Non-relativistic Quantum Mechanics versus Quantum Field Theories
aa r X i v : . [ h e p - ph ] M a y Non-relativistic Quantum Mechanics versus Quantum Field Theories
Antonio Pineda
Grup de F´ısica Te`orica and IFAEUniversitat Aut`onoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain
We briefly review the derivation of a non-relativistic quantum mechanics description of aweakly bound non-relativistic system from the underlying quantum field theory. We highlightthe main techniques used.
In a first approximation, the dynamics of the Hydrogen atom can be described by the solutionof the Schr¨odinger equation with a Coulomb potential. However, it is not always clear how toderive this equation from the more fundamental quantum field theory, QED, much less how toget corrections in a systematic way. A similar problem is faced in heavy quarkonium systemswith very large heavy quark masses. In this situation the dynamics is mainly perturbative andone efficient solution to this problem comes from the use of effective field theories (EFTs) and inparticular of pNRQCD1 a . This EFT takes full advantage of the hierarchy of scales that appearin the system ( v is the velocity of the heavy quark in the center of mass frame and m is theheavy quark mass): m ≫ mv ≫ mv · · · (1)and makes systematic and natural the connection of the Quantum Field Theory with theSchr¨odinger equation. Roughly speaking the EFT turns out to be something like: i∂ − p m − V (0) s ( r ) ! Φ( r ) = 0 + corrections to the potential+interaction with other low − energy degrees of freedom pNRQCDwhere V (0) s ( r ) = − C f α s /r in the perturbative case and Φ( r ) is the ¯ Q – Q wave-function. This a For a comprehensive review of pNRQCD see2. FT b is relevant, at least, for the study of the ground state properties of the bottomoniumsystem, non-relativistic sum rules and the production of t -¯ t near threshold (for some recentapplications see5 , , , , E ∼ mv . In order to derive pNRQCD we sequentially integrate out the larger scales.10 E ∼ mv QCDNRQCDpNRQCD
Integrating out the hard scale (m)Integrating out the soft scale (mv)
In this paper, we would like to highlight the main techniques needed in order to perform effi-ciently high-precision perturbative computations in non-relativistic bound state systems. Theycan be summarized in four points:1. Matching QCD to NRQCD: Relativistic Feynman diagrams2. Matching NRQCD to pNRQCD (getting the potential): Non-Relativistic (HQET-like)Feynman diagrams3. Observable: Quantum mechanics perturbation theory4. Observable: Ultrasoft loopsThe first two points explain the techniques needed to obtain pNRQCD from QCD, whereasthe last two explain the kind of computations faced in the EFT when computing observables.All the computations can be performed in dimensional regularization and only one scale appearsin each type of integral, which becomes homogeneous. This is a very strong simplification of theproblem. In practice this is implemented in the following way:Point 1). One analytically expands over the three-momentum and residual energy in theintegrand before the integration is made in both the full and the effective theory11 , QCD Z d qf ( q, m, | p | , E ) = Z d qf ( q, m, ,
0) + O (cid:18) Em , | p | m (cid:19) ∼ C ( µm )(tree level) | NRQCD
N RQCD Z d qf ( q, | p | , E ) = Z d qf ( q, ,
0) = 0 !! (2)Therefore, the computation of loops in the effective theory just gives zero and one matches loopsin QCD with only one scale (the mass) to tree level diagrams in NRQCD, which we schematicallydraw in the following figure: b It is also possible to study heavy quarkonium systems in the non-perturbative regime with pNRQCD profitingfrom the hierarchy of scales of Eq. (1), see3 , ) + Ο(1/ = C(m/m= C(m/ µ ) m + ..... NRQCDQCD m^2) Point 2) works analogously13. One expands in the scales that are left in the effective theory.We integrate out the scale k (transfer momentum between the quark and antiquark). Againloops in the EFT are zero and only tree-level diagrams have to be computed in the EFT:
N RQCD Z d qf ( q, k, | p | , E ) = Z d qf ( q, k, ,
0) + O (cid:18) Ek , | p | k (cid:19) ∼ δh s (potential) (3) pN RQCD Z d qf ( q, | p | , E ) = Z d qf ( q, ,
0) = 0 !! (4)We illustrate the matching in the figure below. Formally the one-loop diagram is equal to theQCD diagram shown above. The difference is that it has to be computed with the HQET quarkpropagator (1 / ( q + iǫ )) and the vertices are also different. p > p ′ > k = p − p ′ V2 α k =m1m V1m 2 α NRQCD pNRQCDOnce the Lagrangian of pNRQCD has been obtained one can compute observables. A keyquantity in this respect is the Green function. In order to go beyond the leading order descriptionof the bound state one has to compute corrections to the Green Function ( δh s schematicallyrepresents the corrections to the potential and H I the interaction with ultrasoft gluons): G s ( E ) = 1 h (0) s + δh s − H I − E = G (0) s + δG s G (0) s ( E ) = 1 h (0) s − E .
These corrections can be organized as an expansion in 1 /m , α s and the multipole expansion.Two type of integrals appear then, which correspond to points 3) and 4) above. oint 3) . For example, if we were interested in computing the spectrum at O ( mα s ) (forQED see14), one should consider the iteration of subleading potentials ( δh s ) in the propagator: δG pot.s = δh s δh s δh s + · · · + ∼ h (0) s − E δh s h (0) s − E + 1 h (0) s − E δh s h (0) s − E δh s h (0) s − E + · · · At some point, these corrections produce divergences. For example, a correction of the type: δ ( r ) G (0) s ( C f α s /r ) G (0) s δ ( r ), would produce the following divergence h r = 0 | E − p /m C f α s r E − p /m | r = 0 i∼ Z d d p ′ (2 π ) d Z d d p (2 π ) d m p ′ − mE C f πα s ( p − p ′ ) m p − mE ∼ − C f m α s π ǫ + 2 ln( mEµ p ) + · · · ! . Nevertheless, the existence of divergences in the effective theory is not a problem since they getabsorbed in the potentials ( δh s ). The same happens with ultrasoft gluons, point 4) , δG us s = | {z } / ( E − V (0) o − p /m ) ∼ G c ( E ) Z d d k (2 π ) d r kk + p /m + V (0) o − E r G c ( E ) ∼ G c ( E ) r ( p /m + V (0) o − E ) ( ǫ + γ + ln ( p /m + V (0) o − E ) ν us + C ) r G c ( E ) , which also produces divergences that get absorbed in δh s . Overall, we get a consistent EFT. References
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