Predictions for Diffractive φ Meson Production Using an AdS/QCD Light-front Wavefunction
PPredictions for Di ff ractive φ Meson Production Using anAdS / QCD Light-front Wavefunction
Mohammad Ahmady , Ruben Sandapen and Neetika Sharma Department of Physics, Mount Allison University, Sackville, New Brunswick E4L 1E6, Canada Department of Physics, Acadia University, Wolfville, Nova Scotia B4P 2R6, Canada Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, S.A.S. Nagar,Mohali-140306, Punjab, India a) Corresponding author: [email protected] b) [email protected] c) [email protected] Abstract.
We compute the rate for di ff ractive φ electro-production using the Color Glass Condensate dipole model. The modelparameters are obtained from fits to the most recent combined HERA data on inclusive deep inelastic scattering. As for the φ meson, we use an AdS / QCD holographic light front wavefunction. Our predictions are compared to the available data collected atthe HERA collider.
INTRODUCTION
The light-front wavefunction (LFWF) of the vector meson is an input in the QCD colour dipole model for calculatingdi ff ractive vector meson production. In Ref. [1], successful predictions were obtained for di ff ractive ρ productionusing a holographic wavefunction for the ρ meson. The holographic meson wavefunction is predicted in holographiclight-front QCD proposed by Brodsky and de T´eramond recently reviewed in [2]. In this work, we first use the newdeep inelastic scattering (DIS) data from HERA collider, which were released in 2015 [3], to update the parameters ofthe color glass condensate (CGC) dipole model and then make predictions for di ff ractive φ meson production when aholographic LFWF is assumed for this vector meson.[4] CGC DIPOLE MODEL
In the dipole picture, the scattering amplitude for the di ff ractive process γ ∗ p → V p factorizes into an overlap ofphoton and vector meson light-front wavefunctions and a dipole cross-section [5]: (cid:61) m A λ ( s , t ; Q ) = (cid:88) h , ¯ h (cid:90) d r d x Ψ γ ∗ ,λ h , ¯ h ( r , x ; Q ) Ψ V ,λ h , ¯ h ( r , x ) ∗ e − ix r · ∆ N ( x m , r , ∆ ) , (1)where t = − ∆ is the squared momentum transfer at the proton vertex. Ψ γ ∗ ,λ h , ¯ h ( r , x ; Q ) and Ψ V ,λ h , ¯ h ( r , x ) are the light-frontwavefunctions of photon and vector meson respectively while N ( x m , r , ∆ ) is the proton-dipole scattering amplitude. h and ¯ h are the helicities of the quark and the antiquark respectively. r is the transverse size of the color dipole and x isthe fraction of light-front momentum of the photon (or vector meson) carried by the quark. Both wavefunctions arelabeled by λ = L , T which denotes the polarization of the photon or vector meson. The photon light-front wavefunctionis also a function of the photon’s virtuality Q . The dipole-proton scattering amplitude is the amplitude for the elasticscattering of the dipole on the proton and it depends on the photon-proton centre-of-mass energy W via the modified a r X i v : . [ h e p - ph ] M a r jorken variable x m where x m = x Bj + M V Q with x Bj = Q W . (2)The dipole-proton scattering amplitude is a universal object, appearing also in the formula for the fully inclusive DISprocess: γ ∗ p → X . In fact, by replacing the vector meson by a virtual photon in Equation (1), we obtain the amplitudefor elastic Compton scattering γ ∗ p → γ ∗ p , i.e. (cid:61) m A λ ( s , t ) (cid:12)(cid:12)(cid:12) t = = s (cid:88) h , ¯ h (cid:90) d r d x | Ψ γ ∗ ,λ h , ¯ h ( r , x ; Q ) | ˆ σ ( x m , r ) . (3)ˆ σ in Equation (3) is the dipole cross-section defined as follows:ˆ σ ( x m , r ) = N ( x m , r , ) s = (cid:90) d b ˜ N ( x m , r , b ) , (4)where ˜ N is the Fourier transform of N in the b (impact parameter) space. Indeed, the elastic amplitude given byEquation (3) is directly related to the inclusive γ ∗ p → X total cross-section in DIS via the Optical Theorem: σ γ ∗ p → X λ = (cid:88) h , ¯ h , f (cid:90) d r d x | Ψ γ ∗ ,λ h , ¯ h ( r , x ; Q ) | ˆ σ ( x m , r ) , (5)where now [6] x m = x Bj + m f Q with x Bj = Q W . (6)Therefore, one can use the high quality DIS data from HERA to constrain the free parameters of the dipolecross-section section and then use the same dipole cross-section to make predictions for vector meson production.To lowest order in α em, the perturbative photon wavefunctions are given by [7]: Ψ γ, Lh , ¯ h ( r , x ; Q , m f ) = (cid:114) N c π δ h , − ¯ h e e f x (1 − x ) Q K ( (cid:15) r )2 π , (7) Ψ γ, Th , ¯ h ( r , x ; Q , m f ) = ± (cid:114) N c π e e f (cid:2) ie ± i θ r ( x δ h ± , ¯ h ∓ − (1 − x ) δ h ∓ , ¯ h ± ) ∂ r + m f δ h ± , ¯ h ± (cid:3) K ( (cid:15) r )2 π , (8)where (cid:15) = x (1 − x ) Q + m f and re i θ r is the complex notation for the transverse separation between the quark andanti-quark. As is evident from Equation (8), at Q → x → (0 , m f which prevents the modified Bessel function K ( (cid:15) r ) from diverging, i.e.the quark mass acts as an infrared regulator. On the other hand, a non-perturbative model for the meson light-frontwavefunction is used and assumed to be valid for all r .A simple model for the b -integrated dipole-proton amplitude, i.e. the dipole cross-section in Equation (4) hasbeen proposed in Ref. [8]. This is known as the CGC dipole model and is given byˆ σ ( x m , r ) = σ N ( x m , rQ s , , (9)with N ( x m , rQ s , = N (cid:18) rQ s (cid:19) (cid:104) γ s + l n (2 / rQs ) κλ l n (1 / x m) (cid:105) for rQ s ≤ = − exp[ −A l n ( B rQ s )] for rQ s > , (10)where the saturation scale Q s = ( x / x m ) λ/ GeV. The coe ffi cients A and B are determined from the condition that the N ( rQ s , x ) and its derivative with respect to rQ s are continuous at rQ s =
2. This leads to A = − ( N γ s ) (1 − N ) l n [1 − N ] , B =
12 (1 − N ) − (1 −N N γ s . (11)he free parameters of the CGC dipole model are σ , λ, x and γ s which are fixed by a fit to the structure function F data. In 2015, the H1 and ZEUS collaborations have released highly precise combined data sets [3] for the reducedcross-section σ r ( Q , x , y ) = F ( Q , x ) − y + (1 − y ) F L ( Q , x ) , (12)where y = Q / ˆ sx and √ ˆ s is the centre of mass energy of the ep system for 4 di ff erent bins : √ ˆ s =
225 GeV (78 datapoints), √ ˆ s =
251 GeV (118 data points) and √ ˆ s =
300 GeV (71 data points), √ ˆ s =
318 GeV (245 data points). Ourfitted values for the CGC dipole model parameters together with the resulting χ per degrees of freedom ( χ / d.o.f)values are shown in Table 1. The first two rows indicate that the fit is not very sensitive to the variation in the strangequark mass. Comparing the second and third rows, we can see that the data prefer the lower u and d quark masses andthat increasing them give quite di ff erent fit parameters especially for x . TABLE 1.
Parameters of the CGC dipole model extracted from our fits to inclusive DISdata (with x Bj ≤ .
01 and Q ∈ [0 . ,
45] GeV ) using 3 di ff erent sets of quark masses. [ m u , d , m s ] / GeV γ s σ / mb x λ χ / d.o.f [0 . , . .
741 26 . . × − .
219 535 / = . , .
14] 0 .
722 24 . . × − .
222 529 / = . , .
14] 0 .
724 29 . . × − .
206 554 / = HOLOGRAPHIC MESON LFWF
The vector meson light-front wavefunctions appearing in Equation (1) cannot be computed in perturbation theory.Explicitly, the vector meson light-front wavefunctions can be written as [1] Ψ V , Lh , ¯ h ( r , x ) = δ h , − ¯ h (cid:20) + m f − ∇ r x (1 − x ) M V (cid:21) Ψ L ( r , x ) (13)and Ψ V , Th , ¯ h ( r , x ) = ± (cid:20) ie ± i θ r ( x δ h ± , ¯ h ∓ − (1 − x ) δ h ∓ , ¯ h ± ) ∂ r + m f δ h ± , ¯ h ± (cid:21) Ψ T ( r , x )2 x (1 − x ) . (14)Various ansatz for the non-perturbative meson wavefunction have been proposed in the literature, but in recentyears, new insights about hadronic light-front wavefunctions based on the anti-de Sitter / Conformal Field Theory(AdS / CFT) correspondence have been proposed by Brodsky and de T´eramond. [2]. In this framework the vectormeson wavefunctions Ψ λ ( r , x ) , λ = T , L are given as Ψ λ ( x , ζ ) = N λ (cid:112) x (1 − x ) exp (cid:34) − κ ζ (cid:35) exp − m f κ x (1 − x ) , (15)where we have introduced a polarization-dependent normalization constant N λ . We fix this normalization constant byrequiring that (cid:88) h , ¯ h (cid:90) d r d x | Ψ V ,λ h , ¯ h ( x , r ) | = , (16)where Ψ V ,λ h , ¯ h ( x , r ) are given by Equations (13) and (14). COMPARISON WITH DATA
Having specified the dipole cross-section and the holographic meson wavefunction, we can now compute cross-section for di ff ractive φ production. We shall show predictions using three sets of the CGC dipole parameters as givenin Table 1. We shall refer to these three sets of predictions as Fit A (first row), Fit B (second row) and Fit C (third
10 15 20 25 30 35 40 45 50Q [GeV ]0.011100 s [ nb ] ZEUS (2005)H1 (2010) Q [GeV ] s L / s T ZEUS (2005)H1 (2010)
FIGURE 1.
Predictions for the φ production total cross section (left) and longitudinal to transverse cross-section ratio (right) at W =
90 GeV as a function of Q compared to HERA data [9, 10]. Black solid curve: A. Orange dotted curve: B. Blue dashedcurve: C. row) respectively. Recall that all our predictions will be generated using the same holographic wavefunction given byEquation (15) and they di ff er only by the choice of quark masses and the corresponding fitted parameters of the CGCdipole model as given in Table 1.For φ production, our predictions for the Q dependence of the total cross-section at fixed W are shown inFigure 1. Here, it is clear that the Fit A predictions (solid black curves) are not successful. The data prefer slightlythe Fit B (orange dotted curves) over the Fit C predictions (blue dashed curves) although the lack of data in the low Q region prevents us from making a definite statement. At high Q , our predictions tend to undershoot the (ZEUS)data as expected. Our predictions for the longitudinal to transverse cross-sections ratio for φ production are shown inFigure 1. We can see that the ratio data tend to favour the Fit A prediction (solid black curve) although they are notprecise enough to discard the other two predictions. ACKNOWLEDGMENTS
This research is supported by a Team Discovery Grant from the Natural Sciences and Engineering Research Council(NSERC) of Canada.
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