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What Is the Higgs Boson? José Miguel Figueroa-O’Farrill School of Mathematics

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What Is the Higgs Boson? José Miguel Figueroa-O’Farrill School of Mathematics What is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February 2013 (Innovative Learning Week) Suddenly, last summer... Physics: it’s where the action is! Newton’s second law inertial mass d2x F = ma = m dt2 Newton’s second law inertial mass d2x F = ma = m dt2 x(t) is the path followed by the object dx v = dt x(t) Newton’s first law F = 0 = object moves at a ) constant velocity Conservative forces F = V −r V (x) potential function Conservative forces F = V −r V (x) potential function 1 2 Energy E = m v + V (x) is conserved 2 | | E(t2) E(t1) E(t1)=E(t2) derived Newton’s second law from a principle! derived Newton’s second law from a principle! Nature uses as little as possible of anything. Johannes Kepler (1571-1630) {z } The principle of least| action lagrangian t2 action I = 1 m v 2 V (x) dt 2 | | − Zt1 x(t2) x(t1) {z } The principle of least| action lagrangian t2 action I = 1 m v 2 V (x) dt 2 | | − Zt1 The paths with the x(t2) least action x(t1) satisfy Newton’s second law... and conversely! {z } The principle of least| action lagrangian t2 action I = 1 m v 2 V (x) dt 2 | | − Zt1 The paths with the x(t2) least action x(t1) satisfy Newton’s second law... and conversely! Actions change the way we think about Physics!!! Symmetries A transformation x x0 is called a symmetry 7! if the action does not change : I(x)=I(x0) A transformation x x0 is called a symmetry 7! if the action does not change : I(x)=I(x0) Symmetries take a path with least action to another path with least action (perhaps even to the same path!). So symmetries take solutions of Newton’s equation to solutions of Newton’s equation. For example, a translation x x + a • 7! is a symmetry provided that V (x + a)=V (x) For example, a translation x x + a • 7! is a symmetry provided that V (x + a)=V (x) a rotation x Rx • 7! is a symmetry if V (Rx)=V (x) i.e., V (x)=f( x ) | | Of course, even though the action has symmetries, solutions of Newton’s equations may break some or all of that symmetry. Of course, even though the action has symmetries, solutions of Newton’s equations may break some or all of that symmetry. Consider free motion: V (x)=0 The trivial trajectory x(t)=x0 is invariant under arbitrary rotations about the point x0 x0 If we give it a little push so that that particle is now moving at constant (nonzero) velocity x(t)=x0 + tv we have broken the rotational symmetry to those with axis of rotations v Summary Summary ➡ Dynamical systems can be described by an action principle Summary ➡ Dynamical systems can be described by an action principle ➡ Solutions have least action Summary ➡ Dynamical systems can be described by an action principle ➡ Solutions have least action ➡ Symmetries of the action take solutions to solutions Summary ➡ Dynamical systems can be described by an action principle ➡ Solutions have least action ➡ Symmetries of the action take solutions to solutions ➡ A given solution might break some (or all) of the symmetry of the action Relativistic fields And Maxwell said... and there was light! In vacuo, Maxwell equations are @B E =0 E = r · r ⇥ − @t 1 @E B =0 B = r · r ⇥ c2 @t where E(x,t) electric field B(x,t) magnetic field c speed of light An immediate consequence is that the electric and magnetic fields obey the (massless) wave equation : 1 @2E 1 @2B 2E =0 2B =0 c2 @t2 r c2 @t2 r (In fact, that is what light, radio waves, X- rays,... actually are.) The wave equation can also be derived from an action. Let us consider the scalar wave equation: 1 @2φ 2φ =0 c2 @t2 r Solutions of the wave equation minimize the action: 1 @ 2 I = dt d3x φ 2 c2 @t − |r | Z Z ✓ ◆ ! This action has “hidden” symmetries which mix space and time! In fact, if we introduce a 4-vector : µ =0, 1, 2, 3 xµ =(x0, x) x0 = ct the action becomes @ @ I = d4x ⌘µ⌫ = d4x⌘µ⌫ @ @ φ @xµ @x⌫ µ ⌫ µ,⌫=0,1,2,3 Z X Z 1000 −0100 @ ⌘ = 0 1 @µ := where 0010 @xµ B 0001C B C @ A The action has relativistic symmetry: µ µ ⌫ T φ(x) φ(x0) (x0) = ⇤ x ⇤ ⌘⇤ = ⌘ 7! ⌫ Lorentz transformations xµ coordinates in Minkowski spacetime. “The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.” Hermann Minkowski (1908) Maxwell’s equations are also relativistic! 0 E1 E2 E3 E1 0 B3 B2 Fµ⌫ = 0− − 1 Fµ⌫ = F⌫µ E2 B3 0 B1 − − − B E B B 0 C B− 3 2 − 1 C @ A Maxwell’s equations are also relativistic! 0 E1 E2 E3 E1 0 B3 B2 Fµ⌫ = 0− − 1 Fµ⌫ = F⌫µ E2 B3 0 B1 − − − B E B B 0 C B− 3 2 − 1 C @ A µ⌫ Maxwell’s ⌘ @µF⌫⇢ =0 equations () {@µF⌫⇢ + @⌫ F⇢µ + @⇢Fµ⌫ =0 Maxwell’s equations are also relativistic! 0 E1 E2 E3 E1 0 B3 B2 Fµ⌫ = 0− − 1 Fµ⌫ = F⌫µ E2 B3 0 B1 − − − B E B B 0 C B− 3 2 − 1 C @ A µ⌫ Maxwell’s ⌘ @µF⌫⇢ =0 equations () {@µF⌫⇢ + @⌫ F⇢µ + @⇢Fµ⌫ =0 They can also be derived from an action, but for an “electromagnetic potential”! The Bianchi identity @µF⌫⇢ + @⌫ F⇢µ + @⇢Fµ⌫ =0 can be solved by Fµ⌫ = @µA⌫ @⌫ Aµ − for some Aµ =(φ, A) The Bianchi identity @µF⌫⇢ + @⌫ F⇢µ + @⇢Fµ⌫ =0 can be solved by Fµ⌫ = @µA⌫ @⌫ Aµ − for some Aµ =(φ, A) which is only determined up to a gauge transformation A A + @ ✓ µ 7! µ µ The Bianchi identity @µF⌫⇢ + @⌫ F⇢µ + @⇢Fµ⌫ =0 can be solved by Fµ⌫ = @µA⌫ @⌫ Aµ − for some Aµ =(φ, A) which is only determined up to a gauge transformation A A + @ ✓ µ 7! µ µ Relativistic fields correspond to particles and Aµ describes the photon The Maxwell action (for A µ !) 4 µ⇢ ⌫ I = d x ⌘ ⌘ Fµ⌫ F⇢ Z is unchanged under ➡gauge transformations A (x) A (x)+@ ✓(x) µ 7! µ µ ➡Lorentz transformations 1 ⌫ A (x) ⇤− A (⇤x) µ 7! µ ⌫ Mass For relativistic theories, the mass appears as quadratic terms in the action : I = d4x 1 ⌘µ⌫ @ @ φ 1 m2φ2 2 µ ⌫ − 2 Z For relativistic theories, the mass appears as quadratic terms in the action : I = d4x 1 ⌘µ⌫ @ @ φ 1 m2φ2 2 µ ⌫ − 2 Z For relativistic theories, the mass appears as quadratic terms in the action : I = d4x 1 ⌘µ⌫ @ @ φ 1 m2φ2 2 µ ⌫ − 2 Z I describes a massive scalar field obeying the Klein-Gordon equation : µ⌫ 2 ⌘ @µ@⌫ φ + m φ =0 or 1 @2φ + 2φ + m2φ =0 −c2 @t2 r There is also a massive version of Maxwell equations, described by the Proca action: I = d4x 1 ⌘µ⇢⌘⌫F F + 1 m2⌘µ⌫ A A − 4 µ⌫ ⇢ 2 µ ⌫ Z There is also a massive version of Maxwell equations, described by the Proca action: I = d4x 1 ⌘µ⇢⌘⌫F F + 1 m2⌘µ⌫ A A − 4 µ⌫ ⇢ 2 µ ⌫ Z which yields the Proca (=massive Maxwell) equation: µ⌫ 2 ⌘ @µF⌫⇢ + m A⇢ =0 There is also a massive version of Maxwell equations, described by the Proca action: I = d4x 1 ⌘µ⇢⌘⌫F F + 1 m2⌘µ⌫ A A − 4 µ⌫ ⇢ 2 µ ⌫ Z which yields the Proca (=massive Maxwell) equation: µ⌫ 2 ⌘ @µF⌫⇢ + m A⇢ =0 The Proca action is no longer gauge invariant! Summary Summary ➡ The wave equation is relativistic Summary ➡ The wave equation is relativistic ➡ Maxwell’s equations are also relativistic Summary ➡ The wave equation is relativistic ➡ Maxwell’s equations are also relativistic ➡ They are both obtained from Lorentz- invariant actions defined on Minkowski spacetime Summary ➡ The wave equation is relativistic ➡ Maxwell’s equations are also relativistic ➡ They are both obtained from Lorentz- invariant actions defined on Minkowski spacetime ➡ The Maxwell action depends on a field Aμ which is determined only up to gauge transformations Summary ➡ The wave equation is relativistic ➡ Maxwell’s equations are also relativistic ➡ They are both obtained from Lorentz- invariant actions defined on Minkowski spacetime ➡ The Maxwell action depends on a field Aμ which is determined only up to gauge transformations ➡ Mass is the quadratic term (without derivatives) in the action The Higgs mechanism Actions can be combined in order to “couple” fields. Actions can be combined in order to “couple” fields. The abelian Higgs model couples the Maxwell field to a complex scalar: Φ = φ1 + iφ2 Actions can be combined in order to “couple” fields. The abelian Higgs model couples the Maxwell field to a complex scalar: Φ = φ1 + iφ2 d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z 2 1 µ⇢ ⌫ F = 2 ⌘ ⌘ Fµ⌫ F⇢ 2 µ⌫ DΦ = ⌘ D ΦD Φ DµΦ = @µΦ + ieAµΦ | | µ ⌫ Actions can be combined in order to “couple” fields. The abelian Higgs model couples the Maxwell field to a complex scalar: Φ = φ1 + iφ2 d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z 2 1 µ⇢ ⌫ F = 2 ⌘ ⌘ Fµ⌫ F⇢ 2 µ⌫ DΦ = ⌘ D ΦD Φ DµΦ = @µΦ + ieAµΦ | | µ ⌫ electric charge d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z The first term is just the Maxwell action, which is still unchanged under gauge transformations A A + @ ✓ µ 7! µ µ d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z The first term is just the Maxwell action, which is still unchanged under gauge transformations A A + @ ✓ µ 7! µ µ The second term also remains unchanged if, in addition, ie✓ Φ e− Φ 7! d4x 1 F 2 + 1 DΦ 2 V (Φ) − 2 2 | | − Z The first term is just the Maxwell action, which is still unchanged under gauge transformations A A + @ ✓ µ 7! µ µ The second term also remains unchanged if, in addition, ie✓ Φ e− Φ 7! And this is also a symmetry of the third term, provided that V (Φ)=f( Φ ) | | The Maxwell equation is modified by an “electric” current term : ⌘λµ@ F = J = 1 e(Φ@ Φ Φ@ Φ) e2A Φ 2 λ µ⌫ ⌫ 2 ⌫ − ⌫ − ⌫ | | The Maxwell equation is modified by an “electric” current term : ⌘λµ@ F = J = 1 e(Φ@ Φ Φ@ Φ) e2A Φ 2 λ µ⌫ ⌫ 2 ⌫ − ⌫ − ⌫ | | J⌫ =(⇢, j) Potential V (Φ)=f( Φ ) | | v Φ | | Potential unstable critical point V (Φ)=f( Φ ) Φ =0 | | | | v Φ | | Potential unstable critical point
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