Joint resummation for pion wave function and pion transition form factor

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1 Joint resummation for pion wave function and pion transition form factor Yu-Ming Wang RWTH Aachen and TUM Talk based on: Hsiang-nan Li, Yue-Long Shen, YMW, JHEP 01 (2014) 004. QCD Evolution Workshop, Santa Fe, Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

2 What is pion wave function? TMD wave function in k T factorization (Collins, 2003): Φ(x,k T,ζ 2, µ f ) = dy + 2π d 2 y T (2π) 2 e ixp y + +ik T y T 0 q(y)w y (u) I u;y,0 W 0 (u)n/ + γ 5 q(0) π(p). Light-cone divergence regularized by the rapidity parameter ζ 2 = 4(n u) 2 /u 2. Transverse gauge link I u;y,0 to ensure a strict gauge invariance. Does not contribute in covariant gauge, but contributes in light-cone gauge (Belitsky, Ji and Yuan, 2003). Light-cone distribution amplitude in collinear factorization: 0 q(0) [0,y]y/ γ 5 q(y) π + (p) y2 =0 1 = if π p y dxe ixp y φ π (x, µ). 0 ER-BL evolution implies expansion in Gegenbauer-polynomials: + [ ] (0) αs (µ) γ n /2β 0 φ π (x, µ) = 6x(1 x) an (µ 0 )Cn 3/2 (2x 1). n=0 α s (µ 0 ) Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

3 Why pion wave function? Key nonperturbative quantity in k T factorization for many exclusive processes. NLO k T factorization for the γ π 0 γ form factor (Nandi and Li, 2007): [For a different scheme, see Brodsky and Lepage (1981), and Musatov and Radyushkin (1997)] F(Q 2 ) = Φ H S J. Soft contribution suppressed by the Sudakov mechanism (Botts and Sterman, 1989; Li and Sterman, 1992). Transverse momentum dependence becomes important at the end-points. Threshold resummation can suppress the end-point contribution further b x NLO k T factorization for the pion e.m. form factor (Li, Shen, YMW and Zou, 2011) and the B πlν form factors (Li, Shen and YMW 2012). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

4 Structure of Pion wavefunction at NLO Quark-Wilson-line vertex diagrams (Nandi and Li, 2007): 7 (b) (c) (d) (e) (g) (h) FIG. 5: Effective diagrams. Φ (1) d H (0) = α [ sc F 1 4π ε + ln 4πµ2 f kt 2 ζ 2 eγ ln2 E kt 2 Φ (1) e H (0) = α sc F [ln 2 xζ 2 ] 4π kt H (0). (i) lson line direction to regularize the light-cone singularities. The NLO wave Unification the scale ζ1 2 4(n P of rapidity, k 1) 2 / n 2 or ζ2 2 T and 4(n threshold P resummation: joint resummation 2) 2 / n 2 (Li, 1998;, which is regarded as a is dependence, Laenen, entering Sterman, a hard Vogelsang, kernel 2000, when 2001). taking the difference between the ams, can be minimized by adhering to a fixed n 2. Note that the soft subtraction ary here, which is needed for a choice of n 2 = 0. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21 (j) + ln ζ 2 ] kt H (0), The double rapidity logarithm ln 2 ζ 2 in the B meson case is absent here. Mixed logarithm lnx ln(ζ 2 /k 2 T) in the sum (Φ (1) d + Φ (1) e ) H (0).

5 Construction of evolution equation Rapidity derivative (Collins, Soper, 1981; Li, 1998): ζ 2 d dζ 2 Φ = u2 n u n α 2 d du α Φ. Advantage: u dependence appears only through the Wilson line interactions. Rapidity derivative of the Feynman rule associated with the Wilson line: ζ 2 d u β dζ 2 u l + iε = ûβ 2u l, ( ) û β = u2 n l n u u l uβ n β. The rapidity evolution equation: ζ 2 d dζ 2 Φ(x,k T,ζ 2, µ f ) = Γ(x,k T,ζ 2 ) Φ(x,k T,ζ 2, µ f ). The vertex û β contracted to the vertex in Φ. Suppression of collinear dynamics associated with the Wilson link. Soft and hard gluon radiations are dominant in the kernel Γ(x,k T,ζ 2 ). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

6 Soft function Soft gluon radiations: π + π The reducible diagram: K 1 = ig2 C F d 4 l 2 (2π) 4 û n (u l + iε)(l 2 + iε)(n l + iε) = α s C F 4π ( ) 1 ε γ E + ln 4πµ2 λ 2. IR divergence regularized by the gluon mass λ. The irreducible diagram: K 2 Φ = ig2 C F d 4 l û n 2 (2π) 4 (u l + iε)(l 2 + iε)(n l + iε) Φ(x l /P, k T l T,ζ 2, µ f ). Fourier and Mellin transformations of K 2 Φ: K 2 = α [ ( s C F ζ P )] b ( K 0 (λb) K 0 + O 1/ζ 2). 2π N Unrenormalized soft function in the large N limit: K (b) = K 1 + K 2 = α s C F 4π ( 1 ε γ E + ln 4πµ2 N 2 ) ζ 2 P 2. Cancelation of IR divergence! Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

7 Hard function Hard gluon radiations: π π Subtraction to avoid the double counting of soft contribution. Analytical expressions: G 1 = ig2 C F 2 G 2 = K 1. d 4 l ( xp/ + l/)û/ (2π) 4 (u l + iε)(l 2 + iε)[( xp + l) 2 + iε], Unrenormalized hard function: G (b) = G 1 G 2 = α s C F 4π [ 1 ε γ 4πµ 2 ] E + ln ζ 2 ( xp ) 2 4. Infrared finite due to the cancelation of soft divergence! Factorization scale dependence cancels between soft and hard functions. µ independence of the mixed logarithm to be resummed. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

8 RG improved evolution kernel Renormalized soft and hard functions: K (r) (µ) = α s C F 2π ln µ N ζ P, G(r) (µ) = α s C F 2π ( ln µ ) ζ P 2. RGE of soft and hard functions: µ d K (r) dµ = α s(µ)c F, µ dg(r) 2π dµ = α s(µ)c F. 2π RG improvement: µ1 K (r) (µ) + G (r) (µ) = K (r) (µ 0 ) + G (r) d µ α s ( µ)c F (µ 1 ), µ 0 µ 2π µ 0 := µ 0 (ζ ) = ζ P N, µ 1 := µ 1 (ζ ) = e 2 ζ P. Chosen to diminish the initial conditions K (r) (µ 0 ) and G (r) (µ 1 ). Gluon radiations from the spectator quark also contribute. Only contribute to the kernel Γ at NLL level. Do not generate the mixed logarithm lnx ln(ζ 2 P 2 /k 2 T). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

9 Solution in Mellin and impact-parameter spaces Evolution equation in N and b spaces: Solution: RGE for µ f evolution: ζ 2 d dζ Φ(N,b,ζ 2 2, µ f ) = Γ(N,b,ζ 2 ) Φ(N,b,ζ 2, µ f ). µ1 Γ(N,b,ζ 2 ) = K (r) (µ) + G (r) (ζ ) d µ α s ( µ)c F (µ) = µ 2π Φ(N,b,ζ 2, µ f ) = exp { ζ 2 ζ 2 0 d ζ 2 ζ 2 Φ(N,b,ζ 2 0, µ f ). µ 0 (ζ ) [ µ1 ( ζ ) µ 0 ( ζ ) d µ µ α s ( µ)c F 2π d µ f Φ(N,b,ζ 2, µ f ) = 3 α s (µ f )C F Φ(N,b,ζ 2, µ f ). dµ f 2 π Combined evolution: { Φ(N,b,ζ 2, µ f ) = exp ζ 2 d ζ [ 2 µ1 ( ζ ) d µ ζ0 2 ζ 2 µ 0 ( ζ ) µ µf d µ µ i µ α s ( µ)c F π. ]} ] α s ( µ)c F 2π } Φ(N,b,ζ 0 2, µ i). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

10 Evolution of the hard kernel Rapidity evolution: ζ 2 d dζ H(N,b,ζ 2 2,Q 2, µ f ) = Γ(N,b,ζ 2 ) H(N,b,ζ 2,Q 2, µ f ). Factorization scale evolution: d µ f H(N,b,ζ 2,Q 2, µ f ) = 3 α s (µ f )C F dµ f 2 π Combined evolution: H(N,b,ζ 2,Q 2, µ f ) = { ζ 2 1 d exp [ 2 µ1 ( ζ ) ζ 2 ζ 2 µ 0 ( ζ ) 3 2 µf t d µ µ H(N,b,ζ 2,Q 2, µ f ). d µ µ α s ( µ)c F π ] α s ( µ)c F 2π } H(N,b,ζ 1 2,Q2,t). Depends on the final rapidity parameter ζ 1 and the characteristic hard scale t. Choices of boundary conditions: ζ 2 0 = ( an 1/4 P b ) 2, ζ 2 1 = ãn1/2. Eliminate the logarithmic enhancements in Φ(N,b,ζ 2 0, µ i) and H(N,b,ζ 2 1,Q2,t). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

11 Joint resummation improved k T factorization Factorization formula of γ π 0 γ form factor: { ζ 2 F(Q 2 1 d ) = exp ζ [ 2 ] µ1 ( ζ ) d µ α s ( µ)c F + 3 t d µ ζ0 2 ζ 2 µ 0 ( ζ ) µ 2π 2 µ i µ Φ(N,b,ζ 2 0, µ i) H(N,b,ζ 2 1,Q2,t), Φ(N,b,ζ 2 1,t) H(N,b,ζ 2 1,Q2,t). } α s ( µ)c F π Have confirmed that the expansion of the exponential factor reproduces the mixed logarithm and the single logarithm ln(1/n) in the NLO pion transition form factor. A complete treatment of the logarithmic enhancement for an arbitrary rapidity parameter in the pion transition form factor. The conventional k T factorization formula with the Sudakov and threshold resummations is not factorization-scheme independent. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

12 Resummation improved wave functions Inverse Mellin transformation: c+i Φ(x,b,ζ1 2,t) = dn c i 2πi (1 x) N Φ(N,b,ζ1 2,t). No Fourier transformation for the comparison of Sudakov resummation. Factorized model for the initial condition: Φ(x,k T,ζ 2 0, µ i) = φ(x,ζ 2 0, µ i) Σ(k 2 T), Translated into the Mellin and impact-parameter spaces. See Radyushkin s talk for more discussion. Three different models of φ(x,ζ 2 0, µ i): φ I (x,ζ 2 0, µ i) = 6x(1 x) φ II (x,ζ 2 0, µ i) = 1 1 N, φ III (x,ζ 2 0, µ i) = 6x(1 x) 6 (N + 1)(N + 2) Σ(k 2 T) = 4πβ 2 exp( β 2 k 2 T). 6 (N + 1)(N + 2), [ 1 + a 2 C 3/2 ] 2 (2x 1) [ (N 1)(N 2) 1 + 6a 2 (N + 3)(N + 4) Can include the higher-order Gegenbauer moments, but (Agaev et al, 2011; Kroll, 2011.) The contributions from higher moments suppressed by the soft corrections. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21 ].

13 Joint Resummation improved wave functions Inverse Mellin transformation with both frozen and running α s. Analytical parametrization of α s (Solovtsov and Shirkov, 1999): [ ] α s (µ) = 4π 1 β 0 ln(µ 2 /Λ 2 QCD ) Λ2 QCD µ 2 Λ 2. QCD Resummation effect in pion wave function Φ(x,b,ζ1 2, µ i): 2.0 Φ Π I x, b,ζ1 2,Μi solid: initial condition φ I (x,ζ 2 0, µ i); (blue) dashed: b = 2ã for a frozen ap α s = 0.3 (running α s ); 0.5 (blue) dotted: b = 4ã for a frozen ap α 0.0 s = 0.3 (running α s ) x Stronger suppression of the small x region compared to the moderate x region. The suppression strengthens with the transverse separation b at a given x. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

14 Comparison of Joint and Sudakov resummations Joint vs Sudakov resummation: x, b,ζ 1 2,Μi x, b,ζ 1 2,Μi Φ Π I 0.2 Φ Π I bgev 1 bgev 1 Left: Sudakov resummation, solid, dashed, and dotted curves for Q 2 = 5 GeV 2, 10 GeV 2, and 40 GeV 2 at x = 0.2. Right: The same for the joint resummation. Large b region suppressed in both cases, but stronger with the joint resummation. Different phenomenological consequences with the two resummation techniques. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

15 γ π 0 γ form factor Transition matrix element: γ(p,ε) j em µ (q) π 0 (P) = ig 2 em ε µναβ ε ν P α P β F(Q 2 ). Different approaches to compute the pion form factor: Direct approaches: Collinear (k T ) factorization formulae. Indirect approaches: (Light-cone) QCD Sum rules. Status of experimental measurements: 0.35 Q 2 F(Q 2 ) (GeV) Scaling violation? Shape of pion wave function? BaBar 0.1 fit(a) CLEO CELLO 0.05 Belle fit(a) fit(b) Q 2 (GeV 2 ) The onset of QCD factorization? Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

16 Some popular explanations: Non-vanishing pion wave function at the end points (Radyushkin, 2009; Polyakov, 2009). F(Q 2 ) = 2 1 ) ϕ π (x) xq 2 exp ( xq2. 2 xσ }{{} from k T dependence of pion wave function del I 14 Large soft (Feynman) corrections at moderate Q 2 (Agaev, Braun, Offen, Porkert, 2011) Q 2 F π 0 γ γ(q 2 ) soft+hard hard soft The hard and soft contributions to the π 0 γ γ form factor for model I (solid curves) and model III (dashdotted curves). The experimental data are from BaBar (full circles) and CLEO (open triangles) Q 2 FIG. 10: Contributions to the π 0 γ γ form factor from large Threshold ( hard ) and resummation small ( soft ) invariant generates masses power-like in the dispersion [x(1 x)] c(q2) distribution (Li and Mishima representation, cf. Eq. (53), for model I (solid curves) and 2009). modelc(q III (dash-dotted ) is around curves). 1 forthe lowexperimental Q 2, but small data arefor high Q 2. from [1] (full circles) and [9] (open triangles). Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

17 Factorization formula of pion form factor k T factorization with the conventional resummation (Nandi and Li, 2007): F(Q 2 2fπ 1 ) = dx bdb Φ(x,b,t) e S(x,b,Q,t) S t (x,q) K 0 ( [ xqb) 1 α ( )] s(t)c F 3ln t2 b 4π 2 xq + γ E + 2lnx + 3 π2. 3 The rapidity parameter fixed as ζ 2 = 2. k T factorization with the joint resummation: F(Q 2 2fπ 1 ) = dx bdbφ(x,b,ζ1 2 3,t)K 0( xqb) 0 0 [ 1 α ( )] s(t)c F 3ln t2 b 4π 2 xq + ln Choices of the factorization scale t: t 2 = xq/b to eliminate the remaining logarithm. Typical scale of the hard scattering t = max( xq,1/b) [default choice]. Two scenarios do not generate practical difference in our formalism. The joint resummation has suppressed the contribution from the nonperturbative region effectively. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

18 Confronting the data I Results with the asymptotic pion wave function: 0.4 Q 2 FQ 2 GeV Q 2 GeV 2 The experimental data from CLEO (dots), BaBar (triangles), and Belle (squares). The dashed and dotted (dotdashed and solid) curves from LO and NLO predictions with the conventional resummations (joint resummation). The predicted Q 2 F(Q 2 ) with the conventional resummations saturates as Q 2 > 5GeV 2. Can accommodate the Belle data except the first two bins. Fail to describe the BaBar data. The joint-resummation effect decreases the predictions in the conventional approach. Due to the stronger suppression at small x from joint resummation. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

19 Confronting the data II Results with the flat pion wave function: 0.4 Q 2 FQ 2 GeV Q 2 GeV 2 The experimental data from CLEO (dots), BaBar (triangles), and Belle (squares). The dashed and dotted (dotdashed and solid) curves from LO and NLO predictions with the conventional resummations (joint resummation). The form factor Q 2 F(Q 2 ) grows steadily with Q 2 in both resummation formalisms. Tree-level k T factorization formula Q 2 F(Q 2 ) ln(q 2 /k 2 T). The NLO curves from the conventional resummations and from the joint resummation turn out to be similar. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

20 Confronting the data III Results with the non-asymptotic pion wave function: 0.4 Q 2 FQ 2 GeV The experimental data from CLEO (dots), BaBar (triangles), and Belle (squares). The second Gegenbauer moment: a 2 = Q 2 GeV 2 Pion wave function with a small a 2 can describe the data better. The Chernyak-Zhitnitsky model or the Bakulev-Mikhailov-Stefanis model, which involve a large a 2, overshoot the data in our formalism. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21

21 Conclusion and outlook Constructed an evolution equation to resum the mixed logarithm lnx ln(ζ 2 /k 2 T). The moderate x and small b regions more highlighted with joint resummation. The predictions for the pion transition form factor confronted with the data. A small a 2 favored in joint-resummation improved k T factorization. More efforts are in demand on theory side: Better control on the pion wave function. Include the soft contribution in k T factorization. Have checked that our joint-resummation formalism can be extend to k T factorization of pion e.m. form factor. The same ζ0 2 diminishes the large logarithms in the amplitudes of effective diagrams at NLO. Can find ζ1 2 and ζ 2 2 to eliminate the large logarithms in the hard kernel. Yu-Ming Wang (RWTH Aachen and TUM) QCD Evolution Workshop, Santa Fe, /21