Analytic structure of $\phi^4$ theory using light-by-light sum rules

We apply a sum rule for the forward light-by-light scattering process within the context of the $\phi^4$ quantum field theory. As a consequence of the sum rule a stringent causality criterion is presented and the resulting constraints are studied within a particular resummation of graphs. Such resummation is demonstrated to be consistent with the sum rule to all orders of perturbation theory. We furthermore show the appearance of particular non-perturbative solutions within such approximation to be a necessary requirement of the sum rule. For a range of values of the coupling constant, these solutions manifest themselves as a physical bound state and a $K$-matrix pole. For another domain however, they appear as tachyon solutions, showing the inconsistency of the approximation in this region.


Introduction
Sum rules provide a powerful tool to study relativistic quantum field theories, and apply also outside the regime where perturbative expansions hold. As sum rules are consequences of such general principles as analyticity and unitarity, they allow to establish rigorous relations between physical observables, even when the underlying theory is non-perturbative in nature and cannot be solved exactly.
In recent works [1,2], we have derived three sum rules for the low-energy forward light-bylight scattering process. These γγ-sum rules are non-perturbative in origin and demonstrate that the low and high-energy behaviors of the theory are related. We showed e.g. that a sum rule for the helicity-difference total cross-section of the photon-photon-fusion process, γγ → X, reveals in the hadron sector an intricate correlation between contributions of pseudoscalar and tensor mesons. In the charm quark sector, the γγ sum rules reveal an interplay between cc bound states, open charmed meson continuum states, as well as exotic cc resonance states [2]. Several experiments, primarily at e + e − collider facilities, are presently uncovering the rich spectroscopy of such systems, see e.g. [3,4] for some recent reviews.
Besides its relevance to hadron phenomenology, we can use the γγ-sum rules in the same way in model field theories, where in the case of renormalizable models they can be applied perturbatively. When the conditions of applicability are fulfilled these sum rules were shown to hold in leading order calculations [2]. However, the realization of the causality constraints at higher orders as well as in the non-perturbative regime of quantum field theories is still an open issue.
Studies of causality constraints on the basis of different sum rules were carried out in the past in a number of different contexts. Especially the realization of the well-known Gerasimov-Drell-Hearn sum rule [5] within perturbative field theory was analyzed for spin-1/2 targets at the lowest nontrivial order [6] as well as at higher orders in QED [7]. In Refs. [8,9] consequences of the sum rules within asymptotically free theories were considered. In more recent years, they have also been discussed within the context of quantum gravity [10,11].
In the present work, we are using light-by-light scattering sum rule as a tool to study causality constraints within a model field theory, the φ 4 scalar theory. We consider a bubble-chain resummation and demonstrate it to be consistent with causality to all orders of perturbation theory. Furthermore, it is shown that the sum rule strictly defines the non-perturbative structure of the solutions which arise dynamically within this approximation. In a particular regime of the coupling constant the spectrum of solutions contains a dynamically generated bound state and a K-matrix pole. For another domain the solution possesses an unphysical pole with negative invariant mass being a direct sign of the inconsistency of the approximation.
The outline of this letter is as follows. In Sect. 2, we compute the light-by-light scattering sum rule involving the helicity difference cross section for the γγ → X process, within the φ 4 scalar field theory at one-loop level. In Sect. 3, we provide a calculation beyond the oneloop level in the "bubble-chain" approximation. In Sect. 4 we discuss how causality imposes constraints on the solutions for different values of the renormalized self-interaction coupling constant of the φ 4 theory. The summary and outlook are given in Sect. 5.

One loop
In this work, we will focuss on a sum rule for the forward light-by-light scattering, which involves the helicity-difference cross-section for real photons [12,13,1] and reads as: where ∆σ(s) = σ 2 (s) − σ 0 (s) is the total helicity-difference cross section of the two-photon fusion process γγ → X, with the Mandelstam variable s = (q 1 + q 2 ) 2 , where q 1 and q 2 are the two photon 4-momenta, and s 0 is the lowest production threshold of the process.
We will study the above sum rule in a particular model quantum field theory. We take one of the simplest examples: a self-interacting scalar field φ(x) with charge e and mass m as described by the following Lagrangian density, where λ 0 is the self-interaction coupling constant, while the covariant derivatives and electromagnetic field-strength tensor are given as usual by We denote the helicity amplitudes for the process γγ → φφ * by M ++ and M +− , where the subscripts indicate the photon helicities. Given these amplitudes, the cross section for total helicity-0 and 2 are found as: where θ is the angle of one of the members of the φφ * pair w.r.t. the photon in the center-of-mass system, and where we introduced their relative velocity β as: To leading order inλ 0 ≡ λ 0 /(4π) 2 and in the fine-structure constant α ≡ e 2 /4π, the helicity amplitudes are found to be: where the first term in M ++ and the expression for M +− correspond with the tree level amplitudes for γγ → φφ * in scalar QED. The second term in M ++ describes the one-loop production process corresponding with Fig. 1, with the grey blob denoting the four-particle amplitude evaluated to leading order in λ 0 . Furthermore in Eq. (6a), the (dimensionless) form factor F (s) describing the transition of photons to a scalar pair at one-loop order is given explicitely by : It is quite easy now to compute the cross sections for γγ → φφ * , the result for the helicity difference cross-section is: where ∆σ (tree) is the tree-level cross section in scalar QED (cf., e.g., Appendix in [2]). The treelevel cross section weighted with 1/s integrates to zero by itself, and it can easily be verified that Hence, we have shown that the sum rule is obeyed at the one-loop level.

Bubble-chain sum
A class of diagrams naturally arises when one analyzes how the sum rules are realized at higher orders in perturbation theory, and in fact beyond perturbation theory. In the following,  we discuss the contribution of the bubble-chain type diagrams to the γγ-fusion process as shown on Fig. 1, where the shaded blob now denotes a bubble-chain contribution to the fourparticle vertex, as shown in Fig. 2. The bubble-chain approximation arises in many contexts, but most notably as the leading large-N result of the O(N) models. The interest in such an approximation is due to the fact that it preserves much of the non-linear structure of the exact theory [14,15].
The bubble-chain only contributes to the helicity-0 amplitude M ++ , with the nth-bubble contribution being given by In the dimensional regularization scheme the one-loop corrections to the four-particle vertex, corresponding with a single bubble in Fig.2, can be expressed through the scalar integral : with p the total four-momentum, and s = p 2 . This integral has the explicit form : Here is the dimensional regularization factor and µ is the The resulting amplitudes are independent of the renormalization scale µ which is achieved as usual by adding a counter term corresponding to the coupling constant renomalization. We define a renormalized coupling constantλ by the equatioñ This gap equation allows to renormalize the calculated contribution to each order perturbatively. Using such renormalization, leads to the renormalized (subtracted) one-loop four-point function :B where we have also absorbed a factor (4π) 2 in its definition. We notice that our renormalization procedure is conveniently chosen so as to make a subtraction at threshold:B(4m 2 ) = 0. We show the real and imaginary parts of the renormalized functionB(s) versus s in Fig. 3.
The interference of two chain diagrams with total number of (n − 1) bubble loops gives rise to a cross-section correction of the order O(λ n ). For the helicity-difference cross-section, which in the given case is equal to the helicity-0 cross section, we obtain as correction beyond the tree-level : where we used the notation One can check explicitly that the expression of Eq. (15) satisfies the helicity-difference sum rule exactly in each order of perturbation theory, i.e.
In Fig. 4, we plot the integrands of the sum rule at different orders of perturbation theory. One sees that in all cases the low-and high-energy contributions cancel.
It is well-known in quantum field theory that when trying to sum the different orders of perturbation theory, the resulting series is mostly divergent and has at best a meaning as an asymptotic series, a notable exception being the case of theories which have the property of asymptotic freedom [16,17,18]. As a result, the perturbation theory often can not determine the solution uniquely. If we are only interested in perturbative phenomena, it is not a problem to deal with it in the region of small coupling constants. However if we are interested in nonperturbative phenomena then we are faced with the problem to give a meaning to a divergent series.
For example, the photon propagator calculated in the leading logarithmic approximation possesses an unphysical pole associated with negative invariant mass and negative probability, usually called the "Landau ghost" [19]. In QED, however, due to the smallness of the coupling constant the ghost appears at an extremely high energy scale, and the results at the energy scales accessible in experiment are not influenced by it. However, in contrast to the QED case, in the theory under consideration the coupling constant is not constrained to small values, and we are faced with the need to regularize our solution. The appearance of the Landau singularity is usually attributed to non-Borel-summability of the considered series, where some individual Feynman amplitudes are positive and grow like the factorial of the number of vertices producing a singularity on the real positive axis of the Borel transform of the perturbative series [20]. A similar situation arises also in the context of our model. We can easily see how non-Borelsummability manifests itself in the context of sum rules, by analyzing the contribution from the high-energy region to the sum rule integral. In the example under study, the cross section Eq. (15) in the n-bubble approximation behaves at large s as : The contribution of the high-energy region to the sum rule integral can then be approximated by : where Λ 2 ≫ m 2 . As we see for negativeλ the high-energy contribution to the sum rule integral is positive definite and grows factorially with the order of perturbation theory, which amounts to the non-Borel summability of the series As a result we can consider ∞ n=0 ∆σ (n) (s) only in the sense of an asymptotic series. According to Poincaré [21], a divergent series is an asymptotic expansion of a function f (λ) if This definition implies that an asymptotic series does not define a function uniquely. The expansion coefficients of a function of the type e −1/λ n k=0 f k λ k being identically zero, such a function can be added to f (λ) without changing Eq. (21) [22].
Thus it is natural to expect that in order to obtain the correct behavior of I(λ) for negativẽ λ, one has to modify a formal resummation of the geometric series of renormalized bubble-chain corrections to the cross section in Eq. (15), which is given by : where the tree-level cross section satisfies the sum rule by itself. As was discussed above, due to the non-Borel-summability of the series of Eq. (20), we do not have any reasons to expect that the sum rule integral will vanish for the resummed theory. In Fig. 5 we show the dependence of the sum rule integral for the cross section of Eq. (22) on the value ofλ. We indeed notice from Fig. 5 that the sum rule is only valid for positive values ofλ (denoted by region I), but is violated for negative values ofλ (regions II and III on Fig. 5), showing that the naive procedure of the resummation is not applicable. In order to preserve validity of the sum rules beyond the region I, we need to find a way to evaluate the cross section correctly. We will discuss the physical situation for the three regions ofλ in the following.

Region I : convergent perturbative expansion
Though the sum of Eq. (20) is formally undetermined, we can still use a naive resummation at least in the region of positiveλ, as one can see from Fig. 5. Forλ −1 > 0 (region I) the series is alternating-sign, since ReB(s) < 0, and one can expect the series to be resummable. Alternatively we can interpret such a resummation at the level of photon-photon-fusion amplitudes by summing up contributions of bubble-chain diagrams at different orders, which yields the amplitude :

Region II : Bound state and K-matrix pole
Now we proceed to the discussion of the second region, denoted by region II on Fig. 5. In the domain of −2 <λ −1 < 0 the sum rule is not valid for the naive cross section, given by Eq. (22). It is easy to see that the amplitude of Eq. (23) acquires additional singularities below the two-particle production threshold. Indeed from Fig. 3 we can see that the imaginary part ofB(s) vanishes for s < 4m 2 and monotonically increases for s > 4m 2 . The real part satisfies the inequality −∞ < ReB(s) 0, for 0 s < ∞, with the upper limit attained for s = 4m 2 . Thus we see that forλ −1 < 0 there is always a S-matrix pole for s < 4m 2 . Above the two-particle threshold, there is a value of s whereλ −1 = ReB(s), corresponding with a K-matrix pole as will be discussed below.
We first discuss the pole below the threshold, which obviously corresponds to a bound state of the scalar pair. One notices from Fig. 3 that the mass of the bound state varies continuously from M 2 B = 4m 2 forλ −1 = 0 to M 2 B = 0 forλ −1 = −2 as we sweep over the bound state region on Fig. 5. In general, the position of the bound state pole is defined by the equation : It is important to note that the bound state singularity is not described by perturbation theory being essentially of non-perturbative nature. The agreement with the sum rule can now be remedied by treating the bound state as a new asymptotic state of the theory and thus including the channel of its production in γγ collision, see Fig. 6. The corresponding contribution to the helicity-0 cross section is: where the effective coupling of the bound state to φφ * is found as the residue of the pole: It is not difficult to see now that its contribution to the sum rule integral: exactly counter-balances the contribution of the γγ → φφ * channel shown in Fig. 5. Thus the causality is restored in this region ofλ.
We now turn to the position of the singularity above the two-particle threshold. To describe the elastic φφ scattering in φ 4 theory, we only need to consider S-wave scattering in the bubblechain approximation. The (dimensionless) elastic forward scattering amplitude f (s) is expressed through a real phase shift δ(s) as: or through the K-matrix amplitude K(s) ≡ tan δ(s) as: Since the imaginary part of the loop function is for s ≥ 4m 2 we can define the elastic amplitude as : and hence K −1 (s) = 1 λπβ(s) In Fig. 7 we show plots of the phase shift for different values of the coupling constant. Note that for negativeλ the phase-shift starts from π which indicates the presence of one bound state. Also for negativeλ the phase shift crosses π/2 at s > 4m 2 satisfying the following equation: This is the location of the K-matrix pole, corresponding with a scattering amplitude which becomes purely imaginary. Usually this behaviour is attributed to a resonance. Since above the threshold, the imaginary part of B(s) is not zero for all complex s andλ is real, the amplitude (23) does not possess a S-matrix pole for s > 4m 2 . Hence there is no resonance associated with this K-matrix pole. Note that on the right side of region II on

Region III : Ground state instability and tachyonic solution
Ifλ −1 becomes smaller than −2 (corresponding with region III on Fig. 5), the binding energy of the bound state exceeds 2m, and the pole crosses the point s = 0, moving into the unphysical region s < 0, and producing a tachyonic solution (a pole with negative invariant mass). The occurrence of a pole for negative values of s signals that the ground state of the theory is unstable [14,15]. We could formally still include this pole in the γγ-production channel as an asymptotic state and add its contribution to the total cross section in the same way as we did for the bound state, which restores the validity of the sum rule. For small negative values of λ, i.e.λ −1 → −∞, corresponding with the left asymptotic edge of the region III in Fig. 5, the contribution of the tachyon pole is vanishingly small and the sum rule is satisfied approximately. This is consistent with the observations in other models, for example, in the leading logarithmic approximation in QED, where the position of the Landau pole appears at very large scales due to the smallness of the fine structure constant. The position of the tachyon pole in this limit is defined by which shows explicitly the non-perturbative structure of this contribution. The contribution of this pole in this sum rule is asymptotically defined as and vanishes whenλ −1 → −∞. However such a procedure has a number of inconsistencies. Being required by the sum rule such extra contribution spoils the general principles of field theory. The cross section of Eq. (36) is negative, thus indicating that the appearance of a tachyon ghost state contradicts the usual quantum mechanical probabilistic interpretation and spoils unitarity. Moreover, the delta function is located at a space-like squared four momentum −M 2 indicating the occurrence of a tachyonic instability and spoiling the analyticity principle. All these facts show that the bubble-chain approximation is essentially inconsistent in region III of the coupling constant. For the full theory, such unphysical state will not appear among the exact eigenstates.

Conclusions
We have studied consequences of causality constraints imposed by a recently established sum rule for the forward light-by-light scattering process within the φ 4 scalar quantum field theory. Within this theory, we verified the sum rule at the one-loop level as well as to all orders within the bubble-chain approximation. Furthermore, we have performed a resummation of bubble graphs. We have demonstrated that depending on the value of the renormalized selfinteraction coupling constant of the φ 4 theory, three different regimes emerge. In a first regime, the perturbative series is convergent and the sum rule as calculated from the continuum states in the theory holds exactly. In a second regime for the renormalized coupling, the resummed amplitude acquires additional singularities: a dynamically generated bound state below the twoparticle production threshold and a K-matrix pole above the two-particle production. It was shown that when evaluating the light-by-light sum rule, the bound state contribution exactly cancels the continuum contribution, so as to verify the sum rule. Furthermore, we found a third regime of the renormalized coupling where a tachyonic solution with negative invariant mass appears, signaling that in this regime the vacuum is unstable and that the considered bubble-chain resummation is essentially inconsistent.
The results within the considered model relativistic quantum field theory have demonstrated that light-by-light scattering sum rules provide a very powerful tool to quantitatively connect dynamically generated bound states with the continuum region of the theory. As such this can be a first step, to apply such a tool e.g. to the study of mesons produced in the γγ-fusion process presently under study at different e + e − collider facilities.