Field redefinitions and renormalization group equations in R c T

1. Field redefinitions and renormalization group equations in RcT J.J. Sanz-Cillero ( UAB – IFAE ) [arXiv:0905.3676 [hep-ph] ] PSI, July 2nd2009

2. Outline • Formal 1/NCexpansion in RcT • Theexample of thepp-VFF: • One-loopcomputation • EoM, Fieldredef. & redundantoperators • RGE  Solutions IR-fixedpoint • PerturbativeregimePerturbationtheory in 1/NC

3. 1/NCexpansion in RcT

4. RcT=chiralinvariantframework • fortheinteraction of chiral NGB & Resonances • Thestandard EFT expansion in powers of (p2)m , • notvalid in presence of heavy resonanceloops • Analternativepowercountingisrequired • RcTtakesthe formal 1/NCexpansion as a guidingprinciple: • - NCscaling of theoperators in L at LO in 1/NC: • L= … + l O , with k mesonfields scaleslike NC1-k/2 • - Subleadingoperators subleadingscaling in 1/NC • (withrespecttothelatter) [Ecker et al.’89] [‘t Hooft’74, 75] [Witten’79]

5. Forpracticalpurposes, themesoniclagrangian can beorganised in theform • Usually, theoperators are builtwiththelowestnumber of derivatives as • - operatorswith more derivativestendtoviolate • theasymptotichigh-energybehaviour • prescribedby QCD for Green-fuctions and form-factors • - in some cases, higherderivativeoperators • can be removed bymeans of theEoM (fieldredefinitions) [Ecker et al.’89] [Rosell, Pich, SC’04] [Xiao & SC’08]

6. At LO in 1/NC(large NC): • TheoperatorswithonlyGoldstones (no Resonance) are thosefromcPT: • In the case of thevectors (in theantisymmetric tensor form.), • withthe canonical free-fieldkinetictermgivenby [Gasser & Leutwyler’84,85] [Ecker et al.’89] [Ecker et al.’89] [Gasser & Leutwyler’84] [Ecker et al.’89]

7. At NLO in 1/NC(loops + NLO tree-level): [Rosell, Pich, SC’04] [Ruiz-Femenía, Portoles, Rosell’07] [Kampf, Novotny, Trnka’09] … • Thenaive dimensional analysistellsusthat • - The LO amplitudescaleslikeM~ p2 • - The NLO one-loopamplitudeM~ p4ln(-p2) • UV-divergencesM~ l∞p4(NLO in 1/NC) • O(p4) subleadingoperatorstorenormalize • E.g. in the case of thepp-VFF, [Rosell, Pich & SC’04]

8. However, higherpowercorrections ( likee.g.M~ p4 ) • may look potentiallydangerousif p2~ MR2 •  as there no characteristicscaleLRcT • thatsuppressesthe NLO for p <<LRcT • A solutiontothisissuewillbeachieved • bymeans of mesonfieldredefinitions • forthe case of thepp-VFF…

9. pp-Vector Form Factor

10. pp-Vector Form Factor: Basic definitions and one-loopcomputation

11. Thepp-VFF isdefinedbythescalarquantitysF(q2), • The VFF can bedecomposed in 1PI contributions of theform • Example of thetree-level LO form factor

12. Assumptions: • Truncation of theinfinitenumber of large-NChadronicstates • Only up to O(p2) NGB operators at LO • +only up to O(p4) NGB operators at NLO • (which can bejustifiedunder short-distancearguments) • Justthelowestthresholds (pp-cut) willbetakenintoaccount • (higheronessupposedtoberenormalized in a decouplingscheme, • withtheircontributionsupressedbelowtheirproductionthreshold)

13. Explicitone-loopcalculation Loopcontributionstothevertexfunctions,

14. Thisrequirestheintroduction of • tomaketheamplitudefinite • Here ( [Pich,Rosell, SC’04] [SC’09] ) the MS(+constant)schemewasused

15. Thisleavestherenormalizedvertexfunctions [Rosell, Pich & SC’04] withthefinitetrianglecontribution, x0 being x∞

16. Thecouplings in therenormalized amplitudes • Renormalizedcouplings • The NLO running in GV(m) • induces a residual NNLO m-dependence • Thiswillallowusto use the RGE • toresumpossiblelargeradiativecorrections

17. pp-Vector Form Factor: Fieldredefinitions and redundantoperators

18. TheLVNLOcouplings are notphysicalbythemselves • Impossibletofixthe XZ,F,Galone, • justcombinations of them and othercouplings • Thereasonisthatthey are proportionaltotheEoM, • and alsoobeyingthe KG equation, [Rosell, Pich & SC’04] at least 2 mesonfields at least 1 externalsource

19. Thecan betransformedby a vector fieldredefinitionVV+x • intootherresonanceoperatorswithlessderivatives: MV, FV, GV • +theoperator • + otheroperatorsthat do notcontributetothepp-VFF

20. Similarly, higherderivativeLV operatorsforthe VFF can besimplified • Thustheundesired O(p4) resonanceoperators (and their O(p4) contributions) • can be removed fromthetheory • We can removetherenormalizedpart of thisoperatorsthrough V V + x(XZr,XFr,XGr) [Rosell, Pich & SC’04]

21. Thisfieldtransformation: • removesXrZ,F,G • encodestheirrunninginto • Althoughthefieldredefinitionx(XZr,XFr,XGr) • dependsonthe particular munderconsideration (sinceXrZ,F,G do), • theresultingtheoryisstillequivalenttothe original one • AfterremovingXrZ,F,G, • theremainingcouplings are ruledby

22. Ifnowone sets thescalem2= -q2 (≡Q2) , the VFF takesthe simple form • withtheevolution of theamplitudewith Q2 • providedbytheevolution of thecouplings(throughthe RGE) • Forinstace, the MV(m) wouldberelatedtothe pole massthrough

23. pp-Vector FormFactor: RGE solutions

24. The RGE for MV and GVprovide a closesystem of equations, • withsolution of theform, withtheintegration variables k and L

25. The RGE for MV and GVprovide a closesystem of equations, • withsolution of theform, withtheintegration variables k and L • This produces the flux diagram:

26. The RGE for MV and GVprovide a closesystem of equations, • withsolution of theform, withtheintegration variables k and L • This produces the flux diagram:

27. The RGE for MV and GVprovide a closesystem of equations, • withsolution of theform, withtheintegration variables k and L • This produces the flux diagram: • Thereappearsan IR fixedpoint • at MV=0, 3GV2=F2 • whereallthetrajectories converge

28. Thesamehappensfortheremainingcouplings, whichfreezeout at m0 • Foranyinitialcondition, • the vector parametershavealwaysthesame IR fixedpoint (m0), •  istheonlyonethatdependsonthem=0 condition • Thetheoryevolvesthen • fromthe IR fixed-point in thecPTdomain, • up tohigherenergiesthroughthe log runninggivenbythe RGE

29. Remarkablefact #1 • Itisremarkablethat • thevaluesforthe FV and GV IR-fixedpoints • agreewiththoseobtainedat large-NC • afterdemandingtherightbehaviour at Q2∞ : • -Forthepp-VFF  FV GV =F2 • -Forthepp-partial wave scattering  3 GV2 = F2 [Ecker et al.’89] [Guo, Zheng & SC’07]

30. Remarkablefact #2 • Likewise, itisalsointerestingthattherequirement • fortheresummed VFF, • leads tothesamevalues as thosefromthe IR-fixedpoint, • and, in addition, • forallm (beingm-independent and withtheirrunningfrozenout)

31. Theagreement of theresummed VFF and data isfairlygood. • Thus, forthe inputs GV(m0)2=0 (sameother inputs) GV(m0)2=F2 (sameother inputs) • However, • The non-zeropmassisresponsiblefor a 20% decreasing of ther-width • Anaccuratedescription of bothspace-like/time-like data • needstheconsideration of thepNGBmasses [Guo & SC’09]

32. pp-Vector Form Factor: Perturbativeregime

33. Independently of possiblehigh-energymatching, • the RGE show theexistence of a region in thespace of parameters • (closeenoughtothe IR-fixedpoint at m0) • wheretheloopcorrections are small • and one has a slowlogarithmicrunning • Thisfills of physicalmeaningthe formal 1/NCexpansion • basedonthe formal NCscaling of thelagrangianoperators • Thetheory can bedescribedperturbatively as far as one • iscloseenoughtothefixedpoint • ANALOGY: • A fixedorder QCD calculationisformallyvalidforanym • (and independent of it) • RGE perturbationtheoryonlyvalidforlargem

34. In our case, theparameterthatactually rules • thestrength of theResonance-Goldstoneinteraction • in the RGE is • Thisparametergoesto 0 whenm0 • Aroundthe IR-fixedpoint (m0) thecouplingsvaryslowly • Thus, Althoughthe formal expansionis 1/NC • aVisthe actual quantitysuppressingsubleadingcontributions • Since at LO aV≈GV/MV≈ 0.2 form ≈MV, • the 1/NCexpansionmeaningful in theresonanceregion • as far as itisnarrowenough(as itisthe case)

35. Nevertheless, • - In case of broadstates, • - or more complicateprocesses, • Lessintuitiveidentification • of theparameterthatcharacterizes • thestrength of theinteraction • However, PerturbationTheorywillmakesense • as far as we are abletofind a momentumregionwhere • - Thisparameterbecomessmall • - One has a slowrunning of thecouplings

36. Conclusions

37. A priori, higherderivativeoperatorsneededfor 1 looprenormalization • However, there are redundantoperators • (which can be removed throughmesonfieldredefinitions) • RGEexistence of an IR-fixedpoint • + slow log running in theregionaroundm0 • whereaVissmallenough • Thisiswhere a perturbativeexpansion in 1/NCmakessense: • Amplitudegivenbythe RGE evolution of thecouplings • fromthe IR-fixedpoint at 0 up to Q2 • Theseconsiderations are expectedtoberelevant • forother QCD matrixelements • In particular, forScalars (wherewidth and corrections are large) • e.g., thescalarform-factor