LEADING LOGARITHMS FOR THE NUCLEON MASS

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

/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)

/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 () 45-66 [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 + )

4/9 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

5/9 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 + )

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 + )

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 + )

6/9 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 + )

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

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:99.586 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 + )

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 + )

/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

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

/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

/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

/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 + )

/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 + )

/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 + )

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

5/9 Mass to order O(N + )

() () (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 + )

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 + )

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

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

/9 Massive O(N) sigma model: Checks Need (many) checks: use many different parametrizations compare with known results: M phys = M ( L M + 7 8 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

/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 7 8 4 467 4 5 5 88 44 9964667 6 68 7 4 7N 4 + 5 N 8 7 N 5 N + N 4 4 89 44 6 N + 44 695 N 5 N N4 + 48 6 8 66 4 547 N 97587 N 79 N 67 N4 + + 7 N5 4 4 58989/888 879/59 N +468587/7776 N 74967/68 N +7466/576 N 4 775/5 N 5 +79/4 N 6 O(N + ) Masses, decay large N Other work

/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 47/576 +45/864N 477/456N +65/9N 5/8N 4 5 647 584 87/648 +4594/78N 89875/76N +54694/4N 69/6N 4 +/N 5 6 698787 46656 77796/9 + 6879/8664 N 686646/4 N +87999/76 N 86459/456 N 4 +699/56N 5 /4N 6 O(N + ) Masses, decay large N Other work

/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 9 8 9 4 85 8 5 46 N N 4 N 8 N N + N 45N 55N + 5N 5N4 48 8 7N 47N 557 N + 48 4 9 N4 + N5 6 O(N + ) Masses, decay large N Other work Anyone recognize any funny functions?

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/

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

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

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, 5.499 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

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 + )

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 4 467/5 46657/59 5/8 N 4 77/ N + 9/8 + /88 N +4/7 N 4 5 88/44 74759 8664 6 9964667 68 9776 9 8684/5 N 5 + 66/ N 59/8 N 577/44 N 7/4 N 874/456 N 5 7999/4 N 6 659/4 N 4 49/555 N + 56549/97 696876/7776 N + 49/59 N 4 +778/4 N 6 O(N + )

/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 + )

/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 + )

/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

/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

/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 + )

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 4 44 64 ( 7 k 8 8 g A +(c +4c 4c )M ) 9 c M ) ga k 9 ( 6gA 6 + 57g4 A 569g A 4 + 5569 8 ( g A +(c +4c 4c )M ) + 57 c M k 57 ga k ( 95gA 8 + 58747g6 A 6 4499g4 A 945 + 679g A 648 987855 956 g A g A : only even powers k n peculiar structure ) O(N + ) Drop g A then can calculate k

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 7 4 8+5g A 7gA 4 ) 5 r 8 c M ) ga r 9 ( 6gA 6 + 647g4 A 457g A + 7 4 75 r 6 c M ga r (95gA 8 679567g6 A 6 + 45799g4 A 78 557g A 5 + 896467 648 75 r 4 c M everything rewritten in terms of physical pion mass Simpler expression ) O(N + )

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

Numerics M phys -M [GeV].5..5 -.5 -. 4 5 -.5..4.6.8. 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

8/9 Numerics M phys -M [GeV].... e-5 e-6 e-7 O(N + ) e-8 4 6 8 4 6 n

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 + )