/* qq-qq1.mac April 30, 2011 Edwin L. Woollett woollett@charter.net http://www.csulb.edu/~woollett needs ch. 12 dirac package (version2) with addition of qcd1.mac (easiest to insert load(qcd1) into the load section of dirac2.mac). QUARK-QUARK ELASTIC SCATTERING, in first order dominated by a single gluon exchange. CASE 1: quark-quark scattering of different flavored quarks, like u d --> u d. CASE 2: quark-quark scattering of same flavor quarks, such as u u --> u u. Context: sub-parton hard scattering processes where we can ignore the mass of the quarks. In first order, we only include gluon exchange. We interpret the Feynman rules as defining -%i*M. Our notation and conventions agree generally with Peskin and Schroeder, unless otherwise indicated. References: 1. Leader, Elliot and Predazzzi, Enrico, An Introduction to Gauge Theories and the New Physics, Cambridge Univ. Press, 1983, pp. 460-463 2. Peskin, Michael and Schroeder, Daniel, An Introduction to Quantum Field Theory, Addison-Wesley, 1995 Feynman rules: pp. 507-515, pp.802-803 note: g^2 = 4*%pi*alpha_s sub-parton hard scatt: pp. 568-573 3. Griffiths, David, Introduction to Elementary Particles, Harper & Row, 1987 (there are later editions), Chapter 9. 4. Chyla, Jiri, Quarks, Partons and Quantum Chromodynamics, 210 pp lecture notes, available at http://www-hep2.fzu.cz/~chyla/lectures/text.pdf, Feynman rules, etc., pp 201-204 cross sections: pp. 120 - 123. 5. Cutler, Roger, and Sivers, Dennis, Quantum-chromodynamic gluon contributions to large-p_T reactions, Physical Review D, Vol. 17, 198-211, Jan. 1, 1978, Appendix A: Evaluation of QCD Graphs. */ " ======================================================"$ " file qq-qq1.mac "$ " Maxima by Example, Ch. 13 "$ " Quark Gluon Hard Scattering Processes "$ " Edwin L Woollett, woollett@charter.net "$ " http://www.csulb.edu/~woollett "$ print (" ver: ",_binfo%," date: ",mydate )$ " ======================================================"$ " QUARK-QUARK ELASTIC SCATTERING "$ " Case: different flavor quark-quark elastic scattering "$ " such as u d --> u d "$ " Let alpha and beta denote the quark flavor."$ " Here we assume alpha # beta "$ " We only include gluon exchange in our first example "$ " dirac2.mac should load qcd1.mac to define the "$ " SU(3) 3 x 3 color matrices T[a], a = 1,2,...,8 "$ " and the antisymmetric gauge group structure contstants"$ " f[a,b,c] "$ " HIGH ENERGY LIMIT, CENTER OF MOMENTUM FRAME, NEGLECT MASSES "$ " Let i,j,k,l (taking values 1,2,3) be the specified quark color assignments "$ " q(p1,s1,i,alpha) + q(p2,s2,j,beta) --> q(p3,s3,k,alpha) + q(p4,s4,l,beta) "$ " ma and mb, quark masses, set to zero."$ " ------------------------------------------ "$ "NON-POLARIZED DIFFERENTIAL CROSS SECTION: SYMBOLIC METHODS"$ " POPULATE THE LIST invarR OF 4-VEC DOT PRODUCT VALUES, "$ " Using p1 + p2 = p3 + p4, s = (p1+p2)^2 = (p3+p4)^2 ,"$ " t = (p1-p3)^2 = (p2-p4)^2,"$ " and u = (p1-p4)^2 = (p2-p3)^2 "$ " CASE HIGH ENERGY (HE) LIMIT E >> ( ma,mb) "$ " ---------------------------------------------"$ invar (D(p1,p1) = 0, D(p1,p2) = s/2, D(p1,p3) = - t/2, D(p1,p4) = - u/2, D(p2,p2) = 0, D(p2,p3) = - u/2, D(p2,p4) = - t/2, D(p3,p3) = 0, D(p3,p4) = s/2, D(p4,p4) = 0)$ "------------------------------------------------------"$ " Mt = At * Ct = dirac amplitude times a color amplitude "$ "Atsq is < |At|^2 > which includes (1/4)*(sum over all helicities)"$ " multiply by g^4/(4*t^2) later "$ Atsq : factor (scon ( tr (p3,mu,p1,nu)*tr (p4,mu,p2,nu), mu,nu)); Atsq : (g^4/(4*t^2))*Atsq; " Ctsq is < |Ct|^2 > which incudes (1/9)*(sum over all quark colors"$ Ctsq : sum (sum ( mat_trace (T[a] . T[b])^2,b,1,8),a,1,8)/9; Mtsq : Atsq*Ctsq; dsigdt : Mtsq/(16*%pi*s^2)$ (display2d:true,display (dsigdt),display2d:false)$ " CONVERSION TO EXPLICIT FUNCTION OF SCATTERING ANGLE "$ assume ( p > 0, th >= 0, th <= %pi )$ comp_def ( p1( E,0,0,E), p2( E,0,0,-E), p3 (E,E*sin(th),0,E*cos(th)), p4 (E,-E*sin(th),0,-E*cos(th)) )$ s_th : VP (p1+p2,p1+p2); t_th : factor (VP (p1-p3,p1-p3)); u_th : factor (VP (p1-p4,p1-p4)); " sub_stu replaces s by s_th, t by t_th "$ " and u by u_th "$ MtSQ : sub_stu (Mtsq); " convert to (th/2) form "$ MtSQ2 : factor (trigsimp (to_ao2 (MtSQ,th))); dsigdo_CM : MtSQ2/(64*%pi^2*s)$ (display2d:true,display (dsigdo_CM),display2d:false)$ "============================================="$ " CASE ELASTIC SCATTERING OF SAME FLAVOR QUARKS"$ " need to include u channel crossed diagram "$ " including an extra minus sign compared with t channel."$ " Total Unpolarized cross section will now include four "$ " terms: Mtsq + Musq + Mt*conj(Mu) + Mu*conj(Mt) "$ " coming from < (Mt + Mu)*conj(Mt + Mu) > "$ " Mu = Au * Cu = dirac amplitude times a color amplitude "$ "Ausq is < |Au|^2 > which includes (1/4)*(sum over all helicities)"$ " multiply by g^4/(4*u^2) later "$ Ausq : factor (scon ( tr (p4,mu,p1,nu)*tr (p3,mu,p2,nu), mu,nu)); Ausq : (g^4/(4*u^2))*Ausq; " Cusq is < |Cu|^2 > which incudes (1/9)*(sum over all quark colors"$ " and which is the same numerical value as Ctsq "$ Cusq : Ctsq; Musq : Ausq*Cusq; " cross term Mtu = Atub*Ctub "$ Ctub : sum (sum ( mat_trace (T[a] . T[b] . T[a] . T[b]),b,1,8),a,1,8)/9; Atub : tr(p3,mu,p1,nu,p4,mu,p2,nu); Atub : -(g^4/(4*t*u)) * Atub; Mtu : Atub*Ctub; " second cross term Mut = = < Aut*Cut> = Mtu "$ " Autb = "$ Autb : tr(p4,nu,p1,mu,p3,nu,p2,mu); Autb : -(g^4/(4*t*u)) * Autb; Cutb : Ctub; Mut : Autb*Cutb; " total averaged squared amplitude = <|M|^2> "$ Msq : Mtsq + Musq +2*Mtu; Msq : pullfac (Msq,2*g^4/9); dsigdt : Msq/(16*%pi*s^2)$ (display2d:true,display (dsigdt),display2d:false)$ " CONVERSION TO EXPLICIT FUNCTION OF SCATTERING ANGLE "$ " sub_stu replaces s by s_th, t by t_th "$ " and u by u_th "$ MSQ : sub_stu (Msq); " convert to (th/2) form "$ MSQ2 : factor (trigsimp (to_ao2 (MSQ,th))); dsigdo_CM : MSQ2/(64*%pi^2*s)$ (display2d:true,display (dsigdo_CM),display2d:false)$ "============================================="$ /* (%i2) batch("qq-qq1.mac"); read and interpret file: #pc:/work5/qq-qq1.mac (%i3) " ======================================================" (%i4) " file qq-qq1.mac " (%i5) " Maxima by Example, Ch. 13 " (%i6) " Quark Gluon Hard Scattering Processes " (%i7) " Edwin L Woollett, woollett@charter.net " (%i8) " http://www.csulb.edu/~woollett " (%i9) print(" ver: ",_binfo%," date: ",mydate) ver: Maxima 5.24.0 date: 2011-04-30 (%i10) " ======================================================" (%i11) " QUARK-QUARK ELASTIC SCATTERING " (%i12) " Case: different flavor quark-quark elastic scattering " (%i13) " such as u d --> u d " (%i14) " Let alpha and beta denote the quark flavor." (%i15) " Here we assume alpha # beta " (%i16) " We only include gluon exchange in our first example " (%i17) " dirac2.mac should load qcd1.mac to define the " (%i18) " SU(3) 3 x 3 color matrices T[a], a = 1,2,...,8 " (%i19) " and the antisymmetric gauge group structure contstants" (%i20) " f[a,b,c] " (%i21) " HIGH ENERGY LIMIT, CENTER OF MOMENTUM FRAME, NEGLECT MASSES " (%i22) " Let i,j,k,l (taking values 1,2,3) be the specified quark color assignments " (%i23) " q(p1,s1,i,alpha) + q(p2,s2,j,beta) --> q(p3,s3,k,alpha) + q(p4,s4,l,beta) " (%i24) " ma and mb, quark masses, set to zero." (%i25) " ------------------------------------------ " (%i26) "NON-POLARIZED DIFFERENTIAL CROSS SECTION: SYMBOLIC METHODS" (%i27) " POPULATE THE LIST invarR OF 4-VEC DOT PRODUCT VALUES, " (%i28) " Using p1 + p2 = p3 + p4, s = (p1+p2)^2 = (p3+p4)^2 ," (%i29) " t = (p1-p3)^2 = (p2-p4)^2," (%i30) " and u = (p1-p4)^2 = (p2-p3)^2 " (%i31) " CASE HIGH ENERGY (HE) LIMIT E >> ( ma,mb) " (%i32) " ---------------------------------------------" (%i33) invar(D(p1,p1) = 0,D(p1,p2) = s/2,D(p1,p3) = (-t)/2,D(p1,p4) = (-u)/2, D(p2,p2) = 0,D(p2,p3) = (-u)/2,D(p2,p4) = (-t)/2,D(p3,p3) = 0, D(p3,p4) = s/2,D(p4,p4) = 0) (%i34) "------------------------------------------------------" (%i35) " Mt = At * Ct = dirac amplitude times a color amplitude " (%i36) "Atsq is < |At|^2 > which includes (1/4)*(sum over all helicities)" (%i37) " multiply by g^4/(4*t^2) later " (%i38) Atsq:factor(scon(tr(p3,mu,p1,nu)*tr(p4,mu,p2,nu),mu,nu)) (%o38) 8*(u^2+s^2) (%i39) Atsq:g^4*Atsq/(4*t^2) (%o39) 2*g^4*(u^2+s^2)/t^2 (%i40) " Ctsq is < |Ct|^2 > which incudes (1/9)*(sum over all quark colors" (%i41) Ctsq:sum(sum(mat_trace(T[a] . T[b])^2,b,1,8),a,1,8)/9 (%o41) 2/9 (%i42) Mtsq:Atsq*Ctsq (%o42) 4*g^4*(u^2+s^2)/(9*t^2) (%i43) dsigdt:Mtsq/(16*%pi*s^2) (%i44) (display2d:true,display(dsigdt),display2d:false) 4 2 2 g (u + s ) dsigdt = ------------ 2 2 36 %pi s t (%i45) " CONVERSION TO EXPLICIT FUNCTION OF SCATTERING ANGLE " (%i46) assume(p > 0,th >= 0,th <= %pi) (%i47) comp_def(p1(E,0,0,E),p2(E,0,0,-E),p3(E,E*sin(th),0,E*cos(th)), p4(E,-E*sin(th),0,-E*cos(th))) (%i48) s_th:VP(p2+p1,p2+p1) (%o48) 4*E^2 (%i49) t_th:factor(VP(p1-p3,p1-p3)) (%o49) 2*(cos(th)-1)*E^2 (%i50) u_th:factor(VP(p1-p4,p1-p4)) (%o50) -2*(cos(th)+1)*E^2 (%i51) " sub_stu replaces s by s_th, t by t_th " (%i52) " and u by u_th " (%i53) MtSQ:sub_stu(Mtsq) (%o53) -4*g^4*sin(th)^2/(9*cos(th)^2-18*cos(th)+9) +8*g^4*cos(th)/(9*cos(th)^2-18*cos(th)+9) +24*g^4/(9*cos(th)^2-18*cos(th)+9) (%i54) " convert to (th/2) form " (%i55) MtSQ2:factor(trigsimp(to_ao2(MtSQ,th))) (%o55) 4*g^4*(cos(th/2)^4+1)/(9*sin(th/2)^4) (%i56) dsigdo_CM:MtSQ2/(64*%pi^2*s) (%i57) (display2d:true,display(dsigdo_CM),display2d:false) 4 4 th g (cos (--) + 1) 2 dsigdo_CM = ------------------- 2 4 th 144 %pi s sin (--) 2 (%i58) "=============================================" (%i59) " CASE ELASTIC SCATTERING OF SAME FLAVOR QUARKS" (%i60) " need to include u channel crossed diagram " (%i61) " including an extra minus sign compared with t channel." (%i62) " Total Unpolarized cross section will now include four " (%i63) " terms: Mtsq + Musq + Mt*conj(Mu) + Mu*conj(Mt) " (%i64) " coming from < (Mt + Mu)*conj(Mt + Mu) > " (%i65) " Mu = Au * Cu = dirac amplitude times a color amplitude " (%i66) "Ausq is < |Au|^2 > which includes (1/4)*(sum over all helicities)" (%i67) " multiply by g^4/(4*u^2) later " (%i68) Ausq:factor(scon(tr(p4,mu,p1,nu)*tr(p3,mu,p2,nu),mu,nu)) (%o68) 8*(t^2+s^2) (%i69) Ausq:g^4*Ausq/(4*u^2) (%o69) 2*g^4*(t^2+s^2)/u^2 (%i70) " Cusq is < |Cu|^2 > which incudes (1/9)*(sum over all quark colors" (%i71) " and which is the same numerical value as Ctsq " (%i72) Cusq:Ctsq (%o72) 2/9 (%i73) Musq:Ausq*Cusq (%o73) 4*g^4*(t^2+s^2)/(9*u^2) (%i74) " cross term Mtu = Atub*Ctub " (%i75) Ctub:sum(sum(mat_trace(T[a] . T[b] . T[a] . T[b]),b,1,8),a,1,8)/9 (%o75) -2/27 (%i76) Atub:tr(p3,mu,p1,nu,p4,mu,p2,nu) (%o76) -8*s^2 (%i77) Atub:-g^4*Atub/(4*t*u) (%o77) 2*g^4*s^2/(t*u) (%i78) Mtu:Atub*Ctub (%o78) -4*g^4*s^2/(27*t*u) (%i79) " second cross term Mut = = < Aut*Cut> = Mtu " (%i80) " Autb = " (%i81) Autb:tr(p4,nu,p1,mu,p3,nu,p2,mu) (%o81) -8*s^2 (%i82) Autb:-g^4*Autb/(4*t*u) (%o82) 2*g^4*s^2/(t*u) (%i83) Cutb:Ctub (%o83) -2/27 (%i84) Mut:Autb*Cutb (%o84) -4*g^4*s^2/(27*t*u) (%i85) " total averaged squared amplitude = <|M|^2> " (%i86) Msq:2*Mtu+Musq+Mtsq (%o86) 4*g^4*(u^2+s^2)/(9*t^2)-8*g^4*s^2/(27*t*u)+4*g^4*(t^2+s^2)/(9*u^2) (%i87) Msq:pullfac(Msq,2*g^4/9) (%o87) 2*g^4*(2*(u^2+s^2)/t^2-4*s^2/(3*t*u)+2*(t^2+s^2)/u^2)/9 (%i88) dsigdt:Msq/(16*%pi*s^2) (%i89) (display2d:true,display(dsigdt),display2d:false) 2 2 2 2 2 4 2 (u + s ) 4 s 2 (t + s ) g (----------- - ----- + -----------) 2 3 t u 2 t u dsigdt = -------------------------------------- 2 72 %pi s (%i90) " CONVERSION TO EXPLICIT FUNCTION OF SCATTERING ANGLE " (%i91) " sub_stu replaces s by s_th, t by t_th " (%i92) " and u by u_th " (%i93) MSQ:sub_stu(Msq) (%o93) -320*g^4/(27*sin(th)^2)+128*g^4/(9*sin(th)^4)+8*g^4/9 (%i94) " convert to (th/2) form " (%i95) MSQ2:factor(trigsimp(to_ao2(MSQ,th))) (%o95) 8*g^4*(cos(th/2)^2*sin(th/2)^2-3)*(3*cos(th/2)^2*sin(th/2)^2-1) /(27*cos(th/2)^4*sin(th/2)^4) (%i96) dsigdo_CM:MSQ2/(64*%pi^2*s) (%i97) (display2d:true,display(dsigdo_CM),display2d:false) 4 2 th 2 th 2 th 2 th g (cos (--) sin (--) - 3) (3 cos (--) sin (--) - 1) 2 2 2 2 dsigdo_CM = ---------------------------------------------------- 2 4 th 4 th 216 %pi s cos (--) sin (--) 2 2 (%i98) "=============================================" (%o99) "qq-qq1.mac" */