Extension of positivity bounds to non-local theories: IR obstructions to Lorentz invariant UV completions
Junsei Tokuda

0pt
16ptto-20pt
           
KUNS-2750




Population protocols with unreliable communication[]

    Mikhail Raskin 0000-0002-6660-5673
 
 
    December 30, 2023
=====================================================



§ INTRODUCTION


Several theories have been extensively studied as a candidate for UV completed fundamental theory which describes our real world, such as string theory. 
While it must be important to seek for such theories from theoretical aspects, low energy effective field theories (LEEFT) have played an important role from the phenomenological point of view. Many LEEFT have been constructed to explain the observational data. For example, in cosmology, so-called modified gravity models have been extensively studied. 

One expectation is that it may be possible to reveal some nature of UV completion by constraining the LEEFT parameters by observations. However, unless the connection between the UV completion and LEEFT becomes clear, it would be very difficult to obtain some information of the fundamental theory from observations. From this perspective, it will be very important to investigate quantitatively how the information of UV completion is encoded in IR data.




Recently, it has been argued that the 2 to 2 scattering amplitudes of low energy effective field theories must satisfy an infinite number of inequalities, so-called positivity bounds, in order to admit a local, analytic, unitary, and Lorentz invariant UV completion with a mass gap <cit.>, and these bounds are applied to various models (e.g., <cit.>). The existence of such bounds means that locality, unitarity, analyticity, and Lorentz invariance of UV completions are secretly encoded in LEEFT. Then, the following question will naturally arise: 
all the assumptions are really necessary to derive positivity bounds obtained in the literature? 
If one can clarify which conditions on UV completions are necessary to derive positivity bounds, one could extract information of UV completion more precisely, only from IR data. For example, if positivity bounds were successfully derived without assuming locality, such bounds could be useful to test unitarity, analyticity, or Lorentz invariance of UV completion only from IR data.  

This motivates us to investigate whether we can derive positivity bounds 
even when the locality assumption on UV completion is removed. 
It is known that the modulus of the forward scattering amplitude cannot grow faster than s (log s)^2 in the limit s→∞ due to the unitarity bound, assuming the polynomial boundedness of the amplitude at unphysical region and existence of a mass gap. This bound is called Froissart-Martin bound <cit.>. Here, s denotes one of the standard Mandelstam variable, corresponding to the center of mass energy. Because it is often naively argued that the locality implies the polynomial boundedness of the scattering amplitude in s even in the unphysical region, the locality and unitarity are thought to be encoded in the scattering amplitude in the form of the Froissart-Martin bound. This boundedness properties lead to the 2-subtracted dispersion relation of the amplitudes, and is fully utilized in the derivation of positivity bounds <cit.>. Then, how is the scattering amplitude bounded at high energy in non-local theories? It is expected that the non-locality will allow amplitudes to grow more rapidly than the case where locality assumption is imposed. In order to derive how the scattering amplitude is bounded at high energy in the non-local theory quantitatively, 
we consider the 2 to 2 massive scalar scattering in Jaffe's class of strictly localizable/quasi-local/non-localizable field theories. In such theories, the exponential growth of the 
momentum space Wightman functions can grow exponentially at high energy. Non-locality is incorporated in these theories by allowing the momentum space Wightman function to be highly singular at high energy, and it is not necessary to specify an explicit form of Hamiltonian or Lagrangian to define the non-locality. This class of theories has been well investigated: Wightman formulation of these theories has been developed <cit.>, and it is known that one can define unitary S-matrix which has standard properties as it has in standard local quantum field theories, such as cluster decomposition, LSZ construction, crossing symmetry, and CPT symmetry <cit.>.

After obtaining the high energy behavior of the scattering amplitude, we investigate whether the dispersion relation with finite number of subtraction can be derived or not, and clarify under which condition positivity bounds can be obtained.
We will show below that 


  * It is possible to derive positivity bounds even in non-local theory, unless the momentum space WIghtman functions grow too rapidly. This implies the existence of IR obstructions to unitary, analytic, and Lorentz invariant UV completions: some LEEFT which is apparently consistent with unitarity, analyticity, and Lorentz invariance can be in fact inconsistent with these properties.
 

  * Froissart-Martin bound can be violated even in local QFT, although the forward scattering amplitude is still bounded by s^2 as s→∞. Thus, if the theory is local, 2-subtracted dispersion relation of forward limit amplitude holds.












This paper is organized as follows: in sec <ref>, we firstly review the definition and several important properties of Jaffe's class of strictly local/quasi-local/non-localizable theories. In sec. <ref>, firstly we introduce some basics on scattering amplitude. Next, we list up the assumptions we used to derive the results of this study and explain motivations to assume these conditions. Then, we derive the high-energy behavior on the forward scattering amplitude. In sec. <ref>, we derive positivity bounds for α>0 theory by making use of the results obtained in sec. <ref>. Sec. <ref> is devoted to conclusion and discussion of this study.
We adopt the units with c=ħ=1 and the following notation:

    p^2η_μνp^μ p^ν=-(p^0)^2+δ_ijp^ip^j ,  η_μν= diag(-1,+1,+1,+1) .




§ STRICT LOCALIZABILITY, QUASI-LOCALITY, AND NON-LOCALIZABILITY

In this section, we explain the definition of Jaffe's classification of quantum field theories and its physical meaning. In sec <ref>, we introduce the definition of the Jaffe's classification and several essential properties of Jaffe's theories which are relevant to our work, following <cit.>. In sec. <ref>, we explain several important properties of the time-ordered correlation functions in Jaffe's theories.



 §.§ Definition

Following <cit.>, we briefly summarize the definition of the Jaffe's classification and its relation to non-locality. In <cit.>, the criterion which classify QFT into so-called strictly localizable theories and non-localizable theories was given. This criterion was given in terms of the growth rate of the n-point Wightman functions in momentum space for physical time-like momenta {k_i}, which is parameterized by α as 

    |W(k_1,⋯,k_n)|<( constant)×(∑_i=1^nk_i)^2Nexp[σ(∑_i=1^nk_i)^2α] .





Here, σ is some positive constant and X denotes the Euclidean length of a real vector X: X√(∑_i=1^4X_i^2). N is a non-negative constant. W({k_i}) denotes the n-point Wightman function in momentum space, and trivial delta function which expresses total momentum conservation is abbreviated in the above expression. 
For this α parameter, fields are classified into 3 classes as

    0≤α<1/2   :  strictly  localizable field
    α=1/2   :  quasi-local field
    α>1/2   :  non-localizable field .

Note that α=0 case is special: in this case, the growth rate of Wightman functions is polynomially bounded for physical momenta, i.e., Wightman functions are tempered distributions, and thus this case is especially called tempered localizable. 



In this classification, the non-local nature is naturally incorporated in the theory by considering the highly singular Wightman functions in momentum space, and it is not necessary to specify an explicit form of Hamiltonian or Lagrangian to define the non-locality. Then, why are α<1/2 case, α=1/2 case, and α>1/2 case called strictly localizable, quasi-local, and non-localizable, respectively? 
In order to understand the physical meaning of this classification based on the value of α, it would be instructive to consider the Lehmann-Källén spectral density ρ_ϕ(μ) of a scalar field ϕ: 

    ρ_ϕ(μ)∼exp[σμ^α] ,


where ρ_ϕ(μ) is defined by

    ∑_n|⟨0|ϕ(0)|n⟩|^2δ^(4)(p-p_n)=1/(2π)^3Θ(p^0)ρ_ϕ(-p^2) .


Here, we omitted the polynomially growing factor in eq. (<ref>) because it is irrelevant in the discussion below. From this definition, it turns out that ρ_ϕ(μ) is a real and non-negative Lorentz scalar. Θ(z) is the Heaviside's step function. |n⟩ denotes the multi-particle state belonging to the eigenstate of the momentum four-vector P^μ with eigenvalue p_n^μ. The label n includes all the labels which specify the multi-particle states with momentum eigenvalue p_n, and can include both continuous and discrete variables. From the spectral condition, p_n satisfies p^0_n≥0 and -p_n^2≥0. Then, ρ_ϕ (-p^2)=0 for -p^2≤0.  
Let us define a 2-point Wightman function of a scalar ϕ in momentum space, which is referred to as W_ϕ(p) in terms of the spectral density ρ_ϕ as

    W_ϕ(p)
    =(2π)Θ(p^0)ρ_ϕ(-p^2) .

Then, from eq. (<ref>), Fourier transformation of W_ϕ(p) naively gives

    ∫d^4p/(2π)^4 W_ϕ(p) e^ip(x-y)=∑_n|⟨0|ϕ(0)|n⟩|^2e^-ip_n(x-y)=⟨0|ϕ(x)ϕ(y)|0⟩=W_ϕ(x,y) ,

for x≠ y. From the translation invariance, a 2-point Wightman function in position space can be written as

    W_ϕ(x,y)   =⟨0|ϕ(x)ϕ(y)|0⟩=:W_ϕ(x-y) ,

and W_ϕ(z) may be expressed in terms of the spectral density as

    W_ϕ(z)   =∫d^4p/(2π)^4 W_ϕ(p) e^ipz
       =∫d^4p/(2π)^4Θ(p^0)∫^∞_0dμ (2π)δ(p^2+μ)ρ_ϕ(μ) e^ipz=∫^∞_0dμ ρ_ϕ(μ)W^(μ)_ free(z) ,

where W^(μ)_ free(z) denotes a Wightman function for free scalar with mass square m^2=μ, whose explicit form is 

    W^(μ)_ free(z)∫.d^3p/(2π)^32p^0 e^ipz|_p^0=√(μ+δ_ijp^ip^j) .

From the asymptotic behavior of W^(μ)_ free(z) at μ |z^2|≫1, it turns out that the above expression for W_ϕ(z) with z^2≠0 is convergent and well-defined for α<1/2, but ill-defined for α>1/2. This implies that one cannot define Wightman function in position space without smearing for α>1/2. Note that one can define a position-space two-point Wightman function W(z) for sufficiently large but finite |z^2| when α=1/2. In order to obtain a well-defined two-point Wightman function for α≥1/2, it is necessary to introduce a smeared field ϕ [f_x_0] centered at x=x_0:

    ϕ[f_x_0]∫d^4x ϕ(x)f_x_0(x) ,

where f_x_0(x) denotes a smearing test function which is centered at x=x_0 defined by 

    f_x_0(x)∫d^4k/(2π)^4 f̃(k)e^ik(x-x_0) ,

and f̃(k) is an entire function in k. Note that ϕ [f_x_0]=ϕ(x_0) when f̃(k)=1. Then, a position-space smeared two-point Wightman function W_ϕ(f_x_0,g_y_0) is given by

    W_ϕ(f_x_0,g_y_0)   ⟨0|ϕ [f_x_0] ϕ [g_y_0]|0⟩
    =∫d^4p/(2π)^4 Θ(p^0)f̃^*(p)g̃(p)ρ_ϕ(-p^2)e^ip(x_0-y_0) .

Eq. (<ref>) becomes well-defined if one chooses test functions f and g such that f̃(p),g̃(p)<Ce^-σ/2p^2α with some constant C. 
Although quasi-local theories or non-localizable theories have non-local nature as mentioned above, Wightman formulation of these theories has been developed <cit.>, and it is known that one can define unitary S-matrix which has standard properties as it has in standard local quantum field theories, such as cluster decomposition, LSZ construction, crossing symmetry, and CPT symmetry <cit.>. For non-localizable field theories, the standard micro-causality condition which is expressed by local commutativity of fields is replaced by the asymptotic commutativity which expresses the macro-causality (detail discussions in this aspect can be found in e.g., <cit.> and references therein).





 §.§ Time-ordered products

In order to evaluate the scattering amplitudes, it is necessary to know the time-ordered products of fields, or equivalently, retarded products of fields. We only discuss the time-ordered products below. For notational simplicity, let us consider the 2-point time-ordered correlation function, i.e., the Feynman propagator firstly. The formal definition of the Feynman propagator is naively given by

    G_F(x,y)Θ(x^0-y^0)W_ϕ(x,y)+Θ(y^0-x^0)W_ϕ(y,x) .

However, the right-hand side of the above expression is ill-defined at the space-time points where the Wightman function diverges. As a result, one needs to regularize divergences appearing in the position-space Wightman functions in order to define the time-ordered products. Therefore, it is generally impossible to determine time-ordered correlation functions unambiguously from the momentum-space Wightman functions. Regularization scheme dependence will be inevitable. Then, the well-defined Feynman propagator in position space is formally given by <cit.>

    G_F(x,y) g(-i_x,-i_y)[Θ(x^0-y^0)W_g(x,y)+Θ(y^0-x^0)W_g(y,x)] ,

where W_g(x,y) denotes a smeared 2-point Wightman function in position space defined by

    W_g(x,y)∫d^4k_1/(2π)^4∫d^4k_2/(2π)^4 (2π)^4δ^(4)(k_1+k_2)W̃_ϕ(k_1,k_2)/g(k_1,k_2)e^i(k_1x+k_2y) .

Here, W̃_ϕ(k_1,k_2) is the 2-point Wightman function in momentum space which is related to W_ϕ(p) given in eq. (<ref>) as W̃_ϕ(k,-k)=W_ϕ(k). g(k_1,k_2) denotes an entire function which is needed to make the above expression (<ref>) finite for an arbitrary set of points (x,y) in Minkowski spacetime. Such g is called an indicator function. Then, the Feynman propagator in momentum space is simply given by the convolution product of the 2-point Wightman function and a step function:

    G_F(k_1,k_2)   ∫d^4x_1∫d^4x_2 G_F(x_1,x_2)e^-ik_1x_1-ik_2x_2
       =(2π)^4δ^(4)(k_1+k_2)g(k_1,k_2)∫^∞_-∞dp^0/2π .-i/g(p,-p)[W̃_ϕ(p,-p)/p^0-k_1^0-iϵ-W̃_ϕ(-p,p)/p^0-k_1^0+iϵ]|_p⃗=k⃗_1 .

Here, we chose a permutation invariant indicator function which satisfies g(k,-k)=g(-k,k).[As we will mention soon, on-shell quantities such as scattering amplitudes is independent of a particular choice of an indicator.] If we choose a Lorentz invariant indicator function g and write g(-k^2) g(k,-k), and using eq. (<ref>) with W_ϕ(p), the above expression of the Feynman propagator in momentum space is reduced to the well-known Lehmann-Källén spectral representation by changing the integration variable in eq. (<ref>) from p^0 to μ(p^0)^2-|k⃗|^2:

    G_F(-k^2)=g(-k^2)∫^∞_0dμ ρ_ϕ(μ)/g(μ)-i/k^2+μ-iϵ ,

where G_F(-k^2) is defined by G_F(k_1,k_2)=(2π)^4δ^(4)(k_1+k_2)G_F(-k_1^2). 
When ρ_ϕ(μ)∼μ^Nexp[σμ^α], we choose g(μ) which behaves as g(μ)∼μ^N+1exp[σμ^α]. Then, eq. (<ref>) is well-defined. 




Feynman propagator depends on the choice of an indicator function. Such ambiguities are expressed as contact terms, which may become manifest if we rewrite eq. (<ref>) as 

    G_F(-k^2)   =∫^∞_0dμ (-i) ρ_ϕ(μ)/k^2+μ-iϵ(1-g(μ)-g(-k^2)/g(μ))
       =∫^∞_0dμ (-i) ρ_ϕ(μ)/k^2+μ-iϵ[1-1/g(μ)∑_n=1^∞1/n!^n g(-k^2)/(-k^2)^n(μ+k^2)^n] .

The second term in the final line expresses the divergent contact terms. These contact terms cancel on-shell singularity included in the Feynman propagator, and hence do not contribute to on-shell quantities. When α=0, namely, the theory is tempered-localizable, an indicator g(μ) is an real polynomial, and hence the number of contact terms is finite. When α>0, an indicator g(μ) is given by an entire function, and hence the number of contact terms is infinite. We do not expect the appearance of an infinite number of contact terms within a validity of a perturbation theory. Therefore, quantum field theories whose high-energy behavior of them can be captured perturbatively will fall into α=0 theory. However, the appearance of an infinite number of contact terms does not necessarily mean that the number of parameters included in the theory is infinite. Thus, it seems that even the perturbatively renormalizable theory can be α>0 theory generically, although in α>0 case it will be difficult to determine the value of α for some given QFT rigorously, because in order to do so one needs to evaluate the high-energy behavior of the spectral density ρ(μ) non-perturbatively.[It is also expected from perturbative analysis that an infinite number of divergences will appear in perturbatively nonrenormalizable theories. The connection between perturbatively nonrenormalizable theories and non-localizable theories was discussed in <cit.>.] There are several theories which are conjectured to be α>0 theories. For example, it has been conjectured that Little String Theories may be α=1/2 quasi-local theories in <cit.>, and  Galileon theories may be α>1/2 non-localizable theories in <cit.>. 
 
In fact, eq. (<ref>) is also useful for investigating the behavior of the Feynman propagator for large modulus of -k^2∈ℂ. This is because g(-k^2) is an entire function and hence G_F(-k^2) is analytic in -k^2 in the complex -k^2-plane modulo an isolated pole and branch cut. 
Therefore, one can easily analytically continue G_F(-k^2) for -k^2∈ℂ and it turns out that the growth rate of the Feynman propagator G_F(-k^2) for large modulus of -k^2∈ℂ is bounded as[Note that the left-hand side of (<ref>) will be non-zero for time-like k, while it can be zero for space-like k, generally. This is because an indicator g (-k^2) must grow for large -k^2, while it may not grow for different directions in a complex -k^2-plane. In some special case g (-k^2) can decay rapidly for Euclidean direction. The rapid decay of g (-k^2) in Euclidean direction is assumed and used as a key property regulating perturbative UV loops in Efimov's non-local QFT program (see e.g., <cit.>), although this rapid decay might not necessarily be ensured generically in quasi-local/ non-localizable theories in Jaffe's language, as is also pointed out in <cit.>.]

    lim_|-k^2|→∞|G_F(-k^2)/(-k^2)^Ne^σ|-k^2|^α|<∞ ,




when -k^2 is not on singularities. This boundedness property obeys from the growth rate of an indicator function via eq. (<ref>). This means that the growth rate of the Feynman propagator for large  modulus of -k^2∈ℂ is determined by the growth rate of the 2-point Wightman function W_ϕ(k) at large -k^2 with physical time-like k. It is not necessary to specify the properties of the 2-point Wightman function W_ϕ(k) at large modulus of -k^2 with unphysical k to derive eq. (<ref>). 

Next, let us generalize the above discussion to the 4-point time-ordered correlation function in momentum space, because we will consider the 2 to 2 scattering amplitude which is related to 4-point time-ordered correlation functions through the reduction formula. As in the case of the 2-point function, the 4-point time ordered correlation function is also formally defined by 

    G_F(x_1,⋯,x_4)
        g(-i_x_1,⋯,-i_x_4)[∑_IΘ(x_i_1^0-x_i_2^0)Θ(x_i_2^0-x_i_3^0)Θ(x_i_3^0-x_i_4^0)W_g(x_i_1,⋯,x_i_4)] ,

where I denotes the permutations ([   1,   ⋯,    4; i_1,   ⋯,  i_4 ]), and ∑_I denotes the summation over all permutations. Here, W_g(x_1,⋯, x_4) is a smeared 4-point Wightman function defined by

    W_g(x_1,⋯,x_4)[∏_i=1^4∫d^4k_i/(2π)^4](2π)^4δ^(4)(∑_j=1^4k_j)W̃_ϕ(k_1,⋯, k_4)/g(k_1,⋯, k_4) e^i∑_n=1^4k_nx_n ,

and we chose the permutation-invariant indicator function in eq. (<ref>) for simplicity:

    g(k_1,⋯,k_4)=g(k_i_1,⋯,k_i_4)   for   ∀ I .

Then, momentum space 4-point time-ordered correlation function is simply given by the convolution product of 4-point Wightman function and step functions:

    G_F(k_1,⋯, k_4)   [∏_i=1^4∫d^4x_i] G_F(x_1,⋯,x_4)e^-i∑_j=1^4k_jx_j
       =(2π)^4δ^(4)(∑_i=1^4k_i)g(k_1,⋯ ,k_4)∑_I[∏_j=1^4∫dp_j^0/2π](2π)δ(∑_l=1^4p_l^0)
         ×.W̃_ϕ(p_i_1,⋯, p_i_4)/g(p_i_1,⋯, p_i_4)|_{p⃗_i_n=k⃗_i_n}_n=1,⋯,4∏_r=1^3(-i/∑_n=1^r(p_i_n^0-k_i_n^0)-iϵ) .

This is just the generalization of eq. (<ref>) to the 4-point case.



Then, using the fact that 4-point Wightman function can be written in terms of a Lorentz scalar function 𝒲_I as 

    .W̃_ϕ(p_i_1,⋯, p_i_4)|_p_1+⋯+p_4=0=[∏_n=1^3Θ(p_i_n^0)].𝒲_I(-(p_1+p_2)^2, -(p_1-p_3)^2, {p_i^2})|_p_1+⋯+p_4=0 ,

and choosing a Lorentz invariant indicator function g=g(-(k_1+k_2)^2, -(k_1-k_3)^2, {k_i^2}), eq. (<ref>) reduces to

    G_F(-(k_1+k_2)^2, -(k_1-k_3)^2, {k_i^2})/g(-(k_1+k_2)^2, -(k_1-k_3)^2, {k_i^2})
    
    
       =∑_I∫^∞_-|k⃗_1+k⃗_2|^2ds'/2π∫d(p_1^0-p_2^0)/4π∫dp_3^0/2π.𝒲_I(s', -(p_1-p_3)^2, {p_i^2})/g(s', -(p_1-p_3)^2, {p_i^2})|_{p⃗_i_n=k⃗_i_n}_n=1,⋯,4, p_4^0=-∑_j=1^3p_j^0
         ×∑_λ=±1.Θ(λ(p_1^0+p_2^0))/2λ(p_1^0+p_2^0)∏_r=1^3(-i/∑_n=1^r(p_i_n^0-k_i_n^0)-iϵ Θ(p_i_r^0))|_p_1^0+p_2^0=λ(s'+|k⃗_1+k⃗_2|^2)^1/2 ,

where

    G_F(k_1,⋯, k_4)(2π)^4δ^(4)(k_1+⋯+k_4)G_F(-(k_1+k_2)^2, -(k_1-k_3)^2, {k_i^2}) .

Here, {k_i^2} stands for {k_i^2}_i=1,⋯,4. In order to obtain eq. (<ref>), we changed the integration variables by introducing s'(p_1^0+p_2^0)^2-|k⃗_1+k⃗_2|^2. Again, G_F(k_1,⋯,k_4) can be determined only up to contact terms, although they do not contribute to scattering amplitudes. As in the case of the 2-point Feynman propagator, it is expected from eq. (<ref>) that the 4-point time-ordered correlation functions in momentum space will be also bounded for large modulus of -(k_1+k_2)^2∈ℂ with fixed -(k_1-k_3)^2 and {k_i^2}_i=1,⋯,4 as 

    lim_|-(k_1+k_2)^2|→∞|G_F(-(k_1+k_2)^2, -(k_1-k_3)^2, {k_i^2})/(-(k_1+k_2)^2)^Ne^σ'(-(k_1+k_2)^2)^α|<∞ ,

when 
the arguments of G_F are not on singularities. 

Here, σ' is some positive constant whose precise value of σ' may depend on the value of -(k_1-k_3)^2, {k_i^2}, or σ appearing in eq. (<ref>),  but the precise value of σ' is irrelevant in our study.   



It should be noted in advance that this boundedness property (<ref>) of time-ordered correlation functions is an important observation which would support the plausibility of the assumption C introduced in the next sec. <ref>.









§ BOUNDS ON THE HIGH ENERGY BEHAVIOR OF SCATTERING AMPLITUDES FOR PHYSICAL S

In this section, we will obtain the high energy behavior of the forward-limit scattering amplitude under several assumptions. 
In sec. <ref>, we firstly introduce some basics of 2 to 2 scattering amplitudes. Next, we list up the assumptions which are needed to obtain the high energy behavior of the scattering amplitude and the dispersion relation. We also explain the physical meaning or motivations of these assumptions. In sec. <ref>, we derive the high energy behavior of the scattering amplitude for α≥0 case. We also explain the intuitive derivation of the high energy behavior.




 §.§ Basics and assumptions

Let us consider the 2 to 2 scattering of the real scalar ϕ with positive mass square m^2>0. We refer to the incoming momenta and outgoing momenta as p_1,p_2 and p_3,p_4, respectively. From the total momentum conservation, p_4=p_1+p_2-p_3. Then, from the Lorentz invariance, one can express the corresponding scattering amplitudes in terms of the Mandelstam variables s, t and u which are defined by

    s   -(p_1+p_2)^2=-(p_3+p_4)^2 ,
    
    t   -(p_1-p_3)^2=-(p_2-p_4)^2 ,
    
    u   -(p_1-p_4)^2=-(p_2-p_3)^2 .

From this definition, it turns out that s+t+u=4m^2 holds, and hence we refer to the scattering amplitude of this process by F(s,t). In the center of mass frame (CM frame), on-shell momenta p_1, p_2, and p_3 can be parameterized as 

    p^μ_1=(E_ cm/2,0,0,q_s) ,   p^μ_2=(E_ cm/2,0,0,-q_s) ,   p_3^μ=(E_ cm/2,q_ssinθ,0,q_scosθ) .

Here, q_s=1/2√(s-4m^2) denotes the amplitude of the three-momentum in the CM frame. Then, Mandelstam variables can be written in terms of the 4-momenta in CM frame as

    s=E^2_ cm ,   t=2q_s^2(cosθ-1) ,   u=-2q_s^2(cosθ+1) ,

and so s and t are called center of mass energy and momentum transfer, respectively.


From now on, in order to make the discussion clearer, we list up the assumptions which are needed to obtain the results presented in this paper:


  * A: Lorentz invariance, unitarity, and s↔ t↔ u crossing symmetry





  * B1: Analyticity properties in the large Lehmann ellipse

For fixed s∈ℝ+iϵ for sufficiently large Re  s with an infinitesimal positive constant ϵ, F̃(s,z) F(s,t)|_t=2q_s^2(z-1) is assumed to be holomorphic in z on or inside the large Lehmann ellipse 𝒞_0 with foci at z=±1, and the semi-major axis is 1+(t_0/2q_s^2). The ellipse 𝒞_0 can be expressed as 𝒞_0={z'=z(s):  z(s)cosh(β_0(s)+iφ') , -π<φ'≤π} with cosh(β_0(s))=1+(t_0/2q_s^2). Here, t_0>0 is an s-independent constant. We expect that we can take t_0=m^2(1-δ) with 0<δ≪1, although the specific value of t_0 does not change any discussions below. 

  * B2: Analyticity properties in the complex s-plane


The forward scattering amplitude F(s,0) is assumed to be holomorphic in s in the first Riemann sheet of the complex s-plane modulo poles and branch cuts which are predicted by unitarity. F(s,t) is assumed to be analytic in t at t=0 for any s in the first Riemann sheet of the complex s-plane modulo poles and branch cuts.

  * C: Exponential boundedness of the amplitude for fixed t at large s

On the ellipse 𝒞_0, F̃(s,z(s)) is assumed to be exponentially bounded:[This boundedness property in the complex z-plane has been already obtained in <cit.> for α>1/2 case. 
]

    |F̃(s,z(s))|<C̃s^Nexp[Cs^α] ,   N<∞ ,   0<C ,C̃ ,   as    s→∞+iϵ .


Furthermore, we also assume that |^n_t F(s,t)|_t=0 is also exponentially bounded in the complex s-plane: 

    |^n_t F(s,t)|_t=0<C̃_n |s|^N_nexp[C_n|s|^α] ,   N_n<∞ ,   0<C_n ,C̃_n ,   as   |s|→∞ .

Here, C, C̃, C_n, and C̃_n denote some positive constants.


  * C1: Asymptotic behavior at large s∈ℝ+iϵ for fixed t, inside the Lehmann-Martin ellipse

For z inside of the Lehmann-Martin ellipse, |F(s,z)| is assumed to be exponentially bounded:

    |F(s,z)|<s^Nexp[Cs^α] ,   N<∞ ,   0<C ,   as    s→∞+iϵ .



  * C2: Asymptotic behavior at large |s| with s∈ℂ at fixed t=0

|^n_t F(s,t)|_t=0 is assumed to be exponentially bounded: 

    |^n_t F(s,t)|_t=0<|s|^N_nexp[C_n|s|^α] ,   N_n<∞ ,   0<C_n ,   as   |s|→∞ .







  §.§.§ Motivations for assuming conditions B1-C

There are several motivations why it would be meaningful to work under the assumptions B1-C. 


  * Motivation for assuming B1: 

Assumption B1 plays an essential role to constrain the high energy behavior of the scattering amplitude. As we will discuss at the end of sec. <ref>, the holomorphy in the ellipse 𝒞_0 is closely related to the short-range nature of the force which induces the scattering under consideration.[This point is also mentioned in <cit.>.] We expect this holomorphy in the ellipse 𝒞_0 as long as the scatterings are caused by a short-range force. 

  * Motivation for assuming B2: 

It is known that causality and analyticity of the forward scattering amplitudes in complex s plane 
are closely connected to each other, and their connection has been extensively investigated from 1950's (e.g., <cit.>). Roughly speaking, the retarded propagator associated with the scattering process is given by the Fourier component of the forward scattering amplitude, and hence analyticity in the first Riemann sheet of the complex s-plane implies causality <cit.>.[It is argued in <cit.> that analyticity with allowing exponential growth (<ref>) implies macro-causality.] 
We simply assume that these properties hold for generic α because of its close connection to the causality. 


  * Motivation for assuming C:

F(s,t) is written in terms of 4-point time-ordered correlation functions G_F(p_1,p_2,-p_3, -p_4=-(p_1+p_2-p_3)) via the reduction formula. 



Thus, we expect that the boundedness property (<ref>) of G_F in the complex -(p_1+p_2)^2-plane with fixed -(p_1-p_3)^2 and {p_i^2} will be intimately connected to the boundedness properties of F(s,t) for large |s| in the unphysical region with fixed t.[In analytically continuing F(s,t) to the unphysical region, one must maintain the on-shell condition k_i^2=-m^2. 

] 
Then, from (<ref>), it is expected that F(s,t) would be also bounded exponentially for large |s| as (<ref>) or (<ref>).  

 


Although the rigorous derivation of analyticity properties B1-2 are discussed in the context of axiomatic QFT (see e.g., <cit.>), we simply assume these properties as a starting point of this study. From now, we move on to derive several consequences followed from these assumptions. 


 §.§ High energy behavior of the scattering amplitude



Firstly, we develop the method to obtain a high energy behavior of the scattering amplitude at t=0 and its n-th derivative with respect to t at t=0.  That is, we obtain the upper bound on F^(n)(s) at large s, where F^(n)(s) is defined by

    F^(n)(s).1/n!^n/ t^nF(s,t)|_t=0 ,
 
under the assumptions A-C. Note that the method presented in this subsection <ref> is developed based on <cit.>, although the bounds on the F^(n)(s) with n≥1 was not discussed there.














Partial wave expansion of F(s,t) is given by

    F(s,t)=√(s)/q_s∑_l=0^∞(2l+1)f_l(s)P_l(1+t/2q_s^2)

where P_l(z) denotes the Legendre polynomial.
From the S-matrix unitarity SS^†=1, one obtains the unitarity constraints on the partial wave amplitude f_l(s) for s≥4m^2:

    |f_l(s)|≤1 .

 






From the holomorphy of F̃(s,z) on or inside the ellipse 𝒞_0 
f_l(s) can be expressed as

    f_l(s)=q_s/2π i√(s)∮_𝒞_0F̃(s,z')Q_l(z')dz' .

This is known as the Neumann's expansion.
Here, Q_l(z) denotes the Legendre functions of the 2nd kind. By making use of the asymptotic behavior of Q_l(z) at large l, one obtains

    |f_l(s)|<G_l(β_0,s)q_s/√(s)B_0(s)√(e^-β_0coshβ_0)·e^-β_0l/√(l) ,

where

    B_0(s)1/√(8π)∫^π_-πdφ'|F̃(s,cosh(β_0(s)+iφ'))| .

Defining L(s) by G_L<1≤ G_L-1, one obtains 

    G_l'+L(β_0,s)=e^-β_0l'/√(1+l'/L)G_L(β_0,s)<e^-β_0l'/√(1+l'/L) .

Thanks to the bound (<ref>) and from the asymptotic behavior of the Legendre polynomial P_l(z) in the limit l→∞, Weierstrass's theorem ensures that partial wave expansion (<ref>) uniformly converges inside the ellipse 𝒞_0 when s∈ℝ+iϵ. Because P_l(z) is holomorphic in the whole complex z-plane, Weierstrass's double series theorem ensures that the n-th derivative of F(s,z) with respect to z is given by

    ^n/ z^nF(s,z)=√(s)/q_s∑_l=0^∞(2l+1)f_l(s)P^(n)_l(z) ,   P^(n)_l(z)^n/ z^nP_l(z) ,

inside the ellipse 𝒞_0.
Then, when s∈ℝ+iϵ, the n-th derivative of F(s,t) with respect to t at t=0 is given by

    F^(n)(s)=√(s)/q_s1/n!2^nq_s^2n∑_l=0^∞(2l+1)f_l(s)P^(n)_l(1) .







We have obtained two constraints on |f_l(s)| so far: (<ref>) and (<ref>). From the definition of L(s), it turns out that (<ref>) and (<ref>) give the stronger bound for l≤ L(s)-1 and for l≥ L(s), respectively.
Therefore, by using (<ref>),

    |F^(n)(s)|   <√(s)/q_s1/n!2^nq_s^2n[∑_l=0^L-1(2l+1)P^(n)_l(1)+∑_l=L^∞(2l+1)G_l(β_0,s)P^(n)_l(1)]
       <√(s)/q_s1/n!2^nq_s^2n[∑_l=0^L-1(2l+1)P^(n)_l(1)+∑_l'=0^∞ e^-β_0l'P^(n)_l'+L(1)(2L(1+l'/L)^1/2+(1+l'/L)^-1/2)] .

Because

    P_l(z)=_2F_1(l+1,-l;1;1-z/2) ,

and Gauss's hyper geometric function _2F_1(α,β;γ;x) satisfies

    / x _2F_1(α,β;γ;x)=αβ/γ_2F_1(α+1,β+1;γ+1;x) ,

P^(n)_l(1) can be evaluated as

    P^(n)_l(1)=1/2^n(l+n)!/(l-n)!    n≤ l ,
    
    0     n≥ l+1 .

Substituting eq. (<ref>) into eq. (<ref>),

    |F^(n)(s)|
       <√(s)/q_s1/n!4^nq_s^2n[∑_l=n^L-1(2l+1)l^2n(1+𝒪(l^-1))
       +∑_l'=0^∞ e^-β_0l'(l'+L)^2n(1+𝒪((l'+L)^-1))(2L(1+l'/L)^1/2+(1+l'/L)^-1/2)] ,
       ∼√(s)/q_s1/q_s^2n[L^2n+2+2L^1/2/β_0∫^∞_0dy e^-y(y+Y)^2n+1/2/β_0^2n+1/2] ,

with neglecting the irrelevant proportionality constant. Here, yβ_0l' and Yβ_0L, and we assumed that L and Y grows as s increases and neglected the terms which are lower order in L or Y. When L grows as s increases, L and B_0(s) are related as

    B_0(s)≃2e^β_0L√(L) ,

which can be obtained from the definition of L.


  §.§.§ •    α=0 case

By using eqs. (<ref>) and (<ref>), polynomial boundedness (<ref>) with α=0 is converted to the bound on Y=β_0L as

    Y<(N-1/4)ln(s/s_0)-1/2ln Y≈(N-1/4)ln(s/s_0)-1/2lnln(s/s_0) ,

where s_0 is some constant. Applying eq. (<ref>) to eq. (<ref>), and by making use of the approximation 2β_0≈(t_0/s)^1/2 which is valid



at large s, one obtains

    |F^(n)(s)|<( constant)× s(ln(s/s_0))^2n+2 .

n=0 case of eq. (<ref>) gives the bound on the total cross section σ_ tot(s) as 

    σ_ tot(s)<( constant)×(ln(s/s_0))^2 ,

which corresponds to a celebrated Froissart-Martin bound[More precisely, the bound on σ_ tot(s) which can be obtained from this analysis is slightly stronger than a simple log-square form, because of the second term in eq. (<ref>). This fact is pointed out in <cit.>.] <cit.>. Eq. (<ref>) naively suggests that s^-2·|F(s,t)|→0 as s→∞+iϵ for small but positive t. Indeed, the bound on F(s,t) obtained in <cit.> for t>0 also implies the existence of such small but positive t if we assume the polynomial boundedness (<ref>) with α=0. This is in fact crucially important to obtain a dispersion relation of F(s,t) for t>0 with finite number of subtractions <cit.>.













  §.§.§ •    α>0 case

Next, let us derive the bound on |F^(n)(s)| at s→∞ assuming the exponential boundedness (<ref>) with α>0. Then, from eq. (<ref>), B_0(s) is also bounded as

    B_0(s)<C̃√(π/2)s^Nexp[Cs^α] .

From this bound and eq. (<ref>), Y=β_0L is bounded as

    Y<Cs^α+(N-1/4)ln(s/s_0)-1/2ln Y≈ Cs^α+(N-α/2-1/4)ln(s/s_0)-1/2lnln(s/s_0) .

Applying the bound (<ref>) to eq. (<ref>), for n≥0, one obtains


    |F^(n)(s)|<C^2n+2s^1+(2n+2)α/t_0^n+1
     .

Note that the right-hand side diverges in the limit t_0→0, which is consistent with the fact that |F(s,0)| diverges when the massless particle exchange exists. This result shows that |F(s,0)| is polynomially bounded in s when s→∞+iϵ, even if |F(s,t)| can grow exponentially in the unphysical region. The powers of s of the right-hand side of (<ref>) grows as n grows, however. This suggests that |F(s,t)| grows faster than any polynomial in s in the Regge limit when t>0. This is consistent with the assumption C which we imposed to derive this result, and indeed, the bound on F(s,t) obtained in <cit.> for t>0 also implies the exponential growth of F(s,t>0) in the Regge limit if we assume the exponential growth (<ref>) with α>0. This result implies that one cannot obtain dispersion relation with finite number of subtractions for F(s,t) when t>0. (<ref>) also suggests that the Froissart-Martin bound can be violated even in local QFT with 0<α<1/2. We do not deny the possibility that F̃(s,z) were in fact polynomially bounded in s on or inside the ellipse 𝒞_0 even when 0<α<1/2, but we do not expect such boundedness property from the consideration in sec. <ref>. It should be noted that the forward scattering amplitude F(s,0) is still bounded by s^2 for 0≤α<1/2 in spite of the possible violation of the Froissart-Martin bound for α>0:

    lim_s→∞+iϵ|F(s,0)/s^2|=0   for   0≤α<1/2 .




Intuitively, the origin of the existence of the bounds (<ref>) on the partial wave amplitudes f_l(s) can be understood as the short-range nature of the force mediating the scattering under consideration. In the large impact parameter regime b∼(l/q_s)≫1 
and for fixed 0≤-t≪ s, i.e., for small θ, the scattering amplitude will be dominated by the exchange of soft particles. The force mediated by the soft particle exchange with mass M is expected to decay as exp[-Mb] for large impact parameter Mb≫1. 
On the other hand, scattering with fixed b will grow as exp[Cs^α] when the scattering energy increases, because the number of excited intermediate multi-particle states can increase as exp[Cs^α] at most, which is responsible for an exponential growth of the Lehmann-Källén spectral density. With taking into account l∼ q_sb, the above discussion implies the partial wave amplitudes may behave as

    |f_l(s)|∼ e^Cs^αe^-M/q_sl ,

for sufficiently large l. Then, using q_s∼√(s), exponential suppression of f_l(s) for l≳L̃(s) M^-1Cs^α+1/2 is expected. Thus, 

    F^(n)(s)∼∑_l=0^L̃(s)(2l+1)f_l(s)(l/q_s)^2n .

Unitarity in the form of SS^†=1 which leads to the condition |f_l(s)|≤1 further constrains the high energy behavior of the scattering amplitude:

    |F^(n)(s)|∼∑_l=0^L̃(s)(2l+1)(l/q_s)^2n∼(L̃(s))^2n+2s^-n=(M^-1C)^2n+2s^1+(2n+2)α .

This coincides with the obtained bounds (<ref>) assuming the holomorphy of F̃(s,Z) inside the ellipse 𝒞_0. This result indicates the holomorphy of F(s,t) inside the ellipse 𝒞_0 is intimately related to the short-range nature of the force. 

If the domain of holomorphy is smaller in the complex z plane, one will obtain the weaker upper bound on the high energy behavior of F(s,t). Indeed, for non-localizable field theory, it has been shown that F(s,z) is holomorphic inside the Lehmann ellipse with t_0=s^-1d where d denotes some s-independent positive constant, and then weaker bound (so-called Greenberg-Low type bound) was also obtained <cit.>. However, such upper bound seems to be too large from the viewpoint of short-range nature of the force mediated by massive particles <cit.>. 
This is a reason why we assume the holomorphy of F̃(s,z) on or inside the ellipse 𝒞_0. In order to check the consistency with the results obtained in <cit.>, let us investigate how our results (<ref>) changes when we choose 


t_0 as 

    t_0=d/s ,

with d>0 is s-independent constant. Then, it is straightforward to see that the upper bound (<ref>) with n=0 reduces to the Greenberg-Low type bound, which was already obtained in <cit.>. 
In this sense, our results are consistent with the previous work.[For α≥1, one may be able to obtain a stronger upper bound on |F(s,0)| for s→∞ by assuming some specific form of F(s,0) <cit.>. 

However, this difference of the boundedness for s∈ℝ 
would be irrelevant in our study here, because what is important to derive positivity bounds is whether dispersion relation holds or not, and it would be impossible to obtain dispersion relation with finite number of subtractions for α≥1 case as we will discuss in sec. <ref>.]



§ POSITIVITY BOUNDS

In this section, we derive the positivity bounds for α>0 case. 

In sec. <ref>, we review the derivation of positivity bounds for α=0 case based on <cit.>. In sec. <ref>, we derive the positivity bounds for α>0 case. It turns out that it seems impossible to obtain any positivity bounds for α≥1 case, because we fail to obtain the dispersion relation with a finite number of subtractions. In sec. <ref>, we obtain the criterion which allows us to obtain the lower bound on α through the violation of positivity bounds. We also briefly comment on the application of our results to massive Galileon model and its massless limit.


 §.§ Positivity bounds for tempered localizable field

Using Cauchy's integral formula, F(s,t) can be expressed as

    F(s,t)=(s-2m^2+t/2)^2∮_𝒞ds'/2π iF(s',t)/(s'-s)(s'-2m^2+t/2)^2 ,

where F(s,t) is holomorphic in s inside the counterclockwise contour 𝒞. Then, by deforming the contour and fully making use of the s↔ u crossing symmetry, one obtains


    F(s,t)   =[- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u]+∑_k=0^2P-1a_n,k(t)s^k
       +2(s̅+t̅/2)^2P/n!π∫^R_4m^2dμ  F(μ+iϵ,t)/(μ̅+t̅/2)^2P-1[(μ̅+t̅/2)^2-(s̅+t̅/2)^2]
       +(-1)^2P(s̅+t̅/2)^2P/n!π∫^R+4m^2-t_Rdμ  F(μ+iϵ,t)/(μ̅+t̅/2)^2P[(μ̅+t̅/2)+(s̅+t̅/2)]
       +(s̅+t̅/2)^2P∫_𝒞^±_Rds'/2π iF(s',t)/(s'-s)(s̅'̅+t̅/2)^2P ,



    F(s,t)   =[- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u]+∑_k=0^1a_k(t)s^k
       +2(s̅+t̅/2)^2/π∫^∞_4m^2dμ  F(μ+iϵ,t)/(μ̅+t̅/2)[(μ̅+t̅/2)^2-(s̅+t̅/2)^2] ,

where X̅ X-(4m^2/3). Here, we used the fact that |s^-2· F(s,t)|→0 as |s|→∞ for 0≤ t< 4m^2, t≠ m^2 when α=0 <cit.>. We also used the Schwarz reflection principle. We did not write down the explicit expressions of coefficients a_k(t) because they are irrelevant for positivity bounds. 



    ^n_t/n!F̃(v,t)
       =^n_t/n![(- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u+∑_k=0^2P-1a_k(t)s^k)_s=v+2m^2-(t/2)]
       +2v^2P/n!π∫^R_4m^2dμ ^n_t{  F(μ+iϵ,t)/(μ̅+t̅/2)^2P-1[(μ̅+t̅/2)^2-v^2]}
       +(-1)^2Pv^2P/n!π∫^R+4m^2-t_Rdμ ^n_t[  F(μ+iϵ,t)/(μ̅+t̅/2)^2P(μ̅+t̅/2+v)]+v^2P∫_𝒞^±_Rds'/2π i ^n_t[F(s',t)/(s̅'̅+t̅/2-v)(s̅'̅+t̅/2)^2P] ,

for any (s,t) on which F(s,t) is analytic.



    ^n_tF(s,t)   =(s-2m^2+t/2)^2P∮_𝒞ds'/2π i^n_tF(s',t)/(s'-s)(s'-2m^2+t/2)^2P .

{Begin comment} 

Here, we used

    ^r_t[(s-m^2+(t/2))^2P/(s'-m^2+(t/2))^2P(s'-s)]=^r-1_t[P(s-m^2+(t/2))^2P-1/(s'-m^2+(t/2))^2P+1]=⋯=∑_k=0^2P-1a_r,k(t,s)s'^k/(s'-m^2+(t/2))^2P+r ,

where a_r,k(t,s) denotes some polynomial in s and t, and 

    ∮_𝒞ds'/2π is'^k^n-r_tF(s',t)/(s'-2m^2+t/2)^2P+r=0 ,

with k≥0 and 2P+r≥2.

{End comment}
Eq. (<ref>) implies that 2-subtracted dispersion relation holds for 0≤ t< 4m^2, t≠ m^2:





    . F(s,t)|_s=v+2m^2-(t/2)=.(s-2m^2+t/2)^2∮_𝒞ds'/2π iF(s',t)/(s'-s)(s'-2m^2+t/2)^2|_s=v+2m^2-(t/2)
       =[λ/m^2-s+λ/m^2-u+c_1(t)]_s=v+2m^2-(t/2)+2v^2/π∫^∞_4m^2dμ  F(μ+iϵ,t)/(μ̅+t̅/2)[(μ̅+t̅/2)^2-v^2] ,

where v(s-u)/2=s-2m^2+(t/2), λ - Res_s=m^2F(s,t)= Res_u=m^2F(s,t)= Res_t=m^2F(s,t), and X̅ X-(4m^2/3). c_1(t) is some function of t.
Here, we used the s↔ u crossing symmetry, which is equivalent to v↔ -v symmetry. 

Now let us consider what happens if we take the limit R→∞ in eq. (<ref>). Eq. (<ref>) implies that |F(s,t)||s|^-2 goes to 0 as |s|→∞ for |t|<4m^2, t≠ m^2 because F(s,t) combined with the fact that .

 By taking the limit R→∞, and using eqs. (<ref>) and (<ref>), one obtains the following 2P-subtracted dispersion relation with P≥1 for 0≤ t<4m^2, t≠ m^2, : 

    ^n_t/n!F̃(v,t)   =^n_t/n![(- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u+∑_k=0^2P-1a_n,k(t)s^k)_s=v+2m^2-(t/2)]
       +2v^2P/n!π∫^∞_4m^2dμ ^n_t{  F(μ+iϵ,t)/(μ̅+t̅/2)^2P-1[(μ̅+t̅/2)^2-v^2]} .


Then, the pole-subtracted amplitude B(s,t) which is defined by

    B(s,t) F(s,t)-[- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u+ Res_t=m^2F(s,t)/m^2-t] ,

can be expressed as 

    B(s,t)   = Res_t=m^2F(s,t)/m^2-t+∑_k=0^1a_k(t)s^k
       +2(s̅+t̅/2)^2/π∫^∞_4m^2dμ  F(μ+iϵ,t)/(μ̅+t̅/2)[(μ̅+t̅/2)^2-(s̅+t̅/2)^2] ,

for 0≤ t< 4m^2, t≠ m^2. Thus, by making use of the fact that Res_t=m^2F(s,t) is independent of s in scalar theories, one can derive the following expression from eq. (<ref>): 

    B^(2N,M)(t)=∑_k=0^M(-1)^k/k!2^kI^(2N+k, M-k)(t) ,

for 0≤ t< 4m^2, t≠ m^2, with N≥1 and M≥0. Here, B^(2N,M)(t) and I^(q,p)(t) are defined by 

    B^(2N,M)(t).^2N_v^M_t/M!B̃(v,t)|_v=0 ,  B̃(v,t).B(s,t)|_s=v+2m^2-(t/2) ,
       I^(q,p)(t)q!2/p!π∫^∞_4m^2dμ^p_t  F(μ+iϵ,t)/(μ̅+t̅/2)^q+1>0 .

In (<ref>), we used ^n_t  F(s+iϵ,t)>0 for n≥0, when 0≤ t<4m^2, t≠ m^2, and s≥ 4m^2 are simultaneously satisfied <cit.>. Note that the right-hand-side of eq. (<ref>) is finite thanks to the bounds (<ref>). B^(2N,M)(t) is however not ensure to be positive for M≥1 because (-1)^k factor is included in the right-hand side of (<ref>). Then, from eqs. (<ref>) and (<ref>) together with

    I^(q,p)(t)<q/ℳ^2I^(q-1,p)(t) ,

where ℳ^2(μ̅+(t̅/2))_μ=μ_ min, μ_ min is the lower bound value of μ of the integration over μ on the right-hand side of eq. (<ref>), it has been shown that the following infinite number of inequalities which are recursively defined must be satisfied <cit.>:



    Y^(2N,0)(t) B^(2N,0)(t) ,
       Y^(2N,M)(t)∑_r=0^M/2c_rB^(2N+2r,M-2r)(t)+1/ℳ^2∑^(M-1)/2_ even k=0(2(N+k)+1)β_kY^(2(N+k),M-2k-1)(t)>0 ,


for N≥1, M≥0 and 0≤ t<4m^2 , t≠ m^2. 
Coefficients c_k and β_k are defined by

    c_kE_2k/(2k)!2^2k ,  β_k(-1)^k(2^2k+3-2)B_2k+2/(2k+2)! ,

where E_2k and B_2k+2 denote the Euler numbers and Bernoulli numbers, respectively. These bounds are also extended to the scattering of the particles with spin <cit.> and are applied to various models (e.g., <cit.>). An infinite number of inequalities (<ref>) with N≥1, M≥0 are obtained, and they are called positivity bounds. Thanks to the condition v=0 and 0≤ t<4m^2, the left-hand side of (<ref>) can be evaluated within the validity of the LEEFT, meaning that (<ref>) gives the non-trivial bounds on LEEFT.

In fact, one can improve the above positivity bounds eq. (<ref>) when perturbation theory based on LEEFT is valid for s<Λ_ th^2 with m^2≪Λ_ th^2, by evaluating F(μ+iϵ) with 4m^2<μ<Λ_ th^2 which expresses the contributions from loops of light particle ϕ with external momenta satisfying 4m^2<s<Λ_ th^2:



    B^(2N,M)_Λ_ th(t) B^(2N,M)(t)-∑_k=0^M(-1)^k/k!2^k(2N+k)!2/(M-k)!π∫^Λ_ th^2_4m^2dμ^M-k_t  F(μ+iϵ,t)/(μ̅+t̅/2)^2N+k+1
               =∑_k=0^M(-1)^k/k!2^kI^(2N+k,M-k)_Λ_ th(t) ,
       I^(q,p)_Λ_ th(t)q!2/p!π∫^∞_Λ_ th^2dμ^p_t  F(μ+iϵ,t)/(μ̅+t̅/2)^q+1>0 .


From the definition of Λ_ th, one can evaluate B^(2N,M)_Λ_ th(t) perturbatively by using LEEFT . Then, by making use of inequalities

    I^(q,p)_Λ_ th(t)<q/Λ_ th^2+(t/2)-2m^2I^(q-1,p)_Λ_ th(t)
     ,

which is just an analogue of (<ref>), one obtains the improved positivity bounds:



    Y_Λ_ th^(2N,0)(t) B_Λ_ th^(2N,0)(t) ,
       Y_Λ_ th^(2N,M)(t)∑_r=0^M/2c_rB_Λ_ th^(2N+2r,M-2r)(t)
               +1/Λ_ th^2+(t/2)-2m^2∑^(M-1)/2_ even k=0(2(N+k)+1)β_kY_Λ_ th^(2(N+k),M-2k-1)(t)>0 .





begin{comment}

Chain-rule:



    _vB̃(v,t)=._s B(s,t)|_s=v+2m^2-(t/2) ,
       _tB̃(v,t)=[_t B(s,t)-1/2_s B(s,t)]_s=v+2m^2-(t/2) .


From eqs. (<ref>), one obtains

    B^(2N,M)(t)   =∑_k=0^M(-1)^k/k!2^k[^2N+k_s1/(M-k)!^M-k_tB(s,t)]_s=2m^2-(t/2)=∑_k=0^M(-1)^k/k!2^kI^(2N+k,M-k)(t) ,

Here, in order to obtain the second equality, we used

    [^2N+1_s1/(M)!^M_tB(s,t)]_s=2m^2-(t/2)=I^(2N+1,M) ,

for N≥1 and M≥0.

end{comment}







 §.§ Positivity bounds without temperedness assumption

In this subsection, we derive the positivity bounds without temperdness assumption, namely, for theories with α>0. 



  §.§.§ •     0<α<1 case

Using Cauchy's integral formula, F(s,t) can be expressed as

    F(s,t)=(s-2m^2+t/2)^2P∮_𝒞ds'/2π iF(s',t)/(s'-s)(s'-2m^2+t/2)^2P ,

for any P∈ℕ. Here F(s,t) is holomorphic in s inside the counterclockwise contour 𝒞. Then, combined with an analyticity assumption B2, one obtains the following expression by deforming the contour and fully making use of the s↔ u crossing symmetry:

    F(s,t)   =[- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u]+∑_k=0^2P-1b_k(t)s^k
       +2(s̅+t̅/2)^2P/π∫^R_4m^2dμ  F(μ+iϵ,t)/(μ̅+t̅/2)^2P-1[(μ̅+t̅/2)^2-(s̅+t̅/2)^2]
       +(s̅+t̅/2)^2P/π∫^R+4m^2-t_Rdμ  F(μ+iϵ,t)/(μ̅+t̅/2)^2P[(μ̅+t̅/2)+(s̅+t̅/2)]
       +(s̅+t̅/2)^2P∫_𝒞^±_Rds'/2π iF(s',t)/(s'-s)(s̅'̅+t̅/2)^2P ,

where 𝒞^±_R denotes the semi-circle with radius R: 𝒞^+_R{s:s=Re^iθ+iϵ, 0≤θ≤π} and 𝒞^-_R{s:s=Re^iθ-iϵ, π≤θ≤2π}. We did not write down the explicit expressions of coefficients b_k(t) because they are irrelevant for positivity bounds. We will take the limit R→∞ later. For any s∈ℂ in the complex-s plane, except for poles and branch cuts on the real-s axis, F(s,t) is analytic in t at t=0 from the analyticity assumption. Thus, eq. (<ref>) gives rise to

    .^n_t/n!F̃(v,t)|_t=0   =^n_t/n![(- Res_s=m^2F(s,t)/m^2-s+ Res_u=m^2F(s,t)/m^2-u+∑_k=0^2P-1b_k(t)s^k)_s=v+2m^2-(t/2)]_t=0
       +2v^2P/n!π∫^R_4m^2dμ ^n_t{  F(μ+iϵ,t)/(μ̅+t̅/2)^2P-1[(μ̅+t̅/2)^2-v^2]}_t=0
       +v^2P/n!π^n_t[∫^R+4m^2-t_Rdμ   F(μ+iϵ,t)/(μ̅+t̅/2)^2P(μ̅+t̅/2+v)]_t=0
       +v^2P/n!∫_𝒞^±_Rds'/2π i ^n_t[F(s',t)/(s̅'̅+t̅/2-v)(s̅'̅+t̅/2)^2P]_t=0 ,

where

    F̃(v,t).F(s,t)|_s=v+2m^2-(t/2) .


Thus,

    B^(2N,M)(0)=[^2N_v(.^M_t/M!B̃(v,t)|_t=0)]_v=0
    
       =(2N)!2/M!π∫^R_4m^2dμ ^M_t[  F(μ+iϵ,t)/(μ̅+t̅/2)^2N+1]_t=0+._v^2N[._t^M/M!(. Res_t=m^2F(s,t)|_s=v+2m^2-(t/2)/m^2-t-iϵ)|_t=0]|_v=0
       +(2N)!/M!π^M_t[∫^R+4m^2-t_Rdμ   F(μ+iϵ,t)/(μ̅+t̅/2)^2N+1]_t=0+(2N)!/M!∫_𝒞^±_Rds'/2π i ^M_t[F(s',t)/(s̅'̅+t̅/2)^2N+1]_t=0 ,

for any N∈ℕ and M≥0. Note that when Res_t=m^2F(s,t) is independent of s, the second term in the second line of eq. (<ref>) vanishes. In scalar theories, Res_t=m^2F(s,t) is independent of s, and hence one can drop the contribution from t-channel pole in (<ref>). Now, let us consider what happens if we take the limit R→∞ in eq. (<ref>). From (<ref>), s↔ u crossing symmetry, and the Schwarz reflection principle, it turns out that |F^(n)(s±ϵ)|<|s|^1+(2n+2)α along the real axis. If this polynomial boundedness property also holds on the complex s-plane, the terms in the final line of eq. (<ref>) vanish after taking the limit R→∞. In order to show the boundedness property of |F^(n)(s)| in the complex s-plane, the Phragmén-Lindelöf theorem plays a crucial role. Combined with the analyticity assumption B2, the theorem ensures that the polynomial boundedness (<ref>) on the real axis also holds for modulus of s in the upper-half plane if |F^(n)(s)| can grow no faster than exp[σ|s|^γ] with γ<1 as |s|→∞ in the upper-half plane. Combined with the boundedness assumption C, this means that for α<1 case, |F^(n)(s)| is bounded by |s|^1+(2n+2)α in the upper-half plane.[This polynomial boundedness on the scattering amplitudes in the complex s-plane is consistent with the result obtained in <cit.>.] This boundedness in the upper-half plane also ensures the boundedness in the lower-half plane by making use of the s↔ u crossing symmetry. Thus, from eqs. (<ref>) and (<ref>), one can obtain the following expression by taking the limit R→∞, only for 2N>1+(2+2M)α:

    B^(2N,M)(0)=(2N)!2/M!π∫^∞_4m^2dμ ^M_t[  F(μ+iϵ,t)/(μ̅+t̅/2)^2N+1]_t=0=∑_k=0^M(-1)^k/k!2^kI^(2N+k,M-k)(0) .





This means that for 0<α<1 case, one can relate B^(2N,M)(t) to the imaginary part of the scattering amplitude as in eq. (<ref>) only when both 2N>1+(2M+2)α and t=0 are satisfied. Therefore, for 0<α<1 case, positivity bounds (<ref>) with 2N>1+(2M+2)α and t=0 must be still satisfied, while other bounds which must be satisfied in α=0 case could be violated. Indeed, eqs. (<ref>), (<ref>), (<ref>), and (<ref>) imply that for α>0 case, I^(2N,M)(t), B^(2N,M)(t), and Y^(2N,M)(t) are ensured to be finite only when both 2N>1+(2M+2)α and t=0 are satisfied. Note that one can also obtain the improved positivity bounds by introducing Λ_ th which specifies the regime of the validity of the perturbative calculation of LEEFT, and evaluating the contributions from the loops of light particle as is done for α=0 case in sec. <ref>.

The result obtained here is remarkable: there are still infinite number of inequalities which must be satisfied by LEEFT scattering amplitude, even when UV completions are non-local. Existence of such bounds implies the existence of IR obstructions to unitary, analytic, and Lorentz invariant UV completions. In order to demonstrate the importance of this result, let us consider the forward-limit positivity bounds Y^(2N,0)(0)>0 with N≥1, for example. Leading-order forward-limit positivity bound Y^(2,0)(0)>0 can be an IR obstruction for LEEFTs to admit local, analytic, unitary, and Lorentz invariant UV completions, even if LEEFTs are apparently consistent with locality and Lorentz invariance. This statement has been already obtained in <cit.>.
In the literature, sub-leading order forward-limit positivity bounds Y^(2N,0)(0)>0 with N≥2 are also regarded as IR obstructions to local, analytic, unitary, and Lorentz invariant UV completions. However, our results suggest that sub-leading order forward-limit positivity bounds can be IR obstructions to analytic, unitary, and Lorentz invariant but possibly non-local UV completions. 

 


  §.§.§ •    α≥1 case

As is discussed just before obtaining eq. (<ref>), one cannot make use of Phragmén-Lindelöf theorem for α≥1 case, because |F^(n)(s)| can grow as fast as exp[C|s|] in this case. This means that it would be impossible to show the polynomial boundedness of |F^(n)(s)| in the complex-s plane even if |F^(n)(s)| is polynomially bounded on the real axis. This means that it will be impossible to obtain dispersion relation with a finite number of subtractions for α≥1 case, and hence all the positivity bounds can be violated.


Let us summarize the results obtained in this subsection <ref>. Among an infinite number of positivity bounds (<ref>) which must be satisfied in α=0 case, all the bounds with t>0 could be violated for any value of (2N,M) in α>0 case. On the other hand, some of positivity bounds (<ref>) with t=0 must be still satisfied for 0<α<1. In fig. <ref>, we list up which inequalities must be satisfied for generic α. For fixed M and α with 0<α<1, inequalities with 2N>1+(2+2M)α must be satisfied. For fixed N and α with 0<α<1, inequalities with M≥ (2α)^-1(2N-1)-1 could be violated even when t=0.  When α≥1, all the inequalities could be violated. As is emphasized above, there are still infinite number of positivity bounds for α<1, such as sub-leading order forward-limit positivity bounds. This implies the existence of IR obstructions to Lorentz invariant UV completions: our results open the new possibility to falsify unitary, analytic, and Lorentz invariant UV completions via the violation of positivity bounds, even if LEEFT is apparently Lorentz invariant.



 §.§ Lower bound on the α parameter

If we maintain the assumptions A-C which are given in sec. <ref>, the results obtained in sec. <ref>-<ref> allow us to put a lower bound on α in the following manner:


  * If at least one of the positivity bounds (<ref>)  is violated for t>0, then α>0.

  * If the positivity bounds (<ref>) with t=0 are violated for ∀(2N,M)∈{(2N_i,M_i)}_i∈𝒟, where 𝒟 specifies the set of variables (2N,M), then 

    α≥ Min[2N_*-1/2M_*+2 , 1] ,  2N_*-1/2M_*+2 Max[{2N_i-1/2M_i+2}_i∈𝒟] .


This result shows that one can obtain the information of the α parameter only from the low energy data which would be accessible from observations, by investigating which inequalities are violated among an infinite number of inequalities (<ref>). 

 

 




 §.§ Absence of massless spectra and the applicability of the positivity bounds

In order to obtain the positivity bounds, one needs to assume the absence of massless mediator for the scattering under consideration, but it seems difficult to justify this assumption because in our real world, all of the matter fields couple to the graviton due to the nonlinear nature of gravity, and this nonlinearity would become very important at sufficiently high energy, typically above the Planck scale M_ pl. Since one also has to take the limit s→∞+iϵ to obtain the dispersion relation which leads to the positivity bounds, it seems problematic for the validity of the positivity bounds. Nevertheless, the results obtained by assuming the absence of massless mediator, such as Froissart-Martin bound (<ref>) is experimentally verified for some scattering processes (e.g., see <cit.>). 
Then, under which condition these results should be satisfied? We address this issue briefly in this subsection. 








Let us consider the theory described by the Hamiltonian H_ (m), which includes the degrees of freedom of the sector under consideration but does not include the fluctuation of the massless particles, such as graviton. H_ (m) can also include heavy fields which are not included explicitly in the Hamiltonian of LEEFT. Suppose that the quantum field theory defined by H_ (m) is analytic, local, unitary, and Lorentz invariant UV completion. Then, one can obtain the dispersion relation for F_ (m)(s,t) which is evaluated by analyzing the QFT defined by H_ (m):

    .F_ (m)(s,t)|_s=v+2m^2-(t/2)=(light  poles)+2v^2/π∫^∞_4m^2dμ  F_ (m)(μ+iϵ,t)/(μ̅+t̅/2)[(μ̅+t̅/2)^2-v^2] .








Here, the first term on the right-hand side expresses the contribution from poles coming from the s,u-channel exchange of light particles. If F_ (m)(s,t) well coincides with an effective scattering amplitude F_ eff(s,t)  which is evaluated by using LEEFT at s∼ m^2, one obtains

    .F_ eff(s,t)|_s=v+2m^2-(t/2)≈(light  poles)+2v^2/π∫^∞_4m^2dμ  F_ (m)(μ+iϵ,t)/(μ̅+t̅/2)[(μ̅+t̅/2)^2-v^2] ,

for s∼ m^2. Eq. (<ref>) leads to the positivity bounds (<ref>) on LEEFT as we reviewed the result of <cit.> in sec. <ref>. 


Therefore, when there exists some analytic, local, unitary, and Lorentz invariant UV completion which includes no massless particle and can reproduce an effective scattering amplitude F_ eff(s,t) of the sector under consideration at s∼ m^2 in a good approximation, positivity bounds on F_ eff(s,t) must be satisfied even if the sector under consideration couples to massless particles at high energy in the real world. This means that, if positivity bounds are violated, the sector under consideration must be UV completed in combination with massless spectra which are irrelevant at low energy physics, or with the violation of at least one of the essential assumptions, that is, locality, unitarity, analyticity, and Lorentz invariance. 

It should be noted that it is also possible in principle that one can make the LEEFT to satisfy positivity bounds by simply adding the new light states by hand which is coupled to the sector under consideration at low energy, even if the LEEFT does not satisfy positivity bounds before adding new light states. This is simply because F_ eff is modified if one adds the new light states into LEEFT. 




 §.§ Extract the information of UV completion

From the results obtained in sec. <ref> and <ref>, we conclude that one can extract the information of UV completion of the sector under consideration through the violation of positivity bounds in the following manner:


  * Any possible UV completion of the sector under consideration must violate at least one of the essential assumptions imposed on the properties of S-matrix, i.e., unitarity, analyticity, or Lorentz invariance.

  * The sector under consideration is unitary, analytic,and Lorentz invariant but is essentially non-local at high energy, and one can obtain the lower bound on the degree of the non-locality, namely, the lower bound on the α parameter, by checking which inequalities are violated.

  * The sector under consideration must be UV completed in combination with massless spectra which are irrelevant at low energy physics. 


Of course, any combination of them is also possible. As is mentioned in sec. <ref>, if one adopts the second possibility 2, i.e., if one maintains assumptions A-C corresponding to Lorentz invariance, unitarity, causality, and the exponential boundedness of the scattering amplitudes (while connection between causality and assumptions B1-C are not yet rigorously shown for α>0 case as far as we know), one can obtain the lower bound on α parameter characterizing the degree of non-locality by using the criterion (<ref>) as we discussed in sec. <ref>. We should make the connection between causality and assumptions B1-C rigorous as possible in the context of Jaffe theory, as has been done in local QFT. We will study this aspect in our future work.



  §.§.§ Application to massive Galileon: massless limit

Let us briefly comment on the application of our results to massive Galileon models. Positivity bounds are applied to massive Galileon models in <cit.>. Their results suggest that it is impossible to take the massless limit with satisfying the leading-order forward limit positivity bound. From eq. (<ref>), this means that the UV completion of Galileon models which are obtained by taking the massless limit of massive Galileon models cannot be strictly localizable theories: if one try to maintain unitarity, analyticity, and Lorentz invariance, one must give up strict locality. This is consistent with the conjecture proposed in <cit.> that Galileons may fall into non-localizable theory.


    ℒ=∑_n=2^5g_n/n!Λ^3(n-2)πΠ^μ_1_[μ_1⋯Π^μ_n-1_μ_n-1]-1/2m^2π^2 .

In α=0 case, positivity bounds on the parameter space of the massive Galileon model were obtained <cit.>. For simplicity, let us consider the bounds on the coefficients of leading derivative couplings g_3 and g_4 up to the leading order, namely, up to 𝒪 (m^2s^2/Λ^6). In this case, relevant positivity bounds are (2N,M)=(2,0), (2,1):



    Y^(2,0)_Λ_ th(0)   =m^2g_3^2/8Λ^6>0 ,
    
    Y^(2,1)_Λ_ th(0)   =1/16Λ^2(3m^2g_3^2/Λ_ th^2-2m^2+3g_3^2-4g_4)>0 .



As is summarized in fig. <ref>, if 0<α<1/4 both (<ref>) and (<ref>) must be satisfied, leading to 

    [                                     g_4/g_3^2≤3/4 
    
    g_4>0 , or  g_4/g_3^2>3/4 
    Λ_ th^2<2m^2+3m^2/4(g_4/g_3^2)-3 
    
    g_4>0                                                or  
    g_4≤0 . ]

Thus, even if g_4/g_3^2>7/8, the Galileon model can have analytic, unitary, and Lorentz invariant UV completion when α>0, while in that case the cutoff scale of LEEFT becomes very low when α<1/4: Λ_ th≃ 2m^2.[Cutoff scale means the energy scale under which the perturbative LEEFT evaluation is valid.] If 1/4≤α<1/2, only (<ref>) must be satisfied, leading to g_3≠0. One obtains no upper bound on Λ_ th. If α≥1/2, namely, if the theory is quasi-local or non-localizable, then one has no bound on g_3 or g_4.

Now, let us consider the Vainshtein mechanism. The conditions under which Vainshtein mechanism works in the static and spherically symmetric configuration are <cit.>

    g_3>-√(g_4) ,   g_4≥0 .

When 0<α<1/4, parameter region which satisfies both the required positivity bounds and (<ref>) is given by

    [                                                      g_3≥√(4g_4/3)>0 , or  
    -√(g_4)<g_3<√(3g_4/4) 
    Λ_ th^2<2m^2+3m^2/4(g_4/g_3^2)-3 
    
    g_4>0 . ]

When α≥1/4, required positivity bounds are automatically satisfied if eq. (<ref>) is satisfied. Thus, negative value of g_3 is also allowed without suffering from the low cutoff. 
 
The fact that g_3<0 is allowed without suffering from too low cutoff scale  may imply that self-accelerated branch may be allowed to be embedded into unitary, analytic, and Lorentz invariant, α>1/4 UV completion without suffering from non-predictability due to the too low cutoff of the LEEFT. To be confirmed. 




MEMO: The following investigation may be unnecessary.
Let us also assume that Λ satisfies Λ≪ M_ pl. 
Then, the growth behavior of an exact scattering amplitude F_ exact(s,t) at large Λ≪ s≪ M_ pl will be well approximated by that of F_ (m)(s,t) which is evaluated by analyzing the QFT defined by H_ (m). This means that one can apply the positivity bounds with an accuracy 𝒪(m^2/Λ^2). Intuitively, this irrelevance of massless mediator in the case Λ≪ M_ pl can be understood in the following manner. The derivative couplings included in the LEEFT are induced by the radiative correction of the heavy particle or high-energy massless particle in the following manner, formally: 

    ∑_k=0^2nc_2n,k^2n/Λ^2nM_ pl^k≃c_2n,0^2n/Λ^2n .

This means that at all of the modulus of the contributions from the heavy particles to the coupling is much larger than that from massless graviton, while the contribution of derivative couplings are of course suppressed at low-energy.








§ CONCLUSION AND DISCUSSIONS

In this study, we have derived the posivity bounds on LEEFT which admit an analytic, unitary, and Lorentz invariant UV completions with a mass gap. 
Under several reasonable assumptions on the S-matrix, we find that an infinite number of subtractions will be required for t>0, unless α=0. This means that the beyond forward limit positivity bounds obtained in the literature can be violated even in local QFT because standard local QFT can have 0≤α<1/2. Beyond forward limit positivity bounds are obtained only when α=0 case, namely, momentum space Wightman functions are polynomially bounded. 
On the other hand, we have shown that _t^n F(s,t)|_t=0 are polynomially bounded in the complex s-plane when α<1, leading to the dispersion relation with finite number of subtractions. As a result, we obtained an infinite number of positivity bounds for α<1 theories: obtained bounds for generic α are summarized in fig. <ref>. Because α≥1/2 theories are essentially non-local, our results suggest the existence of IR obstructions to analytic, unitary, and Lorentz invariant UV completions. 
So far, it was impossible to conclude that the UV completion must violate at least one of the analyticity, unitarity, or Lorentz invariance, by testing the violation of positivity bounds. This is because it was not known whether positivity bounds could be obtained when the locality assumption is removed. Our results open the very exciting windows to test the analyticity, unitarity, or Lorentz invariance of UV completion only from IR data, even if Lorentz invariance is maintained at low energy. 

As is well known, however, one of the most annoying issue is the necessity of the introduction of a mass gap in the scattering amplitude under consideration to derive positivity bounds. The existence of a mass gap is important for deriving the high energy behavior of the amplitude and the dispersion relation. Both of them are crucially important to derive positivity bounds. Because massless graviton will be coupled to all sectors, this issue must be considered seriously. 
It should be clarified under which situation one can correctly capture the properties of the amplitude by introducing mass terms in the theory as an IR regulator. 
We will study these aspects in our future work.




J. T. would like to thank Andrew J. Tolley and Claudia de Rham for useful discussions and valuable comments, as well as their hospitality during his visits to Imperial College London. The author also would like to thank Takahiro Tanaka for useful discussions.





 




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