Event Generation of String Resonances using CalcHEP

Noriaki Kitazawa, Tokyo Metropolitan University
May 27, 2011

The following is a procedure how to generate events of string resonances at the LHC using CalcHEP (ver.2.5.6).

String resonances contribute the following processes with two-parton finial states. 

pp -> GG,
pp -> uG, pp -> dG,
pp -> UG, pp -> DG,
pp -> uU, pp -> dD, pp -> sS, pp -> cC, pp -> bB, pp -> tT,
where p, G denote proton and gluon, respectively, and u, d, s, c, b, t denote quarks and the corresponding upper case letters denote anti-quarks. Each process includes some subprocess. For example, there are five subprocesses in pp -> GG 
uU -> GG,
Uu -> GG,
dD -> GG,
Dd -> GG,
GG -> GG,
where the different order of letters in initial state describes two possible origin of partons from colliding protons. The contribution of string resonance states to these subprocess are described as squared amplitudes in Ref.[1] (the original Ref.[2] includes some minor numerical errors). The event files which include string resonance effect only can be generated by replacing standard model squared amplitudes by corresponding string squared amplitude in C codes of ClacHEP for each subprocesses. CalcHEP has a function of output the results of symbolic calculations of squared amplitudes to C codes before lunching numerical calculations for event generations.

For example, one output file for subprocess uU -> GG is as follows. 

/*******************************
*    CalcHEP version 2.5.6   *
*******************************/
#include
#define Real double
#include"num_out.h"
#include"num_in.h"
extern void C3_ext(Real*);
extern Real F3_ext(void);
Real F3_ext(void)
{
/*
                u           G    !  G           u                    
              ==>==\     /-3-----!---3-\     /==>==                  
                P1 |     |  P4   !  P4 |     |  P1                   
                   |     |       !     |     |                       
                U  |  G  |  G    !  G  |  G  |  U                    
              ==<==@-1---@-2-----!---2-@---4-@==<==                  
                P2    P5    P3   !  P3   -P6    P2                   
*/
Real N,D,R,Q;
Real* V=va_ext;
static Real C[1],CC[0];
Real S[2];                                                                  
     
int i;
if(CalcConst) C3_ext(C);
S[0]=V[1]*V[1];
S[0]*=S[0];
N=-32*S[0];
D=+3;
S[1]=DP[1]*DP[1];

R=+DP[0]*(DP[0]-DP[1])+S[1];
R*=(N/D)*Q2[2];
 if(cb_coeff_ext)
 {
  cb_coeff_ext[1] += (R*-9)/(-16);
  cb_coeff_ext[2] += (R*-9)/(-16);
 }
 return R;
}
This code define a standard model squared amplitudes. There are 16 of these files which determine squared amplitude including interferences between different Feynman diagrams. Editing this file as follows replace standard model squared amplitude by string squared amplitude. 
/*******************************
*    CalcHEP version 2.5.6   *
*******************************/
#include
#define Real double
#include"num_out.h"
#include"num_in.h"
extern void C3_ext(Real*);
extern Real F3_ext(void);
Real F3_ext(void)
{
/*
                u           G    !  G           u                    
              ==>==\     /-3-----!---3-\     /==>==                  
                P1 |     |  P4   !  P4 |     |  P1                   
                   |     |       !     |     |                       
                U  |  G  |  G    !  G  |  G  |  U                    
              ==<==@-1---@-2-----!---2-@---4-@==<==                  
                P2    P5    P3   !  P3   -P6    P2                   
*/
Real N,D,R,Q;
Real* V=va_ext;
static Real C[1],CC[0];
Real S[2];                                                                  
     
int i;
if(CalcConst) C3_ext(C);
S[0]=V[1]*V[1];
S[0]*=S[0];
N=-32*S[0];
D=+3;
S[1]=DP[1]*DP[1];

R=+DP[0]*(DP[0]-DP[1])+S[1];
R*=(N/D)*Q2[2];
 if(cb_coeff_ext)
 {
  cb_coeff_ext[1] += (R*-9)/(-16);
  cb_coeff_ext[2] += (R*-9)/(-16);
 }

/* to make string resonance effect only */
 R=0;
/* to make string resonance effect only */

/* begin adding string amplitude */
Real Ms=3e3;
Real Gam_g2=45.*(Ms/1e3);
Real Gam_c2=75.*(Ms/1e3);
Real gs4=S[0];

Real s_hat=2*DP[0];
Real t_hat=-2*DP[1];
Real u_hat=-s_hat-t_hat;

Real Ms4=Ms*Ms*Ms*Ms;

Real t_hat2=t_hat*t_hat;
Real u_hat2=u_hat*u_hat;

R+=(56./27.)*(gs4/Ms4)*((5./7.)*u_hat*t_hat*(u_hat2+t_hat2)/((s_hat-Ms*Ms)*(s_hat-Ms*Ms)+(Ms*Gam_g2)*(Ms*Gam_g2)));
R+=(56./27.)*(gs4/Ms4)*((2./7.)*u_hat*t_hat*(u_hat2+t_hat2)/((s_hat-Ms*Ms)*(s_hat-Ms*Ms)+(Ms*Gam_c2)*(Ms*Gam_c2)));
/* end adding string amplitude */

 return R;
}
In the other files which do not define squared s-channel amplitude, the standard model squared amplitude should be simply replaced by zero by inserting 
/* to make string resonance effect only */
 R=0;
/* to make string resonance effect only */
at the end of the file. An example of work directories of CalcHEP is given here.

By lunching CalcHEP and doing numerical calculations with these modified files gives event files which include string resonance effects only.

References
[1] "A Closer look at string resonances in dijet events at the LHC," N. Kitazawa, JHEP 1010:051,2010. 

[2] "LHC Phenomenology for String Hunters." L. A. Anchordoqui, et al., Nucl.Phys.B821:181-196,2009.