mu_s = 1/4 mu_q

From: Michael Murray (murray@CyclotronMail.tamu.edu)
Date: Sat Jun 29 2002 - 23:09:57 EDT

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       Dear Flemming,
                 I was planning to submit the corrected one you sent back to me.
    As for (pbar/p)**1/3 that just comes from quark counting.
    
    In a thermal model 
                dN/dP = (Spin Factor) *(Factor of T/M) * exp Mu/T * phase space 
    density. 
    
    For antiparticle/particle ratios everthing cancels except the chemical 
    potentials. In principle one has 3 chemical potentials for up, down and strange
    quarks (ignoring charm, bottom  and top). The diffent potentials for up and down
    are required to conserver electric charge but in practice the difference between
    them is very small. Let mu_q = the potential for a light (up or down) quark and
    mu_s be the potential for a strange quark. 
    
    
      Then (pbar/p) = exp -3mu_q/exp (3mu_q/T) since protons have 3 light quarks 
                    = exp(-6mu_q/T)                                     (1)
    
      Kaons have one light quark and one strange one so
    
           k-/k+    = exp((mu_s-mu_q)/T)/exp-(mu_s-muq)/T)
                    = exp(2mu_s/T) * exp -2Mu_q/T
                    = exp(2mu_s/T) * (pbar/p)**1/3                      (2)
    
    For e+e- k-/k+=pbar/p=1 ==> mu_q = mu_s = 0
    
    The data in Figure 4 approach (pbar/p)**1/3 as (pbar/p)==>1 but the AGS data
    are way above, implying a positive strange chemical potential. Thus one
    can vu figure 4 as showing that the strange chemical potential decreases 
    with sqrt(s). The fact that there is one universal curve 
    implies local strangeness conservation since this implies a constraint on
    mu_s. Another way of interpreting k-/k+=(pbar/p)**1/4  uses Eqn (2)
    
           (pbar/p)**1/4=exp(2mu_s/T)*(pbar/p)**1/3
    
           (pbar/p)**-1/12=exp(2mu_s/T)
      
    Using (1) and taking logs gives
          -1/12 * -6 mu_q/T  = 2 mu_s/T
    
               or mu_S = 1/4 mu_q
    
    This looks like a nice result.
                  Yours Michael
    
    
               
    
    Quoting Flemming Videbaek <videbaek@sgs1.hirg.bnl.gov>:
    
    > Dear Michael
    > 
    > Could you give me the final version of your abstract as well as indicate
    > if
    > it  has been submitrted.
    > And too thanks for your comment  to the paper. Is it easy to see that a
    > thermal model with constant u give p/p**1/3?
    > The we should amke a point of this in the paper; PRL is 'general' so
    > such
    > simple relsations should be stressed.
    > 
    > Flemming
    > 
    > 
    > ------------------------------------------------------
    > Flemming Videbaek
    > Physics Department
    > Brookhaven National Laboratory
    > 
    > tlf: 631-344-4106
    > fax 631-344-1334
    > e-mail: videbaek@bnl.gov
    > ----- Original Message -----
    > From: "Michael Murray" <murray@CyclotronMail.tamu.edu>
    > To: <brahms-l@bnl.gov>
    > Sent: Saturday, June 29, 2002 6:32 PM
    > Subject: Re: Draft Abstract for Fall DNP Meeting (Michigan State
    > University)
    > 
    > 
    > 
    > 
    > 
    
    
    
    Michael Murray, Cyclotron TAMU, 979 845 1411 x 273, Fax 1899
    



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