aa r X i v : . [ nu c l - t h ] J un π γ invariant mass distribution in the low energy γp → ωp reaction Swapan Das
Nuclear Physics Division, Bhabha Atomic Research CentreMumbai-400085, India
Abstract
We study the reaction mechanism for the correlated π γ emission in thephoton induced reaction on the proton target. Since this reaction is studiedin the GeV region, we assume that it proceeds through the formation of thevector meson in the intermediate state. The vector meson, being an unsta-ble particle, decays into π γ bosons after propagating certain distance. Ouranalysis shows that the π γ event seen in coincidence in the final state domi-nantly arises due to the decay of ω meson. The calculated results reproducethe measured π γ invariant mass distribution spectra very well. The vector meson production in the nuclear and particle reactions has opened upvarious avenues providing ample opportunities for learning many interesting topicsin physics. The leptoproduction of vector meson is a potential tool to investigatethe hadron structure of photon [1]. The dilepton production in the intermediateenergy region is undoubtly understood due to the decay and the interference of thevector mesons [1]. Since the vector meson strongly couples to the nucleon and itsresonances [2, 3], the dynamics of these resonances can be explored by studying thevector meson production process. In fact, there are predictions of many nucleonicresonances (see Ref. [4], and references their in), which are yet to be found. Thevector meson production could be an useful probe to search these missing resonances.It should be added that the sub-threshold production of the vector meson can probethe low lying resonances, like N (1520) [5].In the recent years, the production of vector meson in the nuclear reaction hasdrawn considerable attention to explore its properties in the nuclear medium [6,7]. Indeed, it is a fundamental issue in the nuclear physics. In this context, the π γ invariant mass distribution spectrum was measured, in the recent past, by theCBELSA/TAPS collaboration at the electron stretcher accelerator (ELSA) in Bonn[8] to look for the medium modification on the omega meson in the Nb nucleus.They had also taken data for the above spectrum due to γp reaction. Here, wepresent the reaction mechanism for the ( γ, π γ ) reaction on the proton target in1he GeV region. In this energy region, the π γ in the final state (according to theparticle data group [9]) can arise due to the decay of the low lying vector mesons,such as ρ (768) , ω (782) and φ (1020). Therefore, we consider this reaction proceedsthrough three steps: (i) the formation of vector mesons, (ii) the propagation of thesevector mesons, and (iii) the decay of these mesons into π γ channel. Symbolically,this reaction goes as γp → V p ; V → π γ , where V stands for the above mentionedvector mesons ( i.e., V ≡ ρ , ω and φ ).The qualitative comparison (presented below) amongst the ρ , ω and φ mesons’contributions to the above reaction shows that the ω meson contributes dominantlyto this reaction. The data on the γp reaction show that the production cross sectionfor the ρ meson is the largest compared to those for the other two vector mesonsin the energy region considered here. For example, the measured cross section at1.5 GeV beam energy, as reported by various group, are σ t ( γp → ρ p ) ≃ µ b[10], σ t ( γp → ωp ) ≃ . µ b [11], and σ t ( γp → φp ) ≃ . µ b [12]. Therefore, themeasured σ t ( γp → ρ p ) is about 3.53 times larger than the measured σ t ( γp → ωp ),and the former is about 135.29 times larger than the measured σ t ( γp → φp ). Thevector meson propagator G ω ( m ) is given by G V ( m ) = 1 m − m V + im V Γ V ( m ) , (1)where V stands for ρ , ω and φ mesons as mentioned above. For these mesons, wehave values for their resonance masses and total widths [9]: m V ( ≡ ρ ) ≃
768 MeV,Γ V ( ≡ ρ ) ≃
151 MeV; m V ( ≡ ω ) ≃
782 MeV, Γ V ( ≡ ω ) ≃ .
43 MeV; m V ( ≡ φ ) ≃ V ( ≡ φ ) ≃ .
43 MeV. The data taken at ELSA [8] show the peak at ∼
780 MeV inthe π γ invariant mass distribution spectrum. Around this mass, the propagatorsfor both ρ and ω mesons behave as G V ( m ≈
782 MeV) ∼ − im V Γ V ( V ≡ ρ , ω ), whereas G φ ( m ) goes as G φ ( m ≈
782 MeV) ∼ m − m φ . The decay widths for ρ , ω and φ mesons in the π γ branch (according to the Ref. [9]) are Γ(768) ρ → π γ ≈ .
12 MeV,Γ(782) ω → π γ ≈ .
72 MeV, and Γ(1020) φ → π γ ≈ .
006 MeV. In fact, Γ ρ → π γ ( m ) isnegligibly larger than 0.12 MeV and Γ φ → π γ ( m ) could be much less than 0.006 MeVat m ≈
782 MeV. Therefore, at 1.5 GeV the ratios σ t ( γp → ωp ) σ t ( γp → φp ) ≃ . | G ω ( m ≈ G φ ( m ≈ | ∼ . × , and Γ( m ≈ ω → π γ Γ( m ≈ φ → π γ >
120 show that we can safely ignore the contributionto the cross section originating from the φ meson of mass around 782 MeV. For ρ and ω mesons, the above data show σ t ( γp → ωp ) σ t ( γp → ρ p ) ≈ . | G ω ( m ≈ G ρ ( m ≈ | ∼ . × ,and Γ( m =782) ω → π γ Γ( m =782) ρ → π γ ≈
6. Therefore, this analysis illustrates qualitatively that thecontribution to the π γ emission from the ω meson decay is a factor about 521times larger than that originating due to the ρ meson decay. Similar analysis2hows this factor is about 10 at E γ equal to 1.2 GeV.There were two aspects in the experiment (done at ELSA [8]) to be men-tioned. One aspect in this measurement was the gamma beam of wide energy spread(0 . − .
53 GeV), since it was the tagged photon produced by the bremsstrahlungradiation of the 2.8 GeV electron on the Pb target. This is unlike to the conven-tional experiments where the beam energy is used to be taken almost monoenergetic.In fact, the measured π γ invariant mass distribution spectrum has been reportedfor definite ω meson momentum bin instead of a particular incident energy. Theanother aspect of this experimental set-up was the large width in the detectingsystem (55 MeV), which is about 6.5 times larger than the width of the ω meson(i.e., Γ ω ( m = 782 MeV) = 8 .
43 MeV in the free state) produced in the γp reaction.Therefore, the measured π γ invariant mass distribution spectrum showing its widthabout 55 MeV can be believed undoubtly due to the width of the detector resolu-tion. To compare the calculated results with the data, we incorporate these twoaspects in our formalism presented in sec. 2. The results of this study are discussedin sec. 3, and we give the conclusion in sec. 4 in this manuscript. The formalism for the π γ emission in the γp reaction, as mentioned above,consists of the production, propagation, and the decay of the omega meson producedin the intermediate state, i.e., γp → ωp ′ ; ω → π γ ′ . The primes on p and γ areused at present to distinguish them from the initial state particles, which have beendropped afterward for convenience. The T-matrix T fi for this process can be writtenas T fi = ( π , γ ′ | Γ ωπγ | ω ) G ω ( m ) F ( γp → ωp ′ ) , (2)where G ω ( m ) denotes the propagator for the ω meson. ( π γ ′ | Γ ωπγ | ω ) in this equationis the matrix element for ω → π γ ′ due to intrinsic coordinates. It is governed bythe Lagrangian density [13]: L ωπ γ = f ωπγ m π ǫ µνρσ ∂ µ A ν π ∂ ρ ω σ , (3)where A appearing in this equation represents the photon field. f ωπγ is the constantfor the ωπγ coupling. It is equal to 0.095, as extracted from the width of ω → π γ [9]. In Eq. (2), F ( γp → ωp ′ ) describes the production mechanism for the ω meson3n the γp reaction. It is given by F ( γp → ωp ′ ) = − πE ω " E ω + 1 E p ′ < ω, p ′ | f γp → ωp ′ (0) | γ, p >, (4)where f γp → ωp ′ (0) is the forward amplitude for the γp → ωp ′ reaction.The differential cross section for the reaction described above can be written as dσ = 12 E γ (2 π ) δ ( k i − k f ) < | T fi | > m p E p ′ d k p ′ (2 π ) E π d k π (2 π ) E γ ′ d k γ ′ (2 π ) . (5)The annular bracket around the T fi -matrix indicates the average over the polar-ization and spin in the initial state and the summation over the polarization andspin in the final state. Since the π and γ bosons in the final state are consideredoriginating due to the decay of the ω meson, the above expression can be workedout in terms of the ω meson mass m (i.e., the π γ ′ invariant mass) as dσ ( m, E γ ) dm = Z d Ω ω [ KF ]Γ ω → π γ ′ ( m ) | G ω ( m ) | | F ( γp → ωp ′ ) | . (6)Where d Ω ω is the infinitesimal solid angle subtended by the ω meson momentum k ω (= k π + k γ ′ ). Γ ω → π γ ′ ( m ) denotes the width for the ω meson of mass m decayingat rest into π γ channel. It is expressed in Eq. (14). [ KF ] represents the kinematicalfactor for this reaction, which is given by[ KF ] = 3 π (2 π ) k ω m p m k γ | k ω ( E γ + m p ) − k γ . ˆk ω E ω | . (7)The Eq. (6) illustrates the differential cross section for the ω meson mass distri-bution due to fixed beam (gamma) energy E γ . But the measurement was done bythe CBELSA/TAPS collaboration [8], as mentioned earlier, using a range of inci-dent γ energy instead of fixed beam energy. We incorporate it in our calculation bymodulating the cross section in Eq. (6) with the beam profile function W ( E γ ) [14],i.e., dσ ( m ) dm = Z E mxγ E mnγ dE γ W ( E γ ) dσ ( m, E γ ) dm . (8) E mnγ and E mxγ are equal to 0.64 GeV and 2.53 GeV respectively, as mentioned inRef. [8]. The profile function W ( E γ ) for the γ beam, originating due to bremsstrahlungradiation of the electron, varies as W ( E γ ) ∝ E γ [14].The calculated cross section due to Eq. (8) can’t reproduce the measured distri-bution, since the detecting system in the experimental set-up (as mentioned earlier)4ad large resolution width, i.e., 55 MeV, [8]. This issue is incorporated in the for-malism by folding the differential cross section in Eq. (8) with a Gaussian function R ( m, m ′ ): dσ ( m ) dm = Z dm ′ R ( m, m ′ ) dσ ( m ′ ) dm ′ . (9)The function R ( m, m ′ ) accounts the resolution (or response) for the detector. Theexpression for it [15] is R ( m, m ′ ) = 1 σ √ π e − ( m − m ′ )22 σ . (10)Here, σ is related to the full width at the half-maxima (FWHM) of this function asFWHM= 2 . σ . We take the value for FWHM equal to 55 MeV, so that the function R ( m, m ′ ) can describe properly the resolution for the detector used at ELSA [8]. The amplitude for the γp → ωp reaction, i.e., f γp → ωp (0) needed in Eq. (4), isrelated to the four momentum q transfer distribution dσ ( γp → ωp ) /dq [16] as | f γp → ωp ( q ) | = k γ π dσdq ( γp → ωp ) . (11)Therefore, the energy dependent values for | f γp → ωp (0) | (required to calculate thecross sections in Eqs. (8) and (9)) can be extracted from the four momentum transferdistribution dσ ( γp → ωp ) /dq . In fact, the forward dσ ( γp → ωp ) /dq is obtainedby extrapolating the measured dσ ( γp → ωp ) /dq to q = 0 for E γ ≥ . ω mesonproduction in the γp reaction at lower energy, i.e., E γ ≤ . γp → ωp reaction with SAPHIRdetector at electron stretcher ring (ELSA), Bonn [11]. In this measurement, theyhave reported the measured dσ ( γp → ωp ) /dq vs | q − q min | ( q min is defined inRef. [16, 18]) in the energy region E γ = 1 . − . dσ ( γp → ωp ) /dq behaves as ∼ exp [ b ( q − q min )] in the lowfour momentum q transfer region. Since this energy region is well accord with ourstudy, we extract | f γp → ωp (0) | from the SAPHIR data and use it in our calculation.The ω meson propagator G ω ( m ) in Eq. (6) can be described by the Eq. (1).The total width Γ ω ( m ) appearing in it is composed of widths due to omega mesondecaying into various channels [9]:Γ ω ≈ Γ ω → π + π − π (88 . ω → π γ (8 . ω → π + π − (2 . ω → l + l − ( ∼ − %) . (12)5e ignore here widths due to the leptonic decay (Γ ω → l + l − ), since they are insignif-icant. The typical magnitudes for them are of the order of keV or less, where asthe widths for the ω meson decaying into other channels in Eq. (12) are within therange of 100 keV to 10 MeV.Γ ω → π + π − π ( m ) in the above equation represents the width for the ω → π + π − π channel. We use the form for it as derived by Sakurai [19]:Γ ω → π + π − π ( m ) = Γ ω → π + π − π ( m ω ) mm ω ( m − m π ) ( m ω − m π ) U ( m ) U ( m ω ) , (13)with Γ ω → π + π − π ( m ω = 782 MeV) ≈ .
49 MeV. U ( m ) → m → m π and U ( m ) → . m →
787 MeV. We have also taken U ( m ) equal to 1.6 for m > ω → π γ ( m ) in Eq. (12) arises due to ω → π γ channel. Using theLagrangian density L ωπγ given in Eq. (3), it is evaluated asΓ ω → π γ ( m ) = Γ ω → π γ ( m ω ) " k ( m ) k ( m ω ) , (14)with Γ ω → π γ ( m ω ) ≈ .
72 MeV at m ω = 782 MeV. k ( m ) denotes the momentum ofpion originating due to the ω meson of mass m decaying at rest.In Eq. (12) , Γ ω → π + π − ( m ) denotes the width for the ω meson decaying into π + π − channel. This channel arises due to the small pure isovector ρ meson componentpresent in the physical ω meson [1]. Using the Lagrangian density L ωππ = f ωππ ( ~π × ∂ µ ~π ) · ω µ , the width for this channel is worked out asΓ ω → π + π − ( m ) = Γ ω → π + π − ( m ω ) m ω m " k ( m ) k ( m ω ) . (15)The value for Γ ω → π + π − ( m ω = 782 MeV), according to Ref. [9], is approximatelyequal to 0.19 MeV. k ( m ) represents the pion momentum in the π + π − cm of system.We calculate the cross sections for the ω meson mass distribution in the γ inducedreaction on the proton target. The calculated results have been compared with themeasured π γ invariant mass distribution spectra in the p ( γ, π γ ) p reaction, sincethe π and γ in the final state (as mentioned earlier) are assumed to originate dueto the decay of the ω meson produced in the intermediate state. The measured π γ invariant mass distribution spectra have been reported in the Ref. [8] for four ω meson momentum bins: (i) 0 . < k ω (GeV/c) < .
4, (ii) 0 . < k ω (GeV/c) < . . < k ω (GeV/c) <
1, and (iv) 1 < k ω (GeV/c) < .
4. Therefore, we havecalculated cross section for (i) k ω = 0 . − .
39 GeV/c; (ii) k ω = 0 . − .
59 GeV/c;(iii) k ω = 0 . − .
99 GeV/c; and (iv) k ω = 1 . − .
39 GeV/c.6he ω meson mass distribution spectra calculated using the Eq. (8) are presentedin Fig. 1 by the dash-dot curves. The sharp peak at the mass around 780 MeV andwidth about 8.43 MeV, appearing in these curves, are the characteristics featuresfor the ω meson produced in the free state. The solid curves in this figure illustratethe ω meson mass distribution spectra calculated using the Eq. (9). These curvesarise due to the folding of the calculated cross section given in Eq. (8) with thedetector resolution function R ( m, m ′ ) in Eq. (10). Therefore, the incorporation ofthe detector resolution function in the calculated cross section, as shown in thisfigure, has widen all spectra and simultaneously, it reduces the cross section at thepeak. The enhancement in the width from ∼ .
43 MeV (dash-dot curves) to ∼ R ( m, m ′ ), see Eq. (10).In Fig. 2, we compare the calculated results due to Eq. (9) with the measured π γ invariant mass distribution spectra for all ω meson momentum bins. This figureshows that the calculated ω meson mass distribution folded duly with the detectorresolution function, as described above, reproduce the data very well for all bins ofthe ω meson momentum. We have calculated the differential cross section for the π γ invariant massdistribution in the γp reaction. Our analysis shows that the π γ appearing in thefinal state is occurring dominantly due to the decay of the ω meson produced inthe intermediate state. The calculated results showing sharp and narrow peak char-acterize the production of the ω meson in the free state. The incorporation of theGaussian function (to describe the detector resolution) in the calculation broadensthe ω meson mass distribution spectrum, which is well accord with the data. I gratefully acknowledge L.M. Pant for making me aware about the measure-ment on the omega meson mass distribution at ELSA. The discussion with D.R.Chakrabarty on the detector resolution is highly appreciated. The communicationmade with E. Oset regarding the beam profile function is very helpful. I acknowl-edge D. Trnka and V. Metag for sending the data. I thank A.K. Mohanty, R.K.Choudhury and S. Kailas for their support.7 eferences [1] T.H. Bauer, R.D. Spital, D.R. Yennie and F.M. Pipkin, Rev. Mod. Phys.
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300 (1962).9 igure Captions
1. The ω meson mass distribution spectra are presented for various ω mesonmomentum bins. The dash-dot curves correspond to the calculated resultsdue to Eq. (8), where as the solid curves arise due to the Eq. (9) (see text).2. The calculated ω meson mass distribution spectra for various ω meson momen-tum bins are compared with the data (see text). The solid curves correspondto the calculated results due to Eq. (9). The histograms represent the mea-sured counts for the π γ invariant mass distribution spectra [8], normalised tothe respective calculated peaks. 10 (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) p ( g , w p g ) pk w (GeV/c)=0.21-0.39 p ( g , w p g ) pk w (GeV/c)=0.41-0.59 p ( g , w p g ) pk w (GeV/c)=0.61-0.99 p ( g , w p g ) pk w (GeV/c)=1.01-1.39Eq. (8)Eq. (9) Eq. (8)Eq. (9)Eq. (8)Eq. (9) Eq. (8)Eq. (9) (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) m (GeV) d s / d m ( m b / G e V ) p ( g , w p g ) pk w (GeV/c)=0.21-0.39 p ( g , w p g ) pk w (GeV/c)=0.41-0.59 p ( g , w p g ) pk w (GeV/c)=1.01-1.39 p ( g , w p g ) pk ww