Dispersive approach to non-Abelian axial anomaly and , production - PowerPoint PPT Presentation

Dispersive approach to non-Abelian axial anomaly and η , η ′ production in heavy ion collisions S. Khlebtsov 2 , Y. Klopot 1 , A. Oganesian 1 , 2 and O. Teryaev 1 1 BLTP JINR, Dubna 2 ITEP, Moscow Seminar ”Theory of Hadronic Matter under Extreme Conditions” 4 July 2018, BLTP JINR, Dubna

◮ η and η ′ mesons are known to be deeply related to Abelian and non-Abelian axial anomalies. ◮ We generalize the exact anomaly sum rules to the case of non-Abelian axial anomaly and apply the results to the processes of η and η ′ radiative decays and their production in heavy ion collisions.

Outline Introduction: Axial anomaly Anomaly Sum Rule ASR and meson contributions Low-energy theorem for mixed states Hadron contributions and analysis of the ASR Numerical analysis η/η ′ ratio in heavy ion collisions Conclusions & Outlook

Axial anomaly In QCD, for a given flavor q , the divergence of the axial current J ( q ) µ 5 = ¯ q γ µ γ 5 q acquires both electromagnetic and strong anomalous terms: q γ 5 q + e 2 F + α s ∂ µ J ( q ) 8 π 2 e 2 q N c F ˜ 4 π G ˜ µ 5 = m q ¯ G , (1) An octet of axial currents � λ a J ( a ) √ µ 5 = q γ 5 γ µ ¯ q 2 q Singlet axial current J (0) u γ µ γ 5 u + ¯ 1 µ 5 = 3 (¯ d γ µ γ 5 d + ¯ s γ µ γ 5 s ): √ √ 1 ( m u u γ 5 u + m d d γ 5 d + m s s γ 5 s ) + α em 3 α s ∂ µ J (0) 2 π C (0) N c F ˜ 4 π G � √ µ 5 = F + G , 3 (2)

The diagonal components of the octet of axial currents J (3) 1 u γ µ γ 5 u − ¯ µ 5 = 2 (¯ d γ µ γ 5 d ), √ J (8) u γ µ γ 5 u + ¯ 1 µ 5 = 6 (¯ d γ µ γ 5 d − 2¯ s γ µ γ 5 s ) √ acquire an electromagnetic anomalous term only: 1 ( m u u γ 5 u − m d d γ 5 d ) + α em ∂ µ J (3) 2 π C (3) N c F ˜ µ 5 = √ F , (3) 2 1 ( m u u γ 5 u + m d d γ 5 d − 2 m s s γ 5 s ) + α em ∂ µ J (8) 2 π C (8) N c F ˜ µ 5 = √ F . (4) 6 The electromagnetic charge factors C ( a ) are 1 1 C (3) = ( e 2 u − e 2 √ d ) = √ , 2 3 2 1 1 C (8) = ( e 2 u + e 2 d − 2 e 2 √ √ s ) = , 6 3 6 1 2 C (0) = ( e 2 u + e 2 d + e 2 √ s ) = √ . (5) 3 3 3

Anomaly sum rule for the singlet axial current The matrix element for the transition of the axial current J α 5 with momentum p = k + q into two real or virtual photons with momenta k and q is: � e 2 T αµν ( k , q ) = d 4 xd 4 ye ( ikx + iqy ) � 0 | T { J α 5 (0) J µ ( x ) J ν ( y ) }| 0 � ; (6) Kinematics: k 2 = 0 , Q 2 = − q 2

Anomalous axial-vector Ward identity for the singlet component of axial current: p α T αµν = 2 mG ǫ µνρσ k ρ q σ + C 0 N c 2 π 2 ǫ µνρσ k ρ q σ + N ( p 2 , q 2 , k 2 ) ǫ µνρσ k ρ q σ , (7) where 2 mG ǫ µνρσ k ρ q σ = � 0 | � q = u , d , s m q ¯ q γ 5 q | γγ � , √ 3 α s 4 π G ˜ G | γ ( k ) γ ( q ) � = e 2 N ( p 2 , k 2 , q 2 ) ǫ µνρσ k µ q ν ǫ ( k ) ρ ǫ ( q ) � 0 | σ , (8) � 0 | F ˜ F | γ ( k ) γ ( q ) � = 2 ǫ µνρσ k µ q ν ǫ ( k ) ρ ǫ ( q ) σ . (9)

The VVA triangle graph amplitude presented as a tensor decomposition: F 1 ε αµνρ k ρ + F 2 ε αµνρ q ρ T αµν ( k , q ) = + F 3 k ν ε αµρσ k ρ q σ + F 4 q ν ε αµρσ k ρ q σ (10) + F 5 k µ ε ανρσ k ρ q σ + F 6 q µ ε ανρσ k ρ q σ , F j = F j ( p 2 , k 2 , q 2 ; m 2 ), p = k + q . In the kinematical configuration with one real photon ( k 2 = 0) the anomalous Ward identity can be rewritten in terms of form factors F j as follows ( N ( p 2 , q 2 ) ≡ N ( p 2 , q 2 , k 2 = 0)): ( q 2 − p 2 ) F 3 − q 2 F 4 = 2 mG + C 0 N c 2 π 2 + N ( p 2 , q 2 ) . (11) – G , F 3 , F 4 can be rewritten as dispersive integrals without subtractions. [Horejsi, Teryaev ’94] – N : rewrite it in the form with one subtraction, N ( p 2 , q 2 ) = N (0 , q 2 ) + p 2 R ( p 2 , q 2 ) , (12) where the new form factor R can be written as an unsubtracted dispersive integral.

The imaginary part of AWI (11) w.r.t. p 2 ( s in the complex plane) reads ( q 2 − s ) ImF 3 − q 2 ImF 4 = 2 mImG + sImR . (13) – Divide every term of Eq. (13) by ( s − p 2 ) and integrate: � ∞ � ∞ � ∞ � ∞ ( q 2 − s ) ImF 3 ds − q 2 1 s − p 2 ds = 1 ImF 4 2 mImG s − p 2 ds + 1 sImR s − p 2 ds π s − p 2 π π π 0 0 0 0 (14) – After transformation and making use of the dispersive relations for the form factors F 3 , F 4 , G , R : � ∞ � ∞ ( q 2 − p 2 ) F 3 − 1 ImF 3 ds − q 2 F 4 = 2 mG + p 2 R + 1 ImRds . (15) π π 0 0 Comparing (15) with (11) we arrive at the anomaly sum rule for the singlet current: � ∞ � ∞ 1 ImF 3 ds = C 0 N c 2 π 2 + N (0 , q 2 ) − 1 ImR ( s , q 2 ) ds , (16) π π 0 0

ASR and meson contributions Saturating the l.h.s. of (16) with resonances according to global quark-hadron duality, we write out the first resonances’ contributions explicitly, while the higher states are absorbed by the integral with a lower limit s 0 , � ∞ � ∞ M F M γ ( q 2 ) + 1 ImF 3 ds = C 0 N c 2 π 2 + N (0 , q 2 ) − 1 Σ f 0 ImR ( s , q 2 ) ds , π π s 0 0 (17) where � d 4 xe ikx � M ( p ) | T { J µ ( x ) J ν (0) }| 0 � = e 2 ǫ µνρσ k ρ q σ F M γ ( q 2 ) , (18) � 0 | J ( a ) α 5 (0) | M ( p ) � = ip α f a M . (19) ◮ ”Continuum threshold” s 0 ( q 2 ) [KOT’11],[Oganesian,Pimikov,Stefanis,Teryaev’15] . s 0 � 1 GeV 2 . ◮ If one saturates with resonances the last term in the ASR: the glueball-like states.

Low-energy theorem The matrix element � 0 | G ˜ G ( p ) | γ ( k ) γ ( q ) � ? ◮ No rigorous calculation from the QCD. ◮ Possible to estimate it in the limit p µ = 0. [Shifman’88]. We consider the case of two real photons ( q 2 = k 2 = 0). Supposing that there are no massless particles in the singlet channel in the chiral limit (i.e. no admixture of the η ): p → 0 p µ � 0 | J µ 5 ( p ) | γγ � = 0 , lim � 0 | ∂ µ J µ 5 | γγ � = 0 . Using the explicit expression for the divergence of axial current in the chiral limit (put m q = 0), one can relate the matrix elements of � 0 | G ˜ G | γγ � and � 0 | F ˜ F | γγ � in the considered limits. ◮ Mixing: η spoils the theorem!

Low-energy theorem for mixing states Take into account mixing. J ( x ) µ 5 = aJ (0) µ 5 + bJ (8) µ 5 , � 0 | J ( x ) µ 5 | η � = 0 . (20) µ 5 − f 8 η J ( x ) µ 5 = b ( J (8) J (0) µ 5 ) , (21) f 0 η � 0 | J ( i ) µ 5 (0) | M ( p ) � = ip µ f i M . (22) The current (21) gives no massless poles in the matrix element � 0 | J ( x ) µ 5 | γγ � even in the chiral limit, and therefore p → 0 � 0 | ∂ µ J ( x ) lim µ 5 ( p ) | γγ � = 0 . (23) In the chiral limit, at p µ = 0: √ 3 α s η C (0) ) � 0 | α e G | γγ � = N c η C (8) − f 8 4 π G ˜ 2 π F ˜ ( f 0 � 0 | F | γγ � . (24) f 8 η N c η C (8) − f 8 ( f 0 η C (0) ) . N (0 , 0 , 0) = (25) 2 π 2 f 8 η

Hadron contributions and analysis of the ASR � ∞ � ∞ M F M γ ( q 2 ) + 1 ImF 3 ds = C 0 N c 2 π 2 + N (0 , q 2 ) − 1 Σ f 0 ImR ( s , q 2 ) ds π π s 0 0 The first hadron contributions to the ASR: η and η ′ . For real photons, the transition form factors determine the 2-photon decay amplitudes A M ( M = η, η ′ ): � 64 π Γ M → 2 γ A M ≡ F M γ (0) = . (26) e 4 m 3 M The ASR for the octet channel [KOT’12] for real photons: 1 f 8 η A η + f 8 2 π 2 N c C (8) . η ′ A η ′ = (27)

The ASR in the singlet channel: 1 f 0 η A η + f 0 η ′ A η ′ = 2 π 2 N c C 0 + B 0 + B 1 , (28) where � ∞ � ∞ B 0 ≡ N (0 , 0 , 0) , B 1 ≡ − 1 ImR ( s ) ds − 1 ImF 3 ds . (29) π π 0 s 0 ◮ The B 0 term stands for a subtraction constant in the dispersion representation of gluon anomaly; ◮ The B 1 term consists of two parts: spectral representation of gluon anomaly and the integral covering higher resonances. The latter is proportional to α 2 s : F 3 is described by a triangle graph (no α s corrections) plus diagrams with additional boxes ( ∝ α 2 s for the first box term). The α 2 s suppression of the box graph contribution is due to s > s 0 � 1 GeV 2 . ◮ In the case of both real photons in the chiral limit the triangle amplitude is zero ( ∝ q 2 ). So, B 1 is represented by the integral with the lower limit s 0 ∼ 1 GeV 2 and is suppressed at least as α 2 s on the scale of 1 GeV 2 .

Combining ASRs for the octet and singlet channels, we obtain the 2-photon decay amplitudes: � N c � A η = 1 2 π 2 ( C (8) f 0 η ′ − C (0) f 8 η ′ ) − ( B 0 + B 1 ) f 8 , (30) η ′ ∆ � N c � A η ′ = 1 2 π 2 ( C (0) f 8 η − C (8) f 0 η ) + ( B 0 + B 1 ) f 8 , (31) η ∆ where ∆ = f 8 η f 0 η ′ − f 8 η ′ f 0 η . Making use of the result of the LET for B 0 : B 1 f 8 A η = N c C (8) η ′ − ∆ , (32) 2 π 2 f 8 η A η ′ = B 1 f 8 η ∆ . (33) Note, that low energy theorem leads to the cancellation of the photon anomaly term with subtraction part of gluon anomaly B 0 in (31), so the amplitude η ′ → γγ (in the chiral limit) is entirely determined by B 1 , i.e., predominantly by the spectral part of the gluon anomaly.