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The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units it is:

mT2=m2+px2+py2=E2−pz2displaystyle m_T^2=m^2+p_x^2+p_y^2=E^2-p_z^2,

where the z-direction is along the beam pipe and so

pxdisplaystyle p_x and pydisplaystyle p_y are the momentum perpendicular to the beam pipe and

mdisplaystyle m is the (invariant) mass.

This definition of the transverse mass is used in conjunction with the definition of the (directed) transverse energy

E→T=Ep→T|p→|=EE2−m2p→Tdisplaystyle vec E_T=Efrac vec p_T=frac Esqrt E^2-m^2vec p_T

with the transverse momentum vector p→T=(px,py)displaystyle vec p_T=(p_x,p_y). It is easy to see that for vanishing mass (m=0displaystyle m=0) the three quantities are the same: ET=pT=mTdisplaystyle E_T=p_T=m_T.
The transverse mass is used together with the rapidity, transverse momentum and polar angle in the parameterization of the four-momentum of a single particle:

(E,px,py,pz)=(mTcosh⁡y, pTcos⁡ϕ, pTsin⁡ϕ, mTsinh⁡y)displaystyle (E,p_x,p_y,p_z)=(m_Tcosh y, p_Tcos phi , p_Tsin phi , m_Tsinh y)

Using these definitions (in particular for ETdisplaystyle E_T) gives for the mass of a two particle system:

Mab2=(pa+pb)2=pa2+pb2+2papb=ma2+mb2+2(EaEb−p→a⋅p→b)displaystyle M_ab^2=(p_a+p_b)^2=p_a^2+p_b^2+2p_ap_b=m_a^2+m_b^2+2(E_aE_b-vec p_acdot vec p_b)

Mab2=ma2+mb2+2(ET,apa,x2+pa,y2+pa,z2pT,aET,bpb,x2+pb,y2+pb,z2pT,b−p→T,a⋅p→T,b−pz,apz,b)displaystyle M_ab^2=m_a^2+m_b^2+2left(E_T,afrac sqrt p_a,x^2+p_a,y^2+p_a,z^2p_T,aE_T,bfrac sqrt p_b,x^2+p_b,y^2+p_b,z^2p_T,b-vec p_T,acdot vec p_T,b-p_z,ap_z,bright)

Mab2=ma2+mb2+2(ET,aET,b1+pa,z2/pT,a21+pb,z2/pT,b2−p→T,a⋅p→T,b−pz,apz,b)displaystyle M_ab^2=m_a^2+m_b^2+2left(E_T,aE_T,bsqrt 1+p_a,z^2/p_T,a^2sqrt 1+p_b,z^2/p_T,b^2-vec p_T,acdot vec p_T,b-p_z,ap_z,bright)

Looking at the transverse projection of this system (by setting pa,z=pb,z=0displaystyle p_a,z=p_b,z=0) gives:

(Mab2)T=ma2+mb2+2(ET,aET,b−p→T,a⋅p→T,b)displaystyle (M_ab^2)_T=m_a^2+m_b^2+2left(E_T,aE_T,b-vec p_T,acdot vec p_T,bright)

These are also the definitions that are used by the software package ROOT, which is commonly used in high energy physics.

Transverse mass in two-particle systems

Hadron collider physicists use another definition of transverse mass (and transverse energy), in the case of a decay into two particles. This is often used when one particle cannot be detected directly but is only indicated by missing transverse energy. In that case, the total energy is unknown and the above definition cannot be used.

MT2=(ET,1+ET,2)2−(p→T,1+p→T,2)2displaystyle M_T^2=(E_T,1+E_T,2)^2-(vec p_T,1+vec p_T,2)^2

where ETdisplaystyle E_T is the transverse energy of each daughter, a positive quantity defined using its true invariant mass mdisplaystyle m as:

ET2=m2+(p→T)2displaystyle E_T^2=m^2+(vec p_T)^2,

which is coincidentally the definition of the transverse mass for a single particle given above.
Using these two definitions, one also gets the form:

MT2=m12+m22+2(ET,1ET,2−p→T,1⋅p→T,2)displaystyle M_T^2=m_1^2+m_2^2+2left(E_T,1E_T,2-vec p_T,1cdot vec p_T,2right)

(but with slightly different definitions for ETdisplaystyle E_T!)

For massless daughters, where m1=m2=0displaystyle m_1=m_2=0, we again have ET=pTdisplaystyle E_T=p_T, and the transverse mass of the two particle system becomes:

MT2→2ET,1ET,2(1−cos⁡θ)displaystyle M_T^2rightarrow 2E_T,1E_T,2left(1-cos theta right)

where θdisplaystyle theta is the angle between the daughters in the transverse plane.
The distribution of MTdisplaystyle M_T has an end-point at the invariant mass Mdisplaystyle M of the system with MT≤Mdisplaystyle M_Tleq M. This has been used to determine the Wdisplaystyle W mass at the Tevatron.

References

• 
  J.D. Jackson (2008). "Kinematics" (PDF). Particle Data Group..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em - See sections 38.5.2 (mTdisplaystyle m_T) and 38.6.1 (MTdisplaystyle M_T) for definitions of transverse mass.


• 
  J. Beringer; et al. (2012). "Review of Particle Physics". Particle Data Group. - See sections 43.5.2 (mTdisplaystyle m_T) and 43.6.1 (MTdisplaystyle M_T) for definitions of transverse mass.