# £ With  real, the field  vanishes and our Lagrangian reduces to introducing a MASSIVE Higgs scalar field, , and “getting” a massive vector gauge field.

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£ With  real, the field  vanishes and our Lagrangian reduces to introducing a MASSIVE Higgs scalar field, , and “getting” a massive vector gauge field G  Notice, with the  field gone, all those extra , , and  interaction terms have vanished Can we employ this same technique to explain massive Z and W vector bosons?

Let’s recap: We’ve worked through 2 MATHEMATICAL MECHANISMS for manipulating Lagrangains Introducing SELF-INTERACTION terms (generalized “mass” terms) showed that a specific GROUND STATE of a system need NOT display the full available symmetry of the Lagrangian Effectively changing variables by expanding the field about the GROUND STATE (from which we get the physically meaningful ENERGY values, anyway) showed The scalar field ends up with a mass term; a 2 nd (extraneous) apparently massless field (ghost particle) can be gauged away. Any GAUGE FIELD coupling to this scalar (introduced by local inavariance) acquires a mass as well!

Now apply these techniques: introducing scalar Higgs fields with a self-interaction term and then expanding fields about the ground state of the broken symmetry to the SU L (2)×U(1) Y Lagrangian in such a way as to endow W, Z s with mass but leave  s massless. + 0+ 0 These two separate cases will follow naturally by assuming the Higgs field is a weak iso-doublet (with a charged and uncharged state) with Q = I 3 + Y w /2 and I 3 = ±½  Higgs = for Q=0  Y w = 1 Q=1  Y w = 1 couple to EW U Y (1) fields: B 

 Higgs = with Q=I 3 + Y w /2 and I 3 = ±½ Y w = 1 + 0+ 0 Consider just the scalar Higgs-relevant terms £ Higgs with † †† not a single complex function now, but a vector (an isodoublet) Once again with each field complex we write  + =  1 + i  2  0 =  3 + i  4  †    1 2 +  2 2 +  3 2 +  4 2 £ Higgs ††† ††††

U = ½  2  †  + ¹/ 4 †  ) 2 L Higgs ††† †††† just like before:  1 2 +  2 2 +  3 2 +  4 2 =  2  2 Notice how  1 2,  2 2 …  4 2 appear interchangeably in the Lagrangian invariance to SO(4) rotations Just like with SO(3) where successive rotations can be performed to align a vector with any chosen axis,we can rotate within this  1 -  2 -  3 -  4 space to a Lagrangian expressed in terms of a SINGLE PHYSICAL FIELD

Were we to continue without rotating the Lagrangian to its simplest terms we’d find EXTRANEOUS unphysical fields with the kind of bizarre interactions once again suggestion non-contributing “ghost particles” in our expressions. So let’s pick ONE field to remain NON-ZERO.  1 or  2  3 or  4 because of the SO(4) symmetry…all are equivalent/identical might as well make  real! + 0+ 0  Higgs = Can either choose v+H(x) 0 v+H(x) or But we lose our freedom to choose randomly. We have no choice. Each represents a different theory with different physics!

Let’s look at the vacuum expectation values of each proposed state. v+H(x) 0 v+H(x) or Aren’t these just orthogonal? Shouldn’t these just be ZERO ? Yes, of course…for unbroken symmetric ground states. If non-zero would imply the “empty” vacuum state “OVERLPS with” or contains (quantum mechanically decomposes into) some of  + or  0. But that’s what happens in spontaneous symmetry breaking: the vacuum is redefined “picking up” energy from the field which defines the minimum energy of the system.

0 1 = v a non-zero v.e.v.! This would be disastrous for the choice  + = v + H(x) since  0|  +  = v implies the vacuum is not chargeless! But  0|  0  = v is an acceptable choice. If the Higgs mechanism is at work in our world, this must be nature’s choice.

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