Blog Prophetism | Social Science blog

Awareness - To Face Our True Identity

Thursday, 06 July 2017 06:00 Written by

Men are not fully aware: they think to be so to protect themselves.

Men are ignorant, although they are convinced of their knowledge, and the claim of their alleged knowledge precisely feeds their ignorance.

Furthermore, men are diametrically opposite to their expectations: the ability to recognise this aspect is the first step in starting a great revolution: the internal revolution!

We need courage to face our true identity.

Ideals may be appealing and gratifying, but ideals have just one purpose: hiding the reality.

We keep on believing in great ideals, however we are moved not by the love for these ideas, but rather by the need to hide the cruel reality.

Conversely, aware persons show a stronger and practical concern for real facts.

Aware persons relate to the world without any division: interiority and exteriority are not separate.

But this is precisely the effect of any kind of idealism: the separation of interiority from exteriority, reality from fantasy, the ban of genuineness and spontaneity and the rejection of one’s identity; idealists are dreamers!

Dreaming is no evil, but now let's have a look at practical facts and make them come true!

And so you may start believing in the beauty of your ideals, but these empty words conceal a reality which is quite different from the expectations.

For centuries people have been talking about nonviolence (such a great ideal!), but have never been able to put it into practice.

And they will never succeed in this task, because speeches are dangerous: they make us believe that we have already reached our goals.

Over time, after a series of speeches, illusion spreads among the others and takes the control of our life as well.

After many centuries marked by speeches on nonviolence, we have started to believe that we have eventually become meek.

This is the reason why we are still talking about nonviolence .

Those who want to improve their spirituality must become aware of this ideological concealment.

It is easy to be fascinated by great ideals, but consider men: if you know their ideals, you will notice that their reality is their exact opposite.

Someone's ideology will help you realise that their behaviour is completely different from the underlying ideas.

The belief in ideals just proves the presence of a hidden truth.

Countries are not getting ready for peace, their only aim is war!

The violent wants to become nonviolent, but how can he manage to do that?

He was used to be violent towards the others, now he is violent towards himself.

It is cold and is snowing, but someone is naked out there, what is he doing?

He is only tormenting his body, while people say: "He is so brave!"

Just think about slimming diets: how many people "rape" their body in the name of alleged ideals!

The killer is more eye-catching than the suicide victim, however the difference between them is slight; they are both murderers, they both love violence.


Get rid of all the ideals.

Do not try to be different from yourselves: consider only reality, whatever it may be.

Abide by facts, do not mistake them with dreams or fantasy, otherwise you will always be divided: please remember that ideals and dreams cannot help you, they have never helped anyone so far.




Proton structure functions at small x

Monday, 06 March 2023 17:41 Written by

Proton structure functions at small x

Martin Hentschinski

Brookhaven National Laboratory, Physics Department, Upton, NY, 11973, U.S.A. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.


Abstract. Proton structure functions are measured in electron-proton collision through inelastic scattering of virtual photons with virtuality Q on protons; x denotes the momentum fraction carried by the struck parton. Proton structure functions are currently described with excellent accuracy in terms of scale dependent parton distribution functions, defined in terms of collinear factorization and DGLAP evolution in Q. With decreasing x however, parton densities increase and are ultimately expected to saturate. In this regime DGLAP evolution will finally break down and non-linear evolution equations w.r.t x are expected to take over. In the first part of the talk we present recent result on an implementation of physical DGLAP evolution. Unlike the conventional description in terms of parton distribution functions, the former describes directly the Q dependence of the measured structure functions. It is therefore physical insensitive to factorization scheme and scale ambiguities. It therefore provides a more stringent test of DGLAP evolution and eases the manifestation of (non-linear) small x effects. It however requires a precise measurement of both structure functions F2 and FL, which will be only possible at future facilities, such as an Electron Ion Collider. In the second part we present a recent analysis of the small x region of the combined HERA data on the structure function F2. We demonstrate that (linear) next-to-leading order BFKL evolution describes the effective Pomeron intercept, determined from the combined HERA data, once a resummation of collinear enhanced terms is included and the renormalization scale is fixed using the BLM optimal scale setting procedure. We also provide a detailed description of the Q and x dependence of the full structure functions F2 in the small x region, as measured at HERA. Predictions for the structure function FL are found to be in agreement with the existing HERA data.



1.    Introduction

The description of the proton in terms its elementary constituents, quarks and gluons, remains one of the big unsovled problems of nuclear- and elementary particle physics.  At the

typical energy scale of the proton, which is of the order of ΛQCD ~ 200 MeV, Quantum


Chromodynamics, the Quantum Field Theory description of strong interactions is strongly coupled, and quarks and gluons are subject to confinement. It is however possible to obtain very valuable information about the structure of the proton from collision processes of protons with leptonic projectiles, such as the electron. Due to the point-like structure of the electron and a very good theoretical understanding of electromagnetic interactions, the electron provides the perfect probe to explore the nucleon. To leading order in Quantum Electrodynamics (QED), scattering of the electron and the proton takes place through the exchange of a virtual photon with virtuality q2 = Q2, see Fig. 1. If the photon virtuality is large, the proton is destroyed during the scattering and the process is generally referred to as Deep Inelastic Scattering (DIS). The cross-section for neutral-current DIS on unplolarized nucleons can be written in terms of





























Figure 1. Schematic diagram of the Deep Inelastic Scattering (DIS) process. Measurement of the final state electron allows to determine inclusive nuclear structure functions in terms of the resolution scale Q2 = q2 provided by the photon virtuality and Bjorken x = Q2/(2p q) (Figure taken from [1] ).






two Lorentz invariant structure functions F2 and FL in the following way


d2σ dxdQ2 =








  1 − y + y


F2(x, Q2)



y                      l


FL(x, Q2)








·         ·




Here y = (q p)/(k p) denotes the inelasticity with 0 < y < 1, see also Fig. 1. The structure func- tions themselves depend on only two Lorentz invariants, the photon virtuality Q2 and Bjorken x = Q2/(2p q). Within the parton model, to be discussed below, x denotes the momentum fraction of the parton hit by the virtual photon.




If the photon virtuality Q2 is significantly larger than the non-perturbative energy scale of the proton, asymptotic freedom provides for such processes a weak strong coupling constant αs(Q2) 1 and a description within perturbative theory becomes possible. The conventional theoretical framework for such DIS processes is based on the collinear factorization theorem [2]. At leading order, the essential physics is captured by the parton model [3] of the proton. Within this model, the highly virtual photon interacts not with the entire proton with characteristic

size 1/ΛQCD, but with a single, essentially point-like, parton, i.e., a quark or gluon, with

effective size 1/Q. Interference effects with spectator quarks or gluons are on the other hand suppressed by powers of Q2. To arrive at the complete cross-section, the “partonic” interac- tion of virtual photon and quark, needs to be convoluted with parton distribution functions (PDFs), fi(x, Q2), i = q, q¯, g, which encode the probability to find a parton with a certain pro- ton moment fraction inside the proton. Higher order corrections to such partonic cross-sections, calculated within QCD perturbation theory, possess reveal then a new kind of singularity, apart from the conventional ultra-violet singularity, which is removed through renormalization of the QCD Lagrangian. This new singularity is of infra-red type and can be associated with configu-




rations where an additionally emitted parton is collinear to the proton momentum. In physical terms, this initial state singularity reflects interference between the perturbative calculable par- tonic interactions at the hard scale Q, with spectator quarks and gluons, characterized by the hadronic scale ΛQCD. Collinear factorization provides then a systematic framework to remove

such singularities from the perturbative hard cross-section, resulting into finite Wilson coeffi-

cients [4–7], absorbing them into parton distribution functions, which encode the long-distance, non-perturbative physics; for a recent review see, e.g., [8].


To make the separation between long- and short-distance physics manifest, one needs to introduce some arbitrary factorization scale µf , apart from the scale µr appearing in the renor- malization of the strong coupling αs. The independence of physical observables such as F2,L on µf can be used to derive powerful renormalization group equations (RGEs) governing the scale dependence of PDFs in each order of perturbation theory, known as the Dokshitzer-Gribov- Lipatov-Altarelli-Parisi (DGLAP) evolution equation. The corresponding kernels are the anoma- lous dimensions or splitting functions associated with collinear two-parton configurations [9–11]. DGLAP evolution has been impressively confirmed by experiment, in particular through the very accurate DIS data on the structure function F2 from the DESY-HERA experiment, see

Fig. 2. In particular, parton distribution functions, fitted at some initial scale Q0 ~ 1 2 GeV







10 7



10 6


H1 and ZEUS



HERA I+II inclusive, jets, charm PDF Fit




10 5


10 4







10 3






10 2































1                    10                  10





3                                          4                                          5

10                  10                  10











10-1                      x 1


Q2/ GeV2


Figure 2. Left: Combined HERA data and parton distribution functions. Right: Proton parton distribution functions plotted as functions of Bjorken x. (Source [12])


and evolved with the DGLAP equations, have been shown to describe F2 data over orders of magnitude, both in x and Q2 [13].



Despite of the impressive success of the DGLAP evolution equations, theoretical considerations suggests that at some point in phase space this description is supposed to break down. With decreasing x, logarithms ln 1/x increase and are capable of balancing the strong coupling, αs ln 1/x 1, leading to a break-down of the na¨ıve perturbative expansion, see Fig. 3. The necessary resummation of enhanced terms (αs ln 1/x)n is the achieved by the Balitsky- Fadin-Kuarev-Lipatov (BFKL) evolution equation [14]. One of the main predictions of BFKL evolution is a power-like rise of the gluon density. If continued to ultra-small x, the 1/Q2 expansion, on which collinear factorization is based, will eventually break down. A description













Low Energy                                        High Energy



x0 >> x





Proton (x0, Q2)


many new smaller p

are produced




(x, Q2)






as ~ 1


as << 1


ln Q2



Figure 3. Left: The proton wave-function at small-x contains a large number of gluons as compared to the same wave-function at a larger x = x0. The figure is a projection on the plane transverse to the beam axis. Right: The map of high energy QCD in the (Q2 , Y = ln 1/x) plane. (Source [1])



of proton structure functions in this region of phase space is provided by the Balitsky-Kovchegov (BK) [15] and JIMWLK [16] evolution equations, which provide a non-linear extension of BFKL evolution, resumming corrections due to high gluon densities to all orders. While theoretical arguments suggests the relevance of such corrections already at current collider energies, current data provide no clear evidence for deviations from linear DGLAP evolution, which would provide a signature for the onset of a non-linear kinematic regime dominated by high, or saturated, gluon densities. Definite evidence for such a regime of QCD requires therefore new experiments such as a future high-luminosity electron-ion collider, i.e.., the EIC [1] and the LHeC [17] projects, whose physics case is currently studied. In particular these projects plan to measure both structure functions F2 and FL and their scaling violations very precisely at small x both in electron-proton and in electron-heavy ion collisions.

From the theory side this requires the development of suitable tools which allows to pin down possible deviations from DGLAP evolution. In particular it is necessary to reduce the large freedom in fitting initial conditions of parton distribution functions. With non-linear saturation effects most likely to manifest themselves at small values of Q2, the large number of free parameters used for the description of initial conditions in PDF fits, does not allow to exclude the possibility that saturation effects, while present in reality, are currently hidden in the initial conditions of DGLAP evolution.

In the following we present two approaches which have the potential to restrict this large freedom at low scales. Section 2 is dedicated to the concept of physical evolution kernels, which allows to reduce the number of independent PDFs in DIS fits and eliminates scale- and scheme dependence in their definition. Section 3 contains results of a recently achieved BFKL fit of the combined HERA data. While both theoretically and experimentally less explored than collinear factorization, BFKL evolution has the potential to reveal the emergence of non-linear effects more easily than DGLAP evolution. Unlike DGLAP evolution, BFKL drives the system into the saturated regime, making a detection of high density effects more likely. For details we refer to [18–20]




2.   Physical evolution kernels for DIS observables

Since collinear factorization can be carried out in infinitely many different ways, one is left with an additional choice of the factorization scheme for which one usually adopts the MS prescription. Likewise, the RGE governing the running of αs with µr can be deduced from taking the derivative of F2,L with respect to µr. Upon properly combining PDFs and Wilson coefficients in the same factorization scheme, any residual dependence on µf is suppressed by an additional power of αs, i.e., is formally one order higher in the perturbative expansion but not necessarily numerically small.

Alternatively, it is possible to formulate QCD scale evolution equations directly for physical observables without resorting to auxiliary, convention-dependent quantities such as PDFs. This circumvents the introduction of a factorization scheme and µf and, hence, any dependence of the results on their actual choice. The concept of physical anomalous dimensions is not at all a new idea and has been proposed quite some time ago [4, 21, 22] but its practical aspects have never been studied in detail. The framework is suited best for theoretical analyses based on DIS data with the scale µr in the strong coupling being the only theoretical ambiguity. In addition, F2,L or their scaling violations can be parametrized much more economically than a full set of quark and gluon PDFs, which greatly simplifies any fitting procedure and phenomenological analysis. The determination of αs from fits to DIS structure functions is the most obvious application, as theoretical scheme and scale uncertainties are reduced to a minimum.

Here we largely focus on the practical implementation of physical anomalous dimensions in analyses of DIS data up to next-to-leading order (NLO) accuracy. We shall study in detail potential differences with results obtained in the conventional framework based on scale- dependent quark and gluon densities, which could be caused by the way how the perturbative series is truncated at any given order.


  • Theoretical Framework

dimensions instead of the scale-dependent quark singlet, Σ


q (q + q¯), and gluon distributions


This gist of the factorization scheme-invariant framework amounts to combine any two DIS observables {FA, FB} and determine their corresponding 2 ×L2 matrix of physical anomalous


{           }          {                         }




appearing in the standard, coupled singlet DGLAP evolution equations. Instead of using measurements of F2 and FL (actually their flavor singlet parts), one can also utilize their variation with scale for any given value of x, i.e., dF2,L(x, Q2)/d ln Q2 as an observable. The required sets of physical anomalous dimensions for both F2, FL and F2, dF2/d ln Q2 have been derived in [22] up to NLO accuracy. The additionally needed evolution equations for the non-singlet portions of the structure functions F2,L are simpler and not matrix valued. As we shall see below, the required physical anomalous dimensions comprise the inverse of coefficient and splitting functions and are most conveniently expressed in Mellin n moment space. The Mellin transformation of a function φ given in Bjorken x space, such as PDFs or splitting functions, is defined as





dx x





φ(x) ,                                                                 (2)


where n is complex valued. As an added benefit, convolutions in x space turn into ordinary products upon applying (2), which, in turn, allows for an analytic solution of QCD scale evolution equations for PDFs.    The corresponding inverse Mellin transformation is straightforwardly performed numerically along a suitable contour in n space, see, e.g., Ref. [23] for details. The necessary analytic continuations to non-integer n moments are given in [24,25], and an extensive list of Mellin transforms is tabulated in [26]. We will work in Mellin space throughout this review. Assuming factorization, moments of DIS structure functions FI at a scale Q can be expressed







FI(n, Q2) =







e2 CI,k



Q2         2





µ !








n, αs(µ2),         ,  r

f         f


  • fk



2         2









µ !








n, αs(µ2),    f        r

0        f





where the sum runs over all contributing nf active quark flavors with electric charge squared

e2 and the gluon g, each represented by a PDF fk. For e2, the averaged quark charge factor

e¯ = (1/nf ) L


q                                                                                                                                                        g



q                                   q


2                                      e2 has to be used. µr and µf specify renormalization and factorization scale,

respectively. The scale Q0 defines the starting scale for the PDF evolution, where a set of non-

perturbative input distributions needs to be specified. For simplicity we identify in the following the renormalization scale with the factorization scale, i.e., µr = µf ≡ µ. The coefficient functions CI,k are calculable in pQCD [4–7] and exhibit the following series in as ≡ αs/4π




n, αs(µ ), µ2


as (µ ) CI,k


n, µ2


,                                    (4)


2      Q


L     m       2











where m0 depends on the first non-vanishing order in as in the expansion for the observable under consideration, e.g., m0 = 0 for F2 and m0 = 1 for FL.





Each PDF fk(n, µ2/Q2) obeys the DGLAP evolution equation which reads



d ln µfk





n,     2












n, αs(µ2) fl





n,     2






where the l        k splitting functions have a similar expansion [9–11] as the coefficient functions in Eq. (4):



Pkl(n, αs(µ2)) = L as(µ2)1+m P (m)(n) .                                                                                                                (6)










The Pkl(n) relate to the corresponding anomalous dimensions through γkl(n) = 2Pkl(n) in the normalization conventions we adopt, where we use the leading order (LO) and NLO expressions for γkl(n) given in App. B of the first reference in [10]. We note that the same normalization is used in the publicly available Pegasus evolution code [23]. In practice one distinguishes a 2 2 matrix-valued DGLAP equation evolving the flavor singlet vector comprising Σ(n, µ2/Q2) and g(n, µ2/Q2) and a set of nf  1 RGEs for the relevant non-singlet quark flavor combinations.

The scale-dependent strong coupling itself obeys another RGE governed by the QCD beta function


d ln µ2










das(µ) = β(a ) = L am+2β








with β0 = 11  2nf /3 and β1 = 102  38nf /3 up to NLO accuracy. To compare below with the results for the physical anomalous dimensions in Ref. [22] we also introduce the evolution variable



t ≡ −β ln




a (Q2)


.                                                                  (8)


0                s        0

(n, as(Q )) ·



F (S)(n, Q2)




Instead of studying FI(n, Q2) in (3) in terms of scale-dependent PDFs, which are obtained from solving the singlet and non-singlet DGLAP equations (5) in a fit to data [13], one can also derive evolution equations directly in terms of the observables FI(n, Q2). To this end, we consider a pair of DIS observables FA and FB, to be specified below, whose scale dependence is governed by a coupled matrix-valued equation



F (S)(n, Q2)




d        F (S)(n, Q2)!




 KAA       KAB




2       F (S)(n, Q2)!






for the flavor singlet (S) parts of FA,B and a set of non-singlet (NS) equations


dF (NS)(n, Q2)

d ln Q2




       A,B                        = K(NS)(n, as(Q2)) · F (NS)(n, Q2)                      (10)

for the remainders.



The required physical anomalous dimensions in Eqs. (9) and (10), obey a similar perturbative expansion in as as in (6). The singlet kernels in (9) are constructed by substituting






= e¯f C


  F (S)(n, Q2))           2


2      Q


 Σ(n, µ2/Q2)









n, αs(µ ),








into the left-hand side of Eq. (9) and taking the derivatives. Note that we have normalized the quark singlet part of FA,B with the same averaged charge factor e¯2 which appears in the gluonic sector. Upon making use of the RGEs for PDFs and the strong coupling in Eq. (5) and (7), respectively, one arrives at




Kij(n, αs(Q )) =


β(as(Q ))


2                              2     ∂C(n, αs(Q2), 1)

+ C(n, αs(Q2), 1) · P (n, as(Q2))  · C1(n, αs(Q2), 1) , l                                                                                    (12)



where we have introduced 2 × 2 matrices

CB,q       CB,g


Pgq           Pgg


C CA,q       CA,g  ,                                    P Pqq       2nf Pqg  .                         (13)

for the relevant singlet coefficient and splitting functions, respectively. An analogous, albeit much simpler expression holds for the NS kernel K(NS) in (10). As has been demonstrated in [22], the kernels (12) are independent of the chosen factorization scheme and scale but do depend on µr and the details of the renormalization procedure. We also note that the inverse

C1 in (12), appearing upon re-expressing all PDFs by FA,B, can be straightforwardly computed

only in Mellin moment space.


  • Example I: F2 and FL





(1)                                                          (1)


Let us first consider the evolution of the pair of observables {F2, FL}. A precise determination of FL in a broad kinematic regime is a key objective at both an EIC [1] and the LHeC [17]. Since the perturbative series for FL only starts at O(as), one wants to account for this offset by actually considering the evolution of either {F2, FL/(as)} or {F2, FL/(asC  )}. Both

sets of kernels KAB show a rather different behavior with n, as we shall illustrate below, but



without having any impact on the convergence properties of the inverse Mellin transform needed to obtain x dependent structure functions.  The kernels KAB at LO and NLO accuracy for

{FA, FB} = {F2, FL/(asCL,g )} can be found in [22]. Note that evolution in [22] is expressed in terms of t. Using (8), d/das = 2/(asβ0)d/dt, and (7) to compute to extra terms proportional to β(as), we fully agree with their results.







For {FA, FB} = {F2, FL/(asC(1))} one finds










K(0) = P (0)   L,q P (0),                                                             K(0) =   L,q P (0),                                                                                                   (14)






22              qq


(1) qg L,g


2L               (1)  qg






























KL2 =  C(1) − C(1)  Pqg  − Pgg  + Pqq ,                           KLL = C(1) Pqg  + Pgg










at the LO approximation, i.e., after expanding Eq. (12) up to O(as). Only the off-diagonal entries change if dividing FL by asC(1) ; see Eqs. (41)-(45) in Ref. [22]. For NLO kernels we refer

to [18].







-10                                                                                                                               3
















5       10      15      20      25 n 30 0







5       10      15      20      25 n 30






20                                                                                                                               -40






0        5       10      15      20      25 n 30 0



5       10      15      20      25 n 30







Figure 4.     Physical anomalous dimensions KAB(n) in LO and NLO for {FA, FB} =



{F2, FL/(asCL,q )} assuming αs = 0.2 and nf = 3.  The dash-dotted and dotted lines show





the NLO results where all contributions from C(2) and β0, respectively, to K(1) have been

omitted.     A global factor of αs/4π has been ignored in the perturbative expansion, i.e.,







KAB = K(0) + (as/4π)K(1) + O(a2) is displayed.





In Fig. 4 we illustrate the n dependence of the LO and NLO singlet kernels KAB for the evolution of {FA, FB} = {F2, FL/(asC   )} assuming αs = 0.2 and nf = 3. As can be seen, NLO






3                                                                                                                              200









1                                                                                                                              100


0                                                                                                                                50





0        5       10      15      20      25 n 30 0



5       10      15      20      25 n 30



Figure 5. As in Fig. 4 but now comparing the LO and NLO off-diagonal kernels K2L and KL2





for {FA, FB} = {F2, FL/(asC(1))} and {F2, FL/(asC(1) )}; see text.




corrections are sizable for all singlet kernels, in particular, when compared with the perturbative expansion of the singlet splitting functions Pkl(n) in (6); see Figs. 1 and 2 in Ref. [11]. This is, however, not too surprising given that the known large higher order QCD corrections to the Wilson coefficients CL,g and CL,q [27] are absorbed into the physical anomalous dimensions KAB for the evolution of the DIS structure functions F2 and FL. The impact of contributions




from the NLO coefficients C(2)


and C(2)


on the results obtained for KAB is illustrated by the


dash-dotted lines in Fig. 4. Another source for large corrections are the terms proportional to β0

in the NLO corrections as can be inferred from the dotted lines; note that K(1) and K(1) include

L2                   LL





terms proportional to β0C(2) and β0C(2). In Sec. 2.4 we will demonstrate how the differences

between the LO and NLO kernels become apparent in the scale evolution of F2,L(x, Q2).

Figure 5 compares the LO and NLO off-diagonal kernels K2L and KL2 for {FA, FB} =

(1)                                                       (1)

{F2, FL/(asCL,q )} and {F2, FL/(asCL,g )}. The most noticeable difference is the strong rise with



n for the kernel KL2 governing the evolution of {F2, FL/(asC(1) )}. At LO accuracy, this is readily





2          (0)                            (0)                            (0)


understood by inspecting the n → ∞ limit which yields, see Eq. (41) in [22], K(0) ∼ n ln n,











recalling that asymptotically C(1) 1/n, C          1/n , P         ln n, P        1/n, P                                                                                              1/n,



and P (0) ln n. The NLO kernel KL2 exhibits an even stronger rise with n. In the same way

one obtains, for instance, that K(0) governing the evolution of {F2, FL/(asC(1))} only grows like



ln n, see Eq. (14).








Despite this peculiar n dependence and the differences between the singlet kernels shown in Fig. 5, both sets of observables, {F2, FL/(asC(1))} and {F2, FL/(asC(1) )} can be used

interchangeably in an analysis at LO and NLO accuracy. Results for the QCD scale evolution

are identical, and one does not encounter any numerical instabilities related to the inverse Mellin transform, which we perform along a contour as described in Ref. [23]. In fact, it is easy to see that the eigenvalues















λ± = 1  K(0) + K(0) ±  (K(0) − K(0))2 + 4K(0)K(0) l ,                                                                                                                            (15)






which appear when solving the matrix valued evolution equation (9), are identical for both sets of kernels and also agree with the corresponding eigenvalues for the matrix of singlet anomalous dimensions P (0); see also the discussions in Sec. 2.4


  • Example II: F2 and dF2/dt

{                 }


Of future phenomenological interest could be also the pair of observables F2, dF2/dt , in particular, in the absence of precise data for FL. Determining experimentally the t or Q2 slope of F2 is, of course, also challenging.

Defining FD ≡ dF2/dt, we obtain the following physical evolution kernels

K(0) = 0 ,        K(0) = 2 ,

22                              2D













K(0) = 1 IP (0)P (0) − P (0)P (0)l ,


K(0) = P (0) + P (0)




2D             gg              qq


at LO to be used in Eq. (9); for NLO kernels see [18]. The kernels KAB in (16) exhibit more moderate higher order corrections, mainly through terms proportional to β0,1, than those listed in Sec. 2.2. This shall become apparent in the next Section when we discuss results for the scale

dependence of both {F2, dF2/dt} and {F2, FL}.

  • Numerical Studies

In this Section we apply the methodology based on physical anomalous dimensions as outlined above and compare with the results obtained in the conventional framework of scale-dependent quark and gluon densities and coefficient functions. Due to the lack of precise enough data

for FL or dF2/dt we will adopt the followingrealistic “toy” initial conditions for the standard

DGLAP evolution of PDFs at a scale Q0 =         2 GeV [23]






xuv(x, Q2) = 5.1072 x0.8 (1 − x)3, xdv(x, Q2) = 3.06432 x0.8 (1 − x)4,





xu¯(x, Q2) = (1 x) x d¯(x, Q2),



xd¯(x, Q2) = 0.1939875 x0.1 (1 x)5,

xs¯(x, Q2) = 0.2 x [u¯(x, Q2) + d¯(x, Q2)],

0                                           0                           0



xg(x, Q2) = 1.7 x0.1 (1 − x)5.                                                                                                      (17)

for all our numerical studies. The value of the strong coupling αs at Q0 is taken to be 0.35. For our purposes we can ignore the complications due to heavy flavor thresholds and set nf = 3 throughout. We use this set of PDFs to compute the flavor singlet parts of F2, FL, and dF2/dt

at the input scale Q0 using Eq. (11). For studies of DIS in the small x region, say x ;S 103,

in which we are mainly interested in, the flavor singlet parts are expected to dominate over NS

contributions and, hence, shall be a good proxy for the full DIS structure functions. Results at scales Q > Q0 are obtained by either solving the RGEs for PDFs or by evolving the input structure functions directly adopting Eq. (9). For the solution in terms of PDFs we adopt from now on the standard choice µ = Q.

For completeness and to facilitate the discussions below, let us quickly review the solution of the matrix-valued RGEs such as Eqs. (5) and (9). While one can truncate the QCD beta function and the anomalous dimensions consistently at any given order in as, there exists no unique solution to beyond the LO accuracy. The matrix-valued nature of (9) only allows for







iterations around the LO solution, which at order ak


can differ in various ways by formally


higher-order terms of O(a      ).


l>k s

To this end, we employ the standard truncated solution in Mellin moment space, which can

be found, for instance, in Ref. [24], see also [23], and reads


Γi(n, Q) = Li(n, as, a0i(n, Q0)                                                                                                (18)


where the evolution operator up to NLO is defined as


Li(n, as, a0) = L(0)(n, as, a0) + L(1)(n, as, a0)                                                                                                              (19)

i                                          i






L(0)(n, a , a ) =







as        β0



+ (+) ()


i                 s       0







L(1)(n, a , a ) =








− a )e




 λ "













                as




e−R(1)e+      l




a0 − as        a


β0 λ


− λ




+      (+) () .               (20)






Here, a0 = as(Q0), ΓP =  Σ , and ΓK FA , i.e., the index i = P refers to the coupled RGE

{            }


for the quark singlet and gluon and i = K to the RGE for the pair FA, FB                of DIS structure functions in (9). For i = K one has






1        (0)







=        K


,  R(1)


1        (1)          β1       (0)














=        K              K       ,                               (21)









with a corresponding definition for i = P in terms of the 2 2 matrices of singlet splitting functions P (0) and P (1). λ± denote the eigenvalues given in Eq. (15) and e± the projection operators onto the corresponding eigenspaces; see Refs. [23, 24].

As has been mentioned already at the end of Sec. 2.2, the eigenvalues λ±(n) are identical

when computed for the kernels KAB and Pkl. This in turn implies that as long as, say, F2

and FL are calculated at µ = Q0 with LO accuracy, their scale evolution based on physical anomalous dimensions reproduces exactly the conventional results obtained with the help of scale-dependent PDFs.

Figure 6 shows our results for the scale dependence of the DIS structure functions F2 and FL.

The input functions at Q0 =        2 GeV are shown as dotted lines. While LO results are identical,

starting from NLO accuracy the comparison between the two methods of scale evolution becomes more subtle, and results seemingly differ significantly as can be inferred from the middle panels of Fig. 6.

The origin of the differences between F2,L(x, Q2) computed based on Wilson coefficients and scale-dependent PDFs and physical anomalous dimensions can be readily understood from terms which are formally beyond NLO accuracy. For instance, upon inserting the NLO Wilson



coefficients (4) and the truncated NLO solution (18)-(20) into Eq. (11), F2 at O(as) contains spurious terms of both O(a0as) and O(a2). Since FL starts one order higher in as, similar terms

are less important here. On the other hand, when we evolve F2,L with the help of physical anomalous dimensions we first compute, due to the lack of data, the input at a0 based on Eq. (11), which then enters the RGE solution (18)-(20). Again, this leads to terms beyond










1                                                                                                                                                       -1




-1                                                                                                                                                                                                      -2

10                                                                                                                                                   10


1.2                                                                                                                                                  1.2


1                                                                                                                                                     1


0.8                                                                                                                                                  0.8


2                                                                                                                                                     1





-6             -5             -4             -3             -2             -1

10        10        10        10        10        10     x




-5             -4             -3             -2             -1

10        10        10        10        10     x







Figure 6. Scale dependence of the DIS structure functions F2 (left) and FL (right) at NLO accuracy obtained with physical anomalous dimensions (dash-dotted lines) and in the standard

way through a convolution of PDFs and Wilson coefficients (solid lines). The dashed and dotted

lines show the results obtained at LO accuracy and the input at Q = Q0 =            2 GeV, respectively.

The middle panels give the ratios of the two different methods to evolve F2,L at NLO, and the lower panels illustrate the size of NLO corrections when physical anomalous dimensions are being used; see text.




NLO. In case of F2 they are now of the order O(a0as) and O(a2), i.e., even more relevant than

in case of PDFs since a0 > as.





To test if the entire difference between the two evolution methods shown in Fig. 6 is caused by these formally higher order contributions, one can easily remove all O(a2), O(a0as), and

O(a0) contributions from our results. Indeed, the scale evolution based on physical anomalous

dimensions and the calculation of F2,L from PDFs then yields exactly the same results also at

NLO accuracy. We note that this way of computing properly truncated physical observables from scale-dependent PDFs beyond the LO accuracy has been put forward some time ago in Ref. [28, 29] but was not pursued any further in practical calculation.

Another interesting aspect to notice from Fig. 6 are the sizable NLO corrections illustrated in lower panels, in particular, for F2 in the small x region. For this comparison, LO results refer to the same input structure functions F2,L as used to obtain the NLO results but now



evolved at LO accuracy, i.e., by truncating the evolution operator in Eqs. (18)-(20) at L(0). At first sight the large corrections appear to be surprising given that global PDF fits in general lead to acceptable fits of DIS data even at LO accuracy [13]. However, this is usually achieved by exploiting the freedom to have different sets of PDFs at LO and, say, NLO accuracy. The framework based on physical anomalous dimensions does not provide this option as the input for the scale evolution is, in principle, fully determined by experimental data, and only the value




for the strong coupling can be adjusted at any given order. In this sense it provides a much more stringent test of the underlying framework and perhaps a better sensitivity to, for instance, the possible presence of non-linear effects to the scale evolution in the kinematic regime dominated by small x gluons.







10 2









10                                                                                                                                                        -1

















1                                                                                                                                                     1








-5             -4             -3             -2             -1

10        10        10        10        10     x



-5                -4                -3                -2

10           10          10           10      x







Figure 7. Same as in Fig. 7 but now for the pair of observables F2 and FD ≡ dF2/dt.

In Fig. 7 we show the corresponding results for the scale dependence of the DIS structure function F2 and its slope FD = dF2/dt. Again, any differences between the scale evolution





performed with physical anomalous dimensions and based on PDFs are caused by formally higher order terms O(a2), O(a0as), and O(a2), which can be removed with the same recipe as

above. As for {F2, FL}, NLO corrections are sizable in the small x region due to numerically

large contributions to K(1) and K(1) from the QCD beta function.

D2                    DD


  • Summary

We have presented a phenomenological study of the QCD scale evolution of deep-inelastic structure functions within the framework of physical anomalous dimensions. The method is free of ambiguities from choosing a specific factorization scheme and scale as it does not require the introduction of parton distribution functions. Explicit results for the physical evolution kernels needed to evolve the structure functions F2, its Q2 slope, and FL have been presented up to next-to-leading order accuracy.

It was shown that any differences with results obtained in the conventional framework of scale- dependent quark and gluon densities can be attributed to the truncation of the perturbative series at a given order in the strong coupling. At next-to-leading order accuracy the numerical impact of these formally higher order terms is far from being negligible but, if desired, such contributions can be systematically removed.




A particular strength of performing the QCD scale evolution based on physical anomalous dimensions rather than auxiliary quantities such as parton densities is that the required initial conditions are completely fixed by data and cannot be tuned freely in each order of perturbation theory. Apart from a possible adjustment of the strong coupling, this leads to easily testable predictions for the scale dependence of structure functions and also clearly exposes the relevance of higher order QCD corrections in describing deep-inelastic scattering data. Next-to-leading order corrections have been demonstrated to be numerically sizable, which is not too surprising given that the physical evolution kernels absorb all known large higher order QCD corrections to the hard scattering Wilson coefficients.

Once high precision deep-inelastic scattering data from future electron-ion colliders become available, an interesting application of our results will be to unambiguously quantify the size and relevance of non-linear saturation effects caused by an abundance of gluons with small momentum fractions. To this end, one needs to observe deviations from the scale evolution governed by the physical anomalous dimensions discussed in this work. The method of physical anomalous dimensions can be also used for a theoretically clean extraction of the strong coupling and is readily generalized to other processes such polarized deep-inelastic scattering or inclusive one-hadron production.


3.   F2 and FL at small x using collinearly-improved BFKL resummation

  • Structure functions within the BFKL framework

At small x and center-of-mass energy s = Q2/x, we can apply high energy factorization and write the structure functions FI, I = 2, L as


FI(x, Q )  =


π p2 ΦI


q, Q


2              / d2q / d2p       (


2)      (       2)





p, Q0


F (x, q, p) .                        (22)


where q   q2 ) . ΦP is the non-perturbative proton impact factor which we model using








ΦP (


p, Q2)  =





p2      δ







e      0 ,                                                    (23)





where we have introduced two free parameters and a normalization. ΦI is the impact factor associated to the photon which we treat at leading-order (LO), i.e.


/ d2q




(q, Q2





αs(µ  ) L



e2 cI(ν) ,                                   (24)




Q2                                2π



2      nf














π      I (ν) sech(πν) tanh (πν)                                                 (25) 4 (ν + ν3)




ν = i(1/2 γ), Ω2 = (11 + 12ν2)/8, ΩL = ν2 + 1/4, and the strong coupling αs is fixed at the renormalization scale µ2. In the present work we will also use the kinematically improved impact factors proposed in [30, 31], which include part of the higher order corrections by considering exact gluon kinematics. Its implementation requires to replace the functions cI(ν) by c˜I(γ, ω) where





c˜L(γ, ω) =


4Γ(γ +ξ +1)Γ(1+γ)  (ψ(γ +ξ)          ψ(γ)) 3ω2      ξ2 + 1         6ωξ

                           (                  )          


ξΓ(1 + ω)(ξ4 5ξ2 + 4)                                                          (26)




and c˜2 = c˜L + c˜T , with





Γ(γ +ξ)Γ(γ)

(γ, ω) = ξΓ(1+ω)(ξ4 5ξ2 + 4)


  2ξω (ξ2 + 32 + 6ω + 11)


+ ψ(γ + ξ) − ψ(γ) ξ4 10ξ2 + 3ω2 (ω2 + 2ω + 4) 2ω (ξ2 1) + 9  .                                                                                                                                               (27)




| | ≥


| |


ψ(γ) is the logarithmic derivative of the Euler Gamma function and ξ = 1 2γ + ω, while ω is the Mellin variable conjugate to x in the definition of the gluon Green function , see Eq. (28) below. The main difference between these impact factors is that the LO ones roughly double the value of their kinematically improved counterparts in the region with small ν , while being very similar for ν  1.

The gluon Green function can be written in the form


1 / /


1  q2  γ


−ω           1          


F (x, q, p) = π






q2       p2


x      ω − α¯ Kˆ (γ) ,                                (28)





with α¯s = αsNc. The collinearly improved BFKL kernel as introduced in eq. (28) is an

operator consisting of a diagonal (scale invariant) piece χˆ(γ) with eigenvalue




χ(γ)  =


α¯ χ (γ) + α¯2χ (γ) 1 α¯2χ "(γ)χ (γ) + χ






(α¯ , γ, a, b) ,                       (29)






















12 Nc


36 N 3


where χ0(γ) = 2ψ(1) − ψ(γ) − ψ(1 − γ), a =  5 β0 13 nf



55                              1 β0                  nf




8 Nc


6N 3


      and b =         −


     , plus








a term χˆRC(γ) proportional to β0 which contains the running coupling corrections of the NLO kernel [32]:





χˆ      (γ)  =


α¯2  β0  χ (γ)−∂




χ (γ) + 2 log(µ2) .                                  (30)





s 8Nc










The precise form of the NLO kernel χ1 can be found in [19, 33]. The resummation of collinear logarithms of order α¯3 and beyond is realized by the term [19, 34, 35]



χRG(α¯s, γ, a, b) =


α¯s(1 + ¯s) (ψ(γ) ψ(γ ¯s))





α¯2 ""


2       π2             1 L


2α¯s(1 + ¯s) +  (γ 1 m + ¯ )2 + 4α¯ (1 + ¯ )).                             (31)


2 ψ


(1 γ) ¯s sin2 (πγ) + 2




γ 1 − m + ¯s




1 − γ + m

Our final expression for the structure functions reads


s                    s                    s


FI(x, Q2) /




dν xχ( 1 +




δ      2 − iν


2                  1

 α¯ β0χ0 ( + )


1 +     s              2             log 8Nc









1            l Q2  1 +




×  i(πcoth(πν) 2π tanh (πν) − MI(ν)) − ψ


δ 2 − iν


2       cI(ν),






where M2 and ML can be found in [19]. For the kinematical improved version of FI we replace

cI(ν) by c˜I(1/2 + iν, χ(1/2 + )).






In Eq. (33) the scale of the running of the coupling has been set to µ2 = QQ0. Building on the work of [36] we found in [19] that in order to obtain a good description of the Q2

dependence of the effective intercept of F2, λ, for x < 102, it is very useful to operate with non-

Abelian physical renormalization schemes using the Brodsky-Lepage-Mackenzie (BLM) optimal

scale setting [37] with the momentum space (MOM) physical renormalization scheme [38]. For technical details on our precise implementation we refer the reader to [19] (see also [39] for a review on the subject and [40] for a related work). More qualitatively, in these schemes the pieces of the NLO BFKL kernel proportional to β0 are absorbed in a new definition of the running coupling in order to remove the infrared renormalon ambiguity. Once this is done, the residual scheme dependence in this framework is very small. We also found it convenient [19] to introduce, in order to describe the data with small Q2, an analytic parametrization of the running coupling in the infrared proposed in [41].


  • Comparison to DIS experimental data

In the following we compare our results with the experimental data for F2 and FL. Let us first compare the result obtained in [19] for the logarithmic derivative d log F2/d log(1/x) using Eq. (33) with a LO photon impact factor and our new calculation using the kinematically

improved one. In Fig. 8 we present our results with the values of our best fits for both types of impact factors and compare them with the H1-ZEUS combined data [12] for x < 102. The values of the parameters defining the proton impact factor in (23) and the position of the

(regularized) Landau pole (we use nf = 4) for the strong coupling are δ = 8.4, Q0 = 0.28 GeV, Λ = 0.21 GeV for the LO order case and δ = 6.5, Q0 = 0.28 GeV, Λ = 0.21 GeV for the

kinematically improved (note that the normalization C does not contribute to this quantity).


















1                    5       10                  50     100



Figure 8. Fit to λ for F2 with the LO photon impact factor (solid line) and the kinematically improved one (dashed line). The data set has been extracted from [12].


The LO impact factor generates lower values than the kinematically improved one in the high Q2 region and slightly higher ones when Q2 ;S 2 GeV2. It is interesting to see how the approach presented here allows for a good description of the data in a very wide range of Q2, not only for high values, where the experimental uncertainties are larger, but also in the non-perturbative regions due to our treatment of the running of the coupling.







Encouraged by these positive results we now turn to investigate more differential distributions. We select data with fixed values of x and compare the Q2 dependence of our theoretical predictions with them, now fixing the normalization for the LO impact factor to  = 1.50 and 2.39 for the kinematically improved. Our results are presented in Fig. 9. The equivalent



1                        10                     102


1                        10                      102


1                        10                     102



1.4                                                                                                                                                                                                                    1.4


1.0                                                                                                                                                                                                                    1.0


0.6                                                                                                                                                                                                                    0.6


0.2                                                                                                                                                                                                                    0.2



1.4                                                                                                                                                                                                                    1.4


1.0                                                                                                                                                                                                                    1.0


0.6                                                                                                                                                                                                                    0.6


0.2                                                                                                                                                                                                                    0.2



1.4                                                                                                                                                                                                                    1.4


1.0                                                                                                                                                                                                                    1.0


0.6                                                                                                                                                                                                                    0.6


0.2                                                                                                                                                                                                                    0.2



1.4                                                                                                                                                                                                                    1.4


1.0                                                                                                                                                                                                                    1.0


0.6                                                                                                                                                                                                                    0.6




2                                                                                                                   2





1                        10                     10


1                        10                      10


1                        10                     10



Figure 9. Study of the dependence of F2(x, Q2) on Q2 using the LO photon impact factor (solid lines) and the kinematically improved one (dashed lines). Q2 runs from 1.2 to 200 GeV2.


comparison to data, this time fixing Q2 and looking into the evolution in the x variable, is shown in Fig. 10. We observe that our predictions give a very accurate description of the data for both types of impact factors.

Let us remark that the values for the parameters in this fit are in syntony with the theoretical expectations for the proton impact factor since Q0 is very similar to the confinement scale and


























































Figure 10. Study of the dependence of F2(x, Q2) on x using the LO photon impact factor (solid lines) and the kinematically improved one (dashed lines). Q2 runs from 1.2 to 120 GeV2.


the value of δ sets the maximal contribution from the impact factor also in that region. This is reasonable given that the proton has a large transverse size.


The longitudinal structure function is an interesting observable which is very sensitive to the gluon content of the proton. We will now present our predictions for FL using the best values for the parameters previously obtained in the fit of F2. We will see that the agreement with the data is very good. First, Q2 is fixed and the x dependence is investigated in Fig. 11. The experimental data have been taken from [43]. To present the Q2 dependence it is convenient to calculate, for each bin in Q2, the average value of x, see Fig. 12. In some sense this is a similar plot to the one previously presented for λ in the F2 analysis and we can see that the effect of using different types of impact factors is to generate a global shift in the normalization. Again





we note that we have an accurate description of the transition from high to low Q2, which was one of the main targets of our work.







































Figure 11. Fit to FL with the LO photon impact factor (solid lines) and the improved one (dashed lines). The experimental data are taken from [43].



  • Predictions for future colliders

While our predictions for the structure functions are in agreement with the data from the HERA collider experiments H1 and ZEUS, these observables are too inclusive to provide unambiguous evidence for BFKL evolution (for other recent studies in this context see [42]). Comparable in quality fits can be obtained by both DGLAP evolution and saturation models, see e.g. [43, 44]. In order to distinguish among different parton evolution pictures new collider experiments are needed, such as the proposed Electron-Ion-Collider (EIC) at BNL/JLab (USA) [1] and the Large Hadron Electron Collider (LHeC) at CERN (Switzerland) [45], which will be able to measure both F2 and FL at unprecedented small values of Bjorken x. In Fig. 13 we present two studies

with our predictions for F2 and FL down to values of x = 106.



















2                         10                       50       100




Figure 12. The proton structure function FL as a function of Q2. The average x values for each Q2 of the H1 data (black) are given in Figure 13 of [43]. ZEUS data are taken from [44]. The solid line represents our calculation with the LO photon impact factor and the dashed line using the kinematically improved one.



F2 predictions for LHeC






























FL predictions for LHeC


10-6                 10-5                 10-4                 10-3                 10-2



10-6                10-5                10-4                10-3                10-2




Figure 13. Predictions for F2 (left) and FL (right) for LHeC. On the left plot, the curve with Q2 = 10GeV2 can be compared with Figure 4.13 of [45]. Simulated measurements for FL in the kinematic range plotted here (right) can be found in Figure 3.7 of the same reference.



  • Conclusions

We have presented an application of the BFKL resummation program to the description of the x and Q2 dependence of structure functions as extracted from Deep Inelastic Scattering data at HERA. We have also provided some predictions for these observables at future colliders. In order to obtain the correct dependence on the virtuality of the photon at high values of the scattering energy, we have included in the BFKL kernel the main collinear contributions to all orders. We have also used optimal renormalization and an analytic running coupling in the infrared in order to accurately describe the regions of low Q2.


4.   Summary

DIS scattering experiments allow to explore the proton in terms of its QCD content, i.e., quarks and gluons. At small values of x, the description in terms of linear DGLAP evolution and






parton distribution functions is expected to break down and non-linear effects, associated with high gluon densities, are believed to set in. To pin down such effects in DIS, new collider experiments are needed, which allow to measure both structure functions F2 and FL with high accuracy for both electron-proton and electron-nucleus scattering. DGLAP evolution formulated in terms of physical evolution kernels allows for a direct evolution of structure function doublets. Apart from removing scale- and scheme dependence in the description of structure functions, it further reduces the number of free parameters, used in the parametrization of non-perturbative initial conditions. BFKL evolution describes on the other hand directly evolution in x and hence drives the system into the non-linear regime, promising higher sensitivity to non-linear effects. Being less explored than DGLAP evolution, we took a first step towards such applications by confronting BFKL evolution with the combined HERA data. In particular we demonstrated that NLO BFKL evolution is capable to describe the combined HERA data, if the NLO BFKL kernel is supplemented with collinear resummation and optimal scale setting for the QCD running coupling is used.



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Laughter and Smile – The only real Prayer

Friday, 29 September 2017 06:00 Written by

I have realized that nowadays we laugh and smile no more!

And then I have asked myself the reason why.

We are overwhelmed by facts, stressful conditions and existential issues.

Those who are not affected by these problems and are not subjected to beliefs or the control of masses cannot help laughing.

The whole existence is wonderful, and the only possible answer is laughter.

The only real prayer, the only gratitude for being alive, is laughter.

Leaving seriousness aside, we do not lose anything; actually we become healthier and gain a higher virtue.

Conversely, if we give up laughter, we lose everything.

Suddenly, the joy of our existence, the colours of our life and our vitality are all gone. We become monotonous and eventually die.

In this case, our energy is no longer a vital flow.

To understand it, regardless of your current state of existence, you must be happy, because you are alive and you can still change your life.

You are overwhelmed by theories, concepts, notions, ideologies, theologies, philosophies, and you are not able to grasp the meaning of laughter; then you get angry because you cannot afford a certain thing or you are not trendy enough: you just worry about trifles.

Hotei, a Japanese deity called "the laughing Buddha" led a life completely different from that of traditional religious men.

He just kept on laughing.

The legend goes that he even started to laugh while he was asleep and his big belly used to jolt when he laughed.

When he was awake or asleep, life for him was a comedy.

You have turned life into a tragedy.

You have turned life into an endless series of catastrophes.

Even when you laugh, you are pretending.

Even when you want to laugh, your laughter is unnatural, artificial.

It is not sincere ... it does not originate in your heart; you are faking ... your laughter has nothing to do with genuine laughing.

You laugh just by convention; even laughter has become a compromise: people laugh for a personal gain; this is politics!

Look at children; look at their laughter: it is profound, genuine, hearty.

When they are born, the first social activity that they learn, or better that spontaneously manifests itself, is smiling.

It is their first social activity: as soon as they smile, they become members of society.

Everything follows; but smiling is their first existential evidence.


I just want you to understand that today we laugh only if there is a special reason to do it, only when we are forced.

Have you noticed it?

If you are laughing and  happy, everyone asks you the reason why ... it is a stupid question.

If you are sad, nobody ever asks you why.

It is taken for granted: if you are sad, you are OK!

Everyone is sad ... and if you want to talk about your sadness, nobody pays attention to you; then it is useless to tell your misadventures. Would it help?

They are similar to those of the others ... a nod is enough!

How can we change?

Think of a person whose life is a gift!

Someone who loves sharing: there is no better thing than sharing something.

Have you ever given something to someone?

As an unselfish gesture – it does not matter if it is a valuable thing, but the simple gesture is extremely gratifying! Just like smiles: they do not cost anything, but people have become so mean!

Do not be attached to worldly things, but live in the world: there is no other place where we should be.

This is the only world: there is no other.

Sing, be happy, be contended with what you have, laugh at your mistakes and weaknesses, as you reach the Divine.

You do not have to be specialists to realise the extraordinary demand for psychology, its description, its application in the third millennium, so obscure and chaotic due to the development of modern technologies (Internet, mobile phones, etc.)

More or less explicitly, our contemporaries want to understand and be understood; “understand” because it is the first step towards absolute freedom; “be understood” since it is the necessary condition to be recognized, accepted, integrated.

We can question the origins of these needs and of this contemporary anxiety: once problems were faced by a small group of thinkers - belonging to the so-called "intellectual class"; today they have involved everyone. Why?

The first answer is quantitative: men are more numerous, therefore mutual problems have become more frequent; they also live in big cities, which have nothing to do with those of the past.

Due to demographic over-density, disputes and conflicts are more frequent.

In the empty world of some millennia ago, man was a wolf to another man.

Today, in the crowded spaces of our contemporary world, man is an annoyance to the other men!

After all, men are so numerous and - at least in industrialized societies - more educated and informed.

The civilization era of paper, books, and newspapers has revolutionised the old world, but with an action primarily addressing the ruling classes, that is those individuals who made the effort necessary to acquire, preserve and contain a suitable knowledge.

Today, the galaxy arising from the intelligence of computers is threatened by the collision against these new universes, a world made of supernova and bright stars, which are mass and audiovisual media.

The sound or visual signals that they transmit are easily received and assimilated in different ways, their repetition ensures a long-lasting influence.

The ease of transmission ensures an incredible diffusion among such a large audience that, in some cases, (election results, great scientific discoveries, death of a famous person, etc. ...") the whole mankind, nation or group of nations, is involved in the information process.

This new man is open towards the world, but every opening involves the risk of losing our defences and barricades.

And this is precisely what is happening.

The twenty-first-century man is going through a change, a revolution in his life, habits, beliefs and models, that is, of the patterns that determine his attitudes towards crucial problems.

Therefore, there are some gaps in contemporary existence and, all of a sudden, these gaps and failures lead to a crisis, individual or collective, which bewilders those who cannot grasp the essential meaning of things.

We will study together this modern phenomenon, these questions which modern psychology will try to answer.  

If we want to improve mankind, we must give up judgments.

We must help people become aware.

We must give people awareness instead of a conscience.

If someone tells you stupid ideas, then you will bring them along with you for the rest of your life.

People should follow their awareness and reject any commandment or rule established by the others, otherwise they will act like slaves.

This is the condition of mankind so far.

Therefore, I am asking you to give up your conscience, this is the reason why all the religions can be an obstacle.


Judges serve the society in which men have grown.

As a consequence, there are as many judges as cultures, societies, religions and ideologies.

Conversely, the witness is the same, there are no differences between a Christian, a Muslim or a Buddhist witness.


There is only one witness.

The witness is not the result of society: it is the awakening of your soul, it is awareness.

As a witness, you must not condemn, blame or appreciate: you make no evaluation, you don’t say anything.

Just observe.

There is only one thought in your mind: you look at something like a reflection in a mirror; you do not say if it is good or bad: you make no assessment.

It just arrives, stops in front of you and then goes away.

Do not make any comment on the nature of things: a witness is a reflecting awareness to the greatest extent.

There are many judges, while there is only one witness; the Christian witness is identical to the Muslim or Hindu one.

If you don’t make any effort to awake, you will always be subjected to your ideology and the desire to always judge everything and everyone: this is the strategy adopted by society to rule over you.

You can find the light, thus acquiring more awareness and becoming a smarter witness.

Give up your judgement and become a witness: as soon as you succeed in doing this, you will no longer judge the others, but especially yourself.

You will be more sympathetic and compassionate.

Those who constantly judge themselves will inevitably judge the others too!

They will be cruel and hard-hearted towards the others ... they condemn themselves and tend to be even stricter towards the others.

They will always look for mistakes and will never be able to see the glory of human nature;

They will pay too much attention to trivialities.

And they will focus on negligible actions.

Witnesses act like mirrors: they just observe.

This is a miracle: if you can observe someone’s mind without becoming a judge, you will soon go beyond the mind itself.

If you do not get rid of your judgments, you will be subjected to the mind: if you like something, you will cling to it; if you do not like it, you will push it away.

Thus, you will get trapped and involved by the mind, ending up identifying yourself with it.

Actually, you do not know what truth, good or beauty are; you have borrowed your knowledge, you just know what society has taught you.

Society has been repeating the same things for centuries.

Society is not enlightened: still does not exist an illuminated society, but only illuminated individuals.

Now the imperialism of the USA and the power of South Africa are stronger than ever; meanwhile, true protest movements no longer exist in Europe.

In Europe no one truly opposes the political power.

Every effort seems useless; conversely, in Russia and China, some new anti-imperial initiatives have been observed, but their outcome is still unknown.

Eventually, in Latin America, people have started widespread rebellions.

Thus the global peace or war is at stake.

But what about the European Union? Is Europe really united?

Or is it just an illusion created by the powerful to protect their own money interests?

We must forget our dreams and reject the old beliefs and friendships prior to life. Never mind this Europe, which always celebrates men and yet does not miss an opportunity to wear them down, along its streets throughout the whole world.

For centuries, Europe has stopped the arrival of other men and has enslaved them in the name of its projects and glory; during these centuries, the alleged "spiritual adventure" of few individuals has almost killed mankind.

Now it is going through an atomic and spiritual disintegration, the result of a false union.

Europe is ruling over the world with greed, cynicism and violence.

Just look at the shadows cast by its monuments: how they grow increasingly bigger.

Every movement of Europe has expanded the space and thought limits.

However, Europe has rejected humility, modesty, care and even tenderness. It was never thrifty, but rather carnivorous and murderous.

Then fellows, stop following this Europe.

Europe constantly stresses its concern for men, but the victories of the spirit of unity of mankind are weighing on the migrants, causing them terrible pains.

Europe must stop playing this game, it’s time to change the tune.

We can do anything, provided that we do not imitate and obsessively please this Europe.

Nowadays Europe, with its current management, has become crazy and chaotic (every Country acts in an autonomous and independent way), thus going out of control and being on the verge of falling into the dreadful abyss which should be kept at a safe distance, instead.

However, men need new model, schemes and ideas; the European achievements, technology and current style can no longer tempt us and undermine our balance.

If I look for a human element in the European technology and style, I only find the opposite of men, a horde of murderers.

Two centuries ago, a former European colony wanted to fill the gap with respect to Europe.

It accomplished its target with such a success that now the United States of America has turned into a monster, where Europe's defects, illness and inhumanity have reached critical dimensions.

Dear fellows, why are we looking for a third Europe?


The West was supposed to be an adventure of the Spirit.

We should renew ourselves, develop a new thought and create a new man, and we should do it for Europe, for ourselves and for the whole mankind.

The world is expecting much more of us!  



Concept of Love – The Nirvana

Friday, 09 June 2017 13:27 Written by

The fourth step: be nothing.

When you start thinking about being someone, you block yourselves; thus, love cannot flow.,

Love can flow only from that one who is nothing.

Love only live in nothing.

When you are empty, love is present.

When you are full of ego, love disappears.

Love and the ego cannot live together.

Love can exist with God, but not with ego, because love and God are synonymous. 

Love and ego cannot hand in hand.

So, let's be nothing.

Nothing is the source of everything; in nothing borns the infinite….

Nothing is God.

Nothing means "nirvana".

Be nothing … and in being nothing, you will get.

Being someone, you will miss the point; being nothing, you will come home.

Concept of Love – The Sharing

Friday, 09 June 2017 13:07 Written by

Never as it is in this age it is important to love.

Because when we truly love, we can deeply know other people and ourselves.

The body can surrender to the injuries of the age, but our spirit, when it loves, or better when it knows how to love - always free - will allow us to escape the hellish cycle of unforeseen events, in which generations after generations, the human being makes same mistakes.

Even though thoughts are repeated identical in human life, there is an entity capable of changing destinies: it is Love!

However, we must understand how to love.

So, this is the third step: sharing.

When the negative is present, keep it for you

When the positive is present, share it.

Normally people share their negativity; they never share their positivity.  

Humanity is simply stupid.

When people are happy, they never share their happiness: they are extremely miserly!

Instead, when they are unhappy are extremely generous: in this case, they are very happy to share.

When people smile, they economise; they smile but with reserve.

Conversely, when he is angry, he goes totally angry.

The third step is to share positivity.

This will bring your love to flourish like a river that is born of your heart: and when you share, your love will begin to emerge.

I heard a very strange phrase by Jorge Borges. Listen:


Give is holy for dogs.

I throw pearls to the pigs.

Because the most important thing is giving!


You will have heard the exact opposite: do not throw anything to dogs, do not give pearls to pigs, because they cannot understand.

In fact, the relevant thing is not "what you are giving" pearls, holiness and love, and "to whom" you are giving it: this is not important.

The important thing is giving.  

When you have something, give it!

Love is not a property to be accumulated, it means irradiate, it is a fragrance to share.

More you share and more you possess; less you share and less you possess.

Therefore, the third step towards love is to share your own positivity, share your vitality, share everything you have: whatever you have in your hands, never accumulate it.

Beware I'm not talking about material things, material goods; Then: share your wisdom, share your prayer, your happiness, your love ... share it!

Obviously, if there is no one, share with dogs, with cats, with flowers, trees, stones ... but share.

What matters is giving.

Accumulate poison the heart.

Everything that is accumulated is poisonous.

If you share, your body will be free from poisons.

Then when you give, don’t worry if the giving is or is not one-sided.

Do not even expect a thank you! You will not be disappointed.

Sharing is one of the greatest spiritual virtues, one of the greatest virtues!

The second step towards the love is to learn changing your poison into honey

Many people love, but their love is deeply contaminated by poison: hate, jealousy, anger, possessiveness.

Your love is surrounded by a thousand and a poison!

Love is something delicate: you think of anger, hate, possessiveness, jealousy ... how will it ever survive love?

First people move in their heads and forget the heart: most people do it. 

Secondly a minority still lives a little in the heart, but makes another mistake; that little flame of love is surrounded by jealousy, hate, rage, and a thousand and a poison.

In this way, the whole journey becomes bitter.

Love is the ladder between the paradise and the hell, but that ladder has two directions: you can go up or down.

In case these poisons are present, the ladder will take you down and you will go to the hell, you will never reach the paradise!

Instead of getting a melody, your life will become hell; A constant contrast, a deafening traffic; A rumor that will make you go crazy ... a chaos of noises, no harmony.

You will always live in madness... .


Then, the second thing to remember is to learn how to change your poisons into honey.

And how to transform them?

There is a very simple process.

Actually, defining it “transformation” is not correct, because you it is not necessary to do anything: simply you should sit down in silence and observe.

Do not judge: do not be against or in favor.

Do not favor it, do not repress it.

You just observe it. Being patient, just observe what happens… let it go up.

Remember one thing: never try to change your mood, when poisons possess you, simply wait. 

The time comes when poison begins to turn into something else ... this is one of the fundamental laws of life: everything changes continually in its opposite, you have periodic changes - a good man becomes evil, and a bad man becomes good; the saint has times when he is sinner and the sinner lives moments of holiness.

Ypu should just wait.

Do not ever act when anger is at its peak; otherwise you will regret it, and you will create a chain of reactions that will produce a karma (cause-effect law always working in life).

The idea of falling victim to karma is all here: do something when you are in a negative state and you will find yourself in a chain of endless reactions.

When you are in a negative condition, you always do something, also the other one becomes negative and he is ready to do something: the negative creates additional negative, the anger generates anger, Hostility creates other hostility, in an infinite chain of reactions ... and people are mutually tangled ... and the thing goes on! 


When you are angry is the time to meditate.  

Don’t lose this moment; anger creates in yourself immense energy ... it can be destructive.

However, the energy is neutral; the same energy that destroys can also create.  


The same energy that hat can upset, can give life ... just wait.

Waiting without hurry, one day you will have a great surprise: you will see inner change, anger relaxes, energy is released, and you find yourself in a positive state of mind; is a positive state of mind.


It is the creative state of mind.

Now you can act, you can do something.

You always wait the positive moment.

RemeberAct, when you are positive.

Do not force the positivity, wait for it emerges spontaneously.

This is the secret.

When I say: “Learn to transform your poisons in honey”, I mean this!

Question: "We live in a terrible age, our world is ruled by violence: what is the opinion of modern prophetism? What will happen to us and to the world?" Thank you, you are really smart! Giulia from Modena.

. _ ._ .

Answer: If we consider our history, we will realise that men have always been at war.

Life is impossible: it should not be like this, but there is nothing that we can do about it!

Actually, we tend to think that this part of the century will influence the future and determine the fate of mankind.

This decisive period will lead to the extinction of mankind, followed by the total destruction of life on the Earth, or will result in the birth of a new man.


- A man who does not hate life, as happened in the past; a man truly able to love life.

- A man who is affirmative instead of negative.

- A man who does not aim at life after death, but fully enjoys each moment and regards this life as a gift, instead of a punishment.

- A man who is not be the enemy of his body, but rather respects it as the temple of his soul.

- A man who is able to love and does not fear it; a man who establishes various kinds of relations and yet does not reject his identity.


There is no longer a third alternative.

Under the current conditions, man cannot survive.

He must change and evolve, or he will die and leave the Earth!


But we believe that "something" or "someone" will come, because life appeared only on our tiny planet, a trifle – just think about the Earth’s size: it is just a grain of sand in the universe- and the creator of life will surely protect it!

This must be the luckiest place in the whole universe: the birds sing, the trees grow and bloom, men are alive and, in spite of the haters, love, sing, dance and help one another ...

Something amazing has happened, and we know that it will be always so!

Page 1 of 5

Existence of God

Principles of Modern Prophetism

Modern Prophetism

  • Violence: the Meaning of all its Aspects – the Root of Violence
    Violence: the Meaning of all its Aspects – the Root of Violence In this article, I would like to focus on Violence and provide an insight into all its various aspects. It goes without saying that the approach to violence takes on either an objective or a subjective connotation depending on the subjects who use violence, i.e. the oppressed or the ruling…
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  • The Mistaken Interpretation of Non Violence
    The Mistaken Interpretation of Non Violence I want to evoke the historical personality of Gandhi, so similar to that of Jesus; you should not forget what he actually taught, not what they would make him to say. In his diary, “old as mountains: the truth and non-violence,” he says, “if I had to choose between armed…
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  • Fanaticism-Power and Limits of Fanaticism
    Fanaticism-Power and Limits of Fanaticism Power and limits of fanatic doctrines. The power and the strength of a community does not only depend on the members or components and on their number, but very much on the economic component, but even more on the doctrine that prophesies and that everyone carries out. Let's see therefore…
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  • Dethroning the Powerful
    Dethroning the Powerful Only following the path "of the last who shall be first, and the first last" (thus avoiding the concept of “last”, “first”, “coincidence” and “removal of opposites”), we can feel universal love, which will enable us to love and help both our friends and enemies looking for humanization; just like…
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  • The Importance of Word and Time
    The Importance of Word and Time Over the life cycle of men and books, in particular of those dealing with life like this one, the final chapter and the closing sentence are - perhaps coincidentally - the most important steps. Probably some articles have succeeded, albeit marginally, in addressing or inspiring some readers, therefore I would…
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  • Politics and Religion
    Politics and Religion Politics and Religion should be simple: I do not mean that they are easy, but they are not complicated, life is not complicated; all the difficulties arise from the mind, which – if left free to run – tends to prevail over the others. Politicians and philosophers find it very…
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  • The Technological Era
    The Technological Era We are living in the technological era, but each situation features other three sides. To the detriment of our personality, our technological society has created the ingenious monstrosity of the group - a downside to be constantly controlled – together with the social "slave - master" hierarchy, while socialist, animist,…
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  • The Network and the Modern Prophetism
    The Network and the Modern Prophetism Modern Prophetism circulates on the network, since the network provides a bridge, as evidenced by a modern prophet, Pope Francis, the leader of the Catholic Church. In fact, to use his words: "We are at war, but this is not a religion war"; Well, I think that "In a society…
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  • The Ethics of Power
    The Ethics of Power We know that the words, maybe not all, have a power, for this one must say risky words but calibrated, significant. Each word therefore has a power, so I show you " the ethics of power " for those who will use words to communicate to the people, today the…
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  • Modern Prophets
    Modern Prophets Note: the drug is the cure of the disease, to defeat the disease, the cry is the effect of fright; in this way the Modern Prophet is the effect of entrenched injustice. Without injustices the Prophecy that I give you would remain just the Poet, the Priest- " to the…
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  • The Power of the Organizations
    The Power of the Organizations The feeling that makes up the psychological source of power and tradition in its obsequious appearance for Kings, priests, party bosses are: fear and personal ambition that originate power. The power of the organizations through which power is exercised, considering them first as bodies endowed with a life of its…
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  • Peace Meaning
    Peace Meaning We know how important words are, in the sacred name, in primordial sound would hide the power of being and its possible creation. This means that there would be words that create, there would exist sound images, which create under certain conditions. So let me evoke the word Peace, in…
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  • Be Aware - The Love of Fear
    Be Aware - The Love of Fear The love of fear- the refusal of pleasure - Fear paralyzes will The relation between "rich-poor"; "servant-master" divides all, it divides especially the consciences of those who have an interest in the destruction of that relationship. The Biblical prophetic wisdom expresses this situation in the right way: "In a State…
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  • The Rules of Modern Prophecy
    The Rules of Modern Prophecy The love of power - in its broadest sense- is the desire to cause effects in the world, this belongs to human nature. Since the time of Lao-Tse this has had its supporters: mystics, politicians, people who cared about the holiness of the people, more like a state of mind…
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  • The Prophets of Globalisation
    The Prophets of Globalisation We must consider that there are two kinds of pursuit of power, already identified in the past, and two elements: a ruler and his/her follower (Christ and the apostles, all the prophets and devotees.) We must understand that men tend to adhere to the ideals conveyed by the ruler and,…
    1 comment
  • Globalisation of Thoughts
    Globalisation of Thoughts Globalisation of Thoughts - Social science Men are able to think, and within their thoughts they combine representation and awareness. This union results from two factors: the first one merely relates to the subject and has a subjective connotation, while the second one is independent and necessarily objective. In the…
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  • The Freedom of the Peoples
    The Freedom of the Peoples Peoples will never be free if they do not get rid of their rulers: according to the latter, in fact, the masses must be poorly educated, just to meet the awareness requirements of the State-Country. The counter-revolution, which has always been popular among the peoples, despises the awareness of the…
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  • Law enters the Service
    Law enters the Service Law enters the service of men and not vice versa, it supports the weak, regardless of their identity, however it is not suitable for the current history as well as for the existing, future and unattainable justice (messiahship.) A new force has arisen, or had already arisen, from these writings…
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  • The Canticle of the Promised Land
    The Canticle of the Promised Land In our world, the just do not exist, because there can be no prevailing justice, and the just are often killed, persecuted, deceived (whereas the opposite occurs in the Promised Land): the just, i.e. those who act with liberality and full awareness, are crucified. "... Does not ride a horse…
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Predictions Future of the World

Human Rights

Meaning of Life

Modern Prophets - Principles of Humanistic Logic