Choudhury, D K

Gogoi, Rupjyoti

IACS Institutional Repository

Indian J. Phys. 80 (6), 659-662 (2006) r ^ 4 O t€Some comments on fra<|tality of proton at small jr,'iand Rupjy$pcs. Ciaiihijyoti GogoiD K ChoudhuryDeparlment of I*hy$|^ ati Univcrsily,Guwahati -  014, Assam, IndiaB-mail dkc_plhysC?^yahoo.co inR e c e i v e d  1 D e c e m b e r  2 0 0 5 , a c c e p t e d  22  M a t c h  2 0 0 6Abstract • Using Ihc concept of sclf-sinnlaiily in Ihc slrucUirc ol the prtiU^ n al small \, \vc commcnl on possibility ol a single jiositive hactal dimension ol proton m analogy with classical monofructals Plausible dynamics and physical inicrpietation of tractal dimension are also discussedKeywords • Self similanly, fraclal dimension, deep inelastic scalicnng, strucfure function, low \PACS Nos. 05 45.1)1. 47 53.-fii. 12 38 t. 13.60 HbSelf-similarity is a familiar property in nature 11,21. Many of the seemtngly irregular shapes o f nature have hidden self-similarity in them. It is not the usual symmetry with respect to rotation or translation, but symmetry with respect to scale or size: a small part o f a system is .self-similar to the entire system. Such a system IS defined through its self-similar dimension, which is in general fraction, hence called fractal dimension. Classical fractals discussed in standard references [1,2] are Cantor dust, Koch curve and Sierpinski gasket whose fractional dimensions are 0.63, 1.26 and 1.585, respectively, which lie between Euclidean point and surfaces.Notion o f self-similarity and fractal dimensions are being used in the phase spaces o f hadron multiparticle production processes since nineteen eighties (3-7]. However, these ideas did not attract much attention in contemporary physics o f deep inelastic lepton-hadron scattering till 2002 when Lastovicka 18] developed relevant formalism and proposed a functional form of the structure function F 4,x ,Q ^) at small jc. Specifically, a description o f F^ix.Q ^) reflecting self-similarity is proposed with a few parameters which are fitted to recent HERA data 19,10]. The specific parameterization is claimed to provide an excellent description o f the data which covers a region o f four momentum transferred squared 0.045 ^ 150GeV^ and o fBjorken jc, 6 . 2 x  10"*^  ^ jc S 0.2.Corresponding AuthorMore recently, it was observed 111-13] that the positivity o f fractal dimensions prohibits some o f the fitted parameters o f the structure function o f Ref. f8]. Specifically, out o f the fractal dimensions/),, D^and one is negative (Z):^  == -1.3). Htiwever, the positivity o f fractal dimension forbids such negative value. In order to avoid such possibility, it is suggested that the proton is described by the single self-similarity dimension D. This then facilitates one to compare the self-similarity nature of the proum at small ,v with the classical monofractals which is the aim o f the present note.Under the hypothesis o f  self-similarity o f the proton structure at small ,r, Lastovicka [8] obtained the following form o f the structure function F 4 x ,Q ^ ).(expH - D , +  O , lo g  ) /V /  Qo )(1)by using the following form o f un-integrated quark density fXx.Q^) o f i-th quark flavor :log/.(Jr. Q ^ ) /?, log)/log(^l©20061ACS660 D  K  Choucihiiry and Rupjyoti G a ^o iQ2*o 085 GeV34 D2 log J/ + Z), log|^  I + ^ y Q 2 j'^ ^0 . (2)In eqs. ( 1 ) and (2), Dj is the dimensional correlation relating the two magnification factors lAv and \  ^ while and Z), are theself-similarity dimensions associated with them, being the normalisation constant. Since the magnification factors should be positive, non- ° zero and dimensionless, a choice \-^Q ^ / q I  , rather than Q- has been made, while isarbitrary small virtuality, q ' > (J~ . Explicit ^  confrontation with HERA data [9, 10] yields:«  0 1 -1.0°\o® 15»<‘l0-6 2 0x10'-«Q2=o 11 GeV^x‘ 0 2Q2«0.l5GeV3.VQ2=o 2 GeV32 0x10'^ 4 0«10-® 0 0 3 0x10-® 6 0x10 e>X XQ2=0 25 GoV3  ^  ^ Q2=0 3 GeV^1 qLA, =0.339 ± 0 .1 45,A  =0.073 ±0.001,D, =1.013±0.01, (3)D, = - 1.287 ±0.01,=0.062 ± 0.01 G e f ^ \As the self-similarity dimensions o f fractals are positive [ 1 ,2 ] by its definitions, one expects D ^ > 0 ,  0 2 ^ 0, D3 > 0, a feature absent in the empirical fit o f [8J as far as is concerned. In analogy with other classic fractals [1,2], we therefore assume that proton at small x is a monofractal with just one single fractal dimension, so that= (4)Under such a hypothesis, eq. (1) is rewritten0 8 0x10-® 1 6x10-XQ2*0 4 GeV^.><-0 20 1  ^0 00 4u. »04•  ^ -r—-- - ----- 0 5 ---  ..-..0 4L‘ -- - - ------ 0 ‘^ -0 4ir 0 3 ' — _ U-1 . . 1 . . . - 0 2 . 0 20 25 0‘x 10--* 0.0 5 0x10-  ^ 10x10XQ2=0 66 GeV200 50x10-^  10x10*^  00 20x10-^  4 0x10^  00X XFigure I. versus x in bins of with D, ^ 0 (eq (5))3 0x10® 6 0 x1 0 ^Q2s=0 2 GeV=» Q2=0 26 GeV® Q=»0 3 GeV®1 'F z> 4- D log y '^-Dlosl1±5 (5)Alternately, monofractal ity is attainable also for £), -  0 (zero dimensional correlation) and D2 0 4 == Z?3 = Z), so that eq. (1) takes the alternate form: SQ2=o.4 OeV» Q2=o 5 GeV^ q2«0.85 QeV*[ , - f ] 3.0x10“^  6.0x10XFigure 2. Fpc,Q )^ versus x in bins o f  with 0 (cq.(6 )).Some comments on fractality o f  proton at small x 661In Figure 1, we plot v ersu s x in bins of Q - asmeasured by low Q - data of ZEUS [ 10| using eq. (5). Results of the fit yields:Do =-1.692 ±0.14 , /)= 0.653 ±0.029,= 0.0449 ± 0.0003 G e V ^ .78 504 83 156X 'ld o f1.402 0 621Our analysis thus indicates that only in the limited x  -  range ( 0 ~  <  12 GeV~ ), the notion of monofractality of proton holds. In that range, dimensional coirelation (D j) vanishes and the proton possesses fractality (Z )=  1.107) close to Koch curve (D  «  1.26). Description of F.,U’, (2“) in the entire small jc range in terms of monofractal will result in a continuous x^ Q j  dependent fractal dimension [13] which is a considerable extension of parameter space and contrary to the usual notion of fractal.It is also instructive at this stage to ascertain the physical interpretation of fractal dimension of proton, since the notion is rather recent in literature. As is well known [14], the fractal dimension measures the way, in which distribution of points fill a geometric space on the average. If the distribution is highly inhomogeneous, the set of points have a distribution of fractal dimensions leading to multifractality. Extending the notion to the -r -  plane of the unintegrated quark density, fractal dimension tells, how densely small x  partons Utl the proton in self-similar way on the average. In the special case ofD D 2 »  .fbim:D^, unmtegrated quark density takes the simple(9)(7)^However, this fit (eq. (7)) can not be extrapolated to higher  ^Q- range of HI [9], Even forQ^ > 0.4 GeV- of ZEUS 110] dataj becomes large.In Figure 2, we show the similar analysis using eq. (6) for ZEUS data 110]. Results of the fit yields;Do = -2 .7 13± 0.231,D, =0,D=1.107±0.008,<2(7 = 0.045 ± 0.00012 G eV  ~. (8)This fit (eq. (8)) can be extrapolated to higher Q- range of H 1 |9j upto = 12 GeV“. The x  ~ f^ >r eqs. (7) and (8) arc recorded in Table 1.Table 1. Tlic X tor rqs (7) and (8) Fitcq.(7) (0.045 = = 0 4GcVblc|.(8) (0.045 = = 12ticVband fractal dimension is essentially close to .v-slope 115j oi Pomeron intercept 116-18].Let us conclude this note with a few comments. The limitation of the present approach as indeed with that proposed by Lastovicka |8J is that it provides merely a parameterization i>f structure functions based on fractality m terms (d‘ a few parameters to be determined from data and identify them later as fractal dimensions. Because of the availability of extremely high quality data |9,10) and accurate parameterizations like CTEiQ 119] and MRST |20], such new approach to parametrizing the proton structure function must have strong physical reason forIt.In the present note, we have shown that m the monofractal limit, fractal dimension is closely related U) more familiar v-slope 115] or Pomeron intercept | 16-1 8] as is evident from eq. (9). As fractals can be seen in the context of renormalization group (RG) [21,22], as well, eq. (9) can be interpreted as a solution of RG type of equation for the self-similar structure function F, itself:f/F;(A)d  In 1 (10)where the f j  function obeys a power series [21,22] in F,,,F") =  Oq 4* Z/ I F2 ±  02  f*2 (11)with fljj, r/j, being constants. Retaining terms upto linear m F, and choosing the boundary condition F^  (a ) = 0 at .v = 1, cq.(9) immediately follows if the coefficient rqofthe p  -function is identified as the monofractal dimension D.It is also noteworthy that similar tci the present approach, fractal characters of hadrons have been pursued in Refs. [22,23 ] within a statistical quark model with considerable success. Our results compliment such notion in deep inelastic regime.AcknowledgmentWe thank Dr T JLastovicka for useful correspondences. 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Some Comments on Fractality of Proton at Small x

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Some Comments on Fractality of Proton at Small x

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