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 = {md − mu,e } 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- √2a− 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 corresponding 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 −mˆ 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 mˆ (where mˆ = (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. While the former 2 gin), both of which scale as e (md−mu). These mech- reaction has been measured and the data are not yet anisms fundamentally affect the analytic structure precise enough to analyze the cusp (see e.g. Ref. [28]), of the amplitudes in question: in the charged decay it was pointed out in Ref. [29] that in the isospin limit channel η → π+π−π0, there is a Coulomb pole at the BR(η0 → ηπ+π−)= 2BR(η0 → ηπ0π0) so that the boundary of the physical region (at the π+π− thresh- cusp should be strongly visible in η0 → ηππ decays. old), while in the neutral decay channel η → 3π0, the In that paper, a detailed analysis of the cusp based on pion mass difference induces a cusp behavior at the non-relativistic EFT, which is the most appropriate π+π− thresholds (see next section). At this order, tool to investigate such phenomena, was performed. the neutral and charged amplitudes must be calcu- The central results in that paper are the various decay lated separately. amplitudes, calculated to two loop accuracy. Invok- It turns out that the em corrections are in general ing theoretical information on the coupling constants small, but need to be accounted for if one wants e.g. involved, the size of the cusp effect can be predicted. to determine the Dalitz slope parameters to a high ac- It reduces the decay spectrum below the charged-pion 2 curacy. Also, corrections of order e (md−mu) (DKM) threshold by more than 8%, see Fig. 4. This is a much turn out to be as big (or even bigger) as the ones of or- more sizeable effect than e.g. in η → 3π0 decays. Ap- 2 der e mquark (BKW). In addition, cusps in the neutral proximate isospin symmetry dictates that the cusp amplitude due to rescattering π0π0 → π+π− → π0π0 of O(a2) above threshold is strongly suppressed, and are clearly visible, see Fig. 3 where the em corrections three-loop effects can be estimated to yield a correc- are shown in comparison to the strong one-loop result tion below 1%. Therefore the threshold singularity in of Ref. [20] (GL). For more details, I refer to Ref. [19] η0 → ηπ0π0 is determined to very high precision by (see also [21]). the leading O(a) rescattering effect. Experimental 1166 Chinese Physics C (HEP & NP) Vol. 34 verifications of these predictions at various laborato- 0.56. Most recent determinations of this ratio includ-

ries are eagerly awaited. It was also shown in [30] that ing also lattice data give mu/md = 0.47 0.08 [34], the πη threshold parameters can not be deduced from which is barely compatible with the value extracted the measured decay spectrum. This can be seen as from Eq. (9). It is therefore of fundamental interest to follows: Since the opening of the πη channel is at the understand theoretically the discrepancy between the border of the Dalitz plot there is no one-loop cusp in values of the up-down quark mass ratio determined the physical region as in ππ scattering. Further, there from different sources. In Ref. [35] it was shown that is an exact cancellation between the product of two- the theoretical framework to relate the ψ0 decays into loop and tree graph and the product of two one-loop J/ψπ0(η) is not yet accurate enough for extracting graphs. Thus, without exact theoretical knowledge of the quark mass ratio to the desired precision. In that the tree-level couplings an extraction of πη threshold paper, the contribution from intermediate charmed parameters in this framework is not possible. meson loops (see also the talk by Zhao at this work- shop) was worked out to leading order in HQEFT. Be- cause the ψ0 and J/ψ are SU(3) singlets, it is obvious that the decay ψ0 → J/ψπ0 violates isospin symmetry, and the decay ψ0 → J/ψη violates SU(3) flavor symmetry. Accordingly, the decay amplitudes reflect the flavor symmetry breaking. Here, all the charmed mesons in a flavor multiplet can contribute, and it is the mass differences within the multiplet that gener- ates the isospin or the SU(3) breaking. In fact, for heavy quarks, the velocity is the appropriate expan-

sion parameter, here v ∼ (2MDˆ −Mψˆ )/MDˆ ' 0.5,

where MDˆ is the averaged pcharmed-meson mass, and

Mψˆ = (MJ/ψ +Mψ0 )/2. As it is well-known for charm systems, this expansion parameter is not very small Fig. 4. (color online). The differential decay but for our purposes of discussing the influence of the rate dΓ/ds3 divided by phase space for the tree (dashed) and full (band) amplitude that so far neglected loops, a leading order analysis in v clearly shows the cusp effect. suffices. The loops with virtual intermediate charmed mesons themselves scale as v3 · v2/(v2)2 = v, whereas

the remaining terms proportional to mq (that lead to 6 The light quark mass ratio mu/md the prediction Eq. (9)) are proportional to an energy from ψ0 decays scale of O(v2). Consequently

0 0 M(ψ → J/ψπ ) ∼ (m −m )|~qπ| , The decays of the ψ0 into J/ψπ0 and J/ψη were direct d u 0 0 |q~π| suggested to be a reliable source for extracting the M(ψ → J/ψπ ) − ∼ (m −m ) . (10) D loops d u v light quark mass ratio mu/md, see e.g. Refs. [31, 32]. Based on the QCD multipole expansion and the axial Therefore, the charmed meson loop contribution is anomaly, the relation between the quark mass ratio potentially enhanced compared to the direct term. and the ratio of the decay widths of these two decays This power counting estimate was confirmed by ex- was worked out up to next-to-leading order in the plicit calculations of the various one-loop graphs in chiral expansion. Including leading SU(3) breaking Ref. [35]. One can take an extreme position and effects, one obtains assume that the intermediate charmed-meson loop 0 Br(ψ0 → J/ψπ0) mechanism saturates the decay widths of the ψ → Rπ0/η = = 0 Br(ψ0 → J/ψη) J/ψπ (η). This leads to Rπ0/η = 11%...14% (with an uncertainty of about 50% from higher orders), m −m 2 F 2 M 4 q~ 3 d u π π π which is larger than the experimental number [33], 3 2 4 . (9) md +mu Fη Mη q~η Rπ0/η = (3.880.230.05)%. This findings are simi-

Using the most recent measurement of the decay- lar to the ones of Ref. [36], where it was shown that width ratio [33], one obtains mu/md = 0.40 0.01 meson loops invalidate the direct relation between which diﬀers signiﬁcantly from the LO analysis of the the ratio Γ (η0 → π0π+π−)/Γ (η0 → ηπ+π−) and the

Goldstone boson masses, which leads to mu/md = quark mass ratio (md − mu)/(ms − mˆ ). A NLO cal- No. 9 Ulf.-G. Meißner: Isospin violation, light quark masses, and all that 1167 culation of the charmed meson loop contributions to terference between this term and the other terms that 0 0 ψ → J/ψπ (η) should be performed to find out how drives the mass splittings within the Σc iso-triplet to accurately the ratio mu/md can indeed be inferred have a different pattern compared to any other known from these decays. isospin multiplet. This leads one to expect that the isospin mass splittings in the charm hadrons are al- 7 Mass splittings in heavy baryon ways different from those in the bottom hadrons even if the heavy quark symmetry were exact. Besides the multiplets heavy baryons considered in Ref. [38], the D and B-

meson mass splittings, mD −mD0 = 4.780.10 MeV Mass splittings within isospin multiplets of and mB0 − mB = 0.37 0.24 MeV are a nice exam- hadrons appear due to both the mass difference be- ple for the effect, although the ordering does not get tween u and d quarks and electromagnetic effects. changed here. Since the d quark is heavier than the u, usually the There is no loop contribution to the mass split- hadron with more d quarks is heavier within one 0 ting between the two Ξc baryons, and we predict isospin multiplet. For instance, the neutron (udd) 0+ 00 mΞc − mΞc = −0.2 0.6 MeV. The present data is heavier than the proton (uud), and the K0(d¯s) is 0 for the masses of the Ξc baryons are not accurate heavier than the K+(u¯s). There is only one exception enough yet to test the prediction. For the Σb states, ++ + to this pattern, the Σc iso-triplet (Σ (cuu), Σ (cud) c c the β1h term interferes constructively with the other 0 and Σc (cdd)). The mass splittings within the Σc iso- terms and hence the loop corrections are less impor- triplet are measured 0 tant. The mass of the Σb and the mass difference m 00 −m 0− are predicted to be 5810.31.9 MeV and + 0 Ξ Ξ ∆1c ≡ mΣ −mΣ = −0.90.4 MeV, b b c c −4.0 1.9 MeV, respectively, which can be tested ∆ ≡ m ++ −m 0 = 0.270.11 MeV . (11) 2c Σc Σc in future experiments. For an alternative view, see Remarkably, the state with two u quarks has the Ref. [39]. largest and the one with a u and a d quark has the smallest mass. Only recently some of the bot- 8 Summary and outlook tom cousins of the Σc, Σb , were observed by the CDF Collaboration [37]. Their masses are for the Precision calculations in the light quark sector buu state m + = 5807.8 2.7 MeV and for the bdd lead to important tests of the flavor and the symme- Σb state m − = 5815.2 2.0 MeV, respectively — their try structure of QCD. I have discussed the intricate Σb 0 neutral partner Σb has not been observed yet. Thus, interplay of strong and electromagnetic isospin viola- here the natural ordering of the states seems to be tion for four very different processes, namely elastic restored. On the other hand, heavy quark symmetry pion-nucleon scattering, neutron-proton fusion to a relates baryons containing a b quark to those with a deuteron and a neutral pion, higher order electromag- c quark. netic corrections in η → 3π decays and the cusp in In Ref. [38], we have calculated the mass split- η0 → ηππ. There are also intriguing isospin-violating tings within the heavy baryon isospin multiplets Σc(b) effects in heavy-light systems. He, I focused on the 0 3 0 and Ξc(b) to O(p ) in the chiral expansion. For do- extraction of the light quark mass ratio from ψ de- ing that, we constructed both the strong and the em cays and the generation of mass splittings in baryon Lagrangians at O(p2) which are responsible for the multiplets with one heavy (c,b) quark. Clearly, re- mass corrections. In contrast to mass splittings in fined theoretical calculations are called for. I outlined light quark baryon multiplets, there is an additional a few directions that deserve more attention. I have operator that describes the hard virtual photons ex- pointed out that this field also offers various great changed between the heavy quark and light quarks experimental challenges that should not be missed. accompanied by a LEC β1h. Remarkably, this term has a different sign for the charm baryons and the I thank all my collaborators for sharing their in- bottom baryons. This is due to the fact that the sign sight into the topics discussed here. I am also grateful of the electric charge of the charm quark is different for the organizers for their support and efficient or- from that of the bottom quark. It is the different in- ganization. 1168 Chinese Physics C (HEP & NP) Vol. 34

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