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abstract: |
    The empirical value of the real part of the pion-deuteron scattering length can be well understood in terms of the dominant isovector $\pi N$-double scattering contribution.  We calculate in chiral perturbation theory all one-pion loop corrections to this double scattering term which in the case of $\pi 
    N$-scattering close the gap between the current-algebra prediction and the empirical value of the isovector threshold T-matrix $T_{\pi N}^-$. In addition to closing this gap there is in the $\pi d$-system a loop-induced off-shell correction for the exchanged virtual pion.  Its coordinate space representation reveals that it is equivalent to $2\pi$-exchange in the deuteron.  We evaluate the chirally corrected double scattering term and the off-shell contribution with various realistic deuteron wave functions.  We find that the off-shell correction contributes at most -8% and that the isovector double scattering term explains at least 90% of the empirical value of the real part of the $\pi d$-scattering length. 
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24.cm 16.8cm -1.0cm -0.5 cm -0.5 cm \#1\#2

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N. Kaiser\

[Physik Department T39, Technische Universität München, D-85747 Garching, Germany]{}

To be published in [*The Physical Review C (2002), Brief Reports*]{}

The tool to systematically investigate the consequences of spontaneous and explicit chiral symmetry breaking in QCD is chiral perturbation theory.  Recently, methodology has been developed [@weinberg] which allows one to relate scattering processes involving a single nucleon to nuclear scattering processes.  For instance, one can relate pion-nucleon scattering to pion-deuteron scattering.  The non-perturbative effects responsible for deuteron binding are accounted for by (phenomenological) deuteron wavefunctions.  Although this introduces some model dependence one can evaluate matrix elements of chiral operators in a variety of wavefunctions in order to estimate the theoretical error induced by the off-shell behavior of different NN-potentials. 

In a recent precision experiment the PSI-group has extracted the isoscalar and isovector pion-nucleon scattering lengths $a^{\pm}_{\pi N}$ from the shift and the width of the 1s-level in pionic hydrogen, with the result [@schroeder]: $$T^+_{\pi N} = 4\pi \Big( 1 +{m_\pi \over M_p}\Big)\, a^+_{\pi
N} = (-0.045 \pm 0.088)\, {\rm fm}\,,$$ $$T^-_{\pi N} = 4\pi \Big( 1 +{m_\pi \over M_p}\Big)\, a^-_{\pi
N} = (1.847 \mp 0.086)\, {\rm fm}\,.$$ One concludes from these results that the isoscalar $\pi N$-scattering length $a^+_{\pi N}$ is compatible with zero and that the (present) errors of $a^\pm_{\pi N}$ are anti-correlated.  The aim of future PSI-experiments is to measure the level width $\Gamma_{1s} \sim a^-_{\pi N}$ with the same precision as the level shift $\epsilon_{1s} \sim  a^+_{\pi N}+a^-_{\pi N}$. We have converted in eqs.(1,2) scattering lengths into equivalent threshold T-matrices by multiplying with the appropriate kinematical factor.  $M_p = 938.27$MeV and $m_\pi=139.57\, $MeV denote the proton and charged pion mass. 

In an analogous PSI-experiment the complex-valued pion-deuteron scattering length $a_{\pi d}$ has been extracted from the shift and the width of the 1s-level in pionic deuterium [@chat; @hauser].  Again converted into a threshold T-matrix the (averaged) experimental result of refs.[@chat; @hauser] reads for the real part, $${\rm Re}\, T_{\pi d} = 4\pi \Big( 1 +{m_\pi \over M_d}
\Big)\, {\rm Re}\, a_{\pi d} = (-0.496 \pm 0.016)\, {\rm fm}\,,$$ where $M_d=1875.6$MeV denotes the deuteron mass. 

Because of the large size (and weak binding) of the deuteron it is possible to establish an accurate relationship [@erics] between the real part Re$ T_{\pi d}$ and $\pi N$ threshold T-matrices $T^\pm_{\pi N}$. The single scattering contribution $ T^{(s)}_{\pi d}= 2T^+_{\pi N}$ is small and within present error bars compatible with zero.  The empirical value of the real part Re$T_{\pi d}$ can be well understood in terms of the dominant isovector double scattering term [@erics] which reads, $$T^{(d)}_{\pi d} = -\Big(T^-_{\pi N} \Big)^2 
\int_0^\infty dr\, {\rho(r) \over \pi r} \,, \qquad \rho(r)=u^2(r)+w^2(r) \,.$$ Here, $u(r)$ and $w(r)$ denote the conventional s- and d-state deuteron wavefunctions, normalized to $\int_0^\infty dr \, \rho(r)=1$. The relation eq.(4) can be easily derived from the diagrams shown in Fig.1.  The four-momentum of the in- and out-going pion (marked by arrows) is $(m_\pi,
\vec 0\,)$ and that of the exchanged virtual pion is $(m_\pi,\vec q\,)$. After Fourier-transforming the product of the corresponding pion-propagator $1/\vec 
q\,^2$ and the deuteron momentum-distributions into coordinate space one gets $1/(4\pi r)$ times the deuteron density $\rho(r)$. The remaining factor $-4$ originates from the isospin factors of the diagrams.  The advantage of working with threshold T-matrices (which are directly related to Feynman diagrams) is that there appear no additional kinematical factors in eq.(4).  Other three-body interaction terms induced by $\pi\pi$-interaction (the in- and out-going pion couple to a virtual pion in flight) which are formally of the same chiral order have been evaluated in refs.[@weinberg; @beane] and they were found to be negligibly small.  For extensive earlier works on the pion-deuteron scattering length and related issues, see refs.[@kolybasov; @myhrer; @afnan; @faeldt; @landau; @baru]. 

The shortcoming of the present chiral perturbation theory calculations of Re$T_{\pi d}$ is that the isovector double scattering contribution eq.(4) is (and for reasons of consistency has to be) evaluated with the leading order expression $T^-_{\pi N}({\rm tree})=
m_\pi/2 f_\pi^2= 1.61\,$fm for the isovector $\pi N$ threshold T-matrix.  As usual $f_\pi = 92.4\,$MeV denotes the weak pion decay constant.  Since this Tomozawa-Weinberg prediction $T^-_{\pi N}({\rm tree})=m_\pi/2f_\pi^2$ is about 13% smaller than the empirical (central) value of $T^-_{\pi N}$ (see eq.(2)) the (leading order) chiral isovector double scattering contribution to Re$
T_{\pi d}$ comes out about 25% too small, as can be seen from the numbers given in Table1 of ref.[@beane].  In fact, Weinberg argued in ref.[@weinberg] that because of its dominance the isovector double scattering contribution eq.(4) should be calculated including vertex corrections and he simply took the measured $\pi N$-scattering length $a^-_{\pi N}$. The purpose of the present paper is to study in the systematic framework of chiral perturbation theory the corrections to this simple substitution rule. 

It has been shown in ref.[@pinloop] that the about 13% gap between the current algebra prediction $T^-_{\pi N}({\rm tree})=m_\pi/2f_\pi^2$ and the empirical value of $T^-_{\pi N}$ is closed by chiral one-pion loop corrections at order ${\cal
O}(m_\pi^3)$. All other possible corrections from resonances etc.  are much too small to close this gap.  Keeping only the important pion-loop correction the chiral expansion of the on-shell isovector $\pi N$ threshold T-matrix reads [@pinloop], $$T^-_{\pi N}= {m_\pi \over 2f_\pi^2} + {m_\pi^3 
\over 16 \pi^2 f_\pi^4} \Big( 1-2 \ln{m_\pi \over \lambda} \Big) \,,$$ with $\lambda$ a scale parameter introduced in dimensional regularization and minimal subtraction.  With $\lambda = 1.04\,$GeV the central empirical value of $T^-_{\pi N}$ is reproduced.  The scale parameter $\lambda$ can also be interpreted as a momentum space cut-off $\Lambda$. By comparing with the expression of relevant (logarithmically divergent) loop integral obtained in cut-off regularization one finds the relation $\Lambda =  \sqrt{e} \,\lambda/2=
857\,$MeV. This value of the cut-off $\Lambda$ is comparable to the usual estimate of the chiral symmetry breaking scale $\Lambda_\chi = 2\sqrt{2} \pi
f_\pi \simeq 820\,$MeV.

In order to improve on the isovector double scattering contribution $T^{(d)
}_{\pi d}$ one should therefore include pion-loop corrections at the $\pi N\to 
\pi N$ transition vertices.  The grey disk in the diagrams in Fig.1 symbolizes all one-loop graphs of (isovector) $\pi N$-scattering.  Now, since the exchanged pion with four-momentum $(m_\pi,\vec q\,)$ is off its mass-shell the loop diagrams included in the grey disk will lead for such a half off-shell kinematics to more than the order ${\cal O}(m_\pi^3)$ loop correction to $T^-_{\pi N}$ given in eq.(5). 

In addition there is an off-shell contribution $\delta T^-_{\pi N}$(off) which turns out to be generated entirely by the two graphs shown in Fig.2 involving the chiral $4\pi$-vertex.  This off-shell contribution reads explicitly, $$\begin{aligned}
  \delta T^-_{\pi N}({\rm off}) &=&{m_\pi\over96\pi^2 f_\pi^4} 
\bigg\{ q^2 (1+5g_A^2)\ln{m_\pi \over \lambda}+{q^2 \over 6} (1+17g_A^2)
\nonumber \\ && + \Big[4m_\pi^2(1+2g_A^2) +q^2(1+5g_A^2) \Big]  \bigg[
{s \over q} \ln {s+ q \over 2m_\pi} -1 \bigg] \bigg\} \nonumber \\ &=&  {m_\pi 
\over 2f_\pi^2} \,G^V_E(-q^2 )_{loop}\,, \end{aligned}$$ with the abbreviation $s=\sqrt{4m_\pi^2+q^2}$. As indicated it is proportional to the one-pion loop contribution to the isovector electric form factor of the nucleon $G^V_E(-q^2)$ (normalized to unity at $q^2=0$).  We also note that the off-shell correction $\delta T^-_{\pi N}$(off) in eq.(6) is independent of the choice of the interpolating pion-field even though the (off-shell) $4\pi$-vertex is not. 

With inclusion of chiral pion-loop corrections to the threshold T-matrix $T^-_{\pi N}$ the isovector double scattering term takes the form, $$T^{(d)}_{\pi d}={m_\pi^2 \over 4f_\pi^4 }\bigg[ -1+
{m_\pi^2 \over 2\pi^2 f_\pi^2} \ln{m_\pi \over 2\Lambda} \bigg] \int_{1/
\Lambda}^\infty dr\, {\rho(r) \over \pi r} \,,$$ which is practically equivalent to inserting the empirical value of $T^-_{\pi
N}$ into eq.(4).  The effect of the off-shell correction eq.(6) to Re$T_{\pi
d}$ is obtained by multiplying $\delta T^-_{\pi N}$(off) with the pion-propagator $1/\vec q\,^2$ and Fourier-transforming this product into coordinate space.  The terms in the first line of eq.(6) give this way a delta-function $\delta^3(\vec r\,)$ which exclusively probes the deuteron wavefunction at the origin.  The Fourier-transform of the other non-polynomial terms in eq.(6) is most conveniently obtained with the help of a dispersion relation and one finds $$\delta T^{(o)}_{\pi d} = {m_\pi^2 \over 96\pi^3 f_\pi^6}
\int_{1/\Lambda}^\infty dr \, {\rho(r)\over r}  \int_{2m_\pi}^\infty 
d\mu \, e^{-\mu r} \sqrt{\mu^2-4m_\pi^2} \,\bigg[ 1+ 5g_A^2 -{4m_\pi^2 \over
\mu^2} (1+2g_A^2) \bigg] \,.$$ One observes that the weight function multiplying the deuteron density $\rho(r)$ goes like $r^{-3}$ for $r\to 0$. On the other hand its asymptotic behavior for $r\to \infty$ is $e^{-2m_\pi r} r^{-5/2}$, the typical form of a $2\pi$-exchange potential.  In contrast to the double scattering term eq.(7) the off-shell correction eq.(8) is mainly sensitive to the short distance behavior of the deuteron wavefunction.  The $r^{-3}$-singularity in eq.(8) originates from the large-$q$ behavior of the (one-loop) form factor in eq.(6), which clearly is not realistic for large momentum transfers.  We regularize the $r^{-3}$ short distance singularity (in a minimal way) by starting the radial integration at a nonzero $r_{min} = 1/\Lambda  = 0.23\,$fm, where $\Lambda
=857\,$MeV is the (equivalent) momentum cut-off entering the chiral logarithm in eq.(5). 

              Potential               Paris[@paris]   Bonn-CD[@cdbonn]   Idaho[@idaho]   NL5[@evgeni]   NL6[@evgeni]   NNL5[@evgeni]   NNL6[@evgeni]
  ---------------------------------- --------------- ------------------ --------------- -------------- -------------- --------------- ---------------
       $T^{(d)}_{\pi d}$ \[fm\]         $-$0.479          $-$0.495         $-$0.466        $-$0.478       $-$0.459       $-$0.500        $-$0.499
   $\delta T^{(o)}_{\pi d} $ \[fm\]       0.015            0.025             0.017          0.023          0.008           0.036           0.036

In Table 1, we present numerical values of the chirally corrected double scattering contribution $T^{(d)}_{\pi d}$ and the off-shell correction $\delta T^{(o)}_{\pi d} $ for various realistic deuteron wavefunctions taken from refs.[@paris; @cdbonn; @idaho; @evgeni].  One observes that the chirally corrected isovector double scattering contribution $T^{(d)}_{\pi d}$ is rather stable and close to the empirical value of Re$T_{\pi d}$ in eq.(3).  When starting the radial integration at $r=0$ instead of $r_{min}=0.23\,$fm these numbers are affected at most in the last digit.  The off-shell correction $\delta T^{(o)}_{\pi d}$ which is mainly sensitive to the short distance behavior of $\rho(r)$ varies more with the deuteron wavefunction.  It is however always a small attractive effect whose relative magnitude does not exceed 8%. 

We conclude that the (chirally corrected) isovector double scattering term explains at least 90% of the empirical value of the real part of the $\pi d$ threshold T-matrix.  This is in agreement with the findings of ref.[@tony] were a host of other small corrections to Re$T_{\pi d}$ has been investigated.  For example, in ref.[@tony] an off-shell correction of the form eq.(6) proportional to a phenomenological dipole form factor (minus unity) has been considered.  The off-shell correction eq.(6) obtained in the present work has in fact a well-defined conceptual and physical origin.  It is the unique byproduct of the pion-loop graphs which close the gap between the current-algebra prediction and the empirical value of the isovector $\pi N$ threshold T-matrix $T_{\pi N}$. Its effect on Re$T_{\pi d}$ is comparable to the inherent theoretical uncertainty of the isovector double scattering contribution $T^{(d)}_{\pi d}$.

Acknowledgements {#acknowledgements .unnumbered}
================

I thank E. Epelbaum and R. Machleidt for providing me tabulated deuteron wavefunctions. 

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