OOn the sigma sigma term
Peter C. Bruns
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany (Dated: November 7, 2018)
Abstract
We give some estimates for the light-quark mass dependence of the pole position of the sigma( f (500)) resonance in the complex energy plane, with the help of a chiral Lagrangian for the resonancefield and some input from hadronic models constrained by Chiral Perturbation Theory and elasticunitarity. We also speculate on the fate of the sigma resonance when the quark masses becomeunphysically large. a r X i v : . [ nu c l - t h ] D ec . INTRODUCTION There are basically two ways to implement and study resonance phenomena in Chiral Perturba-tion Theory (ChPT) [1–3], the low-energy effective field theory of the strong interaction. Thefirst way is the most direct one: a chiral effective Lagrangian is constructed which containsexplicit field variables for the particles associated with the resonances. This has lead to theso-called Resonance Chiral Theory [4–7]. The other way is paved by “Unitarized Chiral Per-turbation Theory” (UChPT) [8–12], where an infinite string of higher-order terms in the chiralexpansion is resummed in some or the other way, to guarantee exact coupled-channel unitarity(in the space of the most relevant particle channels) for a given scattering problem. Here, theresonances enter indirectly: the resummed scattering amplitudes can have poles in the complexenergy plane, which are associated with the resonance mass and width. The resonance is saidto be “dynamically generated”. This approach has been very succesful in describing low-energyhadron physics phenomenology, but one should also mention that it has some shortcomings:crossing symmetry and S-matrix analyticity [13] are in general not exactly fulfilled (see e.g.[11, 14–19]), and there is a non-negligible model dependence [14, 15, 20] in particular for en-ergies above the low-energy region, and for large quark masses much above the chiral regimewhere ChPT can be applied (i. e., problems are expected for M π (cid:38) . . .
400 MeV [21]).In [22–25], the quark mass dependence of the σ (or f (500)) (and ρ ) mass and width has beenstudied in a UChPT framework. The σ resonance is of particular interest for the study of low-energy ππ scattering, a key problem of ChPT. The corresponding pole in the complex energyplane is tightly constrained already from the low-energy ππ interaction given by ChPT andthe constraint of elastic unitarity, as pointed out e.g. in Sec. 18 of [26]. In a framework wherethe amplitude is also constrained by partial-wave analyticity and crossing symmetry (the Royequations for ππ scattering [27, 28]), the f (500) pole position can be fixed with an impressiveaccuracy [29, 30]. For a recent comprehensive review on the σ resonance, including a historicaloverview and an extensive list of relevant references, we recommend to consult [31].It is the aim of this work to establish a close contact between the two ways of describing the σ resonance, and to complement the UChPT studies on the quark mass dependence of thisresonance by adding another viewpoint to it, in the hope that this may help to further reduceany model dependence, and to contribute to a better understanding of the resonance physics.As an application, we give a first estimate for the leading quark-mass dependence of the mass2f the resonance, m σ (the phrase “sigma term” is only used in loose analogy to the pion-nucleoncase - we do not evaluate the scalar form factor of the σ ), and also show some tentative extra-polations to higher quark masses.This work is organized as follows: In Sec. II, we construct the one-loop approximation to the σ self-energy from a resonance chiral Lagrangian, in a similar fashion as we did for the vectormesons in [32]. In Sec. III, we exploit the fact that the σ resonance is located close to theenergy region where (two-loop) ChPT is expected to give reliable results, and that its positionis tightly constrained by unitarity and the chiral ππ interaction, to obtain estimates for themost relevant parameters (low-energy constants, or LECs for short) entering the one-loop ex-pression for the self-energy in the chiral limit. In Sec. IV, we present and discuss our numericalresults for the quark mass dependence of the σ pole parameters. The appendix is devoted to ashort discussion of the renormalization procedure for theories with explicit resonance fields, ina slightly simplified field-theoretical model. 3 I. SIGMA SELF-ENERGY
To begin, we have to write down an effective chiral Lagrangian for the resonance field and itsinteraction with the pions (the pseudo-Goldstone bosons, PGBs, of spontaneously broken chiral SU (2) L × SU (2) R symmetry). The leading-order chiral Lagrangian for a massive scalar-isoscalarfield σ is constructed in [4] and reads L σ = 12 ∂ µ σ∂ µ σ − µ σ σ + c σd σ (cid:104) u µ u µ (cid:105) + c σm σ (cid:104) χ + (cid:105) , (1)where the usual chiral-covariant building blocks are used (see also [4]), u µ = iu † ( ∂ µ U ) u † , χ ± = 2 B (cid:0) u † M u † ± u M u (cid:1) , U = exp (cid:18) iF π a τ a (cid:19) , u = √ U . (2)We have set the external (axial-)vector source fields [2] to zero. The isovector pion field isexpanded in Pauli matrices τ a , F is the pion decay constant in the chiral limit, B and c σd,m arefurther low-energy constants, and M = diag( m u , m d ) is the quark mass matrix (we will workin the isospin limit where m u = m d =: m (cid:96) ). The quark masses can be expressed through the(squared) pion mass via M π = 2 Bm (cid:96) + O ( m (cid:96) log m (cid:96) ) [2]. We will also introduce countertermsfor mass and wave-function renormalization (compare Eq. (A.1)). These are needed in additionto Eq. (1) because there is obviously a non-vanishing interaction with pions even in the chirallimit, generating e.g. the width of the sigma resonance (see below). In an effective field theory,an infinite string of higher-order terms with arbitrarily many derivatives is in principle allowed,but we will not need the explicit form of such terms here. The leading quark-mass correctionto the counterterm Lagrangian is constructed in analogy to the corresponding term in thepion-nucleon Lagrangian [3], L (2) χσ := c σ (cid:104) χ + (cid:105) σ + c σ (cid:104) u µ u µ (cid:105) σ , L (4) χσ := − e σ (cid:104) χ + (cid:105) σ + . . . . (3)Chiral Lagrangians involving resonances (instead of particles like pions or nucleons, which arestable under the strong interaction) like the ones given above have to be applied and interpretedwith care. First, quantum (field) theory is based on measurements. For a broad (short-lived)resonance like the σ , there seems to be no obvious or natural concept of a “ σ state”, let alonemultiparticle σ states, and consequently, one might also have doubts to use a σ field . Second,the renormalization process is non-standard for resonances (see also App. A). And third, since One could, however, resort to a formalism as proposed in [33, 34], at least for narrow resonances. σ is generated in the perturbative loop expansion in such a framework,it is possible that the resonance couples so strongly to other states that the perturbative ex-pansion does not converge.Therefore, it is clear that the present study has some exploratory character. We see the reso-nance Lagrangian as a convenient tool to parameterize and describe the phenomenon observedas the resonance, and the σ field as a (largely arbitrary, up to the quantum numbers) integra-tion variable in the path integral which is used in the description of the relevant observations,not implying any assumptions about the “nature” (quark content, etc.) of the resonance. Thetheory at hand can be considered as a natural generalization of the framework designed for(nearly) stable particles, respecting all known symmetries relevant at low energies, but alsoshowing some unusual features which will be encountered in the present work (compare also[35] for a theoretical study of general properties of the σ propagator).We add a remark on the power counting. In the usual ChPT framework [1, 2], PGB masses,momenta and energies are counted as being “small of order O ( p )”, since these energies are smallcompared to a typical hadronic scale ∼ ∼ πF . Even though the mass and width of the σ do not vanish in the chiral limit, the resonance energy region is not far away from the regionwhere the low-energy expansion properly works. On a practical level, an expansion in the en-ergy over 4 πF could still be reasonably effective. For the application of the self-energy intendedhere, it will turn out that the expansion in small energies is of minor importance, since we aremainly interested in quark mass corrections to the pole position of the resonance in the chirallimit. We will rearrange our expression for the self-energy in a way that is convenient for thispurpose, similar to our work on vector mesons in [32], but without some of the approximationsmade therein. We restrict our application of the chiral resonance Lagrangian to the one-looplevel, which is O ( p ) in the usual low-energy counting, but we will mainly be interested in theleading quark-mass dependence (of O ( M π )) of the resonance position here, and will thereforeneglect some higher-order corrections of O ( M π ) in our final numerical estimates, which cannotbe fixed without additional input.The most prominent contribution to the σ self-energy is given by the one-loop graph of Fig. 1,which describes the coupling of the resonance to the ππ continuum states, with a strength givenby some coupling parameter g . From the vertex rules of the Lagrangian (1) it is straightforward5 IG. 1: One-loop contribution to the σ self-energy. Dashed lines: pions, double lines: σ resonance. to compute the contribution to the self-energy Π σ ( s ) due to this Feynman graph,Π σ | ππ ( s ) = 24 F (cid:18)(cid:20) c σd s − M π ) + c σm M π (cid:21) I ππ ( s ) − (cid:0) ( c σd ) ( s − M π ) + 4 c σm c σd M π (cid:1) I π (cid:19) . (4)Here s is the squared four-momentum of the resonance. In dimensional regularization, the loopintegrals entering here are given by I π := (cid:90) d d l (2 π ) d il − M π = 2 M π ¯ λ + M π π log (cid:18) M π µ (cid:19) + O (4 − d ) , where ¯ λ contains the pole in d − µ such that the µ -dependence of the logarithm is cancelled, while I ππ can be obtained from Eq. (A.3) with M φ → M π . - Let us first discuss the representation of the self-energy in the chiral limit where M π →
0. In this limit (indicated by a superscript ◦ ), Π σ | ππ reduces to ◦ Π σ | ππ ( s ) = 3( c σd ) s π F (cid:18) π ¯ λ − − log (cid:18) − µ s (cid:19)(cid:19) . (5)Of course, in the chiral limit, decays into any higher (even) number of PGBs are kinematicallyallowed, but the corresponding contributions to the self-energy are still suppressed in the low-energy counting, and thus the use of a one-loop approximation can still be meaningful.The inverse propagator of the resonance at the one-loop level is then of the form( ◦ D σ ) − ( s ) = s − ◦ µ σ + 3 ◦ g σ π s log (cid:18) − µ s (cid:19) + polynomial( s ) , ◦ g σ := 2 c σd F . (6)The polynomial is due to the analytic piece in the loop function and counterterms from theLagrangian. Applyling the low-energy counting, the polynomial would be restricted to secondorder in s . But the order is not really relevant here, since we are only interested in the behavior6f this function close to the pole position of the propagator on the second Riemann sheet inthe variable s , written as ◦ s σ ≡ ◦ µ σ − i | ◦ µ σ | ◦ γ σ ≡ (cid:18) ◦ m σ − i ◦ Γ σ (cid:19) , (7)so that, for s sufficiently close to the pole,( ◦ D σ ) − ( s ) = s − ◦ µ σ + 3 ◦ g σ π s log (cid:18) − µ s (cid:19) + κ + κ ( s − ◦ µ σ ) + κ ( s − ◦ µ σ ) + . . . . (8)Terms of quadratic and higher order in ( s − ◦ µ σ ) contain higher powers of the width for s → ◦ s σ and are beyond the one-loop order, so we can drop them for our purposes. In the following, wewill fix µ = 1 GeV, and sometimes write the logarithm in the loop function simply as log (cid:0) − s (cid:1) with the unit being understood. Of course, the numerical values of the coefficients κ i will ingeneral depend on the choice of µ . The logarithm has branch points at zero and infinity, andwe choose the pertinent branch cut along the positive real s -axis. The first Riemann sheet ( I )is the one where the log is real on the negative s -axis. On the second Riemann sheet ( II ),log (cid:0) − s (cid:1) | II = log (cid:0) − s (cid:1) | I + 2 πi . The κ -term is a mass counterterm, which has to be adjustedsuch that ◦ µ σ is indeed the real part of the pole position, while the κ -term is essentially awave-function normalization counterterm. Of course, the complex coefficients κ i must be suchthat the counterterm polynomial is real for real s . To proceed further, we must establish aconnection to the ππ scattering amplitude (in the chiral limit). First, we note that, on treelevel, the exchange of a σ resonance in the s − channel gives a pole-graph contribution( t ( s )) (tree) σ − ex . = − π ◦ g σ s s − ◦ µ σ (9)to the isospin I = 0 s-wave ( J = 0) partial-wave ππ scattering amplitude t I =0 J =0 ( s ) (defined asin [2]). This violates elastic unitarity and is certainly not a good approximation for a broadresonance. But, taking the “dressing” of the resonance into account, we can write an improvedamplitude ( t ( s )) σ − ex . = − ◦ g σ π s ◦ D σ ( s ) , (10)which satisfies the constraint of elastic unitarity (stated in Eq. (16) below) and should be agood approximation to the full partial-wave scattering amplitude close to the resonance pole,provided that the pole position and coupling are properly adjusted. Assuming that we know7he pole position and (complex) residue ◦ R σ of the full scattering amplitude, t ( s → ◦ s σ ) = ◦ R σ s − ◦ s σ ≡ − ◦ g σ π ◦ s σ Z + i Z s − ◦ s σ , Z , ∈ R , (11)we can try to adjust our free parameters in (8) and (10) to obtain the required approximation.We will use the renormalization conditions that the mass in the chiral limit is given by ◦ µ σ ,and that the real part of the derivative of the self-energy vanishes, Re ◦ Π (cid:48) σ ( ◦ s σ ) = 0 (whichapproximately fixes Z = 1, compare App. A). We find κ = − ◦ g σ π Re (cid:20) ◦ s σ log (cid:18) − µ ◦ s σ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) II (cid:21) , κ = − ◦ g σ π Re (cid:20) ◦ s σ (cid:18) (cid:18) − µ ◦ s σ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) II − (cid:19)(cid:21) , (12)3 ◦ g σ π = | ◦ µ σ | ◦ γ σ | ◦ µ σ | ◦ γ σ Re (cid:20) ◦ s σ (cid:18) (cid:16) − µ ◦ s σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) II − (cid:19)(cid:21) + Im (cid:20) ◦ s σ log (cid:16) − µ ◦ s σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) II (cid:21) . (13)As usual in ChPT, the quark masses, and therefore the pion masses, are treated as an additionalexternal perturbation. The pole position is shifted by this perturbation, to s σ ≡ µ σ − i | µ σ | γ σ ≡ (cid:18) m σ − i σ (cid:19) , (14)and the one-loop approximation to the inverse propagator is of the form( D σ ) − ( s ) = s − ◦ µ σ + κ + 4 c σ M π (cid:18) − M π π F log (cid:18) M π µ (cid:19)(cid:19) + 3 M π c σ π F log (cid:18) M π µ (cid:19) − e σ M π + (cid:0) κ + 4 κ (cid:48) M π (cid:1) ( s − ◦ µ σ ) + 3 M π π F (cid:0) ( c σd ) ( s − M π ) + 4 c σm c σd M π (cid:1) log (cid:18) M π µ (cid:19) (15)+ 32 π F (cid:20) c σd s − M π ) + c σm M π (cid:21) (cid:18) σ ( s ) artanh (cid:18) − σ ( s ) (cid:19) − log (cid:18) M π µ (cid:19)(cid:19) , where we have absorbed some analytic pieces of the loop integrals in the (renormalized) LECs(also, the leading corrections to the mass formula M π = 2 Bm (cid:96) have been tacitly absorbed in c σ , e σ ). The new LEC κ (cid:48) is due to a quark-mass correction to the wave-function renormalizationconstant. On the unphysical sheet, σ ( s )artanh ( − /σ ( s )) II = σ ( s )artanh ( − /σ ( s )) I + iπσ ( s ), σ ( s ) = (cid:112) − (4 M π /s ) . Eq. (15) is a main result of this work: if the LECs are known, theequation ( D σ ) − ( s σ ) ! = 0 determines the complex pole position s σ of the σ for any prescribedvalue of M π . In the next section, we want to obtain estimates for the most important parametersentering the above equation. 8 II. CHIRAL UNITARY MODEL IN THE CHIRAL LIMIT
The following partial-wave amplitudes t IJ ( s ) in the chiral limit satisfy the requirement of elastictwo-particle unitarity, Im (cid:2) ( t IJ ( s )) − (cid:3) = − , for s > , (16)and agree with the known chiral expansion at the two-loop level [36, 37] , t ( s ) = s πF (cid:20) s (4 πF ) (cid:18) r s −
17 + 64 π (11 l r + 7 l r )12 − log (cid:18) s (cid:19) (cid:32) − s πF ) (cid:16) π (82 l r + 29 l r ) (cid:17)(cid:33) + 175 s πF ) log (cid:18) s (cid:19)(cid:19) − s (4 πF ) log (cid:18) − s (cid:19)(cid:21) − ,t ( s ) = s πF (cid:20) s (4 πF ) (cid:18) r s + 16 π (2 l r − l r ) −
19 + log (cid:18) s (cid:19) (cid:32) − s πF ) (cid:16)
737 + 576 π (34 l r + 33 l r ) (cid:17)(cid:33) − s πF ) log (cid:18) s (cid:19)(cid:19) − s πF ) log (cid:18) − s (cid:19)(cid:21) − ,t ( s ) = − s πF (cid:20) s (4 πF ) (cid:18) r s + 51 + 768 π ( l r + 2 l r )36 + log (cid:18) s (cid:19) (cid:32) s πF ) (cid:16)
883 + 192 π (193 l r − l r ) (cid:17)(cid:33) + 85 s πF ) log (cid:18) s (cid:19)(cid:19) + s πF ) log (cid:18) − s (cid:19)(cid:21) − . The unknown constants r IJ appear at two-loop order and contain the renormalized two-loopLECs r i ( µ = 1 GeV). The one-loop LECs l r , are (roughly) known; for definiteness, we willemploy the values (and errors) of [26], ¯ l = − . ± .
6, ¯ l = 4 . ± .
1, which translates to l r ( µ = 1 GeV) = ( − . ± . · − and l r ( µ = 1 GeV) = (0 . ± . · − . For a discussion ofthese (and other) LECs relevant for ππ scattering we refer to [38]. The pion decay constant inthe chiral limit will be taken as F = 86 MeV [39].There is a resonance pole on the second Riemann sheet of the above model amplitude for t ( s ).For given values of l r , l r and r , we can extract its position with the help of mathematica ® routines. To estimate the uncertainty, we generate ∼ random number sets for ( l r , l r , r ),normally distributed around their central values, with a standard deviation of the correspondingerror. Values for r are taken around 0 with a deviation of 1 GeV − , which is the order ofmagnitude of the numerically fixed terms in the two-loop calculation entering the combination r . This range is quite generous and might also cover some systematic error due to the model-dependence involved in the choice of the unitarization procedure. We show the result of thisenterprise (for the “predicted” Re t ) in Fig. 2(a). The analogous plots for Re t and Re t are shown in Figs. 2(b,c). The red lines show the result for the central values of the inputparameters, the curves due to all other parameter configurations form the gray bands. Theuncertainty of the parameterization seems to be relatively moderate in the σ resonance region,while the situation appears to be worse for the ρ . For the pole position of the σ in the chiral9 .2 0.4 0.6 s (cid:45) (cid:45) (a) (cid:45) (cid:45) (b) (cid:45) (cid:45) (c) FIG. 2: Re t IJ ( s ), over s in GeV (a): IJ=00, (b): IJ=11, (c) IJ=20. limit, ◦ s σ = ( ◦ m σ − i ◦ Γ σ ) , and the residue at the pole, ◦ R σ , we find ◦ m σ = (395 ±
17) MeV , ◦ Γ σ = (630 ± , (17) ◦ R σ = (0 . ± . + i (0 . ± . . (18)The uncertainty in the mass is surprisingly small and one might suspect that it is underesti-mated. Nevertheless, we will adopt the values given above for our estimates of the quark massdependence. For the record, we note that we find the pole position of the lowest resonance in t (the ρ ) at ◦ m ρ = (711 ±
45) MeV , ◦ Γ ρ = (333 ± , (19) ◦ R ρ = ( − . ± . + i (0 . ± . . (20)We point out that the numbers given above, and the curves in Fig. 2, are not due to a fit todata - the data enter only indirectly through the numerical values of F, l r , l r (and to a lesserextent ( O ( p )) also through the fixed µ = 1 GeV). Taking for granted the result of Ref. [29](compare also [31]), m phys σ = 441 +16 − MeV , Γ phys σ = 544 +18 − MeV , (21)10e obtain a first rough estimate of the “ σ sigma term”, m (cid:96) ∂m σ ∂m (cid:96) ≈ m phys σ − ◦ m σ ∼
45 MeV .
IV. NUMERICAL ANALYSIS AND DISCUSSION
Let us now use the estimates obtained in the previous section to study the quark mass depen-dence of the resonance pole position. Since we cannot expect more than first rough estimateswithout analyzing precise lattice data, it makes sense to set the O ( M π ) parameters e σ , c σ inEq. (15) to zero for the moment. Our numerical strategy will be the same as in the previoussection: We generate a large set of random numbers for ( ◦ µ σ , ◦ γ σ , µ σ , γ σ , c σm ) (see Eqs. (7), (14))with central values and errors as specified in Eqs. (17), (21) (while c σm is varied in the range − . . . + 50 MeV, which is the expected range for this coupling, compare e.g. Secs. 4, 5 of[4]), and determine the LECs c σ and κ (cid:48) , for every set, from the equation ( D σ ) − ( s σ ) ! = 0 for M π = 139 MeV.As a first result, we obtain (via Eq. (13)) an estimate for the coupling in the chiral limit,3 ◦ g σ π = (0 . ± . − , (22)or | c σd | ≈
36 MeV. This coupling is quite large and we have to expect that higher-order cor-rections could be sizeable. To provide a test of the validity of our approach, we note that theresidue of our model amplitude in the chiral limit, Eq. (10), is also fixed via Eqs. (22), (12),and results in ◦ R σ ≈ (0 .
169 + 0 . i ) GeV (so that Z ≈ Z ≈ − .
04 in Eq. (11)), whichcan be compared with Eq. (18). It is clear that the given residue in the chiral limit can not bereproduced exactly, because we work only at one-loop accuracy, and because vertex correctionsare missing in the simplistic model used in Eq. (10) (see also App. A). The impact of thesedeficiencies seems to be moderate, however.The values and mean errors for the LECs determined from the procedure described above are c σ = 0 . ± . , κ (cid:48) = (0 . ± .
39) GeV − . (23)As already anticipated, the uncertainties are relatively large. This is reflected by the estimatedquark mass dependence of the mass m σ and the width Γ σ for small M π : Inserting an ansatz m σ = ◦ m σ + a m M π log M π + b m M π , Γ σ = ◦ Γ σ + a Γ M π log M π + b Γ M π (24)11n Eq. (15), expanding everything to order M π , and solving this truncated version of theequation ( D σ ) − ( s σ ) ! = 0 for all generated parameter configurations, we obtain a m = a Γ = 0 , b m = (1 . ± .
65) GeV − , b Γ = ( − . ± .
95) GeV − , (25)resulting in the estimate ∼ M π, phys / GeV, or (38 ±
32) MeV for the “sigma sigma term”. Ofcourse, this result, and the numbers in Eq. (25) do not provide much more than a consistencycheck between the estimates in Sec. III, the framework outlined in Sec. II and the methodsused here. (cid:144)
GeV0.20.40.60.81.0m (cid:144)
GeV (a) (cid:144)
GeV0.51.01.5 (cid:71) (cid:144)
GeV (b)
FIG. 3: Mass (a) and width (b) of the σ resonance for 10 solutions of D − σ ( s σ ) ! = 0. It is probably bold to extrapolate our results so far to unphysically large pion masses. But inorder to see what the present formalism is in principle capable of, we show in Fig. 3 the pionmass dependence of the mass m σ and width Γ σ , for a hundred of our solutions for the zeroes ofEq. (15). The red curves give the leading quark mass dependence of Eq. (24), for the centralvalues of Eqs. (17), (25). Note that our analysis is not complete at O ( p ), because we haveset the parameters c σ , e σ to zero by hand, so that the resulting curves can only give a firstimpression of the full one-loop solution. While the mass follows the description given by theleading quark-mass dependence up to rather high pion masses, the uncertainty in the widthis large beyond the physical point. But at least it seems fair to say that the σ can become abound state not much below M π ∼
300 MeV (if at all). This is in accord with a recent latticestudy [40]. More definite conclusions can only be drawn if the present formalism is appliedto an analysis of precise lattice data. This is a natural next step in the development of this study.12 cknowledgments
I thank Maxim Mai, Andreas Sch¨afer and Philipp Wein for discussions on the manuscript. Thiswork was supported by the Deutsche Forschungsgemeinschaft SFB/Transregio 55.13 ppendix A: Renormalization conditions for resonances
Let us assume that a scalar resonance (called σ ) is observed in the elastic scattering of (stable)spinless particles described by a field φ . Let us also assume that the mass and the width,as well as the residue of the φφ scattering amplitude at the resonance pole have somehowbeen extracted to some satisfying accuracy, and that the effects due to inelastic channels aresuppressed. We would like to describe this situation with the help of a simple Lagrangian, L basic = 12 ∂ ν σ∂ ν σ − µ σ σ + 12 ∂ ν φ ∂ ν φ − M φ φ − g σ φ , (A.1) L ct = δZ σ (cid:0) ∂ ν σ∂ ν σ − µ σ σ (cid:1) − δµ σ σ + δZ φ (cid:0) ∂ ν φ ∂ ν φ − M φ φ (cid:1) − δM φ φ − δg σ φ . This model has also been studied e.g. in [41], and in App. E of [42], with different methods. Asin Sec. II we denote the measured pole position of the σ in the complex Mandelstam plane as s σ = µ σ − iµ σ γ σ , and M φ is the measured mass of the φ particle. The Lagrangian in the secondline contains the counterterms (for the standard procedure in the counterterm approach, seee.g. Chapter 10 in both [43] and [44], or [45]). The counterterms δZ φ and δM φ will be adjustedfollowing the usual renormalization conditions, so that the physical mass (the pole of the fullpropagator) of the φ equals M φ to all orders in the loop expansion, while the residue of thepropagator is fixed to 1. We will not discuss the φ self-energy further. Note that δg starts atorder g , while the other counterterms start at O ( g ). In the following, we will try to fix theremaining counterterms to one-loop accuracy by studying the resonance pole contribution tothe φφ scattering amplitude. This contribution is depicted symbolically in Fig. 4(a). One finds T pole φφ ( s ) = − (cid:20) δgg + g I σφφ ( s ) (cid:21) g s − µ σ − Π σ ( s ) (cid:20) δgg + g I σφφ ( s ) (cid:21) , (A.2)Π σ ( s ) = g I φφ ( s ) + δµ σ − ( s − µ σ ) δZ σ . The loop integrals occuring here are given by I φφ ( s ≡ k ) := (cid:90) d d l (2 π ) d i (( k − l ) − M φ )( l − M φ )= I φφ (0) − s π (cid:90) ∞ M φ ds (cid:48) σ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) (A.3)= I φφ (0) − π (cid:18) σ ( s ) artanh (cid:18) − σ ( s ) (cid:19)(cid:19) , σ ( s ) := (cid:115) − M φ s ,I φφ (0) = 2¯ λ + 116 π (cid:18) (cid:18) M φ µ (cid:19)(cid:19) + O (4 − d ) , a) (b) FIG. 4: (a) General structure of the resonance pole contribution, (b) a one-loop vertex correction.Dashed lines: φ particles, double lines: σ resonance. The squares in (a) contain the vertex corrections,the blob on the resonance line indicates the full dressed propagator. ¯ λ = µ d − π (cid:18) d − −
12 [ln(4 π ) + Γ (cid:48) (1) + 1] (cid:19) ,I σφφ ( s ≡ k ) := (cid:90) d d l (2 π ) d i (( k − l ) − M φ )( l − M φ )(( k − q − l ) − µ σ ) (cid:12)(cid:12)(cid:12)(cid:12) q =( k − q ) = M φ = 116 π (cid:90) ∞ M φ ds (cid:48) log (cid:16) µ σ + s (cid:48) − M φ µ σ (cid:17)(cid:113) s (cid:48) ( s (cid:48) − M φ )( s (cid:48) − s ) , (A.4)employing dimensional regularization. From the dispersive representations in Eqs. (A.3)and (A.4), the imaginary parts (for real s ) can be directly read off. The real part of I φφ ( s )contains a divergent constant for d →
4. Real values of s are to be approached fromthe upper complex plane for s ∈ [4 M φ , ∞ ] on the physical real axis. Note that, in theapplication of this appendix, we have to use the expressions for the loop integrals on theunphysical Riemann sheet of the variable s , which can be obtained by analytic continuation in s .To avoid that the real part of the pole position is shifted from its value µ σ , we have to fix δµ σ = − g Re I φφ ( s σ ). To assure that the imaginary part of the pole position equals − µ σ γ σ , weneed µ σ γ σ (1 + δZ σ ) = − g I φφ ( s σ ) ⇒ γ σ = − g µ σ Im I φφ ( s σ ) + O ( g ) , (A.5)15hich fixes the value for g in our theory. Then, close to the pole, T pole φφ ( s ) → − g s − s σ [1 + ( δg/g ) + ( g / I σφφ ( s σ )] (cid:2) δZ σ − ( g / I (cid:48) φφ ( s σ ) (cid:3) ≈ − g s − s σ (cid:20) (cid:18) δgg − δZ σ (cid:19) + g (cid:18) Re I σφφ ( s σ ) + 12 Re I (cid:48) φφ ( s σ ) (cid:19) + ig (cid:18) Im I σφφ ( s σ ) + 12 Im I (cid:48) φφ ( s σ ) (cid:19)(cid:21) , (A.6)where in the second line we have neglected terms of two-loop order. We see that only a combi-nation of wave-function and coupling counterterms can be fixed from the real part of the residueat the pole, and that the imaginary part of the residue is fixed by g and µ σ at one-loop order.The usual choice for δZ σ would be ( g / I (cid:48) φφ ( s σ ), in which case the real part of the residueof the resonance propagator is fixed to 1. This choice was adopted e.g. in [32, 46, 47] (com-pare also [48] for the “complex mass scheme”). More generally, the residue of the scatteringamplitude at the resonance pole is determined by corrections to wave-function renormalization,and by vertex corrections. The latter are essentially given by the scattering amplitude on thesecond Riemann sheet, with the pole term subtracted , evaluated at the resonance pole. The σ exchange in Fig. 4(b) is a part of this subtracted amplitude. Intuitively it should be the vertexcorrection due to this subtracted scattering amplitude that is strongly related to the conceptof “compositeness” of the resonance [42, 49] (besides the resonance location), since apparentlyit determines the “overlap” matrix element between the “resonance state” and a state of twoparticles interacting via t − and u − channel exchanges (so we should expect resonances with anoteable composite two-particle component to be sensitive to t − and u − channel dynamics).The subtracted scattering amplitude in question could be extracted employing unitarity andanalyticity: it is well-known [29, 50] that the partial-wave S-matrix elements for elastic scat-tering, S (cid:96) = 1 + 2 iσ ( s ) t (cid:96) ( s ) on the first ( I ) and second ( II ) sheet are related by S II(cid:96) = 1 /S I(cid:96) (note that σ ( s ) in Eq. (A.3) also has a branch point at the threshold). So, numerically, thezeros s on the first (physical) sheet agree with the resonance pole positions s σ on the secondsheet. Expanding S I(cid:96) ( s ) = 0 + ( s − s ) a + ( s − s ) a + . . . , we find S II(cid:96) ( s → s σ ) = a − s − s σ − a a , (A.7)which gives us the subtracted S-matrix element at the resonance position on the second sheet.This is an interesting result - however, the page ends here.16
1] S. Weinberg, Physica A (1979) 327.[2] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[3] J. Gasser, M. E. Sainio and A. Svarc, Nucl. Phys. B (1988) 779.[4] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B (1989) 311.[5] P. D. Ruiz-Femenia, A. Pich and J. Portoles, Nucl. Phys. Proc. Suppl. (2004) 215 [hep-ph/0309345].[6] V. Cirigliano, G. Ecker, M. Eidem¨uller, R. Kaiser, A. Pich and J. Portoles, Nucl. Phys. B (2006) 139 [hep-ph/0603205].[7] J. J. Sanz-Cillero, Nucl. Phys. Proc. Suppl. (2010) 236 [arXiv:1009.3446 [hep-ph]].[8] A. Dobado, M. J. Herrero and T. N. Truong, Phys. Lett. B (1990) 134.[9] J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. D (1999) 074001 Erratum: [Phys. Rev. D (1999) 099906] Erratum: [Phys. Rev. D (2007) 099903] [hep-ph/9804209].[10] J. A. Oller and E. Oset, Phys. Rev. D (1999) 074023 [hep-ph/9809337].[11] J. Nieves and E. Ruiz Arriola, Nucl. Phys. A (2000) 57 [hep-ph/9907469].[12] A. Gomez Nicola and J. R. Pelaez, Phys. Rev. D (2002) 054009 [hep-ph/0109056].[13] R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, “The Analytic S-Matrix,”Cambridge University Press (1966).[14] T. N. Truong, Phys. Rev. Lett. (1991) 2260.[15] M. Boglione and M. R. Pennington, Z. Phys. C (1997) 113 [hep-ph/9607266].[16] I. P. Cavalcante and J. Sa Borges, hep-ph/0101037.[17] Q. Ang, Z. Xiao, H. Zheng and X. C. Song, Commun. Theor. Phys. (2001) 563 [hep-ph/0109012].[18] H. Q. Zheng, hep-ph/0304173.[19] C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D (2003)076009 [hep-ph/0210311].[20] I. Caprini, P. Masjuan, J. Ruiz de Elvira and J. J. Sanz-Cillero, Phys. Rev. D (2016) no.7,076004 [arXiv:1602.02062 [hep-ph]].[21] S. D¨urr, PoS LATTICE (2015) 006 [arXiv:1412.6434 [hep-lat]].[22] C. Hanhart, J. R. Pelaez and G. Rios, Phys. Rev. Lett. (2008) 152001 [arXiv:0801.2871 hep-ph]].[23] J. Nebreda and J. R. Pelaez., Phys. Rev. D (2010) 054035 [arXiv:1001.5237 [hep-ph]].[24] J. R. Pelaez and G. Rios, Phys. Rev. D (2010) 114002 [arXiv:1010.6008 [hep-ph]].[25] J. Nebreda, J. R. Pelaez and G. Rios, Phys. Rev. D (2011) 094011 [arXiv:1101.2171 [hep-ph]].[26] G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B (2001) 125 [hep-ph/0103088].[27] S. M. Roy, Phys. Lett. B (1971) 353.[28] B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Phys. Rept. (2001) 207[hep-ph/0005297].[29] I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. (2006) 132001 [hep-ph/0512364].[30] R. Garcia-Martin, R. Kaminski, J. R. Pelaez and J. Ruiz de Elvira, Phys. Rev. Lett. (2011)072001 [arXiv:1107.1635 [hep-ph]].[31] J. R. Pelaez, arXiv:1510.00653 [hep-ph].[32] P. C. Bruns, L. Greil and A. Sch¨afer, Phys. Rev. D (2013) 114503 [arXiv:1309.3976 [hep-ph]].[33] T. Berggren, Nucl. Phys. A (1968) 265.[34] G. Garcia-Calderon and R. Peierls, Nucl. Phys. A (1976) 443.[35] N. N. Achasov and A. V. Kiselev, Phys. Rev. D (2004) 111901 [hep-ph/0405128].[36] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. E. Sainio, Phys. Lett. B (1996) 210[hep-ph/9511397].[37] T. Hannah, Phys. Rev. D (1997) 5613 [hep-ph/9701389].[38] J. Nebreda, J. R. Pelaez and G. Rios, Phys. Rev. D (2013) 054001 [arXiv:1205.4129 [hep-ph]].[39] S. Aoki et al. , arXiv:1607.00299 [hep-lat].[40] R. A. Briceno, J. J. Dudek, R. G. Edwards and D. J. Wilson, arXiv:1607.05900 [hep-ph].[41] F. Giacosa and T. Wolkanowski, Mod. Phys. Lett. A (2012) 1250229 [arXiv:1209.2332 [hep-ph]].[42] T. Hyodo, D. Jido and A. Hosaka, Phys. Rev. C (2012) 015201 [arXiv:1108.5524 [nucl-th]].[43] S. Weinberg, “The Quantum Theory of Fields. Vol. 1: Foundations”, Cambridge, UK: Univ. Pr.(1995).[44] M. E. Peskin and D. V. Schroeder, “An Introduction to Quantum Field Theory,” Reading, USA:Addison-Wesley (1995).[45] J. C. Collins, “Renormalization. An Introduction To Renormalization, The RenormalizationGroup, And The Operator Product Expansion,” Cambridge University Press (1984).
46] A. Denner, Fortsch. Phys. (1993) 307 [arXiv:0709.1075 [hep-ph]].[47] F. Klingl, N. Kaiser and W. Weise, Z. Phys. A (1996) 193 [hep-ph/9607431].[48] D. Djukanovic, J. Gegelia, A. Keller and S. Scherer, Phys. Lett. B (2009) 235 [arXiv:0902.4347[hep-ph]].[49] T. Sekihara, T. Hyodo and D. Jido, PTEP (2015) 063D04 [arXiv:1411.2308 [hep-ph]].[50] J. L. Basdevant, C. D. Froggatt and J. L. Petersen, Nucl. Phys. B (1974) 413.(1974) 413.