Pion-mass dispersion relation in the baryon sector
Vladimir Pascalutsa, Marc Vanderhaeghen, Jonathan M. M. Hall, Tim Ledwig
aa r X i v : . [ nu c l - t h ] M a r Pion-mass dispersion relation in the baryon sector
Vladimir Pascalutsa ∗ † , Marc Vanderhaeghen Institut für Kernphysik, Johannes Gutenberg Universität, Mainz D-55099, Germany
Jonathan M. M. Hall
CSSM, School of Chemistry and Physics, University of Adelaide 5005, Australia
Tim Ledwig
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC,E-46071 Valencia, Spain
By looking at the complex plane of the pion-mass squared we establish a dispersion relationwhich the static quantities, such as baryon masses, magnetic moments, polarizabilities, shouldobey. This dispersion relation yields insight into the differences between the heavy-baryon andrelativistic calculations in the baryon sector of chiral perturbation theory. ∗ Speaker. † Supported by the Deutsche Forschungsgemeinschaft (DFG) through Collaborative Research Center SFB 1044. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ion-mass dispersion relation
Vladimir Pascalutsa
1. Introduction
It is well-known that the chiral perturbation theory ( c PT) is able to predict some ‘non-analytic’dependencies of static quantities (masses, magnetic moments, etc.) on pion-mass squared, or thequark mass ( m p ∼ m q ). It is therefore interesting to examine the origin of these dependenciesarising by considering the analytic properties of chiral expansion in the entire complex m p plane. ( b ) ( c ) ( d )( a ) Figure 1:
Examples of chiral-loop corrections to the nucleon mass. Nucleon (pion) propagators are denotedby solid (dashed) lines. t Figure 2:
The branch cut and the con-tour defining the analyticity domain inthe complex plane of t = m p . Considering the chiral loops with external nucleonson shell, as in the graphs of Fig. 1 arising in the origi-nal (manifestly Lorentz-invariant) formulation of baryon c PT (B c PT ) [1], we observe that they are analytic func-tions of m p everywhere except for the branch cut alongthe negative axis, see Fig. 2. In this case it is possible towrite down a simple dispersion relation in the pion-masssquared [2]: f ( m p ) = − p Z − ¥ d t Im f ( t ) t − m p + i + , (1.1)where f is a chiral-loop correction to a static quantity, orthe static quantity itself; 0 + is an infinitesimal positive number.Several applications of this dispersion relation have been discussed in [2]. Here we shallfocus on a study of large discrepancies between the leading-order heavy-baryon (HB) [3, 4] andB c PT [1, 5] calculations encountered in e.g. Refs. [6–9].
2. B c PT vs. HB c PT at O ( p ) The chiral expansion of a static quantity f is an expansion in the quark mass m q around thechiral limit ( m q → c PT becomes an expansion in p = m p / L c , the mass of thepseudo-Goldstone boson of spontaneous chiral symmetry breaking over the scale of chiral symme-try breaking L c ≃ p f p ≈ m p plane along thenegative real-axis, the chiral expansion is not a series expansion (otherwise, it would have a zeroradius of convergence), but rather an expansion in non-integer powers of m p (cid:181) m q .By writing the dispersion integral as: f ( m p ) = − p Z − L c + − L c Z − ¥ d t Im f ( t ) t − m p , (2.1)2 ion-mass dispersion relation Vladimir Pascalutsa it is evident that the second integral can be expanded in integer powers of m p / L c . Hence this termis of analytic form and can only affect the values of the LECs. Indeed, the physics above the scale L c is not described by c PT and therefore its effect should be absorbable in the LECs.The second integral generates an infinite number of analytic terms, while the number of LECsto a given order of the calculation is finite. The higher-order analytic terms are present and notcompensated by the LECs at this order, but their effect should not exceed the uncertainty in thecalculation due to the neglect of all the other higher-order terms. That is, the second integral canbe dropped, while the resulting cutoff-dependence represents the uncertainty due to higher-ordereffects. We are thus led to examine the cutoff dependence of the pion-mass dispersion relation [10]: f ( m p ; L ) = − p Z − L d t Im f ( t ) t − m p (cid:18) m p t (cid:19) n , (2.2)where n indicates the number of subtractions around the chiral limit. Our main aim is to see atwhich values of the cutoff any deviation occurs between the HB- and B c PT results. If the deviationbegins at L ≪ p , the imaginary parts of the nucleon mass, the proton and neutron AMMs,and the magnetic polarizability of the proton are given by:Im M ( ) N ( t ) = g A ˆ M N ( p f p ) pt (cid:16) t + l (cid:17) q ( − t ) , (2.3a)Im k ( ) p ( t ) = g A ˆ M N ( p f p ) pl (cid:16) t + l (cid:17) h − (cid:16) t + l (cid:17)i × q ( − t ) , (2.3b)Im k ( ) n ( t ) = − g A ˆ M N ( p f p ) pl (cid:16) t + l (cid:17) q ( − t ) , (2.3c)Im b ( ) p ( t ) = − ( e / p ) g A ( p f p ) ˆ M N pt l h − l + ( l − ) t − ( l − ) t + ( l − ) t + t i q ( − t ) , (2.3d)where ˆ M N ≃
939 MeV is the physical nucleon mass, e / p ≃ /
137 is the fine-structure constant,and the following dimensionless variables are introduced: t = t ˆ M N , l = r t − t . (2.4)The corresponding HB expressions at order p can be obtained by keeping only the leading in1 / ˆ M N term (i.e, l ≈ √− t , etc.):Im M ( ) N ( t ) HB = g A ˆ M N ( p f p ) pt √− tq ( − t ) , (2.5a)Im k ( ) p ( t ) HB = g A ˆ M N ( p f p ) p √− tq ( − t ) HB = − Im k ( ) n ( t ) , (2.5b)Im b ( ) p ( t ) HB = ( e / p ) g A ( p f p ) ˆ M N p √− t q ( − t ) . (2.5c)3 ion-mass dispersion relation Vladimir Pascalutsa
The full, renormalized result for a given quantity is obtained by substituting these imaginaryparts into the dispersion relation of Eq. (2.2). The number of subtractions required in each casediffer: n = M N , n = / ˆ M N term, orequivalently, by substituting the corresponding imaginary parts from Eq. (2.5), into the dispersionrelation. In the latter case, the same integral is encountered in all of the examples: J ( m p ; L ) ≡ Z − L d t ( t − m p ) √− t = − m p arctan L m p . (2.6)All of the above quantities to O ( p ) in HB c PT are given by this integral, up to an overall constant,and a factor of m n p . n is the number of subtractions (or pertinent LECs) at this order. In Fig. 3, theresulting cutoff-dependence of the above loop contributions is shown at the physical value of thepion mass: m p ≃
139 MeV. Each quantity (mass, isovector AMM, and polarizability) is presented - - - - - - - M N H L H M e V L HB Χ PTB Χ PT0.0 0.5 1.0 1.5 2.0 - - - - Κ i s o v H L - - - L H GeV L Β p H L H - f m L Figure 3:
The cutoff-dependenceof leading-order loop contributionsto various nucleon quantities (mass,isovector AMM, and proton’s mag-netic polarizability) calculated inHB c PT (blue dashed curves) andB c PT (red solid curves). ion-mass dispersion relation Vladimir Pascalutsa in a separate panel, where the results within the relativistic ChPT and using the HB expansion aredisplayed.The figure illustrates the following two features:1. The HB c PT results have a stronger cutoff-dependence than the B c PT results, indicating alarger impact of the unknown high-energy physics to be renormalized by higher-order LECs.Quantitatively, the residual cutoff-dependence in HB c PT falls as 1 / L in all of the consideredexamples, while the dependence in the case of B c PT behaves as 1 / L for M N , and as 1 / L for both AMMs and b p .2. The HB- and B c PT results are guaranteed to be the same at small values of L , as can be seenby taking derivatives of Eq. (2.2) with respect to L , at L =
0. However, at finite values of L the differences are appreciable. Observing significant differences for L of order m p , as in thecase of b p , indicates that the size of the 1 / ˆ M N terms is largely underestimated in HB c PT.
3. Conclusion and outlook
The HB c PT and B c PT can be viewed as two different ways of organizing the chiral EFTexpansion in the baryon sector. While the heavy-baryon expansion is often considered to be moreconsistent from the power-counting point of view, it appears to be less natural. Certain terms thatare nominally suppressed by powers of m p / M N , and hence dropped in HB c PT as being ‘higherorder’, appear to be significant in explicit calculations.The problem is more pronounced in some quantities and less in others. To quantify this, oneneeds to note the power of the expansion parameter at which the chiral loops begin to contributeto the quantity in question. For the considered examples of the nucleon mass, AMMs, and polar-izability, this power index is 3, 1, and −
1, respectively. The smaller the index, the greater is thedifficulty for HB c PT to describe this quantity in a natural way.The negative index simply means that the chiral expansion of that quantity begins with neg-ative powers of m p . Apart from polarizabilities, the most notable quantities of this kind are thecoefficients of the effective-range expansion of the nuclear force. The non-relativistic c PT in thetwo-nucleon sector [11] failed to describe these quantities [12], thus precluding the idea of ‘per-turbative pions’ in this sector. The present work encourages us to think that B c PT can solve thisproblem in a way similar to the case of nucleon polarizabilities [8].
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