NRQCD HQET Summary Matching Coefficients in NRQCD and HQET P . Marquard Institut for Theoretical Particle Physics University of Karlsruhe Computational Theoretical Particle Physics B AC KA SFB TR9 RADCOR 2007
NRQCD HQET Summary Outline NRQCD 1 Introduction Calculation Results HQET 2 Introduction Calculation Results Summary 3
NRQCD HQET Summary Fermionic corrections to the three-loop matching coefficient of the vector current. Nucl.Phys.B758:144-160,2006; P .M., J.Piclum, D. Seidel, M. Steinhauser
NRQCD HQET Summary Introduction matching coefficients: Connection between QCD and non-relativistic QCD (NRQCD)
NRQCD HQET Summary Introduction matching coefficients: Connection between QCD and non-relativistic QCD (NRQCD) important for threshold behaviour of bottom and top system top production at threshold at a future linear collider ⇒ determination of top quark mass decay of heavy mesons
NRQCD HQET Summary Introduction matching coefficients: Connection between QCD and non-relativistic QCD (NRQCD) important for threshold behaviour of bottom and top system top production at threshold at a future linear collider ⇒ determination of top quark mass decay of heavy mesons fermionic non-singlet contribution with one light fermion loop
NRQCD HQET Summary Calculation j i = φ † σ i χ j µ Q γ µ Q v = ¯ ˜ ↔ j k + d v ( µ ) j k φ † σ k � v = c v ( µ )˜ D 2 χ + . . . 6 m 2 Q Z 2 Γ v = c v ˜ Z 2 ˜ Z − 1 ˜ Γ v + . . . v
NRQCD HQET Summary Calculation j i = φ † σ i χ j µ Q γ µ Q v = ¯ ˜ ↔ j k + d v ( µ ) j k φ † σ k � v = c v ( µ )˜ D 2 χ + . . . 6 m 2 Q Z 2 Γ v = c v ˜ Z 2 ˜ Z − 1 ˜ Γ v + . . . v c v independent of external momenta ⇒ threshold expansion ⇒ evaluate Γ v for s = 4 m 2 ˜ Γ v tree level Z 2 = 1 ˜ Z v calculated ˜ Z 2 n l part recalculated and agreement with known result found
NRQCD HQET Summary Calculation
NRQCD HQET Summary Calculation diagrams generated with qgraf 4 three-loop topologies identified and mapped with the help of q2e and exp calculation done with form reduction to 12 master integrals with crusher master integrals calculated/checked with Mellin-Barnes (MB.m)
NRQCD HQET Summary Master Integrals I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 Some coefficients of the ǫ expansion of the master integrals only known numerically
NRQCD HQET Summary Master Integrals I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 I f I f 9 12 Some coefficients of the ǫ expansion of the master integrals only known numerically Try to avoid expansion of master to more than O ( ǫ 0 ) ⇒ ǫ -finite basis ǫ -inite integrals more complicated ⇒ lower numerical accuracy but helpful to determine the analytical results for some of the "normal" master integrals
NRQCD HQET Summary Bottom System Decay rate of Υ( 1 S ) into leptons v ( m b ) + C 2 � � F α 2 s ( µ s ) Γ(Υ( 1 S ) → l + l − ) = Γ L O ρ 1 c 2 c v ( m b ) d V ( m b ) + 3 � � 12 with α s ( M Z ) = 0 . 118, m b = 5 . 3 GeV and µ s = 2 . 0967 GeV Γ 1 ≈ Γ LO 1 ( 1 − 0 . 446 NLO + 1 . 75 NNLO − 1 . 67 NNNLO ) changes to Γ 1 ≈ Γ LO 1 ( 1 − 0 . 446 NLO + 1 . 75 NNLO − 1 . 20 NNNLO ) convergence improved
NRQCD HQET Summary Top System normalized t ¯ t cross section at a linear collider σ ( e + e − → t ¯ t ) R ( e + e − → t ¯ t ) = σ ( e + e − → µ + µ − ) � v ( m b ) + C 2 � F α 2 s ( µ s ) R ( e + e − → t ¯ t ) = R L O ρ 1 c 2 c v ( m b ) d V ( m b ) + 3 � � 12 with α s ( M Z ) = 0 . 118, m t = 175 GeV and µ s = 32 . 625 GeV R 1 ≈ Γ LO 1 ( 1 − 0 . 243 NLO + 0 . 435 NNLO − 0 . 268 NNNLO ) changes to R 1 ≈ Γ LO 1 ( 1 − 0 . 243 NLO + 0 . 435 NNLO − 0 . 195 NNNLO ) Sum of all corrections ∼ 0 . 003
NRQCD HQET Summary Three-Loop Chromomagnetic Interaction in HQET. arXiv:0707.1388; submitted to Nucl. Phys, B A. Grozin, P .M., J. Piclum, M. Steinhauser
NRQCD HQET Summary Introduction matching coefficient of the chromomagnetic interaction in HQET responsible for breaking of the spin symmetry ⇒ B − B ∗ mass splitting
NRQCD HQET Summary Introduction matching coefficient of the chromomagnetic interaction in HQET responsible for breaking of the spin symmetry ⇒ B − B ∗ mass splitting chromomagnetic matching coefficient known up to two loops charm mass effects at two loops known
NRQCD HQET Summary Introduction matching coefficient of the chromomagnetic interaction in HQET responsible for breaking of the spin symmetry ⇒ B − B ∗ mass splitting chromomagnetic matching coefficient known up to two loops charm mass effects at two loops known three-loop calculation of the matching coefficient and the anomalous dimension byproduct: heavy quark magnetic moments
NRQCD HQET Summary Calculation Lagrangian of HQET � � 1 1 Q v ivDQ v + ( O k + C cm ( µ ) O cm ( µ )) + O L HQET = ¯ 2 m Q m 2 Q chromomagnetic operator O cm = 1 Q v G µν σ µν Q v ¯ 2 scattering amplitude v µ − q µ i � � u v ( − q ) σ µν q ν ( 1 + µ c ) t a u v ( 0 ) ¯ − 2 m Q 2 m Q matched with in HQET v µ − q µ i � � u v ( − q ) σ µν q ν C 0 t a u v ( 0 ) ¯ − cm 2 m Q 2 m Q
NRQCD HQET Summary on QCD side calculate quark-antiquark-gluon vertex with onshell quarks and zero momentum transfer i Γ µ = γ µ F 1 ( q 2 ) − σ µν q ν F 2 ( q 2 ) 2 m Q colour charge Z OS F 1 ( 0 ) − 1 = Z OS F 1 ( 0 ) = 1 ⇒ 2 2 chromomagnetic moment γ cm = Z OS F 2 ( 0 ) C cm ( µ ) = Z cm ( α s ( µ ))[ 1 + µ cm ( α s ( µ ))] ⇒ 2
NRQCD HQET Summary ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) ( i ) diagrams generated with qgraf topologies identified and mapped with the help of q2e and exp calculation done with form reduction to 19 master integrals with crusher master integrals calculated/checked with MB
NRQCD HQET Summary Use projectors to calculate F 1 , F 2 1 F 1 ( q 2 ) = 2 ( d − 2 )( q 2 − 4 m 2 Q ) � � γ µ + 4 m Q ( d − 1 ) � � × Tr p 1 + m Q ) p µ p 2 + m Q ) Γ µ ( / ( / q 2 − 4 m 2 Q 2 m 2 Q F 2 ( q 2 ) = − ( d − 2 ) q 2 ( q 2 − 4 m 2 Q ) � � γ µ + 4 m 2 Q + ( d − 2 ) q 2 � � × Tr p 1 + m Q ) Q ) p µ p 2 + m Q ) Γ µ ( / ( / m Q ( q 2 − 4 m 2 projectors develop pole in momentum q ⇒ expand up to q 2 ⇒ onshell propagators calculation done with arbitrary gauge successfully checked Z OS = F 1 ( 0 ) − 1 2
NRQCD HQET Summary Results matching coefficient C cm ( m c ) = 1 + 0 . 2309 + 0 . 1835 + 0 . 2362 = 1 . 6506 C cm ( m b ) = 1 + 0 . 1492 + 0 . 0676 + 0 . 0497 = 1 . 2664 C cm ( m t ) = 1 + 0 . 0741 + 0 . 0144 + 0 . 0045 = 1 . 0930
NRQCD HQET Summary Results matching coefficient C cm ( m c ) = 1 + 0 . 2309 + 0 . 1835 + 0 . 2362 = 1 . 6506 C cm ( m b ) = 1 + 0 . 1492 + 0 . 0676 + 0 . 0497 = 1 . 2664 C cm ( m t ) = 1 + 0 . 0741 + 0 . 0144 + 0 . 0045 = 1 . 0930 anomalous dimension γ cm ( m c ) = 0 . 1599 + 0 . 0298 + 0 . 0163 = 0 . 2060 γ cm ( m b ) = 0 . 1033 + 0 . 0099 + 0 . 0029 = 0 . 1160 γ cm ( m t ) = 0 . 0513 + 0 . 0018 + 0 . 0002 = 0 . 0533
NRQCD HQET Summary Heavy Quark Mass Splittings mass splittings B = 4 � Λ QCD � m 2 B ∗ − m 2 3 C ( 4 ) cm ( µ ) µ 2 G ( 4 ) ( µ ) + O m b R = m 2 B ∗ − m 2 B ( exp . ) = 0 . 88 m 2 D ∗ − m 2 D resummed logs R = 0 . 8517 − 0 . 0696 − 0 . 0908 +[ − 0 . 1285 ] = 0 . 6914 +[ − 0 . 1285 ] fixed order R = 1 − 0 . 1113 − 0 . 0780 − 0 . 0755 + . . . = 0 . 7352 + . . .
NRQCD HQET Summary Magnetic moments replace external gluon by photon a Q � 2 0 . 2122 α ( n f ) α ( n f ) ( m Q ) + ( 0 . 8417 − 0 . 0469 n l ) � ( m Q ) = s s Q Q � 3 α ( n f ) � � � 4 . 5763 − 0 . 5856 n l + 0 . 0145 n 2 ( m Q ) + O α 4 + s s l a c = 0 . 0478 + 0 . 0533 + 0 . 0758 = 0 . 1770 a b = − 0 . 0153 − 0 . 0103 − 0 . 0084 = − 0 . 0340 a t = 0 . 0152 + 0 . 0047 + 0 . 0017 = 0 . 0215
NRQCD HQET Summary Magnetic moments replace external gluon by photon a Q � 2 0 . 2122 α ( n f ) α ( n f ) � ( m Q ) + ( 0 . 8417 − 0 . 0469 n l ) ( m Q ) = s s Q Q � 3 α ( n f ) � � � 4 . 5763 − 0 . 5856 n l + 0 . 0145 n 2 ( m Q ) + O α 4 + s s l a c = 0 . 0478 + 0 . 0533 + 0 . 0758 = 0 . 1770 a b = − 0 . 0153 − 0 . 0103 − 0 . 0084 = − 0 . 0340 a t = 0 . 0152 + 0 . 0047 + 0 . 0017 = 0 . 0215 a b ( m Z ) = − 0 . 0083 − 0 . 0066 − 0 . 0056 = − 0 . 0206 exp: a b > − 0 . 135 (68% CL)
NRQCD HQET Summary Lattice matching coefficient also needed for lattice calculations rewrite matching coefficient in terms of Λ QCD / M RGI � d α s γ m , 0 m ∗ ) m ∗ ) � � � α s ( ¯ � α s ( ¯ � − β 0 exp � γ m β − γ m , 0 M RGI = ¯ m ∗ 2 β 0 − π β 0 α s 0 1.3 1.2 1.1 improved convergence, 1 C cm 0.9 error ≈ 1 % 0.8 0.7 0.6 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Λ QCD /M RGI
NRQCD HQET Summary Conclusion NRQCD non-singlet three loop contributions calculated generally improves convergence still incomplete
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