-1,M2,-2
  1453. Id,kp(al~)= - p(al)- k(al) - pp(al)
  1454. Al,Dotpr,kp(al~)= - p(al) - k(al)- pp(al)
  1455. Id,Epfred
  1456. Al,Even,NOM,1
  1457.  
  1458. Id,Count,4,M,-1,M2,-2
  1459. Id,pDpp=-pDk-kDpp
  1460.  
  1461. IF NOT NOM(q~,m)
  1462. Id,Count,0,m,1
  1463. ENDIF
  1464.  
  1465. *yep
  1466.  
  1467. Q2RED{M}
  1468. Q2RED{M}
  1469. *yep
  1470. Q2RED{m}
  1471. Q2RED{m}
  1472. *yep
  1473. REDUC{}
  1474. *yep
  1475. REDUC{}
  1476. *yep
  1477. REDUC{}
  1478.  
  1479. Id,Count,4,M,-1,M2,-2
  1480.  
  1481. *yep
  1482.  
  1483. Id,NOM(q~,M~)=NOM(M,q)
  1484. Id,Commu,NOM
  1485. Id,NOM(M~,q~)=NOM(q,M)
  1486.  
  1487. Id,Adiso,pDq^4*NOM(q~,m)*NOM(q1~,m)*NOM(q2~,M)*NOM(q3~,M)=0
  1488.  
  1489. *yep
  1490.  
  1491. Id,Count,x,NOM,1
  1492.  
  1493. IF NOT x
  1494. Id,Addfa,0
  1495. ENDIF
  1496.  
  1497. Id,x=1
  1498.  
  1499. Id,NOM(q~,M)=x*NOM(q,M)
  1500. IF Multi,x^3
  1501. Id,Addfa,Error
  1502. ENDIF
  1503.  
  1504. Id,x=1
  1505.  
  1506. *begin
  1507. Write BoxFF1
  1508. *end
  1509.  
  1510. C WW-scattering 8.  One loop Fi-Fi scattering, part 2. 228 sec.
  1511.   Uses output from 7, file BoxFF1. Produces BoxFF2.
  1512.  
  1513. C Work done:
  1514.  - Expand all Higgs propagators: 1/((q+qx)^2+m^2) => 1/(q^2+m^2)
  1515.  - The assumption at this point is that that there are no more
  1516.    than 2 non-Higgs propagators. If two, take them together in
  1517.    the function Two(qa,qb). If qa not q then shift momentum so
  1518.    that only Two(q,qx) occurs. The Higgs propagators become shifted
  1519.    again. They are expanded again.
  1520.  - Expand Two(q,qx) times any non-zero number of Higgs propagators:
  1521.    Rationalize 1/(q^2+M^2)* 1/(q^2+m^2), the result contains a NOM
  1522.    but no more Two.
  1523.    Reduce any non-zero number of qDq together with Two(q,qx).
  1524.  - Work out NOM(qx,M) with qx not q, and any number of Higgs propagators.
  1525.    Shift qx to q. Expand shifted Higgs propagators.
  1526.  - Rationalize again.
  1527.  - Reduce all qDq occurences.
  1528.  - After this work there are the following types of terms:
  1529.    One Two function and no Higgs propagator;
  1530.    One NOM(q,M);
  1531.    Any number of Higgs propagators.
  1532.  
  1533. P ninput
  1534.  
  1535. Enter BoxFF1
  1536.  
  1537. Read WWb.e
  1538. ASSIGN{}
  1539.  
  1540. *fix
  1541.  
  1542. Names BoxFF
  1543. P stats
  1544.  
  1545. Z BoxFF(a,al,k,b,be,p,c,ga,pp,d,de,kp) =
  1546.   BoxFF(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  1547.  
  1548. C Expand Higgs propagators.
  1549.   There may be two of them.
  1550.  
  1551. Id,NOM(q,m)=Nohm
  1552.  
  1553. IF NOM(q~,m)=Fxx(q)
  1554. Id,Count,Div,q,1,NOM,-2,Two,-4
  1555. Id,Fxx(q~)=Fact^-2*NOM(q,m)
  1556. Id,Fact^n~=Fxx(n,0,0)
  1557. Id,Nohm^n~*Fxx(n1~,0,0)=Nohm^n*Fxx(n1-2*n,0,0)
  1558. Id,m^n~*Fxx(n1~,0,0)=m^n*Fxx(n1,n,0)
  1559. Id,Div^n~*Fxx(n1~,n2~,0)=Fxx(n1,n2,n)
  1560. Id,Adiso,Fxx(n1~,n2~,n3~)*NOM(q1,m)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDp),1)
  1561. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q2,m)=Fdiv(n1,n2,n3)*
  1562.  Exp(n1,n2,n3,(-2*qDp-2*qDpp-2*pDpp),1)
  1563. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  1564.  *Exp(n1,n2,n3,(2*qDk),1)
  1565. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q4,m)=Fdiv(n1,n2,n3)
  1566.  *Exp(n1,n2,n3,(2*qDk+2*qDpp-2*kDpp),1)
  1567. ENDIF
  1568.  
  1569. Id,Fact^n~=1
  1570.  
  1571. B Nohm,Nom
  1572. *yep
  1573.  
  1574. IF NOM(q~,m)=Fxx(q)
  1575. Id,Count,Div,q,1,NOM,-2,Two,-4
  1576. Id,Fxx(q~)=Fact^-2*NOM(q,m)
  1577. Id,Fact^n~=Fxx(n,0,0)
  1578. Id,Nohm^n~*Fxx(n1~,0,0)=Nohm^n*Fxx(n1-2*n,0,0)
  1579. Id,m^n~*Fxx(n1~,0,0)=m^n*Fxx(n1,n,0)
  1580. Id,Div^n~*Fxx(n1~,n2~,0)=Fxx(n1,n2,n)
  1581. Id,Adiso,Fxx(n1~,n2~,n3~)*NOM(q1,m)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDp),1)
  1582. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q2,m)=Fdiv(n1,n2,n3)*
  1583.  Exp(n1,n2,n3,(-2*qDp-2*qDpp-2*pDpp),1)
  1584. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  1585.  *Exp(n1,n2,n3,(2*qDk),1)
  1586. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q4,m)=Fdiv(n1,n2,n3)
  1587.  *Exp(n1,n2,n3,(2*qDk+2*qDpp-2*kDpp),1)
  1588. ENDIF
  1589.  
  1590. Id,Fact^n~=1
  1591.  
  1592. B Nohm,Nom
  1593. *yep
  1594. Id,pDpp=M^2-pDk-kDpp
  1595. Id,Count,4,M,-1,M2,-2
  1596.  
  1597. HCOUNT{}
  1598. *yep
  1599.  
  1600. Id,Adiso,NOM(q~,M)*NOM(q1~,M) = Two(q,q1)
  1601. Id,Commu,NOM
  1602. Id,Two(q1~,q)=Two(q,q1~)
  1603.  
  1604. Id,Two(q1,q2~)=Sh1*Two(q,q2,-p)
  1605. Al,Two(q2,q5~)=Sh2*Two(q,q5,qt)
  1606. Al,Two(q3,q5~)=Sh3*Two(q,q5,k)
  1607. Al,Two(q4,q5~)=Sh4*Two(q,q5,qu)
  1608.  
  1609. Id,Two(q,q3,-p)=Two(q,q5)
  1610.  
  1611. SHIFT{}
  1612. HCOUNT{}
  1613.  
  1614. *yep
  1615.  
  1616. C Reduction of 1/(q^2+M^2) * 1/(q^2+m^2)^n
  1617.  
  1618. IF Nohm^n~*Two(q,q1~)=1/q2M*Fxx(q1)/q2m^n
  1619. Id,2,Ratio,q2M,q2m,[m2-M2]
  1620. Id,q2M^-1*Fxx(q1~)=Two(q,q1)
  1621. Al,Fxx(q1~)=NOM(q1,M)
  1622. Al,q2m^n~=1/Nohm^n
  1623. ENDIF
  1624.  
  1625. HCOUNT{}
  1626.  
  1627. C Elimination of qDq and Two.
  1628.   Should not occur, as qDq and M type propagators were already treated.
  1629.  
  1630. IF qDq^n~*Two(q,q1~)=qq2^n*Fxx(q1)/q2M
  1631. Id,2,Ratio,qq2,q2M,M2
  1632. Id,q2M^-1*Fxx(q1~)=Two(q,q1)
  1633. Al,Fxx(q1~)=NOM(q1,M)
  1634. Al,qq2=qDq^2
  1635. ENDIF
  1636.  
  1637. *yep
  1638.  
  1639. C Integration variable shift of  1/((q+qx)^2 + M^2)
  1640.  
  1641. IF NOT NOM(q,M)=Nom
  1642. Id,NOM(q1,M)=Sh1*Nom
  1643. Al,NOM(q2,M)=Sh2*Nom
  1644. Al,NOM(q3,M)=Sh3*Nom
  1645. Al,NOM(q4,M)=Sh4*Nom
  1646. Al,NOM(q5,M)=Sh5*Nom
  1647. ENDIF
  1648.  
  1649. SHIFT{}
  1650.  
  1651. HCOUNT{}
  1652.  
  1653. *yep
  1654.  
  1655. C Reduction of 1/(q^2+M^2) * 1/(q^2+m^2)^n
  1656.  
  1657. IF Nohm^n~*Nom=1/q2M/q2m^n
  1658. Id,2,Ratio,q2M,q2m,[m2-M2]
  1659. Id,q2M^-1=Nom
  1660. Al,q2m^n~=1/Nohm^n
  1661. ENDIF
  1662.  
  1663. *yep
  1664.  
  1665. C Elimination of qDq.
  1666.  
  1667. IF Nom
  1668. Id,Nom^n~=1/q2M^n
  1669. Al,qDq^n~=qq2^n
  1670. Id,Ratio,qq2,q2M,M2
  1671. Id,q2M^n~=1/Nom^n
  1672. Al,qq2=qDq
  1673. Al,M2=M^2
  1674. ENDIF
  1675.  
  1676. IF Nohm
  1677. Id,Nohm^n~=1/q2m^n
  1678. Al,qDq^n~=qq2^n
  1679. Id,Ratio,qq2,q2m,m2
  1680. Id,q2m^n~=1/Nohm^n
  1681. Al,qq2=qDq
  1682. Al,m2=m^2
  1683. ENDIF
  1684.  
  1685. IF NOM(q~,M~)
  1686. OR Two(q~,q1~)*Nohm^n~
  1687. Id,Addfa,Error
  1688. ENDIF
  1689.  
  1690. Id,Count,4,M,-1,M2,-2
  1691.  
  1692. Id,Count,x,Nohm,1,Nom,1,Two,1,NOM,1
  1693.  
  1694. IF NOT x
  1695. Id,Addfa,0
  1696. ENDIF
  1697.  
  1698. Id,x=1
  1699.  
  1700. C Check dimension 0.
  1701. C
  1702. Id,Count,x,M,1,M2,2,m,1,m2,2,Two,-4,Nom,-2,Nohm,-2,[m2-M2],2,
  1703.  q,1,p,1,k,1,pp,1,kp,1
  1704.  
  1705. IF NOT x^-4=1
  1706. Id,Addfa,Error
  1707. ENDIF
  1708.  
  1709. B Error,Nohm,Nom
  1710.  
  1711. *begin
  1712. Write BoxFF2
  1713. *end
  1714.  
  1715. C WW-scattering 9.  One loop Fi-Fi scattering, part 3. 125 sec.
  1716.   Uses output from 8. Produces BoxFF_comm.
  1717.  
  1718. C Part 3 of BoxFF.
  1719.  
  1720. C One loop diagrams.
  1721.   Four point function. FF scattering.
  1722.   Evaluated in the limit m^2 >> s,t,u >> M^2, where m = Higgs boson
  1723.   mass and M = W boson mass.
  1724.  
  1725. C Work to be done:
  1726.  - Do integrals.
  1727.    1/(qx^2+M^2) with or without Higgs propagators;
  1728.    Higgs propagators;
  1729.    Functions Two and no Higgs propagator.
  1730.  
  1731. P ninput
  1732.  
  1733. Enter BoxFF2
  1734.  
  1735. Read WWb.e
  1736. ASSIGN{}
  1737.  
  1738. *fix
  1739.  
  1740. Names BoxFF
  1741. P stats
  1742.  
  1743. Z Box(a,b,c,d,s,t,u) =
  1744.  BoxFF(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  1745.  
  1746. IF NOT Two(q~,q1~)
  1747. Id,All,q,N,Fq
  1748. ENDIF
  1749.  
  1750. Id,Fq(al~)=0
  1751. Al,Fq(al~,be~,ga~)=0
  1752. Al,Fq(al~,be~,ga~,de~,la~)=0
  1753. Al,Fq(al~,be~,ga~,de~,la~,a~,b~)=0
  1754.  
  1755. Id,Fq(al~,al~)=0
  1756.  
  1757. Id,Fq(al~,be~,be~,be~)=0
  1758. Al,Fq(be~,be~,be~,al~)=0
  1759.  
  1760. Al,Fq(be~,be~,be~,be~,al~,ga~)=0
  1761. Al,Fq(al~,be~,be~,be~,be~,al~)=0
  1762. Al,Fq(al~,ga~,be~,be~,be~,be~)=0
  1763.  
  1764. B Nohm,Nom
  1765. *yep
  1766. IF NOT Two(q~,q1~)
  1767. Id,Fq(al~,be~,ga~,de~)*Nom^l~ = dede(al,be,ga,de)*H(l,M2)
  1768. Al,Fq(al~,be~)*Nom^l~ = D(al,be)*G(l,M2)
  1769. Al,Nom^n~ = F(n,M2)
  1770. Id,Fq(al~,be~,ga~,de~)*Nohm^l~ = dede(al,be,ga,de)*H(l,m2)
  1771. Al,Fq(al~,be~)*Nohm^l~ = D(al,be)*G(l,m2)
  1772. Al,Nohm^l~ = F(l,m2)
  1773. ENDIF
  1774.  
  1775. Id,pDp=-M^2
  1776. Al,kDk=-M^2
  1777. Al,ppDpp=-M^2
  1778. Al,kpDkp=-M^2
  1779.  
  1780. Id,Multi,M^-2=M2^-1
  1781. Al,Multi,m^2=m2
  1782. Al,Multi,m^-2=m2^-1
  1783.  
  1784. *yep
  1785.  
  1786. STINT{}
  1787.  
  1788. Id,N=N_+4
  1789. Id,N_=0
  1790. Al,N=4
  1791. Id,pDpp=-kDp-kDpp+M2
  1792. B i,Pi,N_,Nohm,M2
  1793. Id,Count,4,M,-1,M2,-2
  1794. *yep
  1795.  
  1796. Id,Two(q,q1~)=Two(q1)
  1797. IF Two(q~)
  1798. Al,All,q,N,Fq
  1799. ENDIF
  1800. Id,Adiso,Two(q4)*Fq(al~,be~) =
  1801.  D(al,be)*B22(u,M,M) + (k(al)+pp(al))*(k(be)+pp(be))*B21(u,M,M)
  1802. Al,Adiso,Two(q4)*Fq(al~)=- (k(al)+pp(al))*B1(u,M,M)
  1803. Al,Two(q4)=B0(u,M,M)
  1804.  
  1805. Id,Adiso,Two(q5)*Fq(al~,be~) =
  1806.  D(al,be)*B22(s,M,M) + (k(al)+p(al))*(k(be)+p(be))*B21(s,M,M)
  1807. Al,Adiso,Two(q5)*Fq(al~)=- (k(al)+p(al))*B1(s,M,M)
  1808. Al,Two(q5)=B0(s,M,M)
  1809.  
  1810. Id,Adiso,Two(q2)*Fq(al~,be~) =
  1811.  D(al,be)*B22(t,M,M) + (pp(al)+p(al))*(pp(be)+p(be))*B21(t,M,M)
  1812. Al,Adiso,Two(q2)*Fq(al~)= (pp(al)+p(al))*B1(t,M,M)
  1813. Al,Two(q2)=B0(t,M,M)
  1814.  
  1815. *yep
  1816.  
  1817. Id,B22(u~,M~,m~)=(-0.5*F1(m)+M**2*B0(u,M,m)
  1818.       -0.5*(-u+m^2-M**2)*B1(u,M,m))/[1-N]
  1819. Id,B21(u~,M~,m~)=((0.5*N-1)*F1(m)
  1820.        -0.5*N*(-u+m^2-M**2)*B1(u,M,m)
  1821.        +M**2*B0(u,M,m) )/u/[1-N]
  1822. Id,B1(u~,M~,m~)=- (0.5*F1(M)-0.5*F1(m)
  1823.               -0.5*(-u+m^2-M**2)*B0(u,M,m) )/u
  1824. Id,B0(u~,M~,M~)= - 2*i*Pi^2/N_ - i*Pi^2*Log(u) + 2*i*Pi^2
  1825. Al,F1(M) = 2*i*Pi^2*M2/N_ + i*Pi^2*M2*(-1+Log(M2))
  1826. Al,F1(m) = 2*i*Pi^2*m2/N_ + i*Pi^2*m2*(-1+Log(m2))
  1827. Al,M^n~=M2^(n/2)
  1828. Al,m^n~=m2^(n/2)
  1829.  
  1830. Id,N=N_+4
  1831. Al,[1-N]^-1=-1/3 + N_/9
  1832. Id,N_=0
  1833. Id,N=4
  1834. Al,ExtM = - 2*kDpp/u*{1 + M2/u + M2^2/u^2}
  1835.  
  1836. Id,Count,0,m2,2,[m2-M2],2
  1837. Al,Multi,m2^-1=0
  1838. Id,Count,x,m2,1
  1839. IF x
  1840. Id,x^n1~*[m2-M2]^n~=x^n1*m2^n*DS(K,0,n1+n,(DB(-n+K-1,K)*M2^K*m2^-K))
  1841. Id,x^n1~*Extm^n~=DS(K,0,n1,(DB(n+K-1,K)*(-2*kDpp/m2)^K))
  1842. ELSE
  1843. Id,[m2-M2]^n~=m2^n
  1844. Al,Extm=1
  1845. ENDIF
  1846. Id,Count,0,m2,1
  1847. Al,x=1
  1848.  
  1849. Id,pDk= - 0.5*s
  1850. Al,pDpp = - 0.5*t
  1851. Al,kDpp = - 0.5*u
  1852. Id,pDp=-M2
  1853. Al,kDk=-M2
  1854. Al,ppDpp=-M2
  1855. Al,kpDkp=-M2
  1856. Id,Count,4,s,2,u,2,t,2
  1857. Id,t=-s-u
  1858. Keep Box
  1859. *next
  1860.  
  1861. P input
  1862.  
  1863. Common FTot
  1864. Delete BoxFF
  1865.  
  1866. C Add further diagrams, obtained by crossing. Set labels A0-A3 to 1.
  1867.  
  1868. Z FTot=Box(a,b,c,d,s,t,u) + Box(a,c,b,d,u,t,s) + Box(a,b,d,c,s,u,t)
  1869.  
  1870. B i,Pi,N_,M2
  1871. Id,A0=1
  1872. Al,A1=1
  1873. Al,A2=1
  1874. Al,A3=1
  1875. Al,R1=1
  1876. Al,R2=1
  1877. Al,R3=1
  1878. Al,R3a=1
  1879. Al,R3b=1
  1880. Al,R4=1
  1881. Al,R4a=1
  1882. Al,R4b=1
  1883. Al,R5=1
  1884. Al,R6=1
  1885. Al,T1=1
  1886. Al,T2=1
  1887. Al,T3=1
  1888.  
  1889. IF D(a,c)
  1890. Id,t=-s-u
  1891. ENDIF
  1892.  
  1893. IF D(a,b)
  1894. Id,u=-s-t
  1895. ENDIF
  1896.  
  1897. IF D(a,d)
  1898. Id,s=-t-u
  1899. ENDIF
  1900. P output
  1901. *yep
  1902.  
  1903. Id,R1Z=1
  1904. Al,R5Z=1
  1905.  
  1906. *begin
  1907. Write BoxFF_comm
  1908. *end
  1909.  
  1910. C WW-scattering 10. Fi-Fi Renormalization. Result Fi-Fi amplitude.
  1911.   Uses output from 9.
  1912.  
  1913. C Subtraction terms four-Fi amplitude.
  1914.  
  1915. P ninput
  1916.  
  1917. A M,m,x,qq2,qqM,q2M,s,t,u
  1918. V q,q0,q1,q3,q2,q4
  1919.  
  1920. Enter BoxFF_comm
  1921.  
  1922. Read WWb.e
  1923. VERT{}
  1924. *fix
  1925.  
  1926. Names FTot
  1927.  
  1928. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  1929. I a=3,b=3,c=3,d=3
  1930.  
  1931. C Momenta: all taken to be ingoing.
  1932.   k,p in, pp,kp out. k + p = - pp - kp.
  1933.  
  1934. Z RenFF4 = FFFF(a,k,k,b,p,p,c,pp,pp,d,kp,kp)*FFFFK
  1935.  
  1936. Z RenFFr(a,b,c,d,s,t,u)=
  1937.    DS("F;"F;-J1;TAP,(
  1938.    VIE1("F,a,k,k,"F,b,p,p,"F,c,pp,pp,"F,d,kp,kp,J1) ))
  1939.  
  1940.  + DS("F;"F;-J2;TAP,(DS("F;"F;J3;TAP,(
  1941.    VIE2("F,a,k,k,"F,b,p,p,"F,c,pp,pp,"F,d,kp,kp,J2,J3) )) ))
  1942.  
  1943.  + DS("F;"F;J4;"Z;TAP,(
  1944.    VIE3("F,a,k,k,"F,b,p,p,"F,c,pp,pp,"F,d,kp,kp,J4) ))
  1945.  
  1946.  
  1947. Id,VIE1(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~) =
  1948.    VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  1949.    VE3(K2,K4,J1,*,b,be,p,*,d,de,-kp,*,l2,m0,qu)*
  1950.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  1951.    CONT(K1,K3,J1,/,"K)*CONT(K2,K4,-J1,/,"K)
  1952.    *R3
  1953.  
  1954. Al,VIE2(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~) =
  1955.    VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  1956.    VE3(K2,K4,J2,*,b,be,p,*,d,de,-kp,*,l4,m4,qu)*
  1957.    VE3(J1,-J2,"N,*,l2,m0,qu,*,l3,m3,-qu,*,l5,m5,q0)*
  1958.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  1959.    PROP(J2,-J2,*,l3,m3,qu,*,l4,m4,-qu)
  1960.    *R5
  1961.  
  1962.  + VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  1963.    VE3(K2,K4,J2,*,b,be,p,*,d,de,-kp,*,l4,m4,qu)*
  1964.    VE3(J1,-J2,"Z,*,l2,m0,qu,*,l3,m3,-qu,*,l5,m5,q0)*
  1965.    NNZ(a,al,k,b,be,p,l6,m6,-q0)*
  1966.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  1967.    PROP(J2,-J2,*,l3,m3,qu,*,l4,m4,-qu)*
  1968.    PROP("Z,"Z,*,l5,m5,q0,*,l6,m6,-q0)
  1969.    *T3
  1970.  
  1971. Id,VIE3(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~) =
  1972.    VE4(K1,K3,-J1,"Z,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu,*,l3,m3,-q0)*
  1973.    VE3(K2,K4,J1,*,b,be,p,*,d,de,-kp,*,l2,m0,qu)*
  1974.    NNZ(a,al,k,b,be,p,l4,m4,q0)*
  1975.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  1976.    PROP("Z,"Z,*,l3,m3,q0,*,l4,m4,-q0)
  1977.    *T1
  1978.  
  1979.   +VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  1980.    VE4(K2,K4,J1,"Z,*,b,be,p,*,d,de,-kp,*,l2,m0,qu,*,l3,m3,-q0)*
  1981.    NNZ(a,al,k,b,be,p,l4,m4,q0)*
  1982.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  1983.    PROP("Z,"Z,*,l3,m3,q0,*,l4,m4,-q0)
  1984.    *T2
  1985.  
  1986. Id,Anti,TAP
  1987.  
  1988. Id,Compo,<X>,VE4,VE3,PROP
  1989. Id,Compo,<X>,CONT
  1990.  
  1991. Id,Adiso,CONT(FF~)*CONT(WW~)= FF + WW
  1992. Al,CONT(FF~)= FF
  1993.  
  1994. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  1995.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  1996. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  1997.        FF(l1,al,k,l2,be,q,l3,ga,p)
  1998. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  1999. B i,Pi,N_,M
  2000.  
  2001. Id,qu(al~)=k(al)+pp(al)
  2002. Al,quDqu=2*kDpp
  2003. Al,Dotpr,qu(al~)=k(al)+pp(al)
  2004. Al,Even,NOM,1
  2005.  
  2006. C By definition:
  2007.   Extm = 1/{1 + (2*kDpp + kDk + ppDpp)/m^2}
  2008.        = 1/{1 + 2*(kDpp - M^2)/m^2}
  2009.   This makes the principal behaviour explicit.
  2010.  
  2011. Id,NOM(qu,m)= Extm/m^2
  2012. Al,NOM(q0,M~)= 1/M^2
  2013.  
  2014. Id,NOM(qu,M)=-1/u
  2015. Al,Even,NOM,1
  2016. Id,kpDkp=0
  2017. Al,kp(al~)=-k(al)-p(al)-pp(al)
  2018. Al,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  2019. Id,kDk=0
  2020. Al,pDp=0
  2021. Al,ppDpp=0
  2022. Id,pDk=-0.5*s
  2023. Al,pDpp=-0.5*t
  2024. Al,kDpp=-0.5*u
  2025.  
  2026. ETE1{}
  2027.  
  2028. *yep
  2029.  
  2030. Id,Count,0,m,1,m2,2,[m2-M2],2
  2031. Id,Count,x,m,1,m2,2,[m2-M2],2
  2032. IF x
  2033. Id,x^n1~*[m2-M2]^n~=x^n1*m2^n*DS(K,0,n1/2+n+1,(DB(-n+K-1,K)*M2^K*m2^-K))
  2034. Id,x^n1~*Extm^n~=DS(K,0,n1/2+1,(DB(n+K-1,K)*(-2*kDpp/m^2)^K))
  2035. ELSE
  2036. Id,[m2-M2]^n~=m2^n
  2037. Al,Extm=1
  2038. ENDIF
  2039. Id,M2^n~=M^(2*n)
  2040. Al,m2^n~=m^(2*n)
  2041. Al,kDpp=-0.5*u
  2042. Id,Count,0,m,1,m2,2
  2043. Al,x=1
  2044.  
  2045. Id,Count,4,s,2,t,2,u,2
  2046. *yep
  2047.  
  2048. Id,Epfred
  2049.  
  2050. IF D(a,c)
  2051. Id,t=-s-u
  2052. ENDIF
  2053. IF D(a,b)
  2054. Id,u=-s-t
  2055. ENDIF
  2056. IF D(a,d)
  2057. Id,u=-s-t
  2058. ENDIF
  2059.  
  2060. Keep RenFF4,RenFFr
  2061. *next
  2062. B i,Pi,N_,M
  2063.  
  2064. Z RenFFt=RenFF4
  2065.   + RenFFr(a,b,c,d,s,t,u) + RenFFr(a,c,b,d,u,t,s) + RenFFr(a,b,d,c,s,u,t)
  2066.  
  2067. Id,R5W=1
  2068. Al,R3=1
  2069.  
  2070. IF D(a,c)
  2071. Id,t=-s-u
  2072. ENDIF
  2073. IF D(a,b)
  2074. Id,u=-s-t
  2075. ENDIF
  2076. IF D(a,d)
  2077. Id,s=-t-u
  2078. ENDIF
  2079.  
  2080. *yep
  2081. Id,T1=1
  2082. Al,T2=1
  2083. Al,T3=1
  2084. Al,R5Z=1
  2085. Al,R5=1
  2086. Keep RenFFt
  2087. *next
  2088. P input
  2089. C Renormalized FFFF amplitude.
  2090.  
  2091. Z RenF = FTot - RenFFt
  2092. B i,Pi,N_,M2,M
  2093. Id,Multi,M2^n~=M^(2*n)
  2094. Al,Logm2=Log(m2)
  2095. Id,Log(s)=Log(s,m2)+Log(m2)
  2096. Al,Log(t)=Log(t,m2)+Log(m2)
  2097. Al,Log(u)=Log(u,m2)+Log(m2)
  2098. P output
  2099. *yep
  2100. C Specialize, for computimg purposes, to index a=b, c=d, a not c.
  2101.  
  2102. IF NOT D(a,b)=1
  2103. Id,Addfa,0
  2104. ENDIF
  2105. Id,D(c,d)=1
  2106.  
  2107. *end
  2108.  
  2109. C WW-scattering 11. One loop W-W scattering, part 1. 681 sec.
  2110.   Produces BoxWW1, containing BoxWW.
  2111.  
  2112. C One loop diagrams.
  2113.   Four point function. WW scattering.
  2114.   Evaluated in the limit m^2 >> s,t,u >> M^2, where m = Higgs boson
  2115.   mass and M = W boson mass.
  2116.  
  2117. C Terms in the output are labelled by A0, A1, A2, A3, and R, T for the
  2118.   reducible and tadpole types.
  2119.   The connection is:
  2120.   Rx: reducible diagrams (in u-channel, as A3).
  2121.   R1 type: One propagator.
  2122.   R5 type: Two propagators: selfenergy insertion.
  2123.   Diagrams marked with R1Z and R5Z are Z-exchange diagrams.
  2124.   Tx: tadpole types.
  2125.   A0: Box diagram, a,al,k and c,ga,pp in opposite corners.
  2126.   A1: Inverted Triangle diagram, a,al,k and c,ga,pp on 4-vertex.
  2127.   A2: Triangle diagram, a,al,k and c,ga, pp on triangle basis.
  2128.   A3: Bubble diagram, a,al,k and c,ga,pp on one end.
  2129.  
  2130. P ninput
  2131.  
  2132. C Work done:
  2133.   - Generate the diagrams.
  2134.   - Reduce as much as possible q ocurrences in the numerator.
  2135.  
  2136.   It is assumed that there are no more than two non-Higgs propagators.
  2137.   If there are Error will be attached. Such terms add up to zero,
  2138.   demonstrated elsewhere.
  2139.  
  2140. P ninput
  2141.  
  2142. C This order is of importance when ordering NOM.
  2143. C
  2144. A M,M2,m,m2,x,qq2,qqM,q2M
  2145. V q,q1,q3,q2,q4,q0
  2146.  
  2147. Read WWb.e
  2148. VERT{}
  2149. *fix
  2150.  
  2151. BLOCK REDUC{}
  2152. Id,Count,x,NOM,1
  2153. IF Multi,x^3
  2154. Id,Adiso,qDk^n~*NOM(q,M~)*NOM(q3,m~)=0.5*qDk^(n-1)*
  2155.  {NOM(q3,m) - NOM(q,M) + (kDk-M^2+m^2)*NOM(q,M)*NOM(q3,m)}
  2156. Id,Adiso,qDp^n~*NOM(q1,M~)*NOM(q,m~)=0.5*qDp^(n-1)*
  2157.  {NOM(q,m) - NOM(q1,M) + NOM(q,m)*NOM(q1,M)*(-pDp+m^2-M^2)}
  2158. Id,Adiso,qDpp^n~*NOM(q2,M~)*NOM(q1,m~)=0.5*qDpp^(n-1)*
  2159.  {NOM(q1,m) - NOM(q2,M) + NOM(q1,m)*NOM(q2,M)*(- 2*pDpp -ppDpp+m^2-M^2)}
  2160. Id,Adiso,qDpp^n~*NOM(q4,M~)*NOM(q3,m~)=-0.5*qDpp^(n-1)*
  2161.  {NOM(q3,m) - NOM(q4,M) + NOM(q3,m)*NOM(q4,M)*(- 2*kDpp -ppDpp+m^2-M^2)}
  2162. Id,Adiso,qDkp^n~*NOM(q2,M~)*NOM(q3,m~)=0.5*qDkp^(n-1)*
  2163.  {NOM(q2,M) - NOM(q3,m) + NOM(q2,M)*NOM(q3,m)*(
  2164.  - kpDkp - 2*pDkp - 2*ppDkp + M^2 - m^2)}
  2165. Id,Adiso,qDkp^n~*NOM(q4,M~)*NOM(q1,m~)=0.5*qDkp^(n-1)*
  2166.  {NOM(q1,m) - NOM(q4,M) + NOM(q1,m)*NOM(q4,M)*(- 2*pDkp -kpDkp+m^2-M^2)}
  2167. Id,pDp=0
  2168. Al,kDk=0
  2169. Al,ppDpp=0
  2170. Al,kpDkp=0
  2171. Id,pDpp=-pDk-kDpp
  2172. Id,NOM(q~,M~)=NOM(M,q)
  2173. Id,Commu,NOM
  2174. Id,NOM(M~,q~)=NOM(q,M)
  2175. ENDIF
  2176. Id,x=1
  2177. ENDBLOCK
  2178.  
  2179. BLOCK Q2RED{X}
  2180. Id,NOM(q~,M~)=NOM(M,q)
  2181. Id,Commu,NOM
  2182. Id,NOM(M~,q~)=NOM(q,M)
  2183. Id,qDq=qq2
  2184.  
  2185. C Do only for X.
  2186.  
  2187. Id,qq2^n~*NOM(q~,'X')=qq2^n/qqM*Fxx(q,'X')
  2188.  
  2189. C This works for M and m.
  2190.  
  2191. Id,Ratio,qq2,qqM,q2M
  2192. Id,q2M^n~*Fxx(q,m~)=m^(2*n)*Fxx(q,m)
  2193. Al,q2M^n~*Fxx(q1,m~)={2*qDp-M^2+m^2}^n*Fxx(q1,m)
  2194. Al,q2M^n~*Fxx(q2,m~)={2*qDp+2*qDpp+2*pDpp-2*M^2+m^2}^n*Fxx(q2,m)
  2195. Al,q2M^n~*Fxx(q3,m~)={-2*qDk-M^2+m^2}^n*Fxx(q3,m)
  2196. Al,q2M^n~*Fxx(q4,m~)={-2*qDk-2*qDpp+2*kDpp-2*M^2+m^2}^n*Fxx(q4,m)
  2197. Id,qqM^-1*Fxx(q~,m~)=NOM(q,m)
  2198. Al,Fxx(q~,m~)=1
  2199. Al,qq2=qDq
  2200. ENDBLOCK
  2201.  
  2202. P stats
  2203.  
  2204. Common BoxWW
  2205.  
  2206. V q1,q2,q3,q4
  2207. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  2208. I a=3,b=3,c=3,d=3
  2209.  
  2210. F F4q,F3,F2,F1
  2211.  
  2212. C Momenta: all taken to be ingoing.
  2213.   k,p in, pp,kp out. k + p = - pp - kp.
  2214.  
  2215. Z BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp) =
  2216.     VIER("W,a,k,k,"W,b,p,p,"W,c,pp,pp,"W,d,kp,kp)/M^4
  2217.  
  2218. C  For information: this is to be added to get the full result:
  2219.  
  2220.    + VIER("W,a,al,k,"W,c,ga,pp,"W,b,be,p,"W,d,de,kp)
  2221.    + VIER("W,a,al,k,"W,b,be,p,"W,d,de,kp,"W,c,ga,pp)
  2222.  
  2223. FOUR{}
  2224.  
  2225. Id,pDp=0
  2226. Al,kDk=0
  2227. Al,ppDpp=0
  2228. Al,kpDkp=0
  2229.  
  2230. *yep
  2231.  
  2232. Id,q1(al~)=q(al)+p(al)
  2233. Al,q1Dq1=qDq+2*qDp
  2234. Al,Dotpr,q1(al~)=q(al)+p(al)
  2235. Al,q0(al~)=0
  2236. Al,Dotpr,q0(al~)=0
  2237. Id,q2(al~)=q(al)+p(al)+pp(al)
  2238. Al,q2Dq2=qDq+2*qDp+2*qDpp+2*pDpp
  2239. Al,Dotpr,q2(al~)=q(al)+p(al)+pp(al)
  2240. Id,q3(al~)=q(al)-k(al)
  2241. Al,q3Dq3=qDq-2*qDk
  2242. Al,Dotpr,q3(al~)=q(al)-k(al)
  2243. Id,q4(al~)=q(al)-k(al)-pp(al)
  2244. Al,q4Dq4=qDq-2*qDk-2*qDpp+2*kDpp
  2245. Al,Dotpr,q4(al~)=q(al)-k(al)-pp(al)
  2246. Id,qu(al~)=k(al)+pp(al)
  2247. Al,quDqu=kDk+2*kDpp
  2248. Al,Dotpr,qu(al~)=k(al)+pp(al)
  2249.  
  2250. Al,Even,NOM,1
  2251.  
  2252. Id,Epfred
  2253. *yep
  2254.  
  2255.  
  2256. C By definition:
  2257.   Extm = 1/{1 + (2*kDpp + kDk + ppDpp)/m^2}
  2258.        = 1/{1 + 2*(kDpp - M^2)/m^2}
  2259.   ExtM = - 2*kDpp/(u-M^2). Note ExtM=-1 in lowest order.
  2260.   This makes their principal behaviour explicit.
  2261.  
  2262. Id,NOM(qu,m)= Extm/m^2
  2263. Al,NOM(qu,M)= ExtM/kDpp/2
  2264. Al,NOM(q0,M~)= 1/M^2
  2265.  
  2266. Id,pDp=0
  2267. Al,kDk=0
  2268. Al,ppDpp=0
  2269. Al,kpDkp=0
  2270. *yep
  2271. Id,Count,0,NOM,-2,p,1,k,1,pp,1,kp,1,q,1,m,1
  2272. Id,kp(al~)= - p(al)- k(al) - pp(al)
  2273. Al,Dotpr,kp(al~)= - p(al) - k(al)- pp(al)
  2274. Id,Count,0,NOM,-2,p,1,k,1,pp,1,kp,1,q,1,m,1
  2275. Id,pDpp=-pDk-kDpp
  2276. Id,Count,0,NOM,-2,p,1,k,1,pp,1,kp,1,q,1,m,1
  2277.  
  2278. Id,pDp=0
  2279. Al,kDk=0
  2280. Al,ppDpp=0
  2281. Al,kpDkp=0
  2282.  
  2283. Id,Epfred
  2284. Al,Even,NOM,1
  2285.  
  2286. IF NOT NOM(q~,m)
  2287. Id,Count,0,m,1
  2288. ENDIF
  2289.  
  2290. *yep
  2291.  
  2292. Q2RED{M}
  2293. Q2RED{M}
  2294. *yep
  2295. Q2RED{m}
  2296. Q2RED{m}
  2297. *yep
  2298. REDUC{}
  2299. *yep
  2300. REDUC{}
  2301. *yep
  2302. REDUC{}
  2303. *yep
  2304. REDUC{}
  2305. *yep
  2306. REDUC{}
  2307. *yep
  2308. REDUC{}
  2309. *yep
  2310. Id,qDk=-qDkp-qDp-qDpp
  2311. REDUC{}
  2312. *yep
  2313. REDUC{}
  2314. *yep
  2315. Id,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  2316. Id,pDp=0
  2317. Al,kDk=0
  2318. Al,ppDpp=0
  2319. Al,kpDkp=0
  2320. *yep
  2321. REDUC{}
  2322. *yep
  2323. REDUC{}
  2324. Id,qDpp=-qDkp-qDp-qDk
  2325. *yep
  2326. REDUC{}
  2327. *yep
  2328. REDUC{}
  2329. *yep
  2330. Id,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  2331. Id,pDp=0
  2332. Al,kDk=0
  2333. Al,ppDpp=0
  2334. Al,kpDkp=0
  2335. *yep
  2336. REDUC{}
  2337. *yep
  2338. REDUC{}
  2339.  
  2340. Id,pDpp=-pDk-kDpp
  2341. Id,Count,0,NOM,-2,p,1,k,1,pp,1,kp,1,q,1,m,1
  2342.  
  2343. *yep
  2344.  
  2345. Id,NOM(q~,M~)=NOM(M,q)
  2346. Id,Commu,NOM
  2347. Id,NOM(M~,q~)=NOM(q,M)
  2348.  
  2349. Id,Adiso,pDq^4*NOM(q~,m)*NOM(q1~,m)*NOM(q2~,M)*NOM(q3~,M)=0
  2350.  
  2351. *yep
  2352.  
  2353. Id,Count,x,NOM,1
  2354.  
  2355. IF NOT x
  2356. Id,Addfa,0
  2357. ENDIF
  2358.  
  2359. Id,x=1
  2360.  
  2361. Id,NOM(q~,M)=x*NOM(q,M)
  2362. IF Multi,x^3
  2363. Id,Addfa,Error
  2364. ENDIF
  2365.  
  2366. B Error
  2367.  
  2368. Id,x=1
  2369.  
  2370. *begin
  2371. Write BoxWW1
  2372. *end
  2373.  
  2374. C WW-scattering 12. One loop W-W scattering, part 2. 605 sec.
  2375.   Uses output from 11. Produces BoxWW2, containing BoxWW.
  2376.  
  2377. C Part 2 of BoxWW.
  2378.  
  2379. C The input file contains terms (three-point functions, no Higgs propagators)
  2380.   labelled with 'Error'. They add up to zero; see program 15 for proof.
  2381.   Here only the inifinite part is kept, and shown to add up to zero as
  2382.   no 'Error' label remains.
  2383.  
  2384. C Work done:
  2385.  - Expand all Higgs propagators: 1/((q+qx)^2+m^2) => 1/(q^2+m^2)
  2386.  - The assumption at this point is that that there are no more
  2387.    than 2 non-Higgs propagators. If two, take them together in
  2388.    the function Two(qa,qb). If qa not q then shift momentum so
  2389.    that only Two(q,qx) occurs. The Higgs propagators become shifted
  2390.    again. They are expanded again.
  2391.  - Expand Two(q,qx) times any non-zero number of Higgs propagators:
  2392.    Rationalize 1/(q^2+M^2)* 1/(q^2+m^2), the result contains a NOM
  2393.    but no more Two.
  2394.    Reduce any non-zero number of qDq together with Two(q,qx).
  2395.  - Work out NOM(qx,M) with qx not q, and any number of Higgs propagators.
  2396.    Shift qx to q. Expand shifted Higgs propagators.
  2397.  - Rationalize again.
  2398.  - Reduce all qDq occurences.
  2399.  - After this work there are the following types of terms:
  2400.    One Two function and no Higgs propagator;
  2401.    One NOM(q,M);
  2402.    Any number of Higgs propagators.
  2403.  
  2404. P ninput
  2405.  
  2406. Enter BoxWW1
  2407.  
  2408. Read WWb.e
  2409.  
  2410. ASSIGN{}
  2411.  
  2412. *fix
  2413.  
  2414. Names BoxWW
  2415.  
  2416. P stats
  2417.  
  2418. Z BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp) =
  2419.   BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  2420.  
  2421. IF Error
  2422. C Compute infinite part of three-pount functions not containing m.
  2423.   Supposedly nothing survives. See program 15.
  2424.  
  2425. Id,Count,-4,NOM,-2,q,1
  2426. Id,All,q,N,Fq,"F_
  2427. Id,Adiso,NOM(q~,M)*NOM(q1~,M)*NOM(q2~,M)*Fq(al~,be~)=
  2428.  -2*i*Pi^2/N_*D(al,be)
  2429. ENDIF
  2430.  
  2431. Id,pDp=0
  2432. Al,kDk=0
  2433. Al,ppDpp=0
  2434. Al,kpDkp=0
  2435. Id,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  2436. Id,pDp=0
  2437. Al,kDk=0
  2438. Al,ppDpp=0
  2439. Al,kpDkp=0
  2440.  
  2441. Id,pDpp=-pDk-kDpp
  2442.  
  2443. *yep
  2444.  
  2445. C Expand Higgs propagators.
  2446.   There may be two of them.
  2447.  
  2448. Id,NOM(q,m)=Nohm
  2449.  
  2450. IF NOM(q~,m)=Fxx(q)
  2451. Id,Count,Div,q,1,NOM,-2,Two,-4
  2452. Id,Fxx(q~)=Fact^-2*NOM(q,m)
  2453. Id,Fact^n~=Fxx(n,0,0)
  2454. Id,Nohm^n~*Fxx(n1~,0,0)=Nohm^n*Fxx(n1-2*n,0,0)
  2455. Id,m^n~*Fxx(n1~,0,0)=m^n*Fxx(n1,n,0)
  2456. Id,Div^n~*Fxx(n1~,n2~,0)=Fxx(n1,n2,n)
  2457. Id,Adiso,Fxx(n1~,n2~,n3~)*NOM(q1,m)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDp),1)
  2458. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q2,m)=Fdiv(n1,n2,n3)*
  2459.  Exp(n1,n2,n3,(-2*qDp-2*qDpp-2*pDpp),1)
  2460. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  2461.  *Exp(n1,n2,n3,(2*qDk),1)
  2462. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q4,m)=Fdiv(n1,n2,n3)
  2463.  *Exp(n1,n2,n3,(2*qDk+2*qDpp-2*kDpp),1)
  2464. ENDIF
  2465.  
  2466. Id,Fact^n~=1
  2467.  
  2468. B Nohm,Nom
  2469. *yep
  2470.  
  2471. IF NOM(q~,m)=Fxx(q)
  2472. Id,Count,Div,q,1,NOM,-2,Two,-4
  2473. Id,Fxx(q~)=Fact^-2*NOM(q,m)
  2474. Id,Fact^n~=Fxx(n,0,0)
  2475. Id,Nohm^n~*Fxx(n1~,0,0)=Nohm^n*Fxx(n1-2*n,0,0)
  2476. Id,m^n~*Fxx(n1~,0,0)=m^n*Fxx(n1,n,0)
  2477. Id,Div^n~*Fxx(n1~,n2~,0)=Fxx(n1,n2,n)
  2478. Id,Adiso,Fxx(n1~,n2~,n3~)*NOM(q1,m)=Fdiv(n1,n2,n3)*Exp(n1,n2,n3,(-2*qDp),1)
  2479. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q2,m)=Fdiv(n1,n2,n3)*
  2480.  Exp(n1,n2,n3,(-2*qDp-2*qDpp-2*pDpp),1)
  2481. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q3,m)=Fdiv(n1,n2,n3)
  2482.  *Exp(n1,n2,n3,(2*qDk),1)
  2483. Al,Adiso,Fxx(n1~,n2~,n3~)*NOM(q4,m)=Fdiv(n1,n2,n3)
  2484.  *Exp(n1,n2,n3,(2*qDk+2*qDpp-2*kDpp),1)
  2485. ENDIF
  2486.  
  2487. Id,Fact^n~=1
  2488.  
  2489. B Nohm,Nom
  2490. *yep
  2491. Id,pDpp=M^2-pDk-kDpp
  2492. Id,Count,0,NOM,-2,Two,-4,Nom,-2,Nohm,-2,[m2-M2],2,p,1,k,1,pp,1,kp,1,q,1,m,1
  2493.  
  2494. HCOUNT{}
  2495. *yep
  2496.  
  2497. Id,Adiso,NOM(q~,M)*NOM(q1~,M) = Two(q,q1)
  2498. Id,Commu,NOM
  2499. Id,Two(q1~,q)=Two(q,q1~)
  2500.  
  2501. Id,Two(q1,q2~)=Sh1*Two(q,q2,-p)
  2502. Al,Two(q2,q5~)=Sh2*Two(q,q5,qt)
  2503. Al,Two(q3,q5~)=Sh3*Two(q,q5,k)
  2504. Al,Two(q4,q5~)=Sh4*Two(q,q5,qu)
  2505.  
  2506. SHIFT{}
  2507. HCOUNT{}
  2508.  
  2509. *yep
  2510.  
  2511. Id,Two(q,q3,-p)=Two(q,q5)
  2512. Al,Two(q,q4,k)=Two(q,q6)*Chsi
  2513. Al,Two(q,q4,-p)=Two(q,q7)
  2514. Al,Two(q,q2,k)=Two(q,q7)*Chsi
  2515. Al,Two(q,q2,-p)=Two(q,q6)
  2516.  
  2517. Id,Multi,Chsi^2=1
  2518. IF Chsi=1
  2519. Al,qDq=qDq
  2520. Al,Dotpr,q(al~)=-q(al)
  2521. ENDIF
  2522.  
  2523. *yep
  2524.  
  2525. C Reduction of 1/(q^2+M^2) * 1/(q^2+m^2)^n
  2526.  
  2527. IF Nohm^n~*Two(q,q1~)=1/q2M*Fxx(q1)/q2m^n
  2528. Id,2,Ratio,q2M,q2m,[m2-M2]
  2529. Id,q2M^-1*Fxx(q1~)=Two(q,q1)
  2530. Al,Fxx(q1~)=NOM(q1,M)
  2531. Al,q2m^n~=1/Nohm^n
  2532. ENDIF
  2533.  
  2534. HCOUNT{}
  2535.  
  2536. C Elimination of qDq and Two.
  2537.   Should not occur, as qDq and M type propagators were already treated.
  2538.  
  2539. IF qDq^n~*Two(q,q1~)=qq2^n*Fxx(q1)/q2M
  2540. Id,2,Ratio,qq2,q2M,M2
  2541. Id,q2M^-1*Fxx(q1~)=Two(q,q1)
  2542. Al,Fxx(q1~)=NOM(q1,M)
  2543. Al,qq2=qDq^2
  2544. ENDIF
  2545.  
  2546. *yep
  2547.  
  2548. C Integration variable shift of  1/((q+qx)^2 + M^2)
  2549.  
  2550. IF NOT NOM(q,M)=Nom
  2551. Id,NOM(q1,M)=Sh1*Nom
  2552. Al,NOM(q2,M)=Sh2*Nom
  2553. Al,NOM(q3,M)=Sh3*Nom
  2554. Al,NOM(q4,M)=Sh4*Nom
  2555. Al,NOM(q5,M)=Sh5*Nom
  2556. Al,NOM(q6,M)=Sh6*Nom
  2557. Al,NOM(q7,M)=Sh7*Nom
  2558. ENDIF
  2559.  
  2560. SHIFT{}
  2561.  
  2562. HCOUNT{}
  2563.  
  2564. *yep
  2565.  
  2566. C Reduction of 1/(q^2+M^2) * 1/(q^2+m^2)^n
  2567.  
  2568. IF Nohm^n~*Nom=1/q2M/q2m^n
  2569. Id,2,Ratio,q2M,q2m,[m2-M2]
  2570. Id,q2M^-1=Nom
  2571. Al,q2m^n~=1/Nohm^n
  2572. ENDIF
  2573.  
  2574. *yep
  2575.  
  2576. C Elimination of qDq.
  2577.  
  2578. IF Nom
  2579. Id,Nom^n~=1/q2M^n
  2580. Al,qDq^n~=qq2^n
  2581. Id,Ratio,qq2,q2M,M2
  2582. Id,q2M^n~=1/Nom^n
  2583. Al,qq2=qDq
  2584. Al,M2=M^2
  2585. ENDIF
  2586.  
  2587. IF Nohm
  2588. Id,Nohm^n~=1/q2m^n
  2589. Al,qDq^n~=qq2^n
  2590. Id,Ratio,qq2,q2m,m2
  2591. Id,q2m^n~=1/Nohm^n
  2592. Al,qq2=qDq
  2593. Al,m2=m^2
  2594. ENDIF
  2595.  
  2596. IF NOM(q~,M~)
  2597. OR Two(q~,q1~)*Nohm^n~
  2598. Id,Addfa,Error
  2599. ENDIF
  2600.  
  2601. C Count all but M.
  2602. C
  2603. Id,Count,0,NOM,-2,Two,-4,Nom,-2,Nohm,-2,[m2-M2],2,
  2604.  p,1,k,1,pp,1,kp,1,q,1,m,1
  2605.  
  2606. Id,Count,x,Nohm,1,Nom,1,Two,1,NOM,1
  2607.  
  2608. IF NOT x
  2609. Id,Addfa,0
  2610. ENDIF
  2611.  
  2612. Id,x=1
  2613.  
  2614. C Check dimension 0.
  2615. C
  2616. Id,Count,x,M,1,M2,2,m,1,m2,2,Two,-4,Nom,-2,Nohm,-2,[m2-M2],2,
  2617.  q,1,p,1,k,1,pp,1,kp,1
  2618.  
  2619. IF NOT x^-4=1
  2620. Id,Addfa,Error
  2621. ENDIF
  2622.  
  2623. B Error,Nohm,Nom
  2624.  
  2625. *begin
  2626. Write BoxWW2
  2627. *end
  2628.  
  2629. C WW-scattering 13. One loop W-W scattering, part 3. 447 sec.
  2630.   Uses output from 12. Produces BoxWW_comm, containing Wtot.
  2631.  
  2632. C One loop diagrams.
  2633.   Four point function. WW scattering.
  2634.   Evaluated in the limit m^2 >> s,t,u >> M^2, where m = Higgs boson
  2635.   mass and M = W boson mass.
  2636.  
  2637. P ninput
  2638.  
  2639. C Work to be done:
  2640.  - Do integrals.
  2641.    1/(qx^2+M^2) with or without Higgs propagators;
  2642.    Higgs propagators;
  2643.    Functions Two and no Higgs propagator.
  2644.  
  2645. Enter BoxWW2
  2646.  
  2647. Read WWb.e
  2648.  
  2649. ASSIGN{}
  2650.  
  2651. *fix
  2652.  
  2653. Names BoxWW
  2654. P stats
  2655.  
  2656. Z BoxW(a,b,c,d,s,t,u) =
  2657.  BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  2658.  
  2659. IF NOT Two(q~,q1~)
  2660. Id,All,q,N,Fq
  2661. ENDIF
  2662.  
  2663. Id,Fq(al~)=0
  2664. Al,Fq(al~,be~,ga~)=0
  2665. Al,Fq(al~,be~,ga~,de~,la~)=0
  2666. Al,Fq(al~,be~,ga~,de~,la~,a~,b~)=0
  2667.  
  2668. Id,Fq(al~,al~)=0
  2669.  
  2670. Id,Fq(al~,be~,be~,be~)=0
  2671. Al,Fq(be~,be~,be~,al~)=0
  2672.  
  2673. Al,Fq(be~,be~,be~,be~,al~,ga~)=0
  2674. Al,Fq(al~,be~,be~,be~,be~,al~)=0
  2675. Al,Fq(al~,ga~,be~,be~,be~,be~)=0
  2676.  
  2677. B Nohm,Nom
  2678. *yep
  2679. IF NOT Two(q~,q1~)
  2680. Id,Fq(al~,be~,ga~,de~)*Nom^l~ = dede(al,be,ga,de)*H(l,M2)
  2681. Al,Fq(al~,be~)*Nom^l~ = D(al,be)*G(l,M2)
  2682. Al,Nom^n~ = F(n,M2)
  2683. Id,Fq(al~,be~,ga~,de~)*Nohm^l~ = dede(al,be,ga,de)*H(l,m2)
  2684. Al,Fq(al~,be~)*Nohm^l~ = D(al,be)*G(l,m2)
  2685. Al,Nohm^l~ = F(l,m2)
  2686. ENDIF
  2687.  
  2688. Id,pDp=-M^2
  2689. Al,kDk=-M^2
  2690. Al,ppDpp=-M^2
  2691. Al,kpDkp=-M^2
  2692.  
  2693. Id,Multi,M^-2=M2^-1
  2694. Al,Multi,m^2=m2
  2695. Al,Multi,m^-2=m2^-1
  2696.  
  2697. *yep
  2698.  
  2699. STINT{}
  2700.  
  2701. Id,N=N_+4
  2702. Id,N_=0
  2703. Al,N=4
  2704. Id,pDpp=-kDp-kDpp+M2
  2705. B i,Pi,N_,Nohm,M2
  2706. Id,Count,x,M,1,M2,2
  2707. IF NOT x^-4=1
  2708. Id,Addfa,0
  2709. ENDIF
  2710. C Id,Count,4,p,1,k,1,pp,1,kp,1,q,1,Two,10,Extm,10,ExtM,10
  2711. *yep
  2712. Id,Two(q,q1~)=Two(q1)
  2713. IF Two(q~)
  2714. Al,All,q,N,Fq
  2715. ENDIF
  2716. Id,Adiso,Two(q4)*Fq(al~,be~) =
  2717.  D(al,be)*B22(u,M,M) + (k(al)+pp(al))*(k(be)+pp(be))*B21(u,M,M)
  2718. Al,Adiso,Two(q4)*Fq(al~)=- (k(al)+pp(al))*B1(u,M,M)
  2719. Al,Two(q4)=B0(u,M,M)
  2720.  
  2721. Id,Adiso,Two(q5)*Fq(al~,be~) =
  2722.  D(al,be)*B22(s,M,M) + (k(al)+p(al))*(k(be)+p(be))*B21(s,M,M)
  2723. Al,Adiso,Two(q5)*Fq(al~)=- (k(al)+p(al))*B1(s,M,M)
  2724. Al,Two(q5)=B0(s,M,M)
  2725.  
  2726. Id,Adiso,Two(q2)*Fq(al~,be~) =
  2727.  D(al,be)*B22(t,M,M) + (pp(al)+p(al))*(pp(be)+p(be))*B21(t,M,M)
  2728. Al,Adiso,Two(q2)*Fq(al~)= (pp(al)+p(al))*B1(t,M,M)
  2729. Al,Two(q2)=B0(t,M,M)
  2730.  
  2731. Al,Adiso,Two(q1)*Fq(al~,be~) =
  2732.  D(al,be)*BB22 + p(al)*p(be)*BB21
  2733. Al,Adiso,Two(q1)*Fq(al~)= p(al)*BB1
  2734. Al,Two(q1)=BB0
  2735.  
  2736. Al,Adiso,Two(q3)*Fq(al~,be~) =
  2737.  D(al,be)*BB22 + k(al)*k(be)*BB21
  2738. Al,Adiso,Two(q3)*Fq(al~)= -k(al)*BB1
  2739. Al,Two(q3)=BB0
  2740.  
  2741. Al,Adiso,Two(q6)*Fq(al~,be~) =
  2742.  D(al,be)*BB22 + pp(al)*pp(be)*BB21
  2743. Al,Adiso,Two(q6)*Fq(al~)= pp(al)*BB1
  2744. Al,Two(q6)=BB0
  2745.  
  2746. Al,Adiso,Two(q7)*Fq(al~,be~) =
  2747.  D(al,be)*BB22 + (k(al)+pp(al)+p(al))*(k(be)+pp(be)+p(be))*BB21
  2748. Al,Adiso,Two(q7)*Fq(al~)= -(k(al)+pp(al)+p(al))*BB1
  2749. Al,Two(q7)=BB0
  2750.  
  2751. *yep
  2752.  
  2753. Id,B22(u~,M~,m~)=(-0.5*F1(m)+M**2*B0(u,M,m)
  2754.       -0.5*(-u+m^2-M**2)*B1(u,M,m))/[1-N]
  2755. Id,B21(u~,M~,m~)=((0.5*N-1)*F1(m)
  2756.        -0.5*N*(-u+m^2-M**2)*B1(u,M,m)
  2757.        +M**2*B0(u,M,m) )/u/[1-N]
  2758. Id,B1(u~,M~,m~)=- (0.5*F1(M)-0.5*F1(m)
  2759.               -0.5*(-u+m^2-M**2)*B0(u,M,m) )/u
  2760. Id,B0(u~,M~,M~)= - 2*i*Pi^2/N_ - i*Pi^2*Log(u) + 2*i*Pi^2
  2761. Al,F1(M) = 2*i*Pi^2*M2/N_ + i*Pi^2*M2*(-1+Log(M2))
  2762. Al,F1(m) = 2*i*Pi^2*m2/N_ + i*Pi^2*m2*(-1+Log(m2))
  2763. Al,M^n~=M2^(n/2)
  2764. Al,m^n~=m2^(n/2)
  2765.  
  2766. Id,BB0=i*Pi^2
  2767.   * ( - Log(M2) - 2*N_^-1 - [Pi/Sqrt(3)-2] )
  2768. Al,BB1=i*Pi^2
  2769.   * ( 1/2*Log(M2) + N_^-1 + 1/2*[Pi/Sqrt(3)-2] )
  2770. Al,BB21=i*Pi^2
  2771.   * ( 1/18 - 1/3*Log(M2) - 2/3*N_^-1 )
  2772. Al,BB22=i*Pi^2*M2
  2773.   * ( - 4/9 + 5/12*Log(M2) + 5/6*N_^-1 + 1/4*[Pi/Sqrt(3)-2] )
  2774.  
  2775. Id,N=N_+4
  2776. Al,[1-N]^-1=-1/3 + N_/9
  2777. Id,N_=0
  2778. Id,N=4
  2779. Al,ExtM = - 2*kDpp/u*{1 + M2/u + M2^2/u^2}
  2780.  
  2781. Id,Count,0,m2,2,[m2-M2],2
  2782. Al,Multi,m2^-1=0
  2783. Id,Count,x,m2,1
  2784. IF x
  2785. Id,x^n1~*[m2-M2]^n~=x^n1*m2^n*DS(K,0,n1+n,(DB(-n+K-1,K)*M2^K*m2^-K))
  2786. Id,x^n1~*Extm^n~=DS(K,0,n1,(DB(n+K-1,K)*(-2*kDpp/m2)^K))
  2787. ELSE
  2788. Id,[m2-M2]^n~=m2^n
  2789. Al,Extm=1
  2790. ENDIF
  2791. Id,Count,0,m2,1
  2792. Al,x=1
  2793.  
  2794. Id,pDk= - 0.5*s
  2795. C - 0.5*pDp - 0.5*kDk
  2796. Al,pDpp = - 0.5*t
  2797. C - 0.5*pDp - 0.5*ppDpp
  2798. Al,kDpp = - 0.5*u
  2799. C - 0.5*kDk - 0.5*ppDpp
  2800. Id,pDp=-M2
  2801. Al,kDk=-M2
  2802. Al,ppDpp=-M2
  2803. Al,kpDkp=-M2
  2804. Id,Count,4,s,2,u,2,t,2
  2805. Id,t=-s-u
  2806. Keep BoxW
  2807. *next
  2808. P input
  2809. C Add further diagrams, obtained by crossing. Set labels A0-A3 to 1.
  2810.  
  2811. Common WTot
  2812. Delete BoxWW
  2813.  
  2814. Z WTot=BoxW(a,b,c,d,s,t,u) + BoxW(a,c,b,d,u,t,s) + BoxW(a,b,d,c,s,u,t)
  2815.  
  2816. B i,Pi,N_,M2
  2817. Id,A0=1
  2818. Al,A1=1
  2819. Al,A2=1
  2820. Al,A3=1
  2821. Al,R1=1
  2822. Al,R2=1
  2823. Al,R3=1
  2824. Al,R3a=1
  2825. Al,R3b=1
  2826. Al,R4=1
  2827. Al,R4a=1
  2828. Al,R4b=1
  2829. Al,R5=1
  2830. Al,R6=1
  2831. Al,T1=1
  2832. Al,T2=1
  2833. Al,T3=1
  2834. IF D(a,c)
  2835. Id,t=-s-u
  2836. ENDIF
  2837. IF D(a,b)
  2838. Id,u=-s-t
  2839. ENDIF
  2840. IF D(a,d)
  2841. Id,s=-t-u
  2842. ENDIF
  2843. P output
  2844. *yep
  2845.  
  2846. Id,R1Z=1
  2847. Al,R5Z=1
  2848.  
  2849. *begin
  2850. Write BoxWW_comm
  2851. *end
  2852.  
  2853. C WW-scattering 14. W-W Renormalization. Result W-W amplitude.
  2854.   Uses output from 14.
  2855.  
  2856. C Subtraction terms four-W amplitude.
  2857.  
  2858. P ninput
  2859.  
  2860. A M,m,x,qq2,qqM,q2M,s,t,u
  2861. V q,q0,q1,q3,q2,q4
  2862.  
  2863. Enter BoxWW_comm
  2864.  
  2865. Read WWb.e
  2866. VERT{}
  2867. *fix
  2868.  
  2869. Names WTot
  2870.  
  2871. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  2872. I a=3,b=3,c=3,d=3
  2873.  
  2874. C Momenta: all taken to be ingoing.
  2875.   k,p in, pp,kp out. k + p = - pp - kp.
  2876.  
  2877. Z RenWW4 = WWWW(a,k,k,b,p,p,c,pp,pp,d,kp,kp)/M^4*WWWWK
  2878.  
  2879. Z RenWWr(a,b,c,d,s,t,u)=
  2880.    DS("W;"W;-J1;TAP,(
  2881.    VIE1("W,a,k,k,"W,b,p,p,"W,c,pp,pp,"W,d,kp,kp,J1)/M^4 ))
  2882.  
  2883.  + DS("W;"W;-J2;TAP,(DS("W;"W;J3;TAP,(
  2884.    VIE2("W,a,k,k,"W,b,p,p,"W,c,pp,pp,"W,d,kp,kp,J2,J3)/M^4 )) ))
  2885.  
  2886.  + DS("W;"W;J4;"Z;TAP,(
  2887.    VIE3("W,a,k,k,"W,b,p,p,"W,c,pp,pp,"W,d,kp,kp,J4)/M^4 ))
  2888.  
  2889.  
  2890. Id,VIE1(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~) =
  2891.    VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  2892.    VE3(K2,K4,J1,*,b,be,p,*,d,de,-kp,*,l2,m0,qu)*
  2893.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  2894.    CONT(K1,K3,J1,/,"K)*CONT(K2,K4,-J1,/,"K)
  2895.    *R3
  2896.  
  2897. Al,VIE2(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~) =
  2898.    VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  2899.    VE3(K2,K4,J2,*,b,be,p,*,d,de,-kp,*,l4,m4,qu)*
  2900.    VE3(J1,-J2,"N,*,l2,m0,qu,*,l3,m3,-qu,*,l5,m5,q0)*
  2901.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  2902.    PROP(J2,-J2,*,l3,m3,qu,*,l4,m4,-qu)
  2903.    *R5(J1,J2)
  2904.  
  2905.  + VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  2906.    VE3(K2,K4,J2,*,b,be,p,*,d,de,-kp,*,l4,m4,qu)*
  2907.    VE3(J1,-J2,"Z,*,l2,m0,qu,*,l3,m3,-qu,*,l5,m5,q0)*
  2908.    NNZ(a,al,k,b,be,p,l6,m6,-q0)*
  2909.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  2910.    PROP(J2,-J2,*,l3,m3,qu,*,l4,m4,-qu)*
  2911.    PROP("Z,"Z,*,l5,m5,q0,*,l6,m6,-q0)
  2912.    *T3
  2913.  
  2914. Id,VIE3(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~) =
  2915.    VE4(K1,K3,-J1,"Z,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu,*,l3,m3,-q0)*
  2916.    VE3(K2,K4,J1,*,b,be,p,*,d,de,-kp,*,l2,m0,qu)*
  2917.    NNZ(a,al,k,b,be,p,l4,m4,q0)*
  2918.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  2919.    PROP("Z,"Z,*,l3,m3,q0,*,l4,m4,-q0)
  2920.    *T1
  2921.  
  2922.   +VE3(K1,K3,-J1,*,a,al,k,*,c,ga,-pp,*,l1,m1,-qu)*
  2923.    VE4(K2,K4,J1,"Z,*,b,be,p,*,d,de,-kp,*,l2,m0,qu,*,l3,m3,-q0)*
  2924.    NNZ(a,al,k,b,be,p,l4,m4,q0)*
  2925.    PROP(J1,-J1,*,l1,m1,qu,*,l2,m0,-qu)*
  2926.    PROP("Z,"Z,*,l3,m3,q0,*,l4,m4,-q0)
  2927.    *T2
  2928.  
  2929. Id,Anti,TAP
  2930.  
  2931. Id,Compo,<X>,VE4,VE3,PROP
  2932. Id,Compo,<X>,CONT
  2933.  
  2934. Id,Adiso,CONT(FF~)*CONT(WW~)= FF + WW
  2935. Al,CONT(FF~)= FF
  2936.  
  2937. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  2938.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  2939. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  2940.        FF(l1,al,k,l2,be,q,l3,ga,p)
  2941. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  2942. B i,Pi
  2943.  
  2944. Id,qu(al~)=k(al)+pp(al)
  2945. Al,quDqu=2*kDpp
  2946. Al,Dotpr,qu(al~)=k(al)+pp(al)
  2947. Al,Even,NOM,1
  2948.  
  2949. C By definition:
  2950.   Extm = 1/{1 + (2*kDpp + kDk + ppDpp)/m^2}
  2951.        = 1/{1 + 2*(kDpp - M^2)/m^2}
  2952.   This makes the principal behaviour explicit.
  2953.  
  2954. Id,NOM(qu,m)= Extm/m^2
  2955. Al,NOM(q0,M~)= 1/M^2
  2956.  
  2957. Id,NOM(qu,M)=-1/u
  2958. Al,Even,NOM,1
  2959. Id,kpDkp=0
  2960. Al,kp(al~)=-k(al)-p(al)-pp(al)
  2961. Al,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  2962. Id,kDk=0
  2963. Al,pDp=0
  2964. Al,ppDpp=0
  2965. Id,pDk=-0.5*s
  2966. Al,pDpp=-0.5*t
  2967. Al,kDpp=-0.5*u
  2968.  
  2969. ETE1{}
  2970. Id,Count,0,m,1,m2,2,[m2-M2],2
  2971. *yep
  2972.  
  2973. Id,Count,x,m,1,m2,2,[m2-M2],2
  2974. IF x
  2975. Id,x^n1~*[m2-M2]^n~=x^n1*m2^n*DS(K,0,n1/2+n+1,(DB(-n+K-1,K)*M2^K*m2^-K))
  2976. Id,x^n1~*Extm^n~=DS(K,0,n1/2+1,(DB(n+K-1,K)*(-2*kDpp/m^2)^K))
  2977. ELSE
  2978. Id,[m2-M2]^n~=m2^n
  2979. Al,Extm=1
  2980. ENDIF
  2981. Id,M2^n~=M^(2*n)
  2982. Al,m2^n~=m^(2*n)
  2983. Al,kDpp=-0.5*u
  2984. Id,t=-s-u
  2985. Id,Count,0,m,1
  2986. Al,x=1
  2987.  
  2988. Id,Count,4,s,2,t,2,u,2
  2989. Al,R5("W,"W)=R5W
  2990. Al,R5("Z,"Z)=R5Z
  2991. *yep
  2992.  
  2993. Id,Epfred
  2994.  
  2995. Id,t=-s-u
  2996.  
  2997. Keep RenWW4,RenWWr
  2998. *next
  2999. B i,Pi,N_,M
  3000.  
  3001. Z RenWWt=RenWW4
  3002.   + RenWWr(a,b,c,d,s,t,u) + RenWWr(a,c,b,d,u,t,s) + RenWWr(a,b,d,c,s,u,t)
  3003.  
  3004. C Id,R5W=1
  3005.   Al,R3=1
  3006.  
  3007. IF D(a,c)
  3008. Id,t=-s-u
  3009. ENDIF
  3010. IF D(a,b)
  3011. Id,u=-s-t
  3012. ENDIF
  3013. IF D(a,d)
  3014. Id,s=-t-u
  3015. ENDIF
  3016.  
  3017. *yep
  3018. Id,T1=1
  3019. Al,T2=1
  3020. Al,T3=1
  3021. Al,R5Z=1
  3022. Al,R5W=1
  3023. Al,R3=1
  3024. Keep RenWWt
  3025. *next
  3026. P input
  3027. C Renormalized WWWW (all longitudinal) amplitude.
  3028.  
  3029. Z RenW = WTot - RenWWt
  3030.  
  3031. B i,Pi,N_,M2,M
  3032. Id,Multi,M2^n~=M^(2*n)
  3033. Al,Logm2=Log(m2)
  3034. Id,Log(s)=Log(s,m2)+Log(m2)
  3035. Al,Log(t)=Log(t,m2)+Log(m2)
  3036. Al,Log(u)=Log(u,m2)+Log(m2)
  3037.  
  3038. P output
  3039. *yep
  3040. C Specialize, for computimg purposes, to index a=b, c=d, a not c.
  3041.  
  3042. IF NOT D(a,b)=1
  3043. Id,Addfa,0
  3044. ENDIF
  3045. Id,D(c,d)=1
  3046.  
  3047. *end
  3048.  
  3049. C WW-scattering 15. Verification of part of WW scattering calculation.
  3050.   Uses output from 11.
  3051.  
  3052. C Cross.e: demonstrates that the part labelled with 'Error' in file
  3053.   BoxWW1 is zero.
  3054.   This requires addition of the crossed pieces and working out of the
  3055.   three point functions.
  3056.   The block CCCP contains expressions for the C-functions as needed
  3057.   for the purposes here. Block CCC is somewhat more detailed.
  3058.  
  3059. P ninput
  3060.  
  3061. Enter BoxWW1
  3062.  
  3063. BLOCK CCC{}
  3064.  
  3065. Id,C11(t~) =  + 2*i*N_^-1*Pi^2*t^-1 + i*Pi^2*LogM2*t^-1 + i*Pi^2*
  3066.  [Pi/Sqrt(3)-2]*t^-1 - CC0(t)
  3067.  
  3068.  + B0(t,M,M)
  3069.   * ( t^-1 )
  3070.  
  3071. Al,C12(t~) =  - 2*i*N_^-1*Pi^2*t^-1 - i*Pi^2*LogM2*t^-1 - i*Pi^2*
  3072.  [Pi/Sqrt(3)-2]*t^-1
  3073.  
  3074.  + B0(t,M,M) * ( - t^-1 )
  3075.  
  3076. Al,C24(t~) =  + 1/4*i*Pi^2 - 1/2*M^2*CC0(t) + 1/4*B0(t,M,M)
  3077.  
  3078. Al,C21(t~) =  - 3*i*N_^-1*Pi^2*t^-1 - 3/2*i*Pi^2*LogM2*t^-1 - 3/2*i*Pi^2*
  3079.  [Pi/Sqrt(3)-2]*t^-1 + CC0(t)
  3080.  
  3081.  + B0(t,M,M)
  3082.   * ( - 3/2*t^-1 )
  3083.  
  3084. Al,C23(t~) = 
  3085.   + 2*i*N_^-1*Pi^2*t^-1 + i*Pi^2*LogM2*t^-1 + i*Pi^2*
  3086.  [Pi/Sqrt(3)-2]*t^-1 + 1/2*i*Pi^2*t^-1 - M^2*t^-1*CC0(t)
  3087.  
  3088.  + B0(t,M,M)
  3089.   * ( t^-1 )
  3090.  
  3091. Al,C22(t~) =  + i*N_^-1*Pi^2*t^-1 + 1/2*i*Pi^2*LogM2*t^-1 + 1/2*i*Pi^2*
  3092.  [Pi/Sqrt(3)-2]*t^-1
  3093.  
  3094.  + B0(t,M,M)
  3095.   * ( 1/2*t^-1 )
  3096.  
  3097. Id,B0(u~,M~,M~)= - 2*i*Pi^2/N_ - i*Pi^2*Log(u) + 2*i*Pi^2
  3098.  
  3099. ENDBLOCK
  3100.  
  3101.  
  3102. BLOCK CCCP{}
  3103. C BB0 and BB1 are the two-point functions for equal mass with
  3104.   also pDp = - M^2.
  3105.  
  3106. Id,C11(t~) =  - t^-1*BB0 - CC0(t) + B0(t,M,M) * ( t^-1 )
  3107.  
  3108. Al,C12(t~) =  t^-1*BB0 + B0(t,M,M) * ( - t^-1 )
  3109.  
  3110. Al,C24(t~) =  1/4*i*Pi^2 - 1/2*M^2*CC0(t) + 1/4*B0(t,M,M)
  3111.  
  3112. Al,C21(t~) =  t^-1*BB0 - t^-1*BB1 + CC0(t) + B0(t,M,M) * ( - 3/2*t^-1 )
  3113.  
  3114. Al,C23(t~) =  1/2*i*Pi^2*t^-1 - M^2*t^-1*CC0(t) - t^-1*BB0
  3115.  + B0(t,M,M) * ( t^-1 )
  3116.  
  3117. Al,C22(t~) =  t^-1*BB1 + B0(t,M,M) * ( 1/2*t^-1 )
  3118.  
  3119. Id,BB1 = -0.5*BB0
  3120.  
  3121. ENDBLOCK
  3122.  
  3123. *fix
  3124.  
  3125. F Fq,CC0,C11,C12,C21,C22,C23,C24
  3126.  
  3127. Names BoxWW
  3128. P stats
  3129.  
  3130. Z BoxW(a,k,b,p,c,pp,d,kp,s,t,u,q1,q2,q3,q4) =
  3131.  BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  3132.  
  3133. IF NOT Error=1
  3134. Id,Addfa,0
  3135. ENDIF
  3136.  
  3137. Id,ExtM=1
  3138.  
  3139. Keep BoxW
  3140. *next
  3141.  
  3142. Z Box=
  3143.     BoxW(a,k,b,p,c,pp,d,kp,s,t,u,q1,q2,q3,q4)
  3144.   + BoxW(a,k,c,pp,b,p,d,kp,u,t,s,q6,q2,q3,q5)
  3145.   + BoxW(a,k,b,p,d,kp,c,pp,s,u,t,q1,q4,q3,q2)
  3146. *yep
  3147.  
  3148. IF Adiso,NOM(q,M)*NOM(q1,M)*NOM(q3,M)=Three(k,p,s)
  3149. Al,Dotpr,q(al~)=q(al)+k(al)
  3150. ENDIF
  3151.  
  3152. Id,Adiso,NOM(q,M)*NOM(q1,M)*NOM(q2,M)=Three(p,pp,t)
  3153.  
  3154. Id,Adiso,NOM(q,M)*NOM(q1,M)*NOM(q4,M)=Three(p,kp,u)
  3155.  
  3156. IF Adiso,NOM(q,M)*NOM(q2,M)*NOM(q3,M)=Three(k,kp,t)
  3157. Al,Dotpr,q(al~)=-q(al)
  3158. ENDIF
  3159.  
  3160. IF Adiso,NOM(q,M)*NOM(q3,M)*NOM(q4,M)=Three(k,pp,u)
  3161. Al,Dotpr,q(al~)=-q(al)
  3162. ENDIF
  3163.  
  3164. IF Adiso,NOM(q1,M)*NOM(q2,M)*NOM(q3,M)=Three(pp,kp,s)
  3165. Al,Dotpr,q(al~)=q(al)-p(al)
  3166. ENDIF
  3167.  
  3168. IF Adiso,NOM(q,M)*NOM(q3,M)*NOM(q5,M)=Three(k,p,s)
  3169. Al,Dotpr,q(al~)=-q(al)
  3170. ENDIF
  3171.  
  3172. Id,Adiso,NOM(q,M)*NOM(q6,M)*NOM(q2,M)=Three(pp,p,t)
  3173.  
  3174. IF Adiso,NOM(q,M)*NOM(q6,M)*NOM(q3,M)=Three(k,pp,u)
  3175. Al,Dotpr,q(al~)=q(al)+k(al)
  3176. ENDIF
  3177.  
  3178. Id,Adiso,NOM(q,M)*NOM(q6,M)*NOM(q5,M)=Three(pp,kp,s)
  3179.  
  3180. IF Adiso,NOM(q1,M)*NOM(q3,M)*NOM(q4,M)=Three(kp,pp,s)
  3181. Al,Dotpr,q(al~)=q(al)-p(al)
  3182. ENDIF
  3183.  
  3184. IF Adiso,NOM(q6,M)*NOM(q3,M)*NOM(q2,M)=Three(p,kp,u)
  3185. Al,Dotpr,q(al~)=q(al)-pp(al)
  3186. ENDIF
  3187.  
  3188. Id,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  3189. B i,Pi,M
  3190.  
  3191. C IF NOT D(a,b)
  3192.   Id,Addfa,0
  3193.   ENDIF
  3194.  
  3195. P output
  3196. *yep
  3197.  
  3198. C Id,Three(p~,k~,t~)=Three(p,k,t)*Fxx(p,k,t)
  3199.  
  3200. Id,All,q,N,Fq
  3201. Id,Adiso,Three(p~,k~,t~)*Fq(al~,be~)=
  3202.  p(al)*p(be)*C21(t) + k(al)*k(be)*C22(t)
  3203.  + (p(al)*k(be)+k(al)*p(be))*C23(t) + D(al,be)*C24(t)
  3204. Al,Adiso,Three(p~,k~,t~)*Fq(al~)=p(al)*C11(t) + k(al)*C12(t)
  3205. Al,Three(p~,k~,t~)=CC0(t)
  3206. Id,pDp=0
  3207. Al,kDk=0
  3208. Al,ppDpp=0
  3209. Al,kpDkp=0
  3210. Id,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  3211. Id,pDp=0
  3212. Al,kDk=0
  3213. Al,ppDpp=0
  3214. Al,kpDkp=0
  3215. Id,pDpp=-pDk-kDpp
  3216. P output
  3217. *yep
  3218.  
  3219. Id,A0=1
  3220. Al,A1=1
  3221. Al,A2=1
  3222. Al,R1=1
  3223. Al,R2=1
  3224. P output
  3225. *yep
  3226.  
  3227. CCCP{}
  3228. *end
  3229.  
  3230. C WW-scattering 16. Infinities and Log's of irreducible 4-point W function.
  3231.  
  3232. C One loop diagrams.
  3233.   Four point function: WW scattering.
  3234.   Ln(m^2) and infinities.
  3235.   Notation: N_ = n-4, m = Higgs mass, Log(m) = Ln(m^2).
  3236.  
  3237.   Separate results: 'Long', containing partial result for longitudinal
  3238.   W's, crossed contributions not yet added. 'Tot' includes crossed
  3239.   pieces.
  3240.  
  3241. P ninput
  3242.  
  3243. I al=N,be=N,la=N,de=N,ga=N,la=N
  3244.  
  3245. Read WWb.e
  3246.  
  3247. ASSIGN{}
  3248.  
  3249. VERT{}
  3250.  
  3251. *fix
  3252.  
  3253. V q1,q2,q3,q4,q0
  3254. I m1=N,m0=N,m3=N,m4=N,m5=N,m6=N,m7=N,m8=N
  3255. I a=3,b=3,c=3,d=3,e=3,f=3,g=3,h=3,j=3
  3256. A N,N_
  3257. F Ph,Pw,Fq
  3258.  
  3259. Z BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp) =
  3260.     VIER("W,a,al,k,"W,b,be,p,"W,c,ga,-pp,"W,d,de,-kp)
  3261.  
  3262. C  + VIER("W,a,al,k,"W,c,ga,-pp,"W,b,be,p,"W,d,de,-kp)
  3263.    + VIER("W,a,al,k,"W,b,be,p,"W,d,de,-kp,"W,c,ga,-pp)
  3264.  
  3265. Id,VIER(K1~,a~,al~,k~,K2~,b~,be~,p~,K3~,c~,ga~,pp~,K4~,d~,de~,kp~)=
  3266.    + DS(K1;J4;-J1;TAP,(
  3267.      DS(K2;J1;-J2;TAP,(
  3268.      DS(K3;J2;-J3;TAP,(
  3269.     A0*VIE(K1,a,al,k,K2,b,be,p,K3,c,ga,pp,K4,d,de,kp,J1,J2,J3,J4)
  3270.      * DC("F,TFE,-1,J1,J2,J3,J4) ))) )))
  3271.  
  3272.     + DS(K1;K3;J7;-J5;TAP,(
  3273.      DS(K2;J5;-J6;TAP,(DC("F,TFE,-1,J5,J6,J7)*
  3274.     A1*VIE1(K1,a,al,k,K2,b,be,p,K3,c,ga,pp,K4,d,de,kp,J5,J6,J7) )) ))
  3275.  
  3276.    + DS(K2;K4;J8;-J9;TAP,(
  3277.      DS(K1;JA;-J8;TAP,(DC("F,TFE,-1,J8,J9,JA)*
  3278.    A2*VIE2(K1,a,al,k,K2,b,be,p,K3,c,ga,pp,K4,d,de,kp,J8,J9,JA) )) ))
  3279.  
  3280.    + DS(K1;K3;J0;-JB;Sym;J0;-JB;TAP,(DC("F,TFE,-1,J0,JB)*
  3281.     A3*VIE3(K1,a,al,k,K2,b,be,p,K3,c,ga,pp,K4,d,de,kp,JB,J0) ) )
  3282.  
  3283.  
  3284. Id,Anti,TAP
  3285.  
  3286. Id, VIE(K1~,a~,al~,k~,K2~,b~,be~,p~,
  3287.     K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~,J3~,J4~)=
  3288.    VE3(K1,J4,-J1,*,a,al,k,*,l8,m8,q3,*,l1,m1,-q)*
  3289.    VE3(K2,J1,-J2,*,b,be,p,*,l2,m0,q,*,l3,m3,-q1)*
  3290.    VE3(K3,J2,-J3,*,c,ga,pp,*,l4,m4,q1,*,l5,m5,-q2)*
  3291.    VE3(K4,J3,-J4,*,d,de,kp,*,l6,m6,q2,*,l7,m7,-q3)*
  3292.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  3293.    PROP(J2,-J2,*,l3,m3,q1,*,l4,m4,-q1)*
  3294.    PROP(J3,-J3,*,l5,m5,q2,*,l6,m6,-q2)*
  3295.    PROP(J4,-J4,*,l7,m7,q3,*,l8,m8,-q3)
  3296.  
  3297. Al,VIE1(K1~,a~,al~,k~,K2~,b~,be~,p~,
  3298.     K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~,J3~)=
  3299.    VE4(K1,K3,J3,-J1,*,a,al,k,*,c,ga,pp,*,l6,m6,q4,*,l1,m1,-q)*
  3300.    VE3(K2,J1,-J2,*,b,be,p,*,l2,m0,q,*,l3,m3,-q1)*
  3301.    VE3(K4,J2,-J3,*,d,de,kp,*,l4,m4,q1,*,l5,m5,-q4)*
  3302.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  3303.    PROP(J2,-J2,*,l3,m3,q1,*,l4,m4,-q1)*
  3304.    PROP(J3,-J3,*,l5,m5,q4,*,l6,m6,-q4)
  3305.  
  3306. Al,VIE2(K1~,a~,al~,k~,K2~,b~,be~,p~,
  3307.     K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~,J3~)=
  3308.    VE4(K2,K4,J1,-J2,*,b,be,p,*,d,de,kp,*,l2,m0,q,*,l3,m3,-q4)*
  3309.    VE3(K1,J3,-J1,*,a,al,k,*,l6,m6,q3,*,l1,m1,-q)*
  3310.    VE3(K3,J2,-J3,*,c,ga,pp,*,l4,m4,q4,*,l5,m5,-q3)*
  3311.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  3312.    PROP(J2,-J2,*,l3,m3,q4,*,l4,m4,-q4)*
  3313.    PROP(J3,-J3,*,l5,m5,q3,*,l6,m6,-q3)
  3314.  
  3315. Al,VIE3(K1~,a~,al~,k~,K2~,b~,be~,p~,
  3316.     K3~,c~,ga~,pp~,K4~,d~,de~,kp~,J1~,J2~)=
  3317.    VE4(K1,K3,J2,-J1,*,a,al,k,*,c,ga,pp,*,l4,m4,q4,*,l1,m1,-q)*
  3318.    VE4(K2,K4,J1,-J2,*,b,be,p,*,d,de,kp,*,l2,m0,q,*,l3,m3,-q4)*
  3319.    PROP(J1,-J1,*,l1,m1,q,*,l2,m0,-q)*
  3320.    PROP(J2,-J2,*,l3,m3,q4,*,l4,m4,-q4)
  3321.  
  3322. Id,Compo,<X>,VE4,VE3,PROP
  3323. Id,Stats
  3324. Id,VE4(FF~,l1~,al~,k~,l2~,be~,p~,l3~,ga~,pp~,l4~,la~,kp~)=
  3325.  FF(l1,al,k,l2,be,p,l3,ga,pp,l4,la,kp)
  3326. Al,VE3(FF~,l1~,al~,k~,l2~,be~,q~,l3~,ga~,p~)=
  3327.        FF(l1,al,k,l2,be,q,l3,ga,p)
  3328. Al,PROP(FF~,l1~,al~,q~,l2~,be~,k~)=FF(l1,al,l2,be,k)
  3329.  
  3330. Id,Count,-4,NOM,-2,q,1,q1,1,q2,1,q3,1,q4,1
  3331.  
  3332. C Only divergent pieces.
  3333.  
  3334. Id,q1(al~)=q(al)
  3335. Al,Dotpr,q1(al~)=q(al)
  3336. Id,q2(al~)=q(al)
  3337. Al,Dotpr,q2(al~)=q(al)
  3338. Id,q3(al~)=q(al)
  3339. Al,Dotpr,q3(al~)=q(al)
  3340. Id,q4(al~)=q(al)
  3341. Al,Dotpr,q4(al~)=q(al)
  3342. Id,k(al)=0
  3343. Al,p(be)=0
  3344. Al,pp(ga)=0
  3345. Al,kp(de)=0
  3346. Id,pDp=-M^2
  3347. Al,kDk=-M^2
  3348. Al,ppDpp=-M^2
  3349. Al,kpDkp=-M^2
  3350.  
  3351. Id,Count,-4,q,1,NOM,-2
  3352. Al,Even,NOM,1
  3353. Id,NOM(q~,M)=Nom
  3354. Al,NOM(q~,m)=Nohm
  3355. B Nom,Nohm
  3356. Id,Epfred
  3357. *yep
  3358.  
  3359. IF Nohm^n~*Nom^l~=1/q2M^l/q2m^n
  3360. Id,2,Ratio,q2M,q2m,[m2-M2]
  3361. Id,q2M^n~=1/Nom^n
  3362. Al,q2m^n~=1/Nohm^n
  3363. ENDIF
  3364.  
  3365. Id,All,q,N,Fq
  3366.  
  3367. Id,Fq(al~)=0
  3368. Al,Fq(al~,be~,ga~)=0
  3369. Al,Fq(al~,be~,ga~,de~,la~)=0
  3370. Al,Fq(al~,be~,ga~,de~,la~,a~,b~)=0
  3371. Id,Fq(al~,be~,ga~,de~)*Nom^l~ = dede(al,be,ga,de)*H(l,M2)
  3372. Al,Fq(al~,be~)*Nom^l~ = D(al,be)*G(l,M2)
  3373. Al,Nom^n~ = F(n,M2)
  3374. Id,Fq(al~,be~,ga~,de~)*Nohm^l~ = dede(al,be,ga,de)*H(l,m2)
  3375. Al,Fq(al~,be~)*Nohm^l~ = D(al,be)*G(l,m2)
  3376. Al,Nohm^l~ = F(l,m2)
  3377.  
  3378. STINT{}
  3379.  
  3380. Id,Count,0,m,1,m2,2,[m2-M2],2
  3381. Id,[m2-M2]^n~=m2^n
  3382.  
  3383. Id,M2^n~=M^(2*n)
  3384. Al,m2^n~=m^(2*n)
  3385. Id,N=N_+4
  3386. Id,N_=0
  3387.  
  3388. Id,Count,1,N_,-1,Log,1
  3389. Id,Log(M2)=0
  3390. Al,Log(m2)=Logm2
  3391.  
  3392. B i,Pi,M,N_
  3393. Keep BoxWW
  3394.  
  3395. *next
  3396.  
  3397. C Longitudinal, u-channel.
  3398.  
  3399. Z Long(a,k,b,p,c,pp,d,kp)
  3400.  = BoxWW(a,k,k,b,p,p,c,pp,pp,d,kp,kp)/M^4
  3401.  
  3402. Id,kpDkp=0
  3403. Al,kp(al~)=-k(al)-p(al)-pp(al)
  3404. Al,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  3405.  
  3406. Id,pDp=0
  3407. Al,kDk=0
  3408. Al,ppDpp=0
  3409. Id,pDk= - 0.5*s
  3410. Al,pDpp = - 0.5*t
  3411. Al,kDpp = - 0.5*u
  3412. B i,Pi,M,N_
  3413. Id,t=-s-u
  3414. Keep BoxWW
  3415. *next
  3416.  
  3417. P input
  3418. C The calculation sofar needs addition of further diagrams, obtained
  3419.   by crossing. Also add counterterms.
  3420.  
  3421. Z Total(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  3422.  = BoxWW(a,al,k,b,be,p,c,ga,pp,d,de,kp)
  3423.  + BoxWW(a,al,k,c,ga,pp,b,be,p,d,de,kp)
  3424.  + BoxWW(a,al,k,b,be,p,d,de,kp,c,ga,pp)
  3425.  + DLP*WWWW(a,al,k,b,be,p,c,ga,pp,d,de,kp)*WWWWK
  3426.  
  3427. B i,Pi
  3428.  
  3429. Id,A0=1
  3430. Al,A1=1
  3431. Al,A2=1
  3432. Al,A3=1
  3433.  
  3434. Id,Epfred
  3435. ETE1{}
  3436. P output
  3437. *yep
  3438. C Id,DLP=-1
  3439. Keep Total
  3440. *next
  3441. C Longitudinal, full result.
  3442.  
  3443. Z Tot = Total(a,k,k,b,p,p,c,pp,pp,d,kp,kp)/M^4
  3444. Id,kpDkp=0
  3445. Al,kp(al~)=-k(al)-p(al)-pp(al)
  3446. Al,Dotpr,kp(al~)=-k(al)-p(al)-pp(al)
  3447.  
  3448. Id,pDp=0
  3449. Al,kDk=0
  3450. Al,ppDpp=0
  3451. Id,pDk= - 0.5*s
  3452. Al,pDpp = - 0.5*t
  3453. Al,kDpp = - 0.5*u
  3454. B i,Pi,M,N_
  3455. IF D(a,c)
  3456. Id,t=-s-u
  3457. ENDIF
  3458. IF D(a,b)
  3459. Id,u=-s-t
  3460. ENDIF
  3461. IF D(a,d)
  3462. Id,s=-t-u
  3463. ENDIF
  3464. P output
  3465. *yep
  3466. Id,DLP=-1
  3467. *end
  3468.