The renormalized Hamiltonian truncation method in the large $E_T$ expansion

Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact $g^2$ and some of the $g^3$ contributions for the $\phi^4$ theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and implement them to compute the lowest energy levels at strong coupling. A simple diagrammatic representation of the corrections and various tests are also introduced.


Introduction and review
An outstanding problem in theoretical physics is to solve strongly coupled Quantum Field Theories (QFT). When they are not amenable to analytic calculations one can resort to numerical approaches. The two most used numerical approaches are lattice simulations and direct diagonalization of truncated Hamiltonians. In this paper we further develop the Hamiltonian truncation method recently presented in Ref. [1][2][3], that renormalizes the truncated Hamiltonian H T to improve the numerical accuracy.
The Hamiltonian truncation method consists in truncating the Hamiltonian H into a large finite matrix (H T ) ij and then diagonalizing it numerically. There is a systematic error with this approach that vanishes as the size of the truncated Hamiltonian H T is increased. There are different versions of the Hamiltonian truncation method that mainly differ on the frame of quantization and the choice of basis in which H is truncated. Two broad categories within the Hamiltonian truncation methods are the Truncated Conformal Space Approach [4] and Discrete Light Cone Quantization [5]. A less traveled route consists in using the Fock-Space basis to truncate the Hamiltonian [1,2,[6][7][8][9][10]. Lately there have been many advances in the Hamiltonian Truncation methods, see for instance [3,[11][12][13][14][15][16][17].
We review the truncated Hamiltonian approach following the discussion of Ref. [1,3]. The problem we are interested in is finding the spectrum of a strongly coupled QFT. Therefore we want to solve the eigenvalue equation where H = H 0 + V , H 0 is a solvable Hamiltonian or the free Hamiltonian and V is the potential. H 0 is diagonalized by the states H 0 | E n = E n | E n . Suppose we are interested in studying the lowest energy states of the theory. One way to do it is separating the Hilbert space H into H = H l ⊕ H h , where H l is of finite dimension and it is spanned by the states | E n with E n ≤ E T . Then, the Hilbert space H h is an infinite-dimensional Hilbert space containing the rest of the states E n > E T . The states are projected as P l | x ≡ | x l ∈ H l and (I − P l )| x = P h | x ≡ | x h ∈ H h . Then, the eigenvalue problem can be replaced by where H ef f ≡ H T + ∆H(E), the truncated Hamiltonian is H T = P l HP l and with O ij ≡ P i OP j for i, j ∈ {h, l}. To derive Eq. (2), project Eq. (1) into the two equations and then substitute | E h = (E − H hh ) −1 H hl | E l from the second equation in (4) into the first. Notice that Eq. (2) is an exact equation and that a complete knowledge of ∆H(E) would render the original eigenvalue problem of Eq. (1) solvable by an easy numerical diagonalization. In the limit where E T → ∞ the corrections ∆H to H T can be neglected, but it is computationally very costly to increase the size of H T and then diagonalize it. Therefore it is interesting to calculate ∆H to improve the numerical accuracy for a given E T . A first step to compute ∆H is to perform an expansion of Eq. (2) in powers of V hh (E − H 0 ) −1 , where the matrix elements of ∆H n are given by ∆H n (E) rs = j 1 ,··· ,jn−1: in the H 0 eigenbasis and the sums run over all labels j 1 , . . . , j n−1 of states belonging to H h with r, s denoting the matrix elements (corresponding to eigenstates of H 0 with E s , E r ≤ E T eigenvalues). Naively the truncation of the series in Eq. (5) is justified for V hh /H 0 hh < 1 which for large enough E T and E E T is fulfilled, and allows to go to strong coupling. This is discussed in detail in Sec. 5.3. The operator ∆H depends on the exact eigenvalue and in practice the way Eq. (2) is solved is by diagonalizing iteratively H ef f (E * ) starting with an initial seed E * . It is convenient to take E * close to the exact eigenvalue E, a simple and effective choice is to take the eigenvalue obtained from diagonalizing H T .
In Ref. [1] the φ 4 theory in two dimensions was studied at strong coupling using the Hamiltonian truncation method just presented in the Fock basis. There, the leading terms of ∆H 2 doing a local expansion were computed and shown to improve the results with respect to the ones found by only diagonalizing H T . The main result of our work is to explain a way to calculate the exact corrections to ∆H at any order ∆H n . As an example we calculate the ∆H 2 correction and some of the ∆H 3 terms for the φ 4 theory in two dimensions and present various approximation schemes for a faster numerical implementation. This can be seen as an extension of the method presented in Ref. [1] which we believe to be very promising.
The paper is organized as follows. In Sec. 2 we introduce a general formula to compute ∆H n (E, E T ) at any order n. Then we apply the method to the φ 2 and φ 4 scalar field theories in d = 2 space-time dimensions which we first define in Sec. 3. The method is tested in Sec. 4 by studying the spectrum of the solvable φ 2 perturbation with the calculation of ∆H 2 and ∆H 3 . Other numerical tests are also performed in this section. Next, in Sec. 5 we give the ∆H 2 correction for the φ 4 theory, and discuss the ∆H 3 calculation with some examples. There we also discuss the convergence of the ∆H n expansion and compute the lowest energy levels of the theory at strong coupling. In Sec. 6, we conclude and outline future directions of the method that are left open. In Appendix A we introduce a simple diagrammatic representation to compute ∆H n . Lengthy derivations and results are relegated to the Appendices B and C. All the numerical calculations for this work have been done with Mathematica.

Calculation of ∆H at any order
In this section we present one of the main results of this paper which is the derivation of the nth-order correction ∆H n of Eq. (5) to the Truncated Hamiltonian. We start by defining the operator which in the H 0 eigenbasis is given by ∆ H n (E) rs = ∞ j 1 , ..., j n−1 =1 where the indices j 1 , j 2 , . . . , j n−1 run over the states of the full Hilbert space H. Notice that the only difference between ∆H n and ∆ H n is that the later receives contributions from all the eigenstates of H 0 while ∆H n only from those with E j energies E j > E T . This translates into the fact that each term in ∆H n (E) has all the poles located at E > E T as seen in Eq. (5). From here the derivation of ∆H n follows from the observation that Eq. (7) can be rewritten as the improper Fourier transform of the product of potentials restricted to positive times iH 0 t and T denotes the time ordering operation 4 . Then, our method consists in applying the Wick theorem to Eq. (8) to calculate ∆ H n and obtaining ∆H n by keeping only the terms of ∆ H n corresponding to states with E j > E T , i.e. by keeping only the terms of ∆ H n which have all poles above E T . 5 In the following sections we show how to carry this procedure for the cases of the φ 2 perturbation and φ 4 theory.

Scalar theories
We study scalar theories in two space-time dimensions defined by the Minkowskian action For simplicity we consider the cases where α = 2, 4 and m 2 > 0. The symbol : : stands for normal ordering which for S 0 means that we set the vacuum energy to zero; while the interaction term is normal ordered with respect to S 0 , which in perturbation theory is equivalent to renormalize to zero the UV divergences from closed loops with propagators starting and ending on the same vertex.
To study these theories using the Hamiltonian truncation method we begin by defining them on the cylinder R × S 1 where the circle corresponds to the space direction which we take to have a length Lm 1, and R is the time. We impose periodic boundary conditions φ(t, x) = φ(t, x + nL) for n ∈ Z on S 1 . The compact space direction makes the spectrum of the free theory discrete and regularizes the infra-red (IR) divergences. and integrating over all times t 1 , . . . t n . Also notice that the time ordering operation is trivial because the V operators are time ordered in all the integration domain. The lim →0 is taken at the end of the calculation. 5 This procedure can be formalized as follows.
The first correction can be written as where C is any path than encircles only all the poles above E T . For The generalization to the nth correction is straightforward.
In canonical quantization the scalar operators can be expanded in terms of creation and annihilation operators as where ω k = √ m 2 + k 2 , k = 2πn L with n ∈ Z and the creation and anihilation operators satisfy the commutation relations The Hamiltonian then reads and the potentials for a φ 2 and a φ 4 interaction are given by and respectivley, where g ≡ g 4 and δ k 1 +k 2 ,0 , δ 4 i=1 k i ,0 stand for Kronecker deltas. We implement the Hamiltonian truncation using the basis of H 0 eigenstates while H h is spanned by the rest of the basis. Then, the truncated Hamiltonian is In this basis, the operator ∆H is given by where the labels r, s denote entries with E r , E s ≤ E T and the sum over j, j runs over all states with E j , E j > E T .
The Hamiltonian H can be diagonalized by sectors with given quantum numbers associated with operators that commute with H. These are the total momentum P , the spatial parity P : x → −x and the field parity Z 2 : φ(x) → −φ(x), which act on the H 0 -eigenstates as We work in the orthonormal basis of eigenstates of H 0 , P , P and Z 2 given by where β = 1/2, 1/ √ 2 for P | E i = | E i and P | E j = | E j , respectively. As done in Ref. [1], in the whole paper we focus on the sub-sector with total momentum P | E i = 0, spatial parity P| E i = +| E i and diagonalize separately the Z 2 = ± sectors. 6 In this paper we do not investigate the dependence of the spectrum as a function of the length L of the compact dimension which we leave for future work, and always consider it to be finite. 7 All the numerical calculations are done for m = 1 and L = 10.

Case study φ 2 perturbation
In this section we apply the method introduced in Sec. 2 to the scalar theory H = H 0 + V with a potential This is a simple theory that allows to illustrate various aspects of the calculation of ∆ H in Eq. (8) and its relation to ∆H. Also since the theory is solvable we can compare our procedure with the exact results. The theory is solved by using the eigenstates of H, given by where | Ω = | E 0 is the vacuum of the theory and b † /b are the creation/annihilation operators so that with Ω k = ω 2 k + 2g 2 . Then, one can relate the operators b † /b to the a † /a in H 0 (given in Eq. (13) and Eq. (14)) by the Bogolyubov transformation b k = sinh α k a † −k + cosh α k a k provided that Ω k sinh 2α k = ω −1 k g 2 , Ω k cosh 2α k = ω k + g 2 /ω k . Then, since 0 |H| 0 = 0 we have that [1]: 6 For the V = dt : φ 2 : theory, the matrix element E i |V | E j = 0 with P| E i = | E i and P| E j = | E j . Therefore, one can diagonalize the P| E i = | E i and P| E i = | E i sectors separately. 7 To match the L → ∞ spectrum one has to take into account the Casimir energy difference between the L → ∞ and the finite L theory and inspect how various states converge as L is increased. See Refs. [18,19] and Ref. [1] for a thorough study of the L dependence.
where the sum can be done by means of the Abel-Plana formula, which is the exact vacuum energy of the theory.
A brief summary of the rest of this section is the following. In Sec. 4.1 and Sec. 4.2 we calculate the 2 and 3-point corrections to the operator ∆H. In Sec. 4.3 we perform a numerical test to check that our expressions for ∆H are correct. Then, in Sec. 4.4 we discuss the numerical results and the convergence of the expansion ∆H(E i ) = n ∆H n (E i ) by comparing with the exact spectrum E i .

Two-point correction
Following the steps explained in Sec. 2 we begin the calculation of the two-point correction by first computing ∆ H 2 . From Eq. (8) we have that Then, applying the Wick theorem to Eq. (24) we find where s p = 2 p 2 p! are the symmetry factors and D F (z, t) is the Feynman propagator with discretized momenta. Henceforth we label the terms m = 0, 1, 2 by ∆ H φ 2m 2 so that ∆ H 2 = ∆ H 1 2 + ∆ H φ 2 2 + ∆ H φ 4 2 and similarly for ∆H 2 ; the labels only inform about the total number of fields in each term which do not need to be local. Due to the time integration domain, it is convenient to use half Feynman propagator the momentum of the propagator is discretised due to the finite extent of the space. Next, we proceed to calculate the operators in Eq. (25), starting with the detailed calculation of the coefficient of the identity operator ∆ H 1 2 : where 1 rs ≡ δ rs L/2 −L/2 dz has dimensions of [E] −1 . Then, upon inserting the propagator of Eq. (26) and performing the space-time integrals we find The operator in Eq. (28) has poles from all possible intermediate states and, as explained in Sec. 2, the operator ∆H 1 2 (E) is found by keeping only those terms with poles located at E r + 2ω k > E T , therefore The calculations of ∆H φ 2 2 is similar to the one for Eq. (29), we start by computing where we expand : φ(x + z, t)φ(x, 0) : in modes, as in Eq. (11), and do the simple space-time integrals. For the full expressions of ∆ H φ 2 2 see Appendix B. Then, keeping only the terms with poles at E > E T we get The operator ∆H φ 4 2 is obtained in a similar way, In Appendix A we give a simple way to derive these expressions from diagrams, and for the full expressions of ∆ H φ 2 2 and ∆ H φ 4 2 see Appendix B. Notice that the values of q 1 , q 2 and q appearing in the sums of Eq. (31) and Eq. (32) can take only the momenta of the states | E s ∈ H l on which a and a † act, and therefore are bounded. On the other hand, the values of the k's in Eq. (29) go all the way to infinity. Also, even though the operators in Eq. (31) and Eq. (32) may seem not hermitian due to the E r appearing in the expressions, one can see that the operator (∆H φ 2 2 ) rs is diagonal and therefore E r = E s , while ∆H φ 4 2 is not diagonal, but one can check that E r + 2ω q 2 = E s + 2ω q 1 , making it hermitian as well.
We end this section by noticing that the operator of Eq. (29) can be rewritten as where Φ 2 is the two-particle phase space with discretized momenta, where from Eq. (33) one has that E − E r > 2m. 8 Eq. (33) can be evaluated by means of the Abel-Plana formula, which for LE T 1 is well approximated by its continuum limit 9 . The 8 The lower limit in Eq. (33) should be taken slightly above E T to reproduce the lower limit q : 2ω q + E r > E T in the sum of Eq. (28). 9 The difference between the continuum limit and discrete result ranges from O( depending on the matrix entry.
continuum two-body phase space is given by where P µ = (E, 0) and E > 2m. Therefore (for LE T 1) we find This result is useful for numerical implementation since Eq. (36) can be integrated in terms of logarithmic functions. Finally, we notice that upon expanding the function s 2 /(2π)Φ 2 (E) around m/E = 0 we find agreement with Ref. [1] that computed it by other means (there called µ 220 (E) = 1/(πE 2 )).

Three-point correction
The calculation of the three-point correction ∆H 3 also starts from the expression in Eq. (8) where T k = 3−k n=1 t n . Next we apply the Wick theorem and find that the time ordered product (38) where we have introduced the notation X k = z + 3−k n=1 x n and φ x,t = φ(x, t); while the symmetry factor is given by We use the same notation as in the previous section ∆ H 3 6 3 , and similarly for ∆H 3 . Then, upon performing the space-time integrals in Eq. (38) and only keeping the terms with all the poles above E T we find ∆H 3 . Then, for the term ∆H 1 3 we get The expressions for ∆H φ 2 3 , ∆H φ 4 3 and ∆H φ 6 3 are lengthy but straightforward to obtain and are relegated to Appendix B.
As done in the previous section, Eq. (40) can be written as which for L −1 E T 1 is well approximated by its continuum limit and can be integrated in terms of logarithmic functions. This is useful for a fast numerical implementation.

A numerical test
We perform a numerical check to test our prescription to select the poles of ∆ H n (E) to get ∆H n , i.e. that we can select the desired intermediate states of H 0 by looking at the poles of the terms of ∆ H n . The check consists in computing ∆ H 2 as explained, and then selecting only the terms with all poles at E ≤ E T . We refer to the expression as ∆H ll n to differentiate it with ∆H n that only receives corrections from terms with poles at E > E T . ∆H ll 2 is then compared with the matrix elements of V P l (E − H 0 ) −1 P l V , finding an exact agreement. The same is done for ∆ H 3 (E) by comparing it against This check has been done for all the matrices used in the present work, both for φ 2 and φ 4 . For brevity we only show the check for two matrix entries of the φ 2 theory. These are In Fig. 1 we compare both sides of equations Eq. (43) and (44). The red curves correspond to the right hand side of Eqs. (43)-(44), which are our analytical results, and the blue dots are  given by the product of the matrices in the left hand side of the equations. In the left plot, done for 6 k=0 |∆H ll 2 | 6 k=0 , the first pole arises at the four-particle threshold and subsequent poles appear for higher excited states. Instead, the first pole in the right plot, done for 0 |∆H ll 3 | 0 , occurs at E = 2m. Notice that in both figures there are no poles for E > E T .

Spectrum and convergence
We perform a numerical study of the convergence of the energy levels as a function of the truncation energy E T and their convergence as higher order corrections ∆H n are calculated for a fixed E T . We use the formulas in Eqs. (29)-(32), (40) and (B.5)-(B.8) to numerically compute ∆H 2 and ∆H 3 . 10 We begin by comparing the vacuum eigenstate E i 0 obtained by numerically diagonalizing H T + N n=2 ∆H n (for N = 2 and 3) with the exact vacuum energy E 0 . In Fig. 2 we show a plot of ∆ i 0 = E i 0 − E 0 as a function of the coupling constant g 2 . The plot is done for a truncation energy of E T = 12 and L = 10 (recall that we work in m = 1 units). For an easier comparison with previous work, these plots have been done with the same choice of parameters and normalizations as in Fig. 2 of Ref. [1]. The gray curve in Fig. 2 is obtained by numerically diagonalizing H T , whose lowest eigenvalue is E T 0 . The blue curve is obtained by diagonalizing the renormalized hamiltonian truncation of the series expansion ∆H = ∞ n=2 ∆H n at n = 3 is perturbative in the studied range. The effect is more pronounced for the highest couplings g 2 [0.6, 0.8]. As a benchmark value E 0 (g 2 = 0.8) = −0.351864, see Eq. (23). Therefore the relative error at g 2 = 0.8 is 2%, 0.01% and 0.002% for the Truncated, the V V and the V V V corrections, respectively.   gives better results than ∆ V V V 0 while for larger E T the behavior is reversed. This indicates that for E T 15 (and g 2 = 1.8) the truncation of the series ∆H = ∞ n=2 ∆H n is not a good approximation, and adding more terms will not improve the accuracy. However, as E T is increased it pays off to introduce higher order corrections to get a better result. This is because ∆ V V V 0 has a faster converge rate than ∆ V V 0 to the real eigenvalue. The value is E 0 (g 2 = 1.8) = −1.360719, see Eq. (23). Therefore the relative error at E T = 20 is 1%, 0.04% and 0.009% for the Truncated, the V V and the V V V corrections, respectively.
In Fig. 4 we repeat the plots of Fig. 3 for the first Z 2 -even excited state but taking the absolute value of the ∆ V V 1 curve for clarity. The plots show a similar convergence rate for the three ∆ i 1 curves. However, there is a similar pattern compared to Fig. 3: for E T 15 introducing higher order corrections of the series ∆H = ∞ n=2 ∆H n gives worse results, while for larger values of E T adding higher ∆H n corrections improves them. The value is E 1 (g 2 = 1.8) = 0.784042, hence the relative error at E T = 20 is 0.8%, 0.3% and 0.17% for the Truncated, the V V and the V V V corrections, respectively. Truncation

The φ 4 theory
Next we apply the method presented in previous sections to the φ 4 theory. We start by deriving the exact expressions for ∆H 2 in detail, then we perform various useful approximations for a faster numerical implementation and discuss general aspects of the method. We also discuss the pertubativity of the ∆H n expansion and compute the spectrum of the theory at different couplings while studying its behaviour in E T and g using the results of ∆H 2 . We end the section with some comments on future work and a discussion of the calculation of ∆H 3 .

Two-point correction
Again, we follow Sec. 2 to derive ∆H by first computing ∆ H. From Eq. (8) we have It is convenient to re-write the two-point correction in the following equivalent form where E rs = (E r + E s )/2. Applying the Wick theorem we find where s p = 4 p 2 p! are the symmetry factors. By integrating Eq. (47) and keeping only the contributions from high energy intermediate states E j > E T we obtain the exact expression for ∆H 2 . We use the shorthand notation ∆H 2 = ∆H 1 2 + ∆H φ 2 2 + ∆H φ 4 2 + ∆H φ 6 2 + ∆H φ 8 2 for m = 0, 1, 2, 3, 4, and similarly for ∆ H 2 . For ∆H 1 2 , ∆H φ 2 2 we obtain: and the operator F 2 (k 1 , k 2 , k 3 , q 1 , q 2 , E, E T ) is given by In Eqs. (48)-(49), all q i 's are bounded from above (q i ≤ q max ) because they correspond to the momenta of creation/annihilation operators that act on the light states (i.e. states in H l ). Instead the k i = 2πn i /L run over all possible values n i ∈ Z. Similar expressions for ∆H φ 4 2 , ∆H φ 6 2 , ∆H φ 8 2 are given in Appendix C. As mentioned before, a simple way to derive these expressions from diagrams is given in Appendix A. We have performed the same kind of numerical checks done in Sec. 4.3 for all the operators ∆ H 2 in the φ 4 theory.

Approximations
The exact expressions for ∆H 2 are computationally demanding. Here we present different approximations that speed up the calculations and simplify their analytic structure. These basically consist in approximating the contribution from the highest energy states to ∆H in terms of a local expansion (as normally done in Effective Field Theory calculations), while keeping the contributions from lower energy states in their original non-local form. This is achieved by defining an energy E L and then by separating ∆H 2 into two parts, ∆H 2 + where we only sum over intermediate states with E j ≥ E L and ∆H 2 − where we sum over those with We choose E L E T so that ∆H 2 + is well approximated by local operators 11 . As an example we show how to implement this procedure for the contribution of ∆H φ 2 2 given in Eq. (49) and Eq. (51). We start by examining the term ∆H φ 2 2 + (E, E L ) = ∆H φ 2 2 (E, E L ), which is obtained by replacing E T by E L in Eq. (51). In this case i ω k i E L E T ω q , E rs , and then it can be well approximated by with and V 2 = and, after integrating, keeping only the contributions from those states that produce poles at E > E L , when E rs is neglected. On the other hand ∆H φ 2 2 The same exercise done for ∆H φ 2 2+ can be done for ∆H 1 2+ and ∆H φ 4 2+ and one has that in the limit E L E T where V α = L 0 dx φ α (x) and has dimensions of [E] −1 , and c 2 is given in Eq. (55). On the other hand the operators ∆H φ 6 2 and ∆H φ 8 2 are of the tree-level and disconnected type because they involve one and zero propagators respectively, see Eq. (47). Therefore the operators ∆H φ 6 2+ and ∆H φ 8 2+ are not well approximated by a local expansion, and we do not approximate them. For E L sufficiently big though, ∆H φ 6 2+ = ∆H φ 8 2+ = 0 and all the contribution to ∆H φ 6 2 , ∆H φ 8 2 comes from ∆H φ 6 2− , ∆H φ 8 2− , as can be 11 In the cases where we are only interested in having a good approximation for the lower energy entries r, s of the matrix, then E L can be taken to be similar to E T .
and, after integrating, keeping only the contributions from those states that produce poles at E > E L , when E rs is neglected. The evaluation of the coefficients in Eq. (57) can still be hard to evaluate numerically. In the next section we explain an alternative and simpler derivation of the coefficients c 2m and further approximations to evaluate them.

Local expansion and the phase-space functions
From the first term in the local expansion of Eq. (60) the coefficients of the local operators are given by:ĉ where s 4−n is the symmetry factor and, as explained above, the common E rs -shift on the eigenvalue E is neglected. 12 Next, applying the Kramers-Kronig dispersion relation to c n (E) in Eq. (61)ĉ Next, we compute Imĉ 2n . First we do the space integral which, up to g 2 s 4−n , yields where we have used D F (t, x)θ(t) = D(t, x)θ(t) with D(t, x) = k (2Lω k ) −1 e ikx−iω k t . Therefore we find 13 ,ĉ where Φ m (E) is the m-particle phase space 12 The derivation of the coefficientsĉ 2n (E) in Eq. (61) applies to any φ α theory. 13 Eq. (64) can also be derived from the optical theorem, with careful treatment of the symmetry factors.
Finally, the coefficients in Eq. (57) are obtained by including only the contributions from poles located at It would be interesting to see if in general, higher ∆H n+ corrections can also be written in terms of phase space functions. In the rest of the section we explain useful approximations to evaluate Eqs. (66)-(68).

Continuum and high energy limit of the phase space
We start by approximating the phase space by its continuum limit. 14 Recall that in the continuum limit the relativistic phase-space for n-particles is given by where P µ = (E, 0) and k µ i = (ω k i , k i ). Then, for the 2-body phase space one has Next, solving for the Dirac delta's in Eq. (69), the 3-body phase-space is given by with E ≥ 3m. This integral can be solved by standard Elliptic integral transformations and we obtain, is an elliptic integral.
In general though, finding the exact phase space functions Φ n (E) is difficult but can be simplified in the limit E m. In our case, this limit is justified because the phase space functions are evaluated for E ≥ E L m. Notice that to take the high energy limit of Φ n (E) one can not expand the integrand of Eq. (69) because, after solving for the Dirac delta's constraints, it is of O(1) at the integral limits, see for instance the elliptic integral in Eq. (71). Instead, we use the following relation for the phase space where D E (x, τ ) is the euclidean propagator and Φ n (E) is only non vanishing for E ≥ n m. The Euclidean propagator in d = 2 is given by the special Bessel function of second kind K 0 (mρ) with ρ = √ x 2 + τ 2 and I n (τ ) ≡ ∞ −∞ dx K n 0 (mρ)(2π) −n . At this point we can use a clever trick done in Ref. [1] to find the leading terms of the inverse Laplace transform of I n (τ ) in the limit E → ∞. Since the phase space Φ n (E) is the inverse Laplace transform of I n (τ ), the leading parts of Φ n (E) as E → ∞ come from the non-analytic parts of I n (τ ) as τ → 0. To find the non-analytics parts of I n (τ ) first one notices that where γ is the Euler constant. Then, the contributions to I n (τ ) = ∞ −∞ dx K n 0 (mρ)(2π) −n when τ → 0 are dominated by the region where ρ 1/m and the integrand can be approximated by K 0 (mρ) ≈ − log e γ mρ 2 . 15 This approximation introduces spurious IR divergences in the region of integration ρ 1/m where the approximation of the integrand is not valid. These divergences can be regulated with a cutoff Λ or, equivalently, one can take derivatives with respect to the external coordinate τ to regulate the integral I n (τ ). 16 Hence, approximating K 0 (mρ) ≈ − log e γ mρ 2 and integrating over x one can find the non-analytic terms of ∂ τ I n (τ ) as τ → 0. For instance, for n = 4 ∂ τ I 4 (τ ) = 1 4π 3 log(mτ e γ ) log(mτ ) log(mτ e 2γ ) + γ 2 + where the constant does not depend on τ . Lastly from Eq. (73), ∂ τ I n (τ ) is related to the phase space Φ n (E) by the Laplace transform, so that for n = 4 one has 15 This method is like the method of regions which is used to get the leading terms of multi-loop Feynman diagrams in certain kinematical limits or mass hierarchies. 16 This is similar to the fact that the UV divergences of multi-loop Feynman diagrams are polynomial in the external momenta because taking enough derivatives with respect to the external momenta the integrals are UV finite. Therefore using Eq. (77) and expanding Eqs. (70), (72) at large E, where the error made in the approximations is of the order O (m 2 /E 2 L ). We end this section by noticing that the leading terms of the phase space functions Φ 2 (E) and Φ 3 (E) in the large E expansion agree with the corresponding result of Ref.

Spectrum and convergence
Before starting with the numerical results we first discuss the series ∆H = n=2 ∆H n in more detail. The truncation of the ∆H series in powers of (V hh /H 0 hh ) n is only justified for V hh /H 0 hh < 1. Notice that even for weak coupling g 1 the series does not seem to converge. Let us consider a particular matrix entry where all the terms in the sums have a definite sign depending on whether n is even or odd. For instance, consider a contribution to Eq. (81) from states of high occupation number but low momentum like where | N k N −k is a Fock state with N particles of momentum k and −k that satisfy 2N ω k > E T . The term of Eq. (82) gives a non-perturbative contribution even for small g for high enough N and becomes worse for smaller momentum |k|. Thus the series (∆H) rs = (∆H n ) rs seems to be non-convergent but we will assume that (when the expansion parameter is small) the first terms of the series are a good approximation to (∆H) rs . Notice that the appearance of the non-perturbative contributions (like in Eq. (82)) can be worse for those matrix entries (∆H) rs with energies E r,s closer to E T because the intermediate states in V jj can have lower momentum and high occupation number for a given ∆H n .
For the first terms of the expansion (V hh /E h ) n , a naive estimate of the dimensionless expansion parameter is α rs ∼ g/E T × 1/(Lµ 2 rs ) where the g and L can be read off from the potential; the E −1 T arises because the sums in Eq. (81) are dominated by the first terms, starting at 1/E T (for E << E T ); and by direct inspection of the potential m/N µ rs E T where N is a possibly large occupation number, depending on the matrix entry.
It can happen that entries with energies E r , E s close to E T do not have a perturbative (∆H) rs = (∆H n ) rs expansion and even including the first terms of the series is a worse approximation than setting (∆H) rs → 0; these entries can induce big errors on the computed eigenvalues. Since the eigenvalues we are interested in computing are mostly affected by the lower E r,s -energy matrix entries we will neglect the renormalization of the higher E r,s energy entries where the series (∆H) rs = (∆H n ) rs is not perturbative. One way to select those entries would be to keep only those that satisfy α rs ∼ (∆H 3 ) rs /(∆H 2 ) rs < 1. However, this can be computationally expensive and instead we take a more pragmatic approach and only renormalize those matrix entries (H T ) rs with either E r or E s below some conservative cutoff E W , below which the series is perturbative.
Up until this point the discussion has been done for g 1. However, for those matrix entries where α rs is a perturbative expansion parameter one can increase g to strong coupling 17 by increasing E T at the same time. Increasing E T means enlarging the size of H T and ∆H, and it can happen that the new matrix entries do not have a perturbative (∆H) rs = (∆H n ) rs expansion. As explained above, in those cases we set (∆H) rs to zero. 18

Numerical results
In the rest of the section we perform a numerical study of the spectrum of the φ 4 theory. First we summarize the concrete implementation of the method. We find the spectrum of H by diagonalizing H ef f = H T + ∆H 2 (E T ) where E T is the eigenvalue of H T . 19 As explained in Sec. 5.1, to calculate ∆H 2 we separate it in ∆H 2+ and ∆H 2− defined in Eqs. (52)-(53) and take E L = 3E T . 20 . We found little differences when iterating the diagonalization with E. We also find that increasing E L does not have a significant effect on the result. For ∆H 2− (E, E T , E L ) we use the expressions in Eqs. (C.1)-(C.5) and for ∆H 2+ (E, E L ) we use the ones in Eqs. (78)-(80). We do a conservative estimate of the expansion parameter α rs and set to zero (∆H 2 ) rs for all those entries that are not perturbative.
First we study the lowest eigenvalues of H at weak coupling, where we can compare with standard perturbation theory. The perturbative corrections to the vacuum and the mass are 17 In the φ 4 theory the strong coupling can be estimated to be g 1, see Eqs. (83) and (84). 18 For the φ 2 perturbation studied in Sec. 4 we find that the error in the computed eigenvalues can be decreased by increasing E T even without introducing E W . For the φ 4 we find that E W must be introduced. 19 The  given by [1]: whereḡ ≡ g/m and m ph is the physical mass. In Fig. 5 we show the result for the vacuum energy and m ph . As explained before, only those entries with E r,s energies below a cutoff E W are renormalized. We do the plot for different values of E W = E T /2, E T /5 and we find that the vacuum energy and the physical mass do not depend much on this cutoff. For the left plot the difference between E W = E T /2 and E W = E T /5 is inappreciable. 21 We find that the spectrum is much flatter as a function of E T for renormalized eigenvalues than the ones computed with H T . Since the exact spectrum is independent of the truncation energy E T , a flatter curve in E T indicates a closer value to exact energy levels. However, it could still happen that adding ∆H 3 corrections shifted the spectrum by a small amount, as it happens for the φ 2 perturbation seen in Figs. 3 and 4 for the range 16 E T ≤ 20. In the plots we have superimposed constant dashed black lines that are obtained from the perturbative calculations in Eq. (83) and Eq. (84). We find that the eigenvalues computed with ∆H 2 are much closer to the perturbative calculation than the ones done with H T . The difference between the perturbative result and the one from E V V is of O(10 −4 ) and can be attributed to   higher order corrections in the perturbative expansion. Another source of uncertainty comes from higher order ∆H n corrections not included.
In Fig. 6 we show plots with different energy levels as a function of the truncation energy E T for g = 1, 2, 3. To compare with previous work, these plots have been done with the same choice of parameters and normalizations as in Figs. (9)-(10) of Ref. [1]. In all the plots the dotted lines are computed using the truncated Hamiltonian while the solid and dashed lines are computed using ∆H 2 with E W = E T /2 and E W = E T /3, respectively. The diamonds and the circles correspond to states in the Z 2 -even and Z 2 -odd sectors of the theory. We find that in all the plots, for high enough values of E T , the solid lines for the ∆H 2 are flatter than the truncated ones. The difference between the dotted and dashed lines is bigger for the plot for g = 3 than the one for g = 1. This can be understood because one expects more overlap from higher H 0 excited states with the vacuum for higher coupling. The difference between the solid and dashed lines becomes smaller as E T is increased. This can be understood because as E T is increased bigger parts of (H T ) rs are being renormalized, and eventually the difference between using E W = E T /2 and E T /3 becomes negligible. An intrinsic error of our calculation of the eigenvalues is the difference between the values obtained for different choices of E W . This error could be reduced with a more careful estimate of the expansion parameter α r,s , which would be very interesting for the future development of the method. In fact, it seems that for E T 12 (14) for g = 2(3) the cutoff E W is too high (and might include non-perturbative corrections like the one in Eq. (82)) as the eigenvalues deviate a lot from the computation done with H T . Another small source of uncertainty in our calculation comes from not having included higher order ∆H n corrections; in the next section we explain the calculation of ∆H 3 .
In Fig. 7 we show two plots of the vacuum and first excited states as a function of the coupling constant g for E T = 18 (cf. Fig. 4 of Ref. [1]). There is an intrinsic uncertainty in our procedure in the choice of E W , and as we discussed above it could be lowered by increasing the size of the truncation E T or ideally by refining the determination of E W . Notice that the renormalization of the truncated Hamiltonian matters as the solid lines have a significant difference with respect to the truncated (as seen in Fig. 6 the solid lines show a better convergence as a function of E T ). For g 3.5 the first Z 2 -odd excited state seems to become degenerate with the vacuum which is a signal of the spontaneous breaking of the Z 2 symmetry. This plot can be used to determine the critical coupling, see Ref. [1].

Three point correction and further comments
As explained in the previous section we have performed the numerical study of the φ 4 theory without taking into account the three point correction ∆H 3 . This would be an interesting point for the future and therefore we give a small preview of the type of expressions one obtains when computing the three point correction. As done throughout the paper, to get the expression for ∆H 3 we start by first computing where T k = 3−k n=1 t n . Then we find ∆H 3 by keeping only those terms that have all poles at E > E T . Then, we see that the three point correction can be split into where the subindices denote the number of fields in each term. The correction ∆H 1 3 is given by .
(87) where the symmetry factor is defined in Eq. (39). The rest of the terms ∆H φ 2 3 , · · · , ∆H φ 12 3 can be computed in a similar fashion as explained in previous sections, but we do not present them here since we did not include them in the numerical analysis.
Another interesting thing to study in the future is the local expansion of ∆H 3 and higher orders in ∆H n . Here we present some of the terms for the ∆H 3 case. As done for ∆H 2 , when the local expansion applies the calculation is simplified. We use the diagrammatic representation explained in Appendix A for the expressions at O(t 0 , z 0 ) of the local renormalization. As an example the leading local coefficients that renormalize the operators V 2 , V 4 and V 6 are where for example, = s 131 g 3 (2L) 5 k,l,p i L 2 δ l+k,0 p 1 +p 2 +p 3 +k,0 For the renormalization of the quartic we get where for example, = s 220 g 3 (2L) 4 As final remark, notice that the expression in Eq. (91) is the square of the coefficient of V 4 (in ∆H φ 4 2+ ) up to a numerical factor (see Eq. (59)) It would be very interesting to investigate whether certain classes of diagrams in the ∆H + = n ∆H n+ expansion can be resumed. This would reduce the error in the computed spectrum and its dependence on the arbitrary truncation energy E T . For instance, it could be that the resummation comes only from the leading pieces of the different diagrams. 22 5.5 Summary of the method and comparison with Ref. [1] In this section we summarize our approach to the renormalized Hamiltonian truncation method and briefly comment on the main differences with Ref. [1].
The aim of the renormalized Hamiltonian truncation method is to find the lowest eigenvalues E of H. This is done by diagonalizing H ef f ≡ H T + ∆H, where H T is the truncated Hamiltonian and ∆H encodes the contributions from the H 0 eigenstates with E > E T . Computing ∆H is difficult but the problem is simplified if one expands ∆H in powers of V hh /H hh . One expects that the first terms of the series ∆H = n ∆H n are a good approximation to ∆H if the expansion parameter is small. These terms can be computed as explained in Sec. 2, by first finding ∆ H n and keeping only the contributions from the states with E > E T . Then, we notice that for some entries with E r , E s close to E T , the series (∆H) rs = n (∆H n ) rs is not perturbative (for the chosen parameters g, E T ). We deal with this problem by setting to zero all those entries with E r or E s > E W where E W is chosen appropriately, see Sec. 5.3.
In order to speed up the numerics and gain analytic insight, we perform several approximations to the exact expression of ∆H 2 . First we introduce a scale E L so that ∆H 2 = ∆H 2− + ∆H 2+ where ∆H 2+ only receives contributions of the states with E ≥ E L while ∆H 2− only receives contributions of states with E T < E < E L . The scale E L is chosen such that ∆H 2+ can be well approximated by the first terms of a local expansion. In our case, we only keep the leading terms ∆H 2+ = n=2 n=0 c 2n dx φ 2n (x, t) and we find that the coefficients c i can be written in terms of phase space functions. Lastly, the coefficients c i are approximated by taking the continuum limit and then expanding them in powers of m/E L . On the other hand ∆H 2− is kept exact because its numerical implementation is less costly and it does not admit an approximation by truncating a local expansion. The whole procedure has been described in Sec. 5 and used to do the plots of Sec. 5.3.

Comparison with Ref. [1]
Refs. [1,3] introduced a renormalized Hamiltonian truncation method by diagonalizing H ef f = H T + ∆H and expanding ∆H in a series. As explained, we have used this as our starting point. In Ref. [1] though an approximation to ∆H 2 is calculated using a different approach than in this paper. To get ∆H 2 , Ref. [1] starts by defining the following operator M (E) and then noticing that M (E) is related to the matrix element by a Laplace transform. In Ref. [1], the E → ∞ behavior of M (E) is found by doing the inverse Laplace transform of the non-analytic parts of C(τ ) in the limit τ → 0. This is done in the continuum limit, which is a good approximation. The obtained result for M (E) in this limit is taken to compute ∆H 2 . Ref. [1] differentiates two renormalization procedures, one where the term (E r + E s )/2 in Eq. (96) is approximated to zero (called local), and one where it is taken into account (called sub-leading). In the later case M (E) is given by M (E − E rs ), and therefore for entries with E rs ∼ E T taking the limit E − E rs m is not justified when E ∼ E T . The way in which this problem is dealt with is by neglecting all the contributions of M (E − E rs ) for E ≤ E rs + 5m; in other words, a θ(E − E rs − 5m) is multiplied to the integrand in Eq. (95). 23 With this, we can already find the main differences between the two approaches. In our case we calculate the exact expression of ∆H 2 which, if needed, can be approximated. Instead, Ref. [1] finds the contributions of ∆H 2 that are leading in the limit where E → ∞ (which neglects the tree and disconnected contributions). From our approach we can recover the local result of Ref. [1] if we set E L = E W = E T , neglect the tree and disconnected contributions, take the continuum limit, perform a local expansion to ∆H 2+ , and make an expansion in m/E 1. The choice E L = E T implies ∆H 2 = ∆H 2+ and ∆H 2− = 0, while E W = E T means that no entries (∆H 2 ) rs are set to zero. In a similar way we can recover the sub-leading result taking into account the E rs terms, while introducing by hand a θ(E − E rs − 5m) in the integrals of the coefficients.
Even though the two approaches are quite different, our method and their sub-leading renormalization can still give similar results due to the following. For large enough E T , the low entries of (∆H 2 ) rs only receive contributions from loop-generated operators 24 , and can be well approximated by a local (up to the E rs dependence) expansion even if E L = E T . On the other hand, for high energy entries of (∆H 2 ) rs the tree and disconnected operators are non-zero, and none of the operators can be approximated by a truncated local expansion if E L = E T . However, in many cases these high energy entries become non perturbative and we set them to zero when E r or E s > E W . Therefore we find that if E W is used, it can be a good approximation for large enough E T to neglect the tree and disconnected terms all together and set E L = E T while performing a local expansion. With this we connect with Ref. [1] where the scale E W is not used to get rid of the non-perturbative contributions. Instead the tree and disconnected terms are neglected, all the entries of (∆H 2 ) rs are approximated by the loop-generated local (up to E rs ) operators only and the θ(E − E rs − 5m) is introduced in Eq. (95). As explained, neglecting the tree and disconnected terms is justified, while the introduction of θ(E − E rs − 5m) and truncating the local expansion in practice largely reduce the values of the high energy entries with respect to the exact result. All of these effectively act as our scale E W . Therefore we see that in many cases our approach and the one in Ref. [1] can give similar results.
Even though the numerical results are similar, our approach introduces new tools and insights that we think improve the renormalized Hamiltonian truncation method and can help to develop it further.

Conclusion and outlook
In this paper we have developed further the Hamiltonian truncation method. In particular we have explained a way to compute the corrections to the truncated Hamiltonian at any order in the large E T expansion of ∆H = n ∆H n . We have applied these ideas to scalar field theory in two dimensions and studied the spectrum of the theory as a function of the truncation energy and the coupling constant.
There are various open directions that are very interesting and deserve further investigation. Firstly, it would be a great improvement to the method to find a more precise estimate of the expansion parameter of the series. This estimate should be easy to implement numerically and lead to a precise definition of the cutoff E W . In this work we have been pragmatic in this respect, and investigated the behaviour of the spectrum as this cutoff is modified. It might be that only removing the contribution of certain type of matrix elements (like the ones corresponding to high occupation number and zero momentum) the series is greatly improved.
We have not pushed the numerical aspects of the method very far and all the computations have been done with Mathematica. With more efficient programming languages it would be interesting to further study and check that as the truncation energy E T is increased the uncertainty in the precise choice of E W is reduced.
Another point that should be addressed is the dependence of the spectrum on L as higher ∆H n corrections are added; also it could be relevant to inspect if there are diagrams that dominate for large Lm 1. Another very interesting path to develop further is to apply renormalization group techniques to resum the fixed order calculations of ∆H. Since the exact eigenvalues do not depend on the truncation energy E T , it may be possible resum the calculation of ∆H n . Our analytic expressions for the ∆H n corrections permit a precise study of the possible resummation of the leading corrections at each order in the perturbation theory of the large E T expansion. One could start by studying the resummation of the leading local corrections, and for that the phase space formulation that we have introduced is useful as there are simple recursion relations for the differential phase space.
Another fascinating avenue to pursue is the applicability of the method to other theories with higher spin fields and to increase the number of dimensions. In this regard, we notice that the derivation of Eqs. (66)-(68) seems to be formally valid in any space-time dimension d. Recall that the c i 's are the coefficients of the local operators added to H T to take into account the effect of the highest energetic H 0 eigenstates not included in the light Hilbert space H l . As d is increased beyond d = 2 the UV divergencies appear due to the increasingly rapid growth of the phase space functions Φ i (E). One can then regulate the c i coefficients with a cutoff Λ. For instance, consider the coefficient c 4 of the φ 4 operator in d = 4. Then, requiring that the energy levels are independent of the regulator one finds the following β-function where the E corrections can be neglected in the limit of large Λ E. Redefining g ≡ λ/4! one recovers the known result for the λφ 4 theory β(λ) = 3 16π 2 λ 2 + O(λ 3 ) , where we have neglected the mass corrections that for Λ m decouple as Φ 2 (Λ) = 1/(8π) + O(m 2 /Λ 2 ). A possible way to make contact between the calculation in the renormalized Hamiltonian method and the standard calculation of the beta function is by noticing that the coefficient of the divergent part of the amplitude is proportional to the coefficient of its finite imaginary part which in turn (by the optical theorem) is proportional to the two-particle phase space. It would be very interesting to further study RG flows from the perspective of the renormalized Hamiltonian truncation method approach.
We think that the Hamiltonian truncation method is a very promising approach to study strong dynamics, and that there are still open important questions to be addressed.

A Diagramatic representation
There is a simple and powerful diagrammatic representation that permits to easily find the expression for ∆H n . This can be used to either compute the full operator ∆H n or the leading O(t 0 , z 0 ) coefficients in the local expansion of ∆H n+ defined in Sec. 4. This representation is valid for any φ α theory, but here we give examples only for the φ 4 case for concreteness.

Local coefficients
Imagine that we want to find the local coefficients O(t 0 , z 0 ) for ∆H φ 2 3+ . To find them one puts 3 vertices ordered horizontally 25 and draws all possible diagrams that have only 2 external lines, four lines meeting at each vertex and don't have any lines starting and ending at the same vertex. Next, we assign a momentum for each internal line and draw a vertical line between every pair of vertices. One such diagrams is k5 k1 k2 k3 k4 . (A.1) The expression corresponding to a given diagram with n vertices and N propagators is given by where k j = 2πn j /L with n j ∈ Z. Each of the n − 1 sets of momenta {s p } consist in the momenta of the internal lines that are cut by each vertical line. In (A.1) these would be s 1 = {k 1 , k 2 , k 5 } and s 2 = {k 3 , k 4 , k 5 }. The symbol δ p stands for a Kronecker delta that imposes that the total momentum crossing a cut is zero; s is a symmetry factor that counts all the ways that the lines of the vertices can be connected to form the diagram. Applying this recipe to the diagram in (A.1) one has k5 k1 k2 k3 k4 = s 221 4 g 3 k s L 2 δ k 3 +k 4 +k 5 ,0 where the symmetry factor s mnv p is given in Eq. (39). Another example of a contribution to ∆H φ 2

3+
would be = s 212 4 g 3 k s L 2 δ k 1 +k 2 +k 3 +k 4 ,0 Notice that to include the contributions E rs mentioned at the end of Sec. 5.2 the same diagrammatic representation applies but one must then substitute E → E − E rs in Eq. (A.2) making the coefficients depend on the matrix entry.

Exact ∆H n opertors
A similar diagrammatic representation can be used to calculate the exact ∆ H n operator from which one can easily get ∆H n . The prescription to follow is very similar to the one for the local case, where one starts drawing the same diagrams and putting vertical lines between every pair of vertices. The only difference is that now one extends the external lines to left and right in all possible combinations for each diagram drawn and also assigns a momentum to the external lines. For the diagram in (A.1) this means Lδ α A+B r=A+1 a † qr 2Lω qr A l=1 a q l 2Lω q l , (A.7) where the sums over k s, q s sum over all possible momenta for a given k i , q i . Then, each of the n − 1 sets of momenta {s p } consists in the momenta of the lines that are cut by each vertical line. For the first diagram from the left in (A.7) these would be s 1 = {k 1 , k 2 , k 5 } and s 2 = {k 3 , k 4 , k 5 }, and for the second one s 1 = {q 1 , k 1 , k 2 , k 5 } and s 2 = {q 1 , k 3 , k 4 , k 5 }. The symbol δ α stands for a Kronecker delta that imposes momentum conservation at each vertex α. The symbol ω rs depends on the energy of the states E r |, | E s on which a and a † act i.e. it is different for each entry a † −qr a q l rs , and is given by w rs ≡ E rs − 1 2 A+B i=1 ω q i where E rs = (E r + E s )/2. As before s is a symmetry factor that counts all the ways that the lines of the vertices can be connected to form the diagram. Lastly κ counts all the equivalent ways that the external lines coming out from the same vertex can be ordered left and right, for the diagrams in (A.7) is is always one, since there is only one external line per vertex. Applying this recipe to the first and second diagrams in (A.7) one has k5 k1 k2 k3 k4 q1 q2 = s 221 4 g 3 k 1 ,··· ,k 5 q 1 ,q 2 L 2 δ k 3 +k 4 +k 5 ,q 2 k 1 +k 2 +k 5 ,q 1 √ ω q 1 ω q 2 , (A.8) k5 k1 k2 k3 k4 q1 q2 = s 221 4 g 3 k 1 ,··· ,k 5 q 1 ,q 2 L 2 δ k 3 +k 4 +k 5 +q 1 ,0 k 1 +k 2 +k 5 +q 1 ,0 5 i=1 (2Lω k i ) θ(ω rs + ω k 1 + ω k 2 + ω k 5 + ω q 1 − E L ) E − ω rs − ω k 1 − ω k 2 − ω k 5 − ω q 1 θ(ω rs + ω k 3 + ω k 4 + ω k 5 + ω q 1 − E L ) E − ω rs − ω k 3 − ω k 4 − ω k 5 − ω q 1 Lδ q 1 +q 2 ,0 a † −q 1 a q 2 2L √ ω q 1 ω q 2 , (A. 9) where ω rs = E rs − (ω q 1 + ω q 2 )/2 and the symmetry factor s mnv p is given in Eq. (39). With this set of rules one can easily get the expression for ∆ H 2 and ∆ H 3 for the φ 2 and φ 4 theories. Then one finds ∆H 2 and ∆H 3 by keeping only the contributions with all poles E > E T .
B ∆H for the φ 2 perturbation B.1 Two-point correction In this section we give the full expressions of the ∆ H 2 corrections for the scalar theory with potential V = g 2 dxφ 2 . Recall that the symmetry factor is given by s p = 2 p 2 p!. We will where G 0 = δ k 1 +k 2 ,0 δ k 1 +k 3 ,0 [f 0 ] 12  G 6 = a † q 1 a † q 2 a q 3 a q 4 a q 5 a q 6 δ 2 δ q 1 +q 2 ,0 δ q 3 +q 4 ,0 [f 6 ] 123456 [f 6 ] 1234 + h.c. + 2a † q 1 a † q 2 a † q 3 a q 4 a q 5 a q 6 δ 3 δ q 1 +q 2 ,0 δ q 5 +q 6 ,0 [f 6 ] 12356 [f 6 ] 12456 . (B.13) We have defined w p rs ≡ E rs − 1 j=d+1 q j (the Kronecker delta that imposes momentum conservation to the creation/annihilation operators) and (B.14) C ∆H for the φ 4 theory In this appendix we give the exact two-point correction and the first terms in the local expansion of the three-point correction. Getting the exact three-point correction would be straightforward.