_{1}

^{*}

In the light of their relationships with renormalization, in this paper we associate the scaling transformation with nonlocal interactions. On one hand, the association leads us to interpret the nonlocality with locally symmetric method. On the other hand, we find that the nonlocal interaction between hadrons could be test ground for scaling transformation if ascribing the running effects in renormalization to scaling transformation. The nonlocal interaction Lagrangian turns out to vary under scaling transformation, analogous to running cases in renormalization. And the total Lagrangian becomes scale invariant only under some extreme conditions. The conservation law of this extreme Lagrangian is discussed and a contribution named scalum appears to the spin angular momentum. Finally a mechanism is designed to test the scaling effect on nonlocal interaction.

Nowadays, on account of the developments of string theory [

In this paper we try to use the scaling transformation (dilatation of space-time), which is inspired by renormalization and assumed effective in nonlocal description of hadron physics, to unveil part of running effects in nonlocal interaction. More often than not, previous investigations on nonlocal interaction tried to fit certain results to those of renormalization [

where

Here the vertex

in momentum-space then is

in momentum space and the expansion like

The renormalization group method (RGM) has its intrinsic relationship with scaling transformation if viewing

the differentiating operator

presents a vertex function, a wave function or a propagator, its renormalized form and unrenormalized form are linked as [

whence the form factor

where

In the next section one may note that the operator

apart from a coefficient i. The Equation (4) is a special form of renormalization group equation, and the well known form is [

which is for any Green’s function of massless

that is the essence of RGM. Besides the obvious application of spatial scaling-transformation to the nonlocal form factor

The scaling transformation, i.e. a freedom added to Poincare group to form Wely group [

In this paper we shall use the scaling feature of RGM, but we free us from the detail calculation of renormalization. Since we are looking for a transformation method to interpret the running effect in nonlocal interaction, once we already got an effective form of Lagrangian or Hamiltonian, we would just use tree-level form to do calculations. The further loop calculation will double count something, Born approximation is fine for most cases of interest. The calculation resembles that used in deep inelastic scattering, though we are involved just elastic processes. In summary, in the whole paper we focus more on the properties of scaling transformation/conformal group, and on how to apply them to nonlocal fields.

The rest of the paper is arranged as follows. Sect. II is dedicated to introducing the two representations of scale transformation, i.e. the coordinate/operator representation and spinor/unitary representation. In Sect. III we establish the physical relationship between the two representations, on condition that a scale invariant vertex exists. Subsequently in Sect. IV we discuss the conservation law for the derived scaling-invariant vertex, and the possibility that it relates to the nucleon’s polarizations is posed. In Sect. V according to the characteristics of applying the general vertex-form

It is well known that the scaling transformation belongs to a larger Conformal Group [

will understand the role of operator

rencing to that of Ref. [

Mostly the scaling transformation in 4-dimension is discussed as a subset of conformal group, and in previous literature its applications are seldom considered independently [

reserving which gives the popular definition of conformal group [

together with the differential form

to the definition of 6-dimensional angular-momentum

Then one gets the following generators for conformal group [

The projected form (making K as constant boundary of Minkowski Space [

where

Before using Cartan method to achieve the unitary representation of Conformal Group, let’s review first the steps of Cartan method with

The trace

immediately we have

thus

matrix X acts as a mapping between

by

with the solution

which automatically satisfies

From the above Cartan matrix X we can extract the Pauli matrices

And coincidentally the n-vectors form (defined in Equation (24)) based on Pauli matrices don’t generate new matrices, neither the multiplications nor the commutations among them, they themselves are closed. Now in what follows we would find the corresponding Cartan matrix from

To achieve its unitary/spinor representation in 4-dimension, mimicking the relationship between the metric

by the following matrix [

Count the degrees of freedom of the groups that conserve separately Equation (5) and Equation (18), one finds they are both 15. Next we only need to extract the coefficients before

and

the latter falls into Dirac spinor like

It can be examined that the matrix A in Equation (19) meets the invariant expression

just like the above 3-dimension example, while the

it simultaneously reserves the metric Equation (18). The above method of linking real metric to a matrix is closely analogous to the Cartan method of constructing a spinor representation in any real space. Actually, the true spinor space for 4-d conformal group following Cartan method should be of 8-dimension instead of 4-dimension [

where

where P denotes different permutations. Apply the above formula to 2-vector, and use the corresponding subscripts to denote the 1-vectors involved, then

Similarly, let’s exhaust all possibilities, then obtain other nontrivial 2-vectors

We note that the new ones which are independent of

Computing the 4-vectors and the higher ones would not give new independent matrices. Finally, we can rearrange all above k-vector-produced matrices as follows [

where

kowski to Euclidean spaces while instead requiring

The route of inquiring the concrete matrices following Cartan method as above could be a shortcut that rarely mentioned in literature. It is can be checked that the commutations among

We use

examined by computer. Now we recognize that the role of operator

is equivalent to the scaling operator D, with its unitary form

Enlightened by Lorentz transformation, in this section we try to link physically the spatial form of scaling transformation with its spinor/unitary form, the former representing the realistic expansions and contractions of space-time (dilatation and shrinkage means the same), the latter representing the intrinsic freedom very like spin angular momentum. Considering both representations in a sole frame is the main feature of this paper.

As for a nonlocal interaction

the transformed result

transformation, our goal in this section is to search for the corresponding vertex-form

transformation

erator D, and

Usually we perform the spatial Lorentz transformation on the vectors

where

While looking for

where formally we have used

hermit one. With the relation

Now let’s substitute

From the experience of calculating

we find out a possible form of

The coefficients

By this way we set up the relationship between the operator D and its unitary counterpart

The key element of linking the two operators D and

operator D is responsible for acting on the real vector coupling to

Now we are interested in what if we perform the scaling transformation

from which one notes that the vector vertex arrives at its limits

Apart from these two extremes, the true vertex-form for nonlocal interaction would mostly be of mixture form like

The necessity of studying the conservation law for the extreme vertices is that such vertices might exist for a very short moment in some scattering processes. According to Equation (38), to repeatedly perform the transformation succeedingly until

from the vertex-form

From a classical point of view, while a “soft” body (with definite mass m) rotating, its shrinkage or inflation (like zooming in or out) would not alter its total orbital angular momentum. However for a quantum particle, its shrinkage or inflation occurs only when it absorbs or releases a certain amount of energy. Such kind of energy exchange of course breaks the angular-momentum conservation by intuition. But this intuition is right only partially, since in what follows we recognize that only spin part is varied, and the spatial angular momentum is not varied due to the commutation

A newly similar consideration of the scaling symmetry appears in Ref. [

While discussing conservation law, in Lagrangian there are at least two other additional terms to be involved, namely the kinetic term

here the

In summary the Lagrangian without mass term yields

where

Here we mainly investigate the conserved angular momentum for Equation (41) under the transformation set

the usual spatial transformation, i.e. the translations and rotations. These 4-dimensional spatial transformations with infinitesimal forms are

where

in which the matrices elements

Take

one gets a general conserved current (known as the Nöether current) [

where

and

The current Equation (45) leads to angular momentum operator in four dimensions by

in which

Since the orbital angular momentum is not affected by scaling transformation due to

only add the new ingredient

(43). Thus the spinor part would vary with the change of

with

Since the part

and we should caution that in the second term only the product of the constants

Thus apart from the infinitesimal parameter

the constant

Now it is evident that the angular momentum Equation (46) varies correspondingly with the transformation. In this sense we conclude that the nonlocal interaction entails the new internal freedom and becomes the particle intrinsic local property. Meanwhile it brings about the extrapolation of conventional spin angular-momentum. To put it in other words, the particles with shape and those point-like seem to follow different conservation laws.

For the extensive particles it is necessary to involve this correction term

Thus when an extended particle (like proton) is smashed we shall not evaluate the polarizations of its initial state and its final state (smashed shreds of proton) in conventional way, since the initial state (proton) and the final states (smashed proton) all have their non-point size and thus the initial polarization might not be the sum of its final states (smashed proton) (

It has been a long-standing puzzle that how the nucleon spin originates from its constituent parts, namely, the angular momentum of quarks and gluons. The conflict arose from the estimation of the total spin of proton based on the experimental value of the antisymmetric structure function

where

where the coefficients

People once conceived gluons’ spin may contribute much to proton’s spin, but recent experimental analysis [

Now let’s focus on the coefficients of

in which

In this section we will discuss that after finite steps of scaling transformations, what is the contribution of the evolving vertex

Firstly, let’s carry out the structure function of unpolarized cross-section by averaging over the initial spins and summing over the final spins [

and apart from the coefficient

and the result of the first term is well-known, it is

the second term is

and the third term results in the same

The last term has the same form as the first term,

We note the new structure functions are from Equations (60), (61), whose contribution however, would vanish since the tensor

Secondly, let’s consider the polarized cross-section. Now we do not fix the initial or the final spin states and leave the spin operator in the potential. With the same marks as in

Only the terms with coefficients

where the indices 1, 2 represent respectively the first and the second particles. Substitute the concrete form of

Dirac spinor

tion, and after lengthy calculation, it yields

here we use underlines to denote the inner product and

The second step follows while using the approximation

approximation, one further gets

in the second step of the above equation we have used the following relations

and

Likewise, we obtain

We note that the terms

Lorentz-invariant currents

unless some of them are mixed. Thus we realize that

We would like to present a simple gedanken experiment to test nonlocal effect due to the handedness terms, which are all proportional to helicity

The proposed experiment is to use polarized electrons (or hydrogen atoms) colliding on polarized hydrogen- atom (polarized by magnetic field). Different from the previous calculations on polarized electron-electron (e-e) scattering [

or

where

which has never been investigated in previous theoretical study [

additionally the present interaction. If any experiment gets a nontrivial parameter

viate from former evaluations in literature if involving all of the handedness terms

In this paper we have discussed elaborately the role of scaling transformation in nonlocal interaction. This transformation pertains to describing the relationship of different energy/space-time scales in scattering between (fermion) hadrons. The scaling transformation is recognized/constructed based on the conclusions of RGM and the popular expressions of conformal group. The most significant feature of this paper is to combine its spinor representation

tex

vector vertex,

Based on the knowledge that the transformation

one finds the varying coefficients ahead of

We also call

The extreme states as well as the extreme vertex may not exist in nature. However, by the inquiring and inferring process we recognize that the conformal group exists more like for running properties rather than for invariance of quantum fields. For an extended particle involved in a scattering at certain energy, we have to make corresponding scaling transformations to interpret locally its interaction vertex. Assume a nonlocal interaction interpreted initially/unphysically by exchanging vector bosons, then a general interaction-vertex

Although the dynamics used in this paper mostly stems from the perturbative dynamics, it opens a door for our understanding to nonperturbative dynamics. There have been continuous efforts to study nonperturbative interaction ever since the birth of renormalization [

I am grateful to the hospitality of Prof. Y. B. Dong and Prof. P. Wang when I visited High Energy Institute of Chinese Academy, as well as the fruitful discussions with them. The Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and Fundamental Research Funds for the Central Universities.