Dispersion relation (10 min)

Transcription

1 Indiana chool lecture (black board); prepared by I.V.Danilkin (JLab, June, 5) Literature: [] M. E. Pekin and D. V. Schroeder. An Introduction to Quantum Field Theory. Wetview Pre, 995. [] Batian Kubi lecture [3] F. J. Yndurain. Low-energy pion phyic. arxiv: hep-ph/8,. Diperion relation ( min) Cauchy theorem d ' FH 'L FHL i () ' - G for holomorphic function FHL and G i a rectifiable path. We can alway deform the contour until the cloet ingularity. If FH L on the large emi-circle then d ' FH 'L FHL i d ' FH ' + i ΕL - d ' FH ' - i ΕL ' - i 4 m ' - i 4 m ' - d ' FH ' + i ΕL - FH ' - i ΕL d ' Dic FH 'L i 4 m ' - i 4 m ' - Dic FHL FH + i ΕL - FH - i ΕL G () Analyticity: FHL i 4 m d ' Dic FH 'L (3) ' - Reflection principle: F* H + i ΕL FH - i ΕL (4) allow to relate imaginary part of the amplitude (unitarity) to analytical continuation of the amplitude (i.e. diperion relation) Exercie: Dic FHL FH + i ΕL - FH - i ΕL FH + i ΕL - F* H + i ΕL i Im FHL Subtraction: In general, one can alway introduce ubtraction (5)

2 Documentation_Indiana_chool.nb FHL d ' ' Im FH 'L d ' Im FH 'L d ' Im FH 'L H - L d ' Im FH 'L + 4 m 4 m 4 m ' - ' - ' - ' - ' - ' - (6) A a reult we got an improved convergence at (due to additional power of ' in the denominator); um rule (when the integral converge) d ' Im FH 'L FH L 4 m (7) ' - Exercie: If the integral doe not converge ufficiently fat on the large emi-circle, derive diperive relation for (auming FH L i Real) FHL - FH L (8) - General formula. Note that by introducing ubtraction you improve the convergence at, however there are additional parameter to determine. n- FHL â i FHiL H L H - Li + i! H - Ln d ' Im FH 'L n H ' - L ' - (9) Calculation of the Dic ( min) Intead of making full loop calculation one can ue Cutkoky (cutting) rule: p - m + i Ε - i Ip - m M () Example: d4 l im 4 H L Il - m + i ΕM IHp - ll - m + i ΕM d4 l Dic M H-iL H L4 H- il Il - m M H- il IHp - ll - m M l + m l 4 d l d l l d l dw, Il - m M Il - l - m M IHp - ll - m M Ip + l - p l - m M I - p l p l - p l l l M, l, cm : p I, M, l Hl, ll ()

3 Documentation_Indiana_chool.nb p l p l - p l 4 l, cm : p I, M, l Hl, ll l d l i Dic M d W I - l l M Kl l dl l 3 - m l d l, I - l M 4 m i Dic M O - i Im M 8 Therefore 4 m Im M 6 ΡHL () 6 Pion vector form factor (5 min) We conider the proce of a tranition of a photon into a pair of pion. Thi i an important building block: repoible for a hadronic part of e+ e- + -, Τ- - ΝΤ,... The matrix element: Y+ HpL - HqL J Μ HL ] Hp - ql Μ FV HL (3) where J Μ i the EM current and FV HL i the pion vector form factor (normalized FV HL ). Thi proce doe not have any croed channel exchange and therefore no left-hand cut. Note alo, that the factor Hp - ql Μ inure gauge invariance when the two pion are on-hell (i.e. Hp - ql Μ Hp + ql Μ p - q m - m ). At very low energy the pion vector form factor can be calculated in Chiral Perturbation Theory (ΧPT). At NLO it read ()

4 4 Documentation_Indiana_chool.nb L ~ i A Μ H+ Μ Μ L FV HL + 6 H4 f L HL6 - L + 6 f (4) - I - 4 m M J HL + OI M, L6» 6 Improvement: diperion relation need to calculate the Dicontinuity (Cutkoky rule again) Hp - ql Μ Dic FV HL d4 l H-iL H L4 H- il Il - m M H- il IHp - ll - m M TI* H, zl Hp - ll Μ FV HL (5) p p + q, Hp + ql, z co Θ i c.m. angle note alo that Hp + ql 9,, p q l, p q l ΡHL We obtain Hp - ql Μ Dic FV HL Hp - ql... Exercie: Show that dw TI* H, i 64 ΡHL FV HL dw TI* H, zl Hp - ll Μ zl Hp - ll Μ L Hp + ql Μ + L Hp - ql Μ... then contract with Hp + ql, (6)

5 Documentation_Indiana_chool.nb 5 * * dw TI H, zl Hp - ll Μ Hp - ql Μ â z z TI H, zl (7) - where TI H, zl i the amplitude TI H, zl 3 âh l + L tli HL Pl HzL l (8) l l' â z Pl HzL Pl' HzL - l+ Since Dic FV HL i Im FV HL, we get Im FV HL ΡHL FV HL t* HL ΘI > 4 m M (9) If we conider only elatic cattering: t HL in HL e i HL ΡHL t* HL in HL e -i HL, ΡHL () Waton final tate theorem: the phae of the form factor i determined by the two-particle cattering phae hift: FV HL FV HL e i HL FV HL FV HL e i HL () Arg IFV HLM HL The full Omne Mukhelihvili problem ( min) We want to find the mot general repreentation for a function, FHL, which i the analytic in the complex plane with the cut from A4 m, E, auming that we know it phae on the cut, Arg HFHLL HL, > 4 m () Solution i not unique, if F HL i a olution, then eα F HL i a olution too. Need to know the aymptotic information! We look for a olution in the from FHL PHL WHL HWH + i ΕL - WH - i ΕLL WH + i ΕL in HL e -i HL, i e i HL - e -i HL WH + i ΕL i i e -i HL WH + i ΕL e - i HL WH - i ΕL i W H + i ΕL e- i HL WH - i ΕL, ln W H + i ΕL - i ln W H + i ΕL DicHln WHLL i HL Diperion relation i (3)

6 6 Documentation_Indiana_chool.nb ln WHL d ' DicHln WH 'LL d ' H 'L, 4 m ' - ' - d ' H 'L d ' H 'L WHL Exp, WHL Exp a + 4 m ' - 4 m ' ' - i 4 m WHL Exp d ' H 'L 4 m ' ' -, WHL (4) (5) One ubtraction: normalization WHL. The function W() i known a the Omne function. Many application! Exercie: Show that Arg WHL HL if HL Α, how WH L (6) Α Ue f H 'L d ' - i Ε f H 'L d p.v. ± i f HL (7) ' - If one aume aymptotic: FHL, then FV HL WHL Numerical implementation (5 min) WHL Exp Tangent tretching Let now conider the integral d ' H 'L 4 m ' ' - (8)

7 Documentation_Indiana_chool.nb 7 f HyL â y a In order to account the whole region, the replacement i needed y yhxl, which change the integrating range (a,) into (,) f HyL â y f HyHxLL a dy â x, yhxl a + Cext tg dx dy Cext x ; dx coi xm n f HyL â y â f Hyi L wi. i If the integrand i mooth, it i convenient to ue Gauian weight for integration. Few word about Cext : A you can ee y Hx L a + Cext. It mean that n/ point will be accounted before a + Cext and n/ after. It i very ueful when you know the general behavior of your function. If you do not, jut put it equal to unity, Cext. Example : Ù e-y â y << NumericalDifferentialEquationAnalyi` n ; Cext ; wg GauianQuadratureWeight@n,, D; yn + Table@Cext * Tan@ * wg@@i, DDD, 8i, n<d; Cext wgn TableBwg@@i, DD, 8i, n<f; Co@ wg@@i, DDD ^ Sum@Exp@-yn@@iDD ^ D wgn@@idd, 8i,, n<d Integrate@Exp@-y ^ D, 8y,., Infinity<D If the integral contain a quare root ingularity Iomething like Ùa f Iy, dy yhxl a + Cext tg x ; Cext dx y - a M â ym it i ueful to introduce another replacement tg x coi xm Principle value integral Very often one ha to take integral of the following form FHL 4 m â' f H 'L ΘI < 4 m M ' ' - - i Ε â ' f H 'L 4 m ' ' - â' f H 'L +ΘI > 4 m M 4 m ' ' - - i Ε (9) The latter integral can be written a â' f H 'L p.v. ' ' - -i Ε â ' f H 'L f HL + i (3) ' ' - For the p.v. integral we ue the following trick: p.v. â ' f H 'L ' ' - p.v. â ' f H 'L - f HL + f HL 4 m ' ' - â ' f H 'L - f HL ' ' - 4 m f HL ln m (3)

8 8 Documentation_Indiana_chool.nb p.v. 4 m 4 m â' ln ' H ' - L - 4 m All together (Omne function) WHL Exp d ' H 'L 4 m ' ' - (3) Exit@D; << NumericalDifferentialEquationAnalyi` SetDirectory@NotebookDirectory@DD; StyleLit 8AboluteThickne@.D, AbolutePointSize@5D, ð< & 8Blue, Red, Green, Purple<; SetOption@Plot, Frame -> True, Axe -> 8True, Fale<, PlotStyle -> StyleLit, ApectRatio ->.8, FrameStyle -> Directive@3DD; << PiPi_Madrid.m; mpi.38; mk ; Mrho.77; Ε.; LamPhShift.3; OmneNInt@ig_D@_D : Exp@ Pi * NIntegrate@deltaFinal@D@bD Hb Hb - - I ig ΕLL, 8b, 4 mpi ^, 4 mk ^, Infinity<, AccuracyGoal 5, MaxRecurion DD; nomn ; Cx.; wg GauianQuadratureWeight@nOmn,, D; 4 mpi ^ ; n + Table@Cx * Tan@Pi * wg@@i, DDD ^, 8i, nomn<d; wgn Table@Cx * * Tan@Pi * wg@@i, DDD * wg@@i, DD * Pi HCo@Pi * wg@@i, DDD ^ L, 8i, nomn<d; Deltan Table@deltaFinal@D@n@@iDDD, 8i, nomn<d; Clear@deltaD; OmneTemp@_D Exp@ * Sum@Deltan@@iDD Hn@@iDD Hn@@iDD - LL * wgn@@idd, 8i,, nomn<dd; OmneTemp@_D Exp@ * Sum@HDeltan@@iDD - deltal Hn@@iDD Hn@@iDD - LL * wgn@@idd, 8i,, nomn<dd; Omne@ig_D@_D : Which@ig, OmneTemp@D, ig ÈÈ ig -, delta delta@d@d; Which@ 4 mpi ^, OmneTemp@D, > 4 mpi ^, Exp@I * ig * delta@d@dd * Exp@delta Log@ H - LDD * OmneTemp@DDD; (3)

9 Documentation_Indiana_chool.nb ^ D * 8 Pi, 8par, mpi,.<, PlotPoint 5, MaxRecurion, PlotStyle 8Black, Thick<, FrameLabel " HL">F, DeltaExp@D, ImageSize 85, 5<F HL 5 GraphicRowB:PlotBRe@Omne@D@par ^ DD, 8par, mpi,.<, PlotStyle 8Black<, MaxRecurion, PlotPoint, PlotRange All, ImageSize 85, 5<, FrameLabel "ReHWHLL">F, PlotB8Im@Omne@D@par ^ DD<, 8par, mpi,.<, PlotStyle 8Black<, MaxRecurion, PlotPoint, PlotRange All, ImageSize 85, 5<, FrameLabel "ImHWHLL">F>F ImHWHLL ReHWHLL Omne@D@D OmneNInt@D@D ä