Powered by Foswiki, The Free and Open Source Wiki

Difference: ConstraintDeltaMs (1 vs. 5)

Revision 5

30 Jun 2010 - Main.AdrianBevan

   Constraint from
-<
<
+The  oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system, is proportional, in the Standard Model, to . Neglecting terms giving small contributions,  is independent of  and . 
Instead, the ratio  is proportional to  and  and the advantage of using this ratio instead of only  is that the ratio  is expected to be better determined from the theory than the individual quantities entering its expression. The measurement of  gives a similar constraint as  :
->
>
+The  oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system, is proportional to  in the Standard Model. Neglecting terms giving small contributions,  is independent of  and . Instead, the ratio  is proportional to  and  and the advantage of using this ratio instead of only  is that the ratio  is expected to be better determined from the theory than the individual quantities entering its expression. The measurement of  gives a similar constraint as  :
-<
<
+In practice a problem arise in the (possible) use of the information from  twice in the fit: in the  constraint itself and in the ratio . The correlation has to be taken into account. It should be remembered that  alone would give a constraint on  and so . It should be also considered that due to the error coming from chiral extrapolations the quantities which are best known and which are effectively calculated are  and  while  is derived from the other two. For this reason we write the constraints in the following way :
->
>
+In practice a problem arises in the (possible) use of the information from  twice in the fit: in the  constraint itself and in the ratio . The correlation has to be taken into account. One should recall that  alone would give a constraint on  and so . It should also be considered that due to the error coming from chiral extrapolations the quantities which are best known and which are effectively calculated are  and  while  is derived from the other two. For this reason we write the constraints in the following way :
  \Delta m_d \simeq [(1-\bar{\rho})^2+\bar{\eta}^2] \frac{f_{B_s}^2 B_{B_s}}{\xi^2}
  Set plot = DeltaMsRhoEta?
 
-->
-<
<
+EPS - PDF -  PNG - JPG - GIF
->
>
+  EPS - PDF - PNG - JPG - GIF

Revision 4

09 May 2010 - Main.AdminUser

Line: 1 to 1
	Constraint from $\Delta m_d /\Delta m_s$
Line: 25 to 25

Changed:
< <	[EPS format] [JPG format]
> >	<-- Set plot = DeltaMsRhoEta? --> EPS - PDF - PNG - JPG - GIF

Revision 3

04 Apr 2010 - Main.VincenzoVagnoni

Line: 1 to 1

Added:
> >	Constraint from $\Delta m_d /\Delta m_s$
	The $B^0_s-\bar{B}^0_s$ oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system, is proportional, in the Standard Model, to $\|V_{ts}\|^2$ . Neglecting terms giving small contributions, $\|V_{ts}\|^2$ is independent of $\bar{\rho}$ and $\bar{\eta}$ . Instead, the ratio $\Delta m_d / \Delta m_s$ is proportional to $\bar{\rho}$ and $\bar{\eta}$ and the advantage of using this ratio instead of only $\Delta m_s$ is that the ratio $\xi=f_{B_d}\sqrt{B_{B_d}} / f_{B_s}\sqrt{B_{B_s}}$ is expected to be better determined from the theory than the individual quantities entering its expression. The measurement of $\Delta m_d / \Delta m_s$ gives a similar constraint as $\Delta m_d / \Delta m_d$ :

Revision 2

02 Apr 2010 - Main.VincenzoVagnoni

Line: 1 to 1
Changed:
< <	-- VincenzoVagnoni - 01 Apr 2010
> >	The $B^0_s-\bar{B}^0_s$ oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system, is proportional, in the Standard Model, to $\|V_{ts}\|^2$ . Neglecting terms giving small contributions, $\|V_{ts}\|^2$ is independent of $\bar{\rho}$ and $\bar{\eta}$ . Instead, the ratio $\Delta m_d / \Delta m_s$ is proportional to $\bar{\rho}$ and $\bar{\eta}$ and the advantage of using this ratio instead of only $\Delta m_s$ is that the ratio $\xi=f_{B_d}\sqrt{B_{B_d}} / f_{B_s}\sqrt{B_{B_s}}$ is expected to be better determined from the theory than the individual quantities entering its expression. The measurement of $\Delta m_d / \Delta m_s$ gives a similar constraint as $\Delta m_d / \Delta m_d$ : $\frac{\Delta m_d}{\Delta m_s} & = & \frac{m_{B_d}f^2_{B_d}\hat{B}_{B_d}}{m_{B_s}f^2_{B_s}\hat{B}_{B_s}} \frac{\|V_{td}\|^2}{\|V_{ts}\|^2} =\\ & = & \frac{m_{B_d}f^2_{B_d}\hat{B}_{B_d}}{m_{B_s}f^2_{B_s}\hat{B}_{B_s}} \left ( \frac {\lambda} {1-\lambda^2/2} \right ) ^2 \frac {(1-\bar{\rho})^2+\bar{\eta}^2}{\left ( 1+ \frac{\lambda^2}{1-\lambda^2 /2} \bar{\rho} \right ) ^2 +\lambda^4\bar{\eta}^2}$ In practice a problem arise in the (possible) use of the information from $\Delta m_d$ twice in the fit: in the $\Delta m_d$ constraint itself and in the ratio $\Delta m_d / \Delta m_s$ . The correlation has to be taken into account. It should be remembered that $\Delta m_s$ alone would give a constraint on $\|V_{ts}^2\|$ and so $f_{B_s}\sqrt{B_{B_s}}$ . It should be also considered that due to the error coming from chiral extrapolations the quantities which are best known and which are effectively calculated are $f_{B_s}\sqrt{B_{B_s}}$ and $\xi$ while $f_{B_d}\sqrt{B_{B_d}}$ is derived from the other two. For this reason we write the constraints in the following way : $\Delta m_d \simeq [(1-\bar{\rho})^2+\bar{\eta}^2] \frac{f_{B_s}^2 B_{B_s}}{\xi^2}$ $\Delta m_s \simeq f_{B_s}^2 B_{B_s}$ Here the quantities entering in the expression of $\Delta m_d$ are the calculated ones: $f_{B_s}^2 B_{B_s}$ and $\xi$ . To make the constraint on $\Delta m_d$ more effective both quantities have to be improved. The constraint on $\Delta m_s$ helps in improving the knowledge of one of those: $f_{B_s}^2 B_{B_s}$ . We implemented this bound using directly the information from the experimental likelihood provided by CDF. The representation of this constraint in the $(\bar{\rho},~\bar{\eta})$ plane is given below. [EPS format] [JPG format]

View topic | History: r5 < r4 < r3 < r2 | More topic actions...

Ideas, requests, problems regarding this web site? Send feedback