Decays of $B$ Mesons to Two Light Pseudoscalars

David London; Jonathan L. Rosner; Michael Gronau; Oscar F. Hernandez

REVIEW

Decays of $B$ Mesons to Two Light Pseudoscalars

Michael Gronau, Oscar F. Hernandez, David London, Jonathan L. Rosner

High Energy Physics - Phenomenology · Submitted 18 Apr 1994 · 28 references

Pith's verdict: A clean, honest SU(3) method paper: the new content is the Eq. (23) triangle and the rate-only γ extraction, and the authors are upfront that the whole thing rests on dropping E, A, PA.

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CitationDavid London, Jonathan L. Rosner, Michael Gronau, Oscar F. Hernandez. "Decays of $B$ Mesons to Two Light Pseudoscalars." arXiv:hep-ph/9404283 (1994). https://pith.science/paper/PRSWDZHA

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Worth your time if you work on B → πK or on rate-only CKM phase extraction. The short version: the SU(3) decomposition is not theirs — they say so in Sec. VI and credit Zeppenfeld and Savage-Wise — but the simple linear relations among amplitudes are, and Eq. (23) in particular is the one that lasted. It ties the I = 3/2 πK combination to √2 A(B+ → π+π0) with the CKM ratio ru, and with the charge-conjugate relation it turns four self-tagging rates into a determination of γ. No time-dependent measurements, no tagging. That is the real contribution, and it is derived, not fitted: you can check every entry in Tables 1–3 by hand against the graphical decomposition, and the algebra closes. Sec. III D's construction of the shared sub-triangle from Eqs. (60)–(63) is also nicely done, and the observation that the |P|, |C| < |T| < |P′|/ru hierarchy kills most of the discrete ambiguities is a genuinely useful practical remark rather than decoration. The η/η′ section they themselves label as of limited use, which is the right call. On the stress test: it lands, but it is not news to the authors. Sec. III A gives the fB/mB and helicity arguments, Sec. III C says explicitly that the B+ → π+K0 and B− → π−K̄0 rates should be equal only if A′ can be neglected, and Sec. VI repeats that the whole programme is 'predicated on the relative unimportance of strong rescattering effects.' What is genuinely missing is a number: no estimate of A′/P′ or of the rescattering feed, so the reader cannot judge how biased the extracted γ would be. That is a real gap, not a fatal one, and the paper's own rate-equality check is the experimental handle on it — the stress test does not give them credit for that. Two smaller things. Eq. (66) handles leading SU(3) breaking through fK/fπ but the rt relation (29) is left with only a hand-wave about form factors. And the sentence in Sec. II D 3 around Eq. (38) — 'Here δT′ and δP′ Comparing (38) with (31)' — is a dropped clause that needs fixing. Audience: B phenomenologists, and anyone teaching the SU(3) toolkit. Send it out; the referees will mostly argue about rescattering, which is the right argument to have.

Referee report

Major objections (3)

[Sec. III.C, Eq. (24) and Fig. 4] The γ extraction identifies the weak phase of A(B+ → π+K0) = P′ + A′ as π, which requires A′ ≈ 0. The justification given (fB/mB and helicity suppression) is order-of-magnitude only; no estimate of A′/P′ or of rescattering is provided. Because the bias in the extracted γ is linear in A′/P′, the paper should state the level at which the CP-conjugate rate equality Γ(B+ → π+K0) = Γ(B− → π−K̄0) must hold for a given γ precision, and should give at least a crude bound on A′/P′ or cite one. The authors acknowledge the rescattering caveat in Sec. III, but the headline application in the abstract (s5) would be strengthened by making the quantitative tolerance explicit.
[Sec. III.B, Eq. (23)] The left-hand side of Eq. (23) is −(T′+C′), a difference of two amplitudes that are individually of order P′ if penguin dominance holds. The authors note this cancellation in the text following Eq. (53), but do not quantify the experimental precision required on the two large rates. A short discussion of the relative accuracy needed on Γ(B+ → π0K+) and Γ(B+ → π+K0) to extract the small I = 3/2 combination would make the feasibility claim concrete and would connect the abstract's statement about determining γ to realistic event yields.
[Sec. III.D, Fig. 5] The counting of discrete ambiguities (four-fold for the sub-triangle, eight-fold when the penguin magnitudes are included) is asserted rather than derived. Since the hierarchy argument is then used to reduce the ambiguity to two-fold, a sentence or two showing the circle-intersection construction explicitly would let the reader verify the counting and would clarify which reflection symmetries are being removed by the hierarchy |P|, |C| < |T| < |P′/ru|.

Minor objections (7)

[Eq. (44) vs. Appendix (A.1)–(A.5)] Please verify the normalization of the reduced matrix elements in Eq. (44). A direct substitution of E = A = PA = 0 into (A.1)–(A.2) appears to give {81} = −(√15/4){1} rather than −(√5/4){1}, and (A.4)–(A.5) do not appear to give {83} = {27}/4 at all. If the {1}, {81}, etc. in Eq. (44) are defined with a different normalization than in the Appendix, this should be stated explicitly.
[Sec. II.D.3, Eq. (38) and surrounding text] The sentence beginning 'Here δT′ and δP′ Comparing (38) with (31)...' appears garbled or truncated; please repair it so the definition of the strong phases is unambiguous.
[Sec. III.D, list of five rates] In the list of five rates used to determine |T|, |C|, ΔC, ΔT, and ΔP−γ, the entries 'B0 → π+K0' and 'B0 → π+π0' should presumably read 'B+ → π+K0' and 'B+ → π+π0'; please correct.
[Table 1, B0 → η8η8 entry] The formatting '1/3√2' is ambiguous; please clarify whether the coefficient is 1/(3√2) or √2/3, and check the normalization against the other entries in the row.
[Sec. IV, Tables 4–5] The predictions in Tables 4 and 5 are given for the mixing angle φ = 19.5° only, while the text mentions the φ ≈ 10° case. A footnote or a second column for the alternative mixing would make the scheme-dependence of the predictions (e.g., the vanishing of P in Bs → ηK̄0 and of P′ in B0 → K0η) clearer.
[Sec. V.C, Eq. (66)] The replacement of ru by (fK/fπ)ru in Eq. (66) is a heuristic estimate rather than a derived SU(3)-breaking correction; please state this explicitly so the reader does not over-interpret the corrected relation.
[References [21] and [22]] Both are listed as unpublished or submitted. Since the γ method is the paper's headline application, the priority relative to [21] should be stated clearly in the text, not only in the reference list.

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The paper

The paper claims

The decays of B mesons into two light pseudoscalars obey triangle relations in flavour SU(3), and one of those triangles turns measured decay rates of charged B mesons into a determination of a CP-violating quark-mixing phase.

The reading

The paper asks a practical question: given that CP violation is supposed to come from phases in the quark-mixing matrix, which B-meson decays let you actually read those phases off, and with the fewest and easiest measurements? Its answer is that the charmless two-body decays $B to PP$ (a B meson into two pseudoscalar mesons such as pions and kaons) are tied together by relations that follow from treating the three light quarks as a symmetry triplet. Written as complex numbers, those relations are triangles: three amplitudes whose vector sum closes. The most useful one links the isospin-3/2 part of $B to pi K$ to the single amplitude $B^+ to pi^+ pi^0$, scaled by the small ratio $r_u = V_{us}/V_{ud} approx 0.23$. If the relations hold, a handful of measured rates for charged B decays — all of them self-tagging, so no flavour tagging or time-dependent analysis is needed — fix the weak phase $gamma$, and the fuller set of $pi pi$, $pi K$ and $K bar K$ rates fixes two weak phases together with several differences of strong (final-state) phase shifts.

What carries the argument

The machinery is a graphical decomposition of $B to PP$ amplitudes that is equivalent to an SU(3) reduced-matrix-element expansion: five independent combinations of six diagrams, so that twelve amplitudes per strangeness sector satisfy seven linear relations. The relations come in three kinds — pure isospin triangles within $B to pi pi$ and within $B to pi K$, triangles that mix strange and non-strange final states, and cross-sector relations that use $T'/T = C'/C = E'/E = A'/A = r_u$ for tree-type graphs and $P'/P = PA'/PA = r_t$ for penguins. Neglecting $E$, $A$ and $PA$ translates into two simple statements about reduced matrix elements, ${8_1} = -sqrt{5}{1}/4$ and ${8_3} = {27}/4$, which is what converts the triangles into rate equalities such as $Gamma(B_s to pi^+ K^-) = Gamma(B^0 to pi^+ pi^-)$ and $Gamma(B^0 to pi^- K^+) = Gamma(B_s to K^- K^+)$.

Extended reading notes

Core claim

The central claim is that SU(3) flavour symmetry, organised graphically into a small set of topological amplitudes (tree $T$, colour-suppressed $C$, penguin $P$, exchange $E$, annihilation $A$, penguin annihilation $PA$, and their strangeness-changing primed partners), produces relations among $B to PP$ amplitudes that are simple enough to be used as measurement tools. The load-bearing relation is Eq. (23): $$sqrt{2},A(B^+ to $pi^{0}$ K^+) + A(B^+ to pi^+ $K^{0}$) = r_u, sqrt{2},A(B^+ to pi^+ $pi^{0}$),$$ which says that the combination of $B to pi K$ amplitudes carrying isospin $3/2$ is a pure 27-plet and is therefore proportional, with the known factor $r_u$, to the $B^+ to pi^+ pi^0$ amplitude. Because the $B^+ to pi^+ K^0$ amplitude is dominated by the penguin with weak phase $pi$ while the $pi^+ pi^0$ amplitude carries the tree phase $gamma$, the two triangles formed by this relation and its charge conjugate have a relative orientation that measures $gamma$ — provided the annihilation contribution $A'$ can be dropped, in which case the rates for $B^+ to pi^+ K^0$ and $B^- to pi^- bar K^0$ should be equal. The paper then shows that if the exchange, annihilation and penguin-annihilation graphs are neglected altogether, the surviving relations collapse into ten statements — six rate equalities, three triangles and one quadrangle — from which $gamma$, $alpha$, and the strong-phase differences $delta_P - delta_2$ and $delta_{P'} - delta_{3/2}$ can all be extracted from rates alone.

Load-bearing premise

The extraction of $gamma$, and the ten relations that follow, rest on treating the exchange, annihilation and penguin-annihilation diagrams (and their primed versions) as negligible, a suppression the authors justify by $f_B/m_B$ and helicity arguments but explicitly qualify as valid only if rescattering effects are unimportant.

Editorial extensions

If this is right

Four measured rates — $B^+ to pi^0 K^+$, $B^- to pi^0 K^-$, $B^+ to pi^+ K^0$ and $B^+ to pi^+ pi^0$ — suffice to determine the weak phase $gamma$, with no time-dependent measurement and no tagging of the initial B flavour.
A CP-violating rate difference between $B^+ to pi^0 K^+$ and $B^- to pi^0 K^-$ requires both two distinct weak subprocesses and a nonzero strong-phase difference between the $I = 1/2$ and $I = 3/2$ channels, so observing no asymmetry does not by itself rule out the mechanism.
Rates for the full set of $pi pi$, $pi K$ and $K bar K$ charge states yield two CKM phases and several strong-phase differences, and the overlapping constructions provide independent cross-checks on SU(3) breaking.
Several channels are predicted to be strongly suppressed or absent if the annihilation-type graphs are negligible: $B^0 to K^+ K^-$, $B_s to pi^+ pi^-$ and $B_s to pi^0 pi^0$ should all be near zero, making searches at the $10^{-7}$ level a direct test of the assumption.
The same triangle logic applied to $B to pi bar D$ decays, where a single weak phase governs all channels, isolates final-state phase differences between $I = 1/2$ and $I = 3/2$ amplitudes; present data already bound the difference below about $35^circ$ at 90% confidence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The strategy is a template that outlives the specific SU(3) relations: whenever a symmetry forces one combination of amplitudes to be proportional to a single well-measured mode, that mode becomes a calibration leg for a triangle, and the phase information moves from interference in time to interference in the complex plane.
Because the $gamma$ extraction is over-determined by the four rates, the residual of the triangle closure is itself a quantitative handle on SU(3) breaking — a use the paper notes only in passing when it suggests replacing $r_u$ by $(f_K/f_pi), r_u$ in Eq. (66).
The rate equalities involving $B_s$ decays were not testable with the data available in 1994; they become sharp predictions for any experiment that can produce and identify $B_s$ mesons, and a measured $B_s to pi^+ pi^-$ rate above background would directly contradict the neglected-rescattering premise rather than merely strain it.
The paper's insistence that strong-phase differences are needed for direct CP violation in charged-B decays suggests the natural follow-up is to ask whether the rescattering that would generate those phases can be computed, since that is what separates a rate-based phase extraction from a phase extraction that is robust to it.

Original abstract

The decays $B \to PP$, where $P$ denotes a pseudoscalar meson, are analyzed, with emphasis on charmless final states. Numerous triangle relations for amplitudes hold within SU(3) symmetry, relating (for example) the decays $B^+ \to \pi^+ \pi^0,~ \pi^0 K^+$, and $\pi^+ K^0$. Such relations can improve the possibilities for early detection of $CP$-violating asymmetries. Within the context of a graphical analysis of decays, relations are analyzed among SU(3) amplitudes which hold if some graphs are neglected. One application is that measurements of the rates for the above three $B^+$ decays and their charge-conjugates can be used to determine a weak CKM phase. With measurements of the remaining rates for $B$ decays to $\pi \pi,~\pi K$, and $K \bar K$, one can obtain two CKM phases and several differences of strong phase shifts.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The decays $B \to PP$, where $P$ denotes a pseudoscalar meson, are analyzed, with emphasis on charmless final states. Numerous triangle relations for amplitudes hold within SU(3) symmetry, relating (for example) the decays $B^+ \to \pi^+ \pi^0,~ \pi^0 K^+$, and $\pi^+ K^0$. Such relations can improve the possibilities for early detection of $CP$-violating asymmetries. Within the context of a graphical analysis of decays, relations are analyzed among SU(3) amplitudes which hold if some graphs are neglected. One application is that measurements of the rates for the above three $B^+$ decays and their charge-conjugates can be used to determine a weak CKM phase. With measurements of the remaining rates for $B$ decays to $\pi \pi,~\pi K$, and $K \bar K$, one can obtain two CKM phases and several differences of strong phase shifts.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The decays $B \to PP$, where $P$ denotes a pseudoscalar meson, are analyzed, with emphasis on charmless final states.	contribution 0.88	0.74	0.06	asserted 0.92
s1	Numerous triangle relations for amplitudes hold within SU(3) symmetry, relating (for example) the decays $B^+ \to \pi^+ \pi^0,~ \pi^0 K^+$, and $\pi^+ K^0$.	result 0.58	0.33	0.13	asserted 1.00
s2	Such relations can improve the possibilities for early detection of $CP$-violating asymmetries.	interpretation 0.99	0.11	0.14	hedged possibility 0.96
s3	Within the context of a graphical analysis of decays, relations are analyzed among SU(3) amplitudes which hold if some graphs are neglected.	method 0.67	0.70	0.07	asserted 0.86
s4	One application is that measurements of the rates for the above three $B^+$ decays and their charge-conjugates can be used to determine a weak CKM phase.	interpretation 0.63	0.32	0.20	asserted 0.50
s5	With measurements of the remaining rates for $B$ decays to $\pi \pi,~\pi K$, and $K \bar K$, one can obtain two CKM phases and several differences of strong phase shifts.	result 0.40	0.23	0.26	hedged possibility 0.75

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.56.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Diagrams describing decays of

**Figure 2.** The two triangles are fixed by measurements of all six decay r

**Figure 6.** FIG. 6. Isospin triangle for

Showing 3 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9404283 by the authors.

References

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V. M. Belyaev, A. Khodjamirian and R. R¨ uckl, CERN preprint CER N-TH- 6880-93, May, 1993. 28 FIGURE CAPTIONS FIG. 1. Diagrams describing decays of B mesons to pairs of light pseudoscalar mesons. Here ¯q = ¯d for unprimed amplitudes and ¯s for primed amplitudes. (a) “Tree” (color-favored) amplitude T or T ′; (b) “Color-suppressed” amplitude C orC ′; (c) “...

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Current B-factory data still allow new physics at a few TeV to contribute 30 to 40 percent of the Standard Model amplitude to B and K meson mixing, with arbitrary CP phase, so the flavor scale need not sit far above t...
Charming penguins in B => K* pi, K (rho,omega,phi) decays
hep-ph 2003-07 conditional novelty 5.0 of 10

Adding a computed imaginary charming-penguin amplitude plus a fitted real part to short-distance B decay amplitudes brings theory into rough agreement with measured B to K*pi and K(rho,omega,phi) branching ratios.
Studying the tensor resonance contributions in $B \to PP\ell^+\ell^-$ and $B \to PV\ell^+\ell^-$ decays
hep-ph 2026-07 conditional novelty 4.0 of 10

Tensor-resonance contributions to B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻ are small compared with measured totals once SU(3)-related B → Tℓ⁺ℓ⁻ rates are fixed to Bs → f′₂(1525)μ⁺μ⁻.
Two-body charmed anti-charmed baryonic $B$ decays
hep-ph 2026-04 unverdicted novelty 4.0 of 10

Topological decomposition of amplitudes for B to charmed baryon anti-baryon decays yields predictions for several channels after fitting 35% SU(3) breaking effects to existing data, with large cancellations noted in s...
Anomalies in Hadronic $B$ Decays
hep-ph 2026-08 unverdicted novelty 2.0 of 10

A review of SU(3)_F fits to B->PP decays reports a 3.6 to 5.2 sigma discrepancy with the Standard Model, hinting at new physics, but realistic SU(3) breaking remains uncalculated.
Comment on "QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case''
hep-ph 2026-06 unverdicted novelty 2.0 of 10

The authors argue that ETRs follow automatically from SU(3) group theory and identify weaknesses in a recent QCD-factorization fit that omitted them.

How later papers used it

68 later papers cite it

Evidence: some. 52 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 9 of the 106 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 12 papers with this label

Decomposed the decay amplitudes of B to K* rho into linear combinations of the topological amplitudes in this paper's quark diagram approach.

Generalized analysis on $B\to K^* \rho$ within and beyond the Standard Model $-$ Can it help understand the $B \to K \pi$ puzzle? (2007) · 0.87 · paraphrased: the licence does not allow quoting

How the literature uses it

54 arXiv papers cite this in 91 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 62 sentences without a confident label

Uses its method or tool 10 papers

[ref] , which made use of the well established topological amplitude formalism [ref] and asymptotic relations [ref] in the large $m_b$ limit.

Two-body Baryonic $B_{u,d,s}$ and $B_c$ to Charmless Final State Decays · Introduction · 0.98
[ref] , which was used to study two-body baryonic $B$ decays, $B→'$, making use of the well established topological amplitude formalism [ref] .

Charmless Semileptonic Baryonic $B_{u,d,s}$ Decays · Introduction · 0.99
Within the diagrammatic approach of Refs. [ref] , $B$-decay amplitudes are expressed in terms of six diagrams[The annihilation, exchange and penguin-annihilation diagrams are neglected, as they are expected to be very small in the SM.]: the color-favored and color-suppressed tree amplitudes $T'$ and $C'$, the gluonic penguin amplitudes $P'_tc$ and $P'_uc$, and the color-favored and color-suppressed electroweak penguin amplitudes $$ and $$ (the primes on the amplitudes indicate $$ transitions).

Uniting Low-energy Semileptonic and Hadronic Anomalies within SMEFT · $$ · 0.99

Compares with it 1 paper

Importantly, our results are consistent with [ref] , but differ by a sign from those reported in [ref] .

A Phenomenological Model of Mesons for Charged Current Weak Decays · Consistency checks in hadronic decays via isospin sum rules · 0.94

Cites it as support 1 paper

Yet, we expect an equivalence between the two approaches [ref] .

Flavor $SU(3)$ in Cabibbo-favored $D$-meson decays · Connections between flavor-flow and matrix elements in $SU(3)_F$ · 0.96

Cites it as background 16 papers

In Ref. [ref] , Neubert and Rosner established a relation between the electroweak penguin (EWP) and tree topological diagrams of Ref. [ref] .

Comment on "QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case'' · Introduction · 0.96
The four amplitudes may be written in terms of penguin ($P_tc$ and $P_uc$), tree ($T$), color-suppressed ($C$), annihilation ($A$), color-favored electroweak penguin ($P_EW$), and color-suppressed electroweak penguin ($P_EW^C$) contributions [ref] :

Diagnostic for new physics in $B \to \pi K$ decays · Introduction · 0.80
These decays involve several contributing amplitudes (diagrams) [ref] .

CP Violation in $B$ Decays (and the Search for New Physics) · Direct CP Violation · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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