Comments on high-energy total cross sections in QCD

We discuss how hadronic total cross sections at high energy depend on the details of QCD, namely on the number of colours $N_c$ and the quark masses. We find that while a"Froissart"-type behaviour $\sigma_{\rm tot}\sim B\log^2s$ is rather general, relying only on the presence of higher-spin stable particles in the spectrum, the value of $B$ depends quite strongly on the quark masses. Moreover, we argue that $B$ is of order ${\cal O}(N_c^0)$ at large $N_c$, and we discuss a bound for $B$ which does not become singular in the $N_f=2$ chiral limit, unlike the Froissart-\L ukaszuk-Martin bound.

We discuss how hadronic total cross sections at high energy depend on the details of QCD, namely on the number of colours Nc and the quark masses. We find that while a "Froissart"-type behaviour σtot ∼ B log 2 s is rather general, relying only on the presence of higher-spin stable particles in the spectrum, the value of B depends quite strongly on the quark masses. Moreover, we argue that B is of order O(N 0 c ) at large Nc, and we discuss a bound for B which does not become singular in the N f = 2 chiral limit, unlike the Froissart-Lukaszuk-Martin bound.
The behaviour of hadronic total cross sections at high energy is one of the oldest puzzles of strong interactions. Experimental results, up to the largest energies available at hadronic colliders [1][2][3][4], show a steady rise of total cross sections for √ s 5 GeV [5], where s is the total center-of-mass energy squared. The theoretical challenge is to explain the observed behaviour starting from the first principles of QCD, which is believed to be the fundamental theory describing strong interactions. So far, most of the efforts have focussed on phenomenological approaches, aimed at finding the appropriate parameterisation of experimental data, usually taking inspiration from the Regge-Gribov theory. To date, the majority of the parameterisations agree on the leading energy dependence being of the "Froissart"-type σ tot ∼ B exp log 2 s with universal B exp [5][6][7][8][9][10][11], although alternative behaviours are also considered [12,13]. A universal log 2 s rise, first proposed by Heisenberg [14], has been supported by several theoretical arguments [15][16][17][18][19][20][21], and recently also by numerical results in lattice QCD [22].
A correct prediction (from first principles) of the highenergy behaviour of total cross sections would nontrivially confirm the validity of QCD as the fundamental description of strong interactions, in a largely untested energy-momentum regime. In fact, the main difficulty in attacking this problem in the framework of QCD is its nonperturbative nature, as it is part of the more general problem of soft high-energy scattering, characterised by small transferred momentum squared t (|t| 1 GeV 2 ) and large s. To avoid the shortcomings of perturbation theory in the presence of the soft scale t, a nonperturbative approach to these processes has been developed [23][24][25][26][27][28][29][30], which relates the relevant scattering amplitudes to the correlation functions of certain nonlocal operators, the so-called Wilson loops, in the fundamental theory. To our knowledge, this approach is so far the closest to a systematic derivation from first principles.
In a recent paper [31] we have argued, within the above-mentioned nonperturbative approach [23][24][25][26][27][28][29][30] in Euclidean space [32][33][34][35][36][37][38], that hadron-hadron total cross sections at high energy behave like The prefactor B is determined from the stable asymptotic hadronic spectrum, considering strong interactions in isolation, by maximising the following ratio, where a runs over the particle species, and j (a) and M (a) are the spin and mass of particle a, respectively. Only higher-spin particles (j (a) > 1) have to be considered: if they were absent, then σ tot would be at most a constant at high energy, and B = 0. The parameter κ is bounded by unitarity to be |κ| ≤ 1, but is otherwise undetermined at this stage. In Ref. [31] we remarked that the most natural value yielding the universality observed in experiments is κ = 0, corresponding to a black-disk-like behaviour at high energy. However, we do not have a purely theoretical argument to show that this is actually the case. Furthermore, the phenomenological analyses in the literature give different estimates of the "blackness" of the scatterers in the high-energy limit, see, e.g., Refs. [8,10,11,39,40]. In Ref. [31] we also gave a numerical estimate of B using experimental data for (QCD-)stable mesons, baryons and nuclear states. The "dominant" particle, i.e., the one which maximises B (a) , turns out to be the Ω baryon, and yields B QCD ≃ 0.56 GeV −2 , which compares well to the experimental value B exp ≃ 0.69 ÷ 0.73 GeV −2 [5]. Interestingly enough, our value for 2B QCD is about two orders of magnitude smaller than the analogous prefactor B FLM = π M 2 π appearing in the Froissart-Lukaszuk-Martin bound [41][42][43], and only about 50 ÷ 60% larger than the experimental value, resulting in a much more restrictive "Froissart-like" bound (which is satisfied by B exp ).
It is part of the standard lore that hadronic total cross sections should be mostly governed by the "gluonic sector" of the theory, and this leads to expect that they could be described fairly accurately using the quenched approximation of QCD, i.e., pure SU (3) gauge theory. In this case, and in the framework of the nonperturbative approach discussed above, the relevant spectrum for the computation of the prefactor B would be the stable, higher-spin part of the glueball spectrum. However, using data from Ref. [44], the resulting value of B turns out to be 2 ÷ 3.5 times the one obtained using the physical, unquenched spectrum, suggesting the presence of unexpectedly large unquenching effects [31].
The high sensitivity of B to the presence or not of dynamical quarks raises an interesting question: how much does the actual value of B depend on the details of QCD? More precisely, how much does it depend on the values of its parameters, i.e., the number of colours N c and the quark masses? Since only the stable spectrum enters the maximisation Eq. (2), the crucial point is to understand how the stability of hadrons changes as the parameters are varied, and how this affects the overall scale of total cross sections. This is precisely the purpose of this letter.
We begin by discussing the behaviour of high-energy total cross sections in the 't Hooft large-N c limit [48,49]. The first point we want to clarify is precisely how this limit has to be taken. Eq. (1) describes the asymptotic high-energy behaviour of σ tot , i.e., σ tot for center-of-mass energies much larger than any other mass/energy scale in the problem. Formally, Eq. (1) has to be written as The quantity B is well defined for every finite N c , as the number of stable states is finite, and so it is sensible to consider its large-N c limit [57]. It is therefore clear that we take first the large-s limit, and then the large-N c limit. In Refs. [45,46] the authors consider pion-pion scattering amplitudes at large N c , taking first the large-N c limit and then the large-s limit. According to the usual counting rules, large-N c mesonic amplitudes are expected to be of order O(1/N c ), leading to total cross sections of order O(1/N c ) via the optical theorem. Inspection of the relevant diagrams shows that the processes contributing to this order involve qq exchange between the mesons, corresponding to Reggeon exchange. Although dominant at large N c and fixed s, these processes are suppressed by inverse powers of s at large s and fixed N c . The dominant process at large s is Pomeron exchange, understood here as the exchange of gluons, which leads to σ tot of order O(1/N 2 c ) at large but fixed s, according to the usual counting rules. In Ref. [47] the two limits s → ∞ and N c → ∞ are taken together, as the particles' momenta are scaled proportionally to √ N c as N c is increased. The resulting total cross section is proportional to log 2 N c . In our approach we do not have to scale the momenta, since they are formally taken to infinity before taking the large-N c limit; all that matters is the large-N c behaviour of the spectrum.
The large-N c behaviour of meson and baryon masses is well known [48][49][50]: meson masses are of order O(N 0 c ), while baryon masses are of order O(N c ). Roughly speaking, this is due to the fact that while mesons are always qq states, independently of N c , baryons are made of N c quarks. Concerning higher-spin states, no higher-spin QCD-stable meson is known in the "real world", i.e., for N c = 3, and unless this is a subtle consequence of O(1/N c ) corrections to the meson masses at N c = ∞, there is no reason to expect the situation to change when N c is large (but finite). On the other hand, a QCDstable higher-spin baryon exists for N c = 3, namely the Ω baryon (j (Ω) = 3 2 ). In the baryon sector, large-N c QCD possesses an effective light quark spin-flavour contracted symmetry SU (2N f ) for N f degenerate light quark flavours [51,52]. Real-world QCD is close to have an exact N f = 2 isospin symmetry, so for the physically most interesting case, at large N c the contracted symmetry is SU (4). Here we work with 2 + 1 light flavours (up/down + strange), neglecting isospin breaking effects. Furthermore, the large-N c limit is taken keeping N c odd, so that baryons are fermions as in the real, N c = 3 case.
Dashen, Jenkins and Manohar argued in Refs. [51,52] that in terms of this emergent, large-N c symmetry, baryons can be classified in multiplets corresponding to the irreducible representations of the contracted spinflavour symmetry. These representations are labelled by the isospin i, the spin j, and a further quantum number k, related to the number N s of strange quarks as N s = 2k.
The allowed values of k for given i, j are obtained via the usual composition rule for angular momenta, so that |i − j| ≤ k ≤ i + j. Large-N c consistency conditions, obtained by imposing unitarity in pion-baryon and kaonbaryon scattering processes, constrain the form of the baryon masses as follows [51,52], , which leads to σ tot behaving as expected according to the naïve large-N c counting rules. However, it is also possible that states with even higher spin are stable at large N c , which could change the large-N c behaviour of B. To see this, recall that a state with a given value of k is possible only if k ≤ i + j. Furthermore, if 2k = N s out of N c quarks are strange quarks, one has i ≤ N c /2 − k, and so also j ≥ 2k − N c /2, which is effective if N s ≥ (N c + 1)/2 (as j ≥ 1/2). Consider now the Ω baryon, defined for arbitrary N c as the baryon made of N c strange quarks, therefore having j (Ω) = k (Ω) = N c /2 and i (Ω) = 0. In a hypothetical decay of Ω into a baryon with N s ≥ (N c + 1)/2 strange quarks and spin j, one has from the bound above in a hypothetical decay to a state with N s < (N c + 1)/2, since 2∆k > (N c − 1)/2 and ∆j < (N c − 1)/2, the bound Eq. (5) still holds. As a consequence, a decay to a baryon with a decrease of ∆j in spin has to come with at least a decrease of ∆j in (the absolute value of) strangeness, which requires the emission of ∆j kaons [58]. It is therefore possible that the mass balance between initial and final states remains negative, as it is for N c = 3, therefore making the Ω stable also at large N c .
To make this statement quantitative one should know the exact mass formula, rather than its approximation Eq. (4), which in principle is valid only for j = O(N 0 c ). However, numerical studies on the lattice [53] (up to N c = 7) find good agreement with the mass formula Eq. (4) also for states with j = O(N c ). This indicates that higher-order terms in Eq. (4) give small contributions even for j = O(N c ), so that they can be neglected (in a first approximation), and Eq. (4) can be used to give a sensible quantitative estimate of the stability of the Ω baryon at large N c .
Working in the isospin limit, one can estimate the mass parameters m i at N c = 3 by fitting the (isospin averaged) masses of the physical octet and decuplet baryons with the mass formula Eq. (4). The error on the masses is taken as the sum (in quadrature) of the experimental error and of an extra uncertainty, accounting for isospin breaking and electromagnetic effects. This uncertainty is estimated as the standard deviation of the masses in an isomultiplet, and set to 1 MeV for isosinglets (raising this to 2 ÷ 3 MeV yields similar results). The fit of the baryon masses with Eq. (4) yields effective parameters, which include contributions from higher-order terms neglected in Eq. (4). To estimate the corresponding uncertainty ε Nc , we have repeated the fit including an extra termm (a) in the mass of each baryon, i.e., using the expression M (a) = M 1 (i (a) , j (a) , k (a) )+m (a) to fit the mass of baryon a. The parametersm (a) were constrained to be "small" by means of the usual constrained-fit techniques [54]. In particular, we took these extra parameters to be normally distributed around zero with standard deviation σ = 10 MeV. are reported in Tab. I. Variations of the resulting parameters between the two fits give an estimate of ε Nc , and are at most of 15%.
All the parameters but m 4 are positive at N c = 3; if their sign remains the same at large N c (which is supported by lattice results [53]), it is easy to obtain a bound on the mass balance ∆M 1 , plugging Eq. (5) into Eq. (4), Within our approximations, stability of the Ω baryon at large N c is ensured if (∆M 1 − 2∆k · M K )| Nc=∞ < 0 for all possible channels. Using the bound Eq. (6), this is certainly the case if δm ≡ (m 1 + 2m 3 − 2M K )| Nc=∞ < 0. The numerical results of Ref. [53] indicate that |m i | decreases as N c is increased, so using m i (N c = 3) instead of m i (N c = ∞) should make the bound even more conservative. On the other hand, O(1/N c ) corrections to the meson masses have not been measured in lattice simulations of the full theory. Numerical results for the quenched theory [55] suggest that the variation of meson masses between N c = 3 and N c = ∞ is of the order of 10%. A reasonable upper bound on δm is therefore We remind the reader that this bound is rather loose, since it does not include the negative contribution of m 4 , and it overestimates m 1 and m 3 . Moreover, δm ′ remains negative up to a reduction of around 15% of M K . Our conclusion is that stability of the Ω baryon at large-N c is at least plausible. If it is indeed so, since the corresponding B (Ω) is of order B (Ω) = O(N 0 c ), then one would necessarily have B = O(N 0 c ). This is in contrast with the expected O(1/N 2 c ) from the naïve large-N c counting rules, but not in contradiction, as that expectation holds in the limit N c → ∞ at large but fixed s.
We now turn to the chiral limit. There are two interesting cases: the N f = 2 chiral limit (m u = m d = 0, m s = 0) and the N f = 3 chiral limit (m u = m d = m s = 0). In both cases the spectrum of the theory contains massless pseudoscalar Goldstone bosons due to the spontaneous breaking of chiral symmetry. Note that massless particles of spin 0 leave Eqs. (1) and (2) unchanged [31]. Generally speaking, the chiral limit can only turn stable states into unstable states, and not viceversa, due to the possibility of decaying through the emission of massless particles.
The N f = 3 chiral limit is most easily discussed: since in this case kaons become massless, the Ω baryon is no longer stable. The role of dominant particle will therefore be taken by some higher-spin nuclear state. Notice that some excited nuclear states can become unstable against strong decays as m u,d → 0, since pions become massless.
The N f = 2 chiral limit is less straightforward. In this case kaons remain massive, so the stability or not of the Ω baryon depends on how much its mass, and the masses of the other strange baryons and of the kaon, change as m u,d → 0. It is likely that the difference between the physical masses of these particles and the corresponding masses in the N f = 2 chiral limit is of the order of the current light-quark masses, i.e., a few MeV. On the other hand, M Ω − M X − M K ∆N s is negative and of magnitude O(0.1 ÷ 1 GeV) for all baryons X in the octet and in the decuplet, i.e., at least two orders of magnitude bigger than the expected effect on the kaon and strange baryons masses of taking m u,d → 0. The effect of this limit on the masses of nuclei is again expected to be of a few MeV, so we expect that the Ω remains the dominant particle. As a consequence, our "Froissart-like" bound, Eqs. (1) and (3), is not singular in the N f = 2 chiral limit. Let us finally discuss the limit of large quark masses. For quark masses larger than some critical value, purely gluonic states (glueballs) will become stable, and will enter the set over which B (a) has to be maximised. Eventually, as the quark masses are further increased, at most only a finite number of higher-spin mesons and baryons will remain stable against decays, which can now take place through the emission of glueballs, since these have finite masses in the limit m q → ∞. Of course, the values of B (a) corresponding to mesons and baryons keep decreasing as the quark masses increase. The bottom line is that for large enough quark masses, the relevant part of the spectrum over which one has to maximise B (a) will consist only of stable higher-spin glueballs. Eventually, as m q → ∞, one will end up with the quenched theory, where B has been shown to be at least larger than B Q 1.09 GeV −2 [31].
It is interesting to remark that, according to our results, in the problem at hand the full and the quenched theory are not equivalent in the large-N c limit. This is essentially due to the fact that while baryon masses grow like N c , the stability of glueballs is not improved as N c grows, since they can always decay into light mesons (at least for physical quark masses), whose masses are essentially unaffected by the large-N c limit. Therefore, glueballs do not enter the game, while baryons still play an important role even though they become heavier and heavier. It is however worth noticing that both the quenched and the unquenched theory are expected to have B = O(N 0 c ) at large N c . Indeed, we have argued above that the full theory is likely to show this behaviour due to the stability of the N c -quark Ω baryon. In the quenched theory, glueball masses are O(N 0 c ), and a few higher-spin stable states exist at N c = ∞ according to lattice results [56], and so B Q = O(N 0 c ). In conclusion, in this letter we have discussed how total cross sections at high energy depend on the details of QCD, namely on the number of colours and the quark masses. Our results indicate that while a "Froissart"type behaviour σ tot ∼ B log 2 s is rather general, relying only on the presence of higher-spin stable particles in the spectrum, the value of B depends quite strongly on the details of the theory, and particularly on the quark masses. For example, it is likely to be discontinuous as the N f = 3 chiral limit, or the static limit, are approached. On the other hand, we expect that B behaves smoothly as the large-N c and the N f = 2 chiral limits are approached. In particular, we have argued that B is likely to be of order O(N 0 c ) in the large-N c limit, due to the stability of the Ω baryon, in contrast with the expectation based on the naïve counting rules; and that the more restrictive "Froissart-like" bound of Eqs. (1) and (3) is not singular in the N f = 2 chiral limit.
MG is supported by the Hungarian Academy of Sciences under "Lendület" grant No. LP2011-011.