( 1) β function for the Higgs quartic coupling λ in the standard model (SM) h h. h h. vertex correction ( h 1PI. Σ y. counter term Λ Λ.
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1 funon for e Hs uar oun n e sanar moe (SM verex >< sef-ener ( PI Π ( - ouner erm ( m, ( Π m s fne Π s fne verex orreon ( PI Σ (,, ouner erm, ( reen funon ({ } Σ s fne Λ Λ Bn A n ( Caan-Smanz euaon n n ( (owes orer n n ( n n funon for : n e Fenman-' Hoof aue ( ξ s aoe n e foown auaon In e SM w s aue, ( n n SM ( n SM Cauaon for n For su uaon, a masses an exerna momenn be neee ere are! fferen onraons for e box aram w a o oo: տ ւ ւ ց ր ց Σ ( r ( 8 8 Γ( fne ( ( ( fne 8 Γ( n ( ( n ր տ (
2 ~ ~ ~ fne ( ( Σ ( Γ( 5 5 ( fne ( ( ( Γ( 5 fne 9 n, ( 8 ( ( n Σ ( ( Γ fne ( ( n, ( n Σ ( ( Γ fne ( ( n, ( n Z Z Z Z Z η Z η Z Z ~ ~ ~ η η η η η η ~ fne ( ( Z Z Σ ν ν ( 8 Z Z ( ( Z Γ( fne n, ( ( n Γ( fne ( Σ ( Γ ( ( ν ν ( ( ( fne n, ( n Γ( fne ( (
3 Z, Σ Σ Z Z ( ( ( Z, Z Z Z Γ( fne ( Γ( fne ( ( ν νρ ρ ( ( ( ρ ν ν ρ ( Z Γ( fne n, ( ( n 8 Σ Σ, Γ( fne ( ( ( ν νρ ρ (, ( Γ fne (, Σ ( Σ, Γ( fne ( Γ( fne ( Γ( fne n, ( ( 8 ρ ν ν ρ n (
4 Z \ Z ~ \ ~ Z ( ( fne ZZ Σ Z Z \ ν ρ ν ρ ( ( Γ( fne ( ( ( ZZ Γ( fne n, ZZ ( ( 8 n Σ Σ, \ ( Γ( fne ( ( (, \ Γ( fne ( Γ( ( ( fne ( n ν ρ ν ρ ν ρ ν ρ, n \ ~ \ \ ~ ~ ( ( fne (
5 Z Σ Z \ ( Z ( ( Γ( fne ( Γ( fne n, ( ( ( ( ν ν Z n Σ Σ,, \ ( ( ( ( Γ( fne ( \ ( Γ( fne ( ( ( ν ν n, ν ν n Γ( fne (, ( ( [( ] ( n SM n ( ( 8 Ex, aue-neenen exressons for e SM ( 5 SM n SM n funons of ( ( 8 ( 9 ( ( 8
6 SU ( aue smmer: marx noaon vs ensor noaon Marx noaon Conser a SU( re as an exame aue ransformaon marx U x a x a U ( ex[ θ ( ], ex( θ D ( U, U U U U D U U D U U U UU ( U U U U ( U U U ( U ( U U U UD ransforms as Infnesma ransformaon U θ θ, ( θ [, ] ( θ b b ( D ( ( b b b b ( θ θ [, ] ( a θ b b b ( θ θ θ θ [ b b b a, ] ( θ b b θ θ θ ( θ D b b D D ( ( ν ν ν ν a [ D, D ] ( b b [, ] ν ν ν ν ν ν [, ] ( [ D D a a a b b ab b a ν ν ν ν, ν ] ( ν ν ε ν ab b ν ε ν a ν ν [ D, D ] U[ D, D ] U[ D, D ] U U U U ν ν ν ν bν b r ( s ue nvaran ν ν (
7 ensor noaon V ( x ex[ θ ( x τ ], ( V ex( θ τ, ( D ( ( a τ [ ef: a-pe Cen & n-fon, aue eor of Eemenar Pare Pss, E.(. ] V ( V, ( V ( ( V ( V V Infnesma ransformaon θ τ θ τ V θ ( τ, ( V θ ( τ a a a ( ( θ [( τ ( τ ] a ( θ [( τ ( ( ( τ ] ( θ ( τ ( D ( ( ( ( ( a a ( θ [( τ ( τ ] θ [( τ ( ( τ ] a θ [( τ ( ( ( τ ] ( θ ( τ a a θ [( τ ( ( a a a a ( ] ( ( a θ [( τ ( τ ]( a ( θ [( τ ( τ ] τ θ τ θ ( τ θ ( τ ( θ ( τ ( a a a a a a θ ( ( τ θ ( ( τ θ ( ( τ θ ( τ [ ( θ τ a [( ( D ( D ( ] ( D ransforms as τ θ a a ( ] [ ( ( ]( τ ( D D ν ( ( ( ( ( ν ν ν ν ( [ ( ( ] [ ( ( ]( ([ D, D ] [ ( ν ν ( ν ν ν ν ν [( ( ν ( ] [ ( ( ] ν ν ν ν ν ν ( ( ] [( ( ( ( ] ν ν ν ( ([, ] { ( ( ν ν ν ν [( ( ν ( ν ( ]} D D ( ν ( ν ν ( [( ( ν ( ν ( ] { ( ( [( ( ( ( ]} ν ν ν ν m ( ( V [( m ( ]( V ν ν m ν m ( ransforms as ν ( ( ν ν n ν n V ( ( V V ( V V ( V V ( ( V m ν s ue nvaran n n m n m ν n n ( 7
8 SU ( aue neraons For an SU ( oube D, D D ( τ D, or euvaen, ( D D D ( D ( a a ( τ (, (, ( ( For an SU ( re, ( D ( ( ( (, ( (,,, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( [ ( ( ( ( ( ( ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( [ ( ( ( ( ( ( ] [( ( ( ( ] [ ( ( ( ( ( ( ] ( ( ( ( ( s s euvaen o a oe of SU( eneraors n as,, Noe: an ffer from ose ven n.5 b a mnus sn! ( 8
9 For a ener SU ( mue ψ, we ave ψ ( D ψ ( m m m m m ψ m ψ n n n n ( Noe: n fferena eomer, e aue fe ( s a onneon form. e aue onneon efnes a rna bune wose base sae s e saeme an sruure rou s e aue rou. n n e uer (ower nes of ψ an m ( ψ m mm ( ψ m {,, } m {, }, ( ψ,, ε ε ε ε ε ψ ( ψ ψ ψ ψ ( ( ( ε s n m r ε ( ψ s ε n( mε ε ψ r r s m r s n n m ( ψ ( ψ ( ψ ( D ψ are smmer ( ψ {,, } m m m m m ψ ( m m m m m n n m ε ε n( m ( ψ ε n( mε ψ s s r n m ( ε ( ψ ( ε ε ( ( ε ψ r n ( ψ n ψ ( n n n n {,, } n {, }, {, }, ( ψ ψ ( n m n m m r ( ψ ( ψ ( ψ ( ψ ψ ( r Smfe form: ( ψ ( D ψ ( ψ ψ ( ( ( ψ ψ aue neraons for a SU( uarue e snor uarue ( 9 ( ( ( ( ( ( ( (
10 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
11 ( D ψ ( ( ( ( ( D ( ( ( aue ( ( s s euvaen o a oe of SU( eneraors n Noe: an ffer from ose ven n.5 b a mnus sn! as (
12 enormazaon rou euaons n e re-uarue fermon ar maer (FDM moe FDM moe ef-ane e snors : (,,,,, D ( m.., D D ( m.. H H.. H -omonen snors:,,, ( ( ( ( s a Maorana snor, an e oers are Dra snors,, ( ( ( ( ( ( ( ( (..,..,..,..,..,mas s m m.. m m m ( m.. m m (.., ( ( ( ( ( ( ( ( (,aue ( ea Z ( ( ( ( ( ( ea Z ( ( ea Z [ Noe: e ouns o ffer from ose ven n.5 b a mnus sn! ] (
13 ( ( ( ( ( ( (, ( ( ( ( ( ( ( ( ( ( (, ( (,aue ( B Y ( B Y Z Z ( ( ( s ( s ea Z ( ea Z ea ( ( s ( s Z ( ea Z ( ( s ea Z ( ea ( s Z ( [( ( ] [( ( [( ( ].. ea ( Z s [ ( ( s ( s ].. Noe: e ouns o ffer from ose ven n.5 b a mnus sn! [ ] ] (
14 P P P P P P P,, ( ψ ψ ψ, ( ψ ψ, ( ψp ψ ψ P ψ ψ P ψ ( ( ( P, P ( ( ( C C (, ( ( ( ( ( ( P (, ( ( ( ( ( P ( ( ( H ( v ( v.. P P P ( ( v P P P ( P P ( P ( v P P ( P P P P v ( ( P P ( P P P P v P P ( ( P ( ( P P ( P P ( P P ( ( P P ( P P ( ( P P ( v ( P P ( P P ( P P ( ( v ( P P ( P P ( ( P P Exames for Fenman rues x x x [ a (, s u(, s e b (, s v(, s e ], ~ b ve a ( s x x x x ~ a e b e, ( ~ b e a e a ( P P a a b b ( P P b ( P P ue x (
15 ( P P a b ( P P b b ( P P ( b ue a x x x x ve, ( ~ a ve b ue [Denner, E, Han & Kɺɺubbe, Nu. Ps. B87, 7-8 (99] a ( P P ( b a b a ( P P ( b a b ( ( P P a b ( ( P P a b ( ( ( P P ( P P Conrbuon o e funon of e Hs sef-oun Conrbuon o n r[( ( P P ( P P ] r[( ( P P ] ( r[( ] x( ( x x x ( x x( x K x, K x( x ν ν r[( ] ( ν r( ( ν ( [ ( x ] ( x [ x( x ] ( K x x ( x ( ( x ( K Π, ( ( ( ( ( ( ( P P ( ( P P ( r r[( ( P P ( P P ] ( K x ( ( K Γ( x ( K x x Γ( ( ( K Γ( ( ( K Γ K Γ( x( x, x x( x ( K ( K, ( Π ( ( 5 ( ( x Γ ( K ( Γ ( x ( x x ( x Γ( fne ( fne (
16 r[ ( P P( ( P P] r[( ( P P ( P P], ( ( r ( Π ( P P P P ( ( ( ( (,, ( Π ( Π r [ ( P P( ( P P ] Π, Π (,,, ( Π ( Π ( ( ( ( P P ( P P ( Π r,, Π ( ( ( ( ( ( P P P r P (,, ( Π ( Π Π, Π,5 ( (,5 ( (, Π Π ( ( ( r P P ( ( ( P P Π, ( (, Π, ( Π (,5 Π ( r ( ( P ( P ( ( P P Π,5 ( Π, ( Π Γ( Γ( ( ( ( ( Γ( ( fne ( n ( ( n ( (
17 Conrbuon o n r {[ ( P P ( P P ] } r{ [ ( P P ( P P ] } r [ ( P P ] ( r[ ( P P ( { } P P ] ( r( P P ( ( r( ( (, Σ ( ( P P r ( P P P P ( P P ( ( r [ ( P P ( P P ] ( ( { } 8 8 Γ( ( ( fne ( ( (, 8 Γ( ( fne ( n ( ( 9, 8 ( n 9 C( P C P, C( P C P r[ ( P P C( P P C C( P P C ( P P ] r[ ( P P ( P P ( P P ( P P ] r[ ( P P ( P P ( P P ( P P ] ( r[ ( P P ( P P ] ( r ( (, Σ ( ( P P r ( P C ( P C C( P P P P C ( P P P P r[ ( C( C C( P P C ( P P ] ( ( ( ( Γ( fne (, Γ( 8 fne n ( ( 9, n 9 ( 7
18 { } { } r [ ( P P ( P P ] r [ ( P P ( P, Σ ( ( ( P P P P ( ( ( P ( P P] ( ( ( P P r r [ ( P ] { P ( P P } ( ( ( ( Γ( fne (, Γ( ( fne ( n ( ( 9, ( n 9 Σ, ( ( ( ( ( ( ( r ( P P P P P P P P ( ( ( ( r [ ( P P P P ( ] ( ( (, { } Γ( ( fne ( ( ( fne Γ( ( ( (, ( n ( n ( 8
19 P P P P ( ( C ( ( C ( P P ( C( P P C C( ( P P ( ( ( P P ( C( P P ( ( P P C ( CC ( P P C ( ( P P r[ ( P P ( P P ( P P ( P P ] [ ] [ ] ( r ( P P ( P P ( P P ( P P ( r ( P P ( P P ( r( (,5 Σ ( ( ( ( ( ( ( ( P P ( ( P ( P ( P P ( r P P ( P P ( P P ( P P ( P P r 8 8 ( (,5 8 Γ( ( (,5 8 ( n [ ] Γ( fne ( fne ( n n n, 8 ( ( 9 9 Conrbuon o n 8 ( n 9 ( 9 8 8( ( 9 9 ( 9
20 Cauaon for funons of an - - verex ( P P տ ր For, sef-ener ( PI Π Π Π,, P P Π Π (, Π,, P Π, ( Z Z Z Z, ( * ( ( Z P Z P ( ( * * Z P Z P Z P Z P * * ( ( [( Z Z P ( Z Z P ] * * [( Z Z P ( ] Z Z P * * [( Z Z P ( Z Z P ] * * ( Z Z, ( Z Z,, - ouner erm ( P P Π Π, an,, Π, are fne,, P For, sef-ener PI Π (, Π ( - ouner erm ( P P Π Π, an,,,,, are fne Π Π Π P,, P ր - - verex orreon PI տ - - ouner erm ր ( P P տ Σ an,, Σ,, are fne Σ Σ Σ Σ, (,,,,,, P,, P (
21 ( - ( - ( reen funon: ( ({ } ɶ ( (, ({ } P ɶ, ({ } P ree-eve PI oo Verex E roaaor arams ounererm xerna e orreons P P B P ( ( B P ( P P ( A, P A, P Λ ( A n ( Λ n ( P P Λ n, P A, P P,, P P P ( Λ ( P P ( A, P A P P P, n (,, Λ P P B P B P P ( ( n ( P Λ ( P P ( A, P A, P n ɶ (, ɶ (, ( A Λ n ( (, P A, P P, P, P P Λ ( P P ( A, P A, P n ( P P,, ({ } B n Λ Λ A, n,, ({ } Λ B n A Λ n,,, (,, ɶ (, n n Caan-Smanz euaons (,, ɶ (, n n ( (,, n,, n owes orer (,, n (,, n n n (,, n (, n,,, n n n n,, n n n n (
22 Cauaon for n,, an n ν ν ν (, ( ( ν ( ( x( ( x x x ( x x( x x, K x( x, ( x ( x x ( [ x ( ( x ] x ( K Π ( ( e e ( K ( ( ( ν ν e ( ( ( ( x s ( x ( ( ( ( K x( x ( Π Γ( s( x( x ( K fne s Γ( s( x( x ( K Γ( fne ( Π Z Z ( ( Z ( Π Γ( fne ( ( ( ( ν ν Γ( ( x( x ( K Π ( a ( ( Π Γ( fne ( ( ( ( ( P P ( P P a ( ( x( x ( K ν ν Γ ( P P ( ( P P ( ( P P ( P P ( ( P P, ( Π ( ( P P ( P P ( ( ( ( P P ( P P ( ( ( ( ( x x ( P P ( ( K ( P P Γ( x( x ( P P ( K, ( Π Γ( ( x x ( ( P P fne ( P P Γ fne ( K ( (
23 ( C(, ( ( C ( ( P P ( P P ( ( ( ( ( P P P P C ( P P ( ( ( P P C C ( P P C C( ( CC ( P P C C( P P C ( ( C( P P C a ( ( P P ( P P( a C( a P P C ( ( C( P P C a ( P P ( ( P P ( ( P P, Π ( ( ( (, ( Π ( P P Γ( fne ( C( P P C ( ( P P ( P ( P ( P P ( ( ( ( C( P P C ( (, Π, ( Π ( P P Γ( fne ( P P ( ( ( ( P P Π (, Π, (, ( ( ( Π Γ( ( P P fne ( ( C( C, P P C( P P C Π ( ( Π, (, ( Π ( ( P P Γ fne ( ( ( P P ( P P ( ( Π, ( ( (, ( Π P P Γ( ( fne ( ( C( C P P C( P P C ( ( (
24 ( ( Π Π (,, Γ s ( fne ( Γ( fne ( Γ( (, fne ( n ( (, ( n ( Π ( Π,, Γ( s ( fne Γ( ( ( fne, ( n Cauaon for,, n an n Π ( ( e ( ( ( ν ν ( Π Γ( s Π Z Z ( ( fne Z ( Π ( s Γ( fne ( ( s ( ( ( ν ν Π, ( ( ( ( ( ν ν,, ( Π Γ( fne ( Π, ( ( ( ( ν ν,, ( Π ( Γ fne ( Π ( ( ( Π Γ( ( P P fne ( ( ( P P ( P P ( (
25 Π ( ( Π ( P ( Γ( P fne ( ( P P ( ( P P ( Π Π ( ( Π Γ ( ( P P fne ( s ( ( P P P P ( ( ( ( s ( s s Π ( Π ( Π,, Γ( ( s s fn ( e 5 Γ( fne ( 5 Γ( 5, fne n ( (, 5 n ( Π ( Π Γ,, ( ( s s ( 5 Γ( fne (, 5 n f ne Cauaon for n an n ( P P ( P P ( P P Σ \ ( e, ( P P ( ( ( s ( P s s P P ( ( ( ( P P Γ( fne ( P Σ Z, \ Z ( s P P Γ( ( fne ( ( 5 ( s P P ( (
26 Σ ց ( (, ( ( ( P ( ( P P ( P ( P P Γ fne (, Σ \ P P P ( ( ( P P P ( P P ( ( ( ( ( P P Γ fne ( Σ, \ \ Σ, P P P P ( ( C ( ( C ( P P ( C( P P C C( ( P P( ( ( P P ( C( P P P P ( ( C ( CC ( P P C ( ( P P ( Σ ( \, P P ( ( ( P P ( P P P ( ( ( P P P Γ( ( fne ( ( ( P P ( ( P P ( \ ( ( Σ, P P P P ( ( ( ( ( Γ P P ( P P ( ( P P fne ( Σ ւ \, ( P P P ( ( ( C P C C( P P C ( ( ( P P P P Γ( ( fne ( (
27 Z Σ, ( s \ Z ( s s P P P ( ( ( ( ( ( P P ( ν ν P Γ( fne ( Σ Z, Z \ P P ( ( ν ν P P P P ( ( ( ( Γ( fne ( Σ, ւտ ( ( P P ( ( P P ( P P Γ fne ( ( ( ν ν Σ, րց ( ( P ( ( ( P P ( P P P Γ( ( fne ( ν ν Σ,, ( Σ,, Γ( ( s ( s ( s fne Γ( ( 8 8 Γ( 8 ( Γ( fne ( 8 fne Γ( fne n ( ( n fne ( 7
28 Σ ( Σ,,,, Γ( s ( s ( ( Γ( ( ( s fne Γ( fne n ( ( n fne Exressons for an SM [ ( ] n n n,, n n n n 5 [ ( ] ( ,, n n n n 9 5 [ ( ] ( 9 9 Conrbuon o e funon of e o Yuawa oun V, n n n,,, V ( n n ( ( 8
29 5 Conrbuons o funons of aue ouns SU( : C( r( ( ( ( C r [ C( ], [ C( ] 5 Summar s SM SM SM SM SM s,,,, SM SM 9 SM 7s,, s SM 9 8 ( ( SM s 8,, 8 8( (, ( s resu s onssen w e P@E resu (9
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