Complete set of essential parameters of an effective theory

TL;DR

This paper identifies the complete set of essential parameters in an effective scalar field theory, distinguishing them from redundant ones, to facilitate proper renormalization and theoretical consistency.

Contribution

It systematically determines the full set of essential coupling constants in effective scalar theories, clarifying which parameters are necessary for renormalization.

Findings

Distinguished essential from redundant parameters in effective theories

Provided a method to select parameters relevant for renormalization

Focused on scalar field theories with arbitrary high-dimensional operators

Abstract

The present paper continues the series [V. V. Vereshagin, True self-energy function and reducibility in effective scalar theories, Phys. Rev. D 89, 125022 (2014); A. Vereshagin and V. Vereshagin, Resultant parameters of effective theory, Phys. Rev. D 69, 025002 (2004); K. Semenov-Tian-Shansky, A. Vereshagin, and V. Vereshagin, S-matrix renormalization in effective theories, Phys. Rev. D 73, 025020 (2006)] devoted to the systematic study of effective scattering theories. We consider matrix elements of the effective Lagrangian monomials (in the interaction picture) of arbitrary high dimension D and show that the full set of corresponding coupling constants contains parameters of both kinds: essential and redundant. Since it would be pointless to formulate renormalization prescriptions for redundant parameters, it is necessary to select the full set of the essential ones. This is done in the present paper for the case of the single scalar field.

Equations

$$S=T_W\exp i\int dxL(x)$$

$${\partial }^{[n]}\stackrel{\mathrm{def}}{=}{\partial }^{{\mu }_1}\dots {\partial }^{{\mu }_n},{\partial }_{[n]}\stackrel{\mathrm{def}}{=}{\partial }_{{\mu }_1}\dots {\partial }_{{\mu }_n}$$

$${\partial }^{[a,b,c]}\stackrel{\mathrm{def}}{=}{\partial }^{{\alpha }_1,\dots ,{\alpha }_a;{\beta }_1,\dots ,{\beta }_b;{\gamma }_1,\dots ,{\gamma }_c};{\partial }_{[c]}^{[a,b]}\stackrel{\mathrm{def}}{=}{\partial }_{{\gamma }_1,\dots ,{\gamma }_c}^{{\alpha }_1,\dots ,{\alpha }_a;{\beta }_1,\dots ,{\beta }_b}.$$

$$V=G:\varphi {\partial }^{\mu }\varphi {\partial }_{\mu }\varphi :.$$

$${\partial }^{\mu }{\partial }_{\mu }\varphi =-m^2\varphi .$$

$$V=G:\varphi {\partial }^{\mu }\varphi {\partial }_{\mu }\varphi :\simeq \tilde{V}=-G:\varphi {\partial }^{\mu }\varphi {\partial }_{\mu }\varphi :+G{m}^2:{\varphi }^3:+\dots ,.$$

$$V=G:\varphi {\partial }^{\mu }\varphi {\partial }_{\mu }\varphi :\simeq \frac{G{m}^2}{2}:{\varphi }^3:\equiv g_3:{\varphi }^3:$$

$$V=G^{00n}:\varphi {\partial }^{[n]}\varphi {\partial }_{[n]}\varphi :$$

$$V_3=\sum _{s_1,s_2,s_3=0}^{\infty }{G}^{s_1{s}_2{s}_3}:\left({\partial }^{[s_1]}{\partial }_{[s_2]}\varphi \right)\left({\partial }^{[s_2]}{\partial }_{[s_3]}\varphi \right)\left({\partial }^{[s_3]}{\partial }_{[s_1]}\varphi \right):.$$

$$D^{s_1{s}_2{s}_3}:\left({\partial }^{[s_1]}{\partial }_{[s_2]}\varphi \right)\left({\partial }^{[s_2]}{\partial }_{[s_3]}\varphi \right)\left({\partial }^{[s_3]}{\partial }_{[s_1]}\varphi \right)\simeq g_3:{\varphi }^3:.$$

$$V_3=g_3:{\varphi }^3:.$$

$$V_4=\sum _{s_1\dots s_6=0}^{\infty }{g}^{s_1\dots s_6}\left({\partial }^{[s_1{s}_2{s}_3]}\varphi {\partial }_{[s_1{s}_4{s}_5]}\varphi {\partial }_{[s_2{s}_6]}^{[s_4]}\varphi {\partial }_{[s_3]}^{[s_5{s}_6]}\varphi \right).$$

$$L(s_1,\dots ,s_6)\equiv \left({\partial }^{[s_1{s}_2{s}_3]}\varphi \right)\left({\partial }_{[s_1{s}_4{s}_5]}\varphi \right)\left({\partial }_{[s_2{s}_6]}^{[s_4]}\varphi \right)\left({\partial }_{[s_3]}^{[s_5{s}_6]}\varphi \right)$$

$$L^{p,q,r}\simeq L(0,0,0,p,q,r)=\varphi {\partial }_{[q]}^{[p]}\varphi {\partial }_{[r]}^{[q]}\varphi {\partial }_{[p]}^{[r]}\varphi .$$

$$L^{p,q,r}=m^2{L}^{p-1,q,r}-L^{p-1,q+1,r}-L^{p-1,q,r+1}.$$

$$L^{0qr}=\varphi {\partial }_{[q]}\varphi {\partial }_{[r]}^{[q]}\varphi {\partial }^{[r]}\varphi .$$

$$L^{0qr}=L^{0rq}\equiv L^{qr},$$

$$L^{1,r}\simeq \frac{m^2}{2}{L}^{0,r}-\frac{1}{2}{L}^{0,r+1}.$$

$$V_4=\sum _{N=0,2,3,...}^{\infty }\sum _{q=0,2,3,...}^{E(N/2)}{g}_{q,N-q}{L}^{q,N-q}(\varphi )$$

$$M^{22}$$

$$M^{23}$$

$$M^{24}=-\frac{2}{3}{M}^{33}+\frac{1}{6}{M}^{06}-\frac{2}{5}{M}^{05}+\frac{1}{2}{M}^{04}-\frac{2}{3}{M}^{03}+\frac{1}{2}{M}^{02}-\frac{1}{30}{M}^{00}$$

$$s=(k_1+k_2{)}^2,t=(k_1+k_3{)}^2,u=(k_2+k_3{)}^2,$$

$$\begin{array}{rl}{M}^{qr}(s,t,u)& \equiv ⟨0|\int dx:\varphi {\partial }_q\varphi {\partial }_r^q\varphi {\partial }^r\varphi :|k_1,k_2,k_3,k_4⟩=\\ & =(2\pi {)}^4\cdot 4\cdot (-1/2{)}^{q+r}\left[{\hat{s}}^q\left({\hat{t}}^r+{\hat{u}}^r\right)+{\hat{t}}^q\left({\hat{t}}^r+{\hat{u}}^r\right)+{\hat{u}}^q\left({\hat{t}}^r+{\hat{s}}^r\right)\right].\end{array}$$

$$\sigma \equiv \frac{\hat{s}}{2m^2},\tau \equiv \frac{\hat{t}}{2m^2},\xi \equiv \frac{\hat{u}}{2m^2};\sigma +\tau +\xi =-1.$$

$$M^{qr}(\sigma ,\tau ,\xi )\simeq m^{2N}(-1{)}^N\left[{\sigma }^q\left({\tau }^r+{\xi }^r\right)+{\tau }^q\left({\xi }^r+{\sigma }^r\right)+{\xi }^q\left({\sigma }^r+{\tau }^r\right)\right].$$

$$M^{00}=D_0(+6);$$

$$M^{02}=D_2(-4a_2+2);$$

$$M^{03}=D_3(-6a_3-6a_2+2);$$

$$M^{04}=D_4(+4a_2^2-8a_3-8a_2+2);$$

$$M^{05}=D_5(+10a_2{a}_3+10a_2^2-10a_3-10a_2+2);$$

$$M^{06}=D_6(+6a_3^2-4a_2^3+24a_2{a}_3+18a_2^2-12a_3-12a_2+2);$$

$$M^{07}=D_7(-14a_2^2{a}_3+14a_3^2-14a_2^3+42a_2{a}_3+28a_2^2-14a_3-14a_2+2);$$

$$M^{08}=D_8(+4a_2^4-16a_2{a}_3^2-48a_2^2{a}_3+24a_3^2-32a_2^3+64a_2{a}_3+40a_2^2-16a_3-16a_2+2),$$

Full text

I Introduction and preliminaries

In this paper we continue constructing the renormalization scheme suitable for the single-scalar effective scattering theory. Let us recall that the theory is called effective if the interaction Lagrangian in the interaction picture contains all the monomials consistent with the given algebraic (linear) symmetry (see WeinEFT ) – Weinberg1 ).

It is pertinent to note that at present three different concepts are often confused: 1) effective Lagrangian (should not be mixed with a phenomenological Lagrangian!); 2) effective field theory (EFT111In what follows we use the following abbreviations: EFT - for effective field theory; EST - for effective scattering theory, LSM – for Linear Sigma Model, ChPT - for Chiral Perturbation Theory, RP – for renormalization prescription); 3) effective low-energy expansion (say, ChPT or LSM). All three concepts were first explicitly formulated by S.Weinberg (see WeinEFT ), Wein2009 , Weinberg1 and references therein). Perhaps this circumstance, along with the proximity of the content of these concepts, is the reason for the aforementioned confusion. Let us briefly explain the difference between these concepts.

The effective Lagrangian (more correctly, the Lagrangian density) is nothing but the most general linear combination of local monomials constructed from the interaction picture fields (and their derivatives of arbitrary order) that are presented in the model under consideration. Those terms must be consistent with the Lorentz invariance and other algebraic symmetry requirements. Clearly, the effective Lagrangian contains an infinite number of terms, and – therefore – an infinite number of corresponding coupling constants.

The concept of EFT means that it is just Dyson’s perturbation scheme based on the relation

[TABLE]

(here $T_{\scriptscriptstyle W}$ stands for Wick’s $T$-product and $L(x)$ – for the effective Lagrangian density). The attractiveness of this concept is explained by the obvious renormalizability property of EFT: there is a counterterm to absorb the ultraviolet divergency of any graph. However, this concept also has a shortcoming: it suffers from the ”problem of couplings”. One needs to know an infinite number of renormalization prescriptions (RPs) in order to obtain the well-defined physical predictions (below we discuss this problem in more detail).

Further, the ChPT is nothing but the calculational scheme adapted for fixing the coefficients of the amplitude expansion at small values of the mass of the light boson and its momentum modulo (see WeinEFT , Gasser-L ). This concept is not as general as EFT just because the latter does not imply using data to solve the problem of couplings at every next order of the low energy expansion as it is done in ChPT, which is ”more experimental science”222This expression belongs to J.F.Donoghue Donoghue . as compared to EFT (which is a ”more theoretical” one). This feature should be kept in mind when using the term “effective theory”. EFT claims to describe the entire kinematics of the amplitudes, while the ChPT is only designed for describing the low energy region and small mass. These two concepts differ from each other even by purely technical methods of calculation. When working in any loop order, the EFT is based on the expression 1. In contrast, in the cases of ChTP and LSM the expression 1 in each new order uses the Lagrangian (with only few coupling constants) obtained in the previous stage.

In this paper we study the effective scattering theory (EST) that is just the effective field theory (EFT) only designed for perturbative calculations of the S-matrix elements on the mass shell. Green functions may contain divergent contributions: we are only interested in Dyson’s perturbation scheme of calculating the S matrix333H.Georgi Georgi called it ”on shell effective field theory”. in the framework of EFT.

Though the effective theory is renormalizable by construction (see WeinEFT , AVVV2 , KSAVVV2 and the references therein) it presents no interest until the ”problem of couplings” is solved. Thus it is necessary to point out an infinite number of renormalization prescriptions (RPs) that allow one to fix the finite parts of counterterms. If this is done arbitrarily, the theory loses its predictive power. Unfortunately, we do not have an infinite number of corresponding physical principles needed to avoid the problem. Therefore, one must either indicate new (sufficiently powerful) principles or radically reduce the number of free parameters in the theory. Anyway, one needs to know how to construct the whole list of free parameters which $S$-matrix depends upon.

For this it looks necessary to understand better the general features of the EFT construction. The point is that at present the overwhelming majority of papers in the area of EFT are devoted to the problem of asymptotic safety in gravity. Since gravity is the very complicated theory, this makes it too difficult the study of general regularities of EFT. Such a study requires the systematic investigation of the simplest models – single-scalar and multiscalar EFT. Some features were discussed earlier (see, e.g., Georgi , VV2 , GomisWein , KSAVVV2 ). Here we just continue this line: below we consider the single-scalar effective scattering theory.

In the recent paper VV2 the part of the problem of couplings has been analyzed by the example of a two-leg one-loop graph (conventionally called “self energy”). It was shown that in order to avoid the problem one needs to revise the concept of one-particle reducibility/irreducibility (1PR/1PI) and to redefine the notion of “self energy”. In this case one only needs to rely upon two conventional physical conditions (the pole position and the wave function normalization). This turns out quite sufficient for obtaining the well-defined two-leg function which can be used as the insertion both in external and internal lines of Feynman graphs of the effective scattering theory. The possibility of introducing the alternative definition of reducibility is based on the fact that the analytical expression for an arbitrary Feynman graph of the effective scattering theory can be identically rewritten in terms of the minimal lines and minimal vertices by means of the operation called line reduction.

It seems pertinent to us to briefly recall the definitions of the above terms. The effective vertex is called minimal with respect to its line with momentum $k$ if it does not contain the ”killing” factor $k^{2}-m^{2}$ (here $m$ stands for the particle mass; it is implied that $m\neq 0$). The line is called minimal if the adjacent vertex (or both vertices, when the line in question is internal) is minimal with respect to it. The graph is called minimal if all of its lines are minimal (these definitions along with many examples have been suggested in AVVV2 and KSAVVV2 ). At last, the parameters that appear in the analytic expression for the minimal effective vertex are called the minimal resultant parameters.

Thus one concludes that the S-matrix elements only depend on the minimal parameters and thus do not require introducing RPs fixing the nonminimal ones. In turn, this means that the minimal resultant parameters are the only ones needed to fix the physical content of the EST.

Another problem waiting for its solution is the following. To calculate graphs one needs to construct a convenient form of recording of the n-leg effective Lagrangian monomials in the interaction picture. The point is that the form suggested in VV2 is not convenient for calculations because it is excessively general: it contains many identical terms. So, we need to suggest the more suitable form (without loss of generality) and, hence, to suggest the complete set of essential coupling constants (see WeinAsySafe ) needed to fix the four-leg minimal effective vertices444When writing the term ”vertex” we mean the corresponding Lagrangian monomial (or - simpler - monomial) multiplied by a constant.. To put it another way: it is necessary to point out the complete set of basic four-leg Lagrangian monomials (basis) with certain highest dimension $D$ (see footnote 555When using the term ”dimension” we mean the ”physical dimension” $D$ calculated in units of $m$. Keep in mind, however, that, starting from the next Section, we deviate from this rule and express the physical dimension in units of $m^{2}$.). The linear span of these basic monomials creates the relevant linear space of corresponding physical dimension. The coefficients at the individual basic monomials present the essential coupling constants.

It is the problem of constructing the basic set of monomials (more correctly, of the corresponding matrix elements in the momentum representation) which we solve in the given paper. This issue is important in the renormalization theory, because it would make no sense to formulate the renormalization prescriptions (RPs) that correspond to redundant parameters.

Below we use the same notations as those in Ref. VV2 :

[TABLE]

and

[TABLE]

(Latin letters show the number of Lorentz indices which are denoted by Greek letters.)

II Three-leg effective vertex

In the field theory two Lagrangian vertices (densities) that differ from one another by surface terms arising from the integration by parts in action are considered equivalent. In certain cases this allows one to simplify the form of the vertex record. Let us illustrate this with the simplest example.

Let the three-leg vertex have the form

[TABLE]

Here the sign $:\ldots:$ stands for the normal product and $G$ is the dimensional coupling constant (with $D=1$ in the units of $m$). Recall that inside the normal product the field operators commute.

Integrating by parts we obtain the equivalent vertex (as usually, the surface term is neglected). Since we consider the interaction picture field then

[TABLE]

This allows one to simplify 2 as follows:

[TABLE]

Here we use the symbol $A\simeq B$ to show the field-theoretic equivalence of the operators $A$ and $B$ (the ellipses stand for the neglected term out of integral). From the relation 4 one concludes that the initial vertex $V$ is equivalent to the triple vertex $:\phi^{3}\!:$

[TABLE]

which is more simple than the initial one666Note that the similar trick could not be performed with the vertices of the form

$V=G^{00n}:\phi\partial^{[n]}\phi\partial_{[n]}\phi:$

where $n\geqslant 2$. In the latter case one needs to compare the relevant matrix elements – see below..

Now let us turn to a consideration of the general form of three-leg vertex in effective theory. The method allowing one to construct the $n$-leg effective vertices was presented in VV2 . It is certainly general. For example, in the framework of this method the three-leg vertex takes the form ($G^{s_{1}s_{2}s_{3}}$ stand for the corresponding coupling constants)

[TABLE]

The problem is that this form is excessively general because it contains many identical terms. This makes difficult the interpretation of coupling constants and complicates calculations.

As suggested in VV2 one can make use of the integration by parts (together with the boundary condition at infinity) in order to simplify the form 5. Taking away the derivatives from the first factor and lowering and raising indices one can obtain the expression given in VV2 . Further simplification is also possible but it requires comparison of matrix elements in the space of three-particle states (together with the energy-momentum conservation low). From this comparison one finds that an arbitrary three-leg vertex in the interaction picture is equivalent to the simplest one:

[TABLE]

So, the final form of the effective three-leg vertex reads

[TABLE]

Here $g_{3}$ is the true resultant coupling constant that has the dimension of mass: $D(g_{3})=D(m)=1$

In what follows we shall often omit the sign of the normal ordering though it will be implied.

III Four-leg effective vertices

Let us now analyze the more complicated object – the four-leg effective vertex. Precisely as above we will use two different principles: (a) Zero boundary condition at infinity and (b) two vertices are equivalent if the corresponding matrix elements are the same.

The obviously general form of the four-leg effective vertex reads

[TABLE]

Let us consider the corresponding Lagrangian monomial with fixed dimensionality

[TABLE]

and make use of the integration by parts in order to simplify it. Clearly, integrating by parts $(s_{1}+s_{2}+s_{3})$ many times one can remove all the derivatives from the first factor. This means that the expression 8 is just a sum of more simple Lagrangian monomials of the form

[TABLE]

So, it is shown that integration by parts allows one to use the three-index monomials (and, hence, the three-index coupling constants777This form was used in VV2 , in that case the problem of identical terms was not important.) in place of the six-index ones. However, this is not the result that we would like to get. The point is that the form 9 still contains identical contributions. Therefore it is necessary to further simplify the monomials of this kind. Let us do this.

As above, we may reduce the order of derivative operator in the second factor in 9: $p+q\rightarrow p+q-1$. Integration by parts allows one to get the following relation (the corresponding calculation is simple but tedious; see Appendix A)

[TABLE]

Thus we can make the first index equal zero and need to consider only the two-index monomials of the form

[TABLE]

These are the monomials which could seem to be the basic ones. Nevertheless, this is not true. The thing is that the monomials 11 are obviously symmetric in their indices:

[TABLE]

therefore only those of them are independent which have $q\leqslant r$. If we want (as we do) to work with the monomials that have the dimensionality $D\leqslant N\ ,$ (in the units of $m^{2}$) then forms $L^{r,N-r}$ with $r\leqslant[N/2]$ might be considered independent. Here the symbol $[x]$ stands for the integer part of $x$.

At last, considering the monomials with $q=1$ we see that

[TABLE]

This means that the monomials of the form $L^{1,r}$ should be considered dependent and, therefore, excluded from the set of basic ones.

Does it mean that the most general independent four-field vertex in the effective Lagrangian of the single-scalar theory can be presented as the linear sum of terms of the form

[TABLE]

where $g_{q,N-q}$ stands for the corresponding independent coupling constants888Only the independent coupling constants present a set of essential parameters of a theory.? The answer is no. The reason is that among different interaction picture operators $L^{q,N-q}$ acting on the space of four-particle states $|k_{1},k_{2},k_{3},k_{4}\rangle$ (here $k_{i}^{2}=m^{2}$ and $k_{1}+k_{2}+k_{3}+k_{4}=0$ due to the conservation law) there are many dependent ones. We discuss this point in the next Section.

IV The space of matrix elements.

Let us denote $M^{qr}$ the matrix element of the four-field Lagrangian monomial $L^{qr}$ between the vacuum and four-particle state $|k_{1},k_{2},k_{3},k_{4}\rangle$ under the conditions $k_{1}+k_{2}+k_{3}+k_{4}=0$ and $k_{i}^{2}=m^{2}$.

It is possible to show (see below) that

[TABLE]

These relations demonstrate that the corresponding monomials are linearly dependent and, therefore, the constants $g_{22}$ and $g_{23}$ are not independent parameters of a theory.

It might seem that the elements $M^{0N}$ create a basis in the linear space $M_{N}$ of the above-described four-field monomials $\langle 0|L^{q,r}|k_{1},k_{2},k_{3},k_{4}\rangle$ with $q+r=N$. However, for arbitrary $N$ this is not true. For example, one can deduce the relation

[TABLE]

which demonstrates that $M^{24}$ is just a linear combination of another (linearly independent!) matrix elements with $N\leqslant 6$. Therefore it must be excluded from the set of basic monomials of the index $N\geqslant 6$.

We find it pertinent to note that the number $N$ indicates the physical dimension $D_{N}=N$ (in the units of $m^{2}$) of the relevant matrix element $M^{qr}$ from the space $M_{N}$. This matrix element is a homogeneous polynomial (with the homogeneity degree $N$) that depends on three Mandelstam variables $s,t,u$

[TABLE]

restricted by the conditions $s+t+u=4m^{2}$ and $k_{i}^{2}=m^{2}$:

[TABLE]

Here $\hat{s}=s-2m^{2},\ \ \ \hat{t}=t-2m^{2},\ \ \ \hat{u}=u-2m^{2}$. In fact, we are more interested in these matrix elements than in the monomials $L^{qr}$ in coordinate space. The point is that we are studying the effective scattering theory and thus we need to consider the minimal graphs (see AVVV2 and KSAVVV2 ) while the concept of minimality is well defined only in the momentum representation. Therefore we have to provide the full set of constants that allow to fix the graph as a function of relevant kinematic variables. It is more appropriate to do this in momentum space.

In place of three dependent dimensional variables $(\hat{s},\hat{t},\hat{u})$ it is more convenient to work in terms of three dimensionless ones:

[TABLE]

This gives:

[TABLE]

Here the inessential numerical factor $4\cdot(2\pi)^{4}$ has been omitted. The dimensional factor $D_{N}=N$ turns out separated and the dependence on the kinematical variables is just a dimensionless three-variable symmetric polynomial with the leading degree $N=q+r$.

In this Section we will consider not the linear space of Lagrangian monomials itself but the closely related space $M_{N}$ of matrix elements of these monomials with the dimension $D=N$. In accordance with that written above we need to reveal the basic elements of $M_{N}$; the relevant coefficients will be precisely the essential parameters that completely fix the vertex in question. No redundant parameters can appear in this case.

Let us – for example – show how the relation 14 in this technique can be obtained. For this we need to write down the explicit formulae for matrix elements of the monomials with, say, $N\leqslant 8$ (see 19):

[TABLE]

where two important combinations

[TABLE]

have been introduced. Note that, in accordance with Waring formula (see Appendix B), the leading terms in all matrix elements are different from zero.

These matrix elements are not the only ones that correspond to all interaction picture monomials $L^{qr}$ with $q+r\leqslant 8$, $q\leqslant r$. The clearly dependent ones $L^{1r}$ (or, better, their matrix elements) have been excluded from our list: as shown in Sec.III they can be written as linear combinations of the above-presented monomials.

To prove the relation 14 one needs to write down the general linear dependence/independence condition in the space $M_{N}$ of matrix elements of the vertices with dimension $D=N=q+r\leqslant 4$:

[TABLE]

Here $\alpha_{qr}$ stand for the arbitrary numerical factors999The element $M^{13}$ is absent in accordance with the note above.. Now we need to check if this equality can be fulfilled with some factors $\alpha_{qr}$ that are nonzero. If such a solution exists, we conclude that the elements $M^{22}\ldots M^{00}$ are linearly dependent. Otherwise they should be considered linearly independent. In the latter case we can consider them as the basis elements in the space $M_{4}$.

As known (see Kurosh ), an arbitrary symmetric (in variables $\sigma,\tau,\xi$) polynomial can be uniquely (identically!) rewritten in the form of an ordinary polynomial depending on three elementary symmetric combinations:

[TABLE]

In the case under consideration $a_{1}=-1$, so, in fact, we deal with the ordinary polynomials $M^{ij}$ that depend on two combinations $a_{2}$ and $a_{3}$ only.

The left side of the equation 22 is nothing but the sum of several ordinary polynomials depending on two variables $a_{2}$ and $a_{3}$. To equate this sum to zero one has to equate zero each coefficient of this sum. This gives one the following homogeneous system of equations101010Each equation contains terms corresponding to the fixed dimension $D\leqslant N$; the dimensional factors are omitted.:

[TABLE]

This system has a stepped form. The length of the top step equals the number of terms (in the case in question it is two) with the maximal dimension $D=4$ in the expressions for $M^{04}$ and $M^{22}$ (see 20). The thickness of this top step equals to one line.

Let us now move one of the two columns starting with the top line (for example, that $\sim\alpha_{22}$) from the left to the right side. The matrix of the resulting inhomogeneous system in the left side has triangular form. Thus, its determinant does not vanish, and the system is solvable with regard to $\alpha_{0k},\,k=4,3,2,0.$ Taking $\alpha_{22}=-1,$ one obtains the relation 14 with elements $M^{04},M^{03},M^{02},M^{00}$ forming the basis. Alternatively, we could choose four other elements $M^{22},M^{03},M^{02},M^{00}$ as the basic ones. The relation 15 can be proved similarly.

The above-considered example, in fact, shows the most important steps, that should be taken in order to construct a basis in the space $M_{N}$ of matrix elements of the dimension $D\leqslant N$. It only remains to illustrate the purely technical complications arising when one passages to the case $N\geqslant 6$. For this it is sufficient to analyze the case $N=6$. Let us do this.

Consider the linear dependence/independence test:

[TABLE]

With account taken of equations 20 this equality takes a form of the following system of equations111111As above, each equation contains terms corresponding to the fixed dimension $D\leqslant 6$:

[TABLE]

Here the indices $[l_{1}]-[l_{7}]$ stand for the line numbers while $(D_{N})$ – for the relevant dimensional factors.

The system 25 is stepped and, therefore, solvable. There is no necessity to compute the main determinant. The only problem is that the thickness of the upper step equals two lines: the step is too thick. However, this is just an apparent difficulty that can be easily removed. Indeed, considering the sum of $l_{1}$ and $l_{2}$ (with corresponding coefficients), one obtains the two new top lines corresponding to $D_{6}$:

[TABLE]

This is precisely what we need.

Now, moving the column with $\alpha_{24}$ to the right side, we obtain the quite solvable (the main determinant is nonzero!) system of equations for $\alpha_{ij}\,\,(i+j\leqslant 6)$ which shows that $M^{24}$ is just a linear combination of $M^{33},M^{06},\ldots,M^{02},M^{00}$. It can be easily shown that this latter set of matrix elements presents a basis in the space $M_{6}$.

It is clear that the order of the arrangement of columns containing $\alpha_{24}$, $\alpha_{33}$ and $\alpha_{06}$ is unimportant. It is possible to bring the system to a stepped form in any case. For this reason one can consider dependent any one of three elements: $M^{24}$, $M^{33}$ or $M^{06}$. This is just a matter of taste. By our agreement, we always consider $M^{0r}$ as independent.

The above-considered example can be easily generalized for the case of arbitrary large $N$. Let us formulate the general algorithm how to construct the basis in subspace of the dimension $D=N$ under the condition that this has been done already in the subspace with $D\leqslant(N-1)$:

Construct the linear combination of all $[N/2]$ different matrix elements $M^{qr}$ with $q+r=N,\,\,q\leqslant[N/2]$ and all the basic matrix elements with $q+r\leqslant N-1$. 2. 2.

Equate this combination to zero. This will give a test of linear dependence/independence. 3. 3.

Present the obtained test in the form of a system of linear equations for the coefficients of ordinary two-variable polynomial (with the leading degree $N$ in the aggregate variable $\sigma,\tau,\xi$) depending on two combinations $a_{2}(\sigma,\tau,\xi)$ and $a_{3}(\sigma,\tau,\xi)$ (see 21). 4. 4.

The constructed system of equations will take the stepped form. Every step that corresponds to the dimension factor $D<N$ will have the thickness equal to one line (the basic elements with $D<N$ are considered known!). In contrast, the thickness of the top step may happen to be equal $t$ lines. The lengths of these top lines will be equal to $[N/2]$ elements. Since $[N/2]\geqslant t$ (the total number of all matrix elements $M^{q,N-q}$ with the index $N$ cannot be less than the number of linearly independent ones), it is always possible to rewrite (identically!) the top (thick) step in the completely stepped form. The length of the finally obtained top step may happen to be equal, say, $l+1$. 5. 5.

Move $l$ columns from the left side of the system to the right side. As a result, one obtains an ingomogeneous system with nonzero main determinant because the matrix in the left side is triangular (surely, one has to retain nonzero top step) 6. 6.

The columns corresponding to the coefficients $\alpha_{qr}$ in the left side will present the basic matrix elements, while those in the right side – the linearly dependent ones.

From the above analysis it is clear that the number of independent matrix elements with index $N$ is equal to the number of pairs $[i,j]$ such that $2i+3j=N$ minus one. A specific choice of a set of these elements is arbitrary.

When the basis in the space of matrix elements $M^{qr}$ of the interaction picture monomials $L^{qr}$ with arbitrary fixed maximal index $N$ is constructed, one can point out the complete set of the corresponding essential parameters. Those parameters are nothing but the set of numbers fixing every basic matrix element. As shown above, the basis is just a set of arbitrary two-variable polynomials that depend on two combinations of kinematical variables: $a_{2}$ and $a_{3}$ (see 21). This means that we need to point out the numbers fixing those polynomials. Since an arbitrary polynomial is completely fixed by the values of its coefficients, we can consider those coefficients as the only essential parameters of a theory. So, we arrive at the conclusion that the complete set of essential parameters of the effective single scalar theory is exhausted by the numbers fixing the coefficients of ordinary polynomials depending on two variables.

V Conclusion

The effective theory, by definition, deals with the Lagrangian density that contains all terms consistent with a given linear symmetry. When studying such theories, we are faced with a problem that was not previously encountered: the complete list of free parameters that determine the finite $S$-matrix is unknown. According to the common belief, this problem does not occur in the theories with a finite number of terms in the interaction Lagrangian. In this latter case, all coupling constants, as a rule, are considered as the independent parameters. One of the important results obtained in this paper shows that this is not always true. For example, when the dimensions of the interaction monomials are $6\leqslant D<\infty$ (in units of mass squared), one has to take into account only a certain part of them in order to avoid doubling of some couplings.

One more result has been formulated in the previous Section. Though it strongly restricts the number of the essential parameters of the single-scalar scattering theory only containing the vertices with finite maximal dimension $D_{\rm max}=N\geqslant 6$, the total number of the essential parameters of effective theory (that corresponds to $N\to\infty$) is actually infinite. This supports the common belief that the effective theory itself has no predictive power (see WeinEFT - Weinberg1 ): it requires infinitely many physical conditions to formulate an infinite number of renormalization prescriptions needed to remove the divergences from the amplitudes of physical processes. In subsequent publication we will show that this pessimistic conclusion is wrong. More specific, the unitarity of the full $S$-matrix plays here the fundamental role. A variety of matrix elements contributing to $S_{2\to 2}-$matrix is constructed from three- and four-leg vertices. Therefore, the condition of unitarity of the full $S-$matrix provides definite relations between parameters of these vertices. Different terms have different analytical structure, and correspondingly, different divergent asymptotic behavior. Detail calculations show preliminarily that by a suitable choice of relations between coupling constants and using results obtained in the present paper, one can provide unitarity in the case of the single-scalar EFT. Thus, the principle of unitarity helps to fix the most part of the essential parameters but several ones.

The third important result is the following. We suggest an explicit algorithm for constructing the essential bases in that cases when the number of legs of graphs in question is $l=4$ (and, hence, the number of independent kinematical variables is 3). The algorithm can be generalized for monomials with a greater number of legs. This note (at least partially) gives an answer to the problems discussed in Murayama .

Acknowledgements

V.V.V. is grateful to A. Drachev, S. Paston and K. Semenov-Tian-Shansky for friendly support. The work of M.V.I. was partially supported by RFBR Grant No. 18-02-00264-a.

Appendix A: Derivation of Eq.(10)

In this Appendix we will prove the relation 10. Let us write down the left side with the indices $(p+1,q,r)$. This monomial must be brought to the form in which the first index is reduced by one unit. Below there will be many formulas whose essence reduces to integrating by parts and discarding the boundary terms:

[TABLE]

Thus, due to definition (9) the monomial under consideration takes the form:

[TABLE]

So,

[TABLE]

Therefore, step by step the first index of three-leg monomial $L^{p,q,r}$ can be lowered down to zero, and we may consider independent only $L^{0,q,r}.$

Appendix B: Some properties of symmetric polynomials

Here we recall for the reader some facts from the theory of three-variable symmetric polynomials (see, e.g., Kurosh :

• •

The three-variable polynomial $P(x,y,z)$ is called symmetric if

[TABLE]

• •

An arbitrary symmetric polynomial $P(x,y,z)$ can be presented as just the conventional polynomial depending upon three symmetric combinations

[TABLE]

In the theory of three-variable symmetric polynomials this is known as the main theorem.

• •

Every power sum

[TABLE]

can be calculated step by step with the Newton formula

[TABLE]

and presented as the polynomial in $\sigma_{1},\sigma_{2},\sigma_{3}$ with the help of so-called Waring formula (see, e.g. MacMahon , ZengZhou ) :

[TABLE]

Here the summing runs over all sets of non-negative numbers $(i_{1},i_{2},i_{3})$ such that

[TABLE]