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Isospin Violation, Light Quark Masses, and All That *

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Isospin Violation, Light Quark Masses, and All That * CPC(HEP & NP), 2010, 34(9): 1163{1168 Chinese Physics C Vol. 34, No. 9, Sep., 2010 Isospin violation, light quark masses, and all that * Ulf-G. Meißner1;2;1) 1 HISKP and Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany 2 IKP-3, IAS-4 and JCHP, Forschungszentrum Julic¨ h, D-52425 Julic¨ h, Germany Abstract Isospin violation is driven through the light quark mass difference and electromagnetic effects. I review recent progress in extracting the light quark mass difference and tests of the chiral dynamics of Quantum Chromodynamics in various reactions involving light as well as heavy quarks. Key words quantum chromodynamics, isospin violation, quark masses, chiral Lagrangians PACS 12.38.-t, 12.39.Fe, 11.30.Hv 1 Introduction and motivation ularly sensitive to IV [1]. This problem was read- dressed in the framework of heavy baryon CHPT In the Standard Model (SM), isospin violation has about a decade ago, see e.g. Ref. [2]. Interest was two sources, namely the differences of the light quark rekindled due to the precision measurements of pionic (u, d) masses (QCD) and their charges (QED), hydrogen and deuterium at PSI [3]. In the framework of covariant baryon CHPT, [4] the elastic π−p am- 1 ¯ HQCD(x) = (md −mu)(dd−uu)(¯ x); (1) plitude was calculated at threshold to third (leading 2 ie one-loop) order in the chiral expansion. In Ref. [5], H (x) = − uγ¯ Aµu−dγ¯ Aµd (x) : (2) QED 2 µ µ the extension to all physical channels was given based on the Feynman graphs shown in Fig. 1. Isospin violation (IV) thus offers a unique window to access the light quark mass difference (ratio) in light or heavy-light quark systems as I will show in a va- riety of examples in this talk. In most cases these strong (str) and electromagnetic (em) effects are of the same size, cf. the neutron-proton mass difference discussed below, and must be treated consistently. Furthermore, in many cases the small IV effects sit on top of a large isospin-conserving \background" and thus an accurate theoretical machinery is manda- tory to be able to extract the information encoded in the isospin-violating contributions. This machinery is given by chiral perturbation theory (CHPT) coupled to virtual photons. This will constitute the theoreti- Fig. 1. IV in πN scattering: loop diagrams at cal framework underlying the topics presented here. third order. Solid, dashed and wiggly lines denote nucleons, pions and photons, in order. 2 Isospin-breaking in the pion- nucleon scattering lengths In particular, analytical formulae to leading oder 2 in the isospin breaking parameter δ = fmd − mu;e g Already decades ago, Weinberg pointed out that are given, e.g. the IV shift for the isospin-symmetric systems of (neutral) pions and nucleons are partic- amplitude a+ reads: Received 19 January 2010 * Supported by DFG (SFB/TR-16), EU FP7 HadronPhysics2, HGF VH-VI-231 and BMBF (grant 06BN9006) 1) E-mail: [email protected] ©2010 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 1164 Chinese Physics C (HEP & NP) Vol. 34 2 for CSB was found in reactions involving the produc- + mp 4∆π e ∆a = 2 c1 − 4f1 +f2 − tion of neutral pions. At IUCF non-zero values for 4π(mp +Mπ) F 2 π the dd ! απ0 cross section were established [8]. At 2 gAMπ 33∆π 2 TRIUMF a forward-backward (fb) asymmetry of the 2 2 +e ; (3) 32πFπ 4Fπ differential cross section for pn ! dπ0 was reported where f1;f2 (c1) are dimension two em (strong) low- [9], 2 2 π energy constants (LECs), ∆π = Mπ − Mπ0 (with Mπ =2 dσ dσ the charged pion mass) is the charged-to-neutral pion (θ)− (π−θ) dcosθ Z0 dΩ dΩ mass difference, F the pion decay constant and m Afb = π = π p =2 dσ dσ the proton mass. Note in particular the sizable pref- (θ)+ (π−θ) dcosθ Z dΩ dΩ actor 33=4 that signals the large contribution from the 0 (17:28(stat)5:5(sys))·10−4 : (4) triangle diagram (s5) in Fig. 1. The numerical results for all pion-proton threshold amplitudes are collected Theoretically, Afb is directly related to the interfer- − − in Table 1. The imaginary part in π p ! π p is ence of an isospin-conserving and an IV amplitude, due to the π0n intermediate state in the rescattering dσ A1 graph (s3). In the charge-exchange amplitude, the = A0 +A1P1(cosθπ)+::: ! Afb ' : (5) dΩ 2A0 contribution from the triangle graph is absent and The first calculation utilizing chiral EFT was per- thus the corrections are smaller. Note also that the formed in Ref. [10]. This was improved recently in uncertainty is largely given by our poor knowledge of Ref. [11]. It was shown that in the rescattering graph the dimension two and three em LECs. In particular, not only the IV Weinberg-Tomozawa term but also the IV corrections to the triangle ratio (that vanishes the neutron-proton mass difference on the nucleon in the isospin limit) turn out to be (1:51:1)%, consis- lines plays a role, as shown in Fig. 2. These two LO tent with earlier findings in heavy baryon CHPT [6]. contributions combine in such a way, that A at LO An remarkably large shift due to the cusp effect in the fb in the chiral expansion is entirely given by the strong π0p amplitude is predicted, that might be measured neutron-proton mass difference. in third generation photoproduction experiments at HIγS or MAMI. An extension of this work above threshold is given in Ref. [7]. Table 1. Predictions for the IV shifts to all pion-proton scattering lengths in units of −3 10 =Mπ. isospin limit channel shift a+ +a− π−p π−p 3:4+4:3 +5:0i ! − −6:5 a+ a− π+p π+p 5:3+4:3 − ! − −6:5 Fig. 2. (color online). LO (rescattering) dia- p2a− π−p π0n 0:4 0:9 − ! grams for IV S-waves in pn ! dπ0. Left: IV a+ π0p π0p 5:2 0:2 ! − in the πN vertex. Right: IV contribution due to the neutron-proton mass difference. 3 The strong neutron-proton mass A crucial ingredient for a precise calculation of A difference from np ! dπ0 fb is to take A0 from the precision PSI experiment [12] using isospin invariance. Putting pieces together, one Amongst the IV effects in hadronic reactions the str can express the fb-asymmetry in terms of δmN as ones that are charge-symmetry-breaking (CSB), i.e. LO −4 str that emerge from an interchange of up and down Afb = (11:53:5)·10 (δmN =MeV) : (6) quarks, are of particular interest. Their importance From the measured asymmetry Eq. (4) one can thus is due to the fact that the neutral-to-charged pion deduce mass difference, which is almost entirely of em origin δmstr = 1:50:8(exp)0:5(th) MeV : (7) and usually dominates IV hadronic observables, does N not contribute here. Therefore, the sensitivity to the This is nicely consistent with the extraction based on str quark mass difference md −mu is enhanced in observ- the Cottingham sum rule, δmN = 2:050:30 MeV [13] str ables related to CSB. Recently, experimental evidence and a recent lattice determination, [14] δmN = 2:26 No. 9 Ulf.-G. Meißner: Isospin violation, light quark masses, and all that 1165 0:570:420:10 MeV. This calculation is a particular beautiful example how very different quantities and processes are related through the same LECs { here the dimension two LEC c5 that drives the strong IV in the nucleon sector [15]. It also shows that a refined NLO calculation of Afb will offer a viable alternative to extract this fundamental quantity (the correspond- ing isospin-symmetric calculations of pion production where recently performed [16]). 4 EM corrections to η ! 3π decays The charged (η ! π+π−π0) and neutral (η ! 3π0) Fig. 3. (color online). Real part of the neu- three-pion decays of the η meson are driven by isospin tral amplitudes corresponding to GL (dashed), BKW (dot-dashed), and DKM (full) for t = u. violation, The hatched regions denote the error bands − − mu −md due to a variation of electromagnetic low- Γ (η ! 3π) ∼ Q 4; Q 1 = : (8) ms −m^ energy constants. The vertical lines show the limits of the physical region. The amplitude for η ! 3π has been calculated to two-loop accuracy in CHPT [17] and em corrections 0 0 0 2 ! at O(e mquark) were analyzed in Ref. [18] (BKW). 5 The cusp in η ηπ π Since these turned out to be very small, in Ref. [19] (DKM) the formally subleading em corrections of The discovery of the cusp in K+ ! π+π0π0 by 2 O(e (mu−md)) have been calculated. There are vari- the NA 48/2 collaboration [22] has renewed interest ous reasons why one should consider these subleading in such effects (for early works see e.g. Refs. [23, 24]) IV corrections: by restricting oneself to terms of the since it allows for a precise extraction of the pion-pion 2 2 form e m^ (where m^ = (mu+md)=2) and e ms, one ex- scattering lengths [25{27]. The cusp effect is driven cludes some of the most obvious electromagnetic ef- by the rescattering from two neutral pions through fects: real and virtual photon contributions, as well as an intermediate charged pion pair and should thus effects due to the charged-to-neutral pion mass differ- also be visible in other processes like η ! 3π0 (as ence (which is predominantly of electromagnetic ori- discussed before) or η0 ! ηπ0π0.
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