A Construction of an Effective Hamiltonian from Feynman Diagrams: Application to the Light-Front Yukawa Model

Progress of Theoretical Physics, Jul 2001

We study a similarity transformation to construct an effective Hamiltonian systematically, which does not contain particle-number-changing interactions, by means of the Fukuda-Sawada-Taketani-Okubo method. We show that such a Hamiltonian can be constructed from Feynman diagrams and give rules for constructing it in the light-front Yukawa model. We prove that it is renormalized by the familiar covariant perturbative renormalization procedure. It is very advantageous that the effective Hamiltonian can be obtained from our rules immediately. We also numerically diagonalize it to second order in the coupling constant as an exercise.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://ptp.oxfordjournals.org/content/106/1/99.full.pdf

A Construction of an Effective Hamiltonian from Feynman Diagrams: Application to the Light-Front Yukawa Model

Yuki Yamamoto 0 1 0 Department of Physics, Kyushu University , Fukuoka 812-8581, Japan 1 A Construction of an Effective Hamiltonian from Feynman Diagrams We study a similarity transformation to construct an effective Hamiltonian systematically, which does not contain particle-number-changing interactions, by means of the FukudaSawada-Taketani-Okubo method. We show that such a Hamiltonian can be constructed from Feynman diagrams and give rules for constructing it in the light-front Yukawa model. We prove that it is renormalized by the familiar covariant perturbative renormalization procedure. It is very advantageous that the effective Hamiltonian can be obtained from our rules immediately. We also numerically diagonalize it to second order in the coupling constant as an exercise. - transformations. One is a transformation in momentum space, which is equivalent to integrating out states that exchange energies greater than some energy cutoff, proposed by G:lazek and Wilson, 3) and independently by Wegner. 4) It is interesting that this transformation gives the nonperturbative low energy physics and a logarithmic confining potential in LF QCD, 5) although it is difficult to obtain the effective Hamiltonian, even at lowest order in the coupling constant. The details of this method and applications are discussed in Ref. 6), and recent progress is reported in Ref. 7). The second similarity transformation is a transformation in the particle number space, which reduces the Hamiltonian to one which has no particle-number-changing interactions, so that the transformed one can be solved easily. This transformation was proposed by Harada and Okazaki. 8) It can avoid the problem of sector-dependent counterterms, because its origin is the general property of the Hamiltonian that it has particle-number-changing interactions and its eigenstates need an infinite number of particles. However, applying it in actual calculations is complicated and tedious. Although their method was considered in the LF field theory, it is not new in the ET context and has been used for obtaining the TMO potential of nuclei. 9) We call it the Fukuda-Sawada-Taketani-Okubo (FSTO) method. 10), 11) This method provide an easier method for constructing the effective Hamiltonian and seems to be promising in the LF framework. However, it lacks manifest Lorentz covariance, and therefore it is difficult to tell what sort of divergences the effective Hamiltonian has before doing actual calculations. This makes the renormalization procedure more complicated than that in usual covariant perturbation theory. It is important to determine how the divergences emerge in the FSTO framework. The purpose of this paper is to show that the effective Hamiltonian constructed with the FSTO method can be immediately obtained from Feynman diagrams, as the S-matrix element in covariant perturbation theory, and that one can use the usual renormalization procedure to renormalize it. This makes the construction of the effective Hamiltonian systematic and simpler. In particular, it allows us to perform higher-order calculations. 12) Our strategy is the following. First, we show that the FSTO effective Hamiltonian is a sum of the auxiliary operator G and its products to fourth order in the interaction part of the Hamiltonian. The advantages of using G are that it is constructed from the same Feynman diagrams as the S-matrix elements, and that it has no particle-number-changing interaction, so that one can easily calculate the normal ordering of their products. We give the rules for constructing G from Feynman diagrams. They are slightly different from the familiar Feynman rules in the covariant perturbation theory. This set of rules is one of the main results of the present paper. Our method is more convenient and powerful for constructing the effective Hamiltonian and investigating its renormalization than other similarity methods. Then, we find that there are three types of UV divergence. One is the familiar loop divergence, which can be renormalized by the usual renormalization procedure. The second comes from the difference between our construction rules and those of Feynman. We can make this second divergence harmless by using an ambiguity. These two divergences emerge only in G. The last divergence emerges from the products of the renormalized operators G. We show that this divergence works as a box counterterm, 13) which is needed to cancel the cutoff dependence of the eigenvalue, in diagonalizing the effective Hamiltonian. This paper is organized as follows. In 2, we briefly review the FSTO method. In 3, we show that the effective Hamiltonian constructed from the FSTO method can be written in terms of the auxiliary operators F or G. (F is written in terms of G.) We consider the LF Yukawa model, 13) whose Lagrangian and Hamiltonian are given in Appendix A, as a concrete example, and give the rules for the construction of G from Feynman diagrams and the renormalization of it and the effective Hamiltonian. In 4, we summarize and discuss the validity of our method. In Appendix B, as an exercise, we calculate the eigenvalue of the ground states of the effective Hamiltonian to second order in the coupling constant in the LF Yukawa model. In Appendix C, we explain an ambiguity of the energy integrations. 2. Review of the FSTO method This section briefly reviews the FSTO method, partly following Ref. 11). The FSTO method is to reduce a Hamiltonian to block-diagonal form using a similarity transformation. We can obtain the effective Hamiltonian for the subspace of the Fock space without loss of necessary information. We wish to solve the Schrodinger equation for the second-quantized Hamiltonian H, which consists of the free part H0 and the interaction part H : The eigenstate | can be expanded in terms of the complete set of eigenstates of H0. To divide the Fock space into two, we introduce a projection operator that commutes with H0: H = H0 + H . where both |1 and |2 are states written in terms of the eigenstates of H0. |1 is a state in the target Fock space, which is at our disposal. For our purpose, we choose to be an operator that selects states with definite particle number. In matrix notation, and | are written as and we express an arbitrary operator O as O = = U |1 is completely decoupled from |2 , so that we can concentrate on only the equation for Heff and |1 in the subspace selected by : We assume a form for the similarity (unitary) transformation operator as U = It is convenient to use J , (1 + AA)1/2 A(1 + AA)1/2 A(1 + AA)1/2 (1 + AA)1/2 J = instead of A. In order for the off-diagonal parts in Eq. (2.6) to become zero, J must satisfy (1 ) H J + [H0, J ] J H J where the brackets are defined by O O + (1 )O(1 ) = Using Eqs. (2.6), (2.9), and (2.11), we formally obtain the effective Hamiltonian, Heff = J J 1/2J HJ J J 1/2, = (1 ) H (t)J (t) J (t)H (t)J (t) , where J (0) is identified with J . H (t) and J (t) are defined in this picture as H (t) eiH0tH eiH0tet , J (t) eiH0tJ eiH0tet , where et is an adiabatic factor with 0+. We hereafter omit it in order to keep equations simple. Provided that we set the initial condition as J () = 1, J can be found order by order. Alternatively, if we solve = H (t)V (t) V (t)H (t)V (t), under the initial condition J is given by V () = 1, It is easily found that the solution of Eq. (2.17) is given by V (t) = U (t)U (t) 1 where U (t) is the usual time evolution operator, U111(t) and U221(t) are the inverse operators of U11(t) and U22(t), respectively. J is explicitly obtained from Eqs. (2.19) and (2.20), and then we expand it in terms of increasing order in H to obtain the effective Hamiltonian perturbatively: J = U21(t)U111(t) U (t) = T exp i U21(t)U111(t) = 1 + where n indicates the order of H . The resultant effective Hamiltonian becomes Heff = H0 + H + H J1 + H J2 1 +H J3 + 4 [J1J1, H J1] + h.c. + O(H5 ), to fourth order. It is known that this can produce the TMO potential in the symmetrical pseudoscalar pion theory with pseudovector coupling, which describes the properties of the deuteron very well. Although Eq. (2.23) is general and can be calculated very straightforwardly, the calculation is absurdly tedious, because J is not a time-ordered operator but a product of such operators, and possesses particle-number-changing interactions. In order to renormalize the theory perturbatively, we need to calculate the counterterms, but it is not clear what sort of the divergences emerge in the effective Hamiltonian before doing actual calculations. Neither is it obvious whether it is renormalizable even in the renormalizable theory. 3. Construction and renormalization of the effective Hamiltonian In this section, we describe the construction method of the FSTO effective Hamiltonian from Feynman diagrams and discuss its renormalization. First, we introduce the auxiliary operators F and G for convenience, and then show that the effective Hamiltonian can be written in terms of them. G can be constructed from the same Feynman diagrams as those for S-matrix elements in the covariant perturbation theory. We give rules for its construction and show that it is renormalized by the usual renormalization procedure. Lastly, we show that the effective Hamiltonian has divergent terms even if G is renormalized. We discuss their role in diagonalizing the effective Hamiltonian. 3.1. Definitions of F and G Considering the first few terms in Eq. (2.23), we define the operator We will show that the effective Hamiltonian can be rewritten in terms of H0 and F . By using Eqs. (2.19) and (2.20), we write J as = (1 ) + U (0)U (0) 1. J = U21(0)U111(0) Therefore, F becomes where we define the operator G as H (0) exp i dt H (t) H (0) + (i)n tn1 As we will shortly see, introducing G is crucial for using the the diagrammatic rules. An advantage of using G is that it is diagonal in the particle number space, so that we can normal-order products of Gs more easily than those of J s and H . Note that G is similar to the S-matrix operator, but there is one important difference: the upper limit of the time integration is 0. We find that U (t) can be written in terms of G by integrating t from to t after sandwiching G between eiH0t and eiH0t : i = i dt G(t ) i dt H (t ) + = F2(t) + F2(t), = {H (t) + H (t)J1(t)} + {H (t) + J1(t)H (t)} (3.10) (i)n+1 tn1 (i)n+1 Next, we solve the differential equation which is obtained from Eq. (2.14). It is easily found that J1J1 = i dt {F2(t) F2(t)}, By substituting U (0) in Eq. (3.5) into Eq. (3.3), we can write F as It is important that F depends only on G. Here, we expand F and G in terms of increasing order in H : Then, it is convenient to define G(t) = eiH0tGeiH0t. = G F = G = Fn = Gn = 1 [J1J1, H J1] + h.c. = [J1J1, H + H J1] + [H + J1H , J1J1] 0 = i dt [F2(t) F2(t), F2] + h.c. dt [F2(t) F2(t), F2] + h.c. + O(H5 ). (3.13) All interactions are written in terms of Gn through Fn to fourth order in H . To summarize, the effective Hamiltonian can be easily constructed once Gn is obtained. 3.2. Rules for the construction of G In this subsection, we give rules for the construction of G from Feynman diagrams. We do not need to use the old-fashioned perturbation theory. Knowledge gained from covariant perturbation theory helps us to find them. From Eq. (3.5), we immediately find that where S is the familiar S-matrix operator. Comparing order by order, we can relate the nth order term of S to Gn as dt Gn(t). Gn = (i)n1 (n 1)! dtn1 T(H (0)H (t1) H (tn1)). (3.17) It is important to make the difference between Sn and Gn clearer. For this purpose, it is useful to rewrite them in terms of the T-product and compare them order by order to find the correspondence with Feynman diagrams: Wicks theorem tells us that T-products of the H (t) can be written as sums of normal-ordered products of creation and annihilation operators with amplitudes, which are given by the same Feynman diagrams, acting as the coefficients. The difference between Sn and Gn arises from the time integrations. It is apparent that the role of dt is to give the energy-conserving delta function 2(Eout Ein) to each vertex in the Feynman diagrams, where Eout is the outgoing energy from the vertex and Ein is the incoming energy to it. Because the time integrations end at 0 in Gn, the energy denominators (i)(Eout Ein i )1 appear instead of the delta functions. The sign of the infinitesimal constant must be taken to be positive to ensure the convergence of the integrations at t = . The n factors of H (t) become equivalent and cancel the factor 1/n! in Sn due to n time integrations, while there are (n 1) time integrations and the factor 1/(n 1)! is canceled in Gn. However, there is a H (0) which is not integrated, and therefore there are n terms for each vertex which have (n 1) products of the energy denominators. Let us consider the LF Yukawa model) as a concrete example. In this model, the interaction part of the Hamiltonian has the form H = gH(1) + g2H(2) . The effect of H(2) , which is the instantaneous interaction, is absorbed into the fermion propagator, and we hereafter omit it, because we assume that LF diagrams are equivalent to the covariant one.) We define as the projection operator for the two-body state that consists of a fermion and an anti-fermion. Because the power of g is equal to the number of the scalar fields and ()n = 0 is satisfied for odd m, the relation [H (x1+)H (x2+) H (xn+)](m) = 0, F2 = G2 + O(g4), F4 = G4 + iG2 dx+G2(x+) + O(g6). Also, Gn can be written as where we define Gn = Mn b(p1)b(p2)(2)33(p1 p2) Mn d(p1)d(p2)(2)33(p1 p2) (2)33(p1 + l1 p2 l2), p 3(p q) (p+ q+)2(p q). ) The Lagrangian and Hamiltonian in this model are given in Appendix A. ) This point is explained in the last paragraph of Appendix A. We hereafter call p the energy and (p+, p) the three-momentum. The creation and annihilation operators of the fermion, b(p) and b(p), and those of the antifermion, d(p) and d(p), satisfy the anti-commutation relations i i i i is assigned to each vertex, where pout (pin) is the outgoing (incoming) fourmomentum from (to) the vertex. The energy denominator corresponds to the energy-conserving delta function in the Feynman rules. An independent energy p is assigned for each propagator and integrate it. We do not impose the conservation of the energies. A total energy difference (p1 + l1 p2 l) is assigned as an overall factor. 2 It is obvious that the first two differences come from the domain of the time integrations. The origin of the last one is that there are n terms that have different products of the energy denominators, as mentioned above. For example, in the case of the diagram Fig. 1(a), we consider the energy denominators obtained from Eq. (3.17) with factors of i: (i)3i4 i l1 + k2 q1 i q2 k2 l2 i k1 q2 p2 p1 + q1 k1 q2 k2 l2 k1 q2 p2 p1 + q1 k1 l1 + k2 q1 k1 q2 p2 p1 + q1 k1 l1 + k2 q1 q2 k2 l2 i i i i i i Finally, Vn becomes Vn = (p1 + l1 p2 l2) (iAn), i p1 k1 q1 i l1 k2 + q2 i i dk1 dk2 dq1 dq2 vertices (ig5). In general, the total factor of i is (i)V 1iV , where V is the number of vertices in the diagram. Since the sum in Eq. (3.27) is the combination of removing one from four denominators, we can make it a product of all the energy denominators: = (p1 + l1 p2 l2) i p1 + q1 k1 i q2 k2 l2 i k1 q2 p2 l1 + k2 q1 i This factor corresponds to the elimination of the total energy-conserving delta function. where An is the usual amplitude constructed from the Feynman diagrams, but each energy of the propagators is an independent variable, which is integrated outside of An. Therefore, An is constructed from the propagators SF (k) and F (q) with each independent energy, the vertex g5, the external fermion line u(p1, 1), u(p2, 2), v(l1, 1), v(l2, 2), and an integration of the four-momentum for each loop. The three-momenta are conserved, but energies are not. Although each propagator includes both the loop and independent energy, it is easy to split them. For example, in the box diagram Fig. 1(a), the corresponding energy denominators are i p1 + q1 k1 i i l1 + k2 q1 i i q2 k2 l2 i i k1 q2 p2 i If we consider q2 as the loop momentum, by shifting the momentum as k1 k1 + q2, q1 q1 + q2 and k2 k2 + q2, the denominators become i p1 + q1 k1 i i k1 p2 i i l1 + k2 q1 i i k2 l2 i Fig. 1. An example of the one-loop box diagrams. In (a), energies are assigned to each propagator individually. In (b), energies are assigned to each propagator but the loop four-momentum q2 is specified. All the three-momenta are conserved in both cases. A continuous line corresponds to a fermion propagator, and a broken line correspond to a scalar propagator. whose energies are assigned in Fig. 1(b). Because the loop momenta are not restricted by the usual conservation law of four-momenta in the covariant perturbation theory, it is apparent that the energy denominators do not depend on them. 3.3. Renormalization of G As mentioned in 3.2, An corresponds to the usual amplitude. What is different from the amplitude is that the energies of the propagators in An are independent of each other. Even if the energies are not conserved, we can renormalize An with the usual prescription in the covariant perturbation theory, because UV divergences from the integrations of the loop momenta are local. They emerge as the coefficients of the polynomial of the other momenta. If the zeroth-order term in these expansions includes a divergence, it is renormalized by shifting masses or a coupling constant. The other divergences are logarithmic and depend on momenta. They must be renormalized by the field renormalization. After renormalizing An, we must consider the energy integrations. It is not clear whether the energy integrations are finite, because the energies are not conserved. First, we show that a new divergence arises when we integrate the energies of the renormalized An with the energy denominators. We discuss the field renormalization and show that an ambiguity which may cancel the above-mentioned divergence comes from the energy integrations. 3.3.1. Divergences in G and the three-momentum conservation law for the fermion line, we find that the one-particle states conserve the energies: p1+ = p2+, p1 = p2 p1 = p12 + m2 p22 + m2 = p2. The same conservation law is applied to the anti-fermion line. It is obvious from Eq. (3.14) that G multiplied by the total energy-conserving delta function is S. Therefore the one-particle states of G are equivalent to those of S. If we renormalize the self-energy part under the physical renormalization condition, Mn and Mn in Eq. (3.22) vanish. An example of order g2 is considered in Appendix B. In multi-particle states, even though the external lines satisfy the on-shell p1 p2 condition and the three-momentum conservation law, one cannot say that the total energy is conserved. We will show l1 l2 that such divergences disappear even in the multi-particle states, except for the case that the outgoing external fermion Fig. 2. A diagram of the one-particle state. line has a self-energy part. The blob is a connected diagram. First, we consider the one-particle irreducible part (k1, k2, ) of the renormalized An, where k1, k2, are the fourmomenta of the propagators, except for the loop momentum, and satisfy the threemomentum conservation law. Since (k1, k2, ) is covariant, the analysis for large four-momentum ki , which is used in the operator product expansion, is valid when we investigate the asymptotic behavior for large ki . 14) Therefore, after realizing that the loop momenta associated with the ki are as large as ki , we regard ki as larger than ki+ and ki. By power counting, we find that the leading contribution comes from the self-energy part. For the scalar self-energy part, the leading behavior of (k1, k2, ) is (k1, k2, )scalar self-energy cij(ki+ kj + ki kj+ ), and for the fermion self-energy, (k1, k2, )fermion self-energy where cij and ci may depend on logarithmic factor of ki. It is important that both are proportional to ki . If we multiply (k1, k2, ) by the energy denominators and Fig. 3. (a) includes a scalar self-energy in an internal scalar line. (b) (d) include a fermion selfenergy part in an internal, incoming-external and outgoing-external fermion lines, respectively. In all cases, the three-momenta are conserved. The blobs are some one-particle irreducible graphs. integrate over the energies, it seems that the energy integration is logarithmically divergent. However, such a term cannot exist when the energies are conserved; such a divergence is not possible in S, so it must be proportional to the total energy difference. For the scalar self-energy in the internal line, we regard the blob in Fig. 3(a) as (k1, k2, )scalar self-energy. The contribution from it is dk2 i 2 q2 k1 k2 i (k1, k2, )scalar self-energy i q2 q1 i {i(q1, q2) + iq+(q2 q1) log(q+ ) +(q2 q1)(finite terms)}, where (q1, q2) becomes the usual scalar self-energy if we replace the energy denominator with the energy delta function, and is the UV cutoff of the energy. The second and third terms are proportional to (q2 q1) for the reason stated above. q+ and q+ are proper longitudinal momenta. The divergence only arises in the second term, because Eq. (3.38) is regarded as a Taylor expansion in (q2 q1). The first term is the ordinary renormalized finite term. Although the second term is divergent, it is found that it is logarithmically divergent at most by Lorentz covariance and power counting. The third term converges, because the second term is logarithmically divergent, and differentiation by the energy decreases the power of the divergence by 1. Although Eq. (3.38) includes the divergence, it vanishes when it is inserted in the internal scalar line. Let us consider scalar propagators and energy denominators in vertices around the blob in Fig. 3(a). It is crucial that the second term in Eq. (3.38) dq1 dq2 i 2 k4 k3 q2 i i F (q2) i But the three-momenta are conserved (q1+ = q2+, q1 = q2), and thus it is obvious that Eq. (3.39) vanishes. The fermion self-energy is logarithmically divergent, like Eq. (3.38). For the internal fermion line in Fig. 3(b), the contribution from the divergent part is dk2 dk3 i 2 k4 + q2 k3 i i k3 k2 i SF (k2) k2 k1 q1 i i where k+ is a proper longitudinal momentum. Unlike F (q), SF (k) has a term which does not depend on the energy but is proportional to +: SF (k) = Fortunately, + has the property that (+)2 = 0, so that the first term has no effect in Eq. (3.40). The second term behaves in the same way as in the scalar case, and hence Eq. (3.40) vanishes. Note that the above discussed mechanism only works for internal lines. The diagrams which have self-energy in the external fermion line may have the same divergence. We consider the incoming external fermion line as in Fig. 3(c). The contribution from the divergent part becomes dk1 2 q1 + k2 k1 i i k1 p2 i i i+(k1 p2) log(k+) u(p2, 2) = SF (q1 + k2)(p2+) log(k+)u(p2, 2), ) We have used dq i 2 F (q) (q k) i = dq i i 2 2q+(q q2q++ 2 + iq +) (q k) i where k+ = q+ and k = q. = u(p1, 1)SF (k2 + q1)(p1+) log(k+). Since the outgoing line satisfies p1+ > 0, this does not vanish and is logarithmically divergent. 3.3.2. Field renormalization Fig. 4(a) = dk i (1 Z3) F1(k) i k q1 i (1 Z3) cq1+ + i q2 q1 i F1(q1) , where c is an ambiguous constant that depends on the manner of taking the limit k . More generally, the contribution from n counterterms connected by the scalar propagators is where all the three-momenta are conserved. Since the incoming line has positive longitudinal momentum, p2+ > 0, this vanishes. In the case of the outgoing external fermion line like in Fig. 3(d), the contribution from the divergent part becomes i p1 k1 i Fig. 4(b) = dk1 i i i dk2 dkn dq3 dqn dq2 = (1 Z3)n sn(q1+)q1+ + i qn+1 q1 i F1(q1) , where all the three-momenta are conserved. sn(q+) is defined as sn(q+) = an(q+) + bn(q+), Fig. 4. Diagrams that contribute to the renormalization of the scalar field. Here, X represents a counterterm for the scalar field. (a) has a counterterm for the scalar field between scalar lines. q1 and q2 are four-momenta assigned to the external scalar lines, and k is the four-momentum for the counterterm. (b) has n counterterms connected by the scalar propagators. In (c), (b) is inserted between two vertices. (d) is an infinite sum of the diagrams (c). In all cases, the three-momenta are conserved. where an and bn are ambiguous constants, because there is an ambiguity in the order of the energy integrations.) If we insert Eq. (3.46) into an internal scalar line, the contribution becomes Fig. 4(c) = dq1 dqn+1 ) See Appendix C. (1 Z3)n sn(q1+)q1+ + F (q1) k4 k3 qn+1 i i i qn+1 q1 i F (qn+1) F1(q1) i i i dq (1 Z3)n F (q) (1 Z3)n F (q) i i q + k2 k1 i This does not depend on sn(q+) and is equivalent to a free propagator multiplied by the constant (1 Z3)n. The mechanism responsible for the vanishing of sn(q+) is the same as that responsible for the vanishing of the logarithmic divergence in the self-energy. We take the sum of diagrams as in Fig. 4(d) and obtain the result F (q) dq i Z3 2 k4 k3 q i F (q) (1 Z3)n Fig. 5(a) = dq1 i kn+1 qn i where the three-momenta are conserved. This property is the same as that of the usual field renormalization. Similarly, we can treat the fermion field. If we connect n counterterms for the fermion field with fermion propagators, the contribution becomes i = (1 Z2)n fn(k1+)+ + i kn+1 k1 i SF1(k1) , dqn i i where all the three-momenta are conserved and Z2 is the field renormalization constant for the fermion field. The function fn(k+) is defined as fn(k+) = cn(k+) + dn(k+), where cn and dn are ambiguous constants, due to a similar ambiguity. When we insert Fig. 5(a) into the internal fermion line, its contribution is given Fig. 5(b) = dk1 dkn+1 i SF (kn+1) SF1(k1) (1 Z2)n fn(k1+)+ + i kn+1 k1 i i dk i i 2 q2 + p2 k i (1 Z2)nSF (k) (1 Z2)nSF (k) i i k q1 p1 i which is a free propagator multiplied by the constant (1 Z2)n and the energy denominators. The infinite sum of the diagrams becomes dk Z2 2 q2 + p2 k i i i k q1 p1 i i q1 + p1 k1 i where k12 = m2 and k1+ > 0. Of course, all the three-momenta are conserved. In this case, fn(k+) does not affect the incoming external fermion line. The infinite sum becomes which is 1/Z2 times the tree external line. k is the momentum of the external fermion. i q1 + p1 k1 i For the outgoing external fermion line, the contribution is i kn+1 q1 p1 i = (1 Z2)n(cn + 1)u(kn+1, ) + where kn2+1 = m2 and kn+1 > 0. The infinite sum becomes (1 Z2)n(cn + 1) u(k, ) i i kn+1 q1 p1 i k q1 p1 i 3.3.3. Summary of the renormalization of G We have shown that UV divergences from the loop integrations are renormalized using the usual procedure in the covariant perturbation theory. After renormalizing An, we carry out the energy integrations. As a result, the new UV divergence and ambiguous constant cn arise only from the diagrams in which the fermion self-energy and its counterterm are inserted into the outgoing external line. We fix cn so that it can remove this divergence. This is always possible, because both arise in the outgoing lines and are closely related. Although this is not the only way to fix cn, we believe that it is natural for these to cancel. 3.4. Divergences in Heff In 3.3, we showed that G is not divergent after renormalizing it by the usual prescription in the usual perturbation theory. In this subsection, we show that Heff is divergent even if G is finite. From Eqs. (3.13), (3.20), and (3.21), the effective Hamiltonian can be written i 0 + dx+[G2(x+) G2(x+), G2] + h.c. + O(g6), (3.58) 4 to order g4. We can renormalize G2 and G4, but new divergences arise from the product of G2. One of the examples is the product of two one-scalar exchange diagrams in Fig. 6. The corresponding term in G2 is G2ex = i [ F (l2 l1)(l2+ l1+) + F (p1 p2)(p2+ p1+)] where we have integrated the energy of F (q). The third term in the brackets of Eq. (3.58) is x(1 x)x (1 x ) +(finite terms), where is the cutoff of the transverse momentum of the external fermion. Here, x and x are defined as x = x = Equation (3.60) is logarithmically divergent as . A Feynman box dia- p1 gram, which is finite, as seen from power counting, consists of the sum of various G2ex = time-ordered diagrams which may be divergent individually. The above product is one such diagram, and it is logarith- l1 mically divergent. It is important to recognize that Heff should be divergent. The reason for this can be understood from its perturbative diagonalization: Epert = |(H0 + Heff )| + Fig. 6. The one-scalar exchange diagram that contributes to G2. where Heff = Heff H0 and | is an eigenstate of H0. Because the logarithmic divergence in Eq. (3.60) comes from the integral of the three-momentum of the intermediate state, the divergences of 2i 0 dx+G2(x+)G2 and of 2i G2 0 dx+G2(x+) have opposite signs, and they cancel each other. In general, using the eigenstates of H0, we obtain the matrix element i | 2 G2 1 1 |G2| |G2| 2 p p i + in which the power of the intermediate energy p is decreased by 1, and thus the power of the transverse momentum of the intermediate states is decreased by 2. It is important that although these divergences must be canceled by adding artificial counterterms in other similarity methods, they automatically arise in the higher-order terms in our method. As shown in Appendix B, if we diagonalize the effective Hamiltonian to order g2, the eigenvalue is divergent. In general, such divergences are mainly related to the box diagrams and arise not only in this case but also in the case of the TD approximation in (3 + 1) dimensions. 13) Equation (3.63) shows that such divergences are canceled if we include g4 order terms. In this case, the cutoff dependence in the eigenvalue is weakened. However, new divergences arise in diagonalizing the effective Hamiltonian due to new interactions. We expect that they are canceled by higher-order interactions, because the exact eigenvalue should not depend on the cutoff, and the similarity transformation does not change the eigenvalue. 4. Summary and discussion In this paper, we have shown that the effective Hamiltonian, which is obtained with the FSTO similarity transformation in the particle number space, can be written in terms of only G (or F ) to fourth order in H . Introducing G is crucial for constructing the effective Hamiltonian more easily than in the traditional methods, especially at higher orders. G has the favorable property that it is diagonal in the particle number space. Since it also has the same form as the formula of the S-matrix operator, we can use the Feynman diagrams and the rules mentioned in 3.2 in the LF Yukawa model. By using knowledge obtained from covariant perturbation theory, we can avoid complexity in calculating the effective Hamiltonian and make the renormalization procedure transparent. Note that our construction rules are slightly different from the usual Feynman rules, because energies are not conserved at each vertex. The divergences due to the integrations of the loop momenta are renormalized by a familiar prescription of covariant perturbation theory. The divergences due to the energy integrations can be canceled by terms which come from the ambiguity of the counterterms for the fields. The mechanism of the cancellation of the extra divergences from the energy integrations is valid only in the LF field theory. It will be applied not only to the LF Yukawa model but also to the other LF models. Although we cannot use it in the ET field theory, we expect that more precise integrations and a treatment of the limit are necessary for the cancellation of the divergences. Although G is constructed from Feynman diagrams and renormalized, the effective Hamiltonian has divergent terms which are written in terms of products of the renormalized G. It is very important that such terms do not need counterterms, but act as those which cancel the divergences in diagonalizing the effective Hamiltonian. Although the exact eigenvalue should not depend on the cutoff, our method is perturbative, and then it is not possible to treat nonperturbative divergences in diagonalization and to get the exact eigenvalue. It is important to find the nonperturbative counterterms for them. When we find the low energy states, our method is sufficiently useful in the small coupling region to allow us to ignore the cutoff dependence if the cutoff is much larger than lower eigenvalues. There is also the problem of a vanishing energy denominator. Here, we only consider the LF Yukawa model without a massless particle, so that we may avoid this problem by including the term i in denominators. If there is a massless particle, as in QCD, we must consider it. We proved here that the effective Hamiltonian can be written in terms of G to fourth order by explicit calculation. However, the general proof for higher orders is lacking. Although we do not know how a general proof might be constructed, we think that it is likely that this feature persists to all orders. Recently, Hansper proposed a nonperturbative approach for the FSTO method in LF field theory. 15) Although he applied it only to the parton distributions, it is very useful to solve the mesonic bound states in QCD if it is applicable to our method. As reported in Appendix B, we have shown numerically that in the case that the coupling constant is sufficiently large, the lowest-energy state has positive binding energy, but we do not regard it as a bound state, because it depends on the transverse cutoff . As mentioned above, our method is valid for small coupling, and we expect that it will be improved in higher-order calculations. We are now extending the present work to the next order to confirm the cancellation of the cutoff dependence and to obtain bound states. 12) It is interesting to note that this method might also be applied to the similarity transformation in momentum space, because Eq. (3.13) does not depend on the choice of . If this can be done, it will be easier to obtain the effective Hamiltonian than in the traditional method. The author is grateful to K. Harada for helpful discussions and encouragement, and also thanks A. Okazaki for his useful work. Acknowledgements Appendix A Light-Front Yukawa Model 1 {(x), (y)}x+=y+ = (x y)2(x y)+, 2 where (x) is the dynamical part of the fermion field (x): are the projection operators of the fermion field and are defined as = 1 0. (A.7) 2 From Eq. (A.3), the familiar Legendre transformation gives the LF Hamiltonian, where k+ = k = (k0 + k3)/2, k = k+ = (k0 k3)/2 and ki = (k1, k2). Nonzero elements of the metric tensor are g+ = g+ = g+ = g+ = g11 = g22 = 1. If k satisfies the on-shell condition k2 = m2, k can be written H0 = H = H = P = H0 + H , dxd2x 2 ()2 + 1 22 + 2 + m2 , 1 2 2i + g22 () 21i () 21i (T )() = 0, = 0. In the interaction picture, they become H0(x+) = H (x+) = dxd2x 2 ()2 + 1 22 + + 2 + m2 , 1 2 2i dxd2x ig5 + g22 2i () ( 2i )T()T SF (k) = d4x0|T +((x)(0))|0e ikx = includes the noncovariant term. In Fig. 7, the contributions of the noncovariant term in the first two diagrams are naively canceled by the vertex 2 k3+ + q2+ + k3+ q1+ , in the last diagram. This comes from the instantaneous interaction in the Hamiltonian. For example, let us consider the scalar self-energy in Fig. 8. Using (+)2 = 0, we can see that the noncovariant contribution from the first diagram, dq2 i where all the field operators are defined in this picture. Note that (x) is not the same as that in Eq. (A.3) but, rather, is a new field made from (x) in this picture, and the solution of the free Dirac equation: 16) i +(g5) k/ + /q2 m + i (g5) 2ik++ , i is canceled by the second one, dk2 i i 2 k3 k2 i k2 k1 i is canceled by the second one, i k3 k1 i Fig. 7. The first two diagrams are examples which include an internal fermion line. The propagators include the noncovariant term. The last diagram is the contribution from the instantaneous vertex included in the LF Hamiltonian. Fig. 8. The scalar self-energy. The first diagram is covariant, except for the fermion propagators in the loop, which have a noncovariant term. The second one comes from the instantaneous vertex in the Hamiltonian. Fig. 9. The fermion self-energy. The first diagram is covariant, except for the noncovariant term in the fermion propagator. The second one comes from the instantaneous vertex. Therefore, we regard the fermion propagator as the first term in Eq. (A.15) and effectively covariant, and omit the second term in Eq. (A.15) and the vertex Eq. (A.16) together. As a result, the scalar and fermion propagator are given by F (q) = SF (k) = Appendix B Explicit Calculations andNumerical Result We calculated the ground state energy numerically to second order in g. Of course, the calculations to this order are not so different from those using other methods. 17) The advantage of the present formulation becomes apparent at higher orders. 12) The reason we present the second order calculation here is to demonstrate some features of our method and to clarify what would be expected at the next order. From Eqs. (3.13) and (3.20), the LF effective Hamiltonian is Fig. 10. Feynman diagrams that are necessary to construct the effective Hamiltonian to second order in g. The first diagram is the free part. The next four diagrams are the self-energy parts, and X represents the counterterms for the fermion mass and field. The last two diagrams are the one-scalar exchange part and the fermion annihilation part. to g2 order. Also, we add the flavor of fermions. It is easy to estimate Eq. (B.1) from Feynman diagrams by using our rules. Since possible graphs are the same as those needed in constructing the S-matrix, we can immediately imagine those which contribute to Eq. (B.1). The Feynman diagrams associated with the effective Hamiltonian to second order are shown in Fig. 10. Terms which should be renormalized are only the fermion self-energy terms M2 and M2 at this order. Although we do not give the rules for these, they are similar to those for Vn. The self-energy terms correspond to the first and the third graphs in the second line of Fig. 10 and are written G2self = (p1 p2) dk + (1 Z2i)(2)SFi1(k) imi(2) bi(p1)bi(p2)(2)3(p1 p2), where the m(2) and (1 Z2i)(2) are the masses and field renormalization constants i of order g2, respectively, and the on-shell condition (p21 = p22 = mi2) for the external fermion lines is satisfied. Although there is no energy-conserving delta function, G2self = 1 p 2p+ ui(p, )i(2)(p)ui(p, ) bi(p)bi(p), (p2 = mi2) G2ex + G2ex = g2 (2)33(p1 + l1 p2 l2) ui(p1, 1)5ui(p2, 2)vj(l2, 2)5vj(l1, 1) 1 1 (p1 p2)2 2 + (l2 l1)2 2 ui(p1, 1)5vi(l1, 1)vi(l2, 2)5ui(p2, 2) 1 1 (p1 + l1)2 2 + (p2 + l2)2 2 (p1 p2) p1 k i k p1 i G2an + G2an = g2 (2)33(p1 + l1 p2 l2) which are all second order interactions in the LF effective Hamiltonian. Of course, since it does not have particle-number-changing interactions, the eigenstate is a pure two-body state. ei(m1/22/2)ij(x, ; 1, 2, m) ]0.6 V e [G0.5 y rg0.4 e n e0.3 g idn0.2 n iB0.1 ] eV0.25 G [ yg 0.2 r e ne0.15 indg 0.1 n iB0.05 0.5 ] V [eG0.4 y rg0.3 e n ge0.2 n i d in0.1 B Appendix C Ambiguity in Integrations In this appendix, we show that there is in general an ambiguity in integrations, like that appearing in Eq. (3.46), by considering an explicit example. Let us consider the following integration: i i the residue theorem, I becomes I = k1 k2 i where we keep the radii finite in the second term, because I has various values depending on the manner in which the limits are taken. For example, 1. 1 before 2 , 2. 2 before 1 , I = k1 k2 i I = k1 k2 i I = k1 k2 i The other limits may yield other constants. Note that such an ambiguity arises from the terms in which the total dimension of the variables of the integrations is zero.


This is a preview of a remote PDF: http://ptp.oxfordjournals.org/content/106/1/99.full.pdf

Yuki Yamamoto. A Construction of an Effective Hamiltonian from Feynman Diagrams: Application to the Light-Front Yukawa Model, Progress of Theoretical Physics, 2001, 99-130, DOI: 10.1143/PTP.106.99