LEADING LOGARITHMS FOR THE NUCLEON MASS

Transcription

1 /9 LEADING LOGARITHMS FOR THE NUCLEON MASS O(N + ) Lund University bijnens@thep.lu.se bijnens bijnens/chpt/ QNP5 - Universidad Técnica Federico Santa María UTFSMXI Valparaiso -6 March 5

2 /9 Overview Main question: can we get information on high orders? Leading logarithms () O(N +) Masses, decay large N Other work 4 SU(N) SU(N) O(N + ) 5 6 Work done with Alexey A. Vladimirov (nucleon) and Lisa Carloni, Stefan Lanz and Karol Kampf (various mesonic)

3 /9 References JB, L. Carloni, Nucl.Phys. B87 () 7-55 [arxiv:99.586] Leading Logarithms in the Massive O(N) Nonlinear Sigma Model JB, L. Carloni, Nucl.Phys. B84 () 55-8 [arxiv:8.499] The Massive O(N) Non-linear Sigma Model at High Orders JB, K. Kampf, S. Lanz, Nucl.Phys. B86 () [arxiv:.68] Leading logarithms in the anomalous sector of two-flavour QCD JB, K. Kampf, S. Lanz, Nucl.Phys. B87 () 7-64 [arxiv:.5] Leading logarithms in N-flavour mesonic Chiral Perturbation Theory JB, A.A. Vladimirov, Nucl. Phys. B9 (5) 7-79 [arxiv:49.67], Leading logarithms for the. O(N + )

4 4/9 Leading logarithms () BEWARE: leading logarithms can mean very different things Take a quantity with a single scale: F(M) Subtraction scale in QFT (in dim. reg.) is logarithmic L = log(µ/m) F = F + ( F ( ) L+F + F L +F ) L+F + ( F L + )+ Leading Logarithms: The terms Fm mlm The Fm m can be more easily calculated than the full result O(N + ) µ(df/dµ) Ultraviolet divergences in Quantum Field Theory are always local

5 5/9 Renormalizable theories Loop expansion α expansion F = α+ ( f α L+f α) + ( f α L +f α L+f α) + ( f α4 L + )+ f j i are pure numbers µ df dµ F, µ dα dµ α, µ dl dµ = F = α + ( f α +f α αl+f αα ) + ( f α L+f α α L +f α +f α α L+f α α ) + ( f α L +f 4α α L + )+ α = β α +β α + = F = ( β +f ) α + ( β f +f ) α L + ( β +β f ) +f α + ( β f +f ) α 4 L + f = β, f = β, f = β,... O(N + )

6 5/9 Renormalizable theories Loop expansion α expansion F = α+ ( f α L+f α) + ( f α L +f α L+f α) + ( f α4 L + )+ f j i are pure numbers µ df dµ F, µ dα dµ α, µ dl dµ = F = α + ( f α +f α αl+f αα ) + ( f α L+f α α L +f α +f α α L+f α α ) + ( f α L +f 4α α L + )+ α = β α +β α + = F = ( β +f ) α + ( β f +f ) α L + ( β +β f ) +f α + ( β f +f ) α 4 L + f = β, f = β, f = β,... O(N + )

7 5/9 Renormalizable theories Loop expansion α expansion F = α+ ( f α L+f α) + ( f α L +f α L+f α) + ( f α4 L + )+ f j i are pure numbers µ df dµ F, µ dα dµ α, µ dl dµ = F = α + ( f α +f α αl+f αα ) + ( f α L+f α α L +f α +f α α L+f α α ) + ( f α L +f 4α α L + )+ α = β α +β α + = F = ( β +f ) α + ( β f +f ) α L + ( β +β f ) +f α + ( β f +f ) α 4 L + f = β, f = β, f = β,... O(N + )

8 6/9 Renormalizable theories Leading logs known as soon as β is known F(M) = α ( αβ L+(αβ L) +(αβ L) + )+ α = +αβ L + running coupling constant Generalizes to lower leading logarithms as well Multiscale problems: many other terms possible O(N + )

9 Renormalization Group Can be extended to other operators as well Underlying always µ df dµ =. Gell-Mann Low, Callan Symanzik, Weinberg t Hooft In great detail: J.C. Collins, Renormalization Relies on the α the same in all orders one-loop β N two-loop β O(N + ) In effective field theories: different Lagrangian at each order The recursive does not work 7/9

10 8/9 for leading logarithms Weinberg, Physica A96 (979) 7 Two-loop leading logarithms can be calculated using only one-loop: Weinberg consistency conditions ππ at -loop: Colangelo, hep-ph/9585 General at loop: JB, Colangelo, Ecker, hep-ph/9884 Proof at all orders using β-functions: Büchler, Colangelo, hep-ph/949 with diagrams: JB, Carloni, arxiv: Extension to nucleons: JB, Vladimirov, arxiv:49.67 First mesonic case where loop-order = power-counting (or ) order Later nucleon case where loop-order power-counting (or ) order O(N + )

11 9/9 µ: dimensional regularization scale d = 4 w loop-expansion -expansion L bare = n L (n) = i c (n) i = k=,n n µ nw L (n) c (n) i O i c (n) ki w k c (n) i have a direct µ-dependence c (n) ki k only depend on µ through c (m<n) i O(N + )

12 /9 L n l l-loop contribution at order n Expand in divergences from the loops (not from the counterterms) L n l = k=,l L n w k kl Neglected positive powers: not relevant here, but should be kept in general {c} n l all combinations c(m ) k j c (m ) k j...c (mr) k rj r with m i, such that i=,r m i = n and i=,r k i = l. {cn n} {c(n) ni }, {c} = {c() i,c () i c () k } O(N + ) L (n) = n

13 /9 Mass = + O(N + ) + +

14 /9 : L : w ( µ w L ) ({c} )+L +µ w L ({c} )+L Expand µ w = w logµ+ w log µ+ /w must cancel: L ({c} )+L = this determines the c i O(N + ) Explicit logµ: logµl ({c} ) logµl

15 /9 : L : w ( µ w L ) ({c} )+L +µ w L ({c} )+L Expand µ w = w logµ+ w log µ+ /w must cancel: L ({c} )+L = this determines the c i O(N + ) Explicit logµ: logµl ({c} ) logµl

16 /9 : ( µ w w L ({c} )+µ w L ({c} )+L ) + ( µ w L w ({c} )+µ w L ({c} )+µ w L ({c} ) )+L + ( µ w L ({c} )+µ w L ) ({c} )+L /w and logµ/w must cancel L ({c} )+L ({c} )+L = L ({c} )+L ({c} ) = Solution: L ({c} ) = L ({c} ) L ({c} ) = L Explicit log µ dependence (one-loop is enough) log µ ( 4L ({c} )+L ({c} ) ) = L ({c} )log µ. O(N + )

17 /9 : ( µ w w L ({c} )+µ w L ({c} )+L ) + ( µ w L w ({c} )+µ w L ({c} )+µ w L ({c} ) )+L + ( µ w L ({c} )+µ w L ) ({c} )+L /w and logµ/w must cancel L ({c} )+L ({c} )+L = L ({c} )+L ({c} ) = Solution: L ({c} ) = L ({c} ) L ({c} ) = L Explicit log µ dependence (one-loop is enough) log µ ( 4L ({c} )+L ({c} ) ) = L ({c} )log µ. O(N + )

18 /9 : ( µ w w L ({c} )+µ w L ({c} )+L ) + ( µ w L w ({c} )+µ w L ({c} )+µ w L ({c} ) )+L + ( µ w L ({c} )+µ w L ) ({c} )+L /w and logµ/w must cancel L ({c} )+L ({c} )+L = L ({c} )+L ({c} ) = Solution: L ({c} ) = L ({c} ) L ({c} ) = L Explicit log µ dependence (one-loop is enough) log µ ( 4L ({c} )+L ({c} ) ) = L ({c} )log µ. O(N + )

19 4/9 Mass to : = O(N + ) : = but also needs : =

20 5/9 Mass to order O(N + )

21 () () (45) (67) (89) () () (55) (77) (99) () (4) (65) 4 (87) () (4) (46) (68) (9) () (4) (56) (78) () () (44) (66) (88) () (5) (47) (69) (9) () (5) (57) (79) () () (45) (67) (89) (4) (6) (48) (7) (9) (4) (6) (58) (8) () (4) (46) (68) (9) (5) (7) (49) (7) (9) (5) (7) (59) (8) () (5) (47) (69) (9) (6) (8) (5) (7) (94) (6) (8) (6) (8) (4) 4 (6) (48) (7) (9) (7) (9) (5) (7) (95) (7) (9) (6) (8) (5) 4 (7) (49) (7) (9) (8) () (5) (74) (96) (8) (4) (6) (84) (6) 4 (8) (5) (7) (94) (9) () (5) (75) (97) (9) (4) (6) (85) (7) 4 (9) (5) (7) (95) () () (54) (76) (98) () (4) (64) (86) (8) 4 () (5) (74) (96) () () (55) (77) (99) () (4) (65) (87) (9) () (5) (75) (97) () (4) (56) (78) () () (44) (66) (88) () () (54) (76) (98) () (5) (57) (79) () () (45) (67) (89) () () (55) (77) (99) (4) (6) (58) (8) () (4) (46) (68) (9) () (4) (56) (78) () (5) (7) (59) (8) () (5) (47) (69) (9) () (5) (57) (79) () (6) (8) (6) (8) (4) (6) (48) (7) (9) (4) (6) (58) (8) () (7) (9) (6) (8) (5) (7) (49) (7) (9) (5) (7) (59) 5 (8) () (8) (4) (6) (84) (6) (8) (5) (7) (94) (6) (8) (6) 5 (8) (9) (4) (6) (85) (7) (9) (5) (7) (95) (7) 4 (9) (6) 4 (8) () (4) (64) (86) (8) () (5) (74) (96) (8) (4) (6) 4 (84) () (4) (65) (87) (9) () (5) (75) (97) (9) (4) (6) (85) () (44) (66) (88) () () (54) (76) (98) () (4) (64) (86) 6/9 Mass to order 6 O(N + )

22 7/9 General Calculate the divergence rewrite it in terms of a local Lagrangian Luckily: symmetry kept: we know result will be symmetrical, hence do not need to explicitly rewrite the Lagrangians in a nice form Luckily: we do not need to go to a minimal Lagrangian So everything can be computerized Thank Jos Vermaseren for FORM We keep all terms to have all PI (one particle irreducible) diagrams finite O(N + )

23 8/9 Massive O(N) sigma model O(N + ) nonlinear sigma model L nσ = F µφ T µ Φ+F χ T Φ. Φ is a real N + vector; Φ OΦ; Φ T Φ =. Vacuum expectation value Φ T = (...) Explicit symmetry breaking: χ T = ( M... ) Both spontaneous and explicit symmetry breaking N-vector φ N (pseudo-)nambu-goldstone Bosons N = is two-flavour Chiral Perturbation Theory O(N + ) Masses, decay large N Other work

24 9/9 Massive O(N) sigma model: Φ vs φ Φ = φt φ F φ F. φ N F ( Φ 5 = + φt φ Weinberg Φ 4 = sin 4F = φt φ F φ F φ cos T φ φ T φ F F φ φ T φ ( φt φ F φ F ) CCWZ ) Gasser, Leutwyler Φ = φ T φ F φ T φ φ 4 F F only mass term O(N + ) Masses, decay large N Other work

25 /9 Massive O(N) sigma model: Checks Need (many) checks: use many different parametrizations compare with known results: M phys = M ( L M L M + ), L M = M µ 6π log F M Usual choice M = M. large N but massive results more hidden Coleman, Jackiw, Politzer 974 JB, Carloni, : mass to 5 loops O(N + ) Masses, decay large N Other work

26 /9 Results M phys = M (+a L M +a L M +a L M +...) L M = M 6π F log µ M i a i, N = a i for general N N N N 8 7 N 5 N + N N N 5 N N N N 79 N 67 N N / /59 N /7776 N 74967/68 N +7466/576 N 4 775/5 N 5 +79/4 N 6 O(N + ) Masses, decay large N Other work

27 /9 Results F phys = F(+a L M +a L M +a L M +...) i a i for N = a i for general N / +/N 5/4 / +7/8N /8N 8/4 7/4 +/6N 7/48N +/N 4 /88 47/ /864N 477/456N +65/9N 5/8N / /78N 89875/76N /4N 69/6N 4 +/N / /8664 N /4 N /76 N 86459/456 N /56N 5 /4N 6 O(N + ) Masses, decay large N Other work

28 /9 Results i q i q i = BF (+c L M +c L M +c L M +...) M = Bˆm χ T = B(s...) s corresponds to ūu + dd current c i for N = c i for general N N N 4 N 8 N N + N 45N 55N + 5N 5N N 47N 557 N N4 + N5 6 O(N + ) Masses, decay large N Other work Anyone recognize any funny functions?

29 4/9 Large N Power counting: pick L extensive in N F N, M } {{ } n legs PI diagrams: N L = N I n N n + N I +N E = n nn n F n N n N } N L = n (n )N n N E + O(N + ) Masses, decay large N Other work diagram suppression factor: N N L N N E/

30 5/9 Large N diagrams with shared lines are suppressed each new loop needs also a new flavour loop in the large N limit only cactus diagrams survive: O(N + ) Masses, decay large N Other work

31 large N: propagator Generate recursively via a Gap equation ( ) = ( ) resum the series and look for the pole M = M phys + N F A(M phys ) O(N + ) Masses, decay large N Other work A(m ) = m 6π log µ m. Solve recursively, agrees with other result Note: can be done for all parametrizations 6/9

32 7/9 Other results Bissegger, Fuhrer, hep-ph/696 Dispersive methods, massless Π S to five loops Kivel, Polyakov, Vladimirov, 89.6, 94.8, 4.97, In the massless case tadpoles vanish hence the number of external legs needed does not grow All 4-meson vertices via Legendre polynomials can do divergence of all one-loop diagrams analytically algebraic (but quadratic) recursion relations massless ππ, F V and F S to arbitrarily high order large N agrees with Coleman, Wess, Zumino large N is not a good approximation O(N + ) Masses, decay large N Other work

33 8/9 SU(N) SU(N) SU(N) SU(N) (vector and scalar) Mass, Decay constants, Form-factors Meson-Meson, γγ ππ No luck with guess for general N-dependence either Four different parametrizations of a unitary matrix used O(N + )

34 9/9 Mass i a i for N = a i for N = a i for general N / / N 7/8 7/8 9/N / + /8N /4 799/648 89/N + 9/N 7/4N /N 4 467/ /59 5/8 N 4 77/ N + 9/8 + /88 N +4/7 N / /5 N / N 59/8 N 577/44 N 7/4 N 874/456 N /4 N 6 659/4 N 4 49/555 N / /7776 N + 49/59 N /4 N 6 O(N + )

35 /9 Lagrangian We use the heavy-baryon approach, explicit powercounting LO Lagrangian is order p (mesons p ): L () Nπ = N(iv µ D µ +g A S µ u µ )N Partly checked with other schemes (IR-regularization) Propagator is order /p (mesons /p ) Loops add p just as for mesons Different parametrizations for mesons O(N + )

36 /9 Lagrangian Two different p Lagrangians: Bernard-Kaiser-Meißner L () πn = N [ (v D) D D ig A {S D,v u} v M +c tr(χ + ) ( ) + c g A 8M (v u) +c u u + ( ) ] c 4 + 4M iǫ µνρσ u µ u ν v ρ S σ N v Ecker-Mojžiš L () Nπ = N [ M (D µd µ +ig A {S µ D µ,v ν u ν })+A tr(u µ u µ ) +A tr ( (v µ u µ ) ) +A tr(χ + )+A 5 iǫ µνρσ v µ S ν u ρ u σ ]N O(N + )

37 /9 loops set n p n+ for meson-nucleon set n p n+ for mesons Introduce a RGO renormalization group order max power of /w same p-order can be different RGO, e.g. O(N + ) both p 5, left RGO, right RGO Note: same equations, if no tree level contribution next-to-leading log also calculable For nucleon can have fractional powers of quark masses

38 /9 loops set n p n+ for meson-nucleon set n p n+ for mesons Introduce a RGO renormalization group order max power of /w same p-order can be different RGO, e.g. O(N + ) both p 5, left RGO, right RGO Note: same equations, if no tree level contribution next-to-leading log also calculable For nucleon can have fractional powers of quark masses

39 /9 Results M:, m: pion mass, L = m µ log (4πF) m m M phys = M +k M +k πm (4πF) +k m 4 µ 4 (4πF) ln M m πm 5 µ +k 5 ln (4πF) 4 m +... = M + m k n L n +πm m M (4πF) k n+ L n, n= n= O(N + )

40 4/9 Results k k g A k 4 4 ga k 5 4c M ( g A +(c +4c 4c )M ) c M ( ) 8 6g A k 6 ( 4 g A +(c +4c 4c )M ) ( ) + c M k 7 ga 8gA 4 + 5g A ( 7 k 8 8 g A +(c +4c 4c )M ) 9 c M ) ga k 9 ( 6gA g4 A 569g A ( g A +(c +4c 4c )M ) + 57 c M k 57 ga k ( 95gA g6 A g4 A g A g A g A : only even powers k n peculiar structure ) O(N + ) Drop g A then can calculate k

41 5/9 Results r r g A r 4 4 r 5 6gA r 6 5c M 4c M ( g A +(c +4c 4c )M ) 5c M g r ( A g A 7gA 4 ) 5 r 8 c M ) ga r 9 ( 6gA g4 A 457g A r 6 c M ga r (95gA g6 A g4 A g A r 4 c M everything rewritten in terms of physical pion mass Simpler expression ) O(N + )

42 6/9 Results Conjecture: M = M phys + 4 m4 phys c (4πF) µ m phys log µ m phys (4πF) m 4 phys (µ ) dµ µ. ( g ) A 4c +c +4c M phys O(N + ) Take now known result for pion mass, k 4 and k 6 calculable

43 Numerics M phys -M [GeV] M = 98 MeV, c =.87 GeV c =.4 GeV c = 5.5 GeV g A =.5 µ =.77 GeV F = 9.4 MeV O(N + ) m [GeV ] 7/9

44 8/9 Numerics M phys -M [GeV].... e-5 e-6 e-7 O(N + ) e n

45 9/9 Leading logarithms can be calculated using only one-loop diagrams Results for a large number of quantities for mesons Look at the (very) many tables, we would be very interested in all-order conjectures mass as the first result in the nucleon sector O(N + )