Matching heavy-quark fields in QCD and HQET Andrey Grozin - PowerPoint PPT Presentation

Matching heavy-quark fields in QCD and HQET Andrey Grozin A.G.Grozin@inp.nsk.su Budker Institute of Nuclear Physics Novosibirsk

HQET = non-abelian Bloch–Nordsieck Electron + soft photons (real and virtual) Bloch, Nordsieck (1937) see Bogoliubov, Shirkov § 46

HQET = non-abelian Bloch–Nordsieck Electron + soft photons (real and virtual) Bloch, Nordsieck (1937) see Bogoliubov, Shirkov § 46 Heavy quark + soft gluons and light quarks, antiquarks (real and virtual): HQET (1990)

HQET = non-abelian Bloch–Nordsieck Electron + soft photons (real and virtual) Bloch, Nordsieck (1937) see Bogoliubov, Shirkov § 46 Heavy quark + soft gluons and light quarks, antiquarks (real and virtual): HQET (1990) QED electron propagator in the IR ≈ Bloch–Nordsieck electron propagator, see Bogoliubov, Shirkov § 50.3

HQET = non-abelian Bloch–Nordsieck Electron + soft photons (real and virtual) Bloch, Nordsieck (1937) see Bogoliubov, Shirkov § 46 Heavy quark + soft gluons and light quarks, antiquarks (real and virtual): HQET (1990) QED electron propagator in the IR ≈ Bloch–Nordsieck electron propagator, see Bogoliubov, Shirkov § 50.3 Here we shall discuss this relation in detail, including corrections

HQET p = mv + k

HQET p = mv + k QCD: Q HQET: Q v = / vQ v � 1 Q v iv · DQ v + 1 � L = ¯ 2 m ( O k + C m ( µ ) O m ( µ )) + O m 2 Matching S -matrix elements Reparametrization invariance v → v + δv

HQET p = mv + k QCD: Q HQET: Q v = / vQ v � 1 Q v iv · DQ v + 1 � L = ¯ 2 m ( O k + C m ( µ ) O m ( µ )) + O m 2 Matching S -matrix elements Reparametrization invariance v → v + δv QCD operator � 1 � O ( µ ) + 1 O ( µ ) = C ( µ ) ˜ � B i ( µ ) ˜ O i ( µ ) + O 2 m m 2 i Matching on-shell matrix elements

Q via Q v , . . . Here we shall consider Q Matrix elements of Q are not measurable, why bother?

Q via Q v , . . . Here we shall consider Q Matrix elements of Q are not measurable, why bother? Lattice ◮ QCD a ≪ 1 /m ◮ HQET a ≪ 1 / Λ MS HQET heavy-quark propagator in the Landau gauge ⇒ QCD propagator

Tree level � 1 + i / D ⊥ � Q ( x ) = e − imv · x 2 m + · · · Q v ( x ) C.L.Y. Lee (1991) K¨ orner, Thompson (1991) Mannel, Roberts, Ryzak (1992)

Matching Matching � 1 / 2 u ( p ) Z os � < 0 | Q 0 | Q ( p ) > = Q � 1 / 2 � ˜ Z os < 0 | Q v 0 | Q ( p ) > = u v ( k ) Q

Matching Matching � 1 / 2 u ( p ) Z os � < 0 | Q 0 | Q ( p ) > = Q � 1 / 2 � ˜ Z os < 0 | Q v 0 | Q ( p ) > = u v ( k ) Q Foldy–Wouthuysen transformation � k 2 � �� 1 + / k u ( mv + k ) = 2 m + O u v ( k ) m 2

Matching � 1 � � 1 + i / D ⊥ � �� z 1 / 2 Q 0 ( x ) = e − imv · x Q v 0 ( x ) + O 0 2 m m 2 Q ( g ( n l +1) , a ( n l +1) z 0 = Z os ) 0 0 Q ( g ( n l ) , a ( n l ) ˜ Z os ) 0 0

Matching � 1 � � 1 + i / D ⊥ � �� z 1 / 2 Q 0 ( x ) = e − imv · x Q v 0 ( x ) + O 0 2 m m 2 Q ( g ( n l +1) , a ( n l +1) z 0 = Z os ) 0 0 Q ( g ( n l ) , a ( n l ) ˜ Z os ) 0 0 Reparametrization invariance Luke, Manohar (1992)

Matching � 1 � � 1 + i / D ⊥ � �� z 1 / 2 Q 0 ( x ) = e − imv · x Q v 0 ( x ) + O 0 2 m m 2 Q ( g ( n l +1) , a ( n l +1) z 0 = Z os ) 0 0 Q ( g ( n l ) , a ( n l ) ˜ Z os ) 0 0 Reparametrization invariance Luke, Manohar (1992) Renormalized decoupling Z Q ( α ( n l ) ˜ ( µ ) , a ( n l ) ( µ )) s z ( µ ) = z 0 Z Q ( α ( n l +1) ( µ ) , a ( n l +1) ( µ )) s

m c = 0 ◮ ˜ Z os Q = 1 ◮ Z os Q (single scale m ): Melnikov, van Ritbergen (2000) ◮ ˜ γ Q : Melnikov, van Ritbergen (2000); γ Q : Chetyrkin, Grozin (2003) ◮ γ Q : Tarasov (1982); γ Q : Larin, Vermaseren (1993) Decoupling: Chetyrkin, Kniehl, Steinhauser (1998)

m c = 0 ◮ ˜ Z os Q = 1 ◮ Z os Q (single scale m ): Melnikov, van Ritbergen (2000) ◮ ˜ γ Q : Melnikov, van Ritbergen (2000); γ Q : Chetyrkin, Grozin (2003) ◮ γ Q : Tarasov (1982); γ Q : Larin, Vermaseren (1993) Decoupling: Chetyrkin, Kniehl, Steinhauser (1998) Melnikov, van Ritbergen obtained ˜ Z Q from Z os Q and finiteness of z ( µ )

Result α ( n l ) ( µ ) s z ( µ ) = 1 − (3 L + 4) C F 4 π � 2 � α ( n l ) ( µ ) z 22 L 2 + z 21 L + z 20 s � � + C F 4 π � 3 � α ( n l ) ( µ ) z 33 L 3 + z 32 L 2 + z 31 L + z 30 s � � + C F + · · · 4 π Depends on a ( µ ) starting from 3 loops

Large β 0 � β dβ � γ ( β ) 2 β − γ 0 � z ( µ ) = 1 + β 2 β 0 0 � 1 � ∞ + 1 � du e − u/β S ( u ) + O β 2 β 0 0 0

Large β 0 � β dβ � γ ( β ) 2 β − γ 0 � z ( µ ) = 1 + β 2 β 0 0 � 1 � ∞ + 1 � du e − u/β S ( u ) + O β 2 β 0 0 0 γ Q = − 2 β γ ( β ) = γ Q − ˜ F ( − β, 0) β 0 (1 + β )(1 + 2 3 β ) β = 2 C F β 0 B (2 + β, 2 + β )Γ(3 + β )Γ(1 − β ) γ Q − ˜ γ Q is gauge invariant at 1 /β 0

Borel image S ( u ) = F (0 , u ) − F (0 , 0) u � � e ( L +5 / 3) u Γ( u )Γ(1 − 2 u ) (1 − u 2 ) − 1 = − 6 C F Γ(3 − u ) 2 u

Borel image S ( u ) = F (0 , u ) − F (0 , 0) u � � e ( L +5 / 3) u Γ( u )Γ(1 − 2 u ) (1 − u 2 ) − 1 = − 6 C F Γ(3 − u ) 2 u ∆¯ ∆ z ( µ ) = 3 Λ 2 m

Borel image S ( u ) = F (0 , u ) − F (0 , 0) u � � e ( L +5 / 3) u Γ( u )Γ(1 − 2 u ) (1 − u 2 ) − 1 = − 6 C F Γ(3 − u ) 2 u ∆¯ ∆ z ( µ ) = 3 Λ 2 m z ( µ ) is gauge invariant at 1 /β 0 Expand and integrate

Numerically � 2 α (4) � α (4) z ( m ) = 1 − 4 s ( m ) s ( m ) − (16 . 6629 − 4 . 5421) 3 π π � 3 � α (4) s ( m ) − (153 . 4076 + 42 . 6271 − 61 . 5397) π � 4 � α (4) s ( m ) − (1953 . 4013 + · · · ) + · · · π � 2 � 3 � � α (4) α (4) α (4) = 1 − 4 s ( m ) s ( m ) s ( m ) − 12 . 1208 − 134 . 4950 3 π π π � 4 � α (4) s ( m ) − (1953 . 4013 + · · · ) + · · · π

Gauge dependence of QED propagators µν ( k ) = 1 � g µν − k µ k ν � D 0 k 2 k 2 S ( x ) = S L ( x )

Gauge dependence of QED propagators µν ( k ) = 1 � g µν − k µ k ν � D 0 + ∆( k ) k µ k ν k 2 k 2 0 ( ˜ ∆( x ) − ˜ S ( x ) = S L ( x ) e − ie 2 ∆(0)) ∆( k ) e − ikx d d k � ˜ ∆( x ) = (2 π ) d

Gauge dependence of QED propagators µν ( k ) = 1 � g µν − k µ k ν � D 0 + ∆( k ) k µ k ν k 2 k 2 0 ( ˜ ∆( x ) − ˜ S ( x ) = S L ( x ) e − ie 2 ∆(0)) ∆( k ) e − ikx d d k � ˜ ∆( x ) = (2 π ) d a 0 ˜ ∆( k ) = ∆(0) = 0 in dim. reg. ( k 2 ) 2

Gauge dependence of QED propagators µν ( k ) = 1 � g µν − k µ k ν � D 0 + ∆( k ) k µ k ν k 2 k 2 0 ( ˜ ∆( x ) − ˜ S ( x ) = S L ( x ) e − ie 2 ∆(0)) ∆( k ) e − ikx d d k � ˜ ∆( x ) = (2 π ) d a 0 ˜ ∆( k ) = ∆(0) = 0 in dim. reg. ( k 2 ) 2 Landau, Khalatnikov (1955) Fradkin (1955) Zumino (1960) Fukuda, Kubo, Yokoyama (1980) Bogoliubov, Shirkov § 45.5

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x )

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4 � ε � − µ 2 x 2 = σ L ( x ) + a ( µ ) α ( µ ) e γ E ε Γ( − ε ) 4 π 4

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4 � ε � − µ 2 x 2 = σ L ( x ) + a ( µ ) α ( µ ) e γ E ε Γ( − ε ) 4 π 4 = log Z ψ + σ r

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4 � ε � − µ 2 x 2 = σ L ( x ) + a ( µ ) α ( µ ) e γ E ε Γ( − ε ) 4 π 4 = log Z ψ + σ r log Z ψ ( α, a ) = log Z L ( α ) − a α 4 πε

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4 � ε � − µ 2 x 2 = σ L ( x ) + a ( µ ) α ( µ ) e γ E ε Γ( − ε ) 4 π 4 = log Z ψ + σ r log Z ψ ( α, a ) = log Z L ( α ) − a α 4 πε γ ψ ( α, a ) = 2 a α 4 π + γ L ( α ) d log( a ( µ ) α ( µ )) /d log µ = − 2 ε exactly γ L ( α ) starts from α 2

Gauge dependence of Z ψ , γ ψ Massless electron S ( x ) = S 0 ( x ) e σ ( x ) � ε e 2 � − x 2 0 σ ( x ) = σ L ( x ) + a 0 Γ( − ε ) (4 π ) d/ 2 4 � ε � − µ 2 x 2 = σ L ( x ) + a ( µ ) α ( µ ) e γ E ε Γ( − ε ) 4 π 4 = log Z ψ + σ r log Z ψ ( α, a ) = log Z L ( α ) − a α 4 πε γ ψ ( α, a ) = 2 a α 4 π + γ L ( α ) d log( a ( µ ) α ( µ )) /d log µ = − 2 ε exactly γ L ( α ) starts from α 2 Some Russian paper in the second half of the 1950s ?