 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
iN<p* Dill ... DIlN<P in Visual Studio .NET
iN<p* Dill ... DIlN<P Scanning PDF417 2d Barcode In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. PDF417 2d Barcode Generator In VS .NET Using Barcode printer for .NET framework Control to generate, create PDF 417 image in .NET framework applications. i N  1  D Il2'" DIlNl/I l/IYIlI
PDF417 2d Barcode Recognizer In Visual Studio .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET framework applications. Barcode Maker In .NET Using Barcode creation for VS .NET Control to generate, create barcode image in VS .NET applications. iN  2 F 111 V D /12 ... D /INl Pv IlN
Bar Code Reader In .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. PDF 417 Drawer In C#.NET Using Barcode creator for .NET Control to generate, create PDF417 image in .NET applications. (13170) Draw PDF417 In .NET Using Barcode creation for ASP.NET Control to generate, create PDF 417 image in ASP.NET applications. PDF417 Creator In VB.NET Using Barcode maker for VS .NET Control to generate, create PDF 417 image in VS .NET applications. Symmetrization and extraction of traces is understood. We have only mentioned "physical" fields in this list. For specific subtleties of gauge theories, see some remarks below. To justify Eq. (13167) we relate it to the shortdistance expansion by projecting out a given spin exchange of the crossed Compton amplitude. This quantity depends only on the variable q2, the square virtual photon momentum, and is dominated in the limit  q2 ~ CfJ by the operator of corresponding spin and twist two in the Wilson expansion. This gives a definite information on the moments of the structure functions. For short we continue to ignore the vector character of the currents and the subsequent tensor analysis of the structure functions; we therefore pretend Printing Barcode In VS .NET Using Barcode generator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. 1D Barcode Encoder In .NET Framework Using Barcode creator for .NET Control to generate, create Linear 1D Barcode image in VS .NET applications. ASYMPTOTIC BEHAVIOR
USS Code 39 Printer In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create Code 39 image in VS .NET applications. Make EAN8 In .NET Using Barcode drawer for Visual Studio .NET Control to generate, create EAN8 Supplement 2 AddOn image in Visual Studio .NET applications. that we deal with the scalar amplitude
Barcode Recognizer In VS .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Universal Product Code Version A Printer In None Using Barcode encoder for Font Control to generate, create UPC Symbol image in Font applications. A(q2, V) Creating DataMatrix In VB.NET Using Barcode generation for Visual Studio .NET Control to generate, create ECC200 image in .NET framework applications. Bar Code Generation In None Using Barcode generator for Office Excel Control to generate, create barcode image in Excel applications. Jd4Xeiq'X<pITJ(~)J( ~)iP>
Code 3/9 Encoder In Java Using Barcode generation for Android Control to generate, create Code 3 of 9 image in Android applications. Encoding EAN 13 In Java Using Barcode printer for Android Control to generate, create EAN13 image in Android applications. (13171) EAN 13 Creation In None Using Barcode creation for Software Control to generate, create GTIN  13 image in Software applications. UPC  13 Maker In Java Using Barcode maker for Java Control to generate, create GS1  13 image in Java applications. For finite q2, v it can be expanded on a basis of orthogonal polynomials in the variable z = iv/m~ which play the role of a cosine of the angle between the fourvectors p and q. These polynomials are orthogonal with respect to the measure dzJl  Z2 and generalize to 0(4) the Legendre polynomials relative to 0(3) invariance. We refer to the work of Nachtmann quoted in the notes for details on this projection and the constraints arising from the positivity properties of the structure functions. We shall satisfy ourselves here with a simplified presentation. Inserting (13167) into the definition (13171) we find
A(q2, v) = iq d4x e . x
CN,ix 2  ic)x/l
X/l N
<PIO~,~"I'N Ip>
Some care has been paid to the analytic properties in x space to give an infinitesimal negative imaginary part to the variable X2. Define <P IO'f.J:~ 'I'N IP> = aN,a(plll ... pllN
CN,a(q2) = i (  + trace terms) 2 4 iq d x e . xCN,ix  ic; (13172) in terms of which the dominant term in A(q2, v) is given by
A(q 2 ,v) ~ " ~ N,a
' (2 Pq)NaN,aCN,a(q) 2 (13173) The contributions of the operators ON,a are related to the coefficients of the Taylor series of A in powers of the scaling variable w = XI = 2p' q/ _q2, while experiments measure the absorptive part of A for w larger than one. It is therefore necessary to isolate the coefficient of wN in order to use the shortdistance expansion to study CN a(q2) for large negative q2. In this discussion the variable w rather than its inverse x is more convenient. Since the point w = 0 is outside the experimental reach an analytic continuation cannot be avoided. This is expressed through a forward dispersion relation for the virtual Compton amplitude A, using as discontinuity the structure function W(q2, v). The variable in this dispersion relation is the energy v/m, or what amounts to the same for fixed q2, the quantity w. It is also necessary to use the crossing properties to define the discontinuity for w smaller than 1. In a fictitious scalar case as here W(q, p) =  W(  q, p) or W(q2,  w) =  W(q2,  w). In the realistic vector case W/l.(q, p) =  WV/l(  q, p), which means that all three structure functions WI, W2 , W3 may be considered as odd in w. Further information must be provided concerning the number of subtractions k, uniformly for large negative q2. With 1m A = nW, and Pk  I being a polynomial QUANTUM FIELD THEORY
of degree k  1 in w with q2 dependent coefficients, we find
A(q2, w) = Pk_l(q2, w) r w w JIMI >1 ~ (W,)k W(l, w') (13174) Expanding near the origin this is also
A(q 2 ) = Pk  1 (q 2 ,w) ,w
dw' (13175) ~ W(q 2 , ,w) 1(0'1>1 W For N ~ k we have obtained the desired relation between the moments MN(q2) of the structure function and the Wilson coefficients N2:k
2 M (N) (q)  "N w
dw w+T W(q 2 ,w) 1(012:1 '00 aN,aCN,a(q 2 ) (13176) For fixed N the leading terms on the righthand side correspond to twist two operators. Positivity conditions on the W result in convexity properties of the moments in the variable N. The smaller the N the more sensitive are the M(N) to the asymptotic region since for large N we test mostly the region w ~ 1. According to various hypothesis we have the following possibilities for M(N)(q2) for large l: constant naive scaling M(N)(q2)

