MC4BSM on-site exercise

A toy Standard Model (SM) extension.

1. A model is described HERE

2. We need to do followings,

   • Need to read existing SM model to be added on.

   • Need to define new particles in LanHEP

   • Need to define new Lagrangian

   • Need to compile this code for calchep to import as a Model file.

3. To make this model file be compatible with the one from FeynRules, we will follow notations of the model generated by a FeynRules

A. Adding a model to existing SM file.

1. We will use "newsm.mdl" file in the mdl folder ( lanhep315/mdl directory). A full source code is provided as attachment. (You can download it at the bottom of this page.) It would be better to save the model file (mc4bsm.mdl) at the mdl folder (same folder with 'newsm.mdl') We will explain this code step by step as followings,

   1. • • 1. Standard Model part

             1. • • 1. read newsm.

                    2. keys CKMdim=1.

                    3. keys SMmassless=1.

                    4. keys gauge_fixing=Feynman.

                    5. parameter GG = 1.21977796370 : 'Strong coupling constant (Z pnt, alphaS=0.1184(PDG2012))'.

                    6. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                    7. % Remove the 1st and 2nd generation mass

                    8. do_if SMmassless==1.

                    9. parameter Mm = 0 , Ms =0, Mc=0 .

                    10. end_if.

                    11. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                    12. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                    13. % Make CKM a diagonal Matrix.

                    14. do_if CKMdim==1.

                    15. parameter Vub=0, Vcb=0, Vtd=0, Vts=0, Vtb=1, Vud=1, Vus=0, Vcs=1, Vcd=0.

                    16. end_if.

                    17. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                    18. do_if gauge_fixing==Feynman.

                    19. model 'MC4BSM_feynman'/4.

                    20. do_else_if gauge_fixing==unitary.

                    21. model 'MC4BSM_unitary'/3.

                    22. end_if.

2. read newsm. : This "read" command reads existing a model file called newsm.mdl in the same directory.

3. Reset CKM parameters and masses of SM particles to be consistent with the model from FeynRules.

• • Set the keys to control cases.

  • Using "do_if" and "end_if" commands to divide cases.

  • Here we set the mass of muon and s-quark to be 0. (By default in newsm model file, electron and u,d-quark are massless.)

4. We redefined a default strong coupling in "newsm.mdl" file with updated strong coupling GG with \alphaS = 0.1184 (also this is the default value in FeynRules)

• • • parameter GG = 1.21977796370.

5. Reset the name of this model file as "MC4BSM_gauge_fixing".

B. Defining new particles.

• • • • 1. Properties of new particles

           1. • • 1. % Define new parameters.

                  2. parameter MM1 = 200 : 'Mass parameter for pi1',

                  3. MM2 = 300 : 'Mass parameter for pi2',

                  4. MM12 = 50 : 'Mixing term for pi1 and pi2',

                  5. lam1 = 1 : 'Yukawa interaction for pi1, U and u',

                  6. lam2 = 1 : 'Yukawa interaction for pi2, U and u',

                  7. lam1p = 1 : 'Yukawa interaction for pi1, E and e',

                  8. lam2p = 1 : 'Yukawa interaction for pi2, E and e',

                  9. Muv= 500.0 : 'Mass parameter for uv and v',

                  10. Mev = 250.0 : 'Mass parameter for ev and e'.

                  11. parameter MPe1 = sqrt(MM2**2+MM1**2-sqrt(4.0*MM12**4+(MM2**2-MM1**2)**2))/Sqrt2,

                  12. MPe2 = sqrt(MM2**2+MM1**2+sqrt(4.0*MM12**4+(MM2**2-MM1**2)**2))/Sqrt2.

                  13. parameter tth = (MM2**2-MM1**2+sqrt(4.0*MM12**4+(MM2**2-MM1**2)**2))/(2.0*MM12**2).

                  14. parameter sth = tth/sqrt(1+tth**2), cth = 1.0/sqrt(1+tth**2).

                  15. angle sin=sth,cos=cth,tan=tth, texname='\\theta'.

                  16. % Define new particles.

                  17. spinor ev:('heay electron',mass Mev, width Wev=auto),

                  18. uv:('heavy quark', color c3, mass Muv, width Wuv=auto).

                  20. scalar p1/p1:('LNP',mass MPe1),

                  21. p2/p2:('Heavy scalar',mass MPe2, width Wpe2=auto).

                  23. let pi1= -sth*p1+cth*p2, pi2=cth*p1+sth*p2.

                  24. prtcproperty pdg:( ev=9000009,uv=9000008,p1=9000006,p2=9000007).

• 1. We defined the parameters of new particles: mass parameters, couplings and mixing.

1. Width in the particle definitions

   • When we set the width of particles, we make the width to be calculated in calchep on the fly (for example)

     • width Wev=auto

2. New fermions

   • ev: New heavy right-handed electron

   • uv: New heavy right-handed up-type quark. color c3 means a color triplet as as up-type quark.

3. Scalar particles:

   • Mass eigen states: p1 and p2

   • Flavor eigen states: pi1 and pi2 are defined with mixing angle \theta as followings,

     • let pi1= - sth*p1+cth*p2, pi2=cth*p1+sth*p2

   • The Lagrangian will be written with pi1 and pi2. Through the above definition, in the model file, the Lagrangian will be described in terms of p1 and p2 (mass eigen states) automatically.

   • Mass eigen values (masses of p1 and p2) are defined as MPe1 and MPe2

   • Width of p1 is not defined, because p1 is a stable particle. By default, the width = 0.

4. Set the PDG number to be consistent with a model from FeynRules. In this onsite exercise, we will not change calchep.rc file.

   • • prtcproperty pdg:( ev=9000009,uv=9000008,p1=9000006,p2=9000007).

C. Set the Lagrangian

• • • • 1. Lagrangian

           1. • • 1. % Kinematics terms for new fermion particles

                  2. % right fermion interaction with gauge fields

                  3. let PR = (1+g5)/2.

                  4. lterm anti(psi)*gamma*PR*(i*deriv - Y*g1*B1)*psi

                  5. where

                  6. psi=ev,Y= -1;

                  7. psi=uv, Y= 2/3.

                  8. lterm GG*anti(uv)*lambda*gamma*G*uv.

                  9. % Kinematic terms for scalars

                  10. lterm deriv*pi1*deriv*pi1/2+deriv*pi2*deriv*pi2/2.

                  11. % New mass term

                  12. lterm -MM1**2/2 * pi1*pi1-MM2**2/2 * pi2*pi2-MM12**2*pi1*pi2.

                  13. lterm -Muv*anti(uv)*uv-Mev*anti(ev)*ev.

                  14. % New Yukawa type interaction.

                  15. lterm lam1*pi1*anti(uv)*PR*u+lam2*pi2*anti(uv)*PR*u+lam1p*pi1*anti(ev)*PR*e+lam2p*pi2*anti(ev)*PR*e + AddHermConj.

                  16. CheckHerm.

1. Define the kinematic and interaction terms with gauge bosons of new fermions.

   • We defined right hand projector with g5 (g5 is defined in newsm.mdl as g5=gamma5.)

     • let PR = (1+g5)/2.

   • We used "where" grammar to reduce unnecessary repeats and to give a clear structure.

     • psi is a generic notation for fermions here.

     • anti(psi) is a "\bar{psi}" same as FeynRules convention.

     • Y is a hyper charge of new particles.

     • For uv(colored particle) there is an interaction with G (Gluon, defined in newsm.mdl) with coupling GG: 'Strong coupling constant (Z pnt, alp=0.1172\pm0.002 (PDG2002)'

2. Kinematic terms of pi1 and pi2 (will be replaced with p1 and p2)

   • These particles are SM gauge singlets as we can see in the corresponding lagrangian

     • lterm deriv*pi1*deriv*pi1/2+deriv*pi2*deriv*pi2/2.

   • Mass terms are defined, and these terms will give the mixing between pi1 and pi2.

     • lterm -Muv*anti(uv)*uv-Mev*anti(ev)*ev.

3. Yukawa interaction terms.

   • We need to add hermitian conjugate terms by writing

     • +AddHermConj.

4. Check the hermiticity of your Lagrangian by

   • • CheckHerm.

D. Exporting a compiled model to CalcHEP: HERE.