Cut off dependence in a few basic quantities this into redefinition: A ren. = A 0 (4.2) 1 Z ψ. g 0 Z ψ Z Z T ψψa. g ren. =

Transcription

1 Chapter 4 Renormalization Cut off dependence in a few basic quantities this into redefinition: L t a t aµν + ψ µν0 0 0 (ip 0 m 0 )ψ 0 (4.) 4 A ren. A 0 (4.) ZA ψ ren. ψ 0 Z ψ (4.) Z g ψψa ren. ψ ren. γ µ T ψ ren. A µren. g 0 ψ 0 γ µ ψ 0 A µ0 (4.4) g ren. g 0 Z ψ Z Z T ψψa A (4.5) Z ψψ ψ ren. ψ ren. m ren. ψ 0 ψ 0 m 0 (4.6) m 0 Z ψ m ren. (4.7) Z ψψ The wave-function renormalizations are absorbed into normalizing the particle states. The renormalized couplings are physical particles matrix of renormalized states. Indeed, these are the parameters define the theory, at least perturbatively. We fix these to experiment. Then, other quantities must be expressed finite terms using them (and we remove the cut off). Notice that in a gauge theory, we must use a gauge-invariant regulator to get the simplicity. Otherwise, there are A (no derivatives res is like), AσA, AAA, A 4, etc. coupling appearing. We may set them to zero and entree conspiracies to record gauge symmetry (throwing their a gauge invariant regulation). 0

2 Z A Γ A g ren. g 0 vector pt. (4.8) Z A g 0 vector 4 pt. g ΓA 4 (4.9) A Γ ψψa Z Z ψ (4.0). Z (4.) Z a universal (Ward identity) (4.) Z νma In Abelian theory are equal. In N.A. theory, not generally (+ gauge dependent). Figure 4.: Abelian Theory. To clarify this structure, let s work an example. Correlation function of Bare ψ, ψ (inverse propagator). bare + Figure 4.: Bare. Λ d 4 k ( i)(i)γ k + m)γ µ bare i(/ m) + (ie) µ (/p / p (π) 4 k [(p k) m ] (4.)

3 CHAPTER 4. RENORMALIZATION k 0 k i k 0 E Figure 4.: Bare Diagram. Renormalized: bare + Figure 4.4: Bare. bare iz(/ p m R ) + e i (no cut off) (4.4) k bare + p p k Figure 4.5: Renormalized. Λ d 4 k γ µ (/ p k / + m)γ µ bare i(/ p m) + ie (4.5) (π) 4 k [(p k) m ] }{{} new Eulidec S π (4.6) ie Λ (/ p k) / + 4m bare i(/ p m) + dωdkk (4.7) 8π k [(p k) m ] We are going to keep only the divergent part. We shall subtract by normalizing the log divergence (only) at µ. linear + log divergent (4.8)

4 Λ log : /p + 4m dωdkk (4.9) k [(p k) m ] Λ dωdkk [ /p + 4m] k 4 (4.0) ln Λ[ /p + 4m] (4.) linear : Λ dωdkk /k k 6 pk Λ Λ Λ Λ Putting it all together (as discussed): drdkk /k k [(p k) m ] dωdkk /k k pk + p + m dωdkk k ( /k pk k ( p ) + m + k k (4.) (4.) (4.4) dωdkk /k pk k 6 (4.5) dω/kk µ dωk µ k ν γ ν (4.6) γk (4.7) 4 Λ /p dkk (4.8) k 4 /p ln Λ (4.9) e Λ bare i(/p m 0 ) i ln [ /p + 4m γπ 0 + /p] µ (4.0) iz ψ (/p m r ) (4.) Z ψ Finite matrix d t at p µ e Λ ln 8π µ (4.) Check Ward identify: e Λ m R (µ) m 0 [ ln ] (4.) 8π µ bare Λ ( ie) γ ν (/p /k)γ µ (/p /k)γ ν (i )( i) ieγ µ + k ((k p) ) ie Λ ieγ µ dωdkk /kγ µ /k 8π k 6 (4.4) (4.5)

5 4 CHAPTER 4. RENORMALIZATION µ p k p k p k p Figure 4.6: Ward Identity. dk k 5 ieγ µ ie γ α γ µ γ α 8π 4 k 6 (4.6) ie ieγµ γ µ ln Λ 8π (4.7) Z e + ln Λ T ψψa 8π µ (4.8) e Λ Z T ln ψψa 8π µ (4.9) Check gauge-invariance of m R (µ): Figure 4.7: Added Stuff. added stuff Λ d 4 k αk µ k ν γ µ (/ p k / + m)γ ν (π) 4 k k (p k) (4.40) need p / m (4.4) Λ dkk log : a ( ( )/p + 4m) (4.4) 8π k Λ dωdkk µ k ν ( )(γ µ /kγ ν (pk)k ) linear : (4.4) 8π k 8 Λ dωdk( )/k(pk)k ) (4.44) 8π k 6 Λ pk 4 / dk p + m (4.45) 8π k 4 / Effective mass: m R (µ ) equivalent to m R (µ ) if same m 0, Λ

6 5 Renormalization group: N.B. : e µ too. Physical significance: m R (µ ) m R (µ ) (4.46) e Λ e Λ 8π ln 8π ln µ µ e Λ 8π ln µ e Λ ln µ m R (µ ) (4.47) m R (µ ) 8π! e µ ln 8π µ (4.48) m R (µ) e ln µ 8π (4.49). With normalization at µ, no large loop for processes with parameter a µ.. Internally, not rotate for µ to Ω. Anticipate: dg(µ) b 0 g (b 0 > 0) (4.50) d ln µ d g t 0 (4.5) dt µ b 0 ln (4.5) g (µ) g (µ 0 ) µ 0 b e 0 g (µ) µ e b 0 g (µ 0 ) (4.5) µ 0 b 0 g (µ 0) ]e b 0 g (µ) µ [µ 0 e (4.54) Take µ, dimensional transmutation perturbative coupling. Lattice: Given, determine µ. Measure M fixed, independent of µ as g (µ) a µ g 0. Good news: Check approach to continuous limit. Bad news: Decreasing g(µ) required exponentially bigger lattice. With opposite sign for b 0, coupling blows up as µ (a 0), no limiting theory guaranteed.

7 6 CHAPTER 4. RENORMALIZATION Running coupling and dimensional transmutation: Figure 4.8: Running Coupling Bare charge 0 dg(µ) d ln µ d g b 0 g (b 0 > 0) (4.55) b 0 (4.56) d ln µ µ + b 0 ln (4.57) g (µ) g (µ 0 ) µ 0 Identify from µ limit [finite]. More accurate: µ b e 0 g (µ) e b 0 g (µ 0 ) µ 0 (4.58) b 0 g (µ 0 ) µ 0 e b 0 g (µ) µe (4.59) dg d ln µ 5 b 0 g + b g (4.60) d g b 0 b d ln µ g (4.6) du b b 0 dt u (4.6) u 0 b 0 t (4.6) du b b 0 dt b 0 t (4.64) b b 0 u b 0 t ln t (4.65)

8 7 µ b µ b 0 ln ln ln (4.66) g (µ) µ 0 b0 µ 0 µ b b 0 ln ln (4.67) µ 0 b0 b 0 g b ln µ 0 ln µ + ln (4.68) b 0 g b 0 b 0 g b b µ 0 0 g (µ) [ ] b µe 0 finite as µ (4.69) g (µ) Relation to lattice (fundamental): Hold µ 0 fixed (reference mass) a(g) e b 0 g [ µ b ] b 0 g µ 0 (4.70) Small a corresponds to small g. N.B.: Exponential dependence. Test: Physical masses stay constant as g 0. Explicitly: Measures correlation fall-off e λn, where n is the lattice sparcing. λ This is e ML with L an for M. a e λn λ a an e (4.7) b [ ] b 0 µ 0 e b 0 g g λ (4.7) Fixing µ 0, we see λ 0 as g 0. Large correlation lengths, variating of lattice artifacts. Alternatively, am 0 without asymptotic freedom we would be lost here, in tertudation theory. Long correlation lengths can occur at critical points. To nail things down let s look at this more closely: so we have, g(µ) g C dg d ln µ b 0 (g g C ) (4.7) ln(g g C ) b 0 ln µ (4.74) g(µ) g C ( µ 0 ) b 0 g(µ 0 ) g C µ (4.75) g(µ) g C µ µ 0 ( ) b 0 g(µ 0 ) g C (4.76) λ e a na (4.77) λ M (4.78) a e nλ

9 8 CHAPTER 4. RENORMALIZATION β (g) Fixed Point g Figure 4.9: Fixed Point a(g(µ)) ( µ µ0 g(µ 0 ) g C g(µ) g C ) b 0 (4.79) λ M fixed λ 0, a. Alternatively, IR fixed points, but these do not a M support non-zero masses. Antiscreening as paramagnetism: Vacuum polarization: In reality E.D ɛf oi F oi (4.80) B.H µ F ij F ij (4.8) ɛµ (4.8) either is varying g. Antiscreening (ɛ < ) paramagnetism (µ > ) magnetism is easier to think about, since particles do not get produced. E class. B (4.8) g But zero point energies, hω (Bose), hω (Fermi) would like to throw it out, but it is field-dependent and cut off dependent. Interpretation using running coupling: Wilson approach: Contribution of modes between µ 0 + µ (say µ > µ 0 ) is

10 9 Result and physical interpretation: µ n ln B µ 0 (4.84) E g (µ 0 ) }{{} B (4.85) cut off at µ 0 µ ( n ln ) B g (µ) µ 0 }{{} (4.86) cut off at µ n b 0 (4.87) n [ {T(R 96π 0 ) T(R + T(R ))}] + [ 96π { T(R + 8T(R ))}] (4.88) To aid in physical interpretation, pick a particular direction (e.g., τ ) in internal space for B, and pretend it is electromagnetism. First term from orbital diamagnetism (Lambda). Note for polarizations, for fermion. N.B. This applies to 0 point energy of virtual particles. Real particles do not have this. Second term from spin paramagnetism (Pauli). Note g factor (also for vectors), for fermion, ratio as in solid state. S 4 (4.89) S AF comes from gluon paramagnetism. Reference: F. W., RMP 7 S85 (Ceturean Issue). Learning from QED: With these insights, we could extract the non-abelian β function from the Abelian one. Numerical value: QCD : T(R ) f (4.90) T(R ) (4.9) f f b 0 [+ + ( + 8 )] 96π [66 4f] [ f] (4.9) 96π 6π

11 40 CHAPTER 4. RENORMALIZATION b 0 > 0 for f 6. Gluons heavily dominate for f. Application for unified theories: SU() 8F + 66 (4.9) SU() F( 4 }{{}}{{} 4 ) + 44 fermion factor number of doublets }{{} T doublet 8F + 44 Higgs (4.94) U() normalized charge : (4.95) 60 q F( + ( 4 + 6( ) + ( 6) )) (4.96) 60 equations: constraint, t u, g u W.S.B. F 8F Higgs (4.97) 0 b i t u + (4.98) g u g Note: unchanged by complete multiplets (no F) g g i (b b )t u (4.99) b b b b g g g g Fixing scale: note exponential dependence (4.00) g t g u (4.0) b b b gu b g b g u b g b g b b g g b + b b g ( + ) + ) (4.0) g g b b ( g u b b g g b b

12 4.. SUSY MODIFICATIONS 4 4. SUSY Modifications. Gauge Modification. Higgs Modification R R (4.0) 6R 4R (4.04) 9 R }{{}}{{} factor (4.05) group theory chialitic }{{} majorna (4.06) 44 6 (4.07) doublets (4.08) fermion partner (4.09) ( + 6) like 6 Higgs doublet }{{} (4.0). Matter Modification by same reasoning (4.) Example: N 4 gauge field. 4 adjoint fermions. 6 Higgs. diamagnetism : 8 paramagnetism : 8 8 (4.) Numerical aspects: SUSY: : b b b b + 75 (4.) b b b b (4.4) b b 6 + 6( + )

13 4 CHAPTER 4. RENORMALIZATION 6 (4.5) (4.6) Raising scale looks small but appears in exponential (p decay). SUSY: b + b b b F + F (4.7) b + b b b b : (very little remaining) F + F (4.8) 44 8F 9 6 F (4.9) Other applications: m b from minimal SU(5) (4.0) m τ Vacuum instability and Colman Weinberg: ϕ ϕ 4 λ( ) (4.) µ interpret as energy from extra nodes. Too many fermions. moment µ, field strength ϕ (as in our magnetic calculation). Weak Coupling Massless Figure 4.0: Weak Coupling Massless

14 4.. SUSY MODIFICATIONS 4 g g SU() b SU() b U() b g g t u g u 0 Table 4.: Running of coupling SM MSSM 66 8F 44 8F 8F 0 54 F 6 F F 9 5 nearly flat b b b b agrees with expectation g g b b larger scale b g b g b b important for p-decay close to Planck scale families and extra stuff appear here bigger coupling desirable? Running of coupling summary sheet Other applications: N 4 SUSY, glue, 4 Majorna fermions, 6 scalars. Dia: count DOF. Para: 4 fermion, β 0 Normal SU(5): Normalizes to Vacuum instability m b at unification (4.) m τ m b m τ (4.) µ λ( )ϕ 4 ϕ negates (AF) contribution λ( )ϕ 4 from fermions (4.4) µ 0 µ 0 Variational interpretation? + < ϕ > a k, a k (4.5) mix mode in k dependent manner. C.F. our magnetic field calculation. Colman-Weinberg: Electroweak breaking through leavry stop. t instability ulicrod when tˆ comes in. PQCD: Simplest case: e + e hadrons

15 44 CHAPTER 4. RENORMALIZATION Figure 4.: Vacuum Instability V( φ ) Weak Coupling m0 φ pass through zero Figure 4.: Spontanous Symmetry Breaking σ(p, g (µ), µ) σ(λp, g (µ), µ) σ(λp, g(λµ), λµ) (4.6) σ(p, g(λµ), µ) λ }{{} (4.7) expand perturbutively ( + ) Figure 4.: Expand Perturbatively. Potential problem from soft and collisions divergence. They do not occur in inclusive qualities. Jet phenomena: Observation, heuristic explanation, hard scattering, rate. Sterman- Weinberg comes, energy bite. Stiull no scale same argument. Extremum pattern. Hadmic gets over 6 nodes of reguitode.

16 4.. SUSY MODIFICATIONS 45 Figure 4.4: W Exchange. OPE, DIS: W exchange. Nodes up to k m ω. Simplify to + Different Q #s Figure 4.5: Simplify. Operator versions: X small x J(x)J( ) C i (x)o i (o) (4.8) Singularities of C ordered by dimensions of O i keep low dimensions. C i (x,g,µ) includes renormalization of O. C i (x,g,µ) [µ + β(g) γ θ (λ)]c i 0 (4.9) µ g C i (λ x,g(µ 0 ),µ 0 ) C i (x,g(λµ 0 ))e g dg γ g θ 0 β ḡ c 0 C i (x,g(λµ 0 ))( ) b 0 (4.0) g One can also have operators mixing. For DIS, light cone singularities (tracing quarks and glouns) are important. x J µ ( )J ν x ( ) but not individual component. x 0 C (i) O µνα α n (x) (0)x α x αn (4.)

17 46 CHAPTER 4. RENORMALIZATION Twist dimension spin ( pq q ) is what appears. B j limit: pq q < O > p µ ν p (symmetric) (4.) g x or (4.) g g g (4.4) finite ( ). x QCD has twist tor families of operators (respectively chirality). αn g (L) γ α α q (L) D + n, J n (4.5) G a µα αn α G a D 4 + n, J n (4.6) α nµ First family includes nonsinglets, currents. T µν also appears (but not by itself). Twist J governs J th moment of appropriate structure function. Many testable predictions: Parton model sum rules and g correlations. CG relation and g correlations. E: Max with o th order correlations. B j scalar deviations.